122 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY X/$ IEEE

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1 122 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010 Improved Model-Based Parameter Estimation Approach for Accelerated Periodic Method of Moments Solutions With Application to the Analysis of Convoluted Frequency Selected Surfaces and Metamaterials Xiande Wang, Senior Member, IEEE, and Douglas H. Werner, Fellow, IEEE Abstract An improved smart interpolation approach known as model-based parameter estimation (MBPE) is applied to the wide-band interpolation of periodic method of moments (PMM) impedance matrices for normal and oblique incidence cases. Prior to interpolation, easy to calculate but hard to interpolate, phase terms are removed from the impedance matrices. An efficient spectral-domain PMM formulation is introduced for the accelerated analysis of frequency selective surface (FSS) problems with a large number of unknowns, employing a one dimensional ( log ) FFT-based method to speed up the computation of matrix-vector products within the bi-conjugate gradient (BCG) iterative solver, which is made possible by the asymmetric multilevel block-toeplitz structure of the impedance-matrix. The MBPE interpolation algorithm provides a faster matrix fill time than the brute force method and is comparable or even faster than the 2-D FFT-based method for a large number of unknowns. It also has the advantage that it can be applied to non-uniform gridding cases. The accuracy and efficiency of the proposed techniques for large FSS problems are demonstrated by several design examples for both the normal and oblique incidence cases. We also apply this efficient analysis tool to the design of multiband single-layer FSS filters and artificial magnetic conductors (AMC) comprised of a 2-D periodic arrangement of convoluted metallic strips in the shape of a Hilbert curve. The multiband properties of the Hilbert curve FSS filters are studied for different iteration orders (i.e., different degrees of space-filling). Index Terms Artificial magnetic conductors (AMC), fast Fourier transform (FFT), frequency selective surfaces (FSS), impedance matrix interpolation, metamaterials, model-based parameter estimation (MBPE), multiband FSS, periodic moment method (PMM), space-filling Hilbert curves, spectral-domain periodic method of moments (PMM). I. INTRODUCTION SINCE their introduction, frequency selective surfaces (FSS) have been utilized in a wide range of applications such as electromagnetic filters, radomes, and more recently Manuscript received May 24, 2008; revised December 15, First published November 06, 2009; current version published January 04, This work was supported in part by the Penn State MRSEC under NSF Grant DMR and in part by ARO-MURI Award PH-MUR. X. Wang and D. H. Werner are with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA USA ( dhw@psu.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAP as components of metamaterials. Spatial-domain and spectral-domain formulations of the periodic method of moments (PMM) [1] [3] have been widely used for the analysis of FSS s. PMM formulations only require the discretization of one unit cell in a two-dimensional periodic FSS screen and employ a layered-media Green s function to account for the presence of the dielectric substrate and superstrate. This leads to an important advantage of the PMM; namely, it is very fast compared to other techniques (e.g., the hybrid finite element boundary integral (FEBI)-based method [4]) in the analysis of FSS structures composed of a homogenous layered-media. The most computer-intensive operations in the PMM are filling the impedance matrix and solving the linear matrix equation to obtain the current distribution on the FSS screen. Under certain circumstances of practical importance, these operations can become computationally burdensome, especially when performing frequency and angle sweeps. In order to overcome this shortcoming, impedance matrix interpolation techniques were first proposed in [5] to significantly reduce the large matrix fill times associated with certain antenna analysis problems. A modified three-sampling-point interpolation technique was later applied in [6] to more efficiently predict the response of FSS screens. A reduced-order model was also proposed to generate a broadband approximation of the reflection coefficients for multiple-screen FSS [7]. More recently, a model-based parameter estimation (MBPE) technique was used to effectively interpolate the frequency response of electromagnetic systems [8] and for wide-band interpolation of impedance matrices [9]. However, the MBPE-based technique proposed in [9] was only applied to the normal incidence case for the wideband interpolation of PMM impedance matrices related to the FSS with substrate and superstrate configurations but not to oblique incidence cases. In this paper, we extend it to oblique incidence cases by the development of a robust and efficient algorithm that can be described as a smart MBPE-based approach which removes problematic phase terms from the impedance matrices prior to the interpolation process. We also describe how this approach can be applied to the analysis of large FSS problems with the help of FFT acceleration techniques produced by incorporating a generalized one-dimensional (1-D) implementation of an FFT-based method into the PMM. This method has been originally proposed in [10] for X/$ IEEE

2 WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS 123 Fig. 1. Cross-sectional view of the FSS structure with both a dielectric substrate and a superstrate. analyzing 3-D scattering problems, to speed up computation of multilevel block-toeplitz matrix-vector products within the bi-conjugate gradient (BCG) iterative solver. The validation and the efficiency of the proposed algorithms are demonstrated by comparisons of the CPU times required for the conventional PMM and the improved MBPE-based interpolation algorithm for several design examples. We also describe the application of the proposed acceleration techniques to the design of a polarization dependent/independent multiband single-layer FSS as well as artificial magnetic conducting (AMC) surfaces using a 2-D periodic arrangements of convoluted Hilbert-curve-shaped strips with iteration orders as high as four [16]. In addition to AMC surfaces, the MBPE-based interpolation algorithm could also be employed for the efficient analysis and design of other types of metamaterials. II. IMPROVED MBPE INTERPOLATION AND FFT-BASED ACCELERATION A detailed discussion of the spectral-domain PMM can be found in [3] for the analysis of doubly periodic FFS structures (e.g., an FSS with both a dielectric substrate and a superstrate as shown in Fig. 1 or an FSS screen backed by a dielectric substrate as shown in Fig. 2). By expanding the surface electric current within a unit cell in terms of 2-D roof-top basis functions and then employing Galerkin s procedure to the appropriate boundary electric field integral equation, the following linear matrix equation is obtained: For the sake of illustration consider an expression for the elements in the submatrices given as (similar expressions exist for the other submatrices, and ) (1) Fig. 2. The Jerusalem slot FSS screen backed by a dielectric substrate with unit cell dimensions 2.5 cm cm and a thickness of 0.02 cm. convergence of the double summation. Therefore, we refer to this conventional PMM formulation as the brute-force calculation method. In order to reduce the large matrix fill times that can be associated with conventional PMM formulations, we employ an efficient MBPE interpolation method to estimate the impedance matrix elements following the ideas originally presented in [9], where only the normal incidence case was investigated. Here, a more robust MBPE-based interpolation technique is introduced for both normal and oblique incidence cases by removing phase terms from impedance matrix elements prior to interpolation, which are easy to calculate but difficult to interpolate. For example, an element of the submatrix can be expressed as follows: where is the phase factor associated with the source edge and test edge position and the incident wave vector, and for the normal incidence case. The following fitting model has been found to be very effective [the same MBPE model can be used for fitting all other elements in the matrices in (1)]: (3) (4) where is the spectral-domain Green s function, and denotes the Fourier transform of the basis function. Direct calculation of (2) is very time consuming due to the relatively slow (2) where. The parameters,,, are defined in [3], [9] for the FSS structure with both a dielectric superstrate and substrate. The parameters,, and in (4) denote the interpolation coefficients, while represents the different combinations of the Floquet harmonics and, denotes the total number of Floquet harmonics required, which is dependent upon the total number of

3 124 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010 cutoff frequencies that fall within the range of interest. By sampling at a total of frequencies points, (4) can be converted into the following matrix equation: where is called the interpolation matrix. The interpolation coefficients can be obtained by solving the matrix (5) via LU decomposition (LUD) or by using singular value decomposition (SVD), which is more robust than LUD. Upon obtaining interpolation coefficients, the interpolated impedance matrix elements can be calculated by multiplying the results from (4) by the phase factor, i.e.,. It is well known that the cutoff frequencies are dependent on the periodicity of the FSS structure and the working frequency range and are not impacted by the resonance behavior of the FSS. Hence, the fitting frequencies can be determined by the automated selection procedure described in [9]. In addition, since the interpolation matrix is independent of, only a one time implementation of LUD or SVD is required to solve for. The fast computation of the impedance matrix elements required to fill in (4) and (5), is achieved through a 2-D FFT-based method [11] which can be applied at desired sampling frequencies to calculate the double summation for the appropriate impedance sub-matrix elements of. The uniform discretization of the FSS screen and the roof-top basis functions employed here in conjunction with Galerkin s method lead to an impedance matrix in (1) that has an asymmetric multilevel block-toeplitz structure. An FFT-based method with a 1-D FFT implementation [10] can be applied to speed up the matrix-vector product within the BCG iterative solver. This can be classified as a truly minimal memory method because it stores only nonredundant matrix entries for the asymmetric multilevel block-toeplitz structure with memory requirements. It may be expressed as, where and are 1-D vectors which can be automatically generated from the asymmetric multilevel block-toeplitz matrix and vector, respectively. Here, and denote the 1-D forward FFT and the 1-D inverse FFT, respectively. A detailed discussion of the FFT-based method for a 1-D implementation can be found in [10]. III. MULTIBAND FSS AND AMC SURFACE DESIGNS WITH HILBERT-CURVE SHAPED STRIPS Conventional approaches to the design of multiband FSS typically involve using multilayer surfaces or perturbing the elements or the spacing between elements [12]. Multiband (5) FSS have also been designed by utilizing certain types of self-similar fractal elements [13], or by a combination of genetic algorithm (GA) and geometry-refinement techniques [14], [15]. More recently, spacing-filling curves [16] have been introduced to the microwave engineering community, e.g., space-filling Hilbert curves have been applied in the design of FSS [17], [18] and AMC surfaces [19], [20]. However, utilizing metallic strips in the shape of Hilbert space-filling curves with high iteration order (i.e., iteration orders greater than three) for application in multiband FSS and AMC surface designs have not been adequately investigated primarily due to limitations on conventional PMM modeling tools. In this paper, polarization dependent/independent multiband single-layer FSS and AMC surfaces are investigated using 2-D periodic arrangements of Hilbert-curve-shaped strips. The developed PMM solver has been utilized to efficiently analyze Hilbert-curve FSS and AMC surfaces with a large number of unknowns and higher iteration order than those considered in previous applications. IV. NUMERICAL RESULTS AND DISCUSSION A. Validation and Efficiency of the Proposed Techniques A specific FSS design example consisting of the Jerusalem slot FSS screen shown in Fig. 2 was considered to validate the accuracy and efficiency of the proposed PMM acceleration techniques. Note that nonmagnetic materials are assumed in the following simulations unless otherwise specified. For this example, as presented in Fig. 2, the FSS is assumed to be backed by a dielectric substrate with a thickness and relative permittivity of 0.02 cm and, respectively. The FSS structure is doubly periodic with a periodicity defined by. For this configuration, the total number of Floquet harmonics required to cover the frequency range of GHz was found to be based on the following equation, which can be used to predict the cutoff frequency for each Floquent harmonic denoted by and, shown in (6) at the bottom of the page, where and [9]. The locations of the [see (4)] fitting frequencies were determined by using the automated selection scheme described in [9], and are given by for the normal incidence case. A discretized unit cell consisting of a grid of pixels is used in the examples unless specified otherwise. The MBPE-based interpolation method has been successfully applied to speed up the evaluation of PMM impedance matrix elements over a wide frequency band of interest as compared to the conventional PMM [9] for the normal incidence cases. However, it is well known that the impedance matrix elements are (6)

4 WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS 125 Fig. 3. Comparison of impedance matrix elements calculated by the conventional PMM expressed in (2) and the PMM with the improved MBPE-based interpolation technique described in (4). The plots are for an arbitrarily chosen element in [Z] for the oblique incidence case: =45 and =1. (a) Magnitude in db for an element of [Z ], (b) phase angle for an element of [Z ]; (c) magnitude in db for an element of [Z ], (d) phase angle for an element of [Z ]; (e) magnitude in db for an element of [Z ], (f) phase angle for an element of [Z ]; (g) magnitude in db for an element of [Z ], (h) phase angle for an element of [Z ].

5 126 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010 also dependent upon the incidence angles and we have examined the accuracy and efficiency of the proposed MBPE-based interpolation technique introduced in (4) for both the normal incidence and oblique incidence cases. The technique introduced here is more robust for oblique incidence cases than the method presented in [9], due primarily to the removal of a phase factor (that varies rapidly with respect to frequency) from the impedance matrix elements prior to interpolation. Fig. 3 compares the response over the frequency range of GHz for impedance matrix elements (selected at random from for the FSS configuration illustrated in Fig. 2) evaluated by the conventional PMM and the improved MBPE-based approach for the oblique incidence case:,. Excellent agreement was observed between the two methods as shown in Fig. 3(a) and (b) for, Fig. 3(c) and (d) for, Fig. 3(e) and (f) for, and Fig. 3(g) and (h) for. For the normal incidence case, good agreement has also been observed as demonstrated previously in [9]. The transmission and reflection spectra of the FSS were simulated using three PMM solvers (e.g., conventional PMM, improved MBPE-based interpolation approach proposed in this paper, and the MBPE-based method introduced in [9] with no removal of phase factors) for the validation of the developed code. Very good agreement was obtained for the reflection and transmission properties of the Jerusalem slot FSS shown in Fig. 2 as demonstrated by the comparisons shown in Fig. 4(a) for the co-polarized normal incidence. For the TM-polarized oblique incidence cases, the results computed by the improved MBPE-based interpolation algorithm agreed well with those calculated via the conventional PMM. However, good agreement was not obtained for the MBPE-based technique introduced in [9] as illustrated in Fig. 4(b) and (c), respectively, for the TM-polarized oblique incidence case (the corresponding TE-polarized results are not included here). The total CPU time required to perform each PMM simulation of the FSS unit cell illustrated in Fig. 2 has been recorded in Table I for the brute-force method, the conventional 2-D FFTbased method and the improved MBPE-based interpolation algorithms. The FSS unit cell was discretized by a and a grid of pixels for the normal incidence and the oblique incidence cases, respectively. The corresponding CPU times required by the different methods for nonredundant matrix element filling at a single frequency point are presented in Table II for the normal incidence case. For all of the results presented here, the computations were performed on a Xeon 3.0 GHz processor machine with 8.0 GB of RAM. The BCG iterative solver with the help of the 1-D FFT-based acceleration technique and improved MBPE-based technique provided a faster solver compared to the brute-force method and resulted in a speed-up of for the overall solution time as indicated in Table I. As the bandwidth of interest is reduced, the corresponding number of sampling points used in the MBPE approach decreased, which results in faster matrix filling (see Table II). Alternatively, the performance of the 2-D FFT-based method [11] does not depend on a particular frequency range. The advantage of the MBPE interpolation method is that it provides a faster matrix fill time than the brute-force method and can be comparable or even faster than the 2-D FFT-based Fig. 4. Frequency response (co-polarized) calculated by three different PMM approaches for the Jerusalem slot FSS shown in Fig. 2 at different incidence angles. (a) Reflection and transmission coefficients for = 1 and = 1, (b) reflection coefficients for =45 and =1, (c) transmission coefficients for =45 and =1. method, especially in the case of frequency sweeps when solving large FSS problems with many unknowns as demonstrated in Table I and II. The MBPE-based interpolation technique also has the additional advantage of being independent of the choice of basis functions employed in the PMM formulation. Hence, (4) can still be applied to the case of

6 WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS 127 TABLE I COMPARISON OF CPU TIMES FOR THE PMM CALCULATIONS USING THE BRUTE-FORCE METHOD, THE 2-D FFT-BASED METHOD AND THE IMPROVED MBPE-BASED INTERPOLATION ALGORITHM FOR THE FSS DESIGN PRESENTED IN FIG. 2 TABLE II CPU TIMES REQUIRED FOR MATRIX FILLING VIA THE BRUTE-FORCE ALGORITHM, THE 2-D FFT-BASED METHOD AND THE IMPROVED MBPE-BASED INTERPOLATION METHOD (NORMAL INCIDENCE CASE) FOR THE FSS DESIGN SHOWN IN FIG. 2 non-uniform discretization of the FSS unit cell; however, the 2-D FFT-based method [11] cannot be applied to this type of problem. Further, the MBPE interpolation is not influenced by the overall frequency response of the FSS unlike other scattering parameter-based methods, such as three-point-sampling interpolation techniques [5], [6]. B. Application to Multiband FSS and AMC Surfaces With Hilbert Curve Shaped Strip Elements The Hilbert curve is a continuous fractal space-filling curve, which is an important member of the family of space-filling curves [16]. For a Hilbert curve with side dimension and iteration order, the length of each line segment and the sum of the lengths of all the line segment are given by [16] (7a) (7b) The geometry for a 2-D Hilbert-curve shaped strip is illustrated in Fig. 5(a) and (b) for iteration orders equal to and, respectively. From (7b), it can be seen that the total length of all the line segments becomes longer within the same size unit cell as the iteration order is increased. 1) Polarization Dependent Multiband Hilbert Curve FSS: A multiband FSS was implemented using a 2-D periodic arrangement of metallic strip elements shaped like Hilbert-curves with an iteration order equal to as shown in Fig. 5(a). Here, the FSS was assumed to be backed by a dielectric substrate with a relative permittivity of. The unit cell dimensions were chosen as 2.5 cm 2.5 cm with a dielectric substrate thickness of 0.02 cm. The FSS unit cell was discretized into a Fig. 5. Hilbert space-filling curves with various iteration orders. (a) n =4, (b) n = grid. In this case, a total of only Floquet harmonics were required for the MBPE fitting model, where the

7 128 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010 fitting frequencies were selected over the range of GHz as. These fitting frequencies were determined based on the automated selection scheme described in [9] with The reflection spectra computed via the three PMM solvers are presented in Fig. 6(a) and (b) (for normal incidence cases), and in Fig. 6(c) and (d) (for oblique incidence cases) for the TE and TM modes, respectively. As can be seen, the agreement between these three different PMM solvers was excellent for the normal incidence case. It should be noted that the non-symmetric FSS screen configuration results in polarization dependent multiband filters. For the oblique incidence case, the improved MBPE-based interpolation algorithm generated the correct results, but accurate results were not obtained for the MBPE-based interpolation technique introduced in [9] as illustrated in Fig. 6(c) and (d). Note that the corresponding transmission coefficients are not plotted here. Almost the same frequency response for the oblique incidence case at the lower frequency range was observed for the TE- and TM polarization cases [see Fig. 6(c) and (d)]. Next, the iteration number for the metallic strip Hilbert curve FSS structure was increased from to as illustrated in Fig. 5(b). This case required that a finer mesh consisting of a grid be used to achieve accurate PMM simulations of the more complex Hilbert-curve-shaped metallic strip structure (corresponding computed reflection and transmission spectra for this FSS design are not presented here). For this case we also observed that the improved MBPE-based interpolation algorithm generated the correct results for oblique incidence. It is expected that the TE and TM cases will yield a different response due to the asymmetrical geometry of the Hilbert curve pattern within the unit cell [see Fig. 5(a) and (b)]. 2) Polarization Independent Multiband Hilbert Curve FSS: Multiband FSS structures that incorporate Hilbert curve elements can also be designed that exhibit a polarization independent response. Onewayto accomplishthis is to require that each unit cell of FSS consist of a 2 2 array of Hilbert-curve shaped metallic strips as illustrated in Fig. 7(a) and (b) for iteration order numbers equal to (64 64 grid) and ( grid), respectively. Note that a symmetrical unit cell geometry is constructed by successively rotating the original Hilbert curve pattern by 90 until all four quadrants are filled. The other parameters employed here are the same as used in Fig. 6. The polarization independent reflection and transmission coefficients for normal incidence versus operating frequency were also calculated via three different PMM solvers. The corresponding results are plotted in Fig. 8 for the case, and good agreement was observed between the three different PMM solvers. The corresponding calculated results for the case as shown in Fig. 7(a) are not plotted here. We note that these single-layer Hilbert-curve FSS designs exhibited a polarization independent multiband response in the GHz range for and (e.g., eight bands for ). For the oblique incidence case, the variation of reflection coefficients with respect to frequency were also computed using three different PMM solvers. The corresponding results are illustrated in Fig. 9(a) and (b) for the TE- and Fig. 6. The reflection coefficients (co-polarized) as a function of frequency at different incident angles for polarization dependent Hilbert-curve FSS with iteration order equal to n =4(with a grid) as shown in Fig. 5(a). (a) TE polarization for =1 and =1, (b) TM polarization for =1 and =1, (c) TE-polarization for =1 and =1, and (d) TM-polarization for =45 and =1. TM-polarization, respectively. We observed very good agreement between the results calculated via the conventional PMM

8 WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS 129 Fig. 7. Unit cell composed of a 2 by 2 array of metallic strips in the shape of Hilbert-curves with various iteration orders. (a) n =3, (b) n =4. Fig. 9. The reflection coefficients (co-polarized) with respect to frequency for Hilbert-curve FSS with iteration order n =4and a grid as shown in Fig. 7(b) for oblique incidence: = 45 and = 1. The unit cell dimensions are chosen to be 2.5 cm2 2.5 cm with a dielectric substrate thickness of 0.02 cm and a relative permittivity of 2:0 0 j0:01. (a) TE-polarization, (b) TM-polarization. Fig. 8. The reflection and transmission coefficients (co-polarized) versus frequency for polarization independent Hilbert-curve FSS with iteration orders n = 4 and a grid as shown in Fig. 7(b) for normal incidence: = 1 and = 1. The unit cell dimensions are chosen to be 2.5 cm cm. with a dielectric substrate thickness of 0.02 cm and a relative permittivity of 2:0 0 j0:01. and improved MBPE-based interpolation algorithms as shown in Fig. 9(a) and (b), which further demonstrated the robustness and accuracy of the proposed PMM acceleration algorithm for oblique incidence cases. Note that corresponding results for the case as shown in Fig. 7(a) are not plotted here. 3) Polarization Independent Multiband Hilbert Curve AMC: Finally, we consider a polarization independent AMC surface design that follows the idea introduced in Section B.2, where the unit cell consists of a 2 2 arrangement of Hilbert-curveshaped strip elements. Two designs will be considered, one for (illustrated in Fig. 7(a) on a grid) and the other for (illustrated in Fig. 7(b) on a grid). The corresponding computed reflection magnitude and phase as a function of frequency are shown in Fig. 10 for the case of normal incidence. The unit cell dimension is 0.72 cm 0.72 cm with a dielectric substrate backed by a PEC ground plane and having a thickness and dielectric constant of 0.6 cm and, respectively. The resonant frequency and bandwidth versus the iteration order are listed in Table III. It can be seen that the first resonance shifts down lower in frequency and the corresponding bandwidth decreases as the iteration order is increased. This is because, within a unit cell of fixed dimensions, the total length of the Hilbert-curves become longer as the iteration order is increased. The polarization independent Hilbert-curve AMC designs are seen to exhibit a dual-band response for and a

9 130 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010 TABLE III POLARIZATION INDEPENDENT AMC SURFACE DESIGN. COMPARISON OF THE CENTER FREQUENCY AND PERCENT BANDWIDTH VERSUS THE ITERATION ORDER FOR 2 BY 2 ARRAYS OF HILBERT-CURVE ELEMENTS reported in the literature [17], [18]. These designs possessed a large number of unknowns due to the fine meshing needed to accurately model the higher iteration orders of the embedded Hilbert-curve structures within a unit cell of fixed dimensions and were successfully evaluated by using the very effective PMM acceleration techniques presented in this paper. Fig. 10. The phase angle and magnitude of the reflection coefficient versus frequency for a polarization independent Hilbert-curve AMC with different iteration orders n at normal incidence. The unit cell size is 0.72 cm cm with a dielectric substrate backed by a PEC ground plane and having a thickness of 0.6 cm. The unit cell is discretized by a grid and a grid for n =3[shown in Fig. 7(a)] and n =4[shown in Fig. 7(b)], respectively. quadruple-band response for over the range of GHz (see Fig. 10 and Table III). V. CONCLUSION The improved MBPE approach for efficient wide-band interpolation of PMM impedance matrices is proposed in this paper through the removal of a phase factor from the impedance matrix elements prior to interpolation, producing a more robust technique for oblique incidence cases than the original method introduced in [9]. The advantage of the MBPE interpolation algorithm is that it provides faster matrix fill times than the bruteforce method and can be comparable to the 2-D FFT-based method for matrix filling as well as is independent of the choice of basis functions. The 1-D FFT technique was employed to accelerate the calculation of matrix-vector products within the BCG iterative solver. The design examples presented demonstrated the accuracy and efficiency of the developed methods for both the normal incidence and oblique incidence cases. In addition, multiband single-layer FSS filters and AMC metamaterial surfaces with polarization dependent/independent performances were designed using 2-D periodic arrangements of metallic strips in the shape of space filling Hilbert-curves. We investigated the performance of designs that employed higher iteration Hilbert-curve shaped strips than has been previously REFERENCES [1] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, [2] Frequency Selective Surface and Grid Array, T. K. Wu, Ed. New York: Wiley, [3] R. Mittra, C. H. Chan, and T. Cwik, Techniques for analyzing frequency selective surfaces A review, IEEE Proc., vol. 76, no. 12, pp , Dec [4] T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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10 WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS 131 [15] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, Multiband singlelayer frequency selective surfaces designed by combination of genetic algorithm and geometry-refinement technique, IEEE Trans. Antennas Propag., vol. 52, no. 11, pp , Nov [16] H. Sagan, Space-Filling Curves. Berlin, Germany: Springer-Verlag, [17] E. A. Parker and A. N. A. El Sheikh, Convoluted array elements and reduced size unit cells for frequency-selective surfaces, Proc. Inst. Elect. Eng. H, Microw. Antennas Propag., vol. 138, no. 1, pp , Jan [18] E. A. Parker, A. N. A. El Sheikh, and A. C. de Lima, Convoluted frequency-selective array elements derived from linear and crossed dipoles, Proc. Inst. Elect. Eng. H, Microw. Antennas Propag., vol. 140, no. 5, pp , Oct [19] J. McVay, N. Engheta, and A. Hoorfar, High impedance metamaterial surfaces using Hilbert-curve inclusions, IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp , Mar [20] J. McVay, A. Hoorfar, and N. Engheta, Peano high impedance surfaces, Radio Sci., vol. 40, no. 6, pp. RS6S03 RS6S03, Sep Xiande Wang (M 03 SM 09) received the B.Sc. degree in radio physics from Lanzhou University, China, in 1989, and the M.Eng. and Ph.D. degrees in electrical engineering from Xi an Jiaotong University, China, in 1996 and 2000, respectively. He is currently a Postdoctoral Researcher in the Department of Electrical Engineering, Pennsylvania State University. From 1989 to 2001, he worked as an Assistant Engineer, an Engineer and then as a Senior Engineer in the China Research Institute of Radiowave Propagation (CRIRP). From October 2001 to August 2006, he was a Research Scientist in the Temasek Laboratories, National University of Singapore. His research interests include numerical techniques and fast algorithms for computational electromagnetic, metamaterials, the spatial-domain Green s function for multilayered medium, electromagnetic scattering and radiation in complex media for applications in antennas and frequency selective surfaces, scattering cross section predication for complex objects in complex environments, wave propagation and scattering of random media and rough surfaces and their applications in target detection and microwave remote sensing. Dr. Wang is a Senior Member of Chinese Institute of Electronics and a Member of the Chinese Institute of Space Science. He is a reviewer for papers in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE Antennas and Propagation Magazine, Radio Science, IET Microwaves, Antennas and Propagation and Journal of Electromagnetic Waves and Applications. Douglas H. Werner (S 81 M 89 SM 94 F 05) received the B.S., M.S., and Ph.D. degrees in electrical engineering and the M.A. degree in mathematics from The Pennsylvania State University (Penn State), University Park, in 1983, 1985, 1989, and 1986, respectively. He is a Professor in the Pennsylvania State University Department of Electrical Engineering. He is the Director of the Computational Electromagnetics and Antennas Research Lab (CEARL) psu.edu/labs/dwernergroup/ as well as a member of the Communications and Space Sciences Lab (CSSL). He is also a Senior Scientist in the Computational Electromagnetics Department of the Applied Research Laboratory and a faculty member of the Materials Research Institute (MRI) at Penn State. He edited a book entitled Frontiers in Electromagnetics (Piscataway, NJ: IEEE Press, 2000). He has also contributed a chapter for the book, Electromagnetic Optimization by Genetic Algorithms (New York: Wiley Interscience, 1999) as well as for the book, Soft Computing in Communications (New York: Springer, 2004). He has coauthored Genetic Algorithms in Electromagnetics (Hoboken, NJ: Wiley/IEEE, 2007) and has completed an invited chapter on Fractal Antennas for the popular Antenna Engineering Handbook (New York: McGraw-Hill, 2007). He has published over 375 technical papers and proceedings articles and is the author of eight book chapters. His research interests include theoretical and computational electromagnetics with applications to antenna theory and design, phased arrays, microwave devices, wireless and personal communication systems, wearable and e-textile antennas, conformal antennas, frequency selective surfaces, electromagnetic wave interactions with complex media, metamaterials, electromagnetic bandgap materials, zero and negative index materials, fractal and knot electrodynamics, tiling theory, neural networks, genetic algorithms and particle swarm optimization. Dr. Werner is a Fellow of the IEEE, the IET, and ACES. He is a member of the American Geophysical Union (AGU), URSI Commissions B and G, the Applied Computational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi and Sigma Xi. He was presented with the 1993 Applied Computational Electromagnetics Society (ACES) Best Paper Award and was also the recipient of a 1993 International Union of Radio Science (URSI) Young Scientist Award. In 1994, he received the Pennsylvania State University Applied Research Laboratory Outstanding Publication Award. He was the recipient of a College of Engineering PSES Outstanding Research Award and Outstanding Teaching Award in March 2000 and March 2002, respectively. He was also presented with an IEEE Central Pennsylvania Section Millennium Medal. He was the coauthor of a paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION which received the 2006 R. W. P. King Award. He has also received several Letters of Commendation from the Pennsylvania State University Department of Electrical Engineering for outstanding teaching and research. In March 2009, he received the PSES Premier Research Award. He is a former Associate Editor of Radio Science, and an Editor of the IEEE Antennas and Propagation Magazine.