Floquet wave–based analysis of transient scattering from doubly periodic, discretely planar, perfectly conducting structures

Authors


Abstract

[1] A Floquet wave–based algorithm for solving an electric field time domain integral equation pertinent to the analysis of transient plane wave scattering from doubly periodic, discretely planar, perfect electrically conducting structures is presented. The proposed scheme accelerates the evaluation of fields generated by periodic constellations of band-limited transient currents via their expansion in time domain Floquet waves and use of blocked fast Fourier transforms. The validity and effectiveness of the resulting algorithm are demonstrated through a number of examples.

1. Introduction

[2] This paper presents a marching-on-in-time (MOT) and Floquet wave–based scheme for solving an electric field time domain integral equation (TDIE) pertinent to the analysis of transient plane wave scattering from doubly periodic, perfect electrically conducting (PEC), discretely planar, and freestanding structures. The proposed approach uses blocked fast Fourier transform (FFT) based accelerators [Harrier et al., 1985; Bleszynski et al., 2001; Yilmaz et al., 2002] to efficiently evaluate time domain Floquet wave (TDFW) decomposed electromagnetic fields [Capolino and Felsen, 2002, 2003; Felsen and Capolino, 2000; Marrocco and Capolino, 2002] generated by doubly periodic, discretely planar, and temporally band-limited source distributions.

[3] In the past, transient scattering from doubly periodic structures has been analyzed predominantly using finite difference time domain methods [Veysoglu et al., 1993; Tsay and Pozar, 1993; Harms and Mittra, 1994; Roden et al., 1998; Holter and Steyskal, 2002]. These solvers update fields inside a periodic structure's so-called mothercell using the classical Yee scheme [Yee, 1996] and impose periodic/absorbing boundary conditions on mothercell walls with normal vectors residing in/perpendicular to the plane of periodicity. Unfortunately, for obliquely excited periodic structures, these periodic boundary conditions call for future fields values to update current ones, and therefore cannot be applied directly. Several avenues for tackling this noncausality problem have been suggested [see Maloney and Kesler, 2002, and references therein]. It appears, however, that most fixes proposed to date are either hard to implement or somewhat limited in scope. Transient scattering from periodic structures also can be analyzed using TDIE-based schemes. Indeed, TDIE solvers for analyzing scattering from doubly periodic freestanding or substrate imprinted PEC elements were proposed by Chen et al. [2002, 2003]. Just like in their finite difference counterparts, noncausal terms arise when discretizing periodic structure TDIEs for obliquely incident fields using marching-on-in-time (MOT) procedures. Chen et al. [2002, 2003], removed these noncausal terms through the introduction of time-shifted temporal current basis functions in conjunction with a prolate-based extrapolation scheme. Unfortunately, even though these periodic structure TDIE solvers now efficiently cope with noncausal artifacts, their high computational complexity precludes them from being applied to the analysis of real-world structures. Generally speaking, the computational cost of MOT-based TDIE solvers can be attributed to their need to evaluate, at each and every time step, fields produced by past currents supported by the structure under analysis. The TDIE solvers of Chen et al. [2002, 2003] carry out this operation classically, by direct space-time convolution of the free space Green's function with all currents on the periodic structure. To be more specific, to evaluate the fields due to the past current, there is a double summation over the periodic cells. When the fields are observed on the mothercell, with the marching of time, the region around the mothercell in which the sources have to be take into account becomes larger and larger. This renders the solvers of Chen et al. [2002, 2003] computationally expensive.

[4] Here, an improved MOT-based TDIE solver for periodic structures is proposed. Whereas spectral methods for computing frequency domain Green's functions (Ewald representations [Jordan et al., 1986], off-plane plane wave sums [Jorgenson and Mittra, 1991], etc.) are commonly used in periodic structure frequency domain integral equation solvers, the proposed solver is the first to do so within periodic structure TDIE simulators. The solver relies on a time domain Floquet wave (TDFW) representation of fields produced by periodic transient current constellations [Capolino and Felsen, 2002, 2003; Felsen and Capolino, 2000]. Specifically, the proposed solver exploits the fact that TDFW representations of fields produced by quiescent and band-limited sources only involve “propagating modes” (this fact, to the authors' knowledge demonstrated here for the first time for time domain signals, constitutes another important contribution of this paper). Hence TDFWs provide a natural, compact, and computationally efficient means of representing fields produced by band-limited sources residing on practical periodic structures that only support a finite and small number of propagating waves within their operating band, that is, structures with unit cells of linear dimensions on the order of the wavelength at the highest frequency in the incident field. Because the TDFW propagator is not time-local, costly time domain convolutions are carried out using a blocked-FFT scheme (first introduced in [Harrier et al., 1985] for the purpose of solving one dimensional Volterra integral equations, and interpreted/tuned here within the proposed TDFW-TDIE framework). It will be shown that this decomposition and subsequent TDFW representation of the fields provides a means for computing fields produced by “past” currents in a manner consistent with the classical MOT-TDIE framework that is especially effective when the structure under study is discretely planar, viz. comprising a finite set of metallized layers. The computational cost of the new solver is only a fraction of that of periodic structure TDIE solvers not using TDFW concepts.

[5] This paper is organized as follows. Section 2 outlines the proposed TDFW/FFT-based scheme for rapidly computing transient fields produced by periodic current arrangements and its incorporation into an MOT-based TDIE solver for analyzing scattering from discretely planar structures. Section 3 presents numerical results that demonstrate the capability and accuracy of the proposed method. Section 4 relates our conclusions and avenues for future research.

2. Formulation

[6] Below, following a high-level description of the proposed solver, the periodic structure TDIE solver and TDFW concepts described by Chen et al. [2003] and Capolino and Felsen [2003] are reviewed. Next, the implementation of the proposed algorithm is outlined with emphasis placed on the FFT-based scheme for accelerating temporal convolutions involving TDFW kernels.

2.1. Outline of the Proposed Solver

[7] The proposed solver derives from a periodic structure TDIE solver developed earlier by several of the authors. The solver is accelerated by using a TDFW expansion of scattered fields and blocked-FFT methods. Below, these solver aspects are discussed in turn.

2.1.1. Periodic Structure TDIE Solver

[8] The periodic structure TDIE solver that forms the basis of the proposed simulator is almost identical to that of Chen et al. [2003]. When a periodic structure is excited by a transient plane wave, currents and fields in different cells are related to one another by a simple temporal shift. The TDIE solver considers as unknown the currents in one cell (the “mothercell”) and imposes boundary conditions on the electric field throughout the same cell. To this end, the solver computes the fields generated by all currents on the structure by spatially convolving the mothercell currents with the free space periodic Green's function. Besides being very costly—this procedure amounts to the time domain equivalent of attempting to sum the frequency domain Green's function directly in the spatial domain—several difficulties are encountered when discretizing the TDIE using classical MOT methods, viz. schemes that permit the iterative reconstruction of the currents one time step after another. Chief among these difficulties is the fact that the resulting MOT equations are noncausal. This difficulty however almost entirely can be circumvented by expanding the mothercell currents in a set of time-shifted basis functions that fire in a synchronized fashion with the time of arrival of the incident plane wave. The (few) remaining noncausal terms in the resulting MOT system are then eliminated by expressing “future” currents in terms of past ones through a bootstrapped band-limited extrapolation procedure. These manipulations notwithstanding, the physical interpretation of the resulting MOT equations barely changes: (almost all) “past MOT matrices” represent fields produced by periodic current constellations active at different points in time. To avoid their costly multiplication by current expansion vectors, a new representation of these fields is called for, however. The periodic structure TDIE solver is discussed in section 2.2. The reader is encouraged to study [Chen et al., 2003] as it elucidates and justifies the many parameter choices relating to the current expansion and extrapolation introduced there.

2.1.2. TDFW Expansion of Scattered Fields

[9] The proposed solver relies on a time domain Floquet wave (TDFW) representation of fields produced by periodic transient current constellations [Capolino and Felsen, 2003]. TDFWs are transient fields with fixed in-plane phase progression. While an infinite number of them is required to represent the fields produced by periodic current constellations for all space and time, only so-called propagating modes are required when the sources are quiescent and band limited. Within the proposed solver this fact is exploited by splitting fields produced by periodic transient current constellations into two components. First, there are the instantaneous, direct fields produced by currents in a mothercell, as well as its immediate neighbors through the action of the free space (nonperiodic) Green's function; these fields are evaluated classically for two reasons: their sources are not quiescent at their time of arrival and they act in the MOT system through matrices that are influenced by the extrapolation procedure. Second, there are the fields produced by sources that do not reside in the immediate vicinity of the mothercell: they are evaluated following their expansion in TDFWs. Because our focus here is on periodic structures that only support a finite and small number of propagating waves within their operating band, that is, structures with unit cells of linear dimensions on the order of the wavelength at the highest frequency in the incident field, TDFWs provide a very efficient means for representing these fields. The TDFW representation of fields scattered from periodic structures suitable within a TDIE-MOT context is discussed in section 2.3.

2.1.3. Blocked-FFT Acceleration of Convolutions Involving TDFW Kernels

[10] TDFWs do not constitute standard nondispersive plane waves. Indeed, their fixed transverse behavior along with their broadband character renders them dispersive in the direction perpendicular to the plane of periodicity. As a result, TDFW propagators are not time-local and costly time domain convolutions are required to track the temporal evolution of the TDFW amplitudes. Fortunately, if all sources and observation points reside in the same plane (or a finite set of distinct planes), then these convolutions can be carried out using blocked-FFT schemes [Harrier et al., 1985; Yilmaz et al., 2002]. This topic is discussed in section 2.4.

2.2. Periodic Structure TDIE Solver

[11] Consider a periodic structure consisting of identical freestanding PEC elements Smn, m, n = −∞, ∞ residing in rectangular cells of dimensions Dx by Dy that are anchored to transverse position vectors ρmnc = mDxequation image + nDyequation image and periodically arranged along x and y (Figure 1). Element S00 is said to reside in the mothercell. In what follows, it is assumed that the Smn comprise connected or disjoint but planar patches residing in the xy plane. Later, it will be argued that the proposed scheme easily is generalized to accommodate the analysis of scattering from discretely planar periodic structures comprised planar PEC elements confined to a finite set of parallel screens; other applications of the proposed scheme are highlighted in the conclusion section. The structure is illuminated by a band-limited plane wave pulse propagating along direction equation imageinc = −sin θinc cos ϕincequation image − sin θinc sin ϕincequation image − cos θincequation image with electric field Einc (r, t, equation imageinc) = equation imageincf(tequation imageinc · r/c) where c and equation imageinc denote the free space speed of light and the incident field's polarization, respectively. This incident field's temporal signature f(t) is assumed band limited to angular frequency ωmax and vanishingly small throughout the mothercell for t < 0. In addition, it is assumed that both Dx and Dy are of Omin), where λmin is the free space wavelength at ωmax. This assumption guarantees that only a fixed number of Floquet modes propagate away from the structure irrespective of the geometric features of S00. Finally, to facilitate the description of the proposed FFT-accelerated field evaluator, it is assumed that the origin of the Cartesian coordinate system resides on one of the four corners of the mother cell such that equation imageinc · ρ ≤ 0 for all ρ = xequation image + yequation image on S00.

Figure 1.

Top view of a doubly periodic structure comprising identical PEC elements.

[12] Let Jmn(ρ, t) = J00(ρρmnc, tequation imageinc · ρmnc/c) denote the electric current density induced on Smn in response to excitation of the periodic structure by Einc(r, t, equation imageinc). The current equation imageJmn(ρ, t) generates the scattered field Esca(r, t). The total electric field comprises the sum of the incident and scattered fields. An electric field TDIE for J00(ρ, t) is constructed by forcing the temporal derivative of the total electric field tangential to S00 to vanish:

equation image

The time derivative of the scattered field is expressed as

equation image

In (2), ∇ρ = (∂/∂x)equation image + (∂/∂y)equation image, the Green's function is

equation image

where the asterisk denotes temporal convolution and μ0 and ɛ0 are the free space permeability and permittivity, respectively.

[13] To solve (1) using an MOT procedure, S00 is approximated by triangular facets and J00(ρ, t) is expanded as

equation image

In the above equation, Ik,l is the unknown expansion coefficient associated with space-time basis function fl(ρ)Tk,l(t). In our implementation, the fl(ρ), l = 1, 2,…, Ns are Rao-Wilton-Glisson functions [Rao et al., 1982]; in other words, one zeroth-order divergence-conforming basis function is associated with each interior or cell boundary traversing edge in the S00 mesh. The number of spatial degrees of freedom, Ns, is chosen to ensure adequate spatial resolution of S00 and sampling of the current—lengths of edges in the discretized S00 should be no larger than say λmin/10. The Tk,l(t) are time-shifted approximate prolate spheroidal wave functions (APSWFs) parameterized as Tk,l(t) = P(ttktld, Tp, ω0, Ω). Here tk = kΔt and Δt is the time step size; the latter, and consequently Nt, are chosen such that representation (4) oversamples (from the Nyquist rate) all temporal waveforms by a factor between 5 and 10. The basis function dependent time shift imposed on the lth spatial basis function is tld = equation imageinc · ρlc/c where ρlc is the center of the edge defining fl(ρ); these shifts are introduced to mitigate the appearance of noncausal terms appearing in the MOT equations that result upon discretizing (2) when the structure under analysis is obliquely excited (the procedure can be thought of as the time domain equivalent of frequency domain phase extraction schemes and also is used when testing the discretized integral equation—see below and Chen et al. [2003]). The APSWFs, originally proposed by Knab [1979], are

equation image

where ω0 = 0.5(ωs + ωmax), Ω = 0.5(ωs − ωmax), ωs = π/Δt > ωmax, and Tp = NproΔt. The APSWFs are used as the temporal basis functions because of their interpolatory properties: they are strictly band limited to ωs and virtually time limited as they become vanishingly small for ∣t∣ > (Npro + equation imaget for large enough time bandwidth products ΩTp. As a result, each temporal basis function only covers (2Npro + 1) time steps; typically, to ensure accuracies in line with those associated with the above discussed spatial representations, Npro is chosen between 3 and 5 (this and other parameters along with their typical values are collected in Table 1).

Table 1. Description of the Parameters in the Paper
ParameterFunctionTypical Value
NsNumber of spatial degrees of freedom 
NtNumber of temporal degrees of freedom 
NproAPSWF half width in time steps3–5
NadvNumber of advanced currents in noncausal MOT system (6)0–3
NextNumber of past current coefficients used to extrapolate advanced ones to convert noncausal MOT system (6) to causal MOT system (10)3–5
NdelNumber of past current coefficients whose field is evaluated classically, without relying on TDFW decomposition5–7
NexcMaximum length of all spatial basis functions in time steps measured by the incident field1–2
Np2Half width of the modified APSWFs in time steps5–8
ξ*One of the two parameters to determine Nmod, the number of “propagating modes”1.1–1.3
ϑOne of the two parameters to determine Nmod, the number of “propagating modes”1–4

[14] Substituting (4) into (2) and testing the resulting equation using each of the fl(ρ), l = 1, 2, …, Ns, at time ti + tld, i = 1,…, Nt, results in the following matrix equation:

equation image

Here Nadv denotes the number of future/advanced current vectors that appear in the MOT system because of the structure's periodicity and use of the APSWFs; it was shown by Chen et al. [2003] that Nadv is independent of the mothercell dimensions (in practice, Nadv always is less than 3). The Ii and Vi are vectors of length Ns, [Ii]l = Ii,l, i = 1, …, Nt,

equation image

and Zk, k = −Nadv, −Nadv + 1,…, 0,…, Nt − 1 are matrices of dimensions Ns × Ns with elements

equation image

Equation (6) cannot be solved by a standard MOT procedure because at time step i future coefficients are involved. This difficulty is resolved by extrapolating these future coefficients through a band-limited extrapolator [see Chen et al., 2003]. In essence, the future current vectors (approximately) can be expressed in terms of present and past current vectors as

equation image

where Next denotes the number of past current samples used in the extrapolation and the Aqk are extrapolation coefficients for the kth future current vector associated with the qth past current vector (all relative to the present time step, i); these coefficients can be obtained using the prolate-based extrapolation scheme detailed by Cadzow [1979] and Slepian and Pollak [1961]. Upon inserting (9) into (6), the latter is recast as

equation image

with equation imageq = Zq + equation imageAqkZk for q = 0,…, Next, and equation imageq = Zq for q > Next. Often, just like Npro, Next is chosen in between 3 and 5 [Chen et al., 2003]. In what follows, however, it is assumed that Next < Npro, which implies that matrices equation imageq for q > Npro are unaffected by the extrapolation. In other words, past space-time basis functions that are quiescent at time i enter the MOT system through matrices equation imagei = Zi that describe fields produced by periodic constellations of past currents. This fact will prove to be important in the construction of a TDFW-based scheme. Equation (10) can be solved by a standard MOT procedure to obtain current vectors for all time steps. Since equation image0 is not diagonal, a nonstationary iterative solver such as (TF)QMR [Saad, 1996] is used.

[15] The dominant computational cost in the above scheme arises from the need to repeatedly evaluate the right-hand side (RHS) of (10) for all time steps and scales as O(Ns2Nt3). Indeed, the RHS of (10) is a measure of the field at Ns observers on S00, produced by all sources in the mothercell plus those in surrounding cells whose field at a given time step has reached the mothercell. Obviously, the number of such sources grows larger with time step and Nt; it is easily shown that, at a given time step in the analysis, there are on average a total of O(NsNt2) of them. Evaluating their field at O(Ns) observers in the mothercell thus costs O(Ns2Nt2) CPU resources for one time step, and O(Ns2Nt3) for all time steps.

2.3. TDFW Scheme

[16] This subsection details a TDFW-based scheme to rapidly evaluate the sum on the RHS of (10). The scheme hinges on an expansion of the periodic time domain Green's function in TDFWs [Capolino and Felsen, 2002, 2003; Felsen and Capolino, 2000]. As TDFWs represent efficiently only fields produced by quiescent band-limited sources—see below—the time signature of the source J00(ρ, t) (available at time step i) is decomposed as (Figure 2)

equation image

At time step i, the known signature, that is, the values prior to time step i, is decomposed into two components, instantaneous sources (dashed line in Figure 2, the first term of (11)) and past sources (solid line in Figure 2, the second term in (11)), in temporal dimension according to the delay index Ndel. Note the width of the instantaneous sources is fixed from time step i to time step i + 1. Meanwhile, the width of the past sources is increased by a time step size. The parameter Ndel in (11) is chosen to satisfy Ndel > Npro + Nexc, with Nexc a small integer defined as

equation image

where ⌈·⌉ selects the smallest integer larger than the argument and ρanyl denotes an arbitrary point in the support of fl(ρ). Note that Npro is the half width of the temporal basis functions Tk,l(t), and Nexc measures the maximum size of all the Rao-Wilton-Glisson functions with respect to the incident angle. As a result, the choice of Ndel > Npro + Nexc guarantees that the past sources are in essence quiescent at time step i for all the possible observers. Because Next < Npro < Ndel, it also implies that all past sources enter the MOT update equations through the original matrices Zi, unaffected by the extrapolation procedure. In other words, evaluation of the contribution of the past sources to the RHS of (10) requires the computation of the fields they produced, and nothing more. Note that the choice of Tk,l(t) guarantees that both the instantaneous and past sources are band limited in time. It is noted that Ndel is a constant of O(1) that, just like Npro, Nadv, and Next, does not depend on Ns or Nt. The reason for the presence of the term Nexc in the above inequality for Ndel is subtle and will become clear when discussing the aforementioned blocked FFT accelerator (section 2.4).

Figure 2.

Illustration of the split of the source time signature in the TDFW scheme.

[17] The above definition of instantaneous and past sources prompts a similar decomposition of the sum on the RHS of (10) as

equation image

The first and second terms on the RHS of (13) define the instantaneous and delayed fields at time step i, respectively. Instantaneous fields, viz. fields produced by instantaneous currents, can be computed classically, through multiplication of the sparse matrices equation imagek, k = 1,…, Ndel − 1 with the current vectors Iik, k = 1,…, Ndel − 1. Because Ndel is of O(1) and the linear cell dimensions are of Omin), this operation requires O(Ns2) operations per time step and thus no more than O(Ns2Nt) for the entire simulation. Delayed fields, instead, are computed by casting them in terms of TDFWs. To elucidate the introduction of TDFWs into the MOT framework, consider the following expressions for equation imagel, viz. the lth component of the delayed field vector at time step i:

equation image

This equation follows directly from (8). Note that none of the integrands in (14) possess spatial singularities because they model couplings with (integer) delays greater than Ndel or equivalently, between spatially separated sources and observers.

[18] Capolino and Felsen [2003] show that the periodic time domain Green's function can be expanded in TDFWs as

equation image

where βpq = 1 for p = 0 and βpq = 2 for p ≠ 0, ℜe[·] selects the real part of its argument,

equation image

j = equation image, J0(·) denotes the zeroth-order Bessel function, U(·) is the Heaviside step function,

equation image

and t0 = η equation image1 · (ρρ′)/c. The second equality in (15) is due to the fact that ApqFW(ρ, ρ′, t) and A(−p)(−q)FW(ρ, ρ′, t) are complex conjugates of one another. Before using the above TDFW expansion of the Green's function in (14), two important observations are in order. First, although the summation in (15) comprises an infinite number of terms, upon convolution of the Green's function with the band-limited and essentially time-limited APSWF, only so-called propagating modes should be retained provided that the field is observed no earlier than t0 + ϑΔt seconds after the APSWF (essentially) vanishes with ϑ > 1 a dimensionless parameter. The propagating modes are those with modal indices satisfying min (∣equation imagepq + equation imagepq∣, ∣equation imagepqequation imagepq∣)/ωs < ξ*, – recall that ωs is the bandwidth of the APSWF, with ξ* > 1 another dimensionless parameter. In other words, the convolution of the APSWF with the nonpropagating modes essentially vanishes for times t starting t0 + ϑΔt seconds after the APSWF elapses. This observation is demonstrated in Appendix A, which demonstrates the fast convergence of the truncated TDFW series with increasing ϑ and ξ*; typical values of ϑ and ξ* in line with accuracies of spatial current expansions are 3 and 1.2, respectively. It is also shown in Appendix A that the total number of propagating TDFWs retained in the expansion, denoted as Nmod, is proportional to the area of the mothercell. Because the linear cell dimensions are of Omin), Nmod is of O(1), that is, independent of Ns and Nt. Second, the Floquet wave in (16) can be expressed as

equation image

Artificial as this decomposition might seem at present, it has important computational consequences as demonstrated next.

[19] Indeed, use of (15)–(18) and the realization that only propagating modes are needed in the TDFW expansion of the Green's operator for the signals considered in (14) yields the following expression for the delayed fields at time step i produced by all past sources and tested by fl(ρ), l = 1,…, Ns:

equation image

This equation is the crux of the proposed TDFW scheme for evaluating delayed fields produced by past sources and its interpretation within a computational framework is elucidated next. First, note that the first and second terms on the RHS of (19) describe vector and scalar potential contributions to the delayed field, respectively. The evaluation of each term in (19) comprises three temporal convolutions. As for the vector potential term, these three convolutions are equation image(t) (convolution 1), ApqFW(0, 0, t) * (convolution 2), and equation imagedsfl(ρ)equation image δequation image * (convolution 3), where the asterisk indicates temporal convolution. The three convolutions for the scalar potential term can be similarly identified. These convolutions are carried out on a time step by time step basis, resulting in the three-stage scheme for evaluating equation imagel described next.

[20] 1. Project the sources temporal signatures onto “source rays” (Figure 3a). Vector and scalar source rays Spq,iA(t) and Spq,iϕ(t) for TDFW mode (p, q) and time step i are defined as

equation image

Vector and scalar source rays SpqA(t) and Spqϕ(t) for TDFW mode (p, q) are defined as SpqA(t) = equation imageSpq,iA(t) and Spqϕ(t) = equation imageSpq,iϕ(t), respectively. As seen from (20), vector and scalar source rays are constructed by carrying out convolution 1 in (19) between source signals residing at spatial locations ρ′ and delay operators δ[t + η equation image1 · ρ′/c]. In Figure 3a, the cross section of the mother cell S00 is shown for illustration. The top plots of Figure 3a depict the time signatures of the scalar source signals at points equation image and equation image at time step i. The bottom plot of Figure 3a shows the formation of source ray Spq,iϕ(t) via projecting all the scalar sources onto the point O = (x = 0, y = 0, z = 0) along the direction of equation imageinc. Note that the scalar source ray Spq,iϕ(t) is a complex quantity, although it is drawn as a real-valued function in Figure 3a.

Figure 3a.

Pictorial description of the first stage in a three-stage scheme.

[21] 2. Construct “field rays” (Figure 3b). Vector and scalar field rays Fpq,iA(t) and Fpq,iϕ(t) for TDFW mode (p, q) and time step i are defined as

equation image

Vector and scalar field rays FpqA(t) and Fpqϕ(t) for TDFW mode (p, q) are defined as

equation image

where ⌊·⌋ selects the largest integer smaller than the argument. Field rays represent the time signature of the TDFW-decomposed delayed field observed at the spatial origin. In (22), the upper limit in the summation over i is ⌊tt⌋, because Spq,⌊ttA(t)/Spq,⌊ttϕ(t) is the last available vector/scalar source ray at time t.

Figure 3b.

Pictorial description of the second stage in a three-stage scheme. The scalar field ray Fpq,iϕ(t) is formed via convolving the scalar source ray Spq,iϕ(t) with the TDFW operator ApqFW(0, 0, t).

[22] 3. Project the “field rays” onto the observers (Figure 3c):

equation image

In Figure 3c, the time signatures of the scalar field ray Fpqϕ(t) observed at the points equation image and equation image are shown in the upper right and left insets, respectively. These time signatures are obtained by projecting the scalar field rays observed at point O onto the observation points along the direction of equation imageinc.

Figure 3c.

Pictorial description of the third stage in a three-stage scheme.

[23] Next, the computational implications and complexity of the above three-stage decomposition and scheme are elucidated. Before doing so, however, it is useful to call attention to similarities between the proposed method and the plane wave time domain algorithm [Ergin et al., 1998] or fast multipole methods in general. Indeed, like the latter, the proposed field evaluator realizes computational savings by not having each source and observer communicate directly with one another; instead, they connect through a set of uncoupled though common carriers, TDFWs in the present scheme. Information contained in the delayed fields produced by the O(Ns) sources is compressed/aggregated (step 1) into a data stream of Nmod TDFWs, before being propagated (step 2) and uncompressed/disaggregated (step 3) to obtain O(Ns) observer fields. In view of the properties of the temporal basis functions used — recall that they are in essence time limited — all projections inherent in steps 1 and 3 are strictly local and therefore the computational costs of (a single of) these steps scales linearly in Ns and Nmod and is independent of Nt; because steps 1 and 3 are carried out for all Nt time steps, their total cost for the duration of the analysis thus scales as O(NmodNtNs). The cost of step 2 scales linearly in Nmod and is independent of Ns. Because step 2 requires a costly temporal convolution of source rays with the nonlocal temporal propagator ApqFW(0, 0, t), its cost scales as O(NmodNt2) if the convolution is evaluated classically. Fortunately, the propagator ApqFW(0, 0, t) is invariant with respect to temporal shifts, thereby allowing the above convolutions to be evaluated using blocked FFTs, as described in the next paragraph, at a cost of O(Nt log2Nt); the computational cost of executing step 2 for all time steps thus scales as O(NmodNt log2Nt). The total cost of the proposed scheme therefore scales as O(Ns2Nt + NsNmodNt + NmodNt log2Nt) with the first, second, and third terms in this estimate stemming from the evaluation of the instantaneous fields, steps 1 and 3, and step 2 of the delayed field evaluation, respectively. Because the above assumptions guarantee that Nmod is of O(1), the scheme's cost essentially would scale linearly in both Ns and Nt, were it not for the cost of evaluating instantaneous fields. Fortunately, if Ns becomes large—this means under the current assumptions that the mothercell would be packed with many spatial unknowns to resolve fine geometric features of the scatterer (an unlikely scenario)—then this cost can be reduced to O(NsNt log2Nt) by using a low-frequency plane wave time domain algorithm [Aygün et al., 2000] or time domain adaptive integral method [Yilmaz et al., 2003]. This last option was not exercised in our current implementation, as our focus was on relatively simple mothercells. Also, note that the case of an electromagnetically large unit mothercell, viz. one supporting many propagating modes that in number might scale as Ns, remains problematic. Although even in this scenario the proposed solver outperforms the classical solvers Chen et al. [2003], it remains prohibitively expensive. The situation is no different for frequency domain integral equation-based schemes for analyzing scattering from periodic structures: to the authors' knowledge, at present, no fast solver for such structures exists if the number of unknowns grows proportionally with the number of propagating modes. The usefulness of such a solver is, however, even more questionable as that of one capable of analyzing highly resolved mothercells.

2.4. Blocked FFT-Based Evaluation of the Temporal Convolutions

[24] To describe a cost-effective FFT-based scheme for evaluating temporal convolutions requisite in step 2 of the above scheme, a discussion of the spectral properties and representation of source and field rays is in order. Equation (4) expands currents in terms of essentially time-limited APSWFs of bandwidth ωs > ωmax. These APSWFs permit the discrete representation/interpolation of the current with exponential accuracy. Equation (14) casts source rays in terms of scaled and shifted such APSWFs. It follows that the discrete representation/exponentially accurate local interpolation and manipulation of source rays requires the introduction of an even more resolved interpolant, itself capable of representing the original APSWFs used to expand the current. To this end, source rays are sampled at time intervals Δt/2 and represented in terms of a new set of modified APSWFs P(t, equation imagep, equation image0, equation image) of bandwidth equation images = 2ωs, where equation imagep = Np2Δt/2, equation image0 = 1.5ωs, and equation image = 0.5ωs (Figure 4). Because field rays constitute convolutions of source rays band limited to ωs with ApqFW(0, 0, t), they are band limited to ωs as well and hence can be represented by the same new APSWFs.

Figure 4.

Reconstruction of the scalar source ray Spq,iϕ(t) using a set of modified APSWFs P(t, equation imagep, equation image0, equation image).

[25] With this background, a description of the blocked FFT scheme for evaluating temporal convolutions involving TDFW kernels becomes possible. For the sake of brevity, only the scheme's application to the construction of the scalar field rays is described. With minor modifications, the procedures outlined can be applied to the construction of vector field rays as well. It follows from the above discussion that samples of vector field ray (p, q) can be expressed in terms of a discrete convolution

equation image

for i′ = 1, …, 2Nt, where

equation image

The first equality in (24) follows from (21) and (22). The second equality was obtained by expressing the source rays in terms of their samples associated with APSWF P(t, equation imagep, equation image0, equation image). The summation over k′ in (24) can be split into two parts:

equation image

where Nshift = 2Ndel + 2Npro + 2Nexc. This choice for Nshift guarantees that

equation image

when k′ ≤ i′ − Nshift − 1. (Please refer to Figure 5 for the temporal locations of the source rays at different time steps.) The first summation on the right hand side of (26) requires O(1) operations for each i′, and therefore a total of O(Nt) operations for i′ = 1,…, 2Nt. The second summation in the right-hand side of (26) can be efficiently evaluated using the blocked FFT scheme as detailed next. First, it is noted that this sum can be expressed as a matrix-vector multiplication as

equation image

for i′ = 1,…, 2Nt, where the vector Bpq is of dimension 2Nt and contains elements

equation image

while the equation imagepq is a matrix of dimension 2Nt × 2Nt with elements (Figure 6)

equation image

Samples of Hpq(t) can be calculated and stored before the MOT process starts. Equation (28) should be evaluated in the context of the MOT. This implies that, the convolutional nature of (28) notwithstanding, a straightforward FFT cannot be used to evaluate the Fpqϕ(i′Δt/2) because source rays Spq,⌊i′/2⌋ϕ(t), Spq,⌊i′/2⌋+1ϕ(t), … are not available yet. Therefore the convolution in (28) is evaluated using so-called blocked FFTs (proposed by Harrier et al. [1985], with the purpose of accelerating temporal convolutions when solving Volterra integral equations), that allow fields produced by currents to become available for the purpose of advancing the MOT process without requiring FFTs of full length 2Nt each and every time step. To this end, as shown in Figure 6b, the part of matrix equation imagepq that is covered by nonzero elements is subdivided into blocks, each of which can be multiplied with the corresponding part of the vector Bpq using a simple (nonblocked) FFT, within the framework of MOT. The reader is referred to Yilmaz et al. [2002] for the details of the scheme, and a proof of the fact that the cost of the resulting scheme scales as O(Nt log2Nt).

Figure 5.

Illustration of the scalar source rays at two different time steps. Note that the width of the scalar source ray is fixed at different time steps.

Figure 6.

(a) Illustration of the matrix equation imagepq. (b) Schematic diagram of the acceleration of the matrix vector multiplication using a blocked FFT scheme.

[26] Although the above presentation focused on planar PEC structures, it is easily extended to the case of discretely planar structures, viz. scatterers comprising a finite number of (offset) planar screens. The required modifications to the algorithm involve the construction of TDFW representations and corresponding blocked FFT accelerators for all pairs of interacting screens. This renders the scheme impractical when the number of screens becomes large, or, equivalently, when a continuum of source and observation planes exists, as is the case when studying scattering from substrate imprinted structures.

3. Numerical Results

[27] This section presents several numerical results that demonstrate the capabilities of the above-described Floquet wave–based MOT (FW-MOT) solver. All results obtained with the FW-MOT code were compared against data from a periodic frequency domain method of moments (P-MOM) code following Fourier transformation of the time domain currents/fields to the frequency domain. All periodic structures considered below are illuminated by an electric field Einc(r, t, equation imageinc) = equation imageincf(tequation imageinc · r/c) with f(t) a modulated Gaussian pulse parametrized as

equation image

where fc is center frequency of the incident wave, σ = 3/(2π fbw) and tp = 6σ with fbw termed the “bandwidth” of the signal.

[28] The first structure analyzed comprises periodically arranged rectangular slots in a PEC ground plane (Figure 7). The side length of the square mothercell is 1 cm. The dimension of the slot is shown in the inset of Figure 7. Current on the slotted ground plane is described in terms of Ns = 1032 spatial unknowns. The incident pulse has equation imageinc = −equation image, equation imageinc = equation image, fc = 15 GHz, and fbw = 11 GHz. The time step is Δt = 2.564 ps and the number of time steps Nt = 1024. The number of Floquet modes Nmod of 69 is used for the FW-MOT scheme. The power reflection coefficients [Chen et al., 2003] for the structure obtained via the FW-MOT scheme and P-MOM scheme are compared in Figure 7. Excellent agreement between the two data sets is observed.

Figure 7.

Power reflection coefficients for a slot element at normal incident case (P-MOM, asterisks; FW-MOT, diamonds).

[29] Next, the structure analyzed comprises periodically arranged Minkowski patches studied by Gianvittorio et al. [2001]. This patch is designed to resonate in two separate frequency bands. The side length of the square mothercell is 30 cm. The dimension of the patch is shown in the inset of Figure 8. Current on the patch is discretized in terms of Ns = 2166 spatial unknowns. The incident pulse has equation imageinc = −equation image, equation imageinc = equation image, fc = 1.25 GHz, and fbw = 1.25 GHz. The time step is Δt = 22.22 ps and the number of time steps Nt = 2048. It is noted that in this example the dimensions of the mothercell measured at the sampling frequency ωs = π/Δt are considerably larger than those used in the last example (by approximately a factor of ten). As a result, Nmod = 799 Floquet modes have to be used in the FW-MOT solver. The power transmission coefficients [Chen et al., 2003] for the structure obtained using the FW-MOT scheme and P-MOM scheme are compared in Figure 8. As expected, two nulls corresponding to the resonances of the big and small square patch elements, respectively, are observed. The results obtained using the FW-MOT and P-MOM codes agree very well.

Figure 8.

Power transmission coefficients for a fractal element at normal incident case (P-MOM, diamonds; FW-MOT, solid line).

[30] The third structure analyzed comprises periodically arranged four-legged elements which are loaded with PEC patches and lumped elements [Epp, 1990]. The side length of the square mothercell is 1 cm and the dimension of the four-legged element is shown in the inset of Figure 9. As shown in the inset of Figure 9, each of the center cross' two legs is connected to the PEC patch through a parallel RLC resonant circuit with R = 1000 Ω, L = 1.3 nH, and C = 0.1 pF, which is marked as a shaded square. Current on the four-legged element and the PEC patch is discretized in terms of Ns = 383 spatial unknowns. The incident pulse has equation imageinc = −equation image, equation imageinc = equation image, fc = 15 GHz, and fbw = 15 GHz. The time step is Δt = 2.22 ps and the number of time steps Nt = 1024. The number of Floquet modes Nmod of 61 is used for the FW-MOT scheme. The power reflection coefficients of the structure obtained using the FW-MOT scheme and P-MOM scheme are compared in Figure 9. The two nulls around 14GHz and 25GHz corresponding to the resonances of the structures while the RLC resonant circuits are removed or replaced with PEC patches respectively are observed in the power reflection coefficients plot. Again, TD-MOT and P-MOM results are in good agreement.

Figure 9.

Power reflection coefficients for a four-legged element at normal incident case (P-MOM, asterisks; FW-MOT, dashed line with diamonds).

[31] Finally, a two screen structure comprising periodically arranged identical square loop elements is analyzed. The side length of the square mothercell is 4 cm. The dimensions of the square loop are defined in the inset of Figure 10. Current on the dual square loop elements is discretized in terms of Ns = 224 spatial unknowns. The incident pulse has equation imageinc = −equation image, equation imageinc = equation image, fc = 3 GHz, and fbw = 3 GHz. The time step is Δt = 8.33 ps and the number of time steps Nt = 1024. The number of Floquet modes Nmod of 99 is used for the FW-MOT scheme. The power transmission coefficients plot for the structure obtained via the FW-MOT scheme and P-MOM scheme are shown in Figure 10 and agree excellently.

Figure 10.

Power transmission coefficients for a dual-screen square loop element at normal incident case (P-MOM, asterisks; FW-MOT, solid line with diamonds).

4. Conclusions

[32] A TDFW accelerated MOT-based scheme pertinent to the analysis of transient scattering from doubly periodic, discretely planar, PEC structures was presented. The TDFW concepts were employed to efficiently represent the fields generated by periodic arrangements of discretely planar source constellations. APSWFs were chosen as temporal basis functions in the proposed solver because of their interpolatory and spectral properties, and the convolution of the TDFW kernel with the periodic structure currents was accelerated using a blocked FFT scheme. Even though application of the scheme is restricted to discretely planar structures, it has many applications, some of which were demonstrated in this paper. The proposed scheme was validated through comparison of scattering data for various periodic structures against frequency domain results. Current research focuses on the use of this method in the construction of hybrid time domain boundary integral—finite element schemes for analyzing transient scattering from penetrable periodic structures (in which the boundary integrals are confined to two parallel interfaces and the present scheme is directly applicable) and on developing a Floquet wave–based solver that permits efficient means for analyzing transient scattering from nonplanar doubly periodic structures.

Appendix A: Truncation of the TDFW Series

[33] This appendix demonstrates that, when the periodic time domain Green's function G(ρ, ρ′, t) is convolved with the (approximately) time- and (strictly) band-limited APSWF P(t, Tp, ω0, Ω), then only the “not deeply evanescent TDFWs” significantly contribute to the resulting field “shortly after P(t, Tp, ω0, Ω) vanishes.” A more intuitive, though only qualitative, demonstration was given by Capolino and Felsen [2003] on the basis of the excitation of local instantaneous frequencies of the various ApqFW.

[34] Before firming up the above statement, some preliminary comments are in order. (1) The TDFW ApqFW(ρ, ρ′, t) in (16) can be expressed as

equation image

where

equation image

and γ = equation image/2DxDyequation image. Below, the convolution equation imagepqFW(ρ, ρ′, t) * P(t, Tp, ω0, Ω) will be studied first; conclusion reached will be extended to ApqFW(ρ, ρ′, t) * P(t, Tp, ω0, Ω) thereafter. (2) The Fourier transform of equation imagepqFW(ρ, ρ′, t) is (Figure A1) [Gradshteyn and Ryzhik, 1980]

equation image

The spectrum of equation imagepqFW(ρ, ρ′, t) is nonzero only for ω ∈ (−equation imagepq + equation imagepq, equation imagepq + equation imagepq); it follows from (17) that both ∣−equation imagepq + equation imagepq∣ and ∣equation imagepq + equation imagepq∣ grow and that the spectrum of equation imagepqFW(ρ, ρ′, t) becomes increasingly smooth around ω = 0 as ∣p∣ and/or ∣q∣ increase. Note however that (A3) is not the spectrum of ApqFW. (3) The Fourier transform of P(t, Tp, ω0, Ω) is nonzero only for ω ∈ (−ωs, ωs) with ωs = ω0 + Ω (Figure A1); P(t, Tp, ω0, Ω) itself is vanishingly small outside the temporal interval (−Tp, Tp). (4) The dimensionless parameters equation image = equation image (∣−equation imagepqs + equation imagepqs∣, ∣equation imagepqs + equation imagepqs∣) measure the position of the spectra of equation imagepqFW(ρ, ρ′, t) and P(t, Tp, ω0, Ω) relative to one another. The parameter ξpqmin is especially important. Modes (p, q) with ξpqmin > 1 are termed evanescent; for these modes P(t, Tp, ω0, Ω)'s spectral support is fully contained within that of equation imagepqFW(ρ, ρ′, t) —this situation is depicted in Figure A1. Modes (p, q) with ξpqmin < 1 are termed propagating; for these modes P(t, Tp, ω0, Ω)'s spectral support resides partially outside that of equation imagepqFW(ρ, ρ′, t). The discussion below pertains only to evanescent modes.

Figure A1.

Illustration of the spectra of P(t, Tp, ω0, Ω) and equation imagepqFW(ρ, ρ′, t).

[35] A partially computational demonstration of this appendix' opening statement based on the fact that the spectra of TDFWs are increasingly smooth in the interval ω ∈ (−ωs, ωs) for larger ξpqmin, is presented next. Assume that there exists a time-limited function B(t) with spectrum that approaches that of equation imagepqFW(ρ, ρ′, t) for ω ∈ (−ωs, ωs) and with energy not exceeding that of equation imagepqFW(ρ, ρ′, t) restricted to the same interval by a fixed and small multiplicative constant c1 (typically < 10):

equation image
equation image

The fact that P(t, Tp, ω0, Ω) is band limited to (−ωs, ωs) along with (A4a) ensure that

equation image

Equation (A5), along with the fact that P(t, Tp, ω0, Ω) is approximately time limited and (A4b) guarantee that B(t) * P(t, Tp, ω0, Ω) and equation imagepqFW(ρ, ρ′, t) * P(t, Tp, ω0, Ω) are approximately time limited too – condition (35b) ensures that out-of-band spectral components that arise upon time limiting P(t, Tp, ω0, Ω) to (−Tp, Tp) in a computational setting do not destroy (A5). Next, it is shown that, for modes with large enough ξpqmin > 1, there exist B(t) for which (A4a) holds to arbitrary accuracy. To this end, B(t) is assumed of the form B(t) = equation imageBnδ(ttn); here the tn are sampled uniformly in between −ϑΔt + t0 and ϑΔt + t0t = π/ωs) and the coefficients Bn, n = 1,…, N are determined by enforcing (A4a) in a least squares sense while restricting B(t)'s energy according to (A4b). The relative error in (A4a) is defined as

equation image

and only depends on the dimensionless parameters ξpqmin, ξpqmax, and ϑ. Figure A2 shows that for evanescent modes with sufficiently large ξpqmin > 1 the relative error decreases exponentially fast with ϑ. For all cases shown in Figure A2, the relative error reduces to less than 10−5 when ϑ is larger than 4. For a fixed ϑ, the relative error decreases exponentially fast with ξpqmin and, understandably, is insensitive to ξpqmax. In conclusion, when ξpqmin is sufficiently large, then a well-behaved function B(t) that is nonzero only for t ∈ [−ϑΔt + t0, ϑΔt + t0] can be constructed such that equation image {equation imagepqFW(ρ, ρ′, t)} ≈ equation image {B(t)} for ω ∈ (−ωs, ωs). It follows that B(t) * P(t, Tp, ω0, Ω) is time limited to within t ∈ (−Tp − ϑΔt + t0, Tp + ϑΔt + t0) and therefore that equation imagepqFW(ρ, ρ′, t) * P(t, Tp, ω0, Ω) ≈ 0 when t > Tp + ϑΔt + t0. Using (A1) it follows that

equation image

Hence

equation image

when ξpqmin is sufficiently larger than one and t > Tp + ϑΔt + t0. A direct consequence of (A8) is that, when the time domain periodic Green's function is convolved with an APSWF, the high-order Floquet waves do not contribute if the convolution is observed for t > Tp + ϑΔt + t0. In other words, in

equation image

the error in the TDFW expansion rapidly tends to zero as ξ* increases beyond unity and t > Tp + ϑΔt + t0. It is easily verified that, for a fixed ξ*, the number of TDFWs for which ξpqmin ≤ ξ* is proportional to the area of the mother cell measured in wavelengths at the sampling frequency ωs = π/Δt. To be specific, if the total number of TDFWs that satisfies ξpqmin ≤ ξ* is denoted as Nmod, then NmodDxDy(ξ*ωs/c)2. The convergence of the truncated TDFW series (A9) is verified numerically through one example with Δt = 1 s, ω0 = 0.55π rad/s, Ω = 0.45π rad/s, Tp = 7 s, ωs = ω0 + Ω = π rad/s, Dx = Dy = 7cΔt, ρ = (0.0,0.00001Dx) and ρ′ = (0.0,0.0). Figure A3 show the instantaneous relative error in (A9), for various choices of ξ*. In Figures A3a and A3b, equation imageinc = equation image and equation imageinc = equation image(1/2) + equation image(equation image/2), respectively. The waveform of the APSWF is also plotted in the inset of Figure A3a. It is observed the APSWF virtually vanishes after time Tp. The relative error for t < Tp is always large, irrespective of the choice of ξ*. When ξ* is smaller than 1, the relative error remains large, even for t > Tp. However, when ξ* is chosen greater than 1, the relative error becomes vanishingly small for t > Tp.

Figure A2.

Relative error when approximating the spectrum of equation imagepqFW(ρ, ρ′, t): for line a ξpqmin = 2.5 and ξpqmax = 2.5; for line b ξpqmin = 1.8 and ξpqmax = 2.2; for line c ξpqmin = 1.4 and ξpqmax = 2.6; for line d ξpqmin = 4/3 and ξpqmax = 2; for line e ξpqmin = 8/7 and ξpqmax = 12/7; and for line f ξpqmin = 9/8 and ξpqmax = 11/8.

Figure A3.

Instantaneous error due to the truncation of TDFWs.

Acknowledgments

[36] This research was supported in part by ARO grant DAAD19-00-1-0464, the DARPA VET Program under contract F49620-01-1-0228, MURI grant F49620-01-1-04 “Analysis and design of ultra-wide band and high power microwave pulse interactions with electronic circuits and systems,” and the NSF under CCR: 9988347 and 0306436.

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