IEEE Transactions on Antennas and Propagation [volume 59 number 8]

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AUGUST 2011

VOLUME 59

NUMBER 8

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Dual-Band Patch Antennas Based on Short-Circuited Split Ring Resonators ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .... O. Quevedo-Teruel, M. Ng Mou Kehn, and E. Rajo-Iglesias Hierarchical Curl-Conforming Nédélec Elements for Quadrilateral and Brick Cells ... . R. D. Graglia and A. F. Peterson Compact Dual-Band and Ultrawideband Loop Antennas ......... ......... ........ ......... .. M. K. Mandal and Z. N. Chen Design of the Millimeter-wave Rectangular Dielectric Resonator Antenna Using a Higher-Order Mode ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . Y.-M. Pan, K. W. Leung, and K.-M. Luk Pulsed Electromagnetic Field Radiation From a Wide Slot Antenna With a Dielectric Layer ... ........ ......... ......... .. ˇ .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... A. T. De Hoop, M. Stumpf, and I. E. Lager Reaching the Chu Lower Bound on Q With Magnetic Dipole Antennas Using a Magnetic-Coated PEC Core .. ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... O. S. Kim and O. Breinbjerg Experimental Determination of DRW Antenna Phase Center at mm-Wavelengths Using a Planar Scanner: Comparison of Different Methods ... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ...... P. Padilla, P. Pousi, A. Tamminen, J. Mallat, J. Ala-Laurinaho, M. Sierra-Castañer, and A. V. Räisänen Truncation-Error Reduction in Spherical Near-Field Scanning Using Slepian Sequences: Formulation for Scalar Waves . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ........ K. T. Kim Arrays Fundamental Electromagnetic Properties of Twisted Periodic Arrays .... ........ ......... .. D. Van Orden and V. Lomakin Millimeter-Wave High Radiation Efficiency Planar Waveguide Series-Fed Dielectric Resonator Antenna (DRA) Array: Analysis, Design, and Measurements .. ......... ..... .... ......... W. M. Abdel-Wahab, D. Busuioc, and S. Safavi-Naeini Double-Layer Full-Corporate-Feed Hollow-Waveguide Slot Array Antenna in the 60-GHz Band ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... Y. Miura, J. Hirokawa, M. Ando, Y. Shibuya, and G. Yoshida A Deterministic Two Dimensional Density Taper Approach for Fast Design of Uniform Amplitude Pencil Beams Arrays . ......... ......... ........ ......... ......... .... ..... ......... ......... ........ ......... ....... O. M. Bucci and S. Perna Sparse Antenna Array Optimization With the Cross-Entropy Method ... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... P. Minvielle, E. Tantar, A.-A. Tantar, and P. Bérisset Spherical-Wave-Based Shaped-Beam Field Synthesis for Planar Arrays Including the Mutual Coupling Effects ....... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... J. Córcoles, J. Rubio, and M. Á. González Synthesis of Conformal Phased Arrays With Embedded Element Pattern Decomposition ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... K. Yang, Z. Zhao, Z. Nie, J. Ouyang, and Q. H. Liu Robust Beampattern Synthesis for Antenna Arrays With Mutual Coupling Effect ....... ......... .... T. Zhang and W. Ser RFID Grids: Part II—Experimentations .. ......... ........ ......... . ........ ........ ......... ... S. Caizzone and G. Marrocco

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(Contents Continued on p. 2757)

(Contents Continued from Front Cover) Numerical and Analytical Techniques Efficient Analysis of Large Scatterers by Physical Optics Driven Method of Moments .... M. S. Tasic and B. M. Kolundzija Implementation of Material Interface Conditions in the Radial Point Interpolation Meshless Method ..... Y. Yu and Z. Chen Improving the Accuracy of FDTD Approximations to Tangential Components of the Coupled Electric and Magnetic Fields at a Material Interface .. ......... ......... ........ ......... ......... ........ ...... T. M. Millington and N. J. Cassidy Near-Field PML Optimization for Low and High Order FDTD Algorithms Using Closed-Form Predictive Equations . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... M. F. Hadi Efficient Modeling of Three-Dimensional Reverberation Chambers Using Hybrid Discrete Singular Convolution-Method of Moments .... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .... H. Zhao and Z. Shen A Prescaled Multiplicative Regularized Gauss-Newton Inversion ........ ........ ......... ......... P. Mojabi and J. LoVetri Reconstruction of the Electromagnetic Field in Layered Media Using the Concept of Approximate Transmission Conditions ...... ......... ........ ......... ......... . ........ ......... ......... ........ .... O. Özdemir, H. Haddar, and A. Yaka Spectral Analysis of Relativistic Dyadic Green’s Function of a Moving Dielectric-Magnetic Medium . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... T. Danov and T. Melamed Mastering the Propagation Through Stacked Perforated Plates: Subwavelength Holes vs. Propagating Holes .. ......... .. .. ........ ......... ......... .... M. Navarro-Cía, M. Beruete, F. Falcone, J. M. Illescas, I. Campillo, and M. Sorolla Ayza Design and Free-Space Measurements of Broadband, Low-Loss Negative-Permeability and Negative-Index Media ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ S. M. Rudolph, C. Pfeiffer, and A. Grbic Wireless Frequency Selective Buildings Through Frequency Selective Surfaces .. ........ ......... M. Raspopoulos and S. Stavrou Diffuse Scattering Model of Indoor Wideband Propagation ...... . ...... O. Franek, J. B. A. Andersen, and G. F. Pedersen Building Penetration Loss for Satellite Services at L-, S- and C-band: Measurement and Modeling .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... M. Kvicera and P. Pechac Accurate Modeling of Body Area Network Channels Using Surface-Based Method of Moments ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... A. M. Eid and J. W. Wallace Exact SLF/ELF Underground HED Field Strengths in Earth-Ionosphere Cavity and Schumann Resonance .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... Y.-X. Wang, R.-H. Jin, J.-P. Geng, and X.-L. Liang

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COMMUNICATIONS

Controlling Resonances for a Multi-Wideband Antenna by Inserting Reactive Components .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... H. Choi, S. Jeon, J. Yeom, O. Cho, and H. Kim Band-Notched UWB Antenna Incorporating a Microstrip Open-Loop Resonator ..... J. R. Kelly, P. S. Hall, and P. Gardner Linearly Polarized Radial Line Patch Antenna With Internal Rectangular Coupling Patches .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... J. L. Masa-Campos and M. Sierra-Pérez A Novel Unidirectional Dual-Band Circularly-Polarized Patch Antenna ........ ......... ... C.-H. Chen and E. K. N Yung Circularly Polarized Antenna Having Two Linked Slot-Rings ... ......... ....... .. ......... ...... T.-N. Chang and J.-M. Lin Monofilar Spiral Slot Antenna for Dual-Frequency Dual-Sense Circular Polarization .. . .. X. L. Bao and M. J. Ammann A Bandwidth Improved Circular Polarized Slot Antenna Using a Slot Composed of Multiple Circular Sectors ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ S. H. Yeung, K. F. Man, and W. S. Chan Bunny Ear Combline Antennas for Compact Wide-Band Dual-Polarized Aperture Array ..... Y. Zhang and A. K. Brown Bandwidth Enhancement of a Wide Slot Using Fractal-Shaped Sierpinski ...... ......... ......... ........ ........ Y. J. Sung A Reconfigurable Microstrip Leaky-Wave Antenna With a Broadly Steerable Beam ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... R. O. Ouedraogo, E. J. Rothwell, and B. J. Greetis Integration of Circular Polarized Array and LNA in LTCC as a 60-GHz Active Receiving Antenna .... ......... ......... .. .. ........ ......... ......... ........ ......... .. M. Sun, Y.-Q. Zhang, Y.-X. Guo, M. F. Karim, O. L. Chuen, and M. S. Leong Two-Shell Radially Symmetric Dielectric Lenses as Low-Cost Analogs of the Luneburg Lens . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... .... A. V. Boriskin, A. Vorobyov, and R. Sauleau Closed-Form Green’s Function Representations for Mutual Coupling Calculations Between Apertures on a Perfect Electric Conductor Circular Cylinder Covered With Dielectric Layers ...... ... M. S. Akyüz, V. B. Ertürk, and M. Kalfa Multiwall Carbon Nanotubes at RF-THz Frequencies: Scattering, Shielding, Effective Conductivity, and Power Dissipation ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... . J. A. Berres and G. W. Hanson Sub-mm Wet Etched Linear to Circular Polarization FSS Based Polarization Converters ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ...... M. Euler, V. Fusco, R. Dickie, R. Cahill, and J. Verheggen Reducing the Number of Elements in Linear and Planar Antenna Arrays With Sparseness Constrained Optimization .. .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... W. Zhang, L. Li, and F. Li Antenna Array Optimization Using Dipole Models for MIMO Applications ..... S. Karimkashi, A. A. Kishk, and D. Kajfez Interpolatory Macro Basis Functions Analysis of Non-Periodic Arrays .. ........ .... D. González-Ovejero and C. Craeye Multiband Multipolarization Integrated Monopole Slots Antenna for Vehicular Telematics Applications .... .. P. Mousavi An 800 MHz 2 1 Compact MIMO Antenna System for LTE Handsets .... M. S. Sharawi, S. S. Iqbal, and Y. S. Faouri Mode-Based Estimation of 3 dB Bandwidth for Near-Field Communication Systems .. ... .... Y. Tak, J. Park, and S. Nam

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Digital Object Identifier 10.1109/TAP.2011.2163481

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 8, AUGUST 2011

Dual-Band Patch Antennas Based on Short-Circuited Split Ring Resonators Óscar Quevedo-Teruel, Member, IEEE, Malcolm Ng Mou Kehn, Member, IEEE, and Eva Rajo-Iglesias, Senior Member, IEEE

Abstract—A study of an innovative antenna based on split ring resonators (SSRs) is presented. SSRs have been exhaustively used in the literature as the unit cell of periodic structures for obtaining left-handed media, whose interesting characteristics can be applied to waveguides or antennas. In this work, the authors propose the use of only one unit cell of a double SRR as a radiator. The double SRR is printed on a grounded dielectric slab, acting as the radiating element of a microstrip patch antenna, and grounded pins are used to short-circuit the structure for size reduction. A compact dual band antenna is obtained in this way. For example, a particular design making use of PP ( = 2 2) as substrate allows a size of 0.05 0 0.05 0 for the lower frequency of operation with an acceptable radiation efficiency. Simulated and measured results of return losses, gain and radiation efficiency of this new type of patch antenna are provided. Index Terms—Dual band, microstrip patch antenna, split ring resonator.

I. INTRODUCTION

O

VER the last years, the definition and study of artificially-synthesized materials known as metamaterials have been accorded considerable attention from the scientific community [1]. Basically, these materials have ordinary microscopic characteristics, but macroscopically, they exhibit properties that cannot be found in nature. The interest has typically been mainly focused on structures with unusual electromagnetic properties defined by Veselago [2], which can be achieved from subwavelength periodic repetitions of particular resonant unit cells. These new conditions are normally only obtained in a narrow frequency band, and they depend primarily on the periodicity and dimensions of the elements. One of the most commonly used unit cell is the SRR (Split Ring Resonator) which has been vastly studied in the literature since its inception and measurements of early prototypes ([3], [4]). Manuscript received June 20, 2010; revised November 12, 2010; accepted December 16, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported in part by the Spanish Government TEC2006-13248-C04-04 and in part by the National Science Council of Taiwan (NSC 99-2218-E-009-009). Ó. Quevedo-Teruel is with Department of Theoretical Physics of Condensed Matter at Autonomous University of Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain (e-mail: [email protected]). M. Ng Mou Kehn is with Department of Electrical Engineering National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: [email protected]). E. Rajo-Iglesias is with Department of Signal Theory and Communications at Carlos III University of Madrid, Avda. Universidad 30, 28911, Leganés (Madrid), 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.2011.2158786

Since then, SSRs have been used for numerous purposes and applications. One of the most common is the use for obtaining the propagation of new modes or for filtering propagation bands in guiding structures. These theories are already successfully applied to different types of transmission lines, for which waveguides and microstrip lines are the most widely studied in the literature ([5]–[10]). Therefore, considering that these SSRs can eliminate or provide new bands of operation, they can be used not only in transmission lines, but also in the design of antennas. Particularly, in microstrip technology some authors have developed designs where the SSRs have for instance contributed to notch bands of traditional antennas ([11]–[14]). On the other hand, other authors have used left-handed media based on SSRs as substrates for microstrip patch antennas with different purposes such as reducing the size of the antennas ([15], [16]) or improving the impedance bandwidth ([17], [18]). In addition, other works have used left-handed media as superstrates for increasing the directivity and gain of patch antennas ([19]). Finally, some new patch antennas that use SSRs for defining new operation bands (at lower frequencies) have also been presented ([20]–[22]). These aforementioned radiators combine SSRs or derived structures with traditional antennas for obtaining new resonances, or to modify the radiation pattern or the usual operation frequency. However, in the present paper, the authors propose the use of short-circuited SSRs themselves as main radiators to create a new microstrip patch antenna, and not only as external loads. The antenna is dual-band and highly compact, and consequently can be suitable for applications where there are size constraints. An important fact known to achieve such compactness is the use of pins which short-circuit both SSRs to the ground plane, thus creating similarities with PIFA antennas. Other ways of obtaining compact antennas with dual band performance have been explored in the literature. Most of them use substrates with high permittivities. For example, in [23] but using the authors obtained a compact size of materials with relative permittivity of 16 and 30; and in [24] the in one of its authors propose an antenna with a length of sides, making use of FR4 (that has relative permittivity of 4.2). In this paper, we propose an antenna whose size is compact even with low permittivity materials. Besides, the “self-similar” nature of the proposed concentric type double-SRR radiator is highly elegant and provides the strong compactness. This is a characteristic that is absent from most other previous works, which mainly involved PIFAs that are located at physically disparate locations, each taking care of a certain band. To possess this attribute is vital for applications in modern-day

0018-926X/$26.00 © 2011 IEEE

QUEVEDO-TERUEL et al.: DUAL-BAND PATCH ANTENNAS BASED ON SHORT-CIRCUITED SRRs

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Fig. 2. Measured and simulated s for an antenna with the following dimen: mm; for a mm, r mm, c mm, d mm and p sions: R with 3 mm thickness. substrate of PVC 

=6

=4 ( = 3)

=1

=1

= 0 25

Fig. 1. Top and side view of the proposed antenna.

handheld devices where “real estate” for antenna placement is scarce. All the results obtained in this work have been simulated with CST Microwave Studio and validated with measurements of return loss as well as gain in an anechoic chamber and radiation efficiency in a reverberation chamber. II. DEFINITION OF THE ANTENNA In this Section, the proposed antenna is described. Fig. 1 shows the top and side views of the antenna. Since it has a grounded dielectric and a printed metallization, it can be classified as a microstrip patch antenna. The radiating face is composed of two concentric split rings with opposite gaps and shortcircuited to the ground plane at one end of each ring as for PIFA antennas. The antenna is fed by a microstrip line placed beside the rings as shown in Fig. 1, although another type of excitation could be also possible as will be demonstrated later. The dielectric material of the substrate has a low permittivity, as antennas would have typically in order to provide good radiation characteristics. (simulated and measured) for such an anFig. 2 shows the tenna whose arbitrary dimensions are as follow (with reference mm, mm, mm, to Fig. 1 for the notations): mm and mm; for a substrate of PVC with 3 mm thickness. As the Figure shows for both simulation and measurement, there are two bands of operation related to the resonances established by the SSRs, although the bands are not well matched in this example. The operation frequencies obtained by simulations fit properly with the measurements. The electric field distributions for both modes are shown in Fig. 3. plane, the main field comAssuming the antenna is in the as it would typically be for patch antennas. The ponent is inner ring mainly contributes to the excitation of the higher resonance frequency whereas the outer one to that of the lower frequency. In addition the separation between the inner and outer

Fig. 3. Electric field for both radiation modes shown in Fig. 2. (a) Amplitude of the electric field distribution at the lower frequency (1.4 GHz). (b) Amplitude of the electric field distribution at the higher frequency (2.29 GHz).

rings affects both frequencies. Later on, a parametric study will be presented in Section IV. III. CIRCUIT MODEL An equivalent circuital model of the antenna is now presented. The proposed model is as shown in Fig. 4 (based on the ones presented in [5], [21]) and it is intended to qualitaand tively show the operation of the antenna. Firstly, correspond to the typical equivalent circuit of a transmission line. These two elements are shunt connected to the antenna

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Fig. 4. Equivalent circuital model for the antenna.

Fig. 6. Evaluation of the operation frequency of both resonances when the relative size of the antennas is changed. Crosses correspond to measured prototypes.

Fig. 5. Measurement and circuit-model simulation of s for an antenna with mm, r mm, c mm, d the following dimensions: R mm and , PVC  : c : mm; for the following substrates: PVC  and PVC  , all of them with 3 mm of thickness.

= 0 25 ( = 1)

=6

=4

=1 ( = 3)

=1 ( = 2 2)

which is composed of two series capacitances ( and ) defined by the coupling between the two SRRs (which depends on the distance between them) and also the gaps of both rings, and ) generated by the SSRs two shunt inductances ( and their connections to the ground plane, and two low order and ), characterized by the total surface capacitances ( area of the SRRs and the substrate attributes (permittivity and thickness). By tuning the lumped element values of the equivalent circuit based on guidelines from [5], [21], [25], the frequency behavior was fitted for three particular examples that were designed with the dimensions studied in Section II, but different dielectric maand ), thus achieving very good terials ( agreement between the measurements and simulations as Fig. 5 has the following values: shows. Particularly, the case of nH, pF, pF, pF, nH and nH. The influence of the dielectric materials is principally modeled by the series capacitances ( and ) and the shunt inductances ( and ). However, in principle, the estimation of the values of all these components and how to relate them to the antenna geometry is not trivial. Therefore this model is a good tool for analyzing and understanding the operation of the antenna, but it is not suitable for design purposes. IV. PARAMETRIC STUDY A parametric study of the effects which the different parameters have on the two resonance frequencies is now presented.

The results are obtained by simulations and verified by measurements. An initial antenna will be considered whose dimenmm, mm, mm, mm and sions are: mm; for a substrate with 3 mm thickness and . From this design, some parameters will be varied and their influences studied. Obviously, depending on the total size of the SSRs, the operation frequencies will be modified as in any resonant structure. A study of this size effect was carried out obtaining the results shown in Fig. 6, whereby the two operation frequencies are represented as a function of the relative size of the antennas, which is defined by a scaling factor with respect to the size of the rings in the preliminary configuration. Thereby, when this parameter is equal to unity, we are referring to the initial case. As this Figure shows, when the relative size of the rings is increased the operation frequencies exponentially decrease, as expected. Two other important parameters of the antenna are the thickness of the substrate and its permittivity. The influences of these two parameters on the resonance frequencies are plotted in Figs. 7 and 8, respectively. As seen, the higher and lower resonance frequencies fall considerably with increasing values of both parameters. This concurs with well known characteristics of traditional patches, whose operation frequencies are such that some dimension of the printed part of the antenna is in the material used as subapproximately a multiple of strate [26], [27]. However, in traditional patches the thickness affects mainly the bandwidth and the efficiency of the antenna (Q factor), but not meaningfully the operation frequency of the modes [28] as is the case for the proposed antennas. This can be understood by knowing that the principle which defines the resonant frequencies of these SRR-based patch antennas is different from that of classical ones. Although traditionally, the fundamental mode of patch antennas is defined by the length of one of its sides, the modes for the present case however are defined by the capacitance between rings, the capacitance of the splits, and the inductance of the rings and their short-circuited pins. Thus, if the thickness of the substrate is changed, these inductances will be strongly modified and the resonances will be shifted in frequency. The latter coincides with how the

QUEVEDO-TERUEL et al.: DUAL-BAND PATCH ANTENNAS BASED ON SHORT-CIRCUITED SRRs

Fig. 7. Evaluation of the operation frequency of both resonances when the thickness of the substrate is changed. Crosses correspond to measured prototypes.

Fig. 8. Evaluation of the operation frequency of both resonances when the relative permittivity of the substrate is changed. Crosses correspond with measured prototypes.

substrate thickness affects PIFA antennas due to the similar grounded connection. Figs. 9 and 10 show the influences of the gap of the splits in the rings and the width of the strips . In studying the latter, the outer radii of both rings remain fixed as the inner ones are varied. Modifications of both parameters introduce changes in the series capacitances as well as the shunt inductance. Thus, variations in the operation frequency are obtained. However, these variations are not as strong as the ones arising from the other parameters studied before. Consequently, the gap size and strip width are ineffective parameters for adjusting the operation frequency through a wide range, but they can help in fine tuning the band in a small range if it is required. Future works on this fact open the possibilities of broadband and reconfigurable antennas by connecting lumped elements in between the gaps. Another aspect to be investigated are the positions of the grounded vias attached to the rings. Thus far, it has been

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Fig. 9. Evaluation of the operation frequency of both resonances when the gap c of the outer and inner rings are changed. Crosses correspond to measured prototypes.

Fig. 10. Evaluation of the operation frequency of both resonances when the widths d of the outer and inner rings are changed. Crosses correspond to measured prototypes.

assumed that the grounding via of the external ring is situated above the gap, whereas that of the internal ring is located below its gap (as depicted in Fig. 1). However, it is also possible to place the pins of both inner and outer rings simultaneously above or below their respective gaps. This modification of the pin location produces a change in the operation frequency of the bands as can be seen in Fig. 11, which shows a comparison between two designs: one being the previous placement in opposite sense, and the other with both pins placed above the gaps of the rings which they are grounding. When the placement of both grounded vias are of the same sense, i.e., above the gap, the operation frequency of the lower band decreases whereas that of the upper band increases. As a result, the bands get more separated. Therefore, the location of the pins provides an alternative for more flexibility in the design of the bands. Finally, the distance between the SSRs and the microstrip line ( in Fig. 1), must be small in order to excite the structure. This separation changes the matching of the antenna but the

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Fig. 11. Influence of the position of the grounded vias in the external and inner rings.

Fig. 12. Scheme of a multilayer configuration of the antenna, with the feeding located in a middle substrate.

frequency of operation remains almost unaffected. The dimensions of the microstrip line ( and ) will affect the matching can be approximately deas well. Particularly, the width termined from classical microstrip line theories, for obtaining 50 , although smaller values can be used for low permittivity materials with large thicknesses. V. PROTOTYPE To conclude, a particular example of fabricated prototype will be studied in terms of return losses, gain and radiation efficiency. To this aim an improved feeding based on a multilayer structure is employed for obtaining better matching and more compactness. The microstrip feed line is introduced in a middle slab below the radiant elements which are located in the top layer. The scheme of this new antenna is illustrated in Fig. 12. The selected dimensions for this example are as follow: mm, mm, mm and mm; for two with 3 mm thickness of the lower substrates of PP one and 3 mm thickness of the upper one. The ground plane size is 22 mm 17 mm. A photo of the manufactured prototype is shown in Fig. 13. The simulated and measured return losses are shown in Fig. 14. As seen, the antenna has two radiation bands arising from the two rings as explained in the previous section. The simulations approximate properly the operation frequency of the modes. The antenna is not very directive since its size is small (0.05 0.05 at the lower frequency and 0.08 0.08 at the higher one, not including the ground plane). It is quite compact if we consider that the antenna is made of PP and it could be further reduced by using a higher permittivity material but at the price of a lower radiation efficiency. Consequently, it depends on the requirements of the given application. The total efficiency measured in a reverberation chamber1 has a value of 0.5 for the lower frequency (1.24 GHz) and 0.8 for the higher frequency (1.96 GHz). The maximum gain measured 1It must be noted that inaccuracies in these measurements can be due to the effect of the cable connected to the antenna that can contribute to radiation.

Fig. 13. Photograph of the manufactured prototype.

Fig. 14. Measured and simulated s for an antenna with the following dimensions: R : mm, r mm, c : mm and d mm; for a substrate : with 3 mm thickness of the lower one and 3 mm thickness of of PP  the upper one.

=65 ( = 2 2)

=5

=22

=1

in an anechoic chamber was dBi for the lower band of operation and 1 dBi for the higher one.

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VI. DISCUSSION

Fig. 15. Measured radiation pattern at 1.24 GHz and 1.96 GHz, where respectively are the operation bands of the antenna presented in Section V. (a) Plane  (1.24 GHz), (b) Plane  (1.24 GHz), (c) Plane  (1.96 GHz), (d) Plane  (1.96 GHz).

= 90

=0

=0

= 90

Considering that the relative 3 dB bandwidth of the antenna is 6.3% for the higher operation band, and 0.9% for the lower band; the Q factor of both bands is 15.82 and 112.94, respectively. The practical limitation of antennas in terms of size, bandwidth and efficiency comes from the Chu Limit [29], that has been reviewed by some authors [30], [31]. The lower bound of Q factor has the following expression: (1) where is the radiation efficiency of the antenna. For the evaluation of this limit, the size of the antenna is expressed in terms (where , and is the minimum radius of of a sphere in which the antenna including its ground plane can be contained). Therefore, according to the Chu Limit, for two antennas with same electrical size the one which provides the lowest band of operation can achieve smaller values of radiation efficiency (for the same bandwidth). In our case, mm, since it is defined not only by the diagonal of the planar antenna but also by its thickness.2 For the lower band this term is and for the upper one . Thereequal to and fore, for the lower band, the lower bound is the Q factor of the antenna is almost 10 times this bound, whilst and the Q for the upper band, this lower bound is factor of the antenna is 3 times the lower bound. As reported in [31], planar antennas cannot approach this lower bound. Finally, Fig. 15 shows the measured radiation patterns for the operation bands in an anechoic chamber. The component is seen to be stronger for most observation directions, although the component is generally non-negligible. 2

a=

height + width 2 2

+ length 2

:

After the complete analysis of the proposed antenna, we need to point out what are the main advantages/differences of this design when compared to the many antennas that can be found in the literature with similar properties. Although there had been reported prior studies on PIFA related types of antennas, there still remain major differences between those works and the present one. The following is a comparison with some of the most relevant works: In the paper of [32], the antenna there was claimed to have efficiencies of about 40% and 62% in the lower and upper bands respectively, as compared to 50% and 80% achieved by the present design. Although higher gains were obtained in this [32] and dB in (1.62 and 2.13 dB in the bands, compared to this work), one has to consider the considerably larger ground 0.24 at 900 MHz) there as compared to (0.09 plane (0.76 0.07 at 1.24 GHz) here. The present concentric twin-SRR configuration is far more compact, not taking up space of other displaced locations. Each of the two rings serves one band, a mechanism that is inherently different from that of [32]. Another limitation of the antenna there is its non-coplanar nature as opposed to the herein clean structure which has a gap that allows the use of capacitive lumped elements for tuning or reconfigurability purposes. The work in [33] also dealt with miniaturized dual-band PIFA antennas. Although a dielectric slab with higher permittivity versus 2.2 here), the antenna which we was used there ( propose is still electrically smaller. Besides, that paper made use of a larger ground plane, which moreover, is not located below the metallization, thus not qualifying the configuration there as a patch-type antenna. Considering the fact that the gain in [33] is in the same order as what we have achieved, it is implied that the efficiency there is lower since the directivity is higher. As a consequence, the radiation patterns which we obtain are more omni-directional than those in that work. Nonetheless, while the present design appears to outperform the one in [33], the antennas are after all of different topologies and a completely fair comparison may not be possible. Amongst the most cited papers on PIFA is that of [34], being one of the first reports on dual-band PIFAs. As interesting and pioneering a work as it was during that time, that work however, though understandably, did not venture into the study of stretching this antenna to its limits. Firstly, the PIFA size there is large, being 0.156 along the largest dimension at the lower band, as opposed by a corresponding value of 0.05 in the present paper. The bandwidth there, as in this paper, is modest due to the thin substrate; nonetheless bandwidths in both works are on par with each other. No information about the gain or directivity was provided there though, whereas we have presented such results here. In addition, the radiation patterns of our design are comparable to those in that [34] (compare Fig. 5 there with Fig. 15 here), thus strengthening the credibility of the present work. Another related work may be found in [35], in which a bigger ground plane than the one considered here was used. Even discounting this, the size of that PIFA itself is still larger. Moreover, the radiation patterns there are less omni-directional than

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those of Fig. 15. The PIFA considered in [36] is also large when compared to the present design, despite not even considering the ground plane. However, for the upper band, the design of that work provides a wider bandwidth than conventional PIFAs. Furthermore, the gain there is higher than ours and most other PIFAs, although this is attributable to the lager size of that antenna. Most of the compact PIFAs that can be found in literature had been designed for integration into a mobile phone. This means that a large ground plane might have to be used, which of course contributes to the radiation and also affects the frequencies of operation. As a radical breakout, the presently proposed SRR-type dual-band PIFA antenna could offer itself to other applications, such as implantable antennas, medical sensors, or devices where the frequency bands of operation may be either close or far away from one another, thereby becoming more flexible, instead of the rigid 0.9 GHz and 1.8 GHz as considered by most papers. Another promising advancement of our configuration is its great potential of being made reconfigurable by using active devices. We have established a theoretical equivalent-circuit modeling of our antenna which produces simulated results that are in strong agreement with experimental ones. This adds theoretical radiance to the paper and offers insights into the circuital mechanisms that are in play, something that is lacking in other works. Finally, also amiss from a vast majority of existing papers on PIFAs, a thorough parametric study is herein conducted, which provides understanding of the various factors that influence the antenna properties, thereby serving as important design guidelines. These reasons are in addition to the numerous advantages which the presently proposed antenna has over those studied by existent literature—particularly, in terms of antenna miniaturization, smallness in ground-plane size, higher efficiencies, better omnidirectionality, simplicity of design, manufacture and implementation, and at many occasions, even with regards to gain and bandwidth performances. VII. CONCLUSION In this paper, a strongly miniaturized microstrip patch antenna based on short-circuited SRRs has been presented and studied. In virtue of its two concentric rings, this antenna is able to provide two simultaneous bands of operation. A parametric study of the operation frequency has been performed, showing how it is influenced by the dimensions of the rings. Results have shown that the total size of the rings and the permittivity of the dielectric substrate have a significant effect on these resonances. In addition, other parameters such as the widths or gaps of the rings are relevant but in lower order of magnitude. The thickness of the substrate is a very important parameter, since it can strongly modify the operation frequency. The grounded pins are key parameters of this design, as they are responsible for the antenna compactness. In order to demonstrate the operation of the antenna, simulations were carried out and prototypes were manufactured and measured to validate them. A final prototype was manufactured to show in detail the characteristics of the antennas such as gain, bandwidth and radiation efficiency, the latter being measured in a reverberation chamber. The results are very promising and

some designs using this unit cell as the main radiator could be useful in realistic scenarios where compactness is required, such as for instance in RFID applications or integrated body communications where the employed antennas, according to the literature, have similar or even lower radiation efficiencies for the same antenna sizes. Moreover, by adding rings, further multiplicity of bands can be achieved. The concept of a multifunctional but yet light and compact antenna has thus been mooted. Needless to say, such antennas find vast applications in the mobile wireless age of today. The operation frequency of the antenna bands are quite independent of each other. However, in principle, there are two limitations: the minimum difference between both operation frequencies is limited by the minimum distance between rings; and the maximum difference is limited by the maximum distance between rings that allows the coupling between them. However, even then, other parameters such as the gap of the rings (that can be loaded with lumped capacitors) or the ring widths, can provide more flexibility depending on the requirements of the application. ACKNOWLEDGMENT The authors would like to thank the Antenna Group of Chalmers University of Technology for supporting the measurements of the antennas in the anechoic and reverberation chambers, especially E. Pucci for her help. Also, they want to thank C. J. Sánchez-Fernández for manufacturing the prototypes. Finally, they would like to thank Dr. S. Best for his support and discussion about the estimation of the Chu Limit. REFERENCES [1] A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials, vol. 56, pp. 2–11, Feb. 2007. [2] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Soviet Phys. Uspekhi, vol. 10, pp. 509–514, Feb. 1968. [3] J. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permitivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [5] R. Marques, F. Martin, and M. Sorolla, Metamaterials With Negative Parameters: Theory, Design, and Microwave Applications, ser. Series in Microwave and Optical Engineering. Hoboken, NJ: Wiley, 2008. [6] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, pp. 110–119, Jan. 2005. [7] J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márques-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes-theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1506–1514, Apr. 2005. [8] E. Rajo-Iglesias, O. Quevedo-Teruel, and M. Ng Mou Kehn, “Multiband SRR loaded rectangular waveguide,” IEEE Trans. Antennas Propag., vol. 57, pp. 1571–1575, May 2009. [9] O. Quevedo-Teruel, E. Rajo-Iglesias, and M. Ng Mou Kehn, “Numerical and experimental studies of split ring resonators loaded on the sidewalls of rectangular waveguides,” IET Microw., Antennas Propag., vol. 3, pp. 1262–1270, Dec. 2009. [10] J. Carbonell, L. J. Roglá, V. E. Boria, and D. Lippens, “Design and experimental verification of backward-wave propagation in periodic waveguide structures,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1527–1533, Jun. 2006.

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[11] Y. Zhang, W. Hong, C. Yu, Z.-Q. Kuai, Y.-D. Don, and J.-Y. Zhou, “Planar ultrawideband antennas with multiple notched bands based on etched slots on the patch and/or split ring resonators on the feed line,” IEEE Trans. Antennas Propag., vol. 56, pp. 3063–3068, Sept. 2008. [12] J.-G. Lee and J.-H. Lee, “Suppression of spurious radiations of patch antenna using split ring resonators (SRRs),” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2005, vol. 2B, pp. 242–245. [13] J. Ding, “A harmonic suppression antenna using split ring resonators coupled with microstrip line,” presented at the 7th Int. Symp. on Antennas Propag. and EM Theory, Oct. 2006. [14] A. Ali, M. A. Khan, and Z. Hu, “Applying negative permittivity media to microstrip patch antennas for harmonic suppression,” in Proc. Loughborough Antennas Propag. Conf., Apr. 2007, pp. 245–248. [15] K. D. Jang, J. H. Kim, D. H. Lee, G. H. Kim, W. M. Seong, and W. S. Park, “A small CRLH-TL metamaterial antenna with a magnetodielectric material,” presented at the IEEE Antennas Propag. Society Int. Symp., Jul. 2008. [16] I. K. Kim and V. V. Varadan, “Microstrip patch antenna on LTCC metamaterial substrates in millimeter wave bands,” presented at the IEEE Antennas Propag. Society Int. Symp., Jul. 2008. [17] M. Kärkkäinen and P. Ikonen, “Patch antennas with stacked split-ring resonators as an artificial magneto-dielectric substrate,” Microw. Opt. Technol. Lett., vol. 46, pp. 554–556, Sep. 2005. [18] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 1654–1661, Jun. 2006. [19] S. N. Burokur, M. Latrach, and S. Toutain, “Theoretical investigation of a circular patch antenna in the presence of a left-handed medium,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 183–186, 2005. [20] S.-G. Mao and S.-L. Chen, “Characterization and modeling of lefthanded microstrip lines with application to loop antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1084–1091, Apr. 2006. [21] R. K. Baeel, G. Dadashzadehl, and F. G. Kharakhilil, “Using of CSRR and its equivalent circuit model in size reduction of microstrip antenna,” presented at the Asia-Pacific Microwave Conf., Dec. 2007. [22] M. Palandoken, A. Grede, and H. Henke, “Broadband microstrip antenna with left-handed metamaterials,” IEEE Trans. Antennas Propag., vol. 57, pp. 331–338, Feb. 2009. [23] Y. Zhou, C.-C. Chen, and J. L. Volakis, “Dual band proximity-fed stacked patch antenna for tri-band gps applications,” IEEE Trans. Antennas Propag., vol. 55, pp. 220–223, Jan. 2007. [24] A. Khaleghi, “Dual band meander line antenna for wireless LAN communication,” IEEE Trans. Antennas Propag., vol. 55, pp. 1004–1009, Mar. 2007. [25] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006. [26] C. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ: Wiley Interscience, 2005. [27] J. R. James and P. S. Hall, Handbook of Microstrip and Printed Antennas. New York: Wiley, 1997. [28] C. Martin-Pascual, E. Rajo-Iglesias, and V. González-Posadas, “Invited tutorial: ‘Patches: The most versatile radiator?’,” presented at the IASTED Int. Conf. Advanced in Communications, July 2001. [29] L. Chu, “Physical limitations of omnidirectional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [30] R. Ziolkowski and A. Erentok, “At and below the chu limit: Passive and active broad bandwidth metamaterial-based electrically small antennas,” IET Microw. Antennas Propag., vol. 1, pp. 116–128, Feb. 2007. [31] S. R. Best, Mitre Corporation, “Optimization of the bandwidth of electrically small planar antennas,” Tech. Rep., 2009. [32] H. Li, J. Xiong, and S. He, “Extremely compact dual-band PIFAs for MIMO application,” Electron. Lett., vol. 45, pp. 869–870, Aug. 2007. [33] Y.-S. Wang, M.-C. Lee, and S.-J. Chung, “Two PIFA-related miniaturized dual-band antennas,” IEEE Trans. Antennas Propag., vol. 55, pp. 805–811, Mar. 2007. [34] Z. D. Liu, P. S. Hall, and D. Wake, “High isolation proximity coupled multilayer patch antenna for dual-frequency operation,” IEEE Trans. Antennas Propag., vol. 45, pp. 1451–1458, Oct. 1997. [35] G. K. H. Lui and R. D. Murch, “Compact dual-frequency PIFA designs using LC resonators,” IEEE Trans. Antennas Propag., vol. 49, pp. 1016–1019, July 2001. [36] J.-S. Row, “Dual-frequency triangular planar inverted-f antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 874–876, Feb. 2005.

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Oscar Quevedo-Teruel (S’06–M’10) was born in Spain on October 18, 1981. He received the M.Sc. degree in telecommunication engineering and Ph.D. degree from University Carlos III of Madrid, in 2005 and 2010, respectively. Currently, he is with Department of Theoretical Physics of Condensed Matter at Autonomous University of Madrid in Alianza 4 Universidades Framework. His research activity has been focused in optimization techniques applied to Electromagnetism, analysis and design of compact microstrip patch antennas and metamaterials applied to microwave designs. He has (co)authored more than 20 contributions in international journals and more than 30 in international conferences. Dr. Quevedo-Teruel received the Carlos III University of Madrid Award of Excellence 2010 to the “best professional career of former students who obtained the university degree from February 2004 to November 2006,” and the Spanish National Award (Arquimedes) to the “best supervisor of M.Sc. theses in engineering and architecture 2010.”

Malcolm Ng Mou Kehn (S’02–M’06) was born in Singapore on September 26, 1976. He received the B.Eng. (honors) degree from the National University of Singapore, Singapore, in 2001, and the Licentiate and Ph.D. degrees from Chalmers University of Technology, Gothenburg, Sweden, in Feb. 2004 and Dec. 2005, respectively, all in electrical engineering. During 2006–2008, he was a Postdoctoral Fellow in the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada. Following this, he proceeded to Concordia University, Montreal, QC, Canada, for a year. Currently, he is with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan, serving as an Assistant Professor. He had been actively involved in research projects funded by the Swedish Defense Research Agency between 2002 and 2006. In the autumn of 2004, he spent several months at the University of Siena, Italy, for a research visit. Throughout 2006 to 2009, he worked extensively on numerous projects supported by Canadian industry and national research bodies. Dr. Ng Mou Kehn received the Union Radio-Scientifique Internationale (URSI) Young Scientist Award in 2007. In December 2004, he visited the University of Zagreb, Croatia, as an invited speaker where he gave an IEEE lecture, in connection with the IEEE Croatia Chapter activities.

Eva Rajo-Iglesias (SM’08) was born in Monforte de Lemos, Spain, in 1972. She received the Telecommunication Engineering degree from University of Vigo, Vigo, Spain, in 1996 and the Ph.D. degree in telecommunication from University Carlos III of Madrid, Madrid, Spain, in 2002. From 1997 to 2001, she was a Teacher Assistant at the University Carlos III of Madrid. In 2001, she joined the University Polytechnic of Cartagena as Teacher Assistant for a year. She came back to University Carlos III as a Visiting Lecturer in 2002 and since 2004, she is an Associate Professor with the Department of Signal Theory and Communications, University Carlos III of Madrid. After visiting Chalmers University of Technology (Sweden) as a Guest Researcher, during autumn 2004, 2005, 2006, 2007 and 2008, she is, since 2009, an Affiliate Professor in the Antenna Group, Signals and Systems Department. Her main research interests include microstrip patch antennas and arrays, metamaterials and periodic structures and optimization methods applied to Electromagnetism. She has (co)authored more than 35 contributions in international journals and more than 70 in international conferences. Dr. Rajo-Iglesias received the Loughborough Antennas and Propagation Conference (LAPC) 2007 Best Paper Award and “Best Poster Award in the field of Metamaterial Applications in Antennas” sponsored by the IET Antennas and Propagation Network, at Metamaterials 2009: 3rd International Congress on Advanced Electromagnetic Materials in Microwaves and Optics. She presently serves as Associate Editor for the IEEE Antennas and Propagation Magazine and for IEEE Antennas and Wireless Propagation Letters.

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Hierarchical Curl-Conforming Nédélec Elements for Quadrilateral and Brick Cells Roberto D. Graglia, Fellow, IEEE, and Andrew F. Peterson, Fellow, IEEE

Abstract—New curl-conforming hierarchical vector basis functions are developed for quadrilateral and hexahedral cells. These bases are constructed from orthogonal polynomials and are shown to maintain superior linear independence as their polynomial order is increased. The procedure for constructing these basis functions is described and general expressions are provided for an arbitrary order. Index Terms—Basis functions, boundary integral equations, curvilinear geometry, electromagnetic analysis, electromagnetic diffraction, electromagnetic scattering, Galerkin method, high-order modeling, moment methods, numerical analysis.

I. INTRODUCTION ECTOR basis functions are widely used in electromagnetics for volumetric discretizations of the vector Helmholtz equation in 2D and 3D and surface and volume discretizations of the electric and magnetic field integral equations in 3D. To obtain high accuracy numerical results, favorable approaches employ hierarchical bases and implement some form of adaptive refinement. In the following, new hierarchical bases for quadrilateral and hexahedral cells are proposed. These functions span the reduced-gradient curl-conforming spaces of Nédélec [1], which implies that the number of independent vector-valued polynomials that form a th-order complete hierarchical base on a quadrilateral and hexahedral cell is equal and , respectively. to Several basis families with similar properties have been proposed [2]–[6]. However, hierarchical bases often exhibit poor linear independence as the order of the representation is increased, resulting in an ill-conditioned system of equations. The new basis functions are constructed from orthogonal polynomials, by a process similar to that used to generate vector bases for triangles and tetrahedra in [7]–[9], and are shown to alleviate the loss of linear independence. They were first reported in [10] and [11]. The following sections describe the development of the new basis functions, and provide general expressions that can be

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Manuscript received June 29, 2010; revised November 13, 2010; accepted December 28, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported in part by the Italian Ministry of Education, University and Research (MIUR) under the PRIN grant 20097JM7YR: “Detection and Electromagnetic signature of hidden objects for transportation hubs and infrastructure security.” R. D. Graglia is with the Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy and is also with the ISMB-Istituto Superiore Mario Boella, 10138 Torino, Italy (e-mail: [email protected]). A. F. Peterson is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158789

used to obtain basis functions of any polynomial degree. Results demonstrate that the bases exhibit a very slow growth in matrix condition number, indicating that they are able to maintain excellent linear independence as their polynomial order increases. Readers may find it helpful to review [12] for a detailed introduction to the notation and other background information. II. EDGE, FACE, AND VOLUME-BASED HIERARCHICAL BASES Interpolatory high-order vector bases on brick and quadrilateral cells are constructed in [12] by multiplying the zeroth-order curl-conforming vector functions with Silvester-Lagrange interpolatory polynomials. Linear combinations of those interpolatory polynomials are used here to obtain generating hierarchical polynomials which are subdivided from the outset into three dif-, face -, and volume-based ferent groups of edge functions. In each group, all the hierarchical polynomials are constructed a priori to be mutually orthogonal independent of the definition domain of the relevant inner product, i.e., either the edge, the face, or the volume of the brick. The hierarchical vector bases are then constructed by using the same technique given in [12], where we simply substitute the new scalar hierarchical polynomials for the interpolatory ones of [12]. Thus, for any given polynomial order, the number of the new hierarchical vector basis functions is equal to the number of the interpolatory basis set given in [12], and the space spanned by these two bases is exactly the same. In this section, the quadrilateral and the brick vector-bases are derived at the same time by considering the quadrilateral cell as bounding face of a brick cell described by its six the [12]. The dummy parent variables are assumed to be independent, with variables

(1) and the parent brick-cell of the parent space is of unitary edge-length [12]. In the cube indicates the parent-edge formed by the inthe following, ; whereas and are tersection of faces and the parent-faces, respectively. Simithe larly, a quadrilateral cell is described by four dummy parent where, once again, we assume variables to be independent, with and given as in (1); recall is the face of the brick that the quadrilateral cell ). In the parent-space, is the square (i.e., of unitary edge-length [12], and is the . parent-edge

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GRAGLIA AND PETERSON: HIERARCHICAL CURL-CONFORMING NÉDÉLEC ELEMENTS FOR QUADRILATERAL AND BRICK CELLS

A. Generating Polynomials For the brick element, interpolatory curl-conforming bases complete to th-order are obtained in [12] by multiplying the , zeroth order curl-conforming vector function associated with the edge formed by the intersection of faces , with the following Silvester-Lagrange interpolatory polynomials

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TABLE I CORRESPONDENCE BETWEEN DUMMY AND PARENT VARIABLES

(2)

For the quadrilateral cell, the zeroth-order vector factor associated with the edge is while the Silvester-Lagrange interpolatory polynomials used to build the th-order complete bases are obtained from (2) by setand , which yield , and ting . The number of vector functions provided by this multiplicative process for the quadrilateral and brick cells is and , respectively. Some of these vector functions are linearly dependent on the others. Using a procedure similar to that discussed in [12] for the interpolatory case, the dependent face and volume-based polynomials from the generating scalar set must be eliminated. For the quadrilateral set, face-based vector functions are linearly dependent. face-based vector functions and For the brick set, volume-based vector functions are dependent and must be eliminated [12]. After removing the redundant functions, the total number of vector bases obtained is and for the quadrilateral and the brick cells, respectively; these are the degrees of freedom necessary to span the Nédélec mixed-order spaces [1], [12]. The dummy parent variable set used to form the polynomials (2) is an appropriate permutation that depends of the brick parent variables (or for the on the associated zeroth-order edge factor quadrilateral cell), as reported in Table I. Notice that vanishes on the two brick faces , while vanishes for , although the polynomials (2) do not vanish for those values. Set (2) was first introduced in [12] because of its interpolatory properties and of its symmetry relations which involve polynomials, that is a number of independent polynomials higher than what is sufficient to build th-order comand plete scalar bases. (In fact, a minimum of polynomials is required for the 2D and the 3D case, respectively.) The total order of each polynomial in set (2) is equal to and for the brick and the quadrilateral cell, respectively, [12]. By using the dependency relations (1) to rewrite (2) in terms of the independent parent variables only, and by considering the index summation rules reported

at bottom of (2), one immediately recognizes that the order of these polynomials is in the , the , and the independent variables. Various th-order complete noninterpolatory sets are easily obtained by linearly combining the polynomials of (2). linearly Similarly, a th-order complete set formed by independent noninterpolatory polynomials is equivalent to (2) if the total order of each of its terms is lower or equal to , and if linear combinations of its terms can produce an interpolatory polynomial set on the same interpolation grid of set (2). More precisely, a polynomial set is equivalent to (2) if its linearly independent polynomials comprise: volume-based polynomials that completely • vanish on the and the faces (these polynomials are equivalent to the interpolating ones listed in (2) for ); face-based polynomials with degrees of • face, and that freedom (DoF) associated with the vanish for (these polynomials are equivalent to the ones listed in (2) for ); face-based polynomials based on the • face and that vanish at ; that is, with DoFs associated face (these are equivalent to those listed in with the ); (2) for • edge-based polynomials based on the edge (these are equivalent to the polynomials listed in (2) that interpolate, for and , the edge ). formed by the intersection of faces and if the order of the equivalent polynomials, when written in terms of their independent parent variables only, is lower or equal to in the , the , and the variables; this implies that the total order of each polynomial of the equivalent set cannot be

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higher than (or for the quadrilateral cell). Each equivalent set, including (2), takes the form

which, on

and

simplifies as follows:

(11) (3) and appearing in (3) are different Notice that the indices and in from and do not play the same role as the indices (2). In general, the polynomials of the equivalent sets (3) are not interpolatory and usually have inhomogeneous form [12]. As said, we are only interested in equivalent hierarchical polynomial sets. Set (3) is hierarchical if (zeroth-order) the set contains only the edge• for ; based polynomial , the th-order complete set is obtained by incre• for with the menting the set of order polynomials with subscripts and given by the three double-nested loops (4), (5), and (6)

For these weight functions, the generating hierarchical polynomials turn out to be naturally expressed in terms of the Jacobi , with or , and where is the polynomials polynomial degree. These Jacobi polynomials are either even . In or odd functions of , with associated with the simparticular, the polynomials are equal to the Legendre polyplified weight function . The Legendre and Jacobi polynomials used here nomials are easily obtained from the recurrence relations reported in the Appendix. The normalized hierarchical polynomial set equivalent to set (2) obtained in this manner consists of the following polynomials:

(4) (12)

(5) (6) In the following, the hierarchical polynomials veniently written in terms of the variables

are con-

(7)

which, because of (1), read (8)

B. Volume-, Face-, and Edge-Based Orthogonal Polynomials We now form each group of volume-, face-, and edge-based hierarchical polynomials with mutually orthogonal polynomials. Obviously, different choices of the weight function used to define the orthogonality will produce different polynomial sets. For the brick cell, all the higher-order vector functions obtained in this manner take the form

(9) and are, therefore, conveniently obtained by using scalar generating polynomials orthogonal with respect to the weight function (10)

for in (7), and where

; with

given

(13)

indicates the rescaled Jacobi polynomial of order . The polyassociated with the face (that nomials face) are simply obtained from the polyvanish on the associated with the face by changing to nomials to , and to . As far as the quadrilateral element is concerned, recall that the zeroth-order vector factor associated with the edge is . In this case, it is convenient to make the polynomials mutually orthogonal with respect to the weight function ; it is then readily proved that the polynomials to be used for the quadrilateral element are the polynoand already given in (12). mials Notice also that the order of the hierarchical polynomials in (12) equals the sum of the subscripts used to denote them. To facilitate numerical implementations and code debugging, the edge-, face-, and volume-based polynomials up to the third order are explicitly reported in Table II. Equation (8), together with (31) and (32) of the Appendix, readily prove that, within each group, the polynomials are mutually orthogonal with respect to the weight functions (10). In particular: • the volume-based polynomials are orthogonal over the brick parent-cell ;

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TABLE II EDGE-, FACE-, AND VOLUME-BASED HIERARCHICAL POLYNOMIALS

• the face-based polynomials are orthogonal over , as well and ; as over the two quadrilateral faces • the edge-based polynomials are orthogonal over , over and attached to the both the faces edge, and also over the edge .

In general, however, a polynomial of one group is not orthogonal to those of a different group. Table III reports the inner products of the hierarchical polynomials in matrix form; the normalization used for the polynomials of each group is reported along the main diagonal of the matrices of Table III.

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TABLE III INNER PRODUCTS OF THE HIERARCHICAL POLYNOMIALS OF TABLE II

C. Symmetry Relations of the Hierarchical Bases Used to Guarantee Tangential Continuity

quadrilateral cell

, or associated with the one has

edge of a

For brick cells, the high-order vector functions associated with the face oriented in the direction are [see (9) and (12)]

(18) (19)

(14) with (15) (16) In (14), the zeroth-order factors used to construct the higher order vector functions are reported in square brackets. Clearly, (16) shows that to secure basis independence one has to discard either or from the basis set [12]. Furtherand , one more, since has (17) Equations (16) and (17) simply show that to impose tangential continuity across adjacent elements that share the same quadrilateral face it is sufficient to adjust the sign of the face-based vector functions in use in one of the two adjacent cells. Similarly, for the higher-order vector functions associated with the edge formed by the intersection of the brick faces

Once again, (18) and (19) show that to secure tangential continuity across adjacent elements that share the same edge it is sufficient to adjust the sign of the edge-based vector functions in use in one of the two adjacent cells. In connection with the tangential continuity issue, it important to note that the hierarchical edge-based functions provided in the present paper match those given in [7]–[9] and, along a given common edge, are normalized in the same manner. This permits one to mesh two-dimensional domains with a mixture of triangular and quadrilateral cells, and to use meshes with a mixture of brick and tetrahedral cells in volumetric regions. However, to mesh volumetric regions with elements of different shape, one needs to develop hierarchical vector bases for the triangular prism and possibly the pyramid; these bases will be considered in future papers. D. Equivalent Polynomial Bases With Improved Linear Independence In general, as discussed in Sub-Section II.B and apparent from the Table III results, a polynomial of the edge-, face-, or volume-based group is not orthogonal to the polynomials of a different group. However, appropriate linear combinations of the volume-based polynomials can be added to any given face-based polynomial to obtain a new face-based polynomial orthogonal to the volume based-ones. The same holds for any

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TABLE IV

INDIVIDUAL ELEMENT

T-MATRIX CONDITION NUMBERS FOR 2D AND 3D CELLS OF UNITARY EDGE-LENGTH

given edge-based polynomial, which can be linearly combined with its nonorthogonal face and volume-based polynomials to get a new edge-based polynomial orthogonal to the face and volume-based ones. Linear combinations as such are driven by the analytical results of Table III. , one can obtain a th-order complete set Thus, for with by incrementing the set of order mutually orthogonal polynomials different from those again given by (4), (5), of Table II, but with subscripts and (6). Although the elements of the new hierarchical family order are not completely obtained in this manner for the orthogonal to the elements added to complete this base to th-order, the linear independence of the elements of the new th-order basis is improved relative to the bases given in Table II. This further orthogonalization process produces the equivalent polynomial set

and subscripts. The normalizathe sum of the relevant tion constants used for (20) were chosen to improve the linear independence of the polynomials, established by studying (and lowering) the condition numbers of the Gram matrices having equal to the inner product (over and ) coefficients of the th and th polynomials of the new pth-order-complete family. These polynomials are thus normalized as follows:

(23) that is, the new edge-based polynomials remain normalized to a unitary inner product over their primary edge and the new face-based polynomials remain normalized to a inner product equal to 1/3 over their associated face, as do the polynomials of Table II. III. NUMERICAL RESULTS

(20)

The relative performance of the hierarchical basis functions described above can be evaluated by considering the individual element matrix having entries of the form

that is simply given in terms of the polynomials of Table II, corrected by factors that involve the rescaled Jacobi polynomial

(24)

(21) of order , with (22) are easily obtained by using the recurThe polynomials rence formulas of the Appendix. The volume-based polynomials are obtained by multiplying those of Table II by 3/4 while, in general, the order of the edge- and face-based polynomials in (20) is higher than

is the volume (or surface) of a given 3D (or 2D) cell, where indicates the th vector basis function. The -maand where measures the degree of linear trix condition number independence of the basis functions, and provides an indication of the performance of the basis functions in numerical applications [7]–[9]. Table IV presents element matrix condition numbers for vector bases of increasing order on square and cubic reference cells of unitary edge-length. The proposed hierarchical bases are compared to the interpolatory bases of [12]. The CNH and data are relative to the hierarchical bases of Table II and

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to the hierarchical bases of Section II-D, respectively. The CNI data obtained with the interpolatory basis functions of [12] are reported for reference. The hierarchical condition numbers clearly grow at a much slower rate as their order increases than those of the interpolatory set. It is interesting to observe that the bases of Table II and the alternative bases of Section II-D yield the same condition numbers for a 2D square cell while, for a cubic cell, the alternative bases reduce the condition numbers given by the Table II bases by a factor of two. This suggests that the Table II bases are almost optimal. (In practice, it is not convenient to use the alternative bases of Subsection II.D because the global polynomial-order of each of their terms is higher than that of the equivalent term of the Table II bases.) Preliminary numerical results were reported for skewed quadrilateral-cell bases in [10] and [13], which compared the global -matrix condition numbers arising from basis families [2]–[5], and [6] to those of the proposed functions. Results showed that of the previous families, only the functions of [5] were comparable to the condition numbers of the new set; the other families exhibited higher condition numbers.

In Section II, we use the following two systems of rescaled Jacobi polynomials: (29) (30) that satisfy the following orthogonality relations, obtained from [see (8)]: [14] after setting (31) (32) is the Kronecker delta. Equations (31), (32) together where with (8) yield the inner product results of Table III. used in Section II-D The rescaled Jacobi polynomials [see (21)] are obtained from the lowest order ones (33)

IV. CONCLUSION A new hierarchical family of curl-conforming vector basis functions has been proposed for quadrilateral and hexahedral cells. These functions are constructed from orthogonal polynomials and exhibit excellent linear independence as their polynomial order increases. The process by which they were obtained is described, and preliminary numerical results are discussed.

by use of the following recurrence relations with respect to the degree : (34) with

(35) APPENDIX I AUXILIARY LEGENDRE AND JACOBI POLYNOMIALS The polynomial families used in Section II to build hierarchical vector bases on quadrilateral and brick cells are given , with in terms of the Jacobi polynomials or , and where is the polynomial degree. These polynomials are orthogonal with respect to the weight function on the interval , and are either even . or odd functions of , with In particular, the polynomials are associated with the and are therefore equal to simplified weight function . For , the Jacobi polythe Legendre polynomials nomials are obtained from the lowest order ones (25) (26) by use of the following recurrence relations with respect to the degree : (27) (28)

REFERENCES [1] J. C. Nédélec, “Mixed finite elements in R3,” Numer. Math., vol. 35, pp. 315–341, 1980. [2] M. Ainsworth and J. Coyle, “Hierarchic hp-edge element families for Maxwells equations on hybrid quadrilateral/triangular meshes,” Comput. Methods in App. Mechan. Eng., vol. 190, pp. 6709–6733, 2001. [3] M. M. Ilic and B. M. Notaros, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 1026–1033, Mar. 2003. [4] M. Djordjevic and B. M. Notaros, “Higher-order hierarchical basis functions with improved orthogonality properties for moment-method modeling of metallic and dielectric microwave structures,” Microw. Opt. Technol. Lett., vol. 37, no. 2, pp. 83–88, Apr. 2003. [5] E. Jorgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag., vol. 52, pp. 2985–2995, Nov. 2004. [6] S. Zaglmayr, “High order finite element methods for electromagnetic field computation,” Ph.D. dissertation, Johannes Kepler Univ., Linz, Austria, Jul. 2006. [7] R. D. Graglia, A. F. Peterson, and F. P. Andriulli, “Hierarchical polynomials and vector elements for finite methods,” in Proc. Int. Conf. Electromagn. Adv. Appl. (ICEAA 2009), Torino, Italy, Sep. 2009, vol. 1, pp. 1086–1089. [8] A. F. Peterson and R. D. Graglia, “Scale factors and matrix conditioning associated with triangular-cell hierarchical vector basis functions,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 40–43, 2010. [9] R. D. Graglia, A. F. Peterson, and F. P. Andriulli, “Curl-conforming hierarchical vector bases for triangles and tetrahedra,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 950–959, Mar. 2011.

GRAGLIA AND PETERSON: HIERARCHICAL CURL-CONFORMING NÉDÉLEC ELEMENTS FOR QUADRILATERAL AND BRICK CELLS

[10] A. F. Peterson and R. D. Graglia, “Evaluation and comparison of hierarchical vector basis functions for quadrilateral cells,” presented at the 14th Biennial IEEE Conf. Electromagn. Field Computat., Chicago, IL, May 2010. [11] R. D. Graglia and A. F. Peterson, “Curl-conforming hierarchical vector elements for quadrilateral and brick meshes and their generating orthogonal polynomials,” presented at the CNC/USNC/URSI Radio Sci. Meet., Toronto, ON, Jul. 2010. [12] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics, special issue on “advanced numerical techniques in electromagnetics”,” IEEE Trans. Antennas Propag. , vol. 45, no. 3, pp. 329–342, Mar. 1997. [13] A. F. Peterson and R. D. Graglia, “Evaluation of hierarchical vector basis functions for quadrilateral cells,” IEEE Trans. Magn., vol. 47, no. 5, pp. 1190–1193, May 2011. [14] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1968.

Roberto D. Graglia (S’83–M’83–SM’90–F’98) was born in Turin, Italy, on July 6, 1955. He received the Laurea degree (summa cum laude) in electronic engineering from the Polytechnic of Turin in 1979 and the Ph.D. degree in electrical engineering and computer science from the University of Illinois at Chicago in 1983. From 1980 to 1981, he was a Research Engineer with CSELT, Italy, where he conducted research on microstrip circuits. From 1981 to 1983, he was a Teaching and Research Assistant at the University of Illinois at Chicago. From 1985 to 1992, he was a Researcher with the Italian National Research Council (CNR), where he supervised international research projects. In 1991 and 1993, he was an Associate Visiting Professor with the University of Illinois at Chicago. In 1992, he joined the Department of Electronics, Polytechnic of Turin, as an Associate Professor, and since 1999, a Professor of Electrical Engineering. He has authored more than 150 publications in international scientific journals and symposia proceedings. His areas of interest comprise numerical methods for high- and low-frequency electromagnetics, theoretical and computational aspects of scattering and interactions with complex media, waveguides, antennas, electromagnetic compatibility, and low-frequency phenomena. He has organized and offered several short courses in these areas. Dr. Graglia has been a Member of the Editorial Board of ELECTROMAGNETICS since 1997. He is a Past Associate Editor of the IEEE TRANSACTIONS ON

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ANTENNAS AND PROPAGATION and a Past Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. He is currently an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, and a Reviewer for several international journals. He was the Guest Editor of a special issue on Advanced Numerical Techniques in Electromagnetics for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in March 1997. He has been Invited Convener at URSI General Assemblies for special sessions on Field and Waves in 1996, Electromagnetic Metrology in 1999, and Computational Electromagnetics in 1999. He served the International Union of Radio Science (URSI) for the triennial International Symposia on Electromagnetic Theory as Organizer of the Special Session on Electromagnetic Compatibility in 1998 and was the co-organizer of the special session on Numerical Methods in 2004. Since 1999, he has been the General Chairperson of the International Conference on Electromagnetics in Advanced Applications (ICEAA). He has been a member of the AP-S Administrative Committee for the triennium 2006–2008 and is presently an AP-S Distinguished Lecturer (2009–2012).

Andrew F. Peterson (S’82–M’83–SM’92–F’00) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at UrbanaChampaign, in 1982, 1983, and 1986, respectively. Since 1989, he has been a member of the faculty of the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, where he is now Professor and Associate Chair for the Faculty Development. He teaches electromagnetic field theory and computational electromagnetics, and conducts research in the development of computational techniques for electromagnetic scattering, microwave devices, and electronic packaging applications. He is the principal author of Computational Methods for Electromagnetics (New York: IEEE, 1998). Dr. Peterson is a past recipient of the ONR Graduate Fellowship and the NSF Young Investigator Award. He has served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, as an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, as the General Chair of the 1998 IEEE AP-S International Symposium and URSI/USNC Radio Science Meeting, and as a member of the IEEE AP-S AdCom. He also served for six years as a Director of ACES, and two years as Chair of the IEEE Atlanta Section. He was the President of the IEEE AP-S during 2006. He is a recipient of the IEEE Third Millennium Medal. He is also a Fellow of the Applied Computational Electromagnetics Society (ACES), and a member of the International Union of Radio Scientists (URSI) Commission B, the American Society for Engineering Education, and the American Association of University Professors.

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Compact Dual-Band and Ultrawideband Loop Antennas Mrinal Kanti Mandal, Member, IEEE, and Zhi Ning Chen, Fellow, IEEE

Abstract—Compact planar loop antennas fed by coplanar striplines (CPS) are presented for dual-band and ultrawideband (UWB) operations. The dual-band antenna consists of two resonant radiating parts, namely a strip loop and a slot loop. Based on the dual-band design, a wideband operation can be achieved. Two prototype antennas operating at the 2.4/5 GHz wireless LAN bands with dimensions of 45 2 15 2 0.813 mm3 and for the lower UWB band (3.1–4.8 GHz) with dimensions of 34 2 13.5 2 0.813 mm3 were fabricated on a RO4003 substrate. The measured input reflection loss is lower than 15 dB for the dual-band antenna and 10 dB for the UWB antenna. Both of the antennas show acceptable gain and dipole-like radiation patterns over the operating frequency bands. Index Terms—Coplanar strip line (CPS), loop antenna, uniplanar antenna, ultrawideband (UWB), wireless LAN (WLAN).

I. INTRODUCTION ITH the rapid developments in high-speed wireless communication technology, multiband or wideband printed compact antennas are increasingly desirable as a key component in handheld and small devices. The ultrawideband (UWB) planar monopole antennas fed by microstrip or CPW lines can provide wideband solutions [1]–[5]. However, these monopole-like unbalanced antennas suffer from a strong ground plane effect due to the currents flowing back to the system ground plane. As a result, the impedance and radiation performance strongly depend on the dimensions of feed structure and ground planes. On the other hand, many RF/microwave components such as differential amplifiers, mixers are designed with balanced input/output lines. Thus, a differential wideband antenna fed with a balanced transmission line, such as coplanar stripline (CPS), avoids additional balun for the transition between an unbalanced antenna and balanced components. In this context, the balanced antennas have an attractive advantage [6]–[13]. Loop antennas are one of the popular balanced antennas. Several configurations of slot loop antennas, etched on the ground plane of a microstrip line, to obtain various characteristics such as circular polarization [6], multi-band [7], and dual-band [8] operations have been reported. Reference [7] also used plated

W

Manuscript received July 14, 2010; revised November 16, 2010; accepted December 28, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. M. K. Mandal is with the Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, Canada. Z. N. Chen is with the Institute for Infocom Research, A-STAR, Singapore (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.2011.2158790

Fig. 1. The modified rectangular printed loop antenna.

loops for multi frequency operations. These antennas are excited by an unbalanced feed, usually, by a microstrip line. Balanced antennas acting as a printed dipole [9], [10] and folded dipole [11] were also presented for dual-band operations. However, these antennas have narrow bandwidths, typically a 10-dB impedance matching bandwidth less than 10%. In comparison, antennas using folded patch loops [12], [13] achieved a wide impedance bandwidth, typically more than 50%, although at the cost of fabrication difficulty owing to their three-dimensional configurations. On the other hand, unbalanced CPW fed planar loop antennas for dual-band [14] and wideband [15] operation are simple to fabricate. But, they are associated with the problem of returning ground currents. In [16], a microstrip fed balanced planar antenna by integrating a balun was reported for UWB operation. In this paper, a novel loop antenna combining a resonant strip loop with a slot loop is proposed for a dual-band operation. The slot loop is completely enclosed by the strip loop thus greatly reducing the overall antenna size. The combination is fed by a CPS line so that it avoids the returning ground current effects. The configuration can be easily modified to achieve a UWB operation. Both of the dual-band and UWB designs are experimentally verified. Furthermore, parametric studies are carried out to study the mechanism and provide some design guidelines for other substrate parameters. II. ANTENNA DESIGNS A. Characteristics of Printed Strip Loop Antennas The effective length of the strip loop is taken as one operating wavelength at the fundamental operating frequency . For a loop antenna at the resonant frequencies, most of the field lines lay inside the air above the strips printed on a dielectric substrate. So, an effective loop wavelength is close to that in air. Fig. 1 shows a modified rectangular loop of width and length , fed by a CPS line. This modification provides extra degrees of freedom to simultaneously control the operating frequency and the input impedance. The width of each

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Fig. 2. Equivalent circuit of the (a) rectangular loop antenna and (b) the same in odd-mode excitation.

strip of the CPS feed line is 0.8 mm and separated by a gap of 0.1 mm. It corresponds to a characteristic impedance of 100 on a 0.813 mm thick Roger 4003 substrate with a dielectric con. The loop behaves like a folded dipole if the stant of is small, typically less than 0.1 times the operinner length ating wavelength. In that case, the higher harmonics are repeated is long enough so at each odd multiple of . In this paper, that the loop mode is always excited. An approximate equivalent circuit of the rectangular loop is shown in Fig. 2(a). Each side of the loop is approximated by a transmission line section. The equivalent transmission line impedance increases with the decreasing widths of the loop represents half of the feeding side with sides. Here, and width . and represent the length and widths , respectively. sides with lengths stands for total loss including the radiation loss. The equivalent circuit in the odd-mode excitation is shown in Fig. 2(b). Asfor simplicity. Then input sume impedance from the feeding end is

(1)

Fig. 3. Variation of f with (a) L (L

= 20 mm) and (b) L

(L

= 10 mm).

If the electrical length is at the fundamental operating frequency , then the resonance condition at an arbitrary frequency by setting the input impedance to zero is (2) Full-wave simulations are carried out using IE3D to observe the dependence of the antenna characteristics on the widths and . The inner dimensions are fixed as mm, mm. As expected from (1), the operating frequency increases with increasing and and when the other parameters kept decreases with increasing unchanged. It was found that the impedance matching at the mainly depends on the arm widths and . However, the variation of of the arm opposite to the feed point has a minimum effect on shifting the . So, the impedance matching . is achieved mainly by tuning ( mm) The variation of with the inner lengths and ( mm) for different and are shown in mm is Fig. 3(a) and (b), respectively. In each case, considered for a good impedance matching. As expected from increases with decrease in and in(2), the variation of crease in and . The impedance increases when the width

Fig. 4. The dual-band loop antenna.

decreases. Thus, a compact configuration can be obtained by properly selecting the widths. B. Dual-Band Printed Loop Antennas The proposed antenna configuration for a dual-band operation is shown in Fig. 4. A metallic patch of length and width is inserted symmetrically inside the loop so that the widths of the left-right and top-bottom opposite slots are equal. This configuration forms an additional slot loop confined by the inner patch and outer strip. The slot loop with a circumference of one operating wavelength operates at its resonance frequency . Thus, the overall configuration behaves like a dual-band antenna operating at both and . The inner metallic patch without any ground plane on the other side of the substrate does not support

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Fig. 5. Average and vector current distributions on the loop with and without the inner patch.

a conventional half-wavelength resonant mode. Also the patch length is much smaller than one half-wavelength at . So the arrangement is different from the cases where patches are used as parasitic elements to obtain a dual-band operation or to tune the operating frequency [10], [17], [18]. Due to its different operating principle, the present arrangement enables a wider variation of and compared to the cases where patches are used as a parasitic element. Moreover, the current distributions on the loop antenna with/without the inner patch, shown in Fig. 5, show the excitation of the slot modes at . The antenna dimensions mm, mm, mm, mm, are mm, mm and mm. Without the patch, is 2.63 GHz. However, when the patch is inserted, the coupling between the outer loop and the inner patch results in concentrating more current along the inner edge of the outer loop at . As a result, the effective current path of the original increases to 2.7 GHz. The metallic loop is reduced so that coupling strength depends on the slot widths, i.e., the gap between the inner patch and the outer loop. The corresponding is 4.43 GHz. The slot loop dimensions depend on both of the outer metallic loop and the inner patch. Simulations are carried out to investion the outer loop parameters. Fig. 6 gate the dependence of and for shows the variation of the resonance frequencies and . The outer dimensions of the loop different and are kept fixed as 40 and 13.5 mm, results in higher for the decreased respectively. A larger and , the dependence of the input slot length. At both is minimal. On the other hand, the variation impedance on of is small even if and are changed by ten times their original values. However, the input impedance at greatly deand . So, can be controlled by changing pends on while impedance matching can be obtained by tuning and . The plots show that the variation of due to is negli. So, is used to tune gible in comparison to that due to the input impedance at . and on the Fig. 7 shows the effects of widths impedance matching and resonant frequencies. Narrowing the or by increasing the lengths or , respecslot width increases. tively, makes the coupling stronger. As a result, is less than 5% unless or is Usually, the change of

Fig. 6. Input jS j variation for different (a) W , (b) W , and (c) W for the double loop of Fig. 4.

too small, typically less than 0.5 mm. A smaller value of or with fixed outer dimensions also increases the effective slot length. As a result, decreases. However, this approach of tuning the frequencies is avoided because it simultaneously and the input matching levels. changes In general, the design steps for the dual-band antenna are as follows: 1) Select the outer loop length for from the plots shown in Fig. 3. For other substrate parameters similar plots can be obtained by any full wave simulator.

MANDAL AND CHEN: COMPACT DUAL-BAND AND UWB LOOP ANTENNAS

Fig. 7. Input jS

j

for different width W and W for the double loop.

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Fig. 8. Measured measurements.

S -parameters of a back to back balun used in the antenna

2) Select the inner patch dimension so that the mid slot length [19]. is one wavelength at mainly depends on the width 3) The frequency ratio . So, fine tune for desired and . 4) Tune to obtain good impedance matching at and . with If the frequencies are changed a little, fine tune to bring them back to previous value. fixed for better impedance matching. 5) Fine tune The plots in Fig. 6 suggest that a good impedance matching over the frequency range from to can be obtained by tuning and . Thus a wideband operation can be realized. It is observed by simulation that a 10-dB input matching bandwidth up to 95% can be obtained on the present substrate by tuning the width values. III. FABRICATION AND MEASUREMENTS Two prototype compact loop antennas were designed for the 2.4/5 GHz wireless LAN bands (Antenna A) and the 3.1–4.85 GHz lower UWB band (Antenna B) on a 0.813-mm-thick RO4003 substrate with a dielectric con. The frequencies are selected as stant of and GHz for Antenna A and GHz and GHz for Antenna B. The dimensions of Antenna A mm, mm, mm, are mm, and mm. Those for the Antenna B are mm, mm, mm, mm, other dimensions remain the same as those and of antenna A. An Agilent® N5230A vector network analyzer and a novel UWB planar balun [20] are used for the input impedance measurement. Fig. 8 shows that the back-to-back balun has a measured transition loss less than 1.5 dB over a 2–10 GHz frequency band. of Antenna A are The measured and simulated input shown in Fig. 9. The measured impedance matching is better than 15 dB across the frequency bands from 2.4 to 2.67 GHz and 5.17 to 6.17 GHz, which covers and exceeds both of the wireless LAN bands. The measured gain patterns for Antenna A in the three principal planes are shown in Fig. 10. The radiation pattern measurements were carried out inside an anechoic chamber. The radiation patterns are typical ones as those of a strip loop and slot

Fig. 9. (a) Photograph of the dual-band antenna A with the balun and (b) the measured and simulated input jS j of the Antenna.

loop antennas with one-wavelength circumferences at their respective resonance frequencies. They have a null along axis. The boresight gain including the losses due the feed network using the balun and SMA connector is almost consistent over both the wireless LAN bands with a variation between 1 and 2.5 dBi. However, the balun is printed on the same substrate layer as the antennas so that its broadside is in the - plane. As a result, the radiation patterns in the - plane are mostly affected and deviate from the ideal patterns, especially at the higher frequencies. of Antenna B Next, the simulated and measured input are shown in Fig. 11. It shows a 10-dB impedance bandwidth over 2.6 to 4.9 GHz. Fig. 12 shows that the measured dipole-like radiation patterns for Antenna B in the three Cartesian planes is

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frequency end, the slot loop is slightly greater than one wavelength so that the radiation patterns of the antenna starts to deviate from the dipole-like patterns and becomes slightly directive along the axis. IV. CONCLUSION

Fig. 10. Measured gain patterns for Antenna A in (a) x-y , (b) y -z , and (c) z -x planes (dashed lines: horizontal and solid lines: vertical polarization).

In this paper, a uniplanar modified rectangular loop antenna has been presented for a compact design. The antenna has been further modified by combining a strip loop and a slot loop for a dual-band operation. Design guide lines to change the opand and the frequency ratio erating frequencies have been provided. Thus, the antenna can be used for other dual-band operations and the designs can be repeated for other substrate parameters. Moreover, based on the dual-band design, an UWB operation has been obtained by tuning the widths of the feeding side and the side just opposite to it of the outer loop. As examples, two antennas have been designed and fabricated for the 2.4/5 GHz wireless LAN bands as well as 3.1–4.8 GHz lower UWB. The antenna configuration has been simple and easy to fabricate. Compared to a conventional square loop, the dual-band and UWB antennas are compact with 30% and 22% size reductions, respectively. REFERENCES

Fig. 11. Measured and simulated input jS

j

of Antenna B.

Fig. 12. Measured gain patterns for Antenna B in (a) x-y , (b) y -z , and (c) z -x planes (dashed lines: horizontal and solid lines: vertical polarization).

similar to those for Antenna A. The measured boresight gain including the losses due to the balun and the SMA connector varies between 0.5 to 3 dBi over the UWB band. At the higher

[1] T.-G. Ma and S.-J. Wu, “Ultrawideband band-notched folded strip monopole antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 2473–2479, Sep. 2007. [2] D. H. Kwon and Y. Kim, “CPW-fed planar ultrawideband antenna with hexagonal radiating elements,” in Proc. IEEE AP-S Symp. Dig., Jun. 2004, pp. 2947–2950. [3] A. J. Kerkhoff and H. Ling, “Design of a band-notched planar monopole antenna using genetic algorithm optimization,” IEEE Trans. Antennas Propag., vol. 55, pp. 604–610, Mar. 2007. [4] W. Wiesbeck, G. Adamiuk, and C. Sturm, “Basic properties and design principles of UWB antennas,” Proc. IEEE, vol. 97, pp. 372–385, Feb. 2009. [5] Z. N. Chen, T. S. P. See, and X. Qing, “Small printed ultrawideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, pp. 383–388, Feb. 2007. [6] X. Qing and Y. W. M. Chia, “A novel single-feed circular polarized slotted loop antenna,” in Proc. IEEE AP-S Symp. Dig., Jun. 1999, pp. 248–251. [7] H. Nakano, M. Fukasawa, and J. Yamauchi, “Discrete multiloop, modified multiloop, and plate-loop antennas—multifrequency and wideband VSWR characteristics,” IEEE Trans. Antennas Propag., vol. 50, pp. 371–378, Mar. 2002. [8] N. Anantrasirichai, S. Chanoodhorm, J. Nakasuwan, and T. Wakabayashi, “Designing rectangular slot loop antenna for WLAN application,” in Proc. TENCON, 2005, pp. 1–5. [9] Y.-H. Suh and K. Chang, “Low cost microstrip-fed dual frequency printed dipole antenna for wireless communications,” IEE Electron. Lett., vol. 36, no. 14, pp. 1177–1179, Jul. 2000. [10] P. Nepa, G. Manara, S. Mugnaini, G. Tribellini, S. Cioci, G. Albasini, and E. Sacchi, “Differential planar antennas for 2.4/5.2 GHz WLAN applications,” in Proc. IEEE AP-S Symp. Dig., July 2006, pp. 973–976. [11] J. A. Flint and J. C. Vardaxoglou, “Exploitation of nonradiating modes in asymmetric coplanar strip folded dipoles,” IEE Proc.-Microw. Antennas Propag., vol. 151, no. 4, pp. 307–310, Aug. 2004. [12] S. Tanaka, S. Hayashida, H. Morishita, and Y. Atsumi, “Wideband and compact folded loop antenna,” IET Elec. Lett., vol. 41, pp. 45–46, Aug. 2005. [13] D.-H. Kwon and Y. Kim, “Small low-profile loop wideband antennas with unidirectional radiation characteristics,” IEEE Trans. Antennas Propag., vol. 55, pp. 72–77, Jan. 2007. [14] T. A. Denidni, H. Lee, Y. Lim, and Q. Rao, “Wide-band high-efficiency printed loop antenna design for wireless communication systems,” IEEE Trans. Veh. Technol., vol. 54, pp. 873–878, May 2005. [15] M. C. Mukandatimana, T. A. Denidni, and L. Talbi, “Design of a new dual-band CPW-fed slot antenna for ISM applications,” in Proc. Veh. Technol. Conf., 2004, vol. 1, pp. 6–9.

MANDAL AND CHEN: COMPACT DUAL-BAND AND UWB LOOP ANTENNAS

[16] X. N. Low and Z. N. Chen, “A compact planar dipole antenna with ultrawide band performance,” in Proc. IEEE AP-S Symp. Dig., Jul. 2008, pp. 1–4. [17] S.-Y. Suh, A. E. Waltho, L. Krishnamurthy, D. Souza, S. Gupta, H. K. Pan, and V. K. Nair, “A miniaturized dual-band dipole antenna with a modified meander line for laptop computer application in the 2.5 and 5.25 GHz WLAN band,” in Proc. IEEE AP-S Symp. Dig., Jun. 2006, pp. 2617–2620. [18] W.-S. Chen and Y.-H. Yu, “Dual-band printed dipole antenna with parasitic element for WiMAX applications,” IET Elec. Lett., vol. 44, no. 23, pp. 1338–1339, 2008. [19] K. C. Gupta, R. Garg, and I. J. Bhal, Microstrip Lines and Slotlines, 2nd ed. Norwood, MA: Artech House, 1996. [20] M. K. Mandal and Z. N. Chen, “A compact ultrawideband microstrip-to-coplanar stripline transition,” IEEE Micow. Wireless Comp. Lett., to be published.

Mrinal Kanti Mandal (S’06–M’09) was born in West Bengal, India, in 1977. He received the B.Sc. degree (with honors) in physics and the B.Tech. and M.Tech. degrees in radiophysics and electronics from the University of Calcutta, Kolkata, India, in 1998, 2001, and 2003, respectively, and the Ph.D. degree from the Indian Institute of Technology Kharagpur, Kharagpur, in 2008. Since November 2007 to September 2009, he was a Research Fellow with the Institute for Infocom Research, A-STAR, Singapore. He was a Postdoctoral Fellow with the President Kennedy Campus, UQAM, Monreal, Canada, from October 2009 to September 2010. At present, he is a Postdoctoral Fellow with the Polygrames Research Centre, Ecolepolytechnique de Montreal, Canada. He has authored or coauthored more than 25 journal papers of international repute. His current research interests include the design of passive mm-wave components. Dr. Mandal is a reviewer for the IEEE MICROWAVE AND WIRELESS COMPUTER LETTERS and IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the IIT Kharagpur Institute Fellowship for graduate research. He is listed in Marquis’ Who’s Who.

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Zhi Ning Chen (F’10) received the B.Eng., M.Eng., and first Ph.D. degrees from the Institute of Communications Engineering, China, and a second Ph.D. degree from the 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 IBM T. J. Watson Research Center, as an Academic Visitor. Since 1999, he has been with the Institute for Infocomm Research, Singapore. His current appointments are Principal Scientist and Department Head for RF & Optical. He is concurrently holding Adjunct/Guest Professorships with Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University, University of Science and Technology, China, Dalian Maritime University, and the National University of Singapore. He has published 290 journal and conference papers, as well as authored and edited the books Broadband Planar Antennas, UWB Wireless Communication, Antennas for Portable Devices, and Antennas for Base Station in Wireless Communications. He also contributed chapters to the books UWB Antennas and Propagation for Communications, Radar, and Imaging, as well as Antenna Engineering Handbook. He holds 28 granted and filed patents with 21 licensed deals with industry. His current research interest includes applied electromagnetic engineering, RF transmission over biochannels, and antennas for wireless systems, in particular at mmW, submmW, and THz for medical and healthcare applications. Dr. Chen has organized many international technical events as 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. He serves as an IEEE Antennas and Propagation Society Distinguished Lecturer and Associate Editor of the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION.

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Design of the Millimeter-wave Rectangular Dielectric Resonator Antenna Using a Higher-Order Mode Yong-Mei Pan, Kwok Wa Leung, Fellow, IEEE, and Kwai-Man Luk, Fellow, IEEE

Abstract—At millimeter-wave (mm-wave) frequencies, the size of the dielectric resonator antenna (DRA) may be too small to fabricate precisely. To relax the precision problem of fabrication, it is proposed to obtain a larger DRA by designing it with its higherorder mode. In this paper, the rectangular mm-wave slot-fed DRA excited in a higher-order mode is investigated systematically. It is found that when the slot is centrally located beneath the DRA, a mode of the DRA can be excited only when all of the indices , , are odd numbers. The aspect ratio of the DRA that gives a single (higher-order) 11 -mode operation is found. Like the fundamental 111 mode, the higher-order 11 modes have broadside radiation patterns. To validate our results, two DRAs were designed to operate in the higher-order 115 and 119 modes. In each case, the reflection coefficient, radiation pattern, and antenna gain are studied, and reasonable agreement between the measured and simulated results is observed. The effect of fabrication error on the frequency shift of the DRA was also studied. A design rule for minimizing the frequency shift is suggested. The results should be useful for practical designs of the mm-wave DRA. Index Terms—Dielectric resonator antenna, higher-order mode, millimeter-wave antenna, slot coupling.

I. INTRODUCTION ODAY, the carrier frequency of modern communication system has been gradually shifting upward to the millimeter-wave (mm-wave) band to provide a much wider bandwidth, avoid interference with the overcrowded lower frequency spectrum, and have a better penetration through fog and heavy dust. In the past, the mm-wave system was primarily found in the military, but its application has been extended to the civil sector in recent years. As a result, some research efforts have been paid on investigating mm-wave antennas. Examples include the mm-wave microstrip antenna [1] and its reflectarray [2]. However, the metallic and surface wave loss of these antennas may decrease the antenna gain and efficiency considerably at mm-wave frequencies. In contrast, the dielectric resonator antenna (DRA) is simply made of dielectric, whose loss can be made very small even at mm-wave frequencies [3], [4]. Other advantages of the DRA include its small size, light weight, flexible excitation scheme, and relatively wide bandwidth when

T

Manuscript received August 30, 2010; revised December 10, 2010; accepted January 15, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported in part by a Strategic Research grant from the City University of Hong Kong (Project No.: 7008097) and in part by the National Natural Science Foundation of China (Project No.: 60928002). The authors are with the State Key Laboratory of Millimeter Waves and Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR (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.2011.2158962

compared with the microstrip antenna [4], [5]. All these features make the DRA an excellent antenna candidate for mm-wave systems. Although the DRA was originally proposed for mm-wave applications, studies of the DRA have been concentrated in the microwave band for a long time. Recently, some works on the mm-wave DRA have been reported in [6]–[9], where, as usual, the DRAs were excited in their fundamental modes. As a result, their sizes are much smaller than those in the microwave band. For example, a microwave rectangular DRA with a dihas length 24 mm, width 23.5 electric constant of mm, and height 12.3 mm at 3.2 GHz. When the operating frequency increases to 40 GHz, the respective dimensions are substantially reduced to as small as 1.91 mm, 0.635 mm, and 1.91 mm [6]. As a second example, a 2 GHz cylindrical DRA with has a radius of 20 mm and a height of 19 mm, which are greatly reduced to 1.25 mm and 1.1 mm, respectively as the frequency increases to 35 GHz [7]. The size of the DRA becomes even smaller at 94 GHz [8]. Obviously, precise fabrication of the DRA is not easy when the size is so small, leading to a practical realization problem at mm-wave frequencies. In this paper, the DRA is excited in a higher-order mode instead of the fundamental mode to obtain a larger mm-wave DRA. It should be mentioned that although higher-order modes of the DRA were reported [10]–[17], they were mainly used together with the fundamental modes to give either wideband [10]–[15], dualband [16], or circularly polarized [17] DRAs. Only very recently higher-order modes have been individually used to increase the gain of the DRA [18]. To the knowledge of the authors, thus far, however, no studies of using the higher-order mode have been carried out to address the size issue of the mm-wave DRA. The rectangular DRA has 3 dimensions (length, width, and height), providing more degrees of freedom than for the hemispherical and cylindrical counterparts. In this paper, the slotcoupled rectangular DRA excited in its higher order mode is -mode family will be considered for it investigated. Its mode, which has been used includes the fundamental extensively by DRA researchers. It is known that the mode distribution of a rectangular DRA is affected by its aspect ratios. When the dimensions of the DRA are chosen improperly, other modes that are very close to the operating mode may be excited. In general, these degenerate modes have different radiation patterns and may undesirably increase the cross-polarized field or even distort the radiation pattern of the antenna. In this paper, the DRA is carefully designed in such a way that the (higher-order) operating mode is sufficiently isolated from its neighboring modes, giving a good single-mode operation. A systematic study was carried out and

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PAN et al.: DESIGN OF THE MILLIMETER-WAVE RECTANGULAR DRA USING A HIGHER-ORDER MODE

design curves for obtaining a good single-mode operation are presented for the first time. The DRA can be excited by different excitation schemes, such as the coupling aperture, coaxial probe, microstripline, and dielectric image guide. The excitable DRA modes are strongly dependent on the feeding source and position. They also determine the radiation pattern of the DRA [4]. In this paper, the aperture-coupled source [19] is chosen because its feed network can isolate the antenna from the ground plane and, thus, spurious radiation from the feed network can be avoided. Also similar to the slot antenna, the rectangular DRA radiates like a horizontal magnetic dipole when excited by a centrally fed slot. This minimizes the undesirable cross-polarized fields because the radiated fields of the DRA and slot are similar to each other [20]. More importantly, it is easier to obtain a good single-mode opmodes and half of the modes are eration because all modes eliminated when a centrally fed slot is used (only with odd , , can be excited for this configuration). This will be explained in more detail in the next section. The organization of the paper is as follows. In Section II, we will discuss in detail what modes can be excited using a central feeding slot. Their resonance frequencies with different DRA aspect ratios are found. Two design examples of using and modes are presented in the higher-order Section III. For ease of comparison, both of the modes are excited at 24 GHz. In each example, the reflection coefficient, radiation pattern, and antenna gain are measured and simulated, and reasonable agreement between the measured and simulated results was obtained. The surface areas and volumes of the DRAs excited in different higher-order modes are compared in Section IV. In addition, the effect of the dielectric constant on the higher-order mode is also reported. In Section V, the frequency shift of the DRA due to fabrication errors is studied. A design rule for minimizing the shift is suggested. Finally, a conclusion is drawn in Section VI. II. THEORY A. Excitable DRA Modes Fig. 1 shows the configuration of the slot-coupled rectangular DRA with length , width , and height , which was designed at 24 GHz. The y-directed slot is placed beneath the DRA centrally. It is worth mentioning that the microstrip feedline is used here simply because it can be designed and fabricated easily. If a much higher frequency is used, a coplanar waveguide [21] or a dielectric image guide [22] should be used to reduce the feedline loss. By using image theory, an isolated rectangular dielectric resonator with a height of is analyzed. In this paper, the dielectric waveguide model (DWM) [20] is used to find the resonance frequencies of various DRA modes. The characteristic equations of the wavenumbers , and for the mode can be derived using the Marcatili’s approximation technique [23] as follows:

(1) (2)

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Fig. 1. Configuration of the slot-coupled rectangular DRA.

(3) , and are decay constants of the field outside where the DRA, and is the free-space wavenumber that can be determined from . -mode resoBy solving the simultaneous (1)–(3), the nance frequencies of the isolated DRA can be determined. The results of (1), (3) have the same form as for (2), except that in the argument of the arctangent is replaced by . It is -mode characworth mentioning that (1), (3) are actually teristic equations for a dielectric slab waveguide of thicknesses and , respectively, whereas (2) is the -mode characteristic equation for a dielectric slab waveguide of thickness and dielectric constant [23]. It is well known that there are two kinds of -mode solutions for a slab waveguide. For an even-mode solution, the tangential electric field is symmetric about the middle plane that acts as an equivalent magnetic wall. mode to be an This requires the index of the waveguide odd number . An example for is shown in Fig. 2(a), where a symmetric (even-mode) -field distribution can be observed. For the odd-mode solution, the tangential electric field is antisymmetric about the middle plane, which is now an equivalent electric wall. In this case, the mode index is an even number . An example of the antisym. Conmetric -field distribution is shown in Fig. 2(b) for versely, for modes, the middle plane is equivalent electric and magnetic walls for the symmetric and antisymmetric cases, respectively [24]. From this analysis, it can be deduced that for mode, both the and planes are electric a (magnetic) walls when , are odd (even) numbers. However, plane is a magnetic (electric) wall when is an odd the (even) number. modes of the When the ground plane is present, the DRA cannot be excited if is an even number. It is because , which these modes require the -field to be maximum at is contradictory to the boundary condition that the tangential -field should vanish on the ground plane. Therefore, the index must be an odd number. Next, the nature of the indices , is discussed. It is known that the -directed equivalent magnetic current in the slot will excite -directed magnetic fields in the and planes DRA around the slot. It implies that the support the tangential and normal magnetic fields, respectively.

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TABLE I CALCULATED (DWM) AND SIMULATED (HFSS) RESONANCE FREQUENCIES OF VARIOUS DRA MODES

Fig. 4. Design curves of frequency as functions of the ratio a = b.

Fig. 2. Tangential E -field (H -field) distribution in the dielectric slab waveguide. (a) TE (TM ) mode. (b) TE (TM ) mode.

b=d: "

= 10,

to be expected. Other dimensions and dielectric constants of the DRA were tested and similar phenomena were observed. Table I lists the calculated and simulated resonance frequencies of varmodes. It should be mentioned that the DWM does not ious take into account the actual excitation source and the effect of the ground plane. Therefore, it can only be used to calculate the resonance frequency of a mode but cannot tell whether the mode can be excited properly or not. B. Single Higher-Order Mode Operation

Fig. 3. Simulated reflection coefficient of the DRA as a function of frequency: a = 4:8 mm, b = 6:4 mm, d = 3 mm, " = 10, W = 0:5 mm, L = 2:8 mm, W = 0:78 mm, and L = 2:6 mm.

In other words, the and planes are electric and magnetic walls, respectively, therefore both indices , should be odd numbers from the analysis as given above. In conclusion, all of the indices , , must be odd numbers for a centrally fed slot. To verify our analysis, Ansoft HFSS was used to simulate a , , , and . DRA with Fig. 3 shows the simulated reflection coefficient of the DRA. With reference to the figure, the modes with even mode indices , , ) cannot be excited properly, which is (

mode has broadIt is well known that the fundamental side radiation patterns, with the fields being maximum in the . To obtain similar broadside radiaboresight direction modes with , , tion patterns, the higher-order were investigated, where is an integer. The effect of the aspect ratio on the resonance frequency is studied first. Using the DWM model, the resonance frequencies of the first few higher order modes were calculated for different ratios of , with . Fig. 4 shows various mode curves as a function of . The shaded area is the region where a single-mode operation can be obtained. As can be observed from Fig. 4, the resonance frequencies of these modes decrease with an increase . It can also be observed that the -curve intersects of the -curves at different values of , with the intersec, tion points shown as dots in the figure. At these values of mode is a degenerate mode of the mode. As the mentioned before, the mode degeneracy would cause the radiation pattern of the antenna to deteriorate. Therefore, in general, the operating mode should be isolated from its adjacent modes as far as possible. In this paper, the DRA dimensions are designed to provide at least 10% guard bands to obtain a good

PAN et al.: DESIGN OF THE MILLIMETER-WAVE RECTANGULAR DRA USING A HIGHER-ORDER MODE

Fig. 5. Single T E -mode design curves of frequency as functions of the ratio b=d: " . For each set of curves, there are five curves which from left to , 1.5, 1, 0.67, and 0.5. right correspond to the dimension ratios of a=b

= 10

=2

single -mode operation. In other words, it is required that and , , , and denote the resonance frequencies where mode, and the of the lower-order mode, the operating higher-order mode, respectively. Taking this guard band into ac-mode design curves that determine the DRA dicount, , 5, 7, mensions were generated and are shown in Fig. 5 for and 9. With reference to the figure, four sets of curves are given for the four cases. For each set of curves, there are five curves which from left to right correspond to the dimension ratios of , 1.5, 1, 0.67, and 0.5. Design examples of designing a mm-wave DRA with Fig. 5 will be given in the next section. III. DESIGN EXAMPLES As shown in the last section, the designer has the flexibility of choosing different aspect ratios for a given frequency and operation mode. But since the primary objective of this paper is to mitigate the precision problem of fabricating the mm-wave DRA, it is suggested not choosing aspect ratios that give very was used in the following small dimensions. Therefore, examples. In this paper, the DRAs were fabricated using an ECCOS. The microstrip TOCK HiK dielectric material with feedline was etched on a Duroid substrate having a dielectric , a thickness of , and a constant of size of . The reflection coefficients were measured using an HP8510C vector network analyzer, whereas the radiation pattern and gain were measured using an NSI near field and modes were used system. The higher-order in the demonstration. In each case, the dimensions of the DRA were tuned to give a resonance frequency of 24 GHz. A. Operating in the

Mode

mode is discussed in the first exThe higher-order , it can be found from Fig. 5 that the aspect ample. Given ranges from 0.26 to 0.75 for a single -mode ratio of operation. In this example, we arbitrarily choose and obtain from the -axis of the figure. Since the

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Fig. 6. Simulated and measured reflection coefficients of the DRA excited in mode: a b ,d : : : ," ,W , the T E L : : , and L : . ,W

= 2 2 mm

= = 4 0 mm = 6 1 mm = 10 = 0 78 mm = 1 4 mm

= 0 5 mm

TABLE II COMPARISON OF THE MEASURED, CALCULATED (DWM) AND SIMULATED AND T E MODES (HFSS) RESONANCE FREQUENCIES OF THE T E

operating frequency is 24 GHz, we have and, . For convenience, thus, were used in our design. Fig. 6 shows the simulated and measured reflection coefficients of the DRA, and reasonable agreement between them is observed. With reference to the figure, the simulated and meaare sured 10-dB impedance bandwidths given by 5.75% (23.32–24.70 GHz) and 5.39% (23.30–24.59 GHz), respectively. The measured resonance frequency (min. ) is 23.86 GHz, agreeing well with the calculated (23.91 GHz) and simulated (24.06 GHz) values. Table II summarizes the measured, calculated, and simulated frequencies, along with the errors of the calculation and simulation. Fig. 7 shows the simulated and measured radiation patterns of mode at 24 GHz. The simulated 3D radiation pattern the is provided in Fig. 8. As can be seen from Fig. 7, the antenna has broadside radiation patterns as expected. In the boresight direction, the -plane ( plane) and -plane ( plane) co-polarized fields are stronger than their cross-polarized counterparts by about 20 dB and 30 dB, respectively. The ripples in the measured patterns are primarily due to diffractions from the finite ground plane and the antenna mount. Also, since the SMA connector becomes electrically large at such a high frequency, it can be an effective scatterer that degrades the radiation pattern [4]. Fig. 9 shows the simulated and measured antenna gains of the DRA, which are 5.8dBi at around 24 GHz. It should be mentioned that as discussed in [18], there is more than one equivalent magnetic dipole along the height of the DRA for a higher-order mode. The separation between the equivalent magnetic dipoles

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Fig. 9. Simulated and measured antenna gains of the DRA excited in the T E mode. The parameters are the same as in Fig. 6.

Fig. 7. Simulated and measured radiation patterns of the DRA excited in the T E mode at 24 GHz. The parameters are the same as in Fig. 6. ( ) Simulated 1 1 11 1 1 Measured.

Fig. 10. Simulated and measured reflection coefficients of the DRA excited in mode: a = b = 4:2 mm, d = 10:7 mm, " = 10, W = 0:5 mm, the T E L = 1:8 mm, W = 0:78 mm, and L = 0:9 mm. Fig. 8. Simulated 3D radiation pattern of the DRA excited in the T E at 24 GHz. The parameters are the same as in Fig. 6.

mode

is determined by the DRA dimensions. It directly affects the antenna gain, beamwidth, and sidelobe of the DRA. Our simulation results showed that their values do not increase or decrease monotonically with the order of the mode. B. Operating in the

Mode

Next, the higher-order mode is discussed in the second example. With reference to Fig. 5, for , the aspect ratio ranges from 0.23 to 0.40 for a single -mode opof and eration. In this example, we arbitrarily choose from Fig. 5. Since the resonance frequency obtain is 24 GHz, we have and . Again, for convenience, the dimensions were slightly modified and in our final design. to Fig. 10 shows the simulated and measured reflection comode, with efficients of the DRA operating in the reasonable agreement. With reference to the figure, the simulated and measured 10-dB impedance bandwidths are 3.40% (23.73–24.55 GHz) and 3.87% (23.58–24.51 GHz), respec-

mode, the mode tively. As compared with the has a narrower bandwidth because the effective dielectric constant increases with the order of the mode. The measured resonance frequency is 23.87 GHz, which agrees well with the calculated (24.09 GHz) and simulated (24.05 GHz) values. The mode are shown various frequencies and errors for the in Table II. With reference to the table, the errors are less than 1%. Fig. 11 shows the simulated and measured radiation patterns mode at 24 GHz, whereas Fig. 12 displays the of the simulated 3D radiation pattern of the mode. With reference to Fig. 11, the co-polarized fields of both planes are stronger than their cross-polarized counterparts by more than 25 dB in the boresight direction. Again, the measured cross-polarized fields are stronger than the simulated counterparts as explained above. Fig. 13 shows the measured and simulated antenna gains of the DRA. The measured gain is 6.3 dBi at around 24 GHz. It should be mentioned that the beamwidth, sidelobe, and gain of mode change with the DRA dimensions as discussed the before. These two examples show that it is feasible to design a DRA with its higher-order modes.

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TABLE III COMPARISON OF THE SURFACE AREA AND VOLUME BETWEEN THE DRAS EXCITED IN THE T E , T E AND T E MODES. ALL OF THEM ARE EXCITED AT 24 GHz

Fig. 11. Simulated and measured radiation patterns of the DRA excited in the T E mode at 24 GHz. The parameters are the same as in Fig. 10. ( ) Simulated 1 1 11 1 1 Measured.

Fig. 14. Simulated reflection coefficients of the DRAs excited in the T E mode for " = 5, 10, and 15, with the DRA dimensions given in Table IV. The slot length L and stub length L were tuned to obtain good match. TABLE IV COMPARISON OF THE SURFACE AREA AND VOLUME BETWEEN THE DRAS USING DIFFERENT DIELECTRIC CONSTANTS. THE DRA IS MODE AT 24 GHz IN EACH CASE EXCITED IN THE T E

Fig. 12. Simulated 3D radiation pattern of the DRA excited in the T E at 24 GHz. The parameters are the same as in Fig. 10.

mode

Fig. 13. Simulated and measured antenna gains of the DRA excited in the T E mode. The parameters are the same as in Fig. 10.

IV. COMPARISON OF DRA SIZES FOR DIFFERENT MODES For comparison, a rectangular DRA operating in its fundamental mode was also designed at 24 GHz. The parame-

ters of the DRA are given by , , and . Table III compares the dimensions of the DRAs oper, , and modes. With reference to ating in the the table, considerable increases in the DRA size are obtained by using the higher-order modes. For the DRA excited in the mode, its surface area and volume are 3.9 and 7.4 times mode, respectively. When the larger than those of the mode is used, the surface area and volume of the DRA become 6.4 and 14.4 times larger than for the mode, respectively. By increasing the size, the resonance frequency of the mm-wave DRA can be made less sensitive to fabrication errors. This will be explained in detail in the next section. To see how the DRA size is affected by the dielectric constant, , 10, and 15 were designed to operate at three DRAs of the same frequency of 24 GHz. All of these DRAs were excited mode. Fig. 14 shows the simulated in their higher-order reflection coefficients for the three cases. Their corresponding antenna dimensions are given in Table IV. In each case, the slot were tuned to optimize length and microstrip stub length the impedance match. With reference to the table, the size of the DRA increases with a decrease of the dielectric constant,

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TABLE V COMPARISON OF SIMULATED (HFSS) AND CALCULATED (DWM) FREQUENCY SHIFTS (1f ) OF , TE AND T E MODES FOR DIFFERENT FABRICATION ERRORS THE T E

which is to be expected. The volume of the DRA with is almost 4 times larger than for . Also, it can be seen from the figure that the 10-dB impedance bandwidth increases from 3.74% to 13.47% as decreases from 15 to 5. Therefore, a lower dielectric constant is preferably used to increase the size and bandwidth of the DRA. The dielectric constant, of course, cannot be too low or a DRA mode cannot be excited properly. It is worth mentioning that the radiation pattern of a mode can be distorted significantly when the mode is weakly excited [25]. V. FREQUENCY SHIFT DUE TO FABRICATION ERROR Suppose the dimension of a DRA is changed due to a fabrication error, then the change is electrically larger for a higherorder mode than for a lower-order mode. Thus, it would be expected that the higher-order mode should have a larger frequency shift because of the larger (electrical) change. However, as found from Table III, a higher-order-mode DRA has a larger size and, thus, the change is smaller in terms of percentage. Therefore, it can be conversely expected that the higher-order mode should have a smaller frequency shift because the (percentage) change is smaller. Obviously, these two inferences are different from each other and it is difficult to know the true answer intuitively. To find the true answer, the frequency shifts , , and modes due to three different of the fabrication errors of 0.01 mm, 0.05 mm and 0.1 mm were calculated and are listed in Table V. In each case, the error was added to each dimension of the DRA and the new frequency was calmode for example, the DRA was deculated. Take the and . signed at 24 GHz with It was assumed that the dimensions of the DRA will become , when there is a fabrication error of 0.01 mm. The calculated and simulated results were obtained using the DWM and HFSS, respectively, and they agree reasonably well with each other. With reference to Table V, it is important to note that, for all of the fabrication errors, the frequency shift decreases with increasing the order of the mode. mode is about For each error, the frequency shift of the mode. This re66% lower than that of the fundamental sult is very encouraging for using a higher-order mode. With reference to the table, it is also found that the frequency shift increases with an increase of the fabrication error, which is to be expected. The frequency shift is also dependent on the aspect ratios of the DRA. Fig. 15 shows the percentage shift as a function of for the , , and modes, with . As should be used for found from Fig. 5, a different range of each mode to ensure a single-mode operation; the ranges are

Fig. 15. Calculated frequency shift as a function of b=d for the T E and T E modes, with a fabrication error of 0.05 mm.

, TE

Fig. 16. Calculated frequency shift of the T E mode as a function of a=b for b=d = 0:4, 0.5 and 0.6, with a fabrication error of 0.05 mm.

0.26–0.75 and 0.23–0.40 for the and modes, respectively. Here, a design frequency of 24 GHz and a fabrication error of 0.05 mm were used. With reference to the figure, . This is bethe frequency shift increases with a decrease of cause the dimensions of , are small when is small. Therefore, the percentage change of the dimension can be significant even when the absolute change is small, leading to a larger frequency shift. Again, it is important to note that a smaller frequency shift is obtained for a higher-order mode. Fig. 16 shows mode as a function of for the frequency shift of the , 0.5, and 0.6. As can be observed from the figure, . Also, a the frequency shift decreases with an increase of

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TABLE VI COMPARISON OF THE DRA DIMENSIONS, VOLUMES, AND FREQUENCY SHIFTS (1f ) BETWEEN OUR AND PREVIOUS [18] T E

larger ratio of results in a smaller frequency shift. The result is coincident with that of Fig. 15. The change of frequency is small. It slows down shift is more rapid when the ratio of . It can be observed from the figure significantly when ; the that the slow-down rate is dependent on the ratio of is, the slower the change of the frequency larger the ratio of shift. When , the frequency shift remains almost un. The mode was also studied. Simchanged for ilar results were obtained and the result is not included here for brevity. From the above analysis, it can be inferred that a small DRA dimension should be avoided to minimize the frequency shift. A and the largest value suggested design rule is to choose . As a demonstration, a DRA excited in the mode of is designed at 24 GHz using this design rule, and the result is -mode design [18]. In compared with that of a previous [18], the original DRA was operated at 11.2 GHz with the DRA and . For ease dimensions given by of comparison, the dimensions were scaled down to , for a 24-GHz design. It is understood that the design presented in [18] is to consider the antenna gain instead of fabrication error. The comparison can give an idea of the difference between the optimal and non-optimal designs. and (which Using the design rule, we have -mode operation), from is the largest ratio for a single and . which we obtained Table VI shows the dimensions, volumes, and calculated frequency shifts of the two designs. With reference to the table, our design is 45.8% larger in volume than for the previous one [18], with the frequency shift being 50% less than that of the previous design for the three fabrication errors. VI. CONCLUSION It has been proposed to design the mm-wave rectangular DRA using its higher-order mode to mitigate the tolerance problem of fabrication. Using a higher-order mode, the antenna can have a larger size and therefore, will be more tolerant to fabrication erand rors. Two rectangular DRAs excited in their modes have been designed and fabricated. Compared with the fundamental mode, the higher-order modes also produce broadside radiation patterns but they desirably give a much mode, the DRA size is 7.4 larger antenna size. Using the mode when . It intimes larger than for the creases to over 14 times when the higher-order is used. The size of the DRA can be further increased by using a lower dielectric constant, which also provides another advantage that the impedance bandwidth of the DRA can be broadened.

-MODE DESIGNS AT 24 GHz

The frequency shift of the mm-wave DRA has been investigated for different higher-order modes. It has been found that given the same fabrication error, the frequency shift of a higherorder mode is less than that of a lower-order mode. To demonstrate this, a 24-GHz DRA was designed and three fabrication errors of 0.01 mm, 0.05 mm, and 0.1 mm were introduced. It mode is has been found that the frequency shift of the about 66% lower than for the fundamental mode for all of the three cases. A design rule has been suggested to minimize the frequency shift due to possible fabrication errors. It advises that the ratio of be made as large as possible while keeping . It is hoped that the results presented in this paper would be useful for practical designs of the mm-wave DRA. ACKNOWLEDGMENT The valuable comments of the reviewers are gratefully appreciated. REFERENCES [1] D. M. Pozar, “Consideration for millimeter wave printed antennas,” IEEE Trans. Antennas Propag., vol. 31, no. 5, pp. 740–747, Sep. 1983. [2] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997. [3] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 406–412, 1983. [4] A. Petosa, Dielectric Resonator Antenna Handbook. Norwood, MA: Artech House, 2007. [5] , K. M. Luk and K. W. Leung, Eds., Dielectric Resonator Antennas. London, U.K.: Research Studies Press, 2003. [6] M. G. Keller, M. B. Oliver, D. J. Roscoe, R. K. Mongia, Y. M. M. Antar, and A. Ittipiboon, “EHF dielectric resonator antenna array,” Microwave Opt. Tech. Lett., vol. 17, no. 6, pp. 345–349, Apr. 1998. [7] 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. [8] J. Svedin, L.-G. Huss, D. Karlen, P. Enoksson, and C. Rusu, “A micromachined 94 GHz dielectric resonator antenna for focal plane array applications,” in Proc. IEEE MTT-S Int. Microwave Symp., Jun. 2007, pp. 1375–1378. [9] W. M. A. Wahab, S. Safavi-Naeini, and D. Busuioc, “Low cost low profile dielectric resonator antenna (DRA) fed by planar waveguide technology for millimeter-wave frequency applications,” in Proc. Radio and Wireless Symp., Jan. 2009, pp. 27–30. [10] L. N. Zhang, S. S. Zhong, and S. Q. Xu, “Broadband U-shaped dielectric resonator antenna with elliptical patch feed,” Electron. Lett., vol. 44, no. 16, pp. 947–949, Jul. 2008. [11] A. A. Kishk, “Wide-Band truncated tetrahedron dielectric resonator antenna excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2913–2917, Oct. 2003. [12] A. A. Kishk, Y. Yin, and A. W. Glisson, “Conical dielectric resonator antennas for wide-band applications,” IEEE Trans. Antennas Propag., vol. 50, no. 4, pp. 469–474, Apr. 2002. [13] B. Li and K. W. Leung, “Strip-fed rectangular dielectric resonator antennas with/without a parasitic patch,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2200–2207, Jul. 2005.

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[14] G. Almpanis, C. Fumeaux, and R. Vahldieck, “The trapezoidal dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2810–2816, Sep. 2008. [15] C. S. De Young and S. A. Long, “Wideband cylindrical and rectangular dielectric resonator antennas,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 426–429, 2006. [16] T. H. Chang and J. F. Kiang, “Dualband split dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3155–3162, Nov. 2007. [17] K. P. Esselle, “Circularly polarised higher-order rectangular dielectricresonator antenna,” Electron. Lett., vol. 32, no. 3, pp. 150–151, Feb. 1996. [18] A. Petosa, S. Thirakoune, and A. Ittipiboon, “Higher-order modes in rectangular DRAs for gain enhancement,” presented at the 13th Int. Symp. on Antenna Technology and Applied Electromagnetics and the Canadian Radio Sciences Meeting, 2009. [19] Y. M. M. Antar and Z. Fan, “Theoretical investigation of aperture coupled rectangular dielectric resonator antenna,” IEE Proc.-Microw. Antennas Propag., vol. 143, no. 2, pp. 113–118, 1996. [20] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1348–1355, Sep. 1997. [21] M. S. Al-Salameh, Y. M. M. Antar, and G. Seguin, “Coplanar waveguide fed slot-coupled rectangular dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1415–1419, Oct. 2002. [22] A. S. Al-Zoubi, A. A. Kishk, and A. W. Glisson, “Aperture coupled rectangular dielectric resonator antenna array fed by dielectric image guide,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2252–2259, 2009. [23] R. K. Mongia, “Theoretical and experimental resonant frequencies of rectangular dielectric resonators,” IEE Proc.-H, vol. 139, no. 1, pp. 98–104, 1992. [24] S. K. Koul, Millimeter Wave and Optical Dielectric Integrated Guides and Circuit. New York: Wiley, 1997. [25] K. W. Leung, K. K. Tse, K. M. Luk, and E. K. N. Yung, “Cross polarization characteristics of a probe-fed hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 47, no. 7, pp. 1228–1230, Jul. 1999.

Yong-Mei Pan was born in Huangshan, Anhui Province, China, in 1982. She received the B.Sc. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China (USTC), in 2004 and 2009, respectively. She is currently a Research Fellow at City University of Hong Kong. Her research interests include dielectric resonator antennas, leaky wave antennas, and metamaterials.

Kwok Wa Leung (S’90–M’93–SM’02–F’11) was born in Hong Kong in 1967. He received the B.Sc. degree in electronics and Ph.D. degree in electronic engineering from the Chinese University of Hong Kong, in 1990 and 1993, respectively. From 1990 to 1993, he was a Graduate Assistant with the Department of Electronic Engineering, the Chinese University of Hong Kong. In 1994, he joined the Department of Electronic Engineering at City University of Hong Kong as an Assistant Professor. Currently, he is a Professor and an Assistant Head

of the Department. From Jan. to June, 2006, he was a Visiting Professor in the Department of Electrical Engineering, The Pennsylvania State University. His research interests include RFID tag antennas, dielectric resonator antennas, microstrip antennas, wire antennas, guided wave theory, computational electromagnetics, and mobile communications. Prof. Leung is a Fellow of HKIE. He received the International Union of Radio Science (USRI) Young Scientists Awards in 1993 and 1995, awarded in Kyoto, Japan and St. Petersburg, Russia, respectively. He received Departmental Outstanding Teacher Awards twice in 2005 and 2010. He was the Chairman of the IEEE AP/MTT Hong Kong Joint Chapter for the years of 2006 and 2007. He was the Chairman of the Technical Program Committee, 2008 Asia-Pacific Microwave Conference, Hong Kong, the Co-Chair of the Technical Program Committee, 2006 IEEE TENCON, Hong Kong, and the Finance Chair of PIERS 1997, Hong Kong. He was an Editor for HKIE Transactions and a Guest Editor of IET Microwaves, Antennas and Propagation. Currently, he serves as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and received Transactions Commendation Certificates twice in 2009 and 2010 for his exceptional performance. He is also an Associate Editor for the IEEE ANTENNAS WIRELESS PROPAGATION LETTERS.

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 in 1981 and 1985, respectively. In 1985, he joined the Department of Electronic Engineering, City University of Hong Kong, 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 served two terms as Head of Department of Electronic Engineering from 2004 to 2010. He is currently Chair Professor of the Department of Electronic Engineering and Director of State Key Laboratory of Millimeter Waves (Hong Kong). His recent research interests include design of patch, planar and dielectric resonator antennas, microwave and antenna measurements, and computational electromagnetics. He is the author of three books, nine research book chapters, over 270 journal papers and 200 conference papers, with over 2300 citations by others according to SCOPUS. He was awarded three US patents and over 10 PRC patents on the designs of various printed antennas. Prof Luk is a Fellow of the IEEE, the Chinese Institute of Electronics, the Institution of Engineering and Technology, U.K., and the Electromagnetics Academy, USA. He received best paper awards at the International Symposium in Antennas and Propagation held in Taipei in November 2008 and the Asia Pacific Microwave Conference held in Chiba, Japan, in December 1994. He was awarded the very competitive Croucher Award in Hong Kong in 2000. He received the Applied Research Excellence Award in City University of Hong Kong in 2001. He was the General Co-Chairman of iWAT2011 held in Hong Kong in March 2011. He was Chairman of 2006 IEEE Region 10 Conference held in Hong Kong in October 2006. He was Technical Program Co-Chair of ISAP2010 held in Macau in November 2010. He was the Technical Program Chairperson of PIERS 1997 held in Hong Kong in January 1997. He was the General Vice-Chairperson of the 1997 and 2008 Asia-Pacific Microwave Conference held in Hong Kong. He was a member of the Scientific Board, EC Network of Excellence in Antennas, Europe. He has successfully supervised 17 Ph.D. and 11 M.Phil. students. Most of them have achieved their distinguished careers in Hong Kong, China, Singapore, UK, Finland and USA. He is the Chief Guest Editor of a Special Issue on “Antennas in Wireless Communications” for IEEE PROCEEDINGS to appear in 2012. He is the Deputy Editor-in-Chief for the Journal of Electromagnetic Wave Applications.

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Pulsed Electromagnetic Field Radiation From a Wide Slot Antenna With a Dielectric Layer ˇ Adrianus T. De Hoop, Member, IEEE, Martin Stumpf, and Ioan E. Lager, Member, IEEE

Abstract—Analytic time-domain expressions are derived for the pulsed electromagnetic field radiated by a wide slot antenna with a dielectric layer in a two-dimensional model configuration. In any finite time window of observation, exact pulse shapes for the propagated, reflected and refracted wave constituents are constructed with the aid of the modified Cagniard method (the Cagniard-DeHoop method). Numerical results are presented for field pulse shapes at the dielectric/free-space interface, the pulse time widths of the excitation being chosen such that the separate arrivals from the two edges of the slot can be distinguished. Applications are found in any system whose operation is based on pulsed electromagnetic field transfer and where digital signals are detected and interpreted in dependence on their pulse shapes. Index Terms—Cagniard-DeHoop method, slot antenna, time domain.

I. INTRODUCTION ITH the rapid development of communication systems whose operation is based upon the transfer of pulsed electromagnetic fields and the detection and subsequent interpretation of the pertaining digital signals, there is a need for the mathematical analysis of model configurations where the influence of (a number of) the system parameters on the performance shows up in analytic time-domain analytic expressions that characterize the physical behavior. The present paper aims at providing such a tool with regard to the pulsed radiation behavior of a wide slot antenna covered with a dielectric layer in a two-dimensional setting. The source exciting the structure is modeled as a prescribed distribution of the transverse electric field across a slot of uniform width in a perfectly electrically conducting planar screen. The pulse shape of the exciting field is arbitrary. In front of this slotted plane there is a homogeneous, isotropic dielectric slab of uniform thickness. The structure fur-

W

Manuscript received August 05, 2010; revised January 07, 2011; accepted January 15, 2011. Date of publication June 09, 2011; date of current version Auˇ gust 03, 2011. The work of M. Stumpf was supported in part by COST IC0603 ASSIST and in part by the European Community’s Seventh Framework Program FP7/2007–2013 under Grant nr. 205294, HIRF SE project. A. T. De Hoop is with the Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). ˇ M. Stumpf is with the Department of Radio Electronics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Purkynova 118, 612 00 Brno, The Czech Republic (e-mail: [email protected]. cz). I. E. Lager is with the International Research Centre for Telecommunications and Radar (IRCTR), Faculty of Electrical Engineering, Mathematics and Computer Science, 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.2011.2158959

ther radiates into free space. Using the combination of a unilateral Laplace transformation with respect to time and the spatial wave-slowness representation of the field components that is known as the modified Cagniard method (the Cagniard-DeHoop method), analytic time-domain expressions are obtained for the electric and the magnetic field as a function of position and time. The representation appears as the superposition of a number of propagating, reflecting and refracting wave constituents in the slab and is, within any finite time window of observation, exact. It is immediately clear that the pulse shapes of these constituents (that successively reach a receiving observer) are distorted versions of the activating source signature. Parameters in this respect are: the pulse shape of the excitation (characterized by the pulse rise time and the pulse time width of a unipolar pulse), the width of the slot, the thickness and the dielectric properties of the slab, as well as the position of observation relative to the exciting slot. The analytic expressions are readily evaluated numerically. Results are presented for field pulse shapes at the dielectric/free space interface, in the vacuum half-space and inside the dielectric slab for a variety of parameters, all chosen such that the pulse time width is smaller than the travel time needed to traverse the slab and such that the separate arrivals from the two edges can be distinguished. In this way, the study can focus on the changes in pulse shape that occur in the individual successive wave constituents. In line with the International Electrotechnical Vocabulary (IEV) of the International Electrotechnical Committee (IEC 60050-IEV) [1], the signature of the excitation is taken to be a unipolar pulse characterized by its pulse amplitude, its pulse rise time and its pulse time width. The power exponential pulse provides a convenient mathematical model to accommodate these parameters. Apart from this, the obtained expressions can serve a purpose of benchmarking the performance of purely computational techniques that have to be called upon in the more complicated configurations met in practice, in particular the ones in patch antenna design, where the field calculated in the present paper represents the field ‘incident’ on the geometry of the patches located on the dielectric/free-space interface. The method employed allows for the inclusion of Boltzmanntype relaxation behavior (which includes, for example, Lorentzline and Drude/Debye-absorption behavior) at the expense of having to use more complicated theorems of the time Laplace transformation. For some related problems that have been handled in the case of acoustic waves, see [2], [3]. A similar problem associated with the excitation by a narrow slot (in fact, a line source) has been studied in [16]. The excitation via a wide slot (or a modal one in the radiating aperture

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while the excitation condition is (8) denoting the Heaviside unit step function. It for all with starts to act at and that prior to this is assumed that instant the field vanishes throughout the configuration. III. FIELD REPRESENTATION Fig. 1. Configuration with indication of the excitation and the critical angle  c =c .

= arcsin(

)

of a parallel-plate waveguide) shows additional features in that the corners of the waveguide feed show a separate diffractive behavior with accompanying wavefronts. II. DESCRIPTION OF THE CONFIGURATION AND FORMULATION OF THE FIELD PROBLEM The configuration examined is shown in Fig. 1. In it, position is specified by the right-handed orthogonal Cartesian co. The time coordinate is . Partial differordinates is denoted by is a reserved entiation with respect to symbol denoting partial differentiation with respect to . The configuration consists of an unbounded electrically perfectly conducting screen with a feeding aperture . The covering dielectric slab occupies the domain . The structure radiates into the vacuum half-space . The spatial distribution of electric permittivity and magnetic permeability is

Analytic time-domain expressions for the field components will be constructed with the senior (first) author’s modification of the Cagniard method (the Cagniard-DeHoop method) [4]–[11]. The method employs a unilateral Laplace transformation with respect to time of the type (9) in which is taken to be real-valued and positive relying on Lerch’s uniqueness theorem [12]. The latter theorem more precisely states that uniqueness of the inverse transformation is enis specified at the sured under the weaker condition that sequence of real values . The next step is to use the slowness representation of the field quantities

(10) that involves imaginary values of the complex slowness parameter . Using (9) and (10), the field equations (3)–(5) transform into (11) (12) (13)

(1) The

corresponding electromagnetic wave speeds are and . The antenna aperture is fed by the uniformly distributed, -independent, electric field

the interface boundary conditions (6) and (7) into (14) (15)

(2) is the feeding ‘voltage’. Since the excitation, as well where as the configuration, are independent of , the non-zero comand the ponents of the electric field strength magnetic field strength satisfy in and the source-free field equations

and the excitation condition (8) into (16) The slowness-domain field quantities follow from (11)–(16) by expressing them in the form

(3) (4) (17)

(5) and The interface boundary conditions require that (6) (7)

(18)

DE HOOP et al.: PULSED ELECTROMAGNETIC FIELD RADIATION FROM A WIDE SLOT ANTENNA WITH A DIELECTRIC LAYER

in which

for

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, with

(19)

(28)

for all . Using these expressions in with (14), (15) and (16) it is found that

as the wave travel time from the left edge of the slot to the point of observation and

(20)

(29)

(21)

as the wave travel time from the right edge of the slot to the point of observation.

(22)

V. THE TIME-DOMAIN FIELDS IN THE DIELECTRIC LAYER In this section we focus on the time-domain constituents of the fields components propagating in the dielectric slab, i.e., in . Using the results of Section III we express them as

in which (23) (24) (25) Via the convergent expansion (26) the slowness-domain field quantities can be written as the superposition of constituents each of which admits an analytic time-domain representation attainable via the Cagniard-deHoop method. For the convenience of the reader, the procedure as based on the time Laplace transformation used by Cagniard [4] and De Hoop [6], is briefly reviewed in Appendix A. The corresponding analysis as based on the time Fourier transform can be found in Chew [13]. IV. RADIATED FIELD IN THE ABSENCE OF A DIELECTRIC SLAB To illustrate the pulse distortion that results from the presence of the dielectric slab we compare the relevant results with the ones applying to the radiation in the absence of the slab. The latter can be found in [18]:

(30) Obviously, the expressions represent two sets of upgoing and two sets of downgoing waves. The latter waves, as well as subsequent reflected upgoing waves, disappear as the contrast in and vanishes. The electromagnetic properties between corresponding time-domain expressions of each of these constituents follow upon the application of the Cagniard-DeHoop technique as described in Appendix A. VI. THE TIME-DOMAIN FIELDS AT THE VACUUM/DIELECTRIC INTERFACE In this section we focus on those field components at the inthat are continuous across this interface, i.e., terface and . Using the results of Section III we express them as

(27)

(31)

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. All time convolution integrals contain inverse square-root singularities at one of the end points of the integrals. These are numerically handled via a stretching of the variable of integration according to (33) with the Jacobian (34) for a body-wave constituent with arrival time

and (35)

with the Jacobian Fig. 2. The power exponential excitation signature.

(36) The corresponding time-domain expressions of each of these constituents follow upon the application of the Cagniard-DeHoop technique given in Appendix A. VII. THE TIME-DOMAIN FIELDS IN THE VACUUM HALF-SPACE In this section we focus on the time-domain constituents of the fields radiated into the vacuum halfspace . Using the results of Section III we express them as

(32) Note that, in this case, the Cagniard-DeHoop contour does not have a simple parametrization and has to be determined either algebraically via Cardano’s formula or via the iterative numerical procedure described in [11]. Once the iterative procedure is terminated, the corresponding time-domain body-wave expressions of each of these constituents result from the procedure as applied in Sections V and VI. The iterative numerical procedure provides results within any prescribed accuracy. For details on the subject, see Appendix B. VIII. ILLUSTRATIVE NUMERICAL EXAMPLES This section provides illustrative numerical results for the case of excitation with the power exponential pulse defined in Appendix C. The power exponential excitation signature with is shown in Fig. 2; this pulse is used throughout this section. The first part of this section shows pulse shapes of continuous at the level of vaucuum/dieleccomponents component just tric interface and pulse shapes of below and above the interface (across which it jumps), while the second part provides the time evolution of the Poynting vector within a certain regions of space at two successive observation times. The properties of the slab are taken as

for a head-wave constituent. The integration limits at the time convolution integrals are subsequently adjusted to the corresponding intervals in and . A. Examples of Field Pulse Shapes The mathematics of the Cagniard-DeHoop method makes explicit that the time-domain constituents arise from successive reflections at the two bounding surfaces of the dielectric layer. So, each next constituent arises at a later time than the previous one and this implies that in any finite time window of observation a finite number of constituents leads to a closed-form expression that is exact. The objective of our analysis is to compare the pulse shapes of the different constituents with the ones that the slot antenna would radiate into a half-space with the properties . The latter can be found in [18]. of have been selected: Four positions of observation at , (B) , (C) and (D) (A) (indicated, not to scale, in Fig. 1). The observation point (A) lies right in front of the radiating slot. The observation point (B) is within the range of critical refraction associated with the left edge of the slot and outside the range of critical refraction associated with the right edge of the slot, while the observation points (C) and (D) are within the range of critical refraction associated with both edges of the slot. The time window of observation is taken . The width of the slot and the thickness via of the dielectric slab are interrelated as . The rise time of the excitation pulse is taken as half of the free-space travel time across the slab. As described in Appendix B, the pulse time width is related to the pulse rise time via (B.3) which leads to for . The arrival times of the different contributions are collected in Table I. Figs. 3–6 show the results. In them, the normalized time scale , the normalized electric field is while the is normalized magnetic field is . The observation point (A) lies outside the range of critical refraction range of both radiating edges. Here, the wave motion consists of a superposition of a plane-wave contribution emanating from the radiating slot and cylindrical waves emanating from the edges of the slot. No head-wave contribution occurs. The observation point (B) lies within the range of critical refraction of the

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TABLE I ARRIVAL TIMES OF TIME-DOMAIN CONSTITUENTS

Fig. 3. Normalized E field time-domain response and normalized . time-domain response at x =d

=0

H

field

left edge of the slot and outside the one associated with the right edge. Here, the wave motion consists of a superposition of cylindrical waves emanating from both edges of the slot and head-wave contributions emanating form the right edge of the slot. The observation points (C) and (D) lie within the range of critical refraction of both edges of the slot. Here, the wave motion consists of a superposition of cylindrical waves emanating from both edges of the slot and head-wave contributions emanating form both edges of the slot. In all those regions where head-wave contributions occur pulse shapes show up that drastically differ from the excitation. For the observation of pulse shapes of the

Fig. 4. Normalized E field time-domain response and normalized . time-domain response at x =d

=1

H

field

electric field component just off the interface, six observation points below and above the points (B), (C) and (D) at have been chosen. In the vertical levels plots, the normalized time scale is and the normalized elec. Figs. 7–9 exhibit the jump in magnitude tric field is . that is related to the electric contrast ratio Due to the smoothness of the exciting pulse shape (Fig. 2), the wave constituents show a more or less smooth behavior across their wavefronts. In case one wants to experimentally deduce the arrival times from the observed behavior near the wavefront, a sufficiently sharp excitation should be employed instead.

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Fig. 5. Normalized E field time-domain response and normalized time-domain response at x =d .

=3

H

field

Fig. 7. Normalized and below x =d

(

E

field time-domain responses above

= 0:995) the interface at x =d = 1.

(x =d = 1:005)

B. Time Evolution of the Poynting Vector The (color) vector density plots Figs. 10–11 show two time snaps of the two-component Poynting vector (37) (38) normalized with respect to (39)

Fig. 6. Normalized E field time-domain response and normalized time-domain response at x =d .

=5

H

field

corresponds to the maximum The reference magnitude value of the Poynting vector as it would be carried by a TEMmode in a parallel-plate waveguide that would be feeding the radiating aperture. The spatial domain of observation is taken and two observation as and . The width of the times are chosen: slot and the thickness of the dielectric slab are interrelated . via Fig. 10 shows the evolution of the Poynting vector for the . In power-exponential pulse excitation with Fig. 10(a) the wavefront just reaches the dielectric/vacuum interface. Because of the relatively high ratio of the spatial extent of the excitation pulse compared with the slot width, the radiation of the slot resembles the one that would be radiated from a line source. In Fig. 10(b), the reflected wave constituents as well as head-wave constituent are clearly seen. In order to illustrate the dependence on the relation between the spatial extent of the excitation pulse and the slot width, the

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Fig. 10. Normalized Poynting vector of the field at (a) c t=d = 2; (b) c t=d = d=w = 1 and electromagnetic parameters  ; f4 ;  g. Parameters of the excitation

4. Slab thickness versus slot width is g= of the dielectric slab are f pulse are = 0 9236 = 2.

c t =d

Fig. 8. Normalized E field time-domain responses above (x and below (x =d = 0:995) the interface at x =d = 3.

:

;

=d = 1:005)

Fig. 11. Normalized Poynting vector of the field at (a) c t=d = 2; (b) c t=d = d=w = 1 and electromagnetic parameters f4 ;  g. Parameters of the excitation  ;

4. Slab thickness versus slot width is g= of the dielectric slab are f pulse are = 0 1847 = 2.

c t =d

Fig. 9. Normalized E field time-domain responses above (x and below (x =d = 0:995) at x =d = 5.

=d = 1:005)

:

;

rise time of the excitation pulse has been decreased to one tenth of the free-space travel time across the slab. The corre. sponding normalized pulse time width is For this case, Fig. 11 shows the Poynting vector distribution at and . Overthe two observation times lapping cylindrical waves arising from the radiating edges and a plane-wave contribution emanating from the slot are clearly distinguishable. The electromagnetic power density radiated from

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the slot is now concentrated within the narrower beam propagating above the radiating slot. Our analysis can be used to further study the possibilities of adapting the excitation pulse to design requirements associated with optimum signal transfer [17] and/or time-domain beam shaping of antenna arrays. IX. CONCLUSION Closed-form analytic expressions have been constructed for the head-, body- and plane-wave constituents of the time-domain electromagnetic field radiated by a pulse-excited slot antenna in a perfectly conducting ground plane with a dielectric slab. The expressions provide insight into the radiation mechanism that can be useful in the procedure of antenna synthesis and in the description of the behavior in other parts of communication systems where the quantification of pulse distortion plays the important role. Illustrative examples clearly show the drastic changes in pulse shapes that do arise from the multiple reflections of the time-domain constituents as they are composed of the non-uniform cylindrical waves arising from the edges of the slot and the plane-wave contribution right in front of the slot. The idealized model under consideration (occupying an unbounded domain in space) can serve the purpose of checking and validating numerical codes that are needed to investigate the more realistic configurations met in practice in early-time results where the excited wave constituents have not yet reached the boundaries that were not included in the model. APPENDIX A THE CAGNIARD-DE HOOP METHOD FOR THE GENERIC FIELD CONSTITUENT

the plane, supplemented with the remainder of the imaginary axis. To arrive at the time-domain equivalent of (41), the path of integration is deformed into one along which (42) (the Cagniard-De Hoop path) with real-valued and positive. Solving for and accounting for the presence of the branch cuts, ( denotes complex we attain the hyperbolic arc conjugate), where

(43) as the body-wave path, with (44) as the body-wave arrival time, and the loop around the branch cut , where

(45) as the head-wave path, with (46)

This Appendix discusses the Cagniard-De Hoop method for the generic field constituent

as the head-wave arrival time. The latter arises only in the region . From (42) and (43) it follows that

(40)

(47)

in which

and (48) (41)

From (42) and (45) it follows that

where

is an algebraic function of and and . The only singularities in the right hand side of (40) are associated with and the the branch points associated with . In view of branch points the further needed deformation of the path of integration away from the imaginary -axis, we make, in the cut -plane, and single-valued through the introduction of the branch and cuts and take and . Now, the Cagniard-De Hoop method can only be applied to the separate terms in (40) and each of them has a simple pole as (41) shows. To accommodate this feature, the path at of integration in (41) (the imaginary -axis) is deformed into a semi-circular arc of vanishingly small radius in the right half of

(49) and (50) is, via the application of With this, the body-wave part of Cauchy’s theorem, Jordan’s lemma and Schwarz’s reflection principle of complex function theory, obtained as

(51)

DE HOOP et al.: PULSED ELECTROMAGNETIC FIELD RADIATION FROM A WIDE SLOT ANTENNA WITH A DIELECTRIC LAYER

and the head-wave part of

as

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as the starting values for determining intermediate values of along the path. APPENDIX C THE POWER EXPONENTIAL PULSE

(52) Lerch’s uniqueness theorem of the unilateral Laplace transformation [12] then ensures that the time-domain body-wave contribution of is

(53)

A convenient pulse type to model a unipolar pulse excitation is the power exponential pulse [14] (59) for where is the pulse amplitude, is the rising exponent of the pulse and is the pulse rise time [1]. Note . The pulse time width follows from that

and the time-domain head-wave contribution is

(60) as (54) (61)

we must, in the contour deformation from the imagiFor , take nary -axis and the semi-circular indentation around . This contriinto account the contribution from the pole bution can be interpreted as a plane-wave contribution . From (41) it follows that

The time Laplace transform of (59) is (62) The spectral amplitude of

follows from (62) as

(55) (63) denotes the Heaviside step function. The correwhere sponding plane-wave time-domain contribution of is

From (64)

(56) it follows that both

These results are used in the main text.

(65) APPENDIX B THE CAGNIARD-DEHOOP PATH FOR WAVE CONSTITUENTS TRANSMITTED INTO THE VACUUM HALF-SPACE The Cagniard-DeHoop path for a wave constituent that has (multiply) traversed the dielectric layer and subsequently radiates into the vacuum half-space is of the general shape (57) is the receiver offset (with respect to the transmitter) where is the normal distance traversed in the along the boundary, is the (multiply) traversed normal vacuum half-space and distance in the dielectric layer. A rapidly converging numerical is found in Newton’s method with method for determining two starting values. First, the point of intersection with the real -axis is determined, with and as the starting . The asymptotic value as values. This leads to follows from (57) as (58) which represents a straight line. Note that the relative error with , which implies again rapid respect to the leading term is and are now used convergence. The values

and (66) is plotted against , In the spectral diagram (where both on logarithmic scales), the right-hand sides of (65) and (66) are straight lines that are denoted as the spectral bounds . The two spectral bounds intersect at their of corner point (67) ACKNOWLEDGMENT The authors would like to express their thanks to the reviewers for their careful reading of the manuscript and their constructive suggestions for the improvement of the paper. REFERENCES [1] “Electropedia: The world’s online electrotechnical vocabulary,” [Online]. Available: http://www.electropedia.org [2] A. T. de Hoop, “Impulsive spherical-wave reflection against a planar absorptive and dispersive Dirichlet-to-Neumann boundary—An extension of the modified Cagniard method,” in Proc. 2nd. AIP Conf. on Mathematical Modeling of Wave Phenomena, Växjö, Sweden, 2006, vol. 834, pp. 13–24, American Institute of Physics.

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[3] A. T. de Hoop, “Line-source excited pulsed acoustic wave reflection against the mass-loaded boundary of a fluid,” in Proc. 3rd Conf. on Mathematical Modeling of Wave Phenomena August AIP Conf., 2009, vol. 1106, pp. 118–129, American Institute of Physics. [4] L. Cagniard, Réflexion et Réfraction des Ondes Séismiques Progressives. Paris: Gauthier-Villars, 1939. [5] L. Cagniard, Reflection and Refraction of Progressive Seismic Waves. New York: McGraw-Hill, 1962, (Translation by E. Flinn, C. H. Dix, of Réflexion et réfraction des ondes séismiques progressives, Paris, Gauthier-Villars, 1939). [6] A. T. de Hoop, “A modification of Cagniard’s method for solving seismic pulse problems,” Appl. Sci. Res., no. 8, pp. 349–356, May 1960, Sect. B. [7] A. T. de Hoop and H. J. Frankena, “Radiation of pulses generated by a vertical electric dipole above a plane, non-conducting, earth,” Appl. Sci. Res., no. 8, pp. 369–377, May 1960, Sect. B. [8] K. J. Langenberg, “The transient response of a dielectric layer,” Appl. Phys., no. 3, pp. 179–188, Mar. 1974. [9] A. T. de Hoop, “Pulsed electromagnetic radiation from a line source in a two-media configuration,” Radio Sci., vol. 14, no. 2, pp. 253–268, Mar.–Apr. 1979. [10] A. T. de Hoop and M. L. Oristaglio, “Application of the modified Cagniard technique to transient electromagnetic diffusion problems,” Geophys. J., vol. 94, no. 3, pp. 387–397, Sep. 1988. [11] A. T. de Hoop, “Transient diffusive electromagnetic fields in stratified media: Calculation of the two-dimensional E -polarized field,” Radio Science, vol. 35, no. 2, pp. 443–453, Mar.–Apr. 2000. [12] D. V. Widder, The Laplace Transform. Princeton, NJ: Princeton University Press, 1946, pp. 63–65. [13] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold, 1990. [14] D. Quak, “Analysis of transient radiation of a (traveling) current pulse on a straight wire segment,” in Proc. IEEE EMC Int. Symp., Aug. 2001, vol. 2, pp. 849–854. [15] R. E. Collin, Antennas and Radiowave Propagation. New York: McGraw-Hill, 1985, pp. 164–292. ˇ [16] M. Stumpf, A. T. de Hoop, and I. E. Lager, “Pulsed electromagnetic field radiation from a narrow slot antenna with a dielectric layer,” Radio Sci., vol. 45, Oct. 2010, RS5005, doi:10.1029/2009RS004335. [17] D. M. Pozar, R. E. McIntosh, and S. G. Walker, “The optimum feed voltage for a dipole antenna for pulse radiation,” IEEE Trans. Antennas Propag., vol. 31, pp. 563–569, Jul. 1983. ˇ [18] M. Stumpf, A. T. de Hoop, and I. E. Lager, “Closed-form time-domain expressions for the 2D pulsed EM field radiated by an array of slot antennas of finite width,” in Proc. URSI Electromagnetic Theory Symp., Berlin, Aug. 2010, pp. 786–789.

Adrianus T. De Hoop (M’58) was born in Rotterdam, The Netherlands, on December 24, 1927. He received the M.Sc. degree in electrical engineering (1950) and the Ph.D. degree in the technological sciences (1958) from Delft University of Technology, Delft, The Netherlands, both with the highest distinction. He served Delft University of Technology as an Assistant Professor (1950–1957), Associate Professor (1957–1960) and Full Professor in Electromagnetic Theory and Applied Mathematics (1960–1996). Since 1996 he is Lorentz Chair Emeritus Professor in the Faculty of Electrical Engineering, Mathematics and Computer Science of this University where, in 1970, he founded the Laboratory of Electromagnetic Research which has developed into a world-class center for electromagnetics, having a huge impact on the world’s electromagnetic community and on electromagnetic research and education in The Netherlands. His research interests are in the broad area of wavefield modeling in acoustics, electromagnetics and elastodynamics. His interdisciplinary insights and methods in this field can be found in his book Handbook of Radiation and Scattering of Waves (London,

Academic Press, 1995), with wavefield reciprocity serving as one of the unifying principles governing direct and inverse scattering problems and wave propagation in complex (anisotropic and dispersive) media. He spent a year (1956–1957) as a Research Assistant with the Institute of Geophysics, University of California at Los Angeles, CA, where he pioneered a modification of the Cagniard technique for calculating impulsive wave propagation in layered media, later to be known as the Cagniard-DeHoop technique. This technique is presently considered as a benchmark tool in analyzing time-domain wave propagation. During a sabbatical leave at Philips Research Laboratories, Eindhoven, the Netherlands (1976–1977), he was involved in research on magnetic recording theory. Since 1982, he has been a Visiting Scientist on a regular basis with Schlumberger-Doll Research, Ridgefield, CT, (presently at Cambridge, MA), where he contributes to research on geophysical applications of acoustic, electromagnetic and elastodynamic waves. Dr. De Hoop is a Member of the Royal Netherlands Academy of Arts and Sciences and a Foreign Member of the Royal Flemish Academy of Belgium for Science and Arts. He holds an Honorary Doctorate in the Applied Sciences from Ghent University, Belgium (1981) and an Honorary Doctorate (2008) in the Mathematical, Physical and Engineering Sciences from Växjö University (since 2010, Linnaeus University), Växjö, Sweden. He received grants from the Stichting Fund for Science, Technology and Research (founded by Schlumberger Limited) which supported his research at Delft University of Technology. He was awarded the 1989 Research Medal of the Royal Institute of Engineers in the Netherlands, the IEEE 2001 Heinrich Hertz Gold Research Medal, and the 2002 URSI (International Scientific Radio Union) Balthasar van der Pol Gold Research Medal. In 2003, H.M. the Queen of The Netherlands appointed him “Knight in the Order of The Netherlands Lion.”

ˇ ˇ Martin Stumpf was born in Cáslav, the Czech Republic, on September 22, 1983. He received the B.Sc. degree (2006) and M.Sc. degree (2008) in electrical engineering from Brno University of Technology, Brno, the Czech Republic, where he is currently working toward the Ph.D. degree. In 2008, he was with the Institute for Fundamental Electrical Engineering and Electromagnetic Compatibility, Magdeburg, Germany, and from 2009–2010, with the International Research Centre for Telecommunications and Radar, Delft, The Netherlands. His main research interests include electromagnetic, acoustic and elastodynamic wave phenomena. ˇ Mr. Stumpf received a 2010 Young Scientist Award at the URSI International Symposium on Electromagnetic Theory.

Ioan E. Lager (M’98) was born in Bras¸ov, Romania, on September 26, 1962. He received the M.Sc. degree in electrical engineering (1987) from the “Transilvania” University of Bras¸ov, Bras¸ov, Romania, the Ph.D. degree in electrical engineering (1996) from Delft University of Technology, Delft, The Netherlands, and a second Ph.D. degree in electrical engineering (1998) from the “Transilvania” University of Bras¸ov. He successively occupied several research and academic positions with the “Transilvania” University of Bras¸ov and the Delft University of Technology, where he is currently an Associate Professor. In 1997, he was a Visiting Scientist with Schlumberger-Doll Research, Ridgefield, CT. He has a special interest for bridging the gap between electromagnetic field theory and the design, implementation and physical measurement of radio-frequency front-end architectures. His research interests cover computational electromagnetics and antenna engineering, with an emphasis on non-periodic (interleaved) array antenna architectures. He is actively involved in several antenna engineering related European networks, primarily in the “ASSIST” COST Action, his focus area concerning the specific higher education aspects.

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Reaching the Chu Lower Bound on Q With Magnetic Dipole Antennas Using a Magnetic-Coated PEC Core Oleksiy S. Kim and Olav Breinbjerg, Member, IEEE

Abstract—We analytically solve the radiation problem for a spherical magnetic dipole antenna with a material-coated perfectly electrically conducting core. Using the closed-form expressions derived for the internal and external stored energies as well as for the radiation quality factor , we determine the optimal geometrical and material parameters of the antenna. We show that the optimal permeability of the coating increases as the coating becomes thinner; as a result, the energy stored in it and the radiation quality factor reduce. In the limit of an infinitely thin coating, the optimal permeability tends to infinity, the internal stored energy vanishes, and the reaches the Chu lower bound, irrespective of the antenna electrical size and permittivity of the coating. Index Terms—Chu lower bound, electrically small antennas, fundamental limitations, magnetic dipole, quality factor.

I. INTRODUCTION N his seminal work from 1948 [1] Chu established a lower bound for the radiation quality factor of an electrically small antenna; this can be expressed as [1]–[3]

I

(1) with being the free-space wave number and the radius of the smallest sphere circumscribing the antenna. The Chu lower bound is generally considered to be too optimistic, since it takes into account only the stored energy in the region external to the circumscribing sphere. If the stored energy in the internal will increase. For a region is also taken into account, the spherical electric surface current distribution with free-space internal region—a so-called air-core—it has been shown [4]–[6] that the quality factor is limited from above to for spherical mode) electric dipole antennas (radiating the and for magnetic dipole antennas (radiating the spherical mode) as . A range of practical antennas [7]–[12] confirm these values. There is obviously a strong interest in antennas that overcome the limits of the air-core spherical antennas and approach the Chu lower bound. The lower requires the internal stored energy—electric energy for electric dipole antennas and magnetic energy for magnetic dipole antennas—to be reduced. One way Manuscript received September 11, 2010; revised November 17, 2010; accepted December 10, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. The authors are with the Department of Electrical Engineering, Electromagnetic Systems, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158965

to do it was pointed out by Wheeler in his work from 1958 [4]. Wheeler showed that the Chu lower bound can be reached by a magnetic dipole antenna with a solid spherical magnetic core . For a of infinite permeability—but only in the limit finite-size magnetic dipole antenna the lowest quality factor is proved to be [13], [14] (2) can apOn one hand, the expression (2) suggests that the very closely; e.g., within proach the Chu lower bound 10% for . On the other hand, it excludes that can be reached by a finite-size magnetic dipole antenna with a solid spherical magnetic core. The limiting mechanism is the internal resonances, which start occurring when the electrical size of the core reaches specific values. However, they may be avoided if a metal sphere is introduced into a solid magnetic core, as shown in this paper. Another way to lower the follows from the Huygens-Love principle, which states that equivalent electric and magnetic currents on a surface enclosing a source radiate the same fields as the original source. Furthermore, the equivalent currents produce zero fields internally to the surface they reside on. Thus, equivalent electric and magnetic surface currents being a source itself represent an ideal antenna with no internal stored energy and the lowest possible [15], [16]. The principle is used in [17], [18] to derive lower bounds on the for electrically small antennas of arbitrary shape. In subsequent works by the same authors [19], [20], it is suggested to utilize a thin sheet of highpermeable magnetic material to produce magnetic polarization currents as an alternative to the current of magnetic charges. Indeed, a numerical investigation of an electric dipole antenna [19], [20] illustrates that the air-core limit can be overcome and the Chu lower bound closely approached. The purpose of this paper is to show that the Chu lower bound can be approached even closer than (2) for a finite-size magnetic dipole antenna using a material-coated perfectly electrically conducting (PEC) core. This facilitates electrically small antenna designs of the best possible performace with respect to the quality factor and thus bandwidth. Preliminary results of this work were presented in [21], and similar results were independently reported in [22]. Here, we present the details of the analytical solution for the problem (Section II) and determine the optimal parameters of the antenna ensuring the lowest possible (Section III). ratio The time factor is assumed and suppressed throughout the paper.

0018-926X/$26.00 © 2011 IEEE

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(5c)

(5d)

Fig. 1. Magnetic dipole antenna with a material-coated PEC core. Time-harmonic surface current density J is distributed over the surface of the antenna.

II. THEORY AND RESULTS The geometry of the antenna is sketched in Fig. 1. A spherical PEC core of radius is coated with a concentric magnetodielectric shell of external radius . The material of the shell is linear, isotropic, homogeneous, lossless and dispersion-less with the relative permittivity , relative permeability , and . In a spherical coordinate system wave number with the origin at the center of the core, an impressed electric current density on the surface of the shell is defined as

The radial dependence of the fields in free space is represented by the spherical Hankel function of the first kind , whereas for the shell an appropriate linear combination of spherical Bessel and Neumann functions is introduced, so that the boundary condition at the surface of the PEC core is satisfied. By substituting (5) into (4) and enforcing the boundary con, the coefficients ditions at the surface of the antenna can be determined in closed form as (6a) (6b) where

(3) is the azimuthal unit vector and is the amplitude where (A/m). In free space, externally to the antenna, this current radiates electromagnetic fields equal to those of a -directed elementary magnetic dipole located at the origin. A. Analytical Solution The electric and magnetic fields in the material shell, and , and in free space, and , are expressed in terms of vector spherical wave functions as (4a)

(7)

B. Stored Energy and Quality Factor To determine the stored energies, the fields in (4) are spatially integrated over the respective regions. In free space, the contribution of the propagating field is subtracted, so that only the energy density of the non-propagating field remains [3]. The , internal magresulting expressions for the internal electric , external electric , and external magnetic netic stored energy are found to be

(4b) (4c)

(8a)

(4d) where and are the intrinsic admittances of the shell and free are coefficients to be determined. space, respectively, and The -functions are defined using the notation of Hansen [23] as

(8b)

(5a)

(8d)

(5b)

where . The expressions for the external stored energies, (8c) and (8d), have been reported previously [2], [13], [24], whereas the expressions for the stored energies in the spherical shell enclosing the PEC sphere, (8a) and (8b), are new.

(8c)

KIM AND BREINBJERG: REACHING THE CHU LOWER BOUND ON Q WITH MAGNETIC DIPOLE ANTENNAS

The radiation quality factor

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is determined from (9)

using (6), (8), and the radiated power (10) Expression (9) assumes that a resonant system is established by increasing the lesser of the electric and magnetic energies to equal the larger through use of, e.g., a lumped-component tuning circuit. C. Results The effect of the magnetic-coated PEC core is illustrated in Fig. 2. In Fig. 2(a), the ratio of the total stored electric to the total stored magnetic energy energy is plotted vs. the relative permeability of the shell for three values of the relative radius of the PEC core , and with fixed external electrical radius . For the permeability close to the free space value, . As the magnetic energy clearly dominates irrespective of the permeability increases, the relative part of the stored electric energy becomes larger, and eventually the first resonance is reached. Fig. 2(b) shows the ratio of the internal stored ento the total stored energy ergy vs. the relative permeability . The internal stored energy, which is predominantly passes an magnetic, initially decreases with ; however, as optimum value, it starts rising towards the internal resonance, at which the total stored energy is entirely internal. demonstrates similar beThe radiation quality factor haviour, as seen in Fig. 2(c), where the , normalized by the , is plotted vs. the relative permeability Chu lower bound . The ratio is lowest for some optimal , and it becomes of course infinite at the internal resonances. , the ratio inIn the free space case creases as the PEC core becomes relatively larger (Fig. 2(b)). The source currents on the surface of the antenna induce oppositely directed currents on the surface of the PEC core, and as the PEC surface becomes closer to the source currents, the external fields—and thus the external stored energy and the radiand , ated power—reduce. In the limiting case tends to unity, and the tends to infinity. the ratio The most prominent effect of the magnetic-coated PEC core is that the larger , the higher the value of the permeability , at which the first resonance occurs. The optimal value of the permeability, at which the ratios and are minimal, becomes correspondingly higher. Potentially, an arbitrary high can be used—and an arbitrary low ratio achieved—without the risk of hitting a resonance if the shell is thin enough. In the limit of an infinitely thin magnetic shell , the optimal is infinitely high, the magnetic energy stored in the shell vanishes, and the reaches the Chu lower bound .

Fig. 2. Energy and quality factor as functions of  (" = 1) for fixed ka = 0:5: (a) Ratio of electric and magnetic stored energies; (b) Ratio of internal and total stored energy; (c) Quality factor (normalized by Chu lower bound).

III. OPTIMAL PARAMETERS In this section, we determine the optimal geometrical and mafor a spherical terial parameters minimizing the ratio magnetic dipole antenna with material-coated PEC core. First, the optimal parameters are found from the exact analytical solution presented in Section II. Second, a simple standing wave model that provides a physical insight into the problem of determining the optimal parameters is discussed.

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A. Optimal Parameters Extracted From the Exact Solution Assuming that all parameters are chosen such that the stored magnetic energy exceeds the stored electric energy, , the expression for the radiation quality factor can be written from (6)–(10) as (11) where

(12) and : The minimum of 1) Optimal for given in (11) with respect to the relative permeability of the shell is found by solving the equation (13) Since in (12) is an oscillating function, (13) has infinitely many solutions—values of —for any value of . , the optimal permeability Denoting these solutions as and is determined as of the shell for a given (14) Substituting (14) into (11), we obtain the lowest achievable for each as (15) are plotted in Fig. 3(a) for three values of The functions , and ; the corresponding ratios are shown in . It is observed that the solutions with Fig. 3(b) for higher yield higher ratios , and though the difference , the first solution is of becomes negligible as most interest. As evident from (15), this observation holds for . arbitrary Use of plotted data is not always convenient. Therefore, we applied a least-squares curve fitting of a rational function (16) . In (16), to the numerically found data points of is a constant determined in [14] while solving a problem for a spherical magnetic dipole antenna with a solid material . Constants and are core found to yield a root mean square error of 1.4%; the resulting approximation is compared to the exact curve in Fig. 3(a). vs. The dependence of the lowest achievable ratio is presented in Fig. 3(c) the relative radius of the PEC core . The plot is based on (15) with for various values of . It is observed, that irrespective of the antenna electrical can be reached as the shell besize, the Chu lower bound . For other shell thicknesses, comes infinitely thin depends on . Parthe lowest achievable ratio ticularly, for a fixed , the deteriorates as the relative per-

p =10

p

Fig. 3. Optimal electrical size k a (a) and the corresponding lowest achievable ratio Q=Q : (b) for n ; ; , and ka " : ; (c) for various ka " and n .

=1

=1 2 3

mittivity of the shell increases. The similar dependence was observed for a magnetic dipole antenna with a solid magnetodielectric core [13], [14]. However, with a material-coated PEC core the deteriorating effect of the permittivity can be mitigated ). For instance, by decreasing the shell thickness (increasing an antenna with and yields

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. Changing the relative permittivity from 1 to and brings up to 2.44. By 16 results in , the ratio making the shell thinner, so that can be reduced to its initial value 1.09. Although, in all cases the is assumed to be optimal relative permeability of the shell according to (14), this prerequisite is not very strict, due to the wide minimum seen in Fig. 2(c). If the electrical size of the antenna , the radius of the core and the permittivity of the coating are fixed, the optimum is found from either (16) or Fig. 3(a). The optimum size is then determined using (14), permeability of the coating and the lowest achievable is given by (15) or Fig. 3(c). for given and : As shown above, 2) Optimal the advantage of a material-coated PEC core versus a solid material core is that a very close to the Chu lower bound can be achieved for a finite-size magnetic dipole antenna. This is done using a very thin magnetic coating of a high-permeable material with no restrictions on its permittivity. However, a free choice of material permeability is not usually possible in practice. Therefore, it is useful to determine optimal relative radius of the PEC for selected antenna electrical size , and coating percore meability and permittivity . The problem reduces to solving the following equation (17) Similar to (13), the (17) can have multiple solutions as (18) The first three ones are shown in Fig. 4(a). Note, that is function. For instance, for , not the inverse of the for a solid core is the optimal (Fig. 3(a)), whereas the value of for which the optimal is approximately 2.083 (Fig. 4(a)). Substituting (18) into (11) yields the lowest achievable for and given (19) Again, as in the previous case (Section III-A1), the first solution yields the lowest ratio (Fig. 4(b)). And as in require higher , the previous case, higher values of and , to achieve the same ratio and thus higher (Fig. 4(c)). is found to be A convenient approximation of (20) which ensures the root mean square error of 0.4%; the fitting is compared to the exact solution in Fig. 4(a). The design of a magnetic dipole antenna with a materialand coated PEC core starts with the antenna electrical size parameters of the material. The optimal ratio is determined using Fig. 4(a) or the approximation (20). The lowest achievable is then found using (19) or Fig. 4(c) for the corre. If , a solid material core should sponding be applied, and the lowest is then given by (2).

p = 10

Fig. 4. Optimal relative radius of the PEC core b=a (a) and the corresponding : (b) for n ; ; , and ka " : ; (c) lowest achievable ratio Q=Q for various ka " and n .

p

=1

=123

3) Range of Validity: The expressions (11)–(20) are valid as , that is, until the first resonance is reached. long as Fig. 5 shows the optimal parameters compared to the parameters for the first resonance found numerically for several values of and . Note, that the results are presented as vs. the relative radius , that is, the vertical axis of the plot represents the shell thickness measured in wavelengths . in the coating material

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=p

Fig. 6. Scattering of a spherical wave by a PEC sphere immersed in a medium =k . "  k;  with parameters " ;  ; k

=2

Fig. 5. Optimal shell thickness and shell thickness for the first resonance.

To show the optimal parameters found in Sections III-A1 and III-A2 on the same plot in Fig. 5, the expression (14) is transformed as (21) and the expression (18) is cast in a parametric form as (22) is a parameter. where The optimal parameters are valid as long as the optimal shell is thinner than the shell corresponding to the first resonance. Since the optimal shell thickness found from (18) is always less than the resonance thickness, it is valid in the entire range of . On the other hand, the validity of (14) is parameters conditional. For , the optimal parameters found using (14) can safely be used in the entire region where an antenna , and actually up is regarded as electrically small . In this range of , the shell thickness, at which to the resonance occurs, exceeds the optimal shell thickness. For and permittivity , the applicability of larger electrical sizes . (14) depends on , the optimal parameters (14) are valid irIn the limit respective of and . Indeed, as , the electrical size becomes much larger than 1 (see Fig. 3(a)). Consequently, the fields in the shell are locally plane waves, implying that the , whereas the shell thickness for the internal resonance is optimal thickness found using (14) is approximately . B. A Standing Wave Model Optimal parameters can also be determined approximately by considering the material coating as a spherical waveguide with a PEC sphere of radius centered at the origin (Fig. 6). The PEC sphere is illuminated by an inward-propagating spherical wave, and, due to the spherical symmetry, the reflected outward-propagating wave is also a spherical wave. Since , a spherical standing wave is formed, there is a radius at which the wave impedance is infinite. Placed at this radii an impressed electric current density (3) will radiate minimally inward, and, thus, the internal stored energy and the radiation

Fig. 7. Exact and approximate optimal thickness of the shell.

will be minimized. It can be shown that the radius impedance is given by the following equation

of infinite (23)

For a vanishingly small PEC sphere can be found analytically as

, the solution (24)

is the p-th root of the spherical Neumann function where . For finite , equation (23) is solved nuof first order merically and the result is shown in Fig. 7 along with the exact solutions found in Section III-A. It is observed, that for small the approximate solution is very close to the exact optimal the approxparameters found from (14), whereas for larger imate solution approaches the exact optimal parameters found from (18). Thus, the standing wave model provides a reasonable approximation to the exact solution. IV. CONCLUSION The radiation problem for an electric current density impressed on the surface of a material-coated PEC sphere and spherical mode is solved analytically. radiating the Closed-form expressions for electric and magnetic internal and external stored energies as well as for the radiation quality factor are derived. From those, the optimal parameters of the antenna, which ensure the radiation closest to the Chu lower bound, are determined.

KIM AND BREINBJERG: REACHING THE CHU LOWER BOUND ON Q WITH MAGNETIC DIPOLE ANTENNAS

It is shown that a PEC sphere introduced into a solid magnetic -mode core shifts the core permeability, at which the first resonance occurs, towards higher values. Consequently, the optimal permeability also becomes higher, and this results in lower internal stored magnetic energy and lower . For instance, the is optimum permeability of a solid magnetic core with , which corresponds to , whereas the optimum permeability of the same core with a PEC sphere of rainside is , and thus . dius Furthermore, it is shown that the negative effect of the dielectric permittivity can be compensated by decreasing the thickness of the magnetodielectric coating. For an infinitely thin coating, reaches the optimal permeability becomes infinite, and the the Chu lower bound, irrespective of the permittivity and antenna electrical size . Thus, the realistic lower bound on the radiation for spherical magnetic dipole antennas is indeed the Chu lower bound. ACKNOWLEDGMENT

[17] A. D. Yaghjian and H. R. Stuart, “Lower bounds on Q for dipole antennas in an arbitrary volume,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Toronto, Canada, Jul. 11–17, 2010, pp. 1–4. [18] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3114–3121, 2010. [19] H. R. Stuart and A. D. Yaghjian, “Using high permeability shells to improve the Q of electrically small electric-dipole antennas,” presented at the IEEE Antennas Propag. Soc. Int. Symp., Jul. 11–17, 2010. [20] H. R. Stuart and A. D. Yaghjian, “Approaching the lower bounds on Q for electrically small electric-dipole antennas using high permeability shells,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3865–3872, Dec. 2010. [21] O. S. Kim and O. Breinbjerg, “Decreasing the radiation quality factor of magnetic dipole antennas by a magnetic-coated metal core,” presented at the 20th Int. Conference on Applied Electromagnetics and Communications (ICECom 2010), Dubrovnik, Croatia, Sep. 20–23, 2010. [22] J. S. McLean, H. Foltz, and R. Sutton, “Broadband, electrically-small spherical-wire and generalized loop antennas exploiting inhomogeneous magnetic cores,” presented at the 20th Int. Conference on Applied Electromagnetics and Communications (ICECom 2010, Dubrovnik, Croatia, Sep. 20–23, 2010. [23] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, U.K.: Peter Peregrinus, 1988. [24] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, no. 2, pp. 151–155, Mar. 1969.

The authors thank Dr. A. Yaghjian for numerous fruitful discussions on the physical limits of electrically small antennas facilitating the work of this paper.

Oleksiy S. Kim received the M.S. and Ph.D. degrees from the National Technical University of Ukraine, Kiev, in 1996 and 2000, respectively, both in electrical engineering. In 2000, he joined the Antenna and Electromagnetics Group at the Technical University of Denmark (DTU), where he is currently an associate professor with the Department of Electrical Engineering, Electromagnetic Systems. His research interests include computational electromagnetics, metamaterials, electrically small antennas, photonic bandgap and

REFERENCES [1] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, 1948. [2] R. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 23–27, Jan. 1964. [3] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 672–676, May 1996. [4] H. A. Wheeler, “The spherical coil as an inductor, shield, or antenna,” Proc. IRE, vol. 46, no. 9, pp. 1595–1602, 1958. [5] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [6] R. Hansen and R. Collin, “A new Chu formula for Q,” IEEE Antennas Propag. Mag., vol. 51, no. 5, pp. 38–41, 2009. [7] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 953–960, Apr. 2004. [8] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, Mar. 2005. [9] S. R. Best, “A low Q electrically small magnetic (TE mode) dipole,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 572–575, 2009. [10] H. R. Stuart, H. R. Stuart, and C. Tran, “Small spherical antennas using arrays of electromagnetically coupled planar elements,” IEEE Antennas and Wireless Propag. Lett., vol. 6, pp. 7–10, 2007. [11] O. S. Kim, “Low-Q electrically small spherical magnetic dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2210–2217, July 2010. [12] O. S. Kim, “Novel electrically small spherical electric dipole antenna,” presented at the Int. Workshop on Antenna Technology (iWAT2010), Lisbon, Portugal, 2010. [13] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1898–1906, June 2010. [14] O. S. Kim and O. Breinbjerg, “Lower bound for the radiation Q of electrically small magnetic dipole antennas with solid magnetodielectric core,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 679–681, Feb. 2011. [15] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005. [16] A. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antennas Propag., vol. 55, no. 61, pp. 1495–1505, 2007.

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plasmonic structures.

Olav Breinbjerg (M’87) was born in Silkeborg, Denmark on July 16, 1961. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Denmark (DTU) in 1987 and 1992, respectively. Since 1991, he has been on the faculty of the Department of Electrical Engineering (formerly Ørsted1DTU, Department of Electromagnetic Systems, and Electromagnetics Institute) where he is now Full Professor and Head of the Electromagnetic Systems Group including the DTU-ESA Spherical Near-Field Antenna Test Facility. He was a Visiting Scientist at Rome Laboratory, Hanscom Air Force Base, MA, in the fall of 1988 and a Fulbright Research Scholar at the University of Texas at Austin, in the spring of 1995. His research is generally in applied electromagnetics—and particularly in antennas, antenna measurements, computational techniques and scattering—for applications in wireless communication and sensing technologies. At present, his interests focus on meta-materials, antenna miniaturization, and spherical near-field antenna measurements. He is the author or coauthor of more than 40 journal papers, 100 conference papers, and 70 technical reports, and he has been, or is, the main supervisor of 10 Ph.D. projects. He has taught several B.Sc. and M.Sc. courses in the area of applied electromagnetic field theory on topics such as fundamental electromagnetics, analytical and computational electromagnetics, antennas, and antenna measurements at DTU, where he has also supervised more than 70 special courses and 30 M.Sc. final projects. Furthermore, he has given short courses at other European universities. He is currently the coordinating teacher at DTU for the 3rd semester course 31400 Electromagnetics, and the 7-9th semester courses 31428 Advanced Electromagnetics, 31430 Antennas, and 31435 Antenna Measurements in Radio Anechoic Chambers. Prof. Breinbjerg received a US Fulbright Research Award in 1995. Also, he received the 2001 AEG Elektron Foundation’s Award in recognition of his research in applied electromagnetics. Furthermore, he received the 2003 DTU Student Union’s Teacher of the Year Award for his course on electromagnetics.

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Experimental Determination of DRW Antenna Phase Center at mm-Wavelengths Using a Planar Scanner: Comparison of Different Methods Pablo Padilla, Patrik Pousi, Aleksi Tamminen, Juha Mallat, Juha Ala-Laurinaho, Manuel Sierra-Castañer, Member, IEEE, and Antti V. Räisänen, Fellow, IEEE

Abstract—A study of the phase center position of dielectric rod waveguide (DRW) antennas (silicon and sapphire) for different millimeter-wave (mm-wave) frequencies is presented in this document. Phase center position is determined using data obtained by planar scanning and analyzed by means of different methods: least squares fit method with and without weighting coefficients and plane wave spectrum (PWS) analysis method. A study of the radiation pattern and phase center position for different mm-wave frequencies is provided and the results of the different methods are presented and compared. Index Terms—Dielectric rod waveguide (DRW) antennas, millimeter-wave frequencies, phase center determination, planar measurement system, plane wave spectrum (PWS).

I. INTRODUCTION

I

N the area of antennas at millimeter-wave (mm-wave) frequencies, there is a growing interest for dielectric rod waveguide (DRW) antennas, e.g., [1]–[4]. For their proper application to mm-wave systems (for instance, as reflector/lens feeders), it is essential to determine not only the radiation pattern but also the phase center position of the antenna, together with the variation of its position when the operating frequency of the DRW antenna is changed. This paper provides an approach to DRW antennas radiation pattern characterization and phase center calculation based on measurements carried out with a planar acquisition system at millimeter wavelengths. The most straightforward methods for phase center determination are derived from rotational measurements [5]. In these methods [6]–[8], the phase center can be determined experimentally by finding a spherical equi-phase surface over a range of directions. Then, the center of this surface corresponds to the

Manuscript received February 17, 2010; revised November 16, 2010; accepted January 15, 2011. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported by a cooperation agreement between the Department of Radio Science and Engineering of Aalto University (Finland) and the Radiation Group of the Technical University of Madrid (Spain). P. Padilla and M. Sierra-Castañer are with the Radiation Group (GR), Technical University of Madrid (Universidad Politécnica de Madrid—UPM), Madrid, Spain (e-mail: [email protected]). P. Pousi, A. Tamminen, J. Mallat, J. Ala-Laurinaho, and A. V. Räisänen are with the MilliLab and SMARAD, Department of Radio Science and Engineering, Aalto University, Espoo, Finland (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.2011.2158782

phase center of the antenna. To obtain the phase pattern, the antenna under test is rotated in the test zone of the rotational measuring system. However, rotational measurements at mm-wave frequencies suffer from accuracy problems, yielding inaccurate results [9]. Rotary joints at these frequencies are one critical problem due to the ripples introduced into the amplitude and the uncertainty of the phase pattern. Thus, in this work the main purpose is to explore the phase center position for mm-wave DRW antennas with a planar acquisition system. Some considerations about the planar measurement configuration used and the different applied methods for phase center determination with measurements are given in this paper. Phase center determination for different DRW antennas (silicon and sapphire) is described along with radiation pattern characterization. The paper is organized as follows: Section II introduces DRW antennas with different dielectric rod materials. Section III presents DRW antenna radiation pattern characterization for different frequencies. In Section IV, different analysis methods for phase center determination are provided. Section V exhibits the results in terms of phase center position, by means of the proposed analysis methods. Finally, in Section VI, conclusions are drawn. II. DIELECTRIC ROD WAVEGUIDE ANTENNAS The measured antennas are based on dielectric rod waveguides made of high-permittivity low-loss materials. The antenna rod with a tapered section is inserted in a metal waveguide, thus creating a transition to free space, analogously to the behavior of a horn structure. The relatively high dielectric constant of the rod facilitates the transition of the waveguide modes to the radiating free space modes. A dielectric holder (typically Teflon) is placed to accurately locate the DRW in the waveguide. This mounting scheme provides a very good matching, thus it is not necessary to use horn antennas or horn-shaped ends for the waveguide [2]. These DRW antennas have been previously found to present a fairly constant beamwidth versus frequency and a low return loss [2]. This is promising for their application as feed antennas. . Although higher modes are The mode of propagation is also possible in DRW antennas at high frequencies, in this case these higher order modes are not excited [10], [11]. The antennas measured in this paper are made of silicon , ) and sapphire ( , ( , ). The cross section of the DRW

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Fig. 1. DRW antenna prototype model.

is 0.5 1.0 . Length of the silicon rod is 42 mm. The sapphire rod is 20 mm long. The same DRW antennas have been studied earlier in [1], [2]. Fig. 1 presents the DRW antenna prototype model used in this paper. The waveguides contain a tapering section both in the feed and radiation end. The tapering is only in E plane (vertical) as it is relatively easy to manufacture and it has as good performance as the pyramidal taper [12]. The tapering section in the antennas used in this study is 6 mm in both radiation and feed ends. Different radiation patterns can be obtained by modifying the radiation taper. A longer tapering section provides a more directive antenna. III. RADIATION PATTERN CHARACTERIZATION Radiation patterns are measured with an AB Millimètre MVNA-8-350 vector network analyzer. A NSI 200 V–5 5 planar scanner is used in the far field of the DRW antennas and an open-ended waveguide WR-10 as a measuring probe. – . The planarity of the scanner is approximately Probe compensation [13] is applied to the measured results in order to have the proper radiation pattern of the DRW antennas. These amplitude pattern values are used in the analysis methods provided for phase center determination, as discussed in Section IV. Fig. 2 presents the radiation pattern of the silicon DRW antenna. It is observed from the measurement results that the E-plane patterns are slightly asymmetric. This is due to the small asymmetry in the radiation taper. Another reason is that the matching from the metal waveguide to the DRW is worse at lower frequencies. This causes more additional radiation from the transition region. IV. PHASE CENTER DETERMINATION Different analysis methods for the computation of the phase center of the DRW antennas are applied and they are described in this section. Sections IV-A and IV-B make reference to the least squares fit method, meanwhile Section IV-C makes reference to PWS analysis method. A. Least Squares Fit The calculation of the phase center with the least squares fit method comes from the comparison between the measured phase and the theoretical one for different distances (DRW antenna to probe distances). The distance that provokes the smallest deviation of theoretical results regarding measured ones is the one that establishes the phase center position. As mentioned, to minimize the deviation between theory and measurements, the least squares fit procedure is applied with the theoretical phase value as given in Fig. 3 and in (1). Equation (2) offers the minimum

Fig. 2. Radiation pattern of silicon DRW antenna, for different working frequencies (83.5, 93.5, 103.5, and 113.5 GHz). (a) H plane, (b) E plane, (c) Complete pattern for 83.5 GHz.

phase deviation calculation for the least squares fit, for different distances from the probe. Variables and identify the measurement points in the – measuring grid

(1)

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Fig. 3. Measurement setup. The tip of the DRW antenna is located at the origin of the coordinate system. The measurement grid lays in the x–y plane at z . z

= Fig. 4. Iterative phase center determination. The focal length of the parabolic reflector is varied to find maximum directivity and the corresponding location of the DRW antenna phase center. The curvature of the phase front is exaggerated in the picture.

(2) where is the fixed distance from the tip of the DRW anis the tenna (located at the origin) to the measurement grid, distance from the tip to the phase center, and are the measured and theoretical phase value at each point of the grid. B. Least Squares Fit With Weighting Coefficients This approach should be considered as a variation of the previous one, considering coefficients for the different terms of the summation: the values corresponding to points placed far away from the boresight direction are considered with lower weight. The weighting coefficients are identified with the values of the amplitude in the radiation pattern, normalized to the highest amplitude value. Thus, the weight applied to each point of the grid is directly related to the importance of each point in the phase configuration. Equation (3) yields the modified computation of the phase error considering the weighting coefficients

(3) where is the normalized measured amplitude and are, and again, the measured and theoretical phase values at each point of the grid. C. Plane Wave Spectrum Analysis Method The third analysis method proposed in this paper gives importance to the application of the DRW antenna as a feed of a parabolic reflector. A parabolic reflector antenna transforms a spherical wave front to a planar one, when the source of the spherical wave front is placed in the focus of the reflector. Indeed, it would be possible to measure the position of the DRW antenna phase center by repeated directivity measurement of a well-defined parabolic reflector antenna. The -coordinate of the DRW antenna would be iteratively varied until its apparent phase center and focus of the reflector coincide, indicated by the global maximum of the directivity of the antenna system. The aforementioned iteration for the highest directivity could be performed computationally using measured radiation pattern

Fig. 5. Equivalent array of sources, with the proper choice of phase value in each point of the array.

of the DRW antenna and by a conceptual parabolic reflector antenna. However, the proposed analysis method is based on a planar measurement at a fixed distance and, therefore, the above-mentioned iteration cannot be fully adopted here. The measurement distance defines the focal length of the concep. Fig. 4 illustrates the tual parabolic reflector being iteration of the conceptual paraboloid. Three paraboloids with different focal lengths are shown. Measured DRW source is applied to the focus of the conceptual paraboloid, a planar wave front emerges only when the focus coincides with the phase center of the DRW antenna (solid line, case 3). The consideration of the conceptual reflector antenna yields complex geometric calculations in terms of distances. However, the geometry of the parabolic configuration and its complexity can be significantly simplified with the calculation of the plane wave spectrum (PWS) of the equivalent array of sources, with the proper choice of the amplitude and phase value in each point

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Fig. 6. (a) Measurement setup scheme for DRW antenna. (b) Sapphire prototype. (c) Silicon prototype. The part of the DRW antennas outside the waveguide B is 11.5 mm and 19 mm for sapphire and silicon antennas, respectively.

of the array. Fig. 5 depicts the equivalent configuration to be taken into account. As it is detailed, the DRW antenna and the conceptual parabolic reflector are replaced by the equivalent array of sources placed at the measurement plane. The size of the corresponding antenna array being the same as the measurement grid differs slightly from the size of the aperture of the conceptual parabolic reflector. However, in this case this has no effect on the results of the iteration. The PWS of the equivalent array is as follows:

(4) (5) where, again, and are the measured amplitude and phase at each point of is the distance between points in the the measuring grid, grid, and are the components of the wave vector in Cartesian coordinate system and and are the reference to the location of the different points in the grid. is the expected phase value at each point of the grid, according to (2). In this case, the phase center of the DRW antenna coin, which provides the highest cides with the value of directivity estimation from the PWS, according to (4).

Fig. 7. Details of the acquisition setup for measurements.

Fig. 8. Measurement setup. (a) Dimensions and distances. (b) Measuring grid.

V. PHASE CENTER DETERMINATION: EXPERIMENTAL RESULTS A. Measurement Setup In this section, the results obtained in the planar acquisition system are processed in order to derive the phase center of the DRW antenna with the three methods previously mentioned. The different DRW antennas (sapphire and silicon) are measured at different frequencies (83.5, 93.5, 103.5, and 113.5 GHz). Fig. 6 exhibits the measurement scheme and DRW antenna prototypes. Fig. 7 shows the measurement setup. The planar setup for measurements is fixed as follows (Fig. 8): • and mm, two distances for each antenna and frequency, in order to validate results obtained for the phase center location; • for sapphire antenna, 19 mm for silicon antenna; • both in - and -coordinates. This dimension, together with , results in about field of

view. Please note that this is approximately the area capturing the main beam of the DRW antenna down to the level of about (see Fig. 2). This choice reflects potential use of the DRW antenna as a feed for a reflector. The pattern is acquired for a rectangular grid . B. Probe Compensation The planar measurement setup includes an open-ended WR-10 waveguide, in Fig. 9, as a probe. The DRW antenna phase and amplitude patterns are corrected by means of the subtraction of the probe pattern. As previously referred in Section III, the effects of the probe have to be eliminated from the measurements. The open-ended WR-10 waveguide is selected in order to have low influence of the probe in the pattern measurements due to its inherently wide beam. The probe patterns (amplitude and phase) are measured using two identical open-ended WR-10 waveguide probes. The mea-

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TABLE I CALCULATED PHASE CENTER POSITION (FROM THE ANTENNA TIP) FOR THE DIFFERENT ANALYSIS METHODS, FOR SAPPHIRE AND SILICON DRW ANTENNAS

Fig. 9. WR-10 waveguide as a measuring probe.

Fig. 11. Comparison between phase center positions offered by the three analysis methods at different frequencies, for silicon DRW antenna. (a) 204 mm, (b) 224 mm.

C. Phase Center Location Results

Fig. 10. Comparison between phase center positions offered by the three analysis methods at different frequencies, for sapphire DRW antenna. (a) 204 mm, (b) 224 mm.

sured patterns contain information of both antennas working together and are corrected accordingly. Although the phase center of the open-ended waveguide probe can be assumed with a good accuracy to be located at the aperture plane [14], the open-ended WR-10 waveguide probes are calibrated probes of the measuring system and their phase center positions are accurately known.

Results obtained in the planar scanner acquisition system, with probe compensation, are processed in order to derive the phase center position of the two different DRW antennas, with the three different approaches previously described. Figs. 10 and 11 offer the results for sapphire and silicon DRW to antennas, respectively, for two different distances the probe. Table I summarizes the phase center results for both DRW antennas. The differences appearing in the results are not due to the measuring instrumentation. The effect of mechanical uncertainties in relation to phase center position determination can be estimated to be in the order of 0.1 mm or less.

PADILLA et al.: EXPERIMENTAL DETERMINATION OF DRW ANTENNA PHASE CENTER AT mm-WAVELENGTHS

However, uncertainty in the manually measured distance remains. The estimated uncertainty of manual measure. As a consequence, the vertical scale in ment is Figs. 10 and 11 is uncertain up to this value. The aforementioned results are based on the selection of the field of view. Different choice of the field of view might result in slightly different phase center position. When the tendency of the obtained results for the phase center position is analyzed, it can be seen that the phase center position is moving towards the tip of the antenna when the frequency is increased. The phase center of the probe for each frequency and the distances of the measurement scheme are known and they are taken into account in the results. The measured movement of the phase center towards the tip of the DRW antenna is potentially linked to the approximately constant beamwidth that has been observed for these antennas. For this kind of high-permittivity DRW antennas, further analysis of the relation between radiation pattern parameters and phase center movement is a subject of a separate study. When analyzing the results, it is seen that there is a slight difference between the PWS analysis method and the least squares fit method (either with or without weighting coefficients). It can be noticed that least squares fit methods are not considering any kind of tendency in the summation error: although the phase difference between theory and measurements can be positive or negative, only the square value is considered in the summation, neglecting the positive or negative nature of the errors in the summation. Indeed, simple analytical and numerical analysis shows that these methods give slightly different results in the realistic case of nonspherical wavefronts. When applying to an ideal isotropic antenna, these methods converge to the same results. The authors suggest the PWS analysis method to be applied when designing a reflector system with highest directivity.

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[2] D. Lioubtchenko, S. Dudorov, J. Mallat, and A. V. Räisänen, “Dielectric rod waveguide antenna for W band with good input match,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 1, pp. 4–6, Jan. 2005. [3] S. Kobayshi and R. Mittra, “Dielectric tapered rod antennas for millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 30, no. 1, pp. 54–58, Jan. 1982. [4] J. P. Pousi, D. V. Lioubtchenko, S. N. Dudorov, and A. V. Raisanen, “High permittivity dielectric rod waveguide as an antenna array element for millimeter waves,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 714–719, Mar. 2010. [5] J. Wang, “An examination of the theory and practices of planar nearfield measurements,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 746–753, Jan. 1988. [6] P. A. Beeckman, “Analysis of phase errors in antenna-measurements applications to phase-pattern corrections and phase-centre determination,” IEE Proc. H Microw., Antennas Propag. , vol. 132, no. 6, pp. 391–394, Oct. 1985. [7] J. Richter, J. Garbas, and L.-P. Schmidt, “Mean and differential phase centers of rectangular dielectric rod antennas,” in Proc. 34th Eur. Microw. Conf., 11–15, 2004, pp. 1193–1196. [8] J. Richter, M. Müller, and L.-P. Schmidt, “Measurement of phase centers of rectangular dielectric rod antennas,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 20–25, 2004, pp. 743–746. [9] J. Mallat, A. Lehto, and J. Tuovinen, “Antenna phase pattern measurements at millimeter wave frequencies using the differential phase method with only one power meter,” Int. J. Infrared and Millimeter Waves, vol. 15, no. 9, pp. 1497–1506, 1994. [10] S. Dudorov, Rectangular Dielectric Waveguide and Its Optimal Transition to a Metal Waveguide, Doctoral Thesis. Helsinki, Finland: Radio Lab., Helsinki Univ. Technol., Otamedia, 2002. [11] D. Lioubtchenko, S. Tretyakov, and S. Dudorov, Millimeter-Wave Waveguides. The Netherlands: Kluwer Academic, 2003. [12] S. Kobayashi, R. Mittra, and R. Lampe, “Dielectric tapered rod antennas for millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 30, pp. 54–58, Jan. 1982. [13] D. Paris, W. Leach Jr., and E. Joy, “Basic theory of probe-compensated near-field measurements,” IEEE Trans. Antennas Propag., vol. 26, no. 3, pp. 373–379, May 1978. [14] A. V. Shishkova, S. N. Pivnenko, O. S. Kim, and N. N. Gorobets, “Phase radiation characteristics of an open-ended circular waveguide,” in Proc. Int. Conf. on Math. Methods in Electromagn. Theory, Kiev, Russia, Sep. 2002, pp. 361–363.

VI. CONCLUSION At mm-wave frequencies, a planar scanning system is a respectable alternative to the rotational measuring systems. In this paper, planar scanning is utilized and the experimental results are analyzed with different analysis methods for the phase center determination: the least squares fit method with and without weighting coefficients, and PWS analysis method. Experimental results are obtained at four different frequencies in W band for silicon and sapphire DRW antennas. The results with these methods provide similar values for the DRW antenna phase center location and they let conclude that the phase center is moving towards the tip of the antenna when increasing the frequency. The least squares fit method (either with or without weighting coefficients) is assumed to be the most used method for phase center determination. However, the alternative method proposed in this work, based on PWS analysis, provides valuable results as it gives importance to the application of the antenna under test as a feed of a parabolic reflector. REFERENCES [1] D. Lioubtchenko, S. Dudorov, J. Mallat, J. Tuovinen, and A. V. Räisänen, “Low-loss sapphire waveguides for 75–110 GHz frequency range,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 6, pp. 252–254, Jun. 2001.

Pablo Padilla was born in Jaén, south of Spain, in 1982. He received the Telecommunication Engineer degree and the Ph.D. degree from the Technical University of Madrid (UPM), Spain, in 2005 and September 2009, respectively. In 2007, he was with the Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, as an invited Ph.D. student. In 2009, he carried out a short postdoctoral stay with the Helsinki University of Technology (TKK). Currently, he is an Assistant Professor with the Universidad de Granada (UGR) and collaborates with the Radiation Group of the UPM. His research interests include antenna design and synthesis and the area of active microwave devices.

Patrik Pousi was born in Vantaa, Finland, in August 1976. He received the Master of Science (Tech.) and Licentiate of Science (Tech.) degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 2003 and 2006, respectively, where he is currently working toward the Doctor of Science (Tech.) degree. Since 2004, he has been a Research Engineer with the Department of Radio Science and Engineering, TKK. At the beginning of 2010, TKK merged with two other universities and is now Aalto University. His current research interests include active and passive dielectric rod waveguide components for millimeter wavelengths.

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Aleksi Tamminen was born in Ruotsinpyhtää, Finland, in 1982. He received the Bachelor’s (Tech.) degree and Master of Science (Tech.) degree in electrical engineering (with distinction) from the Helsinki University of Technology (TKK), Espoo, Finland, in 2005 and 2007, respectively. Since 2005, he has been a Trainee, Research Assistant, and currently a Research Engineer pursuing the Doctor of Science (Tech.) degree. He is involved with millimeter-wave measurement projects, especially with THz-imaging, with the Department of Radio Science and Engineering, Aalto University, formerly TKK.

Juha Mallat was born in 1962 in Lahti, Finland. He received the Diploma Engineer (M.Sc.) with honors, Lic. Tech., and Dr. Tech. degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1986, 1988, and 1995, respectively. He has been with the TKK Radio Laboratory (and its Millimetre Wave Group) since 1985, working as a Research Assistant, Senior Teaching Assistant, and Research Associate until 1994. From 1995 to 1996, he was a Project Manager and Coordinator in an education project between TKK and the Turku Institute of Technology. Since 1997, he has been a Senior Scientist in Millimetre Wave Laboratory of Finland—ESA External Laboratory (MilliLab), with the exception of a period of one year during 2001–2002 when he served as a Professor (pro tem) in Radio Engineering with TKK. In January 2010, TKK was part of a merger in the new foundation-based Aalto University. He now works in the Department of Radio Science and Engineering with continued participation also in the activities of MilliLab. His research interests and experience cover various topics in radio engineering applications and measurements, especially at millimeter wave frequencies. He has been involved also in building and testing millimeter-wave receivers for space applications.

Juha Ala-Laurinaho was born in Parkano, Finland, in 1969. He received the Diploma Engineer (M.Sc.) degree in mathematics, and the Lic.Sc. (Tech.) and D.Sc. (Tech.) degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1995, 1998, and 2001, respectively. He has been with the TKK serving in the Radio Laboratory during 1995–2007 and currently (2008-) in the Department of Radio Science and Engineering. At the beginning of 2010, TKK merged with two other universities forming Aalto University. During 1995, he worked as a Research Assistant and, since 1996, he has been a Research Associate. He has been a Project Manager in many millimeter-wave technology related projects including several hologram CATR development projects funded by the European Space Agency (ESA). His current research interests are the antenna measurement techniques for millimeter and submillimeter waves, the lens antennas, and the millimeter wave imaging.

Manuel Sierra-Castañer (M’95) was born in Zaragoza, Spain, in 1970. He received the Telecommunication Engineer degree in 1994 and the Ph.D. degree in 2000, both from the Technical University of Madrid (UPM), Madrid, Spain. Since 1997, he has been with the University “Alfonso X” as a Teaching Assistant, and since 1998, with the UPM as a Research Assistant, Assistant, and Associate Professor. His current research interests are in planar antennas and antenna measurement systems.

Antti V. Räisänen (S’76–M’81–SM’85–F’94) received the Doctor of Science (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1981. He was appointed to the Professor Chair of Radio Engineering at TKK in 1989, after holding the same position pro tem in 1985 and 1987–1989. He has held Visiting Scientist and Professor positions at the Five College Radio Astronomy Observatory (FCRAO) and the University of Massachusetts, Amherst (1978–1979, 1980, 1981), at Chalmers University of Technology, Göteborg, Sweden (1983), at the Department of Physics, University of California, Berkeley (1984–1985), at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena (1992–1993), and at the Paris Observatory and the University of Paris 6 (2001–2002). Currently, he is supervising research in millimeter-wave components, antennas, receivers, microwave measurements, etc., at Aalto University, Department of Radio Science and Engineering and Millimetre Wave Laboratory of Finland—ESA External Laboratory (MilliLab). Currently, he is also Head of the Department of Radio Science and Engineering, Aalto University. He has authored and coauthored approximately 400 scientific or technical papers and six books, e.g., Radio Engineering for Wireless Communication and Sensor Applications (Norwood, MA: Artech House, 2003). Dr. Räisänen has been Conference Chairman of several international microwave and millimeter wave conferences including the European Microwave Conference in 1992. In 1997, he was elected the Vice-Rector of TKK for the period of 1997–2000. He served as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES from 2002 to 2005. He is a member of the Board of Directors of the European Microwave Association (EuMA) for 2006–2008 and 2009–2011. He is Chair of the Board of Directors of MilliLab. He has been a Fellow of Antenna Measurement Techniques Association (AMTA) since 2008. In 2009, he received AMTA’s Distinguished Achievement Award. The Centre of Smart Radios and Wireless Research (SMARAD) that he led at Aalto University has obtained the national status of CoE in Research for 2002–2007 and 2008–2013.

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Truncation-Error Reduction in Spherical Near-Field Scanning Using Slepian Sequences: Formulation for Scalar Waves Kristopher T. Kim, Senior Member, IEEE

Abstract—We discuss the error that results when the far field is reconstructed from spatially truncated near-field samples and present an effective mitigation technique based on the Slepian sequence for acoustic spherical near-field scanning. We show that the truncation error is inevitable whenever the far field is reconstructed using the classical near-field-to-far-field transformation. After discussing the Slepian sequence for a truncated spherical surface and its analytic and numerical properties, we apply it to expand truncated NF samples and derive the near-field-to-far-field transformation of the resulting expansion coefficients, from which the far field can be computed. We demonstrate the efficacy of this transformation by applying it to near-field scanning for bistatic scattering from a sphere and radiation from a current distribution. Index Terms—Antenna measurement, bistatic scattering, nearfield scanning, near-field-to-far-field transformation, Slepian sequence, truncation error.

I. INTRODUCTION

E

LECTROMAGNETIC near-field (NF) scanning [1]–[3] has been an indispensable technique for the antenna measurement community for many decades. Instead of measuring the radiation pattern directly at a far-field (FF) distance, one samples the radiated field in the vicinity of an antenna under test at a sufficient sampling rate over a planar, cylindrical, or spherical surface and computes the FF pattern from the NF samples using, respectively, the planar, cylindrical, or spherical near-field-to-far-field (NF-to-FF) transformation. NF techniques have also been explored to determine the bistatic RCS of a target [4]–[8] and bistatic scattering from a random rough surface [9]. Among the three types of NF scanning, the spherical NF scanning technique [10]–[12] is suitable when it is necessary to determine the FF pattern over a wide range of horizontal and vertical directions. It is also particularly suitable for bistatic scattering [4], since targets, unlike antennas which are designed to focus radiated energies, do not exhibit

Manuscript received September 17, 2010; revised November 23, 2010; accepted January 06, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported in part by the Air Force Office of Scientific Research. Portions of this paper were presented at the IEEE Int. Symp. on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting Toronto, ON, Canada, Jul. 11–17, 2010. The author is with the Electromagnetics Division, Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, MA 01731 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.2011.2158968

focused scattering patterns in general, except for the forward scattering peak. When the NF samples are collected over the entire spherical surface at an appropriate sampling rate, the radiated/scattered field can be determined anywhere outside the minimum sphere that completely encloses the antenna/target under test. In practice, however, the samples are almost always collected over a truncated surface and, as a result, the FF reconstructed from them suffers from the truncation error: the reconstructed FF is accurate over only a subset of the angular directions at which the NF samples are collected. Because of the truncation error, it is always necessary to collect the NF samples at a wider range of angles than that of the angular directions at which the FF is to be determined. This necessity is onerous enough to increase the NF acquisition time. In spherical NF antenna measurements, the truncation error can completely be eliminated by collecting the NF samples over the entire spherical surface, even though this is seldom done in practice as mentioned above. However, as noted in [13], the samples cannot be collected over the entire spherical surface in NF scanning for bistatic scattering. Scattering measurements require an externally generated incident plane wave and the NF probe cannot be placed in the region between the target under test and the source that generates the incident wave—its presence would distort the incident wave. Thus, truncation errors are unavoidable in NF bistatic scattering measurements and limit the range of angles at which bistatic RCS can be reconstructed. It appears that, since the planar and cylindrical scanning techniques are more widely used and since the spherical scanning technique, if desired, can avoid truncation errors in antenna measurement, the existing body of research [5], [14]–[19] on truncation-error reduction is mainly concerned with reducing the FF errors that arise when the linear extent of a scan surface is terminated, as occurs in planar scanning and the vertical truncation in cylindrical scanning. Some of the techniques explored in the papers referenced above are scan-geometry independent and may be applicable to spherical scanning. However, as far as the author is aware, the only work that specifically addresses truncation-error reduction in spherical scanning is the work by Wittmann, Stubenrauch and Francis [20], where a regularized, constrained least-squares method is developed for hemispherical NF data. We discussed, in [13] for 2D cylindrical/spherical scanning, that the truncation error is unavoidable whenever the FF is reconstructed from a truncated data set using the classical NF-to-FF transformation based on the multipole expansion of NF samples. Since the multipole field is not orthogonal over

U.S. Government work not protected by U.S. copyright.

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a truncated circle, we expanded the NF samples using the Slepian sequence [21, and references therein] that is specifically constructed to be orthogonal over a given truncated scan circle. We derived the NF-to-FF transformation of the resulting expansion coefficients and showed that it reconstructs the FF more accurately than the classical transformation, albeit at a higher computational cost. In this paper, we extend the Slepian technique of [13] for 3D spherical scanning for scalar waves. The probe-corrected spherical NF scanning theory for scalar waves is reviewed in [22]. As is commonly done in this type of work, we shall use in this paper the electromagnetics terms, such as “antenna” and “RCS,” instead of “transducer” and “acoustic cross section,” respectively. The layout of this paper closely parallels that of [13]. In Section II, in addition to introducing the notations and the conventions used in this paper, we examine the shortcomings of the classical NF-to-FF transformation and show that the truncation error results invariably whenever the FF is computed from a truncated NF sample set using the classical NF-to-FF transformation. We also show, explicitly and without resorting to geometrical considerations, that the scan radius critically determines the extent of truncation errors. After a general discussion on the spatial band-limitedness of the radiated/scattered wave and its consequences on the expansion of truncated NF samples, we consider the Slepian sequence, originally developed for the 1D band-limited signal [21] and applied to 2D NF cylindrical/spherical scanning in [13], to 3D spherical scanning for scalar waves. Specifically, we show that the sequence may be regarded as a set of ordered functions that are orthogonal over a given truncated spherical surface with their “concentrations” over the truncated surface decreasing exponentially beyond a certain critical order. In Section III, we consider the expansion of a truncated NF sample set using the Slepian sequence and derive the NF-to-FF transformation for the resulting expansion coefficients from which the FF can be computed. Even though we rely on the Slepian-sequence formalism, the FF can be computed without explicitly constructing the Slepian sequence. In Section IV, we apply the new transformation to NF scanning for bistatic scattering from a sphere and show that it reconstructs the FF accurately over a wider range of angles than the classical transformation. We also examine the sensitivity of the reconstructed FF to signal-to-noise ratios and to the number of terms included in the expansion, and discuss the computational cost of the Slepian-sequence-based transformation relative to that of the traditional transformation. In the Appendix, we apply the Slepian formalism to a scan surface that is truncated only in the direction, and show that a substantial reduction in computational cost can be achieved. We conclude with a summary.

II. TRUNCATION ERRORS IN THE CLASSICAL NF-TO-FF TRANSFORMATION In this section, we examine how truncation errors arise in the classical NF-to-FF transformation and show that the scan radius is an important parameter that determines the extent of the truncation error for a given NF scanning scenario. The radiated/scat, may be expanded in terms of spherical tered field,

multipoles outside the minimum sphere of radius pletely encloses the antenna/scatterer under test

that com-

(1) with , a small Here, is the wavenumber; , the spherical Hankel function of the first kind of integer; is the spherical harmonic function of degree order . and order (2) is the associated Legendre function, and the where , is chosen so that normalization constant,

and (3) We note that the practical NF probe does not directly sample the radiated/scattered field at a point. Rather, the complex output of such a probe is a complicated but known function of the radiated/scattered field. The output of an axisymmetric probe, which may be regarded as the acoustic counter part of the electromagprobe” [10], can be expressed in a form similar netic “ to (1) [22]. The utility of spherical NF scanning is to extract the expan, from the NF samples, . If the sion coefficients, NF samples are collected over the entire spherical surface of raat an appropriate sampling rate, the expandius with , can be determined using the orthonorsion coefficients, , (3) mality of (4)

can be determined anywhere and, as a consequence, using (1). If, on the other hand, the NF samples are and collected over a truncated spherical surface, , then it is customary to assume that the field outside the truncated surface is zero and compute the expansion , coefficients (5)

Since , computed from does not agree with computed from over the entire angular region, leading to truncation errors. The angular regions where the two sets of field values diverge from each other may be estimated for a given set of NF measure-

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III. SLEPIAN SEQUENCE FOR A TRUNCATED SPHERICAL SURFACE Since it is the non-orthogonality of over a truncated spherical surface that leads to truncation errors in the classical NF-to-FF transformation, we may seek an alternative expansion of the NF scan data using a basis function set that is orthogonal over the truncated angular domain. It is well-known that the radiated/scattered field is spatially band limited [23]. for Stated in the context of (1), it is index limited: . However, the radiated/scattered field cannot also be angle limited, since a function cannot simultaneously be index and angle limited—just as a function cannot simultaneously be band and time limited [21]. This observation suggests that we may not be able to find a basis function set that satisfies the Helmholtz wave equation and vanishes identically outside the truncated spherical surface [13]. Following the procedure outlined in [21, pp. 205–215] for a 1D band-limited function and extended in [13] to an index-limited function that satisfies the 2D Helmholtz function, we, instead, seek a basis function set that is orthogonal over a truncated spherical surface and that results in the maximum conover the truncated surface. Thus, the centration of basis function set we are seeking will maximize

Fig. 1. Angular distributions, in logarithmic scale, of the jT (r; a; ;  ;  0  )j corresponding to different scan radii, a, for L = 27,  = 90 ,  = 180 and r = 1000. (a) a = 10; (b) a = 15; (c) a = 25; (d) a = 50.

ment parameters. Using (1) and (5), we may express , as a convolution of the truncated scan data,

Using (1), the above equation may be written as

, (7) where

(6) where (8) We note that , which is closely related to the taper function discussed in [5], determines the angular extent of . Compared the NF data that are required to compute corresponding to four in Fig. 1 are the , 15, 25, and ) for , different scan radii ( , , , and . As shown in the figure, the smaller gets, the wider becomes the shoulder and . Therefore, at a smaller scan around radius, one must collect the NF samples over a wider angular extent in order to compute the FF at and . If some of the NF samples are not available in the shoulder region as a result of scan surface truncation, as would occur when and are near the limits of the scan surface, then this leads to errors in computed FF values.

and and are, respectively, the column vector containing elements of and the Hermitian matrix conthe values of . taining the Even though every positive definite matrix is, in theory, invertable, , as we will see below, is numerically rank-deficient. It has full rank only when the scan surface is un-truncated, in which case it reduces to the identity matrix. and be the th eigenvector and eigenvalue of so Let . Or, that (9) where

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= 1 . . . 784 = 15 45   135  0156

Fig. 2. (A): Distribution of the  , n ; ; ; (B): The orthogonality ,  , of the V for the truncated spherical scan surface a  and L . A floor of is introduced to avoid taking the logarithm of zero. (a): Eigenvalue distribution; (b): Othogonality of V.

60   300

= 27

and is the diagonal matrix whose diagonal elements are . is always positive, is a positiveSince definite matrix. As a consequence, is an orthogonal matrix (10) and are real with [21]. for the scan Shown in Fig. 2(a) is the distribution of the and , and . surface, We note that the eigenvalues stay close to 1 for and drop off exponentially for , even though the rate of its drop is slower than that of the 2D case considered in [13]. is due The unexpected presence of the “fishhook” for to numerical errors. The eigenvalue distribution suggests that the truncated NF samples do not possess enough information to modes used in the expansion, uniquely determine the (1), [23]. Fig. 2(b) plots in logarithmic scale, confirming the orthogonality of the . We may use the to construct an expansion function set that is orthogonal over the truncated angular domain. We define (11) In the above equation, is the th element of the th are, respectively, eigenvector, , where the indices and and through related to (12) We shall interchangeably use and in the rest of the paper. The Slepian sequence [21] derived from , spherical surface,

, and and may be regarded as the for the truncated .

j ( )j

Fig. 3. Angular distributions of the s ;  , in logarithmic scale, corresponding to different values of n for the truncated scan surface considered ; (b) for Fig. 2. n is related to l; m through (12). (a) ; (c) ; (d) .

n = 500

n = 700

(

Using (9), the the truncated domain,

)

n = 50

n = 200

can be shown to be orthogonal over

(13)

physically correspond to the concensuggesting that the over the truncated spherical surface. trations of the Plotted in Fig. 3 are the angular distributions of the for , 200, 500, and 700 obtained for the truncated scan surface considered for Fig. 2. For , the is nearly entirely concentrated inside the scan surface with its peaks clus, tered near the middle of the truncated surface. For case, elevated levels of concencompared with the trations are observed outside the scan surface with its highest , significoncentrations occurring near its edges. For cantly higher concentrations are found outside the scan surface and 200 cases, and for compared with the virtually no concentration is observed inside the scan surface. as a function of and In Fig. 4, we plot the in the top figure, and the as a function of and in the bottom figure. Both plots clearly show that the are nearly entirely concentrated inside the truncated . As increases well besurface for those whose , increasingly higher levels of concentrayond where tion are found outside of the truncated surface, and for those

KIM: TRUNCATION-ERROR REDUCTION IN SPHERICAL NEAR-FIELD SCANNING USING SLEPIAN SEQUENCES

Fig. 4. Top Figure: js (;  =  )j as a function of  and n; Bottom Figure: js ( = =2; )j as a function of  and n. A floor of  0156 is introduced to avoid taking the logarithm of zero.

whose , the reside nearly completely outside the truncated surface. This shifting of the concentrations from the inside to the outside of the truncated surface with increasing is consistent with (13) and its interpretation that the correspond to the concentrations of the over the truncated spherical surface. Plotted as a circle in Fig. 2(a) are for computed from (13), which the agree well with the corresponding values obtained directly from the eigenvalue decomposition (EVD), (9). The orthogonality of is shown in Fig. 5. the IV. NF-TO-FF TRANSFORMATION USING THE SLEPIAN SEQUENCE In the previous section we have shown, using the theory of positive matrices and a numerical example, that the may be regarded as a sequence of functions that are orthogonal over the truncated surface and whose concentrations over the truncated surface, as determined by , decrease uniformly as increases. In this section, we explore the possibility of expanding the NF data, , in terms of the and determining the FF from the resulting expansion coefficients. We thus write (14) where we have truncated the expansion at , since increasingly are available over the scan less concentrations of the

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Fig. 5. Orthogonality of the s (; ) over the truncated spherical surface. A floor of  0156 is introduced to avoid taking the logarithm of zero.

surface as increases beyond . Using (13), the expan, can be obtained from the scan data, sion coefficients,

(15) They can also be determined directly from the expansion coefficients, of the truncated scan data, (5), using (11), (8), and (9), (16) where is the diagonal matrix of order with diagonal en; is the matrix obtained by taking tries is the diagonal matrix of the first columns of ; and order with diagonal entries . Thus, are . When the scan surface is una linear combination of , and , , and all reduce to the truncated, identity matrix of order . Thus, , and the Slepian-sequence expansion, (14), is identical to the classical multipole expansion, (1). In order to obtain the FF we need to establish how the transform as . Substituting (4) and (11) into (16) and are related to using (8) and (9), we may show that the

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the (unknown) expansion coefficients, , of the untruncated scan data, (4), (17) Since order

, where and

and when and are the identity matrices of , respectively, we may show that is the

pseudo-inverse [24] of and obtain the least-squares solution for in terms of from (17) (18) Substituting (18) into (17) and setting ((17) is valid for any ), we obtain the following NF-to-FF transformation for

Fig. 6. Diagonal dominance, in logarithmic scale, of for the considered in Fig. 2.

600

V

Q V V

with

P=

(19) where is the diagonal matrix of order whose diagonal entries are . We shall call (19) the Slepian sequence-based NF-to-FF transformation or the SS-NF-to-FF transformation for short. The FF can be computed either using (19) and (14), or (18) and (1). If the latter approach need not be computed explicitly. is taken, If it is necessary to perform probe correction, one may obtain the probe-corrected, classical expansion coefficients, , using, for example, the theory discussed in [22] for axially symmetric probes, from which the probe-corrected, SS-NF-to-FF transfor, can be obtained using mation expansion coefficients, (16). The FF can then be computed using either (18) or (19), as discussed above. We now examine whether (14) satisfies the scalar Helmholtz . Substituting (17) into (14) equation when (20) where . Since and , is a projection matrix, repre, of and senting the coupling between the indices, . (14) satisfies the scalar Helmholtz equation those of rigorously if is a diagonal matrix. Fig. 6 plots the in logarithmic scale for the NF scan scenario considered above. It shows a strong diagonal dominance, suggesting that (14) numerically satisfies the Helmholtz equation. V. NUMERICAL EXAMPLES AND DISCUSSION We consider NF scanning for bistatic scattering from a -radius sphere as shown in Fig. 7. The incident plane wave is traveling in the negative -direction and the forward scattering direction corresponds to and . A total of 1176 scattered field samples are collected by an ideal probe over the truncated scan surface, , and , at approximately every 4.3 in both and —a sampling rate that is one and a half times the Nyquist rate with . Fig. 8 compares two sets of FF solutions

Fig. 7. Scan geometry: The incident plane wave, propagating in the negative . Scattered fields are collected x-direction, is scattered by a sphere of radius and by an ideal probe on the truncated spherical surface, with radius . The angles, and , are defined with respect to the - and - axes, respectively, with the forward scattering direction and . corresponding to

3:5

60    300 z x  = 90

15  = 180



45    135 

computed from the NF samples with reference solutions. The top figure compares the -cut patterns at , while the bottom figure compares the -cut patterns at . The classical NF-to-FF solutions are obtained from the expansion coefficients, , (5). To obtain the FF from the SS-NF-to-FF transformation, we first construct the matrix using the trapezoidal rule to perform the angular integrations. Because of numerical errors, the thus obtained is not exactly Hermitian. , which is Hence, we perform the EVD, (9), on exactly Hermitian. This seems to reduce numerical errors in the computed and . We then construct the Slepian sequence, , using (11). The NF expansion coefficients, , that were obtained are next obtained using (16) from the above for the classical NF-to-FF solutions, and are then transformed to the FF expansion coefficients, , using the SS-NF-to-FF transformation, (19). For the FF solutions shown

KIM: TRUNCATION-ERROR REDUCTION IN SPHERICAL NEAR-FIELD SCANNING USING SLEPIAN SEQUENCES

Fig. 8. Comparison of the FF solutions computed from the truncated NF data for the bistatic scattering scenario shown in Fig. 7.

in Fig. 8, we set , where as can be seen from Fig. 2. (The NF-to-FF solutions shown in Fig. 8 (and in Figs. 9–12) are all computed at every 1 for plotting purposes, while the expansion coefficients that are used to compute them are all determined from the above scan data sampled at every 4.3 .) We have also computed the second set of SS-NF-to-FF solutions using (18) and (1) and found them to be in excellent agreement with the above SS-NF-FF solutions. The classical NF-to-FF transformation solutions deviate from the reference solutions for and in the top figure and and in the bottom figure; near the limits of the truncation angles, the difference is more than 5 dB at some bistatic angles. The geometrical formula, , discussed in [11], predicts approximately 15 for the extent of the angular truncation errors. The SS-NF-to-FF transformation solutions, on the other hand, agree within 0.1 dB with the reference solutions over the entire range of angles where the NF samples are available. In order to examine the sensitivity of the SS-NF-to-FF transformation solutions to , we plot in Fig. 9 the FF solutions obtained with , 551, 590, 622, and 655 (with the corresponding , , , , and , respectively). The respective abscissas of the top and bottom plots are limited to and . Both plots show that the SS-NF-to-FF transformation solutions agree with the reference solutions over a wider range of angles than the classical NF-to-FF solutions. Moreover, the agreement improves for

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Fig. 9. Dependence of the FF solutions on the numbers of terms included in the expansion, (14).

both pattern cuts as increases. (The exception is the solution for the cut shown in the bottom figure: The solution agrees better with the reference solution by a fraction of a dB over .) The sensitivity to observed in the regions near the truncation angles is due to the precipitous rise of the for in those angular regions as shown in Fig. 4 and to the fact that the are sensitive to small errors in the computed and as . We now examine the effects of noise on the SS-NF-to-FF transformation solutions. One of the important advantages of the NF scanning technique is that it is often possible to obtain high signal-to-noise ratio (SNR) samples as measurements are performed in a highly controlled environment: SNRs of 40–60 dB are routinely achieved [25]. We obtain from the exact NF samples considered above two sample sets with approximately 40 dB and 60 dB SNRs. Since the represent the concentrations of the within the truncated surface, we may want to avoid choosing a whose is excessively smaller than the inverse of the SNR. We first plot in Fig. 10 the FF solutions obtained from the 40 dB SNR sample set with , 504, 526, and 551 ( , respectively). Comparison with Fig. 9 shows that the addition of noise has degraded all SS-FF-to-NF transformation solutions. However, it has disproportionately affected the SS-NF-to-FF solutions with higher , as illustrated most noticeably by the solutions. Thus, choosing a that is not commensurate with the SNR leads to poor agreement. We note that the solutions (with ), agree best with

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Fig. 10. Dependence on the number of terms used in the expansion, (14), of the FF solutions obtained from the NF sample set with 40 dB SNR.

Fig. 11. Dependence on the number of terms used in the expansion, (14), of the SS-NF-to-FF solutions obtained from the NF sample set with 60 dB SNR.

the reference solutions. We similarly plot in Fig. 11 the FF solutions obtained from the 60 dB SNR set with , 526, 551, and 590 ( , respectively). Compared with the solutions plotted in Fig. 10, all solutions agree better with the reference solutions because of the higher SNR. Again, high- solutions do not necessarily agree better with the reference solutions. For example, the solution agrees best with the reference solution for the cut. However, it agrees least well with the reference solution for the cut. The choice of , with , results in the best overall agreement with the reference solutions. We observe from Figs. 8–11 that the SS-NF-to-FF solutions, whether obtained from the noise-free data set or from the data sets with finite SNRs, and regardless of the exact values used (as far as they are chosen not too aggressively relative to the SNRs), agree appreciably better with the reference solutions over a wider range of angles. As noted above, our recommendation for choosing according to is based on the theoretical interpretation that the represent the concentration of the over the truncated spherical surface. In addition, it was tested satisfactorily for up to 45 and . The SS-NF-to-FF transformation requires a complete EVD flops; while the clasof , which nominally requires sical transformation can be implemented using flops for each scan set [1]–[3], [11]. is independent of scan radius and scan data; analytically, it is a function of the limits of scan angles only, even though it also weakly depends on the sampling

intervals and because of the errors resulting from the numerical integrations. The EVD, therefore, needs to be performed only once and the resulting and can be recycled over subsequent scan sets that are collected using the same sampling rates and over the same angular extents. When pre-computed and are available, the Slepian-sequence approach requires flops due to the matrix-vector multiplications, in (16) and in (18). In bistatic scattering applications, it is often necessary to determine the RCS of a target corresponding to incident angles. Since each incident angle requires a separate scan, one then needs to scan the target times. In this situation, the Slepian-sequence approach would require a total of flops, regardless of whether one needs to perform the EVD or uses pre-computed and ; while the classical approach would require flops. Nevertheless, it would be extremely costly or unfeasible to perform the EVD for electrically large targets and antennas. In Appendix, we show that the EVD can be performed using flops if the scan surface is truncated only in the direction. As shown above, the truncation errors produced by the Slepian-sequence approach are mild compared with those produced by the classical transformation. The SS-NF-to-FF transformation, therefore, would need a smaller scan surface than the classical transformation to produce FF solutions with comparable truncation errors. This is demonstrated in Fig. 12, which compares the classical transformation solutions obtained from the samples collected over the enlarged scan surface,

KIM: TRUNCATION-ERROR REDUCTION IN SPHERICAL NEAR-FIELD SCANNING USING SLEPIAN SEQUENCES

Fig. 12. Comparison of the classical solutions computed from the extended NF sample set collected over 34  146 and 43  317 with the P = 551 SS-NF-to-FF transformation solutions shown in Fig. 11.

 

 

and with the SS-NF-to-FF transformation solutions shown in Fig. 11. The extended scan surface is approximately 42% larger than the original scan surface. In order to ensure an even comparison, the augmented sample set is obtained using the same sampling rate and has approximately the same SNR. Since NF sample acquisition time is, to first order, proportional to the scan surface and ultimately drives the overall cost of NF scanning, the SS-NF-to-FF transformation offers the potential to make the spherical NF scanning technique even more cost-effective. Having established that the Slepian approach requires a much smaller scan surface to produce comparable FF solutions, it would be useful for practical applications to derive a “rule of thumb” for the “inverse problem”: What is the minimum scan area (MSA) required for the SS-NF-to-FF transformation to accurately reconstruct the FF over a prescribed angular extent of interest? Based on the results presented above, we expect the MSA to depend on the SNR of the scan samples. Since the Slepian technique takes advantage of the fact that the radiated/scattered field is index-limited, it may also depend on the spherical harmonics spectral content of the noise. It appears that a more comprehensive approach needs to be undertaken in order to obtain a reliable and robust expression for the MSA and thus will not be considered in this paper.

whenever the classical NF-to-FF transformation is used to construct the FF from a truncated NF sample set, with the scan radius critically determining their extent. The lack of the orthogonality of the spherical harmonics over a truncated spherical scan surface has prompted us to look for an alternative expansion of NF samples. We have thus considered the Slepian sequence that is derived from the spherical harmonics and is constructed for a given scan surface. We have shown, through the theory of positive matrices and a numerical example, that the Slepian sequence may be regarded as a sequence of functions that are orthogonal over the truncated scan surface and are ordered in descending order of their concentrations over the truncated surface, with the concentrations beyond a critical order dropping off exponentially. We have next considered the expansion of a truncated NF sample set using the Slepian sequence and derived the least-squares-based NF-to-FF transformation for the resulting expansion coefficients, from which the FF can be computed. We have also shown that the FF can be computed without explicitly constructing the Slepian sequence. The numerical examples considered in this paper have demonstrated that, compared with the classical NF-to-FF transformation, this new transformation, which we have named the Slepian-sequence NF-to-FF transformation, significantly reduces FF truncation errors, as shown in Figs. 9–12 and Fig. 14. As a result, while computationally more expansive, it requires a smaller scan surface than the classical transformation to produce the FF with comparable accuracy. Furthermore, it requires neither axillary NF samples that are extraneous to the scan surface nor a densely sampled data set. The Slepian approach presented in this paper can readily be extended to fully electromagnetic spherical NF scanning, where the Slepian version of the transverse vector spherical harmonics can be constructed in a similar manner for a given truncated scan surface from the classical vector transverse spherical harmonics [26]. APPENDIX We show that the cost of performing the EVD, (9), to flops, if the scan can be reduced from surface is truncated only in the direction. We asand sume NF samples are available over . Using and , we may reduce (7) to the following form:

(21) where

VI. SUMMARY We have considered truncation-error reduction in spherical NF scanning for scalar waves. Truncation errors are inevitable

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and the

and

(22) are, respectively, the column vector containing elements of , and the real, symmetric

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Fig. 14. Comparison of the FF patterns for the NF scanning shown in Fig. 13. Fig. 13. NF scanning geometry for the current distribution,  (x; y; z ) = A  (x i=2) (y j=2) (z ), 10 i; j 10. The current amplitudes, A , are tapered in both x and y directions using the Taylor weights.

0

0

0 



matrix containing the values of . Since , we need to perform the EVD only for non-negative values of ,

represents the matrix containing where eigenvectors with non-negligible eigenvalues for a given the . The expansion coefficients, , can be obtained using (25). As shown in (16) for the general truncated surface, they can also be determined directly from the multipole-expansion , (5), of the truncated scan data, and the eigencoefficients, and , information,

(23)

(27)

is the orthogonal matrix containing the where eigenvectors of ; and is the diagonal matrix containing the corresponding eigenvalues. Instead of performing the EVD matrix, , (8), at the expense of on the operations, (21) allows us to perform the EVD on each , whose dimension gets smaller as increases from 0 to . flops. As a result, the total cost of the EVD is reduced to , using the We may construct the Slepian sequence, for each non-negative ,

is the diagonal matrix whose diagonal elements are where the non-negligible eigenvalues of ; is the diagonal values of matrix whose diagonal elements are the ; and is the column vector containing the values of , . Since is the pseudo-

(24) where

(28) from which the FF can be computed using (1). For completeness, we note that the FF Slepian-expansion coefficients, , can be obtained from the NF expansion coefficients,

For the negative the

, we may use the symmetry relation, . If we denote to be th element of the , and similarly for , then they satisfy the orthogonality relation,

, we may obtain the least-squares solution for inverse of in terms of ,

(25) Thus, represent the concentrations of the the truncated scan surface. We may expand the NF scan data using the , non-negligible

over with

(26)

(29) where denotes the diagonal matrix whose diagonal values of , entries contain the . Thus, the FF can also be obtained using (29) and . (26) and letting Fig. 13 schematically shows the NF scanning geometry for the current distribution, , , where the current amplitudes, , are tapered in both and directions using the Taylor weights. The radiated fields are sampled over the scan surface, , , , in steps of , which is one half the Nyquist interval. No noise is used. Fig. 14 is added and the threshold eigenvalue of compares the -cut FF pattern at obtained from (28) and

KIM: TRUNCATION-ERROR REDUCTION IN SPHERICAL NEAR-FIELD SCANNING USING SLEPIAN SEQUENCES

(1) with the pattern obtained from the classical NF-to-FF transformation and the reference pattern. (The two NF-to-FF solutions are computed at every 1 for plotting, using the respective expansion coefficients obtained from the scan samples with the sampling interval specified above.) The geometrical formula , predicts approximately 14 for the extent of [11], truncation error. The SS-NF-to-FF transformation significantly reduces the truncation error. ACKNOWLEDGMENT The author wishes to thank the two anonymous reviewers for their helpful comments. REFERENCES [1] A. D. Yaghjian, “An Overview of near-field antenna measurements,” IEEE Trans. Antenna Propag., vol. 34, no. 12, pp. 435–445, July 1986. [2] J. Appel-Hansen, J. D. Dyson, E. S. Gillespie, and T. G. Hickman, , A. W. Rudge, Ed. et al., “Antenna measurements,” in The Hanbook of Antenna Design. London: Peter Peregrinus, 1986, vol. 1. [3] M. H. Francis and R. C. Wittmann, “Near-field scanning measurements: Theory and practice,” in Modern Antenna Handbook, C. A. Balanis, Ed. New York: Wiley, 2008, ch. 19. [4] M. G. Cote and R. M. Wing, “Demonstration of bistatic electromagnetic scattering measurements by spherical near-field scanning,” in Proc. AMTA Symp., 1993, p. 191. [5] T. B. Hansen, R. A. Marr, U. H. W. Lammers, T. J. Tanigawa, and R. V. McGahan, “Bistatic RCS calculations from cylindrical near-field measurements-Part I: Theory,” IEEE Trans. Antenna Propag., vol. 54, no. 12, pp. 3846–3856, Dec. 2006. [6] R. A. Marr, U. H. W. Lammers, T. B. Hansen, T. J. Tanigawa, and R. V. McGahan, “Bistatic RCS calculations from cylindrical near-field measurements-Part II: Experiments,” IEEE Trans. Antenna Propag., vol. 54, no. 12, pp. 3857–3864, Dec. 2006. [7] B. J. Cown and C. E. Ryan, Jr., “Near-field scattering measurements for determining complex target RCS,” IEEE Trans. Antenna Propag., vol. 37, no. 5, pp. 576–585, May 1989. [8] E. G. Farr, R. B. Rogers, G. R. Salo, and T. N. Truske, “Near-field bistatic RCS measurements,” RADC-TR-89-198, Oct. 1989. [9] D. Zahn and K. Sarabandi, “Near-field measurements of bistatic scattering from random rough surfaces,” in Proc. IEEE Antennas Propag. URSI Symp., Salt Lake City, Utah, Jul. 2001, pp. 1730–1733. [10] P. F. Wacker, “Non-planar near-field measurements: Spherical scanning,” National Bureau of Standards, NBSIR 75-809, Jun. 1975. [11] J. Hald, F. Jensen, and F. Larson, Spherical Near-Field Antenna Measurements, J. E. Hansen, Ed. London: Peter Peregrinus, 1988.

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[12] R. C. Wittmann and C. F. Stubenrauchc, “Spherical Near-Field Scanning: Experimental and Theoretical Studies,” National Institute of Standards and Technology, NISTIR 3955, Jul. 1990. [13] K. T. Kim, “Truncation-error reduction in 2D cylindrical/spherical near-field scanning,” IEEE Trans. Antenna Propag., vol. 58, no. 6, pp. 2153–2158, Jun. 2010. [14] P. Petre and T. K. Sarkar, “Difference between modal expansion and integral equation methods for planar near-field to far-field transformation,” Progr. Electromagn. Res., vol. 12, pp. 37–56, 1996. [15] F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and G. Riccio, “An effective technique for reducing the truncation error in the nearfield-far-field trnasformation with plane-polar scanning,” Progress in Electromagnetcs Research, vol. 73, pp. 213–238, 1996. [16] O. M. Bucci, G. D’Elia, and M. D. Migliore, “A new strategy to reduce the truncation error in near-field/far-field transformation,” Radio Sci., vol. 35, no. 1, pp. 3–17, Jan.–Feb. 2000. [17] O. M. Bucci and M. D. Migliore, “A new method for avoiding the truncation error on near-field antenna measurements,” IEEE Trans. Antenna Propag., vol. 54, no. 10, pp. 2940–2952, Oct. 2006. [18] F. Ferrara, C. Gennarelli, R. Guerriero, G. Riccio, and C. Savarese, “Extrapolation of the outside near-field data in the cylindrical scanning,” Electromagnetics, vol. 28, pp. 333–345, 2008. [19] E. Martini, O. Breinbjerg, and S. Maci, “Reduction of truncation errors in planar near-field aperture antenna measurements using the Gerchberg-Papoulis algorithm,” IEEE Trans. Antenna Propag., vol. 56, no. 11, pp. 3485–3493, Nov. 2008. [20] R. C. Wittmann, C. F. Stubenrauch, and M. H. Francis, “Spherical scanning measurements using truncated data sets,” in Proc. AMTA Symp., Cleveland, 2002, pp. 279–283. [21] A. Papoulis, Signal Analysis. New York: McGraw-Hill, 1977. [22] R. C. Wittmann, “Probe-corrected spherical near-field scanning theory in acoustics,” IEEE Trans. Instrum. Meas., vol. 41, no. 1, pp. 17–21, Feb. 1992. [23] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenna Propag., vol. 37, no. 7, pp. 918–926, Jul. 1989. [24] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore: The Johns Hopkins Univ. Press, 1996. [25] S. Pivnenko, Private Commun. Oct. 2008. [26] K. T. Kim, “Slepian transverse vector spherical harmonics and their application to spherical near-field scanning,” In preparation. Kristopher T. Kim (SM’99) received the B.A. degree from the University of Chicago, Chicago, IL and the Ph.D. degree from Purdue University, West Lafayette, IN. He is with the Electromagnetic Scattering Branch, Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, MA, where he previously served as its Technical Advisor and Branch Chief. Prior to his current position at AFRL, he was a Senior Principal Systems Engineer at Raytheon, responsible for target signature characterization for a major ballistic missile defense integrated flight test program and was on the technical staff of Mitre Corporation, working on computational electromagnetics, target discrimination, and propagation and clutter modeling.

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Fundamental Electromagnetic Properties of Twisted Periodic Arrays Derek Van Orden, Student Member, IEEE, and Vitaliy Lomakin, Senior Member, IEEE

Abstract—Linear periodic arrays of elements sequentially rotated about the array’s axis display interesting polarization properties. These “twisted” arrays, characterized by both discrete and helical periodicities, exhibit features of periodic gratings and bulk chiral media. For arrays with small periodicities, in which the zeroth diffraction (Floquet) mode dominates, two independent, transverse traveling wave modes are identified. One of these mode types is a guided wave that remains bound to the array, while the other is a leaky (radiating) wave mode. The electric fields characterizing each mode consist of two circularly polarized waves with distinct phase velocities. For larger periodicities, higher diffraction orders of these two modes may radiate. The coupling between different modes may result in stopbands of several types but the broadside stopband typical of leaky wave structures is eliminated. A finite twisted array is simulated to verify the predicted properties, and to show that the two transverse modes may be excited by circularly polarized sources with opposite handedness. Results are demonstrated for arrays of plasmonic particles in the optical regime and arrays of half-wave dipoles in the microwave regime. Twisted arrays can find uses in various microwave and optical applications, such as frequency and polarization sensitive waveguides, couplers, filters, and antennas. Index Terms—Chiral structures, Green’s functions, helical structures, leaky wave antennas, periodic structures.

I. INTRODUCTION TRUCTURES for controlling polarization, waveguidance, and radiation of electromagnetic waves are essential in many areas of optics and microwave engineering. Periodic arrays of resonant elements offer a number of important properties for achieving such control. In the microwave regime, arrays of metallic half-wavelength (resonant) dipoles support traveling waves that are bound to the array and propagate without radiation loss. Such arrays (tapered and truncated) are used as Yagi-Uda antennas radiating endfire beams [1]. In the optical regime similar phenomena are supported by arrays of plasmonic particles that are resonant even on a deeply subwavelength scale, thus enabling subwavelength particle array waveguides of Yagi-Uda type. Properly tuning the particle properties, distribution, and ambient environment allows controlling an array’s waveguiding and radiation properties [2]–[7]. Polarization control is another important aspect of

S

Manuscript received July 27, 2010; revised November 13, 2010; accepted January 06, 2011. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported by the DARPA NACHOS program and by the NSF ERC CIAN Center. The authors are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 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.2011.2158792

Fig. 1. (a) A parallel array of elements. (b) A twisted array with each particle sequentially rotated an angle  around the array axis. (c) A “tilted” twisted array, in which the major axis of each particle is tilted at an angle  with the array axis. Arrays of gold ellipsoidal nanoparticles in optics and half wave dipoles in microwave are considered.

manipulating electromagnetic waves. Chiral media, including optically active liquid crystals [8]–[10], and artificial surfaces and media [11]–[14] allow polarization control but are not intended to support waveguidance or radiation. Helical antennas rely on polarization rotation properties to support waveguidance and radiation [15]. Another class of structures resulting in optical activity and waveguidance is chiral waveguides [16] and chiral fibers [17]–[19]. Such configurations lead to optical activity properties similar to those of liquid crystals but with benefits of a guiding structure. In this paper we consider structures comprising linear arrays of particles that are sequentially rotated about the array axis by a constant angle [Fig. 1(b), (c)]. Such “twisted” arrays, first presented in [20], have similarities with cholesteric liquid crystal media, chiral optical fiber gratings, and helical antennas. Twisted arrays are unique, however, in that their symmetry is discrete, their cross-sectional size can be small, they can support modes with a large wavenumber, and they can lead to both waveguidance and radiation. Unlike bulk media, these structures may exhibit leaky-wave radiation at broadside. Twisted arrays retain the properties of general periodic structures, including the existence of diffraction modes that may couple to form stopbands. In contrast, structures with continuous helical symmetry, such as chiral fiber gratings and helical antennas, do not exhibit diffraction modes. Recently, the idea of twisted arrays was used to demonstrate one-way waveguidance using the Faraday effect [15]. [22].

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VAN ORDEN AND LOMAKIN: FUNDAMENTAL ELECTROMAGNETIC PROPERTIES OF TWISTED PERIODIC ARRAYS

This work extends the introductory results of [20] to present a thorough theoretical and computational analysis of the electromagnetic properties of twisted arrays. A theoretical analysis with closed form analytical expressions for modal fields is given for a twisted array whose elements are modeled via their dipole polarizability tensors. Propagation and radiation properties of finite twisted arrays are then analyzed, including “tilted” arrays [Fig. 1(c)] in which the longitudinal modes are coupled to the twist as well as “compound” arrays, which exhibit the radiation properties of sparse arrays while preserving the resonant interaction and waveguidance of dense ones. The analysis of finite twisted arrays is given for elements modeled via their polarizabilities and for half-wave dipole rods modeled as thin wires via an electric field integral equation solver. Untwisted arrays [Fig. 1(a)] support “slow” traveling waves (i.e., source-free modal fields) that propagate without radiation loss with traveling wavenumber greater than the free-space , with wavelength . These waves reprewavenumber sent the propagation of (transverse and longitudinal) polarization states of the array elements. The introduced rotation between the array’s elements couples the otherwise independent transverse modes supported by the array. This coupling leads to new modes with different propagation and radiation behavior. Twisted arrays support leaky wave modes, which radiate, as well as slow modes, which are guided without radiation. The polarization sensitivity is enhanced by the high particle asymmetry, which results from distinct resonances along its major and minor axes. Furthermore, the resonant coupling among the array elements allows efficient wave guidance for large rotation rates, making the phase velocity and radiation properties highly tunable for each mode. These features make twisted array structures a promising prospect for waveguidance, radiation, and polarization control in various areas of microwave engineering and photonics. The investigated structures can be directly realized in the optical regime using plasmonic (e.g., gold or silver) particles and in the microwave regime using metal rods possibly loaded with lumped circuits. The general features of twisted arrays may also be realized in many systems by adding a discrete periodicity to helical antennas, chiral fibers, chiral surfaces, and chiral metamaterials or by adding chiral features to a periodic structure. Understanding properties of the structures presented here is important for the study and design of other related structures characterized by the presence of both helical and non-helical periodicities. II. STRUCTURE CONFIGURATION AND OUTLINE Consider an array of asymmetric elements with spacing orielement is rotated ented along the axis, as in Fig. 1. The such that positive values of about the axis by an angle the rotation increment indicate a right handed twist. Two types of elements are considered including deeply subwavelength particles and (near) half-wavelength dipoles. The subwavelength particles are small so that, in response to an external electric field , they behave as point dipoles with dipole mo, where is the particle polarizability tensor. ment Considering a near-infrared or visible optical regime, each particle has distinct plasmonic resonances corresponding to different principal axes of , and the resonant interaction among

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the particles is responsible for the existence of traveling wave modes. In the microwave regime, similar resonant properties can be supported by subwavelength elements loaded by a resonant lumped circuit. Half-wave dipoles are assumed to be made of perfect electric conductors and they can be modeled via electric field integral equation solvers. Such dipole elements are realized by simple metallic rods in the microwave, infrared, or terahertz regimes. This paper identifies the modes supported by such twisted arrays, finds their dispersion, and characterizes their propagation and radiation behavior. The paper is organized as follows. Sections III–V-B analyze the dispersion relations of twisted arrays in the point dipole approximation for which the analysis can be done in a closed form. In Section III, we review an infinite untwisted resonant array [Fig. 1(a)] and use the dyadic periodic Green’s function to solve for its dispersion relations. In Section III we extend this approach to analyze an infinite twisted array and formulate a dyadic Green’s function adapted to its unique periodicity conditions. Next, in Section IV, we use this Green’s function to analyze an infinite array in which the three principal axes of each particle’s polarizability tensor are oriented parallel and perpendicular to the array axis [Fig. 1(b)]. In Section V-A we identify two independent transverse modes, find their complex propagation wavenumbers, and find expressions for the modal field distribution. In Section V-B we discuss each mode’s unique radiation and propagation characteristics. We further discuss the dispersion relations for the modes and identify stopbands that may result from coupling with higher diffraction orders. In Section V-C we analyze finite twisted arrays to verify the predicted modal behavior, to analyze the source excited fields of twisted arrays, and to demonstrate the results in different frequency ranges (optical and microwave) and for different structure types (point and half-wave dipoles). This section presents results for both nanoparticle arrays and arrays of half-wave dipoles, for which the dipole approximation is not valid. Finally, in Section VI we discuss “compound” arrays which allow higher diffraction orders to radiate while preserving strong resonant interaction. Finally, we consider the more general case of a “tilted” twisted array, in which the particle principal axes are tilted with respect to the array axis [Fig. 1(c)], coupling the transverse modes to the longitudinal mode. III. DISPERSION EQUATIONS OF PARALLEL ARRAYS In this section we review properties of linear, untwisted arrays and introduce an efficient Green’s function analysis to find the complex dispersion relations of the guided and leaky-wave modes supported. This analysis is used and extended in Sections III–V for the study of the twisted arrays. Consider an infinite array of parallel (non-twisted) elements oriented on the axis. The polarization states of each particle in this array must satisfy the (Floquet) periodicity condition, which of the particle, centered requires that the dipole moment at , satisfy (1) accounts for a linear phase shift between elements, Here, a boundary condition allowed by the Floquet theorem. It may be

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introduced, for example, by a periodic excitation, such as a plane along the periodicity diwave with wavenumber component rection. In addition to this periodicity condition, traveling wave modes supported by the array must satisfy the electromagnetic equations in the absence of external sources. This occurs only , denoted by , which reprefor certain discrete values of sent the mode propagation wavenumbers. These values of are found by setting up and solving a self-consistent equation for the zeroth dipole moment of the array, which, under the assumption of small particles, is given in terms of the particle polarizablity as (2) is the electric field resulting from all the other partiwhere cles in the array. It may be found by summing all of their source contributions using the dyadic Green’s function for a single dipole source in free space:

(3) where is the free-space wavenumber. is a diagonal matrix if any two of the three observation coordinates are zero. Using the Floquet condition of (1), (2) and (3) lead to the following self-consistency relation:

(4) is the dyadic periodic Green’s function where for an infinite array of parallel dipole sources with spacing and linear phase shift . It is a diagonal matrix when the fields are found on the array axis, as is the case when finding the interaction among array elements. The modal wavenumbers are found as the solution of the following equation for : (5) Solving this implicit equation yields the modes dispersion relations, which give the dependence of the (generally complex) modal wavenumbers on the free-space wavenumber . The general form of the allowed modes is determined by the , which may be repperiodic Green’s function, resented as a sum of single source dyadic Green’s functions that are sequentially phase shifted by

can be represented in terms of an infinite series of diffraction (Floquet) modes:

(7) and are where the longitudinal and transverse Floquet mode wavenumbers, respectively. It should be noted that the summations in (6) and (7), while correctly defining the equations and fields, do not converge for the special case required in (5). In our computations we use the spectral representation outlined in [22], which converges well for observation points directly on the array axis, as well as for lossless media and complex . The dispersion relation in (5) may be solved computationally plane using variations of Newton’s method in the complex [23] for complex valued functions. Three independent solutions is diagare found. If the particles are oriented such that onal [as in Fig. 1(a), where the principal axes are aligned along the coordinate axes], the three corresponding modes include two transverse modes and one longitudinal mode. The periodicity of the system and the fact that propagation occurs in both directions require that for each modal wavenumber there exists a set of wavenumbers (with an integer) that are also solutions to (5). [To avoid confusion, let denote the solutions of (5) closest to .] Because the fields resulting from , it is clear the system are given by from (7) that the modal fields may be viewed as the sum of an infinite number of diffraction mode contributions. Those with represent higher order diffraction modes which, though not directly observable in dense arrays, may couple to the lower order modes in some cases. In general the solutions for untwisted arrays have real parts that are greater than (but may be close to) , which means that for dense arrays the transverse Floquet wavenumbers are all imaginary. This indicates slow wave modes whose fields decay exponentially away from the array. For large spacings it may occur that one or more , resulting diffraction orders have wavenumbers . In this case the mode in real transverse wavenumbers becomes a leaky wave mode in which these diffraction orders radiate out of the array (even though the zeroth diffraction order remains slow.) In solving (5) one must note that there are an infinite number of square root branch cuts in the complex plane with branch points at , resulting from the dependence of in the periodic Green’s function in (7). These branch cuts must be carefully accounted for, as solutions corresponding to leaky-wave modes may appear on different Riemann sheets (see Appendix). IV. DISPERSION EQUATIONS OF TWISTED ARRAYS

(6) This function satisfies the periodicity conditions required for the polarization in (1), as well as the radiation boundary conditions transverse to the array axis. Alternatively,

For the case of a twisted array, the self-consistency equations and dispersion relations can be obtained by generalizing the derivations in Section II to include the sequential rotation boundary conditions. In a twisted array the induced polarization of each particle must be symmetric under translation by ,

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a phase shift of , and a rotation by about the axis. of the particle must satisfy That is, the dipole moment (8) where is the matrix for rotations about the axis. The symmetry properties in (8) are satisfied by particle dipole moments of the form

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array of dipole sources with spacing , rotation angle , and . linear phase shift With the Green’s function established, the traveling wave modes may be identified, as in Section II, by finding the fields resulting from the array at the origin and matching them to the of the particle centered there. The mode dipole moment wavenumbers are found by solving the resulting self-consistent equation, which is similar to (5) but with a new Green’s function

(12) (9)

particle has dipole moment . The first two terms in (9) describe transverse dipole vectors that rotate at the same rate as the array elements. In cases where the polarizability along one axis is significantly greater than that along the others, as may occur near a resonance, one of these two terms will dominate. The third term represents the longitudinal wave component, and the extent to which it couples to the transverse components depends on the particle orientation, as will be discussed in subsequent sections. The fields resulting from a twisted array can be found, similar to (6), as an infinite sum of dipole source contributions but with the dipole moments rotated according to (9) where the

Similar to the array of parallel elements considered in Section III, each independent solution of (12) has an infinite in the complex number of diffraction orders spaced by plane. This is true despite the fact that the spacing is not the spatial period of a twisted array in a Cartesian reference frame. In a helical coordinate system, however, the spacing is the structure’s period, which leads to the existence of diffraction orders. Note that (12) involves the components of with three different three periodic Green’s functions . From the Floquet linear phase shift factors mode representation of the periodic Green’s function (7) it is clear that the resulting determinant has branch points in the plane at , , complex . The presence of six branch cuts and per Floquet mode is unique to twisted arrays, and must be considered when finding solutions to (12) (see Appendix). V. WAVE PROPERTIES OF TRANSVERSE TWISTED ARRAYS

(10)

Based on this equation, a new Green’s function for twisted arrays may be developed by incorporating the sequential rointo the linear phase shift applied to each array tation of element. This is accomplished by expressing the sine and cosine functions in (10) in their exponential forms, and then abof the apsorbing them into the linear phase shift factor propriate matrix elements. The resulting equation, upon comparison with (6), may be expressed in terms the components of the dyadic periodic Green’s function evaluated with distinct phase shift factors. The expression for the electric field in (10) then becomes [see (11) at the bottom of the page], where . The matrix in (11) is interpreted as the for a twisted dyadic Green’s function

We discuss the properties of different traveling wave modes by considering first the special case of particles arranged such that two principal axes of are transverse to the array axis , as shown in with corresponding principal values Fig. 1(b), where and are the polarizability constants of a particle along its transverse principal axes. In this case the longitudinal traveling waves are independent of the transverse waves, and are not affected by the sequential rotation. To simplify the analysis the longitudinal waves will be ignored, and the component of all fields set to zero. A. Dispersion Equations and Modal Fields To find the dispersion relations of transverse twisted arrays we first note that both the periodic Green’s function and the single source Green’s function are diagonal matrices with and

(11)

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, respectively. Equation (12) then reduces to

(13) This equation may be solved numerically (e.g., using Muller’s method) for specific cases of interest. Based on our simulations (see Section V-B), we find that it has two inde, found near , and , found pendent solutions: , which represent the wavenumbers of two close to transverse modes. With the modal wavenumbers found, the characterizing each modal polarization dipole moment vector may then be found up to a constant factor by solving . The resulting (linearly dependent) system of equations yields the ratio of the transverse components of (14) where each Green’s function component is evaluated at for the mode in question. The dipole moments of all other array elements may then be found by (9), giving a complete description of the traveling wave modes in terms of the array geometry and the periodic Green’s function. The modal transverse polarization wave may be expressed in an alternative form by rewriting (9) as

This equation shows that the axial electric field preserves the polarization properties of the polarization vector as described by (15), i.e., it comprises a RCP and LCP components with distinct propagation wavenumbers. For observation points off the array’s axis, the electric field can be given in terms of a Floquet by evaluating (11) and expansion series, found in terms of using the Floquet series representation of the periodic Green’s function

(17) Here, the fields have been separated into two wave components with longitudinal and transverse wavenumbers and , respectively, ( is an integer). Each wave component has a distinct phase velocity along the array and a distinct decay rate out of the array. B. Modal Field Types: Propagation and Radiation Behavior

(15) has been replaced with . The expressions in (9) where and (15) highlight different interpretations of the traveling wave. The first shows as the propagation wavenumber with respect to the fixed axes of the elements themselves, i.e., with respect to the twisted coordinate system. The second expression, however, shows the fields as they propagate in an untwisted, laboratory reference frame. In the latter expression, the modal polarization wave is represented as a sum of right-hand circularly polarized (RCP) and left-hand circularly polarized (LCP) components. These components have distinct propagation wavenumthat are relevant to the observable excitabers tion and radiation behavior of the supported modes. The radiation and propagation properties of the two indepenand modes become more apparent when studying dent the corresponding modal fields. Recalling the periodic Green’s function is diagonal for observation points on the array’s axis and components ), the (with identical electric field on the array’s axis can be given by the following expression:

(16)

The expression for the electric field distribution (17) helps to elucidate the physical behavior of the traveling wave modes. Consider the case of dense arrays, for which is small enough that no higher diffraction orders radiate. The fields corresponding to the mode consist of two wave components with propagation wavenumbers . In terms of the representation in (15) for the modal polarization, these wave components correspond to the LCP and RCP parts of the mode. The corresponding transverse wavenumbers are imaginary, and from (17) the electric fields decay exponentially away from the array. This mode represents a slow wave that propagates without radiation loss. For dense arrays with no dissipation losses, is therefore a purely real number. Being a slow wave, this mode cannot be coupled to a far-field excitation, but it can be excited by a source with a strong evanescent spectrum. The mode, in contrast, is comprised of a component with wavenumber , corresponding to an evanescent field, and a component for which in the case of small to moderate twist rates (i.e., reasonably small ), corresponding to a propagating field. The former and latter components are RCP and LCP components in the representation (15) for the modal polarization. In such cases this mode is a fast leaky-wave mode whose LCP component radiates out of the array. The radiated wave direction makes an angle with the array axis that satisfies the phase matching condition . This mode can be efficiently excited by a source field that is matched to either its RCP or LCP wave component. That is, an RCP evanescent wave can

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be matched to its slow, RCP wave component, or an LCP propagating wave can be matched to its fast, LCP component. The polarization of the radiated wave may be found from (17) at observation points far from the array at an observation angle with the array axis. At this distance the propagating fields are assumed to dominate, so the twisted Green’s function may be in (17) and including only well approximated by neglecting the zeroth diffraction order of [i.e., the term in (7) and (17)], which is responsible for far-field radiation in this case. The radiated field observed in the plane is given by (18) is perpendicular to the rawith diation direction in the plane. This wave is elliptically polarized with ellipticity . A twisted array may therefore act as an antenna whose radiation angle and polarization are determined by the . We note that for sufficiently large twist rates the twist rate . propagation wavenumber may satisfy In this case the mode is a slow wave and the array does not radiate at all. To better characterize the modes we show the dispersion relations, found by tracking the mode wavenumbers over a range for which of values of the excitation wavenumber waveguidance is supported. Fig. 2 shows the dispersion relations of the and modes, found by solving (12) for an array of gold prolate spheroids with permittivity , assumed to follow the Drude model with parameters given by [24]. The array is embedded in an infinite silicon dioxide medium. The dispersion of the two independent modes is shown for both realistic lossy particles and for lossless particles, for which case the Drude damping constant is set to zero. The array has spacing , and twist angle . Each spheroid has a major axis of length 33 nm and minor axis of length 16.5 nm, chosen to achieve both strong anisotropy and reasonably low loss. The long and short axes of each spheroid have plasmonic resonances at free-space wavelengths of 501 and 298 nm, respectively, and the range of is chosen close to but less than the resonance of the major axis at . The length of each spheroid is small enough compared to the wavelength to justify the dipole approximation for this structure [25]. The dispersion relations of certain eigenmodes may exhibit stopbands of various types. These stopbands result from the coupling between modes that propagate in opposite directions. Mathematically, it occurs when two independent solutions of (13) approach the same point in the complex plane. In Fig. 2 the behavior of the mode results from interaction with the , found at first diffraction order of the mode at . In the lossless case, for which is purely real, this , where interaction results in a clear stop-band at the wavenumbers of the two modes are equal. For the lossy case, in which has a nonzero imaginary part, this condition cannot be satisfied, though the strong coupling is clearly visible in the dispersion curves: as the two mode wavenumbers approach each other, each dispersion curve approaches the

=

Fig. 2. Dispersion relations of an infinite, twisted array with spacing d . The array elements are gold prolate and rotation increment  spheroids with major axes of length 33 nm and minor axes of length 16.5 nm.

72 nm

= 30

Fig. 3. Dispersion relations of the mode of an infinite twisted array identical to that considered in Fig. 2, but with rotation increment  .

= 80

stopband point and then bends back away from it, indicating a change in group velocity and strong coupling (i.e., hybridization) with the counter-propagating diffraction mode. This type of stopband is unique to twisted structures with discrete periodicity. The mode, in contrast, does not couple to the diffraction order at except in the case of (often impractically) large spacings, for which waveguidance is not well supported. However, for large rotation rates it may couple directly , resulting in a stopband. to the mode with wavenumber Fig. 3 shows the dispersion relations of the array considered before, but with . For this large twist rate is a purely real number in the lossless case, so a stopband forms at where the two modes merge. It is noted that the dispersion relations in Figs. 2 and 3 are given for the wavenumber with respect to the rotate coordinate system as discussed after (15). In the laboratory coordinate system the wavenumbers would . One result from here is that no open appear shifted by (i.e., broadside) stopbands are present with respect to the laboratory coordinate system. As is shown in Fig. 5, this property allows smoothly scanning the beams radiated by the twisted arrays through the broadside direction, thus eliminating the open stopband issue of typical leaky wave antennas [26].

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C. Finite Twisted Arrays In this section we study propagation and radiation properties of twisted arrays composed of both nanoparticles at optical frequencies matching those considered for the infinite array analysis and (near) half-wave dipoles in the microwave regime. (The case of half-wave dipoles can be considered as a modification of a Yagi-Uda array with twisted rods.) These results verify the properties of twisted arrays predicted by considering infinite arrays, characterize their source excited fields, and demonstrate that similar phenomena occur in different frequency ranges and for different arrays. The nanoparticle array is excited by a waveguide excitation that consists of a periodic array of nanospheres connected to the twisted array at one end to form a continuous array, with a localized source placed at its opposite end. The nanosphere array has the same spacing as the twisted array, and each sphere has the same resonant wavelength as each ellipsoid’s major axis. (A waveguide source, such as a dielectric waveguide, could also be used to excite this array.) The periodic Green’s function cannot be used to analyze this finite structure, so we use the dyadic Green’s function for a single dipole source, shown in (3), to model the array as collection of mutually interacting elements. A self-consistent matrix equation for each array element’s polarization in response to the excitation field may then be set up and solved. The microwave half-wave dipole array parameters are identical to the nanoparticle arrays (relative to the wavelength) to achieve the same general behavior, except that each microwave and radius element is a perfect conducting rod of length , and is assumed to behave as a thin wire. The microwave array is excited by a pair of (near) half-wave dipole antenna sources at 300 MHz, positioned at right angles and phase shifted next to the left side of the array to create a circularly polarized excitation. The array response is modeled using an electric field integral equation solver in the thin wire approximation, using 11 basis functions per wire segment [27]. Consider an array of 150 gold ellipsoids with the same array parameters as in the infinite case analyzed in the previous section. The source consists of 25 gold nanospheres and a . This dipole source with free-space wavelength waveguide source creates an incident field with a propagation wavenumber slightly greater than the free-space wavenumber . Inspection of (15) and (17) reveals that an RCP waveguide source is therefore well matched to the mode, whose RCP plane wave component has wavenumber close to , while an LCP source is better matched to the mode. Taking the Fourier transform of a fixed, transverse polarization component of each spheroid element allows one to see the wavenumber content of the propagating fields. Fig. 4 shows this spectrum for RCP and LCP source excitations, in which the dipole moment of the localized source is and , respectively. The response to a LCP source is two spectral peaks located near and , as predicted for the mode for infinite arrays. The same structure excited by a RCP source has a spectrum with peaks found near and , consistent with the mode. (The small peak observed near results from imperfect matching of the source to this mode.) In

Fig. 4. Fourier spectrum of the transverse polarization component of a finite twisted array of gold prolate spheroids excited by a waveguide source with (a) LCP and (b) RCP polarization and wavelength 550 nm. The same spectra for an array of half-wave dipoles in the microwave regime, excited by (c) LCP and (d) RCP antenna sources. The spectral peaks are spaced 2 =(k d) apart. The array parameters are the same as in Fig. 2.

Fig. 5. (a) Radiation pattern resulting from the mode of the finite twisted array of gold prolate spheroids considered in Fig. 4 with three distinct rotation increments. For  = 37 , the beam is radiated directly at broadside and is linearly polarized. For  = 30 and  = 45 , the beams are directed in the forward and backward directions relative to the source, and are left and right handed elliptically polarized, respectively. (b) Radiation pattern from a twisted array of half-wave dipoles at 300 MHz with the same rotation increments and spacing relative to the wavelength.

the microwave case, the spectrum is found by taking the Fourier transform of the transverse current moment at each rod’s center. It is evident that the RCP excitation excites both modes in this case, most likely owing to the contribution of the near-field spectrum of the localized antenna source. To verify the radiation properties we show in Fig. 5 the radiation pattern, viz. the scattered far-field intensity as a function of the observation angle made with the array axis, for three values of the twist angle . For , the beam is radiated directly at broadside and is linearly polarized. For and , the beams are directed in the forward and backward directions relative to the source, and are left and right handed elliptically polarized, respectively, with ellipticity approximately given by . The peak directions are consistent with the predicted radiation angle for the mode, given by . An important property is that the radiated beam does not exhibit an open stopband behavior typical of many leaky wave antennas, e.g., it does not

VAN ORDEN AND LOMAKIN: FUNDAMENTAL ELECTROMAGNETIC PROPERTIES OF TWISTED PERIODIC ARRAYS

Fig. 6. Radiation pattern at endfire for the twisted (a) nanoparticle array (at  ) and (b) microwave array considered in Fig. 5, excited by LCP sources.

= 540 nm

change its width and shape. An open stopband behavior is also absent when frequency scanning the beam angle. This property may be useful for applications in leaky wave structures, such as leaky-wave antennas. The twisted arrays can also radiate end-fire beams, similar to mode does not radiate Yagi-Uda antennas. Specifically, the at broadside, but still radiates into an endfire beam at . This radiation is similar to that from conventional Yagi-Uda antennas but it is of the LCP type (for the positive twist angle), similar to helical antennas. Fig. 6 shows the endfire radiation from twisted nanoparticle and microwave arrays excited by LCP sources, confirming this behavior. VI. EXTENSIONS TO COMPOUND AND TILTED TWISTED ARRAYS For the twisted arrays considered up to this point, the periodicity was sufficiently small such that higher diffraction orders did not contribute to radiation. For larger periodicities, however, radiation can occur due to higher diffraction orders, although in such systems the resonant interaction between array elements may be weak. To demonstrate diffraction mode radiation and preserve the strong waveguidance of a dense nanoparticle array, we can periodically intersperse resonant spheres within a sparse array of ellipsoids, creating a compound array that has a large spacing between ellipsoids but can still support efficient waveguidance due to the strong interaction between closely spaced spheres. Similar performance can be achieved by slightly modifying the size or shape of certain equally spaced elements in the array. For the large spacing , higher diffraction orders may contribute to the radiation and field behavior. Fig. 7(a) shows the radiation pattern from a compound array with two spheres placed between each neighboring pair of ellipsoids, and excited by an LCP source. The spacing between ellipsoids is , and the twist angle remains . For this large spacing the first diffraction order of mode, which is a slow wave in dense arrays, becomes the fast. Both its right and left handed components radiate at angles , satisfying and . For the case of microwave arrays, we achieve the same effect by constructing the array with unit cells of three parallel rods that are sequentially rotated and translated. The scattered field pattern, as shown in Fig. 7(b), is very similar to that of the

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Fig. 7. (a) Radiation pattern resulting from the mode of a compound twisted array excited by an LCP waveguide source at 550 nm. The array consists of a sparse array of prolate spheroids, which are spaced 217 nm apart, with 2 spheres, spaced 72 nm apart, interspersed symmetrically in between. (b) Radiation pattern from a twisted microwave array consisting of unit cells of three parallel rods sequentially translated and twisted.

nanoparticle array. These compound structures allow a new dimension of radiation and polarization control. Consider now the more general case of nanoparticles oriented such that the principal axes of the polarizability tensor are not all parallel to or perpendicular to the array axis. In such cases, as shown in Fig. 1(c) (where the major axis of each ellipsoid is “tilted” towards the array axis) the longitudinal mode is coupled to the transverse modes, so there now exist three independent modes that are coupled to the twist. Their wavenumbers may be found by solving (12), noting that the polarizability tensor must be appropriately rotated and is therefore no longer diagonal. The modal polarization wave now becomes

(19)

and modes discussed From (19) it is clear that the above have longitudinal components. However, their general radiation and polarization properties are otherwise largely unthat is changed. The third mode has a wavenumber , and is thereclose to . It generally satisfies fore a leaky-wave mode whose LCP component radiates at an . An expression angle satisfying for the radiation from this component alone may be found from and (11) by including only the longitudinal component of by neglecting all terms in the twisted Green’s function except , , and . Unlike the the zeroth diffraction orders of mode, the radiation from the mode is radiation from the linearly polarized. To demonstrate the radiation properties of these twisted arrays, Fig. 8(a) shows the radiated electric field intensity for a finite array of gold prolate spheroids identical to that first considered in Section V-C but with each spheroid “tilted” so that with respect to the its major axis makes an angle of array axis. Fig. 8(b) shows similar results for the case of a tilted twisted array of half-wave dipoles in the microwave regime.

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Fig. 8. Radiation pattern resulting from the and modes of finite “tilted” twisted arrays of (a) gold prolate spheroids and (b) half-wave microwave dipoles excited by an RCP sources. In each case, the radiated beam on the left is linearly polarized, while that on the right is left handed elliptically polarized. The array instead of 90 . parameters are as given in Fig. 4, but with tilt angle 

= 85

The mode radiates at the same angle as in the purely transverse case shown in Fig. 5, but there is an additional radiated mode. It is observed at an angle beam that results from the . The angular separation between these two beams may be varied by changing . This new radiation behavior results from the the twist rate coupling of the formerly independent longitudinal mode to the transverse modes and their distinct propagation behavior. VII. CONCLUSION In this paper, we have studied twisted resonant arrays characterized by a sequential rotation of each array element around the periodicity axis. Using the point dipole approximation, we derived a dyadic Green’s function for these resonant array structures and used it to find the waveguiding modes and their dispersion. For densely spaced elements, we identified two independent transverse modes with distinct polarization properties: one is generally a slow wave for dense arrays, while the other radiates for small to moderate twist rates. The angle and polarization of the resulting radiated beam are given in terms of the modal wavenumber. For arrays with larger spacing, higher diffraction orders of these modes can have important effects leading to radiation or mode coupling. The dispersion relations of the transverse modes may exhibit stopbands, resulting from coupling to counter propagation modes, including higher diffraction orders. The open stopbands typical of leaky wave antennas, however, are absent. By simulating a finite array we showed that these two modes are efficiently excited by circularly polarized sources of opposite handedness, and that the propagation and radiation properties agree with those predicted for an infinite array. Finally, we introduced “compound” twisted arrays and “tilted” twisted arrays, which lead to additional radiation mechanisms. Twisted arrays offer the possibility of polarization control and they could find applications as waveguides and radiators with tunable phase velocity, and as polarization-sensitive filters and couplers. The results were shown for twisted arrays of plasmonic nanoparticles in the optical regime and twisted (Yagi-Uda) arrays of dipoles in the microwave regime. Similar behavior should also apply to a general class of structures having both discrete and helical periodicities.

Fig. 9. Branch cuts and solutions of Eq. (13) in the complex k plane for the infinite arrays considered in Fig. 2 and Fig. 3, corresponding to rotation increand (b)  , respectively. (To enhance visibility, the ments (a)  branch cuts are plotted for a value of k with very small imaginary part.)

= 30

= 80

APPENDIX BRANCH CUTS AND RIEMANN SHEETS It is clear from the Floquet mode representation in (7), that has an infinite number of branch cuts in the plane that must be accounted for when solving complex (5) or (12). The diffraction modes have longitudinal wavenumand transverse wavenumbers bers , having branch cuts with branch points at . Note that these branch cuts are necessarily found in all representations of the periodic Green’s function. A set of dispersion relations can in general be defined on different Riemann sheets. However, only solutions on specific Riemann sheets that correspond to physically excitable modes should be considered. For the case of untwisted dipole arrays, it can be shown that for small periodicities for which no Floquet modes are radiating ), physical solutions correspond to values (i.e., of on the top Riemann sheet, for all . Such modal solutions correspond to slow traveling waves that are bounded to the array. For larger periodicities, one or more of the Floquet and modes become propagating, with the physical solutions are leaky waves corresponding to on a for a finite number of FloRiemann sheet with quet modes . Such solutions correspond to waves that radiate into a conical beam out of the array at an angle determined by phase matching conditions. For twisted arrays in which the particles have principal axes transverse to the array axis, (13) involves two Green’s functions, , whose linear phase shift factors differ by a translation of . The resulting determinant has plane at branch points in the and , as shown in Fig. 9. For dense ), all arrays with large twist rates (approximately solutions of (13) are found on the top Riemann sheet, on which . Fig. 9(a) shows, in the complex plane, the solutions and branch cuts for the array consid) at a free-space waveered in Fig. 3 (with twist rate length of 550 nm. This array is characterized by a large rotation rate, and all of the traveling wave modes are slow waves

VAN ORDEN AND LOMAKIN: FUNDAMENTAL ELECTROMAGNETIC PROPERTIES OF TWISTED PERIODIC ARRAYS

that do not radiate. Note that in this case all the solutions are defound close to the branch points. As the twist rate creases, however, the two branch points on each side of the , the and origin approach each other. At around modes become leaky-wave modes. For smaller twist rates ) and move to the bottom Rie(roughly branch cuts of and , mann sheets defined for the respectively, while and remain on the top Riemann sheet. This is the case in Fig. 9(b), which shows the branch cuts (analyzed in for the same infinite but with twist angle Fig. 2). Finally, for the case of tilted twisted arrays, there are three plane. In addition to independent solutions in the complex the branch cuts described above for the transverse twisted array, , there are now branch cuts with branch points at resulting from the presence of the periodic Green’s function in (12). The details of which Riemann sheets component each solution is found on are somewhat involved, and may deand the tilt angle . This dispend on both the twist rate cussion is therefore omitted. REFERENCES [1] C. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005. [2] W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B, vol. 70, p. 125429-1, 2004. [3] A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B, vol. 74, p. 205436, 2006. [4] M. Quinten, A. Leitner, R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett., vol. 23, p. 1331-1, 1998. [5] M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B (Condensed Matter), vol. 62, p. 16356-16359, 2000. [6] D. Van Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett., vol. 34, pp. 422–424, 2009. [7] V. Lomakin, L. Meng, and E. Michielssen, “Optical wave properties of nano-particle chains coupled with a metal surface,” Opt. Expr., vol. 15, pp. 11827–11842, 2007. [8] S. Chandrasekhar, Liquid Crystals. Cambridge, U.K.: Cambridge Univ. Press, 1977, 1994. [9] C. Oldano, E. Miraldi, and P. Taverna Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A, vol. 27, p. 32913299, 1983. [10] H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: Exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A, vol. 37, pp. 3520–3529, 1987. [11] A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, “Optical manifestations of planar chirality,” Phys. Rev. Lett., vol. 90, 2003. [12] B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,” Phys. Rev. A, vol. 76, p. 023811-1, 2007. [13] G. Shvets, “Optical polarizer/isolater based on a rectangular waveguide with helical grooves,” Appl. Phys. Lett., vol. 89, p. 141127, 2006. [14] I. V. Semchenko, S. A. Khakhomov, S. A. Tretyakov, and A. H. Sihvola, “Microwave analogy of optical properties of cholesteric liquid crystals with local chirality under normal incidence of waves,” J. Phys. D: Appl. Phys., vol. 32, pp. 3222–3226, 1999.

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[15] R. Mittra, “Wave propagation on helices,” IEEE Trans. Antennas Propag., vol. AP-11, pp. 585–586, 1963. [16] N. Engheta and P. Pelet, “Modes in chirowaveguides,” Opt. Lett., vol. 14, pp. 593–595, 1989. [17] V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Sci., vol. 305, pp. 74–75, 2004. [18] A. Z. Genack, V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, and D. Neugroschl, “Chiral fiber Bragg gratings,” presented at the Proc. SPIE-The Int. Soc. Opt. Eng., Denver, CO, 2004. [19] G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt., vol. 11, p. 074007, 2009. [20] D. Van Orden and V. Lomakin, “Twisted chains of resonant particles: Optical polarization control, waveguidance, and radiation,” Opt. Lett., vol. 35, pp. 2579–2581, 2010. [21] Y. HadadB. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett., vol. 105, 2010. [22] D. Van Orden and V. Lomakin, “Rapidly convergent representations for 2D and 3D Green’s functions for a linear periodic array of dipole sources,” IEEE Trans. Antennas Propag., vol. 57, pp. 1973–1984, 2009. [23] R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. Pacific Grove, CA: Brooks Cole, 2004. [24] E. D. Palik, Handbook of Optical Constants of Solids. New York: Academic, 1998. [25] A. Govyadinov and V. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B, vol. 78, p. 0354031-12, 2008. [26] P. Baccarelli, S. Paulotto, D. R. Jackson, and A. A. Oliner, “A new Brillouin dispersion diagram for 1-D periodic printed structures,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 1484–1495, 2007. [27] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE, 1998.

Derek Van Orden received the B.S. degree in applied physics from Rice University in 2004, and is currently working toward the Ph.D. degree at the University of California, San Diego. His research work is in the field of computational and applied electromagnetics. His interests include periodic systems, surfaces structures, waveguiding structures and plasmonics.

Vitaliy Lomakin received the M.S. degree in electrical engineering from Kharkov National University (Ukraine) in 1996 and the Ph.D. in electrical engineering from Tel Aviv University (Israel) in 2003. From 1997 to 2002, he was a Teaching Assistant and Instructor in the Department of Electrical Engineering, Tel Aviv University. From 2002 to 2005, he was a Postdoctoral Associate and Visiting Assistant Professor in the Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign. He joined the Department of Electrical and Computer Engineering, University of California, San Diego, in 2005, where he currently holds the position of Associate Professor in the Department of Electrical and Computer Engineering. His research interests include computational electromagnetics, computational micromagnetics/nanomagnetics, the analysis of microwave phenomena on structured surfaces, the analysis of optical phenomena in photonic nanostructures, the analysis of magnetization dynamics in magnetic nanostructures

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Millimeter-Wave High Radiation Efficiency Planar Waveguide Series-Fed Dielectric Resonator Antenna (DRA) Array: Analysis, Design, and Measurements Wael M. Abdel-Wahab, Student Member, IEEE, Dan Busuioc, and Safieddin Safavi-Naeini, Member, IEEE

Abstract—A low cost general architecture for a substrate integrated waveguide (SIW) series-fed dielectric resonator antenna (DRA) array, formed by two different slot polarizations, is proposed. In addition, a novel, simple, and generic transmission line (T.L.) circuit model, along with a fast and generic formulation for the new linear array antenna, is developed. The model can be used for reflection coefficient and radiation pattern (gain) calculations. The experimental data from two linear array modules, operating at the millimeter-wave band, are used to verify the simulated results of HFSS and the proposed model results. The measured 1 SIW-DRA array demonstrates a radiation pattern for a 4 broadside beam with a radiated gain of 11.70 dB over an operating impedance bandwidth of 4.70%. Moreover, the simulated radiation efficiency is more than 90%. Index Terms—Dielectric resonator antenna (DRA), millimeterwave, printed circuit board (PCB), radiation efficiency, substrate integrated waveguide (SIW), waveguide.

I. INTRODUCTION HE development of modern microwave and millimeterwave communication systems has attracted a considerable amount of interest in the industry and in academia [1], [2]. Many applications have been assigned to this frequency range including wireless communication networks [3], [4], automotive radar systems, and passive millimeter-wave imaging [5]. Therefore, highly efficient, compact, and inexpensive antennas and RF circuits are critical requirements for these systems. Microstrip antenna arrays exhibit a low radiation efficiency due to the ohmic and dielectric loss in the feed network and the excitation of surface wave in the dielectric substrate [6]. The efficiency limitation is a severe problem in large arrays, where the microstrip lines forming the feed network are long and complicated, and the conduction loss dominates as the frequency increases. Hybrid coupling, based on using a rectangular waveguide (RWG) as the main feed line for the microstrip arrays, is presented in [6]. Although the feed loss is reduced, the overall antenna efficiency of this millimeter-wave array is only 53% and the bandwidth, 3%. However, dielectric resonator antennas

T

Manuscript received January 27, 2010; revised December 26, 2010; accepted December 30, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported in part by the Electronic Research Institute (ERI), Egypt, and in part by the NSERC, Canada. The authors are with the University of Waterloo, Electrical and Computer Engineering, Waterloo, ON N2L 3G1, Canada (e-mail: wmabdelw@maxwell. uwaterloo.ca). 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.2011.2158946

(DRAs) indicate higher radiation efficiency (no conductor loss) and a better bandwidth, compared with microstrip antenna at microwave and millimeter-wave frequency ranges. Most of the planar feeding schemes [7]–[9] demonstrate either considerable feed line losses or a radiation disturbance at millimeter-wave frequency ranges. However, the rectangular waveguide (RWG) provides a low loss excitation mechanism [10], and the applications of the waveguide at millimeter-wave frequencies are still limited by high manufacturing costs, a relatively large size, and difficulties of integration with other components (not compatible with compact and planar technology). The cost, compactness, and efficiency of the antenna at the millimeter-wave are significant considerations, especially for portable systems. Recently, a substrate integrated waveguide (SIW) [11] has been proposed as a solution to these problems. Not only are its characteristics similar to those of the RWG, but also offers additional merits, such as ease of integration with other planar circuits, compact size, and low manufacturing costs where the conventional printed circuit board (PCB) can be used. Many passive components and active devices based SIW are found in [12]–[14]. SIW-based antennas with operation frequencies up to the Ka-band have been developed by standard PCB processes [15]. These studies indicated that the SIW is a good candidate for realizing microwave and millimeter-wave circuits. The goal of this paper is to model and design a high radiation efficiency SIW-series fed DRA antenna array (SIW-DRA array). Moreover, a general, simple, and novel transmission line (T.L.) circuit model, based on a single mode excitation, is introduced to simplify the optimization of the antenna array. Also, the proposed T.L. can be applied to other types of antenna feeding structures, as long as the T.L. characteristics of the propagation mode are known. Simulations, fabrication, and measurements of the designed examples are presented. The simulated results are in a good agreement with the measured results. The array design methodology, presented in this paper, is outlined as follows. Step 1) The SIW-DRA single element such as the one presented in [16] is modeled as a 2 port-network and . represented by S-matrix Step 2) The T.L. model for the SIW is constructed by modeling the discrete SIW structure, excited in its , by its propagation confundamental mode, , and wave impedance, . Depending stant, on the designed array configuration, the T.L. model is formed by cascading a number of 2-port-network blocks via the T.L.

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ABDEL-WAHAB et al.: MILLIMETER-WAVE HIGH RADIATION EFFICIENCY PLANAR WAVEGUIDE SERIES-FED DRA ARRAY

Fig. 1. The schematic layout of the SIW-DRA array (a) a horizontally polarized slot and (b) a vertically polarized (longitudinal) slot.

Step 3) A generic formulation, based on the proposed SIW-DRA array T.L. circuit model, is introduced to calculate the antenna array overall reflection coefficient and radiation pattern (gain) in terms of the antenna array elements excitation voltages and the element pattern. This formulation is used to optimize the antenna array parameters. and , are conStep 4) The optimized parameters, verted into the actual SIW lengths by using the criteria in [16]. To describe the aforementioned newly developed method in more detail, its applications to a typical series-fed array is described in this paper. This paper is organized as follows. The configuration of the proposed antenna array and several design guide lines are described in Section II. A T.L. circuit model for the array is introduced in Section III. An analysis and a discussion of the mutual coupling between any two adjacent array elements are addressed in Section IV, followed by Section V which deals with the fabrication, measurements, and simulations of the proposed antenna array. Finally, Section VI presents the conclusion. II. SIW-SERIES FED DRA ARRAY (SIW-DRA ARRAY) CONFIGURATION Fig. 1 shows the general antenna layout of a SIW-series fed DRA linear array (SIW-DRA array). The optimized single element (SIW-DRA single element) in [16] is used as a building block for the N-element SIW-DRA array by using two different slot orientations, horizontal (transverse slot) and vertical polarization (longitudinal slot), as depicted in Fig. 1(a), (b), respectively. The slots are excited by the SIW’s fundamental mode, , to couple the energy to DRA elements. The primary array parameters are the DRA positions: (1) the distance between the centers of the neighboring elements, , and (2) the distance between the last element and the SIW short circuited end, .

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The distance, D, between any two neighboring elements, is a vital parameter, because it defines the phase difference between two successive elements, which, in turn affects all the radiation characteristics of the antenna. The distance from the last element , not only affects the phase to the SIW short circuited end, difference, but also, changes the antenna input impedance. In and other words, by using the appropriate values for both , the antenna elements can be placed in certain positions to give a prescribed phase distribution such as co-phase or linear and phase excitation. For example, choosing the distances, , in Fig. 1(a) is equal to and respectively, provides a co-phase SIW-fed DRA array excitation. The structure in Fig. 1(a) is a standing-wave (SW) linear array. The slots are positioned, where the maximum axial (parallel to the X-axis) SW currents occur. Therefore, the excitation . slot fields are set in phase On the other hand, in the SIW-fed vertical polarized slots in Fig. 1(b), the DRAs, excited by the vertical polarized (longitudinal) slots, cut on the broad wall of the SIW. They are mutually positioned at the opposite sides of the waveguide centerline, providing an additional phase shift of between any two neighboring slot fields. The spacing between two successive slots and and , respecthe short circuit position are tively. In this case, the slots are positioned where the maximum standing-wave currents are parallel to the Y-axis. The DRA el. ements are co-phased, because III. SIW-DRA ARRAY T.L. CIRCUIT MODEL AND OPTIMIZATION (PARAMETRIC STUDY) To simplify the design and optimization of the antenna arrays in Fig. 1, and to gain more physical insight, a simple T.L. circuit is developed. This circuit does not take the inter-element mutual coupling into account (the coupling effect will be studied in Section IV). Afterwards, the circuit model is used to optimize , (by a parametric study) for the antenna parameters, and the optimum overall antenna reflection coefficient and radiation pattern (gain). The proposed T.L. circuit model is the 2-port-network S-parameters of the SIW-DRAs in Fig. 2(a). They are calculated numerically by HFSS numerical solver referred to the plane . The calculated S-parameters of a typical SIW-DRA , are shown in Fig. 2(b). The other ele2-port-network, mode ment in the proposed model is the T.L. model of the propagation in the straight segment of the SIW. The SIW T.L. , which is the same as the physical model has a length of , and propagalength of the SIW, wave impedance, , where tion constant and are the SIW fundamental model attenuation and phase constants, respectively. Fig. 3 illustrates the T.L. circuit model for a series-fed antenna array with a short-circuited SIW. For an example, an SIW-DRA single element, terminated by a short section of SIW presented in [16], is proposed. The reflection coefficient of the SIW-DRA single element is given by (1)

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(3) (4) and

The proposed T.L. circuit model is used then to determine the overall antenna reflection coefficient and the antenna radiation pattern (gain). The gain is formulated by using the pattern multiplication principle [17] with the SIW-DRA single element radiation pattern and the excitation voltage at each antenna input circuit element. The required excitation coefficient for the array factor is calculated by the proposed T.L. circuit model. The antenna array factor (with the array formed along the -axis direction) is given by (5)

the antenna excitation voltage coefficient; the antenna element (reference voltage);

excitation voltage

for the SIW-DRA array architecture shown in Fig. 1(a); for the SIW-DRA array architecture shown in Fig. 1(b); n is odd; n is even.

Fig. 2. The SIW-DRA 2-port-network S-parameters, calculated at the reference plane A A , of both horizontal polarized (transverse) SIW-slot arrangements and vertical polarized (longitudinal) SIW-slot arrangement (a) equivalent models, (b) magnitude (dB), and (c) phase (degree).

0

Similarly, an N-element SIW-DRA array is modeled as N-cascaded SIW-DRA 2-port-network blocks, connected by an SIW-T.L of length, , as shown in Fig. 3. The overall antenna reflection coefficient is calculated as follows: (2)

Due to the loading effect, caused by the antenna element(s), the SIW T.L., and (the SIW-DRA antenna single and array parameters) deviate from their assigned values, as mentioned in Section II. Therefore, a detailed parametric study should be conducted to analyze the effect of these parameters on the overall antenna performance. The proposed T.L. circuit model is used to perform this study and to optimize the parameters of the SIW-DRA linear array in Fig. 1(a), (b). The circuit model is considered to be a simple and fast way, compared with CADs such as HFSS. A parametric study is conducted on a four-element linear antenna array within the frequency range, 33–40 GHz. Not only are the phase differences among the excitation electromagnetic (EM) fields to the antenna elements defined, but also the ef, is defect of the overall antenna reflection coefficient, on and the impact scribed. For clarity, the impact of of on the radiation pattern (gain) are described in this paper. A. Short Circuit Position: To verify the proposed T.L. circuit model, a simple example, the SIW-DRA single element, is studied. The reflection coefficient is calculated and compared with that of HFSS result, as shown in Fig. 4, for different values close to for the transverse slot and for the longitudinal

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Fig. 3. The SIW-DRA T.L. circuit model of N-elements SIW-DRA array.

Fig. 4. The SIW-DRA single element calculated reflection coefficient (using , T.L. proposed circuit model and HFSS) for different short circuit position, (a) transverse slot and (b) longitudinal slot.

X

slot. The proposed T.L. circuit model is in a good agreement , Fig. 4(a), and with HFSS (full wave solver) at , Fig. 4(b). A similar study is performed on an by using the proposed T.L. cirarray of four elements cuit model for different short circuit positions, , at the same inter-element distance, , as shown in Fig. 5. The short circuit position acts as a matching stub (first degree of freedom) for the antenna input impedance and shifts the DRA resonance frequency (frequency tuning). B. Array Inter-Element Distance: D The antenna inter-element distance, D (second degree of freedom), affects both the antenna input impedance as well as the amplitude and phase differences for the antenna array

S

Fig. 5. The SIW-DRA array (Four elements) reflection coefficient, (dB), (calculated by using the proposed T.L. circuit model) for different short circuit positions (a) with the transverse in Fig. 1(a), for = 7 60 mm, and (b) with the longitudinal slot in Fig. 1(b), for = 3 50 mm.

X

D

:

D

:

element excitation voltages, as shown in Table I. As indicated , control the by (5), the excitation voltage coefficients, antenna array radiation pattern (gain). The proposed circuit model is used to study the impact of changing the inter-element distance, , of a SIW-DRA array, comprised of four elements on the total gain (dB). Table I summarizes this study for the slot configurations in Fig. 1(a), (b) at the frequencies of 34 GHz and 38 GHz, respectively. As the distance, , approaches a specific value , and that corresponds approximately to respectively, the antenna elements are almost co-phased and excited equally.

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TABLE I VOLTAGE EXCITATION COEFFICIENTS A OF SIW-DRA ARRAY

N = 4 ELEMENTS

Fig. 6. The calculated (using the proposed T.L. circuit model) broadside gain (dB) of the SIW-DRA array N for both antenna array configurations in Fig. 1.

( = 4)

Consequently, their radiated far field is added constructively to form a broadside radiation of maximum gains, 12.80 and 12.62 dB, as demonstrated in Fig. 6. Conversely, as the distance deviates from these critical values, the antenna array elements become out of phase and are not excited equally. Therefore, the antenna array radiation pattern (maximum gain) deteriorates. To study the radiation bandwidth of the antenna arrays in Fig. 1 and the frequency-scanning characteristics, the radiation pattern is simulated over the frequency range of 33–40 GHz, as shown in Fig. 7. It is noticed that as the frequency varies from 35.21 GHz to 40.0 GHz, the main beam is steered by 15 degrees with a maximum gain variation of 0.66 dB. IV. SIW-DRA ARRAY INTER-ELEMENTS MUTUAL COUPLING The single (fundamental) mode T.L. circuit model is valid, as long as the discontinuities (antenna, short circuit, junctions, and other types of discontinuities) are sufficiently apart so that higher order mode interactions are negligible. Therefore, it is pivotal to study the mutual coupling and its effect on the proposed T.L. model validity, particularly when the antenna elements are close.

Fig. 7. The SIW-DRA array (four elements) calculated radiation pattern (gain) (dB) (using the proposed T.L. circuit model) for different frequencies : , and (b) (a) E-plane of the transverse slots array in Fig. 1(a) for D . H-plane of the longitudinal slots array shown in Fig. 1(b) for D :

= 7 60 mm = 3 50 mm

The mutual coupling between the adjacent antenna elements is due to two mechanisms: internal coupling (caused by the excitation of higher order mode at the SIW-slot interface discontinuities), and exterior coupling (caused by the electromagnetic space wave interactions). Fig. 8 reflects the geometries for studying the mutual coupling between a two antenna elements for both array configurations shown in Fig. 1. It consists of two identical SIW-DRA elements set close to each other, and excited independently with waveguide ports, , and , to study both the E-coupling (Fig. 8(a)) and H-coupling (Fig. 8(b)). To study the exterior mutual coupling, a row of copper-plated vias (SIW-SC) is inserted between the two DRAs to block any internal coupling between the two ports. In other words, the geometries in Fig. 8(a),(b) allow for space wave only to be coupled between the two ports (antenna exterior mutual coupling). The distance between the two DRA elements are kept at , and , represented in Fig. 1(a), (b), respectively to maintain the far field co-phase radiation. The calculated (simulated) mutual coupling demonstrates that the mutual coupling in both planes E-plane (E-coupling) and H-plane (H-coupling) are less than 15 dB, and 40 dB,

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Fig. 10. The calculated mutual coupling (without SIW-SC) (dB) between two adjacent antenna elements in the array arrangement.

Fig. 8. Model used for the calculation of the mutual coupling between elements within the SIW-DRA array layouts (a) E-coupling, and (b) H-coupling, (c) SIW-SIW back to back.

Another study is performed to investigate the total mutual coupling between the antenna elements by using the setup in Fig. 8(a), (b) except that the SIW-SC is removed. The layout in Fig. 8 is modeled by the T.L. circuit model shown in Fig. 3 (for , without the short circuited termination). The coupling , is calculated by the T.L. model and (insertion loss), HFSS (full wave solver). Fig. 10 shows the results for this study for different , for both slot arrangements at the frequencies 36 GHz and 37.50 GHz, respectively. As the distance increases, the HFSS results converge to that of the T.L. model, indicating that the mutual coupling, due to the higher order modes and radiation coupling, decreases. Conversely, as the elements become closer, the two results diverge, and the aforementioned mutual coupling increases. However, it is noticed in Fig. 10 that the two calculated results (Fig. 8(a)), and (Fig. 8(b)), converge at respectively. At these distances, the effect of SIW higher order modes can be ignored. The effect of mutual coupling on the antenna performance will discuss in Section V. V. FABRICATION MEASUREMENTS AND SIMULATIONS

Fig. 9. The calculated mutual exterior coupling (with the SIW-SC) (dB) between two adjacent antenna elements in the array arrangement.

respectively within the operating bandwidth, as shown in Fig. 9. Therefore, the mutual coupling between the array elements is considered to be weak particularly in H-plane. As a reference, the mutual coupling of the SIW-SIW back to back model, shown in Fig. 8(c), is calculated to be less than 40 dB. Strictly speaking, the calculated mutual coupling verifies that the SIW structure has a negligible leakage loss.

All the modeling and design methodology steps which are applied to a practical example are summarized in the algorithm, illustrated in Table II. The fabricated structures for the antennas in Fig. 1(a), (b) appears in Figs. 11 and 12 with the optimal design parameters. A multi-layer circuit fabrication technique, based on low cost PCB processes, is developed. A single-sided 50-mil Rogers RT/6010 substrate with a 0.5 oz copper plating is attached to the top of the RT/5870-based SIW feed by using a 2-mil bonding material. The top metal layer is then etched away leaving an uncovered dielectric layer. High-speed milling bits are used to clear out most of the RT/6010 dielectric material so that the DRA elements are bonded to the underlying SIW layer. The entire antenna is designed as a three metal-layer PCB with two dielectric layers sandwiched between the metal layers. The dielectric layers represent the SIW feed layer and DR elements, respectively. As a manufacturing process, the

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TABLE II MODELING AND DESIGN ALGORITHM OF THE SIW-DRA ARRAY

N =4 a = " = 10:20 h = 1:27 mm 3:0 mm d = 1:50 mm " = 2:33 h = 0:7874 mm a = 4:80 mm s = 0:60 mm L = 3:20 mm W = 0:30 mm d = 0:30 mm X = 5:0 mm D = 3:50 mm y = 1:50 mm

Fig. 12. The SIW-DRA array prototype using vertical polarized SIWslot (longitudinal) arrangement (a) Front view (b) Rear view), DRA substrate: Roger RT/6010 ( , ), DRA dimensions: , , SIW substrate: Roger RT/5870 ( , ), SIW dimensions: , , , slot dimensions: , , , array parameters: , off-center distance .

N=4 " = 10:20 h = 1:27 mm a = 3:0 mm d = 1:50 mm " = 2:33 h = 0:7874 mm a = 4:80 mm s = 0:60 mm d = L = 3:20 mm W = 0:30 mm 0:30 mm X = 2:50 mm D = 7:60 mm y =0

Fig. 11. The SIW-DRA array, prototype using horizontal polarized (transverse) SIW-slot arrangement (front view), DRA substrate: Roger RT/6010 ( , ), DRA dimensions: , , SIW substrate: Roger RT/5870 ( , ), SIW dimensions: , , , slot dimensions: , , array parameters: . , , off-center distance

bottom layer which contains the SIW feed, is first fabricated from a two-sided 31-mil Rogers RT/5870 Duroid material. The fabricated antennas are measured over the operating frequency band 33–40 GHz. The measured reflection coefficients for both antenna array configurations are depicted in Figs. 13 and 14 with impedance bandwidths 1.60% (around the center frequency 33.87 GHz), and 4.70% (around the center frequency 37.80 GHz), respectively. A good agreement between the measured and the HFSS results are observed, especially near the frequency at which the reflection coefficient is minimum. However, some deviation is noticed between the proposed circuit model and the measured results, especially in Fig. 13 where there is a stronger mutual coupling. This deviation is attributed to the

Fig. 13. The simulated and measured reflection coefficient (dB) of SIW-DRA array using horizontal polarized SIW-slot arrangement prototype shown in Fig. 11.

discrete nature of the SIW structure that affects the short circuit (SC) performance (magnitude and phase), T.L. lengths, and the mutual coupling between any two adjacent antenna elements, caused by the excitation of higher order mode at the SIW-slot interfaces, as discussed in Section IV. All these issues are handled efficiently and are taken into account by the full-wave solver. Therefore, the measured reflection coefficient agrees to that of the HFSS, except for a very small deviation, caused by fabrication tolerances and measurement errors. However, the proposed T.L. circuit model is still valid and can easily predict and estimate the resonance behavior and the impedance bandwidth of the proposed antenna arrays in very short time compared with HFSS. The radiation patterns (gains) of the proposed antenna prototypes are experimentally characterized. The measured gains of the two antennas, arranged along -axis, with a scanned angle varied between and , is described. The simulated and measured XZ- and -plane radiation patterns are shown

ABDEL-WAHAB et al.: MILLIMETER-WAVE HIGH RADIATION EFFICIENCY PLANAR WAVEGUIDE SERIES-FED DRA ARRAY

Fig. 14. The simulated and measured reflection coefficient (dB) of the SIW-DRA array ( = 4) using the vertical polarized SIW-slot arrangement prototype shown in Fig. 12.

N

N

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Fig. 16. The simulated and measured radiation pattern (gain) of the SIW-DRA array ( = 4) using vertical polarized SIW-slot arrangement prototype shown in Fig. 12: H-plane (XZ-plane), E-plane (YZ-plane).

N

Fig. 15. The simulated and measured results of the SIW-DRA array ( = 4) using the horizontal polarized SIW-slot arrangement prototype in Fig. 11: E-plane (XZ-plane) and H-plane (YZ-plane).

Fig. 17. The calculated (HFSS) antenna gain (dB), and radiation efficiency of both the SIW-DRA array (N = 4) configurations in Fig. 1(a), (b).

in Figs. 15, 16. The horizontal polarized (transverse) and vertical polarized (longitudinal) antenna arrays achieve a maximum broadside gain of 11.70 and 10.60 dB, respectively. The measured radiation patterns in both the E and H-planes confirm the simulated ones with less than a 0.8 dB difference in the maximum gain due dielectric, transition, and launcher losses. It is obvious that the proposed circuit model is capable of predicting the radiation pattern (gain), especially in the vicinity of the main beam (broadside). Of all the antenna parameters that characterize millimeterwave antennas arrays, the overall radiation efficiency is highly affected by the feeding structures. Therefore, one of the primary goals of this paper is to propose a novel and appropriate feeding scheme for millimeter-wave applications and to verify its performance. The simulated results demonstrate a stable gain and high radiation efficiency of more than 90%. Fig. 17 denotes the calculated (HFSS) gain (dB) and the radiation efficiency for the proposed SIW-DRA array within the operating impedance bandwidth. A high Q-feeding structure, the SIW structure enhances the overall antenna radiation efficiency and provides a stable gain by minimizing the

conductor and leakage losses, and blocking the undesirable radiation caused by other feeding structures. Typically, these traditional feeding structures degrade the efficiency and disturb the antenna array radiation pattern (unstable gain). Table III provides a comparison of some of the measured antenna characteristics between the measured and simulated results using both HFSS and the proposed circuit model. The antenna prototype in Fig. 12 depicts a wider bandwidth (impedance and radiation) with a more compact and smaller aperture area than the antenna prototype in Fig. 11. However, the latter provides a narrower beam width. All these characteristics demonstrate that the proposed SIW-DRA array structure is a low cost, highly efficient, and compact antenna system which is suitable for portable millimeter-wave applications. As compared with the full-wave electromagnetic (EM) numerical solvers, the proposed circuit model is an easy and fast and , way for optimizing the antenna array parameters, to achieve the best antenna reflection coefficient, , and radiation pattern (gain) performances. This is mainly due to the large size of the mesh required by the EM solver, especially as the number of antenna elements increases.

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TABLE III SIW-DRA ARRAY ( = 4) CHARACTERISTICS

N

[4] H. L. B. K. M. Strohm and J. Wenger, “Development of future short range radar technology,” in Proc. 2nd Eur. Radar Conf., Oct. 2005, pp. 165–168. [5] L. Yujiri, “Passive millimeter wave imaging,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, 2006, pp. 98–101. [6] R. Mailloux, J. McIlvenna, and N. Kernweis, “Microstrip array technology,” IEEE Trans. Antennas Propag., vol. 29, pp. 25–37, 1981. [7] R. A. Kranenburg and S. A. Long, “Microstrip transmission line excitation of dielectric resonator antennas,” Electron. Lett., vol. 24, pp. 1156–1157, 1988. [8] R. A. Kranenburg, S. A. Long, and J. T. Williams, “Coplanar waveguide excitation of dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 39, pp. 119–122, 1991. [9] G. P. Junker, A. A. Kishk, and A. W. Glisson, “Input impedance of dielectric resonator antennas excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 42, pp. 960–966, 1994. [10] K. W. Leung and K. K. So, “Rectangular waveguide excitation of dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 51, pp. 2477–2481, 2003. [11] X. Feng and W. Ke, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 66–73, 2005. [12] C. Xiao-Ping and W. Ke, “Substrate integrated waveguide cross-coupled filter with negative coupling structure,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 142–149, 2008. [13] X. Xinyu, R. G. Bosisio, and W. Ke, “A new six-port junction based on substrate integrated waveguide technology,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 2267–2273, 2005. [14] M. Abdolhamidi and M. Shahabadi, “X-band substrate integrated waveguide amplifier,” IEEE Microw. Wireless Comp. Lett., vol. 18, pp. 815–817, 2008. [15] Y. J. Cheng, H. Wei, W. Ke, Z. Q. Kuai, Y. Chen, J. X. Chen, J. Y. Zhou, and H. J. Tang, “Substrate Integrated Waveguide (SIW) Rotman lens and its Ka-band multibeam array antenna applications,” IEEE Trans. Antennas Propag., vol. 56, pp. 2504–2513, 2008. [16] W. M. Abdel Wahab, D. Busuioc, and S. Safavi-Naeini, “Low cost planar waveguide technology-based Dielectric Resonator Antenna (DRA) for millimeter-wave applications: Analysis, design, and fabrication,” IEEE Trans. Antennas Propag., vol. 58, pp. 2499–2507. [17] C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. New York: Wiley, 2008.

VI. CONCLUSION This work presents the modeling, analysis, characterization, and design procedures of an high radiation efficiency SIW-DRA array. A general layout of N-element linear arrays, based on two different slot configurations, is proposed. Then, a simple transmission line circuit model is developed as a fast method to characterize and optimize the SIW-DRA array for the optimum antenna reflection coefficient and radiation patterns. Finally, SIW-DRA array modules are fabricated and measured to validate the proposed physical, circuit model, and design concepts, respectively. A good agreement is achieved between the simulated (HFSS and the proposed circuit model) data and the measured ones. The high radiation efficiency of the proposed antennas is verified by the simulation results. REFERENCES [1] C. W. James, “Status of millimeter-wave technology and applications in 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,” IEEE MTT-S Digest, vol. 2, pp. 635–638, May 1994. [3] Y. Takimoto, “Recent activities on millimeter wave indoor LAN system development in Japan,” in Proc. IEEE Microwave Systems Conf., 1995, pp. 7–10.

Wael M. Abdel-Wahab (S’09) received the B.Sc. degree in electrical and computer engineering (ECE) from Zagazig University, Zagazig, Egypt, in 1998 and the M.Sc. degree in ECE from Cairo University, Cairo, Egypt, in 2004. He is currently working toward the Ph.D. degree at the University of Waterloo (UW), Waterloo, ON, Canada. He has been a Teaching Assistant/Lecturer (parttime) at UW (2006-2011). He is a member of a newly established Center for Intelligent Antenna and Radio System (CIARS) at UW. His current research interests focus on low-loss planar waveguide technologies for low cost efficient circuits and planar antennas in microwave and millimeter-wave frequency band, including multi-layers passive circuit components modeling, design, and fabrication. He has authored many research papers in highly cited international journals and proceedings of international conferences, such as the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (TAP) and IEEE Antennas and Wireless Propagation Letters. He published a patent application in proposing a novel low cost fabrication method for millimeter-wave antennas using multi-layer process. Mr. Abdel-Wahab is a member of the IEEE Antennas and Propagation Society (AP-S), and also serves as a reviewer for TAP and Progress In Electromagnetics Research (PIER). He was the recipient of several prestigious awards, grants, and scholarships, including international graduate scholarship from The Egyptian Government (Canada 2006-2010), UW Studentship (2010-2011), UW Graduate Scholarship for Excellence in Research (twice), and the UW Completion Thesis Award (Winter 2011).

ABDEL-WAHAB et al.: MILLIMETER-WAVE HIGH RADIATION EFFICIENCY PLANAR WAVEGUIDE SERIES-FED DRA ARRAY

Dan Busuioc received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electrical and computer engineering from University of Waterloo, Waterloo, ON, Canada in 2001, 2002, and 2005, respectively, and the MBA degree from Boston University, Boston, MA, in 2007. From 1996 to 2009, he has held a number of positions in engineering, applications, and marketing, with renowned companies in Germany, Canada, Sweden, and United States. He was one of the original founders of MASSolutions Inc., a Canadian-based company focusing on advanced sensors, microwave packaging, and antenna systems for microwave and mm-wave frequencies. His research interests include novel antenna systems and high-frequency circuitry, including miniaturized, low-cost feed systems. He has further interest in wireless systems and semiconductor test equipment industries. D. Busuioc was the recipient of the National Science and Engineering Research Council of Canada (NSERC) Industrial Postgraduate Scholarship in 2001 (with Ericsson Radio Access, Sweden) and in 2003 (with Winegard Company, Burlington, IA) and was a recipient of the Governor of Canada Medal.

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Safieddin Safavi-Naeini (M’79) received the B.Sc. degree in E.E. from the University of Tehran, Tehran, Iran, 1974 and the M.Sc. and Ph.D. degrees in E.E. from the University of Illinois at Champaign-Urbana, in 1975 and 1979, respectively. He was a faculty member of the School of Engineering, University of Tehran, from 1980 to 1995. He joined the University of Waterloo, Waterloo, ON, Canada, in 1996, where he is now a Professor in the Department of Electrical and Computer Engineering, the holder of the RIM/NSERC Industrial Research Chair in Intelligent Radio/Antenna and Photonics, and is also the Director of a newly established Center for Intelligent Antenna and Radio System (CIARS).

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Double-Layer Full-Corporate-Feed HollowWaveguide Slot Array Antenna in the 60-GHz Band Yohei Miura, Student Member, IEEE, Jiro Hirokawa, Senior Member, IEEE, Makoto Ando, Fellow, IEEE, Yuzo Shibuya, and Goro Yoshida

Abstract—In order to achieve a wide bandwidth characteristic of high gain and high efficiency antennas, a double-layer hollow-waveguide slot array is proposed, where a full-corporate-feed waveguide is arranged in the lower layer. This antenna can be built up easily by the process of diffusion bonding of laminated thin metal etching plates, which has high precision and is possibly a low cost technique. The radiating elements and the feed waveguide are designed to suppress the reflection over a wideband. The predicted bandwidth of the reflection less than 014 dB is 8.3% for a 16 2 16-element array antenna. A test antenna is fabricated in the 60-GHz band, and about 80% antenna efficiency with more than 32 dBi is achieved over 4.8 GHz. Index Terms—Double-layer structure, full-corporate-feed, high efficiency, waveguide slot array, wideband.

I. INTRODUCTION HOLLOW-WAVEGUIDE slot array antenna is advantageous for a larger-size or a higher gain since they neither suffers from dielectric nor radiation loss in comparison with a microstrip line-fed array antenna [1]. On the other hand, a conventional waveguide slot array antenna was unsuitable for mass production due to the complicated three-dimensional feeding structure below the radiating waveguide array [2], [3]. An alternating-phase fed single-layer slotted waveguide array antenna [4]–[6] was proposed which had the feed waveguide in the same layer of the radiating waveguides for the simpler fabrication. It was characterized by high gain, high efficiency, and high mass productivity, and it had been adopted for a subscriber antenna for the fixed wireless access system (FWA) [6]. Unfortunately, these arrays with series-feed have a fundamental problem that the bandwidth is narrowed due to the long-line effect when the array size becomes large. To remedy this bandwidth problem, two types of partially corporate feeds were introduced which kept the single-layer structures. The center feed type [7] halved the long-line effect by arranging the feed waveguide at the center of the antenna in the comparison with the end feed type. The partially corporate feed type [8] quartered the long-line effect

A

Manuscript received May 10, 2010; revised November 17, 2010; accepted December 30, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. Y. Miura is with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan and also with the Japan Radio Co., Ltd., Tokyo 181-8510, Japan (e-mail: [email protected]). J. Hirokawa and M. Ando are with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan. Y. Shibuya and G. Yoshida are with the Development Department, Research and Development Center, Japan Radio Co., Ltd., Tokyo 181-8510, Japan. Digital Object Identifier 10.1109/TAP.2011.2158784

by dividing the antenna into four subarrays. However, the aperture efficiency of both the models was degraded by grating-lobes due to no-slot areas in the aperture just above the feed waveguide newly introduced. A wideband and high efficiency corporate feed antenna was reported in the 12-GHz band [9], but it needed to put a dielectric material on each element to suppress grating lobes due to the element spacing larger than a free-space wavelength. By the way, multiple layer structure has the potential in enhancing the design freedom of feed waveguide which includes corporate feeds for a wider bandwidth. The key challenges here would be the feed waveguide arrangement in association with the low cost fabrication method. A double-layer slot array antenna, which was composed of a parallel feed waveguide and several 2 4-element cavity-backed subarrays, was investigated to obtain a wide bandwidth and high radiation efficiency in the 12-GHz band [10], [11]. The two elements on the center-fed subarray in the E-plane direction were spaced by about a guided wavelength to excite each element in phase. As a result, high side-lobe levels of 8.8 dB were observed in the E-plane pattern. We propose the structure of a double-layer corporate-feed waveguide slot array with constant element spacing less than a free-space wavelength in this paper. This antenna consists of a corporate feed waveguide arranged in the lower layer and 2 2-element small subarray units in the upper layer, the 4 elements in the subarray are almost equally spaced from the coupling aperture between layers at the end of the corporate feed and the array is substantially fed by a fully corporate feed. In general, multilayer structure cannot be fabricated easily by conventional techniques such as machining or die-casting. We adopt the diffusion bonding of laminated thin metal etching plates for the fabrication of a double-layer full-corporate-feed waveguide slot array antenna, which gives high precision and potentially low cost technique [12]. In the 12-GHz band, a cavity-backed planar slot array antenna with 16 16-elements in [10], [11] has 33.7 dBi gain with . However, 85% efficiency and 12% bandwidth for it is difficult to achieve these characteristics in the millimeterwaveband for which high fabrication accuracy is needed. In 40-GHz band, however, a two-layer slotted-waveguide antenna array assembled with the screws with 24 24-elements in [3] has 33.8-dBi gain but with lower efficiency of 51%, and narrower bandwidth of 5.5%. In contrast with these with multiple layer structure antennas in [3], [10], and [11], the efficiency and the bandwidth of the proposed antenna remain remarkable even in the higher frequency band of 60 GHz.

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Fig. 2. Exploded perspective view of the 2

2 2-element subarray.

Fig. 1. Configuration.

In the broadband design of high gain arrays, not the interference control of reflections as in the narrow band design, but the suppression of reflection from every part is indispensable since the reflections are accumulated in phase at some frequency in the operating bandwidth. In the design of corporate feed in this paper, the 2 2-element subarray is intensively discussed first with the aim of a wide bandwidth in term of reflection suppression from the subarrays, where the mutual coupling is taken into consideration assuming the infinite two-dimensional array. After the explanation of the configuration of the antenna, Section III clarifies the relation between the mutual coupling and the reflection bandwidth as functions of the aspect ratio of the slot element. For systematic design of large structure for wider bandwidth, the full-corporate-feed waveguide is divided into three parts, and a wideband design is discussed for each part. Section IV designs a 16 16-element array in the 60-GHz band and compares the calculated results with the experimental ones. II. CONFIGURATION OF THE DOUBLE-LAYER FULL-CORPORATE-FEED ANTENNA Fig. 1 shows the configuration of the double-layer full-corporate-feed hollow-waveguide slot array antenna. The shaded area shows the full-corporate-feed waveguide that consists of several H-plane T-junctions located in the lower layer. The antenna is fed through a feed aperture of the same size as a standard WR15 rectangular waveguide located on the back side. A coupling aperture is arranged at each end of the full-corporate-feed waveguide with offset. Radiating slots are arrayed as the elements both in the and directions with a constant spacing. Fig. 2 shows the exploded perspective view of the 2 2-element subarray as the design unit. The 2 2-element subarray is included in a cavity on the upper layer. The cavity is partitioned into four spaces by two sets of walls extending in the and directions. The coupling aperture is placed at the center of the cavity. In order to realize the strong excitation of cavity, it has an offset from the center axis of the waveguide, which takes

Fig. 3. Analysis model of the 2

2 2-element subarray.

the maximum physically for the given width of coupling aperture and the waveguide. The 2 2-element subarray is excited in phase and with equal amplitude by making the coupling aperture resonant even though it is offset in the full-corporate-feed waveguide. In fabrication by the diffusion bonding, each layer is composed by laminating thin metal etching plates. The number of the etching patterns for this antenna is only five. The etching patterns are for the slotted plate, the cavity, the coupling aperture, the full-corporate-feed waveguide and the feed aperture in order from the top to the bottom in the antenna. III. DESIGN A. 2

2-Element Subarray

The 2 2-element subarray is a design unit of the proposed antenna. Fig. 3 shows the model of the 2 2-element subarray to analyze the frequency characteristic of the reflection to the feed waveguide. Two sets of periodic boundary walls are placed in the external region to simulate the mutual coupling in the infinite two-dimensional slot array. The set of the walls in the direction are placed to suppress unwanted higher modes in the cavity. The design frequency of the antenna is 61.5 GHz, and the slot spacing in the and directions are fixed to be 4.2

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Fig. 4. Slot coupling mechanism.

Fig. 5. Reflection coefficient of the 2 ratio w=l of the radiating slot.

Fig. 6. Frequency behavior of reactance component at the input port.

2 2-element subarray for various aspect

mm (0.86 wavelengths at the design frequency). The model is analyzed by Ansoft HFSS. Fig. 4 shows the slot coupling mechanism of the 2 2-elemode in the feed ment subarray. The magnetic field of the waveguide has mainly the component at the coupling aperture. The magnetic field in the cavity becomes symmetrical distribution with respect to the direction. The design parameters should be reduced as much as possible for the simple design. Here, they are the length and the width of the slot, the width of the feed waveguide and the distance between the cavity center and the wall edge in the direction as shown in Fig. 3. The other parameters are listed in Table I(a). A wideband reflection characteristic with double-tuned impedance matching is achieved in the 2 2-element subarray with the peas shown in riodic boundary walls for various aspect ratio Fig. 5. The circular dot and the triangle dot on each line show the lower and higher matching frequencies, and are defined as and , respectively. The square dot shows the local maximum value of the reflection between and . The frequency separation between and is determined depending on since the frequency behavior of the reactance relating to the resonance at the input port is changed by the mutual coupling influenced by as shown in Fig. 6. The frequency is determined by the resonance of the radiating slot. It is fixed at 60 GHz by controlling properly for each . The reflections at and can be almost the same value by controlling the input impedance by , and controlling reactance by . The design procedure is as follows. is selected for a desired frequency separation between (1) and ; (2) is determined so that may be approximately 60 GHz;

Fig. 7. Local maximum value of the reflection coefficient and reflection bandwidth for VSWR 1.5 according to aspect ratio w=l of the radiating slot. TABLE I ANTENNA PARAMETERS

(3)

and are determined so that the center of the loop of the impedance characteristics on the Smith chart may be placed at the center of the chart; (4) To correct the shift of , the steps (2) and (3) are repeated. Fig. 7 shows the local maximum value of the reflection and . The solid line with square dots the bandwidth for various indicates that the maximum value of the reflection as shown , and it takes the minimum in Fig. 5 decreases for large . The bandwidth for calculated at from Fig. 5 takes the maximum up to 9.2% at . is chosen to be 0.64 for the design of a 16 16-element array antenna that the local maximum value of the reflection is less , and the bandwidth is the maximum. The paramthan eters thus obtained are listed in Table I(b). The bandwidth for

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Fig. 10. H-plane T-junction. (a) T-junction without Iris. (b) T-junction with Iris. Fig. 8. Reflection coefficient at each port.

Fig. 9. E-bend. (a) Conventional structure. (b) Proposed structure.

are 9.1% and

and the local maximum value of the reflection , respectively.

B. Full-Corporate-Feed Waveguide The full-corporate-feed waveguide for the 16 16-element array antenna consists of an E-bend, four T-junctions and an H-junction from the feed aperture to the coupling aperture. In order to suppress the reflection from the array antenna over a wide bandwidth, the reflection bandwidth of each part should be extended. In this section, the full-corporate-feed waveguide is divided into the following three parts, and each of the parts is designed separately. (1) E-bend located at the feed aperture; (2) Each of the first four H-plane T-junctions from the feed aperture; (3) H-plane H-junction that combines the last two T-junctions to the coupling aperture. Fig. 8 shows the frequency characteristics of the reflection of each part. The design of the E-bend is shown in the beginning. A conventional E-bend had a cutout at the corner to reduce the reflection as shown in Fig. 9(a). In fabrication by the diffusion bonding of laminated thin metal plates, the number of the etching patterns increases because the shape of each pattern changes along the thickness of the antenna for this conventional E-bend. Fig. 9(b) shows the proposed structure without such a cutout. The feed aperture and the feed waveguide have different size and are vertically connected as shown in the figure. The reflection can be suppressed by selecting the distance from the center axis of the feed aperture to the short wall of the feed waveguide. The reflection of the proposed E-bend is below over the bandwidth for in the 2 2-element subarray as shown in Fig. 8. The design of each of the first four H-plane T-junctions from the feed aperture is shown. Each of the first four T-junctions is

Fig. 11. H-plane H-junction.

so far from each other not to couple with higher modes that it can be designed independently. Fig. 10 shows the structure of the H-plane T-junction. The reflection of a T-junction shown in over the bandwidth for Fig. 10(a) is below in the 2 2-element subarray as shown in Fig. 8 by choosing . However, this rethe wall length and the window width flection is not enough for the full-corporate-feed circuit. An iris is installed at a distance of a quarter of the guided wavelength away from the window as shown in Fig. 10(b). The reflections from the window and the iris are cancelled with each other since the round-trip phase difference is 180 degrees. As a result, the reflection of the T-junction with the iris as shown in Fig. 10(b) is below over the same bandwidth. Finally, the design of the H-plane H-junction is shown. Fig. 11 shows the structure of the H-plane H-junction. The two T-junctions are so close to each other to couple higher modes that they should be designed as one body by including the higher modes. The reflection of the H-junction is below over the bandwidth for in the 2 2-element subarray as shown in Fig. 8. The amplitude and phase variation among the four output ports are 0.1 dB and 1.4 degrees in the H-junction, respectively. The wideband full-corporate-feed waveguide for the 16 16-element array antenna is achieved to excite all the coupling apertures uniformly both in amplitude and phase. The is 11.8%, and it is bandwidth of the reflection less than over the bandwidth for in the below 2 2-element subarray as shown in Fig. 8. The amplitude and phase variation over this bandwidth are 0.4 dB and 3.6 degrees among all the coupling apertures, respectively. C. 16

16-Element Array Antenna

The full structure of the 16 16-element array antenna that combines the 2 2-element subarrays and the full-corporate-

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Fig. 12. Reflection coefficient of the 16

2 16-element array.

Fig. 14. Frequency behavior of reflection coefficient.

coupling aperture layer of 0.3 mm thickness, and the feed waveguide layer of 1.2 mm thickness is composed by laminating thin copper plates of 0.3 mm thickness. These four kinds of the layers and the feed aperture plate of 6.0 mm thickness are heated up to about 1000 degrees with high pressure in the diffusion bonding. The 6.0 mm thickness for the feed aperture is not required for the antenna operation. It is installed just to connect a standard WR15 waveguide by screws in measurement. B. Reflection

Fig. 13. Picture of the fabricated antenna.

feed waveguide are analyzed by HFSS. Fig. 12 shows the frequency characteristic of the reflection at the feed aperture. The is 8.3% (58.8 GHz–63.9 GHz) bandwidth for that is narrower than 9.1% in the 2 2-element subarray with the periodic boundary walls due to the ripples around 64 GHz. There are two differences between both the models. One is in terms of the periodic boundary walls and the other is in terms of the full-corporate-feed waveguide. The ripples comes from the reflection from the full-corporate-feed waveguide, because similar ripples in the reflection of the 16 16-element array antenna if it is analyzed by assuming the periodic boundary walls around the periphery as shown in Fig. 12. IV. EXPERIMENTAL RESULTS A. Fabrication of an Antenna by the Diffusion Bonding The 16 16-element array antenna in the 60-GHz band is fabricated by the diffusion bonding of laminated thin metal (15.4 wavelengths 15.6 waveplates sized by 75 76 lengths at 61.5 GHz). Fig. 13 shows the picture of the fabricated elantenna. The aperture area is defined as 67.2 mm ( 4.2 mm spacing) square. Each of the slot layer of ements 0.3 mm thickness, the cavity layer of 1.2 mm thickness, the

Fig. 14 shows the frequency characteristic of the reflection at the feed aperture. The solid line with some width shows the measured reflection including the deviation among eight fabricated antennas. The measured reflection is degraded but it is below over the bandwidth for the calculated . The measured frequency characteristic is shifted about 600 MHz lower than the calculated one. In order to identify the factors for the shift, we have done calculations for various values of over-etching. The result assuming 0.02 mm over-etching for all the layer patterns is added into Fig. 14, which agrees with the measured result in whole bandwidth; we can conclude the difference between the original calculation and the measurement comes from the over-etching by 0.02 mm. The reflection coefficient changes sensitively by this fabrication error but we confirmed that other characteristics such as radiation patterns, gain and efficiency are stable and almost unchanged. C. Aperture Field Distributions The aperture field distributions reflect the quality of the diffusion bonding to achieve the uniform excitation of the array antenna. Fig. 15 shows the measured near-field distribution over the aperture at 62 GHz. As for the amplitude and the phase, a symmetric distribution is observed both along the -axis and the -axis. Therefore, it is confirmed that the fabrication by the diffusion bonding have succeeded. The amplitude and phase variation over the aperture area are about 5 dB and 40 degrees, respectively. The dips especially in the amplitude at run in parallel to -axis, which was also numerically predicted as the truncation-effects of the finite array in the E-plane with strong mutual coupling. In the H-plane, on the other hand, we can hardly recognize them since the mutual coupling is weak. The truncation-effects are not included in the present design where periodic boundary conditions are imposed in the two directions in the external region to simulate the mutual coupling in the infinite array of the elements. This assumption is not valid

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TABLE II BEAM WIDTH, FIRST SIDELOBE LEVEL AND CROSS POLARIZATION DISCRIMINATION AS FUNCTION OF FREQUENCY

Fig. 16. Radiation patterns. (a) E-plane. (b) H-plane. Fig. 15. Near-field distributions. (a) Amplitude [dB]. (b) Phase [deg.].

for the elements around the periphery and the field is not uniform around the periphery. Nevertheless, the aperture efficiency is quite high and the uniformity of the aperture field is sufficient to demonstrate the feasibility of the high efficiency in the proposed antenna. D. Radiation Patterns The radiation patterns at 62 GHz are measured in the Eplane and the Hplane. Fig. 16(a) shows the E-plane pattern. Good agreement is observed between the measured pattern and the calculated one. The measured first sidelobe level . It is slightly lower than calculated one of . is Fig. 16(b) shows the H-plane pattern. The measured pattern coincides with ideal one that obtained from uniform illumination. The 3-dB down beam width in both the E- and H-planes is 3.8 degrees. The measured cross-polarization level is well in the both planes even though suppressed less than

Fig. 17. E-plane radiation patterns in operating bandwidth.

the slots are wide. Fig. 17 shows the E-plane radiation patterns in the operating bandwidth at 58, 62, and 64 GHz. The beam width, first sidelobe level, and cross-polarization discrimination as a function of the frequency are shown in Table II. The main

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Fig. 18. Frequency behavior of directivity, gain, and efficiency.

beam direction is always at boresight (0 degrees) in principle, unchanged against the frequency because all the elements are symmetrically fed with respect to the center. E. Gain and Efficiency Fig. 18 shows the frequency behavior of the directivity, gain, and efficiency characteristics. The measured directivity as shown in the figure is calculated by Fourier transforming the near-field distribution. The gain of the eight fabricated antennas is measured in an anechoic chamber and compared with a standard gain horn. The conductor loss and the reflection loss are included in the gain. The solid area indicates the measured gain including the deviation among the eight antennas, which is well predicted by the calculation. The measured directivity is 33.5 dBi at the design frequency (61.5 GHz) and the aperture efficiency is 93.7%. High antenna efficiency of 83.6% is achieved by measuring antenna gain of 33 dBi including the losses. The measured 1-dB down bandwidth of the gain is 11% by reflecting the full-corporate-feed. V. CONCLUSION A double-layer corporate-feed hollow-waveguide slot array antenna with more than 32 dBi gain and about 80% antenna efficiency over 4.8 GHz in the 60-GHz band is designed. The radiating elements and the feed waveguide circuit are designed to suppress the reflection over a wide bandwidth. The array designs to obtain uniform aperture field distribution have been conducted. The diffusion bonding of laminated thin metal plates is adopted as a production method of the antenna for the purpose of high accuracy and mass-producability. High antenna efficiency of 83.6% has been obtained at 61.5 GHz by the measured antenna gain of 33 dBi including the losses. The measured 1-dB down gain bandwidth has been achieved wideband characteristic of 11%. REFERENCES [1] E. Levine, G. Malamud, S. Shtrikman, and D. Treves, “A study of microstrip array antennas with the feed network,” IEEE Trans. Antennas Propag., vol. 37, no. 4, pp. 426–434, Apr. 1989. [2] R. C. Johnson and H. Jasik, Antenna Engineering Handbook. New York: McGraw-Hill, 1984, ch. 9.

[3] 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. 51, no. 5, pp. 393–398, Oct. 2004. [4] N. Goto, A Planar Waveguide Slot Antenna of Single Layer Structure 1988, IEICE Tech. Rep., AP88-39. [5] T. Shirouzu, K. Nidaira, M. Baba, and T. Saitoh, “Wireless IP access system for low cost FWA services,” in Proc. Eur. Conf. Wireless Technol., Oct. 2003, pp. 435–438. [6] 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. [7] S. Park, Y. Tsunemitsu, J. Hirokawa, and M. Ando, “Center feed single layer slotted waveguide array,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1474–1480, May 2006. [8] S. Fujii, Y. Tsunemitsu, G. Yoshida, N. Goto, M. Zhang, J. Hirokawa, and M. Ando, “A wideband single-layer slotted waveguide array with an embedded partially corporate feed,” in Proc. Int. Symp. Antennas Propag., TP-C27-5, Oct. 2008. [9] T. Tsugawa, Y. Sugio, and Y. Yamada, “Circularly polarized dielectric-loaded planar antenna excited by the parallel feeding waveguide network,” IEEE Trans. Broadcasting, vol. 43, no. 2, pp. 205–212, Jun. 1997. [10] K. Jung, H. Lee, G. Kang, S. Han, and B. Lee, “Cavity-backed planar slot array antenna with a single waveguide-fed subarray,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 115.5, Jun. 2004. [11] B. Lee, K. Jung, and S. Yang, “High-efficiency planar slot array antenna with a single waveguide-fed cavity-backed subarray,” Microw. Opt. Technol. Lett., vol. 43, no. 3, pp. 228–231, Nov. 2004. [12] J. Hirokawa, M. Zhang, and M. Ando, “94 GHz single-layer slotted waveguide array by diffusion bonding of laminated thin plates,” in Proc. Int. Symp. Antennas Propag. TP-C03-3, Oct. 2008.

Yohei Miura (S’09) was born in Tokyo, Japan, on November 19, 1976. He received the B.S. and M.S. degrees in electrical engineering from Takushoku University, Tokyo, in 1999 and 2001, respectively. He is currently working toward the D.E. degree with the Tokyo Institute of Technology. He has been with the Japan Radio Co., Ltd., Mitaka, since 2001. His current research area is in slotted waveguide array antennas and millimeterwave antennas. Mr. Miura is a member of IEICE.

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 the 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 with 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 Communication Society. He is a Member of IEICE.

MIURA et al.: HOLLOW-WAVEGUIDE SLOT ARRAY ANTENNA IN THE 60-GHZ BAND

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 Tech), Tokyo, Japan, in 1974, 1976, and 1979, respectively. From 1979 to 1983, he was with Yokosuka Electrical Communication Laboratory, NTT, Japan, and was engaged in the development of antennas for satellite communication. He was a Research Associate with Tokyo Tech 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 interest includes 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 Awards from IEICE Japan in 1993 and 2009. 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 and the Award in Information Promotion Month 2006, 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 was the General Chair of the 2004 URSI EMT Symposium in Pisa, Italy, and of the ISAP 2007 in Niigata. He served as Chair of the Technical Committee of Electromagnetic Theory (2004–2005) and Antennas and Propagation (2005–2007) in IEICE. He was a member of the Administrative Committee of the IEEE Antennas and Propagation Society 2004–2006 and also a member (since 2004) of the Scientific Council for Antenna Centre of Excellence—ACE in EU’s 6th framework program. He served as the Chair of Commission B of URSI 2002–2005. He was the 2007 President of Electronics Society IEICE and the 2009 President of IEEE Antennas and Propagation Society. He is currently serving as the Program Officer for the Engineering Science Group, Research Center for Science Systems, JSPS. He is a Fellow of IEICE.

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Yuzo Shibuya received the B.S. degree in electrical engineering from Chuo University in 1987. In 1987, he joined Japan Radio Co., Ltd., Mitaka. Since then, he has been in charge of the development of microwave antennas and radar systems.

Goro Yoshida was born in Kumamoto, Japan, on November 7, 1952. He received the B.S. degree in electrical and electronic engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1976. He joined the Japan Radio Co., Ltd., Tokyo, in 1976, where he is currently a General Manager of the Research and Development Center. His main works were developments of antennas and RF modules for radar systems.

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A Deterministic Two Dimensional Density Taper Approach for Fast Design of Uniform Amplitude Pencil Beams Arrays Ovidio Mario Bucci, Fellow, IEEE, and Stefano Perna, Member, IEEE

Abstract—The synthesis of electrically large, aperiodic planar arrays with equi-amplitude excitations plays an increasingly relevant role in satellite applications. Unfortunately, for such a kind of synthesis problems, local optimization procedures may be ineffective, whereas global optimization procedures involve a severe computational effort. To circumvent these problems, a new deterministic approach for fast design of aperiodic concentric ring arrays is proposed. The method exploits easily obtained optimal continuous planar solutions, accounts for the geometric properties of the array element and allows, once the central geometry of the array (that is, at least the most internal ring) is fixed, to compute in a deterministic, iterative, very fast way the whole geometry of the density-tapered concentric ring array. Numerical examples show the effectiveness of the method, which provides in a few seconds layouts of hundreds of elements able to produce directivity patterns that satisfy realistic satellite project requirements. Index Terms—Antenna arrays, antenna synthesis, planar arrays, ring arrays, sparse arrays.

I. INTRODUCTION HE synthesis of aperiodic arrays with equi-amplitude excitations, investigated since the sixties of the last century [1]–[12], nowadays is assuming an increasing relevance in satellite applications, wherein active arrays could be a valid alternative to conventional reflector antennas when many high gain, possibly scanned, spot beams are required [13]–[23], provided that complexity, weight, and cost are kept reasonable. For such applications, electrically large antenna apertures are necessary, which makes standard periodic arrays inappropriate, particularly for the transmitting mode, due to the large number of control points, hence of power amplifiers. Moreover, proper tapering of the array excitations would require amplifiers of variable gain thus inducing poor efficiency in DC-to-RF power conversion.

T

Manuscript received April 30, 2010; revised October 13, 2010; accepted December 01, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was partially supported by the European Space Agency under ESA/ESTEC contract no. 21689/08/NL/ST. O. M. Bucci is with the Dipartimento di Ingegneria Biomedica Elettronica e delle Telecomunicazioni (DIBET), Università degli Studi di Napoli “Federico II”, 80125 Napoli, Italy and also with the Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA) of Consiglio Nazionale delle Ricerche (CNR), 80125 Napoli, Italy (e-mail: bucci@unina). S. Perna is with the Dipartimento per le Tecnologie (DIT), Università degli Studi di Napoli “Parthenope”, Centro Direzionale, 80143 Napoli, Italy and also with the Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA) of Consiglio Nazionale delle Ricerche (CNR), 80125 Napoli, Italy (e-mail: perna@uniparthenope). Digital Object Identifier 10.1109/TAP.2011.2158783

Differently, non uniformly spaced planar arrays with uniform amplitude excitations (isophoric) require amplifiers operating all under the same optimal condition thus guaranteeing higher power efficiency [17]–[21]. Moreover, they limit the effects of grating lobes, thus improving the performances in terms of side lobe level (SLL). Finally, for a given aperture, aperiodic arrays can employ a number of elements significantly smaller than that of periodic ones, without noticeable enlargement of the beam-width. This produces reduction of costs as well as of mutual coupling effects due to the increase of the interelement distances. On the other side, this also produces losses of aperture efficiency, hence of maximum reachable directivity [17]–[23] unless more elaborate architectures [24], [25] are exploited. Unfortunately, the synthesis of isophoric non uniform arrays is a much more difficult task as compared to that of periodic, non isophoric ones because it requires, in principle, the minimization of a nonlinear cost functional over a non convex set [16]; moreover, for arrays of large electrical dimensions (as in the case of satellite applications) a large number of unknowns must be dealt with. Thus, global optimization procedures are strongly limited by the problem size [26], whereas local optimization procedures may be trapped into “local minima,” which may be far from the global optimum. To circumvent these problems, in this paper it is presented a new deterministic approach for the synthesis of planar aperiodic arrays, able to exploit continuous reference planar sources. Rotationally symmetric beams are considered; thus, circularly symmetric continuous reference sources defined over circular apertures [21], [27] are exploited. The proposed approach extends to the two-dimensional (2D) case the density taper approach proposed for linear sources by Doyle [5], as reported in [6]. Concentric ring arrays, which well approximate the abovementioned circular field symmetry, are considered. Differently from other 2D deterministic approaches dealing with concentric ring arrays and available in [11], the solution proposed in this paper exploits a Doyle-based rationale for determining the whole geometry of the array, i.e., the minimization of the (properly weighted) mean square difference between the patterns produced by the array itself and the continuous reference source is enforced. This allows substituting the key concepts of linear subintervals and equal areas introduced in [5], [6] for 1D sources, and briefly recalled in Section II, with those of angular sectors and equal volumes, respectively, which are appropriate for circular aperture sources. Such concepts, introduced in [4] and mentioned in [6], are rigorously derived for

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the first time in this work (Section III). Their exploitation is carried out in Sections III and IV where it is shown that, given the continuous reference source, the geometric properties of the array element and the central geometry of the array (that is, at least the most internal ring), then it is possible to obtain in a deterministic, iterative, easy and very fast way the whole 2D geometry of the density tapered concentric ring array, without the need of other a-priori information (as the number of rings or/and elements per ring or/and array elements). Numerical examples are provided in Section V. Conclusions follow. II. 1D DENSITY TAPER APPROACH: MAIN RATIONALE The 1D density taper approach in [5], [6] is now recalled. be the continuous reference source (real and posiLet tive), being the abscissa of the source itself, of length ; and be the normalized cumulative distribution let (1)

subwhich represents the area of the reference source , and normalized to that tended by the interval . The 1D density subtended by the whole interval taper approach in [5] and [6] consists of two steps. First, the is partitioned in subintervals ( being interval the number of array elements) subtending the same area of : these subintervals may be easily the reference source found by considering equal increments of the function in (1) and projecting them onto the axis. As second step, each array element is located within each subinterval according to different strategies proposed in [6] and [16]. Following the same rationale, a two-step strategy is followed also for the proposed 2D case and the two different steps are addressed in Sections III and IV, separately. Similarly to the 1D case, mutual coupling effects are not accounted for by the proposed method. III. FIRST STEP: PARTITION GENERATION To address the first step of the presented method it is necessary to reformulate the concepts of linear source, linear subintervals, and equal areas, not appropriate in this 2D case. A. Problem Formulation For azimuthally symmetric fields, circularly symmetric continuous sources are appropriate and represent the 2D counterpart of linear reference sources in [5] and [6]. Also in this case, reference sources are real and non negative, which always happens for sufficiently narrow pencil beams [27]. Turning to the sampled solution, concentric ring arrays allow well approximating the azimuth symmetry of the field. Thus, given the 2D continuous reference source, the analogous of the 1D first step is a proper subdivision of the available circle in concentric rings, and of each ring into equal angular sectors, which thus represent the 2D counterpart of the linear subintervals introduced in [5], [6] for the 1D case.

Fig. 1. Geometry of the problem relevant to the k

ring.

It is shown now that if all these sectors are forced to subtend the same volume of the 2D reference source, then it can be minimized the (properly weighted) mean square difference between the patterns of the continuous source and the concentric ring array containing one element per each sector. define the angular position of Let us refer to Fig. 1: the observation point in the far field region. The antenna pattern produced by a circular aperture of radius with circularly sym( being the radial antenna cometric continuous amplitude ordinate) is [29] (2) is the zero-order Bessel function of first kind [30], being the wavelength, , and is the scaled by a proper (inessential) constant factor. On source the other side, the array factor of an isophoric concentric ring array (with arbitrarily spaced rings) is where

,

(3) where is the total number of rings; is the number of ring array, and represent, respecelements of the array tively, the azimuth and radial coordinates of the element of the ring (see again Fig. 1). Introduction of the following discrete array illumination function: (4) being the delta of Dirac, leads to (5) Note that the pattern of the azimuthally symmetric continuous source is azimuthally symmetric as well [see (2)], whereas the array pattern is not. However (see Appendix I), for our case of interest, i.e., high directivity focusing antennas,

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such a dependence is negligible in the angular sector wherein the radiated field is significant. Thus, only the zero harmonics of the array pattern can be retained

(6) where (7) Fig. 2. 2D density taper approach.

Use of (2) and (6) leads to (8)

of

being the number of array elements. From (13), enforcement requires the following scaling: (14)

By integrating by parts the integral in (8), introducing the cumulative distribution functions as follows:

to be applied to

(9)

, which is equivalent to let (15)

where the function (10) (16) and enforcing [i.e., (6), and (10)], it turns out that

, see (2), (9),

(11)

Equation (11) establishes a generalized Hankel transform relationship [31] between the (weighted) pattern difference and the cumulative distribution difference; application of the Parseval’s theorem [31] is thus possible, which leads to

subtended represents the volume of the 2D reference source by the circle of radius , normalized to that subtended by the whole circle of radius . Use of (16) allows rigorously deriving the 2D counterpart of the equal areas concept used in [5], [6] for the 1D case. To this aim, we observe that to minimize the right scaled according to (15) hand side of (12), the function should pass trough each step of . By referring to Fig. 2, this ring array (with elements) should means that the generic be placed at a radial coordinate belonging to the interval , where

(12)

Thus, minimizing the mean square difference of the patterns weighting is equivalent to minimizing the mean with square difference between the cumulative distributions in (9) and (10), provided that these latter distributions satisfy the . Further comments are now needed. constraint From (4), (7), and (10), it turns out that is a sum of (with ). In particular, steps, whose height is and (13)

(17) elements of the ring array In other words, to locate the we need to subdivide the ring delimited by the radii and onto adjacent angular sectors, and subsequently locate one element per each angular sector. According to (16), each of these adjacent angular sectors will subtend the following (normalized) volume of the reference source (18) where use of (15) and (17) has been done in the first and second equality, respectively. The volume in (18) is independent of ; thus, application of the same procedure for all the rings leads

BUCCI AND PERNA: A DETERMINISTIC 2D DENSITY TAPER APPROACH FOR FAST DESIGN

to a total number [see (13)] of angular sectors, each of one [see (18)], that is, a fraction subtending the same volume of of the volume subtended by the whole available circular area (of radius ). Thus, these sectors will be referred hereafter to as iso-volume sectors. By summarizing, in the 2D case, to minimize the (properly weighted) mean square difference of the patterns produced by the continuous reference source and the concentric ring array containing one element per sector, the equal areas concept, rigorously derived in [5] and [6], should be substituted by the concept of equal volumes, introduced in [4], mentioned in [6], and now rigorously derived. Further details for computation of these iso-volume sectors are provided in the following sections. B. Computation of the Iso-Volume Sectors According to the previous analysis, given the reference source , the first step of the 2D density taper procedure consists in finding the following parameters:

(19) in such a way to obtain iso-volume angular sectors. In (19), ring and the internal as usual, is the external radius of the ring (note that ), whereas is the radius of the number of adjacent angular sectors of the ring. To find iso-volume angular sectors, let us consider the scaled cumulative distribution in (15), divide the ordinate onto equal unitary increments, and project these points onto the axis. This is equivalent to compute the following set of radii: (20) in such a way that (21) By extracting the searched set , and enforcing for each

from the following condition: (22)

then a number of rings and iso-volume sectors will be achieved. Indeed, (22) allows satisfying the second condition in (17), and thus (18), which guarantees the condition of isomay volume sectors. However, several sets : adoption of a unitary, suitable be extracted from criterion is thus needed. To this aim, it can be enforced the requirement that all the iso-volume sectors should be as “square” as possible, i.e., with equal radial and azimuth extensions. This easily leads to the following iterative rule:

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where the symbol “ ” in (23) means that the equality must be enforced as much as possible and the parameter in (24) takes into account the array element size. Note that generalization to rectangular sectors is in any case straightforward. An iterative application of (23), subject to the constraints in (24), allows computing in deterministic way the radius once the radius is known; subsequent use of (22) (with the ob) allows determining the number of vious substitution adjacent angular sectors of the ring delimited by the radii and previously computed. Some further considerations are now in order. First, the whole geometry of the so achieved concentric ring array is determined according to the iso-volume concept introduced in the previous subsection. Differently, the 2D density taper solution suggested in [11] exploits the equal-areas concept to compute the radial geometry of the array, whereas the azimuth geometry depends on rules fixed a priori, which in any case do not assure minimization of the mean-square difference of continuous and sampled source patterns. Analogously, the 2D density taper solution suggested in [14] computes the radial positioning of the array by reducing the problem into a 1D one, which is then addressed with a Doyle-like approach, whereas subsequent definition of the azimuth geometry is carried out according to rules that in any case are not shown to ensure minimization of the mean-square difference of continuous and whole sampled source patterns. Second, application of the iterative rule in (23) requires comvia (15); thus, it is necessary to have putation of the function a-priori knowledge of the total number of angular sectors to be considered, i.e., the total number of array elements. In addition, the iterative rule in (23) also requires a-priori knowledge and , that is, (at least) the of (at least) the starting values geometry of the most internal ring. As shown in Section III-C, on the other side the choice of on one side and of and are not independent each other. C. Setting the Central Geometry and the Parameter N Choice of the array core geometry depends basically on the shape and size of the array element. For instance, for circular feeds an option may be to locate one feed at the center of the available area. Of course, different solutions can be suitable as well and, more generally, given the geometry of the array element, it is possible to fix a-priori the geometry of a number of , by setting the following parameters: rings, say (25) According to (25), the following number of elements: (26)

(23)

, subtending is placed into a circle with radius equal to equal to a (normalized) volume of the 2D reference source , see (16). Thus, each of these elements (located a priori) belongs to a 2D interval subtending, in average, the following elemental (normalized) volume

(24)

(27)

subject to the following constraints:

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Due to the searched azimuth symmetry of the pattern, azimuth distance between adjacent elements of the same ring should be , where the reference kept constant and so determines in practice the angular positions of all the angle adjacent sectors of the ring. It can be shown (see Appendix I) does not affect significantly the array patthat the value of tern shape in the near bore-sight directions. Accordingly, can be set randomly, thus saving the computational efficiency of the whole procedure. Of course, more refined choices could be instead adopted, for instance by means of local optimization procedures. is now addressed. Computation of the parameter Extension to the 2D case of the strategy adopted in [6] for the 1D case is again appropriate, and consists in finding the paramin (29) in such a way to enforce the minimization of the eter right-hand side in (12), and thus, of the mean square (weighted) difference of the patterns produced by the continuous and sampled sources. By doing so, along the same lines shown in [6] for the 1D case, we easily obtain Fig. 3. Diagram flow of the first step of the proposed approach. Reference source i() and array element dimensions are given; the core geometry ;K ;r ;...r ; ;... ) is set a-priori. (N

Application of the equal volumes concept thus requires that also the remaining angular sectors (to be iteratively computed) subtend the (normalized) volume in (27). Accordingly, the total number of elements to be located in the whole array is

(30) which allows computing in a very simple and fast way the paonce and are known. Note that the solution rameter suggested in [4] for deterministic tapering of concentric ring arrays, although based on the concept of equal-volumes, does not applies the rule (30) for locating the elements, thus not assuring the minimization of the mean-square difference of the patterns produced by the continuous and sampled source.

(28) V. NUMERICAL EXAMPLES depends on both the conFrom (25)–(28) it turns out that tinuous reference source and the core geometry. Thus, once this latter is fixed a-priori as in (25), subsequent application of the iterative rule in (22)–(24) is possible by starting from and substituting (28) in (15). The first step of the proposed approach is summarized in the diagram-flow of Fig. 3, which shows that, given the continuous reference source, the array element geometry, the array core geometry, then it is possible to carry out in a deterministic, iterative and very easy way all the parameters in (19) without the need of other a-priori information (as the number of rings or/and elements per ring or/and array elements), which are instead required by the 2D algorithms proposed in [4] and [11]. IV. SECOND STEP: POSITIONING THE ELEMENTS INSIDE THE ANGULAR SECTORS Once the parameters in (19) have been computed at the first step, the second step of the synthesis strategy, i.e., positioning the array elements inside the iso-volume sectors, basically con, the folsists in finding, for each lowing parameters:

(29) already defined in Section III-A. A discussion on the parameter

is now needed.

A specific problem of interest for the realization of a multibeam satellite array antenna for telecommunications (see [22] for more details) has been considered to show the effectiveness of the proposed method. The requirements for the central beam, to be met with a source of maximum radius of 60 wavelengths, as follows: are given in terms of directivity

(31) where is the angle at the edge of coverage (EOC) is the EOC angle of the nearest “isocolor” zone, beam, i.e., the beam with the same frequency and polarization, is the maximum inside Earth angle. According to the project constraints, a circular aperture of has been considered. The optimal reference radius satisfying at best the concontinuous aperture distribution straints in (31) has been computed as in [27]: it is shown in Fig. 4 along with the corresponding ( -independent) directivity pattern, the main characteristics of which are listed in Table I. As for the sampled solution, a minimum interelement distance of is enforced hereafter [i.e., it is set in (24)]. As first example, we have fixed a-priori the geometry of only the first ring by setting the parameters in (25) as follows: , , , . Under such conditions, it has been obtained the layout of Fig. 5(a),

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Fig. 4. (A) Radial behavior of the (normalized) continuous reference source. (B) Corresponding directivity pattern along with the directivity mask defined by the constraints in (31). Fig. 6. Superposition, at three different scales, of '-cuts of the array factor pattern (continuous line) of the layout in Fig. 5(a) along with the pattern of the continuous reference source (dotted line) of Fig. 4.

TABLE I PERFORMANCES OF THE GENERATED ARRAYS

Fig. 7. Superposition, at two different scales, of '-cuts of the directivity pattern of the layout in Fig. 5(a) along with the mask defined by the constraints in (31). Array elements: uniform circular feeds of radius .

2

Both FSLL and SLL on Earth are calculated with respect to the EOC.

= =1 =5

Fig. 5. Layouts achieved by considering elemental feeds of radius r , , and setting the parameters in (25) as follows: (A) (First Array) K  ,r r ,R r ; (B) (Second Array) K ,  ,r r ,R , and the remaining parameters fixed in such a way to cover at the best the most internal part of the available area. More details are collected under Table I.

2

= 4 = (p2+1) =1 = =0

= p2

characterized by 11 rings and 317 elements. The parameter introduced in Section IV has been set according to a uniform random distribution. The corresponding array factor along many azimuth cuts is shown in Fig. 6 along with the pattern of the continuous source in Fig. 4. Note that has been enforced in Fig. 6 according to the analysis carried out in Section III-A. As can be seen, the pattern is practically -independent up to (and thus up to angles well outside the coverage zone), thus confirming the analysis carried out in Appendix I. The first side-lobe ( 24.3 dB) is about 1.6 dB higher as compared to the pattern of the continuous reference source. Moreover, the separation between the first side-lobe

level and the EOC level, hereafter referred to as FSLL, is equal to 20.7 dB thus fully satisfying the specifications (note that the first side lobe is at ). On the other side, the far outside lobes are significantly increased with respect to the pattern of the continuous reference source. This depends on the weight appearing in (12), and renders the second constraint in (31) not satisfied for . When we pass to evaluate the directivity pattern, the element factor needs to be accounted for. To increase as much as possible the array directivity, uniformly illuminated feeds are appropriate; in particular, according to the minimum interelement distance previously set, we consider in the following uniformly , whose patilluminated circular apertures of radius tern expression is reported in [29]. The so achieved array directivity pattern is plotted in Fig. 7, and the main characteristics are collected under Table I (First Array). As expected, the element pattern lowers the level of the far outside lobes. In particular, the separation between the highest side-lobe level inside Earth and the EOC level (hereafter referred to as SLL on Earth), is equal to 20.5 dB, thus satisfying the second constraint in (31). The third constraint in (31) is satisfied as well. On the other side, a directivity of 43.3 dBi has been reached at the EOC, which is 0.5 dB less than that required by the first constraint in (31). Note that due to the azimuth symmetry of the pattern of the employed feeds, the directivity pattern in Fig. 7 retains the azimuth behavior of the array factor pattern of Fig. 6, that is, it is practically -independent up to angles well outside the coverage zone.

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Fig. 9. Number of array elements obtained by setting a-priori different core geometries. For the core geometry is that of Fig. 5(a). In all the adjacent and fully populated other cases the core geometry is given by concentric ring arrays including one feed in the center.

K

Fig. 8. Main directivity results relevant to the layouts in Fig. 5(a), but with different feeds locations along the rings. In the first trial the orientations of the rings are the same as in Fig. 5(a).

Investigation of the performances achievable with different ring orientations is now in order. The same layout of Fig. 5(a), but for different ring orientations has been considered: 19 trials have been carried out, and (with , see for each trial the parameters Section IV) have been set according to a uniform random distribution. The main directivity characteristics achieved for these 19 trials are collected in Fig. 8. It can be seen that the FSLL is the same in all the trials, which again confirms that different ring orientations do not affect significantly the pattern shape in the near bore-sight region. Differently, the SLL on Earth changes in the different trials (variations up to 1.5 dB can be observed in Fig. 8) because the array pattern in the far outside region (see (32), Appendix I). Notwithstanding, these depends on changes do not have a significant impact on the bore-sight (BS) and EOC directivities, which are constant for all the trials. Investigation of the performances achievable with different core geometries is now addressed. We have considered eight different core geometries represented by fully populated concentric ring arrays with one feed in the center and a number of adjacent rings ranging from 3 to 10. in (25) has been set ranging In other words, the parameter , , from 3 to 10; in all the cases it has been set , whereas the remaining parameters in (25) have been fixed in such a way to cover at best the most internal part of the available area. We explicitly note that with the considered continuous source and elemental feed, for some core geometries different from the eight considered above (for instance, by locating a priori only one feed in the center or one feed in the center and only one fully populated adjacent ring array) the elemental volume carried out via (27) turns out to be such that simultaneous enforcement of conditions (23) and (24) cannot be always achieved. Thus, such core geometries have not been considered. Nonetheless, such kind of problems can be easily circumvented and in (25) by retaining the parameters in such a and setting again the parameters way to increase the elemental volume in (27) [with consequent reduction of the parameter in (28)]. For the eight considered cases, we have achieved eight layouts characterized by different numbers of elements, plotted in Fig. 9 versus the parameter . For it is reported

=1

K

Fig. 10. Directivity results relevant to the layouts obtained by setting a-priori the core geometries considered in Fig. 9. The first and second constraints in (31) are highlighted in the two plots with a continuous line.

the number of elements of the layout of Fig. 5(a). As can be seen, the higher the number of fully populated ring arrays located a-priori in the central part of the array, the higher the total number of elements of the final synthesized array. In all the cases, the third constraint in (31) turns out to be largely satisfied. As for the other constraints in (31), directivity results achieved for the eight geometries listed above are collected in Fig. 10, along with the results relevant to the layout of Fig. 5(a). On the x axis it is reported the total number of array can elements: the corresponding values of the parameter be read in the plot in Fig. 9. The top plot in Fig. 10 shows that both the BS and EOC directivities are increasing functions of the number of elements: this depends on the different array aperture efficiencies achievable with different numbers of elements. It is interesting to note that the difference between the BS and EOC directivities of Fig. 10 remains practically unchanged in the different cases. Moreover, all the layouts with more than 380 elements allow satisfying the first constraint in (31) (represented in Fig. 10, top plot, with a continuous line). On the other side, the FSLL does not remain constant in the different cases and, more important, the achieved performances increase up to around 370 elements and then decrease. Indeed, the higher the number of fully populated rings placed a-priori in the array core, the more similar gets the sparse array with respect to an equi-amplitude fully populated periodic one, which, although ensuring better performances in terms of maximum directivity, presents a FSLL

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Fig. 11. Superposition, at two different scales, of '-cuts of the directivity pattern of the layout in Fig. 5(b) along with the mask defined by the constraints in (31). Array elements: uniform circular feeds of radius 2.

certainly not satisfying the second constraint in (31). Considerations similar to those carried out for the FSLL hold also for the SLL on Earth; however, a more oscillating behavior is observed in the plot of Fig. 10 due to the random ring orientations imposed in the different analyzed cases. According to all these considerations, it turns out that more sophisticated solutions are worth to be investigated to increase the aperture efficiency, and thus the array directivity, without impairing the FSLL. One way could consist in adopting some kind of size-tapering, that is, by locating larger feeds in the outer rings: this would allow also reduction of the whole number of required feeds [25]. Another way could consist in setting core geometries different from concentric ring arrays: this solution is easily compatible with the procedure of Section III-C and is particularly suitable for some kinds of feeds, as square ones, for which core geometries on triangular lattices allow increasing the aperture efficiency of the core area [25]. Analysis of both these solutions, preliminarily addressed in [25], is however beyond the scope of this work. In any case, according to the plots in Fig. 10, different layouts are found to satisfy the second specification in (31) (represented in Fig. 10 with a continuous line). From the set collected in Figs. 9 and 10 we pick up the array ), which alwith 382 elements (which corresponds to lows satisfying all the specifications in (31): the corresponding layout and directivity pattern are shown in Figs. 5(b) and 11, respectively, whereas the main results are collected under Table I (Second Array). Again, the pattern shape up to angles well outside the coverage zone is not affected significantly by the sampling procedure, which, in such a region, introduces only directivity reduction due to the increase of the far outside lobe level. This is better shown in Fig. 12, where the superposition of the directivity patterns of Figs. 4(b), 7, and 11 is plotted in the near bore-sight region. Comparison between the obtained results and those achievable with a fully populated, non isophoric ring array is now needed. Such an array consists of about 700 feeds: by solving the corresponding convex optimization problem, we get the results summarized in Table I (Fully populated Array). An improvement of only 0.3 dB in terms of separation is obtained with respect to the isophoric layout of Fig. 5(b), which uses about one half elements. Obviously, the lower aperture efficiency of

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Fig. 12. Superposition of the directivity patterns in Figs. 4, 7, and 11.

the isophoric layout with respect to the fully populated one implies a reduction of EOC directivity. However, the actual 1.7 dB decrease is less than that one could expect on the basis of the reduction of the overall radiating area, which is equal to about 2.6 dB. Thus, that the solution obtained with the proposed procedure is indeed very near to the optimal one. Few words are now devoted to the scanning performances of the achieved layouts. Indeed, in a multi-spot scenario, the requirements in (31), which refer to the central beam, need to be extended also to scanned beams. By considering the most severe scanning angle required by the 19 spots configuration in [22], that is, 1.12 , and by applying a proper phase control, we obtain with the two layouts of Fig. 5 practically the same maximum and EOC directivities achieved for the central beam: the occurred losses (of the order of 0.05 dB) are in any case equal to those one could expect by considering the pattern decay of the employed array element. With the two considered layouts, FSLL losses of 0.2 and 0.5 dB, respectively, take place, whereas losses of 1 and 0.3 dB occur for the SLL on Earth. Thus, the layout of Fig. 5(b) turns out to satisfy all the requirements in (31), even for the most severe scanned beam. We remark that the so achieved fast solution, suitable per se, could be adopted also as suitable starting point for subsequent local optimization procedures, because it allows avoiding trapping in unsatisfactory “local minima.” Moreover, it is worth underlining that the narrower the beam radiated by the elemental feed and/or the higher the scanning angle, the more critical the scanning losses get. Thus, for multispot scenarios, special care should be taken in choosing array element dimensions and pattern. In our allowed us to reach a case, uniform circular feeds of radius sound trade-off between maximum directivity, number of control points, scanning losses and mutual coupling effects. Smaller feeds are instead more appropriate for applications requiring scanning angles higher than those listed in [22]. Of course, in such cases, increase of the number of control points will occur; moreover, mutual coupling effects could get not negligible and thus worth to be addressed, reasonably during a subsequent local optimization step, since they would be responsible for losses in any case (of practical interest) not severer than those induced by the sampling procedure addressed by the proposed method. Some final few words on computational time are now needed. Computation of the layouts of Fig. 5 took at worst about one second by using a 3.4-Ghz Intel Pentium D CPU with 504 MB RAM. Accordingly, application of the procedure described in

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this section, which consists in the generation of a number (equal, at worst, to the number of rings of the corresponding fully populated ring array) of different layouts characterized by different core geometries and in the subsequent selection of the best one, can be carried out in a very fast way. Similarly, computation of the continuous source in Fig. 4 took a few seconds. Computational time of the proposed method is therefore negligible, even if the continuous reference source needs to be computed. VI. CONCLUSION A new deterministic approach for the synthesis of pencil beams isophoric concentric ring arrays has been presented. It exploits optimal continuous planar reference sources by fully extending to the 2D case the density taper approach proposed for linear sources by Doyle [5], [6]. Similarly to the 1D case, it consists of two steps. In the first step, the available circular aperture is subdivided in concentric rings and each ring into equal angular sectors subtending the same volume of the reference source. As second step, a rule for locating the array elements inside the sectors has been devised. Similarly to the 1D case, the proposed strategy allows minimizing the mean square difference of the patterns of the continuous reference source and the achieved concentric ring array. The proposed procedure accounts for the geometric properties of the employed array element and requires that the central geometry of the array is preliminarily set, without the need of other a-priori information. It provides in a few seconds layouts of hundreds of elements able to produce directivity patterns satisfying realistic satellite project requirements. Based on the concept of iso-volume 2D intervals rigorously derived in this work, more flexible solutions appear worth to be investigated, for instance by employing geometries different from concentric ring arrays, or by pursuing both density and element-size tapering. Such solutions are matter of current study [25] and future work. APPENDIX I The array factor of the

ring array can be written as [29]

(32) is the Bessel function where is defined in Section IV, [30]. As well known, Bessel functions of first kind of order of first kind go rapidly to zero when the argument is smaller than the order. Accordingly, even the first nonzero harmonic in (32) becomes negligible for (33) wherein is the feed diameter. Accordingly, all higher order field harmonics are negligible for values of well inside , which in the case of our interest, that is, large focusing antennas, is much larger than the beam-width, which is of the , being the radius of the whole antenna. Thus, order of in angular sector around the bore-sight direction where the field is significant, higher order field harmonics are negligible and the array factor in (32) gets practically -independent.

REFERENCES [1] H. Unz, “Linear arrays with arbitrarily distributed elements,” IRE Trans. Antennas Propag., vol. 8, pp. 222–223, Mar. 1960. [2] A. Ishimaru, “Theory of unequally-spaced arrays,” IRE Trans. Antennas Propag., pp. 691–702, Jun. 1962. [3] A. Ishimaru and Y.-S. Chen, “Thinning and broadbanding antenna arrays by unequal spacings,” IEEE Trans. Antennas Propag., vol. 13, no. 1, pp. 34–42, Jan. 1965. [4] R. E. Willey, “Space tapering of linear and planar arrays,” IRE Trans. Antennas Propag., vol. 10, pp. 369–377, Jul. 1962. [5] W. Doyle, “On Approximating Linear Array Factors,” RAND Corp. Mem RM-3530-PR, Feb. 1963. [6] M. I. Skolnik, “Nonuniform array,” in Antenna Theory, R. E. Collin and F. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 6, pt. I. [7] Y. T. Lo, “A study of space-tapered arrays,” IEEE Trans. Antennas Propag., vol. 14, no. 2, pp. 22–30, Jan. 1966. [8] B. P. Kumar and G. R. Branner, “Synthesis of unequally spaced arrays using Legendre series expansion,” in Proc. IEEE Antennas Propag., Soc. Int. Symp., 1997, pp. 2236–2239. [9] A. L. Maffett, “Array factors with nonuniform spacing parameter,” IRE Trans. Antennas Propag., vol. 10, no. 2, p. 131136, 1962. [10] A. L. Maffett and C. T. Tai, “Some properties of the gain of uniform and nonuniform arrays,” IRE Trans. Antennas Propag., vol. 18, no. 4, pp. 556–558, 1970. [11] T. M. Milligan, “Space-tapered circular (ring) array,” IEEE Antennas Propag. Mag., vol. 46, no. 3, pp. 70–73, Oct. 2004. [12] M. Vincente-Lozano, F. Ares-Pena, and E. Moreno, “Pencil_beam pattern synthesis with a uniformly excited multi-ring planar antenna,” IEEE Antennas Propag. Mag., vol. 42, no. 6, pp. 70–73, 2000. [13] G. Toso, M. C. Viganó, and P. Angeletti, “Null-Matching for the design of linear aperiodic arrays,” presented at the IEEE Antennas Propag. Soc. Int. Symp. (AP-S 2007), Honolulu, HI, Jun. 10–15, 2007. [14] G. Toso and P. Angeletti, “Method of designing and manufacturing an array antenna,” Eur. patent EP2090995, Feb. 2008. [15] P. Angeletti and G. Toso, “Aperiodic arrays for space applications: A combined amplitude/density synthesis approach,” presented at the 3rd Eur. Conf. Antennas Propag. (EUCAP 2009), Berlin, Germany, 2009. [16] O. M. Bucci, M. D’Urso, T. Isernia, P. Angeletti, and G. Toso, “Deterministic synthesis of uniform amplitude sparse arrays via new density taper techniques,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1949–1958, Jun. 2010. [17] G. Toso, C. Mangenot, and A. G. Roederer, “Sparse and thinned arrays for multiple beam satellite applications,” in Proc. 29th ESA Antenna Workshop on Multiple Beams and Reconfigurable Antennas, Apr. 2007, pp. 207–210. [18] Y. Cailloce, G. Caille, I. Albert, and J. M. Lopez, “A Ka-band direct radiating array providing multiple beams for a satellite multimedia mission,” in Proc. IEEE Int. Conf. Phased Array Syst. Technol., May 2006, pp. 403–406. [19] G. Caille, Y. Cailloce, C. Guiraud, D. Auroux, T. Touya, and M. Masmousdi, “Large multibeam array antennas with reduced number of active chains,” presented at the Proc. 2nd Eur. Conf. Antennas Propag. (EUCAP 2007), Edinburgh, U.K., Nov. 2007. [20] M. C. Viganó, G. Toso, G. Caille, C. Mangenot, and I. E. Lager, “Sunflower array antenna with adjustable density taper,” Int. J. Antennas Propag., to be published. [21] O. M. Bucci, T. Isernia, A. F. Morabito, S. Perna, and D. Pinchera, “A periodic arrays for space applications: An effective strategy for the overall design,” presented at the 3rd Eur. Conf. on Antennas Propag. (EUCAP 2009), Berlin, Germany, 2009. [22] ESA/ESTEC Tender AO/1-5598/08/NL/ST, Innovative Architectures for Reducing the Number of Controls of Multiple Beam Telecommunications Antenna. [23] ESA/ESTEC Tender AO/1-6338/09/NL/JD, Active Multibeam Sparse Array Demonstrator. [24] T. Isernia, M. D’Urso, and O. M. Bucci, “A Simple idea for an effective sub-arraying of large planar sources,” IEEE Antennas Propag. Lett., vol. 8, pp. 169–172, 2009. [25] O. M. Bucci, T. Isernia, A. F. Morabito, S. Perna, and D. Pinchera, “Density and element-size tapering for the design of arrays with a reduced number of control points and high efficiency,” presented at the 4th Eur. Conf. Antennas Propag. (EUCAP 2010), Barcelona, Spain, 2010. [26] A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization. New York: Wiley Interscience Ser. Discr. Math., 1983. [27] O. M. Bucci, T. Isernia, and A. F. Morabito, “Optimal synthesis of directivity constrained pencil beams by means of circularly symmetric aperture fields,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1386–1389, 2009.

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[28] T. T. Taylor, “Design of circular apertures for narrow beamwidth and low side lobes,” IRE Trans. Antennas Propag., vol. 8, pp. 17–22, 1960. [29] C. Balanis, Antenna Theory, Analysis and Design. New York: Harper and Row, 1982. [30] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1964. [31] J. J. Betancor, “A mixed Parseval’s equation and a generalized Hankel transformation of distribution,” Can. J. Math, vol. 41, no. 2, pp. 274–284, 1989.

Ovidio Mario Bucci (F’93) was born in Civitaquana, Italy, on November 18, 1943. He was an Assistant Professor with the Istituto Universitario Navale of Naples, Italy, during 1967–1975, then Full Professor of Electromagnetic Fields with the University of Naples. He was Director of the Department of Electronic Engineering during 1984–1986 and 1989–1990, Vice Rector of the University of Naples, during 1994–2000. He is the author or coauthor of more than 370 scientific papers, mainly published on international scientific journals or proceedings of international conferences. His scientific interests include scattering from loaded surfaces, reflector and array antennas, efficient representations of electromagnetic fields, near-field far-field measurement techniques, inverse problems and noninvasive diagnostics, biological applications of nanoparticles and electromagnetic fields. Dr. Bucci is a Member of the Academia Pontaniana. He was President of the National Research Group of Electromagnetism, of the MTT-AP Chapter of the

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Centre-South Italy Section of the IEEE, Director of the Interuniversity Research Centre on Microwaves and Antennas (CIRMA), and of the CNR Institute of Electromagnetic Environmental Sensing (IREA), since 2001. Among others, he was a recipient of the International Award GUIDO DORSO for Scientific Research, 1996, and of the Presidential Gold Medal for Science and Culture, 1998.

Stefano Perna (S’03–M’07) received the Laurea degree (summa cum laude) in telecommunication engineering and the Ph.D. degree in electronic and telecommunication engineering, both from the Università degli Studi di Napoli “Federico II,” Naples, Italy, in 2001 and 2006, respectively. He was with Wise S.p.A., Naples, from 2001 to 2002. In 2003, 2005, and 2006, he received grants from the Italian National Research Council (CNR) to be spent at the Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA), Naples, for research in the field of remote sensing. In 2003 and 2006, he visited Orbisat Remote Sensing, Brazil, for repeat pass interferometric processing of airborne synthetic aperture radar (SAR) data. Since 2006, he has been with the Dipartimento per le Tecnologie (DIT), Università degli Studi di Napoli “Parthenope,” where he is currently a Researcher in electromagnetics. He currently holds the position of Adjunct Researcher with IREA-CNR, Naples. His main research interests are in the field of microwave remote sensing and electromagnetics: airborne SAR data modeling and processing, airborne differential SAR interferometry, modelling of electromagnetic scattering from natural surfaces and, more recently, synthesis of antenna arrays.

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Sparse Antenna Array Optimization With the Cross-Entropy Method Pierre Minvielle, Emilia Tantar, Alexandru-Adrian Tantar, and Philippe Bérisset

Abstract—The interest in sparse antenna arrays is growing, mainly due to cost concerns, array size limitations, etc. Formally, it can be shown that their design can be expressed as a constrained multidimensional nonlinear optimization problem. Generally, through lack of convex property, such a multiextrema problem is very tricky to solve by usual deterministic optimization methods. In this article, a recent stochastic approach, called Cross-Entropy method, is applied to the continuous constrained design problem. The method is able to construct a random sequence of solutions which converges probabilistically to the optimal or the near-optimal solution. Roughly speaking, it performs adaptive changes to probability density functions according to the Kullback-Leibler cross-entropy. The approach efficiency is illustrated in the design of a sparse antenna array with various requirements. Index Terms—Antenna design, cross-entropy, phased array, stochastic optimization.

I. INTRODUCTION LECTRONICALLY controlled antenna radiation patterns dates back to the forties when it became possible to substitute mechanical steering of the antenna main beam for electronics. Afterwards, arrays of phased antennas have been developed in order to better the antenna capabilities: reduction of the mechanical movement, fast scan of the field of view, control, and possible reconfiguration of the radiation pattern (e.g., beamwidth, sidelobes), etc. Fig. 1 shows a basic phased array, used for radar cross-section (RCS) measurements. Yet, the realization of fully sampled arrays is awkward [1]. It is mainly due to conflicting dimensional requirements, to technical limitations and finally to the cost concern. For example, considering a high resolution purpose, a fully populated phased antenna array require at a time a large spatial dimension (to provide a narrow beamwidth), a low inter-element spacing close to half of the minimum wave length (to prevent the appearance of grating lobes due to spatial under-sampling) and miniaturized antennas (to avoid mutual coupling between adjoining antennas) [2]. These problems account for the growing interest in antenna arrays with fewer elements. From this viewpoint, different kinds of arrays have been proposed [3]–[5]. The two main ones are the thinned arrays [2], [5], [6] (regular arrays where

E

Manuscript received September 07, 2009; revised November 18, 2010; accepted January 06, 2011. Date of publication June 16, 2011; date of current version August 03, 2011. P. Minvielle and P. Bérisset are with CEA, DAM, CESTA, F-33114 Le Barp, France (e-mail: [email protected]; [email protected]). E. Tantar and A.-A. Tantar are with INRIA Bordeaux Sud-Ouest, 33405 Talence cedex, France (e-mail: [email protected]; alexandru-adrian. [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.2011.2158941

Fig. 1. A phased antenna array illuminating a metallic biconical target.

a certain number of elements have been withdrawn) and the sparse arrays, also called nonuniform arrays. For sparse antenna arrays, the number of antennas is reduced and the interelement spacing is increased. Besides, they can lead to improvements in weight, heat dissipation, power consumption, etc. A known pitfall of sparse arrays is high level sidelobes in the radiation pattern. Designing a nonuniformly spaced array consists in controlling the number of antennas, their positions and their associated weights in the aim to reduce the sidelobes and fulfill an expected radiation pattern requirement. Unfortunately, there are no closed form solutions [7]. It can be shown that the sparse antenna array design, according to some application requirements, corresponds to the resolution of a constrained multi-dimensional nonlinear optimization problem [1]. Usually, through lack of convex property, such a multiextremal problem is awkward to solve. Deterministic gradient-based methods, as steepest descent or quasi-Newton methods, do not cope well with many local optima. As they determine their search direction from locally evaluated features, they might get trapped in a locally convex zone, around a local extremum. In such a situation, global stochastic optimization approaches, are known to be more efficient [4]. Their search strategy is based not only on one point but on a large number of random points; it tends to provide a global point of view of the objective function. Well-known random search methods are simulated annealing, based on Markov chain Monte Carlo (MCMC), evolutionary algorithms (EAs), Taboo meta-heuristic search, etc. Some of them have theoretic foundations, associated to asymptotic convergence results; others are just efficient heuristics. In any case, the global random search must be guided in order to isolate progressively the global extremum. These search methods have been widely applied to electromagnetic

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MINVIELLE et al.: SPARSE ANTENNA ARRAY OPTIMIZATION WITH THE CROSS-ENTROPY METHOD

(EM) optimization problems. Thus, [4] and [8] give a large review of EAs applications, from the design of antennas, to microwave and electromagnetic systems. One can mention antenna arrays [4], [8], multilayer microwave absorbers [4], frequency selective surfaces [4], [8], radar target recognition [4], stub-loaded monopoles [9], etc. More specifically, the issues of thinned or sparse arrays have been addressed by simulated annealing [10]–[12], EAs [6], [9], [13] and MCMC importance sampling [7]. Other exploratory approaches are currently emerging, inspired by social biology: particle swarms [5], [14], [15], [33], [34], ant colonies [16], [17], etc. Besides, [13] and [18] emphasize that for a certain class of problems (e.g., fixed geometry arrays), the partial convexity can be exploited, leading to efficient hybrid (i.e., partly deterministic and local, partly stochastic and global) algorithms. Yet, in spite of all these achievements in the EM field, the stochastic methods have drawbacks. Firstly, they are known to be slow. These iterative methods proceed by trials, requiring a large number of explorations before leading to the optimum. Furthermore, as expressed in [17], “the major disadvantage is the many user-defined parameters that govern the algorithm’s search behavior.” A recent stochastic approach, called the cross-entropy (CE) method [19], is particularly appropriate to tackle these difficulties. This versatile approach can cope with both combinatorial and continuous optimization. It constructs a random sequence of solutions which converges probabilistically to the optimal or the near-optimal solution. Roughly speaking, it performs adaptive changes to probability density functions according to the Kullback-Leibler distance, aka cross-entropy. Quite straightforward to implement, the CE method can be controlled with only a few parameters. Concerning the convergence speed, it is rather quick, as shown in [20] for challenging continuous multiextremal optimization cases. In the article, the CE method is especially applied to the design of a planar array for RCS measurement. Considering an object exposed to a radar, the RCS is, in straightforward terms, a fictitious area that quantifies the intensity of the wave that is reflected back to the radar. For many reasons, such as the prediction difficulty, its experimental estimation is crucial. It is usually completed inside indoor test chambers. Compared to outdoor test ranges, these also called anechoic chambers, literally “no echo,” afford convenience, confidence and all-weather test capability. Nevertheless, the RCS evaluation requires getting over various drawbacks. Depending on the wavelength, the radar antenna generates an incident wave which is no longer plane at the surroundings of the object. This near-field effect implies measurement errors, since the RCS definition requires a plane wave. Besides, the reflections from the walls can be significant. Despite many efforts, from the use of absorbing materials to data processing techniques of background suppression, the residual echoes due to interference or electromagnetic coupling may still perturb the measurement, particularly for low RCS. Notice that there can be other error factors, such as random internal thermal noise of the instrumentation, spurious signals, etc. In addition to these efforts, a key point to improve the RCS evaluation and its associated accuracy is the choice of the radar antenna. Indeed, by shaping its radiated EM field, it can be possible to solve some of the aforementioned problems. For ex-

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Fig. 2. Sparse antenna array requirements.

ample, wall reflections can be of lesser importance if the antenna is directive and exclusively illuminates the target. Obviously, sparse antenna arrays have potential to provide such an appropriate illumination, with a reduced number of antennas. In the article, the CE method is applied to the design of a sparse antenna array, presented and detailed as a constrained optimization problem. The requirements are expressed as a multi-objective goal attainment function; several constraints are introduced. The organization of the paper is as follows. In Section II, the problematic of sparse antenna array design is developed, leading to a constrained optimization problem. Section III is dedicated to a brief introduction to the CE method. The next section consists of its application to basic problems and then to the design of a sparse antenna array. The method efficiency is illustrated from simulations and experiments; it is confirmed by an outline of comparisons with other algorithms. II. SPARSE ANTENNA ARRAY DESIGN Consider the sparse antenna array of Fig. 2 where the elementary antennas are identical. If the antenna elements operate independently, i.e. without significant mutual coupling, the total EM field radiated at point M can be seen as the superposition of the elementary fields: (1) is the EM field radiated where is the number of antennas, from antenna and is a complex weight or at point relative feed coefficients of the antenna . (1) is closely related to the pattern multiplication principle [2], [23]. It shows that relative displacements of the antenna elements introduce relative phase shifts in the radiation vectors which, in turn, can add constructively in some directions and destructively in others [23]. Moreover, it points out that the EM field depends on array parameters (array location, orientation, antenna number, location, and weight of each antenna, etc.) and the considered point , i.e., its position regarding the array. Assume that the array requirements (directivity, low sidelobes, multibeams, etc.) can be expressed by a radiation pattern about the field in a domain of the space. The pattern may correspond to required levels, maximum allowed variations, etc. The goal is to determine the array parameters such as the total field is consistent with the radiation pattern . In a way, it is an inverse problem since the array shape and the excitation law of the radiating elements have to be determined from

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the resulting radiation pattern. Furthermore, constraints must be respected, concerning the number of antennas, the array size, the space between decoupled antennas, the array location inside a closed room, etc. Consequently, the design problem is equivalent to the following constrained nonlinear optimization problem: with

(2)

where is the array (meaning the array hyperparameter in the space ), is the objective function taking into account the pattern in the domain and C is the part of the parameter space where the constraints are respected. There are widespread methods to fix antenna weights according to a radiating pattern, especially for a given array geometry [10], [23]. Thus, it can be shown that the design of equally spaced arrays is equivalent to the synthesis of finite impulse response (FIR) digital filters in digital signal processing. When the array geometry is not set, these methods can be incorporated in a global optimization approach. Reference [18] illustrates the inclusion of the common Schelkunoff’s polynomial method inside a genetic evolutionary algorithm. When constraints need to be taken into account, [10] shows how convex programming can be used to reduce the number of unknowns in the global optimization problem. Conversely, when the weights are fixed, deterministic procedures have been developed, from the “density taper” approach [2] to the more recent approach of [24]. Based on Fourier-Bessel series, this analytic approach quickly determines nonuniformly circular spaced arrays from the requirement of antenna radiation patterns. Notice that these arrays are formed of hundreds, or even thousands, of elements with fixed uniform weights. In satellite communication, other deterministic procedures, based on aperiodic tilings and convex optimization, have been developed for the design of directive large planar sparse arrays [3], [25]. Concerning thinned arrays, the ON/OFF state of each element have to be determined by binary or combinatorial optimization [5], e.g., dynamic programming [2], [12]. However, as mentioned above for RCS measurements, the array requirements are more complex than the usual far-field radiation pattern. Moreover, the number of antenna elements may be restricted, therefore involving unknown unequal feed coefficients. In such a context, the problem (2) is much more tricky to solve. Classic direct approaches, such as the “density taper” procedure that would require in any case a large number of elements, are unsuited. And yet, the dimension of the parameter space remains high, roughly a few tens or hundreds. The objective function is globally not convex. That is intuitively understandable, just taking into consideration the invariance by permutation of the antenna locations. Generally speaking, there are many local optima, whose number increases with the dimension of . Moreover, the constraints may make the task even tougher. Being faced with all these difficulties, global stochastic optimization methods are more appropriate than deterministic ones. Refer to [5], [17] for linear array synthesis, to [18] for sparse arrays, to [26] for conformal arrays, to [27] for sidelobe reduction, etc.

III. THE CE METHOD The CE method is an iterative stochastic algorithm method, initiated for simulation and estimation of rare events [19], [28]. Afterwards, the CE method was widened to optimization problems. Roughly speaking, it consists in considering the objective function optima as rare events. A growing number of applications have been carried out for various problems, from combinatorial optimization to machine learning, in various fields such as telecommunication, control and navigation, signal processing, scheduling, etc. [19]–[22], [28] Among them, one can mention planification in [29], a variety of search problems, i.e., optimization of the search efforts, in [30] and game programming by reinforcement learning in [31]. In the EM field, [32] deals with antenna selection in a multiple-input multiple-output (MIMO) system. Concerning continuous multiextremal optimization, [20], [21] demonstrates the ability to solve difficult problems, with or without constraints. Next we present briefly the CE approach principles and finally describe its extension to optimization. For further details, refer to [19] and [20]. A. Adaptive Importance Sampling for Rare Event Simulation Rare event simulation is an important issue for estimation in reliability, telecommunication systems, etc. In this section, we briefly present it before introducing the CE approach. Let be a random vector, taking its values in the space and its probability density function (pdf). We would like to evaluate the probability i.e.,

(3) (4)

where is a real-valued function and is the Heaviside step function. When is low enough (typically lower than ), the event “ ” is called a rare one. In such a case, the estimation of by direct Monte Carlo simulations , where are samples drawn according to , is inefficient. Indeed, a huge number of samples is required to obtain an accurate estimate (each “eventuality” sample has a probability re). To overcome the huge sample lated to the event loss, a natural approach is to resort to importance sampling. Importance sampling is a common technique in Monte Carlo simulation. It consists in drawing samples indirectly from the a so-called importance proposal distribution. Calling and the “importance weight” importance distribution on , the unbiased estimator is (under certain conditions) (5) Again, the convergence speed depends on the importance density. It can be shown that the optimal importance density (to estimate ) is proportional to . Unfortunately, it is practically of no interest. It depends on parameters that are unknown since the goal is to evaluate them. All the same, it

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indicates that sampling is efficient in the most significant parts is high enough. It is crucial where the value of to apply oneself to generate the samples in the significant parts. Sophisticated methods, generally called adaptive importance sampling, deal with this issue. One of them is the CE method. Notice that the “Simulated Annealing” method strives after a similar goal. The CE approach rests on the following principle: is chosen , with the parameter among a parametric pdf family (e.g., the mean and variance of a Gaussian law). The idea is then to determine the parameter so that the distance between the pdfs and are minimal. It is based on the KullbackLeibler (KL) “distance,” also called “cross-entropy,” which is widely found in information theory and processing. Basically, the parameter , linked to the optimal distribution, can not been determined directly; an adaptive and iterative approach must be implemented. For further details, see [19]. In the next section, we detail the extension of the CE method to optimization. B. From Estimation to Optimization Consider the following general optimization problem. We seek to maximize an objective function on the space . Let us note the point of where is maximal and its associated value (6) The extension lies in the expression of the optimization problem as an estimation problem. First of all, it requires to probabilize the space , by introducing a parametric probability density function (pdf) family over the space . Then, an associated stochastic problem (ASP) is defined (7) . The aim is to estimate with a random vector of pdf for a value close to . Of course, due to the very nature (the maximum), there are only few values of that exof ceed . Consequently, “ ” is a rare event. The estimation of is not a trivial problem. It is precisely the goal of the CE method. It solves the ASP stochastic problem by carrying out adaptive changes on pdfs, as shown in Fig. 3. Based on the Kullback-Leibler distance, it creates a sequence of pdfs , etc. that concentrate on the theo(degenerated density at the optimal retic optimal density point ). More precisely, the CE method generates a sequence of couples that converge towards a neighborhood . In other words, it progressively of the optimal couple samples realizations of the rare event that is linked with the optimum. Concerning continuous optimization, the adaptive step can be performed rather straightforwardly [20]. Afterwards, we describe the basic CE algorithm used for sparse array design in next Section IV-C. The parametric importance pdf family conparamsists of Gaussian distributions. At iteration k, the eter consists of the sufficient statistics, i.e. the successive means and the covariance matrices ( is de the dimension of the search space ). In the present case, the covariance matrices are chosen to be strictly diagonal,

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Fig. 3. The CE adaptive approach.

. Then, it can be i.e., shown that the algorithm amounts to the computation of basic estimators, which involves selections of “best samples” [20]. Its pseudocode is given here. • Initialization and (i.e., the initial importance distri— Choose bution, covering a large area of ) — (counter) • Repeat random samples of — Generation: draw (from ) Gaussian pdf — Evaluation: compute the associated values — Selection: order the samples depending on the values (in a decreasing order according to ) and select the best samples , where and is the ratio of retained samples (typically 5%) — Adaptive update: for each component , and — Counter increment: Until a break criterion (e.g. ) is fullfilled as the evaluated • Choose the current “best sample” global optimum Remark: For robustness concerns, smoothing mechanisms and . are introduced in the adaptive updates, for both Thus, the algorithm is less subject to unfortunate draws which could lead to suboptimal solutions. See [20] for further details. Considering constrained optimization, let assume that the constraints can be expressed in the form of equalities and/or inequalities . Former (6) becomes

(8) Generally, local deterministic search methods run up against difficulties with constraints. For example, the projected gradient method requires rather complicated projection on subspaces where the constraints are respected. For global stochastic approaches, it can be more direct. In the CE method scheme of Fig. 3, it is possible to incorporate an “accept-reject” strategy in the sampling step. Common in the Monte Carlo simulation field, it simply consists in drawing samples from the current importance distribution and accept them as soon as they respect

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the constraints. Thus, the constraints are checked for all the generated samples. Subsequently, in the application to sparse array design, this straightforward approach is applied. Remark: The difficulty grows when the importance distribution is too large compared to the area limited by the constraints. The task is then far too long and the acceptance ratio too low. The importance distribution needs to be reworked, from the constraints . On the other hand, it is also possible to use a penalty approach, as in classic optimization. IV. APPLICATIONS Fig. 4. The box illustration.

A. Optimization Benchmark Concerning continuous optimization, the CE method has been evaluated in several works (see [20]–[22]). It has been compared to other methods, such as genetic algorithms [22], on standard high dimensional, possibly constrained, optimization problems: the Rosenbrock function (aka banana function), the Rastrigin function, etc. Numerous test cases indicate that the CE method provides a fast and efficient procedure for solving a great variety of optimization problems, be it continuous multiextremal, multimodal, nonlinear, or constrained functions. To extend these evaluations, we compare to [8], dedicated to electromagnetic optimization and evolutionary programming (EP). [8] considers the minimization of the so-called Ackley function

(9) where is the search space dimension. This severely multiextremal Ackley function has a known global minimum, equal to 0, at the origin . The CE search algorithm is straightforwardly initialized with the following initial importance districomponents between and ones equal to bution: 10. is set to 500. Similarly to [8], we make comparisons for and 20. In both cases, the convergence is quite quick and leads to a solution very close to the global minimum, unlike genetic algorithms [8]. Moreover, compared to the EP algorithms, the required number of “fitness evaluations,” i.e., evaluations of , is smaller. For instance, for , the CE method requires around 60 iterations to converge to a solution whose (evaluated from 100 trials), mean fitness value is around requiring 30 000 evaluations, i.e. three less than EPs. With a limitation of only 15 000 evaluations, the mean CE method solution , compared to and more for EP algorithms. is around B. 1D Nonuniform Array Design Linear array synthesis is a typical EM problem [2]. The design consists in determining the location of the elements of a 1D array in order to cope with requirements linked with the EM radiation pattern. It can be expressed as an optimization problem [5], with constraints such as a minimum space between adjacent elements. Next, we have exactly considered the problem fully described in [5]: the goal is to minimize the lowest peak SLL (sidelobe level) of a symmetrical linear array, where four

Fig. 5. The hybrid array search.

elements have to been located. The CE method is now set with corresponding to equidistant elements and 1000 samples, equal to 1. With 50 independent trials, the average peak SLL goal is around 19.2 dB after ten iterations. The solution, very close to the one in [5], provides a similar radiation pattern and reminds an equal-didelobe design that would be achieved by a Dolph-Chebyshev method [2]. A variation of this problem is described in [17], where 5 elements have to be located. Compared to the ant colony approach of [17], the CE approach leads to similar results with far less computation time. C. Sparse Antenna Array Design In Section II, it has been shown that sparse antenna array design can be formally considered as a constrained optimization problem. In this part, we apply the CE method to the specific application introduced in Section I, i.e. the design of a sparse antenna array dedicated to RCS measurement inside an indoor test chamber. The experimental set-up is represented in Fig. 4: the array is located on the left wall while the target is centered at a reference point . In order to lower the measurement errors, the array design consists in reducing the potential coupling with the chamber while providing more or less a plane wave illumination around . Similarly to [5], the issue is multiobjective, more complex than the usual (far-field) radiation pattern objective [12], [15], [17], [24]. The application of the CE method is described in Fig. 5. It presents the architecture of the sparse antenna array search and its main components, from the computation part, i.e. the stochastic search algorithm, to various elements modeling the problem, i.e. the parametrization, the objective function and the constraints. The array design is based on this optimization process, which leads to the determination of the optimal design. All the architecture components are detailed in the next subsections.

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1) The Parametric Model: A parametric sparse antenna array model and the associated search space of the hyper-paramare to be determined. The search space is , eter where is the space dimension. According to the application, the parametric model can include array location parameters (position, orientation, etc.), array parameters (symmetry level, antenna location, weights , weight pairing, etc.), environment parameters (room size, specific object location, etc.) Based on the superposition model (1), it is able to approximate the EM field in the specification area . It is important to remark that it requires several assumptions, such as no mutual coupling, that are considered here as constraints in the search space . Besides, notice that it requires the knowledge of the elementary antenna radiation pattern. However, it is desirable to reduce the dimension of the param. Coming eter state, i.e., the number of degrees of freedom back to (1), it is important to notice that, for a given array geometry, the EM field depends linearly on the array weights. Consequently, the weights can be tuned with a specific efficient method, while the array geometry requires global optimization. Notice that it is close to the hybrid optimization approaches developed in a number of contributions, such as [10], that take advantage of the problem convexity. The basic idea is to explore the parameter space in a differand the parts of the hyper-paentiated way. Let us call rameter representing respectively the array geometry and the array weights (see Fig. 5). Assuming that there is no specific constraints on the array weighs, the global constrained minimum search of (2) is equivalent to (10) where is the optimum weight parameter for a given array geometry. It is evaluated by a “least mean square” (LMS) approach, based on QR decomposition. More details are given further. Henceforth, we consider the optimization (10), the hyper. parameter being reduced to the geometry component D. The Multiobjective Formulation and Approach The chosen formulation consists of a scalarization method, namely the goal-attainment approach, which is a direction based approach [35]. In this method, the goals to achieve (feasible or unfeasible solutions) are fixed, and they are associated with gains or penalties depending on whether there is success or failure. In order to obtain the complete formulation a direction is also required, along which the search is performed. In the following the direction will be denoted as the slope. In the application, the multiobjective function is related to the potential coupling with the surrounding chamber. Various have been positioned on the chamber boundaries points (walls, ceiling and floor) and on elements inside the chamber. The multiobjective function is computed as the sum of the contributions at the objective points (at discrete frequencies of the frequency band ) (11)

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Fig. 6. Example of elementary objective function.

Fig. 7. Space constraint for uncoupled antennas.

Furthermore, note

the relative level in dB (12)

where and are, respectively, the EM field levels at the and at point . reference point at point is: The elementary objective function if and otherwise, with and being, respectively, the gain and the penalty by (see Fig. 6). Thus, it is comparison to the objective level possible to assign differentiated objectives to various points of the space. Another objective is assigned. It consists in forming a “quiet (for the frequency band ): the zone” at the reference point variation of the EM field around must not exceed an objective level, otherwise a penalty is allocated. Again, the evaluation is performed by discretization at different points. As dealing with a scalarization approach, the multiobjective goal attainment function can include other objectives if needed, as low sidelobes, multiple beams, etc. 1) The Constraints: In the application, we consider various constraints. First of all, global constraints on the antenna array: its limited size and its inclusion inside the indoor chamber. Within the array, there is a constraint linked to the basic EM superposition model (1). It requires that the antennas are not significantly coupled. This implies a minimum space between neighboring antennas [2]. Fig. 7 illustrates the uncoupling constraint, representing an exclusion area for each antenna. Finally, specific constraints need to be added according to the selected parametric model. Some parameters may need to be limited to a certain interval. Thus, the angular parameters of next illustrative section need to be confined to the interval in order to avoid unnecessary multiple modes, which are harmful to the search efficiency.

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Notice that some constraints, such as antenna weight pairing, can be taken into account by an adapted modeling. It reduces the search space dimension and spares time-consuming “acceptation-rejection” sampling. 2) An Illustration: The application of the design method is illustrated in the following basic example. The indoor chamber is the small box represented in Fig. 4. The sparse array antenna . Each is composed of identical bipolar antennas antenna is made of two dipoles that are folded down. Their radiation patterns are close to the one of a dipole. The frequency band of interest B is narrow: [1900–2100 MHz]. The array is to be put on the left wall of the small indoor chamber. Furthermore, a small object stands on the floor. The array optimization lies in two goals. The first one is to produce a “quiet zone” where the EM field (amplitude) variation is 200 . The second is less than 0.2 dB. The penalty one is to limit the coupling with the box and the object. A few tens of points are positioned at the boundaries. The associated are: 40 dB on the walls, 30 dB on the objective levels floor and the ceiling, 35 dB at the edges and 25 dB on the and gain are respectively object surface. The penalty . fixed to 2 and 0.1 The parametric modeling consists in choosing multiple-crown arrays. It consists of circular arrays, akin to [24], but with nonuniform amplitudes or weights. Besides a circles. central antenna, antennas are equally spaced on Antennas from a same crown have the same weight. There are antennas on the crown of radius , the first one located at the angle . The total number of antennas is: . Here, the hyperparameter (associated to the search space ) is

Fig. 8. The small box optimized sparse planar antenna array.

(13) In addition to the box inclusion and the uncoupling constraint, the specific constraints are: Fig. 9. Relative field levels (2 GHz—HH polarization).

and

(14)

After a few variations on and , we finally chose a generic array of 29 antennas, divided in 5 crowns with successively 4, 4, 4, 8, and 8 antennas. The CE method, drawing 500 random samples at every step, converges after 38 iterations. The optimized antenna array is represented in Fig. 8. In addition to the antenna locations, the antenna weights , the amplitude , are figured. (dB) and the phase at the objective points are repThe relative field levels resented in Fig. 9. Most of the objective levels are respected. Naturally, the objectives on the object are the hardest to achieve. Concerning the field variation, Fig. 8 demonstrates that the array manages to generate the specified “quiet zone” around . 3) An Experimental Confirmation: An experimental confirmation has been performed. In order to make it simpler, notably reducing the number of elements, the previous goals are slightly slackened. Now, the optimization process leads to the antenna array of Fig. 11. In addition to the central antenna, 12 antennas are placed on 3 crowns of 4 antennas, with successive 45 shifts. The crown radiuses are, respectively, 8.5, 14.5, and 18.0 cm. Concerning the weights, the internal crown weights are tuned in the same way as the middle antenna, ensuring the “quiet zone”

Fig. 10. Relative field levels (2 GHz—HH polarization).

goal. On the other hand, the external crowns are involved in the directivity.

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Fig. 11. The produced optimized antenna array. Fig. 13. Comparison to prediction and measurement (2 GHz).

Fig. 12. The relative field levels (2 GHz—HH polarization) in different slice planes from the antenna array: 25 cm (down), 50 cm (center), and 65 cm (up).

The comparison to the prediction and measurement of the optimized array is presented in the directivity pattern of Fig. 13. Whatever the polarization, there is a good agreement for the main lobe, when the azimuth is less than 40 . The difference grows bigger for the sidelobes. It is well known that the approximate model (1), related to the pattern multiplication principle, leads to errors on the location and amplitude of remote sidelobes [2]. Notably, the effects of mutual coupling between antennas in close proximity are significant, compared to these low levels. All the same, the required array properties are checked. It is further enlightened in Fig. 12, representing the calculated array directivity in different slice planes. 4) Comparisons With Evolutionary Algorithms: A comparison to different evolutionary algorithms (EAs) [41], [42] was conducted as a complementary part of this study, the considered paradigms and drawn conclusions being outlined hereafter. Note that, as being beyond the scope of the article, no in-depth details are included hereafter. Please refer to the hereafter cited

articles for a thorough description of the algorithms. A detailed comparison is considered to be developed in future work. A standard EA [41], [42] was used as a comparison starting point. EAs adapt well to nonsmooth surfaces with multiple local minima also offering support for different hybrid or co-evolutionary designs. The algorithm evolves a population of initially random candidate solutions, at each generation (i.e., iteration of the algorithm), a set of operators (mutation and crossover) being used to sample new solutions. A selection strategy is used to screen the solutions that undergo mutation and crossover, at the opposite end a replacement strategy controlling how the obtained offsprings are inserted back into the population. Note that due to their parallel intrinsic nature, EAs can be easily adapted for large scale computing on clusters or grids. Nonetheless, a large number of parameters have to be generally tuned, e.g., operators and associated rates, selection and replacement strategies, etc. Besides the standard EA, well-known paradigms like the differential evolution (DE) [43] or the covariance matrix adaptation evolution strategy (CMA-ES) [36] were considered along with a hybrid co-evolutionary parallel design combining the three methods. DE algorithms, while following an approach similar to standard EAs, rely on multi-parent operators for combining the information of a given solution with the one of several, randomly chosen, different other solutions. The nature of the employed operations, defined over a vectorial space, induce a self-adaptive behavior for the algorithm. Finally, CMA-ES extends the EA design by including an ES-like adaptive technique. Along the evolution of the algorithm, a covariance matrix is used to capture the pairwise dependencies between variables. A second-order model of the fitness function is thus described while alleviating the drawbacks of inverse Hessian matrices used in Quasi-Newton methods. At each iteration, candidate solutions are sampled with respect to the modeled distribution, the covariance matrix being subsequently updated. All experimentations were performed on Grid’5000 (https://www.grid5000.fr) [40], a nationwide computational grid including, at the writing time of this article, more than 6000 cores. First, the algorithms were executed independently under

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problem. Such a problem can be addressed by global stochastic optimization approaches, widespread in the EM field. Among them, the emerging Cross-Entropy approach, an adaptive importance sampling approach based on the Kullback-Leibler distance, is especially direct and efficient. Its efficiency has been specially demonstrated for the multigoal design of a sparse antenna array dedicated to RCS measurement. With this approach, other objectives and constraints can be taken into account: compactness, multibeams, far-field low sidelobes, etc. What is more, other arrays can be investigated: arrays without symmetry, conformal arrays, etc. Besides, possible extension could be to introduce discrete optimization in order to optimize simultaneously the array type, the antenna number, the array geometry, the antenna weights, etc. More generally, the CE method can provide an useful optimization scheme to a large number of EM design applications.

Fig. 14. The relative field levels (2 GHz—HH polarization) for the hybrid algorithm in different slice planes from the antenna array.

different setups. Cauchy and Gaussian mutation operators were used for the EA algorithm with different crossovers: -blend, fuzzy recombination, etc. For a survey of the different operators employed in literature please refer to [37]–[39]. Also, different population sizes, operator rates and selection/replacement strategies were examined. In similar manner, the DE algorithm was deployed with different strategies: rand/1/bin, best/1/exp, etc. For all the considered algorithms candidate solutions are evaluated in parallel as to speedup the execution, with a maximum of 19 000 fitness function evaluations. In addition, a hybrid design was set where the three algorithms were launched in parallel as part of a co-evolutionary model (the algorithms exchange solutions in periodical asynchronous manner). While the co-evolutionary setup surpassed the individual algorithms, the obtained results, with respect to the imposed maximum number of evaluations, were largely inferior to the ones provided by the CE method. Compared to Fig. 12, the best obtained solution (see Fig. 14) by the hybrid algorithm is less directive and contains a large number of relative-field hot-spots. As determined by the obtained results, it can be concluded over the efficiency of the CE algorithm on this particular problem. Mind that the algorithms were compared in a context where a fixed number of fitness function evaluations was set and that none of tested EAs attained convergence in the specified interval. Consequently, the advantages of the CE method lie in its straightforward application, with only a few parameters, and in its relatively quick convergence. Moreover, as explained before, it rests on a strong framework, at the junction of optimization and applied probability. V. CONCLUSION Currently, there is a growing interest in sparse antenna arrays, notably for cost reduction. It has been shown that their design is a constrained multidimensional nonlinear optimization

ACKNOWLEDGMENT The authors are grateful to B. Bicrel and J. Lartigau for the construction, the tuning and the measurement of the experimental prototype. REFERENCES [1] C. I. Croman, I. E. Lager, and L. P. Ligthard, “Design considerations in sparse array antennas,” in Proc. 3rd Eur. Radar Conf., Manchester, U.K., 2006. [2] , R. E. Collin and F. J. Zucker, Eds., Antenna Theory—Part 1. NewYork: McGraw-Hill, 1969. [3] A. F. Morabito, T. Isernia, M. G. Labate, M. D’Urso, and O. M. Bucci, “Direct radiating arrays for satellite communications via aperiodic tilings,” Progress in Electromagn. Res. (PIER 93), pp. 107–124, 2009. [4] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag., vol. 39, no. 4, pp. 7–21, 1997. [5] 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, 2007. [6] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 993–996, 2003. [7] S. Kay and S. Saha, “Design of sparse linear arrays by Monte Carlo importance sampling,” IEEE J. Ocean. Eng., vol. 27, no. 4, 2002. [8] A. Hoorfar, “Evolutionary programming in electromagnetic optimization: A review,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 523–537, 2007. [9] P. L. Werner, Z. Bayraktar, B. Rybicki, D. H. Werner, and K. Schlager, “GA optimized stub-loaded monopoles with high gain performance,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 10–15, 2007. [10] M. D’Urso and T. Isernia, “Solving some array synthesis problems by means of an effective hybrid approach,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 750–759, 2007. [11] V. Murino, A. Trucco, and C. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Process., vol. 44, pp. 119–123, 1996. [12] J. Hooker and R. K. Arora, “Unequally spaced antenna arrays synthesized via dynamic programming,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 10–15, 2007. [13] F. J. Ares-Pena, A. Rodriguez-Gonzalez, E. Villuaneva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 506–510, 1999. [14] K. C. Lee and J. Y. Jhang, “Application of particle swarm algorithm to the optimization of unequally spaced arrays,” J. Electromagn. Wave Appl., vol. 20, no. 14, 2006. [15] M. Mussetta, P. Pirinoli, and R. E. Zich, “MetaPSO-based design of the frequency behavior of a beam scanning linear array,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 10–15, 2007. [16] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: Optimization by a colony of cooperating agents,” IEEE Trans. Syst., Man, Cybern., vol. 26, no. 1, pp. 29–41, 1996.

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[17] E. Rajo-Iglesias and O. Quevedo-Teruel, “Linear array synthesis using an ant-colony-optimization-based algorithm,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 70–79, 2007. [18] D. Marcano and F. Dur´n, “Synthesis of antenna arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 3, 2000. [19] P.-T. L. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, “a tutorial on the cross-entropy method,” Ann. Operat. Res., vol. 134, no. 1, pp. 19–67, 2005. [20] D. P. Kroese, S. Porotsky, and R. Y. Rubinstein, “The cross-entropy method for continuous multi-extremal optimization,” J. Methodol. Comput. Appl. Probab., vol. 8, pp. 383–407, 2006. [21] R. Rubinstein, “The cross-entropy method for combinatorial and continuous optimization,” Methodol. Comput. Appl. Probab., vol. 2, pp. 127–190, 1999. [22] J. Liu, “Global optimization techniques using cross-entropy and evolution algorithms,” coursework Masters Thesis, Univ. Queensland, , 2004. [23] S. J. Orfanidis, Electromagnetic Waves and Antennas [Online]. Available: http://www.ece.rutgers.edu/~orfanidi/ewa [24] T. A. Milligan, “Space-tapered circular (Ring) array,” IEEE Antennas Propag. Mag., vol. 46, no. 3, pp. 70–73, 2004. [25] T. Isernia, M. D’Urso, and O. M. Bucci, “A simple idea for an effective sub-arraying of large planar sources,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 169–172, 2009. [26] L. Vaskelainen, “Constrained least-square optimization in conformal array antenna synthesis,” IEEE Trans. Antennas Propag., vol. 55, no. 3, 2007. [27] S. H. Son and U. H. Park, “Sidelobe reduction of low-profile array antenna using a genetic algorithm,” ETRI J., vol. 29, no. 1, 2007. [28] R. Rubinstein and D. Kroese, The Cross-Entropy Method: An Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning. New York: Springer Verlag, 2004. [29] F. Celeste, F. Dambreville, and J. P. Le Cadre, “Optimal path planning using cross-entropy method,” in Proc. 2006 Int. Conf. Inf. Fusion, Piscataway, NJ, 2006. [30] C. Simonin, J. P. Le Cadre, and F. Dambreville, “The cross-entropy method for solving a variety of hierarchical search problems,” in Proc. ISIF Int. Conf. Inf. Fusion, Jul. 2007. [31] S. Istvan, “Learning Tetris using the noisy cross-entropy method,” J. Neural Computat., vol. 18, no. 12, pp. 2936–2941, 2006. [32] Y. Zhang, C. Ji, W. Q. Malik, D. C. O’Brien, and D. J. Edwards, “Cross-entropy optimisation of multiple-input multiple-output capacity by transmit antenna selection,” IET Microw. Antennas Propag., vol. 1, no. 6, pp. 1131–1136, 2007. [33] J. Robinson and Y. Rahamat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, 2004. [34] D. W. Boeringer, “Particle swarm optimization versus genetic algorithms for phased-array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, 2004. [35] F. W. Gembicki and Y. Y. Haimes, “Approach to performance and sensitivity multiobjective optimization: The goal attainment method,” IEEE Trans. Autom. Control, vol. 20, no. 6, pp. 769–771, 1975. [36] N. Hansen and S. Kern, “Evaluating the CMA evolution strategy on multimodal test functions,” in Proc. Parallel Problem Solving From Nature (PPSN VIII), 2004, pp. 282–291. [37] F. Herrera, M. Lozano, and A. M. Sanchez, “A taxonomy for the crossover operator for real-coded genetic algorithms: An experimental study,” Int. J. Intell. Syst., vol. 18, no. 3, pp. 309–338, 2003. [38] D. B. Fogel and A. Ghozeil, “A note on representations and variation operators,” IEEE Trans. Evolut. Computat., vol. 1, no. 2, pp. 159–161, Jul. 1997. [39] N. Saravanan and D. B. Fogel, “Multi-operator evolutionary programming: A preliminary study on function optimization,” in Proc. Int. Conf. Evolut. Program. VI (EP’97), 1997, pp. 215–222, Springer-Verlag. [40] F. Cappello, E. Caron, M. Dayde, F. Desprez, Y. Jegou, P. Primet, E. Jeannot, S. Lanteri, J. Leduc, N. Melab, G. Mornet, R. Namyst, B. Quetier, and O. Richard, “Grid’5000: A large scale and highly reconfigurable grid experimental testbed,” in Proc. IEEE/ACM Int. Workshop on Grid Comput. (GRID’05), Nov. 2005, pp. 99–106. [41] J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor: Univ. Michigan Press, 1975. [42] Z. Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics. New York: Springer-Verlag , 2000. [43] R. Storn and K. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimiz., vol. 11, no. 4, pp. 341–359, 1997.

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Pierre Minvielle was born in Madrid, Spain, in 1969. He graduated from the Ecole Nationale Supérieure en Informatique et Mathématiques Appliquées de Grenoble (ENSIMAG), France, in 1992. After a first working in simulation software development at SAS, Cadarache, France, he joined the French Atomic Energy Commission (CEA) in 1997. For approximately ten years, he had worked on operational analysis and data processing, i.e., target tracking, multitracking, and data fusion. During 2003, he was on secondment with QinetiQ Ltd., Malvern, U.K., working on particle filtering application to video noncooperative target recognition. He is currently interested in inverse problems, stochastic optimization and statistical signal processing. His application fields essentially concern electromagnetics, scattering, radar imaging, and antenna design.

Emilia Tantar received the Master’s degree in 2005 in the field of computational optimization from the Computer Science Faculty, “Al. I. Cuza University,” Lasi, Romania. She received the Ph.D. degree for landscape analysis in multiobjective optimization in 2009 from the University of Lille 1, France. In 2005, she joined the French National Institute for Research in Computer Science and Control (INRIA), Lille. Between 2007 and 2009, she held a lecturer position with University of Lille 1. During her Ph.D. studies, she was also awarded an INRIA Explorateurs scholarship to the CWI, Amsterdam, Netherlands. She developed a strong interest in new challenging aspects regarding landscape analysis in multiobjective, but also in the theoretical foundations of stochastic methods and their scaling to practical problems. Before joining the Computer Science and Communications (CSC) Research Unit, University of Luxembourg, in October 2010, she was an INRIA Postdoctoral Researcher in the Advanced Learning Evolutionary Algorithms (ALEA) team, at INRIA Bordeaux, dealing with performance guarantees factors for multiobjective particle methods, such as evolutionary algorithms and rare event simulation techniques.

Alexandru-Adrian Tantar received the Ph.D. degree in computer science in 2009 from the University of Lille, France. He was a member of French National Institute for Research in Computer Science and Control (INRIA), DOLPHIN Team, Lille, and of the Fundamental Computer Science Laboratory of Lille (LIFL). He was involved in the ANR Docking@GRID and the ANR CHOC projects. From September 2009 until March 2010, he was a Postdoctoral Researcher with the Advanced Learning Evolutionary Algorithms (ALEA) Team, INRIA Bordeaux—Sud-Ouest, France, working on parallel and distributed techniques for interacting Markov chains-based modeling and development. He addressed topics ranging from evolutionary computation and optimization, parallel and distributed algorithms to Monte Carlo-based algorithms with applications to general optimization problems, bioinformatics, and rare events simulation. He collaborated with CEA Life Sciences Division, CEA CESTA, the Biology Institute of Lille (IBL), and the Sea French Research Institute (IFREMER). Since April 2010, he has been a Postdoctoral Researcher with the Computer Science and Communications (CSC) Research Unit, University of Luxembourg (AFR Grant). He is currently involved in the GreenIT (FNR Core 2010–2012) project, which aims at providing a holistic autonomic energy-efficient solution to manage, provision, and administer the various resources of Cloud Computing/HPC centers.

Philippe Bérisset was born in Soyaux, France, in 1970. He received the Dipl. Ing. degree from the Ecole Nationale Supérieure de l’Aéronautique et de l’Espace, Toulouse, France, in 1992 and the Master of Science degree in aeronautics and astronautics from the Massachusetts Institute of Technology, Boston, in 1993. From 1993 to 2002, he was with Délégation Générale pour l’Armement, part of the French MOD, where he worked on the development of electromagnetic and infrared sensor test facilities. In 2002, he joined the CEA-CESTA, France. His research interests are in radar cross-section measurement techniques.

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Spherical-Wave-Based Shaped-Beam Field Synthesis for Planar Arrays Including the Mutual Coupling Effects Juan Córcoles, Member, IEEE, Jesús Rubio, and Miguel Á. González

Abstract—An analytical method to synthesize shaped-beam patterns with planar arrays, based on the handling of spherical waves, is proposed. Translational Addition Theorems will be used here for two different purposes: 1) relating the spherical modes produced by each element in the array to calculate the mutual coupling effects, and 2) expressing the field radiated by each element in terms of spherical modes corresponding to the whole array, to carry out a spherical-wave synthesis procedure based on the orthogonal properties of spherical modes. This field synthesis method is based on the fact that any antenna radiated field can be expressed as a discrete series of weighted spherical vector wave functions and it only requires the a priori knowledge of the Generalized Scattering Matrix of each array element considered as isolated from the rest of the array elements. Index Terms—Generalized scattering matrix, mutual coupling, pattern field synthesis, planar array, spherical wave expansion, translational addition theorems.

I. INTRODUCTION

T

HE problem of pattern synthesis has attracted a lot of attention from the antenna research community since the early days of array design. There are currently a great number of available methods that allow the computation of the feeding excitations necessary to synthesize a prescribed pattern. When referring to pattern synthesis, we must first distinguish power pattern synthesis from field pattern synthesis [1]. In the former, only the modulus of the desired field is specified, while in the latter both the modulus and the phase are given. Although formulating a pure power pattern synthesis problem would constitute the most rigorous approach in most applications, the phase of the field is usually specified for the sake of simplicity at the expense of reducing the number of degrees of freedom and thus the possibility of guaranteeing the best Manuscript received May 10, 2010; revised November 23, 2010; accepted January 06, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by CICYT, Spain, under projects TEC2010-20249-C02-01 and TEC2010-20249-C02-02. J. Córcoles is with the Departamento de Tecnología Electrónica y de las Comunicaciones, Escuela Politécnica Superior, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, C/ Franciso Tomás y Valiente 11, 28049 Madrid, Spain (e-mail: [email protected]). J. Rubio is with the Departamento de Tecnología de Computadores y Comunicaciones, Escuela Politécnica de Cáceres, Universidad de Extremadura, 10071 Cáceres, Spain (e-mail: [email protected]). M. Á. González is with the Departamento de Electromagnetismo y Teoría de Circuitos, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]. es). Digital Object Identifier 10.1109/TAP.2011.2158950

solution is achieved. This way, the problem becomes a field pattern synthesis problem, as is the case in this work. As for shaped-beam field pattern synthesis, two main approaches can be distinguished [1]. In the first one, a set of required specifications for the desired pattern is usually defined. The synthesized pattern must then remain within this set of specifications, usually given in terms of lower and upper bounds (secondary lobes, directive gain, roll-off, etc.); in this case, it is more proper to talk of a prescribed mask rather than a desired pattern. The computation of the pointwise error with respect to the mask is usually necessary in order to fulfill the specifications in all prescribed regions of the radiating space. Eventually, constraints on other performance indices of the array, typically concerning the excitations (such as the dynamic range ratio of the amplitudes, or the maximum phase difference) may also be imposed. As the reader may be aware, the strength this approach provides requires the development of intricate design methods. A flexible, general synthesis technique that effectively copes with this problem for a wide variety of array configurations is the intersection approach proposed by Bucci [1], [2]. Generally speaking, numerical iterative methods are especially suited in this case. Most of them rely on optimization schemes to carry out the design, with an increasing complexity as the different assumptions made on the configuration of the array (regular grid, presence of symmetries, known element patterns, absence of mutual coupling) are set aside [3]–[11]. The main drawback of this approach is in the increased computational time these numerical methods may need, depending on the size of the array. The second approach in shaped-beam field pattern synthesis consists of computing the excitations which provide the best fit to a desired (ideal, not feasible) radiated field. In terms of optimization, this results in minimizing the mean square error between the synthesized field pattern and the desired one [12]–[14]. Further restrictions to this unconstrained formulation can be added and, depending on their nature, they can make the field synthesis problem resemble one having the first approach. One clear advantage of this second approach in its unconstrained form is that it is more amenable to analytical or quasi-analytical methods, making the synthesis problem computationally more efficient. On the other hand, the simplicity of these analytical methods will also depend on the different assumptions made on the array configuration. An elegant and typical analytical technique for this approach is expressing the radiated field as a truncated expansion in orthonormal basis functions from which to obtain the excitations. In this sense, classic analytical methods have been used for

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CÓRCOLES et al.: SPHERICAL-WAVE-BASED SHAPED-BEAM FIELD SYNTHESIS FOR PLANAR ARRAYS

shaped-beam synthesis, such as the Fourier series method or the Woodward-Lawson frequency sampling method, based on the discrete Fourier transform [15]–[17], and its generalization for a wide class of apertures [18], by considering equispaced ideal isotropic uncoupled elements. The major drawback of most of these classic methods is that they are conceived under different assumptions, which can cause the solutions obtained using these procedures not to yield the desired response in real designs. Among these assumptions, one can highlight the use of isotropic sources or very well-known analytical element patterns, the total omission of mutual coupling effects and the constraints applicable to the array configuration, grid, or layout. Recently, we have proposed a Fourier series-based method that takes real antenna element patterns and mutual coupling effects into account [19]; however, this method is only valid for linear arrays with equispaced elements at a distance less than or equal to a half wavelength. A general way of taking all electromagnetic effects present in an arbitrary array into account may require the insertion of a full-wave analysis step in the synthesis procedure, possibly rendering it computationally prohibitive. Characteristic modes and similar approaches to derive a modal expansion of the radiated field or power have been used in array pattern synthesis [20]–[22], making it possible to take all these aforementioned effects into account. However, it should be borne in mind that the computation of these modal formalisms requires a numerical generation process for the whole array. Prolate spheroidal functions [23] and, more recently, spherical Bessel functions [24] have been used as the basis functions to expand the radiated field and then compute the excitation distribution of a line source. In this paper, we will exploit the fact that any radiated field (both from the array and from a single antenna) can be expressed as a discrete (truncated) series of weighted spherical vector wave functions. This issue was not only addressed in array pattern synthesis problems long ago [25]–[28], but also in antenna measurements [29]. Specifically, the idea of synthesizing a field pattern of an antenna array in terms of spherical modes was outlined in [26], where no interelement coupling effects were assumed. In [27], a spherical-mode-based method for pattern synthesis was proposed for planar arrays whose elements are placed on an arbitrary contour, although antennas were assumed to be uncoupled point dipoles. In our case, translational and rotational addition theorems for spherical waves [29]–[33] are used to relate the spherical modes describing the scattered field of each array element with its own coordinate system to the spherical modes of a reference coordinate system belonging to the overall antenna array. Therefore, as long as the conditions for using these theorems hold [29], no restrictions on the array configuration apply, making it possible to have an arbitrary layout. In the reference coordinate system, the application of orthogonal properties of spherical vector waves can be used to synthesize a desired field in terms of spherical modes. This idea has also been recently used for near field reconstruction [34] and for antenna modeling in terms of infinitesimal dipoles [35], [36]. However, in this work, the scattered field of each array element expressed in terms of spherical modes is also related to its feeding coefficient, making it possible to take mutual coupling effects and real element patterns into account.

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For this purpose, the methodology explained in [37] and [38] will be used. It also makes use of translational and rotational addition theorems for spherical waves, in this case to compute the mutual coupling between the array elements. Therefore, the two cases for the translation of spherical vector waves between spheres with a different centre reported in [29] are addressed here. In the second case, the centres of the spheres are situated outside the radius of the other spheres, while in the first case at least one of the centres lies inside the other sphere. This way, the combination of the two cases of the application of translational theorems for spherical modes is carried out to yield a shaped-beam synthesis method for arbitrary planar arrays that includes the mutual coupling effects and the real antenna element pattern. Therefore, the transmission and reflection features of every antenna in the array environment are rigorously incorporated into the synthesis process. The proposed procedure has a practical application in arrays made up of radiating elements whose radiated field can be individually expressed as a spherical wave expansion, such as apertures, arbitrarily shaped monopoles, cavity-backed microstrip patch antennas, horns or dielectric resonator antennas (DRA’s). In addition, it is applicable to the field synthesis of planar arrays with an arbitrary geometry, generating arbitrary shaped-beam patterns. II. THEORY A. Spherical Wave Expansion of the Shaped-Beam Field It is a well known result that the far field of any radiating structure can be broken down into a weighted sum of spherical far-field vector wave functions. In this case, the radiating structure corresponds to an antenna array, whose radiated field will be expressed as (1) is the th spherical far-field vector mode (see where is its asso[29] for the expressions used in this work) and ciated complex amplitude. Spherical far-field wave functions satisfy the orthogonality relationship

(2) Therefore, by multiplying (1) by every spherical far-field wave function, integrating in the whole space and using (2), a value of every expansion coefficient needed to achieve a desired field is obtained as

The integral in the denominator has a well-known analytical expression [29]. The integral in the numerator can be carried out

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where (6) and (7) being the identity matrix. According to Fig. 1, coefficients are related to coefficients , corresponding to the th element’s hemisphere, by means of a General Translation Ma, in the form trix (8)

Fig. 1. Scheme of the relationships between feeding and the different spherical modes used in the proposed spherical-wave-based pattern synthesis procedure.

with the aid of a numerical quadrature routine. In our case, the Lededev-Laikov scheme has been used [39]. It is important to note that, for the proposed synthesis techneed to be nique, coefficients computed only once for a given desired shaped-beam field, no matter which type of antenna elements the array is made up of, their position and even the number of elements. However, needed it should be born in mind that the number of modes must comply with the criterion given in [29] and [40], relating it to the radius of the minimum sphere circumscribing the whole array in this case. For practical purposes, coefficients are calculated only until the radiation pattern fits the specifications, and the rest of them are set to zero to satisfy the aforementioned criterion. These coefficients are the starting point to derive a series of relationships between different spherical wave expansions and antenna features that give rise to the proposed synthesis procedure. B. Computing the Excitations From the Relationship Between Spherical and Feeding Modes A scheme of the overall interactions established both by the antennas and the different spherical wave expansions in which the field is expressed in the proposed methodology is shown in Fig. 1 for two antennas and . In this figure, we assume that the array elements are located on an infinite ground plane so that the spherical mode expansions are related to hemispherical surfaces. There is an overall spherical wave expansion of the raon a sphere surrounding diated field defined by coefficients the array. Analogously, the coefficients corresponding to outgoing spherical waves on the sphere surrounding the th antenna , while the coefficients element of the array are given by corresponding to incoming spherical waves on this sphere are given by . coming We will start by assuming the contribution to antennas in the array is known exactly. from each of the as Therefore, we could express

where the rotational theorems and the theorems of translation for the case where the centre of each element’s hemisphere lies within the array’s hemisphere (second case reported in [29]) are used. On the other hand, the contribution to coefficients of incident spherical waves on the th element’s hemisphere of coefficients of scattered spherical waves coming from the th hemisphere can be established, as shown in [37], by means of another Genas eral Translation Matrix (9) is computed by using the transIn this case, matrix lational additional theorems of the first case in [29], when the centres of the hemispheres are situated outside the radius of the other hemispheres, as well as rotation theorems. Details on the calculation of General Translation Matrices can be found in [29] and [37]. The overall coefficients of incident spherical modes of antenna are given then by the contribution of the scattered field coming from the other radiating elements in the array and the direct field from outside of the array (10)

In (10), are the coefficients of the spherical mode expansion of the incident direct field from outside the array. By substituting (9) in (10) we get (11)

Now, from the GSM of each isolated (single) antenna , given by [29], [37] (12) we can get, by using (11), the following relationship:

(4) (13)

where corresponds to contributions from each array element to the spherical modes in the overall array sphere (see Fig. 1). Rewriting this equation in matrix form would yield (5)

where trix,

is the transmission matrix, is the scattering maare the feeding coefficients for each antenna , and

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is the identity matrix. If we repeat these equations for each one of the antennas in the array and we arrange them, the following matrix equation is found for the array:

from where it follows immediately that the required feeding coefficients to achieve the desired shaped-beam pattern are given by

(14)

(25)

where

are defined as .. .

.. .

.. .

.. .

.. .

.. . (15)

and

are diagonal block-matrices

(16) and

is defined as:

.. . .. .

..

.

..

.

..

.

..

.

..

.

.. . .. .

(17) Now, by assuming, as we are dealing with a radiating array, that there is no incident field coming from outside the array, i.e., , (14) is reduced to (18) from where it immediately follows: (19) By recalling (8) we can multiply both sides in (19) by a diagonal block-matrix given by (20)

denoting the pseudoinverse matrix of , defined with in (24). contains all the reIt is important to note that matrix lationships reported in Fig. 1, both those corresponding to the scattering and transmission features of each of the individual antennas in the array and those corresponding to the interactions of the different spherical wave expansions used in different spheres, which gives rise to take interelement mutual coupling effects inherently into account as well as making the proposed pattern synthesis procedure possible and efficient. Moreover, assuming that the transmission and scattering characteristics of each individual array element are a priori known, matrix is built analytically by making use of the translational and roare then tational addition theorems. Dimensions of matrix given by the number of modes used in the overall array hemisphere, by the number of feeding excitations in the array (usually, equal to the number of antennas, unless considering antennas with more than one feeding port). Therefore, up to several hundreds of antenna elements, the proposed procedure results in a fast synthesis technique (in the order of tens of seconds on a standard PC computer). Note that with this proposed method, the only requirement to incorporate a full-wave characterization of the whole array in the synthesis process is to have the GSM of one single element of the array considered as isolated. Furthermore, the synthesis method is independent of the numerical/analytical procedure used to compute this GSM. The proposed method in this work provides both strength and flexibility, as repeating the synthesis procedure after introducing modifications into the array is easy and computationally affordable. As an example, the positions of the array elements may be varied, the number of elements may be increased or decreased, or the antenna elements themselves can even be changed to check the performance of different antennas. Accounting for these modifications in a new synthesis would only imply the computational each time, in contrast burden of analytically building matrix to classic approaches which, in order to incorporate all electromagnetic effects in the synthesis, would require the previous full-wave analysis (i.e., based on the free-space Green’s functions) of the whole array altogether each time.

so that C. GSM-Analysis of the Array (21) Now, by simply multiplying both sides in (21) by to (5), we get

, according (22)

where coefficients are known, since they are given by the spherical wave expansion of the desired field for the shapedbeam pattern. Therefore, we can express (23) (24)

In this subsection, we briefly outline, for the sake of completeness, the basics (full details can be found in [37]) of the validated procedure used in this work to carry out the analysis of an array of real and coupled antennas, once the excitations have been obtained using the proposed synthesis method. This analysis is based on a rigorous GSM-characterization of the whole array, thus taking into account all interelment coupling effects. The accurate computation of the overall GSM of the whole array is carried out by means of a validated full-wave hybrid and modular procedure based on the GSM of each element of the array considered as isolated, as well as on rotation and translation theorems of spherical modes.

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Assuming no external incident field (the incident field on each antenna comes exclusively from the other antennas in the array), of the GSM of the whole the overall transmission matrix array relates complex coefficients of incident modes on the excitation ports of all the elements of the array (the computed ex, with scattered modes citations in the synthesis method), on the spherical ports of all the elements of the array, computed by imposing a value for the excitation modes as , where and are the same as defined in (15). This matrix rigorously provides the radiating characteristics for any synthesized excitations including the mutual coupling effects. antennas The radiated field of a planar array made up of is then obtained as a superposition of shifted spherical modes, weighted with their complex amplitudes, as

(26) are the coordinates of each antenna in the array, where considering a planar array of elements in the - plane; is the free-space wavenumber; is a vector containing the expressions of the spherical mode far field vectors for each antenna (see [29] for the expressions used in this work); and stands for the standard scalar product between vectors. of the GSM of Analogously, the overall reflection matrix the array can be obtained. As before, in the case where there is no incident field coming from outside the array, this matrix rigorously provides the matching characteristics for any selected excitations including the mutual coupling effects through , where is defined in the same way as (15) and contains the complex amplitudes of the reflected modes at the excitation ports. This information can be used, for example, to set the specifications of a feeding network which achieves computed by the the desired distribution of incident modes proposed synthesis method in this work by taking into account . the values of the reflected modes As this analysis method has been validated through comparison with measurements and previous results from other methods [37], [41], it will be used in this paper to check the capabilities of the proposed synthesis method. It should be noted that, although both (analysis and synthesis) methods are based on the same tools (GSM-analysis, rotation, and translation of spherical waves), they are actually different as the real radiated field of the array is computed from (26) as a sum of the contributions from every coupled element using the array theory, while the desired field to be synthesized is expressed in (1) by means of a weighted spherical wave expansion. III. NUMERICAL RESULTS In this section different array topologies with two different radiating elements are considered to generate several shaped beam patterns in order to illustrate the validity and capability of the proposed synthesis procedure. The first array element under consideration is a coaxial probe-fed circular microstrip antenna with a superstrate enclosed in a circular cross-section cavity recessed in an infinite

Fig. 2. Top and side view of a cavity-backed circular microstrip antenna enclosed in a circular metallic cavity. R = 3 cm, R = 2 cm, c = c = 0:6 cm, x = 0:62 cm, " = 2:5; " = 20. Coaxial feed: 50- SMA connector.

metallic plane. Fig. 2 shows the geometry of the antenna with a single coaxial feed used to generate a linear polarized pattern. The coaxial probe feeding corresponds to a 50- SMA connector with inner and outer radii of 0.65 and 2.05 mm, respectively. The process to obtain the GSM of the single isolated antenna was previously carried out in [42] and the computed resonant frequency, used throughout the different examples, is 1.215 GHz. The second array element used in the examples is the hemispherical DRA with a 50- coaxial probe feeding depicted in Fig. 3 and previously characterized in [37]. The computed resonant frequency is 3.64 GHz. Measurement results for a two-element DRA array are used in this reference to validate the analysis procedure from which the proposed synthesis method is verified. A. Circular-Shaped Footprint, Circular Planar Array. In the first example a circular flat-top pattern covering the has been synthesized using a 325-elesolid sector spacing between element square-meshed array with a ments and a circular contour with a radius of , as shown in Fig. 4. The array element used is the cavity-backed patch of Fig. 2. As described in Section II-A, the synthesis procedure starts with the computation of the expansion in spherical coeffrom (3), of the ideal shaped-beam field. Fig. 5(a) ficients, shows, together with the mask of the footprint, the copolar field patterns, according to Ludwig’s third definition, for -cuts in steps of 45 , which have been obtained by computing the first 575 modes in the expansion. After the application of the synthesis technique the resulting excitations computed from (25) generate the field pattern depicted in Fig. 5(b). As observed, it is practically equal to the objective pattern in the coverage zone.

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Fig. 3. Geometry of the hemispherical dielectric resonator antenna with a coaxial probe feeding. R = 12:7 mm, " = 9:5; s = 6:4 mm, h = 6:5 mm; coaxial feed: r = 0:5 mm, r = 1:505 mm.

Fig. 5. (a) Copolar field patterns for '-cuts in steps of 45 , obtained from the =4, dashed expansion in spherical modes of a circular flat-top pattern ( line). (b) Synthesized circular flat-top pattern with the proposed method for the array in Fig. 4 with radiating elements of Fig. 2. (c) Synthesized flat-top pattern without taking into account the mutual coupling between array elements.



Fig. 4. Configuration of a 325-element array with a circular contour and square meshes of 0:5 -equispaced elements.

The pattern radiated by the same array with the excitations computed without taking mutual coupling between array elements instead of exinto account, obtained by using pression (25), is shown in Fig. 5(c). It is evident that the noninclusion of the coupling into the synthesis procedure results in a different real pattern, degraded with respect to the one achieved by taking all interelement coupling effects into account. The same circular footprint pattern has also been synthesized with the same array topology but a different array element. In this case, the hemispherical DRA shown in Fig. 3 is used. The target footprint, from which the synthesized pattern is obtained, is the same one as the previous example, shown in Fig. 5(a), since, as pointed out in Section II-A, it is independent of the array topology and the type of array elements. The synthesized radiation pattern, shown in Fig. 6, is practically coincident with the one obtained with the array of patches. However, in this case a major influence of the noninclusion of the mutual coupling in the synthesis process, that gives rise to a total degradation of the synthesized pattern, was found.

Fig. 6. Three-dimensional (3D) plot of the result of the synthesis of a circular =4) for the array topology in Fig. 4 with DRA radiating elefootprint ( ments of Fig. 3.



B. Circular-Shaped Footprint, Square Planar Array The circular footprint pattern of the previous example is now synthesized with an 18 18-element square array with a spacing made up of the same cavity-backed microstrip antennas. The target footprint will be again that of Fig. 5(a). It can be

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Fig. 7. Copolar field patterns for '-cuts in steps of 45 of the synthesized =4) for an 18 18-element square array of circular flat-top pattern ( cavity-backed microstrip antennas in Fig. 2.



2

Fig. 8. 3D plot of the result of the synthesis of a rectangular footprint for the array in Fig. 4 with cavity-backed patch antennas.

seen how in this case the synthesized pattern, shown in Fig. 7, has a higher ripple and average slope on the transition region compared with the one obtained with the circular planar array made up of practically the same number of elements. C. Rectangular-Shaped Footprint, Circular Planar Array Now, a shaped-beam pattern with sharper corners is considered. It is a rectangular footprint covering the region synthesized using the circular planar array of Fig. 4 made up of the microstrip antennas of Fig. 2 as array elements. The resulting excitations generate a pattern whose 3D plot is shown in Fig. 8. D. Sectorized Planar Array

-Shaped Pattern, Nonuniform Circular

Next, a more complex example is considered. It is a sectorized cosecant-square shaped-beam pattern generated by a nonuniform random planar array with a circular contour made up of the microstrip antennas of Fig. 2 as array elements. The ideal

Fig. 9. Configuration of a nonuniform random 325-element array used or the synthesis of the sectorized csc -shaped pattern.

Fig. 10. Copolar field patterns for '-cuts in steps of 45 of the synthesized sectorized csc -shaped pattern for the nonuniform array in Fig. 9 made up of cavity-backed microstrip antennas of Fig. 2.

pattern covers the sector defined by , and the layout of the array, shown in Fig. 9, is obtained from the uniform circular array of Fig. 4 varying the position of cm in both the and directions, where each element is a uniformly distributed random variable ( to ). Fig. 10 pattern, the shows, together with the mask of the ideal copolar field patterns obtained with the proposed synthesis technique, for -cuts in steps of 45 . The three-dimensional synthesized radiation pattern is shown in Fig. 11. E. Circular Footprint Pattern With Circular Polarization, Circular Planar Array In this last example, a circular polarized radiation pattern covis synthesized using the ering the circular footprint circular-shaped array of Fig. 4. The array element, depicted in Fig. 12, is now a cavity-backed microstrip antenna but with a dual coaxial feed, used to generate a circularly polarized field.

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Fig. 11. 3D plot of the result of the synthesis of a csc shaped pattern for the nonuniform array in Fig. 9 with cavity-backed patch antennas.

Fig. 13. (a) Co- and cross-polar field patterns (black and grey lines respectively) for '-cuts in steps of 45 , obtained from the expansion in spherical modes of a circular polarized flat-top pattern with a circular contour ( =4). (b) Synthesized circular polarized footprint pattern with the proposed method for the array in Fig. 4 with radiating elements of Fig. 12. (c) Synthesized footprint pattern without taking into account the mutual coupling between array elements.



Fig. 12. Top and side view of the dual-coax feed cavity-backed circular microstrip antenna enclosed in a circular cavity. R = 3 cm, R = 2:475 cm, x = 0:62 cm, y = 0:62 cm, c = 0:1524 cm, c + c = 0:55 cm, " = 3:2 and " = 1:0.

This antenna was proposed as a benchmarking structure in [43]. Both coaxial feeds are 50- SMA connectors as in the previous linearly polarized microstrip antenna. The computed resonant frequency, using the methodology in [42] to calculate its GSM, is 1.958 GHz. Fig. 13(a) shows the co- and cross-polar components of the circularly polarized field patterns, for -cuts in steps of 45 , obtained from the expansion in spherical coefficients of the ideal circular-shaped footprint pattern using expression (3). Fig. 13(b) and (c) compares the field patterns generated by the synthesized excitations computed respectively by taking and without taking mutual coupling between array elements into account. As in Section III-A for the same synthesis pattern problem with linear polarization, an agreement is observed between the target, Fig. 13(a), and synthesized, Fig. 13(b), patterns. Again, a degradation in the achieved pattern, Fig. 13(c), both in ripple and cross-polar component levels, is observed when the coupling is not included in the formulation. For the sake of completeness, some details on the computational times are now given. In all examples, the computation of

matrix and its pseudoinverse, (24)–(25), takes 65 s using a Matlab routine. The computation of the elements of matrices [ ] and [ ], through rotation and translation theorems of spherical modes (implemented in Fortran code), takes 119 s in all examples except for the array of example B, whose computational time is 194 s. Note that once these steps are carried out for a specified array, the result can be stored to use this array to synthesize any desired field pattern. The computation of coef(implemented in Matlab code) takes about 45 s in ficients all presented examples. Please note again than once this step is completed, the result can be stored to use the coefficients corresponding to a desired pattern to synthesize it with any antenna array. All calculations are carried out on a laptop computer with Intel Core Duo T7500 at 2.20 GHz and 3 GB RAM. IV. CONCLUSION A procedure based on the translational and rotational addition theorems of spherical wave functions for the synthesis of arbitrary shaped beam patterns using planar arrays with an arbitrary boundary, geometry or lattice has been developed. The technique provides a closed-form analytical solution from the GSM of the array elements considered as isolated and takes the

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array element patterns and the mutual coupling between them into account. The method is applicable to arrays made up of complex array elements that may be described in terms of spherical waves and is independent of the technique used to obtain the GSM of the array elements. The proposed technique has turned out to be a powerful and flexible shaped-beam field pattern synthesis procedure, which makes it ideal to be incorporated into large scale design that may require repetitive computations of the feeding excitations in an iterative process to check different results. REFERENCES [1] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection approach to array pattern synthesis,” in Proc. Inst. Elect. Eng. Microw. Antennas Propag., Dec. 1990, vol. 137, no. 6, pp. 349–357, Pt. H. [2] O. M. Bucci, G. D’Elia, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, pp. 358–371, Mar. 1994. [3] O. Einarsson, “Optimization of planar arrays,” IEEE Trans. Antennas Propag., vol. 27, no. 1, pp. 86–92, Jan. 1979. [4] D. J. Shpak, “A method for the optimal pattern synthesis of linear arrays with prescribed nulls,” IEEE Trans. Antennas Propag., vol. 44, no. 3, pp. 286–294, Mar. 1996. [5] H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Trans. Signal Process., vol. 45, no. 3, pp. 526–532, Mar. 1997. [6] J. A. Rodríguez, R. Muñoz, H. Estévez, F. Ares, and E. Moreno, “Synthesis of planar arrays with arbitrary geometry generating arbitrary footprint patterns,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2484–2488, Sep. 2004. [7] F. Ares, R. S. Elliott, and E. Moreno, “Design of planar arrays to obtain efficient footprint patterns with an arbitrary footprint boundary,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1509–1514, Nov. 1994. [8] H. J. Orchard, R. S. Elliot, and G. J. Stern, “Optimising the synthesis of shaped beam antenna patterns,” in Proc. Inst. Elect. Eng. Pt. H, Feb. 1985, vol. 132, no. 1, pp. 63–68. [9] W. L. Stutzman, “Synthesis of shaped-beam radiation patterns using the iterative sampling method,” IEEE Trans. Antennas Propag., vol. 19, no. 1, pp. 36–41, Jan. 1971. [10] J. R. Mautz and R. F. Harrington, “Computational methods for antenna pattern synthesis,” IEEE Trans. Antennas Propag., vol. 23, no. 4, pp. 507–512, Jul. 1975. [11] J. Córcoles, M. A. González, and J. Rubio, “Multiobjective optimization of real and coupled antenna array excitations via primal-dual, interior point filter method from spherical mode expansions,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 110–121, Jan. 2009. [12] G. Franceschetti, G. Mazzarella, and G. Panariello, “Array synthesis with excitation constraints,” in Proc. Inst. Elect. Eng. Microw. Antennas Propag., Dec. 1988, vol. 135, no. 6, pp. 400–407. [13] A. K. Bhattacharyya, “Projection matrix method for shaped beam synthesis in phased arrays and reflectors,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 675–683, Mar. 2007. [14] B. P. Ng, M. H. Er, and C. Kot, “A flexible array synthesis method using quadratic programming,” IEEE Trans. Antennas Propag., vol. 41, no. 11, pp. 1541–1550, Nov. 1993. [15] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley , 1981. [16] P. M. Woodward, “A method for calculating the field over a plane aperture required to produce a given polar diagram,” Inst. Elect. Eng. J., vol. 93, no. 10, pt. IIIA: Radiolocation, pp. 1554–1558, 1946. [17] P. M. Woodward and J. D. Lawson, “The theoretical precision with which an arbitrary radiation-pattern may be obtained from a source of a finite size,” Inst. Elect. Eng. J., vol. 95, pt. III, pp. 363–370, Sep. 1948. [18] G. Borgiotti, “A synthesis method for bi-dimensional apertures,” IEEE Trans. Antennas Propag., vol. 6, no. 11, pp. 188–193, Mar. 1968. [19] J. Córcoles, M. A. González, and J. Rubio, “Fourier synthesis of linear arrays based on the generalized scattering matrix and spherical modes,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1944–1951, Jul. 2009.

[20] R. F. Harrington and J. R. Mautz, “Control of radar scattering by reactive loading,” IEEE Trans. Antennas Propag., vol. 20, no. 4, pp. 446–454, Jul. 1972. [21] R. F. Harrington and J. R. Mautz, “Pattern synthesis for loaded N-port scatterers,” IEEE Trans. Antennas Propag., vol. 22, no. 2, pp. 184–190, Mar. 1974. [22] D. M. Pozar, “Antenna synthesis and optimization using weighted Inagaki modes,” IEEE Trans. Antennas Propag., vol. 32, no. 2, pp. 159–165, Feb. 1984. [23] D. R. Rhodes, “The optimum line source for the best mean square approximation to a given radiation pattern,” IEEE Trans. Antennas Propag., vol. 11, no. 4, pp. 440–446, Jul. 1963. [24] H.-P. Chang, T. K. Sarkar, and O. M. C. Pereira-Filho, “Antenna pattern synthesis utilizing spherical Bessel functions,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 853–859, Jun. 2000. [25] P. J. Wood, “Spherical waves in antenna problems,” Marconi Rev., no. 34, pp. 149–172, 1971. [26] J. R. James, “Conformal antenna synthesis using spherical harmonic wavefunctions,” Proc. IEE, vol. 122, no. 5, pp. 479–486, May 1975. [27] M. S. Narasimhan, K. Varadarangan, and S. Christopher, “A new technique of synthesis of the near- or far-field patterns of arrays,” IEEE Trans. Antennas Propag., vol. 34, no. 6, pp. 773–778, Jun. 1986. [28] M. J. Mehler, “Spherical wave expansions for shaped beam synthesis,” in Proc. Inst. Elect. Eng. Microw. Antennas Propag. Pt. H, Oct. 1988, vol. 135, no. 5, pp. 327–332. [29] , J. E. Hansen, Ed., Spherical Near-Field Antenna Measurements. London, U.K.: Peter Peregrinus , 1988. [30] S. Stein, “Addition theorem for spherical wave functions,” Quarter. Appl. Math, vol. 19, no. 1, pp. 15–24, 1961. [31] O. R. Cruzan, “Translational addition theorem for spherical vector wave functions,” Quarterly Appl. Math, vol. 20, no. 1, pp. 33–40, 1962. [32] J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I—Multipole expansion and Ray-optical solutions, and part II—Numerical and experimental results,” IEEE Trans. Antennas Propag., vol. AP-19, no. 3, pp. 378–400, May 1971. [33] A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed. Princeton, NJ: Princeton Univ. Press, 1974. [34] A. Gati, Y. Adane, M. F. Wong, J. Wiart, and V. F. Hanna, “Inverse source characterization for electromagnetic near field reconstruction and interaction with the environment,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 9–14, 2006, pp. 1065–1068. [35] M. Serhir, P. Besnier, and M. Drissi, “An accurate equivalent behavioral model of antenna radiation using a mode-matching technique based on spherical near field measurements,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 48–57, Jan. 2008. [36] M. Serhir, P. Besnier, and M. Drissi, “Antenna modeling based on a multiple spherical wave expansion method: application to an antenna array,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 51–58, Jan. 2010. [37] J. Rubio, M. A. González, and J. Zapata, “Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3-D FEM and spherical mode expansion,” IEEE Trans. Antennas Propag., vol. 53, pp. 1133–1144, Mar. 2005. [38] J. Rubio, J. Córcoles, and M. A. González, “Inclusion of the feeding network effects in the generalized-scattering-matrix formulation of a finite array,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 819–822, 2009. [39] V. Lebedev and D. Laikov, “A quadrature formula for the sphere of the 131st algebraic order of accuracy,” Doklady Math., vol. 59, no. 3, pp. 477–481, 1999. [40] F. Jensen and A. Frandsen, “On the number of modes in spherical wave expansions,” [Online]. Available: http://www.ticra.com [41] J. Rubio, M. A. González, J. Zapata, A. Montesano, F. Monjas, and L. E. Cuesta, “Full-wave analysis of the Galileo system navigation antenna by means of the generalized scattering matrix of a finite array,” in Proc. 1st Eur. Conf. Antennas Propag. (EuCAP 2006), Nice, France, Nov. 2006, pp. 6–10. [42] J. Rubio, M. A. González, and J. Zapata, “Analysis of cavity-backed microstrip antennas by a 3-D finite element/segmentation method and a matrix Lanczos-Padé algorithm (SFELP),” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 193–195, 2002. [43] Antenna Centre of Excellence, ‘Antenna Software Tools’ Activity. EU 6th Framework Research Program [Online]. Available: www.antennasvce.org

CÓRCOLES et al.: SPHERICAL-WAVE-BASED SHAPED-BEAM FIELD SYNTHESIS FOR PLANAR ARRAYS

Juan Córcoles (M’11) was born in Albacete, Spain, in 1981. He received the Ingeniero de Telecomunicación degree in 2004 and the Doctor Ingeniero de Telecomunicación (Ph.D.) degree in 2009, both from the Universidad Politécnica de Madrid, Spain, the Diplomado en Ciencias Empresariales in 2008 from the Universidad Complutense de Madrid, and the Licenciado en Economía degree in 2010 from the Universidad Nacional de Educación a Distancia, Madrid. Since September 2005, he has been collaborating with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. From November 2008 to March 2009 he was a Visiting Researcher with the Institut für Hochfrequenztechnik und Elektronik (IHE), Universität Karlsruhe (TH). At present, he is with the Departamento de Tecnología Electrónica y de las Comunicaciones, Universidad Autónoma de Madrid, as a part-time Assistant Professor. His current research interests include the application of numerical and analytical methods, as well as optimization techniques, to the analysis and design of antennas, especially antenna arrays.

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Jesús Rubio was born in Talavera de la Reina, Toledo, Spain, in 1971. He received the Ingeniero de Telecomunicación degree in 1995 and the Ph.D. degree in 1998, both from the Universidad Politécnica de Madrid, Spain. Since 1994, he has collaborated with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. At present, he is with the Departamento de Tecnología de Computadores y Comunicaciones, Universidad de Extremadura as a Professor. His current research interests are in the application of the finite element method and modal analysis to antennas and passive microwave circuits problems.

Miguel Á. González was born in Madrid, Spain. He received the Ingeniero de Telecomunicación and Ph.D. degrees, both from the Universidad Politécnica de Madrid, in 1989 and 1997, respectively. Since 1990, he has been with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid, first as a Research Assistant, as an Assistant Professor from 1992 to 1997, and Associate Professor from 1997. His main research interests include analytical and numerical techniques for the analysis and design of antennas and microwave passive circuits.

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Synthesis of Conformal Phased Arrays With Embedded Element Pattern Decomposition Kai Yang, Zhiqin Zhao, Senior Member, IEEE, Zaiping Nie, Senior Member, IEEE, Jun Ouyang, and Qing Huo Liu, Fellow, IEEE

Abstract—A novel embedded element pattern decomposition method is proposed to synthesize conformal phased antenna arrays. This method decomposes the embedded element patterns as a product of a characteristic matrix and a Vandermonde structured matrix. This Vandermonde matrix is composed of modes in the modal space at the sampling angles. Because the fast Fourier transform (FFT) algorithm can be used in the Vandermonde structure, the computational cost can be reduced tremendously. In addition to its computational efficiency, the proposed method is also applicable to the synthesis of a pattern with any mainlobe direction and optimized polarization. A modified particle swarm optimization (PSO) method is applied to optimize the weights of the modes. This method is demonstrated through a simulation of a 9-element conformal array. The results show the advantages of the new method in achieving a low peak side lobe, accurate mainlobe scanning, and low cross-polarization. Index Terms—Conformal antenna arrays, particle swarm optimization (PSO), pattern synthesis.

I. INTRODUCTION ANY analytical techniques, including the DolphChebyshev method [1], Taylor method [2], and Fourier method [3], have been developed for antenna pattern synthesis to obtain low side lobes and controllable beam widths. These methods are very effective in pattern synthesis under the assumption of no mutual couplings. For conformal antenna arrays, however, mutual couplings between the elements and the platform are always very strong, especially when there is a ground plane. Therefore, mutual couplings must be taken into account in the design. Under this condition, the performance of conventional methods will degrade dramatically. Recently, many optimization methods have been studied in pattern synthesis, for example, the genetic algorithm [4], the simulated annealing [5], the iterative least square [6], and the adaptive array theory [7]. These methods can take in account of the mutual couplings among the antenna elements, thus they can be applied to the synthesis of more realistic arrays, such as conformal antenna arrays.

M

Manuscript received June 18, 2010; revised December 19, 2010; accepted December 28, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported in part by the National Natural Science Foundation of China by Grants 60771042, 60927002 and 60728101, and China 111 Project by Grant B07046. K. Yang, Z. Zhao, Z. Nie, and J. Ouyang are with the School of Electronic Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu, China (e-mail: [email protected]). Q. H. Liu is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158954

Because an embedded element pattern (sometimes also known as active element pattern) contains the information of mutual coupling between the element and its environment, it has been widely used in array synthesis. The embedded element pattern of a single element is obtained by feeding this element while keeping all the other elements terminated in their own characteristic impedances [8], [9]. When all the embedded element patterns are obtained by simulation or measurement, a desired array pattern can be synthesized by using a proper weighting superposition of all the embedded element patterns. In [10], the genetic algorithm was applied to array beamforming based on the active element patterns under the condition of strong platform effects. Very good results were obtained. Usually the embedded element patterns of all elements are pre-stored in an embedded element pattern method [4]. Each embedded element pattern is simulated or measured separately. The sampling density of each pattern depends on the precision required. Higher precision requires a larger sampling number, thus larger computational cost and storage. Another problem existing in the conventional embedded element pattern method is the control of the mainlobe/null direction. If the design is to steer the mainlobe/null at an angle which is not included in the sampling set, the conventional method will not be able to optimize this problem accurately. This is because that the data of the embedded element patterns at this angle is not included in the sampling set. A class of curve-fitting algorithms known as model-based parameter estimation (MBPE) [11], [12] can be applied in solving this problem. However it requires a large amount of time when the sampling number is large. Aiming to solve this problem efficiently, this paper proposes a new decomposition method in the embedded element pattern method. The manifold separation method and effective aperture distribution function (EADF) [13]–[15] have been widely used in the direction of arrival (DOA) estimation. In the EADF method, the array responses to a far-field source are measured by moving uniform angles. By comthe source around the array with puting the -point inverse Fourier transform of the measured points of each element, the characteristic matrix of the array largest Fourier coefcan be constructed by the ficients of each element. Then the steering vector of the array can be approximately written as a product of the characteristic and are large matrix and a Vandermonde vector. When and the truncation enough, the aliasing error determined by error determined by have little influence on the DOA estimation error [15]. Inspired by this estimation method in DOA, a novel embedded element pattern decomposition method is proposed to

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YANG et al.: SYNTHESIS OF CONFORMAL PHASED ARRAYS WITH EMBEDDED ELEMENT PATTERN DECOMPOSITION

synthesize conformal phased antenna arrays. In this method, the embedded element patterns are decomposed as a product of a characteristic matrix and a matrix with a Vandermonde structure. This new method is applied to synthesize the desired patterns with the lowest peak side lobe level and restricted mainlobe width of conformal antennas. In the proposed method, first the embedded element patterns are transformed into the mode domain by inverse discrete Fourier transform (IDFT). Then, the synthesis of the array pattern is converted to the synthesis of these modes. A steering vector composed of these modes possesses a Vandermonde manifold structure. Fast Fourier transform (FFT) can be applied to improve the computational speed. Meanwhile, in contrast to the conventional embedded pattern methods, this method only needs to store a much smaller characteristic matrix rather than the whole embedded element patterns. Because of the above two advantages, the new method is much faster than existing methods in conformal array pattern synthesis. Because the modes are analytically given, the embedded element patterns at any angle (included or not included in the sampling set) can be approximately obtained by a product of the characteristic matrix and a Vandermonde vector composed by the modes at this angle. This also allows the optimization of those angles which are not included in the sampling set. In some applications, the pattern is required to achieve a desired polarization orientation, and the cross-polarization pattern needs to be as low as possible. The proposed method can be easily used in the polarization optimization when the vertical and horizontal components of each embedded element pattern are separately measured. For the tradeoff between the mainlobe width and the side lobe level, particle swarm optimization (PSO) is applied. PSO has the advantages of requiring fewer parameters and being easier in implementation [16]. It has been shown to outperform other evolutionary algorithms such as genetic algorithm [17] in some instances. In order to make this PSO algorithm suitable for the pattern synthesis, a modification is proposed. Numerical simulation results show that the method proposed in this paper can be effectively applied in pattern synthesis and needs much less computational time than conventional methods. The remainder of the paper is organized as follows. In Section II, the new embedded element pattern decomposition method is proposed. PSO is used to find the optimum virtual weights for minimizing the peak side lobe level with the constraint of a mainlobe width. Some numerical results are shown in Section III to demonstrate the efficiency of the new method. Conclusions are drawn in the last section. II. THEORETICAL ANALYSIS A. Embedded Element Pattern Decomposition elements. For Consider an arbitrary antenna array with simplicity, we fix the elevation angle and only consider the azimuthal angle. (However, the discussion below can be easily extended to 3-D patterns.) Suppose that the embedded pattern sampling angles is exof the th element with , pressed as where are the sampling angles uniformly

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distributed between 0 and . When the array excitations are , the far field pattern is given by obtained by the superposition of those embedded element patterns which can be written as (1) . where The optimization objective is to find a set of complex excitations so that the field pattern meets the requirements. Conventionally, this process can be directly fulfilled by stochastic optimization methods, such as the particle swarm optimization (PSO) [18], which we call the conventional method. In this paper, a novel method is proposed to obtain the excitations . Because each embedded element pattern is a continuous periodic function of , we can use partial sums of the Fourier series to approximate it. And each frequency component of the is called a mode [13]. Therefore, the sampling matrix embedded element patterns can be decomposed into a product of a characteristic matrix of the array itself and a Vandermonde matrix depending on the sampling angles. This decomposition is implemented by IDFT and can be written as (2) is a characteristic matrix. Each row of the where is composed of the main coefficients (i.e., the principal terms) of the Fourier series of the embedded pattern of an is the error matrix with respect to the antenna element. selected number of modes and . A larger leads to in the decomposition, i.e., can be a smaller error . Usually is chosen to be more accurate in expressing but smaller than . an odd number which is larger than is a Vandermonde matrix

.. .

.. .

..

.

.. . (3)

. where Combining (1) and (2) and ignoring the error matrix yields an approximated array pattern (4) where (5) is an approximated pattern of , which is named as virtual and can be made pattern in this paper. The error between and . For any other angle arbitrarily small by increasing that is not included in , the virtual pattern can be obtained by . Therefore, we can optimize instead of to meet the pattern performance requirements. The vector is the virtual weight vector. has the Vandermonde structure. can be realized using DFT. In order to speed up Therefore, the calculation, -point FFT will be applied by zeros padding.

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Then the problem of finding the optimal array excitations is converted into an optimization problem for the virtual weight vector . When the vector is given, the excitations can be obtained by using (5). However, (5) is overdetermined for when is larger than . In order to make sure that an exact solution of exists for (5), the vector should be in the solution , which is noted as . Therefore, space of matrix the excitations are given by (6) where the superscript stands for the Moore-Penrose pseudoinverse. In the following subsection, we will use the PSO to obtain the optimization which satisfies an objective pattern.

Fig. 1. Illustration of a cylindrical-parabolic conformal antenna array model.

B. Particle Swarm Optimization For a given array, one always has to make a tradeoff between the mainlobe width and the side lobe level. We use the PSO to find the optimum virtual weights for minimizing the peak side lobe level and with a constraint on the mainlobe width. PSO is a popular stochastic, population-based algorithm consisting of independent particles with social interactions [18]. Each particle has its own position and velocity . The iterative steps of the algorithm are described as

(7) (8) where the superscript refers to the th iteration time and the subscript refers to the th particle. denotes the Hadamard and are, respecproduct, and is the inertia weight. tively, the best known position of the th particle and the best known position of the entire swarm. and are two posiand are two random vectors where each tive constraints. element is uniformly distributed within the range [0, 1]. More details about this algorithm can be referred to [16]–[19]. In the optimization of the pattern synthesis using PSO, the virtual weight is the optimization target. The solution space of is . However, the particles “fly” in a -dimensional , which is not the same space as the -dispace mensional space . Therefore, if we use the basic PSO algorithm directly, the vector may not be in during the optimization process, which means that (5) has no exact solution. In order to apply the constraint that , the basic algorithm needs to the particles “fly” in be modified. We initialize the particle positions in and adjust (8) as (9) where is the orthogonal projection operator onto . is The orthogonal projection ensures the velocity in ; furthermore, we initialize in . will be Therefore, it is straightforward to conclude that always in the solution space. Then the particles will adjust their flying according to their own experience and the experience of . the current best individual along the space

M

Fig. 2. Magnitudes of the modes of the 1st (o), the 5th (-), and the 9th ( 1 ) element when = 360.

III. NUMERICAL SIMULATION AND DISCUSSION A. Synthesis for Minimizing Peak Side Lobe Level In this subsection, a model of 9-element cylindrical-parabolic conformal antenna array is taken as an example, as shown in Fig. 1. The elements are vertically polarized rectangular microstrip patch antennas with pin-feed 50 impedances. The anGHz. tenna array is designed for a center frequency of The inter-element space is a half wavelength. In the simulais fixed. ( is measured tions, an elevation angle of down from the -axis). The embedded pattern of each element is simulated by using FEKO software with 360 samples over the xoy-plane. The azimuthal angles are measured counterclockwise from the axis in the xoy-plane. As examples the mode distributions of the 1st, 5th, and 9th embedded element patterns after the IDFT are shown in Fig. 2. The vertical axis stands for the magnitude of the coefficients of the modes. A higher amplitude of a mode means a higher importance for this mode. The number of modes to be used depends on the required precision of synthesis. can be obtained In this proposed method, virtual pattern by using FFT to save considerable computation time. This is one

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TABLE I COMPARISONS OF THE CONVENTIONAL METHOD AND THE PROPOSED METHOD , AND WITH DIFFERENT

K;P

M

of the main advantages of the proposed method. The amount of the computation and the precision of the proposed method depend on , i.e., the selected number of modes. Table I gives the comparisons between the conventional method and the proposed method for different and ( is the number used in the FFT). The number should be chosen large enough to describe the pattern. In the simulations, first we choose modes, then zero padding to points. depends on the precision of the synthesized antenna pattern. Usually, a larger gives more accurate peak side lobe level and mainlobe in the PSO optimization process. Too small a value may lead to the failure of selection of the optimal position in the optimization process. Nevertheless, too large a value will incur huge computational cost. In Table I, we chose and , respectively. The table gives the impact of on the computing time and the maximum amplitude of other modes which are not modes. included in the In Table I, the results of the conventional optimization method were obtained by using the PSO and (1) directly. In the condepends on the precision of the syntheventional method, sized pattern. It has the similar function as in the proposed method. On the other hand, in the proposed method, affects will increase the the aliasing error of the IDFT. A smaller as 360 and 36 in the proposed method aliasing error. We set means that samples the embedded element pattern at ( 10 degrees increments between 0 and 359 in the xoy-plane). The iteration number of PSO in the conventional method and in the proposed method was both set at 500. All the codes are programmed using MATLAB. Table I shows that the computation time of the proposed method is much less than that of the conand . The new ventional method even with method has also been investigated in maximum error between and . The error is defined as

(10) In Table I, it is shown that when with , the dB, which is hard to discriminate the error is as small as difference between and . It is also worth pointing out that the proposed method dematrix in the mands much less storage memory, a

Fig. 3. Convergence speed of the proposed method for different .

360

K when M =

matrix in the conventional new method versus a method. (Usually is less than .) In the following, a synthesis testing case is studied, with the dB mainlobe width less than 28 when constraint of the scanning both to 0 and to 30 . Here, and are chosen as 512 and 360 respectively. The PSO is applied in (4) to obtain the virtual weight vector . Fig. 3 gives the convergence curves and 201. The convergence speed with is for faster than that of 201. Obviously from Table I and Fig. 3, there exists a tradeoff for between speed and precision. should be chosen as small as possible under the precision requirement. , the excitations can Due to the fact that be exactly obtained by using (6). Then, we compare three plots obtained by using (4) with , pattern in Fig. 4: pattern using (1) with obtained by (6), and the full-wave numerical computation result using FEKO software with excitations . Fig. 4(a) and (b) are the results when the mainlobe scans to 0 and to 30 , respectively. It can be seen that the impact of on the error between and is so the truncation error small that the two curves are almost identical. The full-wave computation result is very close to the results by using (1) and (4). This validates the effectiveness of the proposed method of this paper. is set as 36, the errors are also As shown in Table I, when small for and . In many real-world applications, dB. Therefore, it is acceptable when the error is as small as , there is no need to choose a large . Compared to . Thus, when a lot of computation can be saved when the method is extended to three-dimensional pattern synthesis, instead of 64800 samples 648 samples are needed with . Therefore, in the following simulations, we will with as 36. set B. Precision of the Mainlobe Direction Control For the conventional method by directly using (1), if the desired mainlobe direction is not at one of the sampling angles , i.e., , the pattern value at the cannot

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Fig. 5. The pattern can exactly scan to the angle of 31.5 , which is not included in the embedded element pattern sampling angles.

where Fitness (dB) ized peak side lobe

Fig. 4. Comparison among the normalized pattern obtained by (4) using PSO, the pattern by (1) using w acquired by (6), and the full-wave simulated pattern using w . (a) Scanning to 0 . (b) Scanning to 30 .

be obtained directly by using (1). As an alternative, one needs in which is closest to . This to choose an angle means that the precision at the desired angle is out of control. The proposed method has an analytic manifold, which can steer the mainlobe direction to any angle. Assuming the mainis desired, the normalized pattern lobe direction of is given by value at

is equal to the normallevel given by , where is the set of angles of the side lobe region. The second term on the right-hand side (RHS) of (12) is to ensure the mainlobe scan to . The pattern value at should be the peak of the pattern, which means that when dB is expected, otherwise dB. Overall, a larger is better for mainlobe scanning to . is larger than zero, and due to the fact that The weight is much smaller than the the absolute value of absolute value of the Fitness, a large should be chosen. Fig. 5 shows the simulation results acquired by FEKO with excitations obtained by (6) using the proposed method. and are chosen. The embedded element patterns are sampled with 36 samples at . The desired mainlobe direction is 31.5 . It can be observed that the peak value of the synthesized radiation pattern is located exactly at the position as desired. In this case, if one wants to obtain a mainlobe directing to 31.5 by using the conventional method, the embedded element pattern sampled at 31.5 has to be prepared. But by using the proposed method, this angle can be synthesized by using the samples at other angles. Similarly, we can obtain a null at any direction with this method. C. Pattern Synthesis With Optimized Polarization

(11) Therefore, a brief modification for the fitness in PSO is given by (12)

In this subsection, the polarization control using the proposed method will be investigated. The model is the same as that used in Subsection A, but each element has two individual feeds to obtain two orthogonal polarization components. The embedded element patterns of and components of each port are simand are ulated by FEKO. The characteristic matrices

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. The full-wave lobe are nearly at the same level because simulations have been done by using the excitations obtained by (6) to demonstrate the effectiveness of the proposed method. In the figure, the solid line and the circled line are the co- and cross-polarization results of using full-wave simulation, respectively. The full-wave simulation results agree very well with that of the proposed method. IV. CONCLUSION

Fig. 6. The co- and cross-polarization results when in the desired pattern the polarization angle is 45 and the beam is scanned to 30 .

obtained by IDFT to and spectively. Therefore, is rewritten as

embedded element patterns re. Equation (4)

(13) where and are and components of the array radiation pattern, respectively. Consider the destination linear polarization angle given by . Then the co-polarization pattern and cross-polarization are given by pattern (14a) (14b) The optimization objective is to make the co-polarization beam have a low peak side lobe with a mainlobe constraint while minimizing the cross-polarization beam especially in the mainlobe area. We adjust the fitness function as

(15) where is a weight determined by actual requirements. A larger means that the low cross-polarization is more important than the side lobe of the co-polarization. In Fig. 6 the optimized co- and cross-polarization patterns are presented for a case where the polarization angle is 45 and the beam is scanning to 30 in the desired pattern. In the fitness, the , which means the cross-polarization has an equal weight importance as the side lobe of the co-polarization. The co-polarization pattern and cross-polarization pattern results using (13) by PSO are given in Fig. 6. The dashed line and the starred line are the co- and cross-polarization results by using the proposed method, respectively. It can be observed that the cross-polarization pattern and the co-polarization side

A novel method is proposed and applied in conformal antenna arrays to synthesize patterns with variable requirements, such as low peak side lobe level, scanning to a specific angle and optimized polarization. This method has the advantages of low computation and storage costs, and ease for implementation. Mutual coupling and other nonideality factors in real conformal arrays are included by using the embedded element pattern. The method is able to synthesize a pattern with mainlobe directing to any angle, even when the angle is not in the sampling angle set of the embedded pattern. Several cases have been studied to validate the effectiveness of the proposed method. The proposed method is not limited to conformal arrays. In fact, the method can be applied in the pattern synthesis of any type of antenna arrays. In some situations when the embedded element patterns are identical or almost identical, the array pattern can be obtained by multiplying embedded element pattern or average embedded element pattern with an array factor, then the array pattern design is equivalent to an array factor design. In other words, both the conventional method and the proposed method can be utilized in this situation. But for more common situations, the proposed method has advantages when the embedded element patterns are greatly different either due to mutual coupling or different radiation direction in conformal arrays. REFERENCES [1] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005. [2] S. J. Orfanidis, Electromagnetic Waves and Antenna 2004 [Online]. Available: http://www.ece.rutgers.edu/~orfanidi/ewa/ [3] A. Schell and A. Ishimaru, “Antenna pattern synthesis,” in Antenna Theory, R. Collin and F. Zucker, Eds. New York: McGraw-Hill, 1969, pt. I. [4] J. OuYang, Q. R. Yuan, F. Yang, H. J. Zhou, Z. P. Nie, and Z. Q. Zhao, “Synthesis of conformal phased array with improved NSGA-II algorithm,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 4006–4009, Dec. 2009. [5] J. A. Ferreira and F. Ares, “Pattern synthesis of conformal arrays by the simulated annealing technique,” Electron. Lett., vol. 33, no. 14, pp. 561–564, 2000. [6] L. I. Vaskelainen, “Iterative least-square synthesis methods for conformal array antennas with optimized polarization and frequency properties,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1179–1185, July 1997. [7] C. Dohmen, J. W. Odendaal, and J. Joubert, “Synthesis of conformal arrays with optimized polarization,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2922–2925, Oct. 2007. [8] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, no. 8, pp. 1176–1178, Aug. 1994. [9] D. F. Kelley, “Embedded element patterns and mutual impedance matrices in the terminated phased array environment,” in Proc. Int. Symp. on Antennas Propagation, Jul. 3–8, 2005, vol. 3A, pp. 659–622. [10] T. Su and H. Ling, “Array beamforming in the presence of a mounting tower using genetic algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2011–2019, June 2005.

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[11] E. K. Miller, “Model-based parameter estimation in electromagnetic: Part I—Background and theoretical development,” IEEE Antennas Propag. Mag., vol. 40, no. 1, pp. 42–52, Feb. 1998. [12] D. H. Werner and R. J. Allard, “The simultaneous interpolation of antenna radiation patterns in both the spatial and frequency domains using model-based parameter estimation,” IEEE Trans. Antennas Propag., vol. 48, pp. 383–400, Mar. 2000. [13] F. Belloni, A. Richter, and V. Koivunen, “Extension of root-MUSIC to non-ULA array configurations,” presented at the IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), France, May 14–19, 2006. [14] A. Richter, F. Belloni, and V. Koivunen, “DoA and polarization estimation using arbitrary polarimetric array configurations,” presented at the IEEE Workshop Sensor Array and Multichannel Processing, Waltham, MA, Jul. 12–14, 2006. [15] F. Belloni, A. Richter, and V. Koivunen, “DoA estimation via manifold separation for arbitrary array structures,” IEEE Trans. Signal Process., vol. 55, no. 10, pp. 4800–4810, Oct. 2007. [16] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” presented at the IEEE Conf.Neural Networks IV, Piscataway, NJ, 1995. [17] W. B. Daniel and H. W. Douglas, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [18] P. J. Bevelacqua and C. A. Balanis, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3442–3449, Dec. 2007. [19] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004.

Zaiping Nie (SM’96) was born in Xi’an, China, in 1946. He received the B.S. degree in radio engineering and the M.S. degree in electromagnetic field and microwave technology from the Chengdu Institute of Radio Engineering [now the University of Electronic Science and Technology of China (UESTC)], Chengdu, in 1968 and 1981, respectively. From 1987 to 1989, he was a Visiting Scholar with the Electromagnetics Laboratory, University of Illinois, Urbana. Currently, he is a Professor with the Department of Electromagnetic Engineering. He is also the author or coauthor of more than 290 journal papers. His research interests include antenna theory and techniques, field and waves in inhomogeneous media, computational electromagnetics and its applications, electromagnetic scattering and inverse scattering, antenna techniques in mobile communications, etc.

Kai Yang was born in 1986. He received the B.S. degree in electronic engineering from Xidian University, Xi’an, China, in 2008. He is working toward the Ph.D. degree at the University of Electronic Science and Technology of China (UESTC), Chengdu, China. His research interests include conformal antenna arrays and array signal processing.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, Fujian Province, China, in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at UrbanaChampaign, in 1989. He was with the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign, as a Research Assistant from September 1986 to December 1988, and as a Postdoctoral Research Associate from January 1989 to February 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield CT, from 1990 to 1995. From 1996 to May 1999, he was an Associate Professor with New Mexico State University, Albuquerque. Since June 1999, he was been with Duke University, Durham, NC, where he is now a Professor of Electrical and Computer Engineering. He is also a Visiting Professor at the University of Electronic Science and Technology of China (UESTC), Chengdu. His research interests include computational electromagnetics and acoustics, inverse problems, geophysical subsurface sensing, biomedical imaging, electronic packaging, and the simulation of photonic and nano devices. He has published more than 350 papers in refereed journals and conference proceedings. Dr. Liu is a Fellow of the Acoustical Society of America, a member of Phi Kappa Phi, Tau Beta Pi, and a full member of the U.S. National Committee of URSI Commissions B and F. Currently he serves as an Associate Editor for Radio Science, and for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, for which he also served as a Guest Editor for a Special Issue on Computational Methods. He received the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation.

Zhiqin Zhao (SM’05) received the B.S. and M.S. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, and the Ph.D. degree in electrical engineering from Oklahoma State University, Stillwater, in 1990, 1993, and 2002, respectively. From 1996 to 1999, he was with the Department of Electronic Engineering, UESTC. From 2000 to 2002, he researched rough surface scattering as a Research Assistant with the School of Electrical and Computer Engineering, Oklahoma State University. In 2003, he was a Research Associate with the Department of Electrical and Computer Engineering, Duke University, Durham, NC. In 2006, he became a Full Professor with the School of Electronic Engineering, UESTC. His current research interests include computational electromagnetics and signal processing. Dr. Zhao is a member of Phi Kappa Phi Honor Society.

Jun Ouyang received the B.S., and Ph.D. degrees in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2004 and 2008, respectively. He is currently with the Department of Electronic Engineering, UESTC. His research interests include antenna theory and computational electromagnetics.

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Robust Beampattern Synthesis for Antenna Arrays With Mutual Coupling Effect Tongtong Zhang and Wee Ser, Member, IEEE

Abstract—An array beampattern synthesis method which use a set of new constraints on magnitude response combined with an compensation method for the mutual coupling effect among array elements is investigated in this paper. The beampattern synthesis problem was solved by two different optimization methods. The proposed method can provide much improved flexibility in the design and has been demonstrated to be effective for the design of various array patterns including superdirective, null forming, and shaped-beam pattern. The non-convex problem of the constraints on magnitude response is solved by using a relaxed linear constraint. With the proposed method, the number of the DOFs consumed for beampattern control is small, so that the synthesized array has an attractive performance on interference cancelation. Simulation results show the effectiveness of the proposed methods. Index Terms—Adaptive array, antenna array, beamforming, beampattern synthesis, mutual coupling.

I. INTRODUCTION

A

NTENNA pattern synthesis is a vast subject in the antenna literature. The amount of publication is extensive. Most papers deal with linear array antennas, the results can then be carried over to planar antennas [1]. Currently, iterative methods based optimization techniques are very powerful tools for pattern synthesis. Arbitrarily shaped beam patterns can be synthesized with minimum effort. The subject of power shaped beam pattern synthesis problem of narrow band conformal arrays was analyzed in [2]. The proposed method iteratively linearizes the non-convex power pattern function to obtain a convex sub-problem in the design variables. The problem is solved by using second-order cone programming. In the design, only the magnitude response is specified. To look back at the history of the beam pattern synthesis, there are several of the classical methods, for instance, the Woodward-lawson synthesis, Dolph-Chebyshev synthesis, Fourier synthesis, and the Taylor line-source synthesis which are both elegant and analytical. The method presented in [3] is to utilize adaptive array theory to extrude the synthesized pattern through arranging an artificial interferer. Iteration coefficients are important in the method, which determine stability and convergence speed. But, the iteration coefficients cannot be chosen easily. The method

Manuscript received April 14, 2010; revised November 02, 2010; acceptedNovember 27, 2010. Date of publication May 10, 2011; date of current version August 03, 2011. T. Zhang is with the School of EEE, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]). W. Ser is with the Centre for Signal Processing, Nanyang Technological University, Singapore 639798, Singapore (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.2011.2152329

in [4] is based on a mainlobe control mechanism. Its magnitude can be easily determined according to the mainlobe shape, but it’s appropriate phase is difficult to choose. Numerical approaches based on convex optimization techniques for array pattern synthesis using vector weighting approaches [5] have received some attention, due to its capability of handling more complicated design specifications. However, the global optimal solutions can be obtained efficiently only for uniform linear array. In [6], Dolphs proposed an approach to obtain the Chebyshev linear array. For a rectangular array or circular planar array, Chebyshev patterns can be obtained or approximated through an iterative numerical process or optimization process. Nevertheless, the Dolphs-Chebyshev method does not have flexibility in specifying array beampattern, such as the locations of nulls, the beamwidth and ripple of the mainlobe. For optimization techniques, there many approaches have been reported in beampattern design. For example, a min-max optimization which is used to design equi-sidelobe array beampattern with some additional constraints applied to control the beampattern. The well investigated Quadratic optimization methods are used to design array beampattern too with more flexibility on beampattern control [16]–[18]. Most of the current antenna beampattern synthesis methods have flexible control on the mainlobe response. However, some of the arrays cannot be designed with the upper and lower bounded response at specified mainlobe. Although this problem was solved by using spectral factorization method [19], the method only works for a uniform linear array (ULA) and cannot work for an arbitrary antenna array. In this paper, we proposed a method with constraints on magnitude response (CMR), so that the mainlobe response can be specified with an allowable ripple. The non-convex constraint is eliminated by using relaxed linear constraint. Unlike any other methods for array pattern synthesis, this paper deals with the practical array imperfections such as MC effect and other array calibration errors. It is known that the performance of an adaptive array is strongly affected by mutual coupling between elements of the antenna array. This effect has been studied by many researchers [7]–[14]. Gupta and Ksieuski [7] investigated the performance of a linear array with mutual coupling being taken into account. They developed an expression for the linear array by considterminal linear bilateral ering the N-element array as a network responding to an outside source and obtained a relationship between the open-circuit voltage (OCV) and the actual voltage at the terminals of the dipole elements. The method is known as OCV method. Leou and Yeh [8] analyzed the OCV

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method and method of moment (MOM) to compensate for the mutual coupling effects. Recently [14] studied the performance of the estimation of the DOA in a linear array by using a receiving-mutual-impedance method for the compensation of the MC effect. The efficient mutual impedance method [9], [10] is based on redefining the mutual impedance matrix which is used to calculate the mutual coupling effect between antenna elements. With redefined mutual impedance, the performance of the array antennas in searching for the directions of arrival of multiple-signals is significantly improved. In this paper, the investigation on the performance of the receiving mutual impedance method on compensation for the MC effect in array beampattern synthesis is presented. The conventional array pattern synthesis methods do not have good performance on null design for practical fabricated arrays. However, the problem is solved in this paper by a robust optimization method which exploits the pre-calculated MC effects matrix which is calculated by efficient mutual impedance compensation method. The method also tolerates the modeling error with a presumed norm. It is also shown that the efficient mutual impedance method combined with an robust array beampattern synthesis method is very effective in the design of various array patterns. The paper is organized as follows: Section II is devoted to the receiving mutual impedance method on compensation for the mutual coupling effect for beampattern synthesis. In Section III, the review on array beampattern synthesis and the proposed method are presented. Numerical results based on the proposed method are given in Section IV. Finally, the conclusion is drawn in Section V.

and . with sponds to the load impedance of the antenna terminal. The elements in matrix I and V are

.. .

.. .

..

.. .

.

.. .

.. .

(1)

.. .

..

.

.. .

(3)

.. .

(4)

.. .

.. .

..

.

.. .

.. .

.. .

(5)

are the decoupled terminal voltages and the measured terminal voltages. is the mutual impedance between the th and the th antennas. In order to include effect of mutual coupling among antenna arrays for array application of beampattern synthesis, we insert a mutual coupling matrix to modify array steering vector . The mutual coupling matrix [12] is transformed (ASV) from mutual impedance matrix which can be obtained by using many mutual coupling compensation methods. The modified which taking into account the efarray steering vector fect of mutual coupling is given by where

(6)

where, the elements in matrix Z is

.. .

.. .

respectively. There is the “single-mode” approximation in theoretical analysis. The detailed description of the assumption can be found in work [9], [10], [14]. By doing so, (1) is simplified and the matrix blocks are replaced by the respective terminal quantities. As we know, in practice, it is not possible to measure the entire current distribution of the antenna. The measurable quantiand terminal voltties are the terminal currents . By using of “single-mode” approximaages tion, the current induced on the each antenna element of the array consist of a single-mode current based on the current at the antenna terminal. Therefore, we can obtain the mutual impedances which have already taken into account the presence of other antenna elements. For a -element array [9], this can be written as

II. DESCRIPTION OF THE METHOD OF COMPENSATION FOR THE MUTUAL COUPLING EFFECT IN ARRAY ANTENNAS The detailed discussion on the mutual coupling compensation method has been presented in [10], [14]. For the integrity of the paper, a brief description on the efficient mutual impedance compensation method is given as follows. In an array antennas, the current on each array element is the summation of the exciting current and coupling current from all the other elements either directly or indirectly. In the analysis, segments, a set of same each antenna element is divided into linear equations can be obtained for an ULA and UCA which both have antenna elements. These linear equations in a matrix form is as below

corre-

(2)

where, is defined as mutual coupling matrix. The mutual coupling matrix obtained by [12], [13] is used to express the relationship between received and decoupled signal at the antenna elements. In this paper, we use the method [9], [10] to model the relationship between ASV and modified ASV. From (5), we can easily observed that, for ULAs, the arrays are linear and thus follow the rotational symmetry. Therefore, the

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coupling matrix for ULA is symmetric Toeplitz matrix [10], [11], and can be described as.

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the noise gain of the array. In [19], [20], the CMR is used in the array processing. It is expressed as (13)

.. .

.. .

.. .

..

.

(7)

.. .

For UCAs [11], the arrays are circular. The coupling matrix is circulant, and given by

.. .

.. .

Therefore, from (5), (7), (8), are given by

.. .

..

.

(8)

.. . in matrix

(9) . where, The application of the mutual coupling matrix for beampattern synthesis is presented in Section III. III. ARRAY PATTERN SYNTHESIS WITH CMRS AND MC EFFECTS BEING TAKEN INTO ACCOUNT

where and are the lower and upper limits of the magare the lower and upper nitude response. The angle and and are debounds of the mainlobe. The values of pendent on . To simplify the notation, we assume that both of them have fixed values, and , respectively. With constraints on magnitude response (13), the array beampattern can be controlled with a great freedom which does not consume too many DOFs. However, it can be proved that the left side of (13) is non-convex over . This problem is solved in [19], [20] by the transformation of the array magnitude reof the sponse to a function of the autocorrelation function is carried out. With array weights, and the optimization of the obtained optimal , the spectral factorization technique [21] is applied to get the optimal array weight. The drawback of this method is that it only works for an ULA. For non-uniform linear arrays, this method does not work. In following section, the method for array beampattern synthesis is modified with constraints on magnitude response. The method is applicable to array with arbitrary structure. Further considering the constraints on magnitude response in (13), the right side of (13) is convex over . The constraints can be applied to the optimization directly. For the left side

The conventional array beampattern synthesis is formulated as the following optimization [16], [18]

(14) can be relaxed as

(10)

(15)

where denotes Hermitian transpose operator, array the array steering vector. The desired weighting vector and response is defined in the interested range . The most typical constraint is the unit gain constraint at a desired direc. The magnitude constraint is used tion , i.e., to limit the sidelobe of the array to be lower than a threshold in the presumed range . The cost function has many choices. Two typical examples are the Dolph-Chebeshev method (equi-sidelobe) and quadratic optimization method. For the Dolph-Chebeshev method

where we constrain the imaginary and real parts of the response simultaneously. This idea works well in robust adaptive beamformer [22] if the constraints are only imposed on a desired direction . However, if the mainlobe is wide, the constraints on the imaginary part of the response consume many DOFs. Lemma 1: The constraints must be satisfied if , where is the feasible set of the array steering vector. The proof of Lemma 1 is very straightforward by using the . The real part constraint is a property sufficient but not necessary condition of the constraint on magnitude response. With some relaxation, the array beampattern synthesis problem with constrains on magnitude response can be formulated as

(11) where is the range of sidelobe, and for the quadratic optimization method, (12)

The beampattern synthesizer in (10) lacks accurate control on the mainlobe response. Although a lot of constraints can be imposed on mainlobe to achieve this goal, too many degrees of freedom (DOFs) are consumed which results in the increase of

(16) where

is the angle set of the mainlobe.

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For the proposed method (16), we can see that all the constraints are convex over . The semi-infinite constraints and cost function can be approximated in a direct way by sampling the angle intervals. The semi-infinite constraints can then be converted into a set of convex constraints. The interior-point method (IPM) can be used to find the optimum solution with high efficiency and accuracy. Some public solvers can be found on websites [23], [24]. In practical applications, the assumption on the array is always violated by the sensor mismatch, the array geometry error, MC effects of antenna array, etc. These kinds of array imperfections may introduce unavoidable errors in the problem of pattern synthesis. In some scenarios, although the fixed beamformer can tolerate array imperfections in some measure, some strict requirements still need to be satisfied on accurately controlling the responses. For example, in the design of response nulls, the desired nulls always disappear in practical fabricated arrays which have imperfections. In the paper, the method is designed to include the pre-calculated MC effect to be taken into account. The error between the pre-calculated and the true MC effects and other array imperfections are also considered in the optimization stage by using worst-case optimization technique. With use of the pre-calculated MC matrix, the mutual coupling effect has been compensated at an easy rate. The error of array steering vector is reduced to a very small value. Assuming that which is described by (7) and pre-calculated MC matrix is (8), and true MC matrix is . The error between the true and compensated array steering vector is . This value is much smaller than the conventional methods. We also can consider other array imperfections. Assume that the true array is steering vector (17) where is the modeling error. The only constraint on this error is that its norm is limited by (18) With this arbitrary array imperfection, a robust beampattern synthesis method is derived based on the following Lemmas. Lemma 2: For any given , is guaranteed if . Lemma 3: For any given , a sufficient condition of is . The robust array pattern synthesis method is

Using the definition of array steering vector and the inequalities (20), it is clear that (21) From (19), one has two inequalities

(22) Hence, we have

(23) That means, (24) In the end, we have the upper bound of (25) (26)

IV. NUMERICAL RESULTS In this section, some numerical results are presented to evaluate the performance of the proposed methods. The MC matrix is calculated based on the array with all elements are of and radius , where equal length is the operating wavelength. All the antenna ele. We consider ments are loaded with a terminal load two kind of array geometries in this section, both are consisting of 32 antenna elements. Generally, with more constraints on array synthesized pattern, it requires more degrees of freedom, i.e., more number of sensors, in array synthesis to achieve high performance. The selection of this number can be changed according to design specifications. In UCA, the antenna elements are uniformly placed along the circumference of a circle of raand in the plane, respectively. dius In ULA, the antenna elements are placed along the x-axis, with and in two simulations, reinter-element spacing of spectively. The first set of simulation study are carried out based on the array, which has no compensation of MC effect being made. The other set of simulations are carried out with array MC effects compensated. A. Simulation Results Based on ULAs

(19) Remark: It should be noted that with Lemma 2 and 3, the optimization problem may have no solution if is not suitably set. From (16), one has (20)

In this subsection, the array beampattern synthesis simulation is carried out to evaluate the performance of the proposed method. The ULA array has a mainlobe with beam centered at 90 and beamwidth 20 at the 30 dB level and with response ripple 2 dB. The value of and are determined by the spec, ifications. For example, when the ripple is given (in dB) as and . The array is required then to attenuate two angularly-extended interferers. Without loss of generality, two nulling areas with different beamwidths are

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Fig. 1. The comparison of the true responses of the methods with/without precalculated MC effects in ULA by using equi-sidelobe cost function. Fig. 3. The comparison of the true responses of the methods with/without precalculated MC effects in ULA by using quadratic optimization cost function with dense (0:3) inter-antenna spacing.

depths reach deeper levels, resulting in much better performance in terms of interferences rejection capabilities. The beamwidth and response ripple are well controlled with the method with pre-calculated MC effects being compensated. For the method without compensation of MC effects, the nulls are disappeared. The proposed MC compensated method without robustness control has better performance than that of the classical method but poor than that of the robust method. Fig. 2 shows the same set of curves as Fig. 1, but using different cost function. All nulls are disappeared in this simulation. To further show that the proposed method can work with smaller inter-antenna element, a is used. dense antenna array with inter-element spacing The results show in Fig. 3 also reveal that the proposed method can produce beampattern with high performance. Fig. 2. The comparison of the true responses of the methods with/without precalculated MC effects in ULA by using quadratic optimization cost function.

B. Simulation Results Based on UCAs

designed. In practice, the numbers and beamwidths of nulling areas can be set freely according to the specification of applications. The center and width of the nulling areas are [20 , 1 and [120 , 6 ], respectively. The nulling areas should have at least 60 dB responses. Following the optimization process, we get the pattern function shown in Fig. 1 and Fig. 2. In the following six figures, the red solid curve represents the proposed robust method with pre-calculated MC effect and the green line represents the non-robust method with pre-calculated MC effect in simulation). The blue dotted curve is produced by ( the classical pattern synthesis methods, like Dolphs-Chebyshev method and quadratic method, without compensation of the MC effects among antenna array. The true responses in Fig. 1 are obtained with the equi-sidelobe cost function (12) and the true responses in Fig. 2 are obtained with the quadratic optimization cost function (12). It can be clearly seen from Fig. 1 that the proposed method with pre-calculated MC effects, the null

In this subsection, the array beampattern synthesis simulations are carried out in UCA. The simulation results shown in Fig. 4 and Fig. 5 with spacing between adjacent array elements is half wavelength. The spacing between array elements in Fig. 6. The array is design to have two is reduced to narrow band nulls at bearings of [30 , 1 ] and [120 , 6 ]. The nulling areas must have at least 20 dB responses. Figs. 4–6 show the nulls pattern for the proposed robust method with and without pre-calculated MC effect, respectively. It can be seen, the proposed method has overall lower sidelobe level compared with the method without pre-calculated MC effect. The null depths for the proposed method reach deeper levels than specified values, whereas no nulls at all for the method without compensation of the MC effect. The true responses in Fig. 4 and Fig. 5 are obtained by using the equi-sidelobe cost function and the quadratic optimization cost function, respectively. Both of them show that the proposed method with pre-calculated MC effects has excellent performance on specified nulls forming. Considering dense inter-element spacing, similar conclusions can be

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Fig. 4. The comparison of the true responses of the methods with/without precalculated MC effects in UCA by using equi-sidelobe cost function.

Fig. 6. The comparison of the true responses of the methods with/without precalculated MC effects in UCA by using quadratic optimization cost function with dense (0:33) inter-element spacing.

response is solved by using a relaxed linear constraint. With the proposed method, the number of the DOFs consumed for beampattern control is small, so that the synthesized array has an attractive performance on interference cancellation. Simulation results show the effectiveness of the proposed methods. ACKNOWLEDGMENT The authors would like to thank Dr. Z.L. Yu for the fruitful discussions on the beampattern synthesis problem. REFERENCES

Fig. 5. The comparison of the true responses of the methods with/without precalculated MC effects in UCA by using quadratic optimization cost function.

obtained as that from the ULA case. The proposed method can still produce high performance beampatterns. V. CONCLUSION An effective compensation method for the mutual coupling effect in arbitrary antenna array employed for the problem of array beampattern synthesis is introduced. An novel array beampattern synthesis method which use a set of new constraints on magnitude response combined with the effective compensation method for the mutual coupling effect among array elements is investigated in this paper. The beampattern synthesis problem was solved by two different optimization methods. The method provides more design flexibility and has been demonstrated to be effective for the design of various array patterns including superdirective, null forming, and shaped-beam pattern. The non-convex problem of the constraints on magnitude

[1] L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design. Piscataway-Hoboken, NJ: IEEE Press-Wiley, 2006. [2] K. M. Tsui and S. C. Chan, “Pattern synthesis of narrowband conformal arrays using iterative second-order cone programming,” IEEE Trans. Antennas Propag., vol. 58, pp. 1959–1970, Jun. 2010. [3] C. A. Olen and R. T. Compton, “A numerical pattern synthesis algorithm for arrays,” IEEE Trans. Antennas Propag., vol. 38, pp. 1666–1676, 1990. [4] P. Y. Zhou and M. A. Ingram, “Pattern synthesis for arbitrary arrays using an adaptive array method,” IEEE Trans. Antennas Propag., vol. 47, p. 962-869, 1999. [5] H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Trans. Signal Processing, vol. 45, no. 3, pp. 526–532, Mar. 1997. [6] C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level,” Proc. IRE, vol. 34, pp. 335–348, Jun. 1946. [7] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. 31, pp. 785–791, 1983. [8] M. L. Leou, C. C. Yeh, and D. R. Ucci, “Bearing estimations with mutual coupling present,” IEEE Trans. Antennas Propag., vol. 37, pp. 1332–1335, 1989. [9] H. T. Hui, “Decoupling methods for the mutual coupling effect in antenna arrays: A review,” Recent Patents Engrng., vol. 1, pp. 187–193, 2007. [10] H. T. Hui, “Improved compensation for the mutual coupling effect in a dipole array for direction finding,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2498–2503, Sep. 2003. [11] T. T. Zhang, Y. L. Lu, and H. T. Hui, “Simultaneous estimation of mutual coupling matrix and DOAs using structured least square method,” in Proc. IASTED, Canada, Jul. 2005, pp. 277–280.

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[12] S. Durrani and M. E. Bialkowski, “Effect of mutual coupling on the interference rejection capabilities of linear and circular arrays in CDMA systems,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1130–1134, Apr. 2004. [13] T. Svantesson, “Modeling and estimation of mutual coupling in a uniform linear array of dipoles,” in Proc. ICASSP99, Phoenix, AZ, Mar. 1999, pp. C2961–C2964. [14] T. T. Zhang, H. T. Hui, and Y. L. Lu, “Compensation for the mutual coupling effect in ESPRIT DOA estimations,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1552–1556, Apr. 2005. [15] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory. New York: IEEE Press, 1994. [16] B. D. Carlson and D. Willner, “Antenna pattern synthesis using weighted least squares,” Proc. Inst. Elect. Eng. Microw., Antennas Propag. H, vol. 139, no. 1, 1992. [17] M. H. Er, S. L. Sim, and S. N. Koh, “Application of constrained optimization techniques to array pattern synthesis,” Signal Processing, vol. 34, 1993. [18] B. P. Ng, M. H. Er, and C. A. Kot, “A flexible array synthesis method using quadratic programming,” IEEE Trans. Antennas Propag., vol. 41, no. 11, pp. 1541–1550, 1994. [19] S. P. Wu, S. Boyd, and L. Vandenberghe, “FIR filter design via spectral factorization and convex optimization,” App. Compu. Contr., Signal Commun., pp. 1–33, 1998. [20] Z. L. Yu, W. Ser, and M. H. Er, “A novel robust adaptive beamformer based on linear optimization,” in Proc. ICICS, Singapore, December 2007. [21] A. H. Sayed and T. Kailath, “A survey of spectral factorization methods,” Numer. Linear Algebra Applicat., vol. 8, no. 6–7, pp. 467–496, 2001. [22] S. A. Vorobyov, A. B. Gershman, and Z. Q. Luo, “Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem,” IEEE Trans. Signal Processing, vol. 51, no. 2, pp. 313–324, Feb. 2003. [23] J. F. Sturm, “Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones,” Optim. Math. Softw., vol. 11, pp. 625–653, Aug. 1999. [24] M. Grant, S. Boyd, and Y. Ye, CVX Users’ Guide, Version 1.0 Feb. 2007 [Online]. Available: http://www.stanford.edu/boyd/cvx/cvx_usrguide.pdf

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Tongtong Zhang received the Ph.D. degree from Nanyang Technological University, Singapore, in 2006. Since 2006, she has worked at Nanyang Technological University as a Research Fellow. She has worked in the field of wireless communications, such as in multi-antenna systems, MIMO systems, diversity for mobile communication systems, smart antennas, adaptive arrays, etc. Her recent research interests include array signal processing, digital signal processing, image processing and signal feature extraction and classification techniques. She has a strong interest in numerical methods in the EM field.

Wee Ser (SM’92) received the B.Sc. (Hon.) and Ph.D. degrees in electrical and electronic engineering from the Loughborough University, U.K., in 1978 and 1982, respectively. He joined the Defence Science Organization in 1982 and became Head of the Communications Research Division in 1993. In 1996, he was appointed Technological Advisor to the CEO of DSO National Laboratories. In 1997, he joined Nanyang Technological University, Singapore, and has since been appointed Director of the Centre for Signal Processing. His research interests include sensor array signal processing, signal detection and classification techniques, and channel estimation and equalization. He has published more than 130 research papers in refereed international journals and conferences. He holds six patents and is a coauthor of six book chapters. He is the Principal Investigator of several externally funded research projects. Dr. Ser is a member of a Technical Committee in the IEEE Circuit and System Society. He was a recipient of the Colombo Plan scholarship and the PSC postgraduate scholarship. He was awarded the IEE Prize during his studies in the U.K. While at DSO, he was a recipient of the prestigious Defence Technology Prize (Individual) 1991 and the DSO Excellent Award 1992. He is currently an IEEE Distinguished Lecturer and an Associate Editor for IEEE COMMUNICATIONS LETTERS and the Journal of Multidimensional Systems and Signal Processing (Springer).

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RFID Grids: Part II—Experimentations Stefano Caizzone and Gaetano Marrocco

Abstract—The RFID Grid is a model for generally coupled multitudes of tags including single-chip tags in close mutual proximity or a single tag with a plurality of embedded microchips. Some properties of this new entity, useful for passive Sensing and for Security, are the possibility to increase the read-range and to provide responses rather insensitive to the interrogation modalities. These recently introduced issues are here experimented for the first time with many real-world examples comprising multi-chip configurations designed for improved power scavenging and for passive sensing of things. Index Terms—Array, grid, MIMO, remote sensing, RFID, sensor, space diversity.

I. INTRODUCTION ERY recently, the paper [1] introduced the concept of RFID Grid, e.g., a generally coupled multitude of UHF (or higher frequency) RFID tags, including single-chip tags in close mutual proximity or a single tag with a plurality of embedded microchips. This model is useful to predict the impedance and the backscattering performances of tags over a dense collection of objects such as items over a shelf, as well as to design new sensing-oriented tags. RFID tags, indeed, may be equipped with more than a single chip to communicate the identity of the object, by one of the chip, and its physical state by means of the other chip or a combination of both [2]–[4]. In [1] the basic equations for the direct and inverse links were derived by using the very general model of multi-port loaded scatterers. The theoretical analysis has permitted to discover new features, some of them common to MIMO and Space-Diversity systems [5], and in particular the possibility i) to handle the inter-antenna coupling in a useful way with the purpose to improve the power harvested by the microchip and hence to enlarge the read range; ii) to achieve measured analog backscattered information which are rather independent on the mutual position between reader and tags themselves (distance and orientation). Such a function, denoted as Analog Identifier (A-Id) as complementation of the Digital Identifier (D-Id) normally stored in the microchip memory, originates from the combined processing of direct- and

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Manuscript received October 16, 2010; revised December 14, 2010; accepted January 24, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by the Italian Ministry of University under PRIN-2008: MULTITAG research program. S. Caizzone was with the DISP—University of Roma Tor Vergata, Via del Politecnico, 1, 00133 Roma, Italy . He is now with the Antenna Group, Institute of Communications and Navigation, German Space Agency (DLR), 82230 Wessling, Germany. G. Marrocco is with the DISP—University of Roma Tor Vergata, Via del Politecnico, 1, 00133 Roma, 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.2011.2158974

inverse-link measurements. A-Id could have significant application in the context of sensing and security. In both cases, indeed, the analog response of the tag (basically the level of the backscattered power or the turn-on power) is processed in addition to the digital microchip identifier, to earn some physical information about the tag’s identity (security) and about the variation of the tagged object of the nearby environment (sensing). At these purposes it is of primary importance that the reading of analog information from the tag is as more independent as possible on the interrogation modality, in order to easily collect the tag’s response at different times without the need to strictly replicate a specific reader-tag set-up. In particular, the availability of reading-robust functions could boost the emerging pervasive sensing and context aware applications within the Internet of Things paradigm [6]. The wireless characterization of RFID tags has been mainly focused to the measurement of the tag’s read range or radiation patterns starting from the experimental evaluation of the threshold (or turn-on) power level [7] of the tag. Another performance parameter, commonly measured, is the backscattered power from the tag to the reader’s detector or, more in general, the tag’s radar cross section [8], [9]. Both turn-on power and RCS are related to gain and input impedance of the tag and therefore they cannot be easily de-embedded by means of conventional wireless measurements techniques. The characterization of tag input impedance is instead generally performed by using balanced probes and de-embedding procedures as in [10]. The response of RFID grids has never been characterized in a systematic way: some studies [11]–[16] have been conducted in order to assess the overall influence of mutual coupling between tags over the degradation of the RFID link and the results are mainly aiming at mitigating this degradation (see [1] for an extended discussion). On the other hand, in the perspective of understanding the coupling between tags to exploit useful features, this paper presents a detailed experimental campaign to prove and discuss some of the concepts introduced in the previous companion paper [1] by the help of real-world examples having incremental complexity. Starting from a tag with a single microchip, the experiments go on through integrated two-microchips tags, up to 3 3 RFID grids. Measurements have been performed in presence of different reader-grid positions, time-varying tagged objects and non-stationary environments. The aim is to demonstrate that the RFID-oriented analytical formulas provided in [1] can be useful to dominate the inter-port coupling within the design task and to verify the true degree of invariance of the analog identifier in uncontrolled conditions. The paper is organized as follows. Section II introduces the measurement procedures for the relevant RFID responses,

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including angle- and distance-invariant analog identifiers, the turn-on power and the realized gain. A single-chip tag in standalone configuration is measured in Section III to understand the quality of responses achievable in absence of inter-tag interactions and of other ambient perturbation. Three examples of two-chips grids are then considered in detail in Section IV, with particular attention to evaluate the improvement in the RFID performances by a systematic design of the grid, and to characterize the angular invariance of the analog identifier also in sensing-like applications. Finally, a 3 3 grid of commercial tags is explored in Section V where the sensitivity of the fingerprint is experienced with respect to the presence of metallic scatterers close to the interrogating reader. II. BACKGROUND AND MEASUREMENT METHODOLOGY According to [1], the power scavenging performances of an RFID Grid are completely characterized in the free space by means of the embedded realized gain of each th port (1) is the th row of the admittance matrix of the grid, being the impedance matrix referred a diagonal mato the RFID microchips’ connections, and trix containing the equivalent impedances of the microchips. The vector of elements embeds all the electromagnetic pa, rameters associated with the ports, e.g., the open circuit gain the polarization mismatch , the self resistance and fiof the embedded field. The gain is the panally the phase rameter to be maximized (in the useful direction) for best power scavenging and hence for longest read distance. Physical information about the Grid and about its interaction with the tagged objects can be extracted from the analog of the th port, independently on identifier or fingerprint the reading modalities. The analog identifier has been defined of the th port and in [1] as a function of the turn-on power collected by of the corresponding backscattered power the reader. For the particular modulation scheme for which the microchip’ impedance is switched between a high impedance , the analog identifier may and the scavenging impedance be written in a very compact form where

(2) where is the sensitivity of the microchip at the th port, and the proportionality factor accounts for the reader’s front-end and the modulation scheme, as discussed later on. The particular case of the single-chip tag yields (3) where is simply the input impedance of the antenna of the tag. Both the above expressions may be evaluated along with the frequency to carry on further informations. The physical meaning of analog identifier, and its calculation from experimental data deserves some clarification. It combines, by definition, measured data from both the direct (power harvesting) and reverse (power backscattering) links. Under the hy-

pothesis of a stationary environment between the reader’s interrogation and the tag’s responses, any interaction with the nearby environment, and the mutual distance and orientation between the reader and the RFID systems, will affect both the links in a same manner, thanks to the antenna reciprocity. Hence, it was demonstrated that the simple combination of turn-on power and backscattered power will mathematically remove any kind of the above mentioned electromagnetic interactions. What remains is of the functional dependence on the th diagonal entry the complete admittance matrix, in the case of multi-chip systems, or the dependence on the series connection between the input impedance of the tag’s antenna and of the microchip in the simpler case of one-chip tag. In this sense, the A-Id is a structural property of the (multi-port) tag. Since the network admittance matrix is sensitive to the object where the tag is attached on, the variation of the A-Id in successive measurements may be related to time-changing chemical-physical properties of the object, therefore giving sensing capability. The orientation-invariance of A-Id will permit to perform this measurement without taking care to replicate a same reader-tag position at successive times. The calculation of A-Id requires the knowledge of the turn-on . At this purpose the power entering into the reader’s power antenna is gradually increased, in a controlled way, till the th chip begins responding. At that moment, such a value of and the corresponding input power is the required turn-on received by the reader is stored backscattered power on a Personal Computer. In case of multi-chips configurations, an inventory task is preliminary performed, e.g., all the chips’ identifiers are collected, and then each chip is interrogated sequentially. Usually, commercial readers return backscattered power in the form of RSSI (received signal strength indication), after a process of demodulation, frequency conversion and equalizaand the RSSI tion. The particular relationship between is hardware-dependent but often takes the form (4) are specific to the reader’s chipset but also to the with modulation scheme. In all the measurements shown in this work, the stored RSSI signal is related to the high-level modulation state. Anyway, the experimentally-evaluated A-Id from (2) and (4) is not an absolute fingerprint of the tag but it is filtered by the reader’s circuitry. The use of A-Id for sensing will therefore requires a same reader family for subsequent interrogations to avoid other kinds of uncertainties. Finally, the A-Id depends on the microchips’ impedance that, as its is well known, exhibits a non-linear dependence on the impinging interrogation power. The proposed procedure to derive the A-Id, however, enforces the turn-on power condition: it means that the microchip will always receive the same amount of power, just corresponding to its sensitivity. Thanks to this procedure, the microchip impedance can be considered quite stable over subsequent interrogations. The phenomenology of RFID GRID systems, as theoretically found in [1], could be experimentally analyzed, under a complete control on the hardware, according to the technical solu-

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Fig. 1. The LabID UHF tag used in the experimentation campaign.

tion proposed in [17], e.g., by emulating the RFID microchips with PIN diode modulators. In this paper we instead follow a more direct approach, e.g., we use commercial tags or ad-hoc designed tags embedding real RFID microchips with the purpose to recognize the RFID-grid features, theoretically argued, even in presence of all the uncertainties of electrical data. In the following several experiments, to fully understand the true invariance of A-Id parameters, also the realized gain is measured. This is a typical parameter commonly used to characterize the radiation performance of a tag and also its interaction with time-variant materials or changing environments (humidity and gas, [18]). To be specific, the estimation of the realized gain is obtained by using the equation

(5) The realized gain is indeed mathematically dependent on the reader-tag mutual position and requires the knowledge of the reader’s antenna gain at the specific orientation. All the measurements are performed by an UHF ThingMagic reader, connected to a 3-dB gain linear polarized PIFA antenna. Tags under test (TUTs) are placed on a rotating platform, in order to get data acquisition at different angles. Both the reader and the tag are 110 cm high from the floor. It is worth noticing that the measurement scene is a real 5 m by 5 m laboratory, with no attempt to reduce the ground reflections as well as any other type of electromagnetic interaction with the surrounding objects. The measurements are repeated for 13 equally-spaced . samples within the frequency band

Fig. 2. Single chip: a) normalized measured analog identifier F=F ; b) mea~) = 0:46. The read dissured realized gain normalized by its maximum (G tance is fixed to 60 cm while the reader-tag orientation is changed in successive measurements according to points P . . . P , as specified in the inset.

III. ONE-CHIP TAG A commercial Lab-ID’s UH-100 tag (Fig. 1) is here considered: a meandered dipole with a T-match impedance transformer. This tag is used to check the fingerprint invariance with the angle in a so particular one-port case. This experimentation provides a reference case for the next experiments with multiports grids, with the purpose to understand the best that can be obtained in absence of other complications such as the inter-port coupling in more complex configurations and time-varying environments. A. Analog Identifier Versus Reader-Tag Orientation The reader-tag distance is fixed to 60 cm. Fig. 2(a) shows of the considered the normalized analog identifier tag obtained from the processing of the measured power at five observation angles according to (2). For comparison, Fig. 2(b) also gives the normalized realized gain in the same conditions, which, as expected by the variable gain of the tag over the vertical plane, is rather sensible to the reader-tag orientation.

Fig. 3. Single chip: normalized A-Id measured for a single tag with variable distances from the reader and fixed angle P : ( = 90 ;  = 0 ).

The invariance of the fingerprint suggested by (3) is particularly apparent within the European RFID band, wherein the tag responds best, due to proper impedance matching. B. Analog Identifier Versus Reader-Tag Distance The measurements are repeated for a fixed reader-tag orienand increasing distance tation . The results in Fig. 3 again show, as theoretically predicted, a substantial invariance of the analog identifier, within all the considered frequency band. IV. TWO-PORTS GRIDS Some examples of two-port grids are here analyzed with particular attention to the improvement of the scavenging capa-

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Fig. 5. Two-tags grid: measured and simulated realized gains for the coupled tags designed to improve gain performances with respect to a single tag of same size, by means of a constructive management of inter-port coupling. Fig. 4. Two-tags grid: prototype of equal-shape dipoles at a close distance. Size in Table I.

The optimized two-tags grid exhibits self and mutual impedances

TABLE I SIZES OF THE DESIGNED SINGLE TAG AND COUPLED TAGS

bility, obtained by means of electromagnetic coupling, and to the angle-invariance of the analog identifier. A-Id is theoretically and experimentally estimated from (2) when the reader-tag position is changed and when the grid is placed onto a timevarying object to mimic a true sensing application. In some of the following geometries the grid is composed by two close-distance dipoles having same or different size, while in a third case a true two-chip tag is analyzed. A. Scavenging-Optimized Two Tags In a first example, two coupled tags have been specifically designed as an interconnected system with the purpose to discuss the enhancement of RFID capabilities which may be achieved when the electromagnetic coupling is handled in a constructive way. The tags have a conventional dipole-like layout, with a T-match impedance transformer [19] whose aspect ratio , is properly selected, by the help of a finite difference time domain model of the coupled dipoles, to maximize the embedded realized gains in (1) of both the ports. A low-impedance NXP microchip transponders has been assumed, having and power sensitivity . The resulting antenna layout is shown in Fig. 4, and the optimized geometrical sizes are reported in Table I. The performance of the two-tags grid are here compared with that of a single port dipole having the same size as the tags of the grid except for the T-match which has been re-optimized (see again Table I) to achieve the maximum realized gain in single-port configuration. The single tag has estimated input impedance and power transfer coefficient . The is almost omnidirectional on the antenna’s realized gain H-plane, with maximum value .

It is worth noticing the negative mutual resistance . As proved in [1], the power transfer coefficient of each port for interrogation along the normal axis of the grid is, under and may Hermite matching condition, , as indeed obtained be higher than 1 provided that in this example. For the actual case of two port grid the value has been predicted by the FDTD model. The embedded realized gain of the grid, as obtained by measurements and theory, are shown in Fig. 5. Due to the proximity between the two tags, the embedded realized gains corresponding to the two ports are sensibly distorted with maximum in the endfire direction and nearly univalue tary gain in the broadside direction. The measurement results are in good agreement with the simulations, verifying the theoretical analysis in [1]. The experiments moreover demonstrate that the concurrent design of the two closely spaced tags permits to improve the realized gain of each tag, with respect to the standalone dipole, of a factor

which definitely enlarges the maximum read distance of 20%. The measured analog identifier associated to chip 1 is shown in Fig. 6. As in the single chip case, the analog identifier is rather independent on the angle except for those frequencies where the realized gain is small, as in the case of observation angles or . In these conditions the reader is processing a low-intensity backscattered signal and the computation in (2) is hence affected by poor accuracy. Nevertheless the measured analog identifier appears considerably much more stable than the measurement of the realized gain shown in Fig. 5. B. Integrated Dual-Chips Tag The two-port tag already presented by the authors in [20] to achieve a temperature sensor is here considered to discuss the analog identifier’s invariance. The tag (Fig. 7) consists of a

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Fig. 6. Two-tags grid: normalized a) A-Id and b) realized gain at some obser= 2 34. vation angles, ( ~ )

G

:

Fig. 8. Two-chips Grid: normalized A-Id and realized gain of one of the ports = 0 7. at some observation angles. Normalization factor for the gain: ( ~ )

G

:

observation angle than the previous example, due to the higher values of the realized gains so that the tag will backscatter fields of similar strength, resulting in a more accurate processing of the received power. C. Sensing Tags

Fig. 7. Two-chip tag: prototype and size from [20].

partly folded dipole provided with two equal-size T-match circuits to control the port impedance. of the double T-Match The size were designed to maximize the active power transfer coeffiof the two-ports grid cients in the broadside direction having assumed the same NXP microchip as before. An FDTD model of the antenna predicted self and mutual impedance at 870 MHz of and , respectively. Fig. 8 shows the measured analog identifiers of port 1 and the corresponding realized gain at three different observation angles over the horizontal plane. The A-Id is even more independent on the

The fingerprint-extraction procedure is here applied to the sensing of things and in particular to the detection of the physical (time-varying) state of the tagged object. For instance, in logistics it is sometimes required to monitor the status of containers (bottles, packages, bags) with the purpose to detect variation of their solid, powder or liquid content. The self-sensing natural capabilities of RFID tags may be exploited to handle these events (see [2] for a complete introduction to RFID self-sensing). The example given here considers the sensing of sugar-powder level inside a plastic container. This is a very challenging problem for RFIDs due to the low-relative , which is very permittivity . The variation of filling level similar to that of the air is hence expected to produce only a slight modification of the tag’s electromagnetic parameters, and in turn of the sensed data. used for the sugarThe perspex box powder experiment is shown in Fig. 9. The sensing RFID Grid is now composed by two T-match dipoles of different size attached at a close distance onto a same face of the box. The two tags, and , have been denoted as optimized for the best conjugate matching to the NXP microchip

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Fig. 9. Two-port grid for sensing the sugar level within a perspex box. Geo, ; , , metrical size (in [mm]): , , , ; , , , .

H = 12 D = 95 TAG L = 142:8 W = 6 a = 28:8 b = 15:6 w = 8:4 TAG L = 119 W = 5 a = 24 b = 13 w=7

when the box is empty and filled with sugar , respectively. The various measurements of the analog identifiers are executed by gradually filling the box with sugar, starting from up to , i.e., roughly corresponding to the . The reader and the tags are height of the longest dipole aligned to have maximum polarization efficiency. The diagrams in Fig. 10 give the theoretical (computer simulated) and experimental analog identifier for both the tags for three different mutual orientations. The analog identifier of is rather insensitive (Fig. 10(a)) to the variation of the target, and could be used as a control information, e.g., it could provide the digital “label” of the object. The Analog identifier (Fig. 10(b)) is instead able to sense the sugar related to variation when the filling level exceeds half the tag size and spans an overall dynamics of about 50%, up to saturation. The collected data are very little dependent on the mutual orientation between the reader and the container, even for rotation of 180 e.g., when the two antennas are completely shadowed with respect to the reader by the sugar itself. V. TWO DIMENSIONAL 3

F

Fig. 10. Two-port grid for Sugar sensing: (Normalized) analog identifiers for: a) and b) . The data acquisition has been repeated for different orientation, on the horizontal plane, between reader and tagged box so that the case of 180 rotation means that the grid is completely shadowed to reader by the box.

TAG

TAG

3 GRID

It is here finally considered a 3 by 3 grid (Fig. 11) made of the commercial tags already used in Section III. The analog identifier of the central (label 22) and the edge (label 11) elements are measured by the same procedure as before. The invariance of the so obtained fingerprints (Fig. 12) respect to the reader-grid orientation is less sharp than in the previous examples, but nevertheless A-Id is still much more insensitive to rotations than the measurement of the realized gain (Fig. 13), especially when edge elements are observed. Moreover it is interesting to note that the fingerprints of the two tags are rather similar, unlike the corresponding measured gains. A. Variable Environment The presence of any object close to the reader-tag system can in principle modify the tag responses (turn-on power and

2 3 RFID Grid: the tags are attached over a foam slab. Grid steps: 1 = 9 cm, 1 = 14 cm. Fig. 11. 3

backscattered power) due to the electromagnetic scattering that it produces. However, since its disturbing effect applies on both the forward and backward links of the RFID communication in the same manner, it can be in principle removed by the calculation of the analog identifier which involves the processing of the turn-on and the backscattered powers.

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Fig. 14. 3 3 RFID Grid in variable environment: Measurement setup, comprising a RFID reader (left), the 3 3 RFID grid (right) and the metallic cylindrical scatterer interposed between the reader and the grid.

2

2

Fig. 12. 3 3 RFID Grid: measured A-Id of a) central and b) edge elements. The grid plane is 80 cm far from the reader. The vertical segments give the residual angular variation of A-Id at each frequency sample as the observation angles are randomly chosen in the range ( = 90; 0 <  < 2 ). The continuous line gives the angle-average fingerprint.

2

Fig. 15. 3 3 RFID Grid: a cylindrical scatterer moved close to the reader/tag system into two different positions A and B: a) analog identifier of the central (22) tag collected for mutual reader-tag orientation  = 0 ,  = 90 and b) normalized realized gain.

2

Fig. 13. 3 3 RFID Grid: normalized measured realized gain of a) central el~ = 0:29) and b) ement (maximum gain used for the normalization: G ~ = 0:53). Diagram’s conventions as in Fig. 12. edge element (G

Assuming that the scatterer is distant enough from the grid so that its input parameters are not significantly modified, the presence of the scatterer will therefore affect only the radiation pattern of the reader and of the grid. However, since the definition of A-Id drops out the reader’s gain, the obtained are expected to be “transparent” to the presence of the scatterer which, in this sense, becomes “invisible” for the grid. To verify this property, a vertical metallic cylinder, 180 cm high and 7 cm in diameter, has been placed in the region between the reader and the grid under test (Fig. 14), close to Line Of Sight condition. It was actually at a distance of 20 cm from the reader antenna and moved 11 cm sideways from the line of sight, first on the left and later on the right. The grid is measured in the . fixed direction The A-Id of the central tag is shown in Fig. 15(a) and compared to the realized gain of the same configuration 15 b). This experiment is considered very challenging since the big metallic

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ever, since the definition of the fingerprint is based on the accurate measurement of the turn-on power and of the backscattered power, the true reliability of the analog identifier, for instance during sensing or security assessment, can not leave apart the quality of the reader’s receiver. It is reasonable to assume that even more robust results could be achieved by using the value of the backscattered power as a quality factor of the measurements. The cost of grid-based devices is expected to be higher than that of standard tags used as simple wireless replacement of barcodes. However multi-chips tags will offer augmented features making them closer to sensors than to radiofrequency labels. Application to sensing appears indeed promising and in future experiments a multi-chip sensing grid could be empowered by a plurality of chemical receptors to achieve multi-variant low-cost remote sensors. The fingerprint concept will give a reasonable confidence that a change in the grid’s responses in successive interrogations will be related to a change of the physical parameters under test, rather than to a change of the mutual reader-tag position. So, there will be many opportunities to develop new devices but also there will be the need of ad-hoc data retrieval algorithms to master specific applications. REFERENCES Fig. 16. Single tag in presence of a variable environment in the same conditions as in the 3 3 array case. a) analog identifier b) normalized realized gain.

2

object is expected to produce a significant scattering and to seriously affect the measurement of the realized gain of grid’s elements, as clearly visible in Fig. 15(b). Instead, the analog identifier in Fig. 15(a) revealed also in this case to be really insensitive to the presence and to the position of the scatterer, and by extrapolation, to the motion of nearby objects or people provided that the characteristic times of such motion are large compared with the RFID interrogation rate. However, it is worth mentioning that the cancellation of the scatterer is not an exclusive property of the GRID arrangement, but it can be achieved also with just a standalone tag, as shown in the experiment in Fig. 16 where a single element of the array is placed close to the pipe scatterer as in the previous configuration. VI. CONCLUSIONS The presented experimentations corroborate the theoretical analysis of RFID grids. The formulas for the active realized gain and the impedance matching can be used together with any standard optimization design procedure. The inter-port coupling, which is generally considered as a negative effect of tags’ proximity, can be instead judiciously handled to enlarge the read range of each microchip, or for the same distance, to reduce the required power. The tag’s fingerprint obtained by the calculation of the analog identifier seems to be valuable in both single and multi-chip configurations. This quantity demonstrated to be rather insensitive to the many uncertain conditions of typical RFID interrogations in a real world, e.g., to its mutual position with respect to the grid, as well as to the variation of the environment itself. How-

[1] G. Marrocco, “RFID grids: Part I—Electromagnetic theory,” IEEE Trans. Antennas Propag.. [2] G. Marrocco, L. Mattioni, and C. Calabrese, “Multi-port sensor RFIDs for wireless passive sensing—Basic theory and early simulations,” IEEE Trans. Antennas Propag., vol. 56, no. N.8, pp. 2691–2702, Aug. 2008. [3] M. Philipose, J. R. Smith, B. Jiang, A. Mamishev, S. Roy, and K. Sundara-Rajan, “Battery-free wireless identification and sensing,” IEEE Perv. Comput., vol. 4, no. N.1, pp. 37–45, Jan.–Mar. 2005. [4] P. Nikitin, K. V. S. Rao, and S. Lam, “RFID Tags With Enhanced Range and Bandwidth Obtained by Spatial Antenna Diversity,” U.S. Patent no. US2009/ 0219158, Sep. 2008. [5] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans. Wireless Commun., vol. 3, no. N.4, pp. 1317–1325, Jul. 2004. [6] D. Preuveneers and Y. Berbers, “Internet of things: A context- awareness perspective,” in The Internet of Things: From RFID to the NextGeneration Pervasive Networked Systems, L. Yan, Y. Zhang, L. T. Yang, and H. Ning, Eds. London: Auerbach Publications, 2008. [7] L. Ukkonen and L. Sydanheimo, “Threshold power-based radiation pattern measurement of passive UHF RFID tags,” in Proc. PIERS, Cambridge, 2010, pp. 87–90. [8] P. V. Nikitin and K. V. S. Rao, “Theory and measurement of backscattering from RFID tags,” IEEE Antennas Propag. Mag., vol. 48, no. N. 6, pp. 5459–5462, Dec. 2006. [9] S. Skali, C. Chantepy, and S. Tedjini, “On the measurement of the delta radar cross section ( RCS) for UHF tags,” in Proc. IEEE Int. Conf. on RFID 2009, pp. 346–351. [10] S. L. Chen, K.-H. Lin, and R. Mittra, “A measurement technique for verifying the match condition of assembled RFID tags,” IEEE Trans. Instrum. Meas., vol. 59, no. 8, Aug. 2010. [11] J. P. Daniel, “Mutual coupling between antennas for emission and reception- application to passive and active dipoles,” IEEE Antennas Propag. Mag., vol. 22, no. 2, pp. 347–349, 1974. [12] F. Lu, X. Chen, and T. T. Ye, “Performance analysis of stacked RFID tags,” in Proc. IEEE Int. Conf. on RFID, 2009, pp. 330–337. [13] K. Lee and T. Chu, “Mutual coupling mechanisms within arrays of nonlinear antennas,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 963–970, 2005. [14] X. Chen, F. Lu, and T. T. Ye, “The weak spots in stacked UHF RFID tags in NFC applications,” in Proc. IEEE Int. Conf. on RFID, 2010, pp. 181–186. [15] V. Rizzoli, A. Costanzo, M. Rubini, and D. Masotti, “Rigorous investigation of interactions between passive RFID tags by means of nonlinear/electromagnetic co-simulation,” in Proc. Eur. Microwave Conf., 2006, pp. 722–725.

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[16] Y. Tanaka, Y. Umeda, O. Takyu, M. Nakayama, and K. Kodama, “Change of read range for UHF passive RFID tags in close proximity,” in Proc. IEEE Int. Conf. on RFID, 2009, pp. 338–345. [17] J. D. Griffin and G. D. Durgin, “Complete link budgets for backscatterradio and RFID systems,” IEEE Antennas Propag. Mag., vol. 51, no. N.2, pp. 11–25, 2009. [18] J. Virtanen, L. Ukkonen, T. Bjorninen, and L. Sydanheimo, “Printed humidity sensor for UHF RFID systems,” IEEE Sensors Applicat., pp. 269–272, 2010. [19] G. Marrocco, “The art of UHF RFID antenna design: Impedance matching and size-reduction techniques,” IEEE Antennas Propag. Mag., vol. 50, no. N.1, pp. 66–79, Feb. 2008. [20] S. Caizzone, C. Occhiuzzi, and G. Marrocco, “Multi-chip RFID antenna integrating shape-memory alloys for detection of thermal thresholds,” IEEE Trans. Antennas Propag., to be published.

Stefano Caizzone received the M.Sc. degree in telecommunications engineering from the University of Rome “Tor Vergata,” in 2009 and is currently part-time working toward the Ph.D. degree. His main research interests concern small antennas for RFIDs and navigation, antenna arrays and grids with enhanced sensing capabilities. He is now with the Antenna group of the Institute of Communications and Navigation of the German Space Agency (DLR), Wessling, Germany, where he is responsible for the development of innovative miniaturized antennas.

Gaetano Marrocco was born in Teramo, Italy, on August 29, 1969. He received the Laurea degree in electronic engineering (Laurea cum laude and Academic Honour) and the Ph.D. degree in applied electromagnetics from the University of L’Aquila, Italy, in 1994 and 1998, respectively. Since 1997, he has been a Researcher at the University of Rome “TorVergata,” Rome, Italy, where he currently teaches Antenna Design and Medical Radio-Systems, manages the Antenna Lab and is Advisor in the Geo-Information Ph.D. program. In October 2010, he achieved the level of Associate Professor of electromagnetic. In summer 1994, he was at the University of Illinois at Urbana-Champain as a Postgraduate student. In autumn 1999, he was a Visiting Researcher at the Imperial College in London, U.K. In 2008, he joined the Ph.D. program of the University of Grenoble (FR). His research is mainly directed to the modeling and design of broad band and ultrawideband (UWB) antennas and arrays as well as of sensor-oriented miniaturized antennas for biomedicine, aeronautics and radiofrequency identification (RFID). He has been involved in several Space, Avionic, Naval and Vehicular programs of the European Space Agency, NATO, Italian Space Agency, and the Italian Navy about the analysis and the design of non conventional antennas and systems. He holds eight patents on broadband naval antennas and structural arrays, and on sensor RFID systems. Prof. Marrocco currently serves as an Associate Editor of the IEEE Antennas and Wireless Propagation Letters, he is a reviewer for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, IEEE PROCEEDINGS, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, and is a member of Technical Program Committee of several International Conferences. In 2008, he was the General Chairman of the first Italian multidisciplinary scientific workshop on RFID: RFIDays-2008: Emerging Technology for Radiofrequency Identification. He was the Co-Chair of the RFIDays-2010 International Workshop in Finland and Chairman of the Local Committee of the V European Conference on Antennas and Propagation.

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Efficient Analysis of Large Scatterers by Physical Optics Driven Method of Moments Miodrag S. Tasic and Branko M. Kolundzija, Fellow, IEEE

Abstract—A new iterative procedure is presented that enables method of moment (MoM) solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients. In each iteration the body is excited by a plane wave and by the currents, which are obtained as an approximate solution in the previous iteration. The physical optics (PO) and modified PO techniques are used to determine the PO and the correctional PO currents, which are expressed in terms of original MoM basis functions and grouped into macro-basis functions (MBFs). Weighting coefficients of all MBFs are determined from the condition that mean square value of residuum of original MoM matrix equation is minimized. The iterative procedure finishes when the residuum decreases below the maximum allowed value. The accuracy and efficiency of the proposed method are illustrated on two examples: cube scatterer and airplane scatterer. Since the construction of MBFs by PO and modified PO techniques ensures fast convergence to the original MoM solution, the method is named PO driven MoM. Index Terms—Electromagnetic scattering, iterative methods, macro-basis functions, method of moments (MoM), physical optics (PO).

I. INTRODUCTION

I

N the last few decades there is continuous interest in analysis of metallic scatterers based on the method of moment (MoM) solution of surface integral equations (SIEs). According to basic MoM theory [1], [2] induced current over metallic surface is approximated by series of known basis functions multiplied by unknown coefficients and the SIE is transformed from the linear operator form into the system of linear equations, which is solved for unknown coefficients. By increasing the number of unknown coefficients, , the memory occupation and the matrix solution time and matrix fill time increase as (in case of direct methods as Gaussian or LU decomposition) in. Hence, the size of the solvable problem in terms creases as of is limited by memory and time resources of the computer used for the simulation. For example, 8 GB of operative memory on personal computers enables solution of systems up to about unknowns. However, the electrical size of the of surface area) is also dependent on solvable problem (in flexibility of basis functions. In case of RWG basis functions Manuscript received May 29, 2010; revised November 16, 2010; accepted February 23, 2011. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported by the Serbian Ministry of Science and Technological Development by Grant ET-11021. The authors are with School of Electrical Engineering, University of Belgrade, 11120 Belgrade, Serbia (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.2011.2158785

, resulting in about 300 typical edge length of triangles is [3]. In particular, if , the maxunknowns per imum electrical size is limited to about 100 . Many techniques have been developed to increase the electrical size of the solvable structure within the limits of computer resources. Generally, these techniques can be grouped into two classes. First class includes techniques, which decrease the memory resources and matrix fill/solution time for given number of unknowns (e.g., iterative techniques extended to fast multipole method (FMM) and further to multilevel fast multipole algorithm (MLFMA) [4]). Second class includes techniques, which decrease the number of unknowns. Being of interest in this paper, in what follows the second class of techniques will be elaborated. The basic way to decrease the number of unknowns is to use higher order basis functions. For interpolatory higher order basis functions defined over triangles [5], the maximum edge length , resulting in 40–70 unof triangles can be extended to [6]; for polynomial higher order basis functions knowns per defined over quadrilaterals [2], [7] the maximum edge can be ex, resulting in 20–35 unknowns per [7]. Thus tended to electrical size of solvable structure within limit of computer resources is increased for an order of magnitude when compared with RWG basis functions. In both cases, the expansion orders are chosen according to electrical size of patches, without taking into account electromagnetic properties of the problem. Further decrease in number of unknowns can be achieved if electromagnetic properties of the problem are exploited. In case of electrically large scatterers, the currents over electrically large and smooth surfaces in the lit region can be predicted using current based asymptotic techniques, such as the physical optics (PO). There are few variants of hybrid PO-MoM method [8]–[12]. The first two are iterative, while other three are direct. The first three use subdomain approximation in the MoM part, while other two use higher order basis functions. Common for all variants is that the MoM currents in the PO region are replaced by the PO currents, which are expressed in terms of the original excitation and the MoM currents in the rest of the structure. Thus hybrid methods retain the same number of unknown coefficient as original MoM problem in the MoM region, while unknown coefficients in the PO region are eliminated. Since in general case the PO region and the MoM region are approximately of the same size, the number of unknowns needed for original MoM problem can be halved by hybridization. In order to achieve more significant reduction in number of unknowns a variety of methods, which exploit electromagnetic properties of the problem in construction of basis functions, is developed. Common for most of these methods (e.g., [13]–[19]) is

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that they: 1) start from triangular/rectangular mesh of the structure and subdomain basis functions (RWG/rooftop) spanned over this mesh; 2) divide the structure into a number of electrically large domains; and 3) construct a new set of entire-domain basis functions over each of the domains as linear combination of subdomain basis functions in the domain. Weighting coefficients of subdomain basis functions are obtained by MoM as response of this domain to specific excitation. All these methods mutually differ regarding the choice of these excitations. In case of subdomain multilevel approach (SMA), in addition to original excitation, the excitations are performed by bridge basis functions of unit magnitude [13], [14]. (Bridge basis functions are basis functions shared by two neighbouring domains.) In case of synthetic function approach (SFA) the excitations are performed by electric dipoles densely placed over the surface surrounding the domain [15], [16]. Similarly, in one variant of characteristic basis function (CBF) approach the excitations are performed by plane waves incoming from different densely distributed directions [17]. In both cases initially large number of entire-domain basis functions is reduced by singular value decomposition (SVD), resulting in minimum number of entiredomain basis functions needed for practically arbitrary excitation. In another variant of CBF approach the primary CBF for a domain is determined by original excitation, and secondary CBFs for the domain are obtained after excitation by primary CBFs of all other domains [18]. In [19], CBFs are combined with MLFMM to further decrease the memory requirements and speed up the calculations. Instead of these MoM derived CBFs, some authors use the PO derived CBFs [20]. After replacing the original set of MoM basis functions by smaller set of their linear combinations, the original MoM matrix equation can not be exactly satisfied, i.e., such replacement results in residuum of original MoM matrix equation. Application of SVD is indirect way to ensure that the residuum of original matrix equation is relatively small. However, being optimal for arbitrary excitation, the number of basis functions obtained after application of SVD is not minimal for specific excitation. The goal of this paper is to solve the problem with minimum number of unknowns for given excitation and given maximum allowed relative residuum of matrix equation. To achieve this goal we propose the method that we named PO driven MoM. In this method the relative residuum of matrix equation is decreased in iterative procedure. In each iteration the previous solution is used to excite the structure and determine correctional PO currents. The correctional PO currents are expressed in terms of original basis functions and grouped into macro-basis functions (MBFs). Then weighting coefficients of all MBFs are determined from the condition that mean square value of residuum of original MoM matrix equation is minimized. The proposed method evolved from the methods given in [21]–[23]. In all these methods the correctional PO currents in each iteration are represented by single MBF. The unknown coefficients of MBFs are determined either by minimizing the mean square error of the tangential component of magnetic field at the body surface [21], [22] or by applying the Galerkin method to MBFs [23]. The crucial improvement of the proposed method is usage of set of new MBFs in each iteration instead of single MBF, and in minimizing the residuum of original MoM solution.

Detailed formulation of PO driven MoM is given in Section II. Implementation details are given in Section III. Numerical examples are given in Section IV. II. FORMULATION OF PO DRIVEN MoM Consider a closed body made of perfect electric conductor (PEC), placed in vacuum, and excited by a time-harmonic incident electromagnetic field of angular frequency and electric and magnetic field vectors and . As a result, electric currents of density are induced over the body surface, so that the incident field inside the body is annihilated. Once these currents are determined, all other quantities of interest can be easily evaluated. Namely, since the total field inside the PEC body is zero, the total field inside/outside the body will not change if the PEC is replaced by a vacuum, while retaining the currents . In this equivalent problem the currents are placed in infinite vacuum medium, so that scattered field due to these currents, and , can be evaluated by closed form expressions. The scattered field outside the body is uniquely determined by its excitation and appropriate boundary condition at the body surface. Starting from the equivalent problem the boundary condition is expressed in terms of the excitation and induced currents, thus resulting in the corresponding surface integral equation (SIE). The SIEs are linear operator equations, for which MoM represents a general solution method. A. MoM Solution of Scattering Problem Let us consider linear operator equation in form (1) where is known vector function (excitation), is linear operator, and is unknown vector function to be determined (response). The unknown function is approximated by a linear combination of known, mutually independent, vector basis functions multiplied by unknown coefficients , as (2) After substituting instead of into (1), the residuum of linear operator equation (1) can be written in the form (3) The inner products of the residuum and known, mutually inde, , are forced pendent, vector testing functions to be zero, i.e. (4) Thus, the system of linear equations in terms of unknown coefficients is obtained as (5) where

and

are inner products of the form (6)

(7) and

is domain of the th testing function.

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Let us focus on a case when unknown function is density of surface electric currents, . Initially, these currents are approximated using the PO solution of the problem.

repwhere represents a position vector of this point and resents the surface inside the body, just below surface . The is obtained as [2] scattered magnetic field at surface

B. PO Driven MoM Solution of Scattering Problem PO solution of the problem states that currents induced over the surface of the body are (8) where is a position vector, is unit vector normal to the surface of the body and directed outwards, and is incident magnetic field vector. These PO currents can be approximately , linear combination of original MoM basis represented as , i.e., as functions multiplied by weighting coefficient (9) Procedure for projecting PO currents onto the set of original MoM basis functions, given by (9), depends on geometrical modeling and type of basis functions, and will be addressed later. For the purpose of this procedure the PO currents are determined in a set of points over the surface in four steps: 1) set of points is specified; 2) outward unit normal vector is determined in each point; 3) for each point it is determined if it belongs to the shadow region or to the lit region; and 4) in each point the PO current is calculated using (8). This procedure represents one variant of classical PO technique. Approximation of unknown currents , which is given by (2), is initially set to , i.e., . [In that case coefficients in (9) represent initial value of coefficient in (2).] The initial approximation is improved using the following iterative procedure. In the th iteration the excitation of the structure is performed by original incident field and approximation of unknown cur. The total incirents obtained in the previous iteration, dent magnetic field in the th iteration is written as (10) is the scattered magnetic field obtained in the th iteration. If the solution for currents in the th iteration would be exact, this incident field inside the body would be equal to zero. However, the solution for currents is approximate and this incident field inside the body slightly differs from zero. We know that the magnetic field vector due to the surface curin the point just bellow the surface element rent element equals . Hence the tangential component of incident magnetic field in a point just bellow the surface can be canceled by adding the surface current element over the surface just above this point, if the current density of this element is given by where

(11)

(12) where integration is applied in the principle value sense, is , , del operator, , and and are position vectors of the field and source point. The correctional currents all over the surface are adopted using (11). However, although each correctional surface current just bellow element annihilates tangential component of the surface , it does not annihilate the additional magnetic field produced by all other correctional surface current elements. . By proper This additional field can be even larger than weighting of correctional currents it can be achieved that the adannihilated. ditional field created is smaller than a part of In what follows we shall deal with proper weighting of correctional currents, so that the solution is maximally improved in each iteration. Note that (11) is formally the same as that used for lit region in (8), but is applied in both, the shadow region and the lit region. Hence, to determine these currents we use the technique similar to that for determination of the PO currents: 1) set of points is specified; 2) outward unit normal vector is determined in each point; and 3) in each point the current is calculated using (11). In what follows this technique will be referred to as modified PO technique. Accordingly, the currents obtained by this technique will be referred to as correctional PO currents. The correctional PO currents can also be approximately rep, linear combination of original MoM basis resented as functions multiplied by weighting coefficient , i.e., as (13) In what follows function will be referred to as correctional current in the th iteration. The correctional current in the th iteration is split into parts, , where each part represents one macro-basis function (MBF), i.e., the correctional current is expressed as a sum of MBFs as (14)

The th macro-basis function is generally defined as linear combination of all original MoM basis functions , , multiplied by weighting coefficients , as (15) Expressing on the left-hand side (LHS) of (14) in terms of according to (13), and substituting on the right-hand

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side (RHS) of (14) by sum in (15), the weighting coefficients of in the correctional current and the weighting coefficients of MBFs are mutually related as (16)

, , of the Obviously, the weighting coefficients MBFs are obtained by splitting the weighting coefficient of the correctional current. Different splitting strategies will be addressed later. After introducing new MBFs in the th iteration, the approxis represented as linear imate solution in the th iteration combination of all MBFs defined in this and previous iterations as (17)

are unknown coefficients that should be determined. where (Since the solution in the 0th iteration cannot be improved much , i.e., by adjusting these coefficients, it is adopted that .) in (17) by sum given in (15) the approximate Substituting solution is obtained in the form (18)

, , given by (20) into (21), After substituting and thus obtained expression for residua into (22), after simple is written in manipulations, the mean square residuum terms of unknown coefficients , , , as (23)

is inner product of linear operator applied to the th where MBF in the th iteration, , and the th MoM testing function, , i.e. (24) The unknown coefficients are determined by minimizing the . The residuum is minimized by immean square residuum posing condition that its first derivative with respect to each of , , , equals zero, i.e. coefficients

(25) After substituting given by (23) into (25) the determined system of linear equations in terms of unknown coefficients , , , is obtained in the form

After changing the order of two sums in (18) the approximate is represented as linear combination of original solution MoM basis functions multiplied by coefficients as (26) (19)

(20) Note that approximate solution given by (19) has the same form as approximate MoM solution given by (2), except that MoM weighting coefficients , , are replaced by weighting coefficients , . After , in (5) by coefficient replacing coefficient , , , the residua of these equations in the th iteration are obtained as (21) The mean square residuum after the th iterations is calculated as (22)

After the system is solved for unknown coefficients (e.g., by using Gaussian elimination or LU decomposition) the mean given by (23) is easily calculated. If this square residuum error is greater than some specified value, the new iteration is started. Otherwise, the iterative procedure is finished. . It is shown by numerical The order of this system is experiments that very good results are obtained in few iterations , if number of new MBFs per iteration is adopted to be comparable with square root of number of unknowns, i.e., . Note that in each iteration the solution is improved, approaching to the original MoM solution. Since MBFs in the 0th iteration are obtained by the PO technique, and improvement of MoM solution in each iteration is based on inclusion of additional MBFs obtained by the modified PO technique, the method is referred to as PO driven MoM. The proposed method is quite general and can be applied to various integral equations, and various basis and testing functions. Besides, the approximation of correctional PO currents by MoM basis functions can be done in different ways. In the next section the specific implementation of the proposed method will be elaborated.

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(31) where and are orders of approximation along and coare unknown coefficients, , , 1, are ordinates, , , are patch basis functions, edge basis functions, , and . The -component expansion is obtained by interchanging the coordinates and in expressions (30) and (31). Since the patch basis functions are equal to zero along the edges, which are on the way of the current flow, the continuity of currents along the junction of two bilinear surfaces is provided by coupling edge basis functions of the same type from different sides of the junction into doublets. , there is single doublet per such junction (For representing the rooftop basis function.) Set of doublets at all junctions and patch basis functions at all patches give the expansion of all currents induced over the body, the expansion whose general form is given by (2).

Fig. 1. Sketch of bilinear surface.

III. IMPLEMENTATION OF PO DRIVEN MoM In this paper the proposed method is applied to entire-domain Galerkin solution of electric field integral equation (EFIE), which is explained in details in [2] and [7]. In what follows, basic steps of this solution will be given. A. Solution of EFIE by Entire-Domain Galerkin Method The EFIE is obtained from boundary condition for tangential component of electric field at the surface of PEC body, i.e., . After expressing in the form of integral of currents , the EFIE is written as

(27) in (27) are represented in terms of The unknown currents known basis function , , in general form (2). , After applying the Galerkin test procedure to (27) ( ), and assuming that basis functions are real inner products (6) and (7) are obtained as (28)

The geometry of the PEC body is modeled by bilinear surfaces. This surface is described by parametric equation

(29) , 2, are position vectors of the four points, and where , and are local parametric coordinates, as shown in Fig. 1. The currents distributed over a bilinear surface are decomcoordinate system. In posed into two components in local particular, the -component is expanded as (30)

B. Construction of MBFs As explained in Section II-B the first step in construction of MBFs is approximation of correctional PO currents by MoM basis functions. The residuum of such approximation can be written as (32) The goal of the approximation is to minimize the residuum in some sense. In this paper the square error of the residuum (33) is minimized, by requiring that its derivatives with respect to coefficients , , are equal to zero, i.e. (34) After substituting (32) into (33) and (33) into (34) the system of linear equation is obtained in the form (35) Most often basis functions and do not overlap, so that above system is highly sparse. In spite of that, its solution can be relatively time consuming, requiring significant memory resources. To reduce resources needed for determination of these coefficients the system (35) is posed separately for each patch. Since doublet basis functions are shared by two patches, the for doublets are determined twice. weighting coefficients Hence in the final approximation of correctional PO currents such coefficient is taken with its arithmetic mean value. The integrals in (35) are calculated using double numerical integration based on Gauss-Legendre qaudrature. It is shown by numerical experiment that sufficient numbers of integration

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points along the and coordinates of a patch are and , which is in total points for the patch. Since number of basis functions per patch is approx, total number of points for the body is about imately . It is also shown that by far the largest part of simulation , time, which is needed for determination of coefficients , is used to calculate magnetic field in the integration points. As explained in Section II-B the second step in construction onto different MBFs. of MBFs is splitting coefficients MBF can be created by random choice of basis functions among all MoM basis functions. However, faster convergence of PO driven MoM is observed when basis functions belonging to one MBF are physically grouped, i.e., if MBF contains all basis functions defined over a set of mutually connected patches. In addition, the faster convergence is found if such set of patches does not contain internal sharp edges. Having all above in mind the following algorithm for grouping basis functions into MBFs is proposed. In the first step contours of sharp edges of an object are identified. It is considered that a sharp edge occurs at junction of two patches if the angle formed by their normal vectors is greater than some specified value (e.g., 30 degrees). In particular, if a contour is longer than some specified maximum (e.g., 5 wavelengths), it is subdivided into minimum number of contours not longer than allowed. Patches along one side of each contour are grouped into one macro-patch (MP), and patches along another side of the contour are grouped into another MP. The remaining patches are grouped into macro-patches (MPs) whose maximum dimension is smaller than some given value (e.g., 5 wavelengths). In each iteration new MBF is formed over each of MPs, from basis functions belonging to this MP. A basis function , which completely belongs to one MP, enters into its MBF with full . A basis function which is shared by two MPs enweight . A basis function ters into their MBFs with half of weight which does not belong to a MP enters into its MBF with zero weight. Since the number of basis functions shared by two MPs is much smaller than the number of basis functions that completely belong to one MP, the average number of basis functions . forming one MBF is approximately Finally, note that almost half of such obtained MPs are placed in the shadow region. Hence, MBFs defined over these MPs in the 0th iteration has 0th value and are omitted from matrix (26). Hence the final order of the matrix equation in the th iteration . is reduced to about C. Storage and Computational Cost (Memory and Analysis Time Requirements) Determination of initial approximation of unknown currents, given by (9), requires negligible resources when compared with improvement of the solution through iterative procedure. Note that in each iteration we need the original MoM matrix only for calculation of inner products , according to (24). It is seen from this equation that for calculation of these products for all , and , we need only the th row in original MoM matrix. It means that original MoM matrix can be calculated row by row in each iteration, with storing single

row at a time. In that sense the iterative procedure can be optimized either for speed (i.e., the simulation time is minimized) or for memory size (i.e., the required memory resources are minimized). In the first case the original MoM matrix is calculated once and stored, just before the iteration procedure started, so that by far the largest part of the required memory resources is used to store this matrix. In the second case the original MoM matrix is calculated in each iteration without storing, so that the required memory resources are mostly used to store the PO , and using driven MoM matrix. Having in mind that 8 bytes for one complex matrix element (single precision), the required memory resources in bytes for these two procedures, and another one optimized one optimized for speed, , are estimated to for memory size, (36) There are three time consuming steps performed in each iteration common for both cases: a) evaluation of the magnetic field in a grid of points over the body surface and construction of MBFs, b) calculation of elements in PO driven MoM matrix needed for this calculation, and c) and all inner products solution of the PO driven MoM system of linear equations (26). Let us estimate the total simulation time. The calculation time of original MoM matrix is given by , where is time needed for calculation of one matrix , where is element. It is found that time needed for one basic operation consisting of one summation and one multiplication. The evaluation time for magnetic , with time field in a grid of points is also proportional to needed for calculation of the magnetic field per point slightly shorter than . In the th iteration all inner products , , , are calculated, except those which cor. Since in average the number respond to omitted MBFs for of non-zero coefficients in (24) is , the evaluation time . in the th iteration is approximately st iteration is PO driven MoM matrix obtained in the submatrix of that obtained in the th iteration. So, the number of additional matrix elements calculated in the th iteration is . According to (26) the number of basic operations needed to calculate one such elthe total calculation ement is . Having in mind that . time needed in each iteration is comparable to Time needed to solve the PO driven MoM system (26) is . Having in mind that it is seen that this time increase with number of unknowns as and is much less time consuming than other two steps. Finally, the simulation time for the procedure optimized for the speed, , and the simulation time for the procedure optimized for the memory size, , is roughly estimated as (37) It is seen from (37) that procedure optimized for speed is only less than twice faster than procedure optimized for memory size. The difference is much smaller if the original MoM matrix is not stored in RAM, but in a hard disk. In that case the procedure optimized for speed uses the additional time to store the

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M=6

Fig. 2. PEC cube scatterer modeled by 7776 patches grouped into: a) macro-patches. b) macro-patches. c) macro-patches.

M = 54

M = 486

matrix and to read the matrix from the disk in each iteration. On the other side, it is seen from (36) that procedure optimized for times less memory than that memory size requires about optimized for speed. For example, if number of unknowns is and number of iterations larger than 100 thousands , than procedure optimized for memory is less than 10 size requires 1000 times less memory than that optimized for speed. Hence, in what follows we shall use the procedure optimized for memory size.

Fig. 3. Residuum for cube scatterer versus number of iterations for different number of macro-patches.

IV. NUMERICAL RESULTS The accuracy and efficiency of PO driven MoM are illustrated on two examples: 1) cube scatterer, and 2) airplane scatterer. As a measure of accuracy the normalized mean square residuum is used, which is defined as (38)

In what follows it will be shortly referred to as residuum. (Note that for weighting coefficients of original MoM basis functions set to zero the residuum is equal to one.) All calculations are performed on Intel Quad core PC with processor clock at 2.66 GHz, with 4 GB of RAM, and 1 TB of space at the hard disk. Calculation of original MoM matrix and of magnetic field in the set of integration points are performed by improved version of commercial software package WIPL-D Pro v8.0 [24]. All PO driven MoM results are obtained using the procedure optimized for memory size. A. Canonical Example—Cube placed in a vacuum, excited Consider PEC cube of side by -polarized plane wave incoming from direction given by angles and ( is measured from plane to axis). Each face of the cube is modeled by 36 by 36 square patches of equal size. The first-order approximation is used for all patches (i.e., rooftop basis functions), so that total number of . original MoM basis functions is For the first set of results obtained by PO driven MoM the , 6, 54, and 486 patches are uniformly grouped into MPs, as shown by different colors in Fig. 2. For the each MP consists of all MP consists of all patches, for each MP consists of 12 patches of one cube face, for each MP consists of 4 by 4 by 12 patches, and for patches.

=0

Fig. 4. Surface current distribution over the cube at the moment t , when the plane wave electric field in the coordinate origin varies as E t E ! , E = , obtained by PO, PO driven MoM (486 MPs), and MoM. a) PO solution. b) First iteration of PO driven MoM. c) Second iteration of PO driven MoM. d) MoM solution.

() =

cos( t)

= 1V m

Fig. 3 shows the residuum versus number of iterations , for different number of MPs. It is seen that in each next iteration the residuum decreases, and that this convergence is faster for larger number of MPs. To better understand how the residuum level is related to the solution accuracy let us consider the current distribution over the top and two back surfaces of the cube, as shown in Fig. 4 (i.e., , ). the cube is inspected from direction Magnitudes of the surface current density are shown over the using various colors, when the plane cube at the moment wave electric field in the coordinate origin varies as , . The results are given for the PO

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Fig. 5. RCS of PEC cube of side 4 in plane ' = 45 for third Iteration PO driven MoM solution, compared to PO and pure MoM solution.

Fig. 6. Three ways of grouping patches of cube scatterer into 216 macropatches: a) random, b) uniform, and c) edged.

solution (a), the first and second iteration of the PO driven MoM solution with 486 MPs (b, c), and the MoM solution (d). It is seen that there are no currents in the shadow region in the PO solution, as expected, and that currents obtained in the second iteration are very much alike to that obtained in the MoM solution. The colored representation of currents obtained in the third iteration can not by distinguished by eye from that obtained by the MoM solution. Note that the residuum obtained after the . third iteration using 486 MPs is about In addition, let us consider the radar cross section (RCS) in versus angle , obtained after dBs in incident plane 3 iterations, as shown in Fig. 5. This solution almost coincides with the pure MoM solution, while the PO solution shows much greater disagreement. From this and similar examples it is concluded that very accurate results for currents distribution and . It is RCS are obtained for residuum lower than also found that such accuracy is obtained in small number of itis comparable with erations, if the number of macro-patches square root of number of unknowns . For the second set of results obtained by PO driven MoM the using three different patches are grouped into ways of grouping: 1) random, with 36 randomly chosen patches for each MP, 2) uniform, with MPs consisted of 6 by 6 patches, and 3) edged, with MPs consisted of 2 by 2 patches in corners, of 2 by 8 patches along edges, and of 8 by 8 patches inside faces, as shown in Fig. 6. The corresponding residua, versus number of iterations , are shown in Fig. 7. It is seen that convergence of the results is faster if the uniform grouping is used instead of the

Fig. 7. Residuum for cube scatterer, versus number of iterations, for different ways of grouping patches into macro-patches (random, uniform, edged), compared with stochastic construction of MBFs.

random one. The fastest convergence is obtained by the edged grouping. If the weighting coefficients in MBFs (for the first and higher iterations) are obtained by stochastic choice, instead of using the modified PO technique, the fastest convergence is again obtained for edged grouping. However, the convergence of these results is very slow, as shown by curve “Stochastic” in Fig. 7, which proves effectiveness of using modified PO technique for construction of MBFs. From this and similar examples it is concluded that faster convergence is achieved if MPs consist of physically grouped patches, if MPs do not contain internal sharp edges, and if MPs along sharp edges are narrow. Based on such conclusions the method for automatic grouping is proposed in Section III-B. obtained by the conjuFig. 7 also shows the result for gate gradient (CG) iterative method applied for solution of the original MoM system of linear equations [2], [24]. (In what follows this method will be referred as CG MoM method.) It is seen that convergence of the results obtained by the PO driven MoM is much faster than that obtained by the CG MoM method. In particular, while edged grouping requires only 3 iterations for , it is shown that the CG MoM method residuum requires 20 iterations. It is also shown that by increasing the electrical size and complexity of the structure, the number of CG MoM iterations significantly increases, while in the case of PO driven MoM this number is practically kept constant by proper choice of MPs. In particular, by increasing the electrical size and complexity of the structure above some point, the original MoM matrix become too large to be stored in RAM. In that case it must be either: a) stored in a disk and then read in each iteration (out-of-core solution), or b) calculated in each iteration. In both cases duration of the CG iteration becomes comparable with duration of the PO driven MoM iteration. For such problems the PO driven MoM simulation is shown to be much shorter than the CG MoM simulation. It can be shown that out-of-core CG solution is slower than out-of-core direct solution [24]. In the next subsection we shall consider problems with MoM matrices which can not be stored in RAM of regular PCs and we will compare the total simulation time with that of out-of-core direct solution.

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Fig. 8. a) PEC airplane excited by circularly polarized wave modeled by 7445 patches. b) Grouping of patches of airplane scatterer into 315 macro-patches.

Fig. 9. Residuum for airplane scatterer versus number of iterations for different number of macro-patches.

Fig. 10. Simulation time for airplane scatterer versus number of iterations for different number of macro-patches.

B. Real-Life Example—Airplane long, placed along -axis Consider PEC airplane, about in Cartesian coordinate system, with wings spanned in plane, as shown in Fig. 8. The airplane is excited by circularly polarized plane wave incoming from direction given by

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Fig. 11. Residuum for airplane scatterer versus number of MBFs for different number of macro-patches.

= 135

Fig. 12. Mean error of airplane RCS in plane " Number labels along the curve are the iteration numbers.

versus residuum.

angles and ( is measured from plane to axis). The airplane is modeled by 7445 patches. The second-order approximation is used for almost all patches, so . Using that total number of MoM basis functions is the automated grouping algorithm, proposed in Section III-B, , 315, 643, 1099, and the patches are grouped into in Fig. 8). The numbers 1721 MPs (as shown for of MPs, which are at least by one part placed in the lit region, are 108, 197, 386, 621, and 922, respectively, representing the number of MBFs after the 0th iteration. and the simulaFigs. 9 and 10 show the residuum tion time, respectively, versus the number of iterations , for different numbers of macro-patches . As in the case of the cube scatterer, it is seen that in each iteration the residuum decreases, and that the convergence is faster for larger number of macro-patches . It is also seen that the simulation time almost linearly increases with the number of iterations and does not depend much on number of groups. Fig. 11 shows the residuum versus number of MBFs, for different numbers of MPs. It is seen that the fastest convergence for small number of MBFs is achieved with , and that by increasing the number of MBFs the faster convergence is obtained by larger number of MPs.

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Fig. 14. RCS of airplane 110 long in plane ' = 135 after first and fifth iteration of PO driven MoM solution compared to PO solution.

cross section (RCS) in dBs. The error is calculated with respect to the full MoM solution. Fig. 12 shows such error in plane degrees versus the residuum for . Very similar curves are obtained for other scatterers. The largest residuum is due to the PO solution and corresponding mean square error is about 5 dB. After the first iteration the error drops bellow 3 dB, after the fourth iteration the error drops to 1.5 dB, and after the 7th iteration the error drops to 1 dB. In particular, let us compare the results for RCS in plane degrees obtained by PO driven MoM with that obtained by the full MoM solution. Fig. 13 shows RCS versus for initial PO solution, and after the first, the fourth and the ninth iteration, respectively. It is seen that even the solution after the first iteration is acceptable from engineering point of view, whereas results after the ninth iteration can be considered very accurate. long (with same shape and Finally, PEC airplane, about excitation as that from Fig. 8) is modeled with 37 163 patches, . Note that 659 MPS of total 1453 grouped in MPS are in the shadow region, so that only 794 MBFs are created in the 0th iteration. The second-order approximation is used for almost all patches, so that total number of MoM basis func347. Results for RCS in plane detions is grees obtained by PO driven MoM after the first and the fifth iteration are compared with the PO solution in Fig. 14. As in previous cases it is seen that the first iteration gives major improvement comparing with the PO solution, while higher iterations enable fine correction. Using only 2247 MBFs the first iteration takes about 7.3 hrs. Higher iterations, from second to fifth, use 3700, 5153, 6606, and 8059 MBFs, respectively, and take 7.6, 8.0, 8.6, and 9.4 hrs, respectively. Note that full MoM solution based on direct out-of-core matrix solution is estimated to take about 15 days. Fig. 13. RCS of airplane 40 long in plane ' = 135 after 0, 1, 4, and 9 iterations of PO driven MoM solution, compared with MoM solution.

To better understand the relation between residuum and accuracy of final results consider the mean absolute error of radar

V. CONCLUSION The PO driven MoM technique presented in the paper enables MoM solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients. The reduced number of unknowns is based on construction of MBFs in each iteration

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using the PO and modified PO technique, so that they fit best to the given excitation. Convergence to the original MoM solution is provided by the request that unknown coefficients multiplying MBFs minimize the residuum of original MoM matrix equation. High accuracy is obtained in small number of iterations, if the number of macro-patches is comparable with square root of number of unknowns. Thus the number of unknowns is reduced for one to two orders of magnitude. If the proposed method is optimized for memory size, the duration of one iteration corresponds to double calculation of original MoM matrix, while required memory resources are mostly used to store the PO driven MoM matrix. For example, very usable results for airplane scatterer, which originally require 296 347 MoM basis functions, are obtained in one iteration using only 2247 MBFs. While the full MoM solution based on direct out-of-core matrix solution is estimated to take about 15 days, the solution based on MBFs is obtained in 7.3 hrs. Significant acceleration of the method can be achieved by exploiting fast methods for matrix fill and near field calculations, which is planed for future work. REFERENCES [1] R. F. Harrington, Field Computations by Moment Methods. New York: MacMillan, 1968. [2] B. M. Kolundzija and A. R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures. Norwood, MA: Artech House, 2002. [3] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [4] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, Oct. 1997. [5] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 329–342, Mar. 1997. [6] K. Donnepudi, J. M. Song, J. M. Jin, G. Kang, and W. C. Chew:, “A novel implementation of multilevel fast multipole algorithm for higher order Galerkin’s method,” IEEE Trans. Antennas Propag., vol. AP-48, pp. 1192–1197, Aug. 2000. [7] B. M. Kolundˇzija and B. D. Popovic´, “Entire-domain Galerkin method for analysis of metallic antennas and scatterers,” IEE Proc.-H, vol. 140, no. 1, pp. 1–10, Feb. 1993. [8] T. J. Kim and G. A. Thiele, “A hybrid diffraction technique-General theory and applications,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 888–897, Sep. 1982. [9] R. E. Hodges and Y. Rahmat-Samii, “An iterative current-based hybrid method for complex structures,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 265–276, Feb. 1997. [10] U. Jakobus and F. M. Landstorfer, “Improved PO-MM hybrid formulation for scattering from three-dimensional perfectly conducting bodies of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 43, no. 2, pp. 162–169, Feb. 1995. [11] E. Jørgensen, P. Meincke, and O. Breinbjerg, “A hybrid PO-higherorder hierarchical MoM formulation using curvilinear geometry modeling,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. Dig., Columbus, OH, Jun. 22–27, 2003, vol. 4, pp. 98–101. [12] M. Djordjevic´ and B. M. Notaroˇs, “Higher order hybrid method of moments-physical optics modeling technique for radiation and scattering from large perfectly conducting surfaces,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 800–813, Feb. 2005. [13] E. Suter and J. R. Mosig, “A subdomain multilevel approach for the efficient MoM analysis of large planar antennas,” Microw. Opt. Technol. Lett., vol. 26, no. 4, pp. 270–277, Aug. 2000. [14] I. Stevanovic´ and J. R. Mosig, “Using symmetries and equivalent moments in improving the efficiency of the subdomain multilevel approach,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 158–161, 2005.

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[15] L. Matekovits, G. Vecchi, G. Dassano, and M. Orefice, “Synthetic function analysis of large printed structures: The solution space sampling approach,” in Proc. IEEE AP-S 2001, Boston, MA, Jul. 2001, pp. 568–571. [16] L. Matekovits, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2509–2521, Sep. 2007. [17] E. Lucente, A. Monorchio, and R. Mittra, “An iteration-free MoM approach based on excitation independent characteristic basis functions for solving large multiscale electromagnetic scattering problems,” IEEE Trans. Antennas Propag., vol. 56, pp. 999–1007, Apr. 2008. [18] V. V. S. Prakash and R. Mittra, “Characteristic basis function method: A new technique for efficient solution of Method of Moments matrix equations,” Microw. Opt. Technol. Lett., vol. 36, no. 2, pp. 95–100, Jan. 2003. [19] E. Garsia, C. Delgado, I. G. Diego, and M. F. Catedra, “An iterative solution for electrically large problems combining the characteristic basis function method and the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2363–2371, Aug. 2008. [20] M. Degiorgi, G. Tiberi, A. Monorchio, G. Manara, and R. Mittra, “An SVD-based method for analyzing electromagnetic scattering from plates and faceted bodies using physical optics bases,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. Dig., Wash. DC, Jul. 3–8, 2005, vol. 1A, pp. 147–150. [21] M. S. Tasic´ and B. M. Kolundˇzija, “A PO driven iterative solution of MFIE for large scatterers,” in Proc. 7th Int. Conf. Papers, Telecommun. Modern Satellite, Cable and Broadcast. Serv., Niˇs, Serbia, Sep. 28–30, 2005, pp. 24–27. [22] M. S. Tasic´ and B. M. Kolundˇzija, “PO driven iterative least square solution of MFIE,” in Proc. IEEE/ACES Conf. on Wireless Commun. Appl. Commput. Electromag., Miami, FL, Mar. 12–16, 2006, pp. 470–475, (CD ROM Edition: s15p06.pdf). [23] M. S. Tasic´ and B. M. Kolundˇzija, “PO driven iterative Galerkin solution of field integral equations,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 4073–4076. [24] Software and User’s Manual. Belgrade, 2010, WIPL-D Pro v8.0, WIPL-D d.o.o.

Miodrag S. Tasic received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Belgrade, Belgrade, Serbia and Montenegro, in 1998 and 2004, respectively, where he is currently working toward the Ph.D. degree. Since 2000, he has been with the School of Electrical Engineering, University of Belgrade, as a Research and Teaching Assistant. His research interests are automatic segmentation of 3D surface models and numerical problems in electromagnetics.

Branko M. Kolundzija (M’92–F’05) was born in 1958 in Zenica, the former Yugoslavia. He received the B.Sc., M.Sc., and D.Sc., degrees from the University of Belgrade, Serbia, in 1981, 1986, and 1990, respectively. He joined the Faculty of Electrical Engineering, University of Belgrade, in 1981, where he is currently a Full Professor in electromagnetics and antennas and propagation. He is the author or coauthor of two monographs, a chapter in a monograph, three software packages, one textbook, 28 journal articles, and 103 articles at international conferences. He has held lectures and short courses in Ottawa, Lisbon, Urbana, Albuquerque, Syracuse, Monterrey, Stuttgart, New York, Copenhagen, Helsinki, Calgary, Zurich, Long Island, Colcata, Tokyo, Washington, and Yokohama. His research interests are numerical problems in electromagnetics, especially those applied to antennas and microwave components. He is a main architect of WIPL-D software packages.

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Implementation of Material Interface Conditions in the Radial Point Interpolation Meshless Method Yiqiang Yu, Member, IEEE, and Zhizhang Chen, Fellow, IEEE

Abstract—We propose a systematic approach to accurate imposition of material interface conditions for the meshless radial point interpolation method (RPIM). A new set of equations for updating fields at interfaces is derived. A piecewise polynomial and a modified radial basis function are proposed and applied to account for field discontinuities near the interfaces. In contrast to the previous work, the proposed approach incorporates material properties and reduces geometrical dependence of the interpolation function. Numerical results show that the proposed approach significantly improves the simulation accuracy of the RPIM at a cost of a small overhead in computational time. Index Terms—Finite-difference time-domain, meshless methods, radial point interpolation method (RPIM), time-domain finite-element method, time-domain modeling.

I. INTRODUCTION

T

HE meshless methods have recently gained attention in computational electromagnetic community as they offer a new approach to solutions of complex electromagnetic problems with irregular geometries. Unlike the conventional numerical methods, these meshless methods do not require predefined meshes or grids to establish a system of algebraic equations. Instead, a set of nodes scattered within a problem domain and a set of nodes scattered on boundaries of the domain are defined to represent the problem. Theoretically, these nodes can be randomly distributed spatially; it therefore unleashes the long-existing dependence of numerical accuracy on connectivity laws of grid nodes and on shape and dimension of numerical cells in classical numerical techniques. This unique property renders the meshless methods with several advantages over classical grid-based methods, such as conformal modeling of arbitrary boundaries, capabilities of the multi-scale solutions, and adapting ability by adding or removing nodes during the simulation.

Manuscript received September 27, 2010; revised December 10, 2010; accepted December 16, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. The work was supported in part by the National Natural Science Foundation of China (NSFC) fund under contract 61061003 and Natural Science and Engineering Research Council of Canada under contract 155230-07. Y. Yu is with the Department of Communication Engineering, East China Jiaotong University, Nanchang, China and also with the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, Canada (e-mail: [email protected]). Z. Chen is with the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, 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.2011.2158969

A number of meshless techniques have been reported so far; they include the element free Galerkin method [1], the moving least square reproducing kernel method [2], the smoothed particle electromagnetic method [3] and the radial point interpolation method (RPIM) [4]–[8]. Among them, the RPIM shows advantages in its consistency and relative simplicity of implementation. With a proper implementation of RPIM, the conformal and multi-scale modeling of arbitrary geometries can be achieved with ease. In addition, since RPIM interpolation functions possess the property of the Kronecker delta function, imposition of boundary conditions of perfect electric and magnetic conductors can be readily carried out. However, this is not true when material interfaces are encountered. This is because the radial basis functions and their derivatives used for construction of the RPIM interpolation functions are always continuous; as a result, they introduce difficulties in modeling fields at material interfaces since the fields can be discontinuous across the interfaces. This has become one of the major hurdles that prevent the meshless method from being effectively and accurately applied to complex structures involving materials and interfaces. To deal with the problem, an elliptical basis function was used in addition to the radial basis functions to model fields near material interfaces [9]; however, such a strategy does not incorporate material properties into the formulations and therefore can be very much geometry dependent. Alternatively, a correction term was introduced in [10] to increase simulation accuracy of RPIM in modeling problems with dielectric material interfaces; nevertheless, the parameters of the correction term are empirical, presenting technical difficulties for general applications. In this work, we propose a systematic approach for solving the above-mentioned difficulties in handling material interfacing conditions. A new set of equations for updating field quantities near interfaces are derived. A piecewise polynomial and a modified radial basis function are proposed and applied in conjunction with an integration scheme to account for discontinuities of the fields near interfaces. The material properties are incorporated in the formulations, and geometric dependence of the boundary treatment is removed. The effectiveness of the proposed approach for improving the simulation accuracy is evaluated through numerical experiments, and the computational overhead of the proposed scheme is also presented. In the following sections, we will first briefly review the RPIM formulation and its application to two-dimensional transverse-magnetic waves without loss of generality. Then, we introduce the special treatments of the RPIM formulations for nodes at a dielectric interface. Numerical examples are finally provided to demonstrate the effectiveness of the proposed treatments.

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values of the fields at the nodes to the unknown interpolation coefficients can then be obtained. With further enforcement of the constraint of the unique solution [11], coefficient vectors and can be related to the field values at the nodes, as given by (6) (7) Fig. 1. Support domain of the point of interest

X and its surrounding nodes.

II. THE RPIM METHOD APPLIED TO THE TWO-DIMENSIONAL TRANSVERSE MAGNETIC (TM) WAVES A. Construction of RPIM Shape Function The RPIM with polynomial reproduction is a node-based numerical technique that fundamentally relies on local interpolation. It interpolates a field variable at the point of interest by its values at surrounding nodes that fall into the support domain of (see Fig. 1); it assumes that there are no or negligible contributions from the nodes that fall outside the domain. For simplicity, consider the two dimensional case. The interpolation function can be written as: (1) Here is the point of interest at which field variis the radial basis function, able is to be interpolated. is the monomial basis function, and and are the associated interpolation coefficients. is the number of points in the support domain of that determines the number of radial is the number of basis functions used for interpolation (1). the monomial basis functions used; inclusion of monomial basis ensures the unique interpolation of linear function in fields. The radial basis function in a Gaussian form is used throughout this work. It is an exponential function of the distance from a node with shape parameter to control its decaying rate. It is expressed as (2) where is the coordinates of the th node surrounding the point of interest . is the radius that defines the area of the support domain. of linear monomial basis functions in Three terms are used to construct the polynomial basis. the form of Thus (1) can be written in vector form as (3) where (4) and (5) Enforcing (1) to pass through each scattered node in the support domain of point , a matrix equation that relates the true

is the vector holding the field values at the nodes Here is the moment matrix in the support domain, and matrix evaluated at the nodes. associated with the radial basis With (6) and (7), interpolation (1) can be expressed as (8) is a vector of where shape functions that collect shape functions related to the nodes in the support domain of . Since is a constant matrix in the support domain, the spatial derivative of shape functions can be analytically obtained as (9) where is the element of matrix

element of matrix .

, and

is the

B. Application of RPIM to Time-Dependant Maxwell’s Equations Implementation of the RPIM meshless technique for the solutions of Maxwell’s equations requires a set of electric field nodes (E-nodes) as well as a set of magnetic field nodes (H-nodes) to be defined and distributed over a problem domain. Because of the coupling nature of electric and magnetic field components, the E-nodes and H-nodes are defined in a way that each E-node is surrounded by H-nodes and vice versa. More specifically, for problems with regular node distributions, E-nodes and H-nodes are defined in a manner similar to the point-matched time-domain finite-element method [12]; For problems with non-uniform node distributions, our strategy is to start with definition of the E-nodes at a set of specific positions and then applies a Voronoi decomposition to find the H-nodes at vertices of Voronio cells that are composed by the E-nodes [13]. Suppose that a TM wave propagates in the -direction. The and , and the nonnon-zero electric fields considered are . All three non-zero field components zero magnetic field is do not vary in the direction. Fig. 2 depicts the geometry of the case under study. It is a wave-guided structure filled with and material 2 two dielectric materials, material 1 of ; the structure is terminated with a PML region at the of right end. The dielectrics are assumed to be linear, lossless, nonmagnetic and frequency-independent. The dielectric interface is slanted across the two perfect electric side walls with an angle . The time-dependant Maxwell’s equations for the TM wave are given by

(10)

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i) Place both the E- and H-nodes along the interface in an interlaced manner; ii) At each node at the interface, set up a local coordinate system - with the node position being the origin, the direction being normal to the interface and the direction being tangential/parallel to the interface. With the new coordinate system, the electric fields at the interface can be decomposed into the tangential electric field and the normal electric field flux density . The permittivity can be conveniently expressed as: (17) Fig. 2. TM wave incident onto the dielectric interface.

Substitution of electric fields, and , into Maxwell’s equations (1)–(3) in the new local coordinate system then reads: (11) (12) For nodes that are placed away from the interface, the same procedures as those in the standard RPIM [4], [5] are applied to (11) and (12) for formulating the equations that update the field quantities. Consequently, a set of leap-frog RPIM equations, where electric fields and magnetic fields are half a time-step staggered from each other, are obtained as follows: (13) (14)

(18) (19) (20) Since and are continuous across the interface according to (16), the continuous radial basis function used in the standard RPIM can now be applied to them at the nodes at the interface in their respective support domains; there and are expanded based on interpolation (1) in terms of the and . The interface boundary conditions new coordinates are thus weakly enforced. and into By substituting the interpolations of (18), (19) and (20), we can obtain a new set of equations for updating the fields at the interface. (21)

(15) However, at the dielectric interface, the standard RPIM formulations cannot be directly applied and a special treatment needs to be developed in accordance with the interface boundary conditions. The treatment is proposed and described in the next section.

(22)

III. THE PROPOSED SPECIAL BOUNDARY TREATMENT AT THE DIELECTRIC INTERFACE

(23)

Based on electromagnetic theory, the boundary conditions at and material the dielectric interface between material 1 2 are: (16) where represents the field component tangential to the interface, represents the field component normal to the interface, and is electric flux density. In order to fulfill the interface boundary conditions, we propose the following procedure to handle the field quantities at the interface (see Fig. 2).

IV. FURTHER IMPROVEMENTS WITH AN MODIFICATION TO THE RADIAL BASIS FUNCTIONS Although a special boundary treatment for a dielectric interface presented in the last section resolves the field discontinuity problem, it has another issue that is not addressed: discontinuity field and field across the of the normal derivatives of both interface due to change of physical properties of materials. Such a discontinuity cannot be accurately accounted for by plain use of the interpolation functions of the standard RPIM; this is because derivatives of the radial basis functions used to construct the interpolation function are continuous across the interface. It

YU AND CHEN: IMPLEMENTATION OF MATERIAL INTERFACE CONDITIONS IN THE RPIM METHOD

is observed in [14] that use of continuous functions to interpolate a non-continuous quantity will unnecessarily smooth solutions by introducing unphysical oscillations near discontinuity points. In order to tackle this problem and to precisely model fields near the interfaces, new degrees of freedom that incorporate changes of the normal derivatives of field quantities have to be of RPIM introduced into the field interpolation function in (1). We examined a series of radial basis functions [11] with various possible modifications by following the guideline given in [14]. Mathematical expressions of suitable basis functions are finalized through a sequence of numerical experiments. Here two approaches are proposed along with an integration scheme. The first approach is to apply a piecewise polynomial function and the second approach is to apply a modified radial basis function. The details are described in the next two subsections. A. The Piecewise Polynomial Function as the Monomial Basis For the nodes with their support domains extending over the dielectric interface, we propose the monomial basis to be in the form of and use it to replace the monomial basis (5) to construct a piecewise like polynomial basis. The spatial derivatives of the new monomial basis functions are given by (24) and (25) As a result, the normal derivatives of the field variables expressed by the new interpolation function can be discontinuous across the interface whereas the tangential derivatives remain continuous across the interface. B. The Modified Radial Basis Function In the second approach, we propose a new radial basis function that is modified from the original radial basis function (2). It is expressed as (26)

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Fig. 3. Radial basis functions: the left figure is the original and the right is the modified radial basis.

where . This parameter is dependent on contrast ratio of the dielectric constants between the two dielectric regions. The spatial derivatives of the proposed radial basis function (26) are shown in (27)-(28) at the bottom of the page. Note that in the above basis function, contrast ratio of the two dielectric regions contributes to the decaying rate of the function along the direction. When the contrast ratio and the proposed basis function (26) returns to the original basis function (2). Fig. 3 plots the original radial basis function and the proposed , and . modified radial basis function with As can be seen, the derivative with the original basis function whereas the modified basis is continuously smooth at function is not. C. Integration Scheme Applied to the Nodes at the Interface Mathematically, the modified basis functions will introduce new degrees of freedom to the field interpolation function to account for changes of the normal derivatives of the fields at the dielectric interface. In practice, however, a plain application of the modified functions to (21)–(23) does not guarantee the solution improvements. This is due to the fact that in the RPIM, the single-point testing scheme (with Dirac delta function) is applied to all the nodes in a problem domain to derive the updating equations. As a result, the interpolation function is always evaluated at the node that lies in the geometrical center of its support domain. For a node at the material interface where the fields become discontinuous, the Dirac delta testing strategy may not

(27)

(28)

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work well; it makes the performance of the constructed interpolation function highly dependent on geometrical shapes of the interface and the position of the nodes at the interface. In order to solve this problem, in this paper, we propose an integration scheme to replace the single point testing scheme and derive the new RPIM equations for updating field quantities at the nodes at the interface. and in (18) are continuous across Since both the interface, the single-point testing scheme can still be applied, . However, for and (21) remains unchanged for updating (19) and (20), we apply the following process: first, the RPIM interpolation function constructed with the proposed modified basis functions are substituted into (19) and (20); then the integration is applied to both sides of the resultant equations over the support domain of the node; it yields

Fig. 4. Nodal distribution over the problem domain of TM wave with normal incidence.

(29)

(30) In deriving above equations, the following approximations are used

Here is the area of the support domain of node and it is computed as , where is the radius of support domain. A local cylindrical coordinate system centered at node and a trapezoidal integration rule are employed to compute the surface integrals in (29) and (30). Take (29), for instance, the surface integral is evaluated as

Fig. 5. Transmission coefficients of normal incidence at various permittivity ratios: simulated vs. theoretical.

presented. Both normal and oblique incidence of a wave onto an material interface were simulated. The effectiveness of the proposed schemes is evaluated through comparison of the transmission coefficients simulated at the observation point (shown in Fig. 2) with analytical solutions. Numerical errors are assessed and quantified as (32) where is the numerically computed transmission coeffiis the analytical transmission coefficient. cient and Numerical results simulated with the standard RPIM of regular radial basis functions and the FDTD based on orthogonal grids are also included for comparison purposes. A. Normal Incidence

(31) is a function that is transformed from the intewhere grand in the left side of (31) into the local cylindrical coordinate system. (29) and (30), together with (21), comprise the updating equations for field variables at nodes at the interface. V. NUMERICAL EXPERIMENTS In this section, a number of numerical experiments in modeling wave transmission between two dielectric materials are

The first numerical example investigated was a transversemagnetic (TM) wave normally incident on the interface of two in Fig. 2). Permitdielectric regions (i.e., is set to be tivity ratio between two materials was varied from 1.2 to 30. Fig. 4 shows the irregular distribution of the E- and H- nodes near the material interface which is twice as dense as the node distributions elsewhere. The largest distance between any two at . adjacent nodes is Fig. 5 presents the transmission coefficients computed with different methods, whereas Fig. 6 gives the associated numerical errors. It is clearly seen from Figs. 5 and 6 that the RPIM with the proposed modified radial basis significantly improves the simulation accuracy: an overall reduction of 10 dB in numerical

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Fig. 7. Nodal distribution over the problem domain of TM wave with oblique incidence (the left half including the material interface is shown). Fig. 6. Numerical errors of simulated transmission coefficients for normal incidence.

errors is observed when the permittivity ratio is over 15 as compared with the RPIM with the conventional radial basis. The RPIM with the proposed piecewise polynomial only (without the proposed improved scheme) yields significant decrease in ; however, for the permittivity numerical errors for contrast less than 2, it actually has the increased error. The inconsistency of the performance of the RPIM with piecewise polynomial might be explained from (24): although the piecewise polynomial accounts for the discontinuous normal derivatives of the interpolation function at the interface, it does not incorporate material properties at the both sides of the interface; in other words, it does not account for the degree of stepping change of dielectric constants across the material interface.

Fig. 8. Numerical errors of simulated transmission coefficients for angular incidence.

B. Oblique Incidence To further assess effectiveness of the proposed approach to modeling fields at the dielectric interface and to demonstrate the advantage of RPIM in conformal modeling of an interface or boundary with arbitrary slant angles, the incidence of the TM wave was set obliquely by varying from 0 to 30 . The perbetween two materials is set to be 20. mittivity ratio Fig. 7 illustrates the nodal placement over the left half of the , where the irregular nodal distriproblem domain for bution had to be employed due to the slanted material interface. , whereas the largest distance The average nodal spacing is and smallest distance is between two adjacent nodes is . Fig. 8 depicts the numerical errors of simulated transmission coefficients. The solutions obtained with the conventional FDTD of the rectangular grids are also included for comparison. To reduce the staircase effect in the FDTD modeling, a fine disis used to produce reliable FDTD results. cretization of As can be observed from Fig. 8, the RPIM with the modified radial basisfunction significantly reduces the numerical errors of the simulation in comparison to the standard RPIM with the conventional radial basis; however, both RPIMs outperform the FDTD method. C. TM Wave Propagation in the Presence of Dielectric Rod The third example computed is the TM wave incident onto an infinitely-long circular dielectric rod. As depicted in Fig. 9, the TM wave was excited with a line current placed along the -direction at the far left end of the problem domain, and the

Fig. 9. Geometry of the TM plane-wave propagation in the presence of dielectric rod.

polarization of the incident TM wave is set to be perpendicular to the axis of the rod. The time-domain profile of the current density along the line current is a ramped sinusoidal function. A line of observation points are placed at the right side of the problem domain to monitor the change of electric field due to the presence of the dielectric rod. The entire problem domain is enclosed by a PML frame. The electrical size of the problem domain and the radius of the rod given in Fig. 9 are scaled with respect to the free space wavelength at 3 GHz. Fig. 10 shows the nodal distribution over the lower left quadrant of the problem domain, where the surface of the rod is precisely modeled with a set of E-field nodes and a set of Hnodes placed in an interlaced manner; such a nodal discretizawith respect to tion yields an average nodal spacing of the wavelength inside the dielectric rod at 3 GHz.

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Fig. 10. Nodal distribution over the problem domain of TM wave propagation in the presence of dielectric rod with " (the lower left quadrant is shown).

=9

Fig. 12. Steady-state electric field distribution when a TM wave is incident onto a dielectric rod; top figure: the field distribution without the dielectric rod; middle figure: field distributions in the presence of a dielectric rod of " ; bottom figure: field distributions in the presence of a dielectric rod of " .

=2 =9

TABLE I COMPUTATIONAL COST FOR DIELECTRIC ROD "

( = 9)

Fig. tion

11. Steady-state E field values at a line of observapoints. (a): absolute values; (b): difference in dB, where jE 0E j=E .

Di = 20log (

)

The nodal discretization with lower resolution is used for the RPIM computation of dielectric rod with lower dielectric con. The time-domain profiles of the electric field stant at the observation points are recorded and Fourier-transformed to compute the steady-state E-field values. Fig. 11 plots the comcomponent of the steady-state electric field at the line puted of the observation points with the presence of dielectric rod of various dielectric constants. The field values in the absence of rod are given as reference. The results computed with the conventional FDTD with very fine discretization are also included for comparison. The results from the proposed RPIM show good agreement with ones solved with the FDTD method of fine spatial discretization. Fig. 12 presents the graphical displays of the electric field distribution over the problem domain with and without the dielectric rod; it shows the expected scattering phenomenon.

D. Computational Expenditure In the proposed RPIM scheme, to handle discontinuities of normal derivatives of fields at a material interface, integrations are required for updating field values at the nodes at the interface; they incur some but small additional computational time in comparisons with the standard RPIM. Table I lists the computational expenditures of the RPIM, both with and without integration. The computational time of the conventional FDTD is also included for comparison. As can be seen, the proposed scheme does not increase memory requirement but small extra computational time; such an extra cost is commendable in light of the significant improvement in the simulation accuracy. If the same accuracy level is required, in comparison with the FDTD, the RPIM requires 50% less CPU time.

YU AND CHEN: IMPLEMENTATION OF MATERIAL INTERFACE CONDITIONS IN THE RPIM METHOD

VI. CONCLUSION A systematic approach to tackling the difficulties of the Radial Point Interpolation Meshless Method on accurate imposition of interface conditions has been presented. A new set of equations are derived for updating field components at material interfaces. Two new basis functions are proposed and applied in conjunction with an integration scheme for the treatment of field discontinuities at dielectric interfaces. Numerical experiments show that the proposed scheme significantly improves the simulation accuracy of the RPIM with little extra overhead in computational time. Of the two basis functions proposed, the modified radial basis function appears to be more attractive than the piecewise polynomial for its consistency and better performance. REFERENCES [1] V. Cingoski, N. Miyamoto, and H. Yamashita, “Element-free Galerkin method for electromagnetic field computations,” IEEE Trans. Magn., vol. 34, no. 5, pp. 3236–3239, 1998. [2] S. A. Viana and R. C. Mesquita, “Moving least square reproducing Kernel method for electromagnetic field computation,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1372–1375, 1999. [3] G. Ala, E. Francomano, A. Tortorici, E. Toscano, and F. Viola, “Smoothed particle electromagnetics: A mesh-free solver for transients,” J. Comput. Appl. Math., vol. 191, no. 2, pp. 194–205, 2006. [4] J. G. Wang and G. R. Liu, “A point interpolation meshless method based on radial basis functions,” Int. J. Numer. Methods Eng., 2001b. [5] T. Kaufmann, C. Fumeaux, and R. Vahldieck, “The meshless radial point interpolation method for time-domain electromagnetics,” in Proc. MTT-S Int. Microwave Symp., Atlanta, GA, Jun. 15–20, 2008, pp. 61–64. [6] S. J. Lai, B. Z. Wang, and Y. Duan, “Meshless radial basis function method for transient electromagnetic computations,” IEEE Trans. Magn., vol. 44, no. 10, pp. 2288–2295, 2008. [7] Y. Yu and Z. Chen, “Towards the development of unconditionally stable time-domain meshless numerical methods,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 3, pp. 578–586, 2010. [8] Y. Yu and Z. Chen, “A three-dimensional radial point interpolation method for meshless time-domain modeling,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 8, pp. 2015–2020, 2009. [9] R. Gordon and W. Hutchcraft, “Using elliptical basis functions in a meshless method to determine electromagnetic fields near material interface,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 3592–3595. [10] T. Kaufmann, T. Merz, C. Fumeaux, and R. Vahldieck, “Modeling of dielectric material interfaces for the radial point interpolation time-domain method,” presented at the MTT-S Int. Microwave Symp., Boston, Jun. 7–12, 2009.

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[11] M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Boundary Elements, vol. 23, pp. 285–296, 1999. [12] J. Lee, R. Lee, and A. Cangellaris, “Time-domain finite-element methods,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 430–442, Mar. 1997. [13] H. Alt and O. Schwarzkopf, “The Voronoi diagram of curved objects,” in Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 89–97. [14] B. Fornberg, T. A. Driscoll, G. Wright, and R. Charles, “Observations on the behavior of radial basis function approximations near boundaries,” Comput. Math. Applicat., vol. 43, no. 3–5, pp. 473–490, 2002. Yiqiang Yu (M’07) received the M.Sc. degree (with distinction) in communication systems and the Ph.D. degree in microwave communications engineering from Swansea University, U.K., in 2003 and 2007, respectively. He is now an Associate Professor with the East China Jiatong University. He is also a Research Fellow with the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, Canada. His primary interest is in the application of computational electromagnetics, in particular the use of finite-difference methods, method of moments, and fast multipole methods in both the time and frequency domains. His interests also include RF/microwave components design, antennas design and measurement, EMI/EMC analysis and testing, and iterative solvers and preconditioning techniques for large-scale matrix computation. Dr. Yu was a recipient of the Overseas Research Scholarship, awarded from the U.K. Overseas Research Award Scheme during 2004–2007.

Zhizhang (David) Chen (S’92–M’92–SM’96) received the Ph.D. degree from the University of Ottawa, Ottawa, ON, Canada, in 1992. From Jan. to August of 1993, he was an NSERC Postdoctoral Fellow with the ECE Department of McGill University, Montreal, Canada. In 1993, he joined Dalhousie University, Halifax, Canada, where he is presently a full Professor and the Killam Chair in Wireless Technology. He has authored and coauthored over 155 journal and conference papers in computational electromagnetics and RF/microwave electronics. He was one of the originators in developing new numerical algorithms (including ADI-FDTD method) and in designing new classes of compact RF front-end circuits for wireless communications. His current research interests include numerical modeling and simulation, RF/microwave electronics, smart antennas, and wireless transceiving technology and applications. Dr. Chen received the 2005 Nova Scotia Engineering Award, a 2006 Dalhousie graduate teaching award, the 2006 ECE Professor of the Year award, and the 2007 Faculty of Engineering Research Award from Dalhousie University.

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Improving the Accuracy of FDTD Approximations to Tangential Components of the Coupled Electric and Magnetic Fields at a Material Interface Tim M. Millington, Student Member, IEEE, and Nigel J. Cassidy

Abstract—The finite-difference time-domain (FDTD) method is one of the most popular tools used for modeling important electromagnetic wave propagation and scattering problems. Of the many variations on the basic formulation, conventional orthogonal schemes remain prevalent because of their high level of accuracy and relative ease of implementation. Inaccuracies do persist, however, and in this paper a simple, computationally efficient remedy to an error that occurs when simulating waves that impinge on a material interface is identified. The detail of when and how the error occurs is illustrated using a simplistic interface scenario, with an analytic electric field being used to drive a single FDTD magnetic field update step. The resulting FDTD approximations are compared to the analytic solution to reveal the extent of the error. The proposed modification is then introduced and its remedial effect shown by repeating the above comparison using the modified FDTD equations. The broader effects of the proposed modification are demonstrated using a practical, 3D, full-wave FDTD forward modeling tool, taking scenario examples from the application area of ground penetrating radar (GPR). It is concluded that the overall accuracy of conventional, orthogonal FDTD schemes is improved, without unduly increasing their computational burden or algorithmic complexity. Index Terms—Error correction coding, finite-difference timedomain (FDTD) methods, ground penetrating radar, interface phenomena.

I. INTRODUCTION

S

INCE the publication of Yee’s seminal 1966 paper [1], the finite-difference time-domain (FDTD) method has grown in popularity as a means of modeling electromagnetic wave propagation. As computer technology advances, the list of application areas to which modeling can be applied grows, with the calculation of specific absorption rates [2] and ground penetrating radar simulations [3]–[5] being among the examples. The accuracy and reliability of the models, therefore, becomes ever more important. This paper discusses a numerical error that occurs in the FDTD calculation of the tangential component of the magnetic field when a wave impinges on a dielectric interface at non-normal angles of incidence and the electric field is Manuscript received October 19, 2010; revised December 03, 2010; accepted December 10, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported the U.K. Engineering and Physical Sciences Research Council under grant EP/004032/1. The authors are with the University of Keele, Keele, Staffordshire ST5 5BG, U.K. (e-mail: [email protected]; [email protected]. uk). Digital Object Identifier 10.1109/TAP.2011.2158975

polarized in the plane of incidence. Although there already exists a significant body of research relating to the accuracy of O(2,2) and higher order FDTD methods at material interfaces [6]–[12], the approach to dealing with the error described here differs from any in the literature in a way that has significance for inversion algorithms such as that described in [13], when accurate field information is required local to interfaces. First, a simple test scenario is described, including a definition of the analytic solutions for the electric fields and also the partial spatial derivatives thereof. These solutions are then used to drive the magnetic field update step of a typical FDTD implementation, this being an O(2,2) finite-difference approximations to Maxwell’s equation for Faraday’s law. The resulting approximations are compared to the analytic solution in order to highlight the error. This comparison is followed with a statement of the proposed modification and then, by repeating the above comparison using modified formulations, an analysis of its remedial effects. The investigation is completed by observing the effects of the proposed modification on two full-wave, 3D, ground penetrating radar (GPR) simulation examples, executed using both O(2,2) and O(2,4) formulations, with scenarios designed to isolate waves in which the electric field is polarized a) perpendicular and b) parallel to, the plane of incidence (only the second case should show any effect). These scenarios are of particular interest because of the growth in the use of GPR for borehole and in-pipe applications, in which the field polarizations are typically parallel and the angles of incidence far from normal. In addition, the accuracy of GPR simulations is becoming increasingly important because of their use in driving inversion and interpretation tools.

II. TEST SCENARIO DESCRIPTION A plane sinusoidal electromagnetic wave of angular frequency, , having the electric field vector linearly polarized in the x-y plane, impinges on a planar dielectric interface which . The dielectric is aligned in the y-z plane and located at and . permittivities are First, expressions for the wave numbers and on either side of the interface are established. The velocity of electromagnetic waves in a dielectric medium is given by

0018-926X/$26.00 © 2011 IEEE

(1)

MILLINGTON AND CASSIDY: IMPROVING THE ACCURACY OF FDTD APPROXIMATIONS TO TANGENTIAL COMPONENTS

being the magnetic permeability and the dielectric permittivity of the medium. Then, from the definition of wave number, , (1) may be re-written as

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It is now possible to derive expressions for the magnitudes of the x- and y-directed vector components of the incident, reflected and transmitted electric fields on both sides of the interface:

which may be combined and simplified to yield (2) Next, expressions for the magnitude of the electric field are given. If the incident wave travels in a direction with radians, the magnitude angle of incidence between 0 and of the time-varying electric field associated with the incident wave, as a function of position and time, assuming unit amplitude, is given as

(11) From these the required partial derivatives may be derived:

(3) From Fresnel’s equations [14] for the case of parallel polarization, the ratio of the amplitude of the reflected wave to that of the incident wave is then given by: (4) and the ratio of the amplitude of the transmitted wave to that of the incident wave by (5) where, [14]:

, the angle of refraction, is derived using Snell’s law (12)

(6) The magnitudes of the reflected and transmitted waves, as functions of position and time, are therefore given as (7)

Finally, the expressions (12) are substituted into (10), applying the principle of superposition where necessary, and simand choosing plifying by placing the interface at to describe the point at which the expressions are evaluated. and On the incident side, with , arbitrarily close to the interface, the result is that

(8) respectively. The analytic solution for the temporal derivative of the magnetic field, against which the FDTD approximations will be compared, may now be derived. Maxwell’s equation for Faraday’s law, expressed in differential form [14], assuming no magnetic sources and “idealized” materials (linear, isotropic, loss-less and non-dispersive), may written as

(13) whilst on the transmission side, with the result is that

and

(9) Since the given incident wave is linearly polarized in the x-y plane, the magnetic field will be linearly polarized in the z-direction and only the z-directed vector component of (10) needs be considered: (10)

(14) It can be shown that the expressions derived from either side , as is required for the of the interface are equivalent when continuity of the (tangential) z-directed component of the magnetic field [14]. With the additional observation that the terms in

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Crucially, since the normal, x-directed component is discontinuous at the interface, a choice must be made as to whether to , as if approaching use the limits of the expressions (11) as from either the incident side or the transmission . From the incident side, using the term side by way of example, the formulation is

(18) whilst on the transmission side, for the same term, the formulation is Fig. 1. Test scenario schematic. A plane incident wave, with the electric field ; direction, impinging polarized in the plane of incidence, travels in the on an x-normal planar interface, with angle of incidence  . The interface is " , resulting in characterized as a discontinuity in dielectric permittivity, " wave numbers k and k on either side of the interface.

+x 0y

:

the above expressions are continuous, it may be concluded that both expressions are valid at the interface. Equations (11), (12), (13) and (14) provide the analytic solution against which the FDTD approximations may be compared. III. STANDARD FDTD APPROXIMATIONS In this section, typical O(2,2) accurate approximations to the differential form of Maxwell’s equation for Faraday’s Law, which relates the temporal rate of change in the magnetic flux density to the spatial rate of change of the electric field, are introduced. Assuming the model materials are linear, isotropic, loss-less and non-dispersive, the approximations for the temporal rate of change in the magnetic field are formulated (using the notation of [15]) as

(15) Simplicity is maintained by aligning the standard, cubic, orthogonal Yee cell grid [1] in a manner consistent with the co-oris lodinate system of Fig. 1 and positioning it so that . cated exactly on the material interface, i.e., at The assumed polarization of the incident wave ensures the xand y-directed components of the magnetic field remain at 0. The terms on the right hand side of (15) are replaced with expressions obtained from the analytic electric field (11). The tangential, y-directed components, on the incident side of the interface, is

(16) and on the transmission side is

(17)

(19) At angles of incidence away from the normal, the alternative substitutions, (18) and (19), give rise to different evaluations of (15) at the interface, with neither corresponding to the results obtained from the analytic solution, that being the limit of both (13) and (14). Error in the successive evaluations of the z-directed components of the magnetic field would obviously follow when iterating through the time-stepping FDTD algorithm. Fig. 2 shows plots of our analytic solution and O(2,2) approxfor angles of incidence 0 , 30 and imations to 60 . The interface was characterized by the ratio and the results obtained with FDTD cell sizes . At normal incidence, it can be seen that the error reduces to the “small” phase error typical of conventional FDTD schemes [7], which produces a numerical dispersion artifact in full simulations. At angles of incidence away from normal, however, a significant magnitude error is introduced. The differences in amplitude between the FDTD approximations and the analytic solution, as a percentage of the analytic solution, are plotted against angle of incidence in Fig. 3. These indicate that, for this permittivity ratio, the error can reach almost 35%. Importantly, comparison of Fig. 3(a) and (b) reveals that the error on an individual cell specific level does not diminish with a reduction in cell size. Fig. 3(b) repeats the plot of Fig. 3(a) . It with the FDTD cell sizes reduced to should be noted, however, that the degree to which a cell specific error translates to the grid as a whole will depend of the number of cells per wavelength of propagating wave so that a 10% error in a single cell calculation, in a scheme with nominally ten cells per wavelength, will dilute to circa 1% error over a wavelength. For higher order schemes, devised to allow a coarser grid and thereby reduce computational load, this diluting effect is lessened, and for a typical O(2,4) scheme, allowing half that number of cells per wavelength, the overall error is doubled. Fig. 4 illustrates the maximum percentage error plotted against the dielectric permittivity ratio that characterizes the interface, and shows that significant error is introduced at relatively low contrasts. This section is concluded by remarking that, via the FDTD algorithm, numerical error introduced at any grid point is propagated through the entire grid. Therefore, an error introduced at

MILLINGTON AND CASSIDY: IMPROVING THE ACCURACY OF FDTD APPROXIMATIONS TO TANGENTIAL COMPONENTS

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Fig. 3. Percentage amplitude difference between the FDTD approximations and the analytic solution for (@Hz )=(@t), plotted against angle of incidence. (a) Percentage Amplitude Error vs Angle of Incidence FDTD cell size: =10, (b) Percentage Amplitude Error vs Angle of Incidence FDTD cell size: =50.

Fig. 2. Analytic and FDTD approximations to (@Hz )=(@t) plotted against the phase angle. Plots are shown for incident angles of 0 (top), 30 (middle) and 60 (bottom).

an interface will be evident in both the transmitted and reflected waves.

Fig. 4. Maximum percentage amplitude error plotted against interface-defining dielectric permittivity ratio " =" .

IV. MODIFIED FDTD APPROXIMATIONS The standard FDTD formulation gives rise to an apparent conflict: the approximations approaching from either side of the

interface result in a discontinuous tangential (z-directed) magnetic field component.

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In the analytic solution, the discontinuity in the spatial derivative of the normal component of the driving electric field is compensated for in Maxwell’s equation by a discontinuity in the spatial derivative of the tangential component. In the FDTD approximations, however, there is no such compensation as the approximation to the derivative of the tangential component “smooths out” the analytic discontinuity. Further inspection of the plots in Fig. 2, suggests that the analytic curve is an arithmetic average of the FDTD approximations calculated on either side on the interface. The difference between the approximations from either side of the interface is due only to the difference in the terms for the derivative of the terms in (15)), normal component of the electric fields, (the and it follows that only these terms would need to be averaged. Fortunately, the average is readily determined because, given the “idealized” materials, the normal components of the electric field on either side of the interface are related: (20)

Fig. 5. Analytic and modified FDTD approximations to (@Hz )=(@t), plotted against the phase of the incident wave at the interface, for an angle of incidence of 60 .

and the required average value is calculated in a computationally efficient way by multiplying appropriate terms of the original approximation, (15), by coefficients to give

(21) where the coefficients

(22) (A similar would be used in the equivalents of (22) for the and .) calculation of For the test scenario, assuming that at the interface, the normal component of the electric field are evaluated as if on the , incident side, i.e., with (18) rather than (19) and the coefficients (22) become (23) The decision as to whether to calculate these time invariant coefficients, should depend on the number of interfaces of different materials in the model. If there are a small number of different material interfaces then the burden of the additional memory required for the coefficients will be small, and the directional coefficients represent no more of an overhead than the modelling anisotropic media. If the number is large, in-line calculation may prove more efficient. The effect of applying this averaging is significant. In Fig. 5, a repeat of Fig. 2(c) with results obtained using the modified

Fig. 6. Percentage error in amplitudes of the modified approximations to (@Hz)=(@t), against angle of incidence. For this result set, the FDTD cell size was  =10 for the O(2,2) formulation.

formulations, the plots of the approximations from either side of the interface are now indistinguishable and the FDTD error diminishes to the familiar small phase error. Fig. 6 shows the percentage amplitude error for the modified formulations and should be compared with Fig. 3, (showing the error without the modification). Note that the y axis of Fig. 6 is rescaled in comparison to that of Fig. 3. The modified scheme th of the unmodified error percentage is approximately error when the angle of incidence is just 30 . V. HIGHER ORDER SCHEMES The discussions so far have concerned themselves with the O(2,2) formulation of the FDTD technique, but is important to note that similar observations and results can be obtained with O(2,4) and other higher order formulations. The figures below show the graphs of similar plots to those of Figs. 2, 3 and 4, obtained using an O(2,4) formulation.

MILLINGTON AND CASSIDY: IMPROVING THE ACCURACY OF FDTD APPROXIMATIONS TO TANGENTIAL COMPONENTS

Fig. 7. Analytic and conventional O(2,4) FDTD approximations to (@Hz )= (@t), plotted against the phase of the incident wave at the interface, for an angle of incidence of 60 .

Fig. 8. Analytic and modified O(2,4) FDTD approximations to (@Hz )=(@t), plotted against the phase of the incident wave at the interface, for an angle of incidence of 60 .

VI. EFFECTS ON A PRACTICAL SIMULATION In this section, the impact of the proposed modification on a practical simulation is demonstrated using an example from the application area of ground penetrating radar (GPR). An advanced 3D, forward modeler, [16], is used to simulate the propagation of the GPR electromagnetic waves, both with and without the proposed correction, so that a comparison of the simulated traces can be made. A transmitting/receiving dipole antenna pair is embedded in a material of relative permittivity 1 (free space), a ‘pulsed’ source signal injected at the transmitter (with a bandwidth of approximately 250–650 MHz) and the simulated “trace” at the receiver ratio was set to recorded. For an O(2,2) simulation the 10 and, for comparison purposes, in particular to see the effects ratio to 5, the simulations are repeated of a lowering the with an O(2,4) scheme. In both cases, with no reflectors present, it is possible to verify that the proposed correction has no effect

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Fig. 9. Percentage amplitude difference between the O(2,4) FDTD approximations and the analytic solution for (@Hz )=(@t), plotted against angle of incidence.

Fig. 10. Percentage error in amplitudes of the modified O(2,4) approximations to (@Hz )=(@t), against angle of incidence. For this result set, the FDTD cell size was  =5.

on the recorded “background” trace. A dielectric ‘slab’ of relative permittivity 9 was then added to the model, positioned so that a 60 degree incident angle is established along the ray path from transmitter to receiver via the slab. The resulting “total” traces are also recorded and the reflector’s signature is isolated by subtracting the background trace from the total trace. Both “perpendicular” and “parallel” polarizations of the incident electric field at the reflector are considered by modeling “broadside” and “end-fire” antenna configurations: A. Broadside Configuration The “broadside” configuration, with transmitter and receiver principal axes parallel, but offset a fixed distance along a line perpendicular to those axes, is typically employed for surface GPR applications such as pavement evaluation, non-destructive testing, etc. Note that the electric field is polarized perpendicular to the plane of incidence:

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Fig. 11. Modeling scenario schematics, showing “broadside” and “endfire” GPR antenna configurations.

The signatures for the broadside case, modeled with , with and without the “correction”, are shown in Fig. 12. As would be predicted, the difference between the traces obtained with and without the correction is barely noticeable, the difference in amplitudes being calculated at less than 0.1%. A simulation made using the effective permittivity technique [17] gave results almost identical to those obtained with the proposed modification, showing a maximum difference of approximately 0.00001%. The difference between the proposed scheme and the effective permittivity technique lies only in the values of the discontinuous normal components of the electric field at interfaces. In the proposed scheme, the value is correct for one side of the interface, whereas in the effective permittivity technique, the value held is the average of the correct values for incident and transmission sides. The difference is significant for applications in which the accuracy of field component values at interfaces are important, such as in the inversion algorithm described in [13] and [18].

Fig. 12. Broadside reflector signatures, with and without the FDTD scheme . Interface permittivity contrast, " =" . modification.  =

1=5

=9

Fig. 13. End-fire reflector signatures, with and without the FDTD scheme mod. Interface permittivity contrast, " =" . ification.  =

1=5

=9

B. End-Fire Configuration The “End-fire” configuration, in which the transmitter and receiver share a principal axis, is typical of bore-hole and in-pipe GPR survey geometries. In this case, the incident electric field is polarized in the plane of incidence at the reflector. Comparison of the reflector signatures for the end-fire mode, , as shown in Fig. 13, reveals a more modeled with pronounced difference. The amplitudes of the signatures obtained without the modification are over 2.5% and 7.75% greater and than “corrected” signatures in the cases that , (shown), respectively. The evaluation of the relevant tangential component of magnetic field at the interface in the uncorrected scheme is over estimated because the spatial derivative of the normal component of the electric field used in this evaluation is made as if approaching from the high speed side of the interface. As would be predicted, increasing the permittivity contrast exacerbated the effect. Repeating the simulations with the relative permittivity of the dielectric ‘slab’ set to 40 revealed that the

Fig. 14. End-fire reflector signatures, with and without the FDTD scheme mod. Interface permittivity contrast, " =" . ification.  =

1=5

= 40

signature obtained without the modification to be almost 20% greater than “corrected” signature, as shown in Fig. 14.

MILLINGTON AND CASSIDY: IMPROVING THE ACCURACY OF FDTD APPROXIMATIONS TO TANGENTIAL COMPONENTS

Again, repeat simulations made using the effective permittivity technique [17] gave results almost identical to those obtained with the proposed modification, showing a maximum difference of approximately 0.00001%. The effect of the modification on computational load is different for the O(2,2) and O(2,4) schemes because of the relative effect of adding the same number of arithmetic operations to a single field vector component calculation. For the O(2,4) scheme, in modeling 1024 time increments over a grid of 96 96 128 cells, execution times of 119 seconds and 131 seconds were recorded for the principal time stepping loops of the original and corrected schemes respectively, an increase of approximately 10%. For the O(2,2) scheme, in modeling 2048 time increments over a grid of 128 128 128 cells, execution times of 291 seconds and 361 seconds were recorded. As would be expected this represents the larger percentage increase of approximately 24%. The times for the corrected scheme were recorded with the extra arithmetic taking place for every time step (as opposed to the calculation and storage of anisotropic update coefficients in advance of the time stepping loops). Times recorded for the effective permittivity technique [17] were similar to those achieved with the proposed modification.

VII. CONCLUSION A significant error, which affects the modeling of EM waves impinging on a dielectric interface, present in typical orthogonal FDTD formulations, can be eliminated by making the a simple, computationally inexpensive modification to the finite difference approximations to Maxwell’s equation for Faraday’s law. This effect of this cell-local error on the overall scheme depends upon the FDTD grid cell size relative to wavelength, so the effects of the correction are more pronounced for higher order schemes that are usually employed to reduce computational load by the use larger cell sizes. The proposed modification is justified in the case of planar interfaces, but further work is required to assess the effect on simulations relating to corner artifacts, stair-casing geometry approximations and oblique interfaces. A comparison to the effective permittivity technique showed practically identical results at the receiving antenna, with the difference being that in the proposed approach the values held for the field vector components local to the interface are “correct” for the incident side of the interface, as opposed to the average of the “correct” values on both sides of the interface. This difference has significance when the correct local values of the fields are important, such as in the inversion algorithm described in [13] and [18]. The symmetry of Maxwell’s equations with respect to the magnetic and electric fields allow us to assert that similar results could be obtained for the case of a discontinuity in magnetic permeability, when calculating the tangential component of the electric field, with a similarly simple modification to the relevant finite-difference approximations.

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REFERENCES [1] K. S. 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. [2] B. B. Beard, W. Kainz, T. Onishi, T. Iyama, S. Watanabe, O. Fujiwara, J. Q. Wang, G. Bit-Babik, A. Faraone, J. Wiart, A. Christ, N. Kuster, A. K. Lee, H. Kroeze, M. Siegbahn, J. Keshvari, H. Abrishamkar, W. Simon, D. Manteuffel, and N. Nikoloski, “Comparisons of computed mobile phone induced SAR in the SAM phantom to that in anatomically correct models of the human head,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, pp. 397–407, 2006. [3] N. J. Cassidy, “The application of mathematical modelling in the interpretation of near-surface archaeological ground-penetrating radar,” 2001. [4] O. Brandt, A. Taurisano, A. Giannopoulos, and J. Kohler, “What can GPR tell us about cryoconite holes? 3D FDTD modeling, excavation and field GPR data,” Cold Regions Sci. Technol., vol. 55, no. 1, pp. 111–119, 2009. [5] T. Kobayashi, X. Feng, and M. Sato, “FDTD simulation on array antenna SAR-GPR for land mine detection,” Syst. Human Sci.—Safety, Security Depend., p. 309-+, 2005. [6] B. Yang and C. A. Balanis, “Dielectric interface conditions for general fourth-order finite difference,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 8, pp. 559–561, Aug. 2007. [7] A. Christ, S. Benkler, J. Frohlich, and N. Kuster, “Analysis of the accuracy of the numerical reflection coefficient of the finite-difference time-domain method at planar material interfaces,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, pp. 264–272, 2006. [8] A. Christ and N. Kuster, “Correction on the numerical reflection coefficient of the finite-difference time-domain method for efficient simulation of vertical-cavity surface-emitting lasers,” J. Opt. Society Amer., vol. 20, no. 7, pp. 1401–1408, July 2003. [9] Q. X. Chu and H. Ding, “Second-order accurate FDTD equations at dielectric interfaces,” Microw. Opt. Techn. Lett., vol. 49, no. 12, pp. 3007–3011, 2007. [10] T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microw. Guided Wave Lett., vol. 10, no. 9, pp. 359–361, 2000. [11] K.-P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 4, pp. 158–160, Apr. 2001. [12] T. T. Zygiridis, T. K. Katsibas, C. S. Antonopoulos, and T. D. Tsiboukis, “Treatment of grid-conforming dielectric interfaces in FDTD methods,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1396–1399, Mar. 2009. [13] L. Crocco, G. Prisco, F. Soldovieri, and N. J. Cassidy, “Advanced forward modeling and tomographic inversion for leaking water pipes monitoring,” in Proc. 4th Int. Workshop on Advanced Ground Penetrating Radar, 2007, pp. 113–117. [14] D. Corson and P. Lorrain, Introduction to Electromagnetic Fields and Waves, 1st ed. New York: Freeman, 1962. [15] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [16] T. M. Millington and N. J. Cassidy, “Optimising GPR modelling: A practical, multi-threaded approach to 3D FDTD numerical modelling,” Comput. Geosci., vol. 36, no. 9, pp. 1135–1144, Sept. 2010. [17] K.-P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 4, pp. 158–160, Apr. 2001. [18] T. M. Millington, N. J. Cassidy, L. Nuzzo, L. Crocco, F. Soldovieri, and J. K. Pringle, “Interpreting complex, three-dimensional, near-surface GPR surveys: An integrated modelling and inversion approach,” Near Surface Geophys. 2010, DOI: 10.3997/1873-0604.2010010.

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Tim M. Millington (S’10) received the bachelor’s degree in mathematics and computer science from Keele University, Staffordshire, U.K., in 1987. After two decades working in industry, he returned to pursue a doctorate and is currently a research student in the School of Physical and Geographical Sciences at Keele University. He has published a number of peer reviewed articles and is looking forward to submitting his thesis “Tomographic reconstruction by inverse scattering of ground penetrating radar with numerical background field and Green’s functions” in 2011.

Nigel J. Cassidy received the Ph.D. degree from Keele University, Staffordshire, U.K., in 2001. Currently, he is a Senior Lecturer of applied geophysics within the School of Physical and Geographical Sciences at Keele University, and has published over 70 international peer-reviewed papers and reports. His research focuses on the application of geophysical methods for the characterization of environmental, geological and geotechnical phenomena and has specific expertise in numerical modelling and ground penetrating radar (GPR). Dr. Cassidy is an Associate Editor of Near-Surface Geophysics and was the recipient of the 2009 Ludger Mintrop Award from the European Association of Geoscientists and Engineers (EAGE).

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Near-Field PML Optimization for Low and High Order FDTD Algorithms Using Closed-Form Predictive Equations Mohammed F. Hadi, Senior Member, IEEE

Abstract—The convolutional perfectly-matched-layer (CPML) absorbing boundary condition is fully capable of handling near-field wave absorption that usually combines near-grazing wave incidence with wave evanescence. The appropriate choice of the various CPML parameters to realize this potential for any given simulation problem is a challenging task that is typically achieved through exhaustive and time-consuming searches that involve large numbers of full-scale simulations. The presented work here uses a previously developed predictive system of equations that accurately determines numerical reflections off the PML interface and embeds it into a global optimization routine that reliably computes the required optimum CPML parameters. This predictive system of equations has also been extended and validated for the M24 and FV24 integral-based high-order FDTD algorithms. With this approach, the task of selecting optimum CPML parameters that would usually take several days of intense computations can now be accomplished within a few minutes on an average personal computer. Index Terms—Absorbing boundary conditions, finite-difference time-domain method, high-order FDTD methods, perfectly-matched layer (PML).

I. INTRODUCTION S the electrical size of the problem domain increases beyond several wavelengths, grid resolution required by the finite-difference time-domain (FDTD) method has to be increased progressively to keep numerical dispersion errors under control. The demand on computing resources under such circumstances quickly takes the FDTD method to the realm of cluster computing. The two-dimensional M24 algorithm [1] and three-dimensional FV24 algorithm [2] were especially designed high-order FDTD variants to effectively control numerical dispersion errors while keeping grid resolutions low, and hence allow modeling electrically large structures using the average high-end PC or workstation. However, the combination of electrically large models and coarse grid resolutions create other modeling challenges. One of these challenges is in the area of absorbing boundary conditions (ABC). Berenger’s perfectly-matched-layer (PML) is the most accurate ABC technique used for FDTD today. During its introduc-

A

Manuscript received June 23, 2010; revised November 11, 2010; accepted November 29, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. The work was supported by Kuwait University under Research Grant EE02/08. The author is with the Electrical Engineering Department, Kuwait University, Safat 13060, Kuwait (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158955

tion [3], a complete PML theory was developed for the continuous space. When implemented within the discrete FDTD grid however, its behavior starts to deviate gradually from continuous PML theory as the impinging wave propagation deviates from normal incidence, causing in the process spurious reflections. Although initially very low, these reflections start to increase appreciably as the outgoing wave begins to impinge on the PML at steep incidence angles (beyond 75 ) [4] and are aggravated by low grid resolutions. The combination of large problem size and low grid resolution which is afforded by highorder algorithms dictates that an uncomfortably large spacing be maintained between modeled objects and PML boundaries to counteract PML ambiguity in this area [1]. On the other hand, a recent publication [5] demonstrated through brute-force optimization that PML is capable of excellent wide-angle performance, beyond 75 and up to near-grazing wave incidence. This procedure required using as an optimization objective function a fully functioning FDTD model that needed to be run in every iteration which can be very costly if three-dimensional optimum PML parameters were desired, especially with the FV24 algorithm. An equally time-consuming PML parameters selection procedure for wide-angle performance was reported by Roden and Gedney [6] through exhaustive testing of the different PML parameter combinations. Clearly, there is a need to understand and quantify PML behavior to efficiently and predictably extend its performance to wide-angle applications. Several works have already been reported in the literature that investigate this approach for standard FDTD. The earliest efforts limited their investigations to a single scatterer-PML interface [7]–[9], though they were critical to understanding the nature and causes of numerical reflections in FDTD-PML implementations. To quantify and predict PML reflections in actual FDTD models, however, one needs to account for all inter-PML reflections. In other words, the problem needs to be approached as a discrete multi-layer oblique angle incidence, mirroring in the process the classical continuous counterpart in electromagnetic theory. This treatment was finally introduced by Berenger in [10] during his investigation of PML behavior with evanescent waves. He introduced in that work a closed-form system of algebraic equations that accurately predicted numerical reflections off -layer PML ABCs. Furthermore, those equations were valid for waves that combine both propagation and evanescence. Later [4], that technique, which was developed for the original split-field PML formulation (SPML), was extended to both uni-axial PML (UPML) [11], [12] and convolutional PML (CPML) [6], [13] variants.

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The update equation for a typical mid-PML

node

(3) produces Fig. 1. Index assignments of E and H

nodes within the PML layers.

(4) The aim of this work is to extend the functionality of Berenger’s reflection analysis and design an optimization procedure that would readily predict optimum PML parameters for any given FDTD simulation problem. The predictive equations will be robust enough to handle FDTD problems that could possibly include a mix of traveling and evanescent waves, near-grazing wave incidence on the PML layer and even ultrawideband frequency operation. The same treatment will also be extended to the M24 and FV24 high-order FDTD algorithms, which would benefit greatly from eliminating the hitherto mandatory and costly large scatterer-PML separations.

II. NUMERICAL REFLECTIONS OFF PML ABCS The following is a brief summary of Berenger’s reflection analysis which will later serve as basis for high-order exten-polarized and obliquely incident plane sions. Assuming an wave within the -plane onto an -Layer PML (see Fig. 1), and field nodes are defined in terms of , the incithe dent wave amplitude as

and are also defined in Appendix A. Rearranging where and defining the PML depth-specific constant (5) simplifies the update equation to (6) When the same substitution is carried out for the remaining field nodes, we find out that the above algebraic equation is valid for . For , (6) reduces to (7) Rewriting (3) for

gives

(8) which reduces to (9)

(1) For

, substituting into the

update equation

(2) is an abbreviation of , the incident wave’s where are unfield polarization factor from Appendix A, knowns to be determined, is the number of PML layers and is the desired overall reflection coefficient off the scatterer-PML is only valid for the inciinterface. Note that although dent wave from the scatterer region, it appears in every field quanexpression within the PML. It is assumed that every tity includes information related to both and terms. Writing field expressions as above eliminates the need for explicitly matching boundary conditions at every half-layer within and . The rethe PML and the requisite calculations of quired equations to uniquely determine these unknowns are and update obtained from substituting (1)–(2) into the , . The incident wave equations at nodes remains unchanged throughout the tangential wavenumber PML as per electromagnetic boundary conditions.

(10) produces

(11) where

is a scatterer region quantity. Here, the used substitution is (2) without the factor. This equation then reduces to (12)

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or (13) The above system of equations can be conveniently presented in the tri-diagonal matrix form

.. .

.. .

(14) (18)

where

..

.

..

.

(15)

The -parameters, which obey , are chosen via an optimization routine that minimizes numerical dispersion over all propagation directions in 3D space. Representative values of these parameters and the procedure to compute them can be found in [2]. Substituting the expressions (1)–(2) into this update equation produces

At this point, [10] proceeded to replace every occurrence of (using (85)) with (16) (19) (17) which eliminates the need to explicitly solve the numerical dispersion relation for .

Rearranging and using definition (5), the above equation reduces to

III. PML REFLECTION ANALYSIS FOR THE FV24 ALGORITHM The FV24 algorithm [2] was developed as a high-order FDTD extension with extreme phase coherence that allowed modeling electrically large problems accurately using coarse FDTD grids. This high-order algorithm suffers as low as one degree of phase error per 3,000 wavelengths of propagation at 20 cells per wavelength resolution, compared to one degree per a single wavelength of propagation for the standard FDTD method. Since its introduction in 2007, the FV24 algorithm received several important capability upgrades. First, a phase-matched hybrid FV24/standard FDTD scheme was developed [14] to allow the FV24 algorithm to model practical electromagnetics problems using existing standard FDTD modeling tools. Those FDTD tools were then upgraded one-by-one to work directly within the FV24 algorithm. Among these tools are modeling transparent point sources [15], modeling exact plane wave sources using the total-field/scattered-field approach [16] and modeling PML ABCs [5], [17]. Following the same procedure outlined in Section II, we start with the update equation of a typical mid-PML field node (arbithis time) [5] trarily choosing

(20) where (21) The above algebraic equation is valid for . For , all 9 right-most nodes (at ) will vanish being tangential to the PML’s PEC back-plane (see Fig. 2), and (20) reduces to (22) , , the corresponding update equations For need to collapse to the standard FDTD equations [14] resulting in (23) and (24) respectively. On the other side of the PML, the reflection coefficient is assigned to node and will appear in the expression of . This is due to the every field node at or to the left of

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(31) with (32) (33) Fig. 2. Required E indices (up to 20 values) to update a typical H node within the PML layers. The locations L 3=2 include matching E nodes to the six shown here at the offsets k 1 (normal to the paper in front and back, not shown). The same requirement is needed for updating E nodes.

6

(34) (35)

6

(36) (37)

extended-stencil FV24 cell which causes the onset of reflections to occur one standard FDTD cell to the left of the scatterer-PML , 0, , , (18) and the matching interface. For update equation produce

(25)

(38) (39) Finally, solving (29) will yield , the desired overall reflection coefficient off the PML ABC within the FV24 algorithm. The is detailed in required FV24 dispersion relation to solve for Appendix A as there is no easy substitution that would eliminate its need.

(26) IV. PML REFLECTION ANALYSIS FOR THE M24 ALGORITHM (27)

(28) respectively. The above system of matrix form as

The M24 algorithm [1] was the two-dimensional precursor to the three-dimensional FV24 algorithm. As with FV24, the M24 algorithm received several upgrades in the areas of absorbing boundary conditions, point and plane wave sources, hybrid models and compact-FDTD extension for modeling waveguiding structures [15], [16], [18]–[20]. The TE M24 update equations within the PML are given by

equations can be written in (29)

where is the banded (40), shown at the bottom of the page, and

matrix given in (30)

..

(41)

. ..

.

(40)

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TABLE I OPTIMIZED CPML PARAMETERS FOR A CLOSELY-BOUND THIN PLATE IN , , IS , A STANDARD FDTD ALGORITHM ( EVANESCENCE FACTOR)

N = 10 1t

n =1 

(42) where (43) (44) A table of these -parameters and the procedure for computing them can be found in [1]. The required node distribution for developing the reflection coefficient is the same used for the FV24 algorithm from Fig. 2. Substituting the appropriate expressions from (1)–(2) into a typnode yields ical mid-PML

(45)

(51) is a constrained optimization vector that includes , , , , , and possibly, . Depending on the case at hand, could either be an input parameter or an optimization variable as will be illustrated in the following validation examples. Furthermore, the constraint on could be relaxed to and a valid (and, at times, markedly superior) optimum CPML set could still be obtained (see for example, Table I in [5]). Extensive experiments however have demonstrated that such optimum sets were, at times, too sensitive to simulation parameter variations and thus deemed unreliable. Due to the complexity of the above optimization problem, the objective function proved to have a great number of local minima and often without a clear and easily found global minimum. Hence, a global optimization routine is warranted. To further aid in reaching optimum results, experience in CPML theory needs to be drawn on to reduce the number of optimization variables whenever possible as will be illustrated later. The chosen global optimization routine for the following examples was MATLAB’s GlobalSearch.m in conjunction with the local constrained optimizer fmincon.m—both member functions of Mathwork’s Global Optimization Toolbox. where

which further reduces to

(46) where (47) Similarly, a mid-PML

reflection coefficient in terms of a set of optimization parameters that include all 6 CPML parameters from (71)–(73) . In computing and, possibly, the evanescence parameter within each iteration, the wave incidence angle needs to be to achieve optimum results for this swept across incidence angle range. Mathematically, the following optimization problem needs to be solved;

node would yield

(48) which reduces to

(49) where

A. Purely Traveling Waves at Moderate Incidence Angles (50)

Continuing the analysis as in Section III would finally yield the same system of equations there with and modified by replacing , with , in odd-numbered , in even-numbered rows. rows and with V. OPTIMIZATION PROCEDURE AND VALIDATION EXAMPLES To compute the optimum CPML parameters for any combination of FDTD grid/source/scatterer parameters, an optimization routine needs to be developed that would minimize the

To simulate this case, a vacuum FDTD test domain of 50 50 51 cells was constructed and surrounded by a 10-layer CPML. Spatial and temporal steps were at 1 GHz and . The source was a hard point-source at the center of the domain [3] (52) that was switched off after and had a 1 GHz first harmonic. Simulation was run for 100 time steps, then all values within the central -plane cut were compared to those

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obtained from a much larger reference domain. The maximum . error between these two data sets was then recorded as An Optimum CPML set for this case at was computed by searching for only and while keeping , and . The optimization procedure which was based on the highest harmonic in the source signal (3 GHz) and . For comparresulted in ison the empirical formula suggested by Gedney [12], (53) was used with , resulting in . When both CPML sets were used in the test domain above, they reand , respectively. In sulted in comparison, the optimized CPML parameters from [5] resulted in . To derive this set, however, the entire FDTD simulation was used as the objective function in the optimization procedure there and all six CPML parameters were , actively searched for which resulted in , , and . B. Purely Traveling Waves at Extreme Incidence Angles At extreme (near-grazing) incidence angles, wave interaction with the PML layers starts to exhibit evanescence as per electromagnetic theory. Under such circumstances, wave evanescence needs to be accounted for in the CPML optimization process. Specifically, the term needs to be included as an optimization variable along with the other CPML parameters. Furthermore, experiments have shown that all six CPML parameters needed to be included in the search algorithm to achieve excellent CPML performance. Using the same experiment above while collapsing the dimension from 51 cells to 1 cell increased the maximum incidence angle to . The CPML optimization process produced the following parameters for this case; , , , , , and . Plugging these values in the above FDTD test domain re. By comparison, the best obtained sulted in CPML parameters when omitting to include in the optimization process resulted in . Care must be taken when all CPML parameters are involved in the optimization search. The optimization problem in this case is complex enough that even with the global optimization routine used, different initial estimates, , resulted in relatively wildly varying “global” minima. Hence, several initial estimates needed to be tested for best results. C. Mixed Traveling/Evanescent Waves at Extreme Incidence Angles To test the viability of the developed CPML optimization procedure, the realistic and challenging problem proposed by Gedney in [6], [21] is tackled next. This problem involves a thin metal plate in free space with dimensions 10 2.5 cm. The plate is modeled by a three-dimensional FDTD grid with and and surrounded by a 10-layer CPML with

only 3 cells of separation between plate and CPML on all directions. The plate is oriented along the -plane and illuminated by a -oriented point current source one cell above one of its probe collects a time series response one corners while an cell away from the diagonally opposite corner for 2,000 time steps. The source function is the differentiated Gaussian pulse (54) and . with There are three challenging aspects in this problem. First, the close proximity of both source and observation probes to the elongated metal scatterer causes appreciable levels of wave evanescence that is not given a chance to dissipate before it reaches the PML layers. Second, the close proximity of the CPML to the elongated scatterer would result in a maximum as in possible wave incidence of approximately the previous case which translates into another cause of wave evanescence. Third and no less challenging, the source function spans a frequency range of approximately 0.5–10 GHz which decidedly classifies it as an ultrawideband signal. To carry out the optimization procedure, the various parameters that populate (14) need to be based on the highest expected frequency in the simulation which corresponds to the coarsest effective spatial resolution. The lone exception here is the pawhich now needs to be involved as an input, rameter rather than an optimization variable, to introduce the evanescence effect of the elongated scatterer. Berenger suggested an estimate for this quantity in [10] (55) where is the unbounded wavenumber of the medium hosting the metal scatterer and is of the order of the scatterer’s largest dimension. It is clear from this estimate that wave evanescence is highest at the lower end of the present frequency spectrum in the simulation. Hence, our estimate for this problem is based on 0.5 GHz with which translates into . Furthermore, sensitivity to this evanescence estimator is tested by multiplying it with a variable factor. Extensive experiments with this problem have shown that with such relatively high wave evanescence, the optimization problem can be simplified while achieving good performing CPML sets by setting and . The corresponding 4-parameter optimization problem was then run for several values of near the above estimate to check its validity and the resulting CPML parameters were listed in Table I. Listed in the table as well are Gedney’s exhaustively searched parameters from [21]. Fig. 3 compares the performances of the various optimized sets in the table in terms of , the absolute difference of collected time series at the observation point from two FDTD runs, one using the above grid parameters and a reference run where the scatterer-CPML spacing was increased by 75 cells on all sides. In the reference run, Gedney’s hand-picked CPML parameters were used for all cases. As can be seen from the figure, performance is moderately sensitive

HADI: NEAR-FIELD PML OPTIMIZATION FOR LOW AND HIGH ORDER FDTD ALGORITHMS

Fig. 3. Comparison of optimized CPML performances for the standard FDTD . algorithm at several values of 

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Fig. 5. Comparison of optimized CPML parameters for the high-order FV24 algorithm with Gedney’s manually optimized set [21] and a generic set from [22].

. As in the standard FDTD algorithm, results confirmed (55) ’s moderate robustness in predicting the necessary evanescence factor for this problem. A typical set of CPML parame) was , ters (at , and . These values varied slightly at different optimization runs due to the inherent random element in the used global optimizer. They also varied slightly with with performance peaking at approximately the above mentioned value. Fig. 5 repeats for FV24 the comparison from the previous figure. E. Computational Efficiency Advantage Fig. 4. Comparison of optimized CPML parameters for the standard FDTD algorithm with Gedney’s manually optimized set [21] and a generic set from [22].

to the choice of the evanescence factor, though all plots were within one order of magnitude of each other. Fig. 4 displays the performance of the best optimized set ) compared to Gedney’s manufrom Table I (at ally optimized set and to the performance of a typical set of generic CPML parameters that were chosen as the mid-points in the suggested ranges in [22, Sect. 8.3], , , , , and . D. FV24 CPML Optimization Performance The last validation test was repeated for the FV24 algorithm using the latter’s tuning parameters that match the grid’s , and the maximum 10 GHz simulation frequency; , and [2]. The FV24 extended-stencil update equations were collapsed around the thin plate and PML back-planes as per the phase matching technique introduced in [14]. Detailed optimization and simulation analysis have shown that, again, no discernible advantage is gained from including all six CPML parameters in the optimization process compared to setting and

A major advantage of the presented optimization procedure over either exhaustive CPML parameters search [21] or brute-force optimization [5], is the fact that no actual simulation runs are needed to compute the required parameters. As an example, computing the optimized CPML parameters for the standard FDTD algorithm at required 716 function evaluation in [5], each running for 215 seconds on a Xeon/Nehalem quad-core workstation. The present procedure, on the other hand, ran 8.5 seconds for an entire local optimizer invocation, translating into (56) Similar analysis for the FV24 algorithm have shown a typical speed-up of 25590:1. VI. CONCLUSION The time-consuming and involved process of selecting optimum CPML parameters for realistic FDTD simulations has been resolved into a computationally efficient and reliable optimization technique that utilizes predictive numerical reflection equations as optimization objective functions for both low and high-order FDTD algorithms. In particular, the presented approach accounts for various degrees and causes of wave evanescence in combination with wave propagation. Due to the

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complexity of the optimization problem, global optimization was warranted and yet it still was not guaranteed to reach global reflection error minima. Derived parameters, however, were always consistently close in performance to those obtained through time consuming exhaustive searches. It was found that some CPML parameters were more detrimental to optimum performance than others which somewhat simplifies the optimization procedure. Reduction in computer run-times was in the range of four orders of magnitude for both standard and high-order FDTD algorithms, compared to previous CPML optimization approaches. Through appropriate substitutions, the parameter selection approach is also extendable to the uniaxial PML as well as the original split-field PML formulations. To further improve the presented technique’s reliability, better and more specific estimators of scatterer-caused wave evanescence need to be researched.

, and, for a matched PML, loss effecting parameters are given by [4]

. The PML

(69)

(70)

and . In the above, , with matching expressions for and are the key PML parameters that are typically polynomially graded as a function of the incremental PML normalized depth, (71) (72)

APPENDIX NUMERICAL DISPERSION RELATION WITHIN PML ABCS A. PML Within the Standard FDTD

(73) Equations (61)–(64) can be written in matrix form as

The numerical dispersion relation is a critically accurate predictor of the (possibly complex-valued) numerical wave number rendered by FDTD algorithms in both lossy and lossless numerical spaces. Following the approach by Hadi and Mahmoud [20], SPML formulation within the -plane

(74)

(57)

The corresponding dispersion relation for the above algorithm can be found by setting the determinant to zero, which eventually produces

(58) (75) (59) the discrete form of [10] (60) (76) is transformed, by assuming pendence, into

dewhere (61) (62) (63) (64)

(77)

where

(78) (65)

For a matched PML

, (75) reduces to

(66) (67) (68)

(79) while for isotropically lossy media duces to

, it re-

(80)

HADI: NEAR-FIELD PML OPTIMIZATION FOR LOW AND HIGH ORDER FDTD ALGORITHMS

Typically, the numerical wave number is written as

and (keeping [20, Appendix I]

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distinct to accommodate a matched PML)

(81) (82) where is the angle of incidence upon the PML within the -plane, with denoting normal incidence. We will adopt, however, Berenger’s definition [10] to account for incident wave evanescence

(90)

where

(83) (84) Note that the imaginary parts’ signs are reversed from [10], though still one of the valid choices as mentioned there. This sign convention assures numerically stable dispersion relation and . In solutions when using positive values for both [10], for example, negative and positive values were used. Readers are referred to [10] for detailed explanation of the evanescence variable . A clear distinction needs to be made here between and on one hand which define the decomposition of and which dictate the wave number , and field polarizations within the FDTD discrete space. The latter are defined as [10], [19], [23]

(91)

(85)

while , are the same as (67)–(68). Setting the determinant to zero, we arrive at the dispersion relation

(92)

(93) (94)

(86) (95) It can be easily shown that the numerical dispersion relation could also be written as

with the matching polarization modifiers (96)

(87)

(97)

B. PML Within the FV24 Algorithm When implemented within the PML, the FV24 dispersion relation within the -plane is the same as (75) with the following discrete operator substitutions [2]

(88)

(89)

C. PML Within the M24 Algorithm Due to the non-symmetry of the M24 update equations [1], the matrix equation form of (74) does not apply. Instead, we get

REFERENCES [1] M. F. Hadi and M. Piket-May, “A modified FDTD (2,4) scheme for modeling electrically large structures with high-phase accuracy,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 254–264, Feb. 1997. [2] M. F. Hadi, “A finite volumes-based 3-D low dispersion FDTD algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2287–2293, Aug. 2007. [3] J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994. [4] J.-P. Bérenger, “Numerical reflection from FDTD-PMLS: A comparison of the split PML with the unsplit and CFS PML,” IEEE Trans. Antennas Propag., vol. 50, no. 3, pp. 258–265, Mar. 2002. [5] M. F. Hadi, “Wide-angle absorbing boundary conditions for low and high-order FDTD algorithms,” Appl. Computat. Electromag. Soc. J., vol. 24, no. 1, pp. 9–15, Feb. 2009. [6] J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett., vol. 27, no. 5, pp. 334–339, Dec. 2000. [7] W. C. Chew and J. M. Jin, “Perfectly matched layers in the discretized space: An analysis and optimization,” Electromagn., vol. 16, no. 4, pp. 325–340, 1996.

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[8] J. Fang and Z. Wu, “Closed-form expression of numerical reflection coefficient at PML interfaces and optimization of PML performance,” IEEE Microwave Guided Wave Lett., vol. 6, no. 9, pp. 332–334, Sep. 1996. [9] Z. Wu and J. Fang, “High-performance PML algorithms,” IEEE Microwave Guided Wave Lett., vol. 6, no. 9, pp. 335–337, Sep. 1996. [10] J.-P. Bérenger, “Evanescent waves in PML’s: Origin of the numerical reflection in wave-structure interaction problems,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1497–1503, Oct. 1999. [11] Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag., vol. 43, no. 12, pp. 1460–1463, Dec. 1995. [12] S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1630–1639, Dec. 1996. [13] M. Kuzuoglu and R. Mittra, “Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers,” IEEE Microwave Guided Wave Lett., vol. 6, no. 12, pp. 447–449, Dec. 1996. [14] M. F. Hadi and R. K. Dib, “Phase-matching the hybrid FV24/S22 FDTD algorithm,” Progr. Electromagn. Res., vol. 72, pp. 307–323, 2007. [15] M. F. Hadi and N. B. Almutairi, “Discrete finite-difference time domain impulse response filters for transparent field source implementations,” IET Microw. Antennas Propag., vol. 4, no. 3, pp. 381–389, Mar. 2010. [16] M. F. Hadi, “A versatile split-field 1-D propagator for perfect FDTD plane wave injection,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2691–2697, Sep. 2009. [17] M. F. Hadi, “PML absorbing boundary conditions for the integralbased high-order FV24 FDTD algorithm,” Intr. J. Numer. Modell.: Electron. Networ. Devices and Fields, vol. 24, no. 1, pp. 2–12, Jan./Feb. 2011 [Online]. Available: 10.1002/jnm.756 [18] A. M. Shreim and M. F. Hadi, “Integral PML absorbing boundary conditions for the high-order M24 FDTD algorithm,” Progr. Electromagn. Res., vol. 76, pp. 141–152, 2007. [19] M. F. Hadi and R. K. Dib, “Eliminating interface reflections in hybrid low-dispersion FDTD algorithms,” Appl. Computat. Electromag. Soc. J., vol. 22, no. 3, pp. 306–314, Nov. 2007.

[20] M. F. Hadi and S. F. Mahmoud, “A high-order compact-FDTD algorithm for electrically large waveguide analysis,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2589–2598, Aug. 2008. [21] S. D. Gedney, “Perfectly matched layer absorbing boundary conditions,” in Computational Electrodynamics, A. Taflove and S. C. Hagness, Eds., 3rd ed. Boston, MA: Artech House, 2005, ch. 7, pp. 273–328. [22] A. Z. Elsherbeni and V. Demir, The Finite-Difference Time-Domain Method for Electromagnetics With MATLAB Simulations. Raleigh, NC: SciTech Publishing, 2009. [23] M. Celuch-Marcysiak and W. K. Gwarek, “On the nature of solutions produced by finite difference schemes in time domain,” Int. J. Numer. Model.: Electron. Networ., Devices Fields, vol. 12, no. 1–2, pp. 23–40, Jan.–Apr. 1999.

Mohammed Hadi (S’87–M’00–SM’07) received the B.Sc. degree in electrical engineering from Kuwait University, in 1988 and the M.Sc. and Ph.D. degrees from the University of Colorado at Boulder, in 1992 and 1996, respectively. Currently, he is an Associate Professor in the Electrical Engineering Department, Kuwait University. His research is currently focused on FDTD development for modeling electrically large structures. He has over ten years experience in governmental work and consultation in the areas of engineering training, higher education planning and Kuwait’s labor profile studies. Since 2004, he has been a Board member of the Kuwait Fund for Arab Economic Development’s prestigious National Engineers Training Program. He is a sworn-in consulting expert at Kuwait’s Court of Appeals since 2007. He also held membership and chair positions in several high-level governmental inquiries at the Kuwait Ministries of Defence, Energy and Commerce. He was a Visiting Research Scholar at Duke University, Durham, NC, during the academic year 2007–2008.

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Efficient Modeling of Three-Dimensional Reverberation Chambers Using Hybrid Discrete Singular Convolution-Method of Moments Huapeng Zhao, Student Member, IEEE, and Zhongxiang Shen, Senior Member, IEEE

Abstract—Efficient modeling of a three-dimensional reverberation chamber (RC) is achieved by combining the discrete singular convolution (DSC) method and the method of moments (MoM). An RC usually consists of a metallic cavity and one or two conducting stirrers, whose size is normally small compared to the chamber size. The large cavity is efficiently modeled by the DSC method, and the stirrer is simulated by the flexible MoM. Exploiting the property of RWG basis, solutions from the two methods are combined together using the equivalence principle. The validity and advantages of the proposed hybrid technique are shown through comparisons with the commercial software FEKO. Employing the high efficiency of the DSC method, the hybrid technique can analyze one stirrer position of a medium-sized RC in a few hundred seconds on a single personal computer, for which FEKO needs thousands of seconds CPU time. The memory requirement of the proposed method is also less than that of FEKO. Furthermore, our hybrid method provides efficient calculation of electric field strength at a large number of field points, which is of great interest in RC analysis. Simulations show that our method only takes 1.7 seconds to compute electric field strength at 4026 field points. Index Terms—Discrete singular convolution, hybrid technique, method of moments, regularization technique, reverberation chamber.

I. INTRODUCTION

S

INCE the establishment of IEC standards in 2003 [1], measurement using reverberation chamber (RC) has become popular. Fig. 1 illustrates the geometry of a three-dimensional RC, which is constituted by a large metallic cavity and a rotating stirrer. A transmitting antenna is usually placed at a corner of the RC to excite it. By rotating the stirrer, the boundary condition is changed and a working volume can be created inside the RC. In the working volume, fields are statistically uniform, isotropic, and polarization-independent. With the same input power, the RC can generate much higher electric field strength than the anechoic chamber. Furthermore, electromagnetic interference may come in any direction in practice, therefore, the RC environment can simulate the real situation better. Hence, the RC

Manuscript received September 17, 2010; revised December 07, 2010; accepted January 15, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore (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.2011.2158966

Fig. 1. Geometry of a typical three-dimensional reverberation chamber.

is very suitable for electromagnetic compatibility (EMC) measurements, and it first attracted much interest in the EMC community. Recently, its applications have been extended to the measurements of antennas [2], multiple-input multiple-output antennas [3], scattering cross section [4], and absorbing cross section [5]. Due to its popularity, RC modeling has also become important for many applications, such as optimizing RC design and characterizing the effectiveness of new stirring methods. Meanwhile, experimental results can be validated by numerical RC modeling. In recent years, RC modeling has been attempted using various numerical methods, such as the method of moments (MoM) [6]–[9], the finite element method (FEM) [10], [11], the finite difference time domain (FDTD) method [12], and the transmission line matrix (TLM) method [13]. However, as noted by Bruns [9], the RC modeling remains a challenging problem. The great computational burden required by RC modeling mostly lies in the large cavity. The main difficulty is that most iterative techniques suffer from the problem of slow convergence in solving the matrix equation arising from RC modeling [14], which prevents the application of fast algorithms. In this case, traditional numerical methods will require long computation time for RC modeling. In [13], it was estimated that around 200 days were required to analyze 200 stirrer positions of an RC using the TLM method with coarse spatial resolution. In [6], a cavity Green’s function was derived based on spectral domain techniques, and the Ewald summation was used to accelerate the computation of the cavity Green’s function [8]. The cavity Green’s function method renders fast modeling of small objects in a large cavity. Nevertheless, as the size of the object increases, the efficiency of the cavity Green’s function method may degrade due to the overhead in filling the interaction matrix.

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Therefore, it is desirable to develop new efficient techniques for RC modeling. In solving complex electromagnetic scattering and radiation problems, hybrid methods [15]–[19] are powerful because they can exploit advantages of different methods. These hybrid methods employed the advantages of integral equation and differential equation methods, and they rendered more efficient and accurate solution to complex electromagnetic problems. However, most existing hybrid methods can not reduce the computational burden from the large cavity of an RC, because the geometrical simplicity of the large cavity has not been taken into account. Therefore, a hybrid method specifically designed for RC modeling is required. On the other hand, high-order methods are powerful in solving large scale electromagnetic problems. With high-order basis, the high-order method can obtain satisfactory accuracy with coarser grids; one of them is the discrete singular convolution (DSC) method [20], [21]. The DSC method is similar to the pseudospectral (PS) method [22], [23], because they both aim to find the accurate approximation of the spatial differentials in Maxwell’s equations. Discussions on the similarities and differences between DSC and PS methods can be found in [24], [25]. Using a higher-order basis, the grid size in the DSC method can be chosen to be one third to one sixth of the wavelength [20], which can reduce the computational effort without sacrificing the accuracy. Hence, it is desirable to develop high-order hybrid techniques to obtain a high efficiency. In [26], a two-dimensional hybrid DSC-MoM technique was recently presented to illustrate the advantage of combining the DSC method and the MoM. With high flexibility, the MoM is a good choice for modeling the stirrer in an RC, which is arbitrary in shape and purely metallic. Adopting structured grids, the DSC method [20] poses high efficiency in modeling the large rectangular cavity. The DSC method has been shown to be rather efficient and accurate for large scale electromagnetic problems [21]. However, using rectangular grids, the DSC method is inflexible in modeling objects of arbitrary shape. This drawback can be complemented by the MoM. In this work, a three-dimensional hybridization of the DSC method and MoM is developed for efficient analysis of a threedimensional RC. The RWG basis is adopted in the MoM, which enables the proposed method to easily model stirrers of arbitrary shape. The DSC method is employed to reduce the computational burden from the large cavity. Hence, this hybrid method only requires a few hundred seconds to analyze one stirrer position of a medium-sized three-dimensional RC on a personal computer, for which thousands of seconds CPU time is required by the commercial software FEKO. It should be mentioned that as the stirrer rotates, geometry of the cavity and shape of the stirrer hold the same; only matrices connecting MoM and DSC models need updating. Using the field expansion in the DSC method, the proposed method also enables very fast computation of electric field strength at a large number of field points in the RC. While substantially reducing the computation time, the hybrid method retains the flexibility of the MoM through expressing current sources on the DSC grids using a regularization technique. The rest of this paper is organized as follows. Section II describes the proposed hybrid technique in detail. Section III demonstrates the validity and efficiency of our hybrid technique

Fig. 2. The MoM model: a metallic stirrer enclosed by a Huygens’ box.

through numerical examples. Finally, Section IV concludes the work descried in this paper. II. FORMULATION Fig. 1 shows a three dimensional RC, which consists of a large , and a mechanical stirrer . A metallic cavity of transmitting antenna is positioned at a corner of the RC. In practice, the transmitting antenna can be a horn or a log-periodic antenna. For simplicity, the transmitting antenna is replaced by a in this work, which doesn’t affect the analcurrent source ysis of the RC itself. From the viewpoint of integral equation method, current will be induced on the surface of the metallic stirrer when it is illuminated, and the total tangential electric field along the stirrer surface should be zero. Equivalently, the metallic stirrer may be replaced by a current sheet , and the tangential electric field along the surface of the stirrer should be forced to zero at the same time. Therefore, the original problem can be decomposed into two subproblems. The first is to force the tangential electric field along the surface of the stirrer to zero, which results in an integral equation. The second one is and a a large cavity excited by the original current source current sheet , and it can be solved efficiently with the discrete singular convolution (DSC) method. The MoM and DSC models are illustrated in Figs. 2 and 3, respectively. As shown in Fig. 2, by enclosing the stirrer with a Huygens’ box , the field impinging the stirrer is exactly the radiated field by the and magnetic current along equivalent electric current and are related to the Huygens’ box. At the same time, the field solution of the second subproblem through the equivalence principle. The two subproblems are thus coupled together, and by solving the two subproblems simultaneously, the induced current as well as the fields in the RC can be obtained. A. The MOM Model To enforce the tangential electric field along the stirrer surface to zero, the following electric field integral equation (EFIE) is applied along

(1)

ZHAO AND SHEN: EFFICIENT MODELING OF THREE-DIMENSIONAL RC USING HYBRID DSC-MoM

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and

(7) Fig. 3. The DSC model: a large cavity excited by the original source J~ and the current sheet J~ . For simplicity, the transmitting antenna has been replaced by a current source J~ . This is reasonable when the main concern is the analysis of the RC itself.

where is the electric field illuminating , is the unit vector tangential to at , is the wavenumber, and represents the free space Green’s function [27]. With the Huygens’ box enclosing , the electric field illuminating the stirrer is ra. is then written diated by the equivalent currents and as

(2) Substituting (2) into (1), one can derive

denotes the RWG basis, whose definition can be In (5)–(7), found in [28]. represents the domain of the RWG basis is the domain of the on the surface of the stirrer , and RWG basis on the surface of the Huygens’ box . Matrix is the same as the interaction matrix in the convenonly has a differtional MoM solution of the EFIE. Matrix ence of negative sign from the interaction matrix in MoM solution of the EFIE. Except for a difference of negative sign, mais the same as the interaction matrix in the conventional trix MoM solution of the magnetic field integral equation [27]. The calculation of matrix elements and the treatment of singularity are thus similar to those in the conventional MoM. In this work, the integral over a triangle is calculated with a Gaussian quadrature. Since the Huygens’ box doesn’t overlap with the stirrer, and . The singularity there is no singularity in matrices encountered when filling matrix is dealt with using the analytical and numerical integrations [27]. More details about the numerical calculation of integrals in (5) to (7) and the treatment of singularity can be found in [27]. B. The DSC Model Maxwell’s equations governing the fields in the DSC model shown in Fig. 3 are

(3)

(8a) (8b)

The stirrer surface and the Huygens’ box surface can be discretized using a set of planar triangles. , , and are then expanded with the famous RWG bases [28]. Invoking Galerkin method upon both sides of (3), one can obtain the following:

where represents the current on the stirrer and is the original current source. Taking curl on both sides of (8a) and elimiusing (8b), one can obtain nating

(4) where , , and are vectors constituted by current expan, and , respectively. Matrices , sion coefficients of , , and are defined as

(9) In (9), there are totally six differential operators, which are expanded as follows. First, arbitrary field component can be expanded with interpolating bases as

(10) where (5)

(6)

represents regularized Lagrange interpolating bases, and is the order of the interpolating bases. The definition of can be found in [20], and it is omitted here for simplicity. is

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a point neighboring to the point , and , , and are in, , and -directions, respectively. dexes of the point in the , these differential operators can be Hence, at a point expressed as

Fig. 4. Illustration of the vector n ^

.

C. Hybridization On the surface of Huygens’ box, the equivalent currents and are related to the field solution in the DSC model by the equivalence principle

(13)

(11) and represent ordinary like derivatives of the first where and second orders, respectively. Differentials of interpolating bases can be numerically computed with a recursive procedure [29]. For numerical implementation of the DSC method, Yee’s grid [30] can be used to descretize the computational domain. may not be conformal with Since the current sources and the structured grids, the regularization technique [31] can be adopted to express the current sources on the structured grids. The validity of the regularization technique has been rigorously proved by Tornberg [31] with more details. Descretizing the computational domain with Yee’s grids, expressing the current sources on the Yee’s grid with the regularization technique, and approximating the differential operators with (11), the partial differential equation (9) is transformed to a set of linear equations for the unknown scalar field quantities (12) , and . is the unknown where spatial expansion coefficients for the electric field. is the difis the ferentiation matrix [29], and is the identity matrix. and are vector constituted by discrete sampling of . regularization matrices for and , respectively. Numerical and are summarized in Appendix A. computations of To satisfy the boundary condition of the conducting rectangular cavity, symmetric extension is used for normal electric or tangential magnetic field components near cavity walls. For tangential electric or normal magnetic field components near cavity walls, anti-symmetric extension is used to force them to be zero along the conducting walls.

where is the inward unitary vector normal to the Huygens’ and denote the magnetic and electric fields box. and on Huygen’s box, respectively. In the MoM model, are expanded using the RWG bases. In order to relate the curand , one may inrent expansion coefficients with voke a Galerkin’s matching process upon (13). However, that will introduce extra computation. On the other hand, an interesting property of RWG basis is that the expansion coefficient basis happens to be the value of the current comfor the shared edge. Utilizing the aforemenponent normal to the tioned property of the RWG basis, one can easily obtain

(14) and are the th current expansion coefficients for where and . is the unitary vector normal to the shared edge, which lies in and points towards the free vertex of , as shown in Fig. 4. can be an arbitrary point on the shared edge. In this work, is chosen to be the middle point of the edge. The matrix form of (14) is

(15) and are, respectively, vectors constituted where by values of magnetic and electric fields at the centers of shared edges. Matrices and represent the translation between fields and currents, and their definition is detailed in Appendix B. Since may not happen to be a grid point in the DSC model, interpolation may be required to obtain the electric and magnetic fields at . By the field expansion (10), -component of and can be calculated as

(16)

ZHAO AND SHEN: EFFICIENT MODELING OF THREE-DIMENSIONAL RC USING HYBRID DSC-MoM

where the superscript represents the direction of the field comand are ponent, and it can be , , or . values of -component of electric and magnetic fields at the )th nearest grid point from . , , and are indexes in , ( , and -directions for DSC grid points surrounding . and represent interpolating weights for -component of electric and magnetic fields, respectively. It should be noted that all field components have mutually different grid points in Yee’s grids. Therefore, with the same index ( , , ), the values of interpolating weight are different for every field component. Equation (15) can be written in the matrix form as

(17) where vector is constituted by the expansion coefficients of magnetic field at DSC grid points. The interpolation matrices and are defined as

denotes the index of the shared edges, and represents the index of the DSC grid point in or . is the coordinate of the middle point for the shared edge. is the coordinate of the corresponding DSC grid point. , , and are values of interpolating , , functions in the x-, y-, and z-directions respectively. and are DSC grid sizes in x-, y-, and z-directions. The superscript and , respectively, denote the electric and magnetic types. Using (17), (15) is rewritten as

(18) Using (18),

and

in (4) can be eliminated, resulting in (19)

in (19) can be eliminated with utilizing (8a). By discretizing the curl operator in (8a) with the DSC method, the following relationship is obtained (20) where is a differentiation matrix arising from the curl operator of (8a). Using (20), (19) can be further simplified as (21) where

Combining (21) with (12) and rearranging the equations, one can easily obtain (22)

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and the induced curBy solving (22), the electric field rent can be obtained. In (22), the differentiation matrix is sparse, and therefore most part of the left-hand-side matrix in (22) is sparse. At the same time, it is known that iterative solvers have the problem of slow convergence in solving matrix equation arising from RC modeling due to its resonant nature. Hence, in this work, the direct sparse matrix solver from UMFPACK [32] is employed for the solution of (22). The derivation so far is based on the assumption that the RC is lossless. In practice, an RC has losses due to the cavity wall, the aperture leakage, a receiving antenna, and the loss in the equipment under test [33]. In order to take the small loss into account, the homogeneous loss model by Kildal [8] is adopted in this work. In the loss model by Kildal, the permittivity is assumed to be complex, and its value is calculated from the quality factor of an RC by (23) where and represent the permittivity of free space and the lossy material, respectively. In order to consider the loss, one only needs to replace with in the previous formulation. It is known that the loss from a receiving antenna dominates the value of quality factor at low frequencies [33]. Therefore, Q can be calculated as [33] (24) where is the volume of the cavity, and the frequency of interest.

is the wavelength at

III. NUMERICAL EXAMPLES Numerical examples are presented in this section to demonstrate the validity and advantages of our proposed method. A C++ code is written to implement our hybrid method. The performance of the hybrid method is compared with the commercial software FEKO. Unless otherwise stated, simulations are conducted on a personal computer with a 2.67 GHz CPU and a 3.25 GB RAM. In FEKO and our MoM model, the maximum length of the triangle edge is chosen to be one-tenth of . For multi-frequency analysis, frequency sweeping with a uniform frequency step size is adopted, and is chosen to be the wavelength at the highest frequency of interest. All integrals are calculated with a four-point Gaussian quadrature. Because iterative solvers have the problem of slow convergence [14], the direct solver is used in FEKO. In simulations that follow, the source is chosen to be a point source. current A. Code Validation Using an RC with a single-plate stirrer, the hybrid code is first validated against FEKO, and its advantages over FEKO are demonstrated. The geometry of the RC is shown in the inset of Fig. 5. The single-plate stirrer is 4.25 m away from the bottom of the cavity, and it is parallel to the bottom surface of the cavity. The longer side of the plate is along the -direction. The current is located at the point (2 m, 2 m, 1.6 m), and all the source three orthogonal components of are set to be . The dimension of the large cavity is 8.5 m 12.5 m 6 m. Starting

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Fig. 5. L error of jE j along a line inside the RC shown in the inset. The cavity is of dimension 8.5 m 2 12.5 m 2 6 m. The stirrer is a rectangular plate of dimension 8 m 2 0.8 m, and its center is located at the point (6.6 m, 6.25 m, 4.25 m). A point current source is located at P (2 m, 2 m, 1.6 m). All three orthogonal components of the current source is set to be 1 A=m . Field strength is observed along the straight line connecting points P (1 m, 10.5 m, 3 m) and P (7.5 m, 10.5 m, 3 m).

from 41 MHz, the mode density of the cavity is above 1 mode per MHz. Therefore, the RC can theoretically operate from 41 MHz. According to [9], the most interesting frequency band for this RC would be 41 MHz to 82 MHz, where its performance needs analysis and optimization. In the present work, the order of the DSC method is fixed is calculated to be twelve. At the frequency of 82 MHz, along the line shown in the inset of Fig. 5. According to (23) and . (24), the permittivity is set to be error of using different spatial samFig. 5 presents the pling densities. The reference results are obtained using the grid , where . , size of , and represent the grid size in x-, y-, and z-directions, respectively. It is seen that the grid size can be chosen to be around five points per wavelength. Therefore, all simulation results that follow are obtained with DSC order of twelve and grid . size around Fig. 6 to Fig. 8 present the field distribution along the straight line indicated in the inset of Fig. 5 at the frequency of 82 MHz. Results from our proposed hybrid method agree well with those from FEKO. Fig. 9 presents the observed electric field strength at the center of the RC from 40 MHz to 82 MHz. The frequency step is 0.1 MHz. Good agreement is again obtained between our proposed hybrid method and FEKO. Table I summarizes CPU time and memory cost comparisons between our proposed method and FEKO. The proposed hybrid method is about ten times faster than FEKO. In FEKO, filling the interaction matrix requires calculation of integrals over the cavity. Furthermore, the LU decomposition of a full matrix is time consuming. In the proposed hybrid technique, integral over the surface of a large cavity is avoided. In modeling the cavity, the differentiation matrix D in this example has a sparsity of 95.55%, and its filling is fast. Meanwhile, LU decomposition of a sparse matrix is efficient as well. More importantly, with coarse grids in the DSC method, the number of unknowns is

Fig. 6. jE j along the straight line between points P and P in the RC shown in the inset of Fig. 5.

Fig. 7. jE j along the straight line between points P and P in the RC shown in the inset of Fig. 5.

Fig. 8. jE j along the straight line between points P and P in the RC shown in the inset of Fig. 5.

reduced to one-fourth of that in FEKO. The memory usage of the proposed method is only one third of that by FEKO. It should

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TABLE I TIME AND MEMORY COST COMPARISONS BETWEEN OUR HYBRID METHOD AND FEKO FOR ONE STIRRER POSITION AND ONE FREQUENCY POINT

Fig. 9. E (E = jE j + jE j + jE j ) at the center of the RC (illustrated in the inset of Fig. 5) from 40 MHz to 82 MHz.

be noted that the memory cost in the proposed hybrid method is mainly caused by the LU solver. This is different from the case in FEKO, where both interaction matrix and LU solver consume a lot of memory. B. Study of Stirring Effect After having validated the proposed hybrid method and demonstrated its high efficiency, the proposed method is used to study the stirring effect of three stirrers. The first stirrer is the one shown in the inset of Fig. 5, which is a single-plate stirrer of dimension 0.8 m 8 m. The second one is a larger single-plate stirrer of dimension 1.6 m 8 m. The third one is a five-paddle stirrer consisting of five equal-sized square plates of side length 1.6 m. Geometry of the third stirrer is shown in Fig. 10. For the single-plate stirrers, the center of the stirrer is located at the point (6.6 m, 6.25 m, 4.25 m). For the five-paddle stirrer, the center of paddle III is located at the point (6.6 m, 6.25 m, 4.25 m). All three stirrers are assumed to be rotated clockwise (looking toward the positive y-direction) around a y-directed straight line passing through the point (6.6 m, 6.25 m, 4.25 m). The angle by which a stirrer has been rotated is named as the rotation angle . Figs. 11–13 illustrate the strength of the three orthogonal electric field components at the point (4.25 m, 6.36 m, 3 m) during one stirrer rotation. In all three cases, a point source is adopted as the excitation. The definition of the point source is the same as in Section III-A. The step size of rotation angle is 1 degree. For single-plate stirrers, 180 stirrer positions are considered, because a single-plate stirrer goes back to its original position after being rotated by 180 degrees. For the five-paddle

Fig. 10. Geometry of an RC with a five-paddle stirrer. Dimension of the five paddles is the same. Angle between two adjacent paddles is 122.09 . The center of paddle III is located at (6.6 m, 6.25 m, 4.25 m). The dimension of the cavity is the same as the one shown in the inset of Fig. 5.

stirrer, 360 stirrer positions are considered. From Figs. 11–13, one can easily observe different stirring effects of the three stirrers. A single-plate stirrer of dimension 0.8 m 8 m has little stirring effect, though its length is several wavelengths. When the width of the single-plate stirrer is increased to 1.6 m, field strengthes change a lot during one stirrer rotation. This phenomena indicates that the width of the stirrer should be large enough to make the stirrer effective. With the five-paddle stirrer, more drastic variations are observed during one stirrer rotation. This is reasonable and expected, since the five-paddle stirrer introduces more irregularities during one stirrer rotation. Figs. 14–16 show the electric field strength distribution along , and ) in a plane ( the RC. The electric field strength distribution is plotted at four , 45 , 90 and 135 . Fig. 14 stirrer positions, which are indicates that the small single-plate stirrer doesn’t cause noticeable variation of the electric field strength distribution. With the larger single-plate stirrer and the five-paddle stirrer, the electric field strength distribution changes a lot as the stirrer rotates, which can be observed in Figs. 15 and 16. When calculating the electric field strength, the spatial step size is 0.1 m in both and -directions. Therefore, for one stirrer position, the electric field strength is calculated at 4026 points. If FEKO is used, it will take a long time to calculate the electric field strength at such a large number of field points. This is because FEKO needs to calculate the integral over a large cavity in computing the electric field strength. In the proposed method, electric field strength can be quite efficiently calculated using the field expansion (10). In calculating the electric field strength at 4026 points, the proposed method only takes 1.7 seconds. Therefore, the proposed method renders efficient calculation of electric

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Fig. 11. Electric field strength at the point (4.25 m, 6.36 m, 3 m) during one stirrer rotation with a single-plate stirrer of dimension 0.8 m 8 m. The excitation frequency is 82 MHz.

Fig. 13. Electric field strength at the point (4.25 m, 6.36 m, 3 m) during one stirrer rotation with the five-paddle stirrer illustrated in Fig. 10. The excitation frequency is 82 MHz.

Fig. 12. Electric field strength at the point (4.25 m, 6.36 m, 3 m) during one stirrer rotation with a single-plate stirrer of dimension 1.6 m 8 m. The excitation frequency is 82 MHz.

Fig. 14. Distribution of E along a plane in the RC at four stirrer positions with excitation frequency of 82 MHz. The stirrer is the single-plate stirrer of dimension 0.8 m 8 m.

field strength at a large number of field points, which is of great interest in RC modeling. Table II presents the memory cost and total CPU time for one stirrer position. For the three RCs, the proposed method achieves the analysis of one stirrer position in a few hundred seconds, while FEKO may take thousands of seconds CPU time. The maximum peak memory cost in the three cases is 0.88 GB. It should be noted that the memory cost and CPU time are affected not only by the number of unknowns, but also by the sparsity of the left hand side matrix of (22). Therefore, the memory cost and CPU time for the second and third stirrers are different, though the numbers of unknowns are similar. However, the left hand side matrix of (22) is always sparse, because its major part is the differentiation matrix. Therefore, the memory costs presented in this work are representative.

Lastly, the proposed method is used to model the RC with five-paddle stirrer at 164 MHz to investigate its ability in modeling a large RC. The simulation is conducted on an IBM x3950M2 server with 48 cores and 256 GB RAM. Every core has a clock speed of 2.66 GHz, and one core is used in this work. For this example, the number of unknowns is 34943. Four stirrer positions are simulated. Fig. 17 illustrates the field , distribution along a plane ( and ) in the RC at four stirrer positions. For one stirrer position, the simulation takes 1.94 hours CPU time and 45.57 GB memory. In this simulation, the memory cost is dominated by the solution of matrix equation. To avoid the solution of matrix equation, the recursive update (RU) technique in frequency domain [34] can be used, and its application to the proposed method is under development.

2

2

2

ZHAO AND SHEN: EFFICIENT MODELING OF THREE-DIMENSIONAL RC USING HYBRID DSC-MoM

Fig. 15. Distribution of E along a plane in the RC at four stirrer positions with excitation frequency of 82 MHz. The stirrer is the single-plate stirrer of dimension 1.6 m 8 m.

2

Fig. 16. Distribution of E along a plane in the RC at four stirrer positions with excitation frequency of 82 MHz. The stirrer is the five-paddle stirrer illustrated in Fig. 10.

TABLE II TIME AND MEMORY COST FOR ONE STIRRER POSITION AND ONE FREQUENCY POINT WITH THE PROPOSED HYBRID METHOD

IV. CONCLUDING REMARKS This paper has described a new hybrid technique combining the DSC method and MoM for efficient modeling of a three-dimensional RC. The RC modeling has been implemented in two

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Fig. 17. Distribution of E along a plane in the RC at four stirrer positions with excitation frequency of 164 MHz. The stirrer is the five-paddle stirrer illustrated in Fig. 10.

steps by replacing the metallic stirrer with a current sheet and enforcing the tangential electric field along the stirrer surface to be zero. In the first step, the tangential electric field along the stirrer surface is enforced to be zero by the application of an EFIE. In the second step, the current sheet and the original current source are taken as the excitation of the large metallic cavity. The MoM is used to solve the EFIE in the first step, and the DSC method is adopted to solve the cavity problem in the second step. The two steps are combined together by the equivalence principle. The proposed hybrid method combines the flexibility of the MoM and the high efficiency of the DSC method. Employing the geometry flexibility of RWG basis, it can be used to model stirrers of arbitrary shape. With the high efficiency of the DSC method, the proposed hybrid method has been shown to be ten times as fast as FEKO. The memory cost by the proposed method is also less than that of FEKO. With a single personal computer, it takes the proposed hybrid method a few hundred seconds in analyzing one stirrer position of a practical medium-sized RC, for which FEKO may need thousands of seconds CPU time. Meanwhile, with the field expansion (10), the proposed method can efficiently calculate the electric field strength at a large number of field points. For a demonstration purpose, only a point source is considered in this paper. However, with the flexibility of the regularization technique, line or surface sources can be easily modeled using the proposed method. Due to the usage of a direct solver, the memory requirement is high in solving a large scale matrix equation. Therefore, the proposed method is particularly suitable for modeling a medium-sized object inside a metallic cavity. If the cavity is very large and the object in the cavity is dipole antenna), the cavity Green’s function small (e.g., a method [6], [8] will be a better choice. Future work is to develop an RU [34] solution technique for the proposed method, which will reduce the memory usage drastically in solving large scale problems.

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APPENDIX A DEFINITION OF MATRICES

AND

For an arbitrary current distributed on a three-dimensional surface , using the regularization technique, the regularized at the DSC grid point can be computed as current

. Since both the magnetic and electric fields where have three components, one can observe from (31) that matrices and are of size , and they are defined as

(25) and . nents of the vector

where

represents the regularized Dirac function defined in [31]. and are coordinates of and , reinto (25), one can spectively. Substituting the expansion of derive (26) Depending on the direction of the field component designated , different components should be extracted in (26). From at as (26), one can obtain the definition of (27) where the unitary vector represents the direction of the electric field component designated to the th DSC grid point. may be a surface or line current source. When is a surface current source, it can be expanded using the RWG basis, and then its regularization matrix will be the same as (27). If is a line current source, its regularized form at is (28) where represents the domain of . can be of arbitrary shape. Expanding with pulse function basis and replacing the integration with a trapezoidal rule, (28) becomes (29) where is the total number of discrete segments for , and is the length of the segment. is the center of . Therefore, matrix has the form of (30) If is a point current source, the integral in (28) will reduce to a product. APPENDIX B DEFINITION OF MATRICES Using the vector identity rewritten as

AND

, (14) is

(31)

, , and are the , , and compo, respectively.

REFERENCES [1] IEC 61000-4-21-Electromagnetic Compatibility (EMC)-Part 4-21: Testing and Measurement Techniques-Reverberation Chamber Test Methods, CISPR/A and IEC SC 77B, International Electrotechnical Commission (IEC), Geneva, Switzerland Int. Std., Aug. 2003. [2] P.-S. Kildal, C. Carlsson, and J. Yang, “Measurement of free space impedances of small antennas in reverberation chambers,” Microw. Opt. Tech. Lett., vol. 32, no. 2, pp. 112–115, Jan. 2002. [3] 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, no. 12, pp. 104–112, Dec. 2004. [4] G. Lerosey and J. Rosny, “Scattering cross section measurement in reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 49, no. 2, pp. 280–284, May 2007. [5] U. Carlberg, P.-S. Kildal, A. Wolfgang, O. Sotoudeh, and C. Orlenius, “Calculated and measured absorption cross sections of lossy objects in reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 46, no. 2, pp. 146–154, May 2004. [6] U. Carlberg, P.-S. Kildal, and J. Carlsson, “Study of antennas in reverberation chamber using method of moments with cavity Green’s function calculated by Ewald summation,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 805–814, 2005. [7] E. Laermans, L. Knockaert, and D. D. Zutter, “Two-dimensional method of moments modeling of lossless overmoded transverse magnetic cavities,” J.Comput. Phys., vol. 198, pp. 326–348, Feb. 2004. [8] K. Karlsson, J. Carlsson, and P.-S. Kildal, “Reverberation chamber for antenna measurements: Modeling using method of moments, spectral domain techniques, and asymptote extraction,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3106–3113, 2006. [9] C. Bruns and R. Vahldieck, “A closer look at reverberation chambers-3-D simulation and experimental verification,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 612–626, Aug. 2005. [10] C. Bunting, “Shielding effectiveness in a two-dimensional reverberation chamber using finite-elment techniques,” IEEE Trans. Electromagn. Compat., vol. 45, no. 3, pp. 548–552, Aug. 2003. [11] G. Orjubin, E. Richalot, S. Mengue, M.-F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat., vol. 49, no. 1, pp. 76–85, Feb. 2007. [12] F. Moglie, “Convergence of the reverberation chambers to the equilibrium analyzed with the finite-difference time-domain algorithm,” IEEE Trans. Electromagn. Compat., vol. 46, no. 3, pp. 469–476, Aug. 2004. [13] A. Coates, H. S. Amd, D. Coleby, A. Duffy, and A. Orlandi, “Validation of a three-dimensional transmission line matrix (TLM) model implementation of a mode-stirred reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 49, no. 4, pp. 734–744, Nov. 2007. [14] C. Bruns, “Three-Dimensional simulation and experimental verification of a reverberation chamber,” Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, 2005. [15] J. M. Jin and J. L. Volakis, “Te scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag., vol. 38, no. 8, pp. 1280–1286, Aug. 1990. [16] X.-Q. Sheng, E. K.-N. Yung, C. H. Chan, J. M. Jin, and W. C. Chew, “Scattering from a large body with cracks and cavities by the fast and accurate finite-element boundary-integral method,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1153–1160, Aug. 2000.

ZHAO AND SHEN: EFFICIENT MODELING OF THREE-DIMENSIONAL RC USING HYBRID DSC-MoM

[17] C. Feng and Z. Shen, “A hybrid FD-MoM technique for predicting shielding effectiveness of metallic enclosures with apertures,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 456–462, Aug. 2005. [18] Z. Huang, K. Demarest, and R. Plumb, “An FDTD/MoM hybrid technique for modeling complex antennas in the presence of heterogeneous grounds,” IEEE Trans. Geosci. Remote Sensing, vol. 37, no. 6, pp. 2692–2698, Nov. 1999. [19] Y. Lin, J.-H. Lee, M. C. J. Liu, J. A. Mix, and Q. H. Liu, “A hybrid SIM-SEM method for 3-D electromagnetic scattering problems,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3655–3663, Sep. 2009. [20] Z. Shao, Z. Shen, Q. He, and G. Wei, “A generalized higher order finite-difference time-domain method and its application in guidedwave problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 856–861, Mar. 2003. [21] Z. Shao, G. Wei, and S. Zhao, “DSC time-domain solution of Maxwell’s equations,” J. Comput. Phys., vol. 189, pp. 427–453, 2003. [22] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett., vol. 15, no. 3, pp. 158–165, Jun. 1997. [23] Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 131–134, 2001. [24] S. Yu, S. Zhao, and G. W. Wei, “Local spectral time splitting method for first- and second-order partial differential equations,” J. Comput. Phys., vol. 206, no. 2, pp. 727–780, Jul. 2005. [25] Y. C. Z. S. Y. Yang and G. W. Wei, “Comparison of the discrete singular convolution algorithm and the Fourier pseudospectral method for solving partial differential equations,” Comput. Phys. Communi., vol. 143, no. 2, pp. 113–135, Feb. 2002. [26] H. Zhao and Z. Shen, “Hybrid discrete singular convolution-method of moments analysis of a two-dimensional transverse magnetic reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 52, no. 3, pp. 612–619, Aug. 2010. [27] W. Gibson, The Method of Moments in Electromagnetics. New York: Chapman and Hall, 2008. [28] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, May 1982. [29] B. Fornberg, A Practical Guide to Pseudospectral Methods. New York: Cambridge University Press, 1998. [30] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 5, pp. 302–307, May 1966. [31] A.-K. Tornberg and B. Engquist, “Numerical approximations of singular source terms in differential equations,” J. Comput. Phys., vol. 200, pp. 462–488, Nov. 2004. [32] T. A. Davis and I. S. Duff, “A combined unifrontal/multifrontal method for unsymmetric sparse matrices,” ACM Trans. Math. Softw., vol. 25, no. 1, pp. 1–19, 1999.

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[33] D. A. Hill, Electromagnetic Fields in Cavities: Deterministic and Statistical Theories. Hoboken, NJ: Wiley, 2009. [34] C. Pflaum and Z. Rahimi, “An iterative solver for the finite-difference frequency-domain (FDFD) method for the simulation of materials with negative permittivity,” Numer. Linear Algebra Appl., 2010.

Huapeng Zhao (S’08) was born in Hebei province, China, in 1983. He received the B. Eng. and M. Eng. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, in July 2004 and March 2007, respectively. He is currently working toward the Ph.D. degree at Nanyang Technological University, Singapore. His research interests include numerical methods and electromagnetic measurement in reverberation chamber.

Zhongxiang Shen (S’96–M’99–SM’04) received the B. Eng. degree from the University of Electronic Science and Technology of China, Chengdu, in 1987, the M. S. degree from Southeast University, Nanjing, China, in 1990, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1997, all in electrical engineering. From 1990 to 1994, he was with Nanjing University of Aeronautics and Astronautics, China. He was with Com Dev Ltd., Cambridge, Canada, as an Advanced Member of Technical Staff in 1997. He spent six months each in 1998, first with the Gordon McKay Laboratory, Harvard University, Cambridge, MA, and then with the Radiation Laboratory, the University of Michigan, Ann Arbor, as a Postdoctoral Fellow. In 1999, he joined Nanyang Technological University, Singapore, where he is presently an Associate Professor in the School of Electrical and Electronic Engineering. His research interests include design of small and planar antennas for various wireless communication systems, design of thin absorbing layers, hybrid numerical techniques for modeling RF/microwave components and antennas. He has authored or coauthored more than 100 journal papers. Dr. Shen is a member of the Antennas and Propagation and Microwave Theory and Techniques Societies of the IEEE. He served as Vice-Chair and Chair of the IEEE MTT/AP Singapore Chapter in 2008 and 2009, respectively. He currently serves as the AP-S Chapter Activities Coordinator.

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A Prescaled Multiplicative Regularized Gauss-Newton Inversion Puyan Mojabi, Member, IEEE, and Joe LoVetri, Senior Member, IEEE

Abstract—A prescaled multiplicative regularized Gauss-Newton inversion (GNI) algorithm is proposed which utilizes a priori information about the expected ratio between the average magnitude of the real and imaginary parts of the true contrast as well as the expected ratio between the average magnitude of the gradient of the real and imaginary parts of the true contrast. Using both synthetically and experimentally collected data sets, we show that this prescaled inversion algorithm is successful in reconstructing both real and imaginary parts of the contrast when there is a large imbalance between the average magnitude of these two parts where the standard multiplicative regularized Gauss-Newton inversion algorithm fails. We further show that the proposed prescaled inversion algorithm is robust and does not require the a priori information to be exact. Index Terms—Gauss-Newton inversion, microwave tomography (MWT), regularization.

I. INTRODUCTION

I

N microwave tomography (MWT), the goal is to reconstruct the complex permittivity of the object of interest (OI) using scattered field data collected outside the OI. Different iterative algorithms such as contrast source inversion [1]–[6], Gauss-Newton inversion (GNI) [7]–[11], and stochastic optimization methods [12]–[14] have been utilized to handle the nonlinearity of the problem. In conjunction with these iterative techniques, different regularization techniques such as additive [15], [16], multiplicative [3], [9], [10], and projectionbased regularization techniques [17], [18] have been used to treat the ill-posedness of the problem. Microwave tomography is of interest for various applications such as oil and gas-multiphase-flow imaging [19] and biomedical imaging [20]–[22]. In some applications of MWT, the magnitude of the real and imaginary parts of the OI’s permittivity can considerably be out of balance [23]. For example, in biomedical imaging, the real part of the permittivity can be much larger than the imaginary part. As a result of this imbalance, blind inversion algorithms inadvertently favor the reconstruction of the real part over the imaginary part. This imbalance usually results in an oscillatory

Manuscript received June 23, 2010; revised November 16, 2010; accepted November 17, 2010. Date of publication June 07, 2011; date of current version August 03, 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.2011.2158788

reconstruction of the imaginary part of the permittivity. To enhance the imaginary-part reconstruction for these cases, Meaney et al. developed a prescaled GNI algorithm based on Tikhonov regularization which optimizes over the imaginary part and an scaled version of the real part [23]. In this paper, we present a prescaled multiplicative regularized GNI algorithm which adjusts the average magnitude of the real and imaginary parts of the OI’s electric contrast throughout the inversion algorithm. In this method, the MWT problem is regularized with a weighted -norm total variation multiplicative regularizer which enforces a larger regularization weight on the imaginary-part reconstruction than the real-part reconstruction. As will be shown using synthetically and experimentally collected data sets, the standard multiplicative regularized Gauss-Newton inversion algorithm fails in reconstructing the imaginary part of the contrasts when a large imbalance exists between the average magnitude of the real and imaginary parts. We show that the proposed prescaled inversion algorithm is capable of reconstructing both real and imaginary parts of such targets. We further show that the algorithm does not require the a priori information to be exact. In general, the proposed algorithm can improve reconstruction results for the applications wherein the real and imaginary parts of the contrast are out of balance such as biomedical imaging applications. Within the framework of this paper, we consider the two-dimensional transverse magnetic (TM) illumination and assume a time factor of . II. PROBLEM FORMULATION containing a Consider a bounded imaging domain outside nonmagnetic OI and a measurement domain of the object of interest. The imaging domain is immersed in a known nonmagnetic homogeneous background medium with the relative complex permittivity of . Denoting the unknown by relative complex permittivity of the OI at the position , the complex electric contrast function is defined as (1) In MWT, the OI is successively interrogated with a number of known incident fields , where . Interacwith the OI results in the total tion of the incident field is then defined as the field . The scattered electric field difference between the total and incident electric fields corre. The sponding to the th transmitter; i.e., total and incident electric fields are then measured by some receiver antennas located on . Thus, the scattered electric field corresponding to the th transmitter is known on and denoted . The MWT problem may then be formulated as the by

0018-926X/$26.00 © 2011 British Crown Copyright

MOJABI AND LOVETRI: PRESCALED MULTIPLICATIVE REGULARIZED GNI

minimization over tional

of the least-squares data misfit cost-func-

(2) is the simulated scattered field at the observawhere tion points corresponding to the contrast and the th transdenotes the 2-norm on . The weighting is mitter, and chosen to be . III. MULTIPLICATIVE REGULARIZED GNI (MR-GNI) The GNI method is based on the Newton optimization, but ignores the second derivative of the scattered electric field with respect to the contrast function. That is, the scattered field due is approximated as to the contrast (3) This approximation, which we refer to as the GNI approximation, utilizes only the first two terms of the Taylor’s expansion. As far as the updating scheme is concerned, the contrast at the th iteration of the GNI method is updated as where is the predicted contrast at the th iterais the correction. tion, is an appropriate step-length, and To treat the ill-posedness of the problem, the data misfit cost-functional may be regularized using different regularizaby the weighted tion techniques [24]. Herein, we regularize -norm total variation multiplicative regularizer. That is, at the th iteration of the inversion algorithm, we minimize [9], [10], [24] (4) The multiplicative regularization term is given as (5) where the gradient is taken with respect to the position vector . Denoting the area of the imaging domain by , the weighting is given as function (6) The choice of the steering parameter is described below. It is worth noting that this multiplicative regularizer provides an edge-preserving regularization where the regularization weight is chosen by the algorithm itself [24]. In the discrete setup, we discretize the imaging domain into cells using 2D pulse basis functions. Thus, the contrast func. Assuming tion is represented by the complex vector the number of measured data to be , the measured scattered data on the discrete measurement domain is denoted by the complex vector . The vector is the stacked version of the measured scattered fields for each transmitter. is then formed by stacking the disThe vector . The matrix represents the discrete forms of crete form of the derivative of the scattered field with respect

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. That is, it repreto the contrast and evaluated at . The Jacobian masents the discrete form of is then formed by stacking matrices trix where . The positive parameter is chosen to be where is the area of a single cell in the uniformly discretized domain [9], [10], [24]. Minimizing over the complex vector , the complex correction vector may be found from [9], [10], and [24] (7) where the discrepancy vector is given as and . The regularization operator represents the discrete form of the operator where is the divergence operator. We note that the weighted Laplacian operator, , provides edge-preserving characteristics for the inversion algorithm [10]. This completes the brief explanation of the MR-GNI method. IV. PRESCALED MR-GNI (PSMR-GNI) Assume that there exist two pieces of a priori information about the OI: i) the expected ratio between the average magnitude of the real and imaginary parts of the OI’s contrast, and ii) the expected ratio between the average magnitude of the gradient of the real and imaginary parts of the OI’s contrast. Denoting the real and imaginary parts of the OI’s contrast by and , respectively, we suppose that the average magnitude of is approximately times larger than the average magnitude of . We further assume that the is approximately times average magnitude of . larger than the average magnitude of To incorporate these two pieces of a priori information into the inversion algorithm, we take three main steps. First, we formulate the problem in the real-domain as opposed to the complex-domain formulation presented in Section III. Second, we utilize a prescaled weighted -norm total variation multiplicative regularizer which enforces a similar weight on the average and . Third, we balance the real and magnitude of imaginary parts of the correction vector at the updating stage of the algorithm. Formulating the optimization problem in terms of the real and , at the th and imaginary parts of the contrast; i.e., iteration, we minimize the cost-functional (8) where is equal to assuming that and belong to the space of real functions defined on . The is chosen prescaled multiplicative regularizer as

(9) where

(10)

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and and are the real and imaginary parts of the predicted contrast at the th iteration of the GNI algorithm. As can is chosen to be times more be seen, the weight of than that of so as to balance the contribution of these two quantities in the multiplicative regularizer. We note that when is chosen to be 1, will be the same as given in (5). Usually, there is no a priori information about the ratio between the average magnitude of the gradient of the real and imaginary parts of the contrast. However, from our numerical experiments, we have found when there is a large imbalance beand , there is a similar imtween the average magnitude of and . Thus, balance between the average magnitude of equal to . Although we for realistic targets, we can set , we set formulate the problem in terms of and in all the numerical results, unless otherwise stated. It is useful to keep them separate in the derivation of the algorithm so that their individual effect on the final algorithm can be clearly identified. In the discrete domain, the real-valued correction vectors and may be found by solving (11) where is the gradient vector at the th iteration of the inverrepresents the approximate sion algorithm. The matrix calculated using the GNI apform of the Hessian matrix proximation (3). The conjugate gradient (CG) algorithm is used to solve (11). However, due to the imbalance between and , we first equilibrate this linear system of equations utilizing the scheme used by Meaney et al. [23]: we introduce ; thus, balancing the average a dummy variable and . We then optimize over and . magnitude of Thus, instead of solving (11), we solve (12) where the prescaled gradient vector is given as

(13)

and the matrix

is the prescaled Hessian matrix

(14)

and the matrix

as

(16) is the discrete form of the operator . Once , and are found, the real and imaginary parts of the contrast are updated as and . As far as the regularization is concerned, the edge-preserving in the comcharacteristics of the regularization operators, plex-domain and in the real-domain, are governed by and , respectively, [24]. When there is a large imbaland , ance between the average magnitudes of is dominated by that of . the average magnitude of Thus, if the complex-domain formulation is used for such cases, the edge-preserving characteristic of the algorithm is effectively . However, in the governed only by the magnitude of can prescaled real-domain approach, the magnitude of and have similar also play a role as long as magnitudes, see (10). This explains why we need such a weight . To justify the presence of in the other factor of in the integrand of (9), we note that we want the value of evaluated at and to be 1 at each iteration of the algorithm as the main goal of the optimization is to minimize not . , solving (11) is equivWe remark that when alent to solving (7). This can be checked by multiplying the second row of (11) by and adding that to the first row of (11), and is a verification that optimizing in the real-domain is equivalent to that in the complex-domain. This completes the description of the prescaled MR-GNI method to which we refer as the PSMR-GNI method. Finally, we note that in practical applications, there is usually some a priori information about the dielectric properties of the object being imaged. For examples, in biomedical applications, we usually know the average dielectric properties of the biological tissues being imaged. Noting that the dielectric properties of the background medium are also known, the average contrast of the tissues being imaged is available which can be used in setting the value of . However, it is usually not straightforward to have a priori information which can be used in setting the . As explained earlier in this section, we can set value of equal to in the practical cases where there the value of is a large imbalance between the real and imaginary parts of the contrast. The operator

V. SYNTHETIC DATA RESULTS calculated under the GNI approximation (3). As shown in the Appendix, the prescaled gradient vector can be conveniently written as (15)

We first consider the target shown in Fig. 1(a) and (b) which has the same geometry as the target used in [25] and [26] for a resolution test study. This target has features of various dimensions 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. The corresponding contrast is about

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Fig. 1. First synthetic target (left: Re( ) and right: Im( )) (a), (b) its true relative complex permittivity, (c), (d) its reconstruction using MR-GNI, and (e), (f) its reconstruction using PSMR-GNI with Q = Q = 10.

Fig. 2. The reconstructed relative complex permittivity (left: Re( ) and right: Im( )) of the first synthetic target using (a), (b) PSMR-GNI with Q = Q = 20, (c), (d) PSMR-GNI with Q = Q = 40, and (e), (f) PSMR-GNI with = 60. Q = Q

at ; thus, the true ratio between the real and imaginary parts of the OI’s contrast is about 40. Noting that the OI is a is very close to the true homogeneous target, the required value of , based on the numerical evaluation of the gradient on the grid. The synthetic scattering data, which includes 16 transmitters and 16 receivers per transmitter, is generated using square. a grid of 150 150 square pulses in a We have also added 3% RMS additive white noise to the synthetic data set using the formula given in [27]. The imaging doand is discretized into main is chosen to be a 71 71 square pulses. The inversion of this data set using the MR-GNI method is shown in Fig. 1(c) and (d). As can be seen, the imaginary-part reconstruction is not satisfactory. Moreover, one of the three upper left details of the target is very blurred in the real-part reconstruction. Using the PSMR-GNI method with , both real- and imaginary-part reconstructions, see Fig. 1(e) and (f), are sucessful. The inversion results using the PSMR-GNI method for three more values of and are shown in Fig. 2. As can be seen, the prescaled inversions and are corresponding to successful in reconstructing the real and imaginary parts of the contrast. However, the prescaled inversion begins to deteriorate . We note that setting and having at (and, vice versa) failed to reconstruct the imaginary part of this target (not shown here). Finally, it should be noted that the sensitivity of the quantitative accuracy of the reconstructed

image is low with respect to and when they change from 10 to 40. for the MR-GNI algorithm is The final data misfit value 0.2%, for the PSMR-GNI with and both equal to 10, 20, 40, and 60, it is 0.1%. It should be noted that a smaller data misfit cost-functional does not necessarily mean a better reconstruction due to the ill-posedness of the MWT problem. We next consider the target shown in Fig. 3(a) and (b). The right-most detail of this target has a relative complex permitand the rest of the target has a relative comtivity of plex permittivity of . The relative complex permittivity . That is, the target conof the background medium is and . sists of two different contrasts: As opposed to the first synthetic case where there was only one ratio between the real and imaginary parts of the contrast, this target consists of two different contrasts with two completely different ratios between the real and imaginary parts: the ratio between the real and imaginary parts of the contrast is about 3 in the right-most detail of the target and is about 40 in the rest of the target. For this target, the true is about 13 is about 10. The noisy synwhereas the true numerical is collected using the same procethetic data at dure used for the first synthetic data set as is the discretized imaging domain. The MR-GNI reconstruction of this target is shown in Fig. 3(c) and (d). As can be seen, the imaginary part of the contrast is very oscillatory. Using the PSMR-GNI method

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Fig. 3. Second synthetic target (left: Re( ) and right: Im( )) (a), (b) its true relative complex permittivity, (c), (d) its reconstruction using MR-GNI, and (e), f) its reconstruction using PSMR-GNI with Q = Q = 5.

with , both real- and imaginary-part reconstructions of the target are shown in Fig. 3(e) and (f). Except for undershooting the imaginary-part of the right-most detail, the reconstruction is satisfactory. The inversion results using the and PSMR-GNI method with are shown in Fig. 4(a)–(d). As can be seen, the inversion results using and are very sim. However, the ilar to the inversion results using . prescaled inversion begins to deteriorate at in the inWe have also utilized the true values for and version algorithm which resulted in a very similar reconstruc. That the reconstructed tion as the case where imaginary part of the rightmost detail of the target undershoots its true value and that the separation between the two rightmost details of the target has not been resolved in the reconstructed imaginary part is probably due to the fact the prescaled inversion algorithm provides an over-regularized reconstruction for that region of the target. However, it provides a reasonable regularization weight for the rest of the target. We note that the standard MR-GNI method provides an underregularized reconstruction for the whole imaginary part of the target. Finally, we remark that we have also inverted these two synthetic data sets using the multiplicative-regularized contrast source inversion (MR-CSI) algorithm as outlined in [3]. The inversion results using the MR-CSI method were very similar to the inversion results using the standard MR-GNI method.

Fig. 4. The reconstructed relative complex permittivity (left: Re( ) and right: Im( )) of the second synthetic target using (a), (b) PSMR-GNI with Q = = 10, (c), (d) PSMR-GNI with Q = Q = 13, and (e), (f) PSMR-GNI Q = 20. with Q = Q

VI. EXPERIMENTAL DATA RESULTS We consider two different MWT systems. The first system is the University of Manitoba air-filled MWT system [28] which utilizes 24 coresident Vivaldi antennas capable of collecting data from 3 to 6 GHz. The single-frequency measured data, 23 measurements, is calibrated for which consists of 24 the TM polarization as explained in [28]. In this paper, we choose the frequency of operation to be 3 GHz for this system. The second system is the Universitat Politècnica de Catalunya (UPC) MWT system. This system is a near-field 2.33-GHz microwave scanner system which consists of 64 water-immersed antennas equi-spaced on a 12.5-cm-radius circular array [29]. In this system, for each case of using one of the 64 antennas as a sole transmitter, field data is collected using only the 33 antennas positioned in front of the transmitting antenna. The measured data is then calibrated for the TM polarization. 1) Wood-Nylon Data Set: We utilize a circular nylon-66 cylinder with a diameter of 3.8 cm and an (approximately) square cross-section wooden block with the side of 0.087 m. The complex relative permittivities of wood and nylon are and at 3 GHz, respectively [28]. Thus, the ratio between the real and imaginary parts of the contrast is about 5 in wood and 67 in the nylon rod. In addition, noting the structure of the target, it is be close to the true . expected that the true numerical The target was placed in the University of Manitoba MWT

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Fig. 5. (a) Dielectric phantom consisting of nylon and wooden cylinders, its reconstruction (left: Re( ) and right: Im( )) using (b), (c) MR-GNI, and (d), = 10. (e) PS-MRGNI with Q = Q

Fig. 6. The reconstructed relative complex permittivity (left: Re( ) and right: Im( )) of the nylon-wood phantom using (a), (b) PSMR-GNI with = 20, (c), (d) PSMR-GNI with Q = Q = 40, and (e), (f) Q = Q = 70. PSMR-GNI with Q = Q

system as shown in Fig. 5(a). The inversion result using the MR-GNI method is shown in Fig. 5(b) and (c). As can be seen, the imaginary-part reconstruction is not satisfactory. Using the , see Fig. 5(d) and (e), PSMR-GNI method with both real- and imaginary-part reconstructions are satisfactory: the reconstructed imaginary part is not oscillatory and the reconstructed value for the real part of the nylon rod is more accurate compared to its reconstructed value using the MR-GNI method. The reconstructed imaginary part of wood using the PSMR-GNI method is about 0.12 which undershoots its true value. We note that the nylon rod is almost lossless; thus, it is very difficult to reconstruct its imaginary part considering the limited signal-to-noise ratio and dynamic range of the system. The inversion results using the PSMR-GNI method for ; namely , three more values of and , and are shown in Fig. 6. It is worth mentioning that the MR-CSI reconstruction of this target, which has been shown in [28], also results in a nonsatisfactory imaginary-part reconstruction. 2) FANTCENT Data Set: We next consider the FANTCENT phantom from the UPC Barcelona experimental data set which is shown in Fig. 7(a). The phantom consists of two thin plexiglass cylinders filled with two different concentrations of ethyl alcohol. The inversion results are constrained to lie within the and , region defined by as in [3]. The MR-GNI inversion of this data set is shown in

Fig. 7(b) and (c). Although the real-part reconstruction is satisfactory, the imaginary-part reconstruction is very oscillatory. The PSMR-GNI reconstructions with four different values for and are shown in Fig. 7(d) and (e) and Fig. 8. As can to be be seen, having equal to 5 and 10 and choosing the same as improves the imaginary-part reconstruction compared to the MR-GNI reconstruction. However, increasing and to 20 starts deteriorating the reconstruction. This is provides an overprobably due to the fact that regularized solution. We note that the ratio between the real and imaginary parts of the contrast is about 9.5 in 96% ethyl alcohol and 1.7 in 4% ethyl alcohol. The thickness of the two plexiglass cylinders are too small to be reconstructed (2 mm thickness for the outer cylinder and 1.5 mm for the inner cylinder); thus, we have not used the ratio between the real and imaginary parts of the plexiglass contrast in the PSMR-GNI algorithm. Having created a numerical phantom similar to the FANTCENT phantom, but excluding the plexiglass containers, we have found to be 5. the true value of to be 3 and the true value of The PSMR-GNI reconstruction for these two values of and are shown in Fig. 8(e) and (f). Finally, we note that the MR-CSI reconstruction of this target, shown in [3], provides a good overall reconstruction for both real and imaginary parts of the phantom. However, the PSMR-GNI algorithm provides a slightly more accurate quantitative reconstruction for the 4% ethyl alcohol. It should also be noted that in all the examples we

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Fig. 7. (a) FANTCENT phantom, its reconstruction (left: Re( Im( )) using (c), (d) MR-GNI, (e), (f) PSMR-GNI with Q = Q

) and right: = 5.

have tried, the quantitative accuracy of the reconstructed image . was not very sensitive to the values of and VII. CONCLUSION We have shown that the PSMR-GNI can provide a good reconstruction for both real and imaginary parts of the relative complex permittivity when there is a large imbalance between the real and imaginary parts of the OI’s electric contrast. It has also been demonstrated that the PSMR-GNI is not very sensitive to the choice of the prescaling parameters.

Fig. 8. The reconstructed relative complex permittivity (left: Re( ) and right: Im( )) of the FANTCENT phantom using (a), (b) PSMR-GNI with Q = = 10, (c), (d) PSMR-GNI with Q = Q = 20, and (e), (f) PSMR-GNI Q = 5. with Q = 3 and Q

To find the derivative operators of the data misfit cost-functional with respect to and at the th iteration, in the diwe start by finding the Gâteaux differential of rection [30, p. 468]

(19) and where which belongs to

APPENDIX

is an arbitrary complex function . Utilizing the little- notation, , (19) can be written as

The required derivative operators in the continuous domain are derived and then presented in their discretized forms. We spaces of complex functions defined on and denote the by and with the norms and inner products defined as and

(17) (20)

and and simplified to and (18) where the superscript denotes the complex conjugate operator.

(21)

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Denoting the adjoint operator by the superscript , (21) may be written as

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(27) may be written as

(22) where now the inner product is on tions, and , we have

. For two complex func-

(23) Thus, (22) may be written as (29) (24)

Therefore

The first term of (24) represents and the second term represents . We note that these to . derivative operators are linear mappings from To find the second derivatives and at the th iteration of the inversion algorithm, we start with finding the limit (30) (25)

and

Utilizing the operators just calculated, and the definition of the adjoint operator, the above limit may be written as

(31) To find the derivatives (26)

and , we start with finding the

limit This can be simplified to (32) After mathematical simplifications similar to the ones presented above, the limit (32) will be

(27) Noting that for two complex functions

and , we have (28)

(33)

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Noting (33) and utilizing (23), it can be concluded that

(34) and

(35) The second-order derivative operators in (30), (31), (34), and to . It should (35), are linear mappings from also be noted that in the Gauss-Newton inversion method, the is neglected and thus (30), (31), (34), and operator (35) are simplified. The PSMR-GNI method also requires the derivatives of with respect to and . Using the same procedure explained above, these derivatives can be derived. The closed-form expressions of these derivative operators are given in [31, Appendix D]. Having found the derivative operators in the continuous domain, the discretized forms of these operators can easily be found. For example, the discretized form of can be written as

(36)

where the superscript denotes the transposition operator. As another example, the discretized form of (35), calculated under the GNI approximation, can be expressed as

(37) REFERENCES [1] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Probl., vol. 13, pp. 1607–1620, 1997.

[2] A. Abubakar, W. Hu, P. van den Berg, and T. Habashy, “A finite-difference contrast source inversion method,” Inverse Probl., vol. 24, p. 065004, 2008. [3] 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. Microw. Theory Tech., vol. 50, no. 7, pp. 1761–1777, Jul. 2002. [4] C. Gilmore, P. Mojabi, and J. LoVetri, “Comparison of an enhanced distorted born iterative method and the multiplicative-regularized contrast source inversion method,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2341–2351, Aug. 2009. [5] P. Mojabi and J. LoVetri, “Eigenfunction contrast source inversion for circular metallic enclosures,” Inverse Probl., vol. 26, no. 2, p. 025010, Feb. 2010. [6] A. Zakaria, C. Gilmore, and J. LoVetri, “Finite-element contrast source inversion method for microwave imaging,” Inverse Probl., vol. 26, no. 11, p. 115010, 2010, (21pp). [7] 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. [8] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imag., vol. 9, no. 2, pp. 218–225, 1990. [9] A. Abubakar, T. Habashy, V. Druskin, L. Knizhnerman, and D. Alumbaugh, “2.5 D forward and inverse modeling for interpreting low-frequency electromagnetic measurements,” Geophysics, vol. 73, no. 4, pp. F165–F177, Jul.–Aug. 2008. [10] P. Mojabi and J. LoVetri, “Microwave biomedical imaging using the multiplicative regularized Gauss-Newton inversion,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 645–648, 2009. [11] 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. [12] M. Pastorino, “Stochastic optimization methods applied to microwave imaging: A review,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 538–548, Mar. 2007. [13] L. Garnero, A. Franchois, J. P. Hugonin, C. Pichot, and N. Joachimowicz, “Microwave imaging-complex permittivity reconstruction by simulated annealing,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 11, pp. 1801–1807, Nov. 1991. [14] S. Caorsi, A. Massa, and M. A. Pastorino, “A computational technique based on a real-coded genetic algorithm for microwave imaging purposes,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1697–1708, 2000. [15] 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. [16] A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov, V. G. Posukh, and Y. E. Sizov, “Three dimensional vector microwave tomography: Theory and computational experiments,” Inverse Probl., vol. 20, pp. 1239–1259, 2004. [17] 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. [18] 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. [19] Z. Wu, H. McCann, L. E. Davis, J. Hu, A. Fontes, and C. G. Xie, “Microwave-tomographic system for oil- and gas-multiphase-flow imaging,” Measure. Sci. Technol. vol. 20, no. 10, p. 104026, 2009 [Online]. Available: http://stacks.iop.org/0957-0233/20/i=10/a=104026 [20] S. Semenov, “Microwave tomography: Review of the progress towards clinical applications,” Phil. Trans. R. Soc. A, vol. 367, pp. 3021–3042, Jul. 2009. [21] 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. [22] S. Poplack, T. Tosteson, W. Wells, B. Pogue, P. Meaney, A. Hartov, C. Kogel, S. Soho, J. Gibson, and K. Paulsen, “Electromagnetic breast imaging results of a pilot study in women with abnormal mammograms,” Radiology, vol. 243, no. 2, pp. 350–359, 2007. [23] 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.

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[24] 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. [25] 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. Microw. Theory Tech., vol. 48, no. 4, pp. 538–544, Apr. 2000. [26] 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. [27] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “A robust iterative method for Born inversion,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 2, pp. 342–354, Feb. 2004. [28] 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. [29] 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. Microw. Theory Tech., vol. 39, no. 5, pp. 836–844, May 1991. [30] L. Debnath and P. Mikusin´ski, Introduction to Hilbert Spaces With Applications. Burlington, MA: Elsevier Academic, 2005. [31] P. Mojabi, “Investigation and development of algorithms and techniques for microwave tomography” Ph.D. dissertation, Univ. Manitoba, Winnipeg, Manitoba, Canada, 2010 [Online]. Available: http:// mspace.lib.umanitoba.ca/handle/1993/3946

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Puyan Mojabi (M’10) received the B.Sc. degree in electrical engineering from the University of Tehran, Iran, in 2002, the M.Sc. degree in electrical engineering from the Iran University of Science and Technology, in 2004, and the Ph.D. degree in electrical engineering from the University of Manitoba, Canada, in 2010. His current research interests are computational electromagnetics, microwave tomography, and inverse problems.

Joe LoVetri (SM’01) received the B.Sc. (with distinction) and M.Sc. degrees, both in electrical engineering, from the University of Manitoba, Canada, in 1984 and 1987, respectively. He received the Ph.D. degree in electrical engineering from the University of Ottawa. Canada, in 1991. He is currently a Professor in the Department of Electrical and Computer Engineering, University of Manitoba. His main interests lie in time-domain computational electromagnetics, modeling of electromagnetic compatibility problems, microwave tomography, and inverse problems.

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Reconstruction of the Electromagnetic Field in Layered Media Using the Concept of Approximate Transmission Conditions Özgür Özdemir, Houssem Haddar, and Alaaddin Yaka

Abstract—We propose a fast and stable method to reconstruct the electromagnetic field in layered media from overdetermined data on the outer boundary. This procedure can be used for instance as a data pre-processing in inversion algorithms to accurately image buried objects under known stratified structures. Our method is based on successive use of so-called approximate transmission conditions that provide accurate reconstructions for layers with sufficiently small width. The feasibility and efficiency of the proposed method are demonstrated by numerical experiments. Index Terms—Approximate transmission conditions, thin layers, rough surface, inverse scattering, multi-layered medium.

I. INTRODUCTION HE present work is motivated by microwave (or ultrasound) non-invasive diagnostics of multi-layered media: mine detection, tumor cells recognition, through wall surveillance, non destructive testing of coatings, etc. Classical imaging algorithms (see e.g., [1]–[5] and references therein) for those applications would require to determine the Green function associated with the complex background (which may be very costly). One possibility to avoid this is to evaluate the electromagnetic field in a homogeneous neighborhood of the target (or the investigated surface) from available data in the outer domain. This problem, known as data completion problem or the Cauchy problem is a classical inverse problem which is severely unstable and requires special treatments. In the present work we propose a fast and robust method to solve it in an approximate way using the concept of approximate transmission conditions (ATC). Our method is particularly suited for configurations where the layers are thin with respect to the wavelength (skin, wall, coatings, etc.) The concept of approximate (or asymptotic) models is commonly used in direct scattering problems in the presence of thin layers (see e.g., [6]–[10] and references therein). Whereas there

T

Manuscript received September 17, 2010; revised November 20, 2010; accepted December 10, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under Grant 108E173. Ö. Özdemir and A. Yaka are with the Electrical and Electronics Engineering Faculty, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey (e-mail: [email protected]). H. Haddar is with INRIA Saclay Ile de France & CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France (e-mail: [email protected]. fr). 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.2011.2158967

is a rich literature for thin coatings (see e.g., [11]–[13]), past work on accurate models for transmission layers is very sparse. This is particularly true when one considers models that accurately and efficiently account for general curvilinear behavior of the transmission layer (see e.g., [14]–[16]). This is why our first interest in this work will be the derivation of higher order conditions for arbitrary (regular interfaces). The main result is the expression of the fourth order ATC. The application to the Cauchy problem would consist into splitting the layer into sufficiently small ones, then apply the transmission conditions successively to each layer in order to construct the field at the lowest one. At each step only a boundary operator has to be inverted and the expression of this operator in independent from the data. The procedure is then particularly efficient in the case of multistatic data (measurements associated with many incident waves). Let us indicate however, that our procedure inherits from the instability of the exact Cauchy problem by obtaining badly conditioned linear systems. The condition number explodes either by letting the thin layers width or the mesh size used to discretize the boundary operator goes to zero. A trade-off between accuracy and stability is then needed. Numerical experiments show that this balance makes the results provided by the third order ATC non satisfactory for relatively thick layers. An appropriate modification of the fourth order ATC expression (without deteriorating the accuracy) enables us to obtain an interface condition that has roughly the same conditioning with respect to the mesh size as the third order one. The obtained ATC is well adapted to solve the inverse problem for thick layers. This is also demonstrated through numerical experiments. Let us quote that for this first study we shall restrict ourselves to the two-dimensional setting (considering for instance TM polarized electromagnetic waves) and only consider layers with constant thickness. For the derivation of ATC in the 3-D electromagnetic case we refer to [15] where the expression of ATC are obtained up to the third order for curved geometries and up to the fourth order for flat ones. The case of layers with non constant thickness can be treated in a similar way as [17] where generalized impedance boundary conditions for thin coatings are derived. These extensions will be the subject of future works. The outline of the paper is as follows. The first section is dedicated to the derivation of ATC for regular arbitrary curved layers. The technical details related to this section are given in the Appendix. The second section explains how these conditions can be used to solve the Cauchy problem for thin and thick layers. We conclude this study with numerical results that demonstrate the efficiency and limitations of this method.

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be a parametrization of defined on scissa . For

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in terms of the curvilinear abwe shall set

Fig. 1. Notation for domains and boundaries.

Consider now the truncated series II. DERIVATION OF ATC Let us consider a thin layer of thickness that encloses and be the outer and a connected domain of . Let inner boundary of the thin layer, respectively and be defined as , i.e., the parallel interface located half way between the two boundaries (see Fig. 1). We shall assume that is a regular surface and denote by the normal vector on such that the outer and inner boundaries of can be expressed . as be the third component of the electric field of a Let and respectively in and . TM polarized wave where We denote by the wave number inside : denotes the relative electric permittivity of the layer (we assume that the permeability is constant over the whole domain). Then it is assumed that

Then the ATC can be written in the form

(3) where denotes some boundary operator. The transmission term conditions obtained from (3) after neglecting the . According to (A.29), will be referred to as the ATC of order the first order ATC corresponds with , which means that we neglect the presence of the thin layer. This condition is therefore useless. The expressions of second and third order conditions respectively follows from (A.29)–(A.31). We then observe that they coincide and correspond to

(1) with the continuity conditions at the interfaces (4) (2) The problem we would like to solve (referred to in the sequel and on as the Cauchy problem) is to compute from the knowledge of and on . It is well known that this problem unstable and that the degree of instability increases with . An exact solution indeed would require the evaluation of . Our goal in the following is to solve this problem in an approximate way when is sufficiently small, by using an asymptotic development of the solution with respect to . More precisely, assuming that expansions of the inner of outer field of the form

where denotes the curvature of (see Appendix B). These expressions are in concordance with the one given in [15] for the 3D electromagnetic case. From (A.29)–(A.32) we deduce that the fourth order condition corresponds with

(5) hold, we shall derive ATCs that are verified by the boundary values of the truncated series up to error terms which are polynomial with respect to . We follow a similar procedure as in [15], based on so-called scaled asymptotic expansions, but we shall carry over computations for rough geometries up to higher orders (which turns out to be necessary as explained in the numerical results sections). The principle of the method and a sketch of the technical details are given in Appendix. The outcome of this procedure is the following. Let

where

denotes the derivative of

with respect to .

III. APPLICATION TO THE CAUCHY PROBLEM A. The Case of a Single Thin Layer Let us consider the configuration presented in the previous and (or an approximasection. We assume that of them) are given on the outer boundary. tion of order Then, according to (3), in order to obtain an approximation of

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order of and of equations posed on

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, one can solve the system for the unknowns and

better explain these concerns and the proposed solution let us introduce two new unknowns

(9) (6) then get an approximation of order from

of

Substituting (9) into (6) we obtain the following equivalent writing of the fourth order condition

and

(7) The system of (6) can be solved using a variational technique. Third order ATC: In the case of the third order condition, one obtain the variational writing of this system by multiplying each equation with a test function then integrate by parts the second order derivatives with respect to . We then seek approximated solutions of and in the form of (8) where and , are the set of unknown cois a set of basis functions chosen as rooftop efficients and functions that are constructed over a regular mesh of . Inherited from the ill-posedness of the exact problem, the inversion of the linear system (6) is unstable as . This can be seen after substituting for instance from the equation verified by for in the first equation of (6). One obtain

We clearly observe that the operator on the left hand side of the latter equation cannot be invertible for all values of and its inverse cannot be continuous as . From the numerical perspectives this implies that the condition number of the matrix associated with the discretized problem would have a condition number that explodes as or . A regularization procedure should then be incorporated. One possibility would be to use classical Tikhonov regularization with a regularization parameter that can be adapted to the noise level in the data and . As we shall numerically observe, this procedure introduce a balance between and that makes the results provided by the third order ATC non satisfactory for relatively thick layers. Fourth order ATC: The case of the fourth order condition needs a special treatment since 1) a variational writing of this condition would involve second order derivatives which are not compatible with rooftop basis functions and 2) the condition number for this condition may be much higher than for the third order ATC as . The latter problem comes from the fact that solving (6) amount for instance to invert the operators . However, the latter operation is unstable since the second order derivatives in comes with the bad sign. To

(10) with

The above mentioned instabilities can be explained with the fact that operators appearing in (9) are not stable approximations . In order to overcome this problem of the identity as and respectively with and we shall replace defined by

(11) and One easily checks that . Therefore making the substitutions and in (10) would not affect the accuracy (with respect to ) of the ATC. The obtained system of equations formed by (10)–(11) formally contains at most second order derivatives with respect to . It can then be solved variationally in a similar way as for the ATC of order three. Here we further need to expand as (12) and are unknown coefficients while for and where , the same expansions (8) are used. Note that the discretizaunknowns whereas tion of (10) leads to a linear system with unknowns are required to determine in the third order ATC. However, we numerically observed that the condition number of the obtained linear system has a similar behavior with respect to and as in the case of the third order ATC. This is of course a good news since the fourth order ATC has better accuracy in terms of and therefore would perform better in solving the Cauchy problem. B. Extension to a Multi-Layered Structure Consider an N-layered structure with rough interfaces as shown in Fig. 2. Assume that each layer has different characteristics, i.e., the thickness of the layer is and the wavenumber is . The ATC formulation explained in previous sections can be easily extended to this configuration by

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Fig. 2. Configuration for N-layered medium.

applying the formalism to each layer separately. This schematically results for each layer into an ATC of the form (13) where is the matrix representation of the ATC given by (6)–(7). Notice that this writing means that we set up by conand for vention, . Therefore the ATC associated with the overall structure can be written as, (14) C. Extension to Thick Layers The accuracy of the th ATC is determined by the order of which means that increment of the thickness of the layer causes deterioration in the accuracy of the ATC. To improve the accuracy of the derived ATC without the need to go for higher order conditions, we propose the application of Cascaded ATC (CATC): In this approach, the “thick” layer with thickness is layers having the same wavenumber but with divided into smaller thickness, then apply the ATC as in the case of multilayered structures, (13), (14). The formal accuracy provided by this procedure is which is indeed better than for large and positive . However the discussion on the stability with respect to the thin layer width indicates that cannot be chosen too large. Also the price for this improvement has to matrices inversion/or be paid as a computational burden of multiplication. IV. NUMERICAL RESULTS This section is devoted to the validation of Approximate Transmission Conditions for both cylindrical and planar media through some numerical experiments. The accuracy is analyzed by comparing the field distributions obtained by ATC with exact data which is synthetically calculated by solving the forward scattering problem (using an integral equation method). In all above the upper examples, a line source located at is the interface is used to generate the total field, where free space wavelength. The source frequency is chosen as . In all examples, a rectangular shaped perfect is electric conducting object with dimensions below from the located inside the lower medium, at lowest interface.

Fig. 3. Configurations for the numerical experiments. (a) Circular cylindrical medium (constant curvature), (b) non-circular cylindrical medium (variable curvature), (c) multilayered non-circular cylindrical medium (variable curvature), (d) planar medium with variable curvature.

CASE I. Cylindrical Mediums As a first example, a simple configuration which is composed of two concentric circular regions is considered, Fig. 3(a). The and discretization radius of outer circle is chosen as . It is worth noting here that the computational number burden of the proposed method (6) which leads to matrix for 3rd order ATC and matrix for th order ATC. The region between the circles (i.e., the thin layer) is filled with a lossy dielectric material having complex permittivity . The thickness of the layer is . The inner circle is assumed to be filled with a lossy dielectric with permittivity . The variations of exact and ATC modeled fields as well as their normal derivatives on the surface of outer cylinder are shown in Fig. 4. As it can be clearly observed that both 3rd order and 4th order ATC yield very accurate results. This is not an unexpected situation since the thickness of the layer is small enough in terms of wavelength in order to obtain accurate results while not too small to maintain stability. In order to analyze the effect of the thickness on ATC, same configuration is considered as in the first example, Fig. 3(a), We observe that while but with higher thickness of the 3rd order ATC provides poor results, the 4th order is almost distinguishable from the exact field, see Fig. 5. In this example, the range of accuracy and stability of cascaded approximate transmission conditions is investigated. To this aim a “thick” circular layer whose thickness and permitand , respectively is contivity are sidered, Fig. 3(a). First, 3rd and 4th order ATC are directly applied to original layer and then in order to apply CATC, we consider four different cascade configuration that we divide the original layer into 2, 3, 4 and 5 layers. Thickness values for cascade experiments are presented in Table I.

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TABLE II TABULATED ERROR OF FIELD AND ITS NORMAL DERIVATIVE FOR ATC AND CATC EXPERIMENTS

Fig. 4. Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig.3(a) and  = 0:1 .

Fig. 5. Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig. 3(a) and  = 0:3 .

TABLE I THICKNESS VALUES FOR CASCADED I AND CASCADED II EXPERIMENTS FOR  = 0:5 THICKNESS LAYER

In order to make precise determination of the accuracy of mean square error is defined by

CATC,

(15)

where is the approximated field by ATC and is the exact field. Error values for direct application of ATC and CATC experiments are presented in Table II. Before analyzing the resulta it is important to note that since the matrices of 3rd and 4th order ATC are badly conditioned, a regularization procedure should be used after the first step (since the application of these conditions introduce artificial errors in the data). As we mentioned earlier we have employed here the classical Tikhonov regularization. The choice of the regularization parameter has been experimentally fixed to obtain optimal results. One can see from Table II that cascading experiments improve both accuracy of 3rd and 4th order conditions. Increment in the layer number ameliorate the accuracy of 3rd condition. However this is not always true for 4th order CATC, increasing the layer number more than three results in deterioration of the accuracy of CATC in comparison the CATC I and CATC II experiments. This is due to the fact that error for and are almost the same in the case of 4th order ATC (due to the regularization effect). Therefore we cannot expect improvement for Cascade III and IV comparing to CATC II. In addition to Table II, we also plotted the comparison of field data for 3rd order and 4th order in Figs. 6 and 7 in order to visualize the effect of the error. For the sake of readability, only CATC I and CATC IV experiments are shown in these figures. The following experiment aim to study the effect of the curvature variation on ATC. We first consider a non-circular cylindrical medium which is defined by parametric equation

(16) . The geometry of the medium is depicted where and the in Fig. 3(b). The thickness of the layer is . We use mesh points permittivity is for this geometry. The field and the normal derivative of the field are compared with exact data as illustrated in Fig. 8. It is clear that while 4th order ATC yields quite accurate result, 3rd order ATC does not give good approximation due to the layer thickness, which is exactly the same case observation as in the second example. These results validates that the curvature and its derivative are treated adequately in approximate transmission conditions. As a last example concerning the cylindrical medium, three non-circular cylindrical mediums are considered in order to demonstrate the effectiveness of the presented method in multilayered media, Fig. 3(c). Parametric equation of the geometry is given by (16) as in the previous example. Permittivities of

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Fig. 6. Exact and 3rd order ATC modeled Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig. 3(a) and  = 0:5 .

Fig. 8. Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig. 3(b) and  = 0:2 .

Fig. 7. Exact and 4th order ATC modeled amplitude of (a) the field u and (b) the normal derivative of field (@u=@n) on the lower interface for configuration Fig. 3(a) and  = 0:5 .

Fig. 9. Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig. 3(c) and total thickness  = 0:3 .

layers are chosen as and respectively and permittivity of inner circle is . Thickness of each individual layer is which means that the total thickness of overall layer . is The field and its normal derivative on the lowest interface are determined by the use of 3rd order and 4th order ATC and their amplitudes are shown in Fig. 9. These results confirm that ATC can be applied to multilayered media. CASE II. Planar Mediums We have examined the accuracy and range of validity of the ATC in cylindrical medium in other words in closed domain.

As a final case, we demonstrated that ATC are not restricted to only closed interface, it is also feasible for planar mediums, i.e., open domains. For this aim, locally rough interfaced layer with is considered, Fig. 3(d). Roughness profile is chosen , while . as Permittivities of the thin layer and lower medium are and , respectively. Comparison of the amplitude of approximated and exact field values are given in Fig. 10. Note that the curvature is also variable for this surface. It can be observed that both 3rd and 4th order ATC exhibit the same characteristics as in the cylindrical medium, that’s to say 4th order ATC almost model perfectly the

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of potentials) that can be used. Our method is expected to maintain the same accuracy provided by this first step. We finally mention that possible applications of ATC are not limited to the determination of missing data. For instance, if the thin layer is accessible from both sides, ATC can also be employed to determine the (possibly variable) thickness and the physical characteristics of the layer.

APPENDIX A PRINCIPLE OF THE DERIVATION OF ATC The principle of the derivation of ATC is based on the use of so-called scaled asymptotic expansion of the solution with respect to the thin layer width . More precisely we assume that

(A.17)

Fig. 10. Amplitude of (a) the field u and (b) the normal derivative of the field (@u=@n) on the lower interface for configuration Fig. 3(d) and  = 0:2 .

and

where

are related by

exact data while 3rd order ATC does not yield sufficiently accurate approximations.

(A.18) is being the orthogonal projection of on (the couple will be referred to as (curvilinear or) parametric coordiwith respect to ) and where nates of is independent of and is defined on with being the length of . Let be a function defined on and be defined by let with

V. CONCLUSION We proposed a new approach to determine the electromagnetic field on the inner boundaries of a layered medium from over determined data on the outer boundary. The method is based on the use of ATC which are constructed using scaled asymptotic expansions of the field inside thin layers. With the aim of better accuracy, higher order conditions up to the 4th order have been derived. Furthermore, we have introduced the CATC in order to extend the feasibility of ATC to “thick” layers. Numerical experiments for both open and closed domain configurations are presented to demonstrate the effectiveness and feasibility of the method. It is shown that the range of applicability of the method is not limited to surfaces with constant curvature, which is important for real applications. Numerical results also show that as expected, the approximation obtained by 4th order ATC is more accurate than that of 3rd order ATC and in the case of cascaded applications of the ATC, only 4th order ATC provide satisfying results (due to the tradeoff between accuracy and stability) for noisy data. Our approach can be effectively used in imaging algorithms related to objects buried under stratified medium with arbitrary interfaces. This topic is the subject of a work under progress. Let us point out two issues related to this application. 1) The proposed method could not be used to determine the field values without the background information. However, in practice, only a rough estimate of the background properties is required (mean values of the physical characteristics) since the used wavelength is assumed to be larger than the layer width. 2) Measurements are usually available only on a limited aperture of the outer domain. In order to reconstruct the data at the outer interface, there are number of continuation methods (based for instance on a representation of the solution in terms

(A.19) where

and

verify (A.18). Then we recall that

(A.20)

, one observes from (A.20) and (A.17) that (1) Setting is equivalent to

(A.21) The continuity conditions at

can be expressed as (A.22) (A.23)

ÖZDEMIR et al.: RECONSTRUCTION OF THE ELECTROMAGNETIC FIELD IN LAYERED MEDIA USING THE CONCEPT OF ATC

Multiplying (A.21) by then substituting the asymptotic expansion (A.17) yields after expanding with respect to and equating the same powers of

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explained in the second section). We summarize here the results and . (of the lengthy calculations) for

(A.29)

(A.30)

(A.31)

(A.24) for and for , with the for negative . The continuity conditions convention (A.22), (A.23) imply

(A.25) (A.26)

for all

. We first observe from (A.25) and (A.26) that (A.32)

(A.27) and

APPENDIX B EXPRESSION OF THE CURVATURE Consider a regular curve

parametrized as (B.33)

(A.28) Applying these two rules and using (A.24) allows us to inducand in terms of tively determine and and . We next observe that (A.25) and (A.26) also show that

Then the curvilinear abscissa is defined as

and the curvature

is given by (B.34)

In the expressions above respect to . Therefore

These two identities allow us then to express and in terms of and . The obtained identities then serves to derive the expression of the ATC (as

denotes the derivative of

with

REFERENCES [1] N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: An iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag., vol. 39, pp. 1742–1752, Dec. 1991.

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[2] D. Colton, H. Haddar, and M. Piana, “The linear sampling method in inverse electromagnetic scattering theory,” Inverse Prob., vol. 19, no. 6, pp. S105–S137, 2003, special section on imaging. [3] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Prob., vol. 13, pp. 1607–1620, 1997. [4] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imag., vol. 9, pp. 218–225, 1990. [5] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Prob., vol. 25, 2009. [6] T. Senior and J. L. Volakis, “Approximate boundary conditions in electromagnetics,” IEE Electromagn. Wave Series, 1995. [7] R. Harrington and J. Mautz, “An impedance sheet approximation for thin dielectric shells,” IEEE Trans. Antennas. Propag., vol. 23, no. 4, pp. 531–534, July 1975. [8] C. Holloway and E. Kuester, “Effective boundary conditions for rough surfaces with a thin cover layer,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 1999, pp. 506–509. [9] A. Karlsson, “Approximate boundary conditions for thin structures,” IEEE Trans. Antennas. Propag., vol. 57, no. 1, pp. 44–148, 2009. [10] S. Moskow, F. Santosa, and J. Zhang, “An approximate method for scattering by thin structures,” SIAM J. Appl. Math., vol. 66, no. 1, pp. 187–205, 2005. [11] S. Chun and J. S. Hesthaven, “High-order accurate thin layer approximations for time-domain electromagnetics and their implementation, part i: Coatings,” J. Comput. Appl. Math., no. 231, pp. 598–611, 2009. [12] J. Shumpert and T. Senior, “Impedance boundary conditions in ultrasonics,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1653–1659, 2000. [13] D. Hoppe and Y. Rahmat-Samii, “Higher order impedance boundary conditions for anisotropic and nonreciprocal coatings,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 1992, pp. 1993–1996. [14] K. Mitzner, “Effective boundary conditions for reflection and transmission by an absorbing shell of arbitrary shape,” IEEE Trans. Antennas. Propag., vol. 16, pp. 706–712, 1968. [15] S. Chun, H. Haddar, and J. S. Hesthaven, “High-order accurate thin layer approximations for time-domain electromagnetics part II: Transmission layers,” J. Comput. Appl. Math., 2010.

[16] B. Delourme, H. Haddar, and P. Joly, “Approximate models for wave propagation across thin periodic interfaces,” INRIA, Res. Rep. RR-7197, 02, 2010 [Online]. Available: http://hal.inria.fr/inria00456200/PDF/RR-7197.pdf [17] B. Aslanyurek, H. Haddar, and H. Sahinturk, “Generalized impedance boundary conditions for thin dielectric coatings with variable thickness,” INRIA, Res. Rep. RR-7145, 2009 [Online]. Available: http://hal. inria.fr/inria-00440159/PDF/RR-7145.pdf

Özgür Özdemir received the B.Sc. and M.Sc. degrees in electronics and communication engineering from Istanbul Technical University, Istanbul, Turkey, in 1998 and 2000, respectively, and the Ph.D. degree from the New Jersey Institute of Technology, Newark, in 2005. From 2007 to 2008, she was a Postdoctoral Researcher at ENSTA/INRIA. She is currently working as an Assist. Prof. Dr. with the Istanbul Technical University. Her research interests are mainly in the areas of direct and inverse scattering problems in electromagnetics.

Houssem Haddar received the Ph.D. degree in applied mathematics from the Ecole des Ponts et Chaussees, France. After a Postdoctoral position at the University of Delaware in 2001, he entered the French Research Institute on Automatic and Computer Science (INRIA) where he became a Director of Research in 2008. He is leading a joint research group on inverse problems and optimization methods (DEFI) between INRIA (center of Saclay Ile de France) and Ecole Polytechnique where he is also holding a part time professor position.

Alaaddin Yaka was born in Samsun, Turkey, in 1986. He received the B.Sc. degree in telecommunication engineering from Istanbul Technical University, Istanbul, Turkey, in 2008, where he is currently working toward the M.Sc. degree.

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Spectral Analysis of Relativistic Dyadic Green’s Function of a Moving Dielectric-Magnetic Medium Tatiana Danov and Timor Melamed, Senior Member, IEEE

Abstract—The present contribution is concerned with obtaining plane-wave spectral representations of the relativistic electric and magnetic dyadic Green’s functions of an isotropic dielectric-magnetic medium (at the frame-at-rest) that is moving in a uniform velocity. By applying a simple coordinate transformation, scalarization of the EM vectorial problem is obtained in which the EM dyads are evaluated from Helmholtz’s isotropic scalar Green’s function. The spectral plane-wave representations of the dyadic Green’s functions are obtained by applying the spatial 2D Fourier transform to the scalar Green’s function. We investigate these spectral representations in the under and over phase-speed regimes, as well as 2D and 3D formulations and discuss the associated wave phenomena. Index Terms—Dyadic Green’s function, moving medium, planewave spectrum, special relativity.

I. INTRODUCTION AND STATEMENT OF THE PROBLEM

M

ATHEMATICAL techniques for obtaining analytic solutions for the dyadic Green’s functions (GFs) of electromagnetic (EM) theory and in particular for wave propagation in a moving medium has been a subject of continuance research for the past decades [1]–[8], as well as in recent years [9]–[12]. A comprehensive review of analytic methods for dyadic Green’s functions can be found in [13], [14]. Green’s function for sources in an infinite domain moving medium has been derived in [1], [2], [4], and [5]. The dyadic Green’s functions of homogeneous isotropic dielectric-magnetic medium (with respect to an observer in an inertial non-co-moving frame of reference) have been derived in closed form in [9] and an alternative simple derivation was presented in [12] for a medium moving in a velocity that does not exceed the medium’s phase-speed of light in the co-moving frame. Plane-wave (PW) spectral decomposition for time-harmonic excitation has been a major tool for solving various scattering and diffraction problems, such as diffraction from different plane screens, scattering by planar bodies, modal expansion of planar waveguides, propagation in different anisotropic media, and many more (see comprehensive examples in [15], [16]). Such spectral decomposition of incident wave has been formulated and applied to the two-dimensional problem of the uniformly moving perfectly conducting cylinder [17], for solving the EM field radiated by an infinitely long thin wire-antenna which uniformly translates in a direction parallel Manuscript received October 03, 2010; revised December 03, 2010; accepted December 03, 2010; date of publication June 09, 2011; date of current version August 03, 2011. The authors are with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158972

to a plane interface [18], for the EM radiation from fractal antennas [19], for scattering of a plane-wave by a uniformly moving half-plane [20], for EM wave scattering from rough surfaces [21], and for obtaining asymptotically exact technique for EM scattering by uniformly moving targets in [22]. Other recent contributions to relativistic scattering which apply PW spectrum are found in [23]–[26]. We are aiming at obtaining the dyadic GFs for medium which , where we is moving in a constant translation velocity, assume with no loss of generality, that the velocity is in the direction of the -axis. Unit vectors in the conventional cartesian coordinate system, , are denoted by hat over bold fonts. The medium is assumed to be a linear isotropic dielectric-magand denoting its relative netic in the frame-at-rest with permittivity and permeability with respect to vacuum, respectively. The corresponding Minkowski constitutive relations can be stated as [12] (compare with PWs propagation with non-instantaneously responding medium in [27])

(1) where

is the speed of light in vacuum,

, (2)

and

is a diagonal matrix (3)

Here and henceforth double underlined boldfaced letters denote matrices/dyads. Assuming time-harmonic fields with a time-dependance, Maxwell’s curl equations which are corresponding to (1) are [12]

(4) In the next section we derive the electric and magnetic dyadic GFs in a compact way by reducing the vectorial problem of EM propagation in bi-anisotropic medium into the isotropic Maxwell’s equations from which the dyadic GFs are obtained by applying simple differential operators to a scalar isotropic GF. This formulation is used for obtaining closed-form expressions for the 3D dyadic GFs whereas 2D problems are explored in Section III. By applying a spatial Fourier transform for the scalar representation we derive in Section IV the PW spectral representations of the dyadic GFs.

0018-926X/$26.00 © 2011 IEEE

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II. 3D DYADIC GREEN’S FUNCTIONS

where, we denote

A. General Formulation and , perThe electric and magnetic Green’s dyads, to form a linear mapping from the current density sources the electric and magnetic vector fields, and , by

(13) . One can readily observe that the curl equawith tions in (12) have the form of Maxwell’s curl equations in linear isotropic medium which are characterized by

(5)

(14)

Following [12], we normalize all fields and sources in the form

is given by (10). The dyadic GFs of where the branch cut of (12) can be found in different textbooks by using the GF of the scalar Helmholtz operator [15] (15)

(6) is the free-space (vacuum) wavenumber. By where inserting the vector fields in (6) into Maxwell’s equations in (4), we obtain

from which the isotropic electric and magnetic dyadic GFs are obtained via

(16) (7) Next, in order to recast equation (7) in a standard (isotropic) form, we define a normalized -coordinate (cf. [28])

Thus, the normalized vector fields due to the source in (12) are given by

(8) and a corresponding del (nabla) operator (9) , and so forth. Note that is positive or negative for medium velocities that are less than or exceed the phase-speed of light in the medium (at rest), ). The branch cut of in (8) is respectively (i.e., for chosen such that

(17) By substituting (6) with (13) into (17) and compare the resulting integrals with (5), we identify (18)

where

(10) in order to ensure the validity of the following derivation for the two velocity regimes in which the medium speed, is under or (see also over the medium’s (at rest) phase-speed of light, discussion in Section II-B and [29]). Using these new definitions, we note that

in (11) into (18), the dyadic GFs can Finally, by inserting be obtained directly from the scalar GF, , via (19) where the constants

are given by (20)

and the differential operators

are given by

(11)

(21)

in (7) into (11) and applying a similar proBy inserting cedure to , we obtain

The procedure of obtaining the EM dyadic GFs is as follows: First the isotropic scalar GF is obtained by solving (15) subject to causality conditions (see for example (25)). By inserting the resulting into (19), we obtain the desired EM dyadic . GFs,

(12)

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B. Velocity Regimes We distinguish two medium velocity regimes which describe a medium (or sources in the co-moving frame) that is moving in a velocity which is either under or over the medium’s (at rest) phase-speed. These velocity regimes are identified by eior for which in (3) and in (8) is positive ther or negative, respectively. Next we follow the procedure in the previous section in order to obtain the 3D dyadic GFs in the two medium’s velocity regimes. 1) Under Phase-Speed Regime: Causality in the under phase-speed regime is manifested by boundary conditions of . The solution outgoing waves with respect to the source at of (15) is identified as the 3D scalar free-space GF (22) where is given in (8). The dyadic GFs, , are obtained in (22) into (19) with in (21). Note by substituting that the resulting dyadic GFs in this speed regime are identical to those which were obtained in [1]. 2) Over Phase-Speed Regime: In the over phase-speed and therefore and are imaginary numbers. regime, In this regime we define real longitudinal coordinate and wavenumber by

Fig. 1. Physical configurations cross sections for the 2D scalar Green’s functions for medium velocity in the z -direction. (a) Transversal GF in which the . (b) Longitudinal GF in which @ j sources are functions of y; z , i.e., @ j . Here the electric current density is identify by J r j x; y jkmz .

0

=0

( ) = ( ) exp(

= )

The scalar Green’s function resides within a conical region in the direction of the medium velocity. By inserting (8) into in (26), we identify this region the condition where is the conventional angle of the to be observation vector with the -axis and is related to the medium velocity by (3). The discontinuous behavior of the ˇ Green’s function in this case is a manifest of Cerenkov radiation. Note that the scalar Green’s function in (26) is consist which propagate towards of two waves . The 3D dyadic GFs are and away from the source at obtained by inserting in (26) into (19) with (29)

(23) . Using these definitions, (15) transforms and therefore into the 2D Klein-Gordon differential equation in coordinates (24) Here the medium velocity exceeds the phase-speed in the co-moving frame so that causality is manifested by

III. 2D DYADIC GREEN’S FUNCTION We identify two kinds of (or 3D in relativistic jargon) (or ), and dyadic GFs, transversal GF in which longitudinal GF in which (see Fig. 1). A. 2D Transversal Dyadic GF 1) Under Phase-Speed Regime: By substituting (15) we obtain

(30)

(25) The solution for

into

The solution of (30) in this velocity regime is identified as the 2D free-space GF

in (24) subject to (25) is [30]

(31)

(26) where

where (27) When applying the differential operators in (21) to in (26), it is convenient to use partial derivatives with respect to . Using and , we obtain

(28)

denotes the Hankel function of the second kind and . The 2D dyads are obtained by inserting (31) into . This procedure yields (19) and using (32) where

are given in (20) and

(33)

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2) Over Phase-Speed Regime: In the over phase-speed , where and (15) transforms regime into the 1D Klein-Gordon differential equation

the transversal coordinates. The Fourier transform pair is define by (39)

(34)

(40)

subject to boundary condition (25). Thus [31] (35) A. 3D Spectral Dyadics where denotes the Bessel function of the first kind, denotes the unit-step (Heavyside) function and is given in (23). The GF in (35) resides within a wedge region in the direction of (see also Fig. 3(a)). The 2D the medium velocity , corresponding transversal dyadic GFs are

(36) where

and

are given in (29) and

By applying the Fourier transform operator in (39) to the differential equation in (15) we obtain (41) where and denotes the Fourier transform of . Equation (41) is subject to causality boundary conditions of outgoing wave in the under phase-speed regime or the one in (25) in the over phase-speed regime. scalar spectral dis1) Under Phase-Speed Regime: The tribution which is corresponding to is obtained by . By applying causality condition of solving (41) with we obtain the conventional (isotropic) an outgoing wave at spectral GF [15] (42)

(37)

where (43)

B. 2D Longitudinal Dyadic GFs By substituting into (15), the scalar GF is identified . Since as the 2D free-space GF in (31) with , this solution holds for both the under and the over here phase-speed regimes. Thus the 2D dyadic GFs are given by (32) with

dyadic GFs are obtained The spectral representations of the by applying the inverse Fourier operator in (40) to (42) and substituting the resulting integral into (19). By differentiating under the integral sign we obtain (44) where the electric and magnetic spectral dyads, by

(45)

(38) Note that according to (6), longitudinal GFs in this context are source -dependence, identified by a specific and therefore this special case is investigated further.

, are given

where are given (20), spectral matrices are given by

for

and the

IV. PLANE-WAVE SPECTRAL REPRESENTATIONS Following the motivation which is outlined in the introduction, in this section we use the general formulation in Section II in order to obtained PW spectral representations for the dyadic GFs in the different velocity regimes. The spectral representations are obtained by applying by the spatial Fourier transform to (15), solving the resulting ODE and following the procedure in Section II. We denote the 2D (spatial) Fourier transform of by an over tilde, , where is the (transversal) spectral wavenumber vector and are

(46) The spectral representations in (44) describe the Dyadic GFs as a continuous superposition of PWs. Each of the columns of the spectral dyads in (45) is a plane-wave in coordinate system (or term is included) in either or in when the half-spaces. These PWs propagates away from the source

DANOV AND MELAMED: SPECTRAL ANALYSIS OF RELATIVISTIC DYADIC GREEN’S FUNCTION OF A MOVING DIELECTRIC-MAGNETIC MEDIUM

with

in (29),

,

=0

Fig. 2. The integration contour is presented for k over complex k plane. The branch points at k are determine by k in (43).

6

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and

(55) Similarly, the magnetic dyadic GF spectral integral is

which is located at . Propagating PWs occur for spectral wavenumbers within the so-called Ewald sphere where . For other values for which is imaginary, the PWs are evanescent and decay exponentially away from the hysource. The integration contour in (44) in the complex perplane is presented in Fig. 2. 2) Over Phase-Speed Regime: In the over phase-speed and in (23) for which regime, we use the definitions of (41) takes the form

(56) where the magnetic spectral dyads are given by

(57) with

in (29) and

(47) (58)

subject to causality condition (48) Solving (47) with boundary conditions (48) yields (49) where (50) In order to obtain PW spectral representation for the dyadic GFs, we recast (49) in the form (51) where we denote (52) The subscripts “a” and “t” stands for “away” or “toward” since, as discussed in Section IV-A-3, these spectral distributions describes PWs that propagates away from or towards the source. The 3D dyadic GFs spectral representations are obtained from (51) in a similar way to (45). The resulting PW spectral representation of the electric field dyadic GF is (53)

3) Spectral Causality: Each of the columns of the integrands in the spectral integrals in (53) and (56) is a PW in half-space. The spectral or consist of PWs that propagate away from or towards the source at , respecare PWs tively. The seemingly “non-causal” PWs of that propagate in the negative -direction. In the co-moving frame these PWs are radiated by the source which is moving faster than the propagation speed of the PWs. Therefore in the laboratory frame the PWs propagate towards the source. Unlike the spectral dyads in the under phase-speed regime, here propagating PWs occur for all spectral wavenumbers , so that no evanescent waves appear in . Note that in the under phase-speed regime, the radios of the Ewald sphere where the propagating spectrum resides, depends on the normalized ve. Thus the locity through and beyond that velocity, in radius goes to infinity as the over phase-speed regime, all spectral components are in the visible (propagating) spectral range. B. 2D Transversal Spectral Dyads Next we examine the spectral representation of 2D which were explored in transversal dyads where Section III-A. By applying the Fourier transform in (39) to (30) we obtain (59) 1) Under Phase-Speed Regime: The 2D dyads in the under phase-speed regime is obtained by solving (59) similarly to the 3D spectral dyadic in Section IV-A-1. Thus,

where the spectral electric dyads are given by

(54)

(60)

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where (61) Following the procedure in (44) for in (60), we obtain the transversal dyadic GFs in the spectral representations of the form (62) where the electric and magnetic spectral dyads, by

, are given

(63)

Fig. 3. The 2D transversal scalar Green’s function in the over phase-speed : and : . (a) The absolute regime. The medium parameters are n value of the exact solution g r in (35). (b) Numerical evaluation of the spectral integral in (40) over the spectral scalar Green’s function g k in (65).

()

where and

= 15

are given in (29),

= 08 ~( )

,

Here the 2D observation and wavenumber vectors are and (for ), respectively, and (69)

(64)

Each of the columns of the spectral dyads in (63) is a plane-wave half-spaces. Propagating PWs occur for spectral in either where for all other values, is wavenumbers imaginary and the PWs are evanescent. 2) Over Phase-Speed Regime: In order to obtain PW spectral representations for 2D GFs in the over phase-speed regime, we apply the Fourier transform (39) to (34). The solution under condition (48) is given by (65) where (66) with in (23). The 2D dyadic GFs spectral representations are obtained by following the procedure in Section IV-A-2. The results are (67) Here the spectral electric and magnetic dyadic GFs are given by

(68)

Fig. 3 plots the 2D transversal scalar Green’s function in the over phase-speed regime. The medium parameters are and . The absolute value of the exact solution in (35) is plotted in Fig. 3(a). The numerical evaluation of the spectral integral in (40) over the spectral scalar Green’s function in (65) is shown in Fig. 3(b). The scalar GF resides within a wedge region in the direction of the medium velocity . This phenomena is clearly demonstrated in the figure. V. SUMMARY We have presented a simple derivation of the dyadic Green’s function of an isotropic dielectric-magnetic medium which is moving in uniform velocity. The EM vectorial formulation in bi-anisotropic medium (in the laboratory frame) was reduced into a simple scalar formulation in which the dyads are obtained by applying differential operators to a scalar Green’s function. The scalar formulation involved solving the conventional isotropic homogeneous medium Green’s function. This formulation includes 3D and 2D sources which were discussed in Sections II and III, respectively. Two medium’s velocity regimes were investigated, under and over phase-speed regimes in which the medium velocity is either less than or exceeds the speed of light in the medium at rest, respectively (see Section II-B). These regimes differ in the manifestation of causality into boundary conditions for the scalar Green’s function (see (25)). Following the scalar formulation, a plane-wave spectral representations of the electric and magnetic dyadic GFs were obtained in Section IV by applying inverse Fourier representation of the scalar Green’s function. We investigate 3D and 2D sources in Sections IV-A and IV-B, respectively. Under and over light speed regimes were discussed, as well as the wave phenomena associated with the spectral representations. These representations can serve as the basis for solving different PW spectrum related scattering problems.

DANOV AND MELAMED: SPECTRAL ANALYSIS OF RELATIVISTIC DYADIC GREEN’S FUNCTION OF A MOVING DIELECTRIC-MAGNETIC MEDIUM

REFERENCES [1] C. T. Tai, “The dyadic Green’s function for a moving isotropic medium,” IEEE Trans. Antennas Propag., vol. 13, pp. 322–323, 1964. [2] K. S. H. Lee and C. H. Papas, “Electromagnetic radiation in the presence of moving simple media,” J. Math. Phys., vol. 5, pp. 1668–1672, 1964. [3] C. T. Tai, “A study of electrodynamics of moving media,” Proc. IEEE, vol. 52, pp. 685–689, 1964. [4] K. S. H. Lee and C. H. Papas, “Antenna radiation in a moving dispersive medium,” IEEE Trans. Antennas Propag., vol. 13, p. 799, 1965. [5] R. T. Compton and C. T. Tai, “Radiation from harmonic sources in a uniformly moving medium,” IEEE Trans. Antennas Propag., vol. 13, pp. 574–577, 1965. [6] H. Berger and J. Griemsmann, “Poynting’s theorem for moving media,” IEEE Trans. Antennas Propag., vol. 15, pp. 490–490, 1967. [7] P. C. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Proc. Inst. Elect. Eng., vol. 110, pp. 101–106, 1963. [8] J. Van Bladel, Relativity and Engineering. New York: Springer, 1984. [9] A. Lakhtakia and W. S. Weiglhofer, “On electromagnetic fields in a linear medium with gyrotropic-like magnetoelectric properties,” Microw. Opt. Techn. Let., vol. 15, pp. 168–170, 1997. [10] T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric-magnetic medium,” J. Phys. A, Math. Gen., vol. 37, pp. 5697–5711, 2004. [11] T. G. Mackay, A. Lakhtakia, and S. Setiawan, “Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium,” Optik, vol. 118, pp. 195–202, 2007. [12] A. Lakhtakia and T. G. Mackay, “Simple derivation of dyadic Green functions of a simply moving, isotropic, dielectric-magnetic medium,” Microw. Opt. Techn. Let., vol. 48, pp. 1073–1074, 2006. [13] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory. Piscataway, N.J.: IEEE Press, 1994. [14] W. S. Weiglhofer, “Analytic methods and free-space dyadic Green’s functions,” Radio Sci., vol. 28, pp. 847–857, 1993. [15] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Piscataway, N.J.: IEEE Press, 1994, Classic reissue. [16] P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields. New York: IEEE Press and Wiley, 1996. [17] P. De Cupis, G. Gerosa, and G. Schettini, “Gaussian beam diffraction by uniformly moving targets,” Atti della Fondazione Giorgio Ronchi, vol. 56, pp. 799–811, 2001. [18] P. De Cupis, “Radiation by a moving wire-antenna in the presence of plane interface,” J. Electromag. Waves Appl., vol. 14, pp. 1119–1132, 2000. [19] W. Arrighetti, P. De Cupis, and G. Gerosa, “Electromagnetic radiation from moving fractal sources: A plane-wave spectral approach,” Progr. Electromagn. Res., vol. 58, pp. 1–19, 2006. [20] M. Idemen and A. Alkumru, “Relativistic scattering of a plane-wave by a uniformly moving half-plane,” IEEE Trans. Antennas Propag., vol. 13, pp. 3429–3440, 2006. [21] P. De Cupis, “Electromagnetic wave scattering from moving surfaces with high-amplitude corrugated pattern,” J. Opt. Soc. Am. A, vol. 23, pp. 2538–2550, 2006.

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[22] P. De Cupis, D. Anatriello, and G. Gerosa, “An asymptotically exact technique for electromagnetic scattering by uniformly moving objects with arbitrary geometry,” in Proc. XXVIIth General Assembly of URSI, 2002, pp. 509–512. [23] P. De Cupis, G. Gerosa, and G. Schettini, “Electromagnetic scattering by uniformly moving bodies,” J. Electromag. Waves Appl., vol. 14, pp. 1037–1062, 2000. [24] P. De Cupis, P. Burghignoli, G. Gerosa, and M. Marziale, “Electromagnetic wave scattering by a perfectly conducting wedge in uniform translational motion,” J. Electromag. Waves Appl., vol. 16, pp. 345–364, 2002. [25] M. Idemen and A. Alkumru, “Influence of the velocity on the energy patterns of moving scatterer,” J. Electromag. Waves Appl., vol. 18, pp. 3–22, 2004. [26] P. De Cupis, “An analytical solution for electromagnetic wave scattering by multiple wedges,” Opt. Commun., vol. 261, pp. 203–208, 2006. [27] T. G. Mackay and A. Lakhtakia, “Positive-, negative-, and orthogonalphase-velocity propagation of electromagnetic plane waves in a simply moving medium: Reformulation and reappraisal,” Optik, vol. 120, pp. 45–48, 2009. [28] A. Lakhtakia and T. G. Mackay, “Dyadic Green function for an electromagnetic medium inspired by general relativity,” Chinese Phys. Lett., vol. 23, pp. 832–833, 2006. [29] S. C. Bloch, “Eighth velocity of light,” Am. J. Phys., vol. 45, pp. 538–549, 1977. ˇ [30] M. H. Cohen, “Radiation in a plasma. I. Crenkov effect,” Phys. Rev., vol. 123, pp. 711–721, 1961. [31] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers. New York: Birkhauser Boston, 1997.

Tatiana Danov was born in Rubtzovsk, Russia, in November 1966. She received the M.Sc. degree (magna cum laude) in physics in 1988 from Lobachevsky University, Gorky, Russia. She is currently working toward the Ph.D. degree at Ben Gurion University of the Negev, Israel. From 1989 to 2000, she was with the Department of Physics and Electrical Engineering, Technical University in Nizhny Novgorod, Russia. From 2002 to 2006, she was with the VLSI center, Ben Gurion University of the Negev, Israel.

Timor Melamed (SM’08) was born is Tel-Aviv, Israel, in January 1964. He received the B.Sc. degree (magna cum laude) in electrical engineering in 1989 and the Ph.D. Degree in 1997, both from Tel-Aviv University. From 1996 to 1998, he held a postdoctoral position at the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA. From 1999 to 2000, he was with Odin Medical Technologies. Currently he is with the Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Israel. His main fields of interest are analytic techniques in wave theory, transient wave phenomena, inverse scattering and electrodynamics.

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Mastering the Propagation Through Stacked Perforated Plates: Subwavelength Holes vs. Propagating Holes M. Navarro-Cía, Member, IEEE, M. Beruete, F. Falcone, Senior Member, IEEE, J. M. Illescas, I. Campillo, and M. Sorolla Ayza, Senior Member, IEEE

Abstract—More insight on the physics underlying the transmission through subwavelength hole arrays prism by comparing it with propagating hole arrays prism is provided. We show the critical role that the size of the holes plays in this electromagnetic propagation, changing the effective index of refraction from negative (backward wave) to positive values (forward wave) as the hole diameter increases. This causes negative refraction for the zeroth order emerging beam in the cut-off holes prism whereas positive refraction in the non-cut-off holes prism. Furthermore, we revisited from the perspective of superposition principle the explanation of these stacks based on the so-called building sub-units: horizontal rods and vertical wires. Our simple analysis reveals the drawbacks of this earlier interpretation, and reinforces the powerful model founded on a inverse transmission line. Experimental results (coand cross-polar measurements) performed at the V-band of the millimeter-waves in the Fresnel zone are well supported by numerical analyses. As expected, higher order diffracted outgoing beams are recorded for the classical prism but not for the cut-off holes prism. Index Terms—Extraordinary transmission, frequency selective surfaces, perforated plates, negative index of refraction, metamaterials.

I. INTRODUCTION RTIFICIAL dielectric materials trace back to 1940s, when they were employed in microwave engineering for beam shaping [1]–[3]. Afterwards, they were used to mimic the electrical properties of plasma (in the absence of dc magnetic fields) [4]. Moreover, stacked structures were employed to synthesize unusual (but positive) refractive indexes [5]. A thorough summary on artificial dielectrics can be found in [6]. New fresh ideas in artificial effective media have recently followed the remarkable discover of the split-ring resonator [7]. This resonator has managed to gather together artificial dielectric and magnetic media disciplines under a new definition:

A

Manuscript received July 14, 2010; revised December 30, 2010; accepted January 06, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by the Spanish Government and E.U. FEDER funds under contracts Consolider “Engineering Metamaterials” CSD2008-00066 and TEC2008-06871-C02-01. M. Navarro-Cía, M. Beruete, F. Falcone, and M. Sorolla are with the Millimeter and Terahertz Waves Laboratory, Universidad Pública de Navarra, Campus Arrosadía, 31006 Pamplona, Spain (e-mail: [email protected]). J. M. Illescas is with TAFCO Metawireless, Pol. Mocholi, Plz. Cein 5, T1, 31110 Noain, Spain (e-mail: [email protected]). I. Campillo is with CIC nanoGUNE Consolider, Tolosa Hiribidea 76, 20018 Donostia, 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.2011.2158957

metamaterials. By overlapping the negative magnetic behavior of this split-ring resonator with a negative electric permittivity media [8] such as a wire mesh [4], [9], [10], the revival of the left-handed media proposed by Veselago [11] in 1968 was attainable. Among others, an intriguing property of left-handed media is negative refraction which comes as a result of the negative index of refraction of these media. The extension of left-handed media to high frequencies such as THz or infrared has become a challenge because of the high losses usually obtained at those frequencies [12]. Consequently, great research effort has been recently devoted to overcome this problem. The most promising double-negative-scheme attempt to elude high losses at such high frequencies relies on the fabrication of a pair of very thin metallic perforated plates (fishnet structures) [13], [14]. Alternatively, another research line has been put forth in electromagnetism with the advent of the extraordinary optical transmission [15]. This phenomenon has opened new ways for controlling electromagnetic wave propagation, and has been successfully expanded to other frequency ranges such as millimeter-waves [16]. Here the structure is reminiscent of dichroic filters, but the basic operation principle is substantially different from them because narrow pass bands appear below the cut-off frequency of the holes whereas the typical frequency of operation of dichroic filters is when the apertures are above cut-off. A noticeable consequence of the result published in [16] and extended to finite size structures in [17] was that the phenomenon appeared also for the case of nearly infinite conductivity. This fact called into question the explanation based upon plasmons suggested in [15] giving therefore more relevance to the periodic structure effect [18]. Therefore, in the microwave spectrum it appears more appropriate the coined term extraordinary transmission instead of the original extraordinary optical transmission. The combination of metamaterials and extraordinary transmission has been proven to be a powerful tool for minimizing losses. Stacked metallic perforated plates working at the extraordinary transmission resonance have been measured at millimeter-waves, showing low-loss backward propagation [19]. This behavior has been confirmed by both interferometric techniques [19] and wedge experiment [20], [21]. Moreover, explanations based on equivalent inverse transmission line circuit [19], [22], [23] alongside full wave analysis [22] support the experiments. Recently, anomalous refraction in dielectric embedded stacked subwavelength hole arrays has been reported at terahertz [24]–[29] and near-infrared [30].

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NAVARRO-CÍA et al.: MASTERING THE PROPAGATION THROUGH STACKED PERFORATED PLATES

Our research on extraordinary transmission metamaterials, has led us in the past to the development of several practical realizations such as demultiplexors [31] or beam shaping structures [32]–[36]. The aim of this work is to highlight the key differences between extraordinary transmission metamaterial and stacked frequency selective surfaces (FSSs) based on propagating holes—perforated plates—. In our prior publication [20] the comparison was done through numerical calculations only. Here we provide an experimental comparison that confirm and extend the simulation results of [20] as well as a further analysis of the extraordinary transmission metamaterial, providing coand cross-polar measurements at two different distances, which complement and also extend our previous work [21] where the experimental study was reduced to one single distance, considered only holes in cut-off and no cross-polarization measurement was presented. We report that the extraordinary transmission metamaterial can be defined by an effective one-dimensional negative index of refraction medium provided that the stack periodicity is subwavelength enough so as not to be in electromagnetic bandgap regime, whereas the structure with holes operating in propagation has an effective positive index of refraction associated to the waveguide mode defined by the holes [5]. In this analysis, we will focus on the wedge experiment because it gives a direct correlation between the angle of refraction of the emerging zeroth order beam and the effective index of refraction of the prism. Moreover, the results presented here can contribute to shed more light on the controversy of the true nature of the observed deflection of the beam in metamaterial prisms because, unlike initial metamaterial-wedge-experiments [8], the transmittance of the outgoing main beam in our experiments reaches relatively high values, refuting the criticism on metamaterials supported on the low transmittance of former experiments [37]. Furthermore, as it is demonstrated in this paper, the negatively refracted outgoing beam cannot be identified with a periodic structure grating lobe as a general explanation to metamaterial wedge experiments, as it was suggested in [37]. A last introductory remark deals with the usefulness of the extraordinary transmission perforated plates [16], [17] for practical FSS applications. Solid interior element type-FSSs (using the nomenclature of [38]) were disregarded since the very beginning of FSS applications due to their angle of incidence sensitivity and early onset of grating lobes [38]. However, perforated plates are currently employed in commercial millimeter and submillimeter wave devices where the term quasioptics was established many years ago [39]. In fact, our research roots are in quasioptics [40] and not in optics as suggested in [37]. Moreover, it is interesting to point out that, to the best of our knowledge, no experimental work has been reported under the conditions described in [16], [17] with the sole exception of the numerical computation reported in Fig. 9.9 of [39].

II. NUMERICAL ANALYSIS A prism made of stacked subwavelength hole arrays and another one composed of plates perforated with standard propagating holes were firstly numerically analyzed by the finite-integration time domain-based commercial software CST

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Fig. 1. Pictures of the prototypes: perspective and top view of extraordinary transmission metamaterial (a) and (c) and propagating holes prism (b) and (d). The angle of each prism is: 26.6 deg and 16.7 deg for SSHA and non-cut-off prism, respectively.

Microwave Studio™ [41] and compared with each other. Subsequently, they were built and measured. The hole pattern in both instances was drilled over an aluminum plate of thickness mm. The dimensions of the extraordinary transmismm, sion metamaterial were: transversal periodicities mm, hole diameter mm; whereas for the propagating hole prism were: transversal periodicities mm, mm, hole diameter mm; Note that the lattice constant was chosen as 5 mm in order to admit a propagating hole of such diameter inside the unit cell. For this particular set of parameters of the extraordinary transmission metamaterial, the first resonance emerges, for a single plate, around 57 GHz although the cut-off frequency of the circular waveguide defined by the hole is 70 GHz. On the other hand, the cut-off frequency of the perforated plate with propagating holes is around 41 GHz. The longitudinal lattice constant of both stacks was mm ( for the extraordinary transmission metamaterial) allows us to deal with metamaterials concepts (in the case of the extraordinary transmission metamaterial prism) rather than electromagnetic bandgap structures (whose dimensions are on the order of the wavelength) within our operating frequency. To achieve the desired prism profile, the blocks of stacked plates were cut in steps of one by one unit cells, removing them gradually along , see Fig. 1. Both prisms comprise 15 stacked perforated plates, which involves different effective apertures of sizes mm . We first compute the dispersion diagram of the infinite structure by using the eigenmode solver of commercial software CST Microwave Studio [41]. We have modeled metal as a perfect electric conductor, which is a reasonable approximation for metals at millimeter-waves. From the dispersion diagram, the effective index of refraction is directly calculated through the relation:

(1)

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Fig. 3. Dispersion diagram of the subunits in which stacked perforated plates can be divided in the case of cut-off holes (green lines), and non-cut-off holes (dark lines). Electric field is vertically polarized. Cut-off hole arrays: d : mm; Non-cut-off hole arrays: d mm, d mm and size hole s d mm. mm and s

=5

=5

=4

= 25

=3 =

Fig. 2. Dispersion diagram and index of refraction derived from the dispersion diagram: (solid lines) cut-off holes, (dashed lines) non-cut-off holes. Data from [20].

Both dispersion diagrams for the first propagating modes and their index of refraction of cut-off and non-cut-off holes are depicted in Fig. 2. Prominent features in this figure are as follows. • The first band of the cut-off hole extraordinary transmission metamaterial starts at 53 GHz and ends at 58 GHz, totally enclosed within the cut-off region (below 70 GHz); on the other hand, the first band of the propagating holes FSS starts at 42 GHz and ends at 58 GHz also, completely contained in the propagation region (above 42 GHz). • The first mode of the extraordinary transmission metamaterial displays a negative slope (backward propagation), i.e., phase velocity runs antiparallel to group velocity, whereas the first band of the standard perforated plates shows a positive slope, i.e., it has right-handed characteristics (forward propagation). • The extraordinary transmission metamaterial dispersion diagram predicts an effective negative index of refraction at 53.5 GHz. • In both cases, the index of refraction does not violate the required condition that for lossless and passive media . • The extraordinary transmission metamaterial mode presents a more marked dispersive behavior than stacked propagating holes, as it could be expected since its operation is based on a strong resonance. Because of the subwavelength stacking, higher order modes play a role in the propagation through these structures. A method to analyze higher order mode interaction for closely spaced diaphragms can be found in [6] in the chapter devoted to periodic structures. In addition, we analyzed in a previous work [22] the relevance of higher order modes for a single extraordinary transmission plate and the extension of the analysis to the subwavelength stacking by an approach based upon finite-integration time domain numerical method [41]. Alternatively, to gain more understanding about the origin of modes of each structure, the unit cell has been divided into two independent subunits as in [30]: An array of metallic wires parallel to the incident electric field direction and an array of metallic discontinuous wires (strips) orthogonal to the electric

Fig. 4. Transmitted power through an array of subwavelength square holes (dark), vertical wires (red), horizontal strips (blue). Sum of the transmitted power of vertical wires and horizontal strips (dashed dark).

field, see Fig. 3. The wire grid aligned with the incident electric field can be modeled as an inductive shunt admittance (inductive grid) and thus has a high-pass filtering behavior allowing propagation of an electromagnetic wave only above a certain frequency (labeled in Fig. 3 as electric resonance) [39]. As a general rule, the wider the air slit, the lower the cut-off frequency see Fig. 3 and [42]. Conversely, a screen of horizontal rods has a low-pass response, and can be modeled as a capacitive shunt admittance [6], [39], [42]. An additional mutual capacitance arises for stacked pairs of parallel rods and provides a local magnetic dipole moment along as a result of the loop-like circuit formed by the antiparallel conductive currents in each pair of stacked rods closed by the displacement currents (the aforementioned electric coupling) [30]. Under the perspective stated in [30], a resonant behavior of the single plate and backward propagation in the stacked structure can be explained by simply combining both independent subunits. Also, this qualitative approach for the single plate case can be found in chapter 9 of [39]. Nevertheless this is only qualitatively correct because after a rigorous analysis of the induced surface currents on one plate, one can determine that a coupling effect arises in the combined structure (hole array). This is manifested in Fig. 4, where the transmittance of subwavelength hole arrays (dark line), the infinite vertical metallic wires (red line) and the horizontal strips (blue line) is plotted. If the previous discussed splitting framework

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Fig. 5. Side view of electric field (a), (b) and surface current (c), (d) for stacked cut-off (left column) and propagating holes (right-column). Perspective view of the y component of the surface current on each face of the plates for stacked cut-off (e) and propagating holes (f). Notice that the back faces of the plates are visible because of the middle cutting plane.

were strictly correct, the superposition principle should be fulfilled already for the case of one layer. However, notice that not only the sum of the transmittance of each subunits does not produce the transmittance of the hole array at low frequencies, but it also does not reach total transmission at the extraordinary transmission peak of the subwavelength hole arrays. The study of the necessary subunits to achieve extraordinary transmission resonance will be published elsewhere since it is beyond the scope of this work. In Fig. 5 we investigate the field and surface currents of the complete stack to correlate its field and current distribution with the description based on equivalent dipoles. For the sake of clarity, for both stacking apertures we have plot the electric field (Fig. 5(a), (b)) and surface current (Fig. 5(c)–(f)) separately. Clearly, in both cases the surface currents run antiparallel between consecutive faces, yet the magnitude of them are higher in the case of cut-off holes. However, what is outstanding is the negligible electrical coupling between consecutive plates in the case of propagating holes, as was already discussed in [23]. Therefore, while stacked subwavelength hole arrays give rise to the abovementioned close loop-like circuit, and then, to a magnetic dipole moment along aforementioned, stacked non-cut-off holes do not. Also this figure stresses the simplicity, yet accurate picture provided by the equivalent inverse transmission line circuit to explain the behavior of stacked subwavelength hole arrays: the series capacitance and shunt inductance arises from the electrical coupling between consecutive plates and the cut-off regime of the hole, respectively [19], [22], [23]. It should be mentioned that the double period array was chosen in the case of the extraordinary transmission metamaterial prism for two purposes: • To increase the number of illuminated holes that has been already shown to be a critical parameter of ET structures [43], [44];

6

Fig. 6. Conditions for the generation of zeroth and firsts (m = 1) higher order diffracted beams; The zone where sin ' has meaning (real solution) is the yellow dashed one, and it demonstrates the absence of diffracted beams in our prototype: (a) extraordinary transmission metamaterial prism, (b) propagating holes prism.

• To prevent the excitation of higher order diffracted modes that could distort the experiment. Assuming that the surface step imposed by the finite transversal size of the unit cell forms a grating, the transmission through such interface is governed by the grating formula: , where is the order of the the effective index of refraction of the prism, the mode, incident angle and the angle of refraction, and we have taken . For our particular set of the second medium as air parameters, the first term of the left hand side is sufficiently large to just prevent any beam, other than the main beam, from forming in visible space. This is confirmed in Fig. 6, in which the region where real solutions for exist has been shadowed in yellow. Only the zeroth order transmittance is within the aforementioned dashed zone. However, a different result is obtained for the non-cut-off holes prism. In this case, the order appears in the visible space from around 50 GHz. The next step to gain physical insight on our structures is to visualize the vertical electric field (notice that this is the total field), together with power flow along the -plane or H-plane (Fig. 7). To simplify numerical effort and without loss of generality, these simulations deal with two-dimensional prisms: lateral absorber along with top and bottom electric walls allows TEM propagation as in a parallel plate waveguide. Thus, it restricts the polarization of the electric field to lie along the direction (vertical direction). This 2D configuration carries most of the information on the transmission characteristics of the addressed wedge experiment.

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Fig. 7. Electric field and power flow (bottom insets) of: (top) a SSHAs prism at 53.5 GHz (left) and 57.5 GHz (right); (bottom) a propagating hole arrays prism at 44 GHz, 53.5 GHz and 57.5 GHz from left to right in that order. The normal of the interface alongside an arrow describing the approximated Poynting vector of the main outgoing beam have been drawn so as to highlight the angle of refraction.

From these numerical results there is no doubt that the subwavelength hole arrays change dramatically the response of the prism, from a positive refraction to a negative refraction. However, there are more features that can be seen in the figure. The non-cut-off holes prism exhibits the expected higher order diffracted beam as well as edge effect, whereas the extraordinary transmission metamaterial prism only displays the latter effect. Furthermore, it is clear that the negatively deflected beam has a higher transmittance than the beam originated by the edge of the prism (which indeed appears at positive angles), reinforcing that the deflection of the main beam is governed by the effective negative index of refraction of the metamaterial, rather than the explanation stated in [37] that identified it with a grating lobe caused by the periodic output face and finiteness of the prism. At lower/higher frequencies, the phase change inside the extraordinary transmission metamaterial/perforated plates prism occurs more often than at high/low frequencies as a result of the greater absolute magnitude of the index of refraction (re). On the contrary, larger values of index member of refraction happen at higher frequencies for the non-cut-off holes prism, which explains the evolution of the beam, gradually moving away from the normal as the frequency increases. III. MEASUREMENT RESULTS The feeding part of the experimental setup consists of parallel aluminum plates with a fixed corrugated horn antenna [39] as the source, see photographs of Fig. 1 for details of the real prototypes and setup. The lateral open sides are covered with absorbers and the parallel plates are ended by the wedge made by stacked perforated metallic plates. Because of the electric walls imposed by the parallel plates, the incoming

electromagnetic wave sees an infinite structure along the vertical direction ( -axis), which further assists the transmission. In addition, another corrugated horn antenna is placed at two , and 600 mm and rotated in different distances angle in order to scan the outgoing beam. Both antennas are connected to the AB-Millimeter Quasioptical Vector Network Analyzer [45]. This setup allows us to measure the refraction angle and estimate the effective index from the grating formula. However, it is worth noting that the frequency band alongside the effective aperture used imply that we are dealing with near mm, field or Fresnel zone due to the fact that where is the largest dimension of the prism (this is the better case and corresponds to the extraordinary transmission metamaterial prism). Besides, for the receiver distance of 100 mm, we are in the limit between the reactive near-field distance mm. and the radiating near-field, since Therefore, the index of refraction derived from this setup is an approximation, but can be considered accurate enough, since the Fresnel region is characterized by the onset of diffusion of the field and the wavefront outside the boundaries defined by the extension of the rays through the effective aperture; the latter, however, still define the propagation of the major portion of the field. The frequency dependence of the spatial power distribution for the extraordinary transmission metamaterial prism is rendered in panel 8 alongside the dispersion diagram so as to correlate the eigenmodes predicted by the numerical analysis with the experimental results. From around 53 GHz up to 60 GHz the emerging beam is recorded at negative angles of deflection for the co-polar measurements, see Fig. 8(a) and (c). The high transmittance observed in these panels (the negative index band dB. Indeed, comparable insertion losses reaches values of for the negative index band have been reported at THz frequencies [24]) could serve to refutes the argument of those who stress the low level of transmission in the former metamaterials experiments to arrive at the conclusion that the reported deflection it is nothing but simply the radiation pattern of a surface wave that can exist only on a finite periodic structure [37]. It is worth mentioning, that rectangular or ellipsoidal apertures can be considered to further increase the transmission [24], [27], [29]. Following with the description of Fig. 8, we can clearly observe at 60 GHz the minimum associated to Rayleigh-Wood’s anomaly and from 63 GHz the second eigenmode appears. In this case, as the band structure predicted (see Fig. 2), the behavior of this mode is right-handed and thus, the outgoing zeroth-order beam suffers positive refraction. Furthermore, this mode is related to the first propagating mode of the waveguide defined by the hole. It is well-known that the index of refraction of this mode is positive and smaller than 1 [5]. Thus, from Snell’s law, the angle of deflection must be smaller than the angle of the prism, which in our extraordinary transmission metamaterial prism case is 26.6 deg. This agrees with the experimental results of Fig. 8 since the angle of refraction of this second mode is below 15 deg all along the band. Relative significant transmittance is also scanned for positive angles of deflection but this stems from the finite size of the aperture, giving rise to edge distortions as it was also observed in the simulation. Finally, the cross-polar values (Fig. 8(b) and (d))

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Fig. 8. Angular power distribution of the extraordinary transmission metamaterial prism as a function of frequency for co-polar measurement (a), (c) and mm (top), and 600 mm (bottom). cross-polar measurement (b), (d) at z The angle of refraction is referenced to the normal surface. (top inset) Band structure. (Note: panel (a) was already presented in [21]).

= 100

remain at least 20 dB below co-polar results within the interest region. On the other hand, the results corresponding to the non-cutoff hole arrays prism are depicted in Fig. 9. Now, the zeroth-order beam displays a positive angle of refraction for both bands. Together with these beams, transmittance spots appear at negative angles of refraction. However, these emerging beams correspond to the higher order diffracted ones as predicted by Fig. 6(b) and also Fig. 7. Besides, the higher order diffracted can be seen at around 70 deg of angle of refraction mode for frequencies above 67 GHz. The previous comment about the near-zero effective index of refraction associated to the normal propagating modes of hollow metallic waveguides is also applicable in this case. Neither the first propagating mode, nor the second one undergoes angle of refractions greater than the angle of this prism (16.7 deg). As in the extraordinary transmission metamaterial prism, the Rayleigh-Wood’s anomaly is also emphasized by the deep minimum at 60 GHz, since this feature depends on the periodicity along the -axis (parallel to the incident electric field), independently of any other dimension or type of propagation inside the stack. The relatively high values of the cross-polar component in the case of non-cut-off hole prism, see Fig. 9, may stem mode (first propagating mode) supported by from the the circular waveguide. This mode has components both in and -axis, and thus, in this regime the prism is susceptible of cross-polarization. It is worth noting that this characteristic of higher cross-polar values when the hole operates above cut-off

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Fig. 9. Angular power distribution of the non-cut-off holes prism as a function of frequency for co-polar measurement (a), (c) and cross-polar measurement (b), (d) at z mm (top), and 600 mm (bottom). The angle of refraction is referenced to the normal surface.

= 100

is also observed in the extraordinary transmission metamaterial prism, see Fig. 8 again. There, the transmittance above the Rayleigh-Wood’s anomaly, which in our particular set of parameters defined the limit between cut-off and propagating regime, increases slightly in the cross-polar case with respect to the one recorded in the backward frequency band. As it has been mentioned before, the wedge experiment allows us to retrieve straightforwardly the index of refraction of the medium via the grating formula particularized to the zeroth order beam (i.e., Snell’s law). The results corresponding only to mm are plotted in Fig. 10 along with the longer distance the experimental error and the predicted index of refraction derived from the dispersion diagram of the infinite structure. The agreement between the numerical value and the experimental one is reasonable. This result is also in agreement with the index mm. In the retrieved in [20], where the distance was present case, the retrieved index can be considered more reliable, since at 600 mm we are in the Fresnel region, as mentioned above. IV. CONCLUSION Further physical insight on the variety of propagation characteristics of stacked perforated metallic plates has been reported in this work. It has been shown that the dimensions of the hole change dramatically the behavior of the stack, moving from a non-conventional negative effective index of refraction medium when the hole is in cut-off towards a positive effective index of refraction when the hole is working in propagation. Dispersion diagrams of the stacked subunit structures have

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REFERENCES

Fig. 10. Evolution of refractive index with the frequency calculated numerically and experimentally for the extraordinary transmission metamaterial prism (a) and the non-cut-off prism (b). Red spots are the experimental points along with the corresponding error bars, the black curve is the polynomial interpolation (order 3) of the experimental data and the blue one is the simulation result.

been presented to stress the topological dependence of the negative refractive index band and connect the resonances of those subunit structures to the electromagnetic propagation behavior of the complete structures: extraordinary transmission metamaterial and stacked non-cut-off hole arrays. This analysis discloses the different mechanisms that govern each stack. The simulation and experimental results show plainly that the negative refraction achieved with subwavelength holes is not caused by grating lobes. Even more, the grating lobes that emerge in the prism of cut-off holes are indeed located at positive angles, supporting the interpretation of left-handed media behavior. For propagating holes, the main beam is positively refracted with grating lobes in negative angles, as usual. There is a very good agreement between simulation and measurement despite the fact that experiment is done in the Fresnel zone. The numerically computed and experimentally derived refractive indices agree well within the experimental error associated to the measurement. The present results could find application in other ranges of the electromagnetic spectrum such as terahertz, infrared or even visible, since extraordinary transmission has been reported there as well. Applications on lenses, volumetric multiplexors, polarizing devices and frequency selective surfaces can be envisaged at those higher frequencies. Finally, we expect that the reported results, mainly the experimental ones, can stimulate the discussion on the explanation of effective negative refractive index.

[1] W. E. Kock, “Metal-lens antennas,” Proc. IRE, vol. 34, no. 11, pp. 828–836, 1946. [2] W. E. Kock, “Path-length microwave lenses,” Proc. IRE, vol. 37, no. 8, pp. 852–855, 1949. [3] S. B. Cohn, “Electrolytic-tank measurements for microwave delay lens media,” J. Appl. Phys., vol. 21, pp. 674–680, 1950. [4] W. Rotman, “Plasma simulation by artificial dielectrics and parallelplate media,” IEEE Trans. Antennas Propag., vol. 10, no. 1, pp. 82–95, Jan. 1962. [5] J. Brown, “Artificial dielectrics having refractive indices less than unity,” Proc. IEE (London), vol. 100, pt. 4, pp. 51–62, 1953. [6] R. E. Collin, Field Theory of Guided Waves, ser. Electromagnetic Wave Theory, 2nd ed. Piscataway: IEEE Press, 1991. [7] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, 1999. [8] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, no. 5514, pp. 77–79, 2001. [9] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, no. 25, pp. 4773–4776, 1996. [10] A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter, vol. 20, no. 29, p. 295222-1-11, 2008. [11] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys. Usp., vol. 10, no. 4, pp. 509–514, 1968. [12] V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon., vol. 1, pp. 41–48, 2006. [13] S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negativeindex metamaterials,” Phys. Rev. Lett., vol. 95, no. 13, pp. 1374041–137404-4, 2005. [14] G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science, vol. 32, no. 5775, pp. 892–894, 2006. [15] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature, vol. 391, no. 6668, pp. 667–669, 1998. [16] M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimetre-wave transmission through subwavelength hole arrays,” Opt. Lett., vol. 29, pp. 2500–2502, 2004. [17] M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimetre wave transmission through quasioptical subwavelength perforated plates,” IEEE Trans. Antennas Propag., vol. 53, pp. 1897–1902, 2005. [18] D. R. Jackson, A. A. Oliner, T. Zhao, and J. T. Williams, “The beaming of light at broadside through a subwavelength hole: Leaky-wave model and open stopband effect,” Radio Sci., vol. 40, pp. 1–12, 2005. [19] M. Beruete, M. Sorolla, and I. Campillo, “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express, vol. 14, no. 12, pp. 5445–5455, 2006. [20] M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express, vol. 16, no. 2, pp. 560–566, 2008. [21] M. Beruete, M. Navarro-Cía, F. Falcone, I. Campillo, and M. Sorolla, “Connection between extraordinary transmission and negative refraction in a prism of stacked subwavelength hole arrays,” J. Phys. D: Appl. Phys., vol. 42, no. 16, pp. 165504-1–165504-4, 2009. [22] M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla Ayza, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1514–1521, 2007. [23] M. Beruete, M. Navarro-Cía, I. Campillo, F. Falcone, I. Arnedo, and M. Sorolla, “Parametrical study of left-handed or right-handed propagation by stacking hole arrays,” Opt. Quantum Electron., vol. 39, no. 4–6, pp. 285–293, 2007. [24] C. Croënne, F. Garet, É. Lheurette, J. L. Coutaz, and D. Lippens, “Left handed dispersion of a stack of subwavelength hole metal arrays at terahertz frequencies,” Appl. Phys. Lett., vol. 94, no. 13, pp. 1331121–133112-3, 2009. [25] P. Weis, O. Paul, C. Imhof, R. Beigang, and M. Rahm, “Strongly birefringent metamaterials as negative index terahertz wave plates,” Appl. Phys. Lett., vol. 95, no. 17, p. 171104-1-3, 2009.

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[26] O. Paul, R. Beigang, and M. Rahm, “Highly selective terahertz bandpass filters based on trapped mode excitation,” Opt. Express, vol. 17, no. 21, pp. 18590–18595, 2009. [27] S. Wang, F. Garet, K. Blary, C. Croënne, E. Lheurette, J. Coutaz, and D. Lippens, “Composite left/right-handed stacked hole arrays at submillimeter wavelengths,” J. Appl. Phys., vol. 107, no. 7, p. 074510-1-6, 2010. [28] O. Benjamin, R. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett., vol. 96, no. 24, p. 241110-1-3, 2010. [29] S. Wang, F. Garet, K. Blary, E. Lheurette, J. L. Coutaz, and D. Lippens, “Experimental verification of negative refraction for a wedge-type negative index metamaterial operating at terahertz,” Appl. Phys. Lett., vol. 97, no. 18, p. 181902-1-3, 2010. [30] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature, vol. 455, no. 7211, pp. 376–379, 2008. [31] M. Navarro-Cía, M. Beruete, I. Campillo, and M. Sorolla, “Millimeter-wave left-handed extraordinary transmission metamaterial demultiplexer,” IEEE Antennas Wireless Propag. Lett., vol. 8, no. 1, pp. 212–215, 2009. [32] M. Beruete, I. Campillo, J. E. Rodríguez-Seco, E. Perea, M. NavarroCía, I. J. Núñez-Manrique, and M. Sorolla, “Enhanced gain by doubleperiodic stacked subwavelength hole array,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 12, pp. 831–833, 2007. [33] M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, “Planoconcave lens by negative refraction of stacked subwavelength hole arrays,” Opt. Express, vol. 16, no. 13, pp. 9677–9683, 2008. [34] M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Converging biconcave lens owing to a left-handed extraordinary transmission metamaterial,” Appl. Phys. Lett., vol. 94, no. 14, pp. 144107-1–144107-3, 2009. [35] M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Viability of focusing effect by left-handed stacked subwavelength hole arrays,” Phys. B, vol. 405, no. 14, pp. 2950–2954, 2010. [36] M. Navarro-Cía, M. Beruete, I. Campillo, and M. Sorolla, “Fresh metamaterials ideas for metallic lenses,” Metamaterials, vol. 4, no. 2–3, pp. 119–126, 2010. [37] B. A. Munk, Metamaterials: Critique and Alternatives. Hoboken, NJ: Wiley, 2009. [38] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [39] P. F. Goldsmith, Quasioptical Systems—Gaussian Beam, Quasioptical Propagation, and Applications. New York: Wiley IEEE Press, 1998. [40] M. Thumm, A. Jacobs, and M. Sorolla, “Design of short high-power TE HE mode converters in highly overmoded corrugated waveguides,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 2, pp. 301–309, Feb. 1991. [41] [Online]. Available: http://www.cst.com [42] N. Marcuvitz, Waveguide Handbook, ser. Electromagnetic Waves Series. London: P. Peregrinus on behalf of IEE, 1986. [43] M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microwave Wirel. Compon. Lett., vol. 15, pp. 116–118, 2005. [44] M. Beruete, M. Sorolla, M. Navarro-Cía, F. Falcone, I. Campillo, and V. Lomakin, “Extraordinary transmission and left-handed propagation in miniaturized stacks of doubly periodic subwavelength hole arrays,” Opt. Express, vol. 15, no. 3, pp. 1107–1114, 2007. [45] [Online]. Available: http://www.abmillimetre.com

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Miguel Navarro-Cía (S’08–M’10) was born in Pamplona, Spain, in 1982. He received the M.Sci. and Ph.D. degrees in telecommunication engineering, and the M.Sc. degree in introduction to research in communications from the Public University of Navarre, Navarre, Spain, in 2006, 2010 and 2007, respectively. From September 2006 to January 2010, he was working as a Predoctoral Researcher in the Electrical and Electronic Engineering Department, Public University of Navarre, where he is currently working

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as an Assistant Researcher. Also, he was working as a Visiting Researcher at the University of Pennsylvania of Philadelphia, from May to August 2010, at the Imperial College London of London, from June to September 2008 and from April to October 2009, and at the Nanophotonics Technology Center of Valencia, from November to December 2008. His current research interests are focused on metamaterials, enhanced transmission phenomena, antennas, complex surface waves, frequency-selective surfaces, and millimeter, terahertz, and infrared frequencies. Dr. Navarro-Cía is member of the Optical Society of America and National Association of Telecommunication Engineers. He was the recipient of an Research Staff Training (FPI) Fellowship, and was awarded the “COIT/AEIT best doctoral thesis in basic principles and technologies of information and communications and their applications” corresponding to the XXXI edition of the Telecommunication Engineers Awards 2010 and of the 2011 Junior Researcher Raj Mittra Travel Grant.

Miguel Beruete was born in Pamplona, Spain, in 1978. He received the M.Sc. and Ph.D. degrees in telecommunication engineering from the Public University of Navarre, Navarre, Spain, in 2002 and 2006, respectively. From January to March 2005, he was working as a Visiting Researcher at the University of Seville, as a part of his doctoral research. From February 2007 to September 2009, he was at the Electronics Department of the technological center CEMITEC in Noain (Navarre), developing, designing and measuring high frequency communication devices. He is currently working as a Postdoctoral Researcher for the Millimeter-waves and Terahertz Laboratory at the Public University of Navarre, Pamplona, Spain. His research interests include quasi-optical devices, microwave, millimeter-wave and terahertz frequency selective surfaces and antennas based on extraordinary transmission and metamaterials. Currently, he is exploring the connection between extraordinary transmission, photonic crystals and left-handed metamaterials.

Francisco Falcone (M’05–SM’09) was born in Caracas, Venezuela, in 1974. He received the M.Sc. and Ph.D. degrees in telecommunication engineering from the Public University of Navarre, Navarre, Spain, in 1999 and 2005, respectively. From 1999 to 2000, he was with the Microwave Implementation Department, Siemens-Italtel, where he was involved with the layout of the Amena mobile operator. From 2000 to 2008, he was a Radio Network Engineer with Telefónica Móviles España. Since the beginning of 2003, he has also been an Associate Lecturer with the Electrical and Electronic Engineering Department, Public University of Navarre. Currently he is working as an Associate Professor at UPNA. His main research interests include electromagnetic-bandgap devices, periodic structures, and metamaterials.

Jesús M. Illescas was born in Talavera de la Reina, Spain, in 1978. He received the M.Sc. degree in electrical and electronic engineering in 2002 and the M.Sc. degree in introduction to research in communications in 2007 from the Public University of Navarre, Spain. In September 2001, he received a two-year scholarship to collaborate as a researcher in the Electrical and Electronic Engineering Department at the Public University of Navarre. From September 2003 to March 2009, he worked in the R&D Department at the company “Consultora Navarra de Telecomunicaciones.”He is currently working in the R&D Department at the technology-based innovative company TAFCO Metawireless. His research interests include passive microwave devices and antennas, and numerical computation of electromagnetic theory.

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Igor Campillo was born in Bilbao, Spain, in 1971. He received the degree in physics and the Ph.D. degree in condensed matter physics from the University of the Basque Country, Bilbao, Spain, in 1994 and 1999, respectively. He spent five years involved in the study of electronic excitations in real solids at the University of the Basque Country, first as a Postgraduate Student and then as an Associate Professor. From 2000 to 2006, he was with Labein Centro Tecnológico, Derio, Spain, as a Researcher and Project Manager. In 2006, he joined the Nanoscience Cooperative Research Center, CIC nanoGUNE Consolider.

Mario Sorolla Ayza (S’82–M’83–SM’01) was born in Vinaròs, Spain, in 1958. He received the Telecommunication Engineer degree from the Politechnical University of Catalonia, Barcelona, Spain, in 1984 and the Ph.D. degree from the Politechnical University of Madrid, Madrid, Spain, in 1991. From 1986 to 1990, he designed very high power millimeter waveguides for plasma heating in the Euratom-Ciemat Spanish Nuclear Fusion Experiment and was an Invited Scientist at the Institute of Plasma Research at Stuttgart University, in Germany, from 1987 to 1988. He worked in microwave integrated circuits and monolithic microwave integrated circuits for satellite communications for industries. From 1984 to 1986, he was a Professor at the Politechnical University of Catalonia, and from 1991 to 1993, at Ramon Llull University in Barcelona. Since 1993, he has been a Professor at the Public University of Navarre, Navarre, Spain. His interest range from high-power millimeter waveguide components and antennas, coupled wave theory, quasioptical systems in the millimeter wave and terahertz range and applications of metamaterials and enhanced transmission phenomena to microwave circuits and antennas.

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Design and Free-Space Measurements of Broadband, Low-Loss Negative-Permeability and Negative-Index Media Scott Michael Rudolph, Member, IEEE, Carl Pfeiffer, Student Member, IEEE, and Anthony Grbic, Member, IEEE

Abstract—We present the design and measurement of broadband, volumetric negative-permeability and negative-refractive-index (NRI) media. Both of these media are fabricated using standard printed-circuit-board techniques and operate at X-band frequencies. The S-parameters of four-cell slabs of the negative-permeability and NRI media are measured, and the material parameters of the NRI lens are extracted. The four-cell-thick ( 0 3) NRI lens exhibits a backward-wave bandwidth of 41.2% and a total loss of 0.67 dB at the operating 1). Super-resolved focusing in free frequency (where space is also demonstrated, and spatial frequencies beyond the free-space wavenumber are recovered over a bandwidth of 7.4%. A focus with a half-power beamwidth of 0 27 0 is achieved at 10.435 GHz. Index Terms—Metamaterials, negative index of refraction, negative permeability, periodic structures, super resolution.

I. INTRODUCTION

S

INCE the initial realization of a negative-refractive-index (NRI) medium [1], volumetric NRI media have been associated with high losses and narrow operating bandwidths. However, this perception should be limited to split-ring-resonator (SRR)/wire arrays, rather than NRI media as a whole. Typical SRR arrays exhibit negative permeability over a 10% bandwidth, limiting the bandwidth of NRI media to 10% as well. This forces NRI structures to operate at frequencies close to the resonant frequency of the negative permeability medium , where the permeability diverges and losses are high. A design of a broadband NRI medium was proposed in [2] and was subsequently analyzed in [3]. This design and others similar in approach [4] use contra-directional coupling between a forward free-space wave and a backward wave guided by a transmission line to achieve negative permeability over bandwidths more than five times that achievable using isolated SRRs. The enhanced negative-permeability bandwidth allows for an equally Manuscript received March 02, 2010; revised November 26, 2010; accepted December 10, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by an Air Force Office of Scientific Research (AFOSR) Young Investigator Research Program Award (FA9550-08-1-0067), an NSF Faculty Early Career Development Award (ECCS-0747623) and through the Multidisciplinary University Research Initiative Program (FA9550-06-01-0279). The authors are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 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.2011.2158948

large backward-wave bandwidth when a wire array (negative permittivity medium) is introduced to create a NRI medium. The wide bandwidth of this structure further separates the operating frequency from , resulting in lower losses at the frequencies of interest when compared to SRR/wire media made from the same materials. The loss reduction is supported by a well established tenet of filter theory, which states that the midband insertion loss of a bandpass filter is inversely proportional to its fractional bandwidth [5]. Since NRI media have a bandpass frequency response, this rule extends naturally to them as well. In an earlier experimental demonstration [6], slabs of broadband negative-permeability and NRI media were fabricated using printed-circuit-board (PCB) technology and measured in a parallel plate waveguide environment. The NRI lens was used to demonstrate subwavelength focusing at 2.45 GHz, indirectly confirming its low-loss performance. While this test showed that volumetric NRI media could be realized using this new topology, the loss and bandwidth of the medium remained unconfirmed by experimental data. Furthermore, the question of how such a lens would perform in free-space was unresolved. In this paper, we present free-space measurements of metamaterial slabs that exhibit a negative permeability and a negative index of refraction over broad bandwidths at X-band frequencies [7]. As in the earlier experimental work [6], these slabs are fabricated using PCB technology, but a more efficient design is employed to allow the construction of electrically large slabs. Both the negative permeability and NRI slabs are characterized through free-space measurements of their normal-incidence scattering parameters and are found to exhibit low loss and unprecedented bandwidths of operation. The measured bandwidth and loss of the NRI medium are compared to other NRI topologies, demonstrating that this design achieves the widest backward-wave bandwidth and the lowest loss published to date. Additionally, the NRI slab is used as a lens to demonstrate free-space focusing beyond the diffraction limit and the recovery of evanescent spatial frequencies over a bandwidth of 7.4%: approximately three quarters of the entire backward-wave bandwidth of a typical SRR/wire medium. The material parameters extracted from free-space measurements are used to predict and corroborate the focal pattern measured at the operating frequency of 10.435 GHz. To the knowledge of the authors, this is the first time experimentally extracted material parameters have been used to predict the results of focusing experiments.

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Fig. 2. Variations in backward-wave bandwidth of the NRI medium through the change of its geometric parameters. In all cases except the nominal, the medium f . is not impedance-matched to free-space, and f

=

Fig. 1. Unit cell of the broadband NRI medium with an enlargement of the interdigitated capacitor footprint. Removing the central wire results in the unit cell of the broadband negative-permeability medium. TABLE I DIMENSIONS OF THE NRI UNIT CELL

II. DESIGN One unit cell of the broadband NRI medium is shown in Fig. 1, with the physical dimensions provided in Table I. In the absence of the central wire, the structure represents one unit cell of the negative-permeability medium, which is formed by the remaining transmission-line cage [3]. As in SRR/wire media, the central wire is used to achieve negative permittivity in the NRI medium for waves with vertically polarized electric fields. Wire arrays exhibit negative permittivity from DC to their elec, making them suitable for broadband tric plasma frequency applications. Consequently, only the SRRs were eliminated, as they represent the limiting factor in achieving a broadband NRI medium. For this structure, maximizing bandwidth is not the only concern. In order for this NRI lens to produce a super-resolved focus, it must be impedance-matched to free-space at the op. The elecerating frequency, necessitating that tric response of the transmission-line cage generally increases the relative permittivity of the medium, similar to structures described in [8]. This higher permittivity has two consequences: one advantageous and one detrimental. The advantage is that thicker wires (i.e. lower inductances) are used to achieve at the design frequency. This is particularly important since or ) exhibit higher gauge wires ( higher losses and are quite fragile. The disadvantage is that

as frequency increases the relative permittivity approaches a , while the relative permeability remains value above . Since at a lower frelower than quency, the slope (with respect to frequency) of the permittivity is larger than the slope of the permeability at these higher frequencies. Consequently, in impedance-matched structures, the electric plasma frequency is significantly lower than the mag, and the high-frequency netic plasma frequency end of the backward-wave band is limited by the permittivity response. This decrease in bandwidth for the impedance-matched structure is illustrated in Fig. 2. The dispersion diagram of the NRI medium with the nominal parameters given in Table I shows that the backward-wave bandwidth of this structure is 49.2%. However, if the impedance of the medium is not a concern, the bandwidth can be made much larger. Increasing the height of increases the electric the unit cell, b, and the wire radius, plasma frequency providing a wider NRI bandwidth. Additional bandwidth can be achieved by lowering the resonant frequency of the negative permeability medium. This is done by extending the length of the interdigital fingers of the capacitor, thereby increasing the capacitance. By optimizing these three parameters, the NRI bandwidth can be increased up to 68.2%. This shows that this topology of NRI media can achieve bandwidths even larger than those measured in the experimental structure, if one is not constrained by the impedance matching criterion. For the design in this paper, we also require that the unit cell width is much smaller than a wavelength (specifically, ) for all frequencies in the backward-wave band, ensuring the validity of effective medium theory at our operating . Despite meeting this requirement, frequency where the effective medium theory breaks down at low frequencies, where the permeability diverges. In this region, the refractive . index has a very large magnitude, such that When this occurs, the NRI medium exhibits spatial dispersion as can be seen in Fig. 3. However, for frequencies closer to the , the medium exhibits two-dimenoperating point sionally isotropic behavior for waves propagating in the plane, as indicated by the circular contours at these frequencies.

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Fig. 5. Diagram showing the MTL circuit model of the broadband NRI medium for on-axis propagation. The dashed lines separate the individual components that must be cascaded to form a unit cell.

Fig. 3. Equifrequency contour plot of the infinite NRI medium.

Fig. 6. Dispersion diagram of the backward-wave mode as calculated by MTL analysis (solid line) and full-wave simulation (circles).

Fig. 4. Unit cell of the broadband NRI medium simplified using image theory. The labels indicate the MTL component variables.

III. CIRCUIT MODEL While Fig. 2 demonstrates how particular parameters affect bandwidth, a circuit model can provide a more complete understanding of the NRI medium. In [9], broadband NRI media similar to the one presented here are analyzed using multiconductor transmission-line (MTL) theory. This same analysis can be applied to the structure in Fig. 1. Since the electric field is vertically polarized, image theory allows the infinite periodicity in the vertical direction to be modeled by placing perfectly-conducting sheets on the top and bottom of the unit cell. Further, due to the symmetry of the structure, the bottom half of the unit cell can be replaced by a perfectly conducting sheet placed in the center of the unit cell, as shown in Fig. 4. In this form, the

structure can be represented by an MTL circuit , as shown in Fig. 5. The MTL circuit consists of three different loading elements , and the central (the interdigitated capacitors, , the vias, ) connected by coupled transmission lines. The inducwire, tance and capacitance matrices of the unloaded transmission lines can be found in the quasi-static limit using Ansoft’s Maxwell, a commercial three-dimensional electromagnetic solver. It should be noted that the unloaded transmission lines are formed by eliminating the vias and central wire from the unit cell and extending the fingers of the capacitor such that they directly connect one side to the other while still preserving the and are footprint of the capacitor. The inductance values found through eigenmode simulations using Ansoft’s HFSS, is obtained in the quasi-static and the loading capacitance limit using Maxwell. The values of all the circuit elements are given in Table II. Once the values of all parameters are known, the effects of the individual components can be integrated together using transfer (ABCD) matrices. To accomplish this, the transfer matrices of individual elements are successively multiplied together to give the transfer matrix of the entire unit cell [9]. Fig. 5 depicts the components associated with each individual transfer matrix and the order in which they are multiplied together. Once the complete transfer matrix of the unit cell is obtained, periodic boundary conditions are applied to calculate the dispersion diagram. The results of this MTL analysis exhibit good agreement with the results obtained from full-wave simulation, both of which are plotted in Fig. 6.

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TABLE II VALUES FOR THE MTL PARAMETERS

Fig. 7. Photograph of the capacitive grid on one of the PCB layers used to construct both the negative-permeability and NRI slabs.

IV. S-PARAMETER MEASUREMENTS Using the design with the physical parameters listed in Table I, NRI and negative-permeability slabs with thicknesses of four cells (1.02 cm) were constructed in order to measure their material properties. The slabs were built using 46 horizontal PCB layers, which were spaced 1.07 mm apart. Each PCB layer (shown in Fig. 7) consisted of a capacitive grid printed on each face of the board. These grids were connected to each other , as shown in Fig. 1. The layers were held by vias of radius with the prescribed spacing by a plastic holder on either end. Once assembled, this structure formed the negative-permeability slab discussed below. The NRI lens was constructed using the same PCB layers which were supported in the same fashion, but vertical wires were threaded through the center of each unit cell. The wires were then attached to ground planes with silver epoxy on both the top and bottom PCB layers. To measure the scattering parameters (S-parameters) of these slabs in free space, the samples were illuminated with a tightly focused, collimated source. In order to produce a collimated beam that was confined within a small radius, a quasi-optical Gaussian beam telescope was used [10]. The telescope consisted of a rectangular horn antenna and a pair of lenses, known as an achromatic doublet [11]. As the name suggests, the achromatic doublet produced a focus whose size and location were independent of frequency. An additional benefit of using a pair of lenses rather than a single lens was that the spot size at the focus was much smaller. Each lens in the doublet was made of Rexolite and was bi-hyperbolic in shape. The diameters of the lenses were 32.5 cm and the input and output focal distances were equal to 45 cm. The measurement plane of the Gaussian beam telescope occurred at the output focus of the second lens, 1.8 m away from the input beam waist of the horn antenna. The S-parameters of the slabs were measured by placing the sample between two Gaussian-beam telescopes, as shown in Fig. 8. Each telescope was placed on a separate linear translation stage, whose position was controlled by a stepper motor

Fig. 8. Photograph of the quasi-optical, free-space measurement system.

with 5 accuracy. This allowed the measurement planes to be exactly aligned with the faces of the slab. The beam radii at the focal planes were approximately 5 cm, while the slabs measured 16 cm high and 16 cm wide. As a result, the amplitude of the Gaussian beam was 20 dB lower at the edge of the sample than at its center, limiting diffraction from the slab. The horn antennas of each telescope were connected to the two ports of an Agilent E8361A network analyzer, which collected the transmission and reflection coefficient data. Before any data were taken, the system was calibrated using a free-space thru-reflect-line (TRL) calibration method. This calibration de-embedded the S-parameters of the slab from those of the entire measurement system. The calibration of the measurement system was particularly important because it corrected for the imperfections inherent in the system’s design. One such issue was that the phase centers of the horn antennas were frequency dependent. As a result, the output beam waist did not coincide with the output focus of the achromatic doublet at frequencies for which the input beam waist (the phase center of the horn) did not coincide with the input focus of the doublet. This resulted in imperfect collimation at the measurement plane for those frequencies. This effect, however, was slight since the deviation in beam waist position was small relative to the Rayleigh range [10]. Nevertheless, the TRL calibration was able to account for the phase variation over the measurement plane and eliminate its effects. A second issue was that the Gaussian beam telescopes were designed to focus the fundamental Gaussian mode. However, the rectangular horn antennas only coupled 88% of their power into the fundamental mode [12], with the rest of the power being coupled into higher-order modes. These higher-order modes have larger effective beam radii than the fundamental, which made them more susceptible to diffraction losses. Consequently, much of the power in higher-order modes escaped the measurement system. Fortunately, these losses increased the purity of the transmitted beam, while the effect of the lost power was eliminated through

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Fig. 11. Real (solid lines) and imaginary (dashed lines) parts of the measured (thick, black lines) and simulated (thin, red lines) index of refraction of the NRI slab. Fig. 9. Simulated and measured magnitudes of S permeability slab.

and S

for the negative-

Fig. 12. Real (solid lines) and imaginary (dashed lines) parts of the measured (thick, black lines) and simulated (thin, red lines) impedance of the NRI slab. The singularities are due to Fabry-Perot resonances of the slab.

Fig. 10. Simulated and measured magnitudes of S

and S

for the NRI lens.

the calibration process. The TRL calibration also accounted for scattering from proximate objects, most notably, the stand which supported the sample. To check the accuracy of the calibration, commercial microwave substrates were measured and their permittivity and permeability values were verified. Due to errors in the manufacturing process, some of the dimensions shown in Fig. 1 were slightly different from those on the fabricated structure. Both the values of the original design as well as those measured on the fabricated boards are presented in Table I. The most significant changes were in the substrate height (designed as 2.34 mm, measured as 2.29 mm) and the via , measured as 330 ). These erdiameter (designed as 254 rors shifted the design frequency from 10 GHz to 10.435 GHz. The simulations were redone to accurately reflect the dimensions of the measured structure. The new simulated results are used in the remainder of the paper, rather than those of the original design. The measured S-parameters of the negative-permeability medium are compared to simulation in Fig. 9. By calculating the effective permeability from the S-parameters, the structure was found to exhibit negative permeability over a bandwidth of 45.3%. In this negative-permeability region, the slab provided ) over a frequency range excellent attenuation ( of 8.86 GHz to 12.3 GHz. Additionally, the medium exhibited low loss throughout this frequency range. At the operating frequency of 10.435 GHz, the four-cell negative-permeability slab experienced material losses of only 0.235 dB. The measured and simulated S-parameters of the NRI lens from are shown in Fig. 10. The slab exhibited 8.99 GHz to 11.91 GHz, indicating that the structure was well

Fig. 13. Real (solid lines) and imaginary (dashed lines) parts of the measured relative permittivity (blue) and permeability (red) of the NRI slab. The singularities are due to Fabry-Perot resonances.

Fig. 14. Measured loss of the NRI (solid line) and negative-permeability (dashed line) four-cell slabs.

matched to free space over this frequency range. The S-parameters were used to calculate the material properties of the NRI lens, such as index of refraction (Fig. 11), impedance (Fig. 12), permeability and permittivity (Fig. 13), and loss (Fig. 14). As shown in Fig. 13, both permittivity and permeability are negative over a bandwidth of 41.2%. The resonant frequency of the limits the bandwidth at low frepermeability quencies, while the high-frequency limit is the electric plasma . The extracted permittivity and frequency permeability curves display unusual behavior below 10 GHz.

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TABLE III COMPARISON OF THE LOSS PERFORMANCE OF NRI MEDIA

3One-dimensional propagation only yQuoted from simulation

Fig. 15. Photograph of the near-field focusing measurement setup. The inset shows an enlargement of the dipole probe with a sleeve balun.

At these frequencies, the unit cells begin to exhibit spatial dispersion, degrading the accuracy of the effective medium theory. Additionally, the material parameter extraction method used to calculate the wave impedance breaks down when the electrical [13]. This phenomlength of the NRI slab is a multiple of enon occurs around 7.8 GHz, 8.2 GHz, 8.7 GHz and 10.0 GHz for the measured slab (see Fig. 12). Since the wave impedance is used to calculate relative permittivity and permeability, the extraction inaccuracies are inherent in these curves as well. As a result, the permittivity curve has a positive imaginary part close to the resonant frequencies, which might raise concerns about the structure exhibiting gain. To verify the passivity of the medium, one should focus on the index of refraction (Fig. 11), for which the extraction method remains accurate even at resonant lengths. The imaginary part of the index of refraction is negative for all frequencies, as is required for a passive structure. The passivity the material is further verified by Fig. 14, which shows that the loss is positive for all frequencies. Additionally, Fig. 14 shows that despite having high losses close to the reso, the losses are less than 1 dB for the entire nant frequency four-cell slab for all frequencies above 9.5 GHz. This means that the majority of the backward-wave bandwidth is low-loss, making this lens suitable for practical applications. The operating frequency of the NRI lens was 10.435 GHz, which corresponded to the frequency at which the relative permeability was closest to 1. The material parameters at this and frequency were found to be . At the operating frequency, the entire four-cell slab exhibited a loss of only 0.67 dB, or 0.17 dB/cell.

A common metric of loss performance is the figure of merit, defined as the real part of the refractive index divided by the . At the design frequency, imaginary part or the measured FOM of the NRI lens was 31.4. The bandwidth, loss and FOM of the NRI lens are compared with other NRI media reported in literature in Table III. These media represent several different topological approaches to achieving a negative index of refraction, including SRR/wire arrays [14], [15], a planar NRI transmission-line (TL) medium [16], volumetric NRI TL media [17], [18] along with the proposed structure. The planar NRI TL medium has a bandwidth conspicuously larger than any of the other topologies. This is primarily because this medium uses chip components, by which much higher capacitance and inductance values can be realized compared to printed elements. This medium also exhibits a FOM slightly higher than the one measured in the proposed structure, however, the value in [16] is taken from simulation. Comparison of this value to the full-wave simulation of the proposed structure is more appropriate. In that case, the FOM of the planar NRI TL medium is less than half of the proposed structure’s FOM, despite operating at a frequency one order of magnitude lower. Compared to all other volumetric structures, the measured structure exhibits lower loss and wider bandwidth regardless of topology or frequency of operation. As is expected, the SRR/ wire media have the narrowest bandwidths compared to other topologies. The bandwidths of the volumetric NRI TL media are better, but still significantly less than that of the proposed structure. Unlike the bandwidth, the loss per cell of these NRI media do not form a predictable pattern. While the proposed structure still exhibits the best loss performance, a relationship between topology and loss remains unclear. Such variation can be attributed to a lack of standardization in the way each medium is designed. In each structure, the substrates, Q-factors of the loading elements, unit cell size (in terms of wavelength), and operating frequency are different. All of these factors influence the loss performance of a NRI medium. Further confusing the matter, the reported losses are often the minimum losses that occur in the backward-wave band, rather than those that occur at the frequency of operation. To exemplify the difference, the proposed structure exhibits losses as low as 0.095 dB/cell at the high-frequency edge of the backward-wave band (11.9 GHz), which is much less than the 0.17 dB measured at the operating frequency. Regardless of these many differences, the proposed structure clearly represents the realization of a volumetric NRI medium with widest bandwidth and lowest loss performance of any design published to date.

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Fig. 16. Plots of the measured vertically polarized electric field at 10.435 GHz. (a) Contour plots of the normalized electric field magnitude on the exit side of the NRI lens. Contour plots of the unwrapped electric field phase on the exit side of the NRI lens.

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Fig. 17. Plots of the normalized electric field magnitude at the image plane. (a) Normalized electric field magnitude at the image plane at 10.435GHz for the measured field with (solid line) and without (dotted line) the lens, as well as the predicted field with the lens (dashed line). (b) Normalized electric field magnitude at the image plane for several frequencies exhibiting sub-diffractionlimited focusing.

V. FOCUSING EXPERIMENTS In addition to measuring the S-parameters of the NRI lens, focusing experiments were also performed. Two dipoles with sleeve baluns were used to excite the lens and probe the fields around it. The dipoles were oriented such that the electric field incident on the lens was vertically polarized. One probe was used as a source and held a fixed distance of 5.08 mm (half the lens thickness) away from the lens. The other probe was attached to a three-dimensional translation stage and used to scan the fields beyond the exit face of the lens. The vertically-polarized electric field was measured on a horizontal plane, such that both the scanning probe and the source probe were at the same height. The half-power beamwidth was measured at the image plane, located 5.08 mm away from the exit face of the lens. Evanescent spatial frequencies were recovered from 10.015 GHz to 10.78 GHz: a fractional bandwidth of 7.4%. The normalized magnitude and phase of the vertically polarized electric field at 10.435 GHz are shown in Fig. 16. Fig. 17(a) shows the normalized field amplitude at the image plane at 10.435 GHz. A half-power beamwidth of was observed, which is significantly narrower than the diffractionlimited beamwidth for a line source of . Fig. 17(b) shows how the focus changes with frequency, plotting curves for several frequencies at which super-resolution is observed. To provide a point of comparison, the field was measured in the absence of the lens as well. The normalized field amplitude for this measurement is also plotted at the operating frequency of 10.435 GHz in Fig. 17(a). Prior to normalization, the electric field magnitude measured with the lens present was 88% higher than without the lens. The dashed curve in Fig. 17(a) represents the theoretical image of a vertically directed line current as predicted by the

impedance and propagation constant of the NRI medium ( and , respectively), which were measured in Section III. The field at the image plane is calculated by first finding the transfer function of the lens in terms of the transverse wavenumber . This is given by the expression

(1) where

Next, the contribution of the continuum of transverse wavenumbers is calculated for each point along the image plane, resulting in the expression

(2)

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REFERENCES

Fig. 18. Normalized Fourier transform magnitude of the field at the image plane for the measured field with (solid line) and without (dotted line) the lens, as well as the predicted field with the lens (dashed line). The values of these curves are normalized by the peak amplitude of the spectrum calculated with the lens present.

where is a constant that normalizes the maximum field amplitude at the image plane to 1. As Fig. 17(a) shows, the focus predicted by this analysis agrees well with the one obtained experimentally. The agreement between the focusing and transmission measurements confirms the results of both experiments. Furthermore, since the predicted curve was calculated under the assumption of a homogeneous slab, its agreement with the measured focal pattern validates the use of effective medium theory in designing the NRI metamaterial. To further demonstrate the resolution enhancement produced by the lens, Fourier transforms of all three curves shown in Fig. 17(a) were calculated and are plotted in Fig. 18. Again, the spectrum of the field measured with the lens present exhibited good agreement with the spectrum predicted analytically using the measured material parameters. Both of these curves have , indicating reFourier components with covery of the evanescent spectrum. However, the spectrum calculated from the field in the absence of the lens only includes , corresponding to propagating waves. This shows that without the lens, the evanescent waves decayed completely before reaching the focal plane, while with the lens, part of the evanescent spectrum was recovered. VI. CONCLUSION We have shown that negative-permeability and NRI media can be designed to exhibit low loss and a broad bandwidth of operation. We designed and fabricated a slab of the NRI medium and measured its material parameters in free space over X-band frequencies. The NRI slab had the highest figure of merit, lowest loss, and widest bandwidth of any volumetric NRI medium reported to date. Additionally, focusing experiments were performed using the NRI lens, and the material parameters obtained from transmission experiments were used to accurately predict the half power beamwidth of the super-resolved focus. Spatial frequencies higher than the free-space wavenumber were observed at the focal plane over a bandwidth of 7.4%, which is close to the entire backward-wave bandwidth of typical SRR/wire media. This low-loss, broadband performance, demonstrated that NRI media are indeed suitable for use in practical microwave-focusing and antenna applications.

[1] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [2] A. Grbic, “A 2-D composite medium exhibiting broadband negative permittivity and permeability,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 4133–4136. [3] S. M. Rudolph and A. Grbic, “A volumetric negative-refractive-index medium exhibiting broadband negative permeability,” J. Appl. Phys., vol. 102, p. 013904, Jul. 2007. [4] M. Stickel, F. Elek, J. Zhu, and G. V. Eleftheriades, “Volumetric negative-refractive-index metamaterials based upon the shunt-node transmission-line configuration,” J. Appl. Phys., vol. 102, p. 094903, Nov. 2007. [5] S. B. Cohn, “Dissipation loss in multiple-coupled-resonator filters,” Proc. IRE, vol. 47, no. 8, pp. 1342–1348, Aug. 1959. [6] S. M. Rudolph and A. Grbic, “Super-resolution focusing using volumetric, broadband NRI media,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2963–2969, Sep. 2008. [7] S. M. Rudolp and A. Grbic, “Broadband, low-loss negative-permeability and negative-Index media for free-space applications,” presented at the IEEE Int. Microwave Symp., Boston, MA, Jun. 9–11, 2009. [8] J. Shin, J. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett., vol. 102, p. 093903, Mar. 2009. [9] S. M. Rudolph and A. Grbic, “The design of broadband, volumetric NRI media using multiconductor transmission-line analysis,” IEEE Trans. Antennas Propag., vol. 58, no. 4, Apr. 2010. [10] P. F. Goldsmith, “Quasi-optical techniques,” Proc. IEEE, vol. 80, no. 11, pp. 1729–1747, Nov. 1992. [11] F. Biraud and G. Daigne, “Achromatic doublets for Gaussian beams,” IEEE Trans. Antennas Propag., vol. 39, no. 4, Apr. . [12] P. F. Goldsmith, Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications. New York: Wiley, 1997. [13] J. Baker-Jarvis, M. D. Janezic, J. H. Grosvenor, and R. G. Geyer, “Transmission/reflection and short-circuit line methods for measuring permittivity and permeability,” Nat. Inst. Standards Technol., vol. Tech. Note 1355-R, pp. 6–19, Dec. 1993. [14] K. Li, S. J. McLean, R. B. Greegor, C. G. Parazzoli, and M. H. Tanielian, “Free-space focused-beam characterization of left-handed materials,” Appl. Phys. Lett., vol. 82, no. 15, pp. 2535–2537, Apr. 2003. [15] K. Aydin and E. Ozbay, “Left-handed metamaterial based superlens for subwavelength imaging of electromagnetic waves,” Appl. Phys. A, vol. 87, pp. 137–141, Jan. 2007. [16] A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett., vol. 92, no. 11, p. 117403, Mar. 2004. [17] A. K. Iyer and G. V. Eleftheriades, “A multilayer negative-refractive-index transmission-line (NRI-TL) metamaterial free-space lens at X-band,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2746–2753, Oct. 2007. [18] A. K. Iyer and G. V. Eleftheriades, “Mechanisms of subdiffraction freespace imaging using a transmission-line metamaterial superlens: An experimental verification,” Appl. Phys. Lett., vol. 92, no. 13, p. 131105, Mar. 2008.

Scott Michael Rudolph (S’06–M’11) received the B.S.E., M.S.E., and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 2006, 2008, and 2011, respectively. His research interests include the design of volumetric metamaterials, frequency-selective surfaces and antennas. Dr. Rudolph received the IEEE Microwave Theory and Techniques Society Graduate Fellowship in 2009. In 2010 he was awarded the Rackham Predoctoral Fellowship.

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Carl Pfeiffer (S’08) received the B.S.E. and M.S.E. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 2009 and 2011, respectively, where he is currently working toward the Ph.D. degree. Mr. Pfeiffer received the IEEE MTT-S Undergraduate/Pre-Graduate Scholarship and the Graduate Assistance in Areas of National Need Fellowship, both in 2009.

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Anthony Grbic (S’00–M’06) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Toronto, ON, Canada, in 1998, 2000, and 2005, respectively. In January 2006, he joined the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, where he is currently an Assistant Professor. His research interests include engineered electromagnetic structures (metamaterials, electromagnetic bandgap materials, frequency-selective surfaces), printed antennas, microwave circuits and analytical electromagnetics. Dr. Grbic received the Best Student Paper Award at the 2000 Antenna Technology and Applied Electromagnetics Symposium and an IEEE Microwave Theory and Techniques Society Graduate Fellowship in 2001. He received an AFOSR Young Investigator Award and an NSF Faculty Early Career Development Award, both in 2008. In January 2010, he was awarded a Presidential Early Career Award for Scientists and Engineers. In 2011, he was the recipient of an Outstanding Young Engineer Award from the IEEE Microwave Theory and Techniques Society, a Henry Russel Award from the University of Michigan, and a Booker Fellowship from the United States National Committee of the International Union of Radio Science.

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Frequency Selective Buildings Through Frequency Selective Surfaces Marios Raspopoulos and Stavros Stavrou, Member, IEEE

Abstract—This paper proposes the deployment of frequency selective surfaces (FSS) in indoor wireless environments and investigates their effect on radio wave propagation. FSS can be deployed to selectively confine radio propagation in indoor areas, by artificially increasing the radio transmission loss naturally caused by building walls. FSS can also be used to channel radio signals into other areas of interest. Simulations and measurements have been carried out in order to verify the frequency selectivity of the FSS. Practical considerations regarding the deployment of FSS on building walls and the separation distance between the FSS and the supporting wall have been also investigated. Finally, a controlled, small-scale indoor environment has been constructed and measured in an anechoic chamber in order to practically verify this approach through the usage of ray tracing techniques. Index Terms—Buildings, frequency selective surfaces (FSS), indoor radio communication, radio propagation, repeaters.

I. INTRODUCTION

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ADIO signals generally propagate through reflections, refractions, diffractions, and scattering. Transmitted signal components arrive at the point of interest with various amplitudes and phases and combine together to produce the received signal. Due to these multipath arriving components, the instantaneous received signal strength can typically vary as much as 20–30 dB over a fraction of a wavelength due to constructive and destructive contributions [1]. In a typical indoor environment, the received signal is mainly attenuated due to reflections and transmissions through building materials. Building penetration loss depends on various variables associated with the building architecture, including but not limited to the building structure, the building walls periodicity and their electrical properties, the angle of incidence, etc. [2]–[5]. Depending on the application, there are cases where the radio signals have to be confined in designated areas of interest, improving in this way wireless security. Such signal confinement can also assist interference management. However, there are also cases where these signals have to be amplified in order to increase wireless coverage and system reliability. The literature proposes various methods to increase coverage. These include leaky feeders, active and passive repeaters, distributed antennas, etc. Leaky feeders principle of operation is based on the leakage fields arising from specially made coaxial Manuscript received April 19, 2007; revised November 16, 2010; accepted January 06, 2011. Date of publication June 07, 2011; date of current version August 03, 2011. M. Raspopoulos is with Sigint Solutions Ltd., 2048 Nicosia, Cyprus (e-mail: [email protected]). S. Stavrou is with Open University Cyprus, 1055 Nicosia, Cyprus. Digital Object Identifier 10.1109/TAP.2011.2158779

cables, which are usually deployed in large pathways (e.g., tunnels) [6], [7]. Despite that they have been mainly used in tunnels and mines [8] they have also found applications in indoor environments, especially in offices with lengthy corridors [9], [10]. The active repeaters are based on the idea of receiving the signal in a particular location, amplifying and retransmitting it to the same or another direction. However, this technique requires power and good isolation between the receiving and retransmitting antennas [11], otherwise the system might become oscillatory. Also, with active repeaters, the received noise and interference is reradiated on both the forward and the reverse link. The technique investigated in this paper is to utilize frequency selective surfaces (FSS) as isolators or even as passive repeaters, in indoor environments [12]. The passive repeater concept is based on the assumption that the mean signal strength received due to a reflected contribution from a FSS would be higher from the one received from any other object/material that does not produce a strong reflection [13]–[16]. Alternatively, FSS can be used to provide radio isolation by selectively rejecting a frequency range, thus reducing interference between adjacent co-channel wireless systems. Wireless security can be also improved by minimizing the spillover of radio waves outside designated areas. Similar behavior can be achieved, if instead of FSS, a metallic surface is used as the reflecting interface. However, the drawback of such a method is that it will unselectively block and/or reflect the frequency range of multiple wireless systems [17]. FSS can be deployed in such a way so as WLAN signals can be confined in an indoor environment while GSM signals can be allowed to enter and exit the building with a zero or minimal effect. The authors of this paper have made a first attempt to demonstrate the principle of the application of FSS in wireless environments to improve and/or restrict coverage in specific areas in 2005 and 2006 [12], [18]. In 2006, Sung has demonstrated through in-situ measurement results over a frequency range of 2.3–7 GHz that a frequency selective wall can be successfully created for UNII (Unlicensed National Information Infrastructure) [19], [20]. The experiments have been carried out in two adjacent rooms, separated by a wall which is transformed into frequency selective by attaching a bandstop FSS. Deployment practicalities such as misalignment and overlapping are investigated. In this paper, we investigate the application of FSS through the use of ray tracing simulations which incorporate the behavior of FSS. This gives the flexibility to predict the behavior of a frequency selective building where FSS are deployed in more than one wall. Practical considerations regarding the deployment of FSS on building walls and the separation distance between the FSS and the supporting wall have

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been also investigated using CFDTD simulations. Section II of this paper presents some of the FSS behavior issues related to this work. These include the performance of various element shapes, the effect of angle of incidence and typical building materials on the transmission and reflection characteristics of FSS, when these are placed at various distances behind the FSS. Typical modeled and measured responses are also provided. Section III presents results from a custom-written Ray Tracing engine, developed and utilized for the purpose of this investigation. Section IV presents a small scale FSS indoor environment that has been constructed and tested in an anechoic chamber. This experiment practically verifies the usefulness of this investigation and the usage of ray tracing for predicting radio propagation in FSS environments. Section V presents a more detailed investigation of radio propagation in a typical FSS indoor environment through the use of the ray tracing engine, by incorporating the FSS interface behavior. Fig. 1. Square-loop FSS dimensions (substrate thickness: 1.6 mm).

II. FSS FSS are planar periodic structures which consist of thin conducting elements, usually printed on a dielectric substrate. They behave as passive electromagnetic filters, selectively reflecting a desired frequency band. One of the most important factors influencing the FSS response is the element shape [21]–[23]. Their response can also be affected by the element size, the permittivity and thickness of the substrate, the angle of incidence and the gap between the elements. Various approaches have been developed for analyzing the FSS behavior. Examples include the equivalent circuit method (ECM) [27]–[29], the method of moments (MoM), the finite difference time domain method (FDTD), and the conformal finite difference time domain method (CFDTD) [24]–[26]. For the purpose of this investigation a square-loop FSS was designed and fabricated on FR4 dielectric . In this paper, the square loop design was selected mainly because of its superiority with regards to its angular insensitivity over the other element shapes. Table I presented by Wu in [31] presents a typical comparison between different elements. Fig. 1 shows the dimensions of the fabricated square-loop FSS used. To characterize the FSS response, measurements were carried out in the anechoic chamber of the Centre for Communication Systems Research in the University of Surrey. Double Ridge Guide linearly polarized Horn Antennas have been used for transmission and reception and a Vector Network Analyzer (Rohde and Schwarz VNA ZVCE 20 KHz–8 GHz) was used to sweep the frequency between 1–4 GHz. These directional antennas have frequency independent characteristics in this frequency range with relatively constant gain at 9 dBi and 40 degrees beamwidth. The measurement setup shown in Fig. 2 consists of a wooden board, covered with aluminium foil and absorbing material, which has a 72 cm 72 cm aperture in the middle for FSS placement. Given that the horn antennas’ maximum dimension is m, have and in Fig. 2 have been selected to be 1.3 m in order to ensure that the FSS lies in the far-field region of the two antennas based on

TABLE I PERFORMANCE OF VARIOUS FSS ELEMENT SHAPES

, where is the wavelength (12.5 cm). In order to ensure that all the contributions arriving at the receiving antenna are all in phase, the 60% of the 1st Fresnel Zone should be kept and unobstructed. This is done by ensuring that where is the radius of the th Fresnel Zone . For this setup given by m which satisfies the aforementioned conditions. To eliminate the effects of any possible edge diffraction around the edges of the aperture that the FSS is placed and the edges of the absorbing wall, these have been covered with absorbing material. To further eliminate these diffraction effects, the results have been time-gated using a 5 ns timing window. The simulations were performed using the CFDTD solver presented in [25] and [26]. For simulation purposes, the FR4 effective dielectric constant was set to 2.775 as suggested by [17], [25]. The assumption in these CFDTD simulations is that the current distribution of the truncated FSS (72 m 72 m) is the same as the one of an infinite structure and zero outside the finite structure [30]. This means that the infinite

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Fig. 2. Anechoic chamber setup for FSS transmission loss measurements.

Fig. 3. Transmission through square-loop FSS at normal incidence.

simulation does not consider the edge effects which have been eliminated in the chamber by using absorbing material and time gating. Fig. 3 presents the simulated and measured transmission characteristics of the square-loop FSS under normal angle of incidence. The measurement result is in close agreement with the simulated result which gives us a degree of confidence on the CFDTD method used in addition to the published results of [25]. Figs. 4 and 5 present CFDTD [25], [26] simulation results and reflection characteristics of the of the transmission square-loop FSS, under four angles of incidence (0 , 18 , 36 , and 54 ), for TE and TM polarizations, respectively. It is clear that as the angle of incidence is varied, the transmission and reflection characteristics will vary as well. Thus, when designing a FSS for a particular scenario, this angular sensitivity has to be taken into account in order to correctly model the radio wave interaction with the FSS deployed in the modelled environment. For the purpose of this investigation the FSS was deployed on the building interfaces. Literature reports that when a FSS is placed on a dielectric medium, its standalone frequency

Fig. 4. TE Transmission (S ) and TE Reflection (S ) for square loop FSS under various angles of incidence.

Fig. 5. TM Transmission (S ) and TE Reflection (S ) for square loop FSS under various angles of incidence.

response will change [17], [32]. Therefore, the interaction of building walls and FSS has to be investigated when considering this kind of application. Fig. 6 shows the building materials effect on the frequency response of the FSS. The results indicate that there is a decrease on the tuning frequency of the FSS when the latter is attached to the building material. This is due to the fact that Floquet modes which decay exponentially with distance from the elements, still have significant amplitudes at the boundary, hence modifying their relative amplitudes and the energy stored close to the array and thus affecting the resonant frequency [32]–[35]. To further investigate this effect, simulations and measurements have been performed by varying the distance between the FSS and the material. Fig. 7 presents simulations and measurement results after varying the air gap between the FSS and a wooden building wall and ). (MDF with the Fig. 8 indicates that when the air gap is bigger than effect of the wooden wall on the tuning frequency of the FSS is eliminated [34].

RASPOPOULOS AND STAVROU: FREQUENCY SELECTIVE BUILDINGS THROUGH FSS

Fig. 6. Effect of wood and plaster when these are attached on the square-loop FSS.

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Fig. 9. Measurement route for validation of the ray tracing tool.

III. RAY TRACING MODEL

Fig. 7. Effect of varying the distance between the square-loop FSS and a wooden board.

Fig. 8. Simulation and measurement of the effect of the air gap between the square-loop FSS and a 9-mm-thick wooden (MDF) board on the tuning frequency.

In order to study the effect of deploying FSS in an indoor environment, a 3D ray tracing model was developed to simulate a simple scenario with and without the FSS. The application of ray tracing falls into the category of deterministic or site-specific modeling [36], [37], which is very well suited for this paper. Ray tracing is based on geometrical optics (GO) and is used to identify all the possible ray paths between the transmitter and the receiver. Typical ray tracing algorithms include the shooting and bouncing ray (SBR) and the image method. For the purpose of this work, a ray tracing algorithm was implemented in MATLAB, based on the image method. For this method, for a given source point and a facet, the reflected rays on the facet can be considered as being directly radiated from a virtual source called the image source, which is symmetrical to the real source with respect to the facet. The first step during the implementation of the algorithm was to unambiguously define the environment under investigation in terms of its geometrical and morphological characteristics. These two descriptions are integrated into the faceted model, where every building interface is represented by a polygon-shaped facet, which geometrically and morphologically describes the interface. In order to validate the basic ray tracing algorithm prior to the investigation of the effect of FSS, a real scenario (second floor of the Centre for Communication Systems Research, CCSR, University of Surrey) was simulated and radio measurements were performed along the measurement route depicted in Fig. 9. The basic ray tracing modeling approach was also compared with results obtained from a commercial ray tracing simulator by RemCom (Wireless Insite), which utilizes the SBR method. Both ray tracing simulation tools consider up to six reflections, all the possible refractions and one UTD diffraction [38]. The electrical parameters of the scenario interfaces, used in both simulating tools, are tabulated in Table II. Typical constitutive parameters for different materials can be found in [3]. Fig. 10 shows the measurement route consisting of 51 (both LOS and NLOS) averaged measurement points. Measurements were carried out with a portable spectrum analyzer from Rohde and Schwarz (FSP 30). The transmitter used was a Rohde

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TABLE II ELECTRICAL PARAMETERS USED FOR CCSR INTERFACE

Fig. 11. Anechoic chamber experiment setup.

Fig. 10. Comparison between measurements and modeling.

Schwarz signal generator transmitting a 10 dBm CW signal at a frequency of 2.4 GHz. Dipole antennas were used for transmitting and receiving. Fig. 10 shows the comparison of the theoretical predictions and the measurement set. The recorded set of measurements has indicated a mean square error of around 4 dB and a model error standard deviation of around 5 dB. One reason for the minor differences between the theoretical predictions and the measurements is the possible inaccurate use of the constitutive parameters and the presence of indoor clutter [2], [3]. Fig. 12. Anechoic chamber experiment setup.

IV. FSS—RAY TRACING IN ANECHOIC ENVIRONMENT In order to verify the applicability of ray tracing for predicting radio propagation in FSS indoor environments, an experiment was carried out in an anechoic chamber as depicted in Figs. 11 and 12. The transmitting antenna was placed outside the chamber so as a LOS component could not be received by the identical receiving antenna (1–8 GHz horn antennas with 40 degrees beamwidth), which was placed inside the chamber. Two wooden boards were placed inside the chamber to direct the ray components to the receiving antenna (Figs. 11 and 12). With this setup, the rays launched by the transmitter reflect on board 1 and 2 and then reach the receiver. The average distance traveled by the rays after the two reflections is 6.3 meters, corresponding to a path loss of 56 dB at 2.4 GHz and the angle of incidence varies between 45 to 60 degrees.

Three cases have been measured; reflectors (1) are made of wood; (2) are covered with metallic surfaces; and (3) are covered with FSS as depicted in Fig. 11. Fig. 13 presents the results of these measurements. These results have been time-gated to isolate only the reflected components and discard any other possible contributions. It is clear that there is a significant increase of the field strength at 2.4 GHz for the FSS case compared to the results obtained from the wooden board case. The received power at 2.4 GHz is the same as the one for the metallic case, but significantly lower for any other frequencies. This result verifies the application of the FSS as a passive repeater, used to increase signal coverage for specific frequencies of operation without affecting the operation of other systems operating on other frequencies. Also this experiment and the comparison with the the-

RASPOPOULOS AND STAVROU: FREQUENCY SELECTIVE BUILDINGS THROUGH FSS

Fig. 13. Anechoic chamber results for the 3 cases under investigation.

Fig. 14. Scenario under investigation (Height

= 3 m).

oretical ray tracing results demonstrates that Ray Tracing can be applied to predict radio propagation in FSS environments. The inclusion of FSS makes the problem not entirely ray-optical and therefore the developed ray tracing model was modified to incorporate the theoretical behavior of FSS by replacing the Fresnel reflection and transmission coefficient calculations with precalculated CFDTD angle-dependent FSS coefficients. Diffraction effects in this case have been neglected and that is why the edges of the boards in the chamber have been covered with absorbing material. V. IMPACT OF FSS IN AN INDOOR ENVIRONMENT After the ray tracing algorithms were verified, a simpler indoor scenario was developed (Fig. 14) in order to study the theoretical behavior of the FSS when these are incorporated on the building interfaces. The facets in the simulated environment

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Fig. 15. Comparison of the transmission loss between the case where the square loop is alone and the case where it is placed 12.5 mm in front of plaster for two angles of incidence.

were also described in such a way as to take into account the FSS behavior, under various angles of incidence, as suggested by the CFDTD method. The square-loop FSS reflection and transmission coefficients for all possible angles of incidence were calculated and incorporated into the Ray Tracing model. The ray tracing model considers up to six reflections, all the possible refractions and one UTD diffraction around the wall edges. It is assumed that the FSS are not placed around the edges of building walls and therefore the diffraction effects are calculated using the UTD solution of [38]. As already mentioned in Section II, when the air gap between the FSS and the building material is , then there is no significant effect on the FSS bigger than response [34]. For this reason the square-loop FSS was placed 12.5 mm away from all the building materials (i.e., brick, concrete and plaster board). An illustrative example is shown in Fig. 15 where the square loop FSS is placed 12.5 mm away from plaster board. The first scenario under investigation deals with the effect of deploying the FSS, on the external walls (including floor and ceiling). Results obtained along the estimation root, are shown in Fig. 16. The spacing between the receiver locations along the estimation route is 10 cm. Two dipole antennas with 2.15 dBi have been used as transmitting and receiving antenna elements. The transmitting power was set to 10 dBm. The obtained results suggest that the field strength inside the building has generally increased whereas the one outside the building has decreased. The power outside the building is reduced by roughly 15–20 dB. This effectively means that any interference caused to other wireless systems operating on the same frequency channel outside the external walls should be minimized. The power inside the building is increased by 10–20 dB depending on the receiver location. It is noted that in areas where radio propagation is dominated by LOS components (e.g., along the corridor), the application of FSS on the external walls had a minimal effect. Results also suggest that the received field inside the meeting room has not been significantly increased. The next step of this work was to find a way to channel the signal along the corridor and increase the signal strength in this area. To achieve this,

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To highlight the effect of the deployed FSS, tuned at 2.4 GHz, a radio transmission at 5.2 GHz was assumed. The transmission and reflection coefficients of the latter frequency were also obtained through the CFDTD method. Fig. 17 suggests that the FSS has very little effect on the radio propagation characteristics on this frequency, since the FSS is not tuned at 5.2 GHz. This effectively means that carefully designed and deployed FSSs will not have any effect on a system operating on another frequency. An example could be the confinement of WLAN 802.11b/g signals within a building without obstructing the transmission of GSM or 802.11a signals through such a frequency selective environment.

Fig. 16. Comparison of different cases at 2.4 GHz. The letters at the top of the figure correspond to the points along the estimation route shown as shown in Fig. 14.

Fig. 17. Comparison of different cases at 5.2 GHz.

FSS were added to the internal walls as well. Results presented in Fig. 16 suggest that there is a significant increase of field strength in the meeting room compared to the non-FSS case. It is also noted that the field strength inside office rooms 3, 4 and 5 has not decreased since the attenuation suffered by the rays while crossing the internal walls of these rooms is compensated by the stronger multiple reflections along the narrow corridor. However, this kind of compensation does not happen on the received power outside the building, for the case that FSS were added only to the external walls. In this case the ray paths which undergo multiple reflections on the external walls are bigger in length, compared to the paths which undergo multiple reflections along the narrow corridor. This effectively means that these components will suffer higher path loss and they will not contribute significantly to compensate the power reduction due to the transmission through the wall. Results suggest that in certain scenarios, FSS will act as passive repeaters, channeling the signal to areas of interest while at the same time restricting coverage to other areas.

VI. CONCLUSION In this paper the basic isolation and passive amplification capabilities of FSS were demonstrated through the use of specially modified ray tracing algorithms and anechoic chamber measurements. Simulations and measurements were carried out to verify the angular sensitivity of the reflection and transmission characteristics of the FSS and the effect on their frequency response when the distance between the FSS and the building material is varied. Based on a modified ray tracing model, a simple scenario was simulated, highlighting typical isolation and amplification figures that can be obtained. The typical amplification figures were also verified through a small scale experiment in an anechoic chamber. These figures will depend on the specific type and setup of the FSS used. The results suggest that proper FSS deployment can be used in indoor wireless environments in order to increase or restrict coverage and that Ray Tracing techniques can be applied to predict radio propagation in such environments. Proper FSS deployment can assist signal channelling or confine coverage in specific areas. REFERENCES [1] H. Hashemi, “The indoor radio propagation channel,” IEEE Proc., vol. 81, no. 7, pp. 943–968, Jul. 1993. [2] S. Stavrou and S. R. Saunders, “Factors influencing outdoor to indoor radio wave propagation,” in Proc. 12th Int. Conf. Antennas Propag. (ICAP 2003), Mar. 31–Apr. 3 2003, pp. 581–585. [3] S. Stavrou and S. R. Saunders, “Review of constitutive parameters of building material,” in Proc. 12th Int. Conf. Antennas Propag. (ICAP 2003), Mar. 31–Apr. 3 2003, pp. 211–215. [4] M. Yang and S. Stavrou, “Investigation of radio transmission losses due to periodic building structures,” in Proc. 11th Eur. Wireless Conf., Nicosia, Cyprus, Apr. 2005, pp. 737–739. [5] M. Yang and S. Stavrou, “Rigorous coupled-wave analysis of radio wave propagation through building structures,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 204–207, 2005. [6] N. Monk and H. S. Winbigler, “Communication with moving trains in tunnels,” IRE Trans. Veh. Commun., vol. VC-7, pp. 21–28, Dec. 1956. [7] D. J. R. Martin, “Leaky-feeder radio communication: A historical review,” in Proc. 34th IEEE Veh. Commun. Conf., May 21–23, 1984, vol. 34, pp. 25–30. [8] Q. V. Davis, D. J. R. Martin, and R. W. Haining, “Microwave radio in mines and tunnels,” in Proc. 34th IEEE Veh. Commun. Conf., May 1–3, 1989, vol. 1, pp. 375–382. [9] K. J. Bye, “Leaky-feeders for cordless communication in the office,” in Proc. 8th Eur. Conf. Electron. (E-UROCON 88), Jun. 13–17, 1988, pp. 387–390. [10] Y. P. Zhang, “Indoor radiated-mode leaky feeder propagation at 2.0 GHz,” IEEE Trans. Veh. Technol., vol. 50, no. 2, Mar. 2001. [11] J. P. Daniel, “Mutual coupling between antennas for emission or reception application to passive and active dipoles,” IEEE Trans. Antennas Propag., vol. 22, no. 2, pp. 347–349, 1984. [12] F. A. Chaudhry, M. Raspopoulos, and S. Stavrou, “Effect of frequency selective surfaces on radio wave propagation in indoor environments,” in Proc. 11th Eur. Wireless Conf. 2005, Nicosia, Cyprus, Apr. 2005, pp. 732–736.

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[13] R. F. H. Yang, “Passive repeater using double flat reflectors,” IRE Int. Convention Rec., vol. 5, pt. 1, pp. 36–41, Mar. 1957. [14] H. D. Hristov, R. Feick, and W. Grote, “Improving signal coverage by the use of through wall passive repeaters,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 17–19, 2001, vol. 2, pp. 158–161. [15] Y. Huang, N. Yi, and X. Zhu, “Investigation of using passive repeaters for indoor radio improvement,” in Proc. IEEE Int. Symp. Antennas Propag., Jun. 20–25, 2004, vol. 2, pp. 1623–1626. [16] C. Colavito and G. D’Auria, “Experimental research on the behavior of passive repeaters,” IEEE Proc., vol. 51, no. 11, pp. 1423–1430, Jul. 1963. [17] G. H. H. Sung, K. W. Sowerby, and A. G. Williamson, “The impact of frequency selective surfaces applied to standard wall construction materials,” in Proc. IEEE Int. Symp. Antennas Propag., Jun. 20–25, 2004, vol. 2, pp. 2187–2190. [18] M. Raspopoulos, F. A. Chaudhry, and S. Stavrou, “Radio propagation in frequency selective buildings,” Eur. Trans. Telecoms, vol. 17, pp. 407–413, Mar. 2006. [19] G. H. H. Sung et al., “A frequency selective wall for interference reduction in wireless indoor envrionments,” IEEE Antennas Propag. Mag., vol. 48, no. 5, pp. 29–37, 2006. [20] G. H. H. Sung et al., “Modeling a low-cost frequency selective wall for wireless-friendly indoor environments,” Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 311–314, 2006. [21] J. C. Vardaxoglou, Frequency Selective Surfaces—Analysis and Design. London, England: Research Studies Press, 1997. [22] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [23] W. Gregorwich, “The design and development of frequency selective surfaces for phased arrays,” in IEEE Aerosp. Conf. Proc., Mar. 6–13, 1999, pp. 471–479. [24] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces—A review,” IEEE Proc., vol. 76, no. 12, pp. 1593–1615, Dec. 1988. [25] W. Yu and R. Mittra, Conformal Finite-Difference Time Domain Maxwell’s Equations Solver. Norwood, MA: Artech House, Nov. 2003. [26] W. Yu and R. Mittra, “A conformal FDTD software package modeling antennas and microstrip circuit components,” IEEE Antennas Propag. Mag., vol. 42, no. 5, pp. 28–39, Oct. 2000. [27] S. M. A. Hamdy and E. A. Parker, “Current distribution on the elements of a square loop frequency selective surface,” Electron. Lett., vol. 18, no. 14, pp. 624–626, Jul. 1982. [28] R. J. Langley and E. A. Parker, “Equivalent circuit model for arrays of square loops,” Electron. Lett., vol. 18, no. 7, pp. 294–296, 1982. [29] C. H. H. Sung, K. W. Sowerby, and A. G. Williamson, “Equivalent circuit modeling of a frequency selective plasterboard wall,” in Proc. IEEE Antennas Propag. Int. Symp., Jul. 2005, vol. 4A, pp. 400–403. [30] W. Yu and R. Mittra, “A look at some challenging problems in computational electromagnetics,” IEEE Antennas Propag. Mag., vol. 46, no. 5, pp. 18–32, Oct. 2004. [31] T. K. Wu, Frequency Selective Surface and Grid Array. New York: Wiley , 1995.

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[32] P. Callaghan, E. A. Parker, and R. J. Langley, “Influence of supporting dielectric transmission properties of frequency selective surfaces,” IEE Proc., vol. 138, no. 5, pp. 448–454, Oct. 1991. [33] T. Cwik and R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propag., vol. 35, no. 12, pp. 1397–1405, Dec. 1987. [34] M. Raspopoulos and S. Stavrou, “Frequency selective surfaces on building material—Air gap impact,” IET Lett., vol. 43, no. 13, pp. 700–702, Jun. 2007. [35] S. Contu and R. Tascone, “Passive arrays in a stratified dielectric medium scattering matrix formulation,” in Proc. IEEE Int. Symp. Antennas Propag., May 1983, vol. 21, pp. 622–625. [36] G. E. Athanasiadou and A. R. Nix, “A novel 3-D indoor ray-tracing propagation model: The path generator and evaluation of narrow-band and wide-band predictions,” IEEE Trans. Veh. Technol., vol. 49, no. 4, pp. 1152–1168, Jul. 2000. [37] Z. Ji, B. H. Li, H. X. Wang, H. Y. Chen, and T. K. Sarkar, “Efficient ray tracing methods for propagation prediction for indoor wireless communications,” IEEE Antennas Propag. Mag., vol. 43, no. 2, pp. 41–49, 2001. [38] R. G. Kouyoumjan and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 42, pp. 1148–1461, Nov. 1974.

Marios Raspopoulos was born in Nicosia, Cyprus. He received the M.Eng. degree in electronics and mobile communications, the M.Sc. degree in communications networks and software, and the Ph.D. degree from the University of Surrey, U.K., in 2003, 2004 and 2008, respectively. He is currently with Sigint Solutions Ltd., Nicosia. His research deals with techniques for improving radio propagation mainly in indoor environments. His work involves deterministic modeling using ray tracing, 3D electromagnetic simulations, and design of frequency selective surfaces. His other interests include antenna design and UWB communications, localization techniques, and MIMO communications.

Stavros Stavrou (M’01) received the B.Eng. degree in computer and communication engineering from the University of Essex, U.K., in 1996 and the Ph.D. degree from the University of Surrey, U.K., in 2001. From 1997 to 2000, he was working as a Research Fellow with the Centre for Communication Systems Research (CCSR), University of Surrey, contributing to a number of international and national research projects. In late 2000, he was appointed to a lecturing position with the University of Surrey, in the area of radio propagation, RF, and antenna design. His main research interests include wireless propagation for terrestrial and satellite systems.

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Diffuse Scattering Model of Indoor Wideband Propagation Ondˇrej Franek, Member, IEEE, Jørgen Bach Andersen, Life Fellow, IEEE, and Gert Frølund Pedersen

Abstract—This paper presents a discrete-time numerical algorithm for computing field distributions in indoor environments by diffuse scattering from the walls. Calculations are performed for a rectangular room with semireflective walls. The walls are divided into 0.5 2 0.5 m segments, resulting in 2272 wall segments in total and approximately 2 min running time on average computer. Frequency independent power levels at the walls around the circumference of the room and at four receiver locations in the middle of the room are observed. It is demonstrated that after a finite period of initial excitation the field intensity in all locations eventually follows an exponential decay with the same slope and approximately the same level for given delay. These observations are shown to be in good agreement with theory and previous measurements—the slopes of the decay curves for measurement, simulation and theory are found to be 18, 19.4, and 20.2 dB per 100 ns, respectively. The remaining differences are further discussed and an additional case of a spherical room is used to demonstrate the influence of the room shape on the results. It is concluded that the presented method is valid as a simple tool for use in indoor radio coverage predictions. Index Terms—Diffuse fields, indoor radio communication, numerical methods, propagation.

I. INTRODUCTION N order to achieve a high degree of quality of service in wireless communication systems, the mobile device needs to maintain sufficient level of signal strength from the base station at all possible locations. This brings about a need for predicting the radio coverage from the base station, so that we can choose adequate radiated power and optimize the position of the base station antenna, or reduce their number if multiple access points are necessary in the given conditions. The character of the problem and its solution depends on the scale and complexity of the propagation environment, and two distinct scenarios, indoor and outdoor, are usually considered when choosing the appropriate method. This paper is focused on prediction of radiowave propagation in indoor environments, with application to personal communication systems in the centimeter-wave frequency range. Various methods have been employed to solve similar problems, the most prominent likely being ray tracing [1]. However, ray tracing accounts only for propagation by means of specular reflections or diffractions, but not diffuse scattering, which we see

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Manuscript received July 15, 2010; revised November 17, 2010; accepted December 28, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported by the Danish Center for Scientific Computing. The authors are with the Antennas, Propagation and Radio Networking section, Department of Electronic Systems, Aalborg University, DK-9220 Aalborg Øst, Denmark (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158791

as more important, if not crucial, taking into consideration the commonly used wavelengths of wireless systems (cm) and comparable sizes of common room obstacles and surface structures. Drawbacks of the ray tracing approach with respect to diffuse scattering prediction are discussed in [2]. There have also been attempts to tackle the coverage problem with the finite-difference time-domain (FDTD) method, most recently in [3]. Nevertheless, the FDTD method is, in spite of steadily increasing computer speed and memory, still very demanding in terms of computational resources. As a result, FDTD studies are usually limited to two-dimensional algorithms and employ frequency reduction techniques in order to keep sufficient sampling per wavelength without memory exploding [3]. In a recent work [4], the concept of room electromagnetics was introduced, in analogy to room acoustics, a well-established discipline of predicting a sound field in a room. The idea is based on similarity of the wavelengths for both audio frequencies and microwave frequencies, whereas the size of the room and the roughness of the walls are expected to produce similar reverberation effects. Trying to obtain a numerical model to support the theory, we chose the radiosity method, which is based on purely diffuse scattering and has been successfully used in the acoustics discipline [5], as well as in computer graphics and architectural lighting [6]. The radiosity approach has also been employed in radio coverage prediction in outdoor studies [7]–[9] and in combination with ray tracing in indoor environment [10]. Among its advantages we would like to highlight the relative simplicity and speed of the algorithm, while the lack of any information on specular reflections might be seen as a disadvantage. It is also a power based method, in the sense that all phase and polarization information is missing. However, these are supposed to have random character anyway in most common scenarios of rooms with rough surfaces. It should be noted that the observed similarity between the acoustic and electromagnetic waves is purely mathematical, not physical. Propagation effects are governed by the wave equation in both cases, and the relation of free space velocities and used frequencies results in comparable wavelengths and thus comparable propagation effects. Nevertheless, media different from air (e.g., walls, floors, water, ground) will, of course, have different propagation properties for either type of wave and cannot be interchanged. Another difference is that electromagnetic waves are transverse and exhibit polarization effects, whereas acoustic waves are longitudinal. However, both polarizations are represented equally in diffuse field and therefore can be treated together. In a real situation, the receiving antenna will usually pick up only

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space radiation from Tx to segment as ray 1 and ray 2 for segment , denoted by dashed lines in Fig. 1. These are the incident fields and they are stored under the relevant delays ‘1’ and ‘2’, which are quantized so that all rays falling within the are added together in power. Segment is same time span , also illuminated by segment via ray 3 with the delay and vice versa. The process keeps going on with diminishing values. When all the segment values have been determined up to a suitable total delay, the intensity at any point in the room may be determined by simple summation over Tx and all the wall segments (solid lines in Fig. 1). The numerical process is fast since it is only simple forward stepping in time. The coupling between the segments is given by a square symmetric matrix with elements Fig. 1. Ray paths in a rectangular room (dashed: direct rays from transmitter to segments, dotted: scattering between segments, continuous: rays from transmitter and segments towards receiver).

one polarization of the incoming wave and the measured field may then be up to 3 dB lower than levels predicted by the radiosity model. The goal of the present paper is to demonstrate that the radiosity method, despite some differences between its original domain and electromagnetics as described above, can be successfully applied to radiowave coverage prediction in indoor environments. The novelty of the work lies in application of the method to the important communication bands in the microwave and millimeter region. The results are compared with measurement around 6 GHz in realistic environment. The paper is organized as follows. In Section II the diffuse scattering model is described and the formulas necessary for implementing the numerical algorithm are given. Section III then presents a simple numerical example together with impulse responses in discrete points in the room, angular responses and responses at the walls along the circumference of the room. Mean power quantities are shown, and measurement results from earlier paper are also included. Section IV proceeds with discussion of the results and their comparison to theoretical expectations from the field of acoustics. The work is concluded in Section V and some remarks about a successful implementation are drawn. II. PROBLEM DESCRIPTION The walls of the room are divided into segments of area . We assume Lambertian diffuse scattering in lack of better information and the same scattering coefficient from each segment, although this is not a condition for the model. Lambertian scattering means that the scattering cross sections are proportional to cosine to the angle measured from the normal. This means that there is no scattering between segments along the same wall. The algorithm may be explained with reference to Fig. 1. , where is the time step and number 1 At time represents the first discrete time instance, the transmitter (Tx) sends out an impulse. The shape of the impulse is not critical, as the radiosity algorithm does not support dispersion and all power contributions are simply added; the pulse width should only be short compared with the length of the time step. Next, the strengths of scattering sources are determined by simple free

(1) where is the power scattering (reflection) coefficient , and are the angles from the normals of the respective is the distance between segment censegments and , and should be small enough to repters. The size of the segments resent the geometry of the room with reasonable accuracy, but too small size would result in a high number of segments and correspondingly in high memory demands and running time. The time stepping algorithm for power of th element in time , , is then

(2) (the where all delays are rounded to the nearest multiple of time steps, time resolution). The algorithm is running for basically until all the powers drop below certain agreed level, which can represent the noise floor. The summation is over all segments in the room—it encompasses segments on all reis the delay between flecting walls and obstacles. The delay segments and . The summation over is performed for each value of and , ensuring that all the multiple interactions are is the power of the direct signal from Tx, taken into account. which is non-zero only at time instant corresponding to time delay between Tx and the segment with distance , assuming a gain of 1

(3) The field at an arbitrary point inside the room is determined analogously to the update scheme (2), where the receiver stands for additional scattering segment (without rescattering though) having the incident directivity constant over all angles, , and equivalent surface of with being the wavelength at the center frequency of the pulse. Note that this is the only place where frequency appears in the theory, and it only influences the levels of the received power, not the shape of the response. Also, a real antenna directivity could be added, if wanted.

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Fig. 3. Impulse responses along the circumference of the room at height 1.25 m.

Fig. 2. Impulse responses at different distances from the transmitter.

To avoid any energy losses, the following inequality must be satisfied: (4) that is, the shortest distance between any two segments must be at least as long as half the distance which the wave travels in one time step. This condition serves to ensure that all delays between segments will be nonzero after rounding to integer number of time steps. Zero delay would give rise to infinite values of power at the affected segments, or would have to be neglected otherwise, leading to nonphysical dissipation. One theoretical limitation of the model is the extent of roughness of the walls. A room with perfectly smooth walls will not be characterized satisfactorily, in such a case ray tracing should be used instead. Also, the scattering cross sections of the walls are not known exactly and we use simplified assumptions of uniform diffuse scattering and absorption coefficient. Last but not least, polarization effects are entirely neglected in the present version of the algorithm. III. NUMERICAL EXAMPLE In the following, a numerical simulation of a rectangular room at frequency 5.9 GHz is performed, with receiver locations chosen in various distances from the transmitter. The room dimensions are width 11 m, length 19 m, and height 2.5 m. Segment size has been chosen 0.5 0.5 m and the timestep is 2 ns, obeying (4) in the corners. Transmitter location is at near the left wall and receiver locations are , (8,6), (12,6), (16,6). In all instances, the (vertical, height) coordinates are 1.5 m. The coordinate origin is at the lower left corner. Both the transmitter and the receivers are omnidirectional in our simulations, although adding the respective radiation patterns into the computation is straightforward. The scattering coefficient is 0.5. Fig. 2 shows the responses at the receiver locations and Fig. 3 shows along the circumference of the room at the

height of 1.25 m.1 All power levels are expressed in dBW with unit reference (1 W Tx output power). The first 60 ns are dominated by the incident fields—all of the curves in Fig. 2 show similar behavior in that there is a gap between the direct path (the first arrival) and the diffuse power reflected from the walls. However, the gap is not so deep at the receivers farthest away, indicating the influence of the floor and ceiling scatter. After that, the power falls off approximately exponentially with approximately the same power level at all receiver locations and also along the complete circumference (Fig. 3). This is in agreement with the theory and the experimental results. The decay rate is about 19 dB/100 ns. The slope could be changed by choosing another effective value of . The simulation involved a total of 2272 wall segments and took approximately 2 min. on an average computer with Pentium 4 at 2.8 GHz. For comparison, in Fig. 4 we also show the power delay profiles at various positions around the room obtained by measurement published in [4]. The room has the same dimensions as in the present numerical model, although it has windows and several obstacles scattered around it. The power distribution at all probe positions is again practically uniform after 60 ns and follows similar decay rate of 18 dB/100 ns. The absolute power levels are not directly comparable, as the calibration for the measurement was slightly different, namely the time integration window was larger resulting in higher magnitudes of power. 1) Power Distributions and Rice Factor: Since the direct line-of-sight (LOS) path is available as well as the diffuse part, it is possible to calculate the Rice factor, which is the ratio of the coherent and the integrated incoherent power. The LOS power is given as the peak value of the power delay profile

(5)

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1This height corresponds to the centers of the 0.5 0.5 m panels, into which the walls are discretized, at approximately the same height as the transmitter. To obtain the responses at the exact height of the transmitter would need some kind of interpolation, which we wanted to avoid for simplicity.

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Fig. 4. Impulse responses at various positions around the room obtained by measurement [4].

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Fig. 5. Rice factor (7) in dB, the diffuse power (6), the total power (8), and the LOS power (5) versus log of distance in meters.

occurring at time instant and the diffuse part is calculated as a sum of all values following (6) The Rice factor

is then their ratio (7)

whereas the total power is their sum (8) These indicators are shown in Fig. 5 in dB scale. The Rice factor is approximately leveled for the three outermost receivers, whereas the LOS and the tail decay at the same rate, which is a direct outcome of the presence of vertical scattering. However, between the first and the second receivers, the Rice factor is a decaying function of distance because the diffuse power (lower curve) decreases more slowly than the LOS power. Overall, the Rice factor is very small, indicating that the propagation channel follows rather Rayleigh fading, or, expressed differently, the distance is larger than the reverberation distance. 2) Angular Response: For multiple-input and multiple-output (MIMO) applications the angular response is relevant, and this is easily found since all the scattering strengths and corresponding angles from the receiver are known from (2), see Fig. 6. It is noted that the angular spreading is almost uniform after about 100 ns. Before that the response is dominated by the single scattering. 3) Mean Power: Finally, Fig. 7 shows the distribution of the mean power, i.e., the power integrated over the whole impulse response, across the room at height 1.5 m from the ground (the same height as the transmitter). As expected, the power decreases monotonically with distance from the transmitter, although there is apparent sign of leveling at the opposite wall, caused probably by multiple reflections. However, this view illustrates the potential of the algorithm for coverage predictions.

Fig. 6. Angular response at one position (12,6) from access point at (2,6).

Fig. 7. Mean power distribution across the room at height 1.5 m from the ground.

The discretization of the walls into 0.5 m segments has been chosen in order to achieve sufficient accuracy of the algorithm while keeping the computational burden low. If we choose smaller double resolution, i.e., 0.25 m, the segments will be

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Fig. 8. Relative distribution of power in dB on the walls of the 19

2 11 2 2.5 m room after 300 ns.

and the total number of segments larger. Hence, the more elements with corscattering matrix (1) will have responding memory demands. Moreover, the update sequence more time as a result of the (2) will be expected to take increased number of elements and the necessity to halve the time step as well due to (4). These projections of the running time are of course purely theoretical and the actual timing might differ, nevertheless it gives the programmer an important idea about the scaling of the algorithm. Apart from the presented algorithm, we also tried a simplified 2.5-dimensional (2.5D) version where only the circumference walls were taken into account and floor and ceiling virtually did not exist. The 2.5D version had even smaller computational demands, but the physical interpretation of the results was problematic, despite quite remarkable qualitative similarities to full 3D. We therefore concluded that the 3D algorithm is preferable, it is reasonably fast and this would only improve with computer developments. IV. DISCUSSION As can be seen in Fig. 2, as well as in experimentally obtained Fig. 4, the slopes of the response curves are everywhere the same inside the room. They are related to reverberation time known from acoustics theory (9) , and Here, stands for energy in the room with initial value is the time variable. The reverberation time can be obtained from (10) which is commonly referred to as Sabine’s law [11]. and are the volume and total surface area of the room, respectively, is the velocity of light and is the absorption coefficient of the

. Generally, (10) is only an approximation for walls, small ; larger values can be accommodated by Eyring formula, which results from (10) when is substituted by (11) Formulas (10) and (11) assume equal probability for all ray paths, which is not generally true for rectangular rooms. Kuttruff proposes further correction by (12) where is a parameter that accounts for the shape of the room. It can be obtained from numerical calculations and its values vary between 0.3–0.6 for rectangular rooms [11]. and the slope of the decay curve in Our room has Fig. 2 is 19.4 dB/100 ns. The closest to this result is Kuttruff’s correction by (12) giving 20.2 dB, while Eyring formula (11) pagives 24.5 dB and Sabine’s law alone (10) 17.7 dB. The rameter was taken 0.51 after [11], where this value has been obtained by Monte Carlo method for room with relative dimensions 1:5:10, similar to our room. Fig. 8 shows the power at three walls of the room (the other three are symmetric) after 300 ns from the initial pulse launch. The contour values are in dB with respect to the reference at the center of the floor, where the power level dropped to . It can be seen that even after a long time from the initial excitation the power distribution around the room is not entirely homogeneous, although it gradually drops with the same decay rate, in this case 19.4 dB/100 ns. In fact, in this simulation, the relative power distribution remained unchanged after capturing the field in Fig. 8, only the overall level was decreasing. We can see that the power at the walls is stronger on the longer ends of the room, whereas the floor and the ceiling have the lowest levels, and the power distribution has its maximum in the centers of the walls and is diminishing towards

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V. CONCLUSION

Fig. 9. Impulse responses at different distances from the transmitter at the center of the sphere with diameter 20 m.

the corners. From here it follows that (10) and (11) can indeed be only approximative, since they rely on homogeneity of the field across the room. It should be noted that Fig. 8 also shows one unphysical artifact, namely that the power levels are elevated in the corners of the room. This effect comes from the approximative nature of (1), in which coupling is strongly overestimated for segments very close to each other. For comparison, an exact solution of the reverberation time is available for a sphere of diameter , given by [12], [13] (13) where . Although this formula was derived for the sound waves, it is as well applicable to electromagnetic wave propagation provided that it obeys Lambertian diffuse scattering and polarization effects are neglected, which is the case that is studied in the present paper. The reverberation time is obtained by solving (13) implicitly and gives 6.5 dB/100 ns. Numerical calculation was carried out for sphere of diameter 20 m, see Fig. 9. The source is positioned in the center of the sphere, and the responses are taken by omnidirectional probes at distances 2, 4, 6, and 8 m from the center. The time step is again 2 ns and the mesh size is at least 0.5 m, and smaller towards the poles as it follows spherical coordinates. The frequency was again 5.9 GHz, but it is in fact irrelevant, because in this example we are interested in the decay rate only. The slope of the decay is 6.4 dB/100 ns, which we consider as a very good match and a proof of validity of the algorithm. The calculations presented in this section are carried out for rooms with constant reflection (and, correspondingly, absorption) coefficient along the walls, which was also the case of the numerical examples. Realistic rooms will, of course, have different coefficients for various materials in the room (carpets, bookshelves, windows) and the responses will show irregularities, but the overall trends in decay will be similar. The conclusions should therefore be understood rather qualitatively, or in the sense of rooms with all coefficients averaged.

It has been shown that the radiosity method is capable of predicting electromagnetic reverberation times which fit well with theory, the difference between the simulation and Kuttruff’s corrected reverberation formula is only 0.8 dB/100 ns. The presented room responses are also noted to be in agreement with previous measurements in an office space of similar dimensions [4], although the equivalent value of the absorption coefficient (0.5) for the practical case is debatable. Validity of the algorithm has been verified by comparing the results to exact solution for a spherical cavity, where excellent match within 0.1 dB has been obtained. However, we conclude that the theoretical values for reverberation given by Sabine and Eyring are only informative when it comes to rectangular rooms and, by generalization, rooms of arbitrary shape. Limited accuracy of the theoretical formulas thus highlights the importance of the presented algorithm for general, complicated shapes of rooms. Even though the numerical experiment involved an empty rectangular room only, the algorithm can be easily applied to more complex indoor and also outdoor scenarios, and the computational burden is not expected to be tremendous. Only 2272 wall segments with 0.5 0.5 m size were used, being quite large with respect to the wavelength of 5 cm at frequency 5.9 GHz, and still achieving very good accuracy. This is a clear advantage to other methods (FDTD for example) which rely on sufficient spatial discretization of the waves and usually need many samples per wavelength. The numerical algorithm is also general enough to accommodate objects of arbitrary shape in the room (people, furniture), which will be represented by additional scattering segments. Nevertheless, the presence of such obstacles is already included in the diffuse characteristics of the walls and, therefore, adding them into the simulation as separate objects might not yield substantially different result. The numerical model is indeed very simple, yet it agrees very well with theory and experimental results, and, therefore, provides useful prediction of radiowave coverage in rooms. ACKNOWLEDGMENT The authors would like to thank the TAP reviewers and Dr. T. Brown for their useful comments on the manuscript. REFERENCES [1] R. Valenzuela, “A ray tracing approach to predicting indoor wireless transmission,” in Proc. IEEE 43rd Veh. Technol. Conf., May 1993, pp. 214–218. [2] R. Vaughan and J. B. Andersen, Channels, Propagation and Antennas for Mobile Communications. London, U.K.: Inst. Elect. Eng., 2003. [3] A. Valcarce, G. D. L. Roche, Á. Jüttner, D. López-Pérez, and J. Zhang, “Applying FDTD to the coverage prediction of WiMAX femtocells,” EURASIP J. Wireless Commun. Netw., vol. 2009, pp. 1–13, 2009. [4] J. B. Andersen, J. Ø. Nielsen, G. F. Pedersen, G. Bauch, and M. Herdin, “Room electromagnetics,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 27–33, Apr. 2007. [5] E.-M. Nosal, M. Hodgson, and I. Ashdown, “Improved algorithms and methods for room sound-field prediction by acoustical radiosity in arbitrary polyhedral rooms,” J. Acoust. Soc. Amer., vol. 116, no. 2, pp. 970–980, 2004. [6] I. Ashdown, Radiosity: A Programmer’s Perspective. New York: Wiley, 1994.

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[7] C. Kloch and J. B. Andersen, “Radiosity—An approach to determine the effect of rough surface scattering in mobile scenarios,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. Dig., Jul. 1997, vol. 2, pp. 890–893. [8] C. Kloch, G. Liang, J. B. Andersen, G. F. Pedersen, and H. L. Bertoni, “Comparison of measured and predicted time dispersion and direction of arrival for multipath in a small cell environment,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1254–1263, Sep. 2002. [9] M. Liang and Q. Liu, “A practical radiosity method for predicting transmission loss in urban environments,” EURASIP J. Wireless Commun. Netw., vol. 2004, no. 2, pp. 357–364, 2004. [10] G. Rougeron, F. Gaudaire, Y. Gabillet, and K. Bouatouch, “Simulation of the indoor propagation of a 60 GHz electromagnetic wave with a time-dependent radiosity algorithm,” Comput. Graphics, vol. 26, no. 1, pp. 125–141, 2002. [11] H. Kuttruff, Room Acoustics, 4th ed. London, U.K.: Taylor and Francis, 2000. [12] M. M. Carroll and C. F. Chien, “Decay of reverberant sound in a spherical enclosure,” J. Acoust. Soc. Amer., vol. 62, no. 6, pp. 1442–1446, 1977. [13] W. B. Joyce, “Exact effect of surface roughness on the reverberation time of a uniformly absorbing spherical enclosure,” J. Acoust. Soc. Amer., vol. 64, no. 5, pp. 1429–1436, 1978.

Ondˇrej Franek (S’02–M’05) was born in 1977. He received the M.Sc. (Ing., with honors) and Ph.D. degrees in electronics and communication from Brno University of Technology, Czech Republic, in 2001 and 2006, respectively. Currently, he is with the Department of Electronic Systems, Aalborg University, Denmark, as a Postdoctoral Research Associate. His research interests include computational electromagnetics with focus on fast and efficient numerical methods, especially the finite-difference time-domain method. He is also involved in research on biological effects of nonionizing electromagnetic radiation, indoor radiowave propagation, and electromagnetic compatibility. Dr. Franek was the recipient of the Seventh Annual SIEMENS Award for outstanding scientific publication.

Jørgen Bach Andersen (LF’92) received the M.Sc. and Dr.Techn. degrees from the Technical University of Denmark (DTU), Lyngby, Denmark, in 1961 and 1971, respectively. In 2003 he was awarded an honorary degree from Lund University, Sweden. From 1961 to 1973, he was with the Electromagnetics Institute, DTU, and since 1973, he has been with Aalborg University, Aalborg, Denmark, where he is now a Professor Emeritus and Consultant. He was head of the Center for Personal Communications, CPK, from 1993 to 2003. He has been a Visiting Professor in Tucson, AZ, Christchurch, New Zealand, Vienna, Austria, and Lund, Sweden. He has published widely on antennas, radio wave propagation, and communications, and has also worked on biological effects of electromagnetic systems. He coauthored a book Channels, Propagation and Antennas for Mobile Communications (London, U.K.: IEE, 2003). Prof. Andersen is a former Vice President of the International Union of Radio Science (URSI) from which he was awarded the John Howard Dellinger Gold Medal in 2005. He was on the management committee for COST 231 and 259, a collaborative European program on mobile communications. He is Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS and Co-Editor of a forthcoming Joint Special Issue of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES “Multiple-Input Multiple-Output (MIMO) Technology.”

Gert Frølund Pedersen was born in 1965. He received the B.Sc.E.E. degree, with honors, in electrical engineering from the College of Technology, Dublin, Ireland, and the M.Sc.E.E. and Ph.D. degrees from Aalborg University, Denmark, in 1993 and 2003, respectively. He has been with Aalborg University since 1993 where he is now full Professor and heads the Antennas, Propagation and Radio Networking Group and is also the Head of the Doctoral School on Wireless, which has close to 100 Ph.D. students enrolled. His research has focused on radio communication for mobile terminals, and especially on small antennas, diversity systems, propagation, and biological effects. He has published more than 75 peer-reviewed papers and holds 20 patents. He has also worked as a consultant for developments of more than 100 antennas for mobile terminals including the first internal antenna for mobile phones in 1994 with lowest SAR, first internal triple-band antenna in 1998 with low SAR and high TRP and TIS, and lately various multiantenna systems rated as the most efficient on the market. He has been one of the pioneers in establishing the over-the-air measurement systems. The measurement technique is now well established for mobile terminals with single antennas and he is now chairing the COST2100 SWG2.2 group with liaison to 3GPP for over-the-air tests of MIMO terminals.

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Building Penetration Loss for Satellite Services at L-, S- and C-band: Measurement and Modeling Milan Kvicera, Student Member, IEEE, and Pavel Pechac, Senior Member, IEEE

Abstract—An extensive measurement campaign, covering a representative set of typical buildings in an urban area and aimed at building penetration loss for satellite services at L-, S- and C-band, was performed in Prague in the summer of 2009. Throughout the measurements, a remote-controlled airship was used as a pseudosatellite carrying a transmitter which provided unmodulated continuous wave left-handed circularly polarized signals at 2.0 GHz, 3.5 GHz, 5.0 GHz, and 6.5 GHz. A description of the measurement campaign is provided here, together with a thorough analysis of the resulting experimental data. A significant dependence on elevation angle is reported for line-of-sight and non-line-of-sight propagation conditions separately. Subsequent steps to introduce elevation dependent empirical models at corresponding frequencies follow. Finally, a comparison to other studies dealing with building penetration loss for high elevation angles is presented. Index Terms—Building penetration loss, modeling, propagation measurements, satellite-to-indoor propagation.

I. INTRODUCTION UILDING penetration loss is a standard characteristic of an outdoor-to-indoor propagation channel and is a key factor when planning indoor wireless services. Generally, it is defined as excess loss due to the presence of a building wall and other building features [1]. Thus, when measuring indoors, it is necessary to subtract a reference level from the experimental data to obtain correct levels of penetration loss. A universal model for building penetration loss for terrestrial systems was presented in [2] as the COST 231 model which was based on measurements taken in the 900 MHz to 1800 MHz range. It distinguished between line-of-sight (LOS) and non-line-of-sight (NLOS) propagation conditions between the transmitter and the receiver with respect to the illuminated wall of a building. Other studies related to terrestrial systems are available in the literature for various frequency bands and scenarios [3]–[5]. The situation is much more complicated when considering satellite services due to long distances resulting in a very tight power budget. However, receiving signals inside buildings may become a reality soon thanks to ever-improving satellite

B

Manuscript received September 02, 2010; revised November 08, 2010; accepted January 26, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This work was supported by the ESA PECS Project No. 98069 “Building Penetration Measurement and Modelling for Satellite Communications at L, S and C-Bands.” The authors are with the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic (e-mail: [email protected]; [email protected]. cz). Digital Object Identifier 10.1109/TAP.2011.2158963

technologies, giving rise to an increasing number of studies on building penetration loss for high elevation angles [6]–[17]. Other factors to consider include the wide range of high elevation angles, the fact that satellites can be non-geostationary, and that the elevation angle between the user and the satellite may vary in time. As the link budget must be carefully planned, relevant models for penetration loss are of great interest. In addition to direct measurements with a geostationary satellite [6], [7], helicopters [8]–[11], aircraft [12], airships [13], [14], crank-up towers, or cranes [12], [15], [16] can also be utilized as pseudo-satellites. Moreover, reference levels need to be obtained in different ways so that propagation conditions can be defined clearly. As few models providing penetration loss as a function of high elevation angles are available, the terrestrial COST 231 outdoor-to-indoor model [2] was chosen and extended to high elevation angles from 55 to 90 in [17]. An empirical model at 2 GHz is provided for pure LOS propagation conditions between a satellite and the building facade. In [13], an empirical model for penetration loss at 2.0 GHz and 3.5 GHz is introduced as a function of the elevation angle ranging from 10 to 90 . The model was developed for mobile services in urban areas regardless of the LOS/NLOS propagation conditions. Results of wideband measurements using a channel sounder covering the frequency range from 2470 MHz to 2670 MHz and elevations from 15 to 60 are presented in [10]. Primarily LOS conditions were addressed and a linear fit of averaged building entry loss was introduced. Unmodulated continuous wave (CW) measurements performed at 5.1 GHz for elevations from almost 0 to 90 are presented in [9]. The elevation dependence was found to be different depending on whether LOS or NLOS propagation conditions were present. A general model for building penetration loss covering the deterministic free space loss, slow fading and fast variations of the received signal is described in [8]. To create such a model, CW measurements at 2420 MHz and 1620 MHz were taken at elevation angles from 30 to 80 . Available models for satellite services address different frequencies, elevations and different propagation scenarios and conditions as well. However, it is necessary to study penetration loss within a wide range of frequencies and elevation angles for clearly defined propagation conditions and scenarios by employing one single methodology. Only in this way it is possible to propose a generally valid elevation dependent model. This is why we carried out an extensive measurement campaign in an urban area of Prague during the summer of 2009 to obtain

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Fig. 3. The front side of building C which is 15 m high. Fig. 1. The rear side of building A which is 34 m high.

Fig. 4. The rear side of building D which is 20 m high.

Fig. 2. The front side of building B which is 13 m high.

a set of statistically relevant experimental data. For this trial, a representative set of typical urban buildings was carefully chosen. Based on the results, we were able to develop elevation dependent empirical models of building penetration loss at L-, S- and C-band for clearly defined propagation conditions as discussed above. II. MEASUREMENT SCENARIOS Five different types of buildings were selected for the measurement campaign so that buildings made of reinforced concrete or bricks with different types of windows were included. A. Building A: Office Building The Faculty of Electrical Engineering of the Czech Technical University in Prague (CTU) is an eight-story building made of reinforced concrete with tinted windows consisting mainly of offices, classrooms and halls. There are no other buildings of the same height in its vicinity. Overall, 21 measurement sites were selected at different positions inside this building including measurements at different distances from the windows, on different floor levels and on the roof with the majority of the measurements being performed on the rear side of this building shown in Fig. 1. B. Building B: Brick Building The CTU Rector’s office building shown in Fig. 2 is a threestory brick building with non-tinted windows consisting of offices, classrooms and corridors. Three measurement positions

at the front side and two measurements at the rear side were selected at different distances from the windows along with one measurement on the roof. C. Building C: Residential House Building C (Fig. 3) is located in the middle of other surrounding houses. This four-story brick building with up to 40-cm-thick walls and standard thermal glass windows consists of nine private flats and a set of stairs on the rear side. Ten different positions were selected so that four measurements at different distances from the window in a flat were performed and another six measurement sites were located on different floors at the stairs 1 meter from the windows. D. Building D: Office Building One section of a typical, modern, office building made of structural slabs with double-layer tinted windows shown in Fig. 4. was also selected. This four-story building, surrounded by a park on the rear side and a main street on the front side, consists of offices, halls and meeting rooms. Except for two measurements at different distances from the window in an office and two measurements on the roof, all the four remaining measurement sites were located on different floors at the stairs on the rear side of this building at a uniform distance of 1 meter from the windows. E. Building E: Shopping Centre The last typical building for an urban area was selected to be a modern two-story shopping centre. The shopping mall area contains 23 stores of various sizes, several restaurants and both a covered and an uncovered parking lot. The whole building is made of a thermal glass facade and a steel construction. Skylights are placed in the roof as well. Two measurements were

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Fig. 5. The front side of the shopping centre, building E.

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the measurements. A battery-powered high sensitivity radio receiver (Rohde&Schwarz PR100) enables the detection of signal (for a 10 kHz bandlevels as low as approximately width). The rf input of this receiver is connected to the antenna by a low-loss cable, whereas the data output is connected via LAN to a laptop recording signal levels at each of the four corresponding frequencies one after the other. As a result, this measurement system is capable of measuring penetration loss of up to 60 dB with a 1 dB level of precision for the measurement distances employed during the campaign. IV. DATA PROCESSING A. Extraction of Penetration Loss

Fig. 6. Planar wideband spiral antennas at the transmitter station.

performed on each of the two floor levels—one at the one-meterdistance from the window or the glass facade and the other almost in the middle of the building. III. MEASUREMENT SYSTEM A. Experimental Setup A remote-controlled nine-meter-long airship carrying the transmitter was utilized as a pseudo-satellite during the measurements. The transmitted signals were received by the receiver station placed at a fixed position inside each of the case-study buildings. Synchronization of the recorded signal levels and flight data provided by the airship’s sensors (differential GPS position, pitch, roll, and compass) was enabled by using measurement time stamps.

In accordance with [10], we opted for a calibration method on a flat open field where the receiver was in a LOS towards the transmitter. By recording received signal levels during flyovers of the airship carried out in a very dense combination of azimuths and elevations with respect to the position of the receiver, a database with reference levels for azimuths and elevations accomplished during the subsequent in-building measurements was obtained. This calibration method makes the calculation of penetration loss very straightforward: the corresponding reference level is subtracted from the experimental data with respect to the azimuth and elevation angle. To eliminate the influence of free space loss for different distances between the transmitter and receiver, all recorded signal level data were recalculated to a sufficiently large and uniform distance, which in our case was 20 km. Based on the knowledge of the real distance in meters, the recalculation to the 20 km distance can in dB from the meabe easily done by subtracting the value sured signal levels:

(1)

B. Transmitter Station One part of the transmitter station consists of CW generators providing approximately 27 dBm unmodulated signals at 2.0 GHz, 3.5 GHz, 5.0 GHz, and 6.5 GHz. The integral parts of these generators are four voltage controlled oscillators stabilized by means of a phase lock loop and four power amplifiers. The other part is formed by three left-handed circularly polarized (LHCP) planar wideband spiral antennas (2.0 GHz and 3.5 GHz signals are combined by a diplexer), Fig. 6. The back volume of the antennas is filled with a polyamide carbon absorber in order to attenuate the right-handed circularly polarized component, which is radiated in the opposite direction. The whole transmitter station is covered by expanded polystyrene (to mechanically protect the unit) and attached to the bottom part of the airship. C. Receiver Station The four frequencies are received by a single LHCP planar wideband spiral antenna, with a design and construction similar to the antennas at the transmitter station. It was placed on a tripod at a constant height of 1.5 m above floor level throughout

B. Data Pre-Processing According to the flight data, only experimental data measured when the pitch and roll of the airship were within degrees interval were further processed since the radiation pattern of the transmitting antenna did not change dramatically within this interval. In addition, measured datasets were cleared of data obtained during takeoff, landing, turning maneuvers of the airship and any noise data with respect to the noise floor of the receiver. C. Measurement Procedure and Analysis At least four flyovers of the airship, which were approximately 1.5 km long and separated by 45 in azimuth, were planned above each measurement site at a constant altitude of approximately 200 m above the ground. This ensured that a wide range of elevation angles (counted with respect to horizontal plane) from 20 up to 90 was covered at different azimuths (0 defined perpendicular to the building facade in front of the receiver) as seen from the position of the receiver station. The speed of the airship was kept between 2 and 6 meters per second based on the speed and direction of the wind.

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objects such as chimneys or antennas, levels of measured penetration loss varied slightly around 0 dB, as expected. Since this was the case for all roof measurements, despite being performed on different days, the time-stability of the system, correctness of the data processing method and reliability of obtained results were validated. V. RESULTS Fig. 7. The criterion for distinguishing between the LOS and NLOS case (receiver placed in front of the window). The borderline between LOS and NLOS is outlined by the building facade.

As the receiver was always placed inside the case-study building in front of a window, a common criterion respecting the illuminated external wall with this window could be applied to define LOS and NLOS propagation conditions, as indicated in Fig. 7. The flyovers parallel to the facade of the investigated buildings resulted in problematically defined LOS/NLOS propagation conditions and were not further analyzed. Within the LOS or NLOS cases, data were processed regardless of azimuth. This respected the fact that the mean of differences between penetration loss recorded during the particular flyovers at different azimuth angles were only up to few dB and that the data did not show any clear azimuthal dependence. Accordingly, penetration loss was obtained for each measurement site as follows: a mean value was calculated separately for the LOS and NLOS cases from all the data within five-degree intervals in elevation, which ranged from 22.5 to 87.5 . This, for simplicity’s sake, enabled us to represent the results as a single value for an elevation angle in the middle of these intervals, i.e., 25.0 , 30.0 etc. The width of these intervals was selected experimentally and guaranteed the mean value to be calculated from a sufficient amount of empirical data, while information about the elevation dependence of penetration loss remained. The elevation interval of 87.5 to 90 was not considered further as it contained a statistically small amount of data and, more importantly, the size of the Fresnel zones of the propagating signals prevented us from accurately distinguishing the LOS and NLOS cases. Using these results, the elevation dependence of average penetration loss at each building could be addressed. For this purpose, only the measurement sites where the receiver was placed at the uniform one-meter distance in front of a closed window were considered. By calculating mean values at five-degree intervals in elevation, we obtained penetration loss for each building for the LOS and NLOS cases. Although statistically insignificant, several measurement sites were excluded from this processing such as when a roof window or shadowing by surrounding buildings was present. D. Data Validation Measurements on the roofs of the case-study buildings were used to validate the measurement methodology as the transmitter was in a constant LOS toward the receiver as during the calibration measurements. Although this scenario differs from the flat open field due to the shape and dimensions of the roof and shadowing and reflections caused by the presence of nearby

Figs. 8 and 9 show average penetration loss for each building under LOS or NLOS conditions, respectively. Not only is penetration loss rising with elevation, but also it is considerably different under these two propagation conditions. Furthermore, the differences between the case-study buildings can be found in the trend and slope of this dependence as well as in the absolute values of penetration loss. It should be noted that the reported penetration loss was always within the 60 dB dynamic range of the measurement system. As far as the absolute values are concerned, one notices that penetration loss was observed to be highest inside both office buildings A and D and lowest inside building B, for both LOS and NLOS cases. This can be explained by the respective structures of these buildings: buildings A and D are modern buildings with steel construction and tinted windows, whereas building B is a brick building with large, non-tinted windows, which is the main cause of exceptionally low penetration loss at 2.0 GHz for the NLOS case. Clearly, the presence and type of windows turned out to be an important factor at the given frequencies. The level of attenuation caused by the two remaining buildings C and E is mostly between the previously mentioned border cases at lower elevations and close to the values for building B at higher elevations both for LOS and NLOS. This can be due to smaller thermal windows in building C and by the presence of the large, thermal glass facade and skylights in building E. In addition, at higher elevations, particularly for the NLOS case, close values of penetration loss inside buildings C and B are given by their similar structure (both are made of bricks). Skylights in building E may have caused lower attenuation despite the presence of the thermal glass facade and steel construction and thus the reported penetration loss is close to penetration loss inside buildings B and C. On the other hand, higher loss inside building C, when compared to building B in terms of NLOS at lower elevations, could be caused by much thicker walls and the wider dimensions of building C. The transition between LOS and NLOS conditions at around 90 elevation results in almost the same maximum penetration loss for each building. In contrast, minimum values of penetration loss can be observed for low elevations and are approximately 10 dB higher for NLOS than LOS. Another difference between the case-study buildings is the slope of the dependence of penetration loss on elevation. Concerning LOS, a sharp rise of loss inside building B is present at low elevations due to the fact that the signal propagates through the large non-tinted windows and starts to propagate more through the brick wall or upper floors with increasing elevation. Regarding NLOS, the trend of each curve in Fig. 9 is almost the same: penetration loss rises significantly with elevation up to approximately 60 and then tends to remain nearly constant. This is presumably given by the fact that starting at 60 of

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Fig. 8. Building penetration loss plotted versus elevation at the four utilized frequencies for each of the case-study buildings A-E, LOS.

Fig. 9. Building penetration loss plotted versus elevation at the four utilized frequencies for each of the case-study buildings A-E, NLOS.

elevation, the signal does not significantly propagate through windows and walls at the side of the building which is opposite

to the position of the receiver. It should be noted that the smallest overall difference between maximum and minimum

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Fig. 10. Penetration loss at 5.0 GHz for the position of the receiver 1 and 4 meters from a window on the sixth floor of building A.

values of penetration loss under LOS conditions was observed inside building D, which corresponds to the assumption that there may be no significant difference if the signal propagates through a tinted thermal glass window or a wall. The results are provided for the one-meter-distance from a window; however, it is worth mentioning that receivers can be placed also at different distances. The one-meter-distance was selected to represent a standard limit case of the position of the receiver in a satellite-to-indoor scenario since the deeper the receiver is placed inside the building, the smaller the difference between LOS and NLOS propagation conditions when applying the definition in Fig. 7. Farther distances from a window should result in penetration loss for LOS to be closer to the penetration loss under NLOS conditions at the one-meter-distance, whereas penetration loss should be nearly indifferent of distance for the NLOS case. Although this is generally dependent on the specific building structure, this proved to be true throughout our measurement scenarios. To support this hypothesis, measurements were taken on the sixth floor of building A with receiver placed 1 and 4 meters from a window (results at 5.0 GHz are shown in Fig. 10), and on the ground floor of building E where the receiver was placed 1 meter from the glass facade and almost in the middle of the building (Fig. 11 shows results at 3.5 GHz). The position in the middle of building E represents only a NLOS condition as the LOS case could not be defined according to Fig. 7. As can clearly be seen in Figs. 10 and 11, practically the same penetration loss is observed in NLOS cases regardless of the distance from the window or the glass facade, whereas under LOS, penetration loss at the four-meter-distance and in the middle of the building is close to the NLOS case at the one-meter-distance. As these results are provided for one particular measurement site only, the absolute values differ from the overall results in Figs. 8 and 9. Generally, it can be stated that providing results for the one-meter-distance from a window for both LOS and NLOS can be used to estimate common limits of penetration loss for other distances from a window. To summarize the results at all four frequencies:

Fig. 11. Penetration loss at 3.5 GHz for the position of the receiver 1 meter from the glass facade and on the ground floor in the middle of building E.

• Building penetration loss is significantly influenced by the geometry of the scenario, that is, the position of the receiver inside a building with respect to the building orientation and satellite position. This is why the LOS/NLOS criterion was utilized rather than precisely considering the azimuth angle and the facade orientation. • There is a strong dependence of penetration loss on the elevation angle. We notice an almost linear rise of penetration loss with elevation for the LOS case and saturation for the NLOS case where maximum loss is reached at approximately 60 of elevation. • Significant differences between the absolute values of penetration loss are found amongst the selected types of buildings with the maximum loss seen inside the modern office buildings, and the minimum loss observed inside the brick building. The presence and type of windows clearly influence penetration loss. • The measurement results for the receiver placed 1 m from a window can be used to estimate building penetration loss for other indoor receiver positions. The minimum and maximum expected values are represented by the LOS and NLOS results, respectively. VI. MODELING Based on the overall results for the case-study buildings, the dependence of penetration loss on the elevation angle was modeled at each of the selected frequencies separately for the LOS and NLOS cases. An important presumption is that the selected buildings form a representative set of buildings and outline an interval within which levels of penetration loss inside any other common urban buildings should fall. As shown in the previous section, the loss reported for the office buildings can be more than 20 dB higher than the values for the brick building. We decided to provide models for overall minimum and maximum penetration loss together with a model for its mean value. The reason for this approach is as follows: as we covered minimum and maximum

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penetration loss frequently observed in an urban area, penetration loss can now be estimated to be closer to these minimum or maximum borderlines based on the structure of a particular building. Further, the model for mean penetration loss predicts an average estimation if no detailed information about the building is available. A. LOS With respect to the almost linear rise of penetration loss in dB with elevation for the LOS case, we implemented a linear model based on the following equation:

(2) in are held constant for a given where in dB and frequency and is the elevation angle in degrees. Theoretically, could stand for penetration loss at an elevation of 0 , however, the model is valid only from an elevation of 25 ; parameter characterizes the slope of the fitted curve. The empirical model was obtained as follows. First, at each frequency, minimum, maximum and mean values of penetration loss within each elevation interval were determined based on the overall mean penetration loss for each of the case-study buildings as shown in Fig. 8. Then, the least-square method was used to fit (2) to these values. It was found that parameter did not significantly vary with frequency for each of the minimum, maximum or mean fits, which enabled us to reduce the number of values representing for the fitted curves in the folfor the minimum fit lowing way: from the four values of at the four corresponding frequencies, an average represented was calculated. The same process further by parameter was applied to the maximum and mean fits so that parameters and were obtained. To obtain the correct values of parameter , (2) had to be fitted again to the minimum, maximum and mean penetration loss at each frequency, but the cor, and , respectively, responding parameters had to be used. Finally, three values of parameter were ob, and . As a result, tained at each frequency: for example, the model for the minimum penetration loss at 2.0 and by the GHz is represented by the value of , which is independent of the frevalue of quency. All the necessary values of the parameters mentioned are clearly summarized in Table I. The fitted curves, together with corresponding experimental data, are shown in Fig. 12 for the frequency of 2.0 GHz. Note that the experimental data shown in Fig. 8 correspond to Fig. 12. B. NLOS For the NLOS case, penetration loss first rises with elevation and then tends to be nearly constant at high elevation angles. To embed this effect in the empirical model of the loss in dB at each frequency, the following equation was used to fit minimum, maximum and mean of the overall penetration loss for all casestudy buildings presented in Fig. 9:

(3)

Fig. 12. Minimum (dashed), maximum (dotted) and mean (solid) linear fits of the experimental data at 2 GHz, LOS.

TABLE I PARAMETERS OF THE EMPIRICAL MODEL FOR LOS

where parameter in dB represents maximum penetration loss represents the difat an elevation of 90 , parameter in ference between minimum and maximum penetration loss at elevation angles 25 and 90 , respectively, and parameter determines the trend of the fitted curve. In contrast to the LOS case, it was noticed that at each frequency the dependence of minimum, maximum and mean penetration loss on elevation was very similar simplifying the modeling in the following way: using the least-square method, (3) was fitted at each frequency to the mean values of penetration loss calculated in the same way as for the LOS case. It was found that the four obtained values of parameter varied only from 2 to 3.1 and the trend of the curve was not significantly sensitive to this parameter. This is why the mean value of was calculated and its value of 2.5 was later used. Next, (3) was fitted again to mean values of penetration loss at each frequency, but this time . As a result, a total of four values of and with were obtained, one at each frequency as four values of before. Based on the previously mentioned similar dependence of minimum, maximum and mean penetration loss on elevation, to model it was necessary to change only the value of minimum and maximum penetration loss, while the value of remained unchanged. At each frequency, we manually increased by the interval in dB so that only one value the value of from the found maximum penetration loss was allowed to exceed penetration loss outlined by the corresponding maximum

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Fig. 13. Mean fit (solid) of the experimental data at 2 GHz together with the minimum (dashed) and maximum (dotted) border lines, NLOS.

TABLE II PARAMETERS OF THE EMPIRICAL MODEL FOR NLOS

fit of more than 1 dB. An analogous approach was applied to the minimum fit. We also found only a single value of at each frequency, which could be subtracted from or added to a value of . For example, at 2.0 GHz, the minimum penetration loss is obtained by subtracting can be modeled by (3), where the value of from the value of , and . All the necessary parameters of the empirical model for the NLOS case are clearly summarized in Table II and an illustration of the fit is shown in Fig. 13, again for the frequency of 2.0 GHz.

VII. DISCUSSION Different definitions of penetration loss, type and structure of a particular building and a strong dependence on elevation angle must be acknowledged when a comparison with results from [8]–[11], [13], [14] and [17] is to be made. As with our study, the models provided in [9]–[11] and [17] employ a linear or nearly linear rise of penetration loss with elevation for the LOS case. In [13], the LOS case can be considered only for high elevation angles, where the linear rise is also noticeable. In addition, the experimental data presented in [8] for lower floor levels show the same linear trend, as do the results for higher elevations in [14].

It is also not easy to compare the absolute values available in the literature. For the LOS scenario, up to 10 dB lower penetration loss is noticed in [17] at 2.0 GHz for a concrete building with common-size windows than is outlined by the minimum fit based on Table I. Moreover, slightly lower loss is predicted for higher elevation angles by the overall model at 2.0 GHz and 3.5 GHz in [13]. In contrast, building A was studied in [13] and the subsequent results presented in [14] are in good accordance to ours: penetration loss of approximately 45 dB at 2.0 GHz and 50 dB at 3.5 GHz was reported for the fifth floor within the range of elevation angles from 70 to 80 . The linear model at 2.5 GHz in [10] predicts a linear rise of penetration loss from 20.7 dB to 28.5 dB for elevation angles from 25 to 90 . This fits well between the minimum and maximum borderlines given by our model at 2.0 GHz. Further, the early work in [8] provided results at 1.62 GHz for a concrete office building with non-tinted windows. For lower floors, penetration loss rose from 13 dB to 25 dB within an elevation interval of 30 to 80 . These values for lower elevations are slightly higher than the minimum fit at 2.0 GHz given by Table I, and, slightly lower for higher elevation angles. We believe that this may have been caused by the different frequency. Measurements performed at an office building at 5.1 GHz [9] report an overall penetration loss ranging from approximately 15 dB to 35 dB within the elevation interval of 25 to 90 , which is close to the minimum values predicted by our proposed model. In addition, another comparison can be made at 5.2 GHz where 30 dB of mean penetration loss is reported for a range of elevation angles from 15 to 60 in [11], which is consistent with our model for the mean value of penetration loss. In compliance with all the comparisons made, it is evident that our approach, based on providing elevation dependent models separately for LOS and NLOS, is correct. Moreover, by introducing models for minimum and maximum penetration loss, we were always able to find a good match with the results presented in other studies, which justifies the large intervals between the minimum and maximum models for penetration loss reported. VIII. CONCLUSION We have presented results of an extensive measurement campaign covering a wide range of elevation angles performed at different representative types of buildings in an urban area of Prague aimed at building penetration loss for satellite services operating in L-, S- or C-band. It was shown that the corresponding signal attenuation depends strongly on the structural characteristics of a particular building and that there is a significant dependence on elevation. Further, we introduced elevation dependent empirical models for building penetration loss at 2.0 GHz, 3.5 GHz, 5.0 GHz and 6.5 GHz separately for LOS and NLOS propagation conditions. These models cover a linear rise of penetration loss with elevation for LOS and a tendency to be close to constant at high elevations for the NLOS case. The comparison with results in six other relevant studies dealing with penetration loss indicates that the proposed models are capable of predicting both extremely low and high penetration loss as are typically reported

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in brick buildings with non-tinted windows or office buildings with tinted windows. Since the models provide minimum, maximum, as well as mean values, they can be used as a universal tool enabling the prediction of building penetration loss for a wide range of satellite-to-indoor scenarios at broad frequency ranges. REFERENCES [1] Propag. data and prediction methods for the planning of short-range outdoor radio communication systems and radio local area networks in the frequency range 300 MHz to 100 GHz, ITU-R Rec. P.1411-3. 03/2005, 2005. [2] J. E. Berg, Building penetration COST 231 Final Rep., Brussels, Belgium, 1999, pp. 167–174. [3] L. Ferreira, M. Kuipers, C. Rodrigues, and L. M. Correira, “Characterisation of signal penetration into buildings for GSM and UMTS,” in Proc. 3rd Int. Symp. on Wireless Communication Systems (ISWCS ’06), Valencia, Spain, Sep. 2006, pp. 63–67. [4] E. F. T. Martijn and M. H. A. J. Herben, “Radio wave propagation into buildings at 1.8 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 122–125, 2003. [5] A. F. Toledo, A. M. D. Turkmani, and J. D. Parsons, “Estimating coverage of radio transmission into and within buildings at 900, 1800 and 2300 MHz,” IEEE Personal Commun., vol. 5, no. 2, pp. 40–47, Apr. 1998. [6] R. Hoppe, T. Hager, T. Heyn, A. Heuberger, and H. Widmer, “Simulation and measurement of the satellite to indoor propagation channel at L- and S-band,” presented at the EuCAP 2006-1st Eur. Conf. on Antennas and Propag., Nice, France, Nov. 2006. [7] P. Veltsistas et al., “Satellite-to-indoor building penetration loss for office environment at 11 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 96–99, 2007. [8] I. Frigyes et al., “Theoretical and experimental characterization of the satellite-to-indoor radio channel,” in Proc. 5th Int. Conf. on Satellite Systems for Mobile Communications and Navigation, May 1996, pp. 47–50. [9] A. A. Glazunov, L. Hamberg, J. Medbo, and J.-E. Berg, “Building shielding loss measurements and modelling at the 5 GHz band in office building areas,” in Proc. 52nd IEEE Vehicular Technology Conf. (VTC 2000-Fall), Sep. 2000, vol. 4, pp. 1874–1878. [10] F. Perez-Fontan et al., “A wideband, directional model for the satellite-to-indoor propagation channel at S-band,” Int. J. Commun. Syst. Network, pp. 28–28, 2009. [11] F. Perez-Fontan et al., “Characterization of the C-band wideband satellite-to-indoor channel for navigation services,” presented at the EuCAP 2010-4th European Conf. on Antennas and Propag., Barcelona, Spain, Apr. 2010.

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[12] W. J. Vogel and N. Kleiner, “Propagation measurements for satellite services into buildings,” in Proc. IEEE Aerospace Conf., Aspen, CO, Feb. 1997, vol. 1, pp. 91–107. [13] J. Holis and P. Pechac, “Penetration loss measurement and modelling for HAP mobile systems in urban environment,” EURASIP J. Wireless Commun. Network., vol. 2008, no. 4, pp. 7–7, Jan. 2008. [14] J. Holis and P. Pechac, “Measurements of building penetration loss as a function of the elevation angle and floor level,” in Proc. EuCAP 2009-3rd Eur. Conf. on Antennas and Propag., Berlin, Germany, Mar. 2009, pp. 3586–3590. [15] W. J. Vogel and G. W. Torrence, “Propagation measurements for satellite radio reception inside buildings,” IEEE Trans. Antennas Propag., vol. 41, no. 7, pp. 954–961, 1993. [16] T. Jost, W. Wang, A. Dammann, U.-C. Fiebig, M. Walter, and F. Schubert, “Satellite-to-indoor broadband channel measurements at 1.51 GHz and 5.2 GHz,” in Proc. EuCAP 2009-3rd Eur. Conf. on Antennas and Propag., Berlin, Germany, Mar. 2009, pp. 3586–3590. [17] D. I. Axiotis and M. E. Theologou, “An empirical model for predicting building penetration loss at 2 GHz for high elevation angles,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 234–237, 2003.

Milan Kvicera (S’08) received the M.Sc. degree in radio electronics from the Czech Technical University in Prague, Czech Republic, in 2008, where he is currently working toward the Ph.D. degree. His research interests are focused on radiowave propagation and satellite communication.

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, Czech Technical University in Prague. His research interests are in the field of radiowave propagation and wireless systems.

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Accurate Modeling of Body Area Network Channels Using Surface-Based Method of Moments Ahmed M. Eid and Jon W. Wallace, Member, IEEE

Abstract—An accurate method for modeling body shadowing in body area networks (BANs) is presented, based on an efficient surface-based method-of-moments (MOM) solution. The method allows the fields radiated by a transmit line current with arbitrary orientation (polarization) in the presence of a lossy dielectric cylinder of arbitrary cross section to be computed with high accuracy. The fields due to a point source are then found using Fourier transform techniques. The utility of the method for BAN modeling is demonstrated by comparing with BAN measurements on a human subject performed in a compact chamber and in an open field. Index Terms—Biological systems, method of moments, modeling, propagation.

I. INTRODUCTION

B

ODY AREA networks (BANs) are an emerging paradigm for wireless communications where communicating nodes are placed near or inside the body, with applications in biomedicine, sports, emergency response, and consumer electronics. Although such sensors can be connected by wired links, wireless BANs provide more freedom of movement for the user or patient. However, designing wireless BAN systems that are robust, efficient, and high performance is challenging since the body forms an integral part of the antennas and propagation channel. The purpose of this paper is to develop a model for on-body communications that is not only simple and accurate, but also as general as possible in accommodating different antenna types and environments. In [1] it was identified that propagation between two shadowed BAN nodes can occur due to waves penetrating through the body or creeping around the body, but that the creeping waves dominate in most cases. BAN characterization has been achieved by measurements [2]–[4], as well as detailed simulations [1]. Although these studies successfully assess BAN communications for a specific type of antenna and environment, one drawback of these approaches is that the observations are antenna and environment specific, possibly limiting the generality of the results. An important recent step toward antenna- and environmentindependent characterization is provided by [5], which models the body as an infinite lossy dielectric cylinder with circular Manuscript received September 30, 2010; revised December 17, 2010; accepted December 28, 2010. Date of publication June 09, 2011; date of current version August 03, 2011. The authors are with the School of Engineering and Science, Jacobs University, 28759 Bremen, Germany (e-mail: [email protected]; wall@ieee. org). 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.2011.2158971

cross section, derives the scattered fields due to a -directed (along axis) current source using conventional modal analysis, and derives the response to a point source using a Fourier transform. The model was extended in [6] for arbitrary polarization and compared with measurements in a compact anechoic chamber, revealing good agreement for some polarizations, but poor agreement for others. The purpose of this paper is to take the approach in [5] one step further by modeling the body as an infinite cylinder with arbitrary cross section, since the body shape and curvature can play a vital role for certain antennas and polarizations. The natural choice taken herein is to consider surface-based method-ofmoments (MOM) [7]–[9], where the only unique unknowns are tangential fields at the air-body interface. Although such developments are slightly more complicated than simple traditional volume-based approaches that segment the interior of the scatterer [10], [11], the surface-based approach only requires segmentation of the air-body surface, leading to significant improvement in efficiency. In the paper, a compact and self-contained derivation of the surface-based MOM approach for BAN is provided, not only to make the method as accessible and complete as possible for the BAN research community, but also to help generalize and simplify existing treatments. Additionally, measurements are performed both in a compact anechoic chamber and in an open outdoor range to check the accuracy of the proposed model. The organization of the paper is as follows: Section II develops the surface-based method-of-moments solution for BAN modeling, and in Section III it is validated and different torso models are presented. Section IV then compares the model with on-body measurements taken in an anechoic chamber and outdoors. Finally, Section V concludes the paper. II. SURFACE-BASED NUMERICAL SOLUTION FOR BAN PROPAGATION Fig. 1 depicts a lossy dielectric cylinder model of the body, plane and is infinite and howhose shape varies only in the mogeneous in the -direction. The perimeter of the object is defined by the contour . It is required to calculate the field at in response to a point current source of observation point arbitrary orientation with density (1) The problem is solved using the approach in [5], where first the fields due to a line source with current density

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(2)

EID AND WALLACE: ACCURATE MODELING OF BAN CHANNELS USING SURFACE-BASED MoM

holds, where verse operator and

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is a two-dimensional trans. The solution to the problem (7)

is given by the usual two-dimensional scalar Green’s function (8) and combining (8) and (6) with Green’s theorem, yields

Fig. 1. Arbitrarily shaped biological body illuminated by an external source.

are obtained, which can then be transformed via Fourier techniques to find the point source solution. We note that surface-based MOM solutions for an infinite dielectric cylinder with arbitrary cross-section are well known [7]–[9], and are closely related to the problem here of solving for fields radiated by the line current (2) in the presence of the lossy cylinder. However, directly applying these previous treatments is difficult due to existing limitations that are not trivial: lack of generality (i.e. single polarization), restriction to plane-wave excitation (usually normal incidence), and the absence of the variation. Instead of trying to remedy or extend existing treatments, the approach taken in this work is to develop a compact and univariation of fied approach that directly exploits the fields as is done in waveguide analysis, and as is shown, this novel approach results in a very compact and elegant solution for BAN modeling.

(9) where is in the th region, is the contour formed by is the normal the boundary of the homogeneous region , vector on the contour at point outward from , and the same . For a region defined to lie inside mulequation holds for tiple unconnected boundaries, (9) still holds where includes all boundaries and is always away from the medium . Note that the observation point in (9) must be strictly inside . Fields in Region 0 (outside the body) are found by writing (9) with closed contours around the line source, around the body, and at infinity (far-field), as depicted in Fig. 1. The contour around the line source just yields fields radiated by the line source in free space, and the contour integral at infinity vanishes, leaving (10)

(11) A. Governing Equations In a homogeneous region without sources, electric magnetic field satisfy

and

(3) where is the wavenumber of the medium. Since the line source has variation and the gemust ometry is homogeneous in , the solutions for and variation. As in waveguide analysis, this also have form allows fields transverse to the direction to be written as (4) (5)

where is the outward normal direction from the scatterer (the body) and the sign change comes from in Region 0. Fields in Region 1 (inside the body) are given by the single contour integral

(12) B. Incident Fields Incident electric and magnetic field due to a line source in free space is given in [6] and for brevity is not repeated here. C. Boundary Conditions

where , which means that and are the only unique unknowns, and in the th homogeneous region, the two-dimensional relationship (6)

Solving the governing equations requires enforcing the proper continuity conditions at the boundary. Although and on , the normal derivatives and may be discontinuous on the boundary.

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Let be the outward normal of the scatterer (orthogonal be the vector tangential to the surface. The to ) and transverse field components satisfy the conditions

(13) Fig. 2. The nth segment for the method-of-moments discretization procedure. Order of the endpoints is chosen so that when moving to the next endpoint, the outward normal of the scatterer n ^ points to the right.

Considering (4), (5), the required operations

(14) are easily derived, where is an arbitrary function. Thus, substituting (4), (5) into (13) gives

(15)

D. Discretization We discretize using the usual MOM procedure, where the consists of straight-line segments, and a closed contour is typical segment is depicted in Fig. 2, where , a coordinate along the contour, and , , and are the length, first endpoint, midpoint, and (body) outward normal, of the th segment, respectively. Segpoints to the right when ments are connected such that to , i.e. the segments go counter-clockmoving from wise around the scatterer. and , these unknowns are expanded in To find Region 0 as

(16) (22)

(17)

(23)

(18)

lies on the contour , and where the point is a pulse function giving 1 when lies on segment and 0 otherwise. Substituting into (10) and applying point matching , at

and must Note that the tangential derivatives be equal on the two sides of the interface, since everywhere and . Using this fact on the contour the tangential derivatives can be eliminated from the equations, yielding

(24)

where

(19) (25) where (20) (21) and . Therefore, we only need to retain -directed fields and normal derivatives on the outside surface as the unique unknowns, and substitute (19) for fields on the inside boundary. Also, note that since governing equations and boundary conand are uncoupled, they can be solved for ditions for separately.

(26)

along the contour segment are given by , with . The derivative of the Green’s function in (25) can be evaluated directly as and

and

(27) where

and

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m = n, where segment is rotated and

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Fig. 3. Integration path for case of translated to the origin for convenience.

Fig. 4. Integration path for observing fields in Region 0 for the singular case of , where segment is rotated and translated to the origin for convenience.

For , we can approximate integrals in (24)–(26) as the integrand evaluated at the midpoint times the length of the segment, or

, and (34) vanishes as does the contribuwhere has a contribution, and tion on . Thus, only the integral on , recognizing

m=n

(28)

(35)

(29) For higher accuracy, multipoint numerical quadrature methods can also be applied. , the observation point is on the source For the case of segment, and the integral must be evaluated. Fig. 3 depicts the where the segment has integration path for the case of been rotated to align with the -axis and translated to have its midpoint at the origin. We have

(36)

where the integral was performed from to 0 to ensure that is positive and the logarithm came from the small argument approximation of the Hankel function. In a similar way, the surface integral inside the body (12) is evaluated according to

(30) (37) (31)

where (38)

which can be evaluated using the indefinite integral

(39) (32) where

and

are the Struve functions [12]. Since , we have

. For where (19) was combined with (23) to obtain , single point numerical quadrature can be used as before. , we have For the case of (40)

(33)

(41)

For the evaluation of , we avoid direct integration of the higher order singularity by changing the contour to be that depicted in Fig. 4, chosen such that the observation point at the middle of the interval is inside the complete contour , and let . Along , the contribution to is

Note that has the opposite sign as , since for the required contour to enclose the observation point, it must circle around the top side of the singularity depicted in Fig. 4, and . Summarizing, (24) and (37) characterize the system, which can be written in matrix form as

(34)

(42)

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where and . Scattered

, and (42) can be inverted to find outside the body is given by

(43) where

(44)

(45) Fig. 5. Channel comparison of E in response to a z -directed line source.

and analogous expressions can be derived for total field inside the body. The procedure for finding is identical to that for , by simply making the substitutions

and

(46) (47) (48) (56)

in the above development. and , transverse fields are given by After solution of (4) and (5), which in component form are

(57)

(49) E. Point Source (50) (51)

The electric and magnetic fields due to a line source near the lossy torso model derived in the previous sections can be transformed to a point source using the Fourier transform technique described in [5].

(52) III. VALIDATION AND TORSO MODELS Transverse scattered field outside the body is found by substituting (43) and the analogous expression for into (49)–(52). In this case, derivatives with respect to or can be transferred to , and , or

(53)

where

,

(54)

(55)

As a validation of the proposed BAN model for arbitrary cross section, the numerical method explained above is compared with the closed form solution for a circular cross section and relative permittivity [6]. A lossy cylinder of radius is used, and the source was placed at and of . The comparison for different orientation of receive/transmit sensor/source is shown in Figs. 5 and 6 where the fields ( and ) are compared at fixed observation radius versus observation angle . In order to correctly quantify path loss around the human body, the human torso shape should be accurately identified. A camera-based measurement system was used, where the subject stands on a circular disk which is turned to different angles in 10 increments and a photo is taken at each position. Comparing the width of the subject to a measured standard in the photo and properly accounting for the camera perspective allows the diameter of the subject versus rotation to be reliably estimated.

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Fig. 6. Channel comparison of E in response to a -directed line source.

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Fig. 8. Model comparison of E in response to a z -directed point source.

Fig. 7. Measured torso of the subject (in wavelengths).

The torso measurement result is shown in Fig. 7 which has axial length and . The human torso may be approximated by a superquadratic ellipse, given by (58) where

and are the axial lengths, is the exponent, and is the center coordinate. In order to show that the specific shape of the body can make a significant difference in BAN modeling for certain polarizations, we also consider simplified circular and elliptical models. For all models, relative permitwas chosen to match average propertivity ties of fat (15%) and muscle (85%) at 2.55 GHz [13]. Also, the and for and polarsource was placed at for polarization. Model parameters are ization and as follows: , and the Circular model: For the circular model, in order to have the radius is chosen to be same perimeter as the subject. and the Elliptical model: The elliptical model has same perimeter length as the measured torso, keeping constant. This leads to , .

Fig. 9. Model comparison of E in response to a -directed point source.

Superquadratic model: For the general superquadratic , and are chosen model to match the measured subject. at The electric field intensity as a function of angle and polarization is shown in 2.55 xGHz for polarization refers Figs. 8 and 9 respectively, where to -directed field in response to a -directed point current. The simulation shows that the propagation around the torso is nearly exponentially with distance (observation angle) and can be sensitive to the shape of the model, especially for cross polarization. Also, these figures shows some fluctuation (partial nulls) in the shadow region near the back of the torso which is explained by the interference of clockwise and counter-clockwise creeping waves.

IV. BODY AREA NETWORKS MEASUREMENTS This section describes measurements conducted in the anechoic chamber in Fig. 10(a) and in an open range, indicating that the proposed method provides reasonable accuracy.

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Fig. 11. Measurement and model comparison of E in response to a -directed point source. Fig. 10. Measurement of BAN channels: (a) anechoic chamber measurement with subject, (b) antenna for ^ polarization and (c) antenna for z^ and ^ polarizations.

A. Network Analyzer-Based System Measurements of the BAN channel were performed with a Rohde&Schwarz ZVB20 vector network analyzer connected to the antennas via 3 m instrument grade SMA cables (Mini-Circuits CBL-10FT-SMS+) and 20 dBm transmit power. During the measurements, the subject placed his hands over his head to reduce the influence of the arms. In the following study, one antenna was placed above the left hip of the subject and the other antenna starting above the right hip and moved in 1 cm increments around the waist toward the other antenna. B. BAN Antenna Fig. 10(b) and (c) depict the antennas that were used for the measurement, which are 1.5 cm monopoles. The monopoles were chosen to be short (approximately ) compared to the wavelength at 2.55 GHz, thus approximating a point transmit current and a point receive field sensor. The matching efficiency of the short monopoles is between 0.1 and 0.5 depending on the orientation relative to the body, but the maximum additional link loss of 20 dB does not significantly hinder the short range BAN measurements. Antennas were sewn onto small Velcro patches attached to an ordinary back-support Velcro band worn around the waist. To allow all combinations of the three transmit and receive polarizations to be measured, two antennas were constructed with right angle cable connections ( polarization) and two with straight connections ( and polarization). A long Velcro measuring tape was attached to the waist strap allowing the antennas to be positioned with 1 mm accuracy. When the antennas were very close ( 3 cm), accurate placement was difficult due to the overlap of the Velcro bands, so some variation from the ideal response is expected. The circumference of the subject from the two extreme points above the hip was 47 cm. The thickness of the clothes and Velcro band together was estimated as 0.5 cm. The distance between the body and the antenna can significantly influence the pathloss

and needs to be carefully determined. The additional displacement of the antennas from the body was 2.5 cm for oriented and 0.5 cm for and oriented monopoles. C. Anechoic Chamber and Outdoor Measurement A small anechoic chamber was constructed for performing on-body measurements having dimensions 2.0 m for the width and length and 2.2 m for the height. The floor was constructed as an open lattice of thin planks, allowing microwave absorber to be placed between the planks and a human subject to stand over the absorber. The walls, floor, and ceiling are covered with EPP-22 absorber material from Telemeter Electronic GmbH, having a normal reflection below 40 dB in the 2–4 GHz band. Due to budget constraints, only the center 1.5 m 1.5 m area of each surface was covered with absorber. Fig. 10(a) depicts the inside of the chamber with the human subject. As shown, the cables were oriented to hang parallel to the body, since having cables encircling the body had a strong influence on the measurements. Although channels were measured with a broadband 2 to 5 GHz sweep, only the results at 2.55 GHz are presented here. Outdoor measurements of the polarization were also carried out at the Jacobs University campus in a large open environment, where the nearest building from the measurement point is 70 m. Measurements were performed by covering the network analyzer with absorber to reduce the effect of reflections from the instrument. In addition, microwave absorber was placed on the ground around the subject. D. Measurement and Model Comparison Measurement results are now compared with the superquadratic model in Section III. Fig. 11 depicts the result for oriented transmitter and receiver (antenna perpendicular to body). The model is able to accurately predict the field decay with increasing separation, as well as the presence of a small dip just before the most extreme separation. Fig. 12 shows the result for oriented transmitter and receiver (antenna parallel to body), which exhibits very high shadowing. A floor is seen in the measured power in the strongly shadowed

EID AND WALLACE: ACCURATE MODELING OF BAN CHANNELS USING SURFACE-BASED MoM

Fig. 12. Measurement and model comparison of E in response to a z -directed point source.

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Fig. 14. Measurement and model comparison of E in response to a -directed point source.

comparison, which cannot predict the two initial decay regions with differing slopes. V. CONCLUSION

Fig. 13. Measurement and model comparison of E in response to a -directed point source.

region, which we believe is due to wall or floor reflections. Comparison of the floor with a simulated wall reflection assuming a 40 dB reflection coefficient for the wall shows reasonable agreement. Unfortunately, outdoor measurements of the same polarization did not improve the accuracy of fit of the model, which could be resulting from the absorber around the subject’s feet. These results reveal that performing BAN measurements for regions with strong shadowing are difficult, since reflections can easily overshadow on-body propagation mechanisms. Fig. 13 compares the measurement and model for a -directed source and sensor. Although an initial discrepancy is seen due to placement constraints of the antennas, the overall trend is well captured, with the exception of an unexpected dip in the shadow region. Fig. 14 shows the result for a cross-polarization measurement, which is expected to be more sensitive to body shape and may be important for sensors that can be placed at arbitrary angle. Although the fit is not perfect, the trend is quite similar, where both the measurement and model predict higher initial decay, followed by a region of more gradual decay, ending with a sharp drop. The circular cylinder model is also shown for

This paper has developed an improved propagation model for body area networks, where the body is modeled as an infinite cylinder having an arbitrary cross section. Expressions for the fields around the body in response to a point source are obtained using an efficient surface-based method-of-moments formulation. The method was validated by comparison to the closed-form model of a circular cylinder. Fields around the torso were simulated for three simple models indicating nearly exponential decay of the fields with distance on the body. Additionally, it was observed that some polarizations were sensitive to the shape of the body. The model was compared with on-body measurements performed in an anechoic chamber as well as an open field. Although reasonable agreement was obtained, results indicate that for certain polarizations, direct measurement is difficult in the shadow region due to weak reflections from chamber walls and other environmental effects. REFERENCES [1] J. Ryckaert, P. De Doncker, R. Meys, A. de Le Hoye, and S. Donnay, “Channel model for wireless communication around human body,” Electron. Lett., vol. 40, pp. 543–544, Apr. 2004. [2] T. Zasowski, F. Althaus, M. Stager, A. Wittneben, and G. Troster, “UWB for noninvasive wireless body area networks: Channel measurements and results,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, Zurich, Switzerland, Nov. 16–19, 2003, pp. 285–289. [3] A. Fort, C. Desset, J. Ryckaert, P. De Doncker, L. Van Biesen, and P. Wambacq, “Characterization of the ultra wideband body area propagation channel,” presented at the IEEE Int. Conf. on Ultra-Wideband, Leuven, Belgium, Sep. 5–8, 2005. [4] A. Alomainy, Y. Hao, X. Hu, C. Parini, and P. Hall, “UWB on-body radio propagation and system modelling for wireless body-centric networks,” IEE Proc. Commun., vol. 153, pp. 107–114, Feb. 2006. [5] A. Fort, F. Keshmiri, G. Crusats, C. Craeye, and C. Oestges, “A body area propagation model derived from fundamental principles: Analytical analysis and comparison with measurements,” IEEE Trans. Antennas Propag., vol. 58, pp. 503–514, Feb. 2010.

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[6] A. M. Eid, N. Murtaza, and J. W. Wallace, “Green’s function models and measurements for body area network (BAN) channels,” presented at the Proc. IEEE Int. Conf. on Wireless Information Technology and Systems (ICWITS’10), Honolulu, HI, Aug. 28–Sep. 3 2010. [7] Y. Leviatan and A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current model,” IEEE Trans. Antennas Propag., vol. 36, pp. 1026–1031, Jul. 1988. [8] E. Arvas and T. K. Sarkar, “RCS of two-dimensional structures consisting of both dielectrics and conductors of arbitrary cross section,” IEEE Trans. Antennas Propag., vol. 37, pp. 546–554, May 1989. [9] M. A. Al-Kanhal and E. Arvas, “Electromagnetic scattering from a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag., vol. 44, pp. 1041–1048, July 1996. [10] J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag., vol. 13, pp. 334–341, Mar. 1965. [11] M. Sadiku, Numerical Techniques in Electromagnetics. Boca Raton, FL: CRC Press, 1992. [12] Abramowitz and Stegun, Handbook of Mathematical Functions. Washington, DC: U.S. Dept. of Commerce, National Institute of Standards and Technology, 1965. [13] “Dielectric Properties of Body Tissues” Italian National Research Council, Institute for Applied Physics [Online]. Available: http://niremf.ifac.cnr.it/tissprop/

Ahmed M. Eid was born in Egypt in 1974. He received the B.S. degree in electrical engineering from Zagazig University, Egypt, in 1997, the M.S. degree in communication engineering from Cairo University, Egypt, in 2006, and the Ph.D. degree in electrical engineering from Jacobs University, Bremen, Germany, in 2010. From 2000 to 2007, he worked as a Vessel Traffic Management System (VTMS) Engineer for the Suez Canal Authority, and from 2007 to 2010 he was a Research Associate of the Wireless and Applied Electromagnetics Laboratory, Jacobs University Bremen. His research interests include antennas and microwave circuits, electromagnetic analysis and modeling, and microwave radar and sensing.

Jon W. Wallace (S’99–M’03) received the B.S. (summa cum laude) and Ph.D. degrees in electrical engineering from Brigham Young University (BYU), Provo, UT, in 1997 and 2002, respectively. From 2002 to 2003, he was with the Mobile Communications Group , Vienna University of Technology, Vienna, Austria. From 2003 to 2006, he was a Research Associate with the BYU Wireless Communications Laboratory. Since 2006, he has been an Assistant Professor of electrical engineering at Jacobs University, Bremen, Germany. His current research interests include MIMO wireless systems, physical-layer security, cognitive radio, and body-antenna interactions. Dr. Wallace is serving as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Exact SLF/ELF Underground HED Field Strengths in Earth-Ionosphere Cavity and Schumann Resonance Yuan-Xin Wang, Rong-Hong Jin, Member, IEEE, Jun-Ping Geng, Member, IEEE, and Xian-Ling Liang, Member, IEEE

Abstract—Electromagnetic wave radiation from a SLF/ELF underground horizontal electric dipole (HED) related to seismic activity is discussed. In order to estimate the effects on the electromagnetic waves associated with the seismic activity, SLF/ELF waves radiated from a possible seismic current source modeled as a electric dipole, are precisely computed by using a speeding numerical convergence algorithm. In 1999, Barrick proposed an algorithm, which was only suitable to solve the electromagnetic problems under the ideal electric conductor condition. To solve the problems under the non-ideal electric conductor condition, we have further developed Barrick’s method and proposed a speeding numerical convergence algorithm. The variations of the electromagnetic field along the propagation distance and the altitude, as well as the frequency during day and night are analyzed and calculated. It takes 0.5 minute to calculate the sum of the series by my algorithm, while it needs 15 minutes by calculating directly the sum of the series. Our algorithm is compared with the calculation directly the sum of the series algorithm, and two algorithms agree with each other very well. Therefore, our algorithm is correct. Schumann resonance is also verified. Index Terms—Earth-ionosphere cavity, Schumann resonance, seismic activity, speeding numerical convergence.

I. INTRODUCTION

E

ARTHQUAKE prediction researchers have discovered the abnormality of the electromagnetic radiation [1], [2] in the super low frequency (SLF) and extremely low frequency (ELF) bands before the earthquakes since 1970 s. Because the electromagnetic radiation often appears just before an earthquake occurs, it is an important tool for the short-term and imminent earthquake prediction. Therefore, many earthquake-prone countries such as Japan, Russia, America and China have carried out the theoretical and experimental study, but it is still at the exploratory stage. To further explore this problem and discover the rule of earthquake electromagnetic radiation as an indicator of earthquake occurrence, it is imperative that the propagation rules [3] of the electromagnetic wave be studied in SLF/ELF [4], [5] bands.

Manuscript received June 03, 2010; revised December 03, 2010; accepted January 15, 2011. Date of publication June 09, 2011; date of current version August 03, 2011. This paper was supported by the National Support Fund for Earthquake Prediction (2008BAC35B05), ‘973’ (2009CB320403), National Science Fund for Creative Research Groups (60821062), National Science and Technology of major projects (2011ZX03001-007-03), the National Science Foundation of Shanghai (10ZR1416600), Doctoral Fund of Ministry of Education of China (No. 20090073120033), Aerospace Key Fund, Aerospace Support Fund in China. The authors are with the Shanghai Jiao Tong University, Shanghai, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158952

In SLF/ELF bands, because of the absorption and attenuation of the stratum, wave can travel through the surface of the earth, and be received by a receiving apparatus on the ground. The electromagnetic wave can propagate to a distant place along the Earth-ionosphere cavity [6], [7], because the earth and the ionosphere are both good reflection walls. Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique. It is considered easy to understand and easy to implement in software. Since it is a time-domain method, solutions can simultaneously model a range of frequencies, not just provide single frequency solutions within a range of frequencies. Our method works also over a range of frequencies, but each frequency may be individually chosen. A description of the first fully 3-D global FDTD models is developed independently by Simpson [8]. The global FDTD model [9] of the Earth-ionosphere cavity has been used for a number of studies, including a propagation attenuation validation study. Three other groups have also developed 3-D FDTD models of the global Earth-ionosphere cavity [10], [11] as described in [12]. Previously, Simpson and Taflove used this latitude longitude FDTD Earth-ionosphere model to study EM propagation from electrokinetic currents occurring in the Earth’s crust below the epicenter of the Loma Prieta earthquake [13], [14]. It is curious that solutions to this seemingly simple exact canonical boundary value problem, excitation of a spherical cavity with non-ideal conducting walls, has not been catalogued in standard handbooks and cast in algorithmic form for numerical solution. After all, the Helmholtz vector wave equation is separable in spherical coordinates, its orthonormal bases being the Bessel, Legendre, and trigonometric functions of , , and . Why are no such simple but accurate solution estimates available for this problem? At higher frequencies this problem was cleverly circumvented beginning with G. N. Watson nearly 80 years ago, followed by seminal works of B. van, W. O. Schumann, and others, as summarized well by Wait [15]. In 1962, Wait proposed an asymptotic approximation approach that assumes , and the sphere size is large in terms of wavelength. Thus it is a high-frequency method, although Wait and others have shown that it can sometimes work surprisingly well when the above parameter is only slightly larger than unity. At the lower Schumann resonances [16], [17] and even below, however, these asymptotic approximations are clearly inadequate. Because the application is widespread in SLF/ELF bands, more and more scholars begin to study it. In 1999 Donald E. Barrick [16] proposed a spherical harmonic series algorithm of SLF/ELF vertical electric dipole

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(VED) on the ground in an ideal Earth-ionosphere cavity, which was able to obtain the solution to the electromagnetic fields problem. To study this problem in the non-ideal Earth-ionosphere cavity, Wang Yuan-xin, et al., studied the electromagnetic fields problem of SLF/ELF VED [17] and horizontal electric dipole (HED) [18] on the ground in 2007. The former studies are mainly restricted in the launch source located on the ground, and the receiving point is located on the ground or under the ground or the sea. But the launch source buried deeply under the ground is not considered, so, in this paper, we will discuss this problem in detail below. The electromagnetic fields can be excited by the launch source buried deeply under the ground and its base equations can be expanded to the spherical harmonic series in Earth-ionosphere cavity. The series include many items, and it would seem a simple matter to sum the series on a computer to obtain quantitative results. In fact, if we calculate directly the sum of the series item by item, it can require a prohibitively large number of items and the speed is also very slow, even though the series remain absolutely convergent. The approach we employ below subtracts and adds appropriate identical terms to the original exact series. The subtraction accelerates significantly its numerical convergence. The added terms sum to simple closed-form expressions. The proposed algorithm, namely speeding numerical convergence modifies the summation item by item algorithm, and decreases the series items and also improves the speed. It is known by the lateral wave propagation theories of R.W.P King [19] et al. that the electromagnetic field strength excited by an underground HED [20], [21] is bigger than that excited by an underground VED. Therefore, the underground radiation source will be idealized as a HED. II. ELECTROMAGNETIC FIELDS IN EARTH-IONOSPHERE CAVITY

Fig. 1. Earth-ionosphere cavity in spherical coordinates.

(2)

(3) Where the potential functions and following differential equations [16]

should satisfy the

(4) (5) Where , a. In Area g, when the earth and the ionosphere do not exist, the potential functions of the primary field excited by the underground HED can be expressed as

A. Model

(6)

The Earth-ionosphere geometry is representable to the first order as two concentric spheres, as shown in Fig. 1, so we select to solve this problem. the spherical coordinate system denotes the earth, and Area Area the air space, and Area the ionosphere. The earth and the ionosphere are both non-ideal reflection walls with certain and surface impedances, which are separately recorded as . is the radius of the earth, and , and are respectively the distances of the dipole, the ionosphere and observation point from the sphere center. The excitation source is idealized as , which is placed in the a HED, whose current moment is , , . position of

(7) where

(8)

B. Spherical Harmonic Series Expansion of Electromagnetic Fields The electromagnetic fields in the cavity can be expressed by two potential functions [16]

(9) and (10)

(1) where

.

WANG et al.: EXACT SLF/ELF UNDERGROUND HED FIELD STRENGTHS IN EARTH-IONOSPHERE CAVITY

When the earth, the air and the ionosphere exist at the same time, the potential functions of the secondary disturbance field in the cavity will be produced due to the reflection of the boundaries, and their general expressions can also be expanded according to the spherical harmonic functions. They may be rewritten as

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(18)

(11) (12)

where is the second-kind Hankel function, which is expressed as the wave reflected from the surface of the earth to the stratum. It is defined by

(19)

(13) The total potential functions should be expressed as follow:

(14)

(20)

(15)

Substituting (14) and (15) into (1)–(3), we get

(16)

(17)

(21)

In Area a, two propagation waves should exist, first up and then down, because the earth and the ionosphere both exist. Their potential functions may be expressed as follows:

(22)

(23)

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where is expressed as the wave reflected from the ionosphere to the surface of the earth, which is defined by (24) . where Substituting (22) and (23) into (1)–(3), it can be obtained readily

In the surface of the earth, the electromagnetic fields should satisfy the following impedance boundary conditions, which are expressed as (30) (31) (32) (33) From (30)–(33), the coefficients of (16)–(21) and (25)–(28) may be solved as (34) (35)

(25)

(36) (37)

(38) (26)

(39) where

(27)

(28) where and are the Legendre functions of is expressed in terms of cylindrical half-order order . Bessel functions of the first kind, while is its corresponding outgoing first-kind Hankel function, and is the second-kind Hankel function. The Bessel function forms employed here satisfy the following differential equation [16]

(29)

and

WANG et al.: EXACT SLF/ELF UNDERGROUND HED FIELD STRENGTHS IN EARTH-IONOSPHERE CAVITY

III. SPEEDING NUMERICAL CONVERGENCE ALGORITHM To (27), it would seem a simple matter to sum the series on a computer to obtain quantitative results. In fact, summation item by item can require a prohibitively large number of items, even though the series remain absolutely convergent. The situation is not improved in the low frequency limit (below Schumann resonance). The reason is that the relevant radii appearing in the Bessel-function arguments here are all very close to each other. The approach employed below subtracts and adds appropriate identical series to the original exact series. The added series have a simple closed form that is given by the Legendre generating function identities. The individual items of the subtracted series reduce to the same respective items of the solution series for the large , so their differences become small even though the individual items are not small. The speeding numerical convergence algorithm of the electric can be expressed as follows field component (40)

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(43) (44) (45) where

When , from the series expression of the circular cylinder function near the zero point, we can achieve (46) (47) where . From (42)–(47), it is seen that

Since the latter item in (40) is an analysis expression which is easy to be calculated and does not need to be summed term by term, while the amplitude of rapidly reduces with the increase of , the series will rapidly converge, and as a result, the fields will be calculated easily with better calculation precision. It can achieve the sufficient accuracy and convergence by taking 200 items in the series, while it needs 1000 items by the calculation directly the sum of the series algorithm. It takes 0.5 minute to calculate the sum of the series by my algorithm, while it needs 15 minutes by calculating directly the sum of the series algorithm. So it can be seen that the speed of speeding numerical convergence algorithm has been enhanced thirty times. A. Approximation Expressions of Electromagnetic Fields Items to be subtracted and added are obtained by observing that the Bessel functions rapidly approach simple powers of or when . For example, at , this happens , while for , it is met when . First, we for look at whether we can obtain the sum of the series when the earth and the ionosphere are both the ideal electric conductor, and both equal to zero [16]. where , the electric field component of (40) can When be expressed as

(48)

(49)

(41) where

where (42)

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because (50) Substituting (46)–(50) into (41), the following equation may be derived

(53) where

B. Numerical Generation of Legendre Functions (51) The following sums of the infinite series can be obtained by the literature [16]

and associated Legendre funcThe Legendre functions versus argument are easily generated by recurtions rence over the order . The polar angle is related to great circle . Bedistance between source dipole and observer as cause recurrence upward over is stable, it is conveniently implemented starting at known values for , 1, and 2 [16]. Define

Then (52) where

Substituting (52) into (51), it is obtained readily

With the recurrence relation

and

With the recurrence relation

C. Numerical Generation of Bessel Functions Separate generation of the required Bessel functions within lower and upper spans over the order is necessary. Over the lower span of the actual Bessel functions are calculated by the recurrence according to the following guidelines. To higher

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, products of Bessel functions must be generated because the functions themselves exceed the range of machine precision [16]. 1) Lower order

(54) 2) Higher order . At some value before the preceding method exceeds machine precision, one must switch to another attack. We calculate directly the products of the Bessel functions required in the series items. These are obtained from the Debye’s asymptotic expansions for Bessel functions whose orders and arguments both may be large. First the following functions are defined [16]

The Bessel functions need to be calculated by the following formula [22]

(55) is very small, It is worth noting that when is large, is very large, and tends and even tends to be zero, while to reach up to infinity, and both of them have exceeded the computer’s precision. However, it should be noted that their product is still within the computer’s precision. Therefore, we must caland . culate the product of Due to , Bessel and Hankel functions equal approximately to [22]

(56)

Fig. 2. Variation of the electric field component along the propagation distance during day.

IV. CALCULATION RESULTS AND DISCUSSION The calculation results have been obtained under the nonideal electric conductor condition. In this case, the altitude of during day, and the low ionosphere is taken as during night. The radius of the earth is . , and the ionosphere The ground conductivity is . The distance of the electric conductivity is dipole from the surface of the earth is d, and its current moment . is It shows the variation of the electric field component along the propagation distance during day in Fig. 2, where the solid line represents the result calculated by the speeding numerical convergence algorithm, and the dashed line represents the result obtained by the calculation directly the sum of the series algorithm. The calculations are both under the condition that the dipole is located at 10 km under the ground, and the receiving point is located on the surface of the earth, and the frequency is , . It can be seen from Fig. 2 that two algorithms agree well. We may clearly see that they can correctly show the “interference” phenomenon of two waves propagating along the short great-circle way and the long great-circle way when they pass through the antipode to the receiver after 10000 km, because the attenuation ratio of the SLF wave propagation in the cavity is very small. along the The variation of the electric field component propagation distance during day and night is given in Fig. 3, which is calculated by the speeding numerical convergence algorithm. It is calculated when the dipole is located at 30 km under the ground and the receiving point is located on the sur, . face of the earth, and the frequency is It can be observed from Fig. 3 that the results during day and night are similar to each other, and their difference is less than 2 dB. The absorption loss during night is less than it during day. The fields between the earth and the ionosphere are “standing wave” because the length of the wave and the perimeter of the earth are comparable.

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Fig. 5. The noise power spectrum which Larsen and Egeland measured in 1968.

Fig. 3. Variation of the electric field component along the propagation distance during day and night.

Fig. 6. Variation of the electric field component with frequency during day and night.

Fig. 4. Variation of the electric field component along the altitude during day.

It shows the variation of the electric field component along the altitude during day in Fig. 4. It is the result calculated by the speeding numerical convergence algorithm. The calculation condition specifies that the dipole be located at 30 km under the ground, while the receiving point should be located between the earth and the ionosphere, and the frequencies are , , , . It is demonstrated in Fig. 4 that as the altitude increases, the increases gradually and then deelectric field component creases slowly. A kind of resonance phenomenon is called as the Schumann’s resonance [21], and the Schumann’s resonance has been confirmed by the atmospheric noise power spectrum of ELF band. Fig. 5 is the noise power spectrum measured by Larsen and Egeland in 1968, where A was obtained by the record of the 30 seconds, and B and C were obtained by the record of the two min-

utes. It may be clearly seen that there is the “resonance” phe, and . nomenon nearby with the freThe variation of the electric field component quency during day and night is given in Fig. 6, which is calculated by the speeding numerical convergence algorithm. It is calculated when the dipole is located at 30 km under the ground and the receiving point is located on the surface of the earth, , . It can be seen from Fig. 6 that the wave crests during night are steeper than those during day, and the changes of the resonance frequencies during day and night are small. The resonance equal approxifrequencies of the electric field component mately to 7.2 Hz, 14.5 Hz and 21.5 Hz during day, and they are 8.2 Hz, 15.4 Hz and 22.5 Hz during night. Moreover, the resonance curves are smooth due to the absorption loss. V. CONCLUSION In order to theoretically evaluate the SLF/ELF emissions in the Earth-ionosphere cavity above seismic zones, we presented theoretical calculations of the SLF/ELF waves radiated from an

WANG et al.: EXACT SLF/ELF UNDERGROUND HED FIELD STRENGTHS IN EARTH-IONOSPHERE CAVITY

underground HED. Former advanced work by Barrick focused on the electromagnetic waves excited by a SLF/ELF VED on the ground in the Earth-ionosphere cavity under the ideal electric conductor condition, while our study focused on the electromagnetic waves radiated from a SLF/ELF HED in the Earth-ionosphere cavity under the non-ideal electric conductor condition. We have quantitatively discussed the amplitude of the electromagnetic waves giving detectable SLF/ELF emissions directly radiated from the underground HED. Under the non-ideal electric conductor condition, we have further developed Barrick’s method and proposed a speeding numerical convergence algorithm. Our result of the calculation coincides with it of the calculation directly the sum of the series and it has verified the correct of our algorithm. Our discussions support the suggestions that our algorithm is very effective to solve the problems of the electromagnetic waves related to the seismic-related SLF/ELF emissions. Schumann resonance is also verified. REFERENCES [1] B. Hisatoshi, “Investigation of electromagnetic radiation associated with earthquakes observational results related to earthquakes,” Tokai Daigaku Sogo Kagaku Gijutsu Kenkyujo Kenkyukai Shiryoshu, vol. 19, pp. 67–74, 2000. [2] S. Adachi, “Signal processing and sonification of seismic electromagnetic radiation in the ELF band,” IEICE Trans. Fundam. Electron Commun. Comput. Sci., vol. E84, pp. 1011–1016, 2001. [3] A. C. Fraser-Smith and P. R. Bannister, “Reception of ELF signals at antipodal distance,” Radio Sci., vol. 33, pp. 83–88, 1998. [4] P. Bannister, “Far-field Extremely low frequency (ELF) propagation measurements,” IEEE Trans. Commun., vol. 22, pp. 468–474, 1974. [5] A. Meloni, P. Palangio, and A. C. Fraser-Smith, “Some characteristics of the ELF/VLF radio noise measured near L’Aquila, Italy,” IEEE Trans. Antennas Propag., vol. 40, pp. 233–236, 1992. [6] P. Bannister, “Some note on ELF earth-ionosphere waveguide daytime propagation parameters,” IEEE Trans. Antennas Propag., vol. 27, pp. 696–698, 1979. [7] V. K. Tripathi, C. L. Chang, and K. Papadopoulos, “Excitation of the earth-ionosphere waveguide by an ELF source in the ionosphere,” Radio Sci,, vol. 17, pp. 1321–1326, 1982. [8] J. J. Simpson and A. Taflove, “Three-dimensional FDTD modeling of impulsive ELF antipodal propagation and Schumann resonance of the earth-sphere,” IIEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 443–451, Feb. 2004. [9] J. J. Simpson, “Current and future applications of 3-D global earth-ionosphere models based on the full-vector Maxwell’s equations FDTD method,” Surveys Geophys., vol. 30, no. 2, pp. 105–130, 2009. [10] H. Yang and V. P. Pasko, “Three-dimensional finite-difference timedomain modeling of the Earth-ionosphere cavity resonances,” Geophys. Res. Lett., vol. 32, p. L03114, 2005. [11] H. Yang and V. P. Pasko, “Three-dimensional finite difference time domain modeling of the diurnal and seasonal variations in Schumann resonance parameters,” Radio Sci., vol. 41, p. RS2S14, 2006. [12] J. J. Simpson and A. Taflove, “A review of progress in FDTD Maxwell’s equations modeling of impulsive sub-ionospheric propagation below 300 kHz,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1582–1590, Jun. 2007. [13] J. J. Simpson and A. Taflove, “Electrokinetic effect of the Loma Prieta earthquake calculated by an entire-Earth FDTD solution of Maxwell’s equations,” Geophys. Re. Lett., vol. 32, p. L09302, 2005. [14] J. J. Simpson, “Global FDTD Maxwell’s equations modeling of electromagnetic propagation from currents in the lithosphere,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 199–203, 2008. [15] J. R. Wait, Electromagnetic Waves in Stratified Media. Oxford: Pergamon Press, 1962. [16] D. E. Barrick, “Exact ULF/ELF dipole field strengths in the Earth-ionosphere cavity over the Schumann resonance region: Idealized boundaries,” Radio Sci., vol. 34, pp. 209–227, 1999. [17] Y. X. Wang, W. S. Fan, and W. Y. Pan et al., “Spherical harmonic series solution of fields excited by vertical electric dipole in earth-ionosphere cavity,” Chinese J. Radio Sci., vol. 22, no. 2, pp. 204–211, 2007.

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[18] Y. X. Wang, Q. Peng, and W. Y. Pan et al., “The fields excited by SLF/ELF horizontal electric dipole in earth-ionosphere cavity,” Chinese J. Radio Sci., vol. 22, no. 5, pp. 728–734, 2007. [19] R. W. P. King, O. Owens, and T. T. Wu, Lateral Electromagnetic Wave. Berlin, Germany: Springer-verlag, 1992. [20] K. Li, Y. Lu, and M. Li, “Approximate formulas for lateral electromagnetic pulses from a horizontal electric dipole on the surface of one-dimensionally anisotropic medium,” IEEE Trans. Antennas Propag., vol. 53, pp. 933–937, Mar. 2005. [21] W. Y. Pan, Long Wave Beyond Long Wave Extremely Long Wave Propagation, C. Du, Ed. New York: Electric Scientific and Technical Univ. Press, 2004. [22] A. Jeffrey, Table of integrals, series, and products. New York: Academic, 1980. Yuan-Xin Wang was born in Shandong, China, in 1978. He received the M.S. degree in electromagnetic fields and microwave technology from the Chinese Research Institute of Radio wave Propagation, Qingdao, in 2007. He is currently working toward the Ph.D. degree in electromagnetic fields and microwave technology at Shanghai Jiao Tong University, Shanghai, China. He is an Engineer at the Chinese Research Institute of Radio Wave Propagation. His current research interests include SLF/ELF wave propagation and applications, antenna theory design and electromagnetic fields theory.

Rong-Hong Jin (M’93) was born in Jiangsu, China, in 1963. He received the B.S. degree in 1983, M.S. degree in 1986, and Ph.D. degree in 1993, all from the Shanghai Jiao Tong University, Shanghai, China. He is a Professor in the Department of Electronic Engineering, Shanghai Jiao Tong University. His current research interests include modern communication antenna theory and design, digital beam forming technology, beam forming antenna, and numerical value analysis technology.

Jun-Ping Geng (M’03) was born in Shanxi, China, in 1972. He received the B.S. degree in 1996, M.S. degree in 1999, and Ph.D. degree in 2003, all from the Northwestern Polytechnical University, Xi’an, China. He is a Professor in the Department of Electronic Engineering, Shanghai Jiao Tong University. His current research interests include Mobile terminal multi-antenna, smart antennas and electromagnetic compatibility.

Xian-Ling Liang (M’11) received the B.S. degree in electronic engineering from Xi’an University of Technology, Xi’an, China, in 2002 and the Ph.D. degree in electric engineering from Shanghai University, Shanghai, China, in 2007. From 2007 to 2008, He worked as a Postdoctoral Research Fellow at the Institut National de la Recherche Scientifique (INRS), Université du Quebec, Montreal, Canada. In December 2008, he joined the Department of Electronic Engineering, Shanghai Jiao Tong University, as a Lecturer. He has authored or coauthored over 80 papers in refereed journals and conference proceedings. His current research interests include microwave and millimeter-wave antennas, integrated antennas, wideband and active antennas, phased arrays, and etc. Dr. Liang received the Award of Shanghai Municipal Excellent Doctoral Dissertation in 2008, the Nomination of National Excellent Doctoral Dissertation in 2009, and the Best Paper Award presented at the 2010 International Workshop on Antenna Technology: Small Antennas, Innovative Structures and Materials.

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Communications Controlling Resonances for a Multi-Wideband Antenna by Inserting Reactive Components Hyengcheul Choi, Sinhyung Jeon, Junghwan Yeom, Oul Cho, and Hyeongdong Kim

Abstract—A novel method is proposed to control the resonance frequencies of an antenna for multi-wideband operation. The parallel addition of reactive components to an antenna shifts the resonance frequencies without increasing the size of the antenna. These resonance frequency shifts can be analyzed by observing the voltage difference via an approach based on an open-circuit transmission line. The designed antenna has 180 MHz (940–1120 MHz) and 630 MHz (1970–2600 MHz) bandwidths under with a high radiation efficiency at dual-band. The lower and the higher bands of the antenna provide coverage for Global System for Mobile communications 900 (GSM900), Universal Mobile Telecommunications System 2100 (UMTS2100), Wireless Broadband (WiBro) and Bluetooth.

VSWR = 2 5 : 1

Index Terms—Multi-band antennas, printed circuit antennas and resonance perturbation.

I. INTRODUCTION Planar inverted-F antennas (PIFAs) for mobile handsets have been frequently investigated in order to reduce antenna size as well as to provide multi-band operation [1]. However, since the low height between the antenna patch and the ground plane narrows the impedance bandwidth [2], a monopole antenna placed at the side of the ground plane has been receiving much interest from antenna engineers, especially when the slim phone is the general design tendency. Usually, a fundamental resonance (at f1 : The first resonance frequency) of a monopole antenna is generated when its length is a quarter wavelength (1 =4). Since the high-order resonance frequencies of a monopole antenna are determined at multiples of f1 , an additional branch structure is utilized for multi-band operation [3]. The branch structure provides a multi-band operation; however, it does require extra space within mobile handsets. In order to obtain multi-band operation without using branch structure, a high-order resonance should be utilized. The high-order resonance frequencies (fn , n = 2; 3; 4; . . .) can be roughly evaluated by multiplying f1 by the resonant mode index (n), where n is an odd and even integer at resonances (i.e., series resonances) and anti-resonances (i.e., parallel resonances), respectively. Series resonance occurs at frequencies where the input reactance changes from capacitive to inductive, and parallel resonance occurs in the opposite case, from inductive to capacitive. Generally, the impedance bandwidth is wider at a series resonance than at a parallel resonance; thus, a series resonance is utilized for an antenna design. Two design techniques Manuscript received November 16, 2009; revised November 25, 2010; accepted December 10, 2010. Date of publication June 07, 2011; date of current version August 03, 2011. This work was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2009-C1090-0902-0003). The authors are with the Department of Electrical and Computer Engineering, Hanyang University, Seoul 133-791, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2158944

have been investigated in order to use single monopole antenna for multi-band mobile handsets because the third resonance frequency (f3 ) is generally fixed at three times the first resonance frequency (f1 ). One method switches the electrical length of a monopole antenna by inserting a chip inductor, which acts like a low pass filter, into the middle of the monopole antenna [4]. The other method is to control f3 by utilizing inter-coupling capacitance and helical inductance [5]. The antennas using the above-mentioned methods actually do not provide enough bandwidth for GSM900 around f1 , and this was the phenomenon that prompted research in the field of multi-wideband operation antenna. It is well known that the impedance bandwidth of an antenna is limited by its size [6], and it is difficult to provide a wide bandwidth in case of mobile handsets due to the very small size of the antenna. The conventional papers concerning multi-band antenna designs utilize return loss or voltage standing wave ratio (VSWR) for verifying the design mechanism, but inspecting the input reactance in order to obtain the insight of a multi-wide band resonance mechanism may be more helpful. Antenna engineers cannot considerably improve the radiation characteristic of mobile handsets through only the antenna design because the chassis of a mobile handset is primarily what contributes to radiation [7], and the dimensions of the chassis are nearly the same in most cases. Therefore, it may be better to provide bandwidth extension in antennas by controlling input reactance rather than input resistance (i.e., radiation). For input reactance curves, slopes at series resonances (i.e., f1 and f3 ) indicate the bandwidth [8]. Note that slopes at series resonances are dependent on their distance from the adjacent parallel resonances (i.e., f2 and f4 ). A good example is a small meander antenna of narrow bandwidth that has a short distance between f1 and f2 . Therefore, it is important to control parallel resonances as well as series resonances when designing multi-wideband antennas for mobile handsets. Reference [9] presents the controlling resonances of a loop- type antenna for wide bandwidth. This communication proposes a novel method involving the insertion of reactive structures into a loop antenna in order to control resonances for wide multi-band operation without changing the size of the antenna. The shifts in resonance frequency are evaluated by observing the voltage difference between connecting points of reactive structures. This communication shows the voltage distribution of an antenna conceptually by using an open-circuit transmission line model because the operation mechanism of an antenna can be explained by employing such a model [10]. The reactive components used in the proposed method are helix inductance and gap capacitance, and whether inductance or capacitance components are used determines the variation direction of the resonance frequencies. This dual property is not shown in papers [4] and [11] about an antenna using a reactive component. The proposed antenna occupies a small volume of 18 mm 2 17:5 mm 2 1:6 mm including a 5 mm clearance between the antenna and the system circuit board (i.e., ground) of dimension 40 mm 2 100 mm. The measured bandwidths of the proposed antenna, which are less than the VSWR of 2.5:1, are 180 MHz (940–1120 MHz) and 630 MHz (1970–2600MHz). The resonance frequencies and bandwidths in this communication are a little higher and wider than the requirements for GSM900/UMTS2100/WiBro/Bluetooth operations considering the resonance shifts by the handset case and battery [12].

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Fig. 1. Overall view of the reference antenna.

Fig. 3. (a) Reference antenna added with the gap capacitor and (b) reference antenna added with the helix inductor.

Fig. 4. Simulated VSWR of the antennas in Figs. 1, 2 and 3.

II. ANTENNA DESIGN A. Reference Antenna

Fig. 2. (a) Three-dimensional view and (b) detailed top view of the proposed antenna.

Fig. 1 shows the reference antenna that was installed on a printed circuit board (PCB). A copper plane of size 40 mm 2 100 mm on PCB was used for a ground plane and FR4 substrate ("r = 4:4) without the ground plane was used for the antenna installation. As shown in Fig. 1, the antenna feed point is set at one corner of the ground plane, which is the optimal feed position of an antenna on a rectangular ground plane [13]. The reference antenna consists of meandering metal lines in order to reduce antenna size, and it has a symmetrical structure to insert helix and gap without changing the overall antenna geometry.

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TABLE I RESONANCE FREQUENCIES AND BANDWIDTHS OF THE SIMULATED ANTENNAS

=first resonance frequency (first series resonance) =third resonance frequency (second series resonance) Bandwidth=relative bandwidth under VSWR = 2:5 f f

Fig. 6. (a) Open-circuit transmission line of a monopole antenna and b) normalized voltage distribution at open-circuit transmission line.

Fig. 7. VSWRs of reference and proposed antennas.

Fig. 5. Input reactances of the antennas in Figs. 1, 2 and 3.

B. Proposed Antenna The proposed antenna differs from the reference antenna by the addition of two reactances with helical and gap structures. As shown in Fig. 2, the helical structure is a type of inductor and is inserted between the input and middle points of the reference antenna. Metallic lines on both the bottom and the top are connected holes for realizing a rectangular helix of three turns. Inductance of the helix can be tuned by varying either the length of the lines or the number of turns. The gap structure is equivalent to a capacitor and is placed between the input and end points of the reference antenna. Capacitance is also adjusted by varying the gap space.

Selecting reactive components and their positions is important for controlling the resonances in this study. When inserting a reactive component into an antenna, the resonance frequencies are perturbed. There is a dual relationship with resonance frequency shifts whether inductance or capacitance is used. The degree of perturbation can be controlled by choosing positions of the reactive components and the amount of reactance. This method is different from a common matching network, which is at antenna input circuit and simply controls input impedance rather than directly controlling the resonance mode of the radiator. C. Simulation Results In order to examine the operation mechanism of the proposed antenna, we consider two additional antennas that have only gap capacitance, as in Fig. 3(a), or only helical inductance, as in Fig. 3(b). In others words, four types of antennas, a reference antenna (Fig. 1), a reference antenna with gap capacitance (Fig. 3(a)), a reference antenna with helical inductance (Fig. 3(b)) and the proposed antenna (Fig. 2), are simulated and analyzed in order to understand the operation mechanism of

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Fig. 8. Simulated and measured radiation patterns in the x-y, z-x and y-z planes at 1020 MHz and 2130 MHz.

the proposed antenna. The simulated VSWR data for the four antenna types are shown in Fig. 4 and are summarized in Table I. As shown in Table I, the resonance frequencies change by the addition of reactance. There are certain rules for the frequency shifting behavior. When the gap capacitance is inserted into the reference antenna, the resonance frequencies decrease for both f1 and f3 , while the resonance frequencies increase for both f1 and f3 when the helix inductance is added. However, the designed antennas have different bandwidth changes for f1 and f3 when the gap capacitance is added. This

phenomenon is caused by non-uniform resonance shifts that are also shown at split ring resonator (SRR) [14]. This feature aids in the wide multi-band characteristic of the proposed antenna. In the simulation, the proposed antenna provides both 170 MHz (940–1110 MHz) and 540 MHz (1945–2485 MHz). This shows a remarkable improvement compared to the bandwidth of the reference antenna at f3 . Note that the bandwidth at f3 is extended and the frequency of f3 is also decreased without increasing the size of the antenna while the resonance characteristics at f1 are almost fixed.

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III. RESONANCE FREQUENCY SHIFT BY REACTANCE When an antenna is structurally disturbed, observing the input reactance rather than the VSWR is more helpful in understanding the resonances of an antenna because the input reactance curve shows both parallel as well as series resonance frequencies. As shown in Fig. 5, adding gap capacitance decreases the resonance frequencies [15], but the amount of frequency shift for each resonance is different. When this phenomenon is utilized in antenna design, we can obtain a lower resonance frequency with a narrow bandwidth at f1 as well as a lower resonance frequency with a wide bandwidth at f3 . Note that these results are due to large shifts in f2 and f3 and small shifts in f1 and f4 . For the case using helical inductance, the higher resonance frequencies with wide bandwidths at f1 and f3 can be predicted by observing the change in the resonance frequencies as shown in Fig. 5. It is shown in Figs. 4 and 5 that the relative position of parallel resonances, such as f2 and f4 , affects the bandwidths of the f1 and f3 resonances. If the two different effects of adding capacitance and inductance are applied simultaneously to the reference antenna, a lower resonance frequency with a wide bandwidth at f3 is obtained while the characteristic of f1 remains almost fixed. Based on the above observation, it is believed that the resonance frequencies can be easily controlled by simply adding reactance in the appropriate positions. The unique non-uniform shifting property of resonance frequencies can be effectively utilized in order to obtain multi-wideband antenna. It is useful to observe the voltage difference between the two points where reactive elements are connected in order to utilize this non-uniform shifting property. The amount of resonance perturbation is proportional to the voltage difference between the two points. The resonance frequency of a large voltage difference is greatly affected while the resonance frequency of a small voltage difference is either not affected or is nearly fixed. Since it is difficult to calculate the voltage distribution of a complex antenna structure, the proposed monopole antenna is replaced with an open transmission line in this communication. The intuitive analysis using a simple equivalent model provides insight into how to control resonances of an antenna using the proposed method. For simplification, the connecting points are chosen from three points such as the starting, middle and ending points that provide the inserting points of the gap and helix in the proposed antenna. Fig. 6 shows the normalized voltage distributions on an open-circuit transmission line for f1 , f2 , f3 and f4 . Open boundary condition and different wavelength of each resonance frequency create maximum voltage at ending point of transmission line and different voltage distribution according to each resonance. Observing the voltage at starting point of transmission line also gives us the simple reason why input impedance is low and high at series resonances (f1 and f3 ) and parallel resonances (f2 and f4 ) respectively. Observing voltage distributions raises the prospect of resonance shifting behaviors by inserting reactive component. For example, antenna resonance does not shift when voltage difference between an added component is zero. As there are no stored energy and current at the inserted component, the component is electrically meaningless. On the other hand, inserting reactive component within large voltage difference greatly changes resonance frequency because of much perturbation of electric and magnetic stored energy. Based on this view, the reactance added between the starting and ending points will change f3 , while f4 will remain almost fixed. In the case of reactance added between the starting and middle points, f4 will change largely, while f3 will shift slightly. When, as in the proposed antenna, reactive components are inserted to the reference antenna, f3 is decreased and f4 is increased according to this property. In other words, the reactance slope variation at f3 is smoothed and the resonance of f3 is lowered as shown in Fig. 5(b). Thus, the shifting behaviors of f3 and f4 contribute to the multi-wideband operation. By locating the appropriate

Fig. 9. Total radiation efficiency of the proposed antenna.

reactance between the proper two points, we can selectively control the resonance frequencies. IV. RESULTS The simulated and measured VSWRs of the proposed antenna are presented in Fig. 7. The measurement data have a good agreement with the simulation data, and the measured bandwidths (under VSWR = 2:5 : 1) are 180 MHz (940–1120 MHz) and 630 MHz (1970–2600MHz). The proposed antenna satisfied the impedance bandwidths required for GSM900/UMTS2100/WiBro/Bluetooth operations in consideration of the usual slight decrease in resonance frequencies due to dielectric phone cases. Fig. 8 shows co- and cross-polarization patterns at the resonance frequency of dual-band (1020 MHz and 2130 MHz) in the x-y plane (i.e., azimuthal plane), z-x plane (i.e., elevation plane 1) and y-z plane (i.e., elevation plane 2). Based on the three introduced radiation patterns, we can roughly estimate an overall radiation pattern. These radiation patterns show that the proposed antenna has almost linear polarization. Cross polarization at 1020 MHz (f1 ) is weaker than that at 2130 MHz (f3 ) because low frequency radiation is affected by a handset ground that generates monopole-like radiation patterns. Since antenna polarization is deformed by multipath propagation in an urban environment [16], radiation efficiency is more important than polarization in handset antenna design. As shown in Fig. 9, the total radiation efficiencies measured are 78% and 92% at the two resonances respectively, and over 67% in operation bands. This total radiation efficiency is three-dimensionally measured in an anechoic chamber by 15-degree sampling. As total radiation efficiency includes antenna radiation efficiency and reflection efficiency, these data show that the proposed antenna can be applied to practical multi-band mobile handsets. V. CONCLUSION This communication proposes a novel method for designing a monopole antenna for multi-band operation. The proposed method not only extends the operation bandwidth of the upper band, but also decreases its operation frequency while maintaining resonance characteristics in the lower band. In addition, this method does not increase the size of the antenna because it only requires the addition of reactive components. The operation mechanism of the proposed method can be understood by observing the voltage distribution of an open- circuit transmission line model for a monopole antenna. The resonance perturbations by the reactive components are expected to be proportional to the voltage difference between connecting points of the reactive components. A variety of antennas can be designed using resonance disturbance by adding reactive components because there are many

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 8, AUGUST 2011

choices in the type and position of reactive components. The proposed method has been successfully applied in designing monopole antenna for a mobile handset satisfying GSM900/UMTS2100/WiBro/Bluetooth requirements.

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Band-Notched UWB Antenna Incorporating a Microstrip Open-Loop Resonator James R. Kelly, Peter S. Hall, and Peter Gardner

ACKNOWLEDGMENT The authors would like to thank RadiNa Inc. Ltd. in Korea and the Brain Korea 21 project for manufacture and measurement support.

REFERENCES [1] K. L. Wong, Planar Antennas for Wireless Communications. Hoboken, NJ: Wiley, 2003, pp. 26–53. [2] Z. N. Chen, Antennas for Portable Devices. Chichester, U.K.: Wiley, 2007, p. 142. [3] P. L. Teng and K. L. Wong, “Planar monopole folded into a compact structure for very-low-profile multiband mobile-phone antenna,” Microw. Opt. Technol. Lett., vol. 33, pp. 22–25, Jan. 2002. [4] S. H. Yeh and K.-L. Wong, “Compact dual-frequency PIFA with a chipinductor-loaded rectangular spiral strip,” Microw. Opt. Technol. Lett., vol. 33, pp. 394–397, May 2002. [5] H. Choi and H. Kim, “Dual-band chip antenna design using intercoupling capacitance,” Microw. Opt. Technol. Lett., vol. 51, pp. 1467–1470, Mar. 2009. [6] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, p. 672, May 1996. [7] P. Vainikainen, J. Ollikainen, O. Kivekas, and K. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [8] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1298–1324, Apr. 2005. [9] H. Choi, S. Jeon, S. Kim, and H. Kim, “Controlling resonance frequencies in antennas to achieve wideband operation,” Electron. Lett., vol. 45, pp. 716–717, Jul. 2009. [10] R. Schmitt, Electromagnetics Explained: A Handbook for Wireless/RF, EMC and High-Speed Electronics. Boston, MA: Newnes, 2002, pp. 229–230. [11] G. K. H. Lui and R. D. Murch, “Compact dual-frequency PIFA designs using LC resonators,” IEEE Trans. Antennas Propag., vol. 49, pp. 1016–1019, Jul. 2001. [12] S. Dong-Uk and P. Seong-Ook, “A triple-band internal antenna: Design and performance in presence of the handset case, battery, and human head,” IEEE Trans. Electromagn. Compat., vol. 47, pp. 658–666, Mar. 2005. [13] J. Rahola and J. Ollikainen, “Optimal antenna placement for mobile terminals using characteristic mode analysis,” in Proc. EuCAP, Nice, France, Nov. 2006, pp. 1–6. [14] M. Makimoto and S. Yamashita, Microwave Resonators and Filters for Wireless Communication Theory, Design and Application. New York: Springer, 2001, pp. 84–106. [15] M. Sagawa, K. Takahashi, and M. Makimoto, “Miniaturized hairpin resonator filters and their application to receiver front-end MIC’s,” IEEE Trans. Microw. Theory Tech., vol. 37, pp. 1991–1997, Dec. 1989. [16] V. Pathak, S. Thornwall, M. Krier, S. Rowson, G. Poilasne, and L. Desclos, “Mobile handset system performance comparison of a linearly polarized GPS internal antenna with a circularly polarized antenna,” in Proc. IEEE AP-S Int. Symp., Columbus, OH, Jun. 2003, vol. 3, pp. 666–669.

Abstract—Ultrawideband (UWB) systems require band notch filters in order to prevent sensitive components, within the front-end of the receiver, from being overloaded by strong signals. Recently, it has been shown that these filters can be integrated into the UWB antenna, to great advantage. This communication presents a new method for forming a notch band within the frequency response of a UWB antenna. An open loop notch band resonator is located on the back of the substrate, used to support the UWB monopole. The act of separating the resonator from the antenna means that they can now be designed in isolation, using the standard approach described in the literature, and then combined. A prototype was constructed and good agreement has been obtained between simulation and measurement. The radiation patterns are consistent over the frequency range of interest. Index Terms—Band-stop filters, coplanar waveguides, monopole antennas, ultrawideband (UWB) antennas.

I. INTRODUCTION There is much interest in the use of ultrawideband (UWB) signals (from 3.1 to 10.6 GHz) for short range, high-data rate communications [3]. UWB radar systems have been used to improve the detection of early stage breast cancer [1], [2]. UWB ground penetrating radar can be used to detect mines and damaged utility pipes. Interference from a strong narrowband signal, within the UWB band, could overload the receiver and band-stop filters have been suggested to mitigate for this. This filter might be a separate component, connected in series with the antenna [4], which will increase the size, weight, and complexity of the system or it could be integrated into the antenna’s feed-line [5]. A substrate integrated waveguide (SIW) cavity filter is used in [5], within the feed-line of an UWB monopole antenna, but antenna performance degradations result. An alternative is to integrate some form of band-stop filter into the radiating element. The majority of designs use a resonant slot within the planar monopole antenna [6]–[15]. Unfortunately most of the current solutions are limited by having: 1) poor return loss, i.e., >1.5 dB [5], [7], [13], [14], [25] or >2.5 dB [9], [10], [12], [16]; 2) poor gain suppression, i.e.,