A discrete-time solution to wave propagation in time-varying media

(1)

ADiserete-Time Solution to Wave Propagation in Time-Varying Medin

Sbyb-Kang Jeng

Department of Electrical Engineering and Graduate Institute of Communication Engineering,

National Taiwan University, Taipei, Taiwan email skiene(aew.ee.ntu.~u.tw

Abstract

A novel approximate solution to wave propagation in time-vruying media is proposed. This closed-form solution is derived using the discrete-time approach. It is w i t t e n down duectly from the corresponding frequency domain solution in time-invariant media with simple duality N ~ S . This approach can be applied to developother analytical as well as numerical solution techniques for wave problems in time-varying media.

Introduction

Time-varying media at least can be found in ferroelectric material, magnetoelastic material, general space-time periodic media, and plasma suddenly created by solar flares, strong laser pulse, or by nuclear explosion. The electromagnetic wave propagation in such media is often regarded as a difficult problem. The solution techniques developed before, as far as the author knows, include analytical solutions to differential equations, the WKB ray approximation, the integral equation approach, and the finite-difference time-domain method. Typical related references are [1][2].

This paper will propose a new solution based on the discrete-time approach, which is an extension of the discrete-time electromagnetic theory [3] and will be introduced in the next section. By this approach, a closed form approximate solution for homogeneous gradually-time-varying media is obtained in a slraightfonvard way. This approach can be developed furlher to derive new analytical as well as numerical methods for solving wave problems

(2)

-.

_.

.

_.

.

_.

-

. . .

, . ,

Ds,

I

...

d o d-, d.2

...

so I - I do d.#

...

S ,

...

d , d, d ,

...

sI Ds d

. .

.

Ds,

.

1 AI

or [Os]= -[DI*], where the maUk [D] is an infinite Toeplitr matrix.

(1) I

Let the electromagnetic field quantities be sampled a1 "At, -m < n < m.

Using the discrete-time approach given above, the Maxwell equations can be represented as

The time-varying constitution relationship

(3)

and so on. It is also found

from

experiments that the w,,,'s corresponding to

the trapezoidal numerical integration, i.e., y,, = 0.4 far i = j , and w,,, = 1

for j < i are good enough in general.

The equations (2) and (4) are similar to the frequency domain Maxwell

equations and the constitution relationship for time-invanant media, respectively. Thus we

can

write down the solutions to

(2)

and

(4)

from

the corresponding frequency domain solutions. The rule is simply to substitute j w by i [ D ] and take care of the order of matrix multiplication. From this mathematical duality, the discrete-time plane wave solution in homogeneous time-varjmg media is obtained directly as

(4)

Here the plane wave is linearly polarized in the y-direction,

[vr

=([DG]+*[u,,,lj’[r]; and

b.1,

If-]

are consfant sequences to be

‘lo

determined by other conditions. Equation ( 5 ) can be also derived by manipulating (2) and (4). The

pm

of solution comsponding’to

b*]

is associated with a wave propagating in the +x direction, while the other part represents a wave propagating in the --x direction. This can be.verified by checking their behaviors in free space. Of couse, solutions for other lunds of excitations and media slruchlres can be handled in a similar way. Note, however, the rate of change of the media is assumed to be small, such that the reflection due to abrupt change of the media in time can be neglected.

Far the special case that the media is time-invariant, including the dispersive media l k e Debye or Lorentz media, and using causal differentiator such that d, = 0, for i

<

0, all matrices in (5) become the “staircase matrix,” which is the intersection of the Toeplitz matrix and the lower triangular matrix. Nice properties and fast algorithms for computing the matrix functions of Staircase mamces in (5) have been given in [4]. Same, numerical computations have been done for wave propagations in dispersive and time-varying media.

References

[I] F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,)(

IRE

Trans. Microwave Theory Tech., vol. MTT-6, pp. 167-172,April 1958. [Z]

W.

Re” and

B.

Q. Gao, “The analysis of 3dB microstrip directional coupler in time-varying media by FDTD method,” Proe.

Zd

lnrernorionnl Contrence on Miemwave ondMillimerer Wove Technologv, pp. 375-378,2000,

[3] S.

K.

Jeng, “Discrete-time electromagnetic theory,” 2002 IEEE AP-S

Inrernarional Symposium, San Antonio, TX, lune 2002.

[4] 1. T. Chiang, Mmir Appmoches for Tronsienr Analysis of Compler Transmisrion Line Circuits Using HWSD and SATD, Ph.D. Thesis, National Taiwan University, Taipei, Taiwan, 2002.

The results will be presented in the conference.