IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 16, NO. 6, JUNE 2006 351

Recursive Asymptotic Impedance Matrix Method forElectromagnetic Waves in Bianisotropic Media

Eng Leong Tan, Senior Member, IEEE

Abstract—A recursive asymptotic impedance matrix method ispresented for simple and stable analysis of electromagnetic wavesin bianisotropic media. The method overcomes the numerical in-stability problem associated with the transition matrix method. Itrequires only elementary matrix operations along with thin-layerasymptotic approximation and bypasses the intricacies of theeigenvalue-eigenvector approach. Exploitation of its self-recursionalgorithm with geometric subdivision of a layer leads to highcomputation efficiency. The method also facilitates the trade-offbetween accuracy and speed for various applications.

Index Terms—Bianisotropic media, impedance matrix, recursiveasymptotic method.

I. INTRODUCTION

THERE has been considerable interest in the study of elec-tromagnetic wave propagation and scattering in stratified

bianisotropic media [1]. One of the celebrated techniques foranalysis of such media is based on the 4 4 transition ma-trix (or exponential matrix) [2]. Many methods are available todetermine the transition matrix including the eigenvalue-eigen-vector approach [3] and the Cayley–Hamilton method [4]. Analternative approach that obviates the need to solve for eigen-values is through the direct application of finite difference ap-proximation [5]. Although the transition matrix method is ap-plicable in principle, its practical implementation may lead tonumerical instability (also known as exponential dichotomy)[6]. In particular, the transition matrix computations may be-come inaccurate or overflow when a layer is electrically thickenough with the presence of inhomogeneous waves within thelayer. Several methods have been developed which can over-come the numerical problem of the transition matrix methodsuch as recursive transformation, eigen-, or Riccati-based ad-mittance, impedance, and reflection matrix methods [6]–[11].Since these methods often rely on the exact solutions to eigen-value problem or nonlinear Riccati differential equation, thesimplicity of previous finite difference approximation is gener-ally lost.

Recently, a recursive asymptotic stiffness matrix methodhas been developed for analysis of acoustic waves in layeredpiezoelectric media [12]. This paper adapts and extends theanalysis for electromagnetic waves in bianisotropic mediathrough recursive asymptotic impedance matrix method. Theuse of impedance matrix overcomes the numerical instabilityproblem associated with transition matrix above. Moreover, the

Manuscript received October 17, 2005; revised February 23, 2006.The author is with the School of Electrical and Electronic Engi-

neering, Nanyang Technological University, Singapore 639798 (e-mail:eeltan@ntu.edu.sg).

Digital Object Identifier 10.1109/LMWC.2006.875620

Fig. 1. Copolarized reflection coefficient versus incident angle for a metal-backed bianisotropic layer.

method has the advantage of simplicity since it requires onlyelementary matrix operations along with thin-layer asymptoticapproximation for computing the impedance matrix of eachlayer. Even in the case of arbitrary (lossy, nonreciprocal) bian-isotropic medium, one does not need to call for comprehensiveeigen packages or Riccati solvers. For each layer the recursiveasymptotic method exploits its self-recursion algorithm withgeometric subdivision of the layer, rather than with equal-thick-ness or arithmetic subdivision. This leads to high computationefficiency and also facilitates the trade-off between accuracyand speed for various applications.

II. RECURSIVE ASYMPTOTIC IMPEDANCE MATRIX METHOD

Consider a homogeneous bianisotropic layer of thicknessembedded in free space with one side optionally grounded

(see Fig. 1). The bianisotropic medium is characterized bypermittivity , permeability , and magnetoelectric (and ) dyadics. In the Fourier spectral domain, the tangentialelectric and magnetic field components derivedfrom Maxwell equations can be shown to satisfy a first-ordercoupled differential system

(1)

Here, is a 4 4 system matrix whose elements are functionsof the spectral variables, frequency and constitutive parameters.The explicit expressions of the matrix elements are availablein concise form from [13], [14].

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352 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 16, NO. 6, JUNE 2006

The solution of (1) can be described conveniently using the4 4 transition matrix defined as

(2)

To determine , a simple approach that obviates the need tosolve for eigenvalues is through the finite difference approxi-mation applied to (1) directly [5]. To that end, the layer is di-vided into sublayers having equal thickness each.Then the transition matrix is obtained by successive multiplica-tions of an approximated exponential matrix for thin layer[5, (27)–(28)]

(3)

(4)

Although simple and straightforward, numerical instabilitiesmay arise in the implementation of the transition matrix methodabove. When the layer thickness increases, the transition ma-trix tends to suffer from loss of precision due to mixture ofexponentially large and small terms. In the extreme case, it mayeven become overflow or give rise to completely erroneousresults. Furthermore, it should be noted that (4) is an accuratefinite difference approximation only for small . For a layerwith large thickness, one may need to use many sublayers andhence many multiplications for (3).

To resolve the aforementioned issues, we shall resort to the4 4 impedance matrix defined as

(5)

It can be demonstrated that will be free from the numericalinstability problem for large layer thickness. Although one maydetermine the matrix using eigensolution (refer to [13] for no-tations)

(6)

it is simpler to adopt an approach like the finite differenceapproximation above, which motivates the present recur-sive asymptotic method. For high efficiency, the layer isdivided into 1 sublayers having unequal thicknesses ingeometric progression, i.e., 2 for 1,2and 2 . With such geometric subdivision of alayer, the amount of computation grows with the logarithmof layer thickness only rather than with the thickness itself as

for equal-thickness subdivision. Then the impedance matrix isdeduced via a self-recursion algorithm

(7)

where .The recursion algorithm (7) is to be initialized with

, where the submatrices of aredetermined based on the thin-layer asymptotic approximationin terms of the submatrices of as

(8)

(9)

(10)

(11)

and . Like the case for in (4),the thickness for should be small enough such that (8)–(11)approximate closely the impedance matrix of initial thin layer.The recursion (7) proceeds until 1 and the impedance ma-trix of (5) is obtained as . Using this one can cal-culate readily various waveguiding characteristics or scatteringresponses as to be illustrated in the next section.

With the impedance matrix determined simply by elementarymatrix operations in (7)–(11), one is free from all intricacies ofsolving the eigenvalues and eigenvectors, which include com-plex root searching, degeneracy treatment and upward/down-ward eigenvector sorting or selection, cf. (6) and [13]. Thismethod also finds direct applications in a stratified medium ap-proximated by cascading thin layers. Instead of using the eigen-solution approach for every layer, it may be more efficient togeometrically-subdivide the layer and calculate its impedancematrix with few self-recursions. When the total thickness of thestratified medium is large, the impedance matrix method re-mains stable unlike the transition matrix method that is proneto numerical instability.

III. NUMERICAL RESULTS

To illustrate the recursive asymptotic impedance matrixmethod above, let us consider a plane wave obliquely incidentupon a metal-backed magnetoplasma-like (chiroplasma) bian-isotropic layer. The permittivity dyadic for a -directed biasingmagnetic field has the form

(12)

TAN: RECURSIVE ASYMPTOTIC IMPEDANCE MATRIX METHOD 353

TABLE ITRADEOFF BETWEEN EFFICIENCY AND ACCURACY

while the other constitutive dyadics are and. By substituting these constitutive parameters into

to be applied in (7)–(11), the impedance matrix can be de-duced simply and stably. The submatrices of are then utilizedto obtain the reflection coefficient matrix corresponding toTE and TM polarizations as

(13)

(14)

(15)

(16)

Fig. 1 shows the TM copolarized reflection coefficient versusincident angle in the 0 plane for 40, 80,

1, 0.3, and 6. The results have been com-puted using the transition matrix method with finite differenceapproximation and the recursive asymptotic impedance matrixmethod. For verification, we also compare with the impedancematrix result using eigensolution. It can be seen that the resultof transition matrix method is unstable and incorrect especiallyfor small incident angles. Such numerical breakdown has beendemonstrated earlier in [6] for the normal incidence 0case with 0 and 5.3. On the other hand, the resultsbased on impedance matrix computed using recursive asymp-totic and eigensolution methods remain to be stable and accu-rate even for large . Moreover, the former method is simplerto implement for involving elementary matrix operations only.

Besides stability and simplicity, the recursive asymptoticimpedance matrix method also facilitates the trade-off betweencomputation efficiency and accuracy. Table I lists a sample ofits average efficiency gain over the eigensolution method inachieving certain average accuracy for various layer thicknessencountered in multilayer modeling. All computations utilizethe same material parameters as above and the averaging istaken over the incident angles. The efficiency gain is acquiredin MATLAB based on the flops count for both recursive asymp-totic and eigensolution methods, while the accuracy is definedin terms of relative norm error for their impedance matrices(assuming the eigensolution one is exact). From the table, onecan see that when low accuracy is tolerable and the layer (ina multilayered stack) is not excessively thick, the impedancematrix can be computed using the recursive asymptotic methodwith only a small number of self-recursions ( denoted inbrackets). This then leads to computation speed that could be afew times faster than the exact method based on eigensolution.

Applications for which accuracy need not be high includesimulations with accurate material data unavailable and alsothose such as Fig. 1 above, whereby having the impedancematrix relative error less than 10 seems to be good enoughto make their reflection plots not distinguishable. In general,the efficiency gain is higher for lower accuracy required andalso for thinner layer with fewer geometric subdivisions. Notethat the efficiency improvement described here is meant forevery layer. Overall one will gain substantial savings in thetotal computation time when there are many layers, such as inthe case of thin-layer approximation of a thick inhomogeneousmedium.

IV. CONCLUSION

This paper has presented a recursive asymptotic impedancematrix method for simple and stable analysis of electromag-netic waves in bianisotropic media. The method overcomes thenumerical instability problem associated with transition matrixmethod. It requires only elementary matrix operations alongwith thin-layer asymptotic approximation and bypasses the in-tricacies of eigenvalue-eigenvector approach. Exploitation of itsself-recursion algorithm with geometric subdivision of a layerleads to high computation efficiency. The method also facilitatesthe trade-off between accuracy and speed for various applica-tions.

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