The electromagnetic modeling of practical finite periodic structures is a topic of growing interest. Due to the truncation of the infinite periodic structures, surface waves will be excited and localized near the discontinuous interfaces leading to the edge effect of finite structures. In this work, surface waves are numerically disentangled from the propagating Bloch waves contributions. Based on the universally exponential decay feature of the surface waves, a novel method is developed by connecting the solution to the large finite periodic structure with that to a relatively small one resulting in low complexity and memory consumption. The method numerically reconstructs propagating Bloch waves and surface waves according to the Bloch-Floquet theorem of periodic structures and translation invariant properties of semi-infinite periodic structures, respectively. Numerical examples are privided to validate the efficiency and accuracy of the newly developed method.postprin

Jiang, L

Meng, L

Sha, W

Xiong, X

HKU Scholars Hub

Title A new efficient method for analysis of finite periodic structuresAuthor(s) Xiong, X; Meng, L; Jiang, L; Sha, WCitationThe ACES 30th International Review of Progress in AppliedComputational Electromagnetics, Jacksonville, FL., 23-27 March2014.Issued Date 2014URL http://hdl.handle.net/10722/202289Rights Creative Commons: Attribution 3.0 Hong Kong LicenseA New Efficient Method for Analysis of Finite Periodic StructuresXiaoyan Y.Z. Xiong, Ling Ling Meng, Li Jun Jiang, and and Wei E.I. ShaDepartment of Electrical and Electronic EngineeringThe University of Hong Kong, Pokfulam, Hong Kongxyxiong@eee.hku.hkAbstract: The electromagnetic modeling of practical finite periodic structures is a topic of growing interest.Due to the truncation of the infinite periodic structures, surface waves will be excited and localized nearthe discontinuous interfaces leading to the edge effect of finite structures. In this work, surface wavesare numerically disentangled from the propagating Bloch waves contributions. Based on the universallyexponential decay feature of the surface waves, a novel method is developed by connecting the solutionto the large finite periodic structure with that to a relatively small one resulting in low complexity andmemory consumption. The method numerically reconstructs propagating Bloch waves and surface wavesaccording to the Bloch-Floquet theorem of periodic structures and translation invariant properties of semi-infinite periodic structures, respectively. Numerical examples are privided to validate the efficiency andaccuracy of the newly developed method.Keywords: Finite Periodic Structures, Surface Waves, Edge Effect, Leaky Wave Antenna1. IntroductionPeriodic structures (PSs) have attracted much attention in the past decades due to numerous applica-tions such as metamaterials, antenna arrays, frequency selective surface (FSS), photonic band gap (PBG) orelectromagnetic band gap (EBG) structures, etc [1, 2]. Infinite PSs are impossible in a realistic fabrication.When truncating infinite PSs, discontinuities occur at the interface and surface waves (SWs) will be excit-ed according to mode conversion [3]. In finite PSs, only allowed Bloch waves (BWs) can propagate, andSWs become evanescent and localized around the discontinuities, which leads to the edge effects of finitePSs. The prediction of wave natures in the edge cells of finite PSs is an indispensable procedure during thedesign process [4, 5].The conventional element-by-element method is rigorous and suitable for small finitePSs. However, it requires heavy computational resources for very large ones. The well-established infiniteperiodic approximation approach neglecting the edge effect, although reducing the computational domainto a unit cell, is not sufficiently accurate for engineering applications.To overcome the difficulties in simulating large finite PSs, two different categories of methods havebe proposed. The first category bypassed the difficulties by improving the computational capabilities of theexisting methods. Among them, the fast multipole method [6] and the domain decomposition method [7]are very powerful for modeling large finite PSs. The other category of methods consider the edge effectwith the help of the information in an infinite array, i.e. the convolution of the infinite array solution with anappropriate window function [8].This paper aims to develop an efficient method for predicting the electromagnetic (EM) response oflarge finite PSs accurately from that of relatively small ones. A leaky wave antenna is investigated by thenewly developed method. The results are compared to those calculated by the element-by-element method.Excellent agreements are obtained and particularly the proposed method saves considerable computer re-sources. The presence of SWs is critical for the low side-lobe antenna design.2. Surface Waves in Finite Periodic StructuresIt is known that BWs are normal modes of an infinite PS. However, in finite structures, the fieldsare given by the superposition of BWs and additional SWs which are excited due to the truncation andlocalized near the discontinuous interfaces. The BWs in infinite PSs are referred as infinite periodic Blochwaves (IPBWs) to distinguish them from BWs in finite structures. To accurately model finite PSs, weneed to take the SWs into account. When the magnitude of SW is large and its decay rate is slow, theedge effect is important and can’t be ignored. In our previous work, we have demonstrated that the SWsdecay exponentially and are invariant with respect to the increasing array size when the elements N of thestructures is sufficiently large for capturing the information of SWs [9, 10].3. Field Reconstruction in Large Finite Periodic Structures(a) (b)Fig. 1. (a) Mechanism for the field reconstruction in large finite periodic structures from that of relativesmall ones. (b) LWA geometry. The dielectric waveguide is perturbed by the grating inside the waveguide.where nh=1.45, nwg = 1:67, np = 3:48, d = 2` = 0:97 um, w = 1 um, h = 0:3 um, s = 20 um andN = 60.The basic idea for efficiently calculating large finite periodic structures is inspired by the universaldecay feature of SWs. For a sufficiently large layer numberN , it is reasonable to assume that the center unitcell only contains IPBWs. As shown in Fig. 1 (a), the center unit cell of the large finite periodic structureis aligned with that of the small one. We assume that the small finite periodic structure is long enough atthe periodicity direction to guarantee invariant (converged) SWs. We only simulate the small structure byusing rigorous full-wave solvers and obtain its field distributions (or equivalent sources). The procedures toreconstruct the field for the large structure are as follows:First, the field distribution of the center unit cell of the large structure will exactly copy from that of thesmall one. Second, the IPBW numbers kmB can be extracted by the eigenmode expansion methodE(r) =m=1Xm= 1Em(r) exp( jkmB x) (1)where E can be electric or magnetic fields and Em are the periodic BW envelope functions: Em(x) =Em(x+ P ). The BW vectors kmB can be retrieved by using a standard minimization algorithmminkBE(r) m=1Xm= 1Em(r) exp( jkmB x)2(2)The IPBW vectors in both structures are the same and can be complex due to the radiation or intrinsic(Ohmic) loss. Third, numerically propagate EM fields forward and backward at the IPBW region of thelarge structure by using the extracted BW vectors based on Bloch-Floquet theorem.E(x+ P ) = E(x) exp( jkmBP ) (3)Finally, reconstructing EM fields of the large structure at the surface wave regions by using the translationinvariant properties of semi-infinite PSs. With respect to the left (right) SW region, the right (left) IPBWregions can be approximately regarded as semi-infinite PSs due to fast decayed SWs. So we haveELsw(x)ELIPBW (x lP )=ESsw(x)ESIPBW (x lP )(4)where ELsw and ESsw are fields at the SW regions for large and small structures, respectively. ELIPBW andESIPBW are fields at the IPBW regions for large and small structures, respectively. lP is the distance betweenthe cell in the SW region and its most adjacent cell in the IPBW region (such as the distance between cell 1and 3 in Fig. 1 (a)).4. Numerical ExamplesAn optical leaky wave antenna (LWA) (See Fig. 1 (b)) is efficiently calculated by the developed method[11]. Instead of calculating the original leaky wave antenna with N = 60, we calculate a relatively smallerone withN = 30 by using the FDTDmethod. The middle 10 elements (n=11 to n=20) are considered as theIPBW region. Two Bloch waves located on the third band propagating forward and backward are consideredin the IPBW region. Thus the field has the form of E(n) = a+E(11)e jk+d(n 1) + a E(11)e jk d(n 1).By employ the least mean square interpolation, we get k+ =    j =  2:48e5   j4:40e4, a+ = 1:097and a  =  0:097. After getting the IPBWs of the small structure with N = 30, the field of the originallarge structure with N = 60 can be reconstructed from the information of the N = 30 structure accordingto Sec. 3.−180 −160 −140 −120 −100 −80 −60 −40 −20 0−45−40−35−30−25−20−15−10−505φ (deg)Normalized Radiation Pattern (dB)  Element by Element (N=60)Small Truncated (N=30)Small Truncated (N=20)Infinite (N=60)Fig. 2. Comparison of the far-field radiation patterns calculated by different methods.The magnitude and phase of the field along the radiation aperture of the large finite PSs agree very wellwith the element-by-element simulation results. Fig. 2 shows that the far-field radiation patterns obtained bythe proposed method are also in good agreement with the element-by-element results. The side-lobe around 52 due to the presence of surface waves is not predicted by the infinite array method. Hence, the surfacewave effects are critical for predicting the side-lobes of LWA. The radiation pattern from the small structurewith N = 20 is also compared. Only small discrepancies exist which means the N = 30 structure is largeenough to capture all the SWs information. Table 1 compares the computational information of the LWAwith different perturbation elements. The proposed method can save lots of CPU time and memory since itonly needs to simulate a relatively small structure. Other examples will be given during the conference.Table 1: The computational information of a LWA with different perturbation elements.Elements N=20 N=30 N=60Domain (m2) 32:37 43:97 42:07 43:97 71:17 43:97Memory (MB) 47 62 104Time steps (t) 21440 27840 34240CPU time (s) 381.65 782.47 1548.845. ConclusionIn sum, based on the universally rapid decay character of SWs, an efficient method for modeling largefinite PSs is developed. From a small finite PSs which contains all the information of SWs and IPBWs, wecan accurately predict the performance of large finite ones without ignoring the edge effect. The proposedmethod is even more efficient for larger structures.AcknowledgmentThis work was supported in part by HK GRF (713011, 712612, and 711511), China NSFC 61271158,and US 120018.References[1] C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development ofthree-dimensional photonic metamaterials,” Nat. Photonics, vol. 5, pp. 523–530, Sep. 2011.[2] B. A. Munk, Frequency Selective Surfaces: Theory and Design, Wiley, New York, 2000.[3] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, 1960.[4] W. L. Ko and R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propagat.,vol. 36, pp. 496–503, Apr. 1988.[5] A. Cucini, M. Albani, and S. Maci, “Truncated Floquet wave full-wave (T(FW)2) analysis of largeperiodic arrays of rectangular waveguides,” IEEE Trans. Antennas Propagat., vol. 51, pp. 1373–1385,Jun. 2003.[6] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in ComputationalElectromagnetics, Norwood, MA: Artech House, 2001.[7] J. B. Manges, J. W. Silvestro, and K. Zhao, “Finite-element analysis of infinite and finite arrays,”International Journal of Microwave and Wireless Technologies, vol. 4, pp. 357–364, Jun. 2012.[8] A. Ishimaru, R. Coe, G. Miller, and P. Geren, “Finite periodic structure approach to large scanningarray problems,” IEEE Trans. Antennas Propagat., vol. 33, pp. 1213–1220, Nov. 1985.[9] X. Y. Z. Xiong, L. J. Jiang, V. A. Markel and I. Tsukerman, “Numerical Methods for Effective MaterialParameters of Periodic Electromagnetic Composites at the Surface and in the Bulk,” in Applied. Comp.Electrom. Society, Monterey, CA, pp. 741–746, 2013.[10] X. Y. Z. Xiong, L. J. Jiang, V. A. Markel and I. Tsukerman, “Surface waves in three-dimensionalelectromagnetic composites and their effect on homogenization,” Opt. Express, vol. 21, pp. 10412–10421, 2013.[11] S. Campione, C. Guclu, Q. Song, O. Boyraz, and F. Capolino, “An optical leaky wave antenna with Siperturbations inside a resonator for enhanced optical control of the radiation,” Opt. Express, vol. 20,pp. 21305–21317, Sep. 2012.

A new efficient method for analysis of finite periodic structures

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A new efficient method for analysis of finite periodic structures

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