IEEE Transactions on Antennas and Propagation [volume 59 number 5]

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MAY 2011

VOLUME 59

NUMBER 5

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Analytical Model for Calculating the Radiation Field of Microstrip Antennas With Artificial Magnetic Superstrates: Theory and Experiment ........ ......... ......... ........ ......... ......... ........ . H. Attia, L. Yousefi, and O. M. Ramahi Multi-Frequency, Linear and Circular Polarized, Metamaterial-Inspired, Near-Field Resonant Parasitic Antennas ..... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... P. Jin and R. W. Ziolkowski Left-Handed Wire Antennas Over Ground Plane With Wideband Tuning ....... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ..... F. J. Herraiz-Martínez, P. S. Hall, Q. Liu, and D. Segovia-Vargas Switched-Band Vivaldi Antenna . ......... ......... . ....... ......... ... M. R. Hamid, P. Gardner, P. S. Hall, and F. Ghanem Focusing Properties of Fresnel Zone Plate Lens Antennas in the Near-Field Region .... .. S. Karimkashi and A. A. Kishk Wideband Millimeter-Wave Substrate Integrated Waveguide Slotted Narrow-Wall Fed Cavity Antennas ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... Y. Zhang, Z. N. Chen, X. Qing, and W. Hong Radiation Theory of the Plasma Antenna ......... ........ ......... ......... ........ ........ H. Q. Ye, M. Gao, and C. J. Tang Self-Consistent Analysis of Cylindrical Plasma Antennas ....... ......... ........ .... P. Russo, G. Cerri, and E. Vecchioni Arrays A Slot Antenna Array With Low Mutual Coupling for Use on Small Mobile Terminals ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... S. Dumanli, C. J. Railton, and D. L. Paul An E-Band Partially Corporate Feed Uniform Slot Array With Laminated Quasi Double-Layer Waveguide and Virtual PMC Terminations ..... ........ ......... ......... ........ ......... ......... ........ ... M. Zhang, J. Hirokawa, and M. Ando Ultrawide Bandwidth 2 2 Microstrip Patch Array Antenna Using Electromagnetic Band-Gap Structure (EBG) ...... .. .. ........ ......... ......... ........ ... D. Nashaat, H. A. Elsadek, E. A. Abdallah, M. F. Iskander, and H. M. El Hennawy Green’s Function Based Equivalent Circuits for Connected Arrays in Transmission and in Reception . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... D. Cavallo, A. Neto, and G. Gerini Rectangular Thinned Arrays Based on McFarland Difference Sets ....... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ........ G. Oliveri, F. Caramanica, C. Fontanari, and A. Massa Electromagnetics Incremental Fringe Formulation for a Complex Source Point Beam Expansion . ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... S. M. Canta, D. Erricolo, and A. Toccafondi Modal Analysis and Wave Propagation in Finite 2D Transmission-Line Metamaterials . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... R. Islam, M. Zedler, and G. V. Eleftheriades Low Frequencies and the Brillouin Precursor .... ........ ......... ......... ..... .... ......... ......... ........ ... N. Cartwright

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(Contents Continued on p. 1437)

(Contents Continued from Front Cover) Mixed-Impedance Boundary Conditions . ......... ........ ......... ......... ........ . H. Wallén, I. V. Lindell, and A. Sihvola Uniform Ray Description for the PO Scattering by Vertices in Curved Surface With Curvilinear Edges and Relatively General Boundary Conditions . ......... ......... ........ ......... ......... ..... M. Albani, G. Carluccio, and P. H. Pathak A Novel Microwave Tomography System Using a Rotatable Conductive Enclosure .... ......... P. Mojabi and J. LoVetri Super-Resolution UWB Radar Imaging Algorithm Based on Extended Capon With Reference Signal Optimization ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... S. Kidera, T. Sakamoto, and T. Sato Numerical Techniques Effective Local Absorbing Boundary Conditions for a Finite Difference Implementation of the Parabolic Equation .... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. S. Özbayat and R. Janaswamy TD-UTD Solutions for the Transient Radiation and Surface Fields of Pulsed Antennas Placed on PEC Smooth Convex Surfaces ........ ......... ........ ......... ......... ........ ......... ......... .. H.-T. Chou, P. H. Pathak, and P. R. Rousseau Non-Conformal Domain Decomposition Method With Mixed True Second Order Transmission Condition for Solving Large Finite Antenna Arrays .. ......... ......... ........ ......... . ........ ........ ......... ......... ... Z. Peng and J.-F. Lee Wireless Low-Cost Dual-Loop-Antenna System for Dual-WLAN-Band Access Points .. ......... ......... .. S.-W. Su and C.-T. Lee A 4-Port Diversity Antenna With High Isolation for Mobile Communications .. ......... ......... ... B. Wu and K.-M. Luk Ground Plane Boosters as a Compact Antenna Technology for Wireless Handheld Devices .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. A. Andújar, J. Anguera, and C. Puente Measured Comparison of Dual-Branch Signaling Over Space and Polarization Diversity ... . A. Morshedi and M. Torlak Channel and Propagation Measurements at 300 GHz .... ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ....... S. Priebe, C. Jastrow, M. Jacob, T. Kleine-Ostmann, T. Schrader, and T. Kürner Channel Simulator for Land Mobile Satellite Channel Along Roadside Trees .. ........ M. Cheffena and F. Pérez Fontán Estimation of the Refractivity Structure of the Lower Troposphere From Measurements on a Terrestrial Multiple-Receiver Radio Link ..... ......... ........ ......... ......... ........ ......... ......... P. Valtr, P. Pechac, V. Kvicera, and M. Grabner

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COMMUNICATIONS

Small Broadband Antenna Composed of Dual-Meander Folded Loop and Disk-Loaded Monopole .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... W.-T. Hsieh and J.-F. Kiang A Dual Band Microstrip-Fed Slot Antenna ....... ........ ......... ... ....... ........ ......... . M. N. Mahmoud and R. Baktur A New Super Wideband Fractal Microstrip Antenna .... ......... ......... ........ ......... ......... ........ ......... . A. Azari Bandwidth Enhancement of Internal Antenna by Using Reactive Loading for Penta-Band Mobile Handset Application . .. .. ........ ......... ......... ........ ........ C.-M. Peng, I.-F. Chen, C.-C. Hung, S.-M. Shen, C.-T. Chien, and C.-C. Tseng Small-Size Loop Antenna With a Parasitic Shorted Strip Monopole for Internal WWAN Notebook Computer Antenna . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... K.-L. Wong, W.-J. Chen, and T.-W. Kang Improvement of Time and Frequency Domain Performance of Antipodal Vivaldi Antenna Using Multi-Objective Particle Swarm Optimization ... ........ ......... ..... ..... ........ ......... . S. Chamaani, S. A. Mirtaheri, and M. S. Abrishamian Optimized Microstrip Antenna Arrays for Emerging Millimeter-Wave Wireless Applications .. ........ ......... ......... .. .. ........ ......... ......... . B. Biglarbegian, M. Fakharzadeh, D. Busuioc, M.-R. Nezhad-Ahmadi, and S. Safavi-Naeini Pareto Optimization of Thinned Planar Arrays With Elliptical Mainbeams and Low Sidelobe Levels .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... J. S. Petko and D. H. Werner Low Cross-Polarization Reflectarray Antenna .... ........ ......... ......... ..... H. Hasani, M. Kamyab, and A. Mirkamali Design and Validation of Gathered Elements for Steerable-Beam Reflectarrays Based on Patches Aperture-Coupled to Delay Lines .... ......... ........ ......... ......... ........ ......... ......... ...... E. Carrasco, M. Barba, and J. A. Encinar Scattering From Complex Bodies of Revolution Using a High-Order Mixed Finite Element Method and Locally-Conformal Perfectly Matched Layer .. ........ ......... ... ....... ........ ..... Y. B. Zhai, X. W. Ping, and T. J. Cui Open-Ended Coaxial Line Probes With Negative Permittivity Materials ........ ........ M. S. Boybay and O. M. Ramahi Investigation of Adaptive Matching Methods for Near-Field Wireless Power Transfer .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... J. Park, Y. Tak, Y. Kim, Y. Kim, and S. Nam Implementation of a Cognitive Radio Front-End Using Rotatable Controlled Reconfigurable Antennas ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ..... Y. Tawk, J. Costantine, K. Avery, and C. G. Christodoulou The HF Channel EM Parameters Estimation Under a Complex Environment Using the Modified IRI and IGRF Model .. .. .. ........ ......... ......... ........ ......... ......... ... Z.-W. Yan, G. Wang, G.-L. Tian, W.-M. Li, D.-L. Su, and T. Rahman

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CORRECTIONS

Corrections to “On Corrections to a z-Transform Method for FDTD Simulations With Frequency-Dependent Dielectric Functions” ...... ......... ........ ......... ......... ........ ...... .... ......... ........ ......... ......... ........ ...... D. B. Miron

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Digital Object Identifier 10.1109/TAP.2011.2150172

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011

Analytical Model for Calculating the Radiation Field of Microstrip Antennas With Artificial Magnetic Superstrates: Theory and Experiment Hussein Attia, Student Member, IEEE, Leila Yousefi, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—A fast analytical solution for the radiation field of a microstrip antenna loaded with a generalized superstrate is proposed using the cavity model of microstrip antennas in conjunction with the reciprocity theorem and the transmission line analogy. The proposed analytical formulation for the antenna’s far-field is much faster when compared to full-wave numerical methods. It only needs 2% of the time acquired by full-wave analysis. Therefore the proposed method can be used for design and optimization purposes. The method is verified using both numerical and experimental results. This verification is done for both conventional dielectric superstrates, and also for artificial superstrates. The analytical formulation introduced here can be extended for the case of a patch antenna embedded in a multilayered artificial dielectric structure. Arguably, the proposed analytical technique is applied for the first time for the case of a practical microstrip patch antenna working in the Universal Mobile Telecommunications System (UMTS) band and covered with a superstrate made of an artificial periodic metamaterial with dispersive permeability and permittivity. Index Terms—Artificial magnetic superstrate, cavity model, metamaterials, microstrip antennas, split ring resonators.

I. INTRODUCTION HE addition of a superstrate layer over microstrip patch antenna (MPA) has been reported to allow for the enhancement of the antenna gain and radiation efficiency [1]–[8]. Furthermore, superstrate layers are often used to protect the MPA from its environment hazards, especially when placed on aircrafts and missiles. Earlier publications were focused on understanding how the superstrate affects the resonant frequency and subsequent power matching concerns [9]. While the gain of MPAs can be increased by using planar arrays; a solution mostly attractive because it does not change the vertical profile of the antenna structure, arrays, however, need a feeding network which introduces losses and design complications [10]. Several configurations of superstrates were used to improve antenna radiation properties, such as dielectric slabs [11], [12],

T

Manuscript received May 02, 2010; revised September 09, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the Ministry of Higher Education, Egypt, Research in Motion (RIM) Inc., and in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under the NSERC/RIM Industrial Research Associate Chair and Discovery Programs. The authors are with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1 Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122295

electromagnetic bandgap (EBG) structures [13], [14], highlyreflective surfaces [8], and the most recently artificial magnetic superstrates [7], [15]. Using magneto-dielectric materials with high positive permeability and permittivity values as the superstrate of MPA decreases the wavelength in the media, leading to a lower profile of the whole structure [2], [7]. In [16], the potential application of magneto-dielectric materials as a superstrate to improve the gain of MPA was investigated without considering physical realization of the artificial superstrate. Latrach et al. [17] used edge-coupled split ring resonator (SRR) inclusions to provide artificial superstrate comprising alternately layers with negative permeability and positive index of refraction materials to increase the gain of patch antenna. In [7], an artificial magnetic material based on the broadside-coupled split ring resonator SRR inclusions was used as a superstrate for a MPA to increase the antenna gain and radiation efficiency. To analyze the composite structure (antenna with superstrate), full-wave electromagnetic simulation tools which utilize numerical methods are usually used. However, using numerical methods to analyze metamaterials, or periodic structures in general, is an expensive computational task which requires considerable computer resources. The primary reason for such large computational burden is the resolution needed to capture the quasi-static resonance behavior in the metamaterial particles which are electrically very small. Therefore, numerical methods may not be practical for real-world designs that require several runs for optimization. Developing a fast analytical method to analyze such structures, which is the focus of this paper, can accelerate the design process, and also provides optimization opportunities. So far several analytical methods for calculating the far-field of Hertzian dipole in multi-layer structure, has been reported in literature [1]–[6] and [18], [19]. In [1], Green’s function and the stationary phase integration approach were used to calculate the gain of an infinitesimal dipole embedded in the top layer of a two-layer dielectric structure. Additionally, in [1], it was shown that a resonance condition may be created, whereby gain and radiation resistance are improved over a significant bandwidth. Later in [2], the reciprocity theorem and transmission line (TL) analogy were used to provide asymptotic formulas for gain and beamwidth of a Hertzian dipole embedded within a grounded substrate and covered with a superstrate. In [18], [19], the TL analogy method was used to compute the radiation patterns of arbitrarily directed simple dipole source that is embedded in a multilayered dielectric structure. Also, a dielectric resonator

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ATTIA et al.: ANALYTICAL MODEL FOR CALCULATING THE RADIATION FIELD OF MICROSTRIP ANTENNAS

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A. Reciprocity Theorem and , are two groups of sources radiAssume , , and ating inside the same medium generating the fields , , , respectively. The sources and fields satisfy the Lorentz Reciprocity Theorem in integral form [21], the formula is repeated here to make the discussion complete

Fig. 1. Microstrip patch antenna covered by a superstrate.

antenna covered with plain dielectric superstrate was analyzed [18]. The straightforward method proposed in this work based on the cavity model avoids the need to use the particle swarm optimization (PSO) method used in [18] to obtain a set of Hertzian dipoles representing the antenna to be analyzed using the reciprocity theorem and TL analogy. Although several attempts have been reported to develop analytical methods, most of those works consider the case of a simple infinitesimal dipole antenna covered with plain dielectric superstrate [1]–[6] and [18], [19]. In this paper, we consider a practical microstrip patch antenna working in the UMTS band and covered with an artificial magnetic (metamaterial) superstrate used for increasing the antenna directivity. The cavity model of a MPA in conjunction with the reciprocity theorem and the transmission line analogy is used to develop a fast analytical solution for the radiation field. The artificial superstrate constituted by split ring resonators SRR printed on both sides of a dielectric slab is characterized analytically by obtaining its effective permeability and permittivity. The organization of this paper is as follows: in Section II, the radiation patterns of a MPA covered with a generalized superstrate layer at some distance in free space are calculated by replacing the MPA with two magnetic current sources based on the cavity model and then using reciprocity theorem and the TL analogy to compute the antenna directivity. In Section III, the proposed analytical formulation is verified through a comparison with numerical full wave simulation results, where the directivity of a MPA covered with different conventional superstrates is calculated. Section IV shows the application of the analytical technique to a MPA loaded with an artificial magnetic superstrate to enhance the directivity of the MPA, experimental results are used to validate the analytical formulation. Finally, summary and conclusion are provided in Section V.

II. ANALYTICAL FORMULATION OF THE ANTENNA’S FAR-FIELD The basic problem to be studied is shown in Fig. 1. A MPA is covered with a superstrate layer at a distance in free space. The MPA is printed on a grounded substrate of thickness having and , respecrelative permeability and permittivity of tively. At distance from the substrate is the superstrate layer of thickness having relative permeability and permittivity of and , respectively. To compute the far-field, first the MPA is modeled as a dielectric-loaded cavity [20] and then the reciprocity theorem and the transmission line analogy are applied to the whole structure (the antenna with the superstrate).

(1) In order to use the reciprocity theorem to compute the radiation field of MPA, one need to consider the fields ( , , , ) and the sources ( , , , ) inside a medium that is surrounded by a sphere of infinite radius. Hence, the left side of (1) is essentially zero and (1) is reduced to

(2) According to (2), two problems need to be established. In the first problem, the original radiating patch at is replaced with an electric current source and/or a magnetic current (using the cavity model [20]) radiating an electric source and at the observation point of . In the field of second problem, a fictitious dipole (reciprocity source) of (choosing is equal to zero) at the same observation point and at the original patch location radiates a far-field of . Our goal is to use (2) to formulate the far-field of the at . MPA as B. Cavity Model of Microstrip Antenna In order to use (2) to compute the far-field of the MPA loaded with a generalized superstrate, the MPA should be replaced by a set of electric and/or magnetic current sources. The cavity model [20] of MPA to is used here to determine these equivalent currents. The volume bound by the microstrip patch (located in the - plane at ) and the ground plane can be modeled as a dielectric-loaded cavity resonator by considering the top and bottom walls of this volume as perfect electric conductor (PEC) and the four side walls as ideal open circuit (magnetic walls). The mode of concern here is the dominant transverse magnetic ) which assumes a zero value of but a non-zero mode ( value of . By using the expression of under ideal magnetic side walls boundary condition, one can formulate the equivalent magnetic current in the cavity’s apertures (side walls) using the . The resultant magnetic curequivalence principle as rents and will be -directed on the two cavity’s side walls parallel to the - plane as shown in Fig. 2. The equivalent electric current is very small (ideally zero) because on the side walls. Hence, in (2) is equal to zero. C. Formulating the Antenna’s Far-Field According to the reciprocity theorem, a fictitious dipole (reciprocity source) of (choosing ) at the observation

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Fig. 2. Equivalent magnetic current sources representing the patch antenna.

point is considered to radiate a far-field of and at the patch antenna location at . Hence, (2) reduces to

(3) The reciprocity source is assumed to have a value of where for TM (parallel polarization) and for TE (perpendicular polarization) incident wave on the whole structure (the patch antenna with the superstrate). Hence, (3) reduces to

Fig. 3. Transmission line equivalent model of the structure of Fig. 1.

For TE wave or perpendicular polarization

(4) The above equation will be used to obtain the radiation field due to one of the two equivalent magnetic current sources which can be the one at the radiation slot (i.e., ). and Then, by treating the two identical radiation slots (at ) as a two-element array, the array factor can be used to of the MPA covered with calculate the total radiated field the superstrate, with the assumption that the existence of the superstrate does not considerably affect the current distributions in those two radiation slots. field is determined at It is observed from (4) that the the magnetic current source location (i.e., ) due to the reciprocity source at the observation point in either or direction. From the reciprocity theorem, is proportional to the required radiation field due to the original patch antenna at . The field near the multilayered structure due to this reciprocity source is essentially a plane wave, and location by modeling each layer therefore can be found at as a transmission line segment (see Fig. 3) having a characteristic impedance and propagation constant which depend on the incident angle of . The propagation constants and the characteristic impedances of the multi-section transmission line equivalent model shown in Fig. 3 are derived from the oblique incidence of a plane wave on a plane interface between two dielectric regions [22] as follows:

For TM wave or parallel polarization

where, , In case of TE incident wave ( ) from the reciprocity source (infinitesimal dipole) located at the observation point (see Fig. 4), the component of the far-field due to is equal to this source at the location of

(5) depends on only and it represents the The functions (as midpoint in the radiation slot at ) in current at the transmission line analogy (Fig. 3) due to an incident current . wave of strength A straightforward method to determine the current at is to define the voltage and current on each transmission line segment shown in Fig. 3 as (6)

ATTIA et al.: ANALYTICAL MODEL FOR CALCULATING THE RADIATION FIELD OF MICROSTRIP ANTENNAS

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where

Similarly, in the case of TM incident wave ( ) from the reciprocity source located at the observation point (see Fig. 4), the total radiated field of the MPA covered with the superstrate can be formulated as

Fig. 4. Coordinate system for computing far-field from the reciprocity electric . source J at the patch antenna’s radiation slot located at x

=0

(7) where is the transmission line segment number. In . Note that as the substrate the model considered here, . Also, of the MPA is backed by a ground plane, for the TE case. Then, by enforcing the continuity of the voltages and current at the boundaries of these transmission line segments, one can calculate all the unknown and ) and hence solve for the required curparameters ( . This straightforward method is extendable rent at easily to the case of MPA covered with any number of layers. Substituting from (5) in (4), the far-field due to can be calculated. Due to the length the radiation slot at and straightforward nature of the derivation only the final result is given here as follows:

(10) depends on only and it represents the The functions (as midpoint in the radiation slot at ) in current at the transmission line analogy (Fig. 3) due to an incident current in the transmission wave of strength 1 A. Hence, can be line equivalent model ((6) and (7)). The functions . computed in the same way explained earlier in the case of Interestingly, the closed forms for the far-field given by (9) and (10) agree well with the ones calculated using the conventional vector potential approach [23] (see [23, Eqs (45) and (46)]) when the superstrate is air (i.e., patch antenna in free and , this agreement space) where verifies the proposed analytical formulation. The far-field of the MPA covered with the superstrate is calculated at the desired frequency using (9) and (10), and integrated as follows to calculate the antenna directivity (11) where

(8) where

III. RADIATION DUE TO A PATCH ANTENNA COVERED WITH CONVENTIONAL SUPERSTRATE

Note that for , Similarly, the far-field due to the other radiation slot at can be obtained. By adding the fields due to the two radiation slots together, the total field (in direction) due to the MPA covered with the superstrate can be obtained. Another interesting way to compute the total field of the MPA in the TE mode is and ) as a treating the two identical radiation slots (at two-element array. Hence, the array factor can be used to calof the MPA covered with the culate the total radiated field superstrate. Adopting the later method, the total field can be formulated as

(9)

In this section, the radiation patterns of a patch antenna covered with a conventional superstrate is investigated using the analytical technique explained in the previous section and verified using the commercial full-wave simulator HFSS. In accordance with Fig. 1, assuming a MPA having dimensions of 36 mm 36 mm, and printed on a substrate of Rogers RO4350 having a relative permittivity of 3.48, loss tangent of and a thickness of 0.762 mm. The patch is covered with a superstrate of thickness 6.286 mm. The spacing between the antenna and the superstrate is 12 mm. The threelayer radiating system is infinitely extended in the - plane to comply with the transmission line analogy in order to calculate the current at the patch location due to the TE and TM incident plane waves. Three superstrates with different values of permeability and , permittivity are studied here as shown in Fig. 5 for ( ), Fig. 6 for ( , ), and Fig. 7 for ( , ). It is observed that the resultant -plane ( ) and

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Fig. 5. The directivity radiation pattern at 2.2 GHz of the patch antenna covered with a superstrate ( and  ) at d ,B : and h : . (a) -plan and (b) -plan.

= 0 762 mm

=1 E

=1 H

= 12 mm

= 6 286 mm

Fig. 8. The directivity of MPA shown in Fig. 1 versus the relative permittivity of the superstrate with  at 2.2 GHz, d ,B : and h : .

= 0 762 mm

=1

= 12 mm = 6 286 mm

Fig. 6. The directivity radiation pattern at 2.2 GHz of the patch antenna covered and  ) at d ,B : with a superstrate ( and h : . (a) -plan and (b) -plan.

= 0 762 mm

= 16 E

=1 H

= 12 mm

= 6 286 mm

Fig. 7. The directivity radiation pattern at 2.2 GHz of the patch antenna covered and  ) at d ,B : with a superstrate ( and h : . (a) -plan and (b) -plan.

= 0 762 mm

=1 E

=6 H

= 12 mm

= 6 286 mm

-plane ( ) directivity radiation patterns agree well with HFSS results in the three cases. Fig. 8 shows a comparison between the analytically and numerically (HFSS) calculated directivity in broadside direction ) at 2.2 GHz of the patch antenna covered with a ( superstrate having a relative permeability of 1 and varying . Good agreement is observed between relative permittivity the two methods. IV. ARTIFICIAL MAGNETIC STRUCTURE AS A SUPERSTRATE FOR PLANAR ANTENNAS The proposed analytical solution is used here to analyze a MPA loaded with an artificial magnetic superstrate (see Fig. 9). Fig. 9(a) illustrates a broadside coupled split-ring resonator (SRR) unit cell acting as a building block of the artificial magnetic superstrate [7], [24]. The SRR inclusion consists of

Fig. 9. Geometry of a patch antenna covered with an engineered magnetic superstrate. (a) SRR unit cell. (b) Photograph of top view. (c) Side view. (d) Experimental prototype (t : ,m ,c and d ).

= 12 mm

= 0 762 mm

= 2 mm

= 85 mm

two parallel broken square loops. The host dielectric is made of Rogers RO4350 with a thickness of 0.762 mm, relative permit, and loss tangent of . A planar tivity of 10 10 array of SRRs is printed on the host dielectric layer to provide the engineered magnetic material. The superstrate used here consists of 3 layers of printed magnetic inclusions. The layers are separated by 2 mm of air layers (see Fig. 9(c)).

ATTIA et al.: ANALYTICAL MODEL FOR CALCULATING THE RADIATION FIELD OF MICROSTRIP ANTENNAS

Fig. 10. Analytically calculated relative permeability of the SRR inclusions.

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Fig. 11. The E-plane directivity radiation pattern at 2.12 GHz of the patch antenna covered with the engineered magnetic superstrate.

The SRR unit cell is analytically modeled by obtaining its effective relative permeability as

(12)

, and where is the surface area of the inclusion are the unit cell sizes in and directions as shown in Fig. 9(a). The dimensions of the designed SRR unit cell are , , , . The width of metallic strips ( ) is equal to 0.3 mm, and the metallic strips are assumed to be made of copper. , and can be found in [24]. Formulas for Since the SRRs are aligned in the - plane, the resultant effective enhanced permeability is provided only in the direction [7]. Hence, the engineered material composed of the SRR inclusions will experience the anisotropic permeability tensor of

(13) The analytically calculated effective relative permeability is shown in Fig. 10. An effective and -directed permittivity is provided by the stored electrical energy in the inter-cell capacitors formed in the gap regions between the metallic SRRs inclusions. In case of a -directed electric field, the metamaterials superstrate will experience an effective permittivity equal to that of its host dielectric as the electric field would be perpendicular to the plane of the unit cell. Therefore, the artificial magnetic material composed of the SRRs inclusions will experience anisotropic electric permittivity of [25]

(14) where

According to the above formulas, the effective relative permittivity of the designed structure in the and directions would be equal to 5.62. Substituting the value of permittivity and permeability from (12) and (14) in the equations presented in Section II, the electric field of the MPA is calculated at the desired frequency using (9) and (10) and then integrated to calculate the antenna directivity using (11). To verify the analytical solution of the far-field, the patch antenna covered with the SRR based artificial magnetic superstrate (see Fig. 9) has been fabricated, simulated using CST, and measured. The reason for using CST in this case is that the time domain simulation module in CST stimulates the structure using a broadband signal, broadband stimulation calculates the -parameters for the entire desired frequency range and the radiation patterns at various desired frequencies from only one calculation run. On the other hand, frequency domain solvers such as HFSS perform a new simulation run for each frequency sample. Hence, the relationship between calculation time and frequency steps is linear unless special methods are applied to accelerate subsequent frequency domain solver runs. Therefore, the time domain solver usually is fastest when a large number of frequency samples need to be calculated. The analytical results are compared to the numerical and experimental results for a MPA having dimensions of 36 mm 36 mm, and is printed on a substrate of Rogers RO4350 having a and relative permittivity of 3.48, loss tangent of a thickness of 0.762 mm. The antenna is designed to operate at the frequency band of 2190–2210 MHz (UMTS) at which the magnetic superstrate has an effective permeability of about 15 (real part) and a magnetic loss tangent of 0.11 (see Fig. 10). Fig. 11 shows a comparison between the analytical, numerical and experimental results of the -plane directivity radiation pattern at the antenna’s resonance frequency of 2.12 GHz, a good agreement is observed between the three methods. The -plane directivity radiation pattern of the same structure is shown in Fig. 12, where the discrepancy in the measured results is believed to be due to the finite size of the superstrate used here. These measurement results show that the antenna directivity equals to 9.6 dB (in the broadside direction) after using the artificial magnetic superstrate compared to 6.2 dB of the patch antenna only (see Fig. 5).

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Fig. 12. The H-plane directivity radiation pattern at 2.12 GHz of the patch antenna covered with the engineered magnetic superstrate.

calculate the far-field of these two magnetic line sources representing the patch antenna. The proposed analytical model was implemented and verified by a comparison with a full-wave simulator for the case of patch antenna covered with different conventional superstrate layers. As a practical application, the radiation pattern of a patch antenna covered with an artificial magnetic superstrate was calculated using the analytical method. The broadside coupled split ring resonator (SRR) inclusions acting as building blocks for the artificial magnetic superstrate were characterized analytically to obtain its effective permeability and permittivity to be used in the analytical model of the whole radiating system. Measurement results were provided to support the analytical solution. The directivity of the patch antenna covered with the artificial superstrate was improved by about 3.4 dB while maintaining a relatively low profile of the whole structure. The analytical formulation introduced here can be extended for the case of a patch antenna embedded in a multilayered artificial dielectric structure. ACKNOWLEDGMENT The authors wish to acknowledge Dr. O. Siddiqui for helping with the experimental part of this work.

Fig. 13. The return loss of the microstrip antenna covered with the artificial magnetic superstrate.

Fig. 13 shows the reflection coefficient in dB of the microstrip antenna with the artificial magnetic superstrate, good agreement is observed between the CST and the measured results. We note that the feed location had to be slightly adjusted to achieve good matching after using the superstrate due to the loading effect of the superstrate. The overall profile of the structure is only where is the free-space wavelength at the resonance frequency. The analysis of the MPA covered with engineered superstrate are performed using Intel(R) Core(TM)2 Quad CPU @2.83 GHz machine, the proposed analytical technique requires 6 minutes and 280 megabytes of RAM using MATLAB, while the CST simulations for the same structure requires about 5 hours and 2 gigabytes of RAM when 45 cells/wavelength with total of 2.6 M mesh-cells were used for the entire structure. However, the analytical technique as presented in this work is not capable of determining the input impedance. We emphasis that the analytical method presented in this paper is not restricted to the case of artificial magnetic superstrate, and can be used for analysis of any superstrate made of engineered structure (positive or negative permeability and/or permittivity) or naturally available material. V. CONCLUSIONS A fast and accurate analytical technique was developed to calculate the radiation field of an enhanced directivity microstrip antenna covered with an artificial magnetic superstrate. The analytical formulation is based on using the cavity model to replace the patch antenna by two magnetic line sources. Then, the reciprocity theorem and transmission line analogy were used to

REFERENCES [1] N. Alexopoulos and D. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 32, no. 8, pp. 807–816, Aug. 1984. [2] D. Jackson and N. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 976–987, Sep. 1985. [3] D. R. Jackson, A. A. Oliner, and A. Ip, “Leaky-wave propagation and radiation for a narrow-beam multiple-layer dielectric structure,” IEEE Trans. Antennas Propag., vol. 41, no. 3, pp. 344–348, Mar. 1993. [4] T. Zhao, D. R. Jackson, J. T. Williams, H. Y. D. Yang, and A. A. Oliner, “2-d periodic leaky-wave antennas-Part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [5] G. Lovat, P. Burghignoli, and D. R. Jackson, “Fundamental properties and optimization of broadside radiation from uniform leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1442–1452, May 2006. [6] A. Foroozesh and L. Shafai, “Effects of artificial magnetic conductors in the design of low-profile high-gain planar antennas with high-permittivity dielectric superstrate,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 10–13, 2009. [7] H. Attia, L. Yousefi, M. M. Bait-Suwailam, M. S. Boybay, and O. M. Ramahi, “Enhanced-gain microstrip antenna using engineered magnetic superstrates,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 1198–1201, 2009. [8] A. Foroozesh and L. Shafai, “Investigation into the effects of the patchtype fss superstrate on the high-gain cavity resonance antenna design,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 258–270, Feb. 2010. [9] O. M. Ramahi and Y. T. Lo, “Superstrate effect on the resonant frequency of microstrip antennas,” Microw. Opt. Tech. Lett., vol. 5, no. 6, pp. 254–257, Jun. 1992. [10] W. R. Deal, N. Kaneda, J. Sor, Y. Qian, and T. Itoh, “A new quasiYagi antenna for planar active antenna arrays,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 6, pp. 910–918, Jun. 2000. [11] H. Vettikalladi, O. Lafond, and M. Himdi, “High-efficient and highgain superstrate antenna for 60-GHz indoor communication,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 1422–1425, 2009. [12] R. Mittra, Y. Li, and K. Yoo, “A comparative study of directivity enhancement of microstrip patch antennas with using three different superstrates,” Microwave Opt. Tech. Lett., vol. 52, no. 2, pp. 327–331, Feb. 2010. [13] Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, “Application of electromagnetic bandgap (EBG) superstrates with controllable defects for a class of patch antennas as spatial angular filters,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 224–235, Jan. 2005.

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[14] H. Attia and O. M. Ramahi, “EBG superstrate for gain and bandwidth enhancement of microstrip array antennas,” in Pro. IEEE AP-S Int. Symp. Antennas Propag., San Diego, CA, 2008, pp. 1–4. [15] H. Attia, L. Yousefi, O. Siddiqui, and O. M. Ramahi, “Analytical formulation of the radiation field of printed antennas in the presence of artificial magnetic superstrates,” in Proc. META’10, the 2nd Int. Conf. on Metamaterials, Photonic Crystals and Plasmonics, Cairo, Egypt, 2010, pp. 587–591. [16] A. Foroozesh and L. Shafai, “Size reduction of a microstrip antenna with dielectric superstrate using meta-materials: Artificial magnetic conductors versus magneto-dielectrics,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., Albuquerque, NM, Jul. 2006, pp. 11–14. [17] M. Latrach, H. Rmili, C. Sabatier, E. Seguenot, and S. Toutain, “Design of a new type of metamaterial radome for low frequencies,” in Proc. META08 and NATO Adv. Res. Workshop: Metamater. Secure Inf. Commun. Technol., Marrakesh, Morocco, May 2008, pp. 202–211. [18] X. H. Wu, A. A. Kishk, and A. Glisson, “A transmission line method to compute the far-field radiation of arbitrarily directed hertzian dipoles in a multilayer dielectric structure: Theory and applications,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2731–2741, Oct. 2006. [19] X. H. Wu, A. A. Kishk, and A. Glisson, “A transmission line method to compute the far-field radiation of arbitrary Hertzian dipoles in a multilayer structure embedded with PEC strip interfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3191–3198, Nov. 2007. [20] Y. Lo, D. Solomon, and W. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, no. 2, pp. 137–145, Mar. 1979. [21] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [22] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [23] K. Carver and J. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 2–24, Jan. 1981. [24] S. Maslovski, P. Ikonen, I. Kolmakov, and S. Tretyakov, “Artificial magnetic materials based on the new magnetic particle: Metasolenoid,” J. Progr. Electromagn. Res. (PIER), vol. 54, no. 9, pp. 61–81, Sep. 2005. [25] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 135–146, Jan. 2006. Hussein Attia (S’99) was born in Zagazig, Egypt, in 1977. He received the B.Sc. (highest honors) and M.A.Sc. degrees from Zagazig University, in 1999 and 2006, respectively, both in electronics and communication engineering. He is currently working towards the Ph.D. degree at the University of Waterloo, Waterloo, ON, Canada. From 1999 to 2006, he was appointed as a lecturer assistant at Zagazig University, where he was involved with teaching, academic advising and research. Since 2007, he has been a teaching assistant and Lecturer (part-time) at the University of Waterloo. His research interests include high-gain handset antennas, analytical techniques for electromagnetic modeling, engineered magnetic metamaterials and electromagnetic interference (EMI). Mr. Attia is the recipient of a Graduate Scholarship (2007–2010) from Zagazig University.

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Leila Yousefi (M’09) was born in Isfahan, Iran, in 1978. She received the B.Sc. and M.Sc. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2000 and 2003, respectively, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2009. Currently, she is working as a Postdoctoral Fellow at the University of Waterloo. Her research interests include metamaterials, miniaturized antennas, electromagnetic bandgap structures, and MIMO systems.

Omar M. Ramahi (F’09) was born in Jerusalem, Palestine. He received the B.S. degrees in mathematics and electrical and computer engineering (summa cum laude) from Oregon State University, Corvallis, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana-Champaign. He held a visiting fellowship position at the University of Illinois at Urbana-Champaign and then worked at Digital Equipment Corporation (presently, HP), where he was a member of the alpha server product development group. In 2000, he joined the faculty of the James Clark School of Engineering, University of Maryland at College Park, as an Assistant Professor, a tenured Associate Professor, and was also a faculty member of the CALCE Electronic Products and Systems Center. Presently, he is a Professor in the Electrical and Computer Engineering Department and holds the NSERC/RIM Industrial Research Associate Chair, University of Waterloo, Ontario, Canada. He holds cross appointments with the Department of Mechanical and Mechatronics Engineering and the Department of Physics and Astronomy. He has authored and coauthored over 240 journal and conference papers. He is a coauthor of the book EMI/EMC Computational Modeling Handbook (Springer-Verlag, 2001, 2nd ed.). Dr. Ramahi serves as an Associate Editor for the IEEE TRANSACTIONS ON ADVANCED PACKAGING and as the IEEE EMC Society Distinguished Lecturer.

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Multi-Frequency, Linear and Circular Polarized, Metamaterial-Inspired, Near-Field Resonant Parasitic Antennas Peng Jin, Member, IEEE, and Richard W. Ziolkowski, Fellow, IEEE

Abstract—Several metamaterial-inspired, electrically small, near-field resonant parasitic antennas are presented. Both electric and magnetic couplings to the parasitic are compared and contrasted. The electric-coupled versions are shown to be more efficient and to have more bandwidth. The evolution of circular polarized designs from their linear counterparts by introducing multiple parasitics having different resonant frequencies is demonstrated. Single L1 and dual band L1/L2 GPS systems are emphasized for practical illustrations of the resulting performance characteristics. Preliminary experimental results for a dual band, circularly polarized GPS L1/L2 antenna are provided and underscore several practical aspects of these designs. Index Terms—Antenna theory, electrically small antennas, parasitic antennas, polarization, Q factor.

I. INTRODUCTION

A

LTHOUGH double negative (DNG) metamaterials (MTMs) were proposed over forty years ago [1], they have been experimentally demonstrated only in recent years [2], [3]. The adaptation of a variety of epsilon-negative (ENG), mu-negative (MNG), and DNG metamaterials or simply metamaterial unit cells to achieve enhanced performance characteristics of antenna systems has since received considerable research attention. This includes studies, for instance, of small antennas [4]–[20], multi-functional antennas [21]–[25], infinite wavelength antennas [26]–[28], patch antennas [29]–[32], and leaky-wave antenna arrays [33], [34]. In the pioneering work [35], split ring resonators (SRRs) were adopted to provide an artificial, negative permeability on the high frequency side of their resonance frequencies. Since then, SRRs have been used to achieve magnetic effects from radio up to optical frequencies. Moreover, they have been applied successfully in several realizations of DNG MTMs. Different from the original SRR, which is composed of two concentric split rings in one cell, the capacitively-loaded loop Manuscript received November 11, 2009; revised October 21, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by DARPA Contract HR0011-05-C-0068 and in part by ONR Contract H940030920902. P. Jin was with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA. He is now with Broadcom Corp., Irvine, CA 92617 USA (e-mail: [email protected]). R. W. Ziolkowski is with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123053

(CLL) is a simplified inclusion. By adjusting the shape of its gap to control its capacitance, a CLL can also be made to be resonant at low frequencies, such as those in the radio frequency bands. The CLLs have been used to realize MNG MTMs [36], which in turn have been used to achieve artificial magnetic conductor (AMC) structures for low profile antenna applications. Moreover, it has been demonstrated that an idealized electrically small MNG MTM shell surrounding a loop antenna can be used to achieve a high radiation efficiency in addition to nearly complete matching to a 50 source [4]. Similarly, it has been shown [4], [8] that an epsilon-negative (ENG) shell can also be designed along with an electrically small dipole antenna to achieve a well matched, highly efficient radiating system. In contrast, it has further been shown that such high overall efficiencies can be achieved in practice with a single MNG-based (ENG-based) unit cell that is designed to act as a resonant parasitic element in the extreme near field of a driven, electrically small loop (dipole) antenna [11]. Despite the fact that either only one MNG or ENG unit cell is used and not a complete metamaterial, the predicted performance characteristics of these metamaterial-inspired antennas, which are comparable to their idealized metamaterial-based shell counterparts, have been verified [11], [15], [18], [20]. In the above MNG-noted efforts, the cut loops were excited by magnetic fields oriented perpendicular to the plane that contains them. This coupling mechanism has been described as a magnetic excitation of the magnetic resonance (MEMR) [37]. Such an MEMR can also be achieved, for example, with a monopole-SRR composite antenna by orienting its parasitic SRR element properly to capture the azimuthal magnetic flux created by the driven monopole [9]. On the other hand, it has also been demonstrated that a magnetic resonance can be excited by an electric field across the gap of the SRR [37]. This behavior is termed an electric excitation of the magnetic resonance (EEMR). In the corresponding metamaterial, the EEMR leads to an effective negative permittivity near the magnetic resonance frequencies. Furthermore, a split-ring antenna has been proposed [14] which combines a coax-fed arc monopole with a broadside-coupled SRR. The coupling mechanism in this case is a mixed type. Because the arc monopole and the SRR both lie in the same plane, it is easy to see that the magnetic field of the arc monopole drives the SRR. On the other hand, because the arc monopole itself is very close to the gap of the SRR, its electric field across the gap also drives the SRR. It will be demonstrated in this paper that CLLs can be designed in conjunction with a monopole that is coaxially-fed

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JIN AND ZIOLKOWSKI: MULTI-FREQUENCY, LINEAR AND CIRCULAR POLARIZED, METAMATERIAL-INSPIRED

through a finite ground plane to be mainly an EEMR-based antenna. However, because it mixes an electric-driven electrically small antenna (ESA) with a magnetic-based near-field resonant parasitic (NFRP), this possibility was not expected to work well from the idealized metamaterial-based shell design approach. Moreover, in comparison to the arc monopole-SRR antenna [14], its has a simpler structure and can be used for the more complicated multi-band and circular polarized antenna designs introduced below. While there are no MTMs involved in any of these designs, i.e., only one or a few MTM-inspired NFRPs will be used, the basic excitation and response concepts derived from MTM analyses will guide the element choices and their configurations relative to each other and to the coax-fed monopole. In particular, it will be demonstrated that an electrically small metamaterial-inspired NFRP CLL element can be excited by an electrically small monopole in such a manner that the overall system is nearly completely matched to a 50 source and radiates with high efficiency. Utilizing several parametric studies, comparisons between the electrically coupled and magnetically coupled CLL-based NFRP antennas will be made. It will be shown that with either excitation, the simple cases have comparable performances. However, when multiple CLL elements are introduced for any of the multi-band designs, the electrical excitation leads to simpler, more manageable configurations. Several electrically coupled CLL-based NFRP antennas will be reported; they include single- and dual-band designs. In particular, electrically small single- and dual-band GPS L1/L2 designs will be emphasized, i.e., designs for the frequencies: 1575.42 MHz (GPS L1) and 1227.6 MHz (GPS L2). Coupling between the requisite, single and multiple CLL elements will be discussed. For the dual-band antennas it will be demonstrated that a design composed of an orthogonal arrangement of two CLL elements leads to the least coupling between them, as well as to the most flexibility for tuning their resonances and impedance properties, and, hence, to the best overall system performance. Based on this orthogonal configuration, both single- and dual-band electrically small CLL-based NFRP antennas that produce linear (LP) and circular polarization (CP) will also be reported. This paper is arranged as follows. Comparisons between the electrically and magnetically coupled CLL-based NFRP antennas will be discussed in Section II. The impact of both symmetrical and non-symmetrical NFRP CLL elements and their excitation by the driven ESAs on the overall system performance will be emphasized. The dual-band and CP designs will be introduced in Sections III and IV, respectively. Preliminary experimental results for the dual-band GPS L1/L2 CP antenna introduced in Section IV are provided to confirm the basic tenets in our design methodology. Conclusions are given in Section V. In all cases, the simulations were performed with ANSOFT’s High Frequency Structure Simulator (HFSS). Consequently, as required by the HFSS radiation condition, the finite ground planes in the simulations are in size, where corresponds to the lowest frequency in the as the given frequency sweep. Throughout, we take electrically small antenna criterion, where is the radius of the smallest enclosing sphere of the entire system and being the wavelength of the resonance frequency of

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Fig. 1. Monopole driven CLL-based NFRP antennas. (a) Magnetic coupling, (b) electric coupling.

the antenna. All of the antennas introduced below are designed using 2 oz, 0.7874 mm (31 mil) thick, Rogers Duroid 5880 high frequency laminate material. II. ELECTRICALLY COUPLED CLL-BASED NFRP ANTENNAS The CLL-based NFRP antennas under consideration could have as their driven elements either a monopole antenna or a semi-loop antenna coaxially-fed through a finite ground plane. The important issue is how the NFRP element is excited, either by the corresponding electric or magnetic fields. A monopole-driven, parasitic-SRR composite antenna system was introduced in [9]; it had a monopole-NFRP coupling mechanism similar to the one shown in Fig. 1(a), i.e., the component of the SRR element is driven directly by the magnetic field generated by the driven monopole. It has a large net magnetic flux through the SRR element while the electric driven gap effects are small. Therefore, those antennas are primarily magnetically-driven and, hence, are MEMR-based. On the other hand, as shown in Fig. 1(b), the CLL element can component of the electric field generated be driven by the by the monopole. For these monopole driven CLL-based NFRP antennas, which are achieved by placing the driven monopole in the immediate neighborhood of a resonant CLL element, the HFSS-predicted overall efficiencies, i.e., the ratio of total % when radiated power to the total input power, are . Similar results are obtained with the loop driven, magnetically- and electrically-coupled configurations shown, respecand are, respectively, tively, in Fig. 2(a) and (b), where the contributing components of the magnetic and electric fields generated by the loop. Therefore, in practice, the single CLL element-based systems can be magnetically- or electrically-coupled to a monopole to achieve an efficient ESA. We note that because the NFRP is a CLL-element in all four cases, the antenna radiates as a magnetic dipole with its directivity being maximum in the direction normal to its finite ground plane. It is important to note that the monopole in the antenna shown in Fig. 1(b) is located in the center of the CLL. As a result, the magnetic field generated by the monopole has a zero net magnetic flux through the CLL element. Consequently, this NFRP element is clearly driven by the electric field generated by the monopole across its capacitive gap and, hence, it is an EEMR element. On the other hand, if the CLL element were symmetric, e.g., if there were gaps at both ends, then this desired loop-mode current behavior on the CLL element can not be obtained when the monopole is centered, rather an electric mode would ensue.

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Fig. 2. Loop driven CLL-based NFRP antennas. (a) Magnetic coupling, (b) electric coupling.

When the CLL element is symmetric, the loop-mode is facilitated by offsetting the monopole. The CLL element is thus driven by both the electric and magnetic fields of the monopole. Nonetheless, the electric coupling remains dominant since 1) the strength of the electric field is dominant in the extreme near field of the monopole, and 2) only a small net magnetic flux occurs. We further note that our HFSS simulation studies of this class of electrically-coupled CLL-based NFRP antennas has revealed that when the driven monopole is located further from the gap of the CLL element, its size had to be increased in order to maintain similar performance levels, i.e., it had to be made larger to generate the same level of electric field in that gap to be resonant at the same frequency. Consider again the monopole-driven CLL-based NFRP antenna shown in Fig. 1(b). If the exterior and interior radii of the CLL element are, respectively, 15.0 mm and 13.0 mm; if the width and height of the monopole are, respectively, 1.4 mm and 13.9 mm; and if the gap height is 0.15 mm, a resonance exMHz. Note that for practical reasons, we ists at reaches its minimum use throughout the frequency where value as the resonant frequency instead of the precise frequency where the total input reactance is zero. While the difference is very small, we make this choice to optimize the accepted power in all cases. The HFSS-predicted performance characteristics at were giving dB, %, and , where if RE is the radiation efficiency of the system, the lower bound on the quality factor is related to the Chu value [38], [39] as . The value is determined values. Note that the RE from the half-power points of the value is always greater than or equal to the OE value, i.e., the OE . value includes any mismatch effects: The corresponding HFSS-predicted gain pattern is asymmetric. in the It has its maximum gain, 4.53 dB, at plane, while its gain is 3.65 dB at . This asymmetric gain pattern is explained by the asymmetric current distribution on the CLL element, i.e., it decreases (increases) as it approaches the gap in the CLL element (the ground plane). The total current creating the radiated field can be separated into the “loop current” on the CLL structure and the current on the monopole. The loop current dominates the behavior. Consequently, to achieve a symmetric pattern, it was realized that a symmetric CLL element would be required. The simplest symmetric CLL element is obtained by creating two equal sized capacitive gaps, one at each end of the CLL element. The resulting antenna, which has the same dimensions

Fig. 3. CLL-based NFRP antenna with two capacitively-loaded gaps.

TABLE I ELECTRICALLY COUPLED CLL-BASED NFRP ANTENNAS WITH TWO CAPACITIVELY-LOADED GAPS

as the single gap version, is shown in Fig. 3. The CLL element and the printed monopole are located on opposite sides of the Duroid sheet. The center of the monopole is offset 7.5 mm from the geometric center of the CLL element. However, with no further modifications, this antenna was found to resonate at a much higher frequency. This higher resonance frequency behavior is consistent with the multi-gap results reported in [40]. In particular, since the two capacitors created by the gaps are then in series, their total capacitance is smaller than the original value. Since the resonance frequency is (1) and the total effective capacitance has been decreased, the resonance frequency increases. To determine how to maintain the resonance frequency near its original value, a parametric study of the two-gap CLL-based NFRP antenna with the offset monopole was undertaken. The lumped element LRC boundary condition available in HFSS was used to mimic a lumped element capacitor placed across the gaps of the CLL element. This allowed us to change the gap capacitance without having to redesign the CLL structure to achieve the same value. These simulation results showed that the copper loop width (W) affects the CLL antenna performance in many ways, . A summary of these results is particularly the antenna’s given in Table I. We utilize the similarity between the Duroid version of the CLL-based NFRP antenna and a small wire loop antenna to explain its performance, referring often to the results in [41]. A thinner loop width yields a larger effective radiation loop area, which in turn produces a correspondingly larger radiation resistance . On the other hand, a thinner loop width also produces a higher ohmic resistance, which in turn . The radiation means that it causes a larger conductive loss and, hence, the depend efficiency on these terms. Unfortunately, we found no straightforward

JIN AND ZIOLKOWSKI: MULTI-FREQUENCY, LINEAR AND CIRCULAR POLARIZED, METAMATERIAL-INSPIRED

Fig. 4. HFSS-predicted results for the two-gap CLL-based NFRP antenna. (a) Magnitude of S , (b) input impedance.

Fig. 5. The HFSS-predicted gain patterns (dB) of the two-gap CLL-based NFRP antenna are symmetric and has their maxima in the direction orthogonal to the ground plane.

relations amongst and , the width of the trace. Our set of simulations showed that when mm, the and are maximum and the smallest width tested, both RE value is the lowest of the set. With a larger , the RE is improved at first and then remains relatively unchanged up to mm. The reaches the maximum width tested, mm to mm. Fair its lowest values around comparisons between the cases were obtained by increasing the capacitances as was increased to maintain approximately the same resonant frequency, i.e., a thicker produces a smaller effective inductance . These trends were found to be the same for both the electrically- and magnetically-coupled CLL-based NFRP antennas. To illustrate the resulting behavior of the electrically-driven, electrically-coupled, two-gap NFRP CLL design, the HFSSpredicted current distribution for the 2.0 mm wide, 15.0 mm outer radius CLL element with 0.777 pF capacitors across their 0.2 mm gaps is shown in Fig. 3. It clearly confirms the presence of the loop mode. The corresponding values of the magniand the input impedance are shown, respectively, in tude of Fig. 4(a) and (b). At the resonance frequency MHz, one has giving dB, % %, . Moreover, the HFSS-predicted and gain pattern, shown in Fig. 5, is now symmetric and exhibits a magnetic dipole character (broadside maximum). Compared to and the corresponding one-gap version whose %, this two-gap CLL-based NFRP antenna with

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Fig. 6. Circuit models for the CLL-based NFRP antenna. (a) Magneticallycoupled version, (b) electrically-coupled version.

mm has a larger and a lower RE value. This behavior occurs because the one-gap version works in a mixed current mode (monopole and loop contributions) while the two-gap version works primarily in only its loop mode. The same parametric study was also performed with the magnetically-driven, magnetically-coupled CLL-based NFRP antennas. One surprising outcome of these parametric study results was that the responses of the electrically- and magnetically-coupled versions were really very similar to each other in every respect. However, the electrically-coupled designs were found in general to perform a little better in both their RE and values. For example, when mm, pF, the results for the magnetically-coupled two-gap CLL-based NFRP MHz, %, and antenna are . From such comparisons it was concluded that the radiation performance characteristics of these single band, LP antennas are determined mainly by the radiating NFRP CLL element itself rather than by the actual coupling mechanism. However, a significant difference was recognized in their input impedances. The electrically-driven, electrically-coupled CLL-based NFRP antennas operate most effectively at their first resonance frequency. On the other hand, the magnetically-driven, magnetically-coupled CLL-based NFRP antennas have very closely spaced first anti-resonance and second resonance frequencies. value occurs somewhere in the frequency inThe lowest terval between these values. As a result, they are more difficult to tune than their electrically-coupled counterparts. It must also be emphasized that this is yet different from the magnetically-driven, magnetically-coupled, 3D magnetic EZ antennas [11], [15], which always radiate most effectively at their first anti-resonance. Consequently, these comparison studies determined that the electrically-driven, electrically-coupled designs would provide improved performance, simpler tuning capabilities, and more degrees of freedom for when they were applied to the more complicated multi-band and CP designs introduced below. Circuit models of the CLL-based NFRP antennas were also developed to help understand the coupling mechanisms and the antenna performances discussed above. Our magnetically-coupled model, Fig. 6(a), relies on related results for the analogous magnetic-based metamaterial inclusions [42]. The source is the , through the inductor. Thus, the flux of the magnetic field, CLL element acts like a series LRC circuit driven by a current source. On the other hand, the electrically-coupled CLL element, Fig. 6(b), which also acts like an LRC circuit, is driven , through by a voltage source, the flux of the electric field, the capacitor.

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Fig. 7. Basic protractor antenna with its design dimensions (mm).

Fig. 8. HFSS predicted results for the protractor antenna design. (a) Magnitude of S , (b) gain patterns (dB).

Our study of the symmetric two-gap CLL-based NFRP antenna using the HFSS RLC boundary element in the gaps reaffirmed the need to produce larger capacitance values to maintain the resonance frequency when multiple gaps are introduced. However, to realize such designs in a distributed element fashion, the gaps themselves, when their size is limited by practical fabrication constraints, do not produce enough capacitance. A larger capacitance element can be implemented by inserting an interdigitated capacitor with multiple fingers or by introducing naturally occurring materials with high permeability in the gap between the cut ends to the CLL element and the ground plane. We found an alternate, more effective distributive version of the CLL element also produces the desired increase in capacitance. By introducing legs to the CLL element parallel to the ground plane that are only slightly raised above it, one can produce substantial additional capacitance. We refer to this CLL-based NFRP antenna as a protractor antenna, i.e., the NFRP structure looks like a standard protractor which one would use in a basic geometry course to measure angles. A simple gap is also introduced into the center of its bottom portion. This allows an extra degree of freedom, i.e., the lengths of the legs can be adjusted independently, which has applications to cases in which the monopole is significantly offset. The basic protractor antenna design is shown in Fig. 7 along values and gain with its dimensions. The HFSS-predicted pattern results at its resonance frequency for this protractor antenna are shown, respectively in Fig. 8(a) and (b). The gain pattern is symmetric and reaches its maximum value of 5.683 dB

Fig. 9. Modified protractor antenna designs. (a) With inter-digitated legs, and (b) with extra capacitive strips.

at . Again, the asymmetric electrical coupling in the near field, i.e., the offset monopole, is needed to produce the loop mode. The symmetric NFRP CLL structure then produces the symmetric gain pattern in the far field. This protractor anMHz with tenna has its resonance at dB, % and . gets It should be noted that the effective capacitance larger with bigger (thicker and longer) protractors legs. However, as seen in Fig. 7, larger protractor legs also mean a smaller effective (flux) area for the radiating semi-loop and, hence, a . Consequently, when the leg smaller effective inductance size is increased from zero, the resonant frequency first gets lower and then gets higher after the leg size is increased to a level where it negatively impacts the inductance value. On the other hand, one can increase the capacitance by making the distance between the ground plane and those legs less. However, in practice, because the Duroid material would have to be etched to achieve the desired dimensions, there is a practical limit to this gap height dimension. This is generally about . This is also the reason we have restricted ) in all of the our smallest gap size to be 0.2 mm ( designs reported here. Consequently, we note that cannot be lowered as much as one might desire simply by monotonously increasing the width of the protractor legs or monotonously decreasing the leg-ground plane gap size. If desired, one could further lower the resonance frequency by modifying directly the protractor legs to increase the effective capacitance. For example, the protractor legs could be augmented by introducing interdigitated capacitors into them. Another possibility would be to add an additional piece of copper connected to the ground plane on the side of the Duroid sheet opposite to the protractor element, i.e., on the same side as the monopole. This would increase the coupling between the ground plane and the protractors legs, which in turn would increase the effective capacitance. The bottom portions of the protractor antennas with these modifications are illustrated in Fig. 9. These modified designs can readily achieve lower resonant frequencies (around 1 GHz) with nearly complete matching to the source. However, without further considerations, their OE values are decreased to around 70% because of their smaller values.

JIN AND ZIOLKOWSKI: MULTI-FREQUENCY, LINEAR AND CIRCULAR POLARIZED, METAMATERIAL-INSPIRED

For the remainder of this paper, we will continue with the Duroid-based protractor antenna design given in Fig. 7. This allows us to maintain a reasonably high OE value for a reasonably value. It also provides us with a means to produce the small additional capacitance needed to achieve lower resonance frequencies. Moreover, the NFRP protractor design improves the electric coupling between it and the driven monopole antenna. The latter is usually smaller in size than the parasitic element, but the choice of its location is more flexible. With better coupling, the resonance frequency can be maintained with smaller sized monopoles. Other than the width of the protractor element, the sizes of its legs are parameters that can be used to tune more finely the resonance frequency of the antenna. This particular tuning approach has proved very effective in the following dual-band and CP applications.

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Fig. 10. Dual-band GPS L1/L2 protractor antenna with perpendicular CLLbased NFRP elements and single coaxial-fed two printed monopoles. (a) 3D view, (b) feed structure with dimensions (mm).

III. DUAL-BAND CLL-BASED NFRP ANTENNAS Multi-band NFRP antennas have been achieved by locating multiple parasitic resonators in the near field of the driven monopole antenna [24]. Those multi-band antennas are electrically-coupled to electric-based NFRP elements. Their far field patterns are variations of the basic electric monopole behavior, i.e., the fields have nulls in the direction orthogonal to the ground plane. The same multiple electrically small resonator idea is applied here for the monopole driven, electrically-coupled CLL-based NFRP antennas. However, in contrast, the far field patterns will be variations of the basic magnetic dipole behavior. With the ground plane present in all of these designs, the magnetic dipole has an advantage over the electric monopole (dipole) if one desires a broadside pattern, i.e., one with its peak orthogonal to the ground plane. Another advantage of the CLL-based NFRPs is that we can design them to have their effective magnetic dipoles oriented along the ground plane in different directions and, in particular, in directions that are orthogonal to each other. In contrast, there is only one direction orthogonal to the ground plane available with the monopole behavior. This multiple magnetic dipole option allows us to introduce the circular polarization (CP) designs in the next section. It is not available to the electric monopole type of designs. To obtain symmetric gain patterns at each resonance frequency, the orthogonal protractor NFRP element design shown in Fig. 10 is proposed. It is chosen over designs utilizing two parallel CLLs because in those designs, the magnetic field generated by one CLL will directly drive the other CLL, resulting in a strong mutual coupling. It was expected that the orthogonal design would minimize the mutual coupling between the two protractor NFRP elements and, hence, they could be nearly independently tunable. Moreover, the two protractor elements have the same outer radius, simplifying the design. The thicker protractor element (shown in red) is responsible for the L1 resonance; the thinner one (shown in blue) then is responsible for the L2 resonance. A partial slot is cut from the top of the L1 protractor element to accommodate a compact winebox-like integration of the L2 protractor element with it. The final dimensions of the GPS L1 protractor element are shown in Fig. 11 with the distance between the two protractor legs being 4.1 mm and the gap between the protractor leg and

Fig. 11. Dimensions (mm) of the GPS L1 protractor element of the antenna shown in Fig. 10.

ground plane being 0.2 mm. The GPS L2 element is a regular NFRP protractor as shown in Fig. 7, where mm, mm, mm, and mm. The radius of the inner coaxial line is 1.5 mm. The slot on the L1 CLL is cut such that there is a 0.5 mm space between the L1 protractor element and the nearby dielectric of the L2 protractor element. In the dual-band design shown in Fig. 10, a rectangular driven monopole was adopted initially to couple the two protractor NFRP elements. This was the simplest feed structure that geometrically fit the two-orthogonal plane design. However, to further the independence between the two NFRPs, the alternate design illustrated in Fig. 10(a) was developed. Instead of one solid rectangular monopole, two printed monopoles that are connected directly to the center conductor of the coax-feedline are used to drive the two protractor NFRP elements individually. This feed structure is shown in Fig. 10(b). A square piece of copper, which has a horizontal cross-section equal to 2.76 2.76 mm and is 0.2 mm thick, is first attached to the top of the inner conductor of the coaxial feedline. Then the two printed monopoles, which are the sides of the Duroid pieces opposite to the protractor NFRP elements, are soldered to it. Besides easier fabrication, such a feeding structure also provides more flexibility to the dual-band design. The HFSS-predicted values, and the gain patterns at current distributions and the resonance frequencies for the above dual-band protractor antenna with two orthogonally-oriented NFRP elements are shown, respectively, in Figs. 12 and 13. Nearly complete impedance matching at the two resonance frequencies is easily achieved by adjusting the monopole lengths individually. As

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Fig. 12. HFSS-predicted jS j values and current distributions for the dualband GPS L1/L2 protractor antenna shown in Fig. 10.

between two of them. However, when there is a ground plane present, there is only one direction available when an electric monopole is oriented orthogonal to it to achieve an efficient radiation process. Moreover, when an electric monopole antenna is electrically small, it only radiates in the monopole mode. There are no other modes available to exploit for the CP process. Consequently, the LP, ground plane-based electric mode NFRP antennas are not suitable for CP situations. In contrast, as noted above, the CLL-based NFRP protractor antennas act as magnetic dipoles oriented parallel to the ground plane and radiate LP fields with their maxima normal to it. Consequently, they do not suffer from those drawbacks and are well-suited to CP applications. A. Electrically Small Single-Band CP Protractor Antenna

Fig. 13. HFSS-predicted gain patterns (dB) for the dual-band GPS L1/L2 protractor antenna shown in Fig. 10. (a) GPS L1 resonance, (b) GPS L2 resonance.

shown in Fig. 10(b), the heights of the two printed L2 and L1 monopoles are 14.2 mm and 14.0 mm, respectively, and their widths are the same, 1.4 mm. At the GPS L2 resonance % MHz (1227.6 MHz), one finds diand a maximum gain of 5.4 dB in the broadside rection. At the GPS L1 resonance (1575.42 MHz), one finds % MHz and a maximum gain of 6.42 dB in the broadside direction. At the shorter GPS L1 ; it has wavelength, the antenna has at the GPS L2 wavelength. Consequently, it is electrically small at both its L1 and L2 resonances. Moreover, as shown in Fig. 13, it has symmetrical, broadside gain patterns at both of those frequencies. Each protractor element acts essentially as an independent LP radiator. Furthermore, because of the orthogonal configuration of the two NFRP elements, it is noted that the current is almost entirely on only one element at each of the resonance frequencies. Thus, as desired, there is little mutual coupling between the two protractor NFPR elements. This property greatly decreased the time and effort required to tune the dual-band design at both resonance frequencies. IV. CIRCULARLY POLARIZED PROTRACTOR ANTENNAS There are two typical modalities to achieve circular polarization (CP). One is to introduce two perpendicular, linearly polarized (LP) sources that have the same magnitude and are driven phase difference between them. The other is to make with a the antenna large enough that several radiating modes are availphase difference able and then to introduce the requisite

If the two protractor NFRP elements and their coupling to the driven monopole are made the same, the resulting two magnetic dipoles will radiate with the same magnitude. Therefore, such a perpendicular structure could be used to implement the desired CP modality. To design a CP ESA, the major challenge phase difference between the then is to achieve the requisite two radiating elements. However, CP ESAs are seldom found phase difin practice because it is difficult to achieve this ference in an electrically small space, i.e., the typical phase shifters themselves would have to be electrically smaller than the ESAs so everything could fit into the appropriately sized spherical volume. There are, however, notable exceptions. For instance, the electrically small folded spherical helix antenna [43] is a CP radiator. It should be noted that for the folded spherical helix antenna, although its overall size is electrically small, the individual wire lengths are not. This provides some design freedom to manage the needed phase differences. Another approach was reported, for example, in [44]. Two perpendicularly polarized electric dipole antennas with different lengths and closely spaced resonance frequencies fed by the same source can produce CP radiation. In particular, in the frequency interval between the two resonance frequencies, the reactance of the longer dipole is inductive and that of the shorter one is capacitive. Since the feed and, hence, the voltage is the same for both elements, then the current on the longer element has a negative phase and the current on the shorter element has positive phase. With proper arrangement of the two resonant frephase difference can be achieved between these quencies, a two dipole antennas. We exploit this latter approach to achieve our CP designs. The CP protractor antenna shown in Fig. 14 was designed with this principle to be resonant at the GPS L1 frequency, 1575.42 MHz. The configuration of the CP protractor antenna and its dimensions are shown in Fig. 15, where the outer radius and are both 15.0 mm, mm, mm. The protractor legs are identical and mm, mm, and for the two CLLs, where mm. On the yz-plane, the printed monopole width mm and the height mm. On the mm and xz-plane, the printed monopole width mm. The slot heights are the height mm and mm on the yz- and xz-planes,

JIN AND ZIOLKOWSKI: MULTI-FREQUENCY, LINEAR AND CIRCULAR POLARIZED, METAMATERIAL-INSPIRED

Fig. 14. 3D view of the GPS L1 CP protractor antenna with two orthogonallyoriented CLL-based NFRP elements.

Fig. 15. Dimensions of the GPS L1 CP protractor antenna shown in Fig. 14. (a) xz-plane, (b) yz-plane.

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Fig. 16. HFSS-predicted values for the GPS L1 CP protractor antenna shown in Fig. 14. (a) Magnitude of S , (b) input impedance Z , and (c) axial ratio.

respectively. The slots are cut such that there is a 0.5 mm space between the two protractor elements and the nearby dielectric of the two protractor elements. By applying slightly different CLL protractor elements, two very close and nearly independent resonance frequencies are achieved. The GPS L1 frequency lies in between those frequencies. The two printed monopoles are adjusted individually to achieve good matching at both of those value also will be small resonance frequencies so that the at the L1 frequency. values are shown in Fig. 16(a). The The HFSS-predicted presence of two resonances is readily apparent. At the L1 reso% and nance frequency, one has MHz. This 10 dB bandwidth is about two times larger than that of either of the single protractor LP antenna themselves. The input impedance is shown Fig. 16(b). One observes that the lower frequency resonance is inductive while the upper one is capacitive. The lower (higher) frequency magnetic dipole is oriented along the y-axis (x-axis). Thus, the phase along with y-axis leads that along the x-axis; the net field is proportional , giving the antenna a left-hand circular polarization to (LHCP) behavior. This LHCP characteristic was confirmed with HFSS simulation results. An exchange of the dipole axes between the low and high frequency elements would produce right-hand circular polarization (RHCP) behavior. The axial ratio of this GPS L1 CP protractor antenna is shown in Fig. 16(c). One observes that the axial ratio is 0.44 dB and it has a 7.4 MHz, 3 dB axial ratio bandwidth at the L1 resonance. The spatial beamwidth for the L1 frequency is shown in Fig. 17; the resulting 3 dB beamwidth range is . These results demonstrate that the electrically small, CP protractor antenna is an efficient CP radiator at the desired GPS L1 frequency. Note also that the antenna bandwidth has been increased by putting two closely spaced resonances together without increasing the ka value. In particular, if one only considers the bandwidth and not the mm and axial ratio and if one simply takes mm, one finds that the value can be increased to 39 MHz, from 1533.9 MHz to 1572.9 MHz. The

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011

Fig. 19. Dual-band LP protractor antenna. xz-plane view with dimensions (mm).

Fig. 17. HFSS-predicted values of the GPS axial ratio for the L1 CP protractor antenna shown in Fig. 14. The 3 dB beamwidth characterization at the GPS  < 360 ; L1 resonance indicates an axial ratio less than 3 dB for 0 45  < 45 .

0





j j

Fig. 20. HFSS-predicted S values and the current distributions at the GPS L1 (right) and L2 (left) frequencies for the dual-band LP protractor antenna shown in Fig. 19.

j j

Fig. 18. The HFSS-predicted S values indicate that the bandwidth of a GPS L1 protractor antenna can be increased simply by introducing two nearby, overlapping resonances.

HFSS-predicted values for this enhanced-bandwidth LP case are shown in Fig. 18. B. Electrically Small Dual-Band CP Protractor Antenna A dual-band CP design would involve four resonances, a pair for each operating frequency. Our design studies have shown that once two protractor elements are oriented perpendicularly to achieve a single-band CP performance, the angular degrees of freedom are expended. There is no way to introduce additional protractors for the second operating frequency without introducing unwanted strong coupling between the protractor elements. The strong coupling between the protractor elements and the desire to have a good axial ratio bandwidth causes this design approach to be highly time and effort consuming, if not totally unrealistic. For these reasons, we designed the dual-band LP CLL antenna shown in Fig. 19. It minimizes the coupling between the two sets of protractor elements. The L1 protractor element (red) remains the same as in the single-band CP design. However,

the L2 protractor element (blue) is modified to contain the additional curved protractor legs. The dimensions of this dual-band LP protractor antenna are shown in Fig. 19. This configuration ensures that the mutual coupling between the two CLL elements values and the current is decreased. The HFSS-predicted distributions at the GPS L1 and L2 resonance frequencies are shown in Fig. 20. At the GPS L1 frequency, 1575.42 MHz, % and At the GPS L2 frequency, 1227.6 MHz, % and . The independence of the two CLLs is demonstrated by the current distributions shown in Fig. 20 at the GPS L1 and L2 frequencies. At the GPS L1 resonance frequency the current is mainly distributed on the smaller (inner) protractor element, while at the GPS L2 resonance frequency it is mainly distributed on the outer protractor element. Our design experience also confirms this independence. It should be noted that, with a finite, but large enough ground plane being present, the phase center of the antenna shown in Fig. 19 remains the same (at the origin) for both the GPS L1 and L2 frequencies. This is a very important property for applications such as positioning and jamming/anti-jamming. The dual-band CP protractor antenna then follows immediately. This design is shown in Fig. 21. Some of its dimensions are given in Fig. 22. All values are in millimeters. The whole structure has a 19.4 mm radius. The inner radius dimensions mm and mm.

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= 0 8p2 = 1 13

Fig. 21. Dual-band GPS L1/L2 CP protractor antenna. (a) 3D view, (b) xy-plane, F : : mm.

The outer radius of both inner (L1) protractor elements is 12.5 mm, and the outer radius of both outer (L2) protractor elements is 19.4 mm. Thus, the widths of the inner and outer protractors are, respectively, 2.12 mm and 4.14 mm. The gap sizes between the L1 and L2 protractor elements and between , are identical, 0.3 mm, the former and the ground plane, on both the xz and yz planes. The width of the extra leg of the L2 protractor element is 0.5 mm on both the xz and yz planes. Both of the printed monopoles have the same width mm. The (gap) distance between the L1 protractor legs is 3.0 mm on both the xz and yz planes. On the xz plane, the (gap) distance between the L2 protractor legs is 2.0 mm, and is 4.0 mm on the yz plane. The width of the vertical slots on the xz and yz planes is 2.0 mm. The height of the monopoles on the xz and yz planes are shown in Fig. 22(c). Because the feeding point of each of the component dual-band LP antennas is not close to the origin, asymmetries could arise in the patterns. To achieve symmetric patterns, the feed structure is designed to be symmetric along plane orthogonal to the xy plane as shown in the Fig. 22(c). The monopoles printed on the Duroid sheets are directly connected to this feedline. The actual location of the center of the center conductor of the coax-feedline is the point ( mm, mm) on the ground plane . To minimize the mutual coupling between the two protractors on each Duroid sheet, they are designed to have a minimal amount of area between the outer radius of the L1 protractor element and the inner radius of the L2 protractor element. and axial ratio results for this dualThe HFSS-predicted band CP protractor antenna are shown in Fig. 23. At the GPS L2 % MHz, resonance frequency, the axial ratio is 0.42 dB, and the 3 dB axial ratio bandwidth is 4.3 MHz. The maximum gain is 5.36 dB in the broadside direction; the 3 dB axial ratio beamwidth is 86 . At the GPS L1 % MHz, resonance frequency, the axial ratio is 0.87 dB, and the 3 dB axial ratio bandwidth is 5.3 MHz. The maximum gain is 6.20 dB in the broadside direction; the 3 dB axial ratio beamwidth is 52 . The antenna has measured at the GPS L2 resonance wavelength. Because the inner protractor element is effectively radiating inmeadependently of the outer one, the antenna has sured at the GPS L1 resonance wavelength. Consequently, it is electrically small at both resonance frequencies.

Fig. 22. Dimensions (mm) of the dual-band GPS L1/L2 CP protractor antenna shown in Fig. 21. (a) xz-plane, (b) yz-plane, and (c) feed structure.

We note that based on the single band CP CLL antenna design, it is concluded that using two nearby resonance frequencies approach to achieve CP, the axial ratio bandwidth is about of the corresponding LP antenna. For our half of the CP design, there was a 4 MHz axial ratio requirement for each frequency, a value frequently used in practice. Consequently, we value of the dual-band LP designed the system to have protractor antenna to be no less than 8 MHz at each resonance frequency. C. Dual-Band CP Protractor Antenna Experiments A preliminary version of the dual-band GPS L1/L2 CP protractor antenna was fabricated by Boeing Research and Technology in Seattle, WA and tested at the National Institute of Standards and Technology (NIST) in Boulder, CO. The fabricated antenna is shown in Figs. 24(a)–(c). Both sets of

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Fig. 25. Measured values of the dual-band GPS L1/L2 CP protractor antenna. (a) Magnitude of S , (b) relative total radiated power, and (c) axial ratio.

Fig. 23. HFSS-predicted results for the dual-band GPS L1/L2 CP protractor antenna shown in Fig. 21. (a) Magnitude of S , (b) axial ratios.

Fig. 24. Fabricated preliminary version of the dual-band GPS L1/L2 CP protractor antenna. (a) Small ground plane view, (b) parasitic element view, and (c) feed and monopole elements view.

protractor elements were constructed with the 2.0 oz., 31 mil, Rogers Duroid 5880 high frequency laminate material specified by the design illustrated in Figs. 21 and 22. They were centered on a 119.5 mm diameter copper plate insert for ease of shipping. This configuration will be referred to as the small

ground plane version. The insert was placed into a larger (18 in 18 in mm 457.2 mm) copper ground plane for the measurements referred to as the large ground plane version. The protractor elements were recessed into two orthogonal slots that were milled approximately to a depth of 2.15 mm in the copper plate. Some variations in the depth were allowed to try to ensure the specified height of the horizontal legs of the protractor elements above the ground plane were correct. The protractor elements themselves were interleaved by introducing “keyhole” cuts. Because of these keyholes, the outer protractor element on one sheet was cut into two pieces and had to be reconnected. The reconnection was accomplished by inserting a thin piece of copper foil through the keyhole and then soldering it to both halves. The split feed design began with a 7254-CC-1 SSMA flange mount connector; it was centered at the specified . The pin had a slightly larger diameter 0.86 point mm, rather than the specified 0.80 mm value. It was extended through the ground plane, soldered to the center of the flat strip feedline that was also soldered at each end to the monopole element on each of the protractor sheets. A MDC3312 SSMA male to SMA female connector was required to then connect the feed to the vector network analyzer (VNA) cable. Total radiated power measurements of the dual-band GPS L1/L2 CP protractor antenna were made relative to a reference antenna, an ETS-LINDGREN model 3106 dual-ridged waveguide horn that is about 94% efficient in its 200 MHz–2.0 GHz frequency band. This approach was used effectively in previous electrically small antenna measurements as described, values for instance, in [11], [15], [18], [20]. The measured and relative total radiated power results for both the small and large ground plane configurations are shown, respectively, in Fig. 25(a) and (b). One clearly sees four important features from this data. One, the small and large ground plane configurations produce essentially the same behaviors. The HFSS simulations indicated that the currents associated with the antenna were highly localized to the region surrounding the protractor elements; these experimental results support those observations. Two, there are two distinct pairs of nearby resonances, i.e., the

JIN AND ZIOLKOWSKI: MULTI-FREQUENCY, LINEAR AND CIRCULAR POLARIZED, METAMATERIAL-INSPIRED

pairs of protractor elements tuned to slightly different frequencies performed as predicted. Three, the antenna exhibits very high overall efficiencies (slightly better than the horn efficiency) at those resonant frequencies. Four, the resonances occur at the wrong frequencies, i.e., they are up-shifted to higher values. In fact, the overall efficiencies were better than predicted because the resonances are at higher than expected frequencies; and, values consequently, the protractor elements are radiating at larger than specified by the designs. The corresponding axial ratio measurements in the main beam direction are shown in Fig. 25(c). These were obtained by irradiating the antenna with a linearly polarized horn antenna, measuring the values of between them as a function of the frequency, and then rotating the horn antenna 180 times by 2 increments and repeating the measurements. The axial ratio at each frequency was then . As predicted, generated as the antenna exhibited CP behavior (axial ratio below 3.0 dB) at the first pair of resonances and approached CP-like behavior at the second pair of resonances. Nonetheless, the axial ratios at both pairs of resonance frequencies were not as small as had been predicted. Because we were offered only this one opportunity for the fabrication and testing of our designs, we have tried to understand completely why the antenna as fabricated did not perform exactly as predicted. It was found from length measurements taken at NIST-Boulder that the largest fabrication error was the height of the protractor legs above the ground plane. The height was measured to be 0.6 mm instead of the design specification of 0.2 mm. As noted in Sections III and IV, this height controls the additional capacitance needed to lower the resonance frequency. The shifts in the resonance frequencies observed in the measured results shown in Fig. 25(a)–(c) were then confirmed with HFSS simulations of the antenna which included the measured lengths and the ground plane slots. Additionally, as can be observed with careful examination of Fig. 24(b), the protractor elements on one of the Duroid sheets were not identically aligned with those on the other sheet. This mismatch impacted the axial ratio values. We believe that because the Duroid sheets were simply press fitted into those slots, they pressure released themselves slightly and moved upwards in the slots by different amounts during handling of the antenna before any soldering occurred to the center conductor of the coax, the feedlines, and the monopole elements. Once soldered, these errors were locked into the structure. This was confirmed recently when we received the antenna from NIST-Boulder. We disassembled the structure, sanded down the bottoms of the protractor elements until the correct heights were obtained, and then reassembled values with a local the structure. We then measured the VNA. These values are included in Fig. 25(a) and are labeled as Small GP UA. The resonance frequencies are now very close to their originally predicted values. Unfortunately, because the solder points on the monopoles remained fixed, the feedline from the coax center conductor developed a significant bow, i.e., it was no longer parallel to the ground plane as designed when the Duroid sheets were pressed deeper into the slots. As also noted in Sections III and IV, any change in the feed structure would negatively impact the input impedances at the resonance frequencies. Nonetheless, one can clearly see that lowering the

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height of the protractor legs relative to the ground plane does decrease the resonance frequencies as predicted. Furthermore, the antenna had very high radiation efficiencies and nearly CP behavior at the resonance frequencies in its original realization. We thus anticipate that another iteration of the antenna, which would have the correct lengths for all of the components, would perform as predicted. Based on the simulated and measured results, our NFRP protractor antenna designs can be summarized as follows. The CLL-based NFRP elements of the protractor antennas are electrically coupled to their driven sources, here a monopole that was coaxially-fed through the ground plane. The overall size of the CLL protractor elements, their trace widths, their capacitive gaps and the height of their legs above the ground plane control their resonant frequencies. These design variables, along with the dimensions of the driven monopole source, provided us with enough degrees of freedom to tune the operating characteristics of the overall electrically small antenna system to achieve very high radiation efficiencies and nearly complete matching to a 50 source at selected single and multiple frequencies with either LP or CP operation. Furthermore, by driving the CLL protractor element with an offset monopole, the electrical coupling excites a loop mode whose currents radiate with maximum directivity broadside to the ground plane. If the driven monopole was not offset from the center of the protractor element, only the monopole mode would be generated which would lead to a null along the broadside direction. The broader bandwidth, multiband, and CP protractor antennas were facilitated by introducing multiple orthogonal NFRP protractor elements. As demonstrated by the experimental results shown in Fig. 25, the ground plane size has little impact on the protractor antenna performance. The currents important to the radiation process occur mainly on the NFRP protractor elements. On the other hand, the measurements showed that some care in their fabrication must be exercised to attain the design dimensions carefully. They illustrate how the design is particularly sensitive to the height and the orientation of the protractor legs and the feedlines above the ground plane. Because all the elements are designed for an electrically small, highly resonant environment, significant variations during fabrication may lead to changes in the operating characteristics of the overall antenna system. Although the initial measurement results were only partially successful, they demonstrate that, with careful fabrication, electrically small, high overall efficiency, multiband, CP antennas can be achieved. V. CONCLUSION A variety of electrically small, electrically-driven CLL-based NFRP antennas were introduced in this paper. With a parametric study and equivalent circuit models, they were compared to the more common magnetically-coupled versions. We showed that the CLL-based NFRP element can be electrically or magnetically coupled to a driven monopole or loop antenna in such a manner that the resulting system radiates efficiently in a magnetic dipole mode. Both one gap and two gap CLL elements were considered. We found that the current distribution and the far-field radiation patterns indicated that the one gap CLL-based NFRP antenna radiated in a mixed electrical and

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magnetic resonance mode and had an asymmetric gain pattern. On the other hand, we also demonstrated that the symmetric two gap CLL-based NFRP antenna with an offset driven element can be designed to radiate in a magnetic resonance mode with a symmetric radiation pattern. The equivalent circuit models revealed that the performance of the CLL-based NFRP antennas depends mainly on the properties of the radiating parasitic element rather than on the coupling method. We also showed with parametric studies that while the electrically- and magnetically-coupled CLL-based NFPR antennas had very similar operating characteristics, the electrically-coupled versions had slightly better performance characteristics, i.e., slightly higher values, and strategic advantages when RE and lower considering multiband and CP designs. The focus of our study then became monopole-driven electrically-coupled CLL-based NFRP antennas implemented with Rogers Duroid 5880 high frequency laminate material. Several methods were proposed to achieve a high enough effective capacitance to lower the resonant frequency without using any high permittivity materials. In particular, the protractor NFRP element was introduced. It was shown to be an advantageous NFRP choice because it is a simple structure with several degrees of freedom that can be used to achieve a high effective capacitance, improve the electric-coupling to the driven monopole, and produce a symmetric gain pattern. Then, by introducing two NFRP protractor elements, dual-band GPS L1 and L2 LP antennas were designed and characterized. It was shown that a parallel configuration of these two protractor NFRPs led to strong mutual coupling between them and, hence, to a degradation in the performance characteristics of the resulting antenna system while being costly to achieve in both time and effort. On the other hand, a winebox configuration in which the two protractor NFRPs were mutually orthogonal led to a design that exhibited only a minimal coupling between those two NFRPs. This orthogonal protractor design was then successfully adopted to achieve electrically small single- and dual-band LP antennas operating at the GPS L1 and L2 frequencies. It also was demonstrated that this orthogonal protractor design radiated as two perpendicular magnetic dipoles. By designing these two orthogonal magnetic dipoles to have the same amplitude and resonance frequencies which are closely spaced, we demonstrated that these two electrically small antennas could be made to be resonant at an intermediate frequency with a 90 phase difference. This led to an electrically small single-band CP protractor antenna operating at the GPS L1 frequency. It was shown that this antenna had very nice performance characteristics, including an attractive axial ratio frequency bandwidth and spatial beamwidth. To achieve a dual-band CP design with this approach, another co-planar dual-band LP protractor antenna design was introduced and its performance was characterized numerically. It was demonstrated that each protractor NFRP element of this design radiated nearly independently, i.e., that there was only a minimal coupling between the two co-planar NFRP protractor elements. Two of these dual-band LP antennas were then combined orthogonally with nearby resonance frequencies to achieve a dual-band CP design at an intermediate frequency. An electrically small dual-band CP protractor

antenna operating at the GPS L1 and L2 frequencies, with and , respectively, at the GPS L1 and L2 frequencies with greater than 70% overall efficiency, more than a 4 MHz axial ratio 3 dB bandwidth, and a 60 beamwidth at both frequencies was demonstrated. A preliminary version of the dual-band GPS L1/L2 CP antenna was fabricated and measurements of its properties were performed. The measured and overall efficiency and axial ratio values confirmed the validity of the basic design methodology introduced with all the NFRP antennas reported in this paper. ACKNOWLEDGMENT The authors would like to thank J. A. Nielsen and Dr. M. H. Tanielian of Boeing Research and Technology for the fabrication of the dual-band GPS L1/L2 CP protractor antenna and Dr. C. L. Holloway of NIST-Boulder for the corresponding measurements. The authors would like to thank the four reviewers and associate editor whose diligent efforts contributed to an improved presentation of our work. REFERENCES [1] V. G. Veselago, “Experimental demonstration of negative index of refraction,” Sov. Phys. Usp., vol. 47, pp. 509–514, Jan.–Feb. 1968. [2] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [3] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, pp. 1516–1529, Jul. 2003. [4] R. W. Ziolkowski and A. Kipple, “Application of double negative metamaterial to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 2626–2640, Oct. 2003. [5] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “Compact and low-profile metamaterial ring antenna with vertical polarization,” IEEE Wireless Propaga. Lett., vol. 4, pp. 333–336, 2005. [6] H. R. Stuart and C. Tran, “Subwavelength microwave resonators exhibiting strong coupling to radiation modes,” Appl. Phys. Lett., vol. 87, p. 151108, Oct. 2005. [7] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, pp. 1664–1653, Jun. 2006. [8] R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2113–2130, Jul. 2006. [9] K. B. Alici and E. Ozbay, “Radiation properties of a split ring resonator and monopole composite,” Phys. Stat. Sol.(b), vol. 244, pp. 1192–1196, Apr. 2007. [10] H. R. Stuart and C. Tran, “Small spherical antennas using arrays of electromagnetically coupled planar elements,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 7–10, 2007. [11] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically-small antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 691–707, Mar. 2008. [12] M. A. Antoniades and G. V. Eleftheriades, “A folded-monopole model for electrically small NRI-TL metamaterial antennas,” IEEE Wireless Propag. Lett., vol. 7, pp. 425–428, 2008. [13] D. H. Lee, A. Chauraya, Y. Vardaxoglou, and W. S. Park, “A compact and low-profile tunable loop antenna integrated with inductors,” IEEE Wireless Propag. Lett., vol. 7, pp. 621–624, 2008. [14] O. S. Kim and O. Breinbjerg, “Miniaturized self-resonant split-ring resonator antenna,” Electron. Lett., vol. 45, pp. 196–197, Feb. 2009. [15] R. W. Ziolkowski, C.-C. Lin, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “Design and experimental verification of a 3D magnetic EZ antenna at 300 MHz,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 989–993, 2009. [16] G. Mumcu, K. Sertel, and J. L. Volakis, “Miniature antenna using printed coupled lines emulating degenerate band edge crystals,” IEEE Trans. Antennas Propag., vol. 57, pp. 1618–1624, Jun. 2009.

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[17] P. Jin and R. W. Ziolkowski, “Low Q, electrically small, efficient near field resonant parasitic antennas,” IEEE Trans. Antennas Propag., vol. 57, pp. 2548–2563, Sep. 2009. [18] R. W. Ziolkowski, P. Jin, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “Design and experimental verification of Z antennas at UHF frequencies,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1329–1333, 2009. [19] P. Jin and R. W. Ziolkowski, “Broadband, efficient, electrically small metamaterial-inspired antennas facilitated by active near-field resonant parasitic elements,” IEEE Trans. Antennas Propag., vol. 58, pp. 318–327, Feb. 2010. [20] C.-C. Lin, R. W. Ziolkowski, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “An efficient, low profile, electrically small, VHF 3D magnetic EZ antenna,” Appl. Phys. Lett., vol. 96, p. 104102, Mar. 2010. [21] E. Sáenz, R. Gonzalo, I. Ederra, J. C. Vardaxoglou, and P. de Maagt, “Resonant meta-surface superstrate for single and multifrequency dipole antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, pp. 951–960, Apr. 2008. [22] F. J. Herraiz-Martnez, L. E. Garca-Muoz, D. Gonzlez-Ovejero, V. Gonzlez-Posadas, and D. Segovia-Vargas, “Dual-frequency printed dipole loaded with split ring resonators,” IEEE Wireless Propag. Lett., vol. 8, pp. 137–140, 2009. [23] M. A. Antoniades and G. V. Eleftheriades, “A broadband dual-mode monopole antenna using NRI-TL metamaterial loading,” IEEE Wireless Propag. Lett., vol. 8, pp. 258–261, 2009. [24] P. Jin and R. W. Ziolkowski, “Multiband extensions of the electrically small, near field resonant parasitic Z antenna,” IET Microw. Antennas Propag., vol. 4, pp. 1016–1025, Aug. 2010. [25] J. Zhu, M. A. Antoniades, and G. V. Eleftheriades, “A compact tri-band monopole antenna with single-cell metamaterial loading,” IEEE Trans. Antennas Propag., vol. 244, pp. 1031–1038, Apr. 2010. [26] A. Sanada, M. Kimura, I. Awai, C. Caloz, and T. Itoh, “A planar zeroth-order resonator antenna using a left-handed transmission line,” in Proce. 34th Eur. Microwave Conf., Amsterdam, The Netherlands, Oct. 12–14, 2004, pp. 1341–1344. [27] A. Lai, K. M. K. H. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopolar radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, pp. 868–876, Mar. 2007. [28] J.-H. Park, Y.-H. Ryu, J.-G. Lee, and J.-H. Lee, “Epsilon negative zeroth order resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 3710–3712, Dec. 2007. [29] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 135–146, Jan. 2006. [30] P. M. T. Ikonen, P. Alitalo, and S. A. Tretyakov, “On impedance bandwidth of resonant patch antennas implemented using structures with engineered dispersion,” IEEE Wireless Propag. Lett., vol. 6, pp. 186–190, 2007. [31] A. Alú, F. Bilotti, N. Engheta, and L. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antennas Propag., vol. 55, pp. 13–25, Jan. 2007. [32] F. Bilotti, A. Alú, and L. Vegni, “Design of miniaturized metamaterial patch antennas with -negative loading,” IEEE Trans. Antennas Propag., vol. 56, pp. 1640–1647, Jun. 2008. [33] G. V. Eleftheriades, M. A. Antoniades, and F. Qureshi, “Antenna applications of negative-refractive-index transmission-line structures,” IET Microw. Antennas Propag., vol. 1, pp. 12–22, Feb. 2007. [34] C. Caloz, T. Itoh, and A. Rennings, “CRLH traveling-wave and resonant metamaterial antennas,” IEEE Antennas Propag. Mag., vol. 50, pp. 25–39, Oct. 2008. [35] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999.

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[36] A. Erentok, P. L. Luljak, and R. W. Ziolkowski, “Characterization of a volumetric metamaterial realization of an artificial magnetic conductor for antenna applications,” IEEE Trans. Antennas Propag., vol. 53, pp. 160–172, Jan. 2005. [37] J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic and electric excitations in split ring resonators,” Opt. Exp., vol. 15, pp. 17 881–17 891, Dec. 2007. [38] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [39] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antenna,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [40] R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Exp., vol. 16, pp. 18 131–18 144, Oct. 2008. [41] C. Balanis, Antenna Theory, 3rd ed. Hoboken, NJ: Wiley, 2006. [42] R. Marques, F. Martin, and M. Sorolla, Metamaterials With Negative Parameters. Hoboken, NJ: Wiley, 2007. [43] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 953–960, Apr. 2004. [44] B. Y. Toh, R. Cahill, and V. F. Fusco, “Understanding and measuring circular polarization,” IEEE Trans. Ed., vol. 46, pp. 313–318, Aug. 2003.

Peng Jin (S’05) received the EE B.Sc. degree from the University of Science and Technology of China, Heifei, in 1999, the EE M.Sc. degree from the North Dakota State University, Fargo, in 2004, and the ECE Ph.D. degree from the University of Arizona, Tucson, in 2010. He is currently with the signal integrity group at Broadcom Corporation, Irvine, CA. His research interests include electrically small antennas and metamaterial applications to antenna designs.

Richard W. Ziolkowski (M’97–SM’91–F’94) received the Sc.B. degree in physics (magna cum laude, with honors) from Brown University, Providence, RI, in 1974, and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1975 and 1980, respectively. He was a member of the Engineering Research Division, Lawrence Livermore National Laboratory, Livermore, CA, from 1981 to 1990 and served as the leader of the Computational Electronics and Electromagnetics Thrust Area for the Engineering Directorate. He currently is serving as the Litton Industries John M. Leonis Distinguished Professor in the Department of Electrical and Computer Engineering, University of Arizona, Tuscon. He holds a joint appointment with the College of Optical Sciences. His research interests include the application of new physics and engineering ideas to linear and nonlinear problems dealing with the interaction of electromagnetic waves with complex media, metamaterials, and realistic structures. Prof. Ziolkowski is an IEEE Fellow and an OSA Fellow. He was the President of the IEEE Antennas and Propagation Society in 2005. He continues to be very active in the IEEE, OSA, and APS professional societies.

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Left-Handed Wire Antennas Over Ground Plane With Wideband Tuning Francisco Javier Herraiz-Martínez, Student Member, IEEE, Peter S. Hall, Fellow, IEEE, Qing Liu, and Daniel Segovia-Vargas, Member, IEEE

Abstract—Tunable left-handed (LH) wire antennas over a ground plane are presented in this paper. These antennas are small and have a wide tuning bandwidth and are matched to 50 within the operation range. Two kinds of antennas have been developed: the monopole antenna and the half-loop antenna over ground plane. In both cases, the antennas are designed, manufactured and measured at fixed frequencies as a first step. After that, a study to replace some LH components with variable capacitors has been carried out for each antenna. Tunability over a wide bandwidth has been achieved. Finally, some prototypes of both antennas have been manufactured and measured. The tunable LH monopole antenna has been measured showing a monopolar radiation pattern with a 28% tuning bandwidth (695–924 MHz). Its radiation efficiency takes values between 50% and 70% within all the tuning bandwidth and the maximum dimension varies only between 0 at 695 MHz and 0 at 924 MHz. The tunable LH half-loop antenna over a ground plane has a radiation pattern with maximum radiation orthogonal to the ground plane. It has a 1.64:1 measured tuning bandwidth properly matched to 50 ). Its measured radiation (considering 11 efficiency is always above 54% within the working bandwidth.



0 11



0 15

10 dB

Index Terms—Left-handed materials, loop antenna, monopole antenna, reconfigurable antennas, tunable antennas.

I. INTRODUCTION

M

ETAMATERIALS can be defined as electromagnetic structures engineered to achieve unusual properties. During the last years, these features have been applied to microwave and antenna engineering to achieve new devices with extraordinary properties [1]–[4]. One interesting application of metamaterial structures is the development of artificial left-handed (LH) transmission lines (TLs) [1], [2]. Manuscript received February 08, 2010; revised July 01, 2010; accepted October 26, 2010. Date of publication March 17, 2011; date of current version May 04, 2011. This work was supported in part by Grants TEC2006-13248-C04-04/TCM and TEC2009-14525-C02-01 from MEC, by CCG06-UC3M/TIC-0803, and in part by European Action COST ASSIST (IC0603). The work of F. J. Herraiz-Martínez was supported by an FPU Grant from the Spanish Education Ministry. F. J. Herraiz-Martínez and D. Segovia-Vargas are with the Department of Signal Theory and Communications, Carlos III University in Madrid, Leganés, Madrid 28911, Spain (e-mail: [email protected]; [email protected] ). P. S. Hall is with the Department of Electronic, Electrical and Computer Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. (e-mail: [email protected]). Q. Liu was with the Department of Electronic, Electrical and Computer Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K.. She is now with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2128851

These TLs present the dual performance of the conventional or right-handed (RH) ones. In the LH TLs, the electric and magnetic field vectors and the propagation vector form a LH triplet. This fact allows the propagation of backward-waves, which means that the phase and group velocities are antiparallel ). This is contrary to the conventional case in which ( the electric and magnetic field vectors and the propagation vector form a RH triplet. The equivalent circuit model of an infinitesimal section of a conventional RH TL is a series inductance and a shunt capacitance. On the other hand, as the LH section is the dual part of the RH one, its equivalent circuit is composed of a series capacitance and a shunt inductance. Recently LH loading has been successfully applied to wire antennas [5]–[8]. In [5], [6] Left-Handed dipole antennas are proposed. These antennas are based on a ladder network whose periodic element is the typical LH unit cell. In those works it was demonstrated that the relationship between the electrical length and the frequency of LH-loaded antennas was the opposite of that of conventional antennas, resulting in a reduced wavelength with decreasing frequency. Moreover, the resonant frequencies and input impedance of the LH antennas depend on the elements of the unit cells instead of the total length of the antenna. This is used in [6] to develop small antennas matched to the source without external networks. A study of the efficiency of these antennas has recently been presented [7]. The use of LH loading also provides new capabilities in the design of wire antennas. An LH dipole with orthogonal polarization compared to the conventional one is achieved in [6]. Furthermore, a LH loop antenna with omnidirectional radiation pattern in the plane of the loop is presented in [8]. This is possible thanks to the exindex, according to the convencitation of a mode with tional numbering. This mode has a uniform current distribution on the loop both in phase and amplitude. The main drawback of these LH antennas is the need of a feeding balun what implies losses and an increase in the antenna size. Loaded wire antennas have been used for a long time. For example in [9] a monopole antenna was loaded with an inductor to improve the gain and the antenna matching. More recently, wire antennas have been loaded with active components, such as PIN or varactor diodes. This kind of loading can be applied to develop tunable antennas [10]. The typical tuning bandwidth that can be achieved with this technique is around 20% (i.e., the application of this technique to microstrip patch [11] or PIFA antennas [12]). During the last years, novel tunable antennas have been developed, achieving larger tuning bandwidths such as 44% in antennas for DVB-H application [13] and up to 1.89:1 in slot antennas [14], [15]. These large tuning bandwidths are necessary

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HERRAIZ-MARTÍNEZ et al.: LEFT-HANDED WIRE ANTENNAS OVER GROUND PLANE WITH WIDEBAND TUNING

to cover the huge number of wireless services and standards used nowadays (GSM, PCS, UMTS, GPS, Bluetooth, WiFi, etc.) [16]. Moreover, small antennas are required to integrate these services into compact handheld devices attractive for the users. However, electrically small antennas are difficult to match to any realistic source because of their very large reactance. Thus, considerable effort must be taken in the design of a matching network that conjugate matches the small antenna’s impedance [17]. Furthermore, the matching network increases the total size of the system which is contrary to the final goal. For these reasons, small antennas with large tuning bandwidth and direct matching the source without an external network are required. The goals of the present work are the development of small antennas with wideband tuning and matched to 50 , achieving high values of efficiency. The use of LH structures allows achieving all these goals simultaneously. Other techniques aim at some of the goals previously stated although they do not reach all of them at the same time. Then, as commented before, conventional techniques [10]–[12] only achieve 20% tuning . Other techniques bandwidth with antennas larger than have proposed very compact antennas for DVB-H to integrate them into handheld devices (e.g., the total size of the antenna at 470 MHz varies between and at 702 MHz [13]) but they only cover up to a 40% bandwidth. Finally, novel designs based on slots [14], [15] have achieved wideband tuning (up to 1.89:1 in [15]), but the size of the antenna is similar to conventional and in [15]). ones (it varies between In this paper tunable LH wire antennas over ground plane are developed. These antennas are based on the principle that the resonance frequencies and input impedance of a LH resonant structure depend on the elements of the unit cell. Then, variable capacitors (i. e. varactor diodes) can be used to achieve small wire antennas properly matched with a wideband tuning. Two LH antennas over a ground plane are developed (monopole antenna and half-loop antenna) which avoids the use of a balun. The development of the monopole antenna, the study to make it tunable and the experimental results regarding the fixed and tunable implementations of this antenna are presented in Section II. The halfloop antenna over a ground plane is described in Section III. Theoretical and experimental results of this antenna are presented. The study of the tuning of this antenna and the experimental results of the manufactured tunable prototype are also presented in this Section. Finally, the paper is concluded in Section IV. II. TUNABLE LH MONOPOLE ANTENNA A. LH Monopole Antenna Configuration and Implementation The proposed LH monopole antenna is a ladder network over a finite ground plane. The ladder network is a periodic structure whose unit element is the typical LH unit cell, formed by and a shunt inductor . This kind of two series capacitors monopole antenna is implemented by printing two parallel strips on a dielectric substrate (as shown in Fig. 1). An interconnection between both strips is also printed for each unit cell. The dielectric substrate is placed orthogonally to the ground plane. The antenna is fed through a SMA connector. The outer conductor of the SMA connector is soldered to the ground plane while the

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Fig. 1. Sketch of the LH monopole antenna. In this example, the number of . unit cells is set to

N =2

inner conductor is soldered to the feeding strip. The capacitors are put on the strip connected to the feed and the inductors are placed in the interconnections between the two strips. The resonant frequencies and the input impedance depend on the elements of the unit cell, as in all the LH resonant structures [1]. Thus, the antenna operation frequency can be set by a proper choice of the unit cell and ). Matching to 50 can be obtained by a elements ( proper selection of these elements. In a practical implementation, the monopole antenna is limited to unit cells. In addition to , the other antenna parameters are the length of the strips , the separation between the strips and the width of the strips . These . parameters result in a period given by the ratio The monopole antenna is a LH resonant structure in which each mode corresponds to a standing wave on the monopole which satisfies the following relation: (1) where is the wavelength of the standing wave on the is the resonance number. In LH monopole antenna and resonant antennas, is a negative integer which decreases with decreasing frequency. unit In general, for a LH resonant structure composed of to [1], [18]. cells, the number takes values from For the particular case of the monopole antenna, image theory can be applied because of the presence of the ground plane. unit cells, Hence, for a LH monopole antenna composed of takes values: in the LH operation region. The structure of the monopole antenna imposes an open circuit boundary condition at the open edges of the strips which implies a minimum in the current distribution at these points. On the other hand, a maximum in the current distribution must occur at the opposite edges because they are connected to the feed and the ground plane. These boundary conditions impose that only odd modes can be excited in the monopole antenna. For example, in the case of the LH monopole antenna composed of , only the modes with are excited.

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Fig. 2. Currents on the LH monopole antenna computed with CST Microwave mode. (b) n mode. Studio. (a) n

= 01

= 03

Considering the previous example, Fig. 2 shows the current distributions on the LH monopole antenna for both modes computed with CST Microwave Studio. In the case of the mode, the current distribution on each of the strips has one quarter-wavelength electrical length and is similar to the fundamental mode of the conventional monopole antenna. It is important to note that the currents on both strips have opposite directions, but their amplitudes have different values which produces radiation in the far field [6]. The current distribution of the mode has three quarter-wavelengths electrical length on each strip, as expected. Once again the currents have opposite directions on each strip, but the amplitudes are different. In both cases, a monopolar radiation pattern with maximum in the plane of the ground plane and null orthogonal to the ground plane is obtained, as shown in Fig. 3. The presence of the horizontal currents has two effects on the radiation patterns. The in the first one is an increase of the crosspolar component ( main planes) with respect to conventional monopolar antennas. The second one is that the null in the direction orthogonal to the ground plane is not very deep in one of the planes ( plane). The resonant frequencies corresponding to the LH monopole antenna can be obtained by considering the relation between the phase constant and the frequency. As a first approximation, the phase constant can be obtained by considering an infinite transmission line composed of the LH unit cells and applying the Bloch-Floquet analysis, as shown in other LH resonant structures [1], [6], [18]. In order to compute this dispersion relation, the most accurate model is the one which takes into account the effect of the host line and the LH loading elements. This model is known in literature as composite right/left-handed (CRLH) [1]. The dispersion relation in this case is computed as

(2)

Fig. 3. Simulated radiation patterns of the LH monopole antenna. (a) n mode. (b) n mode.

= 03

= 01

where and are the inherent capacitance and inductance and are functions of a parallel line with the length of . of the width of wires , their distance and the unit cell length . The resonant frequencies can be obtained by sampling the phase constant at the points (3) Fig. 4 shows the results for the monopole with and the following parameters: , , , , and . The computed resonant frequencies for this LH and 828 monopole are 782.5 MHz for the mode with . MHz for the mode with A prototype of the LH monopole with the previous parameters have been manufactured and measured. SMD commercial components are used to manufacture the components. The , and substrate is Duroid 5080 with . The substrate is orthogonally mounted over a finite aluminum ground plane. The effect of the finite ground plane has been evaluated. It has been concluded that when the size of the finite ground plane is large enough in comparison ) the effect of the size with the free space wavelength ( of the ground plane on the antenna performance is negligible.

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N = = 50 mm, = 20 nH.

Fig. 4. Dispersion diagram of the LH monopole antenna with . The other parameters of the antenna are: L a ,r : ,C : and L ,d

2 unit cells = 25 mm = 10 mm = 0 90 mm

= 0 5 pF

Fig. 6. Measured radiation patterns of the manufactured LH monopole antenna. . (b) f (a) f .

= 751 MHz

Fig. 5. Simulated and measured reflection coefficient of the LH monopole antenna.

The shape of the ground plane has also been studied, but no differences have been observed with different shapes. These two facts have been taken into account in all the designs presented in the paper. Then, a square 107 mm 107 mm ground plane (for both simulation and manufacturing) has been used for this prototype. The simulated and measured reflection coefficient is shown in Fig. 5. The measured resonant frequencies are and . The value of the measured reflection coat and at . The effiefficient is ciency of the antenna has been measured by using the JohnstonGeissler method based on the Wheeler Cap principle [19]–[21]. mode and 10% This value is 57% at 828 MHz for the mode. The electrical length of the monopole is for the which is an important reduction in comparison with the conventional monopole. Fig. 6 shows the measured radiation patterns of the manufacand planes at both resonant fretured prototype in the and ). The angular sector between quencies ( was not measured due to limitations in the anechoic chamber facility. A monopolar radiation pattern is obtained at both frequencies, as expected. The radiation patterns are not completely symmetrical with respect to the ground plane due to the effect of the finite ground plane, although it was not observed in simu-

= 828 MHz

lation. In the case of the mode, the null in the direction plane orthogonal to the ground plane is much deeper in the ) than in the plane ( ). This is not the case ( of the mode, in which the radiation pattern has a null in both planes. Cross polarization is lower deeper than in the plane at . However, this magnitude is than in the plane at that frequency. On the increased until other hand, the cross polar component is lower than in both planes at . The measured gains are at and at . B. Tunable LH Monopole Implementation and Experimental Results As the resonant frequencies of the LH monopole antenna depend on the value of the unit cell elements, a tunable antenna can be developed by replacing the unit cell elements with variable components. In practice, the easiest way to implement this approach is to use variable capacitors, for example varactor diodes, . Then, the resonant freacting as the series capacitances quencies of the monopole antenna will be different for each value of the varactor diodes. Fig. 7 shows the frequency dependence on the value of the sefor the previous LH monopole antenna. In ries capacitances both cases the tuning bandwidth is larger than 50%. The drawback of this approach is that the input impedance strongly depends on the values of the unit cell elements too. The relationship between the value of the series capacitances and the real mode has been simpart of the input impedance for the ulated and is plotted in Fig. 8. This magnitude decreases with the

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Fig. 7. Dependence of the resonant frequencies of the LH monopole antenna on the series capacitances of the unit cells (C ). The other parameters of the ,N ,a , antenna are: L ,d r : . and L

= 50 mm = 2 unit cells = 25 mm = 10 mm = 0 90 mm = 20 nH

R s = 1 = = = 50 mm = 2 unit cells = 25 mm = 10 mm = 0 90 mm = 20 nH = = 0 5 pF

Fig. 9. Dependence of the input resistance, (continuous line) and j j (dashed line) of the LH monopole antenna (n 0 mode) with the variable C C ). The other parameters of the antenna are: capacitors (C L ,N ,a ,d ,r : , C : . and C L TABLE I SIMULATION PERFORMANCE OF VARIOUS CONFIGURATIONS OF THE LH TUNABLE MONOPOLE (F: 0.5 pF FIXED CAPACITOR, V: VARIABLE CAPACITANCE)

R s = 1 = 50 mm = 2 unit cells = 25 mm = 10 mm = 0 90 mm = 20 nH

Fig. 8. Dependence of the input resistance, (continuous line) and j j (dashed line) of the LH monopole antenna (n 0 mode) on the series capacitances of the unit cells (C ). The other parameters of the antenna are: L ,N ,a ,d ,r : and L .

capacitance very quickly which makes it impossible to match this antenna to 50 for most of the operation bandwidth. As the goal is to develop a small tunable antenna with wide range and properly matched to 50 , this approach has to be modified in order to achieve a trade-off between tuning and matching bandwidth. Two different approaches have been investigated: using asymmetric unit cells and mixing unit cells with fixed and variable capacitances. In the case of the LH monopole antenna , there are four capacitances composed of which give six different combinations of two variable capacitors and two fixed. All these possibilities have been simulated and the results are summarized in Table I. The capacitors in have been numbered from the one closest to the feed ( Fig. 1) to the one closest to the open edge ( in Fig. 1). The fixed capacitors keep the value of the previous LH monopole ). The value of the variable capacitance antenna ( is changed between 0.5 pF and 3.5 pF. In most cases, the frequency of the different modes is almost fixed except for the last variable and ), case ( in which an 18% tuning bandwidth has been achieved for the mode, where the reflection coefficient is lower than

Fig. 10. Proposed implementation of the tunable LH monopole antenna. (a) Sketch of the antenna. (b) Picture of the manufactured prototype.

( ). This is possible because the real part of the impedance grows with capacitance up to 70 and then remains almost constant, providing good matching to 50 within all the tuning bandwidth (Fig. 9). Fig. 10(a) shows the sketch of the implemented tunable LH monopole antenna. The dimensions of the monopole are kept unchanged with respect to the previous prototype. The tunable monopole is mounted over a square 149 mm 149 mm aluminium ground plane. MV32003 varactor diodes (from

HERRAIZ-MARTÍNEZ et al.: LEFT-HANDED WIRE ANTENNAS OVER GROUND PLANE WITH WIDEBAND TUNING

Fig. 11. Measured js j parameter of the tunable LH monopole for different values of the control voltage V .

Microsemi) are used to implement the variable capacitances ). The fixed capacitors ( ) and the ( inductors ( ) are implemented with conventional SMD components. A network with high impedance lines (0.20 mm-width) has been attached to bias the varactor diodes. The capacitance . Two of the varactors is controlled by the control voltage ) have been inserted in RF-choke inductors ( ) the biasing network. An isolation capacitor ( has been connected close to the RF port to block the DC component. The proposed prototype has been manufactured (Fig. 10(b)) and measured (Fig. 11). mode has a tuning range between 777 MHz and The 924 MHz, which is 17% approximately. Moreover, the antenna ) within is well matched to 50 (considering the whole bandwidth. The measurement shows a shift towards higher frequencies with respect to the simulation. Nonetheless, the relative bandwidth is almost the same. The frequency of mode changes from 639 MHz to 811 MHz and the ) within the range 695–793 it is well matched ( MHz. Considering both modes, an operation range between 695 MHz and 924 MHz is obtained with a 28% tuning bandwidth and proper matching. It is important to note that the monopole at 695 MHz and electrical length varies between at 924 MHz. Finally the efficiency has been measured with the Johnston-Geissler method. The efficiency for the mode is between 55% at 777 MHz and around 75% at the highest fremode the quency of the operating bandwidth. For the efficiency is between 7% and 15% within the whole frequency bandwidth. III. TUNABLE LH HALF-LOOP ANTENNA OVER GROUND PLANE A. LH Half-Loop Antenna Over Ground Plane Configuration and Implementation The LH half-loop antenna is also a ladder network over a ground plane whose periodic element is the typical LH unit cell. However, in this case the printed strips are folded and short-circuited to the ground plane forming a half-loop. This

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Fig. 12. Sketch of the LH half-loop antenna over ground plane.

means that the outer strip has one end connected to the feed and the other connected to the ground plane, whilst the inner strip has both ends connected to the ground plane. In this way the boundary conditions are changed from open circuits to short circuits. There are interconnections between the outer and are connected to the outer strip, inner strips. The capacitors while the inductors are put on the interconnections between the strips. The feeding scheme is the same as the one in the LH monopole antenna (a female SMA connector connected to the feed point with the outer conductor soldered to the ground plane). This antenna is a LH resonant structure whose resonance frequencies depend on the elements of the unit cells. Each mode has a standing wave which satisfies (1), being the length of the half-loop, the wavelength of the standing wave on the half-loop antenna and the resonance number. It is important to note that the boundary conditions imposed by this antenna are different than the ones in the monopole antenna. In this case, all the strip ends are connected to the feed or the ground plane, which implies maxima in the current distributions at all of these points. For this reason, only even modes are excited in this antenna, contrary to the monopole antenna in which only odd modes were excited. Moreover, a mode satisfying (1) for can be excited in this antenna. This provides interesting features as will be shown below. In conclusion, for a LH unit cells, the modes with half-loop antenna composed of can be excited. For ex, ample, for a LH half-loop antenna with takes value , , , 0. Fig. 12 shows the proposed implemented loop based on 3 unit cells. The dimensions of the antenna are the following: length , width of the horizontal cell of the vertical cells , separation between the strips , width of the printed strips . The elements of the unit and . SMD commercells are cial components are used to manufacture the components. The , and substrate is Duroid 5080 with . The substrate is orthogonally mounted over

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Fig. 13. Manufactured prototype of the LH half-loop antenna over ground plane.

Fig. 14. Simulated and measured reflection coefficient of the LH half-loop antenna.

a finite aluminum ground plane. This ground plane is a square with 150 mm side length. The length of the horizontal cell is different to the vertical ones in order to reduce the total size of the antenna. This modification with respect to the periodic structure does not affect the performance of the antenna as it will be shown below. A prototype of this antenna has been manufactured (Fig. 13) and measured. The simulated and measured reflection coefficients are shown in Fig. 14 where good agreement between both results can be seen. mode resonates at . The meaThe sured reflection coefficient of this resonance is . This mode has the unique feature that the current on the strips is uniform in amplitude and phase (Fig. 15(a)). This leads to a radiation pattern with maximum radiation orthogonal to the ground is the main component in plane (Fig. 15(b)). Furthermore, plane, which is not the typical case of antennas mounted the over a ground plane (see for instance the LH monopole antenna in the previous Section). It is important to note that although from the current distribution (Fig. 15(a)) the proposed antenna might seem as an electrically small loop because of the constant currents, this does not behave so. The reason is that the currents through the vertical strips are cancelled because of their same amplitude and opposite directions. This cancellation is possible because the separation between both vertical strips ). Thus, the main radiating element is is small (always the current through the horizontal strip over the finite ground

Fig. 15. (a) Current distribution on the LH half-loop antenna over ground plane . (b) Simulated radiation pattern of this mode. for the mode with n

=0

plane, which produces the maximum radiation orthogonal to this ground plane. At this point, it is important to note that this current element is not cancelled because the separation of this horizontal strip with respect to the ground plane is large enough to produce an image much smaller than the current source. Modes with negative indices are also present in the structure at frequencies lower than . In particular, the modes with , can be identified in the simulated and measured reflection coefficient (Fig. 14), while the mode with is not matched in this prototype. The frequency of the mode is and its measured reflection coefficient is . This mode has a current distribution with six quarter-wavelengths electrical length on each strip. The radiation pattern of this mode has the typical monopolar shape with a null in the direction orthogonal to the ground plane and maximum radiation in the plane of the ground plane. The other mode) resonates at mode with negative index ( and the measured at this frequency is . This mode has a current distribution with four quarter-wavelengths electrical length on each strip. However, the simulation has shown that this mode has always very poor efficiency ). Finally, the mode, which is not matched in ( this example, has a current distribution with a null in the upper

HERRAIZ-MARTÍNEZ et al.: LEFT-HANDED WIRE ANTENNAS OVER GROUND PLANE WITH WIDEBAND TUNING

Fig. 16. Measured radiation patterns of the manufactured LH half-loop antenna. (a) f . (b) f : .

= 755 MHz

= 1 4 GHz

interconnection of the antenna which gives two in-phase quarter wavelengths. This current distribution has the typical monopolar radiation pattern. The radiation efficiency of this prototype has been measured. is 53%. The measured radiation efficiency at , which On the other hand, it is only 7% at confirms the very low efficiency of this mode. Finally, the meamode with is 87%, which sured radiation efficiency of the proves that this mode has high radiation efficiency. The largest at 755 MHz and dimension of the antenna is only at 1.4 GHz. and planes at The measured radiation patterns in the and are shown in Fig. 16. The angular sector between was not measured due to limitations in the anechoic chamber facility. A monopolar radiation pattern is measured at with a cross polarization below in both planes. The expected radiation pattern with maximum orthogonal to the ground plane is measured at , although an angular displacement is observed in the plane. This displacement has not been observed in other LH half-loops and it is not the generalized shape, as it will be shown in the next Subsection. In this particular example, this shift could be due to the coupling with higher order modes because the resonance frequency of mode is the highest that can be achieved because the . In of the low value of the series capacitances

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Fig. 17. Dependence of the LH half-loop antenna over ground plane performance with respect to the with the series capacitances of the unit cells (C ). (a) Dependence of the resonant frequency of the n mode. (b) Dependence of , bold line) and j j (dashed line) of antenna for the the input resistance ( same mode.

R

s

=0

addition, other factors that can be the cause of the tilt in the measured results are a misalignment in the measurement setup and manufacturing tolerances (in particular, we observed that the dielectric board was not perfectly orthogonal to the ground plane). The measured cross polar component in this plane ( ) is lower than . On the other hand, the plane has the expected radiation pattern (maximum radiation orthogonal to the of cross polarization in the direcground plane) with tion orthogonal to the ground plane. However, there is a higher cross polar component in other directions. The reason could be plane, bethe same to the one which causes the shift in the cause the level of the cross polarization is much lower in this plane and this mode for other realizations of this antenna (see for instance Fig. 21). It is important to note that the main com, as explained before, in contrast to the ponent in this plane is ) and other antennas mounted monopolar modes (e.g., over a ground plane (see for instance the previous Section). Furthermore, the back radiation of the antenna at this mode is very at and low in both planes. The measured gains are 4.8 dBi at . The experimental results of this prototype are summarized criteria has been considered for in Table II. mode in which measuring the bandwidth, except for the has been used.

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Fig. 19. Measured reflection coefficient of the tunable LH half-loop antenna for different values of the control voltage V .

Fig. 20. Dependency of the measured radiation efficiency of the tunable LH half-loop antenna with frequency.

Fig. 18. Tunable LH half-loop antenna over ground plane. (a) Sketch of the proposed implementation. (b) Picture of the manufactured prototype.

B. Tunable LH Half-Loop Antenna Over Ground Plane Implementation In order to make the LH half-loop antenna tunable, an approach similar to the one used in the LH monopole antenna can be applied. In this case, let us consider all LH capacitances as variable. The other parameters of the antenna are kept unchanged with respect to the previous example. Fig. 17(a) shows the dependence of the resonant frequency of the mode with . In this case the tuning range is broader than with the monopole antenna, achieving a potential 2.3:1 tuning bandwidth. The dependence of the input resistance and matching to 50 with respect to for the mode is plotted in Fig. 17(b). The input impedance takes values close to 50 for small values of the LH capacitances; however, it decreases for higher values of . This makes the antenna be properly matched (considering ) only for values smaller than 1.5 pF. Hence, the antenna working at the mode is well matched within the range in which the frequency dependence with has the largest slope, as shown in Fig. 17(a). In particular, the antenna is properly matched within a 1.6:1 bandwidth which is a considerably larger than the previous approaches.

Potentially, the tuning bandwidth of the half-loop antenna is much wider than the monopole antenna (Section II.B). This is produced by the change of the ending conditions from open circuit to short circuit. This involves a flatter slope in the dependence of the input resistance with the series capacitances. Specifically, for the case of the monopole antenna (Fig. 8) the input impedance decreases with the series capacitances very quickly: the ratio between the input resistances is 10.48 for a variation of the capacitances between 0.5 pF and 3 pF. On the other hand, for the case of the half-loop antenna (Fig. 17(b)) the ratio between the input resistances is only 3.46 for the same variation of the series capacitances. This fact is the cause for a wider bandwidth in the second case. The sketch of the proposed tunable LH half-loop antenna is shown in Fig. 18(a). It is based on the LH half-loop antenna presented in the previous Section. The dimensions of the antenna are kept unchanged. The LH variable capacitances ( ) are implemented with MV32003 varactor diodes (from Microsemi). The control voltage of the varactors is supplied through two high impedance lines printed at both sides of the half-loop. RF-choke inductors ( ) and DC-isolation capacitors ( ) are attached to the design. The proposed antenna has been manufactured (Fig. 18(b)) and measured. The total tuning range of the mode is 2.35:1 (584.8–1376 MHz) which is very similar to the one

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TABLE II EXPERIMENTAL RESULTS OF THE LH HALF-LOOP ANTENNA

modes cover the bandwidth 494–804 MHz with good matching ). These modes can be used to extend the ( tuning bandwidth of this antenna. However, it must be taken into account that the radiation pattern of these modes is monopolar in most of the cases. Moreover, the radiation efficiency strongly decreases with decreasing frequency (Fig. 20), which makes the efficiency of these modes be below 30% for frequencies lower than 700 MHz. Finally, the radiation patterns of the tunable prototype have been measured to show that the use of biasing networks does not modify their shapes. As an example, the radiation patterns are shown in of the prototype for a control voltage Fig. 21. The angular sector between was not measured due to limitations in the anechoic chamber facility. The and modes have the expected radiation pattern. In the first case a null deeper than is observed at the direction orthogonal to the ground plane in both main planes. Moreover, the cross polarization is lower than . On the other hand, at the null is around in the plane and the cross in the plane and polar components are below in the plane. The expected radiation pattern with maximum orthogonal to the ground plane is obtained at . In this case the cross polar component is lower than in both planes and the back radiation is low. Again, it is interesting to note that the main component in the plane at this frequency is . The measured gains are at , at and 5.2 dBi at . IV. CONCLUSION

Fig. 21. Measured radiation patterns of the manufactured tunable LH half-loop . (a) n mode (0.65 GHz). (b) n mode antenna for V (0.72 GHz). (c) n mode (0.91 GHz).

= 3V =0

= 06

= 02

obtained in simulation (Fig. 17(a)). Considering good matching to 50 ( ), the measured tuning bandwidth of the mode is 1.64:1 (840–1376 MHz) as it is shown in Fig. 19. The radiation efficiency of this mode has been measured (Fig. 20). This magnitude is always above 54% within the operation bandwidth in which the antenna is well matched (840–1376 MHz). The maximum dimension of the antenna varies between at 840 MHz and at 1376 MHz. There are resonances below the mode which cor). These responds to the modes with negative indices (

Two LH antennas over a ground plane have been presented: the LH monopole antenna and the LH half-loop over ground plane. These antennas can be directly fed through a SMA port, which avoids the use of a balun which was an important drawback of previous LH antennas. The LH monopole antenna provides the typical monopolar radiation pattern with omnidirectional shape parallel to the ground plane and minimum radiation in the axis of the monopole. A prototype working at 828 MHz has been manufactured and measured, achieving good results. The LH half-loop antenna over a ground plane provides a mode which has the unique feature that the currents with on the antenna have a uniform distribution in amplitude and phase. This leads to a radiation pattern with maximum radiation orthogonal to the ground plane. Moreover, it has an orthogonal polarization in one of the main planes with respect to the monopolar case. A prototype of the LH half-loop antenna has been manufactured and measured. Its working frequency for the mode is 1.4 GHz and 87% radiation efficiency. It has been demonstrated that the resonant frequencies of both antennas depend on the capacitance of the LH unit cells which

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compose both antennas. This property has been used to develop small tunable LH antennas with wideband tuning and matched to 50 without external networks. In particular, a tunable LH monopole has been designed, manufactured and measured. In this case, it has been necessary to use asymmetric cells to match the antenna to 50 within all the operation bandwidth. The manufactured prototype has a 17% tuning bandwidth (777–924 MHz) considering only one mode and 28% (695–924 MHz) considering the two LH modes that are excited in the antenna. The measured radiation efficiency takes values between 50% and 70% within all the tuning bandwidth. The maximum dimension varies beat 695 MHz and at 924 MHz. tween The other tunable antenna is based on the LH half-loop antenna over ground plane. In this case, all the capacitances have been implemented with varactor diodes achieving a wideband tuning. A prototype of the antenna has been manufactured and measured. The manufactured prototype has a 1.64:1 tuning bandwidth matched to 50 . The measured radiation efficiency is always above 54% within the operating bandwidth. This antenna also offers the possibility of a larger tuning bandwidth (potentially larger than 2:1) by simultaneously using the mode and the modes with . However, it must be taken into account that the radiation patterns and efficiencies of the are different to those of the mode. modes with It has been demonstrated that Left-Handed antennas over ground plane can be used to develop tunable antennas with wide operation range, small dimensions and matched to the source without external networks. This new kind of small antennas can be used in modern and future communication systems in which reconfiguration over a wide bandwidth is needed. REFERENCES [1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley-IEEE Press, 2006. [2] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. New York: WileyIEEE Press, 2005. [3] N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations. New York: Wiley-IEEE Press, 2006. [4] R. Marqués, F. Martín, and M. Sorolla, Metamaterials With Negative Parameters, Theory, Design and Microwave Application. Hoboken, NJ: Wiley, 2008. [5] H. Iizuka, P. S. Hall, and A. L. Borja, “Dipole antenna with left-handed loading,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 483–485, 2006. [6] H. Iizuka and P. S. Hall, “Left-Handed dipole antennas and their implementations,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1246–1253, May 2007. [7] Q. Liu, P. S. Hall, and A. L. Borja, “Efficiency of electrically small dipole antennas loaded with left-handed transmission lines,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3009–3017, Oct. 2009. [8] A. L. Borja, P. S. Hall, Q. Liu, and H. Iizuka, “Omnidirectional loop antenna with left-handed loading,” IEEE Antennas Wireless Propag. Lett., vol. 556, pp. 495–498, 2007. [9] K. Fujimoto, “A loaded antenna system applied to VHF portable communication equipment,” IEEE Trans. Veh. Technol., vol. VT-17, no. 1, pp. 6–12, Oct. 1968. [10] D. J. Roscoe, L. Shafai, A. Itippiboon, M. Cuhaci, and R. Douville, “Tunable dipole antennas,” in Proc. IEEE AP-S, Ann Arbor, MI, Jun. 28–Jul. 2 1993, vol. 2, pp. 672–675. [11] P. Bhartia and I. J. Bahl, “A frequency agile microstrip antenna,” in Proc. IEEE AP-S, May 1982, vol. 20, pp. 304–307. [12] K. L. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1879–1888, Oct. 1997.

[13] M. Komulainen, M. Berg, H. Jantunen, and E. Salonen, “Compact varactor-tuned meander line monopole antenna for DVB-H signal Reception,” IEEE Electron. Lett., vol. 43, no. 24, pp. 1324–1326, Nov. 2007. [14] C. R. White and G. M. Rebeiz, “A slot-ring antenna with an octave of tunability,” in Proc. IEEE AP-S Int. Symp., Honolulu, HI, Jun. 2007, pp. 5841–5845. [15] C. R. White and G. M. Rebeiz, “Single- and dual-polarized tunable slot-ring antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 19–26, Jan. 2009. [16] D. Manteuffel and M. Arnold, “Considerations for reconfigurable multi-standard antennas for mobile terminals,” in Proc. iWAT, Chiba, Japan, Mar. 2008, pp. 231–234. [17] G. S. Smith, “Efficiency of electrically small antennas combined with matching networks,” IEEE Trans. Antennas Propag., vol. AP-40, no. 5, pp. 369–373, May 1977. [18] F. J. Herraiz-Martínez, V. González-Posadas, F. Iñigo-Villacorta, and D. Segovia-Vargas, “Low-cost approach based on an eigenfrequency method to obtain the dispersion diagram in CRLH structures,” IEEE Microwave Wireless Compon. Lett., vol. 17, no. 1, pp. 13–15, Jan. 2007. [19] P. Miskovsky, J. M. González-Arbesú, and J. Romeu, “Antenna radiation efficiency measurement in an ultrawide frequency range,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 72–75, 2009. [20] M. Geissler, O. Litschke, D. Heberling, P. Waldow, and I. Wolff, “An improved method for measuring the radiation efficiency of mobile devices,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2003, vol. 4, pp. 743–746. [21] R. H. Johnston and J. G. McRory, “An improved small antenna radiation- Efficiency measurement method,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 40–48, Oct. 1998.

Francisco Javier Herraiz-Martínez (S’07) was born in Cuenca, Spain, on May 3, 1983. He received the Engineer (first of his class) and the Ph.D. degrees in telecommunications from Carlos III University in Madrid, Spain, in 2006 and 2010, respectively. His research interests include metamaterial applications for antenna and microwave circuits and reconfigurable and active antennas. Dr. Herraiz-Martínez received the Best Master Thesis Dissertation Award from the COIT/AEIT in 2006. He was recipient of a Spanish Education Ministry official grant for funding his doctoral research activity.

Peter S. Hall (M’88–SM’93–F’01) received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K., in 1973. He spent three years with Marconi Space and Defense Systems, Stanmore, U.K., working on a European Communications satellite project. He then joined The Royal Military College of Science, Swindom, U.K., as a Senior Research Scientist, progressing to Reader in Electromagnetics. In 1994, he joined The University of Birmingham, Birmingham, U.K., where he is currently a Professor of communications engineering and the Head of the Devices and Systems Research Centre in the Department of Electronics, Electrical, and Computer Engineering. He has researched extensively in the areas of microwave antennas, radio wave propagation, and vehicle telematics. He has published five books, more than 250 learned papers, and has been awarded patents. These publications have earned six IEE premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas (Peter Peregrinus, 1989). Prof. Hall is a Fellow of the IEEE and the Institution of Electrical Engineers [(IET), formerly Institute of Electrical Engineers (IEE)], London, U.K. He is a past IEEE Distinguished Lecturer. He is a past Chairman of the IET Antennas and Propagation Professional Group and past coordinator for Premium Awards for the IET Proceedings on Microwave, Antennas and Propagation. He is a member of the IEEE AP-S Fellow Evaluation Committee. He chaired the 1997 IEE ICAP Conference, was Vice Chair of EuCAP 2008 and has been associated with the organization of many other international conferences. He was Honorary Editor of the IEE Proceedings Part H from 1991 to 1995 and is currently on the editorial board of Microwave and Optical Technology Letters. He is a past member of the Executive Board of the EC Antenna Network of Excellence.

HERRAIZ-MARTÍNEZ et al.: LEFT-HANDED WIRE ANTENNAS OVER GROUND PLANE WITH WIDEBAND TUNING

Qing Liu was born in China. She received the Ph.D. degree from Birmingham University, Birmingham, U.K., in 2009. Her dissertation was entitled, “Antennas using Left Handed Transmission Lines.” She is a Research Associate with the Communications Group, Electronic, Electrical and Computer Engineering Department, University of Sheffield, Sheffield, U.K. Her current research is on the investigation and development of elastic antennas and radio frequency components. Her research interests include elastic antennas, antennas with left handed loading, and reconfigurable antennas. Dr. Liu received the Chairman’s Commendation Award for her paper and presentation at EuCAP 2007. She was a co-recipient of the 2008 CST Short Paper Award.

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Daniel Segovia-Vargas (M’98) was born in Madrid, Spain, in 1968. He received the Telecommunication Engineering degree and the Ph.D. degree from the Polytechnic University of Madrid, in 1993 and 1998, respectively. From 1993 to 1998, he was an Assistant Professor at Valladolid University. In 1999, he joined Carlos III University in Madrid, where he is an Associate Professor in charge of the microwaves and antenna courses. Since 2004, he has been the leader of the Radiofrequency Group, Department of Signal Theory and Communications, University Carlos III of Madrid. He has authored and coauthored over 110 technical conference, letters and journal papers. His research areas are printed antennas and active radiators and arrays, broadband antennas, LH metamaterials, Terahertz antennas and passive circuits. He has also been an Expert of the European Projects Cost260, Cost284 and COST IC0603.

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Switched-Band Vivaldi Antenna M. R. Hamid, Peter Gardner, Senior Member, IEEE, Peter S. Hall, Fellow, IEEE, and F. Ghanem

Abstract—A novel Vivaldi antenna with added switched band functionality to operate in a wideband or narrowband mode is presented. The antenna reconfiguration is realized by inserting four pairs of switchable ring slots into the ground plane of the structure. A wide bandwidth mode from 1.0–3.2 GHz and three narrowband modes can be selected. A fully functional prototype with PIN diodes switches has been developed. Measured results shows good performance of the proposed designs. The antenna could be a suitable solution for a multimode application requiring wideband and frequency reconfigurable antennas, such as in military applications and cognitive radio. Index Terms—Antenna, cognitive radio, reconfigurable, Vivaldi, wideband.

I. INTRODUCTION HE design of wideband, multiband, or reconfigurable antennas, for multimode terminal applications [1], involving combination of GPS, GSM, 3G, WLAN, Bluetooth, etc., has received considerable attention recently. However when considering the interference levels at the receiver, reconfigurable antennas are the best option, since a single band can be selected at a given time. A significant number of reconfigurable antennas have been reported such as in [2]–[5]. However most of them were only capable to switch between two particular narrow bands. Wideband-narrowband reconfiguration is essential for multimode applications that include UWB, and for future wireless communication concepts such as cognitive radio (CR), which employs wideband sensing and reconfigurable narrowband communications. A standard for fixed access using cognitive radio concepts has been established in IEEE 802.22 [6] in which two separate antennas are suggested. Requirement for CR mobile communications are less clear. However reconfigurable antennas are likely to be useful to both application areas to reduce size by combining wide and narrowband functionality and also allowing some additional receiver pre-filtering, which

T

Manuscript received March 23, 2010; revised September 24, 2010; accepted October 20, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the EPSRC under Grant EP/FOl 7502/1. The work M. R. Hamid was supported by the Universiti Teknologi Malaysia (UTM). M. R. Hamid is with the School of Electronics, Electrical, and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. and also with the Faculty of Electrical Engineering (FKE), Universiti Teknologi Malaysia (UTM), Johor Bahru Campus 81310, Malaysia (e-mail: [email protected]). P. Gardner, P. S. Hall, and F. Ghanem are with the School of Electronics, Electrical and Computer Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122293

is now considered to be vital to successful cognitive radio operation [7]. Recently, a number of reconfigurable antennas have been demonstrated that combine wideband and narrowband functionality. In [8], a wideband and switched subband antenna were integrated. Two switched subbands were achieved by feeding a different structure through 180 rotational motion. In [9], an inherently wideband disc monopole excited with two ports at opposite sides, one fed with coplanar waveguide and the other with microstrip line is proposed. The first port was kept wideband but the latter was narrow banding by inserting the ground plane with a meandered rectangular slot. The narrowband operation can be reconfigured by varying the slot length. In [10], a planar inverted F antenna (PIFA) integrated with a wideband monopole has been designed. An external matching circuit has been used to reconfigure PIFA to three new frequencies. All of these antennas [8]–[10] have been proposed to meet the separate sensing and communication front end criteria. To meet the second architecture (combined sensing and communication) several new antennas concepts have also been developed [11]–[14]. In [11], an L-shaped slot antenna has been demonstrated. Wideband-narrowband reconfiguration has been achieved by using microelectromechanical switches. In [12], a switched band microstrip patch antenna was developed by turning “on” and “off” the ground plane beneath the patch. Both antennas [11], [12] were demonstrated for mobile phone terminals. For base station antennas, the authors have previously shown that by incorporating switches in the slots of a log periodic aperture fed microstrip antenna some degree of frequency reconfiguration can be achieved [13]. Another form that has been proposed is the switchable quad-band antenna [14]. In this paper, we propose a novel switched band Vivaldi antenna. To demonstrate its functionality, the proposed antenna shows reconfiguration between a single wideband mode (1.0–3.2 GHz) and three narrowband modes. Potentially, it can be designed to cover a very wide bandwidth and can have a very wide range of frequency reconfiguration. To achieve switched band properties, eights ring slots which form filters were inserted to the antenna. The overall operating band can be switched by coupling each ring slot into the slot edges through the gaps controlled by means of PIN diode switches, which stop or pass the edge current to obtain frequency reconfiguration capability. Details of the proposed design are described. Wideband antenna design and narrowband reconfiguration design is explained in Sections II and III. In Section IV, the effect of switch controlled ring slots is discussed through passband-stopband analysis. Prototype antenna is discussed in Section V. Finally the results are presented in Section VI and follow by conclusions in Section VII.

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Fig. 3. Simulated surface current distributions excited at 3 GHz. Fig. 1. Nonreconfigurable Vivaldi antenna. (a) Front view. (b) Rear view showing microstrip feed.

Fig. 4. Simulated surface current distributions after perturbation excited at 3 GHz.

Fig. 2. Simulated wideband non reconfigurable Vivaldi antenna.

II. WIDEBAND ANTENNA DESIGN A Vivaldi antenna has been chosen as a basic structure due to the fact that it can operate over wide bandwidth. In addition, it also has a well defined radiation mechanism where most of the current flow is at the edge of the tapered slot. These characteristic help in designing a wideband-narrowband reconfiguration. The Vivaldi used was based on [15], and has been scaled to operate over a bandwidth of 1.0–3.2 GHz. It is shown in Fig. 1. The antenna is simulated and fabricated on FR4 substrate which has . The height of the substrate is 1.6 mm. The slot consists of a 4.6 mm radius circular hole and an elliptical tapered slot with 40 mm horizontal and 80 mm vertical radius, respecmm wide feed tively. The tapered slot is fed with a mm radius quarter circle. The subline terminated in a mm mm. The aperture size strate size is mm and the antenna length mm. Fig. 2 shows the simulated response of nonreconfigurable Vivaldi antenna. III. NARROWBAND AGILE ANTENNA DESIGN The Vivaldi can be reconfigured by perturbing the edge of the tapered slot, distorting the current flow. Fig. 3 shows the surface current before, and Fig. 4 after, the insertion of a ring slot into the antenna. It is clear that the current along the radiating edges is significantly reduced. There are a number of candidate shapes for the insertion, like ring, triangle, rectangle, and thin rectangle, as shown in Fig. 5(a), (b), (c), and (d), respectively. In terms of frequency band, losses, and size, the slot resonator has to be low Q in order to have wide stopband ranges, have low loss and be small in

Fig. 5. Slot shapes (a) ring-, (b) triangle-, (c) rectangle-, (d) thin rectangle shape.

size. Fig. 6 shows the simulated admittance showing that the Q factor is relatively similar for ring, triangle, and rectangle shape but slightly higher in the thin rectangle. The admittance is taken at the bottom ends of the slot line. A similar behavior is also observed in the insertion loss plot shown in Fig. 7. Therefore, ring, triangle, or rectangle shapes are good candidates. The is taken across the ends of the slot line. Nevertheless, we have chosen the ring shape slot, Fig. 5(a), because it’s relatively easy to design. The Vivaldi can be divided into two regions, namely the propagation and active regions. The propagation region is the area which acts like a transmission line and the active region is the area where the signals start to radiate. The radiation will occurs when slot width is approximately half of the wavelength, which makes each part of Vivaldi radiate at different frequencies. The radiation at lower frequencies will occurs closer to the wider end of the slot whilst higher frequencies at the narrower. This

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Fig. 6. Simulated admittance, jY j of various slot shape.

Fig. 7. Simulated jS(

)

j

of various slot shape.

suggest that the ring slots should be placed at various positions along the tapered slot to control the frequency performance. The way this is done is now described. Wideband to narrowband reconfiguration can be made by integrating a switched ring slot in the Vivaldi. For simplification purposes three narrowband modes were selected, namely low band, midband, and high band. Ideally, the frequency response of ring slot should be chosen depending on which narrowband configurations is designed. For example, a low pass filter response is used in selecting low-band mode while a bandpass response can be used to select mid-band mode. Fig. 8 shows the geometry and simulated response of the ring slots on a uniform slot line. A single ring slot produced a stopband response while cascaded ring slots produced a passband response. In addition, by adding bridges as shown in Fig. 8(c) transformed the bandstop to highstop or low-pass response. The equivalent transmission line models of the ring slot resonator are now described. The average circumference of the single ring slot in Fig. 9(a) is 52.3 mm which is approximately a half wavelength at 2.3 GHz. The differential operation of the slot line creates a virtual short circuit, denoted as sc, at the middle point of the structure as shown in the figure. The short circuit transforms over a quarter wavelength to an open circuit at the excitation point which produces a stopband at 2.3 GHz. On the other hand, a cascaded ring slots in Fig. 9(b) makes the length approximately half wavelength long at 2.2 GHz. This

Fig. 8. Simulated S of ring slot resonator: (a) bandstop; (b) bandpass; (c) low pass; (slot line width = 2 mm, ring outer radius = 10 mm, ring inner radius = 6 mm, small connecting gap = 4 mm 2 3.48 mm, small bridge [case (c) only] = 4 mm 2 2 mm).

Fig. 9. Ring slot resonator: (a) bandstop; (b) bandpass; (c) low pass.

thus makes a short circuit at the excitation point, which produces a passband around 2.2 GHz. Finally, with added bridges, the ring slot shown in Fig. 9(c) becomes a very short slot, an eighth wavelength long. This transforms to a stopband at 4.6 GHz or to an inductance at a lowband 1.2 GHz at the excitation point. Since the Vivaldi radiates from different parts of the structure at different frequencies, the narrower end area will become the radiating region for high frequency signals. It is thus appropriate to locate the ring slot there, in order to get the lowband mode. Fig. 10 shows a pair of low pass ring slots at the lower end. From it is clear that this produces a lowband response. the In order to stop the lowband, the top end of the antenna has the bandstop ring slot. Fig. 11 shows the effect of ring slot radius . With mm good return loss of top on the Vivaldi endband is achieved.

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Fig. 10. Simulated S

of low-band configurations (x

= 9:55 mm).

Fig. 11. Simulated S

of high-band configurations (x

= 56:2 mm).

Fig. 13. Impedances associated in finding optimum ring slot position, condi. tion when

Z =Z

S when resonators at matched position, x = = 43 77 mm.

Fig. 12. (a) Simulated antenna : , (b) resonators at x :

23 69

Midband operation is selected by using a bandpass ring slot configuration. The position of the ring slot section has an effect on the frequency response. Fig. 12 compares the performance for optimally and nonoptimally positioned 2.2 GHz ring slot resonator. To minimize the perturbation of the traveling wave on the tapered slot at the passband frequency, the ring slot section should be positioned at a point on the line where its is equal to the characteristic impedance image impedance of the line, . The image impedance of a short section of line containing the ring slot resonator (Fig. 13) is found from a separate simulation. It is defined as the terminating impedance on the output port (AA’) which leads to an input impedance (at BB’) that has the same value. In Fig. 14, resonators causing four

Fig. (

14. (a)

Vivaldi

with

different

bandpass

ring

slots;

RS1

ring outer radius = 12 mm, ring inner radius = 8 mm, small connecting gap = 4 mm 2 2.5 mm, x = 33:68 mm), RS2 (ring outer radius = 10 mm, ring inner radius = 6 mm, small connecting gap = 4 mm 2 2.88 mm, x = 23:69 mm), RS3 (ring outer radius = 8:5 mm, ring inner radius = 5:5 mm, small connecting gap = 3 mm 2 2.5 mm, x = 23:69 mm), and RS4 (ring outer radius = 7 mm, ring inner radius = 4 mm, small connecting gap = 3 mm 2 2.32 mm, x = 10:37 mm). (b) Simulated S of four bandpass configurations at matched position. different passbands have all been positioned to achieve good passband match using this technique.

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Fig. 15. Four-band reconfigurable antenna, (ringouterradius = 10 mm, ring inner radius = 6 mm, small connecting gap = 4 mm 3.5 mm, x = 9:55 mm, x = 33:73 mm, x = 56:2 mm, small bridge [the lowest slot only] = 4 mm 2 mm).

2

2

Fig. 18. Low-band dispersions curve, (attenuation and phase).

Fig. 16. (a) Low-band; (b) Mid-band; (c) High-band configuration.

Fig. 17. Tapered slot line voltages.

The frequency of the passband can be selected by designing of the ring slot as shown in Fig. 14(a). If reconfiguration from a wideband to a single narrow band is required the narrowband frequency can be chosen in this way. However there are size constraints if, as done in the demonstration described in this paper, wideband and low-, middle-, and high-pass modes are integrated together. IV. FOUR-MODE RECONFIGURABLE CONFIGURATION A wideband to 3 narrowband modes reconfigurable antenna is demonstrated here by inserting all three sets of ring slots into the antenna as shown in Fig. 15. To fit all in, the position of the middle ring slots is shifted up by 10.04 mm from the matched position. The outer radius of the uppermost ring slots have also been reduced to 10 mm to allow integration. This makes all ring slots have an inner radius of 6 mm and an outer radius of 10 mm. The positions of the switches necessary for reconfiguration are also shown in the figure. Whilst CST Microwave Studio is used to simulate the scattering parameters of the proposed antenna for comparison with the prototype, an alternative analysis is now described which gives insight into the performance of each set ring slots in the presence of others. Fig. 16 shows the antenna of Fig. 15 in its three narrowband states. On each figure two right pointing

Fig. 19. Midband dispersions curve, (attenuation and phase).

arrows indicate the points at which the voltage across the tapered slot is extracted from the simulation. Fig. 17 shows an expanded view of the line in the vicinity of one of the resand are meaonators, indicating more precisely where sured in each case. Taking the ratio of the complex voltages and gives the propagation constant where is neper and bd is radians, which can thus be extracted. These are now examined for each of the three cases. A. Narrow Low-Band Mode The low-band mode characteristic is shown in Fig. 18. It can be seen that a very high attenuation occurs from 1.25 to 3.2 GHz, and thus the structure prevents any signals beyond 1.25 GHz propagating up the tapered slot. The minimum attenuation occurs at 1.15 GHz. B. Narrow Mid-Band Mode Fig. 19 shows the characteristic of the mid-band mode structure. Its shows that minimum attenuation occurs from 2.0 to 2.2. A high value of in the passband region reflects the low gain noticed in the Table I. There is also evidence of a passband plots shown in around 3.2 GHz, which is also noticed in the Fig. 27(b).

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Fig. 20. High-band dispersions curve, (attenuation and phase).

Fig. 21. Gain comparison between WR, ID, and PD antennas.

C. Narrow High-Band Mode Fig. 20 shows the characteristic of the high-band mode structure. Its shows a stopband, which occur from 1.0 to 2.9 GHz. The passband around 0.8 GHz is also seen in in Fig. 27(c). The minimum attenuation occurs around 3.2 GHz. In the above analysis, the positive value of , neper means loss and/or standing waves and a minima in or , means a passband or low loss traveling wave. Small negative values of are due to small inaccuracies in the calculation, due to the slot line impedance expressions having width limits below the slot width in the antenna. The effects of the ring slots on the gain of the antenna can be analyzed by comparing three antennas configurations, one without the ring slots (WR), as indicated in Fig. 1, one with the ring slots using ideal switches (ID), as shown in Fig. 15, and one with the ring slots using PIN diode switches (PD), as shown in Fig. 24. Even though the ring slots occupy a substantial part of the antenna surface, it is observed that the effect on the gain between WR and ID antennas are small. Fig. 21 also compares the simulated and measured gain of the WR, ID and PD antennas in wideband mode. It is observed that, the gain degradation is caused mainly by the loss in the PIN diode switches. Further insight may be gained by examining the current distributions in the various modes. The comparison between the simulated surface current distributions in the three narrowband operative states and the wideband mode are shown in Fig. 22. In general a similar current distribution is observed between the

Fig. 22. Simulated surface current distributions for: (a) Wideband configuration excited at 1.1 GHz; (b) Low-band configuration excited at 1.1 GHz; (c) Wideband configuration excited at 2.2 GHz; (d) Mid-band configuration excited at 2.2 GHz; (e) Wideband configuration excited at 3.1 GHz; (d) High-band configuration excited at 3.1 GHz.

Fig. 23. Simulated current distributions for the low-band configuration excited at (a) 2 GHz and (b) 3 GHz.

low-band and wideband configurations when excited at the same frequency. The same behavior is also observed between the midand high-band configuration. The current distributions excited beyond the operating frequencies for low-band configurations is also shown in Fig. 23. It is clearly observed that the current at

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Fig. 24. Prototype.

Fig. 25. Simulated wideband mode S ring slots.

response of antenna with and without

Fig. 26. Simulated and measured wideband mode S . Fig. 27. Simulated and measured S11 of antenna in narrowband modes: (a) Low-band state; (b) Mid-band state; (c) High-band state.

the resonant frequency, 1.1 GHz, propagates through the upper part of the antenna, as shown in Fig. 22(b). However most of the current at 2 and 3 GHz is cut off by the lower ring slots, therefore stopping high frequency band operation. V. PROTOTYPE ANTENNA In the CST simulation, the PIN diode was modeled in its ON state as a 2 ohm resistance and in its OFF state using an S-parameter file supplied by the diode manufacturer. The DC lines and DC decoupling components is included in the simulation. The practical antenna is shown in Fig. 24. To give a minimal effect on the radiation pattern, four very thin DC lines were

printed parallel to the beam direction on either side of the antenna. Therefore if there are any RF leakages through the DC line, the affect will be negligible. The DC line was isolated from the RF signal by using 27nH SMD inductors. The tapered slot is DC isolated into four sections with 0.3 mm width slots. To preserve RF continuity, 22pF SMD capacitors bridged the slots every 10 mm. Each switch in Fig. 15 consists of two in-series Infineon PIN diodes, BAR50-02V. Two diodes are used in series because a single diode having 1.2 mm length is unable to bridge the 4 mm gaps. Gaps of 4 mm are used to reduce spurious resonances at higher frequency due to parasitic capacitance across

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TABLE I GAIN OF THE PROPOSED ANTENNA

VI. RESULTS Fig. 25 compares the simulated return loss for wideband operation with and without the ring slots. It is noticed that the S between 1.8 and 3.0 GHz has been improved by the ring slots. The simulated and the measured wideband responses with ring slots are shown in Fig. 26. Good agreement can be seen between them. The results show that a good matching for wideband operation is achieved over a 1.0–3.2 GHz bandwidth. The return loss for the other three states, low band 1.1 GHz, midband 2.25 GHz, and high band 3.1 GHz are shown in Fig. 27. A frequency shift between the measurement and simulation is presumably due to the effect of parasitics in the PIN diodes, fabrication tolerances and biasing components. The PIN diode parasitics depend upon the PIN diode structure and mounting configuration in the antenna. Full wave simulation would be required to established accurate values. However, a general agreement has been achieved in which the main features of the responses are predicted. The E- and H-plane simulated and measured radiation patterns are presented in Fig. 28. Good agreement between simulation and measurement has been achieved. It can be concluded that the biasing circuit has a small effect on the antenna performance. It is also observed that the 3 dB beam width becomes narrower as the frequency increased, which is usual for the Vivaldi antenna. Finally, Table I compares the measured and simulated gains. There is some difference between measurement and simulation. There is also about 3 dB difference between measured wideband gain at 2 GHz and measured mid-band gain at 2.25 GHz. VII. CONCLUSION

Fig. 28. Radiation pattern. (a) Low-band mode excited at 1.1 GHz, E-plane (left), H-plane (right). (b) Mid-band mode exited at 2.2 GHz, E-plane (left), H-plane (right). (c) High-band mode excited at 3.1 GHz E-plane (left), H-plane (right). (d) Wideband mode excited at 2.0 GHz, E-plane (left), H-plane (right). co-polar simulated

co-polar measured.

the gap sides. However, further work might reveal alternative ways to design the gap that could be bridged by a single diode having 1.2 mm length while avoiding spurious resonance problems. The diodes were forward biased appropriately with DC voltage to obtain 100 mA ON state bias current. To obtain the OFF state, the diodes were left unbiased.

A novel switched band Vivaldi antenna has been proposed by introducing the ring slots. Three modes of reconfiguration are shown, when three pairs of ring slots were employed in three different positions within the Vivaldi. The design guideline has been derived and allows some optimizing of the band positions. More sub-bands could be achieved with additional resonators, in a longer version of the antenna. Nevertheless, in the relatively simple example shown good return loss has been obtained for each of the subbands. The proposed antenna could be a suitable solution for applications requiring wideband sensing and dynamic band switching, such as in military application or cognitive radio. REFERENCES [1] S. Yang, C. Zhang, H. Pan, A. Fathy, and V. Nair, “Frequency-reconfigurable antennas for multiradio wireless platforms,” IEEE Microw. Mag., vol. 10, pp. 66–83, Feb. 2009. [2] J. Cho, C. W. Jung, and K. Kim, “Frequency-reconfigurable two-port antenna for mobile phone operating over multiple service bands,” IET Electron. Lett., vol. 45, no. 20, pp. 1009–1011, Sep. 2009.

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[3] S. L. S. Yang, A. A. Kishk, and F. L. Kai, “Frequency reconfigurable U-slot microstrip patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 127–129, May 2008. [4] T. Y. Han and C. T. Huang, “Reconfigurable monopolar patch antenna,” IET Electron. Lett., vol. 46, no. 3, pp. 199–200, Feb. 2010. [5] A. F. Sheta and S. F. Mahmoud, “A widely tunable compact patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 40–42, Mar. 2008. [6] C. Cordeiro, K. Challapali, D. Birru, and N. S. Shankar, “IEEE 802.22: The first worldwide wireless standard based on cognitive radios,” in Proc. First IEEE Int. Symp. New Frontiers in Dyn. Spectrum Access Netw. DySPAN, Baltimore, MD, Dec. 2005, pp. 328–337. [7] B. A. Fette, Cognitve Radio Technology. Burlington, MA: Elsevier, 2009. [8] Y. Tawk and C. G. Christodoulou, “A new reconfigurable antenna design for cognitive radio,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1378–1381, Dec. 2009. [9] F. Ghanem, P. S. Hall, and J. R. Kelly, “Two port frequency reconfigurable antenna for cognitive radios,” IET Electron. Lett., vol. 45, no. 11, pp. 534–536, May 2009. [10] E. Ebrahimi, J. Kelly, and P. S. Hall, “A reconfigurable narrowband antenna integrated with wideband monopole for cognitive radio applications,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., Charleston, SC, 2009, pp. 1–4. [11] L. Zidong, K. Boyle, J. Krogerus, M. de Jongh, K. Reimann, R. Kaunisto, and J. Ollikainen, “MEMS-switched, frequency-tunable hybrid slot/PIFA antenna,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 311–314, Feb. 2009. [12] J. R. Kelly, P. S. Hall, and P. Gardner, “Integrated wide-narrow band antenna for switched operation,” in Proc. IEEE EuCAP Antennas Propag., Berlin, 2009, pp. 3757–3760. [13] M. R. Hamid, P. Gardner, and P. S. Hall, “Reconfigurable log periodic aperture fed microstrip antenna,” in Proc. LAPC Conf. Antennas Propag., Loughborough, U.K., 2009, pp. 237–239. [14] T. Wu, R. L. Li, S. Y. Eom, S. S. Myoung, K. Lim, J. Laskar, S. I. Jeon, and M. M. Tentzeris, “Switchable quad-band antennas for cognitive radio base station applications,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1468–1476, May 2010. [15] P. Li, J. Liang, and X. Chen, “UWB tapered-slot-fed antenna,” in Proc. IET Seminar on Ultra Wideband Systems, Technol. Appl., 2006, pp. 235–238.

M. R. Hamid received the M.Sc. degree in communication engineering from the Universiti Teknologi Malaysia (UTM), Skudai, Johor, Malaysia, in 2001. He is currently working toward the Ph.D. degree at the University of Birmingham, Birmingham, U.K. He has been with the Faculty of Electrical Engineering (FKE), UTM, since 2001. His major research interest is reconfigurable antenna design for multimode wireless applications. He was awarded a scholarship from the UTM to further study in the U.K.

Peter Gardner (M’99–SM’00) received the B.A. degree in physics from the University of Oxford, Oxford, U.K., in 1980 and the M.Sc. and Ph.D. degrees in electronic engineering from the University of Manchester Institute of Science and Technology (UMIST), Manchester, U.K., in 1990 and 1992, respectively. In 1994, he became a Lecturer with the School of Electronic and Electrical Engineering, University of Birmingham, Birmingham, U.K. He was promoted to Senior Lecturer in 2002 and to Reader in Microwave Engineering in 2009. His current research interests are in the areas of microwave and millimetric integrated active antennas and beamformers, microwave amplifier linearization techniques, and reconfigurable broadband and multiband antennas.

Peter S. Hall (M’88–SM’93–F’01) joined the University of Birmingham, Birmingham, U.K., in 1994, where he is a Professor of communications engineering, leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre in the Department of Electronic, Electrical and Computer Engineering. He has researched extensively in the areas of microwave antennas and associated components and antenna measurements. He has published five books, more than 250 learned papers, and taken various patents. Prof. Hall is a Fellow of the IET and a past IEEE Distinguished Lecturer. He is a past Chairman of the Institute of Electrical Engineers (IEE) Antennas and Propagation Professional Group and past coordinator for Premium Awards for IEE Proceedings on Microwave, Antennas and Propagation and is currently a member of the Executive Group of the IEE Professional Network in Antennas and Propagation. He was Honorary Editor of the IEE Proceedings Part H from 1991 to 1995 and is currently on the editorial board of Microwave and Optical Tech Letters. His publications have earned six IEE premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. He is a member of the Executive Board of the EC Antenna Network of Excellence.

F. Ghanem received the Bachelor degree in electronics engineering from the Ecole Nationale Polytechnique of Algiers, in 1996, and the M.Sc. and Ph.D. degrees from the Institut National de la Recherche Scientifique (INRS), Canada, in 2007. He was an honorary Research Fellow in the Department of Electrical Engineering and Electronics, University of Birmingham, U.K., from 2007 until October 2009. In 2010, he became an Assistant Professor with the Prince Mohammed Bin Fahd University, Al-Khobar, Saudi Arabia. His current research interests are in the areas of antenna and RF passive and active circuits design. He is also interested in wireless signal processing.

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Focusing Properties of Fresnel Zone Plate Lens Antennas in the Near-Field Region Shaya Karimkashi, Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—Some focusing properties of Fresnel zone plate (FZP) lens antennas in the near-field region are presented at the ka-band. Simulated and measured results of the FZP antenna show displacement of the maximum intensity of the electric field along the axial direction from the focal point toward the antenna aperture. This displacement increases as the antenna’s focal length increases. In addition, the focused beam scanning of the FZP lens antennas in the radiation near-field is examined. Index Terms—Focusing, Fresnel zone plate lenses, near fields.

I. INTRODUCTION

F

OCUSED antennas are of interest in many applications including microwave wireless power transmission, remote (non-contact) sensing and medical applications. The concept of microwave power transmission to transmit electric power without transmission lines is among the foremost applications of focused antennas [1], [2]. In this application, it is needed to provide the power of orbiting satellites or space radar systems from a power station on earth or an orbiting power satellite [3]. Moreover, it can be used to provide the power from solar power satellite systems to terrestrial markets [4]. Another application of focused antennas is in remote sensing where a focused beam for a precise sensing is needed. In other words, the focused antenna concentrates the signal energy on the point we want to sense [5]. Focused antennas are also of great importance for achieving microwave-induced hyperthermia in medical applications. In this application, the power deposition should be confined on cancerous tissues not to heat the adjacent healthy tissues [6], [7]. However, usually array antennas are used rather than single antennas for this application. Focused antennas are used to focus the microwave power at a point close to the antenna aperture which is usually in the near-field region of the antenna. It was proven theoretically that in the focal plane of a focused aperture, near the axis, the electric field will have all the properties of the far field radiation pattern if a quadratic phase distribution is adjusted on the aperture of the antenna [8]–[10]. Some attempts have been made Manuscript received June 17, 2010; revised September 26, 2010; accepted November 08, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. S. Karimkashi was with the Department of Electrical Engineering. University of Mississippi. University, MS 38677 USA. He is now with the Atmospheric Radar Research Center, Norman, OK 73072 USA (e-mail: [email protected]) A. A. Kishk is with the Department of Electrical Engineering. University of Mississippi. University, MS 38677 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123069

using array antennas [5], [11] and conventional reflector antennas [12], [13] to generate quadratic phase distributions and focus the power at nearby points. However, precise generation of a quadratic phase distribution for large array antennas is complex, costly and limited due to the difficulties in implementing of beam forming networks [11]. On the other hand, the ability to produce a desired aperture distribution using a conventional reflector like parabolic reflector antenna is limited. Although the problem can be solved using shaped reflectors, their implementation is very costly. Another alternative focused antenna is the Fresnel zone plate (FZP) lens consisting of a set of alternative open and opaque annuli arranged on flat or curvilinear surfaces [14], [30], [15], [16]. The planar FZP lens has the benefit of being lighter and easer to design and manufacture compared to the array antennas and reflectors. Moreover, FZP lenses are lighter and thinner than traditional lens antennas especially when a large antenna aperture is needed. In fact the flatness aspect of the FZP antenna is its benefit in the manufacturing process. The focusing behavior of the planar and spherical FZP antennas was first presented in [17]. The focal shift characteristics of these antennas using diffraction theory was described in [18]. Later, some axial defocusing characteristics of FZP lens antennas were presented in [19]–[21]. In [22]–[24] the focusing behaviors of curvilinear and planar FZP lenses were studied. It was shown that the curvilinear FZP lenses don’t necessarily have superior focusing abilities compared to the planar FZP lenses. In [25] it was shown that zone plate and hyperbolic lens antennas have similar focusing properties except that the FZP antennas deliver less power to their focal points. Some other focusing properties of FZP antennas can be found in [26]–[28]. In this paper, some focusing characteristics of Soret FZP lens antennas [14], [30], [15] are examined. Although it is well known that phase corrected (Wood-type) zone plate antennas are more efficient than Soret zone plate antennas, the later one is much simpler to design and fabricate in order to generally represent the focusing properties of FZP lens antennas. Different FZP lens antennas are designed to investigate the effect of focal lengths on both axial and transverse electric field patterns. Simulation and measurement results show the displacement of the maximum intensity of the electric field along the axial direction. Finally, the scanning characteristics of the FZP antenna focused beam are presented. II. DESIGN PROCEDURE The geometrical optic (GO) method is used in the design of FZP antennas to achieve constructive interference at the focal point. Fig. 1 shows the two dimensional configuration of the

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Fig. 1. Two-dimensional configuration of the FZP fed by a circular horn antenna.

FZP fed by a circular horn antenna. The design is performed in the plane (two-dimensional) and then the shape is completed by revolving it around the axis of symmetry. The FZP is designed as circular concentric zones with radius for the -th zone. is the diameter and and are of , the radius of the focal lengths of the FZP. The values of each zone, should be determined such that the rays emanating from one focal point add up in phase at the other focal point. values can be found by satisfying:

Fig. 2. V and H cuts of the measured radiation pattern of the LHCP feed horn antenna at the frequency of 32 GHz.

TABLE I THE NUMBER OF ZONES AND DIAMETER OF EACH ZONE PLATE

(1) is an integer, is the wavelength and and where are the distances between the -th ring and the first and second focal points, respectively. This equation can be rewritten as:

(2) can be obtained by a simple routine and The exact values of trial and error to satisfy the (2). III. SIMULATED AND MEASURED RESULTS FZP antennas with diameter of 0.16 m and various focal lengths, m, 0.075 m, 0.15 m, 0.30 m, 0.45 m, 0.6 m, 0.75 m and 0.9 m, are designed and modeled by a full wave MoM solution [29]. The diameter and the number of zones of each FZP are presented in Table I. It should be mentioned that the antennas with first three focal lengths are focused in the near-field region and the others are focused in the Fresnel region of the antennas. A corrugated circular horn antenna with left-handed circular polarization (LHCP) is used as the antenna feed. Fig. 2 shows the V and H cut measured radiation pattern of the horn antenna at the frequency of 32 GHz. A subtended angle of 40 degrees is chosen to have an edge taper value of dB on FZP rims with the focal length of about m. In order to verify the simulated results, two FZP antennas m and 0.45 m, are fabricated by etching on an with RT/duroid 5880 laminate. The substrate is thin enough (0.508 mm) to assume the opaque rings of FZPs are suspended in air.

Fig. 3. FZP lens antenna test set-up.

The simulation results show that the effect of substrate on the electric field pattern is negligible. It should be noted that the substrate may cause some defocusing on the axial field pattern if the substrate is thick or it has high dielectric constant. However, our calculations using GO shows that the defocusing effect, due to the small thickness of the substrate and the low value of the dielectric constant, is very small. Fig. 3 illustrates the antenna under test at the University of Mississippi planar near-field set up. The simulated and measured transverse electric field distributions along the axial direction for the FZP with the focal length of 0.15 m are shown in Fig. 4. A good agreement between the simulated and measured results is observed. The maximum inm distance tensity of the electric field occurred at from the antenna aperture. The simulated and measured electric m) field distribution of the antenna at the focal plane (

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Fig. 7. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.45 m along its axial direction. Fig. 4. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.15 m along its axial direction.

Fig. 8. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.45 m at the focal plane (z = 0.45 m). Fig. 5. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.15 m at the focal plane (z = 0.15 m).

Fig. 6. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.15 m at the maximum intensity plane (z = 0.137 m).

and the maximum intensity plane ( m) are shown in Figs. 5 and 6, respectively. Notice the triple representation of the x-axis in terms of position, angular direction and radial distance. It can be seen that simulated and measured results are in agreement especially around the main beam.

Fig. 9. Simulated and measured normalized electric field intensity of the FZP antenna with the focal length of 0.45 m at the maximum intensity plane (z = 0:343 m).

The electric field distribution of the FZP antenna with the focal length of 0.45 m is also measured in different planes. Fig. 7 shows the simulated and measured electric field distribution of the antenna versus the axial distance. The maximum intensity m distance from the anof the electric field occurs at tenna aperture. Figs. 8 and 9 show the simulated and measured m) and the electric field pattern at the focal plane (

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TABLE II COMPARISON BETWEEN THE BEAM WIDTH OF FZAS AT THE FOCAL PLANE AND MAXIMUM INTENSITY PLANE

Fig. 12. Normalized maximum intensity values of the FZP antenna versus the focal length.

Fig. 10. Normalized electric field intensity of the FZP antenna versus the axial direction for different focal lengths. Fig. 13. Normalized electric field distribution of the FZP at the focal plane for different focal lengths.

Fig. 11. Focal displacement of FZP antenna versus the focal length.

maximum intensity plane ( m), respectively. Comparing the beam widths of the patterns at the focal plane and the maximum intensity plane for both FZP antennas in Table II, one can observe that a narrower beam is obtained at the maximum intensity plane compared to the one at the focal plane. In order to evaluate the focusing properties of FZP antenna, the electric field distributions versus the axial distances for FZP antennas with different focal lengths are plotted in Fig. 10. It can

be seen that the maximum intensity of the electric field is displaced from the focal point toward the antenna aperture. Fig. 11 shows this displacement for different focal lengths. The normalized maximum intensity values of the FZP antenna versus the focal length are shown in Fig. 12. It is observed that as the focal point moves away from the antenna aperture the focal displacement increases and the maximum intensity value decreases. It should be noted that by increasing the focal length, the maximum intensity length increases with a lower rate and cannot go beyond a certain distance. The reason for this displacement is the spherical spreading of the wave front away from the source. Since the magnitude of the radiating field decays as , where is the distance from the source, by increasing the focal length, the effect of this spreading factor on the focal displacement increases. Although, all rays emanating from the source contribute in phase at the focal point, they add up partly in phase with higher intensities at closer points to the aperture causing the occurrence of the maximum intensity point and therefore the focal shift. The normalized electric field distributions of the antenna for different focal lengths at the focal planes and maximum intensity planes are shown in Figs. 13 and 14, respectively. It can be observed that by increasing the focal length, the half power

KARIMKASHI AND KISHK: FOCUSING PROPERTIES OF FRESNEL ZONE PLATE LENS ANTENNAS IN THE NEAR-FIELD REGION

Fig. 14. Normalized electric field distribution of the FZP at the maximum intensity plane for different focal lengths.

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Fig. 16. Variations of sidelobe level of the electric field pattern at maximum intensity planes and focal planes versus the focal length of the FZP antenna.

Fig. 15. Variations of depth of focus and half power beam width of the electric field pattern at maximum intensity plane versus the focal length of the FZP antenna.

beam width (HPBW) at both the focal plane and maximum intensity plane increases. It should be noted that since the electric field patterns are plotted at different distances from the antenna aperture, the pattern for the smaller focal lengths are observed in wider range of view angles. The variations of HPBW and depth of focus versus the focal lengths are depicted in Fig. 15. The depth of focused is defined as the distance between the axial dB points about the maximum intensity plane [8]. It is seen that by increasing the focal length, not only the HPBW but also the depth of focus, increases. It should be noted that for small focal shifts, the desired focal points falls within the 3 dB zone. However, by increasing the focal length, the focal point falls out of this range. The variation of the sidelobe levels (SLL) of the electric field patterns at maximum intensity planes and focal planes versus the focal length are shown in Fig. 16. It can be seen that by increasing the focal length in the Fresnel region, SLLs of the electric field patterns at both maximum intensity and focal planes are increased. Sidelobe levels in the maximum intensity plane increase faster than that in the focal plane since by increasing the focal length, the focal shift increases. Therefore, increasing the focal length of the antenna causes a wider focusing beam and higher SLLs.

Fig. 17. (a) Simulated and (b) measured steered focused beam of the FZP antenna with the focal length of 0.45 m at the maximum intensity plane for various feed displacements.

TABLE III THE STEERED FOCUSED BEAM PROPERTIES OF THE FZA WITH THE FOCAL LENGTH OF 0.45 M

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IV. BEAM SCANNING BY FEED MOTION Scanning the antenna’s focused beam is of interest in many applications. Although it is costly and complicated to scan the large antenna beam using phase arrays, it can be done easily in FZP antennas. The beam scanning is accomplished by displacing the feed horn antenna along its transverse axis. Fig. 17 shows the simulated and measured electric field distribution of the FZP with the focal length of 0.45 m at the maximum intensity plane for various feed displacements. These results are summarized in Table III. It can be seen that the beam is steered by approximately 0.015 m for each 0.005 m displacement of the feed. It should be mentioned that the steered beams have approximately the same maximum intensity length, depth of focus and HPBW as the original focused antenna. In other words, although the feed is displaced from the FZP focal point, the scanned beams degradations are negligible. V. CONCLUSION Electric field pattern characteristics of the focused Fresnel zone plate lens antenna in the near-field region were presented. The FZP antenna fed by a circular horn was implemented and the effects of various focal lengths on the electric field pattern of this antenna were examined. It was shown that the maximum intensity occurs closer to the focal point and this displacement increases as the focal point moves away from the antenna aperture. By increasing the focal length of the antenna, HPBW, depth of focus and side lobe levels increase. In addition, the focused beam scanning of the FZP antenna was implemented by displacing the feed from the FZP focal point. ACKNOWLEDGMENT The authors would like to thank Rogers Corporation for providing free dielectric substrates. REFERENCES [1] A. Douyere, J.-D. Lan Sun Luk, A. Celeste, and J.-P. Chabriat, “Modeling and simulation of a complete system of energy transmission,” in IEEE Int. Symp. Antennas Propagation Dig., Jul. 2006, pp. 811–814. [2] W. C. Brown, “The history of power transmission by radio waves,” IEEE Trans. Microwave Theory Tech., vol. 32, no. 2, pp. 1230–1242, Sep. 1984. [3] J. O. McSpadden, F. E. Little, M. B. Duke, and A. Ignatiev, “An in-space wireless energy transmission experiment,” in Proc. Int. Energy Conversion Eng., Aug. 1996, vol. 1, pp. 468–473. [4] J. O. McSpaddan and J. C. Mankins, “Space solar power programs and microwave wireless power transmission technology,” IEEE Microw. Mag., vol. 3, no. 4, pp. 46–57, Dec. 2002. [5] M. Bogosanovic and A. G. Williamson, “Microstrip antenna array with a beam focused in the near-field zone for application in noncontact microwave industrial inspection,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2186–2195, Dec. 2007. [6] C. H. Durney and M. F. Iskandar, “Antennas for medical applications,” in Antenna Hand Book: Theory, Applications, and Design, Y. T. Lo and S. W. Lee, Eds. New York: Van Nostrand, 1988, ch. 24. [7] J. T. Loane, III and S. Lee, “Gain optimization of a near-field focusing array for hyperthermia applications,” IEEE Trans. Microwave Theory Tech., vol. 37, no. 10, pp. 1629–1635, Oct. 1989. [8] J. W. Sherman, “Properties of focused aperture in the Fresnel region,” IRE Trans. Antennas Propag., vol. AP-10, no. 4, pp. 399–408, Jul. 1962. [9] W. J. Graham, “Analysis and synthesis of axial field patterns of focused apertures,” IEEE Trans. Antennas Propag., vol. AP-31, no. 4, pp. 665–668, Jul. 1983.

[10] R. C. Hanson, “Focal region characteristics of focused array antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 6, pp. 1328–1337, Dec. 1985. [11] S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a Dolph-Chebyshev near-field design,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3813–3820, Dec. 2009. [12] P. S. Kildal and M. M. Davis, “Characterization of near-field focusing with application to low altitude beam focusing of the Arecibo tri-reflector system,” IEE Proc., vol. 143, no. 4, pp. 284–292, Aug. 1996. [13] L. Shafai, A. A. Kishk, and Sebak, “Near field focusing of apertures and reflector antennas,” in Proc. IEEE Communications, Power and Computing Conf., May 22–23, 1997, pp. 246–251. [14] J. L. Soret, “Ueber die durch kreisgitter erzeugten diffractionsphänomene,” Ann. Phy. Chem., vol. 156, pp. 99–113, 1875. [15] H. D. Hristov, Fresnel Zones in Wireless Links, Zone Plate Lenses, and Antennas. Boston, MA: Artech House, 2000. [16] O. V. Minin and I. V. Minin, Diffractional Optics of Millimetre Waves. Bristol, U.K.: Institute of Phys. Publishing, 2004. [17] K. K. Dey and P. Khastgir, “Comparative focusing properties of spherical and plane microwave zone plate antennas,” Int. J. Electron., vol. 35, no. 4, pp. 497–506, 1973. [18] Y. Li and E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun., vol. 39, no. 4, pp. 211–215, 1981. [19] H. D. Hristov, R. Feick, W. Grote, and P. Fernandez, “Indoor signal focusing by means of Fresnel zone plate lens attached to building wall,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 933–940, Apr. 2004. [20] D. R. Reid and G. S. Smith, “A full electromagnetic analysis for the Soret and folded zone plate antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3638–3646, Dec. 2006. [21] D. R. Reid and G. S. Smith, “A full electromagnetic analysis of grooved-dielectric Fresnel zone plate antennas for microwave and millimeter applications,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2138–2146, Aug. 2007. [22] H. D. Hristov, L. P. Kamburov, J. R. Urumov, and R. Feick, “Focusing characteristics of curvilinear half-open Fresnel zone plate lenses: Plane wave illumination,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 1912–1919, Jun. 2005. [23] I. V. Minin and O. V. Minin, “Comments on ‘focusing characteristics of curvilinear half-open fresnel zone plate lenses: Plane wave illumination’,” IEEE Trans. Antennas Propag., vol. 54, no. 6, p. 2692, Sep. 2006. [24] D. Hristov, L. P. Kamburov, J. R. Urumov, and R. Feick, “Reply to comments on ‘focusing characteristics of curvilinear half-open fresnel zone plate lenses: Plane wave illumination’,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 2692–2693, Sep. 2006. [25] D. R. Reid and G. S. Smith, “A comparison of the focusing properties of a Fresnel zone plate with a doubly-hyperbolic lens for application in a free-space, focused beam measurement system,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 499–507, Feb. 2009. [26] F. Pfeiffer and C. David, “Nanometer focusing properties of Fresnel zone plates described by dynamical diffraction theory,” Phys. Rev. B., vol. 73, no. 24, 2006. [27] Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Amer. A, vol. 1, pp. 801–808, 1984. [28] S. Karimkashi and A. A. Kishk, “A new Fresnel zone antenna with beam focused in the Fresnel region,” presented at the URSI National Radio Science Meeting, Chicago, IL, Aug. 7–16, 2008. [29] FEKO, 5.2 Stellenbosch, South Africa. [30] , J. Ojeda-Castañeda and C. Gòmez-Reino, Eds., Selected Papers on Zone Plates. Bellingham, WA: SPIE, 1996, pp. 11–25.

Shaya Karimkashi (M’08) received the B.S. degree in electrical engineering from K. N. Toosi University of technology, Tehran, Iran and the M.S. degree in electrical engineering from the University of Tehran, in 2003 and 2006, respectively, and the Ph.D. degree from the University of Mississippi, University, in 2010. He is currently with the Atmospheric Radar Research Center, Norman, OK. His research interests include array antennas, focused antennas, reflector antennas, optimization methods in EM and microwave measurement technique. Dr. Karimkashi is a member of the Sigma Xi and Phi Kappa Phi societies.

KARIMKASHI AND KISHK: FOCUSING PROPERTIES OF FRESNEL ZONE PLATE LENS ANTENNAS IN THE NEAR-FIELD REGION

Ahmed A. Kishk (F’98) received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977 and in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, Canada, in 1983 and 1986, respectively. From 1977 to 1981, he was a Research Assistant and an instructor at the Faculty of Engineering, Cairo University. From 1981 to 1985, he was a Research Assistant at the Department of Electrical Engineering, University of Manitoba, where, from December 1985 to August 1986, he was a Research Associate Fellowt. In 1986, he joined the Department of Electrical Engineering, University of Mississippi, as an Assistant Professor. He was on sabbatical leave at Chalmers University of Technology, Sweden during the 1994–1995 and 2009–2010 academic years. He is now a Professor at the University of Mississippi (since 1995) and Director of the Center of Applied Electromagnetic System Research (CAESR). He was an Associate Editor of the IEEE Antennas and Propagation Magazine from 1990 to 1993, and is currently an Editor. He was a Co-editor of the special issue, “Advances in the Application of the Method of Moments to Electromagnetic Scattering Problems,” in the ACES Journal. He was also an Editor of the ACES Journal during 1997 and was an Editor-in-Chief from 1998 to 2001. He was the Chair of the Physics and Engineering Division of the Mississippi Academy of Science (2001–2002). He was a Guest Editor of the Special Issue on Artificial Magnetic Conductors, Soft/Hard Surfaces, and Other Complex Surfaces of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, January 2005. His research

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interest includes the areas of design of millimeter frequency antennas, feeds for parabolic reflectors, dielectric resonator antennas, microstrip antennas, EBG, artificial magnetic conductors, soft and hard surfaces, phased array antennas, and computer aided design for antennas. He has published over 220 refereed journal articles and 27 book chapters. He is a coauthor of the book Microwave Horns and Feeds (IEE, 1994; IEEE, 1994) and a coauthor of chapter 2 on Handbook of Microstrip Antennas (Peter Peregrinus Limited, 1989). Dr. Kishk is a Fellow of the IEEE since 1998 and the Electromagnetic Academy. He is a member of Antennas and Propagation Society, Microwave Theory and Techniques, Sigma Xi society, U.S. National Committee of International Union of Radio Science (URSI) Commission B, Phi Kappa Phi Society, Electromagnetic Compatibility, and Applied Computational Electromagnetics Society. He received the 1995 and 2006 outstanding paper awards for papers published in the Applied Computational Electromagnetic Society Journal. He received the 1997 Outstanding Engineering Educator Award from Memphis section of the IEEE. He received the Outstanding Engineering Faculty Member of the Year on 1998 and 2009, Faculty research award for outstanding performance in research on 2001 and 2005. He received the Award of Distinguished Technical Communication for the entry of IEEE Antennas and Propagation Magazine, 2001. He also received The Valued Contribution Award for outstanding Invited Presentation, “EM Modeling of Surfaces with STOP or GO Characteristics—Artificial Magnetic Conductors and Soft and Hard Surfaces” from the Applied Computational Electromagnetic Society. He received the Microwave Theory and Techniques Society Microwave Prize 2004.

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Wideband Millimeter-Wave Substrate Integrated Waveguide Slotted Narrow-Wall Fed Cavity Antennas Yan Zhang, Student Member, IEEE, Zhi Ning Chen, Fellow, IEEE, Xianming Qing, Member, IEEE, and Wei Hong, Senior Member, IEEE

Abstract—A technique to design substrate integrated waveguide (SIW) slotted narrow-wall fed cavity antennas is presented and investigated for millimeter-wave systems. The proposed antenna consists of a feeding SIW and a slotted narrow-wall fed high permittivity dielectric loaded substrate integrated cavity (SIC) in a co-planar configuration. First, the cavity modes are analyzed for selecting the feeding structure. Then, the equivalent circuit of the feeding structure is formulated to reveal the mechanism of wideband operation of the proposed antenna. The antennas and arrays operating at 35 and 60 GHz bands fabricated with printed circuit board (PCB) technology are exemplified to validate the concepts. The prototypes with single cavity achieve a bandwidth of 10% for 15-dB return loss and gain of 6 dBi at both 35 GHz and 60 GHz. The 2 2 2 35-GHz SIC antenna array shows a bandwidth of 11.7% for 15-dB return loss and gain of up to 10.8 dBi. The SIW-fed SIC millimeter-wave antenna arrays feature a wideband operation with low loss, low cost, low profile, easy integration configuration. Index Terms—Antenna array, cavity antenna, dielectric loading, millimeter-wave, substrate integrated waveguide (SIW), 60 GHz.

I. INTRODUCTION

L

OW-COST wideband wireless communication systems operating at millimeter-wave (mmW) bands (30–300 GHz) have attracted increasing attention for years [1]–[5]. As a key component, the mmW antennas must be of low-cost, high gain and high integration ability. Although the microstrip antennas have remarkable merits in planar design, they suffer from low antenna efficiency due to severe power loss caused by conductors, dielectric and/or surface waves, in particular in microstrip antenna array [6]–[8]. Dielectric resonator antennas (DRAs) have been shown to radiate efficiently with wideband property over the mmW frequency range [9]–[12]. Many different feeding structures have been proposed to couple energy to dielectric resonators, such as microstrip, CPW and substrate integrated waveguide (SIW) (broad-wall slot coupling), and Manuscript received February 04, 2010; revised August 20, 2010; accepted October 20, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the National 973 project of China 2010CB327400, the National Nature Science Foundation of China (NSFC) under Grant 60921063, by the Agency for Science, Technology and Research (A*STAR), Singapore, Terahertz Science & Technology Inter-RI Program under Grant 082 141 0040, and in part by the Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore. Y. Zhang and W. Hong are with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]). Z. N. Chen and X. Qing are with the Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore 138632 (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123055

so on. However, all of these antennas require the dielectric resonators to be attached to the transmission line, which are not strictly planar configurations and cause the mechanical reliability difficulty. Alternatively, mmW antennas, especially at 60 GHz, can be fabricated using the costly low-temperature co-fired ceramics (LTCC) or the complicated silicon micromachining process [13]–[17]. The SIW has been used as a planar embedded waveguide for microwave and mmW circuits, demonstrating the merits of low loss, high power handling capability, wideband operation, co-planar integration, and so on [18]–[26]. However, most of the antennas/arrays coupled by the broad-wall slots of SIW suffer narrow bandwidth, typically smaller than 7% [22]–[24]. Although a 4 4 SIW slot antenna array fabricated on flexible printed circuit board (PCB) achieved 10.7% impedance bandwidth for 10-dB return loss, the gain performance suffers a large variation of up to 11 dB in the bandwidth [25]. To alleviate such challenges, an SIW narrow-wall fed cavity antenna is proposed in this paper. With a dielectric-loaded substrate integrated cavity (SIC) radiator coupled through the slotted narrow-wall of a feeding SIW, the operating bandwidth of the proposed antenna can be broadened in terms of both impedance matching and gain. Furthermore, the proposed SIC radiator can also be thoroughly co-planar integrated with the SIW and fabricated by easy-to-access and low cost standard PCB process. This paper describes the dielectric-loaded cavity antenna as well as the antenna array design, wherein the operating modes of the dielectric loaded-cavity are explored first. Then SIW feeding structure is designed. After that, an equivalent circuit is derived for understanding the mechanism of wideband impedance matching. As design examples, a single element cavity antenna and a 2 2 antenna array are designed at 35.5 GHz to experimentally validate the analysis and concepts. Last, the antenna arrays for 60-GHz wireless systems are designed and studied. II. ANTENNA ELEMENT DESIGN The proposed antenna consists of a dielectric-loaded SIC and a feeding SIW with a short-circuited (S.C.) end as shown in Fig. 1. The SIC is formed by metallic via-holes, a conductor ground, and an open-ended aperture on the top of the substrate , @ (RT/Duroid 6010, 10 GHz). The cavity is applied in the dominant resonant mode for the wideband performance, which will be discussed later to reveal the operation mechanism. The energy radiates from the upper open-ended aperture of the cavity to free space. The SIC will be coupled with the feeding SIW through an

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Fig. 3. The first three eigenmodes and the E-field in Planes 1, 2, and 3.

Fig. 1. The proposed SIW fed cavity antenna.

Fig. 2. Dielectric-loaded cavity model- simplified resonator: (a) with boundary conditions, (b) with dimensions and plane indications.

inductive window in the narrow-wall between the SIC and SIW. An inductive via is positioned near the SIW narrow-wall for impedance matching. Except the aperture, both sides of the substrate are covered by conductors. A. Dielectric-Loaded Cavity The operating frequency of the proposed antenna is determined by the resonant frequencies of the cavity which can be analyzed as depicted in Fig. 2. The top of the dielectric-loaded cavity is open-ended while the rest of walls are all formed by perfect electrically conducting (PEC) vias. The substrate with high permittivity allows the open-ended side to be modeled as a magnetic wall as that of DRAs [9], [10]. The dimensions of the cavity are denoted as , , and , respectively. In order to facilitate the following is assumed here without any loss of generality. analysis, Indicating the thickness of the substrate, is much smaller than and . Therefore, the first eigenmode of the resonator and the following two eigenmodes are and is or and , respectively. The subscript denotes the number of variations in the standing wave pattern along axis [9], [10]. The eigenmode resonant frequencies can be calculated as follows [27, eq. (6.40)]: (1)

where denotes the eigenmode frequency, and are the perrefer to mittivity and permeability of the material, and the numbers of variations in the standing wave pattern along the -, - and -axis directions, respectively. When , and (thickness of RT/Duroid 6010 PCB), the estimated resonant frequencies corresponding to these eigenmodes together with the electric field in Planes 1, 2 and 3 are exhibited in Fig. 3. The cavity is excited by the energy coupled through the inductive window in the narrow-wall of SIC. The SIW with the width is equivalent, in terms of cutoff frequency, to a convenwhen the widths tional rectangular waveguide with a width and satisfy the fitted equation [20, eq. (9)]:

(2) and are the radius of the via and the spacing between where adjacent vias, respectively, as shown in Fig. 1. and length of the dielecUsing (2), the width tric-loaded cavity can be obtained from the simplified resonator size, and , respectively. The feeding SIW is designed to operate in a single dominant mode under 43.5 GHz ( , , ), since the cut-off frequencies of the first two modes supported by the feeding SIW are 21.75 and 43.5 GHz, respectively. If the window is centered can be in the wider sidewall (parallel to Plane 1), only supported in this case. The mode would not be excited since the E-field of this mode is null in Plane 2 as shown in Fig. 3, while the E-field of the feeding SIW at the center of the window is the maximum. The resonance frequency of the mode is much higher than both of the mode and the cut-off frequency of the second mode of the feeding SIW, out of the dominant mode frequency band. Therefore, the proposed feeding scheme enables the dielectric-loaded cavity opmode, where doesn’t affect the reserating in a single . In practice, by tuning , the onant frequency because of loading impedance of the cavity can be adjusted for impedance matching. As the thickness is much smaller than , it is obvious that slightly affects the resonant frequency of mode. In addition, the feeding inductive window destroys the electric wall of the cavity so that the resonant frequencies of the cavity shift down.

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Fig. 4. The simulated jS j and normalized input impedance: (W R R : ,p : ,W : ,L : ,W : ,L X : ,Y : ,W : ,L : , all in mm.)

= = 02 = 07 = 46 = 19 = 0 27 = 0 3 =93 = 11 9

= 18

= 2:4, = 0:92,

Fig. 5. The equivalent circuit model of the feeding structure.

B. Antenna Element Design and Modeling The feeding scheme includes an inductive window and an inductive via, which ensures the wideband impedance matching. The energy is coupled through the inductive window from the feeding SIW to the SIC. The antenna operating at 35.5 GHz was designed and optimized using the CST, a full wave electromagnetic field simuof the proposed antenna lation tool. From the simulated as shown in Fig. 4, two resonances can be observed. The normalized input impedance also certifies a wideband matching has been achieved. In addition, due to the inductive window, the resonant frequencies of the cavity shifts down so that the simulated and 15 dB, reimpedance bandwidths for spectively, reach 4.6 GHz (33.2–37.8 GHz or 13%) and 3.8 GHz (33.4–37.2 GHz or 10.7%). An equivalent circuit of the feeding structure is used to reveal the mechanism of wideband operation as shown in Fig. 5. Both the inductive window and the inductive via can be modeled as a T-shape network [26], including a shunt inductance and represent the characterand two series capacitances. istic impedance of the feeding SIW and the loading impedance

AA

BB

Fig. 6. The simulated E-field distribution at 35 GHz on Planes and . (The arrow indicates the direction of E-field. In y 0 z plane, the arrows point into the paper).

of the cavity, respectively. Thus the feeding network can be regarded as a second-order impedance matching network with two resonant frequencies which is in accordance with the simulation [28]. As a result, the broadband operation can be achieved when the resonant frequencies are excited closely to each other. Fig. 6 shows the simulated E-field distribution at 35 GHz in and . It is seen that in the cavity, the two main planes the E-field intensity distribution along axis is similar to a cosinusoid, but unchanged along axis, which is the same as that in Fig. 3. The maximum E-field intensity along -axis occurs in the interface, an equivalent magnetic wall, between the substrate and free space. The results show that the cavity operates in mode. Moreover, the covering conductor of the a single antenna can suppress the radiation from the adjacent substrate modes because the substrate around the cavity and SIW only with a bottom ground plane can be considered as a slab waveguide. The cut-off frequencies of the TE-mode of a grounded dielectric slab can be determined from [27, eq. (3.167)] (3) where is the mode number, is the light speed in free space, and are the thickness and permittivity of the substrate, respectively. From (3), it can be seen that the 0th-order mode has a 0-Hz cut-off frequency and always gets excited, which deteriorates radiation performance and lowers the efficiency of the antenna. Thus, it is important to introduce the covering conductor on the top of the substrate to suppress the leakage of the energy through the substrate surrounding the feeding SIW and SIC as plane of Fig. 6. illustrated in The radiation characteristics of the proposed antenna have also been studied. The polarization of the antenna is linear along -axis, which is in accordance with the E-field distribution of the SIC resonance. The simulated radiation patterns for the proposed antenna at 35 GHz are shown in Fig. 7. The 3-dB beamwidths of 92.1 in the E-plane and 76.8 in the H-plane are similar to those of the rectangular DRA [10]. The simulated ratio of co- to cross-polarization levels is about 20 dB. Fig. 8 shows the simulated peak gain of higher than 6 dBi over the operating frequency band of 33.2–37.8 GHz.

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Fig. 9. The geometry of the proposed antenna array.

Fig. 7. The simulated normalized radiation patterns of the proposed antenna at 35 GHz in (a) E (x z )-plane, (b) H (y z )-plane.

0

0

Fig. 8. The simulated gain of the proposed antenna.

III. ANTENNA ARRAY DESIGN AT 35 GHz Usually, antenna arrays can be fed in series or shunt format. Series-fed arrays suffer narrow-band performance and beam squinting [22]–[24]. Therefore, a shunt-feed approach is utilized to incorporate with the proposed antenna for a wideband performance and a fixed beam direction. As shown in Fig. 9, the proposed antenna array is composed 2 SIC elements and a compact four-way tree-shape of 2 power divider. The power divider splits the power into four ways equally and excites all the elements in-phase [21]. The width of the SIW divider is equal to the feeding SIW for each element. The unique design consideration of the divider introduced herein is that the SIW and the elements share part

of via-holes to configure the narrow sidewalls, which keeps the antenna array compact. Since the antenna array is implemented with the via-holes, the distance between the adjacent elements along -axis must increase or decrease discretely to fit the minimum space between the adjacent vias required by the standard PCB process. As a result, the final optimized distance and 7.4 between the adjacent elements are 7.2 mm mm along and axis, respectively, where is the wavelength at 35 GHz in free space. The SIW is excited by an SMA connector through a 50microstrip line. A linear tapered SIW is used for impedance matching, as shown in Fig. 9, and the vias of the tapered SIW are equally spaced by along axis. The overall size of the antenna , and the optimized paramprototype is eters are , , , , , , , , , , , , , , , , , , , , , , , , , , , , all in mm. The mutual coupling between the elements has also been studied. Two antennas as shown in Fig. 1 are used for the study of the mutual coupling. Fig. 10 illustrates the simulated mutual coupling in the E- and the H-planes. The distances between two elements are the same as those used in the 2 2 antenna array, i.e., 7.2 mm in Fig. 10(a) and 7.4 mm in Fig. 10(b). It can be observed that the coupling in the H-plane is around 30 dB and is similar to that in Fig. 5 while the coupling in E-plane is 20 dB. The proposed antenna array was optimized using the tool CST and a prototype was fabricated by a standard PCB process to validate the design. Fig. 11 shows the photograph of the 2 2 35-GHz antenna array prototype and a 3.5-mm SMA connector is used in measurement. of the Fig. 12 compares the measured and simulated antenna array. It can be observed that the frequency range for is 32.7–37.4 GHz (13.4%), and for is 32.9–37.1 GHz (11.7%). A slight frequency shifting ( 3% with respect to the center frequency) is observed, which

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Fig. 10. The simulated mutual coupling between SIC elements in (a) E-plane 2 and (b) H-plane. The spacing between elements is the same as that of 2 array as shown in Fig. 9.

2

Fig. 11. The photograph of the 2

2 2 35-GHz antenna array prototype.

2

Fig. 13. The measured normalized radiation patterns of the 2 2 antenna array at 35 GHz. (a) E-plane (x z plane). (b) H-plane (y z plane).

0

0

The measured and simulated radiation patterns at 35 GHz are illustrated in Fig. 13. The 3-dB beamwidths in the E- and H-planes are 29.1 and 32 , respectively. The side lobes are high, as shown in Fig. 13(a), which are caused by the large space between adjacent antenna elements. The measured ratio of co- to cross-polarization levels is about 15 dB in the main beam of both the E- and the H-planes. The deteriorated cross-polarization performance in the H-plane may be caused by the additional radiation from the feeding microstrip line, SMA connector and/or the edges of the truncated ground planes. The microstrip line usually results in a significant cross radiation loss in such a frequency band because the E-field radiated from the microstrip is orthogonal to that radiated from the antenna. It can be improved when the antenna array is integrated with planar circuits fully implemented by the SIW technology. The measured gain varies from 9.5 to 10.8 dBi over the frequency range of 33–37 GHz, about 1–2 dB lower than the simulated, as shown in Fig. 14, which is caused by the radiation loss of the microstrip line and the higher dielectric loss of the substrate used in the prototype. IV. ANTENNA ARRAY AT 60 GHz

Fig. 12. The

jS j of the 2 2 2 35-GHz antenna array.

may be caused by the fabrication tolerance and the permittivity fluctuation.

An SIW-fed cavity antenna array is designed to operate in an unlicensed frequency range of 57–64 GHz. The geometry of the 2 2 60-GHz antenna array is shown in , Fig. 15. With the RO3006 substrate ( @ 10 GHz, ), the initial geometric dimensions of the cavity can be estimated using (1), e.g.,

ZHANG et al.: WIDEBAND MILLIMETER-WAVE SIW SLOTTED NARROW-WALL FED CAVITY ANTENNAS

Fig. 14. The maximum gain of the 2

2 2 35-GHz antenna array.

Fig. 16. The simulated results of the waveguide to SIW transition.

Fig. 17. The measured

2 W = 1:5 W = 5:94 L = 5:76 W = 4:44 L = 5:59 = R = 0:15 p = 0:6 W = 1:69 W = 1:78 L = 1:14 = 1:16 W = 1:34 L = 1:07 L = 0:8 X = 0:59 X = 0:779 = 0:37 Y = 0:2 Y = 0:5 X = 0:45 Y = 0:36 X = 0:49 = 0:49 X = 0:48 S = 1 S = 1 S = 0:78 W = 10:88 = 17:76

Fig. 15. The geometry of the proposed 2 2 60-GHz antenna array. (a) Antenna array geometry. (b) Photo of the prototype. (Left: Bottom view; Right: Top view.) ( , , , , , , , , , , , , , , , , , , , , , , , , , , , , , all in mm).

R L Y Y L

, the cavity are

and

. The first two eigenmodes of The resonant frequency of the

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jS j of the 2 2 2 60-GHz antenna array.

second modeis 71.2 GHz which is out of the desired bandwidth of 57–64 GHz and ensures the cavity operating only in the dominant mode. The radius of the metalized via-holes is 0.15 mm and the spacing between the adjacent vias is 0.6 mm. The width of the feeding SIW and the power divider are can be supported. selected to ensure only the main mode To achieve a good return loss in the desired bandwidth, the final optimized distances between the adjacent elements are 4.44 and 3.49 mm along and axis, respectively, which are equal to and , where is the wavelength at 60 GHz in free space. The overall antenna array size is 10.88 9.76 mm or 10.88 17.76 mm including the waveguide to SIW transition. The optimized parameters based on CST are listed in Fig. 15. For measurement convenience, a waveguide-to-SIW transition is introduced to incorporate with WR-15 waveguide connector, as shown in Fig. 15. A dielectric-loaded cavity is used in the transition to realize wideband transmission, as well as the inductive via and the inductive window on the bottom of the antenna for matching. The WR-15 waveguide connector is perpendicularly attached to the cavity and the energy is coupled into the SIW through the inductive window as shown in Fig. 16. The simulated transition performance is shown in Fig. 16 and two resonances can be observed. of Fig. 17 compares the simulated and measured the array with and without the waveguide-to-SIW transition. Without the transition, the frequency range for is 56.62–64.25 GHz (12.6%). The simulation shows that the

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Fig. 19. The maximum gain of the 2

2 2 60-GHz antenna array.

V. CONCLUSIONS

Fig. 18. The measured normalized radiation patterns of the 2 58 GHz in (a) E (x z )-plane. (b) H (y z )-plane.

0

0

2 2 array at

transition degrades the overall performance of the array, espeis slightly cially around 62.4 and 63.7 GHz, in which the follows the trend higher than 10 dB. The measured of the simulation well. The frequency shift may be caused by fabrication tolerance and the permittivity fluctuation of the substrate. The design is sensitive to the fabrication tolerance of the window due to the partial radiation from the window. In addition, the measured degrades to 7 dB around 59.3 and 60.8 GHz, which may be caused by fabrication tolerance because the size of the via-holes nearly reaches the process limitation. When the antenna is directly integrated into circuits, it can be expected that the performance of the antenna without transition would be improved. The normalized radiation patterns for the antenna array at 58 GHz are shown in Fig. 18. The 3-dB beamwidths in the Eand the H-planes are 27.8 and 32.2 , respectively. The sidelobe levels in the E-planes are high due to the large spacing between the adjacent elements. The measured ratio of co- to cross-polarization levels is about 17 dB in the main beams in both the E- and the H-planes. Without any feeding microstrip line used in this prototype the measured cross polarization levels in H-planes are as low as the simulation. With the waveguide feeding structure, the measured front-to-back ratios are higher than those in the simulation. Fig. 19 shows that the simulated gain is higher than 9.4 dBi over the bandwidth of 57–64 GHz, whereas the measure is higher than 9 dBi over the frequency band of 54.5–60.3 GHz with a slight frequency shift.

In this paper, a technique to design the SIW slotted narrowwall fed dielectric-loaded cavity antenna has been proposed for millimeter-wave applications. The cavity mode analysis and the equivalent circuits of the feeding structure have been used to study the mechanism of the wideband performance of the an2 tenna. To validate the concept, the single-element and 2 arrays have been designed and measured. Furthermore, the validated concept has been applied to design a 2 2 antenna array operating at 60 GHz bands. The simulated and measured results have showed that this technology is promising for millimeter-wave applications due to the merits of lightweight, compact size, planar form, low-cost, low profile, wideband, high gain, good ability of integration. ACKNOWLEDGMENT The authors are grateful for the support of their colleagues in the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, China, and the Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore. In particular, the authors would like to thank Z. Kuai and T. Shie Ping See for their help in fabrication and measurement. REFERENCES [1] C. W. James, “Status of millimeter wave technology and applications in the US,” in Proc. 21st Eur. Microw. Conf., Oct. 1991, vol. 1, pp. 150–157. [2] J. Burns, “The application of millimeter wave technology for personal communication networks in the United Kingdom and Europe: A technical and regulatory overview,” in Proc. IEEE MTT-S Int. Microw. Symp., May 1994, vol. 2, pp. 635–638. [3] E. C. Niehenke, R. A. Pucel, and I. J. Bahl, “Microwave and millimeterwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 846–857, Mar. 2002. [4] D. Lockie and D. Peck, “High-data-rate millimeter-wave radios,” IEEE Microw. Mag., vol. 10, no. 5, pp. 75–83, Jan. 2009. [5] H. Wang, K. Y. Lin, Z. M. Tsai, L. H. Lu, H. C. Lu, C. H. Wang, J. H. Tsai, T. W. Huang, and Y. C. Lin, “MMICs in the millimeter-wave regime,” IEEE Microw. Mag., vol. 10, no. 1, pp. 99–117, Jan. 2009. [6] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, no. 1, pp. 79–91, Jan. 1992. [7] M. G. Keller, D. Roscoe, Y. M. M. Antar, and A. Ittipiboon, “Active millimetre-wave aperture-coupled microstrip patch antenna array,” Electron. Lett., vol. 31, pp. 2–4, 1995. [8] R. N. Tiwari, P. Kumar, and G. Singh, “2-D photonic crystals as substrate for THz/millimeter wave microstrip patch antenna,” in Proc. Int. Conf. Recent Adv. Microw. Theory Appl., Nov. 2008, pp. 787–789.

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[9] R. K. Mongia, A. Ittipiboon, and M. Cuhaci, “Measurement of radiation efficiency of dielectric resonator antennas,” IEEE Microw. Guided Wave Lett., vol. 4, pp. 80–82, Apr. 1994. [10] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1348–1356, Sep. 1997. [11] Q. H. Lai, G. Almpanis, C. Fumeaux, H. Benedickter, and R. Vahldieck, “Comparison of the radiation efficiency for the dielectric resonator antenna and the microstrip antenna at Ka band,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3589–3592, Nov. 2008. [12] W. O. A. Wahab and S. S. Naeini, “Simple circuit model for millimeter wave substrate integrated waveguide (SIW) series-fed dielectric resonator antenna (RDRA) arrays,” in Proc. IEEE AP-S Int. Symp. Dig., Jun. 2009, pp. 1–4. [13] A. E. I. Lamminen, J. Saily, and A. R. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrates,” IEEE Trans. Antenna Propag., vol. 56, no. 9, pp. 2865–2874, Sep. 2008. [14] J. H. Lee, N. Kidera, S. Pinel, J. Laskar, and M. M. Tentzeris, “Fully integrated passive front-end solutions for a V-band LTCC wireless system,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 285–288, Mar. 2007. [15] T. Mitomo, R. Fujimoto, N. Ono, R. Tachibana, H. Hoshino, Y. Yoshihara, Y. Tsutsumi, and I. Seto, “A 60-GHz CMOS receiver front-end with frequency synthesizer,” IEEE J. Solid-State Circuits, vol. 43, no. 4, pp. 1030–1037, Apr. 2008. [16] A. Babakhan, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “A 77-GHz phased-array transceiver with on-chip antennas in silicon,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2795–2806, Dec. 2006. [17] B. Pan, Y. Li, G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “A 60-GHz CPW-fed high-gain and broadband integrated horn antenna,” IEEE Trans. Antenna Propag., vol. 57, no. 4, pp. 1050–1056, Apr. 2009. [18] H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a “laminated waveguide”,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2438–2443, Dec. 1998. [19] W. Hong, K. Wu, H. J. Tang, J. X. Chen, P. Chen, Y. J. Cheng, and J. F. Xu, “SIW-like guided wave structures and applications,” IEICE Trans. Electron., vol. E92-C, no. 9, pp. 1111–1123, Sep. 2009. [20] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 66–72, Jan. 2005. [21] Z. C. Hao, W. Hong, J. X. Chen, X. P. Chen, and K. Wu, “A novel feeding technique for antipodal linearly tapered slot antenna array,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1641–1643. [22] J. Hirokawa and M. Ando, “Efficiency of 76-GHz post-wall waveguide-fed parallel-plate slot arrays,” IEEE Trans. Antenna Propag., vol. 48, no. 11, pp. 1742–1745, Nov. 2000. [23] Y. J. Cheng, W. Hong, and K. Wu, “Millimeter-wave substrate integrated waveguide multibeam antenna based on the parabolic reflector principle,” IEEE Trans. Antenna Propag., vol. 56, no. 9, pp. 3055–3058, Sep. 2008. [24] P. Chen, W. Hong, Z. Q. Quai, J. F. Xu, H. M. Wang, J. X. Chen, H. J. Tang, J. Y. Zhou, and K. Wu, “A multibeam antenna based on substrate integrated waveguide technology for MIMO wireless communications,” IEEE Trans. Antenna Propag., vol. 57, no. 6, pp. 1813–1821, Jun. 2009. [25] S. Cheng, H. Yousef, and H. Kratz, “79 GHz slot antenna based on substrate integrated waveguides (SIW) in a flexible printed circuit board,” IEEE Trans. Antenna Propag., vol. 57, no. 1, pp. 64–71, Jan. 2009. [26] Y. D. Dong, Y. Q. Wang, and W. Hong, “A novel substrate integrated waveguide equivalent inductive-post filter,” Int. J. RF Microw. Comput. Aided Eng., vol. 18, no. 2, pp. 141–145, Feb. 2008. [27] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2004. [28] H. K. Chen, The Theory and Design of Wideband Impedance Matching Network, 1st ed. Beijing: Post & Telecom, 1982.

Yan Zhang (S’09) was born in Hebei Province, China, 1983. He received the B.S. degree from the School of Information Science and Engineering, Southeast University (SEU), Nanjing China, in 2006, where he is currently working toward the Ph.D. degree He was with the Institute for Infocomm Research (I2R), Agency for Science, Technology and Research (A*STAR), Singapore, as a Research Engineer, from January to July 2009. His current research interests include millimeter-wave passive devices, antennas, ultrawideband antennas and artificial electromagnetic materials. Mr. Zhang is the recipient of International Conference on Microwave and Millimeter Wave Technology Best Student Paper Award, 2008.

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Zhi Ning Chen (F’07) received the B.Eng., M.Eng., and Ph.D. degrees from the Institute of Communications Engineering, China and University of Tsukuba, Japan, all in electrical engineering. During 1988–1997, he was with the Institute of Communications Engineering, Southeast University, and City University of Hong Kong, China, with teaching and research appointments. In 1997, he was awarded a JSPS Fellowship to join the University of Tsukuba, Japan. In 2001 and 2004, he visited the University of Tsukuba under the JSPS Fellowship Program (senior level). In 2004, he was with the IBM T. J. Watson Research Center, as Academic Visitor. Since 1999, he has been with the Institute for Infocomm Research and his current appointments are Principal Scientist and Head for RF and Optical Department. He is concurrently holding the appointments as Adjunct/Guest Professors with Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University, and the National University of Singapore. He has published 280 journal and conference papers, as well as authored and edited books entitled Broadband Planar Antennas, UWB Wireless Communication, Antennas for Portable Devices, and Antennas for Base Station in Wireless Communications. He also contributed to UWB Antennas and Propagation for Communications, Radar, and Imaging, as well as Antenna Engineering Handbook. He holds 28 granted and filed patents with 19 licensed deals with industry. His current research interest includes applied electromagnetics, antennas for applications of microwave, mmW, submmW, and THz in imaging systems. Dr. Chen has organized many international technical events as a key organizer. He is the founder of the International Workshop on Antenna Technology (iWAT). He is the recipient of the CST University Publication Award 2008, IEEE AP-S Honorable Mention Student Paper Contest 2008, IES Prestigious Engineering Achievement Award 2006, I R Quarterly Best Paper Award 2004, and IEEE iWAT 2005 Best Poster Award. He is a Fellow of the IEEE for his contribution to small and broadband antennas for wireless and IEEE AP-S Distinguished Lecturer (2008–2011).

Xianming Qing (S’90–M’02) was born in China in 1965. He received the B.Eng. degree from the University of Electronic Science and Technology of China (UESTC), in 1985, and the Dr.Eng. degree from Chiba University, Japan, in 2010. During 1987–1996, he was with the UESTC where he was teaching and conducting research. He was appointed Lecturer in 1990 and Associate Professor in 1995. He joined the National University of Singapore (NUS) in 1997 as a Research Scientist. Since 1998, he has been with the Institute for Infocomm Research (I2R, formerly known as CWC and ICR), Singapore. He is currently holding the position of Research Scientist and the leader of Antenna Group under the RF and Optical Department. His main research interests are antenna design and characterization for wireless applications. In particular, his current R&D focuses on small and broadband antennas/arrays for wireless systems, such as ultrawideband (UWB) systems, radio frequency identification (RFID) systems and medical imaging systems, microwave, mmW, submmW, and THz imaging systems. He has authored and coauthored more than 80 technical papers published in international journals or presented at international conferences, and has authored five book chapters. He holds eight granted and filed patents. Dr. Qing received six awards of advancement of science and technology in China. He is also the recipient of the IES Prestigious Engineering Achievement Award 2006, Singapore. He served as Organizer and Chair for special sessions on RFID Antennas at the IEEE Antennas and Propagation Symposium 2007 and 2008. He also served as Guest Editor of the special issue “Antennas for Emerging Radio Frequency Identification (RFID) Applications” for the International Journal on Wireless and Optical Communications. He has served as the TPC member and Session Chair for a number of conferences, and the reviewer for many prestigious journals such as the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, AWPL, MWCL, IET Microwaves, Antennas and Propagation, and Electronic Letters.

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Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves and serves for the Director of the lab since 2003, and is currently a Professor and the Associate Dean of the School of Information Science and Engineering, Southeast University. In 1993, 1995, 1996, 1997, and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter wave theory and technology, antennas, electromagnetic scattering, RF technology for

mobile communications, etc. He has authored and coauthored more than 200 technical publications, and authored Principle and Application of the Method of Lines (in Chinese: Southeast University Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems” (in Chinese, Science, 2005). Dr. Hong was thrice awarded the first-class Science and Technology Progress Prizes issued by the Ministry of Education of China and the Jiangsu Province Government. He also received the Foundations for China Distinguished Young Investigators and for “Innovation Group” issued by NSF of China. He is a senior member of CIE, Vice-President of the Microwave Society and Antenna Society of CIE, and has served as a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IET Proc.-H, and Electronic Letters. He also served as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES from 2008 to 2010, and is an Editorial Board Member for IJAP and RFMiCAE.

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Radiation Theory of the Plasma Antenna Huan Qing Ye, Min Gao, and Chang Jian Tang

Abstract—Several basal theoretical problems for the radiation of plasma antenna are investigated. The system equations are obtained according to the rule of the disturbing current, which is excited by surface waves on the interface between a plasma column and a dielectric tube, and the radiation theory of dielectric antennae. The analysis indicates that there are two kinds of mechanisms to explain the radiation of plasma antennae. The first one is the radiation of the disturbing current, which is similar to the radiation of a metal antenna. In the second case, because the electromagnetic wave partly cannot satisfy the total reflection condition in some specific frequency bands, transmission can be excited in a plasma antenna, which is similar to the radiation of a dielectric antenna. The radiation derived from these two kinds of mechanisms can be superposed in certain frequency bands. In this paper, a radiation directivity function is obtained. The radiation characteristics of plasma antennae are compared with those of metal antennae. The radiation properties of plasma antennae are presented based on numerical calculation. The present results provide an important theory for further investigation in the performance of controlling plasma antennae and plasma antenna arrays.

I. INTRODUCTION

A

PLASMA antenna is a type of RF antenna in which the metal elements of a conventional antenna are replaced with plasma elements. Some important properties of plasma antennae contrast with those of conventional antennae: wide carrier frequency, narrow radiation direction, stealth features and controllable antenna shape. This has drawn a lot of attention to broad application of the plasma antenna in communication, military and national defense [1]–[3]. The current studies of plasma antennae mostly consist of analysis of experimental results and some discussions about characteristics. Theoretical studies have been carried out based merely on plasma units or in comparison with metal antennae; some problems still need to be solved, such as noise sources of antenna and the influence of antenna parameters on radiation [4]–[11]. To solve these problems, we need to investigate the plasma antennae in more depth and detail. This paper is organized as follows. In Section II, the interface of current disturbance [2], [12], [19] between plasma and medium is introduced into the model of plasma antennae. Equations for EM fields and disturbing current are established and related theoretical descriptions are presented. In Section III, numerical solutions are provided. From the solutions, it is found that the plasma antenna can work in two different active modes: the radiation mode and the transmission mode. The working

conditions and the transmission frequency bands for the transmission mode are obtained. In Section IV, the mechanisms, conditions and properties of the transmission mode are analyzed in detail. The radiation directivity function is derived, and the properties driven by two kinds of radiation mechanisms are clarified. Analysis demonstrates that the plasma antenna has similarities with not only radiation properties of metal antennae but also with those of dielectric antennae. One of the views is that the transmission mode is one of main noise sources in plasma antennae. The conclusions are established in Section V. II. THE MATHEMATICAL MODEL OF EM WAVES IN PLASMA ANTENNAE The physical model of the plasma antenna is shown in Fig. 1. For a convenient investigation of EM wave distribution, the boundary in the -direction is neglected. Because the plasma density decreases very slowly along the axial direction, Region 1 could be considered to be filled with low-temperature high homogeneous density plasma, and we assume that the length of the plasma column that can carry EM wave propagation is less than the practical length of plasma antennae. Region 2 is a homogeneous medium. Region 3 is free space. Define and as the relative permittivity of plasma and medium, respectively. The diameter of the system is , the diameter of the plasma column is , and the thickness of medium cladding is . in which The plasma relative permittivity is is plasma frequency and is the EM wave frequency. According to the rule of plasma antenna excitation [14], the EM wave cannot propagate in low-temperature high homogeneous . Under this condition, the EM density plasma when wave will propagate along the interface between plasma channel and medium tube as a surface wave. Here we assume that a TM plane wave whose frequency is lower than the plasma frequency is fed into the plasma antenna. All equations of the EM fields in cylindrical coordinates satisfy the Helmholtz equation, and they can be obtained from Borgnis [12], [20]. The equations of the EM fields in Region 1 can be written as

(2.1) The equations of the EM fields in Regions 2 and 3 are

Manuscript received August 11, 2009; revised September 01, 2010; accepted November 01, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported by the National Basic Research Program (Grant No. 2009GB105003). The authors are with the College of Physical Science and Technology, Sichuan University, Chengdu, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2123051 0018-926X/$26.00 © 2011 IEEE

(2.2)

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EM wave will decay along the axial direction. The relationship and can be expressed as between

(2.4) where is the speed of light. All of the EM fields on the interfaces are subject to the following constraints:

(2.5)

is the disturbing surface current density on the inwhere terface between plasma and medium. The surface electronic disturbing current in plasma driven by the high-frequency EM , where the phase factor of the disturbing wave is is , and is the surface electron surface velocity density on the interface. The Lorentz equation for an electron in plasma is in which and represent electric field and magnetic field of the EM wave respectively. The total derivative in above equation can be rewritten as

(2.6)

Fig. 1. The model of a plasma antenna. (a) Simplified physical model, (b) cross section at z .

=0

Furthermore

(2.3)

and are undetermined constants, while and are radial phase constants ( denoting the radial direction) of the plasma, medium and free space, respectively. is an axial phase constant (“axial” meaning along the direction). If is real, total reflection will occur on the interfaces, and the EM wave in Region 3 is a surface wave attenuating along the radial direction and propagating along the axial direction. If is pure imaginary, the wave will transmit to Region 3 from the outer interface, and the EM wave is a wave propagating along both the radial and axial directions in Regions 2 and 3, and since the wave transmits along the radial direction, the amplitude of the

where [15], is the phase velocity of the is the surface electron disEM wave in the -direction, and turbing velocity in the -direction. Because the electron disturbing velocity is far less than the phase velocity of the EM , so we can conclude that and wave, i.e., . Also, due to the low-temperature plasma then electron velocity is also much less than c, the magnetic force is sufficiently small to be neglected compared with the electric . Thus it is deforce, so we can conclude duced that the disturbing surface velocity is while the disturbing current is . Therefore, the disturbing current resulting from the plasma electron oscillation driven by the high-frequency EM wave is

(2.7) To simultaneously solve (2.1)–(2.3), (2.5) and (2.7), we form a matrix equation (2.8), shown at the bottom of the next page. Here, the determinant of the coefficients must vanish to yield a non-zero solution. Therefore, we have

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Simplifying this, we obtain

(2.9) where

Obviously, the expression appearing in is the mathematical element derived from the disturbing current. The system (2.4) furthermore yields

Fig. 2. Allowed radiation modes can be read from the intersections of the solid and dashed curves.

(2.10) (2.11) The relation of and can be deduced from (2.9)–(2.11), and the dispersion relation between and is derived from (2.4). III. NUMERICAL CALCULATION AND PHYSICAL ANALYSIS In order to facilitate the numerical calculation, some physical quantities are normalized as follows, and . The system parameters are and . By calculating (2.9)–(2.11), we obtain the denoted as the solid curve and the expression relationship of (2.9) shown by the dashed curve in Fig. 2. Every intersection of these curves represents a set of real solutions of and with a specific EM wave frequency. The results indicate that total reflection exists on the interfaces in Region 2, and that the EM wave in the plasma antenna is a surface wave which decays along the radial direction and propagates along the axial direction. The disturbing current driven by the EM wave will radiate into free space. In this paper, we define these oper-modes ating states as radiation modes, and mark them as . However, when , the values of are pure imaginary according to (2.11), which means that the EM

wave will transmit into Region 3, In this case, it is useful to rein (2.1)–(2.5) and (2.8)–(2.11). With the same place with parameters as in Fig. 2, the numerical results of (2.9)–(2.11) are is still deplotted in Fig. 3, and the expression noted by solid curve. Then (2.9)–(2.11) with gives the dashed curve in Fig. 3. Again, every intersection of these curves and according to the given sperepresents a set of real cific EM wave frequency, thus we obtain the pure imaginary at the same time. These results indicate that, in results of Region 2, the EM wave reflects on the inner interface and transmits to Region 3. Thus, the radiation in free space consists of the disturbing radiation and transmission currents. Here, these operating states are defined as transmission modes and written -modes as From Figs. 2 and 3, we can see that -modes and -modes can coexist in the plasma antenna for given frequencies, and this phenomenon is called mode degeneracy. and According to the above method used to solve for (2.4), the dispersion relation of the radiation modes with difis shown in Fig. 4. Here one can see that higher raferent diation modes appear gradually as the frequency increases. De. By nufine the cut-off frequency of radiation-modes as merical calculation, we obtain that the cut-off frequencies of -mode, -mode, -mode and -mode as the and , respectively. For one mode, when the EM wave frequency is higher than its cut-off

(2.8)

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Fig. 3. Allowed transmission modes can be read from the intersections of the solid and dashed curves.

Fig. 5. Dispersion relationship of transmission-modes.

Fig. 6. The EM wave transmits in the medium and the longitudinal section of Region 2.

IV. ANALYSIS OF PLASMA ANTENNA RADIATION

Fig. 4. Dispersion relationship of radiation-modes.

frequency, it is a radiation mode; otherwise, when the EM wave frequency is lower than its cut-off frequency, it becomes a transmission-mode. and According to the above method used to solve for (2.4), the dispersion relation of transmission modes with different is shown in Fig. 5. It is obvious that some transmissionmodes can only radiate in some specific frequency bands. The gaps of frequency between transmission modes are manifest. For a given transmission-mode, the lower cut-off frequency and the higher cut-off frequency is defined is defined as , and they are denoted as dashed lines perpendicular as to the horizontal axis in Fig. 5. For example, in the case of -mode, the lower cut-off frequency , the the and the frequency band higher cut-off frequency . Comparing Figs. 4 with 5, we can conclude that the radiation modes and transmission modes will co-exist in specific frequency bands, and all these modes are degenerate modes.

From the analysis above, we know that there are two sorts of radiation mechanisms for plasma antennae. One is caused by disturbing currents which are similar to the radiation derived from current oscillation on the surface of a metal antenna. In this paper, we define this kind of radiation as metal-similarity radiation which is written as m-radiation for short. The other is caused by EM wave transmission. This radiation is similar to the radiation of dielectric antenna, so we define it as dielectric-similarity radiation and rewrite it as d-radiation for short. The energy of the EM wave in a plasma antenna will decay due to d-radiation, therefore the EM wave is an attenuating traveling wave in the z-direction. Fig. 6 shows that the EM wave transmits in the medium. The length of the plasma antenna is , the reflection coefficient of the upper interface is , the transmission coefficient in the upper interface is , and the reflection coefficient of the lower interface is . Assuming that the incident wave at the point in Region 2 , the reflected wave is , and the transmission wave is in Region 3, where is and due to the total reflection at the interface between Region 1 and Region 2. If the order of total reflection is even, after times and times total reflection, the final transmission waves are and respectively. Therefore the average attenuation of transmission waves is

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and respectively, where,

(4.1) The transmission waves are

Now define the attenuation factor

(4.2) , where,

(4.3) is the attenuation factor of the transmission wave after an is the attenuation even number of total reflections, and factor of the transmission wave after an odd number. Likewise, , where, when is odd, the attenuation factor

(4.4) The conclusion above is taken into the calculation of transmission modes in Fig. 5. From (2.7) we know that the amplitudes of the fields in all regions can be linearly expressed by the amplitudes in Region 3. The amplitudes of the EM wave in or , and other Region 3 can be written as and are linear functions of , undetermined constants so the attenuation factor or makes no difference in the calculation of the transmission modes. From the analysis above, we know that both the EM wave and the disturbing currents excited by transmission modes will decay along the axial direction when the plasma antenna operates under the transmission mode condition. In order to research the radiation properties conveniently, some appropriate system parameters are used in the calculation, and . Numerical solutions imply which are -mode and that there are two radiation modes, namely the -mode, and one transmission mode, -mode. Define and the disturbing current amplitude of each mode as (substituting into (4.1), we obtain and the attenuation factor ), and the radial and axial phase constants of -mode are defined as and . Firstly, we investigate the m-radiation. In far-field region, the direction is obtained referring electric field of m-radiation in to the radiation field of a metal telescopic antenna

Fig. 7. pattern.

L = 0:6, m-radiation pattern and metal telescopic antenna radiation

is the disturbing current caused by the transmission mode. is the summation of all disturbing currents, given as

(4.6) where the array . In the , the region can be determined as a region where far-field region if the frequency of the EM wave is above VHF. In the far-field region, we define the radiation directivity func. When tion of m-radiation as meters, numerical solutions of are plotted in Fig. 7, in which the solid line is the m-radiation pattern and the dashed line is the radiation pattern of a metal telescopic antenna. Comparing these two patterns, we can see that the shapes of the main lobes are different. The direction of m-radiation is narrower than that of a metal telescopic antenna, and the maximum direction of m-radiation is smaller than that of a metal telescopic antenna. To illustrate this, we analyze (4.6), and find that the phase of each disturbing current is different, which indicates that in the axial direction is different from the the distribution of current distribution of a metal antenna. That is the reason for the difference between the m-radiation direction and the radiation direction of a metal telescopic antenna. In summary, different working modes and disturbing currents distributions can be excited by an EM wave in different frequency bands, and they both influence the radiation direction of a plasma antenna. The radiation direction can be controlled to a large extent by selecting appropriate plasma antenna parameters, working frequency bands and working modes. -mode is the transmisSecondly, as for d-radiation, the sion mode for these given system parameters according to the and into (2.3), analysis we made above. We put , obtaining the field equations of d-radiation and let

(4.5) is the phase constant in free space, and where tance between the field point and the antenna.

is the dis(4.7)

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decays along the length of plasma antenna and becomes an attenuated traveling wave, so the disturbing currents excited by transmission modes also decay. Additionally, the transmission mode is probably one of the noise sources in plasma antenna. If the frequency of the EM wave fed is located in the frequency bands where some transmission modes do not appear, the noise impact of d-radiation on plasma antenna radiation will vanish or decrease. REFERENCES

Fig. 8.

L = 0:6, plasma antenna radiation pattern.

(4.8) Superposing d-radiation and m-radiation, we get the radiation directivity function of the plasma antenna, (4.9) The radiation pattern of the plasma antenna is shown in Fig. 8, in which the dashed curve denotes the m-radiation pattern. Comparing Fig. 7 with Fig. 8 and according to (4.5)–(4.8), we know that the phase constants of m-radiation and those of d-radiation in free space are different, so that d-radiation influences the radiation property of the plasma antenna. As one can see, the main lobes oscillate intensely. There is already much investigation into several kinds of noise in plasma antennae [1], [17], [18]. In this paper, we think that the existence of the transmission mode is also probably one of the noise sources. If the frequency of the EM wave fed into the antenna is located in the gaps shown in Fig. 5, this kind of noise source will decrease or disappear immediately because this EM wave fails to satisfy the total reflection condition and will not be restricted to the medium cladding. V. CONCLUSION When a homogeneous EM wave is fed into a plasma antenna, radiation modes and transmission modes coexist in some frequency bands. The high radiation modes appear gradually as the frequency of EM wave rises. The transmission modes only exist in specific frequency bands. In the case of transmission modes, we can control these modes by changing the frequency of the EM wave and relevant system parameters. The plasma antenna radiation is excited by two kinds of radiation mechanisms: one is the m-radiation which is caused by the oscillation of disturbing currents on the interface between plasma and medium. The other one is d-radiation caused by EM wave transmission. The d-radiation modes excite interface disturbing currents between plasma and medium, and meanwhile the excited disturbing currents influence the transmission modes. Furthermore, due to transmission, the EM wave of transmission modes

[1] G.-W. Zhao, Y.-M. Wu, and C. Chen, “Calculation of dispersion relation and radiation pattern of plasma antenna,” Acta. Phys. Sin., vol. 56, p. 5298, 2006. [2] A. P. Whichello, J. P. Rayner, and A. D. Cheetham, Plasma Antenna Radiation Patterns. College Park, MD: American Institute of Physics, 2003, 0-7354-0133-0/03. [3] A. H. Adzhiev, V. A. Soshenko, O. V. Sytnik, and A. S. Tishchenko, “Experimental investigation of explosive plasma antennas,” Tech. Phys., vol. 52/6, pp. 765–769, 2007. [4] R. Michael, “Polarizing frequency of a fluid plasma antenna element,” presented at the IEEE Antennas and Propagation Society Symp., 2004. [5] W. L. Kang, M. Rader, and I. Alexeff, “Conceptual study of stealth plasma antenna,” in Proc. IEEE Int. Conf Plasma Sci., 1996, p. 4IP07 261. [6] K. C. Gupta, P. K. Garg, and A. Singh, “Narrow-beam antennas using cylindrical columns of isotropic plasma,” Int. J. Electron., vol. 35/2, pp. 193–210, 1973. [7] Z.-J. Wang, G.-W. Zhao, Y.-M. Xu, Z.-W. Liang, and J. Xu, “Propagation of surface wave along a thin plasma column and its radiation pattern,” Plasma Sci. Technol., vol. 9, pp. 526–529, 2007. [8] G. G. Borg, J. H. Harris, N. M. Martin, D. Thorncraft, R. Milliken, D. G. Miljak, B. Kwan, T. Ng, and J. Kircher, “Plasma as antennas: Theory, experiment and applications,” Phys. Plasmas, vol. 7/5, pp. 2198–220, 2000. [9] I. Alexeff, T. Anderson, S. Parameswaran, E. P. Michael, J. Dhanraj, and M. Thiyagarajan, “Advances in plasma antenna design,” in Proc. IEEE Int. Conf. on Plasma Science, 2005, pp. 350–350. [10] I. Alexeff, T. Anderson, E. Farshi, N. Karnam, E. P. Pradeep, N. R. Pulsani, and J. Peck, “Plasma tubes intercept microwave radiation independent of polarization,” in Proc. IEEE 34th Int. Conf. on Plasma Science, 2007, pp. 378–378. [11] Z.-W. Liang, Z.-J. Wang, G.-W. Zhao, J. Xu, and Y.-M. Xu, “Noise measurement and analysis of plasma antenna,” Chinese J. Radio Sci., vol. 22/6, 2007. [12] D. Su, C.-J. Tang, and P.-K. Liu, “The boundary effect analysis on the electromagnetic mode in the beam-ion channel,” Acta Phys. Sin., vol. 56, p. 2802, 2006. [13] H.-R. Li, C.-J. Tang, and P.-K. Liu, Mode Theory of the Plasma Cladding Waveguide. Bristol, U.K.: IOP Publishing, 2007, vol. 40, pp. 1–8. [14] H. Sugi, I. Ghanashev, and M. Nagatsu, “High-density flat plasma production based on surface waves,” Plasma Sources Sci. Technol., vol. 7, pp. 192–205, 1998. [15] S. A. Yahaya, M. Yamamoto, K. Itoh, and T. Nojima, “Dielectric rod antenna based on image NRD guide coupled to rectangular waveguide,” Electron. Lett., vol. 39/15, 2003. [16] J.-Y. Chung and C.-C. Chen, “Two-layer dielectric rod antenna,” IEEE Trans. Antennas Propag., vol. 56/6, 2008. [17] R. L. Stenzel, “High-frequency noise on antennas in plasmas,” Phys. Fluids B, vol. 1, p. 1369, 1989. [18] N. Meyer-Vernet and C. Percche, “Tool kit for antennae and thermal noise near the plasma frequency,” J. Geophys. Res., vol. 94, no. A3, pp. 2405–2415. [19] E. N. Istomin, D. M. Karfidov, I. M. Minaev, A. A. Rukhadze, V. P. Tarakanov, K. F. Sergeichev, and A. Yu. Trefilov, “Plasma asymmetric dipole antenna excited by a surface wave,” Plamsa Phys. Rep., vol. 32, no. 5, pp. 388–400, 2006. [20] F. E. Borgnis and C. H. Papas, “Electromagnetic waveguides and resonators,” in Handbuch der Physik vol. XVI, 1958. Huan Qing Ye, photograph and biography not available at the time of publication. Min Gao, photograph and biography not available at the time of publication. Chang Jian Tang, photograph and biography not available at the time of publication.

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Self-Consistent Analysis of Cylindrical Plasma Antennas Paola Russo, Member, IEEE, Graziano Cerri, Member, IEEE, and Eleonora Vecchioni

Abstract—Plasma can be used to create antennas: its conductivity is given by free electrons resulting from gas ionization. This process is obtained with the application of an intense electromagnetic field to a glass tube filled with a low pressure gas. The pump signal needed for ionization and gas discharge parameters have to be carefully chosen in order to optimize plasma antenna design and realization; in particular discharge working conditions have to be defined to obtain the desired antenna properties in terms of efficiency and effective length. For this purpose a self-consistent numerical model of a surface wave driven plasma column has been developed. This tool helps to understand the physical aspects involved in the problem and allows a parametric investigation to be carried out. Index Terms—Antenna, FDTD, plasma, surface waves.

I. INTRODUCTION

P

LASMA antennas represent a completely new class of antennas. Their technology relies on ionized gas to be used as a conductive element enclosed in tubes or other enclosures. It is well known that plasma has a specific conductivity given by its free electrons obtained from gas ionization [1]–[3], therefore it can be employed in the development of antennas and reflectors. This is a fundamental change from traditional antenna design which is usually based on metallic wires and surfaces. The idea of using a plasma element as the conductive medium in radio-frequency antennas and reflectors dates back many years [4]–[6], but only recently have several studies demonstrated the feasibility of such devices [7]–[11]. The growing interest in plasma antennas shown by the scientific community is mainly given by their peculiar and innovative properties. Simple linear plasma antennas can be created by applying a power signal to a dielectric tube (usually glass or Teflon) filled with a low pressure gas. This pump power launches a surface wave that propagates along the column creating and sustaining the discharge and providing the conductivity necessary for the useful signal to be radiated. For this reason plasma antennas need two different signals: the “pump” signal, for the ionization, and the “useful” signal. Manuscript received March 01, 2010; revised September 22, 2010; accepted October 07, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. P. Russo and G. Cerri are with the Dip. Ingegneria Biomedica, Elettronica e Telecomunicazioni, Universita Politecnica delle Marche, Ancona I-60131, Italy (e-mail: [email protected]; [email protected]). E. Vecchioni was with Dip. Ingegneria Biomedica, Elettronica e Telecomunicazioni, Universita Politecnica delle Marche, Ancona I-60131, Italy. She is now with the Software R&D Unit, Thermowatt SpA, 60011 Arcevia (AN), Italy. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122292

A plasma antenna can be rapidly switched on or off just by applying bursts of pump power to the discharge vessel. When the gas is energized, the plasma is created and the communication takes place; when plasma is de-energized the plasma antenna is simply a dielectric tube with a very low radar cross-section. This “on-off” behavior makes plasma antennas suitable for use in stealth applications for military communications. However, plasma antennas can also be used in many civilian applications, being particularly suitable for wireless communication systems [9], [10]. They can be used to develop reconfigurable antennas and arrays simply by controlling the input power, or time and space selective shielding reflectors. In fact, electromagnetic waves are reflected by energized plasma surfaces, whereas they pass freely through de-energized plasma. Plasma reflectors also allow a frequency selective shielding, as plasma frequency behaves like a cut-off frequency not dissimilar to ionosphere propagation. At present the state of the art concerning plasma antennas is still very poor and most of the papers available in literature deal with experimental approaches. The availability of a model offers the designer a parametric analysis that a merely experimental approach is not able to provide. This is particularly useful in this case where many degrees of freedom have to be handled. In literature several papers can be found concerning the problem of the interaction mechanism between an electromagnetic surface wave and an ionized gas, [4], [12]–[15], but only a few studies have been developed on the specific topic of plasma antennas [7]–[11]. From these studies it is very difficult to infer all the information required in our specific situation, therefore an ad hoc model is presented in this paper, addressed to the on-going analysis of plasma antennas. A theoretical model is necessary to understand plasma antenna operating behaviour and this is also helpful in their design and realization. For efficient and optimized development many parameters need to be known. In particular the discharge process has to be efficiently optimized because it exhibits a strong and non-linear dependence on the power and frequency of the ionizing pump signal and on the gas used. A merely experimental approach would not be able to handle the numerous degrees of freedom that have to be considered (gas composition, pressure, pump power and frequency, pump signal network set-up, vessel dimension and so on), while the model can be used to carry out an investigation on these parameters. In this paper, a kinetic model of plasma antennas based on the solution of Maxwell curl equations together with the Boltzmann equation for the electron distribution function (EDF) is presented. The EDF is the quantity that univocally defines the plasma state. On the basis of this knowledge any information

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on macroscopic plasma parameters, such as electron density or plasma conductivity, can be self-consistently inferred. A kinetic model differs from the fluid-dynamic one that is usually adopted [15]. The latter approach describes plasma behavior in terms of moment equations for the average macroscopic quantities: plasma is treated as a fluid in which every particle moves with the flow. Particle density, mean velocity and other macroscopic quantities used in the fluid model are determined by the distribution function with some simple averaging procedures. However, in a fluid model the information on the EDF is not generated and the kinetic data necessary to solve the moment equations have to be assumed a priori, being part of the model input [16]. The choice of the kinetic model is the most complete because it provides the EDF directly, thus the plasma state is univocally defined and all the microscopic quantities are self-consistently determined. The numerical solution to the system of Maxwell and Boltzmann equations coupled together, is based on a finite difference time domain (FDTD) method, a technique widely used in computational electrodynamics [17]. Its application for problems involving plasma is not new, but in this paper a more general approach is presented. As an example of its application, in [18]–[21] the temporal evolution of plasma state is not taken into account and a weak electromagnetic wave propagates in a previously created plasma. Therefore the electromagnetic field and the plasma state are not mutually determined, but the first is just a weak perturbation of the latter. On the contrary, in the proposed model the plasma state is strongly affected by the propagating electromagnetic field, that is in turn determined by the plasma itself. This is a key point for the characterization of electromagnetic propagation in a plasma. Many critical aspects are involved in the rigorous but complex kinetic model, mainly due to the strong non-linearity of the problem. All these aspects have previously been widely presented and discussed in [22], [23], where a preliminary 1D kinetic model of the propagation of a plane wave impinging on a semi-infinite plasma slab was described. This model allowed the evaluation of the accuracy and reliability of the adopted method. In the present paper an azimuthally symmetric cylindrical geometry was implemented. Although some physical aspects involved in the plasma-wave interaction are mainly the same as the 1D case, the novelty of this work is based on the realistic geometry that has to be simulated. The use of a glass tube filled with gas, ionized with a proper feeding network, introduces new physical aspects to be considered. The finite dimension of the gas discharge tube involves the propagation of a surface wave that has now to be modelled. Moreover the phenomenon of the diffusion of the particles inside the tube, negligible in infinite spaces, now has to be considered. The simulation of the realistic geometry of a linear plasma antenna allows also an accurate description of both plasma state and antenna parameters.

solving a system of equations based on the Maxwell curl equations

(1) (2) and the Boltzmann equation:

(3) and being the electron charge and mass and the electron distribution function. The Boltzmann equation is a non-linear integro-differential equation in a six-dimensional space that describes the evolution in time, space and velocity of the EDF. The EDF gives the number of electrons in a volume having a velocity in the range at the instant . This quantity allows a complete characterization of the plasma state as any information can be inferred from it. In fact, all the macroscopic quantities involved in the problem, such as electron density, plasma conductivity and current density, can be determined with some simple analytical manipulation from the EDF [1], [23]. Equations (1)–(3) are the controlling equations of kinetic plasma simulation. The role of the Boltzmann equation is to provide a relationship between the electromagnetic field ( , ) and current density for inclusion in the Maxwell equations [24]. In fact, the current density results from the EDF according to:

(4) where

is the electron mean velocity and is the electron density in at time . The main task of the problem is the solution of the Boltzmann (3). The usual method to solve the Boltzmann equation consists in expanding the EDF in a spherical harmonics series; if collisions are strong and the electromagnetic field is in the energy range of gas discharges [25], only a few terms are needed and the expansions can be truncated at the second term with good approximation (Lorentz approximation). Therefore, the EDF can be represented by:

(5) II. FORMULATION OF THE PROBLEM Plasma is an ionized gas where free electrons, ions and neutral particles coexist maintaining a global charge neutrality (quasineutrality hypothesis) [1]. The interaction mechanism between an electromagnetic field and a plasma can be characterized by

and terms are known as isotropic and The anisotropic terms: the first takes into account the chaotic movement of electrons, while the latter is due to the perturbation introduced by the electromagnetic field and represents those electrons having a biased direction. In a thermodynamic

RUSSO et al.: SELF-CONSISTENT ANALYSIS OF CYLINDRICAL PLASMA ANTENNAS

equilibrium state the isotropic term assumes a Maxwellian distribution and the anisotropic term is zero. Some other approximations can be introduced in the case study without affecting the model accuracy. Firstly, we will consider a non-magnetized plasma (no external static magnetic field is applied). The contribution due to the magnetic field of the electromagnetic wave is always some orders of magnitude smaller than that of the electric field, therefore it will be neglected in the Boltzmann equation. As a consequence the medium is isotropic. This hypothesis fits well with surface wave discharge plasmas, while it would not be possible for atmospheric plasmas or fusion plasmas. As regards collisions, only the elastic and first order ionization electron-neutral and electron-ion collisions are considered in the model, being the only ones that are significant in the determination of plasma state in gas discharge plasmas. In fact, electron-electron collisions and higher order ionization collisions are important only when the degree of ionization is very high. Moreover, ions are treated as immobile and the very large ion mass to electron mass ratio allows the electron-ion collision term to assume a simpler form just by adding it to the usual elastic collision term [26]. By inserting the expansion (5) of the EDF into the Boltzmann equation and applying the orthogonality relations of the spherical harmonics, it is possible to derive an equation for each term of the EDF:

(6) (7) The right hand side of (6) and (7) are the collision terms for the isotropic and the anisotropic EDF. Their evaluation is not straightforward and is not reported here, but a well documented procedure can be found in [26]:

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Fig. 1. The case study.

drift velocity appears. If the field is intense enough, several electrons will reach an energy greater than the ionization threshold so that when they collide with a neutral particle this can be ionized. The ionization process changes the plasma state that, in turn, will affect the electromagnetic field propagation. resulting from ionization The increasing electron density can be described by the continuity equation given by:

(10) is the ion density and it is assumed to be equal to the electron density according to the quasi-neutrality assumption; and are the ionization and recombination rates respectively, where:

(11) and can be assumed to be a constant chosen according to the gas composition. The complete Maxwell-Boltzmann system that has to be solved for a plasma kinetic modeling is finally represented by (1)–(2) together with (6)–(7) and the continuity (10). A. The Case Study: Cylindrical Geometry

(8) (9) and are the elastic electron-neutral and elecwhere , tron-ion collisions and the electron-neutral ionization collision , is the first enfrequencies, is the assumed neutral temergy threshold of ionization, perature. The collision frequency is in general defined as , where , , can indicate the elastic, ion or ionization is the cross-section for the type-collision collision and [27]. The Maxwell curl equations solved together with the Boltzmann equation allow the self-consistent evolution of the plasma state to be determined. When an electromagnetic field is applied to a weakly ionized plasma, free electrons are accelerated and a

The simplest antenna that can be considered is a cylindrical dielectric tube filled with plasma. The geometry shown in Fig. 1 describes with a good approximation the power supply network experimentally used [9], [11]. A commercially available tube designed for lighting purposes was used to create the plasma column. The tube was inserted in a metallic box placed under a ground plane and a copper ring placed around the tube was used to apply an intense electric field between the ring and the box. In this way the electric field lines penetrating inside the tube launch a surface wave that creates and sustains the plasma column. A surface wave is an electromagnetic wave that propagates along the interface between different media [4]. When a surface wave propagates along a rod of lossless material surrounded by air, the electromagnetic field intensity in the air decays ex, with ponentially in the radial direction as

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and the Similarly the EDF is given by the isotropic term along and directions; anisotropic term the equations for each term become:

(16) (17) (18)

Fig. 2. Axial distribution of the cross-section average electron density [4].

being the attenuation coefficient. At the interface between two conventional dielectrics with permettivities and with the transverse evanescence of the field occurs only on the side with lower permittivity. Surface waves along the interface between a plasma and a dielectric are peculiar as the exponential decaying occurs on both sides of the boundary. This behavior is due to the negative permittivity exhibited by plasma for frequencies below the cut-off plasma frequency, . A plasma characterized by a permittivity can sustain a sur, with being the permittivity of the face wave when discharge vessel (usually glass). This condition implies that the given electron density has to be greater than a specific value by [4], [9]:

Since the vessel dimensions are big enough, the only diffusion process is ambipolar [2] and we can neglect the diffusion term in the Boltzmann equation. The diffusion term is taken into account in the electron density continuity equation:

(19) , is the vessel inner radius and where is the ambipolar diffusion term calculated as [2]:

(20) is the mean ion velocity, is the mean electron is the Boltzmann kinetic energy, is the neutral density and constant.

(12) III. NUMERICAL MODEL is called “critical density”. This where condition is crucial as it determines the propagation of a surface wave along a plasma column. Basically, when an intense field is applied to the dielectric tube, the electron density in correspondence to the application point starts to increase. When the the electromagnetic field starts electron density approaches to propagate but it falls abruptly where the condition (12) is no longer satisfied (Fig. 2). As the ionization front increases, the plasma column height becomes greater and the surface wave propagates along it. This means that the surface wave creates and sustains step by step the medium in which it propagates. Generally, surface waves are obtained as a superposition of TE and TM modes. However, in a cylindrical configuration, symmetric pure surface waves require a TM mode propagation , , . The [6], therefore the only field components are Maxwell curl equations become:

(13) (14) (15)

The complete Maxwell-Boltzmann system can be solved in an explicit way by applying a finite difference time domain approach. As in the classic Yee algorithm [28], the electromagnetic quantities are calculated in a staggered grid and in an iterative way by a leapfrog time stepping [17]. The novelty of the model is to solve simultaneously the Maxwell equations, the Boltzand and the continuity equamann equation for tion for the electron density . Moreover for each iteration and at each point the EDF is normalized according to:

(21) to ensure that is selfconsistent with the EDF. This approach was well tested for the 1D propagation of a plane wave in a plasma slab [22], [23] and showed good stability. The cylindrical configuration introduced here involves a more complex geometry, even if the physical aspects and the numerical implementation and discretization remain basically the same. Moreover the azimuthal symmetry allows the 3D geometry to assume a simpler 2D form. Therefore, the mesh used to discretize the Maxwell equations is obtained by compressing the 3D cell in the direction [29].

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The singularity at in the evaluation of arising from the cylindrical coordinates can be avoided using the integral form of the Maxwell equation [30] giving:

(22) and being the time and space discretization indexes. The 2D FDTD mesh for the EDF is not dissimilar from the one used for the electromagnetic field. The anisotropic terms (and therefore the current density) are calculated at the same position of the electric field and at the same instant of the magnetic field, while the isotropic term and the electron density are calculated at the same position of the magnetic field and at the and also have to same instant of the electric field. , be calculated in a velocity grid. The geometry shown in Fig. 1 was numerically implemented as reported in Fig. 3(a). The boundary conditions used to absorb the electromagnetic field on the numerical domain are the PML for a rotationally symmetric structure [17]. Moreover, since the PML region is far enough from the tube discharge the evanescent field is strongly attenuated. The Boltzmann equation is solved only in the plasma region inside the glass tube, thus boundary conditions for the distribution function have to be defined along the tube wall. These and along conditions consist in the assumption of null the tube corresponding to null current densities as are to be expected because of the tube wall. were chosen to be at The spatial steps least 1/80 of the wavelength corresponding to the highest frequency simulated and greater than the Debye radius so that the quasi-neutrality hypothesis is well satisfied. The time step was chosen according to the Courant stability limit. Moreover, the time step has to be smaller than the shortest characteristic collision time in order to describe the collisional processes properly, thus:

(23) where is a constant usually taken as 0.25. The glass tube ( ) is filled with argon ( , ) at a specific pressure (40 Pa). The plasma column is characterized by the following parameters: • the initial distribution is assumed to be uniform so that corresponds to a Maxwellian distribution at , while and are null inside the tube , , • • , • , , where , , , are respectively the electron and gas , and are the vessel densities and temperatures and outer and inner radius and height. It is well known [6], [15] that the initial electron density chosen has little or no effect on

Fig. 3. Implemented geometry (a) and details of the surface-wave launcher (b).

Fig. 4. Input voltage and current for the numerical determination of the pump signal network input impedance (a) and input impedance in the real situation (b).

the dynamics of the discharge, because the steady-state distribution is self-consistently determined by the power supplied to the plasma. The surface wave is launched applying an intense field between the copper ring and the metallic box (Fig. 3(b)). The copper ring has thickness 0.2 cm and height 2.6 cm, while the metallic box has a 7.4 cm radius and is 14.8 cm long and 0.2 cm thick. In this way the cylindrical metallic box implemented here has a volume equivalent to the rectangular one used in the experimental investigation described in [11]. The gap between the copper ring and the metallic box is 0.8 cm. The cylindrical geometry enables us to determine not only the plasma state evolution as studied in the 1D case, but also the input impedance seen by the feeding network. The FDTD approach allows the impedance evaluation simply by calculating the input voltage and current in correspondence to the gap where the field is applied (Fig. 4(a)). It can be assumed that the input impedance calculated as:

(24) corresponds to the input impedance seen by the coaxial cable in the real situation as performed in [9], [11] (Fig. 4(b)).

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Fig. 5. Spatial and temporal evolution of electron density for f : = . and E

= 12 5 kV cm

= 500 MHz

n

z

z > z as a function of t = 140 s and r = 8 mm for

Fig. 6. Electron density along the -direction for frequency compared with corresponding at = . E

= 10 kV cm

n

IV. RESULTS AND DISCUSSION The model was applied to a linear plasma antenna. As plasma antenna efficiency strongly depends on the pump signal network, a parametric investigation of plasma parameters with respect to the applied signal frequency and intensity is reported. These input quantities can be varied in order to determine the optimum working conditions in terms of gas discharge efficiency and input impedance. The parametric investigation was also carried out with respect to the gas pressure as this strongly affects plasma state. Since we are interested in the plasma antenna behavior, a greater relevance is given to the macroscopic results and the EDF evolution is not reported for the sake of brevity. The quantities considered are the electron density, the plasma conductivity and the input impedance seen by the pump excitation network. A. Electron Density The macroscopic parameter that best represents the plasma state evolution is electron density since from its temporal and spatial behavior it is possible to infer information about the surface wave discharge evolution. In Fig. 5 the electron density along the tube for a 500 MHz and pump signal at different time steps is reported. This result shows that the gas breakdown starts in corre); as the spondence to the point where the field is applied ( , discharge evolves and the electron density approaches the the surface wave and consequently the ionization front begin to propagate and the column lengthens until a steady-state condition is reached after about 120 . The electron density gradually decreases along the column and falls abruptly when the density is no longer sustained. This is clearly shown in Fig. 6 that illustrates the electron density along the column for and for different input frequencies and fixed intenvalues. sity, compared with the corresponding As shown, the plasma column is sustained until condition (12) a greater frequency coris satisfied. Moreover, with responds to a shorter column or, equivalently, greater power is needed to obtain the same column length for a higher frequency signal. This result is also illustrated in Fig. 7(a) and (b) that report the steady-state electron density along the column with respect

t = 140 s = 10 kV cm = 500 MHz

Fig. 7. Electron density along the column at  as a function of the = (a) and as a function of applied pump signal frequency for E field intensity for f (b)(for the color map refer to Fig. 5).

Fig. 8. Electron density along the column as a function of neutral density (gas : = pressure ) at (for the  for f and E color map refer to Fig. 5).

t = 140 s

= 500 MHz

= 12 5 kV cm

to the input signal frequency and intensity, the other parameters being equal. The figures confirm that by increasing the power the column becomes longer; moreover, with the same field intensity a shorter column is obtained for increasing frequencies. The electron density can also be evaluated with respect to the initial neutral density (corresponding to different pressures) and is reported below (Fig. 8). This result shows that the lower the neutral density, the longer the column: with the initial neutral density corresponding to , the steady-state plasma column

RUSSO et al.: SELF-CONSISTENT ANALYSIS OF CYLINDRICAL PLASMA ANTENNAS

>

Fig. 9. Plasma conductivity along the z-direction for z z and r = 8 mm with respect to different pump signals and for a 227 MHz radiated signal.

is the longest and the electron density is the highest for the same pump signal power and frequency. This behavior is expected according to experimental measurements giving the breakdown field with respect to the gas pressure [2]. In fact, in the range considered here, the field needed for ionization decreases with pressure, thus with a fixed intensity a smaller pressure gives a higher ionization. B. Plasma Conductivity Plasma conductivity is a key parameter that has to be evaluated to characterize the “metallic” plasma behavior. The steady-state plasma conductivity can be evaluated with a good approximation by the well known expression [31]:

(25) where the plasma frequency and the average collision frequency to be substituted in (25) are self-consistently determined from the model. In (25) it can be noticed that conductivity depends on the frequency of the useful signal, but it is the pump signal that ignites the ionization process, determines the final electron density and provides the conductive medium for the useful signal to be radiated. Therefore, we calculated plasma conductivity substituting , that is the useful signal frequency [11], and , , that are self-consistently determined with the model for different pump signals and gas pressures, in (25). Fig. 9 reports plasma conductivity along the tube for different pump signal frequencies, the other parameters being equal. This result shows that the highest conductivity is obtained for the highest pump signal frequency among the ones simulated, with the corresponding electron density also being the greatest. Moreover as the pump signal frequency becomes higher the antenna height decreases. A parametric investigation can be carried out to evaluate the optimum gas pressure that allows the highest conductivity to be obtained. As expected, Fig. 10 shows that the highest conductivity is obtained in correspondence to the neutral gas density that pro. vided the maximum electron density, that is

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>

Fig. 10. Plasma conductivity along the z-direction for z z with respect to different neutral densities, for a 500 MHz pump signal and a 227 MHz radiated signal.

C. Input Impedance The input impedance seen by the pump signal network is calculated by the input voltage and current ratio as shown in Section II. The input impedance seen by the pump signal network strongly depends on losses due to the presence of the ignited plasma and on the capacitive coupling used to apply the signal itself. Moreover, the input impedance changes within the plasma state inside the tube. The input impedance is a quantity that is meaningful in the stationary condition, however if we consider that the temporal evolution of the plasma state is very slow compared with the time scale of the electromagnetic propagation, it can be assumed that during plasma state evolution the signal reaches different stationary states. The input impedance is calculated and reported for these states. When the pump power is applied the plasma is initially off, so we expect the input impedance to be given basically by the copper ring and the metallic box alone. This can be verified in Fig. 11 that shows the real ( ) and the imaginary ( ) parts of the input impedance when plasma is not yet ignited. As can be seen, the input impedance behavior is strongly capacitive in the lower frequency range as it depends on the capacitive coupling between the copper ring and metallic box. When the frequency increases the input impedance approaches the resonant frequency of the cylindrical metallic box that for the chosen dimension is 1562 MHz. As long as the pump signal is applied, the gas becomes more and more ionized and consequently the input impedance changes together with the plasma state inside the tube. Fig. 12 shows the temporal evolution of the input impedance simulated at , and . It can be seen that after a strong mismatching the discharge condition leads to a good impedance matching. It is interesting to investigate the input impedance behavior with respect to different discharge conditions. Fig. 13 reports the input impedance with respect to the pump signal frequency for ) and gas presa fixed applied signal intensity ( sure ( ), at (when plasma is off) and at (when a stationary condition can be assumed). This result shows that with the signal coupler used (in terms of dimension and geometry), the best frequency range for impedance

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Fig. 11. Real (R – red line) and imaginary (X – blue line) parts of the input impedance vs pump signal frequency seen by the pump signal network when the plasma is not ignited.

Fig. 12. Temporal evolution of the real (R) and imaginary (X) parts of the input impedance seen by the pump signal network.

Fig. 13. Real (R) and imaginary (X) parts of the input impedance seen by the pump signal network with respect to the pump signal frequency.

TABLE I INPUT IMPEDANCE VS PUMP SIGNAL INTENSITY.

matching is 400–500 MHz that corresponds to the frequency range chosen for the pump network in the preliminary experimental investigation (430 MHz) [11]. The non-linear behavior of the plasma state with the intensity of the applied pump signal is also highlighted by Table I that for reports the input impedance at and with respect to the applied . The input impedance seen by the pump signal network changes together with the applied field since the plasma state is also different. As was shown in Fig. 7(b), by increasing the power the column obtained is longer and the electron density is higher; consequently losses are lower. This is consistent with the decreasing behavior of . V. CONCLUSION A rigorous and self-consistent model for the description of cylindrical plasma antenna behavior has been developed. This model allows the evaluation of the parameters affecting antenna efficiency which are involved in plasma state evolution. The choice of a kinetic model for the solution of the interaction mechanism between an electromagnetic wave and a plasma was made in order to infer self-consistent information on plasma state and allows the evaluation of the mutual relationship between all the quantities. The numerical technique used to solve the Maxwell-Boltzmann system is robust and able to handle the strong non-linearity of the problem. The parametric investigation carried out with the model highlighted that optimization in plasma antenna design and realiza-

tion can be achieved and plasma antenna efficiency can be maximized by properly choosing the set-up parameters. For example, in the case study the model showed that the best impedance matching is obtained by ionizing the gas with a pump signal in the frequency range [400–500] MHz. Pump signal frequency also affects plasma conductivity. In fact, by increasing the frequency, the radiation efficiency is higher . More generally, as widely illustrated in literature, plasma is an innovative material whose properties make it particularly suitable for a great number of applications. The numerical model introduced here offers the possibility to characterize the interaction mechanism between a plasma region and an electromagnetic wave, whatever its geometry or particular application. Thus the model can describe this phenomenon in several situations thereby investigating the possibility of using plasma in different applications. REFERENCES ˇ [1] A. P. Zilinskij, I. E. Sacharov, and V. E. Golant, Fundamentals Plasma Physics. Moscow: MIR, 1983. [2] Y. P. Raizer, Gas Discharge Physics. Berlin: Springer-Verlag, 1991.

RUSSO et al.: SELF-CONSISTENT ANALYSIS OF CYLINDRICAL PLASMA ANTENNAS

[3] M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing. New York: Wiley, 1994. [4] M. Moisan, A. Shivarova, and A. W. Trievelpiece, “Experimental investigations of the propagation of surface waves along a plasma column,” Phys. Plasmas, vol. 24, p. 1331, 1982. [5] T. J. Dwyer, J. R. Greig, D. P. Murphy, J. M. Perin, R. E. Pechacek, and M. Raileigh, “On the feasibility of using an atmospheric discharge plasma as an RF antenna,” IEEE Trans. Antennas Propag., vol. 32, pp. 141–146, 1984. [6] M. Moisan and J. Pelletier, Microwave Excited Plasma. Amsterdam: Elsevier, 1992. [7] G. G. Borg, D. G. Miljak, and N. M. Martin, “Application of plasma columns to radiofrequency antennas,” Appl. Phys. Lett., vol. 74, pp. 3272–3274, 1999. [8] G. G. Borg, J. H. Harris, N. M. Martin, D. Thorncraft, R. Milliken, and D. G. Miljak, “Plasmas as antennas: Theory, experiment and applications,” Phys. Plasmas, vol. 7, pp. 2198–2201, 2000. [9] J. P. Rayner, A. P. Whichello, and A. D. Cheetham, “Physical characteristics of plasma antennas,” IEEE Trans. Plasma Sci., vol. 32, pp. 269–281, 2004. [10] I. Alexeff, T. Anderson, S. Prameswaran, E. P. Pradeep, J. Hulloli, and P. Hulloli, “Experimental and theoretical results with plasma antennas,” IEEE Trans. Plasma Sci., vol. 33, pp. 166–171, 2006. [11] G. Cerri, R. De Leo, V. M. Primiani, and P. Russo, “Measurement of the properties of a plasma column used as a radiated element,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 242–247, 2008. [12] M. Moisan, Z. Zakrzewski, R. Panel, and P. Leprince, “A waveguidebased launcher to sustain long plasma columns through the propagation of an electromagnetic surface wave,” IEEE Trans. Plasma Sci., vol. 12, pp. 203–214, 1984. [13] M. Moisan and Z. J. Zakrzewski, “Plasma sources based on the propagation of electromagnetic surface waves,” J. Phys. D, Appl. Phys., vol. 24, pp. 1025–2048, 1991. [14] Y. Kabouzi, D. B. Graves, E. Castaños-Martínez, and M. Moisan, “Modeling of atmospheric-pressure plasma columns sustained by surface waves,” Phys. Rev. E, vol. 75, 2007. [15] O. A. Ivanov and V. A. Koldanov, “Self-consistent model of a pulsed air discharge excited by surface waves,” Plasma Phys. Rep., vol. 26, pp. 902–908, 2000. [16] M. Meyyappan and T. R. Govindan, “Radio frequency discharge modeling: Moment equations approach,” J. Appl. Phys., vol. 74, pp. 2250–2259, 1993. [17] A. Taflove and S. C. Hagness, Computational Electrodynamics: The FDTD Method. London: Artech House Boston, 2000. [18] J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency dependent finite difference time domain formulation for transient propagation in plasma,” IEEE Trans. Antennas Propag., vol. 29, pp. 29–34, 1991. [19] S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propag., vol. 45, pp. 392–400, 1997. [20] Y. Lee and S. Ganguly, “FDTD analysis of a plasma column,” in Proc. IEEE Antennas Propagation Society Int. Symp., 2005, vol. 1B, pp. 430–433. [21] Z. H. Qian, K. W. Leung, R. S. Chen, and D. X. Wang, “FDTD analysis of a plasma whip antenna,” in Proc. IEEE Antennas Propagation Society Int. Symp., 2005, vol. 2B, pp. 166–169. [22] G. Cerri, F. Moglie, R. Montesi, P. Russo, and E. Vecchioni, “FDTD solution of the Maxwell-Boltzmann system for electromagnetic wave propagation in a plasma,” IEEE Trans. Antennas Propag., vol. 56, pp. 2584–2588, 2008. [23] G. Cerri, P. Russo, V. M. Mariani, and E. Vecchioni, “FDTD approach for the characterization of electromagnetic wave propagation in plasma for application to plasma antennas,” presented at the 2nd Eur. Conf. on Antennas and Propagation, Edinburgh, UK, Nov. 11–16, 2007. [24] A. R. Bell, A. P. L. Robinson, M. Sherlock, R. J. Kingham, and W. Rozmus, “Fast electron transport in laser-produced plasmas and the KALOS code for solution of the Vlasov-Fokker-Planck equation,” Plasma Phys. Controlled Fusion, vol. 48, pp. 37–53, 2006. [25] T. Holstein, “Energy distribution of electrons in high frequency gas discharges,” Phys. Rev., vol. 70, pp. 367–384, 1946. [26] U. Kortshagen, C. Busch, and L. D. Tsendin, “On simplifying approach to the solution of the Boltzmann equation in spatially inhomogeneous plasma,” Plasma Sources Sci. Technol., vol. 5, pp. 1–17, 1996. [27] A. V. Vasenkov, “Non equilibrium argon plasma generated by an electron beam,” Phys. Rev., vol. 57, pp. 2212–2221, 1998.

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[28] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 302–307, 1966. [29] Y. Chen, R. Mittra, and P. Harms, “Finite-Difference Time-Domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 832–839, 1996. [30] N. Dib, T. Weller, M. Scardelletti, and M. Imparato, “Analysis of cylindrical transmission lines with the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 509–512, 1999. [31] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communications Electronics. New York: Wiley, 1994.

Paola Russo (S’98–M’00) was born in Turin, Italy, in 1969. She received the Ph.D. degree in electronic engineering from the Polytechnic of Bari, Italy, in April 1999. In 1999, she worked with a research contract at the Motorola Florida Research Lab. From 2000 to 2004, she worked with a research contract on the development of numerical tools applied to the coupling of electromagnetic field and biological tissue, and to different EMC problems, in the Department of Electronics, University of Ancona (now Università Politecnica delle Marche). Since January 2005, she has been a Researcher at the Università Politecnica delle Marche, Ancona, Italy, where she teaches EMC and Antenna design. Her main research topics are on the application of numerical modelling to EMC problem, reverberation chamber, and new antenna design. Prof. Russo is member of the IEEE EMC and AP societies and the Italian Society of Electromagnetics SIEM.

Graziano Cerri (M’93) was born in Ancona, Italy, in 1956. He received the degree in electronic engineering from the University of Ancona, in 1981. In 1983, after military service in the Engineer Corp of Italian Air Force, he became an Assistant Professor in the Department of Electronics and Control, University of Ancona, where, in 1992, he became an Associate Professor of microwaves. Currently he is a Full Professor of electromagnetic fields at the Dip. Ingegneria Biomedica, Elettronica e Telecomunicazioni, Università Politecnica delle Marche, Ancona, Italy. His research is mainly devoted to EMC problems, to the analysis of the interaction between EM fields and biological bodies and to antennas. Prof. Cerri is a member of AEI (Italian Electrotechnical and Electronic Association). Since 2004, he is the Director of ICEmB (Interuniversity Italian Center for the study of the interactions between Electromagnetic Fields and Biosystems). He is also a Member of the Administrative and Scientific Board of CIRCE (Interuniversity Italian Research Centre for Electromagnetic Compatibility), member of the Scientific Board of CNIT (Interuniversity National Centre for Telecommunications) and member of the Scientific Board of SIEm (Italian Association of Electromagnetics).

Eleonora Vecchioni was born in Macerata, Italy, in 1981. She received the Laurea degree in electronics engineering and the Ph.D. degree in electromagnetism from the Università Politecnica delle Marche, Ancona, Italy, in July 2006 and December 2009, respectively. From January 2010 to May 2010, she was collaborating with the Dipartimento Di Ingegneria Biomedica, Elettronica e Telecomunicazioni, Univpm, as an external collaborator and since June 2010 she is working in the Software R&D Unit, Thermowatt SpA. Her research interests include computational electrodynamics and plasma physics, in particular the physical and numerical characterization of the electromagnetic properties plasma.

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A Slot Antenna Array With Low Mutual Coupling for Use on Small Mobile Terminals Sema Dumanli, Member, IEEE, Chris J. Railton, and Dominique L. Paul, Member, IEEE

Abstract—In order to take full advantage of the benefits to be obtained by using MIMO techniques for mobile communications, it is necessary to use an antenna array which is both compact and also has low mutual coupling between ports. Generally these requirements are conflicting and to achieve them simultaneously is the subject of much research. In this paper a novel design for a two element cavity backed slot (CBS) array is described which has a measured mutual coupling of less than 15 dB despite an element spacing of only 6. This is achieved by adding a simple and easily manufactured meandering trombone structure to an existing CBS array which carries a portion of the input signal to the feed of the neighboring element. Measured and simulated results are presented for the behavior of the antenna and predictions are presented for the achievable channel capacity in several realistic scenarios. Index Terms—MIMO, slot antenna array.

I. INTRODUCTION

W

ITH the steadily growing demand for information to be delivered to mobile terminals and handsets, there is an increasing need to maximize the use of the available bandwidth. One way of achieving this is to use multiple antenna elements at each end of the communications link. In situations where there is plenty of space, such as at mobile phone base stations or on laptop computers, it is not difficult to accommodate an antenna array. On small terminals, however, such as PDAs and mobile phones, it can be a challenge to fit in even a single antenna element since the size of the unit may be of the order of a wavelength. Any array of elements placed in such an environment must, therefore, by necessity be very closely spaced and is likely, therefore, to have an undesirably high mutual coupling. Various techniques have been proposed in order to mitigate this problem. A comprehensive list of references for these is given in [1]. These include choosing the optimum position of the antennas on the PCB board to minimize the mutual coupling between elements [2] or shaping the PCB in some way either by cutting slots or by adding protrusions [3], [4]. Another approach

Manuscript received February 24, 2010; revised October 26, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The work of S. Dumanli was supported by a postgraduate scholarship from Toshiba Research Europe Limited and The Scientific and Technological Research Council of Turkey (TUBITAK). S. Dumanli was with the Centre for Communications Research, University of Bristol, Bristol BS8 1UB, U.K. She is now with Toshiba Research Europe, Bristol BS1 4ND, U.K. (e-mail: [email protected]). C. J. Railton and D. L. Paul are with the Centre for Communications Research, University of Bristol, Bristol BS8 1UB, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123057

is to add an external decoupling network, such as a rat race hybrid as used in [5]. In this case one of the ports feeds the elements in phase while the other feeds the elements in anti-phase [6]–[10]. While this method has been shown to give good results for the desired low mutual coupling, it has the disadvantage that the ports are asymmetrical and also that there can be problems with low bandwidth for the anti-phase port. A more recent and very promising approach is to add an extra structure to the array in order to intentionally couple a small amount of the energy from one element to another. This can be done in such a way as to cancel the mutual coupling. An example of the use of this general approach for PIFA type elements is given in [11] but the proposed structure is very complicated. Another example is given in [12] but this involves a thin suspended stripline which may cause problems in robustness. In this paper a simple and robust method of achieving mutual coupling cancellation in a pair of closely spaced cavity backed slot (CBS) antennas is described. It is shown that even for a spacing , a measured mutual coupling of less than dB of only dB at was obtained together with a reflection of less than the operating frequency of 5.2 GHz. This can be compared to a dB and reflection of dB measured mutual coupling of for the same array without cancellation. In addition, in contrast to the situation when the rat-race hybrid is used, the symmetry and the bandwidth of the antenna are preserved. II. THE ARRAY ELEMENT It has been shown that CBS antennas are good candidates for MIMO systems since they are as efficient as monopole antennas [13] and arrays of CBS antenna elements have low mutual coupling [14]. They offer good MIMO capacities compared to competing designs such as the planar inverted-F (PIFA), and the dielectric resonator antenna (DRA) because of their high efficiency [13], [15]. In addition, if the cavity walls of the antenna are formed by using a curtain of shorting pins instead of solid copper a more accurate and repeatable manufacturing process is obtained without adversely affecting the mutual coupling [16]. For this reason, it was decided to design an array of two of these elements for use on a small mobile terminal and to include a cancellation structure in order to provide a low mutual coupling without compromising the available bandwidth. It is readily possible to extend this to a four element array by using the configuration of [17]. The geometry of the CBS element is shown in Fig. 1 and a two element array formed by placing two elements side by side is illustrated in Fig. 2. This was the structure developed in [16] which used the fewest shorting pins without degrading the mutual coupling as compared to using solid copper walls. Pins with

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Fig. 1. The single CBS element to operate at 5.2 GHz. Fig. 4. Measured radiation patterns of the two embedded elements on a 15 cm ground plane.

Fig. 2. A two element array of CBS elements. Fig. 5. Simulated radiation patterns of the two embedded elements on a 15 cm ground plane.

Fig. 6. Simulated radiation pattern patterns of the two embedded elements with no ground plane. Fig. 3. Measured S parameters with no ground plane.

radius 0.275 mm with a separation of 4 mm were used. The measured S parameters of this array in Fig. 3 show a mutual coupling dB at 5.2 GHz. While this is still usable it does represent of a power loss of approximately 25% so it is desirable for the coupling to be reduced. The measured and simulated radiation patterns of this array are shown in Figs. 4 and 5 which exhibit very good agreement with each other. To facilitate measurement, the array was placed on a ground plane of radius 15 cm. The effect of this is to make the radiation pattern more complicated due to diffraction effects but not to significantly alter the general features. The expected radiation pattern in the absence of a ground plane is shown in the simulated results of Fig. 6. In each case it can be seen that the effect of mutual coupling is to introduce a small amount of squint in the two patterns. This squinting of the radiation pattern is caused by the asymmetry of the structure.

The slot which is not driven acts as a parasitic element which receives and re-radiates some of the energy thus distorting the radiation pattern. While it has been shown that the squint can be an advantage in MIMO and diversity systems [18], this is more than negated by the reduced radiation due to mutual coupling loss. III. THE DECOUPLING STRUCTURE Several different possible structures were investigated and evaluated by means of extensive finite difference time domain (FDTD) simulations. This was done using the enhanced FDTD software developed at the University of Bristol which includes the facility to calculate S parameters and 3D far field radiation patterns of antennas. Of those tested, it was found that structure shown in Fig. 7 gave the best performance, the greatest ease of manufacture and the least sensitivity to manufacturing tolerances. In this scheme, the feed lines have been extended and

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Fig. 7. Structure of the decoupled array.

Fig. 9. S characteristics for slot lengths of 31 to 35 mm in 1 mm increments and trombone length of 11 mm.

Fig. 8. S characteristics for slot lengths of 31 to 35 mm in 1 mm increments and trombone length of 11 mm.

joined with a meandering “trombone” section which carries a portion of the input signal from the excited feed to the neighboring element. With an appropriate choice of dimensions, it is shown that the signal coupled through the trombone can be made to cancel the original mutual coupling yielding a pair of well isolated ports. In order to find the optimum dimensions for the trombone section, a parametric study was done using FDTD simulations. There are a number of parameters which can be chosen in order to give the required magnitude and phase for the coupled signal. Primarily, these are the width and length of the strip making up the trombone section and the length of the slot. Typical results for different slot lengths and trombone lengths are shown in Figs. 8–11. In Fig. 8 it is shown that the frequency at which the minimum reflection is obtained, is lower as the slot is lengthened. Also it can be seen that the match improves as the slot is lengthened. Fig. 9 shows that the frequency at which the mutual coupling is lowest also reduces monotonically as the slot is lengthened but that the level of the minimum is approximately constant. Also it can be seen that the dependence of the as for . Thus frequency on length is not the same for there exists a slot length at which the two minima are at the same frequency. If this frequency can be made equal to the desired operating frequency, this will be the best choice. The effect of the trombone length is more complex as this affects both the matching of the element and also the magnitude

Fig. 10. S characteristics for trombone lengths of 10 mm to 12 mm in 0.5 mm increments and slot length of 35 mm.

and phase of the coupling between elements. Fig. 10 shows the on trombone length. It can be seen that this dependence of length has a considerable effect both on the center frequency and also on the matching. Finally Fig. 11 shows the effect of trombone length on mutual coupling. In this case the effect on the frequency of the minimum is weaker but more complicated. The effect of varying other parameters such as the width of the trombone line and the size of the cavity were also investigated but in most cases no particular advantage or disadvantage was to be gained by changing these. It was found that best results were achieved under the following conditions. 1. The width of the stripline making up the trombone was similar to the width of the feedline. This led to the antenna characteristic being not unduly sensitive to manufacturing tolerances. Narrower trombone widths could be used but the exact dimensions were more critical. 2. The side arms of the trombone section were centrally placed between the radiating slot and the central row of shorting pins. If they were not placed in this position, the coupling between the side arms and the slot or pins led to the results being sensitive to manufacturing tolerances.

DUMANLI et al.: A SLOT ANTENNA ARRAY WITH LOW MUTUAL COUPLING FOR USE ON SMALL MOBILE TERMINALS

Fig. 11. S characteristics for trombone lengths of 10 mm to 12 mm in 0.5 mm increments and slot length of 35 mm.

Fig. 12. The manufactured decoupled array.

TABLE I DIMENSIONS OF THE FINAL DECOUPLED ARRAY IN MM

Given these constraints, the dimensions for the final design were arrived at by means of the results of the FDTD parametric analysis. IV. THE FINAL ARRAY The final manufactured array is illustrated in Fig. 12 and has the dimensions given in Table I. The feed position is the distance between the bottom of the slot and the bottom of the feed line as shown in Fig. 7. It is noted that the slot length of the final array is considerably longer than in the original un-decoupled array. This was necessary in order to obtain the correct center frequency. It is also noted that whereas the manufactured test antenna is fitted with

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Fig. 13. Measured S parameters of the final array.

SMA connectors, the connections would normally be directly made with the RF circuitry. Measurements of the antenna S parameters and radiation patterns were made using the Department’s anechoic chamber and an Anritsu 37397C Vector Network Analyser. In order to measure the 3D radiation patterns, a Flann DP-240AA horn antenna was used as a reference and a pair or orthogonally mounted stepper motors were used to scan the antenna under test in the azimuth and elevation directions. The resulting data was collected and plotted using in-house MATLAB codes. Full details of the equipment and the test setup can be found in [19] and [20]. The measured S parameters for the final manufactured array are shown in Fig. 13. It can be seen that the mutual coupling has been reduced to less than dB and the reflection has been improved over that of the original single element and is now less dB. Measured and simulated radiation patterns using than a model which includes the 15 cm ground plane are shown in Figs. 14 and 15 where good agreement can be seen. The simulated result for the case where there is no ground plane is shown in Fig. 16. In all cases it can be seen that the radiation pattern exhibits a strong squint despite the low mutual coupling. In this case the asymmetry is introduced because the element which is not driven directly is fed with a small amount of power through the trombone structure. The superposition of the main beam and the radiation from the second element results in an asymmetrical squinted radiation pattern. This behavior can be advantageous in providing pattern diversity or a low envelope correlation when used in MIMO systems. Figs. 17 and 18 show the current distribution on the antenna feeds for the original array of Fig. 2 and for the final array of Fig. 7. It can be clearly seen that the current on the victim feed is much less for the final array than for the original. Moreover, the cancellation effect between the trombone current and the energy coupled through the victim slot is apparent as the current sharply reduces when the feed line crosses the position of the slot. Where the array is to be used in a MIMO system, the envelope correlation coefficient is a relevant characteristic. This property was calculated for the original non-decoupled array and for the

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Fig. 18. Current distribution on feeds for trombone array. Fig. 14. Measured radiation patterns of the two embedded elements on a 15 cm ground plane.

Fig. 15. Simulated radiation patterns of the two embedded elements on a 15 cm ground plane. Fig. 19. Measured envelope correlation of the original and final arrays.

where

(2) Although this method is strictly valid only for lossless antennas, in this case the measured efficiency was high so this method can still be used with good accuracy. In Fig. 19, it can be seen that the envelope correlation is improved over that of the original array and that it remains low over a wider frequency range. Fig. 16. Simulated radiation patterns of the two embedded elements with no ground plane.

Fig. 17. Current distribution on feeds for original array.

final array using the measured S parameters and (1) and (2). The method is based on that described in [21].

(1)

V. EFFICIENCY MEASUREMENTS The efficiency of the final trombone array and the original array were measured. This was done by comparing the radiated power from the test antenna with that obtained from an element which is known to have a high efficiency, in this case a monopole. In order to achieve as much accuracy as possible, the measurements were made on the same day in the same anechoic chamber and three sets of measurements were made. Finally the total radiated powers are averaged and compared. This method is expected to have less than a 5% uncertainty. In addition, the efficiency was calculated from the FDTD results using a perturbation method. The results are given in Tables II and III with mismatch loss excluded and included respectively. It can be seen that good agreement exists between measurement and calculation. The calculated losses from the conductors and the dielectrics are given in Table IV. It can be seen that the extra loss associated with the trombone is due to extra losses in the conductors. Nevertheless, this loss

DUMANLI et al.: A SLOT ANTENNA ARRAY WITH LOW MUTUAL COUPLING FOR USE ON SMALL MOBILE TERMINALS

TABLE II MEASURED AND CALCULATED EFFICIENCIES EXCLUDING MISMATCH LOSS

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It is common practice to normalize the channel matrix to a “unit gain” channel as shown in (5)

(5) TABLE III MEASURED AND CALCULATED EFFICIENCIES INCLUDING MISMATCH LOSS

TABLE IV CALCULATED DIELECTRIC AND CONDUCTOR LOSS

where the summations are taken over all elements of the H matrix. This, however, includes the effects of the antenna so that issues such as return and mutual coupling loss are masked. In this work, following [22], the alternative normalization given by (6) is used (6)

Fig. 20. The channel model for comparing antenna performance.

is more than made up for by the reduction of mutual coupling loss. VI. THE ARRAY AS PART OF A MIMO SYSTEM In order to assess the performance of the antenna in a real situation compared with the original array, the system performance was simulated using a number of different measured and statistically generated channels. The channel model which was used is shown in Fig. 20 which is a 2 N port network where N is the number of receive and transmit elements. This network can be described by an S matrix with the following structure (3) where H is the channel matrix while and are the individual S matrices of the transmit and receive arrays respectively. In each case, the channel was expressed as the summation of paths such that the total received signal was the superposition of a number of plane waves. The H matrix which characterizes the transmission from the terminals of the transmit antenna array to those of the receive antenna array is given by (4) are the elevation and azimuth angles of arwhere is the embedded gain rival and departure respectively. of the element in the direction. is the attenuation of the path. The statistical distributions of these parameters will depend upon the channel being used. For the results presented in this paper, the distributions are given below.

Here, the summations are taken over all paths in the channel model. It is noted that this normalization does not involve the properties of the antenna, such as the radiation pattern, the efficiency and return loss so it allows a realistic comparison between antenna systems. As described in [22], this is equivalent to ensuring unit normalized channel gain when ideal isotropic radiators are used. It is also comparable to the “link capacity” described in [23] as contrasted with the “MIMO capacity” also described in the same paper. The results presented in this paper are all calculated using this normalization. Three different sets of channel data were used for comparison. 1. Artificial channel Firstly, the data for the channel was generated using a specified statistical distribution. This was an idealized test scenario where there was a very rich multi-path environment and a uniform distribution of angles of arrival for the received signals. All angles were uniformly distributed. The path lengths were normally distributed with a mean of 2 km and standard deviation of 200 m. Path loss and phase were calculated for a line of sight path of the same total length. 40 independent paths were assumed to exist and 1000 simulations were used in order to obtain the statistical properties. 2. Measured outdoor channel data [24] Tests were also carried out using real channel data measured in Bristol city center. Only the receive parameters were available so the transmit parameters were estimated. In particular the angle of departure was assumed to be uniformly distributed around the azimuthal plane. The azimuthal angle of arrival was distributed over a wide range but exhibited a peak in the direction of the transmitter. Fig. 21 shows the number of paths which arrive at different angles. The elevation angles were mostly close to the horizontal. A maximum number of 40 independent paths were allowed for, based on the measured information, and 100 simulations were done in order to obtain the statistical properties. Ideally, a greater number of simulations would have been used but this was limited by the available data. 3. Ray tracing data for an office environment [24] An in-house ray-tracing tool was used to obtain channel data for an open plan office at Bristol. In general it was

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Fig. 21. Distribution of azimuth angles of arrival for outdoor channel.

Fig. 23. Comparison of capacity for the two antennas for the artificial channel. The theoretical capacity for a 2 2 IID channel and isotropic antennas is given for comparison.

2

Fig. 22. Distribution of azimuth angles of arrival for indoor channel.

found that there were fewer then 6 significant independent paths. 10 000 simulations were used in order to obtain the statistical properties. For this case the angles of arrival showed strong peaks at angles determined by the furniture near the receiver as shown in Fig. 22. The capacity was calculated using the following formula:

Fig. 24. Comparison of capacity for the two antennas for the ray-traced indoor channel.

(7) The capacities were calculated with the signal to noise ratio, in (7), set to 20 dB. These are shown in Figs. 23–25 for the measured and the simulated channels and for the original array and the final array. Also, a comparison is given with an ideal independently identically distributed (IID) Rayleigh channel of the type studied in [25]. The results were calculated using in-house MathCad software which, for each simulation of the channel, applied (7) to ascertain the capacity. The CDFs in each case were then readily obtained. It can be seen that, in each case, there is a substantial improvement to be gained by using the cancellation network when the spacing between elements is small. It is noted that the results for the outdoor channel data are not as smooth as for the others. This is due to the low number of simulations which were available in this case.

Fig. 25. Comparison of capacity for the two antennas for the measured outdoor channel.

VII. CONCLUSION In this paper a novel slot antenna array with a mutual coupling of less than dB and a reflection of less than dB, despite

DUMANLI et al.: A SLOT ANTENNA ARRAY WITH LOW MUTUAL COUPLING FOR USE ON SMALL MOBILE TERMINALS

a separation of only , has been described. The decoupling has been achieved by adding a meandering trombone structure which is easy to manufacture and couples a small amount of energy from one element to the other so as to cancel the original mutual coupling. The result is a radiation pattern which is highly squinted and exhibits a low envelope correlation over a wide frequency range. This array, and other arrays of this type, are expected to have many applications for MIMO type systems on small terminals where there is not enough room for widely spaced array elements.

ACKNOWLEDGMENT The authors would like to thank their colleagues in the Communications Systems and Networks Group, headed by Prof. J. McGeehan, for helpful discussions and for providing information on the measured and ray-traced channels.

REFERENCES [1] C. Luxey, “Design of multi-antenna systems for UTMS mobile phones,” in Proc. Loughborough Antennas and Propagation Confe., Nov. 2009, pp. 57–64. [2] S. Dumanli, Y. Tabak, C. Railton, D. Paul, and G. Hilton, “The effect of antenna position and environment on MIMO channel capacity for a 4 element array mounted on a PDA,” in Proc. 9th Eur. Conf. on Wireless Technology, 2006, pp. 201–204. [3] M. Karaboikis, C. Soras, G. Tsachtsiris, and V. Makios, “Compact dual-printed inverted-F antenna diversity systems for portable wireless devices,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 9–14, 2004. [4] T. Ohishi, N. Oodachi, S. Sekine, and H. Shoki, “A method to improve the correlation and the mutual coupling for diversity antenna,” presented at the IEEE Antennas and Propagation Society Int. Symp., Washington, DC, Jul. 2005. [5] S. Dumanli, C. Railton, and D. Paul, “A decorrelated closely spaced array of four slot antennas backed with SIW cavities for MIMO communications,” in Loughborough Antennas and Propagation Conf., Nov. 2009, pp. 277–280. [6] S. Dossche, S. Blanch, and J. Romeu, “Decorrelation of a closely spaced four element antenna array,” presented at the IEEE Antennas and Propagation Society Int. Symp., 2005. [7] S. Dossche, S. Blanch, and J. Romeu, “Optimum antenna matching to minimise signal correlation on a two-port antenna diversity system,” IET Electron. Lett., vol. 40, no. 19, pp. 1164–1165, Sep. 2004. [8] S. Dossche, J. Rodriguez, L. Jofre, S. Blanch, and J. Romeu, “Decoupling of a two-element switched dual band patch antenna for optimum MIMO capacity,” in Proc. Int. Symp. on Antennas and Propagations, Albuquerque, NM, Jul. 2006, pp. 325–328. [9] M. Shanawani, D. L. Paul, S. Dumanli, and C. Railton, “Design of a novel antenna array for MIMO applications,” in Proc. 3rd Int. Conf. on Information and Communication Technologies: From Theory to Applications ICTTA, 2008, pp. 1–6. [10] A. Nilsson, P. Bodlund, A. Stjernman, M. Johansson, and A. Derneryd, “Compensation network for optimizing antenna system for MIMO application,” presented at the Eur. Conf. on Antennas and Propagations, Edinburgh, U.K., Nov. 2007. [11] A. C. K. Mak, C. R. Rowell, and R. D. Murch, “Isolation enhancement between two closely packed antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3411–3419, Nov. 2008. [12] A. Diallo, C. Luxey, P. Le Thuc, R. Staraj, and G. Kossiavas, “Study and reduction of the mutual coupling between two mobile phone PIF as operating in the DCS1800 and UMTS bands,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3063–3074, Nov. 2006. [13] G. S. Hilton and H. W. W. Hunt-Grubbe, “Simulation and practical analysis of a cavity-backed linear slot antenna for operation in the IEEE802.11a band,” presented at the 5th Eur. Workshop on Conformal Antennas, Bristol, U.K., Sep. 2007.

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[14] A. Hadidi and M. Hamid, “Aperture field and circuit parameters of cavity-backed slot radiator,” in IEE Proc. Microw., Antennas Propag., 1989, vol. 136, no. 2, pp. 139–146. [15] A. Pal, C. Williams, G. Hilton, and M. Beach, “Evaluation of diversity antenna designs using ray tracing, measured radiation patterns, and MIMO channel measurements,” EURASIP J. Wireless Commun. Network., vol. 2007, no. 1, pp. 61–72, 2007. [16] S. Dumanli, C. J. Railton, D. L. Paul, and G. S. Hilton, “Closely spaced array of cavity backed slot antennas with pin curtain walls,” in IET Proc., to be published. [17] S. Dumanli, C. Railton, and D. Paul, “Decorrelation of a closely spaced antenna array and its influence on MIMO channel capacity,” presented at the EuCAP, Edinburgh, U.K., Nov. 11–16, 2007. [18] P. N. Fletcher, M. Dean, and A. R. Nix, “Mutual coupling in multielement array antennas and its influence on MIMO channel capacity,” Electron. Lett., vol. 39, no. 4, pp. 342–344, Feb. 20, 2003. [19] H. Hunt-Grubbe, “Element and integrated system analysis of a dual feed tuneable cavity backed linear slot antenna,” Ph.D. dissertation, Univ. Bristol, Bristol, U.K., Mar. 2007. [20] G. Hilton, “Basic use of VNAs and the anechoic chamber facility,” Laboratory notes for Antennas and Electromagnetic Compatibility, Univ. Bristol, U.K. [21] S. Blanch, J. Romeu, and I. Corbella, “Exact representation of antenna system diversity performance from input parameter description,” Electron. Lett., vol. 39, no. 9, pp. 705–707, May 1, 2003. [22] A. Pal, C. Williams, G. Hilton, and M. Beach, “Evaluation of diversity antenna designs using ray tracing, measured radiation patterns and MIMO channel measurements,” EURASIP J. Wireless Commun. Network., vol. 2007, 58769, Article ID. [23] V. Jungnickel, V. Pohl, and C. von Helmolt, “Capacity of MIMO systems with closely spaced antennas,” IEEE Commun. Lett., vol. 7, no. 8, pp. 361–363, Aug. 2003. [24] M. Beach et al., “An experimental evaluation of three candidate MIMO array designs for PDA devices,” presented at the Joint COST 273/284 Workshop on Antennas and Related System Aspects I in Wireless Communications, Gothenburg, Sweden, Jun. 2004. [25] G. J. Foschini and M. J. Gans, “On the limits of wireless communications in fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, pp. 311–335, 1998.

Sema Dumanli (M’11) was born in Elazig, Turkey, in 1984. She received the Bachelors degree in electrical and electronic engineering from Middle East Technical University (METU), Ankara, Turkey, in 2006 and the Ph.D. degree from University of Bristol, Bristol, U.K., in 2010. From June 2006 to February 2007, she was a Design Engineer at ASELSAN Inc., Turkey. Since August 2010, she has been a Research Engineer at Toshiba Research Europe, Bristol, U.K. Her current research interests include antenna design, propagation channel modelling, MIMO, mm-wave technology and terahertz communications.

Chris J. Railton received the B.Sc. degree in physics with electronics from the University of London, London, U.K., in 1974 and the Ph.D. degree in electronic engineering from the University of Bath, Bath, U.K., in 1988. During 1974 to 1984, he worked in the Scientific Civil Service on a number or research and development projects in the areas of communications, signal processing and EMC. Between 1984 and 1987, he worked at the University of Bath, on the mathematical modeling of boxed microstrip circuits. He currently works in the Centre for Communications Research, University of Bristol, where he leads the Computational Electromagnetics Team which is engaged in the development of new algorithms for electromagnetic analysis and their application to a wide variety of situations including planar and conformal antennas, microwave and RF heating systems, radar and microwave imaging, EMC, high speed interconnects and the design of photonic components.

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Dominique L. Paul (M’03) received the D.E.A. degree in electronics from Brest University, Brest, France, in June 1986 and the Ph.D. degree from Ecole Nationale Superieure des Telecommunications de Bretagne (LEST-ENSTBr), Brest, France, in January 1990. From 1990 to 1994, she was a Research Associate at the Centre for Communications Research, University of Bristol, Bristol, U.K. During 1995 to 1996, she worked as a Research Associate at the Escuela Tecnica Superior de Ingenieros de Telecomunicacion of

Madrid, Spain, under a grant from the Spanish Government. Since 1997, she has been a Research Fellow in the Centre for Communications Research, University of Bristol, Bristol, U.K., with a permanent position from 2003. Her research interests include the electromagnetic modelling of passive devices such as microwave heating systems, dielectric structures at millimeter wavelengths, MIMO systems, low profile antennas, conformal antenna arrays and textile antennas for wearable applications. Dr. Paul is a member of the IET, IEEE Microwave Theory and Techniques Society and IEEE Antennas and Propagation Society.

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An E-Band Partially Corporate Feed Uniform Slot Array With Laminated Quasi Double-Layer Waveguide and Virtual PMC Terminations Miao Zhang, Member, IEEE, Jiro Hirokawa, Senior Member, IEEE, and Makoto Ando, Fellow, IEEE

Abstract—A partially corporate feeding waveguide located below the radiating waveguide is introduced to a waveguide slot array to enhance the bandwidth of gain. A PMC termination associated with the symmetry of the feeding waveguide as well as uniform excitation is newly proposed for realizing dense and uniform slot arrangement free of high sidelobes. To exploit the bandwidth of the feeding circuit, the 4 2 4-element subarray is also developed for wider bandwidth by using standing-wave excitation. A 16 2 16-element array with uniform excitation is fabricated in the E-band by diffusion bonding of laminated thin copper plates which has the advantages of high precision and high mass-productivity. The antenna gain of 32.4 dBi and the antenna efficiency of 83.0% are measured at the center frequency. The 1 dB-down gain bandwidth is no less than 9.0% and a wideband characteristic is achieved. Index Terms—Crossover, diffusion bonding, double-layer, partially corporate feed, PMC termination, waveguide slot array.

I. INTRODUCTION IDE spectrums around the V-band and the E-band have been allocated in most countries worldwide. Demand for multigigabit wireless applications including the backhaul and distributed antenna systems for the mobile infrastructure, enterprise connectivity et al. is increasing [1]–[3]. The highgain, the high-efficiency and at the same time the wideband antenna is required in those millimeter-wave applications. Waveguide slot arrays, generally with multilayer feeding structures [4], [5] have the advantages of low-profile compared with horn and reflector antennas and low transmission loss even in the millimeter-wave band compared to microstrip [6] and triplate line antennas. However, the drastic cost reduction as well as feasible mass productive structure has been long awaited. Authors have developed various types of single-layer slotted waveguide arrays with simple structure and high mass-productivity. Those arrays have been successfully applied to relatively narrowband commercial services for satellite broadcast reception in 12 GHz band [7] and wireless IP access at home in 26 GHz band [8], etc. However, one weak point of these single

W

Manuscript received June 24, 2010; revised October 06, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The authors are with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8552, Japan (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122301

layer waveguide arrays with the common principle of traveling wave operation is that the bandwidth of antenna gain and reflection is limited by the long line effect resulting from the series feed structure. For wider bandwidth, the partially corporate and corporate waveguide feeding circuits in the same layer with the radiating elements [9], [10] have been developed at the expenses of enlarging the element spacing and the degradation in sidelobe. A multilayer antenna with a feeding circuit installed underneath the radiating waveguides deserves more expectations in terms of not only the design freedom of slot arrangement but also the realization of corporate feed. Here, a cost-effective, low loss fabrication technique for laminating multilayered structures in millimeter-wave band is essential. The number of layers is important parameters not only for antenna thickness, but also for fabrication cost and overall transmission loss. Furthermore, it is also crucial to antenna bandwidth and the design freedom of element arrangement. The electrical performance should be compromised carefully with the cost for individual fabrication techniques. In this paper, a double-layer partially corporate feeding structure to feed only two elements in series is introduced to enhance the bandwidth by reducing the long line effect in a series feed array antenna [11], [12]. A 16 16-element uniform array, which consists of 16 4 4-element subarrays, is realized demonstratively in the E-band as one of the promising compromise between the array performance and the cost, where the unique fabrication technique called “diffusion bonding of laminated thin metal plates” [13], [14] is introduced. The diffusion bonding is a process to realize stable surface bonding by applying plastic deformation and atom diffusion motion under the condition of high pressure and high temperain a protective atmosphere or vacuum. Etching ture of thin plates has the features of high precision around and low cost. The number of etching patterns is only five for the double-layer waveguide slot array [14]. Relatively long processing time of the diffusion bonding is not serious if a large number of antennas are processed simultaneously. In order to realize the dense and uniform slot arrangement over the whole aperture, a unique concept of perfect magnetic conductor (PMC) wall is introduced in between two subarrays. Additionally, the waveguide stair structure is arranged at four positions in the array to realize crossover of the feeding waveguides without increasing the array thickness. The double-layer with the waveguide stair for crossover is named as “quasi-double-layer” here.

0018-926X/$26.00 © 2011 IEEE

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2

Fig. 2. 4 4-element subarray with two elements in series and terminated by PCE or PMC walls.

2

Fig. 1. 16 16-element quasi-double-layer partially corporate feed array with PEC and PMC terminations.

The array adopts the corporate feed for those 16 subarrays and each subarray with 4 4 elements has wideband design with only two elements fed in series. The bandwidth is quite wide while the number of waveguide layers is less than three and the transmission loss is very small. The excellent array performance, that is the gain of 32.4 dBi, the efficiency of 83% and the 1-dB down gain bandwidth of 9%, is experimentally confirmed. II. ANTENNA STRUCTURE AND ARRAY CONFIGURATION Fig. 1 shows the configuration of a 16 16-element doublelayer waveguide slot array in a perspective view. By introducing a partially corporate feed structure, the whole antenna is decomposed into 16 4 4-element subarrays with only two elements in series. The whole antenna is fed by a standard waveguide from the bottom through an input aperture. The 16 subarrays are fed in-phase and in parallel with no frequency dependence, via an input aperture and a series of H-plane T-junctions. Since an in-phase fed array with regular slot arrangement has an advantage of suppressing the second-order beam [14], [15], all the radiating waveguides in the top layer are to be fed in-phase through center-inclined slot couplers [16], [17] in the bottom layer. The 16 16-element array with uniform excitation is to be realized for maximizing the directivity. The wideband design of the reflection suppression in the 4 4-element subarray illustrated in Fig. 2 is the key to realizing this double-layer partially corporate feed antenna. The termination of the last slot coupler in the 4 4-element subarray will be illustrated first in the following chapter. III. PROPOSAL OF PMC TERMINATIONS BY SYMMETRICAL WAVEGUIDE CONNECTION As shown in Fig. 3(a), the right-and-left two 4 4-element subarrays are fed in same amplitude and phase according to the

Fig. 3. Proposal of PMC termination by symmetrical waveguide connection in the partially corporate feed arrays. (a) Slot-free region due to PEC termination. (b) Direct open-circuited connection of two feeding waveguides.

partially corporate feed structure. Generally, the last slot coupler terminated by a perfect electric conductor (PEC) wall has a termination distance of half guided wavelength from a short circuit, which is identical to the period of the radiating waveguide in the transverse direction. A large slot-free region existing in between the adjacent subarrays results in the degradation of sidelobe levels (SLLs) especially in the transverse E-plane [9]. In Fig. 3, the arrowed circles indicate the magnetic field inside the feeding waveguides. The centered two arrowed circles in solid lines being in the same direction indicate that the magnetic fields at the terminations of the left-and-right subarrays are also in same amplitude and phase. The authors propose to remove the PEC walls and to join these two feeding waveguides as is illustrated by arrowed circles

ZHANG et al.: AN E-BAND PARTIALLY CORPORATE FEED UNIFORM SLOT ARRAY WITH LAMINATED QUASI DOUBLE-LAYER WAVEGUIDE

Fig. 4. Cross section view of the last slot couplers with PEC and PMC terminations. (a) PEC termination. (b) PMC termination.

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Fig. 6. Stair structure for the crossing with an underpass.

Fig. 5. Crossover of the feeding waveguides to avoid intersection and the last slot coupler with PMC termination and a stair. Fig. 7. Reflections of the designed waveguide stair structures for crossover.

in Fig. 3(b). Those two 4 4-element subarrays will be densely arrayed without any slot-free region while the inner field distribution as well as the operation of subarrays remains unchanged. In this open-connected structure, a virtual perfect magnetic conductor (PMC) wall is formed at the center according to the symmetrical arrangement of feeding waveguides, though the effects due to the asymmetrical arrangement of radiating slots are neglected here. Consequently, the virtue PMC is newly realized in the waveguide structure by symmetrical connection. The PMC termination is newly introduced to the last slot coupler with a reduced termination distance of one quarter guided wavelength. The structures of the last slot couplers with PEC and PMC termination are illustrated in Fig. 4(a) and 4(b), respectively. The key is that they are equivalent at the center frequency, because electromagnetically the short and open, i.e., PEC and PMC conditions will repeat with the interval of one quarter guided wavelength inside a shorted waveguide according to the principle of transmission lines. Those two last slot couplers will share the same structural parameters except for the terminations. The symmetrical feeding network as well as the uniform excitation is indispensable in the realization of PMC terminations in principle. If steeply tapered aperture illumination is aimed for lower SLL, the PMC condition breaks down in principle. In this general situation, a bit more complicated termination such as an impedance boundary rather than the PMC and PEC should be realized in relation to the specified amplitude taper. In addition, the termination distance must be optimized between one quarter and half guided wavelength. Furthermore, as illustrated in Fig. 1 the feeding waveguides would intersect at a right angle according to the partially corporate feed structure at four positions. Actually, a triple-layer structure was necessary to prevent this intersection. However, to save the total antenna thickness, as well as the fabrication cost, the heights of the feeding waveguides are reduced to less than half locally at the intersection as shown Fig. 5. The total antenna thickness is kept unchanged compared to the standard doublelayer antenna, even though the number of etching patterns is in-

creased from five to seven for realizing this quasi-double-layer antenna. The last slot coupler with PMC termination and a stair due to the crossover of feeding waveguide, is also included in Fig. 5. In total, three types of the last slot couplers are to be realized. As illustrate in Fig. 5, a waveguide stair structure is necessary to realize the crossing with an underpass at the crossover of feeding waveguides. Both the one-stair and two-stair structures and illustrated in Fig. 6 are investigated with comparison. are the dimensions of the original feeding waveguide. Stair heights and are fixed at the integral multiple of the etched metal plate thickness. In one-stair structure, stair width is optimized for the specified value of to match the impedance of . In two-stair the waveguide with the cross-section of structure, width and length of the middle stair which functions as a quarter-wave transformer are optimized for the specified values of , and . By comparison, two-stair structure show wider bandwidth than one-stair one as shown in Fig. 7 and will be adopted in realizing the partially corporate feeding circuit. The key point is that, the number of etching patterns is kept at seven for both structures and is not increased even in the realization of two-stair structure. IV. WIDEBAND DESIGN OF A 4

4-ELEMENT SUBARRAY

The wideband design in reflection of the 4 4-element subarray is the key to realizing the 16 16-element double-layer partially corporate feed antenna. The standing-wave excitation [18] will be adopted in the array design of slot couplers and radiating slots with only two elements in series. Moreover, the wide slots in both the feeding and radiating parts are adopted for wideband characteristics. A. Design of Three Types of Last Slot Couplers Three types of the last slot couplers with PEC termination, PMC termination and PMC termination with stair, whose relative locations are illustrated in Fig. 1, are designed with compar-

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Fig. 8. Reflections of the last slot couplers with PEC, PMC, and PMC+stair termination.

Fig. 10. Reflections of 4

2 4-element subarray with two elements in series.

Fig. 11. Overall reflection of the 16 Fig. 9. Standing-wave fed array with two elements in series. (a) Feeding part. (b) Radiating part.

ison. A three-dimensional electromagnetic field solver HFSS is applied in the design. The inclined angle of the coupling slot is fixed at 45 degrees for all the three structures to realize maximum coupling from a feeding waveguide to a radiating waveguide. The reflections are summarized in Fig. 8 for comparison. Very similar frequency characteristics are realized among them. Especially, the last slot couplers with PEC and PMC terminations share the same design parameters as mentioned above. The bandwidths of the reflection for all the three types are limited by coupling all incident power from the feeding waveguide to the radiating waveguide. By the way, the bandwidth of the travelling-wave fed 4 4 subarrays will also be limited for the same reason. B. Design of Standing-Wave Fed 4

4-Element Subarray

The standing-wave fed two elements in series for both the feeding and radiating parts in a 4 4-element subarray are illustrated in Fig. 9. The element spacing is fixed at half guided wavelength. Identical structural parameters are assumed to realize identical normalized impedances or admittances equal to 1/2 in the two elements for input matching. To enhance the bandwidth, the widths of coupling slots and radiating slots are enlarged to some extent. The parameters are determined by using HFSS. Standing-wave fed slot couplers with a stair are also realized. The widths of the feeding and radiating waveguides and , respectively. The waveguide heights are are in common. The widths of both the coupling and radiand , respectively. The slot thickating slots are nesses are in common. After the design of two-element arrays in both feeding radiating and parts in Fig. 9, they are combined into a 4 4-element subarray. The subarrays with two types of PMC terminations in the last slot couplers, that is with and without the stair, are

2 16-element array.

Fig. 12. Gain and directivity of the 16

2 16-element array.

analyzed by HFSS, while the radiating slot design is common for both. The overall reflections are compared in Fig. 10. Almost identical characteristics are achieved in the PMC terminated subarrays regardless of the stair. The bandwidth of reflection is enhanced largely by applying the stranding-wave excitation compared with the traveling-wave fed one. The characteristic of double resonance is achieved by detuning the resonances of the feeding and radiating parts toward a lower and higher frequency away from the center, respectively. V. DESIGN OF A 16

16-ELEMENT ARRAY IN THE E-BAND

The 4 4-element arrays designed above with the standingwave excitation, which show wideband characteristics in reflection, are combined together into a 16 16-element partially corporate feed uniform array in the E-band (60–90 GHz). The slot , , spacings in E- and H- planes are respectively. The array size is defined as . The full-structure analysis is conducted to the overall antenna by applying HFSS. The bandwidth of reflection for VSWR less than 1.5 is 5.7% as shown in Fig. 11. The frequency characteristic of the calculated directivity and gain are summarized in Fig. 12. The directivity of 32.92 dBi with the corresponding aperture efficiency of 93.4% is estimated at the center frequency. The antenna gain of 32.55 dBi and the antenna

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Fig. 13. Test antenna fabricated by diffusion bonding of laminated thin copper plates in the E-band.

Fig. 15. Aperture illumination measured at the center frequency. (a) Amplitude (dB). (b) Phase (deg).

VI. ANTENNA FABRICATION AND EVALUATION

Fig. 14. Etching patterns of copper plates in the quasi-double-layer antenna.

efficiency of 85.9% are also estimated by HFSS with the finite of copper taken into account. The conductivity calculated conductor loss is 0.27 dB.

The antenna designed above is fabricated by the diffusion bonding of laminated thin copper plates in the E-band. The center frequency of the antenna operation is 83.5 GHz. Fig. 13 shows the picture of the test antenna. It is fed by a standard waveguide WR12 connected at the bottom. The antenna size is and the array size is . The total antenna thickness is 8.9 mm. The upper 2.9 mm thickness is for the antenna part, while the lower 6 mm thickness with the cross section of WR-12 is only needed in measurement for screw connection. The etching patterns of the laminated copper plates in the quasi-double-layer antenna are presented in Fig. 14, where the thickness and the number of copper plates required in diffusion bonding is given for each etching pattern. Three etching patterns are necessary for the feeding waveguide with the crossover. Several small pieces shown with meshed lines are isolated from the frames; they have to be fabricated in another etching process individually. Nonetheless, simultaneous bonding of all etched plates is possible by using pins for alignment and fixing. The measured reflection as included in Fig. 11 is in agreement with the calculated one. The aperture illumination is observed in the near-field measurement system and is presented in Fig. 15. The directivity calculated from the measured near-field distribution is also included in Fig. 12. The directivity of 32.8 dBi with the corresponding high aperture efficiency of 90.1% is realized at the center frequency, where a good uniformity in excitation is achieved. The antenna gain measured by comparing a standard gain horn in an anechoic chamber of 32.4 dBi and the associated antenna efficiency of 83.0% are also given in Fig. 12, which agree beautifully with the prediction and imply that there is no additional loss resulting from the process of diffusion bonding. The good electric contact as well as stable surface

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The 1 dB-down gain bandwidth is 9.0%, where a wideband characteristic is realized. High cross polarization discrimination of is also experimentally confirmed for using wide radiating slots. The realization of an impedance boundary condition for low SLL design is remained as an issue in the future. REFERENCES

Fig. 16. Radiation patterns in principle E- and H-plane at the center frequency. (a) E -plane. (b) H -plane.

bonding is also confirmed in the E-band. The 1 dB-down gain bandwidth is 9.0% and a wideband characteristic is realized. The concept and the operation of the PMC termination need the rigorous structural symmetry, which could be disturbed by the fabrication error. A simple simulation shows that a standard occurring in the position of a coupling etching error of slot closest to the PMC termination would cause the breakdown of PMC condition but the power division error would be only 0.02 dB. This suggests that PMC condition is robust against the fabrication errors. It is also confirmed from a large number of antenna fabrications that the accuracy of the diffusion bonding process as well as that of etching is high enough and the degradation has not been observed up to 100 GHz. Fig. 16 shows the radiation patterns at the center frequency. and , reThe SLLs in the E- and H-plane are spectively. The half-power beam widths in the E- and H-planes are 3.8 and 4.2 degrees, respectively. The experimental results are in good agreement with the simulated ones in both cut at planes. High cross polarization discrimination of the neighborhood of boresight is experimentally confirmed, even though wide radiating slots with the width-to-length ratio at almost 1/2 are adopted.

[1] R. Leyshon, “Millimeter technology gets a new lease on life,” Microw. J., vol. 35, no. 3, pp. 26–35, Mar. 1992. [2] H. H. Meinel, “Commercial applications of millimeterwaves history, present status and future trends,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 884–339, Jul. 1995. [3] [Online]. Available: http://ieee802.org/15/index.html [4] R. C. Johnson and H. Jasik, Antenna Engineering Handbook, Sec. 9-6. New York: McGraw-Hill, 1993. [5] R. C. Hansen, Phased Array Antennas, Sec. 6.3. New York: Wiley, 1988. [6] M. A. Weiss, “Microstrip antennas for millimeter waves,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 171–174, Jan. 1981. [7] J. Hirokawa, M. Ando, and N. Goto, “A single layer slotted leaky waveguide array antenna for mobile reception of direct broadcast from satellite,” IEEE Trans. Veh. Technol., vol. 44, no. 4, pp. 749–755, Nov. 1995. [8] Y. Kimura, Y. Miura, T. Shirosaki, T. Taniguchi, Y. Kazama, J. Hirokawa, and M. Ando, “A low-cost and very compact wireless terminal integrated on the back of a waveguide planar array for 26 GHz band fixed wireless access (FWA) systems,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2456–2463, Aug. 2005. [9] M. Ando, Y. Tsunemitsu, M. Zhang, J. Hirokawa, and S. Fujii, “Reduction of long line effects in single-layer slotted waveguide arrays with an embedded partially corporate feed,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2275–2280, Jul. 2010. [10] T. Sehm, A. Lehto, and A. V. Raisanen, “A large planar 39-GHz antenna array of waveguide-fed horns,” IEEE Trans. Antennas Propag., vol. 46, pp. 1189–1193, Aug. 1998. [11] J. C. Coetzee, J. Joubert, and W. L. Tan, “Frequency performance enhancement of resonant slotted waveguide arrays through the use of wideband radiators or subarraying,” Microw. Opt. Technol. Lett., vol. 22, no. 1, pp. 35–39, Jul. 1999. [12] S. S. Oh, J. W. Lee, M. S. Song, and Y.-S. Kim, “Two-layer slottedwaveguide antenna array with broad reflection/gain bandwidth at millimetre-wave frequencies,” IEE Proc.-Microw. Antennas Propag., vol. 151, no. 5, pp. 393–398, Oct. 2004. [13] M. Zhang, J. Hirokawa, and M. Ando, Fabrication of a Slotted Waveguide Array at 94 GHz by Diffusion Bonding of Laminated Thin Plates IEICE Tech. Rep., AP 2008-35, 2008-6. [14] M. Zhang, J. Hirokawa, and M. Ando, “Design of a double-layer slotted waveguide array with a partially corporate feed circuit installed in the bottom layer and its fabrication by diffusion bonding of laminated thin plates in 38 GHz band,” in Proc. 2009 Int. Symp. Antennas Propag., Session: TB2.2, Bangkok, Thailand, Oct. 2009. [15] L. A. Kurtz and J. S. Yee, “Second-order beams of two-dimensional slot arrays,” IRE Trans. Antennas Propag., vol. 5, no. 4, pp. 356–362, Oct. 1957. [16] S. R. Rengarajan, “Analysis of a center-inclined waveguide slot coupler,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 5, pp. 884–339, May 1989. [17] S. R. Rengarajan, “Higher order mode coupling effects in the feeding waveguide of a planar slot array,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1219–1223, Jul. 1991. [18] Y. T. Lo and S. W. Lee, Antenna Handbook, Sec.12-4–12-6. New York: Van Nostrand Reinhold, 1988.

VII. CONCLUSION A double-layer partially corporate feed structure with only two elements in series is introduced to the waveguide slot array to enhance the bandwidth of antenna gain. A PMC termination originally realized by the symmetrical waveguide connection is proposed in the uniform excited partially corporate feed arrays. A 16 16-element array is fabricated by diffusion bonding of laminated thin copper plates in the E-band. As the experimental results, the antenna gain of 32.4 dBi with the high antenna efficiency of 83.0% is achieved at the center frequency.

Miao Zhang (S’05–M’09) was born in Liaoning, China, on June 5, 1979. He received the B.S., M.S., and D.E. degrees in electrical and electronic engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 2003, 2005, and 2008, respectively. From 2005 to 2008, he was a Research Fellow of the Japan Society for the Promotion of Science (JSPS). His current research interests include electromagnetic analysis and planar waveguide arrays. Dr. Zhang received the Best Letter Award in 2009 from Communication Society of IEICE Japan. He is a Member of IEICE.

ZHANG et al.: AN E-BAND PARTIALLY CORPORATE FEED UNIFORM SLOT ARRAY WITH LAMINATED QUASI DOUBLE-LAYER WAVEGUIDE

Jiro Hirokawa (S’89–M’90–SM’03) was born in Tokyo, Japan, on May 8, 1965. He received the B.S., M.S., and D.E. degrees in electrical and electronic engineering from Tokyo Institute of Technology (Tokyo Tech), in 1988, 1990, and 1994, respectively. He was a Research Associate from 1990 to 1996 and is currently an Associate Professor at Tokyo Tech. From 1994 to 1995, he was with the Antenna Group of Chalmers University of Technology, Gothenburg, Sweden, as a Postdoctoral Fellow. His research area has been in slotted waveguide array antennas and millimeter-wave antennas. Dr. Hirokawa received an IEEE AP-S Tokyo Chapter Young Engineer Award in 1991, a Young Engineer Award from IEICE in 1996, a Tokyo Tech Award for Challenging Research in 2003, a Young Scientists’ Prize from the Minister of Education, Cultures, Sports, Science and Technology in Japan in 2005, a Best Paper Award in 2007, and a Best Letter Award in 2009 from IEICE Communications Society. He is a Member of IEICE.

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Makoto Ando (SM’01–F’03) was born in Hokkaido, Japan, on February 16, 1952. He received the B.S., M.S., and D.E. degrees in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1974, 1976, and 1979, respectively. From 1979 to 1983, he was with Yokosuka Electrical Communication Laboratory, NTT, and was engaged in the development of antennas for satellite communication. He was a Research Associate with the Tokyo Institute of Technology from 1983 to 1985 and is currently a Professor. His main interests have been high-frequency diffraction theory such as physical optics and geometrical theory of diffraction. His research also covers the design of reflector antennas and waveguide planar arrays for DBS and VSAT. His latest interests include the design of high gain millimeter-wave antennas. Dr. Ando received the Young Engineers Award of IEICE Japan in 1981, the Achievement Award, and the Paper Award from IEICE Japan in 1993. He also received the 5th Telecom Systems Award in 1990, the 8th Inoue Prize for Science in 1992, the Meritorious Award of the Minister of Internal Affairs and Communications, and the Chairman of the Broad of ARIB in 2004, the Award in Information Promotion Month in 2006, and the Minister of Internal Affairs and Communications. He served as the Guest Editor-In-Chief of more than six special issues in IEICE, Radio Science, and the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He serves as the Chair of Commission B of URSI 2002–2005 and the member of Administrative Committee of IEEE Antennas and Propagation Society 2004–2006. He was the Chair of the Technical Committee of Electromagnetic Theory (2004–2005) and Antennas and Propagation (2005–2007) in IEICE. He was the General Chair of the 2004 URSI EMT Symposium in Pisa, Italy, and of the ISAP 2007 in Niigata. He was the 2007 President of Electronics Society IEICE and also the 2009 President of IEEE Antennas and Propagation Society. He served as the 2007–2009 Program Officer for the Engineering Science Group in Research Center for Science Systems, JSPS.

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Ultrawide Bandwidth 2 2 Microstrip Patch Array Antenna Using Electromagnetic Band-Gap Structure (EBG) Dalia Nashaat, Member, IEEE, Hala A. Elsadek, Senior Member, IEEE, Esmat A. Abdallah, Magdy F. Iskander, Fellow, IEEE, and Hadia M. El Hennawy, Member, IEEE

Abstract—Four types of EBG structures are proposed and used in the design of a patch antenna array to improve the bandwidth, gain and reduce the overall array size. The four ground plane designs for the 2 2 microstrip patch antenna array (MPAA) are; spiral artificial magnetic conductor (SAMC) ground plane, an SAMC embedded with a large spiral artificial magnetic conductor (LSAMC), an SAMC embedded with small spiral patch cells (SSAMC), and an SAMC embedded with small spiral mushroom-type electromagnetic band-gap patches (ESEBG). Simulation results show that each configuration has its advantages and limitations. For example while LSAMC provides better response in the array size reduction and improved bandwidth, SSAMC provides better response in reflection phase and hence higher gain. The ESEBG design provides better antenna gain and bandwidth. The achieved bandwidth of the 2 2 array antenna extends from 0.5 GHz to 20 GHz with 85% reduction in array size compared with conventional array with normal ground plane. The array gain increased from 6.5 to 10.5 dBi and the radiation patterns are all improved when using EBG structures. Index Terms—Compact size, electromagnetic band gap structure (EBG), large spiral AMC (LSAMC), microstrip patch antenna array (MPAA), small spiral AMC (SSAMC) and embedded spiral EBG (ESEBG), spiral artificial magnetic conductor (SAMC), ultrawide bandwidth (UWB).

I. INTRODUCTION HE use of high performance antenna is critically important in any transceiver system design and microstrip patch antenna array (MPAA) has been particularly popular for use in wireless applications. This is because MPAA has several advantages including light-weight, small size, low profile, planar configuration, conformal to host surface, and low fabrication cost [1]. In spite of these advantages, the MPAA suffers from certain limitations including low efficiency, low power handling ability, high Q, poor polarization purity, spurious feed radiation and very narrow bandwidth. Efforts are being pursued to improve the per-

T

Manuscript received August 13, 2009; revised February 25, 2010; accepted October 26, 2010. Date of publication April 05, 2011; date of current version May 04, 2011. This work was supported by NSF project INF11-001-013. D. Nashaat and M. F. Iskander are with the Hawaii Center for Advanced Communications, University of Hawaii at Manoa, Honolulu, HI 96822 USA (e-mail: [email protected]; [email protected]). H. A. Elsadek and E. A. Abdallah are with the Electronics Research Institute, Cairo, Egypt (e-mail: [email protected]; [email protected]) H. M. El Hennawy is with the Faculty of Engineering, Ain Shams University, Cairo, Egypt. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123052

formance of such an antenna array by various techniques. The development of an Ultra-wide bandwidth MPAA and the miniaturization of the array size present critically important design issues in wireless applications as these characteristics enable high data rates wireless connections using relatively lower power requirements [2]. From the array elements point of view, there has been considerable research effort focused on increasing the bandwidth of microstrip patch antennas (MPA). This includes controlling the dimensions and the dielectric constant of the substrate [3] and the use of aperture (or slot) coupled MPA [4]. These designs while provided some improvement in the bandwidth, they contributed difficulties in the feed system and in some cases compromised the desired low profile feature of this antenna. Recently, electromagnetic band-gap (EBG) materials have attracted much attention among researchers in the microwave and antennas communities. While generally known as photonic band-gap (PBG) structures with origin in the area of optics [5], they have now found wide variety of applications in developing components for microwave and millimeter wave devices, as well as in antenna designs [6]. EBG material, in general, is a periodic structure that forbids the propagation of electromagnetic surface waves within a particular frequency band called the band-gap. EBG structures also permit additional control of the behavior of electromagnetic waves in a different way from conventional guiding and/or filtering structures. EBG has the potential to provide a simple and effective solution to the problems of surface and leaky waves and various types of EBG structures have been studied to address these issues [6]. For example, it is shown that when a planar antenna is mounted onto an EBG substrate the overall radiation efficiency of the device is increased [7]. It is also shown that when a compact spiral EBG structure is used as part of a microstrip antenna array design, the performance of the array and its impedance matching characteristics are improved [8]. There are diverse forms of EBG structures and novel designs [9] but most of the research efforts in this area with the exception of [8] are focused on a single microstrip antenna element. In this paper, we focus on the use of different types of spiral EBG structures to improve the antenna array performance. Fig. 1 shows four types of the spiral EBG structures used in this study. Comparing to other EBG structures such as dielectric rods and holes, the proposed spiral structure has a unique feature of compactness, which is important in wireless communication applications. Specifically, in this study, we investigate the use of four shapes of spiral EBG [9] to help increase the bandwidth of a 2 2 MPAA. We choose the shape of a four-arm

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multi-arm spiral structure can create a large band gap. The low frequency operating point of the spiral is determined theoretically by the outer radius and is given by (1). The high frequency operating point is based on the inner radius and is given by (2) (1) (2)

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Fig. 1. Different prototype shapes of 2 2 MPAA with different EBG configurations (a) conventional array, (b) array with SAMC ground plane, (c) embedded (LSAMC) at h , (d) embedded (SSAMC) and (e) embedded (ESEBG) at h with vias in dielectric.

spiral AMC, Fig. 1(b), for complete cancellation of the antenna cross polarization. This four-arm AMC is then embedded with other EBG structures to further improve the performance of the 2 2 MPAA. First it is embedded with a large four-arms spiral (LSAMC) as shown in Fig. 1(c), then with a small spiral SSAMC patch cells with periodicity P as shown in Fig. 1(d), and finally with a mushroom-type EBG with spiral patches as shown in Fig. 1(e). Obtained results show that the LSAMC design improves the antenna bandwidth and reduces size, while the SSAMC improves the antenna reflection phase as well as reduces the antenna array size. The embedded spiral electromagnetic band gap structure (ESEBG) in Fig. 1(e), on the other hand, was found to improve the antenna array bandwidth and gain. Details of the specific design dimensions and the obtained simulation and experimental results are described in the following sections. II. SPIRAL GROUND PLANE CONCEPT AND CONFIGURATION The concept of spiral ground plane is like spiral antenna, it belongs to a category of broadband structures [8]. The term broadband is a relative measure of bandwidth, and it varies with the circumstances. and represent the upper and lower frequencies of operation for which satisfactory performance is obtained, hence broadband antenna can be characterized if the impedance and pattern of the antenna do not change significantly over about an octave or more. The spiral smooth physical structure tends to produce a radiation pattern and input impedance that are changing smoothly with the frequency change. In the spiral antenna case, the spiral radiates from a region where the circumference of the spiral equals to one wavelength. This is called the active region of the spiral [10]. The active region moves around the antenna with frequency. Since the geometry of a spiral is smooth, as frequency is reduced, the active region shifts to locations further towards the outer region of the spiral. Hence, self-scaling occurs and frequency-independent behavior results [7]–[9]. The four arms spiral shown in Fig. 1(b) is used as a ground plane rather than conventional ground shown in Fig. 1(a). The 50 transmission line approach is used to study the performance of the ground plane in each case. It is found that

where and are the inner and the outer radii, respectively and C is the speed of light. To improve the antenna array bandwidth, the air separation between spiral arms should be small while the number of spiral turns should be increased to obtain better band gap performance [9]. The width of each spiral branch is 2 mm . In contrast to the previous which is equivalent to designs, if this unit cell is rotated 90 , it can exactly recover itself. Therefore, this symmetrical condition guarantees the same scattering response for x-and y-polarized incident waves. As a result, no cross polarization is observed from this structure and compactness is, hence, achieved without generating any cross polarization. III. ELECTROMAGNETIC BAND-GAP STRUCTURE A typical 2 2 patch antenna array with conventional rectangular ground plane is shown in Fig. 1(a), where single patch length mm, width mm, with patch separation and substrate thickness mm. The dimensions of the rectangular ground plane are 50 50 mm . The material used for substrate is RT/D6010 . Fig. 1(b) represents the which has a dielectric constant first type of EBG with four arms spiral AMC ground plane [10]. The dimensions of the ground spiral arms were determined based on the optimization of the transmission coefficient response of the spiral, and for spiral arms rotated in the counter clockwise direction, was found to be equal to the separation between arms and mm. The variation of the transmission coefficients of these structures with frequency is shown in Fig. 2. Based on these and many other similar simulation results we were able to make the following observations: there is an optimum value of a/P and in this paper this ratio is chosen to be around 0.9, where a is the dimension of the embedded patches and P is its periodicity. This ratio is chosen for easy fabrication and does not affect the other optimum characteristics. It was also noted that there should be at least two cell patches under each of the array patch to have an effect on the current distribution under the array. Furthermore, it is observed that the embedded spiral patch cells/or vias under the feed are redistributing the current, hence resulting in an improved impedance matching as well as an increase in the array gain. Fig. 1(c) presents prototype array antenna with second shape of EBG, embedded large spiral AMC (LSAMC) in the middle of the substrate at mm with the previous four arms spiral ground plane. The dimensions of the embedded spiral turns is kept the same as in Fig. 1(b) but less number of spiral turns are centered under the 2 2 array antenna. The transmission coefficient response of this structure is also shown in Fig. 2. To further add improvement in antenna response, small cells of spiral patches (SSAMC) are added to the four arms spiral

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TABLE I CHARACTERISTICS OF VARIOUS ANTENNAS

Fig. 3. The reflection phase response of the four different EBG structures. Fig. 2. The transmission response for SAMC ground, embedded LSAMC, embedded SSAMC and embedded ESEBG.

ground plane as shown in Fig. 1(d). The small patch dimensions are equal to 6 6 mm , and the spiral arm width is equal to the gap separation mm. The periodicity P is equal to 6.5 mm. The transmission mm and the substrate height h coefficient response for this spiral structure design is also shown in Fig. 2. Finally, the four arms spiral was embedded with a mushroom-type EBG structure with spiral patches as shown in Fig. 1(e). The EBG structure has vias of radius mm and overall patch 6 mm with dimensions of spiral arm width and gap between arms mm. The transmission coefficient for this structure response is illustrated in Fig. 2. From Fig. 2, it may be noted that the embedded spiral EBG gives best transmission performance followed by embedded small cells spiral with AMC, then embedded large spiral AMC while the worst response is for the case of spiral arms AMC ground plane. The behavior of all pervious structures are studied from different points of view including MPAA bandwidth, which is the main object in this paper, reflection coefficient phase, gain and the radiation efficiency.

Specifically, to evaluate the reflection coefficient phase of the EBG surfaces, and since the array resides in the center of the ground plane, the normal incidence angle is used in the simulation as a simple approximation. The frequency band of the plane wave model is calculated according to refleccriterion is consistent with tion phase region [10]. The the perfect electric conductor (PEC) and perfect magnetic conductor (PMC) [10]. The PEC has a 180 reflection phase and the array antenna suffers from the reverse image current. The PMC surface has a 0 reflection phase and the array antenna does not match well due to the strong mutual coupling. From the results illustrated in Fig. 3, it may be noted that all EBG configurations used give reflection phase better than the array with conventional ground plane. In addition, the embedded SSAMC gives the best results followed by LSAMC and then ESEBG. The embedded EBG exhibits a reflection phase in the middle, hence a good return loss dB in average is obtained for the array antenna. The four-reflection phases for array antenna with four different EBG structures are studied and results are compared in Fig. 3. Table I, summarizes all results from the point of view of antenna array characteristics.

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Fig. 4. The comparison between measured and simulated reflection coefficient Of SAMC ground plane as shown in Fig. 1(b).

Fig. 5. The comparison between measured and simulated reflection coefficient of embedded LSAMC as shown in Fig. 1(c).

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Fig. 6. The comparison between measured and simulated reflection coefficient of embedded SSAMC as shown in Fig. 1(d).

Fig. 7. The comparison between measured and simulated reflection coefficient of embedded ESEBG as shown in Fig. 1(e).

IV. RESULTS AND DISCUSSION Starting with the conventional 2 2 MPAA with resonant frequency at 5.2 GHz, the array average gain is 6.5 dBi and the first harmonic appears at 7.8 GHz. When the conventional ground plane is replaced by four arms spiral AMC ground plane as shown in Fig. 1(b), the antenna size is reduced by 50% and the bandwidth extended from 2.5 to 19 GHz with bandwidth discontinuities in some sub-bands in the operating region as shown in Fig. 4. As may also be seen from Fig. 3, the number of zero reflection phases increased with the implementation of the spiral ground plane. Fig. 4 shows, the comparison between measured and simulated reflection coefficients for such an antenna array with AMC spiral ground. The average antenna gain increased to 7.8 dBi. For further improvement in antenna bandwidth and for further reduction in the antenna size, another embedded spiral with less mm number of arm turns (LSAMC) is added at height from the spiral ground plane. In this case, the antenna bandwidth extended from 1.25 to 20 GHz with decreased number of subband discontinuities as shown in Fig. 5. The antenna average gain is also increased to 8.8 dBi. When the small spiral patches (SSAMC) were added with mm to further improve the antenna perperiodicity formance, the bandwidth is extended in this case from 0.75 to

2

Fig. 8. Fabricated 2 2 microstrip patch array antenna (a) Conventional array, (b) 4 arms spiral AMC ground plane, (c) embedded SSAMC at substrate height h , (d) embedded SSAMC at substrate height h and (e) ground plane of ESEBG with vias.

20 GHz as illustrated in Fig. 6. The average antenna gain is increased to 9 dBi with about 3 dBi increase from the original gain value. Finally, the antenna gain is increased to 10.5 dBi and the bandwidth is extended from 0.5 GHz-3.5 GHz and from 4–19 GHz when using ESEBG structure as shown in Fig. 7. To help validate the simulation results, the conventional and the four different EBG configurations illustrated in Fig. 1(a)–(e) were fabricated as shown in Fig. 8 by using the photolithographic techniques. The measurements of reflection coefficients, reflection phase and radiation pattern are done

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Fig. 10. The E and H-plane radiation pattern for different EBG Structures at higher frequency f = 15 GHz.

Fig. 9. The E- and H-plane radiation pattern for different EBG Structures at lower frequency f = 1:5 GHz.

using E8364A vector network analyzer and indoor antenna echoic chamber. The E-plane and H plane radiation patterns were measured at two different frequencies for the different GHz and configurations. The lower frequency is the results are shown in Fig. 9 while the higher frequency is GHz and the obtained results are shown in Fig. 10. It is clear from both figures that the array antenna provides fairly acceptable radiation pattern. The embedded spiral EBG (ESEBG), however, provides the most improved radiation pattern performance. The array gain is also studied for the four different configurations. Fig. 11 shows antenna array gain versus frequency. From this figure it may be noted that all configurations give better antenna gain over the entire operating band than the conventional array over a perfectly conducting ground plane. Furthermore, it may be seen that the embedded spiral EBG gives the best performance followed by small spiral embedded AMC (SSAMC) then the large embedded spiral AMC (LSAMC). A maximum gain point 15.5 dBi is observed at 6.5 GHz for the ESEBG structure. These gain results are consistent with previously published data [12]. This could be attributed due to the fact that at this

Fig. 11. Array antenna gain-vs-frequency for different configurations.

frequency the EBG cell is resonating at same frequency and hence the reduction in the back radiations as shown in Fig. 12. V. CONCLUSION An ultra wide-bandwidth 2 2 planar microstrip antenna array is designed using different EBG designs to improve the array performance particularly the bandwidth and gain.

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highest gain. Simulations results are validated by experimental measurements. It is shown that the antenna array bandwidth is extended from 0.5 GHz to 20 GHz, the average antenna gain increased from 6.5 to 10.5 dBi, and acceptable phase reflection and radiation patterns in both E and H planes are achieved. REFERENCES

Fig. 12. The E- and H-plane radiation pattern for SEBG Structures at maximum gain point (15.5 dBi) at frequency 6.5 GHz.

Spiral artificial magnetic conductor (SAMC) ground plane alone and with embedded large and small spiral cells were designed and characterized using simulations and experimental verification. A fourth design which involves embedding the SAMC ground plane with a mushroom-type EBG structure with spiral patches was also investigated. Various applications such as image processing, mitigating multi-path problem and serving mobile satellite communications can benefit from the development of such ultra-wide bandwidth antennas array designs. Also, spiral EBG structure implementation in the design of microstrip antenna arrays reduces the size of the antenna array, enhances the reflection phase and increases antenna gain without sacrificing the bandwidth of the array antenna. Through simulations the advantages and limitations of each of these design options were identified. Specifically it is shown that the LSAMC design improves the antenna bandwidth and reduces size, while the SSAMC improves the antenna reflection phase as well as reduces the antenna array size. The embedded spiral electromagnetic band gap structure (ESEBG) in Fig. 1(e), on the other hand, was found to have the broadest bandwidth and

[1] A. Shakelford, K. F. Lee, D. Chatterjee, Y. X. Guo, K. M. Luk, and R. Chair, “Small-Size wide-bandwidth microstrip patch antennas,” in Proc. Antennas and Propagation Society Int. Symp., 2001, vol. 1, pp. 86–89. [2] FCC NEWS (FCC 02-48), Feb. 14, 2002. FCC News Release, new public safety applications and broadband internet access among uses envisioned by FCC authorization of ultra-wideband technology. [3] K. Anandan and K. G. Nair, “Compact broadband microstrip antennas,” Electron. Lett., vol. 22, no. 20, pp. 1064–1065, 1986. [4] W. Yun and Y. J. Yoon, “A wideband aperture-coupled microstrip array antenna using inverted feeding structures,” IEEE Trans. Antenna Propg., vol. 53, no. 2, pp. 861–862, Feb. 2005. [5] R. Gonzalo Garcia, P. de Maagt, and M. Sorolla, “Enhanced patchantenna performance by suppressing surface waves using photonicband-gap substrate,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 213 1–2138, Nov. 1999. [6] F. Yang and Y. Rahmat-Samii, “Applications of electromagnetic band-gap (EBG) structures in microwave antenna designs,” Microw. Millimeter Wave Technol., pp. 528–553, Aug. 2002. [7] L. Yang, M. Y. Fan, F. L. Chen, J. Z. She, and Z. H. Feng, “A novel compact electromagnetic band-gap structure and its applications for microwave circuits,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp. 183–190, Jan. 2005. [8] A. Yu and X. Zhang, “A novel 2-D electromagnetic band-gap structure and its application in micro-strip antenna arrays,” Microw. Millimeter Wave Technol. Lett., pp. 580–583, Aug. 2002. [9] D. Nashaat, H. A. Elsadek, E. Abdallah, H. Elhenawy, and M. F. Iskander, “Enhancement of ultra-wide bandwidth of microstrip monopole antenna by using metamaterial structures,” presented at the IEEE Int. Symp. on Antenna and Propagation Proceedings, Charleston, Jun. 2009. [10] F. Yang, V. Demir, D. A. Elsherbeni, and A. Z. Elsherbeni, “Enhancement of printed dipole antennas characteristics using semi-EBG ground plane,” J. Electromagn. Waves Appl., vol. 20, no. 8, pp. 993–1006, 2006. [11] G. Cakir and L. Sevgi, “Design of a novel microstrip electromagnetic band-gap (EBG) structure,” Microw. Opt. Technol Lett., vol. 46, pp. 399–401, 2005. [12] A. R. Weily, L. Horvath, K. P. Esselle, B. C. Sanders, and T. S. Bird, “A planar resonator antenna based on a woodpile EBG material,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 216–223, Jan. 2005.

Dalia M. Elsheakh (M’09) was born in Giza, Egypt, in 1976. She received the B.S., M.S., and Ph.D. degrees in electrical and communication engineering from Ain Shams University, Cairo, Egypt, in May 1998, Sept. 2004 and Oct. 2010 respectively. Her master’s thesis was about the design of microstrip PIFA antennas for mobile handsets. Her Ph.D. dissertation was entitled, “Electromagnetic Band-Gap (EBG) Structure for Microstrip Antenna Systems (Analysis and Design).” From 2000 to 2004, she was a Research Assistant and since 2004, she has been an Assistant Researcher with the Microstrip Department, Electronic Research Institute, Cairo. From 2008 to 2009, she was a Assistant Researcher in the Hawaii Center for Advanced Communications (HCAC), College of Engineering, University of Hawaii at Manoa, Honolulu. Since Oct. 2010, she has been an Assistant Professor with the Microstrip Department, Electronic Research Institute, Cairo. She has holds one patent, has published 15 papers in peer-refereed journals and 17 papers in international conferences in the area of microstrip antenna design. Her current research interests are in microstrip antennas theory and metamaterials design and electromagnetic wave propagation. She participates in many research projects at the national and international levels as Egypt-NSF-USA joint funds program and the European Committee Programs FP7 program, STDF, etc.

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Hala A. Elsadek (SM’11) graduated from the Faculty of Engineering, Ain Shams University, Cairo, Egypt, in 1991, and received the Master’s degree from the University of Gunma, Japan, in 1996, and the Ph.D. degree through a collaboration between Cairo University and the University of California, Irvine, in 2002. Currently, she is the Microstrip Department Head and an Associate Professor at the Electronics Research Institute, Cairo. Her research interests are in the field of wireless communications, electromagnetic engineering and microstrip antenna systems. She published three books and holds two patents in wireless communications and antenna systems. She acts as a single author and as a coauthor on more than 70 research papers in highly cited international journals and in proceedings of international conferences in her field, such as the IEEE TRANSACTIONS ON ANTENNA AND PROPAGATION, Microwave and Optical Technology Letters, etc. She participated in more than 10 research projects at the national and international levels as Egypt-NSF-USA joint fund program and the European Committee programs of FP6 and FP7 program. She is a supervisor on Master and Ph.D. theses in different universities in Egypt and abroad (Japan and USA). Dr. Elsadek was awarded the 2006 Egyptian Government Encouragement Prize for Young Scientists in Engineering Science and Technology. She is included in the Marquis Who’s Who Encyclopedia in 2008–2010 as one of the worldwide scientists who demonstrate outstanding achievement in her field. She is also a reviewer for many international societies in the area of antennas and propagation.

Esmat A. Abdallah graduated from the Faculty of Engineering and received the M.Sc. and Ph.D. degrees from Cairo University, Giza, Egypt, in 1968, 1972, and 1975, respectively. She was nominated as Assistant Professor, Associate Professor and Professor in 1975, 1980 and 1985, respectively. In 1989, she was appointed President of the Electronics Research Institute ERI, Cairo, Egypt, a position she held for about ten years. She became the Head of the Microstrip Department, ERI, from 1999 to 2006. Currently, is a the Microstrip Department, Electronics Research Institute, Cairo, Egypt. She has focused her research on microwave circuit designs, planar antenna systems and nonreciprocal ferrite devices, and recently on EBG structures, UWB components and antenna and RFID systems. She acts as a single author and as a coauthor on more than 127 research papers in highly cited international journals and in proceedings of international conferences in her field, such as the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, Microwave and Optical technology Letters, etc. She supervised more than 60 Ph.D. and M.Sc. theses in different universities. She participates in many research projects at the national and international levels as Egypt-NSF-USA joint funds program and the European Committee Programs FP7 program, etc. She is also a reviewer for many international societies.

Magdy F. Iskander (F’91) is the Director of the Hawaii Center for Advanced Communications (HCAC), College of Engineering, University of Hawaii at Manoa, Honolulu, Hawaii http://hcac.hawaii.edu. He is also a Co-director of the NSF Industry/University joint Cooperative Research Center between the University of Hawaii and four other universities in the US. From 1997–99 he was a Program Director at the National Science Foundation, where he formulated and directed a “Wireless Information Technology” Initiative in the Engineering Directorate. He spent sabbaticals and other short leaves at Polytechnic University of New York; Ecole Superieure D’Electricite, France; UCLA; Harvey Mudd College; Tokyo Institute of Technology; Polytechnic University of Catalunya, Spain; University of Nice-Sophia Antipolis, and Tsinghua University, China. He authored a textbook Electromagnetic Fields and Waves (Prentice Hall, 1992; and Waveland Press, 2001); edited the CAEME Software Books, Vol. I, 1991, and Vol. II, 1994; and edited four other books on Microwave Processing of Materials, all published by the Materials Research Society, 1990–1996. He has published over 200 papers in technical journals, holds eight patents, and has made numerous presentations in International conferences. He is the founding editor of the journal Computer Applications in Engineering Education (CAE). His research focus is on antenna design and propagation modeling for wireless communications and radar systems, and in computational electromagnetic. Dr. Iskander was the 2002 President of the IEEE Antennas and Propagation Society, and was a member of the IEEE APS AdCom from 1997 to 1999, and 2003–2006. He was the General Chair of the 2000 IEEE AP-S Symposium and URSI Meeting, and the 2003, 2005, 2007, and 2010 IEEE Wireless Communications Technology Conferences in Hawaii. He was also a Distinguished Lecturer for the IEEE AP-S (1994–97) and during this period he gave lectures in Brazil, France, Spain, China, Japan, and at a large number of US universities and IEEE chapters. He is a IEEE Fellow 1991. He received the 2010 University Of Hawaii Board Of Regents’ Medal for Excellence in Teaching and the University of Utah Distinguished Teaching Award in 2000. He also received the 1985 Curtis W. McGraw ASEE National Research Award, 1991 ASEE George Westinghouse National Education Award, 1992 Richard R. Stoddard Award from the IEEE EMC Society. He was a member of the 1999 WTEC panel on “Wireless Information Technology-Europe and Japan,” and chaired two International Technology Institute panels on “Asian Telecommunication Technology” sponsored by DoD in 2001 and 2003. He co-edited two special issues of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION on Wireless Communications Technology, 2002 and 2006, and co-edited a special issue of the IEICE Journal in Japan in 2004.

Hadia M. El Hennawy (M’00) received the B.Sc. and the M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1972 and 1976, respectively, and the Ph.D. degree from the Technische Universitat Braunschweig, Germany, in 1982. In 1982, she returned to Egypt and joined the Electronics and Communications Engineering Department, Ain Shams University, as an Assistant Professor. She was nominated an Associate Professor in 1987 and then a Professor in 1992. In 2004, she was appointed as the Vice-Dean for graduate study and research. In 2005, she was appointed as the Dean of the Faculty of Engineering, Ain Shams University. She has focused her research on microwave circuit design, antennas, microwave communication and recently wireless communication. She has been the Head of the Microwave Research Lab since 1982. She has published more than 100 journal and conference papers and supervised more than 50 Ph.D. and M.Sc. students. Prof. El Hennawy was the Editor-in-Chief of the Faculty of Engineering, Ain Shams University, Scientific Bulletin from August 2004 to August 2005 and is a member of the Industrial Communication Committee in the National Telecommunication Regulatory Authority (NTRA), Educational Engineering Committee in the Ministry of Higher Education, and Space Technology Committee in the Academy of Scientific Research. She is deeply involved in the Egyptian branch activities.

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Green’s Function Based Equivalent Circuits for Connected Arrays in Transmission and in Reception Daniele Cavallo, Student Member, IEEE, Andrea Neto, Member, IEEE, and Giampiero Gerini, Senior Member, IEEE

Abstract—Connected arrays constitute the only family of antenna arrays for which the spectral Green’s Functions have been derived analytically. This formalism is extended here to the receiving case. When the arrays are assumed to be infinitely extended and periodically excited, rigorous equivalent networks can be derived to represent the electromagnetic field distribution in transmission and/or reception. These equivalent networks are based on Green’s functions, thus each components can be associated with a specific physical wave mechanism. Moreover, all components are evaluated analytically. The total power transmitted, received and/or scattered by a connected array with and without backing reflector is discussed. The full efficiency of an array with backing reflector is demonstrated and explained. Finally, measurements from a dual-band connected array prototype validate the equivalent circuit representation. Index Terms—Antenna array, equivalent circuit, Green’s function, printed dipole, receiving antenna.

I. INTRODUCTION

C

ONNECTED arrays constitute the only family of antenna arrays for which the spectral Green’s function (GF) has been derived analytically. For connected arrays of slots, the GF was presented in [1]–[3]. The extension of the analysis to connected arrays of dipoles was presented in [4], while the inclusion of frequency selective backing reflectors to obtain dual-band operation was discussed in [5]. All these GFs considered arrays operating in transmission (Tx). In this paper, the formalism is extended to receiving arrays, including the presence of loads. The extension to reception (Rx) cases allows the derivation of a novel equivalent circuit for the antenna array cells. Unwise use of the equivalent Norton or Thevenin circuits for antennas in reception [6] can lead to improper physical interpretation, as pointed out in [7]. In fact, it is erroneous to associate the equivalent Thevenin impedance with scattered or reradiated power, which would imply that, under conjugate matching condition, only half of the incoming power can be received by an antenna. Recently, [8] was the first to

Manuscript received April 29, 2010; revised October 21, 2010; accepted November 08, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. D. Cavallo and G. Gerini are with TNO, Defense, Security and Safety, 2597 AK The Hague, The Netherlands and also with the Faculty of Electrical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). A. Neto is with the Telecommunication Department, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123063

demonstrate analytically that all the power incident on an antenna array can be received if the antenna is supported by a backing reflector. Moreover, a few attempts at new equivalent circuits for antennas in reception have been recently arising [9], [10]. For phased arrays, the one retaining the most in depth physical insight was proposed in [11]. However, that circuit was not rigorously derived, accounted only for the energy included in the fundamental propagating Floquet wave. In this paper, a rigorous equivalent circuit for planar connected phased arrays is provided that can be used to quantify the scattered fields as well as the currents in the loads. This circuit is based on the Green’s function formulation and, as a consequence, it accounts for all Floquet waves, including those associated with reactive energy. The circuit can be easily used for all types of planar stratifications such as dielectric layers, backing reflectors and also integrated frequency selective surfaces (FSSs). The extension includes, on one hand, the introduction of generators directly proportional to the amplitude of the incoming plane waves rather than the voltage induced in open circuit condition and, on the other hand, an expansion of the antenna impedance in different components, each with a well defined physical meaning. The paper is structured as follows: the first part is theoretical and presents the derivation of the spectral distribution of the electric currents on the array, as a function of the incident electric field and the load impedance at the feeding gaps of the dipoles. In the second part, the equivalent circuit is presented and its applicability to evaluate the power absorbed by different antenna configurations is demonstrated. Finally, in the third part, measured results from a dual-band connected array demonstrator are used to validate the equivalent circuit representation. II. INTEGRAL EQUATION FOR CONNECTED ARRAY OF LOADED DIPOLES The connected array of dipoles in Fig. 1 is investigated. It is assumed to be composed by an infinite number of -oriented dipoles, of width , periodically spaced by along , in free space. The dipoles are fed at periodic locations displaced by by a transmission line of characteristic impedance . In the transmit case, the incident field propagates toward the array . In the realong the mentioned transmission lines ception case, the incident field is associated with an incoming . In both cases, on each -th feeding point the plane wave incident fields are equal in amplitude and with progressive phase to account for scanning or different directions of incidence. Note that a generic plane wave can be represented as

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latter is the electric field radiated by equivalent electric currents on the surface . Focusing only component of the electric field and the on the longitudinal electric currents, (1) can be expressed in integral form as

(2)

Fig. 1. Geometry of a 2-D connected array of dipoles in transmit and in receive.

where is the free space GF, , and are the -components of the incident field and the equivalent electric surface current density, respectively. The incident electric field is the total electric field on the array in absence of the dipoles: it assumes different forms in the transmit and receive cases. In the transmit case, the absence of the dipoles implies an array of transmission lines terminated in open circuit. Thus for each feeding point (see inset of Fig. 2). In the reception case the presence of the feeding lines should be considered as well. However, if only differential currents are allowed to propagate along the feeding lines, the incident field is unperturbed and can be as. sumed to coincide with the incoming plane wave, Common-mode currents on the feeding lines and their impact on the array performance have been discussed elsewhere [12], [13]. A. Separation of Variables

Fig. 2. Definition of gap region 6 and conductive region 6 in the array plane and equivalent planar problem.

superposition of transverse electric (TE) and transverse magnetic (TM) components, transverse with respect to the incidence . The TE and TM unit vectors are parplane: allel to and , respectively. This starting problem will be transformed into an equivalent completely planar one, in order to deal with planar boundary conditions (b.c.). The equivalent problem is obtained substituting the feeding transmission line with a distributed surface impedance that fills the gap region, as depicted in Fig. 2. Accordingly, b.c. are imposed on the tangential field components as follows:

With little loss of generality, the incident electric field along can be assumed to be expressed as the product of two functions and transverse variables, i.e., of the longitudinal , with . Also the electric current on each dipole, when the dipole width is small in terms of wavelength, can be assumed to be characterized by a separable functional dependence from the transverse and longitudinal dimen. In the following we will sions, assume that the electric current on different dipoles are related , where by is the transverse excitation law when the array is scanning to, defined as ward or receiving from the direction given by in Fig. 1. The transverse -dependence of the electric current in each dipole is assumed to verify the quasi-static edge-sin. This choice gularity, i.e., suggests to assume a surface impedance distribution as follows:

(1) where and represent the total magnetic field for and , respectively. This equation is valid on the entire , if we assume that the surface impedance array surface is a discontinuous function that is equal to zero on , and different from zero the conductive part of the dipoles on the gaps . , The total electric field can be expressed as i.e., the superposition of the incident and the scattered field. This

(3) so that, for every

, we can write

(4)

CAVALLO et al.: GREEN’S FUNCTION BASED EQUIVALENT CIRCUITS FOR CONNECTED ARRAYS IN TRANSMISSION AND IN RECEPTION

B. One-Dimensional Equation

with

When (2) is enforced on the axis of the zeroth dipole , it can be compactly expressed as a one dimensional integral equation in the space domain

(5)

where we used the notation

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, and

(6) dependence of is the spatial expression of the longitudinal the Green’s function, once the dependence from the transverse is accounted for. dimension C. Average Currents on the Gaps Equation (5) is a Fredholm integral equation of the second . In order kind for the unknown spatial current distribution to simplify the solution, it is convenient to assume that the gaps are small in terms of wavelength ( -gap generator/load) and thus the current in the gaps can be approximated with its average value

(10) is expressed as an infinite summaIn the last equation, tion over the transverse Floquet wavenumbers , due to periodicity in the transverse direction. is the spectral domain representation of the electric field GF of an is the zeroth electric source, in absence of the dipoles, and order Bessel function, corresponding to the Fourier Transform . (FT) of the transverse current distribution III. SOLUTION FOR PERIODIC ARRAYS AND THEVENIN CIRCUITS Equation (9) is characterized by a continuous spectral integration and remains valid also for a finite number of feeds along the longitudinal direction. In that case, the infinite sumis truncated to a finite number of terms. However, mation in when the array is assumed to be periodically excited by an infinite number of feeds, the currents in the gaps are related by , with . Accordingly, can be expressed as spectral sumthe spectral integration in . The solution for the current mation in spectrum is then obtained by equating each spectral component with a procedure similar to the one presented in [3], and the result is

(11) (7)

dipole is then The spatial current distribution on the derived by performing an inverse Fourier transform (FT)

(8)

(12)

and

D. Spectral Integral Equation Substituting (7) in (5) and expressing the equation in the spectral domain, the spatial convolution in the left hand side (LHS) becomes a spectral product, so that we obtain

It can be noted that the last two equations are not written in explicit form yet, as they depend on the unknown term . The explicit expressions can be obtained by substituting (12) in (8) in order to evaluate the average current on the gap with . This calculation, after simple but tedious algebraic manipulations, leads to

(13) where we introduced the infinite array antenna impedance . As shown in [4], (antenna admittance) is given by

(14) (9)

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Fig. 3. Equivalent Thevenin circuit of the 0-th element of the array in (a) transmit and (b) receive mode.

A. Transmit Case When the array is studied in transmission, the incident field can be assumed to be concentrated in the dipole gaps and uniformly distributed. The spectral expression for a -gap excitation can be written as

(15) where . When substituting this expression into (13), the electric curbecomes rent

(16) Thus, the equivalent circuit that can be used to describe the current in the feeding gaps of a transmitting connected array can be recognized to be the usual one, Thevenin-like [6], and are Thevenin equivalent in Fig. 3(a). Note that impedance and the voltage generator, just mathematical representations, thus these components should not be interpreted with a physical mechanism [7]. For instance, with reference to the feeding transmission lines in Fig. 1, in the condition of conjugate matching, there would be no real power dissipation in the impedance , as no power is lost in the feeding transmission line.

Thus, also in the receiving case the equivalent circuit that is derived by solving directly the integral equation for the current in the feeding gaps of a connected array can be recognized to be the one, Thevenin-like, typically used in reception is the Thevenin equivalent (see Fig. 3(b)). In this case, impedance, hence should not be blindly associated with scattered power. IV. EXPANSION OF THE THEVENIN CIRCUITS IN EQUIVALENT NETWORKS The equivalent circuits in Fig. 3 give an accurate description of the power that is absorbed in the loads (receive case) or the power radiated in free space (transmit case). However, these two circuits do not provide information about the interactions between the capacitive energy in the feeding gaps, the inductive energy surrounding the dipoles, and the propagating fields associated with the lower order Floquet waves. A particularly interesting case from an application point of view is the one in which only the fundamental Floquet mode is in propagation. This is a low frequency approximation and is valid as long as the array is well sampled. In this case, an expanded rigorous circuital representation that accounts for all the above mentioned interactions can be derived. This circuit representation can be obtained expanding the antenna admittance and isolating different terms with a well defined physical meaning. It is evident from (14) that the antenna admittance is expressed and, implicitly via (see as a double summation in (10)). Let us introduce a notation that implies summation over . The starting point to obnon specified scripts, i.e., tain the expanded equivalent circuit representation is the isolaof the term associated with tion in

B. Receive Case When the array is analyzed in reception, a plane wave excitation can be assumed. Since the -component of the incident field is periodic, with constant amplitude and linear phase (pro) over a unit cell, in the spectral domain it can portional to be expressed as

(20) can be further expressed as the series of two impedances by expanding the explicit expression of

(17) is the amplitude of the incident electric field and is 1 for and 0 otherwise. Substituting this expression of the incident field in (13) leads to where

(21) (18) where

(19)

Eventually, the input impedance of the antenna can be represented via three separate components, , and , arranged as in the circuit in Fig. 4. Each of these components has the following physical interpretation. 1) is the portion of the input impedance associated with the fundamental Floquet mode ;

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Fig. 4. Representation of the constitutive terms composing the antenna impedance.

thus, it accounts for the only fields which can propagate away or toward the array surface. is a term that accounts for the reactive energy (in2) ductance) localized in the surrounding of the dipole, mostly ; this term is in series depending on the dipole width with the fundamental mode impedance. accounts for the reactive (capacitance) energy 3) stored in the feeding gap, mostly depending on the gap ; it is in parallel to the previous two components. width This term has been previously introduced by these authors in [4].

Fig. 5. Circuital interpretation of the term of the antenna impedance associated ): equivalent impedance represenwith the fundamental Floquet mode (Z tation (a) and expanded transmission line model (b) of TE and TM modes.

Note that the higher order Floquet modes are purely imaginary and sum up as positive reactance (inductive) in the second term and positive susceptance (capacitive) in the third term. A. Fundamental Mode Equivalent Circuit Representation The fundamental mode component of the antenna impedance can be further expanded to account for the different components arising from the transversalization of Maxwell equations [14], as well as for the upper and lower half spaces . The spectral GF can be defined by the array plane represented as superposition of transverse electric (TE) and transverse magnetic (TM) contributions (see [15, Appendix])

(22) where are the voltage solutions, for unitary shunt current generator, of the TE, TM transmission lines, with characand ( is teristic impedances the free space characteristic impedance) and propagation con. Since the array is radiating in free space, the voltstant are equal to the impedances seen at the section ages of the generator. These impedances are obtained as parallel of two impedances representing the upper and lower media for the TE and TM modes, respectively (23) where the superscript can refer to TE or TM. Thus, the fundamental component of the input impedance becomes (24) with

(25)

Fig. 6. Transmission line model for the propagating part of the fundamental mode component of the impedance of the connected array in the presence of a backing reflector at distance h from the array.

This component of the input impedance can be represented has in circuital form as in Fig. 5(a), where the transformer been included that accounts for the geometrical parameters of the dipole and the cell. An explicit representation that includes the transmission lines is depicted in Fig. 5(b), where two more transformers, and , are included to account for the TE and TM portion of the field radiated by an electric current oriented along , respectively. When the array is radiating . Different planar in free space, stratifications, dielectric or metallic, only alter the portion of and the equivalent circuit following the sections . For instance, when a backing reflector at a distance is included in the antenna geometry, the transmission line representing the lower half space becomes a short circuit stub, (see Fig. 6). thus V. THE EQUIVALENT CIRCUIT IN RECEPTION The explicit circuit in Fig. 5(b) has been obtained for connected arrays by expanding the analytical expression of the antenna impedance, which has a physical meaning only in transmission. In reception it is just a Thevenin equivalent impedance. However, our circuit expansion can also be used in reception as in Fig. 7. This is explicitly shown in the Appendix, where the admittance matrix characterizing the array as a transition at (see Fig. 8) is evaluated analytically and proved to be equivalent to the circuit in Fig. 7.

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Fig. 7. Explicit equivalent circuit of a connected array unit cell.

Fig. 9. Elements of the admittance matrix calculated with the equivalent circuit and HFSS for broadside incidence.

Fig. 8. Multimode equivalent network of a well sampled connected array in free space.

Since all plane waves can be expressed as superposition of TE and TM waves, the circuit in Fig. 7 can be used to evaluate the reflection, the absorbtion, the transmission as well as the induced cross polarization of the array under plane wave incican be expanded as a feeding dence. Note also that the load transmission line, as in the inset, if needed.

Fig. 10. Elements of the admittance matrix calculated with the equivalent cirand  . cuit and HFSS for incidence at '

= 45

= 45

A. Numerical Validation of the Equivalent Circuit in Reception We now compare the admittance matrix elements (Fig. 8) obtained using our analytical circuit representation with results from full-wave simulations performed via Ansoft HFSS. Fig. 9 shows this comparison for normal incidence, while Fig. 10 refers to . The cell dimensions are , with being the wavelength in free space at . The 10 GHz. The dipole width and the gap are both plots show stable curves over two decades (1:100) frequency bandwidth, highlighting once again the very broadband characteristic of a connected array structure. Very good agreement can be observed between the analytical and simulated results over most of the band. The discrepancy at the lowest frequencies is due to the non convergence of the HFSS solution when dealing with very small electrical dimensions of both the simulation domain and the absorbing boundary volume. Our analytical procedure maintains full accuracy also in the low frequency regime. In order to avoid this low-frequency breakdown in HFSS, a larger periodic domain has to be defined, meshed at lower frequency and including more unit cells (Fig. 11). Eventually, also HFSS converged to our solution. B. Power Absorbed and Scattered by Connected Arrays The equivalent circuit in Fig. 7 is particularly convenient if one is interested in estimating the amount of power that is re-

Fig. 11. Elements of the admittance matrix for broadside incidence calculated via HFSS, with two different size of the simulation domain and two different meshes.

ceived or scattered by the array. Fig. 12 shows the power absorbed and backscattered by connected arrays in free space evaluated using the present formulation. The cell dimensions are , and dipole width and gap are . , which would match the The load is assumed to be array in transmission over an infinite relative bandwidth [2]. It is apparent that an array in free-space can only receive half of the incident power, as the level of absorption in the load is about dBs. The remaining half of the power is scattered equally in the lower and higher half spaces, resulting in a reflected power

CAVALLO et al.: GREEN’S FUNCTION BASED EQUIVALENT CIRCUITS FOR CONNECTED ARRAYS IN TRANSMISSION AND IN RECEPTION

Fig. 12. Absorbed and reflected power in the case of a connected array in free space receiving from broadside (Z = 188 ).

Fig. 15. Picture of the dual-band 32 of dipoles.

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2 32 elements prototype connected array

VI. CONSEQUENCE: RCS OF ANTENNAS AS MEASURE OF TX MATCHING

Fig. 13. Absorbed and reflected power in the case of a connected array with backing reflector, receiving from broadside (Z = 377 ).

Fig. 14. Equivalent circuit for plane wave incidence (Rx) or transmission (Tx) in the main planes.

of dBs. Even if not reported in the graph, the transmitted . power is equal to the reflected power When the same array is operating in the presence of a backing , the best matching load in reflector at a distance [4]. At the frequencies for which the transmission is distance from the ground plane is about a quarter wavelength, dB) and a see Fig. 13, the power absorption is almost total ( reflection lower than dBs is observed. One can further observe that in some cases, and only for backed connected arrays, the self scattering coefficients of the incoming plane wave can be equal to the reflection coefficient of the array analyzed in transmission. This is easy to verify for obor , servation in the main planes. In fact, when and is equal to zero, so one of the two transformer the circuit becomes the 2-port network in Fig. 14. and are purely In this circuit, the impedances reactive, thus they do not dissipate power. The impedance is also a pure reactance when a backing structure is included. Thus in this case, the circuit is a lossless two port . network, for which

An important finding of this article is that, when scanning on the main ( and ) planes, the Radar Cross Section (RCS) of the loaded antenna, that includes a backing reflector, is identically equal to the matching in transmission. As a consequence, in order to evaluate the transmission matching properties of the radiating part of a connected array, one can resort to RCS measurements, which can be interpreted as self-scattering parameters. The advantage is that a simple and planar representative prototype can be manufactured without the inclusion of the lossy and expensive feeding network, but loading the planar array with matched resistors eventually physically implementing the analysis configuration of Fig. 2 rather than Fig. 1. This will on one hand save important costs because of the minimum number of elements that constitute a representative wideband connected array. On the other hand, this procedure allows isolating the characterization of the desirably radiating part of the antenna from the spurious radiation effects due to the feeding lines. Here, the equivalent network presented in Section IV.A is used to interpret the results of a dual-band prototype demonstrator that has been manufactured to validate the design procedures presented in [4] and [5] (Fig. 15). The demonstrator represents a receiving connected array composed of 32 32 elements designed to be well matched in Tx on two separate frequency bands, 8.5–10.5 GHz for radar and 14.40–15.35 GHz for Tactical Common Data Link (TCDL). The antenna RCS, normalized to the RCS of a metal plate with the same physical dimension of the antenna, has been measured. The performance of the array in terms of matching and efficiency has been evaluated for two angles: broadside and 45 in the -plane. Fig. 16 shows a comparison between the measurements and the calculated active reflection coefficient of a connected dipole in infinite array configuration. HFSS simulation are also reported for comparison. In the GF based equivalent circuit the stratification of the backing structure has been modelled using the same procedure used in [5], building up the GF from the equivalent transmission line in Fig. 17, which also shows an exploded view of the unit cell. A qualitatively good agreement between the experimental and theoretical results can be observed, and a matching dB is achieved in the entire below the threshold value of scan range in both the radar and TCDL bands. The discrepancy

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Fig. 16. Reflection coefficients obtained via equivalent circuit method and HFSS compared with measured reflection coefficient, for (a) normal incidence and (b) oblique incidence (45 ) on the E -plane. Operative bands are highlighted in gray.

Fig. 17. Stratification of the unit cell and equivalent transmission line of the backing structure.

is mostly due to the neglect in the calculations of the parasitic capacitance typical of the used resistors (PCF0402). The inclusion of this capacitance [16] in both the simulation and the analytical model provides a better agreement with the measurements (see Fig. 18). This agreement between measured RCS and the equivalent network validates the design strategy presented in [5]. More importantly, it also validates the measurement strategy proposed in this paper for the matching of antennas transmitting in the main planes. VII. CONCLUSION Connected arrays are one of the most effective antenna solution to implement the transition between radiated waves and guided waves, over broad frequency bandwidths. The equivalent circuits presented in this paper provide, for the first time, an analytical and rigorous representation of the electromagnetic fields describing this transition, in the case of a periodically excited, infinite array. The equivalent circuit are evaluated analytically and provide an expansion of the standard circuit for receiving antennas that allows their use also to evaluate the scattered power. Assuming that the array is well sampled and only the fundamental Floquet waves are propagating, the TE and TM field components in the vicinity of the array are described by means of two transmission lines which also account for the presence of dielectric stratifications and/or frequency selective surfaces and/or backing reflector. Both these TE and TM waves contribute to the complete magnetic field in the periodic cell. Accordingly, ad hoc transformers

Fig. 18. As in Fig. 16, but including parasitic capacitance of the resistors.

weight the TE and TM equivalent transmission line, to obtain average magnetic field in the entire cell the total at the plane where the antenna terminals are located. The projection of this magnetic field into the feeding gap provides the electric currents in the terminal. This projection is represented

CAVALLO et al.: GREEN’S FUNCTION BASED EQUIVALENT CIRCUITS FOR CONNECTED ARRAYS IN TRANSMISSION AND IN RECEPTION

via another transformer. The near fields only contribute to the localized reactive energy and are represented via lumped loads. This equivalent network representation provides the same quantitative information of full wave numerical simulations, but much more physical insight, as each component is associated with a specific wave mechanism. Moreover the results have been applied to the analysis of the scattering and absorption of a real connected dipole array backed by a frequency selective ground plane. Thanks also to the fact that the array is large in terms of wavelength, the comparison between measured and equivalent network based simulations are outstanding. The RCS measurements in the main planes can be used to characterize the active matching of the radiating part of the antenna in transmission.

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dipoles. We can write the accessible field as the sum of two components

(29) Also the current can be expanded as sum of two components (as in (3) of [15]) weighted for the same coefficient

(30) Substituting the last two modal expansions into (27), we obtain

APPENDIX ADMITTANCE MATRIX EVALUATION: IEMEN APPROACH This Appendix reports the main algebraic steps that lead to the analytical expression for the elements of the admittance matrix representing the antenna as depicted in Fig. 8, using the same approach as in [15]. In order to validate the circuit in Rx, we will then demonstrate that the equivalent circuit in Fig. 7 gives the same admittance parameters. The boundary conditions are imposed as

(31) Since last equation is valid for all possible values of , we term at both sides, leading to two intecan equate the single gral equations

(26) We proceed as in [15], by expressing the total electric field as the superposition of two terms associated with radiated and localized contributions, respectively. We refer to these two contributions as accessible and non-accessible modes . The total localized electric field can be expressed in integral form by introducing the non-accessible portion of the Green’s function. Considering only the -component of electric field and current, (26) becomes

(32) . As done in [15], from the solutions of for (32), resorting to the explicit expressions, the impedance matrix representing the antenna can be calculated as

(33) (27)

where

is the free space non-accessible GF, defined as

and via . which relates The solution of the integral equation (32) can be obtained in the spectral domain with the same steps described in Sections II and III leading to an expression similar to (11)

(34) (28) Writing explicitly . For a periodic well samIn last equation, pled array of connected dipoles, the accessible modes are the (TE) and (TM) components of the radiated or incident plane wave (fundamental Floquet mode). The non-accessible contribution is instead associated with all the other higher order modes contributing to the reactive energy localized in proximity of the

leads, after few algebraic steps, to

(35)

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Fig. 19. Circuit representation of Y as function of Z and higher modal components of Z .

Fig. 21. Circuit transformation step for the calculation of the y

element.

Following the circuit steps as in Fig. 20 allows obtaining, , and Fig. 21 to obtain from the operative definition, . These expressions are the same as the analytical ones in (38). Analogous steps for are not reported for sake of brevity.

ACKNOWLEDGMENT Fig. 20. Circuit transformation step for the calculation of the y

element.

where . The expression of the vector electric field mode functions given by ([15])

(36) Accordingly, introducing (36) in (33), after a number of algebraic steps, the elements of the admittance matrix can be expressed as (37) where leads to

and

. Substituting (34) in (37)

(38) The equivalent admittance higher modal

is a simple function of components of

and

(39)

with defined as in (25). The 2 2 admittance parameters of give by (38) are consistent with the equivalent circuit proposed in Fig. 7, hence validating it. This can be easily verified recognizing in the analytical expressions of the terms that were highlighted in Section IV.A. can be then represented as in Fig. 19.

The authors would like to thank F. Nennie for his support in testing and measuring the prototype demonstrator.

REFERENCES [1] A. Neto and S. Maci, “Green’s function of an infinite slot printed between two homogeneous dieletrics—Part I: Magnetic currents,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1572–1581, Jul. 2003. [2] A. Neto and J. J. Lee, “Infinite bandwidth long slot array antenna,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 75–78, 2005. [3] A. Neto and J. J. Lee, “Ultrawideband properties of long slot arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 534–543, Feb. 2006. [4] A. Neto, D. Cavallo, G. Gerini, and G. Toso, “Scanning performances of wideband connected arrays in the presence of a backing reflector,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3092–3102, Oct. 2009. [5] M. Pasian, S. Monni, A. Neto, M. Ettorre, and G. Gerini, “Frequency selective surfaces for extended bandwidth backing reflector functions,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 43–50, Jan. 2010. [6] C. A. Balanis, Antenna Theory, Analysis and Design, 3rd ed. Hoboken: Wiley, 2005, p. 85. [7] S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio, 2nd ed. New York: Wiley, 1953, p. 564. [8] D.-H. Kwon and D. M. Pozar, “Optimal characteristics of an arbitrary receive antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3720–3727, Dec. 2009. [9] A. W. Love, “Comment: On the equivalent circuit of a receiving antenna,” IEEE Antennas Propag. Mag., vol. 44, pp. 124–125, Oct. 2002. [10] R. E. Collin, “Limitations on the Thevenin and Norton equivalent circuits for a receiving antenna,” IEEE Antennas Propag. Mag., vol. 45, pp. 119–124, Apr. 2003. [11] B. A. Munk, Finite Antenna Arrays and FSS. New York: Wiley, 2003, ch. 2. [12] S. G. Hay, J. , and D. O’Sullivan, “Analysis of common-mode effects in a dual-polarized planar connected-array antenna,” Radio Sci., vol. 43, Dec. 2008, RS6S04, doi:10.1029/2007RS003798. [13] D. Cavallo, A. Neto, and G. Gerini, “Printed-circuit-board transformers to avoid common-mode resonances in connected arrays of dipoles,” IEEE Trans. Antennas Propag., to be published. [14] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. New York: IEEE Press, 1994. [15] S. Monni, G. Gerini, A. Neto, and A. G. Tijhuis, “Multimode equivalent networks for the design and analysis of frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2824–2835, Oct. 2007. [16] R. S. Johnson, C. Wakeman, and W. Cuviello, “Frequency response of thin film chip resistor,” in Proc. 25th CARTS, Palm Springs, CA, Mar. 21–24, 2005, pp. 136–141.

CAVALLO et al.: GREEN’S FUNCTION BASED EQUIVALENT CIRCUITS FOR CONNECTED ARRAYS IN TRANSMISSION AND IN RECEPTION

Daniele Cavallo (S’09) received the M.Sc. degree (summa cum laude) in telecommunication engineering from the University of Sannio, Benevento, Italy, in 2007. Since January 2007, he is with the Antenna Group at TNO Defence, Security and Safety, The Hague, The Netherlands. He is currently Ph.D. a student in the Telecommunication Technology and Electromagnetics Group (TTE/EM), Eindhoven University of Technology, Eindhoven, The Netherlands. His research interests include the analysis and design of antennas, with emphasis on wideband phased arrays. Mr. Cavallo was co-recipient of the Best Innovative Paper Prize at the 30th ESA Antenna Workshop in 2008.

Andrea Neto (M’00) received the Laurea degree (summa cum laude) in electronic engineering from the University of Florence, Italy, in 1994 and the Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2000. Part of his Ph.D. was developed at the European Space Agency Research and Technology Center, Noordwijk, The Netherlands, where he worked for the antenna section for over two years. From 2000 to 2001, he was a Postdoctoral Researcher at California Institute of Technology, Pasadena, working for the Sub-mm wave Advanced Technology Group. From 2002 to January 2010, was a Senior Antenna Scientist at the TNO Defence, Security and Safety, The Hague, The Netherlands. In February 2010, he was appointed Full Professor of applied electromagnetism at the EEMCS Department, Technical University of Delft, The Netherlands. His research interests are in the analysis and design of antennas, with emphasis on arrays, dielectric lens antennas, wide band antennas, EBG structures and THz antennas. Prof. Neto was co-recipient of the 2008 H.A. Wheeler Award for the Best Applications Paper of the Year in the IEEE TRANSACTIONS ON ANTENNAS AND

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PROPAGATION. He was co-recipient of the Best Innovative Paper Prize at the 30th ESA Antenna Workshop in 2008. He was co-recipient of the Best Antenna Theory Paper Prize at the European Conference on Antennas and Propagation (EuCAP) in 2010. He presently serves as an Associate Editor of the IEEE Antennas and Wireless Propagation Letters (AWPL).

Giampiero Gerini (M’92–SM’08) received the M.Sc. degree (summa cum laude) and the Ph.D. degree in electronic engineering from the University of Ancona, Italy, in 1988 and 1992, respectively. From 1992 to 1994, he was Assistant Professor of electromagnetic fields at the same University. From 1994 to 1997, he was a Research Fellow at the European Space Research and Technology Centre (ESA-ESTEC), Noordwijk, The Netherlands, where he joined the Radio Frequency System Division. Since 1997, he has been with the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands. At TNO Defence Security and Safety, he is currently Chief Senior Scientist of the Antenna Unit in the Transceiver Department. In 2007, he was appointed as part-time Professor in the Faculty of Electrical Engineering, Eindhoven University of Technology, The Netherlands, with a Chair on Novel Structures and Concepts for Advanced Antennas. His main research interests are phased arrays antennas, electromagnetic bandgap structures, frequency selective surfaces and integrated antennas at microwave, millimeter and sub-millimeter wave frequencies. The main application fields of interest are radar, imaging and telecommunication systems. Prof. Gerini was co-recipient of the 2008 H. A. Wheeler Applications Prize Paper Award of the IEEE Antennas and Propagation Society, the Best Innovative Paper Prize of the 30th ESA Antenna Workshop in 2008, and of the Best Antenna Theory Paper Prize of the European Conference on Antennas and Propagation (EuCAP) in 2010.

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Rectangular Thinned Arrays Based on McFarland Difference Sets Giacomo Oliveri, Member, IEEE, Federico Caramanica, Claudio Fontanari, and Andrea Massa, Member, IEEE

Abstract—A new class of analytical rectangular thinned arrays with low peak sidelobe level (PSL) is introduced. The proposed synthesis technique exploits binary sequences derived from McFarland difference sets to design thinned layouts on a lattice of ( + 2) positions for any prime . The pattern features of the arising massively-thinned arrangements characterized by only ( +1) active elements are discussed and the results of an extensive numerical analysis are presented to assess advantages and limitations of the McFarland-based arrays. Index Terms—Difference sets (DS), McFarland sequences, peak sidelobe level (PSL), planar arrays, thinned arrays.

I. INTRODUCTION

A

RRAY systems for frequency-modulated continuous-wave (FMCW) radars and SAR applications usually have to exhibit different total main beam widths (TMBWs) in azimuth and elevation and low PSLs [1], [2]. To meet these requirements and provide suitable resolutions, large rectangular layouts are needed [1], [2]. Since large fully-populated rectangular arrangements can yield to unacceptable high costs, weight, power consumption, and feeding network complexity [3], [4], architectural solutions with a reduced number of elements over large apertures with satisfactory PSLs and TMBWs values are often preferred. Towards this end, thinning techniques are generally exploited [3], [4] even though their main drawback is a lower sidelobe control when compared to their filled counterparts [3], [4]. In order to overcome such a limitation, several approaches have been proposed including the random displacement of the array elements [5], [6], the dynamic programming [7], and the stochastic optimization [8]–[16]. In such a framework, analytical techniques seem to be promising tools because of their numerical efficiency and the PSL control [17], [18]. By exploiting the auto-correlation properties of binary sequences, such as difference sets (DSs) [17]–[19] or almost difference sets (ADSs) [20], [21], a regular and a-priori predictable behaviour of the sidelobes is guaranteed [22]. Unfortunately, only specific geometries and array sizes can be synthesized [18], [23], [21]. Despite the availability of quite large DS-ADS repositories [24]–[26], planar arrays

Manuscript received August 03, 2010; revised October 24, 2010; accepted November 02, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. The authors are with the ELEDIA Research Group, Department of Information Engineering and Computer Science, University of Trento, Povo 38050 Trento, Italy (e-mail: [email protected]; federico.caramanica@disi. unitn.it; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123072

based on DSs and ADSs are usually square [19], [21] or almost square [18], [21], while few examples of DS-based rectangular arrangements with different azimuth and elevation TMBWs are actually used1 [1], [18]. In this paper, thinned rectangular arrays based on McFarland sequences [27], which are a particular class of DSs,2 are analyzed for the first time to the best of the authors’ knowledge, and a suitable synthesis procedure based on a binary Genetic Algorithm (GA) [28] is proposed. It is worthwhile to point out that the exploitation of such a class of DSs enables the extension of the design approach proposed in [17], [18] to rectangular ( being a prime number) with layouts of size different azimuth and elevation TMBWs. The outline of the paper is as follows. Section II introduces McFarland sequences and their application to array thinning. Afterwards, the GA-based synthesis technique for designing McFarland arrays is presented (Section III) and a set of representative numerical results concerned with different apertures and thinning factors is provided (Section IV) to show features, potentialities, and limitations of the proposed thinning strategy. Finally, some conclusions are drawn (Section V). II. MATHEMATICAL FORMULATION Let us consider a two-dimensional regular lattice of positions spaced by and wavelengths along and , respectively. The array factor of a thinned arrangement defined over such a lattice is equal to [4] (1) and cosines. Moreover, quence [21]

being the direction is the McFarland binary thinning se-

(2) where is a prime number, is a McFarland , and DS [27] with indexes . Furthermore, and stand for the reminder of division by and , respectively. 1Following the approach discussed in [18], [29], a rectangular DS array of size

N 2N can be generated only if a 1D DS is available with length N = 2 0 1 such that N = 2 0 1 and N = N=N are coprime and greater than one. Accordingly, only 6 of such sequences exist for N < 30 corresponding to N = f15; 63;255;511;1023g [24], and only 3 of these exhibit strongly different azimuth and elevation TMBWs [i.e., (N 2 N ) = f(3 2 85);(3 2 341);(7 2 73)g]. 2McFarland sequences, likewise two-dimensional DSs [18] and unlike planar ADSs [21], exhibit a two-level autocorrelation function.

0018-926X/$26.00 © 2011 IEEE

OLIVERI et al.: RECTANGULAR THINNED ARRAYS BASED ON MCFARLAND DIFFERENCE SETS

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It is now worth noticing that several McFarland arrays can be generated for each value. From the McFarland generation technique in the Appendix, it follows that a distinct DS, , corre, (b) the sponds to (a) each value of the integer in vectors , and (c) set of the elements used for deriving . As a result, up to different McFarland sets can be generated for each prime . In different layturn, each McFarland set defines up to outs by performing cyclic shifts of the thinning matrix [18]

and being the shift indexes along the array axes. In conclusion, the total number of different McFarland arrangements generated for each turns out to be (3) where indicates the factorial. As for the power pattern, a McFarland array defined over a locations satisfies the following rectangular grid of sampling property [18]

(4) where of

is the two-valued periodic autocorrelation function [27] whose values are (5)

being the delta function [i.e., if and otherwise). As an example, Fig. 1(a) , while the corshows a McFarland array obtained for responding autocorrelation reported in Fig. 1(b). From (4) and (5), it follows that the samples of the power pattern of McFarland arrays are a-priori known. Moreover, it has been proved in [18] that they produce patterns with much lower PSL’s than are typical with cut-and-try random placement. More in detail, Monte Carlo simulations have shown that compared to a random (nonlattice) placement of elements on the same aperture, a DS array has an expected PSL improvement of [dB] [18]. In order to fully exploit the features of McFarland sequences for array thinning, a suitable synthesis procedure is presented in Section 3. III. MCFARLAND ARRAY SYNTHESIS PROCEDURE In order to find the optimal (i.e., with the lowest PSL) McFarland layout for every value, all deducible arrays

Fig. 1. McFarland Rectangular Arrays—Example of (a) a McFarland array and (b) the associated (two-level) autocorrelation function (P = 3).

should be, in principle, analyzed. Unlike other 2D DS-based thinned architectures [19], an exhaustive procedure is here computationally unfeasible due to the extremely wide number of values. As an example, more than layouts even for small McFarland arrays can be defined over ( —Table I). As a cona lattice of size sequence, a different and more efficient selection approach is mandatory to analyze the PSL properties of these arrangements for identifying the optimal layout. Towards this end, the problem of finding the optimal McFarlayouts for a given is land array among all existing recast as an optimization one where the fitness function to be minimized is defined as follows (6) where

(7) being the sidelobe region [19]. Because of the discrete nature of the descriptors of the McFarland sets [i.e., and for ], a binary GA-based approach [10], [28] is exploited. More specifically, the following procedure is iteratively applied —A randomly-chosen initial pop1) Initialization trial solutions (or individuals), ulation of is defined; encodes the values 2) Coding—Each individual of the McFarland integer descriptors

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MCFARLAND RECTANGULAR ARRAYS (P

TABLE I

 29)—FEATURES AND PERFORMANCE INDEXES

and into a binary string (or chromosome); 3) GA-Evolution—At each -th iteration, the genetic evolution takes places through selection, crossover, reproduction, mutation and elitism operators [10], [28] taking into account the fitness values of current trial solutions; 4) Termination—The iterative optimization terminates , is when the optimal fitness value, smaller than an user-defined threshold or when a maximum has been reached. Then, the number of iterations is fittest trial individual assumed as the “optimal solution” (i.e., the optimal setup for the McFarland descriptors). Otherwise, the iteration and goto 3. index is updated It is worth to point out that, unlike [8], [10], the objective of the GA procedure is here not to design an optimally thinned array, but the search of the fittest arrangement in terms of PSL among all available McFarland layouts for a given . IV. NUMERICAL RESULTS AND DISCUSSION This section is aimed at (a) numerically assessing the features and the potentialities of the McFarland rectangular layouts and (b) validating the GA-based synthesis approach for generating optimal PSL arrangements when dealing with both small and large apertures. The GA-based search has been applied with the following setup: cross-over probability equal to , maximum number of 0.7, mutation probability equal to iterations , population size . Moreover, has been assumed.3 The first numerical experiment is concerned with the McFarfor which an exhaustive analysis, land sequence with although computationally cumbersome, can be still performed in a reasonable amount of time. The plot of the PSL values of McFarland arrays inthe whole set of dicate that several DS layouts exhibit PSLs equal or very close 3It is worth remarking that, although deduced for a broadside steering, the final layouts will be optimal for s = s = 0:5 whatever the steering direction [thanks to (4)]. Moreover, since in most cases the highest secondary lobes appear near the mainlobe in DS planar arrays [17], such layouts are expected to represent the optimal ones also for most other steering directions and inter-element spacings.

Fig. 2. GA-Based McFarland Synthesis—Plots of (a) the PSL values of the whole set of McFarland arrays and (b) evolution of the PSL of the GA solution during the iterative (i being the iteration index) sampling of the McFarland solution space.

to the optimal one confirmed by the index

dB [Fig. 2(a)]. This is also given by

(8) and defined as the fraction of McFarland layouts that exhibit a (Fig. 3). PSL equal or below times the optimal value As a matter of fact, although the optimal configurations are —Fig. 3], a non-negliquite rare [ gible portion of the randomly-generated layouts exhibits a PSL

OLIVERI et al.: RECTANGULAR THINNED ARRAYS BASED ON MCFARLAND DIFFERENCE SETS

Fig. 3. McFarland Rectangular Arrays—Behavior of  2 f : ; : ; : ; : g.

07 08 09 10

1() versus P

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when

close to [e.g., ]. This suggests that the GA-based search method should quickly find a sub-optimal configuration, while a larger number of iterations may be required to actually reach convergence to the global optimum. Such a behaviour is pointed out by the plot of the evolution of the optimal GA solution within the solution space of McFarland arrays in Fig. 2(b) where the blue crosses identify the elements of the GA solution set at the -th GA iteration, while the red line is concerned with the overall (ordered) McFarland solution set as a function of the sequence index. Indeed, less than 300 iterations are sufficient to find a McFarland arrangedB, while the convergence is reached ment with after steps. Such an outcome confirms that the GA-based synthesis is able to effectively sample a large solution space finding the optimal McFarland layout characterized by a low PSL value despite only 12 active elements over a lattice of 45 positions [Fig. 1(a)]. Similar conclusions can be drawn from the analysis (non exhaustive, but limited to a percentage of the whole set of McFarand [Figs. 4(a) land configurations) carried out for and (b)], even though a faster convergence of the GA-search is expected when dealing with larger dimensions as suggested by [e.g., for vs. the values of for —Fig. 3]. This is further confirmed by the evolution of the GA solutions in Fig. 4. As a matter and iterations are necessary of fact, only [Fig. 4(a)] and to reach the convergence when [Fig. 4(b)], respectively. For completeness, Fig. 5 gives the corresponding arrangements and power patterns. As expected from DS theory, —Fig. 5(a); the optimal layouts at convergence [ —Fig. 5(c)] exhibit controlled and regular sidelobes —Fig. 5(b); —Fig. 5(d)] despite the massive [ for thinning ( for —Table I). Moreover, thanks to the McFarland distribution, the corresponding architectures give different resolutions in each angular domain as indicated by the vs. in locations of the first nulls of the beam pattern (see Table I).

Fig. 4. GA-Based McFarland Synthesis—Evolution of the PSL of the GA solution during the iterative (i being the iteration index) sampling of the McFarland solution space when (a) P and (b) P .

=5

=7

Fig. 5. GA-Based McFarland Synthesis—Optimal McFarland layouts (a), (c) and the corresponding power patterns (b), (d) when P (a), (b) and P (c), (d).

=5

=7

In order to assess the performances of McFarland thinned arrays also when impractical (for an exhaustive analysis) apertures are at hand, the next experiments are concerned . The results of the GA-based synwith and are provided in thesis when

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Fig. 6. GA-Based McFarland Synthesis—Optimal McFarland layouts when and (b) and P . (a) P

= 11

= 13

Fig. 8. Comparison with Standard GA-Thinned Rectangular Arrays—Optimal layout (a) and corresponding power pattern (b) obtained by GA when P ;Q , and K .

7

Fig. 7. GA-Based McFarland Synthesis—Power patterns of the optimal McFarland layouts deduced for (a) P and (b) P .

= 11

= 13

Figs. 6 and 7. Despite the decreasing thinning factor —Table I), ( and high sidelobe do not appear since (Table I). Moreover, the power patterns in Fig. 7 [ —Fig. 7(a); —Fig. 7(b)] show the sidelobe regularity expected from the two-level autocorrelation McFarland layouts notwithstanding the highly-sparse element —Fig. 6(a); —Fig. 6(b)]. distribution [

= 63

= 56

=

Previous conclusions can be also extended to wider —Table I). As it can be noMcFarland layouts ( ticed, low PSL values are obtained whatever the dimension for —Table I), ( despite the sharp reduction of the thinning factor ( —Table I). As a final numerical validation, a comparison between the performances of the best McFarland array and those of the best sparse array with the same size and thinning factor found by means of a traditional GA-based approach [28], [30] is provided4. More in detail, a state-of-the-art randomly initialized GA method (see [10], [30] for the implementation details) is employed for designing a thinned rectangular array of size with active elements. The stochastic optimization has been carried out by considering a GA population of and a crossover size 10, a mutation probability equal to probability of 0.7. The maximum number of GA iterations has [10], [30]. By comparing the performances been set to obtained by the GA-optimized layout [Fig. 8(a)] with those of the McFarland one [Fig. 5(c)], it turns out that the stochastically optimized architecture does not to reach a PSL value [Fig. 8(b)] as low as that of the proposed layout [Fig. 5(d)] dB vs. dB] aven [ though also non-DS layouts can be synthesized in the former case. Such a result is due to the size of the search space that ), has to be explored by the standard GA methodology (i.e., which is extremely larger than that defined by the McFarland —Table I). descriptors ( 4The GA methodology is applied by assuming standard “binary” descriptors of the geometry [10], [30], rather than the McFarland descriptors introduced above. As a consequence, the obtained design will not be a DS layout.

OLIVERI et al.: RECTANGULAR THINNED ARRAYS BASED ON MCFARLAND DIFFERENCE SETS

V. CONCLUSION In this paper, a new family of analytically-designed thinned arrays with different azimuth and elevation TMBWs has been proposed. Thanks to the properties of McFarland DSs, several massively thinned isophoric architectures have been deduced and the PSLs of the arising layouts, defined over grids of size ( being a prime number), have been numerically analyzed. Towards this end, a GA-based search procedure has been exploited due to the extremely large number of admissible McFarland sequences. The numerical results point out the following issues • the design of McFarland arrays is highly efficient whatever , since up to layouts can be obtained by simply selecting the associated descriptors , and for ; • unlike traditional binary encodings used for thinned array designs [28], the GA-based procedure is able to more efficiently identify optimal McFarland layouts thanks to the discrete nature of the McFarland descriptors and also the large number of optimal solutions available within the search space (Fig. 3); • despite the extremely low number of active elements , McFarland arrays exhibit well-controlled sidelobes especially for large dimensions. This suggests their exploitation for the design of extremely light large arrays as well as of architectures with interleaved functionalities (e.g., multi-function radar arrays in which each function correspond to a highly sparse sub-array [1]). Further studies will be devoted to analyze the effects of the presence of real array elements and/or mutual coupling. Furthermore, it is still a work in progress the exploitation of McFarland sequences for designing interleaved architectures. APPENDIX In this section, a procedure for the generation of a McFarland is presented. Sets Let be a prime number and let us define , and . and choose Select an integer necessarily different) vectors with . , let For every as follows: determine the set

where

is a randomly picked element in

(not

and

.

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From [27], it follows that ) with indexes . and

is a McFarland DS (i.e., ,

REFERENCES [1] I. E. Lager, C. Trampuz, M. Simeoni, and L. P. Ligthart, “Interleaved array antennas for FMCW radar applications,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2486–2490, Aug. 2009. [2] M. Skolnik, Radar Handbook, 3rd ed. New York: McGraw Hill, 2008. [3] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [4] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Norwood, MA: Artech House, 2005. [5] Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propag., vol. 12, no. 3, pp. 257–268, May 1964. [6] B. Steinberg, “The peak sidelobe of the phased array having randomly located elements,” IEEE Trans. Antennas Propag., vol. 20, no. 2, pp. 129–136, Mar. 1972. [7] M. I. Skolnik, G. Nemhauser, and J. W. Sherman, “Dynamic programming applied to unequally-space arrays,” IRE Trans. Antennas Propag., vol. 12, pp. 35–43, Jan. 1964. [8] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, July 1994. [9] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna array using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, pp. 2210–2217, Sept. 2003. [10] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, Apr. 2004. [11] T. G. Spence and D. H. Werner, “Thinning of aperiodic antenna arrays for low side-lobe levels and broadband operation using genetic algorithms,” in Proc. IEEE Antennas Propagat. Int. Symp., Jul. 9–14, 2006, pp. 2059–2062. [12] A. Trucco, E. Omodei, and P. Repetto, “Synthesis of sparse planar arrays,” Electron. Lett., vol. 33, no. 22, pp. 1834–1835, Oct. 1997. [13] A. Trucco, “Thinning and weighting of large planar arrays by simulated annealing,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 46, no. 2, pp. 347–355, Mar. 1999. [14] A. Trucco, “Weighting and thinning wide-band arrays by simulated annealing,” Ultrasonics, vol. 40, no. 1–8, pp. 485–489, Mar. 2002. [15] A. Lommi, A. Massa, E. Storti, and A. Trucco, “Side lobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett., vol. 32, no. 3, pp. 194–196, 2002. [16] M. Donelli, S. Caorsi, F. De Natale, D. Franceschini, and A. Massa, “A versatile enhanced genetic algorithm for planar array design,” JEMWA, vol. 18, pp. 1533–1548, 2004. [17] D. G. Leeper, “Thinned Periodic Antenna Arrays With Improved Peak Sidelobe Level Control,” U.S. patent 4071848, Jan. 31, 1978. [18] D. G. Leeper, “Isophoric arrays—Massively thinned phased arrays with well-controlled sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825–1835, Dec. 1999. [19] L. E. Kopilovich, “Square array antennas based on Hadamard difference sets,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 263–266, Jan. 2008. [20] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference sets,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3800–3812, Dec. 2009. [21] G. Oliveri, L. Manica, and A. Massa, “ADS-based guidelines for thinned planar arrays,” IEEE Trans. Antennas Propag, vol. 58, no. 6, pp. 1935–1948, June 2010. [22] G. Oliveri, L. Manica, and A. Massa, “On the impact of mutual coupling effects on the PSL performances of ADS thinned arrays,” PIER B, vol. 17, pp. 293–308, 2009. [23] M. Donelli, A. Martini, and A. Massa, “A hybrid approach based on PSO and Hadamard difference sets for the synthesis of square thinned arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2491–2495, Aug. 2009. [24] La Jolla Cyclic Difference Set Repository [Online]. Available: http:// www.ccrwest.org/diffsets.html [25] ELEDIA Almost Difference Set Repository [Online]. Available: http:// www.eledia.ing.unitn.it [26] G. Oliveri, M. Donelli, and A. Massa, “Genetically-designed arbitrary length almost difference sets,” Electron. Lett., vol. 5, no. 23, pp. 1182–1183, Nov. 2009.

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[27] R. L. McFarland, “A family of difference sets in non-cyclic groups,” J. Combin. Theory, vol. 15, no. 1, pp. 1–10, July 1973. [28] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: Wiley, 2007. [29] F. J. MacWilliams and N. J. A. Sloane, “Pseudorandom sequences and arrays,” Proc. IEEE, vol. 64, pp. 1715–1729, Dec. 1976. [30] G. Oliveri and A. Massa, “Genetic algorithm (GA)-enhanced almost difference set (ADS)-based approach for array thinning,” IET Microw. Antennas Propag., vol. 5, pp. 305–315, Feb. 2011.

Claudio Fontanari was born in Trento, Italy, in 1976. He received the B.S. degree in mathematics from the University of Trento, in 1999 and the Ph.D. from the “Scuola Normale Superiore di Pisa” Italy, in 2002. He is currently an Assistant Professor of Geometry in the Department of Mathematics, University of Trento, Italy. His main interests are in algebraic curves and their moduli, geometry and topology of moduli spaces, geometrical aspects of multimedia and computer graphics.

Giacomo Oliveri (M’09) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in space sciences and engineering from the University of Genoa, Italy, in 2003, 2005, and 2009 respectively. Since 2008, he is a member of the Electromagnetic Diagnostic Laboratory, University of Trento, Italy. His research work is mainly focused on cognitive radio systems, electromagnetic direct and inverse problems, and antenna array design and synthesis.

Andrea Massa (M’97) received the “Laurea” degree in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields at the Department of Biophysical and Electronic Engineering, University of Genoa, teaching the university course of Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor at the University of Trento. Since 2005, he has been a Full Professor of electromagnetic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the Director of the ELEDIALab, University of Trento and Deputy Dean of the Faculty of Engineering. Since 1992, his research work has been focused on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the IEEE Society, of the PIERS Technical Committee, of the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB) and Italian representative in the general assembly of the European Microwave Association (EuMA).

Federico Caramanica was born in Trento, Italy, in 1984. He received the B.S. and M.S. degrees in telecommunications engineering from the University of Trento, Italy, in 2006 and 2008, respectively. He is currently a Ph.D. Student at the ICT School, University of Trento, Italy, where he is also a member of the ELEDIA Research Group. His main interests are in antenna arrays, MIMO communication systems and wave propagation in urban environment.

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Incremental Fringe Formulation for a Complex Source Point Beam Expansion Stefano Mihai Canta, Member, IEEE, Danilo Erricolo, Senior Member, IEEE, and Alberto Toccafondi, Senior Member, IEEE

Abstract—An incremental fringe formulation (IFF) for the scattering by large metallic objects illuminated by electromagnetic complex source points (CSPs) is presented. This formulation has two main advantages. First, it improves the accuracy of physical optics (PO) computations by removing spurious scattered field contributions and, at the same time, substituting them with more accurate Incremental Theory of Diffraction field contributions. Second, it reduces the complexity of PO computations because it is applicable to arbitrary illuminating fields represented in terms of a CSP beam expansion. The advantage of using CSPs is mainly due to their beam-like properties: truncation of negligible beams lowers the computational burden in the determination of the solution. Explicit dyadic expressions of incremental fringe coefficients are derived for wedge-shaped configurations. Comparisons between the proposed method, PO and the Method of Moments (MoM) are provided. Index Terms—Complex source point, diffraction, geometrical theory of diffraction, incremental fringe formulation, incremental theory of diffraction, method of moments.

I. INTRODUCTION

A

new solution to increase the accuracy and improve the numerical efficiency of PO based methods to estimate fields scattered by large structures is presented. The solution consists of an IFF of the field scattered by edges, or shadow lines, in perfect electrically conducting (PEC) objects when they are illuminated by an arbitrary field, which is represented by a CSP expansion. This work is motivated by the need to efficiently describe directional wave fields in many practical electromagnetic (EM) antenna and scattering problems. In fact, in these problems, a distribution of CSP on a closed regular surface [1]–[3], or collapsed to a single real point, provides an efficient representation

Manuscript received February 23, 2010; revised September 22, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the U.S. DoD/ AFOSR Grant FA9550-05-1-0443. S. M. Canta was with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA. He is now with Space Systems/Loral, Palo Alto, CA 94303 USA (e-mail: [email protected]). D. Erricolo is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail: [email protected]). A. Toccafondi is with the Dipartimento di Ingegneria dell’Informazione, Università di Siena, Via Roma 56, 53100 Siena, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122291

of an arbitrary radiating wave field. In addition, a CSP field representation, when combined with the analytic continuation in complex space of ray techniques such as the geometrical theory of diffraction (GTD) [4] and the uniform theory of diffraction (UTD) [5], may provide a very efficient tool to estimate the fields radiated by large antennas. Moreover, the scattering by wedges may also be computed with the incremental theory of diffraction (ITD) [6]–[9] and the ITD was recently extended in [10] to compute CSP diffraction by metallic objects. The ITD is considered because, in many cases, it overcomes the typical impairments of the GTD/UTD ray techniques associated with possible ray caustics and with the difficulties of ray tracing in complex space. Furthermore, PO field representation is used by many of the existing electromagnetic codes to deal with the description of fields radiated by large apertures or reflectors, even in the case of illumination by an arbitrary field [11]. Therefore, in order to augment the accuracy and the efficiency of the PO radiated field predictions, this article presents an IFF of the field scattered by edges in perfect electrically conducting objects when illuminated by a CSP representation of an arbitrary field. To this end, it is supposed that a canonical scatterer is illumiof nated by a proper linear combination of a finite number . The number tilted and scaled CSPs in the complex space of expansion terms required to represent the directional field radiated by general aperture antennas is significantly lower than the one required for a plane-wave expansion for any observation point. The advantage of this representation is that each CSP is in fact treated separately, and then the final representation is recovered using the superposition principle. This article is organized as follows. Section II provides an overview of the new IFF and its applications. Section III discusses the details of the incremental end-point physical optics (IEPO) coefficients that are introduced in the new IFF formulation. Specifically, the determination of IEPO coefficients proceeds by first identifying the incremental end-point contributions that arise at the truncation of the CSP-PO induced currents over the canonical lit surface [12], [13]. These contributions are obtained by properly applying the generalized ITD localization process [8] to the integral representation of the PO field diffracted by the canonical lit half-plane. Then, the incremental contributions due to the PO end-points are subtracted from the ITD field contributions presented in [10] to obtain the incremental fringe coefficients. Hence, the fringe field to be added to the PO representation is obtained by adiabatically distributing and integrating the incremental fringe contributions along the edge discontinuities of the actual surfaces. Section IV contains

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some numerical examples and additional details of the derivations are presented in the two Appendices. All the derivations in this article are carried out for time-harmonic fields at the angular frequency ; the time convention is assumed and suppressed throughout. II. INCREMENTAL FRINGE FORMULATION: AN OVERVIEW To illustrate this novel IFF, we begin by first describing how it is applied. Then, in Section III, we provide the incremental endpoint physical optics coefficients, which are required to apply the IFF. The PO approximation provides a first order estimate of the field scattered by a metallic object, when illuminated by a single CSP [1], [2], [14], [15] or, more generally, by a discrete distribution of CSPs. PO proceeds by replacing the metallic , which scatterers with appropriate induced currents are computed as the Geometrical Optics (GO) currents on the of the scatterer. These currents are allowed lit portions to radiate freely in space and the corresponding radiated field and are determined as [11]

Fig. 1. Geometry of a locally tangent wedge.

Second, a first order diffraction contribution , obtained by the line integration along the same edge line of the ITD incremental contribution defined in [10], is computed (4)

is more accurate than (3) since it comes from a suitable local canonical configuration that more precisely models the actual edge discontinuity. Third, the scattered field is computed as (1) (5)

(2) where is the free space Green’s function, and denote the position of the observation point and of the current source on the scatterer, respectively. At high frequency [7], [16]–[19], up to the first order, the dominant contributions to the scattered field are given by the stationary phase and end-point PO contributions, these latter associated with the abrupt truncation of the currents at the Shadow Boundary Lines (SBL) on the object (for example, the edge of the scatterer). The end-point PO contributions do not accurately describe the diffracted field. Indeed, the associated locally tangent canonical configuration at each diffraction point is constituted by a half-lit infinite plane and this is not an accurate model of the actual object in the neighborhood of the SBL. Therefore, it is important to estimate the inaccurate end-point PO contributions in order to subtract them from the PO surface integral, thus providing an estimate of the surface GO contribution to which a more accurate diffracted field may be later added. As a consequence, the IFF for the scattering by the edge of a metallic wedge, when illuminated by an electromagnetic CSP, of the scatis computed as follows. First, a PO estimate is computed using (1) by integrating the PO tered field induced currents over the lit portion of the object. As discussed in [7], this finite integration introduces inaccurate diffraction contributions by the edges of the lit portion of the scatterer. Such inaccurate diffraction contributions are estimated, at the first order, by integrating the IEPO coefficients (34) along the actual edge contour of the scatterer, i.e. (3)

where, using (3) and (4), (6) is the incremental fringe contribution. The latter, when distributed and integrated along the actual edge discontinuity, provides a correction of the PO estimate (1) of the scattered field. In fact, the IFF coefficient requires the subtraction of the less accurate estimate (3) and replaces it with the more accurate ITD estimate (4). The field representation (5) is particularly suitable for numerical calculations. This is justified by the fact that the integrand in (5) provides a precise cancellation of the singularities that occur at the shadow boundary aspects of the GO incident and reflected fields for both the IEPO and the ITD incremental coefficients. This will be particularly evident in the sections to follow for the special case of a local canonical half-plane. III. INCREMENTAL END-POINT PO CONTRIBUTIONS FOR CSP ILLUMINATION Consider an edge discontinuity in a perfectly conducting object, whose size is large in terms of the wavelength of the radiation source, and a canonical infinite uniform cylindrical wedge exterior wedge angle, as depicted in tangent to it with a Fig. 1. The shape of the edge is arbitrary, but regular. Let us define also a local rectangular coordinate system, with the axis and the axis tantangent to the canonical edge at the point face. gent to its Suppose the canonical infinite wedge be illuminated by an and electric and magnetic CSP with dipole moments

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spectral integral representation of the free-space Green’s funcin (7) and (8) and taking into account that the diftion ferential operators act only on the observation variables, these currents can be calculated as (see Appendix I) (12) Fig. 2. Geometry for the canonical problem of a half-plane in the IFF formulation.

, respectively. Furthermore, two spherical coordiand are used to identify the nate systems location of the observation point and of the illuminating CSP notation is used to represent dipole, respectively. The tilde complex distances and angles. For the sake of simplicity supface in Fig. 2) be illuminated pose that only one face (the by the CSP. As a consequence, the canonical problem for the , . PO diffraction consists of a diffracting sheet on generThe incident magnetic field on the lit half-plane ated by the CSP dipole can be calculated as

(7) (8)

where subscripts and denote the fields associated with the electric and magnetic CSP, respectively. The -components of the complex vector potentials and are

(9) (10) where is the observation point belonging to the is the complex surface of the lit half-plane and source location related to the shape of the beam of the CSP. is a real vector that defines More precisely, the center of the real position of the CSP, is a real vector whose magnitude defines the beamcollimation distance and the unit vector is the beam direction. In (9), (10) we denote the complex Green’s function as the analytical continuation of its real counterpart

(11)

in which is the complex distance from the source to the observation [1]. As mentioned above, a PO description of the problem is obtained by replacing the metallic wedge with a half-sheet of PO and induced currents, . By using the analytical continuation of the Debye

(13) It is worth noting that the electric PO currents in (12), due to a -directed electric CSP, have only a -directed component. On the other hand, the electric PO currents in (13), due to a -directed magnetic CSP, have both -directed and -directed components. When the currents (12), (13) are used in (1), (2), they provide vector expressions for the PO scattered field by the local canonical half-plane. However, since we are dealing with a canonical problem with a uniform configuration along the -axis, we may restrict our attention only to the -components of and . Indeed, according to a well-established procedure [20], these expressions may be used to find a dyadic formulation for the transverse electric and magnetic field components in the local coordinate system at the diffraction point . It can be shown that (see Appendix II) the -component spectral representation of the total PO scattered field by the local canonical half-plane may be written as

(14) for the electric CSP illumination and

(15)

(16) for the magnetic CSP illumination. Since both fields are present in this configuration, a cross-polar component will be found in the final result. In the previous equations, , , and . It is worth noting that in the above expressions we considered , however the final results apply to as the case well. In order to come to canonical expressions suitable for application of the ITD Fourier Transform (ITD-FT) convolution process, as illustrated in [8], let us introduce the spherical co, , ordinates and in the spectral domain ( and

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) and , , in the spatial domain, with and . This leads to the expressions for the -components of the total PO diffracted field as

and

It is straightforward to show that in (22) the Fourier transforms are of the two functions

(17) (23) where

is defined in (18),

(24) (18) in which

Upon substituting (23) and (24) in (22), and after introducing , we obtain the change of variable

(19) (25) and are defined along The contour integrations in the complex and planes, with clockwise indentations around the relevant poles in order to exclude the GO contributions. Taking into account the symmetry property of the original integration contours, only the even parts of the integrand are retained in the previous expressions. It is clear that (17) together with (19) provides an exact expression of the -component of the PO diffracted field, when a half-plane (or a face of a PEC wedge) is illuminated by an electromagnetic CSP. A uniform asymptotic evaluation of the above equations leads to a PO uniform diffraction coefficient, useful in all those situations where a ray field regime is well established and a ray description of the PO diffraction by large structures may be computationally very efficient [21]. However this is beyond the scope of this work, in which the same expressions will be used to derive incremental PO diffraction coefficients useful for a high-frequency diffracted field description for those situations where the ray field regime is not yet established. To this end, it is worth noting that expression (18) can be cast as the result to the product of two of the application of a linear operator spectral functions as

, and the conwhere tour integrations and are defined along in the complex and planes. The expressions (21) and (25) define the -components of the incremental field diffracted by a uniform half-sheet of PO currents in the form of an angular spectral four-fold integral. Of course, the obtained incremental contribution needs to be asymptotically approximated to provide coefficients usable in engineering applications. For this purpose, the same asymptotic analysis presented in Appendix I of [8] can be applied to expressions in (25) thus obtaining

(26) where

(27) (20) , 2 and , 2. By using the ITD-FT localization where process outlined in [8], it is found that the desired PO incremental contributions associated with the generic edge point are (21)

in which the arguments

,

and

are defined by (28) (29) (30)

where (22)

(31)

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with the branch of the square root in (30) chosen to be . and are eventually used to The -components provide the dyadic closed-form expression for the incremental in the local ray-field coordinate system [5]. For PO field this purpose, it is worth noting that (32) and that the -components of the incident field at due to the -oriented electric and magnetic CSP at may be written as (33) Therefore expressions (21) and (26) may be rewritten in the dyadic form (34)

Fig. 3. Geometry for the scattering by a square plate illuminated by a CSP.

fringe coefficient for the special case of a local canonical halfplane. It is a straightforward matter to prove that this coefficient assumes the useful form (37)

Note that the PO incremental contribution (34) is not reciprocal, due to the presence of the cross-polar component . This is expected since the PO field (14)–(16) exhibits a non-reciprocal behavior. The same expression shows a useful symmetry property with respect to and . Moreover, when the observation and/or the source approach the edge of the local canonical conbecomes large and the incremental figurations, the term contribution in (26) tends to vanish. The total PO diffracted field by the entire canonical configuration is obtained by the spatial integration of the incremental contribution (34) along the straight edge . This representation exhibits appropriate discontinuities that occur at the shadow boundaries of the GO field, and are provided by the singularities of the incremental PO coefficients and . It is a straightforward matter to prove that the incident shadow boundary lines (ISBs) occur when for which two singularity directions ( , 2) exist, where

(35) The tangent function is inverted with branch cuts and on the top Riemann sheet of its argument. Analogous considerations hold for the reflection shadow boundaries (RSBs) for which singularities occur when along the two singularity directions ,( , 4), where

(36) One observes that the above singularities are exactly the same as in the ITD coefficients in [10] associated with the upper face of the wedge. This is evident when computing the incremental

where

(38) An inspection of the above incremental contributions reveals that this latter is free of singularities at the reflection and incident shadow boundaries, thus making the calculations of the total PO correction term in (5) easy to determine. Finally, it is worth noting that, when both faces of a wedge are illuminated, two IEPO coefficients with exterior angle are computed and subtracted from the ITD one: one for the 0-face and one for the -face. The coefficient for the -face with and is calculated by substituting with in (34). IV. NUMERICAL RESULTS Let us consider the scattering from a perfectly conducting square plate illuminated by a single CSP. The geometrical configuration is shown in Fig. 3. A -directed electric CSP dipole is from a square plate with sides placed at a distance , and located in the plane. The vector associated with the axis of the beam is defined by , so the beam points towards one of the edges the plate. Numerical results obtained from the MoM (dash-dotted line), PO (dashed line) and this method (PO+IFF) (solid line) are presented in on the Fig. 4 and Fig. 5 for observations at a distance plane and plane, respectively. In Fig. 4 the -component of the scattered electric field through the IFF formulation is plotted . This curve is found in good agreement with for MoM calculations, except for the region close to grazing observation. As expected, at these observation aspects the first-order

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Fig. 6. Geometry for the scattering by a planar parabolic reflector illuminated by a horn antenna in the focus of the reflector. Fig. 4. Scattered jE j component of the far field for the square plate illuminated by a single CSP electric dipole. Observations are made in the yz -plane.

Fig. 5. Scattered jE j component of the far field for the square plate illuminated by a single CSP electric dipole. Observations are made in the xz -plane.

ITD incremental diffraction coefficients exhibit a discontinuity that can be overcome with the introduction of double diffraction contributions [22]. These latter have not been taken into account in this formulation, but will be the subject of further investigations. It is also worth noting that the accuracy of PO calculations is significantly improved when adding to it the incremental fringe calculations. The same considerations also apply for the -component of the scattered electric field shown in Fig. 5 for . However in this case, at grazing observation aspects, the fringe calculation do not exhibit first-order discontinuity in the ITD field. Around 270 , some discrepancy is noted between MoM and PO+IFF results: at first-order, the strongly-lit edge at provides a spurious and discontinuous contribution at the observation point, where the correct, but weak contribution from the edge at 270 is not sufficient to correct this behavior. Furthermore, the contribution from the vertices in this illumination condition becomes more important, but it is not taken into consideration in this formulation. The second example that we consider is the more realistic scattering from a perfectly conducting parabolic reflector when illuminated by a horn antenna. The geometrical configuration is shown in Fig. 6. The parabolic reflector has a radius of and focal distance , equivalent to a height of the paraboloid . The horn antenna operating at is located at the focus of the paraboloid with the main beam axis pointing to the vertex of reflector, as shown in Fig. 6. The horn antenna radiated field, linearly polarized along the -axis, may be completely represented using a proper CSP expansion

Fig. 7. Total jE j far field for the parabolic reflector illuminated by a horn antenna for the geometry in Fig. 6. Observations are made in a plane cut at  .

= 45

[23], [24] consisting of 1122 single CSPs located at the center of the local horn reference system. The expansion is performed in such a way that the CSPs have a uniform beam vector distribution, so as to cover uniformly a sphere of zero real radius. However, to estimate the scattered field from the reflector, it is found that 373 CSPs directly contribute to the scattered field and the incremental quantities, using the truncation algorithm. The from the center of observation is made at a distance on a plane cut at . In the aperture, for Fig. 7 and Fig. 8 the curves relative to the total electric field obtained from MoM simulations are compared with those obtained from PO+IFF and PO alone, respectively. One observes that the results obtained introducing the CSP fringe corrections to the PO formulation are in a very good agreement with MoM calculations. One also observes that this method provides improved accuracy as compared with only PO results, especially at grazing incidence and in the back-lobe region. In the third example, the same horn antenna illuminating the parabolic reflector is located such that its phase center is at , and tilted so that the main beam axis still points to the center of the parabolic dish, as shown in Fig. 9. The same CSP expansion of the previous example for the horn antenna radiated field has been used. Again, from the center the observation is made at a distance on a plane cut at . of the aperture, for Fig. 10 and Fig. 11 show the total electric field obtained form MoM simulations and compared with the PO+IFF and PO formulations, respectively. As expected, an overall significantly improved accuracy is observed when introducing the fringe corrections in the PO formulations. Only in the backward region,

CANTA et al.: INCREMENTAL FRINGE FORMULATION FOR A COMPLEX SOURCE POINT BEAM EXPANSION

Fig. 8. Total jE j far field for the parabolic reflector illuminated by a horn antenna for the geometry in Fig. 6. Observations are made in a plane cut at  .

= 45

Fig. 11. Total electric field jE in a plane cut at  .

= 45

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j

for geometry in Fig. 9. Observations are made

incremental coefficients have singularities that cancel out those of the ITD coefficients. Furthermore, the IFF representation of the scattered correction terms allows a straightforward interpretation of the physical phenomena involved in this extension, thus permitting IFF to be a good candidate to be widely employed in codes for the description of the scattering by complex antenna systems. The formulation was tested with both GO and ITD methods and the MoM for large reflector antennas and the results show the effectiveness of the method. Fig. 9. Geometry for the scattering by a parabolical reflector illuminated by an off-set horn antenna.

APPENDIX I Starting from expressions (7) and (8), the incident magnetic and at the point on the illuminated field face of the half-plane due to an electric and magnetic CSP with and , respectively, may dipole moments be written as (39)

(40) Fig. 10. Total electric field jE in a plane cut at  .

= 45

j

for geometry in Fig. 9. Observations are made

near the curve calculated with the proposed method deviates from that by MoM. This discrepancy may be attributed to creeping wave effects, caused by the off-axis illumination and not accounted for in this formulation. V. CONCLUSIONS In this paper, an IFF for the field scattered by edges, or shadow lines, in perfect electrically conducting objects when illuminated by CSPs was introduced. This was motivated by the need for a fast and efficient way to obtain more accurate results than the PO simulations for the scattering by metallic objects. The formulation is obtained by a proper analytical continuation in complex space of the relevant geometrical quantities. It is found that the obtained incremental fringe contributions are free of first order GO singularities, since the developed IEPO

where, for the sake of simplicity, the spatial dependence from and has been omitted. We can now calculate the PO currents on the illuminated face of the semi-plane as (41) (42) After introducing in the above expressions the analytical continuation of the Debye spectral integral representation

(43) and performing the derivatives with respect to the observation variables, under the assumption that the source lies in the upper semi-space, it is a straightforward matter to obtain (12) and (13) where .

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APPENDIX II Starting from expression (1) let us consider the elementary contribution

that, introduced in (49), directly leads to (14). The scattered electric field due to the magnetic CSP illumination may be obtained similarly by first introducing (13) in (48), yielding (51).

(44) where

(51)

since the currents are defined over a lit half-plane ( , , ). Let us now introduce in the above the well known spectral representation of the Green’s function

(45)

Next, selecting the -component of the above expression and using the identity (50), after a simple algebraic manipulation one clearly obtains (15). Finally, the scattered magnetic field due to a magnetic CSP illumination may be derived from (51) as (52)

where

and . Performing the derivatives with respect to the observation variables leads to

Taking into account that the derivatives act only on the observation variables, the -component of the scattered magnetic field may be written as

(46) where (53) (47) that, after using again (50), provides (16). and is a dyadic, in which the signs apply for respectively. As a consequence, the PO radiation integral in (1) may be written as

(48) and conLet us now restrict the analysis to the case sider only the -component of the scattered electric field. It is a straightforward matter to prove that the final results apply also as well. By introducing (12) in (48) and interchanging to the spatial and spectral integration one obtains

(49)

The inner integral in the above expression, evaluated over the lit half-plane, provides (50)

REFERENCES [1] L. B. Felsen, “Complex source point solution of the field equations and their relation to the propagation and scattering of Gaussian Beams,” in Proc. Symp. Mathematica, 1976, vol. 18, pp. 39–56. [2] M. Couture and P.-A. Belanger, “From Gaussian beam to complexsource-point spherical wave,” Phys. Rev. A, vol. 24, no. 1, pp. 355–359, 1981. [3] A. N. Norris, “Complex point-source representation of real sources and the Gaussian beam summation method,” J. Opt. Society Amer. A, vol. 3, pp. 2005–2010, 1986. [4] J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Society Amer., vol. 52, no. 2, pp. 116–130, 1962. [5] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, no. 11, pp. 1448–1461, 1974. [6] R. Tiberio and S. Maci, “An incremental theory of diffraction: Scalar formulation,” IEEE Trans. Antennas Propag., vol. 42, no. 5, pp. 600–612, 1994. [7] R. Tiberio, S. Maci, and A. Toccafondi, “An incremental theory of diffraction: Electromagnetic formulation,” IEEE Trans. Antennas Propag., vol. 43, no. 1, pp. 87–96, 1995. [8] R. Tiberio, A. Toccafondi, A. Polemi, and S. Maci, “Incremental theory of diffraction: A new-improved formulation,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2234–2243, 2004. [9] D. Erricolo, S. M. Canta, H. T. Hayvaci, and M. Albani, “Experimental and theoretical validation for the incremental theory of diffraction,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2563–2571, Aug. 2008. [10] A. Polemi, A. Toccafondi, G. Carluccio, M. Albani, and S. Maci, “Incremental theory of diffraction for complex point source illumination,” Radio Sci., vol. 42, no. 6, pp. 1–13, 2007. [11] Y. T. Lo and S. W. Lee, Antenna Handbook. New York: Van Nostrand Reinhold Company, 1993.

CANTA et al.: INCREMENTAL FRINGE FORMULATION FOR A COMPLEX SOURCE POINT BEAM EXPANSION

[12] A. Michaeli, “Elimination of infinities in equivalent edge currents; I: Fringe components,” IEEE Trans. Antennas Propag., vol. 32, pp. 252–258, 1984. [13] A. Michaeli, “Elimination of infinities in equivalent edge currents, Part II: Physical optics components,” IEEE Trans. Antennas Propag., vol. 34, no. 8, 1986. [14] J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Society Amer., vol. 61, no. 1, pp. 40–43, 1971. [15] G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett., vol. 7, no. 23, pp. 684–685, 1971. [16] R. A. Shore and A. D. Yaghjian, “Incremental diffraction coefficients for planar surfaces,” IEEE Trans. Antennas Propag., vol. 36, no. 1, pp. 55–70, 1988. [17] P. Y. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction 1971. [18] P. Y. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics, vol. 11, no. 2, pp. 125–160, 1991. [19] S. M. Canta, “High Frequency Incremental Methods for Complex Source Points,” Doctoral dissertation, Univ. Illinois at Chicago, Chicago, 2010. [20] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [21] H.-T. Chou and P. H. Pathak, “Uniform asymptotic solution for electromagnetic reflection and diffraction of an arbitrary Gaussian beam by a smooth surface with an edge,” Radio Sci., vol. 32, no. 4, pp. 1319–1336, 1997. [22] A. Toccafondi and R. Tiberio, “An incremental theory of double edge diffraction,” Radio Sci., vol. 42, no. 6, p. RS6S30, 2007. [23] E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: Complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Society Amer. A, vol. 18, no. 7, p. 1588, 2001. [24] K. Tap, “Complex Source Point Beam Expansions for Some Electromagnetic Radiation and Scattering Problems,” Doctoral dissertation, The Ohio State University, Columbus, 2007.

Stefano Mihai Canta (S’06–M’10) received the Laurea degree in telecommunications engineering (summa cum laude) and the Laurea Specialistica degree in telecommunications engineering (summa cum laude) from the Politecnico di Milano, Milan, Italy, in 2004 and 2007, respectively, and the M.Sc. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Chicago, (UIC), in 2009 and 2010, respectively. His research interests are in applied electromagnetism, specifically propagation and high-frequency methods. He is currently working for the Antenna Subsystems Operations at Space Systems/Loral. Dr. Canta is the recipient of the 2007 Andrew Fellowship and the 2009 Dean Scholar Award.

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Danilo Erricolo (S’97–M’99–SM’03) received the Laurea degree of Doctor (summa cum laude) in electronics engineering from the Politecnico di Milano, Milano, Italy, in 1993 and the Ph.D. degree in electrical engineering and computer science from the University of Illinois at Chicago (UIC), in 1998. He is currently an Associate Professor in the Department of Electrical and Computer Engineering, UIC, where he is also the Associate Director of the Andrew Electromagnetics Laboratory. His research interests are primarily in the areas of wireless communications, electromagnetic scattering, and electromagnetic compatibility. His research activity has been supported by the Department of Defense and the National Science Foundation. He has authored or coauthored more than 150 publications in refereed journals and international conferences. Dr. Erricolo was twice the recipient of both the Andrew Foundation Fellowship and the Beltrami Foundation Fellowship, and a U.S. Air Force Summer Faculty Fellow in 2009. He is a member of Eta Kappa Nu and was elected a Full Member of the U.S. National Committee of the International Union of Radio Science (USNC-URSI) Commissions B and E. He served for USNC-URSI as Secretary (2004–2005), Vice-Chair (2006–2008) and Chair (2009-2011) of Commission E and Chair of the Ernest K. Smith Student Paper Competition (2009-present). He is an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. He also has served as Vice-Chair of the Local Organizing Committee of the XXIX URSI General Assembly (Chicago, IL, Aug. 7–16, 2008) and has been appointed as General Chair of the 2012 IEEE Antennas and Propagation International Symposium/USNC National Radio Science Meeting (Chicago, IL, July 8–14, 2012).

Alberto Toccafondi (M’93–SM’08) received the Laurea degree (summa cum laude) in electronic engineering and the Ph.D. degree in telecommunications and informatics from the University of Florence, Florence, Italy, in 1989 and 1994, respectively. In 1995, he joined the Department of Information Engineering, University of Siena, where he is currently an Associate Professor. His research interest is concerned with antennas and microwave devices and with analytic and numerical techniques for electromagnetic scattering and multi-path propagation prediction. His research activity is mainly focused on the incremental and asymptotic methods applied to the prediction of the electromagnetic scattering, as well as to the analysis and design of antennas and passive tags for RFID applications. He was the coauthor of the incremental theory of diffraction, which describes a wide class of high-frequency electromagnetic scattering phenomena. He is the principal author or coauthor of more than 130 publications in international journals and peer reviewed symposium proceedings. Dr. Toccafondi served, from 2002 to 2005, as a secretary and treasurer of the IEEE Central and South Italy Section as well as of the AP/MTT Joint Chapter of the IEEE. He served as a treasurer (2006–2007) and then as a secretary (2008–2009) of the recently established IEEE Italy Section.

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Modal Analysis and Wave Propagation in Finite 2D Transmission-Line Metamaterials Rubaiyat Islam, Student Member, IEEE, Michael Zedler, and George V. Eleftheriades, Fellow, IEEE

Abstract—In this paper we examine the propagation of plane waves and Gaussian beams in 2D periodic grids constructed with lumped reactive immitances. We demonstrate the equivalence between the Bloch/Floquet modal description and the multiple coupled-line modal analysis of finite-sized periodic grids. This establishes that the Floquet analysis of 2D metamaterial negative-refractive-index grids and the associated exotic wave propagation/refraction/focusing phenomena, can be deduced from traditional analysis coupled lines (and hence eigenmodes), albeit with conof siderable algebraic complexity. We present simulation results of Gaussian beams through these grids to demonstrate that they can be analyzed using simple geometrical optics along with the index of refraction associated with Bloch/Floquet analysis.

M

M

Index Terms—Bloch/Floquet modes, coupled modes, Gaussian beam propagation, metamaterial, multiconductor lines, negative refractive index. Fig. 1. A 2D NRI-TL unit cell in shunt configuration. Each transmission-line segment has an electrical length (1=2) and a characteristic impedance Z .

I. INTRODUCTION EGATIVE-refractive-index transmission-line (NRI-TL) grids comprise transmission lines loaded periodically with series capacitors and shunt inductors to ground [1] and have been used in the verification of theoretical predictions made by Veselago [2] in the sixties. A linear (1D) version of this structure has been utilized in the demonstration of backward waves and backward end-fire leaky wave radiation [3] while 2D slabs of such grids have been used in subdiffraction imaging experiments [4]. In the long wavelength limit, i.e., at frequencies where the unit cell dimension is small compared to the guided wavelength, such grids allow isotropic plane wave1 propagation whose impedance and wavenumber are deduced by Bloch/Floquet analysis [5]. The plane wave supported by these grids enables one to characterize them using an index of refraction associated with the Bloch propagation constant. Even in a finite size grid, plane waves can be excited using appropriately phased sources along its boundary and in conjunction with passive Bloch terminations. It should be recognized though that a unique set of Bloch terminations is required to support a plane wave propagating in a given direction and that at all

N

Manuscript received June 12, 2010; revised August 19, 2010; accepted November 09, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, Scarborough, ON M5S 2E4, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123068 1In this paper the term “plane wave” will be used to describe a voltage/current distribution of constant magnitude but progressive phase in a single direction when the voltage/current samples are restricted to the outer nodes of each unit cell comprising the grid.

other angles, these waves will bounce off the boundaries with a nonzero reflection coefficient. Fig. 1 shows a unit cell of a 2D NRI-TL in shunt configuration. The electrical response of the unit cell at its terminals is identical to the unit cell consisting solely of lumped components, having the element values (1a) (1b) (1c) Let us consider a finite-size rectangular grid with cells along the axis and cells along the axis made of periodically repeated lumped reactive elements as shown in Fig. 2. The experimental observations outlined above and their interpretation using Floquet modes may raise the following questions when one tries to interpret them in terms of multiconductor line theory [6]–[9]. 1) The 2D grid may be viewed as -lines oriented along the -direction periodically coupled by -directed reactive elements and as such the grid must support -modes with propagation constants which are not necessarily all equal. Floquet analysis on the other hand predicts a single planewave propagation along the -direction with a unique propagation constant (which is the same for all angles of propagation). 2) Floquet analysis applies to infinite periodic structures; therefore does its application to finite grids lead to an incomplete modal description? In particular, how does the

0018-926X/$26.00 © 2011 IEEE

ISLAM et al.: MODAL ANALYSIS AND WAVE PROPAGATION IN FINITE 2D TRANSMISSION-LINE METAMATERIALS

x

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M

y

Fig. 3. Unit block of a finite width grid with unit cells in the -direction and of infinite extent in the -direction. The arrows depict direction of current flow. Fig. 2. Infinite grid of periodic lumped reactive elements.

notion of plane waves propagating in arbitrary directions coupled-modes description? reconcile with the 3) Do NRI-TL grids actually support backward waves whereby power is carried in a direction opposite to phase flow or are these artifacts of the analysis that relies on plane (Bloch) waves in infinite periodic structures? In this paper we will show that the concerns outlined above have relatively simple interpretations using Floquet type analysis and that compared to multiconductor line theory [6], [7], the former is much simpler (and yet complete) when applied to periodic structures (infinite or truncated). We will therefore establish that the wave propagation in electrical networks of the form shown in Fig. 2 (under suitable homogeneous conditions) can be analyzed simply by using Bloch/ Floquet theory or geometrical/wave optics thereby associating it with an effective index of refraction. II. FLOQUET MODES IN INFINITE 2D GRIDS We shall at first examine an infinite periodic 2D grid using Floquet modes so that we may refer to these results when addressing the problem of a finite grid analyzed in terms of multiconductor line theory. The typical unit cell of such a grid is depicted in Fig. 2. We can write down the following circuit equations: (2a) (2b) (2c) and in (2), the resulting eigenmodes their corresponding eigenvectors are given by Letting

,

and

(3d) A regular positive refractive index (PRI) grid can be em, and in (1). On ulated by setting the other hand, a metamaterial negative refractive index (NRI) grid is obtained by loading the grid with series capacitors and shunt inductors , yielding the resonant immitances and . III. MODES IN A GRID OF FINITE WIDTH (

CELLS)

grid as a system of In this section we consider an -lines oriented along the -axis. The grid itself is constructed by cascading the blocks of the form shown in Fig. 3 where we have doubled the -directed impedances and halved the shunt admittances such that the final structure resembles a finite width cut of the infinite periodic grid in Fig. 2. Let and be column vectors of voltages and the associated port currents , respectively, as depicted in Fig. 3. We can define column vectors and in a similar fashion where the subscripts and refer to the right- and left-hand side (RHS and LHS) of the block, respectively. We may now relate these vectors by the following transmission matrix: (4) In (4) above, , , , and are all block matrices. The eigenmodes of the system above are obtained by letting the RHS vector be equal to the product of and the LHS vector, and setting the determinant of the resulting system to zero

(3a)

(5)

(3b)

In (5), is the identity matrix. For reciprocal, lossless and symmetric systems as the one depicted in Fig. 3, it can be shown that the eigenvalue problem in (5) simplifies to

(3c)

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(see Appendix A) where the matrix can be determined with relative ease from Fig. 3 and the resystem: sulting equation for the modes is the following

.. .

.. .

(6a)

(6b) Equation (6a) can be solved in terms of Chebyshev polynomials of order as shown in Apof the second kind pendix B (7) The roots of ranging from 1 to and the expressions for compact form as

are

where is an integer . Hence the roots of (7) are given by which after substituting in and from (6b) can be expressed in a

M + 1) 2 (N + 1) grid excited by phased voltage sources.

Fig. 4. A (

Fig. 4 shows a grid surrounded by ideal voltage sources which are phased according to (10a) (10b) (10c) (10d)

(8) . Now observe that (8) is the same as the with Floquet mode equation (3a) if one makes the following substitutions: and . We offer a very simple explanation for this observation. The grid is obtained from the infinite case by introducing horizontal Perfect Magnetic Conductor (PMC) walls (or open circuits) slightly below rows and such that the currents and are set to zero. Hence the Floquet modes which could propagate freely in all directions in the infinite grid, are now guided by these walls. There are exactly independent combinations of Floquet modes which satisfy the horizontal PMC boundary conditions while simultaneously retaining an exponential (propagating or cut-off) voltage/current variation along the axis. We may combine two possible solutions of (3a), namely and to obtain a modal function with exponential dependence along . In order to satisfy the we choose their ratio boundary conditions to be which gives us (9) Finally we set in (9) so that which can then be substituted into (3a) to obtain the corresponding . It is evident that the choices of or provide the same set of eigenmodes where . IV. PLANE-WAVE EXCITATION IN A

GRID

In this section we will demonstrate plane-wave propagation, excitation and termination in a finite grid using multiconductor line theory, where a plane wave implies a solution of the form .

In (10), the constants (3a)

and

are chosen such that they satisfy (11)

It is possible to determine the voltage/current profile within the grid by inspection if we believe in the uniqueness of the solution to the circuit problem depicted in Fig. 4. If we let and in (3a) to (3d), then the resulting voltages and currents satisfy Kirchhoff’s circuit laws at each node. Moreover, the substitutions just described also satisfy the boundary conditions imposed by the voltage sources and hence by uniqueness, must correspond to the only possible solution. Nevertheless, one may determine the expected solution by actually solving the grid using multiconductor line theory. The main idea in this venture is the realization that the solution of the grid problem consists of two multiconductor line problems. We may turn off the top and bottom rows lines of voltage sources in Fig. 4 to obtain a set of that are -directed and excited at both ends by phased sources. On the other hand, by turning off the LHS and RHS columns of the sources, we obtain lines in the -direction. By superposition, the voltage/current profile in the grid will be determined by summing these two solution sets. The four corner sources can be taken into account separately, but their effects are limited to induced currents in adjacent components only and as such do not alter the solution profile in the interior region of the grid. In contrast to the modes described in the previous section, the eigenmodes in the present problem comprise Floquet modes bouncing between Perfect Electric Conductor (PEC) walls created as a result of switching off either the horizontal rows or the

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vertical columns of the voltage sources. Denoting the th mode in and the th mode in the -directed the -directed lines case as lines as , the voltage eigenvectors in these two cases are

(12a) (12b) The eigenvalues equations:

and

are generated by the following

Fig. 5. Voltage phase (in degrees) corresponding to (a) plane-wave excitation : , at 30 relative to x-axis and (b) plane-wave excitation at 60 (a)   : (b)  : , : .

= 0 0245

(13a)

(13b) We have omitted the line-current eigenvectors for the two cases as they are more difficult to evaluate and are unnecessary in our case where the boundary conditions have been conveniently chosen as ideal voltage sources. Using the modes in (12) we form two series, one containing -directed coupled modes and another containing -directed coupled modes: (14a)

(14b) The total voltage induced by the sources in the grid is the sum of the contributions shown in (14). The expansion coefficients that satisfy the boundary voltages in (10) are given by

(15a)

= 0 0245

= 0 0424

= 0 0424

evaluated using the following excitation sets: ( and ) corresponding to a 30 and ( and ) corresponding to a 60 angle of propagation relative to the axis. The voltage magnitude corresponding to each plot in Fig. 5 is unity and the dotted lines in these two plots are defined by . We have obtained the anticipated the equation: result that the superposition of -directed and -directed multiconductor-line modes form plane waves in the grid. For a given choice of and satisfying (11), some of the sources are injecting power into the grid while others are absorbing net power. The latter ones can be replaced by impedances (Bloch impedances) satisfying the voltage to current ratio of the source without affecting the voltage/current distribution anywhere in the grid. The sources that can be replaced by these impedances are discussed in Section V where we examine both PRI and NRI grids. V. DIRECTION OF POWER-FLOW IN PRI AND NRI GRIDS In order to determine which voltage sources are absorbing net power from the grid we need to evaluate their currents. Having convinced ourselves that the node voltages in the grid shown in Fig. 4 are given by we can now compute the source currents and hence the impedance looking into each source. If the real part of this impedance is negative, we retain the source; otherwise we replace the source with a passive load equal to .2 [10] (17a)

(15b)

(17b)

Letting such that the grid emulates isotropic PRI or NRI media, simplifies (11) to

(17c)

(16) and we can evaluate this sum numerically for various choices of and source parameters keeping the medium parameter product fixed. We present numerical results for an by grid with so that (16) can be satisfied using small values of and . Such element values are usual in practice in emulating isotropic homogeneous media where the phase-shift between adjacent cells is small [4]. In Fig. 5 we show the phase of the series whose coefficients (in (15)) were

(17d) and The grids under consideration are lossless and hence are purely reactive which implies that the second terms in the numerator of (17) correspond to the real parts of the impedances looking into the corresponding sources. We have excluded the 2If the real part of the current to voltage ratio looking into a source is negative, then that particular source is involved in supplying the grid with power and cannot be substituted with a passive load which will leave all other voltages/ currents in the grid unaffected.

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This is accomplished by setting the impedance in (17d) corresponding to the PRI grid edge to the negative of the impedance in (17c) corresponding to the NRI grid edge. (18)

Fig. 6. Cascading a PRI grid with a NRI grid. The bold arrows inside the grid indicate the direction of phase flow corresponding to phased sources indicated by V with the added implication that the remaining two sides of each grid are terminated by matched impedances according to (17).

cases of the four corner sources which can be handled in an identical fashion. Now consider a wave propagating from the top left-hand corner of the grid in Fig. 4 to the bottom right-hand corner in a regular positive-index grid where such that , and are positive reactances. We immediately see from (17) that we have to retain the top and left-side sources (due to the negative conductance) and may replace the bottom and right-side sources with passive impedances. On the other hand, for the same propagating wave in a metamaterial NRI grid where and are negative reactances, the source at the bottom and right-side are retained, while the ones at the top and left-side may be replaced. Hence we have a scenario where the phase is decreasing diagonally from the top left-hand corner while the power is flowing out of the bottom right-hand corner of the grid (see the sample phase profiles depicted in Fig. 5). In other words, the direction of power flow in this case is in the direction of increasing phase (a backward wave) for the plane wave supported by the grid. This is a confirmation, using purely circuit analysis, of the known phenomenon that in a negative-index grid the power flow is contra-directional to the phase progression deduced previously from the negative sign of the group velocity in such structures [1]. VI. NEGATIVE REFRACTION FROM THE CIRCUIT PERSPECTIVE In this section we will use circuit analysis to demonstrate how the phenomenon of negative refraction at the interface between a PRI and an NRI metamaterial grid can be interpreted in terms of simple geometrical optics. Assume that we have solved for the voltages and currents in a positive-index grid excited by phased voltage sources as discussed in Section IV. To maintain the phase flow direction in the PRI grid indicated by the bold arrow in Fig. 6, we need sources only on the top and left-sides of the grid and can terminate the other sides with passive loads. Now the passive loads on the right side can be replaced by another grid which has the same impedance and voltage distribution along the juxtaposed edge (to ensure continuity of both the current and the voltage). As shown in Fig. 6 there are two possible orientations of a NRI grid which we would like to cascade with the PRI grid without disrupting the voltages and currents in either of them.

In (18) the subscript 1 refers to the PRI grid whereas the subscript 2 refers to the NRI grid. The reactance is positive whereas is negative, and hence we see that the choice of the parameters and would satisfy (18). Both NRI grids depicted in Fig. 6 satisfy this requirement but only the lower one (with sources attached to the bottom of the grid) exhibit decreasing voltage phase along the -direction. Hence this particular orientation can be cascaded with the positive-index grid and the resulting heterostructure displays negative refraction. The process of satisfying (18) corresponds to the Brewster’s angle condition in optics while the underlying phase matching is equivalent to Snell’s law [10]. The usage of sources in both PRI and NRI media to emulate plane-wave refraction is not a complication unique to this case: it is also necessary in the case of positive refraction of plane waves traveling from one PRI grid to another. This is not to be interpreted as a fundamental limitation of negative refraction phenomena but merely the consequence of dealing with planewaves in finite media which require sources of infinite extent to support them. The obvious questions which arise include what happens when we retain the sources on only a single edge of one of the grids and remove all others? Will we still observe backward waves, contra-directional power flow and negative refraction? The answer is yes if we are willing to use confined beams that can be excited with negligible distortion with sources limited to a single edge. This will be the subject of the remainder of this work. Nevertheless, it is clear that the usage of Floquet modes to analyze such 2D or 3D networks is much simpler and more intuitive compared to multiconductor line theory employed in traditional network analysis. VII. SIMULATION OF BEAM PROPAGATION IN FINITE TRANSMISSION-LINE METAMATERIAL GRIDS Let us now analyze beam propagation through a finite network of unit cells as depicted in Figs. 1 and 2 [1]. The loading of the cells is symmetric, such that and the dispersion relation is thus given by (16). An unloaded unit cell is obtained by setting in (1) which yields and . Let us assign this unit cell the positive effective refractive index . The refractive index of the loaded unit cell is, assuming that the transmission-line segments for the loaded cell are identical to the unloaded case (19) An effective refractive index of cific frequency if

is obtained at a spe-

(20a) (20b)

ISLAM et al.: MODAL ANALYSIS AND WAVE PROPAGATION IN FINITE 2D TRANSMISSION-LINE METAMATERIALS

plane-wave expansion of a Gaussian beam with waist agating along the -axis is

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, prop-

(22) In the simulations performed here, the beam waists is not small compared to the wavelength, i.e., . Hence the evanescent spectrum of the beam in (22) is negligible, yielding the alternative representation

Fig. 7. Schematic of the Gaussian-beam simulation. Grey crosses represent unit cells, on the upper side a PRI grid is excited by a Gaussian beam; the PRI grid interfaces a NRI grid. The Gaussian-beam excitation is chosen such that the beam waist forms at the PRI-NRI interface. Towards the side edges the unit cells are terminated by a suitable termination, denoted by ’T’.

Fig. 8. Phase of the voltage for Gaussian beam propagation along the y axis, the axis refers to the number of cells. n for y < and n for y > . Electrical length of a unit cell is  = and the y axis is perpendicular to the PRI/NRI interface.

0

=1 20

0

= 01

is fulfilled. This is obtained if the unit cell is loaded with series capacitors and shunt inductors with the element values

(21) The propagation of a Gaussian beam through a finite grid of such unit cells is simulated by computing the overall admittance matrix of the setup. A sketch of such a setup is shown in Fig. 7. The PRI grid consists of unloaded unit cells whereas , the NRI grid has appropriate loading elements such that the desired negative refractive index is obtained. The termination on the edges (denoted by “T” in Fig. 7) is chosen as of (17) such that the dominant plane wave of the Gaussian beam is perfectly absorbed. This implies that other components of the Gaussian beam’s spectrum are not perfectly absorbed and thus will partly reflect back into the simulation domain.3 The complex amplitudes used for the excitation of the ports for obtaining a Gaussian beam are obtained as follows: The 3In order to circumvent this problem for pure circuit theory simulations, we presented in [13] an approach which implements the radiation boundary conditions using the method-of-moments-discretized electric field integral equation.

(23) The excitation (23) is evaluated for each port with its port coordinates where is fixed (corresponding to the top edge of the PRI grid in Fig. 7) and varies along the entire width of the grid. Beam propagation at an angle with the -axis is achieved by substituting in (23). The simulation setup uses 128 128 cells for both the PRI half-space and the NRI half-space, the electrical Bloch length of . each unit cells is 25 , yielding a total electrical size of The frequency of operation is . The NRI unit cell used in the simulation is the one shown in Fig. 1 containing four 50 transmission-line segments that are long at the frequency of operation. The loading elements are and . The PRI unit cells contain the same transmission lines as the NRI-TL, but no loading elements. These unit cells yield an effective refractive index at the operating frequency of . The beam waist in the PRI region is cells, whereas the Rayleigh range is cells. Let us first consider a Gaussian beam propagating at normal incidence in the PRI-NRI half spaces shown in Fig. 7. For this setup, Fig. 8 shows the phase of the voltage across the beam center for a normal angle of incidence on the interface. The complex voltage is tracked not only at the unit cell terminals but also across the transmission-line sections. In the PRI region the wave experiences a phase delay. In the NRI region , however, the wave experiences a phase discontinuity of approximately 20 at the position of the series capacitors. This discontinuity causes the total phase change across a unit cell to yield a phase advance [11], [12]. Fig. 9 shows the simulation results for the case for a Gaussian beam incident at oblique incidence. Due to the choice of the loading in (20) the two media are impedance matched, thus no reflection occurs at the interface. A Gaussian beam with an angle of incidence 25 excites the structure, and as expected from (19), the beam refracts negatively at the interface, reconstructing the image in the PRI half space, such that the interface becomes a symmetry plane. The inset in Fig. 9 shows a magnified view of the voltages, each unit cell has 12 samples. Clearly phase fronts can be identified in the structure, and the inset reveals that the small phase discontinuities due to the reactive loading of the unit cells still retains the clear definition of phase fronts. This confirms that the observations made for normal incidence in Fig. 8 also hold for an oblique angle of incidence. As the evanescent spectrum of the

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Fig. 9. Circuit simulation results for a Gaussian beam propagating in a grid of 128 256 network unit cells. The upper half space has a PRI with n , the lower half space has a NRI with n . Time-domain (steadystate) plot of the voltage throughout the mesh. Inset shows the voltage distribution within the unit cells on the transmission-line segments, using 12 sampling points per unit cell. Black pixels represent space outside of the transmission-lines of the unit cells.

+1

2

= 01

=

Fig. 11. Circuit simulation results for a Gaussian beam propagating in a grid of 128 256 network unit cells. The upper half space has a PRI with n , the lower half space has a NRI with n : . Time-domain (steady-state) voltages are shown, for the scale the reader is referred to Fig. 9.

+1

2

= 00 9

=

Fig. 12. Circuit simulation result for a Gaussian beam propagating in a lossy transmission-line grid. The upper half space has n , the lower half : . All unit cells are lossy, with the electric and magnetic space n  : . loss tangent

= 01 01 tan = 0 04

Fig. 10. Circuit simulation results for a Gaussian beam propagating in a grid of 128 256 network unit cells. The upper half space has a PRI with n , the lower half space has a PRI with n . Time-domain (steadystate) voltages are shown, for the scale the reader is referred to Fig. 9.

+1

2

= +2

=

Gaussian beam used for the excitation is negligible, the surface wave at the PRI/NRI interface is vanishingly small [14, ch. 1]. For comparison, Fig. 10 shows the simulation results where the lower half space cells are loaded with , such that and that the the resulting effective refractive index is impedance boundary condition is met for the dominant plane wave of the incident Gaussian beam. For this setup positive refraction is obtained, as expected. Fig. 11 shows the simulation results if the lower half space has . One notes that a) the angle of propagation in the NRI medium is 28 and hence larger than the angle of incidence 25 , as expected from Snell’s law, b) the effective wavelength is longer, and c) some reflection occurs, because the impedance matching condition is only fulfilled for the dominant plane wave of the exciting Gaussian beam. While Fig. 9 showed the ideal case of lossless media being exactly fulwith filled, Fig. 12 shows the lossy case where furthermore the refractive indices are only nearly identical in magnitude, . All unit cells are lossy, with the electric and magnetic loss tangent being significantly

= +1

. Despite these two nonidealities, negative large, refraction is still observed. In order to verify the circuit simulation results presented so far, Fig. 13 shows full-wave simulation results for a Gaussian beam propagating at a PRI/NRI interface. For the simulation we used the commercial full-wave 3D MoM solver FEKO. The total grid size was 54 106 cells, the electrical length of an unloaded unit cell was 18 at the simulation frequency 1 GHz. The microstrip line height was 1 mm, the width 1.61 mm. In order to accelerate the numerical computation, the substrate was taken as air, as then the simpler free-space Green’s function can be used by the MoM solver. For a single microstrip line this geometry results in a characteristic impedance of 100 , however, due to mutual coupling of adjacent (periodic) microstrip lines the (even) mode characteristic impedance was . The upper half space consists of an unloaded transmission. The lower half space line grid, synthesizing transmission-line grid is loaded with lumped reactive elements according to (21) such that is synthesized. Like in the circuit simulations, the ports at the top were excited such that a Gaussian beam of width 1.5 was incident at the PRI/NRI interface with an angle of incidence of 30 . The grid was terminated according to [4] using the Bloch impedance of the dominant plane wave of the Gaussian beam. This setup resulted in a 120 GB MoM matrix which was stored on a SSD, the total solution time using an out-of-core solver was 70 h on eight cores. Fig. 13 depicts the steady-state time-domain result of the

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We can make (5) block-diagonal in the following manner:

(25)

Fig. 13. Full-wave simulation results for a Gaussian beam propagating in a grid of 54 106 unit cells implemented in microstrip technology. Simulation performed using FEKO, the steady-state time-domain E -field across the ground plane is shown. Similar to the setup in Fig. 9 the upper half space consists of an unloaded transmission-line grid which has thus an effective refractive index . The lower half space is loaded with reactive elements such of n is synthesized. that n

2

= +1 = 01

In (25), is the null matrix. The determinant of the system above is the product of the determinant of the diagonal entries. Hence we obtain the eigenmodes of the system by setting . Here we have assumed that the system which does not support degenerate modes are modes where either the voltage vector or the current vector vanishes. II. EIGENMODES AS ROOTS OF CHEBYSHEV POLYNOMIALS

-field across the ground plane of the simulated grid, verifying the circuit simulation results of Fig. 9. The Gaussian beam simulation results presented in this section show that simple ray optics can be used to predict the voltage/current distributions along these electrical grids. This gives further evidence that the association of effective refractive indices, including negative effective refractive indices, to the NRI-TL is physically sound. Moreover, these results clearly refute recent claims in literature that negative refraction is impossible in periodic structures with subwavelength unit cells [15].

of the second kind of order Chebyshev polynomials are generated by the recursive relation (26) where and letting

and

. Dividing all entries in (6a) by we obtain the following equation in :

.. .

.. .

VIII. CONCLUSION In this paper we have examined propagation of plane waves and Gaussian beams in 2D periodic grids constructed with lumped reactive immitances. We have demonstrated the equivalence of the Bloch/Floquet modal description and the coupled-line modal analysis of finite-sized periodic grids. In particular, we have shown that the excitation of modes in an metamaterial grid is completely compatible with the notion of an index of refraction under suitable homogeneous conditions. To further support this point of view we have presented how Gaussian beams propagate within such grids, demonstrating both negative refraction of the power as well as of the phase fronts throughout the extension of the unit cells. These results show that simple ray optics, by means of a suitable effective index of refraction, can be used to predict the voltage/current distribution in transmission-line metamaterial grids. APPENDIX EIGENVALUES OF SYMMETRIC, RECIPROCAL AND LOSSLESS TRANSMISSION MATRICES If the transmission matrix in (4) corresponds to a system which is symmetric, reciprocal and lossless, it can be shown and para-skew Hermitian that the real block matrices matrices and satisfy the following conditions [7]: (24)

(27) We observe that if the two nonzero corner entries in (27) had instead of , the resulting determinant would be a been Chebyshev polynomial of the second kind of order . Nevertheless, the interior determinants of (27) resembles such Cheby. Hence we may expand the shev polynomials of order determinant in (27) first along the top row followed by subsequent expansion along the bottom row to obtain:

(28)

REFERENCES [1] G. V. Eleftheriades, A. K. Iyer, and P. C. Kramer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [2] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Soviet Phys. Usp., vol. 10, no. 4, pp. 509–514, Jan. 1968. [3] A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative index metamaterial,” J. Appl. Phys., vol. 92, no. 10, pp. 5930–5934, Nov. 2002.

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[4] A. Grbic and G. V. Eleftheriades, “Negative refraction, growing evanescent waves and subdiffraction imaging in loaded-transmission-line metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2297–2305, Dec. 2003. [5] L. Brillouin, Wave Propagation in Periodic Structures, 2nd ed. New York: Dover, 2003. [6] L. A. Pipes, “Steady-state analysis of multiconductor transmission lines,” J. Appl. Phys., vol. 12, no. 11, pp. 782–799, Nov. 1941. [7] J. Shekel, “Matrix analysis of multi-terminal transducers,” Proc. IRE, vol. 42, no. 5, pp. 840–847, May 1954. [8] B. E. Spielman et al., “Metamaterials face-off,” Microw. Mag., vol. 10, no. 3, pp. 8–42, May 2009. [9] Negative-Refraction Metamaterials, G. V. Eleftheriades and K. G. Balmain, Eds. Hoboken, NJ: Wiley, 2005. [10] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag. (Special Issue on Metamater.), vol. 51, no. 10, pt. I, pp. 2604–2611, Oct. 2003. [11] M. Zedler and G. V. Eleftheriades, “Spatial harmonics and homogenisation of NRI-TL metamaterial structures,” in Proc. 39th Eur. Microw. Conf., Rome, Italy, 2009, pp. 504–507. [12] M. Zedler and G. V. Eleftheriades, “Spatial harmonics and homogenization of negative-refractive-index transmission-line-structures,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 6, pp. 1521–1531, Jun. 2010. [13] M. Zedler and G. V. Eleftheriades, “Hybridisation of 2D frequencydomain TLM with the MoM-discretised 2D-EFIE,” in Proc. 40th Eur. Microw. Conf., Paris, France, 2010, pp. 1429–1432. [14] N. Engheta and R. W. Ziolkowski, Metamaterials—Physics, engineering and exploration. Piscataway, NJ: IEEE, 2006. [15] B. A. Munk, Metamaterials: Critique and Alternatives. Hoboken, NJ: Wiley, 2009.

Rubaiyat Islam (S’08) received the B.A.Sc. degree in engineering science (electrical option) from the University of Toronto, Toronto, ON, Canada, in 2002. His thesis was on the beam steering of antenna arrays using parasitic loadings and mutual coupling. He is currently working toward the Ph.D. degree at the University of Toronto. He is currently involved in the theoretical investigation of complex modes in metamaterial couplers and its usage in the realization of RF/microwave passive devices such as power dividers and antenna feed networks. His research interests include metamaterials, couplers, phase-shifters, filters, and electromagnetic theory. Mr. Islam held the NSERC PGS D award from 2005 to 2008.

Michael Zedler received the Dipl.-Ing. degree in electrical engineering from RWTH Aachen University of Technology, Germany, in 2002 and the Dr.-Ing. degree from TU München, Germany, in 2008. He is currently a Postdoctoral Researcher with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he is working in the field of metamaterials.

George V. Eleftheriades (F’09) received the diploma in electrical engineering from the National Technical University of Athens, Greece, in 1988, and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 1989 and 1993, respectively. During 1994–1997, he was with the Swiss Federal Institute of Technology, Lausanne. Currently he is a Professor with the Department of Electrical and Computer Engineering, University of Toronto, Canada, where he holds the Canada Research Chair/Velma M. Rogers Graham Chair in Engineering. His research interests include transmission-line and other electromagnetic metamaterials, small antennas and components for wireless communications, passive and active microwave components, plasmonic and nanoscale optical structures, fundamental electromagnetic theory, and electromagnetic design of high-speed interconnects. Professor Eleftheriades received the Ontario Premier’s Research Excellence Award in 2001 and an E.W.R. Steacie Fellowship from the Natural Sciences and Engineering Research Council of Canada in 2004. He served as an IEEE AP-S Distinguished Lecturer during 2004–2009. Among his other scholarly achievements, he is the recipient of the 2008 IEEE Kiyo Tomiyasu Technical Field Award ”for pioneering contributions to the science and technological applications of negative-refraction electromagnetic materials.” He was elected Fellow of the Royal Society of Canada in 2009. He serves as an elected member of the IEEE AP-S AdCom and as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is a member of the Technical Coordination Committee MTT-15 (Microwave Field Theory). He was the general chair of the IEEE AP-S/URSI 2010 International Symposium held in Toronto, during July 11–17, 2010.

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Low Frequencies and the Brillouin Precursor Natalie Cartwright, Member, IEEE

Abstract—When an electromagnetic step-modulated sine wave propagates through a causal dielectric material, the signal evolves into a Brillouin precursor whose peak amplitude point decays algebraically for large, yet finite, propagation distances. This algebraic decay is an apparent contradiction to the Bouger-Lambert-Beer law, which states that each nonzero frequency component of the pulse decays exponentially with propagation distance. Hence, the Brillouin precursor is commonly attributed to the dc or low frequency content of the initial pulse. However, there have been no studies that give the dependence of the peak amplitude on the low frequency content of the initial pulse. We accomplish this here by application of a cascade of single-pole high-pass filters, each with cut-off frequency , to a step-modulated sine wave with carrier frequency . Saddle point methods are used to provide a closed-form asymptotic approximation to the propagated field in a Debye-type dielectric material. Our results show that the Brillouin precursor exists even with the dc and low frequency content suppressed and that a substantial amount of low frequencies must be removed in order to observe a significant decrease in the decay rate of the peak amplitude point of the Brillouin precursor. Index Terms—Dispersion, electromagnetic propagation in absorbing media, precursors, transient pulse phenomena.

I. INTRODUCTION

I

N classical electromagnetic theory, the temporal evolution of an electromagnetic signal through an infinite, isotropic, homogeneous, locally linear, temporally dispersive dielectric material is solely dependent upon the dielectric permittivity of the material. (Here, we assume a relative magnetic permeability of one). Causality requires the dielectric permittivity of all materials to satisfy the Kramer-Kronigs relations; that is, the dielectric permittivity must be a complex function of angular frequency whose real and imaginary parts form a Hilbert transform pair. Thus, the complex wavenumber is also a complex function of and a plane wave electromagnetic signal that travels in the positive direction with propagation factor (1) experiences exponential attenuation as (2) Manuscript received June 15, 2010; revised August 22, 2010; accepted October 01, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant FA9550-08-1-0051. The author is with the Department of Mathematics, State University of New York at New Paltz, New Paltz, NY 12440 USA (e-mail: cartwrin@newpaltz. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122240

For dielectric materials, the attenuation factor is nonzero and positive for all except when at which . Thus, each nonzero frequency component of the input signal is attenuated at the exponential rate (3) whereas the dc component of the input signal experiences no exponential attenuation. This exponential attenuation of all nonzero frequency components is commonly referred to as the Bouger-Lambert-Beer law, or simply Beer’s law [1]–[3]. Despite the exponential attenuation of each nonzero frequency component, an electromagnetic step-modulated sine-wave signal with turn-on time that travels through a causal dielectric will evolve into a Brillouin precursor pulse whose peak amplitude point decays algebraically with large, yet finite, propagation distances, not exponentially [4], [5]. This apparent contradiction of the Bouger-Lambert-Beer law has caused some researchers to suggest that the Brillouin precursor is due to the dc and low frequency components present in the spectrum of the initial pulse [6], [7], as those are the frequencies that experience the least amount of exponential attenuation. However, there has been no quantification of the term low and no studies have shown the dependence of the peak amplitude point on these low frequencies. That then, is the purpose of this paper. Here, we show that, for large yet finite propagation distances, the algebraic decay of the peak amplitude point of the Brillouin precursor is independent of the dc component of the initial pulse spectrum and that a considerable amount of the low frequency content must be removed to observe variation in the rate of algebraic decay. We accomplish this by passing a step-modulated sinusoid of fixed carrier frequency through a cascade of simple-pole filters in order to remove the dc component and to suppress the low frequency components in the spectrum of the input pulse. We then provide a uniform asymptotic approximation to the propagated field of this filtered pulse. The dependence of the peak amplitude point of the Brillouin precursor on the low frequency content of the initial signal is provided by the asymptotic expressions given. Numerical computations that support the analysis are provided. The conclusions are independent of the type of high-pass filter used, as discussed in Section VI. II. INTEGRAL REPRESENTATION OF PROPAGATED FIELD Consider an infinite homogeneous material whose frequency response may be modeled by the causal Debye model [8] in which the relative dielectric permittivity is given by

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(4)

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where is the high-frequency limit of the dielectric permittivity, and denotes the relaxation time of the material. The Debye model is often used to model the frequency response of material such as distilled water, blood and concrete [9]. Let the electric field component of a plane wave pulse on the plane be a step-modulated sine wave of fixed carrier frequency with frequency spectrum . This pulse is passed through a cascade of high-pass filters each with transfer function (5) Here, is the Heaviside step function and denotes the cut-off frequency of the filter at which . The frequency spectrum of the filtered pulse on the input plane is then (6)

(7) where the coefficients (8a)

(8b) are found by performing the required partial fraction expansion. The (unnormalized) magnitude of the spectrum of a step-modulated sine wave [(6) with ] of fixed carrier frequency is shown in Fig. 1 by the solid curve. The dashed, dotted and dashed-dotted curves show the magnitude of the spectrum after the pulse has passed through two, ten and one hundred high-pass filters [(6) with , 10, 100], respectively, each with cutoff frequency . Notice that for , frequencies less than 1 have been effectively suppressed in the spectrum of the filtered pulse and that the dc-component of the filtered pulse is zero [see (5) and (6)]. The propagated field of the filtered pulse on any plane is given exactly by the integral representation (9) (10) Here, is greater than the abscissa of absolute convergence of the initial signal on the plane , is given in (6), denotes the speed of light in vacuum

Fig. 1. Unnormalized spectrum of a step-modulated sine wave with fixed car(solid curve) and the spectrum after the rier frequency pulse has passed through two, ten and one hundred high-pass filters each with . cutoff frequency

plane and Jordan’s lemma yields for . For values , the integral representation of the propagated electric field component appearing in (10) is ideally suited for asymptotic analysis as . In particular, uniform saddle point methods may be used to produce an approximation to the propagated field whose accuracy increases with increasing propagation distance. Uniform saddle point methods require the deformation of the contour of integration through valleys of the accessible saddle points of the complex phase function (solutions of the equation ) with consideration given to coalescing saddle points, endpoints of integration, and poles and branch points of the amplitude function. III. EVALUATION OF THE FILTERED PULSE In the present case of a Debye-type material there is only one relevant saddle point whose location lies on the positive imaginary -axis at , moves down the imaginary -axis with increasing , crosses the real -axis at the space-time point , then continues down the negative imaginary -axis and approaches the branch point of the complex phase function located at as . The amplitude function possesses two poles, a simple pole located at and a pole of order located at [see (6)]. The saddle point is removed from the simple pole appearing at the real value for all , but it does coalesce with the pole at at the space-time point , provided that . Thus, the uniform asymptotic approximation of the propagated field for consists of the contribution from the saddle point and its interaction with the two poles. To proceed, we first express (12)

(11) is the complex phase function of the dielectric material, and is a dimensionless space-time parameter. For values of , the contour may be enclosed in the upper half

where (13)

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A. Contribution From Simple Pole at

and

(14)

The evaluation of [(13)] gives the contribution to the propagated field due to the saddle point and the simple pole located at the real-valued carrier frequency of the input signal. Substitution of (15)–(16) into the integral representation (13) yields

and is the Bromwich contour in (9). Note here that if the step-modulated sinusoidal signal is not filtered ( ), then and . We follow the analysis of Felsen and Marcuvitz [10], [11] and Felsen [12] to obtain a uniform asymptotic approximation for . Define a change of variable by the equation (15) and define

(22)

The first integral appearing in (22) may be expressed in terms of the complementary error function, viz.

(16) (23) (17) where the sign choose is determined by

and

where (24)

(18) and (19)

The second integral appearing in (22) may be evaluated in terms of the gamma function, viz.

are the locations of the poles and in the -plane, respectively, and are regular at and , respectively, and (25) (20) The first term in this series contains the quantity The coefficients are determined in the following analysis. The expansion of about the saddle point located at due to the change of variable (15) is

(26) The expansion of about [given by (21) in which ] shows that will remain finite as provided that . Thus, the leading term in the uniform asymptotic approximation of is

(21) where the argument of presentation.

has been omitted for ease of (27a)

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(27b) . Here, is the space-time point at which the steepest as descent path emanating from the saddle point crosses the pole at causing the sign of to change from negative to positive and the addition of the residue at , as seen in (27b). If , the step-modulated sine wave is unfiltered so that In this case, (27) with [see (8b)] provides the uniform asymptotic expansion of the propagated field of the unfiltered step-modulated sine wave of fixed carrier frequency . Because remains bounded away from zero for all , the large argument expansion of the complementary error function

(28) may be substituted into (27) when the leading term of (27) is

is large. Thus, for large ,

(29) on has been omitted for ease where the dependence of of presentation. Expression (29) is commonly referred to as the Brillouin precursor [13] and is precisely the Brillouin precursor given in [14] for the Rocard-Powles extension of the Debye model.1 Because the saddle point of the complex phase function crosses the origin at the space-time point , and it follows from (11), (12) and (29) that (30) That is, for large enough , the peak amplitude of the unfiltered step-modulated sinusoidal pulse decays algebraically with propagation distance, not exponentially. As an example, we consider a step-modulated sinusoid with carrier frequency , material parameters and and propagation depth m, where . These material parameters are appropriate for modeling the dielectric permittivity of distilled water for frequencies between 0.5 and 3.0 GHz [9]. The asymptotic approximation (27) [with ] 1Note that (15) of this reference has been written incorrectly with pearing in the numerator of the square root rather than the denominator.

ap-

Fig. 2. The asymptotic (dashed) and numerical (solid) approximations to the propagated field of an unfiltered step-modulated sine wave of fixed carrier freat a distance of five absorption depths quency into a Debye medium with material parameters appropriate for distilled water. . The circle denotes the value of the field at the space-time point

is shown in Fig. 2 along with a fast Fourier transform (FFT) computation of the propagated field, by the dashed and solid curves, respectively. The FFT computation used sample points to sample frequencies at the Nyquist rate where . These sampling parameters are used for all numerical computations given in this paper. Note that these sampling parameters result in a minimum frequency of . The asymptotic method, on the other hand, expands the phase function about the saddle point located on the imaginary axis. Despite these differences, the two results are indiscernible. The Brillouin precursor is clearly evident in the figure as its peak amplitude point located about the space-time point is much greater than the amplitude of the signal contribution, which is characterized by oscillations at the carrier frequency . The fact that the asymptotic method produces the same result as the FFT computation with suggests that frequencies below this value are not required to produce a dominant Brillouin precursor. B. Contribution From Pole at The evaluation of [(14)] gives the contribution to the propagated field due to saddle point and the pole of order located at caused by the application of the cascaded filters each with cutoff frequency . Substitution of (15)–(17) into the integral representation(14) yields

(31) The first integral appearing in (31) may be expressed in terms of the complementary error function as in (23). The asymptotic

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expansion of the integrands containing higher order poles can be obtained from (23) by noting that

The coefficients

are the two solutions of (33) in which

(36) (32) where the argument of has been omitted for ease of presentation. The integrand containing may be evaluated in terms of the gamma function as in (25). The first term in this series contains the quantity

(37) where (38) as determined by (8). Thus, the leading term in the uniform asymptotic approximation of for is

(33) about [provided by (21) ] shows that will remain finite as provided that the coefficients of like-ordered poles sum to zero. This requirement provides the equations to solve for the coefficients Once these coefficients have been appropriately defined, the first nonzero term that remains consists of powers of higher-order derivatives of the complex phase function divided by powers of . For the material parameters used here, this term is negligible for but will be included in the expressions given here for completeness. As an example, we consider the case of two high-pass filters ( ) applied to a step-modulated sine wave of fixed carrier frequency so that The expansion of in which

(39a)

(34) to leading order. The uniform asymptotic expansion of the integrand with a simple pole at is given by (23). The uniform asymptotic expansion of the integrand possessing a second-order pole at is determined by (32) to be

(39b)

(35)

as . Here, is the space-time point at which the saddle point crosses the pole at causing the sign

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Fig. 3. The uniform approximation in which , , and into a Debye medium with material parameters appropriate for distilled water. The circle and x-mark deand , respectively. The note the space-time points subtracts the contribution to the propagated field from contribution those low frequencies suppressed by the cascaded filter.

of to change from negative to positive and the addition of the residue at , as seen in (39b). The residue at may be calculated as

(40) for any

. Thus, the residue is of the form times a polynomial in of degree . A plot of the uniform asymptotic expansion of for the case of two filters ( ) both with cutoff frequency is given in Fig. 3. The same carrier frequency, material parameters and propagation distance used for Fig. 2 are used here. It is now evident that is the negative of the low frequency contribution to the field from those low frequencies suppressed by the cascaded filter so that the addition of to subtracts out the low frequency contribution. C. The Propagated Field The uniform asymptotic approximation of the propagated field of a step-modulated sine wave of carrier frequency passed through two high-pass filters each with cutoff frequency in a Debye material is then the sum of (27) in which and (39), as given by (12). The asymptotic approximation (12) is shown in Fig. 4 along with a FFT computation of the propagated field, by the dashed and solid curves, respectively. The same carrier frequency, cutoff frequency, propagation distance, material parameters and numerical sampling as used for Figs. 2 and 3 are used here. Again, the asymptotic result is indiscernible from the numerical calculation. A comparison of the propagated unfiltered signal given in Fig. 2 to the propagated filtered signal given in Fig. 4

Fig. 4. The asymptotic (dashed) and numerical (solid) approximations to the propagated field of a step-modulated sine wave of fixed carrier frequency passed through two high-pass filters each with cutoff frequency at a distance of five absorption depths into a Debye medium with material parameters appropriate to distilled water. The and , circle and x-mark denote the space-time points respectively.

shows that the peak amplitude point of the filtered pulse is lower than that of the unfiltered pulse and is followed by a dip in the field. Both of these attributes in the propagated filtered signal are due to the addition of the term , which is the negative of the low frequency contribution to the field, as well as the decrease in the absolute value of as increases [see (8b)]. IV. DISCUSSION OF ASYMPTOTIC FORMS The dependence of the peak amplitude of a step-modulated sine wave on the low frequency content of the spectrum of the initial pulse is provided by the asymptotic solutions presented above. For large n, these expressions may become unwieldy. However, the equations given above for the case provide sufficient insight to determine the effects of increasing , as discussed here. There are two space-time points at which the complex phase function evaluated at the saddle point is zero. At the space-time point the saddle point crosses the origin so that , which results in the non-exponential attenuation of the peak amplitude of the Brillouin precursor [(29)] for large . There is also the space-time point at which . This zero value of the phase function is most apparent in the residue at . As seen in (39), the residue at is the exponential times a polynomial in of degree . The first peak of the residue occurs at the space-time point and has an approximate negative amplitude of that decreases with increasing . Because , this first peak experiences little exponential attenuation. Thus, this first peak of the residue will subtract from the peak of the Brillouin precursor causing the peak of the Brillouin precursor in the total field to decrease with increasing . For , there are oscillations in the residue that modulate the propagated field structure. These oscillations then interfere with the Brillouin precursor in both a constructive and deconstructive manner. Examples of these residue oscillations are given in Figs. 5

CARTWRIGHT: LOW FREQUENCIES AND THE BRILLOUIN PRECURSOR

Fig. 5. Numerical computation of the propagated field of a step-modulated sine passed through ten high-pass wave of carrier frequency at five absorption filters with each cutoff frequency depths into the Debye material. The circle and x-mark denote the space-time and , respectively. points

Fig. 6. Numerical computation of the propagated field of a step-modulated sine passed through one hundred wave of carrier frequency at five absorphigh-pass filters each with cutoff frequency tion depths into the Debye material. The circle and x-mark denote the space-time and , respectively. points

and 6, which show numerical computations of the propagated field of a step-modulated sine wave of carrier frequency passed through a cascade of ten and one hundred high-pass filters, respectively, each with cutoff frequency , at five absorption depths into the Debye material. V. DECAY RATES If the peak amplitude point of a pulse decays algebraically with relative propagation distance as , where is a constant, then is the slope of the curve . Here, we estimate as the average slope of this expression, which is calculated as the change in natural logarithm of the peak amplitude between successive data points divided by the change in natural logarithm of between successive data points. All results presented here use FFT computations to approximate the propagated field (10) with the same material and sampling parameters as used above. First, we consider a step-modulated sine wave with fixed carrier frequency passed through a cascade of

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Fig. 7. The real and imaginary parts of the index of refraction of the distilled water model. The x marks, cross mark and circles appear at the minimum and maximum frequencies used in the FFT computations, the carrier frequency , considered, respectively. and at the four values of cutoff frequency

two high-pass filters, each with cutoff frequency . We consider four cases for the cutoff frequency: , , , and . We do not consider because, as noted previously, the asymptotic results do not change significantly for and our computadoes not allow for sampling tional sample size limit of below . The real and imaginary part of the index of refraction of this material are shown in Fig. 7. The x marks appear at the minimum and maximum frequencies used in the FFT computations, the cross mark appears at the carrier frequency , and the circles appear at the considered here. four values of cutoff frequency For each case, the peak amplitude point of the pulse is recorded at distances between and into the Debye material. The average slopes of the logarithm of the data of each filtered pulse are represented by the cross marks in Fig. 8 and have been fit with a cubic spline. The solid black curve in Fig. 8 shows the algebraic decay of the unfiltered ( ) pulse. The dashed black curve shows the exponential attenuation at the predominant carrier frequency of the pulse. Similarly, the average slopes of the logarithm of the data of a step-modulated sine wave with fixed carrier frequency passed through a cascade of either ten or one hundred high-pass filters are shown in Figs. 9 and 10, respectively. The case and has been omitted due to the extremely small amplitudes of the propagated pulses. The change in the absolute value of the spectrum (6) for , 2, 10, 100 for the case is shown in Fig. 1. Figs. 8 – 10 show that all pulses experience the exponential decay for small propagation distances, but that there exists some relative propagation distance above which the pulse decays algebraically or else experiences less exponential attenuation. The point of deviation from increases with increasing . Also evident is that the average slope decreases with increasing . The nonsteady behavior seen in some of the cases presented in Figs. 9 and 10 is due to the constructive and deconstructive interference of the oscillations in the residue at with the Brillouin precursor. These numer-

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Fig. 8. Average slope of the logarithm of the data of the propagated field of passed a step-modulated sine wave of carrier frequency . through two high-pass filters each with cutoff frequency

Fig. 11. The asymptotic (dashed) and numerical (solid) approximations to the propagated field of a step-modulated sine wave of carrier frequency passed through a two-pole Butterworth high-pass filter with at five absorption depths into the Debye cutoff frequency material.

The dependence of the peak amplitude of the propagated pulse on the dc content of the initial pulse may be inferred from the case and shown in Fig. 8. [see (5)], the dc component of this filtered Because pulse is zero and yet it decays algebraically, approximately as for large, yet finite, propagation distances. VI. OTHER HIGH-PASS FILTERS

Fig. 9. Average slope of the logarithm of the data of the propagated field of passed a step-modulated sine wave of carrier frequency . through ten high-pass filters each with cutoff frequency

Fig. 10. Average slope of the logarithm of the data of the propagated field of passed a step-modulated sine wave of carrier frequency . through one hundred high-pass filters each with cutoff frequency

ical findings support the analysis given above: the peak ampliso tude of the Brillouin precursor decreases with increasing that longer propagation depths are needed for the Brillouin precursor to become the dominant contribution to the total propagated field.

Although the expressions given above are specific to the case in which a cascade of single-pole high-pass filters are used to suppress the dc and low frequency content of the initial pulse, the same methods may be used and similar expressions obtained for any other high-pass filter, for which the same conclusions would hold. This is because the transfer functions of other causal high-pass filters, such as Butterworth and Chebyshev filters, are analytic in the upper half plane and on the real axis with poles in the lower half plane. The transformation (15) and asymptotic methods presented above may be used to uniformly account for the contributions from the saddle point and each of these poles. Of course the particular temporal structure of the propagated electric field component on any plane may vary because each filter suppresses the low frequencies differently, but the overall behavior of the peak amplitude point of the Brillouin precursor will remain unchanged. As an example, the transfer function of a two-pole Butterworth high-pass filter is where and . Application of the asymptotic method to account for the saddle point and the three simple poles located at and provides a uniform asymptotic approximation to the propagated electric field component on any plane . The asymptotic and numerical approximations to the propagated field of a step-modulated sine wave of carrier frequency passed through a two-pole Butterworth high-pass filter with at five absorption depths into the Debye material is shown in Fig. 11. The field is similar to that obtained from passing the step-modulated sine wave through a cascade of two single-pole filters with cutoff frequency , as seen

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its own rate, thereby rendering the Bouger-Lambert-Beer law inapplicable to the wideband pulse. REFERENCES

Fig. 12. Average slope of the logarithm of the data of the propagated field of passed a step-modulated sine wave of carrier frequency . through a two-pole Butterworth high-pass filter with various levels of

by comparison of Fig. 4 with Fig. 11. The average slopes of the logarithm of the data of the propagated field of a step-modulated sine wave of carrier frequency passed through a two-pole Butterworth high-pass filter with various levels of are shown in Fig. 12. These results are similar to those obtained using a cascade of two single-pole high-pass filters each with cutoff frequency , as seen by comparison of Fig. 8 with Fig. 12. VII. CONCLUSIONS Uniform asymptotic expansions that describe the electric field component of a step-modulated sinusoid that has passed through a cascade of high-pass filters, each with cutoff frequency , have been provided. These expansions quantify the dependence of the peak amplitude point of the Brillouin precursor of a step-modulated sine wave on the dc and low frequency content of the input pulse spectrum. In particular, removal of the dc component does not affect the algebraic decay of the peak amplitude of the propagated pulse and a substantial amount of the low frequency components must be removed from the input spectrum in order to observe significant changes in the decay rate over large, yet finite, propagation distances. Although this algebraic decay appears to be a contradiction to the Bouger-Lambert-Beer law, it must be remembered that Beer’s law assumes monochromatic or, at most, quasimonochromatic radiation. When a wideband pulse propagates through a causal material, each frequency component experiences both attenuation and phase distortion, each at

[1] J. N. Sweetser and I. A. Walmsley, “Linear pulse propagation in stationary and nonstationary multilevel media in the transient regime,” J. Opt. Soc. Am. B, vol. 13, no. 3, pp. 601–612, 1996. [2] U. J. Gibson and U. L. Österberg, “Optical precursors and Beer’s law violations; non-exponential propagation losses in water,” Opt. Exp., vol. 13, no. 6, pp. 2105–2110, 2005. [3] H. Jeong, M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett., vol. 96, p. 143901, 2006. [4] K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer-Verlag, 1994. [5] N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2006. [6] R. Uitham and B. J. Hoenders, “The electromagnetic Brillouin precursor in one-dimensional photonic crystals,” Opt. Commun., vol. 281, pp. 5910–5918, 2008. [7] D. Lukofsky, J. Bessette, H. Jeong, E. Garmire, and U. Österberg, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3612–3618, 2009. [8] P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909. [9] E. G. Farr and C. A. Frost, Impulse Propagation Measurements of the Dielectric Properties of Water, Dry Sand, Moist Sand, and Concrete. Albuquerque, NM: Air Force Weapons Lab., 1997, vol. Note 52, EMP Measurement Notes. [10] L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. [11] L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [12] L. B. Felsen, “Radiation from a uniaxially anisotropic plasma halfspace,” IEEE Trans. Antennas Propag., vol. AP-11, pp. 469–484, 1963. [13] K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin and L. Felsen, Eds. Berlin: Springer, 1995, pp. 257–272. [14] K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1582–1590, 2005.

Natalie Cartwright (M’04) received the M.S. and Ph.D. degrees in mathematics from the University of Vermont in Burlington, in 2000 and 2004, respectively. Currently, she is an Assistant Professor of mathematics at the State University of New York at New Paltz (see, http://www.newpaltz.edu/~cartwrin). Her research interests include asymptotic analysis and ultrawideband pulse propagation through dispersive material.

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Mixed-Impedance Boundary Conditions Henrik Wallén, Ismo V. Lindell, Life Fellow, IEEE, and Ari Sihvola, Fellow, IEEE

Abstract—A novel set of isotropic electromagnetic boundary conditions generalizing some of the recently introduced boundary conditions is introduced. For a planar boundary it is required that the boundary appears as an impedance boundary for the TE and TM components of the field with different surface impedances in the general case. Thus, such conditions can be dubbed as mixed-impedance boundary conditions. It is shown that the conditions can be alternatively expressed in terms of normal components of the fields at the boundary and in this form they can be defined for non-planar boundaries as well. The conditions generalize those of the isotropic impedance boundary and the more recently considered DB and D B boundaries. As a numerical example, scattering from a mixed-impedance sphere is considered, and it is found that the sphere could be useful as a cloaking device. Index Terms—Boundary impedance.

conditions,

scattering,

surface

• Perfect electromagnetic conductor (PEMC) [4]; • Self-dual impedance boundary with [5]; [6], [7]. • Soft-and-hard surface, previous case with Recently, boundary conditions in terms of field components normal to the boundary have been considered [8]–[10] and two conditions dubbed as DB-boundary and D B -boundary conditions were respectively defined for the planar boundary with as (2) and (3)

I. INTRODUCTION HE impedance boundary condition defines a linear relation between time-harmonic electric and magnetic field components tangential to the boundary surface. The basic form with scalar surface impedance was introduced by Shchukin [1] and Leontovich [2] in the 1940’s [3]. Denoting the outer normal unit vector by , its general form can be expressed as

T

(1) Here the subscript denotes component tangential to the sur, while is the two-dimensional surfaceface, impedance dyadic. Actually, (1) can be interpreted as two-dimensional extension of Ohm’s law if we replace the tangential magnetic field in terms of the effective surface current density . The most general impedance condition involves four scalar in any coordiparameters, the components of the dyadic nate system on the boundary. As special cases of the impedance boundary we may list the following. ; • Isotropic impedance boundary ; • Perfect electric conductor (PEC) ; • Perfect magnetic conductor (PMC)

Manuscript received May 05, 2010; revised August 30, 2010; accepted November 01, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported in part by the Academy of Finland. The authors are with the Department of Radio Science and Engineering, Aalto University School of Electrical Engineering, Espoo, Finland (e-mail: henrik. [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123064

The DB conditions (2) were introduced already in 1959 by Rumsey [11] but applications were suggested only very recently [12]–[18]. It was shown that the planar DB boundary acts as a PEC plane for fields polarized TE with respect to the normal of the plane and PMC plane for the TM field [8], [11]. Similarly, the planar D B boundary acts as PMC and PEC planes for the TE and TM fields, respectively [10]. The purpose of this paper is to study a novel more general set of boundary conditions which follow by assigning separate impedance boundary conditions for the TE and TM components of the field. II. TE/TM DECOMPOSITION It is well known that, in a homogeneous and isotropic medium, outside the sources, any field can be decomposed in partial fields polarized TE and TM with respect to a fixed direction in space [19]. Defining the direction by , we can write (4) (5) The decomposed fields satisfy individually the Maxwell equations (6) (7) (8) and the divergences of the four vectors vanish. From these we obtain the following relations:

0018-926X/$26.00 © 2011 IEEE

(9) (10)

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where one should pay attention that, in the last terms, the total fields are involved.

TABLE I SOME SPECIAL CASES OF MIXED-IMPEDANCE BOUNDARY CONDITIONS

III. MIXED-IMPEDANCE BOUNDARY CONDITIONS Let us now consider a new type of electromagnetic boundary on which the impedance boundary conditions are valid for the , the TE and TM fields separately. At the planar boundary are required field components in an isotropic half space to satisfy the conditions (11) (12) and are two surface impedances. Obviously, where the boundary appears as a regfor the case ular isotropic impedance boundary with the surface impedance . Equation (12) can also be written as (13) Operating (11) by

ever a physical realization for the mathematical boundary can be found. Finding a realization for the MI boundary must be left as a topic of future work. However, it is known that localized sources cannot create TEM fields so that the problem of uniqueness is limited to sources of infinite extent. IV. POWER CONSIDERATIONS An isotropic impedance boundary with surface impedance , where the surface resistance and surare real, is passive when and lossless face reactance . Therefore, it seems reasonable to assume that an when MI boundary with surface impedances (17)

and applying (9) we obtain is passive when both surface resistances are non-negative

(18)

(14) . Similar where at the last step we have applied steps for (13) yield a similar result. In this way the following boundary conditions are obtained for the normal components of the total fields (15) (16) which are valid when each of the TE and TM components sees the boundary as an impedance boundary. To prove the equivalence between the two pairs of conditions we also need to show that (15) and (16) implies (11) and (12), which seems difficult in the general case. It is, however, straightforward to show the equivalence between the conditions for arbitrary TE and TM polarized plane waves. Since in the form (15), (16) the boundary conditions do not depend on the TE/TM decomposition of the fields, they appear more universal and we can call them mixed-impedance (MI) boundary conditions. The MI boundary is an isotropic boundary condition, since there is no preferred direction along the boundary, but this isotropy should not be confused with the isotropy of an ordinary impedance surface. Since the introduction was based on the TE and TM decomposition the conditions (15), (16) are not meaningful for fields which cannot be uniquely decomposed in TE and TM parts with respect to the given direction, i.e., fields which are TEM with respect to the axis. A plane wave incident normally to is an example of such a field. This the boundary plane means that, just like in the case of the DB or D B boundary [10], for such special fields the boundary conditions are not sufficient and require an additional condition which emerges when-

and lossless when . To show that this is indeed the case is straightforward for a plane wave reflecting from a planar MI boundary, and the Mie-computations below also confirm this assumption for the plane-wave scattering from an MI sphere. However, for general fields and boundary shapes the problem appears more complicated, and we leave the question whether (18) is a sufficient condition for passivity for a future study. V. SPECIAL CASES It was already mentioned that for the MI boundary becomes an ordinary isotropic impedance boundary. and PMC This includes the limiting cases PEC . Also, the DB and D B boundaries [8]–[10] are special cases of the MI boundary, as summarized in Table I. In [20] a generalization for the DB and D B boundary conditions (2), (3), dubbed as the generalized DB (GDB) boundary conditions, was introduced. The GDB boundary conditions were expressed in the form (19) (20) where is a parameter. From this we can see that the DB condiwhile the D B conditions tions (2) are obtained for . are obtained for The GDB boundary conditions form a special case of the MI boundary conditions. In fact, (16) corresponds to (19) for

(21)

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and (15) corresponds to (20) for

(22) Actually, one can note that when the two surface impedances satisfy the relation (23) the MI boundary becomes a GDB boundary, where the paramcan be complex. eter It was shown previously that the GDB boundary and its special cases, the DB and D B boundaries are self dual, i.e., there exists a duality transformation in term of which the boundary conditions are invariant [20]. One can show that the MI boundary is self dual when the condition (23) is , valid. In fact, making the duality transformation in (15) and (16) the pair of conditions is invariant only if (23) is valid. Thus, the GDB boundary can also be called as the self-dual mixed-impedance boundary. The two impedance parameters can be expressed in terms of as two other parameters (24) where can be called the self-dual surface-impedance parameter and the anti-self-dual surface-impedance parameter. In the impedances satisfy (23) while for a fact, for minus sign must be added to the right side of (23). The parameters and are real for a resistive MI boundary, and in that case the passivity condition (18) corresponds to (25) For a lossless MI boundary, both and are imaginary. The condition for an isotropic impedance boundary implies the condition (26) which is the equation of a hyperbola. In the resistive case, , we get the one-sided hyperbola shown in the upper part of Fig. 1 when we require the surface resistance to be non-negative , both the upper and (passive). In the reactive case, lower branches of the hyperbola shown in the lower part of Fig. 1 are equally reasonable. , Another interesting special case is when which corresponds to the hyperbolic condition (27) which is also plotted in the lower part of Fig. 1. This condition is reasonable only in the reactive case, since would require an active boundary for one of the polarizations. , while The DB boundary condition corresponds to the PEC, PMC, and D B conditions require infinite values of and/or as indicated in Fig. 1.

Fig. 1. Some interesting special cases of the MI boundary in terms of the selfdual and anti-self-dual impedance parameters s, a for the resistive case (above) and reactive case (below).

VI. NON-PLANAR BOUNDARIES It does not appear straightforward to generalize the conditions by , the unit (11), (13) to non-planar surfaces by replacing normal of the boundary surface. In fact, it requires that at the surface is associated to a coordinate system in the space outside the surface and that there exists a TE/TM decomposition of the field with respect to that coordinate. It is better to start from the definition (15), (16) which is not directly associated with a global TE/TM decomposition. Following [10] we should now write the conditions as (28) (29)

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However, at the boundary, and can be still associated to surface impedances when the field is locally approximated by reflecting plane waves decomposed in TE and TM parts with respect to the direction of the local normal. As a concrete example let us consider the region outside the . It is well known that any field can be spherical surface globally decomposed in TE and TM components with respect [21]. Thus, we can start from to the radial unit vector the MI boundary conditions of the form (11), (13) written as (30) (31) and show that they lead to (28) and (29). we now have Because of (32) (33) whence we can operate (30) as

[24], and applying the boundary conditions (37) and (38), we get the scattering amplitudes (39) (40)

where is the size parameter, but otherwise all formulas from [24] remain the same.1 Of particular interest are perhaps the absorption, total scattering and backscattering of the MI sphere, and their dependence on the parameters and . For this purpose, we computed , the absorption efficiency , the extinction efficiency the scattering efficiency , and the backscattering efficiency for a representative selection of parameters presented in Figs. 2–5. These efficiencies are defined as the corresponding cross section divided by the physical cross section, and extinction is the sum of scattering and absorption (41)

(34)

As predicted by the extinction paradox [24], the extinction efficiency approaches 2 for large spheres

by . After similar operations on and replace thus become (31), the boundary conditions at (35) (36) which equal (28) and (29) written in spherical coordinates. In [22] we considered modes in a spherical resonator with DB boundary and, in [20], [23], plane wave scattering from a spherical object with DB, D B and GDB boundaries. It was found that, due to their self-dual character, there is no backscattering from any of these objects. Obviously this should no longer be true for the MI sphere without the self-dual property, i.e., for . To verify this, some numerical tests were made. VII. SCATTERING FROM A MIXED-IMPEDANCE SPHERE Let us consider the scattering of a plane wave from an MI sphere with radius in free space. In terms of the self-dual and anti-self-dual parameters and , the MI conditions (35) and can be expressed as (36) at (37) (38) It is obvious that, making the duality transformation to the fields, , is equivalent with replacing i.e., interchanging and by . This scattering problem can be solved using ordinary Mieseries with very small modifications, as was done for the DB, D B and GDB spheres in [20], [23]. Following the analysis in

(42) and all efficiencies are small for small spheres. The most inis of the teresting effects happen when the size parameter order one, i.e., when the circumference of the sphere is near one free-space wavelength. Fig. 2 shows all four efficiencies for a resistive MI sphere and varying real parameters with size parameter . As expected, the backscattering is zero, , , and if and only if the anti-self-dual parameter vanishes, the absorption is positive when the passivity condition (25) is satisfied. The largest variations in scattering and , and this absorption happens for a self-dual MI sphere case is also plotted in Fig. 3. The absorption has the maximum when and vanishes when and , which correspond to the DB and D B conditions, respectively. , when both the impedThe absorption is zero, ances and the parameters are purely imaginary. The extinction and backscattering efficiencies, and , for such a lossless reactive MI sphere are plotted . in Fig. 4, where the size parameter is chosen to be As already mentioned, the backscattering is zero if and only if , but we can also see that both and appear to have large and sharp maxima for the reactive MI sphere in Fig. 4. To get a better quantitative view of the efficiencies, we present a few horizontal cuts from Fig. 4 in Fig. 5, where a small real part is added to slightly smoothen the curves without 1Notice, however, that we use the e time convention, which corresponds j compared with [24]. to the replacement i

!0

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=1

05 < s; a < 5. = 0. The same

Fig. 2. Extinction, absorption, scattering and backscattering efficiencies for a resistive MI sphere with size parameter kr as a function of The dashed green line marks the boundary where a s and the black contour lines are drawn at integer values of Q, with a thicker line at Q color-scale is also used in all four sub-figures to make comparisons easier.

=6

( =

Fig. 3. Extinction, absorption and scattering efficiencies for a self-dual a resistive s MI sphere with size parameter kr as a function of the self-dual parameter s. (The backscattering is zero, Q .)

0)

(Im = 0)

=1 0

removing any one of the main peaks. The sharpest maxima in the figures are naturally very sensitive to losses, but it seems that a reactive MI sphere can exhibit fairly large extinction and backscattering efficiencies even when losses are included. It is interesting to notice that the efficiencies plotted in Figs. 2 and 4 are symmetric with respect to the anti-self-dual parameter , while the sign of the self-dual parameter is very important. In the reactive case in Fig. 4, the total scattering is stronger when the TE reactance is capacitive and the TM reactance is inductive

, , ) than in the opposite case. For ( , in Fig. 5 we have the instance, in the self-dual case, at , maximum extinction efficiency while the minimum is at . in Fig. 5 suggests that the MI The very low minimum boundary condition could be very interesting from a cloaking perspective. We could, theoretically, place the object to be cloaked inside a sphere with MI boundary conditions on the outside, and choose the MI parameters and so that the total extinction is minimized. Looking at the results presented above, , it seems that the and also testing various ways to minimize optimal choice is , , with depending of the sphere. In this case we have on the size parameter and no absorption , zero backscattering . The results in Fig. 6 show which implies that that the minimum extinction (and scattering) efficiency is very much smaller than for a PEC sphere of the same size is small, while the extinction paradox (42) implies when for very large spheres. In between, for that around unity, we can significantly reduce the total extinction compared with a PEC sphere, provided that the needed MI boundary condition is somehow realizable. The cloaking is never perfect, but the performance might be competitive with other proposed approaches [25]. The GDB sphere considered in [20] corresponds to the reacand using the tive self-dual MI sphere with present notation.

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Fig. 6. Minimum extinction efficiency Q for a self-dual (a = 0) lossless (Re s = 0) MI sphere as a function of the size parameter kr . The left curve shows Q relative to the extinction efficiency of a PEC sphere Q , and the right curve shows the corresponding optimal choice of Im s. For small size parameters, say kr < 1, the MI sphere looks promising as a cloaking device.

Fig. 4. Extinction and backscattering efficiencies for a reactive MI sphere (Re s = Re a = 0) with size parameter kr = 1:5 as a function of 5 < Im s; Im a < 5. The absorption is exactly zero, and the backscattering is zero only when a = 0.

0

the normal total field components in a form which does not depend on the TE/TM decomposition of the field. In this form they can be expressed for non-planar boundaries as well. The MI boundary is a generalization of the previously defined DB, D B , and GDB boundaries, but also the more common PEC, PMC and isotropic impedance boundaries can be expressed as special cases of the MI boundary. As an example, a spherical MI boundary was considered and its scattering properties for some impedance parameters were studied. In the ideal lossless case, the MI sphere can be either a very strong or very weak scatterer depending on the choice of MI parameters. Choosing optimal parameters, it appears that the MI sphere could have some potential for electromagnetic cloaking purposes. A theoretical implementation of the DB boundary was presented in [9] and some promising steps towards an experimental realization was recently presented in [26], but the realization of the more general MI boundary is still an open problem. REFERENCES

Fig. 5. Extinction and backscattering efficiencies for a reactive MI sphere with size parameter kr = 1:5 and either a = 0 or a = j as a function of Im s. A small real (resistive) part, Re s = 0:02, is added to make the curves smoother, without removing the main peaks.

VIII. CONCLUSION A novel set of electromagnetic boundary conditions was introduced under the name mixed-impedance (MI) boundary conditions. Expanding the field in TE and TM polarized components, a planar MI boundary appears as an impedance boundary for the TE field and for the with surface impedance TM field. It was shown that the conditions can be expressed for

[1] A. N. Shchukin, Propagation of Radio Waves. Moscow: Svyazizdat, 1940. [2] M. A. Leontovich, “Methods of solution for problems of electromagnetic waves propagation along the earth surface,” (in Russian) Bull. Acad. Sci. USSR, Phys. Ser., vol. 8, no. 1, p. 1622, 1944. [3] G. Pelosi and P. Y. Ufimtsev, “The impedance-boundary condition,” IEEE Antennas Propag. Mag., vol. 38, no. 1, pp. 31–35, Feb. 1996. [4] I. V. Lindell and A. H. Sihvola, “Perfect electromagnetic conductor,” J. Electromagn. Waves Appl., vol. 19, no. 7, pp. 861–869, 2005. [5] I. V. Lindell, Methods for Electromagnetic Field Analysis, 2nd ed. New York: IEEE Press, 1995. [6] P.-S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, no. 3, pp. 168–170, 1988. [7] P.-S. Kildal and A. Kishk, “EM modeling of surfaces with stop or go characteristics—Artificial magnetic conductors and soft and hard surfaces,” ACES J., vol. 18, no. 1, pp. 32–40, Mar. 2003. [8] I. V. Lindell and A. H. Sihvola, “Uniaxial IB-medium interface and novel boundary conditions,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 694–700, Mar. 2009. [9] I. V. Lindell and A. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E, vol. 79, no. 2, p. 026604, 2009. [10] I. V. Lindell and A. Sihvola, “Electromagnetic boundary conditions defined in terms of normal field components,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1128–1135, Apr. 2010.

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[11] V. H. Rumsey, “Some new forms of Huygens’ principle,” IRE Trans. Antennas Propag., vol. 7, no. Special supplement, pp. S103–S116, Dec. 1959. [12] B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett., vol. 100, no. 6, p. 063904, 2008. [13] A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 10, p. 115022, Nov. 2008. [14] A. D. Yaghjian and S. Maci, “Corrigendum: Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 11, p. 039802, Mar. 2009. [15] R. Weder, “The boundary conditions for point transformed electromagnetic invisibility cloaks,” J. Phys. A, vol. 41, no. 41, p. 415401, 2008. [16] I. V. Lindell, A. Sihvola, P. Ylä-Oijala, and H. Wallén, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2725–2731, Sep. 2009. [17] P.-S. Kildal, “Fundamental properties of canonical soft and hard surfaces, perfect magnetic conductors and the newly introduced DB surface and their relation to different practical applications including cloaking,” in Proc. ICEAA’09, Torino, Italy, Aug. 2009, pp. 607–610. [18] P.-S. Kildal, A. Kishk, and Z. Sipus, “Introduction to canonical sufaces in electromagnetic computations: PEC, PMC, PEC/PMC strip grid, DB surface,” in Proc. 26th Annual Review of Progress in Applied Computational Electromagnetics (ACES 2010), Tampere, Finland, Apr. 26–29, 2010, pp. 514–519. [19] D. S. Jones, The Theory of Electromagnetism. Oxford: Pergamon Press, 1964, p. 19. [20] I. V. Lindell, H. Wallén, and A. Sihvola, “General electromagnetic boundary conditions involving normal field components,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 877–880, 2009. [21] J. G. Van Bladel, Electromagnetic Fields. Hoboken-Piscataway, NJ: Wiley-IEEE Press, 2007, pp. 314–317. [22] I. V. Lindell and A. H. Sihvola, “Spherical resonator with DB-boundary conditions,” Prog. Electromag. Res. Lett., vol. 6, pp. 131–137, 2009. [23] A. Sihvola, H. Wallén, P. Ylä-Oijala, M. Taskinen, H. Kettunen, and I. V. Lindell, “Scattering by DB spheres,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 542–545, Jun. 2009. [24] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983, ch. 4. [25] P. Alitalo and S. Tretyakov, “Electromagnetic cloaking with metamaterials,” Mater. Today, vol. 12, no. 3, pp. 22–29, Mar. 2009. [26] S. Hrabar, D. Zaluski, D. Muha, and B. Okorn, “Towards experimental realization of DB metamaterial layer,” presented at the META’10, 2nd Int. Conf. on Metamaterials, Photonic Crystals and Plasmonics, Cairo, Egypt, Feb. 22–25, 2010, Poster C17. Henrik Wallén was born in 1975 in Helsinki, Finland. He received the M.Sc. (Tech.) and D.Sc. (Tech.) degrees in electrical engineering from Helsinki University of Technology (which is now part of the Aalto University), in 2000 and 2006, respectively. He is currently working as a Postdoctoral Researcher at the Department of Radio Science and Engineering, Aalto University School of Science and Electrical Engineering, Espoo, Finland. His research interests include electromagnetic theory, modeling of complex materials and computational electromagnetics. Dr. Wallén is Secretary of the Finnish National Committee of URSI (International Union of Radio Science).

Ismo V. Lindell (S’68–M’69–SM’83–F’90–LF’05) was born in 1939 in Viipuri, Finland. He received the degrees of Electrical Engineer (1963), Licentiate of Technology (1967), and Doctor of Technology (1971), from the Helsinki University of Technology (HUT), Espoo, Finland. In 1962, he joined the Electrical Engineering Department, HUT, since 1975 as Associate Professor of Radio Engineering and, since 1989, as Professor of Electromagnetic Theory at the Electromagnetics Laboratory which he founded in 1984. During a sabbatical leave in 1996–2001 he held the position of Professor of the Academy of Finland. Currently he is Professor Emeritus at the Department of Radio Science and Engineering, Aalto University School of Electrical Engineering. He utilized a Fulbright scholarship as a Visiting Scientist at the University of Illinois, Champaign-Urbana, in 1972–1973, and the Senior Scientist scholarship of the Academy of Finland at the Massachusetts Institute of Technology, Cambridge, in 1986–1987. He has authored and coauthored 265 refereed scientific papers and 12 books including, Methods for Electromagnetic Field Analysis (IEEE Press, New York 3rd printing 2002), Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood MA, 1994), Differential Forms in Electromagnetics (Wiley and IEEE Press, New York 2004), and Long History of Electricity (Gaudeamus, Helsinki, Finland 2009, in Finnish). Dr. Lindell received the IEEE S.A. Schelkunoff award (1987), the IEE Maxwell Premium (1997 and 1998) and the URSI van der Pol gold medal in 2005, as well as the State Award for Public Information (2010).

Ari Sihvola (S’80–M’87–SM’91–F’06) was born on October 6, 1957, in Valkeala, Finland. He received the degrees of Diploma Engineer in 1981, Licentiate of Technology in 1984, and Doctor of Technology in 1987, all in electrical engineering, from the Helsinki University of Technology (TKK), Finland. Besides working for TKK and the Academy of Finland, he was a Visiting Engineer in the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, in 1985–1986, and in 1990–1991, he worked as a visiting scientist at the Pennsylvania State University, State College. In 1996, he was a Visiting Scientist at Lund University, Sweden, and for the academic year 2000–2001, he was a Visiting Professor at the Electromagnetics and Acoustics Laboratory, Swiss Federal Institute of Technology, Lausanne. In summer 2008, he was a Visiting Professor at the University of Paris XI, France. Currently he is a Professor of electromagnetics at Aalto University School of Electrical Engineering (before 2010, known as the Helsinki University of Technology) with interest in electromagnetic theory, complex media, materials modelling, remote sensing, and radar applications. Dr. Sihvola is Chairman of the Finnish National Committee of URSI (International Union of Radio Science) and a Fellow of IEEE. He was awarded the five-year Finnish Academy Professor position starting August 2005. Since January 2008, he is Director of the Graduate School of Electronics, Telecommunications, and Automation (GETA).

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Uniform Ray Description for the PO Scattering by Vertices in Curved Surface With Curvilinear Edges and Relatively General Boundary Conditions Matteo Albani, Senior Member, IEEE, Giorgio Carluccio, and Prabhakar H. Pathak, Fellow, IEEE

Abstract—A new high-frequency analysis is presented for the scattering by vertices in a curved surface with curvilinear edges and relatively general boundary conditions, under the physical optics (PO) approximation. Both, impenetrable (e.g., impedance surface, coated conductor) as well as transparent thin sheet materials (e.g., thin dielectric, or frequency selective surface) are treated, via their Fresnel reflection and transmission coefficients. The PO scattered field is cast in a uniform theory of diffraction (UTD) ray format and comprises geometrical optics, edge and vertex diffracted rays. The contribution of this paper is twofold. First, we derive PO-based edge and vertex diffraction coefficients for sufficiently thin but relatively arbitrary materials, while in the literature most of the results (especially for vertex diffraction) are valid only for perfectly conducting objects. Second, the shadow boundary transitional behavior of edge and vertex diffracted fields is rigorously derived for the curved geometry case, as a function of various geometrical parameters such as the local radii of curvature of the surface, of its edges and of the incident ray wavefront. For edge diffracted rays, such a transitional behavior is found to be the same as that obtained heuristically in the original UTD. For vertex diffracted rays, the PO-based transitional behavior is a novel result providing offers clues to generalize a recent UTD solution for a planar vertex to treat the present curved vertex problem. Some numerical examples highlight the accuracy and the effectiveness of the proposed ray description. Index Terms—Asymptotic diffraction theory, electromagnetic scattering, geometrical theory of diffraction (GTD), physical optics (PO), uniform theory of diffraction (UTD).

I. INTRODUCTION new high-frequency analysis is presented for the scattering by vertices in a curved surface with curvilinear edges and relatively general boundary conditions, under the physical optics (PO) approximation. PO is an old and well established approach for calculating the electromagnetic field scattered by an object [1]. Its main feature is that it can be easily applied to an arbitrary shaped structure with relatively arbitrary

A

Manuscript received April 12, 2010; revised October 22, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. M. Albani and G. Carluccio are with the Department of Information Engineering, University of Siena, 53100 Siena, Italy (e-mail: matteo.albani@dii. unisi.it; [email protected]). P. H. Pathak is with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123062

boundary conditions imposed on its surface. Therefore PO represents a valid tool to model the interaction of an incident electromagnetic field with real-world, electrically large, complex geometries; and it is extensively used in many engineering applications (reflector antennas, radar cross section, propagation, etc.) when the computational effort of a numerical full wave method, such as the Method of Moments, becomes too demanding. PO is also useful to calculate the electromagnetic field in those canonical configurations for which an exact solution is not available (e.g., rough surfaces [2]–[4]), thus deriving an analytical result which, unlike numerical simulations, usually helps in providing physical insight and compact description of the scattering phenomenon. In this paper, we address the problem of providing a high-frequency ray description for the scattering by curved smooth surfaces with a curvilinear polygonal contour containing curved edges and vertices, and a relatively general boundary condition. By using a unified approach, both thin impenetrable materials and (partially) transparent thin sheets can be treated in the same fashion, as in [5]. The surface is assumed to be electrically large and slowly varying, and it is characterized by local dyadic reflection and transmission coefficients [6], [7]; the transmission vanishes in the impenetrable material cases. This allows the treatment of various kind of surfaces like impedance surfaces, thin material coated conductors, radar absorbing materials, thin material sheets, or frequency selective surfaces (FSS), etc., once the proper reflection and transmission coefficients are defined. Under the PO approximation, an explicit expression for field scattered by the surface is given in the form of a surface integral, which is asymptotically evaluated by using the general procedure in [8]. The result is cast in the typical ray format of UTD [9]. Besides the standard geometrical optics (GO) reflected and forward-scattered/transmitted rays, edge and vertex diffracted rays are also obtained. These diffracted ray field contributions are expressed in terms of novel edge and vertex diffraction coefficients. Such diffraction coefficients do not satisfy the correct boundary condition on the surface; i.e., they do not exhibit the correct angular diffraction pattern especially at grazing, since they are based on the approximate PO [10]. However, they describe the correct transitional behavior of the rays inside the ray shadow boundary transition regions because that is dictated only by the geometry of the curved surface and the associated ray geometry. The transitional behavior of the edge diffracted field is found to be exactly the same as that derived previously in [9]–[12]. There, the straight wedge diffraction coefficient, derived in [11]

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for the perfectly electric conducting (PEC) case, is extended to the curved wedge geometry by heuristically enforcing the compensation of GO ray field discontinuities across their shadow boundaries (SBs), [9]–[12]. Therefore our result also provides a rigorous justification for the heuristic development of [9]–[12]. The transitional behavior for the field diffracted by a vertex in a curved surface which is formed by the interconnection of curved edges is a new result. It can be considered a generalization of the planar vertex UTD diffraction coefficients obtained previously in [13], [14] and of the planar vertex PO diffraction coefficient in [15] to deal with the curved geometry and to the arbitrary material case, under the PO approximation. As is well-known, the PO formulation we adopted does not reconstruct creeping rays, which are therefore not considered in the present paper. The paper is organized as follows. Section II introduces the integral formulation for the PO scattered field, which is then asymptotically evaluated to provide various ray contributions. Next, Sections III–Section V are specifically devoted to present, in detail, the calculation of the GO, edge diffracted and vertex diffracted ray fields, respectively. Some numerical examples are shown in Section VI to emphasize the accuracy and the effectiveness of the formulation. Lastly, Section VII contains some time dependence for the fields concluding remarks. An is assumed and suppressed.

Fig. 1. Scattering by a curved surface S with curved edges and vertices. The scattered field at the observation point P is calculated as the free-space radiation of equivalent surface currents J and M distributed at any integration point Q on the surface S with the scatterer removed.

II. FORMULATION We consider a curved surface bounded by a curvilinear polygonal contour. The surface is illuminated by an arbitrary astigmatic ray field (see Fig. 1). By assuming the scattering surand face as being electrically thin, and by denoting by the parts of the surface that are in the lit and the shadow regions, scattered by is calculated as respectively (Fig. 2), the field the free-space radiation of the equivalent electric and magnetic surface currents on , where denotes the unit outward normal to (see Fig. 2). Hence, we have

(1) is the vector from the integration point where on to the observation point . In (1), it is assumed that is sufficiently large that only the radiative asymptotically leading term of the dyadic Green’s functions needs to be retained, while and denote the free-space (or a homogenous background ambient) wavenumber and impedance, respectively. Under the PO approximation, the electric and magnetic fields at any directly illuminated point on the scattering surface are approximated by those produced on a locally flat surface which is tangential at that point. Namely, as the sum of the incident and the reflected fields on , and as the transmitted fields on . The reflected and the transmitted electric fields can generally be related to the incident field as and

Fig. 2. Reflection and transmission at a curved surface and associated unit vectors. The scattering surface is treated as infinitely thin from the geometrical (ray tracing) point of view. However its electrical (small) thickness is properly accounted by the Fresnel reflection and transmission coefficients.

, by using the local dyadic reflection and transmission coefficients, [5]. Such coefficients are easily obtained for a large class of materials, either impenetrable (like impedance surfaces or thin material coated metallic surfaces) or transparent (like thin material slabs or FSS), and are conveniently expressed via unit vectors that are fixed in and out of the plane of incidence. The latter unit vectors are associated with field components that are parallel and perpendicular to the incident plane, [5]–[7]; i.e., and (Fig. 2). For impenetrable surfaces . For electrically thin scattering panels, the reflection and transmission coefficients can be conveniently referred to the nominal reference surface midway through and by using a phase compensation factor. In this way the two parts and of the equivalence principle integration surface can be geometrically assumed to . Consebe coincident with . In this limit quently, the scattering surface is geometrically assumed as infinitely thin; i.e., from the ray tracing point of view. The entire reflection mechanism which arises between the two panel and is described by a unique cumulative reboundaries flected ray which includes multiple bounce or bulk effects via the proper dyadic reflection coefficient, accounting for the panel electrical thickness. Analogously, a unique transmitted ray accounts for multiple bounce or bulk transmission effects via a proper dyadic transmission coefficient, accounting for the panel

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denoting the curvature dyad of the with spherical scattered wavefront. C. PO Scattered Field Surface Integral By substituting (3) into (2), the PO scattered field is recognized to be represented by a rapidly oscillating surface integral on of the form

(6) with a slowly varying vectorial integrand

Fig. 3. Astigmatic ray representation of the incident field wavefront.

electrical thickness. Furthermore, for each GO ray field (incident, reflected, transmitted), the magnetic field is readily expressed in terms of the electric field via the local plane wave , where denotes the wave propaassumption . Note that gation direction, for and . Therefore, (1) is rearranged as

(7) and a total phase function , whose quadratic approximation is simply derived by summing up (4) and (5). In what follows, by applying the procedure provided in [8], an asymptotic evaluation of (6) is explicitly carried out. Indeed, the inte) dominated by the gral (6) is asymptotically (i.e., for critical points of the phase function , and it can be asymptotically approximated as the sum of contributions associated with each critical point,

(2) in which

denotes the unit dyad and , for

.

A. Incident Field Local Representation The incident field at any point on the surface resented as an arbitrary astigmatic ray

is rep-

(3) where and define the incident field wavefront and polarizaand , tion, respectively. It is noted that which follow from the eikonal equation, [16]. In the neighboron the surface , as in Fig. 1, the hood of a reference point wavefront function admits the quadratic expansion, [16]

(4) in terms of the incident wavefront curvature dyad involving the radii of curvature in the principal directions , for which (Fig. 3). In (4) is the vector pointing the integration point from the reference point (Fig. 1). B. Propagation Phase Factor Expansion An analogous quadratic expansion around for the Green’s function spherical wave [16]

can be repeated

(5)

(8) Such a mathematical procedure results in a ray description of the scattered field as a sum of rays arising at each critical point on (Fig. 4). Various types of critical points include the following. where , which cori) Stationary phase points respond to GO forward scattered, or reflected ray field . Note that the forward scattered contributions ray comprises the transmitted field (if present) and the negative of the incident field. , i.e., points on the conii) Partial stationary phase points tour of where the phase is stationary along the edge, , which correspond to edge difnamely where originating at . fracted ray field contributions where the contour of has a disconiii) Vertex points tinuous derivative (tangent vector), which produces the , arising vertex diffracted ray field contributions . from According to [8], the approximation of each contribution only includes the local leading term of the asymptotic expansion. In particular, higher order terms in the asymptotic expansion of stationary phase point contributions are not considered here, though of the same asymptotic order as corner contributions since they are generally not important for most engineering applications in comparison with the uniformly valid edge and corner diffracted field contributions. In connection to the fact that the interior stationary or edge stationary phase points may or may not occur on , one uses on each such term a unit in (8) which vanishes within the respective step function ray shadow region, thus bounding the various rays inside their domains of existence. Nevertheless, such abrupt field discontinuities across shadow boundaries are smoothly compensated

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Fig. 5. Local parametric representation of the surface S in the neighborhood of a the stationary phase point Q .

(4) and (5), the dependence of the total phase lated as function of the integration variables

of (6) is calcu, , for which

Fig. 4. Ray description for the field at P scattered by the surface S when illuminated by a point source at P . The shading on S is proportional to cos(k') and visualizes the phase oscillations of the radiation integral (6). Various types of critical points are shown with the respective rays.

by the uniform description of edge and corner diffraction contributions, thus providing a continuous scattered field. It is noted that, as pointed out in [8], other kind of field transitions associated with the coalescence of two or more critical points may occur [17], [18], which are not uniformly described here. Indeed such transitions, corresponding to GO or edge diffraction caustic effects, occur for some aspect angles only at specific near field distances from the scattering surface, [19]. However, in most engineering applications such occurrences will generally be rare for observation points sufficiently far from the surface. On the other hand, the abovementioned edge and corner induced shadow boundaries pertaining to truncations of GO and edge diffracted ray systems, respectively, are always present regardless of the observation distance and are therefore fully accounted for in the present paper. The next three sections report the uniform asymptotic expressions for the various types of ray field contributions. III. GEOMETRICAL OPTICS RAYS First we consider a stationary phase point on which may correspond to either a forward scattered or a reflected GO ray. We choose a suitable local mapping for the position of any integration point belonging to the surface in the neighbor, again by resorting to a quadratic expansion hood of [16]

(10) , therefore stationary In (10), where , phase points are either transmission points where ; in both the cases or reflection points . A. The Forward Scattered Ray Field Let us consider first the case of a transmission point . The Hessian matrix of the phase function reduces to

(11) whence it is found

(12) . Furthermore The stationary phase point contribution yields the asymptotic approximation for the field scattered through the surface ; this contribution is observed in the region below into points, and it is given by which (13) with

(9) (14) where are coordinates in the local tangent plane and is the local curvaalong the ture dyad of involving the radii of curvature respective principal surface direction (Fig. 5). Note that as defined earlier . By using (9) in

denoting the GO field transmitted through the surface . Note denotes the classical (as opposed to GO) that in (13) incident field, thus, in (3), the exists everywhere in space with which exists only the obstacle absent; this is in contrast to

ALBANI et al.: UNIFORM RAY DESCRIPTION FOR THE PO SCATTERING BY VERTICES IN CURVED SURFACE WITH CURVILINEAR EDGES

in the lit region of the incident wave. In (8), the forward scattered which is field (13) is multiplied by the unit step function zero in the region illuminated by the direct field, and is unity in its shadow region behind the surface. Indeed, this stationary lies on only when observing the field in the phase point is complementary to the unit step shadow region. Therefore associated to the direct GO incident field [9]; function i.e., . If is impenetrable so that , then , and which is (13) reduces to the well known GO estimation of the forward scattering in the shadow region [20]. It is worth reemphasizing that (13) represents the GO contribution to the scattered field (8) in the incident field shadow region, and it includes the transmitted ray field and the negative of the incident field. When (13) is added to the incident field (defined everywhere) to provide the total field, one properly obtains which is present only in the incident field shadow region for penetrable materials; for impenetrable materials vanishes everywhere due to the fact that now .

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and the determinant of the [Fig. 6(a)]. Note that is unity at , hence Jacobian of the mapping . The edge is locally mapped by and the integral (6) is locally extended to the half-plane . Since the edge also follows the local surface curvature, the total edge and it is given by curvature is along (18) with being the radius of curvature of in the edge direction [Fig. 6(b)]. Therefore the position of any is point on in the neighborhood of

(19) By using (19) in (4) and (5), the dependence of the total phase of (6) is calculated as a function of the integration variables ,

B. The Reflected Ray Field Second, let us consider the case of a reflection point . Here, the Hessian matrix of the phase function reads (20) (15)

In (20) we assumed that for a diffraction point , the observais such that ; i.e., lies tion direction . Note that in (20) on the Keller’s cone

(16)

(21)

with radii of curvature of the reflected wavefront (see [9, Appendix]) and . The standard formula ([8, Eq. (35)]) for the stationary phase contribution now yields

; indeed is a staand tionary point along the edge. Furthermore, from (20) the Hessian matrix of reads

(17)

(22)

After some manipulations, it is found that

which is the GO reflected field [9], [10], [21], as expected. In (8), the reflected field (17) contains the unit step function , which is unity in the region illuminated by the reflected field and is zero in its shadow region. As a matter of fact, the stationary phase belongs to , and its asymptotic contribution has to point be accounted for, only when lies in the region illuminated by the reflected field.

Hence, it is found that

(23) where is the distance between the caustic at the edge and the second caustic of the astigmatic diffracted ray defined by [9] and [12]

IV. THE EDGE DIFFRACTED RAY FIELD Next, we consider an edge stationary phase point ; i.e., a point on the contour of where the phase variation along the is introduced; edge is stationary. A suitable local mapping at namely, the vector in the plane locally tangent to is remapped in terms of a coordinate along as the edge and a coordinate orthogonal to it, with denoting the radius of curvature of the edge in the local tangent plane

(24) with denoting the radius of curvature of the incident field in the plane containing and . The above quantities allow the calculation of the partial stationary phase point contribution in the asymptotic evaluation of (6) as in [8,

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case”. Indeed, by representing the unit vectors in the edge coorand dinate system with origin at

(28) with the upper/lower sign applying to on approaching the shadow boundaries

case, it follows that

Fig. 6. Local parametric representation of the surface S in the neighborhood of a partial stationary phase point Q . (a) 2D view of the mapping in the local tangent plane; (b) 3D view of the mapping on the surface S .

Eq. (36)]. Thus, the edge diffracted ray field arising at found to be

is

(25) in which the dyadic edge diffraction coefficient is defined as

(26) with the quantities in the numerator evaluated at the diffraction , so that must be assumed in . Expression point (26) is cast in the typical UTD format and comprises a non-uniform dyadic diffraction coefficient, which is singular at the incident and reflection shadow boundaries, multiplied by the UTD transition function which provides the non-singular or uniform description of the field inside the transition regions close to the GO shadow boundaries. The PO diffraction coefficient in (26) is similar to that in [10] for a perfectly conducting wedge. Its non-uniform part retains the geometrical information of a , whereas it is indestraight wedge, locally tangent to at pendent of the surface and edge curvature. The argument of the transition function, which is calculated as in [8, Eq. (38)] is given by

(27) and it also involves local curvatures. Despite the fact that PO-based non-uniform part of the diffraction coefficient is approximate outside the transition regions (e.g., it does not nevertheless fulfill the boundary condition on the surface), correctly reproduces the transitional behavior of the edge diffracted field inside the GO shadow boundary transition region due to the proper uniform asymptotic procedure utilized here. It is interesting to note that at and near the GO incidence and and , reflection shadow boundaries, as the argument of the transition function (27) reduces to the expression that was previously derived in [9] and [12] by heuristically enforcing the compensation of the GO field “for PEC

(29) where are the distance parameters defined in [9, Eq. (56)]. Thus, the present result in (27) lends further justification to that obtained previously in [9] via physically based heuristic arguments. Finally, in (8), the edge diffracted field (25) is multiplied by , which is equal to 1 or 0, if lies the unit step function in the region illuminated by the edge diffracted field or in its shadow region, respectively. In the latter case, no partial stais present along the surface boundary tionary phase point and consequently this ray does not contribute to the scattered field. It can be observed that the edge diffracted field abruptly disappears within shadow boundary cones (corresponding to the Keller edge diffraction cones) which originate at the vertex, where the edges truncate [13]. The role of vertex ray contributions is that of compensating for the discontinuity of the edge diffracted fields at their shadow boundaries, and thereby to provide a continuous scattered field. V. THE VERTEX DIFFRACTED RAY FIELD Lastly, we consider the contribution that arises at a vertex ; i.e., a point on the boundary of where the tangent to the boundary is discontinuous. Again, a suitable mapping is introwhich parameterizes the coordinates in duced at the vertex the locally tangent plane by using two coordinates , along the two edges

(30) with and the unit vectors tangent and edge curvatures in the normal to edge 1,2 (Fig. 7), and local tangent plane [for each edge, the same arrangement as that in Fig. 6(a) is assumed]. The determinant of the Jacobian of the at is , so that mapping , while the point on the surface is given by

(31) The edge 1,2 is parameterized by , and the surface . integration in (6) locally covers the quarterplane Yet, the total curvature of th edge combines its curvature

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in the local tangent plane and the local curvature of the surface in the th edge direction, as in (18). The local parameterization in (31) at the vertex is then used in (4) and (5) to derive the explicit dependence of the phase function in (6) on the integration variables Fig. 7. Local parametric representation of the surface S in the neighborhood of a vertex Q .

(32) Hence, the phase gradient and Hessian matrix become

corresponding ones obtained previously in [13] and [23] for the special case of a planar faceted PEC geometry containing straight edges which terminate at the vertex. To this end we again introduce two spherical coordinate systems with origin at and axes , for , 2. the vertex Next, the incidence and observation directions are expressed in terms of their angular coordinates as in [13], [23], namely

(33) and

(39) (34) respectively. By applying [8, formula (37)], the uniform asymptotic expression for the vertex diffracted field is obtained in the UTD form as

By using (39) in (37), around the edge diffracted field shadow of the th edge ( or 2) where boundary cone and , it is found that (37) reduces to

(40) (35) where the uniform dyadic vertex diffraction coefficient is

(36) Here the PO-based vertex ray contribution (35) has a vertex diffraction coefficient (36) consisting of a non-uniform part that has information only on the locally tangent geometry (flat surface angular sector with straight edges), while the remaining quantity in (36) is the proper UTD vertex transition function [14], [22] that provides a uniform description of the transitional behavior of the vertex ray field since it explicitly contains the information on the actual local geometry (curvatures at the vertex). The arguments of are given by [8, Eqs. (39)–(40)], namely:

(37)

(38) Also in this vertex case, we analyze the behavior of the arguments (37) of the transition function and compare these to the

which is [13, Eq. (25)], or [23, Eq. (3b)]. In (40), is the distance between the caustic at the vertex and the second caustic (i.e., other that at the edge) of the ray diffracted by edge at , as defined in (24). In the vertex double transition, the vertex contribution has to compensate for the appearance/disappearance of the GO, edge 1, and edge 2 diffracted ray fields. In this regime, , simultaneously, and the UTD vertex transition function admits the limit , and the vertex contribution grows to the asymptotic order . The paand the distance rameter, together with the parameters in (35) permits to reconstruct the correct GO astigmatic ray spreading factor thus guaranteeing the continuity of the total field. VI. NUMERICAL RESULTS In this section we present two examples to show the effectiveness and the accuracy of the proposed asymptotic formulation. First we consider a curved thin lossless dielectric panel, with and thickness , relative dielectric constant where is the wavelength. The reflection and transmission dyadic coefficients are calculated according to [7]. The panel has a smooth parabolic shape (see Fig. 8), parameterized by the and coordinates while the surface height coordinate is given by the parabola equation (where the are given in terms of ). The panel surface

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Fig. 8. Scattering by a curved thin lossless dielectric panel; geometrical arrangement. The panel has a parabolic surface shape trimmed by a curved quadrilateral boundary, and it is illuminated by an electric dipole from the convex side. The scattered field is observed on a circular scan. Dimensions are in wavelength.

Fig. 9. PO scattered electric field for the arrangement in Fig. 8. Asymptotic ray field (continuous line) and numerical integration (circles). The asymptotic PO scattered field associated with the various kind of rays is also plotted separately; i.e., GO (dashed line) field, sum of the edge diffracted ray fields (dash-dotted line), sum of the vertex diffracted ray fields (dotted line).

is trimmed on the rectangular domain in plane; hence, the curved panel surface has a curvilinear the , , 2, 3, 4 and quadrilateral boundary with four vertices , , 23, 34, 41. The curved four curved edges panel is illuminated from its convex side by a -oriented, unit strength electric current moment, located at . radius Fig. 9 shows the PO scattered electric field along a circular scan, with center at the origin of the reference system, , while the elevation angle ranges from in the plane to 180 . The field contributions of the various kind of rays are plotted separately besides the total scattered field (continuous line), which is the sum of them all. In particular, the various ray contributions are shown as GO (dashed line), sum of edge diffracted contributions (dash-dotted line), and sum of vertex diffracted contributions (dotted line), respectively. and in the region In the region above the surface , the asymptotic PO scattered below the surface field is mainly described by the reflected and by the transmitted

Fig. 10. Scattering by a curved thin lossless grounded dielectric panel: geometrical arrangement. The panel has a parabolic surface shape trimmed by a curved quadrilateral boundary, and it is illuminated by an electric dipole from the convex side. The scattered field is observed along two circular scans on or(black continuous line) and  (red/gray thogonal planes  dashed line).

= 50

= 140

GO contributions, respectively. When observing at the Incident or at the Reflection Shadow Shadow Boundary on the right side of the scan, the GO Boundary stationary phase (transmission or reflection, respectively) point merges with the edge stationary phase (edge diffraction) point on the edge . Consequently the associated edge diffracted ray contribution undergoes a typical UTD-like edge transition and provides a proper compensation to the GO the stationary phase (edge discontinuity. In addition, at at edge merges into the vertex diffraction) point and abruptly disappears up to , where it appears again at the same vertex. At these aspects, the vertex diffracted experiences a transition [13] ray contribution arising at and compensates for the discontinuity of the edge diffracted contribution. On the opposite side of the scan, at the Incident and Reflection Shadow Boundaries, respectively, the GO and two edge diffracted rays disappear simultaneously because the transmission/reflection point and the two edge diffraction points and , on edges and respectively, all merge into the vertex . Here, the vertex diffracted contribution arising at experiences a proper “double” transition [13], thus providing the total field continuity. The uniformity of our closed form asymptotic representation of the PO integral is verified by the smoothness of the total field (continuous line) across the various individual ray field discontinuities and shadow transitions. Its accuracy is highlighted by the comparison against the numerical evaluation of the same PO integral (circles). The two lines overlap except at the minimum field grazing aspects where the difference is about 2 dB. Next, we consider a curved grounded dielectric panel with and thickness (Fig. 10). Indeed, the lower side of the panel is assumed to be covered by a perfectly conducting ground sheet. As a consequence, according to [12], , whereas ; i.e., the surface is impenetrable. The curved panel is again parabolic and parameterized

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tion of the scattering phenomenon. This UTD solution presents a new PO vertex diffraction coefficient that correctly compensates for the GO and edge diffracted ray field discontinuities at their relevant shadow boundaries. This solution can be used to efficiently characterize the scattering of electrically large surfaces, modeled by Splines or NURBS, with a relatively general boundary condition (modeled by using local Fresnel reflection and transmission dyadic coefficients), under the PO approximation, without performing the time consuming numerical integration. REFERENCES

Fig. 11. Amplitude of the electric PO scattered field for the grounded dielectric slab of smooth convex parabolic shape illuminated by an electric dipole on the and  (dotted line for the numerical solution two scan planes  and continuous line for the asymptotic evaluation).

= 50

= 140

by the and coordinates, while the surface height coordinate is given by the parabola equation (again in ). Now the surface is trimmed on the rectangular domain in the plane. The slab is illuminated by a unit strength -oriented electric current . The scattered field is observed moment at along two perpendicular circular scans in the planes (continuous black line) and (red/gray dashed line). , while the elevation The radius of the two scan circles is to 180 . The amplitude of the elecangle ranges from tric PO scattered field is shown in Fig. 11, along the two scans at (black) and (red/gray), as predicted by both the proposed closed-form asymptotic ray expressions (lines), and by the numerical surface integration (dots) which is the reference solution. Like in the previous example, the uniformity of our closed form asymptotic representation of the PO integral is verified by the smoothness of the total field curves. In both the scans, the asymptotic solution agrees very well with the numerical evaluation of the integral. The two predictions overlap ex, where cept at the minimum field grazing aspects they differ slightly. This difference may be reduced by introducing higher order diffraction effects (e.g., slope diffraction), that are not considered in the present formulation. It is worth noting that, when the surface is electrically large, the conventional numerical PO integration becomes very time consuming, conversely the asymptotic PO solution developed here remains highly efficient since the latter is size independent and in closed form. VII. CONCLUSION In this paper we provide a uniform asymptotic evaluation of the PO integral on curved surfaces with a curvilinear polygonal contour and relatively general boundary conditions. This solution is expressed in the UTD format. The GO, the edge diffracted, and the vertex diffracted fields are expressed in terms of the local radii of curvature of the incident wavefront, of the scattering surface, and of the reflected/diffracted wavefront, thus providing a clear physical interpretation and compact descrip-

[1] H. M. Macdonald, “The effect produced by an obstacle on a train of electric waves,” Phil. Trans. Royal Soc. London, ser. A, Math. Phys. Sci., vol. 212, pp. 299–337, 1912. [2] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. New York: Macmillan, 1963. [3] G. Franceschetti, A. Iodice, A. Natale, and D. Riccio, “Stochastic theory of edge diffraction,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 437–449, Feb. 2008. [4] M. Albani, “Stochastic theory of edge diffraction: An alternative formulation,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2495–2497, Aug. 2009. [5] Y. Rahmat-Samii and A. N. Tulintseff, “Diffraction analysis of frequency selective reflector antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 4, pp. 476–487, Apr. 1993. [6] T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics. London, U.K.: Inst. Elect. Eng., 1995, IEE Electromagnetic Waves Series. [7] W. D. Burnside and K. W. Burgener, “High frequency scattering by a thin lossless dielectric slab,” IEEE Trans. Antennas Propag., vol. AP-31, no. 1, pp. 104–110, Jan. 1983. [8] G. Carluccio, M. Albani, and P. H. Pathak, “Uniform asymptotic evaluation of surface integrals with polygonal integration domains in terms of UTD transition functions,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1155–1163, Apr. 2010. [9] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, no. 11, pp. 1448–1461, Nov. 1974. [10] P. H. Pathak, “Techniques for high-frequency problems,” in Antenna Handbook, Y. T. Lo and S. W. Lee, Eds. New York: Van Nostrand Reinhold, 1988, ch. 4. [11] P. H. Pathak and R. G. Kouyoumjian, “The Dyadic Diffraction Coefficient for a Perfectly-Conducting Wedge” Ohio State Univ., ElectroScience Lab., Dep. Elec. Eng., Columbus, 1970, Rep. 2 1834. [12] R. G. Kouyoumjian and P. H. Pathak, “The Dyadic Diffraction Coefficient for a Curved Edge,” Ohio State Univ., ElectroScience Laboratory, Dep. Elec. Eng., Columbus, 1973, Rep. 3001-3. [13] M. Albani, F. Capolino, G. Carluccio, and S. Maci, “UTD vertex diffraction coefficient for the scattering by perfectly conducting faceted structures,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3911–3925, Dec. 2009. [14] K. C. Hill, “A UTD Solution to the EM Scattering by the Vertex of a Perfectly Conducting Plane Angular Sector,” Ph.D. dissertation, Ohio State University, Dept. Elect. Eng., Columbus, 1990. [15] N. D. Taket and R. E. Burge, “A physical optics version of the geometrical theory of diffraction,” IEEE Trans. Antennas Propag., vol. 39, no. 6, pp. 719–731, Jun. 1991. [16] G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE, vol. 60, no. 9, pp. 1022–1035, Sep. 1972. [17] P. H. Pathak and M. C. Liang, “On a uniform asymptotic solution valid across smooth caustics of rays reflected by smoothly indented boundaries,” IEEE Trans. Antennas Propag., vol. 38, no. 8, pp. 1192–1203, Aug. 1990. [18] J. H. Meloling and R. J. Marhefka, “A caustic corrected UTD solution for the fields radiated by a source on a flat plate with a curved edge,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1839–1849, Dec. 1997. [19] V. A. Borovikov, Uniform Stationary Phase Method. London, U.K.: Inst. Elect. Eng., 1994, IEE Electromagnetic Waves Series, ch. 5. [20] P. Y. Ufimtsev, “Black bodies and shadow radiation,” Transl.:Translated by Scripta Technica Soviet J. Commun. Technol. Electron., vol. 35, no. 5, pp. 108–116, 1990.

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[21] V. A. Fock, Electromagnetic Diffraction and Propagation Problems. Oxford, U.K.: Pergamon Press, 1965, ch. 6 and 8. [22] F. Capolino and S. Maci, “Simplified closed-form expressions for computing the generalized Fresnel integral and their application to vertex diffraction,” Microwave Opt. Tech. Lett., vol. 9, no. 1, pp. 32–37, May 1995. [23] F. A. Sikta, W. D. Burnside, T. T. Chu, and L. Peters Jr., “First-order equivalent current and corner diffraction from flat plate structures,” IEEE Trans. Antennas Propag., vol. 31, no. 4, pp. 584–589, Jul. 1983.

Matteo Albani (M’98–SM’10) received the Laurea degree in electronic engineering (1994) and the Ph.D. degree in telecommunications engineering (1999) from the University of Florence, Italy. He is an Adjunct Professor in the Information engineering Department, University of Siena, Italy, where he is also Director of the Applied Electromagnetics Lab. His research interests are in the areas of high-frequency methods for electromagnetic scattering and propagation, numerical methods for array antennas, antenna analysis and design. Dr. Albani was awarded the “Giorgio Barzilai” prize for the Best Young Scientist paper at the Italian National Conference on Electromagnetics in 2002 (XIV RiNEm).

Giorgio Carluccio was born in 1979 and grew up in Ortelle, Lecce, Italy. He received the Laurea degree in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Siena, Italy, in 2006 and 2010, respectively. Since 2008, he has been collaborating with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, where he was an invited Visiting Scholar from October 2008 to March 2009. Since February 2010, he has been a Research Associate with the Department of Information Engineering, University of Siena. His research interests are focused on asymptotic high-frequency techniques for

electromagnetic scattering and propagation, complex source and Gaussian beam electromagnetic field diffraction.

Prabhakar H. Pathak (F’86) received the Ph.D. degree from the Ohio State University, Columbus, in 1973. Currently, he is a Professor Emeritus at the Ohio State University, where his main area of research is in the development of uniform asymptotic theories (frequency and time domain) and hybrid methods for the analysis of electrically large electromagnetic (EM) antenna and scattering problems of engineering interest. He is regarded as a co-contributor to the development of the uniform geometrical theory of diffraction (UTD). Presently, he is developing new UTD ray solutions, for predicting the performance of antennas near, on, or embedded in, thin material/metamaterial coated metallic surfaces. Recently his work has also been involved with the development of new and fast hybrid asymptotic/numerical methods for the analysis/design of very large conformal phased array antennas for airborne/spaceborne and other applications. In addition, he is working on the investigation and development of Gaussian Beam summation methods for a novel and efficient analysis of a class of large modern radiation and scattering problems including the analysis/synthesis of very large spaceborne reflector antenna systems. He has published over 100 journal and conference papers, as well as authored/coauthored chapters for seven books. Prof. Pathak has presented several short courses and invited lectures both in the U.S. and abroad. He has often chaired and organized technical sessions at national and international conferences. He was invited to serve as an IEEE Distinguished Lecturer from 1991 through 1993. He also served as the chair of the IEEE Antennas and Propagation Distinguished Lecturer Program during 1999–2005. Prior to 1993, he served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for two consecutive terms. He received the 1996 Schelkunoff best paper award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He received the George Sinclair award in 1996 for his research contributions to the O.S.U. ElectroScience Laboratory, and the Lumley Research Award in 1990, 1994 and 1998 from the O.S.U. College of Engineering. In July 2000, he received the IEEE Third Millennium Medal from the Antennas and Propagation Society. He is an elected IEEE Fellow and an elected member of US Commission B of the International Union of Radio Science (URSI). He served as an IEEE AP-S AdCom member during 2008-2010.

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A Novel Microwave Tomography System Using a Rotatable Conductive Enclosure Puyan Mojabi, Member, IEEE, and Joe LoVetri, Senior Member, IEEE

Abstract—A novel microwave tomography (MWT) setup is proposed wherein a rotatable conductive enclosure is used to generate electromagnetic scattering data that are collected at each static position of the enclosure using a minimal antenna array having as few as only four co-resident elements. The antenna array remains fixed with respect to the target being imaged and only the boundary of the conductive enclosure is rotated. To show that non-redundant scattering data can be generated in this way several 2D transverse magnetic imaging examples are considered using single-frequency synthetic data. For each example, the reconstruction of the complex permittivity profile is compared to that obtained using a homogeneous open-region MWT setup having 16 co-resident antennas. The weighted 2 -norm total variation multiplicative-regularized Gauss-Newton inversion (MR-GNI) is used for all inversions and for the new MWT setup the data collected at all positions of the conductive enclosure are inverted simultaneously. The quality of images obtained from the two systems is similar, but the advantage of the new configuration is its use of a fixed minimal antenna array which will put less of a burden on the numerical system model. Index Terms—Microwave tomography, inverse scattering.

I. INTRODUCTION

C

ONTRIBUTIONS to microwave tomography have been made in all aspects of the technology, especially the development of improved inversion algorithms, e.g., [1]–[5]. During the past two decades, the actual physical setup used to collect the required electromagnetic scattering data has not undergone much innovation, other than the diverse antenna or transducer systems that have been reported, e.g., [6]–[13]. Obtaining good images from MWT requires the accurate collection of a substantial amount of electromagnetic scattering data, which, for efficiency, is best performed using a relatively large number of co-resident antennas. For example, in the systems described in [6] and [13] the number of elements in the arrays range from 16 to 24 where small monopoles or Vivaldi antennas have been used. The large arrays facilitate gathering bistatic scattering data at many angles without mechanically repositioning the antennas. The antenna elements themselves are typically not taken fully into account in the electromagnetic system model of the associated nonlinear optimization problem,

Manuscript received May 20, 2010; revised August 17, 2010; accepted November 10, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T5V6, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123066

although this is an important consideration in achieving good images (cf. the antenna compensation schemes in [14]–[16]). Including the antennas in the system model is a way of reducing the modeling error that exists between the numerical system model, used in the inversion algorithm, and the actual system, from which data is collected. Modeling error also occurs when assuming a homogeneous unbounded domain for the numerical system model (i.e., assuming that the matching fluid extends to infinity) because boundary conditions (BCs) for the dielectric discontinuity, at the MWT system’s casing, are actually required to properly account for the finite extent of the matching-fluid region. Both the antenna and the BC modeling errors can be reduced by the use of a lossy matching fluid of sufficiently high loss such that electromagnetic energy returning from the boundary or any passive antenna to any receiving antenna is not appreciable. Although this may reduce the modeling errors, the net effect of using a lossy matching fluid in MWT systems may be to reduce the accuracy of the complex permittivity profile reconstructions because the addition of any loss reduces the dynamic range and achievable signal-to-noise ratio of the system. To achieve as much accuracy and resolution as possible from an MWT system it is important to not rely on matching fluid loss to diminish both types of modeling errors (loss should only be used to reduce the contrast so as to allow more energy to penetrate the target). Thus, unless a complex numerical system model is to be used—one which accurately models the co-resident antennas as well as the boundaries of the system—the only way to reduce modeling error is to either (i) incorporate specialized calibration techniques for the measured data, or (ii) construct MWT systems that retain the capacity to provide large amounts of independent scattered field data but can be modeled accurately and efficiently. The purpose of this paper is to propose a novel MWT system within a rotatable conductive enclosure that uses a minimal antenna array which is fixed with respect to the target being imaged. In this system, scattered-field data is obtained by taking bistatic measurements between each pair of elements of the fixed array at several different static positions of the rotatable enclosure. Note that this configuration is fundamentally different than existing MWT systems where either the antennas are moved with respect to a fixed object of interest or the object of interest is moved with respect to a fixed antenna array. The inverse problem is formulated for the two-dimensional (2D) transverse magnetic (TM) case and the enclosure is chosen to have a triangular shape. Although it is not easily shown with numerical experiments using synthetic data, the practical implementation of this system should reduce both types of modeling error: the BCs at the conductive-enclosure boundary are easily

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modeled and the antenna modeling error will be minimized because, as will be shown, small arrays with as few as four elements can be used. The shape of the enclosure is chosen to be triangular because it is the polygon that allows the greatest number of fixed-angle step-rotations before producing a redundant configuration. We note that recently, Wadbro and Berggren have considered MWT in a rotating metallic hexagonal-shaped container where the object of interest is illuminated by waveguides connected to each side of the metallic container [17]. The container, along with the waveguides, can then be rotated to collect more scattering data and topology optimization techniques were used to invert the data [17]. At each rotation such a system produces the identical incident field with respect to the boundary of the enclosure because the sources (i.e., the waveguides) remain fixed with respect to the boundary. In the system described here, each rotation of the boundary produces a different incident field with respect to the boundary. The paper is organized as follows. The formulation of the mathematical problem is given in Section II. In Section III, we attempt to answer the following question: Can MWT systems with different BCs provide non-redundant scattering information about the object of interest? Based on the observations made in Section III, the proposed system is explained in Section IV. Finally, the results will be summarized in Section V.

The MWT problem may then be formulated as the minimization over of the least-squares (LS) data misfit cost-functional (2) is the simulated scattered field at the observawhere tion points corresponding to the predicted contrast for the th denotes the -norm on . The weighting transmitter, and is chosen to be (3) To treat the ill-posedness of the problem, we form the multiplicative regularized cost-functional at the th iteration of the algorithm as [5], [18], [19] (4) as the Here, we consider the multiplicative regularizer weighted -norm total variation of the unknown contrast, defined as [5], [18] (5) The weighting

II. PROBLEM FORMULATION

is chosen to be (6)

We consider the MWT problem for the 2D TM case where time-harmonic fields are used to interrogate the object of inis implicitly asterest (OI). Thus, a time factor of consumed. Consider a bounded imaging domain taining a non-magnetic OI and a measurement domain outside of the object of interest. The OI is immersed in a known non-magnetic homogeneous matching fluid with a relative complex permittivity of . In this paper, we assume that and are located either within a perfect electric conductor (PEC) of arbitrary shape or within an open region environment. In any case, the boundary condition of the environment is assumed to be known. The complex electric contrast function is defined as

(1) is the unknown where denotes a position vector in and relative complex permittivity of the OI at . In MWT, the OI is successively interrogated with a number of known incident , where denotes the number of the acfields tive transmitter. Interaction of the incident field with the OI results in the total field . The total and incident electric fields are then measured by some receiver antennas located on . Thus, the scattered electric field, , is known at the receiver positions on . The goal is to find the electric contrast in a bounded imaging domain , which conon . tains the OI, from the measured scattered fields

denotes the reconstructed contrast at the th itwhere eration of the algorithm, denotes the spatial gradient operator with respect to the position vector , and is the area of . The choice of the positive parameter is explained below. Herein, we consider the discrete form of the MWT problem is discretized into a complex where the contrast function vector . The measured scattered data on the discrete measure. The ment domain is denoted by the complex vector is the stacked version of the measured scattered vector fields for each transmitter. Assuming that the th transmitter is active, the simulated scattered field corresponding to the predicted contrast at the th iteration of the GNI algorithm, , is . The vector is then formed by stacking denoted by . The positive parameter is chosen to be where represents the discrete form of and is the area of a single cell in the uniformly discretized domain . Applying the Gauss-Newton Inversion (GNI) algorithm to the discrete form of (4), the contrast vector at the th iteration of the inversion algorithm is then updated as where is an appropriate step length and is the correction vector. In this work, we utilize the line search algorithm described in [18], [20]. The correction vector can be found from (7) represents the discrete form of the operwhere is the Jacobian matrix which contains the derivatives ator, of the simulated scattered field with respect to the contrast and evaluated at . The discrepancy vector is given as

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and [5], [18], [21]. We note is a weighted Laplacian opthat the regularization operator erator which provides an edge-preserving regularization [22]. Throughout this paper, we use the outlined GNI algorithm, sometimes referred to as the multiplicative regularized GaussNewton inversion (MR-GNI) algorithm, to invert the synthetic data sets. All synthetic data sets are generated on a different grid than the ones used in the inversion algorithm. We also add 3% RMS additive white noise to the synthetic data set using the formula [23]

(8) where is the scattered field on the measurement domain due to the th transmitter obtained by the forward solver, and are two real vectors whose elements are uniformly and 1, and distributed zero-mean random numbers between . The vector , constructed by stacking the vec, is then used to test inversion algorithms against tors synthetic data sets. III. DIFFERENT BOUNDARY CONDITIONS FOR MWT As mentioned earlier, in most MWT systems currently in existence [6], [10]–[13], [24] the OI and the antennas are contained within an enclosed chamber, usually made from a dielectric material such as plexiglass. The dielectric chamber is usually filled with a lossy matching fluid, e.g., a 87:13 glycerin:water solution is used in the Dartmouth College microwave tomography system [1], [25]. Most MWT algorithms used to invert data from these systems assume that the matching fluid extends to infinity, not to the boundary of the dielectric casing. That is, they assume that the scattering data is collected in a homogeneous embedding. In other words, the BC for the problem will be the Sommerfeld radiation condition. We will refer to the scattering data collected in such systems as the open-region scattering data. More recently, researchers have considered MWT where the OI and the antennas are enclosed by a circular metallic enclosure [26]–[32]. We have also considered microwave tomography inside conducting cylinders of arbitrary shapes [33]. The use of conducting enclosures imposes a zero boundary condition on the total field which can be easily modeled within the utilized inversion algorithm. We will refer to the scattering data collected in such systems as the PEC-enclosed-region scattering data. In this section, we show inversion results from the open-region and PEC-enclosed-region scattering data. For the PEC-enclosed-region scattering data, we consider PEC enclosures of two different shapes. Calculation of the Jacobian matrix and the simulated scattered field require repeated forward solver calls. For the openregion case, we utilize the method of moments (MoM) with the conjugate gradient algorithm accelerated by the fast Fourier transform (CG-FFT). The CG-FFT forward solver is also accelerated by employing the marching-on-in-source-position technique [34]. Motivated by the desire to model arbitrary PEC boundaries with both straight and curved edges, we use a finite element method (FEM) based on triangular elements for

Fig. 1. Relative complex permittivity of the synthetic target I: (a) Re( (b) Im( ).

) and

the PEC-enclosed systems. The FEM provides an accurate and fast forward solver, and in fact, is easier to implement with a PEC boundary than with absorbing boundary conditions, which are required for a homogeneous embedding. As the FEM mesh is based on triangles, and the inverse solver based on rectangular pulse-basis functions, we interpolate as required between the two meshes with a bi-linear interpolation algorithm [33]. As the first case, we consider the target shown in Fig. 1. The target consists of three circular regions. Two of these circular regions have the same radius of 0.015 m and their relative complex and permittivities are at the frequency of 1 GHz. These two circular regions are surrounded by another circular region with radius of 0.06 m and . The relative comrelative permittivity of , and represent the relaplex permittivities tive complex permittivity of human breast tumor, fibroglandular, and adipose tissue respectively based on the single-pole Debye model [35]. The target is immersed in a background medium of . We take three different relative complex permittivity of configurations for collecting the scattering data; namely, openregion, equilateral triangular PEC-enclosed-region, and square PEC-enclosed-region. In all cases, the transmitters and receivers are evenly spaced on a circle of radius 0.1 m and the frequency of operation is 1 GHz. These configurations are shown in Fig. 2. Two different scenarios are used to collect the scattering data. In the first scenario, 7 transmitters and 7 receivers are used for collecting the scattering data on the measurement circle. The

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Fig. 2. (a) configuration for the open-region case, (b) configuration for the equilateral triangular PEC-enclosed-region case (the equilateral triangle is the PEC enclosure), and (c) configuration for the square PEC-enclosed-region case (the square is the PEC enclosure). The dotted circle, which has the radius of 0.1 m, shows the transmitter/receiver location (measurement domain S ).

inversion results for these three cases are shown in Fig. 3. As can be seen, all three inversions result in poor reconstructions. We now attempt to answer the following question: do scattering data sets collected under different BCs provide non-redundant information about the OI? To answer this question, we have developed an inversion algorithm to simultaneously invert the scattering data collected in different configurations. For example, for the case where there are two sets of scattering data, one collected in an open-region configuration and the other one in a PEC-enclosed-region configuration, we construct the following regularized cost-functional (9) The cost-functionals and represent the data misfit cost-functional, see (2), for the open-region and PEC-enclosedis given in (5) region cases, respectively. The regularizer and the steering parameter , in the discrete domain, is given as (10) where

and and

are the discrete forms of . The correction is then found by

solving (11) The complex matrix

is constructed as (12)

Fig. 3. 1st scenario: 7 transmitters and 7 receivers (left: Re( ) and right: Im( )) (a)–(b) inversion of the scattering data collected in the open-region embedding, (c)–(d) inversion of the scattering data collected inside the equilateral triangular PEC-enclosed embedding, and (e)–(f) inversion of the scattering data collected in the square PEC-enclosed embedding.

where and are the Jacobian matrices for the open-region and PEC-enclosed-region cases at the th iteration of the inversion algorithm respectively. The normalization factors for the open-region and PEC-enclosed-region scattering and , are also given by (3). The vector data, is given as (13) where and are the complex vectors containing the simulated scattered field at the observation points correfor the open-region and sponding to the predicted contrast PEC-enclosed-region cases. The complex vectors and represent the measured data for the open-region and PEC-enclosed-region cases. The discrete regularization ophas been described in Section II and the weight of this erator regularization, i.e., , is (14) Using this inversion algorithm, we simultaneously invert the three data sets described above (where only 7 transmitters and 7 receivers are used). In Fig. 4, we show the simultaneous inversion of (i) open-region and triangular PEC-enclosed-region

MOJABI AND LOVETRI: A NOVEL MICROWAVE TOMOGRAPHY SYSTEM USING A ROTATABLE CONDUCTIVE ENCLOSURE

Fig. 4. 1st scenario: 7 transmitters and 7 receivers (left: Re( ) and right: Im( )); simultaneous inversion of (a)–(b) scattering data collected in the openregion and triangular PEC-enclosed region configurations, (c)–(d) scattering data collected in the open-region and square PEC-enclosed region configurations, and (e)–(f) scattering data collected in the square PEC-enclosed region and triangular PEC-enclosed region configurations.

scattering data, (ii) open-region and square PEC-enclosed-region scattering data, and (iii) square PEC-enclosed-region and triangular PEC-enclosed-region scattering data. As can be seen, the simultaneous inversion results are very close to the true profile. Comparing Figs. 4 and 3, it can be easily seen that the simultaneous inversion has resulted in a more accurate reconstruction compared to the separate inversions of each data set. That being said, and noting that these data sets are distinguished by their corresponding BCs, it can be concluded that these three BCs have provided non-redundant information about the OI. We now consider the second scenario for collecting the scattering data in these three configurations, where we increase the number of transmitters and receivers to 16; thus, having 256 measurements. The inversion of each data set is shown in Fig. 5. The simultaneous inversion of these data sets are shown in Fig. 6. In this scenario, the separate inversion of each data set and the simultaneous inversions result in similar reconstruction. From these two scenarios and other similar inversion results (not shown here), it can be concluded that different BCs, at least when utilizing very few transmitters and receivers, provide non-redundant information for the reconstruction. We note that the necessary condition to obtain non-redundant information is to use a lossless or low-loss background medium so as to not suppress the reflection from the PEC enclosure. For example, if a lossy background medium with the relative complex

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Fig. 5. 2nd scenario: 16 transmitters and 16 receivers (left: Re( ) and right: Im( )) (a)–(b) inversion of the scattering data collected in the open-region embedding, (c)–(d) inversion of the scattering data collected in the triangular PEC-enclosed embedding, and (e)–(f) inversion of the scattering data collected in the square PEC-enclosed embedding.

permittivity of at 1 GHz (as the one used in the Dartmouth College MWT system [1]) is utilized, the reflection from the PEC enclosure is suppressed significantly; thus, the simultaneous inversion of open-region and PEC-enclosed-region data cannot provide a good reconstruction for the target shown in Fig. 1. In addition, from our numerical trials, we have found that if the OI has high loss, it can also suppress the reflection from the PEC enclosure; thus, reducing the amount of non-redundant information significantly. IV. MWT SYSTEM USING A ROTATABLE CONDUCTIVE ENCLOSURE Based on the idea that collecting scattering data using few transceivers and under different BCs yields different usable information, we now consider a rotatable equilateral triangular metallic casing, , which encloses the OI and a few transceivers, see Fig. 7. The OI is located in the bounded imaging domain . The transceivers are located on the measurement do, which is outside the OI. We assume that the main metallic casing is a PEC and is filled with a lossless or low-loss matching fluid with a known relative complex permittivity of . To obtain more scattering data by changing the BCs of the MWT system, the enclosure is rotated at angles , with respect to the fixed and fixed as depicted

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Fig. 7. The geometrical configuration of the MWT problem with a rotatable conductive triangular enclosure. The equilateral triangle, ABC , represents the metallic casing, which encloses the imaging domain and the measurement domain . The dotted black circle is the circumscribing circle of the triangle. The triangular enclosure can rotate on within a circumscribing circle for  degrees where  [0 ; 120 ).

S

Fig. 6. 2nd scenario: 16 transmitters and 16 receivers (left: Re( ) and right: Im( )); simultaneous inversion of (a)–(b) scattering data collected in the openregion and triangular PEC-enclosed region configuration, (c)–(d) scattering data collected in the open-region and square PEC-enclosed region configurations, and (e)–(f) scattering data collected in the square PEC-enclosed and triangular PEC-enclosed region configurations.

in Fig. 7. At the th configuration of the enclosure , the OI is successively illuminated by some incident electric field, where denotes the transmitter index . Interac. tion of the incident field with the OI results in the total field Note that the resulting field depends not only on the transmitter location, but also on the orientation of the enclosure. The total and incident electric fields are then measured by the receiver antennas located on . Thus, the scattered field at the observation points, contaminated by measurement noise, is known and de. noted by The inversion problem for this system may then be formulated as the minimization over of the following nonlinear least-squares data misfit cost-functional

(15) where is the simulated scattered field on due to a predicted contrast when the th transmitter is active at the th configuration of the triangular enclosure. The normalization is given by (3) where needs to be replaced factor . with

D

4

2

We regularize (15) by the weighted -norm total variation multiplicative regularizer given in (5). Thus, at the th iteration of the inversion algorithm, we minimize the regularized costfunctional (16) in (5) is chosen to be The positive parameter where is the discrete form of . The correction vector is found by solving

(17) The matrix is the Jacobian matrix corresponding to the th rotation of the enclosure and at the th iteration of the inversion algorithm, which is calculated using an FEM forward solver. is equal to . The discrepancy The weight vector is (18) Inversion results are shown for two synthetic data sets that have been created with a frequency-domain FEM forward solver. In both cases, we use the equilateral triangular PEC enclosure shown in Fig. 7 and assume that the radius of the circumscribing circle of the triangle is 0.24 m. The radius of the measurement circle is chosen to be 0.1 m for both data sets. The first synthetic data set is collected from the target described in Section III and shown in Fig. 1. Similar to the inversion results shown in Section III, the frequency of operation is chosen to be 1 GHz. We consider only 4 transmitters and 4 receivers per transmitter which are evenly spaced on .

MOJABI AND LOVETRI: A NOVEL MICROWAVE TOMOGRAPHY SYSTEM USING A ROTATABLE CONDUCTIVE ENCLOSURE

Fig. 8. Target I’s reconstructed relative complex permittivity when the scattering data is collected inside the rotatable triangular conductive enclosure using 4 transmitters and 4 receivers and 12 rotations of the enclosure. (a) Re( ), (b) Im( ).

Therefore, for the th rotation of the PEC enclosure, we have . The PEC enclosure is rotated 12 times with a step of 15 . Therefore, the number of measured data will . The inversion of this scattering data, which is be collected in the rotatable PEC enclosure, is shown in Fig. 8. The inversion of the scattering data collected from the same target in the open-region configuration using 16 transmitters and 16 reis shown in Fig. 5(a) and (b). As can be ceivers seen, the reconstruction inside the rotating PEC enclosure with only 4 transceivers and the reconstruction inside the open-region configurations with 16 transceivers are very similar for this target and both provide a reasonable reconstruction for both the real and imaginary parts of the target’s relative complex permittivity. Finally, we consider the target shown in Fig. 9. This target has the same geometry as the target used in [36] and [37] for a resolution test study. This target has different distances between its details ranging from 8 mm to 20 mm. The relative complex and that of the background permittivity of the target is at the frequency of operation which is chosen medium is to be 2 GHz. To collect scattered field data, we consider 6 transmitters and 6 receivers per transmitter; thus, . The

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Fig. 9. Synthetic target II: true relative complex permittivity profile of the target. (a) Re( ), (b) Im( ).

PEC enclosure is then rotated 48 times with a step of 2.5 ; thus, scattering measurements. The inverproviding sion of the scattering data collected inside the rotating PEC enclosure is shown in Fig. 10(a) and (b), while the inversion of the scattering data collected in the open-region embedding using 16 transmitters and 16 receivers is shown in Fig. 10(c) and (d). In both cases, the real part of the permittivity is reconstructed well but the imaginary part is poorly reconstructed. This is due to the fact that the real and imaginary parts of the OI’s contrast are out of balance [38] and also the imaginary part of the contrast is very . To get a better reconstruction for this target, we small apply the image enhancement method, as outlined in [39], to the final reconstructions of both inversions. The enhanced reconstructions for both cases are shown in Fig. 11. We note that this image enhancement algorithm does not use any a priori information about the OI and is effectively a deblurring algorithm. To have the same number of data points as the open-region configuration, we also tried to reconstruct the target shown in Fig. 9 using 4 transceivers and 16 rotations (thus, providing 256 data points), but the inversion could not resolve all the features of this target. Therefore, the number of data collected does not equal to the amount of useful information for reconstruction purpose. Investigating the amount of information available by rotating or moving BCs with fixed sources is a topic for future study.

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information about the OI. Considering that the modeling error can be thought of as part of the manifest noise, and noting that the achievable resolution limit is affected by the signal-to-noise ratio [36], [40], the proposed MWT system may offer an enhanced spatial resolution over the existing MWT systems. Finally, we note that the rotatable triangular conductive enclosure is just one practical implementation of a system which uses different PEC BCs to obtain non-redundant information about an OI. ACKNOWLEDGMENT The authors thank A. Zakaria for providing the FEM solver. REFERENCES

Fig. 10. Target II: (left: Re( ) and right: Im( )) (a)–(b) reconstructed relative complex permittivity when the scattering data is collected inside the rotating triangular conducting enclosure using 6 transmitters and 6 receivers and 48 rotations of the enclosure (c)–(d) reconstructed relative complex permittivity when the scattering data is collected in the open-region embedding using 16 transmitters and 16 receivers.

Fig. 11. Target II: (left: Re( ) and right: Im( )) reconstruction results after applying the enhancement algorithm inside (a)–(b) the rotatable triangular conductive enclosure with 6 transmitters and 6 receivers and 48 rotations, and (c)–(d) the open-region embedding with 16 transmitters and 16 receivers.

V. CONCLUSION Using synthetic data sets, the possibility of imaging inside a rotatable triangular conductive enclosure using a minimal antenna array having as few as only four or six co-resident elements immersed in a low-loss background medium has been demonstrated for the 2D TM case. This study may result in the development of MWT systems which introduce less modeling error to MWT algorithms compared to the existing MWT systems while maintaining the ability to collect sufficient scattering

[1] T. Rubæk, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton’s method and the CGLS inversion algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2320–2331, Aug. 2007. [2] A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 7, pp. 1761–1777, Jul. 2002. [3] J. D. Zaeytijd, A. Franchois, C. Eyraud, and J.-M. Geffrin, “Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method—Theory and experiment,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3279–3292, Nov. 2007. [4] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging, vol. 9, no. 2, pp. 218–225, 1990. [5] P. Mojabi and J. LoVetri, “Microwave biomedical imaging using the multiplicative regularized Gauss-Newton inversion,” IEEE Antennas and Wireless Propag. Lett., vol. 8, pp. 645–648, 2009. [6] P. Meaney, M. Fanning, D. Li, S. Poplack, and K. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 11, pp. 1841–1853, Nov. 2000. [7] A. Franchois, A. Joisel, C. Pichot, and J.-C. Bolomey, “Quantitative microwave imaging with a 2.45-GHz planar microwave camera,” IEEE Trans. Med. Imag., vol. 17, no. 4, pp. 550–561, Aug. 1998. [8] A. Broquetas, J. Romeu, J. Rius, A. Elias-Fuste, A. Cardama, and L. Jofre, “Cylindrical geometry: A further step in active microwave tomography,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 5, pp. 836–844, May 1991. [9] S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, V. G. Posukh, A. Pavlovsky, P. N. Repin, A. N. Starostin, B. A. Voinov, M. Taran, G. P. Tastis, and V. Y. Baranov, “Three-dimensional microwave tomography: Initial experimental imaging of animals,” IEEE Trans. Biomed. Eng., vol. 49, no. 1, pp. 55–63, Jan. 2002. [10] A. Fhager, P. Hashemzadeh, and M. Persson, “Reconstruction quality and spectral content of an electromagnetic time-domain inversion algorithm,” IEEE Trans. Biomed. Eng., vol. 53, no. 8, pp. 1594–1604, Aug. 2006. [11] C. Yu, M. Yuan, J. Stang, E. Bresslour, R. George, G. Ybarra, W. Joines, and Q. H. Liu, “Active microwave imaging II: 3-D system prototype and image reconstruction from experimental data,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 4, pp. 991–1000, Apr. 2008. [12] T. Rubaek, O. Kim, and P. Meincke, “Computational validation of a 3-D microwave imaging system for breast-cancer screening,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2105–2115, Jul. 2009. [13] C. Gilmore, P. Mojabi, A. Zakaria, M. Ostadrahimi, C. Kaye, S. Noghanian, L. Shafai, S. Pistorius, and J. LoVetri, “A wideband microwave tomography system with a novel frequency selection procedure,” IEEE Trans. Biomed. Eng., vol. 57, no. 4, pp. 894–904, Apr. 2010. [14] K. Paulsen and P. Meaney, “Nonactive antenna compensation for fixed array microwave imaging. I. Model development,” IEEE Trans. Med. Imag., vol. 18, no. 6, pp. 496–507, Jun. 1999. [15] P. Meaney, K. Paulsen, J. Chang, M. Fanning, and A. Hartov, “Nonactive antenna compensation for fixed-array microwave imaging. II. Imaging results,” IEEE Trans. Med. Imag., vol. 18, no. 6, pp. 508–518, Jun. 1999. [16] O. Franza, N. Joachimowicz, and J.-C. Bolomey, “SICS: A sensor interaction compensation scheme for microwave imaging,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 211–216, Feb. 2002.

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[17] E. Wadbro and M. Berggren, “Microwave tomography using topology optimization techniques,” SIAM J. Sci. Comput., vol. 30, no. 3, pp. 1613–1633, 2008. [18] A. Abubakar, T. Habashy, V. Druskin, L. Knizhnerman, and D. Alumbaugh, “2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements,” Geophysics, vol. 73, no. 4, pp. F165–F177, Jul.–Aug. 2008. [19] P. Mojabi and J. LoVetri, “Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2658–2665, Sept. 2009. [20] T. M. Habashy and A. Abubakar, “A general framework for constraint minimization for the inversion of electromagnetic measurements,” Progr. Electromagn. Res., vol. 46, pp. 265–312, 2004. [21] P. Mojabi, “Investigation and Development of Algorithms and Techniques for Microwave Tomography” Ph.D. dissertation, Univ. Manitoba, Winnipeg, Manitoba, Canada, 2010 [Online]. Available: URL: http://mspace.lib.umanitoba.ca/handle/1993/3946 [22] P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Processing, vol. 6, no. 2, pp. 298–311, Feb. 1997. [23] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “A robust iterative method for Born inversion,” IEEE Trans. Geosci. Remote Sensing, vol. 42, no. 2, pp. 342–354, Feb. 2004. [24] A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov, R. H. Svenson, A. G. Nazarov, Y. E. Sizov, and G. P. Tastis, “Three dimensional microwave tomography. Theory and computer experiments in scalar approximation,” Inverse Probl., vol. 16, pp. 863–875, 2000. [25] P. M. Meaney, M. W. Fanning, T. Raynolds, C. J. Fox, Q. Fang, C. A. Kogel, S. P. Poplack, and K. D. Paulsen, “Initial clinical experience with microwave breast imaging in women with normal mammography,” Acad Radiol., Mar. 2007. [26] L. Crocco and A. Litman, “On embedded microwave imaging systems: Retrievable information and design guidelines,” Inverse Prob., vol. 25, no. 6, p. 17, 2009, 065001. [27] C. Gilmore and J. LoVetri, “Enhancement of microwave tomography through the use of electrically conducting enclosures,” Inverse Prob., vol. 24, no. 3, p. 21, 2008, 035008. [28] C. Gilmore and J. LoVetri, “Corrections to the ‘enhancement of microwave tomography through the use of electrically conducting enclosures’,” Inverse Prob., vol. 26, no. 1, p. 7, Jan. 2010, 019801. [29] A. Franchois and A. G. Tijhuis, “A quasi-Newton reconstruction algorithm for a complex microwave imaging scanner environment,” Radio Sci., vol. 38, no. 2, 2003. [30] R. Lencrerot, A. Litman, H. Tortel, and J.-M. Geffrin, “Measurement strategies for a confined microwave circular scanner,” Inverse Prob. Sci. Engrg., pp. 1–16, Jan. 2009. [31] R. Lencrerot, A. Litman, H. Tortel, and J.-M. Geffrin, “Imposing Zernike representation for imaging two-dimensional targets,” Inverse Prob., vol. 25, no. 3, p. 21, 2009, 035012. [32] P. Mojabi and J. LoVetri, “Eigenfunction contrast source inversion for circular metallic enclosures,” Inverse Prob., vol. 26, no. 2, p. 23, Feb. 2010, 025010. [33] P. Mojabi, C. Gilmore, A. Zakaria, and J. LoVetri, “Biomedical microwave inversion in conducting cylinders of arbitrary shapes,” in Proc. 13th Int. Symp. on Antenna Technology and Applied Electromagnetics and the Canadian Radio Science Meeting, Feb. 2009, pp. 1–4. [34] A. G. Tijhuis, K. Belkebir, and A. C. S. Litman, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 6, pp. 1316–1330, 2001.

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[35] M. Lazebnik, M. Okoniewski, J. Booske, and S. Hagness, “Highly accurate Debye models for normal and malignant breast tissue dielectric properties at microwave frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 12, pp. 822–824, Dec. 2007. [36] S. Semenov, R. Svenson, A. Bulyshev, A. Souvorov, A. Nazarov, Y. Sizov, V. Posukh, A. Pavlovsky, P. Repin, and G. Tatsis, “Spatial resolution of microwave tomography for detection of myocardial ischemia and infarction-experimental study on two-dimensional models,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 4, pp. 538–544, Apr. 2000. [37] C. Gilmore, P. Mojabi, A. Zakaria, S. Pistorius, and J. LoVetri, “On super-resolution with an experimental microwave tomography system,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 393–396, 2010. [38] P. Meaney, N. Yagnamurthy, and K. D. Paulsen, “Pre-scaled two-parameter Gauss-Newton image reconstruction to reduce property recovery imbalance,” Phys. Med. Biol., vol. 47, pp. 1101–1119, 2002. [39] P. Mojabi and J. LoVetri, “Enhancement of the Krylov subspace regularization for microwave biomedical imaging,” IEEE Trans. Med. Imag., vol. 28, no. 12, pp. 2015–2019, Dec. 2009. [40] P. Meaney, K. Paulsen, A. Hartov, and R. Crane, “Microwave imaging for tissue assessment: Initial evaluation in multitarget tissue-equivalent phantoms,” IEEE Trans. Biomed. Eng., vol. 43, no. 9, pp. 878–890, Sept. 1996. Puyan Mojabi (M’10) received the B.Sc. degree in electrical and computer engineering from the University of Tehran, Tehran, Iran, in 2002, the M.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2004 and the Ph.D. degree in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2010. His current research interests are computational electromagnetics, microwave tomography and inverse problems.

Joe LoVetri (SM’00) was born in Enna, Italy, in 1963. He received the B.Sc. (with distinction) and M.Sc. degrees, both in electrical engineering, from the University of Manitoba, Winnipeg, MB, Canada, in 1984 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada, in 1991. From 1984 to 1986, he was an EMI/EMC Engineer at Sperry Defence Division, Winnipeg, Manitoba. From 1986 to 1988, he held the position of TEMPEST Engineer at the Communications Security Establishment in Ottawa. From 1988 to 1991, he was a Research Officer at the Institute for Information Technology, National Research Council of Canada. From 1991 to 1999, he was an Associate Professor in the Dept. of Electrical and Computer Engineering, The University of Western Ontario. In 1997/98, he spent a sabbatical year at the TNO Physics and Electronics Laboratory, The Netherlands. Since 1999, he has been a Professor in the Department of Electrical and Computer Engineering, University of Manitoba, and was Associate Dean, Research, from 2004 to 2009. His main interests lie in time-domain computational electromagnetics, modeling of electromagnetic compatibility problems, microwave tomography and inverse problems.

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Super-Resolution UWB Radar Imaging Algorithm Based on Extended Capon With Reference Signal Optimization Shouhei Kidera, Associate Member, IEEE, Takuya Sakamoto, Member, IEEE, and Toru Sato, Member, IEEE

Abstract—Near field radar employing ultrawideband (UWB) signals with its high range resolution has great promise for various sensing applications. It enables non-contact measurement of precision devices with specular surfaces like an aircraft fuselage and wing, or a robotic sensor that can identify a human body in invisible situations. As one of the most promising radar algorithms, the range points migration (RPM) was proposed. This achieves fast and accurate surface extraction, even for complex-shaped objects, by eliminating the difficulty of connecting range points. However, in the case of a more complex shape whose variation scale is less than a pulsewidth, it still suffers from image distortion caused by multiple interference signals with different waveforms. As a substantial solution, this paper proposes a novel range extraction algorithm by extending the Capon method, known as frequency domain interferometry (FDI). This algorithm combines reference signal optimization with the original Capon method to enhance the accuracy and resolution for an observed range into which a deformed waveform model is introduced. The results obtained from numerical simulations and an experiment with bi-static extension of the RPM prove that super-resolution UWB radar imaging is accomplished by the combination between the RPM and the extended Capon methods, even for an extremely complex-surface target including edges. Index Terms—Capon method, frequency domain interferometry (FDI), range points migration (RPM), reference signal optimization, super-resolution imaging, ultrawideband (UWB) radars.

I. INTRODUCTION

U

ltrawideband (UWB) pulse radar with high range resolution has promise for near field sensing techniques. As such, it is applicable to non-contact measurement for manufacturing reflector antennas or aircraft bodies that have high-precision surfaces, or to robotic sensors that can identify a human body, even in an optically blurry vision such as dark smog in

Manuscript received March 19, 2010; revised October 12, 2010; accepted November 02, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the Grant-in-Aid for Scientific Research (A) (Grant 17206044), the Grant-in-Aid for JSPS Fellows (Grant 19–497), and in part by the Grant-in-Aid for Young Scientists (Start-up) (Grant 21860036), promoted by Japan Society for the Promotion of Science (JSPS). S. Kidera is with the Graduate School of Informatics and Engineering, University of Electro-Communications, Tokyo, Japan (e-mail: [email protected]. jp). T. Sakamoto and T. Sato are with the Department of Communications and Computer Engineering, Graduate School of Informatics, Kyoto University, Kyoto, Japan. Digital Object Identifier 10.1109/TAP.2011.2123059

disaster areas. In addition, it is suitable for surveillance or security systems for intruder detection or aged care, where an optical camera has the serious problem of privacy invasion in the case for living places. While various kinds of radar algorithms have been developed based on the aperture synthesis [1], the time reversal approach [2], [3], the range migration [4], [5] or the GA-based solution for the domain integral equations [6], they are not suitable for the above applications because of the large amount of calculation time or inadequate image resolution. To concur the problem in the conventional techniques, we have already proposed a number of radar imaging algorithms, which accomplish real-time and high resolution surface extraction beyond a pulsewidth [7], [8]. As a high-speed and accurate 3-D imaging method applicable to various target shapes, the range points migration (RPM) algorithm has been proposed [9]. This algorithm directly estimates an accurate direction of arrival (DOA) with the global characteristic of observed range points, avoiding the difficulty in connecting them. The RPM is based on a simple idea, yet, it offers an accurate target surface including the complex-shaped target that often creates an extremely complicated distribution of range points. However, this algorithm suffers from non-negligible image distortion in the case of a more complicated target which has a surface variation less than a pulsewidth, or has many convex and concave edges. This distortion is caused by the richly interfered signals scattered from the multiple scattering centers on the target surface. These components are received within a range scale smaller than a pulsewidth, and are hardly separated by the conventional range extraction methods, such as the Wiener filter. In addition, there are small range shifts due to deformed scattered waveforms from the transmitted wave. As the conventional solution for this, the spectrum offset correction (SOC) method has been proposed [8], which directly compensates the range shift by using the center frequency offset between the scattered and transmitted waveforms. However, the SOC is rarely applied to complex-shaped targets, because it requires a completely separated waveform from other interference signals to calculate an accurate center frequency. To overcome this difficulty, this paper proposes a novel range extraction algorithm by extending the frequency domain Capon method. While the Capon is useful for enhancing the range resolution based on FDI [10]–[12], the resolution and accuracy of this method significantly depend on a reference waveform such as the transmitted waveform. In general, the scattered waveform from the target with a wavelength scale differs from the one transmitted [8], and the range resolution given by the original

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KIDERA et al.: SUPER-RESOLUTION UWB RADAR IMAGING ALGORITHM BASED ON EXTENDED CAPON

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Fig. 1. System model in the 2-dimensional model.

Capon method distorts due to this deformation. To outperform the original Capon, this paper extends the original Capon so that it optimizes the reference signal based on the simplified waveform model. The extended Capon significantly enhances the range resolution and accuracy, and brings out the utmost performance of the RPM algorithm. The results obtained from numerical simulations in the 2-dimensional (2-D) and 3-dimensional (3-D) models are presented in Sections II and III. Finally, an experiment using the UWB module, where the bi-static extension of the RPM is applied, verifies that super-resolution imaging is accomplished by the combination between the RPM and the extended Capon, when using a simple radar constitution. II. 2-D PROBLEM

Fig. 2. True range points (upper) and extracted target points with the RPM (lower).

accurate DOA (Direction of Arrival in Fig. 1) estimation by making use of the global characteristics of the observed range is calculated as, map. The optimum

A. System Model Fig. 1 shows the system model in the 2-D model. It assumes the mono-static radar, and an omni-directional antenna is scanned along the -axis. It is assumed that the target has an arbitrary shape with a clear boundary. The propagation speed of the radio wave is assumed to be a known constant. A mono-cycle pulse is used as the transmitting current. The real space in which the target and antenna are located, is expressed . The parameters are normalized by , by the parameters is assumed which is the central wavelength of the pulse. for simplicity. is defined as the received electric field , where is at the antenna location a function of time . B. RPM Algorithm Various kinds of radar imaging algorithms based on an aperture synthesis, time reversal or range migration methods, have been proposed [1]–[6]. As a real-time imaging algorithm, the SEABED has been developed, which uses a reversible transform boundary scattering transform (BST) between the observed ranges and the target boundary [7]. In addition, another high-speed imaging algorithm termed Envelope has been developed aiming at improving the image stability of SEABED, by avoiding the range derivative operations [8]. While these algorithms accomplish fast and accurate imaging for a simple shaped object, such as trapezoid, pyramid, or sphere shapes, it is hardly applicable to complex-shaped or multiple targets because it requires correct connection of range points. As one of the most promising algorithms applicable to various target shapes, the RPM algorithm has been proposed [9]. This exists on method assumes that a target boundary point a circle with center and radius , and then employs an

(1) where , and denotes the amplitude of the received signal at the range and the antenna . is the number of the range points. location denotes the angle from the axis to the intersection point of and . The constants the circles, with parameters and are empirically determined. The detail of this algofor each rithm is described in [9]. The target boundary range point is expressed as and . This algorithm ignores range points connection, and produces accurate target points, even if extremely complicated range distribution is given. Thus, the inaccuracy occurring in the SEABED and Envelope can be substantially avoided by this method. Fig. 2 shows the example of this algorithm under the assumption that the true range points are given as in the upper side of for every is set for simplicity. this figure. Here, The lower side of Fig. 2 shows a distinct advantage for this algorithm in that it can accurately locate the target points, even if a quite complicated range map is given. C. Performance of RPM Using the Wiener Filter The performance example of the RPM is presented here, where the received electric field is calculated by the FDTD (Finite Difference Time Domain) method. The former study [9] employs the Wiener filter in order to extract a range point are extracted from for each location. The range points the local peaks of which are beyond the determined

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Fig. 5. Estimated target points with the RPM and the Wiener filter. Fig. 3. Output of the Wiener filter and extracted range points.

Fig. 6. Waveform comparisons for each antenna location at polygonal target. Fig. 4. Estimated target points with the RPM and the Wiener filter.

threshold. The procedure is detailed in [9]. The example of this method for the target shape shown in Fig. 1 is presented. Fig. 3 shows the output of the Wiener filter, and the extracted range points. The received signals are calculated at 101 locations be. A noiseless environment is assumed. tween Fig. 4 presents the comparison between the true and extracted range points in this case. It shows that the range points suffer from the inaccuracy due to the multiple interference echoes within a range scale of less than a pulsewidth. Fig. 5 shows the target points, when the RPM is applied to the range points in Fig. 4. This figure indicates that the inaccuracy of range points distorts the target image, which is totally inadequate for identifying its actual shape, especially for convex and concave edges. Furthermore, this range map includes a small error caused by scattered waveform deformation, whose characteristics are detailed in [13]. To enhance the accuracy for range points extraction, the spectrum offset correction (SOC) algorithm has been developed aiming at compensating the range shift due to the waveform deformation [8]. It is, however, confirmed that the range accuracy of the SOC is entirely inadequate in such a richly interfered situation. This is because the range errors in this case are dominantly caused by the peak shift of the output of the Wiener filter due to the interferences of multiple scattering echoes. Furthermore, the SOC is based on the center periods estimation of the scattered signal, when each signal should be correctly resolved in the time domain. This is, however, difficult when the multiple interfered signals are mixed together in a time scale less than its center period.

D. Proposed Range Extraction Algorithm To overcome the difficulty described above, this paper proposes a novel algorithm for range points extraction by extending the frequency domain Capon method. The Capon algorithm is one of the most powerful tools for enhancing range resolution based on FDI. It is confirmed, however, that the scattered waveform deformation distorts the range resolution and accuracy of the original Capon method. As a solution for this, the proposed method optimizes the reference signal used in the Capon. This method introduces a reference waveform model, based on the fractional derivative of the transmitted waveform as (2) is the angular frequency domain of the transmitted where signal and denotes a complex conjugate. is a variable which . satisfies The waveform comparison using this simplified model is demonstrated as follows. Fig. 6 shows the scattered waveform from the polygonal target received at the different locations, and the estimated waveforms with the optimized in (2). This figure indicates that a scattered waveform differs depending on an antenna location, or a local shape around the scattering center [13]. This deformation distorts the resolution and accuracy of the original Capon method, because it employs a phase and amplitude interferometry in each frequency between the reference and scattered waveforms. It is confirmed that an estimated waveform with the optimized in the previous model accurately approximates an actual deformed waveform,

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where the range accuracy is estimated of the order of 0.01 when using the matched filter. Based on this waveform model, the observed vector is defined as (3) where denotes the received signal in angular frequency domain at each antenna location , and denotes . Here, in order to suppress a range the dimension of sidelobe caused by the coherent interference signals, frequency is averaging is used. The averaged correlation matrix defined as

Fig. 7. Output of the original Capon method and extracted range points.

(4) where denotes the Hermitian transpose. is the total number of the frequency points, and is determined by the maximum holds. The frequency band of the transmitted signal. output of the extended Capon is defined as (5) where quency

denotes the steering vector of

for each fre-

Fig. 8. Estimated range points with the original Capon method.

(6) is defined as (7) The normalization with enables us to compare the amplitude with respect to . Then, the local maximum of of for and offers an optimized range resolution in the Capon method. That is, performs the highest range resolution, where the reference waveform most coincides with the actual scattered waveform. Finally, it determines the , which satisfies the following conditions range points

(8)

where is empirically determined according to the evaluations in [9]. Equation (8) is numerically solved by searching for and . This algorithm the local maxima of selects an accurate range point by enhancing the range resolution of the Capon method with the optimized reference signal. is calculated from the group of range Each target point points in (1), that is the RPM. This extraction algorithm achieves accurate range point estimation by compensating for the waveform deformation. This is a distinct advantage compared with the original Capon method.

Fig. 9. Estimated target points with the RPM and the the original Capon method.

E. Performance Evaluation in Numerical Simulation This section presents the examples for each range extraction method, where the same target as in Fig. 1 is assumed. Here, and are set. Fig. 7 shows the output of the original Capon method and the extracted range points, which correin (8), that is, the waveform deformation is not sponds to considered. Fig. 8 shows the comparison between the true and extracted range points in this case. In this figure, the number of the accurate range points increases compared to Fig. 4, because the original Capon enhances the range resolution. Fig. 9 shows the estimated target points by using the original Capon method as in Fig. 8. This figure also shows that it enhances the accuracy of the location of imaging points, and the convex edge region is also accurately located. However, the inaccuracy around the concave edge region is recognized, and some parts of the target boundary are still not reconstructed. This is because of the distorted resolution and accuracy of ranges caused by the reference and actual scattered waveform being in-coincidence.

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Fig. 13. Estimated image with the synthetic aperture radar (SAR). Fig. 10. Output of the extended Capon method and extracted range points.

Fig. 14. Number of the target points for each . Fig. 11. Estimated range points with the extended Capon method.

determined by a half of the pulsewidth is substantially inadequate for recognizing the concave or convex edges. This result also proves the advantage for the proposed method, in terms of high-resolution imaging. Here, the quantitative analysis is introduced by defined as (9)

Fig. 12. Estimated target points with the RPM and the extended Capon method.

In contrast, Fig. 10 shows with the optimized , and the range points extracted. Fig. 11 offers the same view as in Fig. 8. and and are the same as in the previous example. This figure verifies that the extracted range points are accurately located, and the number of those points increases compared with the original Capon method. Fig. 12 shows the estimated target points obtained by the RPM. This figure shows these points accurately reconstruct the convex or concave edge region, and offer substantial information for identifying the complicated target shape, even with convex or concave edges. This is because the proposed method enhances the resolution of with respect to the scattered waveform deformation. Thereby, the peaks, which are regarded as the trivial value in the output of the original Capon, can be detected by optimizing the reference waveform. As the comparison for the other methods not specified to the clear boundary extraction, the SAR (Synthetic Aperture Radar) method is introduced. This algorithm is the most useful tool for radar imaging [1], and the near field extension of the SAR is applied here [9]. Fig. 13 shows the example of the SAR. While the image produced by the SAR is stable, its spatial resolution

where and express the locations of the true and estimated target points, respectively. is the total number of . Fig. 14 plots the number of the estimated points for each value of . This figure verifies that the number of the accurate target points with the proposed method significantly increases around 0.01 , compared with other conventional algorithms. , that is defined as the mean values of , for each method is for the Wiener filter, 2.18 for the orig5.66 inal Capon, and 1.23 for the proposed extended Capon, respectively. This result quantitatively proves the effectiveness of the proposed range extraction algorithm. In addition, the examples in noisy situation are investigated, where the white Gaussian noise is added to each received signal . Fig. 15 shows the estimated points with the RPM as and the extended Capon, where the mean S/N is 35 dB. The S/N is defined as the ratio of peak instantaneous signal power to the averaged noise power after applying the matched filter with the transmitted waveform. This figure shows that the target points around the convex edge region are scarcely extracted, and the accuracy of the points near the concave edges is distorted. This is because this algorithm uses the inverse filtering in creating the in (3), which is sensitive to the white observed vector noise. Furthermore, the scattered signals from the convex edges are relatively smaller than those from the concave boundaries. Then, the significant range peaks around the convex region are

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Fig. 15. Estimated target points with the RPM and the extended Capon method in S=N = 35 dB. Fig. 18. System model in 3-dimensional problem.

Fig. 16. Number of the target points for each  in S=N = 35 dB.

proposed method accomplishes most accurate target imaging. While this method requires a high S/N to hold the accuracy, the actual UWB radar system can achieve this level of S/N. This is because we assume the near field measurement, where each receiver obtains an intensive echo from objects even under the spectrum mask of the UWB signal [14], and random noises in received signals can be considerably suppressed using coherent , the conaveraging. In contrast, in the case of ventional Wiener filter holds its accuracy within 0.08 , while the both original and extended Capon methods deteriorate their accuracies over 0.1 . This is because these methods are based on the inverse filtering in creating the observation subvector in (3). It is our future work to enhance the accuracy in the noisy regarding to the situation to modify the definition of S/N. Moreover, the proposed method employs an optimization process for each received signal and requires around 100 sec for the range points extraction, whereas the algorithm with the original Capon method requires only 5 sec. We need to select an appropriate range extraction method for the different kinds of applications, whether they require real-time operation or extremely accurate surface extraction in the high S/N. III. 3-D PROBLEM A. System Model

Fig. 17. Relationship between  and S/N for each method.

not detected in this case. Fig. 16 plots the number of the estimated points for each value of in this noisy case. for each for the Wiener filter, 4.37 for method are 6.00 for the extended Capon. the original Capon, and 4.44 While the accuracy for the both original and extended Capon methods deteriorates due to the noises, the superiority to the Wiener filter is maintained. This is because the range resolution of the Wiener filter also becomes more inadequate than that in the noiseless case. In addition, it is a substantial characteristic of the proposed method that the higher S/N provides the higher resolution for the obtained images. Fig. 17 illustrates the relationship between and S/N for each , the method. This figure shows that, in the case of

Fig. 18 shows the system model in the 3-D problem. The target model, antenna, and transmitted signal are the same as those assumed in the 2-D problem. The antenna is scanned on . It assumes a linear polarization in the direction the plane, . of the -axis. -space is expressed by the parameter for simplicity. is defined as We assume the received electric field at the antenna location . B. Extension of Proposed Method to 3-D Model The extension of the RPM to the 3-D model has been derived in [9]. This assumes that a target boundary point exists on a sphere with center and radius . It calculates by investigating the distribution of the intersection circles between the spheres determined with and . Each intersection circle projected to the

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Fig. 19. Estimated target points with the RPM and the Wiener filter for 0:75 x 0:75.

0

 

plane becomes a straight line, which is defined as as method determines the target location

Fig. 20. Estimated target points with the RPM and the original Capon method for 0:75 x 0:75.

0

 

. This

(10) Fig. 21. Estimated target points with the RPM and the extended Capon method x 0:75. for 0:75

where , , and denotes the amplitude of the received signal at the range and the antenna location . denotes the minimum distance between the line and . , and are empirically determined. Under the assumption , the coordinate of each target point is given by . This algorithm ignores the connecting procedures of a large number of range points, and can avoid instability due to the failure of range connections. Thus, it produces accurate target points, even if an extremely complicated 3-D range map is given. The detailed characteristic is described in [9]. Furthermore, in the proposed range extraction, each range is calculated in (8), where is repoint is calculated from the group defined. Each target point of range points in (10). C. Performance Evaluation in Numerical Simulation This section presents an example of the proposed method using numerical simulation. The mono-static radar is scanned for , , where the number of locations on each axis is 51. The target boundary is assumed as in Fig. 18. Fig. 19 shows the target points obtained by the RPM in the use . of the Wiener filter for range extraction at and are set. This figure shows that the estimated target points suffer from severe inaccuracy, and the produced target boundary is far from the actual one. This is because the obtained image with the RPM seriously depends on the accuracy for the range points, which have non-negligible errors due to multiple interfered signals in the same range gate, or scattered waveform deformation [13]. In contrast, Fig. 20 shows the estimated target points with the is set in (8). RPM and the original Capon method, where and are set. This figure proves that Here, accurate target points estimation is achieved only by enhancing the range resolution. However, it should be noted that the target

0

 

points around the convex edge region are still inaccurate, because this method does not consider the waveform deformation which distorts the resolution and accuracy in range extraction. Fig. 21 shows the same view as in Fig. 19, when the exand tended Capon method is used for range extraction. are the same as in the previous example. It is confirmed that this method improves the accuracy for target point extraction, especially for the edge region. This is because the range resolution of the Capon is significantly enhanced by optimizing the reference waveform. Here, the deep-set concave region is not reconstructed, because the direct scattered signals from this region are not received at any antenna location. This is an inherent problem under the assumption that a single scattered signal is used for imaging. Another study offers a promising solution for this problem by using the multiple scattered waves for imaging [15]. In addition, Fig. 22 shows the number of the target points for each sampled , which is defined in (9). This figure proves that the proposed method enhances the number of the accurate target . for each method are 0.070 for the points around Wiener filter, 0.044 for the original Capon, and 0.035 for the extended Capon. The RPM with the original Capon creates , and these points increase a few target points with compared with that of the proposed method, even if the same in (8) is used. This is because the extended Capon method avoids with the peak lowering around the actual ranges in the optimized . D. Performance Evaluation in Experiment This section investigates the experimental study of the proposed algorithm. We utilize a UWB pulse with a center frequency of 3.3 GHz and a 10 dB-bandwidth of 3.0 GHz. The center wavelength of the pulse is 91 mm. The antenna has an elliptic polarization, of which the ratio of major to minor axes is about 17 dB, and the direction of the polarimetry axis

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Fig. 25. Estimated target points with the RPM and the original Capon in the experiment for 0:5 y 0:5.

0

 

Fig. 22. Number of the target points for each .

Fig. 26. Estimated target points with the RPM and the extended Capon in the experiment for 0:5 y 0:5.

0

Fig. 23. Arrangement for the multiple targets and the small UWB microstrip patch antenna.

Fig. 24. Estimated target points with the RPM and the Wiener filter in the exy 0:5. periment for 0:5

0

 

of the antenna is along the -axis. The 3 dB-beamwidth of the antenna is about 90 . One trapezoid and two triangle prismatic targets are set, and are covered with 0.2 mm thick aluminum sheet. Fig. 23 illustrates the arrangement of antennas with respect to the multiple targets. The transmitting and receiving anplane, for and tennas are scanned on the , respectively, with both sampling intervals set to 0.11 . The separation between the transmitting and receiving antennas is 1.4 in the -direction. The data are coherently averaged 1024 times. The direct scattered signals from the targets are obtained by eliminating the direct signal from the transmitting antenna. Fig. 24 shows the extracted target points using the RPM for , when the Wiener filter is used for range extraction. The mean S/N is around 30 dB. Here, the bi-static extension of RPM is applied, which is detailed in Appendix A.

 

and are set. This figure proves that the target points have non-negligible errors, especially around the trapezoid edge region. In addition, the unnecessary image appears above the actual boundary due to the range sidelobe of the Wiener filter. Fig. 25 shows the same view as in Fig. 24, when the original Capon method is applied to the same received data. and are set. In this figure, it is confirmed that the target points accurately express an actual target surface, including the edge region, and the false image due to the range sidelobe in the Wiener filter is considerably suppressed. This proves that the frequency domain interferometry in the Capon method is sufficiently effective in terms of accurate surface extraction. Furthermore, Fig. 26 shows the same view as in Fig. 24, when the extended Capon is applied for range extraction. While the image obtained using the proposed method creates an accurate target boundary including edges, there is not a significant difference between the images with the both original and extended Capon. Fig. 27 shows the number of the target points with the sampled defined in (9) for each method. The methods with the original and extended Capon obtain an increase in the number of points with the accuracy around 0.04 . for each method are 1.49 for the Wiener filter, 4.53 for the original for the extended Capon. The effecCapon, and 5.55 tiveness of the extended Capon is not obvious in this case. This is because this experiment does not offer the sufficient S/N to recognize the significant discrepancies between the original and extended Capon. According to the discussion in Section II-E, the higher S/N is required to confirm the effectiveness of the extended Capon, and this is an inherent characteristic of this method. However, the both Capon methods accomplish accurate imaging including edge region by suppressing the false images

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APPENDIX BISTATIC EXTENSION OF RPM The bi-static extension of RPM in 3-dimensional model is derived here. At the experimental model assuming in this paper, the transmitting and receiving antenna locations are defined as and , where is a constant. In the is defined as the modified RPM algorithm, minimum distance between and the curve , which plane from the intersection curve between is projected on and the following two ellipsoids with

Fig. 27. Number of the target points for each  in the experiment.

(11) is formulated as

(12) Fig. 28. Intersection curve C between two ellipsoids determined by (X; Y; Z ) and (X ; Y ; Z ).

due to the range sidelobe, which are not avoided by the Wiener filter.

In corresponding to the left term in (12),

is defined as, (13)

where

(14)

IV. CONCLUSION This paper proposed a novel range extraction algorithm as the extended frequency domain Capon method, known as FDI. To enhance the image quality of the RPM method, including the case for complicated shaped objects with concave or convex edges, this method extends the original Capon so that it optimizes the reference signal with a simplified and accurate waveform model. It has a substantial advantage that the range resolution is remarkably enhanced, even if the different scattered waves are mixed together within the range scale less than a pulsewidth. The results from numerical simulations verified that the extended Capon method created accurate range points of the order of 0.01 , where the RPM offered an utmost performance and achieved super-resolution imaging even for the complex-shaped 3-D objects. Finally, in the experiment employing the UWB module, the RPM with the bi-static extension and the extended Capon method significantly improved the accuracy of target surface extraction including the edge boundaries. These results prove that the proposed method has a great potential for super-resolution radar imaging with a non-parametric approach.

The point , which minimizes the distance to the curve , is calculated by solving the constrained optimization problem as (15) is defined with the Lagrange multiplier

as (16)

Then,

satisfies the following conditions

(17)

KIDERA et al.: SUPER-RESOLUTION UWB RADAR IMAGING ALGORITHM BASED ON EXTENDED CAPON

Solving these equations, the following cubic equation of derived,

is

(18) With the solution lated as

in (18), the minimum distance is calcu-

(19) This derivation is readily extended to the general bi-static or multistatic model by modifying the definition of the ellipsoids in (11). REFERENCES [1] D. L. Mensa, G. Heidbreder, and G. Wade, “Aperture synthesis by object rotation in coherent imaging,” IEEE Trans. Nucl. Sci., vol. 27, no. 2, pp. 989–998, Apr. 1980. [2] D. Liu, G. Kang, L. Li, Y. Chen, S. Vasudevan, W. Joines, Q. H. Liu, J. Krolik, and L. Carin, “Electromagnetic time-reversal imaging of a target in a cluttered environment,” IEEE Trans. Antenna Propag., vol. 53, no. 9, pp. 3058–3066, Sep. 2005. [3] D. Liu, J. Krolik, and L. Carin, “Electromagnetic target detection in uncertain media: Time-reversal and minimum-variance algorithms,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 4, pp. 934–944, Apr. 2007. [4] J. Song, Q. H. Liu, P. Torrione, and L. Collins, “Two-dimensional and three dimensional NUFFT migration method for landmine detection using ground-penetrating radar,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 6, pp. 1462–1469, Jun. 2006. [5] F. Soldovieri, A. Brancaccio, G. Prisco, G. Leone, and R. Pieri, “A kirchhoff-based shape reconstruction algorithm for the multimonostatic configuration: The realistic case of buried pipes,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp. 3031–3038, Oct. 2008. [6] A. Massa, D. Franceschini, G. Franceschini, M. Pastorino, M. Raffetto, and M. Donelli, “Parallel GA-based approach for microwave imaging applications,” IEEE Trans. Antenna Propag., vol. 53, no. 10, pp. 3118–3127, Oct. 2005. [7] T. Sakamoto and T. Sato, “A target shape estimation algorithm for pulse radar systems based on boundary scattering transform,” IEICE Trans. Commun., vol. E87-B, no. 5, pp. 1357–1365, 2004. [8] S. Kidera, T. Sakamoto, and T. Sato, “High-resolution and real-time UWB radar imaging algorithm with direct waveform compensations,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 11, pp. 3503–3513, Nov. 2008. [9] S. Kidera, T. Sakamoto, and T. Sato, “Accurate UWB radar 3-D imaging algorithm for complex boundary without range points connections,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 4, pp. 1993–2004, Apr. 2010. [10] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE, vol. 57, no. 8, pp. 1408–1418, Aug. 1969.

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[11] J. Capon and N. R. Goodman, “Probability distributions for estimators of frequency wavenumber spectrum,” Proc. IEEE, vol. 58, no. 10, pp. 1785–1786, Oct. 1970. [12] C. D. Richmond, “Capon algorithm mean-square error threshold SNR prediction and probability of resolution,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 2748–2764, Aug. 2005. [13] S. Kidera, T. Sakamoto, and T. Sato, “A high-resolution imaging algorithm without derivatives based on waveform estimation for UWB radars,” IEICE Trans. Commun., vol. E90-B, no. 6, pp. 1487–1494, Jun. 2007. [14] Federal Communications Commission (FCC), Office of Engineering and Technology (OET) Technology (OET) Bulletin No. 65, Aug. 1997, no. Supplement C, pp. 35–35. [15] S. Kidera, T. Sakamoto, and T. Sato, “Experimental study of shadow region imaging algorithm with multiple scattered waves for UWB radars,” Proc. PIERS, vol. 5, no. 4, pp. 393–396, Aug. 2009. Shouhei Kidera (A’08) received the B.E. degree in electrical and electronic engineering and the M.I. and Ph.D. degrees in informatics from Kyoto University, Kyoto, Japan, in 2003, 2005, and 2007, respectively. He is an Assistant Professor in the Graduate School of Informatics and Engineering, University of Electro-Communications, Japan. His current research interest is in advanced signal processing for the near field radar, UWB radar. Dr. Kidera is a member of the Institute of Electronics, Information, and Communication Engineers of Japan (IEICE) and the Institute of Electrical Engineering of Japan (IEEJ).

Takuya Sakamoto (M’04) was born in Nara, Japan, in 1977. He received the B.E., M.I., and Ph.D. degrees from Kyoto University, Kyoto, Japan, in 2000, 2002, and 2005, respectively. He is an Assistant Professor in the Department of Communications and Computer Engineering, Graduate School of Informatics, Kyoto University. His current research interest is in signal processing for UWB pulse radars. Dr. Sakamoto is a member of the Institute of Electronics, Information, and Communication Engineers of Japan (IEICE), and the Institute of Electrical Engineering of Japan (IEEJ).

Toru Sato (M’92) received the B.E., M.E., and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan, in 1976, 1978, and 1982, respectively. He has been with Kyoto University since 1983 and is currently a Professor in the Department of Communications and Computer Engineering, Graduate School of Informatics. His major research interests have been system design and signal processing aspects of atmospheric radars, radar remote sensing of the atmosphere, observations of precipitation using radar and satellite signals, radar observation of space debris, and imaging with UWB pulse radars. Dr. Sato was awarded the Tanakadate Prize in 1986. He is a fellow of the Institute of Electronics, Information, and Communication Engineers of Japan, and a member of the Society of Geomagnetism and Earth, Planetary and Space Sciences, the Japan Society for Aeronautical and Space Sciences, and the American Meteorological Society.

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Effective Local Absorbing Boundary Conditions for a Finite Difference Implementation of the Parabolic Equation Selman Özbayat, Student Member, IEEE, and Ramakrishna Janaswamy, Fellow, IEEE

Abstract—Domain truncation by transparent boundary conditions for open problems where parabolic equation is utilized to govern wave propagation are in general computationally costly. We utilize two approximations to a convolution-in-space type discrete boundary condition to reduce the cost, while maintaining accuracy in far range solutions. Perfectly matched layer adapted to the Crank-Nicolson finite difference scheme is also verified for a 2-D model problem, where implemented results and stability analyses for different approaches are compared. Index Terms—Crank-Nicolson finite-difference scheme, discrete transparent boundary condition, parabolic equation, perfectly matched layer.

I. INTRODUCTION

R

ADIOWAVE propagation governed by Helmholtz equation can be approximated by parabolic equation (PE) under certain circumstances if the spectral content of waves is narrow around the propagation axis and refractive index inhomogeneity of the atmosphere is smooth [1]. Conventional narrow-angle PE approximation of wave propagation assumes maximum ray angle to be within with respect to the axis of propagation (range). The main distinction between the Helmholtz equation and PE is the reduction of 2nd order derivative to a 1st order derivative along the range, which is based on ignoring back-scattering in the domain. This facilitates a marching-in range technique for numerical computation. Most practical PE applications are concerned with far-range wave behavior, e.g., in tropospheric calculations, where the grazing angles are already very small, which makes the error due to large propagating angles affordable. Another application of PE is propagation prediction in tunnels with lossy walls, where the wave content in long ranges is dominated by the small grazing angles as well [2]. Two schemes are available for the solution of PE, first of which is the split-step Fourier technique that is applicable when analytical eigenfunctions exist for the underlying geometry. Although this technique is numerically very efficient, it is inconvenient for boundary modeling [1].

Manuscript received June 16, 2010; revised October 01, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was funded in part by Army Research Office under Grant W911NF-10-1-0305 The authors are with the Center for Advanced Sensor and Communication Antennas, University of Massachusetts Amherst, MA 01003 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122300

A purely numerical scheme (such as the one based on finite differences (FD)) on the other hand is applicable for more general geometries with boundaries. The subject matter of the current paper is applicable to the latter and Crank-Nicolson FD scheme is assumed to be employed in particular. Numerical solution of wave propagation in open geometries requires domain truncation at a designated reference. Thus an absorbing boundary needs to be placed for typical PE applications. Formulated first by Berenger [3] for Maxwell’s Equations and used for several different electromagnetic problems, perfectly matched layer (PML) adapted to the Crank-Nicolson scheme is one way of domain truncation in open PE problems. Collino [4] implemented PML for a variational solution to PE, while Levy [5] proposed it as a straightforward truncation for FD and split-step Fourier techniques for PE. However, there are no numerical implementation results for PML adapted to FD schemes of PE so far. The first goal of this paper is to provide some numerical comparison results for PML, when adapted to a narrow-angle PE. A question that arises with the use of PML to PE is how effective it is, given that the PE is most accurate in the region of validity where the PML is most reflective, i.e. for zero grazing angle. Domain truncation can also be achieved by placing transparent boundary conditions (TBC) and there are two approaches to implementing the TBCs. A TBC produces zero boundary reflections into the geometry it is defined for. Several authors [1], [6] introduced the discretized version of a continuous TBC. Reference [7] showed that this approach cannot assure unconditional stability when the FD discretization does not match the discretization of the continuous TBC. It can also be shown that this boundary condition (BC) is not reflection-free. Alternatively, one could start directly from the Crank-Nicolson FD discretization of PE, done by Ehrhardt and Arnold [7], and derive numerically exact discrete transparent boundary condition (DTBC), that involves all the boundary field values starting from the initial plane. For example, if the field is desired at the 1,000th range step, the exact discrete BC will involve convolution of field values on the boundary layer at all the previous 999 range steps. Although accurate, DTBC will increase the CPU time, particularly for long ranges and there is a need for considering approximate BCs that are local. The second goal of the paper is to propose some local boundary conditions and show numerical comparisons for their performance. The first localization approach to DTBC will be to truncate the boundary layer convolution by relatively small number of terms, and utilize for convolution not only the boundary layer values but also some already computed interior field values,

0018-926X/$26.00 © 2011 IEEE

ÖZBAYAT AND JANASWAMY: EFFECTIVE LOCAL ABSORBING BCs FOR A FD IMPLEMENTATION OF THE PE

thereby resulting in what we will call the localized DTBC (LDTBC). This is achieved by employing a rational approximation to the spectral transfer function. The second localization is also in the form of a rational approximation, namely an approximation by partial fractions using simple poles in the complex -transform domain [8]. Both localizations will prove to be more accurate than simple truncation of the boundary layer convolution. The stability analysis for the first approximation will be carried out using a 3-layer reduced domain relying on pole locations in -domain, which will turn out to be a sufficient check for the stability of the whole geometry. Simple pole locations in -domain will determine whether the true discrete problem is stable for the second approximation. The main task of numerical verification and simulation of different domain truncations for the discrete PE will be carried out on a simple model problem, i.e. 2-D free-space truncated by the absorbing layer of interest at some elevation and by Perfect Electric Conductor (PEC) as terrain ground plane. True terrain may be uneven which will tend to increase the grazing angles of waves upon reflection. We use flat PEC terrain for simplicity, but waves of larger grazing angles are included by modifying the parameters of the Gaussian source. Crank-Nicolson scheme adapted to the model problem is formulated in Section II-A, whereas in Section II-B the PML absorption will be verified for and applied to the problem. The exact DTBC, corresponding to an exact transfer function and its Taylor series in -domain, is presented in Section II-C. The two approximate localizations to DTBC with stability analyses, implemented results for the model problem and efficiency comparisons will take place in Sections II-D, II-E and III.

Fig. 1. Computation domain, top layer being absorbing boundary and bottom layer PEC.

where is a unit-less parameter. Note that are avoided by averfield values at half-integer range and this scheme is of second aging same fields at and order convergence [11]. As it will be discussed later, the solution to this system of equations will require updating by two tri-diagonal matrices after each march-in range. B. PML Implementation The two-dimensional PML is constructed by replacing the height with complex stretched-coordinate [4] given by (3)

II. THEORY

We use a parabolic profile for the normalized conductivity (unit-less)

A. Standard PE The standard PE in the reduced variable

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is (1)

which is obtained after narrow-angle approximation to the Helmholtz wave equation [9], , is the wave-number in free space at the radian frequency and is the modified for the refractive index of the medium. Here we use simple free space domain, represents the range coordinate variable and the elevation coordinate variable of the compuas the domain tation domain depicted in Fig. 1. We denote as the elevation at which the excitation will be height and placed. This will be referred to as the mean height of the excited Gaussian source that will be discussed later in the paper. The , can more general case of non-zero but constant V for in the be handled by making the transformation end result as in [10]. and exUtilizing the discrete notation for panding central differences around discrete range the derivatives in (1), the Crank-Nicolson discretization for (1) is

(2)

(4) where is the wave impedance in free space and is the true has units conductivity (S/m). Recall that the product thus remains unit-less. The coordinate transformation in (3) will be stable for the Sommerfeld radiation condition, since time convention in time is used. The standard PE in an (1) in the stretched-coordinates becomes (5) Equation (5) discretized utilizing Crank-Nicolson FD scheme becomes [11] with

(6)

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Note that basically reduces (6) to (2) (that governs the domain in Fig. 1 without PML above). The quadratic deon height given in (4) and the associated pendence of wave absorption provides the freedom to place a PEC on top of the PML, since the rays in the layer experience a controlled loss twice as they reflect back into the domain. The PML of thickis backed with a PEC. We ness , a few wavelengths above have control over and to meet a desired reflection coefficient . The reflection coefficient from the PML for a ray with paraxial angle is [3], [4] (7) For the parabolic profile this results in an equation for the true conductivity in terms of the desired reflection coefficient as (8) One aspect to note about reflection is that, it is smaller for wider angle , which leads to the fact that PML simulates a better absorption for waves propagating at wider angles than those at shallower angles [1]. The reflection coefficient is one . That is why is chosen by setting in magnitude for the reflection for a small grazing angle, i.e. a small , so that higher angle content will reflect even less back into the domain. One restriction on layer thickness is that it is not desirable to set it too high, since it will increase the matrix sizes for each marching-in range computation. Computation of (6) obviously involves two tri-diagonal matrices to update the field vector after each marching in range, and entries of both matrices are constants. Therefore, the matrix equation representation of (6) for the domain in Fig. 1 is of the form (9) The length of the vector is and since we need in (6), we discretize the conductivity half-integer indices of two times finer than we do the field .

Fig. 2. Branch cut on z -plane for j (z)j = 1, R = 0:63. C is the inversion circle around origin.

is counter-clockwise as indicated in The inversion contour is assumed for . Taking Fig. 2. Also, as in [7], the -transform on both sides of (2) results in

(11) . More specifically, the transfer function relating where field on the boundary layer in -domain to the field on the layer just below it is

(12) The transfer function in general is multi-valued, therefore the operator above is defined to ensure , for . Let which reflects decaying nature of denote a modified transfer function defined as

C. DTBC Derivation We will now derive and approximate the DTBC directly for the Crank-Nicolson scheme. The original derivation of DTBC is given in [7]. However, we derive a newer form that is amenable to approximations. Solving the discrete PE on and above the boundary layer while assuming a decaying nature of the fields above this layer results in [7] (10) and are weights of convolutions on a layer and , below it. Such a convolution-type on the layer relation of fields leads us to use -transforms, which translates the relation of field values on successive layers on or above the boundary layer to a polynomial in the -transform as complex -domain. We will define the where and the inverse -transform as [10].

(13) with . We introduce a new variable and take the first derivative of (13). This is so that the resulting function can be expressed in terms of where Legendre polynomials as are the Legendre polynomials. The derivative of with respect to is (14) where

(15)

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Using the Legendre function representation of the quantity under the radical sign gives

(16) with written as

. The Taylor-series expansion of

can be

(17)

Fig. 3. Exact DTBC convolution stencil. The two layers of nodes correspond to the two convolutions in (10). This stencil is valid at all altitudes greater than J x.

1

with coefficients terms are non-zero in the second summation. This is because of . the second degree polynomial in the denominator of (18) operator in (12) is As already indicated the multi-valued . Using (17), the modified defined such that and exact transfer functions can finally be written as

D. LDTBC Approximation The first approximation we propose is replacing the lengthsummation in (20) by a Padé approximant , of order

(21) (19) and

(20) . The region of convergence of the with series depends on the singularities of . This leads to the investigation of properties of the exact transfer function in -plane. The branch cut in -plane for the function given in (12) is depicted in Fig. 2, as a bold dashed curve separating the two on it. It can be shown that the branch sheets with and . The point points are at in general could be in either of 1st or 2nd quadrants depending on . There is a practical reason why (12) or (20) is not suitable for direct use in the Crank-Nicolson scheme. Because we are interested in a bounded computation domain that starts at range , the infinite summation in (20) reduces to a convolution of size where is the range of the present marching step. The upper limit in (20) will be referred as being henceforth. This is usually a large number, especially for fine discretizations and far ranges. To have an idea of how this transfer function affects the boundary computation in the spatial domain and how ineffective the convolution is, the nodes used in convolution are depicted in Fig. 3. There, the boundary is at discrete layer and all the nodes on that layer are involved in the first summation in (10). The two nodes on layer contribute to the second summation in (10), i.e. only the first two

where ’s and ’s are determined by conventional Padé approximant procedure [12]. The Taylor series expansion derived in (20) is very convenient in this regard. We will set the leading for convenience. The relations denominator coefficient also in a rational form (20) and (21) result in

(22)

where

for , and . in (22) is plotted in Fig. 4 for The transfer function , . It should be stressed at this point that the Padé approximation above is obtained by enforcing continuities is at the origin in -domain, thus an approximation of order less valid at points closer to the unit circle than near the origin. circle are utilized above for the sake of Points on demonstrating severity of the approximation (the approximation ). It is seen that an approximation of will be worst for higher order is always favorable and that an approximation of is good enough to mimic the exact transfer order function for this case. To appreciate the effect of using interior domain points, i.e. on one layer below the absorbing layer, the case and coefficient magnitudes for the two cases, case, are shown in Fig. 5 for the same value. Clearly, the dominance of low order terms in vector against higher order terms in it is much more significant for approximations with

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where

(25)

(26)

(27) Fig. 4.

j

 (z )j for the z values on the jz j = 0:9 circle, R

(28)

= 0:63.

and for

Fig. 5. Pade coefficient magnitudes of different LDTBC approximations, R = 0:63. d and e are respective vectors of numerator and denominator coefficients of the corresponding approximation in each case.

higher Q. In other words to realize a given accuracy, a smaller is needed with than with , which will result in a smaller convolution in boundary terms. More simulation results pairs are shown in Section III. for different in (22) back to the spatial domain Transforming from through inverse -transform, the boundary value at arbitrary range index is determined in terms of already computed ) and values boundary values (convolution of length on the uppermost computation layer (convolution of length ). Inverse -transform of (22) gives the boundary value at arbitrary discrete range , , as

,

,

(29) The unconditional stability of Crank-Nicolson scheme in free space is well known [11]. However, when the exact transfer function (20) is approximated as in (22), the scheme is not unconditionally stable anymore and it may not be trivial to derive a stability condition for a given geometry and discretization. Instead we study the reduced geometry depicted in Fig. 6, i.e. for a 3 layer problem—the top layer being absorbing boundary, the mid-layer being the computation domain and the bottom layer being the PEC. Although we do not prove that stability of this reduced geometry leads to stability of the larger domain in Fig. 1, it still provides a stability check since the instability comes only from associated approximate boundary conditions. The excitation we use in this case is a point source of magnitude , located in the middle layer, i.e. at . Since an impulse in spatial domain will have components in all angular spectra, such an excitation is a worst-case test for the narrow-angle PE. For this small geometry, (2) reduces to (30) where , layer. Recalling

is enforced by the PEC bottom , the -domain version of (30) becomes

(31) where , , 2 and Furthermore, substitution of (22) for

. gives

(23) (32) The matrix representation of (6) with an absorbing boundary at the top and PEC boundary at the bottom will be of the form (24)

being residues and poles of the function. Equation (32) is true if . This is always the case here, because pairs were obtained through truncating a very long convolution by

ÖZBAYAT AND JANASWAMY: EFFECTIVE LOCAL ABSORBING BCs FOR A FD IMPLEMENTATION OF THE PE

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simple poles in the -domain, assuming they all lie outside the unit circle (to assure stability). The polynomial corresponding to the spatial boundary layer convolution in (20) is approximated as (34) Fig. 6. Reduced geometry to study the approximation stability.

Namely, the approximation is

(35) We retain the original coefficients for the first

terms so that (36)

Fig. 7. Inverse pole locations 1=z for reduced-domain LDTBC solutions on complex Z -domain, l = 0; 1; . . . ; P . Stars are for the unstable case and inverted triangles are for the stable case.

terms and padding new terms to a length-2 convolution (revisit Fig. 3). Transforming (32) back to spatial domain gives (33) It is sufficient for the poles of the system to all lie outside should lie inside the the unit circle for stability (or each unit circle). Fig. 7 depicts two cases, where the approximation is stable in one case and unstable in the other. One point to emphasize about this stability analysis is that it pair for every may not always be trivial to find a stable and , especially if and are desired to discretization be significantly large. E. Approximation by Partial Fractions: LDTBC formulated above introduces tolerable inaccuracy while avoiding a convolution involving all boundary values. However, the numerical results for quite long ranges will show pair will have to be used in LDTBC apthat a large order proximation. The immediate question of how easy is it to find a pair for moderate orders comes up. Moreover, such stable a stable LDTBC approximation still requires two convolutions at each march-in range, i.e. one on boundary layer values of and another of order on one layer below the order boundary. This may be undesirable. A second approximation to the exact and modified transfer functions in (19) is still in the form of a Padé approximant, but . This case has been proposed in [8] and of order corresponds to approximation by partial fractions with

with and , . The standard Padé approximant that gives unique set of ’s involves a linear system to solve [12] and ’s are deteris satisfied for mined by back-substitution; thus . The higher order approximate coefficients given by (36) are desired to mimic exact at as many points as possible, and the quality of the approximation at orders higher will be the deciding factor for choosing . A than scheme with very large is however prone to instabilities due to numerical roundoff. The stability condition of this approximais tion is directly given by the system pole locations, i.e. necessary for stability. However the original function will have as already demonstrated in branch point singularities at Fig. 2, this implies that the approximation could be marginally solution that turns out to be unstable at best. Any stable is expected to possess almost all of its poles in the vicinity of the unit circle but some just inside. In other words, when apis expanded by partial fractions, all or proximate most of the poles of the solution are outside the unit circle, the rest are very close to and inside the unit circle if any. approximaPractical calculations reveal that the tion with more than 30-to-40 simple poles (depending on ) turn out to be unstable. Approximation with fewer poles on the other hand cannot ensure the approximate coefficients in (36) to mimic high order exact coefficients. Such an approximation results in inaccuracy in long-range simulations. For instance, approximation of order is necessary for to accurately mimic in (36), for . The poles of the approximation the given that are inside the unit circle for this case are depicted in Fig. 8. approximation with Furthermore, given an unstable approxigiven number of poles and an parameter, mation with more poles for the same is also another unstable approximation. Hence appropriate pole location modifications are necessary for the sake of stability. Because the poles making the approximation unstable are close to the unit circle, inverting their magnitudes, while their arguments kept unchanged, results in a new stable approximation with undetermined residues. The

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Fig. 8. The poles of in the case of L

LDTBC

approximate solution lying inside unit circle

= 100, M = 0 and R = 0:63.

way to set new residues for the “re-located” poles relies on enforcing continuity of the exact transfer function and its deriva, up to the order to give a linear system to uniquely tives at determine these new residues. This way of stabilizing is summarized as follows. with partial fractions (Padé approximation • ) is applied to approximate the exact of order solution, of the simple poles lying inside the unit circle. , meaning the approximation is unstable, the • If poles inside are pushed out of the unit circle through , . Note here that and are reordered for convenience. • is enforced for giving unique solution of new residues corresponding to re-located poles . Here are also reordered. , Although the two are in the same rational form, providing recursive computation, is superior to LDTBC. Namely there will not be a whole new boundary layer convolution at each march-in range. Instead, the history on the boundary layer will be cumulative, thus could be recursively computed by adding a new term after each march-in range. The approximation in (35) translates in the spatial domain as

coefficients for small . The low order terms of the Legendre polynomials with specific arguments make the first few coefficients very large compared with higher order coefficients are set to be equal to . This [8]. This is why the first makes the dynamic range of the coefficients that rational approximation is enforced on smaller, resulting in a more accurate . The fact that approximation for orders higher than the first few coefficients are very large compared with higher order coefficients is depicted in Fig. 5, where reduces to in case). Choice the case of simple convolution truncation ( will improve the coefficient approximation to an exof tent where it cannot be pushed further, due to the fact that and are those big coefficients making the dynamic range of coefficient magnitudes huge [8]. The dynamic range of in Fig. 5 ( in blue) will be more severe for smaller , thus makes the need to shift the approximation termination by terms inevitable. III. NUMERICAL RESULTS Numerical results for the Crank-Nicolson FD scheme outas well as the PML are lined above for LDTBC and compared in this section. As mentioned previously, the initial , where the abdomain truncation enforces layer. We generate a Gaussian sorbing boundary is placed at source with a standard deviation and centered at height , by considering an infinite sum of eigenfunctions of an underlying geometry (parallel plate waveguide with PEC walls at and ). The approximate Gaussian function is obtained by retaining only the first number of eigenfunctions

(40) is chosen so that the effective The standard deviation beam-width remains well within the PE approximation. The solution of the standard PE in (1) at a range for the excitation in (40) and a PEC boundary at could be shown to be

(37) where .

(38)

in (37) is to be computed at each The convolution of order range step; on the other hand are updated at each step recursively, that results in the boundary layer value at discrete range

(41) where

is the Fresnel Integral and (42)

(39) The reason for keeping the first -term convolution and not associating it in the approximation is revealed in the nature of

for Note that there is no PEC boundary at The model geometry (2-D free space) is truncated at

.

ÖZBAYAT AND JANASWAMY: EFFECTIVE LOCAL ABSORBING BCs FOR A FD IMPLEMENTATION OF THE PE

= 10 000

=1

=

Fig. 9. Field magnitudes (V/m) at range t ; , with  , x for the excitation. The dashed DTBC solution curve is x = and K indistinguishable from the solid analytical solution curve.

2

= 400

Fig. 10. L -norm errors. The dashed curves show LDTBC errors for varying P. Straight solid lines with squares, pentagrams and circles denote error levels for other numerical solutions. Simulation parameters are the same as in Fig. 9.

above the ground plane. This deliberate choice of a short geometry helps simulate far-range solutions quickly, while each excited mode in (40) is assured to experience lot of reflections from both boundaries before arriving at the observation range. The standard deviation of the excitation in (40) is set as , which results in half power points of approximately around the paraxial axis [13]. Fig. 9 depicts solutions at the given range. The exact DTBC solution (dashed) and the PML solution (pentagrams) are obviously very accurate, they match the analytical solution (solid curve) at every altitude of the model geometry. The two approximations, i.e. inaccurate LDTBC solutions are selected on purpose, just to demonstrate that LDTBC approximation is indeed . In that case it is just truncation of severe and useless if the boundary layer convolution in Fig. 3 by nodes. We choose an -norm to quantify magnitude of errors (43)

Fig. 11. Field magnitudes (V/m) at range t ters are the same as in Fig. 9.

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= 10; 000. Simulation parame-

where at discrete altitude , is the analytical field solution is the numerical field solution computed (could be one and ). This error is depicted of PML, DTBC, LDTBC or -norm in Fig. 10, where the three dashed curves show the errors for LDTBC approximations with different . The horizontal axis is varying from 10 to 150, and the three errors expectedly decay with increasing . One thing to recall is that pair, thereLDTBC is not necessarily stable for a given pairs for fore while generating the dashed curves, 10 stable each are first selected based on the 3-layer model discussed in Section II-D, then the curves are formed by linear interpolation of errors defined in (43). These errors are denoted as . Because it avoids the computational burden of convolutions, on the other hand offers for a fast and accurate approximate solution as depicted in Fig. 11. The approximation with 21 poles (brown squares) for given discretization and range is not as accurate; on the other hand the approximation with 101 poles (pink circles) was not stable originally. However, inverting the poles that are inside the unit with 101 simple poles becomes stable and circle, accurate as shown in Fig. 11. It avoids the stability problem through re-approximating the residues of re-located poles as outlined in Section II-E, and accurately mimics the analytical solution while being efficient in CPU time. These two solutions produce those -norm errors shown in Fig. 10 with respective color curves, denoted as . These curves denote nothing but the error level for the respective approximations in Fig. 11, and they are independent of . in Fig. 10 denotes the error level for The quantity the PML solution (green pentagrams) which was also depicted in Fig. 9. At this range, PML results are always accurate and robust. Having control over , and hence over , excellent absorption could be achieved with PML even though a significant spectrum of the excitation contains low grazing angle waves. Another factor in favor of PML over DTBC and thus over LDTBC for long ranges is the computational effort, where the convolution in (23) is avoided altogether. To achieve the in Fig. 10, a LDTBC approxivery low error level of mation of order would be needed, which is

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Fig. 12. Field magnitudes (V/m) for two different problems. The solid curves range for a x domain and dashed lines are solutions at t . The simulation parameters are x at t range for x , t , , . ,x ,L

= 20 km = 50 m = 100 m = 50 km 20 cm 1 = 5 m = 10 m = 30 cm = 3 m = 150

Fig. 13.

L

-norm errors  . All parameters are the same as in Fig. 12.

1 =

neither as accurate nor as fast at this range even if stable. Thus are the two efficient and it is clear that PML and accurate numerical solutions for domain truncation at moderate ranges. Because the grazing angle content becomes very narrow for longer ranges, the large ratio between observation range and domain height can render the PML solution inaccurate, i.e. PML would fail to absorb dominant propagating ray [1]. The next set of simulation results will depict such a more realistic case, where the domain (in free space) sizes will be in metric units and larger along both altitude and range. The Gaussian source and the operating frequency will be will reside at . Two different cases are depicted in Fig. 12. The dashed curves are solutions at 50 km range for a 100 m domain truncation and the solid curves are solutions at 20 km range for 50 m domain truncation and both are cases of far-range and high-domain simulations for which the LDTBC approximation are subseemed to fail. For this reason only PML and ject to comparison with respective -norm errors as marked in the figure. Although both are perfectly absorbing at moderate generally produces a smaller error and small ranges, than PML does at far ranges. This is mainly due to the fact that PML cannot be pushed further, i.e. using a thicker matching layer or setting an extremely small reference reflection does not help beyond a certain limit. Indeed, the PML parameters used in Fig. 12 are the limits for this case which do not improve further. The parameter on the other hand determines the order , thus could be set to reasonably and the accuracy of large values to simulate accurate absorption. However, there is also a limit to , thus there is a range-to-height ratio limit to accurate absorbing solutions. This limit will be discussed later. Fig. 13 in this regard analyzes the effect of observation range and domain height with respect to the error. The range that PML with fixed parameters start to fail at is closer when or than when . This is obvious since each undesired boundary reflection adds inaccuracy to the solution, making smaller altitude boundary problems harder to

simulate at far ranges. The error comparison is held up to a range where the -norm errors become comparable to 1 (which error, i.e. in Fig. 13 is huge in this case). The . Therefore is always smaller than its PML counterpart with 151 simple poles for our case is clearly more accurate than any PML solution at far ranges. Is it possible then to improve it further by setting a bigger ? As indicated before, there also is a certain limit to how big would be chosen. Firstly, solving for the residues and poles in (38) involves inverting a full-matrix. Moreover this matrix to be inverted contains the derivatives of the transfer function , . Therefore depending or of its exact Taylor on the nature of the transfer function coefficients , this matrix could turn out to be ill-conditioned. Practical simulations for problems of our interest show that the condition number of this matrix becomes extremely large if and MATLAB cannot handle such ill-conditioned matrices. That is why our choices of in above results represents approximation orders close to the practical limit. Fig. 14 shows the CPU time for DTBC and convolutions with increasing range. Computing a whole new convolution of all the boundary layer values at each march-in range, exact DTBC solution has quadratic dependency on time for increasing order, whereas recursive computation in solution is linearly dependent on range. This comdepicted in Fig. 14 is parison between DTBC and only for the boundary computation, i.e. the CPU time comparison does not consider time for matrix inversion and update operations, which is not as much as convolutions times at far ranges. Since PML is totally local, i.e. does not employ a boundary layer convolution, there really is no significant added cost to implement it even though the matching layer enlarges the computation domain height by . Therefore, we can surely say that PML is the fastest in CPU time among all techniques discussed here. IV. CONCLUSIONS The exact DTBC verified is unconditionally stable and presents reflection-free boundary layer for open PE problems. However, its computational deficiency depicted in Fig. 3 and in

ÖZBAYAT AND JANASWAMY: EFFECTIVE LOCAL ABSORBING BCs FOR A FD IMPLEMENTATION OF THE PE

Fig. 14. CPU time taken by the boundary computation (convolution in DTBC case).

Fig. 14 motivates localization of the boundary layer convolution. Although they are conditionally stable and not always as are computationally accurate as DTBC, LDTBC and more efficient thus more practical. The FD and narrow-angle PE constraints apply for both exact and approximate transfer functions, therefore the approximations would be preferable at not so close ranges, because they reduce the convolution time especially reduces computation time significantly. dramatically and is assured to be stable due to the modifications to residue/pole locations suggested in the paper. PML on the other hand is robust, fast and is alternative to at moderate ranges. We were able to verify that the degree of freedom to set the layer thickness and layer loss in makes it easily according to the desired maximum reflection verifiable and applicable to truncate open geometries governed by PE using Crank-Nicolson scheme. Fig. 10 summarizes a are the appropriate truncacomparison, that PML and tion techniques for computations at moderate ranges. The study of these techniques at typical far ranges for high geometries in Fig. 12 and in Fig. 13, where PML fails, is interesting. There, remains as the best solution among all. The practical limit to makes its operation range also limited. We do not delve to numerically improve this ill-condition, and avoiding this restriction would help simulate PE truncation at even farther ranges. The procedure detailed here for the derivation of approximate boundary conditions is also applicable to the wide-angle PE as long as a suitable discretization scheme is available in the interior domain. This study will be taken up in the future. REFERENCES [1] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. London, U.K.: IEE Press, 2000. [2] R. Martelly and R. Janaswamy, “Modeling radio transmission loss in curved, branched and rough-walled tunnels with the ADI-PE method,” IEEE Trans. Antennas Propag., vol. 58, pp. 2037–2045, Jun. 2010. [3] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, pp. 185–200, 1994. [4] F. Collino, “Perfectly matched absorbing layers for the paraxial equation,” J. Comput. Phys., vol. 131, pp. 164–180, 1997.

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[5] M. F. Levy, “Perfectly matched layer truncation for parabolic wave equation models,” Proc. R. Soc. Lond. A, vol. 457, pp. 2609–2624, May 2001. [6] V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion, vol. 14, pp. 123–128, Jan. 1991. [7] M. Ehrhardt and A. Arnold, “Discrete transparent boundary conditions for the Schrödinger equation,” Riv. Mat. Univ. Parma, vol. 6, pp. 57–108, 2001. [8] A. Arnold, M. Ehrhardt, and I. Sofronov, “Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability,” Commun. Math. Sci., vol. 1, no. 3, pp. 501–556, 2003. [9] M. F. Levy, “Transparent boundary conditions for parabolic equation solutions of radiowave propagation problems,” IEEE Trans. Antennas Propag., vol. 45, pp. 66–72, Jan. 1997. [10] R. Janaswamy, “Transparent boundary condition for the parabolic equation modeled by the 4RW,” IEEE. Antennas Wireless Propag. Lett., vol. 8, pp. 23–26, May 2009. [11] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 2008. [12] G. A. J. Baker and P. Graves-Morris, Padé Approximants, II ed. New York: Press Syndicate of Univ. Cambridge, 1996. [13] R. Janaswamy, “Radio wave propagation over a nonconstant immittance plane,” Radio Sci., vol. 36, pp. 387–405, May–Jun. 2001. Selman Özbayat (S’08) was born in Gebze, Turkey in 1986. He received Bachelor’s degree in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 2008. In September 2008, he joined the Antennas and Propagation Laboratory, University of Massachusetts, Amherst, where he is currently working toward the Ph.D. degree. His research interests include parabolic equation methods for radio wave propagation, random walk methods for electrodynamics and computational electromagnetics. Mr. Özbayat is a student member of the Applied Computational Electromagnetic Society (ACES).

Ramakrishna Janaswamy (F’03) received the Ph.D. degree in electrical engineering from the University of Massachusetts, Amherst, in 1986, the Master’s degree in microwave and radar engineering from IIT-Kharagpur, India, in 1983, and the Bachelor’s degree in electronics and communications engineering from REC-Warangal, India, in 1981. From August 1986 to May 1987, he was an Assistant Professor of electrical engineering at Wilkes University, Wilkes Barre, PA. From August 1987 to August 2001, he was on the faculty of the Department of Electrical and Computer Engineering, Naval Postgraduate School, Monterey, CA. In September 2001, he joined the Department of Electrical and Computer Engineering, University of Massachusetts, where he is a currently a Professor. He is the author of the book Radiowave Propagation and Smart Antennas for Wireless Communications (Kluwer Academic, November 2000) and a contributing author in the Handbook of Antennas in Wireless Communications (CRC Press, August 2001) and Encyclopedia of RF and Microwave Engineering (Wiley, 2005). His research interests include deterministic and stochastic radio wave propagation modeling, analytical and computational electromagnetics, antenna theory and design, and wireless communications. Prof. Janaswamy is a Fellow of IEEE and was the recipient of the R. W. P. King Prize Paper Award of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in 1995. For his services to the IEEE Monterey Bay Subsection, he received the IEEE 3rd Millennium Medal from the Santa Clara Valley Section in 2000. He is an elected member of U.S. National Committee of International Union of Radio Science, Commissions B and F. He served as an Associate Editor of Radio Science from January 1999 to January 2004, and was an Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY from 2003 to 2006. He is currently an Associate Editor of IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and of IETE Technical Reviews.

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TD-UTD Solutions for the Transient Radiation and Surface Fields of Pulsed Antennas Placed on PEC Smooth Convex Surfaces Hsi-Tseng Chou, Senior Member, IEEE, Prabhakar H. Pathak, Fellow, IEEE, and Paul R. Rousseau, Member, IEEE

Abstract—A time-domain formulation of the uniform geometrical theory of diffraction (TD-UTD) is developed for predicting the transient radiation and surface fields of elemental pulsed antennas placed directly on a smooth perfectly conducting, arbitrary convex surface. The TD-UTD solution is obtained by employing an analytic time transform (ATT) for inverting into time the corresponding frequency domain UTD (FD-UTD) solution. An elemental antenna on the convex surface is excited by a step function in time and a TD-UTD solution is obtained first. The TD-UTD response to a more general pulsed excitation of the elemental current is then found via an efficient convolution of the TD-UTD solution for the step function excitation with the time derivative of the general pulsed excitation. In particular, this convolution integral is essentially evaluated in closed form after representing the time derivative of the general pulsed excitation by a small sum of simple signals whose frequency domain description is a sum of complex exponential functions. Some numerical examples are presented to illustrate the utility of these TD-UTD solutions for pulsed antennas on a convex body. Index Terms—Antenna theory, electromagnetic diffraction, electromagnetic radiation, time domain analysis.

I. INTRODUCTION TIME DOMAIN (TD) version of the uniform geometrical theory of diffraction (UTD), or TD-UTD, has been developed previously for analytically describing the transient electromagnetic (EM) scattering by perfectly conducting arbitrary curved wedges [1], [2], and arbitrary smooth convex surfaces [1], [3], respectively, when each of these is excited by a time impulsive astigmatic wavefront. These TD-UTD solutions have been constructed from the corresponding frequency domain (FD) UTD solutions [2], [4], [5] by an application of the analytic time transform (ATT) [1]–[3], [6], [7]. The advantage of the ATT is that it avoids some of the serious difficulties that result in the use of conventional Laplace/Fourier inversion techniques as discussed in [1]–[3], [6], [7]. Furthermore, the ATT allows one to essentially obtain a closed form (as opposed to numerical) convolution when the TD-UTD response to a realistic finite energy pulse excitation is to be obtained from the TD-UTD response to a time impulsive excitation [1]–[3]; the

A

Manuscript received August 31, 2010; accepted October 20, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. H.-T. Chou is with the Department of Communications Engineering, Yuan Ze University, Chung-Li 320, Taiwan, R.O.C. (e-mail: [email protected]). P. H. Pathak is with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA. P. R. Rousseau is with the Aerospace Corporation, El Segundo, CA 90245, USA. Digital Object Identifier 10.1109/TAP.2011.2122235

latter is a significant advantage which maintains the efficiency of the TD-UTD ray analysis. In addition, TD-UTD employs the same ray paths as in the FD-UTD, and like the FD-UTD it remains uniformly valid across the geometrical optics (GO) shadow boundary transition regions. In this paper, the TD-UTD is extended to analyze another important situation, namely that of analytically predicting the transient radiation and surface fields of elemental pulsed antennas located directly on a perfectly conducting arbitrary convex surface. While the previous TD-UTD scattering solutions [1]–[3] were restricted to sources and observers both sufficiently far from the wedges or smooth convex surfaces, the present TD-UTD solution allows for the pulsed source to lie directly, and hence conformally, on the convex surface while the observer is located either sufficiently far from the convex surface in the radiation problem or directly on the convex surface for the surface field problem. By reciprocity, the radiation solution or the surface field solution also provide the surface fields (currents [8]–[10] and charges) induced on the convex surface by a pulsed source which is off or on the surface, respectively. This new TD-UTD solution is also obtained from the corresponding FD-UTD solution in [4], [11], [12] via the use of ATT when the elemental (point) antennas are excited with a step function in time. Each new UTD solution corresponds to another building block for solving more complex situations than possible with any one of these UTD papers. Thus, all these TD-UTD papers have a common goal in that they increase the variety of potential applications. The major difficulty in the development of the TD-UTD analysis here is due to the presence of the radiation and mutual coupling (surface field) type Fock integrals [11], [12] in the FD-UTD fields which are to be inverted into the TD via the ATT. However, this latter step in the analysis can be performed in a manner somewhat analogous to that developed earlier for the TD-UTD solution for the scattering by a smooth convex object [1], [3]. It is noted that the UTD Fock functions for the scattering [1], [3] are different from those of the UTD radiation and mutual coupling Fock functions which provide a uniform ray solution for radiation and surface field problems of interest in this paper. The present TD-UTD radiation solution is first developed as a response to the case when the excitation by infinitesimal impressed electric or magnetic current elements (or current moments) is a step function in time, and where these current moments are placed on the convex surface. The response to the step function excitation is developed because it appears to be the simplest one to obtain for the problems of interest in this paper. It is required to differentiate this solution if a response to a time impulsive excitation of the current element is desired. Such a

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CHOU et al.: TD-UTD SOLUTIONS FOR THE TRANSIENT RADIATION AND SURFACE FIELDS OF PULSED ANTENNAS

time impulsive excitation is useful because the TD-UTD solution for this excitation can be used to obtain a TD-UTD solution for a more general pulsed excitation via convolution. However, the resulting TD-UTD solution for time impulsive excitation will become complicated if it is obtained by differentiating the step response, or even if it can be obtained directly. Hence, in practical utilization of the present TD-UTD solution for the step function excitation case to obtain the TD-UTD for arbitrary excitation, it is convenient and equivalent to instead differentiate the arbitrary pulsed excitation function first and then perform its convolution with the present TD-UTD solution for the temporal step excitation as shown in Appendix A. An efficient convolution algorithm has been previously developed in [1]–[3] for utilizing the TD-UTD time impulse response to arrive at a TD-UTD for a general pulsed excitation in which the convolution is done essentially in closed form, thereby leading to a relatively efficient TD-UTD for arbitrary pulsed excitation. The above efficient convolution algorithm of [1]–[3] will be utilized in this work as well. The development of a TD-UTD solution for the surface fields induced by the pulsed source in the same surface follows similar to the radiation case. The TD-UTD analysis of more realistic conformal antennas on a convex body can thus be obtained via an appropriate superposition of the response to the elemental (infinitesimal impressed current) antennas which now make up the equivalent quantized sources of these realistic conformal antennas and radiation apertures of finite extent. This paper has the following format. Sections II and III summarize the TD-UTD formulations for the radiation and surface fields problems of interest in this paper, where the field points are off and on the convex surface, respectively. The algorithms for efficiently computing the TD-UTD radiation and surface fields or mutual coupling type Fock integrals required in these solutions are developed in Appendices B–D. Numerical examples illustrating the behavior of the TD-UTD Fock functions as well as the accuracy of the TD-UTD solutions are presented in Section IV by considering a pulsed magnetic current element located tangentially on the surface of a two-dimensional (2-D) circular cylinder. Finally a short discussion and conclusion is presented in Section V. Due to limited space, the notations of FD-UTD expressions used in this paper are made consistent with those in [4] unless otherwise specified for simplification. II. TD-UTD RADIATION FIELD SOLUTIONS FOR OBSERVATION POINTS OFF THE SURFACE A. TD-UTD Solution for Field Points in the Shadow Region of a Current Moment The TD-UTD electric field radiated at located in the shadow region of an infinitesimal magnetic (m) or electric (e) current element, , placed at as in Fig. 1(a) can be expressed by

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Fig. 1. Ray paths of the radiated fields as in (a) and (b). Ray path for the surface field shown in (c). The unit vectors fixed in the ray coordinates are also illustrated. (a) Field point in the shadow region, (b) Field point in the lit region, (c) Field point on the surface.

associated surface- diffracted ray path is shown in Fig. 1(a), and is the spreading factor for the surface diffracted ray propagating into free space given by (2) where is the caustic distance of the diffracted ray as shown in Fig. 2. Also in (1), and

is given by

(1) with and being the geodesic arc where length of the diffracted ray from to a diffraction point, , and the free space ray distance between to , respectively as shown in Fig. 2, with being the speed of light. In (1), the

(3)

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TABLE I TERMS OF (9) AND (10)

Fig. 2. Surface-diffracted ray tube associated with the surface-diffracted ray from to .

Q

P

and

B. TD-UTD for Field Points in the Lit Region of a Current Moment (4)

located in the lit region of The TD-UTD radiation at as in Fig. 1(b) can be expressed as

where the unit vectors and are defined at and , respectively, and shown in Fig. 1(a). In (3) and (4), – are defined in the corresponding FD-UTD solutions [4], [11]. Basically is the ATT of the FD-UTD of [4], [11]. They are shown in Table II in [4], and will not be repeated in this paper for brevity. Also the and are the surface radii of curvature in the direction at and along at , respectively, and the quantity as well as the angles at and are defined in Fig. 2. The functions and are the ATT of the corresponding FD-UTD radiation type Fock functions and in [4], [11], i.e.,

(8) where the associated ray path is also illustrated in Fig. 1(b). The spreading factor, , with being the distance between

and

. Thus

behaves like a

spherical wave in the near and far zone. In (8)

are the

ATT of which are previously defined in the corresponding FD-UTD solution of [4], [11], where

(5a) (9) (5b)

and (10)

where denotes the ATT operation, with and being the acoustic hard and soft Fock functions [4], [11]. The computations of (5a) are described in Appendix B. The parameters in (5) are defined by

and are also defined In which the unit vectors in Fig. 1(b). The analytic TD functions in (9) and (10) are given in Table I. These above are the ATT of the which are defined previously in the corresponding FD-UTD radiation solutions in [4], [11]; they contain radiation Fock functions for the lit region given by

(6a);(6b) (11a) with the FD parameters

and

in (5) defined by (11b) (7a);(7b)

CHOU et al.: TD-UTD SOLUTIONS FOR THE TRANSIENT RADIATION AND SURFACE FIELDS OF PULSED ANTENNAS

with the parameters defined by

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(15) (12a)

and

(12b) is defined in (6a), is the angle of the direct where ray path measured from the surface normal direction at , and with and as previously defined in Section II.A and Table II in [4]. The computation of these TD radiation type Fock functions are shown in Appendix B. However, at the shadow boundary where in (6b) and in (12b), (11) will become identical to (5) thus making the field variation continuous across the shadow boundary. Also in the deep lit region it can be shown from (A.10) and (A.11) in conjunction with Table I that and

, so that (9)

and (10) will reduce to (13) and (14)

In this case, when , the result in (8) with as in (13) and (14) will be equivalent the radiation in free space with a factor of 2 to automatically account for the effects of sources located on the surface of an electrically large perfect electrically conducting body. It is noted that surface diffracted rays may also exist in the lit region for a closed surface due to the multiple encirclements of surface rays around the closed body.

(16)

and fixed in the surface where the unit vectors is the admittance of the ray are shown in Fig. 1(c), free space, is the arc length of the propagation ray path from to on the convex surface as in Fig. 1(c). In (15) and (16), , and at and at between a pair of adjacent rays are as defined in FD-UTD solution of [4], [12] and are shown here in Fig. 2. The general torsion term where with and defined on Table II in [4]. The “-” sign for is chosen if or . Let and be the principal surface radii of curvatures at , in (15) and (16) and as defined in FD-UTD [4]. The TD surface field (or mutual coupling) type Fock functions and in (15) and (16) are obtained directly by applying the ATT operator onto their corresponding FD-UTD Fock functions, namely:

III. TD-UTD SOLUTIONS FOR FIELD POINTS ON THE SURFACE A. TD-UTD Surface Field Solution for Elemental Magnetic Current Sources The TD-UTD solution for EM surface fields at Q produced by at Q can be expressed as

(17)

and are the generalized where is an integer, and Fock integrals for the surface fields on an arbitrary convex surface given by [4], [12]

(18)

with and being the mutual coupling type Fock functions [4], [12]. In (18), and are defined in (6). In (17) the index, , indicates the higher order terms of the asymptotic series in FD and is not to be confused with above. Substituting (18) into (17) gives

(19)

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with

(20) The computation of (20) is described in Appendix D. B. TD-UTD Surface Field Solution for Elemental Electric Current Sources The TD-UTD solution for EM surface fields at Q produced at Q can be expressed as by

H ;t

Fig. 3. Variation of (4 ) with 4 = 1. Comparison between (a) “early time” creeping wave mode series in (A.4), and (b) the “late time” representation in (A.7).

(21) and

(22)

S ;t; M

M

Fig. 4. (4 ) with 4 = 1 and = 1. Comparison between (a) “early time” creeping wave mode series in (A.5), and (b) the “late time” representation in (A.8).

IV. NUMERICAL EXAMPLES Numerical validations on the TD-UTD formulations are examined in this section. In part A, one first examines the behaviors of the TD Fock functions and . In part B, one demonstrates the utilization of these functions for calculating the TD-UTD solutions given in the present work. A. Behavior of the Fock Functions

and

One first considers the characteristics of and with results shown in Figs. 3–6 with and assumed because the solutions are primarily functions of and as discussed in Appendix B. Different values of and will simply cause a change in scale. Also in the , the and are arbitrarily set to unity here for convenience because they only represent an amplitude scalar constant. Figs. 3 and 4 show a comparison between the late time representation of (A.7) and (A.8) versus the early time creeping

wave (residue series) mode representation of (A.4) and (A.5) for the when the observer is in the shadow region scalar (acoustic) hard and soft boundary cases, respectively. The two representations overlap very closely, except when is very small such as when . Thus the formulations can be switched at . On the other hand, Figs. 5 and 6 show a comparison between the late time representation of (A.9) versus the approximate early time power series representation of (A.10) and (A.11) for the hard and when the observer is in the lit region soft cases, respectively. Again the two representations overlap very closely when and more generally they overlap very closely whenever . Also the early time representation breaks down violently as and the approximate early time representation breaks down as . Thus the criterion proposes to switch the formulation at . One next considers the behavior of with results shown in Figs. 7 and 8 with also assumed. In particular,

CHOU et al.: TD-UTD SOLUTIONS FOR THE TRANSIENT RADIATION AND SURFACE FIELDS OF PULSED ANTENNAS

H l ;t

0

Fig. 5. (4 ) with 4 = 1. Comparison between (a) “early time” representation in (A.10), and (b) the “late time” representation in (A.9). (unit: ns).

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Fig. 8. The characteristics of TD mutual coupling type Fock integral

V

(4

;t) at 4 = 1.

tends to diverge when It can be observed from Fig. 7 that becomes large as increases. Only the case of exhibits a finite energy distribution, which reaches the maximum in the region of ns. In the case of , only the imaginary part tends to be finitely distributed and has a maximum around ns. On the other hand, in the case of both imaginary and real parts tend to diverge as increases. However, this will not cause difficulties in practical applications because represents the first order contribution and it tends to and represent higher order dominate, while corrections, unless the point of observation is extremely close to the source element. It is also observed from Fig. 8 that for . Behavior similar to that for

S l ; t; M

0

M

Fig. 6. (4 ) with 4 = 1 and = 1. Comparison between (a) “early time” representation in (A.11), and (b) the “late time” representation in (A.9). (unit: ns).

has been observed.

B. Numerical Examples Based on the TD-UTD Solutions Numerical examples of the TD-UTD solutions are presented , of in this section. It is assumed that the time derivative, a general pulsed TD waveform chosen here for a magnetic current element excitation is shown in Fig. 9 where its corresponding FD waveform is also shown. Note that where is as defined in (A.2) and (A.3). The TD pulsed waveform of the current element, if desired, can be obtained by performing an integration over its time derivative function . In particular, as in [1], [3] the FD signal waveform in Fig. 9 is assumed to be expressed as (23) where is the center frequency. The corresponding TD wave form is

Fig. 7. The characteristics of TD mutual coupling type Fock integral

U

(4

;t) at 4 = 1.

the cases of are examined because the Fock functions for these cases are primarily employed in the TD-UTD solution for surface field prediction presented in this paper.

(24) in Fig. 9 where is the analytic delta function, and the TD is obtained by taking the real part of (24). Thus the total field in terms of TD-UTD solutions can be described as in (A.3). The real response is the real part of (A.3) with .

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Fig. 9. The derivative of the current element’s excitation function h(t) and its distribution in frequency domain. Note that w (t) = (dh(t))=(dt).

The demonstration examples first consider the far zone radiation from a magnetic line current element when it is placed and on a two dimensional (2-D) perfectly at conducting circular cylinder of radius a. In this case only the first two terms in (3) and (9) exist, which allows one to examine the characteristics of the hard and soft radiation Fock functions, respectively. The reference eigenfunction solutions based on an inverse fast Fourier Transform (IFFT) to obtain the corresponding TD solutions are employed for comparison of accuracy. The time reference is selected at the center of the subtracted where is the distance becylinder with tween the cylinder center and observation point. The radius of meter which makes the radius one wavethe cylinder is length at the peak frequency of the window function in Fig. 9. Note that since the circular cylinder has a closed surface, three dominant creeping waves will be included in the computation even if the observation point may be in the lit region of the source. Fig. 10(a)–(c) show the comparisons between TD-UTD solutions and the reference solutions for polarized magnetic point current element to examine the behavior of the hard Fock function. The source and field points are in the plane. On the other hand, Fig. 11(a)–(c) show that for a directed magnetic point current element at to examine the soft Fock function. The values of the fields in all figures are scaled with the same factor to show a large scale comparison. Again, the source and field points are both in the plane. In those figures, both the magnitudes of the analytic time functions and the actual time signals (i.e., the real part) are shown. In particular, Figs. 10(a) and 11(a) show the cases when the observation point is in the deep lit region at . In this case the radiation behavior tends to be dominated by (13), where, as shown in Figs. 10(a) and 11(a), the TD waveform is similar to Fig. 9 except for an amplitude difference. It is noted that Fig. 11(a) has a small amplitude due to the factor of in (13). Figs. 10(b) and 11(b) show the cases when the observation point is at the shadow boundary at . The behavior of the Fock functions are dominated by the late time power series representations in (A.7)–(A.9). The hard case has an impulsive radiation in (A.7). Thus the TD waveform is similar to Fig. 9 with an amplitude factor. Figs. 10(c) and 11(c) show the cases where

Fig. 10. The far field radiation from a 2-D circular cylinder for dP = z^dP . (a)  = 35 in a deep lit region, (b)  = 90 at shadow boundary, (c)  = 135 in a deep shadow region.

the observation point is in the deep shadow region of the source element. In this case the creeping wave modes (and their surface diffracted fields) will exist for the hard and soft Fock functions, and cause significant power attenuation in the shadow due to the surface diffraction phenomena. One next considers the normal electric surface field produced from a directed elemental magnetic current moment when it is placed on an infinitely long perfectly conducting circular cylinder. In this case only the first term in (16) exists and results in a -polarized electrical field. Fig. 12 shows the case when on the cylinder’s surface the field point is located at plane. In this where this source and observer are in the case, the delay between the first two pulses is simply 0.58 ns (0.175 meter in distance). Thus the superposition of these two pulses will result in Fig. 12. Again Fig. 12 shows good agreement between the TD-UTD and the reference TD eigenfunction solution.

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transient radiation and surface field of pulsed antennas placed directly on smooth convex surfaces. Even though the basic TD-UTD solution developed here is a response to the elemental antenna excitation with a step function in time, the TD-UTD for the radiation by a general finite energy pulsed excitation of the current element on the convex body can be obtained from the step response via convolution. Since an efficient (essentially closed form) convolution procedure has already been developed in [1]–[3], it is utilized in the present calculations and hence the efficiency and advantages of the TD-UTD solutions of [1]–[3] are retained here as well. Also efficient algorithms are developed to calculate the radiation and surface type TD Fock functions that are an essential part of the TD-UTD solutions given here. APPENDIX A. TD-UTD Response for a Pulse Excited Impressed Current Moment The TD-UTD solutions are developed in this paper by assuming the electric (e) or magnetic (m) current moment has the form (A.1) where and are the Dirac delta function and the Heav(or iside unit step function, respectively and if is a surface current moment) is even that is taken to be independent related to a source strength with a temporal step function excitation genof time. This

Fig. 11. The far field radiation from a 2-D circular cylinder for dP = ^ dP . (a)  = 35 in a deep lit region, (b)  = 90 at shadow boundary, (c)  = 135 in a deep shadow region.

. Thus the TD-UTD field, , erates an electric field , which exhibits a general produced by a current moment (with finite energy), (arbitrary) transient pulsed excitation where (A.2) is given via convolution as

(A.3) where is the analytic time function corresponding to . The details of this convolution will not be given because of space limitations. However, one may find the convolution procedure in [1]–[3] which does it essentially in closed form within the ATT framework. Fig. 12. The surface field due to a magnetic current element dP = z^dP placed on the surface of 2-D circular cylinder for a field point also located on the surface at  = 175 .

V. CONCLUSION This paper extends the previous works in [1]–[3], which presented the TD-UTD solutions for predicting the scattering from PEC curved wedges and smooth convex surfaces, respectively, to now describe the TD-UTD solution for predicting the

B. Computation of Functions

and

The computation of the TD radiation type Fock functions and in (3) and (11) are summarized. : 1) Deep Shadow Region or Early Time Response In this case, in (5a) is large and the residue theorem [13] can be applied to give (A.4)

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COEFFICIENTS OF B

TABLE II

and (A.5) where roots of

is the Airy function with

and

USED IN

(A.10) AND (A.11)

the time center of the asymptotic series as shown in [1], [3] to except that only the leading term remains centered at . Thus

being the th

and , respectively. is a general creeping wave function [1], [3] given by (A.6)

for is real and . In (A.4) and (A.5), and are in general sufficient to obtain accurate results. 2) In the Shadow Boundary Transition Region: For an observer located on the shadow side and near the shadow boundary, where is sufficiently small, the power series exand in (5b) can be employed to give pansions of

(A.7) and (A.8) and are for and , rewhere the coefficients spectively, and can be found in Logan’s report [13]. In general is sufficient to achieve accurate results. For an observer located on the lit side and near the shadow , comparing (5b) and (11b) gives boundary where (A.9) It is noted that (A.9) shows the continuity at the shadow boundary where and . Also the functions are . Apparently (A.8) and (A.9) can be also centered at employed for calculating the late time response where or are sufficiently small. : In this 3) Deep Lit Region or Early Time case, one can use an early time asymptotic series with shifting

(A.10) and

(A.11) where the coefficients can be found by matching with the late time inverse time power series in (A.9) using a least square error method or extended Prony’s method [14], [15]. One may . Table II shows the switch the solutions at . It is noted that as the observer moves further coefficients into the deep lit region, the leading term or the geometrical optics (GO) component, which is exhibited as (A.11), will dominate.

in (A.10) and

C. Computation of Efficient algorithms are developed to compute the generalized creeping wave function, , defined in (A.6). . Note that (A.6) reduces to (A.1) in [1], [3] when An efficient algorithm can be developed by following the same procedure as developed in [1], [3]. Thus, without going through the details, the formulations are summarized in the following for different parameter ranges: 1) An early time power series representation is described by

(A.12)

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TABLE III THE EXPONENTIAL FACTORS AND COEFFICIENTS IN (A.14)

which will converge and gives . Typically when three terms of (A.12) are sufficient.

for , first

2) A late time (or large scribed by

) series representation is de-

(A.13) which, in a practical stand point, has good convergence . When , the first three terms are for sufficient. if only three terms in (A.12) 3) and (A.13) are employed. Following the procedure in the appendix of [1], [3], for the (A.4) can be approximated by intermediate values of

Fig. A.1.

F

; t; ) with  = 1.

(

and continuous. In fact, the real part of is infinsince all of its time derivatives go to zero itely smooth at as shown in (A.12). D. Computation of and 1) Late Time Representation : In the case of late time ( is large), or for an observer within the paraxial region of an elongated convex body ( is small),

and

can be expressed as a sum of inverse time power (A.14) where

series:

with

and as defined in [1], [3]. Notice for . The first 10 terms of and that are given in Table III, which are sufficient in this computation. The behaviors of with are shown in and in the vicinity of . Figure A.1 for These two cases are needed to find (A.4) and (A.5). The algorithm developed here was employed in the computation. However, the agreement with the results obtained by a direct numerical integration has been achieved via our numerical experimentation, which has also shown that when they are plotted on a large time scale, they appear to be very sharp and almost sinas previously exhibited in [1], [3]. gular in nature near tends to be more obvious as The singular behavior near increases from to . The “zoom in” plot of is shown in the figures; also, the function is seen to be smooth

(A.15)

where the coefficients and can be found in Logan’s report [13] and are shown in Table IV for reference.

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TABLE IV COEFFICIENTS OF THE POWER SERIES EXPANSION OF U ( ) AND V ( ) IN (A.15) [13]

TABLE V USED IN (A.19) COEFFICIENTS OF B

TABLE VI COEFFICIENTS OF C USED IN (A.19)

2) Early Time Representation case: Shadow Region: (a)

or in a Deep

(A.16)

case, Note that (A.16) and (A.17) are valid only for the because the summations in (A.16) and (A.17) tend to diverge or , which can be observed from an early time when response (i.e., when shadow region via

) in

within the

and

(A.17)

(A.18)

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when and will increase as increases; this occurs in (A.16) and (A.17) for large , rendering them divergent. Thus the cases of and will be treated separately below. In practical applications, and are, in general, sufficient to obtain accurate results with (A.16) and (A.17). cases: (b) In this case, an alternative representation, similar to that in [3], [4] consisting of a time power series is employed, thus

[11] P. H. Pathak, N. Wang, W. D. Burnside, and R. Kouyoumjian, “A uniform GTD solution for the radiation from sources on a convex surface,” IEEE Trans. Antennas Propag., vol. 29, no. 4, pp. 609–622, Jul. 1981. [12] P. H. Pathak and N. Wang, “Ray analysis of mutual coupling between antennas on a convex surface,” IEEE Trans. Antennas Propag., vol. 29, 6, pp. 911–922, Nov. 1981. [13] A. Logan, Lockheed Missiles and Space Division, General Research in Diffraction Theory Tech. Rep. LMSD-288087, 1959. [14] R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd ed. Kluwer: Dover, 1973. [15] M. Kay, Modern Spectral Estimation. Englewood Cliffs, NJ: Prentice-Hall, 1987. [16] J. Struik, Differential Geometry, 2nd ed. Reading, MA: Addison-Wesley Publishing, 1961.

(A.19)

Dr. Hsi-Tseng Chou (SM’95) was born in Taiwan, in 1966. He received the B.S. degree in electrical engineering from National Taiwan University, in 1988, and the M.S. and Ph.D. degrees in also electrical engineering from Ohio State University, in 1993 and 1996, respectively. He is currently a Professor in the Department of Communications Engineering, Yuan Ze University. His research interests include antenna design, antenna measurement, electromagnetic scattering, asymptotic high frequency techniques such as uniform geometrical theory of Diffraction (UTD), novel Gaussian Beam techniques, and UTD type solution for periodic structures. Dr. Chou is an elected member of URSI International Radio Science US Commission B.

which is conjectured from (A.15) with its late time replaced by its inverse. The coefficients and are found by matching with late time inverse time power series in (A.15) in a least square error sense in the region of for all , and are shown in Tables V and VI for reference. The point for switching the computation to the late time inverse time power series in (A.15) is at . REFERENCES [1] P. R. Rousseau, “Time Domain Version of the Uniform Geometrical Theory of Diffraction,” Ph.D. dissertation, ElectroScience Laboratory, The Ohio State University, Columbus, 1995. [2] P. Rousseau and P. H. Pathak, “Time-domain uniform geometrical theory of diffraction for a curved wedge,” IEEE Trans Antennas Propag., vol. 43, no. 12, pp. 1375–1382, Dec. 1995. [3] P. R. Rousseau, H.-T. Chou, and P. H. Pathak, “A time domain formulation of the uniform geometrical theory of diffraction for scattering from a smooth convex surface,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1522–1534, June 2007. [4] P. H. Pathak, , Y. T. Lo and S. W. Lee, Eds., “Techniques for high frequency problems,” in Antenna Hand- Book: Theory, Application and Design. New York: Van Nostrand Reinhold, 1988, ch. 4. [5] P. H. Pathak, W. D. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag., vol. AP-28, pp. 631–642, Sep. 1980. [6] E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT). Part I: Formulation and interpretation,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 80–86, Jan. 1987. [7] E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), Part II: Evaluation of the spectral Integral,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 574–580, May 1987. [8] R. H. Schafer, The Ohio State Univ. ElectroSci. Lab., “Transient currents on a perfectly conducting cylinder illuminated by unit-step and impulsive plane waves,” Tech. Rep. 2415-2, 1968. [9] R. H. Schafer and R. G. Kouyoumhian, “Transient currents on cylinder illuminated by an impulsive plane wave,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 627–638, Sept. 1975. [10] J. Ma and I. R. Ciric, “Early-time currents induced on a cylinder by a cylindrical electromagnetic wave,” IEEE Trans. Antennas Propag., vol. 39, pp. 455–463, Apr. 1991.

Prahhakar H. Pathak (M’76–SM’81–F’86) received the B.Sc. degree in physics from the University of Bombay, India, in 1962, the B.S. degree in electrical engineering from the Louisiana State University, Baton Rouge, in 1965, and the M.Sc. and Ph.D. degrees in electrical engineering from Ohio State University, Columbus, in 1970 and 1973, respectively. Since 1973, he has been working at Ohio State University, Columbus, where he is currently a Professor. Dr. Pathak is a member of Sigma Xi and a member of the U.S. Commission B of URSI. He was named an EEE AP-S distinguished lecturer for a three-year term beginning in 1991. He is a former Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Paul R. Rousseau (S’88–M’95) received the B.S. degree (summa cum laude) in engineering from California State University, Northridge, in 1989, and the M.S. and Ph.D. degrees in electrical engineering from Ohio State University, Columbus, in 1992 and 1995, respectively. His research interest is the application of analytical and numerical techniques for solving electromagnetic scattering and radiation problems in the time domain.

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Non-Conformal Domain Decomposition Method With Mixed True Second Order Transmission Condition for Solving Large Finite Antenna Arrays Zhen Peng, Member, IEEE, and Jin-Fa Lee, Fellow, IEEE

Abstract—A non-overlapping and non-conformal domain decomposition method (DDM) is presented for modeling large finite antenna arrays. Two major benefits of the proposed DDM: (a) A mixed true second order transmission condition (SOTC) with corner edge penalty terms is developed to facilitate fast convergence in the DDM iterations. Numerical experiments indicate that the convergence of DDM with the aforementioned SOTC is insensitive to the sizes of arrays; and, (b) The non-conformal property of the proposed DDM permits the use of completely independent discretization for each of the sub-domains. We demonstrate the performance of the proposed approach through several large-scale problems. Index Terms—Domain decomposition, large antenna array, Maxwell’s equation, second order transmission condition (SOTC).

I. INTRODUCTION

N

OWADAYS, large finite antenna arrays are commonly used in wireless communication systems and radars to transmit and receive signals through spaces. As a result, an accurate and efficient numerical modeling of such arrays is required for antenna designs and electromagnetic compatibility (EMC) analyses. However, the full wave electromagnetic (EM) analysis of large finite antenna array is still considered to be a grand challenge. The associated technical difficulties include large problem sizes because of the large number of antenna elements, multi-scale geometrical features and complex material properties. Moreover, many of these real-life antenna arrays require the use of exotic meta-materials, covered with frequency-selected-surfaces (FSS) to further enhance the performance. All these complications add significant computational challenges to numerical simulations. Domain decomposition methods (DDMs) [3]–[7] have been employed as effective preconditioners to improve upon the vector finite element methods [2] for much larger electrical-size EM problems. These approaches have enjoyed considerable success in many real-life EM applications. However, most of the DDMs [8]–[12] incorporated 1st order complex Robin transmission condition (FOTC), which unfortunately fails to

Manuscript received May 24, 2010; revised August 31, 2010; accepted November 01, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported in part by Northrop Grumman Corporation. The authors are with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]; lee.1863@osu. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123067

provide converging mechanism for evanescent modes across domain interfaces. Ergo, the performance of the DDMs deteriorates significantly once the mesh sizes on the interfaces are electrically small. This could potentially be a major handicap in modeling large finite antenna arrays since the multi-scale nature of the antenna arrays makes small mesh sizes often unavoidable. Recently, higher order transmission conditions (TCs) [13]–[19] have been proposed to further improve the operations of the DDMs. One such TC, which we shall refer to as the true second order TC (SOTC), provides convergence for both propagating and evanescent modes. The effectiveness and efficiency of the SOTC was first demonstrated and witnessed for conformal DDMs in [18]. Moreover, it is found in [18], the performance of the DDM depends significantly on the function space where the cement variables reside. Through many electrically large and multi-scale EM examples, we have concluded in [18], the proper function space for the cement variables is the so-called “discontinuous surface curl-conforming space”. Simply put it, the discontinuous surface curl-conforming space corresponds to surface vector functions that are curl-conforming (tangentially continuous) on each of the domain interfaces, however, not tangentially continuous across the corner edges. In summary for conformal DDMs, we have found in [18], the SOTC together with the discontinuous curl-conforming cement variables result in superb convergence compared to other alternatives. Nevertheless, unlike the conformal DDMs, by extending the SOTC to non-conformal DDMs, we have found only marginal improvements over the use of FOTC. The root cause of this defect is due to the enlargement of the function space for the auxiliary cement variables on the domain interfaces. Specifically, it is due to the use of the discontinuous curl-conforming basis functions [18], [19] for the cement variables. Consequently, the non-conformal DDMs allow for eigen-modes, whose magnetic fluxes do not satisfy the needed divergence-free condition on the corner edges. To mitigate such a malady, [19] adopts the interior penalty (IP) formulation [20], and subsequently, introduces additional corner penalty terms relating to the divergence free constraint for the cement variables. The introduction of the corner edge penalty terms in the IP formulation restores the full benefits of the SOTC in the non-conformal DDMs. Regardless of these achievements, we have found in many real-life EM applications, where the convergence of the DDMs still has rooms for improvements. In this paper, we shall present a general mixed true second order transmission condition and discuss the effects of

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PENG AND LEE: NON-CONFORMAL DDM WITH MIXED TRUE SOTC FOR SOLVING LARGE FINITE ANTENNA ARRAYS

employing different parameters in practical applications, such as electrically large finite antenna arrays. Furthermore, the proposed non-conformal DDM herein differs from previous non-overlapping and non-conformal DDMs in yet another noticeable aspect. That is: it does not require two adjacent sub-domains to be shape-conforming. There are certain advantages in doing so. Particularly, in many applications, there might exist different types of repetitions and periodicities within a single EM problem. For example, large finite antenna arrays covered with meta-materials and/or FSS. The shape non-conformal nature of the proposed DDM provides unprecedented flexibility and convenience for partitioning the original global domain into sub-domains, where the partitions can be wrought to exploit local repetitions and also local rotational and translational symmetries. The rest of the paper is organized as follows: Section II introduces the proposed non-conformal domain decomposition scheme and boundary value problems to be solved. We then elucidate a general mixed TC, which has two second-order tangential derivatives. The convergence analysis and the eigenspectrum of the DD matrix using the new mixed TC are presented next. The discussions are followed by the weak and discrete formulations of a finite element implementation. In Section III, we illustrate the performance of the proposed method on several large-scale antenna radiation problems. A summary and conclusion are included in Section IV. II. FORMULATION A. Domain Decomposition With Second Order Transmission Condition Non-overlapping DDM has been shown effective in solving time-harmonic electromagnetic wave problems [4], [8], [9]. The procedure starts by dividing, or partitioning, the original problem domain into many smaller non-overlapping sub-domains. Since it only requires factorizations of sub-domain matrices, the computational resources needed in general are modest. Moreover, during the partition, we may further exploit the local repetitions of sub-structures, which are quite common in many EM applications. For simplicity, and without loss of generality, we consider a and its decomposition smooth domain (see Fig. 1). Furthermore, we define the exterior boundary of as , and the interior side of the interface as seen from . In what follows, we somewhat , abuse the notations and split into is the number of the neighboring sub-domains of . where For instance, the interior interface of in Fig. 1 is split into . We do so to emphasize the fact that not only the inand are allowed to be non-matching, but terface meshes on also the adjacent sub-domains can be shape non-conformal (geometrically non-conformal). represents the outward-directed . We shall also use the surface tangential trace unit normal to , and the twisted tangential trace opoperator, , and the function spaces erator,

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Fig. 1. Notations for decomposition of the domain.

where for is the usual Sobolev space [1]. Note is the space of curl-conforming functions in that that satisfy essential boundary conditions on . is the collection of PEC surfaces on which we enforce the tangential components of the electric field to be zero. Subsequently, we write the time harmonic Maxwell equations for the decomposed domain of Fig. 1 as (1) (2) (3) where represent the electric field in a sub-domain , is the wave gives the radial frequency number in free space, and and are the permittivity of operation with frequency . and permeability of the free space, respectively. We also define and as the relative permittivity and permeability of the material. Equations (2) and (3) enforce the necessary continuities of the tangential electric and magnetic fields on the interfaces between sub-domains. We omit the discussions on the exterior boundary and other boundary conditions simply because they can be accounted for as in the conventional finite element formulations. It is well known that the transmission conditions, which impose the needed field continuities across domain interfaces, greatly affect the convergence of non-overlapping DDM algorithms. In the literature, the 1st order complex Robin transmission condition is commonly employed. Nonetheless, a number of authors have shown that by including higher

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order tangential derivatives in the transmission conditions, the convergence of the DDMs can be drastically improved [13], [17]–[19]. In particular, [18], [19] introduced a true second order transmission condition (SOTC), which is of the form

, where and , with on complex values , , , , and to be determined. By a judicious choice of these parameters, we can further accelerate the convergence of the propagating modes and provide convergence mechanism for both TE and TM evanescent modes. B. Convergence Analysis

on interface between two sub-domains. In contrast to the widely used complex FOTC, this SOTC uses two second-order transverse derivatives, which are essential in addressing the convergence of both TE and TM evanescent modes. The enhancements in performance provided by this SOTC are documented in [18] for conformal DDMs and [19] for non-conformal DDMs. However, the incorporation of SOTC in non-conformal DDMs is not yet fully examined. One very important feature of the SOTC is that the parameters, and , can be chosen to further accelerate the convergence of propagating modes. For many real-life electromagnetic radiation and scattering problems, where propagating modes are predominant, this particular choice results in much better performance than other alternatives, as indicated in [17]. In the following, we propose a mixed true second order transmission condition (mixed SOTC), which strikes a balance between propagating and evanescent modes. Through many numerical examples, we have found that the proposed mixed SOTC in general outperforms the SOTC described in [18], [19]. Let us introduce three surface variables as follows

Assuming an idealized decomposition of with defined , and . to be the plane To further simplify the analysis, we consider plane waves in free space traveling in the -plane with dispersion relation . Here and denote the -directed and the -directed wave numbers, respectively. A real, positive corresponds to a propagating mode, whereas an imaginary refers to an evanescent mode. The mixed SOTC is imposed on and an iterative solution process is considered. While the analysis is performed using a simplified problem, the results obtained herein correspond very well to the performance seen when applied to finite problems, even with general non-conformal domain partitions. We shall express the convergence factors of the iterative algorithm in terms of by decomposing the solution into TE and TM waves. To begin with, we have on :

(6)

(7) direction and the Consider first a TE mode traveling in the following solutions in each of the sub-domains:

We wish to find the rate at which decays to zero as a function of . To determine the convergence factor for TE modes, , we note that where is the tangential electric fields represents the electric currents on the interand is introface. Additionally, a scalar variable duced. These auxiliary variables are defined on interior interface of sub-domain . We do so to emphasize the fact that the interfaces of adjacent sub-domains are allowed to be non-conformal. The mixed SOTC can then be written as and (4) on

and

(5)

PENG AND LEE: NON-CONFORMAL DDM WITH MIXED TRUE SOTC FOR SOLVING LARGE FINITE ANTENNA ARRAYS

As a consequence, we obtain the convergence factor it is written as

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, and

(8) In what follows, we will show that the proper choices of , and accelerate the convergence of TE propagating modes (as compared to FOTC) and also provide convergence for TE , evanescent modes. By simply setting , and , the convergence factor, , can be written as

(9) , and convergence is obtained Note that for in two iterations. Furthermore, we are now free to choose two and . We choose an additional zeros at to provide convergence for the TE evanescent imaginary to be a real number modes. Moreover, by selecting , the convergence of the TE propagating modes can be further accelerated. Thereby, by using this mixed SOTC, all TE propagating and evanescent modes can be made convergent, with the propagating modes converging much faster than those of the FOTC. As for the TM modes, we define the magnetic and electric fields as

and

Fig. 2. A WR-90 waveguide (a) cross section; (b) one way decomposition.

Once again, there are two additional zeroes of the convergence factor in (11) can be chosen for TM propagating and evanescent modes. to account Here, we choose for the TE and TM propagating modes, and and to address the TE and TM evanescent modes. Moreover, and denote the chosen maximum transverse wave number supported on the interfaces for TE and TM modes, and depend respectively. The values of upon both the mesh size and the polynomial order of the field expansion. As the mesh size, , decreases the maximum supported wave number increases at a rate proportional to . Additionally, we assume a linear dependence on the order of the polynomial representation. Thus both and are in proportion to . C. Eigenspectrum

Similar to the TE case, the convergence factor of the TM waves is

(10) and Analogously to the TE case, we may now choose to improve the convergence of TM modes. By selecting and , we may now write

(11)

The convergence factors derived here also predict the eigenvalue distribution of the DDM algorithm. In what follows, we examine the eigenvalue distributions of the mixed SOTC with both the one-way domain decomposition and the non-conformal domain decomposition configurations. 1) One-Way Decomposition: We first examine the eigenvalue distributions that result from the mixed SOTC with the segone-way domain decomposition. To do so, we use a ment of an X-band (WR-90) rectangular waveguide operated mode as the propagating mode, above cutoff, with only the is the wavelength in free space. Both ends at 10 GHz. Here of the waveguide are terminated with a first-order absorbing boundary condition (ABC). We partitioned the waveguide, by a transverse plane, into two equally sized sub-domains. The two sub-domains are meshed independently, and hence the interface triangulations do not match across domain boundary. The geometry and partition of the waveguide are shown in Fig. 2. To gain further insights, we applied four TCs to model the

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TABLE I THE CHOICES OF PARAMETERS IN VARIOUS TCS

Fig. 3. (a) Convergence factor of FOTC; (b) Eigenspectra of FOTC for a WR-90 waveguide.

Fig. 4. (a) Convergence factor of mixed SOTC-TE; (b) Eigenspectra of mixed SOTC-TE for a WR-90 waveguide.

mode propagating along this X-band waveguide. The first one is the widely used first order Robin TC (FOTC), which is oband . The tained by setting second one, denoted as the mixed SOTC-TE, which corresponds (omitting the term) and to therefore allows convergence of TE evanescent modes but not TM ones. Similarly, for the mixed SOTC-TM, we set (omitting the term). Consequently, mixed SOTC-TM allows convergence of TM evanescent modes but not TE ones. Table I includes the values of parameters employed and for these four TCs, where we have set . In Figs. 3–6, we plot both the convergence factors and the discrete eigenspectra of these four TCs employed by the one-way decomposition (Fig. 2(b)). By comparing FOTC of Fig. 3(b) to the mixed SOTC-TE of Fig. 4(b), we see that TE evanescent modes are brought within the shifted unit circle. Also, comparing Fig. 3(b) to the mixed SOTC-TM of Fig. 5(b), a different set of eigenvalues, the TM evanescent modes, are now convergent. Finally, Fig. 6(b) clearly demonstrates that the mixed SOTC combines the desirable properties of both the mixed SOTC-TE and SOTC-TM and wrought all evanescent modes convergent.

Fig. 5. (a) Convergence factor of mixed SOTC-TM; (b) Eigenspectra of mixed SOTC-TM for a WR-90 waveguide.

Fig. 6. (a) Convergence factor of mixed SOTC; (b) Eigenspectra of mixed SOTC for a WR-90 waveguide.

Fig. 7. (a) Convergence factor of SOTC; (b) Eigenspectra of SOTC for a WR-90 waveguide.

Furthermore, we also list the eigenspectra of SOTC that was used in [18], [19] by setting and . Table I gives the values of parameters and , and again we . The convergence have selected factor and eigenspectra of the SOTC are plotted in Fig. 7. By comparisons between Fig. 6 and Fig. 7, we have the following observations: a. both mixed SOTC and SOTC offer converging mechanisms for both propagating and evanescent modes; b. the mixed SOTC provide better performance for propagating modes

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Fig. 8. Non-conformal decomposition for a WR-90 waveguide.

and worse for the evanescent modes than the SOTC. In reallife multi-scale electromagnetic applications, the mixed SOTC should be preferred for problems where EM wave propagation phenomena are predominant. 2) Non-Conformal Decomposition: In this section, we decompose the same waveguide into five sub-domains , as shown in Fig. 8. As can be seen from the figure, the sub-domains are not shape conformal. is now touching all Specifically, we note that sub-domain the other four sub-domains, and results in one interface in sharing with four interfaces from the other sub-domain four neighboring sub-domains. Subsequently, not only the triangulations on the interfaces are non-matching, but also the adjacent sub-domains are non-conformal. The eigenspectra of the DDM matrices using the FOTC and the mixed SOTC are shown in Fig. 9(a) and Fig. 9(b), respectively. By comparing eigenspectra of the FOTC and the mixed SOTC, we observe that most of eigenvalues, in Fig. 9(a), that are on or very close to the shifted-unit-circle have moved toward the center in Fig. 9(b). These are the evanescent modes affected by the mixed SOTC. Nevertheless, we also note another two clusters of eigenvalues: one is at 0 and the other at 2. The cluster of eigenvalues nears zero is of great concern since this indicates the DDM matrix is either singular or nearly singular. This is due to the fact that the auxiliary variables, , which represent the electric currents on the interfaces, were defined discontinuously over the domain interfaces. For example, when four sub-domains meet at a corner edge as shown in Fig. 10, the DOFs of the variable defined redundantly on this particular corner edge. As explained in [19], these redundant basis functions result in a number of eigenmodes, which do not satisfy the condition at the corner edges. In order to mitigate this difficulty, as documented in [19], we propose corner edge penalty terms in the formulation to enforce the divergence free condition on the corner edges. Take the domain partition in Fig. 8 as an example, there are meet at one set of corner four sub-domains edges. The auxiliary variables, , on these corner edges were defined discontinuously within each of the sub-domains. Fig. 10(a) shows the number of variables defined on the corner edges for

Fig. 9. Eigenspectra for a WR-90 waveguide with non-conformal decomposition (a) FOTC; (b) mixed SOTC.

Fig. 10. Corners edges in Fig. 8, (a) numbering of the corner edges (front view); (b) virtual cylinder enclose the set of corner edges (side view).

DDM partition. Note that they are defined discontinuously over the interfaces. Assuming a virtual cylinder, which encloses the set of corner edges shown in Fig. 10(b), the divergence free condition can be written as a surface integral over the cylinder . Taking the limit of goes to zero, we can approximate the divergence free condition by

(12) is the magnetic field on the corner edges, is the where normal of the interface as indicated in Fig. 10(a), is the angle associated with each sub-domain , and is the direction along the corner edge . , is a sign function, whose value is 1

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(16) . Lastly, we choose and to test the definition equations of the scalar auxiliary variables. We have (after again an integration by parts) (17) (18) Fig. 11. Eigenspectra of mixed SOTC for a WR-90 waveguide with non-conformal decomposition (after inclusion of corner-edge penalty).

for odd and 1 for even numbered corner edges. Straightforward manipulation shows that (12) is equivalent to [19] (13) Fig. 11 shows the eigenspectrum of the DDM matrix with the mixed SOTC after inclusion of the corner edge penalty terms of (13) [19]. Clearly we note that the clustering of singular eigenvalues moves away from zero to inside the shifted-unit-circle. D. FEM Implementation

We consider the integration by part operation legitimate since the natural boundary conditions implied are the normal component of to be continued. The final corner edge penalty terms for the corner edges, inspired by the Interior Penalty method [20], are included to penalize the divergence free condition. For each set of corner edges, we test divergence free condition with where denote the sub-domains, which share this set of corner edges (19)

1) Weak Formulation: Here in this section, we formally state the weak formulation of the DDM to solve for (1), (4) and (5). The following notations are employed throughout our discussion on the weak formulation. They are volume, surface and line sesquilinear forms, which are defined as

where are auxiliary variables defined on this set of corner edges, and is the angle with sub-domain which reside in. Note that (19) involves integrations along the corner edge. Take the domain partition and the corner edges in Fig. 10 as an example, the divergence free condition, (13), are tested by , which gives

Equation (1) is tested using curl-conforming functions to give (after integration by parts)

. Moreover, as indicated in Fig. 10, we have that , and the sign functions and . The corner edge penalty terms are then given as

(14) where . and We test TCs (4) and (5) with , and after performing an integration by parts, we have

(15)

. 2) Discrete Formulation: Let denotes the tetrahedral , and and are the surface triangulations mesh of and , respectively. The set of corner edges induced on . On each of the sub-domains, we define is collected as discrete trial and test functions, and , respectively. Here, is taken to be the space spanned by the mixed order curl-conforming . On the basis functions defined in [21], with order interfaces, the discrete trial and test functions are and

PENG AND LEE: NON-CONFORMAL DDM WITH MIXED TRUE SOTC FOR SOLVING LARGE FINITE ANTENNA ARRAYS

, respectively, with . is taken to be the space of interpolatory scalar basis functions, with . Also, the discrete vector trial and test functions order and , with for the domain interfaces are . And, will be taken as the surface curl-conforming on each of the interfaces. However, it is discontinuous across the corner edges. The discrete system can be cast into a matrix equation of the following form

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The vectors of DOFs and represent the electric field basis functions with zero and nonzero tangential component represents the electric current on the interface, respectively. variable, and , denotes the DOFs associated with the expanand are restriction sion of the scalar auxiliary variable. operators, which involve nothing but Boolean operations. Furthermore, the system matrix (20) can be rewritten as (21) represents the dithrough a block-Jacobi splitting, where denotes the off-diagonal blocks. Subseagonal blocks and quently, we write the Jacobi preconditioned system as (22)

(20)

with

and

We remark that in the DDM theory, the convergence factors are closely related to the eigenvalue distributions of the ma. Since for the mixed SOTC, both propagating and trix evanescent modes are converging, the eigenvalues of the matrix are therefore inside the unit circle centered at one. This holds true for both the one-way and the non-conformal DDM algorithms, as demonstrated in Section II-C. III. NUMERICAL RESULTS In this section, we study the performances of the proposed non-conformal DDM via numerical experiments. We validate the accuracy and convergence of the proposed DDM with the mixed SOTC using a WR-90 rectangular waveguide example. A Vivaldi antenna array and an ultra wide band (UWB) antenna array are employed to examine the scalability of the proposed method with respect to the sizes of the arrays. Finally, we demonstrate the superiority of non-conformal DDM with the mixed SOTC by solving several electrically large problems of practical interest. Throughout our numerical examples, we employed the mixed-order curl-conformal elements from [21], . We solve (20) via a preconditioned Krylov with order subspace method, which is the generalized conjugate residual (GCR) method [23]. To facilitate faster convergence, the preis chosen to be the symmetric Gauss-Seidel preconditioner conditioner [9], [24]. It is worth noting here that the performance of the Gauss-Siedel preconditioner is similar to that of the Jacobi preconditioner; except, in general, the Gauss-Siedel preconditioner is much more efficient than the Jacobi preconditoner. A relative residual defined as

is used to terminate the DDM iterations, with smaller than a specified tolerance; where is the DDM matrix, and is the excitation vector, and is the solution at the current iteration. All simulations are performed on a 64-bit AMD Opteron workstation with 16 GB of RAM. A. Rectangular Waveguide In this example, we study the accuracy and convergence of the proposed non-conformal DDM with the mixed SOTC, compared against FOTC and SOTC using an X-band (WR-90) rectangular waveguide operated at 10 GHz. The geometry of the

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Fig. 12. Error in transmission coefficient for a WR-90 waveguide.

Fig. 13. Iterative solver convergence for a rectangular waveguide with nonconformal domain partition.

waveguide is depicted in Fig. 2 and non-conformal domain partition is shown in Fig. 8. A GCR(10) [23] solver is used. Fig. 12 ,a gives the convergence of the transmission coefficient measure of the transmitted power. We see that the proposed DDM with the mixed SOTC gives almost identical solutions to the conventional single-domain FE method. The numbers of iterations required for DDM with FOTC, SOTC and mixed SOTC are given in Fig. 13. The effect of propagating and evanescent modes can also be seen in the character of the convergence for the three TCs. The FOTC and SOTC damp propagating modes to the same degree. This is what gives the similar initial convergence in Fig. 13. The convergence of FOTC is then slowed by the solution of the evanescent portion of the spectrum. Comparing to SOTC, the mixed SOTC improves the portion of propagating modes in the spectrum, which is the reason why it gives the more rapid initial convergence. Also, it damps evanescent modes as well, and therefore the convergence curves of mixed SOTC maintain a relatively constant rate. B. Vivaldi Antenna Array Next, we examine the numerical scalability of the proposed approach for solving a broadband single-polarized Vivaldi antenna array operated at 5 GHz. Fig. 14 shows the geometry

Fig. 14. Geometry of a Vivaldi array. (a) dimension and material parameters of a single Vivaldi array element; (b) 7 7 linear-polarized array arrangement.

2

Fig. 15. Iterative solver convergence for 100

2 100 Vivaldi array.

and arrangement of Vivaldi antenna array used in the simulation. The number of iterations required for DDM with FOTC and mixed SOTC for the 100 100 Vivaldi antenna array are compared in Fig. 15. The GCR(10) solver with stopping-criteria is used. Note that very low tolerance herein is used to study the robustness of the method. The results show a significant improvement in the convergence using the mixed

PENG AND LEE: NON-CONFORMAL DDM WITH MIXED TRUE SOTC FOR SOLVING LARGE FINITE ANTENNA ARRAYS

TABLE II COMPUTATIONAL STATISTICS OF VIVALDI ARRAY ( = 10

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SOTC. The computational statistics of DDM with FOTC and mixed SOTC versus different size of Vivaldi arrays are given in Table II. The new DDM with mixed SOTC provides a good numerical scalability with respect to the array size. Note the simulation here is using first order absorbing boundary condiaway from the array. It may tion (ABC), which is placed not be sufficient for array scanning case. Higher order ABC or boundary integral equation may be needed to truncate the computational domain. This is beyond the scope of this paper but the interested reader may consult [25].

Fig. 16. Aircraft radome with Vivaldi array operated at f = 2 GHz (a) domain partition; (b) electric field magnitude.

C. Vivaldi Antenna Array With Radome To further study the performance of the proposed method for real-life applications, the next example we simulated is 13 vivaldi array in the presence of a conical dieleca 13 . The array and radome are tric aircraft radome with shown in Fig. 16(a) along with a shape-conforming truncation surface away from the radome’s exterior. The simulation uses 174 sub-domains, 169 sub-domains with repetition block for the antenna array and 5 layer shaped sub-domains without repetition, as depicted in Fig. 16. Note that this domain partition may not be the most efficient, but it is a good example to examine the performance of the proposed non-conformal DDM. The simulation requires 9,981,356 DOFs for the FOTC and 10,000,961 DOFs for the mixed SOTC. The mesh size is and the GCR(10) solver (with a stopping criteria of ) is used. Mixed SOTC requires 193 minutes for 21 iterations in contrast to FOTC requires 897 minutes for 94 iterations. For comparison, we also list the convergence curve of SOTC in Fig. 17. We note that the number of iterations significantly reduced by using the SOTC and the mixed SOTC because they deal effectively with the evanescent modes. Furthermore, the mixed SOTC shows even better performance than the SOTC as it improves the convergence of propagating modes, which in this case may be more prominent than the evanescent modes. Fig. 18(a) and Fig. 18(b) show, respectively, the radiation field patterns of the Vivaldi array alone and in the presence of the radome. The cross-polarization is significant introduced because of the effect of the radome.

Fig. 17. Iterative solver convergence for an aircraft radome with Vivaldi array with non-conformal domain partition.

D. Ultrawideband Array The design concept of the UWB antenna is given in [22] where dipole elements are configured to operate at broad range of frequencies in 2–20 GHz. A front view of 3 3 array and the geometry of the unit cell are shown in Fig. 19. In this study, the array is operated at 9 GHz and the size is varied from 3 3 to 50 50. Again the GCR(10) solver with stopping-criteria is used to study the scalability and robustness of the proposed non-conformal DDM. The number of iterations required for DDM with the FOTC and the mixed SOTC are given in Fig. 21(a) and Fig. 21(b), respectively. The results show

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Fig. 20. The electric field distribution at 9 GHz 50

2 50 array.

Fig. 18. Aircraft radome with Vivaldi array (a) radiation pattern (co-polarization); (b) radiation pattern (cross-polarization).

Fig. 19. UWB antenna array (a) the unit cell geometry; (b) the front view of a 3 3 array.

2

significant improvements in the convergence using the mixed SOTC. We note also that the new DDM with the mixed SOTC provides a good numerical scalability with respect to the array size. The computational statistics for 50 50 UWB array is presented in Table III. This simulation uses 2,964 sub-domains.

Fig. 21. Iterative solver convergence for different size of UWB array (a) FOTC; (b) mixed SOTC.

TABLE III COMPUTATIONAL STATISTICS OF 50 50 UWB ARRAY ( = 10

2

)

E. UWB Array With Frequency Selective Surface In many real-life examples, the antenna array is usually covered with Frequency Selected Surfaces (FSS) to further enhance the antenna performance. The last example we analysed here is

a 50 50 UWB array in the presence of a 45 29 slot FSS. The geometry and material properties of the unit cell of FSS

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TABLE IV UWB ARRAY WITH SLOT FSS AT 9 GHz ( = 10

)

TABLE V UWB ARRAY WITH SLOT FSS AT 15 GHz ( = 10

)

Fig. 22. Geometry and material properties of Slot FSS element.

Fig. 24. Iterative solver convergence for UWB array with slot FSS at 9 GHz.

Fig. 23. Slot FSS (a) Dimension; (b) Transmission coefficients.

are shown in Fig. 22. The dimension and transmission coefficients of FSS are given in Fig. 23. The FSS is placed along the broadside direction of the array and distance from the tip of the . Here is the wavelength array to the bottom of FSS is in free space. This distance is chosen to be sufficiently far away to allow for the higher order Floquet modes to die out. We now simulate the UWB with slot FSS at two frequencies, and . The simulation uses 7,767 sub-domains. -adaptive mesh refinement is then applied to all sub-domains, resulting in 183,367,885 DOFs at 9 GHz and 257,038,643 DOFs at 15 GHz. The computational statistics are shown in Table IV and Table V. We also include the statistics of DDM with the FOTC for comparison. , we see that mixed SOTC converges much For faster than the FOTC, requiring only 33 iterations versus the

134 iterations needed for the FOTC. The iterative solver convergences are plotted in Fig. 24. Next, we compare the radiation field patterns of the UWB array with and without the FSS, as shown in Fig. 25(a). From Fig. 25(a) we observe a close similarity, which validates the nearly total-transmission property of ,a the FSS at 9 GHz. Lastly, for the simulation at reduction from 179 to 42 iterations is achieved using the mixed SOTC vs the FOTC. At this frequency, the transmission coeffiof the FSS is 12.6 dB to incident plane waves. Again, cient we plot the radiation field patterns of the UWB array with and without the FSS, as shown in Fig. 25(b). The effect of the FSS is apparent in the decreased level of the lobes, with the main one decreasing from 36.3 dB to 22.8 dB relative to the UWB array without FSS. IV. CONCLUSION In this paper, we have introduced and analyzed a new mixed SOTC for the non-overlapping and non-conformal DDM for the time-harmonic Maxwell equations. Through the convergence analysis and the plotting of eigenspectra of DDM matrices, we verify that the proposed mixed SOTC not only renders convergence for both the TE and TM evanescent eigenmodes, but also improves performance for propagating waves. Furthermore, it is shown that, due to the enlargement of the function space

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Fig. 25. Radiation field pattern comparison of UWB array with and without presence of FSS in XZ-plane (a) operated at 9 GHz; (b) operated at 15 GHz (Red solid line: with FSS; blue dash line: without FSS).

for the auxiliary cement variables on the interfaces, the incorporation of TCs in the finite element implementation of nonconformal DDMs will result in a nearly singular DDM matrix. To successfully implement and enjoy the full benefits of the mixed SOTC for the non-conformal DDMs, we have integrated a corner-edge penalty method. The corner-edge penalty method has been shown to effectively remove the singular eigenvalues by enforcing the divergence free condition. Numerical results verify the analysis and demonstrate the effectiveness of the proposed method on a few canonical problems and several complex large-scale simulations.

[8] Y.-J. Li and J.-M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55, pp. 2803–2810, Oct. 2007. [9] K. Zhao, V. Rawat, S.-C. Lee, and J.-F. Lee, “A domain decomposition method with nonconformal meshes for finite periodic and semi-periodic structures,” IEEE Trans. Antennas Propag., vol. 55, pp. 2559–2570, Sep. 2007. [10] Y.-J. Li and J.-M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 54, pp. 3000–3009, Oct. 2006. [11] M. N. Vouvakis, Z. Cendes, and J.-F. Lee, “A FEM domain decomposition method for photonic and electromagnetic band gap structures,” IEEE Trans. Antennas Propag., vol. 54, pp. 721–733, Feb. 2006. [12] S.-C. Lee, M. Vouvakis, and J.-F. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys., vol. 203, no. 1, pp. 1–21, 2005. [13] V. Dolean, M. J. Gander, and L. Gerardo-Giorda, “Optimized Schwarz methods for Maxwell’s equations,” SIAM J. Sci. Comput., vol. 31, no. 3, pp. 2193–2213, 2009. [14] A. Alonso-Rodriguez and L. Gerardo-Giorda, “New non-overlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput., vol. 28, no. 1, pp. 102–122, 2006. [15] M. J. Gander, F. Magoulès, and F. Nataf, “Optimized Schwarz methods without overlap for the Helmholtz equations,” SIAM J. Sci. Comput., vol. 24, no. 1, pp. 38–60, 2002. [16] P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the onoverlapping domain decomposition for the Maxwell’s equation,” Comput. Methods Appl. Meth. Engrg., vol. 148, pp. 195–207, 1997. [17] Z. Peng, V. Rawat, and J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys., vol. 229, pp. 1181–1197, Feb. 2010. [18] V. Rawat, “Finite Element Domain Decomposition With Second Order Transmission Condition for Time Harmonic Electromagnetic Problem,” Ph.D. dissertation, The Ohio State Univ., Columbus, 2009. [19] Z. Peng and J.-F. Lee, “Non-conformal domain decomposition method with second order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys., 10.1016/j.jcp.2010.03.049. [20] P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau, “Interior penalty method for the indefinite time-harmonic Maxwell equations,” Numer. Math., vol. 100, no. 3, pp. 485–518, 2005. [21] D.-K. Sun, J.-F. Lee, and Z. Cendes, “Construction of nearly orthogonal Nedelec bases for rapid convergence with multilevel preconditioned solvers,” SIAM J. Sci. Comput., vol. 23, no. 4, pp. 1053–1076, 2001. [22] B. Munk, Finite Antenna Arrays and FSS. New York: Wiley, 2003. [23] C. P. Jackson and P. C. Robinson, “A numerical study of various algorithms related to the preconditioned conjugated gradient method,” Int. J. Numer. Methods Eng., vol. 21, pp. 1315–1338, 1985. [24] G. H. Golub and F. F. Van Loan, Matrix Computations. Baltimore, MD: John Hopkins Univ. Press, 1996. [25] J. M. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays. Hoboken-Piscataway, NJ: Wiley-IEEE Press, 2008.

REFERENCES [1] R. A. Adams, Sobolev Spaces. New York-London: Academic Press, 1975, vol. 65, Pure and Applied Mathematics. [2] P. Monk, Finite Element Methods for Maxwell’s Equations, ser. Numerical Mathematics and Scientific Computation. New York: Oxford Univ. Press, 2003. [3] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory. Berlin: Springer, 2005. [4] B. Després, P. Joly, and J. E. Roberts, “A domain decomposition method for the harmonic Maxwell equation,” in Iterative Methods in Linear Algebra (Brussels, 1991). Amsterdam: North-Holland, 1992, pp. 475–484. [5] B. Stupfel, “A fast domain decomposition method for the solution of electromagnetic scattering by large objects,” IEEE Trans. Antennas Propag., vol. 44, pp. 1375–1385, Oct. 1996. [6] C. T. Wolfe, U. Navsariwala, and S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antennas Propag., vol. 48, pp. 278–284, Feb. 2000. [7] B. Stupfel and M. Mognot, “A domain decomposition method for the vector wave equation,” IEEE Trans. Antennas Propag., vol. 48, pp. 653–660, May 2000.

Zhen Peng (M’09) received the B.S. degree in electrical engineering and information science from the University of Science and Technology of China, in 2003, and the Ph.D. degree from the Chinese Academy of Science, in 2008. From 2008 to 2009, he was a Postdoctoral Fellow at the ElectroScience Laboratory, Ohio State University, where, since 2009, he has been working as a Senior Research Associate. His research interests are in scientific computing, specifically in the area of full-wave numerical methods in computational electromagnetic. Recently research directions include the domain decomposition methods for both finite element method and boundary integral method, the hybrid finite element-boundary integral method, and the multilevel fast multipole method. Applications of his research include: novel antennas for wireless communication systems, electromagnetic compatibility and interference analysis of multiple antenna systems on military and commercial aircrafts, signal integrity and package analyses for modern ultra-large integrated circuits, and the design tolls for energy efficient LCD back-light unit.

PENG AND LEE: NON-CONFORMAL DDM WITH MIXED TRUE SOTC FOR SOLVING LARGE FINITE ANTENNA ARRAYS

Jin-Fa Lee (F’05) received the B.S. degree from National Taiwan University, in 1982, and the M.S. and Ph.D. degrees from Carnegie-Mellon University, in 1986 and 1989, respectively, all in electrical engineering. From 1988 to 1990, he was with ANSOFT Corp., where he developed several CAD/CAE finite element programs for modeling three—dimensional microwave and millimeter-wave circuits. From 1990 to 1991, he was a Postdoctoral Fellow at the University of Illinois at Urbana-Champaign. From 1991 to 2000, he was with Department of Elec-

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trical and Computer Engineering, Worcester Polytechnic Institute. Currently, he is a Professor at ElectroScience Lab., Dept. of Electrical Engineering, Ohio State University. His research interests mainly focus on numerical methods and their applications to computational electromagnetics. Current research projects include: analyses of numerical methods, fast finite element methods, fast integral equation methods, three-dimensional mesh generation, domain decomposition methods, hybrid numerical methods and high frequency techniques based on domain decompositions approach, LCD modeling, large antenna arrays and co-design for signal integrity and packagings.

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Low-Cost Dual-Loop-Antenna System for Dual-WLAN-Band Access Points Saou-Wen Su, Member, IEEE, and Cheng-Tse Lee, Member, IEEE

Abstract—A low-cost, printed three-antenna system suitable to be embedded inside a wireless access point (AP) for multiple-input multiple-output (MIMO) applications in the 2.4 and 5 GHz WLAN bands is presented. The antenna system comprises three circular dual-loop antennas occupying a moderate area and printed on the same layer of a circular antenna substrate. Each dual-loop antenna further includes a large outer loop and a small inner loop, both operating at one-wavelength resonant mode and sharing common antenna feeding and grounding. The antennas are set in a sequential, rotating arrangement on the substrate with an equal inclination angle of 120 to form a symmetrical multiantenna structure. The antenna substrate is further stacked above the system circuit board of an AP by a small distance. In this case the design can fully integrate the circuit board of the AP as an efficient reflector for loops into an internal multiantenna solution. The results showed that well port isolation were obtained together with high-gain and dual-polarized radiation characteristics over the 2.4 and 5 GHz bands. Details of a design prototype are described and discussed in this paper. Index Terms—Antennas, internal access-point (AP) antennas, multiantenna system, printed loop antennas, WLAN antennas.

I. INTRODUCTION

A

NTENNAS with dual-polarized radiation characteristics have been very attractive to base-station antenna designs for mobile communications [1] and WLAN [2]–[4] applications. The dual-polarized antennas retain considerable advantage of combating the complex propagation of the transmit/receive waves in the WLAN environment. In addition, the polarization sensitivity of the antenna is mitigated by utilizing the dual-polarized antennas. These features certainly benefit wireless access points (APs) or routers in practical applications. Moreover, for WLAN communications, the majority of the present-day APs are “11n” or “pre-n” compatible on the open market. Particularly, since the IEEE Standard Association ratified the IEEE 802.11n standard in September 2009 [5], the multiple-input multiple-output (MIMO) technology adopting multiple transmit/receive antennas to get higher throughput has become enormously popular. For the mainstream AP products, the 3 3 (representing ) dual-band multiple antennas are usually demanded, especially for the requirements Manuscript received July 14, 2010; revised October 15, 2010; accepted October 20, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. The authors are with the Network Access Strategic Business Unit, Lite-On Technology Corp., Taipei County 23585, Taiwan (e-mail: stephen.su@liteon. com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123070

of internal AP antennas from an esthetic point of view that external antennas are not very pleasing to the user. Thus, the antenna design becomes substantially important when multiple antennas are integrated into APs with similar device size remained. In general, the antenna design consideration can fall into two categories for 3 3 dual-band APs. The first one is for the concurrent, dual-band dual-radio APs, which are usually requested by high-end, enterprise AP applications. In this case six single-band (three 2.4 GHz and three 5 GHz) antennas are employed [6], [7], and the 2.4 and 5 GHz radio signals are simultaneously transmitted or received. The other one is for dual-band single-radio APs, which are of lower cost and include three dual-band (2.4 and 5 GHz) antennas in either 11 b/g (2.4 GHz) or 11 a (5 GHz) single-radio mode [8], [9]. These designs, however, mainly utilize the monopole- and PIFA-type antennas in 3-D, metal-plate structure and require a large antenna ground plate to accommodate the antennas and induced image currents thereof. That large ground plate is expensive and definitely increases the overall material cost. Also, these mentioned internal AP antennas only generate linearly polarized radiation and cannot provide dual-polarized operation. In this paper, we demonstrate a promising, low-cost design concept that is capable of integrating printed multiple antennas and the AP’s system circuit board into a WLAN AP and at the same time, targeting on dual-polarized radiation in the two major elevation planes for each antenna. The proposed design comprises three identical dual-loop antennas, all printed on an inexpensive FR4 substrate and placed in a sequential, rotating arrangement on the antenna substrate with an equal inclination angle to form a symmetrical multiantenna structure. The antenna substrate is further stacked above the system circuit board of an AP. The ground plane of the circuit board can be utilized as an efficient reflector for the antennas to achieve high gain and directional radiation. In this case the design integrates the AP’s circuit board into an internal AP-antenna solution. The dual-loop antenna further includes a large 2.4-GHz (2400–2484 MHz) outer loop and a small 5-GHz (5150–5825 MHz) inner loop, both operating at its own one-wavelength resonant mode. The one-wavelength loop produces bidirectional radiation in free space and is known as a self-balanced antenna, which excites less surface currents on the system or antenna ground plane [10]. Accordingly, the ground plane of the circuit board can even better suppress backward direction (see Fig. 1), which in turn leads radiation in the to more directional radiation. Further, by properly rotating the circular dual-loop antenna, it was found that the dual-polarized radiation can be achieved in the two major elevation planes of and cuts (see Fig. 10). A design example was the

0018-926X/$26.00 © 2011 IEEE

SU AND LEE: LOW-COST DUAL-LOOP-ANTENNA SYSTEM FOR DUAL-WLAN-BAND ACCESS POINTS

Fig. 2. Photo of a fabricated design prototype;

Fig. 1. (a) Top view of the proposed dual-loop-antenna system backed by a circular system PCB of a wireless access point for 2.4/5 GHz WLAN operation. (b) Sectional view of the proposed design. (c) Detailed dimensions of the printed, dual-polarized dual-loop antenna (antenna 1).

constructed and tested. Details of the design consideration are described and discussed along with the analyses of surface current distributions and radiation characteristics. II. ANTENNA CONFIGURATION AND DESIGN CONSIDERATION Fig. 1(a) shows the configuration of the three dual-loop antennas formed on a low-cost, single-layered, circular FR4 substrate of thickness 1.6 mm and diameter 100 mm and backed by a circular system circuit board of a wireless AP for dual-polarized and dual-WLAN-band operation. A sectional view of the antenna substrate stacked above the ground plane of the system circuit board (or usually refer to system PCB) by an air separation of 8.4 mm is shown in Fig. 1(b). This system circuit board or PCB does not have to be circular in shape; however, the circular PCB was selected because the three antennas can be set symmetrically on the system circuit board to simplify the studies and the findings thereof. Also, the ground plane of the PCB can

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.

be utilized as an efficient reflector for the antennas, aiming more radiation in the direction to achieve directional radiation and higher antenna gain. In this case the design concept integrates the system circuit board into an internal AP-antenna solution. Furthermore, compared with the internal, stand-alone AP antennas constructed from stamping metal plates [6]–[9], the proposed design does not require any antenna ground plane to accommodate image currents induced by the monopole/PIFA-type designs. Each of the three identical dual-loop antennas (denoted as antenna 1, 2, and 3) occupies a moderate area and is equally spaced right next to the perimeter of the circular antenna substrate. The diameter of the substrate is required to be less than that of the system PCB such that all the antennas fit within the boundary of the system PCB. All the antennas are located 12.5 mm away from the substrate center with equal inclination angles (formed by the two adjacent antennas and the substrate center) of 120 . The distance between the two antennas is only 11 mm, which is about one-eighth of the distance 80 mm to achieve port isolation below 15 dB between two parallel 2.4 GHz dipole antennas (minimal space of 0.65-wavelength between two parallel dipoles [11]). The antenna signal feeding (point A, B, C) and cable grounding (point D, E, F) face the same direction (same rotation) with respect to the substrate center. Notice that the ground point is closer to the center hole of the substrate compared with the feed point. This way, the routing of the mini-coaxial cable can go smoothly toward the center (see Fig. 2). The three antennas are obviously in a sequential, rotating arrangement and of a symmetrical structure. This configuration not only provides ease of studying and analysis but also allows each antenna to gain equal, 3-D space coverage. Detailed dimensions of a single antenna are presented in Fig. 1(c). The proposed antenna further comprises two circular loops: a large 2.4-GHz outer loop and a small 5-GHz inner loop, which also share common feed and ground points on the outer loop. Both loops operate at one-wavelength resonant mode; the central operating frequency of each loop is determined and controlled by the diameter of the loop. The outer loop dominates the overall antenna size and also encompasses the inner loop. The width of the loop affects the antenna bandwidth, and in this study both loops are of uniform width of 3 mm. Between the outer and the inner loops is there a smallest distance of 2 mm, in which the two loops are connected therein through a thin pair of parallel strips of width 1 mm. The length of the strips has the effects on the input matching of the 5 GHz loop and

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the operating frequencies thereof (because the strips are also part of the resonant path for 5 GHz operation). The distance in between the connecting strips is set the same as the feed gap over points A and D. This parameter largely influences the impedance bandwidth of the 2.4 and 5 GHz bands. The near optimal dimensions of the dual-loop antenna were attained by means of a rigorous analysis with the aid of the electromagnetic simulator, Ansoft HFSS [12]. Notice that for simplicity, the 2.4-GHz outer loop was first designed and then the 5-GHz inner loop was added on. Also, the 0.5-wavelength mode of the inner loop does not affect the 2.4 GHz band whereas the 1.5- and 2.0-wavelength modes of the outer loop impact the 5-GHz band very little (see Fig. 6). Furthermore, the dual-loop is further rotated by an degree rotation with respect to the center of the antenna substrate for dual-polarized radiation in the plane cut along the diameter and the plane perpendicular to the plane in the case of antenna 1. A photo of a constructed prototype is given in Fig. 2 for demonstration and better understanding. As can be seen, the three antennas are right next to the perimeter of the circular antenna substrate and fed by utilizing three 50- mini-coaxial cables of length 9 mm. For practical, industrial applications, each coaxial cable is further connected to the 2.4- and 5-GHz WLAN module on the AP’s PCB through a pair of I-PEX connectors. The routing of the cables passes through a small hole of diameter 8 mm in the center of the antenna substrate. In this case each coaxial cable neither overlaps the others nor the antennas, and is of the same length with the same phase for each antenna. The inner conductor of the cable is connected to the feed point; the outer braided shielding is soldered to the ground point opposite to the feed point over the small feed gap of 2 mm.

Fig. 3. Measured -parameters for the dual-loop antennas; . (a) for antenna 1, 2, 3). (b) Isolation Reflection coefficients ( between two antennas.

III. RESULTS AND DISCUSSION A. Reflection Coefficients, Isolation, and Envelope Correlation A prototype of the proposed antenna as shown in Fig. 2 was first constructed and measured based upon on the design and dimensions thereof described in Fig. 1. Fig. 3(a) and (b) shows the measured reflection coefficients and isolation between the dual-loop antennas, whose simulated counterparts are given in Fig. 4(a) and (b). The reflection coefficients are plotted by the curves of for the three antennas. The isolation between any two of the three antennas is only presented by the curves of for brevity due to the symmetrical structure of the proposed design. On average, the experimental data agree with the simulation results, which were based on the finite element method (FEM). However, discrepancies were found due largely to PCB manufacture tolerance and effects of the mini-coaxial cables in the experiments. The impedance matching of the three antennas over the 2.4 and 5 GHz bands is all below 10 dB (even well below 14 dB in the 5 GHz band), which satisfies the demanded bandwidth specification for WLAN operation. Notice that the impedance bandwidth also covers the Japanese 5 GHz (4900–5090 MHz) band [13]. In general, the parameters are about the same as each other, and the variation between and (or between and ) is small because of the symmetry of the three antennas.

Fig. 4. Simulated -parameters for the dual-loop antennas; . for antenna 1, 2, 3). (b) Isolation (a) Reflection coefficients ( between two antennas.

Second, for the isolation between any two antennas, it is found to be below 15 and 20 dB over the 2.4 and 5 GHz bands. Fig. 5(a) and (b) presents the calculated envelope correlation among the three antennas operating in the 2.4 and 5 GHz

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Fig. 6. Simulated reflection coefficients for antenna 1 (dual loop), the 2.4-GHz outer loop only, and the 5-GHz inner loop only.

Fig. 5. Calculated envelope correlation. (a) For the 2.4-GHz band. (b) For the 5-GHz band.

bands, respectively. The envelope correlation here was determined by the use of parameters in (1), which was derived in [14], for sufficiently accurate results in many practical cases [15] [see (1) at the bottom of the page]. The magnitude and phase of the parameters were collected from the simulation data in Fig. 4(a) and 4(b). The complex conjugate denoted by an asterisk of the parameters was given by multiplying the phase part by 1. A brief description of the calculation was discussed in [9]. From the results, the envelope correlation values remain under 0.02 in the 2.4 GHz band and under 0.0003 in the 5 GHz band. These values are much smaller than 0.3 demanded widely by industry specification and 0.7 at the base station as suggested in [16]. The in-band correlation is also seen to be less than the out-of-band correlation. In addition to this method applied here, the evaluation of the envelope correlation can be carried out as described in [16] or simply measured in a reverberation chamber [17]. B. Parametric Studies on the Dual-Loop Antenna, the Separation Gap, and the Size of the System Ground Plane To understand the wavelength state of the outer and the inner loops, the antenna was separated into two loops. The corresponding results on the reflection coefficients are shown in Fig. 6 for comparison; other dimensions were ketp the same as those set in Fig. 3. In the case of the 2.4-GHz outer loop

Fig. 7. Simulated reflection coefficients for antenna 1 as a function of the distance g of the air separation.

only, the 1.0- and 1.5-wavelength resonant modes at about 2.47 and 4.65 GHz can be spotted. As for the 5-GHz inner loop only, only 1.0-wavelength loop mode at about 5.14 GHz is seen in the frequency range of 2 to 6 GHz; the 0.5-wavelength loop mode was found to be a little below 2 GHz but actually not responsive due to very large resistance (larger than 10 000 ohms, like loading an open-circuit termination) in that resonant mode (recognized by zero reactance). The results suggest that the operating modes of the outer and the inner loops do not much interfere with each other within the bands of interest. Further, although the upper band is able to cover the frenquency range of 6 to 7 GHz (not shown for brevity), this operating band is mixed with the higher-order resonant mode of the outer loop, which may change the directional radiation characteristics and then worsen the antenna gain. Therefore, this part of the impedance bandwidth is not considered using. Finally, despite the 2.4 GHz band was mainly controlled by the outer loop while the 5 GHz band by the inner loop as designed in Section II and agreed here by the reflection-coefficient analysis, the radiation characteristics and surface-current behavior are still required to be examined in the following section in order to verify the studies. Simulation studies of the effects of the air separation distance on the dual-loop antenna impedance bandwidth were also conducted. Fig. 7 presents the reflection coefficients for antenna 1 as

(1)

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Fig. 9. Simulated (a) reflection coefficients and (b) peak antenna gain as a function of the diameter of the system ground plane of the system PCB.

Fig. 8. Simulated input impedance on the Smith chart for antenna 1 as a function of the distance g of the air separation. (a) For frequency range 2 to 3 GHz. (b) For frequency range 5 to 6 GHz.

a function of the distance g of the small air gap. The step of distance 2 mm was taken in analysis. The input-impedance curves on the Smith chart in the frequency range of 2–3 and 5–6 GHz are separately shown in Fig. 8(a) and (b). It is first seen that for 2.4 GHz operation, the impedance bandwidth quickly deteriorates while that at around 4.65 GHz at the 1.5-wavelength resonant mode of the outer loop becomes better. This is because the central operating frequency of the 1.0-wavelength loop mode for the outer (2.4 GHz) loop lies at the axis (plotting resistance only) and moves (to the left) toward the short-circuit termination as the antenna gets closer to the system ground plane [see Fig. 8(a)]. As for 5 GHz operation, the achievable bandwidth defined by reflection coefficients below 10 dB worsens and gets narrower. In this case the curve shifts to more inductive impedance and at the same time, its diameter becomes larger (that is, narrower bandwidth) as can be seen in Fig. 8(b). As a general rule of thumb, decreasing the air separation leads to less matched operating bands of the antenna. The effects of the size of the system PCB on the antenna impedance bandwidth and the peak gain were also analyzed. Fig. 9(a) and (b) shows the simulated reflection coefficients and the peak antenna gain as a function of the diameter, denoted

by , of the system ground plane. It can first be seen that regardless of the size of the system ground plane, the impedance bandwidth is almost the same in various diameters of the PCB. For the antenna gain, the peak gain in the 2.4-GHz band was found to increase when a larger system ground plane was used. However, for the 5-GHz band, the peak gain could further decrease in the case of (about 3.25-wavelength at 5490 MHz). The results indicate that the size of system ground plane impacts more on the gain of the proposed antenna than the antenna operating bands. In addition, the diameter 120 mm used in the proposed design can provide largest peak gain in the 5-GHz band, which is beneficial because the free-space path loss at the same distance from the antenna is higher for 5-GHz operation than that for 2.4- GHz operation. Notice that it is not practical for the case with the diameter less than 100 mm because the system PCB (the reflector) will be smaller than the antenna system. Also, the diameter of the PCB exceeding 180 mm would be too large to use inside the AP. IV. CURRENT DISTRIBUTION ANALYSIS CHARACTERISTICS

AND

RADIATION

The surface-current distributions for antenna 1 excited at 2442 and 5490 MHz, the central frequencies of the 2.4 and 5 GHz bands, are given in Fig. 10. The surface currents are plotted in the form of vectors (by arrow shape) in order to identify the current nulls, and the thicker the arrow is, the stronger (more magnitude) the current is. As expected, the density of currents at 2442 MHz is mostly distributed on the outer loop, and by contrast, the inner loop exited at 5490 MHz is thickly populated by the surface currents. This behavior again, shows that the two resonant modes of the lower and the upper bands are attributed by separate resonant paths in the

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Fig. 10. Simulated surface current distributions for antenna 1 excited at 2442 and 5490 MHz.

Fig. 12. Simulated 2-D radiation patterns at 2442 and 5490 MHz for antenna . 1 studied in Fig. 3 with

Fig. 11. Simulated 2-D radiation patterns at 2442 and 5490 MHz for antenna 1 studied in Fig. 3.

outer and the inner loops. Furthermore, the maximum currents are seen pointing in the direction of on both loops, and accordingly the E-field components in the and planes are almost the same. The maximum-current vector can be considered lying on the hypotenuse of a “virtual,” isosceles right triangle and can be decomposed into two vectors of the same magnitude lying in the two legs of the triangle. Therefore, in each elevation plane (that is, the or cut), there exist two orthogonal fields (in the two “virtual” triangle legs) of similar field strength such that dual-polarized radiation characteristics are attained in the plane cut along the diameter of the antenna substrate and the plane perpendicular to the plane in the case of antenna 1 under condition of 45 degree rotation. The radiation characteristics of the proposed design were also studied, and for brevity the radiation patterns are given at the central operating frequency in each band. Also, due to the symmetrical arrangement of the three antennas, only the results of a single dual-loop antenna are presented. Antenna 1 was chosen to suit the convenience of defining the antenna coordinates. Figs. 11, 12, and 13 plots the far-field, 2-D radiation patterns at 2442 and 5490 MHz, the central frequencies of the 2.4 and 5 GHz bands, in and fields for the proposed and the reference antennas. Except for the degree rotation

Fig. 13. Simulated 2-D radiation patterns at 2442 and 5490 MHz for antenna . 1 studied in Fig. 3 with

on the dual-loop antenna, other dimensions are the same as studied in Fig. 3. For practical, indoor AP applications, the APs are usually mounted on the ceiling or wall such that the radiation in the two elevation planes (the and cuts) is most concerned (not horizontal plane of cut). It can first be seen that maximum field strength is generally in the direction away from the antenna system with the front-to-back ratio larger than 15 and 20 dB at 2442 and 5490 MHz in the and planes. As for the for the reference antennas of and , similar radiation, in which the peak gain occurs in the normal direction to the substrate away from the antennas, can be seen in Figs. 12 and 13. But the reference antennas show more polarization purity in the axial direction and have larger cross-polarization level (XPL). In contrast, the proposed antenna yields dual-polarized radiation in the and planes with the 3-dB XPL range about 97 and 101 for 2.4-GHz operation and about 137

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Fig. 16. Measured peak antenna gain and radiation efficiency against frequency for antenna 1 studied in Figs. 14 and 15.

Fig. 14. Measured 3-D radiation patterns at 2400, 2442, and 2484 MHz for antenna 1 studied in Fig. 3.

Fig. 15. Measured 3-D radiation patterns at 5150, 5490, and 5825 MHz for antenna 1 studied in Fig. 3.

and 73 for 5-GHz operation. The 3-dB XPL in this study is defined for and field variations within 3 dB. In this case each antenna has dual-polarized radiation characteristics in the plane along the diameter of the antenna substrate and also in the plane perpendicular to the former. When the three antennas are functioning as a MIMO-antenna group, it is expected that the design can radiate dual-polarized waves in at least three directions evenly spaced by 120 . Figs. 14 and 15 plots the measured far-field, 3-D radiation patterns at 2400, 2442, and 2484 MHz and at 5150, 5490, and 5825 MHz. Compared with the other frequencies tested within the bands of interest, good consistency in the radiation patterns was also observed. The measurement was taken by the

ETS-Lindgren OTA test system using the great-circle method in a CTIA authorized test laboratory [18]. Clearly, both 2.4- and 5-GHz radiation has maximum field strength in the direction above the plane and away from the system circuit board. Also, high-gain and directional patterns like broadside radiation in the 2.4 and 5 GHz bands are observed too. This phenomenon is expected simply because the one-wavelength loop antenna has bidirectional radiation in free space, backward radiation in the direction is much suppressed and reflected in this study by the system ground, which in turn leads to more directional radiation patterns. The measured peak antenna gain and radiation efficiency for the proposed design are presented in Fig. 16. The peak gain in the 2.4-GHz band is in the range of 6.4 to 7.6 dBi with radiation efficiency exceeding 67%. For the 5 GHz band, the peak gain is at a constant level of about 8.7 dBi with radiation efficiency of about 80%. The gain measurement here took account of the mismatch of the loop antenna, and thus, the “realized gain” [19] was actually measured. The radiation efficiency was obtained by calculating the total radiated power (TRP) of the antenna under test (AUT) over the 3-D spherical radiation first and then dividing that total amount by the input power of 0 dBm (default value) given to the AUT in the test laboratory. The antenna diversity gain was also investigated. To avoid the laborious drive tests, the measurements were performed via QuieTek [20] in the Bluetest reverberation chamber [21], which emulates a rich scattering and also fading environment following a Rayleigh distribution. The of the three dual-loop antennas were measured simultaneously by connecting each antenna (presented as a branch in the test) to a four-port vector network analyzer (VNA). The subscript “1” and “j” of means port 1 connecting to the three transmitting monopoles for three polarization (perpendicular to each other) and port j connecting to each corresponding antenna in the test (see [22, Fig. 12.10]). The cumulative distribution function (CDF) of the measured power-transmission samples for the three branches is about the same and very close to the theoretical Rayleigh distribution observed at 2400, 2442, 2484, 5150, 5490, and 5825 MHz. At a cumulative probability level of 1% (that’s, the sufficient quality 99% of the time), the difference between the CDF of selection combining and the best CDF among the three antennas represents the apparent diversity gain (see [17, Fig. 2]). In this case the diversity gain was found to be about 13.5 and 13.6 dB over the 2.4- and 5-GHz band, respectively.

SU AND LEE: LOW-COST DUAL-LOOP-ANTENNA SYSTEM FOR DUAL-WLAN-BAND ACCESS POINTS

V. CONCLUSION A printed, 3 3 dual-loop-antenna system able to provide good radiation properties in the 2.4 and 5 GHz bands for low-cost, internal AP-antenna applications has been proposed and tested. Each antenna has two one-wavelength, self-balanced circular loops, providing separate and controllable frequency bands. The three dual-loop antennas were formed on a low-cost FR4 substrate with diameter 100 mm within the boundary of the AP’s system circuit board, above which the antennas were stacked by a small air gap. The proposed design in this case integrates the system circuit board as an efficient reflector inside the AP and makes use of it to achieve high gain and directional radiation. The antennas showed 10-dB return-loss bandwidth with port isolation below 15 and 20 dB over the 2.4 and 5 GHz bands. The 3-dB XPL in and planes is about 97 and 101 for 2.4 GHz loops versus 137 and 73 for 5 GHz loops. Directional radiation patterns with the front-to-back ratio exceeding 15 and 20 dB and with peak gain larger than 7 and 8 dBi, respectively, for 2.4- and 5-GHz operation were obtained. Several advantages of the proposed design over the previous paper in [9] have also been noticed, including the lower material/assembly cost, more board space available, better radiation characteristics with separate, controllable resonances, and so on. The proposed design makes it possible for the 11n AP to be equipped with a low-cost, integrable multiantenna system and at the time, is also a viable alternative to conventional, high-gain patch and microstrip antennas. The miniaturization of the antenna system by using the meander line for the circular loop to further reduce the overall 2-D dimensions will be studied in the future. REFERENCES [1] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003, ch. 4, pp. 173–193. [2] Y. X. Guo and K. M. Luk, “Dual-polarized dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 1120–1124, May 2003. [3] T. S. P. See and Z. N. Chen, “Design of dual-polarization stacked arrays for ISM band applications,” Microw. Opt. Technol. Lett., vol. 38, pp. 142–147, Jul. 2003. [4] F. S. Chang, H. T. Chen, K. C. Chao, and K. L. Wong, “Dual-polarized probe-fed patch antenna with highly decoupled ports for WLAN base station,” in IEEE Antennas Propag. Soc. Int. Symp. Dig., Monterey, CA, 2004, pp. 101–109. [5] IEEE Ratifies 802.11n, Wireless LAN Specification to Provide Significantly Improved Data Throughput and Range The IEEE Standard Association [Online]. Available: http://standards.ieee.org/announcements/ieee802.11n_2009amendment_ratified.html [6] S. W. Su, “Very-low-profile monopole antennas for concurrent 2.4- and 5-GHz WLAN access-point applications,” Microw. Opt. Technol. Lett., vol. 51, Nov. 2009. [7] S. W. Su, “Concurrent dual-band six-loop-antenna system with wide 3-dB beamwidth radiation for MIMO access points,” Microw. Opt. Technol. Lett., vol. 52, pp. 1253–1258, Jun. 2010. [8] J. H. Chou and S. W. Su, “Internal wideband monopole antenna for MIMO access-point applications in the WLAN/WiMAX bands,” Microw. Opt. Technol. Lett., vol. 50, pp. 1146–1148, May 2008. [9] S. W. Su, “High-gain dual-loop antennas for MIMO access points in the 2.4/5.2/5.8 GHz Bands,” IEEE Trans. Antennas Propag., vol. 58, Jul. 2010, to be published. [10] H. Morishita, Y. Kim, and K. Fujimoto, “Design concept of antennas for small mobile terminals and the future perspective,” IEEE Antennas Propag. Mag., vol. 44, pp. 30–34, 2002.

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[11] S. W. Su, J. H. Chou, and Y. T. Liu, “Realization of dual-dipole-antenna system for concurrent dual-radio operation using polarization diversity,” Microw. Opt. Technol. Lett., vol. 51, pp. 1725–1729, Jul. 2009. [12] Ansoft Corp. HFSS [Online]. Available: http://www.ansoft.com/products/hf/hfss [13] IEEE 802.11j-2004, Wikipedia the free encyclopedia [Online]. Available: http://en.wikipedia.org/wiki/IEEE_802.11j-2004 [14] J. Thaysen and K. B. Jakobsen, “Envelope correlation in MIMO antenna array from scattering parameters,” Microw. Opt. Technol. Lett., vol. 48, pp. 832–834, May 2006. [15] V. Plicanic, Z. Ying, T. Bolin, G. Kristensson, and A. Derneryd, “Antenna diversity evaluation for mobile terminals,” in Proc. Eur. Conf. Antennas Propag. (2006), Nice, France, pp. 1–3. [16] R. G. Vaughan and J. B. Andersen, “Antenna diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 36, pp. 149–172, Nov. 1987. [17] P. S. Kildal and K. Rosengren, “Correlation and capacity of MIMO systems and mutual coupling, radiation efficiency and diversity gain of their antennas: Simulations and measurements in a reverberation chamber,” IEEE Commun. Mag., vol. 42, pp. 104–112, Dec. 2004. [18] CTIA Authorized Test Laboratory, CTIA, the wireless association [Online]. Available: http://www.ctia.org/business_resources/certification/ test_ labs/ [19] J. L. Volakis, Antenna Engineering Handbook, 4th ed. New York: McGraw-Hill, 2007, ch. 6, pp. 16–19. [20] QuieTek Corporation [Online]. Available: http://www.quietek.com/ index_en.htm [21] Reverberation Test Systems Bluetest [Online]. Available: http://http:// www.bluetest.se/products/reverberation-test-systems [22] B. Furht and S. A. Ahson, Long Term Evolution: 3GPP LTE Radio and Cellular Technology. Boca Raton, FL: CRC, 2009, ch. 12, pp. 441–443.

Saou-Wen Su (S’05–M’08) was born in Kaohsiung, Taiwan, in November 11, 1977. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2001, 2003, and 2006, respectively. Since April 2006, he has been with the Technology Research and Development Center (TRDC), Lite-On Technology Corporation, Taipei, Taiwan, and is currently with the Network Access Strategic Business Unit (NASBU) at the same company. He built up the first RF Antenna Design team at Lite-On Technology Corporation and contributed numerous cutting-edge designs to the company’s ODM projects, including wireless AP router, Bluetooth headset/car kit, home entertainment device, RF module, etc. Many customized antenna designs were successfully mass produced. Currently, he has published more than 70 refereed SCI journal papers and several international conference articles. He has 19 U.S. and 20 Taiwan patents granted, and many patents are pending. His expertise is in the industrial antenna designs for wireless AP router, Bluetooth, WLAN and MIMO applications, and previous researches prior to Lite-On Technology Corporation included mobile-phone and UWB antenna designs.

Cheng-Tse Lee (S’08–M’10) was born in Yilan, Taiwan, in 1983. He received the B.S. degree in electronic engineering from National Changhua University of Education, Changhua, Taiwan, and the M.S. and Ph.D. degrees in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2005, 2007, and 2010, respectively. His main research interests are in antenna designs for wireless communications, especially for the planar antennas for mobile phone, laptop computer, access-point, WLAN, and MIMO applications, and also in microwave and RF circuit design. He is currently with the Network Access Strategic Business Unit, Lite-On Technology Corporation, Taipei, Taiwan.

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A 4-Port Diversity Antenna With High Isolation for Mobile Communications Biqun Wu and Kwai-Man Luk, Fellow, IEEE

Abstract—A novel wideband four-port diversity antenna that is capable of exciting four different radiation patterns is presented. The antenna consists of four magnetoelectric dipoles arranged in a ring configuration above a ground plane and a vertical electric monopole at the center of the ground plane. A matching network is used to combine the signals from the four magnetoelectric dipoles to produce two orthogonal broadside modes and one conical mode with horizontal polarization. The electric monopole is used to generate a conical beam with vertical polarization in electric feed. The performance of the antenna is studied computationally. A prototype operated at around 2.4 GHz was constructed and tested. Experimentally, the antenna exhibits about 22.2% overlapped bandwidth of the four ports. The two orthogonal broadside modes have about 11 dBi average gain. Whereas for the two orthogonal conical modes, an average gain of 6 dBi is achieved. The measured radiation patterns of the four degenerate modes are stable within the operating band. The measured isolations between different ports are generally less than 026 dB within the overlapped bandwidth. The measured radiating efficiencies of the four modes are all over 80% within the operating band. Index Terms—Dipole antennas, microstrip antennas, monopole antennas, shaped beam antennas.

I. INTRODUCTION

T

HE great success in the mobile communication industry has fostered the development of various wireless communication systems, which require complex antenna systems to achieve high quality performance. For this, the multiple antenna approach has received much attention in both antenna and wireless communications sectors. In particular, the diversity antenna has been investigated to increase the system capability [1]. There are different types of diversity antennas including spatial diversity antenna, frequency diversity antenna [2], polarization diversity antenna [1], [3] and pattern diversity antenna [4], [5]. The pattern and polarization diversity techniques are very effective to solve multipath-fading effects in complex environments [2]. Comparing with classical antennas operated with a single radiation mode, the pattern diversity antennas, which are capable of radiating and receiving signals through different radiation modes simultaneously, has the advantage of performing Manuscript received March 25, 2010; revised August 18, 2010; accepted October 25, 2010. Date of publication March 10, 2011; date of current version May 04, 2011. This work was partially supported by a grant from the Research Grant Commerce of the Hong Kong Special Administration Region, China, [Project No. CityU 119008]. The authors are with the State Key Laboratory of Millimeter Waves, Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TAP.2011.2123060

higher effective gain while maintaining the same installation space [4]. Several antenna structures offering pattern and polarization diversities have been proposed in the literature [4], [5]. Unfortunately, the available pattern and polarization diversity antennas suffer from narrow overlapped bandwidth of the excitation ports [4] and incomplete pattern diversity modes [5]. Thus their applications are limited. Microstrip antennas are widely used for achieving directional radiation patterns due to their low profile, light weight, low costs, and flexible structure. However, they have the weakness of narrow bandwidth and unstable radiation pattern across the operating band. Recently, a wideband complementary antenna designated as the magnetoelectric dipole was proposed [6], [7]. The antenna is composed of a vertically oriented quarter-wave shorted patch antenna and a planar electric dipole, which is equivalent to a combination of a magnetic dipole and an electric dipole. Excellent electrical characteristics such as low back radiation, stable antenna gain across the operating band, and symmetric E- and H-plane radiation patterns were demonstrated. Dual-polarized broadside patterns can be produced by microstrip patch antennas with two orthogonal L-probe feeds [8], stacked microstrip antenna with aperture coupling [9], and two orthogonal magnetoelectric dipoles with two L-probe feeds [7]. Comparing their performance, the magnetoelectric dipoles can radiate stable broadside patterns over much wider bandwidth. It impedance bandwidth and its isohas about 67% lation is more than 36 dB over the entire impedance bandwidth [7]. The conical beam with vertical polarization can be produced by a quarter-wave wire antenna or a monopolar plate-patch antenna [10]. On the other hand, the conical beam with horizontal polarization can be generated by a circulating current loop on a radiator. In [11], an approach to produce a rotationally symmetric horizontally polarized conical beam was achieved by arranging a group of conventional patch radiators in a ring configuration. This concept was further developed using slot antennas instead of microtrip patches [12]. However, these designs suffer from narrow impedance bandwidth and low antenna efficiency. By arranging four magnetoelectric dipoles in a ring configuration, a wideband antenna which can radiate stable conical beam with horizontal polarization was resulted [13]. In this paper, a detailed study on a wideband four-port diversity antenna which can radiate with four different radiation patterns, broadside or conical at overlapped frequency band, is presented. The proposed antenna is a combination of a vertical electric monopole and four magnetoelectric dipoles. The excitation of the four ports can be controlled by a switch or excited simultaneously.

0018-926X/$26.00 © 2011 IEEE

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Fig. 2. Photo of proposed antenna. (a) Proposed antenna. (b) Magnetoelectric dipole.

Fig. 1. Geometry of the proposed antenna. (a) 3D-view, (b) Perspective view, (c) Side view.

II. ANTENNA DESCRIPTION AND GEOMETRY The geometry of the proposed antenna operated at around 2.4 GHz is shown in Fig. 1–2, with detailed dimensions shown in Table I. The dimensions were selected after a detailed parametric study for good performance. The antenna consists of four magnetoelectric dipoles arranged in a ring configuration. The configuration of each magnetoelectric dipole is a combination of an electric dipole and a magnetic dipole. The upper part of each magnetoelectric dipole, which is an electric dipole, is implemented by a pair of sectorial-shape horizontal plates with inner radius IR and outer radius OR. The electric dipole is connected to the lower part of the antenna, which radiates

like a folded magnetic current. The equivalent magnetic current is due to a vertically-oriented shorted patch antenna realmm and a portion ized by two vertical walls of height of the ground plane between the two vertical walls. The separation G between them is 10 mm. Each magnetoelectric dipole is excited by an L-shaped strip which is connected to an air microstrip transmission line located in parallel to the vertical walls between a nearby vertical wall of the dipole. The distance and the transmission line is 1.5 mm. The transmission line is actually a quarter-wave transformer which transforms 112 to 50 . The L-shaped strip can be adjusted to achieve good impedance matching [8]. Referring to Fig. 1, the four magnetoelectric dipoles are arranged in a ring configuration and are , 90 , 180 , 270 and radius equal to 50 mm located at from center of the ground plane. By exciting a vertical copper wire at the centre of the four magnetoelectric dipoles, the antenna which functions as a vertical electric monopole can radiate a conical beam with vertical polarization. The feeding network consisting of two hybrid rings and a Wilkinson power divider is located under the ground plane. It is fabricated on a microwave substrate with thickness of 1.57 mm and relative permittivity of 2.33. The feeding network consists of two stages of excitation as shown in Fig. 3. In first stage, both output ports of the two hybrid rings are connected to the vertical transmission strips of the four magnetoelectric dipoles.

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TABLE I DIMENSIONS OF PROPOSED ANTENNA

Fig. 4. Radiation pattern of the combination of an electric dipole and a magnetic dipole.

ally in the H-plane. In the E-plane, the electric dipole radiates bidirectionally, whereas the magnetic dipole radiates uniformly in the E-plane. When the electric and magnetic dipoles are excited simultaneously with proper amplitudes and phases, a circularly symmetric cardiac-shaped radiation pattern with very low back lobe radiation both in the E- and H-planes can be achieved. A. Conical Pattern With Vertical Polarization

Fig. 3. The feeding network.

One isolated input port of the hybrid ring is used to excite a broadside mode. In second stage, the other input port of the hybrid ring is connected to an output port of a Wilkinson power divider, the input port of which can excite a conical mode with horizontally polarized radiation. The four excitation ports are soldered to 50 SMA connectors.

If we want the proposed antenna to radiate a conical beam with vertical polarization, it is simple to excite the vertical wire at the center of ground plane. The four magnetoelectric dipoles act as parasitic elements without excitations. Comparing with a conventional wire monopole, the vertical wire surrounded by four magnetoelectric dipoles exhibits wider bandwidth. Fig. 5(a) and (b) illustrate the current distribution of the antenna and respectively, where T represents the at time period of time. It is found that, when the phase of excitation source equals to 0 degree, the current densities are very high at the vertical copper wire, whereas the current densities are high at the vertical walls of the four magnetoelectric dipoles when the phase equals to 90 degree. It can be observed that, the vertical walls of the four magnetoelectric dipoles can help to shield the coupling from the vertical copper wire into the four inputs of dipoles and enhance the isolation between the two orthogonal conical modes. B. Conical Pattern With Horizontal Polarization

III. OPERATING PRINCIPLE In this section, the operating principle of the diversity antenna will be discussed. The proposed antenna consists of four magnetoelectric dipole elements arranged in a ring configuration. The basic design of the magnetoelectric dipole is composed of a vertically oriented quarter-wave shorted patch antenna and a planar electric dipole, which is equivalent to a combination of a magnetic dipole and an electric dipole to form a radiating element. The radiation pattern due to the combination of an electric dipole and a magnetic dipole is shown in Fig. 4. Two radiating sources are placed perpendicularly to each other to radiate in a complementary manner. The solid (black) and dotted (red) lines describe the field patterns of the electric and the magnetic dipoles, respectively. The electric dipole radiates uniformly in the H-plane, whereas the magnetic dipole radiates bidirection-

When the four magnetoelectric dipoles are excited in phase, an approximately horizontal constant current loop is produced above the ground plane. This can generate a rotationally symmetric conical beam with horizontal polarization. To provide in-phase excitation for the four input ports of the magnetoelectric dipoles, a feed network is required. Fig. 5(c) and 5(d) illusand trate the current distribution of the antenna at time , respectively. It is found that, when , the current densities are very high on the surfaces of the vertical walls of the four magnetoelectric dipoles, which indicates a magnetic , the current dipole mode is strongly excited. When densities are very high on the surfaces of the horizontal plates of the four magnetoelectric dipoles, which shows that an electric dipole mode is strongly excited. The excitation of a magnetic dipole mode and an electric dipole mode simultaneously

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Fig. 6. SWR and gain versus frequency for electric monopole.

current densities are very high on the surfaces of the vertical walls of the excited magnetoelectric dipoles, which indicates a magnetic dipole mode is strongly excited. When , the current densities are very high on the surfaces of the horizontal plates of the magnetoelectric dipoles, which indicates an electric dipole mode is strongly excited. Referring to the coordinate polarized broadside system shown in Figs. 1 and 2, the radiation patterns can be generated by exciting the two magnetoelectric dipoles located along the X or Y axis with 180 out of phase. When each group of the magnetoelectric dipoles is excited, it acts like a two element array radiating a broadside radiation pattern along the Z- direction. IV. SIMULATION AND MEASUREMENT RESULTS Simulation results of SWR, radiation pattern and gain were obtained by HFSS Ver.12 [14]. The performance of the prototype was measured by Agilent E5071C Network Analyzer and SATIMO Near-field Measurement System. Detailed results are as follows.

Fig. 5. Current distribution of four radiation modes. (a) t = 0. (b) t = T=4. (c) t = 0. (d) t = T=4. (e) t = 0. (f) t = T=4. (g) t = 0. (h) t = T=4.

provides a wideband impedance matching and a stable radiation pattern across the operating band. C. Two Orthogonal Broadside Patterns To generate a broadside pattern, a pair of oppositely located magnetoelectric dipoles should be excited. For each pair, the two L-shaped strips should be fed with 180 degree out of phase. Theoretically, this configuration could reduce the unwanted cross polarization from the four vertical strips. 180 degree hybrid rings are required for the design. Fig. 4(e)–(h) illustrate the current distribution with two orthogonal broadside modes at time and . It can be observed that, when , the

A. Electric Monopole Type Radiation (Conical Beam With Vertical Polarization) As shown in Fig. 6, the simulated impedance bandwidth of the electric monopole type radiation is about 39% ranging from 1.92 to 2.85 GHz. And the measured impedance ranging from 1.9 to bandwidth is about 38.6% 2.81 GHz. Good agreement between simulated and measured results can be observed. The figure also shows the measured and simulated gains of the electric monopole. Within the frequency range from 2.15 to 2.65 GHz, the simulated and measured gains are about 6 dBi with 0.1 dB variation. Fig. 7(a) and (b) shows the simulated and measured radiation patterns of the electric monopole at 2.4 GHz. At the elevation plane, they are stable and symmetric over the operating and band. The levels of cross polarization are less than dB, for simulation and measurement, respectively. The back lobe radiation by simulation and measurement are both less dB. The simulated and measured radiation patterns at than

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Fig. 8. SWR and gain versus frequency for magnetic monopole.

Fig. 7. (a) Simulated and measured radiation patterns (elevation plane) for electric monopole at 2.4 GHz. (b) Simulated and measured radiation patterns (azimuth plane) for electric monopole at 2.4 GHz.

the azimuth plane are almost omnidirectional with about 4 dB variation.

B. Magnetic Monopole Type Radiation (Conical Beam With Horizontal Polarization) As shown in Fig. 8, the simulated and measured impedance bandwidth of the magnetic monopole type radiation is about ranging from 2.12 to 2.66 GHz. Good 22.2% agreement between simulated and measured SWR can be observed. The figure also shows the measured and simulated gains of the equivalent magnetic monopole. Within the frequency range from 2.15 to 2.65 GHz, the measured gain is slightly higher than the simulated gain. The average measured gain is about 6 dB with 0.3 dB variation. Fig. 8–9 shows the simulated and measured radiation patterns of the magnetic monopole at 2.4 GHz. The simulated and measured radiation patterns at the elevation plane are stable and symmetric over the operating band. The levels of simudB. The lated and measured cross polarization are less than back lobe radiation by both simulation and measurement is less dB. The simulated and measured radiation patterns at than the azimuth plane are almost omnidirectional with about 5 dB variation.

Fig. 9. (a) Simulated and measured radiation patterns (elevation plane) for magnetic monopole at 2.4 GHz. (b) Simulated and measured radiation patterns (azimuth plane) for magnetic monopole at 2.4 GHz.

C. With

Polarized Patch Type Radiation (Broadside Pattern Linear Polarization)

Referring to the coordinate system shown in Figs. 1 and 2, polarized patch type radiation can be generated the by exciting the two magnetoelectric dipoles located along the X or Y axis with 180 out of phase. As shown in Fig. 10, the simulated impedance bandwidth is about 44.8% ranging from 1.89 to 2.98 GHz. And the measured impedance

WU AND LUK: A 4-PORT DIVERSITY ANTENNA WITH HIGH ISOLATION FOR MOBILE COMMUNICATIONS

Fig. 10. SWR and gain versus frequency for tion.

045

polarized broadside radia-

Fig. 11. Simulated and measured radiation patterns (elevation plane) for polarized broadside radiation at 2.4 GHz.

045

bandwidth is about 43.1% within the frequency range from 1.87 to 2.90 GHz. The figure also shows the measured and simulated gains of the antenna. Within the operating band, the simulated measured gain is slightly below the simulated gain. The antenna exhibits about 11 dBi. Fig. 11 shows the simulated and measured radiation patterns of the antenna over the operating band. The simulated and measured radiation patterns at the elevation plane are stable and symmetric over the operating band. The simulated and meadB. The low crosssured cross polarizations are less than polarization level of the antenna is mainly due to the use of antiphase feeding technique. The back lobe radiation by both simdB. ulation and measurement is less than D. With

Polarized Patch Type Radiation (Broadside Pattern Linear Polarization)

As shown in Fig. 12, the simulated impedance bandwidth polarized patch type radiation is about 39.8% for the ranging from 1.97 to 2.95 GHz. And the measured impedance bandwidth is about 39.2% within the frequency of

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Fig. 12. SWR and gain versus frequency for +45 polarized broadside radiation.

Fig. 13. Simulated and measured radiation patterns (elevation plane) for +45 polarized broadside radiation at 2.4 GHz.

1.95 and 2.90 GHz. The figure also shows the measured and simulated gains of the antenna. Within the operating band, the measured gain is slightly below the simulated gain. The antenna exhibits about 11 dBi in measurement. Fig. 13 shows the simulated and measured radiation patterns of the antenna over the operating band. The simulated and measured radiation patterns at the elevation plane are symmetric and stable over the operating band. The simulated and measured dB. The back lobe radiacross polarizations are less than dB. tion by both simulation and measurement is less than The simulated and measured SWR of the four modes is listed in Table II. The overlapped impedance bandwidth of the four modes by both simulation and measurement is about 22.2% ranging from 2.12 to 2.66 GHz. As shown in Fig. 14, the simulated and measured isolation between the two orthogonal broadand dB over the operating side modes are less than band, respectively. The simulated and measured isolation beand tween the two orthogonal conical modes are less than dB over the bandwidth, respectively. The simulated and measured isolation between a broadside radiation and a conical and dB within the overlapped radiation are less than

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Fig. 15. Measured radiating efficiency versus frequency.

the conical pattern with vertical polarization (electric monopole mode) is 87%–92% over the operating band, whereas the measured efficiency of the conical pattern with horizontal polarization (magnetic monopole mode) is 81%–89.5% within the operating band. The measured efficiencies of the degree polarized patch type radiation (broadside mode with degree polarized) are 82%–94% and 80%–94%, respectively. To summarize, the measured radiation efficiencies of the four modes are all over 80% within the operating band. V. CONCLUSION

Fig. 14. Simulated and measured isolations versus frequency.

TABLE II LIST OF SIMULATED AND MEASURED SWR WITH DIFFERENT MODES

A novel wideband four-port diversity antenna which is capable of exciting four different radiation patterns has been simulated and tested successfully. The antenna consists of four magnetoelectric dipoles arranged in a ring configuration above a ground plane and a vertical copper wire at the center of the ground plane. By optimizing the antenna parameters, the resonant frequency bands of the four radiations could be shifted to overlap each together. Thus the two orthogonal broadside modes and conical modes could be excited at an overlapped operating band. Comparing with most conventional wideband pattern diversity antennas available in [1]–[5], [8], [10], and [11], this antenna can greatly improve the system performance by providing a wide range of coverage in radiation with different polarizations. It also introduces additional degrees of freedom in beam steering or beam forming when implemented in array configurations. REFERENCES

bandwidth of the four modes, respectively. To summarize, the simulated and measured isolation of the antenna are generally and dB, respectively, within the frequency less than range from 2.12 to 2.66 GHz. The measured radiation efficiencies versus frequency are shown in Fig. 15. It can be seen that, the measured efficiency of

[1] E. A. Soliman, W. De. Raedt, and G. A. E. Vandenbosch, “Reconfigurable slot antenna for polarization diversity,” J. Electromagn. Waves Appl., vol. 23, no. 7, pp. 905–916, 2009. [2] N. Behdad and K. Sarabandi, “Dual-band reconfigurable antenna with a very wide tunability range,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 409–416, Feb. 2006. [3] F. Ferrero, C. Luxey, G. Jacquemod, R. Staraj, and V. Fusco, “Polarisation-reconfigurable patch antenna,” in Proc. Int. Workshop Antenna Technol.: Small and Smart Antennas Metamater. Appl. (IWAT’07), Mar. 21–23, 2007, pp. 73–76, vol., no.. [4] S. L. S. Yang, A. A. Kishk, K. F. Lee, K. M. Luk, and H. W. Lai, “The design of microstrip patch antenna with four polarizations,” in Proc. IEEE Radio and Wireless Symp. , Jan. 22-24, 2008, pp. 467–470.

WU AND LUK: A 4-PORT DIVERSITY ANTENNA WITH HIGH ISOLATION FOR MOBILE COMMUNICATIONS

[5] S. L. S. Yang and K. M. Luk, “Design of a wide-band L-probe patch antenna for pattern reconfiguration or diversity applications,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 433–438, Feb. 2006. [6] K. M. Luk and H. Wong, “A new wideband unidirectional antenna element,” Int. J. Microw. Opt. Technol., vol. 1, no. 1, pp. 35–44, Jun. 2006. [7] B. Q. Wu and K. M. Luk, “A broadband dual-polarized magnetoelectric dipole antenna with simple feeds,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 60–63, 2009. [8] H. Wong, K. L. Lau, and K. M. Luk, “Design of dual-polarized L-probe patch antenna arrays with high isolation,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 45–52, Jan. 2004. [9] S. B. Chakrabarty, F. Klefenz, and A. Dreher, “Dual polarized wideband stacked microstrip antenna with aperture coupling for SAR applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2000, vol. 4, pp. 2216–2219, no., vol. 4. [10] J.-S. Row, S.-H. Yeh, and K.-L. Wong, “A wide-band monopolar plate-patch antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1328–1330, Sep. 2002. [11] N. J. McEwan, R. A. Abd-Alhameed, E. M. Ibrahim, P. S. Excell, and J. G. Gardiner, “A new design of horizontally polarized and dual-polarized uniplanar conical beam antennas for hiperlan,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 229–237, Feb. 2003. [12] I. Shtrikman and N. Azzam, “Conical Beam Cross-Slot Antenna,” US Patent 7 064 725 B2. [13] B. Q. Wu and K. M. Luk, “A wideband, low-profile, conical-beam antenna with horizontal polarization for indoor wireless communications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 634–636, 2009. [14] ANSYS, Inc. [Online]. Available: http://www.ansoft.com/ Biqun Wu was born in Guangdong, China. He received the B.Eng. degree (first class honors) in electronic engineering from City University of Hong Kong, in 2007. Currently, he is working toward the Ph.D. degree. His research interest focuses on patch antenna and diversity antenna design. Mr. Wu was awarded the Second Prize in the IEEE Region 10 Student Paper Contest (Postgraduate Category) 2010 and the First Prize in 2009 IEEE Hong Kong Section (Postgraduate) Student Paper Contest.

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Kwai-Man Luk (M’79–SM’94–F’03) was born and educated in Hong Kong. He received the B.Sc. (Eng.) and Ph.D. degrees in electrical engineering from the University of Hong Kong (UHK), in 1981 and 1985, respectively. He joined the Department of Electronic Engineering, City University of Hong Kong, in 1985 as a Lecturer. Two years later, he moved to the Department of Electronic Engineering, the Chinese University of Hong Kong, where he spent four years. He returned to the City University of Hong Kong in 1992, and he is currently Chair Professor of Electronic Engineering and Director of State Key Laboratory in Millimeter waves (Hong Kong). His recent research interests include design of patch, planar and dielectric resonator antennas, and microwave measurements. He is the author of three books, nine research book chapters, more than 260 journal papers, and 200 conference papers. He was awarded 2 US and more than 10 PRC patents on the design of a wideband patch antenna with an L-shaped probe feed. Dr. Luk was the Technical Program Chairperson of the 1997 Progress in Electromagnetics Research Symposium (PIERS 1997), and the General ViceChairperson of the 1997 and 2008 Asia-Pacific Microwave Conference, and the General Chairman of the 2006 IEEE Region Ten Conference. He received the Japan Microwave Prize, at the 1994 Asia Pacific Microwave Conference held in Chiba in December 1994 and the Best Paper Award at the 2008 International Symposium on Antennas and Propagation held in Taipei in October 2008. He was awarded the very competitive 2000 Croucher Foundation Senior Research Fellow in Hong Kong. He is a Deputy Editor-In-Chief of JEMWA. He is a Fellow of the Chinese Institute of Electronics, PRC, a Fellow of the Institution of Engineering and Technology, U.K., and a Fellow of the Electromagnetics Academy, USA.

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Ground Plane Boosters as a Compact Antenna Technology for Wireless Handheld Devices Aurora Andújar, Student Member, IEEE, Jaume Anguera, Senior Member, IEEE, and Carles Puente, Member, IEEE

Abstract—The increasing demand for multifunctional wireless devices has fostered the need to reduce the space devoted to the antenna in order to favor the integration of multiple and new functionalities. This fact becomes a challenge for the handset antenna designers who have to develop antennas capable of providing multi-band operation constrained by physical limitations. This proposal consists in a radiating system capable of providing multi-band operation without the need of an antenna element, by properly exciting the efficient radiation modes associated to the ground plane structure. In this sense, the typical volume devoted to a handset antenna is reduced by a factor of 20. The electrical model approximation of the radiating structure leads to the radiofrequency system design able to provide multi-band operation. The feasibility of the proposal has been tested by electromagnetic simulations as well as by experimental measurements regarding the main antenna parameters: reflection coefficient, efficiency, and radiation patterns. Furthermore, the human head interaction concerning biological compatibility in terms of SAR (Specific absorption rate) has been measured and a solution for its reduction is presented. As a result, a promising standard solution for a radiating system capable to operate in the main communication standards GSM850, GSM900, DCS, PCS and UMTS is provided with a volume reduction factor around 20. Index Terms—Coupling element, electrical models, ground plane modes, handset antennas, low-volume antenna, multi-band, non-resonant antennas.

I. INTRODUCTION HE primary handset phones initially conceived with a limited number of functionalities have evolved to a novel concept of multifunctional wireless devices or smart phones, which must integrate a great number of services in a limited space in order to satisfy the user demands. Furthermore, new frequency bands appear for allocating new communication standards for which the antenna has to guarantee operability. In this sense, the antenna design becomes more and more a challenging effort

T

Manuscript received June 14, 2010; revised October 13, 2010; accepted October 16, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the ACC1Ó. A. Andújar is with the Technology and Intellectual Property Rights Department, FRACTUS S.A., 08174 Barcelona, Spain (e-mail: [email protected]). J. Anguera is with the Technology and Intellectual Property Rights Department, FRACTUS S.A., 08174 Barcelona, Spain and also with the Electronics and Telecommunications Department, Universitat Ramon Llull, Barcelona, Spain (e-mail: [email protected]). C. Puente is with the Technology and Intellectual Property Rights Department, FRACTUS S.A., 08174 Barcelona, Spain and also with the Signal Theory and Communications Department, Universitat Politècnica de Catalunya, Barcelona, Spain. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122299

since it has to provide multi-band operation while dealing with the constraints associated to physical limitations. On one hand, small size antennas are required for allowing the integration of other components in the phone platform, such as big displays enabling touch screens, cameras, batteries, speakers, etc. On the other hand, a low profile and low weight design is preferable in order to ensure slim multifunctional platforms. Nowadays, internal antennas such as patch/PIFAs (planar inverted F Antenna) and monopoles are the most common designs for handsets [1]. Nevertheless, they are constrained by the fundamental limits of small antennas that imply an inherently narrow bandwidth (BW) [2]. For PIFAs, several well-known techniques are used to provide dual-band or multi-band operation such as inserting slits in the radiating path or using slotted ground planes. This fact increases the complexity of the design and difficult the integration in slim platforms, since to guarantee good performance, the antenna has to be arranged at a certain height with respect to the ground plane occupying a considerable volume mm ). Handset monopole antennas are an alternative ( to provide multi-band operation in slim platforms mainly due to its characteristic low profile [3]–[5]. Until relatively recently, the efforts on the antenna design were mainly addressed to the antenna geometry and not to the ground plane, since its relevance in the radiation process was underestimated. Accordingly, the antenna element was typically a self-resonant element that provided an efficient radiation independently from the ground plane structure. Nevertheless, the ground plane is progressively acquiring relevance since several studies have demonstrated its strong contribution to the radiation properties [6]–[21]. In this way, the study presented in [6], proposes an equivalent circuit model that provides a quantitative view of the effect of the combination of a single-resonant antenna and chassis over the most significant antenna parameters. The theoretical effect of the coupling factor and the resonant frequencies is demonstrated through simulation regarding self-resonant antennas such as patch antennas and PIFA antennas. At the same time, [7] presents a folded radiating ground plane fed with a bowtie-shaped planar monopole specially selected to properly excite the desired ground plane modes. However, the folded ground plane can be understood as a PIFA antenna over a finite ground plane (100 mm 40 mm) with significant physical dimensions (49.5 mm 35 mm 10 mm), which are too large for practical purposes in modern handheld wireless devices. Again the radiation becomes a combination of the PIFA antenna and the ground plane modes. In the present study, the efficient radiation is entirely provided by the proper excitation of the ground plane modes since no antenna is regarded. In [8], resonant elements are used for simultaneous tuning of two different ground plane modes. On one hand, the reso-

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ANDÚJAR et al.: GROUND PLANE BOOSTERS AS A COMPACT ANTENNA TECHNOLOGY FOR WIRELESS HANDHELD DEVICES

nance of the first ground plane mode is adjusted by strategically loading the ground plane with a resonant screen acting as a quarter-wave slot resonator for the DCS and PCS bands. In this sense an electrical enlargement of the ground plane is achieved for the low frequency region (0.84–0.96 GHz). On the other hand, the resonant frequency at the high frequency region (1.71–2.17 GHz) is obtained by reducing the electrical length of the chassis for this frequency region. Similar proposals are found in [9]–[14] where a slot is used for tuning the resonant mode to lower frequencies by providing a longer current path, whereas in [15] the resonant mode is tuned to the high frequency region. At the same time, in [16] a distributed antenna system is proposed for enhancing the radiation efficiency of the ground plane mode while providing robustness to the performance degradation caused by the human interaction. Regarding [17], two antenna structures based on coupling elements designed to transfer energy to the ground plane mode are presented. They are intended for covering the communication standards GSM900 and GSM1800 separately by means of a single-resonant matching circuit based on distributed matching elements. Other reference based on coupling elements is given in [18] where an antenna structure consisting in two coupling elements and two resonant circuits is proposed. The proposal achieves a quad-band behavior. Nevertheless, the coupling elements presented for covering each frequency region (624 mm and 64 mm respectively), and specially that in charge of providing operability in the low frequency region, still presents a considerable volume compared with the 250 mm disclosed herein for providing penta-band operation. In [19], the pentaband behavior is achieved by means of two small antenna elements and two matching networks capable to provide multiband operation at each frequency region. Accordingly, in the present study the self-resonant antenna element is replaced by non-resonant ground plane boosters. In this sense, a challenge appears since the ground plane resonance is not coupled to the antenna resonance. Thus, the present research is focused on providing a multi-band wireless handheld device architecture based on the proper excitation of the ground plane without the need of an antenna element [20], [21]. This paper demonstrates that no handset antenna is required for effectively exciting the radiation modes of the ground plane. On the contrary, the novel architecture introduced here only requires small ground plane boosters featured by a high quality factor for the low frequency region and for ( the high frequency region) and extremely poor stand-alone radiation properties in combination with a matching network for providing simultaneous operability in the main communication standards (GSM850/900, DCS, PCS, and UMTS). The paper is structured in the following manner: firstly, the characteristic modes theory as a base of this study is briefly reviewed in Section II. Subsequently, the radiating system comprising both the radiating structure electrical model and the matching network design is presented in Section III. Secondly, the proposal is evaluated through simulations using the software IE3D based on MoM (Section IV). In a third stage, a prototype is built for the sake of validating the simulations with the experimental results (Section V). Finally the conclusions are disclosed (Section VI).

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Fig. 1. Eigenvalues (1) and modal significance versus frequency associated to the first and second predominant modes ( and  ) regarding a PCB with dimensions 100 mm 40 mm. Current distribution (A/m) at the frequency of f = 892 MHz provided by the first radiation mode J .

2

II. THEORETICAL BACKGROUND Characteristic modes theory [22]–[24] becomes a useful tool to understand the ground plane contribution and can be used to perform systematic analysis and design of handset antennas. They are described as the electrical current eigenfunctions linked to the boundary conditions established by an arbitrary shaped conducting body. Thus, they provide a physical insight into the radiation properties of a specific radiating structure and are only dependent on the shape and size of the conducting object. Accordingly, the behavior of the radiating structure can be described as a combination of the radiation and impedance characteristics of the wave modes associated to the antenna element and the phone chassis. In this sense, once the shape of the radiating structure is determined, the radiation modes can be computed and the optimum feeding configuration can be selected in order to correctly excite the desired radiation mode [25]. A predominant longitudinal mode is provided by a conventional ground plane layer with typical handset dimensions (100 mm 40 mm) in the operation range (Fig. 1), which is perfectly aligned with the results shown in [26]. From (Fig. 1) it is possible to determine that a mode is in resonance when its associated eigenvalue is zero. At the same time, the smaller the magnitude of the eigenvalue, the higher the contribution to the total surface current density. In addition, the sign of the eigenvalue determines whether the mode contributes to store magnetic energy or electric energy . It is important to notice that the modal significance of the eigenvalue predominates in the entire frequency range of interest. In the low frequency region, the ground plane is featured by a longitudinal radiation mode characterized by a low quality factor provided by its resonant dimensions at this frequency range (Fig. 1). If a self-resonant antenna element is used, the radiation becomes a combination of the wave mode associated the antenna element and said longitudinal radiation mode. Consequently, the resulting radiation strongly depends on the effi1Note that  in Section II is defined as the eigenvalue and must not be confused with the wavelength.

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Fig. 2. Geometry and dimensions associated to the radiating structure comprising two non-resonant ground plane boosters located at a 2 mm distance from the ground plane edge, each one in charge of the ground plane excitation for each frequency region. Current distribution referred to the excited ground plane mode at f = 0:892 GHz.

ciency of the radiation mode provided by the antenna element if the ground plane mode is not correctly excited. In this sense, the main idea of this proposal consists in maximizing the energy transfer to the efficient radiation mode of the ground plane by means of a non-resonant ground plane booster with a high around 2250 for the low frequency region and 265 for the high frequency region. This non-resonant element merely acts as a ground plane booster for both frequency regions. III. CONCEPT DEFINITION A. Electrical Model of the Radiating Structure The excitation of the radiation modes of the ground plane can be made either via its magnetic or electrical fields. The suitable location of the non-resonant ground plane booster is only dependent of the fields distribution associated to the radiation modes. The predominant mode of the proposed radiating structure presents a current distribution (Fig. 2) similar to that prothick dipole, having its maximum at the center duced by a of the ground plane not only for the low frequency region but also for the high frequency region. In this sense and in order to correctly excite the mode, an electric element should be located at the shorter edge of the ground plane where the maximum of the electric field distribution takes place. Accordingly, the radiating structure is designed following these considerations and high radiation efficiency around 80% is attained at both frequency regions. In this regard, the radiating structure comprises two non-resonant ground plane boosters featured by their reduced dimensions of just 5 mm 5 mm 5 mm, which entails a very low volume (125 mm ), and a ground plane (100 mm 40 mm) (Fig. 2). Said non-resonant ground plane boosters characterized by a high (2250 for the low frequency region and 265 for the high frequency region) are used to couple energy to the ground plane in order to correctly excite the predominant radiation modes of the ground plane [20], [21]. Thus, the ground plane acts as a main radiator taking profit of its high radiation efficiency for a wide range of frequencies (Fig. 1), concerning both the low and high frequency regions.

Fig. 3. ture.

Q

and related BW versus frequency referred to the radiating struc-

However, the proper excitation of the predominant mode is not enough for providing penta-band behavior and a matching network is required in order to guarantee operability in the aforementioned communication standards. In this sense, two non-resonant ground plane boosters are preferable in order to match each one to each one of the frequency regions of interest. Hereafter, the radiating structure and the matching network as a whole will constitute the radiating system. B. Matching Network Design The input impedance associated to the radiating structure presents a capacitive behavior regarding the whole frequency range (0.8–3 GHz). The of the structure (Fig. 3) and its incan be calculated from its input impedance herent BW according to (1) and (2) derived in [27]. for referred to the central frequency of The both operation regions is around 8.9% and 11.6% respectively (Fig. 3), which is not enough for covering the bandwidth associated to the GSM850/900 (15.2%), DCS/PCS/UMTS (23.7%) communication standards

(1) (2) of RLC cirA systematic method for broadening the cuits in a factor around one half of Fano’s limit [28] is proposed in [29] for parallel RLC circuits and in [30] for circuits featuring RLC series input impedances. Thus, before applying the method a previous step is required. A series inductor is used to compensate the capacitive reactance of the radiating structure. In this sense, the input impedance can be modeled as an RLC series circuit and the broadband mechanism developed in [30] can be applied.

ANDÚJAR et al.: GROUND PLANE BOOSTERS AS A COMPACT ANTENNA TECHNOLOGY FOR WIRELESS HANDHELD DEVICES

time, the value of into (3)

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can be easily obtained substituting (8)

(8) (9) Where and respectively

are computed according to (10) and (11),

(10)

(11) Theoretically, the obtained with the addition of the proposed broadband matching network can be defined according to (12) as Fig. 4. (a) Equivalent circuit regarding the input impedance referred to the radiating structure, the reactance cancellation (series inductor (L )) stage and the broadband matching network. (b) Condition required for achieving a BW enhancement around one half of Fano’s limit.

Accordingly and in order to achieve the enhanced bandwidth , a parallel capacitor and inductor is used as a broadband matching network. The proper values of these reactances are readily obtained through an accurate mathematical analysis developed using the associated electrical model (Fig. 4). The suitable values, that , are those that satisfy the condition provide the expected represented graphically in Fig. 4 and defined mathematically in (5), (6), and (7). The imaginary and real part of the input admittance (Fig. 4) is (3) (4) It is important to note that if the imaginary part of the input admittance is equated to 0, three resonant frequencies appear , and ). In order to maximize the BW, the input ( impedance locus has to be forced to fulfill the following and values requirements that will condition the (Fig. 4) (5) (6)

(12) is calculated as the ratio between The enhancement factor of the structure and the potential that the inherent can be achieved with the addition of the two stages matching network. In this sense, is defined as (13) The enhancement factor given by (13) presents the same shape as that given by the Fano’s limit for an ideal network. obtained in this case is not as high since However, the lesser number of components is used. Equation (13) allows that can be achieved. In this determining the theoretical equal to 8.9% sense and as given by Fig. 3, the inherent and 11.6% regarding low and high frequency region respec(13), tively, can be enhanced in a factor 2.45 for for each frequency region, which is enough for covering the communication standards GSM850/900/DCS/PCS and UMTS. As aforementioned, two non-resonant ground plane boosters are proposed in order to provide penta-band behavior. In this sense, each booster is matched separately since one is used to provide operability in the low frequency region while the other is in charge of the high frequency region. Thus, the value of the series inductor is adjusted for tuning the resonance of the radiating structure at a frequency belonging on one hand to the low frequency region (Fig. 5(a)) and on the other hand to the high frequency region (Fig. 5(b)), according to [20], [21]. The addition of this reactance cancellation inductor tunes the resonant frequency whereas the broadband matching network allows tuning the impedance locus at the centre of the Smith (Fig. 7). chart (Fig. 6) providing the expected

(7) IV. MULTI-BAND DESIGN: SIMULATED RESULTS The first solution to (3) gives in a straightforward manner and (8). At the same the relationship required between

Previous section demonstrates the feasibility of providing operability in both frequency regions separately. This section goes

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Fig. 7. Reflection coefficient associated to the radiating structure without any components (dashed line). Reflection coefficient associated to the schematic shown in (Fig. 5(a)) designed for covering the low frequency region (solid line with square markers). Reflection coefficient associated to the schematic shown in (Fig. 5(b)) designed for covering the high frequency region (solid line with triangular markers).

Fig. 5. (a) Reactance cancellation and broadband matching network required for the low frequency region. (b) Reactance cancellation and broadband matching network required for the high frequency region. The one-port box corresponds to the simulated impedance of the radiating structure.

Fig. 6. (a) Impedance associated to the radiating structure (triangular markers). Impedance after the addition of the series inductor as a reactance cancellation element for the low frequency region (square markers). Impedance according with the schematic shown in (Fig. 5(a)) (rhombus marker). (b) The same sequence but regarding the high frequency region (Fig. 5(b)).

beyond and proposes a matching architecture suitable to attain penta-band behavior simultaneously using a single input/output port [21]. Besides the reactance cancellation element and the broadband matching network, the radiating system also comprises a notch filter for the low frequency region as well as for the high frequency region. The integration of the notch filters provides high isolation between both frequency regions, which is preferable in order to avoid coupling effects that would difficult the matching network process (Fig. 8). The isolation achieved allows applying the matching network principle proposed in Section III.B. This systematic network design provides a standard solution for matching all those

Fig. 8. (a) Radiating system designed for achieving a penta-band behavior consisting in a reactance cancellation element, a broadband matching network, and a notch filter for each frequency region. Note that the first stage of the notch filter and the broadband matching network can be simplified using only two components. The two-port box is the simulated input impedance of the radiating structure shown in Fig. 2. (b) Detailed view of the radiating structure obtained from the simulated layout comprising two non-resonant ground plane boosters with dimensions 5 mm 5 mm 5 mm, and a ground plane with dimension 100 mm 40 mm. The ground plane boosters are located at a 2 mm distance from the edge of the ground plane.

2

2

2

non-resonant ground plane boosters featuring capacitive input impedance.

ANDÚJAR et al.: GROUND PLANE BOOSTERS AS A COMPACT ANTENNA TECHNOLOGY FOR WIRELESS HANDHELD DEVICES

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Fig. 9. Reflection coefficient associated to the schematic shown in Fig. 5(a) (dashed line with square markers) and in Fig. 5(b) (dashed line with triangular markers) for providing operability in both frequency regions separately. Reflection coefficient to provide penta-band operation simultaneously achieved with the radiating system proposed in Fig. 8 (solid line).

Fig. 11. Reflection coefficient to provide penta-band operation simultaneously achieved with the radiating system proposed in Fig. 8 (solid line). Reflection coefficient associated to the same radiating structure as shown in Fig. 8(b) but with the difference that the size of the ground plane boosters (previously 5 mm 5 mm 5 mm) has been reduced to only 3 mm 3 mm 3 mm (dashed line). The topology of the radiofrequency system remains equal as disclosed in Fig. 8(a), whereas the values of the reactive components have been adjusted.

Fig. 10. Radiation efficiency ( ), antenna efficiency ( ) and current distribution associated to the radiating structure at the central frequencies of both frequency regions (f = 0:89 and f = 1:94 GHz). The antenna efficiency takes into account the mismatch losses since it is defined as  =  (1 S ).

Fig. 12. Single-band prototype including the reactance cancellation inductor and the broadband matching network.

0j j

The topology remains equal and just the component values are modified according to the frequency of operation. Thus, the simulated analysis demonstrates the feasibility of the proposal since the designed radiating system provides operability in the main communication standards such as GSM850/900, DCS, PCS, and UMTS (Figs. 9 and 10). It is important to remark the relevance of the ground plane boosters since their nature, location, and size clearly conditions the correct excitation of the ground plane mode and consequently, the performance of the radiating system. In this sense, if the size of the ground plane boosters is reduced from 5 mm 5 mm 5 mm to 3 mm 3 mm 3 mm, the radiating system limits the frequency range of operation from a penta-band radiating system to a tri-band radiating system (GSM900/PCS/UMTS) (Fig. 11), still becoming encouraging results taking into account the reduced volume of the ground plane boosters used (54 mm ). V. MULTI-BAND DESIGN: EXPERIMENTAL RESULTS The previous simulated results are validated through an experimental procedure. Accordingly, several prototypes have

2

2

2

2

been built and tested for the sake of demonstrating the effectiveness of the proposal. On one hand, a ground plane (100 mm 40 mm) is etched over a 1 mm thickness FR4 piece ( and tan ) and several pads are arranged in the upper side of the long edge in order to facilitate the integration of the matching network components. On the other hand, a non-resonant ground plane booster featured by its low physical dimensions (solid cube of 5 mm 5 mm 5 mm made of brass) is soldered at a 2 mm distance from the edge of the ground plane layer (Fig. 12). The radiating system consisting in a single element provides operability in a specific frequency region and consequently no filter is required. However, the challenge of the proposal lies in achieving a penta-band behavior and for this reason a prototype integrating two non-resonant ground plane boosters featuring the same characteristics previously described have been manufactured (Fig. 13). The radiofrequency system comprises the reactance cancellation elements, the broadband matching networks, the filtering stages and a transmission line that interconnects both non-resonant ground plane boosters in order to provide a single port system. In this configuration (Fig. 13) the radiofrequency

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Fig. 15. Radiation efficiency ( ) and antenna efficiency ( ) related to the penta-band prototype (Fig. 13). The antenna efficiency takes into account the mismatch losses since  =  (1 S ).

0j j

Fig. 13. Penta-band prototype designed according to the schematic shown in Fig. 8 including besides the reactance cancellation inductor and the broadband matching network, the notch filters required for providing isolation between both frequency regions.

Fig. 16. Set-up for radiation measurement in the Fractus’ anechoic chamber Satimo Stargate-32 showing the coordinate system.

Fig. 14. Reflection coefficient related to the penta-band prototype (Fig. 13).

system occupy a certain space in the PCB. However, such integration can be further improved to minimize it. With this aim, the soldering pads can be arranged parallel to the shorter edge of the PCB diminishing not only the required PCB space but also the transmission line length. Furthermore, it is important to remark that lumped components can be arranged in a reduced space according to current soldering techniques. In order to completely characterize the radiating performance of the proposed prototype the main antenna parameters have been measured and they are gathered in the following subsections. A. Reflection Coefficient and Efficiency A network analyzer has been used for measuring the reflection coefficient associated to the penta-band prototype shown in Fig. 13. As expected, the proposed radiating system is able to provide operability in the main communication standards (GSM850/900/DCS/PCS/UMTS) since the reflection coefficient remains below dB for both operating regions (Fig. 14). As seen, the measured results (Fig. 14) are in good agreement with the simulations (Fig. 9).

The antenna efficiency has been measured by integration of the 3D radiation pattern by means of the anechoic chamber Satimo Stargate-32 (Fig. 15). Regarding the high frequency region, the reduction in the antenna efficiency with respect to the simulated case is negligible since it remains around the 70% for all the frequency range. However, in the low frequency region the difference between the simulated results and the measured ones becomes significant (the simulated values around 70% drop to approximately 40% in the measured case). This reduction is mainly due to the fact that in the simulated case the matching network components are considered to be lossless. However, in practice they present a value which is smaller for the low frequency region than for the high frequency region. In this sense, their effect becomes more significant in the low frequency region. Nevertheless, this antenna efficiency value is still acceptable for mobile communications [31]–[34]. B. Radiation Patterns As aforementioned, the radiation patterns have been also measured using the anechoic chamber Satimo Stargate-32 located in the Fractus lab (Fig. 16). The main cuts normalized to the maximum gain ( and ) are obtained for the frequencies ( MHz, MHz, MHz). They show an omnidirectional behavior for both frequency regions (Fig. 17), which is desirable for a handset antenna. The radiation is associated to that produced by a conventional dipole antenna having a null in the y-axis, especially in the low frequency region. The gain is computed regarding the antenna efficiency. It means that, the losses introduced by the matching network as well as the mismatching losses are considered.

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Fig. 18. SAR measurements for the low frequency region at the specific frequencies of f = 835 MHz and f = 900 MHz regarding both positions: left (antenna up) and right (antenna down). Right cheek position is tested.

Fig. 17. Main cuts (Phi = 0 and Phi = 90 ) of the radiation pattern provided by the penta-band prototype (Fig. 13) measured at the frequencies of f = 900 MHz, f = 1800 MHz, and f = 2000 MHz.

C. SAR Measurement Once the feasibility of the proposal is demonstrated, the biological compatibility of the prototype in terms of SAR is analyzed. The SAR is a measure of the localized maximum value of the power absorbed by the human head. It is defined as the absorbed RF energy by unity of volume, and its dimensions are mW/g. Due to the fact that this absorption is produced in the near field, SAR can be computed from the electric near field ac, and are the human tissue effective cording to (14), where conductivity and the tissue volumetric density, respectively (14) In this sense the SAR values associated to the prototype under study (Fig. 13) have been measured using the DASY4 equipment located at Fractus Lab. The prototype is arranged in the right cheek of the phantom head and the ground plane is spaced apart 1 mm from it thanks to the use of a methacrylate piece. For this location two different positions have been evaluated. On one hand, the non-resonant ground plane boosters are placed near the right ear of the phantom head (antenna up position) and

Fig. 19. SAR measurements for the high frequency region at the specific frequencies of f = 1700 MHz, f = 1800 MHz, f = 1900 MHz and f = 2000 MHz regarding both positions: left (antenna up) and right (antenna down).

on the other hand, the prototype is rotated 180 (antenna down position). Two main conclusions can be extracted from previous results (Figs. 18 and 19). On one hand, SAR values are strongly dependent not only on the geometry and the distribution of the radiation mode excited in the ground plane but also on the way that other handset components are connected to the PCB. For instance, a plastic back-cover may absorb some radiated power decreasing as a consequence the SAR value. On the other hand, as the excited mode presents a localized maximum field value placed in the vicinity of the shorter ground plane edge at a certain distance from its center, the rotation reduces considerably the SAR values for both frequency regions but in a significant way for the high frequency region. In this sense, at high frequencies the hot spot is located near the non-resonant ground plane boosters. This fact produces higher SAR values in the antenna up position mainly due to the proximity between such maximum field value and the human head. Accordingly, the antenna down position considerable diminishes these SAR values since in this context the hot spot is located at a larger distance from the human head. In the case of the

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low frequency region the SAR values regarding both positions are located below or close to the standards (American standard (ANSI/IEEE): 1.6 mW/g (1 g) and European standard (ICNIRP) 2 mW/g (10 g)). However, for the high frequency region the antenna down position is preferred. VI. CONCLUSION An ultra-compact radiating system capable of providing operability in the main communication standards (GSM850/900/DCS/PCS and UMTS) has been presented. The conventional handset antenna featured by a considerable mm ) has been replaced by two low-volume volume ( non-resonant ground plane boosters (250 mm ) and a matching topology with a systematic design. These elements are in charge of properly exciting the efficient radiation mode of the ground plane, which presents high radiation efficiency and low at the frequencies of interest, especially in the low frequency region. The systematic matching network design enables the operability in the desired frequency regions. The radiation contribution provided by such a small boosters is negligible and they should not be considered antennas. Consequently their integration in the handset platform removes the need of including a dedicated antenna in the wireless handheld device [20], [21]. This proposal becomes an alternative to the traditional antenna technology and appears as a promising standard solution for being integrated in the new multifunctional wireless devices. In this regard, the available space in handset platforms for integrating new functionalities is further increased while the radiating performance is preserved. REFERENCES [1] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [2] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. AP-44, pp. 672–676, May 1996. [3] C. Puente, J. Anguera, J. Soler, and A. Condes, “Coupled Multiband Antennas,” Patent App. WO2004/025778, Sep. 10, 2002. [4] C. Lin and K. L. Wong, “Printed monopole slot antenna for internal multiband mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3690–3697, Dec. 2007. [5] S. Risco, J. Anguera, A. Andújar, A. Pérez, and C. Puente, “Coupled monopole antenna design for multiband handset devices,” Microw. Opt. Technol. Lett., vol. 52, no. 2, pp. 359–364, Feb. 2010. [6] P. Vainikainen, J. Ollikainen, O. Kivekäs, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1433–1444, Oct. 2002. [7] M. Cabedo-Fabrés, E. Antonino-Daviu, A. Valero-Nogueira, and M. F. Bataller, “The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 52–68, Oct. 2007. [8] W. L. Schroeder, C. T. Famdie, and K. Solbach, “Utilization and tuning of the chassis modes of a handheld terminal for the design of multiband radiation characteristics,” IEEE Wideband Multi-Band Antennas Arrays, pp. 117–121, Sept. 2005. [9] J. Anguera, I. Sanz, A. Sanz, A. Condes, D. Gala, C. Puente, and J. Soler, “Enhancing the performance of handset antennas by means of groundplane design,” presented at the IEEE Int. Workshop on Antenna Technology: Small Antennas and Novel Metamaterials (iWAT 2006), New York, Mar. 2006.

[10] J. Anguera, A. Cabedo, C. Picher, I. Sanz, M. Ribó, and C. Puente, “Multiband handset antennas by means of groundplane modification,” presented at the IEEE Antennas and Propagation Society Int. Symp., Honolulu, HI, June 2007. [11] C. Picher, J. Anguera, A. Cabedo, C. Puente, and S. Kahng, “Multiband handset antenna using slots on the ground plane: Considerations to facilitate the integration of the feeding transmission line,” Progr. Electromagn. Res. C, vol. 7, pp. 95–109, 2009. [12] A. Cabedo, J. Anguera, C. Picher, M. Ribó, and C. Puente, “Multi-band handset antenna combining a PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2526–2533, Sep. 2009. [13] R. Quintero and C. Puente, “Multilevel and Space-Filling Ground Planes for Miniature and Multiband Antennas,” Patent Appl. WO2003/023900, Sep. 13, 2001. [14] J. Anguera and C. Puente, “Shaped Ground Plane for Radio Apparatus,” Patent Appl. WO2006/070017, Dec. 29, 2005. [15] C. Puente and J. Anguera, “Handset With Electromagnetic Bra,” Patent Appl. WO2005/083833, Feb. 28, 2005. [16] J. Anguera, A. Camps, A. Andújar, and C. Puente, “Enhancing the robustness of handset antennas to finger loading effects,” Electron. Lett., vol. 45, no. 15, pp. 770–771, July 2009. [17] J. Villanen, J. Ollikainen, O. Kivekäs, and P. Vainikainen, “Coupling element based mobile terminal antenna structures,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2142–2153, July 2006. [18] S. Ozden, B. K. Nielsen, C. H. Jorgensen, J. Villanen, C. Icheln, and P. Vainikainen, “Quad-Band Coupling Element Antenna Structure,” U.S. Patent 7 274 340, Sep. 25, 2007. [19] J. Anguera, I. Sanz, C. Puente, and J. Mumbru, “Wireless Device Including a Multiband Antenna System,” Patent Appl. WO2008/119699, Mar. 26, 2008. [20] J. Anguera, A. Andújar, C. Puente, and J. Mumbru, “Antennaless Wireless Device,” Patent Appl. WO2010/015365, Jul. 31, 2009. [21] J. Anguera, A. Andújar, C. Puente, and J. Mumbru, “Antennaless Wireless Device Capable of Operation in Multiple Frequency Regions,” Patent Appl. WO2010/015364, July 31, 2009. [22] R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated and scattered fields,” IEEE Trans. Antennas Propag., vol. AP-19, pp. 348–358, May 1971. [23] R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-19, no. 5, pp. 622–628, Sep. 1971. [24] R. F. Harrington and J. R. Mautz, “Computation of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-19, no. 5, pp. 629–639, Sep. 1971. [25] E. H. Newman, “Small antenna location synthesis using characteristic modes,” IEEE Trans. Antennas Propag., vol. AP-27, no. 4, pp. 530–531, Jul. 1979. [26] C. T. Famdie, W. L. Schroeder, and L. Solbach, “Numerical analysis of characteristic modes on the chassis of mobile phones,” presented at the 1st Eur. Conf. on Antennas and Propagation—EuCAP 2006, Nice, France, 2006. [27] S. R. Best, “The inverse relationship between quality factor and bandwidth in multiple resonant antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2006, pp. 623–626. [28] R. C. Hansen, “Fano limits on matching bandwidth,” IEEE Antennas Propag. Mag., vol. 47, no. 3, pp. 89–90, Jun. 2005. [29] J. Anguera, C. Puente, C. Borja, G. Font, and J. Soler, “A systematic method to design single-patch broadband microstrip patch antennas,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 185–188, Nov. 2001. [30] A. Andújar, J. Anguera, and C. Puente, “A systematic method to design broadband matching networks,” presented at the Eur. Conf. on Antennas and Propagation—EuCAP 2010, Barcelona, Spain, 2010. [31] M. Martínez, O. Letschke, M. Geissler, D. Heberling, A. M. Martínez, and D. Sánchez, “Integrated planar multiband antennas for personal communication handsets,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 384–391, Feb. 2006. [32] B. Kim, S. Park, Y. Yoon, J. Oh, K. Lee, and G. Koo, “Hexaband planar inverted-F antenna with novel feed structure for wireless terminals,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 66–69, 2007. [33] H. Hsieh, Y. Lee, K. Tiong, and J. Sun, “Design of a multiband antenna for mobile handset operations,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 200–203, 2009. [34] Z. Li and Y. Rahmat-Samii, “Optimization of PIFA-IFA combination in handset antenna designs,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1770–1778, May 2005.

ANDÚJAR et al.: GROUND PLANE BOOSTERS AS A COMPACT ANTENNA TECHNOLOGY FOR WIRELESS HANDHELD DEVICES

Aurora Andújar (S’11) was born in Barcelona, Spain, in 1984. She received the Bachelor degree in telecommunication engineering (specializing in telecommunication systems), in 2005, and the Master degree in telecommunications engineering and the Master of Science in telecommunication engineering and management, both in 2007, from the Polytechnic University of Catalonia (UPC), Barcelona, Spain. In 2005, she worked as a Software Test Engineer for applications intended for handset wireless devices. In 2006, she worked as an SW Engineer designing a load simulation tool for testing digital campus in academic environments and developing improvements in the performance of web servers referred to the management of static and dynamics contents. Since 2007, she is working as an R&D Engineer at Fractus, Barcelona, Spain where she has contributed to the maintenance and growth of the patent portfolio of the company. She is also involved in several projects in the field of small and multiband handset antenna design. Since 2009, she is leading research projects in the antenna field for handheld wireless devices in the collaborative university-industry framework. She is doing her Ph.D. in the field of small and multiband antennas for handset and wireless devices. She has published more than 18 journals, international and national conference papers. She has authored five invention patents in the antenna field. Mrs. Andújar is member of the COIT (Colegio Oficial de Ingenieros de Telecomunicación) and AEIT (Asociación Española de Ingenieros de Telecomunicación). She was the recipient of a research fellowship in the field of electromagnetic compatibility from the Signal Theory and Communications Department, UPC, 2004–2005.

Jaume Anguera (S’99–M’03–SM’09) was born in Vinaròs, Spain, in 1972. He received the Technical Engineering degree in electronic systems and the Engineering degree in electronic engineering, both from the Ramon Llull University (URL), Barcelona, Spain, in 1994 and 1998, respectively, and the Engineering and Ph.D. degrees in telecommunication engineering, both from the Polytechnic University of Catalonia (UPC), Barcelona, in 1998 and 2003, respectively. From 1998 to 2000, he was with the Electromagnetic and Photonic Engineering Group, Signal Theory and Communications Department, UPC, as a Researcher in microstrip fractal-shaped antennas. In 1999, he was a Researcher at Sistemas Radiantes, Madrid, Spain, where he was involved in the design of a dual-frequency dual-polarized fractal-shaped microstrip patch array for mobile communications. In the same year, he became an Assistant Professor at the Department of Electronics and Telecommunications, Universitat Ramon Llull-Barcelona, where he is currently teaching antenna theory. Since 1999, he has been with Fractus, Barcelona, Spain, where he holds the position of R&D Manager. At Fractus he leaded projects on antennas for base station systems, antennas for automotion, and currently managing handset and wireless antennas. His current research interests are

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multiband and small antennas, broadband matching networks, diversity antenna systems, electromagnetic dosimetry, and handset antennas. From September 2003 to May 2004, he was with Fractus-Korea (Republic South of Korea) were he was managing projects for miniature and multiband antennas for handset and wireless applications. Since 2005, he has been leading research projects in the antenna field for handset and wireless applications in a frame of Industry-University collaboration: Fractus company and the Department of Electronics and Telecommunications of Universitat Ramon Llull-Barcelona, Spain. He holds more than 30 invention patents in the antenna field. He is the author of more than 120 journal, international, and national conference papers and he has directed more than 50 bachelor and master thesis. Dr. Anguera was a member of the fractal team that received the 1998 European Information Technology Grand Prize from the European Council for the Applied Science an Engineering and the European Commission for the fractalshaped antenna application to cellular telephony. He was the 2003 Finalist to the Best Doctoral Thesis Fractal and Broadband Techniques on Miniature, Multifrequency, and High-Directivity Microstrip Patch Antennas on UMTS. The prize has been promoted by “Technology plan of UMTS promotion” given by Telefónica Móviles España. He was named a 2004 New faces of Engineering by the IEEE and IEEE foundation. In the same year he received the Best Doctoral Thesis (Ph.D.) in “Network and BroadBand Services” (XXIV Prize Edition “Ingenieros de Telecomunicación”) organized by Colegio Oficial de Ingenieros de Telecomunicación (COIT) and the Company ONO. He is reviewer for the IEEE TRANSACTION AND ANTENNAS AND PROPAGATION, IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, Progress in Electromagnetic Research (PIER), IEE Electronics Letters, and ETRI journal (Electronics and Telecommunications Research Institute, South Korea). His biography is listed in Who’s Who in the World, Who’s Who in Science and Engineering, Who’s Who in Emerging Leaders and in IBC (International Biographical Center, Cambridge-England).

Carles Puente (M’97) received the M.Sc. degree from the University of Illinois at Urbana-Champaign, in 1994 and the Ph.D. degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1997. He is a co-founder of Fractus and leads its antenna technology research team, with responsibility for the company’s intellectual property portfolio development and antenna development. He is a Professor at UPC where he started researching fractal-shaped antennas while a student in the late 1980s. From 1994 to 1999, he worked with the faculty of Electromagnetic and Photonic Engineering, UPC, on pioneering developments of fractal technology applied to antennas and microwave devices. He has authored more than 50 invention patents and over 90 scientific publications in fractal and related antenna technologies. Dr. Puente was awarded with the Best Doctoral Thesis in Mobile Communications by the COIT in 1997, the European Information Society Technology Grand Prize from the European Commission in 1998, and the Premi Ciutat de Barcelona in 1999. He and his team at Fractus were awarded with Technology Pioneer distinction by the World Economic Forum, in 2005.

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Measured Comparison of Dual-Branch Signaling Over Space and Polarization Diversity Ali Morshedi, Member, IEEE, and Murat Torlak, Senior Member, IEEE

Abstract—Polarization diversity is an efficient alternative to space diversity as it cuts the antenna spacing requirements in half, but comparable results to spatially separated antennas need to be realized for practical implementation to take place. A dual feed square patch antenna has been chosen to demonstrate a complete system evaluation from the antenna design to symbol error rate (SER) performance of polarization diversity. The patch antenna has been designed with vertical and horizontal polarizations, which are well isolated at 2.4 GHz. The beam patterns were measured in the anechoic chamber on The University of Texas at Dallas (UTD) campus. The nature of the beam patterns greatly influenced the results, which have been reported in line-of-sight (LOS) and nonline-of-sight (NLOS) indoor environments in the Erik Jonsson Engineering and Computer Science North (ECSN) building at UTD. Performance evaluation has been developed for 1 2 1, 2 2 1, and 2 2 2 systems based on channel gain, channel correlation, and SER. The dual feed patch antenna performs more consistently as the environment changes from LOS to NLOS due to the low channel correlation and the complementary nature of the vertical and horizontal beam patterns. Index Terms—Microstrip Antennas, multiple-input-multiple-output (MIMO), polarization diversity.

I. INTRODUCTION

M

ULTIPLE-INPUT-multiple-output (MIMO) technology has been developed to provide better coverage and improved performance over multipath wireless channels by using multiple antennas at the transmitter and receiver. The main problem facing the implementation of MIMO technology is the limited space available at each end of the communication link. Polarization diversity has the ability to solve this problem by replacing two spatially separated linearly polarized antennas with one single dual polarized antenna. Each polarization shares the same physical antenna, but act as two separate transmitting or receiving elements. Saving this extra space may cause performance degradation; therefore, polarization diversity needs to be investigated in true wireless channels. Simulations may provide promising results, but experimental Manuscript received March 09, 2009; revised October 19, 2009; accepted October 07, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The work of M. Torlak was supported in part by Semiconductor Research Corporation. This paper was presented in part at the IEEE SAM – The 5th IEEE Sensor Array and Multichannel Signal Processing Workshop 2008,Darmstadt, Germany. A. Morshedi was with the University of Texas at Dallas, Richardson, TX 75080 USA. He is now with Qualcomm Inc., San Diego, CA 92037 USA (e-mail: [email protected]). M. Torlak is with the University of Texas at Dallas, Richardson, TX 75080 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122210

verification is still required. It is difficult to accurately model the channel coefficients and various impairments such as cross talk between the antennas, multipath, signal fading, noise, and phase noise. The authors in [1] assume a different antenna configuration / degrees in their channel model. They have chosen for their antenna polarizations, which creates simple simulation conditions. In this paper we offer a more practical antenna configuration, 90 and 0 degrees, which we call vertical and horizontal polarizations, respectively. The details of the design and naming convention are shown clearly in Fig. 1 and they are discussed in Section II. Additionally the authors in [1] switch between transmit diversity and spatial multiplexing based on channel conditions developed in their simulations while we have a frame structure that sends subframes of both Alamouti packets and spatial multiplexing packets in a time interleaved fashion. The frame structure is discussed in Section V. We decode each scheme individually and compare the channel conditions, channel correlation and symbol error rate (SER) in Section VI. Experiments were conducted in an indoor office environment as in [2] and in [3] in line-of-sight (LOS) and nonline-of-sight (NLOS). It has been shown in [6] that the power imbalance between antenna ports may cause performance degradation in the channel gain while the authors in [7] take a close look at channel correlation in outdoor environments for a 1 2 receive diversity system. To the best of the author’s knowledge, this is the first experimental work which investigates the antenna beam patterns, SER, channel gain and channel correlation in a complete system analysis from the design of the antenna to the measured results. This paper has been devoted to understanding how polarization diversity impacts the experimental results compared to the traditional space diversity. The beam patterns are a key to understanding the performance of each patch antenna polarization, and have been measured in the UTD anechoic chamber. The multiple antenna testbed provides a platform to test the antenna performance from a signal processing standpoint. The results are included for 1 1 single antenna transmission, 2 1 and 2 2 Alamouti space-time block codes (STBC), and 2 2 spatial multiplexing. By measuring and analyzing the antenna patterns, the channel coefficients and the SER, a strong case is made for utilizing polarization diversity in indoor environments at 2.4 GHz. The rest of the paper is organized as follows: The design and measurement of the dual feed microstrip patch antenna is presented in Section II. The system model used throughout the paper is introduced in Section III. The development of the experimental testbed is explained in Section IV. The frame structure

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MORSHEDI AND TORLAK: MEASURED COMPARISON OF DUAL-BRANCH SIGNALING OVER SPACE

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TABLE I ANTENNA PARAMETERS

Fig. 1. This figure illustrates the dual feed patch antenna. Notice that the same physical antenna is fed from two orthogonal ports, labeled vertical and horizontal. This naming convention is shown with an axis and text box to show what is meant by vertically polarized and horizontally polarized as we discuss the results throughout the paper. This dual feed structure cuts the antenna space in half. The patch antenna was designed to minimize leakage from the vertical port to the horizontal port at 2.4 GHz. Polarized transmission is denoted [V H ] – [V H ] for 2 2 multiple antenna schemes. Spatially separated vertically polarized transmission is denoted [V V ] – [V V ] while only using the vertically polarized ports from the two separate patch antennas. Similarly, spatially separated horizontally polarized transmission is denoted [HH ] – [HH ] while only using the horizontally polarized ports from the two separate patch antennas.

2

and data collection protocol are presented in Section V, while the experimental results are shown in Section VI. Finally, the paper is concluded in Section VII. II. ANTENNA DESIGN AND BEAM PATTERN MEASUREMENTS A dual feed square patch antenna array was designed to be used as the transmitter and receiver. Each patch antenna has and horizontal the capability of transmitting on vertical polarizations independently or simultaneously. The antenna is shown in Fig. 1 with the vertical and horizontal axes labeled. This dual feed patch antenna will be compared to spatially separated antennas, which only use one of the feed lines for each patch antenna. The vertically polarized spatially separated patch antennas only transmit on vertical polarization while the horizontal ports are terminated in 50 Ohms. The horizontally polarized spatially separated patch antennas only transmit on horizontal polarization while the vertical ports are terminated in 50 Ohms. Notice the dual feed antenna only uses one patch antenna and transmits from orthogonal ports. This cuts the antenna size in half when comparing to the spatially separated antennas. A. Antenna Design A dual feed microstrip patch antenna was designed for this study. The goal was to incorporate this antenna into the wireless testbed for performance analysis. The design of the dual feed patch antenna follows the procedure in [8]. The resulting patch antenna parameters are given in Table I and are summarized in the Appendix. The design of the patch antenna was verified using the Advanced Design System (ADS) Method of Moments simulator. The ADS simulation is compared to actual measurements of the taken from the network anforward transmission coefficient from the vertical port to the horialyzer. Fig. 2 illustrates zontal port. A signal from the network analyzer excites the vertical port of the patch antenna. The goal is to have as much power

Fig. 2. Forward transmission coefficient from the vertical port to the horizontal port of the patch antenna.

as possible radiated from the patch without leaking into the horizontal port of the same patch antenna structure. The energy that is not radiated from the patch antenna will leak into the horizontal port. The patch antenna has been designed to have the lowest possible leakage from the vertical port to the horizontal at the frequency of 2.4 GHz. The same result would occur when transmitting from the horizontal port. There would be minimal leakage into the vertical port with most of the energy being radiated from the patch. B. Antenna Measurements in Anechoic Chamber This paper has explored the radiation patterns from each patch antenna to understand the different multipath characteristics resulting from each polarization. UTD has a newly constructed anechoic chamber with a planar scanner manufactured by Nearfield Systems Incorporated (NSI) which was used to measure the patch antenna patterns. The planar scanner is equipped with a waveguide which operates from 2.2 GHz to 3.3 GHz in the nearfield of the patch antenna. A network analyzer excites the patch antenna polarization under test with energy at 2.4 GHz while the waveguide collects the emitted energy. During this process the waveguide mechanically scans the plane facing the patch antenna and then reconstructs the far field pattern in software. The far field co-polarized beam – , and patterns were measured for vertical to vertical, – polarizations. The cross horizontal to horizontal, polarized beam patterns were also measured for vertical to – , and horizontal to vertical, – horizontal, polarizations, by rotating the waveguide. The measurements recorded in the azimuth plane are shown in Fig. 3 and greatly influence the experimental results, which are discussed in Section VI.

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Fig. 3. Normalized azimuth beam patterns for the co- and cross-polarized patch antennas.

Fig. 4. Simulated results which characterize the expected symbol error rate versus transmitted power results while transmitting 8PSK data. This shows the 1 1 single antenna case with no diversity behaves similar to the 2 2 spatial multiplexing algorithm. Recall that spatial multiplexing is able to decode 2 symbols for each time slot. The 2 1 and 2 2 Alamouti STBCs have a better error rate performance, but the data rate is less than spatial multiplexing due to the redundancy introduced in the encoder. The Alamouti schemes discussed in this paper only decode 2 symbols for every 2 time slots.

2

2

2

III. POLARIZED SYSTEM MODEL Alamouti, in [5], has discovered a remarkable scheme for 2 antenna systems which encodes 2 symbols over 2 time slots before the receiver is able to decode the data. The decoding simplicity of the Alamouti STBC makes this code one of the primary transmit diversity schemes for 3G and 4G systems. Unlike STBC, spatial multiplexing [4] does not introduce redundancy in the transmitted data for an improved error rate. Instead, it increases the data rate by sending independent streams of data. The data rate for spatial multiplexing is double the data rate of the Alamouti STBC, but the error rate suffers. The data is sent over the same carrier frequency by using a successive interference cancelation algorithm in the decoder. The Alamouti STBC has been implemented for all different possible pairs of polarizations to see which channel conditions benefit the most while spatial multiplexing has been implemented for the 2 2 dual polarized case. This has allowed an in depth comparison of polarization diversity to space diversity with different encoding and decoding techniques. The testbed is configured with 4 antennas at the transmitter and 4 antennas at the receiver as will be described in Section IV, but the MIMO system has been reduced to a 2 2 system to easily compare spatially separated antennas to dual polarized antennas over quasi static channel conditions. The 1 1, 2 1, and 2 2 polarized system models are shown in this section while the general cases for 2 1 and 2 2 spatially separated system models are shown in [5] for the Alamouti scheme and [4] for the spatial multiplexing scheme. Fig. 4 shows the expected symbol error rate vs signal to noise ratio (SNR) for the coding schemes described in this section based on a simulation done in MATLAB.

2

– , respectively, while and are modeled as ador ditive white Gaussian noise. Under these assumptions, the verat time is given by tically polarized received signal (1) Similarly, the horizontally polarized received signal is given by

at time (2)

B. 2

1 Alamouti [5]

The following transmission matrix, , has polarization rows has been used to encode the signals and time columns. to the polarizations over 2 time intervals during a coherent transmission (3) The first column of is transmitted from polarizations and at time slot 1, and the second column is transmitted at the time slot 2. A symbol is transmitted from vertical polarization and a symbol from horizontal polarization at two time slots so the overall rate is 1. The vertically polarized received signal at time slots 1 and 2 in a 2 1 Alamouti transmission may be expressed as (4) (5)

A. 1

1 System Model

is transmitted from At each time slot an encoded signal any single antenna with a chosen polarization. The transmitted while the transsignal from the vertical feed is denoted . The mitted signal from the horizontal feed is denoted and are the path gains from – channel coefficients

The received signals at time slots 1 and 2 are linearly combined to decouple the signals that are transmitted at the same time. The combiner outputs may be expressed as follows:

(6)

MORSHEDI AND TORLAK: MEASURED COMPARISON OF DUAL-BRANCH SIGNALING OVER SPACE

The combiner outputs are then decoupled and only depend on the transmitted signals

(7) The transmitter has now remained a dual polarized antenna, but the receiver has become a horizontally polarized antenna. In this case, the horizontally polarized received signal at time slots 1 and 2 in a 2 1 Alamouti transmission may be expressed as (8) (9) where and are the channel coefficients from vertical polarization and horizontal polarization, respectively. The received signals at time slots 1 and 2 are linearly combined to decouple the signals that are transmitted at the same time. The combiner outputs may be expressed as follows:

(10)

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polarizations as well as the system model for the Alamouti STBC in 2 1 and 2 2 dual polarized systems. The encoding and decoding schemes have also been presented. In the next section spatial multiplexing will be introduced. D. 2

2 Spatial Multiplexing [4]

Spatial multiplexing, like the Alamouti scheme presented in Sections III-B and III-C, uses the same carrier frequency to transmit two streams of data, but there is no redundancy introduced in the encoder. The Alamouti scheme encodes dependent data over two time slots in order to decouple the data at the decoder, while spatial multiplexing transmits independent streams of data which doubles the data rate, but the decoding algorithm is more complicated. The following transmission vector is used to encode the independent spatially multiplexed symbols to the antennas over one time interval (16) The dual polarized channel model for spatial multiplexing remains the same as the 2 2 Alamouti channel model given in Section III-C and is shown as (17)

The combiner outputs are decoupled and once again only depend on the transmitted signals. The dual polarized received signal is given as

(18)

(11) and the noise vector as C. 2

2 Alamouti [5]

If another receive polarization is added to make this a 2 2 system, the decoding simplicity still remains clear. The received signal is now expressed as

(19) The system model may be written as (20)

(12) (13) where and are the channel coefficients from the dual polarized transmit antenna to the vertically polarized receive anand are the channel coefficients from the dual tenna and polarized transmit antenna to the horizontally polarized receive antenna. In this case, 2 time slots are still required. The combiner outputs may now be expressed as follows:

(14)

(15) In this section the polarized system model has been reviewed for a 1 1 single antenna system with vertical and horizontal

Spatial multiplexing requires only 1 time interval to decode 2 symbols resulting in a rate 2 code. Recall the Alamouti scheme was a rate 1 code requiring 2 time intervals to decode two symbols. To make a valid comparison of the Alamouti scheme to spatial multiplexing, the power per symbol must be kept the same. Because the rate is doubled for the case of spatial multiplexing, the total transmitted power is normalized to per antenna whereas the total transmit power for the Alamouti scheme per antenna. is normalized to The spatial multiplexing zero forcing algorithm is a recursive process that decodes the elements of the transmitted signal, , one at a time according to a specific ordering procedure. This ordering procedure works on the channel with the smallest noise variance. The noise variance is found by computing the pseu, and then looking at the doinverse of the channel matrix, squared norm of the each row. Because this is a 2 2 system, will have two rows. The row of with the minimum noise , is then multiplied by the original revariance, denoted ceived signal . The subscript i in this case denotes the th iteration. Now is assumed to be the signal transmitted along the corresponding channel with the minimum noise variance.

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TABLE II SPATIAL MULTIPLEXING ALGORITHM WITH ZERO-FORCING RECEIVER

is decoded and then the column of , which corresponds to that has minimum noise variance, is filled with the row of zeros. This new channel matrix is then updated for the second iteration. Then the effects of the decoded transmitted signal and the channel gain it experienced are subtracted from the received signal for the next iteration. The spatial multiplexing algorithm is given in Table II. is the pseudoinverse In the algorithm given in Table II, is the updated channel matrix with of the channel matrix, the zero forcing receiver applied after the th iteration, and is the quantizer which decodes the estimates of the transmitted signals to the nearest constellation point for subchannel interference cancelation. After performing the spatial multiplexing algorithm, it is possible to construct the estimate of the vector of transmitted data, . which is then sent to the decoder

Fig. 5. A block diagram is shown of the entire transmit and receive MIMO system.

Fig. 6. The transmitter and receiver are shown facing each other in the line of sight environment chosen for this study.

(21) IV. EXPERIMENTAL SETUP Theoretical work and computer simulations are able to quickly characterize the promise of MIMO technology under predetermined conditions as shown in Fig. 4, but experimental testbeds are crucial in validating these results over real wireless channels subject to implementation impairments. With this motivation, a multiple antenna testbed operating at 2.4 GHz has been designed in order to measure the performance of various signal processing algorithms and validate their performance in non ideal conditions. For this paper, 1 1 single antenna transmission, 2 1 and 2 2 Alamouti STBC and 2 2 spatial multiplexing have all been implemented in the multiple antenna testbed in an indoor office environment while using dual polarized antennas versus spatially polarized antennas at the transmitter and the receiver. In this section, a brief description of the hardware setup is presented while a block diagram of the experimental setup can be seen in Fig. 5. A. Multiple Antenna Testbed The multiple antenna testbed developed at UTD has been designed to implement a variety of array processing techniques. Parameters such as frame structure, data rate, modulation order and antenna array size are all determined by software and can be easily modified. The transmitter unit consists of a PC, ICS660 digital-to-analog (DAC) board, and 4 custom made RF transmitters. In the PC, all the baseband processing such as frame creation, data encoding and pulse shaping are done using MATLAB. Then, this digital data is fed into the ICS660 board to create an analog signal with an intermediate frequency (IF)

of 10 MHz. The ICS660 board is a software defined 4-channel 14-bit PCI Bus DAC card that operates at a fixed 100 MHz conversion frequency. The ICS660 board software has been modified such that the stored digital data is transmitted continuously. When the transmission process begins, the same data is resent for decoding at the receiver. For upconversion, the output of ICS660 is fed to the RF front end. The RF front end consists of 4 identical RF transmitter boards. B. Mobile Unit The mobile receiver consists of the receiver hardware, a PC and an antenna platform. The patch antenna array is mounted on the platform and placed in LOS and NLOS indoor office environments. The transmit antennas were at a height of 2 m while the receive antennas were at a height of 1 m. In the LOS case the antennas were set facing each other 4 meters apart. In the NLOS case the antennas were set facing each other 25 meters apart with 3 partitions between them. each antenna output is connected to the input of a MAXIM 2829 transceiver chip. A custom designed backplane provides power and control signals to 4 MAXIM transceiver chips. The baseband output of each chip is then passed to the ICS645 analog-to-digital (ADC) board. The data is sent into MATLAB and stored for processing and displaying test results. The stored data is used to find SER and channel characteristics. Fig. 8 displays the receiver hardware. V. DATA COLLECTION METHODOLOGY In each environment, the receive antenna array was placed on a grid-like system of rails which allowed it to be moved at inter-

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TABLE III FRAME STRUCTURE

Fig. 7. The receiver is shown in the non-line of sight environment chosen for this study. The transmitter remains stationary in the line of sight environment.

VI. EXPERIMENTAL RESULTS Even though the actual co-polarized and cross-polarized beam patterns of the vertical and horizontal patch antennas were measured, different environments, channel characteristics and different encoding and decoding techniques may benefit one polarization over the other. To understand the antenna performance in different environments, the receiver was placed in LOS and NLOS environments. The LOS environment is in a typical university lab setting as shown in Fig. 6. The transmitter is placed facing the receiver with no objects obstructing the line of sight path. The NLOS environment is shown in Fig. 7. The receiver was moved two hallways away with multiple partitions obstructing the direct path between transmitter and receiver. Fig. 8. The MAXIM 2829 transceiver chips downconvert the received signals to baseband. The signals are then sampled by the ADC and then further processing is handled in MATLAB.

vals of one half the carrier wavelength, (6.15 cm). Thus, at each receiver location inside of the ECSN building, ten frames worth of data were collected at fifty different positions along the grid. Square-root raised cosine pulse shaping filters were used at the transmitter and receiver, effectively forming a raised-cosine filter end-to-end. By pulse shaping the data, the effective bandwidth of the data is limited allowing the intersymbol interference caused by the channel to be kept under control. The frame structure for the transmitted signals of this experimental system was designed to allow many schemes to be tested in a rapidly time interleaved fashion, in the same format as in [9]. A guard interval subframe is inserted in between each frame and time synchronization subframes are placed after the guard interval. The frame structure then consists of 6 data subframes each 155 symbols in length and 8PSK modulated. Pilot symbols inserted in between the data subframes as shown in Table III. The synchronization subframes are random BPSK modulated frames that allows high performance symbol time synchronization for each polarization. Pilot subframes, which consist of all ones are sent from vertical polarization every two subframes for carrier offset estimation. The first two data subframes are used for single antenna transmission where one subframe is sent from each transmit polarization. Three other subframes are set up using the Alamouti STBC. The final data subframe is set up for spatial multiplexing.

A. Symbol Error Rate SER measurements were taken in both environments, but before the measurement campaign took place, a wireline test was recorded to check the hardware limitations without the actual wireless channel impairments. The wireline test is shown in Fig. 9. It was expected that the 1 1 single antenna transmission should perform the same as 2 1 Alamouti because no diversity is given by the wire. In addition, a 3 dB increase in performance by the 2 2 Alamouti case is expected, which is indeed the case. To form the SER curves in LOS and NLOS environments, measurements were taken at five different transmit power levels and then averaged over all grid positions as discussed in Section V. The results in Fig. 10 are shown comparing 1 1 single antenna transmission to 2 2 spatial multiplexing in a LOS envi– has a better SER than – in LOS. This ronment. is due to the dominance of the vertical beam pattern shown in Fig. 3. The dual polarized antenna is tested for 2 2 spatial multiplexing in LOS. Typically LOS does not support spatial multiplexing due to the high channel correlation. This causes the successive interference cancellation algorithm to break down, but with dual polarized channels, spatial multiplexing performs as well as a single antenna while doubling the rate. The 2 2 Alamouti codes were compared in LOS in Fig. 11. – , perform better The vertically polarized antennas, than the horizontally polarized antennas, – . But the – , performs as well as dual polarized antenna, – . In the NLOS environment the trend changes. Fig. 12 shows – performs better than the 2 2 dual polarized that

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Fig. 9. Wireline SER curves to show the hardware only performance.

Fig. 10. Empirical SER curves under LOS conditions varying the transmit power with a built in gain controller. The curves compare [V]-[V], [H]-[H], and [VH]-[VH] which uses spatial multiplexing to double the data rate.

Fig. 12. Empirical SER curves under NLOS conditions varying the transmit power with a built in gain controller. The curves compare [V]-[V], [H]-[H], and [VH]-[VH] which uses spatial multiplexing to double the data rate.

Fig. 13. Empirical SER curves under NLOS conditions varying the transmit power with a built in gain controller. The curves compare the 2 2 spatially separated Alamouti STBC, [VV]-[VV] and [HH]-[HH], to the dual polarized 2 1 Alamouti STBC, [VH]-[VH].

2

2

B. Channel Gain

Fig. 11. Empirical SER curves under LOS conditions varying the transmit power with a built in gain controller. The curves compare the 2 2 spatially separated Alamouti STBC, [VV]-[VV] and [HH]-[HH], to the dual polarized 2 1 Alamouti STBC, [VH]-[VH].

2

2

spatial multiplexing – , while – has the worst performance. The 2 2 Alamouti schemes in Fig. 13 show that the hori– and zontal polarization is dominant in NLOS. – perform very similarly, while – has the worst SER.

The corresponding channel gains are also examined for the single antenna case and for the Alamouti STBC. The channel estimates are initially recorded in amplitude and phase, but the gains are evaluated in power. A summary of the calculation of the corresponding channel gains for 1 1, 2 1 and 2 2 transmission schemes are shown. Recall that the total transmit power is normalized to . • 1 1 Single Antenna – ; – ; • 1 1 Single Antenna • 2 1 Alamouti – ; – ; • 2 1 Alamouti – • 2 2 Alamouti . The channel gains shown in Figs. 14 and 15 also reflect that the dual polarized antennas perform more consistently in both environments. In the LOS environments the channel gains experienced by the vertically polarized channels are stronger, but in the NLOS environment it is the horizontally polarized channels which are dominant. Dual polarized antennas seem to experiand channel conditions as the enence the benefits of the

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TABLE IV LOS CHANNEL CORRELATION

TABLE V NLOS CHANNEL CORRELATION

Fig. 14. Empirical channel gains measurements under LOS conditions computed for all single antenna data and Alamouti data where the total transmit power is fixed while the received channel gain has been normalized by the mean of the best performing single antenna case in LOS.

between these two channels is by first finding the average of the estimate of the sample covariance matrix. The estimate is shown for channel of the sample covariance matrix measurements (22)

Fig. 15. Empirical channel gains measurements under NLOS conditions computed for all single antenna data and Alamouti data where the total transmit power is fixed while the received channel gain has been normalized by the mean of the best performing single antenna case in NLOS.

vironment changes. The LOS channel gains show that the 1 1 – single channel data is clearly worse than 1 1 – , which performs as well as the 2 1 – Alam– and outi. In terms of the 2 2 Alamouti channels, – have similar performance 50% of the time, but at higher channel gains, – improves. Unlike the LOS channel gains, the NLOS channel gains illustrate that the 1 1 – performance is worse than 1 1 – . Noticing – is the 2 2 Alamouti system performances, the clearly better than – , but is clearly – , which performs even worse than outperforming – Alamouti. the 2 1 C. Channel Correlation MIMO systems depend on the uncorrelated fading of the transmitted signals in the multipath environment to receive the expected increase in channel gains and more reliable error rates. The channel measurements are recorded in amplitude and phase. So for each 2 1 transmission, the single receive antenna experiences two different channels, one from each transmit antenna. The process for calculating the correlation

After normalizing by the diagonal elements, the anti-diagonal elements represent the channel correlation while the diagonal elements represent the power imbalance between the channels. Tables IV and V summarize the channel correlations in LOS and NLOS. The horizontally polarized channels experience channel correlations below 0.4 in both environments, but the vertically polarized channels have a channel correlation of 0.886 in LOS. This may seem inconsistent with the results reported in Section VI dealing with SER. The reason is that the – channel gains are performing so well is that it is rare when any of the vertically polarized signals experience bad channel conditions. Diversity in the LOS case for the vertically polarized signals is not needed. This was suspected from the antenna polarization patterns shown in Fig. 3 where the vertical beam had a more direct gain, and the horizontal beam had a more dispersed pattern. The dispersed horizontal beam along with the low channel correlation supports the dominance of the – channels gains in NLOS. The true benefit of the dual polarized antennas is that they experience correlation values as low as 0.3289 in LOS and 0.1067 in NLOS. This is beneficial due to the fact that the vertical channel gains are stronger in LOS while the horizontal channel gains are stronger in NLOS. As long as the channels from the dual polarized antenna fade independently as the receiver changes environments from LOS to NLOS, the mobile user will experience the best diversity, channel gain and error rate from the vertical and horizontal combination. VII. CONCLUSION A wireless multiple antenna test platform has been designed which has enabled the WISLAB at UTD to conduct an experimental study on the performance of single antenna transmission, transmit diversity and spatial multiplexing under real world indoor wireless channel conditions. The main objective was to compare space diversity to polarization diversity at 2.4 GHz. Based on the resulting SERs and channel characteristics in the

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tested indoor environments, polarization diversity offers a more robust solution than space diversity because of its ability to perform consistently in LOS and NLOS. The results motivate the use of polarization diversity for indoor environments due to outperforming the spatially separated schemes in terms of SER while maintaining lower channel correlation. The channel gains are more beneficial for vertical polarization in LOS and the horizontal polarization in NLOS, but to experience the best channel gains while changing environments, the dual polarized patch antenna seems to provide more consistent results. APPENDIX Antenna Design: The design of the dual polarized microstrip patch antenna followed the procedure in [8]. The microstrip patch antenna to be designed is laying on top of a dielectric substrate. Therefore, the fringing electric field components are not limited to free space. It is then necessary to know the substrate wavelength (23) where is the effective dielectric constant, which depends on the relative dielectric constant , and the width to height ratio . This ratio explains how to set the width of the microstrip transmission line relative to the height of the substrate. The input impedance of the patch antenna is found by (24) where is the characteristic impedance of the medium and is the relative dielectric constant due to the type of substrate the patch antenna lies on. Since the patch will be transmitting a signal from orthogonal sides, a square shape is desired. This where and are the length and the width of forces the patch antenna, respectively. Knowing that the impedance of , it can be verified that both impedances the connector are real, meaning a quarter wave transformer may be used for the microstrip transmission line matching network. Using a quarter wave transformer, the impedance of the transmission line can be obtained by (25) Then using Sobel’s equations (26) and (27) the width to height ratio tric constant is given as

is defined and the effective dielec-

(28)

This gives a relation to the desired substrate wavelength . It is which allows all of the uncommon to have known parameters to be found. These parameters are summarized in Table I. It is also worth mentioning that the width, of the microstrip transmission line has been set by the designer ratio, but the lengths of the transmission when setting the to three quarlines were extended from a quarter wave ters of a wave for fabrication convenience. By realizing intervals it can be seen that that the Smith Chart repeats in these lengths produce similar results. In addition, when adding another patch antenna to the same substrate, it is necessary to to minimize coupling separate the two patch antennas by between patch elements.

REFERENCES [1] R. U. Nabar, H. Bolcskei, V. Erceg, D. Gesbert, and A. J. Paulraj, “Performance of multi-antenna signaling strategies in the presence of polarization diversity,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2553–2562, Oct. 2002. [2] D. Ramirez, “A comparative study of STBC transmissions at 2.4 GHz over indoor channels using a 2 2 MIMO testbed,” Wireless Commun. Mobile Comput., vol. 8, no. 9, pp. 1530–1567, Oct. 2007. [3] J. Wallace and M. Jensen, “Measured characteristics of the MIMO wireless channel,” in Proc. IEEE Trans. Veh. Technol. Conf., Oct. 2001, vol. 2, pp. 2038–2042. [4] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Lab. Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [5] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [6] J. P. Kermoal, “Polarization diversity in MIMO radio channels: Experimental validation of a stochastic model and performance assessment,” IEEE Trans. Signal Process., vol. 44, no. 2, pp. 22–26, Oct. 2001. [7] A. M. D. Turkmani, “An experimental evaluation of the performance of two-branch space and polarization diversity,” IEEE Trans. Veh. Technol., vol. 44, no. 2, pp. 318–226, May 1995. [8] L. Reinhold and B. Pavel, RF Circuit Design Theory and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000. [9] A. T. Koc, “Experimental Investigations of Four Transmit Antenna Space-Time Block Codes,” MS, The University of Texas at Dallas, Richardson, 2005. [10] H. Bolcskei and A. J. Paulraj, “Performance of space-time codes in the presence of spatial fading correlation,” in Proc. IEEE Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Oct. 2000, vol. 1, pp. 687–693. [11] R. U. Nabar, V. Erceg, H. Bolcskei, D. Gesbert, and A. J. Paulraj, “Performance of multi-antenna signaling strategies using dual polarized antennas: Measurement and analysis,” Wireless Pers. Commun., vol. 23, no. 1, pp. 31–44, May 2002. [12] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [13] C. Oestges, “Channel correlations and capacity metrics in MIMO dualpolarized Rayleigh and Ricean channels,” IEEE Trans. Signal Process., vol. 2, pp. 1453–1457, Sep. 2004. [14] B. Lindmark and M. Nilsson, “On the available diversity gain from different dual-polarized antennas,” IEEE J. Sel. Areas Commun., vol. 19, no. 2, pp. 287–294, Feb. 2001. [15] R. Vaughan, “Polarization diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 39, no. 3, pp. 177–186, Aug. 1990. [16] Y. Deng, A. Burr, and G. White, “Performance of MIMO systems with combined polarization multiplexing and transmit diversity,” in Proc. IEEE Veh. Technol. Conf., May 2005, vol. 2, pp. 869–873. [17] W. C. Y. Lee, “Effects on correlation between two mobile radio base station antennas,” in Proc. IEEE Veh. Technol. Conf., Nov. 1973, vol. 22, pp. 130–140. [18] Rappaport, Wireless Communications. Upper Saddle River, NJ: Prentice-Hall, 2001.

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Ali Morshedi, (S’05–M’08) received B.S. and M.S. degrees in electrical engineering from the University of Texas at Dallas, in 2005 and 2008, respectively. He is currently working at Qualcomm Inc., San Diego, CA, as an RF hardware design engineer. His interests include antenna design and simulation, experimental multiple antenna systems and signal processing.

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Murat Torlak (SM’05) received M.S. and Ph.D. degrees in electrical engineering from The University of Texas at Austin, in 1995 and 1999, respectively. He spent summers 1997 and 1998, in Cwill Telecommunications, Inc., Austin, TX, where he participated in the design of a smart antenna SCDMA system. In Fall 1999, he joined the Department of Electrical Engineering, The University of Texas at Dallas, where he is currently an Associate Professor. He held a visiting position at University of California Berkeley, in 2008. He has been an active contributor in the areas of smart antennas and multiuser detection. His current research focus is on experimental platforms for multiple antenna systems, millimeter wave systems, and wireless communications with health care applications. Dr. Torlak is an Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He was the Program Chair of the IEEE Signal Processing Society Dallas Chapter during 2003-2005

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Channel and Propagation Measurements at 300 GHz Sebastian Priebe, Christian Jastrow, Martin Jacob, Student Member, IEEE, Thomas Kleine-Ostmann, Thorsten Schrader, Member, IEEE, and Thomas Kürner, Senior Member, IEEE

Abstract—Ultrabroadband Terahertz communication systems are expected to help satisfy the ever-growing need for unoccupied bandwidth. Here, we present ultra broadband channel measurements at 300 GHz for two distinct indoor scenarios, a point-to-point link of devices on a desktop and the connection of a laptop to an access point in the middle of an office room. In the first setup, measurements are taken with regard to distance, different antenna types and device displacements. Additionally, an interference constellation according to the two-ray model is examined. In the second setup, the focus is on the detection and characterization of the LOS- and the NLOS-paths in an indoor environment, including a maximum of two reflections. Temporal channel characteristics are examined with regard to maximum achievable symbol rates. Furthermore, ray obstruction due to objects in the transmission path is investigated. Index Terms—Channel modeling, channel sounding, diffraction, submillimeter channel measurements, THz communications, THz propagation measurements, THz system.

I. INTRODUCTION

I

N the past years it has become obvious that wireless data rates exceeding 10 Gbit/s will be required in several years from now [1]. The opening up of carrier frequencies in the Terahertz range has been identified as most promising to provide sufficient bandwidths required for ultra fast and ultra broadband data transmissions [2], leading to the initiation of the IEEE 802.15 THz Interest Group (IG THz). A suitable frequency window can be found around 300 GHz, offering a currently unregulated bandwidth of 47 GHz [2]. As technological advance keeps up with the ever increasing demand for wireless data transmission capacity, electrical components for the generation of THz frequencies are already commercially available [3]. Based on subharmonic Schottky diode mixers, a 300 GHz transmission system has been set up at the Physikalisch-Technische Bundesanstalt, Braunschweig [4]. Experiments within the Terahertz Communications Lab (TCL) have proved the feasibility of an analog video transmission over 22 m [4] and a digital video data transmission over 52 m

Manuscript received April 20, 2010; revised September 28, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. S. Priebe, M. Jacob, and T. Kürner are with the Institut für Nachrichtentechnik, Technische Universität Braunschweig, Braunschweig 38106, Germany and also with the Terahertz Communications Lab, Braunschweig 38106, Germany (e-mail: [email protected]; [email protected]; kuerner@ifn. ing.tu-bs.de). C. Jastrow, T. Kleine-Ostmann, and T. Schrader are with the PhysikalischTechnische Bundesanstalt Braunschweig, Braunschweig 38116, Germany and also with the Terahertz Communications Lab, Braunschweig 38106 (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122294

with a data rate of 96 Mbit/s [5]. A different approach to THz communication applying optical generation and modulation of frequencies between 300 and 400 GHz is reported in [6]. Primarily, three different scenarios can be identified for data communication at 300 GHz. The first one involves short distances up to 1 m as required for the wireless interconnection of different devices on a desktop or for ultrafast kiosk downloads. The second use case refers to THz indoor pico cells for wireless access as, e.g., imaginable in conference-, office-, or living rooms, whereas the third scenario is the wireless extension of wired backbone networks over ranges greater than 10 m [7]. Private applications will include high speed wireless data exchange with flash drives, the almost instant download of large files (e.g., HD films) and uncompressed video streaming of future ultra high resolution video formats [8]. Professional applications cover, e.g., fast file exchange on conferences, telemedicine, and the provision of interference-free wireless high speed networks in trade fair halls by dividing the huge available bandwidth into multiple subbands. THz waves can also be utilized for a wide range of different other applications apart from data transmission like, e.g., remote sensing, radio astronomy or security. A good overview can be found in [9]. Future THz WLANs will primarily rely on line-of-sight (LOS) conditions and high gain antennas with gains beyond 25 dBi to overcome the high free-space [10] and additional high reflection losses [11]. Sophisticated antenna designs are required to provide both high gain and the ability to serve multiple users at different positions. Here, a very promising approach are multibeam antennas (e.g., dielectric rod or ring-slot antennas) as shown in [12]. In case of LOS paths blocked by persons or objects, the concept of directed nonline-of-sight (NLOS) communication has been proposed [2]. Hence, not only LOS, but also NLOS channels will have to be characterized. Employing bandwidths from several GHz up to several 10 GHz, channels at 300 GHz cannot be described with existing narrowband channel models intended for current WLAN systems at carrier frequencies of up to 5 GHz with bandwidths not exceeding 40 MHz [13], [14]. Therefore, the ultra broadband channel characterizations as presented in this paper will be necessary to develop empiric channel models required for system simulations of such upcoming THz data transmission systems [15]. The remaining paper is structured as follows. In Section II the measurement setup will be introduced in detail. Absorption coefficients of typical building materials will be given in Section III. The additional attenuation due to an object obstructing the direct ray will be examined experimentally as well as theoretically with the knife edge diffraction model [16]. Short range and indoor channel measurements based on the identified use cases are the central topics in the Sections IV-A

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Fig. 1. The 300 GHz measurement setup.

and IV-C, respectively. Furthermore, the interference between the direct and a once reflected ray will be investigated including simulations according to the two-ray model [16] in Section IV-B. The measurement results are finally used as input for a first empiric path loss model before the paper is concluded in Section VI.

Fig. 2. Amplitude response of the 300 GHz measurement system including the antenna gain of twice 26 dBi.

TABLE I MEASUREMENT PARAMETERS

II. THE MEASUREMENT SETUP The measurement setup consists of the core components of the 300 GHz transmission system [4] and a Rohde & Schwarz vector network analyzer (VNA) [17]. Subharmonic Schottky diode mixers are used to up-convert a baseband test signal from a Rohde & Schwarz ZVA 40 VNA to 300 GHz, which is transmitted over the channel and downconverted at the receiver. As perfect phase coherence is obligatory for correct vectorial network analysis, a common local oscillator signal generated by a is tripled twice dielectric resonator oscillator and then fed to the subharmonic mixers at both the TX and RX. The corresponding block diagram is shown in Fig. 1. By recording the frequency dependent scattering parameter [18] for the test signal frequencies at the VNA, the is measured. channel transfer function at parameters specified in dB always imIn the following, all plicitly mean the magnitude of the transfer function whereas all corresponding impulse responses are given as the relative received power over time. Since a system error correction could only be performed at the baseband coaxial in-/output of the mixers by means of a TOSM calibration, just including the cables between VNA and the mixers, additional reference measurements allowing to correct the influence of the transmission system during postprocessing were taken with direct interconnection of the transmitter (TX) and receiver (RX) waveguides. Apart from the system losses, the depicted amplitude response (Fig. 2) also includes the antenna gains of two times 26 dBi [4] for both the TX and RX horn antenna. In the following, the antennas are always considered as a part of the transfer path so that the antenna characteristics are not discussed in more detail. Regarding the phase response (not shown), an almost perfect linear phase was observed, corresponding to an electrical . As double sideband mixers length of are used, the homodyne downconversion leads to an insignificant amplitude distortion between 0 dB at 300 GHz and 0.28 dB at 310 GHz due to different free-space losses in the lower and

upper sideband, which needs to be corrected. The phase response is not influenced in any way. More details can be found in [17]. Multiple reflections between the transceiver modules have already been identified as a source for strong frequency selective channel transfer functions in case of direct antenna alignment [17], causing an uncorrectable error in measurement. Therefore, absorber panels have been mounted at both TX and RX for all measurements presented in this paper. In order to achieve the best possible spatial and temporal resolution of approximately 3 cm or 0.1 ns, respectively, all measurements were taken with the full available bandwidth of 9.99 GHz. The start frequency was bound to a minimum of 10 MHz by the VNA and the maximum frequency could not exceed the system limitation of 10 GHz. Due to input power restrictions of the mixers, a test signal with a power of 5 dBm was used, providing a dynamic range of approximately 85 dB for the chosen . intermediate frequency filter bandwidth of The number of sweep points was chosen to 801, resulting in a maximum detectable path length of 24 m. A complete overview of all measurement parameters is given in Table I. III. TRANSMISSION AND DIFFRACTION MEASUREMENTS Applying THz time domain spectroscopy (TDS), the TCL has already quantified the transmission losses of typical building materials [19]. Here, we determine the transparency of objects in the LOS link with the 300 GHz measurement system in transmission geometry and compare the resulting absorption was coefficients with the results obtained by THz TDS. recorded at a module spacing of 10 cm with and without the

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TABLE II ABSORPTION COEFFICIENTS OF DIFFERENT MATERIALS

Fig. 4. Schematic drawing of the diffraction investigation setup including parameters for the knife edge model in top view.

Fig. 3. Measuring the diffraction on an edge.

different samples brought into the ray path. The transmission from loss was determined by subtracting . Based on the material thickness the attenuation was converted into the absorption coefficient (Table II). Both different methods yield similar outcomes, indicating that absorption coefficients obtained in a spectrometer can be used to estimate the attenuation in an obstructed transmission path. Assuming, e.g., a medium density fiberboard (MDF) door or a window with with a thickness of , absorption attenuations of approximately 65.5 and 86.7 dB are calculated. This result shows that THz WLANs will allow the reuse of the same frequency bands even in neighboring rooms without any co-channel interference to be expected. At the same time, high data security is achieved simply due to the low signal levels outside the intended service area of an access point. Contrasting to those advantages it becomes obvious that LOS- or directed NLOS-links will be required for reliable data communication at 300 GHz. In order to systematically investigate the additional attenuation due to a gradual ray obstruction by objects, a copper plate with a thickness of 1 mm, corresponding to a sharp edge, was brought into the direct ray path at a distance of 35 cm from the TX and 15 cm from the RX (Fig. 3). This way, a worst case estimation of the obstruction attenuation, receiving only a diffracted and no transmitted ray, with given. By moving the plate orthogonally to the ray path being recorded in the degree of shadowing was varied, with steps of 2.5 mm between and (see solid arrow in Fig. 3). The resulting diffraction attenuation is calculated as the difference between for an obstructed and an unobstructed path as depicted in Fig. 5. The parameter refers to the distance of the edge from the direct ray referring to the center of path (dashed line) with the beam so that half of the first Fresnel zone [16] is shadowed. For positive values of h the direct path is blocked whereas it is increasingly unshadowed for increasingly negative .

Fig. 5. Measured and simulated diffraction attenuation for different degrees of shadowing.

In the latter case, the interference of both the direct and the diffracted ray can be observed, leading to a slight gain. As expected, even small positive values of cause a considerable additional attenuation of up to 30.7 dB, hindering ultra broadband data transmission due to seriously deteriorated signal-to-noise ratios (SNR). Therefore, fast switching from the LOS- to a directed NLOS-path becomes obligatory even for marginal ray obstructions. In addition to the measurements, the ray obstruction by the sharp edge is simulated according to the knife edge model as described in [16] based on the geometry visualized in Fig. 4 in top view. There, the copper plate is represented by the bold red , and , line. With the Fresnel parameter is calculated as (1) which is then used to determine the diffraction attenuation

(2) and denote the decomposition of the Fresnel integral and are computed as described in [20, eq. (8.3a)]. A good match between simulation and measurements can be and . For negaobserved between tive , the simulation results partly overestimate the measured diffraction attenuation whereas it is underestimated beyond . In the first case, the difference can be explained by a non-perfect placement of the copper plate in the exact distance

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Fig. 6. Transfer functions for different module distances, misalignments and antenna types. (a) Horn antenna, d and PE-lense, d .

= 100 cm

relative to the direct ray because no optical bench has been used. Moving the copper plate out of the direct path also increasingly uncovers the edge of the table on which the RX is placed, so that a second diffracted ray is received. This ray superimposes with the direct as well as with the ray diffracted at the copper plate, but has not been included in the simulation for the reason that the focus is on modeling the mere influence of the shadowing object represented by the knife edge. Moreover, the ray is received under a comparatively large incidence angle beyond the 3 dB HPBW of the horn antenna and, hence, is of minor importance. Accordingly, the measurement setup corresponds to a realistic scenario where, e.g., a person walks through the ray path, passing by the table. Here it is noteworthy that the knife edge approach has already been validated and applied for modeling the ray shadowing by moving persons between 4 and 10 GHz in [21]. onwards, the increasing deviation of up to From 9.9 dB for mainly results from the directivity of the RX horn antenna. As increases, so does the angle of incidence of the diffracted ray at the RX. The incident diffracted ray is attenuated additionally due to the narrow main lobe for as the modules have remained fixed and have not been pointed directly at the absorbing edge. This aspect has to be considered especially for systems with highly directional antennas by not only calculating the diffraction attenuation, but also respecting the antenna diagram of the RX antenna even for small angles of incidence. If done so, the knife edge model provides a simple and valid means for simulating the ray obstruction at 300 GHz. However, as the intention of the scenario is the investigation of the practically expected additional attenuation, the antenna influence is considered as part of the transmission path and hence has not been corrected here. IV. CHANNEL CHARACTERIZATIONS In this central chapter of the paper, we first show distance-dependent short range channel transfer functions as required for the simulation of communication systems operated on a desktop. For the reason that a perfect module positioning cannot be expected from users, each channel is investigated with regard to a certain transceiver displacement. Additionally, the influence of different antenna types is examined. Next,

= 40 cm. (b) Waveguide, d = 10 cm. (c) Horn

the frequency selectivity of ultra broadband channels caused by interference of the direct and a once reflected ray will be analyzed in detail in the second subsection. Finally, indoor channel measurements in an office environment are presented as obligatory for system simulations of future THz WLANs. Both LOS and NLOS paths will be characterized. A. Desktop Scenario (Short Range) Typical short range applications on a desktop can be expected to operate at distances between 20 cm and 100 cm. Suitable antenna types include open-ended waveguides, horn antennas and optional polyethylene (PE) lenses for further beam collimation. Here, the open-ended waveguides as available at the Schottky mixer outputs have been measured to provide a gain of 9.9 dBi with a 3 dB half power beamwidth (HPBW) [22] of 100 . Alternatively, 26 dBi horn antennas with 10 HPBW are used. Mounting PE lenses in front of the horn antennas enhances the gain by additional 13 dBi, each. In this case the 3 dB HPBW is reduced to 1 . Channel transfer functions have been recorded at specific distances for each antenna type using the measurement setup shown in Fig. 7. The module spacing has been varied by moving the RX (solid arrow). By shifting the modules orthogonally to each other (dashed arrow), a certain module displacement has been simulated. The module misalignment, subsequently denoted with , always refers to the position of best module alignment for a given distance . In all cases the horn antenna has been mounted at the RX, whereas the TX antenna has been changed. To explicitly demonstrate the influence of the different antennas, each transfer function has been corrected by the system response, only. For the setup with two horn antennas, a module distance is chosen with module displacements of of , 2 cm, 4 cm, and 8 cm [Fig. 6(a)]. Regardless of the displacement, a slight frequency-dependent channel behavior with an amplitude variation of up to 2.6 dB peak-to-peak can be observed. This is a consequence of nonperfect absorber panels since a ray reaches the RX after having been reflected between the RX and TX modules as described in [17]. Compared to other ultrawideband channel characterizations at 60 GHz in extreme multipath environments like a

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of service and only the antenna of the mobile terminal needs to be adjusted. Additional PE lenses provide the best possible SNR even over large distances but are critical regarding any displacement so that they will primarily be used for fixed radio links, e.g., for the extension of backbone networks. B. Interference Examination Based on the Two-Ray Model

Fig. 7. Measurement setup for short range channel characterizations.

car with amplitude variations of more than 30 dB peak-to-peak (pp) [23], the measured variation still is almost negligible. After considering the influence of the system loss and the anparameter is expected to equal tenna gains, the recorded the free-space loss (FSL) [16]. The FSL amounts to 74 dB for whereas a loss of approximately 75.5 dB averaged over the whole bandwidth is measured. Reasons for the deviation are the antenna mismatch at the waveguide outputs of the mixers which could not be included in the reference measurement of the system response as well as a nonperfect module . alignment at leads to adEven a slight antenna mispointing of ditional attenuations of at least 4 dB. Whereas such a comparatively low impact on the signal-to-noise ratio (SNR) can already be taken into account in the system design by introducing a link margin, higher additional attenuations of up to 20 dB can cause a serious deterioration of the achievable data rates. Apart from the different path losses, the variation of the misalignment does not influence the general channel behavior. This does not apply for the open-ended waveguide. Here, with , the transfer functions depicted for 1 cm, 2 cm and 3 cm show a stronger frequency dependence as the absorber panels could not be adapted to optimally fit the waveguide output [Fig. 6(b)]. Regarding the misalignment, the broad beamwidth of the RX open-ended waveguide allows comparatively high displacement lengths at short module distances while the RX horn antenna still limits the tolerable mispointing. with ,3 The PE lense is applied at cm, 5 cm, and 8 cm [Fig. 6(c)]. Apart from the provided gain it serves as a spatial filter, suppressing reflections between the modules. But then, an extremely high sensitivity to displacements is induced due to the strong beam collimation. The unexpected frequency dependence that can be observed for increasing displacements is a consequence of a reflection between the RX module and the circular metal fixture of the lense, which for the reason of a flat incidence angle cannot be eliminated by the absorber panel. In summary, two open-ended waveguides can only be operated at shortest ranges of up to several 10 cm due to the lack of a high SNR. Longer range wireless links will have to rely on highly directive antennas, accordingly. Here, a precise antenna alignment is required which is rather feasible for stationary links like the replacement of wired high speed connections or nomadic scenarios where a fixed transceiver unit has a preset area

The short range channel measurements presented in the previous subsection were only taken under almost free-space conditions and did not include a reflecting plane close to the ray path. In case of placing devices on tables, near walls or nearby other reflecting objects, a strong interference of the direct and a once reflected ray can be expected despite the use of highly directive antennas. Here the focus is on a desktop scenario with two horn antennas and module distances of up to 100 cm. Apart from the measurements the scenario is simulated by applying a two-ray model [20]. Both results are compared. To estimate the maximum possible constructive/destructive interference, the table board was simulated by a polished copper and plate, having a nearly ideal reflection factor close to hence providing the maximum amplitude of the reflected path. The copper plate was placed 3.1 cm below the direct path so that the reflected ray could be received under a flat incidence angle smaller than the 3 dB HPBW of the RX horn antenna. For the measurements, typical module distances between 60 cm and 100 cm in steps of 1 mm were chosen. Fig. 8 shows the corresponding setup. Two fading dips caused by destructive interference can be found at 65.7 cm and 98.9 cm. The transfer function is demonstrated in Fig. 9, exemplarily. Here, the at system loss and antenna gains are included so that the mere path loss is depicted. The multipath propagation leads to a strong frequency selectivity of the channel transfer function over the whole bandwidth, ranging from 136.7 dB to 91.4 dB. For frequencies above 306 GHz no severe fading occurs. The measured parameter at those frequencies still exceeds the theoretical free-space loss of about 78 dB by at least 15 dB. For the highest attenuation was detected. Accordingly, this discrete frequency is chosen to demonstrate the distance dependence of the path loss around the first fading dip at 65.7 cm as shown in Fig. 10. For the simulation, is calculated as a superposition of the direct and a once reflected ray according to [20, eq. (6.22)] (3)

and . Qualitatively, a good match of measurements and simulation can be observed. The slight deviation between the measured and simulated position of the fading dip can be explained by the limited spatial resolution of the measurement setup. Still, the two-ray model proves well suited for modeling the given interference situation. In case of simulating other scenarios with incidence angles beyond the 3 dB HPBW of a given antenna, the antenna diagram has to be considered, additionally.

with

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Fig. 8. Setup for interference investigations; direct ray represented by solid line, reflected ray represented by dashed line. Fig. 11. The investigated office room.

C. Office Scenario (Longer Range)

Fig. 9. Channel transfer function with copper plate (d = 65:7 cm).

Fig. 10. Distance-dependent path loss for f = 304:27 GHz.

In conclusion, highly frequency selective channels may occur despite the spatial filtering by the horn antennas as soon as reflecting objects are found near the direct path, particularly within the first Fresnel zone. Regarding the prospective applications, such interference effects seriously influence the performance of ultra broadband data transmission systems and therefore will have to be considered although severe fading is expected for few discrete distances, only, as discussed in [24].

Ray tracing at 300 GHz has already been applied by the TCL in [13] to derive a power level map and to estimate achievable data rates in a virtual indoor scenario. Still, no channel measurements required for calibration of ray tracing algorithms have been presented so far. Therefore, this sub-section focuses on the characterization of all detectable paths including once and twice reflected rays in TM polarization within a small office room with limited to the horizontal plane of the a size of TX and RX modules (Fig. 11). A schematic, full-scale illustration of all measured of all measured paths is given in Fig. 13. The room is furnished with a wardrobe and two tables on which the transceiver modules are placed. This simplified setup corresponds to a mobile device on a desktop wirelessly connected to a fixed access point in the middle of the room. Nevertheless, the presented measurements basically also apply to a situation where the access point is mounted underneath the ceiling if both modules are aligned accordingly. In this case, only the orientation of the plane of incidence, the plane of polarization and the path lengths are changed whereas the general channel behavior remains the same. As high gain antennas with small beamwidths are required to provide a sufficient dynamic range especially for twice reflected paths, no impulse response simultaneously containing all different multipaths can be recorded. Instead, the complex channel transfer functions of the different paths are measured consecutively. If required, e.g., for the comparison of different antenna types, the complete impulse response of the scenario can then be composed of the single multipath components (MPCs). Still, highly directive antennas are most likely to be used as reasoned before so that several of the MPCs will be suppressed in any case. For the measurements, horn antennas and PE lenses are mounted at both the TX and RX. Here, the beam collimation of the lenses also helps to avoid reflections at the two table tops (cf. Section IV-B). By turning the TX and RX, the room has been scanned for detectable paths at different combinations of angles of arrival (AoA) and angles of departure (AoD). As soon as an impulse has been found above the noise level using the time domain option of the VNA, the orientation of both modules has been

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Fig. 12. Transfer functions (upper row) and relative received powers (lower row) of the direct, a once and a twice reflected path; corrected by the system loss. (a) Path (a): LOS; (b) path (c): reflection at wardrobe; (c) path (f): reflections at wall and door; (d) path (a): LOS; (e) path (c): reflection at wardrobe; (f) path (f): reflections at wall and door.

iteratively optimized in steps of for the particular path according to the magnitude of the simultaneously displayed channel transfer function. In this setup, no reflected rays from the upper area of the room could be measured for the reason of too high path losses. The best possible module orientation for the direct path was chosen as the reference position and (cf. straight line in Fig. 11). with A positive AoA/AoD denotes mathematically positive angles and , respectively. Based on the best possible alignment for the respective path the module orientation of either and the TX or RX is varied between while the other module remains fixed. Additionally, both modat the same time. This way, scatules are turned by tering processes are implicitly included in the measurements. In practice, such slight module displacements are even more likely over longer than over shorter distances as the best possible alignment becomes increasingly difficult to achieve. A complete overview of the path losses for all measured paths given by AoA and AoD is shown in Fig. 14 for the exemplary frequency of . Here, the depicted magnitude of was corrected by the system loss and two times the antenna gain of 39 dBi. All letters refer to the labels in Fig. 13 so that the paths can be identified easily. In addition, the path characteristics are summarized in Table III for the respective best possible module alignment. Apart from the measured path loss, the theoretical free-space loss is given. The difference includes the reflection losses as well as the influence of the nonperfect module alignment and of the nonperfect positioning of the lenses in front of

TABLE III CHARACTERISTICS OF ALL MEASURED PATHS FOR f BEST POSSIBLE MODULE ALIGNMENT

= 300:372 GHz AND

the horn antennas. Nevertheless, as the deviation between the FSL and the measured loss only amounts to 3.9 dB for the direct path, similar values can be assumed for the misalignment losses in the NLOS case. Exemplary detailed channel transfer functions and power delay profiles of the LOS, a once and a twice reflected path are given in Fig. 12. For lack of angle-dependent antenna frequency responses, all depicted transfer functions include the inherent system losses, only. The labels of the different graphs refer to , where the AoA and AoD which are denoted with the best possible alignment is indicated by the cross marker. (downwards Displacements of oriented triangle marker), (upwards oriented triangle marker), and (square marker) with regard to the respective best alignment are chosen for demonstration.

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Fig. 13. All paths measured in the indoor scenario.

Fig. 14. Path losses of the detected paths by angle of arrival and departure for f : .

= 300 372 GHz

Almost flat transfer functions are observed over the whole bandwidth in case of no misalignment for all paths. In contrast to the short range measurements, the impulse responses do not show a multipath due to reflections between the TX and RX module apart from the LOS case. This can be explained by high reflection losses. Turning either the TX or the RX by causes additional attenuations of at least 17.1 dB in the LOS case [Fig. 12(a)] and approximately 20 dB for the once reflected paths [Fig. 12(b)]. Corresponding values cannot be given for

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twice reflected rays as the amplitude of the transfer functions has already been too close to the noise floor [Fig. 12(c)]. does not equal Here it is noteworthy that [Fig. 12(a)], which is a consequence of asymmetric lense fixtures partly shadowing the incident ray. Concerning the maximum values, the attenuation exceeds 40 dB for certain AoAs/AoDs. On the one hand, such high attenuations result from the narrow antenna beamwidth, on the other hand they are caused by high scattering losses [25] which occur for nonspecular reflections. The influence of scattering processes can in particular be seen in Fig. 12(c) for reflections at the rough plaster wall where similar transfer functions can be observed regardless of a certain mispointing. For the temporal channel characterization, the RMS delay [26] has been determined from the power delay spread profiles (PDP) applying noise clipping with a fixed threshold of 30 dB below the detected maximum of each respective PDP. The delay spread basically increases with the number of reflections as the narrow beam is broadened by scattering processes around the point of specular reflection. Especially, this holds true for the rough surface of the plaster wall. If the cri[16], [27], where denotes a terion , is applied to avoid high bit error constant with rates (BER) caused by intersymbol interference, symbol rates of up to 60 and 7.8 GSymbols/s can be transmitted over the LOS and the NLOS path (c), respectively. For these upper bounds and is asof possible symbol rates, sumed. Here, path (c) was chosen exemplarily as it provides the least attenuation of all NLOS rays. In general, cannot be determined analytically as the intended BER, the modulation scheme, equalizers and the specific channel impulse response have to be considered. A threshold found in literature for BPSK [27]. is Contrastable AoA/AoD measurements at with two 20 dBi horn antennas in a conference room scenario as presented in [28] have yielded delay spreads of up to 1.2 ns. Compared to such delay spreads, the measured channels at 300 GHz can be considered as flat. Depending on the modulation scheme and based on the considerations above, even data rates beyond 100 Gbit/s can be achieved in the LOS case regarding the temporal channel characteristics. However, it needs to be stressed that such low delay spreads rest upon the suppression of MPCs by the antennas. In summary, ultra broadband channels at 300 GHz provide nearly flat channel transfer functions under both LOS and NLOS conditions. Directed NLOS paths can be utilized in case of direct ray blockage with the drawback of higher attenuations. Still, antenna displacements must be avoided as much as possible especially in the NLOS case. In the future, employing beamforming [29] in combination with planar antenna arrays could offer both high gains and easy adaptation to module displacements. V. AN EMPIRIC PATH LOSS MODEL The channel measurements presented previously have shown almost flat channel transfer functions (CTFs) in all cases. Even though reflected path components have been detected in the indoor environment, the pointing of the narrow beams of the TX

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Fig. 15. Theoretical free space loss as well as measured and linearly approximated path losses averaged between 300 and 310 GHz.

and RX antennas are always optimized either for the LOS or a directed NLOS path, suppressing any other multipath component. It is noteworthy that a certain broadening of the respective received path component is observed in time domain (cf. Fig. 12), which can mainly be attributed to the computation of the inverse discrete Fourier transform of the CTF and is not caused by multipath propagation. Therefore, a simple narrowband path loss model seems reasonably sufficient as a first step towards channel modeling in the investigated scenarios. Accordingly, applying an empiric one-slope model for the path loss PL [22, eq. (4.4.1)], (4) is proposed. Here, denotes the module distance in m, refers to the path loss at and means the propagation coefficient. Both, and are to be derived from experiments. Always assuming the best possible module alignment under LOS conditions, the mere path loss is determined from the measurements for several module distances and different antennas in the first [cf. Fig. 6] as well as for the direct path [cf. Fig. 12(a)] in the second scenario. This is done by correcting the measured parameter by the system influence and antenna gains. To account for the narrowband channel characterization, the CTFs are averaged over the whole bandwidth from 300 to 310 GHz. Then, a linear regression is applied in logarithmic domain to and . Basically, a good agreement between the avobtain eraged theoretical free space loss, the measurements and the reand amount gression line is observed in Fig. 15. Here, to 85.21 dB and 2.17, respectively. Nonperfect module alignments and further unconsidered losses like antenna mismatches at the mixer outputs cause path . losses exceeding the FSL by, e.g., 1.41 dB at Understandably, as the precise module pointing becomes increasingly difficult with larger module distances, increasing mispointing losses lead to a propagation constant slightly above 2. Furthermore, the wavelength of 1 mm can be considered small compared to the room dimensions so that no waveguiding effects are expected.

To finally evaluate the quality of the linear regression regarding the actual measurement points, the standard deviation between the measured and approximated path loss is calculated to amount to 1.44 dB. In conclusion, the simple path loss model proves well suited to describe the 300 GHz LOS channel characteristics as long as no MPCs are present. This condition can be assumed to be fulfilled, if highly directive antennas are employed. However, especially if antenna arrays with almost omnidirectional single array elements are investigated, the knowledge of the respective AoA, delay, phase, amplitude etc. of each MPC arriving at the array becomes necessary. Furthermore, multipath propagation and fading effects may also occur in rare module constellations as has been demonstrated in Section IV-B. Accordingly, the proposed model will have to be extended to a full three-dimensional broadband channel model in the near future. VI. CONCLUSION We have presented ultra broadband channel characterizations at 300 GHz using a Schottky diode mixer based measurement system. The system capabilities have been investigated. Measurements of absorption coefficients of typical building materials have been demonstrated. The diffraction at an edge has been examined and modeled as an estimate of the additional ray attenuation caused by ray blockage. It has shown, that even a slight shadowing of the direct ray will lead to high additional attenuations so that the utilization of a directed NLOS path becomes necessary. Regarding the measurement setup, the precision of the measurements compared to the simulations is limited due to the fact that no optical bench has been employed. In addition, a short range desktop scenario and a small indoor office scenario have been considered for the channel measurements. In both cases, channel transfer functions have been presented also with regard to module misalignments. Additionally, interference investigations have been carried out according to the two-ray model. As highly directive antennas have been used, it has proven necessary to avoid even slight displacements. The analysis of the temporal path characteristics in the indoor scenario has shown that symbol rates of up to several 10 GSymbols/s can be achieved for the channel still to be considered flat. This finding basically allows for a rather simple design of THz transceivers because only rather simple or even no equalizers are required. As a concluding result, an empiric narrowband path loss model has been proposed based on the observed channel transfer functions. All measurements obtained from the isolated propagation investigations, as well as from the channel characterizations can now be used as a guideline for the design of an appropriate 300 GHz PHY layer. For integrated system simulations, the recorded channel impulse responses can also be implemented as filters to account for realistic propagation conditions in the respective scenario. Future work based on the outcomes presented in this paper can be, e.g., the validation of a ray tracing algorithm with the measurements or the development of a more complex and realistic 300 GHz channel model, which especially requires a further extensive measurement campaign.

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REFERENCES [1] S. Cherry, “Edholm’s law of bandwidth,” IEEE Spectrum, vol. 41, no. 7, pp. 58–60, 2004. [2] R. Piesiewicz, T. Kleine-Ostmann, N. Krumbholz, D. Mittleman, M. Koch, J. Schoebel, and T. Kürner, “Short-range ultra-broadband terahertz communications: Concepts and perspectives,” IEEE Antennas Propag. Mag., vol. 49, no. 6, pp. 24–39, 2007. [3] [Online]. Available: http://www.virginiadiodes.com/ [4] C. Jastrow, K. Münter, R. Piesiewicz, T. Kürner, M. Koch, and T. Kleine-Ostmann, “300 GHz transmission system,” Electron. Lett., vol. 44, pp. 213–214, 2008. [5] C. Jastrow, S. Priebe, B. Spitschan, J. Hartmann, M. Jacob, T. KleineOstmann, T. Schrader, B. Spitschan, and T. Kürner, “Wireless digital data transmission at 300 GHz,” IEE Electron. Lett., vol. 9, pp. 661–663, 2010. [6] T. Nagatsuma, H. Song, Y. Fujimoto, K. Miyake, A. Hirata, K. Ajito, A. Wakatsuki, T. Furuta, N. Kukutsu, and Y. Kado, “Gigabit wireless link using 300–400 GHz bands,” in Proc. Int. Topical Meeting on Microw. Photon. (MWP), Oct. 2009, pp. 1–4. [7] D. Britz, Evolution of Extreme Bandwidth Personal and Local Area Terahertz Wireless Networks IEEE P802.15-15-10-0162-00-0thz [Online]. Available: https://mentor.ieee.org/802.15/dcn/10/15-100162-00-0thz-evolution-of-extreme-bandwidth-personal-and-localarea-terahertz-wireless-networks-white-paper.pdf, 2010 [8] F. Okano, M. Kanazawa, K. Mitani, K. Hamasaki, M. Sugawara, M. Seino, A. Mochimaru, and K. Doi, “Ultrahigh-definition television system with 4000 scanning lines,” in Proc. NAB Broadcast Eng. Conf., 2004, pp. 437–440. [9] T. Bird, “Terahertz radio systems: The next frontier?,” presented at the Workshop on the Appl. Radio Sci., Leura, Australia, Feb. 15–17, 2006. [10] R. Piesiewicz, M. Jacob, J. Schoebel, and T. Kürner, “Influence of hardware parameters on the performance of future indoor THz communication systems under realistic propagation conditions,” in Proc. Radar Conf. 2007 (EuRAD 2007), 2007, pp. 327–330. [11] C. Jansen, R. Piesiewicz, D. Mittleman, T. Kürner, and M. Koch, “The impact of reflections from stratified building materials on the wave propagation in future indoor terahertz communication systems,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1413–1419, 2008. [12] T. Bird, A. Weily, and S. Hanham, “Antennas for future veryhigh throughput wireless LANs,” in Proc. IEEE Antennas Propag. Soc. Symp., San Diego, CA, 2008. [13] C. Chong, C. Tan, D. Laurenson, S. McLaughlin, M. Beach, and A. Nix, “A new statistical wideband spatio-temporal channel model for 5-GHz band WLAN systems,” IEEE J. Sel. Areas Commun., vol. 21, 2003. [14] IEEE Standard for Local and Metropolitan Area Networks - Telecommunications and Information Ixchange between Systems - Local and Metropolitan Area Networks - Specific Requirements Part 11: Wireless LAN Medium Access Control (MAC)and Physical Layer (PHY) Specifications Amendment 5: Enhancements for Higher Throughput, IEEE Std. 802.11n-2009, WG802.11 - Wireless LAN Working Group. [15] T. Kürner, M. Jacob, R. Piesiewicz, and J. Schoebel, An Integr. Simul. Environ. Investigat. Future THz Commun. Syst.: Extended Version, vol. 84, no. 2–3, 2008. [16] A. Molisch, Wireless Communication. New York: Wiley, 2007. [17] S. Priebe, M. Jacob, C. Jastrow, T. Kleine-Ostmann, T. Schrader, and T. Kürner, “A measurement system for propagation measurements at 300 GHz,” presented at the Progress in Electromagn. Res. Symp. (PIERS), Cambridge, U.K., 2010. [18] D. Pozar, Microwave Engineering. New York: Wiley, 2007. [19] R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J. Infrared and Millimeter Waves, vol. 28, no. 5, pp. 363–371, 2007. [20] M. Hall, L. Barclay, and M. Hewitt, “Propagation of Radiowaves,” in Propagation of Radiowaves. Inst. Elect. Eng.: London, U.K., 1996. [21] J. Kunisch and J. Pamp, “Ultrawideband double vertical knife-edge model for obstruction of a ray by a person,” in Proc. IEEE Int. Conf. Ultra-Wideband (ICUWB), 2008, vol. 2. [22] R. Vaughan and J. Andersen, Channels, Propagation and Antennas for Mobile Communications. London, U.K.: IET, 2003.

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[23] M. Peter, R. Felbecker, W. Keusgen, and J. Hillebrand, “Measurementbased investigation of 60 GHz broadband transmission for wireless in-car communication,” presented at the IEEE 70th Veh. Technol. Conf. (VTC2009-Fall), Anchorage, AK, Sep. 2009. [24] M. Jacob, S. Priebe, C. Jastrow, T. Kleine-Ostmann, T. Schrader, and T. Kürner, “An overview of ongoing activities in the field of channel modeling, spectrum allocation and standardization for mm-wave and THz indoor communications,” presented at the IEEE Globecom, 2009. [25] R. Piesiewicz, C. Jansen, D. Mittleman, T. Kleine-Ostmann, M. Koch, and T. KÜrner, “Scattering analysis for the modeling of THz communication systems,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pt. 1, pp. 3002–3009, 2007. [26] T. Sarkar, Z. Ji, K. Kim, A. Medouri, and M. Salazar-Palma, “A survey of various propagation models for mobile communication,” IEEE Antennas Propag. Mag., vol. 45, no. 3, pp. 51–85, 2003. [27] T. Rappaport, Wireless Communications: Principles and Practice. Upper Saddle River, NJ: Prentice-Hall PTR, 2001. [28] M. Jacob and T. Kürner, “Radio channel characteristics for broadband WLAN/WPAN applications between 67 and 110 GHz,” presented at the EuCAP, 2009. [29] B. Van Veen and K. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Acoust., Speech, Signal Process. Mag., vol. 5, no. 2, pp. 4–24, 1988.

Sebastian Priebe was born in 1985. He received the Diploma degree in electrical engineering (with highest honors) from the Technische Universität (TU) Braunschweig, Germany, in 2009. During his diploma thesis, he had already begun working on propagation mechanisms at THz frequencies, and is pursuing the Ph.D. degree in the field of future THz communication systems. He is a Research Assistant with the Institut für Nachrichtentechnik, TU Braunschweig. Mr. Priebe was awarded a scholarship by the German National Academic Foundation.

Christian Jastrow was born in Ostercappeln, Germany, in 1981. He received the Diploma degree in electrical engineering from the Technische Universität Braunschweig, Germany, in 2008. During his diploma thesis, he had already begun working on a 300-GHz transmission system, which was characterized and used for first data transmission by him. Currently, he is pursuing the Ph.D. degree, while working as a Research Assistant with the Electromagnetic Fields Group at Physikalisch-Technische Bundesanstalt, Braunschweig. He is primarily concerned with field exposition experiments dealing with possible nonthermal effects of THz radiation. Furthermore, he is involved in channel and propagation measurements, as well as high data rate demonstration experiments at 300 GHz. Mr. Jastrow is a member of the VDE.

Martin Jacob (S’06) was born in Bielefeld, Germany, in 1982. He received the Diploma in electrical engineering from the Technische Universität (TU) Braunschweig, Germany, in 2007. Currently, he is pursuing the Ph.D. degree, while working as a Research Assistant with the Institut für Nachrichtentechnik, TU Braunschweig. He is the author of more than 20 technical journal and conference papers in the field of system and channel modeling for UWB, GPS, mm-wave, and THz systems. His current research interest lies in the field of wireless communication systems at frequencies of 60 GHz and above. His work mainly focuses on channel and propagation modeling, as well as propagation measurements. Dr. Jacob is a contributor to the IEEE 802.11ad 60-GHz WLAN channel model and is a member of the COST 2100 initiative.

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Thomas Kleine-Ostmann was born in Lemgo, Germany, in 1975. He received the M.Sc. degree in electrical engineering from the Virginia Polytechnic Institute and State University, Blacksburg, in 1999, and the Dipl.-Ing. degree in radio frequency engineering and the Dr.-Ing. degree, both from the Technische Universität (TU) Braunschweig, Germany, in 2001 and 2005, respectively. He was a Research Assistant with the Ultrafast Optics Group, Joint Institute of the National Institute of Standards and Technology and the University of Colorado, Boulder, (JILA) and with the Semiconductor Group, Physikalisch-Technische Bundesanstalt, Braunschweig, before he started working on the Ph.D. degree in the field of THz spectroscopy. Since 2006, he has been working as a permanent Scientist with the Electromagnetic Fields Group, PhysikalischTechnische Bundesanstalt. Currently, he is working on realization and transfer of the electromagnetic field strength, electromagnetic compatibility, and THz metrology. In 2007, he has become head of the Electromagnetic Fields Group. Dr. Kleine-Ostmann is a member of the VDE and the URSI. He received the Kaiser-Friedrich Research Award in 2003 for his work on a continuous-wave THz imaging system.

Thorsten Schrader (M’97) was born in Braunschweig, Germany, in 1967. He received the Dipl.-Ing. and Dr.-Ing. degree in electrical engineering from the Technische Universität (TU) Braunschweig, in 1992 and 1997, respectively. In 1998, he was with the EMC Test Systems LP, Austin, TX, (now ETS-Lindgren, Cedar Park, TX). In 1999, he joined the Working Group “High-frequency Measurement Techniques” of Physikalisch-Technische Bundesanstalt (PTB), Braunschweig. During 2000, he was a member of the Presidential Staff Office at PTB. In 2004, he became the Head of the Working Group “Electromagnetic Fields and Electromagnetic Compatibility.” Since 2005, he has been the Head of the Department “High-frequency and Fields” and since 2006, he is also responsible for the Working Group “Antenna Measurement Techniques.” His current interest is the metrology for RF quantities in the mmand sub-mm-wave range. Dr. Schrader is a member of VDE/VDE-GMA and VDI .

Thomas Kürner (S’91–M’94–SM’01) received the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree from the Universität Karlsruhe, Germany, in 1990 and 1993, respectively. From 1990 to 1994, he was with the Institut für Höchstfrequenztechnik und Elektronik (IHE), Universität Karlsruhe, working on wave propagation modeling, radio channel characterization, and radio network planning. From 1994 to 2003, he was with the Radio Network Planning Department at the headquarters of the GSM 1800 and UMTS operator E-Plus Mobilfunk GmbH & Co KG, Düsseldorf, Germany, where he was Team Manager of Radio Network Planning and Support, where he was responsible for radio network planning tools, algorithms, processes, and parameters. Since 2003, he has been a Professor of Mobile Radio Systems at the Institut für Nachrichtentechnik (IfN), Technische Universität Braunschweig. His working areas are propagation, traffic, and mobility models for automatic planning of mobile radio networks, planning of hybrid networks, car-to-car communications, as well as indoor channel characterization for high-speed short-range systems including future terahertz communication systems. He has been engaged in several international bodies such as ITU-R SG 3, UMTS Forum Spectrum Aspects Group, and COST 231/273/259/2100. Dr. Kürner has been a participant in the European projects IST-MOMENTUM and ICT-SOCRATES. Currently, he is chairing IEEE802.15 IG THz. He has served as Vice-Chair Propagation at the European Conference on Antennas and Propagation (EuCAP) in 2007 and 2009 and has been Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY since 2008. He is a member of VDE/ITG and VDI.

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Channel Simulator for Land Mobile Satellite Channel Along Roadside Trees Michael Cheffena and Fernando Pérez Fontán

Abstract—We study the signal fading for the land mobile satellite channel caused by roadside trees. A channel simulator is developed which takes into account the signal fading caused by position-dependent tree scattered fields and by swaying tree components. In the model, the tree canopy is modeled as a vertically oriented cylindrical volume containing randomly distributed and oriented leaves and branches. The tree trunk is modeled as a finite lossy dielectric cylinder. Leaves are modeled as thin lossy dielectric disks and branches as finite lossy dielectric cylinders. The scattering pattern of the new model has a narrow forward lobe with an isotropic background which is comparable to the one found using the radiative energy transfer theory. In addition, the variation of the specific attenuation with frequency of the model is fairly similar to the one given in the ITU Recommendation. The model is validated using measurements at 2 GHz in terms of the cumulative distribution functions of the received signal and the Ricean -factor, average fade duration and level crossing rate. Good agreement is found between the measured and simulated statistics.

K

Index Terms—Land mobile satellite (LMS) channel model, signal fading, vegetation.

I. INTRODUCTION

S

IGNAL propagation in the land mobile satellite (LMS) channel is affected by different propagation impairments. Among them are signal fading caused by vegetation. Channel models are needed for dimensioning LMS systems. Fade mitigation techniques (FMTs) such as diversity might be used to counteract the signal fading caused by vegetation. To design, optimize and test FMTs, data collected from propagation measurements are needed. However, such data may not be available at the preferred frequency, location, wind speed conditions, etc. Alternatively, time series generated from simulation models can be used. In this case, the simulated time series need to have similar statistical characteristics as those obtained from measurements. The signal attenuation by vegetation depends on a range of factors such as tree type, whether trees are in leaf or without leaf, whether trees are dry or wet, frequency, and path length through foliage [1], [2]. The ITU Recommendation P.833 [3] provides a Manuscript received June 11, 2010; revised September 21, 2010; accepted October 20, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the French Space Agency (CNES), the French Aerospace Lab (ONERA), and in part by the Thales Alenia Space-France. M. Cheffena is with CNES, F-31401 Toulouse, France (e-mail: [email protected]). F. Pérez Fontán is with the University of Vigo, E-36200 Vigo, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122297

model for predicting the mean signal attenuation through vegetation. As the mobile terminal travels along a road sided by trees, path length variations of the tree scattered multipath components will result in fading of the received signal [4], [5]. In [6], a model for predicting the position (angular) dependent coherent and incoherent scattered fields by a tree for urban terrestrial microcell environments was reported. In addition, the temporal variations of the relative phase of multipath components due to swaying of the tree (by wind) results in fading of the received signal as reported in, for example, [7]–[9]. The severity of the fading depends on the rate of phase changes which further depends on the movement of the tree components. A model for generating signal fading caused by a swaying tree was presented in [9]. In this paper we present a simulator for the LMS channel along roadside trees which takes into account the signal fading caused by the position-dependent tree scattered fields and by the swaying of tree components due to wind. The model is validated using available measurements at 2 GHz. Unlike the one reported in [5], the new model is applicable for different polarizations and elevation angles. The paper begins in Section II by discussing the LMS signal propagation along a road lined by trees and presents the developed channel simulator. Section III discusses the modeling of irregular shaped canopies. Comparison with models and measurements are given in Section IV. Finally, conclusions are presented in Section V. II. THE MODEL A. Fading Due to Position-Dependent Tree Scattered Fields As the mobile terminal travels along a road sided by trees, the signal is attenuated and scattered. In addition, the path length variations of the tree scattered components result in fading of the received signal. In [6], a model for predicting the position-dependent coherent and incoherent scattered fields for urban terrestrial microcells was reported. The tree canopy is modeled as a vertically oriented cylindrical volume in a rectangular coordinate system defined by the orthonormal vectors, , , and . The canopy volume contains randomly distributed and oriented and branches . Leaves are modeled as thin lossy leaves dielectric disks and branches as finite lossy dielectric cylinders with size and dielectric parameters shown in Table I. In the model, trees were assumed to have an infinite height, and the transmitter and receiver were set to the same height. This simplified the model and resulted in an horizontal (parallel to the ground) path between the transmitter and the receiver.

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TABLE I MEASURED SIZE AND DIELECTRIC (AT ABOUT 2 GHz) PARAMETERS OF BRANCHES AND LEAVES [6]

can be calculated as in [10]. Parameters where and are uniformly distributed within the range, and . The equivalent scattering amplitude per unit volume of the canopy is given by [6]

(4) where and are the number densities of leaves and branches, and refers to the th branch category. Similarly, the equivalent scattering cross-section per unit volume of the canopy is defined as [6]

(5) Fig. 1. Tree model for the LMS channel. x, y and z are the orthonormal vectors of rectangular coordinate system. R and H are the radius and height of the canopy. k and k are the incident and scattered propagation vectors.  and  are the elevation and azimuth directions of the incident field.  and  are the elevation and azimuth directions of the scattered field.

In this paper we extended the above model to include a tree trunk. In addition, a finite tree height with different transmitter and receiver heights are considered. This is applicable for the LMS case and results in an inclined path through the tree as shown in Fig. 1. The propagation vectors of the incident and fields are given by (see Fig. 1) scattered (1) (2) where , and are the orthonormal vectors of rectangular coordinate system. Angles and are the elevation and azimuth directions of the time varying incident field. Angles and are the elevation and azimuth directions of the scattered field. The mean scattering amplitude of a single branch or leaf is obtained by integrating and multiplying the scattering amplitude , by the probability densities of the tensor element, orientation of leaves and branches (which are independent of , and elevational, , each other) in the azimuthal, directions [6]

The signal scattered by a tree has a coherent and incoherent component. The coherent scattered field can be calculated using Foldy-Lax’s multiple scattering theory [11], [12]. For a uniform density of scatterers, the coherent scattered field is given by [13] (6) is an operator which relates the mean scattered where field at caused by a unit volume of random medium located at to the field, , incident on the volume, and is given by (7) where is the free-space function. For an observation point located far away from the canopy, it can be approximated by [13]

(8) where is the distance from the canopy center to the observation point. The incident field in the canopy volume is given by (9)

(3)

is the incident field at the center of the canopy volume where and is the wavenumber. Parameters and are the principal

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radii of curvature of the incident wavefront. Using (9), the mean incident field inside the canopy can be calculated as (10) where is the length of the path to through the tree canopy is an effective propagation constant inside the canopy, and expressed as [14] (11) Using (11) the specific attenuation through the tree canopy in decibels per meter (dB/m) can be obtained [6], [14] (12) The incoherent (diffuse) component is due to the macroscopic dielectric heterogeneity of the tree canopy [6], and can also be found using the multiple scattering theory [15]. For a uniform density of scatterers, the incoherent field intensity is given by [13]

(13) and which are defined in (1) and (2), where respectively. The trunk is modeled as a large vertically oriented dielectric cylinder with radius of 0.4 m, height of 10 m and relative permit(at about 2 GHz). It is reported in [16] that for tivity of such large element, most of the incident power is either attenuated or scattered at an angle close to the incident angle. Thus, its effect on the overall scattered field may be approximated to that of simple attenuation which can be found using the above described theory [16]. In general due to signal loss by the scatterers, the mean field in the canopy decays in its respective propagation direction. A measure of this decay is the skin depth, defined as [14] (14) If the depth of the tree is greater than the skin depth, then the coherent scattered component dominates over the incoherent scattered component in the forward direction. However, if the depth of the tree is less than the skin depth, the incoherent scattered component dominates over the coherent scattered component in the forward direction [14]. Examples of predicted coherent and incoherent scattered powers (relative to the power of the wave incident on the canopy center) for the LMS channel using (6) and (13) are shown in Fig. 2. Reported measured size and dielectric parameters of branches and leaves are used in the simulations, see Table I. Note from Fig. 2 that the scattered power increases with increasing canopy radius. This is because there are more scatterers (branches and leaves) in a canopy with a larger radius than with a smaller one which results in increasing scattered power. We can also observe that, the coherent scattered field is dominant over the incoherent scattered field in a small region that is approximately identical to the shadow region of the

Fig. 2. Coherent and incoherent scattered powers (relative to the power of the , , wave incident on the canopy center) for the LMS. f  ,R ,R ,H r d , and r .

= 60

=1m =5m

= 10 m

= 2 GHz = 30 = = = 50 m

canopy (i.e., approximately between 0 to 8 for and 0 to 18 for , see Fig. 2). Outside this region, the incoherent scattered field becomes much stronger than the coherent scattered field. Furthermore, the calculated skin depth and is greater than the depth of the tree (for both ) which suggests the dominance of the incoherent over the coherent scattered component in the forward direction. The results shown in Fig. 2 are similar to the ones reported in [13]. B. Fading Due to Swaying Tree Components The temporal variations of the relative phase of multipath components due to swaying of the tree components by wind results in fading of the received signal. The severity of the fading depends on the rate of phase changes which further depends on the movement of the tree components. A model for generating signal fading due to a swaying tree by utilizing a multiple mass-spring system to represent a tree and a turbulent wind model was presented in [9]. To reduce the computation time, a simplified model for signal fading due to swaying vegetation is presented in this paper. The swaying of branches and leaves are modeled as masses attached to springs. These masses sway according to the following differential equation (15) where is the time. Parameters , and are the mass, spring constant and damping factor of tree component , re, describes the stiffness of spectively. The spring constant, the wood material, while the damping factor, , describes the energy dissipation due to swaying (aerodynamic damping) and dissipation from internal factors such as root/soil movement and , internal wood energy dissipation [17]. Parameters and are the acceleration, velocity and position (displacement) of tree component , respectively. Parameter is the time varying induced wind force on tree component , and is given by [18] (16)

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TABLE II

k VALUES FOR DIFFERENT TERRAIN TYPES AT 10 m HEIGHT [19]

Fig. 3. Model for simulating wind speed. n(t) is a white Gaussian noise with zero mean and unite variance, H is the lowpass filter defined in (17), n (t) is a colored noise, k is a model parameter, w is the mean wind speed,  = w k , and w(t) is the resulting wind speed time series.

where is the drag coefficient, is the air density, projected surface area of the tree component, and wind speed. is given by [9] The transfer function of

is the is the

Fig. 4. Simulated signal fading due to swaying tree components during low average wind speed conditions (3 m/s). Simulation parameters: f = 2 GHz, N = 100, c = 20, k = 8 10 N=m, m = 2 kg, C = 0:35,  = 1:226 kg=m , A = 5 m , k = 0:285, and w = 3 m=s.

2

(17) and . and are the gain and where time constant of the filter, respectively, and are defined as (18) (19) where is the mean wind speed and the turbulence length scale that corresponds to the site roughness. The turbulence length can be found from the height, , above the ground, ex[19]. The standard deviation of the turpressed as , where is a bulent wind can be calculated as constant which depends on the type of the terrain, see Table II. is the sampling period and designates the beta function which is given by (20) Fig. 3 shows the model for simulating wind speed. The complex envelope of the signal fading due to swaying , can then be obtained by solving (15) for the vegetation, (see Appendix A), displacement of the tree components, then (21) where is the total number of scattering tree components, is the phase which is uniformly distributed within the range , and is the wavelength. Figs. 4 and 5 show examples of simulated signal fading caused by swaying tree components during low (3 m/s) and high (10 m/s) average wind speed conditions.

Fig. 5. Simulated signal fading due to swaying tree components during high average wind speed conditions (10 m/s). Simulation parameters: f = 2 GHz, N = 100, c = 20, k = 8 10 N=m, m = 2 kg, C = 0:35,  = 1:226 kg=m , A = 5 m , k = 0:285, and w = 10 m=s.

2

C. The Overall Fading The overall fading is due to position-dependent tree scattered fields (discussed in Section II-A) and due to swaying tree components by wind (discussed in Section II-B). Fig. 6 shows the proposed simulator for the LMS channel along a road sided by trees which is based on the Loo model reported in [20]. In the developed simulator, a complex white Gaussian noise is passed to a filter (such as the one reported in [21]) to produce the classical Jakes Doppler spectrum. The filter outputs are then multiplied to by the standard deviation of the incoherent component obtain the small scale (multipath) fading. Similarly, a real white Gaussian noise is low-pass filtered by a shadowing filter to obtain the appropriate autocorrelation function. The shadowing filter is defined as [22] (22)

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Fig. 6. LMS channel simulator for a road sided by trees.  (t) and q (t) are the standard deviation of the position-dependent incoherent and the mean total coherent (which is the sum of coherently scattered and free-space field) component, respectively.  (t) is the standard deviation of the total coherent component. v (t) (normalized by its mean value) is the fading due to swaying tree by wind.

Fig. 7. Irregular shaped tree model using multiple cylindrical volumes. Fig. 8. Scattering patterns of the RET ( new model.

with (23) is the sampling distance and is the correlation where distance. The outputs of the shadowing filter are first weighted by the standard deviation of the total coherent component and then added to the mean total coherent component, , (which is the sum of coherently scattered and free-space field) of the channel. The time series are then converted to linear scale to obtain the large scale fading (shadowing). The small and large scale fading time series are summed and then weighted by (normalized by its mean value) to account for the signal fading caused by wind swaying. Finally, the Doppler shift of the direct signal component is accounted for by multiplying the time , where is the phase corresponding to the series by traveled distance. The output from the simulator is a complex signal envelope which incorporates the total fading caused by vegetation.



= 7:5 ) and the

Section II-A can be applied to calculate the coherent and incoherent scattered fields. However, note that this will increase the computation time as the triple integrals given in (6) and (13) (used for estimating the coherent and incoherent scattered fields) have to be performed for each cylindrical volume in the tree canopy. This has been implemented in the developed LMS channel simulator. IV. COMPARISONS WITH MEASUREMENTS A. Comparison With Radiative Energy Transfer Theory A tree scattering model based on the theory of radiative energy transfer (RET) was reported in [23]. In the model, the vegetation is modeled as a statistically homogeneous random , is medium of scatterers. The RET scattering pattern, , with characterized by a narrow Gaussian forward lobe, , given by an isotropic background,

III. IRREGULAR SHAPED CANOPY The model presented in Section II could be used to describe the scattering characteristics of irregular shaped canopies. Any canopy shape could be constructed by using multiple cylindrical volumes of different sizes, see Fig. 7 for an example. The size and density of the leaves and branches in each volume can be chosen to be different. This allows in constructing an irregular shaped tree with varying size and density of scatterers. For each cylindrical volume in the tree canopy, the theory presented in

= 0:19 and

(24) with

expressed as (25)

where is the beamwidth of the forward lobe and is the ratio of the forward scattered power to the total scattered power. Fig. 8 shows the normalized scattering pattern of the RET model

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Fig. 9. Specific signal attenuation (dB/m) through vegetation obtained using (12) and the one reported in ITU-R P.833 [3] for vertical and horizontal polarizations.

Fig. 10. Measured total and mean attenuation time series at 2 GHz.

given in (24) and the new model (the sum of the coherent and incoherent scattered components given in (6) and (13)). Parameters of the RET scattering pattern are found by fitting (24) to the scattering pattern of the new model, and are estimated to and . We can observe that the two be models have similar scattering characteristics, i.e., a narrow forward lobe with isotropic background. B. Comparison With the ITU Recommendation We compared also the specific signal attenuation (dB/m) through vegetation obtained using (12) with the one reported in ITU-R P.833 [3] for vertical and horizontal polarizations, see Fig. 9. The frequency dependency of the dielectric constant of leaves, , is taken into account by using a semiempirical model which is valid in the frequency range of 1 to 100 GHz and for a salinity of about one parts per hundred (1%), defined as [24] (26) is the gravimetric moisture content. For leaves of a where was shown in [25] to be time invariant and have walnut tree, is the dielectric a value of approximately 0.8. Parameter constant of saline water. At room temperature (22 ) and for less than 11 parts per thousand [26] (27) where is the frequency in gigahertz and is the ionic conductivity of the aqueous solution in siemens per meter. Parameter is related to by [26]. For branches, the dielectric constants reported in [27] are used by performing linear interpolation to the desired frequency and moister content and average moisture con(a dry wood density of 0.7 tent of 40% [28] is assumed). Note that the specific attenuation given in ITU-R P.833 [3] is only a typical value, and may vary with tree species and scatterer density. Generally, we can observe that there is a fairly good agreement between the models. The deviation between the two

Fig. 11. Simulated total and mean attenuation time series. Simulation parameters are given in Tables I and III.

curves might be due to differences in angle of incidence, type, density, electrical and physical characteristics of the scatterers. C. Comparison With Measurements S-band measurements were performed in Gaillac, France, in 2006 by the French Space Agency (CNES) in order to characterize the narrowband propagation channel in well identified canonical elements such as tree sided roads for fixed positions of the transmitter. The measurements were taken at 2 GHz with an elevation angle of 30 . In the campaign, the mobile terminal is moving along a straight line over a distance of about 70 m. The distance between the mobile trajectory and the trees was equal to 6 m. In both sides of the road, trees were spaced by 9 meters which results in the mobile being shadowed by seven pairs of trees. The transmitter is positioned at an elevation angle of 30 in the perpendicular direction relative to the mobile trajectory. See [5] for more information on the measurements. These measurements are used here to validate the model on a short distance for this particular environment. Figs. 10 and 11 show examples of measured and simulated total and mean attenuation time series, respectively. The simulation parameters used are given in Tables I and III. The peri-

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TABLE III SIMULATION PARAMETERS

Fig. 14. Measured and simulated LCRs. Simulation parameters are given in Tables I and III.

Fig. 12. CDFs of the total and mean attenuations of the measured and simulated signals. Simulation parameters are given in Tables I and III.

Fig. 15. Measured and simulated AFDs. Simulation parameters are given in Tables I and III.

fading crosses a given threshold in the positive-going direction [30]. The AFD quantifies how long the signal spends below a given threshold i.e., the average time between negative and positive level-crossings [30]. Figs. 14 and 15 show the measured and simulated LCRs and AFDs. We can observe from Figs. 12 to 15 that the agreements between the measured and simulated first and second order statistics are fairly good.

K

Fig. 13. CDFs of the Ricean -factors of the measured and simulated signals. Simulation parameters are given in Tables I and III.

odic fading in the measured and simulated time series indicate the presence of trees along the road. The cumulative distribution functions (CDFs) of the measured and simulated total and mean attenuations are shown in Fig. 12. Comparison between the measured and simulated CDFs of the Ricean -factor (estimated using the moment-method reported in [29]) is presented in Fig. 13. Second order statistics such as the level crossing rate (LCR) and the average fade duration (AFD) describe the dynamic characteristics of the channel. The LCR measures the rapidity of the signal fading, and it determines how often the

V. CONCLUSION The LMS channel through roadside trees is subjected to signal fading caused by position-dependent tree scattered fields and by swaying tree components. These effects need to be taken into account for a realistic characterization of the channel. In this contribution we studied the signal fading due to vegetation in LMS channels. A channel simulator was developed for roadside trees which takes into account the signal fading caused by position-dependent tree scattered fields and by swaying tree components. The model is applicable for different polarizations and elevation angles. In the model, a tree canopy is modeled as a vertically oriented cylindrical volume containing randomly distributed and

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oriented leaves and branches. The tree trunk is modeled as finite lossy dielectric cylinder. Leaves are modeled as thin lossy dielectric disks and branches as finite lossy dielectric cylinders. In addition, by using multiple cylindrical volumes of different size, a method for modeling irregular shaped canopies with varying size and density of scatterers was presented. The scattering pattern of the new model has a narrow forward lobe with isotropic background which is similar to the one found using the RET theory. In addition, the variation of the specific attenuation with frequency is fairly similar to the one given in the relevant ITURecommendation. The model was validatedin terms of the CDFs of the received signal and the Ricean -factor, AFD and LCR using measurements at 2 GHz. The agreement found between the measured and simulated statistics was quite good. APPENDIX A DISPLACEMENT OF TREE COMPONENTS BY WIND of each tree component can be obThe displacement tained by solving (15) using state-space modeling (A.1) (A.2) where on (A.1) and (A.2): equal to

is the state vector. To satisfy (15) based , , and has to be (A.3)

Note that (A.1) and (A.2) are for continuous time and can be converted to discrete time using, e.g., bilinear transformation. REFERENCES [1] M. O. Al-Nuaimi and A. M. Hammoudeh, “Measurements and predictions of attenuation and scatter of microwave signals by trees,” IEE Proc. Microw. Antennas Propag., vol. 141, no. 2, pp. 70–76, Apr. 1994. [2] I. J. Dilworth and B. L. Ebraly, “Propagation effects due to foliage and building scatter at millimeter wavelengths,” in Proc. IEE Antennas Propag. Conf., Apr. 4–7, 1995, pp. 51–53. [3] Recommendation ITU-R P.833-6, Attenuation in Vegetation ITU. Geneva, Switzerland, 2007. [4] T. Sofos and P. Constantinou, “Propagation model for vegetation effects in terrestrial and satellite mobile systems,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1917–1920, Jul. 2004. [5] A. Abele, F. Perez-Fontan, M. Bousquet, P. Valtr, J. Lemorton, L. Castanet, F. Lacoste, and E. Corbel, “A new physical-statistical model of the land mobile satellite propagation channel,” presented at the COST Action IC0802 Workshop, Toulouse, France, Nov. 4–6, 2009. [6] Y. L. C. de Jong and M. H. A. J. Herben, “A tree-scattering model for improved propagation prediction in urban microcells,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 503–513, Mar. 2004. [7] M. H. Hashim and S. Stavrou, “Dynamic impact characterization of vegetation movements on radiowave propagation in controlled environment,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 316–318, 2003. [8] M. Cheffena and T. Ekman, “Modeling the dynamic effects of vegetation on radiowave propagation,” in Proc. IEEE ICC, Beijing, China, May 19–23, 2008, pp. 4466–4471. [9] M. Cheffena and T. Ekman, “Dynamic model of signal fading due to swaying vegetation,” EURASIP J. Wireless Commun. Netw.. Special Issue on Adv. Propag. Modeling for Wireless Syst., 2009, 10.1155/ 2009/306876, Article ID 306876. [10] M. A. Karam, A. K. Fung, and Y. M. M. Antat, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sens., vol. 27, no. 6, pp. 687–697, Nov. 1989. [11] L. L. Foldy, “The multiple scattering of waves,” Phys. Rev., vol. 67, no. 3, pp. 107–119, 1945. [12] M. X. Lax, “Multiple scattering of waves,” Rev. Mod. Phys., vol. 23, no. 4, pp. 287–310, 1951.

[13] Y. L. C. de Jong, “Measurement and Modelling of Radiowave Propagation in Urban Microcells,” Ph.D. dissertation , Techn. Univ. Eindhoven, The Netherlands, 2001. [14] S. A. Torrico, H. L. Bertoni, and R. H. Lang, “Modeling tree effects on path loss in a residential environment,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 872–880, Jun. 1998. [15] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: IEEE Press and Oxford Univ. Press, 1997. [16] P. Ferazzoli and L. Guerriero, “Passive microwave remote sensing of forests: A model investigation,” IEEE Trans. Geosci. Remote Sens., vol. 34, no. 2, pp. 433–442, Mar. 1996. [17] K. R. James, N. Haritos, and P. K. Ades, “Mechanical stability of trees under dynamic loads,” Amer. J. Botany, vol. 93, no. 10, pp. 1522–1530, 2006. [18] H. Peltola, S. Kellomäki, H. Väisänen, and V. P. Ikonen, “A mechanistic model for assessing the risk of wind and snow damage to single trees and stands of Scots pine, Norway spruce, and birch,” Can. J. For. Res., vol. 29, pp. 647–661, 1999. [19] European Standard for Wind Loads, Eurocode EN 1991-1-4, WIND ACTION. [20] C. Loo, “A statistical model for a land mobile satellite link,” IEEE Trans. Veh. Technol., vol. VT-34, no. 3, pp. 122–127, Aug. 1985. [21] T. Aulin, “A modified model for the fading signal at a mobile radio channel,” IEEE Trans. Veh. Technol., vol. 28, no. 3, pp. 182–203, Aug. 1979. [22] F. Perez-Fontan and P. M. Espineira, Modelling the Wireless Propagation Channel: A Simulation Approach With MATLAB. Chichester, U.K.: Wiley, 2008. [23] N. C. Rogers, A. Seville, J. Richter, D. Ndzi, N. Savage, R. Caldeirinha, A. K. Shukla, M. Al-Nuaimi, K. Craig, E. Vilar, and J. Austin, “A Generic Model of 1–60 GHz Radio Propagation Through Vegetation—Final Report,” Radio Agency, U.K., May 2002. [24] C. Matzler, “Microwave (1–100 GHz) dielectric model of leaves,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 6, pp. 947–949, Sep. 1994. [25] K. C. McDonald, M. C. Dobson, and F. T. Ulaby, “Using MIMICS to model L-band multiangle and multitemporal backscatter from a walnut orchard,” IEEE Trans. Geosci. Remote Sens., vol. 28, no. 4, pp. 477–491, Jul. 1990. [26] F. T. Ulaby and M. A. El-Rayes, “Microwave dielectric spectrum of vegetation-Part II: Dual-dispersion model,” IEEE Trans. Geosci. Remote Sens., vol. GE-25, no. 5, pp. 550–557, Sep. 1987. [27] G. I. Torgovnikov, Dielectric Properties of Wood and Wood-Based Materials. Berlin, Germany: Springer-Verlag, 1993. [28] J. D. Dell and C. W. Philpot, “Variations in the Moisture Content of Several Fuel Size Components of Live and Dead Chamise,” 1965, Res. Note PSW-RN-083. Berkeley, CA, U.S. Dep. Agriculture, Forest Service, Pacific Southwest Forest and Range Experiment Station. [29] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment-method estimation of the Ricean K-factor,” IEEE Commun. Lett., vol. 3, no. 6, pp. 175–176, 1999. [30] S. R. Saunders, Antennas and Propagation for Wireless Communication Systems. New York: Wiley, 2003.

Michael Cheffena was born in Eritrea, Asmara. He received the M.Sc. degree in electronics and computer technology from the University of Oslo, Norway, in 2005, and the Ph.D. degree from the Norwegian University of Science and Technology (NTNU), Trondheim, in 2008. He was a Visiting Researcher with the Communications Research Centre (CRC), Canada. From 2009 to 2010, he conducted Postdoctoral studies with the University Graduate Center (UNIK), Kjeller, Norway. Currently, he is a Postdoctoral Fellow with the French Space Agency (CNES), Toulouse. His research interests include modeling and prediction of radio channels for both terrestrial and satellite links.

Fernando Pérez Fontán was born in Villagarcía de Arosa, Spain. He received the Diploma in telecommunications engineering in 1982, and the Ph.D. degree in 1992, both from the Technical University of Madrid, Spain. He is a Full Professor with the Telecommunications Engineering School (University of Vigo). He is the author of a number of international magazine and conference papers. His main research interest is in the field of mobile fixed radiocommunication propagation channel modeling.

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Estimation of the Refractivity Structure of the Lower Troposphere From Measurements on a Terrestrial Multiple-Receiver Radio Link Pavel Valtr, Pavel Pechac, Senior Member, IEEE, Vaclav Kvicera, Senior Member, IEEE, and Martin Grabner, Member, IEEE

Abstract—A matched field technique is used for remote sensing of refractivity profile. The method relies on electromagnetic propagation simulation to estimate the height profile of radio refractivity in the lower atmosphere by comparing theoretical predictions with measurements on a terrestrial radio link. The objective is to discover whether small variations of the air’s refractive index can be detected in the lowest part of atmosphere, up to 150 m above the ground. The resultant estimations of the height profile of refractivity are compared with the measured refractivity profile obtained from meteorological sensors at different heights. The results show a close agreement between the estimated and measured refractivity profiles during periods of deep fading events. Index Terms—Electromagnetic refraction, microwave propagation modeling, microwave radio propagation meteorological factors, radio refractivity estimation.

I. INTRODUCTION HE lowest part of the atmosphere, formed by the troposphere, is a propagation medium that has an impact on various radio systems including terrestrial radio communication links. To a certain extent, the refractive properties of the atmosphere influence the performance of these systems. The troposphere, reaching up to 8–16 km above the earth’s surface, is an inhomogeneous medium with a refractive index spatially dependent on air temperature, pressure and humidity. This dependence makes the troposphere a predominantly horizontally stratified environment. Inhomogeneities of the refractive index of air may result in anomalous propagation such as ducting, where electromagnetic energy is trapped in an atmospheric layer that then behaves as a waveguide. In this way, the electromagnetic wave can travel far beyond the radio horizon formed by the earth’s curved surface. The duct can either occur at higher altitudes where the wave travels in an elevated trapping layer (elevated duct) or close to the ground where the wave is reflected from the earth’s surface (surface duct). A specific propagation mechanism known as the evaporation duct is created by the rapid

T

Manuscript received January 21, 2010; revised June 28, 2010; accepted October 13, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the Czech Science Foundation under Project No. 102/07/0955. P. Valtr and P. Pechac are with the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague, Czech Republic (e-mail: [email protected], [email protected]). V. Kvicera and M. Grabner are with the Czech Metrology Institute, 148 01 Prague, Czech Republic (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2011.2122234

decay of humidity above a large body of water. For most of the time, the evaporation duct is present above the surface of the sea. The refractive properties of air and its influence on electromagnetic wave propagation can be found in [1]–[3]. The ability to describe the behavior of the refractivity structure in the troposphere helps to predict the reliability of communication systems in terms of link availability and interference caused to other services. The usual methods of direct refractive index measurement by means of radiosounding and refractometers can usually only give a rough idea of the refractivity distribution. The basic disadvantage of the radiosonde, resulting from its occasional although regular launching, is that it is not able to provide a detailed model of refractivity structure time development. Another drawback of radiosondes is their low resolution, particularly in lower altitudes. However, high resolution radiosondes do exist, and are able to record atmospheric parameters with a spatial resolution of approx. 10 m [4]. New methods of refractivity structure sounding have been developed as an alternative to direct refractivity structure measurements. These methods rely on extensive propagation simulations and on solving an inverse problem of electromagnetic wave propagation, from the known field at the receiver, to retrieve the parameters of the propagation medium. The purpose of the experiment presented here is to measure received signal levels that are influenced by various refractivity conditions. The refractivity structure of the lowest atmospheric layer up to 150 m from the ground is then inferred from these measurements. The objective is to obtain the height dependence of the refractivity gradient with a high level of detail in order to be able to detect anomalous refractivity layers several tens of meters thick. II. REFRACTIVITY STRUCTURE ESTIMATION TECHNIQUES In recent years, matched field (MF) methods representing indirect methods of refractivity structure sounding have turned out to be a suitable alternative to direct measurement methods. MF was originally used in underwater acoustics to locate sources of acoustic waves and to determine oceanic environmental parameters [5]. Later, it was used to estimate the refractivity structure of the troposphere [6]. [7] reports estimated refractivity results in a coastal area at VHF and UHF frequency bands. More recently, [8], [9] performed estimation of refractivity profile from radar clutter concentrating on surface based ducts over sea area. The principle of the method for refractivity estimation is as follows: a field radiated by a distant transmitter and influenced

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by the propagation environment is measured. A number of simulations are run having specific known parameters (such as terrain profile or antenna parameters) and with other unknown parameters (refractivity profile) being set ’randomly’ for each simulation. The combination of environmental parameters that fits best with the measured values then represents the estimate of the unknown parameters. In this experiment the unknown parameters represent the refractivity profile, as will be explained later, and all other environmental parameters, i.e., position of the transmitter and terrain profile between the transmitter and the receiver are known. No other possible environmental influences (troposcatter, rain scattering, etc.) are taken into consideration. The principal problem with this approach is that it is ill-posed. It means that several different refractivity profiles can give results that are close to measured values. This effect can however be suppressed (at least in principle) by increasing the number of measurement points. When using MF techniques to estimate the refractivity structure from the radio link measurement, several prerequisites are required: — a model of refractivity as a function of the spatial coordinates; — a propagator – that is a procedure by which received field at the receiving antennas is simulated in the presence of the refractivity profile; — a criterion, or fitness function, that shows how closely the simulated field matches the measured field; — a selection process that drives the selection of suitable refractivity profiles from the set of all possible profiles to arrive at an acceptable solution in a reasonable time. The propagator used here is based on a parabolic equation method which is a numerical approach widely used in tropospheric propagation. The fitness function, a term coming from the field of genetic algorithms [10], gives the distance between the measured and simulated fields with the propagation environment characteristics as its input parameter. The principle of the estimation technique is to search through -dimensional parameter space where is the number of unknown parameters of the propagation environment, to find a parameterization of the environment that minimizes the output of the fitness function, i.e., the distance between measurement and simulations. It would in most cases be extremely time-consuming to test all possible combinations of the environmental parameters. Therefore a proper procedure must be applied to make the estimation process more time-efficient. The optimization problem requires finding a global extreme (maximum or minimum) of the -dimensional function. This task is typically solved by evolutionary algorithms. A brief description of the prerequisites mentioned above is provided below.

A. Refractivity Model

one and therefore for convenience the refractive properties of atmosphere are usually expressed in terms of refractivity , rather than the refractive index. Refractivity is defined as (1) and, although it is a dimensionless quantity, it is convenient to denote it using -units. A key aspect in radio refraction is the height gradient of refractivity, rather than its absolute value. In the so-called standard atmosphere, the height dependence of the refractive index and hence refractivity shows an almost exponential dependence. This dependence, at lower altitudes, can be approximated by a linear decrease [11]. For standard atmosphere the long-term mean value of the refrac. The standard refractivity tivity gradient is close to gradient causes the rays representing the electromagnetic wave to bend downwards, but the curvature of the earth exceeds the curvature of the rays, which prevents trans-horizon propagation and creates a shadow area behind the radio horizon range. A ray propagating under refractivity gradient equal to is parallel to the surface of the earth. produce ducting where the Gradients less than curvature of the rays exceeds the curvature of the earth and the wave can travel for a long distance behind the horizon. To account for the earth’s curvature, the modified refractivity is introduced defined as (2) where is the height above the earth’s surface in km. The modified refractivity clearly indicates ducting conditions; if the rays are bent back to the earth, whereas if , the rays run parallel to the earth’s surface. Because of the mostly height dependence of atmospheric pressure, temperature and humidity, atmosphere is predominantly horizontally layered, resulting in the height dependence of refractivity with much smaller gradients in a lateral direction [12]. Height profiles of refractivity in the lower atmosphere can often be represented by piece-wise linear models [13]. This type of model is shown in Fig. 1. It consists of three linear segments forming a total of five parameters: the gradient of the lowest layer , the height of the bottom of the ducting layer , the thickness of the ducting layer and its gradient . The last parameter is the gradient of the upper layer . This refractivity model can represent a wide range of refractivity profiles including an elevated or surface duct or constant gradient of refractivity at all heights. The modified refractivity . Since the received field at zero height is set to amplitude is only influenced by the gradient of refractivity and not by its absolute value, the estimation method is not . Its real value, if required, has to sensitive to changes in be determined by other means. B. Propagator Function

The refractive index of air, which is a spatial function of atmospheric pressure, temperature and humidity, is a key factor for refraction effects. The refractive index of air is close to

The parabolic equation (PE) method represents a numerical approach to solving propagation-related problems. PE is a reduction of the Helmholtz wave equation enabling efficient

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source signal. It is obtained from the propagation simulator. The objective is to find an environmental parameterization that minimizes the fitness function value (3) where the following definitions were adopted (4a) (4b)

Fig. 1. Refractivity model.

modeling of wave propagation problems involving propagation along one predominant direction. It can easily cope with the refractive effects of the atmosphere and terrain diffraction. The properties of PE make it a suitable approach to handle long-range radiowave propagation problems where the paraxial propagation requirement can easily be met. PE is mainly used to model wave propagation in the troposphere [14], [15], and to solve scattering problems [16]. The two common approaches for a numerical solution to parabolic equation are the finite difference and the Fourier transform methods, the latter being far more suitable for problems involving long-range propagation calculations due to its numerical efficiency. A comprehensive review of parabolic equation techniques and their application to various propagation problems can be found in [13]. The parabolic equation code implements the split-step algorithm using a perfectly conducting ground, because it is relevant at the frequency and polarization used (10.7 GHz, ) and also saves computational time. The radiation pattern of the transmitting antenna was approximated by a Gaussian beam. The profile of the terrain was modeled by staircase approximation with a range step of 100 m. The maximum height of the computational domain of the split-step algorithm was 500 m. The absorbing layer in the top-most 200 m was used to prevent reflections from the non-existing upper boundary of the computational domain. The refractivity profile was taken from the measurements at the receiving mast and is assumed to be approximately constant over the whole link. C. Fitness Function Central to the MF technique is the ability to properly express the suitability of a particular trial solution. The fitness function expresses the measure of distance between the measured and simulated results for a particular environmental parameterization . The measurement is represented by a vector of complex numbers representing complex received signals at the -element receiver array. The simulated result with environmental parameterization can be expressed as where is a transfer function for a given parameter and represents the complex source signal. The transfer function is a complex vector of received fields on the array with a unitary

Two fitness functions were used in the optimization algorithm to compare their performance. The first is given by (3) where is the measurement vector of magnitudes of received signal is the result given levels at individual receiving antennas; by PE for the parameterization vector consisting of the five parameters of the above-mentioned refractivity model and for the unitary source signal. The second fitness function is the Bartlett fitness function [7]

(5) The value of this fitness function ranges from zero to one. It will be equal to zero if and only if the simulated vector values are a complex multiple of the measured vector. This means that the Bartlett fitness function discards information about the absolute magnitude of the received field. Since no phase information is available from the measurements, only relative amplitudes at the receiving antennas are considered. D. Optimization Procedure Evolutionary algorithms are particularly suitable for finding the global extreme of high-dimension functions with a number of local extremes. The algorithms are stochastic search procedures that use the concept of a population of possible solutions. The population develops so that the members of the population that are more suitable from the point of view of the fitness function are preferred as the population proceeds towards future generations. The whole process is supposed to tend towards a global extreme after a certain number of generations or loops. Typical examples of evolutionary algorithms are genetic algorithms, now also used in the electromagnetic domain [17]. The optimization procedure used here is the self organizing migrating algorithm (SOMA). In contrast to genetic algorithms, which use mapping of the searched space into a binary string, SOMA employs direct mapping of the search space using a population of individuals to search for the global extreme of a function [18]. The principle of the SOMA algorithm for searching the global maximum of a one-dimensional function is shown in Fig. 2. First, an initial random population is generated. The individual with the best fitness function value is identified as the ’leader’ and the other members in the population approach the leader by steps, one after the other, as shown in Fig. 2 and occupy

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Fig. 3. Terrain profile with the positions of transmitter and receiver, path loss (dB) spatial distribution in standard atmosphere calculated using PE. Fig. 2. One migration cycle of SOMA algorithm.

the positions with the higher value of the investigated function. This procedure is called the migration cycle. At the beginning of each migration cycle, a new leading individual is identified. The schema with one leading individual in each migration cycle is called the all-to-one schema. An all-to-all schema where the leading individuals in each migration cycle are sequentially represented by all individuals in the population is also possible. The principle described can also be directly applied to multi-dimensional functions. Several parameters must be chosen when working with SOMA. Parameter mass determines how far behind the leader , for example, a particular individual gets. When the the individual stops at the position of the leader. The mass parameter should be greater than one, i.e., the individual should be behind the leader to avoid ending up in a local extreme. A is usually sufficient. Another parameter is value of the step; it determines the density of mapping of the search space. The number of individuals in the population should be chosen appropriately. A higher number enables more detailed mapping of the search space, but increases the optimization time, especially in the case of all-to-all searching schema. III. MULTIPLE-RECEIVER RADIO LINK EXPERIMENT An experimental multiple-receiver radio link was set up in Central Bohemia, the Czech Republic, to provide input measurement data for refractivity estimation in mid-latitude continental climate zone. Latitude and longitude of the transmitter were 50.08 and 14.45 , respectively. Corresponding coordinates of the receiver were 50.14 and 15.14 . The length of the link was 49.8 km. Output power of the transmitter was 20 dBm at a frequency of 10.671 GHz. Transmitting antenna was a horizontally polarized 0.65 m parabolic dish with 3 dB beam width of 1.7 and gain of 33.6 dBi. Transmitted signal was amplitude modulated with frequency of 1 kHz. On the receiver side, the received signal was down-converted to intermediate frequency of 221.6 MHz before amplification and detection. Sampling frequency of the output of the signal strength

detector is 10 Hz. The measurement system can detect signals to 10 dB relative to the long-term mean in the range of received level. The transmitting antenna was placed 126.3 m above local ground level. Five receiving antennas of the same type as the transmitting antenna were placed on the receiver mast at heights of 51.5 m, 61.1 m, 90.0 m, 119.9 m and 145.5 m above the local ground level. The height of the receiving antennas was chosen so that the second antenna from the ground was approximately on a line connecting the transmitter and the top of the dominant hill 33 km from the transmitter. The lower-most antenna is below this line and other antennas operate under radio visibility. Fig. 3 shows the terrain profile between the transmitter and the receivers assuming standard refractivity. The positions of the transmitting antenna and the receiving antennas are indicated. The inclination of the path between the transmitter and the top-most antenna is 0.09 . The inclination of the path between the transmitter and the lower-most antenna is 0.20 . The readings in terms of received power magnitude were taken each second. Along with received power measurements, atmospheric pressure, temperature and air humidity values were taken from the sensors each minute to reconstruct the refractivity profile corresponding to the measured power levels. The sensors were placed on the receiver mast at heights of 5.1 m, 27.6 m, 50.3 m, 75.9 m, 98.3 m and 123.9 m above ground level. IV. METHOD VALIDATION As a validation of the presented approach of refractivity structure sounding, estimations were carried out using simulated received levels on a set of randomly selected refractivity profiles. Using simulated field instead of measurements removes the inaccuracy caused by propagation prediction errors and shows the performance of the inverse algorithm itself. Two sets of estimations are shown here. The first one has receiving antennas placed at heights of the actual antennas, i.e., 51.5 m, 61.1 m, 90.0 m, 119.9 m and 145.5 m above the local ground level. The other one uses simulations at heights 20 m, 50 m, 150 m, 200 m and 250 m above local ground. The original refractivity gradient profiles were generated randomly using tri-linear refractivity gradient

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Fig. 4. Profile estimations. Original profiles, thick solid line and its estimation, dotted line (lower part) and root mean square error (upper part). Simulated received levels taken at 51.5 m, 61.1 m, 90.0 m, 119.9 m and 145.5 m. Fig. 6. Refractivity profile constant with the sea level (top) and copying the terrain profile (bottom).

V. RESULTS

Fig. 5. Profile estimations. Original profiles, thick solid line and its estimation, dotted line (lower part) and root mean square error (upper part). Simulated received levels taken at 20 m, 50 m, 150 m, 200 m and 250 m.

model in Fig. 1. To produce realistic profiles, the values of graand were taken from measured statistical disdients , tributions of refractivity gradients at respective heights [19]. In total, some 100 profiles were tested. In the estimation procedure the population size was set to 16; the number of generations was 5. The fitness function (3) was used. Original and estimated profiles of vertical refractivity gradient were compared by means of a root mean square error. For error (M/km) was calculated as: every vertical profile, the

(6) , (M/km) are estimated and original where in the interval (measured) vertical gradients at the height of 0 m – 200 m from the ground. Fig. 4 shows selected refractivity profiles and their estimations for the first set of profiles. On top error (6) is shown. Fig. 5 presents the result for of that, the the second set of profiles with higher range of antenna heights. It can be observed, by visual inspection, that better estimation is obtained by using antennas covering higher range of heights from 20 m to 250 m from local ground. In this case, 69% of below 150 M/km in the lowest profile estimations have 200 m where the value of 150 M/km was identified as the value where there is very good visual fit between original of and estimated profiles. In the case of receiving antennas in the below 150 M/km is obtained range of 51 m to 145 m, the in 60% of cases.

Estimates of the refractivity profile were made for days with no rain events to eliminate any possible propagation mechanisms apart from atmospheric refraction and terrain diffraction. The estimates were made for two interpretations of the measured refractivity profile as shown in Fig. 6. The first possible interpretation is the refractivity constant with sea level; the other interpretation is the refractivity profile copying the terrain. On top of that, the results shown use two fitness functions given by (3) and (5). Table I gives an overview of the parameters used by the optimization technique and the intervals of the parameters of the refractivity model. The boundary values of these parameters were chosen by compromise so that sufficiently broad intervals were searched to achieve realistic profiles on one hand but so that the searched parameter space was not too big for efficient optimization on the other hand. The particular values of the vertical gradients are related to the refractivity gradient statistics obtained at the receiver site [19]. The parameters of the SOMA algorithm used are as follows: the population size was set to 12; the dimension of the fitness function is 5, which is the number of parameters of the refractivity profile. For each profile estimation the number of generations reached is 5. Fig. 7 shows measured received power in terms of excess attenuation relative to free space loss on individual receiving antennas. The measurements shown are averaged values using a sliding window filter with a length of 30 min. In the lower part of the figure hourly averages of the measured refractivity profiles are shown for reference. The estimations of refractivity profile were obtained every hour using averaged measurements of measured power at the respective times. Simulated attenuations at the receiving antennas corresponding to estimated refractivity profiles are also shown. The significant difference between the measured attenuation and the attenuation corresponding to the estimated refractivity is caused by the fact that the estimations were obtained using the Bartlett fitness function (5), which fits relative and not absolute magnitudes. It can be observed that for almost the same combination of measured attenuations at two different times the estimations can differ significantly. That is caused by the fact that several different refractivity profiles can give the same simulated attenuation at the particular heights.

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TABLE I PARAMETERS OF ESTIMATIONS.

Fig. 7. Refractivity profile estimation on 04.11.2008. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

This effect can be avoided by using higher number of receiving antennas. Fig. 8 shows estimations on the same day using a different fitness function (3). Better fit between simulated attenuations corresponding to estimated refractivity and measurements can be observed. Fig. 9 shows estimation results that differ from Fig. 8 in the interpretation of the refractivity profile, see Table I. It can be seen from Figs. 7 and 8 that good agreement between estimated and measured refractivity profile can be achieved when the attenuation at the receiving antennas is substantially higher than free space loss as between 01:00 and 03:00, at 20:00, 22:00 and 23:00, i.e., when the attenuation at the lowest antennas is higher than approx. 10 dB relative to free space loss. In other cases the estimations are not reliable. Results in Fig. 9 corresponding to the refractivity profile constant with sea level show no systematic agreement between estimated and measured refractivity profiles. The fading on 04.11.2008 between 01:00 and 04:00 was caused by change of refractivity gradient at particular range of heights combined with diffraction by path obstacle, namely subrefractive gradient up to 75 m from ground and standard gradient above at 01:00 and standard gradient up to 50 m and subrefractive gradient above at 03:00. The fading at 20:00 at the two lowest antennas was caused by subrefractive gradient

Fig. 8. Refractivity profile estimation on 04.11.2008. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows the corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

Fig. 9. Refractivity profile estimation on 04.11.2008. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows the corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

up to 75 m from ground while above 75 m the gradient is close to standard atmosphere.

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Fig. 10. Refractivity profile estimation on 09.10.2008. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows the corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

Fig. 12. Refractivity profile estimation on 03.05.2009. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

Fig. 11. Refractivity profile estimation on 09.10.2008. The thick lines in the top part show measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows the corresponding measured refractivity profile (thick line) and its estimation (thin line). Crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

Fig. 13. Refractivity profile estimation on 03.05.2009. The thick line in the top part shows measured power at 145 m, 120 m, 90 m, 61 m and 51 m (from the top); the bottom part shows the corresponding measured refractivity profile (thick line) and its estimation (thin line). The crosses in the upper part show simulated attenuation corresponding to the estimated profiles.

Fig. 10 shows refractivity estimation on 09.10.2009 using the Bartlett fitness function and the refractivity height profile copying the terrain. The estimated profile matches the measurements at times where deep fading events occur between 01:00 and 05:00. The fading was caused by diffraction on the path obstacle in combination with multipath propagation influencing mainly the three upmost receivers. Fig. 11 shows estimated results from the same day using a different fitness function and a different interpretation of the refractivity height profile. Less systematic agreement between the measured and estimated refractivity profile is observed here. However sub-refractive gradients are well estimated at least in the upper part of the vertical profiles in morning hours. Figs. 12 and 13 show refractivity estimations from 03.05. 2009. No attenuation at any of the receiving antennas exceeds

10 dB relative to free space loss and the estimations are not generally reliable, although at times the measured and estimated refractivity profiles are very close. The fading of the received power between 01:00 and 06:00 and between 20:00 and 24:00 were caused by irregularities in refractivity profile. Measured and estimated profiles of vertical refractivity gradient are compared by means of a root mean square error. error is calculated using (6). For every vertical profile, the and meters above Specifically in this case, the ground. The values of measured refractivity are linearly . The results are depicted in interpolated to obtain Fig. 14 and also in Table I, where the daily mean and maximum errors are specified, see the third last column values of and the second last one. The poor fit at 3:00 in Fig. 9 seems to be related to the following facts. The fading that occurred

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the two fitness functions used in terms of estimation accuracy. It can be also observed that the estimated refractivity profile shows better agreement with measured profile if the refractivity profile is considered to be copying the terrain. A good agreement between estimated and measured refractivity profiles was observed during deep fading events caused by diffraction on the path obstacle during sub-refractive atmospheric conditions. In the case of the measured power being close to free space loss the estimations are less reliable because the height dependence of received signal is not very sensitive to vertical profiles of refractivity.

Fig. 14. Time dependence of the root mean square error of measured and estimated vertical refractivity gradient, see Table I for daily mean rms values.

Fig. 15. Estimation of several selected profiles, measured profiles (thick line) and their estimation (thin line).

on the two lowest receivers was likely caused by diffraction on the path obstacle. This fading was greater at 2:00 where a better estimation of sub-refractive gradients was achieved. However at 3:00, the smaller diffraction fading was combined with multipath propagation causing the attenuation of about 5 dB on the three highest paths. This fact and also the assumption of refractivity profile constant with sea level is a possible reason why the conditions of propagation medium are worse defined at 3:00 and the estimated profile is not clearly a sub-refractive one. Fig. 15 shows estimations of several refractivity profiles in selected moments of high measured attenuation. Estimations were done using Bartlett fitness function and refractivity height profile copying terrain. Fig. 15 shows, in agreement with other presented results, that deep diffraction fading events are better identifiable by the estimation method.

VI. CONCLUSION A method for the remote sensing of refractivity was presented. The method uses measurement data from a terrestrial radio link and extensive propagation simulation to estimate the height profile of radio refractivity. Estimated results were given as a time development of the refractivity height profile over three different days. The results were obtained using two different fitness functions and two different interpretations of the measured refractivity height profile. From the results presented it can be concluded that there is no significant difference in

REFERENCES [1] H. V. Hitney, J. H. Richter, R. A. Pappert, K. D. Anderson, and G. B. Baumgartner, “Tropospheric radio propagation assessment,” Proc. IEEE, vol. 73, pp. 265–283, Feb. 1985. [2] M. P. M. Hall, Effects of the Troposphere on Radio Communication. London, U.K.: Peter Pelegrinus, 1979, ch. 2. [3] D. C. Livingstone, The Physics of Microwave Propagation. Englewood Cliffs, NJ: Prentice Hall, 1970. [4] S. L. Lystad, “Improved clear-air refractivity parameters and atmospheric description using high resolution radiosondes,” presented at the XXVIIth URSI General Assembly, Maastrich, The Netherlands, 2002, F1.O.5. [5] A. B. Baggeroer, W. A. Kuperman, and P. N. Mikhalewsky, “An overview of matched field methods in ocean acoustics,” IEEE J. Ocean. Eng., vol. 18, pp. 401–424, Oct. 1993. [6] D. F. Gingras, P. Gerstoft, and N. L. Gerr, “Electromagnetic matchedfield processing: basic concepts and tropospheric simulations,” IEEE Trans. Antennas Propag., vol. 45, pp. 1536–1545, Oct. 1997. [7] P. Gerstoft, D. F. Gingras, L. T. Rogers, and W. S. Hodgkiss, “Estimation of radio refractivity structure using matched-field array processing,” IEEE Trans. Antennas Propag., vol. 48, pp. 345–356, Mar. 2000. [8] C. Yardim, P. Gerstoft, and W. S. Hodgkiss, “Estimation of radio refractivity from radar clutter using Bayesian Monte Carlo analysis,” IEEE Trans. Antennas Propag., vol. 54, pp. 1318–1327, Apr. 2006. [9] C. Yardim, P. Gerstoft, and W. S. Hodgkiss, “Tracking refractivity from clutter using Kalman and particle filters,” IEEE Trans. Antennas Propag., vol. 56, pp. 1058–1070, Apr. 2008. [10] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. New York: Addison-Wesley, 1989. [11] B. R. Bean and E. J. Dutton, Radio Meteorology. New York: Dover Publications, 1968. [12] K. H. Craig, “Clear-air characteristics of the troposphere,” in Propagation of Radiowaves, L. Barclay, Ed., 2nd ed. London, U.K.: Inst. Elect. Eng. Press, 2003, pp. 103–128. [13] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. London, U.K.: Inst. Elect. Eng. Press, 2000. [14] D. Dockery, “Modeling electromagnetic wave propagation in the troposphere using the parabolic equation,” IEEE Trans. Antennas Propag., vol. 36, pp. 1464–1470, Oct. 1988. [15] J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci., vol. 26, pp. 381–393, 1991. [16] M. F. Levy and P. P. Borsboom, “Radar cross-section computations using the parabolic equation method,” Electron. Lett., vol. 32, pp. 1234–1236, Jun. 1996. [17] Y. Rahmat-Samii and E. Michielsen, Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [18] I. Zelinka, “SOMA – Self-organising migrating algorithm,” in New Optimization Techniques in Engineering, G. C. Onwubolu and B. V. Babu, Eds. Berlin, Germany: Springer-Verlag, 2004, pp. 167–218. [19] M. Grabner and V. Kvicera, “One-year statistics of atmospheric refractivity in the lowest troposphere,” in Proc. 6th IASTED Int. Conf. on Antennas, Radar, and Wave Propagation, Banff, Alberta, Canada, 2009, pp. 49–53.

VALTR et al.: ESTIMATION OF THE REFRACTIVITY STRUCTURE OF THE LOWER TROPOSPHERE

Pavel Valtr received the Ing. (M.Sc.) and Ph.D. degrees both in radio electronics from the Czech Technical University in Prague, Czech Republic, in 2004 and 2007, respectively. From 2007 to 2009, he was a Research Fellow at the University of Vigo, Vigo, Spain, working on various topics in electromagnetic wave propagation including rough surface and vegetation scattering and land mobile satellite channel modeling. In 2009, he joined the European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands, as a Postdoctoral Research Fellow. His research interests include wireless and satellite communications and computational methods in wave propagation. Dr. Valtr received the Young Scientist Award of XXVIII General Assembly of the International Union of Radio Science (URSI) in 2005.

Pavel Pechac (M’94–SM’03) received the M.Sc. degree and the Ph.D. degree in radio electronics from the Czech Technical University in Prague, Czech Republic, in 1993 and 1999, respectively. He is currently a Professor in the Department of Electromagnetic Field at the Czech Technical University in Prague. His research interests are in the field of radiowave propagation and wireless systems.

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Vaclav Kvicera (M’93–SM’05) was born in Podebrady, Czech Republic, in 1948. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Czech Technical University in Prague, Czech Republic, in 1971 and 1986, respectively. He is a Senior Researcher at the Department of Frequency Engineering, Czech Metrology Institute in Prague. His research activities are in the field of radiometeorology and electromagnetic wave propagation for terrestrial communications in the frequency band from 1 GHz to nm wavelengths of free space optical lines.

Martin Grabner (M’01) was born in Prague, Czech Republic, in 1976. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Czech Technical University in Prague, Czech Republic, in 2000 and 2008, respectively. He is a Researcher at the Department of Frequency Engineering, Czech Metrology Institute in Prague. His research activities are in the field of electromagnetic wave propagation for terrestrial communications in the frequency band from 1 GHz to nm wavelengths of free space optical lines.

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Communications Small Broadband Antenna Composed of Dual-Meander Folded Loop and Disk-Loaded Monopole Wang-Ta Hsieh and Jean-Fu Kiang

Abstract—A small broadband antenna composed of a dual-meander folded loop and a disk-loaded monopole is designed and measured. Measurement results confirm a fractional bandwidth of 10.8% with the voltage standing-wave ratio (VSWR) less than 2. The measured radiation efficiency exceeds 65% over the whole band. This antenna has an omnidirectional radiation pattern of vertical polarization in the horizontal plane. The ground plane size is also optimized to achieve the above radiation characteristics. Index Terms—Broadband, disk-loaded monopole, meander, small antenna.

I. INTRODUCTION With the raised demand for wireless devices, small antennas with low profile, light weight and easy fabrication have been widely explored. As the antenna size decreases, its bandwidth tends to reduce because of higher quality factor [1], rendering broadband characteristic a challenging task. Different techniques have been used to reduce the antenna size, including top loading element [2], folded meander-line [3], multi-folded tapered monopole [4], inductive or capacitive loading elements [5], [6]. Space-filling geometry has also been useful to reduce antenna size [7]–[9]. When applying these miniaturization techniques, the antenna efficiency usually degrades and the impedance bandwidth also becomes narrower. Recently, bandwidth enhancement techniques have been studied, including mode coupling between proximity elements [10], [11], adding vertical lines to a meander line [12], dual meander sleeves [13]–[15]. In [16], a 3-D meander dipole antenna with dimensions of =11 is proposed, its fractional impedance bandwidth is about 2.5%. A small spherical wire antenna with diameter of =5 has also been designed, its fractional bandwidth is about 6.7%. In [17], a low-profile metamaterial ring antenna with dimensions of =11 2 =11 2 =28 is proposed, its fractional bandwidth is about 6.8% and its measured efficiency is about 54%. In this work, a small antenna composed of a dual-meander folded loop and a disk-loaded monopole is proposed. A wide impedance bandwidth is achieved by merging the resonant bands of the loop and the monopole with proximity coupling. Design considerations and geometry of the proposed antenna are described in Section II, simulated and measured results are presented and discussed in Sections III and IV, followed by the conclusion.

Manuscript received February 12, 2010; revised September 17, 2010; accepted October 16, 2010. Date of current version May 04, 2011. This work was supported by the National Science Council, Taiwan, ROC under Contract NSC 98-2221-E-002-050. The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122236

Fig. 1. Panoramic views of the proposed antenna from different looking angles.

II. DESIGN CONSIDERATIONS Fig. 1 shows the panoramic views of the dual-meander folded loop antenna and a disk-loaded monopole from two different looking angles. The connection to the 50 coaxial cable and the disk-loaded monopole can be seen in Fig. 1(b). Fig. 2(a) shows the layout of the unfolded dual-meander loop, Fig. 2(b) and 2(c) show the side view and top view, respectively, of the complete design. The impedance bandwidth of a small antenna is roughly proportional to 1=Q, where the Q factor of a small antenna with dimension a can be estimated as [1]

Q'

1 (ka)3

which implies that a smaller antenna is endowed with a larger quality factor. To increase the impedance bandwidth of a dual-meander folded loop, a disk-loaded monopole is placed inside the former. A resonant mode of the disk-loaded monopole is excited by properly adjusting the gap width g between the disk and one surface of the dual meanders. The resonant bands associated with the dual-meander and the diskloaded monopole, respectively, merge into one if the width of the square disk, Ld , and the gap width g are properly adjusted. Note that larger Ld gives a larger overlapping area between the disk and the dual-meander strips, hence stronger capacitive coupling.

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Fig. 3. Measured ( ) and simulated ( ) reflection coefficients, L = 20, L = 20, L = 19:8, G = 130, G = 110, a = 7:5, b = 1, a = 5, a = 4:5, b = 4:7, d = 1, w = 2, g = 1, D = 1, L = 14:7, all in mm.

Fig. 2. (a) Layout of unfolded dual meanders with five loops, (b) side view, and (c) top view of proposed antenna.

From another point of view, the disk-loaded monople serves as a parasitic element to the dual-meander loop. Over the band of interest where two resonant bands merge, the electrical length of the dualmeander loop is about one wavelength. The disk-loaded monople is coupled to an adjacent meander strip, effectively extending its electrical length to about one-quarter wavelength. Hence, the physical size of the monopole plus the disk can be much shorter than one-quarter wavelengths. III. RESULTS AND DISCUSSIONS Fig. 3 shows the measured and simulated reflection coefficients of the proposed antenna using HFSS. The antenna takes a cubic space of 20 2 20 2 19:8 mm3 , mounted on a 110 2 130 mm2 horizontal

Fig. 4. Simulated surface current distribution at (a) 1.445 GHz and (b) 1.545 GHz.

ground plane. The center frequency is about 1.495 GHz, and the 10-dB bandwidth is about 161 MHz, hence its fractional bandwidth is about 10.8%. The measurement and simulation results match fairly. Fig. 4(a) and (b) show the side view of the simulated surface current distribution at 1.445 and 1.545 GHz, respectively. The surface current on the meander of the opposite side is the same due to symmetry. The one-wavelength resonance of the dual-meander folded loop appears at about 1.445 GHz. At 1.545 GHz, the quarter-wavelength resonance occurs, the current flows uniformly along the vertical part of the monopole to the disk, then is capacitively coupled to the meander strips underneath, as shown in Fig. 4(b). Fig. 5 shows the comparison of simulated reflection coefficients with and without the disk-loaded monopole. In the absence of the disk-

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Fig. 5. Comparison of simulated reflection coefficients with ( without (

Fig. 6. Simulated reflection coefficients with different disk sizes,

:

: L = 14:2 mm, ( 1 ): L = 13:7 mm, = 13:2 mm, other parameters are the same as in Fig. 3.

14 7 mm,

L

) and

) disk-loaded monopole, parameters are the same as in Fig. 3.

Fig. 7. Simulated reflection coefficients with different gaps between disk and patch, : g = 1 mm, : g = 1:5 mm, 1 : g = 2 mm,  : g = 2:5 mm, other parameters are the same as in Fig. 3.

:L = 

:

loaded monopole, the fractional impedance bandwidth of the dual-meander folded loop is about 4.1%. In the presence of the disk-loaded monopole, the impedance bandwidth is increased to 10.8%. The total length of the dual-meander folded loop is 260 mm, about 0.82 wavelengths at its resonant frequency of 1.4 GHz. The total length of the second resonant path is 43 mm, about 0.22 wavelengths at its resonant frequency of 1.785 GHz, which is measured from the contact point at the ground to the center of the disk, then to the edge of the disk. As shown in Fig. 5, when these two parts are combined, the first resonant frequency increases from 1.4 to 1.445 GHz and the second mode decreases from 1.785 to 1.545 GHz. Such shifts can be explained by the strong capacitive coupling between the disk and the meander strips underneath. Similar concept of tight or strong coupling has also been used to provide broad bandwidth in handset antennas [18]. Fig. 6 shows the simulated reflection coefficients versus the width of the square disk, Ld . With a smaller Ld , the second mode is shifted to a higher frequency because the capacitive coupling becomes less and the effective length becomes shorter. Intended mode coupling can be achieved by properly adjusting the value of Ld . Fig. 7 shows the simulated reflection coefficients versus the gap width g . The resonant frequency of the two modes moves farther apart and the impedance matching become worse with increasing g . In this design, we choose g = 1 mm. Fig. 8 shows the measured and simulated radiation pattern at 1.495 GHz, which match reasonably well. The E pattern in the x 0 y plane

Fig. 8. Radiation pattern at 1.495 GHz: (a) x 0 y plane, (b) x 0 z plane,

:

: simulated E ,  : measured E , 2 : simulated measured E , E , 10 dB per division on radials, all parameters are the same as in Fig. 3.

is nearly omnidirectional. The cross-polarization is about 10 dB lower than the co-polarization. The measured peak gain is 0.7 dBi. Fig. 9 shows the measured antenna gain which is positive over the whole band. The gain is lower than that of conventional dipoles because part of the currents along the folded lines flow in opposite direction, and their radiated fields partially cancel each other. Fig. 10 shows the measured radiation efficiency of the proposed antenna exceeds 65% over

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Fig. 9. Measured antenna gain, other parameters are the same as in Fig. 3.

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Fig. 12. Simulated reflection coefficients with different ground plane sizes, , , , : : , 1

G = 130 G = 110 mm G = 50 G = 50 mm : G = 400, G = 400 mm, other parameters are the same as in Fig. 3

except the ground plane size.

Fig. 10. Measured total radiation efficiency, including impedance mismatch effect, other parameters are the same as in Fig. 3.

E in the x z plane at 1.6 GHz with different ground plane G = 130, G = 110 mm, : G = 50, G = 50 mm, : G = 400, G = 400mm, 10 dB per division on radials, all parameters

Fig. 13. Simulated : sizes, 1

0

are the same as in Fig. 3.

Fig. 11. Photograph of the proposed antenna beside a dime.

the whole band, which is obtained using the full 3-D pattern integration in the anechoic chamber of our laboratory. Fig. 11 shows the photograph of the proposed antenna. Four paper pads are inserted to make the structure rigid. A slight frequency shift may be contributed by these paper pads. IV. EFFECT OF GROUND PLANE SIZE The effect of ground plane size on the antenna performance has been analyzed. The bandwidth of the antenna with a larger ground plane, for example, 400 mm 2 400 mm or 2 2 2, is about 13.8%, and the peak gain appears at  = 45 . The bandwidth of the antenna with a smaller

ground plane, for example, 50 mm 2 50 mm or 0:25 2 0:25, is about 2.5%, and the peak gain appears at  = 90 . With a smaller ground plane, it is difficult to tune the input impedances of the two resonant modes to 50 at the same time although the impedance bandwidth with a larger ground plane is much wider than that with a smaller one, the radiation pattern is not dipole-like and the gain turns negative in the x 0 y plane. An electrically small meander-line antenna of dimension 0:08 over a finite ground plane of 0:71 2 0:71 is proposed in [19], the peak gain appears at  = 45 . A dual-meander-line antenna of dimension 0:12 placed at the center of a 0:73 2 0:73 ground plane is proposed in [20]. It seems that by nomenclature, a small antenna is referred to the radiator itself, not concerning the size of the ground plane. In summary, the ground plane does affect the radiation pattern significantly. The proposed antenna can be used for FM broadcasting if the operating frequency is scaled down to 88–108 MHz. For FM broadcasting, the ground plane is usually implemented beneath the soil surface, hence only the structure above the ground is of more concern. This antenna can also be installed in a wireless device with a vertical ground plane. The bandwidth in terms of VSWR remains wide over which the radiation pattern remains dipole-like. In this design, the ground plane does affect the radiation pattern and the input impedance. The ground plane size over the range from

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0:55 2 0:65 to 0:65 2 0:75 meets the three design goals: (1) The radiation pattern is dipole-like with positive gain in the x 0 y plane, (2) the antenna structure above the ground plane is smaller than one-tenth of a wavelength, and (3) the 010 dB impedance bandwidth is more than 10%. If the ground plane is smaller than 0:55 2 0:65, the impedance matching becomes poor and the impedance bandwidth is reduced. If the ground plane is larger than 0:65 2 0:75, the main beam is steered toward  = 45 , violating the first design goal. In this work, the ground plane size of 0:55 2 0:65 gives a broad band in reflection coefficient and a dipole-like radiation pattern. V. CONCLUSIONS A small antenna with broad bandwidth has been designed and measured. Its dimension is about 0:1 2 0:1 2 0:098 and is mounted on a horizontal ground plane size of 0:55 2 0:65 at the center frequency of 1.495 GHz, its fractional bandwidth is 10.8% with VSWR of less than 2, its radiation pattern in the x 0 y plane is nearly omnidirectional. The ground plane size is optimized to meet the three design goals: Dipole-like radiation pattern with positive gain in the x0y plane, smaller than =10 of structure size above ground, and more than 10% of impedance bandwidth. The measured radiation efficiency exceeds 65% over the whole band. The broadband performance is achieved by proximity coupling between the dual-meander folded loop and the disk-loaded monopole.

[15] M. Ali, G. J. Hayes, H. S. Hwang, and R. A. Sadler, “Design of a multiband internal antenna for third generation mobile phone handsets,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1452–1461, July 2003. [16] C. M. Kruesi, R. J. Vyas, and M. M. Tentzeris, “Design and development of a novel 3-D cubic antenna for wireless sensor networks (WSNs) and RFID applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3293–3299, Oct. 2009. [17] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “A compact and low-profile metamaterial ring antenna with vertical polarization,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 333–336, 2005. [18] S. Risco, J. Anguera, A. Andujar, A. Perez, and C. Puente, “Coupled monopole antenna design for multiband handset devices,” Microw. Opt. Technol. Lett., vol. 52, no. 2, pp. 359–364, Feb. 2010. [19] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 691–707, Mar. 2008. [20] K. Noguchi, M. Mizusawa, T. Yamaguchi, Y. Okumura, and S. Betsudan, “Increasing the bandwidth of a small meander-line antenna consisting of two strips,” Electron. Commun. Jpn, Part. 2, vol. 83, no. 10, pp. 35–43, 2000.

A Dual Band Microstrip-Fed Slot Antenna Mahmoud N. Mahmoud and Reyhan Baktur

REFERENCES [1] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antenna,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 672–676, May 1996. [2] H. D. Foltz, J. S. Mclean, and G. Crook, “Disk-loaded monopoles with parallel strip elements,” IEEE Trans. Antennas Propag., vol. 46, no. 12, pp. 1894–1896, Dec. 1998. [3] W. Dou and W. Y. M. Chia, “Compact monopole antenna for GSM/DCS operation of mobile handsets,” Electron. Lett., vol. 39, no. 22, Oct. 30, 2003. [4] I. F. Chen and C. M. Chiang, “Multi-folded tapered monopole antenna for wideband mobile handset applications,” Electron. Lett., vol. 40, no. 10, May 13, 2004. [5] N. Behdad and K. Sarabandi, “Bandwith enhancement and further size reduction of a class of miniaturized slot antennas,” IEEE Trans. Antennas Propag., vol. 22, no. 8, pp. 1928–1935, Aug. 2004. [6] R. Azadegan and K. Sarabandi, “A novel approach for miniaturization of slot antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 421–429, Mar. 2003. [7] J. Anguera, C. Puente, E. Martinez, and E. Rozan, “The fractal Hilbert monopole: A two-dimensional wire,” Microwave Opt. Technol. Lett., vol. 36, no. 2, pp. 102–104, Jan. 2003. [8] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal antennas: A novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag., vol. 44, no. 1, pp. 20–36, 2002. [9] J. Anguera, C. Puente, C. Borja, and J. Soler, “Fractal-shaped antennas: A review,” Wiley Encycl. RF Microwave Eng., vol. 2, pp. 1620–1635, 2005. [10] W. Dou and W. Y. M. Chia, “Small broadband stacked planar monopole,” Microw. Opt. Technol. Lett., vol. 27, no. 4, pp. 288–289, Nov. 2000. [11] T.-H. Chang and J.-F. Kiang, “Broadband dielectric resonator antenna with metal coating,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1254–1259, May 2007. [12] J. W. Jung, H. J. Lee, and Y. S. Lim, “Broadband flexible meander line antenna with vertical lines,” Microw. Opt. Technol. Lett., vol. 49, no. 8, pp. 1984–1987, Aug. 2007. [13] M. Ali, S. S. Stuchly, and K. Caputa, “A wideband dual meander sleeve antenna,” J. Electromagn. Waves Appl., vol. 10, no. 9, pp. 1223–1236, 1996. [14] H. D. Chen, “Compact broadband microstrip-line-fed sleeve monopole antenna for DTV application and ground plane effect,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 497–500, 2008.

Abstract—A simple new design method to achieve a dual band microstrip-fed slot antenna is presented. It is shown that when two slot antennas are placed in series, the spacing between the two antennas can be adjusted to achieve an effective secondary resonance. The new resonance is found to be due to the mutual coupling between the two slot antennas. An approximate circuit model for the dual band antenna is presented to explain the dual band mechanism and to provide a design guideline. The model is validated with a prototype antenna that operates at 4.22 GHz and 5.26 GHz, which are commonly used as the downlink and uplink in satellite communications. Measured results show good return loss at both frequencies, and radiation patterns agree well with the simulations. The proposed antenna has a simple geometry can be easily produced using printed circuit board techniques for applications where compactness and multiband operation are of interest. Index Terms—Microstrip, multifrequency antennas, mutual coupling, slot antennas.

I. INTRODUCTION Slot antennas have appealing features such as low profile, low cost, and ease of integration on planar or non-planar surfaces [1]. An example application is integrating slot antennas with solar panels of small satellites to save surface real estate [1], [3]. While slot antennas are valuable for space applications and self-powered ground sensors [4], most designs are limited to single frequency operation. This communication presents a very simple design where one can achieve a dual band antenna by utilizing coupling between two adjacent slots. Manuscript received May 11, 2010; revised August 23, 2010; accepted October 07, 2010. Date of publication March 07, 2011; date of current version May 04, 2011 The authors are with the Electrical and Computer Engineering Department, Utah State University, Logan, UT 84341 USA (e-mail: reyhan.baktur@usu. edu). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123065

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A slot antenna, when not backed by a cavity, radiates to both sides of the ground plane and can be a good substitute for a dipole antenna [5]. In applications such as solar panel integration, it is important to limit the radiation to only the front side of the slot by utilizing a cavity backing [6]–[8]. The cavity can be loaded with dielectrics to improve antenna performance, such as enhancing the impedance bandwidth. Due to the conformal nature and versatility in choosing the cavity geometry and material, cavity backed antennas have found popularity in both single element implementations and array configurations [9]–[12]. When there is more than one slot element, it is necessary to study the coupling between elements. It has been found that cavity backed slot antennas have small mutual coupling [9]. Further studies have been reported to compute the coupling using numerical techniques [10] and analytical methods [11]. Although there is abundant literature on cavity backed slot antennas and their coupling, most studies have focused on studying the slot elements in parallel alignments. When the slots are placed in series, the spacing between the elements is usually large enough (e.g., at least a half wavelength) for one to ignore the resonance due to coupling. But when two series slot antennas are placed close to each other, we found that one can achieve two resonances. This suggests a dual band antenna design. The design principle is simple and can be conveniently implemented using printed circuit board techniques. This communication presents the design method, analysis using an equivalent circuit modal, and a prototyped dual band antenna. The antenna is studied using Ansoft’s HFSS, and measured results agree well with the simulations, validating the dual band antenna design that can be potentially implemented on solar panels as communication links or sensor nodes. II. DUAL BAND ANTENNA ANALYSIS The configuration of the proposed dual band cavity backed slot antenna is shown in Fig. 1. The antenna and the feed lines are composed of two circuit board substrates (Fig. 1(a)). Two radiating slots are etched on the top layer, which is a copper layer, of the first substrate (Fig. 1(b)). The feed lines are printed on the top layer of the second substrate (Fig. 1(c)). The bottom layer of the second substrate is the ground plane. The two substrates are then assembled together with antennas on the topmost layer and the feed lines sandwiched between the two substrates (Fig. 1(a)). Also, the antenna elements are designed and assembled to be orthogonal to the feed lines (Fig. 1(a)). It should be noted that one does not have to choose the same substrates to etch antennas and to print feed lines. Depending on applications and practical demands, the excitation method can be simple probe feed [13], coplanar waveguide (CPW) feed [14], or microstrip line feed [5], [15]. This communication demonstrates the microstrip line feed due to its simplicity and ease in matching the lines to the slot antenna by adjusting the position and length of the feed lines. After assembling the two substrates, the four side-walls of the substrates and the top plane (i.e., slot antenna and the metal plane) are shorted to the ground plane with either conductive pastes or conductive tapes. When prototyping the antenna, it was necessary to cut a rectangular notch on the top substrate in order to solder a SMA connector (Fig. 1). The existence of the notch affects the resonance frequency and impedance matching, and we have included it in the full wave simulation. The feed design is a 50 microstrip line divided into two 100

lines to excite the two slot elements that resonate at the same frequency (Fig. 1). We found that when the two elements are placed close, a new resonance appears due to the strong coupling between the two slots. The explanation for the second resonance is that the coupling between the two slots acts as if there were an equivalent slot antenna that is longer than the individual slot but shorter than the total length of the two slot elements.

Fig. 1. The proposed dual band slot antenna. (a) An illustration of the antenna geometry, (b) top layer of the fabricated antenna, (c) middle layer, i.e., the microstrip feed lines, of the fabricated antenna.

Fig. 2. Illustration of the location of feed lines and the two slot antennas.

In order to analyze the mechanism of the dual band resonance and provide some insight for designing effective antennas for both frequencies, an approximate circuit model to study the input impedance of the slot antenna is established. The important parameters of the slot geometry are marked in Fig. 2, where Le1 and Le2 are critical for matching the impedance of the two slots, and the spacing between the two slots (d in Fig. 2) has been seen to affect the impedance of the equivalent slot. The approximate model is presented in Fig. 3. It is derived by modifying Syahkal’s circuit model for a single slot antenna fed by a microstrip line [16]. Each slot is modeled as two short-circuited slot lines in parallel with a radiation conductance Gr that represents the radiated power from the slot [16]. The parameters Le1 and Le2 (Fig. 2) correspond to the length of the two short-circuited slot lines, and are marked on Fig. 3 for ease of reading. The characteristic impedance of the slot line is Zcs , and L1 , L2 are inductance of the microstrip feed line and

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measurements. We chose the second approach in this communication since there are several well tested antenna design software tools. Ansoft’s HFSS is used to perform the simulations for two series slot antennas on substrates with different thicknesses and relative permittivity, and different ground plane sizes. It is found that after matching two identical series slots to a resonant frequency 1 , a secondary resonance 2 ( 2 < 1 ) appears when the spacing (Fig. 2) between the two slots is less than 0.20 wavelength. This resonance is generally and we need to continue to move weak with an 11 higher than 0 two slots closer to achieve a reasonable 11 at 2 . It is observed that changing the spacing only changes the level of the resonance, and does not have much effect on 2 . It is also observed that the spacing affects the resonance at 1 too, and one has to adjust the matching microstrip lines to achieve a good 11 at 1 . Same as the case for 2 , the spacing between two slots does not affect the location of 1 . These observations are consistent for different substrates and ground plane sizes, and they are also consistent with the model in Figs. 3 and 4. When the dimension of the slots is fixed, the input impedance of the equivalent slot is mainly determined by couple (Fig. 4), which is affected by the mutual inductance 2 (Fig. 3). It is also seen from Fig. 3 that 2 affects the input impedances of two series slots, and therefore the spacing between them will affect 11 value at 1 . The ratio of 1 = 2 is found to be close to 1.3, the fluctuation is less than 10% for different substrates and ground planes. In other words, the ratio between the length of the equivalent slot and one of the series slots is about 1.3. We studied three substrates (air, teflon, and RO 4003c) with the same thickness of 3.38 mm. The ratio of 1 = 2 for the three substrates is found to be 1.41, 1.32, and 1.27, respectively. Increasing the thickness of the substrate seemed to increase 1 , but the increase is found to be less than 10% for most commonly used substrates.

f f

f

S

3 dB

f S

f

M

Fig. 3. Approximate circuit model of the dual band slot antenna.

f

f

f d

S

f

f

f

Z

S

f

f f

M

f f

III. PROTOTYPE AND MEASUREMENT RESULTS

Fig. 4. Circuit model of the equivalent slot antenna that is due to the coupling of the two original slot antennas.

M

slot line, respectively. The mutual inductance 1 represents the coupling between the microstrip line and slot line. The mutual inductance 2 represents the coupling between two series slots. Because of the coupling, there appears an equivalent slot that radiates at a frequency lower than the two slots. The circuit model for the equivalent slot is presented in Fig. 4. The length of the equivalent slot is eq e1 e2 and total , and is expected to be between e1 e2 . The coupling between the two slots appears as an added impedance couple to the slot line (Fig. 4). It is straightforward to expect that changing the spacing (Fig. 3) between the two slots will change 2 , and accordingly change couple , which is the dominant factor for the input impedance of the equivalent slot. The other factors (characteristic impedance cs , radiation conductance re , and the eq of the equivalent slot line) do not vary much with respect length total to . Therefore, after matching the impedance of the two slots, one can adjust the spacing between the two slots to achieve a reasonable return loss for the equivalent slot antenna. Two methods can be employed to validate the model shown in Figs. 3–4. One can determine the values of cs , 1 , 2 , 1 , 2 following Syahkal [16], then compute the input impedance of the equivalent slot, and finally compare the computed 11 value of the equivalent slot at the input port (i.e., the SMA connector) with experiments. Or, using the model as guidelines, one can perform a parametric study using simulation software, and then validate the design with

M

L 2(L +L ) Z M d

L

(L + L )

d

Z

Z

G

Z L L M M S

Using the approximate model and observations from HFSS studies in Section II as guidelines, we designed and fabricated a dual band slot antenna prototype. The substrates used are Rogers’ high frequency : , : ) laminates RO 4003c ( and the two slots are designed and matched to operate at 5.26 GHz. The equivalent slot operates at 4.22 GHz. The spacing between the two slots and the position of the microstrip feed line are adjusted to and e1 : to achieve good 11 values at be : both frequencies. The antennas and the feeding microstrip lines were fabricated using a LPKF circuit board milling machine, and the ground 2 . plane or the size of the substrate was chosen to be 100 2 100 The two slots on the upper plane have the same width of 1 mm and the same length of 25 mm. After assembling the two substrates, the four side-walls were shorted with conductive copper tape. It is known that reducing the size of the ground plane results in reduction in the resonant frequency and gain of a slot antenna. We found that the resonance and gain stabilize for both bands when the ground plane reaches 3 2 3 in-air 2 . The ground plane of the prototyped antenna is less than 2 222 , which means we lose about 1 dB gain for each band. But this size is chosen because the antenna is for integration on 2 . a cube satellite which has a solar panel of 100 2 100 The simulated and measured results of frequency response are plotted in Fig. 5. The -parameters were measured using a vector network analyzer (Agilent 8510C). The agreements between simulation and measurements are good for both frequencies. The slight shift in the frequency is likely due to the fabrication accuracy when milling the two slots and cutting the rectangular notch for soldering the SMA connector (Fig. 1), and the possibility of having some air between two substrates when they were assembled. The normalized radiation patterns for both bands were measured using a far-field range in an anechoic chamber. The simulated -plane,

thickness = 0 831 mm permittivity = 3 38

d = 0 5 mm

L = 1 5 mm

S

mm

wavelength wavelength

S

mm

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Fig. 5. Comparisons between the simulated and measured S

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parameter.

Fig. 7. Simulated and measured radiation patterns of the antenna at 4.22 GHz. (a) Simulated E and H plane patterns. (b) Measured co- and cross- polarization E and H plane patterns.

IV. DISCUSSIONS AND CONCLUSIONS

Fig. 6. Simulated and measured radiation patterns of the antenna at 5.26 GHz. (a) Simulated E and H plane patterns. (b) Measured co- and cross- polarization E and H plane patterns.

H-plane patterns for 5.2 GHz antennas were plotted in Fig. 6(a). The

measured co- and cross-polarization patterns at 5.2 GHz were plotted in Fig. 6(b). The agreement between E-plane patterns is good and the cross-polarization level is than 020 dB in the principal plane. The measurement facility is not ideal for measuring H-plane patters, and it is reasonable to expect some distortion on the H-plane pattern from the measurement. Taking this into account, the agreement in overall shape in H-plane patterns is reasonably good. The radiation patterns for the band at 4.22 GHz are plotted in Fig. 7. Agreements between simulations and measurements are good, and the measured cross polarization level is less than 020 dB in the principal plane. The cross polarization level was calculated using HFSS for the entire unit sphere and the maximum was found to be in the principal plane. The notch for soldering the connector (Fig. 1) was not found to have a visible effect on the cross-polarization level. The gain of the dual band antenna was measured using a NSI 2000 near-field scanner, and was found to be 3.5 dB and 4.2 dB for the lower and upper bands respectively. The result is reasonable considering the dielectric loss in the two layers of substrates.

A simple geometry to obtain a dual band cavity backed slot antenna is presented. The antenna was designed using two dielectric substrates and was fed by microstrip lines. The antenna can be easily designed to operate effectively at 4.22 GHz and 5.26 GHz, which are downlink and uplink frequencies for satellite communication at C band. The antenna can be integrated with solar panels to save surface real estate of small satellites, and to replace traditional deployed dipole antennas. When a higher gain is needed for communication links, the proposed antenna can be designed in array configurations to achieve the required gain. The dual band operation is achieved simply by utilizing the mutual coupling between two closely placed series slot antennas, and the design procedure is straightforward. In order to understand the mechanism of the dual band operation, an approximate circuit model was presented. The model was validated by parametric study from Ansoft’s HFSS and by a prototype antenna. It is found that when two slot antennas were placed within 0.15 wavelength (i.e., d  0:15 wavelength), there was a reasonably strong secondary resonance due to the coupling between the two antennas. The coupling results in an equivalent longer slot antenna that operates at a frequency about 1.25 times lower than the original antenna resonance. The dual band resonance can be tuned by adjusting spacing between the two slots and by modifying the matching microstrip feed lines. When an optimum spacing d has been found to produce strong resonances for both bands, it is seen that varying d slightly does not severely affect S11 levels of two bands. For example, the frequency response in Fig. 5 is for when d = 0:5 mm. When d was changed to 0.4 mm or 0.6 mm, only a 2 dB variation was observed in one of the two bands. One prototype antenna was fabricated on high frequency laminates and the measured results agree well with simulations. In this study, we shorted the side-walls of the substrates to obtain a cavity. When the walls were not shorted, we found that more resonances appear, and the front-to-back ratio in the antenna pattern was severely degraded. In the prototype, the walls were shorted with conductive copper tapes.

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The fabrication at Utah State University was performed using a circuit milling machine, but the proposed antenna can be easily produced using printed circuit board techniques, and can be conveniently adapted to communication links where multiband operations are required.

A New Super Wideband Fractal Microstrip Antenna

REFERENCES

Abstract—The commercial and military telecommunication systems require ultrawideband antennas. The small physical size and multi-band capability are very important in the design of ultrawideband antennas. Fractals have unique properties such as self-similarity and space-filling. The use of fractal geometry in antenna design provides a good method for achieving the desired miniaturization and multi-band properties. In this communication, a multi-band and broad-band microstrip antenna based on a new fractal geometry is presented. The proposed design is an octagonal fractal microstrip patch antenna. The simulation and optimization are performed using CST Microwave Studio simulator. The results show that the proposed microstrip antenna can be used for 10 GHz –50 GHz frequency range, i.e., it is a super wideband microstrip antenna with 40 GHz bandwidth. Radiation patterns and gains are also studied.

[1] G. John, D. Kraus, and R. J. Marhefka, Antennas for All Applications. New York: McGraw-Hill, 2002. [2] S. Vaccaro, P. Torres, J. R. Mosig, A. Shah, J.-F. Ziircher, A. K. Skrivervik, F. Gardiol, P. de Maagt, and L. Gerlach, “Integrated solar panel antennas,” Electron. Lett., vol. 36, no. 5, pp. 390–391, Mar. 2000. [3] S. Vaccaro, C. Pereira, J. R. Mosig, and P. de Maagt, “In-flight experiment for combined planar antennas and solar cells (SOLANT),” IET Microw. Antennas Propat., vol. 3, no. 8, pp. 1279–1287, 2009. [4] T. Wu, R. L. Li, and M. M. Tentzeris, “A mechanically stable, low profile, omni-directional solar-cell integrated antenna for outdoor wireless sensor nodes,” presented at the IEEE Antenna and Propag. Society Int. Symp., Charleston, SC, Jun. 2009. [5] Y. Yoshimura, “A microstrip slot antenna,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, pp. 760–762, Nov. 1972. [6] T. Adams, “Flush mounted rectangular cavity slot antennas–Theory and design,” IEEE Trans. Antennas Propag., vol. 15, pp. 342–351, May 1967. [7] C. R. Cockrell, “The input admittance of the rectangular cavity-backed slot antenna,” IEEE Trans. Antennas Propag., vol. 24, no. 3, pp. 288–294, May 1976. [8] A. Hadidi and M. Hamid, “Aperture field and circuit parameters of cavity-backed slot radiator,” IEE Proc., vol. 136, no. 2, pp. 139–146, Apr. 1989. [9] H. G. Akhavan and D. Mirshekar-Syahkal, “Study of coupled slot antennas feed by microstrip lines,” in Proc. IEE 10th Int. Conf. on Antenna and Propagation, Apr. 1997, pp. 290–293. [10] T. Hikage and K. Itoh, “FDTD analysis of mutual coupling of cavity backed slot antenna arrays,” IEICE Trans. Electron., vol. E81-C, no. 12, Dec. 1998. [11] D. Pozar, “A reciprocity method of analysis for printed slot and slot coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. 34, no. 12, pp. 1439–1446, Dec. 1986. [12] S. Long, “Experimental study of the impedance of cavity-backed slot antennas,” IEEE Trans. Antennas Propag., vol. 23, pp. 1–7, Jan. 1975. [13] D. Sievenpiper, H.-P. Hsu, and R. M. Riley, “Low-profile cavity-backed crossed-slot antenna with a single-probe feed designed for 2.34-GHz satellite radio applications,” IEEE Trans. Antennas Propag., vol. 47, no. 1, pp. 58–64, Jan. 1999. [14] S. Sierra-Garcia and J.-J. Laurin, “Study of a CPW inductively coupled slot antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 873–879, Mar. 2004. [15] B. Zheng and Z. Shen, “Effect of a finite ground plane on microstrip-fed cavity-backed slot antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 862–865, Feb. 2005. [16] H. G. Akhavan and D. Mirshekar-Syahkal, “Approximate model for microstrip fed slot antennas,” Electron. Lett., vol. 30, no. 23, pp. 1902–1903, Nov. 1994.

Abolfazl Azari

Index Terms—Bandwidth, fractal microstrip antenna, fractals, ultrawideband.

I. INTRODUCTION Modern communication systems require antennas with more bandwidth and smaller dimension. One of the main components of ultrawideband (UWB) communication systems is an UWB antenna. Customarily, wideband antennas need different antenna elements for different frequency bands. If antenna size is less than a quarter of wavelength, antenna will not be efficient. Fractal geometry is a very good solution to fabricate multi-band and low profile antennas. Applying fractals to antenna elements allows for smaller size, multi-band and broad-band properties. Thus, this is the cause of spread research on fractal antennas in recent years [1]–[4]. Fractals have self-similar shapes and can be subdivided in parts such that each part is a reduced size copy of the whole. The self-similarity of fractals is the cause of multi-band and broad-band properties and their complicated shapes provides design of antennas with smaller size. Fractals have convoluted and jagged shapes such that these discontinuities increase bandwidth and the effective radiation of antennas. The space-filling property of fractals leads to curves which have long electrical length but fit into a compact physical volume. [5]–[9]. Several UWB antenna configurations based on fractal geometries have been investigated including Koch, Sierpinski, Minkowski, Hilbert, Cantor, and fractal tree antennas in recent years. The numerical simulation and experimental results of these antennas are available in literature to date. In this communication, a fractal microstrip antenna is presented. This new fractal geometry is based on an iterative octagon. The huge bandwidth is the main advantage of this fractal antenna over conventional fractal antennas. The commercially available simulation software CST Microwave Studio has been used for the design and simulation of the proposed microstrip antenna. According to the results, this new fractal antenna is applicable in 10 GHz–50 GHz frequency range and the gain of this fractal microstrip antenna is reasonable in entire bandwidth. Manuscript received February 16, 2010; revised September 29, 2010; accepted January 24, 2011. Date of manuscript publication March 17, 2011; date of current version May 04, 2011. The author is with the Young Researchers Club, Islamic Azad University – Gonabad Branch, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2128294

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Fig. 1. The geometry of octagonal circular subarray generator.

This communication is arranged in four sections. Design of proposed antenna is discussed in Section II. Simulated and measured results are presented in Section III and the conclusions are summarized in Section IV. II. ANTENNA DESIGN There are several types of fractal microstrip patch antennas. The most well-known fractal patch antenna is Sierpinski patch antenna such that the various types of it are used greatly in telecommunication systems. The quadrilateral and hexagonal fractal microstrip antennas were discussed in some papers. In initial process of this project, many designs and simulations are performed over fractal patch antennas. The various types of quadrilateral, pentagonal and hexagonal fractal patch antennas are considered and finally the octagonal fractal shape is selected because of its good performances in bandwidth and gain. The standard octagonal arrays are formed by placing elements in an equilateral triangular grid. These arrays can also be viewed as consisting of a single element at the center, surrounded by several concentric eight element circular arrays. To investigate the designs for octagonal arrays via a recursive application, we consider the eight-element circular generating subarray of d = =2, shown in Fig. 1. According to the octagonal properties, the interior angle is 135 and the exterior angle is 45 . Thus we can conclude

cos 1352



=

d 2

! r = 1:306 r

d or



2

:

(1)

The array factor may be expressed in the form

^ p (; ') = AF

1 p 8 ej 8p p=1 n=1

[1:306 sin  cos('0' )+ ]

(2)

where

= (n 0 1) 4 n = 0 1:306 sin 0 cos('0 0 'n ):

'n

(3) (4)

The parameter  is a scale factor that controls the largeness of the array with each recursive generation application and P is the number of concentric octagons in the array. Therefore, the total number of elements with P octagons is

Np

= 4P (P + 1) + 1:

(5)

Fig. 2. Iterations of the proposed fractal geometry.

The array factor expression may also be written in the form

^ p (; ') = AF

1 p 8 ej 8p p=1 n=1

(;')

:

(6)

Where n

(; ') = 1:306[sin  cos(' 0 'n ) 0 sin 0 cos('0 0 'n )]:

(7)

If the expansion factor of the recursive octagonal array is assumed to be  = 1, the equation is expressed in form

^ p (; ') = AF

1 8 ej 8 n=1

(;')

p

:

(8)

The geometric construction of this fractal shape starts with an octagon, called the base shape, which is shown in Fig. 2 (Base Shape). By adding another octagon inside the base shape, the first iterated version of the new fractal geometry, shown in Fig. 2 (First Iteration) is created. The process is repeated in the generation of the second iteration which is also shown in Fig. 2 (Second Iteration). In this communication, the second iteration of the octagonal fractal geometry is considered since higher order iterations do not make significant affect on antenna properties. For this antenna, the length of each side of octagon is 2 cm. The proposed fractal geometry is placed on the Rogers TMM substrate with relative permittivity "r = 4:5 and thickness = 1:524 mm. The dimension of the ground plane is chosen to be 6 6 cm2 . The appropriate feeding location is in the maximum of electric filed. The position of coaxial probe to match the input impedance Z = 50

is founded exactly using simulation by this fact that the E-field must be maximum. Thus the location of the coax feed is placed on the patches which is 26.5 mm from the center at the corner. Fig. 3 shows the structure of antenna on the substrate.

2

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Fig. 3. Antenna Structure.

Fig. 6. Real part of the input impedance.

Fig. 4. The simulated S

for the base shape and first iteration.

Fig. 7. Imaginary part of the input impedance.

Fig. 5. The simulated and measured antenna).

S

for the second iteration (proposed

Fig. 8. The current distribution.

III. SIMULATION AND MEASUREMENT RESULTS This microstrip antenna is simulated using commercial software CST Microwave Studio. The ground plane and all conductors are assumed perfect. The simulation frequency range is from 10 GHz –50 GHz. The measurement results are performed till 40 GHz. Fig. 4 shows the simulated S11 for the base shape and first iteration of the octagonal fractal antenna and Fig. 5 shows the simulated and measured S11 . Also, Figs. 6 and 7 show real and imaginary parts of the input impedance. According to the results, the input reflection coefficient is improved in first iteration than base shape. It is also obvious that the bandwidth is maximum in the proposed design or second iteration. Thus, this fractal microstrip antenna can be used for 10 GHz-50 GHz frequency band. Fig. 8 shows the current distribution in some selected frequencies. According to the figure, the wideband behavior is due to the fact that the currents along the edges introduce additional resonances, which produce an overall broadband frequency response characteristic.

Fig. 9. Measured maximum gain.

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Based on fractal geometry, a super wideband fractal microstrip antenna is implemented. The proposed design is a second iteration of an octagonal fractal geometry. The proposed structure has a dimension of 6 2 6 cm2 . The simulated results are obtained using CST Microwave Studio simulator. The results show that it is a super wideband antenna which is applicable for frequencies between 10 GHz–50 GHz. This microstrip antenna is simple to design and easy to fabricate.

REFERENCES [1] A. Azari and J. Rowhani, “ultrawideband fractal microstrip antenna design,” Progr. Electromagn. Res. C, vol. 2, pp. 7–12, 2008. [2] D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Trans. Antennas Propag. Mag., vol. 45, pp. 38–57, Feb. 2003. [3] K. J. Vinoy, J. K. Abraham, and V. K. Vardan, “On the relationship between fractal dimension and the performance of multi-resonant dipole antennas using koch curves,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2296–2303, Sep. 2003. [4] J. P. Gianvittorio and Y. R. Samii, “Fractal antennas: A novel antenna miniaturization technique and applications,” IEEE Antennas Propag. Mag., vol. 44, no. 1, Feb. 2002. [5] D. H. Werner and M. Raj, Frontiers in Electromagnetics. New York: IEEE Press, 2000. [6] D. H. Werner, R. L. Haupt, and P. L. Werner, “Fractal antenna engineering: The theory and design of fractal antenna arrays,” IEEE Antennas Propag. Mag., vol. 41, no. 5, pp. 37–59, 1999. [7] N. Cohen, “Fractal antenna application in wireless telecommunications,” in Proc. Electronics Industries Forum of New England, 1997, pp. 43–49. [8] J. Gouyet, Physics and Fractal Structures. New York: Springer, 1996. [9] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. New York: Wiley, 1990. [10] J. Romeu and J. Soler, “Generalized Sierpinski fractal multiband antenna,” IEEE Trans. Antennas Propag., vol. 49, no. 8, pp. 1237–1239, Aug. 2001. [11] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [12] D. M. Pozar and D. H. Schubert, Microstrip Antennas—The Analysis and Design of Microstrip Antennas and Arrays. New York: IEEE Press, 1995. [13] A. Azari, “Super Wideband Fractal Antenna Design,” presented at the IEEE MAPE 2009, Beijing, China. [14] A. Azari, “A new fractal monopole antenna for super wideband applications,” presented at the IEEE MICC 2009, Kuala lumpur, Malaysia. [15] A. Azari and J. Rowhani, “Ultra wideband fractal antenna design,” presented at the IASTED ARP 2008, Maryland.

Fig. 10. (a) Radiation patterns (x–y plane). (b) Radiation patterns (x–z plane). (c) Radiation patterns (y –z plane).

To study the radiation pattern, Fig. 9 shows the measured maximum gain versus frequency. Also, Fig. 10 shows the radiation patterns (E-Field) for 10 GHz, 20 GHz, 30 GHz, and 40 GHz for x–y , x–z and y–z planes.

IV. CONCLUSION The concepts of fractals can be applied to the design of ultrawideband antennas. Applying fractals to antennas allows for miniaturization of antennas with multi-band and broad-band properties.

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Bandwidth Enhancement of Internal Antenna by Using Reactive Loading for Penta-Band Mobile Handset Application Chia-Mei Peng, I-Fong Chen, Ching-Chih Hung, Su-Mei Shen, Chia-Te Chien, and Chih-Cheng Tseng

Abstract—A metal-wire-cutting bended monopole antenna (BMA) fed by mini-coaxial cable jointly with a thin printed ground-line which protrudes from ground-plane to be a balance-feed structure for bandwidth enhancement is proposed. A prototype of the proposed antenna with 40 mm in length, 6 mm in height and 5 mm in width has been fabricated and experimentally investigated. The human’s hand effect on impedance bandwidth and radiation characteristics of the proposed antenna is studied. A prototype is designed to cover penta-band: CDMA (824–894 MHz), GSM (880–960 MHz), DCS (1710–1880 MHz), PCS (1850–1990 MHz) and WCDMA (1920–2170 MHz). Experimental results are shown to verify the validity of theoretical work. Index Terms—Balance-feed structure, CDMA, GSM, DCS, PCS, WCDMA.

I. INTRODUCTION Nowadays, the wireless product is small, attractive, lightweight, and curvy. There is a remarkable trend towards the development of integrated antenna for portable communication device. An integrated antenna with high performance is characterized by large bandwidth, low volume and nearly omni-directional radiation pattern. Designing an integrated antenna for wireless product with dual or multi-band operation is straightforward, but it is difficult to satisfy the bandwidth requirement for the respective communication bands. Further complication arise when the antenna has to operate in close proximity to objects like shielding cans, speaker, screws, battery, vibrator, digital camera, and other metallic objects. After 2004, there are many publications on the multi-band antenna for mobile handset [1]–[4]. The planar inverted-F antenna (PIFA) has become the main candidate for the above application since its performances reasonably well compared to other alternatives when operated close a ground plane. Therefore, the bandwidth of PIFA is found to be largely dependent on the antenna height from the printed circuit board on which it is mounted and the size of ground plane. Therefore new techniques are needed in order to achieve a maximum bandwidth with an antenna that occupies the smallest space possible. In literature [5]–[8] proposed the insertion of slots on the ground-plane of handset antenna to enhance the operating bandwidth. Manuscript received January 07, 2010; revised September 08, 2010; accepted October 11, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the National Science Council, R.O.C., under Contract 97-2221-E-228-004. C.-M. Peng, I.-F. Chen, C.-C. Hung, and C.-T. Chien are with the Department of Electronic Engineering and Institute of Computer and Communication Engineering, Jinwen University of Science and Technology, Taipei, Taiwan, R.O.C. (e-mail: [email protected]). S.-M. Shen is with the Department of Electronic Engineering and Institute of Computer and Communication Engineering, Jinwen University of Science and Technology, Taipei, Taiwan, R.O.C. and also with the Graduate Institute of Computer and Communication Engineering, National Taipei University of Science and Technology, Taipei, Taiwan, R.O.C. C.-C. Tseng is with the Department of Electronic Engineering and Institute of Computer and Communication Engineering, Jinwen University of Science and Technology, Taipei, Taiwan, R.O.C. and also with the Department of Electrical Engineering, National Ilan University, Taiwan, R.O.C. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122232

Such slots tune the PCB to resonate at a frequency similar to that of the PIFA, in order to finally obtain a broadband behavior. Nevertheless, a drawback of the design is the fact that the ground plane size and shape are generally decided upon at the inception of a product, which can not be changed as far as possible. Furthermore, by removing fully or partially the ground conductor beneath the main radiator to increase operating bandwidth, the impact of human body on antenna efficiency is large [9]. When a cellular phone is used in close proximity to a human head, dielectric-loading effects can be expected. There may also be a detuning issue for narrowband antennas. In [10], a printed planar structure is adopted, its radiating patch is jointly designed with the shape of the ground-plane to obtain wide-band antenna performance. Although those multi-band antenna types can be well used for certain applications, they are lack of flexibility to overcome the human body effect compared with wide-band antenna operating bandwidth. In this communication, a novel penta-band antenna structure was proposed, which comprises metal-wire-cutting bended monopole antenna (BMA) fed by mini-coaxial cable jointly with a thin printed ground-line to operate at penta-band: CDMA, GSM, DCS, PCS and WCDMA bands. The printed thin ground-line which protrudes from ground-plane is like a sleeve-balun for BMA, to let the proposed antenna with a balance-feed structure to overcome the interaction between the antenna and the human body [11]. This antenna combines omni-directional, balance-feed structure, broad bandwidth and low profile in an easy to fabricate structure. Details of the design considerations of the proposed designs and the experimental results of constructed prototype are presented and discussed. II. ANTENNA STRUCTURE AND DESIGN Fig. 1(a) shows the geometry of the proposed penta-band antenna which is put on the top of the ground-plane (the system circuit board of the mobile phone). It comprises metal-wire-cutting BMA fed by minicoaxial cable jointly with a thin printed ground-line which protrudes from ground-plane. The feed and ground points are connected to a 50

RG-178 coaxial-cable (the length is 10 cm) with a SMA connector. The detail configuration is illustrated in the Fig. 1(b), the dimension of proposed BMA is : 38 mm, : 40 mm and : 6 mm. The radius () and length (`) of feeding mini-coaxial cable is 1.13 mm and 46 mm, which is put upon tight the BMA body to be a reactive loading element, in other words, there is no gap between mini-coaxial cable and BMA body. The dimension of the feed point is 3 mm by 3 mm on the ground plane, and the width of the slit between the feed point and the ground plane is 0.5 mm. There are two connecting parts in the figure: part A and part B. In part A, the inner- and outer-conductor of mini-coaxial cable are connected to feed- and ground-point of PCB. In part B, the inner-conductor is connected to the corner of BMA, but the outer-conductor is insulated against the BMA body. For comparison, the geometry of a conventional bended monopole antenna (denoted as a No. 1 reference antenna in this study) is depicted in Fig. 1(c), which without printed ground-line behind the antenna. The effect of reactive loading element will reduce the electrical length of monopole antenna and increase the lower- and upper- operating bandwidth. The resonant mode of total shape ( + + ) is designed to occur at about 892 MHz, and the dimension is designed to resonate at about 1800 MHz. The length of radiating elements can be determined from the quarter-wave length at the resonant frequencies. In Fig. 1, the dimension is the slit-space between bended monopole traces, is the width of printed thin ground-line that is behind the BMA and is the space-distance between BMA and ground-plane; note that the sizes of these dimensions are not identical. By selecting appropriate dimensions ( ; ; ) of the

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Fig. 2. The Return Loss against frequency results of proposed antenna.

III. EXPERIMENTAL RESULTS AND DISCUSSION

Fig. 1. (a) Dimensions of the prototype antenna. (b) Geometry of the proposed antenna and PCB board. (c) Configuration of the No. 1 reference antenna.

antenna structure, good impedance matching of the proposed antenna can be obtained across an extended bandwidth. The printed ground-line structure is behind the BMA to be a parasitical element as shown in Fig. 1, which was found to be effective in obtaining a wider impedance bandwidth in the antenna’s lower operating band [10]. As a “good” embedded antenna, it should not only meet the technical specifications like bandwidth, efficiency, gain, hand effect, etc., but also meet the requirements like compact, light weight, low cost, attractive appearance and easy to fabricate. The antenna should be small in order to be mounted in the limited space provided by the mobile handsets. The desired radiation pattern should be primarily omni-directional in the horizontal plane. In addition, it should be noted that the ground-plane dimension could also affect the resonant frequency and operating bandwidth of the lower operating band. Use of the handset body as a part of the radiator is beneficial particularly when a small antenna element is used. It enhances the handset antenna performance as the handset body is usually larger than the antenna element and effectively enlarges the antenna dimensions so that the gain and the bandwidth of the antenna system are increased [7]. Thus the ground-plane dimension should also be taken into account in determining the proper parameters of the proposed design to achieve the desired band operation.

In the experiment, the proposed antenna has compact dimension 40 mm 2 5 mm 2 6 mm and is mounted on the topside portion of an FR4 substrate (thickness of 1.5 mm, loss tangent of 0.023, relative permittivity of 4.3, and dimension is 80 2 40 mm2 , front- and back-side is shorted by via hole), which can be treated as the circuit board of a practical mobile handsets. By using the described design procedure, a penta-band antenna was constructed to operate at the range = 0:5 of global wireless system. In the case with a = 2 mm, = 5 mm, it was found that the measured Return Loss mm and 7:4 dB can reach nearly 200 MHz (22%) in the lower operating band (CDMA-GSM) and 540 MHz (30%) in the upper operating band (DCS, PCS and WCDMA), as shown in Fig. 2. Detailed effects of varying the three parameters of ; and are studied in Fig. 3. Fig. 3 shows the simulated return loss as a function of the space ( ) between bended monopole trace, the printed thin ground-line width ( ) and the space-distance ( ) between BMA and ground-plane, respectively. In Fig. 3(a), the results for the slit-space ( ) between bended monopole trace varied from 0 to 4 mm are shown; other dimensions of the antenna are the same as given in Fig. 1. Strong effects on the excited resonant modes in both the lower and upper operating bands are seen, indicating that proper selection of the slit-space ( ) is important in the resonant frequencies of the proposed antenna. In this design, the preferred slit-space ( ) is determined to be 2 mm from the obtained simulated results. Fig. 3(b) shows the results for the printed thin ground-line width ( ) varied from 0.1 to 5 mm, the effect on the excited resonant modes is seen to be smaller than that in Fig. 3(a), the preferred ground-line width ( ) is chosen to be 0.5 mm in this study. Effects of the space-distance ( ) between BMA and ground-plane on the antenna operating bandwidth are also studied. The simulated results for the varied from 1 to 7 mm are presented in Fig. 3(c). The bandwidth in the operating bands can be adjusted by varying the space-distance , and the preferred space-distance is 5 mm. Fig. 4 present the measured radiation patterns at 960 MHz and 1880 MHz for free space in the xyand zx-plane, respectively. For the two cases, monopole-like radiation patterns are observed and that is good omni-directional radiation in the zx-plane are seen. The maximum gain in the xy-plane (Vertical-plane) at 960 MHz and 1880 MHz are 1.87 dBi and 0.91 dBi as shown in Fig. 3(a), and zx-plane (Horizontal-plane) at 960 MHz and 1880 MHz are 0.7 dBi and 2.07 dBi as shown in Fig. 3(b). It was found that the E' radiation increases with increasing operating frequencies. Table I shows the measured antenna gains and 3D pattern efficiency within the operating bands of the proposed antenna. Stable radiation patterns are observed. The effect of adding printed thin ground-line and using mini-coaxial feeding-cable, and the radiation characteristics of proposed antenna have been studied and the results are described below.

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Fig. 4. Measured radiation patterns for the proposed antenna at (a) 960 MHz E'). and (b) 1880 MHz. (E;

020

TABLE I THE MEASURED ANTENNA GAINS WITHIN THE OPERATING BANDWIDTH OF THE PROPOSED ANTENNA. THE SYMBOL’S MEANING: “MAX.: MAXIMUM”

Fig. 3. Simulated return loss as a function of (a) the slit-space ( ) between bended monopole trace, (b) the printed thin ground-line width ( ) and (c) the space-distance ( ) between BMA and ground-plane. Other dimensions are the same as given in Fig. 1.

A. Effect of Adding Printed Thin Ground-Line To add the printed thin ground-line (which protrudes from the ground-plane) behind the No. 1 reference antenna, that was denoted as the No. 2 reference antenna. The comparison results of measured Return Loss are shown in Fig. 5, it is seen that the ground-line behind BMA will increase the bandwidth of lower and upper operating band. The length of printed ground-line is nearly =4 (centered at 1800 MHz) shorted at the base which presents an infinite impedance at the top is like a sleeve balun of the upper operating band, but it cannot let the lower operating band has the same performance. The effect of adding printed thin ground-line which protruded from ground-plane is to form a balance-feed structure [12]. By experimental study, the resonant frequency of the lower operating band increases with a decrease in the printed ground-line length, but the upper operating band is slightly affected.

Fig. 5. The Return Loss against frequency results of No. 1 reference antenna compares with No. 2 reference antenna.

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TABLE II MEASURED RESULTS OF THE ANTENNA BANDWIDTH AS A FUNCTION OF CABLE’S DIAMETER

Fig. 6. The Return Loss against frequency results of No. 2 reference antenna compares with proposed antenna.

B. Effect of Using Mini-Coaxial Feeding-Cable According to the above analyses, it is obvious that the bended monopole antenna is the main radiating element for lower and upper operating band. The bended monopole antenna is designed jointly with the printed thin ground-line will increase the lower operating bandwidth, but the upper operating bandwidth is also not enough to meet the requirements of mobile handset. Hence, based on the No. 2 reference antenna structure, using mini-coaxial feeding cable to feed at the corner of bended monopole and to let the cable was set upon tight the bended monopole antenna trace, and then we can obtain the proposed antenna structure. Fig. 6 shows the measured Return Loss plot of the proposed antenna compares with the No. 2 antenna. The bandwidth of proposed antenna in lower operating band is larger than the No. 2 antenna, and the upper operating bandwidth is also increased. Thus it can be seen, the bandwidth enhancement is obtained mainly through the reactive loading element effect of the mini-coaxial feeding cable. Fig. 7 shows impedance characteristics with various diameter of mini-coaxial cable, which control the operating bandwidth. The value of cable’s diameter is varied from 0.8 mm to 1.74 mm, the corresponding antenna’s resonant frequency and impedance bandwidth were shown in Table II (the printed thin ground-line width of 0.5 mm and length of 40 mm, the mini-coaxial feeding-cable length of 46 mm, Return Loss 7:4 dB). Results show the impedance bandwidth does not vary linearly with the cable’s diameter. The optimum cable’s diameter is 1.13 mm, that is because of the mini-coaxial feeding cable masks part of antenna area, to cancel inductance from cable and proximity in the feed-pad area, and therefore the input impedance is more close to the 50 system to make it easy to achieve wide bandwidth for the lower and upper operating band, as shown in Fig. 7(b).

Fig. 7. Impedance characteristics with various diameter of mini-coaxial cable. (a) Real part. (b) Imaginary part.

C. Radiation Characteristics of Proposed Antenna Typical measured radiation patterns of the proposed antenna and the No. 1 antenna in the lower- and upper-operating band are plotted in Fig. 8. From these results, it is seen that the frequencies across the impedance bandwidths of the proposed antenna and the No. 1 antenna have same polarization planes and similar broadside radiation patterns. When a mobile phone is used in close proximity to a human body, dielectric-loading effects can be expected. Fig. 9 shows the comparison results between the proposed and No. 1 reference antenna with and without human hand. The results indicate that the mismatch loss due to human hand can be made small in the balance-feed antenna structure [11]. During normal mobile phone use, the human hand is inside the antenna sphere and is exposed to near field. The near-field energy increases the interaction with the human hand, and can alter the input impedance. The measurement results of the human’s hand effect on the antenna performance are shown in Table III.

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TABLE III THE EFFECT OF THE HUMAN’S HAND ON THE ANTENNA PERFORMANCE

020

Fig. 8. Measured radiation patterns of the proposed antenna (E; E') E; E'). (a) 960 MHz. compares with No. 1 reference antenna ( (b) 1880 MHz.

000

0 0

Fig. 9. The Return Loss against frequency results of proposed antenna compares with No. 1 reference antenna due to human hand effect.

Fig. 10. The normalize simulated current distribution on the ground-plane of proposed antenna and No. 1 reference antenna. (a) 960 MHz. (b) 1880 MHz.

IV. CONCLUSION The comparison results show the proposed antenna leads a small distortion in the antenna radiation efficiency; the total gain reduction can be also made small in the balance-feed system [11]. The normalize simulated current distribution on the ground-plane of the proposed antenna and No. 1 reference antenna at 960 MHz and 1880 MHz are shown in Fig. 10. The simulated results show low current density on the ground-plane of proposed antenna, the feature of balance-feed is verified.

It has been demonstrated that a Metal-wire-cutting bended monopole antenna (BMA) that fed with mini-coaxial cable jointly with printed thin ground-line structure provides penta-band operation. By correctly choosing the suitable dimensions of the bended monopole, feeding mini-coaxial cable and printed thin ground-line structure, two bandwidths defined for a Return Loss 7:4 dB, respectively, 22% and 30%, can be obtained. The contribution of this communication is to implement a simple and low profile antenna for practical mobile handsets

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application. Measurements show that the structure indeed offers very broad bandwidth characteristics. This communication has discussed the design and implementation of such a penta-band antenna accordingly with integrated mobile handsets, which is recognized as critical elements that can either enhance or constrain system performance. The interaction between the proposed antenna and human’s hand is also studied in this communication. The balance-feed antenna system can overcome the human’s hand effect is proved. More detail analysis for proposed antenna on mobile handsets such as Specific Absorption Rate (SAR) and Hearing Aid Compatibility (HAC) are the next subjects for research. ACKNOWLEDGMENT The authors would like to thank Prof. C.-Y. Wu (Department of E.E, Jinwen University of Science and Technology) and Prof. C.-W. Hsue (Department of E.E, National Taiwan University of Science and Technology) for their assistance in the technical discussion. The authors are also grateful to the reviewers for their valuable comments and suggestions.

REFERENCES [1] C.-H. Chang and K.-L. Wong, “Printed /8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [2] Y. S. Shin, B. N. Kim, W. I. Kwak, and S. O. Park:, “GSM/DCS/IMT2000 triple-band built in antenna for wireless terminals,” IEEE Antenna Wireless Propag. Lett., vol. 3, no. 1, pp. 104–107, Dec. 2004. [3] K. L. Wong, Y. C. Lin, and B. Chen, “Internal patch antenna with a thin air-layer substrate for GSM/DCS operation in a PDA phone,” IEEE Antennas Propag., vol. 55, pp. 1165–1172, 2007. [4] A. Bynads, R. Hossa, M. E. Bialkowski, and P. Kabacik, “Investigation into operation of single multi-layer configuration of planar inverted-F antenna,” IEEE Antennas Propag. Mag., vol. 49, no. 4, pp. 22–33, Aug. 2007. [5] A. Cabedo, J. Anguera, C. Picher, M. Ribó, and C. Puente, “Multi-band handset antenna combining a PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2526–2533, Sep. 2009. [6] R. Hossa, A. Byndas, and M. E. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, June 2004. [7] J. Anguera, I. Sanz, A. Sanz, A. Condes, D. Gala, C. Puente, and J. Soler, “Enhancing the performance of handset antennas by means of groundplane design,” presented at the IEEE Int. Workshop on Antenna Technology, Small Antennas and Novel Metamaterials (iWAT 2006), New York, Mar. 2006. [8] C. Picher, J. Anguera, A. Cabedo, C. Puente, and S. Kahng, “Multiband handset antenna using slots on the ground plane: Considerations to facilitate the integration of the feeding transmission line,” Progr. Electromagn. Res. C, vol. 7, pp. 95–109, 2009. [9] K. L. Wong and C. I. Lin, “Internal GSM/DCS antenna backed by a step-shaped ground plane for a PDA phone,” IEEE Trans. Antennas Propag., vol. 54, pp. 2408–2410, 2006. [10] Z. Du, K. Gong, and J. S. Fu, “A novel compact wide-band planar antenna for mobile handsets,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 613–619, 2006. [11] H. Morishita, H. Furuuchi, and K. Fujimoto, “Performance of balanced-fed antenna system for handsets in the vicinity of a human head or hand,” IEE Proc.—Microw. Antennas Propag., vol. 149, pp. 85–91, 2002. [12] J. D. Kraus and R. J. Marchefka, Antennas, 3rd ed. New York: McGraw-Hill, 2002, pp. 804–805.

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Small-Size Loop Antenna With a Parasitic Shorted Strip Monopole for Internal WWAN Notebook Computer Antenna Kin-Lu Wong, Wei-Ji Chen, and Ting-Wei Kang

Abstract—A simple meandered loop antenna with a parasitic shorted strip monopole, both printed on a thin dielectric substrate and folded to 10 4 mm , for the internal penta-band occupy a small size of 50 WWAN notebook computer antenna is presented. The antenna can provide two wide operating bands to cover the GSM850/900 operation (824–960 MHz) and the GSM1800/1900/UMTS (1710–2170 MHz) operation. The lower band is formed by two resonant modes, one half-wavelength loop mode contributed by the meandered loop and one quarter- wavelength monopole mode contributed by the parasitic shorted strip. The upper band is mainly formed by two higher-order loop resonant modes, with their good impedance matching obtained owing to the parasitic shorted strip. Further, in order to decrease the effects of the display ground and keyboard ground of the notebook computer on the antenna performances, two linear slits are embedded in the keyboard ground near the two connection hinges between the two grounds. The two linear slits function as current traps and can effectively reduce the excited surface currents on the keyboard ground. This leads to decreasing size of the effective notebook computer ground plane seen by the antenna. Thus, degrading ground plane effect on the performances of the internal WWAN antenna can be greatly reduced. Index Terms—Loop antennas, mobile antennas, notebook computer antennas, parasitic shorted monopoles, WWAN antennas.

I. INTRODUCTION Some promising internal penta-band WWAN (wireless wide area network) antennas suitable for laptop or notebook computer applications have been reported recently [1]–[7]. These internal WWAN antennas cover the penta-band operation of the GSM850 (824–894 MHz), GSM900 (880–960 MHz), GSM1800 (1710–1880 MHz), GSM1900 (1850–1990 MHz) and UMTS (1920–2170 MHz) systems. To achieve a small size yet wideband operation, the applied antenna structures include the coupled-fed PIFAs (planar inverted-F antennas) or shorted monopoles [1]–[3], the shorted monopole with multi parasitic radiators [4], the multi monopole slot radiators [5], [6], the loop/monopole combo antenna [7]. For these antennas, however, it is still required that the antenna length along the top edge of the display ground should be at least about 60 mm for the penta-band WWAN operation. This length requirement is needed for these reported antennas to generate the desired operating band at about 900 MHz to cover the GSM850/900 operation. This behavior is owing to the presence of the large notebook computer ground plane (the display ground and keyboard ground together) connected to the internal WWAN antenna. Because of the large size of the ground plane in the notebook computer, it cannot assist in the generation of a wide lower band at about 900 MHz for the internal WWAN antenna as it does in the mobile phone [8]–[11]. A longer length or Manuscript received June 12, 2010; revised August 10, 2010; accepted October 16, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The authors are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]. edu.tw; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122298

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larger size for the internal penta-band WWAN antenna is hence required in the notebook computer than in the mobile phone. For an example, the size of the penta-band WWAN mobile phone antenna can be as small as 31 2 15 mm2 printed on the main circuit board of the mobile phone [12]. In this communication, we present a simple meandered loop antenna with a parasitic shorted strip monopole for the penta-band WWAN operation in the notebook computer. The antenna is printed on a thin dielectric substrate and folded to occupy a small size of 50 2 10 2 4 mm3 , allowing it to be employed along the top edge of the display ground of the notebook computer as an internal antenna. Note that the antenna length along the top edge of the display ground is 50 mm only, which is smaller than those of the reported internal pentaband WWAN notebook computer antennas [1]–[7]. With an antenna height of 10 mm only, the small antenna length of 50 mm obtained here is owing to the use of the loop strip in the proposed antenna. Although the loop strip has a long length of about 188 mm for generating a half-wavelength loop mode at about 900 MHz, it can be of narrow strip width for wideband operation [13]–[20] and is suitable to be greatly meandered to achieve a compact size. With the aid of a parasitic shorted strip monopole, which is closely coupled by the meandered loop antenna, three loop resonant modes can be excited with good impedance matching. The first or half-wavelength loop resonant mode [13]–[20] occurs at about 900 MHz, which combines the quarterwavelength monopole mode contributed by the parasitic shorted strip to form the antenna’s lower band to cover the GSM850/900 operation (824–960 MHz). The two higher-order loop resonant modes are formed into the upper band for the GSM1800/1900/UMTS operation (1710–2170 MHz). Both the display ground and keyboard ground are considered in the study. Since the total size of the notebook computer ground plane (the display ground and keyboard ground together) is very large, it usually shows degrading effects on the impedance matching of the internal WWAN antenna to operate in the 900 MHz band. To reduce this ground plane effect, two linear slits are embedded in the keyboard ground near the two connection hinges between the two grounds. The two linear slits function as current traps [21] and can effectively reduce the surface currents on the keyboard ground excited by the proposed antenna. In this case, the total size of the effective notebook computer ground plane seen by the internal WWAN antenna can be decreased. This can greatly reduce the degrading effects of the large notebook computer ground plane on the performances of the internal WWAN antenna in the 900-MHz band [21]–[23]. This also helps in achieving small size yet wideband operation for the proposed WWAN antenna.

Fig. 1. (a) Geometry of the proposed internal WWAN notebook computer antenna. (b) Side view of the antenna mounted at the shielding wall along the display ground. (c) Dimensions of the antenna in its unfolded structure.

II. PROPOSED NOTEBOOK COMPUTER ANTENNA Fig. 1(a) shows the geometry of the proposed antenna. The side view of the antenna mounted at the shielding wall along the display ground is shown in Fig. 1(b) and dimensions of the printed metal pattern of the antenna in its unfolded structure are provided in Fig. 1(c). The notebook computer is modeled to include the display ground and keyboard ground, both connected through two connection hinges and separated by an angle of 90 in this study. Note that different possible angles of 80 to 120 lead to very small effects on the obtained return loss. Also, for the closed lid case (angle of 0 ), since the display ground and keyboard ground are of the same size in this study, the antenna will not “see” the keyboard ground for the closed lid case. In this case, small effects on the obtained return loss of the antenna are expected. The dimensions of the display ground and keyboard ground are selected to be 220 2 130 mm2 , which are reasonable dimensions for the notebook computer with a 10-inch display panel. By considering both of the two grounds, the total size of the notebook computer ground

plane is as large as 440 2 260 mm2 . By embedding two linear slits of length (d) 80 mm in the keyboard ground near the two connection hinges, the excited surface currents on the keyboard ground can be greatly suppressed [21]–[23] and the excited surface current distributions on the display ground become similar to those of the case without the keyboard ground (that is, the case with the display ground only or the tablet personal computer [24]). In this case, effects of the presence of the keyboard ground on the performances of the antenna will be decreased. This improves the impedance matching of the antenna, especially for frequencies in the 900-MHz band. The antenna is mounted along the shielding metal wall of width 6 mm at the top edge of the display ground. To accommodate the lens of the embedded digital camera which is usually placed at the center of the top edge along the display ground, the antenna is positioned near the corner of the top edge with a distance (s) of 15 mm. The antenna itself has a length of 50 mm along the top edge and occupies a volume

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Fig. 3. Measured and simulated return loss for the proposed antenna.

Fig. 2. Photos of the fabricated antenna in the front, back and top views.

of 50 2 10 2 4 mm3 . This small volume allows the antenna promising to be employed inside the notebook computer as an internal antenna. As shown in the unfolded metal pattern shown in Fig. 1(c), the antenna is a meandered loop antenna (strip AC) closely coupled with a parasitic shorted strip monopole (strip BD), both printed on a 0.4-mm thick FR4 substrate and then folded to form a rectangular block as shown in Fig. 1(a). The meandered loop strip has a length of about 188 mm, allowing the antenna to generate a half-wavelength loop resonant mode as the lowest or fundamental resonant mode [13]–[20]. However, this resonant mode alone cannot cover the GSM850/900 operation (824–960 MHz). With the presence of the closely coupled parasitic shorted strip having a length of about 81 mm, an additional quarterwavelength monopole mode at about 850 MHz can be excited and combined with the half-wavelength loop resonant mode to form a wide lower band for the GSM850/900 operation. The fine-tuning of the excited quarter-wavelength monopole mode is easily controlled by adjusting the end-section length t of the parasitic shorted strip. While the front section of the parasitic shorted strip has a small coupling gap of 0.5 mm to the widened front section of the loop strip. This arrangement provides sufficient capacitive coupling between the loop strip and the parasitic shorted strip such that good excitation of the desired quarter-wavelength monopole mode can be obtained. In addition, the presence of the parasitic shorted strip can lead to impedance matching improvement of the higher-order loop resonant modes excited in the desired upper band to cover the GSM1800/1900/UMTS operation (1710–2170 MHz). III. RESULTS AND DISCUSSION Fig. 2 shows the photos of the fabricated antenna in its front, back and top views. Results of the measured and simulated return loss for the antenna are presented in Fig. 3. The antenna is tested with the laptop model shown in Fig. 1. The simulated results are obtained using HFSS version 12 [25]. Agreement between the simulation and measurement is generally obtained. Some deviations between the simulated and measured data are very likely due to the small inaccuracy in fabricating the prototype of the antenna in the experiment. The measured impedance bandwidths defined by 3:1 VSWR, which is widely used for the internal WWAN antenna design specification, are 220 MHz (810–1030 MHz) and 675 MHz (1540–2215 MHz) for the lower and upper bands, respectively. The two wide operating bands cover the penta-band WWAN operation. Note that when the antenna is enclosed by a plastic casing, the excited resonant modes of the antenna will be shifted to lower frequencies by about 3% from the simulation (the plastic casing in this

Fig. 4. Simulated return loss of the proposed antenna and the case without the parasitic shorted strip monopole.

Fig. 5. Simulated input impedance of the proposed antenna and the case without the parasitic shorted strip monopole.

simulation study is made of a material of relative permittivity 3.0, loss tangent 0.02 and thickness 1 mm and the casing has a width of 10 mm to enclose the antenna and the computer ground plane). In this case, the antenna size can be smaller to achieve the desired operating bands. Fig. 4 shows the simulated return loss of the proposed antenna and the case without the parasitic shorted strip monopole. The corresponding simulated input impedance is shown in Fig. 5. It is clearly seen that an additional resonant mode at about 850 MHz is generated when the shorted strip monopole is present. Also, the impedance matching of the loop resonant mode occurred at about 950 MHz is improved owing to the presence of the shorted strip monopole. These two modes contributed by the shorted strip monopole and the meandered loop antenna form the antenna’s wide lower band. The two higher-order modes in the desired upper band contributed by the meandered loop antenna also show improved impedance matching when the shorted strip monopole is added. This leads to a wide operating bandwidth for the antenna’s upper band. Fig. 6 shows the simulated return loss as a function of the end-section length t of the parasitic shorted strip monopole. Results for the length t varied from 43 to 47 mm are presented. It is seen that the first mode in the lower band is shifted to lower frequencies with an increase

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Fig. 6. Simulated return loss of the proposed antenna as a function of the endsection length t of the parasitic shorted strip monopole.

Fig. 8. Simulated surface current distributions at 950 and 1900 MHz on the ground plane of the notebook computer for the three cases studied in Fig. 7.

Fig. 7. Simulated return loss of the proposed antenna, the case without embedded slits in the keyboard ground and the case without keyboard ground.

in the length t, which also causes some impedance matching variations on the second mode in the lower band. For the upper band, very small impedance matching variations are seen. This behaviour is reasonable since the upper band is mainly contributed by the meandered loop antenna. Fig. 7 shows the simulated return loss of the proposed antenna, the case without the embedded slits and the case without the keyboard ground (that is, the tablet personal computer). When the embedded slits are not present, the impedance matching of the second mode in the lower band is degraded, while the impedance matching of the upper band is almost not affected. On the other hand, for the case without the keyboard ground, the impedance matching of the lower band is in general the same as that for the proposed antenna, while there are some small impedance matching variations in the upper bands of the two cases. This indicates that with the embedded slits in the keyboard ground, the proposed antenna can show similar impedance matching as the case without the keyboard ground, especially in the lower band for the GSM850/900 operation. These effects can be explained more clearly from the simulated surface current distributions at 950 and 1900 MHz on the ground plane of the notebook computer shown in Fig. 8. Similar excited surface current distributions at 1900 MHz for the three cases are seen, which supports the observations of the return loss in Fig. 7. At 950 MHz, the excited surface current distributions on the display ground for the proposed antenna and the case without the keyboard ground are similar to each other. This is mainly because the embedded slits function as current traps [21] and effectively decreases the excited surface currents on the keyboard ground, which in turn decreases the effects of the keyboard ground on the performances of the antenna. On the other hand, the surface current distributions of the proposed antenna are different from those of the case without the embedded slits. This causes large variation in the return loss of the lower band seen in Fig. 7. Simulated results of the return loss for the length d varied from 70 to 100 mm are presented in Fig. 9. Other dimensions are the same as given in Fig. 1. The results for the upper band are slightly affected. For

Fig. 9. Simulated return loss of the proposed antenna as a function of the length d of the embedded slits.

Fig. 10. Simulated return loss of the proposed antenna as a function of the distance w between the embedded slits and the connection hinges.

the lower band, when the length d is selected to be at least 80 mm, the impedance matching over the band can be better than 3:1 VSWR. It indicates that a longer embedded slit can lead to better results of reducing the keyboard ground effects. Fig. 10 shows the effects of the distance w between the embedded slit and the connection hinge. Results of the simulated return loss for the distance w varied from 1 to 5 mm are presented. Very small variations for frequencies over the lower and upper bands are seen. This indicates that the distance w is not a sensitive parameter in the proposed design. Fig. 11 shows the simulated return loss for the proposed antenna positioned at different locations along the top shielding metal wall of the display ground. In this study, the results of the simulated return loss for the distance s are varied from 15 mm (proposed in Fig. 1, near the right corner of the top shielding metal wall) to 155 mm (near the left corner

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Fig. 11. Simulated return loss for the proposed antenna positioned at different locations along the top shielding metal wall of the display ground.

Fig. 12. Simulated return loss for the proposed antenna with various sizes of the display ground and keyboard ground. The display ground and keyboard ground are of the same size. Fig. 13. Simulated 3-D radiation patterns at typical frequencies for the proposed antenna and the case without embedded slits.

of the top shielding metal wall). Although there are some effects on the impedance matching of the upper band, the upper-band bandwidth can still cover the desired GSM1800/1900/UMTS operation. For the lower band, however, the impedance matching is degraded to be lower than 6-dB return loss when the antenna is positioned close to the left corner of the top shielding metal wall (s = 140 and 155 mm). The results indicate that the distance s is also an important parameter of the proposed antenna in the practical applications. Fig. 12 shows the simulated return loss for the proposed antenna with various sizes of the display ground and keyboard ground. Both the display and keyboard grounds are of the same dimensions and other dimensions are the same as given in Fig. 1. Results of the simulated return loss for the 9-inch display (display ground 110 2 200 mm2 ), 10-inch display (display ground 130 2 220 mm2 ), 12-inch display (display ground 150 2 265 mm2 ), 15-inch display (display ground 1852335mm2 ) and 17-inch display (display ground 2102380mm2 ) are presented. In this case, although the effects of the keyboard ground are suppressed as discussed in Fig. 7 by embedding two slits in the keyboard ground, some effects on the impedance matching over both the lower and upper bands are seen, which is mainly owing to the different display ground sizes in the three cases. The variations in the display ground size lead to degraded impedance matching of some frequencies in the lower and upper bands. Effects of the ground plane size on the performances of the internal WWAN antenna cannot be ignored for the notebook computer applications. Further, owing to the variation of antenna directivity with different ground sizes, the antenna peak gain will be affected when the ground size varies. Also note that, for a larger display size (for example, 17-inch display case), since the display ground size is larger, effects of embedding the slits in the keyboard ground will become small. That is, the return-loss results for the cases with embedded slits, without embedded slits or without keyboard ground for the proposed antenna will be about the same. Fig. 13 shows the simulated three-dimensional (3-D) radiation patterns at typical frequencies for the proposed antenna and the case without embedded slits. Some variations in the radiation patterns in

the lower band are seen, which is because the excited surface current distributions are varied as shown in Fig. 8. The radiation patterns are generally about the same in the upper band, which is owing to similar surface current distributions seen in Fig. 8 at 1900 MHz. Finally, the antenna gain and radiation efficiency are measured in a far-field anechoic chamber (ETS-Lindgren measurement system, http://www.ets-lindgren.com]. The antenna is tested with the display and keyboard grounds shown in Fig. 1. The radiation efficiency including the mismatching loss is about 52–70% and 56–72% for the GSM850/900 and GSM1800/1900/UMTS bands, respectively. The efficiencies are all better than 50% for the five operating bands. The antenna gain is about 0.6–2.3 dBi and 1.5–3.2 dBi for the GSM850/900 and GSM1800/1900/UMTS bands, respectively. The obtained radiation characteristics are generally acceptable for practical notebook computer applications. IV. CONCLUSION A small-size internal WWAN notebook computer antenna has been proposed, fabricated and tested. The antenna is formed by a meandered loop antenna closely coupled with a parasitic shorted strip monopole. The antenna occupies a rectangular block of 50 2 10 2 4 mm3 only and is promising to be embedded inside the notebook computer as an internal antenna. The presence of the embedded slits in the keyboard ground can also lead to improved impedance matching for the proposed antenna. Good radiation characteristics for frequencies over the five WWAN operating bands have also been observed for the proposed antenna.

REFERENCES [1] C. T. Lee and K. L. Wong, “Study of a uniplanar printed internal WWAN laptop computer antenna including user’s hand effects,” Microw. Opt. Technol. Lett., vol. 51, pp. 2341–2346, Oct. 2009.

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[2] K. L. Wong and S. J. Liao, “Uniplanar coupled-fed printed PIFA for WWAN operation in the laptop computer,” Microw. Opt. Technol. Lett., vol. 51, pp. 549–554, Feb. 2009. [3] C. H. Chang and K. L. Wong, “Internal coupled-fed shorted monopole antenna for GSM850/900/1800/1900/UMTS operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 56, pp. 3600–3604, Nov. 2008. [4] X. Wang, W. Chen, and Z. Feng, “Multiband antenna with parasitic branches for laptop applications,” Electron. Lett., vol. 43, pp. 1012–1013, Sep. 2007. [5] K. L. Wong and L. C. Lee, “Multiband printed monopole slot antenna for WWAN operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 324–330, Feb. 2009. [6] K. L. Wong and F. H. Chu, “Internal planar WWAN laptop computer antenna using monopole slot elements,” Microw. Opt. Technol. Lett., vol. 51, pp. 1274–1279, May 2009. [7] T. W. Kang and K. L. Wong, “Internal printed loop/monopole combo antenna for LTE/GSM/UMTS operation in the laptop computer,” Microw. Opt. Technol. Lett., vol. 52, pp. 1673–1678, Jul. 2010. [8] P. Vainikainen, J. Ollikainen, O. Kivekas, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [9] J. Villanen, J. Ollikainen, O. Kivekas, and P. Vainikainen, “Compact antenna structures for mobile handsets,” in Proc. IEEE 58th Vehicular Technology Conf., vol. 1, pp. 40–44. [10] M. F. Abedin and M. Ali, “Modifying the ground plane and its effect on planar inverted-F antennas (PIFAs) for mobile phone handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 226–229, 2003. [11] K. L. Wong and L. C. Lee, “Bandwidth enhancement of small-size internal WWAN laptop computer antenna using a resonant open slot embedded in the ground plane,” Microw. Opt. Technol. Lett., vol. 52, pp. 1137–1142, May 2010. [12] C. H. Chang and K. L. Wong, “Printed =8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [13] Y. W. Chi and K. L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PCS/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 2541–2547, Sep. 2009. [14] Y. W. Chi and K. L. Wong, “Compact multiband folded loop chip antenna for small-size mobile phone,” IEEE Trans. Antennas Propag., vol. 56, pp. 3797–3803, Dec. 2008. [15] K. L. Wong and C. H. Huang, “Printed loop antenna with a perpendicular feed for penta-band mobile phone application,” IEEE Trans. Antennas Propag., vol. 56, pp. 2138–2141, Jul. 2008. [16] Y. W. Chi and K. L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, pp. 1457–1462, May 2007. [17] W. Y. Li and K. L. Wong, “Surface-mount loop antenna for AMPS/ GSM/DCS/PCS operation in the PDA phone,” Microw. Opt. Technol. Lett., vol. 49, pp. 2250–2254, Sep. 2007. [18] C. I. Lin and K. L. Wong, “Internal meandered loop antenna for GSM/ DCS/PCS multiband operation in a mobile phone with the user’s hand,” Microw. Opt. Technol. Lett., vol. 49, pp. 759–765, Apr. 2007. [19] B. K. Yu, B. Jung, H. J. Lee, F. J. Harackiewicz, and B. Lee, “A folded and bent internal loop antenna for GSM/DCS/PCS operation of mobile handset applications,” Microw. Opt. Technol. Lett., vol. 48, pp. 463–467, Mar. 2006. [20] B. Jung, H. Rhyu, Y. J. Lee, F. J. Harackiewicz, M. J. Park, and B. Lee, “Internal folded loop antenna with tuning notches for GSM/GPS/ DCS/PCS mobile handset applications,” Microw. Opt. Technol. Lett., vol. 48, pp. 1501–1504, Aug. 2006. [21] C. T. Lee and K. L. Wong, “Internal WWAN clamshell mobile phone antenna using a current trap for reduced groundplane effects,” IEEE Trans. Antennas Propag., vol. 57, pp. 3303–3308, Oct. 2009. [22] W. Y. Li and K. L. Wong, “Internal wireless wide area network clamshell mobile phone antenna with reduced ground plane effects,” Microw. Opt. Technol. Lett., vol. 52, pp. 922–930, Apr. 2010. [23] H. Liu, A. Napoles, and B. O. White, “Wireless Device With Distributed Load,” U.S. patent 7,199,762 B2, Apr. 3, 2007. [24] Tablet Personal Computer [Online]. Available: http://en.wikipedia.org/ wiki/Tablet_PC [25] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/ products/hf/hfss/

Improvement of Time and Frequency Domain Performance of Antipodal Vivaldi Antenna Using Multi-Objective Particle Swarm Optimization Somayyeh Chamaani, S. Abdullah Mirtaheri, and Mohammad S. Abrishamian

Abstract—The design of an antipodal Vivaldi antenna for ultra wideband (UWB) applications is presented. The main purpose of this design is the reduction of three parameters of transient distortion, reflection coefficient and cross polarization level. Multi-objective particle swarm optimization (MOPSO) is applied to handle these objectives simultaneously. All antenna simulations are performed using the CST MWS software. Several design cases of the resultant Pareto fronts are compared, and finally, a sample of the Pareto front is fabricated. To assess the time domain performance of the antenna, two identical antennas are placed in an end-to-end orientation. The simulated correlation factor in this scenario is more than 93%. of the developed antenna is less Measurement results show that the than 10 dB, the cross polarization level is below 11 5 dB and the gain is between 4.5 dBi and 9 dBi over the whole UWB frequency band. Index Terms—Antipodal Vivaldi antenna, correlation factor, multi-objective particle swarm optimization, ultrawideband antenna.

I. INTRODUCTION The Vivaldi antenna is the most popular directive antenna for commercial UWB applications due to its simple structure and small size. Recently, some methods have been proposed to improve the impedance matching bandwidth [1], gain [2], cross-polarization level [3], and transient distortion [4] of Vivaldi antennas. Most of these improvements are obtained at the cost of the change in the configuration of the antenna and leads us to manufacturing complexities by the use of dielectric rods for higher gain [2], additional substrate layer for cross-polarization reduction [3], and vias to decrease the transient distortion [4]. In ultra short pulse communications, the time domain response as well as the conventional frequency domain performance of the antenna must be considered. Therefore, the design of antenna for UWB communications is inherently multi-objective. In this communication, multi-objective particle swarm optimization (MOPSO) was applied to handle the multiple objectives through the design of a UWB antipodal Vivaldi antenna. These objectives are the minimization of transient distortion, reflection coefficient and cross polarization level. The antenna analysis was performed via commercially available CST MWS software. A sample of the resultant Pareto front was manufactured. The results show satisfactory time and frequency domain characteristics for the fabricated antenna. The main advantage of this design is the improvement of the correlation and the cross polarization level by only tuning the antenna geometry without any fabricating complexities. II. MOPSO ALGORITHM The basic version of the PSO, introduced by Kennedy and Eberhart in 1995, was devised for single objective problems. Recently, some Manuscript received September 10, 2009; revised September 20, 2010; accepted October 16, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The authors are with the K. N. Toosi University of Technology, Tehran 14317-14191, Iran (e-mail: [email protected]; [email protected]. ac.ir; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122290

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methods have been proposed to handle multi-objective problems and find related Pareto fronts using the PSO [5]. Compared to (non-dominated sorting genetic algorithms-II (NSGA-II) and multi-objective evolutionary algorithms (MOEA), MOPSO has produced better results for the same population size and for the same number of generations [5], [6]. Among several existing multi-objective algorithms, the MOPSO, proposed by Coello et al. [7], was selected due to its ability to maintain diversity between the solutions. The non-dominated particles are stored in a memory space called repository. At the each iteration, the repository is updated for the new non-dominated solutions. At the end of the process, the repository contents form the Pareto front. The main parts of this algorithm are given as follows: Let D be the number of optimization parameters. A D-dimensional vector X presents the position of a particle in the swarm. Each particle also has a velocity vector V. An initial N-size population with a random position and zero velocity is generated. Each of the particles is evaluated and the repository is initialized with the non-dominated solutions. For each particle located in the repository, a fitness sharing value is calculated. The fitness sharing for the i-th particle fsi is given by i

fs

= 10=n

(1)

i

where ni is the number of particles in the vicinity of the i-th particle. To determine ni , the objective space explored so far, is meshed using hypercubes. The number of particles located in the same hypercube as the i-th particle, is ni . A high value of fsi indicates that the vicinity of i-th particle is not highly populated. The velocity and position of the i-th particle are updated by the following relations. t

vi

+1 (d) = ! 1 vt (d) + c1 1 r1 1 (pbestt (d) 0 xt (d)) i

i

i

(2) + c2 r2 (REP (d) x (d)) +1 (d) = x (d) + v +1 (d) x (3) where v (d) and x (d) denote the current velocity (V) and the position 1

t i

t i

t h

1

t i

0

t i

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Fig. 1. General antenna geometry.

In this study, the Vivaldi antenna design problem is defined by three objectives. The first objective is to minimize the reflection coefficient over the operating frequency band (R:C:), the second is to minimize the transient distortion of the radiated pulses (T :D:) and the third is to minimize the cross polarization level of the antenna (X:P:). To deal with transient distortion, both the frequency and time-domain methods are applicable. To obtain a low distortion antenna using frequency domain method, two distinct objectives must be optimized: variations of jS21 j (magnitude of transfer function between transmitter and receiver antenna) and variations of group delay between them [8]–[10]. By contrast, in time-domain method, just one objective of correlation factor between the radiated pulses and a template pulse must be optimized [11]. Therefore, the frequency domain approach increases the complexity of the optimization algorithm and may slow the optimization process. The term considered for the reflection coefficient is given by

t i

t i

(X) of the d-th dimension of the i-th particle, respectively. Furthermore, r1 and r2 are random numbers between (0,1), ! is the inertia weight, and c1 and c2 are the acceleration constants. Finally, pbestit is the best experience of the i-th particle and REPht is the leader particle chosen from the repository. The leader is chosen by applying a roulettewheel selection using fitness sharing values assigned to each particle of the repository. In other words, the particles located in the highly populated areas in the objective space are less likely to be followed. The generated particles must be maintained in the valid search space. Each of the new particles is evaluated. The memory of each particle, pbest, and the repository contents are updated using the dominance criterion.

R:C:

=

f

T :D:

=1

= Ae + B: ay

(4)

The exponent coefficients for the upper curve (au ) and the lower curve (al ) participate in the optimization. The feed section is composed of a short microstrip line with length  = 3 mm and width g1 = 1:73 mm that exhibits 50 ohm impedance over the RF35 substrate with "r = 3:5 and 0.76 mm thickness and a tapered section with length f . The ground side of the tapered feed section is an exponential function with an exponent coefficient of ag . g2 has an indispensable effect on the VSWR value and participates in the optimization. Also, f and W are the tunable parameters of the tapered section.

(5)

=n =1 CF (i ; 'i )

i i

0

(6)

n

where C F (; ') is the correlation factor calculated between the radiated impulse e(t; ; ') and the template signal T (t). The minus of derivative of the excitation signal is selected as template. The correlation factor is given by

(

)

C F ; '

1

= max 

1

01

01

The primary profile of the antenna is formed by the parameters indicated in Fig. 1. The upper and lower edges of the tapered slot are exponential functions that defined by the following equation:

11 (f )jg:

fjS

The term considered for the transient distortion is given by

III. ANTENNA DESIGN

x

max

2[3:1 GHz; 10:6 GHz]

(

) ( +  )dt

e t; ; ' T t

2 je(t; ; ')j dt

1 01

2 jT (t)j dt

:

(7)

Virtual probes placed at the far zone of the antenna record the electric field intensity of the radiated impulses. Three probes (n = 3) are placed at 1 = 2 = 3 = 90 ; '1 = 75 ; '2 = 90 ; '3 = 105 . During the optimization, the fifth derivative of Gaussian pulse GM5 (t) is used as the excitation signal [12]. This signal is given by

5 (t) =

GM

5

0p

t

211

3 t + 102 9 p

15t exp t2 22 27 0

0 p

:

(8)

By setting  = 51 ps, (8) complies with Federal Communication Commission (FCC) mask for indoor systems. The term considered for the cross polarization is given by X:P:

=

f

max

2[3:1 GHz; 10:6 GHz]

cross co '=90 =90

E

E

:

(9)

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Fig. 4. Transfer function of two identical antennas in an end-to-end scenario: (a) jS j, (b) group delay. Fig. 2. Design 1 antenna (top layer). TABLE I DIMENSIONS OF DESIGN 1

Fig. 3. Measured and simulated reflection coefficient of the Design 1 antenna.

These three objectives are handled by MOPSO algorithm implemented in MATLAB. At the each iteration, to evaluate these objectives for an antenna structure, CST MWS is called from MATLAB, executes a VBA program that builds the antenna structure, simulates it, saves the desired parameters in the text files and finally is closed. MATLAB can access the antenna simulation results via text files. This procedure is repeated for all of the particles in all of the generations. IV. NUMERICAL AND EXPERIMENTAL RESULTS For this study, the population size is Npop = 10, the maximum number of generation is Tmax = 80, and the repository size Nrep = 100. On a single Intel 2.4 GHz quad processor, simulation of an antenna takes almost 15 minutes. Thus the whole optimization takes approximately 10 2 80 2 15 min = 200 hr. All designs belonging to the Pareto front are superior in terms of one or two objectives. However, empirical considerations lead us to select a design that all of its parameters remain below an acceptable value. Therefore, a sample from Pareto front namely Design1 is selected for fabrication (R:C: = 011:3 dB, T:D: = 0:011; X:P: = 019 dB). Design1 antenna is shown in Fig. 2 and its corresponding dimensions are listed in Table I. The electrical parameters of Design1 were measured using an HP8410C network analyzer outside a chamber that contains multi-path fading). Fig. 3 shows a jS11 j less than 010 dB in the frequency band 2.6–11 GHz. As shown, there is an acceptable agreement between the simulation and the measurement results. Fig. 4 illustrates the measured and simulated values of the transfer function of the two identical antennas in a Tx/Rx end-to-end scenario

Fig. 5. Simulated (dashed line), measured (solid line) Co-pol component (E ) and measured (dotted line) Cross-pol component (E ).

with a 100 cm distance between them. As shown, the maximum fluctuation of jS21 j is about 14 dB, which is very low in comparison with the UWB monopole antennas [13], [14] as well as in comparison with similar antipodal Vivaldi antennas [15]. Furthermore, the multi-path fading effect in real environments may be alleviated using proper gating which is very popular in pulse receivers. As shown in Fig. 4, if a gating width of Tg = 5:5 ns is applied to impulse response, the new jS21 j and group delay will be flatter. The measured and simulated radiation patterns at two sample frequencies of 4 GHz and 7 GHz for the E-plane (xy plane) and the H-plane (yz plane) are displayed in Fig. 5. A good agreement is observed between the measurement and the simulation results. Fig. 6 shows the simulated and measured cross polarization in main beam direction. The simulated result is better than simulated cross polarization of elliptically tapered antipodal Vivaldi antenna [16]. However, in test of small antenna characteristics, usually peripheral equipments such as crystal detector which has a size in the order of antenna lead us to an inaccurate measurement. This phenomenon is more obvious when a weak level of signal like cross polarization is measured. This is the reason of discrepancy between the simulated and measured one. However, it still shows an improvement in comparison with reported measurements for similar antipodal Vivaldi antennas [3]. Fig. 7 shows the gain of the Design1 antenna. As can be observed, the measured gain varies between 4.5 dBi and 9 dBi which is better than typical Vivaldi antenna gain reported in the literature [2], [4].

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Fig. 6. Cross polarization in main beam direction.

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Fig. 9. Fidelity factor pattern in the E-plane and the H-plane.

beam direction and 2) between the derivative of excitation signal and the received signal (Tx/Rx). The correlation factor in the main direction is more than 0.988, which has been increased in comparison with the value of 0.965 for optimized Vivaldi antennas reported in [4]. Also, the correlation factor in the Tx/Rx system is 0.934 that has been noticeably increased compared with the value of 0.84 reported in [15]. Fig. 9 shows the fidelity factor pattern in the E-plane and the H-plane. V. CONCLUSION

Fig. 7. Measured and simulated gain of the Design1 antenna.

The optimization of the UWB antipodal Vivaldi antenna using MOPSO was introduced. Three objectives of reflection coefficient, transient distortion and cross polarization level were minimized. MOPSO provided the optimum trade-off between solutions in a 3D Pareto front. A sample of Pareto front was fabricated. Experimental results indicated good impedance matching in the desired frequency band for this antenna. Furthermore, there was an improvement in the developed antenna’s time-domain dispersion characteristics, cross polarization level, and frequency domain gain in comparison with similar antennas reported in the literature. ACKNOWLEDGMENT The authors gratefully thank the Iran Telecommunication Research Center for its helpful support.

REFERENCES

Fig. 8. Waveform of different signals.

The transient behavior of the two identical Vivaldi antenna system in an end-to-end scenario was investigated. The antennas are placed 100 cm apart. Kanda has shown that the transmitting transient response is, in general, proportional to the time derivative of the receiving response [17]. On the other hand, the antipodal Vivaldi antenna in transmitting mode, differentiate the excitation and multiplies it by 01. Therefore, based on Kanda’s argument, in receiving mode, it just multiplies the incident signal by 01. Consequently, in this scenario, the received signal must be similar to the derivative of excitation. Fig. 8 shows the excitation impulse, the derivative of excitation impulse, the radiated impulse in main beam direction ( = 90 ; ' = 90 ), and the received signal. The correlation factor is calculated in two cases: 1) between minus of the derivative of the input signal and the radiated impulse in the main

[1] H. Oraizi and S. Jam, “Optimum design of tapered slot antenna profile,” IEEE Trans. Antennas Propag., vol. 51, pp. 1987–1995, Aug. 2006. [2] A. Elsherbini, C. Zhang, S. Lin, and M. Kuhn, “UWB antipodal Vivaldi antennas with protruded dielectric rods for higher gain, symmetric patterns and minimal phase center variations,” in Proc. IEEE Antennas and Propag. Society Int. Symp., 2007, pp. 1973–1976. [3] J. D. S. Langley, P. S. Hall, and P. Newham, “Novel ultrawide-bandwidth Vivaldi antenna with low crosspolarisation,” Electron. Lett., vol. 29, no. 23, pp. 2004–2005. [4] P. Cerny, J. Nevrly, and M. Mazanek, “Distortion minimization of radiated impulses of tapered slot Vivaldi antenna,” Automatika, vol. 49, pp. 45–40, 2008. [5] S. Mostaghim, “Multi-Objective Evolutionary Algorithms,” Ph.D. dissertation, Dept. Elect., Paderborn Univ., Paderborn, 2004. [6] S. K. Goudos, Z. D. Zaharis, D. G. Kampitaki, I. T. Rekanos, and C. S. Hilas, “Pareto optimal design of dual-band base station antenna arrays using multi-objective particle swarm optimization with fitness sharing,” IEEE Trans. Magn., vol. 45, pp. 1522–1525, Mar. 2009. [7] C. A. Coello Coello, G. T. Pulido, and M. S. Lechuga, “Handling multiple objectives with particle swarm optimization,” IEEE Trans. Evol. Comput., vol. 8, pp. 256–279, May 2004. [8] L. Lizzi, F. Viani, R. Azaro, and A. Massa, “A PSO-driven spline-based shaping approach for ultrawideband (UWB) antenna synthesis,” IEEE Trans. Antennas Propag., vol. 56, pp. 2613–2621, Aug. 2008.

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[9] F. Viani, L. Lizzi, R. Azaro, and A. Massa, “A miniaturized UWB antenna for wireless dongle devices,” IEEE Antennas Propag. Lett., vol. 7, pp. 714–717, 2008. [10] L. Lizzi, F. Viani, R. Azaro, and A. Massa, “Optimization of a splineshaped UWB antenna by PSO,” IEEE Antennas Propag. Lett., vol. 6, pp. 182–185, 2007. [11] M. Benedetti, L. Lizzi, F. Viani, R. Azaro, P. Rocca, and A. Massa, “A linear antenna array for UWB applications,” in Proc. Antenna and Propag. Soc. Int. Symp., 2008, pp. 1–4. [12] H. Sheng, P. Orlik, A. M. Haimovich, L. J. Cimini, and J. Zhang, “On the spectral and power requirements for ultra-wideband transmission,” in Proc. IEEE Int. Conf. Commun., 2003, pp. 738–742. [13] J. Liang, L. Guo, C. C. Chiau, X. Chen, and C. G. Parini, “Study of a printed circular disk monopole antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 53, pp. 3500–3504, Nov. 2005. [14] J. Liang, L. Guo, C. C. Chiau, X. Chen, and C. G. Parini, “Study of CPW-fed disk monopole antenna for ultra wideband applications,” in IEE Proc. Microwave Antenna Propag., Dec. 2005, vol. 152, pp. 520–526. [15] A. Mehdipour, K. Mohammadpour-Aghdam, and R. Faraji-Dana, “Complete dispersion analysis of Vivaldi antenna for ultra wideband applications,” Progr. Electromagn. Res. pp. 85–96, 2007 [Online]. Available: http://ceta.mit.edu/PIER [16] X. King, Z. N. Chen, and M. Y. W. Chia, “Parametric study of ultrawideband dull elliptically tapered antipodal slot antenna,” Int. J. Antenna Propag., vol. 2008, Article ID: 267197. [17] M. Kanda, “Time domain sensors and radiators,” in Time Domain Measurement in Electromagnetics, E. K. Miller, Ed. New York: Van Nostrand Reinhold, 1986, ch. 5.

Optimized Microstrip Antenna Arrays for Emerging Millimeter-Wave Wireless Applications Behzad Biglarbegian, Mohammad Fakharzadeh, Dan Busuioc, Mohammad-Reza Nezhad-Ahmadi, and Safieddin Safavi-Naeini

Abstract—Two compact planar antennas operating in the unlicensed 60 GHz frequency band are presented based on the physical layer specifications of IEEE 802.15.3c and ECMA 387 standards for different classes of wireless applications. Each antenna is an array of 2 2 microstrip antennas covering at least two channels of the 60 GHz spectrum. The first antenna is optimized to achieve the highest gain, while the second antenna is optimized to give the largest beamwidth. The maximum measured radiation gain of the first antenna is 13.2 dBi. The measured beamwidth and gain of the second antenna are 76 and 10.3 dBi, respectively. The areas of these two antenna are only 0.25 and 0.16 cm . The variation of radiation gain of each antenna over the frequency range of 57–65 GHz is less than 1 dB.

2

Index Terms—IEEE 802.15.3c, microstrip patch antenna, wireless communication, 60 GHz.

I. INTRODUCTION Recently antenna design for 60 GHz short-range wireless communication has become a subject of great interest [1]–[11]. Many of these antennas do not satisfy some of the requirements enforced by the recent Manuscript received March 16, 2010; revised September 14, 2010; accepted October 26, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the National Science and Engineering Research Council of Canada (NSERC) and Research in Motion (RIM). The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L3G1 Canada (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123058

60 GHz standards such as IEEE 802.15.3c [12] released in December 2009, and ECMA 387 [13]. Therefore, antenna design for 60 GHz applications will remain a hot research area for next few years. For emerging mass market millimeter-wave radio networks, system-on-chip radio modules integrated with compact planar antenna seem to be an attractive solution. A number of on-chip or in-package 60 GHz antennas have been reported so far [2], [3]. They are generally low-gain/low-radiation efficiency structures. Given very low transmitted power and poor receiver sensitivity of low-cost millimeter-wave technologies, such antennas can not mitigate the severe path loss, which is close to 88 dB for 10 m range at 60 GHz. Some of the designed antennas have a narrow beam [4] (30 ), which makes them suitable only for point-to-point applications. Compatibility with the emerging standards for the 60 GHz spectrum is the starting point for the antenna design. Narrowband or low-gain antennas do not comply with the aforementioned standard requirements. For example, the bandwidth of an active antenna for WPAN applications in [1] is about 0.8 GHz which is below the bandwidth requirement for 60 GHz devices (around 2 GHz). Finally, fabrication cost and complexity of the antenna is another major issue which affects its usability. In [8] a high gain and an acceptable bandwidth have been achieved by a CPW-fed integrated horn antenna with a simulated gain of 14.6 dBi. However the size of this end-fire antenna is 32 mm 2 20 mm and requires a complex fabrication process. For the first time, based on the analysis of the two proposed standards for 60 GHz wireless systems, IEEE 802.15.3c and ECMA 387, this work attempts to develop low-cost optimal antenna structures, which can deliver the required antenna gain, bandwidth and the beamwidth. Thus, two optimized, 2 2 2 compact microstrip array antennas are presented. The first antenna is optimized for point-to-point applications which require a high gain antenna. A maximum gain of 13.2 dBi has been measured for an area of 0.25 cm2 . The second antenna is optimized for point-to-multipoint applications requiring a wide beam. The measured beamwidth of this antenna exceeds 76 for an area of only 0.16 cm2 . Both antennas cover at least two channels of the 60 GHz spectrum. The developed antennas can be considered as low-cost and low-profile (planar) solution for various classes of services described in IEEE 802.15.3c and ECMA 387 standards. The organization of this communication is as follows. Section II briefly reviews the IEEE and ECMA standards to extract the required parameters for 60 GHz antenna design. Section III describes the design and optimization of high-gain and fan-beam patch arrays. Section IV presents the measured results, and Section V concludes this communication. II. REQUIRED ANTENNA CHARACTERISTICS ECMA STANDARDS

FOR

IEEE

AND

For 60 GHz short-range wireless communications two standard drafts, IEEE 802.15.3c and ECMA 387, have been released so far. IEEE 802.15.3c defines three classes for different wireless (single carrier) applications [12]: • Class 1 addresses the low-power low-cost mobile market with a relatively high data rate of up to 1.5 Gb/s; • Class 2 supports for data rates up to 3 Gb/s; • Class 3 supports high performance applications with data rates in excess of 5 Gb/s. Similar to IEEE 802.15.3c, ECMA 387 defines three types of devices (Type A, B and C ) for 60 GHz spectrum [13] based on the operational requirements such as maximum range, bit rate and system complexity. Several operational modes have been proposed for each type. Table I shows the requirements of the basic and highest-rate modes of each

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TABLE I MODE DEPENDENT PARAMETERS OF TYPE A, B, AND C DEVICES IN ECMA STANDARD

TABLE II mm-WAVE PHYSICAL CHANNELIZATION

Fig. 1. Transmitter power versus antenna gain for different operational modes at 57 GHz.

device type. In the following, the main parameters of the physical layer of these two standards, which must be considered in the antenna design, are discussed. A. Antenna Bandwidth In both standards, the 60 GHz spectrum has been divided into four channels, shown in Table II. Each channel has a bandwidth of 2.16 GHz. Any device must support at least one channel. In North America, the first three channels have been released. Channel four has been released in Japan and Europe. B. Antenna Gain In a communication link, if the same antennas are used for both transmitter and receiver (GR = GT = Ga ), then the transmitter power (PT ) is related to the antenna gain, receiver sensitivity (Smin ) and path loss (PL) by

PT + 2Ga  Smin 0 PL

(1)

where PT and Smin are in dBm and Ga and PL are in dB. The righthand side (RHS) of (1) is determined by the standard. But the left-hand side (LHS) gives a freedom to the designer to choose the suitable power amplifier and antenna. Fig. 1 demonstrates the transmitter power versus the antenna gain at 60 GHz for the basic modes described in Table I, for LOS links. This figure also shows the FCC limit on the indoor EIRP (document 47 CFR 15.255). For an integrated RFIC the transmitted power is limited by the semiconductor processing technology. The horizontal lines in Fig. 1 show the typical technology limits for CMOS technology (0–10 dBm). Fig. 1 shows that for CMOS-compatible Type C devices a low gain antenna (Ga  2:7 dBi) is sufficient. For Type B devices in CMOS the antenna gain can vary from 3.3 to 8.3 dBi. However for Type A, which is considered as the high end-high performance device [13], a high-gain antenna is necessary. An antenna with an affective gain of 13.7 dBi at 57 GHz for PT = 0 dBm is ideal for mode A0 operating at the maximum range of 10 m. For transmitting higher bit-rates at the maximum range either an array of such antenna can be used or the transmitter power can be increased. Therefore, one of the objectives of this communication is to design a compact antenna with a maximum gain above 13 dBi. The RX-TX antenna polarization mismatch or misalignment can be compensated by a small increase (1–3 dB) in the transmitter power.

Fig. 2. Finding the beamwidth of the antenna. (a) Simulated scenario. (b) Antenna angular coverage versus room size for uniform user distribution.

C. Antenna Beamwidth For 60 GHz applications antennas with higher gains (e.g., 13.7 dBi for Type A) are required, which have directional patterns. For certain receivers connecting to an Access Point (AP) in a room, such as cell phone or laptop, an omni pattern is not required. Assume a square room with a size of l 2 l and the height of 3.5 m. An AP has been installed at the center of ceiling with a hemispherical pattern. It is also assumed that the user distribution along the z-coordinate is uniform ranging within 1.5 to 2.5 m from ceiling (AP). To find the required coverage angle, 20 000 random user locations were generated. The angle between the line connecting the user to AP and z axis was calculated for each user, and the histogram of all calculated angles was plotted. Fig. 2 shows the required values for beamwidth to cover 90% and 100% of the indoor users versus room size (2  l  10) for uniform user distribution. For the maximum room size (10 m), the RX antenna coverage must be respectively 72 and 77.5 to include 90% and 100% of users. III. ANTENNA DESIGN AND OPTIMIZATION Although employing a high-gain antenna relaxes the signal to noise requirement of the front-end RF system, its narrow beam can not cover

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Fig. 3. Antenna array gain versus patch spacing.

the entire zone for point-to-multipoint wireless connections. In this section two antennas for different 60 GHz applications are proposed based on the IEEE 802.15.3c and ECMA 387 standards. The first antenna is designed for the maximum gain for point-to-point applications over the maximum distance of 10 m. The second antenna is optimized to give the maximum beamwidth for point-to-multipoint applications such as WPAN. Each antenna is an array of four patch elements. The distances between patch elements are optimized to obtain the maximum gain or beamwidth. Patch antenna is a planar, high-gain and efficient radiator that can be integrated easily with the rest of the system. A single rectangular patch antenna typically provides 7 dB gain and more than 100 beamwidth when its fundamental mode (TM10) is excited [14]. In this communication, a very low loss substrate (RT/duroid-5880) with 10 mil height, tan  = 1 2 1003 and dielectric constant of r = 2:2 is used as the substrate for microstrip antennas and the feeding lines. The patch and the ground plane are printed on the top and bottom of the dielectric substrate, respectively. The dimensions of the single patch antenna on such substrate is optimized to achieve the maximum gain at 60 GHz. The size of the patch element is 1:85 2 1:45 mm2 . The simulations in Ansoft HFSS show that this single patch has 7.7 dBi gain at 60 GHz. The simulated HPBW in E -plane exceeds 96 , but the HPBW in H -plane is limited to 68 .

Fig. 4. (a) Top view of the high-gain 2 cient of this antenna.

2 2 patch array. (b) Reflection coeffi-

A. High-Gain Antenna Design In this section an array of four patch antennas is formed and the x and y spacings between antennas are optimized to achieve the highest gain. Fig. 3(a) shows the 2 2 2 patch array gain versus patch spacing at 60 GHz obtained by HFSS simulation. It is seen that the maximum gain of 13.5 dBi is achieved for x = 3:5 mm and y = 3:2 mm. The effect of feed network on the array gain is also studied. After optimization, it is seen that the maximum gain is obtained for a1 = 0:2 mm and b1 = 0:45 mm. Parameters a1 and b1 are shown in Fig. 4(a). As shown in Fig. 4(a) the size of the patch array for the optimized spacings is 5.35 mm 2 4.65 mm. Fig. 4(b) shows the reflection coefficient of this array. It is seen that this antenna covers two channels of the 60 GHz spectrum from 57.24 to 61.46 GHz. The return loss is always more than 7.5 dB. The resonance frequency is at 59.4 GHz, the common edge of the two adjacent channels, therefore the most efficient use of the antenna bandwidth has been achieved. Fig. 5(a) and (b) shows the 2D and 3D radiation patterns of the high-gain 2 2 2 patch array. The 3 dB beamwidth of the array in  = 0 plane (x 0z plane) and  = 90 (y 0z plane) are 41 and 36 , respectively. The first pair of nulls happen

Fig. 5. 2D radiation pattern of the high-gain 2

2 2 patch array.

approximately at 650 . Fig. 5(b) shows that the radiation pattern is symmetric around the normal axis. B. Fan-Beam Antenna Design

To achieve the required beamwidth of 77.5 , discussed in Section II-C, another patch array is designed and the element spacing is optimized to obtain the maximum beamwidth. Table III shows the maximum gain of 2 2 2 array at 57 GHz for different element spacings. It is seen that the gain increases with the element spacing since the effective aperture size increases. For

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TABLE III MAXIMUM GAIN OF FAN-BEAM PATCH ARRAY VERSUS ELEMENT-SPACING

TABLE IV BEAMWIDTH OF FAN-BEAM PATCH ARRAY IN E -PLANE

TABLE V BEAMWIDTH OF FAN-BEAM PATCH ARRAY IN H -PLANE

Fig. 6. (a) Top view of the fan-beam 2 cient of this antenna.

y  2:6 mm the maximum gain is always above 10 dBi. We choose Ga  10 dBi as a criterion to find the optimum patch spacing. If a higher gain was required two of these fan-beam antennas can be arrayed in the y -direction to give around 13 dBi gain while the maximum beamwidth at  = 0 surface is unaffected. Table III indicates that the element spacing along x axis must be equal or larger than 2.6 mm to obtain 10 dBi gain. Table IV and Table V present the patch array beamwidth in E -plane and H -plane versus element spacing. The beamwidth in E -plane reduces as the element spacing increases; however, the beamwidth in H -plane is almost insensitive to x-spacing for the values shown in Table V. We choose the spacings which lead to the largest E -plane beamwidth while the gain is above 10 dBi. Table IV shows that for x = 2:6 mm and y = 2:3 mm a beamwidth of 78 is achieved. The size of the patch array, shown in Fig. 6(a), for the optimized spacings is 4.45 mm 2 3.7 mm. Fig. 6(b) shows the reflection coefficient of the fan-beam 2 2 2 patch array. The resonant frequency of the antenna is at 59.5 GHz, and the 10 dB bandwidth covers channel 1 and 2 of the 60 GHz spectrum. Fig. 7 shows the 2D and 3D radiation patterns of the fan-beam 2 2 2 patch array. The 3 dB beamwidths of the array in E -plane and H -plane are 78 and 46 , respectively. The first pair of nulls are far away from the maximum gain direction at 690 . Hence, any possible rotation or misalignment of the antenna cannot nullify the received signal. Fig. 7(b) shows that the radiation pattern in y 0 z plane is symmetric, while the pattern in x 0 z plane is asymmetric due to the feed line effect. IV. FABRICATION AND MEASUREMENTS This section describes the test set-up and presents the measured results of both patch antenna arrays. A. High-Frequency Waveguide to Microstrip Transition Accurate measurement requires minimal interconnect loss between various parts of the test setup. Most of the measuring systems at mm-wave frequency have WR-15 waveguide ports, therefore in this

2 2 patch array. (b) Reflection coeffi-

Fig. 7. 2D radiation pattern of the fan-beam 2

2 2 patch array.

work a low insertion/reflection loss waveguide to microstrip transition (WMT) at 60 GHz band is developed for characterization of the patch antenna arrays. To lower the fabrication cost, a two-part machined metallic structure made out of Aluminum was designed and fabricated for the transition. For the microstrip antenna connection, a perpendicular coaxial transition was developed. This configuration was preferred to an end-launch type due to the easier implementation and better narrowband matching. The cost is also reduced by using simple Corning glass-based 50-ohm coax probes that connect the waveguide section to the microstrip antenna. The narrow air-gap between the two parts of the waveguide assembly creates a discontinuity in the surface current of dominant TE10 mode. This could potentially increase the loss. As a remedy a larger number of sealing screws and a raised ridge was used. The drawing

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Fig. 8. Transition design features and manufacturing implementation.

Fig. 10. Fabricated high-gain and fan-beam 2

2 2 patch array antennas.

Fig. 9. Performance of the back-to-back configuration of two microstrip to waveguide transitions. (a) S . (b) Insertion loss S .

2

and the picture of the developed WMT are illustrated in Fig. 8. The channel length of WR-15 waveguide section is chosen for mechanical convenience. Note that the channel length (L) must include a 0:25g (the quarter guided wavelength in the waveguide) section between the probe location and the shorted section of the waveguide optimized for matching. In our case, the total waveguide length is close to 7 (or 35.2 mm). To determine the performance of this WMT, two identical structures are connected in a back-to-back configuration and tested. The measurement setup involves two WR-15 type waveguides, which are calibrated in a 2-port setup, and a coaxial two-sided probe to mate these two identical structures. Fig. 9(a) and (b) compares the simulated and measured S11 and S21 of the back-to-back configuration. The measured transmission S21 illustrates that at 60 GHz the insertion loss of this configuration is close to 2.0 dB, which indicates that a single WMT has around 1.0 dB loss. The information provided by Fig. 9(b) is used to de-embed the insertion loss of WMT from the measured antenna gain in next section.

Fig. 11. S Simulation and measurement results of the 2 2 patch array antennas connected to the waveguide transitions. (a) High-gain antenna. (b) Fanbeam antenna.

B. Measurement Results

Fig. 12. Measured co-polarization and cross-polarization radiation patterns of the high-gain 2 2 patch antenna arrays at 60 GHz frequency.

Fig. 10 shows the fabricated 2 2 2 patch antenna arrays. The antennas were mounted and attached to the transitions described earlier. The antennas combined with WMT were simulated in HFSS to see the effects of the transition on the reflection coefficient of the antennas. Fig. 11(a) and (b) shows the measured and simulated S11 of the high-gain and fan-beam 2 2 2 patch antenna arrays, respectively. The bandwidth of the high-gain and fan-beam antennas are 3.5 and 3.6 GHz around 60 and 61.70 GHz, respectively. As it is shown in both figures, there are more than one resonances over the frequency range 50–75 GHz. These resonances show the nonradiating cavity modes of

the waveguide section of WMT. This was experimentally verified by using absorbers to suppress the radiating fields into free space. While the resonances of the radiating modes were affected significantly, the non-radiating cavity modes did not change, enormously. Comparing these figures to Fig. 4(b) and Fig. 6(b) and considering the good agreement of HFSS simulations and the measurement results, we can conclude that the array antennas without WMT meet the required bandwidth specifications for the aforementioned wireless standards.

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fan-beam array, the maximum gain varies from 9.5 dBi at 57 GHz to to 10.5 dBi at 61 GHz. So, the variation of the radiation gain of both antennas over 7 GHz bandwidth (57–64 GHz) is less than 1 dB. V. CONCLUSION

Fig. 13. Measured co-polarization and cross-polarization radiation patterns of the fan-beam 2 2 patch array at 60 GHz frequency.

2

In this communication, the two optimized 2 2 2 patch array antennas for emerging millimeter-wave radio networks were presented. Based on IEEE 802.15.3c and ECMA 387 standards, the radiation characteristics of the antenna such as gain, bandwidth and beamwidth have been derived. The developed antennas fully meet the requirements for the point-to-point line-of-sight applications and maximum signal coverage. The measured results of the high-gain antenna show more than 13 dBi gain, and 3.5 GHz of impedance bandwidth including the microstrip to waveguide transition. Furthermore, the developed fan-beam antenna has more than 10 dBi gain with 76 beamwidth in E -plane and 3.6 GHz impedance bandwidth. The variation of the maximum radiation gain of both antennas over the whole 7 GHz spectrum is less than 1 dB. These two antennas can be easily integrated with phase shifters to form high-gain array antennas with beam-scanning capabilities. ACKNOWLEDGMENT The authors would like to acknowledge Dr. G. Rafi for his invaluable support regarding the antenna measurements.

REFERENCES

Fig. 14. Measured maximum gain of patch antenna arrays over the frequency range of 55–65 GHz. (a) High-gain array. (b) Fan-beam array.

Fig. 12 shows the measured co-polarization and cross-polarization radiation pattern of the high-gain 2 2 2 patch antenna array at 60 GHz frequency. The maximum measured gain at this frequency is 13 dBi after de-embedding the insertion loss of WMT (see Fig. 9) and considering 0.3 dB loss for the coaxial probe which feeds the antenna. The measured beamwidth is 42 . Comparing this figure to Fig. 5(a) which shows the simulated gain and radiation pattern, it is found that the maximum gain at 60 GHz has dropped by only 0.5 dB. The measured beamwidth and overall pattern shape are in very good agreement with the simulated values. The maximum cross-polarization gain of the high-gain array with respect to its co-polarization gain is at least 010 dB which is at  = 40 . Fig. 13 depicts the measured co-polarization and cross-polarization radiation pattern of the fan-beam antenna at 60 GHz. The HPBW of the fan-beam antenna is 76 and its maximum gain is 10.3 dBi. Comparing this figure to Fig. 7 shows that the agreement between simulated and measured patterns is very good. The beamwidth has slightly decreased (by about 2 degrees) which is within the fabrication tolerances. The cross polarization gain of the fan-beam array antenna is at least 25 dB less than co-polarization term at any direction. Fig. 14 depicts the measured maximum gain of patch antenna arrays over the frequency range of 55–65 GHz after de-embedding the insertion loss of WMT. The maximum gain of the high-gain array varies from 12.25 dBi at 57 GHz to 13.25 dBi at 65 GHz. For the

[1] C. Karnfelt, P. Hallbjorner, H. Zirath, and A. Alping, “High gain active microstrip antenna for 60-GHz WLAN/WPAN applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2593–2603, Jun. 2006. [2] I.-S. Chen, H.-K. Chiou, and N.-W. Chen, “V-band on-chip dipolebased antenna,” IEEE Trans. Antennas Propag. , vol. 57, no. 10, pp. 2853–2861, Oct. 2009. [3] Y. Zhang, M. Sun, and L. Guo, “On-chip antennas for 60-GHz radios in silicon technology,” IEEE Trans. Electron Devices, vol. 52, no. 7, pp. 1664–1668, Jul. 2005. [4] Y. P. Zhang, M. Sun, K. M. Chua, L. L. Wai, and D. Liu, “Antenna-in-package design for wirebond interconnection to highly integrated 60-GHz radios,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2842–2852, Oct. 2009. [5] K.-C. Huang and D. Edwards, “60 GHz multibeam antenna array for gigabit wireless communication networks,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3912–3914, Dec. 2006. [6] A. Rosen, R. Amantea, P. Stabile, A. Fathy, D. Gilbert, D. Bechtle, W. Janton, F. McGinty, J. Butler, and G. Evans, “Investigation of active antenna arrays at 60 GHz,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pp. 2117–2125, Sep. 1995. [7] A. Lamminen, J. Saily, and A. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrates,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2865–2874, Sep. 2008. [8] B. Pan, Y. Li, G. Ponchak, J. Papapolymerou, and M. Tentzeris, “A 60-GHz CPW-fed high-gain and broadband integrated horn antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1050–1056, Apr.l 2009. [9] A. Patrovsky and K. Wu, “Active 60 GHz front-end with integrated dielectric antenna,” Electron. Lett., vol. 45, no. 15, pp. 765–766, 2009, 16. [10] Y. Murakami, T. Kijima, H. Iwasaki, T. Ihara, T. Manabe, and K. Iigusa, “A switchable multi-sector antenna for indoor wireless LAN systems in the 60-GHz band,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 841–843, Jun. 1998. [11] P. Smulders and M. Herben, “A shaped reflector antenna for 60-GHz radio access points,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1013–1015, Jul. 2001. [12] IEEE, IEEE 802.15 WPAN Task Group 3c (TG3c) Sept. 2009 [Online]. Available: http://www.ieee802.org/15/pub/TG3c.html [13] ECMA, Standard ECMA-387: High Rate 60 GHz PHY, MAC and HDMI PAL Dec. 20081st ed. [Online]. Available: http://www.ecma-international.org/publications/files/ECMA-ST/Ecma-387.p df [14] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, U.K.: Inst. Elect. Eng. (IEE), 1989.

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Pareto Optimization of Thinned Planar Arrays With Elliptical Mainbeams and Low Sidelobe Levels Joshua S. Petko and Douglas H. Werner

Abstract—A multi-objective Pareto genetic algorithm design methodology is applied to thinned planar arrays to simultaneously minimize peak side-lobe levels and target an elliptical mainbeam with specific minimum and maximum half-power beamwidths. This new radiation pattern synthesis technique for thinned planar arrays provides antenna engineers with a set of tradeoffs between low side-lobe levels and close adherence to mainbeam design objectives (i.e., the specified half-power beamwidths corresponding to the major and minor axes of an elliptical mainbeam). One Pareto optimization example is presented for a thinned low side-lobe planar array with a desired minimum and maximum beamwidth of 8.4 and 12 respectively. Two designs from the Pareto front are discussed, one with a 20 92 dB side-lobe level and beamwidths between 11.5 and 7.5 and a second with a 18 97 dB side-lobe level with beamwidths between 12 and 7.93 . Index Terms—Antenna arrays, genetic algorithms (GAs), Pareto distributions.

I. INTRODUCTION Genetic algorithms (GA) and particle swarms (PS) are two nature-inspired optimization techniques that have received a considerable amount of recent attention in the electromagnetics community [1]–[6]. These methods have often been applied to the optimization of antenna array radiation patterns either through the modification of current excitations or through the manipulation of the array geometry [6]–[14]. For instance, one of the initial contributions in this area used a genetic algorithm to selectively remove individual elements from the lattice of a uniformly excited periodic antenna array in order to achieve reduced peak side-lobe levels (SLL) [6]. This process is known as array thinning. Nevertheless, while standard GAs and PSOs are extremely well adept at evolving solutions based on one parameter; many practical antenna array design problems require the optimization of more than one parameter. When faced with these problems, it is desirable to be presented with the set of solutions that are not dominated by any other solution and choose the one which best fits the design goals. This set of non-dominated solutions is referred to as the Pareto front. In recent years, several unique multi-objective genetic algorithms have been implemented with the goal of finding this Pareto front [15]–[23]. Typically, these algorithms perform similarly to conventional genetic algorithms except the overall fitness of a population member is based on its position in the solution space in relation to the other population members rather than based on performance alone. In this way, multi-objective genetic algorithms attempt to maintain diversity along the Pareto front as it evolves while achieving convergence toward the final Pareto front. These optimization techniques have been applied to many different design problems, including some in the field of electromagnetics [10], [14]. In one such example, a Pareto based Manuscript received July 02, 2009; revised December 16, 2009; accepted October 16, 2010. Date of publication May 04, 2011; date of current version May 04, 2011. J. S. Petko is with Northrop Grumman Electronic Systems, Linthicum, MD 21090 USA (e-mail: [email protected]). D. H. Werner is with the Department of Electrical Engineering and Applied Research Laboratory, Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122212

genetic algorithm was used to evolve thinned planar arrays with both narrow beamwidths and low peak sidelobe levels [10]. In addition to the peak side-lobe levels, another important antenna array design requirement characteristic is the overall size and shape of the mainbeam. However, the most common metrics, half-power beamwidth and first null beamwidth, are only valid when the mainbeam is well-behaved. It is not uncommon when such metrics are employed in array optimization problems, to obtain solutions which satisfy the desired half-power and/or first null beamwidths but at the same time possess deformed or otherwise undesirable radiation pattern characteristics. One design methodology that has proven useful in synthesizing radiation patterns with well-behaved mainbeams is based on minimizing the error between the current pattern and a desired ideal cosine pattern [24]. However, this technique alone is relatively ineffective in reducing the peak side-lobe level since the radiation pattern errors are small in this region unless a grating lobe is present. Hence, this communication examines the optimization of a thinned planar antenna array based on two design criteria; namely, minimizing the peak side-lobe level and matching the mainbeam of the radiation pattern to an ideal cosine pattern with specified minimum and maximum half-power beamwidths. These two design criteria are combined together through the application of a multi-objective genetic algorithm to create a new and more flexible radiation pattern synthesis methodology for thinned planar arrays. This new radiation pattern synthesis technique is capable of presenting engineers with a set of tradeoffs between low side-lobe levels and close adherence to mainbeam design objectives (i.e., the specified half-power beamwidths in the two principal planes). This methodology possesses practical benefits when applied to the area of satellite communication, where engineers have a limited power budget to send information to receivers on the ground. In addition, extra constraints on the spurious side-lobe radiation may be necessary when unsecured broadcast channels are present or when neighboring regions have laws prohibiting interference at operating frequencies (for example, the National Radio Quiet Zone in Eastern West Virginia). For this reason it is desirable to tailor an antenna to radiate power only over the area where the receivers are located. By employing the methodology introduced in this communication, engineers can either pre-fabricate or adaptively control the antenna performance by switching on or off the individual elements of the array. Therefore, this procedure can focus the transmitted radiation to a specified area without the use of amplitude attenuators or costly amplifier networks. In this communication, several designs are discussed, each based on an optimization that targets a relatively large ratio between the maximum to minimum beamwidths of the radiation pattern. II. MULTI-OBJECTIVE OPTIMIZATION Many design problems require the optimization of more than one parameter, however single parameter optimizations are often employed in the study of these types of problems. The most common modification involves first specifying a line in the solution space and then defining fitness as the orthographic projection of the solution on that line. This method is achieved by using a weighted sum of the solution parameters and can be very effective if a specific ratio between the design parameters is desired. However, the main issue with creating single objective optimization problems from multiple objective problems is that one limits their search to a small sector of the solution space. This restricts the possibilities that can be used for the final design and can limit the intermediate steps needed to make an optimization effective. Therefore, ideally, the goal of the optimization is to find the set of solutions of which no other solution dominates (i.e., the Pareto front). Domination

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of a member of the population, A by another solution, B is defined by the relation

tnessi (A) < tnessi (B ) for every value i

(1)

where i is the index used to reference each member of the population of solutions. While scientists and engineers have recognized the value of finding the Pareto front of solutions, the methods used to find the Pareto front are constantly improving and continue to be part of a very active area of research. The first attempt at a multiple objective optimization algorithm was the vector evaluated genetic algorithm (VEGA), introduced by Schaffer in 1984 [15]. VEGA had difficulty in finding the entire Pareto front, often finding groups of solutions along the front. The next major advance in this field was the development of the concepts of non-dominated sorting and fitness sharing, introduced by Goldberg in 1989 [16]. Non-dominated sorting and fitness sharing are used together to ultimately judge population members by a single fitness parameter related to their dominance over and proximity to other members in the solution space. Development of these concepts sparked a fury of research in the field leading to several closely related multi-objective optimization techniques, two of which are the niched Pareto genetic algorithm (NPGA) [16]–[18] and the non-dominated sorting genetic algorithm (NSGA) [16]–[18]. In the time after the concepts of non-dominated sorting and fitness sharing were first introduced, several unique multi-objective genetic algorithms have been reported [19]–[23]. One optimization, the strength Pareto evolutionary algorithm (SPEA) [19] and [20], has become of particular interest because of its simplicity and effectiveness. The strength Pareto evolutionary algorithm was developed by Zitzler and Thiele in 1998 [19] and [20]. Like the NSGA and NPGA, SPEA also ultimately evaluates members by a single fitness value, which is called the strength; however, this strength is not based on individual fitness values or proximity to other members but solely on the domination of members of the population over others. Therefore, strength in the SPEA is completely independent of the fitness parameters, whereas the NSGA and NPGA required the optimization designer to define the proximity distance in the solution space to enable fitness sharing. In this communication, we use the SPEA to optimize large thinned planar arrays; however, in our implementation we specifically carry from one generation to the next the entire Pareto front and a number of the best-performed non-Pareto solutions. This stopgap measure is not always indicative of these optimization techniques, but because these solutions are the most desirable from the previous generation, the calculation of the fitness function (cost function in the case of minimization) is computationally very expensive and the size of our population is not very large, it makes sense for the optimization to keep these solutions. The following section details the specific considerations made when evaluating the cost parameters. III. COST PARAMETERS The antenna arrays in this communication are optimized using several different cost parameters. The goal of the first cost function is to minimize the peak side-lobe level (SLL). While this direct approach is effective in reducing the peak side-lobe level, it has no impact on the size or shape of the mainbeam. It is possible that arrays optimized considering only peak SLL can have mainbeams that become distorted, such as possessing a shoulder or other inflection in the mainbeam structure. To overcome this problem, one approach is to fit the radiation pattern to an ideal cosine pattern, which can be represented by [24]

f () =

cos 2 0;

 

;

where  2 [0;  ] elsewhere

(2)

Fig. 1. Application of the ideal cosine functions in the evaluation of a radiation pattern. Note that two cosine functions are implemented such that an elliptical mainbeam can be achieved in the radiation pattern.

Fig. 2. Error of a sample radiation pattern cross section. The radiation pattern cross-section is compared with an ideal cosine function where the shaded portion of the graph represents this error.

where 2 is the first null beamwidth of the array. In this optimization, the maximum and minimum cross-sections of the radiation pattern mainbeam are first identified. Then each of these radiation pattern cross-sections are compared to ideal cosine patterns, one based on a maximum desired beamwidth and one based on a minimum desired beamwidth producing two error values, ef; and ef; . The manner in which these ideal cosine functions are applied to a radiation pattern is illustrated in Fig. 1. Moreover, the error between a specific ideal cosine function and an example radiation pattern cross section is illustrated as the shaded portion of Fig. 2. From this point, the mean-squared errors for both minimum and maximum radiation pattern cross-sections are evaluated. We evaluate these parameters by averaging the square of the difference between the ideal and actual radiation patterns at every sample along the radiation pattern. These error values now become the second and third cost parameters for the optimization. Therefore the set of fitness parameters that must be reduced by such an optimization is given by

SLL

Scost = Minimize e e

f;

:

(3)

f;

IV. RESULTS The SPEA has been used here to evolve thinned periodic planar arrays. The optimized arrays are based on a 20 2 20 square periodic lattice, with 0:5 spacing between the lattice sites. The strength values are based on three cost parameters. The first cost value is equal to the

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Fig. 3. Final Pareto front for a population of thinned planar arrays. Costs are based on the peak side-lobe level, beamwidth conformity of the mainbeam at its maximum and beamwidth conformity of the mainbeam at its minimum.

Fig. 5. Layout (a) and radiation pattern (b) for a 164-element SPEA optimized thinned periodic planar array.

Fig. 4. Layout (a) and radiation pattern (b) for a 178-element SPEA optimized thinned periodic planar array.

peak SLL of the array. The second cost value is based on the conformity of the widest mainbeam cross-section to a cosine pattern with a

12 half-power beamwidth, while the third value is based on the conformity of the narrowest mainbeam cross-section to another cosine pattern with an 8.4 half-power beamwidth. The smallest value of cross-section is 70% the width of the maximum cross-section. SPEA optimizes the population of arrays over 100 generations, adding 300 new arrays to the population on each generation and keeping only 100 dominated solutions for the next generation. The arrays are compared against a maximum of 75 clusters. The final population is illustrated in Fig. 3. From the population of thinned periodic planar arrays shown in Fig. 3, two Pareto optimal solutions are chosen for discussion. The first example is an optimized 178-element array with a 020:92 dB peak SLL, a maximum half-power beamwidth of 11.5 and a minimum half-power beamwidth of 7.5 . This solution had the best SLL suppression in the population. The geometry of this array and its respective radiation pattern are shown in Fig. 4 and the performance parameters of this array are summarized in Table I. The second example is an optimized 164-element array with a peak SLL of 018:97 dB, a maximum half-power beamwidth of 12.0 and a minimum half-power beamwidth of 7.93 . This array was chosen because the shape of the mainbeam is closest to the desired design goals. In fact, the maximum half-power beamwidth is exactly equal to the design goal of 12 . This array geometry and its respective radiation pattern are shown in Fig. 5, while the performance parameters are listed in Table II. Directivity analysis shows that this methodology not only provides an effective way to regulate the radiation patterns but also is com-

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TABLE I PERFORMANCE OF THE 178-ELEMENT SPEA OPTIMIZED THINNED PERIODIC PLANAR ARRAY

TABLE II PERFORMANCE OF THE 164-ELEMENT SPEA OPTIMIZED THINNED PERIODIC PLANAR ARRAY

parable in efficiency to standard array geometries. To calculate the directivity, we apply the following relationship derived for arrays of isotropic sources [25]

2 N I n n=1 Dmax = N (4) N n01 sin( kjr 0r j) 2 + 2 I I I n m n kjr 0r j n=1 n=2 m=1 where In and Im represent element excitations, rn and rm represent

k

element positions and is the free space wavenumber. The directivity of the 178 and 164 element SPEA optimized thinned arrays were found to be 23.58 dB and 23.33 dB respectively. These directivities are comparable to the 23.98 dB directivity found for a 13 2 13, 169 element periodic array. In addition, considering the application of adaptive arrays for satellite communication links, this approach proves to be an efficient method for transmission beamforming. The directivity of the full 20 2 20 periodic array is 27.85 dB, but the peak side-lobe level is only 013 2 dB. When a Taylor amplitude weighting [26] is applied to suppress the peak side-lobe level to be 020 dB (nbar = 4), the directivity of the array remains high at 27.55 dB. However, the overall power radiated from the antenna is reduced to be 40% (04 dB) of its original value. Considering that adaptive amplitude tapering most commonly is achieved through attenuation, this approach would require more sophisticated thermal cooling management while only achieving a level of efficiency comparable to the SPEA optimized thinned array approach. Moreover, because Taylor weighting does not control the array half-power beamwidths, additional beamforming will need to be performed to achieve the desired radiation pattern.

:

V. CONCLUSION A robust multi-objective global optimization technique has been introduced to design large thinned planar arrays. This methodology is capable of producing antenna arrays that meet desired mainbeam design specifications and allow an engineer to trade the benefits between the half-power beamwidths (maximum and minimum values) and the overall peak side-lobe level. These designs have been shown to be efficient and practical when compared to other beamforming methods and comparable arrays. Finally, this methodology has been discussed in the context of space-borne communications. It has been shown that this approach provides an efficient way to control the shape of a radiation pattern without costly integration and thermal cooling schemes. ACKNOWLEDGMENT The authors would like to thank John Wojtowicz for his helpful insights and advice during the completion of this communication.

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REFERENCES [1] D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Reading, MA: Addison-Wiley, 1989. [2] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms. New York: Wiley, 1998. [3] Electromagnetic Optimization by Genetic Algorithms, Y. RahmatSamii and E. Michielssen, Eds. New York: Wiley, 1999. [4] J. Kennedy and R. C. Eberhart, Swarm Intelligence. San Francisco, CA: Morgan Kaufmann, 2001. [5] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004. [6] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [7] R. L. Haupt, “Interleaved thinned linear arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2858–2864, Sep. 2005. [8] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: Wiley, 2007. [9] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 556–567, Mar. 2007. [10] D. S. Weile and E. Michielssen, “Integer coded Pareto genetic algorithm design of constrained antenna arrays,” Electron. Lett., vol. 32, pp. 1744–1745, Sep. 1996. [11] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [12] J. S. Petko and D. H. Werner, “The evolution of optimal linear polyfractal arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3604–3615, Nov. 2005. [13] J. S. Petko and D. H. Werner, “An autopolyploidy based genetic algorithm for enhanced evolution of linear polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 583–593, Mar. 2007. [14] J. S. Petko and D. H. Werner, “The Pareto optimization of ultra-wideband polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 1, Jan. 2008. [15] J. D. Schaffer, “Multiple objective optimization with vector evaluated genetic algorithms,” Ph.D. dissertation, Vanderbilt Univ., Nashville, TN, 1984. [16] J. Horn, N. Nafpliotis, and D. E. Goldberg, “A niched Pareto genetic algorithm for multiobjective optimization,” in Proc. 1st IEEE Conf. Evolutionary Computation, IEEE World Congress on Computational Intelligence, Piscataway, NJ, Jun. 1994, vol. 1, pp. 82–87. [17] N. Srinivas and K. Deb, “Multiobjective optimization using nondominated sorting in genetic algorithms,” Evol. Comput., vol. 2, no. 3, pp. 221–248, 1994. [18] C. Fonseca and P. Fleming, “An overview of evolutionary algorithms in multiobjective optimization,” Evol. Comput., vol. 3, no. 1, pp. 1–16, 1995. [19] E. Zitzler and L. Thiele, “An Evolutionary Algorithm for Multiobjective Optimization: The Strength Pareto Approach,” Swiss Federal Inst. Technol., Zurich, Switzerland, TIK-Rep., 1998. [20] E. Zitzler and L. Thiele, “Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach,” IEEE Trans. Evol. Comput., vol. 3, no. 4, pp. 257–271, Nov. 1999. [21] J. Knowles and D. Corne, “The Pareto archived evolution strategy: A new baseline algorithm for Pareto multiobjective optimisation,” in Proc. Congress on Evolutionary Computation, Washington, DC, Jul. 1999, pp. 98–105. [22] D. A. Van Veldhuizen and G. B. Lamont, “Multiobjective evolutionary algorithms: Analyzing the state-of-the-art,” Evol. Comput., vol. 8, no. 2, pp. 125–127, 2000. [23] M. Laumanns, G. Rudolph, and H. Schwefel, “A spaitial predatorprey approach to multi-objective optimization: A preliminary study,” in In Parallel Problem Solving From Nature—PPSN V, A. E. Eiben, M. Schoenauer, and H. Schwefel, Eds. Amsterdam, Holland: SpringerVerlag, 1998, pp. 241–249. [24] D. H. Werner and A. J. Ferraro, “Cosine pattern synthesis for single and multiple main beam uniformly spaced linear arrays,” IEEE Trans. Antennas Propag., vol. 37, no. 11, pp. 1480–1484, Nov. 1989. [25] Y. Rahmat-Samii and S.-W. Lee, “Directivity of planar array feeds for satellite reflector applications,” IEEE Trans. Antennas Propag., vol. AP-31, pp. 463–470, May 1983. [26] T. T. Taylor, “Design of line-source antenna for narrow beamwidth and low side-lobes,” IRE Trans. Antennas Propag., vol. AP-3, pp. 16–28, Jan. 1955.

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Low Cross-Polarization Reflectarray Antenna H. Hasani, M. Kamyab, and A. Mirkamali

Abstract—A low cross-polarization center-fed microstrip reflectarray antenna is presented. The cross-polarization elimination is accomplished by a particular arrangement of the elements. Using this technique, a center-fed reflectarray antenna has been designed at center frequency of 11.7 GHz. Measurement results are compared with that of a previously reported reflectarray antenna. Both antennas are exactly the same in every aspect except for the elements arrangement. Measurement results show 1 to 12 dB reduction in cross-polarization level and a notable enhancement in the new antenna’s overall gain. Index Terms—Cross-polarization, microstrip, reflectarray.

I. INTRODUCTION Parabolic reflector antennas have been widely used as high gain antennas in wireless communication systems like radars, satellites and etc. High gain property of a parabolic reflector always demands a large diameter, which makes the reflector bulky and costly, especially when incorporated in satellite missions which require parabolic reflectors with large curvature. A phased-array antenna with its planar surface seems to be a suitable alternative for a parabolic reflector antenna by being much less bulky. But using a phased-array antenna would be costly again due to the incorporation of large amounts of phase-shifters. A novel type of antenna, namely reflectarray antenna, has combined the properties of a parabolic reflector antenna and a phased-array antenna, resulting in a planar high gain and low cost antenna. A microstrip reflectarray antenna [1] is composed of a primary feed and a planar surface consisting of printed elements which by changing their geometry (dimension [2], rotating angle [3], etc.) compensate the phase difference caused by feed to element distance, thus forming a cophased radiated field in the desired direction. Microstrip reflectarray antenna combines the technologies of parabolic reflector and phasedarray antennas, having much less size and cost. These properties have made the reflectarray antenna a suitable choice in radar and wireless communication systems. There are, however, some drawbacks associated with microstrip reflectarray antennas, like narrow bandwidth performance and low efficiency. Narrow band performance of reflectarray antennas are attributed to the narrow band behavior of the elements themselves and also to the differential spatial delays due to the feed-to-element path-length delays which is more prominent in large size reflectarray antennas. In recent years, lot of work has been done to overcome this shortcoming by using a thick substrate, stacked patches [4] or elements with Manuscript received June 28, 2010; revised September 14, 2010; accepted October 26, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported by the Iran Telecommunication Research Center (ITRC), Tehran, Iran. H. Hasani is with the Electrical and Computer Engineering Department, K.N. Toosi University of Technology, Tehran 16315-1355, Iran (e-mail: [email protected]; [email protected]). M. Kamyab is with the Electrical Engineering Department, K.N. Toosi University of Technology, Tehran 16315-1355, Iran (e-mail: [email protected]. ir). A. Mirkamali is with the Electrical and Computer Engineering Department, Zanjan University, Zanjan, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123071

Fig. 1. Element structure.

phase-delay lines [5]. Of all these methods, elements with attached phase-delay lines, being single layer, may provide a low-cost method to enhance the antenna bandwidth. However, using this type of element requires bending in delay-lines to increase element phase range which would cause a high level of cross polarization and inefficiency as reported in [6]. A method was developed for minimizing the cross-polarization level for this kind of element by separating the antenna plane into four quadrants with mirror symmetry arrangement [11], which resulted in a null at cross-polarization pattern at broadside direction and efficiency improvement. This method was also used in design of a reflectarray antenna with phase delay lines in [7]. As reported, although this technique resulted in reduction of antenna cross-polarization level to about 016 dB at center frequency of 11.7 GHz, but, due to the small size of the antenna plate, cross-polarization level remained notable. This communication presents a more effective method for cross-polarization elimination in reflectarray antennas using non-self symmetrical elements which mainly include elements with phase-delay lines. To show the effectiveness of the proposed technique, the reflectarray antenna reported in [7] is re-fabricated but this time with a novel elements arrangement. Measurement results show the reduction of 1 to 12 dB in cross-polarization level and an improvement of 1.3 dB in antenna’s overall gain. II. ELEMENT DESIGN AND UNIT CELL The selected element is shown in Fig. 1. In this element phase shifting is accomplished by varying the stub-length i.e., by varying s . Using this element, a center-fed reflectarray antenna with plate size of 19 2 27 cm2 was designed and fabricated at center frequency of 11.7 GHz. The antenna is composed of 650 elements with element spacing of 0:33 etched on a Rogers RO4003 laminate with "r = 3:38 and thickness of 0.83 mm. Fig. 2 shows the first fabricated reflectarray antenna with its four mirrored quadrants to reduce the cross-polarization level, as proposed in [11]. As shown in Fig. 3, the technique employed in the first antenna has caused cross-polarization reduction by creating a null in cross-polarization pattern at broadside direction. However, due to the small size of the antenna plate, antenna cross-polarization level is still notable at 12.5 GHz. Fig. 3 shows the advantage of the technique proposed by [11] in lowering cross-polarization level at broadside direction as well as its shortcoming in the emergence of high side-lobe levels in cross-polarization pattern which are about 09 dB below the main beam. This disadvantage could be more severe with either feedhorn displacement or transmitter-receiver antenna misalignment. The cross-polarization level in the first antenna could be reduced substantially at any observation point in the both E and H planes in the whole frequency band by using a method which is described in the following section.

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Fig. 5. Unit cell excitation showing current direction in (a) Conventional unitcell. (b) New unitcell.

Fig. 2. First fabricated reflectarray antenna incorporating the method proposed in [11].

Fig. 6. Normalized magnitude of reflected co-polar electric field for each unit cell.

Fig. 3. Measured H-plane farfield pattern of the first antenna at 12.5 GHz.

Fig. 4. Current direction on reflectarray patch elements; (a) Conventional design. (b) New design.

III. CROSS-POLARIZATION REDUCTION Cross-polarization level in the first antenna can be eliminated by a particular element arrangement which is best shown by Fig. 4. As shown, in this new arrangement, the elements are mirrored repeatedly in both x and y directions so that, each element is the immediate mirror

of its neighbor elements in both x and y directions. In other words, the new technique employs the method of Chang et al. in every four adjacent elements shown in Fig. 5(b) As a result, as suggested by Fig. 4, the radiated fields of every two adjacent elements viewed as a pair, would eliminate the farfield cross-polar component since current components responsible for cross-polarization i.e., Jx components, are all directed in opposite directions. It seems however, that the incorporation of new arrangement would disturb the elements phases since the element to element periodicity would no longer hold. To investigate the validity of this phenomenon, the phase and magnitude response of the new unitcell, shown in Fig. 5(b), which represents the new arrangement, is compared with that of shown in Fig. 5(a), representing conventional element arrangement. Simulation results for magnitude and phase responses obtained by HFSS software are shown in Figs. 6–8. As shown in Fig. 6, the co-polar electrical field magnitude for the new unitcell configuration is greater than that of conventional one. Fig. 6 shows that, there is a 1 to 5 dB improvement in co-polar magnitude response in the new unitcell for s values between 50 to 110 . Due to the fact that this range includes about 30% of the antenna elements, enhancement in overall gain and efficiency is expected in the new antenna. Fig. 7 demonstrates the prominent characteristic of the new unitcell, by comparing cross-polar components of reflected electric fields for each unitcell. As shown, cross-polarization reduction reaches to 50 dB in the new unitcell in comparison with the conventional one for different values of s . This phenomenon guarantees a substantial reduction in antenna cross-polarization level. The reason, as described earlier, is attributed to the current directions in the new unitcell. As shown

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Fig. 7. Normalized magnitude of reflected cross-polar electric field for each unit cell.

Fig. 9. Second reflectarray antenna with incorporation of the proposed method.

TABLE I MEASURED MAXIMUM CROSS-POLARIZATION LEVEL OF THE TWO ANTENNAS AT FARFIELD REGION

Fig. 8. Phase responses of the two unitcells with different angles for incident wave.

in Fig. 5(b), Jx components of any two neighbor elements in both x and y directions, are all directed in opposite directions. Although magnitude responses of the two unitcells are of great difference, but as shown in Fig. 8, even with different angles of incident wave, phase responses of both unitcells have remained unchanged, meaning that the new arrangement does not disturb the elements phases. As a result, the phase response obtained in [7] for a single element, may be used to design the antenna with the novel technique. IV. MEASUREMENT RESULTS Based on the new method, the second centre-fed reflectarray antenna having the plate area of 27 2 19 cm2 , as shown in Fig. 9, with a blown up section to demonstrate the details, has been designed at center frequency of 11.7 GHz, and is fabricated. Elements are etched on a Rogers RO4003 laminate with thickness of 0.83 mm and "r = 3:38. Due to the small size of the antenna plate, the reflectarray feedhorn is chosen with small aperture size of 2.38 2 2.4 cm2 so to minimize the feed-blockage effect. The feed horn 03 dB beamwidth is 62.5 for H-plane pattern and 53.4 for E-plane pattern at 11.7 GHz and is placed about 16.5 cm above the antenna plane. The antenna has been tested in a compact anechoic chamber of 7.2 m long 2 3.6 m wide 2 3.6 m high in the frequency range of 30 MHz to 18 GHz. In the second antenna, all the parameters, including the feed horn, are chosen just the same as the first antenna described in [7], except for the arrangement of the elements. Table I shows the superiority of the

new antenna performance over the previous one, by comparing their maximum cross-polarization levels measured at farfield region. The improvements are evident for the second antenna in cross-polarization level. This enhancement varies between amounts of 1 dB at 12.5 GHz to 12 dB at center frequency of 11.7 GHz. Figs. 10–11 show the measured E-plane and H-plane far-field patterns of the second antenna with co-polar and cross-polar components at center frequency of 11.7 GHz. As shown, for the second antenna, the cross-polar component is almost completely eliminated in the H-plane pattern. One may conclude that the existing cross-polarization is attributed to the scattering from antenna plate edges and the feedhorn antenna. In addition to the maximum value, cross-polarization level of the two antennas has also been compared at broad side direction. Figs. 12–13 best illustrate the effect of the proposed method by comparing measured E-plane and H-plane cross-polarization level of the two antennas at broadside direction versus frequency. Efficiency has also been improved in the second antenna. Fig. 14 compares the gain versus frequency for the two reflectarray antennas demonstrating 1.3 dB improvement in overall gain of second antenna while the 1 dB bandwidth has remained unchanged. These improvements in antenna cross-polarization level and efficiency will, of course,

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Fig. 10. Measured E-plane farfield radiation pattern of second reflectarray at 11.7 GHz.

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Fig. 13. Measured H-plane cross-polarization level at broadside direction.

Fig. 14. Measured Gain versus frequency of the two reflectarrays. Fig. 11. Measured H-plane farfeild radiation pattern of second reflectarray at 11.7 GHz.

V. CONCLUSION A novel method is proposed for elimination of cross-polarization in a reflectarray antenna using elements with phase-delay lines. It has been shown that by an arrangement in which each of elements is mirrored image of immediate neighboring patch element, very low cross-polarization level is achievable. Measurement results for two similar reflectarray antennas merely different in element arrangement are compared showing 1 to 12 dB reduction in antenna cross-polarization level in the whole 10.7–12.5 GHz bandwidth. In addition, antenna efficiency has also been improved.

REFERENCES [1]

Fig. 12. Measured E-plane cross-polarization level at broadside direction.

be more pronounced for those reflectarray antennas with larger plate size or in case of off-set design.

Munson , “Microstrip Reflectarray for Satellite Communication and Radar Cross-Section Enhancement or Reduction,” U.S. Patent 4,684,952, Aug. 4, 1987. [2] D. Pozar and T. A. Metzler, “Analysis of a reflectarray antenna using microstrip patches of variable size,” Electron. Lett., vol. 29, no. 8, pp. 657–658, Apr. 1993. [3] J. Huang and R. J. Pogorzelski, “A Ka-band microstrip reflectarray with variable rotation angles,” IEEE. Trans. Antennas Propagl, vol. 46, pp. 650–656, May 1998. [4] J. A. Encinar, “Design of two-layer printed reflectarray using patches of variable size,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1403–1410, Oct. 2001.

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[5] M. E. Bialkowski, H. J. Song, K. M. Luk, and C. H. Chan, “Theory of an active transmit/reflected array of patch antennas operating as a spatial power combiner,” in IEEE Antennas and Propagation Society Int. Symp. Digest, Jul. 2001, vol. 4, pp. 764–767. [6] D. C. Chang and M. C. Huang, “Microstrip reflectarray antenna with offset feed,” IEE Electron. Lett., vol. 28, no. 16, pp. 1489–1491, Jul. 1992. [7] H. Hasani, M. Kamyab, and A. Mirkamali, “Broadband reflectarray antenna incorporating disk elements with attached phase-delay lines,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 156–158, 2010. [8] J. Huang, “Microstrip reflectarray,” in Proc. IEEE AP – S/URSI Symp., London, Canada, Jun. 1991, pp. 612–615. [9] Y. Zhang, K. L. Wu, C. Wu, and J. Litva, “Microstrip reflectarray: Full – wave analysis and design scheme,” in IEEE AP – S/URSI Symp., Ann Arbor, Michigan, Jun. 1993, pp. 1386–1389. [10] R. D. Javor, X. D. Wu, and K. Chang, “Beam steering of a microstrip flat reflectarray antenna,” in Proc. IEEE AP – S/URSI Symp., Seattle, WA, Jun. 1994, pp. 956–959. [11] D. C. Chang and M. C. Huang, “Multiple polarization microstrip reflectarray antenna with high efficiency and low cross – polarization,” IEEE Trans. Antennas Propag., vol. 43, pp. 829–834, Aug. 1995.

Design and Validation of Gathered Elements for Steerable-Beam Reflectarrays Based on Patches Aperture-Coupled to Delay Lines Eduardo Carrasco, Mariano Barba, and José A. Encinar

Abstract—The concept of grouping reflectarray elements made of patches aperture-coupled to delay lines is presented in this communication. The combination of two or four elements turning into a sub-array is presented as an effective way of reducing the number of control devices required to reconfigure a beam in reflectarray antennas, with the consequent reduction in cost and manufacturing complexity. It has been shown that the impact of the element-grouping on the antenna radiation patterns is very small, when compared with a reflectarray made of individual phasing elements. The concept has been validated by comparing the simulation results of phase-shift and losses with measurements in waveguide simulator for sub-arrays with two and four elements. Index Terms—Aperture-coupled patch, reflectarray, reflectarray cell, steerable-beam, sub-array.

I. INTRODUCTION Reflectarrays are a very attractive architecture for implementing reconfigurable- or steerable-beam antennas by using, for example, microelectro-mechanical systems (MEMS) [1]–[3], PIN diodes [4]–[6] or liquid crystal substrates [7], [8]. Reflectarray elements based on a square patch coupled through a rectangular slot to a delay line with variable length [9], [10], like that shown in Fig. 1(a) have demonstrated a significant improvement in

Fig. 1. Reflectarray elements based on patches aperture-coupled to delay lines. (a) Expanded view of an individual element. (b) Expanded view of a two-element sub-array.

the bandwidth of large reflectarrays [11], through the introduction of true-time delay (TTD) compensating the effects of the differential spatial phase delay [12]. These kinds of elements have also been used to produce shaped-beams, like those required by local multipoint distribution system (LMDS) base stations [13]. However, for a large reflectarray with a reconfigurable-beam, the cost and manufacturing complexity increases considerably because hundreds of control elements with their respective biasing lines are required. Combining two or more elements by turning into a sub-array can significantly reduce the number of switches required to achieve the beam configuration [14], [15] simplifying the manufacturing process and allowing the use of modules for the biasing circuits [16]. The geometry of the proposed element allows several radiating elements connected by a common delay line to be grouped, in a similar way to the microstrip beam forming networks used to feed conventional arrays. With this configuration the available room for producing TTD is also increased. Since the grouping reduces the degrees of freedom in the phase distribution and consequently the accuracy in the shaping of the beam, the sub-array implementation makes an accurate synthesis of the required phase-shift in the reflectarray elements necessary to obtain the desired beam with minimum distortion or a reduction in gain. In this communication, the concept of grouping elements in sub-arrays made of two or four elements, like those shown in Fig. 2(b) and (c), is presented as an effective way of reducing the manufacturing cost and complexity in steerable-beam reflectarrays. The impact of the element-grouping on the antenna radiation patterns has also been evaluated for a pencil-beam reflectarray which points to different directions. In order to validate the concept experimentally, both the two- and the four- elements proposed sub-arrays have been manufactured and tested in a waveguide simulator (WGS), and the results of phase-shift and losses are within a good concordance with the simulations. II. DESIGN OF GATHERED ELEMENTS

Manuscript received December 04, 2009; revised September 22, 2010; accepted October 07, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by the Seventh Framework of the European Union under the ARASCOM project: FP7-ICT-2007-3. 6:222620, and in part by the Spanish Ministry of Science and Innovation under the project CICYT TEC 2007-63650. The authors are with the Electromagnetism and Circuit Theory Department, Universidad Politecnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122222

For the analysis of sub-arrays with gathered elements, firstly, the individual cell has been designed by a full-wave approach. The next step has been the design of the gathered elements by two approaches: a full-wave approach and a circuit approach. A. Single Element The individual reflectarray element is shown in Fig. 1(a). The scattering matrix of the reflectarray element has been obtained using the transient solver of the CST Microwave Studio software [17]. The boundary conditions have been defined as electric walls in the Xmin

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TABLE I DIELECTRIC SUBSTRATES

Fig. 3. Reflection coefficient in free space for the individual reflectarray element. (a) Phase compared with the case of an ideal phase-shifter ( 2 L). (b) Amplitude (reflection).

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Fig. 4. Schematic of the full-wave approach for the complete sub-array.

Fig. 2. Reconfigurable-beam reflectarray with sub-array modules by grouping the elements. (a) Expanded view of the reflectarray geometry. (b) Top view of the two-element sub-array. (c) Top view of the four-element sub-array.

Xmx planes, magnetic walls in the Ymin and Ymx planes, Et = 0 in the Zmin plane and open in the Zmax plane. Zmax is called and

Port 1 with the phase referred to the top surface of the dielectric layer d(4), while the end of a segment of the microstrip line is the Port 2. Table I shows the thickness, permittivity value and dissipation factor for the different substrates. After the adjustment of all the dimensions for obtaining a quasi-linear phase response [18], the period of the individual element is defined as 18 2 18 mm2 , the patches as a square of 9.00 mm side, the slot as a rectangular of 8.6 2 1.0 mm2 . The delay line width is 0.7 mm, which means a characteristic impedance of 50 . With this geometrical data, the scattering matrix which characterizes the element was obtained at 10.40 GHz. The length of the microstrip line was varied from port 2 to obtain the curve of Fig. 3(a), which is compared with the case of an ideal phase shifter, with a phase variation equal to 02 L, = 357:97 rad=m, being the propagation constant for the microstrip line, at the corresponding frequency, and L the line length. The losses are in the order of 0.3 dB and can be reduced by using low-loss materials. This element will be used in the two- and four-element proposed sub-arrays. The two- and four-element sub-arrays are obtained by replicating the individual element; the delay lines are combined using a T-junction, similar to the microstrip beam forming networks used in conventional arrays, with the aim of using a common delay line as is shown Fig. 1(b). In this configuration, the control devices are shared by the two or four elements as shown in Fig. 2(b) and (c). For the analysis of the sub-arrays two approaches are considered and verified and are briefly exposed in the following paragraphs.

aforementioned boundary conditions, an infinite array has been analyzed for a normal incidence of the impinging wave with the electric field oriented parallel to the x-axis. In this first approach, the two elements are analyzed together loading the sub-array with two ports. The first port corresponds to the planar wave which impinges on the top of the sub-array (once again the phase reference is on the top surface of the dielectric d(4), defined in Fig. 1). The second port is set at the common microstrip delay line, at 1.63 mm from the T-junction. With this full-wave computation, the S matrix which characterizes the whole two-element sub-array is computed. The reflection response produced by the gathered reflectarray element is then obtained by loading port 1 with the value of the vacuum intrinsic impedance and port 2 with an open-ended microstrip line taking into account losses. The length of this microstrip line is varied to obtain the curve of phase delay. This approach called complete analysis is summarized in Fig. 4 and can be extended to a four- or more element sub-array. C. Analysis of Gathered Elements by a Circuit Approach The second approach to analyze the sub-arrays consists of computing, using the same full-wave tool, the S matrix which characterizes the individual element, as in Section II.A. For the sub-array, both individual elements are equally oriented and have the same dimensions. Therefore, the S matrix can be duplicated and connected through each port number 2, which corresponds to the microstrip line of each single element, by a circuit approach which characterizes the line network (including the T junction, the bends and extra line segments). The reflection coefficient of the reflected wave on the double element is obtained by loading, in parallel, port 1 of each individual S matrix with the intrinsic impedance of vacuum. The schematic of this connection is shown in Fig. 5, where the circuit approach is carried out using ADS software [19] and can be extended to a four-element sub-array.

B. Analysis of Gathered Elements by a Full-Wave Approach

D. Gathered Elements Results

The first approach, for the analysis of the two-element sub-array, consists of a full-wave simulation of the whole sub-array. With the

The phase and loss curves for a two-element sub-array have been computed at 10.40 GHz by varying the length of the microstrip line and

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Fig. 5. Schematic of the combined technique for the two-element sub-array resulting from combining two elements with a circuit approach.

Fig. 7. Reflection coefficient in free space for the four-element sub-array obtained by the circuit approach. (a) Phase compared with the ideal phase-shifter ( 2 L). (b) Amplitude.

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Fig. 6. Reflection coefficient in free space for the two-element sub-array obtained by both techniques: complete and circuital-combined, at 10.40 GHz. (a) Phase compared with the ideal phase-shifter ( 2 L). (b) Amplitude (losses).

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using the two previously described approaches. The phase and amplitude (losses) corresponding to each approach are shown in Fig. 6. The phase curves are practically the same for both methods and agree very well with those obtained for an ideal phase-shifter. The differences in the losses obtained by the two approaches are also practically negligible providing a validation of the technique that combines the two cells using a circuit approach. Note that this assumption is valid because the strong coupling between elements happens in both the patches and the slots which are analyzed by a full-wave technique assuming a periodic environment. In a reconfigurable-beam reflectarray, a phase change can be implemented, for instance, by the introduction of PIN diodes or MEMS devices along the line in parallel or series configuration. The scattering matrix which characterizes the control devices can be easily connected to the delay line of the gathered element simplifying the design of the antenna. Once the effectiveness of using the circuital approach has been proven, a four-element sub-array has been designed. For this case, two T-junctions are necessary to split the line, as shown in Fig. 2(c). The individual element geometry remains unchanged. Fig. 7 shows the reflection of the impinging wave on the four-element sub-array, for both phase and amplitude at 10.40 GHz. The phase is compared with the ideal case. As expected, the losses produced by the gathered element are increased because the signal propagates along a longer path. Also the two T-junctions increase the reflections, giving rise to a deterioration in the phase linearity. The increase in losses added to the phase errors produced by spatial resolution, since the grouped elements produce the same phase-shift, can make this configuration less attractive than the two-element sub-array.

Fig. 8. Required phase distribution on the reflectarray surface to generate a pencil beam pointing to 5 in the switching plane and 18.32 in elevation ( = 19 ,  = 15 ), with 256 phase values.

0

0

III. INFLUENCE OF THE GATHERING IN THE RADIATION PATTERN As expected, the grouping reduces the resolution of the phase required to produce the desired beam with effects on the radiation pattern. For this reason, a trade-off between the advantages and disadvantages of gathering must be made. For example, for a reflectarray with 256 individual elements arranged in a circular grid of 18 2 18 elements (diameter 324 mm), the phase distribution required on each element to generate a pencil-beam in the direction  = 19 ,  = 015 (which means 05 in the steering plane, with a tilt of 18.32 in the elevation plane) is shown in Fig. 8. The angles  and  are defined as in conventional spherical coordinates. The phase center of the feed has been located in the spatial point (0100, 0, 330) in mm, in accordance with the coordinate system shown in Fig. 2(a). When the elements are gathered in the y-direction by pairs, the phase distribution becomes that shown in Fig. 9. Fig. 10 shows a comparison between the radiation patterns when all the phase values are taken into account and when only the half of these values are considered, which is equivalent to grouping the elements by pairs, in the y-direction. The radiation patterns have been obtained by ignoring the losses of the elements, but considering the illumination produced by a feed-horn modeled as a cosq function, with q = 8, which means a border illumination of 011 dB with the aim of showing the effects of grouping, independently of the type of element. As can be seen, for a pencil-beam reflectarray, the main effect of the gathering in a two-element sub-array, in the y-direction, is the generation of a grating lobe in the steering pattern without significant effects

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Fig. 9. Required phase distribution on the reflectarray surface to generate a pencil beam pointing to 5 in the switching plane and 18.32 in elevation ( = 19 ,  = 15 ), with 130 phasing elements, by the gathering of elements by pairs in the y -direction.

Fig. 11. Comparison of the radiation patterns, in the switching plane, with element gathering for different scan angles.

Fig. 10. Comparison of the radiation patterns, in the steering plane, both with and without element gathering.

Fig. 12. Comparison of the radiation patterns, in the elevation plane, with element gathering for different scan angles.

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in the elevation pattern. For a steerable-beam reflectarray, a progressive phase can be also introduced by pairs with the finality of control the phase-shifter with a switch attached to the delay line of the real sub-array element. For the same reflectarray configuration, the phase distribution required to point the beam to 0 and 11 in the switching plane, with a constant beam in elevation, has been obtained. Once again, the phase values have been discretized to 130 values instead of the 256 initial elements, by grouping the elements by pairs in the same y-direction. The switching plane and elevation radiation patterns comparing the 3 beams are shown in Figs. 11 and 12. As expected, the impact of gathering the elements are negligible in the elevation plane. For the switching plane, the grating lobes are more intense as the scan angle is increased. However, for many applications, the 16 dB separation between the main beam and the grating lobe is enough bearing in mind that this undesirable beam is produced far from the pointing direction. The effect of these grating lobes can be reduced by grouping the elements in a non-periodic pattern [20]. A trade-off between the pattern degradation and the savings in control elements, biasing lines and manufacturing complexity must be evaluated carefully. IV. VALIDATION IN WAVEGUIDE SIMULATOR With the aim of validating the use of gathered elements, based on patches aperture-coupled to a delay line, as reflectarray elements, the well known waveguide simulator technique (WGS) [21] was applied,

Fig. 13. Waveguide simulator setup. (a) Two-element sub-array. (b) Four-element sub-array, including the transition from WR75 and a taper transition to the square waveguide.

respectively, to a two-element sub-array and a four-element sub-array. In both cases, the corresponding radiating elements were inserted into a waveguide section, and the reflection was measured in the excitation port for different lengths of the common delay line. The elements were manufactured using the same materials as in Table I. The size of the patches was 8 2 8 mm2 , while the size of the slots was 7.0 2 0.8 mm2 . The line was 0.7 mm wide for the delay line and coupling sections, and 0.39 mm for the =4 matching sections. For the two-element sub-array a WR90 waveguide (22.86 2 11.43 mm2 ) was used. The reference plane for the phase was located 27.49 mm over the d(4) substrate. The manufactured sub-array is shown in Fig. 13(a).

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Fig. 14. Comparison between measurements and simulations for the waveguide simulator with two-element sub-array. (a) Phase. (b) Amplitude.

Fig. 14 shows the comparison between measurements and simulations for the manufactured waveguide simulator at 10.40 GHz. A non linear response is expected because the period of the element 2 ) in this case is imposed by the standard wave(11.83 2 10.16 guide WR90, but the dimensions of the patch, aperture and stub were 2 optimized for a different period (18 2 18 ). The measurements correspond to discrete values of the delay line which were obtained by complex value each cutting the metal-strip manually, saving the time. To simplify the test setup, the ground plane was not included. The simulations were obtained by defining the corresponding geometry in CST Microwave Studio. As can be seen, the measured phase values are practically equal to those obtained by simulation. The amplitude response presents small discrepancies which can be attributed to the real losses of the materials which can differ from the nominal values, and also to the tolerances in the manufacturing process. These losses can be reduced if a ground plane is included below the microstrip line, as in the case of Fig. 6. For the four-element sub-array a square waveguide (22.86 2 22.86 2 ) was used to carry out the measurements. Fig. 13(b) shows the manufactured sub-array, including a taper transition between the square 2 ). The simulation correwaveguide and a WR75 (19.05 2 9.52 sponds to a two-element WGS connected by the circuit approach defined in Section II.C, but with the boundary conditions which correspond to the waveguide cavity. The phase reference is located 7.14 mm substrate corresponding with a waveguide section. The above the phase and amplitude of the reflected wave is shown in Fig. 15 for both: measurements and simulations. Once again, by simplicity, the ground plane was not considered. The phase response is practically equal for the measurements and simulations. The losses curve presents the same shape, but small discrepancies, are around 0.5 dB, which can be attributed to tolerances in the dimensions and in the losses of the materials. The losses can be significantly reduced by eliminating the back radiation that appears at some particular line lengths.

mm

mm S11

mm

mm

d(4)

V. CONCLUSIONS The gathering of reflectarray elements based on patches aperturecoupled to delay lines is proposed as a solution to reduce the number of electronic devices and their biasing lines in large reconfigurable-beam reflectarrays. Two approaches for analyzing the sub-arrays have been presented, one based on the full-wave analysis of the sub-array and the other on a circuit connection of the individual cells. Both approaches

Fig. 15. Comparison between measurements and simulations for the waveguide simulator with four-element sub-array. (a) Phase compared with the ideal phase-shifter. (b) Amplitude.

provide similar results, but obtaining a significant simplification in the analysis, more flexibility and a reduction of CPU time when the circuit approach is used. To evaluate possible drawbacks of the element gathering, the impact of the phase control at sub-array level in the radiation patterns has been studied for a pencil-beam reflectarray. The result is an increase in the side-lobe levels far away from the main beam. The proposed gathered elements concept has been validated by comparing simulations and measurements in WGS within a good concordance.

REFERENCES [1] H. Aubert, N. Raven, E. Perret, and H. Legay, “Multi-scale approach for the electromagnetic modeling of MEMS-controlled reflectarrays,” presented at the 1st Eur. Conf. on Antennas and Propag., Nice, France, Nov. 2006. [2] J. Perruisseau-Carrier and A. K. Skrivervik, “Monolithic MEMS-based reflectarray cell digitally reconfigurable over a 360 phase range,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 138–141, 2008. [3] R. Sorrentino, R. Vincenti, and L. Marcaccioli, “Recent advances on millimetre wave reconfigurable reflectarrays,” in Proc. 3rd Eur. Conf. on Antennas and Propag., Berlin, Germany, Mar. 2009, pp. 2527–2531. [4] M. Barba, E. Carrasco, J. E. Page, and J. A. Encinar, “Electronic controllable reflectarray elements in X band,” Frequenz, J. RF-Engrg. Telecommun., no. 9–10, pp. 203–306, 2007. [5] H. Kamoda, T. Iwasaki, J. Tsumochi, and T. Kuki, “60-GHz electrically reconfigurable reflectarray using p-i-n diode,” in Proc. Int. Microwave Symp., Boston, MA, Jun. 2009, pp. 1177–1180. [6] A. E. Martynyuk, A. G. Martinez-Lopez, and J. Rodriguez-Cuevas, “Spiraphase-type element with optimal transformation of switch impedance,” Electron. Lett., pp. 673–675, May 2010. [7] H. Wenfei, R. Cahill, J. A. Encinar, R. Dickie, H. Gamble, V. Fusco, and N. Grant, “Design and measurement of reconfigurable millimeter wave reflectarray cells with nematic liquid crystal,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3112–3117, Oct. 2008. [8] A. Moessinger, R. Marin, S. Mueller, J. Freese, and R. Jakoby, “Electronically reconfigurable reflectarrays with nematic liquid crystals,” Electron. Lett., vol. 45, pp. 686–687, Jun. 2009. [9] A. Bhattacharyya, “Slot-Coupled Patch Reflect Array Element for Enhanced Gain-Band With Performance,” U.S. 6,388,620, Jun. 2000. [10] M. E. Bialkowski and H. J. Song, “Dual linearly polarized reflectarray using aperture coupled microstrip patches,” in Proc. IEEE Int. Symp. Antennas Propag., Boston, MA, Jul. 2001, vol. 3, pp. 486–489. [11] E. Carrasco, J. A. Encinar, and M. Barba, “Bandwidth improvement in large reflectarrays by using true-time delay,” IEEE Trans. Antennas Propag., vol. 56, pp. 2496–2503, Aug. 2008. [12] J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray concept,” in Proc. IEEE Int. Symp. Antennas Propag., Newport Beach, CA, Jun. 1995, pp. 582–585.

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[13] E. Carrasco, M. Arrebola, J. A. Encinar, and M. Barba, “Demonstration of a shaped beam reflectarray using aperture-coupled delay lines for LMDS central station antenna,” IEEE Trans. Antennas Propag., vol. 56, pp. 3103–3111, Oct. 2008. [14] H. Legay and B. Salome, “Low-Loss Reconfigurable Reflector Array Antenna,” U.S. Patent 7,142,164 B2, 2006. [15] E. Carrasco, M. Barba, and J. A. Encinar, “Design of gathered elements for reconfigurable-beam reflectarrays based on patches aperture-coupled to delay lines,” in Proc. Progress in Electromagnetics Research Symp., Moscow, Russia, Aug. 2009, pp. 886–889. [16] K. W. Brown, G. Jones, A. K. Brown, and W. E. Dolash, “Reflectarray Antennas Having Monolithic Sub-Arrays With Improved DC Bias Current Paths,” U.S. Patent 7,423,601, 2008. [17] CST Microwave Studio, [Online]. Available: www.cst.com [18] E. Carrasco, M. Barba, and J. A. Encinar, “Reflectarray element based on aperture-coupled patches with slots and lines of variable length,” IEEE Trans. Antennas Propag., vol. 55, pp. 820–825, Mar. 2007. [19] Advanced Design System, [Online]. Available: www.agilent.com [20] E. Carrasco, M. Barba, and J. A. Encinar, “Switchable-beam reflectarray with aperiodic-gathered elements based on PIN diodes,” presented at the 32nd ESA Antenna Workshop Noordwijk, The Netherlands, Oct. 2010. [21] P. W. Hannan and M. A. Balfour, “Simulation of a phased-array in waveguide,” IEEE Trans. Antennas Propag., vol. 13, pp. 342–353, May 1965. Fig. 1. Slice of a scatterer with the PML enclosure.

Scattering From Complex Bodies of Revolution Using a High-Order Mixed Finite Element Method and Locally-Conformal Perfectly Matched Layer Yong Bo Zhai, Xue Wei Ping, and Tie Jun Cui

Abstract—An efficient finite-element method (FEM) is presented to analyze the scattering from complex bodies of revolution (BOR) made of perfectly electric conductors and/or dielectrics. In the proposed method, high-order edge-based vector basis functions are used to expand the transverse field components, and high-order node-based scalar basis functions are used to expand the angular component. The FEM mesh is truncated using a locally-conformal perfectly matched layer (PML) by the complex coordinate stretching. Such a kind of PML is very easy to implement in the numerical process and is able to enclose an arbitrarily-shaped convex object in the spatial domain. Numerical examples are presented to demonstrate the accuracy and efficiency of the presented method. Index Terms—Body of revolution (BOR), electromagnetic scattering, finite-element method (FEM), perfectly matched layer (PML).

I. INTRODUCTION Electromagnetic scattering from bodies of revolution (BOR) by the finite-element method (FEM) has been studied extensively for its capability of modeling complex geometries with arbitrary materials. The rotational symmetry of the scatterer allows the problem to be solved efficiently using a two-dimensional (2D) computational technique. When Manuscript received December 17, 2009; revised August 20, 2010; accepted October 16, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The authors are with the Institute of Target Characteristics and Identification and the State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]. cn). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122224

applied to the open-region problems, the FEM mesh must be truncated by an artificial boundary to obtain a bounded computational domain, and to render the problem manageable. Greenwood and Jin [1], [2] have implemented the cylindrical perfectly matched layers (PML), which is more efficient for the elongated BOR than a spherical PML. Recently, the conformal truncation boundary has been widely used for mesh truncation to minimize the computation domain as much as possible [3]. Teixeira and Chew derived an exact conformal PML through a complex coordinate stretching approach [4]. Another version of the conformal PML was obtained in [5], which is a very good approximation of exact conformal PML [4]. The above PMLs are anisotropic material-based using a local coordinate system. The implementation of theses anisotropic PMLs may be easy for some smooth geometries (such as a spherical or cylindrical shell). However, it may be very difficult to design an anisotropic PML for geometries with abrupt changes in curvature. Ozgun and Kuzuoglu introduced a locally-conformal PML for three-dimensional (3D) electromagnetic scattering without using artificial anisotropic materials [6], [7], which was based on the analytic continuation of the frequency-domain Maxwell’s equations in the complex space. Such a conformal PML was easy to design to enclose an arbitrarily-shaped convex spatial domain, possibly with abrupt changes and even discontinuities in the curvature of the free space-PML interface. In this communication, we present an efficient FEM algorithm using high-order mixed vector and nodal basis functions and the locally-conformal PML based on the complex coordinate stretching. We first give the complex coordinate transformation for the locally-conformal PML. Then, we derive the FEM formulations in the complex PML space using triangular elements to analyze the scattering from BOR. Finally, numerical examples are presented to show the validity and efficiency of the proposed method, followed by the conclusion. II. THE LOCALLY-CONFORMAL PML The conformal PML domain is a shell which encloses a convex volume containing the BOR scatterer at a small distance, as shown in Fig. 1. It is assumed here that the axis of revolution of the scatterer

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and PML shell is the z -axis. Let L1 and L2 be the generatrix of PML’s inner and outer surfaces, respectively. In the  = 0 plane, each P in the PML region is mapped to P~ by the complex coordinate transformation T , which is similar to the transformation in [6]: k~ r0~ r0 kn01 ~ r 0~ r0 jk0 nk~r1 0 ~ r0 kn01

~ = T (~r) = ~r + 1

~ r

(

)

(1)

where ~r, ~r0 and ~r1 are position vectors of points P , P0 and P1 , respectively;  is a positive parameter (practically, 5k0 <  < 15k0 for a PML thickness between =4 and =2 where  is the wavelength), and n is a positive integer (typically 2 or 3) related to the decay rate of the field strength inside PML. The point P0 is located on L1 , and ~r0 2 L1 is the solution of

min

~r 2L

k~r 0 ~r0 k:

(2)

Such a ~r0 exists and is uniquely defined because L1 is the boundary of the convex set, which can be simply performed by some searching techniques in the mesh coordinate of the FEM program. P1 is the intersection of the line passing through P0 and P , and L2 is shown in Fig. 1. The transformation in (1) can be implemented in a FEM program by replacing the real-valued node coordinate inside PML with their complex-valued counterparts calculated by (1). Because of the rotational symmetry of PML, the transformation (1) is also applicable for other  = constant planes, which is only the function of  and z . As derived in [6], the fields inside the PML region can be easily obtained by noting that the already existing outgoing field in the real space is directly mapped to the complex space through the above analytical continuation. The transformation (1) yields the analytical continuity even in the case of curvature discontinuities on L1 , and ensures that the PML-air interface is reflectionless and waves which propagate through PML is attenuated.

Fig. 2. Triangular element in complex space.

normal vector on Ssc , pointing from the free space region into the scatterer region. To take advantage of the rotational symmetry of the problem, the fields are expanded in the Fourier modes as [1], [2]:

Es =

F

(Es ) = 21 +

V

V

0 S

(r~ 2 Es ) 1 0r 1 1 (r~ 2 Es )

s E; 0 for m

r =

0

0

z

(r~ 2 Es ) 1 0r 1 1 (r~ 2 Ei ) 0k02 Es 1 r 1 Ei dV

0

zz

;

 =

r

0

z

(5)

i=1

=0 s E; 61

=

ee;i Nie ;

6

i=1

Et;s 0 =

i=1

e et;i

Nie

(6)

e N e; e;i i

Et;s 61 = 7 j ~^E;s 61 + for m

8

8

i=1

~N

e  e et;i i

(7)

= 61 s E;m

=

6

i=1

e N e; e;i i

s = Et;m

8

i=1

~N

e  e et;i i

(8)

for jmj > 1, in which

(9)

(10)

(3)

0

z

0 : (4) 0 zz r = I and r = I, where I denotes the unit For the air and PML,  ^ in the dyad. Ssc is the surface of the penetrable scatterer Vsc , and n z



0



6

=

= ; N2e = ; N3e = N4e = 4; N5e = 4 ; N6e = 4  N1e = r~ t  0 r~ t ; N2e = r~ t 0 r~ t N3e = r~ t  0 r~ t ; N4e = ( 0 )N1e N5e = ( 0 )N2e ; N6e = ( 0 )N3e Ne7 = Ne1 ; N8e = N2e :

p r and r are in which k0 = ! 0 "0 is the free space wavenumber,  the relative permittivity and permeability of the medium, which have the following symmetry form: 

m=01

ejm :

N1e

0k02 Es 1 r 1 Es dV

Es 1 (^n 2 r~ 2 Ei )dS

s (~ s (~ Et;m ; z~) + ^E;m ; z~)

The unknown field Es is expanded as:

III. FEM FORMULATIONS Because the real space including the scatterer and intermediate air region are subset of the complex space, the FEM formulations will be derived in the complex space. According to the generalized variational principle [8], the functional for scattering problem is given by the following equation:

+1



Equations (9) and (10) are the second-order hierarchical scalar shape functions [9] and vector shape functions [10], respectively.  ,  and are called the area coordinates [8], and  +  + = 1. In each element shown in Fig. 2, ~ and z~ are approximated with the area coordinates as

~=



3

j =1

~

Nje j ;

~=

z

3

j =1

~

Nje zj ;

(11)

where ~j and z~j are the cylindrical coordinates at the nodal point j (j = 1, 2, 3) within each element. Substituting the expansions in (6), (7) and (8) into (3), the functional is differentiated with respect to the unknown coefficients and then the

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Fig. 3. Slice of a metallic conesphere with a gap and the enclosed PML.

result is set to zero. This process yields a sparse, symmetric matrix equation

Amtt Amt Amt Am

etm = Btm ; m = 0; 61; 62; 1 1 1 : em Bm

(12)

In the process of forming above FEM matrix for a given mode number

m, each element of the matrix needs to be calculated and the following relations are used

ds = d~dz~ = det(Je )dd @ = @ z~ = det(Je ); @ = 0 @ ~ = det(Je ) @ ~ @ @ z~ @ @ = 0 @ z~ = det(J ); @ = @ ~ = det(J ) e @ z~ @ e @ ~ @

Fig. 4. RCS of the metallic conesphere with a gap at 9 GHz. (a)  -polarization. (b) -polarization.

(13)

(14)

TABLE I COMPUTATIONAL STATISTICS TO COMPUTE MONOSTATIC SCATTERING FOR TWO POLARIZATION AND 181 INCIDENCE ANGLES

where

@~ @z~ ~3 z~1 0 z~3 : Je = @@~ @@z~ = ~~1 0 0 ~3 z~2 0 z~3 2 @ @

(15)

Consequently, the FEM formulations can be easily implemented in the complex space by using the complex-valued node coordinates obtained via the complex coordinate transformation. The matrix equation can be solved according to the techniques described in [11]. The above procedure should in principle be carried out for each of the Fourier modes, m = 0; 61; 62; 1 1 1. In practice, however, a rule of truncating the infinite Fourier modes is Mmax = k0 max sin  + 6 [1], where max is the largest cylindrical radius of the scatterer. Such a rule is valid for k0 max sin  > 3. Furthermore, as shown in [1], the solution for each negative modes (m < 0) is simply related to that of the corresponding positive modes (m > 0). Hence the solutions need to be computed for the nonnegative modes only (m = 0; 1; 2; 1 1 1). IV. NUMERICAL EXAMPLES In this section, numerical results are presented to show the validity and accuracy of the proposed FEM technique. In all examples, the mesh

length is chosen as =7, n = 3, and the distance between PML and the scatterer is approximately =4. First, a metallic cone-sphere with a gap shown in Fig. 3 is considered for computation. The sphere has a radius of 2.947 inches, the cone tip has a half-angle of 7 degrees, the cone is 23.821 inches tall, and the cone is tangent to the sphere at the junction. A gap with 0.25 inches wide and 0.25 inches deep located at the junction between the cone and sphere. The monostatic RCS of the metallic cone-sphere at 9 GHz is computed and a comparison to the measured value [12] are illustrated in Fig. 4. In this example, the PML shell is about =2 thick and  = 10k0 . A good agreement is observed in Fig. 4 between the computed and measured results. Table I also shows the computational requirements to compute monostatic scattering results of metallic cone-sphere with a gap for two polarization and 181 incidence angles. To demonstrate the capability of the method to deal with complex dielectric BORs, we consider an anisotropic sphere [13] in Fig. 5, characterized in spherical coordinates by a relative permittivity of (16) which is transformed into cylindrical coordinates shown in (17). The sphere

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locally-conformal PML based on the complex coordinate stretching is extended to truncate the FEM mesh for the scattering from complex BOR geometries. This kind of PML is very easy to implement by replacing the real-valued node coordinates inside the PML region with the complex-valued node coordinates obtained by the complex coordinate transformation. Hence it can be used to enclose an arbitrarily-shaped convex spatial domain. The numerical results from the FEM algorithm using high-order mixed finite elements and conformal PML show good agreements to the measurement values.

REFERENCES

Fig. 5. Slice of an anisotropic sphere with the PML enclosure. The sphere has : , and is characterized by  : j : and  a radius of a : j : .

300 60

= 03m

= 250 50

=

Fig. 6. RCS of an anisotropic sphere at 3 GHz for  -polarization and -polarization.

has a radius of a = 0:3 m, and is characterized by R = 2:50 j 5:0 and  = 3:00 j 6:0. The incident elevation angle is 180 . Fig. 6 shows the bistatic RCS at 3 GHz using the Locally-Conformal PML and the cylindrical PML [1], respectively. As is shown in Fig. 6, very good agreement is obtained between the results using Locally-Conformal PML and the cylindrical PML

r = R r^r^ +  ^^ +  ^^  = R sin2  +  cos2  z = (R 0  ) sin  cos  zz = R cos2  +  sin2 

(16)

(17)

V. CONCLUSIONS In this communication, we have presented the FEM algorithm for BOR scatterers using high-order mixed finite elements and locally-conformal PML. The high-order FEM achieves the same accuracy as its first-order counterpart with fewer degrees of freedom which results in the saving of memory requirement and computational time. The

[1] A. D. Greenwood and J. M. Jin, “A novel efficient algorithm for scattering from a complex BOR using mixed finite elements and cylindrical PML,” IEEE Trans. Antennas Propag., vol. 47, pp. 620–629, Apr. 1999. [2] A. D. Greenwood and J. M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag., vol. 47, pp. 1260–1266, Aug. 1999. [3] P. Liu, J. Xu, and W. Wan, “A finite-element realization of a 3-D conformal PML,” Microwave Opt Technol Lett., vol. 30, no. 3, pp. 171–173, 2001. [4] F. L. Teixeira and W. C. Chew, “Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves,” Microwave Opt Technol Lett., vol. 17, pp. 231–236, Mar. 1998. [5] M. Kuzuoglu and R. Mittra, “Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation,” IEEE Trans. Antennas Propag., vol. 45, pp. 474–486, Mar. 1997. [6] O. Ozgun and M. Kuzuoglu, “Non-Maxwellian locally-conformal PML absorbers for finite element mesh truncation,” IEEE Trans. Antennas Propag., vol. 55, pp. 931–937, Mar. 2007. [7] O. Ozgun and M. Kuzuoglu, “Recent advances in perfectly matched layers in finite element applications,” Turk. J. Elect. Engrg., vol. 16, pp. 57–66, 2008. [8] J. M. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993. [9] J. P. Webb and S. McFee, “The use of hierarchical triangles in finite-element analysis of microwave and optical devices,” IEEE Trans. Magn., vol. 27, pp. 4040–4043, Sep. 1991. [10] L. S. Andersen and J. L. Volakis, “Development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics,” IEEE Trans. Antennas Propag., vol. 47, pp. 112–120, Jan. 1999. [11] R. S. Chen, D. X. Wang, E. N. Yung, and J. M. Jin, “Application of the multifrontal method to the vector FEM for analysis of microwave filters,” Microw Opt Tech Lett., vol. 31, pp. 465–470, Jun. 2001. [12] A. C. Woo, H. T. G. Wang, M. J. Schuh, and M. L. Sanders, “Benchmark radar targets for the validation of computational electromagnetics programs,” IEEE Trans. Antennas Propag., vol. 35, pp. 84–89, Feb. 1993. [13] K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” Proc. Inst. Elect. Eng., Pt. H, vol. 139, pp. 314–318, Aug. 1992.

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Open-Ended Coaxial Line Probes With Negative Permittivity Materials Muhammed S. Boybay and Omar M. Ramahi

Abstract—This work presents analysis of the behavior of open-ended coaxial near-field probes in the presence of -negative material. The effect of using negative material layers on the sensitivity is studied. It is shown that optimal conditions related to the thickness and constitutive parameters of the negative medium give rise to significant enhancement in probe sensitivity which allows the probe to have higher material and detection resolutions. The analytical formulation is validated using three-dimensional full-wave simulations of the coaxial probe in the presence of the negative medium.

Fig. 1. An open-ended coaxial line terminated by a two layered medium. The structure is backed by perfect electric conductor.

Index Terms—Coaxial line sensors, evanescent field amplification, near field detection, subwavelength resolution.

I. INTRODUCTION Microwave near-field probes have a wide range of applications ranging from material characterization to buried target detection [1]–[5]. The technique is based on the interaction between evanescent fields generated by an electrically small probe and the material in the close proximity of the probe. Alterations to the material composition or to the shape of the object near the probe change the near-field distribution. Consequently the reflection coefficient from the probe changes. Scanning the probe over an area yields an image corresponding to the material composition around the probe with a sub-wavelength resolution. Recently, developments in the area of double and single negative materials revealed that properties of near-field probes can be improved considerably. Materials with negative permittivity and/or negative permeability are referred to as negative materials. These materials and their unusual properties have been investigated extensively in the last ten years (see for example [6]–[11] and references therein). Among the extraordinary properties of negative materials, evanescent field amplification has the ability of perturbing the field distribution of near-field probes. It was shown that the sensitivity of near-field probes can be improved using negative materials in [12]. In [13], properties of openended waveguides were studied, and both the sensitivity and the image quality were improved using negative materials. In addition, it was shown by numerical analysis that the sensitivity of open-ended coaxial lines can be improved [14]. Finally in [15], sensitivity improvement using negative materials was demonstrated experimentally. Due to the importance of open-ended coaxial line probes in dielectric spectroscopy, several methods were proposed to model open-ended coaxial lines. Aperture admittance models based on lumped elements and quasi static models were reported [16]–[20]. Full wave solution for an open-ended coaxial line terminated by a two-layered dielectric Manuscript received February 14, 2010; revised July 08, 2010; accepted November 02, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported in part by Research in Motion (RIM) Inc., and in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under the NSERC/RIM Industrial Research Associate Chair and Discovery Programs. The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123056

medium was reported in [21]. Although models based on lumped elements provide simple expressions, full-wave solutions are more accurate and can be used for structures that involve negative material layers. In [22], a full-wave solution for open-ended coaxial line probes terminated by stratified media with arbitrary number of layers was presented. In this work, effects of using an -negative material on the sensitivity are investigated. For the first time, the sensitivity of coaxial line probes is analyzed analytically in the presence of -negative layers by using the equations presented in [22]. The preliminary findings presented in [14] are substantiated by theoretical analysis. The analytical formulas [22] enable effective design of high-sensitivity coaxial line probes enhanced by multilayered negative and positive material layers, without the need for expensive and time-consuming full-wave simulations. The dependence of optimum -negative layer thickness and optimum standoff distance on the material properties are studied. Analytical results are verified using full wave simulations. II. SENSITIVITY ANALYSIS OF COAXIAL PROBES Our objective is to gauge the sensitivity of the probe when a target is placed in its close proximity. In a manner analogous to the analysis in [12] where the target was modeled as an infinite width dielectric layer, here, for convenience and without loss of generality, we consider the target to be a PEC surface. We insert an -negative material in the immediate proximity of the probe as shown in Fig. 1. The -negative layer can be considered as an integral part of the probe. For different negative layer thicknesses, the reflection coefficient is analyzed as a function of the standoff distance. The analytical solution derived in [22] is used to calculate the reflection coefficient. The formulation presented in [22] can be used for any type of coaxial lines as long as the only propagating mode within the coaxial line is the dominant TEM mode. The dielectric layer and -negative layer are assumed to be infinite in x-y plane and the permeability of all media is assumed to be equal to the free-space permeability, 0 . For all test cases, the coaxial line parameters are selected as rin = 1:18 mm, rout = 3:62 mm and c = 2:07. The operation frequency is 5 GHz. These values were selected since they were used in [22] and can be used for comparison purposes. As the first test case, the effect of a conductive plate on the reflection coefficient is analyzed when 2 = 0 . Therefore the target is assumed to be in vacuum. The relative permittivity of the -negative material is chosen to be 1 = 01 0 j 0:1. Since negative materials are realized by metamaterials which are lossy structures, the negative material is assumed to have a loss typical of metamaterials, albeit the choice is on the conservative side.

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Fig. 2. Magnitude of the reflection coefficient as a function of target distance for different -negative layer thicknesses.  is assumed to be 1 j 0:1 and the second layer is assumed to be vacuum. The case without -negative layer has a reflection magnitude above 1 dB. Therefore excluded from the plot.

0 0

Fig. 4. Optimum -negative layer thickness and optimum standoff distance as a function of  .

0

Fig. 3. Phase of the reflection coefficient as a function of target distance for different -negative layer thicknesses.  is assumed to be 1 j 0:1 and the second layer is assumed to be vacuum.

0 0

Figs. 2 and 3 show the reflection magnitude and phase as a function of target distance d for different negative media thickness. Fig. 2 shows that at each -negative layer thickness, a minimum reflection magnitude is observed. In addition, the phase of the reflection coefficient has the highest slope at the target distance at which the minimum reflection is observed. At these target distance and -negative layer thickness combinations, the reflection coefficient is sensitive to the target location such that a small change in the target location causes the maximum change in the phase of the reflection coefficient. As a result the minimum reflection magnitude or the highest phase slope conditions correspond to the most sensitive probe configurations. For thicker -negative layers, the standoff distance at which the maximum sensitivity is observed is larger. In addition, there is an optimum -negative layer thickness and standoff distance combination where the sensitivity is maximized. This condition is defined as the optimum condition. The optimum condition for cases presented in Figs. 2 and 3 is t = 0:8 mm and d = 0:87 mm. In the case of a conventional coaxial line probe where no -negative layer is employed, the optimum standoff distance is equal to zero. Therefore the closer the target to the probe, the more sensitive the probe becomes. On the other hand, when an -negative layer is employed, it is possible to achieve increase in the sensitivity while increasing the standoff distance. These findings are consistent with the predictions reported in [12], where plane wave analysis was used to investigate the effects of single and double negative materials on the sensitivity of a generic near field sensors.

Next, the relationship between the behavior of the probe and the properties of the -negative material is studied. For this purpose, the optimum standoff distance and the optimum negative layer thickness are plotted as a function of the real part of the permittivity of the -negative layer. In order to have same effect of electromagnetic loss in all cases, the loss tangent of the negative layer is assumed to be tan  = 00 =0 = 00:1 for all permittivity values. Fig. 4 shows that as the permittivity decreases, the optimum standoff distance decreases. By using a negative layer with a permittivity close to zero, the optimum standoff distance can be increased. Increasing the standoff distance has a practical importance since the standoff distance corresponds to the range of the probe. By increasing the optimum standoff distance, targets that are further away from the probe can be detected. The behavior of optimum -negative layer thickness is also important since, from a practical point of view, obtaining a specific negative layer thickness may not be possible due to fabrication aspects of metamaterials in general [23], [24]. Fig. 4 shows that the optimum -negative layer thickness is a function of the permittivity of the negative material. Therefore, although the thickness of the negative layer in practical applications is limited by the unit cell size of the inclusions that are used in the metamaterial design, the optimum layer thickness can be changed by using a different permittivity values. Next we consider the sensitivity of the probe when the target is covered with or buried within a dielectric medium. Figs. 5 and 6 show the magnitude and phase of the reflection coefficient when 2 = 20 . Compared to the case when the target was in vacuum, for the same -negative layer thickness, the standoff distance at which the sensitivity is maximized is larger. This means that when the target is buried in a dielectric medium, the range of the probe is increased. For this configuration, optimum distances are t = 0:7 mm and d = 2:18 mm. Note that compared to the case of target in vacuum, the maximum slope achieved in the phase of the reflection plots is slightly smaller in the case of a buried target. Therefore the sensitivity of the probe for a buried target is expected to be smaller compared to the target in vacuum case. The effect of 2 is analyzed further by calculating the optimum -negative layer thickness and target distance for different 2 values. Fig. 7 shows that increasing the permittivity of the medium in which the target is buried, increases the optimum target distance and reduces the optimum -negative layer thickness. Therefore employing an -negative layer in coaxial line probes can increase the sensitivity of the probe in the applications of detecting delamination in IC packages [25] and detecting precursor pitting on aluminum surfaces covered by paint [14]. Note that negative materials are intrinsically highly dispersive. Within a small frequency band, negative materials with different

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Fig. 5. Magnitude of the reflection coefficient as a function of target distance for different -negative layer thicknesses.  is assumed to be 1 j 0:1 and  is assumed to be 2.

Fig. 8. Magnitude of the reflection coefficient as a function of target distance for different -negative layer thicknesses obtained by HFSS.

Fig. 6. Phase of the reflection coefficient as a function of target distance for different -negative layer thicknesses.  is assumed to be 1 j 0:1 and  is assumed to be 2.

Fig. 9. Phase of the reflection coefficient as a function of target distance for different -negative layer thicknesses obtained by HFSS.

0 0

0 0

Fig. 7. Optimum -negative layer thickness and optimum standoff distance as a function of  .

permittivity values can be obtained without changing the physical structure. Therefore achieving the optimum thickness combinations which have to be precise according to the Figs. 2, 3, 5 and 6 is not impractical. By changing the operation frequency slightly, a different permittivity value for the negative material can be obtained which changes the optimum parameters as shown in Fig. 4. III. NUMERICAL VERIFICATION Numerical simulations are conducted to support theoretical findings by using Ansoft HFSS, a finite element full-wave simulation tool. The

same structure described in Section II is constructed in the numerical study. The -negative layer has a relative permittivity of 01 0 j 0:1 and the space between the negative layer and the conductive plate is vacuum. Teflon is used as the insulator material for the coaxial line and the structure is excited using a waveport. The waveport excitation assumes that the coaxial line is matched in the 0z direction. The only difference between the theoretical model and the simulated structure is the size of the dielectric layers and the target in the x-y plane (see Fig. 1). Since an infinite layer cannot be modeled for such a configuration in the numerical domain, dielectric layers and the conductive plate is assumed to be circular with a radius of 20 mm. Because of the interaction of the probe field with the material is highly confined to the close proximity of the probe opening, finite models (in the x-y plane) of the probe flanges and media has negligible effects on the results. Figs. 8 and 9 show the reflection magnitude and phase obtained by HFSS simulations. Similar to the theoretical findings, at each -negative layer thickness, a standoff distance at which the sensitivity is maximized is observed. Target distances at which the reflection coefficient is minimized correspond to the target locations at which the reflection phase has the highest slope. Maximum sensitivity conditions are close to the theoretical results. The optimum negative layer thickness was observed to shift slightly from 0.8 mm to 0.6 mm. Fig. 9 shows that when there is no negative layer, the coaxial line probe generates a 70 phase shift for a conductive plate when the standoff distance is 0.1 mm. On the other hand, when a 0.2 mm negative layer is employed, the phase shift due to a conductive plate with a standoff distance of 0.1 mm was 307 . Increasing the negative layer thickness increases the highest sensitive standoff distance, which

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Fig. 10. The dissipated energy in -negative layer calculated by HFSS using the same configuration used for Figs. 8 and 9.

placed immediately at its opening. The reflection coefficient from the probe is calculated as a function of the standoff distance or target distance. It is shown that using a negative layer increases the sensitivity of open-ended coaxial line probes with strong implication for sensor applications in detection of delamination in ICs or detection of cracks on surfaces covered by paint. Unlike previous works where full-wave simulation was used to analyze the effectiveness of negative media on generic probes, the analysis presented here allows for the design of open-ended coaxial probes with high degree of flexibility. The agreement between numerical and analytical results show that the analytical formulation presented in [22] is able to solve open-ended coaxial line structures that involves negative material layers. The analytic formulation allows the designer to avoid the use of full-wave simulation tools which are expensive to develop and use, and require intensive computational resources and a lot of time to provide simulation results with high accuracy and material resolution. In fact, the analytic formulation opens the door to cost-effective optimization of probe designs.

REFERENCES

Fig. 11. Loss density plotted over the cross section of the coaxial line probe with -negative material. The loss is concentrated immediately next to the inner conductor of the probe in the negative layer.

practically means detecting targets buried deeper in the dielectric medium. Note that although the reflection coefficient is very low at some specific probe configurations, e.g., 017 dB for t = 0:6 mm and d = 0:6 mm, the radiation efficiency, nevertheless, is very low for these configurations (the calculation of the radiation from the probe is not shown here for brevity). This implies that most of the energy transmitted by the probe is dissipated in the negative medium. Fig. 10 shows the energy dissipation in the -negative layer when the incident power is 1 W. The dissipated energy is calculated by integrating the volume loss density over the -negative layer volume. The volume loss density is defined as pv

=

1 Re 2

E J 1

0

0 r 2

E H 1

0

(1)

where E is the electric field, J0 and H0 are complex conjugates of the current density and the magnetic field, respectively. Therefore we conclude that, although the loss tangent of the -negative layer is not high, almost all energy is dissipated by the -negative layer; an interesting finding implying that the negative medium, in such configuration, acts as an efficient heat probe. Fig. 11 shows the volume loss density over the cross section of the probe structure. The energy dissipation is observed to be confined within the -negative layer next to the inner conductor of the coaxial line. IV. CONCLUSION In this work, we presented analytical and numerical analysis of the problem of open-ended coaxial line probe with an -negative layer

[1] M. Tabib-Azar, N. S. Shoemaker, and S. Harris, “Non-desctrutive characterization of materials by evanescent microwaves,” Meas. Sci. Technol., vol. 4, pp. 583–590, May 1993. [2] L. Diener, “Microwave near field imaging with open ended waveguide—Comparison with other techniques of nondestructive testing,” Re.s Nondestr. Eval., vol. 7, pp. 137–152, Jun. 1995. [3] T. Wei, X. Xianga, W. G. Wallace-Freedman, and P. Schultz, “Scanning tip microwave near-field microscope,” Appl. Phys. Lett., vol. 68, pp. 3506–3508, Jun. 1996. [4] M. T. Ghasr, S. Kharkovsky, R. Zoughi, and R. Austin, “Comparison of near-field millimeter-wave probes for detecting corrosion precursor pitting under paint,” IEEE Trans. Instrum. Meas., vol. 54, pp. 1497–1504, Aug. 2005. [5] M. Tabib-Azar, P. S. Pathak, G. Ponchak, and S. LeClair, “Nondestructive superresolution imaging of defects and nonuniformities in metals, semiconductors, dielectrics, composites, and plants using evanescent microwaves,” Rev. Sci. Instrum., vol. 70, pp. 2783–2792, Jun. 1999. [6] J. Pendry, “Negative refraction makes perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 2000. [7] D. R. Smith, W. J. Padilla, D. C. Vire, S. C. Nemat-Nasser, and S. Schults, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [8] N. Fang, Z. Liu, T. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express, vol. 11, pp. 682–687, Apr. 2003. [9] S. A. Ramakrishna, “Physics of negative index materials,” Rep. Prog. Phys., vol. 68, pp. 449–521, Jan. 2005. [10] J. D. Baena, L. Jelinek, R. Marques, and F. Medina, “Near-perfect tunneling and amplification of evanescent electromagnetic waves in a waveguide filled by a metamaterial: Theory and experiments,” Phys. Rev. B, vol. 72, pp. 075116–075116, Aug. 2005. [11] M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett., vol. 87, pp. 034102–034102, Jul. 2005. [12] M. S. Boybay and O. M. Ramahi, “Near-field probes using double and single negative media,” Phys. Rev. E, vol. 79, pp. 016602–016602, Jan. 2009. [13] M. S. Boybay and O. M. Ramahi, “Waveguide probes using single negative media,” IEEE Microwave Wireless Compon. Lett., vol. 19, no. 10, pp. 641–643, 2009. [14] M. S. Boybay and O. M. Ramahi, “Improved sensitivity in coaxial line probes using materials with negative permittivity,” in Proc. IEEE AP-S Int. Symp., Charleston, SC, 2009, pp. 1–4. [15] M. S. Boybay and O. M. Ramahi, “Experimental and numerical study of sensitivity improvement in near-field probes using single-negative media,” IEEE Trans. Microwave Theory Tech., to be published. [16] M. Stuchly, T. Athey, G. Samaras, and G. Taylor, “Measurement of radio frequency permittivity of biological tissues with an open-ended coaxial line: Part II—Experimental results,” IEEE Trans. Microwave Theory Tech., vol. 30, no. 1, pp. 87–92, Jan. 1982. [17] T. Athey, M. Stuchly, and S. Stuchly, “Measurement of radio frequency permittivity of biological tissues with an open-ended coaxial line: Part I,” IEEE Trans. Microwave Theory Tech., vol. 30, no. 1, pp. 82–86, Jan. 1982.

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[18] D. Misra, “A quasi-static analysis of open-ended coaxial lines (short paper),” IEEE Trans. Microwave Theory Tech., vol. 35, no. 10, pp. 925–928, Oct. 1987. [19] C. Sibbaldand and S. Stuchly, “A new aperture admittance model for open-ended waveguides,” in IEEE MTT-S Int. Microwave Symp. Digest, Jun. 1992, vol. 3, pp. 1549–1552. [20] C. L. Pournaropoulos and D. K. Misra, “The co-axial aperture electromagnetic sensor and its application in material characterization,” Meas. Sci. Technol., vol. 8, no. 11, pp. 1191–1202, 1997. [21] J. Baker-Jarvis, M. D. Janezic, P. D. Domich, and R. G. Geyer, “Analysis of an open-ended coaxial probe with lift-off for nondestructive testing,” IEEE Trans. Intrum. Meas., vol. 43, no. 5, pp. 711–718, Oct. 1992. [22] S. Bakhtiari, S. I. Ganchev I., and R. Zoughi, “Analysis of radiation from an open-ended coaxial line into stratified dielectrics,” IEEE Trans. Microwave Theory Tech., vol. 42, no. 7, pp. 1261–1267, Jul. 1994. [23] J. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, pp. 4773–4776, Jun. 1996. [24] J. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [25] Y. Ju, M. Saka, and H. Abe, “Ndi of delamination in IC packages using millimeter-waves,” IEEE Trans. Intrum. Meas., vol. 50, no. 4, pp. 1019–1023, Aug. 2001.

Investigation of Adaptive Matching Methods for Near-Field Wireless Power Transfer Jongmin Park, Youndo Tak, Yoongoo Kim, Youngwook Kim, and Sangwook Nam

Abstract—Adaptive matching methods for a wireless power transfer system in the near-field region are investigated. The impedance and resonant frequency characteristic of a near-field power transfer system are analyzed according to coupling distance. In the near-field region, adaptive matching is necessary to achieve an effective power transfer. We compare the power transfer efficiencies of several schemes including simultaneous conjugate matching and frequency tracking. It is found that effective adaptive matching can be easily achieved by tracking the split resonant frequency. In addition, a modified frequency tracking method is proposed to extend the range over which the power is transmitted with high efficiency. The experimental results are in agreement with the theoretical results. Index Terms—Adaptive matching, frequency tracking, near-field coupling, wireless power transfer.

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reported [1]–[10]. The authors of [1] used the coupled mode theory to analyze the characteristics of near-field wireless power transfer. They demonstrated efficient power transfer between two coupled antennas at the resonant frequency. The measured efficiency was about 40% at a distance of 2 m. They also demonstrated that the efficiency could be improved by putting a mediating resonant antenna between the transmitting and receiving antennas [5]. The frequency characteristic of nearfield coupled antennas that are used for wireless power transfer has been studied using the coupled mode theory [6]. The authors of the paper explain the resonant frequency splitting phenomenon in strongly coupled regions and study the characteristics of input impedance and power transfer efficiency at split modal resonant frequencies. An alternative analysis method was presented using spherical modes and an addition theorem [7]. This method showed that maximum power transfer efficiency and optimum load impedance can be obtained using the characteristics of a single antenna. The frequency characteristic was also derived using mode-based analysis [8]. A wireless power transfer system requires high efficiency power transmissions anywhere in the near-field range. However, there are many unresolved issues that remain in the implementation of highly efficient wireless power transfer systems. First of all, it is known that the optimum source and load impedance vary drastically with the coupling distance and the orientation of the antennas [7]. Hence, it is important to realize an adaptive matching system for efficient power transfer in the near-field region. To improve power transfer efficiency in cases of varying distance between two antennas, some methods were suggested [9], [10]. One method achieved an adaptive matching by tuning the antenna and matching network [9]. But this method is difficult to accomplish practically. The frequency tracking method was suggested in [10] using a model of a resonant coupling circuit. However, in that study, the limitation of the method and the effect of the load impedance were not shown. In this communication, we study the characteristics of several adaptive matching methods, such as conventional simultaneous conjugate matching and frequency tracking. During the investigation, we discovered the limitation of the frequency tracking method and proposed a modified frequency tracking method that makes efficient power transfer possible without simultaneous conjugate matching. The modified frequency tracking method is demonstrated using center-fed spiral antennas with a self-resonant frequency of 10.02 MHz. Potential applications of this idea include wireless power transfer systems, domotics, WBAN (Wireless Body Area Network), OLEV (On-Line Electric Vehicle), and NFC (Near Field Communication). II. IMPEDANCE CHARACTERISTIC OF COUPLED ANTENNAS

I. INTRODUCTION Wireless power transfer has long been a topic of interest. Research on wireless power transfer using near-field coupling has recently been Manuscript received April 01, 2010; revised September 25, 2010; accepted November 02, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the National Research Foundation of Korea (NRF) Grant (No. 2010-0018879) funded by the Korea government (MEST). J. Park, Y. Tak, and Y. Kim are with the School of Electrical Engineering and INMC, Seoul National University, Seoul 151-742, Korea (e-mail: city814@ael. snu.ac.kr). Y. Kim is with the Department of Electrical and Computer Engineering, California State University at Fresno, Fresno, CA 93740-8030 USA (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123061

Power can be transferred efficiently when two antennas are located in close proximity to one another. This near-field coupling phenomenon between antennas can be explained effectively by the coupled mode theory [2], [11]. From [2], the optimum load resistance for maximum power transfer efficiency is represented as

RLopt = RR 1 + k2 4RLTRLR T

R

(1)

where the subscripts T , R, and L mean the transmitter, the receiver, and the load, respectively. RT and RR , which are the respective resistances of the transmitter and the receiver, are composed of the radiation and ohmic resistance of the antenna. LT and LR are the inductance at the transmitter and the receiver, respectively. k is the coupling coefficient between the antennas. The coupling coefficient varies according to the distance between the antennas. Because the optimum

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Fig. 1. Center-fed spiral antenna (outer radius = 30 cm, inner radius = 20 cm, wire thickness = 5 mm, 5.7-turns, self resonant frequency = 10:02 MHz).

0

Fig. 3. The optimum load resistances versus the distance between the antennas.

load impedance versus the distance between the two spiral antennas. When the two antennas get closer, the optimum resistance for the maximum power transfer varies quite drastically. Therefore, it is very difficult to realize an adaptive simultaneous matching circuit on both the transmitting and receiving antennas. Fig. 3 shows the maximum power transfer efficiency obtained using the coupled mode theory, EM simulation, and the power transfer efficiency with a fixed source and a load impedance of 50 ohm. The simulation results agreed with the power transfer efficiencies from the coupled mode theory. The power transfer efficiency with a fixed port impedance of 50 ohm is inefficient almost everywhere, except at the approximate distance of 0.4 m. This is explained by the fact that the optimum load impedance is almost 50 ohms only at a distance close to 0.4 m and that the optimum load impedance sharply changes when the distance recedes from 0.4 m, as shown in Fig. 2. Fig. 2. The optimum load resistances versus the distance between the antennas.

load impedance is a function of the coupling coefficient, it also varies based on the distance between the antennas. Therefore, it is necessary to simultaneously satisfy the matching condition at both the transmitting and receiving ports in order to achieve maximum power transfer efficiency. However, simultaneous matching is difficult to implement. As an example, we consider center-fed 5.7-turn electrically small spiral antennas made of copper wire, as shown in Fig. 1. The outer and inner radii are 30 and 20 cm, respectively. The wire diameter is 5 mm and the self-resonant frequency of the antenna is approximately 10.02 MHz. The receiving and transmitting antennas are identical. In this case, magnetic coupling is dominant, so that the formula for the coupling coefficient is given as k = !M=[2(0T 0R )]1=2 , where M is the mutual inductance between the two antennas. The intrinsic decay rate is 0m = Rm =2Lm , where Rm and Lm are the resistance and the inductance of antenna m, respectively. Then, the optimum load resistance is represented as opt RL

)2 = (Ro + Rr ) 1 + (R(o!M + Rr )2

(2)

where Ro and Rr are the ohmic resistance and the radiation resistance of the antenna, respectively. The mutual inductance was obtained using the electromagnetic theory [2], and the resistances of the single antennas were obtained using an EM simulation. The simulation was performed using FEKO, a commercially available software that is based on the Method of Moments (MOM) technique. Fig. 2 shows the optimum

III. FREQUENCY CHARACTERISTIC OF COUPLED ANTENNAS When two resonant antennas are strongly coupled with each other in a near-field region, the resonant frequency is split [6], [10]. The split resonant frequencies are determined by the amount of coupling between the antennas. Input impedances at the split resonant frequencies for coupled small antennas have recently been investigated [6]. We noticed that the input impedance at the split resonant frequency is almost equal to the load impedance in the strongly coupled region, provided that the load impedance is much greater than the ohmic resistance and the radiation resistance, as in practical systems. Therefore, it is conceived that input matching and efficient power transfer can be achieved with fixed port impedances by adjusting the frequency of the source to a desired split resonant frequency. For the spiral antennas shown in Fig. 1, the split resonant frequencies of the coupled antennas were obtained by a simulation using FEKO, a commercially available software. Fig. 4 shows the simulation results of the power transfer efficiencies for the conventional simultaneous conjugate matching method and the frequency tracking method. The power transfer efficiency is calculated from the S-parameter as jS21 j2 . The simultaneous matching result is obtained on the condition that the frequency is fixed at the self-resonant frequency of the single antenna and that simultaneous conjugate matching is achieved whenever the distance between the antennas is varied. Maximum power transfer efficiency can be achieved using simultaneous matching. On the other hand, the result of the frequency tracking method is based on the assumption that the system can trace the odd mode resonant frequency. The frequency tracking method can achieve almost the same efficiency as the simultaneous matching case in the strongly coupled region. Compared to adaptive

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Fig. 4. Comparison of the power transfer efficiency of two coupled small spiral antennas.

Fig. 5. Inductively coupled fed spiral antenna.

simultaneous matching, it is easy to achieve input matching with the frequency tracking method, because only the source at the transmitting antenna needs to be controlled. However, the shortcoming of the frequency tracking method is the drastic decrease of power transfer efficiency outside of the strongly coupled region. When the port impedance of the antenna is 50 ohms, the frequency tracking method is useful only in the strongly coupled region. IV. THE EFFECT OF PORT IMPEDANCE If it is necessary to stably transmit power outside of a strongly coupled region, then a frequency tracking method with a 50 ohm port impedance is not suitable for the system. Therefore, a modified frequency tracking method that could achieve stable power transfer over an extended range was investigated. For this purpose, the port impedances are not set to 50 ohms, but to the optimum impedance at the target distance. Then adaptive matching is performed using the frequency tracking method. In the demonstration system shown in Fig. 5, inductive coupling is chosen to feed the antenna instead of a direct feed since it is an easy way to realize the matching circuit. The port of 50 ohms is directly connected at the feeding loop. The port impedance at the feeding loop can be converted into the optimum impedance of the spiral antenna by adjusting the inductive coupling between the feeding loop and the spiral antenna. The amount of coupling can be controlled by the size and position of the feeding

Fig. 6. Comparison of the return loss and the power transfer efficiency for two coupled small spiral antennas: (a) power transfer efficiency, (b) reflection coefficient at the input port (case 1: fixed frequency with simultaneous matching condition; case 2: frequency tracking with 50 ohm load impedance; case 3: fixed frequency with fixed load impedance (optimum impedance at 1.5 m); case 4: frequency tracking with fixed load impedance (optimum impedance at 1.5 m)).

loop. The optimum impedance is 2:25 + j1:23 ohms at the target distance of 1.5 m. To optimize the port impedance at the target distance, the loop and the spiral antenna are in the same plane and have the same axis. The radius of the loop is 8 cm. Fig. 6 shows the simulation results of the power transfer efficiency and the return loss of several adaptive matching methods. In case 1, the frequency is fixed with the simultaneous matching condition. In case 2, the frequency tracking method is used with a 50 ohm load impedance. In case 3, the frequency is fixed with the optimum load impedance at 1.5 m. Case 4 uses the modified frequency tracking method with optimum load impedance at 1.5 m. As shown in Fig. 6(a), case 1 performs very well but it is too difficult to realize the simultaneous matching condition for the variation of distance. The system in case 2 is efficient only in the strongly coupled region. The power transfer efficiency outside the strongly coupled region drastically decreased. Case 3 also only performs well close to the 1.5 m target distance. Case 4, the modified frequency tracking method, shows good performance up to the target distance of 1.5 m. Fig. 6(b) shows the reflection coefficient at the input port. The impedance mismatching at the input port is the primary cause for the decrease in power transfer efficiency. Impedance matching can be achieved by the modified frequency tracking method anywhere within the 1.5 m target distance. In the strongly coupled region, the efficiency of case 4 is smaller than that

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Fig. 7. Photograph of the wireless power transfer system using the frequency tracking method: (a) d = 15 cm, (b) d = 150 cm.

Fig. 8. Effect of the plastic structure on the power transfer efficiency for case 4 (dielectric constant: 3.5).

of case 1 and case 2 because of the reactance of the optimum load impedance. It causes a slight impedance mismatch. Even though the power transfer efficiency is not optimal at very close distances, this method is useful to achieve stable high power transfer efficiency in the target region. V. MEASUREMENTS Fig. 7 shows photographs of the experimental setup for the demonstration of the proposed modified frequency tracking method. The structure of the antennas and feeds are the same as in the simulation model. A plastic structure is used to fix the antenna. An LED is used to check the power transmission phenomena. Fig. 7(a) and (b) show that the system successfully works using the modified frequency tracking method at 15 cm and 150 cm, respectively. The LED, which is located at the lower right of the receiving antenna, was activated by the transferred power. In order to fix the antenna, the dielectric loss of the plastic structure should be considered in order to estimate the transfer efficiency correctly. Fig. 8 shows the power transfer efficiency incorporating the dielectric loss. The permittivity of the general plastic is found to vary from 2.5 to 5, and the loss tangent ranges from 0.0001 to 0.035 [12]. In our case, the dielectric constant of the plastic structure is 3.5, and the power transfer efficiency was simulated with various loss tangents. When the loss tangent of the plastic structure increased, the power transfer efficiency decreased. When the loss tangent of the plastic was 0.009, the power transfer efficiency decreased by 30%. To precisely predict the power transfer efficiency, the plastic structure’s dielectric loss should be considered. In addition, a dielectric with a low dielectric

Fig. 9. Comparing the simulation with the measurement: (a) resonant frequency with fixed load impedance (optimum impedance at 1.5 m), (b) power transfer efficiency (loss tan.: 0.009).

loss should be used to fix the antenna. Fig. 9 is a comparison of the measurements and the simulation results. Fig. 9(a) shows the resonant frequency versus the distance. The measured self resonant frequency of the antenna is approximately 9.63 MHz, which is about 4% lower than the simulation result. This seems to be due to the dielectric structure, which causes the resonant frequency to be lower, and to manufacturing error. When the two antennas become closer, the resonant frequency changes into two split resonance frequencies at 1.3 m. The measured results agreed with the simulation results. Fig. 9(b) shows the results of the measured and the simulated power transfer efficiency for case 4. The simulation included a dielectric structure with a dielectric constant and loss tangent of 3.5 and 0.009, respectively. The measured power transfer efficiency agreed with the simulation results. VI. CONCLUSION This communication investigates several adaptive matching methods used to achieve efficient wireless power transfer over a near-field region. Although simultaneous conjugate matching demonstrates the best performance, it seems to be very difficult to implement. The performance of the frequency tracking method with a 50 ohm load is nearly the same as that of the simultaneous conjugate matching in the strongly coupled region. However, its transfer efficiency drops drastically as the distance between the two antennas increases beyond the strongly coupled region. We proposed a modified frequency tracking method with a complex load matched at the target distance to achieve a stable efficiency beyond the strongly coupled region. A wireless power transfer system using the proposed adaptive matching method

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was implemented and tested. The experimental results agreed with the theoretical results. It was found that the effect of the dielectric loss of the plastic structure for fixing the antenna should be considered in the wireless power transfer system.

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Implementation of a Cognitive Radio Front-End Using Rotatable Controlled Reconfigurable Antennas Y. Tawk, J. Costantine, K. Avery, and C. G. Christodoulou

REFERENCES [1] M. Soljacic, “Wireless Energy Transfer Can Potentially Recharge Laptops, Cell Phones Without Cords” 2006, Report in San Francisco Massachusetts Institute of Technology. [2] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Sciencexpress, Jun. 7, 2007. [3] M. Soljacic, E. H. Rafif, and A. Karalis, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev., vol. 75, no. 5, pp. 1–5, 2007. [4] A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Efficient wireless non- radiative mid-range energy transfer,” Ann. Phys., vol. 323, pp. 34–48, Jan. 2008. [5] R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys., vol. 324, pp. 1783–1795, Aug. 2009. [6] Y. Kim and H. Ling, “Investigation of coupled mode behaviour of electrically small meander antennas,” Electron. Lett., vol. 43, no. 23, Nov. 2007. [7] J. Lee and S. Nam, “Fundamental aspects of near-field coupling antennas for wireless power transfer,” IEEE Trans. Antennas Propag., submitted for publication. [8] Y. Tak, J. Park, and S. Nam, “Mode-based analysis of resonant characteristics for near-field coupled small antennas,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1238–1241, Nov. 2009. [9] T. S. Bird et al., “Antenna impedance matching for maximum power transfer in wireless sensor networks,” in Proc. IEEE Sensors Conf., Christchurch, New Zealand, Oct. 2009, pp. 916–919. [10] W. Fu, B. Zhang, and D. Qiu, “Study on frequency-tracking wireless power transfer system by resonant coupling,” in Proc. IEEE 6th Int. Power Electronics and Motion Control Conf., pp. 2658–2663. [11] H. A. Haus, Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984. [12] Omnexus website [Online]. Available: http://www.omnexus.com/

Abstract—This communication presents a new antenna system designed for cognitive radio applications. The antenna structure consists of a UWB antenna and a frequency reconfigurable antenna system. The UWB antenna scans the channel to discover “white space” frequency bands while tuning the reconfigurable section to communicate within these bands. The frequency agility is achieved via a rotational motion of the antenna patch. The rotation is controlled by a stepper motor mounted on the back of the antenna structure. The motor’s rotational motion is controlled by LABVIEW on a computer connected to the motor through its parallel port. The computer’s parallel port is connected to a NPN Darlington array that is used to drive the stepper motor. The antenna has been simulated with the driving motor being taken into consideration. A good agreement is found between the simulated and the measured antenna radiation properties. Index Terms—Cognitive radio, reconfigurable antenna, stepper motor, UWB.

I. INTRODUCTION A cognitive radio system is able to communicate efficiently across a channel by altering its frequency of operation based on the constant monitoring of the channel spectrum. This system is able to continuously monitor gaps (white spaces) in the finite frequency spectrum occupied by other wireless systems, and then dynamically alter its transmit/receive characteristics to operate within these unused frequency bands; thereby minimizing interference with other wireless systems and maximizing throughput [1]. This capability requires a “sensing antenna” that continuously monitors the wireless channel searching for unused carrier frequencies, and a “reconfigurable transmit/receive antenna” to perform the data transfer [2]. The cognitive radio communication schemes have begun to receive a lot of attention with the advent of 3G and 4G mobile communication standards. Various designs and architectures have emerged. In [3], a quad-antenna with a directional radiation pattern is presented. The operating frequency can be adjusted by the use of MEMS switch making it suitable for cognitive radio applications. A reconfigurable slot antenna for cognitive radio applications is presented in [4]. The antenna can be switched between any one of three discrete states. In [5], it is shown that once a cognitive design manages to learn the RF environment, one can use the collected data to train reconfigurable antennas to adapt to any change in the RF environment. The authors in [6] incorporate both the sensing and the reconfigurable antennas into the same substrate. Manuscript received June 05, 2010; revised August 25, 2010; accepted October 12, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. This work was supported by the Air Force Research Lab/RVSE under Contract FA9453-09-C-0309. Y. Tawk and C. G. Christodoulou are with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM USA (e-mail: [email protected]; [email protected]). J. Costantine is with the Electrical Engineering Department, California State University Fullerton, Fullerton, CA, USA (e-mail: joseph.costantine@gmail. com). K. Avery is with the Air Force Research Laboratory, Space Vehicles Directorate, Kirtland AFB, NM, USA. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122239

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Fig. 1. The antenna structure (dimensions in mm).

The reconfigurable antenna is able to tune between 3 GHz–5 GHz and 5 GHz–8 GHz. A coupling of less than 010 dB was achieved between the sensing and the reconfigurable antenna. A reconfigurable C-slot microstrip patch antenna is proposed in [7]. Reconfigurability is achieved by switching on and off two patches using PIN diodes. The antenna can operate in a dual-band or in a very wide band mode. In [8], a combination of wideband and narrowband antennas into the same volume is presented. The wideband antenna is a CPW fed printed hour-glass shaped monopole which operates from 3 to 11 GHz. The narrowband antenna is a microstrip patch printed on the reverse side of the substrate, and connected to the wideband antenna via a shorting pin and designed to operate from 5.15 to 5.35 GHz. Another design was presented in [9] where the antenna reconfiguration was achieved through slot rotation. Each slot rotation redirects the surface current distribution creating an indirect reconfigurable feeding. This antenna achieves a different resonant frequency for each slot position. In this communication a new reconfigurable antenna design is presented. The antenna structure incorporates both a sensing and a reconfigurable antenna module into the same substrate. The sensing antenna covers the band from 2 GHz to 10 GHz, while the reconfigurable antenna is able to tune its operating frequency through the entire band covered by the sensing antenna. Reconfigurability is obtained by feeding at different instances, different antenna patches. This reconfiguration is achieved by a rotational motion. A detailed explanation of the reconfiguration is discussed in Section II. The control mechanism via a stepper motor is shown in Section III and the measurement data are shown in Section IV.

Fig. 2. UWB antenna radiation pattern at f line), and 9 GHz (dotted line).

= 3 GHz

(thin line), 6 GHz (thick

f = 3 GHz, 6 GHz and 9 GHz in the X-Z plane are shown in Fig. 2. The antenna possesses an omni-directional radiation pattern and is able to radiate above and below the substrate due to the fact that it has a partial ground.

B. Reconfigurable Antenna Reconfigurable antennas have been implemented so far using RF MEMS [10], PIN diodes [11], lumped elements [12], or photoconductive switches [13], [14]. In this work, we propose a new technique which is based on the rotational motion of the antenna structure. The advantage of this technique is that no bias lines are needed for the activation/deactivation of the switches. In fact the use of bias lines might affects the EM performance of the antenna and adds further complexity to the antenna structure. In this work, the frequency tuning is achieved by physically altering the patch shape. A circular substrate section holding five different antenna patches is rotated via a stepper motor. A 50 stripline overflows the rotating section in order to guarantee contact between the rotating circular patch and the feeding line. At each rotation stage, the stripline excites a different patch and a different frequency is achieved. The rotation mechanism is described briefly in Fig. 3. The stepper motor is modeled in HFSS with the antenna structure to account for its effect. The stepper motor’s characteristics are extracted from [15].

II. ANTENNA STRUCTURE

III. STEPPER MOTOR CONTROLLER

The sensing antenna is a modified egg-shaped printed monopole antenna. It has a partial ground of dimensions 32 mm 2 7 mm. A tapered stripline is feeding the antenna for better impedance match over the entire bandwidth of interest. This antenna is able to scan the spectrum from 2 to 10 GHz. The computed antenna radiation patterns at

The stepper motor used in this work rotates in 7.5 degree steps, and for each step 2 coils should be activated simultaneously [15]. The sequences of coil activation are summarized in Table I, where “1/0” denotes whether the corresponding coil is “activated/deactivated.” The amount of steps needed to go from one shape to another is summarized in Fig. 4. The stepper motor is attached to the back of the rotatable circular patch. Two plastic screws are used to hold the stepper motor to the antenna substrate as shown in Fig. 5. The motor rotating part consists of a metallic cylinder of length 1 cm and diameter 1 cm. This part is soldered to the back of the rotating circular patch in order to achieve the required rotation.

The antenna is printed on a 70 mm 2 50 mm Rogers Duroid 5880 substrate with a dielectric constant of 2.2 and a height of 1.6 mm. The corresponding antenna structure is shown in Fig. 1. The left module is the sensing antenna while the right part is the reconfigurable section. A. Sensing Antenna

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Fig. 5. The stepper motor used in this work. Fig. 3. Antenna reconfigurability process.

TABLE I COIL ACTIVATION SEQUENCE

Fig. 6. The flowchart of the LABVIEW algorithm.

Fig. 4. The number of steps needed for each rotation.

The stepper motor is connected to a computer via a parallel port and the control of the motor is achieved by using LABVIEW. A LABVIEW code was implemented to send to the pins of the parallel port (pin 2 till 5) one of the four sequences shown in Table I. Each of the four outputs from the parallel port (0 V/5 V TTL signal) is connected to a high voltage, high current Darlington array. The Darlington array, considered as the driver of the stepper motor, consists of two pairs of transistors for higher gain. Each output from the Darlington array is connected to one of the four coils of the stepper motor. In this work, the ULN2003 seven open collector Darlington pairs is used [16]. A 12 V power supply is needed for the stepper motor and the ULN2003. A flowchart of the LABVIEW code is shown in Fig. 6.

Fig. 7. The fabricated antenna.

IV. FABRICATION AND RESULTS A prototype antenna is fabricated and tested. The stepper motor incorporated in the back of the reconfigurable rotating antenna section is connected to the controlling circuit as discussed in the previous section. The fabricated antenna is shown in Fig. 7.

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Fig. 8. The sensing antenna return loss.

Fig. 9. A zoomed view of the connection between the feed line and the rotating part.

TABLE II FREQUENCY RECONFIGURABILITY

A. Sensing Antenna Results As shown in Fig. 1, the sensing antenna has a total length of 38 mm  0:25 2  (where  corresponds to the lowest frequency at “2 GHz”). The comparison between the simulated and the measured return loss for the sensing antenna is shown in Fig. 8. This comparison corresponds to the case when the reconfigurable section is at the initial position shown in Fig. 7. It is noted that the UWB performance of the antenna remains constant for all the reconfigurable section positions. B. Reconfigurable Section Results The reconfigurable section consists of a rotating 18 mm radius circular substrate section that carries five different patches. Each patch on the rotating section resonates at a different band from 2 GHz up to 10 GHz. The five different patches cover collectively the whole band (2–10 GHz). The dimensions and the position of the different shapes were optimized using HFSS. All the shapes are fed via a 10 mm 2 5 mm feeding line and they share a 20 mm 2 10 mm partial ground. The covered band for each shape is summarized in Table II. To ensure a good connection between the rotating circular patch and the feed line, a 50 stripline overflows the rotating circular section. It is soldered to the feed line; a zoomed view of the connection between the feed line and the rotating circular part is shown in Fig. 9. The comparison between the simulated and the measured return loss for the rotating section is shown in Fig. 10. Each band is labeled by the corresponding antenna shape number. An agreement is noticed between both data. C. Coupling Between the Two Antenna Sections For any cognitive radio application the sensing and the reconfigurable antennas must be isolated. In order to quantify the amount of mutual-coupling induced between the two antenna sections, we should

Fig. 10. A comparison between the measured and simulated return loss for the reconfigurable antenna section.

look at the transmission between the two antenna ports. The comparison between the simulated and the measured coupling (jS21j2 ) is shown in Fig. 11. This plot corresponds to the case when the reconfigurable antenna is at the position shown in Fig. 7. A coupling of less than 020 dB is obtained due to the fact that the two antenna structures are fed from the opposite edges of the substrate. The min/max values of the measured coupling for the other positions of the reconfigurable antenna are summarized in Table III. These values correspond to the frequency band where each shape of the reconfigurable antenna operates as summarized in Table II. D. Radiation Pattern of the Reconfigurable Antenna The comparison between the simulated (thick line) and the measured (thin line) radiation pattern for the reconfigurable antenna section in the X-Z plane is shown in Fig. 12 for different frequencies. A good agreement is noticed and the antenna preserves its omni-directional radiation pattern for all the different stages of the rotating section making it very convenient for cognitive radio applications. It is noticed that the stepper motor adds constructively to the antenna radiation pattern.

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TABLE IV ANTENNA PEAK GAIN

The peak antenna gain values for the 5 different patches of the rotating section, at the same frequencies used in the radiation pattern measurement shown in Fig. 12, are summarized in Table IV. V. CONCLUSION

Fig. 11. The coupling between the two sections. TABLE III MEASURED COUPLING

This communication presents an antenna design suitable for cognitive radio applications. The antenna is composed of a sensing section, achieved by an UWB printed monopole antenna and a reconfigurable section, represented by a rotating circular substrate section carrying five different patches. Each patch operates at a different frequency band when fed by a stripline that excites each particular shape upon rotation. The antenna system is fabricated and measured. A stepper motor controlled by LABVIEW through a computer’s parallel port rotates the circular section carrying the different patches. The comparison between the measured and the simulated data is satisfactory. To our knowledge, this is the first antenna designed for cognitive radio that is able to tune throughout the whole band covered by the sensing antenna (2 GHz–10 GHz).

REFERENCES

Fig. 12. The simulated (thick line) and the measured (thin line) radiation pattern for the reconfigurable antenna.

[1] J. Mitola, “Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio,” Ph.D. dissertation, Royal Institute of Technology (KTH), Stockholm, Sweden, 2000. [2] C. G. Christodoulou, “Cognitive radio: The new frontier for antenna design?,” IEEE Antennas Propag. Society Feature Article [Online]. Available: www.ieeeaps.org [3] G. T. Wu, R. L. Li, S. Y. Eom, S. S. Myoung, K. Lim, J. Laskar, S. I. Jeon, and M. M. Tentzeris, “Switchable quad-band antennas for cognitive radio base station applications,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 14668–1476, 2010. [4] J. R. Kelly and P. S. Hall, “Reconfigurable slot antenna for cognitive radio application,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2009, pp. 1–4. [5] C. G. Christodoulou, “Reconfigurable antennas in cognitive radio that can think for themselves,” in Proc. 3rd IEEE Int. Symp. in Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, 2009, pp. K1–K3. [6] Y. Tawk and C. G. Christodoulou, “A new reconfigurable antenna design for cognitive radio,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1378–1381, 2009. [7] H. F. AbuTarboush, S. Khan, R. Nilavalan, H. S. Al-Raweshidy, and D. Budimir, “Reconfigurable wideband patch antenna for cognitive radio,” in Proc. Loughborough Antennas and Propagation Conf., 2009, pp. 141–144. [8] E. Ebrahimi and P. S. Hall, “A dual port wide-narrowband antenna for cognitive radio,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, 2009, pp. 809–812. [9] J. Costantine, S. al-Saffar, C. G. Christodoulou, K. Y. Kabalan, and A. El-Hajj, “The analysis of a reconfiguring antenna with a rotating feed using graph models,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 943–946, 2009. [10] D. E. Anagnostou, G. Zheng, M. T. Chryssomallis, J. C. Lyke, G. E. Ponchak, J. Papapolymerou, and C. G. Christodoulou, “Design, fabrication, and measurement of an RFMEMS-based self-similar reconfigurable antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 422–432, Feb. 2006.

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[11] M. I. Lai, T. Y. Wu, J. C. Hsieh, C. H. Wang, and S. K. Jeng, “Design of reconfigurable antennas based on an L-shaped slot and PIN diodes for compact wireless devices,” IET Microw. Antennas Propag., vol. 3, pp. 47–54, 2009. [12] N. Behdad and K. Sarabandi, “Dual-band reconfigurable antenna with a very wide tunability range,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 409–416, Feb. 2006. [13] Y. Hawk, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Demonstration of a cognitive radio front-end using optically pumped reconfigurable antenna systems (OPRAS),” IEEE Trans. Antennas Propag., submitted for publication. [14] Y. Hawk, A. R. Albrecht, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Optically pumped frequency r reconfigurable antenna design,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 280–283, Mar. 2010. [15] [Online]. Available: http://www.digikey.com [16] [Online]. Available: http://www.datasheetcatalog.com/datasheets_pdf /U/L/N/2/ULN2003A.shtml

The HF Channel EM Parameters Estimation Under a Complex Environment Using the Modified IRI and IGRF Model Zhao-wen Yan, Gang Wang, Guo-liang Tian, Wei-min Li, Dong-lin Su, and Toyobur Rahman

Abstract—The international reference ionosphere (IRI) model is a generally accepted standard ionosphere model. It describes the ionosphere environment in quiet state and predicts the ionosphere parameters within a certain precision. In this communication, we have made a breakthrough in the application of the IRI model by modifying the model for Chinese region. Construct the ionosphere parameters foF2 and M(3000)F2 by using the Chinese Reference Ionosphere (CRI) coefficients, appropriately increase hmE, hmF2 height, reduce the thickness of F layer, validate the parameter by the measured values and solve the electron concentration distribution with QPS (quasi-parabolic segments). In this communication, 3D ray tracing algorithm was constructed based on the modified IRI model and IGRF geomagnetic field model. In short-wave detection, it can be used to predict the electromagnetic parameters of the receiving point, such as, the receiving area, maximum available frequency and the distribution of the group delay etc., which can help to determine the suitability of the communication. As an example, we estimated the short wave EM parameters around Qingdao in the detection from Zhengzhou to Qingdao using the modified IRI and IGRF model and provided technical support for the communication between the two cities.

(IRI) is the most effective and widely accepted empirical model of the ionosphere [1]. The IRI model is the global ionospheric model developed by the IRI Working Group based on a large number of ground observation data and the ionospheric research results for many years since 1960 under the joint funding of the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). Since 2000, IRI Working Group has studied how to introduce the global ionospheric model (Global Ionosphere Model, GMI) and other space radio observations results into IRI model to improve its accuracy. Now, the latest model is IRI2007, released in 2007. However, the process of IRI lacks of the observational data in China, which caused deviation to some extent in the region of China [2]. Reference [3] adapted the International Reference Ionosphere model IRI to improve determination of maximum usable frequency MUF of one European path and proposed the analytical approximation for the IRI model residual error. In [4], the ray tracing approach uses an electronic density profile estimated by using the latest revision (2007) of IRI model. Its work is to show a technique to determine the coordinate registration by jointly using ray tracing and Breit and Tuve theorem. In [5], the group relied on the International Reference Ionosphere (IRI) and Geomagnetic Reference Field models (IGRF) for the electron density profile and implemented the ray-tracing code to devise an optimum choice of operating frequencies for maximum coverage and to locate geographically the backscattering regions of super dual auroral radar network (Super DARN). In paper [6], an artificial-neural-network (ANN) method is employed to predict the HF-communication MUF (maximum useable frequency) in the region of the South China Sea. The comparison of ANN predictions and International Reference Ionosphere (IRI) models with the observations shows that the ANN predictions are closer to the observed results than the IRI predictions. In this communication, the IRI model was amended, using CRI coefficient instead of CCIR coefficient to compute the ionospheric parameters foF2 (F2 layer critical frequency) and M (3000) F2 (F2 layer propagation factor, that is, when the transmission distance is 3000 km, the ratio of the highest frequency to the critical frequency in F2 layer propagation), validating the hmE, hmF1, hmF2 calculated by the IRI and other parameters with the measured values and calculating the multilayer electron density distribution in the reflectivity space by QPS model. And on the basis of the electronic density distribution, we realize 3D ray tracing simulation by ignoring the collision effect and considering the geomagnetic field. As an example, we analyze the group delay parameters etc. in the area near Changchun within a certain range and the simulation results provide the technical parameters for shortwave communications and detection applications.

Index Terms—Distance resolution, group delay, IGRF, international reference ionosphere (IRI), ray tracing, short-wave communication.

I. INTRODUCTION Presently, the empirical or semi-empirical physical parameters ionosphere model including Bent, Chiu, Penn State Mk III, SLIM, FAIM, etc. are commonly used, while the international reference ionosphere Manuscript received April 06, 2010; revised June 26, 2010; accepted October 13, 2010. Date of publication March 03, 2011; date of current version May 04, 2011. The authors are with the School of Electronic and Information Engineering, Beijing University, Beijing, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2122237

II. MODIFICATION OF IRI MODEL In the IRI, the CCIR or URSI coefficients are recommended to calculate the ionospheric parameters foF2 and M (3000) F2, while we use “F2 ionosphere forecast method in Asia-Pacific region” which was derived from 39 years observation data in China and its neighboring observation stations to calculate foF2 and M (3000) F2. That is the CCIR coefficients which are substituted with CRI. After foF2 and M (3000) F2 are acquired, the hmE, hmF1, hmF2, etc. can be calculated by using IRI model and then compared with the measured value. When the error is large, the equation coefficients for hmE, hmF1 and hmF2, are calculated and then the new values of hmE, hmF1 and hmF2 are compared with the measured values until the precision is met. In general, in China take hmE = 115 km. After the amendment, more accurate hmE, hmF1, hmF2 parameters can be found. After obtaining the accurate hmE, hmF1, hmF2 parameters, we put these parameters into the QPS model, using multi-layer quasi-parabolic

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Fig. 1. Algorithm block diagram of modified IRI model.

present six different time profile control of electron density map, reflecting the diurnal variation of electron density in the height from 0 to 500 km above sea level. The location for Beijing is (longitude 116.47 , latitude 39.9 ) and the time is March 18, 2009. III. RAY TRACING ALGORITHM

Fig. 2. The electron density—time profiles. Six curves stand for the electron density distribution of the region above Beijing at six different times.

model to construct the electron density distribution in the reflected region for the use of Appleton-Hartree formula to prepare a spatial refractive index. The block diagram of the algorithm for computing the electron density distribution by using the corrected IRI model is shown in Fig. 1. Fig. 2 is the re-obtained ionosphere electron density profile chart showing the daily changes in electron concentration, where six curves

The magnetic particle is anisotropic under the geomagnetic field, so the electromagnetic waves in the ionosphere propagate in two modes: the o-wave and the x-wave. Thus, the multipath effect will occur and the differences between the locations where the two waves reach the ground are so distinctive. The intensity of geomagnetic field, predicted by the IGRF model, is about 0.5 gauss and it changes with the altitude, longitude and latitude [7]–[9]. Thus, we can construct the propagation environment according to the Appleton-Hartree equation along with the IRI model and the IGRF model. On the basis of the IRI model and the IGRF, the color visual display of three-dimensional ray tracing is realized by using ray tracing algorithm combining with geographic information systems. The model input parameters of the IRI model are simpler compare to other model input parameters, so it has more universal adaptability.Fig. 3 is the block diagram of 3D ray tracing algorithm implemented by using the IRI model and IGRF model. To illustrate the accuracy of the simulation results, as an example, in measurement we select the actual received frequency signal of 5.109 MHz with the group path for the 981 km (without multi-path effect). Using the simulation program, we set the transmitter frequency as 5.109 MHz and the launch elevation angle range from 5 to 35 with

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TABLE I COMPARISON OF THE GROUP PATHS (DATE: 28/10/2008)

TABLE II COMPARISON OF THE GROUP PATHS (DATE: 31/10/2008)

Fig. 3. Block diagram of 3D ray tracing algorithm.

Fig. 4. Simulated ray path trajectory at 14:15. The frequency is 5.109 MHz. There is no multi-path effect.

a step of 2 according to the direction of the antenna. Then, we get a group of trajectory curves as shown in Fig. 4. From the simulated result we find that the group path is 979.45 km when the geodesic distance is 937.53 km and that means the rays can reach at the receiving point in Changchun exactly. The error between the simulation and the measured values is less than 2 km.

measuring data, “—” means the rays cannot reach the receiving point and all the time and date is Beijing standard time. V. EM PARAMETERS COMPUTATION OF THE HF CHANNEL NEAR THE RECEIVING POINT A. Spatial Differentiation

IV. NUMERICAL VERIFICATION OF THE 3-D RAY TRACING METHOD Based on the ray tracing method, we can estimate the characteristics of the rays at the receiving point more accurately by scanning, finding and interpolating the elevation angle. In order to compare with the measured values, we choose Qingdao as the launch point and Changchun as the receiving site and select the forecast time on October 28, 31 and November 1, 2008. Tables I–III gives some comparisons between the simulation group paths and the measured group paths at certain times. The results show that the simulation accuracy is high. Here are some notes about the tables: Freq stand for the frequency of the rays, GD stand for the geodesic distance, GP stand for the group path, SIMU stand for simulation results, MEAS stand for

In order to ensure the uniqueness of the obtained group path parameters within a certain range on the ground, we need to research the changes of the ray trace or propagation path, that is, in other words, the spatial differentiation of the group path. For example, assume that the rays are reflected by the same layer of the ionosphere and reach at two different points along the transmitter-receiver radial direction. If the difference of the group path between two points is larger than the group path resolution of the receiver and if the difference can be identified by the receiver, then the wave paths have spatial differentiation. On the contrary, if the path difference is less than the resolution of the receiver and if the receiver can not distinguish the changes of the group paths, then the radio propagation path has no difference.

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TABLE III COMPARISON OF THE GROUP PATHS (DATE: 01/11/2008)

Fig. 6. Distribution of the difference of the group path reflected by the E layer over time and radial distance.

Fig. 7. Distribution of the difference of the group path reflected by the F2 layer over time and radial distance. Fig. 5. Distribution of the receiver locations: (a) Three dimensional schematic diagram. (b) Two dimensional schematic diagram.

Hence, we make the transmitter-receiver radial direction as the reference direction and choose the group path difference between the main receiver and some sub receivers, which are 60:1 km, 60:2 km, 60:5 km, 61 km, 62 km, 65 km, 68 km, 610 km, 620 km, 650 km away from the main receiver along the reference direction, as a criterion of the spatial differentiation. As shown in Fig. 5, we choose Qingdao as the main receiver point (marked by ’M’ in Fig. 5) and select some reference points, which are 60:1 km, 60:2 km, 60:5 km, 61 km, 62 km, 65 km, 68 km, 610 km, 620 km, 650 km away from the main receiver, as the locations of the sub receivers. And we use “+” and “-” symbol for the direction from transmitter to receiver and the opposite direction respectively. In the unknown position the group paths can be calculated according to the group path values of the adjacent two points in the same direction by linear interpolation. Since the E layer disappears at night and the frequencies that can be reflected are very small, we select daytime 9:15–16:15, July 27, 2009 (Beijing time) as the simulation time and select 8 MHz as the frequency of the rays and calculate the group paths and the path changes around the receiving position every 1 hour. And we choose Zhengzhou (E113:63 ; N34:80 ) as the location of the transmitter and Qingdao

(E120:30 ; N36:10 ) as the receiver respectively. The azimuth angle that we calculate is 74.62 and the geodesic distance is 621.01 km according to the locations. In the subsequent simulations, the location of the transmitter and receiver is same as here. The results of the simulation are shown in Figs. 6 and 7 respectively. Fig. 6 shows the difference of the group path by the E layer over time and radial distance. Fig. 7 shows the difference of the group path by the F2 layer over time and radial distance. B. The Characteristics of the Group Path Difference Around the Main Receiving Point In order to investigate the group path difference between the receiving point and the points around it, we selected the main receiving point as the center and choose 2 km, 5 km, 8 km, 10 km, 20 km as the radius respectively, thus we got a series of concentric circles. To get the distributions of the group path differences between the main receiving point and the points in the circles, we calculate the group path of the discrete points in the circles and compare them with the group path of the main receiving points. Here we choose the same simulation parameters as that in Section V-A. The results of the simulation are shown in Figs. 8 and 9. Fig. 8 shows the contour of the group path reflected by the E layer over the latitude and longitude and the two horizontal axes stand for

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TABLE IV DISTRIBUTION OF E LAYER RANGE RESOLUTION OVER TIME

TABLE V DISTRIBUTION OF F2 LAYER RANGE RESOLUTION OVER TIME

Fig. 8. Contour of the group path reflected by the E layer over the latitude and longitude.

Fig. 9. Contour of the group path reflected by the F2 layer over the latitude and longitude.

the latitude and longitude of the points on the concentric circles respectively, while the vertical axis stands for the group path. Fig. 9 shows the contour of the group path reflected by the F2 layer over the latitude and longitude.

In order to ensure uniqueness of  , referring to the definition of the range resolution in radar systems, we define the ground range resolution L as the smallest distance between the main receiver and the subreceiver when the two receivers can receive different ray paths from the same transmitter and distinguish between them. And the ground range resolution L is the corresponding value of the group path resolution  on the ground. 2) Characteristics of Range Resolution: According to the height of the wave reflection surface, the ground range resolution L can also be divided into the E layer ground range resolution and F2 layer ground range resolution when the reflection occurs at different layers. According to (1), the group path resolution  is determined when the bandwidth of the receiver is given. If we substitute  into the space difference data table of the group path calculated in Section V-B, we can calculate the corresponding radial distance difference by linear interpolation method. The radial distance difference is just the ground range resolution L according to the given bandwidth. Using this method, we get the results of the ground range resolution over time and frequency in the given bandwidth, as shown in Tables IV–VI.

C. Characteristics of the Range Resolution 1) The Definition of Range Resolution: In accordance with the Rayleigh definition of range resolution in radar systems, we define the resolution of the group path in a receiver as:

= Bc : (1) Where  is the group path resolution, c is the velocity of the rays and B is the bandwidth of the receiver. 

Since the rays may be reflected by the E layer and F2 layer respectively, we define the range resolution of the group path reflected by the E layer as E layer resolution and the range resolution of the group path reflected by the F2 layer as F2 layer resolution. (The F1 layer does not appear in the vast majority).

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TABLE VI DISTRIBUTION OF E LAYER RANGE RESOLUTION OVER FREQUENCY

VI. CONCLUSIONS In this communication, the international reference ionosphere (IRI) model has been modified, which is used to obtain the distribution of the ionospheric electron density. It is easier and valuable in practical applications since only international time, longitude and latitude and other parameters need to be provided to predict the parameters of the ionosphere. Considering the effects of the geomagnetic field, the 3-D ray tracing simulation method based on the modified IRI model is constructed and can give a result that is in consistence with the facts. And it can also be used to predict the spread of signal coverage area, which helps to choose the suitable firing frequency in short wave communication according to time and communication objectives. At the receiving point, we can calculate the group path difference according to the geodesic distance and group path to take measures to improve the reliability of short-wave communication when we encode. In short-wave detection, it can be used to predict the electromagnetic parameters near the receiving point like the spread of coverage of the received signals, the maximum available frequency, distribution of the group delay, etc. And these parameters can help to determine the suitability of the short-wave detection and communication. In this communication, we estimated the short wave EM parameters around Qingdao in the detection from Zhengzhou to Qingdao and provided the technical support for the short wave communication of the two cities.

REFERENCES [1] D. Bilitza, N. M. Sheikh, and R. A. Eyfrig, “Global model for the height of the F2-peak using M3000 values from CCIR(J),” Telecommun., vol. 46, no. 9, pp. 549–553, 1979. [2] C. Hongyan and S. Xianru, “A new method of predicting the ionosphoric F2 layer in the Asia Oceania region,” Chinese J. Space Sci., vol. 29, no. 5, pp. 502–507, 2009. [3] B. G. Barabashov, O. A. Maltseva, V. T. Rodionova, and A. S. Shlyupkin, “Evaluation of the IRI model efficiency for operational forecast of HF propagation conditions,” in Proc. 10th IET Int. Conf. on Ionospheric Radio Systems and Techniques, London, U.K., 10, 2006, pp. 253–257. [4] A. Cacciamano, A. Capria, F. Berizzi, M. Martorella, and E. D. Mese, “A ray tracing based method for coarse coordinate registration in HF skywave OTH radar,” in Radar Conf. EuRAD, 2009, pp. 204–207. [5] D. Monselesan and P. Wilkinson, “First step towards TIGER frequency management: A ray tracing exploration using both IRI and IGRF models,” Adv. Space Res., vol. 27, no. 1, pp. 167–174, 2001. [6] W. Zeng and X. J. Zhang, “Predictions of HF communication MUF in the region of the South China Sea,” IEEE Antennas Propag. Mag., vol. 41, no. 04, pp. 35–38, 1999.

[7] S. Macmillan and J. M. Quinn, “The 2000 revision of the joint UK/US geomagnetic field models and an IGRF 2000 candidate model,” Earth Planets Space, vol. 52, pp. 1149–1162, 2000. [8] S. Macmillan and S. Maus, “International Geomagnetic reference field—the tenth generation,” Earth Planets Space, vol. 57, pp. 1135–1140, 2005. [9] N. Olsen, F. Lowes, and T. J. Sabaka, “Ionospheric and induced field leakage in geomagnetic field models and derivation of candidate models for DGRF 1995 and DGRF 2000,” Earth Planets Space, vol. 57, pp. 1191–1196, 2005. [10] D.L. Xi, J.R. Li, Y.M. Liu, and Z.W. Zhao, “A new method for longterm prediction of ionosphere F f2 and M(3000) F2,” Chinese J. Radio Sci., vol. 23, no. 5, pp. 946–949, 2008. [11] C. J. Coleman, “A ray tracing formulation and its application to some problems in over-the-horizon radar,” Radio Sci., vol. 33, no. 04, pp. 1187–1197, 1998. [12] Y.C. Suo, “Ionospheric short-wave ray tracing,” J. Space Sci., vol. 13, no. 04, pp. 306–312, 1993. [13] D. J. C. Mackay, “Good error correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory, vol. 45, no. 02, pp. 399–431, 1999. [14] W. Liu, P.N. Jiao, S.K. Wang, and J.J. Wang, “Short wave 3D ray tracing in the ionosphere and its application,” Chinese J. Radio Sci., vol. 23, no. 01, pp. 31–48, 2008. [15] S.G. Xie, Z.Y. Zhao, and X.Q. Shi, “A fast-grid ionospheric ray-tracing method,” Chinese J. Radio Sci., vol. 14, pp. 39–43, 1999. [16] X.Z. Huang, Q.Y. Lu, and Z.Q. Zhang, “Spherically stratified atmosphere in the accurate calculation of radiation transmission path,” Chinese J. Radio Sci., vol. 16, no. 02, pp. 203–207, 2001. [17] X.H. Li and D.Z. Guo, “Modeling and prediction of ionospheric total electron content by time series analysis,” in Proc. 2nd Int. Conf. Advanced Computer Control (ICACC), China, 2, 2010, vol. 2, pp. 375–379. [18] A. Senior, N. D. Borisov, M. J. Kosch, T. K. Yeoman, F. Honary, and M. T. Rietveld, “Multi-frequency HF radar measurements of artificial F-region field-aligned irregularities,” Ann. Geophys., vol. 22, pp. 3503–3511, 2004. [19] M. Abdullah, D. A. A. Mat, A. F. M. Zain, S. Abdullah, and S. S. N. Zulkifli, “Analysis of ionospheric models during ionospheric disturbances,” in Proc. Asia-Pacific Conf. Applied Electromagnetics, Melaka, 2007, pp. 1–4. [20] A. Komjathy, G. H. Born, and D. N. Anderson, “An improved high precision ionospheric total electron content modeling using GPS,” in Proc. IEEE Int.Geosci. Remote Sensing Symp., Honolulu, HI, 2000, vol. 7, pp. 2858–2860. [21] G. D. Earle and R. I. Desourdis, Jr, “Advanced modeling of HF radio propagation,” in Military Communications Conf. Record, MILCOM ’94, 1994, vol. 3, pp. 895–899. [22] C. J. Coleman, “A propagation model for HF radiowave systems,” in Military Communications Conf. Record, MILCOM ’94, 1994, vol. 3, pp. 875–879.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011

Corrections Corrections to “On Corrections to a z-Transform Method for FDTD Simulations With Frequency-Dependent Dielectric Functions” Douglas B. Miron In [1], the denominator in the third terms on the right hand side of both (16) and (17) should read

This has no effect on the data presented because the software was correct. The column headings for Table II should be changed. In the original version of the paper, there was an equation (18) that was similar to (17) but had two quasi-Lorenz terms. To save space, this equation was removed. The headings could be changed as follows. Eq. (16) ! "DL , Eq. (17) ! "DLz , Eq. (18) ! "DLz2 . The Table title could be “PARAMETER VALUES FOR THREE SILVER PERMITTIVITY CURVE FITS.” The author regrets these errors.

1 + 2dj!=!0 0 (!=!0)2 :

REFERENCES Manuscript received December 31, 2010; accepted January 07, 2011. Date of current version May 04, 2011. The author is at Solway, MN USA 56678 (e-mail [email protected]). Digital Object Identifier 10.1109/TAP.2011.2128295

[1] D. B. Miron, “On corrections to a z-transform method for FDTD simulations with frequency-dependent dielectric functions,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 4100–4104, Dec. 2010.

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