Abstract
A method is presented for generating a broadband rational interpolant approximation of the reflection coefficient of multiple screen Frequency Selective Surfaces (FSSs). The technique is structured around a linearization of the system provided by a spectral domain moment method based analysis of the FSS, followed by a model-order reduction of the linearized system using the dual rational Arnoldi method. This process creates a rational interpolant of the linearized system that matches its transfer function and its derivatives at several expansion points in the Laplace domain. Numerical results indicate that a reduced-order model with a system matrix of dimension less than can accurately reproduce the broadband behavior of multiscreen FSSs originally modeled with several hundreds or thousands of unknowns.

Presentation

Multiscreen Frequency Selective Surfaces (FSSs) are used as frequency and angular filters or polarizers over a broad band of the electromagnetic spectrum [1-7]. Composed of thin sheets of periodic, perfectly conducting planar patches or apertures in a thin perfectly conducting sheet, FSSs are often designed to reflect or transmit plane waves at the resonances of the patches or apertures. A sample FSS structure is depicted in Figure 1. It is situated in the z plane and is infinite and periodic along the transverse directions x and y with given periodicities. This study outlines a fast method for approximating the plane wave reflection coefficient of multiscreen FSSs over a large band of frequencies.

Figure 1: Three-screen FSS with a typical mesh inset

The most popular method for analyzing plane wave reflection from FSSs is the method of moments (MoM) [8] or, more specifically, its spectral Galerkin variant [1, 2]. In its most general incarnation, this method involves expansion of the screen currents in terms of subdomain basis functions, though full-domain basis functions are available for a large variety of commonly used simple elements [1, 2]. Application of this technique results in a linear system of equations for the basis function weighting parameters and a linear relationship between these parameters and the FSS plane wave reflection coefficient. For most FSS scattering problems, hundreds or thousands of subdomain basis functions must be used to model the screen currents accurately.

Because of the summations involved and the number of bases required, the application of the spectral Galerkin technique to the analysis of FSSs can be rather tedious, especially when broadband information is required. This heavy computational load arises from a variety of sources. First, the construction of the MoM matrix requires the summation of slowly convergent infinite series, which must be recomputed for each frequency and incidence angle of interest. Second, because the MoM system is usually large, the matrix inversions required to reconstruct the screen currents are costly. Third, FSS reflection characteristics may exhibit very narrow-band resonances that can only be discovered with a very dense frequency sweep, necessitating the construction and inversion of the MoM matrix at hundreds of different frequencies.

We introduce here a rational Krylov Model Order Reduction (MOR) technique that accelerates the broadband characterization of FSS plane wave reflection coefficients. Specifically, the method reduces the previously mentioned MoM system to a small state-space model with fewer than twenty degrees of freedom. To arrive at a model of this type, the MoM matrix is first approximated by a Hermite Polynomial Interpolant (HPI) [9, 10], so that the system can be expressed in a companion form with a linear frequency dependence. This step both facilitates the MOR and accelerates the construction of the matrix. The reduced order model is constructed to match both the reflection coefficient of the HPI model and its derivatives (moments) at selected Laplace domain interpolation points [9-13]. The method presented here will be seen to remedy all three contributions to the heavy computational load described above.

The explicit construction of the spectral Galerkin matrix at each frequency is eschewed by the introduction of the HPI approximation and the subsequent construction of the reduced order model.

The computational cost inherent in the inversion of the large MoM matrices needed to characterize the reflection coefficient over a broad frequency band is all but avoided as inversion of these matrices is only required at a small number of frequency points to create the reduced order model.

The computational cost associated with the required frequency sweep is made negligible due to the low cost of constructing and inverting small matrices. In spite of the approximations introduced by the HPI and the MOR in the frequency response calculation, the reduced-order system will be seen to reproduce faithfully the behavior of the FSS, including narrow-band resonances. While the examples shown here are limited to subdomain models, the technique is fully applicable to any MoM system.

It is emphasized that MOR methods based on moment matching are not new to electromagnetics. Asymptotic waveform expansion and Padé via Lanczos techniques have been used quite successfully for a number of years in circuit modeling [14-16], and such methods have also been employed to construct reduced-order models of interconnect structures. This study differs from these previous works in several ways. First, unlike [15], the method described here relies on HPIs in place of Taylor series expansions for linearizing the frequency dependence of the MoM matrix. This not only leads to a more broadband linearization of the system, but also avoids the computation of derivatives of very high order, which can be extremely expensive or even impossible. Second, unlike previous Krylov-based MOR procedures used in electromagnetics, the proposed technique results in a rational interpolant that matches the system transfer function and its derivatives at several points in the band of interest, not just at one point as do models resulting from a Padé approximation. Third, the right-hand side of the matrix equation provided by the spectral Galerkin method varies with frequency. This frequency-dependent right-hand side results from the peculiar periodic nature of the FSS-thus, it does not occur in circuit modeling. Fourth, to the authors' knowledge, this study is the first to apply model reduction to the characterization of FSSs.

Numerical Results

The algorithm described above was applied to several different test cases to demonstrate its speed and accuracy. In all of the following examples, a square periodic cell is assumed. Furthermore, all results are computed in terms of a normalized frequency related to the Laplace domain frequency parameter s and all other FSS dimensions are given as normalized lengths related to the true length l.

Because all results presented here are normalized, the subscript "norm" is dropped in subsequent expressions. Finally, reflection coefficients are calculated assuming an incident TM (to z) plane wave.

The algorithm was first applied to the computation of the reflection coefficient of the single-screen Jerusalem cross FSS with unit cell shown in Figure 2.

Figure 2: Single-screen Jerusalem cross FSS

Two interpolation points were used in the MOR. Figure 3 shows the HPI and three reduced-order models constructed to show the convergence of the algorithm.

Figure 3: Comparison of MoM coefficient to that produced by (a) a polyomial interpolant, (b) second-order model, (c) fourth-order model, and (d) sixth-order model for theta=40deg. and phi=0deg. TM incidence

The results shown in Figure 3 demonstrate that the proposed method converges quickly, giving very reasonable results for just a fourth-order model. This establishes a need for a multipoint rational Krylov method such as described here.

The next example applies the MOR algorithm to dual screen FSSs composed of identical screens. The screens analyzed are shown in Figure 4. Results in the format described above are shown in Figure 5.

Figure 4: Six different meshed screens to be used in two screen FSSs composed of identical screens and separated by a normalized distance of 0.45

Figure 5: Power reflection coefficients generated by MoM, polynomial interpolant, and reduced-order model for the six screens of Figure 4 for theta=20deg., phi=0deg. TM incidence

Finally, the algorithm was applied to the FSS composed of two identical screens (shown in Figure 6). This represents a more practical design than the preceding examples; the FSS has a definite passband and stopband, and could be used in a satellite dish subreflector. Results are shown in Figure 7.

Figure 6: Meshed gridded square patch screeen to be used in a two screen FSS with screens separated by a normalized distance of 0.45

Figure 7: Reflection coefficients generated by MoM, polynomial interpolant, and reduced-order model for the screen of Figure 6 excited by an incident, TM polarized wave with phi=0deg., and (a) theta=0deg., (b) theta=10deg., (c) theta=20deg., (d) theta=30deg., and (e) theta=40deg.

A few comments are in order about the results in general. First, due to the use of polynomial interpolation in the linearization of the system, the frequency band to which this method is applied cannot include either zero frequency or the blazing frequency where either poles or branch point singularities exist in the Green's function. (This limitation does not imply that models cannot be constructed above the blazing frequency, only that the domain of each reduced-order model can not pass through this frequency.) Specifically, the locus of the onset of propagation of the lowest order modes for a square cell FSS for a wave incident with is shown in Figure 8.

Figure 8: Locus of the onset of propagation of higher-order modes of square-cell screens situated in free space, excited by waves with phi=0deg.

For a given angle of incidence, a band of frequencies can be represented on this graph by a vertical line. If the point on this line representing the highest frequency in the band is close to the curve representing the blazing locus, a very high-order HPI is required to accurately approximate the system. Because the blazing locus (for the lowest order mode) is concave up, this problem is most severe for incidence angles near normal. Thus, for a given FSS, if all the HPI and MOR parameters are fixed, and reduced-order models are constructed for several incidence angles, the models for those incidence angles closest to normal will be accurate for a smaller percentage of the usable band between zero frequency and the onset of blazing. Figure 8 also shows where models may be constructed above the first blazing frequency: the method will successfully model the behavior of a screen in frequency regimes corresponding to vertical lines on Figure 8 uninterrupted by (and not too close to) the blazing loci. Note that if some of these lines are very short (as they would be for angles near normal above blazing), no reduced-order model should be constructed-the standard MoM will be fast enough.

Finally, note that in some instances, the reflection coefficient at the location of thin resonances given by the reduced-order model does not exactly match the MoM result. At these points, however, the MoM is not very accurate as changing its parameters may also alter the resonance size. The important behavior mimicked by the reduced-order model is not the magnitude of the reflection coefficients at small resonances, but their location in the frequency band.

Extensions to the Model Order Reduction Technique

While all of the examples of the preceding Section apply to FSSs discretized on a grid amenable to the use of FFT matrix multiplication methods, this discretization is not prerequisite for the application of the algorithm. The MOR technique is applicable to MoM systems created using any basis functions, though the resulting acceleration provided may differ from the examples presented here.

For instance, the technique can be applied, without modification, to FSSs discretized with full-domain bases. However, because the matrices produced by such a discretization are typically small, any noticeable acceleration provided by the MOR technique would no doubt spring from the interpolation step which obviates the need for the construction of the MoM system at each frequency. This may be helpful if the infinite summations required are very tedious, but is unlikely to produce the kind of speedups shown in the Tables.

A more important application of the MOR technique would be to FSSs discretized with subdomain bases that are not located on a grid amenable to the FFT technique. Such problems arise whenever such a grid is too restrictive for the fine-tuning of a design, or when the axes of periodicity of the FSS are not perpendicular. In these cases, the MOR method will achieve an acceleration of about the order shown in the examples.

The method can also be extended to frequencies where the FSS blazes. As presented, the method is already applicable to such problems, as long as a new model is constructed after the onset of propagation of each new higher-order Floquet mode. Because of the HPI approximation to the MoM matrix, however, the method discussed here cannot be used for models that span a frequency realm where a mode changes from evanescent to propagating. Additionally, the introduction of new modes requires the model to produce extra information, resulting in a multiple-output system. Extensions of the current technique that solve all of these problems are possible, and will be reported by the authors elsewhere.

Perhaps most interestingly, the MOR method can be extended to extract the dependence of the problem on incidence angle. The method is based on the concept of generalized Krylov subspaces, introduced by the authors in [17], and eliminates the need for the creation of a new model for each angle of incidence.

In addition to the rather fundamental extensions to the work discussed above, many minor extensions to it are underway, including the following:

Linearizing the MoM system with an orthogonal polynomial expansion instead of an HPI to reduce the size of the system,

Using Lanczos procedures in place of the DRA to accelerate the MOR [13, 18] and

Matching different numbers of moments at each interpolation point in the MOR to allocate computational resources more wisely. This could also lead to an adaptive algorithm where interpolation points in the MOR are chosen in band regions where the frequency response has not converged.

Conclusions

An MOR technique has been outlined for reducing systems generated by the application of the MoM to FSS scattering problems. While not a strict requirement on the algorithm, the use of a regular discretization saves enormous amounts of memory by exploiting the Toeplitz-block-Toeplitz-like nature of the MoM matrix. Given the MoM system, the process consists of two main steps: system linearization and MOR using the DRA algorithm. While, the system linearization step does increase the problem order temporarily, the large matrices generated are very sparse, and the MOR speeds the characterization of the FSS enough to make this setup cost acceptable. To elucidate the process, the algorithm was applied to a host of FSS scattering problems that demonstrated its accuracy and efficiency. Many extensions to the work were proposed, and will be discussed in greater detail elsewhere.

References

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[17] D. S. Weile, E. Michielssen, E. Grimme, and K. Gallivan, "A method for generating rational interpolant reduced order models of two parameter linear systems," Applied Mathematics Letters, to be published, 1998.

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The above work is a collaboration between Daniel S. Weile, Prof. Kyle Gallivan, and Prof. Eric Michielssen. Please send suggestions, comments, and inquiries to: dsw@decwa.ece.uiuc.edu.