IEEE Transactions on Antennas and Propagation [volume 59 number 10]

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OCTOBER 2011

VOLUME 59

NUMBER 10

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Pattern Equalization of Circular Patch Antennas Using Different Substrate Permittivities and Ground Plane Sizes ..... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . S. I. Latif and L. Shafai A New Triple-Band Circular Ring Patch Antenna With Monopole-Like Radiation Pattern Using a Hybrid Technique . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... M. Niroo-Jazi and T. A. Denidni A Wideband Circularly Polarized Conical Beam From a Two-Arm Spiral Antenna Excited in Phase .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... H. Nakano, H. Oyanagi, and J. Yamauchi A Low-Profile Printed Drop-Shaped Dipole Antenna for Wide-Band Wireless Applications .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ...... G. Cappelletti, D. Caratelli, R. Cicchetti, and M. Simeoni The Impact of Vehicle Structural Components on Radiation Patterns of a Window Glass Embedded FM Antenna ..... .. .. ........ ......... ......... ........ .... J. H. Schaffner, H. J. Song, A. Bekaryan, H.-P. Hsu, M. Wisnewski, and J. Graham Design, Realization and Measurements of a Miniature Antenna for Implantable Wireless Communication Systems ... .. .. ........ ......... ......... ........ .. F. Merli, L. Bolomey, J.-F. Zürcher, G. Corradini, E. Meurville, and A. K. Skrivervik Design of an Implantable Slot Dipole Conformal Flexible Antenna for Biomedical Applications ...... ......... ......... .. .. M. L. Scarpello, D. Kurup, H. Rogier, D. V. Ginste, F. Axisa, J. Vanfleteren, W. Joseph, L. Martens, and G. Vermeeren Temporary On-Skin Passive UHF RFID Transfer Tag ... ......... ......... ........ ......... .. M. A. Ziai and J. C. Batchelor Carbon Nanotube Composites for Wideband Millimeter-Wave Antenna Applications .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ........ A. Mehdipour, I. D. Rosca, A. R. Sebak, C. W. Trueman, and S. V. Hoa Creation of a Magnetic Boundary Condition in a Radiating Ground Plane to Excite Antenna Modes .. ......... ......... .. .. ........ ......... ......... M. Sonkki, M. Cabedo-Fabrés, E. Antonino-Daviu, M. Ferrando-Bataller, and E. T. Salonen TE Surface Wave Resonances on High-Impedance Surface Based Antennas: Analysis and Modeling . ......... ......... .. .. ........ ......... ......... . F. Costa, O. Luukkonen, C. R. Simovski, A. Monorchio, S. A. Tretyakov, and P. M. de Maagt Miniaturized Self-Oscillating Annular Ring Active Integrated Antennas ......... ....... Y.-Y. Lin, C.-H. Wu, and T.-G. Ma Packages With Integrated 60-GHz Aperture-Coupled Patch Antennas ... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . D. Liu, J. A. G. Akkermans, H.-C. Chen, and B. Floyd 3D-Antenna-in-Package Solution for Microwave Wireless Sensor Network Nodes ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ....... A. Enayati, S. Brebels, W. De Raedt, and G. A. E. Vandenbosch High-Gain Silicon On-Chip Antenna With Artificial Dielectric Layer ... ........ ......... ........ K. Takahagi and E. Sano A Compact Printed Filtering Antenna Using a Ground-Intruded Coupled Line Resonator ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. C.-T. Chuang and S.-J. Chung A Circuit Model for Spherical Wheeler Cap Measurements ..... ......... ........ ....... ... ......... ........ ........ H. L. Thal Arrays and Periodic Structures Edge-Born Waves in Connected Arrays: A Finite Infinite Analytical Representation . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... A. Neto, D. Cavallo, and G. Gerini

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(Contents Continued on p. 3501)

(Contents Continued from Front Cover) Mismatch of Near-Field Bearing-Range Spatial Geometry in Source-Localization by a Uniform Linear Array ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... Y.-S. Hsu, K. T. Wong, and L. Yeh Modified Dielectric Frequency Selective Surfaces With Enlarged Bandwidth and Angular Stability ... ....... L. Zappelli Double Periodic Composite Right/Left Handed Transmission Line and Its Applications to Compact Leaky-Wave Antennas ....... ......... ........ ......... ......... ........ ......... ......... ........ ... C. Jin, A. Alphones, and M. Tsutsumi 1D-Leaky Wave Antenna Employing Parallel-Plate Waveguide Loaded With PRS and HIS ..... ........ ......... ......... .. .. ........ ......... ......... ........ ... M. García-Vigueras, J. L. Gómez-Tornero, G. Goussetis, A. R. Weily, and Y. J. Guo Transmission Line Modeling and Asymptotic Formulas for Periodic Leaky-Wave Antennas Scanning Through Broadside ...... ......... ........ ......... ......... ........ ......... ......... S. Otto, A. Rennings, K. Solbach, and C. Caloz Inverse Scattering and Sensing Constrained Inverse Near-Field Scattering Using High Resolution Wire Grid Models .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... B. Omrane, Y. Goussard, and J.-J. Laurin A Shape-Based Inversion Algorithm Applied to Microwave Imaging of Breast Tumors ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... R. Firoozabadi and E. L. Miller An Inverse Scattering Approach to Soft Fault Diagnosis in Lossy Electric Transmission Lines .. .. H. Tang and Q. Zhang Evaluation of Dielectric Resonator Sensor for Near-Field Breast Tumor Detection ..... ....... K. S. Ryu and A. A. Kishk Wideband Antenna With Conductive Textile Radiators for a Dual-Sensor Subsurface Detection System ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . A. O. Salman, E. Biçak, and M. Sezgin Numerical and Analytical Techniques An Unconditionally Stable Radial Point Interpolation Meshless Method With Laguerre Polynomials .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... X. Chen, Z. Chen, Y. Yu, and D. Su Integral Equations Physically-Based Preconditioner for Two-Dimensional Electromagnetic Scattering by Rough Surfaces ........ ......... ........ ......... ......... ........ ......... ......... ..... S. Tournier, P. Borderies, and J.-R. Poirier Kirchhoff’s Laws as a Finite Volume Method for the Planar Maxwell Equations ........ ........ H. S. Bhat and B. Osting The WLP-FDTD Method for Periodic Structures With Oblique Incident Wave . ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ Z.-Y. Cai, B. Chen, Q. Yin, and R. Xiong Hybrid Ray Tracing Method for Microwave Lens Simulation ... ......... ........ ......... ...... J. Dong and A. I. Zaghloul The Wiener-Hopf Solution of the Isotropic Penetrable Wedge Problem: Diffraction and Total Field ... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... V. Daniele and G. Lombardi Rotational Vector Addition Theorem and Its Effect on T-Matrix ......... .. M. S. Khajeahsani, F. Mohajeri, and H. Abiri A Technique for Real-Time Shadowing Adjustment of RCS Scattering Center Models . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ J. L. Wilson, B. W. Rybicki, L. E. Johnson, and D. M. Koltenuk Huygens’ Principle for Complex Spheres ......... ........ ......... .... ...... ........ ......... ..... T. B. Hansen and G. Kaiser Maxwellian Circuits of Conducting Circular Loops ..... ......... ......... ........ .. W. Shen, C. Xue, K. K. Mei, and J. Lin Wave Propagation and Wireless Time-Dependent Tilted Pulsed-Beams and Their Properties ..... ......... ........ ......... ....... Y. Hadad and T. Melamed Pulse Distortion and Mitigation Thereof in Spiral Antenna-Based UWB Communication Systems ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... M. A. Elmansouri and D. S. Filipovic On the EM Degrees of Freedom in Scattering Environments .... ......... ........ ......... ......... ........ ... R. Janaswamy From Wideband to Ultrawideband: Channel Diversity in Low-Mobility Indoor Environments .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... R. Cepeda, C. Vithanage, and W. Thompson Performance of Site Diversity Investigated Through RADAR Derived Results . ....... J. X. Yeo, Y. H. Lee, and J. T. Ong Comparison of L- and C-Band Satellite-to-Indoor Broadband Wave Propagation for Navigation Applications . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... T. Jost, W. Wang, U.-C. Fiebig, and F. Pérez-Fontán On-Body Transmission at 5.2 GHz: Simulations Using FDTD With a Time Domain Huygens’ Technique ..... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... S. Dumanli and C. J. Railton

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COMMUNICATIONS

A Printed Wide-Slot Antenna With a Modified L-Shaped Microstrip Line for Wideband Applications ......... .. Y. Sung Miniaturization of Slot Antennas Using Slit and Strip Loading .. ......... ........ . B. Ghosh, SK. M. Haque, and D. Mitra Octagonal Ring Antenna for a Compact Dual-Polarized Aperture Array ........ ......... ....... Y. Zhang and A. K. Brown Symmetric-Aperture Antenna for Broadband Circular Polarization ...... ........ .. Nasimuddin, Z. N. Chen, and X. Qing Implementation of Broadband Isolator Using Metamaterial-Inspired Resonators and a T-Shaped Branch for MIMO Antennas ....... ......... ........ ......... ......... . ........ ......... ......... ........ ...... C.-C. Hsu, K.-H. Lin, and H.-L. Su Linear Equidistant Antenna Array With Improved Selectivity ... ......... ........ ......... ......... ........ .. P. S. Apostolov Algorithm for the Computation of the Generalized Fresnel Integral ...... . ...... G. Carluccio, F. Puggelli, and M. Albani

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Is the leading international engineering journal on the general topics of electromagnetics, antennas and wave propagation. The journal is devoted to antennas, including analysis, design, development, measurement, and testing; radiation, propagation, and the interaction of electromagnetic waves with discrete and continuous media; and applications and systems pertinent to antennas, propagation, and sensing, such as applied optics, millimeter- and sub-millimeter-wave techniques, antenna signal processing and control, radio astronomy, and propagation and radiation aspects of terrestrial and space-based communication, including wireless, mobile, satellite, and telecommunications. Author contributions of relevant full length papers and shorter Communications are welcomed. See inside back cover for Editorial Board.

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Digital Object Identifier 10.1109/TAP.2011.2170531

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011

Pattern Equalization of Circular Patch Antennas Using Different Substrate Permittivities and Ground Plane Sizes Saeed I. Latif, Member, IEEE, and Lotfollah Shafai, Life Fellow, IEEE

Abstract—A very unique phenomenon of circular patch antennas (CPA) for obtaining equal E- and H-plane co-polarization patterns, using varying substrate permittivities and ground plane sizes, is investigated in this paper. Pattern equalization is studied for the dominant and the first few higher order modes of the circular patch antenna, using different substrate permittivities ( r ) for the infinite ground plane case, and also by varying the ground plane sizes for the finite ground plane case. The angular range, at which the E- and H-plane patterns can be made equal, decreases with increased mode numbers. Also, for a given r and mode number, there exists a specific ground plane size, which provides nearly equal E- and H-plane co-polarization patterns. For higher order modes, larger ground planes and higher dielectric permittivities are required to obtain pattern equalization with a greater angular range. Index Terms—Circular microstrip antennas, higher order modes, pattern equalization, radiation patterns.

I. INTRODUCTION HE microstrip patch antenna, because of its geometrical simplicity and other attractive features, has been popular in many diverse applications, especially in mobile and satellite communications [1]–[3]. The circular patch antenna (CPA) is one of its fundamental shapes, and offers a unique property in which the current distribution depends on separate functions of its coordinates: the azimuth angle and the radial coordinate. Consequently, the radiated field is also dependant on separate functions of the far field coordinates, the azimuth and elevation angles [4]. This allows the opportunity for shaping its radiation patterns, in somewhat systematic way, to generate suitable patterns for specific applications. It is therefore important to identify the key parameters for such pattern shaping, i.e., in the azimuth and elevation directions. The solutions of the fields in terms of the azimuth coordinate are Fourier harmonics of simple trigonometric functions with well defined behavior. The solutions in terms of the radial coordinate, however, are known only for infinite ground planes, but in general depend on the size of the patch and ground plane. The former depends on the

T

Manuscript received October 18, 2010; revised January 17, 2011; accepted February 17, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with the Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2163775

effective permittivity of the substrate, which is used between the patch and the ground plane to give the mechanical support and strength to the antenna. Since the patch antenna is a resonant structure and the electromagnetic wave is trapped inside the dielectric material, the size of the patch for each resonant mode depends on the permittivity of the substrate. Then, it is obvious that the radiation patterns of a microstrip antenna are affected by its substrate permittivity [4], [5]. The ground plane size, on the other hand, does not depend on the resonant mode, and is selected primarily to match that of the mounting structure, or because of the packaging requirements. However, when there is a finite ground plane, its surface current has finite extent and there is diffraction from its edges, thereby affecting the back radiation. Thus, ground plane size has a significant influence on the radiation patterns of the CPA. In fact, since the substrate height is usually negligible in terms of the wavelength, the size of the ground plane appears to be the dominating factor in controlling the patch radiation patterns. This has been observed in previous investigations, and used in controlling the Eand H-plane co-polarization patterns of a CPA [4], [6]–[8]. The ground plane size, however, is not the only parameter for controlling the patch radiation patterns. The patch size, through the substrate permittivity, has a significant role as well and should be considered together with the ground plane size to tailor the radiation patterns for specific applications. In this paper, a detail study is conducted, with the aim of equalizing the radiation patterns in two principal planes, where, for geometrical simplicity, the ground plane shaping is not considered, and the pattern control parameters are limited to the substrate permittivity and the ground plane radius. The principal plane pattern equalization is highly desirable for many applications, especially in the design of circularly polarized antennas and reflector antenna feeds. In some important applications, such as the GPS and land mobile satellite communications, circularly polarized antennas with broad angular coverage in elevation are desirable. In addition, some applications, like mobile vehicular communications with satellites, require circularly polarized conical antenna patterns, again with broad angular coverage. These pattern types can be generated with higher order circular microstrip patch antenna modes [9]–[12]. However, without shaping the microstrip antenna ground plane geometry generating such circularly polarized radiation patterns is a difficult task [8]. The requirements for equalizing the principal plane radiation patterns of reflector antenna feeds are similar, except that it must be maintained over the illuminated angular range of the reflector [13]. In addition, the pattern role off beyond the reflector edge

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LATIF AND SHAFAI: PATTERN EQUALIZATION OF CPAs USING DIFFERENT SUBSTRATE PERMITTIVITIES AND GROUND PLANE SIZES

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TABLE I ROOTS OF J (kR) = 0 AND CORRESPONDING RADII FOR A RESONANT FREQUENCY OF 3.5 GHz ON A SUBSTRATE (" = 1:2, AND h = 1:59 mm), AND THE PROBE LOCATIONS FOR EXCITING THOSE MODES

Fig. 1. (a) Circular patch antenna with the feeding probe and (b) its reflection coefficient.

must be rapid to achieve a high gain factor, low spillover efficiency and high beam efficiency, and low sidelobes from the reflector antenna [14]. The latter requires that the feed diameter, i.e., the microstrip antenna ground plane size, must be small to reduce the aperture blockage, which is the dominating factor for raising the reflector near-in sidelobes [15]. Instead of using array feeds to achieve these requirements, a single CPA can be used, where pattern equalization can be obtained simply by using engineered materials for controlling the substrate permittivity and the ground plane size. Section II discusses the effects of the substrate permittivity on the pattern equalization from the circular patch antenna on an infinite ground plane. There is a merit for this study. If the principal plane patterns of the patch antenna can be equalized for an infinite ground plane, it can be true for large ground plane sizes, at least approximately. It is shown in this section that this can be done by selecting substrate permittivities near unity. In this case the antenna size will be large, but the ground plane effect will be negligible. Such antennas are ideal for applications, where the antenna platform size cannot be controlled, and may vary depending on the application. The opposite is true for small handheld devices, where the ground plane size will determine the actual device size, an important design parameter. Therefore, effects of the ground plane size on the pattern equalization of the circular patch antenna are also studied in Section III. A full-wave method of moments (MoM) analysis and mathematical formulation are conducted to understand these features of the circular patch antenna. The MoM-based results are verified by using a representative FEM (finite element method)-based simulation [16] in Section IV.

II. CPA PATTERN EQUALIZATION USING SUBSTRATE PERMITTIVITY The geometry of a typical circular patch antenna is shown in Fig. 1, where is the radius of the patch. The resonance fremode can be deterquency of a circular patch for the mined using the following equation [17]: (1)

where is the zero of , is the velocity of light, is the effective radius of the patch, which can be calcuand lated from the following expression:

(2) For each mode, a radius can be found that results in a resonance corresponding to the zeros of the Bessel function. For example, is the zero of , the resonance occurs at if and This relationship can be used to determine the radius of a circular patch for a given resonance frequency. First few modes are listed in Table I. For a resonance frequency of 3.5 GHz, and with substrate parameters , and , patch radii are calculated and shown also in Table I, with corresponding probe location, which is important to excite the desired mode. It can be noticed that as we move to the higher order modes, the patch radius increases as the corresponding eignevalue increases. Moreover, for higher ratio becomes larger, as mentioned in [4], order modes, the [18], where is the distance of the probe from the center. A. Effects of Substrate Permittivity on the Principal Plane Patterns At first, the effects of the substrate permittivity on the radiation patterns of the circular patch antenna with infinite ground plane size are investigated. For the circular patch antenna in Fig. 1(a), co-polarization patterns in two principal planes are is the co-polar component shown in Figs. 2–5. In all cases, is the same in the H-plane. It can be in the E-plane, and noticed that E- and H-plane co-polarization patterns are nearly equal, having a similar beamwidths, for a large range of theta and 1.3. In the case of mode, Fig. 2, for for , in almost the entire upper hemisphere. As increases, the beamwidth of the E-plane co-polar pattern increases, and that of the H-plane co-polar pattern decreases. For , E- and H-plane co-polar patterns are nearly equal for . It is to be noticed that both patterns are exa range of actly equal to each other at least at two locations, one at , where they tangentially meet, and the second one at a large angle near horizon where they intersect each other, at least for small

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Fig. 2. Co-Polarization patterns at TM mode of a circular patch (R = 22:6 mm;  = 6 mm) antenna (using infinite ground plane) on a substrate, having parameters " = 1:0 to 2.0, h = 1:59 mm. Simulation tool: Ansoft Designer, version 4 (a) " = 1:0, (b) " = 1:1, (c) " = 1:2, (d) " = 1:3, (e) " = 1:5, (f) " = 2:0.

values. This value, where the intersection occurs, decreases , both with increasing . It can also be noticed that for patterns are perfectly equal for range. However, for or larger), , i.e., larger values of (e.g., they deviate from each other, and theoretically, there is only one . location when these two patterns are equal, which is The same discussion applies for other higher order modes as well. It is well-known that for those modes, the antenna radiates conical patterns, and as the mode number increases, the peak of the beam moves away from the boresight. It is evident from Figs. 3–5 that the range of , at which the E- and H-plane co-polar patterns are nearly equal, decreases with increasing mode number. However, in this full-wave MoM simumode, due lation using Ansoft Designer version 4.0, for to the excitation of other modes, especially mode which mode [see Table I], the value increases. is very close to Later, we will show that it consistently decreases with mode number, when pure modes are excited. For the same patch sizes, inthe resonance frequencies go to the lower frequency as creases. Thus, it is evident that when the antenna patterns are not significantly disturbed because of the presence of a large ground

TM mode of a circular patch (R = 38:4 mm;  = 19 mm) antenna (infinite ground plane) on a substrate, having parameters " = 1:0 to 2.0, h = 1:59 mm. Simulation tool: Ansoft Designer, version 4 (a) " = 1:0, (b) " = 1:1, (c) " = 1:2, (d) " = 1:3, (e) " = 1:5, (f) = 2:0. Fig. 3. Co-Polarization patterns at

plane, pattern equalization can be achieved from circular patch antennas simply by using the substrate material with an appropriate permittivity. B. Mathematical Interpretation The simulation results obtained from Ansoft Designer, version 4 showed that with increased , the value, at which the second intersection occurs i.e., when , decreases. This can also be explained mathematically, which is shown below. For infinite ground plane size, the expressions for the far-field modes of a circular disk components, corresponding to antenna, are given by [17]: (3) (4) where is known as the edge voltage at , is the radius of the patch, and is the free space propagation constant.

LATIF AND SHAFAI: PATTERN EQUALIZATION OF CPAs USING DIFFERENT SUBSTRATE PERMITTIVITIES AND GROUND PLANE SIZES

Fig. 4. Co-Polarization patterns at TM mode of a circular patch (R = 53 mm;  = 32 mm) antenna (infinite ground plane) on a substrate, having parameters " = 1:0 to 2.0, h = 1:59 mm. Simulation tool: Ansoft Designer, version 4. (a) " = 1:0, (b) " = 1:1, (c) " = 1:2, (d) " = 1:3, (e) " = 1:5, (f)" = 2:0.

In the the E-plane,

plane, , as . Therefore, in is the co-polar component. Then (3) reduces to:

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TM mode of a circular patch (R = 64:25 mm;  = 43 mm) antenna (infinite ground plane) on a substrate, having parameters " = 1:0 to 2.0, h = 1:59 mm. Simulation tool: Ansoft Designer, version 4 (a) " = 1:0, (b) " = 1:1, (c) " = 1:2, (d) " = 1:3, (e) " = 1:5, (f) " = 2:0. Fig. 5. Co-Polarization patterns at

in (5) and (6). Then (7) becomes (9)

(5) , plane, , as Similarly, in the . i.e., in the H-plane, is the co-polar component, which reduces (4) to the following:

Using the following relationships: (10) we can have, from (9)

(6) (11) The condition imposed on this section is the equality of co-polar patterns in E- and H- planes for different modes to obtain the substrate permittivity, and the value at which they are equal, i.e.,

Here, (12)

(7) For small argument approximation, when

Let, (8)

, (13)

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TABLE II ROOTS OF J (kR) = 0 AND CORRESPONDING RADII FOR A RESONANT FREQUENCY OF 3.5 GHz ON A SUBSTRATE (" = 1:2, AND h = 1:59 mm), AND THE PROBE LOCATIONS FOR EXCITING THOSE MODES

and (11) becomes [using (8)]:

(18) Again, is the unknown in (17), which is a function of . When we consider three terms in the series expansion, i.e., in (12),

(19) and

(20) Fig. 6. Calculated relative permittivity versus  for different eigenvalues with small argument approximation for a circular patch antenna (a) TM mode, (b) TM mode, (c) TM mode (d) TM mode.

Therefore, (11) can be written as follows [using (8)]:

and

(14) After simplification, (11) becomes, using (8): (15) is the unknown in (15), which is a function of . The solution of this equation gives the value for a given , when the E, and H-plane patterns are equal. Obviously, at of two principal planes. The second value, at which , is calculated, and plotted in Figs. 5(a)–5(d) for different eigenvalues. We now consider two terms in the series expansion of the Bessel functions, i.e., in (12), then (16) and (17)

(21) Now, (12), containing the Bessel functions can also be solved exactly, i.e., without the series expansion for various values vs. is plotted in using MATLAB. For the four cases of , Figs. 6(a)–6(d). It appears that a small argument approximation and . However, when , is suitable for the approximated calculation gives erroneous results. Therefore, an exact calculation is used in this paper, and Fig. 7 shows the values of for different , when the E- and H-plane patterns are equal. These values are tabulated in Table II, and compared with the simulated ones obtained in the previous section. It shows that or 1.3), the in the case of smaller permittivity (e.g., value, at which two patterns intersect with each other, is larger. In other words, for smaller substrate permittivity, the range up is larger, but decreases with increased to which mode number. The amount of decrease is also larger for higher . It can be noticed that when is much higher , the co-polar patterns are deviated from each other, and therefore, the only equality we can get is along the boresight direction. These theoretical results agree well with the trend obtained in the simulated co-polar patterns in Figs. 2–5, at least for small

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Fig. 7. Calculated relative permittivity for different eigenvalues using exact equation containing Bessel functions for a circular patch antenna.

Fig. 9. Co-Polarization patterns at TM mode of a circular patch antenna (Finite ground plane size: 1:0 ) on a substrate, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:3 (f = 3:37 GHz), (c) " = 1:5 (f = 3:167 GHz), and (d) " = 2:0 (2:778 GHz), h = 1:59 mm. Antenna parameters: : R = 22:6 mm,  = 6 mm. Simulation tool: Ansoft Designer, version 4. Fig. 8. (a) Circular patch antenna with finite ground plane.

mode numbers. In the case of higher order modes, and larger , the calculated ones are different from the computed ones using Ansoft Designer, which is because of the presence of the other modes, excited along the dominant mode. III. EFFECTS OF FINITE GROUND PLANE SIZE ON THE CPA PATTERN EQUALIZATION AT DIFFERENT MODES FOR DIFFERENT SUBSTRATE PERMITTITIVITIES The study in the previous section was conducted based on an infinite ground plane. In practical applications, the ground plane determines the actual size of the antenna, and as it was mentioned earlier it affects the radiation patterns of microstrip patch antennas. In this section, the effects of having a finite ground plane on the pattern equalization of CPA, as shown in Fig. 8, are discussed. A. Effects of Finite Ground Plane on the Pattern Equalization At first a fixed ground plane size of is chosen, and the substrate permittivity is varied as before. In the case mode, when , mostly in the upper hemiof sphere, , as shown in Fig. 9(a). As increases, moves closer to , and when is much larger, , , the co-polar patFigs. 9(b)–9(d). In this case, for terns of E- and H-planes are nearly equal for almost the entire upper hemisphere, Fig. 9(b). However, in the case of and modes, the equalization for and in the upper , as can be noticed in hemisphere is observed for , co-polar component in the E-plane Figs. 10–11. As

Fig. 10. Co-Polarization patterns at TM mode of a circular patch antenna (Finite ground plane size: 1:0 ) on a substrate, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:3 (f = 3:37 GHz), (c) " = 1:5 (f = 3:167 GHz), and (d) " = 2:0 (2:778 GHz), h = 1:59 mm. Antenna parameters: : R = 38:4 mm,  = 19 mm. Simulation tool: Ansoft Designer, version 4.

becomes larger than that in the H-plane for these modes. In the , the nearly-equal co-polar patterns are observed case of for the given ground plane size, Fig. 12. This for the case

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Fig. 11. Co-Polarization patterns at TM mode of a circular patch antenna (Finite ground plane size: 1:0 ) on a substrate, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:3 (f = 3:37 GHz), (c) " = 1:5 (f = 3:167 GHz), and (d) " = 2:0 (2:778 GHz), h = 1:59 mm. Antenna parameters: : T M mode: R = 53 mm,  = 32 mm. Simulation tool: Ansoft Designer, version 4.

Fig. 12. Co-Polarization patterns at TM mode of a circular patch antenna (Finite ground plane size: 1:0 ) on a substrate, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:3 (f = 3:37 GHz), (c) " = 1:5 (f = 3:167 GHz), and (d) " = 2:0 (2:778 GHz), h = 1:59 mm. Antenna parameters: R = 64:25 mm,  = 43 mm. Simulation tool: Ansoft Designer, version 4.

study suggests that, pattern equalizations can be achieved for a particular mode and a given substrate dielectric constant by adjusting the ground plane size. This is studied later. B. Effects of Finite Ground Plane Size on the Pattern Equalization of the CPA for Different Substrate Permittitivities at Different Modes In this section, for a given mode, a fixed substrate is considered, and the ground plane size is varied to obtain pattern mode and , equalization. In the case of mostly in the upper hemisphere for the ground plane size , as can be seen in Fig. 13. As increases, they become nearly equal to each other, but when increased fur, they become unequal. For , nearly ther equal co-polar patterns have been observed in two planes for . For almost the entire upper hemisphere with ), for small ground higher values (e.g., plane radius. They become equal for a smaller angular range for . If the ground plane size is much becomes larger than . larger, mode, pattern equalization is observed In the case of with the ground plane size of in the case for up to of . However, in the case of , equal E- and H-plane co-polar patterns have been obtained for almost the en. As tire upper hemisphere for the ground plane size of the ground plane size increases, the E-plane co-polar pattern be, comes larger than the H-plane co-polar pattern. With with the pattern equalization is observed only for up to [Fig. 14]. ground plane size of

Fig. 13. Co-Polarization patterns at TM mode of a circular patch antenna (R = 22:6 mm;  = 6 mm) for different ground plane sizes on substrates, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:5 (f = 3:612GHz), and (c) " = 2:0 (f = 2:778GHz), h = 1:59mm. Simulation tool: Ansoft Designer, version 4.

LATIF AND SHAFAI: PATTERN EQUALIZATION OF CPAs USING DIFFERENT SUBSTRATE PERMITTIVITIES AND GROUND PLANE SIZES

Fig. 14. Co-Polarization patterns at TM mode of a circular patch antenna (R = 38:4 mm;  = 19 mm) for different ground plane sizes on substrates, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:5 (f = 3:162 GHz), and (c) " = 2:0 (f = 2:778GHz), h = 1:59mm. Simulation tool: Ansoft Designer, version 4.

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Fig. 15. Co-Polarization patterns at TM mode of a circular patch antenna (R = 53 mm;  = 32 mm) for different ground plane sizes on substrates, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:5 (f = 3:162GHz), and (c) " = 2:0 (f = 2:778GHz), h = 1:59mm. Simulation tool: Ansoft Designer, version 4.

For mode, good pattern equalization cannot be found , as shown in for different ground plane sizes when , and , E- and H-plane Fig. 15. However, for co-polar patterns are almost equal for the whole upper hemi. When the sphere with the ground plane size of around , E-plane co-polar pattern ground plane size is less than is less than the H-plane co-polar pattern. Whereas, when it is , E-plane co-polar pattern is larger than the greater than H-plane co-polar pattern. mode, a full upper hemisphere pattern equalization For , and with [Fig. 16]. is observed only with For lower permittivities, E- and H-plane co-polar patterns are equal for a small angular range. The ground plane size changes this range less significantly. IV. VERIFICATION BY ANSOFT HFSS, VERSION 12 In order to verify the simulation results obtained from Ansoft Designer, version 4, the case with is simulated in Ansoft HFSS, version 12, which is a finite-element-method (FEM)-based commercial software. However, in this case, infinite ground plane is assumed. With a few discrepancies, the results agree well with the co-polar patterns in Figs. 2–4, obtained from Ansoft Designer, for the same patch sizes. The resonance frequency is somewhat smaller than 3.5 GHz as can be noticed in Fig. 17, as obtained in Ansoft Designer. This difference is normal, as methods of computation are different for these two software.

Fig. 16. Co-Polarization patterns at TM mode of a circular patch antenna (R = 64:25 mm;  = 43 mm) for different ground plane sizes on substrates, having parameters (a) " = 1:2 (f = 3:5 GHz), (b) " = 1:5 (f = 3:162GHz), and (c) " = 2:0 (f = 2:778GHz), h = 1:59mm. Simulation tool: Ansoft Designer, version 4.

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with small value . However, for higher around order modes, it is larger with larger values, e.g., for mode, nearly equal patterns were achieved for the entire upper hemisphere with the ground plane size of and with . Although substrates with arbitrary permittivity values are not readily available other than some standard ones, with the current advancement in substrate technology, it is possible to manufacture engineered substrates with a desired permittivity. The circular patch antenna with equal patterns can be used as feeds for reflector, in vehicular applications, and GPS devices. REFERENCES

Fig. 17. S plot for TM mode and co-polar patterns of the circular patch antenna at first four modes (a) TM mode: R = 22:6 mm,  = 6 mm, (b) TM mode: R = 38:4 mm,  = 19 mm, (c) R = 53 mm,  = 32 mm, and (d) R = 64:25 mm,  = 43 mm. Substrate parameters: " = 1:2, h = 1:59 mm. Simulation tool: Ansoft HFSS, version 12.

V. CONCLUSION Pattern equalization from circular patch antenna was demonstrated in this paper. Radiation patterns from a circular patch antenna were shown to depend primarily on the size of the patch and its ground plane size. The permittivity of the substrate material, which gives the mechanical support to the antenna, determines the effective size of the patch. It was found in this paper that the E- and H-plane co-polarization patterns could be nearly equalized by controlling the substrate permittivity in the case of an infinite ground plane. For first four modes, it was observed gives good pattern equalization for a large that angular range. In practical applications where a large ground plane is available, e.g., the vehicle rooftop, pattern equalization can be achieved simply by using substrate with the desired permittivity. The elevation angular range at which they are equal decreases for the higher order modes of the antenna. Radiation patterns are also affected by the presence of a finite ground plane. However, it was shown in this paper that the ground plane size can be varied to obtain nearly equal co-polar patterns for different modes when a particular substrate permittivity is used. Pattern equalizations were found for almost the entire upper hemisphere with larger values and ground plane sizes when the mode number is higher. For lower order modes, the co-polarization patterns are nearly equal for the ground plane size of

[1] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London: Peregrinus, 1981. [2] A. G. Derneryd, “Analysis of the microstrip disk antenna element,” IEEE Trans. Antennas Propag., vol. 27, no. 5, pp. 660–664, Sep. 1979. [3] L. Shen, S. Long, M. Allerding, and M. Walton, “Resonant frequency of a circular disc, printed-circuit antenna,” IEEE Trans. Antennas Propag., vol. 25, no. 4, pp. 595–596, Jul. 1977. [4] A. Kishk and L. Shafai, “The effect of various parameters of circular microstrip antennas on their radiation efficiency and the mode excitation,” IEEE Trans. Antennas Propag., vol. 34, pp. 969–976, Aug. 1986. [5] L. Shafai, “Design of multi-arm multi-mode spiral antennas for directional beams using equivalent array concept,” Electromagnetics, vol. 14, no. 3-4, pp. 285–304. [6] J. Huang, “The finite ground plane effect on the microstrip antenna radiation patterns,” IEEE Trans. Antennas Propag., vol. 31, no. 4, pp. 649–652, Jul. 1983. [7] A. K. Bhattacharyya, “Effects of finite ground plane on the radiation characteristics of a circular patch antenna,” IEEE Trans. Antennas Propag., vol. 38, no. 2, pp. 152–159, Feb. 1990. [8] S. Noghanian and L. Shafai, “Control of microstrip antenna radiation characteristics by ground plane size and shape,” IEE Proc. Microw. Antennas Propag., vol. 145, no. 3, pp. 207–212, Jun. 1998. [9] R. G. Vaughan, “Two-port higher mode circular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 36, no. 3, pp. 309–321, Mar. 1988. [10] J. Q. Howell, “Microstrip antennas,” IEEE Trans. Antennas Propag., vol. 23, no. 1, pp. 90–93, Jan. 1975. [11] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment in microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, no. 2, pp. 137–145, Mar. 1979. [12] C. Wood, “Analysis of microstrip circular patch antennas,” IEE Proc. Microw. Antennas Propag., vol. 128, no. 2, pp. 69–76, Apr. 1981. [13] A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds. New York: IEEE, 1994. [14] V. Jamnejad, A. L. Riley, and P. T. Swindlehurst, “Reflector antenna systems for the high altitude MMIC sounding radiometer (HAMSR),” in Proc. IEEE Aerospace Con., Mar. 2000, vol. 5, pp. 113–117. [15] M. S. A. Sanad, “Aperture blocking of a symmetric parabolic reflector antennas,” M.Sc. thesis, The Univ. Manitoba, Winnipeg, MB, Canada, 1982. [16] Ansoft Corporation. Boulder, CO, USA. [17] R. Garg, P. Bhartia, I. J. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001. [18] K. Antoskiewicz and L. Shafai, “Impedance characteristics of circular microstrip patches,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 942–946, Jun. 1990. Saeed I. Latif (S’03–M’08) received the B.Sc. degree in engineering (electrical and electronics) from Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 2000, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2004 and 2009, respectively. During his graduate studies, he was a research assistant and involved in various microstrip and array antenna projects. He was also a teaching assistant during that period and contributed to the undergraduate teaching in the Department of Electrical and Computer Engineering at the University of Manitoba. From 2009 to 2010, he was a Postdoctoral Fellow in the same department. Currently he is an NSERC Postdoctoral Fellow with CancerCare Manitoba, Winnipeg, MB.

LATIF AND SHAFAI: PATTERN EQUALIZATION OF CPAs USING DIFFERENT SUBSTRATE PERMITTIVITIES AND GROUND PLANE SIZES

Dr. Latif is a registered professional engineer with APEGM. He was one of the fifteen finalists in the Student Paper Competition at the 2004 IEEE Antennas and Propagation Society (AP-S) International Symposium in Monterey, CA. He was the recipient of the Young Scientist Award at the International Symposium on Electromagnetic Theory (EMTS), held in Ottawa, ON, Canada in 2007.

Lotfollah Shafai (F’87) received the B.Sc. degree from the University of Tehran in 1963 and the M.Sc. and Ph.D. degrees from the University of Toronto, in 1966 and 1969, all in electrical engineering. In November 1969, he joined the Department of Electrical and Computer Engineering, University of Manitoba as a Sessional Lecturer, Assistant Professor 1970, Associate Professor 1973, and Professor 1979. Since 1975, he has made special effort to link the University research to the industrial development, by assisting industries in the development of new products or establishing new technologies. To enhance the University of Manitoba contact with industry, in 1985 he assisted in establishing “The Institute for Technology Development” and was its Director until 1987, when he became the Head of the Electrical Engineering Department. His assistance to industry was instrumental in establishing an Industrial Research Chair in Applied Electromagnetics at the University of Manitoba in 1989, which he held until July 1994. Dr. Shafai has been a participant in nearly all Antennas and Propagation symposia and participates in the review committees. He is a member of URSI Com-

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mission B and was its chairman during 1985–88. In 1986, he established the symposium on Antenna Technology and Applied Electromagnetics, ANTEM, at the University of Manitoba that is currently held every two years. He has been the recipient of numerous awards. In 1978, his contribution to the design of a small ground station for the Hermus satellite was selected as the 3rd Meritorious Industrial Design. In 1984, he received the Professional Engineers Merit Award and in 1985, “The Thinker” Award from Canadian Patents and Development Corporation. From the University of Manitoba, he received the “Research Awards” in 1983, 1987, and 1989, the Outreach Award in 1987 and the Sigma Xi, Senior Scientist Award in 1989. In 1990 he received the Maxwell Premium Award from IEE (London) and in 1993 and 1994 the Distinguished Achievement Awards from Corporate Higher Education Forum. In 1998 he received the Winnipeg RH Institute Foundation Medal for Excellence in Research. In 1999 and 2000 he received the University of Manitoba, Faculty Association Research Award. He is an elected Fellow of IEEE since 1987 and was elected a Fellow of The Royal Society of Canada in 1998. He was a recipient of the IEEE Third Millenium Medal in 2000 and in 2002 was elected a Fellow of The Canadian Academy of Engineering and Distinguished Professor at The University of Manitoba. In 2003 he received an IEEE Canada “Reginald A. Fessenden Medal” for “Outstanding Contributions to Telecommunications and Satellite Communications”, and a Natural Sciences and Engineering Research Council (NSERC) Synergy Award for “Development of Advanced Satellite and Wireless Antennas.” In 2009, he was elected a Fellow of the Engineering Institute of Canada, and was the recipient of an IEEE Chen-To-Tai Distinguished Educator Award. He holds a Canada Research Chair in Applied Electromagnetics and was the International Chair of Commission B of the International Union of Radio Science (URSI) for 2005-2008.

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A New Triple-Band Circular Ring Patch Antenna With Monopole-Like Radiation Pattern Using a Hybrid Technique Mahmoud Niroo-Jazi, Student Member, IEEE, and Tayeb A. Denidni, Senior Member, IEEE

Abstract—Investigation results of a new conical-shape radiationpattern antenna based on a hybrid technique are presented. A circular-ring patch antenna excited with the dominant mode is integrated with a monopole antenna, offering two distinct operating bands. The coupling effect of the higher-order modes of the monopole and the circular-ring patch create a third operating broad band. Using this hybrid resonant configuration, the expected ripples in the E-plane pattern due to the monopole higher-order modes are alleviated over the third band, providing a good radiation performance. The measurement results performed for the proposed structure represent a good agreement with the simulated ones. Index Terms—Circular ring patch, conical pattern, hybrid technique, multi-band, wideband.

I. INTRODUCTION

T

HE growing demands in delivering different tasks with low-cost and compact wireless communication systems have piqued much research interest in realizing multi-function systems, handling all the operating system requirements. Therefore, holding a paramount role in achieving the desired performances in these systems, an antenna should qualify both the required electromagnetic spectrum (EMS) frequency bands and all the azimuth coverage areas in wireless communication networks, including satellite navigation systems, cellular systems, wireless LANs or combination of these systems [1], [2]. To meet these goals, either the antenna can work in a broad/ultra widefrequency range or it can reconfigure its frequency response [3]–[5]. Planar monopole-shape radiation-pattern antennas are the most practical choices for these applications due to being low profile, low cost and simple design and easy fabrication [6]. However, the availability of wide electromagnetic spectrum (EMS) may increase the probability of interference and also reduces the overall system signal-to-noise ratio. On the other hand, the reconfigurable narrow-band antennas can, in part, alleviate these problems by choosing the appropriate band. However, the operating system can be more effective by assigning

Manuscript received July 19, 2010; revised December 09, 2010; accepted March 04, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. The authors are with INRS, Université de Québec, Place Bonaventure, 800, de la Gauchetière Quest, Bureau 6900, Montreal, QC H5A 1K6, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2163768

different bands of EMS instantaneously, handling various functions at the same time in each band. Therefore, multi-band antennas are an alternative effective solution to the broad band and reconfigurable antenna systems [7]–[10]. In recent years, various multi-band planar antennas have been proposed in which by tailoring the current distribution on the antenna, the desired operating frequency bands have been achieved. For instance, applying parasitic elements [7], using meandered-line structures [8] and integration of some slots within a resonator as a hybrid configuration in planar monopoles [9], [10] are some typical reported multi-band structures. However, because of not representing a conventional antenna configuration with an explicit design procedure, these antennas are usually designed using simulator packages. Most important, usually the radiation patterns of these antennas are not completely omni-directional, and they introduce a high-cross polarization level as well. It is well known that, being in consistent with the monopole antenna, a circular ring patch (CRP) resonator can be ably designed to provide different modes with a conical radiation pattern [11], [12]. Moreover, by adjusting the dimensions of the CRP, the higher order modes of the same family can be excited by this resonator [13], [14]. Therefore, in this paper, by taking advantages of combination of a monopole [15] and a CRP antenna [13], [14], [16], a new hybrid configuration is proposed. This antenna offers an efficient monopole-shape radiation pattern across different frequency bands. In spite of the tailored planar antennas, as an interesting point, the initial values of the antenna dimensions are easily calculated according to the center frequency of the desired operating bands. Furthermore, the antenna provides a good radiation pattern performance across the achieved bands in terms of omni-directionality, cross-polarization level and peak gain. In addition, the higher order modes of the same family either can be effectively coupled together to create broadband operation or even can be decoupled to provide two distinct bands. Therefore, the proposed configuration not only produces better performances compared to the above mentioned planar configurations, but also its design procedure is more easier than the tailored planar structures. In the next sections, the design procedure and the achieved results of the proposed antenna are presented and discussed. II. ANTENNA CONFIGURATION AND DESIGN PROCEDURE Fig. 1 shows the proposed hybrid resonant configuration constructed with a grounded-substrate CRP, which is excited by a monopole antenna. The monopole is designed at an operating

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NIROO-JAZI AND DENIDNI: A NEW TRIPLE-BAND CIRCULAR RING PATCH ANTENNA WITH MONOPOLE-LIKE RADIATION PATTERN

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Fig. 1. Configuration of the proposed antenna.

frequency , while the CRP is excited with the dominant mode at . This mode offers a conical radiation pattern [11]–[13]. Moreover, by adjusting the inner and outer radii of the CRP, a higher-order mode of the same family can be excited [14], [15]. Therefore, the CRP modes can be effectively coupled with the higher harmonics of the monopole to create an operating broad band or two distinct operating narrow bands. However, in this paper, as an application for a hybrid ultra-wideband dielectric resonator [17], the multi-band structure is designed to create triple operating frequency bands in which the last band offers an ultra-wideband performance. According to this design strategy, by individually considering each isolated resonator, their crucial dimensions are calculated using the theoretical formulas for the dominant modes of the monopole and the CRP. Then, CST Microwave Studio is used to optimize the antenna dimensions, considering the interaction between the two resonators [18]. The quarter-wavelength monopole antenna is designed for a resonant frequency , while the CRP antenna is designed to excite the desired mode around . The outer radius of the CRP, is estimated based on the cavity model of an equivalent circular patch without considering the fringing effect around the resonator. This value is calculated using the following formula [16]: (1) where is the root of the following characteristic equation of the circular patch ring for mode. (2) In this equation, is the n order of Bessel function of the first kind. Then, considering the interaction between the monopole and the CRP, the inner radius of the CRP is calculated through the optimization process. III. SIMULATION RESULTS AND PARAMETRIC STUDIES Fig. 2 shows the achieved reflection coefficient of the proposed antenna optimized using Microwave CST Studio simulator. As it can be observed, the hybrid structure introduces three distinct matched radiation bands centered around

Fig. 2. Simulated and measured results of the reflection coefficient.

Fig. 3. Simulation and measurement results of the realized gain for the proposed HCRP antenna.

the frequencies , , and . Because of close interaction of the excited higher-order modes by both the monopole and the CRP resonator, this result predicts a broad matching band centered around the two last matching frequencies. This antenna offers 475 MHz, 412 MHz and 4.25 GHz effective operating frequency ranges for the first, second and third achieved bands, respectively. Assessing the antenna radiation performance, Fig. 3 depicts the simulated realized gain of the antenna calculated at two different points of ( , ) and ( , ), where the angles and are measured from Z and X axes, respectively. These results show that the proposed configuration provides a peak gain with an average value about 4 dB over the all matching bands. Furthermore, to examine the antenna radiation performance in different angles, the simulated radiation patterns of the proposed hybrid circular ring patch (HCRP) antenna are shown

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Fig. 5. Parametric study performed for various monopole lengths.

Fig. 4. Simulated radiation pattern of the proposed HCRP antenna at different frequencies. (a) E-plane. (b) H-plane.

in Fig. 4. The H-plane pattern predicts a low cross-polarization level and an omni-directional pattern across the matched bands. Being out of the cross-polarization null, the E-plane radiation pattern of the antenna is sketched for the plane of , depicting the cross-polarization pattern. As another advantage of the proposed hybrid configuration, the higher-order modes of the CRP resonator with conical radiation pattern are expected to contribute in the radiation along with the higherorder modes of the monopole. This results in improving the E-plane radiation pattern due to the presence of higher harmonics of the monopole resonator. Furthermore, these modes are effectively coupled with the monopole harmonics, considerably enhancing the matching bandwidth. Because of the diffraction of surface waves from the antenna edges and also domination of monopoles’ harmonics in radiation, some ripples are appeared in the E-plane radiation pattern at high frequencies. In the next section, the physical interpretation of the dominant modes excited by the HCRP antenna is completely examined by sketching the electric field patterns of the CRP and monopole resonators. In order to assess the contribution of each resonator on the antenna performances in terms of matching and radiation pattern, a comprehensive parametric study was performed for the crucial antenna dimensions. Here, for brevity, just the effects of the monopole length, the matching step dimensions and

the inner and outer radius of the CRP on the antenna reflection coefficient are presented. Fig. 5 shows the effect of the monopole length on the antenna reflection coefficient. This figure reveals that the variation of the monopole length, shifts the positions of the four minimums. On the other hand, as it can be observed from Fig. 6, the matching step impacts the input impedance across all four bands centered around the frequencies , , and . Indeed, the radius of the matching step, considerably changes the matching at all frequencies, while the length of the matching step, mainly affects both the positions of and and the matching at these frequencies. Alternatively, Fig. 7 represents the CRP relevance to the antenna reflection-coefficient performance over the simulated band. As it can be expected, the CRP mainly affects the resonant modes excited under the circular microstrip patch resonator. However, the simulation results demonstrate that the CRP has a significant impact on the higher-order modes of the monopole resonator without any effect on its first mode at . Therefore, the inner and the outer radii of and dominantly dictate the resonant frequencies of and . The antenna matching is also altered at these frequencies by changing the radii of the CRP. Furthermore, the outer radius also affects the second resonant mode of the monopole antenna . According to these parametric studies, it can be concluded that the four frequencies , , and , more or less individually, can be easily controlled by the antenna dimensions, giving the designer enough flexibility to tailor the desired frequency response. IV. RADIATION MECHANISM OF THE PROPOSED HYBRID CONFIGURATION The simulation results presented in the previous section predict coupled different resonant modes excited by both the monopole and the CRP resonators. In order to explore the nature of each mode, in this section, the near-zone electric field pattern of the HCRP antenna is presented and analyzed. These investigations reveal that at all frequencies except ,

NIROO-JAZI AND DENIDNI: A NEW TRIPLE-BAND CIRCULAR RING PATCH ANTENNA WITH MONOPOLE-LIKE RADIATION PATTERN

Fig. 6. Parametric study performed for step matching dimensions. (a) Simula. (b) Simulation results for different values tion results for different values of . of

both the monopole and the CRP are contributing in radiation. However, the monopole is the dominant resonator at frequencies , and , providing first, second and third harmonics of a quarter-wavelength monopole antenna, respectively. The electric field patterns of the HCRP in just xy-plane are shown in Fig. 8 for each matching frequency which clarifies the type of the modes excited by CRP. The electric field distribution at frequency 5.5 GHz confirms the existence of mode under the CRP resonator. Furthermore, the pattern at frequency 6.8 GHz shows that the CRP is also excited by the other mode, which is from the same family of mode, generating a conical radiation pattern. Therefore, the close couplings of the modes excited by both the monopole and the CRP are the main reason of providing a broad matching band at higher frequencies. In addition, the CRP considerably contributes in the radiation at high frequencies. This leads to alleviate the expected nulls in the E-plane pattern introduced by higher harmonics of the monopole. Furthermore,

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Fig. 7. Parametric study performed for CRP dimensions. (a) Simulation results . (b) Simulation results for different values of . for different values of

not provided here, the animated near electric-field distribution demonstrates that the substrate interface effectively contributes in launching the monopole and the CRP modes, enhancing the smoothness of the radiation pattern. V. EXPERIMENTAL RESULTS AND DISUSSION Fig. 9 shows a photograph of the fabricated antenna prototype. A square piece of RO3003 substrate with dimensions of 24 mm 24 mm is used to etch the CRP resonator. The monopole is created by a piece of cylindrical brass rod. An RF-SMA connector is soldered to the rectangular substrate, exciting the hybrid resonator. The antenna parameters of the fabricated prototype were measured at INRS Antenna-laboratory. The measured reflection coefficient is compared to the simulation result in Fig. 2. The achieved result confirms the operation mechanism of the structure discussed in the previous sections. Although completely confirming the basic concept of the hybrid resonator

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Fig. 9. Photo of the fabricated HDRA prototype.

Fig. 10. Measured radiation pattern of the proposed HCRP antenna. (a) E-plane. (b) H-plane. Fig. 8. Electric field of the proposed antenna in xy plane at different frequencies. (a) Electric field at 2 GHz. (b) Electric field at 5.5 GHz. (c) Electric field at 6.8 GHz. (d) Electric field at 9.6 GHz.

discussed earlier and even providing the desired broad effective radiation band, the third antenna matching band is not as broad as the predicted one by the simulation. A comparison between the measured and simulated realized gain of the antenna for two distinct radiation angles is provided

in Fig. 3. This result also proves the expected radiation performance of the proposed antenna both in the azimuth and elevation planes over the all achieved matching bands. The slight difference between the simulated and measured curves is mainly due to the design, fabrication and measurement accuracy. Furthermore, Fig. 10 shows the achieved normalized co- and cross-polarization patterns of the antenna at four matching frequencies. As it can be noticed, the cross polarization level is less

NIROO-JAZI AND DENIDNI: A NEW TRIPLE-BAND CIRCULAR RING PATCH ANTENNA WITH MONOPOLE-LIKE RADIATION PATTERN

than 10 dB across the entire operating frequency bands. This level is more noticeable at higher frequencies, which it is due to the excitation of the higher-order modes of the CRP, offering a stronger cross polarization component. It is also believed that, by implementing the monopole antenna with a copper piece of thin pipe shape in except of solid one, a better cross-polarization level can be achieved. The proposed antenna provides a quite good omni-directional radiation pattern at the measured frequencies. There is a slightly variation in the received power in one side of the H-plane pattern, which is caused by the measurement errors and the implementation misalignment. In addition, depending on the type of the dominant radiation mode, there is a little variation in the shape of the E-plane radiation pattern. This is more noticeable at higher frequencies because of the excitation of the higher-order modes. Furthermore, the radiation pattern also confirms that because of contributing the higher-order mode of the CRP resonator in radiation, the expected nulls in the E-plane due to the higher harmonics of the monopole are alleviated. VI. CONCLUSION A new multi-band conical-pattern antenna using a hybrid resonator configuration has been presented in this paper. The achieved results show that using hybrid technique different modes of the same family can be efficiently exited, offering a multi-band operating response. Furthermore, the close interaction of the excited hybrid modes not only enhances the matching band, but also improves the radiation pattern of the antenna. Indeed, the higher order modes of the two resonators are effectively coupled each other, offering the third broad band while enhancing the antenna radiation performances as well. The achieved results show that the antenna can be a good candidate for multi-band communication systems. REFERENCES [1] S. R. Saunders and A. A. Zavala, Antennas and Propagation for Wireless Communications. Hoboken, NJ: Wiley, 2007. [2] B. Allen, M. Dohler, E. E. Okon, W. Q. Malik, A. K. Brown, and D. J. Edwards, Ultra-Wideband Antennas and Propagation for Wireless Communications, Radar and Imaging. Hoboken, NJ: Wiley, 2007. [3] K. K. Kang, J. W. Lee, S. C. Choon, and K. T. Lee, “An improved impedance bandwidth of modified UWB antenna with staircased parasitic rings,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 521–524, 2007. [4] M. J. Ammann and Z. N. Chen, “Wideband monopole antennas for multi-band wireless systems,” IEEE Antennas Wireless Propag. Mag., vol. 45, no. 2, pp. 146–150, April 2003. [5] T. Y. Han and C. T. Huang, “Reconfigurable monopolar patch antenna,” Elect. Lett., vol. 46, no. 3, pp. 199–200, Feb. 2010. [6] Z. N. Chen, M. J. Ammann, X. Qing, X. H. Wu, T. S. P. See, and A. Cai, “Planar antennas,” IEEE Microwave Mag., vol. 7, no. 6, pp. 63–73, Dec. 2006. [7] W. C. Liu, M. Ghavami, and W. C. Chung, “Triple-frequency meandered monopole antenna with shorted parasitic strips for wireless application,” IET Microw. Antennas Propag., vol. 3, no. 7, pp. 1110–1117, Oct. 2009. [8] S. C. Kim, S. H. Lee, and Y. S. Kim, “Multi-band monopole antenna using meander structure for handheld terminals,” Elect. Lett., vol. 44, no. 5, pp. 331–332, Feb. 2008. [9] X. C. Lin and C. C. Yu, “A dual-band CPW-fed inductive slot-monopole hybrid antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 282–285, Jan. 2008. [10] C. M. Su, H. T. Chen, and K. L. Wong, “Printed dual-band dipole antenna with U-slotted arms for 2.4/5.2 GHz WLAN operation,” Elect. Lett., vol. 38, no. 22, pp. 1308–1309, Oct. 2002.

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[11] A. A. Zoubi, Y. Fan, and A. Kishk, “A broadband center-fed circular patch-ring antenna with a monopole like radiation pattern,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 789–792, Mar. 2009. [12] Y. J. Guo, A. Paez, R. A. Sadeghzadeh, and S. K. Barton, “A circular patch antenna for radio LAN’s,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 177–178, Jan. 1997. [13] Y. S. Wu and F. J. Rosenbaum, “Mode chart for microstrip ring resonators,” IEEE Trans. Microwave Theory Tech., vol. 21, no. 7, pp. 487–489, Jul. 1973. [14] A. Das, S. P. Mathur, and S. K. Das, “Radiation characteristics of higher-order modes in microstrip ring antenna,” IEE Proc. Microw. Opt. Antennas, vol. 131, no. 2, pp. 102–106, Apr. 1984. [15] C. A. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ: Wiley, 2005. [16] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, U.K.: Peter Peregrinus, 1989. [17] M. N. Jazi and T. A. Denidni, “Ultra-wideband dielectric-resonator antenna with band rejection,” presented at the IEEE Int. Conf. on Antennas Propag., Toronto, Canada, Jul. 11–17, 2010. [18] CST Microwave Studio Simulator 2009.

Mahmoud Niroo-Jazi (S’04) received the B.S. degree in electrical engineering from Azad University of Najafaband, Isfahan, Iran, in 2000, and the M.S. degree in electrical engineering from Urmia University, Urmia, Iran, in 2004. From April 2004 to June 2007, he was with the Information Communication Institute & Technology (ICTI) at Isfahan University of Technology (IUT) as a Research Engineer. During this period, he was involved in the Antennas & Microwave Research Group developing and implementing practical communication systems. He is currently working toward the Ph.D. degree at Institute National de la Recherché (INRS), Montreal-Canada. His research areas of interest broadly include antennas, passive and active microwave circuits, numerical method in electromagnetic and radar systems. His major subjects are focused on ultra-wideband dielectric-resonator antennas, and particularly on periodic structures for the application of reconfigurable antennas.

Tayeb A. Denidni (M’98–SM’04) received the B.Sc. degree in electronic engineering from the University of Setif, Setif, Algeria, in 1986, and the M.Sc. and Ph.D. degrees in electrical engineering from Laval University, Quebec City, QC, Canada, in 1990 and 1994, respectively. From 1994 to 1996, he was an Assistant Professor with the Engineering Department, Université du Quebec in Rimouski (UQAR), Quebec, Canada. From 1996 to 2000, he was also an Associate Professor at UQAR, where he founded the Telecommunications laboratory. Since August 2000, he has been with the Personal Communications Staff, Institut National de la Recherché Scientifique (INRS), Université du Quebec, Montreal, Canada. He founded the RF laboratory, INRS-EMT, Montreal, for graduate student research in the design, fabrication, and measurement of antennas. He possesses ten years of experience with antennas and microwave systems and is leading a large research group consisting of two research scientists, five Ph.D. students, and three M.S. students. Over the past ten years, he has graduated numerous graduate students. He has served as the Principal Investigator on numerous research projects on antennas for wireless communications. Currently he is actively involved in a major project in wireless of PROMPT-Quebec (Partnerships for Research on Microelectronics, Photonics and Telecommunications). His current research interests include planar microstrip filters, dielectric resonator antennas, electromagnetic-bandgap (EBG) antennas, antenna arrays, and microwave and RF design for wireless applications. He has authored over 100 papers in refereed journals. He has also authored or coauthored over 150 papers and invited presentations in numerous national and international conferences and symposia. Dr. Denidni is a member of the Order of Engineers of the Province of Quebec, Canada. He is also a member of URSI (Commission C). From 2006 to 2007, he was an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. From 2008 to 2010, he served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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A Wideband Circularly Polarized Conical Beam From a Two-Arm Spiral Antenna Excited in Phase Hisamatsu Nakano, Life Fellow, IEEE, Hiroshi Oyanagi, and Junji Yamauchi, Senior Member, IEEE

Abstract—Analysis is performed for a two-arm round spiral structure excited in phase, where the arms near the antenna center are backed by a small round disc. This spiral antenna has a frequency range where the radiation is bidirectional, having patterns symmetric with respect to the antenna plane. Subsequently, the bidirectional radiation is transformed into a unidirectional conical beam by backing the spiral with a cavity, where the cavity height is chosen to be extremely small (0.077 wavelength at the lowest design frequency 3.3 GHz). It is found that, as the distance between the spiral plane and the small disc inside the cavity is decreased, variations in the axial ratio and VSWR become smaller, as desired. Other antenna characteristics over a wide frequency range of 3.3 GHz to 9.6 GHz, including the gain and radiation efficiency, are also discussed. Index Terms—Conical pattern, in-phase excitation, low-profile wideband antenna, spiral antenna.

I. INTRODUCTION

A

two-arm spiral antenna, isolated in free space and excited out of phase (balanced-mode excitation), radiates a circularly polarized (CP) wave in the two directions normal to the antenna plane [1]. In other words, this radiation is bidirectional. The active regions of this balanced-mode spiral are located at ring regions on the spiral plane, each having a circumference of , where m is an odd number and is the opapproximately erating wavelength. It is possible to transform the bidirectional radiation from the isolated balanced-mode spiral antenna into unidirectional radiation by backing the spiral with a conducting plane as a reflector [2]. This radiation is called a unidirectional axial beam, whose maximum radiation is on the antenna axis normal to the spiral plane. It is noted that, as the frequency decreases (and hence the electrical antenna height, measured from the reflector, decreases), the polarization of the radiated wave becomes elliptical. In addition, the input impedance varies remarkably. Thus, the inherent wideband characteristics of the axial ratio and input impedance observed for the isolated spiral deteriorate. However, this deterioration is mitigated by connecting resistors to the antenna arm ends [3], [4]. It has also been revealed that use of absorbing material [5]–[8] reduces the deterioration in the antenna

Manuscript received June 05, 2010; revised January 25, 2011; accepted March 09, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. The authors are with the College of Engineering, Hosei University, Koganei, Tokyo 184-8584, Japan (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.2011.2163759

characteristics. Detailed discussion on the mitigation/reduction of this deterioration is found in [7], [8]. A new question arises as to whether a spiral antenna backed by a shallow conducting cavity can produce a conical CP beam (whose maximum radiation intensity is off the antenna axis and whose minimum radiation field is on the antenna axis) over a wide frequency range. An answer related to this question is found in [9], where a two-arm spiral is excited in phase [10]. However, this conical beam is produced only within a narrow frequency range (9.5% bandwidth from 1.5 GHz to 1.65 GHz). In addition, the antenna is not low-profile; the antenna height is one quarter wavelength at the center frequency (1.57 GHz). This paper solves the abovementioned issues (of being narrowband and not having a low-profile structure) by investigating a cavity-backed two-arm spiral antenna, where the cavity height (antenna height) is extremely small: approximately 0.077 wavelength at the lowest design frequency. The frequency response for the antenna characteristics (the input impedance, radiation pattern, axial ratio, gain, and radiation efficiency) is investigated to realize wideband operation (approximately 98% bandwidth from 3.3 GHz to 9.6 GHz). Five sections, including this section, constitute this paper. Section II describes the radiation pattern when an in-phase two-arm spiral antenna having a small conducting disc is located in free space. In addition, formation of a unidirectional beam using a shallow cavity is discussed. Section III reveals the VSWR, current distribution, and radiation efficiency for the cavity-backed spiral. Section IV presents additional findings of the antenna characteristics, and Section V summarizes the obtained results. II. RADIATION A. Bidirectional Radiation Fig. 1 shows a two-arm spiral antenna above a small conducting disc, designated as the SD. The radial distance from the center of the spiral to a point on the spiral arm is defined , where is the by the Archimedean function spiral growth rate and is the winding angle . The conducting arm width and the spacing between neighboring arms are chosen to be the same. The two arms are connected to each other at the center point of the spiral. and the spacing beThe disc behind the spiral has a radius tween the disc and the spiral, referred to as the disc height, is . The fixed configuration parameters throughout this paper are listed in Table I. The disc radius and the disc height are varied subject to the objectives of the analysis. Note

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NAKANO et al.: A WIDEBAND CIRCULARLY POLARIZED CONICAL BEAM FROM A TWO-ARM SPIRAL ANTENNA EXCITED IN PHASE

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Fig. 1. Spiral with a small disc, designated as the SD. Note that this figure is used to illustrate the structural parameters and the relative proportions of the components are not accurately depicted.

TABLE I CONFIGURATION PARAMETERS

that the antenna circumference, defined by , is two wavelengths at a frequency of 2.8 GHz. Analysis throughout this paper is performed using programs developed on the basis of the finite-difference time-domain method (FDTDM) [11], [12], where the curves appearing in the antenna structure are approximated using staircases (one stair has a length of 1/200 wavelength at 3 GHz). The antenna analysis space is truncated by applying the second order Liao’s absorbing condition [13]. The radiation field is calculated and magnetic using the equivalent electric current density on a surface enclosing the antenna, where current density and are calculated using the obtained frequency-domain FDTDM magnetic and electric fields, respectively. First, the effect of a conducting disc (a ground plane) on the radiation pattern/field is analyzed. For this, a frequency range of 2 GHz to 10 GHz is used, where the disc radius and disc height and , respectively. are fixed to be The analysis reveals that the radiation field has zero intensity on the antenna axis (z-axis) and maximum intensity off the antenna axis, as shown in Fig. 2, where the solid and dotted curves show and left-handed CP components, the right-handed respectively. Note that the direction of the maximum radiation and field in the positive-z space is expressed by . the -axis is the x-axis rotated around the z-axis by The radiation mechanism is explained qualitatively using current-band theory [1], as follows. The in-phase currents at the starting point of the two arms travel toward the arm ends, gradually radiating the power input to the antenna into free space. These currents are referred to as the out-going currents. When the out-going currents reach a ring region on the spiral plane where the circumference is two wavelengths, the currents on the neighboring arms form four current bands (in-phase currents rotating on the ring region), as shown in Fig. 3. These current bands, designated as the -current bands, form a zero radiation field on the z-axes and a maximum radiation field off the z-axes, as seen from the direction of the current bands. Note that,

Fig. 2. Frequency response of the radiation pattern for a disc radius of r = 1 mm.

=

10 mm and a disc height of h

for the circumference of the present spiral , the -current bands begin to appear at 2.8 GHz. Note also that the cross-polarization component ( in the positive-z space and in the negative-z space in Fig. 2) observed at lower frequencies is due to the finite length of the spiral arms; as the frequency decreases, the arm length becomes shorter in terms of the operating wavelength. This electrically shorter arm length increases undesirable in-coming currents that travel back from the arm ends to the feed point, resulting in an increase in the cross-polarization component. As the frequency is increased from 2.8 GHz, the -current bands shift toward the inner region of the spiral plane. When the -current bands are formed on a ring region on the spiral above the conducting disc, the radiation toward the negative-z space is reduced, due to the presence of the disc. As a result, the radiation in the positive-z space and the radiation in the negative-z space become asymmetric with respect to the antenna plane (x-y plane). This occurs at frequencies greater than 9.55 GHz at which the disc circumference is two wavelengths.

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Fig. 3. Current bands in a ring region whose circumference is two wavelengths.

Fig. 4. Frequency response of the maximum field directions  for r = 20 mm.

and 

It should also be mentioned that, as the frequency increases, the out-going currents form new current bands, in addition to -current bands. Thus, higher-order -current bands the appear as long as the out-going current is not completely zero. These higher-order current bands form a zero field along the antenna axis and maximum radiation in the direction off the antenna axis, as with the -current bands. However, their contribution to the radiation field is not large, compared with that of the -current bands, due to gradual decay of the out-going currents traveling toward the arm ends. So far, we have discussed the radiation mechanism for a disc . For completeness, we use different radius of (15 mm and 20 mm) and analyze the frequency response (measured from the posiof the maximum field directions (measured from the negative-z axis in the x-z plane) and tive-z axis in the x-z plane). The symmetry starts at the frequency where the spiral’s electrical circumference is and approximately two wavelengths ends at the frequency where the disc’s electrical circumference is approximately two wavelengths for and for . Fig. 4 shows this fact, where a value of is used and frequencies and are marked with black triangles. In reality, a coaxial line is used to feed the SD, where the outer conductor of the coaxial line is connected to the small disc. In such a case, the current on the small disc flows out along the surface of the outer conductor, resulting in distortion of the radiation pattern. In the next Section II-B, the discussion includes the use of a practical antenna feed system.

Fig. 5. Spiral backed by a cavity. (a) Perspective view. (b) Side view. (c) Inner view.

B. Unidirectional Radiation A bidirectional beam is not necessary for most applications. This subsection investigates the transformation of the bidirectional beam of the SD described in Section II-A into a unidirectional conical beam. For this, a conducting cavity of height/ depth is used as a reflector, as shown in Fig. 5, where is chosen to equal the distance from the bottom of the cavity to the spiral antenna plane, (i.e., ). This cavity-backed spiral, referred to as the CSD, is excited using a coaxial line, where the inner and outer conductors of the coaxial line are connected to the center point of the spiral plane and the small disc, respectively, thereby exciting the spiral arms in phase. (In reality, the small disc is rigidly soldered to the outer conductor of the coaxial line in parallel to the spiral plane.) When the frequency increases and the antenna height becomes one-half wavelength, the radiation in the direction normal to the antenna plane becomes zero, as undesired. Therefore, it is recommended that the antenna height be small, in order to increase the high frequency edge of the operating band. However, as the antenna height is decreased, the in-coming currents increase and the wideband characteristics of the spiral deteriorate. For reducing these undesirable in-coming currents, a ring-shaped absorbing strip (ABS) is placed behind the outer arm region of the spiral [5]–[8]. The reduction in the in-coming currents decreases the generation of cross-polarized radiation and makes the polarization circular.

NAKANO et al.: A WIDEBAND CIRCULARLY POLARIZED CONICAL BEAM FROM A TWO-ARM SPIRAL ANTENNA EXCITED IN PHASE

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TABLE II PARAMETERS FOR THE CAVITY AND ABSORBING STRIP (ABS)

Table II shows the parameters for the ABS and the cavity used in this subsection. The ABS has an inner circumference of , which electrically corresponds to two wavelengths at 3.3 GHz. This frequency is referred to as . It is emphasized that the cavity the lowest design frequency (and hence the antenna height ) is chosen to height be extremely small for realizing a low-profile antenna: 0.077 . Note that the electrical characteristics for the wavelength at ABS used for the theoretical and experimental investigations are the same as those for a commercially available non-conducting absorber (ISFA EM by TDK); within a design frequency range (3.3 GHz) to (9.6 GHz) varies from 1.98 to 1.77 of and tan varies from 0.71 to 0.39, where is the real part of the and is the dielectric relative permittivity loss angle . These parameters are discussed in [7], [8]. is varied, while the disc In Section II-A, the disc radius height is fixed at . Conversely, in the following discussion, the disc height is varied, while the disc radius is fixed at . The other parameters are listed in Tables I and II. An increase in the disc height means that the radiation from the vertical line between the spiral and the disc increases. This increased radiation deteriorates the radiation from the horizontal spiral arms. Fig. 6 shows this fact, where the radiation patterns at 8 GHz are presented for four values of the disc height (1, 3, 5, and 7 mm; the disc is in contact with the bottom of the cavity when ). For checking the validity of the theoretical radiation pattern, an experiment is performed for . It is observed that a conical radiation field with a low cross-polarization component is obtained, where the maximum radiation is in the direction of . Fig. 7 shows more detail regarding the direction of the maximum radiation field within the design frequency range of to , where the disc height is held at . The solid line depicts the FDTDM result and the white dots depict approximated results obtained using the -current bands. Both the FDTDM result and the approximation reveal that variation in is small. Note that the approximated radiation pattern using the -current bands from which is obtained is expressed as

(1) where ),

wave number for an operating wavelength of (distance between the -current bands). The

Fig. 6. Radiation pattern at 8 GHz with the disc height h

Fig. 7. Frequency response of the beam direction 

for h

as a parameter.

= 1 mm.

value of in the first term is 2. The second term is derived from the effect of the image of the current bands caused by the bottom plane of the conducting cavity. Figs. 8 and 9 show the frequency response of the axial ratio (AR) and the gain for the right-handed CP radiation relative to an isotropic antenna , respectively, where Figs. 8(a) and

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Fig. 8. Frequency response of the axial ratio with disc height h as a param= 1 mm. (b) For h = 3, 5, and 7 mm (theoretical value eter. (a) For h only).

9(a) are for , presenting both the theoretical and experimental results, while Figs. 8(b) and 9(b) are for , 5, and 7 mm, showing only the theoretical results. These results are observed in the direction, based on the fact that the maximum radiation appears in the vicinity of . It is found that, as the disc height is decreased, a good axial ratio of less than 3 dB is obtained over a wide frequency range, because the effect of the radiation from the vertical line (of length ) decreases. It is also found that, as the frequency decreases, the gain decreases. This is attributed to the fact that, as the -current bands approach the ABS (due to the fact that the frequency decreases), more electromagnetic power is absorbed by the ABS. Note that the gain in Fig. 9 is calculated using the input current, the input impedance, and the radiation field component , where the input current is obtained by integrating the FDTDM magnetic field around the arm at the antenna input, and the input impedance is calculated as the ratio of the applied voltage to the obtained input current. III. VSWR, CURRENT DISTRIBUTION, AND RADIATION EFFICIENCY OF A CSD The CSD is further analyzed in this section. For the following discussion, a disc radius of and a disc height of are used, based on the investigation in Section II-B. The other parameters used in this section are shown in Tables I and II. Fig. 10 shows the theoretical frequency response of the VSWR (solid curve), together with experimental results (white

Fig. 9. Frequency response of the gain with disc height h as a parameter. (a) For h = 1 mm. (b) For h = 3, 5, and 7 mm (theoretical value only).

Fig. 10. Frequency response of the VSWR for the CSD.

dots). It is found that the CSD has a wideband VSWR characteristic. Note that this wideband characteristic is realized with the help of the ABS [in contrast, the use of the ABS for cavity-backed dipole, patch, and slot antennas does not widen the operating bandwidth due to their resonant structures]. If the ABS is not used for the CSD, non-decaying in-coming currents superimpose on the out-going currents at the feed point. This results in a total feed-point current (as a function of frequency) that is not constant, resulting in a variation in the input impedance (and hence the VSWR). The effect of the ABS on the current distribution is clearly shown in Fig. 11, where two situations are presented: one with the ABS, shown in Fig. 11(a), and one without the ABS, shown in Fig. 11(b). When the ABS is present, the spiral has a traveling current distribution, which smoothly decays from the feed point F to the arm end . On the other

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Fig. 12. Frequency response of the radiation efficiency.

Fig. 11. Comparison of current distributions. (a) With ABS. (b) Without ABS.

hand, when the ABS is removed, the spiral has a standing-wave current distribution; this means that the current is composed of an out-going current (traveling from feed point F to arm end point ) and an in-coming current (traveling from to F). The in-coming current deteriorates the wideband VSWR characteristic, as well as the purity of the circular polarization, as mentioned in Section II-B. A disadvantage of the CSD having the ABS is that, as the frequency is decreased, the absorption of the currents at the arm ends increases, resulting in a decrease in the radiation efficiency , defined as the ratio of the radiated power to the input power. Fig. 12 shows this fact, where the calculation is performed assuming that all conducting materials are lossless. The radiation efficiency at the lowest design frequency of is 20% and is more than 90% at the highest design frequency of . Note that, for confirmation, the radiation efficiency obtained using a commercially available program (Microwave Studio by CST) is also presented in Fig. 12 (represented by white dots). IV. OTHER FINDINGS This section presents some findings, which would be useful for designing the CSD. 1) The minimum value of used in Sections II and III is 1 mm. Analysis reveals that, as is decreased from 1 mm, the VSWR deteriorates [the input resistance decreases, while the input reactance remains unchanged]. However, it is found that the axial ratio in the beam direction and the directivity are not remarkably affected by the decreased . It follows that a disc height of

is the optimum value for the antenna characteristics. 2) If the small disc in Fig. 5 is replaced by a metal cylinder of the same radius continuing all the way down to the bottom of the cavity, then omni-directionality in the radiation pattern is deteriorated at high frequencies (above 8 GHz). This is due to the fact that the radiation from the spiral toward the cavity is scattered by the cylinder and this scattered wave is destructively superimposed onto the direct radiation from the spiral into free space. 3) The CSD is mounted on a mobile (for example, cars, ships, helicopters). Irrespective of the mobile’s movement, the omnidirectional radiation of the CSD makes it possible to communicate between the mobile and a base station. Therefore, it is worthwhile to investigate the situation where the CSD is placed on a ground plane. It is found that the frequency response of the VSWR is not remarkably affected by the ground plane, as shown in Fig. 13, where the ground plane is modeled by a round conducting plate . It is also found that, as of diameter is decreased, the radiation pattern at lower frequencies below 3 GHz becomes remarkably different from that for , as shown in Fig. 14. V. CONCLUSIONS A spiral antenna located in free space, which has a small disc and is excited in phase (designated as the SD), radiates a bidirectional beam. It is found that the current-band theory qualitatively explains the behavior of the radiation pattern of the SD. Subsequently, the bidirectional beam from the SD is transformed into a unidirectional conical beam using an extremely shallow cavity, whose height is 0.077 wavelength at the lowest . This cavity-backed spiral design frequency of antenna with a small disc (designated as the CSD) forms a maximum radiation field (of circular polarization) off the antenna axis and a zero field on the antenna-axis, within a design frequency range of to . It is also found that, as the distance between the disc and the spiral plane, , is decreased, variation in the axial ratio as a function of frequency becomes smaller, as desired. When a small value of is selected, the VSWR shows a good value of less than 2 within the design frequency range of to . It is

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Fig. 13. VSWR with the ground plate diameter

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D

as a parameter. (a)

Fig. 14. Radiation pattern with the ground plate diameter

D

D = 250 mm. (b) For D = 80 mm, 150 mm, and infinity (theoretical value only).

as a parameter.

D = 80, 150, 250, and infinity [mm].

NAKANO et al.: A WIDEBAND CIRCULARLY POLARIZED CONICAL BEAM FROM A TWO-ARM SPIRAL ANTENNA EXCITED IN PHASE

concluded that the CSD realizes a circularly-polarized conical beam over a 98% frequency band with a low-profile structure. ACKNOWLEDGMENT The authors thank V. Shkawrytko for his assistance in the preparation of this manuscript. REFERENCES [1] J. A. Kaiser, “The Archimedean two-wire spiral antenna,” IRE Trans. Antennas Propag., vol. 8, no. 3, pp. 312–323, May 1960. [2] H. Nakano, K. Nogami, S. Arai, H. Mimaki, and J. Yamauchi, “A spiral antenna backed by a conducting plane reflector,” IEEE Trans. Antennas Propag., vol. 34, no. 6, pp. 791–796, Jun. 1986. [3] H. Nakano, H. Mimaki, J. Yamauchi, and K. Hirose, “A low profile Archimedean spiral antenna,” in IEEE Int. Symp. Digest on Antennas Propag., Ann Arbor, MI, Jun. 1993, pp. 450–453. [4] S. Yamaguchi, H. Miyashita, K. Nishizawa, K. Kakizaki, and S. Makino, “Tow resistors loaded ultra-wideband small spiral antenna,” IEICE Trans., vol. J90-B, no. 9, pp. 830–836, Sep. 2007. [5] J. J. H. Wang and V. K. Tripp, “Design of multioctave spiral-mode microstrip antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 332–335, Mar. 1991. [6] J. J. H. Wang, “The spiral as a traveling wave structure for broadband antenna applications,” Electromagnetics, vol. 20, no. 4, pp. 323–342, Jul. 2000. [7] H. Nakano, S. Sasaki, H. Oyanagi, and J. Yamauchi, “Cavity-backed Archimedean spiral antenna with strip absorber,” IET Proc. Microw., Antennas Propag., vol. 2, no. 7, pp. 725–730, October 2008. [8] H. Nakano, K. Kikkawa, Y. Iitsuka, and J. Yamauchi, “Equiangular spiral antenna backed by a shallow cavity with absorbing strips,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2742–2747, Aug. 2008. [9] J. Yamauchi, K. Hayakawa, and H. Nakano, “Second-mode operation of an Archimedean spiral antenna backed by a conducting plane reflector,” Electromagnetics, vol. 14, no. 3 & 4, pp. 319–327, Jul. 1994. [10] J. R. Donnellan, “Second-mode operation of the spiral antenna,” IRE Trans. Antennas Propag., vol. 8, no. 6, p. 637, Nov. 1960. [11] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, pp. 302–307, May 1966. [12] A. Taflove, Computational Electrodynamics: The Finite-Difference Time Domain Method. Norwood, MA: Artech House, 1995. [13] Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin., ser. A, vol. 27, no. 10, pp. 1063–1076, Oct. 1984.

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Hisamatsu Nakano (M’75–SM’87–F’92–LF’2011) received the Dr. E. degree from Hosei University, Tokyo, in 1974. Since 1973, he has been a member of the faculty of Hosei University, where he is now a Professor of EE Department. His research topics include numerical methods for low- and high-frequency antennas and optical waveguides. In 1994, he received the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION H. A. Wheeler Award. He was also the recipient of the Chen-To Tai Distinguished Educator Award (from the IEEE Antennas and Propagation Society) in 2006 and the recipient of the Prize for Science and Technology (from Japan’s Minister of Education, Culture, Sports, Science, and Technology) in 2010. Prof. Nakano was elected an IEEE fellow in 1992. He is an Associate Editor of several journals and magazines, such as Electromagnetics, IEEE Antennas and Propagation Magazine, IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Hiroshi Oyanagi was born in Niigata, Japan, on September 22, 1986. He is currently working toward the M.E. degree in electronic informatics from Hosei University, Tokyo, Japan. Mr. Oyanagi is a Member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan.

Junji Yamauchi (M’84–SM’08) was born in Nagoya, Japan, on August 23, 1953. He received the B.E., M.E., and Dr. E. degrees from Hosei University, Tokyo, Japan, in 1976, 1978, and 1982, respectively. From 1984 to 1988, he served as a Lecturer in the Electrical Engineering Department of Tokyo Metropolitan Technical College. Since 1988, he has been a member of the faculty of Hosei University, where he is now a Professor of Electronic Informatics. His research interests include optical waveguides and circularly polarized antennas. He is the author of the book Propagating Beam Analysis of Optical Waveguides (Baldock, Hertfordshire, U.K.: Research Studies Press, 2003). Dr. Yamauchi is a member of the Optical Society of America and the Institute of Electronics, Information and Communication Engineers of Japan.

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A Low-Profile Printed Drop-Shaped Dipole Antenna for Wide-Band Wireless Applications Giovanni Cappelletti, Diego Caratelli, Renato Cicchetti, Senior Member, IEEE, and Massimiliano Simeoni, Member, IEEE

Abstract—A class of printed antipodal drop-shaped dipole antennas for wideband wireless communication systems is presented. A suitable shaping of the feeding lines and radiating arms is adopted to achieve an operating bandwidth larger than 10 GHz useful to meet the requirements of several wireless communication standards. A thin, low permittivity dielectric substrate is used to reduce the excitation of surface waves which are responsible for a degradation of the radiative characteristics. The proposed antenna structures present a reduced occupation volume which allows an easy integration in mobile terminals, as well as in radio base stations. A locally conformal FDTD numerical procedure has been adopted to analyze the radiating structures. An equivalent circuit, useful to predict the frequency-domain behavior of the scattering parameters of a two-element array formed by the proposed structures, is also presented. The numerical results concerning the antenna parameters are found to be in good agreement with the experimental measurements. Index Terms—Antenna array equivalent circuit, circular dipole antenna, drop-shaped dipole antenna, locally conformal FDTD scheme, printed antenna, time-domain response UWB antenna.

I. INTRODUCTION

I

N recent years, the demand for wideband wireless communication systems and radars for special applications is significantly increased. In particular, the development of high-performance radio systems for multi-standard local area networks (LANs) [1]–[4], the monitoring of the automotive transportation [5], [6], the remote screening of biological activities [7]–[9], as well as the analysis of the ground subsurface [10]–[12], and through-the-wall radar imaging [13], require the adoption of broad-band antennas, having low profile, high efficiency and reduced realization costs. Printed antennas, by virtue of their well known properties, can satisfy most of these requirements, although the inherent limitations of this technology, such as the excitation of surface waves, the reduced radiation efficiency, and the typical narrow-band behavior, need innovative design solutions aimed at improving their electromagnetic performances. To this end, several solutions, based on the adoption of multi-resonant cavity antennas, horn and ultrawideband (UWB) dipoles or monopoles have been proposed in the scientific literature [14]–[24]. However, horn Manuscript received July 31, 2010; revised February 03, 2011; accepted March 12, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. G. Cappelletti and R. Cicchetti are with the Department of Electronic Engineering, University of Rome “La Sapienza”, 18-00184 Rome, Italy (e-mail: [email protected]). D. Caratelli, and M.Simeoni are with Delft University of Technology, IRCTR, Delft, The Netherlands. Digital Object Identifier 10.1109/TAP.2011.2163767

antennas, although characterized by a wide operating bandwidth, are difficult to integrate into mobile communication systems, whereas multi-resonant antennas, typically consisting of coupled cavities [1]–[4], are generally characterized by a larger occupation area compared to antennas specifically designed for UWB applications, and feature a stronger dispersive behavior which may be responsible for a significant signal distortion not suitable to pulsed communications [25], [26]. Even though UWB dipole and monopole antennas can operate over a broad frequency range, their integration with a host platform can result in a number of issues related to the intrinsic radiation characteristics. In fact, while dipole antennas featuring an omnidirectional radiation patterns do not prevent the energy emission in the direction of the radio frequency circuitry, monopoles do not radiate along their axes, making an array of monopoles not suitable to work in that direction. Moreover, most of these antennas are realized on thick substrates having high dielectric permittivities, which generally make these structures not conformal, more difficult to integrate on curved surfaces of mobile devices, and affected by dispersive phenomena because of a higher excitation level of surface waves [27]. To overcome these limitations, in this work a new class of low-profile printed drop-shaped dipole antennas for wireless applications, useful to meet the demanding requirements of different communication standards, such as WLAN/WiMAX and UWB protocols, is proposed. The antennas, which are realized on a thin flexible substrate, can be usefully employed into portable devices (such as mobile pc, printers, scanners, DVD players, and digital projectors) [19], radio base stations, or into new generation textile tissues [28]. The paper is organized in five sections. In Section II, the electromagnetic performances of the proposed structures are analyzed, in Section III the time-domain characteristics are derived and discussed, while in Section IV an equivalent circuit, consisting of frequency-independent lumped elements, describing the scattering matrix of a two-element array based on the proposed radiating structures, is presented. Finally, some concluding remarks are given in Section V. II. FULL-WAVE ANALYSIS OF THE ANTIPODAL CIRCULAR AND DROP-SHAPED DIPOLE ANTENNAS In this section, the electromagnetic characteristics of two different antenna geometries consisting of antipodal dipoles having circular and drop-shaped arms, are analyzed. Balanced/unbalanced feeding structures have been considered in both cases. The geometry of the proposed antennas is depicted in Fig. 1. The radiating arms, having circular and drop-like shape, are

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TABLE I GEOMETRICAL PARAMETERS DESCRIBING THE TAPERED FEEDING STRUCTURE

Fig. 1. Geometry of the circular (a) and drop-shaped (b) antipodal dipole antenna employing an unbalanced feeding line. Structure characteristics: mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, . The reference system used to analyze the structures is also shown.

printed on a dielectric substrate having dimensions mm and mm, relative dielectric permittivity , and thickness mm equal to about 0.6% of the free-space wavelength at the working frequency GHz. A commercially available low-permittivity substrate has been selected in order to reduce the level of the surface waves which are responsible for a degradation of the antenna performance. Since the radiation properties of the antenna are strongly affected by the feeding structure, a dedicated study has been carried out in order to determine the optimal shaping of the input line useful to achieve a good performance of the radiating structure in terms of 50 impedance matching. To this end, a suitably tapered transmission line, whose -profile is described by (1) [shown at the bottom of the page] has been specifically designed. In (1), and denote the width of the initial and final sections of

the tapered feeding line, the relevant length, and the width of the ground plane (see Fig. 1). An extensive parametric analysis, useful to determine the optimal value of the aforementioned parameters, as well as the optimal distance between the radiating arms has been carried out using a locally conformal finite-difference time-domain FDTD technique [29]. In Table I the optimized values of these parameters for all the proposed radiating structures are listed, while in Fig. 2, the frequency behavior of the antenna input reflection coefficient for the drop-shaped printed dipole against variations of the parameters and considered in the aforementioned optimization process is reported. It should be observed that, because of the electromagnetic coupling, the optimization process requires the radiator and the balun structures to be analyzed together. Finally, Fig. 3 shows the frequency behavior of the reflection coefficient of the circular and the drop-shaped printed dipole antennas derived using the mentioned FDTD technique, a commercial electromagnetic field solver based on the finite integration technique FIT (CST Microwave Studio), and by experimental measurements on physical prototypes. As it appears in Fig. 3, the proposed antennas feature a bandwidth larger than 10 GHz at 10 dB return-loss level , and a bandwidth of about 12 GHz at 6 dB return-loss level .A good agreement between the FDTD numerical results and the experimental measurements is observed. Using the computed electromagnetic field values, the current density distribution excited on the dipole arms, as well as the radiation patterns at the working frequencies GHz, GHz and GHz, have been evaluated for each antenna structure. The maps relevant to the surface current densities, not reported here for the sake of brevity, show that the surface currents are mainly confined along the edges of the metallic patches forming the radiators, and present a global maximum in proximity of the excitation region. In addition, at the working

(1)

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Fig. 2. Magnitude of the input reflection coefficient versus frequency for the drop-shaped dipole antenna: (a) behavior with respect to parameter, (b) behavior with respect to parameter.

Fig. 3. Magnitude of the input reflection coefficient versus frequency for the circular (a) and for the drop-shaped (b) dipole antenna. A good agreement between the FDTD numerical results and the experimental measurements is observed.

frequency GHz, the real and imaginary parts of the current density excited on each dipole arm exhibit an essentially uniform phase distribution, resulting in a half-wavelength dipole-like radiation pattern (see Fig. 4), whereas at higher frequencies a more complex current distribution results in a modification of the radiation patterns as shown in Figs. 5–6. This behavior is inherently related to the diffraction processes occurring in the structure. As it is well know, such processes strongly depend on the size of the antenna/scatterer compared to the wavelength of the harmonic components of the signal used to excite the structure [27], [30]. The spectral variation of the surface currents excited on the radiating arms of the device caused by the aforementioned phenomena, and the surface waves due to the presence of the dielectric substrate, determine the level of the reactive energy storage, and consequently the frequency band of the antenna. However, the special geometry adopted for the proposed antennas leads to a reduced storage of such energy resulting in a wideband behavior of the considered radiating structures. From Figs. 4–6 it appears that the circular dipole exhibits a conventional radiative behavior, whereas the drop-shaped antenna is characterized by a more interesting wave radiation mechanism due to the particular geometry of the dipole arms resulting in a quasi-travelling behavior of the radiated electromagnetic field. This phenomenon, which can be clearly observed at the frequency GHz (see Fig. 5), is confirmed by the analysis of the spatial distribution of the real part of the

Poynting vector excited along the principal cut-planes of the antenna (see Fig. 7). As it is well known, the spatial directions along which the real part of the Poynting vector features local maxima are those where the far-field radiation patterns present the main and the secondary lobes [30]. As it can be noticed in Fig. 7, the adoption of the drop-shaped arms leads to a remarkable guiding effect of the electromagnetic energy along the back-side contour of the antenna arms resulting in a higher broadside radiation level, as well as in a reduced energy emission in the backward direction. This in turn is useful to achieve a stronger electrical insulation between the antenna feeding structure and the radio frequency (R.F.) circuitry, which may be exploited in the realization of compact mobile R.F. terminals. At higher frequencies, the vortexes and reflections of the surface currents partially suppress the quasi-travelling field behavior as it appears in Fig. 6. Finally, the analysis of the electromagnetic field maps in the -plane, not shown here for the sake of brevity, has outlined that the modal conversion of the surface waves into volume waves, responsible for a degradation of the radiation patterns, occurs at the truncation of the dielectric substrate. The performance featured by the antenna in realistic operative scenarios has been found to be very robust, and only marginal changes compared to the antenna operation in free-space have been noticed. To this end, the radiation pattern of the droplike dipole antenna integrated on a WLAN communication card

CAPPELLETTI et al.: A LOW-PROFILE PRINTED DROP-SHAPED DIPOLE ANTENNA FOR WIDE-BAND WIRELESS APPLICATIONS

Fig. 4. Antenna gain patterns in the -plane (a) and in the -plane (b) for the circular (dashed line) and the drop-shaped dipole antenna (continuous line). Working frequency: 5.25 GHz. A slight quasi-traveling wave behavior is observed in the radiation diagrams relevant to the drop-shaped dipole antenna. For the sake of clarity, only the measured radiation patterns of the drop-shaped dipole are reported.

installed on the laptop computer depicted in Fig. 8 has been evaluated (see Fig. 9). In doing so, the geometry and electrical properties of the laptop have been selected according to [31]. From Fig. 9 it appears that the laptop is responsible for an increase of the antenna gain along the broadside direction, as well as for a reduction of the spurious radiation level in the direction of feeding line, although no degradation of the antenna return loss has been noticed.

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Fig. 5. Antenna gain patterns in the -plane (a) and in the -plane (b) for the circular (dashed line) and the drop-shaped dipole antenna (continuous line). Working frequency: 8 GHz. A quasi-traveling wave behavior is observed in the radiation diagrams relevant to the drop-shaped dipole antenna. For the sake of clarity, only the measured radiation patterns of the drop-shaped dipole are reported.

signal processing and/or modulation [14], [19]. To this end, the suitability of the proposed antennas to work with impulsive signals has been assessed by evaluating the signal integrity [32], and the relative group delay of the radiated field [33]. The antenna excitation signal has been selected to be the fifth derivative of the Gaussian pulse

III. TIME DOMAIN ANTENNA BEHAVIOR The antenna time-domain response is an important feature of a pulsed radio communication system [14], [19], [25], [26], [32]–[34]. Typically, the antenna response is affected by ringing, spreading, and distortion phenomena resulting in a degradation of the system performance in terms of bit rate and signal-to-noise ratio. Although in some applications such processes may be of secondary importance, it is to be pointed out that the adoption of antennas characterized by a reduced signal distortion is important in order to drastically reduce the complexity of the radio frequency circuitry devoted to the

(2) The evaluation of the signal integrity has been carried out using the first derivative of such signal. In (2), defines the time at which the Gaussian pulse exhibits the peak value, whereas denotes the relevant variance. It is worth mentioning that the signal in (2) allows satisfying the requirements in terms of equivalent isotropically radiated power (EIRP) proposed by the Federal Communications Commission (FCC) for indoor UWB wireless communications [34].

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Fig. 7. Spatial distribution of the real part of the Poynting vector in the -plane at the air-dielectric interface (a), and in the -plane along the feeding line axis (b). Working frequency: 8 GHz. An interesting guiding effect of the electromagnetic energy along the back-side contour of the antenna arms and a quasi-traveling field behavior is observed in the radiated power density flux. Fig. 6. Antenna gain patterns in the -plane (a) and in the -plane (b) for the circular (dashed line) and the drop-shaped dipole antenna (continuous line). Working frequency: GHz. For the sake of clarity, only the measured radiation patterns of the drop-shaped dipole are reported.

The signal integrity, the relative group delay, and the time-domain response of the considered unbalanced fed antennas have been evaluated at a distance of 30 cm from the center of the dipole arms, under the assumption that the input signal, with parameters ps and ps, is characterized by a peak value of 1 V. In this way, it has been found that the drop-like dipole exhibits a signal integrity of about 93% along a wide angular sector centered on the positive -direction. This behavior is confirmed by the analysis of the relative group delay , defined as

Fig. 8. Schematic showing a laptop computer on which the drop-dipole antenna mm, mm, is installed. Laptop characteristics: mm, mm, mm, mm, mm, . The reference system adopted to define the field quantities is also shown.

(3) where

(4) is the group delay and is the phase of the radiated electromagnetic field. In (3), denotes the mean value of the group delay within the frequency band of interest. As it is well known,

the group delay properly quantifies the propagation properties of the so-called wave packet [30]. Fig. 10 shows the frequency behavior of for the considered antennas, as well as for the double side rounded bow-tie antenna (DSRBA) [18] in the frequency band 3–11 GHz, while the corresponding time-domain responses are reported in Fig. 11. From the analysis of this figure it appears that, the drop-like dipole presents enhanced time-domain characteristics with re-

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Fig. 11. Time-domain behavior of the co-polarized component of the electric field radiated by the unbalanced fed antipodal circular, drop-like and bow-tie printed antennas along the -axis. The impulse signal used to excite the structures is also shown.

Fig. 12. Two elements drop-shaped dipole antenna array geometry.

Fig. 9. Antenna gain patterns in the -plane (a) and in the -plane (b) for the drop-dipole installed on the laptop computer whose geometrical and electrical parameters are indicated in Fig. 8. Working frequency: 5.25 GHz.

Fig. 10. Frequency-domain behavior of the group delay featured by the antipodal circular, drop-like and bow-tie antennas. A better performance in terms of group delay is observed for the drop-like dipole antenna.

Fig. 13. Equivalent circuit of two-elements drop-like antenna array: quasistatic (a) and dynamic (b) sub-networks. The frequency-independent elements forming the circuit take into account for the metallic and radiation losses as well as for the mutual coupling phenomena.

IV. EQUIVALENT CIRCUIT OF A DROP-SHAPED DIPOLE TWO ELEMENTS ARRAY spect to the circular and antennas, showing reduced and slower oscillations within the considered frequency band. As discussed in [33], the nonlinearities in the group delay indicate the resonant behavior of the device.

The proposed radiating structures can be successfully employed to realize arrays useful to reduce the width of the antenna main beam, preserving at the same time a low profile and a reduced volume occupation.

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effects between the array elements, can be usefully employed to design the relevant feeding network. Moreover, it provides a meaningful insight into the natural resonant modes describing the electromagnetic field distribution excited in the considered structure. The procedure employed to extract the equivalent circuit is based on a heuristic modification of the Cauer’s network synthesis technique [35], [36]. Resistive elements are introduced to model metal, dielectric and radiation losses. The scattering matrix of the antenna pair is modeled by means of a suitable equivalent network consisting of shunt -like networks as shown in Fig. 13, where ideal transformers have been adopted in order to guarantee the physical realizability of the network. In particular, the RLC -like sub-networks are required to properly model the high-frequency resonant phenomena taking place into the structure, whereas the capacitive -like sub-network is needed to describe the circuital behavior of the antenna pair in quasi-static regime. A simple analysis of the network shown in Fig. 13 allows determining the admittance matrix of the equivalent circuit whose scattering matrix can be easily computed as follows: (5) where

is the identity matrix, and (6)

denoting the reference resistance at the input ports of the antenna elements. The synthesis procedure is based on an iterative non-linear fitting procedure [37] aimed at the minimization, over a given frequency band , of the mean-square error functional

Fig. 14. Magnitude of the scattering parameters (a), (b), (c), of two elements drop-shaped antenna array versus frequency. An excellent agreement between the numerical computations based on the conformal FDTD technique and those evaluated by means of the equivalent circuit is observed.

This feature can be easily achieved since the radiating arms of each dipole are printed on different sides of the dielectric substrate. In this way, the spacing between adjacent antennas can be reduced to zero, maintaining at the same time their electrical insulation. As an example, a two-element drop-like dipole array, whose geometry is sketched in Fig. 12, has been considered. The radiation properties of the considered array, realized on a dielectric substrate having dimensions mm and mm, have been carried out by means of the locally conformal FDTD numerical technique. In addition, an equivalent circuit, formed by frequency-independent lumped elements, describing the frequency behavior of the array scattering matrix has been derived (see Fig. 13). This circuit, suitable to model metal, dielectric, and radiation losses, as well as the mutual coupling

(7) being the frequency-dependent scattering matrix of the structure computed by means of the locally conformal FDTD technique. In this way, it has been possible to numerically evaluate the circuital parameters of the array consisting of two identical drop-shaped dipole antennas. Fig. 14 shows the behavior of the scattering parameters in the frequency range 0–14 GHz. An excellent agreement between the numerical results obtained by using the equivalent circuit model and the locally conformal FDTD scheme is observed in Fig. 14. It is worth noting that each -like sub-network is related to a specific resonant process occurring in the structure. In addition, the values of the maximum directivity, pointing angles, and array beam-widths in the -plane at different working frequencies, are listed in Table II, while in Fig. 15 the angular behavior of the absolute gain featured by the considered two-element array at the frequency GHz, corresponding to the WLAN upper bandwidth, is shown.

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Fig. 15. Gain pattern in the -plane (a) and in the -plane (b) of the drop-shaped dipole antenna (continuous line) and the two-element antenna array (dashed line). Working frequency: 5.8 GHz.

TABLE II PARAMETERS DESCRIBING THE RADIATIVE CHARACTERISTICS OF A SINGLE AND A TWO ELEMENT ARRAY ANTENNA

Fig. 16. Time-domain behavior of the co-polarized component of the electric field radiated by the two-element drop-like antenna array: scan angle (a) and (b) . Characteristics of the excitation signal: ps.

Since the antenna elements have been assumed to be excited in phase the maximum gain occurs along the broadside direction . Finally, in Fig. 16 the time-domain response of the co-polarized component of the electric field excited in the -plane of

the array at a distance cm is shown for two different scan angles ( and ) under the hypothesis that the excitation signals in (2), having peak value of 0.707 V and a suitable time delay, are adopted. A good steering property of the

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radiated energy, which can be usefully exploited in order to optimize the quality of service (QoS) in the radio link, is observed. V. CONCLUSION A new class of printed antipodal drop-shaped dipole antennas for wideband wireless applications has been presented. The proposed antennas exhibit an operational bandwidth in terms of input impedance matching at 10 dB return-loss level suitable for multi-protocol WLAN/WiMAX/UWB wireless communications. The electromagnetic characterization has been performed by means of a locally conformal FDTD numerical scheme. The proposed structures, printed on a thin, low-permittivity dielectric substrate, feature an operating bandwidth larger than 10 GHz. This performance has been achieved by means of a suitable shaping of the radiating arms and feeding lines. The analysis of the spatial distribution of the radiated electromagnetic field has shown that the energy emission substantially takes place along the edges of the radiating surfaces, whereas the modal conversion of the surface waves into volume waves occurs at the substrate truncation. In particular, while the circular dipole exhibits a more conventional radiative behavior, the drop-shaped antenna is characterized by an interesting quasi-traveling wave radiation mechanism. Such behavior results in a reduced parasitic coupling with the radio frequency circuitry which is especially useful where the proposed radiating structures are employed to realize communication systems and radio base stations. In this context, the adoption of the proposed equivalent circuit model of the antenna array not only provides a physical insight into the antenna radiation phenomena and mutual coupling effects, but also makes the design of the R.F. front end easier. In conclusion, the reduced volume occupation and the wideband behavior make the class of the proposed antennas suitable for the emerging applications of the wireless technology. REFERENCES [1] R. K. Raj, M. Joseph, C. K. Aanandan, K. Vasudevan, and P. Mohanan, “A new compact-microstrip fed dual-band coplanar antenna for WLAN applications,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3755–3762, Dec. 2006. [2] C. T. Lee and K. L. Wong, “Uniplanar printed coupled-fed PIFA with a band-notching slit for WLAN/WiMAX operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1252–1258, Apr. 2009. [3] R. A. Bhatti, Y. T. Im, and S. O. Park, “Compact PIFA for mobile terminals supporting multiple cellular and non-cellular standards,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2534–2540, Sept. 2009. [4] S. Y. Suh and S. L. Ooi, “Challenges on multi-radio antenna system for mobile devices,” in Proc. IEEE Antennas Propag. Soc. Symp., 2009, pp. 1221–1224. [5] J. Li and T. Talty, “Channel characterization for ultra-wideband intravehicle sensor networks,” in Proc. IEEE Military Commun. Conf., Oct. 2006, pp. 1–5. [6] W. Xiang, Y. Huang, and S. Majhi, “The design of a wireless access for vehicular environment (WAVE) prototype for intelligent transportation system (ITS) and vehicular infrastructure integration (VII),” in Proc. IEEE Vehicular Tech. Conf., Sept. 2008, pp. 21–24. [7] I. Oppermann, L. Stoica, A. Rabbachin, Z. Shelby, and J. Haapola, “UWB wireless sensor networks: UWEN-a practical example,” IEEE Commun. Mag., vol. 42, no. 12, pp. S27–S32, Dec. 2004. [8] M. Klemm and G. Troester, “Textile UWB antenna for wireless body area network,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3192–3197, Nov. 2006.

[9] A. Alomainy, A. Sani, A. Rahman, J. G. Santas, and Y. Hao, “Transient characteristics of wearable antenna and radio propagation channels for ultrawideband body-centric wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 875–884, Apr. 2009. [10] X. Zhuge, T. G. Savelyev, A. G. Yarovoy, and L. P. Ligthart, “Subsurface imaging with UWB linear array: Evaluation of antenna step and array aperture,” in Proc. IEEE Ultra-wideband Int. Conf., Sep. 2007, pp. 66–70. [11] M. Soliman and Z. Wu, “Design, simulation and implementation of UWB antenna array and its application in GPR systems,” in Proc. IEEE Antennas Propag. Eur. Conf., Nov. 2007, pp. 1–5. [12] A. A. Lestari and A. G. Yarovoy, “Adaptive wire bow-tie antenna for GPR applications,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1745–1754, May 2005. [13] C. Liebe, A. Gaugue, X. Zhao, J. Khamlichi, and M. Menard, “A through wall UWB RADAR with mechanical sweeping system,” in Proc. IEEE 39th Eur. Microw. Conf., Rome, Italy, 2009, pp. 1634–1637. [14] H. Schantz, The Art and the Science of Ultrawideband Antennas. Boston/London: Artech House, 2005. [15] J. Powell and A. Chandrakasan, “Differential and single ended elliptical antennas for 3.1–10.6 GHz ultra wideband communication,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2004, pp. 2935–2938. [16] J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, “Study of a printed circular disc monopole antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3500–3504, Nov. 2005. [17] N. Telzhensky and Y. Leviatan, “Planar differential elliptical UWB antenna optimization,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3400–3406, Nov. 2006. [18] T. Karacolak and E. Topsakal, “A double-sided rounded bow-tie antenna (DSRBA) for UWB communications,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 446–449, 2006. [19] B. Allen, M. Dohler, E. E. Okon, W. Q. Malik, A. K. Brown, and D. J. Edwards, Ultra-Wideband Antennas and Propagation for Communications, Radar and Imaging. New York: Wiley, 2007. [20] A. M. Abbosh and M. E. Bialkowski, “Design of ultrawideband planar monopole antennas of circular and elliptical shape,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 17–23, Jan. 2008. [21] M. Goprikrishna, D. D. Krishna, C. K. Anandan, P. Mohanan, and K. Vasudevan, “Design of a compact semi-elliptic monopole slot antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1834–1837, Jun. 2009. [22] X. N. Low, Z. N. Chen, and T. S. P. See, “A UWB dipole antenna with enhanced impedance and gain performance,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2959–2966, Oct. 2009. [23] M. Karlsson and S. Gong, “Circular dipole antenna for mode 1 UWB radio with integrated balun utilizing a flex-rigid structure,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2967–2971, Oct. 2009. [24] H. Nazli, E. Bicak, B. Turetken, and M. Sezgin, “An improved design of planar elliptical dipole antenna for UWB applications,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 264–267, 2010. [25] W. Soergel and W. Wiesbeck, “Influence of the antennas on the ultra wideband transmissions,” EURASIP J. Applied Signal Process., no. 3, pp. 296–305, Mar. 2005. [26] W. Wiesbeck, C. Sturm, W. Soergel, M. Porebska, and G. Adamiuk, “Influence of antenna performance and propagation channel on pulsed UWB signals,” in Proc. Int. Conf. Electromagnetic in Advanced App. (ICEAA), Torino, Italy, Nov. 2007, pp. 915–922. [27] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [28] C. Hertleer, H. Rogier, L. Vallozzi, and L. Van Langenhove, “A textile antenna for off-body communication integrated into protective clothing for firefighters,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 919–925, Apr. 2009. [29] D. Caratelli and R. Cicchetti, “A full-wave analysis of interdigital capacitors for planar integrated circuits,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1598–1601, May 2003. [30] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. New York: Prentice-Hall/IEEE Press, 1994. [31] C.-C. Lin, L.-C. Kuo, and H.-R. Chuang, “A horizontally polarized omnidirectional printed antenna for WLAN applications,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3551–3556, Nov. 2006. [32] D. Lamensdorf and L. Susman, “Baseband-pulse-antenna techniques,” IEEE Antennas Propag. Mag., vol. 36, no. 1, pp. 20–30, Feb. 1994. [33] W. Wiesbeck, G. Adamiuk, and C. Sturm, “Basic properties and design principles of UWB antennas,” Proc. IEEE, vol. 97, no. 2, pp. 372–385, Feb. 2009.

CAPPELLETTI et al.: A LOW-PROFILE PRINTED DROP-SHAPED DIPOLE ANTENNA FOR WIDE-BAND WIRELESS APPLICATIONS

[34] Z. N. Chen, X. H. Wu, H. F. Li, N. Yang, and Y. W. Chia, “Considerations for source pulses and antennas in UWB radio systems,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1739–1748, Jul. 2004. [35] M. E. Van Valkenburg, Introduction to Modern Network Synthesis. New York: Wiley, 1960. [36] D. Caratelli, R. Cicchetti, G. Bit-Babik, and A. Faraone, “Circuit model and near-field behavior of a novel patch antenna for WWLAN applications,” Microw. Opt. Technol. Lett., vol. 49, no. 1, pp. 97–100, Nov. 2006. [37] R. Fletcher, Practical Methods of Optimization. New York: Wiley, 1980.

Giovanni Cappelletti was born in Rome, Italy, in April 1983. He received the degree in electronic engineering (cum laude) from “La Sapienza” University of Rome, Italy, in 2007, where he is currently working toward the Ph.D. degree. His main research activities include the design of microwave passive devices and wideband antenna systems, and the analysis of the related EMC problems.

Diego Caratelli was born in Latina, Ital,y on May 2, 1975. He received the Laurea (summa cum laude) and Ph.D. degrees in electronic engineering from “La Sapienza” University of Rome, Italy in 2000 and 2004, respectively. In 2005, he joined as a Contract Researcher the Department of Electronic Engineering, “La Sapienza” University of Rome. Since 2007, he is with the International Research Centre for Telecommunications and Radar (IRCTR) of Delft University of Technology, The Netherlands, as a Senior Researcher. His main research activities include the design, analysis and experimental verification of printed microwave and millimeter-wave passive devices and wideband antennas for satellite, WLAN and GPR applications, the development of analytically based numerical techniques devoted to the modeling of electromagnetic field propagation and diffraction processes, as well as the analysis of EMC/EMI problems in sensitive electronic equipment. Dr. Caratelli was the recipient of the Young Antenna Engineer Prize at the 32nd European Space Agency Antenna Workshop. He received the 2010 Best Paper Award from the Applied Computational Electromagnetics Society (ACES). He serves as reviewer for several international journals, and is a member of ACES and the Italian Electromagnetic Society (SIEm).

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Renato Cicchetti (S’83–M’83–SM’01) was born in Rieti, Italy, in May 1957. He received the Laurea degree in electronics engineering (summa cum laudee) from the University of Rome “La Sapienza,” Rome, Italy, in 1983. From 1983 to 1986, he was an Antenna Designer at Selenia Spazio S.p.A. (now Thales Alenia Space S.p.A.), Rome, Italy, where he was involved in studies on theoretical and practical aspects of antennas for space application and scattering problems. From 1986 to 1994, he was a Researcher, and from 1994 to 1998, he was an Assistant Professor at the Department of Electronics Engineering, University of Rome “La Sapienza,” where he is currently a Full Professor. In 1998, 2002, and in 2006 he was Visiting Professor at the Motorola Florida Corporate Electromagnetics Research Laboratory, Fort Lauderdale, where he was involved with antennas for cellular and wireless communications. In 2004, he coordinated the research program “EMC/EMI characterization of an airport environment in presence of complex electromagnetic sources” of the MIUR/CNR-ENEA Italian national project (2001–2004) devoted to the protection of people and environment from EM emissions, while in 2004–2005 he was the coordinator of a research program concerning the development of a satellite link with a WLAN communication system operating within a train. He also serves as a Reviewer of several scientific journals. His current research interests include electromagnetic field theory, asymptotic techniques, electromagnetic compatibility, wireless communications, microwave and millimeter-wave integrated circuits, and antennas. Dr. Cicchetti is listed in Marquis Who’s Who in the World and Who’s Who in Science and Engineering.

Massimiliano Simeoni (S’96–M’02) was born in Rieti, Italy, on July 4, 1974. He received the Laurea degree (summa cum laude) from the University of Perugia, Perugia, Italy, in 1999, and the Ph.D. degree in microwave and optical communications from the University of Limoges, Limoges, France, in 2002. From June 2002 to April 2004, he joined the European Space Research and Technology Center (ESTEC), European Space Agency (ESA), Noordwijk, The Netherlands, as Research Fellow. In May 2004, he joined the International Research Centre for Telecommunications and Radar at Delft University of Technology, Delft, The Netherlands, where he is currently Assistant Professor in Antenna Systems. His research interests include the numerical characterization of guiding structures, waveguide discontinuities, microwave passive filter design, ultrawideband antennas, antenna measurement techniques, dielectric resonator antennas and phased-array antenna systems. Dr. Simeoni serves as reviewer for several international journals and conferences.

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The Impact of Vehicle Structural Components on Radiation Patterns of a Window Glass Embedded FM Antenna James H. Schaffner, Senior Member, IEEE, Hyok J. Song, Member, IEEE, Arthur Bekaryan, Member, IEEE, Hui-Pin Hsu, Member, IEEE, Mark Wisnewski, and Janalee Graham

Abstract—The relative long wavelength of FM radio signals compared to a typical vehicle length means that a window glass-embedded FM band antenna couples strongly to vehicle structural components. Numerical simulations have been conducted to identify which key structural components in a full-size SUV have an effect on the average antenna gain patterns. In particular, it was found that the rear hatch and the seats of the SUV had a large effect on FM antenna gain patterns and thus required more careful modeling for the electromagnetic simulation. Also, it was found that the sunroof status, opened or closed, had a moderate effect on FM antenna gain patterns. Comparison with experimentally obtained FM antenna gain patterns demonstrated the importance of using the most detailed model possible when simulating patterns in the FM band. Index Terms—Automotive antennas, FM antennas, glass embedded antennas, vehicle electromagnetic.

I. INTRODUCTION REATING a vehicle model for engineering simulation starts by collecting three-dimensional parts from a component library. A critical preliminary task for simulation is selecting only those few parts, out of the thousands of vehicle parts, that will lead to acceptable simulation results; otherwise, the problem becomes either too large for the available computing resources or else the time for computation becomes too long. Unlike mechanical or thermal analysis, where energy flows from one part to adjoining parts, electromagnetic energy couples to noncontiguous components of a vehicle through the space in and surrounding the vehicle. This causes the far-field FM antenna gain patterns generated by an antenna mounted on or in a vehicle to have much more variation than would be expected from the antenna mounted on a simple ground-plane. It can take many design iterations and in-situ tests to get acceptable FM antenna gain patterns. Any of these iterations that can be performed through simulation would translate into a faster design-to-manufacturing timeline.

C

Manuscript received November 02, 2010; revised January 24, 2011; accepted February 17, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. J. H. Schaffner, H. J. Song, A. Bekaryan and H.-P. Hsu are with the HRL Laboratories, Malibu, CA 90265 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). M. Wisnewski is with the OnStar/General Motors, Milford, MI 48380 USA (e-mail: [email protected]). J. Graham is with the General Motors, Milford, MI 48380 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163778

The detailed features of FM antenna gain patterns depend on where the antenna is integrated into the vehicle and the frequency of operation. The roof provides a good approximation to a simple ground plane for antennas mounted on top of a vehicle. Even for antennas near the edge of the roof, the simulated antenna gain patterns generally agree with measurements, especially at higher frequencies [1]–[5]. Because FM antennas are relatively long, automotive designers now prefer to embed them in vehicle windows. In this case, the antenna receives radiation incident from the inside of the vehicle as well as from the outside. The dimension of the vehicle itself is generally a few wavelengths for FM frequencies; thus, one would expect significant electromagnetic coupling between the antenna and the vehicle structure. Ideally, enough structural detail would be included in the vehicle model for the simulation to provide an acceptable match between the simulated and measured FM antenna gain patterns. Some recently reported simulations of in-glass FM antennas have used very simplified wire grid models of a sedan with rear window wire antennas [6], [7]. This type of modeling is useful for parametric studies of the general effect of vehicle structure dimensions on FM antenna gain patterns. However, automobile interior structures and body details (such as gaps and contours) were not included in these simulations. Other papers presented sedan antenna simulations that used more sophisticated surface mesh elements rather than wire grids [8]–[10]. The surface models used in [9] and [10] had their origin with automaker computer-aided design (CAD) models modified to allow meshing for electromagnetic simulation, and metal seats were included in the simulation in [10]. In the study presented here, we investigated the impact on FM antenna gain patterns of a window glass-embedded FM antenna caused by some of the large structural components inside the vehicle and on the outer vehicle surface. The vehicle model used in these simulations was a full size sport utility vehicle (SUV). Since we had access to the vehicle component models derived directly from the original solid model CAD files, our goal was to determine which components and what level of detail needed to be included in the model so that the simulated FM antenna gain patterns could agree with the measured FM antenna gain patterns. We looked at many scenarios; presented below are the simulations with the model structures that we found to have the largest impact on the shape of the FM antenna gain patterns ( at any given azimuth angle). These were the rows of seats, the sun-roof status (opened/closed), and the detailed structure

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of the rear hatch. Interestingly, some of the structures that did not cause much variation in FM antenna gain patterns ( at any given azimuth angle) were the presence of window glass, the engine bulkhead, the tire rims and the conductivity of the turntable and surrounding ground. II. VEHICLE MODEL AND SIMULATION VALIDATION A. Vehicle Mesh Models The components of the vehicle included in the simulation model were obtained from the solid model data library. A surface mesh of each component was created using the program AltairHyperMesh, and then these meshed components were assembled into the vehicle model to be simulated, converted to a NASTRAN formatted file, and imported into EMSS FEKO, a commercial moment method EM simulation software package. Proper modeling in FEKO required that the mesh nodes coincide at the touching boundaries between the components. This approach allowed us to selectively include/exclude individual vehicle structures. Fig. 1 shows one of the meshed models of the SUV that was used for EM simulation and the structural detail of the AM and FM antenna in the passenger side rear window. For all of the simulation results presented here, the vehicle was placed over a five-meter-diameter, perfectly conducting turntable, which was surrounded by “earth” of dielectric constant 5.0 and conductivity 0.1 S/m. These conditions were used to model the actual far-field FM antenna gain range, located at the General Motors Milford Proving Ground, Milford, MI, USA [11], on which the FM antenna gain pattern measurements were performed. The tire rims are shown in the figure of the model, although as mentioned above, their presence had no impact on the FM antenna gain patterns. Also seen in the figure, the window glass was not included, again because of its insignificance in shaping the FM antenna gain patterns. The main vehicle features that were included in this study were the seats, the sun-roof and the detailed hatch. We would have liked to also include the steering column in this model, but unfortunately it wasn’t included in the CAD model available during simulations. A typical laminated window glass integrated antenna is made from baked conductive paste traces of widths that range from a few tenths of a millimeter to a few millimeters. In our simulations, the window glass integrated antenna was modeled as perfectly conductive wires which were excited with a 1 V source segment at the location shown in Fig. 1(b). The wire antenna also included the AM window glass antenna. Due to the small size, in wavelengths, of the AM antenna, it is connected to an to LNA with an input impedance that can vary from a few 10’s of . We found that there was no discernible difference in the simulated FM antenna patterns when we terminated the AM antenna with an impedance above a few hundred ohms, so for for the AM antenna load termination. this study we used The total number of triangular segments used for the model shown in Fig. 1 was 13,232; this number was derived mostly from the vehicle body and could vary a bit depending on the particular vehicle structures in the model. In addition, the FM and AM antennas accounted for 617 wire segments. A planar

Fig. 1. (a) A surface meshed model of the SUV used in FEKO for numerical pattern simulations and (b) a close-up view of the wire AM and FM antenna structure used in the model.

multi-layer structure Green’s function was used to model the earth, and this prevented us from using the multilevel fast multipole method (MLFMM) for the solution. The solution time for each frequency was approximately nine hours on a 3 GHz Dell Precision 490 with a single processor and 16 GB of RAM and the Microsoft Windows XP x64, v. 2003 operating system. Azimuthal FM antenna gain patterns (at the horizon) are the most important for FM radio system development, so patterns at other elevation angles will not be presented here. The orientation of the vehicle with respect to the azimuth pattern angles is shown in Fig. 2. In the figure, the viewpoint is from the automobile zenith looking down. The figure also shows that the window glass antenna location used for the simulations is on the passenger side of the vehicle. B. Monopole Mast Antenna Mounted on the Roof As a check of our simulation procedure, we compared simulated patterns with the measured FM antenna gain patterns from a monopole mast antenna mounted on the roof of the SUV at the approximate roof center. The roof provided a relatively flat monopole ground plane. Most of the structural components, such as the roof racks and ridges, were horizontal, which should have had little effect on the monopole’s predominant vertically

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Fig. 2. Azimuth angle pattern orientation with respect to the vehicle.

Fig. 3. A photograph of the magnetic mount mast antenna located near the center of the SUV roof top. The sun-roof has been covered with aluminum foil to improve the ground plane for the mast monopole.

polarized FM antenna gain patterns. A photograph of the magnetic mount mast antenna is shown in Fig. 3. Also, for the measurements, the sunroof was covered with aluminum foil; other than that modification, the SUV was outfitted as a typical vehicle from the assembly line. The measured and simulated far-field vertical and horizontal radiation patterns are shown in Fig. 4 for frequencies of 88, 96, and 106 MHz. The average gains over 360 of both the simulated and measured patterns were normalized for the plots. It can be seen that the simulated patterns replicate the salient lobes and nulls found within the measured patterns. It will be seen later that pattern lobes and their positions for the window glass-em-

bedded FM antenna are greatly dependent on the detail of major internal structures. There are a number of approaches to quantify the closeness in comparison between the measured and the simulated FM antenna gain patterns [12]–[14]. We chose to use the correlelogram [15], which is a very simple technique to implement and which gives two parameters that can be directly related to the observed measured and simulated patterns. For a given pair of FM antenna gain patterns, the outputs of the correlelogram are the maximum correlation coefficient between the measured and simulated patterns and the angle that the simulated pattern must be rotated to give the maximum correlation. The normalized FM antenna gain patterns were measured and simulated at 1 increments, so the maximum correlation coefficient is found from (1a) and (1b), shown at the bottom of the page, and n and K are integers and are the normal(representing degrees), is the offset ized pattern gains linearly scaled at angle n, angle (positive is counterclockwise) that the simulated FM antenna gain pattern is rotated, and R is the correlation coefficient. for the mast FM antenna gain patterns of The values of Fig. 4 are given in Table I. The maximum correlations of the horizontally polarized patterns have more variation over frequency than the maximum correlations of the vertically polarized patterns. This may be attributable to the roof rack and roof ridges seen in Fig. 3. These features are primarily elongated in the horizontal plane and thus could be expected to have some impact on the horizontally polarized fields but little impact on the vertically polarized fields. These features were not included in the model used for simulation. As one method for reducing the pattern correlation of a model to a single value for a given set of measurement, we define the average correlation error distance (ACED) as

(2) where (3) is the distance between the tip of the lation vector

measurement correand the perfect

(1a)

(1b)

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Fig. 5. The (a) unibody rear hatch meshed surface model and (b) the detailed rear hatch meshed surface model.

III. VEHICLE STRUCTURES’ IMPACT ON THE AZIMUTH FM ANTENNA GAIN PATTERNS OF A WINDOW ANTENNA A. Detailed Hatch Vs. Unibody Hatch Models

Fig. 4. Simulated and measured horizontally and vertically polarized FM antenna gain patterns of a monopole mast antenna on the center of the SUV roof.

TABLE I MAXIMUM CORRELATION COEFFICIENT AND THE OFFSET ANGLE BETWEEN MEASURED AND SIMULATED ROOF MOUNTED MAST FM ANTENNA GAIN PATTERNS

correlation vector (1,0), and N is the total number of simulations performed. The value of ACED can range between 0 and 2. When comparing correlations between measurement and simulations between two models, the model with the smaller ACED would suggest better overall correlation to the measured patterns. The ACED will be used later to rank overall simulation pattern correlations with measured patterns for models with the window glass embedded FM antenna and various vehicle structural components.

It is not surprising that some of the larger structures on and inside the vehicle would have a significant impact on the far-field FM antenna gain patterns of the rear window glass-embedded antenna. In this section we present the results of the analysis of three structures that caused the largest variations in the simulated FM antenna gain patterns: the rear hatch, the seats, and the sunroof status. The simplest model of the SUV body included all of the vehicle doors and the rear hatch as a single perfect conducting surface; the detail of the rear of this “unibody” vehicle model is shown in Fig. 5(a). The actual vehicle rear hatch is connected to the vehicle body through the two hinges and the latch, and there is a gap between the metal of the hatch and the body with the overlapping edges forming an approximate parallel plate boundary around the hatch perimeter. In the SUV model shown in Fig. 5(b), the rear hatch included multiple surfaces that were derived from the actual vehicle solid model hatch components. We designated this as the “detailed hatch” model. Simulations were performed for these models with and without seats. The simulated normalized average FM antenna gain patterns for the two models with sheet seats are shown in Fig. 6 for 88, 98, and 108 MHz. It can be observed from these figures that there are substantial differences in the FM antenna gain patterns for the two models at identical frequencies, particularly in the presence or absence of nulls in the patterns. Fig. 7 shows the currents at 98 MHz at the rear of the vehicle for the unibody model and the detailed hatch model. It can be seen in Fig. 7(a) that for the unibody case there is are significant currents along the rear pillar and the edges of the hatch window, and even in the inside of the rear of the vehicle near the hatch window. In fact, these currents lead us to consider implementation of the more detailed hatch model. The currents in the detailed hatch model, shown in Fig. 7(b), are still significant along the rear pillar and the edges of the hatch window, although they don’t extend as far away from the windows as for the unibody case. Also, in the detailed hatch model, there is significant current on the vehicle body in the parallel plate boundary and on the hatch hinges that is of the same order of magnitude as the current around the rear window edge. These additional currents in the detailed hatch model are the likely cause of the difference in the patterns between the two models.

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Fig. 6. The azimuth patterns of the window glass antenna for the unibody hatch model and the detailed hatch model with sheet seats included.

B. Strip Seat Vs. Sheet Seat Models The metallic components that make up a seat are rather complex and difficult to readily reduce to a seat surface model. Thus, we used two different approximations to model the seats. The simplest approximation, designated the “sheet seat” model, was to use sheets of perfect conductor shaped to follow the general outline of the seats. The other seat model, the “strip seat” model, was a bit more sophisticated and used perfect conductive sheets and strips to approximate the regions of the seat that had the most metal. The two models of the seats are shown in Fig. 8 with the vehicle body removed for clarity. Both models had the seat legs electrically connected to the vehicle body and were meshed such that the touching body and seat mesh triangles shared common edges and nodes. The normalized azimuth average FM antenna gain patterns with the strip seats and the sheet seat models at 88, 98, and 108 MHz are shown in Fig. 9. The models used for these simulations also included the detailed rear hatch. From the plots, it can be seen that the vertical patterns at 88 MHz and 108 MHz, and the horizontal patterns at 88 MHz had the largest variations. The patterns at 98 MHz showed minimal differences. As with the detailed hatch, a study of the currents flowing on the seats revealed that there was quite a bit of coupling at 88 MHz, some coupling at 108 MHz, but little coupling at 98 MHz. The currents at 88 MHz are shown in Fig. 10, again with the vehicle body invisible. One could expect that even more detail of the seat models

Fig. 7. Surface currents at the rear of the vehicle at 98 MHz for (a) the unibody model and (b) the detailed hatch model. Increasing current levels have increasing shading. There is significant current around the rear window edges and also between the hatch and the vehicle body.

Fig. 8. Two approximations to the seats of the SUV made from sheets and strips of perfect electrical conductors (a) the sheet seat model and (b) the strip seat model.

that included the seat springs and bar supports would lead to additional coupling and resonances in the simulated patterns. However, due to the time investment required to add all of these metallic seat components; further refinement of the seat models was not attempted.

SCHAFFNER et al.: THE IMPACT OF VEHICLE STRUCTURAL COMPONENTS ON RADIATION PATTERNS

Fig. 9. The normalized average azimuth FM antenna gain patterns for the strip seats and sheet seats models of the SUV. The detailed hatch was included in these simulations.

Fig. 10. Surface currents on the (a) sheet seats and (b) strip seats at 88 MHz. Increasing current levels have increasing shading. It is apparent that the seats closest to the antenna (upper left) have the most coupling to the antenna.

C. Sunroof Status: Open Versus Closed The sunroof is a popular option in many vehicles. Since the sunroof effectively adds another opening in the vehicle that is on the order of a wavelength in the FM band, it would be expected that the FM antenna gain patterns would be different depending on sunroof status, that is, whether it is open or closed. The position of the sunroof for the SUV can be seen in Fig. 1(a), just above the front seats. So far, all of the modeled (and measured)

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Fig. 11. Simulated horizontally and vertically polarized normalized average FM antenna gain patterns for models with the sunroof opened or closed.

patterns presented for the SUV with the window glass embedded antenna had the sunroof open. The model with the sunroof open was created by cutting out a section of the roof with dimensions that matched the outline of the sunroof. This section could then be included in the SUV model with a closed sunroof, or not included in the model for an open sunroof. The resulting normalized antenna FM antenna gain patterns with the sunroof open or closed are shown in Fig. 11. The detailed hatch and strip seats were used for the simulations in Fig. 11. Although the pattern differences between the open and closed sunroof models aren’t as prominent as for the other structural components presented above, perhaps due to the distance between the sunroof and the antenna, there are still significant pattern differences particularly for the vertical polarizations at 98 and 108 MHz. IV. COMPARISON BETWEEN SIMULATED AND MEASURED PATTERNS We measured the azimuth vertical and horizontal linear polarized patterns for the SUV window glass-embedded antenna at the outdoor FM band antenna test range located at the General Motors Milford Proving Grounds. The actual FM window glass antenna has an impedance matching network and amplification that was not taken into account during the simulations. However, since the FM receiver circuit does not impact the shape of the FM antenna gain patterns, the use of normalized average gains was the appropriate way to compare the patterns.

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TABLE II CORRELATION COEFFICIENT, OFFSET ANGLE, AND AVERAGE CORRELATION ERROR DISTANCE BETWEEN MEASURED AND SIMULATED ANTENNA FM ANTENNA GAIN PATTERNS FOR VARIOUS MODELS WITH THE WINDOW GLASS EMBEDDED FM ANTENNA.

The trend of the correlation coefficients suggests that closer correlations between measured and simulated patterns might be possible with more detail in the seat model that includes the metallic springs and support bar components. However, the simulated results with the strip seat model may be good enough to provide design guidance for the window glass-embedded FM antenna and possible elimination of one design iteration. V. CONCLUSION

Fig. 12. Simulated normalized average FM antenna gain patterns with the detailed hatch and strip seat model compared to measured normalized average FM antenna gain patterns.

Examples of overlays of measured patterns and simulated normalized average patterns that include the detailed hatch and the strip seat model are presented in Fig. 12. Table II presents the maximum correlation coefficient, pattern angle offset, and average correlation error distance, as defined in (1a) and (1b) and (2), for the 6 models. From the table it can be seen that the detailed hatch models have smaller ACEDs for a given seat component model. In addition, for both the unibody and detailed hatch models, the strip seats lead to smaller ACEDs than the other configurations. It’s interesting to note that models using sheet seats had higher ACEDs than the models with no seats.

Window glass-embedded FM antennas couple strongly to the metallic structures throughout a vehicle. Thus, care must be taken to model these structures as realistically as possible for accurate FM antenna gain pattern simulations. Even the results described in this paper indicate that more detailed models of the structures would be beneficial to the simulated pattern validation with the measured patterns. However, the question of how much detail is needed in a model depends on the goals of the simulations set by the antenna design engineer. If the goal is to reduce one or two design iteration cycles, then perhaps capturing just the major features of the FM antenna gain patterns, as was described in this paper, is good enough. Also, the simulated results presented from this study are for one style of vehicle. Some previous studies suggested there will likely be significant pattern variations for different automobile styles and antenna placements, so design generalizations should wait. The results of the simulations and measurements presented here could lead one to expect that FM antenna gain patterns would be sensitive to vehicle loading, that is, passengers and cargo. Thus, a statistical approach to window glass-embedded FM antenna design may need to be employed. This is where FM antenna gain pattern simulations could really reduce cost in product development by conducting such a study through simulation rather than physical measurements, especially where many parametric variations are needed to build up the statistical model. In this way, robust FM window glass-embedded antennas could be designed to mitigate the effects of the varying electromagnetic environment in the vehicle. ACKNOWLEDGMENT The authors would like to extend a special thanks to M. Deering for his help in obtaining CAD models for EM simulation. The authors would also like to thank D. Martin and E. Yasan for their support of this project.

SCHAFFNER et al.: THE IMPACT OF VEHICLE STRUCTURAL COMPONENTS ON RADIATION PATTERNS

REFERENCES [1] R. Kronberger, A. Stephan, and M. Daginnus, “3D antenna measurement and electromagnetic simulation for advanced vehicle antenna development,” in Proc. IEEE Antennas and Propagation Symp., 2001, Boston, MA, vol. 3, pp. 342–345. [2] R. Leelaratne and R. Langley, “Multiband PIFA vehicle telematics antennas,” IEEE Trans. Veh. Technol., vol. 54, no. 2, pp. 477–485, Mar. 2005. [3] A. Ruddle, “Simulation of far-field characteristics and measurement techniques for vehicle-mounted antennas,” in Proc. IEEE Colloq. on Antennas Automotives, 2000, pp. 7/1–7/8. [4] L. Low, R. Langley, R. Breden, and P. Callaghan, “Hidden automotive antenna performance and simulation,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3707–3712, Dec. 2006. [5] Y.-C. Hsu, K.-H. Lin, and Y. C. Huang, “A scaled-sized model for analysis of vehicular antennas,” in Proc. IEEE Antennas and Propagation Symp., Honolulu, HI, 2007, pp. 521–524. [6] J. C. Batchelor, R. J. Langley, and H. Endo, “On-glass mobile antenna performance modelling,” IEE Proc.-Microw. Antennas Propag., vol. 148, no. 4, pp. 233–238, Aug. 2001. [7] B. A. Austin and R. K. Najm, “Wire-grid modeling of vehicles with flush-mounted window antennas,” in Proc. IEE 7th Int. Conf. on Antennas and Propagation, 1991, vol. 2, pp. 950–953. [8] R. Abou-Jaoude and E. K. Walton, “Numerical modeling of on-glass conformal automobile antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 845–852, Jun. 1998. [9] G. Gottwald, “Numerical analysis of integrated glass antenna systems,” SAE IBEC ’98, vol. 3, Sep. 1998. [10] S. Savia, R. Langley, and A. Walbeoff, “Automotive antenna simulation,” in Proc. 2nd Eur. Conf. on Antennas and Propagation, 2007, pp. 1–4. [11] J. Graham, D. Hibbard, D. Martin, S. Yencer, D. Novotny, C. Grosvenor, N. Canales, R. Johnk, L. Nagy, and T. Roach, “Outdoor vehicular test range turntable impact on electric-field uniformity study,” in Digest of the IEEE Int. Symp. on Electromagnetic Compatibility, Aug. 18–22, 2008, pp. 1–6. [12] J. McCormick, S. F. Gregson, and C. G. Parini, “Quantitative measures of comparison between antenna pattern data sets,” IEE Proc.-Microw. Antennas Propag., vol. 152, no. 6, pp. 539–550, Dec. 2005. [13] A. P. Duffy, A. J. M. Martin, A. Orlandi, G. Antonini, T. M. Benson, and M. S. Woolfson, “Feature selective validation (FSV) for validation of computational electromagnetic (CEM): Part I—the FSV method,” IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp. 449–459, Aug. 2006. [14] A. Bekaryan, H. J. Song, H. P. Hsu, J. Schaffner, and R. Wiese, “Objective metric for antenna patterns comparison using Mahalanobis-Taguchi-Gram-Schmidt method,” in Proc. IEEE 66th Vehicular Technology Conf., 2007, pp. 902–904. [15] A. Duffy, D. Coleby, A. Martin, M. Woolfson, and T. Benson, “Progress in quantifying validation data,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, 2003, vol. 1, pp. 323–32.

James H. Schaffner (S’74–M’79–SM’97) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1978, 1979, and 1988, respectively. He is currently a Senior Research Scientist with HRL Laboratories LLC, Malibu, CA, where he has been since 1988. From 1978 to 1988, he was with the Missile System Group, Hughes Aircraft Company, where he was involved with microwave and millimeter-wave slot antenna arrays, antenna feed networks, and MMIC circuits for transmit/receive modules. At HRL Laboratories LLC, he has developed microwave and millimeter wave lithium-niobate electrooptic (EO) modulators and high dynamic-range EO modulators. He has also been involved with spread-spectrum millimeter-wave communications and broadband wireless digital TV links at 60 GHz, including RF microelectromechanical systems (MEMS) switched reconfigurable antennas and circuits for wireless and radar applications. He has also developed wideband modules for passive millimeter wave imaging. His current research interests are in optical beamforming, RF photonics, phased arrays, and conformal and hidden antennas. He has authored or coauthored over 70 technical papers, and holds 51 patents in microwave, millimeter-wave, and photonics components and systems

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Hyok J. Song (S’95–M’01) received the B.E., M.Eng.Sc., and Ph.D. degrees all in electrical engineering, respectively in 1995, 1998, and 2001, from the University of Queensland, Australia. He is currently a Sr. Research Staff Engineer with the Applied Electromagnetics Laboratory at HRL Laboratories LLC, where he performs research in the field of antennas, applied electromagnetic and EMC/EMI. His primary research interests are small antennas, tunable antennas, transparent film antennas and active matching circuits. His other research interests include EMC and EMI problems in vehicles with a hybrid or electric propulsion architecture. He holds 3 U.S. patents and is a (co)author of two book chapters and more than 40 refereed journal/conference papers in the fields of antennas and applied electromagnetic. Dr. Song was the co-recipient of the 2009 Piergiorgio L.E. Uslenghi Prize Paper Award from IEEE Antennas and Propagation Society. Arthur Bekaryan (M’10) was born in Yerevan, Armania, in 1980. He received the B.Sc. degree in engineering from California State University, Northridge, in 2003 and the M.Sc. degree in engineering from University of California, Los Angeles, in 2009. From June 2002 to January 2004, he held an engineering position with Shell Solar Industries in Camarillo, CA. He assisted in transforming the complete assembly process to fully automated robotic systems. He also improved the assembly process by mitigating assembly defects by identifying and implementing improved assembly techniques. From January 2004 to present he has held a senior development engineer position with HRL Laboratories, LLC in Malibu, CA. He is involved in research in automotive antennas and radars in addition to configurable, hidden and steerable antennas. Hui Pin (H. P.) Hsu (M’05) received the D.Sc. and M.S. degrees in electrical engineering from Washington University, St. Louis, MO, and the B.S. degree in electrophysics from National Chiao Tung University, Hsinchu, Taiwan. He was a Research Program Manager in HRL Laboratories, LLC. He joined HRL in 1994. He was the principal investigator (PI) and project manager of GM/HRL Directed Research program to conduct research in the areas of vehicle antennas, mobile wireless communications and vehicle safety systems in support of GM vehicle product development from 1999 to 2009. He was the principal investigator of High-speed Digital Wireless Battlefield Network Technology Reinvestment Program (TRP) sponsored by DARPA to develop a rapidly deployable wireless/satellite hybrid communication network for the 21st century battlefield. Previously, he was the project leader of another DARPA’s TRP that developed low cost, ruggedized optical transceivers for commercial and military dual-use optical communication systems. Prior to joining HRL, he worked at Hughes Missile System Company (HMSC) from 1982 to 1994 as Principal Investigator and Project Leader in several fiber optic guided missile and active/passive optical missile seeker programs. He also conducted research on fiber optics, integrated optics, and laser communication systems at Bell Laboratories, Allentown Pa. (1978–1982) and U. S. Naval Research Laboratory, Washington D.C. (1974–1978). He has authored more than 40 journal and conference papers, and has over 30 U.S. and international patents issued. Mark Wisnewski is a project engineer with 26 years of electromagnetic validation experience for General Motors.

Janalee Graham, photograph and biography not available at the time of publication.

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Design, Realization and Measurements of a Miniature Antenna for Implantable Wireless Communication Systems Francesco Merli, Member, IEEE, Léandre Bolomey, Jean-François Zürcher, Giancarlo Corradini, Eric Meurville, and Anja K. Skrivervik

Abstract—The design procedure, realization and measurements of an implantable radiator for telemetry applications are presented. First, free space analysis allows the choice of the antenna typology with reduced computation time. Subsequently the antenna, inserted in a body phantom, is designed to take into account all the necessary electronic components, power supply and bio-compatible insulation so as to realize a complete implantable device. The conformal design has suitable dimensions for subcutaneous implantation (10 32.1 mm). The effect of different body phantoms is discussed. The radiator works in both the Medical Device Radiocommunication Service (MedRadio, 401–406 MHz) and the Industrial, Scientific and Medical (ISM, 2.4–2.5 GHz) and bands. Simulated maximum gains attain in the two desired frequency ranges, respectively, when the radiator is implanted subcutaneously in a homogenous cylindrical body phantom (80 110 mm) with muscle equivalent dielectric properties. Three antennas are realized and characterized in order to improve simulation calibration, electromagnetic performance, and to validate the repeatability of the manufacturing process. Measurements are also presented and a good correspondence with theoretical predictions is registered. Index Terms—Biocompatible antenna, dual band, implanted antennas, Medical Device Radiocommunication Service (MedRadio), miniature antenna, multilayered PIFA, spiral antenna.

I. INTRODUCTION

W

IRELESS implantable systems promise large improvements in patients’ care and quality of life. Pacemaker communication, glucose monitoring, insulin pumps and endoscopy are just a few examples of medical treatments that could take advantage of wireless control [1]. The Medical Device Radiocommunication Service band (MedRadio, 401–406 MHz) has been recently allocated [2] to this purpose. Among all the components necessary for implanted telemetry applications, the antenna plays a key role in obtaining robust communication

Manuscript received July 01, 2010; revised December 20, 2010; accepted February 28, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. F. Merli, J.-F. Zürcher and A. K. Skrivervik are with the Laboratoire d’Electromagnétisme et d’Acoustique (LEMA), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail: francesco.merli@epfl. ch). L. Bolomey, G. Corradini and E. Meurville are with the Laboratoire de Production Microtechnique (LPM2), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (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.2011.2163763

links and a significant miniaturization of the whole device. In fact, the design of an electrically small radiator (with reduced efficiency and bandwidth) in the presence of a “hostile” environment such as the human body is a challenging task. Several implantable radiators, with different characteristics and target applications, have been recently presented. Among others, an ultra-wideband compact design (3.5–4.5 GHz) is discussed in [3], whilst a conformal design for an ingestible application at 1.4 GHz, of limited lifetime, is introduced in [4]. Cavity slot radiators without integration with active components are presented for the Industrial, Scientific and Medical band (ISM, 2.4–2.5 GHz) for on-body and in-body applications in [5] and [6], respectively. In the same frequency range, in-vitro and in-vivo tests of an implantable antenna for intracranial pressure monitoring are performed in [7], [8], whereas retinal prothesis applications are investigated with miniature antennas in [9], [10]. Focusing on designs operating within the MedRadio band, planar meander and spiral structures are compared in [11], [12]. Extensive analysis of this kind of antenna typology (including the effect of different dimensions and materials) is also reported in [13], [14]. Finally, a dual band antenna, including the MedRadio band, is presented in [15] with possible subcutaneous rat implantation in [16]. The latter radiator, although planar with sharp edges, shows a remarkable high gain, achieved with the use of the particle swarm optimization method. This work focuses on the design procedure and the realization of an implantable dual band antenna, working in both the MedRadio and the ISM, (2.4–2.5 GHz) bands. The radiator is designed to be integrated with all the required active components [17] and bio-sensors or actuators so as to form a complete generic implantable wireless telemetry system [18]. An example of a potential application is an implantable glucose monitoring system with a 1-year life time [19]. The proposed design overcomes several issues, further described in Section II, leading to the realization and measurement of a successful implantable telemetry system for in-vitro (biology/chemistry) and in-vivo (medicine) applications. The manuscript is organized as follows: Section II describes the physical constraints and electromagnetic specifications required for the antenna design. The equivalent human body model is also introduced. The design procedure and the solution adopted are reported in Section III, along with the investigation of the robustness of the radiator against the presence of the active components and the insertion in different body models.

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Sections IV and V detail the fabrication and the measurements of the prototypes, respectively. The latter pays particular attention to the effect of the feeding cable used for testing purposes. Finally, the obtained results are summarized in Section VI. II. ANTENNA REQUIREMENTS AND BODY MODELING In order to perform the antenna design for a complete active implantable system, several aspects must be taken into account. These aspects may be classified into physical constraints and electromagnetic (EM) specifications. Such classification will be helpful to understand how we developed the design procedure and achieved the final structure. Moreover, the equivalent human body model is chosen at the initial stage of the work, as it strongly influences the antenna design. A. Physical Constraints — The volume of the entire implant (i.e., the antenna, insulation, electronics, batteries and biosensors) must fit in a cylindrical housing. A maximum external diameter of 10 mm is set to allow subcutaneous implantation in rats [20]; — bio-compatible insulation must embed the implantable device. The insulation avoids adverse tissue reaction [21], short-circuiting effects due to highly conductive human body tissues, and it is also very valuable for telemetry applications [3], [7], [13], [22], [23]; — a dense packaging is required to reduce the implant volume [17], hence the antenna feeding point is placed in a small predefined area to minimize the circuitry complexity; — the actual process of antenna manufacturing has to be taken into account in order to avoid extremely tight tolerances and to maximize the repeatability of the construction process itself. B. Electromagnetic Specifications — Dual band capability must be obtained to operate with the transceiver produced by Zarlink Semiconductors [24]. Such capability minimizes power consumption as the sensor can be awoken from sleeping state by receiving a signal in the 2.4–2.5 GHz band. Subsequently, data transmission occurs only in the MedRadio frequency range; — a single antenna feeding point is targeted to minimize the assembly complexity; — antenna gains higher than and are targeted in the MedRadio band and the ISM band, respectively. Note that these gain values include both the antenna radiation characteristics and the simulated body phantom presence. The corresponding radiation efficiencies are rather small, but they provide operational ranges wider than 10 m in the MedRadio and 5 m in the ISM bands, with the use of the Zarlink transceiver and base station. These performances will result in an excellent communication in the targeted 2 m distance [24]. — EM performance has to be robust against the battery and electronics presence; — implant location in the human body must have little effect on the EM performance.

Fig. 1. Implanted sensor conception. Maximum dimensions are indicated. Placements of battery and electronics are indicated to estimate the overall volume occupancy; two possible geometries of the circuitry (in light colors), and consequently of the antenna (solid-dashed lines), are shown.

C. Human Body Modeling Several possibilities of body phantoms, i.e., equivalent human body models, are nowadays available with different complexities; examples can be found in [9], [25]–[27]. Nevertheless, the realization of a single body phantom for frequency ranges as different as the MedRadio and ISM bands, is still remarkably hard [15]. For instance, wide band phantoms are available for a higher frequency spectrum [28] and a broad frequency range is realized only with skull equivalent properties in [29]. In this work we decided to follow the recommendations of [30] using a simple homogeneous cylinder with muscle-like dielectric properties whose values are taken from [31], for each frequency band. The approximation is rather rough, but it does allow to reduce the simulation time, and it provides standard and easy to realize conditions for the radiator measurements. III. ANTENNA DESIGN In order to overcome all constraints and to meet all requirements, the design procedure follows the next steps: — Design the antenna in free space [4], [32] in the absence of dielectric substrates and body phantom. Two main reasons can be given. First, assuming there is no surrounding medium reduces significantly the computation time. Therefore it allows a faster rough design to select an efficient antenna typology, given the available volume. Second, it is important to ignore the body losses at this stage in order not to optimize the antenna bandwidth by just increasing the power lost in the body; — set the allowed excitation area, and consequently arrange the chosen antenna typology; — add the dielectric substrate, bio-compatible insulation and body phantom, and then tune accordingly the design;

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Fig. 2. First investigated typologies in free space: (a) spherical, (b) spiral, (c) meander and (d) multilayered. The ground plane (in light color), always consists of a rectangular section and a conformal part (half hollow cylinder), as indicated for the (a) case. Delta-gap excitations are indicated with dashed lines.

— modify the design in order to obtain the dual band performances. This procedure, performed with Ansoft HFSS, aims at selecting the most efficient radiator in the MedRadio band as the medical data transmission occurs there. We decided to focus on the radiation properties in the lower frequency range despite reduced, but still adequate, performances in the ISM band. Indeed, higher radiation performances can be obtained at 2.45 GHz, as presented in [15], with different constraints. Section III-A presents the free space analysis, while the selected typology is described in Section III-B. In accordance with the EM requirements, Section III-C discusses the design robustness against the active components presence, and we analyzed the use of other two body phantoms. The latter provides insights about the sensitivity of the proposed design versus different implant locations. A. Selection of the Antenna Typology in Free Space The overall implant concept is depicted in Fig. 1. It has a cylindrical volume (diameter: 10 mm, height: 32 mm). The battery and two circuitry placements, indicated in Fig. 1, are offcentered with respect to the cylinder axis. In order to reduce the interference due to the active components on the antenna’s behavior, we considered a ground plane made of a rectangular planar section and a half hollow cylinder part (both depicted in light color in Fig. 2). At the same time this solution, together with its off-centered design, enhances the directivity of the radiator towards the desired out-of-the body direction, and it has the advantage of reducing the power absorbed by the biological tissues. For an easier understanding, the half hollow cylinder part has been referred to as conformal ground plane in Sections IV–VI of the manuscript. Four different typologies, depicted in Fig. 2, were investigated: spherical (conceived for the circuitry placement 1 in Fig. 1), spiral, meander and multilayered (for the circuitry

TABLE I COMPARISON OF THE RADIATION PERFORMANCES FOR FOUR ANTENNA TYPOLOGIES IN FREE SPACE. MAX VALUES ARE REPORTED FOR BOTH GAIN AND DIRECTIVITY

placement 2 in Fig. 1). All designs have a ground connection (like in PIFA antennas) in order to further reduce the resonant frequency. An extensive study of the indicated typologies was performed to choose an efficient antenna structure, given the available volume, with fast computation performance. Radiation characteristics and current vector alignments concept [10], [33], [34] were considered in this study. Table I reports the performance of these four typologies; only one result per typology that can be considered as a reference for all the evaluated designs is given. The single resonant frequency is chosen at around 1 GHz. The dielectric loading effect (due to the presence of dielectric substrates, bio-compatible insulation and body phantom that are considered in the second step of the design procedure) will consequently shift the resonant frequency towards lower frequencies, close to the MedRadio band. Only ohmic losses are taken into account (copper 17 thick). As previously discussed, bandwidth performances are not considered at this stage of the design procedure. The spherical case provides a more accentuated omnidirectional radiation (lowest directivity) without any remarkable efficiency. Thus, we considered the other three typologies, sharing the same antenna dedicated volume indicated in Fig. 1, so as to facilitate the radiation in the desired direction.

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Fig. 5. Antenna off-centered placement in (a) the homogeneous cylindrical body phantom (at the middle of its height) to mimic subcutaneous implantation. The study of (b) a larger box phantom is discussed in Section III-C-III. Fig. 3. Assembling of the 4 ROGER TMM substrates to create a pyramidal are the widths of the verstructure. Spiral metallization is in dark color. tical metallizations whose dimensions are given in Table II. The four substrates are separated for a better comprehension.

Fig. 4. Conformal ground plane: half hollow cylindrical structure to house the batteries. Metallization of the PEEK piece is made of copper foil 0.05 mm thick (dark color). Dimensions are given in Table II.

The spiral design increases the radiation efficiency and directivity compared to the meander case, in agreement with [9], [12], [13]. However, both metallizations of these two typologies (depicted in dark gray in Fig. 2) reach over the region dedicated to the battery housing. The very small distance between the ground plane and the metallizations does not facilitate the radiation. On the other hand, the multilayered solution achieves the desired resonance by utilizing only the uppermost part of the volume. This aspect, combined with the spiral conception, provides the most performing typology, both in directivity and efficiency, for the targeted application. Furthermore, the multilayered structure presents the simplest realization process; therefore this typology was selected for the making of the design.

Fig. 6. Design of the proposed prototype. The conformal antenna and its housing allow for the placement of all the necessary components for the sensor control, data processing, communication and power supply.

B. Multilayered Antenna Design The multilayered spiral model is built by pyramidal assembling to comply with the physical constraints. The structure, illustrated in Fig. 3, is made of four stacked dielectric substrates. Roger TMM 10 (alumina) with 35 copper metallization was chosen because of its high relative dielectric constant and low loss . The pyramidal structure is united with the conformal ground plane, depicted in Fig. 4. The latter consists of a metallized half hollow cylinder made of PEEK (Polyetheretherketones, , ). The choice of this material was mainly dictated by practical requirements, as it is very easy to manufacture as well as to metallize.

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Fig. 7. Top view of Roger TMM substrates: (a) first (double layer) with the excitation area, (b) second (single layer), (c) third (single layer) and (d) fourth (no metallization). Metallic lines are depicted in dark color (blue). Parameter dimensions are given in Table II. Thick black lines indicate where the welding (with the vertical metallizations) occurs.

At this stage of the design procedure, the antenna was integrated with the bio-compatible insulation and inserted into the body phantom. The former is made of PEEK because of its bio-compatibility and excellent mechanical, thermal and chemical resistance. Its thickness was set to just 0.8 mm to maximize the volume for the antenna design, in accordance with [22]. The human model has dielectric properties similar to muscle tissue whose values are given in Table IV. Its radius and height are equal to 40 and 110 mm, respectively. To mimic the targeted subcutaneous implantation, the antenna placement in the cylindrical body phantom was off-centered, as illustrated in Fig. 5(a). Once the excitation area was set by the physical constraints, we optimized (with a trial and error procedure) the dimensions of the multilayered spiral metallization in order to achieve the desired performance. Fig. 6 depicts the final radiator, including its bio-compatible housing. The overall volume of the implantable device is equal to 2477 ; only approximately 24% of it is allocated to the radiator. This results in an electrically very small antenna in the MedRadio band ( referring to the insulation dielectric properties [35, chapter 5]). For a better comprehension of the design, the four substrates are illustrated in Fig. 7, while Table II reports the values all the geometrical parameters. The limited area for excitation, due to the electronics constraints, is clearly indicated in Figs. 6 and 7(a). C. Simulated EM Characteristics Several results are presented in the following parts. For an easier understanding, Table III summarizes the obtained performances. 1) Matching and Radiation: Reflection coefficients versus frequency, , are plotted in Fig. 8. It can be noted that, while the antenna is well matched in the MedRadio frequency

TABLE II VALUES OF THE DESIGN PARAMETERS INDICATED IN FIGS. 3–7

range (with a 2.3% relative band), the higher resonance is slightly lower (i.e., 2.387 GHz) than the targeted 2.45 GHz. Nevertheless, the simulated shows a wide working band (6% at points) that includes a sufficient part of the desired ISM frequency spectrum. 3D polar plots are reported in Fig. 9. The maximum gain values are, taking the body phantom into account, and at MedRadio and at ISM bands, respectively.

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TABLE III SUMMARY OF THE SIMULATED RESULTS OF THE PROPOSED RADIATOR IN DIFFERENT BODY PHANTOMS

Fig. 9. Simulated 3D gain polar plot at: (a) 404.5 MHz and (b) 2.387 GHz. Radiation efficiencies are 0.058% and 0.530%, respectively. Coordinate system is the same as in Fig. 5(a).

Fig. 8. Simulated against frequency. The effect of the presence of the battery (and electronics) is compared with the reference design (no battery): (a) MedRadio and (b) ISM bands.

Despite the higher attenuation of the electromagnetic field in muscle at higher frequency, the radiator is much more efficient in the ISM band, as its size is electrically larger than in the MedRadio range. Surface current distributions are illustrated in

Fig. 10; as expected, almost the whole multilayered structure is relevant in the MedRadio range while some cold zones can be identified when the antenna dimensions are electrically larger, i.e., in the ISM band. 2) Design Robustness Against the Batteries and Electronics Presence: The volumes allocated to the batteries and the circuitry placement “2”, depicted in Fig. 1, were filled with dielectric and metallic materials [4]. A FR-4 substrate (with microstrip lines) and a copper cylinder were included in the model to mimic the presence of the electronics and power supply, respectively. Matching performances are reported in Fig. 8. Only 0.5% and 0.17% frequency shifts resulted in the two working bands, with realized gain variations limited to 0.2 dB (due to minimal radiation efficiency increase, as reported in Table III). These performances confirm the valuable antenna conception and they are completely satisfying for the integration of the whole sensor. 3) Effect of Different Body Phantoms: In order to obtain insights about the sensitivity of the proposed design versus different implant locations, two cylindrical body phantoms with the following characteristics were additionally investigated: — height: 110 mm; — homogeneous composition, radius equal to 40 mm, with equivalent head model dielectric properties ( ,

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Fig. 10. Surface current distributions at (a) 404.5 MHz and (b) 2.387 GHz.

in the MedRadio band and , at 2.45 GHz) [36]; — 3-layered structure made of muscle-fat-skin (dry), inspired from [11], [15], whose radii are 32, 36, and 40 mm, respectively. Dielectric properties are taken from [31] for both frequencies. Despite the variation of the analyzed phantoms, only a maximum difference of 2.3% of the resonant frequency emerged in the MedRadio band, as reported in Fig. 11(a). We also computed the realized gain, i.e., , at 404.5 MHz. Values of , and were found for the muscle, head and multilayered phantoms, respectively. The affordable 2.5 dB reduction in the multilayered case is mainly due to the resonant frequency shift. Giving explanation for the decreased radiation efficiencies in the two latter models is not an easy task, especially when the radiator is electrically very small. In fact, radiation efficiency substantially depends on the nearest surrounding of the antenna (near field coupling), and the transition between the body and the free space as discussed in [22]. On the other hand, the antenna size is electrically larger in the ISM band compared to the MedRadio case. This involves a lower coupling of the near field with the lossy tissues, which results in a reduced sensitivity to the surrounding environmental conditions; the resonant frequency variation is within the 0.3% as shown in Fig. 11(b). Realized gains attain , , for the muscle, head and multilayered phantoms, respectively. Explanation about the improvement of realized gain versus different body phantoms in the ISM frequency range can be obtained by paying attention to the dielectric characteristics of the investigated models. First of all, dielectric losses: note that the maximum realized gain (and highest efficiency) is found for the multilayered model when the antenna is in close contact with fat tissue that presents the lowest among the selected tissues [31]. Furthermore, let us pay attention to the fixed distance between the antenna and the external free space . Given a constant resonant frequency (i.e., 2.387 GHz), this distance is electrically smaller when considering the head and multilayered models, as they are constituted by materials

Fig. 11. Comparison of simulated against frequency considering the antenna inserted in different body phantoms in the (a) MedRadio and the (b) ISM bands.

with lower permittivities. Thus, the radiated wave undergoes minor attenuation. As previously mentioned, these considerations do not hold true for the proposed radiator in the MedRadio range, where the higher near field coupling phenomena are the main responsible factors of the radiation efficiency. As the cylindrical body phantom dimensions might be too small for a correct assessment of the performance of the radiator in the MedRadio range, a muscle rectangular cuboid (box) phantom with dimensions inspired from [11] (i.e., 290 584 320 [mm]) was considered. In this case the antenna was also placed at the middle of the phantom height and off-centered, 5 mm away from the external surfaces. This antenna location, modeling a targeted placement in the human torso above the hips, is depicted in Fig. 5(b). Electromagnetic performances are reported in Figs. 11 and 12 and in Table III. Tolerable differences in directivity and efficiency are found between the box and the cylindrical body phantoms. A similar realized gain (0.4 dB variation) and pattern (especially in the desired direction of radiation) are obtained between the two different modeled geometries. It is worth noting that the numerical analysis of the cylindrical body phantom is six times more efficient than the box case. The above results show the satisfactory robustness of the design versus the variation of the surrounding environmental

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Fig. 13. Manufactured prototype: (a) multilayered spiral design with the conformal ground plane and (b) final prototype assembling including housing and closing cap. The feeding coaxial cable is present only for testing purpose.

Fig. 12. Simulated radiation patterns [dBi] on the horizontal plane when the radiator is inserted in the box and cylindrical body phantoms at 404.5 MHz. The origin of the coordinate system (shown in Fig. 5(a)) is always placed at the center of the antenna [38].

conditions, as well as the suitability of the chosen cylindrical phantom dimensions for the targeted application. Hence, quite stable performances can be expected for realistic implant locations of the whole integrated telemetry system. Specific absorption rate (SAR) was computed at the transmitting frequency, i.e., in the MedRadio band. With a mass density equal to 1.04 [11], we evaluated the peak spatial averaged 1-g SAR at the resonance frequency of 407 MHz in the equivalent head phantom, in accordance with [36]. The obtained value, with a 1 W input power, is 289 W/kg. This implies that the antenna can be fed with signal up to 5.5 mW (7.4 dBm) and still meet the IEEE recommended value of 1.6 W/kg per 1 g averaging [37]. It is worth noting that the input power of the entire system [17], [18] is less than 0 dBm, thus complying with safety requirements. IV. REALIZATION Fig. 13 shows the built implantable antenna and its housing. The manufacturing of ROGER TMM substrates followed standard microstrip fabrication procedures. The assembling (stacking) of the pyramidal geometry of Fig. 3, connected by vertical copper-beryllium pieces, was ensured by a two-component epoxy adhesive. Note that the resonance frequency of the antenna is also affected by the adhesive ( , , thickness of 35 ) [32]. The same glue was used to fix the multilayered structure to the conformal ground plane. The latter consists of a coated PEEK piece with the use of a copper foil 50 thick. The whole construction process is rather delicate but it does still allow good repeatability, as discussed in Section V-B. The prototype weights 2.54 g including housing, without batteries. Following [23] and [39], we realized two liquid solutions to mimic the equivalent body models. Dielectric properties were measured with the HP dielectric probe kit 85070E. Table IV

TABLE IV DIELECTRIC PROPERTIES OF THE EQUIVALENT MUSCLE BODY PHANTOMS

presents targeted and experimental values showing a satisfactory agreement between them. V. MEASUREMENTS Experimental results of three realized antennas are described in this section. All the radiators were measured in the absence of the active components. After the realization and characterization of the first prototype, two more radiators were built to improve the calibration of the electromagnetic simulation and, consequently, the EM performance of the model itself. Finally, the importance of the presence of a feeding coaxial cable (and possible erroneous results) is described. A. First Prototype Fig. 14(a) depicts the satisfactory match between prediction and experiment in the MedRadio band. The comparison between simulated and measured performances in the ISM band is illustrated in Fig. 14(b). The experimental result shows how the frequency behavior is still useful ( equals to at 2.40 GHz) despite a remarkable difference from the targeted characteristics. As presented in Section IV, discrepancy was mainly caused by the difference between the dielectric properties of simulated and real materials. The exact placement of the antenna inside the body phantom, the feeding coaxial cable (as further detailed in Section V-C) and the liquid phantom itself do influence and increase the difficulty of radiation measurement in the anechoic chamber. Therefore, only horizontal radiation pattern measurement was performed

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Fig. 15. Simulated and measured normalized radiation patterns [dB] on the horizontal plane (vertical polarization) at the measured resonant frequency (i.e., 2.3945 GHz). Maximum value corresponds to 4.2 dBi. Cylindrical body phantom and antenna relative positions are indicated at the center of the diagram.

TABLE V VALUES OF THE PARAMETERS, INDICATED IN FIGS. 3–7, THAT DIFFER FROM TABLE II FOR THE FINAL REALIZATIONS

Fig. 14. Simulated and measured : (a) MedRadio and (b) ISM bands. Measurement in the MedRadio band considers the de-embedding of the feeding cable effect.

in the ISM band, as reported in Fig. 15, obtaining an acceptable agreement with theoretical predictions. B. Improved Prototypes The first experimental results allowed the improvement of the calibration of the electromagnetic simulation. The numerical analysis was modified as follows. First, different adhesive dielectric properties and thickness (30 ) were considered in the ISM band. The conductivity of vertical copper-beryllium metallizations (see Fig. 3) was set to while copper conductivity was reduced to . Table V reports the geometrical modifications. The final design has a total length of 32.1 mm. Simulated and measured are reported in Fig. 16. Maximum frequency deviations of 1% and 1.3% are found in the MedRadio and the ISM bands, respectively. Desired matching, that is better than , is achieved for the two working frequencies. The close match found in the ISM band shows the importance of the aforementioned modifications. The new design differs only slightly from the first one (gain directivity , rad. efficiency

in the MedRadio and ISM band, respectively), hence results about radiation performances, robustness against of the presence of batteries and effect of different body phantoms are not reported to avoid redundancy. C. Matching Measurements A feeding coaxial cable with ferrite bead choke was used as illustrated in Fig. 17. While the cable presence was not found to influence the radiator in the ISM band, its de-embedding was mandatory when the antenna is electrically smaller, i.e., for the MedRadio measurements, [40], [41]. In order to to better control the unwanted effects due to the cable presence, let us define four different setups, namely: — case 1: the coaxial cable is in direct contact with the body phantom; — case 2: a vacuum cylindrical shell surrounds the cable; — case 3: the depth of insertion of the antenna into the body phantom is reduced (this frees the cable from the body phantom); — case 4: the desired internal excitation is simulated. For an easier comprehension case 2 and case 3 are depicted in Fig. 17. Simulated reflection coefficients for the four setups are reported in Fig. 18. These results show that there exists a large coupling between the currents flowing on the external metallization of the cable [40] and the body phantom. In fact, by inserting the antenna and the cable directly into the body phantom, both

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Fig. 18. Simulated for case 1 2, 3 and 4 setups (first prototype). In the case 2 condition, the feeding cable is surrounded by a vacuum cylinder 0.3 mm larger.

Fig. 16. Simulated and measured : (a) MedRadio and (b) ISM bands. Model 2 and 3 are measured results. Measurements in the MedRadio band avoid the feeding cable effect as described in Section V-C.

Fig. 19. Comparison of simulated and measured in the MedRadio band (for the first prototype) considering the feeding coaxial cable in direct contact with the body model (i.e., case 1).

simulated and measured depicted in Fig. 19 are far from the desired performance reported in Fig. 8(a). One can appreciate that case 3 is in close agreement with the desired case 4. The former is the setup corresponding to the results reported in Figs. 14 and 16. VI. CONCLUSION

Fig. 17. Description of the different setups in order to understand how to mitigate the effect of the feeding cable. Dashed lines indicate the cylindrical body phantom.

The design procedure, realization and measurements of a miniature conformal antenna for implantable telemetry applications were described. The radiator has dual band capability working in both the MedRadio and the ISM (2.4–2.5 GHz) bands. Physical constraints and EM requirements were discussed in order to set the goals for a successful design. The design procedure included first numerical analysis in free space to properly choose the antenna typology given the available volume. Further on, the excitation area was set to comply with the physical constraints and we arranged accordingly the selected multilayered spiral typology. Finally, all the dielectric materials and the body phantom were introduced in the numerical analysis. Optimization of the stacked spiral design allowed

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to achieve the targeted performances. The design procedure, albeit focusing on the MedRadio frequency spectrum, provided adequate characteristics for the ISM band as well. Fifty Ohms input impedances were targeted for both working frequencies. Different values could be selected to match specific integrated circuit requirements at the early stage of the design procedure. Obviously, this would call for a re-arrangement of the chosen antenna design. The radiator takes into account the presence of the bio-compatible insulation, the electronics and the power supply. Its housing includes a specific volume for the monitoring sensors, or bio-actuators. The impact of the presence of active components was computed to confirm the robustness of the design. Indeed, the design is suitable for the realization of a complete implantable device with wireless telemetry capability. A homogeneous cylindrical body phantom was used for the numerical and experimental analysis. The rough approximation does reduce the simulation time and it provides standard (and easy to realize) conditions for the radiator measurements. We also evaluated the effects of three different body phantoms. This investigation provided insights about the sensitivity of the design versus different implant locations. Building issues and measurement aspects were described. The feeding coaxial cable effect was found to deeply affect the measured performance in the MedRadio band. A Solution to mitigate this effect was presented. The realization of a first prototype allowed a better calibration of the electromagnetic simulation and, thus, the improvement of the model itself. Two more antennas were designed and fabricated obtaining the desired performances and confirming the good repeatability of the manufacturing process. The final structure (diameter: 10 mm, height: 32.1 mm) occupies a volume of 2477 ; it is worth noting that less than fourth of it (approximately 24%) is allocated to the antenna. Radiation performances (maximum gain equal to and in the MedRadio and the ISM band, respectively) fulfill the initial requirements providing robust communication in a 2 m distance [24]. The antenna is currently being integrated with the power supply and the necessary electronic components [17] representing a complete implantable telemetry system for in-vitro and in-vivo testing. Results, including medical considerations, will be presented in the future. Preliminary experiments confirm the radiation characteristics of the proposed radiator and they show encouraging performances with reading distances up to 10 and 5 m in the MedRadio and the ISM band, respectively, [42]. ACKNOWLEDGMENT The authors would like to thank P. Vosseler and M. Leitos (ACI-EPFL) for all the useful discussions about the mechanical aspects and the realization of the presented prototypes and A. Merli for proofreading the manuscript. REFERENCES [1] D. Panescu, “Emerging technologies [wireless communication systems for implantable medical devices],” IEEE Eng. Med. Biol. Mag., vol. 27, no. 2, pp. 96–101, Mar. 2008. [2] Medical Implanted Communication System (MICS), FCC Std. CFR, Part 95, 2009. [3] T. Dissanayake, K. P. Esselle, and M. R. Yuce, “Dielectric loaded impedance matching for wideband implanted antennas,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 10, pp. 2480–2487, Oct. 2009.

[4] P. M. Izdebski, H. Rajagopalan, and Y. Rahmat-Samii, “Conformal ingestible capsule antenna: A novel chandelier meandered design,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 900–909, Apr. 2009. [5] N. Haga, K. Saito, M. Takahashi, and K. Ito, “Characteristics of cavity slot antenna for body-area networks,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 837–843, Apr. 2009. [6] W. Xia, K. Saito, M. Takahashi, and K. Ito, “Performances of an implanted cavity slot antenna embedded in the human arm,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 894–899, Apr. 2009. [7] R. Warty, M. R. Tofighi, U. Kawoos, and A. Rosen, “Characterization of implantable antennas for intracranial pressure monitoring: Reflection by and transmission through a scalp phantom,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 10, pp. 2366–2376, Oct. 2008. [8] U. Kawoos, M.-R. Tofighi, R. Warty, F. A. Kralick, and A. Rosen, “In-vitro and in-vivo trans-scalp evaluation of an intracranial pressure implant at 2.4 ghz,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 10, pp. 2356–2365, Oct. 2008. [9] K. Gosalia, G. Lazzi, and M. Humayun, “Investigation of a microwave data telemetry link for a retinal prosthesis,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 8, pp. 1925–1933, Aug. 2004. [10] K. Gosalia, M. S. Humayun, and G. Lazzi, “Impedance matching and implementation of planar space-filling dipoles as intraocular implanted antennas in a retinal prosthesis,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2365–2373, Aug. 2005. [11] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: Simulations, designs, and characterizations,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 8, pp. 1934–1943, Aug. 2004. [12] J. Kim and Y. Rahmat-Samii, “Planar inverted-f antennas on implantable medical devices: Meandered type versus spiral type,” Microw. Opt. Technol. Lett., vol. 48, no. 3, pp. 567–572, 2006. [13] P. Soontornpipit, C. Furse, and Y. C. Chung, “Design of implantable microstrip antenna for communication with medical implants,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 8, pp. 1944–1951, Aug. 2004. [14] P. Soontornpipit, C. M. Furse, and Y. C. Chung, “Miniaturized biocompatible microstrip antenna using genetic algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 1939–1945, Jun. 2005. [15] T. Karacolak, A. Z. Hood, and E. Topsakal, “Design of a dual-band implantable antenna and development of skin mimicking gels for continuous glucose monitoring,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 4, pp. 1001–1008, Apr. 2008. [16] T. Karacolak, R. Cooper, and E. Topsakal, “Electrical properties of rat skin and design of implantable antennas for medical wireless telemetry,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2806–2812, Sep. 2009. [17] L. Bolomey, E. Meurville, and P. Ryser, “Implantable ultra-low power DSP-based system for a miniature chemico-rheological biosensor,” in Proc. Eurosensors XXIII conf., 2009, pp. 1235–1238. [18] L. Bolomey, F. Merli, E. Meurville, J.-F. Zürcher, and A. Skrivervik, “Telemetry system for sensing applications in lossy media,” France Patent Request 00335/10, Mar. 11, 2010. [19] “Automated characterization of dextran/concanavalin a mixtures-a study of sensitivity and temperature dependence at low viscosity as basis for an implantable glucose sensor,” Sensors Act. B: Chem., vol. 146, no. 1, pp. 1–7, 2010. [20] A. Barraud, “Molecular selective interface for an implantable glucose sensor based on the viscosity variation of a sensitive fluid containing Dextran and Concanavalin A,” Ph.D. dissertation, EPFL, Lausanne, 2008. [21] Handbook of Materials for Medical Device. Materials Park, OH: ASM international, 2003, ch. 1. [22] F. Merli, B. Fuchs, J. R. Mosig, and A. K. Skrivervik, “The effect of insulating layers on the performance of implanted antennas,” IEEE Trans. Antennas Propag., vol. 59, no. 1, pp. 21–31, 2011. [23] P. S. Hall and Y. Hao, Antennas and Propagation for Body-Centric Wireless Communications. Norwood, MA: Artech House, 2006, ch. 9. [24] P. D. Bradley, “An ultra low power, high performance medical implant communication system (MICS) transceiver for implantable devices,” in Proc. IEEE Biomedical Circuits and Systems Conf. BioCAS 2006, Nov. 2006, pp. 158–161. [25] W. G. Scanlon, B. Burns, and N. E. Evans, “Radiowave propagation from a tissue-implanted source at 418 MHz and 916.5 MHz,” IEEE Trans. Biomed. Eng., vol. 47, no. 4, pp. 527–534, Apr. 2000. [26] L. C. Chirwa, P. A. Hammond, S. Roy, and D. R. S. Cumming, “Electromagnetic radiation from ingested sources in the human intestine between 150 MHz and 1.2 GHz,” IEEE Trans. Biomed. Eng., vol. 50, no. 4, pp. 484–492, Apr. 2003.

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[27] K. Ito, “Human body phantoms for evaluation of wearable and implantable antennas,” in Proc. of the 2nd Eur. Conf. on Antennas and Propagation (EuCAP 2007), Edinburgh, Scotland, U.K., Nov. 2007. [28] T. Takimoto, T. Onishi, K. Saito, S. Takahashi, M. Uebayashi, and K. Ito, “Characteristics of biological tissue equivalent phantoms applied to UWB communications,” Electron. Commun. Jpn., vol. 90, no. 5, pp. 48–55, 2007. [29] Y. Okano, K. Ito, I. Ida, and M. Takahashi, “The sar evaluation method by a combination of thermographic experiments and biological tissueequivalent phantoms,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 11, pp. 2094–2103, Nov. 2000. [30] Electromagnetic Compatibility and Radio Spectrum Matters (ERM); Ultra Low Power Active Medical Implants (ULP-AMI) Operating in the 401 MHz to 402 MHz and 405 MHz to 406 MHz Bands; System Reference Document, ETSI Std. TR 102 343 V1.1.1, 2004. [31] C. Gabriel, “Compilation of the Dielectric Properties of Body Tissues at RF and Microwave Frequencies,” Brooks Air Force Base, TX, 1996, Tech. Rep. Report N.AL/OE-TR-. [32] J. Abadia, F. Merli, J.-F. Zürcher, J. R. Mosig, and A. K. Skrivervik, “3D-spiral small antenna for biomedical transmission operating within the MICS band,” Radioengineering, vol. 18, no. 4, pp. 359–367, Dec. 2009. [33] S. Best and J. Morrow, “The effectiveness of space-filling fractal geometry in lowering resonant frequency,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 112–115, 2002. [34] S. Best and J. Morrow, “On the significance of current vector alignment in establishing the resonant frequency of small space-filling wire antennas,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 201–204, 2003. [35] R. W. P. King and G. S. Smith, Antennas in Matter: Fundamentals, Theory, and Applications, 1st ed. Cambridge, Massachusetts, and London: MIT Press, 1981. [36] IEEE Recommended Practice for Determining the Peak Spatial-Average Specific Absorption Rate (SAR) in the Human Head From Wireless Communications Devices: Measurement Techniques, IEEE Std. Std 1528, 2003. [37] Standard for Safety Levels With Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEE Std. Std C95., 1999. [38] R. Moore, “Effects of a surrounding conducting medium on antenna analysis,” IEEE Trans. Antennas Propag., vol. 11, no. 3, pp. 216–225, May 1963. [39] OET Bulletin 65, Edition 97-01, FCC Std. Supplement C, Jun. 2001. [40] A. K. Skrivervik, J. F. Zürcher, O. Staub, and J. R. Mosig, “PCS antenna design: The challenge of miniaturization,” IEEE Antennas Propag. Mag., vol. 43, no. 4, pp. 12–27, Aug. 2001. [41] F. Merli and A. K. Skrivervik, “Design and measurement considerations for implantable antennas for telemetry applications,” in Proc. 4th Eur. Conf. on Antennas and Propagation EuCAP 2010, Apr. 12–16, 2010. [42] F. Merli, L. Bolomey, E. Meurville, and A. K. Skrivervik, “Dual band antenna for subcutaneous telemetry applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp. (APSURSI), 2010, pp. 1–4. Francesco Merli (M’07) received the Laurea degree (cum laude) in telecommunication engineering from the University of Florence, Florence, Italy, in 2006, and the Ph.D. degree in electrical engineering from Ecole Polytechnique Fédérale de Lausanne (EPFL), in 2011. His research interests include antenna theory with particular focus to implantable and UWB antennas, spherical wave analysis, biomedical applications, wireless sensing and atomic watch cavities.

Léandre Bolomey was born in 1980 at Morges (Switzerland). He received the Master and Ph.D. degrees in electrical sciences at the Ecole Polytechnique Fédérale de Lausanne (EPFL) in 2003 and 2010, respectively. In 2004 he worked at Xemics (Semtech) as an ASIC Test Engineer and at DspFactory (Onsemi) for the design of a hearing aids analogue front-end. From 2004 to 2005, he worked as an Assistant at the Laboratory of Microengineering for Manufacturing (EPFL). During this period, he participated in a

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European project (SPARC) in which he developed an embedded dual-core platform for vision application in the automotive area. He has also worked for a CTI project in which he designed a wearable RF transceiver for a glucose sensor. In 2006, he began his thesis work on “Generic implantable Body Senor Node.” Jean-François Zürcher was born in Vevey, Switzerland, in 1951. He graduated with the degree of Electrical Engineer from Ecole Polytechnique Fédérale de Lausanne (Lausanne Institute of Technology) in 1974. He is presently employed as a permanent Scientific Associate with the Laboratoire d’Electromagnétisme et d’Acoustique EPFL, where he is the Manager of the microwave laboratory. His main interest lies in the domain of microstrip circuits and antennas. In 1988, he invented the SSFIP concept (“Strip Slot Foam Inverted Patch antenna”), which became a commercial product. He is presently developing instrumentation and techniques for the measurement of near fields of planar structures and microwave materials measurement and imaging. He is the author or coauthor of about 125 publications, chapters in books and papers presented at international conferences. He is coauthor of the book Broadband Patch Antennas (Artech, 1995). He holds 8 patents. Giancarlo Corradini is a technician in electronics specialized in surface mounted devices. He has worked 18 years in the industry where he was manager of a production unit in the field of hybrid circuits. In 1999, he was hired at the Laboratory of Production for Microtechnics, of the EPFL. He is responsible for the realization of experimental microsystems in thick-film technology as well as microelectronics back-end.

Eric Meurville holds a Master degree in electronics and digital signal processing from the “Conservatoire National des Arts & Métiers” (CNAM), Paris, France. Since 1999, he has been working as head of the Product Design Group at the “Laboratoire de Production Microtechnique” of the EPFL and is responsible for advanced research projects in the field of wearable and implantable biomedical devices. During the last 8 years, he has been particularly active in bringing long-term implantable medical devices concepts to commercial realization. From 1995 to 1999, at the Institute of Microtechnology of the University of Neuchâtel, Switzerland, his main field of research was multi-modal biometric access control systems. He was also Project Manager at the “Laboratoire d’Etude des Transmissions Ionosphériques” (LETTI), France, from 1992 to 1995 in the field of over the horizon radars. As software and hardware developer of airborne electronic warfare subsystems, he spent 6 years at Thalès (formerly Dassault Electronics), France, from 1986 to 1992. Anja K. Skrivervik received the Electrical Engineering degree and Ph.D. degree from Ecole Polytechnique Fédérale de Lausanne in 1986 and 1992, respectively. After a passage at the University of Rennes and the Industry, she returned to EPFL as an Assistant Professor in 1996, and is now a “Professeur titulaire” at this institution. Her teaching activities include courses on microwaves and on antennas. Her research activities include electrically small antennas, multi-frequency and ultra wideband antennas, numerical techniques for electromagnetic and microwave and millimeter wave MEMS. She is author or coauthor of more than 100 scientific publications. She is very active in European collaboration and European projects. She is currently the chairperson of the Swiss URSI, the Swiss representative for COST action 297 and a member of the board of the Center for High Speed Wireless Communications of the Swedish Foundation for Strategic Research.

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Design of an Implantable Slot Dipole Conformal Flexible Antenna for Biomedical Applications Maria Lucia Scarpello, Divya Kurup, Hendrik Rogier, Senior Member, IEEE, Dries Vande Ginste, Member, IEEE, Fabrice Axisa, Jan Vanfleteren, Member, IEEE, Wout Joseph, Member, IEEE, Luc Martens, Member, IEEE, and Gunter Vermeeren

Abstract—We present a flexible folded slot dipole implantable antenna operating in the Industrial, Scientific, and Medical (ISM) band (2.4–2.4835 GHz) for biomedical applications. To make the designed antenna suitable for implantation, it is embedded in biocompatible Polydimethylsiloxane (PDMS). The antenna was tested by immersing it in a phantom liquid, imitating the electrical properties of the human muscle tissue. A study of the sensitivity of the antenna performance as a function of the dielectric parameters of the environment in which it is immersed was performed. Simulations and measurements in planar and bent state demonstrate that the antenna covers the complete ISM band. In addition, Specific Absorption Rate (SAR) measurements indicate that the antenna meets the required safety regulations. Index Terms—Bent antenna, implantable antennas, industrial, muscle tissue sensitivity, scientific and medical (ISM) band, specific absorption rate (SAR).

I. INTRODUCTION

I

MPLANTABLE devices are becoming widely researched for different fields of applications, both for humans and animals. Some examples of applications are: monitoring blood pressure and temperature, tracking dependent people or lost pets, wirelessly transferring diagnostic information from an electronic device implanted in the human body for human care and safety, such as a pacemaker, to an external RF receiver [1]. Small implantable biomedical devices placed inside the human body may improve the lives of numerous patients. Patients with the antenna implanted in the body regularly return to the hospital for checkups, where their status and the status of the implant are verified. With the use of RF technology, data recorded by the implanted antenna can be transmitted wirelessly to the receiving station, while the patient is waiting in the lounge. Some patients may require checks every day. In such case a home care unit can be placed in the patient’s home. The unit can communicate with the medical implant and can Manuscript received June 10, 2010; revised November 23, 2010; accepted April 04, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. M. Scarpello, H. Rogier, and D. Vande Ginste are with the Electromagnetics Group, Department of Information Technology (INTEC), B-9000 Gent, Belgium (e-mail: [email protected]). D. Kurup, W. Joseph, L. Martens and G. Vermeeren are with the Department of Information Technology (INTEC), UGent-WiCa, B-9050 Gent, Belgium. F. Axisa and J. Vanfleteren are with the ELINTEC-TFCG, Technologiepark 914 B-9052 Gent, Belgium. 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.2011.2163761

be connected to the telephone system, or the internet, and send data regularly to the responsible person at the hospital [2]. The state-of-the-art of research in implantable antennas shows that microstrip or planar-inverted F antennas (PIFA), operating in the 402–405 MHz Medical Implant Communications Service (MICS) band, were simulated [3] and also fabricated and measured [4]–[6]. The main issue with this kind of antennas, operating at 403 MHz and thus corresponding to a wavelength of 744.4 mm in free space, is that it is not practical to put them into a living human body without performing thorough miniaturization. Indeed, taking the effect of the body into account during the design, reducing the antenna size by around seven times, without miniaturization it still remains too big to be implanted. In [5], the designed antenna has a cylinder height of 22.72 mm and an external radius of 10.5 mm, and so it is a small size antenna. However, the reflection coefficient value shown in the paper is only simulated and not measured and the simulated MICS bandwidth is only partially covered. In [6], the designed PIFA antenna is also small, i.e., 22.5 mm 18.5 mm 1.9 mm, but its fractional bandwidth is 20% lower than the one of the antenna we propose and it is not embedded in any insulating biocompatible material during measurements. Choosing a higher resonance frequency, corresponding to the Industrial, Scientific, and Medical (ISM) band (2.4–2.4835 GHz), is one way to reduce the antenna size and making it available to be implanted. Another advantage has to do with the radio communication link: the larger bandwidth allows for higher bitrates. Implantable H-shaped slot cavity antennas are studied for 2.45 GHz applications in [7], [8]. In [7], the antenna was simulated, whereas in [8] the same antenna was reduced in size. But since the size was too small to be fabricated (2.8 mm 4.0 mm 1.6 mm), the antenna was rescaled to larger dimensions to be manufactured and measured in order to be able to compare measurements with the finite-difference time-domain (FDTD) simulations. Radiation patterns, gain pattern, and radiation efficiency value are related to the rescaled antenna and to a rescaled resonance frequency equal to 980 MHz. Moreover in [8], no biocompatible material was used to embed the antenna and the cable during the measurements. In [9], [10], two dual band implantable antennas are presented, working properly in the MICS band and in the ISM band. Measurements were performed in a human skin mimicking gel tissue and in a rat skin mimicking gel tissue, respectively. In [10], the gain pattern is simulated and its maximum value is dBi in the ISM band. However, the two antennas are not embedded in any biocompatible material and they are not flexible. In [11] a cardiovascular stent, working at 2.4 GHz

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SCARPELLO et al.: DESIGN OF AN IMPLANTABLE SLOT DIPOLE CONFORMAL FLEXIBLE ANTENNA FOR BIOMEDICAL APPLICATIONS

has been designed, fabricated, and implanted in a live porcine subject. The stents are left without any insulation material and they are in direct contact with the tissue. In [12] three different inhomogeneous digital phantoms are considered to check the different radiation performances of wireless implants. Here again only simulations are performed but no measurements. In this paper, we present the design, characterization, and measurements of an implantable antenna operating in the 2.45 GHz ISM band, recommended by the European Radiocommunications Committee (ERC) for ultra-low-power active medical implants [13]. The antenna is a flexible slot dipole. To make the antenna suitable for implantation, it is embedded in biocompatible Polydimethylsiloxane (PDMS). The reflection coefficient was simulated and measured in the MSL2450 liquid, provided by Speag (Zurich, Switzerland) [14], mimicking a 100% human muscle tissue, having well-defined dielectric values at 2.45 GHz. To investigate how different human tissues, surrounding the implanted slot dipole, affect its radiation characteristics, a study on the sensitivity of the liquid mimicking the human muscle tissue was performed. Thereto its dielectric nominal values were varied from 50% larger to 50% smaller. These simulations were performed to ensure that the antenna is functioning properly in any type of body environment. The radiation characteristics of the antenna in terms of E-field and gain were simulated by means of FDTD calculations. For the evaluation of performances and safety issues related to implanted antennas, the 10-g and 1-g averaged specific absorption rate (SAR) are measured and compared with the ICNIRP [15] and with the FCC guidelines [16]. First, in Section II, the antenna design and its fabrication are presented. Next, in Section III, the performance in terms of reflection coefficient is reported. Good agreement is demonstrated between the simulated and measured reflection coefficient. In Section IV, the sensitivity of the antenna as a function of the dielectric properties of the muscle tissue is analyzed, verifying that the antenna can be placed close to different kinds of tissue. In Section V, the radiation characteristics of the antenna, including the measured SAR distribution, are shown. In Section VI conclusions are summarized. II. BIOCOMPATIBLE FOLDED SLOT DIPOLE ANTENNA DESIGN AND MANUFACTURING PROCESS The antenna, presented in this paper, is a flexible folded slot dipole embedded in biocompatible PDMS, as folded slot dipole geometries can provide significantly larger bandwidths than patch antennas [17]. The top and frontal view of the antenna are shown in Figs. 1 and 2, respectively. The dimensions of the folded slot dipole antenna are shown in Table I. The antenna is designed by means of the 2.5-D EM field simulator Momentum of Agilent’s Advanced Design System (ADS). The antenna design procedure consists of three steps. First, the folded slot dipole antenna was designed, using ADS’s optimization routines, to operate in the 2.45 GHz ISM band in free space. Second, one superstrate and one substrate of PDMS were added to the design and, after the characterization of the PDMS, we redesigned and reoptimized the antenna embedded in silicone so that it covers the ISM band. Third, in a last optimization step, on top of the superstrate and below the substrate one layer

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Fig. 1. Top view of the coplanar waveguide-fed antenna.

Fig. 2. Frontal view of the coplanar waveguide-fed antenna.

of liquid, mimicking the dielectric characteristics of human muscle tissue at 2.45 GHz, was added. Finally, the antenna so designed have good simulations results, working properly in the ISM band. To check more accurately, the antenna was also simulated with the 3-D simulator CST Microwave Studio and the simulation results were still satisfactory. To manufacture the antenna we rely on a flexible electronic technology. A photoresist film was spin-coated on a copper foil and patterned by UV radiation through a photomask; the patterned shape is shown in Fig. 1; two PDMS layers are used as substrate and superstrate, each with a thickness of 2.5 mm, to mould the antenna [18], [19]. The dielectric properties of the PDMS were characterized at 2.45 GHz, to be and . The feeding structure of the slot dipole antenna consists of a coplanar waveguide (CPW) with a 50 mode impedance. Matching the mode impedance of the CPW to 50 is obtained by tuning the distance between the tracks and , as well as the

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TABLE I SIZE OF THE FOLDED SLOT DIPOLE ANTENNA

TABLE II DIELECTRIC VALUES OF MSL2450, AT DIFFERENT FREQUENCIES

TABLE III DIELECTRIC VALUES OF HUMAN MUSCLE TISSUE, AT DIFFERENT FREQUENCIES, AS REPORTED IN [21]–[23]

Fig. 3. Top view of flex antenna without PDMS.

Fig. 4. Side view of antenna and cable embedded in PDMS.

width of the tracks (Fig. 1). The CPW is fed by a U.FL connector and an ultra-fine Teflon coaxial cable supplied by Hirose [20]. The U.FL connector is chosen (specifically for measurements purposes) because it is compact and it suits the small CPW size. Both the connector and the cable are also embedded in PDMS. Fig. 3 shows the antenna prototype before being embedded in the PDMS. Fig. 4 shows a side view of the antenna prototype with its connector and cable after being embedded in the PDMS. III. SIMULATION AND MEASUREMENTS In real-life applications, the antenna is intended to be implanted into the human body, subcutaneously, particularly inside the muscle. Hence, the measurement setup, using a phantom, is as follows. The antenna is placed at the center of a plastic container of dimensions 80 cm 50 cm 20 cm filled with 30 liters of the Human Muscle Tissue liquid MSL2450 [14]. This liquid

Fig. 5. S11 values using the MSL2450 dielectric properties [14] (dotted line) and the human muscle tissue dielectric properties [21] (dashed line), compared to the return loss value of the antenna immersed in a medium with fixed values of permittivity and conductivity, i.e., those of MSL2450 at 2.45 GHz (full line).

mimics the dielectric characteristics of human muscle tissue at 2.45 GHz, standardized in [21] to be S/m, kg/m . Dielectric values of the liquid at 2.45 GHz measured by the manufacturer result to be: S/m, kg/m [14]. In Tables II and III permittivity and conductivity values of the liquid MSL2450 at different frequencies, measured by the manufacturer, and of the human muscle tissue [21]–[23] are listed, respectively. The values in Tables II and III cover the band (2.0 GHz and 3.0 GHz) in which the measurements were performed, and as such also encompass the ISM band (2.4–2.485 GHz). Fig. 5 reports the values for each couple of dielectric values reported in Tables II and III, valid at the specified frequency. It can be observed that the small differences in , reported in Tables II and III, do not lead to very different antenna behavior. A third simulation, where a fixed relative permittivity and a fixed conductivity S/m were used within the complete band, also indicates that the antenna is rather insensitive to changes of the surrounding medium. This will be further illustrated by considering other human tissues (Figs. 9 and 10). Moreover, a detailed study of the sensitivity of the antenna as a function of the dielectric properties of the liquid is reported in Section IV.

SCARPELLO et al.: DESIGN OF AN IMPLANTABLE SLOT DIPOLE CONFORMAL FLEXIBLE ANTENNA FOR BIOMEDICAL APPLICATIONS

Fig. 6. Schematic representation of the reflection measurement setup for the designed implantable antennas using muscle tissue simulating liquid.

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Fig. 8. Antenna in planar state: reflection coefficient, simulations versus measurement.

Fig. 9. Three layers geometry for the design of the slot dipole antenna, placed in between fat and muscle.

Fig. 7. SAR and reflection measurement setup with the implantable antenna inside the liquid during a measurement.

The antenna is connected to a Rhode and Schwarz ZVR Network Analyzer, as shown in Figs. 6 and 7. Inside the phantom, measurements are performed when the antenna is both in planar and bent state. First, Fig. 8 displays a comparison between the measured and simulated reflection coefficient of the antenna in planar state. The simulations are performed using the EM field simulator Momentum of Agilent’s Advanced Design System (ADS) and CST Microwave Studio simulator. Since ADS Momentum is a 2.5-D simulator, it does not account for the finite size of the PDMS layers. This finite size typically results in a shift of the resonance frequency to lower frequencies, so for the initial design in Momentum, to cover the ISM band, the antenna is designed to resonate at 2.5 GHz. Once fabricated and measured, this design ensures that our antenna will cover the complete

Fig. 10. Five layers geometry for the design of the slot dipole antenna, placed in between two layers of muscle.

requested bandwidth, as also verified by CST simulations in Fig. 8. The required dB impedance bandwidth of the antenna is 83.5 MHz in the 2.45 GHz ISM band. Simulations and measurements satisfy the requirements: (i) the antenna was simulated in planar state with ADS. Its bandwidth is very wide, 1.23 GHz (2.22–3.45 GHz), it includes different resonances of the antenna and the fractional bandwidth at the target frequency (2.45 GHz) is approximately 50.2%. (ii) The antenna was also simulated in planar state with CST: its bandwidth is 270 MHz (2.30–2.57 GHz) and the fractional bandwidth at the target frequency (2.45 GHz) is approximately 11.0%. (iii) Antenna measurements are performed to validate the simulations: the mea-

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TABLE IV DIELECTRIC VALUES OF FOUR DIFFERENT HUMAN TISSUES, AT 2.45 GHZ, AS REPORTED IN [21]–[23]

Fig. 12. Cylinders used to bend the antenna: 8 cm and 4 cm diameter.

Fig. 11. Simulation of the return loss of the antenna placed in the structures of Figs. 9 (dashed line) and 10 (dotted line), compared to the antenna placed in the MSL2450 (full line).

sured bandwidth in planar state is 350 MHz (2.20–2.55 GHz) and the fractional bandwidth at the target frequency (2.45 GHz) is approximately 14.2%. Second, to demonstrate that the antenna can work within different human tissues, the antenna is placed into two human body structures, as reported in [24], [25] and shown in Figs. 9 and 10, and simulated by means of ADS Momentum. The dielectric properties ( ), for each layer of the two structures, at GHz, were obtained from [21], [22] and [23], and shown in Table IV. In Fig. 11, the return loss value of the antenna placed in the structures of Figs. 9 and 10 is shown. A slight shift of the resonance frequency towards higher frequencies can be observed, but the return loss value still remains below dB in the whole ISM band. Third, the performance of the antenna when it is bent, as such making it conformal to curved parts of the body, is verified. Thereto, the antenna is bent around its x-axis using two cylinders with different diameters, 8 cm and 4 cm, and with a hole in the middle, as shown in Fig. 12. The measured reflection coefficient of the bent antenna is compared to the planar antenna and shown in Fig. 13. The reflection coefficients of the bent antenna exhibits a resonance frequency at 2.22 GHz. So it is observed that the resonance has shifted to a somewhat lower frequency, compared to GHz for the planar antenna. Still, the bandwidth of 350 MHz is maintained and the complete ISM bandwidth is covered. IV. SENSITIVITY OF THE ANTENNA TO DIELECTRIC PROPERTIES OF THE MUSCLE TISSUE The use of a muscle tissue liquid is widely accepted as a standard, as it is very important to experimentally verify the charac-

Fig. 13. Antenna bent around x-axis versus antenna in planar state: reflection coefficient measurements.

teristics of electromagnetic (EM) propagation inside the human body. Muscle tissue exhibits typical anisotropic electric properties [26]: in the low frequency range, the longitudinal conductivity is significantly higher than the transverse conductivity [27]. A variation of conductivity can influence the performance of antennas in real life applications. Fortunately, the muscle tissue anisotropy is frequency dependent: if the frequency of the current is high enough (i.e., in the MHz range), the anisotropic properties disappear [28]. In this paper, the operation frequency of the antenna is from 2.4 GHz to 2.485 GHz, high enough to neglect the effect of anisotropy. Moreover, the effect of tolerances in the dielectric properties of the different tissues can be significant, potentially influencing communication performance of the implantable slot dipole antenna. Using ADS Momentum, a parametric study is performed that determines the influence of the muscle tissue’s permittivity and conductivity on the reflection coefficient of the antenna, in terms of resonance frequency and fractional bandwidth. This is to ensure that the antenna can be implanted at different locations inside the body, also close to tissues having different dielectric characteristics. The electrical properties of the muscle tissue liquid at 2.45 GHz were standardized in [21] to be S/m, kg/m .

SCARPELLO et al.: DESIGN OF AN IMPLANTABLE SLOT DIPOLE CONFORMAL FLEXIBLE ANTENNA FOR BIOMEDICAL APPLICATIONS

TABLE V RESONANCE FREQUENCY [GHZ] AND REFLECTION COEFFICIENT [DB] FOR SOME COUPLES OF

AND

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VALUES

For the MSL2450 liquid, slightly different values were measured by the manufacturer, resulting in S/m, kg/m at 2.45 GHz [14]. The dielectric characteristics are now varied as follows:

(1) where and are the nominal values of the liquid, corresponding to the values measured by the manufacturer. Varying and , according to (1), results in simulations. 25 relevant samples are shown in Table V. At the dielectric nominal values, the fractional bandwidth equals 50%, the antenna resonates at 2.51 GHz and dB. For all the other simulations the fractional bandwidth is always between 48.5% and 51.5%. In the worst case dB, for . For all considered variations in dielectric properties, the complete ISM band remains covered and the dB bandwidth results to be very large, showing the lowest frequency equals to 2.2 GHz and the highest one equals to 3.5 GHz. As stated before (Section III), the antenna was designed in ADS-Momentum to resonate at 2.5 GHz to account for a possible shift to lower frequencies due to the finite size of the antenna. As can be seen from the Table V, the largest shift can be expected when the relative permittivity of the muscle tissue is drastically lower than its nominal value and when the losses are much higher . Still, the shift of the resonance frequency is limited. Therefore, we conclude that the antenna works properly in various human bodies, with considerably different dielectric properties. V. RADIATION CHARACTERISTICS OF THE ANTENNA Using CST Microwave Studio simulations, the radiation characteristics of the antenna inside the liquid simulating muscle tissue are determined in terms of radiation patterns and gain. To simulate, for example, the implanted antenna in the human arm, the dimensions of the box containing the liquid are chosen to be 180 mm 60 mm 60 mm, as shown in Fig. 14. The antenna is directed toward the surface of the skin (surface of the box), and along the z-direction the distance to the surface of the skin is set to 4 mm. In the x-y plane, the antenna is placed in the center of the surface of the human model arm (center of the box). This setup is the same as in [8]. The computed radiation patterns in the -plane and the -plane

Fig. 14. CST numerical calculation model: the box simulates a human arm.

Fig. 15. Far-field pattern at 2.45 GHz in the -plane, y-z plane.

are shown in Figs. 15 and 16. The patterns are computed at 2.45 GHz, at a reference distance of 1 m, and using an input power of 1 W. Fig. 17 shows the antenna gain. The maximum gain is equal to dBi for and and the radiation efficiency is 0.14%. These values are comparable to other results in literature, such as in [8]. The radiation efficiency value is very low because the antenna is not in free space, but embedded inside a human arm, simulated as a very lossy medium. The SAR is measured in a 3 cm 3 cm 2.5 cm cube above the antenna, as reported in Fig. 18. The measurement setup is provided by Speag (Zurich, Switzerland). The measurement system used is DASY 3 [29] and the phantom is filled with the liquid MSL2450 [14]. The size of the box phantom [30] has been

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Fig. 16. Far-field pattern at 2.45 GHz in the -plane, x-y plane.

Fig. 19. SAR distribution for an input power of 2 mW.

0.079 W/Kg. These values fulfill the ICNIRP [15] and FCC [16] guidelines for general public exposure. VI. CONCLUSION

Fig. 17. Gain pattern at 2.45 GHz.

Fig. 18. Setup for SAR measurements.

chosen to correspond to the average trunk of an adult man. Because sharp edges can cause field modifications, the edges are rounded with a radius of 7 cm. The SAR measurement probe is EX3DV4 [31]. The SAR measurement procedure is described in [32]. Fig. 19 shows the SAR distribution on the x-y plane of the antenna at when the input power is 2 mW: the peak SAR value is 0.308 W/Kg. The 10-g averaged SAR peak value is 0.032 W/Kg and the 1-g averaged SAR peak value is

The design, manufacturing, measurement, and sensitivity study of a flexible folded slot dipole antenna embedded in PDMS for implantation into the human body was performed. The human body was replaced by a human muscle tissue liquid with known dielectric nominal values. The main issues addressed in this paper are: 1) design of a slot dipole antenna suited for implantation into the human body; 2) evaluation of the characteristics of the antenna in terms of reflection coefficient in planar and bent state, E-field and gain; 3) study of the sensitivity of the liquid mimicking the human muscle tissue, varying its nominal dielectric values; 4) checking the SAR limitations, by means of SAR measurements. Measurements and simulations of the reflection coefficient in planar and bent state in the 2.45 GHz ISM band demonstrate a very large bandwidth in both states, fully covering the ISM band. A good agreement is found between simulations and measurements for the planar state antenna. In bent state, a shift of the resonance towards lower frequencies is verified during measurements. The simulated far-field pattern and gain at 2.45 GHz show a dBi gain at and . The study of the sensitivity with respect to the dielectric properties of surrounding tissue shows that the EM characteristics of the antenna are stable for a wide range of tissue properties in the neighborhood of the antenna. In fact, only for extreme changes of the dielectric properties, the resonance frequency starts shifting. The measured SAR values with an input power of 2 mW averaged in 1-g and 10-g tissue show that the antenna respects the ICNIRP and FCC guidelines for general public exposure. In the future, integration of the required transceiver and power supply is envisaged to realize an implantable system for biotelemetry applications, completely embedded in biocompatible silicone and

SCARPELLO et al.: DESIGN OF AN IMPLANTABLE SLOT DIPOLE CONFORMAL FLEXIBLE ANTENNA FOR BIOMEDICAL APPLICATIONS

fabricated with a flexible technology, as shown in [18], [19] by one of the coauthors of this paper. At the current level of our on-going research project, no specific active electronics have yet been developed. The antenna presented in this paper might be slightly large for an immediate implant but it is an important contribution to implantable systems because it is flexible, conformal and completely embedded in biocompatible silicone. It is, important to first simulate and measure a good antenna structure, such as the one presented in this paper, usable as an innovative starting point for future miniaturized design.

REFERENCES [1] B. M. Steinhaus, R. E. Smith, and P. Crosby, “The role of telecommunications in future implantable device systems,” in Proc. 16th IEEE EMBS Conf., Baltimore, MD, 1994, pp. 1013–1014. [2] R. F. Weir, P. R. Troyk, G. De Michele, and T. Kuiken, “Implantable myoelectric sensors (IMES) for upper-extremity prosthesis control,” in Proc. IEEE Eng. Med. Biol. Soc. 25th Annu. Int. Conf., 2003, pp. 1562–1565. [3] K. Y. Yazdandoost and R. Kohno, “An antenna for medical implant communications system,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 392–395, 2007. [4] J. A. Von Arx, W. R. Mass, S. T. Mazar, and M. D. Amundson, “Antenna for an Implantable Medic Device,” U.S. Patent 6708065, Mar. 16, 2004. [5] F. Merli, L. Bolomey, E. Meurville, and A. K. Skrivervik, “Implantable antenna for biomedical applications,” in Proc. IEEE Antennas and Propag. Soc. Int. Symp., San Diego, CA, 2008, pp. 584–587. [6] C. M. Lee, T. C. Yo, F. J. Huang, and C. H. Luo, “Bandwidth enhancement of planar inverted-F antenna for implantable biotelemetry,” Microwave Opt. Tech. Lett., vol. 51, no. 3, pp. 749–751, Mar. 2009. [7] H. Usui, M. Takahashi, and K. Ito, “Radiation characteristics of an implanted cavity slot antenna into the human body,” in Proc. IEEE Antennas and Propag. Soc. Int. Symp., Albuquerque, NM, 2006, pp. 1095–1098. [8] W. Xia, K. Saito, M. Takahashi, and K. Ito, “Performances of an implanted cavity slot antenna embedded in the human arm,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 894–899, Apr. 2009. [9] T. Karacolak, A. Z. Hood, and E. Topsakal, “Design of a dual-band implantable antenna and development of skin mimicking gels for continuous glucose monitoring,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 4, pp. 1001–1008, Apr. 2008. [10] T. Karacolak and E. Topsakal, “Electrical properties of nude rat skin and design of implantable antennas for wireless data telemetry,” in IEEE MTT-S Int. Microwave Symp. Digest, Atlanta, GA, 2008, vol. 1-4, pp. 1169–1172. [11] E. Y. Chow, Y. Ouyang, B. Beier, W. J. Chappell, and P. P. Irazoqui, “Evaluation of cardiovascular stents as antennas for implantable wireless applications,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 10, pp. 2523–2532, Oct. 2009. [12] A. Sani, A. Alomainy, and Y. Hao, “Numerical characterization and link budget evaluation of wireless implants considering different digital human phantoms,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 10, pp. 2605–2613, Oct. 2009. [13] “ERC recommendations 70–03 relating to the use of short range devices (SRD),” in Proc. Eur. Postal Telecommunications Administration Conf., Tromso, Norway, 1997, CEPT-ERC 70–03, Annex 12. [14] [Online]. Available: http://www.speag.com/measurement/liquids/ [15] ICNIRP, “Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields,” Health Phys., vol. 74, pp. 494–5229, 1998. [16] FCC, “Evaluating compliance with FCC guidelines for human exposure to radiofrequency electromagnetic fields,” Supplement C to OET Bulletin 65, Jun. 2001, Washington, DC, 20554. [17] H. S. Tsai, M. J. V. Rodwell, and R. A. York, “Planar amplifier array with improved bandwidth using folded slots,” IEEE Microwave Guided Wave Lett., vol. 4, no. 4, pp. 112–114, Apr. 1994.

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[18] R. Carta, P. Jourand, B. Hermans, J. Thoné, D. Brosteaux, T. Vervust, F. Bossuyt, F. Axisa, J. Vanfleteren, and R. Puers, “Design and implementation of advanced systems in a flexible-stretchable technology for biomedical applications,” Sensors Actuat., pp. 79–87, Mar. 2009. [19] J. Govaerts, W. Christiaens, E. Bosman, and J. Vanfleteren, “Fabrication process for embedding thin chips in flat flexible substrate,” IEEE Trans. Adv. Packag., vol. 32, no. 1, pp. 77–83, Feb. 2009. [20] [Online]. Available: http://www.hirose-connectors.com [21] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: I. Literature survey,” Phys. Med. Biol., vol. 41, no. 11, pp. 2231–2249, 1996. [22] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, no. 11, pp. 2251–2269, 1996. [23] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, no. 11, pp. 2271–2293, 1996. [24] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: Simulations, design, and characterization,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 8, pp. 1934–1943, Aug. 2004. [25] G. Collin, A. Chami, C. Luxey, P. Le Thuc, and R. Staraj, “Small electrical antenna for saw sensor biotelemetry,” Microw. Opt. Technol. Lett., vol. 51, no. 10, pp. 2286–2293, Jul. 2009. [26] J. P. Reilly, “Applied Bioelectricity,” in From Electrical Stimulation to Electropathology. New York: Springer-Verlag, 1998. [27] F. X. Hart, N. J. Berner, and R. L. McMillen, “Modelling the anisotropic electrical properties of skeletal muscle,” Phys. Med. Biol., vol. 44, pp. 413–421, 1999. [28] D. Miklavčič, N. Pavšelj, and F. X. Hart, “Electric properties of tissues,” Wiley Encyclopedia of Biomedical Engineering, 2006. [29] [Online]. Available: http://www.speag.com/measurement/dasy5/ index .php [30] Basic Standard for the Calculation and Measurement of Electromagnetic Field Strength and SAR Related to Human Exposure From Radio Base Stations and Fixed Terminal Stations for Wireless Telecommunication Systems (110 MHz Ű 40 GHz), CENELEC EN50383, Sep. 2002. [31] [Online]. Available: http://www.speag.com/measurement/probes/ex3. php [32] Recommended Practice for Determining the Spatial-Peak Specific Absorption Rate (SAR) in the Human Body Due to Wireless Communications Devices: Measurement Techniques, IEEE 1528/D1.2, Apr. 2003.

Maria Lucia Scarpello was born in 1983. She received the B.S. and M.S. degrees in telecommunication engineering from the Politecnico di Torino, Torino, Italy, in 2006 and 2008, respectively. She is currently working towards the Ph.D. degree at Ghent University, Ghent, Belgium. Her research interests include design and characterization of implantable antennas and textile antennas.

Divya Kurup received the B.E. degree in electronic and telecommunication engineering from Mumbai University, Mumbai, India, in 2001 and M.S. degree in telecommunication and network management from Syracuse University, NY, in May 2007. She then joined the Department of Information Technology (INTEC) of Ghent University in January 2008 where she is currently working as a research assistant in the Wireless and Cable Research group. Her scientific work is focused towards wireless body area networks in particular investigation of in-body propagation. Ms. Kurup received the Best Student award from Syracuse University for the academic year 2006–07.

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Hendrik Rogier (SM’09) was born in 1971. He received the electrical engineering and the Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Postdoctoral Research Fellow of the Fund for Scientific Research Flanders (FWO-V), Department of Information Technology, Ghent University, where he is also Associate Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist at the Mobile Communications Group of Vienna University of Technology. He authored and coauthored about 65 papers in international journals and about 100 contributions in conference proceedings. His current research interests are the analysis of electromagnetic waveguides, electromagnetic simulation techniques applied to electromagnetic compatibility (EMC) and signal integrity (SI) problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier was twice awarded the URSI Young Scientist Award, at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly. He is serving as a member of the Editorial Board of IET Science, Measurement Technology and acts as the URSI Commission B representative for Belgium.

Dries Vande Ginste (M’05) was born in 1977. He received the M.S. degree and the Ph.D. degree in electrical engineering from Ghent University, Gent, Belgium, in 2000 and 2005, respectively. From October 2000 until March 2006, he was with the Department of Information Technology (INTEC), Ghent University, as a Doctoral and Postdoctoral Researcher, where his research focused on fast techniques for the modeling of layered media. In June and July 2004, he was a Visiting Scientist at the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign (UIUC). From April 2006 to May 2007, he was active as a Senior Consultant in a private company, i.e., Applied Logistics N.V., where he was mainly involved in the modeling of material handling systems, feasibility studies, technical-economical evaluations for clients, and where he also had commercial responsibilities. In June 2007, he re-joined the Department of Information Technology (INTEC), first as a Technology Developer in the field of high-frequency technologies for ICT applications and since July 2009 as a Postdoctoral Researcher and Lecturer. His current research interests comprise computational electromagnetics, electromagnetic compatibility, and antenna design.

Fabrice Axisa received the M.S. degree in engineering and the Ph.D. degree in biomedical microsystem from Ecole Centrale de Lyon (France), in 1996 and 2004, respectively. He worked as an Electronic Designer at ARM and as a Researcher at INSA Lyon (France) and at IMEC (Belgium). He is currently responsible for the Microsystem Packaging Laboratory Microsys at the University of Liege. His main focus is on microsystem development and packaging, biomedical electronic system and packaging, and in polymer based and stretchable electronic system. He is author or coauthor of more than 60 articles and 5 patents.

Jan Vanfleteren (M’88) received the Ph.D. degree in electronic engineering from Ghent University (Belgium) in 1987. He is currently a Senior Engineer at the IMECCMST group and is involved in the development of novel interconnection, assembly and substrate technologies, especially in wearable electronics technologies. As a project manager for CMST he has a long standing experience in coordination and cooperation in EC funded projects. In 2004, he was appointed part time Professor at the Ghent University. He is the (co)author of over 200 papers in international journals and conferences and he holds 14 patents/patent applications. Dr. Vanfleteren s a member of IMAPS. Wout Joseph (M’05) was born in Ostend, Belgium, on October 21, 1977. He received the M.Sc. degree in electrical engineering from Ghent University (Belgium) in July 2000. From September 2000 to March 2005, he was a Research Assistant at the Department of Information Technology (INTEC) of the same university. During this period, his scientific work was focused on electromagnetic exposure assessment. His research work dealt with measuring and modelling of electromagnetic fields around base stations for mobile communications related to the health effects of the exposure to electromagnetic radiation. This work led to a Ph.D. degree in March 2005. Since April 2005, he is postdoctoral researcher for IBBT-Ugent/INTEC (Interdisciplinary institute for BroadBand Technology). Since October 2007, he is a Postdoctoral Fellow of the FWO-V (Research Foundation–Flanders). Since October 2009, he is a Professor in the domain of “experimental characterization of wireless communication systems.” His professional interests are electromagnetic field exposure assessment, propagation for wireless communication systems, antennas and calibration. Furthermore, he specializes in wireless performance analysis and quality of experience. Luc Martens (M’92) was born in Gent, Belgium, on May 14, 1963. He received the M.Sc. degree in electrical engineering from Ghent University (Belgium), in July 1986. From September 1986 to December 1990, he was a Research Assistant at the Department of Information Technology (INTEC) of the same university. During this period, his scientific work was focused on the physical aspects of hyperthermic cancer therapy. His research work dealt with electromagnetic and thermal modelling and with the development of measurement systems for that application. This work led to a Ph.D. degree in December 1990. Since January 1991, he is a member of the permanent staff of the Interuniversity MicroElectronics Centre (IMEC), Ghent, and is responsible for the research on experimental characterization of the physical layer of telecommunication systems at INTEC. His group also studies topics related to the health effects of wireless communication devices. Since April 1993, he is a Professor in electrical applications of electromagnetism at Ghent University. Gunter Vermeeren was born in Zottegem on March 9, 1976. He received the M.Sc. degree in industrial engineering from the KAHO Sint-Lieven (Ghent, Belgium), in July 1998 and the M.Sc. degree in electrical engineering from Ghent University (Belgium), in July 2001. From September 2001 to September 2002, he joined the Research and Development Department of the network integrator Telindus (Leuven, Belgium). Since September 2002, he has been a Research Engineer in the WiCa group of Prof. Luc Martens. There, he is involved in several projects in the field of radio frequency dosimetry, electromagnetic exposure, and on-body propagation. His research is in the area of numerical modeling as well as measurements of electromagnetic fields in the proximity of humans.

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Temporary On-Skin Passive UHF RFID Transfer Tag Mohamed Ali Ziai and John C. Batchelor, Senior Member, IEEE

Abstract—A passive UHF RFID tag design is presented in the form of a transfer patch similar to a temporary tattoo that is mountable directly onto the skin surface. The transfer tag is suitable for monitoring of people over time in mission critical and secure environments. The antenna reactance is first calculated to conjugate match the measured RFID chip reactance and then full wave simulation is used to design the tag with good performance on a human flesh model. Finally the tag read range is measured on different parts of a volunteer’s body and compared to simulated read range values for the entire RFID bands.

TABLE I LF, HF AND UHF RFID SYSTEM CHARACTERISTICS [5], [6]

Index Terms—Body centric communications, conducting ink, RFID.

of up to 10 m. The long interrogation range, along with low cost tags and relatively high data rate as shown in Table I make UHF RFID systems suitable for human monitoring, [5], [6].

I. INTRODUCTION ITH the emergence of distributed and wireless sensor technologies readable tags will be able to collect a vast sea of data that can be processed to provide new information. Such information could be extremely important in mission critical environments such as power plants, airports, military bases and depots, refineries, oil rigs, and access restricted areas to provide the highest quality of security to record trends and take immediate required actions. In these environments as well as health care, monitoring and identifying people is vital to interface different services to create a more resilient system. Passive RFID is emerging as particularly useful in monitoring, identifying and tracking people in work environments [1]–[3]. For example, employees or visitors could be located and their action monitored in defined environments over a moderate distance without requiring a deliberate read action from the tagged person [4]. This would allow convenient fast access to restricted areas and a safer working environment for the employee, while benefiting the employing organizations and general public by enhancing security and reliability in the workplace. More established RFID systems such as LF and HF RFID can be used for the monitoring of people but the person to be identified must come very close to, or even touch, the reader. This is not particularly attractive for many applications where fast access or continuous monitoring is required. UHF RFID systems on the other hand can use relatively high gain standard matched antennas and electromagnetic wave propagation as a coupling mechanism and provide an identification distance (read range)

W

Manuscript received December 14, 2010; revised February 22, 2011; accepted April 04, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. The authors are with the School of Engineering, University of Kent, Canterbury, Kent CT2 7NZ, U.K. (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.2011.2163789

II. RFID TAGS FOR HUMAN MONITORING Human tagging, external to the body, is usually based on wrist bands or ID badges which could be removed and given to other people. There is little work reported on skin mounted RFID tags that cannot be removed without destroying the tag, and [7]–[11] describe work which requires a substantial gap of several millimeters between the tag and the skin to provide an acceptable read range. These designs are also complicated by multiple layers either with or without cross layer connections, and therefore they are not suitable for ultra low profile tags which can be directly mounted on skin. Being a passive system, the power collected by the tag antenna is used to activate the tag IC. The transmitted read power available to the tag antenna is constrained by electromagnetic compatibility regulations and thus it is essential that the maximum collected power is transferred to the IC as shown by the Friis equation [12]:

(1) where, is the effective isotropic radiated power of the reader; is the tag antenna gain; is the ASIC circuit activation threshold power; is the efficiency of the rectifier; is impedance matching coefficient between the antenna and the ASIC; and is the polarization coefficient between the reader and tag antenna. The polarization loss factor is given as [13]:

(2) where, is the incoming wave polarization orientation and is the polarization of the receiving antenna.

0018-926X/$26.00 © 2011 IEEE

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Even though most RFID reader antennas are circularly polarized, the orientation plane of the receiving antenna or tag on the body is unknown (as it varies with body position), and therefore polarization loss could potentially be high causing a reduction in read range. Also coupling to the human body adversely affects the tag antenna gain, and consequently reduces the power available to the chip. Since in designing a tag for human identification, the tag designer has no control over tag IC rectifier efficiency and has little control over tag polarization loss, therefore maximizing the power transmission coefficient to the chip is very important. The power transmission coefficient is given as [14]: (3) Fig. 1. Nested slotline RFID tag schematic.

where: (4) where is the reflection coefficient between the antenna and the and represent the antenna and ASIC input impedances; the chip input resistances; and and are the antenna and the chip input reactance, respectively. Therefore, a conjugate is required for maximum power transfer match from the tag antenna to the IC which is significantly capacitive. Being in close proximity to body tissues which have significant conductivity and permittivity the tag is capacitively loaded and the antenna radiation resistance is reduced. This presents a challenge to the designer to achieve high tag input inductance with a small antenna which is suitable for use on a human body. Currently reported designs have a thickness of about (1.8–4 mm) to achieve an acceptable efficiency [7], [10], [15]. This paper describes a substrate-less nested slotline antenna with design guidelines suitable for use directly on the skin as a transfer tattoo, i.e. no dielectric substrate or air space is required between the tag metallization and the surface of the skin. The tag is formed from a single conducting layer and therefore requires no via connections. III. TRANSFER RFID TAG DESIGN Fig. 1 shows a nested slotline antenna [15] that produces a surface current on a rectangular conductive patch. The antenna is designed to radiate at the RFID UHF band. The slot is formed mid-length along the patch and a small distance from the upper edge. Two short coplanar lines separated by gap extend from the slot center point out of the edge of the patch and the RFID ASIC is connected across the two lines as shown in Fig. 1. An electric field is induced in the slot which causes an electric current loop to flow around it. This thin loop provides an inductance to cancel the negative reactance of the capacitive ASIC. Since the slot width is very narrow compared to wavelength, and the current is not confined to the edge of the slot, it spreads out over the conductive patch and the antenna effective aperture size is large enough to offer an improved radiation efficiency, [16]. The tag acts as an electric dipole, linearly polarized along the -axis.

Fig. 2. Slotline equivalent circuit.

The nested slotline antenna resonant frequency and port reactance is controlled by the electrical length of the slot line, where the slotline resonant length and reactance can be approximated using Cohn’s closed form expressions [17] as shown below. The equivalent circuit of the packaged IC connected to the antenna is shown in Fig. 2, where the IC is represented as the voltage source “ ” and capacitive impedance “ ”. The conjugate matched impedance between the antenna and the RFID chip is represented by:

where is the input impedance for a transmission line of length , and characteristic impedance of , terminated on load , given by [18]: (5) where can be calculated for slotline of width on a substrate [18]: of height . If the ratio of

(6)

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TABLE II NESTED SLOTLINE DIMENSIONS ON HOMOGENOUS HUMAN TISSUE BLOCK

TABLE III LAYERED HUMAN MODEL ELECTRICAL PARAMETERS AT 900 MHz, [20]

Fig. 3. RFID transfer tag mounted on multilayer human model.

The effective dielectric constant of the slotline is less than the substrate permittivity , and can be calculated using the following equation:

TABLE IV SIMULATED SLOTLINE DIMENSIONS OBTAINED ON LAYERED PHANTOM

(7) The propagation constant losses is given by [17]:

for an ideal system without any

(8) and where and the ratio

denote the free space and slotline wavelength is given by [1], [17]:

the patch size can also be approximated on an homogenous substrate. As the conductive patch length affects the current distribution it should be long enough to avoid disturbing the high density of the currents around the slotline. It also affects the antenna gain, and higher gains are achieved for resonant patch size. Therefore the conductive patch resonant length at 915 MHz was calculated using a first order approximation for a half wave patch length given by [19]:

(10) (9) To ascertain appropriate values for in (7) and in (6) and (9) a simulated study using CST Microwave Studio was carried out using the transient solver finite difference time domain (FDTD) method. Simulation of a slotline on a dielectric block at 915 MHz indicated that a very simple homogenous flesh model and conductivity was sufficient with to represent the human surface which is in agreement with the results of [7]. Simulated fields were observed to penetrate no deeper than 10 mm, so this value was used for in the closed form equations. The above expressions are then used to calculate the slotline dimensions that provide a conjugate match to the RFID IC reactance. As the tag is mounted directly onto skin with no intervening dielectric sheet, the electrical parameters of flesh are required for the substrate calculation. Here we design the tag for the NXP G2XL flip chip strap package with quoted . typical port impedance Therefore, assuming no losses, a slotline antenna on a 10 mm homogenous substrate with parameters given in Table II gives at 915 MHz which is apan input impedance of propriate to cancel the manufacturer’s quoted input reactance. Having calculated the approximate value of the slotline antenna,

where is the speed of light in vacuum and is the resonant frequency of the antenna. The patch width is chosen to be wide enough not to curtail the surface currents and is at least 3 times the slot width. The slotline parameters obtained in Table I and the calculated patch size are then modeled in CST using the transient FDTD solver to guide the design of a matched and tuned tag directly onto skin with no intervening substrate. For improved accuracy, rather than use the homogenous model which gave the initial slotline and patch length, a 4 layer model of human tissue is created in CST using the electrical data given in Table III, [20]. Table IV lists the simulated values of the antenna parameters optimized for power transfer, gain, and bandwidth on the human torso model. IV. TRANSFER TAG DESIGN PROCEDURE The fields in the slot dominate the reactive impedance presented to the chip. Therefore the input impedance is mostly influenced by the slot length and width . The resistive part of the tag impedance is significantly affected by the slot size, and the thickness of the feed lines which significantly alters

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Fig. 4. Effect of slot length l and width w on antenna port impedance and resonant frequency. Broken lines=resistance, Solid lines=reactance, dotted lines=resonant frequency.

Fig. 6. Effect of patch width W and length L (mm) on antenna port impedance and resonant frequency. Broken lines=resistance, solid lines=reactance, dotted lines=resonant frequency.

Fig. 7. Effect of patch width W and length L (mm) on the antenna gain. Fig. 5. Effect of feedline thickness t on antenna port impedance and resonant frequency. Broken line=resistance, solid line=reactance.

the current flow into the chip. The half wave dipole patch currents are not strong compared to the current intensity around the ASIC feed point and slot transformer and therefore, providing the patch remains relatively wide, the patch size does not strongly affect the tag performance other than to have a secondary influence on resonant frequency. The gain of the tag is determined by the patch size. To inform the general design process, Figs. 4–6 show parameter curves obtained from simulation. For a given chip input match and desired frequency, Fig. 4 allows a suitable slot length to be obtained and combined with a related slot width . Fig. 5 relates the line thickness to input impedance, while Fig. 6 relates patch dimensions and to input impedance and operating frequency. The antenna reactance is primarily determined by the slotline length meaning the following steps can be taken to design the antenna: 1) Based on the quoted or measured capacitance of the ASIC, the length of the slot line is calculated using (5)–(9) to generate the required inductance to cancel the capacitive reactance of the ASIC and any parasitic effects of the chip fixture. 2) Using (10), patch length is calculated for resonance at 915 MHz. The patch width is selected to be equal to the slot length to reduce disruption to the fields around the slotline.

3) For a given slot length , the slot line width is then selected from Fig. 4 to resonate at close to the center resonant frequency (915 MHz). The slot acts as an input transformer and strongly affects the tag terminal impedance and as a consequence, the resonant frequency. 4) The Feed line width is selected to tune the real part of the antenna impedance (Fig. 5). The antenna resonant frequency is not sensitive to this parameter and the effect on tag reactance is only secondary. 5) Further tuning can be performed by adjusting the patch size to tune the antenna impedance (Fig. 6). The patch is essentially a wide half wave dipole with distributed current surface currents. As shown in Fig. 6, these low density currents on the patch do not greatly affect the resonant frequency providing the patch is not too narrow, in which case it becomes a thin dipole with an input impedance more strongly affected by its length. The antenna gain variation with patch dimensions is shown in Fig. 7. Higher gains are obtained for those lengths where half wave modes are supported and provided the patch is not too narrow, the gain is not strongly influenced by patch width (gain variations of 1 dB or less are simulated for a 25 mm change in width). Using the design curves of Figs. 4–7, a value of the port can be achieved by selecting the paimpedance rameters listed in Table II. This provides a reasonable match to . the quoted NXP G2XL impedance of

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Fig. 10 Simulated reflection coefficient in dB against frequency in GHz for tag on human model.

Fig. 8. Polar plot of simulated antenna directivity in y-z plane when mounted on human model.

Fig. 11. RFID chip impedance measurement kit.

VI. FABRICATION AND IMPEDANCE MEASUREMENT

Fig. 9. Simulated surface current distribution of tag on human model.

V. RADIATION PATTERN AND SURFACE CURRENT DISTRIBUTION A simulated polar plot of the slotline antenna directivity on the layered human model is shown in Fig. 8 where the maximum directivity is squinted around 40 from bore-site. This squint arises because the feed is not located on the center line of the finite sized human tissue block. The squint should not be evident when the tag is mounted on a continuous skin surface. The maximum directivity of the tag on the human model block is 4 dBi but radiation efficiency is very low on the human phantom ( 20.7 dB) which as expected reduces the gain and consequently tag read range according to the Friis free space (1). Fig. 9 shows the surface current distribution of the antenna on the human model. Strong surface currents are located close to the edge of the slotline and they spread out on the patch surface creating a half wave resonance along the patch. Fig. 10 illustrates the simulated slotline antenna resonant frequency and bandwidth on the human model. The slot line antenna has an input return loss of 20 dB at the center resonance frequency (915 MHz), with enough bandwidth (119 MHz from 864–983 MHz) to cover the entire FCC RFID frequency bands at an input return loss of 10 dB.

A prototype tag was fabricated using the parameters given in Table II. The antenna structure pattern was first etched as a negative in a thin metal stencil. The metalized layer was deposited on temporary transfer tattoo inkjet paper [27] using electrically conductive silver paint [28] which was profiled using the stencil. The paper covering layer of the Temporary Tattoo Transfer material was removed leaving the electrically conducof plastic. tive silver paint antenna on a thin layer The transfer tattoo tag was then placed on a volunteer’s forearm to measure the impedance at the port contacts. The port impedance was measured using a Bazooka balun [21] tuned at 910 MHz to transform the unbalanced vector network analyzer (VNA) coaxial cable to the balanced input required at the antenna port. In this measurement the VNA [22] was calibrated to the end of the coaxial cable using a standard calibration kit, and then the calibrated reference plane was offset to the tip of the . probe. The measured tag port impedance was To maximize the power transfer from the antenna to the chip, the input impedance of the chip with its fixing straps was also measured. This was achieved using five bespoke test calibration boards, [23], as shown in Fig. 11. Three of the boards were designed to calibrate the VNA using the short-open-load technique. The load board contained a 50 resistor. The fourth test board was used to measure the resistivity of a known resistor to evaluate the calibration (in this case 30 ), while the fifth test board was used to measure the RFID chip impedance. The VNA was calibrated using the calibration kit and verified using the 30 load with 1%. The RFID chip with contact strap impedance was fixed to the test board using electrically conductive adhesive

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TABLE V MEASURED AND QUOTED NXP G2XL ASIC PORT IMPEDANCE AT 915 MHz

Fig. 13. Simulated and measured read range for tattoo tag mounted on human. Solid blue bar = simulated on human model, crossed blue bar=measured on volunteer forearm, dotted green line = measured on the volunteer stomach and blue dimond bar = measured on volunteer chest. 867 MHz, 910 MHz, 918 MHz, 924 MHz, 954 MHz represent center FCC RFID frequency bands for Europe, USA, Korea, China and Japan respectively.

Fig. 12. Tattoo transfer tag mounted on volunteer arm. Conducting paint separated from skin by a 10 micrometer layer.

the thorax. The prototype tag durability was tested by wearing the prototype tattoo transfer tag on the volunteer’s forearm throughout a working day and regularly testing its functionality by measuring read range for the duration of 5 hours. The read range measured was found to be consistent during the experiment period with the volunteer engaged in normal office activities. VIII. CONCLUSIONS AND FURTHER DEVELOPMENTS

transfer tape [24] and the measured chip impedance is compared to the quoted typical manufacture’s value in Table V. The RFID IC was then attached to the transfer tattoo tag ports using electrically conductive adhesive transfer tape and the prototype tag was transferred directly to a volunteer’s skin as shown in Fig. 12. VII. READ RANGE MEASUREMENTS The tattoo transfer tag read range was measured on the volunteers forearm, stomach and chest, using a monostatic reader with 27.5 dBm ERP using an AS3990 reader chipset, [25] connected to a 6.5 dBic circularly polarized patch antenna via a short coax cable. All tag read ranges were measured in a lab environment and are compared to simulated read ranges assuming tag and the reader antennas have matching polarization. The prototype tattoo tags were placed on the forearm, stomach and chest of a volunteer and the maximum ranges at which tags read in the bands for different countries were measured on many occasions and the average distances calculated to mitigate against multipath effects. As illustrated in Fig. 13 the measured prototype tag read range on the forearm was 80 cm, lower than simulated (about 100 cm) while the measured read range on the chest (110 cm) and stomach (120 cm) was higher than simulation indicated. This is because the forearm has a thinner layer of skin and fat than assumed in simulation. The tag antenna was initially tuned on a model representing the thorax with thicker skin and fat layers so the prototype tag was less efficient on the forearm than simulation predicted. Higher losses due to higher conductivity and permittivity of the dense muscles and bone in the forearm could also reduce the antenna radiation efficiency compared to mounting sites on

This paper has introduced a new concept of placing ultra thin RFID tags directly onto the skin in the form of a transfer tattoo. The tag is a novel topology of a slotted patch which requires no artificial substrate layer between the metallization and the human skin surface. Results indicated that useful read ranges of 80 cm are possible when mounted on a forearm and increasing to 120 cm when mounted on the thorax. These read ranges are impressive considering the tag is mounted directly on the skin thick adhesive layer. A design procedure with only a 10 for the tag is provided and with refinement the tag can be optimized to give higher read ranges specific parts of the human body. Applications are proposed for temporary security clearance, ticketing and military use or where it might not be desirable to insert sub-skin RFID capsules. Although RFID has been used to illustrate the concept, it is envisaged that skin transfer antennas could also be designed with the terminals connected by surgical plaster. The technology could also be applicable for medical sensing and as a pick up for implanted devices. Inkjet conducting ink processes could make this a cheap and widespread technology, [26]. ACKNOWLEDGMENT The authors thank Austriamicrosystems for providing the RFID reader development kit. REFERENCES [1] Intelleflex_RFID, Personnel Monitoring [Online]. Available: http://www.intelleflex.com/Solutions.PM.asp [2] Motorola_RFID, “Motorola’s healthcare mobility solutions,” Improving Patient Safety at the Point of Care in Theatre and in Hospitals [Online]. Available: http://www.motorola.com/web/Business/Solutions/Federal%20Government/_Documents/Static%20files/ Fed_Healthcare_AB_FINAL.pdf?localeId=33

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[3] H. Lehpamer, RFID Design Principles. Norwood, MA: Artech House Microwave Library, 2007, p. 293, 13: 978-1596931947. [4] C. Chao et al., “Improving patient safety with RFID and mobile technology,” Int. J. Electron. Healthcare, vol. 3, pp. 175–192, 2007. [5] Z. Chen, Antennas for Portable Devices. New York: Wiley, 2007, 13: 978-0470030738. [6] Atmel, Considerations for RFID Selection [Online]. Available: http:// www.atmel.com/dyn/resources/Prod_documents/secrerf_3_04.pdf [7] G. Marrocco, “Body-matched RFID antennas for wireless biometry,” in Proc. 1st Eur. Conf. on Antennas and Propagation, 2006, pp. 1–5, 92-9092-937-5. [8] C. Occhiuzzi et al., “Body-matched slot antennas for RadioFrequency identification,” presented at the XXIX URSI General Assembly, Chicago, Aug. 7–16, 2008. [9] G. Marrocco, “RFID antennas for the UHF remote monitoring of human subjects,” IEEE Trans. Antennas Propag., vol. 55, pp. 1862–1870, 2007. [10] H. Rajagopalan and Y. Rahmat-Samii, “Conformal RFID antenna design suitable for. Human monitoring and metallic platforms,” in Proc. 4th Eur. Conf. on Antennas and Propagation (EuCAP), , Apr. 12–16, 2010, pp. 1–5, 978-1-4244-6431-9. [11] M. Ziai and J. Batchelor, “Thin ultra high-frequency platform insensitive radio frequency identification tags,” IET Microw. Antennas Propag., vol. 4, pp. 390–398, 2010. [12] K. Finkenzeller, RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification. New York: Wiley, 2003, 13: 978-0471988519. [13] C. Balanis, Antenna Theory. New York: Wiley, 1997, 13: 978-0471592686. [14] K. V. S. Rao, P. V. Nikitin, and S. F. Lam, “Impedance matching concepts in RFID transponder design,” in Proc. Automatic Identification Advanced Technologies. Fourth IEEE Workshop on, Oct. 17–18, 2005, pp. 39–42, 0-7695-2475-3. [15] G. Marrocco, “RFID antennas for the UHF remote monitoring of human subjects,” IEEE Trans. Antennas Propag., vol. 55, pp. 1862–1870, 2007. [16] J. Kraus and R. Marhefka, Antennas. New York: McGraw-Hill, 1988, 0070354227. [17] K. C. Gupta, Microstrip Lines and Slotlines 2nd Ed, 2nd ed. Norwood, MA: Artech House Publishers, 1996, 13: 978-0890067666. [18] B. Wadell, Transmission Line Design Handbook. Norwood, MA: Artech House Boston, 1991, 13: 978-0890064368. [19] A. Ittipiboon, R. Garg, I. Bahl, and P. Bhartia, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001, 13: 978-0890065136. [20] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: I. Literature survey,” Phys. Med. Bio., vol. 41, p. 2231, 1996.

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[21] K. Fujimoto, Mobile Antenna Systems Handbook. Norwood, MA: Artech House, 2008, 13: 978-1596931268. [22] Anritsu, Anritsu ME7808 Broadband Vector Network Analyzer [Online]. Available: http://www.anritsu.co.uk/ [23] L. Mayer and A. Scholtz, “Sensitivity and impedance measurements on UHF RFID transponder chips,” presented at the 2nd Int. EURASIP Workshop on RFID Technology, Budapest, Hungary, Jul. 7–8, 2008. [24] M.E.CT . 9. 703,3M Electrically Conductive Tape 9703 [Online]. Available: http://solutions.3m.com/wps/portal/3M/en_WW/Graphics/Scotch print/Prod-Info/Catalog/?PC_7_RJH9U5230GE3E02 LECIE20S7M3 _nid=23L97PJH4CbeGX9B4321ZWgl [25] Austriamicrosystems, UHF RFID and HF RFID Reader IC Manufacturer [Online]. Available: http://www.austriamicrosystems.com/ [26] J. Batchelor et al., “Inkjet printing of frequency selective surfaces,” Electron. Lett., vol. 45, pp. 7–8, 2008. [27] TheMagicTouch, Temporary Tattoo Transfer Paper [Online]. Available: http://www.themagictouch.co.uk/transfer/tattoo.htm [28] Electrolube, Silver Conductive Paint [Online]. Available: http://www. electrolube.com/docs/catalog.pdf

Mohamed Ali Ziai received the M.Sc. degree in broad band and wireless communication engineering from the University of Kent, Canterbury, U.K., where he is currently working toward the Ph.D. degree. His main research activities are aimed at the electrically thin and platform insensitive ultra high frequency radio frequency identification tags. His other research interests are linked to the field of antennas and UHF RFID.

John C. Batchelor (S’93–M’95–SM’07) received the B.Sc. and Ph.D. degrees from the University of Kent, Canterbury, U.K., in 1991 and 1995, respectively. From 1994 to 1996, he was a Research Assistant with the Electronics Department, University of Kent, and in 1997, became a Lecturer of electronic engineering. He now leads the Antennas Group at Kent and is a Reader in Antenna Technology. His current research interests include UHF RFID tag design, body-centric antennas, printed antennas, compact multiband antennas, electromagnetic bandgap structures, and long-wavelength FSS (frequency-selective surfaces).

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011

Carbon Nanotube Composites for Wideband Millimeter-Wave Antenna Applications Aidin Mehdipour, Member, IEEE, Iosif D. Rosca, Abdel-Razik Sebak, Fellow, IEEE, Christopher W. Trueman, and Suong V. Hoa

Abstract—In this paper, we explore using carbon nanotube (CNT) composite material for wideband millimeter-wave antenna applications. An accurate electromagnetic model of the composite antenna is developed using Microwave Studio for numerical analysis. Good agreement between computed and measured results is shown for both copper and CNT antennas, and their performance is compared. The CNT antenna shows stable gain and radiation patterns over the 24 to 34 GHz frequency range. The dispersion characteristics of the CNT antenna show its suitability for wideband communication systems. Using a quarter-wave matched T-junction as feed network, a two-element CNT antenna array is realized and the performance is compared with a copper antenna. The housing effect on the performance of the CNT antenna is shown to be much lower than for the copper antenna. Index Terms—Antenna, carbon nanotube (CNT) composites, millimeter-wave (mm-wave), wideband antenna.

I. INTRODUCTION

M

ETALS are commonly used in antenna structures for the radiating elements, feed lines, and ground plane. However, for some applications, cost, fabrication procedure, weight or corrosion resistance can limit the usefulness of metal antennas. Some recent studies have used various composite materials as replacements for metals [1]–[5]. In [1], a conductingpolymer patch antenna is proposed. Silver nanoparticle ink [2], [3] and metallo-organic conductive ink [4] have been used to prepare a highly-conductive antenna. In [5], metalized foam is used to make microstrip-patch antenna. Advanced carbon-fiber composite (CFC) materials are being used in the aerospace industry as a replacement for metal [6], [7] because of their higher strength, lower cost and lighter weight. There are various kinds of CFCs: reinforced continuous carbon fibers (RCCF) [6], short carbon fibers (SCF), carbon black (CB), and carbon nanotube (CNT) [7]. In [8], we investigated the use Manuscript received April 27, 2010; revised January 31, 2011; accepted March 17, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). A. Mehdipour was with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G2W1, Canada. He is now with SDP Telecom Inc., Montreal, QC H9P 1J1, Canada (e-mail: [email protected]; [email protected]). I. D. Rosca and S. V. Hoa are with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC H3G2W1 Canada, (e-mail: [email protected]; [email protected]). A.-R. Sebak and C. W. Trueman are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G2W1, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2163755

of carbon fiber materials as radiating elements of an RFID antenna. It was observed that CFC can be efficiently used in such a resonant antenna. Very recently, we explored the use of RCCF for ultra-wideband (UWB) applications [9]. Due to the favorable mechanical and electrical properties, CNTs have been of interest in nanoelectronics and nanoantenna applications [10]–[17]. The density of CNT composites is around 1.4 g/cm , half that of aluminum and more than five times lower than copper. Showing high thermal conductivity of about ten times that of copper, CNTs are attractive for high heat-transfer applications. CNT composites can be made using single-wall nanotubes (SWCNT) or multi-wall carbon nanotubes (MWCNT). Details about CNT types and models of materials are given in [12], [13]. However, CNT dipoles show for microwave extremely low efficiency in the order of applications [14], [15], due to their high resistance per unit m [14]. Therefore, CNT arrays and length, of about 10 k composites are proposed to improve the efficiency [17]–[21]. The electrical conductivity of CNT composites depends on the properties and loading of the CNTs, the aspect ratio of the CNTs, and the characteristics of the conductive network throughout the matrix [7], [22]. Millimeter wave (mm-wave) communication systems are increasingly used in many commercial and military applications, such as imaging systems, automotive radars, medicine, high resolution radars and mobile communication systems [23], [24]. In mobile and military applications where the antenna may be used in harsh environments, replacing metals with a more suitable material increases the system reliability. In this work, we have explored the use of SWCNT composite material for wideband mm-wave antennas. A low-profile wideband microstrip-fed monopole antenna operating over 24 to 34 GHz is designed and investigated by measurement and by numerical simulation. Then, a two-element array with a matched T-junction feed network is realized and the performance of the CNT antenna is compared with a copper one. Since in reality it is likely that the antenna will be close to other devices or be integrated with different components in the circuit, the housing effect on the copper and CNT antennas performance is investigated. II. SWCNT MATERIAL AND METHOD OF COMPOSITE ANTENNA PREPARATION The composite samples are produced at the Concordia Center for Composites (CONCOM) [25]. To develop a model of the composite material for use in computer simulations of mm-wave antennas, we use standard Ka-band rectangular waveguide with cross-section dimension of 7.11 3.55 mm . The thickness of

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MEHDIPOUR et al.: CARBON NANOTUBE COMPOSITES FOR WIDEBAND MILLIMETER-WAVE ANTENNA APPLICATIONS

Fig. 2. (a) The geometry of mm-wave monopole antenna, (b) simulated versus conductivity.

Fig. 1. (a) Buckypaper preparation, (b) Substrate plating with buckypapers; 1—impregnated buckypaper-patch; 2—substrate; 3—vacuum bag; 4—breather; 5—release film; 6—sealant.

composite is 0.2 mm. Then the waveguide setup is modeled with CST-Microwave Studio (CST MWS) [26]. By minimizing the difference between the simulated and measured scattering parameters over a frequency range, the complex permittivity can be extracted [9], [27]. The sample used to build the antennas and described below has effective relative permittivity S/m, over the desired frequency range. conductivity Individual CNTs have outstanding mechanical and electrical properties but the transfer of those properties into the composite is hindered by the difficulties related to their homogenous dispersion and high contact resistances at the nanotube joints [22]. Currently, thin films made of SWCNTs, called buckypapers, display high electrical conductivity [20]. The high conductivity recommends them as an efficient replacement for metals in various electronic applications. For antenna fabrication, we printed CNT on a substrate and then cut out the desired antenna pattern using a high precision milling machine. Buckypaper is a flexible and soft material that needs to be hardened by resin infiltration in order to be processed on a milling machine. Manufacturing buckypaper antennas comprises the following three steps: buckypaper preparation, plating the substrate with buckypaper impregnated with epoxy resin, and cutting out the antenna.

A. Buckypaper Preparation Buckypaper preparation is schematically shown in Fig. 1(a). SWCNTs from Nikkiso Co. (0.5 g) are dispersed in N, N dimethylformamide (DMF) by a horn sonicator (Misonix 3000) at 42 W of power for 30 min. Next the SWCNT suspension is filtered on nylon membrane-filter with pore size of 45 micron. After filtration the buckypaper and the membrane are placed between several filter-papers (Whatman no. 1) and lightly pressed between two aluminum plates to absorb the excess solvent. The wet buckypaper is then separated from the filter membrane and dried at 130 C for 12 hours to form a sheet of 140 140 mm and 50 m thickness.

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S

B. Substrate Plating With Buckypaper We use Rogers 4350 as the substrate of composite antenna. The copper was removed from the Rogers 4350 substrate by hydrogen peroxide/hydrochloric acid etching from both sides. Patches of 50 50 mm were cut out from a buckypaper sheet. As shown in Fig. 1(b), the patches were impregnated with a mixture of epoxy resin (Epon 862) and hardener (26.4 wt% Epikure W, Hexion Specialty Chemicals) in a vacuum oven at 80 C for 30 min. Next the impregnated patches were placed on each side of the substrate, vacuum-bagged and cured in an autoclave at 140 C for 4 hours under 40 psi of pressure. Then, the geometry of antennas is cut out on the substrate using a milling machine.

III. SINGLE ELEMENT ANTENNA AND MEASUREMENTS Fig. 2(a) shows the EM model of monopole antenna developed in MWS. The antenna is fed through a 50-ohm microstrip line. The antenna geometrical parameters are mm, mm, mm, mm, mm, and mm. The substrate is Rogers 4350 with a thickness of 0.508 mm, , and . A. Composite Conductivity An antenna made of composite is basically a lossy antenna, because the ohmic loss of composite material is higher than that of metal. The impedance bandwidth of a monopole antenna for copper and composites with various conductivities is shown in Fig. 2(b). In the composite antenna both the radiating element and the ground plane are made of CNT. It is observed that the bandwidth of the composite antenna is wider than that of the copper antenna operating over the 24 to 34 GHz frequency range. In fact, as the ohmic loss increases, the bandwidth increases and the -factor decreases, and the composite antenna shows this behavior. Due to the lower conductivity, the gain and radiation effiof composite antennas are lower than that of the ciency copper antennas. The radiation efficiency is defined as

(1)

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TABLE I RADIATION EFFICIENCY OF THE MONOPOLE ANTENNA AT REF. PLANE 2

Fig. 4. (a) Fabricated copper and CNT monopole antennas, (b) connector : ;r : ;r : mm), (c) S of copper antenna, model (r (d) S of CNT antenna.

= 0 18

Fig. 3. The radiation efficiency of the copper and CNT ( monopole antennas.

= 0 38

= 1 22

= 10 kS/m)

where and are the radiated power and the ohmic losses. The overall ohmic loss is the sum of the power loss in the microstrip feed line plus the power loss in the antenna itself. The antenna efficiency in (1) excludes the loss of the feed line, and is calculated at the input of the antenna, reference plane 2. Table I shows the effect of the conductivity on the composite antenna’s radiation efficiency at four different frequencies, 24, 28, 32 and 34 GHz. Note that using the parameters of the sample kS/m, the values can be comdescribed in Section II, pared with the copper antenna’s radiation efficiency. Fig. 3 shows the radiation efficiency of the copper and CNT antennas, at both reference planes 1 and 2, over the whole frequency range of interest. The curves labeled [2] give the radiation efficiency of the antenna alone. The curves labeled [1] give the efficiency of the system made up of the lossy feed line and the antenna, much lower than that of the antenna alone. The lossy feed line behaves as an attenuator, giving control over the amount of power reaching the antenna itself. B. Experimental Results The copper and CNT monopole antennas are fabricated as shown in Fig. 4(a). The composite antenna is fabricated using the method described in Section II. The antennas are fed by solder-free 2.92 mm Southwest Microwave connectors [28]. Using an Agilent-E8364B network analyzer, the reflection coefficient of each antenna is measured and shown in Fig. 4(c) and (d). The connector is included in the simulation model in CST-MWS to better simulate the measurement setup. The radiation pattern of both the copper and the CNT antennas in both the and the planes were measured in an anechoic chamber. The normalized radiation

Fig. 5. Normalized radiation pattern of monopole antenna at: (a) 26.5 GHz, (b) 29 GHz, (c) 33.5 GHz.

pattern at 26.5, 29, and 33.5 GHz are reported in Fig. 5. It is observed that the CNT antenna shows a stable radiation pattern over the frequency range of interest. The radiation pattern is found to be almost omnidirectional in the -plane, which is of most interest for wireless communication systems. It should be noted that the loss associated with the length of the microstrip line can play an important role on the antenna gain. The scattering parameters of a 2 cm CNT microstrip line were measured and are shown in Fig. 6. After excluding the dB in total), we found that loss within two connectors ( the CNT microstrip line shows about 2.6 dB/cm loss over the 24 to 34 GHz frequency range. The loss of copper microstrip line is about 0.35 dB/cm. ) in The realized gain at boresight angle ( reference plane 1, at the connector in Fig. 4(a), is shown in

MEHDIPOUR et al.: CARBON NANOTUBE COMPOSITES FOR WIDEBAND MILLIMETER-WAVE ANTENNA APPLICATIONS

Fig. 6. (a) Prototype of CNT microstrip line, (b) S

and S

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parameters.

Fig. 8. Transfer function of monopole antenna in Tx/Rx configuration for R mm, (a) face-to-face, and (b) side-by-side setups.

30

=

Fig. 7. Boresight gain of single monopole antenna. (a) measured and simulated gain at Ref. plane 1. (b) measured gain at Ref. plane 2.

Fig. 7(a). By excluding the loss of the connector and the microstrip line, the measured antenna gain at reference plane 2, at the input to the radiating element, is shown in Fig. 7(b). As expected, it is observed that the average gain of the CNT antenna is lower than the copper one, which is due to the lower conductivity of the composite material. The transfer function of the antenna is measured by making Tx/Rx setups of two identical antennas at a separation distance mm. Fig. 8 shows the transfer function of CNT and of copper antennas for both face-to-face and side-by-side configurations. If the group delay of antennas shows a highly frequencydependant behavior, the time domain pulse is considerably distorted due to the nonlinearity of phase. But the fairly-constant group delay of Fig. 8 shows that the CNT antenna has low dispersion, indicating that it is useful for UWB radios. IV. TWO-ELEMENT ANTENNA ARRAY AND MEASUREMENTS As shown in Fig. 9(a), the two-element array fed by matched T-junction is designed to operate from 24 to 34 GHz. The sepamm which is about ration distance between elements is at 26.5 GHz. The geometrical parameters are , and mm. The ground plane height is 15.5 mm. The radiation efficiency of the copper and CNT array antennas is calculated over 24 to 34 GHz frequency range as displayed in Fig. 9(b). The curves labeled [2] give the radiation efficiency of the antenna alone. The curves labeled [1] give the efficiency of the system made up of the lossy feed structure plus the antenna. The ohmic

Fig. 9. (a) The geometry of two-element array antenna. (b) radiation efficiency of the copper and CNT array antennas.

loss of the matched T-junction and the feed lines reduces the efficiency of the system greatly compared to the efficiency of the antenna alone. The copper and CNT array antennas were fabricated and was measured as shown in the reflection coefficient Fig. 10(c)–(d). Good agreement is observed between the measurement and the simulation, specifically when the effect of connectors are considered in EM model of antenna. Due to the bandwidth limitation of the T-junction feed network, the impedance match is degraded at the upper edge of the frequency range for the copper antenna, and the maximum frequency of operation is decreased to 32 GHz. An impedance match with dB is desired. The normalized radiation pattern of the copper and the CNT array antennas was measured and compared at two frequencies, 26.5 and 30 GHz, as shown in Fig. 11. It is observed that CNT antenna shows a stable radiation performance over the frequency range of interest. The gain at the boresight angle in reference plane 1, at the connector in Fig. 10(a), is shown in Fig. 12(a). Simulations show that the loss of composite T-junction between the connector and reference plane 2 is about 3.5–4.4 dB from 24 to 34 GHz. Taking

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Fig. 13. Mutual coupling of CNT antenna elements at Ref. plane 2.

Fig. 10. (a) Fabricated CNT and copper array antenna, (b) S antenna, (c) S of the CNT antenna.

of the copper

mm. By removing the losses the monopole antennas is in the microstrip line and the connectors, the mutual coupling between two antennas is obtained at reference plane 2. It is obdB over alserved that the mutual coupling is lower than most the entire frequency range of interest. V. HOUSING EFFECT Usually the antenna is integrated in a circuit which includes various components and modules. In mobile and vehicular applications, the antenna could be used in proximity to a variety of materials, such as metal objects or the human body. When the antenna radiates in the vicinity of an object, the backscattered fields produced by the object induce electric current on the antenna, affecting the antenna performance [29], [30]. The induced current on the antenna, , caused by backscattered mag, can be expressed as, netic field from nearby scatterer,

Fig. 11. Normalized radiation pattern of array antenna at: (a) 26.5 GHz, (b) 30 GHz.

Fig. 12. Boresight gain of array antenna, (a) measured and simulated gain at Ref. plane 1. (b) simulated gain at Ref. plane 2.

out the loss of connector and feed network, the simulated boresight gain of antenna array at reference plane 2 is shown in Fig. 12(b). The mutual coupling between the adjacent elements of the array is investigated in Fig. 13. The separation distance between

(2) where . Due to the lower conductivity, the composite antenna produces weaker radiated fields than the . Moreover, is larger copper antenna, leading to lower for the composite, making the ratio in (2) lower. As a result, the CNT antenna is less affected by nearby conductive objects than is the copper antenna. The housing effect setup is shown in Fig. 14(a). A 5 cm 5 cm steel sheet is placed very closely in the front of the antenna with a separation distance of . The reflection coefficient mm, as of CNT and copper antennas is measured for displayed in Figs. 14(b) and (c). It is observed that the performance of copper antenna deteriorates significantly, whereas the CNT antenna still operates over the entire frequency range with dB. We have also investigated the housing effect for different values of , not shown here. It was observed that CNT antenna performance shows much lower sensitivity to the steel sheet than does the copper antenna. The far-field radiation pattern of the antennas in the vicinity of steel sheet is evaluated as shown in Fig. 15. Since the metal

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TABLE II RADIATION EFFICIENCY OF THE COMPOSITE ANTENNA WITH/WITHOUT THE PRESENCE OF METALLIC SHEET

The radiation efficiency of the composite antenna at reference plane 2 in the vicinity of metal sheet is calculated as shown in Table II. It is observed that the radiation efficiency of the antenna changes slightly in the presence of the metal sheet. Therefore, since the matching of composite antenna is still good, the total efficiency does not change significantly. VI. CONCLUSION Fig. 14. (a) Housing effect setup, (b) S antenna.

of copper antenna, (c) S

of CNT

In this paper, a single-wall carbon nanotube composite material is used to fabricate an antenna for millimeter-wave applications. The operation and design guidelines are presented. Measured and simulated results show that the CNT composite antenna has good performance over a bandwidth of 24 to 34 GHz. A two-element array antenna shows similar results. Moreover, the CNT antenna shows low dispersion characteristics over the frequency range of interest and so may be used for ultra-wideband radios. The housing effect on the antenna performance is investigated and it is shown that the CNT antenna is much less affected than the copper antenna. The antenna performance such as gain can be adjusted by changing the conductivity of composite, while it is not possible for materials with fixed conductivity such as copper. REFERENCES

Fig. 15. Normalized radiation pattern in the vicinity of metal sheet at: (a) 25 GHz, (b) 34 GHz.

sheet behaves like a good reflector, the main beam of the -plane radiation pattern moves toward and the -plane radiation pattern is not omnidirectional anymore. It is observed that as frequency goes up, the number of notches in radiation pattern increases which is due to reflections and diffractions from the metal sheet. However, because of the lower EM coupling with nearby metal sheet, the CNT antenna shows much weaker notches at -plane than copper antenna as shown in Fig. 15(b). We found that the CNT antenna shows weaker notches than copper antenna at -plane at frequencies above 30 GHz as well, making the radiation pattern more stable. At lower frequencies, both antennas show almost the same radiation pattern.

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[12] G. Y. Slepyan, S. A. Maksimenko, A. Lakhtakia, O. Yevtushenko, and A. V. Gusakov, “Electrodynamics of carbon nanotubes: Dynamic conductivity, impedance boundary conditions, and surface wave propagation,” Phys. Rev. B, vol. 60, no. 24, pp. 17136–17149, 1999. [13] G. Y. Slepyan, S. A. Maksimenko, A. Lakhtakia, and O. Yevtushenko, “Electromagnetic response of carbon nanotubes and nanotube ropes,” Synth. Metals, vol. 124, pp. 121–123, 2001. [14] P. J. Burke, S. D. Li, and Z. Yu, “Quantitative theory of nanowire and nanotube antenna performance,” IEEE Trans. Nanotechnol., vol. 5, no. 4, pp. 314–334, 2006. [15] G. W. Hanson, “Fundamental transmitting properties of carbon nanotube antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3426–3435, 2005. [16] G. W. Hanson, “Current on an infinitely-long carbon nanotube antenna excited by a gap generator,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 76–81, 2006. [17] C. Rutherglen and P. Burke, “Nanoelectromagnetics: Circuit and electromagnetic properties of carbon nanotubes,” Small, vol. 5, no. 8, pp. 884–906, 2009. [18] J. Song, J. Kim, Y. Yoon, B. Choi, and C. Han, “Inkjet printing of singe-walled carbon nanotubes and electrical characterization of the line pattern,” Nanotechnol., vol. 19, no. 9, p. 095702, 2008. [19] P. J. Burke and C. Rutherglen, “Carbon nanotube based variable frequency patch antenna,” U.S. patent 2009/0231205 A1, Sep. 17, 2009. [20] A. Mehdipour, I. D. Rosca, A.-R. Sebak, C. W. Trueman, and S. V. Hoa, “Advanced carbon-fiber composite materials for RFID tag antenna applications,” Appl. Comput. Electromagn. Society (ACES) J., vol. 25, no. 3, pp. 218–229, 2010. [21] Y. Zhou, Y. Bayram, J. L. Volakis, and L. Dai, “Conformal load-bearing polymer-carbon nanotube antennas and RF front-ends,” presented at the IEEE Antenna and Propagation Symp. (APS 2009), Charleston, SC, Jun. 1–5, 2009. [22] I. D. Rosca and S. V. Hoa, “Highly conductive multiwall carbon nanotube and epoxy composites produced by three-roll milling,” Carbon, vol. 47, pp. 1958–1968, 2009. [23] M. Sun, Y. P. Zhang, G. X. Zheng, and W.-.Y. Yin, “Performance of intra-chip wireless interconnect using on-chip antennas and UWB radios,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2756–2762, 2009. [24] R. A. Alhalabi and G. M. Rebeiz, “High-efficiency angled-dipole antennas for millimeter-wave phased array applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2756–2762, 2009. [25] Concordia Center for Composites (CONCOM), Concordia University, QC, Canada, 1979 [Online]. Available: http://concom.encs.concordia.ca [26] CST—Microwave Studio, Computer Simulation Technology 2009. [27] R. K. Challa, D. Kajfez, V. Demir, J. R. Gladden, and A. Z. Elsherbeni, “Characterization of multiwalled carbon nanotube (MWCNT) composites in a waveguide of square cross section,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 3, pp. 161–163, 2008. [28] Southwest Microwave, Inc.. Tempe, AZ, US. [29] C.-C. Lin, S.-W. Kuo, and H.-R. Chuang, “A 2.4-GHz printed meander-line antenna for USB WLAN with notebook-PC housing,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 546–548, 2005. [30] K. Bahadori and Y. Rahmat-Samii, “A miniaturized elliptic-card UWB antenna with WLAN band rejection for wireless communications,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3326–3332, 2007.

Aidin Mehdipour (S’09–M’11) received the B.S. degree from Amirkabir University of Technology, Tehran, Iran, in 2003, the M.S. degree from the University of Tehran, in 2006, and the Ph.D. degree from Concordia University, Montreal, QC, Canada, in 2011, all in electrical engineering. He is currently a Research Engineer with SDP Telecom Inc., Montreal, QC, Canada. His main research interests include advanced carbon fiber composites electromagnetic analysis, novel material development for shielding and antenna/microwave applications, electromagnetic compatibility (EMC), microwave circuits, small antenna design, RFID, multiband, ultrawideband, and millimeter wave antennas. Mr. Mehdipour was the recipient of the David J. Azrieli Graduate Fellowship, as the highest ranking Concordia University Fellowship Award, and the France and André Desmarais Graduate Fellowship in 2010, the Howard Webster Foundation Doctoral Fellowship and Doctoral Thesis Completion Award in 2011. He is a member of IEEE AP and EMC societies and of the Applied Computational Electromagnetics Society (ACES).

Iosif Daniel Rosca received the Engineer Diploma in polymer science and technology from Polytechnic University of Bucharest, Romania and the Ph.D. degree in chemistry (with Prof. J. M. Vergnaud) from Jean Monnet University, Saint-Etienne, France. Currently, he is an Associate professor of polymer science and technology at the Polytechnic University of Bucharest, Romania. He spent two years as an JSPS Postdoctoral Fellow at Hokkaido University, Japan. He is currently a Research Associate at Concordia University, Canada. His present research focuses on fabrication and application of polymer composites based on carbon nanomaterials.

Abdel Razik Sebak (F’10) received the B.Sc. degree (with honors) in electrical engineering from Cairo University, in 1976 and the B.Sc. degree in applied mathematics from Ein Shams University, in 1978. He received the M.Eng. and Ph.D. degrees from the University of Manitoba, in 1982 and 1984, respectively, both in electrical engineering. From 1984 to 1986, he was with the Canadian Marconi Company, working on the design of microstrip phased array antennas. From 1987 to 2002, he was a Professor in the Electrical and Computer Engineering Department, University of Manitoba. He is a Professor of electrical and computer engineering, Concordia University. His current research interests include phased array antennas, computational electromagnetics, integrated antennas, electromagnetic theory, interaction of EM waves with new materials and bioelectromagnetics. Dr. Sebak received the 2000 and 1992 University of Manitoba Merit Award for outstanding Teaching and Research, the 1994 Rh Award for Outstanding Contributions to Scholarship and Research, and the 1996 Faculty of Engineering Superior Academic Performance. He is a Fellow of IEEE. He has served as Chair for the IEEE Canada Awards and Recognition Committee (2002–2004) and the Technical Program Chair of the 2002 IEEE-CCECE and 2006 ANTEM conferences.

Christopher W. Trueman received the Ph.D. degree from McGill University in 1979. He has applied the methods of computational electromagnetics to problems such as aircraft antenna performance, antenna-to-antenna coupling and EMC on aircraft, aircraft and ship radar cross-section at HF frequencies, suppression of scattering of the signal of a commercial radio station from power lines, dielectric resonators, unconditionally-stable formulations for the finite-difference time-domain method, and the fields of portable radios such as cellular phones held against the head. Recently, his research has investigated the radar cross-section of ships at VHF frequencies, composite materials for aircraft, indoor propagation, and EMC issues between portable radios and medical equipment. He is currently the Associate Dean for Academic Affairs in the Faculty of Engineering and Computer Science at Concordia University.

Suong V. Hoa is a Professor at the Department of Mechanical and Industrial Engineering at Concordia University, Montreal, Quebec, Canada. He was also Chair of the Department for nine years from 1994–2000 and 2003–2006. He has been working on composites for the past 31 years (since 1979). Dr. Hoa is President of the Canadian Association for Composite Structures and Materials. He received the Synergy award for collaborative work on composites with Bell Helicopter from the Natural Sciences and Engineering Research Council of Canada in 2006. He also received the Prix partenariat from Association des Directeur de Recherche Industrielle du Quebec in 2006 and in 2009. He was the recipient of the Nano Academia prize from Nanoquebec in 2008 and was given the title of Research Fellow from Pratt & Whitney Canada Ltd., in September 2008. He together with Dr. H. Hamada of Kyoto Institute of Technology, initiated the series of Canada-Japan workshop on composites which have been held in Canada and Japan every two years since 1996.

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Creation of a Magnetic Boundary Condition in a Radiating Ground Plane to Excite Antenna Modes Marko Sonkki, Student Member, IEEE, Marta Cabedo-Fabrés, Member, IEEE, Eva Antonino-Daviu, Member, IEEE, Miguel Ferrando-Bataller, Member, IEEE, and Erkki T. Salonen

Abstract—The creation of a magnetic boundary condition into the plane of symmetry in a radiating ground plane of a portable device is proposed to enhance its radiation efficiency and bandwidth. This magnetic boundary condition is achieved by exciting the antenna through symmetrical feeding. A novel antenna consisting of a symmetrical folded dipole is presented in order to exemplify how a broad bandwidth and very good radiation properties can be obtained with the proposed technique. The Theory of Characteristic Modes is used to analyze and compare a folded dipole with the symmetrical folded dipole. This study shows how the magnetic boundary condition favors excitation of antenna modes in a broad bandwidth while avoiding excitation of transmission line modes. Two different prototype antennas, a monopole and a symmetrical -balun, were implemented and measured. folded dipole with an The measured 6 dB impedance bandwidth of the monopole extends from 0.95 GHz to 2.15 GHz. In turn, the bandwidth of the symmetrical folded dipole extends from 0.90 GHz to 1.96 GHz. The average of the measured total efficiencies at the aforementioned bandwidth is 1.6 dB for both prototypes. The measured maximum total gain at 0.95 GHz is 2.6 dBi for the monopole and 1.9 dBi for the symmetrical folded dipole. Index Terms—Folded dipole, handset antenna, magnetic boundary condition, radiating ground plane, symmetrical antenna feeding, theory of characteristic modes.

I. INTRODUCTION HE use of radiating ground planes in mobile devices as part of the antenna was proposed some years ago as an alternative to increase the antenna radiation efficiency and bandwidth while maintaining compact size [1]. An analysis of the characteristic modes on a typical rectangular Printed Circuit Board (PCB) or ground plane of a mobile handset was later presented to provide physical understanding of the radiating behavior of the ground plane [2], [3]. Based on this original idea and the understanding of the radiation phenomena, some compact antennas were proposed for mobile terminals and wireless

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Manuscript received November 18, 2010; revised February 04, 2011; accepted March 14, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. This work was supported by the Finnish Funding Agency for Technology and Innovation, Nokia Devices Oulu, and Pulse Finland. The work of M. Sonkki was supported by the Universidad Politécnica de Valencia and COST IC0603 ASSIST. M. Sonkki and E. T. Salonen are with the Centre for Wireless Communications (CWC), P.O.B. 4500, FI-90014, University of Oulu, Finland (e-mail: [email protected]). M. Cabedo-Fabrés, E. Antonino-Daviu, and M. Ferrando-Bataller are with the Institute of Telecommunications and Multimedia Applications (iTEAM), Edificio 8G, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, 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.2011.2163783

devices [4]–[6]. Moreover, some antennas for mobile devices having slots in the radiating ground plane for the purpose of tuning or increasing the operational bandwidth of the antenna were proposed lately [7]–[9]. This paper presents a novel technique for enhancing the radiation efficiency and bandwidth of the radiating ground plane of portable devices. This technique is based on the creation of a magnetic boundary condition in the plane of symmetry of the structure by using symmetrical feeding. A slotted ground plane consisting of a symmetrical folded dipole is presented in order to show the validity of this technique. As shown, this feeding configuration favors excitation of symmetrical currents in a metallic structure, in turn enhancing the radiating behavior of the antenna, at the cost of slightly increasing the complexity and volume of the structure. Another possibility to physically create a magnetic boundary condition is to use periodical structures like high-impedance electromagnetic surfaces or photonic bandgaps (PBG) [10], [11]. Both are usually referred to as artificial magnetic conductors (AMC) or artificial magnetic ground planes (AMG). By using this kind of solutions, an antenna can be laid very close to the ground plane of a portable device without disturbing its radiation properties [12]–[14]. However, AMC structures are relatively narrow-band, complicated, and expensive structures compared with the structure presented in this paper. A modal analysis of the slotted structure is performed by means of the Theory of Characteristic Modes to illustrate the interest of creating this magnetic boundary condition in the ground plane. This analysis reveals that two kinds of modes can be excited in the slotted ground plane: Antenna modes and transmission line modes. As demonstrated, creation of a magnetic boundary condition in the plane of symmetry of the structure is necessary to force excitation of antenna modes that exhibit a broader radiation band and enhanced radiating efficiency. Moreover, excitation of some higher order modes, which disturb the radiating behavior of the antenna, is avoided by means of this feeding technique. The paper is organized as follows. Section II presents the basic idea of creating an artificial magnetic boundary condition and proposes a slotted ground plane consisting of a symmetrical folded dipole. In Section III the theory of characteristic modes is briefly described and applied to analyze the structure proposed in the previous section. A folded dipole and a symmetrical folded dipole are studied and compared in order to explain the behavior of modes in the different structures. Antenna prototypes and measurements are presented in Section IV. Finally, Section V contains conclusions and a discussion of the proposed technique.

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Fig. 1. (a) The ground plane is divided into symmetrical planes, where a magnetic boundary condition is applied along the central symmetry line. (b) A magnetic boundary condition is imposed by applying double symmetrical feeding at the longer edges of the ground plane.

Fig. 2. Two symmetrical folded planar dipoles are combined to create an artificial magnetic boundary condition between the dipoles. The result is a symmetrical folded dipole. The white area is air, whereas gray indicates the conducting material. (Units: mm).

II. ANTENNA STRUCTURE A magnetic boundary condition does not exist naturally, so it needs to be created artificially. In this section we propose the creation of a magnetic boundary condition in a radiating ground plane in order to enhance its radiation efficiency and bandwidth. A. Magnetic Boundary Condition in a Radiating Ground Plane Fig. 1 presents the basic mechanism for creating an artificial magnetic boundary condition into a radiating mobile ground plane by using symmetrical excitation. The black line in the middle of the ground plane in Fig. 1(a) represents the central symmetry line of the ground plane where the magnetic condition is created. The dashed lines represent the symmetry lines of the two separate planes, dividing them both into halves. When the ground plane is separated into parts of uniform width, the surface impedance of the separate parts is equal, thus leading to the best antenna performance, as shown later. As depicted in Fig. 1(b), having sources at the longer edges of the ground plane provokes a vertical current distribution at the center of the radiating ground plane, i.e. a magnetic boundary condition is imposed along the central symmetry line. To create the magnetic boundary condition, both sources need to be excited symmetrically, so that the symmetrical excitation introduces simultaneous amplitude and phase in both sources. The arrows pointing upward correspond to the directions of the currents when the magnetic boundary condition is valid. As explained in Section III, application of this magnetic condition improves the radiation bandwidth and efficiency of the antenna. B. Symmetrical Folded Dipole In order to physically implement the magnetic boundary condition in a radiating ground plane, the structure shown in Fig. 2 is proposed. It consists of a slotted ground plane, where two symmetrical excitations are used to create the magnetic condition along the central symmetry line. As observed, the structure is the result of a combination of two symmetrical planar folded dipoles with the overall dimensions of a typical mobile

Fig. 3. Simulated reflection coefficient of the Symmetrical Folded Dipole with slot lengths of 106 mm and 50 mm. The performance of the 106 mm slot is compared with different terminations (short, open, load).

handset. The symmetry criterion for dividing the ground plane is the same as presented in Fig. 1(a). The slot length is 106 mm and the input impedance of Source 1 and 2 is 300 . As explained in detail in the next section, creation of a magnetic boundary condition along the central symmetry line of the ground plane favors the excitation of those current modes whose radiating properties improve the behavior of the antenna. The length of the slot can be adjusted to optimize the impedance bandwidth of the antenna. obtained with Fig. 3 shows the Active S-parameter CST Microwave Studio [15] for the structure shown in Fig. 2, with slot lengths of 106 mm and 50 mm. This Active S-parameter has been calculated considering a reference impedance of 300 for all cases. As previously mentioned, the impedance bandwidth for Active S (dB) 6 dB of the antenna is increased through the application of the magnetic boundary condition along the central symmetry line of the ground plane. It is also notable that by using a 50 mm slot length, the 6 dB relative impedance bandwidth can be considerably increased from 53% to 80%. For comparison, the slotted ground plane structures in [7]–[9] present a 25–50% relative impedance bandwidth. In [16], a folded dipole exhibits a 6 dB relative bandwidth of 70%.

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Fig. 4. Normalized current distribution at resonance for the first six characteristic modes of the planar folded dipole.

With the purpose of highlighting the effectiveness of the magnetic boundary condition created with symmetrical feeding, the symmetrical folded dipole with a 106 mm slot length is compared in Fig. 3 with cases where Source 2 in Fig. 2 is short, open, or load (300 ) terminated. As a conclusion, the 6 dB relative impedance bandwidth can be increased from 14% to 53% by using symmetrical feeding. In order to get an insight into the physical behavior of the radiating ground plane by creating the magnetic boundary condition along the central symmetry line, the next section is devoted to performing a modal analysis of the proposed structure. III. MODAL CHARACTERIZATION OF ANTENNAS In this section the Theory of Characteristic Modes is presented and used to analyze the proposed antenna structure. The first part of the section briefly discusses the basics of this modal theory. In the second part a planar folded dipole is fully analyzed based on the theory. To better understand the behavior of the proposed antenna structure, the third part of the section analyzes and compares the planar symmetrical folded dipole with a planar folded dipole. A. Modal Theory The Theory of Characteristic Modes (TCM), first developed by Garbacz [17] and later refined by Harrington and Mautz in the seventies [18], can be used to obtain the radiating modes of any arbitrarily shaped metallic structure. These radiating modes, known as characteristic modes, not only present really attractive orthogonality properties, they also bring physical insight into the radiating phenomena taking place on the antenna. Because of these advantages, the TCM is extremely useful for systematic analysis and design of antennas. Recently, the TCM has been used in the design of diverse wire and planar antennas, with excellent results [3], [19]. In this communication, the TCM is used to provide a clear explanation of the operating principle of the proposed planar symmetrical folded dipole antenna. The presented results have been obtained using a method of moment code based on the Mixed Potential Integral Equation (MPIE) [20], and Rao-Wilton-Glisson (RWG) basis functions [21]. As explained in [18], characteristic modes or characteristic currents can be obtained as the eigenfunctions of the following particular weighted eigenvalue equation (1)

are the eigenvalues, are the eigenfunctions or where eigencurrents, and and are the real and imaginary parts of the impedance operator (2) This impedance operator is obtained after formulating an integro-differential equation. It is known from the reciprocity theorem that if is a linear symmetric operator, then its hermitian parts, and , will be real and symmetric operators. From this in (1) are real, and all the it follows that all the eigenvalues can be chosen as real or equiphasal over the eigenfunctions surface on which they are defined. Note that the modes obtained are independent of any specific source or excitation. The resonance frequency of the current modes can be determined using the information provided by its associated charac. The characteristic angles can be defined as teristic angles (3) are the eigenvalues associated to each characteristic where mode. From a physical point of view, the characteristic angle models and the the phase angle between a characteristic current . Hence, a mode is at resoassociated characteristic field is 180 . The closer the nance when its characteristic angle characteristic angle is to 180 , the better radiating behavior the mode presents. Additionally, the sign of the eigenvalue determines whether the mode contributes to storing magnetic energy or electric energy . B. Modal Analysis of Planar Folded Dipole Let us begin analyzing a planar folded dipole from a modal point of view. The structure consists of a 110 mm 20 mm radiating ground plane with a 106 mm 2 mm centered slot that divides the ground plane into two parts or strips. Fig. 4 shows the normalized current distribution at resonance of the folded dipole for the first six characteristic modes described above. Arrows are included in the figure to facilitate visualization of current flow. The current nulls are marked with vertical solid lines. Note that since these modes are the eigencurrents of the structure, they exist independent of any specific source or excitation. are resonant and exhibit siAll the modes except mode presents currents forming a closed nusoidal currents. Mode loop around the structure, and it is a special non-resonant mode with an inductive contribution at all frequencies. Modes

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Fig. 6. Contribution of each mode to the total radiated power of the folded dipole when it is fed at the center of the lower dipole. Fig. 5. Variation with frequency of the characteristic angles associated with the first six characteristic modes of the planar folded dipole.

and correspond to the antenna and transmission line current modes defined by Balanis in [22] for the classical folded wire dipole. The rest of the currents are higher order modes that exhibit an increasingly oscillatory nature as the order of the mode increases. In general, the characteristic modes can be classified as antenna modes and transmission line modes. In Fig. 4, antenna exhibit a magnetic boundary condition at modes , , and the slot, with currents flowing in phase on both sides of the slot. present Conversely, the transmission line modes , , and an electric condition at the slot with a 180 phase difference in the currents. As demonstrated next, the antenna modes are efficient radiators, whereas the transmission line modes are characterized by poor radiating performance. Fig. 5 illustrates the variation with frequency of the characteristic angles associated with the first six characteristic modes of the folded dipole. From Fig. 5, not only the resonant frequency of the modes can be determined, but also the radiating bandwidth of the modes can be deduced by studying the slope of the . Notice that the characteristic angles curves close to associated with mode remain below 180 at all frequencies. does not resonate and presents This means that the mode inductive behavior in the considered frequency range. Also observe that antenna modes , , and , in which currents flow in phase, present a soft slope at 180 and remain close to 150 after resonance. In contrast, the curves associated with transmission line modes and have a steep slope at 180 , taking values close to 90 just after the resonance. If these modes were excited, they would yield poor radiating performance, mostly generating reactive energy at the considered frequency band. Let us now evaluate the power radiated by each mode when the structure is excited. Fig. 6 shows the contribution of each mode to the total radiated power of the antenna when the folded dipole is center-fed. The modes that are excited and contribute to radiate power are , , and . These modes represent nonzero current at the feeding location. From Fig. 6 it is inferred that each peak of the total radiated power can be attributed to a mode or a combination of modes. Antenna mode is dominant at the lowest frequencies (1.2 GHz), and it presents impressive radiating performance,

Fig. 7. Active S-parameter of the folded dipole calculated with different reference.

keeping its contribution to the total radiated power long after resonance. The narrow radiation peak observed at 2.5 GHz is due to the resonance of transmission line mode , and the radiation maximum that occurs at 3.8 GHz is caused jointly by and . modes Therefore, using different reference impedances we can identify which modes are responsible for the matching bandwidth of the active S-parameter calculated in Fig. 7. As observed, the and improves as matching bandwidth of antenna modes the reference impedance approximates to 300 [22]. In conclusion, both antenna and transmission line modes are excited when the planar folded dipole is fed at the center. The excitation of the antenna modes is recommended, since they are effective radiators with broadband impedance matching characteristics. In contrast, the excitation of transmission line modes should be avoided. These modes are poor radiators with a narrow impedance matching bandwidth. In the next section, a technique based on the use of the symmetrical feeding is proposed in order to increase the impedance bandwidth by only exciting the antenna modes. C. Modal Analysis of Planar Symmetrical Folded Dipole This section presents a modal study of the symmetrical folded dipole described in Fig. 2. Fig. 8 shows the normalized current distribution at resonance for the first six characteristic modes of

SONKKI et al.: CREATION OF A MAGNETIC BOUNDARY CONDITION IN A RADIATING GROUND PLANE TO EXCITE ANTENNA MODES

Fig. 8. Normalized current distribution at resonance for the first six characteristic modes of the symmetrical folded dipole with slots of 106 mm

Fig. 9. Variation with frequency of the characteristic angles associated with the first six characteristic modes of the symmetrical folded dipole with 106 mm 2 mm slots.

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the symmetrical folded dipole. This structure is more complicated than the previous one, but the characteristic modes present similar current distribution. exhibit a magnetic boundary Antenna modes , , and condition at the axis of symmetry of the radiating ground plane, so all the currents over the structure are in phase at the resonance frequency. On the contrary, transmission line modes , , and present currents flowing in the opposite phase, generating poor radiation. Fig. 9 depicts the characteristic angles curves for the first six characteristic modes of the symmetrical folded dipole shown in Fig. 8. Again, there is a special non-resonant inductive mode , whose currents form a closed loop around the structure. The present the softest slope at 180 , antenna modes , , and and hence, the largest radiation bandwidth. Fig. 10 illustrates the contribution of the each mode to the total power radiated by the antenna when it is fed symmetrically, as depicted in Fig. 2. As observed, the symmetrical feeding only favors the excitation of those modes exhibiting a magnetic condition in the symmetry plane of the ground plane. and are precisely As confirmed by Fig. 10, the modes the ones that present the widest radiating bandwidth (together with mode , which is not excited as it presents zero current at plays an essential the feeding points). Once more, the mode role in the structure. It dominates at the lowest frequencies and it keeps its contribution to the total radiated power long after to create the the resonance. This mode combines with mode radiation maximum observed at 4 GHz.

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Fig. 10. Contribution of each mode to the total radiated power of the symmetrical folded dipole with 106 mm 2 mm slots, when it is fed simultaneously at the center of the upper and lower dipoles.

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Fig. 11. Active S-parameter, calculated for different reference impedances of the symmetrical folded dipole excited with symmetrical feeding.

Fig. 11 shows the Active S-parameter computed with different reference impedances when the symmetrical folded dipole is symmetrically (equally) fed. Again, the best matching and is obtained for the for the excited antenna modes reference impedance of 300 . The modal analysis of the symmetrical folded dipole demonstrated that it is possible to excite only those modes whose currents flow symmetrically with respect to the symmetry plane of the ground plane (i.e. they exhibit a magnetic condition in the

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Fig. 13. (a) Prototype antenna structure with a 300 mm 300 mm ground plane, and (b) a transmission line under the ground plane to feed the antenna structure symmetrically. (Units: mm). Fig. 12. Active S-parameter, calculated for different reference impedances, of the symmetrical folded dipole with 50 mm 2 mm slots, excited with symmetrical feeding.

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symmetry plane). By using the symmetrical feeding, the appearance of some transmission line modes, whose excitation would ruin the matching bandwidth, can be avoided. Therefore, since the excitation of transmission line mode is avoided by means of the magnetic condition imposed by the symmetrical feeding, an increment in the impedance bandwidth can be obtained if a smooth transition between the mode and is achieved. This can be accomplished if the length mode of the slot is decreased. In this case, the modes excited in the antenna are the same as in the previous case, so neither the resonance frequency nor the radiating bandwidth of these modes is altered by the length of the slots. However, the transition from to the mode can be improved by increasing the the mode impedance bandwidth of the antenna. The length value required to obtain the maximum impedance bandwidth is 50 mm, and the results obtained for the Active S-parameter, computed for different reference impedances, are plotted in Fig. 12. As observed, a larger 6 dB impedance bandwidth is obtained (80%) with the 50 mm slot length than with the 106 mm slot length, where an impedance bandwidth of 53% was achieved. This increment of the operating bandwidth favors the antenna to cover different mobile standards.

Fig. 14. Picture of (a) the prototype of the monopole antenna structure, and (b) the symmetrical folded dipole prototype with LC -balun.

Fig. 15. Block diagram of the implemented LC -balun with the component values for the symmetrical folded dipole prototype.

IV. PROTOTYPE ANTENNAS AND RESULTS In this Section the measurements of the two antenna prototypes are introduced. The first prototype consists of a monopole structure with simplified symmetrical feeding. The second prototype antenna is a symmetrical folded dipole implemented with -balun. This latter approach is closer to the structure prean sented in Section II and further analyzed in Section III. The results measured with both prototype antennas are compared. A. Monopole Prototype and Symmetrical Folded Dipole Prototype With LC-Balun In the first approach, to simplify its practical implementation, the symmetrical folded dipole is split in half as a monopole by using the image theory [22], as shown in Fig. 13(a) and Fig. 14(a). The ground plane size is chosen to be approximately (300 mm 300 mm). The 50 mm slot length is chosen to test the antenna performance to better cover the mobile standards from 900 MHz to 2100 MHz for the 6 dB impedance bandwidth. Additionally,

a switchable antenna structure could be used to vary the slot length from 106 mm to 50 mm to extend the bandwidth for higher frequencies. The feeding line is placed under the ground plane, as presented in Fig. 13(b) and Fig. 14(a). It is optimized to 150 , which is half of the input impedance of a folded dipole. The gap between the transmission line and the ground plane is 2 mm, and the feeding structure is air-insulated. An SMA connector is used to feed the structure from the point of the signal source, as shown in Fig. 13(b). In the second step, the symmetrical folded dipole has been -balun for the double symmetrical implemented by using an feeding, as shown in Fig. 14(b). The symmetrical folded dipole prototype is as presented in Fig. 2, but with a 50 mm slot length. -balun is known as a lattice-type balun. The design of the The height of the transmission line is 5 mm from the ground -baluns are cascaded for both sources plane and a totally two to widen the impedance bandwidth, as described in Fig. 15. The

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Fig. 16. Simulated and measured frequency responses of monopole and symmetrical folded dipole prototypes with measured total efficiency.

-balun consists of two capacitors and two inductors, which produce 90 degree phase shifts at the output port. At the same time, the balun itself carries out the impedance matching from 100 to 300 , after dividing the 50 input signal for both -balun can sources. The basic design rule for designing an be found in [23]. B. Measured Results In Fig. 16 the simulated and measured -parameter performances of the prototype antennas are presented. It can be noticed that the measurements correlate well with the simulation results depicted earlier in Fig. 3. The lower frequency of the measured 6 dB impedance bandwidth for the monopole structure is 964 MHz. The measured upper frequency is 2.2 GHz against 2.1 GHz in the simulation. This corresponds approximately to a total bandwidth of 1.2 GHz. In the prototype with -balun, the lower frequency is 900 MHz, corresponding the to a 6 dB impedance bandwidth of 1.06 GHz to 1.96 GHz. In Fig. 16 the measured total efficiency is also presented. A commercial Satimo StarLab antenna measurement system was used to measure the total efficiency [24]. The measured total efficiency over the 6 dB bandwidth is between 3.0 and 0.9 dB for the monopole, and between 3.3 and 0.7 dB for the symmetrical folded dipole. The symmetrical folded dipole -balun presents better matching than the monopole, with the but the total efficiency remains the same. This is due to the losses -balun. of the lumped components used in the In conclusion, the symmetrical folded dipole is better matched but has a slightly narrower bandwidth, which cannot be considered a critical disadvantage. On the contrary, lower -balun. The frequencies can be achieved by using an measured total efficiency shows that the losses of the feeding network of both prototypes are not very high, especially at the 6 dB impedance bandwidth. In Fig. 17, the radiation patterns of the prototype antennas at 0.95 GHz and 1.95 GHz are presented in terms of total gain. The measurements were carried out with the Satimo system, as previously mentioned. It can be observed that the radiation is symmetrical and dipole-like with both of the prototypes. Still, the monopole presents a slightly higher directivity in Fig. 17(a) and Fig. 17(c) due to the ground plane.

Fig. 17. Measured radiation patterns (total gain) of the prototype antennas at different frequencies: (a) monopole with maximum gain of 2.7 dBi at 0.95 GHz; (b) symmetrical folded dipole with 1.9 dBi at 0.95 GHz; (c) monopole with maximum gain of 3.6 dBi at 1.95 GHz; (d) symmetrical folded dipole with 0.2 dBi at 1.95 GHz.

The maximum radiation is observed at in the XZ- and YZ-cuts with the monopole. The symmetrical folded dipole in Fig. 17(b) and Fig. 17(d) radiates the 1.9 dBi maximum at 0.95 GHz and 0.2 dBi at 1.95 GHz, omni-directionally around the XY-cut. It can be concluded that both prototypes radiate omni-directionally, especially in the XY-cut, which is the main radiation direction for a dipole. V. CONCLUSION AND DISCUSSION This paper proposed a novel technique for enhancing the radiation efficiency and bandwidth of a radiating ground plane. A slotted ground plane that behaves as a symmetrical folded dipole was chosen to demonstrate the validity of this technique. The Theory of Characteristic Modes was applied in order to identify the radiating modes of the antenna. These modes can be classified as antenna and transmission line modes due to the boundary condition they present at the axis of symmetry of the radiating ground plane. Excitation of the antenna modes is recommended, since they exhibit a broader radiation band and enhanced radiating efficiency. Symmetrical excitation creates the magnetic boundary condition in the plane of symmetry of the structure that is necessary to force excitation of the antenna modes. As a result, a 6 dB impedance bandwidth of 80% is achieved when using a 50 mm slot length, at the cost of slightly increasing the complexity and volume of the structure. As a discussion, slots in a mobile ground plane can be a possible problem from the standpoint of practical implementation. On the other hand, the SAR-values might be relatively high, as the antenna is omni-directional. Still, both problems can be

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avoided by considering a proper and comprehensive antenna design. One solution could be a so-called metal bezel structure; slots in the ground plane can be spaced closer to the edge of the ground plane while at the same time keeping the structure symmetrical. On the other hand, the bezel operates as a directive element to decrease the SAR-value. ACKNOWLEDGMENT M. Sonkki thanks A. Vila-Jimenéz for helping in practical problems related to the prototype antenna. REFERENCES [1] P. Vainikainen, J. Ollikainen, O. Kivekas, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1433–1444, Oct. 2002. [2] E. Antonino, M. Cabedo, M. Ferrando, and J. I. Herranz, “Analysis of the coupled chassis-antenna modes in mobile handsets,” in IEEE Antennas Propag. Society Int. Symp., Monterey, Jun. 2004. [3] M. Cabedo, E. Antonino, A. Valero, and M. Ferrando, “The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 52–68, Oct. 2007. [4] J. Villanen, J. Ollikainen, O. Kivekäs, and P. Vainikainen, “Compact antenna structures for mobile handsets,” in Proc. Vehicular Technology Conf., Oct. 2003, vol. 1, pp. 40–44. [5] J. Villanen, J. Ollikainen, O. Kivekäs, and P. Vainikainen, “Coupling element based mobile terminal antenna structures,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2142–2153, July 2006. [6] E. Antonino, C. A. Suárez, M. Cabedo, and M. Ferrando, “Wideband antenna for mobile terminals based on the handset PCB Resonance,” Microw. Opt. Technol. Lett., vol. 48, no. 7, Jul. 2006. [7] R. Hossa, A. Byndas, and M. E. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” IEEE Microwave Wireless Compon. Lett., vol. 14, no. 6, pp. 283–285, June 2004. [8] A. Cabedo, J. Anguera, C. Picher, M. Ribó, and C. Puente, “Multiband handset antenna combining a PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2526–2533, Sept. 2009. [9] J. Anguera, I. Sanz, J. Mumbrú, and C. Puente, “Multiband handset antenna with a parallel excitation of PIFA and slot radiators,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 348–356, Feb. 2010. [10] K.-P. Ma, K. Hirose, F.-R. Yang, Y. Qian, and T. Itoh, “Realisation of magnetic conducting surface using novel photonic bandgap structure,” Electron. Lett., vol. 34, no. 21, pp. 2041–2042, Oct. 1998. [11] D. Sievenpiper, L. Zhang, R. F. Jimenez-Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [12] R. F. Jimenez-Broas, D. F. Sievenpiper, and E. Yablonovitch, “A highimpedance ground plane applied to a cellphone handset geometry,” IEEE Trans. Microwave Theory Tech., vol. 49, no. 7, pp. 1262–1265, Jul. 2001. [13] J. Redvik, “PIFA Antenna With HIGP Structure,” Int. Pub. Number WO 02/087012 A1, Apr. 12, 2002. [14] Z. Du, K. Gong, J. S. Fu, B. Gao, and Z. Feng, “A compact planar inverted-F antenna with a PBG-type ground plane for mobile communications,” IEEE Trans. Veh. Technol., vol. 52, no. 3, pp. 483–489, May 2003. [15] CST—Computer Simulation Technology 2009 [Online]. Available: http://www.cst.com/ [16] S. Tanaka, Y. Kim, H. Morishita, S. Horiuchi, Y. Atsumi, and Y. Ido, “Wideband planar folded dipole antenna with self-balanced impedance property,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1223–1228, May 2008. [17] R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated and scattered fields,” IEEE Trans. Antennas Propag., vol. AP-19, pp. 348–358, May 1971.

[18] R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-19, no. 5, pp. 622–628, Sep. 1971. [19] E. Antonino-Daviu, M. Cabedo-Fabres, M. Ferrando-Bataller, and V. M. Rodrigo-Peñarrocha, “Modal analysis and design of band-notched UWB planar monopole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1457–1467, May 2010. [20] R. F. Harrington, Field Computation by Moment Method. New York: MacMillan, 1968. [21] 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, 1982. [22] C. A. Balanis, Antenna Theory—Analysis and Design, 2nd ed. New York: Wiley, 1997. [23] W. Bakalski, W. Simburger, H. Knapp, H. D. Hohlmuth, and A. L. Scholtz, “Lumped and distributed lattice-type LC-Baluns,” in IEEE MTT-S Int. Microwave Symp. Digest, Seatle, WA, Jun. 2–7, 2002, pp. 209–212. [24] Starlab, Satimo SA, 2006 [Online]. Available: http://satimo.com

Marko Sonkki (S’07) was born in Kalanti, Finland, in 1972. He received the M.S. degree in electrical engineering from the University of Oulu, Oulu, Finland, in 2004, where he is currently working toward the Ph.D. degree in radio telecommunications engineering. He is currently with the Centre for Wireless Communications (CWC),Department of Electrical and Information Engineering, University of Oulu. His research interests are in designing and analyzing wideband single and multielement antennas for mobile terminals, including MIMO and diversity systems.

Marta Cabedo-Fabrés (M’07) was born in Valencia, Spain, on June 8, 1976. She received the M.S. and Ph.D. degrees in electrical engineering from Universidad Politécnica de Valencia, Spain, in 2001 and 2007, respectively. In 2001, she joined the Electromagnetic Radiation Group at Universidad Politécnica de Valencia (UPV), as a Research Assistant. In 2004, she became a Lecturer in the Communications Department, Universidad Politécnica de Valencia. Her current scientific interests include numerical methods for solving electromagnetic problems, and design and optimization techniques for wideband antennas for mobile terminals, and MIMO antennas for Personal Area Networks.

Eva Antonino-Daviu (M’00) was born in Valencia, Spain, on July 10, 1978. She received the M.S. and Ph.D. degrees in electrical engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 2002 and 2008, respectively. In 2002, she joined the Electromagnetic Radiation Group, Universidad Politécnica de Valencia, and in 2005 she became a Lecturer at the Escuela Politécnia Superior de Gandia, Gandia, Spain. During 2005 she stayed for several months as a guest researcher at the Department of Antennas & EM Modelling of IMST GmbH, in Kamp-Lintfort, Germany. Her current research interests include wideband and multi-band planar antenna design and optimization and computational methods for printed structures. Dr. Antonino-Daviu was awarded the “Premio Extraordinario de Tesis Doctoral” (Best Ph.D. thesis) from the Universidad Politécnica de Valencia in 2008.

SONKKI et al.: CREATION OF A MAGNETIC BOUNDARY CONDITION IN A RADIATING GROUND PLANE TO EXCITE ANTENNA MODES

Miguel Ferrando-Bataller (S’81–M’83) was born in Alcoy, Spain, in 1954. He received the M.S. and Ph.D. degrees in telecommunication engineering from the Universidad Politecnica de Catalunya, Barcelona, Spain, in 1977 and 1982, respectively. From 1977 to 1982, he was a Teaching Assistant with the Antennas, Microwave, and Radar Group, Universidad Politécnica de Catalunya, and in 1982 he became an Associate Professor. In 1990, he joined the Universidad Politecnica de Valencia, Valencia, Spain, where he was Director of the Telecommunication Engineering School and Vice-Chancellor. He is currently Director of Long-life learning Office and Professor of antennas and satellite communications. His current research interest includes numerical methods, antenna design and e-learning activities.

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Erkki T. Salonen received the M.S., Licentiate of Science, and Doctor of Science in Technology degrees from Helsinki University of Technology (TKK), Espoo, Finland, in 1979, 1986, and 1993, respectively. From 1984 to 1996, he was in charge of radio wave propagation studies at the Radio Laboratory, Helsinki University of Technology. Since 1997 he has been a Professor of radio engineering in the Centre for Wireless Communications (CWC), University of Oulu. His main research interests include antennas and propagation in radio communications.

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TE Surface Wave Resonances on High-Impedance Surface Based Antennas: Analysis and Modeling Filippo Costa, Member, IEEE, Olli Luukkonen, Constantin R. Simovski, Agostino Monorchio, Senior Member, IEEE, Sergei A. Tretyakov, Fellow, IEEE, and Peter M. de Maagt, Fellow, IEEE

Abstract—Low-profile antennas comprising a horizontal dipole above a high-impedance surface are analyzed. The emphasis of this paper is on the additional resonances of the radiating structure caused by surface waves propagating on the high-impedance surface. It is shown that such resonances can be favorably used for broadening the bandwidth of the antenna. The phenomenon is thoroughly modeled by exploiting a parallel between the HIS structure and a waveguide resonator. In the second part of the paper we discuss homogenized approaches for modeling the radiating properties of the antenna with emphasis to the phenomenon discussed in the first part. As it turns out, it is necessary to take into account the spatially dispersive properties of high-impedance surfaces, and most of the simplified models commonly used for analyzing high-impedance surface based antennas fail in predicting the discussed resonance mode. Index Terms—Artificial magnetic conductor (AMC), electromagnetic bandgap (EBG), frequency selective surfaces (FSS), high impedance surfaces (HIS), metamaterials.

I. INTRODUCTION

H

ORIZONTAL wire antennas over a ground plane suffer from a fundamental limitation concerning the distance from the antenna to the ground plane. The optimal distance for the antenna is a quarter wavelength from the ground plane, high radiation resistance being the figure of merit. If the separation is reduced, the image currents on the ground plane eventually cancel out all radiation from the antenna. In 1999 Sievenpiper et al. [1] proposed to use high-impedance surfaces to overcome Manuscript received March 02, 2010; revised September 28, 2010; accepted March 18, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. F. Costa is with the Department of Information Engineering, University of Pisa, 56122—Pisa, Italy, and also with the Department of Radio Science and Engineering/ SMARAD CoE, School of Science and Technology, Aalto University, FI-00076 Aalto, Finland (e-mail: [email protected]). O. Luukkonen is with the Department of Radio Science and Engineering/SMARAD CoE, Aalto University, School of Science and Technology, FI-00076 Aalto, Finland, and also with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104-6314, USA (e-mail: [email protected], [email protected]). C. R. Simovski and S. A. Tretyakov are with the Department of Radio Science and Engineering/SMARAD CoE, Aalto University, School of Science and Technology, FI-00076 Aalto, Finland (e-mail: [email protected], [email protected]). A. Monorchio is with the Department of Information Engineering, University of Pisa, 56122—Pisa, Italy (e-mail: [email protected]). P. M. de Maagt is with the Electromagnetics Division, European Space Agency, Noordwijk NL-2200, The Netherlands (e-mail: Peter.de.Maagt@esa. int) 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.2011.2163750

this limitation. Since the publication of this seminal paper, the interest in the high-impedance surfaces has increased tremendously in the electromagnetic community and a large number of papers has been devoted to optimization and modeling of antenna systems over high-impedance surfaces [2]–[10]. High-impedance surfaces are commonly characterized in the Fourier domain (i.e., for plane waves) in terms of reflection phase diagrams. At the resonance, the reflection phase of the surface equals zero degrees. However, since high-impedance surfaces are spatially dispersive, the reflection properties of the surface vary with respect to the different spectral components [11]–[14]. Moreover, the study of the plane-wave reflection diagram often assumes an infinite HIS, even if the reflection phase properties of a finite-size HIS depend on the screen size [15]. The aforementioned characteristics of the high-impedance surfaces make the modeling of antennas with high-impedance surfaces very demanding. For instance, in [7] and [8], the interaction of the dipole antenna with the high-impedance surface has been modeled by replacing the high-impedance surface with an infinite-extent equivalent impedance boundary valid for normal incidence. These models neglect both the effects of the spatial dispersion in the surfaces and the effects of the finite size of the screen. In [9] a similar infinite-extent impedance boundary is used, but the effect of the nearby impedance surface through the exact image theory [16] is taken into account. In [10] Paulotto et al. found good agreement comparing the electric near-field values obtained by a full wave analysis and by using the spatially dispersive Green’s function for a homogenized high-impedance surface. The rigorous approach is still applied to an infinite structure, while in practical antenna design the size of the high-impedance surface is commonly only a few unit cells. The size of the meta-surface, as it happens for all small and resonant antennas [17], [18], can be of crucial importance since the surface waves propagating in a finite HIS can be used to create an additional resonance for the antenna system. It is worth noticing that usually the presence of surface waves is considered as a harmful phenomenon and a large body of literature treats HIS as a tool for suppressing surface waves (electromagnetic band-gap structures) [19]. Here, on the other hand, we want to excite surface waves in order to improve the performance of the antenna. In the first part of the present work the effect of TE surface waves excited by a dipole antenna in presence of a finite HIS is discussed and it is shown that they can be used to enhance radiation from the antenna. As matter of fact, the bandwidth where the antenna presents a good return loss and broadside

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COSTA et al.: TE SURFACE WAVE RESONANCES ON HIGH-IMPEDANCE SURFACE BASED ANTENNAS: ANALYSIS AND MODELING

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Fig. 1. Side view of the analyzed structure.

patterns extents from the HIS resonance to the first TE surface wave resonance. In the second part of the paper we will discuss the possibility to model the low-profile antenna in presence of this phenomenon by considering homogenization models with different degrees of accuracy. As a test object for this study we have chosen a recently published antenna configuration [20]. II. THE ANTENNA STRUCTURE The structure under investigation is almost the same as the one presented in [20] apart from the feeding network realized in the practical design. In the present study, for simplicity, the antenna is fed by a lumped voltage source. The layout of the analyzed structure is depicted in Fig. 1. The HIS is composed of a patch array, characterized by the period and a gap between the adjacent patches , printed on a dielectric substrate with thickand permittivity . The dipole antenna is printed over ness the patch array on an additional thin dielectric layer with thickness and with the same permittivity . The choice of a patch array with a very small gap between the adjacent elements allows to achieve the widest possible bandwidth around the HIS resonance [21]. In our case the values for the design parame; ; ; ters read: , , . With the chosen parameters, the high impedance surface resonates at 1.25 GHz at normal incidence. The dipole printed on the top of the second dielectric, and analyzed without the high-impedance surface, resonates at 2.2 GHz. III. RADIATION MECHANISMS OF THE ANTENNA: THE ROLE OF RESONANCES DUE TO THE GEOMETRICAL DIMENSIONS When a dipole antenna is placed in close vicinity of the grounded patch array, it radiates efficiently in the proximity of the HIS resonance. A large portion of the radiated energy towards the orthogonal direction experiences constructive interference with the radiation reflected from the HIS. In our example, the first resonance of the antenna placed near the resonant structure (the first zero of the reactive part of the impedance) occurs approximately at 1.1 GHz with a full-wave analysis. The matching condition can be qualitative explained by a simple transmission line model where the capacitive impedance of the short dipole in free space is placed in parallel to the input impedance of the HIS structure computed by averaged formulas valid at normal incidence [12]. The parallel connection between the inductive impedance of the HIS and the capacitive impedance of the dipole causes, slightly before the HIS resonance, a parallel resonance in the antenna input impedance which determines the matching. The impedance of the short dipole in free space and the surface impedance resulting from parallel connection between the antenna impedance and the HIS

Fig. 2. Qualitative explanation of the fundamental resonance frequency of a dipole located on a HIS surface. Real and imaginary part of a short dipole impedance in free space (Z ) and the impedance Z which is the parallel and the load representing the high-impedance surface. The connection of Z figure reports also the HIS reflection phase.

impedance is reported in Fig. 2. The simple transmission line model used to obtain the curves is reported in the inset of the figure. This radiation mechanism would be the only useful one if the high-impedance surface were infinite. However, in a realistic configuration the size of the structure is inevitably finite and in practical designs the surface commonly contains only a few unit cells. In general, the effects of the finite-sized ground plane (distorted radiation pattern, backward radiation etc.) are considered harmful and a number of publications has been devoted to suppression of surface waves on antenna substrates (see e.g., [1]–[3]). Alternatively, here we wish to use the surface waves favorably by enhancing the radiating currents on the antenna structure at a certain frequency. This can be done by introducing an additional resonance to the antenna structure. In our example the total length of the panel equals to a resonant length for the TE waves that travel along the surface. This phenomenon can be used to create one or more resonances in the return loss profile. By considering the finite-sized HIS as a cavity [22], we can qualitatively determine the TE resonances as: (1) where represents the propagation constant of the first TE surface wave mode and is the length of the cavity given by the number of unit cells multiplied by the cell periodicity . Let us consider the HIS structure composed of five unit cells as of the antenna is in the original design proposed in [20]. The reported in Fig. 3 together with the curves obtained for other dimensions of the HIS structure. For the 5 5 cells configuration, the size resonance is observed at 1.56 GHz and it is the lowest order one. Larger sizes cause the shift of the first resonance towards lower frequencies according to (1). For 7 7 structure it reduces to 1.42 GHz and the second geometrical resonance appears at 1.68 GHz. For 9 9 cells the first size resonance falls close to the HIS resonance and at 1.52 GHz we observe the second size resonance. In Fig. 4 the radiation patterns of the

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Fig. 3. S of the dipole antenna on top of a finite size HIS with a different number of unit cells.

Fig. 6. Surface current distribution on the bottom side of the patches comprising the HIS with the dimensions 5D 5D at 1.56 GHz.

2

TABLE I RESONANCE FREQUENCIES OF THE FIRST TE SURFACE WAVE FOR DIFFERENT SIZE OF THE HIS SURFACE. THE RESONANCE FREQUENCY IN THE CAVITY MODEL IS GRAPHICALLY OBTAINED BY THE INTERSECTION BETWEEN THE TE DISPERSION CURVE AND THE VERTICAL LINES REPRESENTING THE QUANTITY ). ON THE RIGHT SIDE OF THE RELATION (1) (2=l

Fig. 4. Radiation patterns of the dipole on top of the 5

2 5 HIS.

Fig. 5. Magnitude of the tangential electric fields at 1.25 GHz (a) and at 1.56 GHz (b).

radiating structure are shown at three different frequencies in the analyzed band. In the range 1.2 GHz–1.5 GHz the current distribution on the antenna and the surface currents on the patch array generate rotationally symmetric and wide-beam radiation patterns (Fig. 4). As soon as the electrical dimensions of the HIS be(the TE surface wave wavelength) and larger, come close to grating lobes appear in the radiation pattern in the -direction (Fig. 4). In Fig. 5 the amplitude of the tangential electric field is plotted at the HIS resonance frequency, namely 1.25 GHz, and

at 1.56 GHz in correspondence to the first surface wave resonance. The field distribution within the HIS cavity is remarkably different: while at 1.25 GHz the electric field is confined to the center of the HIS, at 1.56 GHz the mentioned surface wave mode is excited and the field distribution has a sinusoidal distribution with a period close to . The surface current distribution on the bottom side of the patches comprising the high-impedance surface at 1.56 GHz is also reported in Fig. 6 in order to show that the currents on the central patches are in opposition of phase with the currents on the peripheral patches. The current distribution is characterized by a sinusoidal distribution with a period of . The presence of higher-order resonances due to the increase HIS dimensions can also be represented on the dispersion diagram shown in Fig. 9 by tracing vertical lines for different corresponding to the solutions of (1). The graphical solutions of the (1) are determined by the intersection between the TE mode curve and the vertical lines . The resonance frequencies representing the quantity obtained both by the full-wave approach and by the simplified cavity model are summarized in Table I. The resonance frequencies (second maxima of the real part of the input impedance) obtained by CST Microwave Studio and by HFSS are compared against the ones obtained by the simplified cavity model im. When the dipole is strongly coupled to the HIS, posing a small frequency shift between the results obtained by the two full-wave codes is observed.

COSTA et al.: TE SURFACE WAVE RESONANCES ON HIGH-IMPEDANCE SURFACE BASED ANTENNAS: ANALYSIS AND MODELING

Fig. 7. Radiation patterns of the dipole on top of the 7

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2 7 HIS.

The resonance of the TE surface wave mode approaches the HIS resonance as the width of the artificial ground plane raises. Moreover, the increase of the index in (1) leads to higher-order resonances positioned very close to each other. These higherorder resonances result in a poor radiation pattern and cannot be considered as beneficial in antenna applications. In Fig. 7 the radiation patterns of the dipole antenna on top of the 7 7 cells HIS are shown in order to demonstrate that, as in the previous case, the radiation pattern above the second resonance (1.4 GHz) is no longer broadside but it is split into two lobes due to the surface current distribution of the HIS. This is the same phenomenon already described for the structure made of 5 5 cells but, in this case, it happens at a lower frequency due to the larger electrical size of the HIS ground plane. In the remaining part of the paper we will concentrate on the aspects that make the modeling of this phenomenon an intricate task. IV. HOMOGENIZED APPROXIMATIONS OF THE FINITE HIGH-IMPEDANCE SURFACE A high-impedance surface can be modeled by an averaged boundary condition (see e.g., [11], [12], [23]). This approximation can be used both for the normal incidence and also for oblique incidence depending on the accuracy of the expressions [12]. The normal incidence expression of the surface impedance is independent of the tangential wave number . The normal incidence surface impedance can be adequate also for the cases of oblique incidence under some hypotheses. If the refractive index of the substrate material is high and the source is located far away from the surface, the response of this surface for normal and for oblique incidence is approximately the same. This is known as the Leontovich approximation or the impedance boundary condition. However, if we position a source very close to the modeled interface, the amplitudes of evanescent modes launched by the source are comparatively high at the surface since the evanescent spatial spectrum in the near field of dipole antenna strongly dominates. Hence, it is appropriate to question whether the hypotheses for the applicability of the Leontovich boundary condition are valid when this approximation is used in the context of dipole antennas on top of high-impedance surfaces. The averaged surface impedance of a high-impedance surface can be represented as a parallel connection of the surface impedance of a grounded dielectric slab and the surface impedance of a capacitive grid [11]. Both of these impedances are dependent on the incidence angle in the rigorous formulation, i.e., they are spatially dispersive. From a more practical point of view, it is therefore possible to increase the accuracy

Fig. 8. Four different homogenization approximations of the real structure.

of the model by gradually taking the spatially dispersive terms into account one by one. Basically, four different models of the structure can be proposed for the full-wave simulations: • The simplest approximation, which neglects the spatial dispersion completely. Here the whole structure is replaced by a Leontovich boundary condition. It will be referred to below as non dispersive model. • An intermediate approximation, where the capacitive grid (patch array) is replaced by a homogenized impedance sheet representing the capacitive grid (for the normal incidence). The grounded dielectric slab is modeled in the simulation model without approximations. This model will be referred to as partially dispersive model. • The most accurate solution, where the FSS surface is represented by an angular dependent surface impedance and the grounded dielectric is modeled in the simulation model without approximations. This model (dispersive model) which can be adopted to analyze the infinite HIS, cannot be implemented in a full-wave HFSS analysis since the incidence angle of the spatial harmonics is unknown. • An equivalent approximation of the FSS as an anisotropic dielectric as proposed by Clavijo et al. in [24] with the grounded dielectric modeled without approximations. This model allows to consider the spatial dispersion within the FSS and can be implemented in HFSS. The model will be referred to as anisotropic dispersive. The different approximations are illustrated in Fig. 8. In order to highlight the differences among the previous models in terms of surface wave propagation, we have analytically computed the dispersion diagram of the high-impedance surface under these approximations. The dispersion diagram has been computed by employing the averaged analytical expressions [12], [17], [24] for the surface impedance of a HIS and by imposing the standard transverse resonance condition on the top of it: (2) Here is the free space impedance. Equation (2) is used to calculate the propagation constants for both TE and TM-polarized surface waves for the four models discussed above.

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Fig. 9. Dispersion diagram of the high impedance surface obtained by four different homogenized approximations and by full-wave approach.

The calculated dispersion diagrams are reported in Fig. 9 and compared against the full-wave results obtained by simulating the actual structure using Ansoft HFSS. Comparing the results obtained according to the non dispersive and partially dispersive models, we see that the TE mode is similar for two different homogenization models, whereas the TM mode shows a completely different behavior. This effect is due to the choice of a thick dielectric substrate whose surface impedance has a strong dependence on the transverse wave number mainly for the TM polarization. It is important to notice that the partially dispersive approximation provides gives an excellent agreement for TM-waves, whereas for TE-waves both homogenization models are not adequate. Next we include in the model also the angular dependence of the grid impedance. The results of the fully spatially dispersive model are plotted (solid line) in Fig. 9. The result agrees fairly well with the full wave results (at least for the first two modes existing below 2.5 GHz). The fully spatially dispersive model and the simulation results show that the propagation of slow TE surface waves is not allowed in the operating band of the antenna, adversely to what is predicted by the two simplistic models. Unfortunately, the third approximate analysis (referred to, in Fig. 9, as dispersive) cannot be employed in existing fullwave simulator tools for the analysis of a dipole antenna located on top of a finite-size HIS. A discussion of angle-dependent surface impedance models for FDTD codes can be found in [25]. However, by recurring to the anisotropic model described in [24], similar dispersion curves are obtained since the anisotropic model takes into account the spatial dispersion within the FSS. The TE curve obtained by the anisotropic approximation of the FSS is slightly steeper than the one computed through the fullwave approach. It is therefore expected that, in accordance to the theory described in Section III, the TE surface wave resonances are slightly shifted towards higher frequencies. A refinement of the anisotropic model (it will be discussed later) leads to a dispersion TE curve characterized by a better agreement with the full-wave result. The accuracy of the described homogenized

Fig. 10. Real part of the input impedance of the dipole antenna on a 1.45 mm dielectric on top of the HIS. Comparison between the full-wave results and the results simulated using the two different models.

Fig. 11. Imaginary part of the input impedance of the dipole antenna on a 1.45 mm dielectric on top of the HIS. Comparison between the full-wave results and the results simulated using the two different models.

models is tested in Section V for the previously described HISbased antenna structure. V. RESULTS OF HOMOGENIZATION MODELS In this section a comparison between the results obtained by full-wave HFSS simulations of the actual radiating structure and the results derived by replacing the high-impedance surface with the three aformentioned, namely the non dispersive, partially dispersive and anisotropic dispersive models, is presented. In Fig. 10 and in Fig. 11, the input impedance of the dipole antenna printed on a 1.45 mm dielectric substrate placed on top of the HIS is shown. Both the real and the imaginary part obtained by a full-wave simulation and by the three different models are reported. It is apparent that the results of the two less accurate models are totally different from the full wave ones due to the close vicinity of the dipole to the reactive surface. It is anyway

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Fig. 12. Radiation patterns in correspondence of different frequencies obtained by the partially dispersive model. The antenna is printed on top of the HIS.

worth noticing that the partially dispersive model shows more reasonable results with respect to the plane-wave non dispersive approximation since it takes into account the spatial dispersion in the substrate. The sharp resonance in the input impedance of the dipole obtained by the non dispersive model around the HIS resonance (1.1 GHz) in Figs. 10 and 11 occurs due to the wrong interpretation of the dipole-HIS interaction. The accuracy of such approximation is comparable to the simplified TL approach employed to introduce the qualitative explanation of the HIS resonance in Fig. 2. On the other hand, the anisotropic dispersive model turns out to have acceptable agreement with the full-wave simulations apart from a moderate frequency shift. The frequency shift of the first resonance is mainly due to the close vicinity between the dipole and the HIS and consequentially to the small size of the HIS. A drawback of this model resides in the longer computation time with respect to the surface impedance approximation. The accuracy of the anisotropic model improves as dielectric substrate representing the FSS is made thinner and thinner, but computation time with thin substrates increases. For this reason an adequate compromise between the computation time and the accuracy has to be found. Regarding the position of the first TE surface wave resonance computed both by the anisotropic dispersive model and by the partially dispersive model, it is evident that in the former case the shift with respect to the full-wave model is towards higher frequencies because the slope of the TE surface wave mode is overestimated (see Fig. 9) and in the latter case the shift is towards lower frequencies with respect to the actual one since the slope of the TE surface wave is underestimated (see again Fig. 9). This is consistent with the cavity mode approach described earlier. The incorrect estimation of the slope of the TE surface wave propagation constant is due to the approximate calculation of the normal permeability of the FSS layer [24]. As the authors argue in their paper, this expression is empirically obtained from the expression valid for an array of wires. Therefore, we can reasonably suppose that this approximate expression underestimates the normal component to the magnetic permeability and we can empirically modify such value up to . In this case, the TE surface wave propagation constant agrees better (see Fig. 9) with the full-wave results. In particular, it is clear that the TE surface wave resonance moves back in frequency in accordance to our predictions. The curves obtained with this modification are identified as anisotropic dispersive, refined. In Fig. 12 the radiation patterns obtained by the anisotropic dispersive model are reported. They compare well with the patterns computed by the full wave approach (Fig. 4). It can be noticed that the radiation towards -direction starts at 1.7 GHz,

Fig. 13. Real part of the input impedance of the dipole antenna on a 5 mm dielectric on top of the HIS. Comparison between the full-wave results and the results simulated using the two different models.

Fig. 14. Imaginary part of the input impedance of the dipole antenna on a 5 mm dielectric on top of the HIS. Comparison between the full-wave results and the results simulated using the two different models.

right after the first TE surface wave resonance (the second maximum of the real part of the impedance). In order to study if the accuracy of the models improves when the antenna is moved away from the HIS, we have increased the distance between the dipole antenna and the loading structure. In this way, we reduce the amplitude of the incident evanescent waves at the HIS surface, which is expected to improve the agreement between the approximate models and the simulation results. Also, the increase of the thickness , reduces the coupling between the dipole antenna and the high-impedance surface, but it does not affect the position of the TE surface wave resonances that are determined by the HIS size. In Figs. 13 and 14 the real and imaginary parts of the input impedance of the dipole antenna printed on a 5 mm dielectric substrate placed on top of the HIS is shown. The results of the full-wave simulation and the partially dispersive model agree up to the first HIS

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dispersive model can be considered a valid tool only if the source is not coupled to the HIS and up to the first resonance. If the dipole is tightly coupled to a HIS surface, even the most accurate homogenized models present some residual inaccuracies in reproducing the behavior of the antenna. Anyway, as soon as the exciting dipole is moved away from the HIS, the fully dispersive model allows to obtain the input impedance rather accurately. The remaining discrepancies can be ascribed to the field scattered from the edges since the homogenization of the periodic structure could not work properly at the border of the FSS where the field is rapidly varying. VI. CONCLUSION

Fig. 15. S of the dipole antenna on a 5 mm dielectric on top of the HIS. Comparison between the full-wave results and the results simulated using the two different models.

resonance but, again, the first TE surface wave resonance computed by the model is encountered just after the HIS resonance because of the inaccurate modeling of TE surface waves. The accuracy of the anisotropic dispersive model is further improved with respect to the previous case, that is, the behavior of both the real and imaginary part of the impedance agree well. The position of the TE surface wave resonance is again dependent on the correct estimation of the TE surface wave propagation constant. Indeed, by applying the aforementioned correction to the normal component of the magnetic permeability, an almost perfect agreement is obtained both for the real and the imaginary part of the input impedance. , shown in Fig. 15, clearly demonstrates that The antenna the result of the partially dispersive model is accurate only up to the HIS resonance. The behavior obtained by the anisotropic dispersive model does not perfectly match the full-wave results but if the normal component of the magnetic permeability of the FSS slab is refined, an almost perfect agreement is obtained. On the contrary, it is evident that the normal-incidence non dispersive model leads to incorrect results characterized by a sharp resonance in correspondence of the HIS resonance. In order to perform a complete analysis of the studied structure, we compared also the results of the two models against full-wave results when the number of the unit cells of the HIS is increased. The analysis performed for the 7 7 patch array and for the 13 13 one reveals the same behavior as found for the 5 5 array, namely the agreement of the two most simplified models start to be reasonable if the distance between the antenna and the HIS is 5 mm and only up to the HIS resonance. The increase of the screen size does not mitigate the effect of the TE surface wave resonances but, conversely, the number of resonances encountered after the HIS resonance is directly proportional to the size of the structure. It can be concluded that the most simple non dispersive model based on the plane-wave approximation of the HIS cannot be used to analyze this kind of near-field problems. The partially

It has been shown that the presence of a dipole antenna on top of a high-impedance surface generates a complex system in which two different radiating mechanisms can be identified: the former contribution is the classical resonance of the HIS-based antenna system, while the second one is due to the propagation of TE surface waves along a finite structure. In the latter case the size of the structure turns out to be a key factor in defining the bandwidth and the radiation pattern of the antenna. The phenomenon is first explained by theoretical speculations and then discussed by homogenization models. The use of the homogenization models is rather challenging since they are used for a finite size panel positioned in close proximity of a radiating source. We have compared three possible homogenization models for replacing the high-impedance surface. The models are derived by progressively taking into account the spatial dispersion of the HIS. Through a careful comparison between approximate and full-wave results of a benchmark antenna configuration, we have shown that the simplistic homogenized models, which do not take into account the spatial dispersion both of the FSS and of the grounded substrate, cannot accurately reproduce the presence of the TE surface wave resonances. On the other hand, a fully dispersive model turns out to be a sufficiently accurate tool for modeling the finite extent antenna. In conclusion, the correct estimation of the dispersion properties of the first two propagating modes is essential for achieving correct results in full-wave computations. REFERENCES [1] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovich, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2059–2074, 1999. [2] S. Best and D. Hanna, “Design of a broadband dipole in close proximity to an EBG ground plane,” IEEE Antennas Propag. Mag., vol. 50, no. 6, pp. 52–64, 2008. [3] G. Bianconi, F. Costa, S. Genovesi, and A. Monorchio, “Optimal design of dipole antennas backed by a finite high-impedance screen,” Progr. Electromagn.s Res. C, vol. 18, pp. 137–151, 2011. [4] H. Mosallaei and K. Sarabandi, “Antenna miniaturization and bandwidth enhancement using a reactive impedance substrate,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2403–2414, 2004. [5] F. Yang and Y. Rahmat-Samii, “Reflection phase characterizations of the EBG ground plane for low profile wire antenna applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2691–2703, 2003. [6] F. Costa, A. Monorchio, S. Talarico, and F. M. Valeri, “An active high impedance surface for low profile tunable and steerable antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 676–680, 2008. [7] R. C. Hansen, “Effects of a high-impedance screen on a dipole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 46–49, 2002. [8] M. F. Abedin and M. Ali, “Effects of EBG reflection phase profiles on the input impedance and bandwidth of ultra-thin directional dipoles,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3664–3672, 2005.

COSTA et al.: TE SURFACE WAVE RESONANCES ON HIGH-IMPEDANCE SURFACE BASED ANTENNAS: ANALYSIS AND MODELING

[9] S. A. Tretyakov and C. R. Simovski, “Wire antennas near artificial impedance surfaces,” Microwave Opt. Technol. Lett., vol. 27, no. 1, pp. 46–50, 2000. [10] S. Paulotto, P. Baccarelli, P. Burghignoli, G. Lovat, G. Hanson, and A. B. Yakovlev, “Homogenized Green’s functions for an aperiodic line source over planar densely periodic artificial impedance surfaces,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 7, pp. 1807–1817, Jul. 2010. [11] S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics. Norwood, MA: Artech House, 2003. [12] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, 2008. [13] O. Luukkonen, P. Alitalo, C. R. Simovski, and S. A. Tretyakov, “Experimental verification of an analytical model for high-impedance surfaces,” Electron. Lett., vol. 45, no. 14, pp. 720–721, 2009. [14] C. R. Simovski, P. de Maagt, S. A. Tretyakov, M. Paquay, and A. A. Sochava, “Angular stabilisation of resonant frequency of artificial magnetic conductors for TE-incidence,” Electron. Lett., vol. 40, no. 2, pp. 92–93, Jan. 2004. [15] Y. Kawakami, T. Hori, M. Fujimoto, M. Yamaguchi, and K. Cho, “Reflection characteristics of finite EBG structures on finite ground plane,” presented at the Int. Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, 2008, iWAT 2008. [16] I. V. Lindell, Methods for Electromagnetic Fields Analysis. Oxford, U.K.: Oxford Univ. Press, 1992. [17] J. Huang, “The finite ground plane effect on the microstrip antenna radiation patterns,” IEEE Trans. Antennas Propag., vol. 31, no. 4, pp. 649–653, Jul. 1983. [18] M.-C. Huynh and W. Stutzman, “Ground plane effects on planar inverted-F antenna (PIFA) performance,” Proc. Inst. Elect. Eng. Microwaves, vol. 150, no. 4, pp. 209–213, 2003. [19] A. B. Yakovlev, O. Luukkonen, C. R. Simovski, S. A. Tretyakov, S. Paulotto, P. Baccarelli, and G. W. Hanson, S. Zouhdi, A. Sihvola, and A. P. Vinogradov, Eds., “Analytical modeling of surface waves on high impedance surfaces,” in Proc. Metamaterials and Plasmonics: Fundamentals, Modelling, Applications, 2009, pp. 239–254, NATO Science for Peace and Security Series B. [20] J.-M. Baracco, L. Salghetti-Drioli, and P. de Maagt, “AMC low profile wideband reference antenna for GPS and GALILEO systems,” IEEE Trans. Antennas Propag., vol. 56, no. 8, 2008. [21] F. Costa, S. Genovesi, and A. Monorchio, “On the bandwidth of high-impedance frequency selective surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1341–1344, 2009. [22] R. Collin, Foundations of Microwave Engineering, 2nd ed. New York: IEEE Press, 2001. [23] S. A. Tretyakov and C. R. Simovski, “Dynamic model of artificial reactive impedance surfaces,” J. Electromagn. Waves Appl., vol. 17, no. 1, pp. 131–145, 2003. [24] S. Clavijo, R. E. Diaz, and W. E. McKinzie, III, “Desing methodology for Sievenpiper high-impedance surfaces: An artificial magnetic conductor for positive gain electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2678–2690, Oct. 2003. [25] M. K. Kärkkäinen and S. A. Tretyakov, “Finite-difference time-domain model of interfaces with metals and semiconductors based on a higher order surface impedance boundary condition,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2448–2455, 2003.

Filippo Costa (S’07–M’10) was born in Pisa, Italy, on October 31, 1980. He received the M.Sc. degree in telecommunication engineering and the Ph.D. degree in applied electromagnetism in electrical and biomedical engineering, electronics, smart sensors, nano-technologies from the University of Pisa, Pisa, Italy, in 2006 and 2010, respectively. From March to August 2009, he was a Visiting Researcher at the Department of Radio Science and Engineering, Helsinki University of Technology, TKK (now Aalto University), Finland. Since January 2010 he is a Postdoctoral Researcher at the University of Pisa. His research is focused on the analysis and modelling of Frequency Selective Surfaces and Artificial Impedance Surfaces with emphasis to their application in electromagnetic absorbing materials, antennas, radomes, waveguide filters and techniques for retrieving dielectric permittivity of materials.

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Olli Luukkonen received the M.Sc. (Tech.) and D.Sc. (Tech.) (with distinction) degrees in electrical engineering from the TKK Helsinki University of Technology, Espoo, Finland, in 2006 and 2009, respectively. From 2005 to 2009, he was an Assistant Researcher and a Researcher at the Radio Laboratory, TKK. In 2009–2010, he was a Visiting Researcher and a Postdoctoral Researcher with a Fulbright scholarship at the Department of Electrical and Systems Engineering, University of Pennsylvania. Currently, he is a Postdoctoral Researcher at the Department of Radio Science and Engineering, Aalto University, Espoo, Finland. His current research interests include electromagnetic theory, artificial electromagnetic materials and surfaces, plasmonics, plasmas, and their applications.

Constantin R. Simovski (M’92) was born on December 7, 1957 in Leningrad, Russian Republic of Soviet Union (now St. Petersburg, Russia). He received the Diploma of Engineer Researcher in radio engineering, the Ph.D. degree in electromagnetic theory, and Doctor of Sciences degree, in 1980, 1986, and 2000, respectively, all from the St. Petersburg State Polytechnic University (formerly the Leningrad Polytechnic Institute, and State Technical University), St. Petersburg. From 1980 to 1992, he was with the Soviet scientific and industrial firm “Impulse.” In 1986, he defended the thesis of a Candidate of Science (Ph.D.) thesis (a study of the scattering of Earth waves in the mountains) in the Leningrad Polytechnic Institute. In 1992, he joined the St. Petersburg University of Information Technologies, Mechanics and Optics, as an Assistant where, from 1994 to 1995, he was an Assistant Professor, from 1995 to 2001, he was an Associate Professor, and since 2001, he has been a Full Professor. In 2000, he defended the thesis of Doctor of Sciences (a theory of 2-D and 3-D bianisotropic scattering arrays). Since 1999, he has been involved in the theory and applications of 2-D and 3-D electromagnetic band-gap structures for microwave and ultrashortwave antennas. Currently, he is with the Helsinki University of Technology where he pursues research in the field of metamaterials for microwave and optical applications including optics of metal nanoparticles.

Agostino Monorchio (S’89–M’96–SM’04) received the Laurea degree in electronics engineering and the Ph.D. degree in methods and technologies for environmental monitoring from the University of Pisa, Pisa, Italy, in 1991 and 1994, respectively. During 1995, he joined the Radio Astronomy Group, Arcetri Astrophysical Observatory, Florence, Italy, as a Postdoctoral Research Fellow, in the area of antennas and microwave systems. He has been collaborating with the Electromagnetic Communication Laboratory, Pennsylvania State University (Penn State), University Park, and he is an Affiliate of the Computational Electromagnetics and Antennas Research Laboratory. He has been a Visiting Scientist at the University of Granada, Spain, and at the Communication University of China in Beijing. He is currently an Associate Professor in the School of Engineering, University of Pisa, and Adjunct Professor at the Italian Naval Academy of Livorno. He is also an Adjunct Professor in the Department of Electrical Engineering, Penn State. He is on the Teaching Board of the Ph.D. course in “Remote Sensing” and on the council of the Ph.D. School of Engineering “Leonardo da Vinci” at the University of Pisa. His research interests include the development of novel numerical and asymptotic methods in applied electromagnetics, both in frequency and time domains, with applications to the design of antennas, microwave systems and RCS calculation, the analysis and design of frequency-selective surfaces and novel materials, and the definition of electromagnetic scattering models from complex objects and random surfaces for remote sensing applications. He has been a reviewer for many scientific journals and he has been supervising various research projects related to applied electromagnetic, commissioned and supported by national companies and public institutions. Dr. Monorchio has served as Associate Editor of the IEEE Antennas and Wireless Propagation Letters. He received a Summa Foundation Fellowship and a NATO Senior Fellowship.

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Sergei A. Tretyakov received the Dipl. Engineer-Physicist, the Candidate of Sciences (Ph.D.), and the Doctor of Sciences degrees (all in radiophysics) from the St. Petersburg State Technical University (Russia), in 1980, 1987, and 1995, respectively. From 1980 to 2000, he was with the Radiophysics Department of the St. Petersburg State Technical University. Presently, he is Professor of radio engineering at the Department of Radio Science and Engineering, Aalto University, Finland, and the president of the Virtual Institute for Artificial ElectromagneticMaterials and Metamaterials Metamorphose. His main scientific interests are electromagnetic field theory, complex media electromagnetics and microwave engineering. Prof. Tretyakov served as Chairman of the St. Petersburg IEEE ED/MTT/AP Chapter from 1995 to 1998.

Peter de Maagt (S’88–M’88–SM’02–F’08) was born in Pauluspolder, The Netherlands, in 1964. He received the M.Sc. and Ph.D. degrees from Eindhoven University of Technology, Eindhoven, The Netherlands, in 1988 and 1992, respectively, both in electrical engineering. From 1992 to 1993, he was a Station Manager and Scientist with an INTELSAT propagation project in Surabaya, Indonesia. He is currently with the European Space Research and Technology Centre (ESTEC), European Space Agency (ESA), Noordwijk, The Netherlands. His research interests are in the area of millimeter and submillimeter-wave reflector and planar integrated antennas, quasi-optics, electromagnetic bandgap antennas, and millimeter- and submillimeter-wave components. He spent the summer period of 2010 as a visiting research scientist at the Stellenbosch University in Stellenbosch, South Africa. Dr. de Maagt was corecipient of the H. A. Wheeler Award of the IEEE Antennas Propagation Society (IEEE AP-S) for the Best Applications Paper of 2001 and 2008. He was granted an ESA Award for Innovation in 2002 and an ESA award for Corporate Team Achievements for the Herschel and Plnack Programme in 2010. He was corecipient of Best Paper Awards at the Loughborough Antennas Propagation Conference (LAPC) 2006 and the International Workshop on Antenna Technology (IWAT) 2007. He served as an associate editor for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION from 2004–2010 and was co-guest editor of the November 2007 “Special Issue on Optical and Terahertz Antenna Technology”.

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Miniaturized Self-Oscillating Annular Ring Active Integrated Antennas Ya-Yun Lin, Cheng-Hsun Wu, Student Member, IEEE, and Tzyh-Ghuang Ma, Senior Member, IEEE

Abstract—With a direct integration of a field effect transistor (FET) into a C-shaped ring radiator, a novel miniaturized self-oscillating annular ring active integrated antenna is proposed and investigated in this paper. The direct integration approach provides a solid basis for the miniaturization while the slow-wave loading effect from the unstable transistor further enhances the reduction ratio. It makes the proposed design show an extraordinarily compact size of , which is 40% the size of the second smallest one in literature using printed circuit board technology. The proposed active antenna, providing overall comparable performance to the previous designs, has an EIRP of 7.30 dBm, DC-to-RF efficiency of 25.9%, phase noise of dBc/Hz@100 kHz offset, dBc/Hz. By introducing a reand an oscillator FOM of verse-biased varactor diode, this miniaturized active antenna also demonstrates promising voltage controllability with a high VCO gain of 48.7 MHz/V. The antenna geometry and modeling, design concept, simulation scheme, and experimental results are discussed thoroughly in this paper. Index Terms—Active antennas, microwave FET oscillators, microstrip antennas, slow wave structures.

I. INTRODUCTION

T

HE active integrated antenna (AIA), an integration of the active device and the radiating element, has received considerable attention in the past decades [1], [2]. Upon the applications and configurations, the active integrated antennas can be categorized into four groups: the low-noise active antennas [3], [4], power amplifying active antennas [5]–[8], self-oscillating active antennas [9]–[23], and self-mixing active antennas [24], [25]. Daniel et al. [3] reported an active receiving path, providing the benefits of improved effective gain, noise figure, and figure of merit G/T; a low-noise active circular patch antenna was proposed for X-band applications in [4]. This class of active antennas is realized by integrating the antenna with a low noise amplifier, resulting in a low noise path in the receiver front-end. In the transmitter front-end, on the other hand, active integrated antennas having the function of power amplifying the transmitting signal have been reported in literature by utilizing the Manuscript received November 16, 2010; revised February 08, 2011; accepted March 14, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported by the National Science Council, Taiwan (R.O.C.), under Grants 99-2628-E-011-001 and 100-2628-E011-004. The authors are with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan, (R.O.C.) (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.2011.2163782

push-pull class-B power amplifiers [5], [6] and the high efficiency class-E/class-F power amplifiers [7], [8]. The self-oscillating active integrated antenna, an integration of the antenna with an oscillating transistor, has also received intensive attention in recent years. The most straightforward way to realize a self-oscillating active antenna is the cascade connection of a potential unstable microwave transistor and a radiator [9]–[12]. Nevertheless, rigorously speaking, this sort of design does not fully comply with the original intention of active integrated antenna, as the radiator and oscillator are still designed as two separate circuit blocks. The only integration is to determine the impedances in-between the circuit blocks to maximize the oscillation power. This approach generally results in a bulky size, preventing it from many practical applications. In the meantime, the phase noise performance has become the major bottleneck in developing high performance self- oscillating active antenna design because the radiator, working also as the load of the oscillator network, is inherently a high loss component with a low quality factor (Q-factor). A low-Q load is prone to produce significant phase noise in oscillator design [26]. To tackle this problem, a variety of approaches, including the installation of back cavity [13], phase locked loop [14], injection-locking network [15], [16] and feedback loop [17]–[20], have been studied to improve the noise performance of self-oscillating active antennas. Among the techniques, the feedback loop approach provides an elegant way to develop self-oscillating active antennas with simple architecture and good noise performance. In [17], [18], the coupling aperture beneath the patch provided the feedback signal to the gate terminal of the transistor, while the microstrip line inside the TM -mode ring radiator served as the feedback path in [19]. In [20], the feednon-radiating microstrip line back loop was formed by a with an AC-coupled capacitor. The integration of an oscillator with a feedback high-Q bandpass filter results in extremely low phase noise design, as well [26], [27]. Benefitting from the promising feedback loop approach, in this paper, we investigate the miniaturization techniques for developing a self-oscillating active antenna with a very compact size. The miniaturization is achieved by directly integrating a microwave transistor into a radiator to form a single circuit block, making the radiator become a part of the oscillator network. By utilizing the slow-wave loading due to the parasitic capacitance of the unstable transistor, the required feedback loop path for establishing a stable oscillation is diminished; it therefore further reduces the antenna size. For demonstration purpose, a C-shaped annular ring antenna, with a nature resonance mode at 6.95 GHz, is applied to the miniaturized design. A simple linear model, based on the small

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As shown in Fig. 1, the C-shaped radiator lies in the xy-plane with the -axis being the broadside direction. The open-circuited condition at the ends of the annular ring makes the effective current path be approximately equal to one-half the guided wavelength at its nature resonance frequency. Taking the nonuniform current distribution across the strip into account, the effective current path can be estimated by (1)

Fig. 1. Circuit layout of the proposed miniaturized self-oscillating annular ring antenna.

signal -parameters, is first used to explain the slow-wave loading as well as the oscillation condition of the proposed antenna. In order to accurately predict the oscillation of such a fully integrated active antenna, an alternative simulation scheme, deeming the radiator as a two port network in-between the gate and drain terminals of the transistor, is then applied to the harmonic balance analysis. Due to the capacitive loading, the oscillation frequency of the proposed design shifts downward to 5.383 GHz. The direct integration approach, together with the slow-wave loading, make the new design show an extraordinarily compact size when compared with previous works in literature using slot loop antenna [21], circular patch [22], and dielectric resonator [23]. The electrical performances, on the other hand, remain acceptable in practical low-power applications. Moreover, by utilizing a simple varactor diode, the proposed active integrated antenna shows excellent voltage controllability, as well. The antenna configuration and modeling, miniaturization techniques, simulation scheme, and experimental results are discussed intensively in the following sections. II. ANTENNA CONFIGURATION AND MODELING A. Antenna Geometry Fig. 1 shows the layout of the proposed miniaturized self-oscillating active integrated antenna. The geometric parameters of the C-shaped passive ring radiator are shown in Fig. 2(a). The antenna was developed on a 0.508-mm Rogers RO4003C substrate with a dielectric constant and a loss . The transistor is an n-channel hetero-junction FET from NEC Corporation with a part number NE3512S02. The radii and are, respectively, the outer and inner radii of the annular ring, and is the strip width. is about one-half the outer circumference of the ring, and is the gap width between the two open ends. and are the length and width of the RF choke lines. In the proposed design, mm, mm, mm, mm, mm, mm, and mm. The inner and outer circumferences of the ring are 9.2 and 21.5 mm, respectively, and the size of the ground plane mm .

In (1), the weighting factors reflect the fact that the surface current shows higher density along the inner side of the annular ring. The effective current path can be viewed as the equivalent circumference of ring resonator. The transistor is integrated directly into the radiator by connecting its gate and drain terminals to the two open ends of the ring. The source terminals, on the other hand, are short-circuited to ground. The DC power lines are fed through via holes from the bottom layer of the substrate to reduce the potential nearfield scattering and coupling. Guard rings are etched around the via holes to prevent the DC signals from grounding. Three high impedance lines, serving as the RF chokes, are added in between the DC pads and the ring radiator to guarantee good DC-to-RF isolation. Although the high impedance lines may be replaced by surface mounted inductors to further reduce the size of the bias network, it complicates the design process since the parasitic effects of the lumped inductors cannot be ignored at the oscillation frequency. A DC block capacitor, with a value of pF, is added to the annular ring to isolate the gate and drain bias. It also serves as an AC-coupled capacitor to ensure the RF connection of the ring. The parasitic capacitance of the 0.5-mm microstrip gap for mounting the DC block is relatively small and could be neglected for brevity. B. Modeling In addition to serving as a radiating element, the annular ring provides a feedback path between the gate and drain terminals of the transistor to improve the noise performance of the active antenna. For the designers, nevertheless, due to the direct integration of the transistor with the radiator, it is not an easy task to determine the antenna electrical parameters using a standard simulation tool. To tackle this difficulty, in this paper, the radiator is deemed as a two-port network while the -parameter analysis and nonlinear circuit analysis are both performed to evaluate the oscillation characteristics. First, we use a simple linear model to explain the circuit miniaturization due to slow-wave loading as well as to interpret the oscillation condition of the proposed active antenna. The harmonic balance analysis, based on a nonlinear circuit emulator, is then applied to accurately predict the antenna parameters including the oscillation frequency, output power, and radiation gain. The modeling begins with the explanation of the circuit miniaturization due to slow-wave loading. As shown in Fig. 2(a), the C-shaped radiator is viewed as a two-port network in the simulator HFSS. The port definitions are shown in the figure, and the system impedance is 50 ohms. Fig. 2(b) and (c) shows, respectively, the simulated two-port -parameters and

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Fig. 2. (a) Geometric parameters of the unloaded annular ring along with the port definitions for full-wave simulation; (b) -parameters of the unloaded ring radiator; (c) input impedance looking into port 1 of the unloaded ring radiator (50-ohm termination); (d) current distribution at the nature resonance frequency (6.95 GHz).

the associated input impedance looking into the port 1 of the radiator. Here, the termination at port 2 is 50 ohms. Clearly from Fig. 2(b), the nature resonance frequency of the passive radiator is 6.95 GHz, with the real part of the input impedance approximately reaching the maximum and the imaginary part approaching to zero. The simulated current distribution in Fig. 2(d) further demonstrates that the effective current path in (1), which is 13.3 mm, is approximately one-half the guided wavelength at the resonance frequency ( mm). In real integration, nevertheless, the port 2 of the radiator is actually connected to the drain terminal of the transistor, as indicated in Fig. 1. Owing to the parasitic drain-to-source/ drain-to-gate capacitances, the transistor would provide additional loading on the radiator. It is known that the capacitive loading, with or without periodicity, is a common slow-wave approach to reduce the required length of a transmission line [28]. For example, by applying capacitive loading to both ends of a transmission line, the required physical length for fulfilling a specific electrical length, say 90 degrees, can be effectively reduced (Chap. 8, [28]). In the same sense, if the length of a loaded resonator remains unchanged, the capacitive loading, in turn, lowers down the resonance frequency. This idea, though has been well developed in the microwave community, has not been exploited in the design of miniaturized active integrated antennas yet.

In the current design, to account for the parasitic capacitive loading on the radiator, a simple model, based on the linear -parameters of the transistor, is established using the circuit topology in Fig. 3(a). This model helps explain the oscillation condition of the proposed active antenna, as well. As indicated in Fig. 3(a), the feedback loop is broken up at the gate terminal of the transistor in order to determine the impedances looking into the port 1 of the radiator and the gate terminal of the transistor, i.e., and , respectively. To this end, the two-port -parameters of the C-shaped radiator derived by HFSS are imported into the ADS as a data item, with the ports 1 and 2 being connected, respectively, to the gate and drain terminals of the transistor. The whole circuit is then analyzed using the -parameter analysis in ADS, with the transistor model provided by the manufacturer. Here, the bias conditions are V and V, the same as that in the active antenna design. Under the small-signal assumption, the -parameter analysis provides a linear approximation of the oscillating circuit. The simulated input impedances looking into the port 1 of the radiator and the gate terminal of the transistor, based on the -parameter analysis, are shown in Fig. 4(a) and (b), respectively. To explain the capacitive loading more clearly, we further replace the transistor by a simplified lumped model, as illustrated in Fig. 3(b). At the drain terminal, the transistor is

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Fig. 3. (a) Simplified linear circuit model for small signal -parameter analysis. (b) Equivalent circuit model for explaining the loading effects of the transistor.

represented by a simple RC network consisting of a resistor in parallel with a capacitor . For brevity, the transistor is assumed to be a unilateral device to simplify the analysis. Based on the -parameter analysis, the lumped values of the RC network, pF and ohms, are determined by curve-fitting the response of the RC network to that of the transistor looking into the drain terminal. In Fig. 3(b), the dotted line part accounts for the equivalent model for the remaining part of the transistor; it provides the negative resistance for establishing oscillation. Using Fig. 3(b), the input impedance looking into the port 1 of the antenna network can be determined by taking into consideration the loading effect of the transistor, (2) where (3)

Fig. 4. (a) Input impedance at port 1 of the capacitively loaded ring resonator. Solid lines: the linear model (Fig. 3(a)); the dash lines: the equivalent model (Fig. 3(b)). (b) Input impedance looking into the gate terminal of transistor in Fig. 3(a).

electrical length of the radiator is reduced by 20% at the resonance. It, in turn, reduces the occupied antenna size. The circuit models in Fig. 3(a) and (b) provide a supportive validation of one of the core miniaturization techniques used in this work, that is, the slow-wave loading due to the parasitic capacitance of the transistor on the passive radiator. On the other side, the criterion for establishing a stable oscillation at requires the loop gain of the closed feedback loop becomes unity while the overall phase shift of the loop is exactly equal to ; that is, (6) and (7)

(4) (5)

It is known that (6)–(7) are equivalent to (8)

With (2)–(5), the input impedance looking into the port 1 of the capacitively loaded ring radiator can be extracted; the results are plotted in Fig. 4(a) at the same time for comparison. Clearly from the figure, the impedance responses acquired from the linear model in Fig. 3(a) agree reasonably well with those calculated using the simplified RC model in Fig. 3(b). In addition, the resonance frequency of the capacitively loaded radiator shifts downward from 6.95 to 5.65 GHz. As the physical dimension of the antenna remains the same, the corresponding

and (9) For the proposed design, as indicated in Fig. 4(a) and (b), the oscillation should be stabilized at the lower side of the resonance frequency of the capacitively loaded ring resonator. It is because that in this region, the antenna input reactance is inductive, capable of compensating for the parasitic capacitance at the gate terminal of the transistor; in addition, the real part of

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Fig. 5. Schematic circuit diagram for harmonic balance analysis.

the antenna input impedance is low enough to be cancelled out by the negative resistance provided by the unstable transistor. According to the impedance relationship in Fig. 4(a) and (b), the predicted oscillation frequency is around 5.10–5.15 GHz, at which the criteria (8) and (9) hold simultaneously. As will be shown in the next section, when compared to the simulated oscillation frequency using harmonic balance analysis, the error of this predicted result is less than 5%, despite of the fact that the linear model in Fig. 3(a) is relatively simplified.

Fig. 6. Simulated output power spectrum of the proposed miniaturized selfoscillating active antenna.

III. EXPERIMENTAL RESULTS AND DISCUSSION The oscillation condition discussed using the simplified model in Section II can be validated by the harmonic balance analysis with Agilent ADS. The schematic circuit diagram for nonlinear simulation is shown in Fig. 5. To accurately determine the oscillation frequency and the output power of the active integrated antenna, the simulated two-port antenna -parameters are again imported into the ADS, with the ports being connected to the terminals of the transistor as the feedback path. In the harmonic balance analysis, the output power spectrum is evaluated at the drain terminal of the transistor ( node) with a port impedance of (10) The port impedance in (10) can mimic the wave impedance of the radiation field in the presence of a dielectric slab. In our experience, it provides a good approximation of the oscillation output power. A variation of the port impedance from 250 to 377 ohms results in a power variation of less than 1.5 dB. The simulated output power spectrum is shown in Fig. 6; the simulated oscillation frequency is 5.393 GHz, with an output power of 11.21 dBm. The second and third harmonics are 18.1 and 35.3 dB, respectively, lower than the oscillation power at the fundamental frequency. At the oscillation frequency, the effective current path in (1) is about . The size of the integrated antenna is mm , or equivalently, . is referred to as the guided wavelength of a 50-ohm microstrip line on the same substrate at the oscillation frequency. Without utilizing a three-dimensional dielectric resonator antenna, the proposed active integrated antenna shows a very compact size when compared with the designs in literature [21]–[23]; it is less than 40% the size of the active antenna in [23]. Accordingly, by applying direct integration and slow-wave loading,

Fig. 7. Photograph of the fabricated sample.

the proposed design shows an effective way to develop miniaturized self-oscillating active antennas. To experimentally investigate the performance of the proposed active antenna, a prototype sample was fabricated; a photograph is shown in Fig. 7. The emission characteristics of the active integrated antenna were measured using the measurement setup in Fig. 8. The active antenna, supported by a foam stand, is placed in an anechoic chamber. The power supply for biasing is properly shielded by absorbers. The bias conditions are V and V, with a drain current of 26 mA. A rectangular patch, with a size of 36.4 18.2 mm , was fabricated and served as the receiving antenna. The gain of the patch antenna is 7.16 dBi, which was determined in advance using a three-dimensional spherical near-field measurement system from the Nearfield Systems Inc. (NSI 700S-90 scanner) at National Taiwan University of Science and Technology. To guarantee the far-field conditions of both antennas at the oscillation frequency, the distance between the active integrated antenna and the receiving patch is mm. The measured received power spectrum is shown in Fig. 9. The measurement was performed by an Agilent spectrum analyzer E4446A with a resolution bandwidth of 100 kHz. The measured power peak is at 5.383 GHz , with a well-behaved spectrum around the oscillation frequency; it is very close to the simulated result. The received power at the

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Fig. 8. Measurement setup for acquiring the effective isotropic radiated power (EIRP) of the proposed miniaturized self-oscillating active antenna.

Fig. 10. Measured radiation patterns of the proposed miniaturized active antenna at the oscillation frequency.

Fig. 9. Measured received power spectrum of the proposed miniaturized selfoscillating active antenna.

oscillation frequency is dBm, from which the effective isotropic radiated power (EIRP) can be calculated, (11) is the gain of the C-shaped ring radiator, and In (11), is the oscillation output power. is the insertion loss of the connecting cable, which is 0.8 dB in the measurement. Using (11), the EIRP of the miniaturized active antenna is 7.30 dBm. To determine the oscillation output power from (11), the absolute gain of the passive radiator should be estimated in advance. However, it is known that for an active integrated antenna with a feedback loop, the radiation gain of the passive radiator can be evaluated only in an indirect way. The most common approach for evaluating the radiating element gain requires the efficiency of the passive radiator to be relatively high. With this assumption, the antenna gain can be approximately equal to its directivity with the well-known formulation [20], (12) In (12), and are the half-power beamwidths in the Eand H-planes of the integrated antenna. Owing to the circuit miniaturization, the proposed design actually belongs to the category of electrically small antennas. According to the small antenna theory, the radiation efficiency of the C-shaped ring radiator is expected to be low. Due to the low

efficiency, (12) fails to provide a good estimation of the radiation gain, and an alternative estimation approach is definitely required. This issue has not been dealt with in literature, probably owing to the fact that the previous designs are not especially small when compared to the guided wavelength. To get a good estimation of the radiation gain of the proposed C-shaped ring radiator, in this paper, the complex voltages acquired at the gate and drain terminals in the harmonic balance analysis are substituted back into the Ansoft HFSS, serving as the scaling factors of the excitations at two lumped ports. According to the harmonic balance analysis, at the gate and drain terminals, and , which give rise to an antenna gain of dBi in the HFSS simulation. Since the EIRP of the design is 7.30 dBm, the oscillation output power of the active antenna can be calculated from (11) as 11.90 dBm; it agrees well with the simulation in Fig. 6. As the DC power consumption is 2.3 V mA mW, the DC-to-RF conversion efficiency is about 25.9%. By using a passive radiating element with high radiation efficiency at the oscillation frequency [29], it can be shown that the calculated gain using this alternative approach agrees well with that estimated from (12). The radiation patterns of the active antenna were measured in a 7 4.9 4.6 m anechoic chamber. An electrically switched dual-polarized log periodic antenna was connected to the spectrum analyzer E4446A as the field probe. The power level at each individual reception angle was recorded for pattern measurement. The radiation patterns are then normalized to its maximum value; the results in the yz- (E-) and xz- (H-) planes are shown in Fig. 10. Referring to the figure, the radiation patterns of the proposed self-oscillating active antenna are, in essence, similar to those of a passive ring radiator, with the main lobe directing toward the broadside direction; in addition, the cross polarization at the broadside direction is 10 dB less than the co-polarized component. The 3-dB beamwidths in the E- and H-planes are both approximately 100 degrees. According to (12), the directivity (D) of the radiating element is about 6.1 dBi. Since the absolute gain of the passive radiator is dBi, the radiation efficiency, without the power generated

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TABLE I COMPARISONS OF THE PERFORMANCES OF VARIOUS SELF-OSCILLATING ACTIVE INTEGRATED ANTENNAS

Fig. 11. Measured phase noise of the proposed miniaturized self-oscillating active antenna.

by the oscillating transistor, is 8.5%. The result complies with the image of an electrically small antenna. The phase noise of the miniaturized active antenna was measured using the spectrum analyzer E4446A in an anechoic chamber; the result is shown in Fig. 11. At 100-kHz and 1-MHz offset from the oscillation frequency , the measured phase noises are and dBc/Hz, respectively. From 10 kHz to 1 MHz, the phase noise shows a delaying slope of 15 dB/decade, deviating from the theoretical value of 20 dB/decade in the Lesson’s model with a low Q load (1/f region) [30]. The possible reasons for this deviation include the uncertainty in the measurement owing to the parasitic coupling, as it involves a receiving path whose dimension is considerably larger than that of the active antenna under test; the settings of the resolution bandwidth and attenuation in the spectrum analyzer could be possible reasons for the uncertainty, as well. In addition, the 1/f noise also plays an important role since the transistor is biased near the cutoff to increase the nonlinearity [31]. Moreover, the proposed active antenna does not contain a

phase locked loop, making the oscillation frequency inevitably drift with time. It is known that the spectrum analyzer becomes less reliable when measuring the phase noise of a free-running oscillator or an unlocked active antenna [15]. For more accurate results, a phase noise measurement system with the frequency discriminator method, such as the setup in [20], should be applied. Despite some uncertainty in the measurement, in a general sense, the phase noise performance of the proposed design is still well behaved and comparable to those in literature; this is a fruitful result of the feedback loop scheme. Finally, the oscillator FOM, a common figure of merit for evaluating the performance of an oscillator, is calculated. The FOM is defined as [27] (13) where is the phase noise at an offset frequency from the carrier . Using (13), the FOM of the proposed miniaturized active antenna is dBc/Hz with kHz. In Table I, the performances of the proposed self-oscillating active antenna, including the occupied size, EIRP, output RF power, DC power consumption, DC-to-RF efficiency, phase noise, FOM, and etc., are summarized. The performances of various previous designs in [9]–[12], [15], [16], [18]–[23] are tabulated for comparison purpose, as well. Some of the data are estimated from the figures in literature. Due to the lack of information regarding the DC power consumption and/or phase noise in [16], [19] and [22], the oscillator FOM of these three designs cannot be calculated. In addition, in the comparison table, we only focus on the sizes of the radiators themselves; for each design, the bias network is therefore excluded from the size calculation. Clearly from the table, the proposed design

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has the smallest size on printed circuit board; it is only 40% the size of the second smallest one. In the meantime, the electrical performances such as the DC-to-RF conversion efficiency, phase noise, and FOM are still comparable with those in literature. The somewhat low EIRP is a result of the miniaturization techniques applied, making it suitable for low power applications such as the 5-GHz radio-frequency identification (RFID) systems. The voltage controllability of the proposed design will be discussed in the following section. IV. VOLTAGE CONTROLLABILITY In practical communication systems, the carrier frequency actually hops over a set of predefined channels in order to improve the data throughput and transmission quality; in addition, the allocated spectrum for a specific application may vary from one nation to the other. It is therefore crucial for a self-oscillating active integrated antenna to demonstrate proper voltage-controlled frequency tunability to comply with the demands in real applications. A self-oscillating active antenna with frequency tunability can be realized by introducing a voltage-controlled component, for example, a varactor or a transistor, to the radiator to alter the oscillation condition upon the extra loading. This sort of design, therefore, can be viewed as an integration of a voltage-controlled oscillator (VCO) with a radiator. To demonstrate the voltage controllability of the proposed miniaturized self-oscillating active antenna, the dependence of the oscillation condition on the circuit parameter is first investigated. Fig. 12 shows the input impedance of the network in Fig. 3(b) as the AC-coupled capacitor in the feedback path varies from 1.5 to 2.5 pF with a 0.5-pF step. Clearly from the figure, as increases, the resonance frequency of the loaded C-shape resonator moves steadily downward to the lower frequency side; the oscillation frequency , satisfying with the criteria in (8)–(9), follows the same trend accordingly. With the help of harmonic balance analysis, the oscillation frequencies with , and 2.5 pF are 5.39, 5.23, and 5.13 GHz, respectively. The vertical and horizontal dotted lines in Fig. 12 indicate the corresponding impedance values at the frequency of oscillation with a specific . Referring to the figure, the corresponding input impedance at the frequency of oscillation is roughly the same for all three cases, a good indication that the oscillation mechanism remains consistent despite the variation of the circuit parameter. To exploit the concept in Fig. 12, a VCO version of the proposed active integrated antenna is developed; the antenna topology is shown in Fig. 13. The original DC block capacitor is replaced by a varactor diode , and an additional bias line for reverse-biasing the varactor is added to the annular ring. The varactor diode is Infineon Technologies’ BB857. From the datasheet [32], the tunable capacitance range is approximately from 0.5 to 6.5 pF as the bias voltage decreases from 28 to 1 V. To isolate the bias and , an additional gap is etched on the radiator with a DC block mounting across the gap. Differing from the of the original design, this DC block has a very large value of 120 pF; it is nearly short circuited at the oscillation without influence on the active integrated antenna.

Fig. 12. Simulated input (a) resistance and (b) reactance of the capacitively varies from 1.5 to 2.5 pF. loaded ring radiator as the AC-coupled capacitor

Fig. 13. Circuit topology of the proposed miniaturized voltage-controlled selfoscillating annular ring active integrated antenna.

By steadily increasing the bias voltage from 1 to 28 V, the oscillation frequency as well as the EIRP of the miniaturized voltage-controlled self-oscillating active antenna is measured and illustrated in Fig. 14. The measurement setup is the same as that in Section III. Clearly from the figure, the oscillation frequency steadily increases from 4.41 to 5.43 GHz as the bias voltage increases, or equivalently, the junction capacitance of the reverse-biased diode decreases. In the linear tunable

LIN et al.: MINIATURIZED SELF-OSCILLATING ANNULAR RING ACTIVE INTEGRATED ANTENNAS

Fig. 14. Measured oscillation frequency and EIRP of the miniaturized voltageof the varactor diode. controlled active antenna versus the reversed bias

region, the oscillation frequency varies from 4.41 to 5.29 GHz as changes from 1 to 19 V; this corresponds to a VCO gain of 48.7 MHz/V. In the meantime, the EIRP shows a trend of decrease from 8 to 5 dBm as the oscillation frequency decreases from 5.4 to 4.4 GHz. The decline in EIRP is a result of the decrease in the antenna gain as the oscillation frequency goes down, making the antenna become smaller and smaller when compared to the guided wavelength. This observation is consistent with the discussion of the antenna gain in Section III. In summary, the demonstration clearly shows the promising voltage controllability of the proposed miniaturized self-oscillating active antenna, making it attractive to a variety of applications. V. CONCLUSION A novel miniaturized self-oscillating annular ring active integrated antenna, by directly integrating the transistor into the C-shaped ring radiator, has been experimentally validated and discussed in this paper. The direct integration, together with the slow-wave loading due to the parasitic capacitance of the unstable transistor, make the proposed active antenna show a very compact size, which is 40% the size of the second smallest one in literature using printed circuit board technology. The electrical antenna performances, in the meantime, remain wellbehaved with an EIRP of 7.30 dBm, DC-to-RF efficiency of 25.9% and oscillator FOM of dBc/Hz. By replacing the AC-coupled capacitor with a reverse-biased varactor diode, the proposed design shows promising frequency tunability with a VCO gain of 48.7 MHz/V. This new active antenna is especially suitable for low power applications such as the 5-GHz RFID systems. REFERENCES [1] K. Chang, R. A. York, P. S. Hall, and T. Itoh, “Active integrated antennas,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 937–944, Mar. 2002. [2] Y. Qian and T. Itoh, “Progress in active integrated antennas and their applications,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1891–1900, Nov. 1998. [3] D. Segovia-Vargas, D. Castro-Galan, L. E. Garcia-Munoz, and V. Gozalez-Posadas, “Broadband active receiving patch with resistive qualization,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 56–64, Jan. 2008.

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[4] A. S. Andrenko, I. V. Ivanchenko, D. I. Ivanchenko, S. Y. Karelin, A. M. Korolev, E. P. Laz’Ko, and N. A. Popenko, “Active broad X-band circular patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 529–533, 2006. [5] C. Y. Hang, W. R. Deal, Q. Yongxi, and T. Itoh, “High-efficiency pushpull power amplifier integrated with quasi-Yagi antenna,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1155–1161, Jun. 2001. [6] Y. Qin, S. Gao, and A. Sambell, “Broadband high-efficiency circularly polarized active antenna and array for RF front-end application,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 2910–2917, Jul. 2006. [7] H. Kim and Y. J. Yoon, “Wideband design of the fully integrated transmitter front-end with high power added efficiency,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 916–924, May 2007. [8] C. H. Tsai, Y. A. Yang, S. J. Chung, and K. Chang, “A novel amplifying antenna array using patch-antenna couplers-design and measurement,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1919–1926, Aug. 2002. [9] D. H. Choi and S. O. Park, “A varactor-tuned active-integrated antenna using slot antenna,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 191–193, 2005. [10] H. P. Moyer and R. A. York, “Active cavity-backed slot antenna using MESFETs,” IEEE Microw. Guided Wave Lett., vol. 3, pp. 95–97, 1993. [11] F. Giuppi, A. Georgiadis, A. Collado, M. Bozzi, and L. Perregrini, “Tunable SIW cavity backed active antenna oscillator,” Electron. Lett., vol. 46, no. 15, pp. 1053–1055, Jul. 2010. [12] J. Shi, J. X. Chen, and Q. Xue, “A differential voltage-controlled integrated antenna oscillator based on double-sided parallel-strip line,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 2207–2212, Oct. 2008. [13] M. Zheng, P. Gardener, P. S. Hall, Y. Hao, Q. Chen, and V. F. Fusco, “Cavity control of active integrated antenna oscillators,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 148, pp. 15–20, 2001. [14] J. W. Andrews and P. S. Hall, “Phase-locked-loop control of active microstrip patch antennas,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 201–206, Jan. 2002. [15] K. H. Y. Ip and G. V. Eleftheriades, “A compact CPW-based single-layer injection-locked active antenna for array applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 2, pp. 481–486, Feb. 2002. [16] Y. Chen and Z. Chen, “A dual-gate FET subharmonic injection-locked self-oscillating active integrated antenna for RF transmission,” IEEE Micorw. Wireless Compon. Lett., vol. 13, pp. 199–201, Jun. 2003. [17] W. J. Tseng and S. J. Chung, “Analysis and application of a two-port aperture-coupled microstrip antenna,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 530–535, May 1998. [18] K. H. Y. Ip, T. M. Y. Kan, and G. V. Eleftheriades, “A single-layer CPW-FED active patch antenna,” IEEE Microw. Guided Wave Lett., vol. 10, no. 2, pp. 64–66, Feb. 2000. [19] J. A. Hagerty and Z. Popovic, “A 10 GHz active annular ring antenna,” in IEEE Int. Antennas Propag. Symp. Dig., San Antonio, TX, Jun. 2002, pp. 284–287. [20] C. H. Mueller, R. Q. Lee, R. R. Romanofsky, C. L. Kory, K. M. Lambert, F. W. V. Keuls, and F. A. Miranda, “Small-size X-band active integrated antenna with feedback loop,” IEEE Trans. Antenna Propag., vol. 56, pp. 1236–1241, May 2008. [21] G. Forma and J. M. Laheurte, “Compact oscillating slot loop antenna with conductor backing,” Electron. Lett., vol. 32, no. 18, pp. 1633–1635, Aug. 1996. [22] D. Bonefacic, J. Baartolic, and Z. Mustic, “Circular active integrated antenna with push-pull oscillator,” Electron. Lett., vol. 38, no. 21, pp. 1238–1240, Oct. 2002. [23] E. H. Lim and K. W. Leung, “Novel utilization of the dielectric resonator antenna as an oscillator load,” IEEE Trans. Antenna Propag., vol. 55, no. 10, pp. 2686–2691, 2007. [24] R. Flynt, L. Fan, J. Navarro, and K. Chang, “Low cost and compact active integrated antenna transceiver for system applications,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pp. 1642–1649, Oct. 1996. [25] C. M. Montiel, L. Fan, and K. Chang, “A novel active antenna with selfmixing and wideband varactor-tuning capabilities for communication and vehicle identification applications,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2421–2430, Dec. 1996. [26] J. Choi, M. Nick, and A. Mortazawi, “Low phase-noise planar oscillators employing elliptic-response bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1959–1964, Aug. 2009. [27] J. Choi and C. Seo, “Microstrip square open-loop multiple split-ring resonator for low-phase-noise VCO,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3245–3252, Dec. 2008.

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[28] R. K. Mongia, I. J. Bahl, P. Bhartia, and J. S. Hong, RF and Microwave Coupled-Line Circuits, 2nd ed. Norwood, MA: Artech House, 2007. [29] C.-H. Wu and T.-G. Ma, “Self-oscillating dual-ring active integrated antenna,” in IEEE Int. Symp. Antennas and Propagation Symp. Digest, Spokane, WA, Jul. 3–8, 2011, pp. 2457–2460. [30] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, pp. 329–330, Feb. 1966. [31] A. Jerng and C. G. Sodini, “The impact of device type and size on phase noise measurements,” IEEE J. Solid-State Circuits, vol. 40, no. 2, pp. 360–369, Feb. 2005. [32] Datasheet of Infineon BB837/BB857 series [Online]. Available: http://www.infineon.com/cms/en/product/findProductTypeByName.html?q=BB857

Ya-Yun Lin was born in Taichung, Taiwan, in 1986. She received the B.S. degree in communication engineering from Feng-Chia University, Taichung, Taiwan, in 2008 and the M.S. degree in electrical engineering from National Taiwan University of Science and Technology, Taipei, Taiwan, in 2010. In 2010, she joined the HTC Corporation, Taipei, Taiwan, where she is now an RF Engineer. Her current research interests include small size active integrated antennas designs and their applications.

Cheng-Hsun Wu (S’11) was born in Yunlin, Taiwan, R.O.C., in 1986. He received the B.S. degree in electrical engineering from National Taiwan University of Science and Technology, Taipei, Taiwan, in 2009, where he is currently working toward the Ph.D. degree. His research interests include active integrated antennas and microwave passive circuit designs. Mr. Wu was the recipient of the Honorable Mention award in the student paper competition at the 2011 IEEE International Symposium on Antenna and Propagation, Spokane, WA.

Tzyh-Ghuang Ma (S’00–M’06–SM’11) was born in Taipei, Taiwan, in 1973. He received the B.S. and M.S. degrees in electrical engineering and the Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, in 1995, 1997, and 2005, respectively. In 2005, he joined the faculty of the Department of Electrical Engineering, National Taiwan University of Science and Technology, where he is now an Associate Professor. His research interests include miniaturized microwave circuit designs, ultrawideband antennas, antenna arrays, and radio frequency identification (RFID). Dr. Ma was the recipient of the Poster Presentation Award at the 2008 International Workshop on Antenna Technology (iWAT), Chiba, Japan, and the recipient of the Best Paper Award at the 2011 International Workshop on Antenna Technology (iWAT), Hong Kong, China. In 2010, he received the Dr. Wu Da-Yu Award from National Science Council, the most outstanding research award for young researchers in Taiwan. In the same year, he received the certificate from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for his exceptional performance as an article reviewer during 2009–2010.

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Packages With Integrated 60-GHz Aperture-Coupled Patch Antennas Duixian Liu, Fellow, IEEE, Johannes A. G. Akkermans, Ho-Chung Chen, and Brian Floyd, Senior Member, IEEE

Abstract—This paper presents balanced-fed and fork-fed aperture-coupled patch antennas and 16-element arrays suitable for broadband millimeter-wave communications. The antennas are realized in a multi-layer organic package structure, to which RF integrated circuits can be integrated. To improve antenna bandwidth and radiation efficiency, an air cavity is used, resulting in a superstrate planar patch-antenna structure. Additionally, resonating apertures are used to further increase the antenna bandwidth. Measured results at 60 GHz for the antennas show good performance in terms of peak gain (about 8 dBi for a single element and 17 dBi for a 16-element array), bandwidth ( 10 GHz for 10-dB return loss bandwidths are achievable), and radiation efficiency (80% for single-element from simulation). Index Terms—60-GHz antennas, antenna arrays, antenna-in-package, aperture-coupled patch antennas, balanced-fed, fork-fed, high efficiency, wide bandwidth.

I. INTRODUCTION HERE is an increasing demand for low-cost wireless communication systems that operate in the 60-GHz frequency band supporting gigabit-per-second (Gbps) data rates. Example applications include wireless uncompressed high-definition video transfer and ultra-fast file transfer. The success of these types of applications very much depends upon the cost of the radio. Highly-integrated transceivers and phased-arrays have already been demonstrated at 60 GHz [1]–[3] using low-cost silicon technology. For these to be useful, though, a manufacturable and high-performance antenna and package solution is required. Preferably, the antenna structures and package approach developed should support both single-antenna and multi-antenna (e.g., beamforming) applications. This paper presents such an antenna-in-package solution. A manufacturable packaging process and material set is used and a high level of performance is achieved at 60 GHz in terms of antenna gain, efficiency, and bandwidth for both single-element antennas and 16-element antenna arrays. There have been various examples of millimeter-wave (mmWave) antennas implemented either within the package

T

Manuscript received June 08, 2010; revised December 09, 2010; accepted March 17, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. D. Liu is with the IBM T. J. Watson Research Center, Yorktown Heights, New York, NY 10598 USA. J. A. G. Akkermans was with the IBM T. J. Watson Research Center, Yorktown Heights, New York, NY 10598 USA and also with the University of Technology, Eindhoven, The Netherlands. He is now with ASML Research, The Netherlands. H.-C. Chen is with MediaTek Inc., Taiwan. B. Floyd was with the IBM T. J. Watson Research Center, Yorktown Heights, New York, NY 10598 USA. He is now with North Carolina State University, Raleigh, NC USA. Digital Object Identifier 10.1109/TAP.2011.2163760

[4], [5] or on the actual integrated circuit [6], [7]. Of these, the on-chip antennas typically achieve very low efficiencies and they also consume large amounts of on-chip area; hence, the antenna-in-package approach appears to be most promising from both a performance and cost standpoint. An early prototype antenna-in-package solution was developed by a subset of our team and presented in [4], [8]. Although this module achieved excellent performance, the assembly was complicated, expensive, and the overall approach was not easily scalable to larger arrays of antennas. As a result, we have been exploring planar antenna structures implemented in multi-layer organic substrates. From a performance standpoint, the main challenge of planar antenna design is the trade-off between radiation efficiency and bandwidth. For 60 GHz applications, a 15% fractional bandwidth is desirable to allow for world-wide coverage. There are two ways to achieve wide bandwidth, namely, to use a very low dielectric constant, preferably close to one [9], or a relatively thick dielectric layer [10]. Since a thicker dielectric layer introduces more losses due to surface-wave excitation in the dielectric, most attention has been directed toward the realization of an effective low dielectric constant. In [4], an air cavity is used radiation efficiency and beneath the antenna to achieve more than 20% bandwidth. The use of cavities has also been examined in [11], where a fractional bandwidth of 9.5% has been return loss using a modified achieved at 60 GHz for low-temperature co-fired ceramic (LTCC) technology. Alternatively, in [12], a balanced-feed patch antenna design has been proposed which does not employ cavities. Instead, resonating slots are used in a typical aperture-coupled patch antenna with , and the antenna eleTeflon-based substrate material ments are designed to cancel part of the surface-wave excitation. A radiation efficiency of more than 80% is obtained for a 10% fractional bandwidth. Although Teflon-based substrate material has low dielectric constant and loss, it is not suitable for package applications due to its high coefficient of thermal expansion. From a manufacturing standpoint, the realization of a robust, efficient, and broadband mmWave antenna within a plastic or multilayer organic (MLO) package is challenging due to assembly difficulties and material parameter tolerances. Additionally, the introduction of embedded air cavities poses difficult manufacturing challenges, particularly if mainstream printed circuit-board (PCB) processes are to be used along with high-performance RF laminates. Despite these difficulties, the performance benefits conferred by the cavity in terms of bandwidth and efficiency are significant, warranting the required process development. In this paper, we present an antenna-in-package approach which combines the benefits of both the air cavity in [4] and

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Fig. 2. Feed structures used in the antenna designs. (a) Dipole feed for BFACP, (b) fork feed for TFACP.

Fig. 1. Cross-sectional sketch of the proposed package with an integrated antenna showing various functional components.

the antenna feed-line designs in [12] and [13]. For this purpose, a novel PCB stack-up is used. We will show that this results in a superstrate antenna topology [14] that has a bandwidth of more than 15%, covering the 57–66 GHz frequency range and minimizing manufacturing variation impact on antenna performance, and a radiation efficiency of more than 80%. Moreover, a package design is proposed that allows for the integration of a phased-array transceiver chip with minimal insertion loss between the antenna feed-lines. This paper begins with a description of the millimeter-wave RFIC package with integrated antenna structures (see Section II). Next, two types of aperture-coupled patch antennas and related feed-line structures in the package environment are discussed in Section III. Simulated and measured patch array performances are given in Section IV. Section V discusses substrate material, manufacturing and measurement issues. Finally, a summary is given in Section VI. II. PACKAGE WITH INTEGRATED ANTENNA CONCEPT One key requirement for a 60-GHz antenna is that it can be embedded in a package that contains active electronics [15]. A cross-sectional view of the antenna-in-package is shown in Fig. 1. The stack consists of five substrate layers (one 2 mils and four 10 mil thick) with adhesive layers in between. The thinner and ) layer is a low dielecSub3 ( tric constant and low-loss substrate, making it suitable to support the RF feed network (on metal layer M3) with well-defined transmission lines having adequate width-to-height ratio. Below the Sub3 layer, two more low-loss layers are placed (Sub4 and and ). The additional metal Sub5 with layers that are built up in this way can be used for the layout of interconnects and for the realization of the reflector elements (on metal layer M4) that are used by the antenna array. At the lowest metal layer M5, package pads are realized that allow connection of the package to the application board. Portions of two lower layers can be removed locally to create an open cavity into which active ICs can be placed. This way, the need for 60-GHz vias and the associated matching networks has been eliminated. and Two additional layers (Sub1 and Sub2 with ) are placed on top of the Sub3 layer. This al-

lows the realization of an air cavity for the antenna, which is discussed further in Section III. To improve the feed line efficiency, a low-loss adhesive is used close to the Sub3 substrate, while relatively lossy adhesive material is used elsewhere to reduce cost and to enhance the package manufacturability. These higher loss adhesives are located further away from the radiating elements and do not affect the RF performance significantly. All adhesive layers are 1 mil thick. Typical metal thickness used in reflector, pad, and patch copper (about 18 (on M1 layer) structures is 1/2 thick), however, 1/4 copper (about 9 thick) thickness is used for the critical M2 (antenna ground plane) and M3 layers where 60-GHz structures are populated to improve the manufacturing resolution. III. SINGLE ANTENNA Two antenna structures have been selected that can be used either as stand-alone single elements themselves or as antenna elements in an array configuration. Both antennas are aperturecoupled patch antennas. This type of antenna has significant advantages compared to other types of planar antennas. The main advantage is that the feed line of the antenna is located on a different substrate than the radiating part of the antenna. This allows one to choose the feed line substrate such that the feed-network performance is optimized, while the substrate that supports the radiating patch can be chosen such that the radiation efficiency and antenna bandwidth are optimized. A. Antenna Feed Design Two kinds of antenna feed structures for the aperture-coupled patch antennas have been used. The first one [12] is shown in Fig. 2(a). We refer to this type of antenna as a balanced-feed aperture-coupled patch antenna (BFACP). This structure is primarily targeted for RFIC transceivers that require a 100-Ohm differential feed, although baluns can be used to convert to a single-ended feed line, as shown in Fig. 3(b). The second antenna feed structure is shown in Fig. 2(b). We refer to this type of antenna as a traditional-feed aperture-coupled patch antenna (TFACP). Instead of using a single microstrip (MS) feed line to couple the energy to the patch, a fork-type feed structure [13] is used. These forks are realized with 50-ohm MS, since 100-ohm MS is difficult to implement in our stack-up. As a result, a quarter-wavelength transformer is used to convert the antenna impedance from 25 to 50 . Note that the main dif-

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TABLE I TRANSMISSION LINES USED IN THIS STUDY

Table I summarizes the transmission line dimensions used for the antenna prototypes, including MS, CPW and CPS. B. Balanced-Feed Aperture-Coupled Patch Antenna Fig. 3. Feed line transitions used in the design with grounded vias. (a) Fork feed line, (b) dipole feed line.

ference between the two antennas is the feed type. Depending on the transceiver electronics, a differential or single-ended feed can be preferred. Therefore, both antenna types are discussed in this work. The probe pads (used for antenna evaluation) for all implemented antenna prototypes use three coplanar strips with ground (simply referred as CPW here). Transitions are therefore needed from CPW to the antenna feed line, which is MS in the case of TFACP and co-planar stripline with ground (simply referred as CPS here) in the case of BFACP. The transition from CPW to MS consists simply of grounded vias, as shown in Fig. 3. To ensure that the conversion is working properly, the overall length of the CPW line, including the tapered section, has to be multiples of half wavelengths in the substrate. For the MS to CPW transition (from the via to the end of the CPW section 1/2 wavelength long), the simulated insertion loss is less than 0.25 dB while the simulated return loss is better than 25 dB in the 50–70 GHz range [16]. The transition from CPW to CPS is accomplished in two steps: first, by converting from CPW to MS as described above, and second, by converting from MS to CPS using a balun. This is shown in Fig. 3(b). Such a two-step conversion is required since a direct CPW to CPS transition is difficult to implement. Note that there is a width change in the CPW line. This is due to the fact that the narrow CPW line section is completely buried in the package substrate while the wide CPW line section is not buried. The MS to CPS transition was studied in [17]. For this study, the transition (including the quarter wavelength matching section) has a simulated insertion loss about 0.5 dB in the 50–70 GHz range, but the phase difference has about 8 GHz bandwidth. An MS line to CPW line conversion can also be realized with quarter-wavelength stubs [18], [19]. The stub option was also used in both balanced and traditional-feed prototypes. The performance of the stub transition is similar to that of the via transition. The vias are avoided in the stub case, but the stub design occupies more surface area than the via-based transition option. Because of this and because the via-based transition works properly, we will concentrate on the via transitions.

The antenna shown in Fig. 2(a) is based on the balanced-feed aperture-coupled patch antenna [12] with a stack-up shown in Fig. 1. This is essentially an aperture-coupled patch antenna with two apertures (slots) and a balanced feed. Table II summarizes the critical dimensions of the BFACP antenna together with the traditional-feed aperture-coupled patch antenna (TFACP) presented next. The two slots play an important role in the antenna design. First, they are used to reduce the surface-wave excitation in the substrate. The two slots are positioned such that the surface waves generated by the slots and the patch interfere destructively and therefore the radiation efficiency of the antenna is improved. Second, the resonant slots are also used to improve the antenna bandwidth. As a result, the bandwidth is increased significantly since the antenna now has two resonant elements, i.e., the patch and the slots, each having slightly different resonance frequencies. Since the slots are designed to radiate, back radiation becomes an issue. To reduce this back radiation, a reflector element is placed behind the ground plane [20]. The antenna bandwidth can be increased further by implementing an air cavity between the ground plane and the patch. Proper selection of cavity depth, superstrate thickness and superstrate dielectric constant can enhance the radiation efficiency to more than 90% (see also [14]). Fig. 4 shows a top view of the layout with major dimensions of the designed antenna. Since the antenna feed line is buried in the substrate, a cavity is used to probe the feed line for measurements. Fig. 5 shows the manufactured antenna with FR4 test fixture (partially shown) which is used to hold the antenna within our probe-based antenna chamber. A summary of the measured performance of the BFACP antenna is shown in Table III. The antenna was measured in an anechoic chamber using a probe-based measurement setup that has a frequency limit of 65 GHz [21]. The complete antenna has been simulated in HFSS as well. The measured and simulated reflection coefficients, S11, are shown in Fig. 6. This figure shows that the antenna is well matched to 50 in a bandwidth that is adequate for 60-GHz applications. The simulated 10-dB return loss bandwidth is about 13.5 GHz. The measured 10-dB return loss bandwidth is at least 9 GHz but the measured frequency range is limited to 65 GHz. Figs. 7 and 8 show the measured and simulated antenna radiation patterns in the E and H planes, respectively, at 60 GHz

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TABLE II ANTENNAS USED IN THIS STUDY

TABLE III SINGLE-ELEMENT ANTENNA AND BROADSIDE ARRAY PERFORMANCE SUMMARY

Fig. 4. 2D view of the balanced-feed antenna with grounded vias.

Fig. 6. Measured and simulated antenna reflection coefficients of the singleelement BFACP antenna.

weak interaction with the antenna radiators. No connectors are used; thus, de-embedding is not necessary. Due to the limited dynamic range of the chamber measurement system, however, the weak cross-polarization pattern measurement is not as reliable as the co-polarization pattern measurement. C. Traditional-Feed Aperture Coupled Patch Antenna Fig. 5. Photos of the balanced-feed antenna with grounded vias.

frequency. Due to the measurement setup limitation, the radiation patterns can only be measured for a 180-degree span in elevation. The measured and simulated peak gains are 7.8 dBi and 7.9 dBi, respectively. The front-to-back ratio is simulated to be 26.9 dB compared to a design target of 18 dB. In a real application, when the package is mounted on a PCB, the back-directed radiation will be further reduced. Note that the agreement between the cross-polarized simulated and measurement patterns is weak, due to limitations in our measurement setup. This setup is targeted for antenna-inpackage applications in which the antenna radiator and the feed line are always on opposite sides of the ground plane. One benefit of this setup is that the measurement system probe has very

The patch antenna designs using a fork-type feed line (see Fig. 2(b)) provide increased tuning flexibility compared to those using single MS feed line. Once again, by letting the aperture/ slot resonate, the antenna bandwidth is increased compared to the non-resonating aperture-coupled patch antenna. Similar to the balanced-feed patch antenna discussed above, the resonating aperture will result in strong back radiation. Therefore, a reflector is required for this design as well. Fig. 9 shows the top view of the TFACP antenna. The sizes of the substrate, probe and patch cavities are the same as those of the balanced-feed patch antenna. Note that for the same antenna orientation, the TFACP antenna has different polarizations. The E-plane for the traditional-feed antenna is on the Y-Z plane, whereas for the balanced-feed antenna it is on the X-Z plane. A summary of the measured performance of the TFACP antenna is shown in Table III. Fig. 10 shows the measured

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Fig. 10. Measured and simulated antenna reflection coefficients of the singleelement TFACP antenna. Fig. 7. Measured and simulated E-plane radiation patterns of the single-element BFACP antenna, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

Fig. 11. Measured and simulated E-plane radiation patterns of the single-element TFACP antenna, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol. Fig. 8. Measured and simulated H-plane radiation patterns of the single-element BFACP antenna, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

Fig. 9. 2D view of the fork-fed antenna with grounded vias.

and simulated antenna reflection coefficients. These are in relatively good agreement, both showing two resonances in the 50–70 GHz frequency span but with some frequency shifting. Figs. 11 and 12 show the measured and simulated

Fig. 12. Measured and simulated H-plane radiation patterns of the single-element TFACP antenna, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

radiation patterns on the E and H planes, respectively, for the traditional-feed antenna at 60 GHz. Again, these are in good

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Fig. 14. 2D view of the BFACP array radiating in the broadside (0-degree) direction.

Fig. 13. A possible phased-array configuration for the 60-GHz receiver.

agreement for the main lobe. The measured and simulated peak gains are 8.4 dBi and 8.3 dBi, respectively. The front-to-back ratio is measured to be 27.0 dB at 60 GHz. Fig. 10 shows a “W-shaped” reflection coefficient curve. The double resonances are due to the interaction between the antenna input impedance and MS-to-CPW transition. The antenna itself has only one resonance (The resonant frequencies from the slot and the patch overlap). The antenna impedance locus at the CPW transition point is offset from the center of the Smith chart, while the BFACP antenna impedance locus is around the center of the Smith chart. So changing the MS line length or optimizing the antenna design can improve the TFACP antenna match.

Fig. 15. The reflection coefficient of the BFACP array radiating in the broadside direction.

IV. ANTENNA ARRAY Multiple antenna arrays have been designed and manufactured based on the single antennas presented in Section III. The antenna array consists of 16 antenna elements arranged in a circular configuration and sharing a single annular antenna cavity structure. This arrangement allows room in the center of the array to place the RF chip, as shown in Fig. 13, which allows for relatively uniform feed-line designs to each of the 16 antennas. In this particular design, the circular ring cavity for the patches has a 12-mm inner diameter and 20-mm outer diam. eter, and the overall package dimension is 28 28 1.18 The patches are almost uniformly located on a circle with a 16-mm diameter; therefore, the spacing between the patches is , or 0.628 wavelength at 60 GHz. With this patch separation, the simulated coupling between the patch antennas is at least 17 dB. For a phased-array implementation, individual feed-lines would be connected between each antenna element and the RFIC in which active electronics would provide the necessary phase and amplitude shifting. In this paper, to allow initial array prototyping, multiple fixed-beam antenna arrays have been implemented (without any active electronics), and the feed lines were designed to excite the antennas at different

scan angles—namely broadside and 30 degrees for both E- and H-plane directions. Figs. 14–17 show the 2D view, reflection coefficient, E-plane, and H-plane radiation patterns for a BFACP type array pointed in the broadside (0-degree) direction. Similarly, Figs. 19–22 show the 2D view, reflection coefficient, E-plane and H-plane radiation patterns for a TFACP type array pointed in the broadside (0-degree) direction, respectively. Key metrics for both types of broadside arrays are listed in Table III together with the single-antenna performance. The reflection coefficient of the BFACP array, shown in Fig. 15, illustrates a relatively good match to 50 . It does from 58 to 61 GHz. show a hump which is greater than Comparing the broadside BFACP fixed beam (shown in Fig. 16) to the 30 E-plane (Y-Z plane) fixed beam shown in Fig. 18, the measured peak gain changes from 15.5 dBi (16.2 simulated) to 13.4 dBi (13.5 simulated) at 60 GHz. In both cases, good correlation is seen on the main lobes between measurement and simulation. Over the same scan range, the side lobe level grows broadside to at 30 degrees. For an ideal, from uniformly-excited linear array consisting of isotropic radiators radiating in the broadside direction, the minimum side lobe

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Fig. 16. E-plane radiation patterns of the BFACP array radiating in the broadside direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

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Fig. 19. 2D view of the TFACP array radiating in the broadside (0-degree) direction.

Fig. 20. The reflection coefficient of the TFACP array radiating in the broadside direction. Fig. 17. H-plane radiation patterns of the BFACP array radiating in the broadside direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

Fig. 18. E-plane radiation patterns of the BFACP array radiating in the 30 direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

level (SLL) is for a large number of array elements for 16 elements. Our circular array achieves and which is 5 dB higher than the 16-element linear case. This difference was expected and is due to the ring configuration which does not include antenna elements in the center [23]. Although the ring configuration has relative higher side lobe levels, the configuration makes the array layout simple in the package environment and easy to design equal-phase feed lines to antenna elements. Finally, for comparison, the BFACP cross-polarization patterns are also included on Figs. 16 and 17. The reflection coefficient of the TFACP array, shown in Fig. 20, shows a good match to 50 . The main lobes of the measured and simulated TFACP radiation patterns, shown in Figs. 21 and 22, are in good agreement. Comparing the broadside TFACP fixed beam to the 30 (H-plane or Y-Z plane) fixed beam shown in Fig. 23, the measured peak gain changes from 16.6 dBi (16.8 simulated) to 15.6 dBi (16.5 simulated) at 60 GHz. Finally, the TFACP broadside array yields about side lobe level, about 1.5 dB better than that of the BFACP broadside array.

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Fig. 21. E-plane radiation patterns of the TFACP array radiating in the broadside direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

Fig. 22. H-plane radiation patterns of the TFACP array radiating in the broadside direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

Fig. 23. H-plane radiation patterns of the TFACP array radiating in the 30 direction, solid line for measured co-pol, dashed line for measured x-pol, dash-dot line for simulated co-pol, and dotted line for simulated x-pol.

V. MATERIAL, MANUFACTURING AND MEASUREMENT ISSUES In previous sections, some non-negligible differences between simulation and measurement results have been observed, particularly for the return loss of the antenna arrays. The main reasons for these discrepancies are discussed in this section. Two sources of uncertainly are the material parameters for the 1-mil thick adhesive layers used in the stack-up and the post-lamination thickness of this layer. For patch antenna designs operating below 10 GHz, researchers have concentrated on antenna structure novelty and low dielectric constant substrate materials (e.g., foam) to improve antenna performance, particularly return loss bandwidth. Adhesive (or prepreg) materials between core substrates are seldom discussed. For 60-GHz patch antenna design, the adhesive layers have to be included during the design and simulation process. It is currently difficult to reliably evaluate the dielectric properties of adhesive materials at mmWave frequencies. As a result, during the design of these antennas, dielectric properties specified at low frequencies by suppliers have been used at 60 GHz. Nominal adhesive layer thickness (1 mil) have been used in simulations, but the adhesive layer thickness after lamination can change as much as 50% depending on the presence of metal structures on the core dielectrics. Two additional sources of uncertainly are the dielectric properties and thickness of the core substrates used in the design. Both the dielectric constant and loss tangent have been measured at 60 GHz using an open resonator system. The open resonator system requires an extremely flat substrate sample. The size and a meanominal 2-mil thick sample used had a 4 4 sured thickness variation of 2 to 2.3 mil, depending on the measuring location. As a result, the measured dielectric constant has tolerance. According to the suppliers, the 2-mil thick a thickness tolerance and the 10-mil thick substrate has thickness tolerance. substrate has Uncertainty in substate thickness and dielectric properties will obviously change the antenna performance. As an example, Fig. 24 shows the antenna reflection coefficient variations when with 10 mil the Sub2 thickness changes from 9 to 11 mil ( nominal). The antenna cavity is formed by locally removing portions of Sub2 along withe the 1-mil adhesives used above and below Sub2. As a result, the antenna cavity thickness can vary from 11 to 13 mil. The simulations clearly show the shift in the antenna impedance, which is qualitatively in line with the measured results shown in Figs. 6 and 10. Manufacturing patch antennas with complicated feed lines that operate in the 60-GHz band is also very challenging. Most PCB facilities require a minimum width and spacing of 4 mil on metal structures. A high-end PCB facility can handle 3 mil spacing and line width. In this study, the spacing for the CPW line is only 2 mil. The reflection coefficient is very sensitive to antenna feed line variations and this is why we see more disagreement between measurement and simulation results for the antenna arrays than for the single-element antennas. Fortunately, the arrays studied here are for array gain, beam width and side lobe value evaluation only. The final phased array designs will have much simpler feed line layout. Table IV lists some tolerance ranges of the TFACP antenna design parameters such that the antenna will maintain 10 dB or better return loss over

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REFERENCES

Fig. 24. Cavity depth effect on antenna reflection coefficient. TABLE IV ANTENNA TOLERANCE STUDY

the 60 GHz band. To obtain these tolerance ranges, only one parameter has been varied, while all other parameters have been fixed at their nominal values. As a final note, a reliable way to handle material and manufacturing uncertainty at 60 GHz is to design the antennas with wide performance margins. VI. SUMMARY Single-element antenna and 16-element antenna array prototypes have been demonstrated in a multi-layer organic package structure. The package utilizes an antenna cavity to increase antenna bandwidth, increase efficiency, and decrease antenna-to-antenna coupling. The package also allows for an optional RFIC chip cavity to eliminate the need for lossy 60-GHz vias and compensation networks. Measured results for the antennas show good performance in terms of peak gain (8 dBi for single-element and 17 dBi for 16-element), 10-dB return loss bandwidths are bandwidth ( achievable), and radiation efficiency (80% for single-element). Also, reasonable correlation is seen between measurement and HFSS simulations. The antennas are suitable for use in either fixed-beam or steered-beam (i.e., phased-array) millimeter-wave applications and can be manufactured using printed-circuit board technology. ACKNOWLEDGMENT The authors would like to acknowledge D. Kam who evaluated transmission line loss; C. Baks who made the test fixtures; R. John who helped with antenna assembly; S. Gowda, M. Soyuer, and J. Zhang for management support.

[1] S. Reynolds, B. Floyd, U. Pfeiffer, T. Beukema, J. Grzyb, C. Haymes, B. Gaucher, and M. Soyuer, “A silicon 60-GHz receiver and transmitter chipset for broadband communications,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2820–2831, Dec. 2006. [2] E. Cohen, C. Jakobson, S. Ravid, and D. Ritter, “A bidirectional TX/RX four element phased-array at 60 GHz with RF-IF conversion block in 90 nm CMOS process,” in Proc. IEEE RFIC Symp., Jun. 2009, pp. 207–210. [3] A. Valdes-Garcia, S. Nicolson, J.-W. Lai, A. Natarajan, P.-Y. Chen, S. Reynolds, J.-H. Zhan, and B. Floyd, “A SiGe BiCMOS 16-element phased-array transmitter for 60 GHz communications,” IEEE ISSCC Dig. Tech. Papers, pp. 218–219, Feb. 2010. [4] J. Grzyb, D. Liu, U. Pfeiffer, and B. Gaucher, “Wideband cavity-backed folded dipole superstrate antenna for 60 GHz applications,” in Proc. IEEE AP-S Int. Symp., Jul. 2006, pp. 3939–3942. [5] Y. P. Zhang, M. Sun, K. M. Chua, L. L. Wai, and D. Liu, “Antenna-in-package design for wirebond interconnection to highly integrated 60-GHz radios,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2842–2852, Oct. 2009. [6] Y. P. Zhang, M. Sun, and L. H. Guo, “On-chip antennas for 60-GHz radios in silicon technology,” IEEE Trans. Electron Devices, pp. 1–5, Jul. 2005. [7] E. Öjefors, E. Sönmez, S. Chartier, P. Lindberg, A. Rydberg, and H. Schumacher, “Monolithic integration of an antenna with a 24 GHz image-rejection receiver in SiGe HBT technology,” in Proc. 35th Eur. Microwave Conf., Paris, France, Oct. 2005. [8] U. Pfeiffer, J. Grzyb, D. Liu, B. Gaucher, T. Beukema, B. Floyd, and S. Reynolds, “A chip-scale packaging technology for 60-GHz wireless chipsets,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3387–3397, Aug. 2006. [9] M. M. Faiz and P. F. Wahid, “A high efficiency L-band microstrip antenna,” in Proc. IEEE Int. URSI Conf., Orlando, FL, Jul. 1999, pp. 272–275. [10] N. Alexópoulos, P. Katehi, and D. Rutledge, “Substrate optimization for integrated circuit antennas,” IEEE Trans. Microw. Theory Tech., vol. 83, pp. 550–557, Jul. 1983. [11] A. E. I. Lamminen, J. Säily, and A. R. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrates,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2865–2874, Sep. 2008. [12] J. Akkermans, M. van Beurden, and M. Herben, “Design of a millimeter-wave balanced-fed aperture-coupled patch antenna,” in Proc. EuCAP 2006, ESA SP626, Nice, France, Nov. 2006, pp. 1–6. [13] S. Targonski, R. Waterhouse, and D. Pozar, “Design of wide-band aperture-stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, pp. 1245–1251, Sep. 1998. [14] N. Alexópoulos and D. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 32, no. 8, pp. 807–816, Aug. 1984. [15] D. Liu and B. Floyd, “Radio Frequency (RF) Integrated Circuit (IC) Packages With Integrated Aperture Coupled Patch Antenna(s),” U.S. patent number 7692590, Apr. 6, 2010. [16] D. Liu and B. Floyd, “Microstrip to CPW transitions for package applications,” in Proc. IEEE Int. Symp. on Antennas and Propagation and CNC-USNC/URSI Radio Science Meeting, Toronto, Canada, Jul. 11–17, 2010. [17] Y. Qian and T. Itoh, “A broadband uniplanar microstrip-to-CPS transition,” in 1997 Asia Pacific Microwave Conf. Dig., Dec. 1997, pp. 609–612. [18] J.-P. Raskin, G. Gauthier, L. P. Katehi, and G. M. Rebeiz, “W-band singlelayer vertical transitions,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 161–164, Jan. 2000. [19] J. A. G. Akkermans, R. van Dijk, and M. H. A. J. Herben, “Millimeterwave antenna measurement,” in Proc. EuMC Eur. Microwave Conf., Oct. 2007, pp. 83–86. [20] S. Targonski and R. Waterhouse, “Reflector elements for aperture and aperture-coupled microstrip antennas,” in Proc. Antennas Propag. Soc. Int. Symp., Jul. 1997, vol. 3, pp. 1840–1843. [21] T. Zwick, C. Baks, U. Pfeiffer, D. Liu, and B. Gaucher, “Probe based MMW antenna measurement setup,” in Proc. AP-S Int. Symp., Jun. 2004, pp. 747–750. [22] J. Akkermans, B. Floyd, and D. Liu, “Radio frequency (RF) integrated circuit (IC) Packages with integrated aperture-coupled patch antenna(s) in ring and/or offset cavities,” US patent number 7696930, Apr. 13, 2010. [23] J. D. Kraus and R. J. Marhefka, Antennas for all Applications. New York: McGraw Hill, 2002.

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Duixian Liu (S’85–M’90–SM’98–F’09) received the B.S. degree in electrical engineering from XiDian University, Xi’an, China, in 1982, and the M.S. and Ph.D. degrees in electrical engineering from the Ohio State University, Columbus, in 1986 and 1990, respectively. From 1990 to 1996, he was with Valor Enterprises Inc. Piqua, Ohio, initially as an Electrical Engineer and then as the Chief Engineer, during which time he designed an antenna product line ranging from 3 MHz to 2.4 GHz for the company, a very important factor for the prestigious Presidential “E” Award for Excellence in Exporting in 1994. Since April 1996, he has been with the IBM T. J. Watson Research Center, Yorktown Heights, NY, as a Research Staff Member. He was named IBM Master Inventor in 2007. He has edited a book titled Advanced Millimeter-wave Technologies—Antennas, Packaging and Circuits (2009, Wiley). He has authored or coauthored approximately 80 journal and conference papers. He has 36 patents issued and 19 patents pending. His research interests are antenna design, EM modeling, chip packaging, digital signal processing, and communications technologies. Dr. Liu has received three IBM’s Outstanding Technical Achievement Awards and one Corporate Award, the IBM’s highest technical award. He is an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and a Guest Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Special Issue on Antennas and Propagation Aspects of 60–90 GHz Wireless Communications (October 2009). He has been an Organizer or Chair for numerous international conference sessions or special sessions and served as a technical program committee member for many international conferences. He was the General Chair of the 2006 IEEE International Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, White Plains, New York. He has served as an external Ph.D. examiner for several universities and external examiner for some government organizations on research grants.

Johannes A. G. Akkermans received the M.Sc. degree in electrical engineering from Eindhoven University of Technology (The Netherlands) in 2004. The research for the final project of his M.Sc. thesis was performed within TNO Science and Industry and was concerned with rectenna design for wireless low-power transmission. In March 2009 he finalized his Ph.D. degree at Eindhoven University of Technology. His Ph.D. research involved the development of planar beam-forming antenna arrays for 60-GHz broadband communication. From September 2005 to May 2006, he did his research on millimeter-wave antennas at TNO Defence, Security and Safety; and from January 2008 to May 2008, he contributed to the development of 60-GHz beam-forming antenna arrays at IBM T. J. Watson Research Center. Results of his M.Sc and Ph.D. research have been published in international journals and conferences. Moreover, he contributed to several patent applications. Currently, he is employed at ASML (The Netherlands) within the Research department where he investigates high-speed wireless communication.

Ho-Chung Chen received the B.S. and M.S. degree in communication engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 2001 and 2003, respectively. From 2003 to 2007, he joined Atheros, where he was responsible for WiFi antennas. Since 2008, he transferred to Mediatek and expatriated to IBM T. J. Watson Research Center to joint the 60 GHz band antenna and package development. In July 2009, he returned to Mediatek headquarter, Hsinchu, Taiwan to continue the 60 GHz band antenna and package research and building mmWave laboratory.

Brian Floyd (SM’11) received the B.S. degree (with highest honors), M. Eng., and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 1996, 1998, and 2001, respectively. While at the University of Florida, he held the Intersil/Semiconductor Research Corporation Graduate Fellowship and the Pittman Fellowship, working on CMOS RFIC design for on-chip wireless clock distribution. During the summers of 1994–1996, he worked with the Motorola Paging Products Group, Boynton Beach, FL, working in the areas of RF product development and IC design. In 2001, he joined the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, where he worked as a research staff member and then later as the manager of the wireless circuits and systems group. His work at IBM included the development of silicon-based millimeter-wave receivers, transmitters, frequency synthesizers, phased-arrays and imagers; millimeter-wave antennas and packages; and WCDMA receivers. In 2010, he joined the Department of Electrical and Computer Engineering at North Carolina State University as an Associate Professor. He has authored or coauthored over 60 technical papers and has 12 issued patents. Dr. Floyd serves on the Technical Program Committee for the International Solid-State Circuits Conference and both the steering and technical program committees for the RFIC Symposium. From 2006–2009, he served on the technical advisory board to the Semiconductor Research Corporation integrated circuits and systems science area. He was a winner of the 2000 SRC Copper Design Challenge; a recipient of the 2006 Pat Goldberg Memorial Award for the best paper in computer science, electrical engineering, and mathematics within IBM Research; and a two-time recipient of the IEEE Lewis Winner Award for the best paper at the International Solid-State Circuits Conference in 2004 and 2006.

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3D-Antenna-in-Package Solution for Microwave Wireless Sensor Network Nodes Amin Enayati, Student Member, IEEE, Steven Brebels, Walter De Raedt, and Guy A. E. Vandenbosch, Senior Member, IEEE

Abstract—A three-dimensional packaging solution is introduced for wireless sensor network nodes at microwave frequencies. The package has a cubic geometry with radiating antennas on its surrounding faces. The cube which is called e-CUBE is designed in such a way that its radiation pattern resembles that of a simple dipole. The antenna is designed in a modular way and the final structure has a return loss better than 10 dB and a dipole-shape pattern at 17.2 GHz. Moreover, the measurement and simulation results show very good agreement. Index Terms—Dipole antenna, RF and microwave packaging, three-dimensional packaging, wireless sensor networks.

I. INTRODUCTION HE recent advances in very large scale integration (VLSI) technology, micro-electromechanical systems (MEMS) and wireless communications have driven many researchers towards innovative solutions for wireless sensor networks (WSNs) [1]–[3]. WSNs are composed of different sensor nodes to feel the physical characteristics of the surrounding environment. By controlling the flow of information in this network in a smart manner a smart sensor network implementing the concept of ambient intelligence [4], [5] can be achieved. One of the candidates for the nodes of a smart WSN is the so-called electronic cubes (e-CUBEs) topology [6] with envisaged applications such as: 1- distributed smart monitoring for Aeronautics and Space applications; 2- WSNs for health and fitness; 3- distributed intelligence for Automotive Control. The nature of the applications requires that the e-CUBEs are miniaturized and cost-effective. Among technologies that provide high integration capabilities, 3-dimentional systemIn-Package (3D-SiP) technologies play a central role [7], [8], especially when antennas are to be implemented in package.

T

Manuscript received November 08, 2010; revised January 24, 2011; accepted March 07, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. A. Enayati is with both the Inter-university Microelectronic Center (IMEC), and the Electrical Engineering Department (ESAT), Katholieke Universiteit Leuven, Leuven, Belgium (e-mail: [email protected], [email protected]). S. Brebels and W. Deraedt are with the Inter-university Microelectronic Center (IMEC), Leuven, Belgium ((e-mail: [email protected]; [email protected]). G. A. E. Vandenbosch is with the Electrical Engineering Department (ESAT), Katholieke Universiteit Leuven, Leuven, Belgium (e-mail: [email protected]@nrim.go.jp). 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.2011.2163779

Fig. 1. Schematic description of the topology.

From an architectural viewpoint, there are several 2and 3-dimensional technologies available to implement antenna-in-package solutions for various applications [9]–[17]. In this paper a conformal antenna array that covers a cube is designed, manufactured, and measured. This array is a first-step design to investigate the applicability of the Flex-rigid PCB technology for e-CUBES nodes. The next step would be to implement an antenna array with beam steering capability based on an RF-MEMS technology. The systematic goal of the design is to have a 3D topology which has a dipole-like pattern while including all the electronics inside. This means that each node will have an omni-directional pattern in the azimuth plane, which makes it capable of communicating with the surrounding nodes with low sensitivity to the architecture of the network, i.e., the placement of the wireless nodes. In the literature, there are three main PCB technologies used to manufacture 3D-like topologies: 1- Rigid PCB Technology [18], [19]; 2- Flex PCB Technology [20], [21]; 3- Flex-Rigid PCB Technology [22]–[24]. The topology chosen introduces a 3D packaging solution which works at the same time as a conformal antenna array. The packaging structure forms an impenetrable cube on the faces of which the antennas of the transceiver system are positioned. It is designed in such a way that all the electronics needed for the power generation/distribution and data processing can be placed within the volume of the cube. A schematic overview is given in Fig. 1. The antenna array should transmit and receive RF signals at 17.2 GHz. It should have a form factor of less than 1 cm in each of the 3 spatial dimensions.

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Fig. 2. The layer build-up for the flex-rigid PCB chosen; (a) 2D layer build-up including islands and metalized vias. (b) Materials used for different layers and their thicknesses. (c) The metallic cube containing all the electronics needed for the functionality of the e-CUBE. (d) A cross-sectional cut of the finalized eCUBE.

The layer build-up for the multi-layer flex-rigid PCB technology is explained in the next section. Then, the antenna elements and the array architecture are brought to view. Finally, a comparison of the measurement and simulation results concludes the paper. II. LAYER BUILD-UP FOR THE FLEX-RIGID PCB The layer build-up chosen to implement the 3D antenna array is shown in Fig. 2. It contains a ceramic-based microwave laminate (RO4003C) with two different thicknesses as core material. This material has a relative permittivity of 3.38 and a loss tangent of 0.0027. Sandwiched between these two layers, there is a polyimide flexible laminate with a relative permittivity of 3.3 and loss tangent of 0.005. Moreover, a no-flow Polyimide Pre-preg (Arlon 38N) was used to fabricate the multilayer PCB. This material has a relative permittivity of 4.8 and loss tangent of 0.01. To prevent probable damage at the locations where PCB is not covered with the rigid laminates, i.e., at the corners of the cube, the flexible material is covered with a compatible material. As shown in Fig. 2, the folded PCB that carries the antenna elements on four of its five sides covers a metallic box. This metallic box acts as an impenetrable shield needed for the electronics in an autonomous wireless node. Hence, the electronics inside the cube are electromagnetically de-coupled from the fields radiated by the antennas. III. ANTENNA ELEMENT DESIGN The antenna element chosen is a dipole antenna. Fig. 3 shows the proposed structure, the definition of the design variables, and their optimized values. The RF signal is coupled through a microstrip line to the dipole antenna which is placed on layer L1. This microstrip line

Fig. 3. The chosen dipole antenna fed by a via-less balun as the coupling structure. (a) 3D view. (b) Optimized values of the design variables. (c) Top view of the layout and definition of design variables.

has its signal line on layer L2 and its ground line on layer L3. This fully metal layer L3 also acts as a back plane reflecting the signal radiated backwards from the dipole towards the front side. Fig. 4 shows the simulated (in HFSS) return loss and radiation pattern for the optimized dipole antenna. The simulated radiation gain of the dipole is around 4 dBi. This value is larger than for a simple dipole antenna because of the ground plane placed at its back side. It prevents the radiated field from penetrating into the negative z half space and directs it towards the positive z half space. As shown in Fig. 5 the asymmetry in the current distribution, a consequence of the asymmetric feeding topology, leads to an asymmetry (although not very dominant) in the patterns shown in Fig. 4(b). This figure shows total radiation patterns because the cross-polar level in all directions is at least 10 dB smaller than the co-polar level. In order to study the polarization purity of the antenna, the cross and co-polar radiation patterns in yz plane are plotted in Fig. 4(c). The co-polar and cross-polar levels in the broadside direction differ less than 20 dB. This ratio is improved by the arraying technique described in the next section. IV. MODULAR DESIGN OF ANTENNA ARRAY As stated previously, the antenna should be designed in such a manner that it yields an omni-directional pattern in the azimuth plane (xy plane in Fig. 1). If instead of a metallic cube, a metallic cylinder was used, the optimum locations for the dipoles to minimize the ripple and the grating lobes in the total radiation pattern would be the ones shown in Fig. 6(a). However, in the presence of the metallic cube, the optimum locations are slightly

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Fig. 4. Simulation results of the optimized dipole; (a) Return loss. (b) Total radiation patterns in xy and yz planes at 17.2 GHz. (c) Co- and cross-polarization radiation patterns in yz plane at 17.2 GHz.

Fig. 6. (a) The optimum location for the optimized dipoles to yield minimum riple and grating lobe in the radiation pattern. (b) 3D view of each face of the e-CUBE. (c) Radiation pattern of the total radiated power in the xy and yz planes. (d) co- and cross-polarization patterns in the yz plane. (e) Magnitude of the surface current in layer L2. (f) Magnitude of the surface current in layer L1.

Fig. 5. Current distribution on different layers of the single dipole antenna. (a) Layer L2. (b) Layer L1.

different. Hence two antenna elements should be placed on each side of the cube as shown in Fig. 6(b). There are only two design variables to be optimized in this array. The first design variable is L1feed shown in Fig. 3(c) and the second one is W4feed shown in Fig. 6(b). The optimized values for these variables are 2.25 mm and 0.5 mm respectively. By designing the 1 2 array in this way, the antenna possesses full symmetry with respect to the yz plane. Hence, the total radiation pattern, shown in Fig. 6(c) has a symmetric shape in the xz plane. The maximum gain of the 1 2 array is 4.7 dB in the broadside direction which is due to the short distance between adjacent elements and the rotated angle of the main beam in the yz plane. It should be emphasized that by increasing the distance between adjacent elements on each face the gain of each 1 2 array could have been higher. However,

this would result in an increase in the grating-lobe and ripple levels. The ratio of co-polar to cross-polar patterns in yz plane, shown in Fig. 6(d), is now better than 40 dB in the broadside direction thanks to the array configuration. Moreover, the pattern of the 1 2 array is rotated in the yz plane due to the radiation from its feeding network. This does not have a destructive effect because it still allows the radiation pattern in the azimuth plane of the final cube (xy plane in Fig. 1) be omni-directional. Moreover, the rotation is in such a direction ( in Figs. 1 and 6(b)) that reduces the probable interference between the radiating power form the e-CUBE and its connector or the ground plane on which it will be installed. In order to feed 4 similar 1 2 arrays on each of the surrounding faces of the cube, a coax-to-microstrip power splitter is designed, see Fig. 7. The power is guided by the coaxial transmission line in the vertical direction. The coaxial line is composed of two co-centered cylindrical conductors with radii r1 and r2. The outer conductor is fabricated by drilling a cylindrical hole in a metallic cube and the central conductor is the central conductor of a commercial SSMA connector.

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2

Fig. 7. 1 4 Coaxial-to-microstrip splitter; (a) Schematic view. (b) Optimized values of the design variables. (c) Return and insertion losses.

There are 3 main design variables in the splitter structure to be optimized: wu1, wu2, and Lu1. The optimized values are shown in Fig. 7(b). Fig. 7(c) shows the return loss at the coaxial port (s11) and the magnitude of the insertion losses from the coaxial port to the four microstrip lines fabricated in PCB (s21, s31, s41, and s51). The latter parameters have a mismatch of less than 0.05 dB, which is due to simulation uncertainties. Moreover, the insertion losses to the microstrip ports are very close to 6 dB which indicates that the radiation loss of the splitter is considerably low. After being split, the power should feed the 1 2 antenna arrays on each of the four surrounding faces. To implement this, a transition should be designed between the top and the surrounding faces. As mentioned earlier, a flex-rigid PCB technology was chosen to implement the PCB part. Hence, as shown in Fig. 8, a rounded shape in the flex part of the PCB was used to model the finalized structure at the corners. The matching is done using two pieces of microstrip line in series. Note that the signal lines of these microstrip lines are implemented in layer L2 while the ground plane is realized by a face of the metallic box. The dielectric part (RO4003C 20 mil) is removed from underneath the curved part of the PCB. Hence, the dielectric of the curved microstrip line is mainly air. In order to compensate for the lower dielectric constant and the higher distance between the signal line and the ground of the microstrips at the corners, the signal line width (wts1) should be greater than the width in the rest of the matching circuit (wts2). Note that the sum of rts) and the effective electrical lengths of the curved part ( the vertical microstrip line (Lts), shown in Fig. 8(a), is approximately a quarter of the guided wavelength at 17.2 GHz.

Fig. 8. Microstrip-to-microstrip transition from the top face of the cube to one of the surrounding faces; (a) Schematic view and definition of design variables. (b) Optimized values of the design Variables. (c) Return and insertion losses.

Fig. 8(b) shows the optimized values for the design variables obtained with HFSS. As shown in Fig. 8(c), the return loss is more than 12 dB for both ports while the total insertion loss is around 0.55 dB for the entire band. More than two thirds of this insertion loss is due to dielectric and conductor losses while the rest is because of radiation loss. Although the transition has a small curvature radius the radiation insertion loss is less than 0.2 dB, which is very promising. V. OPTIMIZED ECUBE BY FINE TUNING OF THE 3D MODEL Since the design in the previous steps was done in a modular manner, and different modules were considered as multi-port linear networks, the electromagnetic coupling between different modules has not been taken into account yet. Hence, the final 3D structure should be tuned to compensate for the effect of these couplings. As a full 3D model needs a huge amount of memory and processing power, during the tuning phase, symmetry planes of perfect-magnetic-conductor nature, shown in Fig. 9(a), were used to reduce the size of the problem. After final tuning the complete 3D structure fed by a realistic model for the modified SSMA connector was simulated in HFSS (Fig. 9(b)). A very good agreement between the simulation results of onequarter of the structure and the full 3D model closed the design procedure. Note that the optimum distance between the adjacent dipoles on each of the faces was found to be 4 mm which is very close to the optimum hypothetic value shown in Fig. 6(a).

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Fig. 9. The final design for the eCUBE antenna; (a) One-fourth of the structure with two symmetry planes used for final tuning.(b) 3D model of antenna and SSMA connector.

VI. SIMULATION VERSUS MEASUREMENT RESULTS Fig. 10(a) shows the simulation versus measurements results for the return loss. The measured return loss has a minimum at 17.3 GHz, which shows a shift of 0.1 GHz with respect to the simulated return loss. This may be because of manufacturing inaccuracies, imperfections in the assembly procedure and the influence of the SSMA-to-SMA adapter. However, the return loss is less than 10 dB at 17.2 GHz, with an acceptable bandwidth. Moreover, the radiation patterns were measured in three different planes. The results for the xy plane are shown in Fig. 10(b). The measured and simulated co-polar patterns are in good agreement in the xy plane. Moreover, the pattern is omni-directional, with a ripple of less than 0.9 dB for the measured pattern. However, the measured cross-polar pattern has an average amplitude higher than the simulated one and degrades at angles around and degrees. Due to the small dimensions of the antenna under test, the rotating holders commercially available for the antenna measurement systems are unsuitable. These holders have a major part made of metal which greatly interacts with the radiation pattern. Hence a custom-designed antenna holder made of wood was used to perform pattern measurements. However, this holder prevented the measurements to be performed in all azimuth angles. That is why the measured patterns are shown from to degrees. Fig. 10(c) shows the radiation patterns in the yz plane. For the co-polar field, the measured pattern shows good agreement with the simulated one. There is a mismatch of about 1–2 dB between these two patterns at angles around 90 degrees. This mismatch is due to the presence of the connector, adapter, and also the cables used to excite the antenna. Note that in this plane the level of the cross polarization is very low, i.e., more than 45 dB below the peak gain for the measured pattern.

Fig. 10. Simulation versus measurement results for the finalized antenna; (a) Return loss. (b) Radiation patterns in the xy plane; Theta = 90. (c) Radiation pattern in the yz plane; Phi = 90. (d) Radiation pattern in the plane of Phi = 45.

The radiation patterns in a plane rotated by 45 degrees relative to the xz plane are shown in Fig. 10(d). The measured co-polar radiation pattern is again in good agreement with the

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Fig. 11. Peak radiation gain at 42 degrees in the yz plane.

simulated one, except for angles close to 90 degrees. However, the cross-polar level is only 17 dB lower than the co-polar one in all directions. This is worse than in the zy and xy planes. Finally, the simulation and measurement results of the peak gain are shown in Fig. 11. Note that the peak gain is located at 42 in the yz plane. The difference between the simulation and measurement results is less than 0.5 dB, which seems promising for such a small antenna. VII. CONCLUSION It is shown that with a careful design procedure, it is possible to implement a 3D antenna-in-package solution for microwave wireless sensor networks. As the package contains all the electronics inside, these electronic parts are prohibited from interfering with the radiated power. Due to its dipole-shape radiation pattern, this antenna-in-package can be used in applications where the sensor nodes need to communicate with all the surrounding ones in an omni-directional way. It is also possible to implement steerable patterns by using RF-MEMS phase shifters or switches integrated on the same flex-rigid platform. ACKNOWLEDGMENT The authors would like to thank M. Libois for his help with the assembly process and T. Webers for his help with preparation of the final layouts. REFERENCES [1] F. L. Lewis, “Wireless sensor networks,” in Smart Environments: Technologies, Protocols, and Applications, D. J. Cook and S. K. Das, Eds. New York: John Wiley, 2004. [2] Y. Zhang, Y. Gu, V. Vlatkovic, and X. Wang, “Progress of smart sensor and smart sensor networks,” in Proc. 5th World Congress on Intelligent Control and Automation (WCICA 2004), Jun. 2004, vol. 4, pp. 3600–3606. [3] C. Townsend and S. Arms, “Wireless sensor networks: Principles and applications,” in Sensor Technology Handbook, J. S. Wilson, Ed. Oxford, U.K.: Elsevier Inc., 2005, ch. 22. [4] H. Nakashima, H. Aghajan, and J. C. Augusto, Handbook of Ambient Intelligence and Smart Environments. New York: Springer, 2010, pp. 3–30. [5] C. Ramos, J. C. Augusto, J. Carlos, and D. Shapiro, “Ambient intelligence: The next step for artificial intelligence,” IEEE Intell. Syst., vol. 23, pp. 15–18, Mar. 2008.

[6] P. Ramm, A. Klumpp, J. Weber, M. Taklo, N. Lietaer, W. D. Raedt, T. Fritzsch, T. Hill, P. Couderc, C. Val, A. Mathewson, K. Razzeb, and F. Stam, “The European 3D technology platform (e-CUBES),” Future Fab. Intern. vol. 34, pp. 103–116, July 2010 [Online]. Available: http:// www.future-fab.com [7] W. Chen, W. R. Bottoms, K. Pressel, and J. Wolf, “The next step in assembly and packaging: System level integration in package (SiP),” International Tech. Roadmap for Semicon. 2008 [Online]. Available: http://http://www.itrs.net [8] P. Ramm, A. Klumpp, R. Merkel, J. Weber, R. Wieland, A. Ostmann, and J. Wolf, “3D system integration technologies,” in Proc. Mat. Res. Soc. Symp., 2003, vol. 766, pp. 1–12. [9] X. Jiang and H. Shi, “Effective die-package-PCB co-design methodology and its deployment in 10 Gbps serial link transceiver FPGA packages,” in Microwave Symp. Digest., Jun. 2009, pp. 793–796. [10] S. Stoukatch, C. Winters, E. Beyne, W. De Raedt, and C. Van Hoof, “3D-SIP integration for autonomous sensor nodes,” in Proc. Electron. Compon. Tech. Conf., Jul. 2006, pp. 404–408. [11] J. H. Lee, G. Dejean, S. Sarkar, S. Pinel, L. Kyutae, J. Papapolymerou, J. Laskar, and M. M. Tentzeris, “Highly integrated millimeter-wave passive components using 3-D LTCC system-on-package (SOP) technology,” IEEE Trans. Mirow. Theroy Thech., vol. 53, no. 6, pp. 2220–2229, Jun. 2005. [12] T. Brbier, F. Mazel, B. Reig, and P. Monfraix, “A 3D wideband package solution using MCM-D technology for tile TR module,” in proc. EGAAS Symp., Oct. 2005, pp. 549–552. [13] A. Enayati, S. Brebels, W. Deraedt, and G. A. E. Vandenbosch, “Vertical vial-less transition in MCM technology for millimeter-wave applications,” Electron. Lett., vol. 46, no. 4, pp. 287–288, Feb. 2010. [14] P. Ramm and A. Klumpp, “Through-silicon via technologies for extreme miniaturized 3D integrated wireless sensor networks (e-CUBES),” in Proc. Intercon. Tech. Conf., Jun. 2008, pp. 7–9. [15] N. Khan, V. S. Rao, S. Lim, H. S. We, V. Lee, X. Zhang, E. B. Liao, R. Nagarajan, T. C. Chai, V. Kripesh, and J. H. Lau, “Development of 3D silicon module with TSV for system in packaging,” IEEE Trans. Compon. Packag. Thechnol., vol. 33, no. 1, pp. 3–9, Mar. 2010. [16] E. C. W. D. Jong, L. A. Ferreira, and P. Bauer, “3D integration with PCB technology,” in Proc. APEC, Mar. 2006, pp. 857–863. [17] S. Yu, C. Chen, C. Wei, C. Tsia, U. Jow, C. Shyu, S. Lay, and M. Lee, “Embedded wide-band filter on rigid/flex multilayer PCB for SiP,” in Proc. IMPAT, Oct. 2006, pp. 1–4. [18] C. Chiu, J. B. Jang, and R. D. Murch, “24-port and 36-port antenna cubes suitable for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1170–1176, Apr. 2008. [19] B. Bonnet, P. Monfraix, R. Chiniard, J. Chaplain, C. Drevon, H. Legay, P. Couderc, and J. L. Cazaux, “3D packaging technology for integrated antenna front-ends,” in Proc. Eur. Microwav. Technol. Conf., Oct. 2008, pp. 1–4. [20] S. Y. Y. Leung, P. K. Tiu, and D. C. C. Lam, “Printed polymer-based RFID antenna on curvilinear surfaces,” in Electron. Material. Packag. Conf., Dec. 2006, pp. 1–6. [21] J. Jung, H. Lee, and Y. Lim, “Broadband flexible comb-shape monopole antenna,” IET Microwav. Antennas Propag., vol. 3, no. 2, pp. 325–332, Mar. 2009. [22] R. N. Das, F. D. Egitto, B. Wilson, M. D. Poliks, and V. R. Markovich, “Development of rigid-flex and multilayer flex for electronic packaging,” in Proc. Electron. Compon. Tech. Conf., Dec. 2010, pp. 987–994. [23] M. Karlsson and S. Gong, “Circular dipole antenna for mode 1 UWB radio with integrated balun utilizing a flex-rigid structure,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2967–2971, Oct. 2009. [24] N. Altunyurt, R. Rieske, M. Swaminathan, and V. Sundaram, “Conformal antennas on liquid crystalline polymer based rigid-flex substrates integrated with the front-end module,” IEEE Trans. Advanced Packag., vol. 32, no. 4, pp. 797–808, Nov. 2009.

Amin Enayati (S’08) was born in Kerman, Iran, on January 20, 1979. He received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 2001 and the M.Sc. degree in electrical engineering from K.N.T. University of Technology, Tehran, 2004. Since 2008, he has been working toward the Ph.D. degree at the Katholieke Universiteit Leuven, Leuven, Belgium. Since then, he has been with the RF Component Design and Modeling (RFCDM) Group at the Interuniversity Microelectronics Center (IMEC), Leuven, Belgium. His main research interests include antenna and packaging solutions for microwave and millimeter wave frequency ranges as well as imaging and spectroscopy at sub-millimeter wave and terahertz wavelengths.

ENAYATI et al.: 3D-ANTENNA-IN-PACKAGE SOLUTION FOR MICROWAVE WIRELESS SENSOR NETWORK NODES

Steven Brebels received the M.S. degree in electrical engineering from the University of Leuven, Belgium and is currently working toward the Ph.D. degree at the Interuniversity Microelectronics Center (IMEC), Leuven, Belgium. His research within the RF Component Design and Modeling Group, IMEC, is directed to integrated circuits and antennas in thin-film and 3D stacked modules. His research interests include microwave and millimeter-wave components and integrated antennas. Mr. Brebels is co-recipient of the IEEE 2003 Microwave Prize.

Walter De Raedt received the M.Sc. degree in electrical engineering from the Katholieke Universiteit Leuven, Leuven, Belgium, in 1981. He subsequently joined the Electronics, Systems, Automation and Technology (ESAT) Laboratory, where he was a Research Assistant involved directly with electron beam technology. Since 1984, he has been with the Interuniversity Microelectronics Centre (IMEC), Leuven, where he has been involved with research on MICs and sub micrometer technologies for advanced high electron mobility transistor (HEMT) devices. Since 1997, he has been with the RF Component Design and Modeling (RFCDM) group, IMEC, where he coordinates projects on integrated passives and interconnections for RF front-end systems.

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Guy A. E. Vandenbosch (M’92–SM’08) was born in Sint-Niklaas, Belgium, on May 4, 1962. He received the M.S. and Ph.D. degrees in electrical engineering from the Katholieke Universiteit Leuven, Leuven, Belgium, in 1985 and 1991, respectively. He holds a certificate of the postacademic course in electro-magnetic compatibility at the Technical University Eindhoven, The Netherlands. He was a research and teaching assistant from 1985 to 1991 with the Telecommunications and Microwaves section of the Katholieke Universiteit Leuven, where he worked on the modeling of microstrip antennas with the integral equation technique. From 1991 to 1993, he held a postdoctoral research position at the Katholieke Universiteit Leuven. Since 1993, he has been a Lecturer, and since 2005, a Full Professor at the same university. He has taught, or teaches, courses on “Electrical Engineering, Electronics, and Electrical Energy,” “Wireless and Mobile Communications, part Antennas”, “Digital Steer- and Measuring Techniques in Physics,” and “Electromagnetic Compatibility.” His research interests are in the area of electromagnetic theory, computational electromagnetics, planar antennas and circuits, electromagnetic radiation, electromagnetic compatibility, and bio-electromagnetics. His work has been published in ca. 120 papers in international journals and has been presented in ca. 200 papers at international conferences. Prof. Vandenbosch has convened and chaired numerous sessions at many conferences. He was Co-Chairman of the European Microwave Week 2004 in Amsterdam, where he Chaired the TPC of the European Microwave Conference. He was a member of the TPC of the European Microwave Conference in 2005, 2006, 2007, and 2008. He has been a member of the Management Committees of the consecutive European COST actions on antennas since 1993, where he is leading the working group on modeling and software for antennas. Within the ACE Network of Excellence of the EU (2004–2007), he was a member of the Executive Board and coordinated the activity on the creation of a European antenna software platform. Since 2001 he has been President of SITEL, the Belgian Society of Engineers in Telecommunication and Electronics. Since 2008, he is a member of the board of FITCE Belgium, the Belgian branch of the Federation of Telecommunications Engineers of the European Union. In the period 1999–2004, he was Vice-Chairman, and in the period 2005–2009 Secretary of the IEEE Benelux Chapter on Antennas en Propagation. Currently he holds the position of Chairman of this Chapter. In the period 2002–2004 he was secretary of the IEEE Benelux Chapter on EMC.

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High-Gain Silicon On-Chip Antenna With Artificial Dielectric Layer Kazuhiro Takahagi and Eiichi Sano, Member, IEEE

Abstract—On-chip antennas are demanded to further lower the cost of wireless CMOS ICs. The low resistivity of silicon substrates is a major obstacle to fabricate high-gain on-chip antennas. We placed an artificial dielectric layer (ADL) between an antenna and Si substrate to improve the antenna gain. A half-wave dipole-antenna that has ADL was designed and fabricated using a CMOS-compatible process with one poly-Si and two metal layers. Using the ADL enhanced gain by 3-dB. The measured gain was the highest ever achieved for the antennas operating at around 10 GHz on low-resistivity Si substrates. A method for further improvement is discussed Index Terms—Artificial dielectrics, antenna gain, dipole antennas.

I. INTRODUCTION ECENTLY, there has been significant interest in researching and developing wireless communication circuits for 3.1–10.6 GHz ultra-wideband, 24-GHz band, and 60-GHz band applications [1]. Modern CMOS technologies enable high-level integration of RF, physical layer (PHY), and medium access control (MAC) layer processors, thus reducing the size and cost of wireless equipment. In recent years, several extensive attempts have been made to fabricate antennas on Si substrates for inter-chip or intra-chip transmissions [2]–[5]. However, a high-resistivity float zone (FZ) and proton-bombarded Si substrate of more than 5 k cm have been used to fabricate high-gain antennas [4], [5]. This is because the low-resistivity Si substrates commonly used in CMOS processes degrade the antenna gain. When standard 10- cm substrates are used, transmission gains range from to dB for transmissions over a few centimeters [4], [5]. Thus, an ingenious development is needed to increase the gains of antennas fabricated on low-resistivity Si substrates. A metal reflector and a high-impedance-surface reflector have been reported to improve antenna gain. When a metal reflector is inserted between an antenna and substrate, the radiated wave

R

and reflected wave should be in-phase. This means that the distance between the antenna and reflector should be a quarter of the wavelength. However, the total thickness of silicon oxide layers for typical CMOS LSIs is less than 10 m, while a quarter wavelength is about 1 cm at 10 GHz. Therefore, no metal reflector can be made with the CMOS fabrication process. As for the high-impedance-surface (HIS) reflector, its cell should be much smaller than the wavelength [6]. Calculated resonance frequency, at which the surface operates as a HIS reflector, is higher than 200 GHz for the cell smaller than 1 mm when a 10- m SiO layer is used. Therefore, it is difficult to fabricate a HIS reflector operating at around 10 GHz using CMOS fabrication process. An artificial dielectric layer (ADL) produces an extremely high permittivity [7], [8]. An ADL has been developed to shield coplanar waveguides (CPWs) from Si substrate [9], [10]. The purpose of this study is to enhance the gain of a Si on-chip antenna using an ADL inserted between the antenna and lowresistivity Si substrate. A half-wavelength dipole antenna that has ADL shielding was designed and fabricated using a CMOScompatible process with one poly-Si and two metal layers. The design method and characterization for antennas with ADL are reported. II. ARTIFICIAL DIELECTRICS An ADL, composed of a dielectric layer sandwiched between two metal layers, achieves a high dielectric constant along the [7], [8]. The high dielectric consurface of the metal layer, stant prevents electromagnetic waves from penetrating into the substrate [10]. Fig. 1 shows a schematic of the ADL. Two identical metal patch arrays are formed on the two metal layers. One patch array is shifted by one-half cell pitch to the other patch and a array. An insulating layer with a dielectric constant thickness is sandwiched between the two metal layers. The dielectric constant along the surface of the ADL is given by

(1) Manuscript received May 27, 2010; revised November 17, 2010, February 03, 2011; accepted March 07, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology, in part by the University of Tokyo in collaboration with Cadence Design System, Inc., and in part by Agilent Technologies Japan Ltd. The authors are with the Research Center for Integrated Quantum Electronics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan (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.2011.2163758

where is a quarter the area of a square metal patch and is the total thickness of the ADL [10]. is increased by decreasing the layer As found in (1), thickness and and/or by increasing the patch area. Fig. 2 shows potential distributions demonstrating the shielding effect of ADL. The CPWs with and without ADLs were simulated with a finite-difference time-domain (FDTD) electromagnetic simulator (MAGNA/TDM). The ADL was modeled as a uniof . The signal source was sine waves form sheet with

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Fig. 3. Cross section of antenna on ADL.

Fig. 1. Schematic of artificial dielectric layer. (a) Top view. (b) Schematic eddy current flows for patches with and without slots. (c) Cross-sectional view.

Fig. 4. FDTD model. (a) Actual model (1 cell: top metal 1 patch and bottom metal 1/4 patch overlapped). (b) Simplified model. Boundary conditions: xy -plane; PML, yz -plane; electric wall, and zx-plane; magnetic wall.

Fig. 2. Electromagnetic field response for CPW (a) with ADL and (b) without ADL. Signal line width = 10 mm and gap = 5 mm.

of 10 GHz. As shown in Fig. 2, the electric field was terminated in the ADL. When an electromagnetic wave illuminates an placed above a Si substrate, the electric field ADL with high strength in the Si substrate reduces more than when no ADL is on the Si substrate. This means that inserting the ADL between the microwave components and Si substrate decreases the propagation loss for microwave components, such as CPWs, caused by the conductive Si substrate. Thus, placing an ADL between the antenna and Si substrate is expected to increases the antenna gain. When current flows in the antenna, eddy current is induced in the metal patch, as shown in Fig. 1. This eddy current loss can be suppressed by forming slots in the metal patches [10]. An ADL can be constructed using a poly-Si layer and Al (M1) layer in the CMOS process, as shown in Fig. 1. The elemental patch of designed and fabricated ADL was 30 30 m, and the width of the slot was 1 m. The area was 11.5 11.5 m. The dielectric layer between the poly-Si and M1 was a 50-nm-thick for SiN layer. The designed value of the dielectric constant the ADL was about , when of 7.5 was assumed. The position of the ADL and the antenna is shown in Fig. 3. A 1-mm wide ADL was laid under the antenna. Antennas were constructed with a 5- m-thick Au layer (M2) separated from

the Al layer by a 2- m-thick SiO . The 5- m-thick Au layer with a resistivity much smaller than that of Al was used for the antenna to reduce the conductor loss and simplify the fabrication method. A huge amount of computer memory and CPU time is required to simulate an antenna that has ADL when the actual ADL structure is used in the FDTD electromagnetic simulator. Therefore, a simplified model for the ADL is needed to design antennas that have ADLs. A simplified model was investigated with the reflection and transmission characteristics of the ADL using the FDTD simulator. Fig. 4 shows the ADL models used in the simulations: Fig. 4(a) is the actual geometry model, and Fig. 4(b) is a simplified model. The boundary conditions of electric wall, magnetic wall, and perfect matched layer (PML) were imposed to -planes, -planes, and -planes, respectively. These boundary conditions enable us to simulate infinite ADL with only a unit cell. Nonetheless, a huge amount of CPU time is required to simulate the ADL with a 50-nm-thick SiN layer shown in Fig. 1. When a space discretization size of 10 nm is used, a time step should be about 0.01 fs to assure numerical stability. This is much smaller than a time step to be treated with reasonable CPU time. Therefore, a 0.5- m-thick SiN layer was used in the actual geometry model. The metal was treated as a perfect conductor. In the simplified model, the ADL was treated of 6300 (calculated by as a uniform sheet with a permittivity (1)). Two measuring lines were placed at appropriate positions so that incident and reflected pulses did not interact. Fig. 5 shows the simulated S-parameters. The actual and simplified models differed little at the frequencies below 10 GHz. These results

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Fig. 7. Microphotograph of fabricated antenna with ADL. Fig. 5. Comparison between actual and simplified models.

Fig. 6. Top view of geometry of half-wavelength dipole antenna.

Fig. 8. Measurement setup.

TABLE I DIMENSIONS OF DESIGNED DIPOLE ANTENNAS

IV. MEASUREMENTS Antennas were fabricated on 6-inch Si wafers, and the unit chip was 20 20 mm. Fig. 7 is a microphotograph of fabricated antenna with the ADL. On-wafer measurements were performed using a RF probe. A. System

indicate that the simplified model can be used to design an antenna that has an ADL when the conductor loss is neglected. The effect of the conductor loss will be discussed later. III. ANTENNA DESIGN Fig. 6 depicts the geometry of designed half-wavelength dipole antenna. An unbalanced configuration was used for both design and experiment due to the lack of appropriate balun available. Three kinds of antennas were designed and fabricated: an antenna with ADL shielding, an antenna without ADL fabricated on a low-resistivity ( 10 cm) Si substrate, and an k antenna without ADL fabricated on a high-resistivity ( cm) Si substrate. Impedance matching design was performed by changing the length and height under a constant width of 10 m with ADS Momentum for each antenna operating at around 9 GHz. The ADL was modeled as a uniform sheet of . The designed values for and are with summarized in Table I.

The fabricated antennas were measured using a standard gain horn antenna (7–11 GHz) with a gain of 15 dB and a vector network analyzer (VNA). Fig. 8 shows the measurement setup. One of the fabricated chips was mounted on a probe station. Port 1 was connected to one of the signal pads through an RF probe, and Port 2 of the VNA was connected to the horn antenna. The other signal pad of the half-wavelength dipole antenna was connected to a 50- load. The differential mode is generally used for dipole antenna measurements. However, the measurement configuration shown in Fig. 8 was used due to the lack of appropriate wideband balun available. This configuration is the same as that used in the antenna design. The two-port Short-Open-Load-Through (SOLT) method was used to calibrate RF cables and probes. First, the SOLT calibration was performed by using RF probes with G-S-G-S-G (G: ground and S: signal) configuration for port 1 and port 2 along with a calibration kit. Then, port 2 was connected to the horn antenna. A measured transmission gain between a fabricated antenna and horn antenna was about 0.3 dB (RF probe loss) larger than the actual transmission gain.

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Fig. 11. Relationship between distance and transmission gain for antenna using low resistivity Si substrate with ADL.

Fig. 9. Measured return loss.

Fig. 10. Relationship between distance and transmission gains for each antenna.

B. Impedance Matching Fig. 9 shows the measured return losses for each antenna. The impedance matching frequencies were shifted to slightly higher frequencies, in particular for the antenna with ADL. However, dB at 9 GHz and might all the return losses were about have had only a minor impact on the antenna gain. C. Distance Dependence A distance dependence of the transmission gain for a fabricated antenna was measured by changing the distance between the horn antenna and the fabricated antenna. Fig. 10 shows the relationship between the distance and transmission gain for each antenna at 9 GHz. The solid lines in Fig. 10 were fitted to the measured gains with the Friis law given by

(2) where is the measured transmission gain, is the respective gain of the horn antenna, and is a wavelength at the operating frequency. The last term in (2) corresponds to the propagation loss in free space. Fig. 11 shows the relationship between the distance and transmission gain for the antenna with ADL at three frequencies around 9-GHz. The measured gains and Friis law agreed well.

Fig. 12. Comparison between measured and simulated antenna gains.

dB were observed at around Transmission gains of about 9 GHz. Fig. 12 shows the antenna gain calculated with (2) along with simulated gains. The measured antenna gain on low resistivity Si substrate with the ADL was slightly lower than simulated gain. The difference might have been caused by simulated and fabricated ADL geometries. The ADS Momentum assumed for infinite flat sheet with a calculated surface permittivity the ADL. On the other hand, the fabricated ADL had a finite size (1 mm wide), as shown in Fig. 7. The gains of the antenna on low-resistivity substrate, on high-resistivity substrate, and with , and dB, respectively, at 9 GHz. SevADL were eral dips can be found in Fig. 12. These might have been caused by unwanted reflection in the experimental setup. Since the standard horn antenna used here covered the 7–11 GHz range, the reason gain reduced at frequencies higher than 10 GHz was unclear. At 9 GHz, the ADL improved the antenna gain by 3-dB more than the antenna on the low-resistivity Si substrate. This is the higher gain ever archived for the antennas fabricated on a low-resistivity Si substrate operating at around 9 GHz. Fig. 13 compares the antenna gains achieved in this work with those reported in previous literature [6], [11], [12]. D. Directivity A directivity of the antenna fabricated on low resistivity Si substrate with the ADL was measured by changing the degree from 0 to 80 deg. with 5 deg. step in the -plane (E-plane) and

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Fig. 13. Comparison of fabricated antenna gains with those reported in the literature [4], [5], [13]–[20]. Fig. 15. Geometry for ADL characterization.

Thus the surface dielectric constant for the ADL is given by

(4)

Fig. 14. Comparison between measured and simulated directivities for E- and H-planes.

-plane (H-plane) in Fig. 8. Fig. 14 compares measured and simulated directivities. A fairly good agreement was obtained from 0 to 60 deg. The gain degradation above 60 deg was caused by the possible influence of the measurement environment, such as wafer stage and RF probe. V. DISCUSSION A characterization method described by Peuzin and Gay [8] was used for ADL. A test structure for characterizing the ADL was fabricated on the same wafer as the antennas. Fig. 15 shows a schematic top view of the test structure. The test structure had an area as large as could possibly be mounted on the chip. The numbers of the bottom and top patches for the ADL between two strip lines were 21 21 and 20 21, respectively. The dependence of the patch number on was not examined. S-parameters for the test structure were measured with VNA and de-embedded with the open and short calibration patterns. On the other hand, the admittance between two strip lines is given by [8]

(3)

The surface dielectric constant calculated by (3) and (4) was averaged from 110 to 510 with the measured MHz, which is slightly smaller than the designed value. Here, to eliminate the effect of the delay in the strip line, a low frequency . region, which was not around 9 GHz, was used to determine The ADL theoretically has no dispersion under the condition in which the patch size is much smaller than the wavelength and the condition was almost satisfied for the fabricated ADL (30 m corresponds to 10 THz in free space). Almost the same value as that mentioned above might be achieved at 9 GHz. The gain of the antenna with the ADL was simulated using . The solid and dotted lines in Fig. 12 are the measured the simulated gains, respectively, without and with the loss of poly-Si layer (200 /sq). The loss was introduced into the 0.22- m thick layer corresponding to poly-Si. The effect of the loss increased as the frequency increased. Measured and simulated characteristics agreed fairly well. The gain reduced at the frequencies higher than 10.5 GHz, but the reason for the reduction is unclear at this time. As is found in Fig. 12, the antenna gain can be improved by reducing the loss of poly-Si layer. Replacing the poly-Si to Al is an effective method for by thinning the reducing this loss. In addition, increasing dielectric layer will improve the antenna gain. The typical CMOS process will need to be slightly modified to introduce these methods. Antennas and MIM structures for ADL should be placed in the three upper-most layers due to a design rules like metal density for other metal layers. In doing so, CMOS circuits without MIM capacitors will be able to be placed underneath the ADL, which will increase the efficiency of the area usage in a CMOS chip. VI. SUMMARY In this study, we placed an artificial dielectric layer (ADL) between an antenna and Si substrate to improve the antenna gain.

TAKAHAGI AND SANO: HIGH-GAIN SILICON ON-CHIP ANTENNA WITH ARTIFICIAL DIELECTRIC LAYER

Dipole antennas with and without ADL were designed and fabricated using a CMOS-compatible process. The ADL was constructed with poly-Si and first metal layers. ADL shielding is improved gain by 3-dB. Measured gain was the higher gain ever achieve for the antennas on a low-resistivity Si substrate operating at around 9 GHz. Further improvement will be possible by reducing the dielectric layer thickness to increase dielectric constant and reducing the loss of poly-Si layer. REFERENCES [1] B. Razavi, T. Aytur, C. Lam, R. Yang, K.-Y. Li, R.-H. Yan, H.-C. King, C.-C. Hsu, and C. C. Lee, “A UWB CMOS transceiver,” IEEE J. SolidState Circuits, vol. 40, pp. 2555–2562, Dec. 2005. [2] B. A. Floyd, C.-M. Hung, and K. K. O, “Intra-chip wireless interconnect for clock distribution implemented with integrated antenna, receivers, and transmitters,” IEEE J. Solid-State Circuits, vol. 37, pp. 543–552, May 2002. [3] Y. Su, J.-Jr. Lin, and K. K. O, “A 20 GHz CMOS RF down-converter with an on-chip antenna,” in Proc. ISSCC, 2005, pp. 270–271. [4] A. B. M. Harun-ur Rashid, S. Watanabe, and T. Kikkawa, “Characteristics of Si integrated antenna for inter-chip wireless interconnection,” Jpn. J. Appl. Phys., vol. 43, pp. 2283–2287, 2004. [5] Y. P. Zhang, M. Sun, and W. Fan, “Performance of integrated antennas on silicon substrates of high and low resistivities up to 110 GHz for wireless interconnects,” Microwave Opt. Technol. Lett., vol. 48, pp. 302–305, 2003. [6] D. Sievenpiper, L. Zhang, R. F. Jimenez Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [7] I. Awai, H. Kubo, T. Iribe, D. Wakamiya, and A. Sanada, “An artificial dielectric material of huge permittivity with novel anisotropy and its application to a microwave BPF,” IEEE MTT-S Digest, pp. 1085–1088, Jul. 2006. [8] J. C. Peuzin and J. C. Gay, “Demonstration of the waveguiding properties of an artificial surface reactance,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 1695–1699, Sep. 1994. [9] Y. Ma, B. Rejaei, and Y. Zhuang, “Artificial dielectric shield for integrated transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 7, pp. 431–433, Jul. 2008. [10] Y. Ma, B. Rejaei, and Y. Zhuang, “Low-loss on-chip transmission lines with micro-patterned artificial dielectric shields,” Electron. Lett., vol. 44, no. 15, pp. 913–915, Jul. 2008. [11] M. Ikebe, D. Ueo, K. Takahagi, M. Ohuno, Y. Takada, and E. Sano, “A 3.1–10.6 GHz RF CMOS circuits monolithically integrated with dipole antenna,” in Proc. IEEE Int. Conf. Electronics Circuits and Systems, Dec. 2009, pp. 17–20. [12] D. Ueo, H. Osabe, K. Inafune, M. Ikebe, E. Sano, M. Koutani, M. Ikeda, and K. Mashiko, “7-GHz inverted-F antenna monolithically integrated with CMOS LNA,” in Proc. Int. Symp. on Intelligent Signal Processing and Communication Systems, Dec. 2006, pp. 259–262. [13] A. Shamim, L. Roy, N. Fong, and N. G. Tarr, “24 GHz on-chip antennas and balun on bulk Si for air transmission,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 303–311, Feb. 2008.

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[14] J.-Jr. Lin, L. Gao, A. Sugavanam, X. Guo, R. Li, J. E. Brewer, and K. K. O, “Integrated antennas on silicon substrates for communication over free space,” IEEE Electron Device Lett., vol. 25, no. 4, pp. 196–198, Apr. 2004. [15] Y. P. Zhang, L. H. Guo, and M. Sun, “High transmission gain inverted-F antenna on low-resistivity Si for wireless interconnect,” IEEE Electron Device Lett., vol. 27, no. 5, pp. 374–376, May 2006. [16] Y. P. Zhang, M. Sun, and W. Fan, “Performance of integrated antennas on silicon substrates of high and low resistivities up to 110 GHz for wireless interconnects,” Microwave Opt. Technol. Lett., vol. 48, no. 2, pp. 1695–1699, Feb. 2006. [17] Y. P. Zhang, M. Sun, and L. H. Guo, “On-chip antennas for 60-GHz radios in silicon technology,” IEEE Trans. Electron Device, vol. 52, no. 7, pp. 1664–1668, Jul. 2005. [18] M. Singer, K. M. Strohm, J.-F. Luy, and E. M. Biebl, “Active SIMMWIC-antenna for automotive applications,” IEEE MTT-S Digest, pp. 1265–1268, 1997. [19] A. Babakhani, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “A 77-GHz phased-array transceiver with on-chip antennas in silicon: Receiver and antennas,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2795–2806, Dec. 2006. [20] K. Takahagi, M. Ohno, M. Ikebe, and E. Sano, “Ultra-wideband silicon on-chip antennas with artificial dielectric layer,” presented at the Int. Symp. on Intelligent Signal Processing and Communication Systems, Dec. 7–9, 2009, MP1-A-1.

Kazuhiro Takahagi was born in Ibaraki, Japan, in 1987. He received the B.S., M.S. degrees in electronic engineering from the Hokkaido University, Sapporo, Japan, in 2009 and 2011, respectively, where he is currently pursuing the Ph.D. degree. His current research interests are applications of metamaterials to the on-chip antennas, electrically small antennas and integrated circuits.

Eiichi Sano (M’84) was born in Shizuoka, Japan, in 1952. He received the B.S., M.S., and Ph.D. degrees from the University of Tokyo, Tokyo, Japan, in 1975, 1977, and 1998, respectively. From 1977 to 2001, he was with NTT laboratories, where he worked on MOS device physics, mixed analog/digital MOS ULSIs, ultrafast MSM photodetectors, electrooptic sampling, high-speed electronic and optoelectronic ICs. In 2001, he joined the Research Center for Integrated Quantum Electronics, Hokkaido University, Japan, as a Professor. His current research interests include high-speed devices and circuits. He has published over 200 papers in major journals and conference proceedings related to these research areas. Dr. Sano is a member of the Institute of Electronics, Information and Communication Engineers of Japan and the Japan Society of Applied Physics.

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A Compact Printed Filtering Antenna Using a Ground-Intruded Coupled Line Resonator Chao-Tang Chuang and Shyh-Jong Chung, Senior Member, IEEE

Abstract—A compact printed filtering antenna with high bandedge gain selectivity is presented. Occupying about the same substrate area as a conventional antenna, the proposed structure not only serves as a radiator but also a second-order bandpass filter, with one filter pole provided by a 0-shaped antenna and the other by a newly proposed coupled line resonator. High band-edge selectivity is achieved due to two additional stop-band transmission zeros provided by the coupled line resonator. To minimize the required area and reduce the spurious radiation, a coupled line structure composed of a microstrip line and a coplanar waveguide by broadside coupling is adopted and intruded into the 0-shaped antenna area. According to the filter specifications, a design procedure for the proposed filtering antenna is depicted in detail. One example at 2.45 GHz with a second-order Chebyshev bandpass filter of 0.1 dB equal-ripple response is tackled. As compared to the conventional 0-shaped antenna, the proposed filtering antenna not only possesses a similar antenna gain but also provides better bandedge gain selectivity and flat passband gain response. The measured results, including the S-parameters, total radiated power, and antenna gains versus frequency, have good agreement with the designed ones. Index Terms—Band-edge selectivity, bandpass filter, coupled line resonator, filtering antenna.

I. INTRODUCTION

W

ITH the rapid development of wireless communication systems, the requirements for compact, low-cost, and low profile passive components are demanded in recent years. To achieve these purposes, various efforts could be done in a single circuit module. Integration of antenna and bandpass filter in one module is one of the ways to achieve miniaturization and improved performance of microwave front ends. There have been numerous studies in the literature for integrating the filter and antenna into a single microwave device [1]–[4]. In order to reduce circuit area, a pre-design bandpass filter with suitable configuration was directly inserted into the feed position of a patch antenna [1]. Also to increase the bandwidth, the bandpass filter can be integrated properly with the antenna [2]–[4] by using an extra impedance transformation structure in between the filter and the antenna. Nevertheless, the tran-

Manuscript received November 01, 2010; revised February 15, 2011; accepted March 18, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by the National Science Council, R.O.C., under Contract NSC 97-2221-E-009-041-MY3. The authors are with the Institute of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, R.O.C. (e-mail: sjchung@cm. nctu.edu.tw). 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.2011.2163777

Fig. 1. The proposed compact second-order filtering antenna.

sition structure needs extra circuit area, and the designs did not give good filter responses in the frequency ranges. In more recent years, the integration approach that the last resonator and the load impedance of the bandpass filter were substituted by an antenna, which is called the filtering antenna, has been discussed. The filtering antennas designed following the synthesis process of the bandpass filter have been presented in [5]–[8]. Although they have been done based on the co-design approach, these filtering antennas did not show good filter performance, especially the band-edge selectivity and stop-band rejection. This is due to the lack of the extraction of the antenna’s equivalent circuit over a suitable bandwidth. Only that at the center frequency was extracted and used in the filter synthesis. Moreover, the total radiated powers and the antenna gains versus frequency, which are the important characteristic of the filtering antenna, were not examined in these studies. In this study, a new compact printed filtering antenna, which is composed of a miniaturized coupled line resonator and a -shaped antenna, as shown in Fig. 1, is proposed. The coupled line resonator in the filtering antenna can provide one pair of transmission zeros with tunable frequencies to achieve the purpose of the high band-edge selectivity. This paper is organized as follows: Section II illustrates the equivalent circuit and performance of the coupled line resonator; Section III shows the design procedure for the filtering antenna. A structure is synthesized as an example. The design and measurement of the proposed compact filtering antenna are then presented in Section IV, followed by a conclusion in Section V. II. ANALYSIS OF THE COUPLED LINE RESONATOR In the design of a narrow-band bandpass filter, one needs to use shunt resonators with high capacitances. Although these

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Fig. 3. (a) Equivalent circuit of the coupled line resonator with equal lengths. (b) The composite shunt resonator with two series LC resonators in parallel. (c) The equivalent shunt LC resonator around the resonant frequency f . Fig. 2. Geometry of the coupled line resonators with (a) equal lengths and (b) unequal lengths.

high-capacitance resonators can be implemented by using short-circuited microstrip stubs, it needs quite large circuit area because of the very wide microstrip line used. In [9], [10], the dual-behavior resonators are designed by associating two different parallel open-circuited stubs. Each stub brings its own transmission zero depending on its fundamental resonant condition. Even so, these resonators also suffer from large circuit area because of the use of the two open-circuited stubs. Recently, the authors proposed the coupled line resonators as shown in Fig. 2(a) [11] to solve this problem, which provides high equivalent capacitance while occupies a reasonable circuit area. In addition, without using the technique of cross-coupling between non-adjacent resonators [12], the proposed coupled line resonator itself produces one pair of tunable symmetric transmission zeros at the two sides of the passband, which greatly increase the band-edge selectivity. In this study, detail analyses of the coupled line resonator are presented. Also, an extended version with different lengths of the coupled lines as shown in Fig. 2(b) is introduced. As will be shown later, this new structure has the advantage of producing two asymmetric transmission zeros that may be contributive to the design flexibility. The coupled line resonator with equal lengths shown in Fig. 2(a) is composed of a quarter-wavelength open-circuited stub and a quarter-wavelength short-circuited stub. These two resonant microstrip stubs couple with each other through the gap between them. Physically, the open-circuited stub is resonator, and the short-circuited stub equivalent to a series resonator. And the coupling gap, which mainly a shunt offers electrical coupling between the resonators, functions as an admittance inverter (or -inverter) [13]. Therefore, the proposed structure can be equivalent to the circuit shown in Fig. 3(a). Note that due to the presence of the coupling, the and have different resonant two resonators frequencies, although the two coupled stubs are with equal lengths. This equivalent circuit can be further transformed into the circuit shown in Fig. 3(b), which consists of two series resonators connected in parallel, with resonant frequencies at and , respectively. It is thus evident that the proposed structure provides two transmission zeros at these two resonant frequencies. We now set up the quantitative equivalence between the coupled line resonator (Fig. 2(a)) and the equivalent circuit (Fig. 3(b)) in the following three steps: First, derive the input admittances of the circuits; secondly, equate the two admittances and their derivatives, respectively, at the center frequency, and

also let the two circuits with the same transmission zeros; thirdly, extract the equivalent circuit components from the above-obtained equations. In the first step, one first derives the admittances of these two circuits as [13]: (1a) (1b) where and are the admittances of the coupled line resonator and the equivalent circuit in Fig. 3(b), respectively. and are the even-mode and odd-mode impedances of the , with coupled line. is the electrical length, the corresponding resonant frequency. , . circuit, the couNote that, like the equivalent pled line resonator has a transmission pole at (or ) and two transmission zeros at and . It is noticed that the two zeros are symmetric to the transmission pole. In the second step, the circuit shown in Fig. 3(b) is equivalent , to the coupled line resonator by letting , and also making their transmission zeros the same. One then obtains the following four equations: (2a)

(2b) (2c) (2d) Once the dimensions of the coupled line resonator are given, , , and will be determined. In the last step, one can thus in Fig. 3(b) by get the four circuit parameters solving these four equations. Furthermore, near the resonant frequency , the coupled resonator, line resonator can be approximated as a shunt resonators in as shown in Fig. 3(c), since one of the series

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even- and odd-mode characteristic impedances of the coupled line resonator as: (5a) (5b)

Fig. 4. Circuit simulation of the coupled line resonator with equal lengths in comparison with the ones of its corresponding equivalent circuits (Figs. 3(b) and (c)). The line width and gap size of the coupled line resonator are 0.5 mm and 0.2 mm.

Fig. 3(b) shows inductive property and the other capacitive at and . The equivalence between the frequency in between these two circuits are established by equalizing their resonant frequencies and the admittance derivatives, which yield

(3) and

(4) can thus be calculated while the The circuit components and are given. impedances Fig. 4 shows the circuit simulation results by AWR [14] for a coupled line resonator with line width of 0.5 mm and gap 0.2 mm, fabricated on a 0.508 mm thick Rogers 4003 substrate with a dielectric constant 3.38 and loss tangent of 0.0027. Here, the ideal transmission lines for the coupled line resonator are con, sidered. The electric length of the coupled lines is which corresponds to a resonant frequency at 2.5 GHz. To get the equivalent circuits, one first calculates the impedances and as 100 and 57 respectively. Then, based on (2c) and (2d), the two transmission zeros, and , are calculated to be 2.05 GHz and 2.95 GHz. Finally, the circuit components in Figs. 3(b) and (c) can be obtained by using (2a), (2b), (3), , and (4), which are , and . It is seen from the figure that, the frequency responses of the equivalent circuit in Fig. 3(b) match very well with those of the coupled line resonator over the full frequency band. Not only with the same transmission pole and zeros, the curves are completely overlapped in the frequency range between and . Also, the responses of the circuit in Fig. 3(c) agree well with the ones of the coupled line resonator near the resonant frequency . As will be shown later, when designing a filtering antenna, the equivalent circuit components are first obtained based on the filter specifications, from which the structure dimensions have to be decided. To this end, one may use (2c) and (4) to derive the

where , . These impedances together with the given resonant frequency can then be used to determine the resonator’s dimensions [13]. In many applications, the required transmission zeros of a bandpass filter are not symmetric to the passband. It is thus necessary to modify the proposed resonator structure shown in Fig. 2(a) that possesses symmetric zeros. One of the solutions is shown in Fig. 2(b), in which the short-circuited stub remains the at the resonant frequency, while the same length, i.e., or shorter open-circuited stub has a length longer . Fig. 5 shows the circuit simulation results for the coupled line resonators with different . The structure parameters are the same as those in Fig. 4 except the length of the open-circuited stub. It is observed that, as is decreased or in, the resonant frequency remains the same but creased from the two transmission zeros move toward the higher or lower frequencies, as shown in Figs. 5(a) and (b), respectively. Furthermore, for the reason that a smaller circuit area is usually required and the fact that the higher transmission zero moves much faster than the lower one, the case of shorter is chosen in this study. In application, one may first design a coupled line resonator and with equal length to determine the resonant frequency the approximate the frequency of the lower transmission zero, and then adjust the frequency of the higher transmission zero to the desired one by tuning the open stub’s length. Note that as depicted in Fig. 5(a), the characteristics of the coupled line resonator keep the same at frequencies near , which means that, when incorporating this resonator into a bandpass filter, the filter’s passband performance will not change due to the adjustment of the higher transmission zero. Another solution, that the while the short-ciropen-circuited stub has a length cuited stub has a variable length can also provide asymmetric transmission zeros but would have different resonant frequency , which is thus not considered here. III. SYNTHESIS OF THE FILTERING ANTENNA In this section, the synthesis of a filtering antenna by using the coupled line resonator is to be presented. The proposed filtering antenna is constructed by directly connecting the coupled line resonator to a -shaped antenna, as shown in Fig. 6(a). Since the antenna is a variety of a monopole antenna, it has a series equivalent circuit as shown in Fig. 6(b) [15]. Here, and express the equivalent inductance and capacitance, respeccorresponds to the antenna radiation resistance. It tively, and is noted that an extra shunt capacitance is incorporated in the equivalent circuit here so that the whole circuit would have the same impedance behavior as the antenna itself in a wider frequency range [11]. This parasitic capacitance comes from the

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Fig. 7. (a) Equivalent circuit of the second-order filtering antenna. (b) Equivalent circuit of the filtering antenna while the resonant frequency of the coupled line resonator is near f . (c) Equivalent circuit of the typical second-order bandpass filter.

Fig. 5. Circuit simulation of the coupled line resonator with unequal lengths for different  . (a)  decreases from =2. (b)  increases from =2.

Fig. 6. (a) The geometry of the second-order filtering antenna. (b) The corresponding equivalent circuit of the 0-shaped antenna.

accumulation of charges around the antenna feed point due to the truncation of the ground plane. By using the equivalent circuits of the coupled line resonator and the -shaped antenna, the second-order filtering antenna can be expressed by the equivalent circuit shown in Fig. 7(a). It can be observed that not only second-order response but also two extra transmission zeros near the filter’s band-edge can be produced. Fig. 7(b) is a further equivalent circuit of the filtering

antenna while the resonant frequency of the coupled line resonator is near . This circuit can then be transformed to a typical second-order bandpass filter as shown in Fig. 7(c), where , , , and . Note , the rethat due to the existence of the small capacitance of the couquired resonant frequency pled line resonator is slightly larger than the operating frequency of the bandpass filter. The design procedures of the proposed filtering antenna are listed below. 1) Specify the requirements of the bandpass filter to be synthesized, including the operating frequency , the frequencies of the transmission zeros ( and ), the fractional bandwidth , and the type of the filter (e.g., bandpass filter with equal ripple). Therefore, the circuit of the typical bandpass elements filter in Fig. 7(c) can be calculated [13]. 2) Choose the suitable dimensions of the -shaped antenna of and then extract the circuit elements its equivalent circuit [11]. These circuit elements are used to function as the last (second) stage of the filter. Therefore, the antenna dimensions should be determined so as to sat, , and isfy the following requirements: . obtained 3) Use the component values of in the above procedures, together with the required resoand lower transnant frequencies mission zero , the characteristic impedances of the coupled line resonator with equal lengths can be calculated by using (5), and then its corresponding dimensions can be obtained. in the cou4) Properly shorten the open stub’s length pled line resonator with unequal lengths so as to adjust . Note the higher transmission zero to the required one that although the position of the lower transmission zero is changed in this step, the variation is quite limited as shown in Fig. 5(a). If a more precise lower transmission zero is required, the frequency used in the above procedure can

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be set a little smaller than the required one to compensate the effect when shortening the stub’s length. As an example, a second-order Chebyshev bandpass filter with a 0.1 dB equal-ripple response is tackled. The filtering antenna is designed on a 0.508 mm thick Rogers 4003 substrate with a dielectric constant of 3.38 and loss tangent of 0.0027. The . In the design, the ground plane size full-wave simulation solver HFSS [16] is used for fine tuning in the last procedure. According to the design procedure, the specifications of the filter should be firstly chosen. Here, the bandpass , a fractional filter has a operating frequency bandwidth , and . The lower transmisis set at 2 GHz, and the higher transmission zero sion zero is 3.3 GHz. Based on these requirements, the circuit compo, nents of the filter can be calculated, which are , , , and . Secondly, we have to determine the antenna sizes. After numerous simulations, we choose the dimensions of the -shaped , , , and antenna as , which correspond to a set of extracted compo, , , nent values (i.e., ) nearest to required one. and the resonant frequency Thirdly, the circuit component of the coupled line resonator are calculated, which are 7.45 pF and 2.51 GHz, respectively. The characteristic impedances can then be gotten by using (5) and the given lower transmission zero, with which the dimensions of the coupled line resonator of equal lengths are obtained. The resultant line width, gap size, and length of the coupled line are 0.64 mm, 0.15 mm, and 18.9 mm, respectively. Finally, to reach of the a transmission zero at 3.3 GHz, the electric length open-circuited stub is shortened from 90 to 70 by simulation. To this end, all the structure dimensions have been determined. And the filtering antenna can be implemented by connecting the coupled line resonator and the -shaped antenna, as shown in Fig. 6(a). Fig. 8(a) shows the full-wave simulated return loss of the designed filtering antenna. The simulated result of the -shaped antenna only is also shown for reference. The proposed filtering antenna exhibits a return loss larger than 15 dB over the passband, and provides better band-edge selectivity as compared to the -shaped antenna only. Also observe that, with the center frequency at 2.45 GHz, the filtering antenna possesses two transmission poles at 2.33 GHz and 2.57 GHz, which are caused from the repel of resonant frequencies due to mutual coupling between the coupled line resonator and -shaped antenna [17]. Although not shown here, it has been found that the lower and higher transmission poles can be tuned independently by the coupled line resonator and the antenna, respectively. Fig. 8(b) depicts the impedance behavior on the Smith chart of the filtering antenna, and Fig. 8(c) illustrates the current distributions at the center frequencies and the two transmission poles. At these three in-band frequencies, as observed from Fig. 8(b), the input impedance possesses negligible reactance but has similar resistances around 30 . Due to the high radiation efficiency in the passband, the input resistance of the proposed antenna is nearly equal to the radiation resistance. Consequently, the currents distributed on the -shaped antenna

Fig. 8. (a) The full-wave simulated return loss of the filtering antenna in comparison with the simulated one of the 0-shaped antenna. (b) The impedance behavior on the Smith chart of the filtering antenna. (c) The simulated current distributions of the filtering antenna at the center frequency and two transmission poles.

have about the same level at these frequencies, as can be observed from Fig. 8(c). Also notice from Fig. 8(c) that the current on the coupled line resonator at 2.33 GHz is relatively large as compared to those at 2.45 GHz and 2.57 GHz, which is apparent since the coupled line resonator is the main contributor for the lower transmission pole at 2.33 GHz. Fig. 9 shows the full-wave simulated total radiated power of the filtering antenna in comparison with the simulated one of the -shaped antenna. Here, the total radiated power has been normalized to the input power. It is seen that the filtering antenna exhibits a constant radiation power over the required frequency

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Fig. 9. The full-wave simulated total radiated power of the filtering antenna in comparison with the simulated ones of the 0-shaped antenna.

Fig. 11. Full-wave simulated S-parameters of the miniaturized coupled line : . (b) difresonator with unequal lengths. (a) different l with w : . (w : ,S : and f ferent w with l : ).

2 5 GHz Fig. 10. Geometry of the proposed miniaturized coupled line resonator.

bandwidth, and has two radiation nulls at 2.13 GHz and 3.33 GHz which are close to the design ones. Also, the proposed filtering antenna has much better stop-band suppression and band-edge selectivity. The corresponding radiation efficiency of the filtering antenna is about 88% in the passband, but is greatly reduced to 0.9% and 0.7% at the frequencies of the two radiation nulls. The input power is mostly reflected back due to the low return losses (about 0.7 dB) at these two nulls. IV. COMPACT SECOND-ORDER FILTERING ANTENNA Although the proposed filtering antenna shown in Fig. 6(a) exhibits good radiation and filtering performances, it still occupies extra circuit area due to the use of the coupled line resonator. To realize a compact filtering antenna, a miniaturized coupled line resonator composed of a microstrip line open-circuited stub and a coplanar waveguide short-circuited stub by broadside coupling is proposed, as shown in Fig. 10, to intrude into the -shaped antenna area. Herein, an extended ground is employed for designing the folded coupled plane line resonator. Fig. 11 shows the full-wave simulated scattering parameters for different dimensions of the miniaturized coupled line resonator. Note that the simulation result does not include the effect of the -shaped antenna. It considers only the two-port folded coupled line resonator between ports 1 and 2 (see Fig. 10). The of the extended ground plane is size

= 5 5 mm

= 1 2 mm

= 0 3 mm = 0 2 mm

=

and the distance between the extended ground plane and verof the -shaped antenna is 1 mm. Fig. 11(a) tical strip line depicts the effect of the length of the microstrip line open-circuited stub. The coplanar waveguide short-circuited stub has a fixed total length of about a quarter wavelength and a line width and gap . As shown, the miniaturized resonator has the same properties as the original one (Fig. 2(b)). It possesses a transmission pole and two side transmission zeros. Also, when the open-circuited stub is shortened (i.e., decreasing ), the behavior near the center frequency is unchanged, and the two transmission zeros move toward the higher frequencies, with the higher transmission zero moving faster than the lower one. of the Fig. 11(b) illustrates the effect of the line width increases, the center frequency coplanar waveguide stub. As remains the same due to the same short-circuited stub length . While the two transmission zeros move away from the causes center frequency. This is because the increase of a stronger mutual coupling between the microstrip line and the coplanar waveguide, which thus results in a larger ratio of . Therefore, from (2c) and (2d), the lower transmission zero appears at a lower frequency and the higher zero at a higher frequency. In summary, the intruded miniaturized coupled line resonator follows the behaviors depicted in Section II. Thus, the design methodology described in the previous section can be applied directly to the proposed compact filtering antenna (Fig. 1). In the design, the filter specifications to be achieved and the dimensions of the -shaped antenna are set the same as those in

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TABLE I DIMENSIONS OF THE MINIATURIZED COUPLED LINE RESONATOR WITH UNEQUAL LENGTHS IN THE COMPACT FILTERING ANTENNA

Unit: mm

Fig. 13. The simulated and measured antenna gains versus frequency of the compact filtering antenna in comparison with the simulated ones of the : 0-shaped antenna. (a) In the z direction and (b) in the x direction. [ : simulation of compact measurement of compact filtering antenna; filtering antenna; : : : :: simulation of antenna only.]

+

+

Fig. 12. The simulated and measured results of the compact filtering antenna in comparison with the simulated ones of the 0-shaped antenna. (a) Return : measurement of compact filtering losses and (b) total radiated powers. [ antenna; : simulation of compact filtering antenna; : : : :: simulation of antenna only.]

Section III. And the final dimensions of the miniaturized coupled line resonator are listed in Table I. Fig. 12(a) and (b) shows the full-wave simulated and measured return losses and total radiated powers, respectively, of the compact filtering antenna. The simulated results of the -shaped antenna are also shown for reference. It is seen that the full-wave simulated results of the compact filtering antenna are almost the same as those of the original design (Fig. 6(a)), which show two filter poles at 2.3 GHz and 2.6 GHz and two radiation nulls at 2.11 GHz and 3.31 GHz. The simulated radiation efficiency at operating frequency is about 82%, slightly lower than the original design, and those at the two transmission zeros are 0.7% and 1.1%. The measured results agree quite well with the simulations. It is evident that, without suffering from the need of extra circuit area, the proposed compact filtering antenna shown in Fig. 1 has a flat passband radiation power as a function of frequency, high band-edge selectivity, and good stop-band suppression. Figs. 13(a) and (b) show the full-wave simulated and meaand directions versus fresured antenna gains in the quency for the compact filtering antenna. The simulated antenna gains of the -shaped antenna in these two directions are also shown for reference. As expected, the antenna gains for the compact filtering antenna are flat in the passband. Also, two clear radiation nulls in the stop band can be observed in the

Fig. 14. Measured radiation patterns in the xz , yz , and xy planes for the compact filtering antenna. E E : E . f : .

[00 :

; 000 : ; . . . : ]

= 2 45 GHz

simulation results, which make the band-edge selectivity and stop-band suppression better than those of the reference structure. The first null of the antenna gain locates at a frequency exactly equal to that (2.11 GHz) of the first null in the total radiation power response (Fig. 12(b)). While the frequency location of the second null deviates a little from that in Fig. 12(b) and depends on the observation angle, due to the influence of the antenna’s radiation pattern. The measurement matches well to the simulation. The ripples in the stop band probably come from the

CHUANG AND CHUNG: A COMPACT PRINTED FILTERING ANTENNA USING A GROUND-INTRUDED COUPLED LINE RESONATOR

Fig. 15. Photograph of the compact filtering antenna. (left-side: top view; rightside: bottom view).

non-ideal anechoic chamber condition. The measured antenna in the and directions, including gains at the circuitry loss, are 1.03 dBi and 0.18 dBi, respectively. Fig. 14 depicts the measured radiation patterns of the comin the three principal pact filtering antenna at planes. The radiation pattern in the -plane is nearly omni-directional with peak gain of 1.2 dBi. Although not shown here, these patterns are about the same as the simulations and the measured results of the -shaped antenna alone. Fig. 15 shows the photograph of the finished compact filtering antenna. V. CONCLUSION A new printed filtering antenna with a very compact configuration has been demonstrated. By incorporating a ground-intruded miniaturized coupled line resonator, the proposed filtering antenna occupied about the same substrate area as a conventional printed antenna, while exhibited flat high band-edge selectivity, and good stop-band suppression. Thorough analysis and design of the coupled line resonator and the filtering antenna have been described. The measured results, including the return loss, total radiation power, and antenna gains versus frequency, have good agreement with the design ones. REFERENCES [1] F. Queudet, I. Pele, B. Froppier, Y. Mahe, and S. Toutain, “Integration of pass-band filters in patch antennas,” in Proc. 32th Eur. Microw. Conf., 2002, pp. 685–688. [2] J.-H. Lee, N. Kidera, S. Pinel, J. Laskar, and M. M. Tentzeris, “Fully integrated passive front-end solutions for a V-band LTCC wireless system,” Antennas Wireless Propag. Lett., vol. 6, pp. 285–288, 2007. [3] N. Yang, C. Caloz, and K. Wu, “Co-designed CPS UWB filter-antenna system,” in Proc. IEEE AP-S Int. Symp., Jun. 2007, pp. 1433–1436. [4] C.-H. Wu, C.-H. Wang, S.-Y. Chen, and C. H. Chen, “Balanced-tounbalanced bandpass filters and the antenna applications,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 11, pp. 2474–2482, Nov. 2008. [5] H. Blondeaux, D. Baillargeat, P. Leveque, S. Verdeyme, P. Vaudon, P. Guillon, A. Carlier, and Y. Cailloce, “Microwave device combining and radiating functions for telecommunication satellites,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 137–140.

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[6] T. L. Nadan, J. P. Coupez, S. Toutain, and C. Person, “Optimization and miniaturization of a filter/antenna multi-function module using a composite ceramic-foam substrate,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, pp. 219–222. [7] A. Abbaspour-Tamijani, J. Rizk, and G. Rebeiz, “ Integration of filters and microstrip antennas,” in Proc. IEEE AP-S Int. Symp., Jun. 2002, pp. 874–877. [8] S. Oda, S. Sakaguchi, H. Kanaya, R. K. Pokharel, and K. Yoshida, “Electrically small superconducting antennas with bandpass filters,” IEEE Trans. Appl. Supercond., vol. 17, no. 2, pp. 878–881, Jun. 2007. [9] C. Quendo, E. Rius, and C. Person, “Narrow bandpass filters using dual-behaviors resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 734–743, Mar. 2003. [10] Z. Ma and Y. Kobayashi, “Design and realization of bandpass filters using composite resonators to obtain transmission zeros,” in Proc. 35th Eur. Microw. Conf., 2005. [11] C.-T. Chuang and S.-J. Chuang, “New printed filtering antenna with selectivity enhancement,” in Proc. 39th Eur. Microw. Conf., 2009, pp. 747–750. [12] N. Yildirim et al., “A revision of cascade synthesis theory covering cross-coupled filters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 6, pp. 1536–1543, Jun. 2002. [13] D. M. Pozar, Microwave Engineering, 3nd ed. New York: Wiley, 2005, ch. 8. [14] AWR Microwave Office (MWO). AWR Corporation. Segundo, CA, 2010. [15] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1998. [16] High Frequency Structure Simulator (HFSS). Ansoft Corporation. Pittsburgh, PA, 2001. [17] G. L. Mattaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Network, and Coupling Structure. Norwood, MA: Artech House, 1980. Chao-Tang Chuang was born in Hualien, Taiwan, R.O.C., on June 2, 1981. He received the B.S. degree in physics from the National Chung Hsing University, Taichung, Taiwan, in 2004 and the M.S. degree in electrical engineering from National Central University, Jungli, Taiwan, in 2006. He is currently working toward the Ph.D. degree in communication engineering in National Chiao Tung University, Hsinchu, Taiwan, R.O.C. He current research interests include design of microwave circuits and antennas.

Shyh-Jong Chung (M’92–SM’06) was born in Taipei, Taiwan, R.O.C. He received the B.S.E.E. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, R.O.C., in 1984 and 1988, respectively. Since 1988, he has been with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C., where he is currently a Professor and serves as the Director of the Institute of Communication Engineering. From September 1995 to August 1996, he was a Visiting Scholar with the Department of Electrical Engineering, Texas, A&M University, College Station. His areas of interest include the design and applications of active and passive planar antennas, LTCC-based RF components and modules, packaging effects of microwave circuits, vehicle collision warning radars, and communications in intelligent transportation systems (ITSs). Dr. Chung received the Outstanding Electrical Engineering Professor Award of the Chinese Institute of Electrical Engineering and the Teaching Excellence Awards of National Chiao Tung University both at 2005. He served as the Treasurer of IEEE Taipei Section from 2001 to 2003 and the Chairman of IEEE MTT-S Taipei Chapter from 2005 to 2007.

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A Circuit Model for Spherical Wheeler Cap Measurements Herbert L. Thal, Jr., Life Fellow, IEEE

Abstract—The efficiency of an electrically-small antenna may be measured conveniently by placing it in a conducting Wheeler Cap. This paper presents a composite equivalent circuit that models an antenna both inside a spherical cap and also radiating into free space. The circuit is used to show how the measurements are affected by the antenna size, efficiency, and polarization (TE or TM), the cap radius, and experimental errors. Methods for using a single cap or multiple caps are analyzed and compared. Antennas with typical properties are represented in a general form rather than by either series or parallel circuits as is commonly done. Loops and spherical dipoles are used as examples. Index Terms—Antenna measurements, antenna theory, electrically-small antennas, equivalent circuits, loop antennas, Wheeler cap. Fig. 1. Circuit models for spherical Wheeler cap measurements of single-mode antennas.

I. INTRODUCTION

A

new circuit model for an antenna in a conducting sphere is introduced in order to study the characteristics of Wheeler cap efficiency measurements [1]. It merges an exact circuit for the propagation between concentric spherical surfaces with a two-port representation of the dominant radiating mode of an electrically-small antenna; the perturbation due to higher-order modes is discussed. The concentric sphere circuit provides the reflection phase from the cap and illustrates the different behavior of TE and TM modes. It allows the calculation of the antenna detuning as a function of the cap radius and the radiating mode, independent of details of the antenna structure. Equations for determining the efficiency from reflection coefficient measurements with one or two caps are derived and related to previously published methods. The assumptions are defined and sensitivity coefficients are given. Some of the results apply to non-spherical caps, and the differences between spherical and non-spherical caps are considered. The procedures are illustrated by examples for a TE and a TM mode antenna using circuit models for these antennas that are known to represent them accurately II. CIRCUIT MODEL

Fig. 1(a) shows the circuit model for the lowest-order TE either radiating into space or within a conducting mode spherical cap. There are three regions defined by two spherical surfaces of radii “a” and “b” where “a” is the smallest sphere that completely encloses the antenna structure and “b” is the Manuscript received December 21, 2009; revised December 16, 2010; accepted February 26, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. The author at Wayne, PA 19080 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.2011.2163746

radius of the cap. The circuit elements within “a” characterize the physical antenna structure. The elements between port 2 and “a” model the propagation of the TE mode [2] and are a necessary and critical part of the composite antenna circuit. Port 1 is where the reflection coefficient (impedance) is measured. All impedances are normalized to the characteristic impedance of free space. The unit impedance transmission line combined with the positive and negative inductances and capacitances between “a” and “b” properly accounts for the propagation of the spherical wave between the two surfaces [3]. A short circuit at terminals “b” is used to calculate the reflection coefficient at port 1 with the cap installed. (Cap losses can be taken into account by using the surface impedance of the cap in place of a perfect conductor.) The positive and negative elements on both sides of “b” exactly cancel when the circuit is radiating. The resulting circuit is just the composite antenna circuit, including the essential external elements at “a,” plus a unit impedance transmission line of length [R-a] where R is the radius of a sphere in the far-zone. The complete transmission and reflection characteristics of the antenna are defined by a reciprocal scattering matrix with co, , , and . Port 2 does not represent a efficients physically accessible port since it lies within the near zone; however, for convenience it can be considered mathematically as just port R with its reference plane transformed inward by removing , , and will the line length [R-a]. Thus, coefficients in place of due to be used in subsequent analyses, with reciprocity. mode antenna. In Fig. 1(b) shows the circuit for a circuit the radiating configuration it is the dual of the in Fig. 1(a). However, the conducting cap upsets the duality and creates different conducting cap responses for the two different type antennas. A magnetic-wall cap (i.e., open circuit)

0018-926X/$26.00 © 2011 IEEE

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would be needed to yield a dual response. (Note that the first and second indices relate to the radial and azimuthal variations, respectively; some references use the reverse order.) A. Concentric Sphere Resonances The spherical region circuit can be validated by using it to compute the resonances of the volume between concentric conducting spheres. These can be found from the circuit representation by placing a short circuit at terminals “a” and solving for the frequencies that have zero reactance at “b”. For the circuit the two inductances have no effect and resonances occur for values of k that satisfy:

(1) . Equations for and 2 resonances have where been derived in a similar manner and solved numerically. The results agree exactly with published data [4]. B. Cap Reflection Phase For the mode the negative elements at “b” modify the effective length of the transmission line, [b-a], by terminating it in a (negative) capacitive reactance, 1/(kb); the shunt inductance in parallel with the short circuit has no effect. Converting this reactance into an equivalent stub and adding it to [b-a] yields an , given by: effective length,

Fig. 2. Normalized TE delta susceptance and TM delta reactance vs. kb for ka = 0:5.

III. IMPACT OF CAP ON ANTENNA TUNING The basic concept of the Wheeler cap is that it eliminates the radiation resistance, leaving only the resistance due to losses, cirwithout significantly detuning the antenna. For the cuit, consider the series combination of the unit radiation resistance and the capacitance just inside port 2. These elements yield a susceptance in parallel with the shunt inductance, (a/c), equal to:

(5) (2) For the mode the parallel combination of the inductance and capacitance at “b”, both having the value of has a susceptance, , which yields an effective length: (3)

Replacing the radiation resistance with a shorted line of length, L, in series with the capacitance yields a susceptance: (6) The difference between these two susceptances, , is a circuit measure of the detuning. In the radiating state the so that . Within the cap, is the dual of the (7)

The arc tangent terms represent near-zone corrections which decrease as kb increases. The reflection phase is related to the lengths of these shorted stubs by: (4) There are six modes—three and three —corresponding to three orthogonal, infinitesimal loops and three dipoles. For the commonly-encountered case of a hemispherical cap over a conducting ground plane only three has a null on the axis normal to the ground of these exist. plane; it could be produced by a monopole or a microstrip patch mode. The two degenerate modes driven in the have orthogonally polarized on-axis peaks; a constant-current mode are half-loop and a microstrip patch in the mode. The mode examples of antennas that can excite a radiated by an antenna can normally be determined from its computed or measured pattern and/or its symmetry planes.

The difference between these two reactances, , is a measure of the detuning. Corresponding equations were demodes. rived for the Fig. 2 shows these susceptance and reactance differences as . The mode is significantly a function of kb for detuned for small values of kb, but the detuning decreases to zero and then increases in the opposite sense. This pattern repeats as kb continues to increase although the shape of the curve mode has negative detuning for changes somewhat. The small kb; the detuning starts to decrease as kb increases, but re. For larger values of kb the curves are esverses at curves. The normalization sentially interlaced with the of the ordinate scale and the marked points on the curves are explained below. and higher index modes are The characteristics of the modes at this ka. completely different from the dominant Their fields decay rapidly away from the antenna surface, and their susceptance or reactance is perturbed from the radiating

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(free space) value only when the cap is close to the surface (ka) or near a resonance. As the index, n, increases, the curves move axis and the resonances progressively closer to the shift to larger values of kb, off the scale of the present graph, resonance at 4.9734, too narrow to show. except for a radiated power is generally at least 40 dB below The the dominant mode in antennas this size due to its small radiated power relative to its stored energy, i.e., its high Q, so that it should have little effect. Only certain higher-order modes are excited by an antenna; they can be determined from the symmetry or pattern measurements; see example VI-B. Of particular interest are cap radii that do not detune the antenna and also ones that result in a 90 degree difference in the cap phase from the no-detuning phase since this phase differmode ence relates to the measurement accuracy. For a antenna there is no de-tuning regardless of the internal antenna so that: configuration when (8)

single mode, changes in the cap phase manifest themselves directly as changes in the reflection coefficient measured at port 1, totally independent of the details or complexity of the internal structure. This condition should be approximately true for well-matched, high-efficiency antennas. Table I summarizes the desirable range of kb values that yield less than 90 degrees of phase change as a function of ka and mode; it shows that the appropriate cap radii are different for TE and TM antennas. IV. ANTENNA REFLECTION COEFFICIENT, , IN CAP

and suitable cap radii, kb, can be computed from (2): (9) Using the

TABLE I OPTIMUM RANGE OF CAP RADIUS, KB, AS A FUNCTION OF KA AND POLARIZATION

reactances in a corresponding manner gives: (10)

This section describes how the reflection coefficient at port 1 depends on the scattering matrix and the cap dimensions. It presents equations that can be used for determining the matrix coefficients, and thus the efficiency, from the measured reflection coefficient. When port 2 of a two-port junction is terminated in a reflection coefficient, , the net reflection, , at port 1 is:

(11)

(17)

The smallest cap that yields an exact solution to (11) has kb greater than four. However, the left-hand side has a minimum which results in a phase error of: for kb equal to

where the denominator accounts for multiple reflections between and port 2. For a lossless cap the amplitude of is unity so that it may . Then, be represented by

(12)

(18)

A 90 degree phase difference is created when the effective radians from the no-detuning length: length is changed by

The unknowns in this equation are and . and are the measured reflections (impedance) at port 1 with the antenna radiating and in the cap, respectively; is the angle of the cap reflection at port 2, known from (2)–(4). If were varied through all phases, would trace a perfect circle on the complex plane determined by the denominator in (18).

(13) For a

mode this equation in (6) and (8) yields: (14)

The net external susceptance, , at the interface with the inplus the shunt inductance, (a/c). That is: ternal structure is (15) Thus, a 90 degree deviation in the cap reflection phase from the no-detuning phase is related to the normalized change in susceptance by: (16) It can be shown that the same relation applies to the reactances mode. for a The 90 degree cap phase deltas occur when the normalized susceptance/reactance values are 1. If an antenna is matched and , and radiates a and lossless, i.e.,

(19a) (19b) (19c) (19d) where C is the complex center, R is the real radius. The asterisk superscript, , indicates complex conjugation. For small , equal increments in yield equally spaced points on the circle, the points are clustered where is a but for large minimum. V. DETERMINING EFFICIENCY AND SCATTERING COEFFICIENTS Various techniques can be used to determine the efficiency and scattering coefficients from the radiating and capped reflec-

THAL: A CIRCUIT MODEL FOR SPHERICAL WHEELER CAP MEASUREMENTS

tion coefficient, depending on the characteristics of the antenna and the number of spherical (or non-spherical) caps.

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Eliminating

yields: (24)

A. Double Spherical Caps (or Triple) Equation (18) is linear in the two unknowns, may be written as:

and

, and (20)

If measurements are made with two different spherical caps, the scattering coefficients and efficiency, , may be found from:

where is the reflection coefficient measured at the matched frequency, i.e., at resonance, in a cap that does not detune the anis the diametrically opposite point on the circle which tenna. would be measured with maximum cap detuning or at frequencies far off resonance. If the antenna is not matched, it can be matched numerically using the following transformation: (25)

(21a) (21b) (21c) and are the measured port 1 reflections with the where and are computed from (2)–(4) based on two caps and the antenna pattern, i.e., mode. For best accuracy the two caps should provide significantly different values of . An optimum choice is for one cap to provide little detuning and the other to produce maximum detuning. From Table I caps with kb equal to 2.8 and 4.5 would approximate this situation for ka up to 1.0. This procedure yields the complex scattering coefficients as a function of frequency across a band. No circuit assumption is involved, and it can be applied to a multiply-tuned antenna as long as there is just a single dominant radiating mode. This technique could be applied, for example, to a folded conical helix [5] which has patterns similar to a spherical helix [6] but has a double-tuned response. Equation (20) can be rearranged to be linear in three un, , and . Therefore, as an alternative knowns: all of the scattering coefficients could be determined by substituting a third spherical cap measurement for the free space test and solving the three linear equations. B. Single Cap When only one cap is used, it is necessary to make some assumptions or auxiliary measurements to determine the other two unknowns provided by a second cap. For a lossless antenna the unitary condition of the scattering matrix provides the additional constraints; although this case may be of academic interest, it is not useful in determining antenna efficiency. Practical techniques can be realized by assuming certain general properties typical of many small antennas. 1) Determining Efficiency: Consider the following amplitude equation derived from (18) for a matched antenna. (22) has unit amplitude, and the maximum For a lossless cap, occur when it is in phase and out of and minimum values of giving: phase with (23a) (23b)

Then, if the cap phase could be varied by changing kb, it would have to be adjusted to find the minimum and maximum values of the transformed and apply these to (24) to find the efficiency. However for small antennas with a single resonant response, changes in the cap dimension should essentially just shift the resonant response. Therefore, as an alternative to changing kb, the frequency can be varied to find the that minimizes the magnitude of the transformed from (25), using the resonant , and this transformed value can be used as in value of (24). This step, of course, represents an approximation which is most accurate when there is little cap detuning. The coefficients of the matching transformation may be taken to be independent of frequency over the narrow bands of interest. (Frequency dependence could be included by multiplying in (25) by The far off resonance response could be used to find an equiv. On the other hand, small antennas can generally be alent represented by an equivalent circuit in which all of the losses are simulated by a single resistance. In this case the circle due to varying the cap phase must be tangent to the unit circle (i.e., the rim of the polar impedance chart) at one point. The proof follows from a general theorem for lossless three-port junctions [7]: There is always a phase of total reflection at one port (the cap phase) that decouples the other ports (the loss resistance is and port 1). At this phase there is no dissipation and unity. Both series and parallel circuits commonly used in describing the Wheeler cap procedure, e.g. [8]–[11], fall within this class. Subject only to these assumptions, the efficiency is determined from at the frequency, , where the transformed value is a minimum. (26a) (26b) 2) Sensitivities: If it further assumed that at resonance has the same phase as at the frequency of (over-coupled) or the opposite phase (under-coupled) the efficiency may be written as: (27) must be reversed for the over-coupled case; the sign of for the under-coupled case. This condition is satisfied exactly for series and parallel circuits and is convenient for deriving the

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sensitivities to experimental errors in tivities are:

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and

. The sensi-

(28a) (28b) Although the procedure defined by (26) assumes that is unity, it is useful to know the sensitivity of this assumption.

(28c) C. Multiple Arbitrary Caps The amplitudes of and can be determined from the with three (or more) different caps of unmeasurement of known phase. The points lie on a perfect circle, defined by (18), regardless of the form of the circuit and the location of the port 1 reference plane (subject to the single radiating mode assumption), and standard equations can be used to find the complex center and radius of the circle from the measured ’s. This technique was described by Johnston and McCory [12] using a somewhat different formulation. For good accuracy the points should be widely separated around the circle, and one should be in the vicinity of the minimum , i.e., where there is little detuning. Then, the scattering coefficients and efficiency follow from the radius and the complex center of the circle defined in (19): (29a) (29b) (29c) 1) Frequency Variation: Under certain circumstances the data obtained with multiple caps may be approximated with a single cap by sweeping the frequency. Geissler et al. [13], [14] and Lai et al. [15] form circles and McKinzie [9] creates conductance circles in this manner. These techniques may not produce a true circle [16], depending on the internal circuitry and the port 1 reference plane. The most sensitive portion of is the “circle” in determining the efficiency is where minimum. If the cap significantly detunes the antenna, the frequencies near this minimum may lie well outside the intended operating range of the antenna and have a dominant influence in determining the apparent efficiency by these techniques as well as by the one described in Section V-B-1. The constant-current loop example illustrates this effect. Fig. 4 shows how the distribution of on a polar chart is affected by the cap radius, kb; they are widely spaced near the critical minimum and compressed near the rim of the chart which may affect how the frequencies are weighted in determining the “circle.” D. Non-Spherical Caps and Losses When an antenna radiates a single mode into a conducting spherical cap, the same mode is reflected with little loss. Multiple reflections between the cap and the antenna increase the

Fig. 3. Loop circuit (not including the cap circuit or the radiation resistance).

losses somewhat, but as long as the antenna is well matched is much less than one, and the increase is small (see (18)). If the cap is not spherical (or hemi-spherical), the outward traveling wave is reflected in other modes in addition to the radiare “cutoff” before ated mode. The higher-order modes reaching the antenna and reflect back towards the cap, continuing multiple reflections until they are eventually dissipated in the cap or converted back into the original radiated mode. These trapped modes introduce notches in at their cap resonant frequencies. In the case of a “half” cap over a ground plane, additional loss may be introduced by the ground plane and its contact with the cap [17]. The amount of cross-coupling of modes depends on the symmetry of the cap, its orientation relative to the phase center of the antenna, and the radiated pattern. A cylindrical cap with its height equal to its diameter and a cube should have relatively low mode conversion; an offset cap would have more. Even a spherical cap will have some conversion if its origin is not at the phase center [8], which can be determined from some combination of symmetry, analysis, or measured pattern. VI. EXAMPLES The procedures described above can be demonstrated by using circuits that are known from references to predict the radiation Q and patterns of actual antennas to generate simulated experimental data from which the “measured” efficiency may be derived. The efficiency, ka, and tuning configuration are easily varied to facilitate studying the effects of the cap radius, the sensitivity to experimental errors, and the influence of a second weakly excited mode since the exact efficiency is known. A. Constant-Current Loop—TE Mode Fig. 3 shows the circuit for a constant-current loop (or half mode. The series capacitance loop) radiating a single and shunt inductance, both (a/c), at the port 2 end represent the external circuit elements. The shunt inductance, w(a/c), and resistance, r, model the magnetic energy and losses due to higher-order modes around the wire which have mode index and decay rapidly away from the loop., The third magnetic inductance, (1/2)(a/c), accounts for the internal energy [18]. The capacitances p(a/c) and s(a/c) provide options for resonating the loop. Port 2 can radiate into free space or into a spherical cap as represented by the circuits in Fig. 1. Port 1 is connected to a transmission line with characteristic impedance, .

THAL: A CIRCUIT MODEL FOR SPHERICAL WHEELER CAP MEASUREMENTS

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TABLE II SIMULATED MEASURED AND REDUCED DATA FOR LOOP IN EIGHT DIFFERENT SPHERICAL CAPS: ka

r

= 1:848

0

Fig. 4. Reflection coefficients, , for loop radiating and in 2 caps; each capped response is centered on its ; the relative frequency separation of the points is 0.0025 with a total spread of 0.050 for each case.

f

Table II gives simulated “measured” data in spherical caps with eight different radii, kb, for a loop with a resonant , , , and , over-coupled. The radiation Q is representative of tape loops having a width-to-diameter ratio between 0.3 and 0.5 [19]. The intentional mismatch at resonance is intended to demonstrate all of the processing steps. The table gives the reflection coeffidefined in (26a). cients at resonance and at the frequency of , is shown on the bottom The radiating reflection coefficient, line. Fig. 4 displays a polar chart for two of the caps. in phase at resonance due Column 4 illustrates the change to cap detuning, relative to the phase with no detuning at ; note that the detuning phase variation with kb generally follows Fig. 2 which was derived independently of any internal antenna structure. Column 7 gives the relative frequency shift for each cap; the maximum delta is approximately three at . times the delta to the nominal band edge, and the efficiency, , from The last two columns give (26) which applies since the losses are represented by a single

= 0:5; w = 2=3; r = 0:00556; p = 16:68; s = 1:465;

resistance, even though this antenna model is not ostensibly a simple series or parallel circuit. The “measured” efficiency varies by approximately 1.5 percent from the exact efficiency of 0.80, given on the last line, over the range of cap diameters. Since the simulated experimental data are perfect, this variation varies with detuning rather must be due to the fact that than simply shifting in frequency as it would for an ideal series or parallel tuned circuit. The impact of measurement errors can be estimated from the sensitivity coefficients in (28). For no detuning they are: ; ; , with relatively little variation over the range of kb. As as example, an at (column 5) from 0.75 to 0.77 ( 0.23 dB) increase in would cause an error in efficiency of percent. A decrease in from the assumed value of unity to 0.98 ( 0.18 dB) would yield an error of 0.46 peris close to the phases at cent. (Note that the phase of which is the assumption made in (27) and in the derivation of the sensitivities.) The preceding analyses assume that the loop is excited at two points, easily accomplished with a half loop over a ground plane, or is tuned with a capacitance diametrically opposite a single feed point, so that a virtually uniform current can be maintained up to ka equal to 0.5 [19, table 4]. On the other hand, if the loop is excited at a single point, the current distribution becomes nonuniform at relatively small values of ka and power is radiated in mode as well as in the . Since the two modes are the affected differently by the cap, the Wheeler cap procedure fails instead of because [20]. (The mode is designated the z axis is defined in the plane of the loop and not along its axis.) B. Spherical Dipoles—

Mode

Two designs consisting of wires on a spherical surface can mode. In be used to illustrate antennas that radiate the the first design the wires are wound on the sphere in a helical mode is excited to resonate manner so that the inductive mode [6], [21]. The performance of this antenna can the be modeled by coupling the circuits for the two spherical modes as shown in Fig. 5 which is a modified version of [18, Fig. 4]. This circuit accurately predicts the radiation Q and the pattern mode. including the cross-polarization lobes due to the The dissipation is modeled by inserting a resistance in series inductance where most of the magnetic energy with the

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TABLE III “MEASURED” AND REDUCED DATA FOR SPHERICAL DIPOLES IN SPHERICAL CAPS; N

Fig. 5. Circuit for spherical dipole with coupled TM =TE cluding the cap circuits or the radiation resistances).

modes (not in-

is stored. The measured in a spherical cap is simulated by replacing the radiation resistance of each mode by its respective modal cap circuit. In the second design, the tuning inductance is provided by coils following lines of longitude [22]. These coils excite higher-order modes that are essentially non-radiating so that mode. In order to allow the antenna radiates only the a controlled comparison, port 3 can be terminated in a unit resistance and the entire lower half of the circuit, i.e., the portion, can be considered to be internal tuning circuitry so that as a function of frequency. the two designs have identical cap circuit on The capped is modeled by placing the port 2. Consider both designs with ka equal to 0.5, a match at res, and a radiated efficiency of 0.90. onance, i.e., The total efficiency of the helical design is 0.9074 when the radipower is included. The simulated data are processed ated in the same manner as for the loop, using (26); the results are summarized in Table III in the same format used for the loop. Column 4 shows that a small cap always detunes the mode; the phase is negative as predicted by Fig. 2 for kb less than 2.8; the smallest cap that does not detune has .

= 3 344; r = 0 1116 :

:

There are errors in the efficiency of the helical version for small values of kb where there is significant detuning of the mode as shown in Fig. 2. The mode is 21.3 dB mode in this design with . Decreasing below the the pitch of the helix changes the transformer ratio in the circuit , representation and lowers the resonant frequency. At for example, the power is 27.0 dB down, and its influence levels represent an extreme case of reduced. These power, the result of equal energy in the two modes; most dipole antennas would have lower levels. (The three-port circuit for the helical antenna has 6 unique complex coefficients and would require 5 spherical caps for a rigorous measurement.) The data for the longitudinal-coil design give an accurate efficiency result over the entire range of kb. The average values of the sensitivity coefficients over the range of the table are: ; ; . VII. CONCLUSIONS A circuit model of an antenna either radiating or in a spherical cap has been presented. The model applies to antennas that radiate a single dominant mode as is characteristic of electrically-small antennas. It allows the detuning of the antenna by the cap to be estimated quite accurately, totally independent of the physical structure located within a specified enclosing spherical surface. It demonstrates that the detuning behavior is different for loop-type (TE) or dipole-type (TM) radiators so that the optimum cap diameter depends on the polarization of the antenna being measured. The efficiency and complete scattering matrix of an antenna can be determined from its reflection coefficient measured in two spherical caps and radiating into free space or with a third cap in place of the free space measurement. When only one cap measurement is available, assumptions about the antenna are needed to compensate for the missing data. If the losses are concentrated in a single element, as in any antenna that can be represented by a simple resonant circuit, special equations for the efficiency and the scattering and sensitivity coefficients in terms of the single measurement apply. Equations for these measurement options have been provided.

THAL: A CIRCUIT MODEL FOR SPHERICAL WHEELER CAP MEASUREMENTS

Simulated test data have been generated using accurate circuit representations of loop and spherical dipole antennas to illustrate the data reduction steps and the sensitivities and to demonstrate that the detuning follows the predictions made independent of the actual antennas. The spherical dipole example shows mode on the results is that the contaminating effect of an relatively small when a suitable cap diameter is used. REFERENCES [1] H. A. Wheeler, “The radiansphere around a small antenna,” Proce. IRE, vol. 47, pp. 1325–1331, Aug. 1959. [2] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl Phys, vol. 19, pp. 1163–1175, Dec. 1948. [3] H. L. Thal, “Exact circuit analysis of spherical waves,” IEEE Trans. Antennas Propag., vol. AP-26, no. 2, pp. 282–287, Mar. 1978. [4] D. L. Rode, “Concentric spherical resonator eigenfrequencies,” IEEE Trans. Microw. Theory Tech, pp. 369–372, Jun. 1968. [5] J. A. Dobbins and R. L. Rogers, “Folded conical helix antenna,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 1777–1781, Dec. 2001. [6] 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. [7] C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits, ser. Radiation Laboratory Series. New York: McGraw-Hill, 1948, vol. 8, sec. 9.1. [8] D. M. Pozar and B. Kaufman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. AP-36, no. 1, pp. 136–139, Jan. 1988. [9] W. E. McKinzie, III, “A modified Wheeler cap method for measuring antenna efficiency,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 13–18, 1997, vol. 1, pp. 542–545. [10] H. Choo, R. Rogers, and H. Ling, “On the Wheeler cap measurement of the efficiency of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2328–2332, Jul. 2005. [11] R. H. Johnston, “Evaluation of Wheeler cap antenna efficiency measurement methods using numerical EM simulation data,” presented at the IEEE Int. Symp. Antennas Propag., Jul. 5–11, 2008, paper 234.9. [12] R. H. Johnston and J. G. McCory, “An improved small antenna radiation-efficiency measurement method,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 40–48, Oct. 1998. [13] M. Geissler, O. Litschke, D. Heberling, P. Waldow, and I. Wolfe, “An improved method for measuring the radiation efficiency of mobile devices,” in Proc. IEEE Int. Symp. Antennas Propag., Jun. 22–27, 2003, vol. 4, pp. 743–746. [14] M. Geissler, O. Litschke, D. Manteuffel, and M. Arnold, “Antennas for mobile terminals and their exact measurement characterization,” URSI Gen Assembly. New Dehli, Oct. 2005 [Online]. Available: http://www.ursi.org/Proceedings/ProcGA05/pdf/CB.4(01478).pdf

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[15] Q. Lai, C. Fumeaux, G. Almpanis, H. Benedickter, and R. Vahldieck, “Simulation and experimental investigation of the radiation efficiency of a dielectric resonator antenna,” presented at the Proc. IEEE Int Symp/ Antennas Propag., Jul. 5–11, 2008, paper 114.9. [16] T. Salim and P. S. Hall, “Efficiency measurement of antennas for on-body communications,” Microw. Opt. Technol. Lett., vol. 48, no. 11, pp. 2256–2259, Nov. 2006. [17] E. H. Newman, P. Bohley, and C. H. Walter, “Two methods for the measurement of antenna efficiency,” IEEE Trans. Antennas Propag., vol. AP-23, no. 4, pp. 457–461, Jul. 1975. [18] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [19] H. L. Thal, “A reevaluation of the radiation Q bounds for loop antennas,” IEEE Antennas Propag. Mag., vol. 51, no. 3, pp. 47–52, Jun. 2009. [20] G. S. Smith, “An analysis of the Wheeler cap method for measuring the radiating efficiency of antennas,” IEEE Trans. Antennas Propag., vol. AP-27, no. 4, pp. 552–556, Jul. 1977. [21] 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. [22] J. J. Adams and J. T. Bernhard, “A low Q electrically small spherical antenna,” presented at the IEEE Int. Symp. Antennas Propag., Jul. 5–11, 2008, paper 209.1.

Herbert L. Thal, Jr. (A’53–M’58–SM’82–F’89– LF’96) was born in Mount Vernon, NY. He received the B.E.E., M.E.E., and Ph.D. degrees in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1953, 1955, and 1962, respectively. From 1953 to 1956, he was a Research Associate at Rensselaer Polytechnic Institute, Troy. In 1956, he joined the General Electric Co., in Schenectady, NY, where he performed research and development on circuits and beam interactions in fixed-frequency and voltage-tunable magnetrons, multiple-beam klystrons, and distributed amplifiers. From 1967 to 1989, he was with GE in King of Prussia, PA, where he worked for on the prediction, control, and measurement of radar cross section, the analysis and measurement of reflector and lens antennas and their feed systems, and the analysis, development and computer-aided testing and tuning of microwave filters and multiplexers. From 1989 to 1992, he was involved with filter and multiplexer development at Microlab/FXR, Livingston, NJ. Since 1992, he has worked in Wayne, PA, as an independent consultant on antennas and passive microwave components. He has been an Adjunct Professor at Drexel University and the University of Pennsylvania. Dr. Thal is a Registered Professional Engineer in Pennsylvania and is a member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi.

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Edge-Born Waves in Connected Arrays: A Finite Infinite Analytical Representation Andrea Neto, Senior Member, IEEE, Daniele Cavallo, Student Member, IEEE, and Giampiero Gerini, Senior Member, IEEE

Abstract—Connected arrays constitute one of the most promising options for wideband phased arrays. Like most phased arrays, they are designed using infinite array theory. However, when finiteness is included, edge effects perturb their behavior. These effects are more severe when the arrays are designed to operate over very broad frequency ranges, since the mutual coupling between the elements facilitates the propagation of edge-born waves that can become dominant over large portions of the arrays. Finite array simulations, which would predict these behaviors, are computationally unwieldy. In this paper we present a Green’s function based procedure to assess edge effects in finite connected arrays. First the electric current distribution on the array is rigorously derived. Later on, the introduction of a few simplifying assumptions allows the derivation of an analytical approximation for the current distribution. This latter provides meaningful insights in the induced dominant edge-wave mechanism. The efficiency of connected arrays as a function of their dimension in terms of the wavelength and of the loading feed impedances is investigated.

Fig. 1. Two-dimensional connected array of dipoles with infinite number of feeds.

2 finite

I. INTRODUCTION N the last years, connected arrays have attracted a growing attention for wideband applications ranging from wide-angle scanning arrays [1], [2], to focal plane arrays for multi-beam imaging [3]. Their wideband performance is due to the fact that the connections between neighboring elements allow currents to remain nearly constant with frequency [4] and each element is effectively larger than its physical dimension. The connections also support the propagation of guided waves from one element to the other. However, as discussed in [5]–[7], these guided waves can be very strongly excited at the edges of the array. As a consequence, the overall behavior of a finite wideband array can be dramatically different with respect to the design based on infinite array analysis. Even if not in the context of connected arrays, [8] and [9] also investigated in detail the effects of strong guided waves associated to the finiteness of wideband dipole arrays. In this paper, we present an accurate and analytical procedure to accurately assess edge effects, already in the preliminary design phase of connected arrays. This appears to be particularly

I

Manuscript received December 07, 2010; revised February 23, 2011; accepted April 04, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. A. Neto is with the Telecommunication Department of the EEMCS Faculty, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). D. Cavallo and G. Gerini are with the Netherlands Organization of Applied Scientific Research (TNO, Defense, Security and Safety), 2597 AK Den Haag, The Netherlands and also with the Faculty of Electrical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163788

needed because finite array full-wave simulations with general purpose tools, even when possible, are too demanding in terms of computational resources to be a useful design tool. Specifically, this paper will focus only on finiteness effects along the longitudinal direction (along which the dipoles are connected), -axis in Fig. 1, to present properties that specifically characterize connected arrays. In fact, the effects of finiteness associated with the transverse direction, in Fig. 1, are dominated by space wave coupling. These effects have been extensively discussed in the dated literature [10] (and there cited references), more recently resorting to windowing type of approximations [11], and lately with analytically enhanced full-wave solutions in [12]–[14]. The latter works heavily relied on the ray field representations introduced by [15], and then refined in a number of more detailed works [16]–[19]. Here we investigate an infinite number of dipoles along the transverse direction, with each dipole fed at a finite number of points in the longitudinal direction, as in Fig. 1. The starting base for the analysis is the availability of transmission lines Green’s Functions (GF) of infinitely extended connected arrays. The derivation of these GF was initiated for the slotted case in [5], then extended to dipoles in [20], and generalized for both transmitting and receiving arrays including loads in [21]. Here, the effects of the array finiteness are explicitly addressed for the first time. In the first part of the paper, the current distribution is rigorously derived resorting to the transmission line GF formalism. The global current distribution is obtained via a MoM-like numerical inversion procedure, which requires only one unknown per elementary cell, independently from the cell geometrical

0018-926X/$26.00 © 2011 IEEE

NETO et al.: EDGE-BORN WAVES IN CONNECTED ARRAYS: A FINITE

INFINITE ANALYTICAL REPRESENTATION

parameters. This is possible thanks to the use of an integral equation with Kernel characterized by the appropriate connected array GF. Results obtained using this methodology are compared with full-wave simulations using commercial software, showing excellent agreement at much lower computational costs. For practical designs, there is no limit to the longitudinal number of elementary cells that can be studied with this method. Both the cases of connected dipole arrays with and without backing reflectors are considered. Using this procedure, important design considerations regarding the role of the loads in the propagation of edge waves are provided. The method allows one to estimate the efficiency of connected arrays that are large or small in terms of the wavelength at very limited computational cost. In the second part of the paper, in order to gain a deeper physical insight into the wave mechanisms occurring in connected arrays, a different approach is proposed. This latter method is based on the representation of the electric current along each long dipole as the superposition of an infinite array contribution plus edge-born waves. While infinite array current components are rigorously derived resorting to the full GF formalism, edge-born waves are approximated as a staircase distribution. It is important to note that this approximation would be totally inadequate if referred to the entire current distribution. However, it leads to small errors in absolute terms when applied only to the edge born contributions. Thanks to this simplification, a single spectrally analytical approximation of the edge currents is obtained. The singularities of this spectrum can be investigated and the pertaining inverse Fourier integrals can be asymptotically evaluated to provide the analytical expressions for the spatial currents. These latter analytical steps are performed only in the cases of arrays in free space and scanning in the -plane, in order to maintain the analytical formulation as simple as possible, while still highlighting the main mechanism. Important potentials remain for future developments of the theoretical formulations.

Fig. 2. Equivalent problem of a connected array of dipoles with infinite number of feeds.

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2 finite

this hypothesis in the actual solution is minimal, just like the infinite length of the dipoles themselves, and it is only retained for the sake of a clearer and simpler formulation. The problem could also be set up with the dipole assumed to be backed by an infinite ground plane at distance . In the present modelling, we assume that both the ground plane and the dipoles are infinitely extended and thus the finiteness of the metallizations is not accounted for. The derivation of the GF for doubly infinite, periodically excited, connected arrays with the inclusion of the loads was pre,a sented in [21]. In the case of a finite number of feeds along the similar integral equation for the unknown current zeroth dipole can be used. One should only take care that in the right hand side (RHS) of (9) in [21], the incident field is now defined over a finite number of feeds, to . The incident field can be assumed to be concentrated in the dipole gaps and uniformly distributed ( -gap excitations). Thus, the relevant integral equation is given by:

II. SET UP OF THE SPECTRAL EQUATION: THE FINITE INFINITE ARRAY CASE The geometry of the problem under analysis is depicted in Fig. 1, for arrays of connected dipoles operating in transmission (Tx). The reception (Rx) case will be discussed in Appendix B, since it does not present particular difficulties, but requires a somewhat different notation. The dipoles, of width and sepalong , are electrically connected along arated by distance . When the array is transmitting, the longitudinal direction points , spaced by period each dipole is fed at . The excitations on the zeroth dipole are realized by lumped voltage generators with internal source impedance and voltages . For all other dipoles along the direction, a progressing phase is imposed: , where is the free space wave number, and indicate the pointing direction of the main beam. The equivalent planar problem is shown in Fig. 2 for an array in free space. Note that, even if the dipoles are fed at a finite number of points , it is assumed that the loads are periodically distributed over the entire length of the infinitely extended dipoles. The impact of

(1) •

is the space domain Green’s function once the dependence from the transverse dimension is accounted for ((6) in [21]); • the incident electric field at each feeding gap is expressed via , with if the array is scanning towards . • is the average current flowing in the -th gap (see (3)); for and • elsewhere. Resorting to the same techniques shown in [21], (1) can be solved in the Fourier domain leading to an expression for the spectral current distribution along the dipoles. The spectrum of the current can be written as follows:

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(2)

is given in (18) and (17) where the explicit expression of of Appendix A, for the cases with and without backing reflector, respectively. The expression of the current spectrum in (2) is given only . In the implicitly, since it depends on the unknown terms remainder of this paper, we propose two different methods to derive an explicit expression for the spatial current distribution over the dipoles. In Section III, a rigorous numerical solution that involves a matrix inversion is presented. Finally, in Section V, we discuss the analytical approximations that allow to explicitly highlight the finiteness effects. III. NUMERICAL SOLUTION A simple numerical procedure to solve (2) is provided here. can be expressed as functions of the spectrum at The terms the left hand side (LHS). To this goal, let us recall the definition of the average currents on the gaps

(3)

Substituting in (3) the spatial current distributions , expressed as inverse Fourier Transform of (2), after a few simple algebraic manipulations, leads to (4) where the infinite summation of loads has been restricted to elements, including two dummy elements at each edge of the array. These are typically sufficient, for non negligible values of , to replace the infinite summation in the RHS of (2). The mutual admittance term in (4) is defined as

(5)

Equation (4) can be written in matrix form, leading to a system of linear equations that can be solved by matrix inversion as (6) is the identity matrix, and is the vector of the impressed voltages. The inversion leads to the exact solution for the average currents on the gaps including the effects of the loads. The elements of the admittance matrix in (5) can be evaluated numerically by performing the spectral integral with convenient where

Fig. 3. Deformation of the integration path in (5): (a) highly-coupled elements; (b) low-coupled elements.

deformations of the original integration path on the real axis. Fig. 3 shows the complex topology and the branch cuts asso) square root of the GF that ciated with the first (for appears in (17) and (18) of Appendix A. The branch points are in the case an array scanned in the longitudinal direction in , or ). ( ), an inteFor highly-coupled elements (small factors gration path deformation as in Fig. 3(a) has been used to avoid , the integrands the branch cuts. For large distances present faster oscillations on the real axis. Thus, a path deformation as in Fig. 3(b) usually guarantees faster convergence in the free space cases and whenever poles of the stratification’s GF are not captured in the deformation. In the case of the array operating in the presence of a backing reflector, the height of the array from the reflector is typically such that no poles are exaxis. Further poles could arise pected to be found in the real in the case the array is printed on a grounded dielectric slab. The presence of these poles would correspond to possible excitation of surface and leaky waves. However, these configurations are not considered useful from a design point of view, regardless of the theoretical interest their analysis can arise.

A. Results of the Numerical Solution The active currents at the gaps calculated via (6) can be used to evaluate the active impedances of the finite array, given by [21]. Fig. 4 shows a comparison between the numerical solution presented here and simulated result obtained via Ansoft HFSS [22]. Fig. 4(a) refers to an array in free space that is excited at 15 feeding points along and is infinite along . The array periods , where is the wavelength at 10 GHz. are and The other geometric parameters of the array are . Fig. 4(b) refers to the same array where a backing from the dipoles. reflector is included at a distance Curves in Figs. 4(a) and (b) are shown for broadside radiation and scanning to 45 on the -plane, while Fig. 4(c) refers to the same array in free space scanning to 45 on the diagonal plane . A very good agreement can be observed when comparing full-wave HFSS simulations and the numerical solution presented here. Note that, once the average currents on the gaps have been obtained, the total current on the array can be expressed using (2), and consequently all important parameters of the array, including the radiation patterns, can be obtained.

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Fig. 5. Efficiency of a connected arrays with backing reflector as a function of the scanning angle on the E -plane: Comparison between infinite array approx. imation and finite array analysis Z

( = 400 )

2

Fig. 4. Active input impedances of a 15 infinite array of connected dipoles and  on in (a) free space and (b) with backing reflector, for  the E -plane, and (c)  on the diagonal plane ' in free space: comparison between the numerical solution and Ansoft HFSS.

= 45

=0 = 45 ( = 45 )

IV. EFFICIENCY OF FINITE CONNECTED ARRAYS In the previous section we have derived a reliable and fast solution for the current distributions at the feeds of a finite connected array. The main advantage of this formalism is that the efficiency of a scanning connected array can be evaluated much more accurately than would be possible with only infinite array solutions. In the present context, by the term array efficiency feeding we refer to the impedance mismatch at each of the points of the array. The array is assumed to be fed by transthat ensures the mission lines with characteristic impedance widest usable BW at broadside. For each feed we can define an , active reflection coefficient is the active impedance at the element. We in which can also associate with the same element a mismatch efficiency . Clearly, the matching of each element will depend on the frequency and the scanning angle. As a conse, is defined as quence, the overall efficiency of the array, the average efficiency of the array as follows: (7)

Especially for array scanning to wide angles and for arrays composed by only a few elements, the current distributions over the finite arrays are significantly different from the infinite array ones. As a consequence, the active impedances are different from those that would be expected only on the base of infinite array designs. Fig. 5 presents the overall array efficiency, defined as in (7), as a function of the scanning angle on the -plane, for different frequencies. The array under analysis is composed of 8 elements and is operating in the presence of a backing reflector. The specific dimensions are taken from an array design discussed in [20] , ( being the wavelength at the frequency ) and refer to with . It is apparent that, for larger a load resistance of scanning angles, the finite array simulations show important differences with respect to the infinite array ones. In practice, the array differences between the exact and approximate modelling are significant when the arrays are not perfectly matched. The availability of an accurate and rapid finite array modelling tool is key in real designs especially if the threshold of acceptable – . functionality is defined for scanning toward Fig. 5 presents the resulting overall array efficiency, as a function of the number of elements of the array, for different scanning angles. The figure presents results for two different arrays, and , both with backing reflector at designed in such a way that the active impedances are well and 100 characteristic impedance lines, refed by 400 spectively. A first predictable consideration is that, when the number of elements of the array tends to be large, the simulations assuming infinite or finite arrays imply similar efficiencies. A second non obvious design aspect emerges from these simulations. For arrays designed to operate well when fed by low impedance feeding lines, the edge effects are more important than for arrays designed to be fed by high impedance lines. Thus

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a designer should avoid antenna design that apparently (with infinite array simulations) require low input impedances. Because in reality in these cases the edge effect dominate a much larger portion of the array and the asymptotic behavior of a large array is only achieved for unrealistically large arrays. This can only be explained by digging deeper into the physics of finite connected arrays. V. SPECTRAL INTEGRAL APPROXIMATION Although the numerical solution presented in Section III is efficient (one unknown per array element) and accurate, it does not provide physical insight on the nature of the edge-waves. In order to obtain an alternative, more insightful, representation it is useful to recall how the infinite array auxiliary problem is can be set up. By simple extension of (1), the current represented as the solution of

Fig. 6. Efficiency of a connected arrays with backing reflector as a function of the number of elements, for different scanning angles on the E -plane. Also reported are the asymptotic values assumed in the infinite array cases. Geometrical parameters are such that array is well matched to (a) 400 loads and (b) to 100 loads.

(8) is assumed to be Once the solution for the current known, as shown in [20], the electric currents in a finite connected array can be expressed in a form that highlights edge effects as follows:

Fig. 7. Comparison of the numerical solution in Section III and the spectral analytical integration, for arrays pointing toward broadside: (a) free space, (b) with backing reflector at distance h = 0:1 .

(9) The edge term represents the perturbation induced by finiteness effects. Proceeding as in Appendix C and assuming the validity of the following conditions (10) it is possible to achieve a rigorous representation of the spatial current distribution. In these cases, the edge currents can be expressed as a single spectral integral, avoiding the necessity to perform the matrix inversion in (6)

(11) This last spectral integral can be performed numerically along the path defined in Fig. 3(a). A validation of the procedure is shown in Figs. 7 and 8, which show the magnitude of the average currents in the gaps, normalized to the infinite array solution. The results are for connected arrays in free space and in the presence of a backing reflector, when scanning toward , respectively. The arrays are fed broadside and toward . For the array in free at 31 points, spaced by ) source space and the one backed by a reflector (at

Fig. 8. As in Fig. 7, but for arrays pointing toward 

= 60

on the E -plane.

impedances of and are assumed, respectively. The remaining geometrical parameters of the arrays . considered are An excellent agreement is obtained between the results predicted by the integration procedure and the fully numerical inversion when (10) are verified. It can be noted that in the considered cases the finiteness of the array can cause variations of the current distribution, with respect to the infinite array solution, corresponding to unity in the graphs, of up to 70% for large scanning angles. These variations are still well represented with the integral solution since this latter does not include any important approximations in the simple cases. This accuracy is maintained also for much smaller or larger arrays. Similar curves, describing edge effects in finite arrays have been first observed in [23], [24] and interpreted with a heuristic Gibbsian model.

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the gap dimension, and accordingly defining the edge-born currents.

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play only a minor role in

VI. FREE SPACE CASE: UNIFORM ASYMPTOTIC EVALUATION OF THE INTEGRAL APPROXIMATION Equation (12) represents a single integral expression for the edge-born currents in a connected array excited at a finite number feeding points. The asymptotic evaluation of this integral can provide the physical insight that is now missing. In order to maintain the formulation as simple as possible, only the case of connected arrays operating in free space will be considered. The first step to the evaluation is the recognition of two separate terms, each associated with one edge of the array. A. Contributions From the Two Edges

Fig. 9. Comparison of the numerical solution and the spectral integral solution for a connected array in free space at four different frequencies: (a) 0:2f , (b) 0:4f , (c) 0:8f , (d) f ; the load impedance is equal to 200 .

The summation over the auxiliary contributions from sources external to the array can be expressed in closed form leading to two contributions associated with the left and right edges of the array as follows:

A. Extrapolation of the Simple Case Solution Equation (11) was obtained thanks to the simplifications in (10). Specifically the second hypothesis is instrumental for the algebraic operations on the spectra. If one assumes that it makes sense to have the load impedance distributed over the entire , while the feeding is only applied to a region , with cell these two parameters being different, (11) can be extrapolated as follows

(13) Note that the first summation is convergent for , while the second one for . The introduction of (13) in (12) leads to

(12) where and , possibly different from , is now in the argument of the sinc function. Fig. 9 shows a comparison of the numerical solution and the spectral integral solution for the currents in the gaps normalized to the infinite array solution. The connected array is in free space with 15 feeds along and loaded by 200 impedances and the is scanning toward 45 on the -plane. The four graphs refer to four different frequencies at which the array periodicity and and the gap dimensions are , respectively. The agreement in Fig. 9 is excellent at low frequencies and shows only minor deviations at higher frequencies where the critical small period approximation that justifies the stair case distribution begins to fail. Overall, it appears that while the low frequency approximation is instrumental to the spectral expression to work, the non verification of the second hypothesis in is not important from the point of view of the re(10) sults accuracy. A possible explanation is that, to the first order, a good approximation of the reactive energy associated with the cell is already included in the infinite array approximation and

(14) This representation highlights the presence of a number of poles, which emerge from the zeroes of associated with the Floquet Waves (FWs) in . Besides FW poles, the integrand also possesses other types of singularities, specifically the branch points as in Fig. 3 and complex poles associated to the disper. Their location in the complex sion equation that characterize the plane depends on the actual loads feeding lines. The approximate solution of this dispersion equation is reported in Appendix D. The real and imaginary part of the dominant pole are plotted in Fig. 10, for a case and . The characterized by figure compares the analytical solution provided in (38) with a numerical solution based on a simple descent along the gradient

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Fig. 10. Real and imaginary part of the pole k array in free space.

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versus load resistance for an

following an accurate starting point. For small loads and unattenuated waves are supported by connected arrays. However for large loads, the imaginary part of this propagating mode is highly negative. It should be noted that these poles correspond to purely attenuating waves due to losses (feed absorption). These are not leaky waves. After the discussion on the singularities, also the approximate analytical evaluation of the two integrals in (14) is derived in Appendix D. The analytical expressions for the current contribution born from the left edge of the array is given by Fig. 11. Comparison between the analytical solution and the numerical method . for an array of 101 elements in free space: (a) broadside and (b) 

= 60

(15)

. In (15) the slope Fresnel function is with , where is the Kouyintroduced: oumjian Fresnel function [26]. This latter is defined in (49) in Appendix D. Also in (15)

(16)

and defined in (45). The with current contribution born at the right edge of the array can be similarly expressed and is reported in the Appendix D. B. Comments on the Analytical Solution A comparison between the results obtained resorting to the analytical expressions in (15), or (50) and (52), and the numerical method is shown in Fig. 11. The array is radiating toward on the -plane. The array is assumed to broadside and feeding points, and the geometrical be composed by , where is the calculation parameters are

frequency, and . It is apparent that the proposed analytical solution is extremely accurate also for arrays that are scanning at very wide angles. The availability of an analytical expression allows one to give qualitative considerations on the nature of the electric current distribution. For high values of the loads , and observation points close to broadside, and the arguments of the functions are large. This means that the current distribution from , which each edge is dominated by the spreading factor is associated with a rapid decay as a function of the distance from the end points. For lower values of , the load induced can be close to the branch point . Also, for obserpole is close to . When vation toward wide scanning angles, either of the two situations occurs, the transition functions argument tends to zero, and the Fresnel function can be approximated by . This implies that the domor inant term to the current distribution is of the type , which present no geometrical spreading and only a or no attenuation at small exponential attenuation all . dependence of the current distribuSpecifically the tion is shown here to emerge from an analytical Green’s function for the first time. It express the idea that the load/source impedances attenuate the edge waves consuming their energy. This mechanism is probably occurring in all arrays, not only in connected arrays, but to our knowledge was never given explicit evidence or demonstrated for any array.

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VII. CONCLUSION This paper has presented a novel analytical methodology to asses edge effects in connected arrays, which provides important guidelines to broad band phased array design. Analysis: Starting from the knowledge of the connected arrays GF, this paper has first presented the derivation of a general purpose numerical procedure for the assessment of the finiteness effects in connected arrays. This procedure is of general applicability in terms of types of array and scanning conditions. In fact, arrays with and without backing reflectors or dielectric stratifications and scanning in the -, - or diagonal planes can be analyzed. The numerical cost of the analysis is only the inversion of a matrix of dimension , where is the number of feeding points in the array along the longitudinal direction. The availability of such numerical procedure can provide unique design opportunities. It is particularly convenient when the performance of wideband wide-angle scanning arrays needs to be assessed in advance of measurements or full-wave, all inclusive, numerical simulations. In a second step, the representation of the total current in terms of the infinite array plus edge-born waves has been introduced. Thanks to this representation, simplifications that would otherwise be unreasonable can be adopted. These simplifications lead to a single analytical expression for the spectrum of the edge-born waves in cases of general applicability. Finally for the specific case of a connected array of dipoles array operating in free space, and scanning only in the -plane, a rigorous analytical approximation of the pertinent inverse Fourier transforms allows an analytically rigorous expression of the edge-born waves. The expression is given in terms of standard Fresnel functions which highlight similarities between the edge-induced currents in connected arrays and the edge-born currents in the canonical problems of diffraction from half planes. Design: From the design point of view the main findings of this investigation are as follows. 1) Edge effects are fundamental to assess the behavior of connected array in wide angle scanning situations. Especially when the arrays are composed of a small number of elements , infinite array simulations are just not good enough to predict the performance. The free space case, treated here analytically, gives good physical understanding and guidelines. 2) The intensity of the edge-born waves is only mitigated by the source load impedances. The information of the load impedance is crucial to assess finite array effects. This was previously anticipated by Hansen in [2] and Munk [8]. The origin of this phenomenon is believed to be explained here for the first time. 3) The intensity of the edge waves is more important for low loading and source impedances. High impedance (400 Ohms) reduces edge-born waves in connected arrays with a backing reflector.

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For arrays in free space, the explicit expression of the GF is given by (17) while in the case of a backing reflector at a distance array, the expression becomes as follows:

from the

(18)

APPENDIX B RECEIVING MODE In the receiving mode, the source is assumed to be an incident , since the response of the plane wave from the direction structure to any other source can be represented as the superposition of responses to a spectrum of plane waves. For plane wave incidence and arrays in free space the incident electric field can expressed spectrally as (19) is the amplitude of the incoming plane wave. A more where general expression of the incident field will be dependent on the specific stratification considered. The case of connected arrays in the presence of backing reflector is of particular interest in this paper and would imply a multiplying factor in (19). The average currents in the gaps can be expressed as (20) where (21) The solution for the spatial current distribution in the rein (6), ceiving mode is obtained substituting are defined as in (21). where the elements of the vector APPENDIX C SPECTRAL REPRESENTATION OF THE EDGE CURRENT Using (9), the integral (1) for the finite array can be re-expressed as

APPENDIX A CONNECTED ARRAY GF’S This appendix reports the explicit expression of the GF for connected arrays, for both cases in which the array is operating in free space and in the presence of a backing reflector.

(22)

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where the following notation was used

(23) Subtracting (8) from (22) leads to Fig. 12. Original the integration path for the evaluation of (29).

APPENDIX D UNIFORM EVALUATION OF THE INTEGRAL

(24) and in genIn this equation the only unknown is eral it cannot be solved in a simple spectral way. However, a particular geometrical case can be solved exactly in a spectrally analytical form. Simple Case: Low Frequency and Generator Distributed Over the Entire Cell The simpler case that can be solved in a closed form is obtained with the configuration in which (25) The first hypothesis leads to the possible use of a stair case approximation for the edge born currents (26)

The integral defining and in (14) are converging over different integration paths. The convergence of the requires a counter-clock circling of the poles integral (see Fig. 12). On the contrary, the integration path of should be deformed in the half plane . These different complex plane topologies suggest two specular uniform asymptotic evaluations for the two integrals. In the following , since the evaluation of can the focus will be on be performed in essentially the same way, once the change of is introduced. variable Dominant Singularities Before proceeding with the asymptotic evaluation, it is useful to isolate the zeroth order from the higher order FWs in the transverse GF terms pertaining to the free space as follows: case (17). This is obtained by representing

The second hypothesis, which implies that the load is distributed over the entire elementary cell, allows one to identify on the LHS and the RHS of (24). This leads the same to the following simplified integral equation

(30)

(27)

Using this representation it is simple to express the loaded transverse GF in terms of as follows:

This equation can be simply solved analytically once it is expressed in the spectral domain (28) The spatial current distribution is then given by an inverse Fourier integral. By projecting the spatial current distribution on the gaps, assuming Volt, we can evaluate the expression of the currents at each feeder of the finite array as follows

(29)

(31) where (32)

Since for well sampled arrays the function is slowly varying with , it is legitimate to approximate . This approximation helps to recognize (31) as a second degree polynomial function of , which can be expressed highlighting its roots (33)

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where (34) Using (33), the contribution due to the first edge in (14) can be then expressed as follows: Fig. 13. Deformation of the integration path for the evaluation of (29): Steepest Descent Path.

(35) is expressed with explicit roots In (35) the denominator in . These roots define the branch cuts of the complex plane, . in particular the top/bottom Riemann sheet, i.e., These roots are also associated with poles in plane, as in will be discussed in the following.

Steepest Descent Path (SDP) in order to perform an uniform asymptotic evaluation of the integral as in Fig. 13. Multiplying and dividing the integrand of (35) for the factor we obtain, after a few algebraic steps, the following expression:

Load Dependent Pole It is simple to verify that is associated with values of far from the the branch points for any values of . It is then plane the poles associated with , useful to locate in the , from (34), as a function of the load impedance . For and consequently . For small values it results of a second order approximation of the square root function in (34) for small argument leads to

(36)

(39) This integral can be deformed into the SDP around the saddle as in Fig. 13. Note that in the deformation none of point the poles associated with the FWs are captured since the original integration path Fig. 12) surrounds all poles counter-clockwise. and in (34) are also not captured The poles defined by in the deformation. From (39) we can then define two contributions as follows: (40)

which implies where (37)

(41)

where and are very small real positive functions of the geometrical parameters at play. Consequently, the approximate is now as follows: expression of

(42) The term (43)

(38) which explicitly shows that, for small values of the load impedance , the dominant poles are located close to the and show small imaginary parts that tend to branch point increases. The pole has a become more negative as was negative imaginary part. The corresponding root in chosen with negative imaginary part, implying that the pole represented in (38) is not associated with a leaky-wave, but with a damped wave. Since the damped wave is located -plane, is in the top Riemann sheet of the complex not captured when deforming the integration path along the

includes both the constants and the slower varying portions of . The intethe integrand from (39), approximated in grand in (42) presents no square root type of branches. Accordingly, in the upward and downward path that define the SDP the integrand is the same, so that the two half paths contributions cancel out. The integral in (41), instead, requires an uniform asymptotic and in can be close to the evaluation since the poles in for particular geometrical, loading or scanning branch point configurations. Before performing the evaluation, it is convenient to express the integrand in a form where the mentioned and in are shown explicitly [25]. This can be poles in

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achieved by adding and subtracting the quantity as follows: (44)

. Proceeding in the same way the and expression for the current born at the right edge of the array can be expressed as

where we defined the function (45) is a smooth regular function in the The function vicinity of the SDP and consequently can be approximated with . The integral in its value at the saddle point (41) can be split into two contributions as follows:

(52) where the substitution the exponential

and are defined with and in the definition of in (43) is replaced by . REFERENCES

(46) While the first of the two integrals in (46) is already in a canonical form, the second one can be brought to the same form by recognizing that

(47) After these manipulations, the three terms composing (41) can be all expressed analytically resorting to the following mathematical identity [16]:

(48) where the slope Fresnel function is introduced: . Here is the Kouyoumjian Fresnel function [26], which is defined as

(49) Using (48), after a few simple algebraic manipulations, the final expression of the current contribution born from the left edge of the array is given by

(50) with

. In (50)

(51)

[1] J. J. Lee, S. Livingston, and D. Nagata, “A low profile 10:1 (200–2000 MHz) wide band long slot array,” presented at the IEEE Antennas and Propag. Int. Symp., San Diego, CA, Jul. 5–11, 2008. [2] R. C. Hansen, “Linear connected arrays,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 154–156, 2004. [3] S. Hay, F. Cooray, J. O’Sullivan, N.-T. Huang, and R. Mittra, “Numerical and experimental studies of a dual-polarized planar connectedarray antenna for the Australian square kilometer array pathfinder,” presented at the IEEE Antennas and Propag. Int. Symp., Charleston, SC, Jun. 1–5, 2009. [4] R. C. Hansen, Phased Array Antennas, 2nd ed. Hoboken: Wiley, 2009, ch. 12. [5] A. Neto and J. J. Lee, “Ultrawide-band properties of long slot arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 534–543, Feb. 2006. [6] J. L. Volakis, Antenna Engineering Handbook, 4th ed. : McGrawHill, 2007. [7] C. Craeye and X. Dardenne, “Element pattern analysis of wide-band arrays with the help of a finite-by-infinite array approach,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 519–526, Feb. 2006. [8] D. S. Janning and B. A. Munk, “Effects of surface waves on the currents of truncated periodic arrays,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1254–1265, Sep. 2002. [9] O. A. Civi and P. H. Pathak, “Array guided surface waves on a finite planar array of dipoles with or without a grounded substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2244–2252, Aug. 2006. [10] N. Amitay, V. Galindo, and C. P. Wu, Theory and Analysis of Phased Array Antennas. New York: Wiley, 1972. [11] A. K. Skrivervik and J. R. Mosig, “Finite phased array of microstrip patch antennas: The infinite array approach,” IEEE Trans. Antennas Propag., vol. 40, no. 5, pp. 579–582, May 1992. [12] F. Capolino and M. Albani, “Truncation effects in a semi-infinite periodic array of thin strips: A discrete Wiener-Hopf formulation,” Radio Sci., vol. 44, no. RS2S91, Apr. 2009, DOI: 10.1029/2007RS003821. [13] A. Neto, S. Maci, G. Vecchi, and M. Sabbadini, “A truncated Floquet wave diffraction method for the full wave analysis of large phased arrays—Part I: Basic principles and 2-D cases,” IEEE Trans. Antennas Propag., vol. 48, no. 3, pp. 594–600, Mar. 2000. [14] H.-T. Chou, H.-K. Ho, P. H. Pathak, P. Nepa, and O. A. Civi, “Efficient hybrid discrete Fourier transform-moment method for fast analysis of large rectangular arrays,” IEE Proc. Microwaves, Antennas Propag., vol. 149, no. 1, pp. 1–6, Aug. 2002. [15] L. Carin and L. B. Felsen, “Time harmonic and transient scattering by finite periodic flat strip arrays: Hybrid (ray)-(Floquet mode)-(MoM) algorithm,” IEEE Trans. Antennas Propag., vol. 41, no. 4, pp. 412–421, Apr. 1993. [16] F. Capolino, M. Albani, S. Maci, and R. Tiberio, “High-frequency analysis of an array of line sources on a truncated ground plane,” IEEE Trans. Antennas Propag., vol. 46, no. 4, pp. 570–578, Apr. 1998. [17] O. A. Civi, P. H. Pathak, and H.-T. Chou, “On the Poisson sum formula for the analysis of wave radiation and scattering from large finite arrays,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 958–959, May 1999. [18] F. Capolino, M. Albani, S. Maci, and L. B. Felsen, “Frequency domain Greens function for a planar periodic semi-infinite phased array. Part I: Truncated Floquet wave formulation,” IEEE Trans. Antennas Propag., vol. 48, no. 1, p. 6774, Jan. 2000. [19] F. Capolino, M. Albani, S. Maci, and L. B. Felsen, “Frequency domain Greens function for a planar periodic semi-infinite phased array. Part II: Diffracted wave phenomenology,” IEEE Trans. Antennas Propag., vol. 48, no. 1, p. 7585, Jan. 2000.

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INFINITE ANALYTICAL REPRESENTATION

[20] A. Neto, D. Cavallo, G. Gerini, and G. Toso, “Scanning performances of wide band connected arrays in the presence of a backing reflector,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3092–3102, Oct. 2009. [21] D. Cavallo, A. Neto, and G. Gerini, “Equivalent circuits for connected arrays in transmission and in reception,” IEEE Trans. Antennas Propag. , accepted for publication. [22] Ansoft HFSS, Ver. 11. Pittsburgh, PA: Ansoft Corporation [Online]. Available: http://www.ansoft.com [23] R. C. Hansen and D. Gammon, “A Gibbsian model for finite scanned arrays,” IEEE Trans. Antennas Propag., vol. 44, no. 2, pp. 243–248, Feb. 1996. [24] R. C. Hansen, “Finite array scan impedance Gibbsian models,” Radio Sci., vol. 31, no. 6, pp. 1631–1637, Mar. 1996, doi:10.1029/ 96RS01366. [25] B. L. Van der Waerden, “On the method of saddle points,” Appl. Sci. Res., vol. B2, p. 3345, 1951. [26] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, pp. 1448–1461, Nov. 1974.

Andrea Neto (M’00–SM’10) received the Laurea degree (summa cum laude) in electronic engineering from the University of Florence, Italy, in 1994 and the Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2000. Part of his Ph.D. was developed at the European Space Agency Research and Technology Center, Noordwijk, The Netherlands, where he worked for the antenna section for over two years. In 2000–2001, he was a Postdoctoral Researcher at California Institute of Technology, Pasadena, working for the Sub-mm wave Advanced Technology Group. From 2002 to January 2010, he was Senior Antenna Scientist at TNO Defence, Security and Safety, The Hague, The Netherlands. In February 2010 he has been appointed Full Professor of Applied Electromagnetism at the EEMCS Department, of the Technical University of Delft, the Netherlands. His research interests are in the analysis and design of antennas, with emphasis on arrays, dielectric lens antennas, wideband antennas, EBG structures and THz antennas. Dr. Neto was co-recipient of the 2008 H.A. Wheeler award for the best applications paper in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the 2008 Best Innovative Paper Prize at the 30th ESA Antenna Workshop, and the 2010 Best Antenna Theory Paper Prize at the European Conference on Antennas and Propagation (EuCAP). He presently serves as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

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

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

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Mismatch of Near-Field Bearing-Range Spatial Geometry in Source-Localization by a Uniform Linear Array Yu-Sheng Hsu, Kainam Thomas Wong, and Lina Yeh

Abstract—Many near-field source-localization algorithms intentionally simplifies the exact spatial geometry among the emitter and the sensors, in order to speed up the signal-processing involved. For example, the Fresnel approximation is a second order Taylorseries approximation. Such intentional approximation introduces a systemic error in the algorithm’s modeling of the actual objective reality from which the measured data arise. A mismatch thus exists between the algorithm’s presumptions versus the data it processes. This modeling-mismatch will introduce a systematic bias in the bearing-range estimates of the near-field source-localization algorithm. This bias is non-random, and adds towards the random estimation-errors due to the additive and/or multiplicative noises. The open literature currently offers no rigorous mathematical analysis on this issue. This proposed project aims to fill this literature gap, by deriving explicit formulas of the degrading effects in three-dimensional source-localization, due to approximating the source/sensor geometry by any order of the Taylor’s series expansion. Index Terms—Acoustic interferometry, array signal processing, direction of arrival estimation, interferometry, linear arrays, nearfield far-field transformation, phased arrays, sonar arrays, underwater acoustic arrays.

I. INTRODUCTION HE wavefront, emitted from a point-source, is necessarily spherical. Only if this emitter lies sufficiently far from the receiving sensors, then the wavefront may appear to the sensorarray as effectively planar. Otherwise, for an emitter in the near field of a sensor-array, the emitted wavefront’s curvature must be affronted by any near-field source-localization algorithm to estimate the emitter’s angle-of-arrival and range. Many near-field measurement-models choose to simplify the exact geometric relations among the near-field source and sensors, in order to speed up the signal-processing, even

T

Manuscript received January 31, 2010; revised January 17, 2011; acceptedMarch 09, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported by the Hong Kong Polytechnic University’s Internal Competitive Research Grant #G-YF22. Y.-S. Hsu is with the Department of Mathematics, National Central University, Chung-Li 32001, Taiwan (e-mail: [email protected]). K. T. Wong is with the Department of Electronic & Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: [email protected]). L. Yeh is with the Department of Mathematics, Soochow University, Taipei, Taiwan. 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.2011.2163749

though the precise spatial geometry can be easily derived. These approximations use some finite-order Taylor-series, to “expand” the phase-delay with respect to the emitter-sensor range. For example, the “Fresnel approximation” [4] represents a second-order Taylor-series expansion of the phase-delay among identical isotropic sensors in a uniformly spaced linear array (ULA), to produce a second-order polynomial with respect to the sensor-index [3], [5], [7]–[13], [15]–[23], [25]–[40]. Similarly, [6], [24] discard the exact ULA array-manifold for an approximation consisting of a sum of approximate array-manifolds, each corresponding to a different order of a Taylor-series expansion of the exact array-manifold. A systemic model-mismatch is thus intentionally introduced by the above references into their bearing-range measurementmodel, in order to facilitate their algorithms’ signal processing. This model-mismatch is non-random in nature, in contrast to the customary additive/multiplicative noise. Source-localization performance would be degraded by this deliberately introduced model-mismatch, besides the usual degradation by the additive/ multiplicative noises. No comprehensive mathematical analysis exists on how such model-mismatch degrades near-field bearing-range estimation: Much of the uniform linear array bearing-range estimators are assessed via limited Monte Carlo simulations, as in [3], [5], [7], [9], [11], [13], [15], [17]–[26], [28], [29], [31]–[34], [36], [39], [40]. Also, the performance analysis and/or Cramer-Rao bound in [10], [12], [16], [19], [20] are derived as if the mismatched measurement-model were the true measurement-model; hence, such analysis has not accounted for any model-mismatch effect. [3] directly affronts the measurement-model mismatch errors, but with very limited Monte Carlo simulations, for ad hoc scenarios, and only for the second-order approximation. No systematic and rigorous analysis is available in [3] nor anywhere in the open literature. This paper fills this literature gap, by rigorously deriving the bearing-range estimation-errors due to any order of a Taylor-series expansion of the exact near-field array-manifold of a uniform linear array of identical isotropic sensors.1 1To design near-field DOA-estimation algorithms, the receive-antennas’ array manifold differs from the far-field case on at least two levels: (a) the antenna-electromagnetics (including mutual coupling effects), and(b) the spatial geometry that inter-relates the transmit-antenna and the receive-antennas, apart from antenna-electromagnetics. Factors (a) and (b) are each highly complicated to analyze, but both are essential considerations in designing a practical DOA-estimation algorithm for the near field. The present paper offers analytical insights on (b), as one useful step towards constructing better near-field DOA-estimators.

0018-926X/$26.00 © 2011 IEEE

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would equal , . A one-to-one corresponand the -element dence exists between the ordered pair .3,4 vector B. Approximate Measurement-Models Obtained From the Taylor’s Theorem The Taylor’s theorem can approximate a differentiable funcaround a given point , by a polynomial whose coeftion ficients depend only on that function’s derivatives. More mathematically,

(3)

Fig. 1. An emitter impinging from the near field upon a linear array of uniformly spaced sensors.

II. NEAR-FIELD BEARING-RANGE MEASUREMENT-MODELS Consider an immobile point-source impinging from a direction-of-arrival (DOA) of (measured from the positive -axis) and a range of from the origin of the Cartesian coordinates. The emitted signal is narrowband, of a known wavelength , with a complex-value envelope constant at unit-power.2 The above emitting source impinges upon an array of iden, for tical isotropic sensors, located at { } in two-dimensional Cartesian space, where denotes the inter-sensor uniform spacing, does not exceed , and is a priori known. This source lies in the sensor-array’s near field, i.e., the source-sensor distance is sufficiently short such that the emitted signal’s wavefront cannot be regarded as planar (as in the far-field case), but circular.

symbolizes the th derivation of with respect where , and . to Apply the th-order Taylor’s expansion to , with respect to the defined in (1). The resulting approximate would account for terms up to only the th power of .5 Explicit expressions of will be derived in the subsequent sections. Inciden, tally, the exact measurement model in Section II-A has . whereas the Fresnel approximation has To obtain the th-order approximate phase-difference, apply (3) to (2) (4) The approximate, , in (4) leads to an approximate , with its th element near-field array-manifold vector equal to

A. True Measurement-Model Based on the Exact Source-Sensor Spatial Geometry

(5)

Denote the distance between the source and the th sensor as . Geometric considerations in Fig. 1 give:

where symbolizes the th element of the vector inside the square brackets. C. The Model-Mismatch Problem in Bearing-Range Estimation

(1)

The th sensor’s temporal phase-difference (relative to an hypothetical sensor at the Cartesian origin) equals

(2)

array-manAll sensors’ data may be collected into an . The th isotropic sensor’s data, if noiseless, ifold vector 2Most direction-finding papers would model s(t) as zero-mean, Gaussian, and temporally uncorrelated. To avoid unnecessary distraction from this present work’s analysis of the near-field array-manifold model-mismatch problem, the incident signal’s complex-valued envelope is herein modeled as a unit constant.

The model-mismatch problem occurs under these circumarises in the real world stances: The collected data as described in Section II-A, but the estimation algorithm to equal the of wrongly presumes this being the Section II-B, with the consequence of wrong estimate. Here, and is each a bivariate function of and . This model-mismatch produces an estimation bias , even under noiseless conditions. The of present paper will rigorously derive analytical expressions of and , , each as an explicit function of , , , and . To avoid distraction from the present focus 3The measurement model below ignores antenna-electromagnetics, such as mutual coupling between antennas. Please see [14], especially chapter 7 therein. 4The spherical spread factor has also been overlooked in the measurementmodels of [3], [8]–[13], [16]–[18], [20], [22], [23], [25]–[36], [38]–[40] for near-field direction-finding. 5Recall from (1) that x depends on, but does not equal to, `. Hence, more than the first k derivatives are needed to obtain the first k powers of `.

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on the data-model mismatch problem, the propagation-channel gain has been set at unity in (1). III. ANALYSIS OF THE SECOND-ORDER (FRESNEL) MODEL-MISMATCH

(8)

A. Explicit Formulas for the Second-Order Mismatched Measurement Model

of

This second-order approximation known “Fresnel approximation”.

The Taylor’s theorem stipulates that for some

is the well

B. Explicit Formulas for the Second-Order Model-Mismatched and Approximates, This subsection will derive the model-mismatched approxand , explicitly in terms of the measured data imates, of Section II-A. Here, the model-mismatched estimation-algorithm takes the measured data,7 computes (9) but erroneously equates it to

(10)

(6) represents a number whose magnitude is where , with referring to a positive upper-bounded by constant roughly equal to .6 Note that the term containing has been absorbed into at the second equality above. To have applied . Taylor’s theorem to (1) as above, it is necessary that This condition is equivalent to

where symbolizes the real part of the complex number inside the curly brackets. and in terms of the measured data To express , set in (10) and then to obtain two equations, subject to the condition of , which must hold under the physically true model of (1).8 From (8) to (10), , , 2. Solving these two equations for the two unknowns for and , of

(11) (7) (12) Consequently, the th sensor’s temporal phase-difference (relative to an hypothetical sensor at the Cartesian origin) equals

C. Derivation of the Second-Order Model-Mismatched and Estimation Errors, Equations (11) and (12) of Section III-B have related the second-order model-mismatched estimators and to the 7To focus on the model-mismatch, the subsequent analysis will assume noiseless data and an unbiased estimator. 8If other values of ` are used here, (11) and (12) would change, but the order of error would remain the same. More precisely, for ` = ` in 1; 2; ;L

6

0

follows: Out of (1=16)x there emerge terms of the form =C (`(1=r )) [ 2 cos ] , where i and j are positive integers such that 2i + j = 4, and C denotes some constant for each i; j . Therefore, O (1 =r ) = Q(1 =r ), where Q is a sum C )L . with terms of the form C ` [ 2(cos )] , with Q 16(max If L is large, 16(max C )L = O(L ). 6The

justification

is

as

(5=128)(1=[1 + x ] )x in (6), C (`(1=r)) [ 2`(cos )(1=r)]

0

j

0 j

0



j

j



= cos

r 

=

` `

` ` `

0` ` 4

0` 0` `

` `

111 g

 2 1

(2 )

0`

f

`

0

` 0` (` 0 ` )

where = a ` + a ` and = a ` + a ` .

:

HSU et al.: MISMATCH OF NEAR-FIELD BEARING-RANGE SPATIAL GEOMETRY IN SOURCE-LOCALIZATION BY A ULA

Fig. 2. a. The second-order model-mismatched estimation-error j

0

j. b. The second-order model-mismatched estimation-error (jr

true values and , but only implicitly. This subsection will rigorously derive explicit relationships for the second-order modeland . mismatch estimation errors, 1) An Explicit Expression for : The entity in (11) needs to be expressed explicitly in terms of and . From (8) and (9)

(13) where the under-braced equality is obtained from (10) and from and in (8). the definitions of Substituting (13) in (11),

(14)

Hence, the second-order model-mismatch estimation-error equals:

(15) The above derived equation is plotted in Fig. 2(a), with a sup. port-region subject to (7) for

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0

rj)=1.

Below are some qualitative observations on (15): {A} From the trigonometric dependence on in (15) and veras . ified in Fig. 2(a): This concurs with the specification in [4] (on page 74) of “the region of validity for the Fresnel approximation” as . versus {B} To investigate how (15) behaves over over , define to facilitate the subsequent discussion. , if and if {B-1} It holds that . However, , and satisfying . {B-2} The curly-bracketed term in (15) is anti-symover . Conmetric with respect to sequentially, , and . {B-3} As the curly-bracketed term in (15) is positive but negative , it holds that , and . , the square bracketed term {C} From (15): As , approaches 1, the curly bracketed term approaches hence . This is expected because the near field approaches the far field in such case. Fig. 2(a) also confirms this and furthermore shows a monotonically deas increases. creasing 2) An Explicit Expression for : To derive an explicit in terms of and , express of expression of (12) explicitly in terms of and . From (8) and (9),

(16)

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where the under-braced equality is obtained from (10) and from the definitions of and in (8). From (12), (13), and (16) (18)

To obtain the th-order approximated measurement model, , thereby apply Taylor’s theorem to the above-defined giving the following expansion:9

All above lines use the fact that and have the same order . of Hence, the second-order model-mismatch estimation-error is given by

(19) with

, and

(17) (20) The above derived equation is plotted in Fig. 2(b), with a sup. port-region subject to (7) for Below are some qualitative observations on (17): , like , is independent of but {D} and . dependent on only {E} From the trigonometric dependence on in (17) and verified in Fig. 2(b), as though not near 0 or . would monotonically decrease with {F} From (17), , and generally would not an increasing approach zero even as . : {G} To investigate the sign of , it may be shown that . {G-1} For : {G-2} For , iff . As and have the same sign, iff .

Multiply out each term in (19). Next, , (i) sum all terms containing to give , for some constants dependent on and ; into a new variable (ii) sum all terms containing ; and into another new (iii) sum all terms containing . variable Then, the exact phase-delay of (18) and (19) may be rewritten as

(21) 9The

following intermediate steps give (19):

IV. ANALYSIS OF THE (GENERAL) TH-ORDER MODEL-MISMATCH

+

A. The th-Order Mismatched Measurement-Model From (1) and (2), the phase-delay equals exactly:

g

g (x) = g (0) +

) rp

1+x

0r

=r

(0)

i! (x )

g

g

(k + 2)! (0)

i! +r

x +

g

(x ) (k + 2)!

Multiply each term by 2=, to give (19).

(0)

(k + 1)!

x

x +r

g

g

(0)

(k + 1)!

x

:

x

x

HSU et al.: MISMATCH OF NEAR-FIELD BEARING-RANGE SPATIAL GEOMETRY IN SOURCE-LOCALIZATION BY A ULA

The in (21) denotes the th-order approximation of (18). Moreover, and of the exact temporal delay are already given in (8) for the Fresnel approximation. (It is well known in mathematics that lower-order and higher-order Taylor series expansions share the same low-order coefficients except the remainder.) Appendix I shows explicitly in terms of the model-parameters. B. Explicit Formulas for Any th-Order Model-Mismatched and Approximates, The th-order model-mismatched algorithm takes the meaof Section II-A, but mistakes it as sured data

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The above (29) and (30) express the th-order approximates explicitly in terms of the measured data:

(31) (32)

C. Derivation of Any th-Order Model-Mismatched Estimation Errors, and 1) An Explicit Expression for physically arise from (21), which gives

: The collected data

(22) This subsection will derive the th-order approximates, and , explicitly in terms of the true parameters . Towards , this objective, first derive explicit expressions of , in terms of . Express the number of linear equations in (22) in terms of the number of scalar unknowns in matrix form:

(33) where (34) (35) (36)

(23) where represents a non-singular matrix with its th row , and equal to

(37) From (33), (38)

(24) (25) with , and lution

However, the (21) gives

th-order mismatched measurement-model of

being known values represents the transposition operator. The sofollows immediately from (23). Moreover,

(39) Hence, (38) becomes

(26) where

refers to the th row of

. Appendix II shows that (40) (27) (28)

where the subscript refers to the th element of the vector in [ ]. (which Hence, under the condition must hold for the physically true model in (1)),

The last equality holds by the definition of Similarly,

(41) Therefore, (31) and (40) imply that

(42)

(29) (30)

and by (39).

10Other and (32).

m , with ` 6= 1, 2, may be used instead to give results similar to (31)

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Fig. 3. a. The third-order model-mismatched estimation-error j

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011

0

j. b. The third-order model-mismatched estimation-error (jr

0

rj)=1.

(43) with the elements in defined in (27) and the elements defined in (48). In the above, (42) holds because in , where both terms are bounded in absolute value, by a product of some . Moreover, (43) holds due to the constant and Taylor-series Theorem. Thus, the th-order mismatched-model estimation-error equals: (44) with the elements in and already defined in (27) and , is (48), respectively. This derived formula in (44), for , 4, with plotted in Figs. 3(a) and 4(a) respectively for the support-region subject to (7) at . Below are some qualitative observations: number of nulls in , located {H} There are , for . Specifically, the at Fresnel approximation null at disappears for an odd . {I} Like point {C} in Section III-C-1 for the Fresnel approxas . Moreover, this imation, drop is more rapid as increases. , or {J} From Figs. 3(a) and 4(a), . : From (32), (40), and 2) An Explicit Expression for (41),

In the above, the exponent-order inside increments or is multiplied or divided by or by . The decrements as above has also used the fact that and are of the . Consequentially, same order as

For the th-order mismatched model, the estimation-error is therefore

(45)

, , and already defined in with the elements in and (27), (28), and (48), respectively. (Recall that both implicitly depend on , and This derived formula in , is plotted in Figs. 3(b) and 4(b) respectively (45), for for , 4, with the support-region subject to (7) at . Below are some qualitative observations: number of nulls in , located at {K} There are , for . Specifically, the

HSU et al.: MISMATCH OF NEAR-FIELD BEARING-RANGE SPATIAL GEOMETRY IN SOURCE-LOCALIZATION BY A ULA

Fig. 4. a. The fourth-order model-mismatched estimation-error j

0

j. b. The fourth-order model-mismatched estimation-error (jr

Fresnel approximation null at disappears for an odd . {L} Like point {C} in Section III-C-2 for the Fresnel approxias . Moreover, this drop mation, is more rapid as increases. and small is expected. {M} Fig. 3(b)’s spike at , which in turn This is because . Hence, as implies that , . D. Degeneration to the If

,

0

rj)=1.

V. CONCLUSION Analytically derived herein are the bearing-range estimation errors for a linear array of identical isotropic sensors, if the emitter/sensors’ near-field spatial geometry is approximated by a th-order Taylor expansion. These derived expressions are explicitly in terms of the model parameters, in closed form, and applicable for any general . Besides this spatial geometry factor, antenna-electromagnetics need also be considered in constructing the near-field measurement model.

Case ,

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APPENDIX I ,

.

Hence, the th-order approximates of (31) and (32) reduce to the second-order approximates of (11) and (12) respectively. Similarly, the th-order model-mismatched estimation errors of (44) and (45) become the second-order results in (15) and (17), respectively.

TO EXPRESS

OF (21) EXPLICITLY IN MODEL-PARAMETERS

TERMS OF THE

To derive

, of interest would be all terms of order . One such term exists in the second term of (19):

E. Convergence to the Exact Near-Field Measurement-Model as By (44) and (45) that , . Therefore, both model-mismatched estimation-errors can be rendered as small as needed, by choosing a sufficiently large . F. Convergence of the General th-Order Approximate Near-Field measurement Model to the Far-Field Measurement-Model Case as , , because . Likewise for for some positive constant

(46) , expand To identify other terms of order to obtain all th-order terms with :

For any particular

,

, be, for some positive cause constant . , any th-order Hence, as the source-sensor range approximate near-field measurement model will converge to the exact far-field model.

(47) where denotes the binomial coefficient . Consider the cases of the natural number being even or odd, separately:

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even : Here, (46), and (47),

for some positive integer

. From (22)

(48) where that odd : Here,

. Recall has already been defined in (20). . (48) remains true, but with replacing .

APPENDIX II EXPLICIT EXPRESSIONS OF Dilate

to a

AND

Vandermonde matrix,

.. .

.. .

.. .

.. .

.. .

where denotes a -element row-vector of all zeroes, and denotes a -element column-vector of all ones. Then, , where

represents a -element column-vector.

gives the solution of the matrix linear The th column of , where refers to a column-vector of all equation zeroes except a one at the th position. The solution has its elements equal to the ascending coefficients of the polynomial, [1]

Therefore, the elements of the th column of ascending coefficients of the polynomial

equal the

As a result, for any , the th entry of equals , where , with the sum taken over all distinct natural between 1 and but unequal numbers to . For , it is true that . The above gives: and . This produces the desired results in (27) and (28). ACKNOWLEDGMENT The authors would like to thank Y. Song, for plotting the graphs based on equations supplied to him.

REFERENCES

[1] N. Macon and A. Spitzbart, “Inverses of Vandermonde matrices,” Amer. Math. Month., vol. 65, no. 2, pp. 95–100, Feb. 1958. [2] B. D. Steinberg, Principles of Aperture and Array System Design. New York: Wiley, 1976, pp. 7–8. [3] A. L. Swindlehurst and T. Kailath, “Passive direction-of-arrival and range estimation for near-field sources,” in Proc. Annu. ASSP Workshop on Spectrum Estimation and Modeling, 1988, pp. 123–128. [4] L. J. Zoimek, “Three necessary conditions for the validity of the Fresnel phase approximation for the near-field beam pattern of an aperture,” IEEE Journal of Oceanic Engineering, vol. 18, no. 1, pp. 73–76, Jan. 1993. [5] D. Starer and A. Nehorai, “Passive localization of near-field sources by path following,” IEEE Trans. Signal Processing, vol. 42, no. 3, pp. 677–680, Mar. 1994. [6] J. C. Mosher and P. S. Lewis, “Taylor series expansion and modified extended prony analysis for localization,” in Proc. Asilomar Conf. on Signals, Systems and Computers, 1994, vol. 1, pp. 667–670. [7] J.-H. Lee, Y.-M. Chen, and C.-C. Yeh, “A covariance approximation method for near-field direction-finding using a uniform linear array,” IEEE Trans. Signal Processing, vol. 43, no. 5, pp. 1293–1298, May 1995. [8] A. Satish and R. L. Kashyap, “Multiple target tracking using maximum likelihood principle,” IEEE Trans. Signal Processing, vol. 43, no. 7, pp. 1677–1695, Jul. 1995. [9] R. N. Challa and S. Shamsunder, “High-order subspace-based algorithms for passive localization of performance of near-field sources,” in Proc. Asilomar Conf. on Signals, Systems and Computers, 1995, pp. 777–781. [10] S. Shamsunder and R. N. Challa, “Performance of near-field localization algorithms based on high-order statistics,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 1996, vol. 5, pp. 3010–3013. [11] M. Haardt, R. N. Challa, and S. Shamsunder, “Improved bearing and range estimation via high-order subspace based unitary ESPRIT,” in Proc. Asilomar Conf. on Signals, Systems and Computers, 1996, vol. 1, pp. 380–384. [12] K. Abed-Meraim, Y. Hua, and A. Belouchrani, “Second-order nearfield source localization algorithm and performance analysis,” in Proc. Asilomar Conf. on Signals, Systems and Computers, 1996, vol. 1, pp. 723–727. [13] K. Abed-Meraim and Y. Hua, “3-D near field source localization using second order statistics,” in Proc. Asilomar Conf. on Signals, Systems and Computers, 1997, vol. 2, pp. 1307–1311. [14] R. C. Hansen, Phased Array Antennas. New York: Wiley, 1998. [15] R. N. Challa and S. Shamsunder, “Passive near-field localization of multiple non-Gaussian sources in 3-D using cumulants,” Signal Processing, vol. 65, pp. 39–53, 1998. [16] N. Yuen and B. Friedlander, “Performance analysis of higher order ESPRIT for localization of near-field sources,” IEEE Trans. Signal Processing, vol. 46, no. 3, pp. 709–719, Mar. 1998. [17] M. Jian, A. C. Kot, and M. H. Er, “DOA estimation of speech source with microphone arrays,” in Proc. IEEE Int. Symp. on Circuits and Systems, 1998, vol. 5, pp. 293–296. [18] J.-H. Lee, C.-M. Lee, and K.-K. Lee, “A modified path-following algorithm using a known algebraic path,” IEEE Trans. Signal Processing, vol. 47, no. 5, pp. 1407–1409, May 1999. [19] J.-H. Lee and C.-H. Tung, “Estimating the bearings of near-field cyclostationary signals,” IEEE Trans. Signal Processing, vol. 50, no. 1, pp. 110–118, Jan. 2002. [20] E. Cekli and H. A. Cirpan, “Unconditional maximum likelihood approach for localization of near-field sources: Algorithm and performance analysis,” Int. J. Electron. Commun., vol. 57, no. 1, pp. 9–15, 2003. [21] Y. Adachi, Y. Iiguni, and H. Maeda, “Second-order approximation for DOA estimation of near-field sources,” Circ., Syst. Signal Processing, vol. 22, no. 3, pp. 287–306, Mar. 2003. [22] J.-F. Chen, X.-L. Zhu, and X.-D. Zhang, “A new algorithm for joint range-DOA-frequency estimation of near-field sources,” EURASIP J.Appli. Signal Processing, vol. 2004, no. 3, pp. 386–392, 2004. [23] B. A. Obeidat, Y. Zhang, and M. G. Amin, “Range and DOA estimation of polarized near-field signals using fourth-order statistics,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 2004, vol. 2, pp. 97–100.

HSU et al.: MISMATCH OF NEAR-FIELD BEARING-RANGE SPATIAL GEOMETRY IN SOURCE-LOCALIZATION BY A ULA

[24] E. Boyer, A. Ferreol, and P. Larzabal, “Simple robust bearing-range source’s localization with curved wavefronts,” IEEE Signal Processing Lett., vol. 12, no. 6, pp. 409–412, Jun. 2005. [25] E. Grosicki, K. Abed-Meraim, and Y. Hua, “A weighted linear prediction method for near-field source localization,” IEEE Trans. Signal Processing, vol. 53, no. 10, pp. 3651–3660, Oct. 2005. [26] Y. Wu, L. Ma, C. Hou, G. Zhang, and J. Li, “Subspace-based method for joint range and DOA estimation of multiple near-field sources,” Signal Processing, vol. 86, no. 8, pp. 2129–2133, August 2006. [27] S. Yu, W. Shuxun, and Z. Zijing, “An algorithm for near field source localization based on multistage wiener filters,” in Proc. Int. Conf. on Signal Processing, 2006, vol. 1. [28] J.-C. Huang, Y.-W. Shi, W.-D. Zhang, and J.-W. Tao, “Joint DOA, range and polarization estimation of near-field sources using second order statistics,” in Proc. Int. Conf. on Machine Learning and Cybernetics, 2006, pp. 3470–3477. [29] Z. Huang, S. Wang, and B. Wang, “A method for 4-D parameter estimation of near-field sources,” in Proc. Int. Conf. on Networking, Sensing and Control, 2006, pp. 997–1000. [30] J. Liang, S. Yang, J. Zhang, L. Gao, and F. Zhao, “4D near-field source localization using cumulant,” EURASIP J. Adv. Signal Processing, vol. 2007, no. 1, p. 126, Jan. 2007. [31] W. Zhi and M. Y. Chia, “Near-field source localization via symmetric subarrays,” IEEE Signal Processing Lett., vol. 14, no. 6, pp. 409–412, Jun. 2007. [32] Y. Wu, H. C. So, C. Hou, and J. Li, “Passive localization of nearfield sources with a polarization sensitive array,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2402–2406, Aug. 2007. [33] Y. Shi, Z. Huang, and S. Wang, “An algorithm for 4-D parameters jointly estimating of near-field sources,” in Proc. Int. Conf. on Wireless Communications, Networking and Mobile Computing, 2007, pp. 1012–1015. [34] K. Deng, Q. Yin, and H. Wang, “Closed form parameters estimation for near field sources,” in IEEE Int. Symp. on Circuits and Systems, 2007, pp. 3251–3254. [35] H. He, Y. Wang, and J. Saillard, “Near-field source localization by using focusing technique,” EURASIP J. Adv. Signal Processing, vol. 2008, 2008. [36] X. Yan, S. Wang, K. Wang, and H. Jiang, “Localization of near field cyclostationary source based on fourth-order cyclic cumulant,” in Proc. Int. Conf. on Signal Processing, 2008, pp. 1629–1632. [37] H. Xu, Y. Su, C. Huang, M. Lu, and H. Wang, “Estimation of direction-of-arrival by an active array,” Congr. Image and Signal Processing, vol. 5, pp. 277–280, 2008. [38] W.-J. Zeng, X.-L. Li, H. Zou, and X.-D. Zhang, “Near-field multiple source localization using joint diagonalization,” Signal Processing, vol. 89, no. 2, pp. 232–238, Feb. 2009. [39] X. Yan, H. Jiang, K. Wang, and S. Wang, “Broadband near-field range and bearing estimation based on fourth-order cumulants,” in Proc. Int. Conf. on Communications and Mobile Computing, 2009, vol. 1, pp. 43–46. [40] K. Deng, Q. Yin, and H. Wang, “Blind ranges, frequencies and DOAS estimation for near field sources,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 2009, pp. 2125–2128. [41] H. He, Y. Wang, and J. Saillard, “Focusing-based approach for wideband source localization in near-field,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 2009, pp. 2129–2132.

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[42] K. Deng, Q. Yin, H. Wang, and W. Wang, “Range and DOA estimation of near-field sources via the fourth-order statistics,” in Proc. IEEE Int. Symp. on Circuits & Systems, 2009. Yu-Sheng Hsu received the B.S. degree in education from National Taiwan College of Education (Chang-Hua, Taiwan) in 1980, the M.S. degree in mathematics from National Central University (Chung-Li, Taiwan) in 1982, and the Ph.D. degree in statistics from Rutgers University (New Brunswick, NJ), in 1990. He has been an Associate Professor at National Sun Yat-Sen University (Kaohsiung City, Taiwan) from 1990 to 1992, and at the National Central University since then. His research interests are probability and statistics.

Kainam Thomas Wong (SM’01) received the B.S.E. degree (chemical engineering) from the University of California (Los Angeles) in 1985, the B.S.E.E. degree from the University of Colorado (Boulder) in 1987, the M.S.E.E. degree from Michigan State University (East Lansing) in 1990, and the Ph.D. in electrical and computer engineering from Purdue University (West Lafayette, IN) in 1996. He was a Manufacturing Engineer at the General Motors Technical Center (Warren, MI) from 1990 to 1991, and a senior professional staff member at the Johns Hopkins University Applied Physics Laboratory (Laurel, MD) from 1996 to 1998. Between 1998 and 2006, he had been a faculty member at Nanyang Technological University (Singapore), the Chinese University of Hong Kong, and the University of Waterloo (Canada). He was conferred the Premier’s Research Excellence Award by the Canadian province of Ontario in 2003. Since 2006, he has been with the Hong Kong Polytechnic University as an Associate Professor. His research interest includes sensor-array signal processing and signal processing for communications. Dr. Wong was an Associate Editor of the IEEE SIGNAL PROCESSING LETTERS in 2006–2010, and Circuits, Systems, and Signal Processing in 2007–2009. He has been an Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY since 2007, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING since 2008.

Lina Yeh received the B.S. degree in mathematics from Soochow University (Taipei, Taiwan) in 1974 and the M.S. degree in mathematics from the State University of New York (Buffalo, NY) in 1981. She has been on the faculty of Soochow University (Taipei, Taiwan) since 1981, presently as an Associate Professor. Her research interest includes numerical analysis and matrix theory.

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Modified Dielectric Frequency Selective Surfaces With Enlarged Bandwidth and Angular Stability Leonardo Zappelli, Member, IEEE

Abstract—The analysis of a dielectric stop band filter is performed in terms of modified dielectric frequency selective surfaces (MDFSS). In fact, an MDFSS can totally reflect the power if a plane wave impinging on it excites a leaky wave, acting thereby as a stop band filter. Its relative bandwidth is about 0.5–0.7% and it can be enlarged by cascading MDFSSs. The development of such a cascade is analyzed in order to obtain a larger relative bandwidth. Unfortunately, the frequency behavior of the cascade is influenced by , the angle of incidence of the impinging plane for normal wave. In fact, if the stop band filter is developed at incidence, not only does it not maintain the same bandwidth for but it also moves its central frequency toward other values with lower bandwidth and loses its stop band characteristic in a range of a few angles. This effect, known as angular stability, can be mitigated by cascading MDFSSs resonating at different angles of incidence. In this case, the choice of the MDFSSs plays an important role and it will be shown that the relative bandwidth can be maintained in the angular range 0–30 . Index Terms—Angular stability, dielectric waveguide grating, frequency-selective surfaces, stop band filters.

I. INTRODUCTION

D

IELECTRIC frequency selective surfaces (DFSS), like the one shown in Fig. 1(a), are structures made up of a periodic combination of two dielectric media which have a frequency selective behavior. From the early ’70s onwards many research papers have concentrated on studying these surfaces [1]–[3]. Recently, DFSS are being investigated for use as stopband and bandpass filters in the microwave and infrared regions [4] with application to low-cost, high-performance devices including beam-splitters, filters, radomes, and polarizers. Currently, there is increasing interest in extending the filtering capabilities of single layer infrared FSS using fractal and genetic algorithm techniques [5]. Stacked DFSS were also used to achieve dual stopbands in the mid-infrared [6]. Infrared filters having many different wavelength, polarization, and angle dependent responses have been proposed or demonstrated using two-dimensional and three-dimensional metallodielectric and DFSS, also referred to as photonic crystal slabs [7], [8]. The intrinsic loss of metals at infrared and optical wavelengths limits

Manuscript received July 12, 2010; revised December 22, 2010; accepted April 04, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. The author is with the Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, 60131 Ancona, Italy (e-mail: l.zappelli@univpm. it). 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.2011.2163765

the minimum bandwidth and absorption losses that can be realized for metallodielectric devices, while DFSS structures can be used to overcome these limitations. Similarly, DFSS based on low-loss high-resistivity silicon can be designed for submillimeter-wave application [9]. Thin-film dielectric structures containing a periodic variation along the film have been of considerable interest in integrated optics [10], because of the important role they play in applications such as beam-to-surface-wave couplers, filters, distributed feedback amplifiers and lasers, nonlinear generation of second harmonics, and beam reflection on steering devices of the Bragg type. Recently, modified dielectric frequency selective surfaces (MDFSS), shown in Fig. 1(b), have been proposed in [11], to overcome the problem of the development of the classic DFSS shown in Fig. 1(a). In fact, in MDFSS two media are (here named the not needed: the medium with the higher “core”) is unchanged in the MDFSS, while the other medium (here named the “cladding”) is substituted with a proper number of dielectric sub-gratings of width , made with the same dielectric, as can be seen by comparing the unit cells in Fig. 1(a) and 1(b). The main property of the “cladding” is that its “equivalent” dielectric constant can be changed by acting on the number of the sub-gratings (two in Fig. 1(b)) and on the dielectric and air widths, and . The “cladding equivalent” . dielectric constant varies in the range Obviously, both the “core” and the sub-gratings are fixed to sustain their weight. The to a dielectric bulk of width construction of the MDFSS is quite simple: for example, it can be obtained by mechanical etching of a dielectric block or by chemical etching/deposition, depending on the application of the MDFSS in microwave, infrared or optic frequency band. medium with a grating made The replacement of a low can be applied in the synthesis of by a medium with higher all-dielectric filters at microwave frequencies proposed in [12]. In fact, complex structures like that reported in [12], made by three, four or five different dielectric media, can be replaced by a MDFSS made by a block of the medium with the higher etched to obtain a bulk with two or four different sub-gratings on the opposite side, each of which replaces the original medium. The MDFSS (or DFSS) has a frequency selective behavior when a plane wave impinging on it excites a leaky wave [1], [2], [13], [14]. The two waves interact, giving total reflection, as will be briefly shown in the following section. The MDFSS shows a better frequency behavior with respect to the classic DFSS, as discussed in [11]. In fact, with the widths being of the unit cell, the “core” and the “cladding” fixed, a larger bandwidth can be achieved by acting only on the geometric parameters of the MDFSS sub-gratings, without acting on the dielectric materials, as should be done on a classic

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Fig. 1. (a) A classic periodic dielectric frequency selective surface (DFSS) and its unit cell ; (b) The unit cell of a modified dielectric frequency selective and are the same as in Fig. (a); (c) 3D view of the unit cell of a cascade of two MDFSSs; (d) 2D view of the unit cell of a surface (MDFSS): the dimensions cascade of three MDFSSs.

DFSS. In fact, the ratios and influence the “cladding equivalent” dielectric constant and they can be chosen to improve the filtering behavior of the MDFSS. In this paper, the combination of two or more MDFSSs in order to obtain more pronounced filtering behavior in terms of bandwidth and angular stability, as requested by present applications [5], [6], will be discussed. In fact, the bandwidth can be enlarged by cascading two or for normal incimore MDFSSs, each of which resonates at dence of a linearly polarized wave. This paper will show that each MDFSS can be replaced, at , by an equivalent circuit, made by means of a shunt load embedded in two lines of proper length. Hence, we can refer to the classic filter theory to improve the filtering behavior for normal incidence. As regards the angular stability, it is well-known [15] that the incidence angle of the impinging plane wave plays a critical role in the frequency response: total reflection is ensured only for a few values of incidence angle. A solution to this angular dependence lies in the use of dielectric materials with high permittivity and/or in a z-periodic cascade of different dielectric layers [16]–[20]. Recently, a solution based on DFSS optimized with genetic algorithms was proposed and an angular stability up to 10 has been obtained [21]. These solutions are obtained without the effort of a mathematical approach like that used in the development of classic filters in the frequency domain. In fact, the problem lies in the fact that such novel mathematical approach must be based on a two dimensional space and, to our knowledge, it has not yet been developed. The approach used here to ensure angular stability is based on the cascade of two or more MDFSSs, each resonating at a different angle of incidence. In fact, it will be shown that the cascade can mitigate the angular dependence of the overall response if a proper choice of the MDFSSs separations and of the characteristics of each MDFSS are performed. II. THEORY The main characteristic of a dielectric frequency selective surface is the total reflection which occurs when an impinging

plane wave excites a leaky wave which interacts with the incoming wave in phase [1], [2], [13], [14]. The excitation of a leaky wave [1] can be approximately predicted as follows [14]. the propagation constant If we denote with along of an “equivalent” slab waveguide, with effective dielectric constant , replacing the periodic unit cell of Fig. 1(a), an incident wave having the same propagation constant as the first spatial harmonic of the “equivalent” will excite this spatial harmonic and slab through it the dielectric waveguide mode [14]. Hence, if (1) the waveguide mode is excited and will reradiate plane waves into air regions above and below the dielectric layers through the same space harmonic, thereby acting as a leaky wave [1]. The reradiated plane wave above the layer adds to the reflected plane wave generated directly on the top surface of the layer to give the total reflected wave. When the two components are in phase, total reflection occurs for the wave impinging from the air. The same excitation of the leaky wave holds for the MDFSS shown in Fig. 1(b), where the dielectric with lower is replaced [11]. by a proper number of sub-gratings with ratio polarized plane wave impinges In the hypothesis that a on an MDFSS with an incidence angle , by recalling the Floquet theorem, the MDFSS is analyzed by applying the continuity of the e.m. fields at the interfaces, with the help of the Multimode Equivalent Network approach. In so doing, three representing respectively the disscattering matrices , , in Fig. 1(b) are continuities at obtained [11]. The ports of the three S matrices are the Floquet accessible modes (propagating or not) of the regions before and and after each discontinuity. By properly inserting in the matrices the lines of length and , we can perform a “cascade” of obtaining the scattering matrix representing . the overall MDFSS, named The total reflection can be obtained at the frequency choosing a proper combination of geometric and dielectric

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Fig. 2. (a): The equivalent circuit of the MDFSS; (b): its particular case at the resonance frequency , when the inductance and are the propagation constants and the impedances of the modes at the input and output ports.

and the capacitance

resonate.

Fig. 3. Equivalent circuit of the cascade of two MDFSSs with an interconnecting rod of length . The presence of the interconnecting dielectric rod imposes , . The load represents the air filled unit cell after the two MDFSSs, as shown . The inductances/capacitances and resonate at and and respectively. The equivalent line in Fig. 1(c), and , , 2 are obtained from (3). lengths

parameters and angle of incidence, as discussed in [11]. The of the MDFSS is limited to relative bandwidth a maximum of about 0.5–0.7% with an accurate choice of the parameters. A solution to increase the bandwidth consists in cascading two or more MDFSSs, as shown in Fig. 1(c) and 1(d). It should be noted that the two/three MDFSSs are joined by a dielectric rod of section ensuring positioning and alignment. In fact, the rod can be placed between the MDFSSs by etching a base of the same dimensions as the rod on the bulk regions leading to a perfect alignment. Moreover, the last MDFSS is placed upside down in order to obtain an overall structure with the gratings on the opposite side. With two or more MDFSSs, we can enlarge the bandwidth as is usually done in filter theory. In fact, we can achieve this goal by cascading equal MDFSS separated by lines of length (henceforth named “ filter”) [16] or by developing a Chebyshev or a maximally flat stop band filter by cascading a proper number of MDFSSs separated by lines of length [22]–[24]. The only problem lies in finding, at the resonance frequency , an equivalent circuit of a single MDFSS which can be used to develop a filter with an enlarged bandwidth. However, the solution is quite simple. In fact, for a plane wave which impinges with an angle , with a proper choice of the geometric parameters, the and its scattering matrix MDFSS exhibits total reflection at can be written as

are included to satisfy the presence of a non-zero phase in and of (2). and are the propagation constants of the modes at the two ports. The minus signs in (2) are due to the presence of the short circuit in the equivalent circuit. A more complex equivalent circuit, defined on three independent paand with ) rameters ( , [16], can be obtained for frequencies other than , as shown in represents the effect of Fig. 2(a), where the circuit element . the MDFSS out of resonance. Obviously, This equivalent circuit can be used to model the cascade of the two MDFSSs in Fig. 1(c) obtaining the overall circuit shown and , , 2 are in Fig. 3, where obtained from (3). The equivalent circuit in Fig. 3 shows that the interconbetween the MDFSSs must be chosen necting rod length taking into account the lengths and of each MDFSS, as is well-known in microwave filters [16]. filter” with two identical MDFSSs, For example, for a “ both resonating at , the distance between the two short cirmust be , with being the equivalent cuits at wavelength of the fundamental mode of the rod partially filled unit cell [16]. This value ensures that the input impedance seen of the MDFSS #2 toward the load is still a from terminal short circuit, at . Hence, due to the equivalent circuit defined, is the real length at

(2) (4) Hence, we can represent the MDFSS at resonance with the equivalent circuit shown in Fig. 2(b): the short circuit yields total reflection and the two lines of length (3)

being an integer and for a two MDFSSs cascade. To reduce the interactions between the higher order modes below can be chosen in (4) to obtain cutoff, a proper value of a larger separation . Finally, with the two MDFSSs placed as . in Fig. 1(c) being equal, it should be noted that

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TABLE I , , 1 SUB-GRATING, . “KIND” REFERS TO THE KIND OF MDFSS: MDFSS PARAMETERS (SEE FIG. 1(b)). “AIR-ROD” REFERS TO A MDFSS LIKE THE FIRST IN FIG. 1(d) (AIR FILLED UNIT CELL AT THE INPUT, ROD PARTIALLY FILLED UNIT CELL AT THE OUTPUT), “ROD-ROD” TO A MDFSS LIKE THE CENTRAL ONE IN FIG. 1(d), AND “ROD-AIR” TO A MDFSS LIKE THE LAST IN FIG. 1(d)

The bandwidth can be further enlarged by increasing the number of MDFSSs , each resonating at . For three MDFSSs, the overall equivalent circuit is obtained by placing a line of length , representing the interconnecting rod, at the input of the equivalent circuit in Fig. 3, and placing the equivalent circuit of the upper MDFSS #3 (Fig. 2(a) with lines ) at the input of the line of length . The overall circuit is not shown here for the sake of brevity. Hence, the real lengths and of the rods in Fig. 1(d) at are obtained from (4) with , 2. In (4), and are the lines at the input and output of the k-th MDFSS equivalent circuit, obtained from (3). For more than three MDFSSs, the lengths can still be obtained from (4) with . The previous approaches were developed in the hypothesis of a plane wave impinging with an angle of incidence in Fig. 1(b). Obviously, the angle of incidence is a parameter that can alter the frequency behavior of the overall filtering structure. In fact, the equivalent wavelength and the equivalent circuit in Fig. 2 change with the incidence angle and the cascade of the MDFSSs no longer ensures the required filtering properties at . This problem is known as angular stability and it has been thoroughly analyzed by several authors, like [15], in the realization of band pass frequency selective surfaces. To avoid this problem, a dielectric medium with high could be used for MDFSSs and the interconnecting rods in Fig. 1(d), because it “straightens” the plane wave in the MDFSSs and in the rod, according to Snell’s law. Hence, if a “ ” filter is developed at , its bandwidth could also be maintained for not normal incidence, up to a certain value of depending on the real value of . This solution can be expensive due to the cost of dielectric material with high and it can be difficult to realize because too high media can introduce a large number of propagating modes which may alter the stop band characteristics of the overall structure. The solution proposed here takes into account that a cascade of stop band MDFSSs, each resonating at the same but at a different frequency , totally reflects the power at each , because the overall equivalent circuit of the cascade contains short circuits, each of which acts at . Starting from this consideration, we can improve the angular stability of a stop band filter by cascading some MDFSSs, each of which resonates at the same but at different incidence angles . In this way, the overall frequency behavior can be maintained over an “angular” band which depends on the number of “angular” MDFSSs placed in the cascade. For example, the cascade of four MDFSSs resonating at and , 10 , 20 , 30 could enlarge the “angular” band of the overall structure up to 30 . As regards the frequency behavior, the cascade of such “angular” MDFSSs should ensure a

good bandwidth, because the MDFSS resonating at , also resonates at a different frequency for , as shown in Fig. 10 of [11]. Hence, we can expect all the resonating “angular” MDFSSs to combine their resonating frequencies (at which total reflection occurs) for any incidence angle in order to obtain a good bandwidth. To clarify this aspect, in Fig. 4 we report the behavior of the reflected power in the plane (frequency—angle of incidence plane) for the MDFSSs used here (named in Table I: “0 ”, “10 ”, “20 ”, “25 ”, “30 ”, “0 ”, “10 ”, “25 ”, “30 ”). The dark red surfaces represent areas where the reflected power is greater than 99%, which, for a stop band filter, can be assumed as the lowest limit which is effective. As clearly shown, the behavior of the reflected power is quite complex. For example, referring to Fig. 4(b), we can assume that the “10 ” MDFSS totally reflects the power at 10 with a bandwidth of about 100 MHz around 15 GHz (14.95 GHz–15.05 GHz, highlighted with a double arrowed line). On the other hand, this MDFSS is an effective stop band filter for any combination lying in the dark red area with a different local bandwidth. The same holds for the other MDFSSs. The aim of this paper is to cascade some of these MDFSSs to “combine” their reflected powers and to ensure a good overall frequency bandwidth over a large “angular” band, because we can expect the cascade to contain almost all the dark red areas of each MDFSS, where total reflection occurs. The main problem is the choice of the length of the dielectric rods connecting the MDFSSs. Let us consider two MDFSSs interconnected with a rod of length , as shown in Fig. 1(c): the two adjacent MDFSSs have the same resonance frequency , but they resonate at different angles and , with . The total equivalent circuit is shown in Fig. 3. To evaluate the length , (4) can be used with and . Such a length ensures that, only at and , the input impedance seen from terminal in Fig. 3 is a short circuit, obtaining total reflection. In fact, at and the inductance and capacitance of the MDFSS #1 resonate, giving a short circuit. Hence, the same short circuit is present at , the total length between and the short circuit at being equal to from (4). At the same time, total reflection occurs also at and , due to the presence of MDFSS #2 which exhibits a short circuit at this combination of frequency and angle of incidence, the inductance and the capacitance of this MDFSS resonating at and . Hence, total reflection in ensured at and and while, with other angle values, the same effect is not obtained at this time, because (4) does not produce an equivalent length equal to for . However, if is not too different from , (4) will be close to yielding a very low input impedance at section .

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Fig. 4. Nominal reflected power for the MDFSSs used here versus frequency and angle of incidence in degrees (named in Table I: “0 ” (a), “10 ” (b), “20 ” (c), “25 ” (d), “30 ” (e), “0 ” (f), “10 ” (g), “25 ” (h), “30 ” (i)). The dark red surfaces represent values of reflected power greater than 99% (defined here as “nominal” bandwidth). The double arrowed lines represent the nominal bandwidths at the resonance frequencies at which each MDFSS was developed, as reported in Table I.

Moreover, two other effects must be considered when choosing the length . Firstly, referring to the circuit in Fig. 3, if we define as the normalized input impedance at and similarly for , reflected which is greater than 99% can be power at terminal is a perfect short circuit but also if achieved not only if is obtained has a low value of resistance. In fact, . by the parallel between the normalized reactive load and Hence, if (5)

is approximately a pure reactive load we can assume that and the reflected power can be assumed greater than 99%. It should be noted that (4) is a condition on just one value of , while (5) is a condition on a range of angles. Hence, the length can be set to a value satisfying (5) over a -range, other than (4), to ensure a better “angular” bandwidth. For example, to , maximize the bandwidth for any angle in the range very close to and (5) in (4) could be used in the case of all other cases. Secondly, let us define the “nominal” reflected power of a single MDFSS as the reflected power if the MDFSS is loaded with the modal impedances of the input/output ports. Some examples of “nominal” reflected power are shown in Fig. 4. By re-

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Fig. 5. (a) Reflected power, or , vs. frequency for normal incidence of a polarized plane wave on: (#1: continuous line) a single MDFSS resonating at 15 GHz (MDFSS named “0 ” in Table I); (#2: dashed line) a cascade of two identical MDFSSs (both named “0 ” in Table I), connected with a , as predicted in (4); (#3: continuous line with triangles) a cascade of three MDFSSs (named “0 ”, “0 ”, “0 ” in dielectric rod of length , , as shown in Fig. 1(d) and as predicted in (4). (b) The “nominal” bandwidth Table I), connected with two dielectric rods of length of cases #1, #2, #3. The results obtained with commercial software CST are shown with lines with markers for the three cases, respectively.

calling that we consider as effective a stop band filter which reflects almost 99% of the incident power, we define as the “nominal” bandwidth at the frequency values at which the “nominal” reflected power is greater than 99% (the dark red areas in Fig. 4). Such a nominal bandwidth is principally due to the presence of a very low shunt reactance representing the inductance and capacitance of the equivalent circuit in Fig. 2 which resproducing a short circuit. If the load impedance onate at is not the modal impedance of the output port, we can expect , being the real bandwidth (corresponding to the reflection coefficient seen at the input port) not to change significantly with respect to the nominal one, because the very low shunt reactance will reflect all the incident power whatever the value of the load impedance. The same considerations can be made when two MDFSSs and with nominal bandwidths and resonating at are connected as shown in Fig. 1(c) with the equivalent circuit shown in Fig. 3. The real reflected power at the input of MDFSS #2 does not significantly change with respect to the nominal one while the shunt reactance is very low, i.e. while with bandwidth . When the shunt it resonates at reactance is no longer resonating, the input impedance at secalters the real reflected power with respect to the nomtion inal one, but the total reflection can also be reached when (5) is satisfied. Hence, we can suppose that the two MDFSSs cascade produces an overall frequency behavior which “contains” both the at and about resonances with bandwidth of about at , due to the two short circuits of the equivalent circuits of the MDFSSs. Obviously, there is a transition region bewhich can tween these two bandwidths in the range be maximized by optimizing the separation length , starting from (4) or (5). Of course, this combination will be more effective in terms of bandwidth the closer the two angular MDFSSs and are. resonances Similarly, the cascade of three “angular” MDFSSs, as shown in Fig. 1(d), can be designed as discussed above.

III. NUMERICAL RESULTS An accurate parametric analysis of the MDFSS is discussed in [11] and will not repeated here. Those results are used here as the starting point to analyze how the frequency bandwidth and the angular stability can be improved. A. “

Filters”

In this sub-section some examples of MDFSS cascades will be discussed in the hypothesis that the MDFSSs are separated at , as in (4). by dielectric rods of equivalent length Let us first consider an MDFSS resonating at like the upper one shown in Fig. 1(c), whose geometric and dielectric parameters are reported in Table I (named: “0 ”). The MDFSS can totally reflect an impinging plane wave if a leaky wave is excited [1] and its analysis is performed in terms of the scattering matrix as discussed at the beginning of the Theory section. The power reflected from the “0 ” MDFSS, expressed in terms of (with being the reflection coefficient at the input port of the MDFSS), is shown in Fig. 5(a) for normal incidence (continuous line: 1 MDFSS): total reflection occurs at 15 GHz and the “nominal” bandwidth (frequency range where if the ports are “terminated” on their modal impedances) is about 91 MHz, as shown in Fig. 5(b). The result obtained with the current approach (S matrices cascade) is compared with that obtained using commercial software CST (continuous line with squares): the agreement is very good over the whole band. As discussed in the previous section, the two joined MDFSSs in Fig. 1(c) can improve the bandwidth. The matrix of each MDFSS is exactly as in (2) at the resonance frequency and we can extract from (3) the and lengths of the equivalent circuit shown in Fig. 2(b), which can be inserted in (4), to evaluate the rod length , obtaining for . To cancel the effects of the non-propagating accessible modes, we choose in (4)

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Fig. 6. Contour plot of the reflected power in the plane relative to (a): the “25 ”–“30 ” MDFSSs cascade, with . The net bandwidth at 15 GHz is (14.96 GHz–15.04 GHz in the range 23.8 –30 ) and is highlighted with a rectangle; (b): the “20 ”–“25 ”–“30 ” MDFSSs , (14.89 GHz–15.04 GHz) in the range 16.5 –30 ; (c): the “10 ”–“20 ”–“25 ”–“30 ” cascade with , cascade with (14.86 GHz–15.03 GHz) in the range 10 –30 ; (d): the “0 ”–“10 ”–“20 ”–“25 ”–“30 ” cascade with , (14.92 GHz–15.03 GHz) in the range 0 –30 . The “0 ”, “10 ”, “20 ”, “25 ”, “30 ” MDFSSs resonate at 15 GHz and , respectively (see Table I).

obtaining . The result of the cascade of these two identical MDFSSs is shown in Fig. 5(a) (dashed line: 2 MDFSS) and the bandwidth has been increased to about 350 MHz, i.e. by about 2.3% (Fig. 5(b)). Once again the comparison with commercial software CST (dashed line with white dots) is very good: the curves overlap. Finally, another MDFSS (named in Table I: “0 ”) is placed between the two equal “0 ” MDFSSs, as shown in Fig. 1(d): it differs from the other two (#1 and #3 in Fig. 1(d)) because its input and output ports both coincide with the Floquet accessible modes of the rod partially filled unit cell, while the input port of #3 and the output port of

#1 coincide with the Floquet accessible modes of the air filled unit cell. The kinds of ports of the MDFSSs are also reported in the last line of Table I as “kind”. The “nominal” bandwidth of this central MDFSS is about 113 MHz. From (4), the lengths of the dielectric rods are , , having chosen in (4) to cancel the effects of the non-propagating accessible modes. It is of no surprise that . In fact, the central MDFSS is not geometrically symmetric and its equivalent circuit (Fig. 2(a)) at contains two different line lengths ( and in (4)), which produce two different rod lengths and .

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The reflected power for the three MDFSSs cascade is shown in Fig. 5(a) (continuous line with triangles: 3 MDFSS) and the bandwidth has been increased to about 830 MHz (Fig. 5(b)), i.e. by about 5%. Once again the comparison with commercial software CST (continuous line with black dots) is very good. B. Filters With Angular Stability As previously discussed, the response of the overall filter changes drastically with , the angle of incidence of the plane wave impinging on it. To solve this problem, in the Theory section we analyzed the development of stopband filters with angular stability and, in this sub-section, some examples will be discussed. The first case attempts to develop a filter centered at with 5 MDFSSs, each resonating at for different angles of incidence: 0 , 10 , 20 , 25 , 30 . The first step is the choice of the geometric parameters of the five MDFSSs which are reported in Table I (named: “0 ”, “10 ”, “20 ”, “25 ”, “30 ”). Each MDFSS was designed for a specific angle of incidence and with a bandwidth of about 100 MHz at . The “nominal” reflected powers of these MDFSSs are reported in Fig. 4 in the plane (subfigures a,b,c,d,e). As defined in the Theory section, the dark red areas represent the “nominal” bandwidth of each MDFSS, i.e. the combination of angle of incidence and frequency values which satisfy . The double arrowed lines are the nominal bandwidths at , as reported in Table I. As previously discussed, each MDFSS resonates not only at and , but also at other frequencies. To validate the results shown in Fig. 4 relative to these MDFSSs, some comparisons between the frequency behaviors obtained with our simulation software, developed as discussed in the previous sections and in [11], and with commercial software CST are shown online at the url [25]. They are not reported here for the sake of brevity, because they are not significant for the purpose of this paper. In the hypothesis that the five MDFSSs sequence is “0 ”, “10 ”, “20 ”, “25 ”, “30 ” and that the last (“30 ”) is placed upside down to have the overall structure with the gratings on the opposite side (as shown in Fig. 1(d) for three MDFSSs), the first step is the optimization of , the rod length between the last MDFSSs (“25 ”, “30 ”). The length can be obtained by applying (4) at and , obtaining , being , , . Such a length was used as a starting point in an optimization process, which takes into account (5), in order to maximize the frequency bandwidth for any angle of incidence in the range 25 –30 : the result is . The behavior of the nominal reflected power, or , with being the reflection coefficient seen at the input port of the two MDFSSs cascade, is shown in the plane in Fig. 6(a). The nominal reflected power of the single “25 ” MDFSS and “30 ” MDFSS are shown in Fig. 4(d) and 4(e). As the reader can see, the nominal reflected power of the cascade has a larger nominal bandwidth than the simple “sum” of the two single MDFSS bandwidths (even if the “sum” is not correct in terms of electromagnetic fields). This is due to the superposition of levels (red and dark red areas in Fig. 4(d) and 4(e)) in the overlapping areas and in their neighborhood which can generate a stronger overall

Fig. 7. Real (dashed line) and modulus of the imaginary part (continuous line) of the input impedance relative to the “25 ”, “30 ” MDFSSs cascade seen at (Fig. 3) vs. the angle of incidence at . Modulus of the terminal of the normalized shunt reactance in Fig. 3 (continuous single “20 ” MDFSS, corresponding to the series line with dots).

reflection in the level . Moreover, a considerable area characterized by appears in Fig. 6(a). This is due to the “combination” of the areas with –0.7 of the single “25 ” and “30 ” MDFSSs, as shown in Fig. 4(d) and 4(e). The net bandwidth is highlighted with a rectangle in Fig. 6(a) and it is about 80 MHz at 15 GHz (14.96 GHz–15.04 GHz) in the range 23.8 –30 . The superposition of the nominal reflected powers relative to the “25 ” and “30 ” MDFSS and to the other MDFSSs cascade discussed in this paper are shown online at the url [25] (filter#1) for any readers who may be interested. They are not reported here for the sake of brevity. The second step is the choice of , the length of the rod interconnecting the third MDFSS (named in Table I: “20 ”) with the cascade of the two previous MDFSSs. Starting once again from (4) and (5), after an optimization process, the length was chosen: the nominal reflected power of the three MDFSSs cascade is plotted in Fig. 6(b). From Fig. 4(c), we can expect the superposition of some parts of the plots to cancel some reflection “holes” in the range 14.8–14.95 GHz, 26 –27 in Fig. 6(a) and to produce a higher reflection for angles less than 20 at 15 GHz (filter#1 at [25] for any readers who may be interested). These effects are shown in Fig. 6(b), where we can observe the disappearance of the “holes” and notice that the total reflection area ( , dark red) has been remarkably increased with respect to the “25 ”–“30 ” cascade.The net bandwidth at 15 GHz is highlighted with a rectangle and it has been increased to about 150 MHz (14.89 GHz–15.04 GHz) in the range 16.5 –30 , with the insertion of the “20 ” MDFSS. It is interesting to discuss the choice of the length by the optimization process. First of all, let us consider the two ports scattering matrix representing the “25 ”–“30 ” MDFSSs cascade. From microwave theory, this cascade can again be viewed as a two ports circuit, with the first being the input port of the “25 ” MDFSS and the second the output port of the “30 ” MDFSS. Hence, it can be represented by a circuit with three parameters like the one in Fig. 2(a), where a new shunt

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Fig. 8. Contour plot of the reflected power in the plane relative to: (a) the “10 ”–“20 ”–“25 ”–“30 ” MDFSSs cascade, with . The is (14.86 GHz–15.03 GHz) in the range 7 –30 and (14.91 GHz–15.03 GHz) in the range 0 –30 . net bandwidth (Fig. 4(g)). (b) the “0 ”–“10 ”–“20 ”–“25 ”–“30 ” cascade with , obtained with The “10 ” MDFSS resonate at 14.8 GHz and (14.82 GHz–15.03 GHz) in the range 0 –30 . The “0 ” the combination of the “0 ” MDFSS and the four MDFSSs cascade in Fig. (b). (Fig. 4(f)). (c) the “0 ”–“10 ”–“20 ”–“25 ”–“30 ” MDFSSs cascade. (14.78 GHz–15.08 MDFSS resonate at 14.91 GHz and and 14.89 GHz, , respectively (see Fig. 4(h) and 4(i)). GHz) in the range 0 –30 . The “25 ”, “30 ” MDFSSs resonate at 14.9 GHz,

reactance with more than one resonance replaces the reactive shunt . In fact, each resonance represents the combination of the resonances of the two MDFSSs constituting the cascade. If the “20 ” filter is connected to the input of the “25 ”–“30 ” MDFSSs cascade with a dielectric rod of length , the equivalent circuit can be considered very similar to the one shown in Fig. 3 where the equivalent circuit #2 is just the “20 ” MDFSS and the equivalent circuit #1 represents the “25 ”–“30 ” cascade with replaced with the new multi-resonances reactance. Finally, the length in Fig. 3 is replaced by , the length we are searching for. Moreover, in the hypothesis that the interconnecting rod cross section is always the same in the cascade (changing only its length to connect the MDFSSs), we can set: . Let us evaluate the input impedance of the “25 ”–“30 ” cascade: the normalized input impedance seen at terminal in Fig. 3 vs. the incidence angle at 15 GHz is shown in Fig. 7. The real (dashed line) and the modulus of the imaginary (continuous line) part of are shown in a log axis of the plot in Fig. 7. A horizontal dotted line represents the limit at which the the “25 ”–“30 ” cascade reflects power at 99%, or : if , the normalized input impedance is almost entirely imaginary and the nominal reflected power is greater than 99%. This occurs in the range 23.6 –30 at 15 GHz, as can be confirmed by studying Fig. 6(a). In the same Fig. 7, the modulus of the normalized shunt reactance (corresponding to the series in Fig. 3) of the single “20 ” MDFSS is plotted with a continuous line with dots vs. the angle of incidence at . It should be noted that this reactance is negligible at where the “20 ” MDFSS becomes just a short circuit. On the other hand, at , the reactance tends to a very large value, approaching an open circuit. Hence, at this angle of incidence, the normalized input impedance at terminal in Fig. 3, , relative to the “20 ”–“25 ”–“30 ” cascade, is a parallel between a quasi open circuit, ,

and the input impedance seen at terminal , . The effect is that almost coincides with . Hence, must have a very low real part to ensure total reflection at this critical angle of incidence where the “20 ” filter is almost an open circuit. Therefore, the length chosen by the optimization process satisfies this requirement exactly. In fact, this value of ensures that the equivalent output line of the single “20 ” MDFSS ( in Fig. 3) and the equivalent input line of the “25 ”–“30 ” cascade ( in Fig. 3) satisfy at , being , , , . Hence, at this angle of incidence. Since has a very low real part at , as shown in Fig. 7, total reflection is also ensured at the input of the “20 ”–“25 ”–“30 ” MDFSSs cascade at , even if the “20 ” filter may not be effective at this angle. Let us now place the fourth MDFSS (named in Table I: “10 ”) resonating at 15 GHz and 10 at the input of the “20 ”–“25 ”–“30 ” cascade: from the optimization process, the separation between them is . The contour plot of the reflected power relative to the “10 ”–“20 ”–“25 ”–“30 ” cascade is shown in Fig. 6(c). The net bandwidth (highlighted with a rectangle) has again increased and is about 170 MHz (14.86 GHz–15.03 GHz) in the range 10 –30 . Comparing Fig. 6(b) and 6(c), it can be noticed that the introduction of the “10 ” MDFSS (Fig. 4(b)) has produced effects in the upper part of the plot, i.e. for and , and in a little descending band around 15 GHz (filter#1 at [25] for any readers who may be interested). Hence, the choice of the “10 ” MDFSS could probably be improved to produce major benefits on the area around and below 15 GHz and we will discuss this aspect below. For the moment, let us continue with this choice and let us place in front of the four MDFSSs cascade the last one, resonating at 0 and 15 GHz (named in Table I: “0 ”). The contour plot relative to the reflected power of the “0 ”–“10 ”–“20 ”–“25 ”–“30 ” cascade is shown in Fig. 6(d)

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for . The global effect is not so interesting. In fact, the net bandwidth is about 110 MHz (14.92 GHz–15.03 GHz) in the range 0 –30 . This is due to the not optimal superposition of the MDFSSs around 14.9 GHz and 0 , where the dark red area of the “0 ” MDFSS (Fig. 4(a)) overlaps with a low reflection area of the four MDFSSs cascade (areas in Fig. 6(c) where ) (filter#1 at [25] for any readers who may be interested). This result can be enhanced by choosing MDFSSs with a different central resonance frequency . By putting at the input of the “20 ”–“25 ”–“30 ” cascade an MDFSS resonating at 14.8 GHz and 10 (named in Table I: “10 ”), with , we can enhance the global effect of the four MDFSSs cascade, as shown in Fig. 8(a): the improvement is evident if Fig. 8(a) is compared with Fig. 6(c). In fact, the new “10 ” MDFSS (Fig. 4(g)) covers part of the low reflection area of the “20 ”–“25 ”–“30 ” cascade in Fig. 8(a) and overlaps part of the area (filter#2 at [25] for any readers who may be interested). Hence, the net bandwidth obtained with four MDFSSs is about 170 MHz (14.86 GHz–15.03 GHz) in the range 7 –30 and 120 MHz (14.91 GHz–15.03 GHz) in the range 0 –30 . Finally, an MDFSS resonating at 0 and (named in Table I: “0 ”), with , can cover the area around 14.9 GHz in Fig. 8(a), obtaining the final effect shown in Fig. 8(b) where the net bandwidth is about 210 MHz (14.82 GHz–15.03 GHz) in the range 0 –30 . From this discussion, it is clear that the combination of MDFSSs with different central resonance frequency at can improve the overall behavior of the cascade. Moreover, it is evident from Fig. 8(b) that the bandwidth changes from very large values at 0 to lower values at 30 . Hence, we can improve the cascade by properly choosing the most critical MDFSSs, “25 ” and “30 ”, as shown in Fig. 8(c), where a stop band filter has been developed with a five MDFSSs cascade, with central resonance frequencies , 15, 15, 14.9 14.89 GHz, respectively (named in Table I: “0 ”, “10 ”, “20 ”, “25 ”, “30 ”). The distances between MDFSSs are: , , , . The net bandwidth is about 300 MHz (14.78 GHz–15.08 GHz) in the range 0 –30 . The main difference between the MDFSSs named “30 ” and “30 ” in Table I consists in the nominal bandwidth, which is larger for “30 ”, as can be seen by comparing Fig. 4(e) and 4(i). Therefore, the “25 ” MDFSS in Fig. 4(h) has been chosen to maximally overlap the red area of the “30 ” MDFSS around 14.9 GHz (filter#3 at [25] for any readers who may be interested). IV. CONCLUSIONS Stop band filters with enlarged bandwidth and angular stability have been analyzed by cascading MDFSSs. The results show that, if the filter is to be used only for normal incidence of the wave impinging on it, the cascade can be very effective in terms of bandwidth (up to 800 MHz at 15 GHz with three MDFSSs, about 5%). On the other hand, if the stop band filter has to be developed with angular stability, i.e. with a frequency bandwidth extending over an angular range, the MDFSS cascade can reach the scope by properly choosing the single

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MDFSS resonating at different frequency/angle combinations. In this case, the frequency bandwidth is smaller than in the previous case (up to 300 MHz, about 2%) but it has an angular extension of up to 30 . REFERENCES [1] T. Tamir and F. Kou, “Varieties of leaky waves and their excitation along multilayered structures,” IEEE J. Quant. Electron., vol. 22, no. 4, pp. 544–551, Apr. 1986. [2] L. Lewis and A. Hessel, “Propagation characteristics of periodic arrays of dielectric slabs,” IEEE Trans. Microwave Theory Tech., vol. 19, no. 3, pp. 276–286, Mar. 1971. [3] A. Coves, B. Gimeno, J. Gil, M. Andres, A. Blas, and V. Boria, “Fullwave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2091–2099, 2004. [4] S. Tibuleac, R. Magnusson, T. Maldonado, P. Young, and T. Holzheimer, “Dielectric frequency-selective structures incorporating waveguide gratings,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 4, pp. 553–561, Apr. 2000. [5] J. Bossard, D. Werner, T. Mayer, J. Smith, Y. Tang, R. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1265–1276, 2006. [6] J. A. Oswald, B. Wu, K. A. McIntosh, L. J. Mahoney, and S. Verghese, “Dual-band infrared metallodielectric photonic crystal filters,” Appl. Phys. Lett., vol. 77, pp. 2098–2100, Oct. 2000. [7] S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B, vol. 65, no. 23, p. 235112, Jun. 2002. [8] K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B, vol. 73, no. 11, pp. 115 126–115 139, Mar. 2006. [9] S. Biber, M. Bozzi, O. Gunther, L. Perregrini, and L.-P. Schmidt, “Design and testing of frequency-selective surfaces on silicon substrates for submillimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2638–2645, 2006. [10] S. S. Wang and R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. vol. 19, no. 12, pp. 919–921, Jun. 1994 [Online]. Available: http://ol.osa.org/ abstract.cfm?URI=ol-19-12-919 [11] L. Zappelli, “Analysis of modified dielectric frequency selective surfaces under 3-d plane wave excitation using a multimode equivalent network approach,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1105–1114, April 2009. [12] C. Zuffada, T. Cwik, and C. Ditchman, “Synthesis of novel all-dielectric grating filters using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 657–663, May 1998. [13] S. Peng, T. Tamir, and H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 1, pp. 123–133, Jan. 1975. [14] H. Bertoni, L.-H. Cheo, and T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag., vol. 37, no. 1, pp. 78–83, Jan. 1989. [15] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [16] R. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 2001. [17] F. Chiadini, V. Fiumara, I. Gallina, I. M. Pinto, and A. Scaglione, “Filtering properties of defect-bearing periodic and triadic cantor multilayers,” Opt. Commun., vol. 281, no. 4, pp. 633–639, 2008. [18] F. Chiadini, A. Scaglione, and V. Fiumara, “Transmission properties of perturbed optical cantor multilayers,” J. Appl. Phys., vol. 100, no. 2, pp. 023 119.1–023 119.5, 2006. [19] C. Mulenga and J. Flint, “Planar electromagnetic bandgap structures based on polar curves and mapping functions,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 790–797, March 2010. [20] D. Kern, D. Werner, A. Monorchio, L. Lanuzza, and M. Wilhelm, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 8–17, Jan. 2005. [21] S. Yun, J. A. Bossard, T. S. Mayer, and D. H. Werner, “Angle and polarization tolerant midinfrared dielectric filter designed by genetic algorithm optimization,” Appl. Phys. Lett., vol. 96, no. 22, pp. 223 101–223 101-3, May 2010.

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[22] E. Jones, L. Young, and G. Matthaei, “Microwave band-stop filters with narrow stop bands,” IRE Trans. Microwave Theory Tech., vol. 10, no. 6, pp. 416–427, November 1962. [23] B. Schiffman and G. Matthaei, “Exact design of band-stop microwave filters,” IEEE Trans. Microwave Theory Tech., vol. 12, no. 1, pp. 6–15, Jan. 1964. [24] D. C. Park, G. Matthaei, and M. S. Wei, “Bandstop filter design using a dielectric waveguide grating,” IEEE Trans. Microwave Theory Tech., vol. 33, no. 8, pp. 693–702, Aug. 1985. [25] [Online]. Available: http://digilander.libero.it/plza081/MDFSSfilters/ index.html

Leonardo Zappelli (M’97) received the M.S. degree (summa cum laude) and the Ph.D. degree in electronic engineering from the University of Ancona, Italy, in 1986 and 1991 respectively. Since 1988, he has been with the Dipartimento di Ingegneria Biomedica, Elettronica e delle Telecomunicazioni, Università Politecnica delle Marche, where he is currently Assistant Professor. His research interests are microwaves, electromagnetic compatibility, phased array antennas and frequency selective surfaces.

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Double Periodic Composite Right/Left Handed Transmission Line and Its Applications to Compact Leaky-Wave Antennas Cheng Jin, Student Member, IEEE, Arokiaswami Alphones, Senior Member, IEEE, and Makoto Tsutsumi, Life Fellow, IEEE

Abstract—A double periodic composite right/left handed transmission line (DP-CRLH TL) is proposed and analyzed based on transmission line theory. Besides the composite right/left handed property, a new leaky-wave radiation is observed at a lower frequency with narrow bandwidth. Dispersion characteristics of the structure are obtained and the cutoff frequencies are investigated by changing the ratio of inductances ( 1 2 ) or capacitances ( 1 2 ). Experiments are performed using microstrip lines with double periodically loaded shunt shorted-stub inductors and series interdigital capacitors. Frequency response, dispersion behavior and radiation characteristics are measured from the prototype structure and they display the expected novel forward leaky-wave = +3 and gain of 4.12 dBi at radiation in the direction of 1.6 GHz, the left-handed backward radiation in the direction of = 19 and gain of 8.62 dBi at 3.3 GHz, and the conventional right-handed forward radiation in the direction of = +12 and gain of 8.78 dBi at 3.9 GHz. Index Terms—Double periodic composite right/left handed transmission line (DP-CRLH TL), left-handed, microstrip line, triple band leaky-wave antenna.

I. INTRODUCTION

R

ECENTLY composite right/left handed (CRLH) transmission lines (TLs) realized by periodic loading of series capacitance and shunt inductance in a host medium are becoming an interesting and promising topic of research [1]–[3]. Since the resonant frequency of CRLH TLs is independent of the physical length [1], [2] these TLs are used to design many compact devices for communication system applications, like small leaky-wave antennas (LWAs) [2] and compact filters [4]. Most of the well-known traditional CRLH TLs are periodically loaded with single inductance and capacitance until now, and may be named as single periodic composite right/left handed (SP-CRLH) TLs in this paper. A novel CRLH TL, which is

Manuscript received March 25, 2010; revised February 27, 2011; accepted April 04, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. C. Jin and A. Alphones are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). M. Tsutsumi is with the Department of Space Communication Engineering, Fukui University of Technology, Fukui 910-8505, Japan (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.2011.2163752

double periodically loaded by a pair of different capacitances and inductances, is proposed in this work and this new TL is named as double periodic composite right/left handed transmission line (DP-CRLH TL) to differentiate it from the SP-CRLH TL. The series capacitances as well as the shunt inductances are modulated within the double periodic unit cell to obtain the proposed DP-CRLH TL. In addition to the CRLH characteristics as SP-CRLH TLs exhibit, a novel leaky-wave radiation is obtained below the left-handed (LH) region with narrow bandwidth [3]. This novelty of the DP-CRLH TLs raises intriguing questions on the excitation and physical meaning of spatial harmonics in such structures. Although the bandwidth of the new leak-wave radiation is narrow at very low frequency, the new method to design compact LWAs at a fixed low frequency based on DP-CRLH TLs is still attractive and promising since its resonant frequency is independent of the physical length and is even below the LH region. The purpose of this paper is to propose a new structure of DP-CRLH TLs. Previously, many researchers have made investigations on SP-CRLH TLs based on the transmission line theory, Floquet theory [1], [2] and equivalent circuit theory [6]. In the same way, the analysis for DP-CRLH TLs is aimed at determining the leaky-wave radiation characteristics through the dispersion behavior of TLs based on the complex propagation constant (phase and attenuation constants). Based on the Floquet Theory and the transmission line theory, dispersion characteristic of DP-CRLH TLs is analyzed, and compared with SP-CRLH TLs. From the dispersion behavior, three bands are observed in the fast wave region which indicates that DP-CRLH TLs are able to radiate in those bands. It has been also demonstrated that the group velocity and phase velocity are parallel in the new-found leaky-wave region. An antenna is designed for experiment based on the equivalent circuit of DP-CRLH TLs and SP-CRLH TLs using microstrip lines with double/single periodically loaded shorted-stub inductors and interdigital capacitors. The frequency response and radiation characteristics are measured and the dispersion characteristics are extracted from the measured results. The measured frequency response shows that large power loss occurs in a narrow band below the LH region for the DP-CRLH TL while it is only stopband below the LH passband for the SP-CRLH TL. Radiation characteristics of the DP-CRLH TL are measured at different frequencies and the results display the expected radiation property of the new forward leaky-wave radiation at a low frequency, backward direction radiation in the LH region and the forward direction radi-

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ation again in the conventional right-handed region. These results verify the CRLH properties of the proposed DP-CRLH TL along with a novel right-handed leaky-wave radiation below the LH region with narrow bandwidth. II. TRANSMISSION LINE APPROACH A transmission line with periodically loaded capacitance and inductance can be considered as an effective transmission medium provided that the dimensions of the unit cell are small compared to the operating wavelength [5]. The model of SP-CRLH TLs consists of a periodic uniform shunt inductance and a periodic uniform series capacitance featuring as shown in Fig. 1(a). the host medium with periodicity The model represents the general form of a structure with CRLH attributes [5], [7]. Many applications including filters and LWAs have been designed based on SP-CRLH TLs [2], [4], [8], [9]. The model of proposed DP-CRLH TLs consists and double of double periodically loaded shunt inductances featuring periodically loaded series capacitances the host medium with periodicity as shown in Fig. 1(b). The and are normally small compared to wavelength. period and capacitances are The loaded inductances different within one unit cell, and the unit cell of DP-CRLH TLs can be seen as combination of two unit cells of different SP-CRLH TLs. Therefore the proposed structure is named as DP-CRLH TL. The periodicity of DP-CRLH TL is chosen as for simple double the periodicity of SP-CRLH TL analysis and without loss of generality, so that the total length of the DP-CRLH TL and the SP-CRLH TL are comparable when the same amount of capacitances and inductances are loaded in the host medium. The transmission matrix [1], [2], [10] of the DP-CRLH TL can be written as

shunt inductances . The series capacitances and shunt induc, which is explicitly tances are repeated with periodicity of and refer to provided by the transmission line segments. the characteristic impedance and wave number of the transmission line used. The parameters and are given by the (3) and (4).

(3)

(4) is a fundamental matrix Based on the Floquet theorem, if , then solution of the periodic TL with periodicity (5) so that, the complex propagation constant derived as [10], see (6) at the bottom of the page, where

is

(1) where

(2) are the matrixes of one part of unit cell of DP-CRLH TLs, which is the same as the transmission matrix as of unit cell of SP-CRLH TLs with periodicity shown in [5]. and represent the impedances of double periodically loaded series and admittances of double periodically loaded capacitances

Here and are the attenuation and phase constants of the DP-CRLH TL, respectively. Fig. 2 shows the dispersion behavior of the proposed DP-CRLH TL using the average phase shift incurred per unit with periodicity of mm (being small cell compared to wavelength at 1.6 GHz) (red solid line) and the of the SP-CRLH TL with perioddispersion behavior icity of mm (blue dashed line). It shows the typical – and conventional right-handed passband LH passband for both the DP-CRLH TL and the SP-CRLH TL.

(6)

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Fig. 1. Model of the structures. (a) Unit cell of L-C network featuring the host transmission line medium (Z ; k ) of SP-CRLH TLs [5]. (b) Unit cell for the L-C network featuring the host transmission line medium (Z ; k ) of DP-CRLH TLs. (c) Line implementation in the form of a periodic network.

When the dispersion characteristic of DP-CRLH TL is in the , the structure is able to radiate, fast wave region which can be used for potential LWAs design [1], [2], [11], [12], where the is the free space wave number. Frequencies, and in Fig. 2 mark the limits of various passbands so and leaky-mode radiation occurs just above , just below above . It is noted that the slope of line for DP-CRLH TL shown in Fig. 2 is twice that of SP-CRLH TL, since length of per unit cell incurring the average phase shift of the DP-CRLH mm) is twice as much as the periodicity of TL ( mm). Fig. 2(a) shows the DP-CRLH TL SP-CRLH TL ( radiates leaky-waves in LH mode and right-handed like the SP-CRLH TL. mode In addition to the CRLH property as SP-CRLH TLs, the – DP-CRLH TL provides a new right-handed passband below the LH region as shown in Fig. 2. The enlarged dispersion characteristic in the new leak-wave region due to the double periodically loaded capacitances and inductances is shown in Fig. 2(b). The new leaky-wave occurs at very low frequency – with narrow bandwidth. The DP-CRLH TL is expected to radiate forward leaky-wave in this region. The cutoff frequencies between the new leaky-wave region, the typical LH region and the conventional right-handed region are affected by the difference between the loaded series and the difference between the loaded capacitances . Fig. 3 shows the effect on dispershunt inductances sion behavior by changing the ratio of capacitances while the loaded inductances are equal . It is noted that TLs with double periodically loaded different capacitances and uniform inductances work as a and can provide a DP-CRLH TL with periodicity of new leaky-wave radiation at the lower frequency. When , the structure works as a SP-CRLH TL with periodicity of , and no passband and leaky-wave characteristic occur below the LH passband as shown in Fig. 3 (bold blue dashed line). The inset of Fig. 3 explains the changes of the cutoff frequencies depending on the ratio of the loaded capacitances. As the ratio increases, the lower cutoff frequency of the new leaky-wave decreases (bold red solid line) while the

Fig. 2. Dispersion characteristics of the proposed structures. (a) Over the frequency range 0 to 10 GHz, and (b) in the vicinity of the new leaky-wave region. DP-CRLH TL (red solid line): C = 0:5 pF, C = 1 pF, L = 3:3 nH, L = 6:8 nH, Z = 50 ; 3 = 10 mm. SP-CRLH TL (blue dashed line): C = 0:5 pF, L = 3:3 nH, Z = 50 ; 3 = 5 mm.

upper cutoff frequency is unchanged when . It (blue dashed line). The then decreases slightly when lower cutoff frequency of the LH passband decreases when , and then remains unchanged when (thin black solid line) while the upper cutoff frequency is (thin green unchanged and independent of the ratio dotted line). The lower cutoff frequency of conventional rightdecreases distinctly (bold pink dotted handed passband , no leaky-wave occurs line). It is noted that when below the LH passband and the structure works as SP-CRLH and drop to zero when TLs. The cutoff frequencies as shown in the Fig. 3. Fig. 4 shows the effect on the dispersion behavior of DP-CRLH TLs by changing the ratio of loaded inducwhile keeping the loaded capacitances equal tances . The effect is almost similar with the variation by the difference of capacitances except that the upper-cutoff decreases with an increase frequency of the LH passband in the ratio as shown in the inset of Fig. 4. It is noted that the TLs with double periodically loaded inductances and uniform capacitances also provides the new leaky-wave radiation. It is a good way to get a

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Fig. 3. Dispersion characteristics of DP-CRLH TLs for different ratio of capacitances (C =C ) while keeping the loaded inductances equal, (C = 0:5 pF, L = L = 3:3 nH).

Fig. 5. Layouts of the unit cell of the periodic structures. (a) DP-CRLH TL. (b) SP-CRLH TL. Substrate: " = 4:47; loss tangent = 0:016; thickness = 1:6 mm. l = 4:35; l = 3:7; l = 4:5; l = 7:9; l = 1; l = 1; l = 0:5; S = 0:15; S = 0:15; w = 0:2; 3 = 10; 3 = 5. All are in mm.

different inductances . The DP-CRLH TL consists of mm. The SP-CRLH TL 4 unit cells with periodicity of mm. The is printed with 8 unit cells and a periodicity of overall length of both the DP-CRLH TL and SP-CRLH TL is cm. The output ports of the TLs are terminated with 50 . The CST Microwave Studio [15] was employed to optimize the dimensions of the TLs and they are listed in Fig. 5. The double and single periodic microstrip lines were fabricated on an inexpen, sive FR-4 substrate with relative dielectric constant loss tangent of 0.016, and thickness of 1.6 mm. Although FR4 is known to be highly lossy, and may lead to high insertion loss in the passband of the structure, the new leaky-wave radiation is still observed with considerable gain for the proposed DP-CRLH TL fabricated on this substrate. Fig. 4. Dispersion characteristics of DP-CRLH TLs for different ratio of inductances (L =L ) while keeping the loaded capacitances equal, (L = 3:3 nH, C = C = 0:5 pF).

better performance of the DP-CRLH TLs by combining a pair of different series capacitances and a pair of different shunt inductances together double periodically loaded in the TL. III. DESIGN AND EXPERIMENTS Fig. 5 shows the configurations of the proposed DP-CRLH TL and SP-CRLH TL using microstrip line with double/single periodically loaded interdigital capacitors and shorted-stub inductors according to the equivalent circuits shown in Fig. 1. The interdigital capacitors implement the series capacitances while stubs are shorted to the ground to obtain the shunt inductances as proposed in [7]. When all the interdigital capacitors and shorted-stubs are uniform as shown in Fig. 5(b), the structure works as the SP-CRLH TL with periodicity of , and it has been experimentally demonstrated for CRLH property [1], [2], [7], [13], [14]. In the proposed DP-CRLH TL, one of the interdigitated structures is constituted by 11 fingers while the other is . realized by 7 fingers to obtain unequal capacitances The length of shorted-stubs is also kept unequable to obtain the

A. Simulation Results Numerical simulations have been carried out on both the DP-CRLH TL and the SP-CRLH TL. Fig. 6 shows the simulated scattering parameters of the SP-CRLH TL, which present the frequency response of the CRLH features as predicted by [1], [2], [7], [13], [14]. The typical LH passband to , followed by a stopband – , is obtained from – , which are qualitatively and right-handed passband matched with the dispersion characteristic analysis as shown in Fig. 2 (blue dashed line). Fig. 7 shows the simulated scattering parameters of DP-CRLH TL. Besides the CRLH characteristic like SP-CRLH TL, the new leaky-wave radiation is obtained below the LH passband – as shown in just above Fig. 7(b). Figs. 6 and 7 also show the frequency dependence of the group delay for the wave propagating along the TLs. The group is defined as the negative rate of change of the indelay with frequency . sertion phase Phase delay is the time delay of a signal at a specific frequency, and is related to the phase of the system by [16]. The group velocity and group delay have , where is the length of system. the same sign since For the case of LH materials, the reversal of phase velocity can

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Fig. 6. Simulated frequency response and group delay of the SP-CRLH TL.

Fig. 8. Prototype, measured frequency response and group delay of the SP-CRLH TL. (a) Prototype. (b) Frequency response and group delay.

the Vector Network Analyzer. The backward wave characteristics can be seen in the frequency region from to as shown in Figs. 6 and 7. Because the accuracy of the group delay is dependent on the data being free of noise and sufficient number of data points, the group delay prediction below the narrow band leaky-wave region may not be accurate. However confirmation of right-handed leaky-wave region at this band is done through dispersion analysis and measurements. B. Measurement Results

Fig. 7. (a) Simulated frequency response and group delay of the DP-CRLH TL. (b) Enlarged version in the new leaky-wave region.

be used as an indication of LH behavior: where the group velocity is positive, the phase velocity becomes negative decreases [2], [17], [18]. This means that the group delay with increasing frequency [3], [19]–[21]. It is convenient to get the group delay directly from the CST Microwave Studio and compare with the measurement results obtained directly from

To confirm the characteristics of the new leaky-wave radiation of the DP-CRLH TL, the LWAs based on both the DP-CRLH TL and the SP-CRLH TL are fabricated and their frequency response, group delay, dispersion behavior and radiation characteristics are measured. The fabricated prototypes are shown in Fig. 8(a) and Fig. 9(a), corresponding to the SP-CRLH TL and DP-CRLH TL, respectively. The measured scattering parameters using the Agilent N5230A PNA-L network analyzer are shown in Fig. 8(b) and Fig. 9(b),(c). The LH passband, – stopband and right-handed passband are found at – and – , respectively in the measurement results. The novel leaky-wave radiation is found just above , which is located below the LH passband in the DP-CRLH TL. It is noted that the satisfactory return loss better than dB is achieved in the new leaky-wave region while the insertion loss is around and below dB as shown in the inset figure of Fig. 9(c). Small difference between the measured and simulated results may be

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Fig. 10. Dispersion characteristics (extracted from measured scattering parameters) of the DP-CRLH TL and the SP-CRLH TL.

Fig. 11. Normalized measured radiation patterns at three different leaky-wave frequencies.

Fig. 9. Prototype, measured frequency response and group delay of the proposed DP-CRLH TL. (a) Prototype. (b) Frequency response and group delay. (c) Frequency response in the new leaky-wave region.

attributed by the effects of manufacturing tolerances and the SMA-connectors mismatch with the circuit. The fingers of the interdigital structures are thinner, which decreases the capacitance contributed by the interdigital fingers. Due to soldering via-holes, some spurious resonances have also been observed which may be improved by better fabrication approaches. In order to validate the CRLH characteristics of TLs, the dispersion characteristics of the proposed DP-CRLH TL (red solid line) and the SP-CRLH TL (blue dashed line) extracted from the measured complex scattering parameters using the average

phase shift incurred per unit cell are shown in Fig. 10. mm for the SP-CRLH TL and mm for the DP-CRLH TL. The new right-handed leaky-wave characteristic is expected to appear in the region of – around 1.6 GHz that supports a forward leaky-wave radiation. The LH around 3.3 GHz leaky-wave is expected to be obtained at that supports the backward leaky-wave. The conventional righthanded leaky-wave is expected to be at around 3.9 GHz. The regions – and – correspond to two stopbands. A comparison with the dispersion characteristics of SP-CRLH TL is also shown in Fig. 10 (blue dashed line) with periodicity of mm. The LH leaky-wave radiation and the conventional right-handed leaky-wave radiation are also observed in the SP-CRLH TL. However, it is noted that no leaky-wave performance occurs at the frequency band below the LH leakywave region. The measured far-field radiation characteristics of the proposed DP-CRLH TL at 1.6 GHz, 3.3 GHz, and 3.9 GHz are shown in Fig. 11. As expected, the antenna radiates forward in the direction of at GHz in the new leaky-wave region just above (red solid line), backward in the direction of at GHz around the LH

JIN et al.: DP-CRLH TL AND ITS APPLICATIONS TO COMPACT LEAKY-WAVE ANTENNAS

leaky-wave frequency (blue dashed line), and forward again at GHz around the conin the direction of ventional right-handed leaky-wave frequency (purple dotted line), where is the angle of main lobe measured in broadside direction. The gain is 4.12 dBi at the new leaky-wave frequency, 8.62 dBi at the LH leaky-wave frequency and 8.78 dBi at the conventional right-handed leaky-wave frequency. IV. CONCLUSION A double periodically loaded LWA has been introduced based on the proposed DP-CRLH TLs. A new leaky-wave radiation is obtained below LH region with narrow bandwidth. The new structure is expected to contribute toward the design of compact LWAs for communication system. Transmission line theory has been proposed to analyze the complex propagation constant and dispersion behavior of the novel DP-CRLH TL. An implementation of microstrip line with double periodic loaded interdigital capacitors and shorted-shunt inductors has been presented to demonstrate the novel leaky-wave radiation of the DP-CRLH TL. Frequency response, dispersion behavior and radiation characteristics are measured for the proposed antenna, and they display the new forward leaky-wave radiation below the LH leaky-wave region with narrow bandwidth in the DP-CRLH TL. The new right-handed leaky-wave is obtained at 1.6 GHz with gain of 4.12 dBi and main lobe radiating at to broadside direction. REFERENCES [1] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. New York: Wiley, 2005. [2] T. Itoh and C. Caloz, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley, 2006. [3] M. Tsutsumi, C. Jin, and A. Alphones, “Leaky wave phenomenon from double periodic left handed waveguide,” in Proc. Asia Pacific Microwave Conf., 2009, pp. 1238–1241. [4] C. Jin and A. Alphones, “Compact interdigital microstrip band pass filter,” Microw. Opt. Technol. Lett., vol. 52, pp. 2128–2132, 2010. [5] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 2702–2712, 2002. [6] F. Aznar, M. Gil, J. Bonache, and F. Martin, “Revising the equivalent circuit models of resonant-type metamaterial transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., 2008, pp. 322–325. [7] T. Itoh and C. Caloz, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag., vol. 52, pp. 1159–1166, 2004. [8] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 161–173, 2005. [9] M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials, vol. 2, pp. 186–197, 2008. [10] D. Pozar, Microwave Engineering. New York: Wiley, 1998. [11] M. Tsutsumi, “Negative refractive index transmission media and its applications to the microwave circuits,” J. IEICE, vol. 88, pp. 23–27, 2005. [12] M. Tsutsumi, “Applications of left handed techniques to microwave circuits,” IEICE Trans. Electron., vol. J89C, pp. 191–197, 2006. [13] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 980–992, 2004.

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[14] C. Caloz and T. Itoh, “A novel mixed conventional microstrip and composite right/left-handed backward-wave directional coupler with broadband and tight coupling characteristics,” IEEE Trans. Microw. Theory Tech., vol. 14, pp. 31–33, 2004. [15] Computer Simulation Technology. Wellesley Hills, MA: CST Microwave Studio, 02481. [16] I. A. Ibraheem, J. Schoebel, and M. Koch, “Group delay characteristics in coplanar waveguide left-handed media,” J. Appl. Phys., vol. 103, 2008. [17] K. Aydin, K. Guven, C. M. Soukoulis, and E. Ozbay, “Observation of negative refraction and negative phase velocity in left-handed metamaterials,” Appl. Phys. Lett., vol. 86, 2005. [18] 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. 312, pp. 892–894, 2006. [19] M. Tsutsumi and T. Ueda, “Nonreciprocal left-handed microstrip lines using ferrite substrate,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, pp. 249–252. [20] M. Tsutsumi, “Left handed microwave ferrite circuit and devices,” in Proc. Eur. Microw. Conf., 2006, pp. 943–946. [21] J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E, vol. 70, 2004.

Cheng Jin (S’07) was born in Hebei, China. He received the B.Eng. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2007. He is currently working toward the Ph.D. degree at the School of Electrical and Electronics Engineering, Nanyang Technological University (NTU), Singapore. His research interests include composite right/left handed transmission lines, substrate integrated waveguides, periodic structures, leaky-wave antennas and millimeter-wave passive components. Mr. Jin is the recipient of Research Scholarship at Nanyang Technological University from 2008 to 2012.

Arokiaswami Alphones (S’82–M’88–SM’98) received the B.Tech. degree from Madras Institute of Technology in 1982, the M.Tech. degree from the Indian Institute of Technology Kharagpur in 1984, and the Ph.D. degree in optically controlled millimeter wave circuits from Kyoto Institute of Technology (Japan) in 1992. He was a JSPS Visiting Fellow from 1996–97 at Japan. During 1997–2001, he was with Centre for Wireless Communications, National University of Singapore, as Senior Member of Technical Staff, involved in the research on optically controlled passive/active devices. Currently he is at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is also Deputy Director of Network Technology Research Centre and Program Director of MSc Communication Engineering. He has 25 years of research experience. He has published and presented over 170 technical papers in international journals/ conferences. His current interests are electro-magnetic analysis on planar, periodic RF circuits and integrated optics, microwave photonics, metamaterial based antennas and hybrid fiber-radio systems. His research work has been cited in the book Millimeter Wave and Optical Integrated Guides and Circuits (Wiley Interscience). Dr. Alphones is on in the editorial review board of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and Microwave and Wireless Components Letters. He has delivered tutorials and short courses in international conferences. He wrote a chapter on “Microwave Measurements and Instrumentation” in Wiley Encyclopedia of Electrical and Electronic Engineering 2002, and a chapter on “Optically Controlled Phased Array Antennas for UWB RFID Reader” in Wiley’s Handbook of Smart Antennas and RFID Systems in 2010. He was involved with the following organizations APMC’99, ICCS 2000, ICICS 2003, PIERS 2003, IWAT 2005, ISAP 2006, ICICS 2007, ICOCN 2008, APMC 2009 and MWP 2011 conferences.

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Makoto Tsutsumi (LF’05) was born in Tokyo, Japan in 1937. He received the B.S. degree in electrical engineering from Ritsumeikan University, Kyoto, in 1961 and the M.S. and Ph.D. degrees in communication engineering from Osaka University, Osaka, Japan in 1961 and 1971, respectively. From 1984 to 1987, he was an Associate Professor of communication engineering at Osaka University. From 1988 to 2000, he was a Professor at Kyoto Institute of Technology, Department of Electronics and Information Science, Kyoto, Japan. From 2000

to 2010, he was a Professor at Fukui University of Technology, Department of Space Communication Engineering, Fukui, Japan. His research interests are primarily millimeter and microwave ferrite and left handed microwave circuits and devices. Dr. Tsutsumi is a Life Fellow of IEEE.

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1D-Leaky Wave Antenna Employing Parallel-Plate Waveguide Loaded With PRS and HIS María García-Vigueras, Student Member, IEEE, José Luis Gómez-Tornero, Member, IEEE, George Goussetis, Member, IEEE, Andrew R. Weily, Member, IEEE, and Y. Jay Guo, Senior Member, IEEE

Abstract—A new type of one-dimensional leaky-wave antenna (LWA) with independent control of the beam-pointing angle and beamwidth is presented. The antenna is based on a simple structure composed of a bulk parallel-plate waveguide (PPW) loaded with two printed circuit boards (PCBs), each one consisting of an array of printed dipoles. One PCB acts as a partially reflective surface (PRS), and the other grounded PCB behaves as a high impedance surface (HIS). It is shown that an independent control of the leaky-mode phase and leakage rate can be achieved by changing the lengths of the PRS and HIS dipoles, thus resulting in a flexible adjustment of the LWA pointing direction and directivity. The leaky-mode dispersion curves are obtained with a simple Transverse Equivalent Network (TEN), and they are validated with three-dimensional full-wave simulations. Experimental results on fabricated prototypes operating at 15 GHz are reported, demonstrating the versatile and independent control of the LWA performance by changing the PRS and HIS parameters. Index Terms—Artificial magnetic conductors (AMC), electromagnetic bandgap structures (EBG), frequency selective surfaces (FSS), high impedance surface (HIS), leaky-wave antennas (LWA), partially reflective surfaces (PRS), periodic surfaces.

Fig. 1. (a) Configuration of the proposed LWA, (b) Transverse Equivalent NetH ,S : work of the structure (a ,D ,  : ,L : : ,L ,P ,Q ).

=22

= = 9 mm

= 11 mm = 5 mm = 1 13 mm = 9 mm = 1 5 mm = 0 5 mm

that is related to the pointing angle ciency and the 3 dB beamwidth approximate expressions [1]:

, the radiation effithrough the following

(1) (2)

I. INTRODUCTION

(3)

L

EAKY-WAVE antennas (LWAs) provide a simple mechanism to obtain highly-directive frequency scanned radiation patterns from a simple feed [1]. One-dimensional (rectilinear) LWAs allow the scanning of a fan beam in the elevation direction. The leaky-mode (LM) time-space harmonic depen, where is the complex londence is of the form gitudinal propagation constant. For simplicity, the conventional time dependence will be omitted hereinafter. It is well known Manuscript received April 29, 2010; revised January 14, 2011; accepted March 25, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by Spanish National project TEC2007-67630-C03-02/TCM, Regional Seneca project 08833/PI/08, by Spanish scholarship “Salvador de Madariaga” (ref. PR2009-0336) and in part by regional scholarship PMPDI-UPCT-2009. M. García-Vigueras and J. L. Gómez-Tornero are with the Department of Communication and Information Technologies, Technical University of Cartagena, Cartagena 30202, Spain (e-mail: [email protected]; [email protected]). G. Goussetis is with the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, Belfast BT3 9DT, Northern Ireland (e-mail: [email protected]). A. Weily and Y. Jay Guo are with the CSIRO ICT Centre, Epping, NSW 1710, Australia. 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.2011.2163756

(4) where is the longitudinal direction of the LWA (see Fig. 1), is the LM phase constant, is the LM leakage rate, is the is the LWA length and a radiation free-space wavenumber, has been assumed to derive the term efficiency on the right side of (4) [1]. and leakage The independent control of the LM phase constants is of key importance for the synthesis and the flexible adjustment of the radiation pattern of a practical LWA. To this end, many different structures have been proposed in the recent decades to design rectilinear LWAs [1]. Most of them are based on waveguides which are asymmetrically perturbed to make the initially bounded mode leaky [2]–[6]. In these cases, the leakage rate is controlled by the degree of asymmetry introduced in the waveguide, whereas the change of the waveguide cross-sectional dimensions are used to adjust the leaky-phase constant. Leakage can also be induced and controlled by shortening the stubs of a non-radiative dielectric (NRD) waveguide [7]. All the above waveguide LWAs require the modification of the cross sectional dimensions of a bulky waveguide in order to achieve independent control of and , thus resulting in

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a complicated and expensive structure. The introduction of a transverse asymmetric strip in the groove guide was proposed in [8], [9] to provide a simpler mechanism to tune the leakage rate. However, the control of the pointing angle was still achieved by modifying the groove waveguide cross section. A hybrid technology combining a dielectric rectangular waveguide loaded with a printed-circuit was first proposed in [10], and later studied in detail in [11]. This hybrid technology presents simple control of the LM phase and leakage rates by only modifying the planar dimensions of a printed slot, thus avoiding the complicated mechanical fabrication of the waveguide. However, the use of a dielectric waveguide introduces higher losses and increased cost compared to LWAs based on hollow waveguides [11], [12]. Recently 2-D LWAs formed between a partially reflective surface (PRS) and a ground plane have been proposed [14]–[18] as broadband radiators. The operation of these antennas is based on establishing a Fabry-Perot resonance in the cavity formed between the PRS and the ground plane. A variation where the bottom grounded PCB behaves as a High-Impedance Surface (HIS) [19], also known as an artificial magnetic conductor (AMC) [20] has also been proposed to reduce the cavity profile [21], [22] as well as electronically reconfigure the LWA operating frequency [23]. At frequencies higher than the Fabry-Perot resonance, these antennas typically produce conical radiation patterns and therefore their use is limited. To the authors’ knowledge, however, the PRS-HIS structure has never been studied in the proposed 1-D topology, where it presents original and interesting features as a rectilinear LWA with flexible control of its scanning direction and directivity. Beyond tailoring the radiation pattern of LWA, this flexibility would allow the implementation of other microwave devices, such as leaky lenses [24] and couplers [25]. In this paper, the design and experiments of a new type of hybrid LWA which makes use of a hollow waveguide and two arrays of printed dipoles are presented [13]. The LWA working mechanism is described in Section II, and it is shown how the LM leakage rate and pointing angle can be flexibly controlled by simply modifying the printed dipole lengths. The synthesis procedure which relates the radiation specifications with the PRS and HIS dipole lengths is also provided. Experimental results of fabricated prototypes operating at 15 GHz are reported in Section III to demonstrate the independent control of the LWA pointing direction and directivity, while keeping high radiation efficiency. Finally, Section IV presents the conclusions of this work. II. ANALYSIS OF PRS-HIS 1-D LEAKY-WAVE ANTENNA The proposed configuration of the LWA is shown in Fig. 1(a), together with its main geometrical parameters. A cavity backed parallel-plate waveguide (PPW) is loaded with two printed circuit boards (PCBs) separated by distance . Each PCB is formed from a periodic array of metallic dipoles. The cavity determines the operating frequency of the antenna height (which is 15 GHz, corresponding to ). A careful study of the natural modes in the LWA has been performed using a specific full-wave Method of Moments

L

L

Fig. 2. (a) Dispersion of natural modes in the LWA ( = = 10 mm) (b) Transverse electric fields in the cross-section of the LWA for each mode.

technique [26]. The leaky-mode dispersion results for the case are plotted in Fig. 2. From this study, three different modes are present in our structure: the horizontally polarized channel-guide mode supported by the and PPW (mode 1 in Fig. 2), and the perturbed horizontal vertical modes of the cavity (modes 2 and 3 in Fig. 2). In the operating band (15 GHz), only modes 2 and 3 are in the fast-wave regime, while mode 1 is a nonradiative slow-wave [1]. Due to the symmetry of the structure, the vertically polarized mode 3 does not leak power to free space [26]. Also, single mode operation is assured by using a horizontally polarized feeding, as it is shown in Section III. As a result, the proposed leaky-mode (directed LWA operates with the perturbed along the x axis in Fig. 1) in the operating frequency band. The is defined in the H-plane ( elevation radiation angle plane in Fig. 1) and it is measured with respect to the -axis. Thus, the dispersion characteristics are calculated considering plane. TE polarization in the leaky-mode can be analyzed with The dispersion of the the Transverse Equivalent Network (TEN) shown in Fig. 1(b). The complex propagation constant (1) is obtained by solving the following transverse resonance equation (TRE): (5) In the TEN, the PRS and HIS printed circuits are modand eled by equivalent admittances, , which can be obtained as described in [27], [28]. This enables one to efficiently obtain the dispersion curves of the TE leaky-mode as a function of frequency and and , see the length of the PRS and HIS dipoles [ Fig. 1(a)]. In the next subsections, the effect of and on the pointing angle and normalized radiation is studied. All the leaky-wave dispersion results are rate

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Fig. 3. Leaky-mode frequency dispersion curves for the LWA in Fig. 1 for dif(L = 9 mm). ferent values of L

obtained from this simple TEN, and they are validated with a finite element method (FEM) full-wave simulator [29]. A. Effect of the Top PRS Dipole Length The transparency of the PRS determines the amount of energy that reaches the top aperture of the LWA, thus controlling the leakage rate [13]. Fig. 2 shows the LM frequency dispersion curves for different lengths of the PRS dipoles (the other dimensions of the LWA are summarized in the caption of Fig. 1). has a minimal effect on From Fig. 3, it is observed that , but it has a strong the pointing angle dispersion curves . The influence on the LM normalized leakage curves is approxPRS dipoles resonate at the frequency where imately half a wavelength [15]. At this frequency, the PRS behaves as an effective Perfect Electric Conductor (PEC), thus premode, which presents nil venting any radiation from the although the wave is fast . In particleakage ular, as is reduced from 9 mm to 7 mm, the frequency at which nil leakage is produced (FSS resonance [20]) is increased from 20 GHz to 22 GHz. It is also shown that the analytical results agree well with the simulated ones using FEM shown in circles in the figure. The above feature can be applied to the control of the radiation rate of the LWA for a given design frequency. At a fixed frequency and according to the bouncing ray model for waveguide propagation [27], the PRS reflectivity experienced by the inciin Fig. 1(b)] is a function of the length of its dent wave [ [15], [16], as shown in Fig. 4(a) where resonant dipoles the frequency is chosen as 15 GHz. The LM dispersion curves at 15 GHz are shown in Fig. 4(b). As can be seen in with , being Fig. 4, the PRS dipoles resonate when approximately half a wavelength at 15 GHz. At this length, the , a FSS PRS behaves as a totally reflective sheet resonance occurs [20] and the leakage rate vanishes . When is decreased from 11 mm, the PRS becomes more transparent to the incident LM, thus leading to a progressive increase of the radiation rate. This phenomenon is also illustrated in Fig. 5, where the leakymode electric field inside the LWA cross section is plotted for . It is seen that as is reduced different values of from 11 mm, more energy illuminates the top aperture of the antenna, thus increasing the leakage level. Consequently, the

at 15 GHz (L = 9 mm) (a) Reflection coefficient Fig. 4. Effect of L at the PRS seen by the incident leaky-mode (b) Leaky-mode pointing angle and normalized leakage rate.

Fig. 5. Transverse electric field of the leaky-mode inside the PRS-HIS cavity in Fig. 1 obtained from the TEN at 15 GHz.

possibility of controlling the radiation rate by varying the PRS dipoles length is verified. Yet, when is varied, a second is also altered to order effect occurs: the pointing angle some extent as shown in Fig. 4(b). This deviation is due to the on the phase of , which also varies dependence of with as illustrated in Fig. 4(a). This issue is addressed in the next sub-section. B. Effect of the Bottom HIS Dipoles Length A metal-backed dipole-based FSS acting as a HIS has been placed at the bottom of the antenna in order to modify the effective height of the resonant cavity formed by the two PCBs [13], [20]–[24]. This effective height strongly affects the freleaky-mode, thus proquency dispersion curves of the viding the control of the pointing angle at a fixed frequency. In a bouncing ray model for waveguide propagation, the length , determines the reflection phase exof the HIS dipoles, perienced by the wave propagating in the waveguide [13], [19], [20], thus changing the effective cavity height. Fig. 5(a) shows affects the LM frequency dispersion curves, shifting how

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Fig. 6. Leaky-mode dispersion curves for the LWA in Fig. 1 for different values (L = 9 mm). (a) Dispersion with frequency. (b) Dispersion with of L L at 15 GHz.

the cutoff frequency from 11 GHz to 15 GHz. For a fixed frequency, the pointing angle is increased and the leakage rate is reduced when the cutoff frequency is decreased (higher effective cavities) [1]. Fig. 5(b) shows the LM dispersion curves with at the operation frequency of 15 GHz. In this figure it is it is possible to scan the pointing clear that by changing angle in a wide range. In order to understand this effect, Fig. 7(a) shows the variation of the reflection phase seen by an incoming plane-wave at the in Fig. 1(b)] with at 15 GHz, while HIS interface [ Fig. 7(b) illustrates the near field patterns inside the LWA. When (approximately half a wavelength at 15 GHz), a FSS resonance occurs and the HIS behaves as a Perfect Electric in Fig. 7(a)], providing a Conductor (PEC) [19] [ [see Fig. 6(b)]. As is given pointing angle of reduced, increases, producing a smaller effective resonant in Fig. 7(b)], and reducing cavity [see close to broadside [ for in Fig. 6(b)]. is reached Maximum pointing angle at endfire for in Fig. 6(b). When , the HIS provides a Perfect Magnetic Conductor (PMC) resonance [19] in Fig. 7(a)]. In this case, the electric field is max[ in Fig. 7(b)], proimum at the HIS interface [see ducing an effective cavity of double height and pointing angle . Further decrease of continues increasing of and reduces both the effective cavity height and the cor[see and 4 respondent pointing angle mm in Fig. 6(b) and Fig. 7]. Therefore, one concludes that the pointing angle of the proposed LWA can be tuned by changing the length of the dipoles in the HIS. The modification of the pointing angle also involves

Fig. 7. Effect of L at 15 GHz (L = 9 mm). (a) Reflection phase seen by the incident leaky-wave at the HIS. (b) Transverse electric field of the leaky-mode inside the PRS-HIS cavity obtained from TEN.

the inherent inverse variation in the LM radiation rate shown in Fig. 6(b). As it is well known [1], as the pointing angle of a leak-wave is increased, the associated leakage rate decreases. C. Synthesis of Radiation Pattern In the previous sections it has been demonstrated that changing the lengths of the printed dipoles of the PRS and the HIS allows the variation of the leakage rate and the pointing angle of the leaky-mode which propagates in the proposed LWA. The synthesis of a LWA consists of the selection of the antenna geometry (in our case, and ) which provides the desired radiation pattern specifications: pointing angle , 3 dB beamwidth , and radiation efficiency . According to (1)–(4), all these parameters can be related to the LWA length and the leaky-mode pointing angle and leakage rate ( and ). As an example, seven LWAs with independent values of the pointing angle and beamwidth , all of them presenting radiation efficiency , are designed in this section as shown in Table I. Four of these antennas point at a fixed angle while having different beamwidths from to ( , , and in Table I). Another set of four antennas have fixed 3 dB beamwidth and scan at different angles from to ( , , and in Table I). From each set of values of , and , the corresponding and can be extracted using (1–4), as summarized in Table I. The last step in the synthesis procedure involves the extraction of the pair of values and which provide the desired values of and . For this purpose, a two-dimensional dispersion map similar to those used in [11], [12] is

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TABLE I ELECTRICAL AND GEOMETRICAL PARAMETERS OF THE LWAS DESIGNED AT 15 GHz

Fig. 9. Photographs of the manufactured LWA prototype.

Fig. 8. 2D contour curve plots of  Fig. 1(b) at 15 GHz.

and  =k obtained from the TEN in

obtained and shown in Fig. 8. and are simultaneously varied so that the functions and are numerically obtained from the TEN. The required and for any pair of and can then be extracted from this dispersion map, where the contour curves of constant and constant are plotted as continuous lines and dashed lines in Fig. 8, respectively. Fig. 8 shows that and are correlated in this antenna, so that in order to maintain constant and vary (or vice and . It versa), it is necessary to simultaneous vary must be noticed that the use of an efficient leaky-mode analysis technique is of essential importance to efficiently compute the 2D dispersion data shown in Fig. 8. As demonstrated in [11], the computational cost is dramatically reduced when the proposed TEN is employed, compared to the use of generic commercial full-wave simulators. The dipole lengths and for each of the seven LWA designs were extracted using the synthesis procedure described, resulting in the values summarized in Table I. The rest of dimensions of the LWA are kept fixed to the values given in the caption of Fig. 1. In this way, no modification of the host waveguide structure is required, and standard photolithographic processes can be used

Fig. 10. Photographs of (a) Radiation pattern measurements (b) S-parameter measurements.

to fabricate different PCB geometries that determine the LWA radiation features, as it is common in hybrid LWAs [12], [26]. III. EXPERIMENTAL RESULTS The seven LWAs designed in Section II were fabricated from Taconic TLY-5-0450 substrates. As can be seen in the photographs of Fig. 9, a single aluminum waveguide structure can be used to host all the PCB designs. The host waveguide is a cavity backed parallel-plate, which contains two pairs of slots to position the PRS and the HIS PCBs at the appropriate location. Two coaxial-to-waveguide transitions are used at the input and output of the LWA, to excite the mode of the resonant cavity and to match the antenna output, as can be seen in the photographs in Fig. 9 and Fig. 10(b). The feeding has been carefully designed to discard any direct radiation from the source, and discontinuities have been minimized to avoid diffraction. These facts ensure that the leaky-mode is the main source of radiation

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Fig. 11. Theoretical and measured normalized radiation patterns for the LWAs . in Table I showing the control of the beamwidth

1

from the antenna. The radiation pattern of the constructed prototype was measured in an anechoic chamber [see Fig. 10(a)] and also the S-parameters were measured using a vector network analyzer [see Fig. 10(b)]. The normalized radiation patterns of the four LWAs presenting a constant pointing angle at and different beamwidths at the design frequency of 15 GHz, are plotted in Fig. 11. The theoretical radiation diagrams obtained from the corresponding leaky-mode propagation constant are compared to the measured results. Good agreement is observed between theory and measurements, showing the ability to control the beamwidth while keeping a constant pointing angle. Fig. 11 presents the theoretical and experimental radiation diagrams at 15 GHz for the four LWAs with constant beamwidth and different scan angles . Again, good agreement is observed between leaky-mode theory and measurements, confirming the ability of the proposed LWA to control the pointing direction without affecting the directivity. It should be pointed out, however, that special care must be taken when radiation angles close to broadside or endfire are required. In the first case, the leaky-mode approaches the cut-off regime, decreasing both the efficiency and directivity of the antenna [1]. This can be observed in Fig. 12 for the case of LWA5, where the beamwidth has increased for scan angle . The maximum scanning angle of Fabry-Perot antennas is limited by the appearance of higher-order modes [17], which increase both the ripple and sidelobe levels of the radiation pattern, as can be seen in Fig. 12 for LWA7 (which points at ). This is a well-known disadvantage of hollow LWAs compared to dielectric filled LWAs, which can scan to higher angles [1]. Finally, Fig. 13 compares the theoretical and measured scan angle and beamwidth obtained for the seven LWAs designed in Table I. As shown in Figs. 10–12, the agreement between the specifications and the obtained measurement results is very good.

Fig. 12. Theoretical and measured normalized radiation patterns for the LWAs . in Table I showing the control of the scan angle



Fig. 13. Theoretical and measured scan angle and beamwidth for the LWAs designed in Table I.

Fig. 14. Measured S-parameters of the LWAs.

The measured S-parameters of one of the fabricated LWAs (all the LWAs present similar S parameters) are shown in Fig. 14. As can be seen, good input matching and a transmission level of is observed at the frequency of design (15 GHz). The ohmic losses were evaluated by measuring the waveguide under no radiation

GARCÍA-VIGUERAS et al.: 1D-LEAKY WAVE ANTENNA EMPLOYING PARALLEL-PLATE WAVEGUIDE LOADED WITH PRS AND HIS

conditions (closing its top aperture), obtaining a transmission value of 1.5 dB. These S-parameter values approximately correspond to a radiation efficiency , as was requested for all the designs. The advantage of working with a hollow waveguide is clear when comparing these losses with the ones associated with the dielectric-filled LWA proposed in [11], , [12]. In this case, the Teflon dielectric waveguide ( [12]) working at 15 GHz and the same length would introduce 6 dB losses. From the experimental results shown in Figs. 10–13, it is seen that the proposed PRS-HIS 1D LWA technology allows synthesis of high-gain radiation patterns with flexible control of scanning angle and directivity, whilst keeping high radiation efficiency in all the designs. IV. CONCLUSION A novel one-dimensional leaky-wave antenna with flexible control of the scanning angle and the leakage rate has been proposed and studied. The new LWA is based on a hollow parallel-plate waveguide loaded by two PCBs, which are formed by periodic arrays of printed metallic dipoles. This structure has been analyzed using a simple Transverse Equivalent Network, from which the leaky-mode dispersion curves are obtained. In addition, the TEN gives physical insight into the operating mechanism of this antenna. It has been demonstrated that, once the cavity dimensions are chosen to operate at a desired frequency, the length of the dipoles allow control of the leaky-mode propagation constant. Specifically, one PCB forms a PRS, whose transparency controls the leakage rate of the antenna. The second grounded PCB creates a HIS, whose equivalent reflection phase determines the effective cavity height, and the pointing angle of the antenna. This enables high-gain radiation patterns to be synthesized using standard photolithographic processes, without the need for modifying the waveguide structure. Compared to previous hybrid LWAs based on PCBs in dielectric waveguides, the proposed antenna avoids dielectric losses since a hollow waveguide is used as the host medium. To verify the concept and design theory, several LWA antenna prototypes operating at 15 GHz have been fabricated. Measured results agree with the predicted ones. It is demonstrated that it is indeed feasible to independently control the scanning angle and the directivity of the antenna at a fixed frequency, whilst keeping 90% radiation efficiency. It should be pointed out that, since the response of the resonant dipoles is determined by their effective length, varactor diodes could be used to enable the electronic control of the radiation pattern. This would lead to an electronically controlled reconfigurable LWA, which is the subject of our on-going research. ACKNOWLEDGMENT The authors would like to express their thanks to R. GuzmánQuirós for his help obtaining some of the full-wave dispersion curves needed for validation of the results of this paper, and C. Holmesby for fabricating the waveguide and transition. REFERENCES [1] A. A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Johnson, Ed., 3rd ed. New York: McGraw-Hill, 1993, ch. 10.

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[2] W. Rotman and A. A. Oliner, “Asymmetrical trough waveguide antennas,” IRE Trans. Antennas Propag., vol. 7, no. 2, pp. 153–162, Apr. 1959. [3] Q. Han, A. A. Oliner, and A. Sanchez, “A new leaky waveguide for millimeter waves using nonradiative dielectric (NRD) waveguide—Part II: Comparison with experiments,” IEEE Trans. Microwave Theory Tech., vol. 35, no. 8, pp. 748–752, Aug. 1987. [4] F. Frezza, M. Guglielmi, and P. Lampariello, “Millimetre-wave leakywave antennas based on slitted asymmetric ridge waveguides’,” IEE Proc. H, vol. 141, no. 3, pp. 175–180, 1994. [5] C. Di Nallo, F. Frezza, A. Galli, G. Gerosa, and P. Lampariello, “Stepped leaky-wave antennas for microwave and millimeter-wave applications,” Ann. Telecommun., vol. 52, pp. 202–208, Mar.–Apr. 1997. [6] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji, and A. A. Oliner, “A versatile leaky-wave antenna based on stub-loaded rectangular waveguide: Part I—Theory,” IEEE Trans. Antennas Propag., vol. 44, no. 7, pp. 1032–1041, July 1998. [7] A. Sanchez and A. A. Oliner, “A new leaky waveguide for millimeter waves using nonradiative dielectric (NRD) waveguide—Part I: Accurate theory,” IEEE Trans. Microwave Theory Tech., vol. 35, no. 8, pp. 737–747, Aug. 1987. [8] P. Lampariello and A. A. Oliner, “A new leaky-wave antenna for millimeter waves using an asymmetric strip in groove guide. Part I: Theory,” IEEE Trans. Antennas Propag., vol. 33, no. 12, pp. 1285–1294, Dec. 1985. [9] Z. Ma and E. Yamashita, “Leakage characteristics of groove guide having a conductor strip,” IEEE Trans. Microwave Theory Tech., vol. 42, no. 10, pp. 1925–1931, Oct. 1994. [10] P. Lampariello and A. A. Oliner, “A novel phased array of printedcircuit leaky-wave line sources,” in Proc. 17th Eur. Microwave Conf., Rome, Italy, Sep. 7–11, 1987, pp. 550–560. [11] J. L. Gómez, A. de la Torre, D. Cañete, M. Gugliemi, and A. A. Melcón, “Design of tapered leaky-wave antennas in hybrid waveguide-planar technology for millimeter waveband applications,” IEEE Trans. Antennas Propag, vol. 53, no. 8, pp. 2563–2577, Aug. 2005. [12] J. L. Gómez, G. Goussetis, A. Feresidis, and A. A. Melcón, “Control of leaky-mode propagation and radiation properties in hybrid dielectric-waveguide printed-circuit technology: Experimental results,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3383–3390, Nov. 2006. [13] M. García-Vigueras, J. L. Gómez-Tornero, G. Goussetis, D. Cañete, F. Quesada, and A. Álvarez-Melcón, “Leaky-mode dispersion analysis in parallel-plate waveguides loaded with FSS and AMC with application to 1D leaky-wave antennas,” in Proc. IEEE AP-S Int. Symp, Charleston, SC, Jun. 1–5, 2009, pp. 1–4. [14] G. V. Trentini, “Partially reflective sheet arrays,” IRE Trans. Antennas Propag., vol. 4, no. 4, pp. 666–671, Oct. 1956. [15] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” IEE Proc. Microw., Antennas and Propag., vol. 148, no. 6, pp. 345–350, Dec. 2001. [16] S. Maci, R. Magliacani, and A. Cucini, “Leaky-wave antennas realized by using artificial surfaces,” in IEEE AP-S Int. Symp. Dig., Columbus, OH, Jun. 23–27, 2003, pp. 1099–1102. [17] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas-part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [18] P. Kosmas, A. P. Feresidis, and G. Goussetis, “Periodic FDTD analysis of a 2-D leaky-wave planar antenna based on dipole frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2006–2012, Jul. 2007. [19] D. Sievenpiper, L. Zhang, F. J. Broas, N. G. Alexopulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [20] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG Characteristics of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006. [21] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [22] J. R. Kelly, T. Kokkinos, and A. P. Feresidis, “Analysis and design of sub-wavelength resonant cavity type 2-d leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2817–2825, Sep. 2008. [23] A. R. Weily, T. S. Bird, and Y. J. Guo, “A reconfigurable high-gain partially reflecting surface antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3382–3390, Nov. 2008.

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[24] J. L. Gómez-Tornero, F. Quesada-Pereira, A. Alvarez-Melcón, G. Goussetis, A. R. Weily, and Y. J. Guo, “Frequency steerable two dimensional focusing using rectilinear leaky-wave lenses,” IEEE Trans. Antennas Propag., to be published. [25] J. L. Gomez-Tornero, S. Martinez-Lopez, and A. Alvarez-Melcon, “Simple analysis and design of a new leaky-wave directional coupler in hybrid dielectric-waveguide printed-circuit technology,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 9, pp. 3534–3542, Aug. 2006. [26] J. L. Gómez-Tornero, F. D. Quesada-Pereira, and A. Alvarez-Melcón, “Analysis and design of periodic leaky-wave antennas for the millimeter waveband in hybrid waveguide-planar technology,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2834–2842, Sep. 2005. [27] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [28] M. García-Vigueras, J. L. Gómez-Tornero, G. Goussetis, J. S. GómezDiaz, and A. Álvarez-Melcón, “A modified pole-zero technique for the synthesis of waveguide leaky-wave antennas loaded with dipole-based FSS,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1971–1979, Jun. 2010. [29] High Frequency Structure Simulator vol. 11, Ansoft Corporation. [30] D. M. Pozar, “Transmission lines and waveguides,” in Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 3.

George Goussetis (SM’99–M’02) graduated from the Electrical and Computer Engineering School, National Technical University of Athens, Greece, in 1998, and received the Ph.D. degree from the University of Westminster, London, U.K. In 2002 he also received the B.Sc. degree in physics (first class) from University College London (UCL), U.K. In 1998, he joined the Space Engineering, Rome, Italy, as RF Engineer and in 1999 the Wireless Communications Research Group, University of Westminster, U.K., as a Research Assistant. Between 2002 and 2006, he was a Senior Research Fellow at Loughborough University, U.K. Between 2006 and 2009, he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology at Queen’s University Belfast, U.K., in September 2009 as a Reader (Associate Professor). In 2010, he was a Visiting Professor in UPCT, Spain. He has authored or coauthored over 100 peer-reviewed papers three book chapters and two patents. His research interests include the modelling and design of microwave filters, frequency-selective surfaces and periodic structures, leaky wave antennas, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis foundation scholarship in 2001. In October 2006 he was awarded a five-year research fellowship by the Royal Academy of Engineering, U.K.

María García-Vigueras (S’09) was born in Murcia, Spain, in 1984. She received the telecommunications engineer degree from the Technical University of Cartagena (UPCT), Spain, in 2007, where she is currently working towards the Ph.D. degree. In 2008, she joined the Department of Communication and Information Technologies, UPCT, as a Research Assistant. She has been a visiting Ph.D. student at Heriot-Watt University in Edinburgh (Scotland, United Kingdom) and at the University of Seville (Spain). Her research interests focus on the development of equivalent circuits to characterize periodic surfaces, with application to the analysis and design of leaky-wave antennas.

Andrew R. Weily (S’96–M’01) received the B.E. degree in electrical engineering from the University of New South Wales, Australia, in 1995, and the Ph.D. degree in electrical engineering from the University of Technology Sydney (UTS), Australia, in 2001. From 2000 to 2001, he was a Research Assistant at UTS. He was a Macquarie University Research Fellow then an ARC Linkage Postdoctoral Research Fellow from 2001 to 2006 with the Department of Electronics, Macquarie University, Sydney, NSW, Australia. In October 2006, he joined the Wireless Technology Laboratory at CSIRO ICT Centre, Sydney. His research interests are in the areas of reconfigurable antennas, EBG antennas and waveguide components, leaky wave antennas, frequency selective surfaces, dielectric resonator filters, and numerical methods in electromagnetics.

José Luis Gómez Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and the Ph.D degree (laurea cum laude) in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999 he joined the Radio Communications Department, UPV, as a Research Student, where he was involved in the development of analytical and numerical tools for the automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders for space applications. In 2001, he joined the Technical University of Cartagena (UPCT), Spain, as an Assistant Professor. From October 2005 to February 2009, he held de position of Vice Dean for Students and Lectures affairs in the Telecommunication Engineering Faculty at the UPCT. Since 2008, he has been an Associate Professor at the Department of Communication and Information Technologies, UPCT. His current research interests include analysis and design of leaky-wave antennas and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. Dr. Luis Gómez Tornero received the national award from the foundation EPSON-Ibérica to the best Ph.D project in the field of Technology of Information and Communications (TIC), in July 2004. In June 2006, he received the Vodafone foundation-COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicación) award to the best Spanish Ph.D. thesis in the area of advanced mobile communications technologies. This thesis was also awarded as the best thesis in the area of Electrical Engineering, by the Technical University of Cartagena, in December 2006. In February 2010, he was appointed CSIRO Distinguished Visiting Scientist by the CSIRO ICT Centre, Sydney.

Y. Jay Guo (SM’96) received the Bachelor and Master degrees from Xidian University, China, in 1982 and 1984, respectively, and the Ph.D. degree from Xian Jiaotong University, China, in 1987. He was awarded an honorary Ph.D. degree in 1997 by the University of Bradford, U.K., for his world leading research in Fresnel antennas. Currently, he is the Research Director of Broadband for Australia Theme in CSIRO, Australia, and the Director of the Australia China Research Centre for Wireless Communications. From August 2005 to January 2010, he served as the Research Director of the Wireless Technologies Laboratory in CSIRO ICT Centre. Prior to joining CSIRO, he held various senior positions in a number of international companies including Fujitsu, Siemens and NEC, managing the development of advanced technologies for the third generation (3G) mobile communications systems. He is an Adjunct Professor at Macquarie University, Australia, and a Guest Professor at the Chinese Academy of Science (CAS) and Shanghai Jiaotong University. He is a Fellow of IET and a Senior Member of IEEE. He has published three technical books Fresnel Zone Antennas, Advances in Mobile Radio Access Networks, and Ground-Based Wireless Positioning, 58 journal papers and over 80 refereed international conference papers. He holds 16 patents in wireless technologies. Dr. Guo is the recipient of Australian Engineering Excellence Award and CSIRO Chairman’s Medal. He has served on the organizing and technical committees of numerous national and international conferences. He was Chair of the Technical Program Committee (TPC) of 2010 IEEE Wireless Communications and Networking Conference (WCNC) and 2007 IEEE International Symposium on Communications and Information Technologies (ISCIT), and is the TPC Chair of IEEE ISCIT2012. He has been the Executive Chair of Australia China ICT Summit since 2009. He was a Guest Editor of the special issue on “Antennas and Propagation Aspects of 60–90 GHz Wireless Communications” in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. Currently, he is serving as a Senior Guest Editor for IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS (JSAC), Special Issue on “Challenges and Dynamics for Unmanned Autonomous Vehicles.”

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Transmission Line Modeling and Asymptotic Formulas for Periodic Leaky-Wave Antennas Scanning Through Broadside Simon Otto, Student Member, IEEE, Andreas Rennings, Member, IEEE, Klaus Solbach, and Christophe Caloz, Fellow, IEEE

Abstract—It is shown, using three specific examples—a series fed patch (SFP) array, a phase reversal (PR) array and a composite right/left-handed (CRLH) antenna—that one-dimensional periodic leaky-wave antennas scanning through broadside build a class of leaky-wave antennas sharing qualitatively similar and quantitatively distinct dispersion and radiation characteristics. Based on an equivalent transmission line (TL) model using linearized series and shunt immittances to approximate the periodic (Bloch) antenna structure, asymptotic TL formulas for the characteristic propagation constant, impedance, energy, power and quality factor are derived for two fundamentally different nearand off-broadside radiation regimes. Based on these formulas, it is established that the total powers in the series and shunt elements are always equal at broadside, which constitutes one of the central results of this contribution. This equal power splitting implies a severe degradation of broadside radiation when only one of the two elements series or shunt efficiently contributes to radiation and the other is mainly dissipative. A condition for optimum broadside radiation is subsequently established and shown to be identical to the Heaviside condition for distortionless propagation in TL theory. Closed-form expressions are derived for the constitutive (LCRG) parameters of the TL model for the specific SFP, PR and CRLH antenna circuit models, and quantitative information on the validity range of the TL model is subsequently provided. Finally, full-wave simulation and measurement LCRG parameter extraction methods are proposed and validated. Index Terms—Bloch-Floquet theorem, broadside radiation, composite right/left-handed (CRLH) metamaterial, Heaviside condition, leaky-wave antennas, periodic structures, phase-reversal (PR) array, quality factor, series-fed patch (SFP) array, transmission line (TL) theory.

I. INTRODUCTION ERIODIC leaky-wave antennas have been extensively studied and widely used for over five decades [1], [2]. Particularly, planar leaky-wave antennas configurations, with their advantages of low cost, light weight, simple fabrication and integrability with electronic components have found

P

Manuscript received November 11, 2010; revised February 24, 2011; accepted April 07, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. S. Otto, A. Rennings, and K. Solbach are with the University of DuisburgEssen, Duisburg, Germany (e-mail: [email protected]). C. Caloz is with the École Polytechnique de Montréal, 2500, ch. de Polytechnique, H3T 1J4, Montréal, Quebec, Canada. 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.2011.2163781

applications in various radar and communication systems at microwave frequencies. While the first planar periodic leaky-wave antennas were reported back in the 1980s, they experience a regain of interest nowadays following recent contributions on novel broadside optimum designs [3]–[5] and on designs with novel features or system functionalities [6]–[12]. An overview of recent advances in this field is available in [13]. This paper presents a lattice-network based TL model of periodic leaky-wave antennas characterized by continuous backward to forward radiation through broadside and derives subsequent asymptotic formulas for the regimes of near-broadside and off-broadside radiation either toward forward or backward directions. The proposed approach applies to all leaky-wave antennas of this class. It is illustrated by three examples, a series fed patch (SFP) array, a phase reversal (PR) array and a composite right/left-handed (CRLH) antenna, and supported by full-wave and experimental results. The asymptotic formulas are the central result of the paper. They provide physical insight and electromagnetic understanding of the behavior of leaky-wave antennas around the broadside frequency. They also explain the issue of inefficient radiation of one-dimensional periodic leaky-wave antennas at broadside. When the unit cell is symmetric with respect to its transverse axis, as assumed in this work, the radiative modes decouple at the broadside radiation frequency [1] and it may be designed to exhibit a seamless transition between the forward and backward frequency ranges by closure of the open stopband. Transversally asymmetrical unit cells, which exhibit an open stopband as a result of mode coupling [1], [2], [4], [13], form a different class of antennas and are out of scope of the present paper. In this work we show that two conditions are required to overcome the broadside radiation issue. In addition to frequency-balancing for closing the band gap, Q-balancing, which is based on the Heaviside condition (distortionless TL), is also required for optimum broadside radiation. The paper is organized as follows. Section II defines the class of antennas under consideration and describes their common qualitative features and their quantitative differences. Section III derives the transmission line (TL) model, based on linearized lattice-network series and shunt immittances, for this class of antennas. Section IV develops a simple equivalent circuit for each of the considered antennas (SFP, PR and CRLH) and compares the periodic (Bloch) propagation constant and the impedance with the TL model results for the validation and estimation of the frequency validity range of the latter. Section V proposes LCRG parameter extraction methods and validates the TL model by

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Fig. 1. Frequency scanning periodic leaky-wave antennas under consideration in this paper. (a) Series-fed patch (SFP) array antenna. (b) Phase reversal (PR) array antenna. (c) Composite right/left-handed (CRLH) antenna with shunt stub inductors and different series capacitors (left: interdigital, right: metal-insulatormetal).

full-wave simulation and experimental results for the SFP case. Upon the basis of Section III to V, Section VI derives asymptotic formulas for the phase constant, leakage factor, impedance, quality factor, energy and power. Finally, conclusions are given in Section VII.

Fig. 2. Typical dispersion diagrams for the three antennas shown in Fig. 1 obtained by full-wave analysis using periodic boundary conditions (FEM Ansoft-HFSS). (a) SFP antenna [Fig. 1(a)]. (b) PR antenna [Fig. 1(b)]. (c) CRLH antenna [Fig. 1(c)]. Only the lowest few modes are shown and the diagram is restricted to the first Brillouin zone (j8j = j jp <  ). The angular frequency ! is normalized with respect to the frequency of broadside radiation, i.e. ! = ! (8 = p = 0). The white areas indicate the slow-wave regions (v = != < c = !=k ), where radiation may potentially exist in the substrate [22] but not in free space, while the shaded regions correspond to the fast-wave leaky-wave region (v = != > c = !=k ) of interest. (d) Superposition of the (a)–(c) dispersion curves of interest, i.e. the positive group velocity (v = @!=@ > 0) leaky-wave curves around the broadside frequency region [thicker parts in (a)–(c)], corresponding to a specific (n ) space harmonic.

with

II. CLASS OF ANTENNAS CONSIDERED The class of antennas considered in this paper covers a wide range of one-dimensional periodic1 leaky-wave antennas [13] characterized by a full or partial frequency beam scanning response from backward to forward angles including broadside. Fig. 1 depicts three antennas which belong to this class and which will be specifically analyzed: a series fed patch (SFP) array antenna [14]–[16], a phase reversal (PR) array antenna [17]–[19] and a composite right/left-handed (CRLH) antenna [6], [20], [21]. While such antenna structures maybe short-ended or open-ended to operate in a standing-wave regime as resonant antennas, with fixed radiation beam, we consider here the case where the antenna structures are terminated by matched loads, so as to operate in a traveling-wave regime as leaky-wave antennas, following the standard frequency-angle scanning law [2], [13]

(1a) 1In fact, although only periodic structures are discussed in this paper, this class could be extended to uniform (non-periodic) antennas, such as the ferrite-loaded open waveguide CRLH leaky-wave antenna recently reported in [10].

(1b) where represents the main-beam radiation angle, denotes the space harmonic, is the free-space wavenumber, : angular frequency, : speed of light in free space and is the period of the structure [1]. Despite their seemingly very different configurations, the three antennas of Fig. 1, as well as all other antennas belonging to the aforementioned class, exhibit a qualitatively similar leaky-wave behavior following from their qualitatively similar [(1)]. Fig. 2 shows the exact leaky dispersion responses dispersion diagrams computed by full-wave analysis for these three antennas, where each antenna is designed to have a closed band gap at . While the overall dispersion diagrams substantially differ between the three cases, in particular in the guided wave regions, the leaky-wave curves around the , are qualbroadside frequency , occurring at itatively identical, they all display a seamless and quasi-linear , allowing transition from negative to positive values of frequency scanning from backward to forward angles according to (1) [13]. All three antennas have been designed to exhibit a

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TABLE I _ ) and SCANNING SENSITIVITIES ELECTRICAL PERIODS (p), PHASE SLOPES (8 (s) FOR THE THREE ANTENNAS IN FIG. 1

single radiating space harmonic so as to provide single-beam full-space scanning [1]. Beyond their aforementioned fundamental similarities, the three antennas also exhibit some important differences. The SFP space harmonic to radiate [2]. For the antenna uses its PR antenna, the radiating space harmonic may be considered eior the [19]. The CRLH antenna ther to be the [21]. Moreradiates in the fundamental space harmonic,2 over, the three antennas have very different quantitative parameters, as shown in Table I, which has an important consequence on the frequency range validity of the derivations performed in the reminder of the paper, as will be shown next. The first column of Table I shows that the electrical periods, , where is the free space wavelength at , vary by a 3.4 factor between SFP and CRLH antennas while the PR antenna has an intermediate value of . For radiation angles near , the inverse sinus in (1) may be approxibroadside mated by its argument, i.e. , where may itself be approximated by its dominant Taylor expansion term, (2) where

is the phase slope at

. Therefore, (3)

from which it follows that the scanning sensitivity of the antennas may be written as (4) Since is inversely proportional to the frequency bandwidth over which the antenna’s main beam scans from broadside to backfire and endfire and since these two extreme directions apgrounded-slab mode (folproximately correspond to the lowing the light line at lower frequencies), the antenna with the largest will have the smallest frequency bandwidth immune of mode. coupling to this Table I shows that the phase slopes for the PR and CRLH antennas are fairly close, while the phase slope for the SFP is twice larger. Incorporating the electrical periods along with into (4), yields the scanning sensitivities, which the slopes are listed in the third column of the table. Thus, the CRLH antenna exhibits the highest scanning sensitivity, as a result of its extremely small electrical period. As a consequence, it exhibits 2The mode starting from ! = 0 in Fig. 2(c) is the light line mode (TM ), so that the mode around ! is the fundamental mode (n = 0) of the CRLH structure.

Fig. 3. Two-port network modeling of a periodic leaky-wave structure in terms of the ABCD parameters of its unit cell and application of Bloch-Floquet theorem to relate its input and output currents and voltages.

the smallest frequency range immune of perturbation related to -mode, i.e. the smallest frequency range coupling to the with a linear dispersion curve, as seen in Fig. 2(c) and Fig. 2(d). has the The PR antenna, with its relatively large and small lowest scanning sensitivity, and therefore displays the broadest range of linear phase-frequency response. The modeling and formulas of the next sections will not acmode, and will therefore be recount for coupling to the stricted to the regions of nearly constant group velocity corresponding to a quasi linear phase in Fig. 2(d). Nevertheless, the proposed analysis will prove to be very efficient within a reasonable bandwidth for all the cases, including the CRLH case. III. PERIODIC NETWORK MODELING AND EQUIVALENT TRANSMISSION LINE MODEL Due to the qualitative similarity of their leaky-wave dispersive responses, the three antennas of Fig. 1 can be described by a common network model. This model will be derived as an approximation of the periodic or Bloch solution, characterized by and impedance , and will be rethe propagation constant ferred to as the transmission line (TL) model, with characteristic and impedance . This propagation constant section establishes the TL model. It will serve as a foundation for the derivation of asymptotic formulas in Section VI. A. Bloch Model, Propagation Constant and Impedance The periodic antenna structures depicted in Fig. 1 may be modeled by their unit cell equivalent two-port network, which is shown in Fig. 3. Applying Bloch-Floquet theorem [23] to the terminals of this two-port network, described in terms of its ABCD matrix, and , yields assuming that the unit cell is symmetrical, i.e. the dispersion relation for the Bloch propagation constant [24] (5) where is the phase constant, which determines the angle of main beam radiation according to (1) (replacing by ), while is the attenuation constant [2], which includes a radiation leakage contribution, determining the directivity of the antenna, and a dissipation contribution. By the is obtained for the symsame token, the Bloch impedance as metrical case (6)

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Fig. 5. Specific lattice equivalent circuits corresponding to the general lattice of Fig. 4(a). (a) Lattice incorporating the equivalent lumped elements given by (13) and (14). (b) Approximate equivalent lattice obtained from the linearized and Y as defined in (15). immittances Z

Fig. 4. Lattice type equivalent circuit model for the unit cell of periodic leakywave antennas. (a) General circuit. (b) Equivalent circuit under odd input-output excitation. (c) Equivalent circuit under even input-output excitation.

Inside the “black box” equivalent network of Fig. 3, the physical unit cell of the periodic structure is modeled by the lattice circuit [26] shown in Fig. 4(a). This topology offers and the advantage of decoupling the series impedance under odd and even input-output termishunt admittance nation conditions [27], respectively, as shown in Fig. 4(b) and Fig. 4(c). While the lattice topology of Fig. 4(a) was actually implemented by distributed and lumped elements to exploit the broadband all-pass nature of this circuit in [28], it is used here as a generic model for the antennas under consideration in of the lattice Section II. The impedance matrix parameters circuit of Fig. 4(a), which is assumed to be symmetric, are given by [29] (7a) (7b)

The difference of these two expressions yields (8a) while the sum yields , which leads, using standard network transformation formulas [24], to

Using (5) and assuming reciprocity in addition to symmetry and ), we obtain the following rela( and tion for the Bloch propagation constant in terms of (10) Similarly, we seek an expression for the Bloch impedance . For this purpose, we first write the and parameters in (6) in terms of the matrix elements using standard network and conversion formulas, i.e. [24], which yields . Inserting next the converand sion formulas into (8b) yields , from which we find (11) B. Transmission Line Characteristic Propagation Constant and Impedance Following From Immittance Linearization The formulas for the Bloch propagation constant and the Bloch impedance given by (10) and (11) may be simplified as a series resonance circuit with resonance by modeling and as a parallel resonance circuit with frequency resonance frequency , respectively. In a leaky-wave antenna, these resonances are typically designed to be equal, i.e. , so as to provide seamless beam scanning from backward to forward angles across broadside [20]. In the remainder of this paper, we will consider that this equality, called frequency-balancing,3 always holds. Fig. 5(a) depicts the lattice circuit with its explicit series and shunt resonators. The series resonance frequency and the shunt resonance frequency are defined by

(8b) We now seek an expression for the Bloch propagation constant in (5) as a function of the series impedance and shunt admittance . For this purpose, we write the and parameters in (8) in terms of the ABCD parameters, using , again standard network conversion formulas, i.e. , and [24], which leads to (9)

(12) The corresponding lumped elements are determined by (13a) (13b) (13c) 3as

opposed to Q-balancing which is addressed in Section VI-D.

OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS

where by

, for the series resonance circuit and

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TABLE II SUMMARY OF THE RESULTS DERIVED IN SECTION IV-A TO SECTION IV-C

(14a) (14b) (14c) , for the shunt resonance circuit, rewhere spectively. and may next be linearized around The immittances their respective resonances (15a) (15b) , as previously where it will be assumed that announced. Here, the LCRG parameters (without any subscripts) have emerged as the parameters of the equivalent linearized model, which is represented in Fig. 5(b). The LCRG parameter can be calculated analytically based on the specific circuit model of the SFP, PR or CRLH unit cell [Section IV-A to Section IV-C] or they can be extracted from full-wave simulation or measurement data [Section V]. The linearized lattice model of Fig. 5(b) will be used as the TL model in the forthcoming evaluations and derivations. Around , it exhibits the broadside frequency, where simple characteristic parameters. By approximating both sides and , of (10) with a Taylor expansion with respect to respectively, and truncating higher order terms one obtains (16) from which the linearized-immittance propagation constant is defined as (17) with the parameters given by (15). On the other hand, the linearized-immittance impedance is directly (without approximation) obtained from (11) as (18) Due to their similarity with TL expressions, in (17) and in (18) will be subsequently referred to as the characteristic propagation constant and the characteristic impedance (hence the subscript “c”) of the antenna structures, respectively. These quantities may be interpreted as the characteristic parameters of a uniform TL approximating the actual periodic structure with and the charthe characteristic propagation constant . acteristic impedance

IV. VALIDATION OF THE TL MODEL WITH SFP, PR AND CRLH CIRCUIT MODELS This section validates the TL model by broadband periodic (Bloch) distributed/lumped circuit models specific to the SFP, PR and CRLH antennas, and thereby also determines the frequency validity range of this model. We introduce a perturbaand yielding simple tional approach for the immittances closed-form expressions for , , and . For each unit cell circuit, the TL model is validated by comparing the characteristic propagation constant and characteristic impedance [(17) and (18)] with Bloch parameters [(5) and (6)]. The following procedure is used: matrix of 1) Compute the lossless4 overall sub-matrices the unit cell by cascading the of its successive equivalent elements ignoring the lossy elements. (e.g. SFP unit cell in Fig. 6(a): ). 2) Compute the lossless and matrices corresponding to matrix obtained in 1) using standard network the conversion formulas [24]. and terms of these matrices 3) Insert the appropriate and , respectively. into (8a) and (8b) to determine using (13b) and 4) Compute the frequency derivatives of using (14b), and insert the resulting expressions, evaluated at , into (15) to obtain the linearized immittances and , which are purely imaginary since the losses are not taken into account so far. 5) Determine the , from (15), as they are the coefficients and . in 6) Introduce the lossy elements in the respective circuit model and apply odd [Fig. 4(b)] and even [Fig. 4(c)] excitations at the resonance frequency to isolate the resistance and conductance , respectively. found in 5) and found in 6), insert the 7) With expressions obtained for and into (17) and (18) and . to compute The LCRG formulas obtained from this seven-step procedure and based on the forthcoming SFP, PR and CRLH distributed/ lumped element equivalent circuits, are given in Table II. For each unit cell circuit, a numerical example comparing Bloch and TL model is presented next. 4This initial lossless approximation is reasonable, as will be shown, because the slopes of the immittance parameters are not significantly affected by loss and radiation.

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Fig. 6. SFP antenna: (a) Layout of a microstrip SFP unit cell and corresponding circuit model with normalized electrical period p = 0:54. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re( p) = p is multiplied by a factor of 5 for better visualization) and (c) impedance Z ; Z .

Fig. 7. PR antenna: (a) Layout of a microstrip PR unit cell and the corresponding circuit model with normalized electrical period p = 0:36. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re( p) = p is multiplied by a factor of 5 for better visualization) and (c) impedance Z ; Z .

A. Series Fed Patch (SFP) Antenna Fig. 6(a) shows the layout and the equivalent circuit model for the SFP antenna [Fig. 1(a)] unit cell [15], [16]. The microstrip patch, of length , is modeled by a low-impedance TL section of impedance , while the interconnecting microstrip , are modeled by lines on both sides of the patch, of length . The radiating high-impedance TL sections of impedance edges of the patch are modeled by the conductances . Distributed losses i.e. conductor loss and dielectric loss are lumped and . The electrical line lengths of the connecinto at the broadtion line and the patch TL have to be side frequency . Therefore, we write the electrical length as where . follows , , A set of realistic parameters— and —has been selected to generate Fig. 6(b) and Fig. 6(c), which will be described in Section IV-D. B. Phase Reversal (PR) Antenna Fig. 7(a) shows the layout and the equivalent circuit model for the PR antenna [Fig. 1(b)] unit cell. The black and gray traces represent the top and bottom metal layers at either side of the substrate, respectively, and radiation occurs from the cross-over region [18], which is modeled by an ideal transformer with a transformation ratio to provide the 180 phase reversal. and , are used In addition, two independent resistors, to model radiation and dissipation loss (conductor and dielectric . loss), respectively. The overall length of the unit cell is at the broadside freThe electrical length of the unit cell is with The following set of paramequency , ters has been chosen for validation: and . The TL-Bloch comparison is shown in Figs. 7(b) and 7(c) and will be discussed in Section IV-D.

Fig. 8. CRLH antenna: (a) Layout of a microstrip CRLH unit cell (left: MIM, right: interdigital) and corresponding circuit model with normalized electrical period p = 0:16. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re( p) = p is multiplied by a factor of 5 for better visualization) and (c) impedance Z ; Z .

C. Composite Right/Left-Handed (CRLH) Antenna Fig. 8(a) shows the layout and the equivalent circuit model for the CRLH antenna [Fig. 1(c)] unit cell, where both the MIM and interdigital implementations are considered. The length of . The set of parameters , the unit cell is , , , and has been selected for the comparison in Figs. 8(b) and 8(c). This example will be further discussed in Section IV-D. D. Comparison and Validity Range The circuit models for the SFP, PR and CRLH antennas in Figs. 6(a), 7(a) and 8(a), respectively are inherently broadband,

OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS

since they are based on the exact physical nature of the corresponding structures. In contrast, the TL model is a much simpler (and therefore narrower band) model; however, this model has the fundamental interest to be common to all one-dimensional periodic leaky-wave antennas scanning through broadside while still being very accurate around the broadside radiation frequency. The degree of accuracy of the TL model away from the broadside frequency strongly depends on the particular antenna structure. The TL model for the SFP antenna, with its large electrical unit cell period, has the highest deviation in the propagation constant [Fig. 6(b)] and impedance [Fig. 6(c)] compared to the PR antenna, which is in excellent agreement with the TL model [Figs. 7(b) and 7(c)]. This excellent agreement is due to the homogeneous TL nature of the PR antenna, which is essentially composed of TL sections of constant impedance. The approximation accuracy of the CRLH circuit antenna [Fig. 8(a)] is in between the two former cases, with a good agreement for the impedance and a reasonable agreement for the propagation constant. Another factor limiting the frequency range validity, quite far from the broadside frequency region, is the coupling to the light line (Fig. 2), not taken into account in the model. The shaded areas in the dispersion diagrams of Figs. 6(b), 7(b) and 8(b) indicate the fast-wave regions, where the model is applicable. The formulas presented in Table II provide design guidance and insight into the influence of the different structural parameters for the development of the specific leaky-wave antenna of interest. For instance, they provide, via Table II, parametric expressions for the Bloch impedance near broadside and off broadside for each of the three discussed antennas using the formulas and , respectively, which will be derived in Section V. Finally, it is important to note that the results in Table II are based on a perturbational (lossless) approach (step 4 above) for and . Such an approach the series and shunt immittances or can clearly not be applied to the propagation constant directly, as is done in TL theory and as will be impedance exploited in Section V, since this would not capture the dominant resistive effects at the broadside frequency.

V. FULL-WAVE AND EXPERIMENTAL PARAMETER EXTRACTION AND VALIDATION This section validates the formulas derived in Sections III and IV by full-wave and measurement results. The SFP antenna is selected for this purpose. This is essentially an arbitrary choice, except for the fact that this antenna is particularly simple to fabricate and test. In the full-wave case, two completely different methods are used in parallel for the extraction of the LCRG circuit model parameters, a driven-mode network method using immittance slope parameters and an eigenmode electromagnetic method using energies and powers computed by integrating electromagnetic fields, while only the former is applicable and applied in the experimental case. Fig. 9 shows the layout and dimensions of the test SFP antenna, which was . designed to operate at

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Fig. 9. Layout and dimensions (unit cell) of the selected SFP antenna. The substrate has a relative permittivity of " : and a height of h : .

=22

= 1 5 mm

Fig. 10. Simulation setup of the 10-cell structure with microstrip TL ports in EMPIRE XCcel. The structure is simulated with a finite ground plane and is fully enclosed by absorbing PML boundaries.

A. Driven-Mode Network LCRG Parameter Extraction The network LCRG parameter extraction consists in the following simulation steps: 1) set up a parameterized (in and ) antenna structure composed of a sufficient number of unit cells to effectively take into account mutual coupling effects [4], [30], [31] and to sufficiently dilute edge (termination) aperiodic effects (here is set to 10); for this -cell 2) compute the transmission matrix structure; -root of this matrix, , to obtain 3) take the the corresponding matrix for one unit cell, , also ) or using an esevaluating a different length (e.g. and in order timation of the broadside frequencies solutions to select the correct physical solution among [31]; 4) use standard conversion formulas to compute the corresponding (one unit cell) impedance matrix and admittance matrix ; 5) insert the appropriate elements of the resulting matrices and , respectively, into (8a) and (8b) to determine and apply (12) to find initial (approximate) values for and ; 6) optimize the parameters and to obtain exactly in (12); 7) insert the optimized impedance and admittance parameters and , respecinto (8a) and (8b) to obtain the exact tively, and compare these results with (15) to obtain the final LCRG parameters. The corresponding structure (FDTD EMPIRE XCcel) used in this procedure for the forthcoming results is shown in Fig. 10. B. Eigenmode Electromagnetic LCRG Parameter Extraction The electromagnetic LCRG parameter extraction was developed and applied to a CRLH antenna in [32] and is recalled here for self-consistency. It employs an eigenmode solver (here FEM Ansoft-HFSS) with periodic boundaries along the axis of

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Fig. 12. Equivalent circuit models of Fig. 5(a) of the structure at 8 = 0. (a) PEC-wall (short-circuited) mode ! , where the series impedance is isolated. (b) PMC-wall (open-circuited) mode ! , where the shunt admittance is isolated.

Fig. 11. Simulation setup for parameter extraction. (a) Periodic boundaries with a constant null phase shift (8 = 0) are applied at both sides of the unit cell along the axis of the antenna structure (x direction). The sides of the computational domains are terminated by PML boundaries in the transverse direction (y direction) as well as the opening to free space on the top (z direction). While the unit cell shown is that of the SFP antenna, the same setup can be used for any antenna of the class considered in the paper. (b) Voltage and current integration paths for (25) and (26), respectively.

the antenna structure, set to a constant null periodic phase shift, , and determines the circuit parameters for the circuit model of Fig. 5(a) from energies and powers computed by integration of the electric and magnetic fields over appropriate domains. Fig. 11 shows the simulation setup used for this purpose. The required electromagnetic quantities are [24]

(19) (20)

: one corresponding to PEC walls and another one corresponding to PMC walls at the positions of the periodic boundaries5 [32]. With such walls, the lattice circuit model of Fig. 5(a) splits in the two simplified circuits shown in Fig. 12, which allows to separate the elements of the series impedance and of the shunt admittance. The corresponding current and voltage are obtained from field quantities as (25) (26) where the integration contours are indicated in Fig. 11(b). and ) for the circuit The reactive parameter values ( components of each resonator are obtained by relating the stored magnetic or electric energy to the current or voltage for the corresponding mode, while the resistive parameter values ( and ) are computed by relating the power loss for each type of loss to the terminal current or voltage. The formulas for the series resonator, corresponding to the circuit model of Fig. 12(a) read

(21) (22)

(27)

(23)

while the formulas for the shunt resonator, corresponding to the circuit model of Fig. 12(b) read

where and represent energies and powers, respectively, the subscripts and refer to electric and magnetic quantities, respectively. The subscripts , and refer to radiation, dielectric and repand conductor losses. The vector quantities , resent the electric field, the magnetic field and the conduction current density, respectively. Fig. 11 indicates the appropriate and . integration domains , , The overall quality factor of the unit cell can be expressed in terms of the above quantities as (24) where represents to total power loss . Assuming that the unit cell is symmetrical ( direction as symmetry axis), which is the case in all the antenna structures considered here, two modes are supported by the structure at

(28) The field calculator in HFSS provides a simple interface for the computation of the integral expression (19)–(23) and the current and voltage path integration (25) and (26). C. Prototype and Measured LCRG Parameters Fig. 13 shows the measurement setup with the 10-cell SFP antenna prototype corresponding to the layout and design parameters of Fig. 9. The LCRG parameters are experimentally determined by using the same driven-mode network method as in the full-wave analysis case (Section V-A), with a careful setup 5Therefore, the problem could also be analyzed without periodic boundaries by alternatively using PEC and PMC walls in two distinct simulations.

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Fig. 13. Measurement setup with the 10-cell SFP antenna prototype based on the layout and design parameters of Fig. 9. TABLE III COMPARISON OF EXTRACTED CIRCUIT PARAMETERS FOR THE SFP UNIT CELL

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Fig. 14. Compared Bloch propagation constant and characteristic (TL) propagation constant [(17)] with the LCRG parameters taken from Table III. (a) FDTD driven mode, following Section V-A. (b) Measurement, following Section V-C. The attenuation constant Re( p) = p is multiplied by a factor of 5 for better visualization.

Fig. 15. Compared Bloch impedance Z and characteristic (TL) impedance Z [(18)] with the LCRG parameters taken from Table III. (a) FDTD driven mode, following Section V-A. (b) Measurement, following Section V-C.

of the measurement reference planes using the built-in port extension function of the vector network analyzer Agilent PNA E8363B. D. Comparison of the Extraction Methods and Validation of the Transmission Line Model The LCRG circuit parameters extracted by the three approaches described above are listed in Table III. All the parameters are in close agreement, except for the shunt quality factor and corresponding conductance. This discrepancy may be due to the higher sensitivity of the shunt resonator following from its lower loss compared to the series resonator. The Bloch analysis (determination of and ) in the driven FDTD simulation and the measurement is based on a ficells yielding a close approximation nite number of of the exact Bloch solution in the limit of an infinite number of cells. Fig. 14(a) compares FDTD full-wave simulation results for the real and imaginary parts of the Bloch propagation conand of the characteristic propagation constant restant sulting from the TL modeling of Section III. The Bloch propagation constant is computed by (5) using the parameter of the unit cell transmission matrix obtained from the first four steps of the procedure given in Section V-A from FDTD simulations. is computed by (17) using the The characteristic constant LCRG parameters following all the seven steps of procedure of Section V-A leading to the parameters listed in the first column

of Table III. Excellent agreement is observed between the two types of results over the frequency range from 4.8 to 6.8 GHz and over the frequency range from 4.8 to 6.4 GHz for for . Similarly, Fig. 14(b) compares measurement results for the Bloch propagation constant and for the characteristic propagation constant, using the experimentally determined parameter in the former case and LCRG parameters in the second column of Table III for the latter case. The experimental dip at the broadside frequency [Fig. 14(b)] is less pronounced than the simulated one [Fig. 14(a)], due to the already mentioned higher encountered in the measurement, but very good shunt loss agreement between Bloch and characteristics results are again observed. Figs. 15(a) and 15(b) provide the same comparisons as Figs. 14(a) and 14(b) but for the Bloch and characteristic impedances, where (5) and (17) are replaced by (6) and (18), respectively. As for the propagation constant, good agreement is observed in both graphs over the frequency range from 4.8 GHz to 6.8 GHz. The ripples in the Bloch results are related of cascaded unit cells and to the finite number correspond to the frequencies where the overall phase across the 10-cell structure is a multiple of and is thus an effect of the finite number of unit cells. Around the broadside frequency, the maximum Bloch impedance (real part) is higher in the simulation [Fig. 15(a)] than in the measurement [Fig. 15(b)]. This is again due to the different shunt losses in Table III.

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N

Fig. 16. Equivalent TL model for an -cells periodic leaky-wave antenna, in particular for the 10-cell prototype shown in Fig. 13. TL analysis based on the propagation constant and the characteristic impedance is applied to compute and compare the Bloch and characteristic scattering matrices.

Z

, with an almost perfect of approximately 15% around accuracy near the broadside frequency, where the immittances were linearized. In conclusion, the TL model developed in Section III has been fully-confirmed by full-wave and experimental results, beyond the circuit based validations provided in Sections IV-A to IV-C. Consequently, the TL model derived in Section III and validated in Section V, is really applicable for all the leaky-wave antennas belonging to the class described in Section II. VI. ASYMPTOTIC TL FORMULAS Based on the TL model established in Section III, this section derives analytical formulas describing the asymptotic response of the considered antennas in two regimes, the regime near the broadside frequency and the regime off the broadside frequency. In all the subsequent formulas, the LCRG parameters may be substituted by the analytical expressions given in Table II, or may be extracted from full-wave simulation, as described in Sections V-A and V-B or from measurement as presented in Section V-C.

S

Fig. 17. Comparison of the parameters obtained by the TL model [(29)] and cells antenna of by direct FDTD simulation and measurement for the Fig. 9. (a) FDTD and TL (using Section V-A LCRG parameter extraction for the latter). (b) Experiment and TL (using Section V-C LCRG parameter extraction for latter).

N = 10

A. General Case For the case of a frequency-balanced antenna unit cell with (30) the angular frequency

In order to obtain a final validation immune of any possible artifacts, the TL model is compared in terms of raw scattering data with full-wave and experimental results for the 10-cell antenna of Fig. 13. From TL theory, the scattering matrix for the and by TL model is given in terms of

is written as (31)

spans a relatively narrow frequency range around the where with . broadside frequency We rewrite (15) as (32a)

(29) where the -referenced TL scattering matrix has been re-normalized to the measurement impedance reference using standard conversion formulas. The LCRG parameters needed in the computation of and are determined from the simulated (Section V-A) or measured (Section V-C) scattering matrices, respectively. Fig. 16 shows the TL model for the reference. The range of validity of the TL model is shown in Fig. 17, where the scattering parameters of the model are compared with full-wave and measurement data. The antenna is poorly matched , due to the relatively high impedance ( to , see Fig. 15). This high value of may be understood from the SFP relations of Table II, corresponding to the circuit model of Fig. 6(a), which show that the SFP lossless char, typiacteristic impedance is due to the alternating high-impedance cally yielding and low-impedance TL sections of the structure. This strong mismatch is beneficial to the present validation purpose as it leads to very clear ripples in the scattering parameters, allowing optimal comparison. The proposed TL model is validated by both full-wave and measurement results over a substantial frequency range

(32b) using the general quality factor definition [23] (33) and the corresponding specific series and shunt quality factors (34a) (34b) for the equivalent series and shunt resonators, with corresponding stored energies and , and power losses and , [24]. Inserting these expressions into (17), the characteristic propis given in terms of the LCRG parameters agation constant and the quality factors with

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(39) (35) The corresponding TL quality factor may be computed [23] from this relation as

Finally, the off-broadside characteristic impedance is oband in (37), tained by neglecting the terms and by reusing the aforementioned Taylor approximation, which yields

(36) Finally, the characteristic impedance is obtained by insertion of the same expressions into (18) as

(40) Assume the loss contributions

and , so to fully neglect and in (32), (40) further simplifies to (41)

(37)

where, and are the TL voltage and current, respectively. Rearranging (41), we obtain the relation

(42) B. Off-Broadside Regime (still Consider the frequency range ), which extends on both sides of , assuming excluding a small region around . Note that the condition may be satisfied even very close to , because the product represents a very small quantity, essentially modeling the radiation loss per unit length of the leaky-wave antenna. In this range, the general propagation constant in (35) reduces to

and are equal. Using showing that the stored energies (33) along with (34a) and (34b), we obtain for the off-broadside quality factor

(43) which is identical to (39). However, the energy and power based derivation which lead to (43) provides deeper physical insight than the other one. C. Near-Broadside Regime

(38a) with

Now consider the complementary frequency range (still assuming ), which corresponds to an extremely narrow frequency region centered at , where according to (31). In this range, the general propagation constant of (35) becomes

(38b) and (44a) (38c) where the subscript has been introduced to indicate the offbroadside regime. The last approximate result in (38a) was obfor tained using the Taylor approximation , with . The condition safely holds here due to the aforementioned low per-unit-length leaky-wave radiation loss. An explicit expression for the off-broadside TL quality factor is then obtained by inserting (38b) and (38c) into (36), which yields

with (44b) as presented in [4] and (44c) where the subscript 0 has been introduced to indicate the nearbroadside regime.

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An explicit expression for the near-broadside TL quality factor is then obtained by inserting (44b) and (44c) into (36), which yields

leads to identical formulas for the off-broadside broadside (0) regimes,

and near-

(51) (45)

(52)

Finally, the near-broadside characteristic impedance is obin the nominator and tained by neglecting the term denominator of (37), and by reusing the aforementioned Taylor approximation, which yields

(53)

(46)

(54) as may be easily verified by substituting (50) into (35) to (37). Furthermore, the following energy and power relations hold for the Q-balanced case in both off- and near-broadside regimes:

The characteristic impedance of (46) may be simplified to (55) (47) and now assuming and the reactive contributions ranging (47), we obtain the relation

and

, so as to neglect in (32). Rear-

(48) revealing that the powers dissipated in the series and shunt resonators are equal. Finally, using (33) along with (34a) and (34b), we obtain the near-broadside (0) quality factor

(49) which is identical to (45). The power equality in (48) together with the quality factor of (49), and their implications in broadside radiating periodic leaky-wave antennas, are the major result of this work. At broadside, the wave propagation and radiation mechanism is essentially resistive and follow these unusual formulas. The TL theory based on the conventional propagation constant and line impedance fails to model this phenomenon, whereas the folparameters lowed TL approach based on the derived and fully captures the dominant resistive effects and explains the physics of the phenomenon. D. Q-Balancing An interesting situation arises when the series and shunt quality factors are equal

(50) which in a conventional homogeneous TL is referred to as the Heaviside condition, corresponding to distortionless transmission [24]. In the present context, this condition simplifies the general formulas for the propagation constant (35), the quality factor (36) and the characteristic impedance (37) in a way that

(56) Thus, in the Q-balanced case, the wave propagation in the offand near-broadside regimes is distortionless and the leaky-wave structure is modeled by the same set of equations [(51) to (56)]. In [4] a CRLH antenna has been optimized for broadside radiation by using a single-sided open-ended stub producing an orthogonally polarized radiation from the shunt resonator. The condition of Q-balancing in (50) will ultimately lead to a distortionless propagation in longitudinal antenna direction with and a constant impedance over a constant leakage factor frequency and hence results in an antenna gain which is constant while the beam is scanned across broadside. It may be deduced from the above developments that the empirically designed antenna in [4] fairly well meets the condition (50). E. Summary and Validation Example Table IV summarizes the off-broadside and near-broadside asymptotic formulas derived in this section. At the broadside frequency the Bloch wave propagation forces the overall power dissipation (radiation together with power loss) to be exactly the same in the series resonator and in the shunt resonator, which has a major impact on the antenna performance and it has, to the authors best knowledge, not been discovered so far. If a periodic leaky-wave antenna has one resonator, either series or shunt with a poor radiation efficiency then the antenna performance will be strongly degraded at broadside due to the equal power split into the two resonators. In case one resonator is not radiating at all and only dissipating power, the radiation efficiency cannot exceed more than 50% at broadside, independent of the amount of power dissipated in the non-radiating resonator. In the off broadside regime, where the energies in the two resonators are equal (42), the overall radiation efficiency depends on the amount of power loss in the dissipative, non-radiating resonator, so with its quality factor approaching infinity a radiation efficiency of ideally 100% can theoretically be achieved. A numerical example is given to verify the formulas derived in Section VI-B and Section VI-C and to visualize their asymptotic behavior. Figs. 18, 19 and 20 compare these formulas with

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TABLE IV SUMMARY OF THE ASYMPTOTIC OFF- AND NEAR-BROADSIDE TL FORMULAS DERIVED IN SECTION VI

Fig. 18. Complex propagation constant comparison between the exact formula of the TL model (35) and the asymptotic formulas of (38) and (44).

Fig. 20. Characteristic impedance comparison between the exact TL model formula of (37) and the asymptotic formulas of (40) and (46).

VII. CONCLUSION

Fig. 19. Quality factor comparison between the exact formula of the TL model (36) and the asymptotic formulas of (39) and (45).

their exact counterparts given by (35) to (37) for the following set of parameters: , , , and . In all the cases, the asymptotic formulas are found to accurately model the off-broadside and near-broadside behaviors of the structures.

Network and transmission line analyzes of periodic leaky-wave antennas scanning through broadside with frequency have been performed and novel asymptotic formulas have been subsequently derived. It has been shown that, despite seemingly very different configurations, these antennas exhibit a qualitatively similar response, and therefore can be described by the same transmission line model. This has been specifically demonstrated for the examples of a series fed patch array antenna, a phase reversal array antenna and a composite right/left-handed antenna. Based on the theory of periodic structures, a simple lattice network model has been proposed and a TL model with linearized immittances has been subsequently developed. This model has been validated by comparison with periodic Bloch results for the three aforementioned antennas, and confirmed by full-wave and experimental results in the particular case of the series fed patch array. The asymptotic formulas, following from this simple model and corresponding to broadside and off-broadside radiation regimes, are the central result of the paper. These formulas provide physical insight and electromagnetic understanding of the behavior of leaky-wave antennas around the broadside frequency. An important result is that

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the total power radiated from or dissipated in the unit cell at broadside equally splits between the series and shunt paths, which explains poor broadside radiation when only one of the two paths efficiently contributes to radiation. REFERENCES [1] A. Hessel, Antenna Theory, Part II, R. E. Collin and R. F. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 19. [2] A. A. Oliner and D. R. Jackson, Antenna Engineering Handbook, J. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2007. [3] P. Burghignoli, G. Lovat, and D. R. Jackson, “Analysis and optimization of leaky-wave radiation at broadside from a class of 1-D periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2593–2604, Sep. 2006. [4] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [5] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “A novel technique for open-stopband suppression in 1-D periodic printed leakywave antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1894–1906, Jul. 2009. [6] L. Liu, C. Caloz, and T. Itoh, “Dominant mode (DM) leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [7] S. Lim, C. Caloz, and T. Itoh, “A reflecto-directive system using a composite right/left-handed (CRLH) leaky-wave antenna and heterodyne mixing,” IEEE Microwave Wireless Compon. Lett., vol. 14, no. 4, pp. 183–185, 2004. [8] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 161–173, Jan. 2005. [9] F. P. Casares-Miranda, C. Camacho-Peñalosa, and C. Caloz, “High-gain active composite right/left-handed leaky-wave antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2292–2300, Aug. 2006. [10] T. Kodera and C. Caloz, “Uniform ferrite-loaded open waveguide structure with CRLH response and its application to a novel backfire-to-endfire leaky-wave antenna,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 4, pp. 784–795, April 2009. [11] T. Kodera and C. Caloz, “Integrated leaky-wave antenna—Duplexer/ diplexer using CRLH uniform ferrite-loaded open waveguide,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 8, pp. 2508–2514, Aug. 2010. [12] T. Kodera and C. Caloz, “Low-profile leaky-wave electric monopole regime of a ferrite-loaded open waveloop antenna using the guide,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3165–3174, Oct. 2010. [13] C. Caloz, D. R. Jackson, and T. Itoh, Frontiers in Antennas, F. Gross, Ed. New York: McGraw-Hill, 2010. [14] J. James, P. Hall, and C. Wood, Microstrip Antenna Theory and Design (Electromagnetic Waves). London, U.K.: Institution of Engineering and Technology, 1981. [15] M. Danielsen and R. Jorgensen, “Frequency scanning microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, no. 2, pp. 146–150, Mar. 1979. [16] A. Derneryd, “Linearly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 24, no. 6, pp. 846–851, Nov. 1976. [17] N. Yang, C. Caloz, and K. Wu, “Fixed-beam frequency-tunable phase-reversal coplanar stripline antenna array,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 671–681, Mar. 2009. [18] N. Yang, C. Caloz, and K. Wu, “Wideband phase-reversal antenna using a novel bandwidth enhancement technique,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2823–2830, Sep. 2010. [19] N. Yang, C. Caloz, and K. Wu-, “Full-space scanning periodic phasereversal leaky-wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 10, p. 1, Oct. 2010. [20] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Piscataway,NJ: Wiley-IEEE Press, 2005.

=0

[21] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Magn., vol. 50, no. 5, pp. 25–39, Oct. 2008. [22] 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, no. 7, pp. 1484–1495, 2007. [23] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991. [24] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2004. [25] R. E. Collin, Foundations for Microwave Engineering. Piscataway, NJ: Wiley-IEEE Press, 2000. [26] F. E. Terman, “Network theory, filters, and equalizers,” Proc. IRE, vol. 31, no. 5, pp. 233–240, 1943. [27] S. Wane and D. Bajon, “Broadband equivalent circuit derivation for multi-port circuits based on eigen-state formulation,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest MTT’09, 2009, pp. 305–308. [28] F. Bongard, J. Perruisseau-Carrier, and J. R. Mosig, “Enhanced CRLH transmission line performances using a lattice network unit cell,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 7, pp. 431–433, Jul. 2009. [29] J.-S. G. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley-Interscience, 2002. [30] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1350–1362, Jun. 2006. [31] T. Liebig, S. Held, A. Rennings, and D. Erni, “Accurate parameter extraction of lossy composite right/left-handed (CRLH) transmission lines for planar antenna applications,” in Proc. Metamaterials, Karlsruhe, Germany, Sep. 2010, pp. 456–458. [32] S. Otto, A. Rennings, T. Liebig, C. Caloz, and K. Solbach, “An energy-based circuit parameter extraction method for CRLH leaky-wave antennas,” presented at the EuCAP, Barcelona, Spain, Apr. 2010. Simon Otto (S’10) received the Diplom-Ingenieur degree from Duisburg-Essen University in 2004, where he is working toward the Ph.D. degree. He was a visiting student at the Microwave Electronics Laboratory of the University of California at Los Angeles (UCLA). Currently, he is with the Antenna and EM Modeling Department at IMST in Kamp-Lintfort, Germany. He has authored or coauthored more than 25 conference and journal papers related to antennas, filter designs, RF components, simulation techniques, magnetic-resonance imaging (MRI) systems and filed three patents. His research interests include array antennas, transmission line metamaterials, EM-theory and numerical modeling. Mr. Otto received the VDE prize (Verband der Elektrotechnik Elektronik Informationstechnik e.V.) for his Diplom-Ingenieur thesis and the second prize of the Antenna and Propagation Symposium (AP-S) student paper award 2005 in Washington.

Andreas Rennings (M’08) studied electrical engineering at the University of Duisburg-Essen, Germany. He carried out his diploma work at the Microwave Electronics Laboratory of the University of California at Los Angeles (UCLA). He received the Dipl.-Ing. and the Dr.-Ing. degrees from the University of Duisburg-Essen, in 2000 and 2008, respectively. From 2006 to 2008, he was with IMST GmbH in Kamp-Lintfort, where he worked as an RF Engineer. Since then he is a Scientist in the Department of Theoretical Electrical Engineering (ATE) of the University of Duisburg-Essen, where he leads the Bio-Electromagnetics/Med-Tech group. His general research interests include all aspects of theoretical and applied electromagnetics, currently with a focus on medical applications. He has authored and coauthored over 50 conference and journal papers, one book chapter, and filed six patents. Dr. Rennings has received several awards, including the second prize at the student paper competition of the 2005 IEEE Antennas and Propagation Society (AP-S) International Symposium and the VDE-Promotionspreis 2009 for his doctoral thesis.

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Klaus Solbach was employed at the University of Duisburg, from 1975 to 1980, as a Junior Researcher in the field of integrated dielectric image line circuits. In 1981, he joined the Millimeterwave Research Laboratory at AEG-Telefunken in Ulm and in 1984 changed to the Radar Systems Group of Daimler-Benz Aerospace (now part of EADS) where he engaged in the design and production of microwave-subsystems for ground based and airborne Radar, EW and communication systems including phased array and active phased array antenna systems. His last position was manager of the RF-and-Antenna-Subsystems Department. In 1997 he joined the faculty of the University of Duisburg as Chair of RF and Microwave Technology. He has authored and coauthored more than 200 national and international papers, conference contributions, book chapters and patent applications. Prof. Solbach was Chairman of the VDE-ITG Fachausschuss “Antennen,” Executive Secretary of the Institut für Mikrowellen-und Antennentechnik (IMA), and Chair of the IEEE Germany AP/MTT Joint Chapter. He was General Chair of the International ITG-Conference on Antennas INICA2007 in Munich and the General Chair of the European Conference on Antennas and Propagation EuCAP2009 in Berlin.

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Christophe Caloz (F’10) received the Diplôme d’Ingénieur en Électricité and the Ph.D. degrees from École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory of University of California at Los Angeles (UCLA). In June 2004, he joined École Polytechnique of Montréal, where he is now a Full Professor, a member of the Poly-Grames Microwave Research Center, and the holder of a Canada Research Chair (CRC). He has authored and coauthored over 380 technical conference, letter and journal papers, 3 books and 8 book chapters, and he holds several patents. His research interests include all fields of theoretical, computational and technological electromagnetics engineering, with strong emphasis on emergent and multidisciplinary topics. Dr. Caloz is a Member of the Microwave Theory and Techniques Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory) and MTT-25 (RF Nanotechnology), a Speaker of the MTT-15 Speaker Bureau, and the Chair of the Commission D (Electronics and Photonics) of the Canadian Union de Radio Science Internationale (URSI). He is a member of the Editorial Board of the International Journal of Numerical Modelling (IJNM), of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE), of the International Journal of Antennas and Propagation (IJAP), and of the journal Metamaterials of the Metamorphose Network of Excellence. He received several awards, including UCLA Chancellor’s Award for Post-doctoral Research in 2004 and the MTT-S Outstanding Young Engineer Award in 2007. He is an IEEE Fellow.

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Constrained Inverse Near-Field Scattering Using High Resolution Wire Grid Models Badr Omrane, Yves Goussard, Member, IEEE, and Jean-Jacques Laurin, Senior Member, IEEE

Abstract—The microwave inverse problem is addressed using a wire grid model representation with capacitors loaded in parallel with resistors to respectively replace the permittivity and the conductivity of the device under test. A new approach is presented to embed the properties of isotropy and positiveness of the constitutive parameters without additional penalty terms or weighting parameters. An edge-preserving regularization technique is used to better estimate the discontinuities present in the device under test (DUT) and to decrease the sensitivity to noise during the reconstruction process. The optimization algorithm makes use of the conjugate gradient method to minimize the objective function. Synthetic data are used to assess the reconstruction speed of the new method. Simulation results show a five-fold reduction of the computation time compared to what had been presented previously. Experimental near-field measurements at 2.45 GHz on thin plate DUTs are used to assess the validity of the proposed reconstruction method. Satisfactory results are obtained and a 20 is achieved. spatial resolution of Index Terms—Conjugate gradient, edge-preserving regularization, inverse scattering, lumped loads, microwave tomography, near-field measurements, wire grid models.

I. INTRODUCTION

I

N the past three decades, microwave tomography (MT) has stirred up interest in many technical fields and was found very useful for the characterization of material properties [1]–[3], owing to the possibility to retrieve spatial distributions of permittivity and conductivity in a nondestructive manner. Microwave tomography and imaging have recently received more attention due to their potential in biomedical applications, such as breast cancer imaging [2]–[6]. However, the ill-posed nature of the tomography equations is an important difficulty that has been addressed in the past decades [1]–[10]. Recently, a new approach to MT, based on wire-grid models augmented with lumped loads [1], has been proposed. This technique leads to an equivalent electrical circuit where material properties are embedded using capacitors and resistors loaded in parallel to represent the permittivity and conductivity of the material [11], [12], respectively. This wire grid formulation can be easily integrated into equivalent electric or magnetic current source inversion problems and this approach has led to accurate characterization of electrically small structures [13], [14]. In [14],

Manuscript received June 08, 2009; revised April 02, 2010, November 29, 2010; accepted February 15, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. B. Omrane is with the Department of Engineering Science, Simon Fraser University, Vancouver, BC V5A 1S6, Canada. Y. Goussard and J.-J. Laurin are with the Department of Electrical Engineering, Ecole Polytechnique, Montreal, QC H3C 3A7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163740

the wire grid approach showed its potential to capture intricate subwavelength features, as the smallest discretization used was (here is the wavelength in free space). Compared to source inversion, contrast inversion is a nonlinear problem that is computationally more demanding. To the authors’ knowledge, the only published contrast inversion solution based [1]. Wire on wire grid models used a grid resolution of grid models used in these references and in this paper use overlapping piecewise sinusoidal basis (PWS) functions to represent the polarization currents. As noticed by Rubin et al. [15], overlapping current expansions (e.g., rooftop or PWS) are free of discontinuities and therefore do not introduce fictitious charges causing errors in the near field, as in models based on pulse basis functions. As pointed out by Richmond in [16], the PWS function is approaching the currents that are naturally occurring on conducting wires. Also, it was observed in [17] that sinusoidal functions are convenient for interpolating currents as they lead to faster convergence than pulse basis functions. The choice of PWS functions in tomography problems can therefore be advantageous to reduce the number of unknowns when the electrical size of the structure under test increases. Our previous results showed that using a small wire-grid stepsize is a key point for obtaining high resolution reconstructions. However, in real world applications, the selection of a small enough stepsize is limited primarily by the performance of the optimization algorithm used to perform the inversion and by its ability to cope with the resulting increase in the number of unknowns. In order to address this difficulty, we propose a method for solving the MT inverse problem for wire grid models that exhibits the following features: (i) the isotropy of the unknown medium is implicitly accounted for in the model, thereby reducing the number of unknowns and avoiding the introduction of constraints; (ii) the positiveness of the unknowns is guaranteed through appropriate changes of variables, which results in significant a speed up of the inversion process; (iii) the inversion is regularized with a non-convex edge-preserving penalty term in order to enhance the accuracy of the reconstruction near sharp discontinuities within the DUT, while smoothing out small variations. This results in improved computational efficiency and reconstruction accuracy with respect to the technique presented in [1]. For example, experiments performed with near-field measurements on thin inhomogeneous DUTs of various shapes incan be achieved with our dicate that a spatial resolution of method when the size of the wire grid lattice is chosen appropriately. II. FORWARD PROBLEM Formulation of the forward problem using a loaded wire grid model was described in details in [1]. It is briefly summarized

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OMRANE et al.: CONSTRAINED INVERSE NEAR-FIELD SCATTERING USING HIGH RESOLUTION WIRE GRID MODELS

here. Let us consider a time-harmonic incident electric field illuminating a test domain including an unknown object occupying volume . The DUT is an inhomogeneous object with relative permittivity and effective conductivity , surrounded by a homogeneous background medium with known , and . In such media, the diselectromagnetic properties cretized tomography equations take the following form: (1) (2) where the contrast

is defined as: (3)

and denote the Green’s function comand repreputed in domain and , respectively. sent the discretized total, scattered and incident fields. Domain can be modeled by an equivalent electrical network using the thin-wire Richmond-Tilson method of moments code (RICHT) network referred [18], [19] and lumped loads. A parallel is mounted at each segment end to as embed the DUT material properties into the wire grid model. In a method of moments (MoM) presentation, (2) and (3) respectively become [1]: (4) (5) and respectively reprewhere complex voltage vectors sent the excitation of the incident field on the wire basis functions immersed in an unbounded and homogeneous material of and , and the set of near-field measurements. properties is calculated from using the same approach as Here, and are respectively rein [1]. The complex matrices ferred to as the mutual impedance matrix between each basis function of the DUT, and the mutual impedance matrix between the basis functions of the wire grid model and the measurement probe. These terms are computed using RICHT. Here, both unare related to the physical properties of the known and DUT as follow: the entries of vector are the coefficients of the basis functions simulating the currents in the DUT, while contains loads inserted on each basis the diagonal matrix function that are implementing the constitutive parameters of . In this paper, wire grid modeling is applied to test objects consisting of thin dielectric plates. Therefore a simple 2D planar wire grid was sufficient. In terms of spatial resolution achievable in the forward problem, it appears that this approach is primarily limited by arithmetic precision. In [15], models of 3D objects based on plates instead of wires were accurate down to . In this paper, the thin plate exa discretization of . In order amples presented have an area of approximately to limit computing time, a discretization of /20 was used. III. INVERSION TECHNIQUE The MT inverse problem consists of retrieving the impedance matrix from a set of near-field measurements (4) and the

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computed reaction of the model basis functions to the incident (5). This is usually not a straightforward problem confield and the nonlinearity sidering the ill-conditioning of matrix of system (4)–(5) with respect to . Furthermore, noise and . The measurement errors preclude exact reconstruction of solution of the inverse problem is then satisfying (4) and (5) only approximately. In [1], the solution of the MT inverse problem was sought as the minimizer of a contrast source inversion (CSI) objective function defined as follows: let and respectively denote the residuals as a function of the number of illuminations associated with (4) and (5). Here, index refers to one of the illuminations that can be applied to the DUT. (6) (7) The general form of the cost function can then be written as: (8)

(9)

(10) where is a column vector formed with the diagonal ele, the inverse of matrix , and where ments of matrix and act as normalizing factors even though they depend on does not conunknown quantity . It will be assumed that tain perfectly conducting objects. Consequently, the entries of will not go to infinity. Usually a regularization funcmatrix tion is added to (8) in order to decrease the sensitivity of the reconstructed profiles to noise. Moreover, to prevent non physical solutions such as negative permittivity and conductivity, positiveness constraints can be enforced as in [1] with a logarithmic barrier [20] acting like a boundary between the positive and the negative admittances. An isotropy constraint can also be added using a quadratic penalty function. This approach generated good results for reconstruction of material properties. However, the presence of the barrier parameter greatly increased the total number of iterations and thus the computation time. In this paper, we propose a new approach in which the constraints are implicitly implemented in the definition of the optimization variable in order to speed-up the reconstruction process by getting rid of the barrier parameter. Isotropy is built into the model by assigning the material propof the wire grid model inerties to each node stead of to each segment . This is illustrated in Fig. 1 for the case of a two-dimensional grid. In RICHT, lumped loads used to implement permittivity and conductivity are associated with grid wire segments. The relationship between the model in which the loading impedances are associated to wire segments (Fig. 1(a)) and the intrinsically isotropic model in which the loading impedances are associated with node connection points (Fig. 1(b)) takes the following form:

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Fig. 1. Material physical properties attribution (a) by segment and (b) by node. In the latter case, the isotropy criteria is built into the wire-grid model.

(11) and are respectively the nodal admittance where into . This column vector and the matrix mapping approach has already been studied in our previous work [1] and has shown a lack of computational efficiency due to the necessity of handling the barrier parameter used to enforce the and . This is due to the fact that embedpositiveness of ding the isotropy criterion into the objective function without embedding the positiveness criterion does not eliminate the barrier parameter from the reconstruction process. Therefore, this involves driving this parameter from a high value toward 0, which requires additional internal iterations during the optimization process. To overcome this issue, the positiveness criterion can be embedded into the optimization process as an absolute value function respectively on the real and imaginary , where and are parts of the admittance: respectively the conductance and the susceptance. However, if a minimization process based on gradients is used, a continuity problem occurs because such a function is not differentiable at 0. In order to overcome this difficulty, we make the following continuously differentiable change of variable:

[26], [27] or differentiable [28] form, remain convex while allowing for better preservation of discontinuities. Yet, even better preservation of discontinuities can be obtained with nonconvex penalty terms (see, e.g., [29], or [30] for a brief review) at the expense of potential additional difficulties during the optimization process. Here, our goal is to reconstruct objects with marked discontinuities, hence the choice of an edge-preserving penalty term. is not a pressing issue as may In addition, convexity of not be globally convex due to the nonlinearity of the MT equations. For these reasons, we selected a nonconvex penalty func[29] and the penalty term was obtained tion by applying to the real and imaginary parts of the admittance as: (13)

and respectively denote the first derivatives along where and are scaling factors indexed for the and axes. the real and imaginary parts of the admittance, respectively, and refers to the number of unknowns forming the intrinsic and isotropic model. In this work, the threshold parameters were determined empirically by trial and error. Finally, , the global objective function emletting bedding both constraints and the edge-preserving regularization takes the following form: (14) The selected optimization procedure was the CSI, which makes use of the conjugate gradient technique to optimize iteratively the objective function with respect to and then to . The first step consists of updating the current vector in the following manner: (15) (16)

(12) where is a real parameter taking a small value to correct the non differentiability of the absolute function at 0 and is used and to change the slope of the positiveness function. are the new dummy optimization variables. Microwave tomography is an ill-posed inverse problem [21] that should be regularized, e.g., by introducing a priori information on the solution through addition of a penalty term to the cost-function defined in (8). Generally, choosing amounts to setting a trade-off between the accuracy of the prior information and the complexity of the resulting optimization procedure [22], [23]. Quadratic penalty terms, either in homogeneous or weighted form, are convex and straightforward to implement. They have been used successfully in several areas including MT [24], but tend to smooth large discontinuities too much. Edge-preserving penalty terms based, e.g., on generalized Gaussian functions [25] or on total variation in its original

and where is the index of iterations, whereas are respectively the complex gradient vector with respect to , the Polak-Ribière conjugate direction and the step-size. Details of the procedure on how these parameters are computed can be found in [1]. Once is updated, the same steps are used to update the admittances: (17) (18) The complex gradient vector with respect to the conjugate of , denoted , can be expressed as (19), shown at the bottom of the following page, where is the Hadamard product. The complex gradient of the cost function with respect to takes the following form:

OMRANE et al.: CONSTRAINED INVERSE NEAR-FIELD SCATTERING USING HIGH RESOLUTION WIRE GRID MODELS

(20) , whereas superscripts and where denote the conjugate transpose and the transpose operators respectively. The complex gradient vectors of the positiveness criteria are written as: (21)

(22)

whereas the complex gradient of the edge-preserving regularization take the following form:

(23) According to whether the gradient is computed with respect to the real part or the imaginary part of the admittance, and are set respectively to and , or to and . The positiveness criterion is then embedded into (23) by using the substitution defined in (12). In this particular case, cannot be expressed in a closed form due the step-size to the non linearity of the forward problem and thus will be evaluated with a numerical procedure using a line search method satisfying the strong Wolfe conditions (SWC) [18]. Following the approach used in [31], we choose constant initial currents that minimize the data error term in (8), that is: (24) Once the initial currents are determined, the initial admittance in (8) [2]: is chosen to minimize (25) Finally,

can be initialized as

.

IV. EXPERIMENTAL RESULTS Experimental data as well as synthetic data were used to highlight the performance of the new technique using embedded

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constraints and an edge-preserving regularization. At first, synthetic data were used to assess the numerical efficiency of the new method based on (14) compared to the one with external constrains [1], whereas experimental data were used to validate the quality of the reconstruction process when dealing with DUTs of arbitrary geometries. It should be pointed out that in order to reduce complexity, the DUTs considered were thin was tanplates. Also, since the incident electrical field component of the contrast gential to the plates, the normal current was neglected. Consequently, only a two-dimensional (along and ) wire grid model was required. The test bench in this experiment consisted of a planar nearfield setup [1] using a horn antenna operating at 2.45 GHz as a source of incident fields and a short dipole connected to a Marchand balun [32] acting as an E-field probe. A calibrated vector network analyzer (VNA) was used to measure the voltage trans) between the horn and the mission coefficient (parameter values when the DUT is present were probe. The measured entries. Measurements of in absence taken directly as of the DUT, with the probe scanning region , were used to , according to a procedure given in compute the entries of [1]. All measurements were done well above the noise floor of the VNA. High-precision motors were used to move the probe with an accuracy of one micrometer in the x and y directions. Very good repeatability was observed in all measurements. The distance between the DUTs and the probe should be set carefully, since the dipole must scan an area near the DUT where the scattered field presents strong variations, while remaining sufficiently far from the emitter to minimize the mutual coupling between the scan between these two devices. A distance of plane and the DUT was found to be an adequate compromise, where denotes the free-space wavelength. Moreover, two illumination polarizations were respectively considered, that is, with the E-plane of the horn antenna parallel to the and axes of the wire grid model, respectively. A total of 2550 measurements were taken for each illumination. In the case of synthetic data, the setup was similar to the experimental one with the exception that a short dipole was used as an emitter instead of a horn antenna. A. Reconstruction Using Synthetic Data We first validated the proposed algorithm with simulated near-field data obtained for a DUT consisting of a -thick , rectangular plate of dielectric of dimensions with and S/m surrounded by vacuum . The probe scanned an area in the plane while the dimensions of the grid of . The emitter was set in five different positions: were i) one position right below the center of the DUT and ii) the remaining ones near each corner of the DUT. For each position

(19)

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Fig. 2. Computation time for the reconstruction process using (– –) the objective function presented in [1] and (–  –) the new objective function (14). The latter shows a five-fold increase in the convergence speed.

both polarizations were used. Therefore the DUT was exposed to a total of ten illuminations. For both methods, the initial was set to . Based on numerical experiments, the convergence threshold was set to (26) A comparison between the computation times of the objective function presented here and the previous one presented in [1] is depicted in Fig. 2. It can be noticed that the new objective function with embedded isotropy criteria is about five times faster than the previous one. The latter uses smaller step-size increments in the linear search method involving SWC in order to maintain positive real and imaginary parts of the admittances. This condition is necessary for the logarithmic barrier function to be defined and computable. On the other hand, with the new objective function, no such limitation exists since the change of variables ensures positiveness of the real and imaginary parts of the admittances. However, it should be noted that, in both cases, a large step-size eventually leads to convergence issues. The robustness of the reconstruction algorithm was evaluated by adding Gaussian noise to the simulated probe voltages and . Typical results from the reconstruction algorithm applied to the square plate DUT are shown in Fig. 3 for the cases with and without the edge-preserving penalty term respectively. The optimization process was carried out using and , while was set respectively to 0.1, 0.7, 1 and 5 for SNR of 50, 40 30 and 20 dB. The actual shape of the DUT is clearly visible for signal-to-noise-ratio (SNR) values of 30 dB and higher, while at an SNR 20 dB a systematic bluring of the reconstructed object’s edges and corners is observed. The mean square errors (MSEs) are reported in Table I and were evaluated according to:

Fig. 3. Reconstruction of the " distributions in the presence of noise.

TABLE I MSE FOR " AND 

where and refer to values 2.33 and 0.204 S/m within the square plate region, and to 1 and 0 S/m for the vacuum. As it can be seen in Table I, the error reduction due to regularization becomes more apparent as the noise level increases. In order to assess the robustness of the solution with respect to the choice of initial values, the simulations were , with repeated with the initial guess modified by a factor . The SNR was set to 30 dB and the same value than for the cases of Table I was used. The MSEs for permittivity and conductivity are reported in Table II, and show the same order of magnitude than the values of Table I for the 30 dB SNR level. Surprisingly, all the MSEs for permittivity are slightly better that of Table I (0.029), whereas the MSEs for conductivity exhibit very small variations compared to the original value.

(26)

B. Reconstruction Using Experimental Data

(27)

In order to evaluate the quality of the new objective function, the reconstruction process was applied using at first experimental data from a thin square shape inhomogeneous dielec-

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TABLE II MSE FOR " AND  UNDER PERTURBED INITIAL GUESS

Fig. 4. DUT made of (a) three inhomogeneous media labeled respectively " ; " and " and, (b) I-shape Duroid characterized by the relative permittivity " . The grid discretization shown is of =10.

TABLE III MSE FOR " AND  FOR VARIOUS SCALING FACTORS APPLIED TO THE “BEST” VALUE OF 

tric material as depicted in Fig. 4(a). The outer part of the DUT was made of 3-mm thick Duroid with nominal properties of and , while the inner part was a dielectric material with nominal properties and . The optimization process was carried out using two different ; case 2) sets of regularization parameters: case 1) and . The regularization parameter is selected by first using a set of seven guess ranging from to in the reconstruction process, and subsequently the “best” value was selected by considering the quality of the reconstructed permittivities/conductivities and their respective profiles. Even though, the selected regularization parameters give satisfactory results, it should be noted that no effort was made to optimize the selection process of the regularization parameter at this stage. The sensitivity of the solution to the choice of can be observed in Table III showing the MSE obtained is scaled at different levels. when the retained value This data, which was obtained with a SNR of 30 dB, shows that the MSE is not very sensitive to the choice of near its optimal sets the transition from the quadratic to value. Parameter the asymptotic regime of the penalty scheme acting on ,and was set according to the assumption that variations higher than 0.25 on the relative permittivity are considered as discontinu-

Fig. 5. Relative permittivity reconstruction from experimental data for the inhomogeneous DUT using edge-preserving regularization: (a)–(b) using no reg= 3 10 . ularization and, (c)–(d) using and  = 7 and  = 

2

ities. However, high values should be avoided as they excessively smooth edges and thus hinder the purpose of using any edge-preserving regularization. On the other hand, due to the can be set to fact that the DUT is essentially an insulator, any arbitrary value without affecting the reconstruction process. The reconstructed profiles are depicted in Fig. 5. The conductivity profile is irrelevant for insulator characterization and is not presented. The reconstructed permittivity profile in the second case is in good agreement with the actual one shown in Fig. 4(a). We notice that the new method is suitable for wire grid model MT as was the case for [1]. In both cases, the permittivity of the inner part of the DUT is closer to the actual value: 5.8 for the new objective function and 5.6 for [1]. This slight difference is mainly due to the handling of the positiveness and isotropy constraints. The fact that these criteria are embedded into the objective function makes both of them act strongly at the beginning of the optimization process, which obviously yields to a different final solution. In addition, no negative values of and are noted, which confirms the effectiveness of the technique used to impose positiveness into the cost function. enOne can notice that the optimization process with , while for hances the reconstructed profiles the map of is not clearly defined . In case 2, the threshold parameter has been chosen carefully in order to preserve discontinuities at the edges as stated previously. Moreover, based on these results, the spatial resolution of the reconstructed profile is clearly limited to one cell from as is the case for the reconstructed the edge of the DUT profiles presented in [1]. C. High Resolution Microwave Tomography Using Experimental Data The 5-fold gain in speed depicted in Fig. 2 makes it possible to use a higher resolution wire grid model for reconstruction of

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Fig. 6. Relative permittivity reconstruction at =20 from experimental data for the inhomogeneous DUT using edge-preserving regularization: (a)–(b) using no = 3 10 . regularization and, (c)–(d) using  = 2 and  = 

2

Fig. 7. Cross-section of the reconstructed profiles along x axis for: (dashed)  = 0 of case 1 and, (solid)  = 0 of case 2.

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material properties using (14) while maintaining a reasonable computation time. The lattice size was decreased from to , which corresponds to a spatial resolution of 6.1 mm. at 2.45 GHz. We also sought to reproduce a realistic situation where a higher resolution may be required while the number of measurements remains constant. The reconstruction process for was performed with the same number of data points as for , i.e., 2550 measurements for each illumination. Even though the isotropy criterion using nodes (11) reduces the number of unknowns, the MT inverse problem using wire grid model remains a MoM formulation using basis functions resolution in both and axis. From this point of view, a yields to an increase of the matrix from 2550 278 to 2550 1118 . Experimental data from the DUT were . used to assess the reconstruction process using a lattice of The optimization process was carried out using: case 1) and case 2) and . The reconstructed profiles are shown in Fig. 6. The reconstructed profile can be better appreciated in Fig. 7, where a cross-section along the axis is shown.

Fig. 8. Relative permittivity reconstruction at =20 from experimental data for the I-shape DUT using edge-preserving regularization: (a)–(b) using no reg=  = ularization (MSE = 0:095) and, (c)–(d) using  = 7 and  3 10 (MSE = 0:031).

2

In spite of the decreased number of data points per unknown which makes reconstruction more difficult, the estimated profiles are in good agreement with the original profile. The relative permittivity value of the inner insulator and the shape of the DUT are well reconstructed. As expected, the resolution of the map lies within one cell of the boundary of the DUT, which . However in case 1, we can notice oscillations inside the is , DUT profile when no regularization, is applied to the reconstruction process. The edge-preserving regularization is very efficient for smoothing small variations , while allowing discontinuities as it is shown by case 2. In addition, it makes it possible to recover the DUT profile using a high resolution wire grid model. The next reconstruction example uses an I-shape DUT. This more complex geometry is of great interest owing to the presence of many discontinuities and edges, which are useful to assess the reconstruction process using an edge-preserving reguand are crucial for larization. Some edges are smaller than verifying whether the spatial resolution can be enhanced solely by changing the size of the unit cells, or if the resolution is limited by measurement inaccuracies. This I-shape DUT was made . The lattice size depicted of 3-mm thick Duroid but the reconstructed model had a lattice in Fig. 4(b) is . The optimization process was carried out using two of different sets of regularization parameters: case 1) ; case and . Regarding the choice 2) , the same hypothesis employed for the three inhomoof geneous media DUT was used here: any variation higher than 0.25 on the relative permittivity was considered as a discontinuity The reconstructed profiles are shown in Fig. 8. As for the inhomogeneous plate presented above, the profiles show a good reconstruction using edge-preserving regularization, even if the map of is above the expected value of 2.33. The final result is satisfactory and the shape of the DUT is clearly visible. The achieved resolution for this complex DUT . lies within one cell from the edges

OMRANE et al.: CONSTRAINED INVERSE NEAR-FIELD SCATTERING USING HIGH RESOLUTION WIRE GRID MODELS

Fig. 9. Relative permittivity and conductivity reconstruction at =20 from experimental data for the fourth DUT using edge-preserving regularization: (a). relative permittivity and (b) conductivity map using no regularization, and (c) relative permittivity and (d) conductivity map using  = 1;  = 7 10 and  = 5 10 .

2

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Fig. 10. Cross-section of the conductance map at y = 0 using edge-preserving = 7 10 , and  = 5 10 . The regularization with  = 1;  nominal value of the conductance is 2:65 10 S in the centre region of the DUT.

2

2

2

2

D. Permittivity and Conductivity Reconstruction Using Experimental Data So far, we have limited our experimental investigation to insulators. The purpose of the next experiment is to assess the capability of the proposed method to precisely reconstruct both permittivity and conductivity profiles from experimental data. mm) square plate of RO4000 The DUT was made of a ( series (Rogers Corp.) used as printed circuit board substrate . Actually, the DUT consists of two superimposed plates 1.5 mm in thickness between which a “space cloth” membrane with nominal surface of 377 per square is inserted. This emulates an resistance S/m for the 3-layer comeffective conductivity of posite sample. The optimization process on experimental data was carried out using two different sets of regularization pa; case 2) and rameters: case 1) . Here the regularization process is achieved by using different threshold values for the real and the imaginary parts of the admittances as the DUT is no longer an insulator. The reconstructed profiles are shown in Fig. 9. In the non-regularized case 1, one can barely identify the shape of the DUT from the surrounding medium. In this case, the permittivity and conductivity are far from the actual values and exhibit some sharp peaks inside the material ( and ). However, in the regularized cases, the shape of the DUT is clearly visible with a resolution limit of ( and ), which confirms the capability of the optimization process and of the new objective function to carry-out reconstruction of material properties with high fidelity. The quality of the reconstruction process can be appreciated better by mapping a cross-section of the surof the DUT at as shown in face conductance Fig. 10. The enhanced reconstruction process using the new objective function generates a solution within approximately 10% of the nominal value of 1/377 Siemens.

V. CONCLUSION In this paper, the microwave tomography inverse problem using embedded isotropy and positiveness criteria in the optimization variable definitions instead of into the cost function was investigated. The objective function for the wire grid model representation was enhanced with an edge-preserving regularization favoring smoothing of small variations while preserving sharp discontinuities. An optimization process based on the conjugate gradients was used to minimize the cost function. The proposed cost function led to a gain of a factor of 5 in computation speed compared to the previous approach using external constraints. This was a key factor for increasing the resolution of the wire grid while keeping the computations at a tractable level. The proposed method was validated experimentally using plate samples and a test frequency of 2.45 GHz. DUTs of complex lattice were imaged with a good shapes discretized on a accuracy, the resolution being limited by discretization. All the reconstructed profiles show that the edge-preserving regularization technique is suitable for material characterization. In fact, discontinuities are always preserved when the threshold parameter is carefully considered, while small variations within the DUT are smoothed. An experimental validation on a non-insulating DUT was performed and the reconstructed profiles were in good agreement with the actual ones. REFERENCES [1] B. Omrane, J.-J. Laurin, and Y. Goussard, “Sub-Wavelength resolution microwave tomography using wire grid models and enhanced regularization techniques,” IEEE Trans. Microwave Theory Tech., vol. 54, pp. 1438–1450, Apr. 2006. [2] A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contract source inversion method,” IEEE Trans. Microwave Theory Tech., vol. 50, pp. 1761–1771, Jul. 2002. [3] A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov, V. G. Posukh, and Y. E. Sirov, “Three-dimensional vector microwave tomography: Theory and computational experiments,” Inverse Prob., vol. 20, pp. 1239–1259, 2004. [4] D. W. Winters, E. J. Bond, B. D. Van Veen, and S. C. Hagness, “Estimation of the frequency-dependent average dielectric properties of breast tissue using a time-domain inverse scattering technique,” IEEE Trans. Antennas Propag., vol. 54, pp. 3517–3528, Nov. 2006.

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[5] 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. Rad., vol. 14, no. 2, pp. 207–218, Feb. 2007. [6] P. M. van den Berg and A. Abubakar, “Inverse scattering and its application to medical imaging and subsurface sensing,” Radio Sci. Bulletin, no. 303, pp. 13–26, Dec. 2002. [7] A. Franchois, A. Joisel, C. Pichot, and J.-C. Bolomey, “Quantitative microwave imaging with a 2.45 GHz planar microwave camera,” IEEE Trans. Med. Imag., vol. 17, pp. 550–560, Aug. 1998. [8] Z. Q. Zhang and Q. H. Liu, “Two nonlinear inverse methods for electromagnetic induction measurements,” IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 1331–1339, June 2001. [9] I. T. Rekanos, M. S. Efraimidou, and T. D. Tsiboukis, “Microwave imaging: inversion of scattered near-field measurements,” IEEE Trans. Magn., vol. 37, pp. 3294–3297, Sept. 2001. [10] K. Belkebir, R. E. Kleinman, and C. Pichot, “Microwave imaging—Location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 469–476, Apr. 1997. [11] K. G. Balmain, C. C. Bantin, M. A. Tilson, A.-M. Chung, and T. D. Pham, “The moment method and extensions of wire-grid modelling for microwave heating,” in Proc. 28th Int. Microwave Power Inst. Symp., Montreal, QC, 1993, pp. 158–169. [12] A. M. Chung and K. G. Balmain, “Tray-shape effect in a computation model of microwave heating,” Can. J. Elect. & Eng., vol. 20, pp. 173–178, Dec. 1995. [13] J. Colinas, Y. Goussard, and J.-J. Laurin, “Application of the Tikhonov regularization technique to the equivalent magnetic current near-field technique,” IEEE Trans. Antennas Propag., vol. 52, pp. 3122–3132, Nov. 2004. [14] J.-J. Laurin, J. F. Zürcher, and F. Gardiol, “Near-field diagnostics of small printed antennas using the equivalent magnetic current approach,” IEEE Antennas Propag. Mag., vol. 49, pp. 814–828, May 2001. [15] B. J. Rubin and S. Daijavad, “Radiation and scattering from structures involving finite-size dielectric regions,” IEEE Trans. Antennas Propag., vol. 38, pp. 1863–1873, Nov. 1990. [16] E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Computational Electromagnetics—Frequency Domain Methods. Piscataway, NJ: IEEE Press, 1991, pp. 156–169. [17] Y. S. Yeh and K. K. Mei, “Theory of conical equiangulair-spiral antennas Part I—Numerical techniques,” IEEE Trans. Antennas Propagat, vol. 15, pp. 634–639, Sept. 1967. [18] M. A. Tilston and K. G. Balmain, “A multiradius, reciprocal implementation of the thin-wire moment method,” IEEE Trans. Antennas Propag., vol. AP-38, pp. 1636–1644, Oct. 1990. [19] M. A. Tilston and K. G. Balmain, “On the suppression of asymmetric artifacts arising in an implementation of the thin-wire method of moments,” IEEE Trans. Antennas Propag., vol. AP-38, pp. 281–285, Feb. 1990. [20] J. Nocedal and S. J. Wright, Numerical Optimization. Berlin: Springer Verlag, 1999, Springer Series in Operations Research. [21] T. Isernia and R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sensing, vol. 35, no. 4, pp. 910–922, 1997. [22] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia, PA: SIAM, 2005. [23] Bayesian Approach to Inverse Problems, J. Idier, Ed. Hoboken, NJ: ISTE Ltd. and Wiley, Apr. 2008. [24] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “Two- and threedimensional algorithms for microwave imaging and inverse scaterring,” J. Elecromagn. Waves Appl., vol. 17, no. 2, pp. 209–231, 2003. [25] C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Processing, vol. IP-2, no. 3, pp. 296–310, 1993. [26] C. R. Vogel, Computational Models for Inverse Problems. Philadelphia, PA: SIAM, 2002.

[27] T. F. Chan and X.-C. Tai, “Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients,” J. Comput. Physics, vol. 193, pp. 40–66, 2003. [28] P. M. van den Berg and R. E. Kleinman, “A total variation enhanced modified gradient algorithm for profile reconstruction,” Inverse Prob., vol. 11, pp. L5–L10, 1995. [29] S. Geman and D. E. McClure, “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst., vol. LII-4, pp. 5–21, 1987. [30] M. Nikolova, “Regularization functions and estimators,” in Proc. Int. Conf. Image Processing, Lausanne, Switzerland, 1996, pp. 457–460. [31] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Prob., vol. 13, pp. 1607–1620, Dec. 1997. [32] R. A. Elasoued, J. J. Laurin, and Y. Goussard, “Design and characterization of a broadband near-field probe,” in Proc. Int. Symp. Antennas Techniques Applied Electromagnetics (ANTEM), Ottawa, Ontario, Canada, Jul. 2004.

Badr Omrane received the B.Eng. and M.A.Sc. degrees in electrical engineering from École Polytechnique, Montreal, QC, Canada, in 2002 and 2005, respectively, and the Ph.D. degree in electrical engineering from University of Victoria, BC, Canada, in 2009. He is currently with the CIBER lab at Simon Fraser University in Vancouver, Canada and his research interests include organic/printable electronics, nano-optics and nano-electronics.

Yves Goussard (M’90) was born in Paris, France, in 1957. He graduated from the École Nationale Supérieure de Techniques Avancées in 1980 and he received the Docteur-Ingénieur and Ph.D. degrees from the Université de Paris-Sud, Orsay, France, in 1983 and 1989, respectively. From 1983 to 1985, he was a Visiting Scholar at the EECS Department of the University of California, Berkeley. In 1985, he was appointed as a Chargé de Recherche at CNRS, Gif-sur-Yvette, France, and in 1992, he joined the Biomedical Engineering Institute and the Electrical Engineering Department of the École Polytechnique, Montreal, Canada, where he is now a Full Professor. After some work on nonlinear system identification and modeling, his interests moved toward ill-posed problems in signal and image processing.

Jean-Jacques Laurin (S’87–M’91–SM’98) received the B.Eng. degree in engineering physics from Ecole Polytechnique de Montreal, Montreal, QC, Canada, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1983, 1986, and 1991, respectively. In 1991, he joined the Poly-Grames Research Centre, Ecole Polytechnique de Montreal, where he is currently a Professor. He was an invited professor at Ecole Polytechnique Fédérale de Lausanne (EPFL) in 1998–1999 and a visiting scientist at ESA/ESTEC in 2008. His research interests include antenna design and modeling, near-field antenna measurement techniques, microwave tomography, and electromagnetic compatibility.

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A Shape-Based Inversion Algorithm Applied to Microwave Imaging of Breast Tumors Reza Firoozabadi, Member, IEEE, and Eric L. Miller, Senior Member, IEEE

Abstract—We propose a new shape-based inversion algorithm to identify an anomaly embedded in an inhomogeneous layered geometry. We apply our approach to microwave breast imaging where the geometry consists of several inhomogeneous layers and the potential tumor is embedded in the innermost layer. In addition to the tumor identification, we estimate the irregular transition layer between the breast inner layers. Our inversion algorithm is based on a low-dimensional parametric form of the relevant geometries, and uses multiple-frequency multi-source data to estimate the unknowns. Several numerical examples are provided to evaluate the effectiveness of our approach and demonstrate its robustness both to the initial choice of parameter values and uncertainty in the complex dielectric properties of the media even with the complex geometry and the simplified inverse model. Index Terms—Antenna arrays, dispersive media, inverse problems, microwave imaging, nonlinear estimation, parametric modeling, spline functions.

I. INTRODUCTION ASED on the latest American Cancer Society report [1], breast cancer is the second most common cancer and the second cause of cancer deaths in US women. Early detection of breast tumors will critically reduce the mortality rate in women. Currently, standard screen film mammography [2], MRI [3], [4] and ultrasound [5] are the imaging modalities used for early breast cancer detection. Microwave tomography is a complementary imaging technique which uses the contrast between the tissues. High contrast between normal breast tissue and the malignant tissue, as well as its accessibility and transparency to microwaves, makes this modality a good option for breast cancer detection. Low power non-ionizing radiation and suppression of breast compression are the other advantages of this method [6]. A novel tomography modality is current-injection electrical impedance tomography (EIT) where the shape-based inversion solutions are utilized [7]. In microwave imaging, an array of antennas sequentially illuminate the breast in a multistatic approach. The consequent

B

Manuscript received September 22, 2010; revised January 07, 2011; accepted February 26, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC9986821). R. Firoozabadi is with the Advanced Algorithm Research Center (AARC), Philips Healthcare, Thousand Oaks, CA 91320 USA (e-mail: [email protected]). E. L. Miller is with the Department of Electrical and Computer Engineering, Tufts University, Medford, MA 02155 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.2011.2163773

scattered fields are recorded by the receiving antennas and used to identify the tumor by reconstructing its boundary or creating the spatial profile of dielectric properties in the region of interest containing the tumor. There are two categories of imaging techniques in literature. In the first category, called inverse scattering methods, a complete or partial map of dielectric properties in the desired region is reconstructed from the measured scattered fields by fitting the data to the predicted scattered fields. A clinical group in Dartmouth College [8]–[10] has been working on this category of breast imaging algorithms. Their imaging system consists of a tank filled with a coupling medium, with the breast immersed in it through a hole. The circular array of transmitter/receiver antennas surrounding the breast illuminates it and collects the scattered fields to be utilized in the reconstruction algorithm. Another research group in University of Wisconsin is also performing comprehensive research on 3-D microwave breast cancer detection [11]–[13]. A group in Duke University [14]–[16] and another one in Carolinas Medical Center [17], [18], have also been working on this category of imaging methods. In the second category of imaging techniques known as radar methods, the location of the strongly scattering anomaly is estimated directly, rather than recovering a detailed map of the pixels. The first system of this category for breast cancer detection was developed by Hagness and colleagues [19]–[21]. Radar-based approaches use ultra-wideband signals with a bandwidth of several gigahertz. A number of locations surrounding the breast are scanned by an antenna. The same antenna collects the back-scattered waves and this process is repeated. The reflections at different locations are then focused by computing the travel time between the sensor and the focal point and application of a time-shifting and summing method. This method is based on the idea that reflections from the tumor add coherently, but the reflection from clutters add incoherently. This category of methods takes advantage of the simpler imaging algorithms and is thus less computationally complex. Our iterative approach uses an inversion technique similar to the first category of imaging methods where the optimization is done in the frequency domain. However, since our primary objective is describing the geometry of the tumor (should one exist) and the intervening tissue layers, we focus on direct estimation of the unknown boundaries, instead of complete estimation of the dielectric profile. In this respect, our method is similar to the second category of imaging methods in that we localize the tumor and additionally characterize its spatial structure. We estimate the boundary of the anomaly inserted in a breast cross-section consisting of several inhomogeneous layers as well as the smooth transition layer between the inner

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Fig. 1. Simplified 3D geometry: Breast layers with the tumor located in the innermost layer. A circular array of uniformly distributed antennas encloses the entire breast.

layers. The entire geometry is surrounded by an array of transmitting/receiving microwave antennas which operate in multiple-frequency multi-source mode: Each antenna radiates at any single frequency, while the other non-adjacent antennas collect the electromagnetic field data. Our iterative inversion algorithm optimizes the unknowns, parameterized in low-dimensional form, by fitting the measured field data to the ones predicted by a forward model at each step until the convergence is met. Our approach is particularly adapted to processing coronal cross sections of the breast. A coronal slice of the breast consists of multiple layers where the tumor is typically located in the innermost layer. The breast with denser geometry is more adapted to our approach as we can distinguish the layers more accurately. The numerical results prove that our approach performs very well in presence of all the inverse model discrepancies with the actual geometry. These mismatches include the breast layered inhomogeneity estimated by homogenous layers, transition layer between inner layers estimated by a sharp boundary, the arbitrary irregular tumor boundary estimated by a circle, and the possible perturbation of mean dielectric properties from their actual values. The paper is organized as follows. In Section II we describe the problem and the physical solution in detail. In Section III, we present the formulation to parameterize the boundaries in low-dimensional form. In Section IV, we present the parameterization of the dielectric properties in a microwave range of frequencies. Inversion algorithm is introduced and described in Section V. Numerical examples are presented and the results are discussed in Section VI. Concluding remarks are given in Section VII. II. PROBLEM STATEMENT In this paper we study our inversion approach applied to breast tumor detection problem where the geometry consists of breast coronal slices. Each slice is composed of several inhomogeneous layers including skin, adipose (fat), and fibroglandular. The breast anatomy is characterized by a smoothly varying transition between the inner layers (adipose and fibroglandular). The tumor is assumed to have an arbitrary irregular shape and to be located in the fibroglandular layer. A simplified 3-D geometry of breast is illustrated in Fig. 1. The

Fig. 2. The geometry of a breast slice consists of skin layer, adipose layer, and fibroglandular layer from the outer to the inner. The tumor is embedded in the fibroglandular layer. The entire breast is immersed in the coupling liquid and surrounded by antennas.

Fig. 2 displays a sample coronal slice. As shown in Fig. 2, the breast is taken to be submerged in a coupling liquid. A circular array of transmitter/receiver antennas embraces the breast. Each antenna operates in either transmit or receive mode. At any moment, one antenna operates as transmitter and the other non-adjacent ones acquire the field data. Electromagnetic field components satisfy Maxwell’s equations and radiation condition in each breast layer which is characterized by its inhomogeneous complex dielectric properties. With the antennas located in direction, a TE wave is propagated in the - plane and the Helmholtz equation in source-free space is [22] (1) and (2) where is the 2-D Laplacian operator, is the wave number, is the electric field in direction, is the angular frequency, is the free space permittivity, is the free space permeis the relative permittivity of medium, and is the ability, medium conductivity. We make use of FDFD forward model to discretize the Helmholtz equation and simulate the propagation of electromagnetic waves in breast layers [23]. In order to solve the open-region problem, we use an eight-grid cell perfect matched layer (PML) absorbing boundary condition [24]. Our ultimate objective is to identify the tumor location and its size in a breast slice from the observed field data collected by the antenna array. The fields scattered by breast are functions of the geometry and dispersive dielectric properties of the breast tissues. In our approach, we attempt to find the unknown parameters by reducing the misfit between the observed field data and the field data predicted by the forward model in an iterative

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procedure. Specifically, our inverse algorithm minimizes a cost function based on the least-squares formulation as follows (3) where is the field vector calculated by forward solver at the receivers for the predicted geometry characterized by the unknown vector , and is the noisy observed field vector. Mathematically we seek a vector to satisfy the following equation (4) III. PARAMETERIZATION OF GEOMETRY Many shape parameterization options are available in literature such as Fourier descriptors [25], Lagrange interpolation [26] and parametrically defined shapes [27], [28] based on global representation of the geometry, as well as B-splines [29]–[31] where the control points determine the properties of the curve. We choose B-splines to model the irregular transition layer between adipose and fibroglandular layers where the control points determine the shape of boundary. While we could model the tumor as well using a B-spline approach, we have found that for small tumors, one cannot stably recover fine scale geometric structures especially in the face of uncertainty in the geometry and electromagnetic contrasts of the adipose and fibroglandular layers. Hence in this paper, we characterize the tumor as a circular object and recover the two coordinates of the center as well as the radius. The geometric parameters that we estimate are then comprised of the interface control point coordinates, and the three parameters describing the tumor. B-splines are piecewise polynomial functions providing local approximation to curves using a small number of parameters called control points. A curve is defined in B-spline parametric form with basic functions associated with control points as [30] (5)

Fig. 3. (Upper-left) basis functions (thin lines) and their summation (bold line) vs. knots, (Upper-right) basis functions weighted by x component of control points (thin lines) and x-variation of final curve vs. knots (bold line), (Lowerleft) basis functions weighted by y component of control points (thin lines) and y -variation of final vs. knots (bold line), (Lower-right) final closed curve. In all figures, knots are indicated by ‘’ and control points by ‘2’.

fined by

with where is the number of interface boundary control points. For our approach the control point positions are converted to polar coordinates where the coordinates center is the determined by the average of control points coordinates. Control points are separated by equal angles . The and are identified only by the radius interface is then parameterized by unknown sub-vector (7) The tumor is modeled by the unknown center coordinates and its radius, parameterized by the unknown sub-vector (8)

with the basis functions tions

defined by the recurrence rela-

The geometry unknown vector is then defined as (9)

(6a) (6b) where

is called the knot vector with . There is a linear relationship between the coordinates of curve points and their associated control points. To change the curve shape, the control point locations are adjusted while the other parameters are fixed. The ends of a curve can be joined to make continuity of the closed a closed loop. In order to keep the control points must repeat at the end. Fig. 3 curve, the first shows the steps to create a closed boundary by B-spline functions. The interface between inner layers is modeled by closed cubic B-Splines where the 2-D control points are de-

In order to discretize the problem geometry for implementation of FDFD forward model in a manner which respects the B-spline representation of the object boundaries, we make use of a filling algorithm which searches in the pixels and marks the ones inside the region enclosed by each boundary including the pixels on the boundary, starting from the outermost boundary. This approach is basically a staircase approximation to the B-spline boundary. By using a very small grid-size (less than 1/20 the lowest wavelength), the desired discretization accuracy is achieved. IV. PARAMETERIZATION OF DIELECTRIC PROPERTIES In microwave breast imaging, knowing the dielectric properties of breast tissues and malignant tumors over the desired frequency range is necessary. A series of publications have docu-

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mented the dielectric properties of biological tissues, including breast tissues, extracted from measurements over various frequency ranges [32]–[42]. Most of the reported studies suggest that there is a large contrast between the normal and malignant breast tissues. Water content level plays the major role in the value of dielectric properties of tissue. Low water content tissues (such as fat) have lower conductivity and permittivity values, while the ones with high water content (such as skin and muscle) show higher dielectric properties. A number of approaches have been proposed in the literature to model the frequency-dependent dielectric properties of dispersive media from the measured data, including recursive convolution [43], auxiliary differential [44], and -transform [45]. These approaches are different in implementation, accuracy and the computational cost. Debye formulation of the first or second order has been widely used to model the dispersive media [44]. This model fits the equation coefficients to the measured data and determines the relation between the electric displacement and electric field. First-order Debye model is of the form (10) where is the relative complex permittivity, is the angular frequency, is the relative permittivity, is the conductivity, is the free space permittivity, is the relative permittivity at infinite frequency, is the static relative permittivity, is the is the static conductivity. relaxation time constant, and We use the first-order Debye model in our work as despite its simplicity, it models the dielectric properties very well in one frequency decade. Fig. 4(a) and (b) illustrate the typical values for dielectric properties of breast tissues versus frequency in skin, adipose, fibroglandular and tumor using first-order Debye model [39], [46]. V. INVERSION ALGORITHM Our inverse problem is posed in a variational context requiring the solution to a nonlinear least squares optimization problem. A variety of methods are present in the literature to solve such problems including descent type methods. In this category of solutions, by knowing the object function in analytical form, the solution is achieved iteratively using the gradient information. Steepest decent, Gauss-Newton, conjugate gradient, and Levenberg-Marquardt methods are in this category. Among all these methods, we choose Levenberg-Marquardt [47], [48] which is basically a trade-off between steepest descent approach with slow convergence and Gauss-Newton method which can be unstable if the Jacobian matrix is poorly conditioned. Regularization is dealt with in an adaptive manner through the choice of the Leveberg-Marquardt parameter at each iteration. In many cases, this algorithm is more efficient than some other numerical methods such as steepest descent and conjugate gradient [49]. Our inversion method minimizes the cost function

Fig. 4. Typical values for dielectric properties of breast tissues (skin, adipose, fibroglandular) and tumor. (a) Relative permittivity. (b) Conductivity.

the predicted geometry, is the unknown vector, is the noisy is the transposed complex observed field data vector and . conjugate vector of Levenberg-Marquardt iterative algorithm uses the following th update formula to calculate the unknown vector in [50] step, starting from an initial guess (12) (13) where and are gradient vector and Hessian matrix of is the number the cost function defined in (11), respectively, is the identity matrix of dimension , and of unknowns, is the Levenberg-Marquardt parameter being updated in each iteration. Complex-valued matrix calculations lead to (14) (15)

(11)

(16)

is the residual vector, is the field where vector calculated by forward solver at the receiver locations for

To solve the above equation, we need to calculate the Jacobian matrix. In the Appendix, we summarize the procedure to cal-

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unknown vector and updated . If the cost function increases in the next step, we increase by a factor of ten, run the inverse algorithm and find the updated unknown vector using this larger parameter, and continue the algorithm. The algorithm is terminated if the cost function becomes less than a pre-defined or the maximum number of iterations threshold is reached. VI. NUMERICAL RESULTS

Fig. 5. The inversion algorithm block diagram.

culate the Jacobian matrix in a multiple-frequency multi-source problem. A block diagram of our inversion algorithm is illustrated in Fig. 5. The algorithm starts with an initial guess of unknown . The forward solver is run with the iniparameters vector tial parameters and computes the initial predicted field vector . The initial cost function is computed using the predicted field vector and the observed data from using (11). In the next step , the actual geometry the Jacobian matrix for the predicted unknown parameters is calculated as explained in the Appendix. The inversion algorithm is then run using (14)–(16) starting with a small initial and updates the unknown vector using (12). We decrease by half and repeat the algorithm using the updated

In this section, we present several numerical examples to support the concept of our inversion algorithm. We study the identification of tumors of different sizes, the behavior when there is no tumor, reconstruction with low contrast between tumor and fibroglandular layer, and sensitivity to perturbation of dielectric properties from their mean values. Although the simultaneous reconstruction of the geometry and the dielectric properties is quite possible (for an example refer to [51]), it will remain as a future work. In this paper we focus on the geometry reconstruction and sensitivity study of dielectric properties. Breast coronal cross-section, consisting skin layer, adipose layer, fibroglandular layer and the tumor, immersed in a coupling liquid, was illustrated in Fig. 2. Typical average values for relative permittivity and conductivity of the breast tissue versus frequency were also plotted in Fig. 4(a) and (b), respectively. In all following examples, for the actual geometry, inhomogeneity in dielectric properties of the layers is modeled by a standard normal distribution around the average values in Fig. 4. A standard deviation of 10% is applied to the relative permittivity and conductivity of skin, adipose and fibroglandular layers, and 5% is applied to the tumor. There is a smooth transition layer between adipose and fibroglandular layers which is modeled by a Gaussian blur filter. Matching liquid is assumed to be a 60% and S/m in all freglycerine solution with quencies. The breast external surface is assumed to be known and the skin layer width is taken to be 2 mm. The unknown boundaries to reconstruct are the transition layer between fibroglandular and adipose layers estimates by a closed B-spline curve using a limited number of control points, and tumor modeled by a circle with an unknown center and radius. A circular array of 16 multiple-frequency transmitter/receiver antennas illuminates the breast by one antenna radiating at a single-frequency at a time while the other 13 non-adjacent antennas collect the fields. In our examples, we use only four transmitting antennas 90 degrees apart. Noisy data are simulated by adding Gaussian noise to the real and imaginary parts of the noise-free data generated by the forward solver using a finer finite difference grid. Simulations are performed in multiple-frequency mode at 600, 900, and 1200 MHz. Signal to noise ratio is assumed to be 40 dB in all examples. In Examples 1–3, the data are generated using the actual geometry mentioned above. In our inverse problem, we use the frequency-dependent average dielectric properties of the tissues extracted from the typical values shown in Fig. 4(a) and (b). In these examples, 15 control points are used to reconstruct the interface between adipose and fibroglandular layers. With 3 tumor unknowns and 15 interface unknowns, the total number of unknowns is 18. Also with 3 operating frequencies and 4 different transmitting antennas, there are 12 sets of data.

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Fig. 6. Example1: Breast coronal cross-section geometry includes the tumor with average diameter of 1 cm, the interface between adipose and fibroglandular layers and the skin layers. Tumor initial guess (inner dotted line) and interface initial guess (outer dotted line) are also shown. In this figure, one antenna is in transmitting mode (bold antenna) and the other non-adjacent antennas in receiving mode (encircled antennas).

Fig. 8. Example 1: Electric field plot at 900 MHz. The magnitude is truncated in order to provide a better illustration of the fields. (a) Real Part. (b) Imaginary part.

Fig. 7. Example 1: Spatial distribution of dielectric properties at 900 MHz. (a) Relative permittivity distribution. (b) Conductivity distribution (S/m).

In Example 1, the tumor with average diameter of 10 mm is reconstructed along with the interface between adipose and

fibroglandular layers in just a few iterations. Fig. 6 shows the ground truth geometry. The spatial distributions of relative permittivity and conductivity at 900 MHz are plotted in Fig. 7 for the entire cross-section. Fig. 8 displays the real and imaginary parts of the electric fields at 900 MHz where one antenna is radiating as shown in Fig. 6. The magnitude of the electric field is truncated in order to provide a better illustration of the fields. The reconstructed geometry and the cost function vs. the iterations are shown in Fig. 9. The results state that starting from a far initial guess for boundaries (shown in Fig. 6), the convergence to actual boundaries is achieved only in 7 steps. A smaller tumor is studied in Example 2 where the average diameter is 5 mm. Fig. 10(a) shows the ground truth geometry and the initial guesses for both boundaries. The reconstructed geometry is shown in Fig. 10(b). As shown in this figure, still we can estimate the interface boundary and localize the tumor with proper accuracy. We study the case with no tumor in Example 3. The inverse problem is started from the initial boundaries shown in Fig. 11(a). By performing our inversion algorithm, we observe that the predicted tumor vanishes by converging into

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Fig. 9. Example 1: Inversion results. (a) Reconstructed boundaries for tumor and the interface (solid lines) vs. true geometry starting from the initial boundaries shown in Fig. 6. (b) Cost function vs. iteration steps.

a point while the other interface is still reconstructed well. Fig. 11(a) shows the geometry including the initial guesses for both boundaries. The reconstructed geometry is displayed in Fig. 11(b). In example 4, we study the capability of algorithm in reconstruction of the geometry with low contrast between the tumor and the fibroglandular layer. We use the same geometry in Example 1, but reduce the frequency-dependent average dielectric properties of the tumor very close to those of the fibroglandular layer, as shown in Figs. 12(a) and (b). The reconstructed geometry is illustrated in Fig. 13. As shown in the figure, even with very low contrast, we can estimate the interface boundary very well and localize the tumor with a good accuracy. In the remaining two examples we examine the sensitivity of our approach to perturbation of the dielectric properties. The observed field data are again simulated using the inhomogeneous geometry mentioned before. But the inverse algorithm uses perturbed complex permittivity values by deviating the typical average vales in Fig. 4(a) and (b). We use a very rough interface between the breast inner layers in these examples. By using a limited number of control points, we still get a good estimate

Fig. 10. Example 2: (a) Breast coronal cross-section geometry and the initial guess, where the tumor has average diameter of 5 mm. (b) Reconstructed boundaries for tumor and interface (solid lines) vs. true geometry starting from the initial boundaries shown in Fig. 10(a).

of the interface, and the tumor is identified well. We make use of 20 control points to estimate the interface between the breast inner layers. With 3 unknowns for the tumor and 20 for the interface, the total number of unknowns is 23. Also with 3 operating frequencies and 4 different transmitting antennas, there are 12 sets of data. In Example 5, the typical complex permittivity average values of skin and tumor are perturbed by 15% increase for use in the inversion algorithm. Adipose and fibroglandular layers are also perturbed by 15% decrease in their complex permittivity from the typical average values. The actual geometry and the initial guesses for both boundaries are shown in Fig. 14(a). Reconstructed geometry is illustrated in Fig. 14(b) where it is observed that good convergence is achieved by our inversion algorithm in 9 steps for both the interface between inner layers and the tumor. The cost function versus the iteration steps is also plotted in Fig. 14(c).

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Fig. 12. Dielectric properties of breast tissues (skin, adipose, fibroglandular) and tumor, with low contrast between tumor and the fibroglandular layer, used in Example 4. (a) Relative permittivity. (b) Conductivity.

Fig. 11. Example 3: (a) Breast coronal cross-section geometry in absence of tumor. (b) Reconstructed boundaries for tumor and interface (solid lines) vs. true geometry starting from the initial boundaries shown in Fig. 11(a).

In Example 6, the typical average values of dielectric properties of the breast layers and the tumor are perturbed by 10% for use in the inversion algorithm where the perturbation is positive for skin and tumor layers and negative for adipose and fibroglandular layers. The geometry and initial guesses for both boundaries are shown in Fig. 15(a). Fig. 15(b) displays the good convergence achieved in 11 steps by our inversion algorithm for both the tumor and the interface between the inner. The cost function versus the iteration steps is also plotted in Fig. 15(c). VII. CONCLUSION A new shape-based inversion algorithm was proposed for detection of anomaly in a multi-layer sliced geometry and was applied to the problem of breast cancer detection successfully. In our approach, breast geometry and dielectric properties are modeled in low-dimensional parametric form. We provided examples to prove that despite the complex geometry and imperfections in the inverse model, our algorithm is robust and can

Fig. 13. Example 4: Reconstructed boundaries for tumor and interface (solid lines) vs. true geometry starting from the initial boundaries in Fig. 12.

identify the tumor and the unknown interface very well. Geometry variations were studied including the tumors of different sizes, the breast slice lacking the tumor, and perturbation of dielectric properties in inversion algorithm from their actual mean values. The results suggest that the tumor is identified pretty well in all studied examples. Our method is more adapted to denser breasts as the inner layers can be distinguished more accurately.

FIROOZABADI AND MILLER: A SHAPE-BASED INVERSION ALGORITHM APPLIED TO MICROWAVE IMAGING OF BREAST TUMORS

Fig. 14. Example 5: (a) Breast coronal cross-section geometry; the initial guess for dielectric properties of all layers is perturbed by 15%. (b) Reconstructed boundaries for tumor and interface (solid lines) vs. true geometry from the initial boundaries shown in Fig. 14(a). (c) Cost function vs. iteration steps.

The future work includes the extension to 3-D reconstruction of anomaly, use of clinical data as the input to our approach, fusion with MRI-extracted geometry, and simultaneous estimation of the geometry and the dielectric properties of the breast tissues.

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Fig. 15. Example 6: (a) Breast coronal cross-section geometry; the initial guess for dielectric properties of all layers is perturbed by 10%. (b) Reconstructed boundaries for tumor and interface (solid lines) vs. true geometry from the initial boundaries shown in Fig. 15(a). (c) Cost function vs. iteration steps.

Although the numerical results demonstrate a degree of robustness to the choice of initialization, we may consider global or hybrid optimization approaches in which a global method is used to get a good initial guess for local method. This is left as a future work.

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APPENDIX A. Derivation of Jacobian Matrix and Residue Vector operating In a multi-frequency multi-source problem with receiving antennas, the residue sub-vector frequencies and and the Jacobian sub-matrices at frequency and for the radiating antenna are as follows (A-1)

.. .

.. .

.. .

.. .

(A-2)

.. .

..

.

.. .

(A-3)

where is the unknown vector, is the predicted field vector, is the observed field vector, and the partial derivatives are obtained using finite differences by perturbing the predicted unknown parameters and calculating the corresponding residue value. Hence

(A-4) (A-5) The global Jacobian matrix is (A-6) The global residue matrix is (A-7) Equations (A-6) and (A-7) are utilized in calculation of the gradient vector (14) and Hessian matrix (15). REFERENCES [1] American Cancer Society, Breast cancer facts & figures 2009–2010. [2] S. W. Fletcher and J. G. Elmore, “Mammographic screening for breast cancer,” New England J. Med., vol. 37, pp. 1672–1680, 2003. [3] S. Heywang-Kobrunner and R. Beck, Contrast Enhanced MRI of the Breast, 2nd ed. Berlin: Springer, 1996. [4] T. Helbich, “Contrast enhanced magnetic resonance imaging of the breast,” Eur. J. Radiol., vol. 34, pp. 208–219, 2000. [5] T. S. Mehta, “Current uses of ultrasound in the evaluation of the breast,” Radiolog. Clinics North Amer., vol. 41, pp. 841–856, 2003. [6] E. C. Fear, “Microwave imaging of the breast,” Technolo. Cancer Rese. Treat., vol. 4, no. 1, pp. 69–82, Feb. 2005, (invited).

[7] S. Babaeizadeh and D. H. Brooks, “Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions,” IEEE Trans. Med. Imaging, vol. 26, no. 5, pp. 637–647, May 2007. [8] P. M. Meaney, S. A. Pendergrass, M. W. Fanning, D. Li, and K. D. Pausen, “Importance of using a reduced contrast coupling medium in 2D microwave breast imaging,” J. Electromagn. Waves Applicat., vol. 17, pp. 333–355, 2003. [9] D. Li, P. M. Meaney, and K. D. Paulsen, “Confocal microwave imaging for breast cancer detection,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1179–1186, Apr. 2003. [10] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1841–1853, Nov. 2000. [11] E. Zastrow, S. C. Hagness, and B. D. Van Veen, “3D computational study of non-invasive patient-specific microwave hyperthermia treatment of breast cancer,” Phys. Med. Biol., vol. 55, pp. 3611–3629, 2010. [12] J. D. Shea, P. Kosmas, S. C. Hagness, and B. D. Van Veen, “Contrastenhanced microwave imaging of breast tumors: A computational study using 3D realistic numerical phantoms,” Inverse Probl., vol. 26, 2010. [13] J. D. Shea, P. Kosmas, S. C. Hagness, and B. D. Van Veen, “Threedimensional microwave imaging of realistic numerical breast phantoms via a multiple-frequency inverse scattering technique,” Med. Phys., vol. 37, no. 8, pp. 4210–4226, Aug. 2010. [14] Z. Zhang and Q. Liu, “Three-dimensional nonlinear image reconstruction for microwave biomedical imaging,” IEEE Trans. Biomed. Eng., vol. 51, pp. 544–548, 2004. [15] Z. Zhang, Q. Liu, C. Xiao, E. Ward, G. Ybarra, and W. Joines, “Microwave breast imaging: 3-D forward scattering simulation,” IEEE Trans. Biomed. Eng., vol. 50, pp. 1180–1189, 2003. [16] J. A. Bryan, Q. Liu, Z. Zhang, T. Wang, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging I—2-D forward and inverse scattering methods,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 123–133, 2002. [17] A. E. Souvorov, A. E. Bulyshev, S. Y. Semenov, R. H. Svenson, and G. P. Tatsis, “Two-dimensional analysis of a microwave flat antenna array for breast cancer tomography,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1413–1415, 2000. [18] A. E. Bulyshev, S. Y. Semenov, A. E. Souvorov, R. H. Svenson, A. G. Nazarov, Y. E. Sizov, and G. P. Tatsis, “Computational modeling of three-dimensional microwave tomography of breast cancer,” IEEE Trans. Biomed. Eng., vol. 48, pp. 1053–1056, 2001. [19] S. C. Hagness, A. Taflove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna-array sensors,” IEEE. Trans. Biomed. Eng., vol. 45, pp. 1470–1479, Dec. 1998. [20] S. C. Hagness, A. Taflove, and J. E. Bridges, “Three-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Design of an antenna-array element,” IEEE. Trans. Antennas Propag., vol. 47, pp. 783–791, May 1999. [21] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal microwave imaging for breast tumor detection: localization in three dimensions,” IEEE. Trans. Biomed. Eng., vol. 49, pp. 812–822, Aug. 2002. [22] D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves. Englewood Cliffs, NJ: Prentice-Hall, 1994, pp. 157–160. [23] E. Marengo, C. M. Rappaport, and E. L. Miller, “Optimum PML ABC conductivity profile in FDFD,” IEEE Trans. Magn., vol. 35, pp. 1506–1509, May 1999. [24] J. Berenger, “A perfectly matched layer for the absorption of electromagnetics waves,” J. Computat. Phys., vol. 114, no. 2, pp. 185–200, Oct. 1994. [25] K. Arbter, W. E. Snyder, H. Burkhardt, and G. Hirzinger, “Application of affine-invariant Fourier descriptors to recognition of 3-D objects,” IEEE Trans. Pattern Analy. Machine Intell., vol. 12, no. 7, pp. 640–647, Jul. 1990. [26] E. T. Whittaker and G. Robinson, “Lagrange’s formula of interpolation,” in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, 1967, pp. 28–30. [27] W. C. Karl, G. C. Verghese, and A. S. Willsky, “Reconstructing ellipsoids from projections,” Comput. Vision, Graphics, Image Processing: Graphical Models and Image Processing, vol. 56, no. 2, pp. 124–139, Mar. 1994.

FIROOZABADI AND MILLER: A SHAPE-BASED INVERSION ALGORITHM APPLIED TO MICROWAVE IMAGING OF BREAST TUMORS

[28] M. E. Kilmer, E. L. Miller, D. Boas, and A. Barbaro, “3D shape-based imaging for diffuse optical tomography,” Applied Optics, 2002. [29] C. Boor, “On calculation with B-splines,” J. Approx. Theory, vol. 6, pp. 50–62, 1972. [30] C. Boor, A Practical Guide to Splines. New York: Springer-Verlag, 1978. [31] L. L. Shumaker, Spline Functions: Basic Theory. New York: Wiley, 1981. [32] H. P. Schwan, Electrical Properties Measured With Alternating Currents; Body Tissues. Handbook of Biological Data. Philadelphia: Saunders, 1956. [33] M. A. Stuchly and S. S. Stuchly, “Dielectric properties of biological substances—Tabulated,” J. Microw. Power, vol. 15, no. 1, pp. 19–26, 1980. [34] R. Pethig, “Dielectric properties of biological materials: Biophysical and medical appliications,” IEEE Trans. Elect. Insulation, vol. 19, pp. 453–474, Oct. 1984. [35] W. T. Joines, Y. Zhang, C. Li, and R. L. Jirtle, “The measured electrical properties of normal and malignant human tissues from 50 to 900 MHz,” Med. Phys., vol. 21, pp. 547–550, 1994. [36] F. A. Duck, Physical Properties of Tissue: A Comprehensive Reference Book. London: Academic Press, 1990. [37] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: I. Literature survey,” Phys. Med. Biol., vol. 41, pp. 2231–2249, 1996. [38] R. W. Lau, S. Gabriel, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, pp. 2251–2269, 1996. [39] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, pp. 2271–2293, 1996. [40] A. J. Surowiec, S. S. Stuckly, J. R. Barr, and A. Swarup, “Dielectric properties of breast carcinoma and the surrounding tissues,” IEEE Trans. Biomed. Eng., vol. 35, pp. 257–263, 1988. [41] S. S. Chaudhary, R. K. Mishra, A. A. Swarup, and J. M. Thomas, “Dielectric properties of normal & malignant human breast tissues at radiowave & microwave frequencies,” Indian J. Biochem. Biophys., vol. 21, pp. 76–79, 1984. [42] A. M. Campbell and D. V. Land, “Dielectric properties of female human breast tissue measured in vitro at 3.2 GHz,” Phys. Med. Biol., vol. 37, no. 1, pp. 193–210, Jan. 1992. [43] R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, “A frequency-dependent finite difference time domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat., vol. 32, pp. 222–227, Mar. 1990. [44] O. Ghandi, “A frequency-dependent finite difference time domain formulation for general dipsersive media,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 658–665, Apr. 1993. [45] D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag., vol. 55, pp. 28–34, Jan. 1996. [46] M. Lazebnik et al., “A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries,” Phys. Med. Biol., vol. 52, pp. 2637–2656, 2007. [47] K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math., vol. 2, pp. 164–168, 1944. [48] D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math., vol. 11, pp. 431–441, 1963.

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[49] C. T. Kelley, Iterative Methods for Optimization. Philadelphia: SIAM, 1999. [50] J. Nocedal and S. J. Wright, Numerical Optimization. New York: Springer, 1999. [51] M. Li, A. Abubakar, T. M. Habashy, and Y. Zhang, “Inversion of controlled-source electromagnetic data using a model-based approach,” Geophy. Prospect., vol. 58, no. 3, pp. 455–467, May 2010.

Reza Firoozabadi (S’02–M’11) received the B.S. degree from Sharif University of Technology, Tehran, Iran, in 1996, the M.S. from Amirkabir University of technology, Tehran, in 1999, and the Ph.D. degree from Northeastern University, Boston, MA, in 2007, all in electrical engineering. From 1996 to 2002, he was a Research Assistant in the Microwave/MM-wave and Wireless Communication Lab at Amirkabir University of Technology. From 2002 to 2006, he was a Research Assistant in the Center for Subsurface Sensing and Imaging Systems (CenSSIS) at Northeastern University. He also worked on development of image and signal processing algorithms for limited-view X-ray tomography and 3-D visualization applied to breast cancer detection utilizing clinical data. He is currently a research scientist in Advanced Algorithm Research Center (AARC) at Philips Healthcare, Thousand Oaks, CA. His current research interests include development of signal and image processing algorithms applied to biomedical applications, inverse problems, and computational electromagnetics.

Eric L. Miller (S’90–M’95–SM’03) received the B.S., M.S., and Ph.D. degrees in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, in 1990, 1992, and 1994, respectively. He is currently a Professor in the Department of Electrical and Computer Engineering and adjunct Professor of Computer Science at Tufts University. Since September 2009, he has served as the Associate Dean of Research for Tufts School of Engineering. His research interests include physics-based tomographic image formation and object characterization, inverse problems in general and inverse scattering in particular, regularization, statistical signal and imaging processing, and computational physical modeling. This work has been carried out in the context of applications including medical imaging, nondestructive evaluation, environmental monitoring and remediation, landmine and unexploded ordnance remediation, and automatic target detection and classification. Dr. Miller is a member of Tau Beta Pi, Phi Beta Kappa and Eta Kappa Nu. He received the CAREER Award from the National Science Foundation in 1996 and the Outstanding Research Award from the College of Engineering at Northeastern University in 2002. He is currently serving as an Associate editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING and was in the same position at the IEEE TRANSACTIONS ON IMAGE PROCESSING from 1998-2002. He was the co-general chair of the 2008 IEEE International Geoscience and Remote Sensing Symposium held in Boston, MA.

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An Inverse Scattering Approach to Soft Fault Diagnosis in Lossy Electric Transmission Lines Huaibin Tang and Qinghua Zhang

Abstract—In this paper, the diagnosis of soft faults in lossy electric transmission lines is studied through the inverse scattering approach, extending a recently reported study on lossless transmission lines. The considered soft faults are modeled as continuous spatial variations of distributed characteristic parameters of transmission lines. The diagnosis of such faults from reflection and transmission coefficients measured at the ends of a line can be formulated as an inverse problem. The relationship between this inverse problem and the inverse scattering theory has been studied by Jaulent in 1982 through transformations from the telegrapher’s equations of transmission lines to Zakharov-Shabat equations. The present paper clarifies and completes the computation of the theoretic scattering data required by the inverse scattering transform from the practically measured engineering scattering data. The inverse scattering method is then applied to numerically simulated lossy transmission lines to confirm the feasibility of the studied approach to soft fault diagnosis. Index Terms—Inverse scattering, lossy transmission line, soft fault diagnosis, telegrapher’s equations, Zakharov-Shabat equations.

I. INTRODUCTION

T

ODAY’S engineering systems are heavily equipped with electric and electronic components, and consequently, the reliability of electric connections becomes a crucial issue. Among the efforts of developing reliable electric systems, a promising technology for transmission line fault diagnosis is the reflectometry, which consists in analyzing the reflection and the transmission of electric waves observed at the ends of a transmission line. For hard fault (open or short circuits) diagnosis, efficient reflectometry-based methods have been reported, for instance, [1]–[3]. However, the diagnosis of soft faults in transmission lines remains an open problem. The weakness of the reflections caused by soft faults constitutes one of the difficulties of this problem, as investigated in [4]. For the purpose of soft fault diagnosis, this paper extends the results of [5] on lossless transmission lines to the case of lossy transmission lines. The considered faults correspond to smooth variations of characteristic parameters in lossy transmission lines. Because of the absence of discontinuities in transmission line characteristic parameters, the diagnosis of such faults

Manuscript received September 01, 2010; revised January 25, 2011; accepted March 25, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work has been supported by the ANR INSCAN and 0-DEFECT. The authors are with INRIA, Campus de Beaulieu, 35042 Rennes, France (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.2011.2163772

cannot be based on conventional reflectometry methods that detect and locate impedance discontinuities. This particular difficulty is addressed in this paper with the inverse scattering approach: given the reflectometry measurements made at the ends of a transmission line, namely the reflection and transmission coefficients, what are the distributed characteristic properties of the transmission line? To our knowledge, the inverse scattering transform (IST) for transmission lines is usually studied under the assumption of lossless transmission lines ([5]), or by neglecting either ohmic loss or dielectric loss ([6], [7]). As losses have significant impacts on most real transmission lines, the lack of studies on the lossy case has been probably a major obstacle to the application of the IST to transmission lines. As the present paper is about an extension to lossy transmission lines of the study on lossless lines recently reported in [5], it will focus on differences between the lossless and the lossy cases. For general information about the inverse scattering theory and its application to transmission lines, the reader may read [5] and the references therein, notably [8]–[10] as referenced at the end of this paper. Compared to the lossless transmission lines related to the single-potential Zakharov-Shabat equations, the lossy transmission lines are related to the two-potential Zakharov-Shabat equations. As shown in this paper, the IST computation in the lossy case requires both the left and the right reflection coefficients and also the transmission coefficient, whereas in the lossess case a single reflection coefficient is sufficient ([5]). Though similar concepts of scattering data have been developed both in engineering practice and in inverse scattering theory, with the same terms reflection and transmission coefficients, their difference was not pointed out in classical publications like [10]. As a matter of fact, the engineering scattering data are measured on finite length transmission lines, whereas the scattering data in the inverse scattering theory are related to the limiting behaviors of the Zakharov-Shabat equations defined . The relationship between on the infinite interval the engineering and theoretic scattering data, recently studied for the lossless case in [5], will be extended to the lossy case in this paper. This result makes possible the computation of the theoretic scattering data from engineering scattering data. Though the results of [10] seemed to indicate that such a computation would involve some unknown constants, it will be shown in this paper that the computation is straightforward and does not involve any unknown constant. In order to illustrate the efficiency of the proposed method for transmission line soft fault diagnosis, results of numerical simulation are presented in this paper. This paper is organized as follows: In Section II, we present the formulation of the inverse scattering problem of lossy elec-

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TANG AND ZHANG: AN INVERSE SCATTERING APPROACH TO SOFT FAULT DIAGNOSIS IN LOSSY ELECTRIC TRANSMISSION LINES

Fig. 1. The lossy transmission line is connected to an alternating voltage source of angular frequency diagram is for measuring the left reflection coefficient and transmission coefficient.

tric transmission lines. Section III presents our simulation results. Concluding remarks are made at Section IV. II. THE INVERSE SCATTERING PROBLEM TRANSMISSION LINES

FOR

LOSSY

In this section, we first shortly recall the transformations from telegrapher’s equations (a classical model of transmission lines) to Zakharov-Shabat equations with two potential functions, and describe the formulation of the related inverse scattering problem. This result initially derived by [10] is the theoretical basis for the inverse scattering problem of transmission lines. Considering its practical aspects, we then clarify and complete the computation of the theoretic scattering data required by the IST from the practically measured engineering scattering data. A. From the Telegrapher’s Equations of Lossy Transmission Lines to Zakharov-Shabat Equations Consider a lossy transmission line which is connected to an alternating voltage source of angular frequency and to a load (see Fig. 1). Let and be the space coordinate values corresponding to the left and right ends of the lossy transmission line. Denote with the source internal impedance and with the load impedance, both assumed to have real values. The voltage and the current at any point are governed by the frequency domain telegrapher’s equations [10]:

at the left end and to a load at the right end. This circuit

will be simply written as , and similarly and . Let and be the -coordinate values corresponding to the left and right ends of the line illustrated in Fig. 1. The value of is known as the wave propagation time over the transmission line. With notation, for

(2) we define two new variables (3a) (3b) which are respectively known as the reflected wave propagating in the negative direction and the incident wave propagating in the positive direction. Then (1) lead to the following Zakharov-Shabat equations:

(4a)

(1a) (1b) where is the imaginary unit, , and are respectively its series resistance, distributed inductance, capacitance, and shunt conductance (RLCG parameters for short) along the longitudinal axis . The angular frequency is strongly related to the (angular) wave number of the electric wave traveling along the transmission line. As the telegrapher’s equations will be transformed to Zakharov-Shabat equations studied in the inverse scattering theory, the notation will be replaced by following the usual convention in inverse scattering theory (see, e.g., [10]–[13]). Following [10], the Liouville transformation

will be applied to replace the space coordinate by the wave propagation time in the telegrapher’s equations. By abuse of

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(4b) with the three potential functions1

(5a) (5b) As a comparison, in the case of lossless transmission lines, as , then the two different postudied in [5], tential functions are equal and , and the corresponding Zakharov-Shabat equations are characterized by a single potential function. As the Zakharov-Shabat equations with two potential functions have been studied in the inverse scattering theory [9], these 1For a better agreement with the engineering definition of scattering data and the potential functions differ which will be reminded later, by a negative sign from the corresponding notations in [10].

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three-potential equations are further reduced to two-potential equations by defining (6a) (6b)

3) Compute the potential functions

from

which satisfy the following Zakharov-Shabat equations (referred to as ) (11) (7) with the two potential functions (8) The inverse scattering problem in this context consists in retrieving the two potential functions from the scattering data (reflectometry measurements). In this paper, the IST as formulated in [9] will be applied to solve this inverse problem. Though it is impossible to retrieve each of the distributed physical parameters , and from the scattering data, the two potential functions computed through the IST can provide useful information for the diagnosis of transmission line faults related to variations of , and . B. The Inverse Scattering Algorithm for To retrieve the potential functions from the scattering data only, it is assumed that the Zakharov-Shabat (7) have no bound state (square integrable solution for ). The IST computation as formulated in [9] requires the left reflection coefficient2 of , and also the left reflection coefficient of the auxiliary Zakharov-Shabat equations (9) which are obtained by interchanging the two potential functions . of The IST consists of the following steps: 1) Compute the Fourier transforms (10a) (10b) 2) Solve the Gel’fand-Levitan-Marchenko (GLM) integral equations for the unknown kernels , , , and in the half plane

2Reflection coefficients will be introduced in Appendices A and B of this paper and discussed in Section II-C.

For more details about the above IST, we refer the readers to [9].3 For the numerical simulation presented in this paper, the numerical algorithm implementing this IST for two-potential Zakharov-Shabat equations is described in [14]. Efficient numerical IST methods are better known for the single-potential Zakharov-Shabat equations, as those developed in [11]–[13]. Our method for two potential functions presented in [14] can be seen as an extension of [13], and is faster than that of [7]. C. Computation of Required Theoretic Scattering Data From Practical Reflectometry Measurements The contents of the two Sections II-A and II-B mostly originated from [10]. However, the pioneer work of [10] has left an important gap between theory and practice: the scattering data and at the input of the IST algorithm (see (10)) are not exactly the same as the scattering data measured in engineering practice, despite their similar names. As a matter of fact, the reflection and transmission coefficients required by the IST are usually defined on the infinite interval , whereas the engineering reflection and transmission coefficients are measured on finite length transmission lines. To distinguish theoretic and engineering scattering data, different notations will be adopted: , and are the theoretic left reflection coefficient, right reflection coefficient and transmission coefficient defined with the limiting behavior of the Jost solutions of the three-potential Zakharov-Shabat (4), whereas , , and are the engineering coefficients of the same names. The definitions of these coefficients can be found in Appendixes A and B of this paper. The relationship between the theoretic scattering data and the engineering scattering data has been studied in [5]. The main idea is to establish the equivalence between the considered finite length transmission line and a fictive infinitely long line, which is obtained by extending the finite line with uniform lines of characteristic impedances equal to the source internal impedance (at the source side) and to the load impedance (at the load side). See the Proposition 1 of [5] for more details. This result, though established for lossless lines in [5], is also valid for lossy lines, by noticing that the extensions made at the two sides of a lossy transmission line are lossless lines, exactly like 3It is noted that our GLM integral equations are with finite integral intervals, differing to the GLM equations in [9] with infinite integral intervals. The transmission lines that we consider in this paper are causal systems, thus and corresponding to the time domain reflectograms are equal . In this case, the GLM equations in [9] are reduced to to zeros for finite integral intervals.

TANG AND ZHANG: AN INVERSE SCATTERING APPROACH TO SOFT FAULT DIAGNOSIS IN LOSSY ELECTRIC TRANSMISSION LINES

in the lossless case. See also [15] for a complete description of the result in the lossy case. The result reported for lossless lines in [5] concerns only the left reflection coefficient. Its extension to lossy lines involves the reflection coefficients at both sides of the transmission line, as well as the transmission coefficient, as stated in the following proposition. Proposition 1: The relationship between the engineering scattering data and theoretic scattering data to the Zakharov-Shabat (4) are as follows:

to the scattering data of the three-potential Zakharov-Shabat (4), namely and through the following equalities (13a) (13b) (13c) and

(14)

(12a) (12b) (12c) where the parameter appearing in the two exponential functions is the wave propagation time over the transmission line and

with being the length of the transmission line as illustrated in Fig. 1. The proof of this proposition, similar to that of the Proposition 2 of [5], is omitted here, but is fully presented in [15]. To ensure the continuity of the potential functions, as formulated by (5), of the extended infinitely long circuit, the lossy parameter functions , , and the derivative of should be smooth at the two connection points and , which need the following assumption. Assumption 1: The RLCG parameters of the transmission line at its left and right ends satisfy that

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where “ ” represents the complex conjugate. It is thus possible to compute and from , and , which are in turn related to the practically measured engineering scattering data , , and through (12). However, apparently, these computations require the knowledge of transmission line length (also known as the wave propagation time over the transmission line) and the integral value , which are unknown values in practice. Fortunately, as shown in the following proposition, and can be directly linked to the engineering scattering data, without requiring the two unknown values. Proposition 2: The theoretic scattering data for IST computations, namely, and , can be calculated from the engineering scattering data as follows (15a) (15b) Proof: Equation (15a) can be proved directly by combing (13a) with (12a). Now let us prove (15b). Substituting (12) and (13) into (14), see equation at the bottom of the following page, thus the result expressed in (15b) is obtained. Proposition 2 indicates that the theoretic scattering data and required for the IST computation, can be computed directly from the engineering scattering data. This proposition completes the result of [10]. III. SIMULATION STUDY

Remark 1: It is noted that Assumption 1 is not necessary for Proposition 1, but it ensures potential function regularities for all required by the IST. In practice, this assumption is not satisfied, since the ohmic and dielectric losses do not tend to zero at the ends of transmission lines. However, as shown in our simulation results (see Section III), the discontinuities caused by typical losses are well tolerated by our numerical IST algorithm. Remark that feeding the IST in (10) is the left reflection coefficient of the two-potential Zakharov-Shabat (7), which may be different from the left reflection coefficient of the three-potential Zakharov-Shabat (4). Moreover, the auxiliary Zakharov-Shabat equations (9) do not physically exist. Hence the reflection coefficient cannot be directly measured. In [10], these two reflection coefficients and are linked

In this section, after presenting a numerical simulator to generate the scattering data of lossy transmission lines, we present some simulation results to confirm the validity of our numerical method. A. Numerical Simulator for Scattering Data For the transmission line shown in Fig. 1, if the value of at the load end, say , was known, then , and the telegrapher’s equation (1) could be integrated reversely from to . It is clear that and depend linearly on . Then it can be readily seen that the left reflection coefficient computed through (16) (see Appendix A) is independent of the actual value of . Hence the arbitrary value of can always be used for computing . Similarly, the transmission coefficient and the right

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reflection coefficient can also be computed with (17) and (18). The above computations were made for a given value of . We can repeat these computations with different values of to cover a sufficiently large spectrum. B. Simulation Results Consider a lossy transmission line of length 1 km with and (constant values). After being converted to the -coordinate, , , and . To simulate soft faults affecting the lossy properties of the line, variations of and are introduced such that the and ratios are as depicted in Fig. 2. The numeric simulator as described in Section III-A is first implemented to generate the engineering reflection and transmission coefficients , and . Then the scattering data and are computed through (15). After that, the numeric IST algorithm [14] is applied to compute the two potential functions from and . As shown in Fig. 5, the result of IST accords well with the “true” potential functions directly simulated from (5) and (8). Because the curves of the “true” potential functions are almost identical to those computed by the numeric IST algorithm, their differences are also plotted in Fig. 6. The above simulation example has been made under Assumption 1, i.e., the resistance and conductance vanish at the two ends of the transmission line. As it is not realistic to assume vanishing and parameters, let us modify the simulation example so that and are as depicted in Fig. 7 and the parameters and remain the same as in the above example. The simulated engineering scattering data are shown in Figs. 8 and 9. As presented in Figs. 10 and 11, though Assumption 1 does not hold, the result of IST accords well with the “true” potential functions directly simulated from (5) and (8) except for slight oscillations close to the two ends. This simulation result shows that our numerical method is tolerant of weak violation of the continuity condition on and parameters stated in Assumption 1.

Fig. 2. Simulated transmission line.

and

when

and

vanish at the ends of the

Fig. 3. Simulated engineering reflection coefficients and depicted in Fig. 2.

and

for

TANG AND ZHANG: AN INVERSE SCATTERING APPROACH TO SOFT FAULT DIAGNOSIS IN LOSSY ELECTRIC TRANSMISSION LINES

Fig. 4. Simulated engineering transmission coefficient depicted in Fig. 2.

for

and

Fig. 5. computed by IST, and compared with the direct simulation for and depicted in Fig. 2. The results of IST accord well with the potential functions computed by direct simulation.

Remarks From the simulation results, the values of the engineering reflection coefficients and are close to 0 after 5 MHz, thus we can use the scattering data truncated at 5 MHz. As the potential functions are real, , and . It is then sufficient to simulate reflection coefficients and for positive values of , though the integrals (10) range from to . IV. CONCLUSION In this paper, based on the theoretic basis for the IST of the general lossy electric transmission lines established in [10], which relates the telegrapher’s equations to the Zakharov-Shabat equations with two potential functions, we have studied the soft fault diagnosis of such lines by clarifying and

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Fig. 6. Errors between computed by IST, and by the direct simulation for and depicted in Fig. 2.

Fig. 7. Simulated and when the ends of the transmission line.

and

tend to positive constants at

completing the computation of the theoretic scattering data required by the IST from the practically measured engineering scattering data. Also, our simulation results confirm the feasibility of this approach. Since the potential functions represent two functional relations of the three quotients of the parameters , , and , these quotients cannot be uniquely determined from the two potential functions. However, the knowledge of can reveal most faults causing distributed variations of RLCG parameters. The results of this paper extending previous works on lossless transmission lines to the case of general lossy lines constitute an important step towards practical applications of the IST to transmission line fault diagnosis. If it is assumed that (resp. ), then and (resp. and ) can be determined from these data. We refer interested readers to [6], [7], [16].

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Fig. 8. Simulated engineering reflection coefficients and depicted in Fig. 7.

Fig. 9. Simulated engineering transmission coefficient depicted in Fig. 7.

and

for

for

Fig. 10. computed by IST, and compared with the direct simulation for and depicted in Fig. 7. The results accord well except for slight oscillations close to the ends.

and Fig. 11. Errors between computed by IST, and by the direct simulation for and depicted in Fig. 7.

APPENDIX The Engineering Scattering Data of Lossy Transmission Lines: For the circuit shown in Fig. 1, the measured left reflection coefficient is

the load end, i.e., connecting the lossy transmission line to the source at the right end and to the load at the left end. For this experiment, we denote with and the source internal impedance and the load impedance, and with , the voltage and current values for . Then the measured right reflection coefficient is

(16) is the input impedance of the where transmission line. And the measured transmission coefficient is

(17) For the measurement of the right reflection coefficient, we make another experiment by inverting the source end and

(18) is the (right end) input where impedance of the transmission line. As the left reflection coefficient , transmission coefficient , and right reflection coefficient expressed in (16)–(18) correspond to the measurements used in engineering practice,

TANG AND ZHANG: AN INVERSE SCATTERING APPROACH TO SOFT FAULT DIAGNOSIS IN LOSSY ELECTRIC TRANSMISSION LINES

they are referred to as engineering scattering data (to be distinguished from the theoretic scattering data used in the inverse scattering theory, as introduced in Appendix B). Their definitions are related to the S-parameters of transmission lines ([17], chapter 13). The Theoretic Scattering Data of Zakharov-Shabat Equations: Let , , and denote the left reflection coefficient, transmission coefficient, and right reflection coefficient (referred to as scattering data) for the Zakharov-Shabat equations with three potential functions (4). As stated in [10], it is known that these scattering data can be expressed as follows (19a) (19b) (19c) where and to (4) satisfying

are the right and left Jost solutions (20) (21)

Similar expressions hold for the scattering data of two-potential Zakharov-Shabat equations. ACKNOWLEDGMENT The authors wish to thank M. Sorine for fruitful discussions.

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[7] P. V. Frangos and D. L. Jaggard, “Analytical and numerical solution to the two-potential Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag., vol. 40, no. 4, pp. 399–404, 1992. [8] G. L. Lamb, Elements of Soliton Theory. New York: Wiley, 1980. [9] W. Eckhaus and A. Van Harten, The Inverse Scattering Transformation and the Theory of Solitons: An Introduction. The Netherlands: Elsevier, 1981. [10] M. Jaulent, “The inverse scattering problem for LCRG transmission lines,” J. Math. Phys., vol. 23, pp. 2286–2290, Dec. 1982. [11] P. V. Frangos and D. Jaggard, “A numerical solution to the ZakharovShabat inverse scattering problem,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 74–79, 1991. [12] P. V. Frangos and D. Jaggard, “Inverse scattering: Solution of coupled Gel’fand-Levitan-Marchenko integral equations using successive kernel approximations,” IEEE Trans. Antennas Propag., vol. 43, no. 6, pp. 547–552, 1995. [13] G. Xiao and K. Yashiro, “An efficient algorithm for solving ZakharovShabat inverse scattering problem,” IEEE Trans. Antennas Propag., vol. 50, no. 6, pp. 807–811, 2002. [14] H. Tang and Q. Zhang, “An Efficient Numerical Inverse Scattering Algorithm for the Zakharav-Shabat Equations With Two Potential Functions,” 2009 [Online]. Available: http://hal.inria.fr/inria-00447358/ [15] H. Tang and Q. Zhang, “Lossy Electric Transmission Line Soft Fault Diagnosis: An Inverse Scattering Approach,” 2010 [Online]. Available: http://hal.archives-ouvertes.fr/inria-00511353/en/ [16] P. V. Frangos, “One-dimensional inverse scattering: exact methods and applications,” Ph.D. dissertation, Univ. Pennsylvania, Philadelphia, 1986. [17] S. J. Orfanidis, Electromagnetic Waves and Antennas 2008 [Online]. Available: http://www.ece.rutgers.edu/orfanidi/ewa/ Huaibin Tang received the B.S. and Ph.D. degrees from Shandong University, in 2003 and 2008, respectively. Since 2009, she has worked as a Postdoctoral Fellow at the Institut National de Recherche en Informatiqueet en Automatique (INRIA), Rennes, France. Her main research interests are in stochastic system, fault diagnosis and signal processing.

REFERENCES [1] P. Smith, C. Furse, and J. Gunther, “Analysis of spread spectrum time domain reflectometry for wire fault location,” IEEE Sensors J., vol. 5, pp. 1469–1478, Dec. 2005. [2] F. Auzanneau, M. Olivas, and N. Ravot, “A simple and accurate model for wire diagnosis using reflectometry,” in PIERS Proc., Aug. 2007. [3] A. Lelong, L. Sommervogel, N. Ravot, and M. O. Carrion, “Distributed reflectometry method for wire fault location using selective average,” IEEE Sensors J., vol. 10, pp. 300–310, 2010. [4] L. A. Griffiths, R. Parakh, and C. F. B. Baker, “The invisible fray: A critical analysis of the use of reflectometry for fray location,” IEEE Sensors J., vol. 6, pp. 697–706, Jun. 2006. [5] Q. Zhang, M. Sorine, and M. Admane, “Inverse scattering for soft fault diagnosis in electric transmission lines,” IEEE Trans. Antennas Propag., vol. 59, no. 1, pp. 141–148, 2011. [6] M. Jaulent, “Inverse scattering problems in absorbing media,” J. Math. Phys., vol. 17, pp. 1351–1360, 1976.

Qinghua Zhang received the B.S. degree from the University of Science and Technology of China, in 1986, the D.E.A. (Diplôme d’Etude Approfondie) from the Université de Lille 1, France, in 1988, and the Ph.D. degree and H.D.R. (Habitation à Diriger des Recherches) from the Université de Rennes 1, France, in 1991 and 1999, respectively. During 1992, he was a Postdoctoral Fellow at Linköping University. Since 1993, he has worked at the Institut National de Recherche en Informatique et en Automatique (INRIA) and also at Institut de Recherche en Informatique et Systémes Aléatoires (IRISA), Rennes, France, as a Research Scientist. His main research interests are in nonlinear system identification, fault diagnosis and biomedical signal processing.

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Evaluation of Dielectric Resonator Sensor for Near-Field Breast Tumor Detection Kenny Seungwoo Ryu, Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—An H-shape ultrawideband (UWB) dielectric resonator (DR) mounted on a vertical ground plane edge with broadside radiation patterns is evaluated for breast tumor detection. In order to scan the whole breast, the proposed DR sensor can be attached to the skin without a need for matching medium. The sensor has a wide half power beamwidth, therefore path loss is reduced. In addition, the DR sensor provides very good pulse-preserving performance such as low distortion and constant gain characteristics with high efficiency. The small footprints of the sensor over the skin allows for more sensors nearby or in contact with the breast skin. Suitability of the compact DR sensor for breast tumor detection can be seen from frequency domain analyses as well as time domain analyses. Accurate tumor response is obtained due to the compact size and very good sensor characteristics. Index Terms—Breast tumor detection, dielectric resonator antennas (DRAs), ultrawideband (UWB).

I. INTRODUCTION

B

REAST cancer is one of the most common cancers for women [1]. Early diagnosis is the key to survive from breast cancer. The radar-based UWB microwave imaging techniques have attracted great attention since Hagness et al. proposed the possibility of breast tumor detection using UWB signal at 1998 [2], [3]. This technique uses a broadband microwave pulse and reconstructs the backscattered energy inside the breast based on the significant contrast in dielectric properties between normal breast tissue and malignant tumors. Radar-based UWB microwave imaging technology research can be divided into two parts. One is the imaging formulation techniques, such as the DAS, MIST, RCB, APES, ATAPES and DCRCB methods [4]–[9]. The other is UWB sensor design [10]–[15]. Optimized ideal UWB antennas have sufficient impedance matching bandwidth to transmit short impulse signals, high radiation efficiency, constant gain, constant group delay, and consistent uniform radiation pattern to avoid undesirable distortions of the radiated and received pulse [16]. Unfortunately, no UWB antenna can achieve all of the ideal UWB performance. While the UWB impedance matching bandwidth based on FCC regulations is mainly concerned with communication systems, a high quality UWB antenna is required to transmit and receive Manuscript received December 15, 2010; revised March 04, 2011; accepted March 07, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. The authors are with the Department of Electrical Engineering, University of Mississippi, University, MI 38677 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2163790

short pulses with low distortions and high efficiency in order to obtain reliable data from tumors. Although many sensors were proposed before, it is not clear what would be the most appropriate sensor design for breast tumor detection. First criterion is related to the sensor bandwidth requirement. It expresses the tradeoff relation between resolution and tissue penetration. Therefore, we need to design sensor, which has appropriate bandwidth for breast tumor detection. Second criterion is related to the dispersion/distortion. In order to reduce signal dispersion/distortion, the constant gain and group delay are important characteristic for the pulse-preserving performance. Third criterion is the sensor efficiency. This is important to compensate low gain characteristics of UWB antennas. However, the sensor efficiency within the UWB operating frequencies is from 60% to 90% [17]–[19]. Recently proposed UWB dielectric resonator antennas (DRAs) mounted on a vertical ground plane edge [20], [21] achieve higher than 95% antenna efficiency within the operating UWB band, because one of the attractive of the DRAs characteristics is their high radiation efficiency. Fourth criterion is the compactness of the sensor size. All of the imaging formulation techniques assume that the sensor is a point source, which originally comes from the phase center concept. However, no real antenna is small enough to be assumed a point source, so the radiation must appear to emanate from a larger area. Even though the point source assumption is common for Global Positioning System (GPS) radar imaging, sensors for Global Network Satellite System (GNSS) applications still suffer due to the phase center variation (PCV) to obtain more suitable position information [22], [23]. The reason is that the phase center is not a fixed position. This implies that the point source has a varied position too. Compared with GNSS applications, it may be more difficult to apply the point source assumption in the breast cancer detection application because the sensor size is quite large compared with the scanned area for tumor detection and the distance between the sensor and the target is small. Therefore, a very compact sensor is more desirable for breast tumor detection and the small aperture of the compact sensor can provide more nodes or footprints near or in contact with the breast skin. The main difference between proposed sensor and previous sensors for the breast tumor detection is not only the compact size and the high efficiency, but also the ability to use it without a need for matching medium, because of the DR characteristics. Most of the metallic-type sensors are required to be immersed into a matching medium with permittivity, which is varied from 3 to 36 for reducing the reflection between free space and breast [24]. A lot of additional work is required to implement the matching medium to surround the patient’s breast, and it

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Fig. 1. Geometry of the DR sensor (a) front view (b) isometric view.

causes inconvenience for the patients. However, the proposed sensor is made up of the DR with a dielectric constant of 10.2, which is similar to the permittivity of the fatty tissue. Therefore, our proposed DR sensor does not need matching medium for reducing the reflection between free space and breast. The DRA mounted on a vertical ground plane edge is a very good candidate for the breast tumor detection, considering its constant gain, and high antenna (total) efficiency, which are two of the requirements of the sensor for the breast tumor detection. To show the suitability of the compact DR sensor for the breast tumor detection, we provide not only frequency domain analysis but also time domain analysis. II. SENSOR CONFIGURATION AND CHARACTERISTICS As shown in Fig. 1, the antenna is slightly modified compared with the one in [21] to obtain slightly wider impedance matching bandwidth and reduce the footprints in contact with the breast skin. The size of the DRA is 14 mm width, 18.3 mm length, and 5.08 mm thickness with a dielectric constant of 10.2, and it is supported by 30 25 mm RT6002 substrate with a dielectric constant of 2.94 and a substrate thickness of 0.762 mm. The ground plane is partially printed on the substrate under the DRA. The size of ground plane is 11 30 mm . We use the same parameters stated in [21] with mm, mm, mm, and mm. At first, we consider system transfer function of the breast is comtumor detection. The system transfer function posed of the antenna as transmitter, , the antenna as re, and the channel, ceiver, (1)

Fig. 2. Measured and simulated (a) reflection coefficient (b) group delay (c) gain.

, is Unfortunately, the transfer function of the channel, in the human breast, which is made up of inhomogeneous materials. In addition, every woman has different density, electric property, size, and shape. It implies that we could not obtain the system transfer function accurately, due to the ambiguity of channel transfer function in a human breast. Therefore, we only consider the dispersion/distortion caused by the sensor because it is the only known factor that can be controlled to improve the pulse-preserving performance by sensor designer for the breast tumor detection. Fig. 2 shows the reflection coefficients, the group delay, and gain characteristics of the DR sensor. It is well known that the phase variation occurs at the resonant frequen-

cies as the reflection changes the phase from positive to negative. As it can be seen from the figure, there are a small number of the resonant frequencies and the variation of the reflection coefficient is minimal within the operating frequency band. This indicates that the amount of the phase variation vs. frequency is also small and the phase variation is minimized near the resonant frequency. This is one of the reasons for the group delay variation to be minimized. Minimizing the group delay variation is extremely important as the dispersion/distortion significantly affects beamforming, especially delay-and-sum beamforming over large portion of the bandwidth [25]. The measure-

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TABLE I SENSOR SIZES FOR BREAST TUMOR DETECTION

Fig. 3. Sensor type I and II dependence on the HPBW.

ments are performed using the HP8510c network analyzer. The simulated and measured matching frequency bands of the prodB reflection coefficients are from 3.5 posed antenna for GHz to 9.9 GHz with a bandwidth of 96% in simulation and from 3.8 GHz to 10.8 GHz with a bandwidth of 96% in measurement. The standard deviation of the group delay within the operating frequency band is only 0.14 ns in simulation and 0.18 ns in measurement. Fig. 2(c) shows the simulated and measured gain of the DR sensor. It is generally known that constant gain is also one of the important factors for the pulse-preserving performance [26]. The standard deviation of the gain within the operating frequency band is only 0.44 dBi in simulation and 0.5 dBi in measurement. Generally, the RF cable from the Vector Network Analyzer significantly affects the measurement of small antennas. Therefore, one would anticipate slight differences between simulated and measured results. Other parameters such as very consistent radiation patterns and high efficiency characteristics are very similar to the sensor from [21]. The beamwidth is the one of the parameters to measure the qualification of the sensor for breast detection [15]. Many sensor designs for breast tumor detection can be found in the literature. The position of some sensors require some distance between sensor and breast skin [10]–[12], [15], while other sensors can be used both near and in contact with the breast skin [13], [14]. Previously, the position of the sensor was determined based on the radiating material. However, as mentioned in [15], the half power beamwidth (HPBW) is an important parameter. Fig. 3 shows the expected position of the sensor depending on the beamwidth. To scan whole of the breast, some sensors (type I) need some distance from the breast skin due to narrow HPBW, and other sensors (type II) do not require distance between sensor and breast skin due to very wide HPBW. The main advantage of senor type II is low path loss between sensor and breast. If the position of the sensor having narrow HPBW is close to the breast skin, then the received signal strength (RSS) information is varying despite having the same distance with different angular position of the target. That may cause error in distinguishing between tumor and glandular tissue and

also in detecting accurate position of the tumor. Therefore, it is important to get information with same power. For example, the overall HPBW of the sensor described in [11] is 32 degrees in the E plane and 60 degrees in the H plane, and the overall HPBW of the sensor in [15] is 34 degrees in the E plane and 51 degrees in the H plane. Therefore, the sensor in [11] requires 131 mm distance from the sensor to the breast skin for scanning of the whole breast and the sensor in [15] requires 121 mm distance from sensor to the breast skin for scanning of the whole breast, if the radius of the breast is 50 mm. The optimum beamwidth of sensor type II is around 180 degrees to scan whole breast. The overall HPBW of the sensor in [13] is 60 degrees in the E plane and 60 degrees in the H plane. Even though the radiating material of the sensor in [13] can be in contact with the breast skin, it requires 50 mm distance from the sensor to the breast skin to operate effectively. The overall HPBW of our proposed DRA is 190 degrees in the E plane and 182 degrees in the H degree. Thus, our proposed DRA can be used near or in contact with the breast skin directly to get tumor response with same power. Table I shows the sizes of the sensors. Among them, the sensors in [13] and [14] can contact the breast skin directly because the materials of their radiating elements are dielectric materials. The area contacting the breast skin (the footprint) is also important. If the contacted area is small, more sensors can be placed around the breast to obtain more data for breast tumor detection. The footprint of sensor in [13] is 23.8 mm by 23.8 mm and the footprint in [14] is 20 mm by 35 mm. The total footprint of the proposed sensor is only 14 mm by 5.08 mm. III. ANALYSIS OF TUMOR RESPONSE In this section, we investigate the tumor response in terms of both frequency and time-domain analysis. Table II shows the dielectric properties of the breast model at 6 GHz [5]. Recently, new electrical constants for UWB microwave dielectric properties are reported [27]. A Modulated-Gaussian pulse with the frens is quency range from 3.2 GHz to 9.8 GHz and used for the signal excitation. Fig. 4 shows the simple numerical model configuration for the breast tumor detection to figure out the tumor response with various situations under mono-static data acquisition. It should be mentioned that the distance of the sensor and the breast skin is 1 mm due to the difficulty of modeling the contacting skin at the circle shaped realistic model in Fig. 11. First, to remove the undesired early time content such as reflections from the breast skin, we use the removal method in

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Fig. 4. Simple model for the breast tumor detection.

TABLE II DIELECTRIC PROPERTIES OF THE BREAST MODEL

[7], so that we obtain only the late time ringing signal from the tumor. This is usually called the calibrated backscattered signal. Fig. 5 shows the tumor responses of three different tumor sizes (3 mm, 4 mm, and 5 mm diameter), when the tumor depth is at 20 mm and with a 1 mm breast skin thickness. This figure shows that the tumor response decreases with the tumor size and the pulse shape is slightly different depending on the size of the tumor. Fig. 6 shows the tumor responses at three different depths (20 mm, 30 mm, 40 mm) when the tumor diameter is 3 mm and with a 1 mm skin thickness. It shows that the tumor response decreases with depth and that each tumor response is very similar for the different depths. We can obtain the point source reference position using the time of arrival (TOA) technique. From the results in Figs. 5 and 6, we find that the distortion in the tumor response pulse is more affected by size or shape of the tumor than by the depth of the tumor location. Fig. 7 shows the tumor responses with different breast skin thicknesses (1 mm and 2 mm) when the tumor diameter is 3 mm and the depth is 30 mm. Even though the breast skin effect is removed using the early time removal method [5] it is still slightly affects the late time tumor response, because the tumor response is from the ringing signal. The difference of the averaged tumor response between the 1 mm and 2 mm breast skin thickness is around 2 dB. Fig. 8 compares the tumor response and glandular tissue response when the tissue diameter is 5 mm and the depth is 30 mm with a 2 mm skin thickness. The difference between tumor and glandular tissue response is around 15 dB. This indicates

Fig. 5. Tumor responses of three different size tumors (3 mm, 4 mm, and 5 mm diameters).

that our sensor can distinguish very clearly the differences between tumor and glandular tissue. Also we can see the difference between the tumor itself and the tumor inside glandular tissue, when the diameter of the tumor is 3 mm and the diameter of the glandular tissue is 5 mm. This result indicates that the response of the tumor inside glandular tissue is slightly higher than the response from the tumor itself. Fig. 9 shows the tumor responses for the 10 mm off-set cases when the tumor diameter is 3 mm and the depth is 30 mm as a reference. Due to the asymmetric structure of the sensor and radiation patterns, the tumor responses for off-set cases are not exactly the same. However, when we investigate the time of the peak amplitude of the time domain response, the difference of the peak time is quite small. That is, even though the point source position is moved depending on the tumor position, the variation of the point source position is very small. It should be mentioned how we calibrate the backscattered signal numerically. As we can see from the above results, the tumor response is very small because it is the late time ringing response. The small numerical error causes unreliable data for the tumor response. Therefore it is really important to reduce numerical errors as much as possible. One of the biggest numerical

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Fig. 6. Tumor responses of three different depths (20 mm, 30 mm, and 40 mm).

Fig. 7. Tumor responses of different skin thickness (1 mm and 2 mm).

errors is from the different mesh structure between the models with and without the presence of the tumor. In such case, the response from the numerical error is bigger than the response from the breast tumor. To avoid this numerical error, we should use the same mesh structure whether the tumor exists or not. In addition, the use of a dense mesh structure is also one of the factors to reduce the numerical error, because a dense mesh usually provides very good convergence rate. From the above preliminary results, we have obtained useful information. First, we obtain received signal strength (RSS). As the energy of a signal changes with distance, the RSS at a target conveys information about the distance between that target and the sensor that has transmitted the signal. Two factors that affect the signal energy are the path-loss, which refers to the reduction of signal energy as it propagates through space, and shadowing effects, which represent signal energy variations due to the obstacles in the environment. Second, we obtain the time of arrival (TOA) of a signal traveling from sensor to target, and the time difference of arrival (TDOA) information between the target and two known reference positions. In our breast tumor detection, a combination of position parameters can be utilized in order to obtain more information about the position of the tumor location. The above hybrid method using a combination of position parameters specifies the range between a sensor and a target, which defines a circle for the possible target positions.

Usually, the peak location of the received signal is considered as the center of the tumor position in the breast tumor detection and tumor size is quite a small compared with the sensor size. Therefore, we use geometric techniques for position estimation to determine the position of the source point from the known target positions. Therefore, in the presence of simulations ( mm and mm of yz plane and xz plane with 20, 30, and 40 mm depths), the position of the point source variation can be calculated by the triangulation method, as shown in Fig. 10. The calculated point source variation on the z axis is only 0.095 mm, on the y axis is 0.53 mm, and on the x axis is 1.94 mm. The biggest point source variation is on the x axis because the E plane pattern of this sensor is slightly asymmetric and the geometric radiating element of the x axis is longer than the y axis. IV. IMAGE OF BREAST TUMOR To process the signals coherently, we shift the time by a number of samples to align the returns from the focal point. After that, we apply a time-window to the time-shifted signals. The window is given by

(2)

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Fig. 9. Shift tumor position responses at

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010 mm, 0 mm, and 10 mm.

Fig. 8. Comparison of the tumor response, glandular tissue response, and tumor in glandular tissue response.

The size of time window is one of the parameters that control the quality of the image. If we choose a wide window, we can obtain robust data to avoid several errors, such as the point source variation. If we choose a narrow window, we can obtain high resolution data with the cost of robustness. From the previous results, we obtain approximately the point source variation of the DR sensor using the triangulation method via several known possible tumor location simulations. Therefore, we are able to choose the time-window appropriately to avoid the error from the point source variation to get reliable breast tumor detection. After that, we can apply any kind of the imaging formulations techniques. Fig. 11 shows a realistic homogeneous model of the breast. We assume that the sensor moves around the breast to form a synthetic aperture, and obtains data every 45 degrees. Dielectric properties of the chest wall are chosen as 50 dielectric constant and conductivity 7 S/m, and dielectric properties of the nipple are chosen as 45 dielectric constant and conductivity 5 S/m [5]. Anatomically, a breast tumor is most commonly inside of the ducts (milk passages) and lobules (milk producing glands at the ends of lobes). In our breast model, we make several lobules, which are considered as glandular tissues. Fig. 12 shows the tumor response image. We can observe very compact tumor

Fig. 10. Position estimation via triangulation.

image response. The tumor location is detected at the true location (50 mm, 40 mm), even though tumor location is not centered in a breast. V. CONCLUSION We evaluated a DR sensor mounted on a vertical ground plane edge for use in breast tumor detection. Compared with other sensors, the proposed DR sensor provides very high total (antenna) efficiency, and low distortion/dispersion with compact size. In addition, DR sensor does not need matching medium to reduce reflection between free space and breast because of DR characteristic. Furthermore, DR sensor provides wide beamwidth

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Fig. 11. Isometric transparent view of a realistic breast model.

Fig. 12. Tumor response image.

due to the low elevation radiation characteristic. Therefore, it induces accurate tumor information, such as RSS. We analyzed the tumor response in both the time and frequency domains to show the performance of our proposed DR sensor. It may be the good guide for antenna engineer to compare the antenna performance for the breast tumor detection. In addition, we calculated the point source variation using known several tumor positions. Due to the very small point source variation of our proposed sensor, the errors from the point source can be complemented by a time-windowing technique, which is a basic technique for the image processing. Therefore, our proposed DR sensor provides very good performance for the location problem, which is an important factor for tumor response localization in practical scenarios when the scan pattern is not centered on the tumor location. REFERENCES [1] “Cancer facts and figures 2010,” in American Cancer Society, 2010. [2] S. C. Hagness, A. Taflove, and J. E. Brides, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna array sensors,” IEEE Trans. Biomed. Eng., vol. 45, pp. 1470–1479, 1998. [3] S. C. Hagness, A. Taflove, and J. E. Brides, “Three-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Design of antenna array element,” IEEE Trans. Antennas Propag., vol. 47, pp. 783–791, 1998.

[4] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal microwave imaging for breast cancer detection: Localization of tumors in three dimensions,” IEEE Trans. Biomed. Eng., vol. 47, pp. 812–822, 2002. [5] X. Li and S. C. Hagness, “A confocal microwave imaging algorithm for breast cancer detection,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 130–132, 2001. [6] X. Li, E. J. Bond, and S. H. B. Veen, “An overview of ultra-wideband microwave imaging via space-time beamforming for early-stage breast-cancer detection,” IEEE Antennas Propag. Mag., vol. 47, pp. 19–34, Feb. 2005. [7] B. Guo, Y. Wang, J. Li, P. Stoica, and R. Wu, “Microwave imaging via adaptive beamforming methods for breast cancer detection,” in Proc. Progress in Electromagnetics Research Symp. (PIERS’05), Hangzhou, China, Aug. 2005. [8] Y. Xie, B. Guo, L. Xu, J. Li, and P. Stoica, “Multi-static adaptive microwave imaging for early breast cancer detection,” IEEE Trans. Biomed. Eng., vol. 53, pp. 1647–1657, 2006. [9] K. S. Ryu, M. Sepehrifar, and A. A. Kishk, “High accuracy peak location and amplitude spectral estimation via tuning APES method,” Digital Signal Processing, vol. 2, pp. 552–560, 2010. [10] M. A. Campbell, M. Okoniewski, and E. C. Fear, “TEM horn antenna for near-field microwave imaging,” Microw. Opt. Technol. Lett., pp. 1164–1170, 2010, vol.. [11] J. Bourqui, M. Okoniewski, and E. C. Fear, “Balanced antipodal Vivaldi antenna for breast cancer detection,” in Proc. 2nd Eur.Conf. on Antennas and Propagation, Nov. 2007, pp. 1–4. [12] X. Li, S. C. Hagness, M. K. Choi, and D. W. van der Weide, “Numerical and experimental investigation of an ultrawideband ridged pyramidal horn antenna with curved launching plane for pulse radiation,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 259–262, 2003. [13] W. Huang and A. A. Kishk, “Compact dielectric resonator antenna for microwave breast cancer detection,” IET Microw. Antennas Propag., vol. 3, pp. 638–644, 2009. [14] J. Bourqui, E. C. Fear, and M. Okoniewski, “Versatile ultrawideband sensor for near-field microwave imaging,” presented at the 4th Europ. Conf. Antennas Propagation, , Spain, Apr. 12–16, 2010. [15] J. Bourqui, M. Okoniewski, and E. C. Fear, “Balanced antipodal Vivaldi antenna with dielectric director for near-field microwave imaging,” IEEE Trans. Antennas Propag., vol. 58, pp. 2318–2326, 2010. [16] Z. N. Chen, Antennas for Portable Devices. New York: Wiley, 2007. [17] J.-F. Pintos, P. Chambelin, A. Louzir, and D. Rialet, “Low cost UWB printed dipole antenna with filtering feature,” in Proc. IEEE Antenna and Propagation Society Int. Symp., Jul. 2008, pp. 1–4. [18] S. J. Boyes, Y. Hwang, and N. Khiabani, “Assessment UWB antenna efficiency repeatability using reverberation chambers,” in Proc. IEEE Ultra-Wideband Symp., Sep. 2010, pp. 1–4. [19] E. G. Lim, Z. Wang, C. Lei, Y. Wang, and K. L. Man, “UWB antennas past and present,” IAENG Int. J. Comput. Sci., vol. 37, no. 3, Aug. 2010, available online. [20] K. S. Ryu and A. A. Kishk, “UWB dielectric resonator antenna mounted on a vertical ground plane edge,” in Proc. IEEE Antenna and Propagation Society Int. Symp., Jun. 2009, pp. 1–4. [21] K. S. Ryu and A. A. Kishk, “Ultra-wideband dielectric resonator antenna with broadside patterns mounted on a vertical ground plane edge,” IEEE Trans. Antennas Propag., vol. 58, pp. 1047–1053, Apr. 2010. [22] D. A. Aloi, A. Rusek, and B. A. Oakley, “A relative technique for characterization of PCV error of large aperture antennas using GPS data,” IEEE Trans. Instrum. Meas., vol. 54, pp. 1820–1832, 2005. [23] J. F. Zumberge, M. B. Heflin, D. C. Jefferson, M. M. Watkins, and F. H. Webb, “Precise point positioning for the efficient and robust analysis of GPS data from large networks,” J.Geophys. Res., vol. 102, pp. 5005–5017, 1997. [24] J. Sill and E. Fear, “Tissue sensing adaptive radar for breast cancer detection: Study of immersion liquids,” Electron. Lett., vol. 41, no. 3, pp. 113–115, 2005. [25] S. Ellingson, “Dispersion by antennas,” LWA Memo Series Dec. 11, 2007 [Online]. Available: http://www.ece.vt.edu/swe/lwa [26] D. H. Kwon, “Effect of antenna gain and group delay variations on pulse-preserving capabilities of ultrawideband antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2208–2215, 2006.

RYU AND KISHK: EVALUATION OF DIELECTRIC RESONATOR SENSOR FOR NEAR-FIELD BREAST TUMOR DETECTION

[27] M. Lazebnik, L. McCartney, D. Popovic, C. B. Watkins, M. J. Lindstrom, J. Harter, S. Sewall, A. Magliocco, J. H. Booske, M. Okoniewski, and S. C. Hagness, “A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries,” Phys. Med. Biol., vol. 52, pp. 2637–2656, 2007.

Kenny Seungwoo Ryu (S’08–M’11) received the B.S. degree in radio communications engineering from Yonsei University, in 1999, the M.S. degree in electrical engineering from Mississippi State University, and the Ph.D. degree from the University of Mississippi in 2010. From 2005 to 2010, he worked as a Research Assistant and Teaching Assistant at the Department of Electrical Engineering, University of Mississippi, University. Since 2010, he has been a Research Engineer at the Mobile Communication R&D Center, LG Electronics, Korea. His research interest includes the area of design and analysis of ultrawideband antenna, dielectric resonator antenna, dual-polarized antenna, computational electromagnetic, artificial magnetic conductors, microwave passive and active circuit design, sensor design for detection, spectral estimation, biomedical applications of signal processing and microwave imaging, and WWAN and LTE antennas for mobile phone applications. He has also served as a reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Antennas and Propagation Magazine, the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the International Journal of Numerical Modeling, the Progress In Electromagnetics Research, and the Journal of EletroMagnetic Waves and Application.

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Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977 and in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, Canada, in 1983 and 1986, respectively. Currently he is a Professor at Concordia University, Tier 1 Canada research Chair, Montréal, Québec, Canada (since 2011). He has published over 220 refereed journal articles and 380 conference papers. He is a coauthor of four books and several book chapters and the editor of one book. He offered several short courses in international conferences. His research interest includes the areas: millimeter wave antennas, dielectric resonator antennas, microstrip antennas, EBG surfaces, artificial magnetic conductors, soft and hard surfaces, phased array antennas, small antennas, and optimization techniques for electromagnetic applications. He is an Editor of Antennas and Propagation Magazine. Dr. Kishk is a member of the Antennas and Propagation Society, Microwave Theory and Techniques Society, Sigma Xi society, U.S. National Committee of International Union of Radio Science (URSI) Commission B, Phi Kappa Phi Society, Electromagnetic Compatibility, and Applied Computational Electromagnetics Society. He is a Fellow of IEEE since 1998, the Electromagnetic Academy, and the Applied Computational Electromagnetic Society (ACES). He and his students are the recipient of many awards. He received 1995 and 2006 Outstanding Paper Awards for papers published in the Applied Computational Electromagnetic Society Journal. He received the 1997 Outstanding Engineering Educator Award from the IEEE Memphis section, the Outstanding Engineering Faculty Member of the Year in 1998 and 2009, and the Faculty Research Award for Outstanding Performance in Research in 2001 and 2005. He received the 2001 Award of Distinguished Technical Communication for the entry in the IEEE Antennas and Propagation Magazine. He also received the Valued Contribution Award for outstanding Invited Presentation, “EM Modeling of Surfaces with STOP or GO Characteristics - Artificial Magnetic Conductors and Soft and Hard Surfaces” from the Applied Computational Electromagnetic Society. He received the 2004 Microwave Theory and Techniques Society Microwave Prize.

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Wideband Antenna With Conductive Textile Radiators for a Dual-Sensor Subsurface Detection System A. Oral Salman, Member, IEEE, Emrullah Biçak, and Mehmet Sezgin

Abstract—A wideband antenna with conductive textile radiators (WACTR) is investigated for a pulsed dual-sensor subsurface detection system, which is composed of a ground penetrating radar (GPR) sensor and an electromagnetic induction (EI) sensor. It is demonstrated that undesired clutter of the EI sensor arising from the metallic mass and conductivity of the GPR antenna radiators can be removed by choosing textile material for the antenna without degradation of the antenna’s performance. To the best of our knowledge, this is the first application of textile material used in antennas for such an aim. The antenna also has a new feeding structure to excite the textile patches, which offers a mechanically more rigid solution and a wider excitation surface. Some measured and simulated antenna characteristics are presented for the antenna located in the dual-sensor search head (DSSH). The ground effects on the antenna performance are additionally investigated as a complementary study. The detection results of some selected targets buried in soil by means of the proposed WACTR are presented. The targets are detected even in the non-metallic and deep-buried cases and it is also demonstrated that the GPR and EI sensors are complementary sensors. Index Terms—Broadband antennas, buried object detection, electromagnetic induction, ground penetrating radar, multi-sensor systems, textile industry.

I. INTRODUCTION

S

MART textiles or e-textiles have recently received great interest in the areas of electronics, computer, and biomedical engineering. One of the important applications of this technology is the sensing of bodily functions of a person such as heart beat, respiration, and body temperature using wearable or implantable textile sensors [1]–[3]. In this context a flexible and wearable antenna is also needed to transmit data, which are collected from the textile or any kind of sensor on the body, to the communication center. Antennas made of textile material are the best choice for this aim. There are numerous recent Manuscript received December 29, 2009; revised August 05, 2010; accepted April 04, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. A. O. Salman was with Hacettepe University, Ankara, Turkey. He is now with TUBITAK, BILGEM, Information Technologies Institute, Sensor Systems Group, 41470 Gebze, Kocaeli, Turkey (e-mail: [email protected]). E. Biçak is with the TUBITAK, BILGEM, Information Technologies Institute, Sensor Systems Group, 41470 Gebze, Kocaeli, Turkey (e-mail: emrullah. [email protected]). M. Sezgin was with the Istanbul Technical University, Istanbul, Turkey. He is now with TUBITAK, BILGEM, Information Technologies Institute, Sensor Systems Group, 41470 Gebze, Kocaeli, Turkey (e-mail: mehmet.sezgin@bte. tubitak.gov.tr). Digital Object Identifier 10.1109/TAP.2011.2163745

studies in the literature about designs and applications such as UWB WBAN [4], Bluetooth [5] and ISM band [6], [7] applications, dual-band [8], [9], dual-polarized [10], and aperture-coupled [11] textile antennas. Aerospace applications of textile antennas and passive microwave textile components, in particular, are examined in [12], [13]. Since textile antennas are made of flexible fabric materials, the effects of antenna bending [14] and crumpling [15], [16] and moisture [17] on the antenna performance play a critical role. The proximity effects of a human body are important in on-body applications of textile antennas and are investigated in [18], [19]. Moreover, determination of electromagnetic parameters of textile material is crucial in the antenna design. Several techniques have been proposed for electrical characterization of conductive [20], [21] and non-conductive [22] textiles. The antennas made from textile material investigated in the literature are commonly in planar form and can be classified in two main groups according to the material type of their radiator patch. A conductive textile material [1], [2], [4]–[6], [8], [10], [11], [13], [15], [18], [21] is used as the radiator patch in the first group. The majority of the studies about textile antennas are addressed in this group, where purely textile antennas are involved. In the second group, a metallic thin Cu foil [9], [12], [14], [19], [23] is used instead of textile material as the radiator patch. In both groups, a non-conductive textile material (i.e., acrylic textile, felt fabric, fleece fabric, or cloth itself) is used as the antenna substrate. The common property of the textile antennas in both groups is their flexibility and wearability due to their on-body applications. In this study, we propose to use conductive textile material in antenna radiators on a standard FR-4 substrate for a wideband GPR antenna used in a dual-sensor subsurface detection system [24]–[27]. This system is composed of a pulsed ground penetrating radar (GPR) sensor and a pulsed electromagnetic induction (EI) sensor. The wideband GPR antenna is a well-known planar elliptical dipole which operates in a frequency range from 645 MHz to 4.87 GHz (in the dual-sensor search head (DSSH)). We note that the GPR antenna of the dual-sensor system has to be mechanically strong in target detection operations,1 and therefore flexibility of the antenna is not a requirement for this type of application. Thus, rigid FR-4 is used as the substrate material instead of soft non-conductive textile. An antenna of this form can be described as a semi-textile antenna due to the usage of textile material in its patches. In the design, a new feeding 1It is a hand-held system and an operator sweeps the dual-sensor search head in a detection operation by swinging it.

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SALMAN et al.: WACTR FOR A DUAL-SENSOR SUBSURFACE DETECTION SYSTEM

structure is applied to the proposed antenna to provide a mechanical rigidity at the feed points of the antenna and a wider interconnection surface area to transfer RF signal from coax-cable to the textile patches. The main reason for combining two sensors in the inspection of underground objects is to obtain a better detection and identification (discrimination) performance. In this context, the greater the number of sensors used, the more information about the probed object is obtained. If the information obtained from the sensors is processed in a proper manner, identification of a buried object can be accomplished successfully. The most popular pair of sensors in hand-held buried object inspection applications is an electromagnetic induction2 (EI) and GPR sensors pair. In our subsurface detection system, the same pair of sensors is utilized, thus providing a higher discrimination and decision performance [25], [28], [29]. One important issue is the detection and identification of metallic as well as non-metallic objects by such a dual-sensor subsurface detection system. Metallic and non-metallic objects can be detected by one or both of these two sensors most of the time. While the dielectric content of a buried object is detected only by using the GPR sensor, the metallic content may be detected by using both the GPR and EI sensors. However, there are some conditions under which only one of the sensors can sense an underground target in a detection operation. For example, when the amount of metallic content is absent or not high enough for the EI sensor’s detection capability, it may not be detected by the EI sensor, but the GPR sensor may detect the underground object in this case. On the other hand, when the physical dimensions of the buried object are not large enough in terms of electromagnetic wavelength or the orientation of the object is not proper for GPR sensing, the GPR sensor does not detect such a buried object, even it is metallic. The EI sensor may detect this target in this case. Such kinds of scenarios show that the EI and GPR sensors can be considered as complementary sensors for underground object inspection. Moreover, as mentioned in the paragraph above, the usage of a multi-sensor system in subsurface detection gives an opportunity to discriminate buried objects. For example, the EI sensor may provide the same metallic density and the same detection signals for a metallic target at a specific depth compared to another underground object which has a higher metallic content and is buried deeper than the first one. In this case, burial depths and physical dimensions can be differentiated from GPR data of the targets and it is possible to determine that these targets are different. There is an important problem in the integration of these two sensors in subsurface detection. A classical entirely metallic (Cu) GPR antenna which has to operate very close to the coils of the EI sensor creates a clutter for the EI sensor and prevents detection of targets by the EI sensor. In this case, the sensitivity of the EI sensor decreases severely. Although the problem can be solved partly by additional hardware, it still seriously affects the EI sensor’s sensitivity performance. Another solution to remove this clutter, which arises from the large metallic mass and conductivity of the GPR antenna, is to reduce the surface area of the antenna by designing the antenna in a notched[30] or gridded form. When the surface notching is applied to a planar 2This

sensor is sometimes called a “metal detector”.

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elliptical dipole, stop-bands in the operating frequency band and impedance mismatches may arise [31]. Similarly, the meshing of the antenna surface reduces the antenna gain and directivity [32]. More importantly, the notching or gridding of the antenna surface only helps to reduce the metallic mass the antenna. However, the EI sensor is sensitive to both the metallic mass and the conductivity of the antenna material, and the choice of textile material as antenna radiators promises to alleviate this undesired cluttering problem by offering reduced metallic mass and conductivity at the same time. To the best of our knowledge, this is the first application of the usage of textile material in a wideband GPR antenna for a dual-sensor subsurface detection system to remove the EI clutter caused by the metallic mass and the conductivity of the antenna material. The outline of the paper is as follows. The properties of the conductive textile material used for the GPR antenna are presented in Section II-A. The dual-sensor search head (DSSH) employed for the antenna element is described in Section II-B and the performance of the WACTR in the DSSH is presented in Section II-C. Subsequently, the ground effects on the antenna performance and the removal of the EI clutter are discussed in Sections II-D and III, respectively. Finally, a detailed discussion of target detection using the proposed antenna structure in the dual-sensor subsurface detection system is given in Section IV. II. THE DESIGN OF WACTR FOR THE PULSED DUAL-SENSOR SUBSURFACE DETECTION SYSTEM A. The Properties of the Textile Material The first step in designing a wideband antenna with conductive textile radiators (WACTR) is the correct selection of textile material. A good conductive textile material should have a low , and stable electrical sheet resistance3 which is less than and this resistance must be homogeneous over the antenna surface [5]. In our application, the fabric does not need to be flexible because it is placed on an FR-4 substrate with a solid planar surface. There are a number of conductive textiles commercially available in the market. We have preferred conductive polyester woven taffeta manufactured by TECKNIT Europe Ltd as the conductive textile material for the antenna radiators and cavity walls of the dual-sensor search head (DSSH). The microscopic optical view of the conductive textile material with magnification is given in Fig. 1. As seen, the conductive textile is made of a dense woven material where there is almost no empty space between two neighboring meshes. The polyester . The total metallic mass of fibers are plated with the antenna made of the described conductive textile material is much lower than the metallic mass of the one made of a classical Cu material due to its rather thin metallic plating. The sheet resistance specified by the manufacturer is typically and the sheet thickness is , and thus the conductivity of the textile material is , which is approximately 300 times smaller than the conductivity of . The effect of the electrical conductivity 3The sheet resistance is a measure of electrical resistivity of thin materials with a uniform thickness [33]Rs =t, where  is conductivity (in S/m) and t is sheet thickness (in m).

=1

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2 magnification) of the con-

Fig. 1. The microscopic optical view (with 100 ductive textile material.

of textile material on antenna gain and field pattern has been investigated by other researchers [12], [23], [34]. In [12], [23], textile antennas made of a conductive fabric and solid copper were compared experimentally and no significant reduction in antenna performance was observed. In [34], S-parameters and field patterns of several textile antennas with different conductivity values were simulated. It was shown that a conductivity is convenient for reasonable value of not less than gain and field pattern.4 We also simulated the antenna gain for different conductivity values. Comparing the simulated antenna gains5 of an elliptical planar WACTR and a solid Cu antenna of the same size, the maximum difference in gain was found to be 0.6 dBi, which is not significant. These results allow us to use this low loss and low metallic mass textile material instead of solid metallic material (Cu) in the design of the proposed GPR antenna without reducing the performance of the antenna. The given properties of the conductive textile material above offer a reduction in the metallic mass and the conductivity, which is necessary for removing the EI clutter from the dual-sensor subsurface detection system (this topic will be discussed in Section III in detail). B. The Dual-Sensor Search Head (DSSH) Having discussed the properties of the textile material, we now present a description of the dual-sensor search head (DSSH), which contains the wideband GPR antenna with conductive textile radiator (WACTR). Note that the GPR sensor requires a wideband antenna to send a short pulse signal to the underground target and the antenna has to provide low frequency components as well as high frequency ones for better penetration into the ground. In this study, a linearly polarized elliptical planar dipole antenna on an FR-4 substrate is chosen as the wideband antenna element for the GPR sensor. This antenna structure is very wellknown in the literature [31], [35]–[37]. So, we will not give a full description of the antenna element; instead some necessary tips for the design will be given and, in particular, the performance of the antenna in DHSS will be described in the next section. If more details about the planar elliptical dipole antenna are required, readers can refer to the references given above. The planar structure of the antenna is crucial in this application because the antenna is required to have a negligible height 4Our

textile material meets this criterion.

5The

graphical results of the simulations are not given here due to lack of

space.

in the DSSH as depicted in Fig. 2. As shown in Fig. 2(a), the GPR antenna with conductive textile radiators consists of a receiver (Rx) antenna and a transmitter (Tx) antenna, which are completely identical. The distance between the Tx and Rx antennas is 145 mm, which is measured from the centers of the two antennas. The coils of the EI sensor are embedded into the frame of the DSSH. Note that the plastic bottom cover is not shown in the figure, and the details of the EI sensor are not presented here because it is out of the scope of this article. A detailed sketch of the individual parts of the antenna with dimensions (in mm) is shown in Fig. 2(b). The elliptical radiators of the antennas are made of conductive textiles whose properties were explained in the previous section. The dielectric substrate of the antenna is an FR-4 with 1.6 mm thickness. The axial ratio, which is the ratio of major to minor axes of an ellipse, is chosen as 1.5. This optimum value was reported in [35] as giving the best antenna matching and gain. The conductive textile radiators are attached to the substrate with two-sided Scotch tape. The outer case of the DSSH is made of plastic. Each WACTR is placed on a cavity. The cavity is constructed from the same conductive textile material, which is also attached using two-sided Scotch tape to the inner wall of the DSSH plastic case. The conductive textile cavity is connected to the ground electrically by a standard electric cable. An LS-24 [38] absorber layer produced by Emerson & Cuming is within the cavity and underneath the patches. The cavity along with the absorber in the DSSH has several important functions. The first one is to decrease the back-lobe level of the antenna, which may cause false detections from the upper hemisphere of the DSSH. The second one is to reduce the coupling between the receiver and the transmitter antennas, which is a requirement for a GPR system. Another important function is to protect the operator and the RF circuitry of the system from the high power of the short pulse.6 The absorber layer additionally reduces the ringing effect occurring between the cavity walls and the antenna. The details of the antenna feed can also be seen in Fig. 2(a). We remark that the feeding structure of the antenna employed here is new. The textile material of the antenna is electrically connected to the feeding using a bolt, a nut, and a washer (see front view in Fig. 2(a). The bolt and nut are attached to a stripline via an intermediate conductor strip (see back view in Fig. 2(a). This stripline connects coaxial cable to the intermediate conductor strip. The advantage of this feeding type is the rigidity of the connection between the textile material and the coaxial cable. It is a more solid solution than soldering or adhering coaxial cable directly to the textile material. The pressure between washer and textile material is fixed at the same level by means of a torque limited wrench to tighten up the bolt. Moreover, an electrical connection through a wider surface is supplied using the washer. Also note that the bolts, nuts, and washers used here are made from inox, which is a specific stainless steel alloy composed of carbon steel and chromium. C. The Performance of the WACTR in the DSSH To give the reader some key points of the GPR antenna design, we now assess the performance of the WACTR located in 6The device was tested in the operational mode according to EMC standard rules [39] and it passed successfully.

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0



Fig. 4. The E-plane (y z plane) measured ( ) and simulated (— —) normalized Ez field patterns of WACTR (a) in free space and (b) in the DSSH at 1 GHz.

Fig. 2. (a) Photographic view of the WACTR (Tx and Rx), which are connected to the DSSH (b) a detailed sketch of the individual parts of the antenna with dimensions (in mm).

Fig. 3. The measured (——) and simulated (— — —) SWR curves of the WACTR in free space and in the DDSH.

the dual-sensor search head (DSSH) in the light of the measured and simulated results. The corresponding free-space results are also given for comparison to demonstrate the improvement of

antenna performance in the DSSH, that is, better matching and gain, the back-lobe reduction in antenna pattern, and the reduction of the coupling between Tx and Rx antennas in DSSH, which are the crucial points for a GPR antenna of a subsurface sensing system. The measured and simulated7 SWR curves of the proposed WACTR in free space and in the DSSH are displayed in Fig. 3. For the antenna in free-space, the operating frequency occurs between 928 MHz and 4750 MHz. band8 On the other hand, the lower operating frequency of the antenna in the DSSH is 645 MHz, which is 283 MHz lower than that of the antenna in free-space. Comparing SWR characteristics of the antennas in free space and in the DSSH, the bandwidth ratios9 are 5.12 and 7.55, respectively. That is, the bandwidth increases as the lower edge of the operating band shifts to the lower frequencies. This is because the antenna matching is improved by placing the antenna in the absorber and cavity. To demonstrate the back-lobe reduction of the antenna in the plane) measured and simulated norDSSH, the E-plane ( field patterns of the WACTR in free space and in the malized DSSH at 1 GHz are shown in Fig. 4. It can be seen that there is a good match between the simulated and measured patterns. The antenna has a wide pattern giving 85 and 98 half power beam widths for the antenna in free space and in the DSSH, respectively. Thus the pattern of the antenna in the DSSH widens by approximately 15% compared to the antenna in free space. As for expected, the back-lobe of the antenna decreases to the antenna placed within the absorber and the cavity as compared to the free-space case. This result agrees with the role of the absorber and cavity, which were explained in the previous section. plane) patterns of the antenna in the DSSH The H-plane ( were also measured (not given in the text due to lack of space) and it is observed that the H-plane patterns are very similar to the E-plane patterns (i.e., in Fig. 4(b) but a little narrower than E-plane patterns due to the geometry of the antenna (a larger 7All simulations presented in the paper were performed using CST Microwave Studio. 8It is noted that the electrical length of the minor axis of the planar elliptical dipole antenna is critical to achieve such a bandwidth. Schantz reported in [35]–[37] that the minor axis is predicted to be approximately 0:14, where  is the wavelength of the lower operating frequency of the elliptical dipole. In our application, the minor axis length is almost 0:15, which is very close to that prediction. 9This

is the ratio of upper to lower operating frequencies.

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Fig. 5. The measured (——) and simulated (— —) gains of the WACTR in free space and in the DSSH.

aperture in the H-plane with reference to the dimensions of the antenna given in Fig. 2(b). In addition, the gain of the WACTR was measured using the three-antennas method.10 The measured and simulated gains of the antenna in free space and in the DSSH are displayed in Fig. 5. The measured maximum gains are 3.54 dBi and 4.32 dBi, respectively. As can be seen from the results, the WACTR in the search head has more broadband as well as higher gain characteristics as compared to the antenna in free space. The simulated current distributions can be useful to explain the radiation mechanism and the behavior of WACTR in the DSSH, which are displayed in Fig. 6(b) for several frequencies (on the gray magnitude scale and in vector form). The corresponding simulated current distributions in free space are also given in Fig. 6(a) for comparison. It can be observed that the current magnitude level is generally higher along the edges of the radiators and at the feed points of the antenna. For the frequencies below 3 GHz, the current is homogeneously distributed over the conductive textile patches; all parts of the textile radiators contribute radiation intensely. The current also has the highest level for the lower frequency range, where the gain has a higher profile as well (see the gain curves in Fig. 5 for comparison). However, the current is not homogeneously distributed over the antenna radiators and is low-level above 3 GHz. Some parts of the radiators hardly contribute to the radiation at all in the upper frequency band, which corresponds to the low-gain region (Fig. 5). It can be observed from Fig. 6(b) (for WACTR in the DSSH) that the outer part of the radiators (i.e., the cavity walls made of the same conductive textile material) has a current distribution as well. The current arrows can also be seen at the side walls of the cavity. The more the frequency increases, the more the magnitude of the current on the cavity decreases. This is because the absorber (invisible in Fig. 6(b) is not thick enough to absorb all electromagnetic energy at the lower frequencies. Moreover, the lowest operation frequency of the LS-24 absorber recommended 10An EMCO 3115 double ridged horn antenna was used as the standard antenna, of which gain is known and the operational frequency range is between 1 GHz and 18 GHz.

Fig. 6. The simulated current distribution (in magnitude and direction) of the WACTR (a) in free space and (b) in the DSSH for several frequencies.

Fig. 7. The measured (——) and simulated (— —) coupling between Rx and Tx WACTRs in free space and in the DSSH for the coupling distance of 145 mm.

by the manufacturer is 1 GHz [38]. The magnitude of the current on the cavity is lower than that on the antenna radiators due to the absorber layer between the antenna and the cavity. When the current distributions over the antennas at a fixed frequency are compared for both cases (in free space and in the DSSH, Figs. 6(a) and 6(b) respectively), it is observed that the existence of the cavity along with the absorber does not significantly affect the current distribution on the patches. The coupling between Tx and Rx antennas is another important issue for the GPR system. High coupling causes difficulty in sensing the targets and thus results in clutter. The measured and simulated couplings between Rx and Tx WACTRs (in free space and in the DSSH) are given in Fig. 7, where the coupling distance (center-to-center distance between Tx and Rx antennas) is 145 mm. As seen in the figure, the coupling is decreased by

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Fig. 8. The measured (a) return loss, (b) coupling between Tx and Rx WACTRs, and (c) simulated gain in free space and 5 cm above the soil.

placing the WACTRs in the DSSH, and as a result, a low coupling between Tx and Rx WACTRs is achieved. This is a desired decrease in coupling and varies between 5 dB and 30 dB with respect to the frequency depending on antenna gain. D. The Effects of the Ground on Antenna Performance For a hand-held subsurface sensing system, when the operator sweeps the dual-sensor search head (DSSH) in the vicinity of the ground, the GPR antenna is forced to operate in the presence of the ground. So, the possible effect of the ground on the antenna performance becomes critical and needs to be investigated as well. For this purpose, the corresponding measurements were performed in the test site. The electromagnetic properties of the , soil in which the target was buried are presumed to be (on average over a 0.6–2.2 GHz band), and [40]. In this group of measurements, return loss and coupling between Tx and Rx WACTRs located in the DSSH was measured while the search head was 5 cm above the soil surface. The measurements when there is no soil close to the DSSH are also included here for comparison. It can be observed from the results in Figs. 8(a) and 8(b) that the return loss and the coupling between Tx and Rx WACTRs both increase to some degree when the antenna approaches the soil. The reason for this increase in return loss and coupling can be explained by the reflection of electromagnetic waves from the soil. Hence, the proximity of the ground to the antenna makes return loss and coupling parameters worse. It is not possible to measure antenna patterns and gains when the soil is close to the antenna because the antenna beam faces to the ground. Thus, only simulations for these antenna parameters were performed and the simulated gain curves of the antenna in free space and 5 cm above the soil are given in Fig. 8(c). Although the measured return loss and coupling parameters get worse, the antenna gain improves remarkably in proximity to the ground, achieving almost 3 dB enhancement compared to the average over the band of interest. III. THE PREVENTION OF THE ELECTROMAGNETIC INDUCTION (EI) CLUTTER As mentioned in the Introduction, the main aim of this study is to design an antenna for the GPR sensor of a pulsed dual-sensor subsurface detection system where the antenna does not create clutter for the electromagnetic induction (EI) sensor. This is

Fig. 9. The solid Cu (up) and gridded Cu (down) GPR antennas which are the same size as the WACTR given in Fig. 2(a). The antennas are shown out of the DSSH.

Fig. 10. The electromagnetic induction (EI) sensor detection signals of Target #1 (see Table I) when (a) the WACTR (b) the gridded Cu of the same size, and (c) solid Cu antennas are used as the GPR antenna, respectively.

achieved by choosing conductive textile as the antenna material with a reduced metallic mass and conductivity. To demonstrate the capability of the proposed antenna in avoiding clutter, the EI detection signals of a target will be compared when the WACTR (Fig. 2(a)) and solid Cu antenna of the same size (Fig. 9, upper antenna) are used as the GPR antenna, respectively. The target used in the measurement involves a small metallic content and is named Target #1 (see Table I). The EI detection signals of the target when each antenna is connected to the system are displayed in Figs. 10(a) and 10(c). In the graphs, the vertical axes represent the detection signal strength and the horizontal axes represent the movement direction index; both are in arbitrary units. The length and the total area of the dark region in the

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TABLE I THE REPRESENTATIVE TARGETS AND BURIAL DEPTHS

Fig. 11. The pulse signal delivered to the WACTR.

the WACTR as the GPR antenna of the dual-sensor system instead of the solid or gridded Cu antenna. The inner surface of the cavity walls is also covered with the same conductive textile material for the same reason. That is, the textile-made cavity does not generate clutter for the EI sensor as compared to the case of the Cu-made cavity. IV. THE TARGET DETECTION graphs depend on the metallic mass, conductivity, shape, burial depth, and orientation of the target. It can be seen from Fig. 10(a) that Target #1 is rather easily detected by the EI sensor without parasitic coupling of the GPR antenna when WACTR, which is made of conductive textile radiators, is used as the GPR antenna. However, the EI sensor detection signal level increases tremendously when the solid Cu antenna is connected to the GPR sensor (Fig. 10(c)). The detection signal of Target #1 is shadowed and it is not possible to detect the target by means of the solid Cu antenna. Actually, this is a false detection signal generated entirely because of the highly metallic mass and conductivity of the Cu material of the antenna. Does gridding of the metallic antenna surface help to overcome this cluttering problem, providing only a reduced metallic mass? For this aim, a gridded Cu antenna (Fig. 9, lower antenna) was fabricated. In this antenna structure, approximately 40% of the metallic content was removed from its surface. The EI sensor detection signal of Target #1 was also obtained after the gridded antenna was connected to the system as the GPR antenna and it is shown in Fig. 10(b). This measurement shows that the clutter signal level is reduced to some degree; however the reduction is still insufficient to obtain a clear detection signal like the one obtained from WACTR as shown in Fig. 10(a). Moreover, the gain of the gridded antenna is reduced by around 2 dBi compared with the solid one.11 If one attempts to reduce the surface area of the gridded Cu antenna further, the gain will also decrease and the antenna will become a very low-gain antenna, showing that this is an impractical method. In this manner, the gridding process of a solid Cu antenna surface is not as efficient as the use of textile radiators in antenna and does not help to overcome the EI cluttering problem. As a result, the reduced metallic mass along with the reduced conductivity of the conductive textile material allows us to use

We now present some target detection examples while the proposed WACTR are located in the search head along with the coils of the EI sensor, which is employed for subsurface detection. In Fig. 11 the pulse signal delivered to the WACTR is depicted. The detection results of representative targets are given in Fig. 12, where the raw GPR data, the processed GPR images (buried object spatial signatures),12 and the detection signals (in a.u.) of the GPR and the EI sensors are shown in the first to fourth rows, respectively. The horizontal and vertical axes of the GPR images are both in arbitrary units and represent the movement direction index and the depth index, respectively. The physical properties and burial depths of the targets are given in Table I. The targets, which are surrogates of real land-mines and triggers, were buried in the test field composed , , of soil with electromagnetic parameters of [40], which are the same values as those used and in Section III. It can be observed from Figs. 12(a)–12(c) that the GPR spatial signatures of the buried objects were made visible via the proposed WACTR, even for Target #4, which is an entirely dielectric cylinder with a large burial depth (20 cm). Note that all of those targets are detected by the EI sensor as well due to their metallic content. As discussed in the Introduction, for some kinds of targets, one of the sensors may not give a response while the other one may. Two examples are given to demonstrate this situation in Figs. 12(d) and 12(e), for Targets #4 and #5, respectively. No response is obtained by the EI sensor from Target #4, which is entirely dielectric, due to its non-metallic structure, while the GPR sensor gives a detection signal and image for this target. Similarly, for Target #5, which is a very thin metallic triggering mechanism, the GPR sensor does not give a detection signal or

11The decrease in the gain of the WACTR was only 0.6 dBi, which is not significant.

12To obtain the buried object spatial signatures, the background subtraction method [30] is used.

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Fig. 12. The raw GPR images (first row), the processed GPR images (second row), the GPR sensor detection signals (third row), and the EI sensor detection signals (fourth row) of the representative Targets (a) # 1 (b) # 2 (c) # 3 (d) # 4 (e) # 5.

image because of the target’s orientation, even though it has a metallic structure; on the other hand, the EI sensor obtains a detection signal from the same target. Those two examples demonstrate that the two sensors operate as complementary sensors. The GPR images and the detection signals of the targets given in the second and the third rows of Fig. 12 show that the proposed WACTRs exhibit a good performance in the GPR sensor. Moreover, as discussed previously, the EI sensor works properly and the GPR antenna does not mask the EI sensor’s detection signal by means of the conductive textile material used in the GPR antenna (and also as the covering material of the inner surface of the cavity), which has low loss and low metallic mass. V. SUMMARY AND CONCLUSIONS A wideband antenna with conductive textile radiators (WACTR) has been proposed for a dual-sensor subsurface detection system composed of pulsed GPR and EI sensors. The most challenging problem of a combined GPR and EI sensor subsurface detection system is the EI clutter arising from the metallic mass and conductivity of the GPR antenna. In our study, this clutter is overcome by replacing conventional solid (or gridded) Cu material of the antenna with a conductive textile

material without reducing the performance of the antenna significantly. The measurements demonstrated that the EI sensor signal is not masked by the conductive textile material of the antenna due to its reduced mass as well as reduced conductivity. The antenna used in the dual-sensor search head (DSSH) is a linearly polarized planar elliptical dipole with conductive textile patches. In the DSSH, the antenna is inserted in a cavity also made from the same conductive textile material to prevent the clutter. There is an absorber layer inserted between the antenna and the cavity. The cavity along with the absorber reduces the back-lobe level of the antenna, thus preventing possible false detections from the upper hemisphere and protecting the operator and RF circuitry from a possible high power effect of the short pulse. The absorber layer is additionally responsible for alleviating the ringing effect. Some antenna characteristics were measured and compared with the simulated data. The corresponding measurements and simulations show that the antenna performance, that is, increased bandwidth, gain, and operational frequency band and decreased coupling between the Rx and Tx antennas, is improved by inserting the antenna in the DSSH as compared to the case in free space.

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The introduced feeding style of the antenna is also new. A bolt, a nut, and a washer made from inox were used in the feeding structure of the antenna, which can easily be implemented by means of a torque limited wrench. Such a feeding structure is mechanically stronger than the classical method of soldering or adhering cables directly to the textile material of the antenna, and it provides a wider excitation surface. Considering the simulated current distributions of the antenna, the current magnitude level is higher along the edges and at the feeding points of the antenna. Also, the current is homogeneously distributed for the frequencies lower than 3 GHz, which corresponds to the high-gain region. In addition, the current distributions on the textile cavity show that the current magnitude level increases with decreasing frequency due to the rather poor performance of the absorber layer at low frequencies. The existence of the absorber and cavity, on the other hand, does not significantly change the current distribution on the radiators of the antenna. The performance of the WACTR was also tested in a real test site and satisfactory results were obtained. In those tests, representative targets buried in soil could be detected even in nonmetallic and deep burial cases. It was also demonstrated that the EI and GPR sensors are complementary and using them together provides a higher discrimination and decision performance. It was observed that the proximity of the antenna to the ground changes the antenna parameters to some degree. While the return loss and coupling worsen, antenna gain is improved by the presence of the soil. As a result, the proposed WACTRs have been applied successfully to the pulsed dual-sensor subsurface detection system by overcoming the EI clutter. ACKNOWLEDGMENT The authors would like to thank O. Kaya for his assistance in the antenna measurements, our technicians N. Pelitçi, N. Kavakli, M. Çaliskan, and O. Akgun for their assistance in the fabrication of the antennas, and O. Baykan for his assistance in the field measurements of the dual-sensor search head. Also, the authors would like to thank Dr. Y. E. Erdemli from Kocaeli University, Turkey as well as the anonymous reviewers for their invaluable comments during the review process of the paper REFERENCES [1] M. Catrysse et al., “Towards the integration of textile sensors in a wireless monitoring suit,” Sens. Act. A—Phys., vol. 114, no. 2–3, pp. 302–311, Sep. 2004. [2] J. Coosemans, B. Hermans, and R. Puers, “Integrating wireless ECG monitoring in textiles,” Sens. Act. A—Phys., vol. 130, no. Special Issue, pp. 48–53, Aug. 2006. [3] G. Lorigia, N. Taccini, M. Pacelli, and R. Paradiso, “Flat knitted sensors for respiration monitoring,” IEEE Ind. Mag., vol. 1, no. 3, pp. 4–7, 2007. [4] M. Klemm and G. Troester, “Textile UWB antennas for wireless body area networks,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pt. 1, pp. 3192–3197, Nov. 2006. [5] I. Locher, M. Klemm, T. Kirstein, and G. Troster, “Design and characterization of purely textile patch antennas,” IEEE Trans. Adv. Pckg., vol. 29, no. 4, pp. 777–788, Nov. 2006. [6] A. Tronquo, H. Rogier, C. Hertleer, and L. V. Langenhove, “Robust planar textile antenna for wireless body LANs operating in 2.45 GHz ISM band,” Electron. Lett., vol. 42, no. 3, pp. 142–143, Feb. 2006.

[7] D. L. Paul, M. Klemm, C. J. Railton, and J. P. McGeehan, “Textile broadband e-patch antenna at ISM band,” in Proc. IET Seminar on Antennas Propag. for Body-Centric Wireless Communications, Apr. 2007, pp. 38–43. [8] Z. Shaozhen and R. Langley, “Dual-band wearable textile antenna on an EBG substrate,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pt. Part 1, pp. 926–935, Apr. 2009. [9] P. Salonen, K. Jaehoon, and Y. Rahmat-Samii, “Dual-band E-shaped patch wearable textile antenna,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 3–8, 2005, vol. 1A, pp. 466–469. [10] L. Vallozzi, H. Rogier, and C. Hertleer, “Dual polarized textile patch antenna for integration into protective garments,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 440–443, 2008. [11] C. Hertleer, A. Tronquo, H. Rogier, L. Vallozzi, and L. V. Langenhove, “Aperture-coupled patch antenna for integration into wearable textile systems,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 392–395, 2007. [12] T. F. Kennedy, P. W. Fink, A. W. Chu, and G. F. Studor, “Potential space applications for body-centric wireless and E-textile antennas,” in Proc. IET Seminar on Antennas Propag. for Body-Centric Wireless Communications, Apr. 2007, pp. 77–83. [13] T. F. Kennedy, P. W. Fink, A. W. Chu, N. J. Champagne, G. Y. Lin, and M. A. Khayat, “Body-worn E-textile antennas: the good, the low-mass, and the conformal,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pt. Part 1, pp. 910–918, Apr. 2009. [14] P. Salonen and Y. Rahmat-Samii, “Textile antennas: effects of antenna bending on input matching and impedance bandwidth,” IEEE Aerosp. Electron. Syst. Mag., vol. 22, no. 12, pp. 18–22, Dec. 2007. [15] Q. Bai and R. Langley, “Crumpled textile antennas,” Electron. Lett., vol. 45, no. 9, pp. 436–438, Apr. 2009. [16] Q. Bai and R. Langley, “Crumpled integrated AMC antenna,” Electron. Lett., vol. 45, no. 13, pp. 662–663, Jun. 2009. [17] C. Hertleer, H. Rogier, and L. V. Langenhove, “The effect of moisture on the performance of textile antennas,” in Proc. 2nd IET Seminar on Antennas Propag. for Body-Centric Wireless Communications, Apr. 20, 2009, p. 1. [18] C. Hertleer et al., “A textile antenna for off-body communication integrated into protective clothing for firefighters,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 919–925, Apr. 2009. [19] P. Salonen, Y. Rahmat-Samii, and M. Kivikoski, “Wearable antennas in the vicinity of human body,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jun. 20–25, 2004, vol. 1, pp. 467–470. [20] D. Cottet, J. Grzyb, T. Kirstein, and G. Troster, “Electrical characterization of textile transmission lines,” IEEE Trans. Adv. Pckg., vol. 26, no. 2, pp. 182–190, May 2003. [21] R. K. Shaw, B. R. Long, D. H. Werner, and A. Gavrin, “The characterization of conductive textile materials intended for radio frequency applications,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 28–40, Jun. 2007. [22] F. Declercq, H. Rogier, and C. Hertleer, “Permittivity and loss tangent characterization for garment antennas based on a new matrix-pencil two-line method,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pt. 2, pp. 2548–2554, Aug. 2008. [23] P. Salonen, Y. Rahmat-Samii, H. Hurme, and M. Kivikoski, “Effect of conductive material on wearable antenna performance: A case study of WLAN antennas,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jun. 20–25, 2004, vol. 1, pp. 455–458. [24] M. Sezgin, “Development of dual sensor hand-held detector,” in Proc. SPIE Detection and Sensing of Mines, Explosive Objects, and Obscured Targets XV, Orlando, Apr. 5–9, 2010, vol. 7664, p. 76641Q. [25] M. Sezgin, G. Kaplan, S. M. Deniz, Y. Bahadirlar, and O. Icoglu, “Identification of metallic objects with various sizes and burial depths,” Proc. SPIE Detection and Sensing of Mines, Explosive Objects, and Obscured Targets XIV, vol. 7303, no. 730319, 2009. [26] M. Sezgin et al., “Buried metallic object identification by EMI sensor,” in Proc. SPIE Conf. on Detection and Remediation Technologies for Mines and Minelike Targets XII, Orlando, Apr. 11–12, 2007, vol. 6553, Art. no.65530C. [27] M. Sezgin et al., “Real time detection of buried objects by means of GPR,” in Proc. SPIE Conf. on Detection and Remediation Technologies for Mines and Minelike Targets IX, Parts 1–2, Orlando, Apr. 12–16, 2004, vol. 5415, pp. 447–455. [28] K. C. Ho, L. M. Collins, L. G. Huettel, and P. D. Gader, “Discrimination mode processing for EMI and GPR sensors for hand-held land mine detection,” IEEE Trans. Geosci. Remote Sensing, vol. 42, no. 1, pp. 249–263, 2004.

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[29] A. H. Gunatilaka and B. A. Baertlein, “Feature-level and decision-level fusion of noncoincidently sampled sensors for land mine detection,” IEEE Trans. Pattern Analy. Machine Intell., vol. 23, no. 6, pp. 577–589, 2001. [30] A. S. Turk, “Ultra-UWB Vivaldi antenna design for dualsensor adaptive ground-penetrating impulse radar,” Microw. Opt. Technol. Lett., vol. 48, no. 5, pp. 834–839, May 2006. [31] H. G. Schantz, G. Wolenec, and E. M. Myszka, “Frequency notched UWB antennas,” in IEEE Conf. on Ultra Wideband Systems and Technologies, VA, 2003, pp. 214–218. [32] T. W. Turpin and R. Baktur, “Meshed patch antennas integrated on solar cells,” IEEE Ant. Prop. Lett., vol. 8, pp. 693–696, 2009. [33] [Online]. Available: http://en.wikipedia.org/wiki/Sheet_resistance [34] E. Yilmaz, D. P. Kasilingam, and B. M. Notaros, “Performance analysis of wearable microstrip antennas with low-conductivity materials,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 5–11, 2008, pp. 1–4. [35] H. G. Schantz, “Planar elliptical element ultra-UWB dipole antennas,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jun. 16–21, 2002, vol. 3, pp. 44–47. [36] H. G. Schantz, “Introduction to ultra-UWB antennas,” in Proc. IEEE Conf. on Ultra-UWB Systems and Technologies, Nov. 16–19, 2003, pp. 1–9. [37] H. Schantz, Ultrawideband Antennas. Norwood, MA: Artech House, 2005. [38] [Online]. Available: http://www.eccosorb.com/america/english/prod uct/44/eccosorb-ls [39] [Online]. Available: http://www.emccompliance.com/new_ page_7.htm [40] H. Nazli, E. Bicak, and M. Sezgin, “Experimental investigation of different soil types for buried object imaging using impulse GPR,” in Proc. XII Int. Conf. on Ground Penetrating Radar, Lecce, Italy, Jun. 2010, pp. 21–25.

A. Oral Salman (S’06–A’07–M’08) was born in Ankara, Turkey, in 1969. He received the B.Sc. and M.Sc. degrees in physics engineering from Hacettepe University, Ankara, Turkey, in 1993 and 1996, respectively, and the Ph.D. degree in electronics and communication engineering from Kocaeli University in Kocaeli, Turkey, in 2006. In February, 2011, he obtained the Associate Professorship title in the area of electrical and electronics. He has been working as a Senior Researcher at Marmara Research Center, UEKAE (National Research Institute of Electronics and Cryptology), and BILGEM, all of which belong to TUBITAK in Kocaeli, Turkey, since 1999. He has been engaged

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in many military, government, and industrial projects as Researcher and Co-Project Leader. Before joining TUBITAK, he worked as a Research and Teaching Assistant in Hacettepe University from 1994 to 1998. He visited ElectroScience Laboratory, in Columbus, OH, between 2002 and 2003 as a visiting scholar. His research areas of interest include antenna theory, design, and measurements, subsurface detection, microwave diffraction tomography and imaging, electromagnetic theory, wave propagation, electromagnetic properties of materials, RF coil and resonator design, and magnetic resonance. He is the main author of a chapter entitled “Diffraction Multi-View Tomography Method in Microwave and Millimeter Wave Bands” in the book Subsurface Sensing (Wiley, 2011). He is also the author of 14 papers printed in refereed journals and 21 oral presentations at national and international conferences. Dr. Salman obtained the 2000 and 2005 MRC awards for his success in two projects and “The Best Young Scientist Paper Award” at the MSMW’01 (The Fourth International Kharkov Symposium—Physics and Engineering Millimeter and Sub-millimeter Waves) conference, in 2001

Emrullah Býçak was born in Istanbul, Turkey, in 1982. He received the B.S. degree in electronics engineering from Gebze Institute of Technology, Kocaeli, Turkey, in 2005, where he is working toward the M.S. degree. He currently works as a Researcher at TUBITAK BILGEM in Kocaeli, Turkey. His research interests include RF and microwave antenna design, HF radar systems, numerical methods in electromagnetic wave scattering, GPR, and electromagnetic launchers

Mehmet Sezgin was born in Bursa, Turkey, in 1965. He received B.Sc., M.Sc., and Ph.D. degrees from Istanbul Technical University, Istanbul, Turkey, in 1986, 1990, and 2002, respectively. He worked at Istanbul Technical University between 1987 and 1990 as a Research Assistant. He has been working as Chief Researcher at the TUBITAK Marmara Research Center, UEKAE (National Research Institute of Electronics and Cryptology), and BILGEM since 1991. He has completed numerous industrial, governmental, and military projects as researcher and project leader. His research area is signal processing, image processing, and multi-sensor system design. He is also the author of several highly cited papers printed in SCI journals and numerous scientific conference papers

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An Unconditionally Stable Radial Point Interpolation Meshless Method With Laguerre Polynomials Xiaojie Chen, Zhizhang (David) Chen, Fellow, IEEE, Yiqiang Yu, Member, IEEE, and Donglin Su, Member, IEEE

Abstract—The time-domain radial point interpolation (RPI) meshless method has the advantage of conformal modeling of a structure with non-uniformly distributed nodes; however, the time step used is still restricted by the stability condition or distances between the nodes. The time step has to be small and computational time can be long when the distances between nodes are small. In this paper, an unconditionally stable scheme using the weighted Laguerre Polynomials is introduced into the (RPI) meshless method. The result is an always-stable RPI meshless method; it retains the advantages of both the node-based meshless method and the unconditionally stable scheme with the weighted Laguerre Polynomials. The unconditional stability, numerical accuracy and efficiency of the proposed method are verified and confirmed through numerical examples. In the case of the numerical examples computed, CPU time saving can be more than 99% in comparisons with the CPU time used with the conventional conditionally stable meshless method. Index Terms—Laguerre polynomials, meshless method, non-uniform nodal distribution, radial point interpolation (RPI), time-domain method, unconditionally stable scheme.

I. INTRODUCTION HE local radial point interpolation (RPI) meshless method has recently emerged as an attractive alternative numerical approach to solving high-frequency electromagnetic structures in the time domain [1]–[3]. Unlike conventional grid-based numerical methods, it uses a set of scattered nodes (instead of a numerical grid) to represent a spatial solution domain of interest and to formulates numerical equations around these nodes. As a result, conformal modeling of arbitrary boundaries and removal/ addition of nodes to a part of a solution domain become possible. In other words, the meshless method is flexible and powerful for transient electromagnetic modeling and simulation. Another advantage of the meshless method is that nodes can be placed uniformly or non-uniformly. In a region that has fast changes of fields, more nodes can be placed in order to capture these changes and to improve modeling accuracy. In a region

T

Manuscript received August 05, 2010; revised November 18, 2010; accepted December 13, 2010. Date of publication August 08, 2011; date of current version October 05, 2011. This work was supported in part by the Natural Science Foundation of China and Natural Science and Engineering Research Council of Canada through its Discovery Grant program. X. Chen and D. Su are with the School of Electronic and Information Engineering, Beihang University, Beijing, China (e-mail: [email protected]). Z. Chen is with the Electrical Engineering Department, Dalhousie University, Halifax, Nova Scotia B3J 2X4, Canada (e-mail: [email protected]). Y. Yu is with the Electrical Engineering Department, Dalhousie University, Halifax, Nova Scotia B3J 2X4, Canada and also with East China Jiaotong University, Jiangxi, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163769

that has slow changes of fields, a fewer nodes can be deployed in order to reduce computational expenditures required. However, like many other time-domain numerical methods, the time-domain RPI meshless method has the issue of numerical stability. The maximum allowable time step to guarantee numerical stability is limited [3]: the maximum time step is restricted by the distances between nodes. The smaller the distances are, the smaller the time step is. This may lead to a very small time step and consequently a long simulation time in the case of a highly non-uniformly distributed node placement that has a very small distance between nodes. Therefore, it is highly desirable to remove such a stability constraint on the time step and to develop an unconditionally stable meshless method where time step can be chosen independently of the distances between nodes. Towards this end, an unconditionally stable RPI meshless method was developed [4]; it was derived and based on the incorporation of the alternative-direction-implicit (ADI) time-step marching scheme into the conventional node-based meshless method. On the other hand, another unconditionally stable scheme were intelligently developed [5], [6] for the conventional gridbased finite-difference time-domain (FDTD) method; there the weighted decaying Laguerre polynomials are used as temporal basis and testing functions to expand unknown field quantities and to minimize residual errors in the time domain; then a procedure of the Method of Moments or Method of Weighted Residuals as described in [6] is applied to find the expansion coefficients for the approximate solutions. Because the weighted La, field quantities guerre polynomials tend to zero as time expanded in terms of these Laguerre polynomials will converge to zero as the time progresses. The obtained solutions are therefore always stable. The method has been shown to be efficient due to the fact that its solutions can be computed in a recursive march-in-order manner [5]. In this paper, we apply the weighted Laguerre polynomials approach to the meshless method and develop a new unconditionally stable meshless method this is not based on the alternating-directional implicit (ADI) scheme but on the incorporation of the weighted Laguerre polynomials. As a result, time step of the meshless method is no longer dependent on distances between nodes. The method then retains not only the high efficiency associated with the march-in-order Laguerre polynomial approach but also the advantages of the meshless method. This paper is organized as follows: formulations of the proposed method are first described; then, numerical examples are presented to verify it effectiveness and efficiency; finally, the conclusions are made in the end.

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CHEN et al.: AN UNCONDITIONALLY STABLE RADIAL POINT INTERPOLATION MESHLESS METHOD WITH LAGUERRE POLYNOMIALS

II. THE PROPOSED UNCONDITIONALLY STABLE RPI MESHLESS FORMULATIONS USING THE WEIGHTED LAGUERRE POLYNOMIALS

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and stand for the basis functions evaluated at the Matrix nodes with the support domain:

The proposed unconditionally stable meshless method is built on the conventional meshless method. Therefore, in the following paragraphs, we will describe the conventional meshless method first and then the modification of the method that leads to the proposed unconditionally stable meshless formulations with the weighted Laguerre polynomials.

.. .

.. .

.. . (6)

.. .

.. .

.. . (7)

A. The Conventional RPI Meshless Method In the conventional RPI meshless method, a field quantity is approximated by the radial point interpolation (RPI) function centered at the nodes within a pre-selected area that is centered at the point of interest . As described in [1], [2], this pre-selected area is called the support domain or influence is approximated by [1]: domain of . That is,

Based on (5), unknowns and can be expressed in terms of the function values at the nodes within the support domain: (8) Substitution of (8) into (2) reads:

(1) (9) is the point of interest at which is to be where is the radial basis function centered at interpolated and node with being located within the support is the monomial basis function which ensure domain of . and are the expansion coefficients, uniqueness of (1), is the number of nodes within the support domain of and represents the order of the monomial basis. Equation (1) can be rewritten in vector form:

is then a vector of where . That is, at a spatial point of interest shape functions can be expressed in terms of the field values at the nodes in the support domain of as follows: (10) Its spatial derivation can then be analytically found as:

(2) (11) where with

,

and .

(3) B. The Proposed Unconditionally Stable RPI Meshless Formulations With the Weighted Laguerre Polynomials A Laguerre polynomials (of order ) is defined as: (4) (12) Equation (2) is applied at all the node positions within the support domain; together with the constraint of the solution uniqueness as described in [1], (2) can then be rewritten in a matrix form:

The polynomials of different order are orthogonal to each : other with respect to the weighting function of (13)

(5) Therefore, an orthogonal set of basis functions can be formed as [5]: where vector value of function

with being the at the th nodes within the support domain.

(14)

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where is a time scaling factor. As , these weighted Laguerre basis functions converge to zero. In this paper, for the sake of simplicity and clarity, the timedomain Maxwell’s equations of TM-to- waves in two dimensions are considered:

is chosen in such a way that the waveform of Upper limit has practically decayed to zero after [5]. source Then the RPI meshless formulations of (10) and (11) are applied to (22)–(24) in the spatial domain and the following equations are obtained:

(15) (16) (26) (17) is reduced to in two dimensions. Here Now the field quantities are expanded in the time domain with the weighted Laguerre polynomials of (14):

(27)

(18) (19) (28) (20) Here , and are the expansion coefficients associated with the basis function of (the weighted Laguerre polynomial of order ); they are functions of spatial positions. Scaling factor is used to increase or decrease the support provided by the above expansion as described in [5]. For any expanded field component function (with being , and ), its first-order temporal derivative with respect to time is [5]:

where , and are the expansion coefficients associated with the weighted Laguerre polynomial of order for , and at the th node in the support domain, respectively. By inserting (27) and (28) into (26), we have the final meshless equations:

(21) By inserting (18)–(21) into (15)–(17), and applying the error minimization procedure in the time domain as described in [6], we can have:

(29) is the total number of electric field nodes in the supwhere port domain of the th magnetic field node, and

(22) (23)

(24) with

(25)

(30) The generation of magnetic field and electric field nodes are detailed in [3], [4]. It is not repeated here due to limitation of space. Equation (29) is applied at each node position and a system of can then be obtained. Once they are obtained, equations for . (29) is solved as the approximate solution of Magnetic fields can then be calculated directly via (23) and (24) once (29) is solved.

CHEN et al.: AN UNCONDITIONALLY STABLE RADIAL POINT INTERPOLATION MESHLESS METHOD WITH LAGUERRE POLYNOMIALS

To model open space, the first-order dispersive absorbing boundary condition [7] is adapted for the proposed meshless , or method. The absorbing boundary is set at , with the terminating equation given by: (31) where is phase velocity and is the unit vector normal to the absorbing boundary. By Inserting (18) and (21) into (31) and utilizing the orthogonal property of the weighted Laguerre polynomials, we can have the following meshless equation of the absorbing or terminating condition: (32)

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can be obtained from (35) and (34) to node . Then , and so can and . In other words, an with efficient recursive process is acquired for solving all the field expansion coefficients by marching in order . is the maximum order (or the number of the Note that weighted Laguerre polynomial functions used for expansion). can be determined by [5], The value of [8], where is the temporal duration to be simulated and is the bandwidth of the signal of interest. However, A recent inshould be determined by applying vestigation [9] shows that , especially for energy analysis and error minimization at a long time duration. can be decomposed To reduce computational time, matrix with the lower-upper (LU) matrix technique right at beginning of the computations. Then it can be stored for the subsequent computations as described in [5]. III. NUMERICAL EXAMPLES

By applying (32) at the magnetic field nodes at the absorbing boundary and inserting (10) into (32), we then have the following equation: (33)

Two examples were chosen to verify the meshless method proposed in this paper, one being a 90-degree H-plane waveguide bend and the other being a parallel waveguide with a PEC slot. In both examples, the modulated Gaussian pulse below is chosen as an exciting electric current source, the same as the one used in [5]:

where (34) Note that if (31) is implemented with the conventional FDTD formulations, the instability will occur. However, with the proposed meshless method, the instability will not happen because of the decaying nature of the weighted Laguerre polynomials. By enforcing (29) and (30) to pass through each electric field nodes within the solution domain and enforcing (33) and (34) to pass through each electric field nodes at the absorbing boundary, we have a final matrix equation:

with and . . The computations were carried out In addition, in a Think Pad X200s notebook computer that has an Intel Duo CPU of 1.86 GHz and a RAM of 1.96 GB. In the two examples, electric field nodes and magnetic field nodes are staggered in space as described in [3]. There are four magnetic field nodes in the support domain of every electric field nodes, and two electric field nodes in the support domain of every magnetic field nodes. The Gaussian function was chosen as the radial basis function, again the same as the one used in [3]:

(35) are determined by , Elements of the sparse matrix and the shape functions. They are independent of order of the . is an one-column weighted Laguerre polynomial array formed by . is an one-column array formed by , and is an one-column array formed by . is obtained by applying (25) at node , while is obtained by applying (30) or (34) at node . Although (35) looks computationally expensive with different order , it can actually be computed efficiently in a , we first solve recursive manner. For (36) . Then and can be obtained by for , , and applying (23) and (24) at node . After are obtained, can be computed by applying (30)

(37) where is a shape parameter, is the coordinates of the is the point of interest in the support domain of nodes, , is the radius of the support domain. In both examples, the value of was chosen as 0.01 as recommended in [3], [4]. A. 90 Degree H-Plane Waveguide Bend The geometry of the bend is depicted in Fig. 1, where , , , and . is the observing point. The solution domain is modeled in a conformal manner by a set of discrete nodes as depicted in Fig. 2. The smallest nodal distance between two neighboring nodes is 0.005236 m, and the largest is 0.010472 m. The ratio of the largest distance between the nodes to the smallest is 2. The CFL

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TABLE I CPU TIME AND MEMORY USED IN COMPUTING THE H-PLANE WAVEGUIDE BEND

Fig. 1. Geometry of the 90 degree H-plane waveguide bend.

Fig. 2. Node distribution for the H-plane bend.

Fig. 4. The parallel plate waveguide with the thin PEC slot tilted with angle . (a) computational domain. (b) node distribution in the vicinity of the slot.

TABLE II COMPUTATIONAL TIME AND MEMORY USED IN COMPUTING THE PARALLEL PLATE WAVEGUIDE

Fig. 3. The electric field recorded at observation point P .

stability condition of the conventional RPI mesheless RPI meshless method was computed and found to be [3]. at observation The simulated time-domain electric field point is plotted in Fig. 3. The agreement between the results obtained with the conventional RPI meshless method and those obtained with the proposed method is very good and the differences between them are very small. This verifies the effectiveness of the proposed method. Table I gives the memory and the CPU time used. Its significance will be explained in the paragraphs after the second example. B. Parallel Plate Waveguide With a Thin PEC Slot Fig. 4 shows the structure of the second example. It is a parallel plate waveguide with a thin tilted PEC slot. , the distance between the two The thickness of the slot is 1 plates is 0.2 m, the tilt angle of the slot is , ,

and . The two observation points are and and they are at a distance of 0.2 m apart. The distance between the position of the electric current and is also 0.2 m. Non-uniform node placements were made and more nodes were placed around the slot (as illustrated in Fig. 4). The smallest distance between two nodes, occurring near the PEC slot opening, along the direction, and 0.5 cm along the diare 0.5 rection. The largest distance between nodes is 1 cm in both the and direction. The ratio of the largest distance between the nodes to the smallest is 20,000. The numbers of electric field and magnetic field nodes to be computed are 1382, and 2662, for the respectively. The CFL stability condition is conventional RPI meshless method. recorded at and , Fig. 5 shows the electric fields respectively. Table II shows the memory requirement and the computational time of the two methods. The agreement between the results of the conventional RPI meshless method and the proposed method is again very good. This again verifies the effectiveness of the proposed method.

CHEN et al.: AN UNCONDITIONALLY STABLE RADIAL POINT INTERPOLATION MESHLESS METHOD WITH LAGUERRE POLYNOMIALS

Fig. 5. The electric fields recorded in the parallel waveguide (a) at p1 (b) at p2.

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Fig. 6. The electric fields recorded with higher node density (a) at p1 (b) at p2.

TABLE III COMPUTATIONAL TIME AND MEMORY USED IN COMPUTING THE PARALLEL PLATE WAVEGUIDE FOR HIGHER NODE DENSITY

To show influence of node density on the accuracy of the proposed method, the largest distance between nodes was changed from 1.0 cm to 0.5 cm in both the and directions; this amounts to nearly doubling of the node densities in both the and directions, respectively. The smallest distances between the nodes remained unchanged. The total numbers of electric field and magnetic field nodes to be computed are then 4513, recorded at and and 8828 respectively. The electric field are shown in Fig. 6. The associated computational time and memory used is shown in Table III. As can be seen from Figs. 6 and 7, the nearly doubled node density in the and directions present almost the same results as those obtained with the original node density. This indicates the original node distributions are good enough for accurate results. However, as indicated in Table III, with the doubled node density, the CPU time increases by more than 10 times and

Fig. 7. The electric field recorded at P2 with N

= 150.

memory requirement increases by more than 2.5 times in comparisons with those with the original node density. (the number of the Laguerre To investigate the impact of was also taken for polynomials used), a smaller the simulation with the double node density. Fig. 7 shows the . electric field recorded at As can be seen from Fig. 7, rippling differences between the results obtained with the conventional RPI meshless method and those with the proposed method can be observed after 9 ns in time. This phenomenon has been observed and thoroughly investigated in [9] where an approach to selecting a proper value

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of was proposed. A technique to handle a long simulation with the weighted Laguerre polynomial was also proposed by the same authors in [10]. Our simulations have confirmed their claims. C. Discussions About Tables I–III From Tables I–III, it can be seen that in the two examples, the memory requirement of the proposed method is generally larger than that of the conventional RPI meshless method; this is need to be computed and stored because the LU matrices of during computations. However, in comparison of the CPU times used, the situation is different. In the first example, the proposed method took more time than the conventional RPI meshless method; this is because the largest distance between node nodes is just two times of the smallest. As a result, the time step used with the proposed method is only about 6.56 times larger than that of the conventional RPI meshless method. Although a smaller number of iterations is required with the proposed method, each iteration with the proposed method takes much longer time than that with the conventional method. The number of iterations with the proposed method is not smaller enough to outweigh the longer computation time. In the second example, on the other hand, the CPU time with the proposed method is significantly smaller, only about 0.13% of that used by the conventional RPI method. This is due to the use of highly non-uniform node distribution that forced the conventional RPI meshless method to take a very small time step due to the CFL stability condition; it leads to a large number of iterations of 7500000. At the same time, the number of iterations for the proposed method is only 201, 0.0238% of 7500000. It is small enough to make the computational time with the proposed method much shorter than that with the conventional meshless method. Therefore, we can draw the conclusion that the proposed unconditionally stable meshless method is especially suitable and highly efficient for structures that require non-uniform node placements or distributions to resolve different degrees of field variations in different regions. Note that in the above two examples, the results with the proposed method are found to be always stable because of the decaying natures of the weighted Laguerre polynomials. Even with the first-order dispersive absorbing boundary condition, the proposed method is still stable. However, when the conventional RPI meshless method is used, the situation is different. Fig. 8 shows the electric field recorded at of the second example with the conventional RPI meshless method when . As can be seen, the simulated electric field quickly becomes unstable. This late-time instability phenomenon has been known to the computational electromagnetics community. Further exin the radial basis funcperiments show that when tion, the instability will arise very quickly only after two or three time steps. More studies in this regard can be found in [11]. IV. CONCLUSION In this paper, an unconditionally stable RPI meshless method using the weighted Laguerre polynomials has been developed and presented. With the meshless method, electromagnetic

Fig. 8. The electric field recorderd at p1 in the parallel waveguide with  = 6 .

structures can be modeled in a conformal manner with their geometries and boundaries with high simulation accuracy and computational efficiency. Numerical examples suggest a significant reduction in CPU time when the proposed method is used to deal with structures that require highly non-uniform node distributions. In the example computed, the saving in CPU time is more than 99% in comparisons with the conventional meshless method. REFERENCES [1] J. G. Wang and G. R. Liu, “A point interpolation meshless method based on radial basis i functions,” Int. J. Numer. Methods Eng., vol. 54, pp. 1623–1648, 2001. [2] T. Kaufmann, C. Fumeaux, and R. Vahldieck, “The meshless radial point interpolation method for time-domain electromagnetics,” in Digests of IEEE MTT-S Int. Microw. Symp., Atlanta, GA, Jun. 15–20, 2008, pp. 61–64. [3] Y. Yu and Z. Chen, “A 3-D radial point interpolation method for meshless time-domain modeling,” IEEE Trans. Microwave Theory Tech., vol. 57, pp. 2015–2020, Aug. 2009. [4] Y. Yu and Z. Chen, “Towards the development of an unconditionally stable time-domain meshless method,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 3, pp. 578–586, Mar. 2010. [5] Y. S. Chung, T. K. Sarkar, B. H. Jung, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microwave Theory Tech., vol. 51, no. 3, pp. 697–704, Mar. 2003. [6] Z. Chen and M. Ney, “Method of weighted residuals: A general approach to deriving time- and frequency-domain numerical methods,” IEEE Antennas Propag. Mag., pp. 51–70, Feb. 2009. [7] Z. Bi, K. Wu, C. Wu, and J. Litva, “A dispersive boundary condition for microstrip component analysis using the FD-TD method,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 774–777, Apr. 1992. [8] J. Lacik, “Laguerre polynomials’ scheme of transient analysis: Scale factor and number of temporal basis functions,” Radioengineering, vol. 18, no. 1, pp. 23–28, 2009. [9] M. Ha, K. Srinivasan, and M. Swaminathan, “Transient chip-package cosimulation of multiscale structures using the Laguerre-FDTD scheme,” IEEE Trans. Adv. Packaging, vol. 32, no. 4, pp. 816–830, Nov. 2009. [10] K. Srinivasan, M. Swaminathan, and E. Engin, “Overcoming limitations of Laguerre-FDTD for fast time-domain EM simulation,” in IEEE Int. Microwave Symp. Digest, Honolulu, HI, Jun. 3–8, 2007, pp. 891–894. [11] T. Kaufmann, C. Engstrom, C. Fumeaux, and R. Vahldieck, “Eigenvalue analysis and longtime stability of resonant structures for the meshless radial point interpolation method in time domain,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 12, pp. 3399–3408, Dec. 2010.

CHEN et al.: AN UNCONDITIONALLY STABLE RADIAL POINT INTERPOLATION MESHLESS METHOD WITH LAGUERRE POLYNOMIALS

Xiaojie Chen received the B.S., M.S. and Ph.D. degrees from Xidian University in 2003, 2006 and 2008 respectively, all in electromagnetic field and microwave technology. From December of 2008 to January of 2011, she was a Postdoctoral Research Fellow with the School of Electronic and Information Engineering of Beihang University. Presently, she works with No. 54 Institute of China Electrics Technology Group. Her primary interest is in computational electromagnetic, especially high-frequency approximate method, finite difference time domain method and moment method. Her research interests also include electromagnetic compatibility, scattering and antenna technology.

Zhizhang (David) Chen (S’92–M’92–SM’96–F’10) received the Ph.D. degree from the University of Ottawa, ON, Canada. He was a NSERC Postdoctoral Fellow with the ECE Department of McGill University, Montreal, Canada. He joined Dalhousie University, Halifax, Canada, in 1993, where he is presently a Professor. He has authored and coauthored over 190 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, 2006 ECE Professor of the Year award and the 2007 Faculty of Engineering Research Award from Dalhousie University.

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Yiqiang Yu (M’09) received the M.Sc. degree (with distinction) in communication systems in 2003 and the Ph.D. in microwave communications engineering in 2007 from Swansea University, U.K. He is presently an Associate Professor with East China Jiaotong University. He is also a Research Fellow with the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, Canada. His primary interest is in applications of computational electromagnetics, in particular 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 from the United Kingdom Overseas Research Award Program during 2004–2007.

Donglin Su (M’96) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Beihang University (BUAA), Beijing, China, in 1983, 1986, and 1999, respectively. She completed her Ph.D. work at the University of California at Los Angeles (UCLA) from 1996 to 1998, through a joint BUAA-UCLA Ph.D. program in electrical engineering. In 1986, she joined the School of Electronics and Information Engineering, BUAA, where she is currently a Professor and the Deputy Dean of the School. Her research interests include numerical methods for microwave and millimeterwave integrated circuits. Currently, she is involved in the research of “top to down, systemic, quantificational EMC design methodology” for aircrafts. She is the author of more than 80 papers and the coauthor of several books. Dr. Su is a senior member of the Chinese Institute of Electronics. Presently, she serves as the Chair of IEEE Antennas and Propagation Chapter of Beijing Section, the Chair of URSI-B China and the Vice Chair of Antennas Society of Chinese Institute of Electronics. She received a National Award for Science and Technology Progress in 2007.

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Integral Equations Physically-Based Preconditioner for Two-Dimensional Electromagnetic Scattering by Rough Surfaces Simon Tournier, Pierre Borderies, and Jean-René Poirier

Abstract—The use of a physically-based preconditioner for the iterative solution of linear system arising from the method of moments is investigated. The preconditioner is built with a modal approach solving a problem physically close to the one we are interested in. Numerical tests about the spectral information are given. Convergence behavior of the iterative solution is studied and the efficiency of the approach illustrated for practical range of values of interest. Although the preconditioner has been designed for periodical surfaces, it was shown to be efficient in the case of truncated surfaces illuminated by plane wave. Index Terms—Floquet mode, integral equation in electromagnetism, preconditioner, rough surface scattering.

I. INTRODUCTION

it is useful to introduce a preconditioned problem which is better conditioned. Let us consider the linear system deduced from MoM (1) where , and are respectively the impedance matrix, the induced current vector and the excitation vector. The matrix is complex, dense and for this application large. For some rough surfaces, this matrix may be poorly conditioned, and the convergence of the iterative solution may become very slow [13], and then, the use of a preconditioning matrix is of great benefit. The linear system (1) is transformed into

S

CATTERING of electromagnetic waves by rough surfaces is a subject of great interest. Applications include, but are not limited to, long-range communications, radio astronomy and remote sensing [1]. Several analytical and numerical techniques have been developed for the efficient analysis of scattering by rough surfaces. Well-known approaches are small perturbation method [2], Kirchoff technique [3], IEM [4] or effective impedance boundary condition [5], [6]. These techniques have restricted regions of validity in terms of slope and roughness of the surface. Rigorous numerical methods have been proposed [7], [8], and an efficient method for the scattering problem by rough surfaces is Integral Equation [9]–[11]. The traditional approach to solve electromagnetic Integral Equations with the method of moments (MoM) is to use direct solvers for dense problems, such as LU decomposition. However, the computational complexity and storage requirement of these methods grow significantly with the problem size with respect to wavelength. Thus, large-scale scattering problems are very expensive to solve. Nevertheless, iterative solvers based on the Krylov subspace method [12] are considered to be the most effective currently available. The convergence rate of an iterative method is often governed by the spectrum of the involved matrix. To improve the convergence rate of the iterative method and reduce computation time,

Manuscript received October 07, 2009; revised December 08, 2010; accepted February 05, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. S. Tournier and P. Borderies are with ONERA-DEMR, 31055 Toulouse Cedex 5, France (e-mail: [email protected]). J.-R. Poirier is with LAPLACE-ENSEEIHT, 31071 Toulouse Cedex 7, France. Digital Object Identifier 10.1109/TAP.2011.2163744

(2) The aim of the preconditioning scheme is to make a new system more amenable to converge than the original one; i.e., has better convergence properties than the the new matrix original matrix. It can be illustrated by the fact that the eigenvalues are more closely distributed around the unit number. The basic strategy to construct an efficient preconditioner is to find a good approximation of the inverse of the original impedance matrix . The preconditioning scheme has to be as inexpensive as possible in computing and solving the linear system in comparison with the iterative resolution. Some mathematical properties of spectral preconditioners based on an approximation of eigenvalues and eigenvectors have been studied in [14]. An algebraic procedure to build approximate eigenpairs (eigenvalues and eigenvectors) based on GMRES algorithm is proposed in [15]. This approach uses only some algebraic properties of Krylov iterative solver (GMRES). Note that other strategies [16]–[18] modify the Integral Equation what leads to wellconditioned formulation. In this paper, a new physically based preconditioner is developed for the fast computational analysis of 2D scattering from perfectly conducting rough surfaces in E-polarization; i.e., Electrical field is parallel to the direction of invariance. The quasiplanar nature of the rough surfaces is exploited to build the preconditioner. In the particular case of an infinite plane, the electromagnetic problem can be solved analytically and the spectral information of the continuous integral operator can be deduced. This information can be used for the construction of an approximate inverse of the impedance matrix corresponding to any problem electromagnetically close to the infinite plane one. This matrix is a good candidate to be introduced as a preconditioner of the kind of linear system we are interested in.

0018-926X/$26.00 © 2011 IEEE

TOURNIER et al.: INTEGRAL EQUATIONS PHYSICALLY-BASED PRECONDITIONER FOR 2-D ELECTROMAGNETIC SCATTERING

where the period and

is divided into if else

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intervals

,

(7)

B. Preliminary Step

Fig. 1. The scattering problem by a perfectly periodic surface.

The use of a right-preconditioning scheme is preferred because this scheme does not modify the residual, and the convergence rate between original linear system and preconditioned one can be easily compared. II. THEORETICAL RESULTS A. MoM Formulation for Periodic Surfaces We are interested in the solution of the scattering from periodic perfectly electric conducting (PEC) surfaces [19], [20], (see Fig. 1). The surface, noted and -periodic, is illuminated by an incident (i.e., E-polarization), and we consider wave: an invariance along the axis, so the problem is scalar and in 2-dimension. The solution of this problem in 2-dimension with E-polarization leads to solving the following integral representation of the field on the vicinity of the surface

In this subsection, we are interested in the trivial problem of the periodic flat plane to illustrate the properties of the physical preconditioner. So, we consider the -periodic surface defined . Note that the boundary by: is now and is simply noted . It is usual [21] to decompose the solution of the scattering problem by a periodic surface on the basis of outgoing Floquet modes (8) This choice is motivated by the consideration of the surface periodicity: any -quasiperiodic wave respecting Helmholtz equais tion can be expanded on the Floquet’s modes. Note that if a pure real (resp. imaginary), then the mode is considered propagative (resp. evanescent). In this work, we introduce as incident wave the ingoing -Floquet’s mode which is a pure mathematical object: . It is the trivial case of an ingoing plane wave, the reflected wave is then the corresponding ; i.e., outgoing wave with a reflection coefficient . The electric surface current density defined by leads, in this particular case, to

(3) where the integral operator on the periodic surface described by the boundary is defined by (4)

with the periodic Green function, , and

is is the free-space wavenumber, is the angle of incidence, , and if (5) else

(9) Equations (3) and (9) allow to build the eigenvalues and the eigenvectors of the integral operator (10) These vectors, associated to the scattering problem by a perfectly conducting periodic flat plane, are actually the trace of the -Floquet’s modes on this plane. Therefore, the following eigenpairs can defined: (11)

The unknown is homogeneous to an electric surface current density , such as . Remark that, in the folby . lowing, we note Using variational form and applying the Galerkin method for discretization with piecewise constant basis functions , we deduce the linear system resulting from moment equation with

Let us now show that the eigenspace we introduced is a good candidate to build an approximate eigenspace of the impedance matrix . Introducing the discretization, and in this particular , the projection vector of case, we have onto the piecewise constant basis functions is built

(6)

(12)

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TABLE I DIFFERENT SURFACES FOR NUMERICAL ILLUSTRATIONS

Fig. 3. Induced Current by the 2nd order Floquet’s mode 8 at normal incidence on different types of surface: Plane (Case 1), Sinus with h = 0:1125 (Case 2) and Sinus with h = 0:2251 (Case 3) (see Table I).

Fig. 2. Induced Current by the Plane Wave 8 at normal incidence on different types of surface: Plane (Case 1), Sinus with h = 0:1125 (Case 2) and Sinus with h = 0:2251 (Case 3) (see Table I).

with and is the size of the problem; i.e., the number of unknowns. , we can suppose that Then, with will be close to with an appropriate mesh step. And it is easy to remark that

which leads, considering an average sampling, to (13) The basic idea of this work is based on the assumption that when rough surfaces are a perturbation of PEC flat plane, their corresponding impedance matrix are also a perturbation. In other words, the eigenpairs of the corresponding PEC flat plane are an approximation of the eigenpairs of the corresponding rough surfaces. Namely, their responses to ingoing Floquet modes to study their spectrum can be simulated. To illustrate this point, the following examples (see Table I) are treated in Fig. 2 and Fig. 3. Fig. 2 plots the current density (absolute value and phase) corresponding to a incident plane wave at normal incidence and the frequency is , and Fig. 3 plots the current density corresponding to an in, and also going 2-Floquet’s mode at normal incidence and the frequency is the same. Note that

these figures also present numerical illustrations of (13): if the incident wave is a -Floquet mode, then the density current is the trace of this -Floquet mode and its amplitude is given by . For a plane wave (see Fig. 2), it is well-known that the current , so in the Case density on a periodic plane is and the phase is constant. Fig. 2 1, shows also that the current density on a perturbed periodic plate (rough surface) is also a perturbation of the current on the plate. Note that the small error between the theoretical absolute value and the numerical absolute value is due to numerical MoM error. Fig. 3 shows that it is also true for the 2-Floquet’s mode. The , so the current density on the plane is absolute value is less important and the phase has a variation of . Fig. 3 shows again that the current density on a perturbed periodic plate has similar properties that the periodic flat plate. So, these figures illustrate that the trace of Floquet’s modes is a reasonable approximation of the eigenspace associated with an object close to the periodic flat plate. In summary, we have expressed a numerical approximation of eigenvalues and eigenvectors (eigenpairs) for the impedance matrix associated with a periodic flat plate. If we are interested now in objects with electromagnetic properties close to this case, in particular those with geometries close to this ideal case, we can use such analysis as a rough approximation. As examples, rough surfaces of small height can be considered. Note that we just need here an approximation of the solution to help the convergence of the Krylov iterative solver and not a precise solution of the involved problem. The precision required for preconditioning is lower than the one required for solving the problem. C. Building of the Preconditioner The preconditioner is based on the approximation of the eigenpairs of the impedance matrix which have been derived in the previous section. The discrete inner product is defined . The goal is to build an approximate by

TOURNIER et al.: INTEGRAL EQUATIONS PHYSICALLY-BASED PRECONDITIONER FOR 2-D ELECTROMAGNETIC SCATTERING

inverse of based on the approximated spectral-information. , , of Let us define a subspace of is an index list of elements. In the dimension , and following, the index denotes an element of the subspace . can be the natural order: In practice, the index list , or it can be the physical order: the first chosen index are all the propagative modes ( is a pure real) completed by the first evanescent modes; i.e., those with pure imaginary and the biggest index . Note that at normal incidence, the natural order and the physical order are the same. A third index list modes with the largest modcan be the algebraic order: the ulus. In the following, the physical order named strategy 1 and the algebraic order named strategy 2 are studied. Let us define the decomposition according to (14) (14) First, note that is a vector in the orthogonal subspace of , denoted by . Second, the vectors are orthogonal, , they are not exactly orthogbut in their discretized forms onal, however this decomposition proves to be numerically satisfactory. is defined by Then, the approximate inverse (15) and it is completed to cover the full-space. So, the new preconditioning matrix-vector product is

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TABLE II DIFFERENT SURFACES FOR THE HEURISTIC STUDY

III. HEURISTIC RESULTS In this section, a numerical study is presented to determine and the number of modes to have the best the index list efficiency. Different kinds of surfaces are proposed now for the same analysis. Will be considered flat plane, sinus functions, Gaussian random processes with a Gaussian correlation function, and Weierstrass functions defined by (18) Table II summarizes the different parameters. mean respectively normalized rms height Note that and and normalized correlation length. Although the preconditioning scheme can be applied to any iterative method, the GMRES algorithm is used in this paper. A. Validity of the Approximation In this subsection, the goal is to verify numerically the validity of the assumption that the eigenpairs of the perfectly conducting periodic plate are a good approximation of the eigenpairs of the perturbed case, with defining the following error criterion:

(16) (19) We have a control on the complementing vector with the parameter . In the case of a periodic perfectly conducting plane, it is clear that the best theoretical choice, which insures the optimal reduction of the condition number, is the following eigenvalue which has the highest modulus. in the list The precoditioning matrix-vector product (16) can be written in piecewise constant basis as a matrix (17) is the change-of-coordinate matrix between Floquet’s with (resp. ) is the idenmodes and piecewise constant basis, (resp. ), and is a diagonal matity matrix of dimension trix . Note that a preconditioning approach based on Fourier Transform can be found in [22]. However, the frame is different: the surface is truncated and it is illuminated by a tapered plane wave. And, contrarily to the Fourier-Transform-based preconditioner, the physically-based preconditioner investigated here is built with an analytical modal approach.

where is a natural physical norm. Note that are normalized. the vectors Fig. 4 shows the relative error versus the index of modes considered for different surfaces (see Table II) at normal incidence. The assumption ((13)) that the trace of Floquet’s modes approximates correctly the eigenspace is illustrated for these different rough surfaces. Note that, at normal incidence, the modes are , so only the relative error versus the symmetric; i.e., positive index of modes is plotted. Firstly, the effect of the mesh sampling is studied on a plane: the number of eigenpairs which are correctly approximated is the same for all sampling ratio (Case 1 and 4 overlay), and it can be evaluated grossly at 30%. Indeed, if the size of the plane is fixed and the number of unknowns increases which that means the sampling ratio decreases, then the first eigenpairs are more correctly approximated, but the discrete problem has more numerous small eigenvalues which can not be precisely approximated. That is why only the first 30% can be considered as correctly discretizing the theoretical eigenspace. On the other hand,

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Fig. 4. Validity of the approximate eigenpairs for different types of surface: Plane with a sampling ratio of =15:22 (Case 1), Plane (bis) with a sampling ratio of =30:44 (Case 4), Sinus with h = 0:2251 (Case 5), Gaussian random with rms height 0:3183 and correlation length 1:273 (Case 6), Weierstrass with fractal Dimension 1.5 (Case 7) (see Table II).

the normalized vectors must be correctly defined in the discretized problem. In practice, we still have an accurate representation when the sampling ratio is 3 points per wavelength, that means that the first 30% modes present an acceptable relative error. Secondly, Fig. 4 shows also the limits of the approximation evaluated by the error criterion ((19)). The effect of different level of perturbations of the flat plane may be seen. The rougher the surface is, and the bigger relative error is; i.e., the physical approximation becomes less reasonable. Furthermore, one peak (i.e, an index of is seen for a number of modes in the order of modes of in these cases). This value is a singular eigenvalue in the spectrum of the matrix associated with the periodic surface. This figure suggests another improvement method to determine the index list . The method would consist in the selection of the modes which are correctly represented, and that are good approximations of the eigenpairs, controlled by the error criterion. But this new strategy is not an efficient solution considering the total resolution CPU time because the evaluation of the error criterion would be very expensive, with at least matrix-vector products. B. Number of Modes In this subsection, the number of modes and the strategies to select it are studied. The goal is to evaluate if the reduction of the needed iterations leads to a time reduction in spite of the additional computation time when more and more modes are considered. iteration by The residual is defined at the (20) The number of iterations required to reach a fixed residual is studied. The parameter , expressed in percentage, is the number of modes that is considered in the preconditioning scheme, and it is relative to the matrix size. The value

versus Fig. 5. Number of iterations required to reach the residual of 10 the number of Modes M = (%)N=100 for different types of surface: Plane (Case 1), Sinus with h = 0:2251 (Case 5), Gaussian random with rms height 0:3183 and correlation length 1.273 (Case 6) (see Table II).

corresponds to the full-GMRES without preconditioning means, in strategy 1, that scheme, and for example all the modes with an index between 0.10 N and 0.10 N are considered. Note that in the case of periodic flat plane, the iterative method converges very quickly, due to a lucky breakdown (see [12] p. 155–156). So in this case only, the initial vector is chosen randomly instead of the classical choice that is the zero vector. Fig. 5 plots the number of iterations needed for a residual versus the number of modes considfixed at ered in the preconditioning scheme for different surfaces (see Table II). As expected, the case of the plane requires more iterations than the sinus case because the initial vector in GMRES is not the same. It is clear from this graph that the convergence is improved if the number of modes considered increases. In addition, when more than 30% of modes are considered, some are badly-evaluated but they do not modify the convergence. Despite of the fact that strategy 2 improves the convergence when the set of propagative modes is not taken, it is not a decisive choice. Indeed, the number of modes to be considered for a strong efficiency is roughly greater then 20% of the matrix size. In this case, the two strategies have the same efficiency, in terms of iterations number. Moreover, if more modes are taken, improvement is significant but implies more computational effort. It will be the purpose of computations times study to see the efficiency in these cases and which is the best trade-off. versus Fig. 6 plots the CPU time needed for a residual of the number of modes considered. Note that Fig. 6 may be compared with Fig. 5. Following the previous test, a strong improvement can be seen on the figure, when taking a number of modes greater than 20% of the matrix size. Furthermore, the strategy 2 is more efficient than the strategy 1 when few modes are considered, due to a smaller number of iterations required (see Fig. 5). However, additional cost, in term of CPU time, for sorting the modes by larger modulus is significant. Moreover, in the region of interest, this additional cost penalizes the strategy 2. Then, on this figure, the cost of the projection onto Floquet modes is

TOURNIER et al.: INTEGRAL EQUATIONS PHYSICALLY-BASED PRECONDITIONER FOR 2-D ELECTROMAGNETIC SCATTERING

Fig. 6. CPU time required to reach the residual of 10 versus the number of Modes = (%) 100 for different types of surface: Plane (Case 1), Sinus with = 0 2251 (Case 5), Gaussian random with rms height 0 3183 and correlation length 1 273 (Case 6) (see Table II).

h

M

:

N=  : 

:



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h 

:



Fig. 7. Convergence history for Sinus surface with = 0 2251 (Case 5), Gaussian random surface with rms height 0 3183 and correlation length 1 273 (Case 6) (Table II). GMRES means classical full-GMRES method, and PGMRES means preconditioned full-GMRES method with physical preconditioning scheme (30% of modes).

:

: 

TABLE III DIFFERENT CASES FOR THE EVALUATION OF PERFORMANCES

seen. When the number of modes is more than 30% and it increases, the number of iterations is constant (see Fig. 5) but the CPU time increases also. If too many modes are taken, these supplementary modes do not decrease the performance, in term of iterations, of the preconditioner, but the further inner products in the preconditioning matrix-vector product decrease the performance, in term of CPU time. In summary, the optimal choice is the physical order named in the precondistrategy 1 with a number of modes tioning scheme.

Fig. 8. Number of iterations required to reach the residual of 10 versus rms height ( ) for Gaussian random surfaces with different correlation lengths: 0 7957 (Case 8), 1 273 (Case 9) and 1 591 (Case 10) (see Table III). GMRES means classical full-GMRES method, and PGMRES means preconditioned full-GMRES method with physical preconditioning scheme.

:

ks 

: 

: 

i.e., a residual smaller than , and more the utilization of preconditioner is significant. In the case of a Gaussian random profile, the benefit of the preconditioner is clear.

IV. NUMERICAL RESULTS In this section, a numerical study to evaluate the performances of the proposing preconditioning scheme is presented. The Table III summarizes the different parameters of the surfaces studied. is the physical order named strategy 1, and The index list is fixed at 30% of the matrix size. the number of modes The convergence history of two rough profiles reaching a is compared. Fig. 7 shows the residual versus residual of the number of iteration for a sinus surface and a Gaussian random surface. In the case of a sinus profile, the convergence is already fast for an approximated solution; i.e., a residual , and the benefit of the preconditioner is greater than negligible. Nevertheless, the more the solution is precise;

A. Performance for Gaussian Random Surfaces The effect of the roughness is studied for Gaussian random surfaces. The normalized correlation length is fixed at , or , and the normalized rms height increases until . Fig. 8 shows the number of iterations required to have the versus the normalized rms height. residual smaller than As expected, the bigger the normalized rms height is, and the more iterations are necessary. When the normalized rms height is small to moderate, the physically-based preconditioner is benefit. Moreover, the validity of the physical hypothesis is seen. As expected, the physically-based preconditioner is affected by the size of the roughness. For example, at rms height fixed, the

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Fig. 9. Number of iterations required to reach the residual of 10 versus rms height (ks) for truncated Gaussian random surfaces with different correlation lengths: 0:7957 (Case 8), 1:273 (Case 9) and 1:591 (Case 10) (see Table III). GMRES means classical full-GMRES method, and PGMRES means preconditioned full-GMRES method with the proposed physical preconditioning scheme.

surface with a normalized correlation length of 5 is rougher than the surface with a normalized correlation length of 10. When the normalized rms height increases, the iterative method GMRES is affected, and it is more affected all the more the surface is rough. The behavior of the GMRES with the physically-based preconditioner is the same. However, the ratio between GMRES and GMRES with the preconditioning scheme depends on the roughness. More roughness a surface has, and less the physically-based preconditioner is efficient. Notwithstanding, the physically-based preconditioner is always efficient for practical range of values. The physically-based preconditioner has been built to improve the solving of the scattering problem from periodic surface illuminated by plane wave. Now, this physically-based preconditioner is tested in the case of the scattering problem from truncated surface illuminated by plane wave. Only the impedance matrix is changed in the method. Fig. 9 shows the same parameters that Fig. 8 but they are for truncated Gaussian random surfaces. In the both case, periodic or truncated, the general behavior is similar. When the rms height is small to moderate, the truncated case (see Fig. 9) converges slower than the periodic case (see Fig. 8), with or no preconditioner. Nevertheless, when the rms height is big, the truncated case converges obviously faster than the periodic case. The difference between the truncated surfaces and the periodic surfaces is the same in the both case, with physically-based preconditioner or without preconditioner. The physically-based preconditioner built for periodic surfaces is also efficient for the truncated surfaces, and its region of validity is the similar in the both case. B. Performance for Weierstrass Surfaces The surfaces with multiscale properties describe some natural soils [23]. The physically-based preconditioner is tested here in the case of Weierstrass profile which has multiscale behavior. The height in the Weierstrass function (see (18)) is fixed at

Fig. 10. Number of iterations required to reach the residual of 10 versus rms height (ks) for truncated Weierstrass surfaces with different height: =3 (Case 11), =6 (Case 12) and =10 (Case 13) (see Table III). GMRES means classical full-GMRES method, and PGMRES means preconditioned full-GMRES method with the proposed physical preconditioning scheme.

, and . For the fractal dimension between 1.1 and 1.6, the corresponding normalized rms height are respectively between 0.272 and 0.502, 0.453 and 0.837, 0.816 and 1.506. At fixed rms height, the Weierstrass profile is rougher than the Gaussian random surface. Note that the slimness of the multiscale is the same for all the heights, only their magnitude is different. Fig. 10 plots the number of iterations required to reach the versus the fractal dimension. The convergence residual of of GMRES without preconditioner is not affected by the multiscale behavior. Comparing Fig. 9 and Fig. 10 for the same and value of normalized rms height; i.e., between , the number of iterations required to reach the residual of is grossly similar. On the other hand, the GMRES with physically-based preconditioner is affected by the multiscale behavior. the limit of the physical hypothesis can be seen to be reached. The parameters which limit the efficiency of the physically-based preconditioner are clearly the rms height combined with the multiscale effect. These limits need to be further investigated, but for the practical range of values of interest the physically-based preconditioner has proven to be efficient. V. CONCLUSION A preconditioning scheme based on a modal approach and physical considerations has been developed in the canonical example of quasi-planar surface. Numerical results show a good efficiency of this technique for appropriate applications guided by physical knowledge. Although the preconditioner has been designed for periodical surfaces, it was shown to be efficient in the case of truncated surfaces illuminated by plane wave. The other polarization (H-polarization) and the dielectric case does not present any a priori difficulty, just rewriting (9) with appropriated quantities. Then, an extension of this work will be now to apply this method to more complicated or 3-dimension problems. Namely a best approximation of the modal approach in the case of rough surface with an homogenization technique [5] is currently under study.

TOURNIER et al.: INTEGRAL EQUATIONS PHYSICALLY-BASED PRECONDITIONER FOR 2-D ELECTROMAGNETIC SCATTERING

REFERENCES [1] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: From Theory to Applications. Norwood, MA: Artech House, 1986. [2] I. A. , Wave Propagation and Scattering in Random Media. New York: IEEE Press, 1978. [3] F. T. Ulaby and C. Elachi, Radar Polarimetry for Geoscience Applications. Norwood, MA: Artech House, 1990. [4] A. K. Fung, Z. Li, and K. S. Chen, “Backscattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sensing, vol. 30, no. 2, pp. 356–369, Mar. 1992. [5] J.-R. Poirier, A. Bendali, and P. Borderies, “Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 995–1005, Mar. 2006. [6] A. Bendali, P. Borderies, and J.-R. Poirier, A two scale expansion for the scattering of a two dimensional time harmonic wave by a highly oscillating surface, unpublished. [7] A. K. Fung and G. W. Pan, “A scattering model for perfectly conducting random surfaces I. Model development,” Int. J. Remote Sensing, vol. 8, no. 11, pp. 1579–1593, 1987. [8] K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on monte carlo simulations,” J. Opt. Soc. Am. A, vol. 12, no. 11, pp. 2491–2499, 1995. [9] V. Jandhyala, B. Shanker, E. Michielssen, and W. C. Chew, “Fast algorithm for the analysis of scattering by dielectric rough surfaces,” J. Opt. Soc. Am. A, vol. 15, no. 7, pp. 1877–1885, 1998. [10] V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 36, no. 3, pp. 738–748, 1998. [11] P. Spiga, G. Soriano, and M. Saillard, “Scattering of electromagnetic waves from rough surfaces: A boundary integral method for low-grazing angles,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 2043–2050, Jul. 2008. [12] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2002. [13] K. Inan and V. Erturk, “Application of iterative techniques for electromagnetic scattering from dielectric random and reentrant rough surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 11, pp. 3320–3329, Nov. 2006. [14] L. Giraud and S. Gratton, “On the sensitivity of some spectral preconditioners,” SIAM J. Matrix Analy. Applicat., vol. 27, no. 4, pp. 1089–1105, 2006. [15] J. Baglama, D. Calvetti, G. H. Golub, and L. Reichel, “Adaptively preconditioned GMRES algorithms,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 243–269, 1998. [16] H. Contopanagos, B. Dembart, M. Epton, J. Ottusch, V. Rokhlin, J. Visher, and S. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1824–1830, Dec. 2002. [17] S. Borel, D. Levadoux, and F. Alouges, “A new well-conditioned integral formulation for maxwell equations in three dimensions,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2995–3004, Sep. 2005.

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[18] F. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderon preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2398–2412, Aug. 2008. [19] K. Zaki and A. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag., vol. 19, no. 2, pp. 208–214, Mar. 1971. [20] J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: One dimensional perfectly reflecting surfaces,” Waves Random Complex Media, vol. 8, no. 4, pp. 385–414, 1998. [21] R. Petit and L. C. Botten, Electromagnetic Theory of Gratings. Berlin: Springer-Verlag, 1980. [22] P. Naenna and J. Johnson, “A physically-based preconditioner for quasi-planar scattering problems,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2421–2426, Aug. 2008. [23] S. Rouvier, P. Borderies, and I. Chênerie, “Ultra wide band electromagnetic scattering of a fractal profile,” Radio Sci., vol. 32, no. 2, pp. 285–293, 1997. Simon Tournier received the Engineering degree in electrical engineering from ENSEEIHT, Toulouse, France, and the M.Sc. degree in microwave and optoelectronic from the Institut National Polytechnique de Toulouse, University of Toulouse, both in June 2007. He is currently working toward the Ph.D. degree at ONERA, Toulouse, France. His research interest are centered around numerical electromagnetism.

Pierre Borderies was born in France in 1953. He received the Dipl. Eng. degree from Ecole Superieure d’Électricite, Paris, France, in 1975. After a period of teaching in Venezuela, he joined the Centre d’Études et de Recherches de Toulouse, part of the Office National d’Études et de Recherches Aérospatiales (CERT-ONERA) in 1979. Since then, he has been working as a Research Engineer in the Microwaves Department. He has worked in the fields of large reflector antennas, microwave devices, and radar targets imaging, characterization, and identification. From 1990 to 1991, he spent a sabbatical year at New York University, Farmingdale. His current research interests include radiation and scattering of antennas, frequency selective surfaces, ultrawideband scattering, subsurface targets identification, natural targets scattering, and remote sensing.

Jean-René Poirier received the Engineer diploma from the National Institute of Applied Sciences (INSA), Toulouse, France in 1996. He was a Ph.D. student at the French National Aeronautics and Space Research Centre (ONERA) from 1997 to 2000 and received the Ph.D. degree in applied mathematics from INSA in December 2000. From 2001 to 2003, he joined the Institute of Analysis and Scientific Computing, EPFL, Lausanne, Switzerland. He is currently working as an Associate Professor at INPT-ENSEEIHT, Toulouse, and joined the Laboratoire PLAsma et Conversion d’Énergie (LAPLACE), in January 2009. His current research interests focus on computational electromagnetics and include rough surface scattering, boundary impedance condition and preconditioning techniques for boundary element method.

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Kirchhoff’s Laws as a Finite Volume Method for the Planar Maxwell Equations Harish S. Bhat and Braxton Osting

Abstract—Beginning with Maxwell’s equations for the polarized mode in an inhomogeneous planar medium, we derive a finite volume method that we recognize as Kirchhoff’s laws for a corresponding circuit consisting of inductors, capacitors, and resistors. This association automatically gives local charge and energy conservation. The method is implemented and used to find the steady-state solution for two test problems. By comparison with the exact solution for the homogeneous medium problem, the method is shown to be linearly convergent. Index Terms—Circuit modeling, equivalent circuits, finite volume methods, Helmholtz equation, Maxwell equations.

I. INTRODUCTION

M

tivity

AXWELL’S equations for the polarized mode in a planar, inhomogeneous medium with permitand permeability are (1a) (1b)

be the rectangular region occupied by Let the medium. Let be the left boundary of . Suppose that on , we have harmonic forcing at frequency : (1c) On the remaining three sides of the boundary, we impose impedance or Leontovich boundary conditions: (1d) , , and denote the boundary of , the unit where outward normal, and the conductance on the boundary. In this paper, we accomplish the following goals: 1) We show that a finite volume discretization of (1) results in Kirchhoff’s laws of voltage and current for a particular circuit consisting of inductors, capacitors, and resistors. Manuscript received December 07, 2010; revised February 06, 2011; accepted March 12, 2011. Date of publication August 18, 2011; date of current version October 05, 2011. This work was supported by the National Science Foundation (NSF) under Grants DMS09-13048 and DMS06-02235, EMSW21-RTG: Numerical Mathematics for Scientific Computing. H. S. Bhat is with the School of Natural Sciences, University of California, Merced, Merced, CA 95343 USA (e-mail: [email protected]). B. Osting is with the Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 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.2011.2163787

2) By comparing finite volume solutions of (1) for constant and with exact solutions obtained via separation of variables, we numerically establish first-order convergence. A. Relationship to Previous Work The idea of demonstrating equivalent circuits whose continuum limit yields Maxwell’s equations is quite old [1]–[4]. These early works predate the widespread use of digital computers to solve differential/integral equations. Since then, when a new numerical method for Maxwell’s equations has been introduced, the corresponding equivalent circuit has been explored, often as a way to gain physical insight useful for modeling purposes [5, Ch. 1]. One of the first papers proposing an equivalent circuit for the FDTD discretization was [6]. Equivalent circuits for the finite element method and the method of moments have been described in [7] and [8, Ch. 5], respectively. For the transmission line matrix method [5], [9], [10] and the spatial network method [11], [12], equivalent circuits feature prominently. The finite volume (FV) method appeared in computational electromagnetics in the late 1980s [13]–[16]. More recent work indicates that FV methods may hold an advantage over other methods for problems with large variations in the material parameters and sub-grid scale variations in the fields [17]–[20]. Note that the convergence of at least two versions of the FV method has been proven rigorously [21]–[24]. Despite the fact that the FV method has been employed successfully for over 20 years, and unlike the situation for any of the other popular methods for solving Maxwell’s equations, there has to date been no discussion in the literature of an equivalent circuit for the FV discretization. We find two main benefits of carefully deriving an equivalent circuit formulation of the FV discretization. First, we obtain precise formulas that relate the local inductance, capacitance, and boundary conductance of the circuit to spatial averages of their continuum coun, and , respectively. Second, reterparts: lating the FV discretization to Kirchhoff’s laws for a circuit automatically yields local energy and charge conservation, in addition to global energy and charge functionals that are natural discretizations of the continuum energy and charge functionals for Maxwell’s equations. From the point of view of using FV to analyze a two-dimensional case of Maxwell’s equations, our work is most similar to [25]. We solve (1) in steady-state for an arbitrary frequency in (1c); this amounts to finding the frequency-domain solution, which is exactly the goal of the frequency-domain FV method proposed in [20]. In the present work, we are not concerned with issues related to unstructured, adaptive, or hybridized meshes

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BHAT AND OSTING: KIRCHHOFF’S LAWS AS A FINITE VOLUME METHOD FOR THE PLANAR MAXWELL EQUATIONS

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[26]–[28], though we note here that our derivation can be generalized in this direction. II. FINITE VOLUME DISCRETIZATION MAXWELL’S EQUATIONS

OF

PLANAR

In this section, we derive a finite volume method discretization for (1). We tile our rectangular domain with small square cells, in the vertical direction and in the horizontal direction. We define the charge in the cell :

(2) We define the capacitance the cell :

Fig. 1. Cell diagram of finite volume discretization for an interior cell blue dashed lines indicate the dual graph.

. The

to be the scaled permittivity in

(3)

Note that this equation says that at each lattice node, the sum of incoming currents must equal the sum of outgoing currents, implying local charge conservation. We define the inductance of the segment as

where is a characteristic length scale in the out-of-plane direction. Next, we define the voltage , so

(4) Let us assume for a moment that is an interior cell, so that has no intersection with . As shown in Fig. 1, has four neighboring cells labeled , , , and , the right, up, left, and down neighbors, respectively. We label the four corners of as , , , and . With this notation,

(9) Let us check how the currents evolve in time. We compute (10a) (10b) (10c)

(5) Now let us define the currents. In general, when we have two neighboring cells and that are separated by a vertical segment , if is to the left of , then we define the horizontal current

(6) and that In general, when we have two neighboring cells are separated by a horizontal segment , if is below , then we define the vertical current

Let us explain the sequence of approximations made above: • In (10a), we replace by its segment average . • To approximate the flux between cells in (10a), the integral is approximated by the value of evaluated at the midpoint of , a second-order accurate finite-difference formula is applied using the values of at the center of the cells and , and then these values are replaced by the cell averages. This is the main finite volume approximation [29]. • To go from (10b)–(c), we replace by its cell average, which gives us

(11) (7) Note that the right-hand sides of (5), (6), and (7) all involve line integrals of scalar fields, which are all independent of parametrization. With these conventions, we have (8)

Using analogous approximations, we compute

(12) For an interior cell , (8), (10), and (12) are Kirchhoff’s laws of voltage and current for a regular square lattice of inductors and capacitors [30], [31].

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A. Boundary Conditions

Remark

To handle boundary condition (1c) on , we use a column of ghost cells. Each ghost cell, where the electric field is prescribed, is directly to the left of a cell in the first column, where the electric field is an unknown. Let be the right boundary of the ghost cell. We compute the voltage of the ghost cell using (11): (13) with

In Appendix A, we show that the polarized mode described by (1) is an exact solution of the fully three-dimensional Maxwell’s equations for a physical system described by two horizontally infinite parallel plates that are separated vertically by the distance . All the definitions made above (e.g., charge, capacitance, resistance, etc.) can be derived in a physically consistent fashion using the setup in Appendix A. One may also make these definitions on the grounds that the quantities being derived have the correct units. B. Assembling the Discretized System

The ghost cells yield new horizontal currents , each of which satisfies an equation of the form (10). Each such equation involves one unknown and one prescribed voltage. If the top, right, or bottom boundary of intersects , then we apply the other boundary condition (1d). Let . Then going back to (4), we find

(14) The second line integral in (14) can be evaluated in the same way as (5) above; we focus on the first line integral. We write

where

is the conductance

(15) Note that if , then (1d) and (1a) imply , a perfectly insulating boundary condition. On the other hand, if , then (1d) implies , a perfectly conducting boundary condition. In this paper, we choose to approximate outgoing boundary conditions, which are obtained as follows. Dotting (1a) with and then using (1d), we find that on ,

At each , the value of for which this equation is the Engquist-Majda outgoing condition [32] is

(16)

Discretization gives us a two-dimensional rectangular lattice with rows and columns, which we represent as an oriented graph, cf. [33, Ch. 13]. This graph is the dual graph of the finite volume mesh as shown in Fig. 1. Nodes represent capacitors and edges represent inductors. The direction or orientation of the edge represents the direction of positive current flow through the associated inductor. In a lattice of size , there are nodes and edges, horizontal ones and vertical ones. We let denote the set of all nodes, and denote the set of all edges. Let be a vector of size such that is the capacitance at node . Let be a vector of size such that is the inductance at edge . We partition into horizontal and vertical inductances by writing . At time , and are, respectively, the voltage across capacitor and the current through inductor . By and we denote the vectors of all voltages and currents, respectively. Of the horizontal edges, there are boundary edges that form a subset , each of which is incident upon only one node and corresponds to a ghost cell to the left of the domain . Specifically, is the left-most column of horizontal edges. All other edges in the graph are incident upon two nodes. In general, we think of an edge as an ordered pair , where . The direction of the edge is given by the ordering of these numbers, so that is the tail and is the head of . For a boundary edge that is incident only upon node , we write . We let denote the incidence matrix of the oriented graph for our circuit. We have if for some if for some otherwise. In addition to the structure described already, the lattice also has resistors and forcing along the boundary. We represent the set of nodes connected to resistors by , and let be the conductance of node . We then extend by defining for all , so that is a vector of size . Let . Then we define the projection matrix by if and otherwise. Note that because ,

BHAT AND OSTING: KIRCHHOFF’S LAWS AS A FINITE VOLUME METHOD FOR THE PLANAR MAXWELL EQUATIONS

the final edges is

columns of

are all zero. The forcing applied at

The frequency is the same in the boundary condition (1c). The vector is arranged as follows: each edge is of the form for some . We set equal to as defined in (13), where is the ghost cell to the left of cell . The finite volume scheme from the previous section, which we have already noted is equivalent to Kirchhoff’s Laws on an inductor-capacitor lattice, can now be written in the following matrix-vector form: (17a)

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4) Inserting (16) into (15), we find that at a boundary node , we have the impedance-matched value of the conductance,

(20) is the edge incident on node that is normal where to the boundary, and is the segment that is dual to edge . In the case where all cells are identical squares, we have . III. CONSERVATION PROPERTIES OF THE CONTINUOUS AND DISCRETE SYSTEMS It is instructive to calculate the time evolution of the total energy for the Maxwell system (1):

(17b)

C. Steady-State Solution of the Discretized Equation Define

so for each ,

(21) . Define This says that the rate of change of energy equals the power forced in through the left boundary minus the power dissipated through the other three sides of the medium. It is clear that power is dissipated at a rate proportional to . We also compute the time-evolution of the total charge of the system

Then the system (17) can be written in the form

(18) Consider the steady-state solution into (18), we derive

. Inserting this

(19)

(22) The interpretation of this equation is that the rate of change of charge equals the current entering the domain on the left boundary minus the current exiting the domain on the other three sides. Again, the outgoing current is proportional to . The association of the finite volume discretized system as Kirchhoff’s laws for a circuit allows for natural definitions of discrete energy and charge. The rate of change of the total energy of the discrete system can be calculated using (17):

D. Discussion 1) The matrix will be invertible if and only if is not an eigenvalue of . Note that if all nodes are resistive, i.e., if for for all , then the spectrum of has strictly negative real part, implying that (19) can be computed for all real . 2) Using Matlab on a desktop computer with 4 GB of RAM, (19) can easily be solved for , . 3) We have formulated the circuit as an oriented graph in order to write the equations compactly and take advantage of the graph-theoretic interpretation of the incidence matrix , which appears naturally in Kirchhoff’s laws. Though we have formulated the problem for an rectangular lattice, the graph-theoretic framework easily accommodates other topologies.

The right hand side, which has the form of power in minus power out, corresponds perfectly with the right hand side of (21). The calculation shows that the dynamics of energy for the entire lattice can be understood by observing boundary phenomena only; this implies that, locally, in the interior of the lattice, energy is conserved. We also compute the time evolution of the total charge of the discrete system (17):

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This has the form of current in minus current out, corresponding perfectly with the right hand side of (22).

Note that

(27)

IV. SEPARATION OF VARIABLES SOLUTION In this section, we use separation of variables to develop the exact, steady-state solution of (1) for constant and . We begin by assuming harmonic time-dependence of the fields,

is an eigenfunction of (25) as long as dental equation

solves the transcen-

(28)

in which case system (1) reduces to (23a) (23b)

Using this and the above properties, we can derive the solution of (23). We expand the left-hand side boundary condition via

(23c) where and . We now assume a solution of the form . Inserting this into the Helmholtz (23a), we split the problem as follows:

Taking inner products, we find

(24) This yields a non-selfadjoint problem for a complex eigenfunction and a complex eigenvalue :

We return to (24) and see that

must satisfy (29a)

(25a)

(29b)

(25b)

(29c)

(25c) We say that

The solution of (29) is

solves the adjoint problem if it satisfies: (26a) (26b) (26c)

We list without proof the properties of the eigenvalue problem that are most relevant to developing a separation of variables solution. For details, refer to [34], [35]. 1) Equation (25) is not a Sturm-Liouville problem because the boundary conditions are not self-adjoint. 2) If the eigenpair solves (25), the eigenpair solves (26), and , then and are orthogonal with respect to the inner product:

The solution of (23) is then

(30)

A. Solving (28) for the Eigenvalues . Taking the square root of both sides of Let (28) and then splitting the resulting equation into its real and imaginary parts leads us to the fixed point iteration scheme

3) If the eigenpair solves (25), then the eigenpair solves (26). In this case, . 4) The eigenvalues are discrete, simple, and live in the first quadrant of . 5) The set is a complete basis of . 6) As , the eigenfunctions are increasingly oscillatory and alternatingly even and odd about .

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Fig. 2. The left panels show results from simulations of a homogeneous medium with and as in Section V-A. The right panels show results from ; otherwise, simulations of an inhomogeneous medium as in Section V-B. The black circles in the upper-right panel mark the parts of the medium within which . Again, everywhere. The same Gaussian boundary forcing at angular frequency is applied in both problems. The numerically computed real part of the electric field is indicated by the color contour plots. Note that the effect of the periodic array with linear defect is to confine the electromagnetic field and not allow it to diffract into the rest of the domain as in the homogeneous medium. The lower panels show the results of a convergence study in the form error between numerically computed solutions and a reference solution, as a function of the number of lattice rows. In the lower-left of log-log plots of the panel, the reference solution is a 50-mode truncation of the exact solution, while in the lower-right panel, the reference solution is the finite volume solution on an 800 800 lattice. The lower panels both show four curves, one for each indicated value of the angular frequency . All eight curves have best-fit slope less than , indicating first-order convergence.

Let be the disc value satisfies mapping principle that

. If the eigen, it can be shown using the contraction has a unique fixed point where . In practice, we find that this means that applying for , one obtains all eigenvalues with real parts in the interval except possibly for one eigenvalue that can be found by applying Newton’s method in the disc . The eigenvalues found in this way constitute the full spectrum of (25). Note that as , the eigenvalues have the asymptotic form (31) B. The Transfer Function on the Rectangle

We define the transfer function to be the mapping from the left boundary condition to the solution on the right boundary, i.e., (32)

Let

. Using (30), we derive

(33) Combining this with (31), we see that for large , (34) Since , the upshot of (34) is that the transfer function (32) does not conserve energy, since the large modes of are severely damped. Also, the solution on the right boundary will be much smoother than . V. NUMERICAL IMPLEMENTATION AND CONVERGENCE In this section, we discuss the application of the finite volume method to two test problems. Throughout the finite volume so. We use to denote the components of the lution, we set finite volume solution that represent voltages at lattice nodes.

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A. Homogeneous Medium For the separation of variables solution, we use (30), truncated at modes, to produce a function . When we compare against , we average over the , following (11)—we denote the averaged separation of cell variables solution by . We focus on Gaussian boundary data with on the square domain with . We set and . The finite volume solution of this problem at on a 400 400 lattice is given in the upper-left panel of Fig. 2. For four different values of , we compare the separation of on an variables solution to the finite volume solution lattice where , 32, 40, 64, 80, 100, 160, 200, 320, 400, and 800. The lower-left panel of Fig. 2 shows a log-log plot error versus . When equals 0.25, of the 0.5, 1.0, and 1.9, the least squares fit to the data gives slopes , , , and , indicating of, respectively, first-order convergence. B. Periodic Medium With a Linear Defect We now consider a medium, modeled after a photonic crystal device [36], that consists of a periodic array of low index circular inclusions with a linear defect. The permittivity outside and inside, . The domain is the the inclusions is . The distance between the centers of square with and each circle has radius . The linear the circles is defect is created by simply removing a row of inclusions. The fion a 400 400 nite volume solution of this problem at lattice is given in the upper-right panel of Fig. 2. As expected, the mode is confined to the defect, rather than diffracting as in the homogeneous medium. We study the convergence of the finite volume method for this inhomogeneous medium by first obtaining a fine-scale fion an 800 800 lattice. For the four nite volume solution values of mentioned above, we compare this solution to the on an lattice for , 100, finite volume solution versus is given in 200, 400. A log-log plot of the lower-right panel of Fig. 2. A least squares fit of the error , , , and , gives slopes of, respectively, indicating first-order convergence.

parameters can be handled readily. A goal for future work is to extensively test how roughness and/or short-wavelength oscillations in the coefficients and affect the performance of the finite volume method, and to compare the finite volume method to other frequency-domain methods for such problems. APPENDIX AN IDEALIZED PHYSICAL CONFIGURATION In this section, we describe an idealized physical configurapolarized mode is an exact sotion in which the lution of Maxwell’s equations and interpret the finite volume method derived in Section II in this context. The idea of formulating more systematic relationships between circuit-and fieldtheoretic concepts stems from [37]. We consider two perfectly conducting plates that are infinite in extent in the and directions and separated by a distance in the direction. Between the plates is a medium with parameters and that may vary in the and directions, but are constant in the direction. Between the plates, the electric and magnetic fields satisfy Maxwell’s equations with no free charge or currents. The boundary conditions on the plates are: (35a) (35b) where

and are the surface charge and surface current. Here, for the upper surface and for the lower surface. polarized mode autoThe matically satisfies (35a). The last two boundary conditions reduce to

where, as before, . Evaluating the line inteconnecting and , we find that gral of are equal in the charge density on the two plates at fixed magnitude but have opposite signs. We now identify the charge , defined by (2), with the area integral over of the surface charge on the top plate. For constant , (3) agrees with the separated capacitance between two parallel plates of area . The electrostatic potential difference by a distance : between the two plates at position can be defined by

VI. CONCLUSION We have derived a physically motivated finite volume method for a planar Maxwell system. The method is easy to implement. Here we have done so to obtain the frequency domain solution of two problems with harmonic time-dependence. However, the system (17) obtained after spatial discretization could be used with a time-stepping scheme to solve an initial value problem in the time domain. Note that to find a steady-state solution through time stepping that is as accurate as the solutions we obtained, one would require a temporal discretization with first-order global error. This typically means that the time-stepping scheme must be at least second-order accurate. To demonstrate convergence, we compared numerical solutions with a separation of variables solution for constant and . Note that it is possible to generalize the separation of variables solution in Section IV to handle separable and . The choice of discretization in Section II does not require smoothness of and . In other words, an advantage of the first-order method proposed here is that discontinuous material

The approximation of the quantity in (11) is on . Thus the apprecisely the average value proximation made in (11) can be interpreted as an electrostatic approximation. Continuity of charge requires that for any rectangular region on the top conducting plate, we must have

Thus the line integral of the surface current over one segment is equivalent to the current defined by (7), and the conof tinuity equation is equivalent to Kirchhoff’s law (8). Suppose there is a surface current between two cells in the direction. This induces a magnetic field in the direction just below the top plate. If the current increases (resp. decreases), then the field will also increase (resp. decrease). By Faraday’s

BHAT AND OSTING: KIRCHHOFF’S LAWS AS A FINITE VOLUME METHOD FOR THE PLANAR MAXWELL EQUATIONS

law of induction, this increasing (resp. decreasing) field will indirecduce an electromotive force in the direction (resp. tion) proportional to . This is Kirchhoff’s law (12). ACKNOWLEDGMENT The authors thank the NSF Institute for Pure and Applied Mathematics (IPAM) for its hospitality. REFERENCES [1] J. R. Whinnery and S. Ramo, “A new approach to the solution of highfrequency field problems,” in Proc. I. R. E., 1944, vol. 32, pp. 284–288. [2] G. Kron, “Equivalent circuit of the field equations of Maxwell—I,” Proc. I. R. E., vol. 32, no. 5, pp. 289–299, 1944. [3] J. R. Whinnery, C. Concordia, W. Ridgway, and G. Kron, “Network analyzer studies of electromagnetic cavity resonators,” in Proc. I. R. E., 1944, vol. 32, pp. 360–367. [4] L. Brillouin, Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices, ser. International Series in Pure and Applied Physics. New York, NY: McGraw-Hill, 1946. [5] C. Christopoulos, The Transmission-Line Modeling Method (TLM). Piscataway, NJ: IEEE Press, 1995. [6] W. K. Gwarek, “Analysis of an arbitrarily-shaped planar circuit a timedomain approach,” IEEE Trans. Microw. Theory Tech., vol. 33, pp. 1067–1072, 1985. [7] K. Guillouard, M. F. Wong, V. F. Hanna, and J. Citerne, “A new global time-domain electromagnetic simulator of microwave circuits including lumped elements based on finite-element method,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2045–2049, 1999. [8] L. B. Felsen, M. Mongiardo, and P. Russer, Electromagnetic Field Computation by Network Methods. New York: Springer-Verlag, 2008. [9] W. J. R. Hoefer, “The transmission-line matrix method—Theory and applications,” IEEE Trans. Microw. Theory Tech., vol. 33, pp. 882–893, 1985. [10] C. Christopoulos, The Transmission-Line Modeling (TLM) Method in Electromagnetics. San Rafael, CA: Morgan & Claypool, 2006. [11] Y. Ko, N. Yoshida, and I. Fukai, “Three-dimensional analysis of a cylindrical waveguide converter for circular polarization by the spatial network method,” IEEE Trans. Microw. Theory Tech., vol. 38, pp. 912–918, 1990. [12] H. Satoh, N. Yoshida, S. Kitayama, and S. Konaka, “Analysis of 2-D frequency converter utilizing compound nonlinear photonic-crystal structure by condensed node spatial network method,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 210–215, 2006. [13] N. K. Madsen and R. W. Ziolkowski, “Numerical solution of Maxwell’s equations in the time domain using irregular nonorthogonal grids,” Wave Motion, vol. 10, no. 6, pp. 583–596, 1988. [14] V. Shankar, W. F. Hall, and A. H. Mohammadian, “A time-domain differential solver for electromagnetic scattering problems,” Proc. IEEE, vol. 77, no. 5, pp. 709–721, 1989. [15] N. K. Madsen and R. W. Ziolkowski, “A three-dimensional modified finite volume technique for Maxwell’s equations,” Electromagnetics, vol. 10, no. 1, pp. 147–161, 1990. [16] V. Shankar, A. H. Mohammadian, and W. F. Hall, “A time-domain, finite-volume treatment for the Maxwell equations,” Electromagnetics, vol. 10, no. 1–2, pp. 127–145, 1990. [17] I. E. Lager, E. Tonti, A. T. de Hoop, G. Mur, and M. Marrone, “Finite formulation and domain-integrated field relations in electromagnetics—A synthesis,” IEEE Trans. Magn., vol. 39, pp. 1199–1202, 2003. [18] C. Fumeaux, D. Baumann, and R. Vahldieck, “Advanced FVTD simulation of dielectric resonator antennas and feed structures,” ACES J., vol. 19, pp. 155–164, 2004. [19] D. Baumann, M. Gimersky, C. Fumeaux, and R. Vahldieck, “Accuracy considerations of the FVTD method for radiating structures,” in Proc. Eur. Microwave Conf., 2005, vol. 2. [20] K. Krohne, D. Baumann, C. Fumeaux, E.-P. Li, and R. Vahldieck, “Frequency-domain finite-volume simulations,” in Proc. Eur. Microwave Conf., 2007, pp. 158–161. [21] E. T. Chung, Q. Du, and J. Zou, “Convergence analysis of a finite volume method for Maxwell’s equations in nonhomogeneous media,” SIAM J. Numer. Anal., vol. 41, no. 1, pp. 37–63, 2003. [22] E. T. Chung and B. Engquist, “Convergence analysis of fully discrete finite volume methods for Maxwell’s equations in nonhomogeneous media,” SIAM J. Numer. Anal., vol. 43, no. 1, pp. 303–317, 2005.

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[23] F. Hermeline, “A finite volume method for solving Maxwell equations in inhomogeneous media on arbitrary meshes,” C. R. Acad. Sci. Paris Sér. I Math., vol. 339, no. 12, pp. 893–898, 2004. [24] F. Hermeline, S. Layouni, and P. Omnes, “A finite volume method for the approximation of Maxwell’s equations in two space dimensions on arbitrary meshes,” J. Comput. Phys., vol. 227, no. 22, pp. 9365–9388, 2008. [25] F. Edelvik, “Analysis of a finite volume solver for Maxwell’s equations,” in Finite Volumes for Complex Applications II. Paris: Hermes Sci. Publ., 1999, pp. 141–148. [26] S. D. Gedney, J. A. Roden, N. K. Madsen, A. H. Mohammadian, W. F. Hall, V. Shankar, and C. Rowell, “Explicit time-domain solutions of Maxwell’s equations via generalized grids,” in Advances in Computational Electrodynamics, ser. Artech House Antenna Lib.. Boston, MA: Artech House, 1998, pp. 163–262. [27] Z. J. Wang, A. J. Przekwas, and Y. Liu, “A FV-TD electromagnetic solver using adaptive Cartesian grids,” Comput. Phys. Comm., vol. 148, no. 1, pp. 17–29, 2002. [28] E. Abenius, U. Andersson, F. Edelvik, L. Eriksson, and G. Ledfelt, “Hybrid time domain solvers for the Maxwell equations in 2D,” Internat. J. Numer. Methods Engrg., vol. 53, no. 9, pp. 2185–2199, 2002. [29] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, ser. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 2002. [30] H. S. Bhat and B. Osting, “Diffraction on the two-dimensional square lattice,” SIAM J. Appl. Math., vol. 70, no. 5, pp. 1389–1406, 2009. [31] H. S. Bhat and B. Osting, “Discrete diffraction in two-dimensional transmission line metamaterials,” Microw. Opt. Tech. Lett., vol. 52, no. 3, pp. 721–725, 2010. [32] B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proc. Nat. Acad. Sci., vol. 74, no. 5, pp. 1765–1766, 1977. [33] L. R. Foulds, Graph Theory Applications, ser. Universitext. New York: Springer-Verlag, 1992. [34] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955. [35] D. S. Cohen, “Separation of variables and alternative representations for non-selfadjoint boundary value problems,” Comm. Pure Appl. Math., vol. 17, pp. 1–22, 1964. [36] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. Princeton, NJ: Princeton Univ. Press, 2008. [37] R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission. New York: Wiley, 1960. Harish S. Bhat received the A.B. degree in mathematics from Harvard University (Cambridge, MA) in 2000, and the Ph.D. degree in control and dynamical systems from the California Institute of Technology (Pasadena) in 2005. From 2005 to 2007, he held a postdoctoral position as the Chu Assistant Professor in the Department of Applied Physics and Applied Mathematics at Columbia University (New York, NY). From 2007 to 2008, he was on the faculty in the Department of Mathematics at Claremont McKenna College (Claremont, CA). He joined the School of Natural Sciences at the University of California, Merced (Merced, CA) as an Assistant Professor in July, 2008. His research interests include developing perturbative and numerical methods to understand linear and nonlinear wave propagation in a variety of media, e.g., networks of electrical oscillators. He is also interested in applying optimization methods to synthesize networks with prescribed properties.

Braxton Osting received B.S. degrees in mathematics, physics, and applied mathematics from the University of Washington (Seattle) in 2005 and the Ph.D. degree in applied mathematics from Columbia University (New York, NY in 2011. His research interests include analytical and computational methods to study mathematical models of wave phenomena in media where inhomogeneity, nonlinearity, dispersion, or geometry affect propagation. He is particularly interested in the strategy of using such a model together with an efficient optimization method to design the media to control some aspect of the wave propagation for novel applications.

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The WLP-FDTD Method for Periodic Structures With Oblique Incident Wave Zhao-Yang Cai, Bin Chen, Member, IEEE, Qin Yin, and Run Xiong

Abstract—A new unconditionally stable finite-difference timedomain (FDTD) method for periodic structures is presented, which is based on the field transformation method and the weighted Laguerre polynomials FDTD (WLP-FDTD). The proposed method uses a field transformation to remove the time gradient across the grid, and then uses the concept of the WLP-FDTD to get the implicit relationship between the transformed field variables. It holds the advantages of the WLP-FDTD, can eliminate the restriction of the Courant-Friedrich-Levy (CFL) stability condition. Compared with other field transform methods, the new method needn’t to do special treatment for the additional terms. It appears to be much more efficient than the other field transformation FDTD method for solving periodic structures with fine structures and large incident angle. To verify the accuracy and the efficiency of the proposed method, we compare the results of the Split-Field FDTD method with the proposed method. Index Terms—Finite-difference time-domain (FDTD), oblique incident wave, periodic structures, unconditionally stable method, weighted Laguerre polynomials (WLPs).

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method is an efficient scheme for solving Maxwell’s equations, and has been widely used in solving periodic structures, such as frequency selective surface (FSS), antenna arrays and so on. Instead of analyzing the entire structure, only a single-unit cell needs to be analyzed by incorporating the periodic boundary condition (PBC). Several methods were used to implement the PBC of the oblique incident wave, such as Sine-Cosine method [1], field transformed method [1], [2], spectral finite-difference time-domain (SFDTD) method [3] and so on. However, all the methods above for periodic structures are explicit time marching technique, its time step is constrained by the Courant-Friedrich-Levy (CFL) condition. This drawback results in a long solution time for solving problems with fine periodic structures and oblique incident wave problem. In order to eliminate the CFL stability condition, an alternating- directionimplicit (ADI) FDTD solution for periodic structures was proposed in [4], but an obliquely incident wave on periodic structures can not be solved in this way [5]. Another unconditionally stable method for periodic structures is called ADI-SFDTD [5],

but this method can be applied only for a single frequency in general [6]. Recently, a new unconditionally stable FDTD method with weighted Laguerre polynomials (WLP-FDTD) [7]–[9] was introduced. Time steps don’t need to be dealt with by using this method, which appears to be much more efficient than the conventional FDTD method in solving problems with fine structures. In addition, the WLP-FDTD method provides sufficient computational accuracy than that of the ADI-FDTD method. In this paper, the WLP-FDTD technique is introduced to the field transformation method for periodic structures. For simplicity, we call it as Periodic WLP-FDTD. The field transformation method for modeling periodic structures uses a field transformation to remove the time gradient cross the grid. However, the transformed equations have additional terms that require special treatment and stability limit is considerably more stringent for large steering angles [1]. The Periodic WLP-FDTD in this paper is based on the field transformation method and weighted Laguerre polynomials FDTD. It can eliminate the restriction of the CFL condition, without the special treatment for the additional terms, and only a limited number of iterations are needed to achieve a satisfactory accuracy [7]. In order to perfect the new method, the total-field/scattered-field (TF/SF) formulations and perfectly matched layer (PML) absorbing boundary condition (ABC) are proposed in the Periodic WLP-FDTD method. Numerical examples verify that the presented method can solve the oblique incident wave on periodic structures conveniently, efficiently and accurately. Numerical simulations show that when the incident angle , the CPU time of the Periodic WLP-FDTD is reduced to about 0.12% of the Split-Field FDTD method. II. MATHEMATICAL FORMULATION A. Formulations of the WLP-FDTD for the Periodic Structures With simple and lossless media, using the transformed vari, , the time-domain ables [1] Maxwell’s equations for the 2-D periodic structures TMz case are

(1) Manuscript received January 05, 2011; revised February 27, 2011; accepted April 04, 2011. Date of publication August 08, 2011; date of current version October 05, 2011. This work was supported by the Chinese National Science Foundation under Grant 60971063. The authors are with the Engineering Institute of Corps of Engineers, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163791

(2) (3) where is the speed of light, and are the electric permittivity and magnetic permeability of free space, respectively. is

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the propagation direction, and when the incident wave is normal to the structures. The periodic boundary conditions for the transformed field variables are now (4) (5) is the length of the spatial period in the direction. where A set of orthogonal basis functions can be constructed using Laguerre Polynomials [7], given by

(6) where is the Laguerre polynomial of order , is is a time-scale factor. the scaled time, and Using these basis functions, the field components, taking for example, can be expanded as

Fig. 1. Position of the electric and magnetic fields of the order q .

(13) (14) Contrary to the conventional FDTD difference, (12)–(14) have an implicit relation. Therefore, inserting (13) and (14) into, we have

(7) and the first derivative of

with respect to time is (15) (8)

Applying the central-difference scheme introduced by Yee to (15), we can obtain discrete space equation for the 2-D TMz Periodic WLP-FDTD method as

According to the derivational procedure in [7], (1)–(3) can be written as

(9)

(10)

(16) where , , and are grid sizes in the and direction, respectively. From (16), we can get that each electric field variable has the relationship with the adjacent four electric fields and four magnetic fields, as shown in Fig. 1. Writing (16) as a matrix equation, we have

(11) (17) where is a time-scale factor, and is the number of weighted Laguerre polynomial function. Equations (9)–(11) can further be written as

(12)

where is the coefficient matrix, is the summation term . from the order 0 to However, in the case of the periodic boundary conditions, we need a different procedure. From (4) and (5), and (7), we can get the PBC formulations as follow (18) (19)

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Contrary to conventional periodic FDTD method, the proposed method has an implicit relationship between the transformed field variables, which results in a sparse system matrix , and there are only five nonzero elements in the each row of it. Since at each recursion the proposed method uses the same system matrix, we can perform the lower-upper (LU) decombefore the iteration process only once, and then position of solve (17) by using the back-substitution routine repeatedly. The magnetic fields can be obtained form (13) and (14). Meanwhile, only a limited number of iterations are needed to achieve a satisfactory accuracy [7]. Then one can reconstruct the field components in the time domain with (7). B. TF-SF Boundary of Periodic WLP-FDTD Method Consider an example of TF/SF boundary configuration as at the left TF/SF face for exshown in Fig. 2. Taking ample, the components and are assumed to be stored as scattered fields, whereas the other seven components in (16) are assumed to be stored as total fields. According to the derivational procedure in [9], the update of the left TF/SF formulation can be given: See (20) at the bottom of the page, denotes incident-field, and where (21) (22)

Fig. 2. TF/SF boundary configuration in a 2-D TM FDTD Grid 1 (a) i (b) i I .

= 01

=I

The upper limit of infinity can be replaced by a finite time [7]. The other TF/SF formulations can be got in a interval similar way. C. PML ABC of Periodic WLP-FDTD The time-domain equations of PML [1], [10] for periodic structures (2-D TMz) are (23)

Fig. 3. Computational domain of 2-D periodic structures

(h = 3 mm).

(24) (27) (25) (26)

and are the conductivity profile different from Where zero only in the PML region to provide attenuation for propagating waves.

(20)

CAI et al.: THE WLP-FDTD METHOD FOR PERIODIC STRUCTURES WITH OBLIQUE INCIDENT WAVE

Fig. 4. Transient electric fields of z component at P , P , P and P (a) ( = 45 ) (b) ( = 85 ).

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Similar to the derivational procedure in the part A, using Garlerkin temporal testing procedure, we can get the further equations as:

TABLE I SIMULATION RESULTS FOR THE 2-D PERIODIC STRUCTURES

(28) (29)

(30)

(31) where (32)

III. NUMERICAL RESULTS In this section, numerical examples are implemented to validate the formulations of the Periodic WLP-FDTD, PML ABC and the TF/SF connecting boundary for the periodic structures. The Split-field FDTD method [1] is chosen as the contrast method. The configuration consists of four parallel foursquare metallic poles with thin slots, whose width is 3 mm, and the size of slots are 3 mm and 0.2 mm, as shown in Fig. 3. The cell size is 1 mm 0.1 mm, there are 100 and 182 subdivisions along the and direction, respectively. The sinusoidal modulated Gaussian pulse is excited through the TF/SF boundary shown as the dashed line in Fig. 3. Computational domain is truncated along the -direction by the 20 additional PML layers and the PBC along the -direction. The incident wave is sinusoidal modulated Gaussian pulse:

(33) , , where , [7] and choose For Split-Field FDTD, the time step the CFL condition [1]

. And we . should correspond to

(34)

when the incident angle and , the CFL stability and condition of this model are . Fig. 4 shows the components of electric fields at four measurement points with the incident angle and . The agreement between the Split-Field FDTD method and the proposed method is very good. Thus the proposed formula in of the Periodic WLP-FDTD method is valid. Table I shows the computational time for the numerical simand . We can ulations with the incident angle find that the proposed method is much more efficient than the Split-FDTD, and with the increase of the incident angle, SplitFDTD will cost much more time, but the time cost by Periodic , the simulaWLP-FDTD is always the same. When tion takes 3167.3 s with the Split-FDTD method and takes 3.96 s with the Periodic WLP-FDTD method totally, this includes the time of LU decomposition procedure (1.56 s). The CPU time of the Periodic WLP-FDTD is reduced to about 0.12% of the Split-FDTD method. All calculations in this paper have been performed on a Core2 2.4-GHz machine, and the unsymmetric-pattern multifrontal method [11] is used to perform the lower-upper (LU) decomposition. IV. CONCLUSION In this paper, a novel scheme with the WLP-FDTD method applied to the periodic structure is presented. As a result, an unconditionally stable FDTD method for periodic structures with oblique incident wave is introduced. The solution is independent of the time discrimination in the method, thus compared with the other field-transformation methods, the new method is much more efficient with the increase of the incident angle. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Similarly, it can be extended to the three-dimensional to solve more complicated periodic structures problems. REFERENCES [1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite- Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000, pp. 569–625. [2] M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference timedomain analysis of wave scattering from periodic surfaces: Oblique incidence case,” J. Electromag. Waves Appl., vol. 7, pp. 1595–1607, 1993.

CAI et al.: THE WLP-FDTD METHOD FOR PERIODIC STRUCTURES WITH OBLIQUE INCIDENT WAVE

[3] A. Aminian and Y. R. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1818–1825, Jun. 2006. [4] S. Wang, J. Chen, and P. Ruchhoeft, “An ADI-FDTD method for periodic structures,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2343–2346, Jul. 2005. [5] Y. Mao, B. Chen, and H. Chen, “Unconditionally stable SFDTD algorithm for solving oblique incident wave on periodic structures,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 5, pp. 257–259, 2009. [6] K. Toyoda, Y. Uno, and T. Arima, “Comparison of FT-FDTD and spectral domain FDTD for periodic structures,” in Proc. iWAT2008, Chiba, Japan, pp. 191–194. [7] Y. S. Chung, T. K. Sarkar, B. H. Jung, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microwave Theory Tech., vol. 51, no. 3, pp. 697–704, 2003. [8] W. Shao, Bing-Zhong, Wang, and X.-H. Wang, “Efficient compact 2-D time-domain method with weighted Laguerre polynomials,” IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp. 442–448, Aug. 2006. [9] Y. Yi, B. Chen, H.-L. Chen, and D.-G. Fang, “TF/SF boundary and PML-ABC for an unconditionally stable FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 91–93, 2007. [10] J. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Comp. Phys., vol. 114, no. 2, pp. 185–200, 1994. [11] Timothy A. Davis and Iain S. Duff, “A combined unifrontal/multifrontal method for unsymmetric sparse matrices,” Computer and Inf. Sci. and Engrg. Dept., Univ. Florida, 1997, Tech. Rep. TR-97-016. Zhao-Yang Cai was born in Henan province, China, in 1985. He received the B.S. degrees in electric systems from Nanjing Engineering Institute, Nanjing, China, in 2007, where he is currently working toward the M.S. degree. His research interests include computational electromagnetics.

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Bin Chen (M’98) was born in Jiangsu, China, in 1957. He received the B.S. and M.S. degrees in electrical engineering from Beijing Institute of Technology, Beijing, China, in 1982 and 1987, respectively, and the Ph.D. degree in electrical engineering from Nanjing University of Science and Technology, Nanjing, China, in 1997. Currently, he is a Professor at Nanjing Engineering Institute. His research includes computational electromagnetics and EMP.

Qin Yin was born in Jinagsu province, China, in 1982. He received the B.S. and M.S. degrees in electric systems and automation from Nanjing Engineering Institute, Nanjing, China, in 2005 and 2007 respectively, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics and electromagnetic tracking.

Run Xiong was born in Sichuan province, China, in 1983. He received the B.S. and M.S. degrees in electric systems and automation from Nanjing Engineering Institute, Nanjing, China, in 2005 and 2010 respectively, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics and EMP.

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Hybrid Ray Tracing Method for Microwave Lens Simulation Junwei Dong, Member, IEEE, and Amir I. Zaghloul, Life Fellow, IEEE

Abstract—Microwave lenses such as the Bootlace/Rotman lenses are designed by placing physical ports of lens input on the theoretical phase centers. These phase center positions are calculated using geometrical optic method under the assumptions of perfect cylindrical waves and true time delay. A real physical lens does not satisfy these conditions due to different port implementation approaches and mutual coupling effects. Full wave investigations and measurements have indicated strong variation at both phase and amplitude couplings between the input and output ports. Efficient theoretical models predicting both phase and amplitude performances are still in great demand to perform advanced lens optimization. The full wave simulation demonstrates accurate results. However, it is not convenient in optimization iterations due to its high computational cost and sophisticated programming process. Based on a ray tracing concept recently explored by the authors, this paper extends its design and formulate a suitable approach for general lens simulation. A microwave lens is systematically treated by hybrid of a flexible tapered port model and multiple-ray-path coupling approach. This method leads to designing the minimum return loss tapered port and fast lens simulation of reasonable accuracy. The predicted results of amplitude, phase couplings, array factors are validated by both full wave simulation and measurement. The comparison shows that the proposed method is fast, accurate and sufficient to predict various microwave lens parameters. This concept can be extended to designing stripline and waveguide lenses as well. Index Terms—Fast simulation, microwave lens, ray tracing, Rotman lens.

I. INTRODUCTION

M

ICROWAVE lens is a component for broadband true-time-delay beam-forming. It was implemented in many phased array systems on different platforms such as satellites and radar. With the advancement of printed circuit technology, new applications based on microwave lens have been proposed recently, such as the vehicular sensor design [1], terahertz imaging [2] and ultra-wide band communications [3]. Typical planar microwave lens uses few radiation elements (beam ports) to perform the initial illumination of a receiving

Manuscript received June 11, 2010; revised February 09, 2011; accepted March 12, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. J. Dong is with Microwave Engineering Corporation (MEC), North Andover, MA 01845 USA (e-mail: [email protected]). A. I. Zaghloul is with Bradley Department of Electrical and Computer Engineering, Virginia Tech, Falls Church, VA 22043 USA and also with the US Army Research Laboratory, Adelphi, MD 20873 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2163762

Fig. 1. Microwave lens as BFN.

contour that consists of a number of receiving elements (ports), as shown in Fig. 1. The receiving elements then direct the energy to a phased array through transmission lines. The effective path length between the radiation element and phased array feed point is crucial in forming a desired phase delay property. Due to the beam ports’ phase center variation and mutual couplings, the effective path length varies as port profile and frequency change. Besides, the amplitude couplings between the beam port and receiving port affect the far field side lobe level (SLL) and scanning quality. Microwave lens models based on geometrical optics in [4]–[7] use phase centers for the beam ports and transmission line lengths between receiving array and radiating array as the main parameters in the lens initial geometry formulation. The amplitude behavior across the receiving array aperture and the multiple reflections within the lens cavity are secondary parameters. The initial investigation on amplitude couplings in the microwave lens regime was carried out in [8]. Some further research was presented in [9] and [10]. These models adopt two apertures coupling theory with no assumption on either the adjacent elements or profiles of the tapered ports. In recent years, accurate full wave simulation methods such as FDTD [11], FEM [12] and MoM [13] have been applied to predict both phase and amplitude performance of the microwave lens. Because of the high computation cost, the full wave methods are somewhat cumbersome to perform lens optimization. Currently, the full wave solvers are primary validation tools for several existing lens designs [11]–[13]. Thus, the need to develop a simulation method that is fast and accurate still exists. Using a ray tracing method is a possible such way of simulating an entire lens structure, including the mutual coupling effect within the lens cavity. This concept was originally proposed by the authors in [14], where its limitation of resolving

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Direct coupling and secondary coupling expressions are derived and combined in the ray tracing process. The transmission lines will be readily taken into account afterwards. The tapered port models are emphasized here, not only because it is important for the ray tracing simulation, but also it provides significant information for the return loss of the beam port, which usually determines the lens’ lowest operation bandwidth. B. Tapered Port Models

Fig. 2. The basic idea of ray tracing in microwave lens.

various type of ports still exists and the accuracy is highly questionable due to lack of validations. In this paper, a full formulation and validation process are hence first time presented. The contents of this article are organized as follows. The presentation in Section II starts with a ray tracing concept for the whole lens structure, and then discuss in detail the sub-models of tapered port, two ports couplings, model combination and the ray tracing process. The validation is addressed in Sections III-A and III-B, which include two examples emphasizing two aspects: 1) apply ray tracing simulation to predict the performance of a single port excitation, leading to the amplitude and phase shift across the receiving aperture; 2) use ray tracing simulation to evaluate lens performance over frequency. Two fabricated lenses are used in and measurement results are used to assess the proposed design concept. II. METHODOLOGY A. Ray Tracing Concept for Microwave Lens Design Fig. 2 shows a general Rotman lens structure, without the transmission lines connected to the tapered ends on both sides of the lens. The ports on the left side (in black) indicate the beam ports, the ones on the right hand (in blue) illustrate the receiving ports, while the ones in between (in red) stand for the sidewall dummy ports. To estimate the coupling between two ports, e.g. points A and D, we need to consider at least three contributors, 1) the return loss due to the tapered beam port, denoted by AB, 2) the direct line of sight coupling between B and C, 3) the reflection due to the receiving port taper CD. The three contributors are combined into a direct coupling factor between A and D. Besides the direct coupling, there are reflections off other ports that result in signals in the direction of the receiving port, e.g. reflections off ports QZ and TN represented by coupling factors and , respectively. The reradiated or reflected rays couple with all ports causing secondary couplings. The ultimate coupling result between A and D is the sum of the direct coupling and all secondary couplings. We will first devise a method of modeling the tapered port, and further discuss the aperture to aperture coupling models.

The tapered port is an essential transmitting and receiving element in the printed microwave lens design, as shown in Fig. 2. The port is necessary to be tapered for two reasons. 1) The transmission line that guides energy in/out the lens usually has characteristic impedance of 50 , which more or less constrains the width of the port input/output given the specific microstrip material. The cavity junction usually has lower impedance due to the large surface area; hence the tapered line functions as an impedance transformer. 2) At the receiving contour, it is necessary to be enclosed by large port size in order to reduce the spillover loss, in other word, to increase the power efficiency. The reflection coefficients that affect the coupling factors are functions of the impedance model at the radiating/receiving elements, which are in turn functions of the physical taper of the elements. Several types of impedance models can be adopted in the Rotman lens design, such as linear, triangular, exponential, Klopfenstein, Chebyshev and other types of tapers. Different models have different reflection patterns versus frequency. Existing approaches of analyzing the microstrip tapers are: small reflection theory [15], contour integral method [16], and general non-uniform line theory [17]. We use the small reflection theory as our basis to form a binomial tapered model for the current microwave lens design. A triangular physical geometry taper is adopted in Fig. 2. Because of its ease of fabrication, it is commonly seen in the microwave lens models. It is worthwhile pointing out that the geometrical triangular shape does not necessarily represent a triangular impedance tapering. We shall build a model suitable to simulate different type of tapers, hence in Fig. 3 a general geometry taper trace is assumed, denoted by , where the x axis is the taper length and y represents its y coordinate. The taper is symmetric with respect to x axis. W(x) is the width of the taper cross-section, and L is the total length. Given the microstrip material’s permittivity , substrate height , the effective permittivity single cross-section can be approximated by (1). The impedance model of the taper can be calculated by (3) [15] (1) where (2)

(3)

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Fig. 6. Optimized binomial tapered line impedance model. Fig. 3. Tapered line geometry model.

The aforementioned linear, triangular, exponential, Klopfenstein and Chebyshev tapers are named after their specific impedance curves. They are not suitable for simulating an arbitrary taper geometry. For an arbitrary taper geometry given in Fig. 3, the reflection coefficient can be solved by the procedures described below. For a given geometry model , use (1)–(3) to calculate its impedance curve . To facilitate the integration of (6), we assume the impedance curve to follow an expression of binomial expansions (7). It is found that 3 orders are sufficient for most smooth taper models for the microwave lens design

Fig. 4. Tapered lines impedance model.

(7) Substituting (7) in (6), the return loss at the input is found as (8)

Fig. 5. Triangular shape tapered line impedance model.

(8)

Suppose the taper in Fig. 3 has an impedance model shown in Fig. 4, where and stands for the input and output impedances. Assume the continuous impedance line is made up of a number of constant impedance segmentations with length of . The incremental reflection coefficient due to the impedance deduction can be calculated as follows: (4)

It is also possible to design tapered line from a given frequency response. In this case, the coefficient parameters have to be firstly solved from (8), then use the impedance model of (7) to solve for the taper geometry function or . Given the microstrip material property, the relation between the impedance model and the geometry parameters are solved in (9) (10) [15]. for

As form:

approaches to zero, (4) gives an exact differential for (5)

Summing all the little reflections with proper phase shift yields the total reflection coefficient at : (6) where k is the wave number

.

(9) where (10) (11) We now apply the theory described above to simulate two tapering port models shown in Fig. 5 and Fig. 6. Fig. 5 is a

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Fig. 9. Aperture-to-aperture coupling.

shown in (12)–(14) respectively. Each of the three models may be used as the port-to-port coupling model in the ray tracing process. Initial ray tracing investigation in this paper adopt the first model given in [18], and represented by (12)

Fig. 7. Impedance curves for tapers in Fig. 5 and Fig. 6.

(12)

(13)

(14) D. Combination Process Fig. 8. Return loss of the tapers in Fig. 5 and Fig. 6.

triangular shape tapered line that is used in the lens described in Section III-B. Appling (1)–(3), the impedance model is solved as shown in Fig. 7. The taper line shown in Fig. 6 is achieved by optimizing the coefficients in (8). Its impedance model is determined by the optimized coefficients, whose result is shown in Fig. 7 as well. Fig. 6 shows the geometry of a binomial taper that is optimized using (9)–(11). The frequency response of both tapers across 2–6 GHz are given in Fig. 7 for the impedance and in Fig. 8 for the return loss. The results demonstrate that the optimized binomial taper has yielded a nonlinear locus in the geometry; however, it has much lower return loss than the geometrical triangular taper.

The complete port-to-port coupling involves direct coupling and secondary couplings, as illustrated in Fig. 2. Take the coupling between port AB and CD in Fig. 2 for example; the final coupling coefficient is expressed in (15). The direct coupling combines the taper coefficient and the port to port aperture coupling as well, and it is expressed in (16), where is a function of port size and pointing direction and stands for the aperture to aperture (B-to-C) model described previously. The secondary coupling combines all reflections from all other ports, which is expressed in (17), where port QZ of Fig. 2 has been considered as any arbitrary port beside of the transmitting and receiving ports. In (17), it is assumed that the total port number in the microwave lens is P (15)

C. Aperture to Aperture Coupling Models The port-to-port coupling mentioned in Section II-A, determines how much energy is coupled from beam port to receiving port, which is a function of the port sizes, port pointing directions and port-to-port distance, as shown in Fig. 9. This model has been studied by using the two dimension aperture theory [18], the mode matching method [19], and the simplified ray equation model [20]. The results of these three models are

(16)

(17)

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Fig. 12. Phase shift across aperture for port 20 excitation at 8 GHz. Fig. 10. X-band Rotman lens configuration; 1 stands for beam port, 0 for dummy port, and 2 for receiving port.

Fig. 13. Amplitude across aperture for port 20 excitation at 10 GHz. Fig. 11. Amplitude across aperture for port 20 excitation at 8 GHz.

In (16), (17), is the wave number in the dielectric medium. Given a loss tangent of , the wave number is calculated in (18) (18) It is noted that the transmission lines in the microwave lens have not been taken into account during the ray tracing formulation. Given a transmission line length of T at either the transmitting port or receiving port, the line effect results in multiplying (15) with a phase term of , where and the effective permittivity is obtained from (1) for a line width of W. III. RAY TRACING VALIDATION BY FULL WAVE SIMULATION AND MEASUREMENT A microwave lens is a multiple port beam forming device used in array applications. To validate the simulation of such device, electrical performance over different ports at different frequencies are considered. Two lenses previously constructed and measured were used to serve such purpose. The first lens is an X-band lens designed and measured by IAI Elta Electronics Industries (private communications) and the second one is a C-band lens built and tested by the US Army Research Lab [21].

In a different research effort [13], the full-wave FEKO simulation results of the second lens have been reported by the authors as well. These provide us, for the first time, wealthy resources to conduct validation of the proposed ray tracing simulation. A. Validation of X-Band Microwave Lens The lens shown in Fig. 10 is designed at C-band and has 16 beam ports and 32 receiving ports, with dimensions 37 cm 44 cm. The beam ports are denoted as 1, receiving port are denoted as 2, and all dummy ports have signs of 0, marked at the narrow ends of the ports. The measurement was conducted by two port network analyzer under the condition that all other ports are terminated by 50 ohm loads. Because the performance of the microwave lens relies on the phase shift and amplitude couplings across the aperture, the measured data is usually post processed into two formats: the phase shift and amplitude couplings across the receiving aperture (port 42–73 output) at single frequency, and the amplitude and phase couplings between two ports across the band. The array factors can be further studied by using the simulated phase and amplitude information. In this sub-section, the phase, amplitude and array factor performance for single port excitation are presented. Performance across frequency band is further studied in the second lens validation in the next sub-section.

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Fig. 14. Phase shift across aperture for port 20 excitation at 10 GHz. Fig. 16. Phase shift across aperture for port 4 excitation at 8 GHz.

Fig. 15. Amplitude across aperture for port 4 excitation at 8 GHz.

Typical frequencies of 8 and 10 GHz, and center and edge ports #20 and #4 are chosen as the representatives in the following analysis. The amplitude distributions and phase shifts across the aperture due to the illuminations of these ports are demonstrated in Figs. 11–18. Note that, in the phase diagram, we purposely plot the phase shift across receiving ports to explicitly visualize the phase linearity across the aperture, because it is a critical factor affecting the scanning angles of the array to be fed. The phase errors between the predicted and measured are indicated by a separate axis on the right hand side of each figure. Besides, two examples of the array factors due to these feeding information result in Figs. 19 and 20. The comparisons show that the amplitude couplings for the center beam exhibit better agreement than the ones for the edge element. This may be due two reasons: 1) Current ray tracing model assumes that the port has symmetric structure, which does not hold true for the edge ports in Fig. 10 due to its bending geometry. 2) The aperture-to-aperture coupling model relies on the correct port pointing direction and correct radiation pattern. For the edge port, the phase center as well as the aperture pattern may encounter higher variations than the ones in the center. The phase shift comparisons show good agreement between the simulation and measurement for both ports’ excitations. The center beam indicated smaller phase variations in degrees, which is due

Fig. 17. Amplitude across aperture for port 4 excitation at 10 GHz.

Fig. 18. Phase shift across aperture for port 4 excitation at 10 GHz.

to the fact that the aperture illuminated by the center beam possesses a much slower phase progression. In general, the error due to phase variation is acceptable. For example, given the material permittivity of 2.5, 10-degree variation shown in Fig. 12 implies a predicted physical distance error of about 0.65 mm. For large angle scanning beams, this error is hardly seen due to

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Fig. 19. Array factor for port 20 excitation at 8 GHz.

Fig. 21. Port notations of C-band microwave lens.

Fig. 20. Array factor for port 4 excitation at 10 GHz. Fig. 22. Port 2-port 15 amplitude couplings across band.

the large phase shifts. These results are demonstrated for port 4 excitations in Fig. 16 and Fig. 18. In these two figures, receiving port #50 indicates excessive phase errors over all others. This might be due to the connector matching at the particular frequencies. The array factor resulting from using the ray tracing model predicted comparable scanning angle, beam width, pattern shapes with the measurements. The higher side lobe at the edge port is due to the high amplitude errors discussed before (Fig. 17). The ray tracing simulation has demonstrated ability of simulating both amplitude and phase couplings. The lens measurement comparison shows that the current model is accurate in terms of phase shift, amplitude coupling for center ports, beam scanning angles, etc. To accurately predict the side lobe level (SLL), more accurate aperture coupling model would be required. B. Validation of C-Band Microwave Lens To further verify the preliminary results described above as well as to create comparison with another simulation tool, we conduct another case study using an individual lens that has been simulated using commercial software FEKO and measured. In this section, the true time delay (TTD), phase, and amplitude performances across the frequency band are emphasized.

The microwave lens discussed in this section was designed at 4–5 GHz, and it has 8 beam ports to feed an 8-element linear array, as shown in Fig. 21. Technical data of the lens can be found in [13]. Similar to the one presented in Fig. 10, this lens has dummy ports in between the beam ports and at sides (notation of ‘0’). The dummy ports between the beam ports are primarily used to increase the isolation between the adjacent beam ports and create similar environment for multiple beam operations at a time. Dummy ports at the side wall function as absorbers that minimize the energy reflected back into the cavity. The dummy ports sometimes are necessary to be incorporated because of the following reasons: 1) the reflection from the side wall may affect the amplitudes and phases at the receiving ports, hence degrading the lens performance; 2) the reflected wave and the forward wave may create standing waves that become a source of heating up the lens system. Unlike the model in Fig. 10, the transmission lines in Fig. 21 are confined to ‘Gaussian’ type curves to yield a constant output spacings. This is a favored configuration for most uniformly spaced linear arrays. Each transmission line was bent to certain extent so that proper line lengths can be satisfied, which is a typical design factor in the microwave lens formulation. Two port network measurements were done between every beam port and receiving port. Similar to the previous lens,

DONG AND ZAGHLOUL: HYBRID RAY TRACING METHOD FOR MICROWAVE LENS SIMULATION

Fig. 23. Port 2-port 15 phase couplings across band.

Fig. 26. Array factor for port 1 excitation at 4 GHz.

Fig. 24. Port 4-port 10 amplitude couplings across band.

Fig. 27. Array factor for port 1 excitation at 5 GHz.

Fig. 25. Port 4-port 10 amplitude couplings across band.

50-ohm loads have to be connected to all other ports while the two ports are being. The scattering matrix was further post processed into amplitude and phase information across the aperture and frequencies. The full wave simulation was carried out by the planer Green’s function MoM solver in FEKO. Beam ports 1–8 were separately excited across frequency 4–5 GHz. Eleven discrete frequency steps were registered for each beam port. The entire simulation took 8.965 hours on a 64 bit workstation, using 4 core Intel(R) Xeon(R) 3.0 GHz CPUs. The peak memory consumption of all processes was 2.136 GByte. The

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ray tracing simulation was executed on a personal computer with Intel(R) Celeron(R) 1.6 GHz CPU (412M RAM). The same scattering matrix across the frequency was registered. The entire simulation took 11 seconds, indicating thousands of times accelerations versus the full wave simulation. This is mainly because it was formulated using hybrid models that asymptotically approach the field behaviors, whereas, the full wave simulation is able to approach the exact solutions of the Maxwell’s equations. For a single beam produced by the microwave lens, the amplitude distribution and phase shift across the aperture are the key parameters. We emphasized this point in the X-band lens validation process. In this sub-section, the simulated and measured results are presented from the frequency sweep perspective. The amplitude coupling between two ports across the frequency implies how much gain variations occur when wide band signal is being sent through the channel. In general, constant amplitude coupling is expected. However, due to the facts that 1) the tapered port has a varied frequency response as indicated in Fig. 8, 2) different frequency has different path length loss, 3) due to the multiple reflections within the lens cavity, the amplitude couplings across the frequency always have certain level of deviations. As examples, we compare the ray tracing results for amplitude and phase couplings between ports 2 and 15 and between ports 4 and 10 with full-wave FEKO simulation and with measurements. The results are displayed in Figs. 22 to 25.

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TABLE I RAY TRACING SIMULATION ACCURACY QUANTIFICATION FOR C-BAND LENS V.S. FEKO FULL-WAVE SIMULATION AND MEASUREMENTS

Fig. 28. Array factor for port 3 excitation at 4 GHz.

Fig. 29. Array factor for port 3 excitation at 5 GHz.

The array factor performance across the frequency reflects the true time delay properties of the microwave lenses, as the true time delay device implies that the far field scanning beams’ pointing directions do not vary with frequency. Consequently, we have plotted the array factors of one edge beam (port #1)

and one central beam (port #3) using the simulated and measured amplitude and phase information at the lowest (4 GHz) and highest (5 GHz) frequencies in Figs. 26 to 29. Both amplitude and phase couplings demonstrate agreeable results. The full wave simulation has confirmed the accuracy of

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TABLE II GEOMETRIC DATA FOR C BAND MICROWAVE LENS

the measurement. The ray tracing simulation has accurately captured the phase variations. Although the ray tracing amplitude prediction has slightly higher errors than the full wave simulation in comparison with the measurements, acceptable trend and variations have been achieved. Both simulations and measurement have yielded good true time delay properties for the port #1 and port #3 excitations. It is observable that the ray tracing has predicted lower array factor gain at 5 GHz, which is probably due to the higher amplitude variations across the aperture. This resembles the results of the X-band lens, namely: the edge port generally has higher error due to the uneven port geometry and pointing directions. Limited number of comparisons has been demonstrated above, considering the multiple port profile of the microwave lens, and comprehensive ray tracing performance for all beam and receiving ports. The accuracy of the proposed algorithm is quantified in the Table I for the C-band lens. In the table, the gain, , side-lobe amplitude, side-lobe position, main beam scanning angle and beam width have been compared respectively from Port 1 to Port 4 excitations at 4 GHz and 5 GHz. Taking the measurement as a reference, the results show that the accuracy of ray tracing represented in average errors

for the performance parameters parameter: gain 0.8 dB, 1.5 dB, side-lobe amplitude 1.5 dB, side-lobe position 0.5 degree, scanning angle 0.4 degree, and beam width 0.4 degree. Comparably, the full wave simulation average errors are: gain 0.2 dB, 0.4 dB, side-lobe amplitude 0.4 dB, side-lobe position 0.2 degree, scanning angle 0.1 degree, and beam width 0.1 degree. We provide the necessary geometry data for this lens in Table II so that interested readers can replicate the presented results. IV. CONCLUSION A fast ray tracing algorithm was proposed for the microwave lens simulation. The proposed method treats the tapered ports, port to port couplings, and transmission lines individually and adopts a ray tracing process to combine different models. Two simulated and measured Rotman lenses have been used to validate the design concept. The results were investigated from the aspects of 1) the amplitude and phase distributions across the receiving aperture for single beam port excitations at a single frequency; 2) the amplitude and phase couplings between the transmitting (beam) and receiving ports across the frequency band; 3)

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array factors for different beam port excitations at different frequencies. Comparison of results demonstrate that the ray tracing program is accurate in predicting both the phase variations and scanning beam properties. The aperture amplitude distribution for the edge ports indicates slightly higher errors than the full wave simulations and measurements, which is primarily due to the uneven edge port structures and pointing directions. The latter is expected to be improved by further edge ports’ geometry variation and pattern investigation. In general, as a simulation tool that is considerably faster than the full wave simulation, very prominent results have been achieved. This also offers possibility of incorporating the proposed method into current microwave lens optimization processes. REFERENCES [1] A. Babakhani, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “A 77-GHz phased-array transceiver with on-chip antennas in silicon: Transmitter and local LO-path phase shifting,” IEEE J. Solid-State Circuit, vol. 41, pp. 2807–2819, 2006. [2] S. Clark, C. Martin, V. Kolinko, J. Lovberg, and P. J. Costianes, “A real-time wide field of view passive millimeter-wave imaging camera,” in Proc. 32nd Applied Imagery Pattern Recognition Workshop, Washington DC, 2003, pp. 250–254. [3] J. Dong, A. I. Zaghloul, and C. J. Reddy, “Quadruple bandwidth true time delay printed microwave lens beam former for ultra wideband multifunctional phased array applications,” presented at the IEEE Int. Symp. on Antennas and Propagation, North Charleston, SC, 2009. [4] W. Rotman and R. Turner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propag., vol. 11, pp. 623–632, 1963. [5] R. C. Hansen, “Design trades for Rotman lenses,” IEEE Trans. Antennas Propag., vol. 39, pp. 464–472, 1991. [6] C. Rappaport and A. Zaghloul, “Optimized three-dimensional lenses for wide-angle scanning,” IEEE Trans. Antennas Propag., vol. 33, pp. 1227–1236, 1985. [7] J. Dong, A. I. Zaghloul, and R. Rotman, “Phase error performance of multi-focal and non-focal 2D Rotman lens designs,” IET Microw. Antennas Propag., vol. 4, pp. 2097–2103, 2010. [8] M. S. Smith, “Amplitude performance of Ruze and Rotman lenses,” Radio Electron, vol. 53, pp. 329–336, 1983. [9] M. S. Smith, “Design considerations for Ruze and Rotman Lens,” Radio Electron. Eng., vol. 52, pp. 181–187, 1982. [10] M. Maybell, “Ray structure method for coupling coefficient analysis of the two dimensional Rotman lens,” in Proc. Antennas Propag. Society Int. Symp., 1981, pp. 144–147. [11] C. W. Penney, R. J. Luebbers, and E. Lenzing, “Broad band Rotman lens simulations in FDTD,” in Proc. IEEE Int. Symp. on Antennas Propag., 2009, vol. 2B, pp. 51–54. [12] N. Yuan, J. S. Kot, and A. J. Parfitt, “Analysis of Rotman lenses using a hybrid least squares FEM/transfinite element method,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 148, pp. 193–198, 2001. [13] J. Dong, A. I. Zaghloul, R. Sun, C. J. Reddy, and S. Weiss, “Rotman lens amplitude, phase, and pattern evaluations by measurements and full wave simulations,” J. Appl. Comput. Electromagn. Society (ACES), vol. 24, 2009. [14] J. Dong, A. I. Zaghloul, and R. Rotman, “A Fast Ray Tracing Method for Microstrip Rotman Lens Analysis,” presented at the XXIXth URSI General Assembly Chicago, 2008.

[15] D. M. Pozar, Microwave Engineering. New York: Wiley, 1998. [16] A. Mittal, K. K. Gupta, G. P. Srivastava, P. K. Singhal, R. D. Gupta, and P. C. Sharma, “Contour integral analysis of planar components,” J. Microw. Optoelectron., vol. 3, p. 15, 2003. [17] G. Razmafrouz, G. R. Branner, and B. P. Kumar, “Formulation of the Klopfenstein tapered line analysis from generalized nonuniform line theory,” Circuit Syst., vol. 3, p. 18, 1996. [18] R. H. Clarke, Diffraction Theory and Antennas. E. Horwood: Halsted Press, 1980. [19] J. L. Cruz, B. Gimeno, E. A. Navarro, and V. Such, “The phase center position of a microstrip horn radiating in an infinite parallel-plate waveguide,” IEEE Trans. Antennas Propag., vol. 42, p. 4, 1994. [20] P. S. Simon, “Analysis and synthesis of Rotman lenses,” presented at the 22nd AIAA Int. Communications Satellite Systems Conf. Exhibit, May 2004. [21] O. Kilc and S. J. Weiss, “Rotman lens applications for the U.S. Army—A review of history, present and future,” Radio Sci. Bull., Jun. 2010. Junwei Dong was born in August 1983, in Luquan, Hebei, China. He received the B.Sc. degree in measurement & control technology and instrumentation from Jilin University, China, in 2006, and the M.Sc. and Ph.D. degrees in electromagnetics from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 2008 and 2009, respectively. After graduation, he joined the Virginia Tech Antenna Group (VTAG) and started his study and research on microwave and antennas. In September 2009, he became a Research Scientist at Microwave Engineering Corporation (MEC) in North Andover, MA, where he researches the physics of antennas and filters using various computational methods.

Amir I. Zaghloul (S’68–M’73–SM’80–F’92– LF’11) received the Ph.D. and M.A.Sc. degrees from the University of Waterloo, Canada, in 1973 and 1970, respectively, and the B.Sc. degree (Honors) from Cairo University, Egypt in 1965, all in electrical engineering. He is with Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, and the US Army Research Lab (ARL) on an IPA (Inter-Governmental Personnel Act) arrangement. He has been with the Bradley Department of Electrical and Computer Engineering at Virginia Tech since 2001, prior to which he was at COMSAT Laboratories for 24 years performing and directing R&D efforts on satellite communications and antennas. Dr. Zaghloul is a Life Fellow of the IEEE, Fellow of the Applied Computational Electromagnetics Society (ACES), Associate Fellow of The American Institute of Aeronautics and Astronautics (AIAA), and Member of Commissions A, B & C of the US national Committee (USNC) of the International Union of Radio Science (URSI). He was the General Chair of the 2005 “IEEE International Symposium on Antennas and Propagation and USNC/URSI Meeting,” held in Washington, DC, and served as an Ad Com member of the IEEE AP Society in 2006–2009. He also served on the IEEE Publication Services and Products Board and on the Editorial Board of “The Institute.” He is a Distinguished Lecturer for the IEEE Sensors Council. He received several research and patent awards, including the Exceptional Patent Award at COMSAT and the 1986 Wheeler Prize Award for Best Application Paper in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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The Wiener-Hopf Solution of the Isotropic Penetrable Wedge Problem: Diffraction and Total Field Vito Daniele and Guido Lombardi, Member, IEEE

Abstract—The diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized Wiener-Hopf equations, and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind. Mathematical aspects are described in a unified and consistent theory for angular region problems. The formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique. The solutions consider engineering applications in terms of GTD/UTD diffraction coefficients and total fields. Index Terms—Analytical-numerical methods, electromagnetic diffraction, Fredholm integral equations, geometrical and uniform theory of diffraction, geometrical optics, isotropic media, spectral factorization, wedges, Wiener-Hopf method.

I. INTRODUCTION OWADAYS, accurate and efficient solutions of diffraction problems are of great interest in engineering, mathematical and physical communities. This paper presents a general solution of the diffraction by an isotropic penetrable wedge, see Fig. 1. The diffraction by a penetrable wedge has constituted in the last century and constitutes an important and challenging problem. Several attempts to find the solution have been reported in literature [1]–[31], where different formulations and analytical and/or numerical approaches have been presented. All the cited papers are of great interest, however some of them proposed incorrect methods and/or solutions. One of the most interesting attempts to solve the penetrable wedge problem was proposed in 1964 by Radlow [1]. This author provided a solution for the diffraction by the right-angled dielectric wedge solving a multidimensional Wiener-Hopf equation. In 1969, Kraut et al. ascertained that this solution was wrong [3]. Moreover the methods proposed by Zavadskii [2], and Aleksandrova-Khiznyak [5] were also wrong, see [7]. In 1977 Rawlins [8] provided a solution of the dielectric wedge problem using a general Integral Equation formulation, which is based on a standard perturbation technique.

N

Manuscript received November 08, 2010; revised February 03, 2011; accepted March 25, 2011. Date of publication August 04, 2011; date of current version October 05, 2011. This work was supported in part by the Italian Ministry of Education, University and Research (MIUR) under PRIN Grant 20097JM7YR. V. Daniele is with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Torino, Italy and also with the Istituto Superiore Mario Boella (ISMB), Torino, Italy (e-mail: [email protected]). G. Lombardi is with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2163780

Fig. 1. The isotropic penetrable wedge.

Its method was generalized in 1991 by Kim et al. These authors proposed an approximate solution of an arbitrary-angled dielectric wedge, which is obtained by performing physical optic approximation to the dual integral equation in the spatial frequency domain [12], [13]: the results are presented in terms of diffraction coefficients and far-field patterns. In 1995 Budaev proposed the application of the popular and effective technique known as Sommerfeld-Malyuzhinets (SM) method (for instance see [19] and [32]), to deal with dielectric wedge problems [18]. The difference equations that arise from this formulation are originally reduced to singular integral equations. A regularization method reduces them to Fredholm equations. Budaev focused his monograph on the correct mathematical formulations avoiding engineering speculations such as the evaluation of the diffraction coefficients. In the authors’ opinion, the most significant results obtained for the penetrable wedge geometries arise from works using formulations in the one-dimensional spectral domain as in [18]. Some of these works produced very important contributions. In particular, in 1999, the monograph about elastic wedge by Croisille and Lebeau [21] presents a formulation of the problem in terms of singular integral equations in the Fourier domain: these equations were successfully solved by using the Galerkin collocation method. Theoretical and numerical aspects of Budaev’s work were discussed in several papers, see for example [22] and references therein. In particular the paper by Kamotski et al. [22] has investigated the diffraction phenomena in an elastic wedge. Formulations in a one-dimensional spectral domain based on the Kontorovich-Lebedev (KL) transform have been proposed in [9] and [15]. In particular, in 2006, Salem et al. estimate the electromagnetic field excited by a line source in the presence of an infinite dielectric wedge [29]: the solution is given in terms of asymptotic approximations for the near and far fields inside and outside the dielectric wedge. In this paper we present a new method in the one-dimensional spectral domain based on the Wiener-Hopf (WH) technique. It

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constitutes an extension of the Wiener-Hopf formulation for impenetrable wedge problems that has been exhaustively considered in the past by Daniele and Lombardi [33]–[36]. The central problem of the proposed method is the factorization of matrix kernels in the WH formulation. Even though this problem has been considerably studied in the past, a general method to analytically factorize an arbitrary matrix is not known up to now. Since the isotropic penetrable wedge with arbitrary aperture angle is a problem where no closed-form general WH factorization is available, we propose optimal approximate factorizations by using the Fredholm method introduced in [37] and [34] (inspired by [38]) that reduces the factorization problem to the solution of Fredholm integral equations of second kind. For the sake of simplicity, in this work, we first present the WH formulation of the isotropic penetrable wedge’s diffraction problem of a plane wave at skew incidence, and then we solve the diffraction by a dielectric wedge at normal incidence focusing the paper on all the mathematical/physical properties to get the solution. This paper is organized as follows: Section II deduces the Generalized Wiener-Hopf Equations (GWHE) of the isotropic penetrable wedge at skew incidence. After introducing useful mappings, the GWHE are reduced to two systems of equations with classical WH unknowns. These Classical Wiener-Hopf Equations (CWHE) are solved using the Fredholm factorization method [37] that reduces the factorization problem to the solution of systems of Fredholm integral equations of second kind. Section III shows the numerical implementation of the proposed method for the case of an E-polarized plane wave incident on a dielectric wedge. In particular we present the numerical solution of the Fredholm equations for the normal incident case and we provide approximate representations of an analytical element of the WH unknowns in the angular complex plane . The same section addresses the analytical continuation of the approximate representations. Section IV deals with the evaluation of the electromagnetic far-field in the whole spatial domain for the normal incident case. In particular this section presents the solution in term of total field by estimating the field components: the Geometrical Optics (GO) component, the diffracted component, possible surface and lateral waves. Note that the geometrical optics contribution can be deduced from the WH formulation without the necessity of solving the Fredholm equations. Finally, numerous significant test cases are presented in Section V to validate our technique and practical discussions are included. We conclude the paper with three Appendices which are fundamental from an implementation point of view. The first is devoted to the evaluation of the source term in the Fredholm equation in the case of a plane wave incident to the wedge. The second Appendix concerns the special mappings used in the analytical continuation of the approximate solutions, and the third one is focused on spectral properties of the solution. For the sake of brevity, we have omitted several mathematical proofs that the reader can find in [28], [35], [37], [41], [42]. We assert that an important advantage of using the GWHE formulation (as well as the spectral method proposed in [21]) is the possibility to solve wedge problems immersed in anisotropic or bianisotropic media. Apparently this extension is not possible in the framework of the Sommerfeld-Malyuzhinets formulations.

II. THE WIENER-HOPF FORMULATION Fig. 1 illustrates the problem of the diffraction of a plane wave at skew incidence by an isotropic penetrable wedge and permeability having permittivity immersed in the free space (permittivity and permeability ). We consider the cylindrical coordinate system and time harmonic electromagnetic fields with a time dependence which is omitted. The incident field specified by the factor is constituted by plane waves having the following longitudinal components: (1) where: and are respectively the zenithal and the azimuthal angles which define the direction of the plane is the wave number and, wave, and are respectively the longitudinal component and the transverse component of the wave vector. Fig. 1 shows two media and four angular regions: 1) , 2) , 3) , and 4) . The first two regions are in free space, the second two are in the isotropic penetrable medium that constitutes the wedge. To facilitate the readability of the paper, we will extensively use the and for the supplementary angles definition of quantities inside the wedge. According to geometrical optics, the field inside the isotropic penetrable medium is , the loncharacterized by: the wave number gitudinal component and the transverse component of the wave and where is detervector (the electromagnetic properties of mined by enforcing the wedge are independent of ). The Wiener-Hopf technique for angular regions’ problems [33], [35] is based on the Laplace transforms of the longitudinal and tangential components of the electromagnetic field:

(2) indicates plus (minus) functions, where the subscript i.e., functions whose regular half-plane is the upper (lower) half -plane. To avoid the presence of singularities on the real axis, we assume with a small negative imaginary part. In the following we will use multiple complex planes ( plane and plane) for the definition of Laplace transforms along different directions :

(3) According to the theory presented in [28], [33], [35], [39]–[42] the generalized Wiener-Hopf equations for the

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

diffraction of a plane wave at skew incidence by a penetrable wedge are reported in (4)–(7), respectively, at the bottom of the page, for the angular regions of Fig. 1 numbered in ascending order. The GWHE are written in terms of the following quantities (8). Note that, in the expression we define the proper branch for of the square root the one that assumes the value

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TABLE I DEFINITIONS

(8)

In order to obtain a compact formulation of the problem we introduce the generalized factorization of the functions , , and reported in [33]: generalized factorization means that and . The use of these factorizations and mathematical manipulations (sum and subtraction) of (4)–(7) yield a new system of (i.e., the axial equations where the -plus functions for , , and ) are despectra -spectral functions defined for termined in terms of - and (facial spectra). It yields a system of eight functional , 3, 5, 7. equations reported in (9) for (9) For the sake of brevity Table I reports all the plus/minus unknowns of (9) defined in terms of: (10) (11) where , and . Equation (9) are GWHE since the unknown functions are de. fined in different complex planes, i.e., and or This system of equations are the Wiener-Hopf formulation of the problem under investigation.

A. Reduction of GWHE to CWHE In order to solve the system of GWHE (9) where multiple complex planes coexist, we introduce the special mapping (12)

(4)

(5)

(6)

(7)

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defined in [33] and used in [36] to solve the equations for impenetrable wedge. This mapping is used to obtain CWHE in anand transverse component gular regions with aperture angle of the wave vector . The mapping is used in each equation of (9) depending on the appropriate angular region (12) With reference to Fig. 1, in regions 1 and 2 the correct mapping is the first of (13) and is applied to the first equations of (9), on the contrary in regions 3 and 4 the mapping is the second one of (13) and is applied to the second equations of (9).

(13)

presents singularities in the standard regularity half plane . The non-conventional singularities are typically poles arising from geometrical optics contributions. Since we suppose that there are no sources in the interior of the penetrable wedge, the unconventional unknowns are only the one defined in the exterior region, i.e., the unknowns defined in the complex plane. We can intuitively deduce if a plus/minus Laplace transform (2) of a plane wave is standard or not, by examining the direction and the orientation of its flow. If the Laplace transform is performed along a certain direction i.e., positive axis) and the plane wave (for instance is flowing along the same direction but opposite orientation direction) we obtain a spectrum with a pole in the upper ( half-plane when the medium is supposed with small losses . In this case we obtain standard minus functions and non-standard plus functions. B. Fredholm Factorization

This procedure yields a new system of eight equations (14) with , 3, 5, 7 and where the following notations have been used: , , , and , , . In and functions, consti(14) the terms, that combine the tute the matrix WH kernel of the system of equations whose elements are defined in two complex planes ( and ). Note that from (8) and (13) (15) (16) We recall that the factorizations of functions , are studied in [33] and for the sake of readability we report them below: (17) (18) In (14) some of the Wiener-Hopf unknowns are non-conventional. We define non-conventional or non-standard plus that (minus) Laplace transform, the functions

To obtain approximate solutions of the system of (14) we apply the Fredholm factorization method described in [34], [37]. This method reduces the WH equations to Fredholm integral equations using the contour integration and the Cauchy formula. The integral equations of the Fredholm factorization are written only in terms of conventional plus (minus) unknowns [37]. We recall that only the unknowns defined in the complex plane can be non-conventional, since there are no sources in the interior of the penetrable wedge. The geometrical optic pole , see [36]. This pole is related to is three waves: the incident wave, the face reflected wave and in the complex the face reflected wave. The location of plane depends on . If the is located in the upper (lower) half of the complex plane yielding . unconventional plus (minus) unknowns with The extraction of non-conventional parts on the non-conventional WH unknowns yields the source terms in the Fredholm equations, see [37]. The source terms in the Fredholm equations are related to incident field and/or reflected fields as the associare located in the proper or improper sheet of the ated poles -plane (see Appendix I of [36]). The associated poles can be captured by contour integration only if they are located in the proper sheet of . A complete discussion on the source terms of Fredholm equations is reported in Appendix I. Using the Fredholm factorization method we obtain the system of integral equations (19), at the bottom of the page, , 3, 5, 7 and where the source terms are with described in Appendix I.

(19)

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

The complete solution of Wiener-Hopf problem is obtained in terms of the spectral unknowns by numerically solving the minus unknowns in (19). We note that the plus unknowns can be obtained through (14) or by using the equivalent integral representation available from (19):

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may represent three different waves (the inWe assert that cident wave, the face reflected wave and the face reflected wave) that have three different representation in -plane: see Appendix I. Given (8), (17), (18) the following representations hold in the and ( and ) planes:

(20) (23)

C. Approximate Solutions of the Fredholm Equations The numerical solution of the Fredholm integral (19) is obtained in several steps. Taking inspiration from the scheme already used in other problems, see [36] and [37], the steps are: • formulation of the Fredholm equations in the angular complex plane ; • introduction of contour deformation to enhance the convergence of the Fredholm equations; • introduction of mapping to relate the unknowns defined the inner and the outer of the wedge [41]; • numerical discretization of the equation and numerical representation of the solution in the angular plane ; • analytic continuation of the approximate solutions through recursive equations in the angular complex plane. The angular complex plane is particulary useful to estimate the far field components (Section IV) as already shown in [36] for the impenetrable wedge case. Since the Fredholm integral (19) are written into two complex planes ( and ), we need to define two angular complex planes and two modified angular complex planes (the overlined ones) respectively related to quantities defined in the free space region and in the isotropic penetrable region:

(21) with and . The properties and the inverse transformations of the mappings are reported in Appendix I of [36]. We recall that in the angular complex planes ( or ), all the plus functions are even functions [33], [42]. From (13) and (21) we obtain the Snell law in the spectral domain

In the following we will use the notations (24) for the axial defined in different spectral domains: spectral unknowns outer (inner) axial refers to direction

(24) A similar notation is applied to quantities with second argument (spectral unknowns for arbitrary direction ), for example . The second step of the procedure allows a fast convergence of the Fredholm equations by contour deformation. The real axis contours in the first and second equations of (19) are warped into and respectively in the straight lines that join the and planes. These straight lines and correspond to the planes parameterized by two lines respectively in the and (25) with real , . Therefore we have:

(26) The system of equations (19) become (27), at the bottom of the page, with , 3, 5, 7 and where , , , are defined as follows:

(28)

(22)

(27)

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We observe that the second equations in (27) are valid in the whole complex -plane through analytical continuation. By en(third step) we obtain the folforcing the constraint lowing complex mapping between -plane and -plane:

TABLE II DEFINITIONS FOR E-POLARIZED PLANE WAVE AT NORMAL INCIDENCE

(29) . The mapping (29) enforces in (27) the where same parameterization of the spectral quantities (10) and (11) defined on the faces and in particular the parameterization of the : spectral voltages and currents for and . The purpose of this procedure is to obtain a solvable system of equations whose unknowns are consistent: for example in terms of avoiding . the Since the integral term in the second equations of (27) is performed along the real axis of the -plane (that corresponds to a curve in the complex -plane), we need to estimate the quantities in terms of the functions . This requirement is achieved through the application of the Cauchy formula:

i.e., explicitly

(33) (30) Similarly we obtain:

where (31)

(34) The fourth step is efficiently implemented using simple quadrature rules as demonstrated in [36] for impenetrable wedges. Finally the analytic continuation of the solution is achieved using recursive equations obtained from the WH formulation (4)–(7) written in the angular domains and . For the sake of simplicity, in the following, we develop the procedure to obtain the numerical solution for the particular case of diffraction of an E-polarized plane wave by a dielectric wedge at normal incidence. III. NUMERICAL IMPLEMENTATION: E-POLARIZED PLANE WAVE AT NORMAL INCIDENCE Since we are dealing with an E-polarized plane wave at , , normal incidence on a dielectric wedge ( , i.e., acute wedge), all the equations reported above are simplified, although the procedure to derive the solution remains similar. In particular the equations reported in the previous sections are valid with the simplified explicit , , definitions reported in Table II. Note that , , , , , , are null for , 6, 7, 8 while , 3. and therefore (27) are not trivial only for , 3. Using (28), the definitions Let us consider (27) for of Table II, (23) and (25) we obtain that: (32)

(35) (36) As reported in step 4 of the previous section, in order to obtain a solvable system of equations from (27), we need to estimate the quantities in terms of the functions . We recall that this procedure is required for the evaluation of the integral term in the second equations of (27). This requirement is achieved through the application of the Cauchy formula (30), explicitly reported below: (37) where

(38) By substituting (33)–(36) in (27) we obtain, with the use of (37), two decoupled explicit systems (39), at the bottom of the next page, amenable to be solved numerically in terms of functions.

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

Note that the quantities and are related to the E-polarized incident wave (see Appendix I for details), and when the plus unknowns are non-standard

(40) and defined in Appendix I. with Efficient approximate methods for the solution of Fredholm equations of second kind are widely available in literature, see for example [43]. Since the kernel of (39) presents a well suited behavior, we use a simple sample and quadrature scheme to obtain accurate and stable numerical solutions. We apply uniform sampling with and modified left-rectangle numerical integration formula where and are respectively the truncation parameter and the step parameter for the integrals in . This rule has been successfully applied for the impenetrable wedge case [36]. The . We observe that as total number of samples is and , the numerical solution of the Fredholm integral equation converges to the exact solution [43]; consequently has to be chosen as small as possible and has to be chosen as large as possible. For instance, according to our experience, we assume and ( and are related to the second and fourth equations in (39)) to get stable solutions and which provide very accurate values in terms of samples, voltages’ and currents’ spectra and field components, see Section V. The discretized form of (39) is reported below:

(41)

where we need to use the discretized form of (37): (42) With reference to the system (39), in (41): • is the column vector containing the samples of function, • is the identity matrix, is the diagonal matrix that represents the function mul• tiplying in equation number ,

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is a full matrix that represents the kernel in equation or , number that multiplies is a full matrix that represents the operators (38), • is the column vector containing the samples of the source in equation number . All the matrices and vector quantities are of dimension . Simple algebraic manipulations allow to obtain two linear and systems of dimension where the unknowns are only . , The physical plus WH spectral unknowns , and are reconstructed in the and planes (respectively , , and ) via the sampling of (20) and using the definitions of Table II and (23): see (43)–(46), at the bottom of the next page, are obtained through for the explicit formula where (42) and non-standard plus unknown are considered, i.e., . Note that the discretization of kernel in (39) yields artificial poles in (43)–(46) due to the zeros of when . These poles correspond to spurious singularities on the axial spectra for and . Since the solution is obtained via numerical proand (25), i.e., the vercedure along the lines i.e. and tical lines i.e. , the two pairs of (43)–(46) provide only analytical elements of the axial spectra. We define the starting spectra as the axial spectra respectively in the regularity strips and . Note that the starting spectra show only the pole singularity of the incident field. Another important property is that the regularity segment belongs to the proper sheet as defined in Appendix I of [36]. To apply the above procedure it is important to study the behavior at infinity of the spectra. This was accomplished in [41] and for the sake of brevity it is not reported here. planes we In order to obtain the global spectra in and need analytical continuations of the numerically approximated analytical elements. We note that if the problem were solved analytically, the closed form solution would be valid in the encomplex planes. However this is not possible in tire and the general case of an isotropic penetrable wedge with arbitrary aperture angle. The analytical continuation of the numerical results is an old and cumbersome problem of applied mathematics that can be approached in various ways. In this work we resort to recursive •

(39)

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equations obtained representing the GWHE (9) of the problem planes using Table II and (22) and (23). By in the and using the continuity relation (99) and eliminating the spectral , we obtain (53), at unknowns defined at the interface the bottom of the page, where we have defined the functions (47) (48) and and where we have introduced the derived from (22) and described in Appendix II:

Note that the recursive formula (53) of the axial unknowns with real argument requires the evaluation of with complex arguments , see the axial unknowns Appendix II and test case 1 in Section V. The use of rotating waves [44] enables us to represent in the angular complex planes ( and ) the Laplace transforms of the spectral unknowns for a direction in terms of the axial spectra

functions

(49) (50) Since plus functions are even functions in the angular planes or [42], we assert that and are odd. The symmetry properties of plus/minus functions together with (53) , , ensure the analytical continuation of and . For instance, let us consider . Its is obtained through the apcorrect evaluation for each valid proximate analytical element : in

(51)

(54) where we have defined the auxiliary quantities:

(55) We observe that the quantities defined inside the dielectric can be derived using symmetry from wedge using the quantities defined outside the wedge the following substitutions:

(52) (56)

(43)

(44)

(45)

(46)

(53)

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IV. FAR-FIELD EVALUATION This section is devoted to the evaluation of the far-field pattern inside and outside the wedge. For the sake of simplicity we refer to a dielectric wedge structure illuminated by an E-polarized plane wave at normal incidence, generalization to skew incidence can be extrapolated from [36] where the impenetrable wedge is discussed. In this section we make reference to the evaluation of the exact field components in the exterior region . Note that the procedure can be extended to the evalthrough the uation of the field in the interior region symmetry relations (56). The exact total field is given by the following inverse Laplace transforms: Fig. 2. Horizontal (Im[] = cost:) Bromwich contours and SDP contour 0:1j . The positive in the w -plane with branch points at k and k = 1 Re[ ] direction in B corresponds to the direction of (B ) towards  + arctan( Im[k ]=Re[k ]) j = 3:042 j in the w plane. The symbols are geometrical optics poles of the outer axial spectra referred to the test case 1 of Section V. From the left to the right side the symbols correspond respectively to the face b reflected wave, the face a reflected wave, incident wave, see also Fig. 6 of test case 1 in Section V.

6

(57) is the Bromwich contour for , . where We recall that the singularities of standard plus functions are is any arbitrary located in the lower half-plane. In this case horizontal line located in the upper half plane. we obtain: By introducing the -plane

0

01 0

0

0

01

The saddle point of the function steepest descendent path is:

is

and the

(60)

(58) where is the mapping of the contour into the -plane. Fig. 2 reports, in the -plane, possible choices of Bromwich . These contours contours, i.e., horizontal lines are consistent with Figs. 13 and 14 of [36] where the properties of the two complex planes and are described. Far-field components (59) are obtained applying the steepest descent path (SDP) method to (58): (59) is the geometrical optics (GO) contributions (see [12] where the diffracted field, the possible contribufor details), tions of the surface waves, the possible contributions of the and in lateral waves. Equation (59) introduces the field the total field. These possible components derive from structural singularities: respectively poles and branch points of the recursive equations (53). In particular the branch points are sinand . At first sight, the gularities of the functions spectral content of the solution shows that surface waves are not excited at normal incidence for E-polarization, on the contrary in the general case of skew incident are possible. The evaluation and as well their interaction with the UTD contribuof tion requires further studies. Interesting considerations on the mathematical existence of the branch line contributions as well as on the radiation conditions in the elastic wedge problems are reported in [22].

denotes the Guderwhere mann function. Fig. 2 reports the SDP contour, too. To integrate (58), the con. tour is deformed to the SDP passing over the saddle point contour to the SDP, we assume In order to deform the with where small loss assumption is con. This choice also avoids the influence of the sidered branch line cuts of the function (the branch points are ) on the approximated numerical solution, see Fig. 14 of [36] for details. The contour deformation process can capture singularities of as poles and branch points, located in the region beand the SDP. On the SDP the extween the two contours is equal to ponential argument where is a continuous real function that goes to toward the end points of the path. The total far field assumes the following form (61), where poles are related to geometrical optics fields’ components (nonstructural singularities) and possible surface waves (structural singularities), whereas branch points are related to lateral waves and functions (structural singularities) due to the

(61)

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In (61) are the poles of , are , and is the conthe poles of the axial spectral unknown tour deformation to consider the possible contribution of branch originated by the function in the application of points recursive equations.

B. Diffracted Fields The SDP integral in (61) represents the diffracted field:

(64)

A. Geometrical Optics Fields The contribution of geometrical optics field arises from the residues of the poles when the poles are captured by the contour deformation from to the SDP in the -plane (see Fig. 2):

where is the Geometrical Theory of Diffraction (GTD) coefficient. As , the major contribution in (64) because of the exponential is located near the saddle point , therefore the GTD diffraction coefficient is: decay of (65)

(62) where . The number of GO poles (non-structural singularities) deand the observation angle . pends on the incident angle For certain ranges of the two angles we can have contribution from incident plane waves, reflected plane waves, transmitted plane waves and multiple reflected/trasmitted plane waves. Besides, the poles relevant to the plane waves could be complex in presence of total reflections inside the wedge with complex trasmission/reflection coefficients. The GO terms assumes the following form:

. The external GTD diffrac) assumes the explicit form (66) where is an odd function for real value of while the plus functions and are even where tion coefficient

(66) Equation (66) is consistent with the definition in terms of Sommerfeld’s functions presented in [36] (67)

(63)

(68)

is the unit vector of the where associated plane wave. . When we vary Let us suppose real poles of the observation angle , some of the poles can cross to the left the SDP contour centered in . In this context these poles are not anymore captured by the contour deformation and their contributions disappear in the total field creating shadow regions for the corresponding GO waves. We are the poles of the axial spectrum recall that the for and is the saddle point in SDP. The shadow regions are generated by the poles located in the interval as . Shadow regions are related to diffraction component to obtain continuous field passing through the shadow boundaries. Since the Fredholm factorization provides the approximate , we solution of the spectra only in the strip must resort to the recursive equations (53) in order to obtain the requested spectra. We observe that the integral term in Fredholm integral equations contributes only to the diffracted fields since it does not contain any poles. Therefore we can obtain the poles and the relevant residues for the non structural poles by ignoring the integral term in the Fredholm factorization, hence it is not necessary to solve the integral equations to estimate the GO components. This property is well known in the literature as reported in [18], [21], [23]. Similar considerations can be applied for the interior region. An excellent discussion of geometrical optics’ contributions is reported in [12] where multiple reflected and transmitted waves are treated.

Uniform expressions of the diffraction component are obtained using the uniform theory of diffraction (UTD) [45]–[48] (69)

(70) where , and are the Fresnel’s reflection coefficients respectively due to the first reflection on face and , and the multiple transmissions/reflections through the wedge (see also [12] for the evaluation of the coefficients). Uniform expressions are near the saddle point are required when GO poles . In particular we recall that shadow regions are possible for singularities of the axial spectra located in . The uniform expression ensures the continuity when the observation angle crosses the shadow boundaries.

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The function is the Kouyoumjian-Pathak transition function defined in [47] and its application in the framework of Wiener-Hopf formulations is reported in (63) of [36]. , we need to Concerning the interior region slightly modify the equations and the quantities involved in the definition of the diffracted field. Using the symmetry relations (56) it yields (71) (72) with explicit expression reported as follows:

(73) As for the exterior region, uniform expressions are required when GO poles (with poles of the axial spectra and ) are near : shadow region are possible for poles in the saddle point . Consequently uniform expressions of the diffracted field (74) are of the same kind of the one is continuous for exterior region (69) and when it crosses the shadow boundaries inside (as outside) the dielectric wedge (74)

Fig. 3. Test case 1: the GO field, the UTD component and, the total far-field . pattern at k

= 10

All the test cases make reference to Fig. 1. In particular, the wedge is illuminated by a plane wave impinging from a di(leaving the wedge with direction ), rection see (76). In this paper we denote the azimuthal direction of the where the subscripts are in upper case GO waves with (lower case) if referred to a wave that leaves (approaches) the wedge: for instance, the face reflected wave propagates as with and , see Fig. 1.

The complete GTD diffraction coefficient is defined by A. Test Case 1 (75)

Note that the complete UTD diffraction coefficient assumes the same form of . V. VALIDATION AND NUMERICAL RESULTS The efficiency, the convergence and the validation of the proposed approximate solutions is illustrated through several test problems. The quantities used in this section are explicitly defined in the previous section: Far-field evaluation. Some of the following numerical results and figures show the comparison between the solution of the dielectric wedge test case and the solution of the perfect conducting (PEC) wedge with the rest of physical parameters unchanged. The first test case is investigated in detail, reporting the whole procedure to solve the problem: from the definition of Wiener-Hopf spectral unknowns to the evaluation of the total field. Moreover, the test cases show the convergence properties of the proposed method and some physical properties of the diffraction by a dielectric wedge. The last test compares our solution with the one of [12], [13] and shows the computational efficiency of our method. The first three tests consider non-standard plus unknowns while the fourth non-standard minus unknowns.

The first test case analyzes all the properties of our solution in terms of spectral quantities, diffraction coefficients, total fields. With reference to Fig. 1 the physical parameters of the problem , , , and . are: According to GO, the E-polarized incident plane wave impinges on the dielectric wedge and generates two reflected waves and two transmitted waves. The two transmitted wave are not reflected in the interior region. This configuration allows to define four geometrical optics shadow boundaries: face reflected shadow boundary, face reflected shadow boundary, face transmitted shadow boundary and, face transmitted shadow boundary. No incident shadow boundary exists. As shown in Fig. 3 with different gray color backgrounds, there are six GO regions: incident wave region , incident and face reflected waves , incident and face reflected region , face transmitted waves region wave region , face transmitted wave and, face and transmitted region . waves region Fig. 3 reports the GO field, the UTD component and, the total from the edge of the wedge. far-field at the distance According to GO, the problem under examination shows: • a face reflected wave angle ;

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0  

Fig. 4. Test case 1: (a)–(b) Imaginary parts of spectral unknowns V^ (w ) and I^ (w ) in the regularity segment 8 w 0; (c)–d) relative error in log of Im[k V^ (w )] and Im[k I^ (w )]: the reference solution is obtained with discretization parameters A = 12, h = 0:015.

• a face

reflected wave angle

• a face

; transmitted wave angle

• a face

transmitted wave angle

and . The required analytical continuation is obtained through the recursive equations reported in (53). Fig. 6 shows the behavior of the absolute value of the specand . The tral unknowns figure highlights the spectral regions necessary to evaluate the GTD diffraction coefficients (66) and (73), i.e., and . The figure shows also the GO poles relevant to the GTD for test case 1. In show peaks for the reflected waves, particular while for the transmitted waves. The location of the poles agrees with the standard GO theory, for inand : using stance consider (66) and (73) and, considering that the spectral unknowns are , we obtain that even function in and (since ). Similar considerations hold for the other GO poles. The GO components can be obtained by using standard techniques or by applying (62). Notice that the study of the axial spectra is is a singularity of , fundamental. In fact, if presents the singularities the spectrum of that can be captured by the integration contour deformation from to SDP, see Section IV-A. The singulariare reported in Fig. 2 together with difties of and are alferent integration contours. While (purely imaginary most purely imaginary functions in

spectively in the interval

; . The solution of the problem is obtained applying the discretization method reported in Section III where the Fredholm factorization method is applied to the GWHE with discretiza, . tion parameters Fig. 4 shows the behavior of the numerical solution in terms and in the regularity of the spectral unknowns : the outer axial starting spectra (as desegment fined in Section III) is purely imaginary (Appendix III). Relative scale by considering as reference soerrors are reported in lution the one obtained for discretization parameters , . For the numerical solution we have chosen different values of the integration parameter and in order to confirm the convergence of our technique. However, an excessive value yields ill-conditioned matrices in the discretizaof tion process. Fig. 5 shows the behavior of the numerical solution in terms and in the reguof the spectral unknowns larity segment . As reported in Section IV the evaluation of the GTD coefficients requires the analytic continuation of the spectral unknowns and re-

scale

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

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0 



Fig. 5. Test case 1: (a)–(b) Imaginary parts of spectral unknowns V^ (w ) and I^ (w ) in the regularity segment 8 w 0; (c)–(d) relative error in linear scale of Im[k V^ (w )] and Im[k I^ (w )]: the reference solution is obtained with discretization parameters A = 12, h = 0:015.

in , see Appendix III); and are complex functions as shown in Fig. 7, where for the sake of simplicity we have reported only the voltage spectra. The approximate total GTD diffraction coefficients are estimated substituting the approximations of the spectral unknowns in (66) and (73). Fig. 8(a) reports the absolute value of the total GTD diffraction coefficient (in dB) for each observation angle . The peaks of the GTD diffraction coefficients occur for the GO angles: reflected and transmitted waves. The convergence is shown in Fig. 8(b) for different integration parameters through the evalscale with respect to the refuation of the relative error in erence solution obtained for , . The scale measures the level of precision in term of digits for each observation angle . Fig. 8(c) reports the phase of the total GTD diffraction coefficient (in dB) for each observation angle . Fig. 8(a) and 8(c) show also the plots for the PEC wedge. The complete solution is reported in Figs. 9 and 10. The first figure reports the total field, GO field component, UTD field . Gray regions are inside component at the distance the wedge. The second figure shows the comparison between the total field of the dielectric wedge with the one of the PEC wedge. We notice that Fig. 9 shows a small loss of convergence and : the reasons for the corner for behavior are different. In the first case, when , the problem is due to the spectral reconstruction of the Wiener-Hopf unknowns in , i.e., and . In fact, for

the UTD/GTD field component is related to the evaluation of and in , see (73) and (75). We recall that the recursive equations are used to estimate the spectral unknowns out of the regularity strips ( and ) in particular in the GTD intervals ( and ). Fig. 11 shows the mapping used in the estimation of functions out of the regularity segment for real value of (see (53)), i.e., . The map starts from that yields and goes all over the gray line as long as is mapped into . Uniform sampling for real is mapped into non-uniform sampling in complex . In particular the significant point is mapped into where the mapping shows a change in slope and highly non uniform sampling. This is the cause for the loss of convergence in UTD/GTD for . The second direction where we experience loss of converand it is due to the UTD uniform expresgence is is very sion of the GTD field component. Since , the close to the interface between the two materials Kouyoumjian-Pathak (KP) transition function [47] is not adequate to model the problem: the uniform diffraction component is a cylindrical wave whose intensity should vanish at the interface, on the contrary, at first sight, the KP transition function’s slope does not consider the change of materials.

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Fig. 6. Test case 1: (a)–(d) Absolute value of the spectral unknowns V^ (w ), I^ (w ), V^ evaluate the GTD coefficients.

B. Test Case 2 The second test case shows the convergence properties of our solution in terms of GTD diffraction coefficient. The closed form solution of the PEC wedge, see for example [36], allows to compare the dielectric wedge solution when the relative permittivity is with high imaginary part and real part equal to 1. Fig. 12 , shows the GTD diffraction coefficient (in dB) when , , and with , . By increasing discretization parameters the solution converges to the PEC wedge. C. Test Case 3 The third test case highlights the capabilities of our method to model the scattering and diffraction by a dielectric wedge in presence of multiple reflections and transmissions. With reference to Fig. 1 the physical parameters of the problem are: , , , and . The E-polarized incident plane wave impinges on the dielectric wedge and generates one face reflected wave and one face transmitted wave. The transmitted wave is totally reflected inside the wedge for two times and generates two evanescent transmitted waves through the two interfaces. This configuration allows to define three geometrical optics shadow boundaries (omitting the ones for the evanescent waves): incident shadow boundary, face reflected shadow boundary, face transmitted and double totally reflected shadow boundary. As a consequence, there are five GO regions:

(w ) and I^ (w ) in (02; 0). Dark gray regions are not used to

• region 1: incident wave; • region 2: incident wave, face reflected wave, evanescent wave through face ; • region 3: face transmitted wave, face reflected wave from face transmitted wave, double reflected wave from face transmitted wave; • region 4: face transmitted wave, face reflected wave from face transmitted wave; • region 5: evanescent wave through face . The GO field, the UTD component and, the total far-field patfrom the edge tern are reported in Fig. 13 at the distance of the wedge. The solution of the problem is obtained applying the discretized method reported in Section III where the Fredholm factorization method is applied to the GWHE with discretization , . Fig. 14 shows the behavior parameters of the absolute value of the spectral unknowns and . The figure highlights the spectral regions necessary to evaluate the GTD diffraction coefficients (66) and and (73), i.e., . The figure shows also the GO poles relevant to the GTD for test case 3. In particular show peaks for the face reflected wave (RA) and incident wave (I), while for the transmitted and double reflected wave (TARR). Notice that the lobes reported in Fig. 14(a) and (b) are related to evanescent waves (complex poles in ).

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Fig. 7. (a) Im[k V^ (w )] for the test case 1 and the PEC wedge, (b) real and imaginary part of k V^ (w ).

The complete solution is reported in Fig. 15 where the total field, GO field component, UTD field component are evaluated . Gray regions are inside the wedge. at the distance Comparison between the complete solution of the dielectric wedge with the one obtained with PEC wedge is also shown. Note that Fig. 15(a) shows loss of convergence in the dif. This spurious local fracted component for corner behavior of the solution is due to the effect of the mapin the spectral reconstruction ping through the recursive of the WH unknowns equations (53) as already discussed for the WH unknowns at the end of test case 1. D. Test Case 4 The fourth test case shows the validation of our method through the comparison of our solution with the one proposed in [12], [13]. In this test case we shows the capabilities of our method to model the scattering and diffraction by a dielectric wedge in presence of multiple reflections and transmissions and, the performance in terms of computational time. With reference to Fig. 1 the physical parameters of the problem are: , , , and . The E-polarized incident plane wave impinges on the dielectric wedge and generates one face reflected wave and one face

Fig. 8. Test case 1: (a) absolute value of the total GTD diffraction coefficient (dB), (b) GTD diffraction’s relative error in log scale using different set of integration parameters (reference solution A = 12, h = 0:015), c) phase of the total GTD diffraction coefficient.

transmitted wave. The transmitted wave is reflected and transmitted through face . The reflected part is then totally reflected on face and generates an evanescent transmitted wave. This configuration allows to define four geometrical optics shadow boundaries (omitting the one for the evanescent wave): incident shadow boundary, face reflected shadow boundary, double transmitted shadow boundary, transmitted-double reflected shadow boundary. As a consequence, there are six GO regions: • region 1: incident wave;

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Fig. 9. Test case 1: Total field (solid line), GO field component (squares), UTD . field component (triangles) at k

= 10

Fig. 12. Test case 2: the GTD diffraction coefficient (dB).

Fig. 10. Test case 1: comparison between the total field of the dielectric (gray line) and the PEC wedge (black line)at k .

= 10

Fig. 13. Test case 3: the GO field, the UTD component and, the total far-field pattern at k .

= 10

= 8+ ( +8 ) 08   0 = 08 = 8 0 arccos(p ) (arccos(p ) ' 1 1462) = 0 08 = 08 + arccos(p ) } 4 = 08 8 0 08 0 arccos(01 p ) 0 08

Fig. 11. Test case 1: w g w mapping used in the estimation of  functions out of the regularity segment w . The map starts from w that yields w " " | : and goes all over the gray line as long as w  is mapped into w " . Uniform sampling for real w corresponds to non-uniform sampling in complex w . The symbols , , , respectively are the mapped value of w , , = " ,  .

• region 2: incident wave, face wave through face ;

reflected wave, evanescent

• region 3: face transmitted wave, face reflected wave from face transmitted wave, double-reflected wave from face transmitted wave; • region 4: face transmitted wave, face reflected wave from face transmitted wave; • region 5: double transmitted wave through face and ; • region 6: no GO components. The GO field, the UTD component and, the total far-field patfrom the tern are reported in Fig. 16 at the distance edge of the wedge. The figure is obtained using the Fredholm , factorization method with discretization parameters and the results can easily be compared with the figures reported in [12] and [13]. the Fig. 16(b) shows a corner beNote that near havior due to the use of in the recursive (53). Table III shows the computational speed of our implementation in Mathematica on an Intel Core 2 Duo CPU ([email protected] GHz 3 GB RAM). Note that the use of Mathematica let us handle and verify all the mathematical details of the procedure. The use of a full numerical implementation of our method would speed up the entire evaluation.

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

Fig. 14. Test case 3: (a)–(d) absolute value of the spectral unknowns V^ (w ), I^ (w ), V^ evaluate the GTD coefficients.

VI. CONCLUSION In this paper we present a new method to study the diffraction by an isotropic penetrable wedge using the WH technique. The solution is presented in terms of GTD diffraction coefficients, UTD diffraction coefficients and total fields. Further work will be focused on the study of the contribution of the structural singularities and on the computational aspects for the general skew incidence case. APPENDIX I SOURCE TERM This appendix is devoted to study the source term (firstorder pole function) of the Fredholm integral equation formulation (19) when the isotropic penetrable wedge is illuminated by a general plane wave at skew incidence from the outer region. Without loss of generality, let us consider an E-polarized plane wave. The GO field of the outer region can be evaluated by solving the simple problem of reflection/transmission of plane waves at skew incidence:

(76)

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(w ) and I^ (w ) in (02; 0). Dark gray regions are not used to

is the unit step function, are the reflection where is the field re-transcoefficient of the two faces and and, mitted from inside the wedge (see [12] for a deep discussion), when it is present. we have identified In order to establish the source term the following strategy based on engineering and mathematical considerations: derives from A) assuming that the GO is valid, the Laplace transform of the known GO field for respectively in the complex planes , and ; B) we evaluate the source term using the residue theorem applied to the non-standard spectral unknowns (see Section II for definitions) only for the singularities located in the proper sheet of the complex plane that contains the segment , see [36]. In the following we consider the case of incidence , the opposite case is obtained using symmetry. By ignoring in a first moment the existence domain of the GO components (76), the application of the Laplace transform (3) yields several first-order pole terms with poles: to (76) for for the incident wave, for the face reflected wave, for for the re-transmitted waves. the face reflected wave and The first three poles have the same representation in the -plane: .

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Fig. 16. Test case 4: a) the GO field, the UTD component and, the total far-field .  , b) the UTD component at k pattern at k

= 10

= 10

Fig. 15. Test case 3: (a) Total field (solid line), GO field component (squares), , (b) total field for the dielectric UTD field component (triangles) at k and PEC wedge.

= 10

By ignoring in a first moment the existence domain of the GO Laplace transform components (76), the application of the yields several first-order pole terms (3) to (76) for with poles: for the incident wave, for the face reflected wave, for the face reflected wave and for the re-transmitted waves. Note that the first two poles have the same representation in the -plane and poles have the through (15) we obtain that the first three . same representation in the -plane: Similar considerations hold for the third case, i.e., : , the face , , and respectively for the incident, the face reflected, the face reflected and the re-transmitted waves. In this case and the first three poles have the same . representation We denote with the poles of the re-transmitted waves in . the complex planes for observation angles These poles are obtained using the first relation of (13) for and (15) for . is obtained through the We assert that the source term residue theorem applied to the non-standard spectral unknowns

TABLE III COMPUTATIONAL SPEED

only for the singularities located in the proper sheet of the complex plane that contains the segment [36]. complex plane Although the spectral unknowns in the present the poles and , we consider only the contribution of since: either 1) is associated to a re-transmitted wave with existence domain that excludes directions or 2) is located in the improper sheet of the plane for or 3) both the previous conditions are simultaneously satisfied. The second condition is easily verified using the -plane: the condition becomes with real for where corresponds to using the expressions of and in , see (22) and (23). Starting from (14) we extract the source term related to the from the non-standard source pole in the complex plane depends unknowns. The location of , if on . With small loss assumption , is located in the lower (upper) half complex plane yielding unconventional plus (minus) unknowns . The pole can be captured by contour integration only if it is located in the proper sheet of . This propis associated erty is well illustrated in the -plane: the pole

DANIELE AND LOMBARDI: THE WIENER-HOPF SOLUTION OF THE ISOTROPIC PENETRABLE WEDGE PROBLEM

to three different waves with same representation in -plane but different in other complex planes such as , and [36]. (22) and (23), we obtain that the incident Using wave, the face reflected wave, the face reflected wave show respectively the following poles in the -plane: : , , • for ; • for : , and , . Note that the reader can extrapolate the following other cases and using a similar procedure: . Only some of the above poles are locate in the proper , see -plane properties in Appendix I of sheet ( [36]) and they correspond to existing waves along the associated . direction Let us, now, suppose , the unconventional plus for , 3, 5, 7 can be decomposed in: function (77) where are standard plus functions representing the axial spectra without the GO components (in this case the incident only the GO pole is located in wave). Since the proper sheet , therefore the incident wave . According to is the only one that contributes to the residue (14) and the procedure described in [37] we define the source term: (78) is the residue of

in

:

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and with . In this second case, from (14), we define the source term: (82) and therefore using (81): (83) Notice that, for , the functions reported in (19) must be substituted with , which is the minus unknowns purified from the unconventional singular term via decomposition (in the Fredholm factorization the unknowns are always standard [37]). are obtained by using the Laplace transforms along the directions in the plane of the GO field components having singularities in the proper sheet of (see Table I and (10) and (11)). case: the to-beLet us focus the attention on the considered GO components are the incident wave and the face reflected wave. These waves must be considered only if face is not in shadow. Similar considerations hold for where we need to consider the incident wave face and the face reflected wave. Using Table I and (10) and (11) in terms of longitudinal and tangential for definition of field components we obtain and . assumes the folIn the general case, the source term lowing form

(84) (79)

with

and . are obtained by using the Laplace transform of the GO field components along the direction in the -plane. In this case the GO field is constituted by only the incident wave, in terms of lonsee (76). Using Table I for definition of and gitudinal and tangential field components we obtain . On the contrary, if , only the pole is located in the proper sheet . In this case the unconventional minus function for , 3, 5, 7 can be decomposed in:

where the explicit expressions of all non-zero and are reported in Table IV when the GO poles are related to the . incident wave and the face reflected wave Note that the signs of the residue terms in are decided according to the orientation of the integration contour in the Fredholm factorization procedure, see [37] and Section II-B. and are cumbersome coefficients of the face reand flected wave available from GO. At normal incidence are simple expressions:

(85) (80) where are standard minus functions representing the facial spectra without the GO components (in this case the inciis the residue of dent wave and the face reflected wave). in : (81)

, where is defined in (49). with Note that the reader can extrapolate the case where we need to consider the reflected wave from face . Example: For the sake of readability, we present some exwhen a dielectric wedge is illumiplicit expressions of nated by an E-polarized plane wave at normal incidence. for The first explicit expression under examination is thus we need to evaluate . In this case,

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011

TABLE IV DEFINITIONS FOR THE SOURCE TERM n  () FOR 0

We recall from the previous subsection that only the facial spectra of face have singularities in the proper sheet of plane, therefore we obtain that: