An Unconditionally Stable FDTD Method for Electromagnetic Wave Propagation Analysis in Bi-isotropic Media

Licinius D. S. Alcantara

Department of Electrical and Computer Engineering, Federal University of Pará, P.O. Box 8619, Zip Code: 66073-900 Belém, PA, Brazil

Abstract — An unconditionally stable Finite-Difference Time-Domain (FDTD) scheme, which employs a similar technique to the Alternating Direction Implicit (ADI), is proposed to study the electromagnetic wave propagation in bi-isotropic media. In the proposed scheme, the unidimensional Maxwell’s equations in the time domain for bi-isotropic media were solved numerically. All transversal field components were included in the formalism to properly account special phenomena that may occur, such as polarization rotation and nonorthogonality between the electric and magnetic fields. The presented results are focused on chiral media.

Index Terms — Implicit FDTD, bi-isotropic media, chiral media, polarization rotation, propagation characteristic.

I. INTRODUCTION

The Finite-Difference Time-domain method was proposed by Yee in 1966 [1] and quickly became a standard for numerically solving the most general electromagnetic problems. However, its applicability on large scale problems depends greatly on the availability of expressive computational resources, which may be impracticable due financial matters or even due the present technological stage. Some improvements were proposed for the FDTD formalism to minimize or eliminate some numerical drawbacks that are present in the original scheme and, by doing so, the method could be conformed to more affordable computational resources.

The Alternating Direction Implicit (ADI) technique, for instance, was incorporated into the FDTD method for the purpose of eliminating the courant stability restraint on the time step value. Therefore, by applying the ADI technique, the time step becomes restricted only by accuracy, and larger time steps could be assigned even for meshes with very small cell sizes compared with the wavelength.

The ADI-FDTD method is applied for two or three-dimensional meshes, which leads to more simple linear systems. By the application of this method, each time step is split into two. In the first step, one spatial derivative that appears in the curl is fowarded in time whereas the other is backwarded. In the second step, the relative time

positioning of the curl terms is reversed to yield a second order error in the overall scheme. Therefore there is only one direction of implicitness for each ADI step, leading to simpler linear systems for implicit resolution. The method presented here is similar to the ADI technique, except for the fact that the implicitness is alternated in relation with the components associated with the bi-isotropic parameters. Therefore, it is applied to ease the development of a implicit numerical solution for bi-isotropic media.

Bi-isotropic media are classified into Tellegen and chiral media. Theoretical research performed over Tellegen media affirms them as nonreciprocal, where a propagating plane wave exhibits a lack of ortogonality between the electric and magnetic fields. The wave also suffers a change on the phase velocity, depending on the Tellegen Parameter. Bi-isotropic chiral media also produces some interesting effects on electromagnetic propagation such as rotation of polarization and circular dichroism [2]. The analysis of such special materials may bring forth new options for the telecommunications devices design.

Some recent efforts to model Bi-isotropic media with the FDTD method were presented in [3]-[4]. The explicit approach proposed in [3] adopts a modified disposal for the electric and magnetic field components in time and space, where several results for one-dimensional problems involving also lossy and dispersive bi-isotropic media were shown. However, it is not clear about the robustness of the method when extended to two or three-dimensional problems, since the time centralization of the equations will be broken due the consideration of another terms in the curl of the electric and magnetic fields, leading to a potentially unstable method. As another proposal, the formalism developed in [4] is based on the wavefield (or circular polarization) decomposition in bi-isotropic media [2], which is a simple but useful approach that holds for cases when the two wavefields does not couple, as occurs in a homogeneous chiral media, for example. This work, on the other hand, concerns mostly on the robustness of the extended FDTD method to bi-isotropic media, but

0-7803-9342-2/05/$20.00 © 2005 IEEE 661

without conducting to very complex systems for numerical resolution.

II. FORMULATION

The constitutive relations for bi-isotropic media can be written as

HD , (1)

HB , (2)

where

00j (3)

and

00j . (4)

The symbol is the Tellegen parameter and is the chirality parameter. Substituting (1) into the Maxwell’s curl equations in the time-domain and assuming a propagating transversal electromagnetic (TEM) wave along the z axis that may exhibit elliptical polarization, one obtains

zE

zH

tE yyx

c , (5)

zE

zH

tE xxy

c , (6)

zH

zE

tH yyx

c , (7)

zH

zE

tH xxy

c , (8)

where = r 0, = r 0,

c (9)

and

c . (10)

By using a similar procedure that characterize the ADI technique, each time step is solved in two substeps.The discretized forms for (5)-(8) are

,2

2

21

21

21

21

21

21

21

)()()()(

)(

)()()()(

naKx

nbKy

nbKyKc

K

nKy

nKyKc

naKx

EEEzt

HHz

tE

(11)

,2

2

21

21

21

21

21

21

212

1

)()()1()(

)(

)()1()(

)(

naKy

nbKx

nbKx

Kc

K

nKx

nKx

Kc

naKy

EEEz

t

HHz

tE

(12)

,2

2

21

21

21

21

21

21

21

)()()()()(

)(

)()()()(

naKx

nbKy

nbKyKcK

K

nKy

nKyKc

naKx

HHHz

t

EEz

tH

(13)

.2

2

21

21

21

21

21

21

21

212

1

)()()1()()(

)(

)()1()(

)(

naKy

nbKx

nbKx

KcK

K

nKx

nKx

Kc

naKy

HHHz

t

EEz

tH

(14)

where t is the time step, z is the propagation step, K is the index along the propagation axis (z = K z) and n is the time index (t = n t). The indexes na = n+½ and nb = n for the first substep, and na = nb = n+1 for the second substep within a time step. Spatially, the electric and magnetic field were disposed as described in [3].

Since (11)-(14) cannot be solved directly, it is still necessary to derive a new set of equations where each one contains only one unknown component. For the first substep, by using the appropriate na and nb, and substituting (14) into (11), results in (15), which can be used to assemble a tridiagonal matrix system for solving Ex, by varying K from 1 to nz, where nz is the number of cells along the propagation axis. By substituting (13) into (12), a tridiagonal expression for Ey is also obtained. The magnetic field components can be now obtained directly from (13) and (14).

A similar procedure is carried out for the second substep, where Ex, Hx were implicitly solved and Ey, Hywere directly computed from (12) and (14).

662

..

24

11

141

)1()()()(

)(

)()1()()(

)(

)()()(

)()()(2)(

2

)1()(

)()(

)(2)(

2

)1()(

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

nKx

nKx

KcK

K

nKx

nKx

KcK

K

nKy

nKy

K

nKy

nKy

nKx

Kc

nKx

Kc

nKx

Kc

Kc

KcnKx

Kc

HH

HH

EE

HHtzE

t

z

EE

t

zE

(15)

III. RESULTS

To validate the proposed approach, simulations concerning the propagation of a electromagnetic wave in a chiral medium were performed, and its propagation characteristics were computed and compared with the theoretical and numerical analysis available in the literature.

The spatial domain has 22500 cells with z = 1/3 mm. Similar as was done in [3], the mesh boundaries were sufficiently remote so that no reflection appeared in the simulation.

Fig. 1 shows the rotation of the electric field suffered by a linear polarized wave that propagates in a chiral media with r = 2, r = 1, = -9.42 10-3 and = 0 at f = 3GHz. The theoretical rotation angle is = e( )k0z, where

e( ) is the real part of the chirality parameter, k0 is the wavenumber in free space, and z is the distance traveled by the wave in the medium [2]-[3]. The rotation angle was measured at z=1500 z and z=3000 z from the source with t = 1,0 ps in order to compare the obtained results with

the results from the explicit FDTD developed for bi-isotropic media in [3], as is shown in TABLE I. The method proposed in this work showed very accurate results.

Agreeing to what was observed in [2]-[3], the polarization rotates clockwise when looking at the direction and sense of propagation (+z direction), because of the negative value of the chirality parameter.

Fig. 1. Polarization rotation of a propagating wave in a chiral medium.

The propagation length was set sufficiently large to compute the phase constant by performing Discrete Fourier Transform (DFT) [5] on the longitudinal distribution of an arbitrary field component. Figure 2 shows the dispersion characteristics (nr = /k0)identification via DFT for a chiral medium. The twin peaks indicates that the linear polarized wave decomposes into two counter rotating circular polarized waves with distinct phase velocities in a chiral medium, fact that justifies the phenomenon observed in Fig. 1. The superscript “+” (in nr ) refers to a right-handed circularly polarized wave, whereas the superscript “ ” refers to a left-handed one. For comparison, the computed effective index for the correspondent isotropic material ( = 0) is also shown, which coincides with the refraction index of the material for an unguided propagation.

TABLE I ANGLE OF ROTATION

Distance 1500 z 3000 z

Theoretical 16.965 33.929 Explicit FDTD [3] 17.127 34.158

Angle of rotation

(degrees) This work 16.891 34.107 Explicit FDTD [3] +0.95 +0.67 Relative

error (%) This work -0.44 +0.52

663

1.30 1.35 1.40 1.45 1.50 1.550.00.10.20.30.40.50.60.70.80.91.0

Nor

mal

ized

fiel

d in

tens

ity, |

Ex|

Effective index, nr

Chiral Isotropic

Fig. 2. Intensity response as a function of the effective index.

To verify the accuracy and stability of the method, Fig. 3 shows the computed effective index for several time step values related to the Courant time step ( tC), which is the upper limit for the traditional FDTD method, in order to sustain numerical stability. Observe that the proposed method does not take this threshold into account and the time step limit is now posed by accuracy and not by stability matters. The theoretical values, for chiral media, are given by rr nn [2]. The maximum relative error observed for the computed values in Fig. 3 was 0.18%.

VII. CONCLUSION

This work proposed an ADI-like solution to model the propagation of electromagnetic waves on bi-isotropic media with the FDTD method. The developed method is unconditionally stable and the results agreed well with the theoretical expectances. The aim for the next works is to extend the developed method to bidimensional analysis by combining it with the original ADI method, also implementing the perfectly matched layer (PML) for bi-isotropic media to avoid reflection at the spatial boundaries.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.405

1.410

1.415

1.420

1.425

Theoretical n+r

Computed n+r

Computed n-r

Theoretical n-r

Ref

ract

ion

inde

x, n

r

Normalized time step, t/ tCFig. 3. Measured refraction indexes in a chiral medium as a function of the normalized time step. The Courant time step is tC 1.5724 ps.

ACKNOWLEDGEMENT

The author wishes to acknowledge the support of the CNPq, grant no. 300002/2004-1 and to Prof. Dr. Carlos Leônidas S. S. Sobrinho for useful suggestions.

REFERENCES

[1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, pp. 302–307, May 1966.

[2] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Norwood, MA: Artech House, 1994.

[3] A. Grande, I. Barba, A. C. L. Cabeceira, J. Represa, P. P. M. So, and W. J. R. Hoefer, “FDTD modeling of transient microwave signals in dispersive and lossy bi-isotropic media,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 3, pp. 773-784, March 2004.

[4] A. Akyurtlu and D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propagat., vol. 52, no. 2, pp. 416-425, February 2004.

[5] C. M. Furse, O. P. Ganghi, “Why the DFT is Faster than the FFT for FDTD Time-to-Frequency Domain Conversions,” IEEE Microwave Guided Wave Lett., vol. 5, no. 10, pp. 326–328, October 1995.

nr = 1.41438

nr = 1. 42464 nr = 1. 40397

664