Cellular automaton approach to electromagnetic wave ... · Cellular automaton approach to...

Cellular automaton approach toelectromagnetic wave propagation

in dispersive media

BY M. W. JANOWICZ1,3,*, J. M. A. ASHBOURN

2, ARKADIUSZ ORŁOWSKI3

AND JAN MOSTOWSKI3

1Fakultat 5, Institut fur Physik, Carl-von-Ossietzky Universitat,26111 Oldenburg, Germany

2Department of Engineering Science, University of Oxford, Parks Road,Oxford OX1 3PJ, UK

3Instytut Fizyki PAN, Al. Lotnikow 32/46, 02-668 Warsaw, Poland

Extensions of Białynicki-Birula’s cellular automaton are proposed for studies of the one-dimensional propagation of electromagnetic fields in Drude metals, as well as in bothtransparent, dispersive and lossy dielectrics. These extensions are obtained byrepresenting the dielectrics with appropriate matter fields, such as polarization togetherwith associated velocity fields. To obtain the different schemes for the integration of theresulting systems of linear partial differential equations, split-operator ideas areemployed. Possible further extensions to two-dimensional propagation and for thestudy of left-handed materials are discussed. The stability properties of the cellularautomaton treated as a difference scheme are analysed.

Keywords: cellular automata; electromagnetic waves; propagation; dispersive media

1. Introduction

This paper is concerned with the simulation of light propagation in dispersivematerials with the help of suitably chosen cellular automata (CA). CA in a broadsense form an idealization of a physical system in which space and time arediscrete. A multitude of applications has already been found for these, includingdynamical systems theory, formal languages, statistical mechanics, theoreticalbiology and neural networks (cf. Ilachinski (2001) and references therein).Techniques of the so-called lattice-gas CA, as developed in Hardy et al. (1976)and Frisch et al. (1986), have been successfully applied to several systems ofpartial differential equations (cf. Rothman & Zaleski (1994) and Chopard & Droz(1998)). The fact that there have only been a few attempts to adapt the CA-related methods in order to model the electromagnetic wave propagation could

Proc. R. Soc. A (2006) 462, 2927–2948

doi:10.1098/rspa.2006.1701

Published online 18 April 2006

The electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2006.1701 orvia http://www.journals.royalsoc.ac.uk.

*Author and address for correspondence: Instytut Fizyki PAN, Al. Lotnikow 32/46, 02-668Warsaw, Poland ([email protected]).

Received 2 December 2005Accepted 21 February 2006 2927 q 2006 The Royal Society

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probably be attributed to the great efficiency of more traditional methods, suchas finite differences or finite elements in problems of this kind. Among the mostwidespread methods for the integration of the Maxwell equations in dispersiveand lossy media (which are of most interest to us), in addition to finite-differencetime-domain methods, there are approaches based on fast Fourier transformation(e.g. Dvorak & Dudley 1995, 1996), various numerical integration methods,asymptotic techniques (e.g. Oughstun & Sherman 1989a,b, 1990; Oughstun &Balictsis 1996) and hybrid analytical–numerical methods, as described, forexample, in Dvorak et al. (1998). Nevertheless, the standard Boltzmann lattice-gas methods have been applied to Maxwell’s equations in three spatialdimensions in Simons et al. (1999).

In two other approaches to CA-based integrators of the wave equations, theauthors decided to relax one of the most important defining requirements of CA,that of the discreteness of the dependent variables. In Chopard et al. (1997) andChopard & Droz (1998), the Boltzmann method was still retained. On the otherhand, in Sornette et al. (1993), the S-matrix formulation of the wave propagationwas used, which turned out to be equivalent to a finite-difference discretizedversion of various wave equations. In addition, Vanneste & Sebbah (2001) as wellas Sebbah & Vanneste (2002) have discussed numerical propagation in randomdispersive media, which is very much in the spirit of our present work. In thispaper, we develop the approach initiated by Białynicki-Birula (1994), who hasbeen mostly interested in problems of a theoretical nature. One of our aims is todemonstrate that in the case of one-dimensional propagation at least, theBiałynicki-Birula algorithm (BBA) is also practically very useful. We show thisby extending it to the problem of the dynamics of signals in dispersive materials.

2. Białynicki-Birula’s cellular automaton in (1D1) dimensions

Let us start with the first time-dependent pair of Maxwell’s equations ina homogeneous dispersionless dielectric in the absence of charges and currents(in SI units):

V!E ZKvB

vt; V!BZm0e0me

vE

vt; ð2:1Þ

where e and m are the dielectric constant and the magnetic permeability ofthe dielectric, respectively. In one spatial dimension, it is sufficient to representthe electric field vector as EZð0; 0;EðxÞÞ and the magnetic field vector asBZð0;BðxÞ; 0Þ (the other polarization may be considered in an analogous way).It is convenient to introduce a new variable F, such that EZcF.

Then Maxwell’s equations take the form

vF

vtZ

c

n2

vB

vx;

vB

vtZ c

vF

vx; ð2:2Þ

where n2Zem. If we introduce the vector GZðFðxÞ;BðxÞÞT, (2.2) takes the formvG

vtZ c sKC

1

n2sC

� �� vG

vx; ð2:3Þ

M. W. Janowicz and others2928

Proc. R. Soc. A (2006)



where

sKZ0 0

1 0

!; sCZ

0 1

0 0

!:

Together with sK and sC, we shall also use the matrices,

s11 Z0 0

0 1

!; s22 Z

1 0

0 0

!;

as well as the standard Pauli matrices sx and sz ,

sx Z0 1

1 0

!; sz Z

1 0

0 K1

!:

Using the identity (e.g. Louisell 1973)

eiðusCCvsKÞ Z cosffiffiffiffiffiuv

pC i

sinffiffiffiffiffiuv

pffiffiffiffiffiuv

p ðusCCvsKÞ;

we obtain as the solution of (2.3)

Gðx; tCDtÞZ coshða 0vxÞCsinhða 0vxÞ nsKC1

nsC

� �� Gðx; tÞ; ð2:4Þ

where a 0Za=nZcDt=n. For any real-analytic function f(x), we have

coshðavxÞf ðxÞZ 1

2ðeavx CeKavx Þf ðxÞZ 1

2ðf ðxCaÞC f ðxKaÞÞ;

as well as

sinhðavxÞf ðxÞZ 1

2ðeavxKeKavx Þf ðxÞZ 1

2ðf ðxCaÞKf ðxKaÞÞ;

so that if we write xZja0, tZ lDt with integer j, l, we obtain the followingdifference scheme to integrate Maxwell’s equations:

Fðj; lC1ÞBðj; lC1Þ

!Z

1

2

FðjC1; lÞCFðjK1; lÞC 1

nðBðjC1; lÞKBðjK1; lÞÞ

nðFðjC1; lÞKFðjK1; lÞÞCBðjC1; lÞCBðjK1; lÞ

0B@

1CA: ð2:5Þ

It is clear that equation (2.5) is an exact consequence of the Maxwell equations.We now perform a check of validity of the above simple difference scheme,namely, let us consider the initial-value problem for the fields propagating in avacuum (nZ1):

Fðx;0ÞZF0 cosðkxÞ; Bðx;0ÞZKF0 cosðkxÞ; ð2:6Þfor 0!x!l and Fðx;0ÞZBðx;0ÞZ0 for all other x. Contrary to appearances, thisproblem is not entirely trivial, since it involves discontinuities at two spatialpoints while we have derived (2.5) under the assumption of real analyticity of thefields. We would like to check numerically whether the algorithm is able to treatthe discontinuities in the Cauchy data properly and that the rounding errors do

2929Cellular automaton approach




not accumulate in a significant way. The exact solutions are, naturally,

Fðx; tÞZF0 cosðkðctKxÞÞ½QðtKðxKlÞ=cÞKQðtKx=cÞ�;

Bðx; tÞZKF0 cosðkðctKxÞÞ½QðtKðxKlÞ=cÞKQðtKx=cÞ�:The results of comparison of the above exact solution with the numerical one arepresented in figure 1 for F0Z100, kZ0.1. In this figure, the relative error of ouralgorithm with respect to the above exact solution has been plotted with respectto the coordinate x after 200 000 time-steps.

The two solutions are completely indistinguishable even for large numbers oftime-steps—the relative error has not exceeded 10K7. Thus, there are nodifficulties with discontinuities in the Cauchy data. We have also observed thatthe rounding errors do not accumulate.

If we want to write down a similar difference scheme for dielectrics with anx-dependent refractive index, we have to sacrifice the exact nature of theautomaton and introduce approximations.

Let n(x) be a positive (the case of left-handed materials is not covered here)refractive index varying in space, let A denote the operator vx and let B denoteð1=n2ðxÞÞvx . We have the following formal solution for the vector G:

Gðx; tCDtÞZ exp½cDtðAsKC BsCÞ�Gðx; tÞ: ð2:7ÞWe do not know of any simple way to represent explicitly the action of an

operator of the type

expðconst:!A BÞ;on a function of x. Operators of this form appear explicitly when we evaluate theexpression on the right-hand side of equation (2.7). Therefore, we must apply anapproximation to this to obtain a difference scheme. The simplest one relies on

0

1×10–8

2×10–8

3×10–8

4×10–8

5×10–8

6×10–8

7×10–8

199800 200000 200200 200400 200600 200800 201000 201200

rela

tive

erro

r

x/a

Figure 1. Relative error in the numerical solution shown for the propagation of an initiallydiscontinuous signal (2.6) after 200 000 time-steps.





the requirement that n(x) is a slowly varying function of x, so that we can ignorethe x-dependence of n(x) when we evaluate the exponential function in (2.7).Then we end up with a scheme (2.5), where the refractive index n is not constantbut is taken at the point x corresponding to the cell j.

In order to estimate the error and discover the limitations of the aboveprocedure, we calculate the expression RZGðx; tCDtÞKGðx; tÞ as well as asimilar expression Rneg obtained from R by neglecting the derivatives of n(x)over x, both in the lowest order with respect to Dt. We find

R Z cDt vxsKC1

n2ðxÞ sC� �

CðcDtÞ2 vx1

n2ðxÞ vxs11C1

n2ðxÞ v2xs22

� ��

COððDtÞ3Þ�Gðx; tÞ

and

Rneg Z cDt vxsKC1

n2ðxÞ sC� �

CðcDtÞ2 1

n2ðxÞ v2xs11C

1

n2ðxÞ v2xs22

� ��

COððDtÞ3Þ�Gðx; tÞ:

The difference RKRneg is equal to

ðcDtÞ2½K2ðn 0ðxÞ=n3ðxÞÞ�s11vGðx; tÞ

vx;

which gives conditions on the time increment Dt in terms of the variation of therefractive index and the fields themselves. In particular, one might think that fora stepwise change in the n characteristic for physically interesting systems likephotonic crystals, BBA is not applicable at all, because there are points wheren0(x) does not exist. However, we will provide numerical evidence to argue thatthe situation is not that bad, at least for systems with piecewise constantrefractive indices. In that case, our CA propagation in each particular layer isstill exact. In addition, we have observed that the field values at neighbouringcells on both sides of any boundary are reasonably close to each other for smalland moderate values of the difference of refractive indices. This suggests that thecontinuity conditions are approximately fulfilled at the boundary. Moreimportantly, the amplitudes of the reflected waves and of the transmittedwaves agree remarkably well with the Fresnel expressions for those amplitudes ifthe difference of indices is smaller than about 5. In table 1, we have shown therelative errors for the amplitude of reflected and transmitted waves when aninitial sinusoidal pulse falls on a single boundary between a vacuum andtransparent dielectric with refractive index n. We have performed simulationswith the wavelength of the pulse equal to 200 cells, i.e. lZ200a, and with thelength of the pulse equal to 10 wavelengths.

The data shown in table 1 are the ratios

Arr Z ArKnK1

nC1

� ��nK1

nC1





for the reflected wave and

Atr Z AtK2

nC1

� ��2

nC1

for the transmitted wave, where Ar and At are the amplitudes of the reflected andtransmitted wave read from the numerical data provided by a FORTRAN 77implementation of our algorithm (see electronic supplementary material).

We note that the error in the ratios of the reflected and transmitted amplitudecoefficients behaves even better, e.g. the relative error of that ratio for nZ4 is assmall as 1.17!10K10. Nevertheless, we have observed that for n larger thanabout 5 the algorithm breaks down, producing very big numbers for nz6. Thus,in any dispersionless layered system with strong differences between therefractive indices of the neighbouring layers, our cellular automaton is unreliable.The simplest and most natural remedy would be to introduce thin ‘buffer’ layersat the boundaries between two dielectrics with indices n1 and n2 in such a waythat the refractive index in the ‘buffer’ varies relatively slowly and interpolatesbetween the values n1 and n2. It should be noted that the ‘buffer’ must be as thinas possible in order to avoid the distortion of the wings of the reflected andtransmitted waves; moreover, the thinner the buffer, the less the relative error inthe amplitudes of these waves will be. We have found that with the buffer presenteven for the dielectric constant of the medium as large as 900, the error does notexceed 1%. Therefore, one might conclude that if n is smaller than 4, it isadvisable to retain the algorithm in its original form, while for larger n a buffer isnecessary.

The above considerations pertain to the least favourable case from the point ofview of our cellular automaton, namely, that which involves a discontinuousinitial-value function and a discontinuous parameter n(x). However, if the initialvalue is a smooth function, the situation is better. Let us again consider awavetrain falling from the vacuum side on the border between a vacuum and thedielectric with the refractive index nZ5. The initial F and B functions are givenby (2.6), but modulated with a Gaussian function. Figure 2 illustrates thecomparison of the numerical and exact results for that case with figure 2ashowing almost indistinguishable plots of the field F for the two cases, whilefigure 2b displays the relative error that does not exceed 3.5%.

We now turn to the case of homogeneous conducting media for which wepropose the following BBA. Let s be the conductivity and let hZcm0s (where m0

Table 1. Relative errors in the amplitudes of reflected and transmitted waves for various refractiveindices of the dielectric.

n Arr (10K4) Atr (10

K4)

1.33 0.40 0.381.5 0.73 0.732.0 1.71 1.713.0 2.47 2.474.0 4.16 4.165.0 4.81 4.81





is the magnetic permeability of the vacuum). Then Maxwell’s equations in (1C1)dimensions can be written as

vG

vtZ c vxsxK

h

2ð1CszÞ

h iGðx; tÞ; ð2:8Þ

with the meaning of G the same as before. We can thus write the formalsolution

Gðx; tCDtÞZ eKð1=2ÞhcDt exp cDt vxsxK1

2hsz

� �h iGðx; tÞ: ð2:9Þ

The difficulty in obtaining the exact difference scheme follows from thedifficulty in evaluating the expression

exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2=4Cv2x

q� �gðxÞ;

–400

–300

–200

–100

0

100

200

300

400

5500 5550 5600 5650 5700

E/c

x /a

numerical resultsexact results

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

rela

tive

erro

r

l=1000a, k =0.1

l = 1000a, k = 0.1

(a)

(b)

Figure 2. (a) Comparison of numerical and exact solutions shown for the propagation of atransmitted Gaussian-modulated wavetrain after crossing the boundary between the vacuum(x!d ) and the dielectric (xOd ) with a refractive index of nZ5. The Gaussian envelope has awidth 50a and was centred at xZ500a. The boundary was located at dZ5000a. (b) Relative errorfor solutions displayed in (a).





where g(x) is any smooth function. It would be tempting to invoke theLie–Trotter formula here (see ch. VIII of Reed & Simon (1974)). In brief, theLie–Trotter theorem states that if A, B and ACB are self-adjoint operators, then

sK limm/N

½expðiAðt=mÞÞexpðiBðt=mÞÞ�m Z expðitðACBÞÞ; ð2:10Þ

where sKlim denotes the limit in the strong (norm) sense. This means that if wemanage to write the evolution equation for G in the form

vG

vtZ iðACBÞG;

with self-adjoint A, B and ACB, we can be sure that by taking the limit Dt/0ðDtZ t=mÞ, the approximate solution given by

GðtÞZ ½expðiADtÞexpðiBDtÞ�mGð0Þ;

i.e. that obtained by applying m times the operator expðiADtÞexpðiBDtÞ,converges to the exact solution. This fact has been used as a basis for what iscalled the ‘split-operator’ or ‘splitting operator’ method (see Feit et al. 1982).This method is often used in the symmetric form, that is

eiðACBÞDtzeð1=2ÞiBDt eiADt eð1=2ÞiBDt;

and when combined with the fast Fourier transformation, it has recently beenextensively used in the context of the physics of atomic interactions with stronglight pulses (e.g. Pont et al. 1991 or Dimou & Kull 2000), the physics of atomiccollisions (e.g. Schultz et al. 2002), in molecular processes (Chu & Chu 2001) andthe theory of Bose condensates (Jackson & Zaremba 2002).

The difficulty which we immediately encounter, however, is that the operatormultiplying the vector G to produce the right-hand side of equation (2.8) is notself-adjoint (due to the presence of damping associated with the non-zeroconductivity). Therefore, we shall use the thesis of the Lie–Trotter theorem onlyas a kind of heuristic device to write

Gðx; tCDtÞzeKð1=2ÞhcDt expðcDtsxvxÞexp KcDth

2sz

� �Gðx; tÞ; ð2:11Þ

and postpone the proof of stability (which implies convergence in the consideredcase) of the difference scheme obtained in this way to the first part of appendixA. If we compute the difference between the exact and approximate evolutionoperators (i.e. the operators which multiply G on the right-hand sides ofequations (2.9) and (2.11)), we obtain

KihðcDtÞ2vxsy COððDtÞ3Þ:

Accurate results may, therefore, be obtained only if Dt is much smaller than thesquare root of 1=ðc2hjkjÞ, where k enumerates spatial harmonics of the Fourierdecomposition of G. That is, the propagation of harmonics with wavenumbersgreater that 1=ðc2ðDtÞ2hÞ will not be accurately covered by the algorithm.





3. Propagation in Drude-like materials

In this section, we apply the cellular automaton method to the description ofpropagation in (the layers of) homogeneous metals. The dielectric function of themetal is modelled according to the Drude prescription, i.e. we have

eðuÞZ 1Ku2p

u2C igu: ð3:1Þ

The above dielectric function results from the following system of partialdifferential equations for the scaled electric field FðxÞZð0; 0;FðxÞÞ, the magneticinduction BZð0;BðxÞ; 0Þ and an additional velocity field VZð0; 0;V ðxÞÞ:

v

vt

Fðx; tÞBðx; tÞV ðx; tÞ

0B@

1CAZ

cvBðx; tÞ=vxKcm0gðx; tÞV ðx; tÞ

cvFðx; tÞ=vxc

M gðx; tÞFðx; tÞKgV ðx; tÞ

0BBBB@

1CCCCA; ð3:2Þ

where the constant M is equal to ðe0u2pÞK1. Indeed, by taking the Fourier

transformation of equation (3.2) with respect to time for a homogeneous time-independent medium (i.e. for gðx; tÞZ1), and by eliminating B and V, we obtaina ‘one-dimensional Helmholtz equation’:

v2F

vx2C

u2

c21K

m0c2

MuðuC igÞ

� �F Z 0;

so that the quantity

1Km0c

2

MuðuC igÞ ð3:3Þ

can be interpreted as a dielectric function of the medium (cf. Jackson (1982)). Infact, the system of partial differential equations above was introduced preciselyin order to model a medium with that dielectric function.

If we now rescale the time and space variables and introduce the new velocityfield W according to

tZaupt; xZaup

cx; W Z ðcm0=upÞV ;

where the constant a specifies the length of the time-step, we obtain the followingnew system of equations, in which all the dependent variables have the samedimension (of the magnetic induction) while the independent variables aredimensionless:

v

vt

Fðx; tÞBðx; tÞW ðx; tÞ

0B@

1CAZ

vBðx; tÞ=vxKðgðx; tÞ=aÞW ðx; tÞvFðx; tÞ=vx

ðgðx; tÞ=aÞFðx; tÞKðG=aÞW ðx; tÞ

0B@

1CA; ð3:4Þ





where GZg=up and where we have introduced the function gðx; tÞZgðx; tÞ tospecify the regions in space and time where the metal resides, i.e. where thecoupling between the velocity field and the electromagnetic field takes place.

To deal with the above system of differential equations, we propose thefollowing three-step scheme to obtain the triple (F, B, W ) at time tCDt from(F, B,W ) at time t.

First, we propagate only the electromagnetic field:

F1ðxÞB1ðxÞW1ðxÞ

0B@

1CAZ

1

2½FðxCDt; tÞCFðxKDt; tÞCBðxCDt; tÞKBðxKDt; tÞ�

1

2½FðxCDt; tÞKFðxKDt; tÞCBðxCDt; tÞCBðxKDt; tÞ�

W ðx; tÞ

0BBBB@

1CCCCA:

ð3:5Þ

Second, we couple the electric field with the velocity field according to

F2ðxÞB2ðxÞW2ðxÞ

0B@

1CAZ

F1ðxÞcosðgðx; tÞDt=aÞKW1ðxÞsinðgðx; tÞDt=aÞB1ðxÞ

F1ðxÞsinðgðx; tÞDt=aÞCW1ðx; tÞcosðgðx; tÞDt=aÞ

0B@

1CA: ð3:6Þ

The third step consists of multiplying W2(x) with the functionexpðKgðx; tÞGDt=aÞ:

Fðx; tCDtÞBðx; tCDtÞW ðx; tCDtÞ

0B@

1CAZ

F2ðxÞB2ðxÞ

W2ðxÞexpðKgðx; tÞGDt=aÞ

0B@

1CA: ð3:7Þ

The measure of the time-step in our scheme is given by a; this parameter mustbe much larger than k, where again k is the wavenumber enumerating the spatialFourier components of G (harmonics with k larger than a will not be propagatedproperly by our automaton).

It is interesting to compare the results obtained with the use of our CA withthose produced by other methods. As a benchmark we have chosen an algorithmfor the inversion of the Laplace transformation proposed by Hosono (see Hosono1980) and improved in Wyns et al. (1989) (cf. Wyns et al. (1989) for discussion ofthat algorithm).

An example of the comparison of the two methods above is shown in figure 3,where the snapshots of solutions of the following initial boundary-value problemhave been displayed for tZ20 000. The space was assumed to be occupied by thehomogeneous (g(x)Z1) medium with the response function given by equation(3.3) with gZ0. The fields at xZ0 are fixed functions of t, tO0:FðtÞZsin ðnt=aÞ, B(t)Z0. The other parameters of the system were chosen tobe: nZ1.2, aZ100. The solid line in figure 3 represents the results obtained withthe use of our automaton, while the dashed line represents those obtained fromthe inversion of the Laplace transformation.





So far we have only been able to prove the stability (which in our caseimplied convergence) of the above scheme in the case of homogeneous media,i.e. for gðx; tÞZ1. One may object that in view of the above difficulties indescribing layered systems with large n without a ‘buffer’ layer, a similarbuffer should be introduced in the case of plasmas. This may be the case,although in all our numerical simulations we have observed excellentbehaviour of electromagnetic fields at the boundaries—the fields are as‘continuous’ as possible for a system on a grid. In addition, there is a principaldifference between the plasma systems of this section and the perfectlytransparent dielectrics considered earlier. Whereas the response of thosetransparent dielectrics to the incoming fields is ‘stiff’, instantaneous and static,the Drude-medium can build its response in a dynamic way with the dielectricfunction changing smoothly over space and time before it takes a well-established value in the bulk material, even though the region with non-zerocoupling between F and V has a well-defined beginning and end. In the case ofnecessity, one can easily create buffer layers by selecting groups of cells withthe values of g(x, t)Zg(x) interpolating between 0 and 1. The above commentsalso apply to the case of the dispersive dielectrics discussed in §4.

4. Dispersive and lossy dielectrics

We now address the problem of dispersive dielectrics. We wish to obtain a simplealgorithm which would reduce to BBA in the absence of the dielectric. This cannaturally be achieved by adding two new fields representing the matter, namely,the polarization field XZ(0, 0, X) and the associated velocity VZ(0, 0, V ). TheMaxwell equations together with the Newton equations for the polarization field

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

0 5000 10000 15000 20000

E/c

x

t=20000

CA results

Laplace-transformresults

Figure 3. Comparison of results obtained with the use of CA with the results of theLaplace-transform inversion, according to the algorithm of Wyns et al. (1989).





in one spatial dimension read

v

vt

Fðx; tÞBðx; tÞXðx; tÞV ðx; tÞ

0BBBB@

1CCCCAZ

cvBðx; tÞ=vxKcm0gðx; tÞV ðx; tÞ

cvFðx; tÞ=vx

V ðx; tÞ

Ku20Xðx; tÞC c

M gðx; tÞFðx; tÞKgV ðx; tÞ

0BBBBBBB@

1CCCCCCCA: ð4:1Þ

The function g(x, t) now specifies the location of our dispersive dielectric. It isequal to 1 in the space-time regions where the dielectric is present and zero whereit is absent. The above model of the dielectric is usually associated with the nameof Lorentz (cf. Jackson (1982)). The constant M is equal to ðbe0u2

0ÞK1, where b isthe static polarizability of the material while u0 is the frequency of a singleresonance line (there is no problem in describing more complicated lineardielectrics—the number of polarization fields has to be larger then). Thedamping constant g characterizes the width of the resonance line.

The model considered here leads to the following dielectric function for themedium:

eðuÞZ 1Kbu2

0

u2Ku20C igu

: ð4:2Þ

One can see this by performing the Fourier transformation of both sides ofequation (4.1) with respect to time (with g(x, t)Z1)), and eliminating all fields infavour of F. The resulting equation is

v2F

vx2C

u2

c2eðuÞF Z 0;

where e(u) is given by (4.2), so that e(u) acquires its natural interpretation of thedielectric constant or dielectric function. We immediately note that we can takea limit: u0/0 with bu2

0Zu2p kept constant to obtain a dielectric function

characteristic for the Drude metals considered in §3. In terms of equation (4.1)this means that X and V remain constant in the absence of the electric field.

Taking advantage of the existence of the natural time-scale associated with u0,we introduce new dimensionless time and coordinate variables:

tZau0t; xZau0

cx;

where a is an arbitrary positive parameter to set the time and space scales. Thus,our time-step is equal to T0=ð2paÞ, where T0 is the period of free oscillations ofthe polarization field. Most of our trial numerical simulations have been obtainedfor aZ100; however, we have checked that the results do not depend on a, atleast for 25!a!200. We now rescale the polarization variables in such a waythat all dependent variables have the dimensions of the magnetic induction:

X Z1

cm0

Y ; V Zu0

cm0

W :





As a result, we obtain a system of equations simpler than (4.1):

v

vt

Fðx; tÞBðx; tÞY ðx; tÞW ðx; tÞ

0BBBB@

1CCCCAZ

vBðx; tÞ=vxKgðx; tÞW ðx; tÞ=avFðx; tÞ=vxW ðx; tÞ=a

KY ðx; tÞ=aCbgðx; tÞFðx; tÞ=aKðG=aÞW ðx; tÞ

0BBBB@

1CCCCA; ð4:3Þ

where GZg=u0.Owing to the favourable 4!4 matrix structure of equation (4.3), we can

propose the following scheme (‘automaton’) to integrate the above system. Wefirst write (cf. (2.5) and (3.5))

F1ðxÞB1ðxÞY1ðxÞW1ðxÞ

0BBBB@

1CCCCAZ

1

2½FðxCDt; tÞCFðxKDt; tÞCBðxCDt; tÞKBðxKDt; tÞ�

1

2½FðxCDt; tÞKFðxKDt; tÞCBðxCDt; tÞCBðxKDt; tÞ�

Y ðx; tÞcosðDt=aÞCW ðx; tÞsinðDt=aÞ

KY ðx; tÞsinðDt=aÞCW ðx; tÞcosðDt=aÞ

0BBBBBBBBBB@

1CCCCCCCCCCA;

ð4:4Þso that we obtain in the first step the exact dynamics of the uncoupledsubsystems (except those for the damping of polarization fields). We believe thatthis is quite an important feature; although the details of the dynamics of thewhole system are reproduced only approximately, the propagation of anyelectromagnetic pulse from one spatial cell to another is fully causal.

In the second step, we take into account the interactions which result in

F2ðxÞB2ðxÞY2ðxÞW2ðxÞ

0BBBB@

1CCCCAZ

F1ðxÞcosðffiffiffib

pgðx; tÞDt=aÞKW1ðxÞsinð

ffiffiffib

pgðx; tÞDt=aÞ=

ffiffiffib

p

B1ðxÞY1ðxÞ

F1ðxÞffiffiffib

psinð

ffiffiffib

pgðx; tÞDt=aÞCW1ðx; tÞcosð

ffiffiffib

pgðx; tÞDt=aÞ

0BBBBB@

1CCCCCA:

ð4:5ÞEquation (4.5) is valid, provided that bR0. However, if b!0, the trigonometricfunctions should be replaced with hyperbolic functions and b with jbj with theappropriate sign changes. This case corresponds to that of pumped media, whichwill be discussed in more detail in a forthcoming paper.

To complete our algorithm, we take into account the damping of thepolarization field, and hence we write

Fðx; tCDtÞBðx; tCDtÞY ðx; tCDtÞW ðx; tCDtÞ

0BBBB@

1CCCCAZ

F2ðxÞB2ðxÞY2ðxÞ

W2ðxÞexpðKGDt=aÞ

0BBBB@

1CCCCA: ð4:6Þ





The above three steps are fully analogous to those of §3; now, however, thereexist non-trivial, oscillatory dynamics of the matter fields in the absence ofcoupling with the electromagnetic field.

The positive static polarizability b corresponds to the standard Lorentziandielectric. Dynamics of fields in such media have been the subject of severalfundamental papers by Oughstun and co-workers. To make our developmentmore specific and also more reliable, we provide examples of comparison ofresults obtained by us with the use of our CA with those obtained using thenumerical procedure described in Wyns et al. (1989), as well as in comparisonwith Oughstun’s asymptotic results. In this section, we follow Oughstun’s choiceof independent variable, which in virtually all his work is taken as qZct=xZt=xwith fixed x (x). In Oughstun & Sherman (1989b), the following parameterscharacterizing media have been chosen:

u0 Z4!1016 sK1; bu20 Z20!1032 sK2; gZ0:28!1016 sK1; uc Z1!1016 sK1;

where uc is the frequency of the applied field. The above values correspond to ahighly dispersive and absorptive medium.We again consider the initial boundary-value problem as in §3 but for the Lorentzian medium with the above parametersand for fixed x0Z13 343, which corresponds to the fixed value of x0Z10K4 cmfound in Oughstun & Sherman (1989b). The boundary values at xZ0 areFð0;tÞZF0 sinðntÞUðtÞ, Bð0;tÞZ0, where U(t) is the unit-step function, nZuc=ðau0Þ and F0Z2. This value of F0 requires explanation, because the ‘secondcanonical problem’ of Oughstun and co-workers involves the same form of F(0, t)but with F0Z1. Oughstun and co-workers do not provide an explicit expression forB(0, t), but it is given implicitly by their requirement that there is no propagationfor x!0 (x!0)—cf. a comment below eqn (9) in Wyns et al. (1989). The resultingB(0, t) is fairly complicated, given by a Laplace transform which cannot beinverted analytically. Therefore, in order to avoid the introduction of additionalerrors due to the numerical inversion of the Laplace transformation, we havechosen boundary values that give precisely the same Laplace-transformed electricfield for xO0. That is, we have propagation for x!0, but for xO0 our results for theelectric field are directly comparable with those of Oughstun.

In figure 4, we have shown the dynamics of the Sommerfeld forerunner(Brillouin 1960) as obtained from our CA and from the numerical Laplacetransformation inverting algorithm developed in Wyns et al. (1989).

As can be clearly seen, excellent agreement has been found. Figure 4 canbe compared to both fig. 3 of Oughstun & Sherman (1989b) and fig. 5 ofWyns et al.(1989) to see that very good agreement withOughstun’s and Sherman’s asymptoticresults has also been obtained for the Sommerfeld forerunner.

We now perform the same comparison for the Brillouin forerunner (Brillouin1960) as well as for an initial part of the main signal just after the arrival of thelatter. The parameters characterizing the signal and the medium are the same asbefore, but now x0Z10K3 cm, which corresponds to x0Z133 426. The abovecomparison is illustrated in figure 5.

It can once again be clearly seen that the agreement between the two numericalmethods is again very good, except for the amplitudes of the first few peaks. It isvery difficult to judge which method gives values closer to reality. This isespecially so because curiously enough the asymptotic results by Oughstun &





Sherman—cf. fig. 10 of Oughstun & Sherman (1989b)—seem to give almostexactly the arithmetic mean of the amplitudes of the main peak visible in ourfigure 5. This interesting though not very important feature may be investigatedfurther in the future, but we still note the most obvious thing in figure 5: for thearrival time of the Brillouin forerunner, the arrival time of the main signal andthe frequencies and phase relations, we obtain excellent agreement; the same istrue about the comparison of our figure 5 with fig. 10 of Oughstun & Sherman(1989b).

5. Possible improvements and extensions

In this section, we would like to outline very briefly how BBA can be extended orimproved in several directions. First, we observe that it can be employed to

–0.04–0.02

0

0.02

0.04

0.06

0.08

0.10

0.12

1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80

E/c x=133426

CA resultsLaplace-transform results

q =t /x

Figure 5. Brillouin forerunner and a part of the main signal just after its arrival, as obtained fromour CA and by inverting the Laplace transformation using the algorithm by Wyns et al. (1989).

–0.004

–0.002

0

0.002

0.004

1.00 1.05 1.10 1.15 1.20 1.25

E/c

x=13343

CA resultsLaplace-transform results

q=t /x

Figure 4. Time-dependence of the Sommerfeld forerunner as obtained from our CA and byinverting the Laplace transformation using the algorithm by Wyns et al. (1989).





study (phenomenological models of) metamaterials with effective negativerefractive index (i.e. ‘left-handed materials’) in the spirit of Ziolkowski &Heyman (2001). This is done quite simply by coupling the magnetic field with themagnetization field (and associated velocity) in the same way as the electric fieldhas been coupled here with polarization; then an appropriate splitting of theevolution operator follows. In the case of homogeneous materials, the proof ofstability and convergence can again be provided without much difficulty.

Second, one can consider the extension to include the nonlinear Maxwell–Bloch equations. Splitting of the time-evolution operator can be performed inseveral ways, all of which are based on physical intuition. However, we have haddifficulty in proving the stability of our cellular-automaton difference schemes,and this subject is still far from having any convincing treatment.

Third, following Białynicki-Birula (1994), we can extend CA to the case oftwo- and three-dimensional propagation. For the sake of brevity, we consider twospatial dimensions and are restricted to transverse electric (TE) polarization.Namely, now let the field F have only one non-zero component depending on xand y, so that

F Z ð0; 0;Fzðx; yÞÞ;while the magnetic induction has two components:

BZ ðBxðx; yÞ;Byðx; yÞ; 0Þ:Then the time-dependent pair of Maxwell’s equations take the form

v

vt

Fz

Bx

By

0B@

1CAZ c

vBy

vxK

vBx

vy

!

KvFz

vy

vFz

vx

0BBBBBBBBB@

1CCCCCCCCCA; ð5:1Þ

which can be conveniently rewritten in matrix (or spinorial) form:

vG

vtZ cðsxvx CsyvyÞG; ð5:2Þ

where

G ZBy C iBx Fz

Fz ByKiBx

!: ð5:3Þ

Equation (5.2) holds if and only if the condition of vanishing divergence of the Bfield is fulfilled. This equation forms the basis of a CA difference scheme on a‘body-centred’ grid in the xKy plane, obtained by using the Lie–Trotter theoremto disentangle the exponential time-evolution operator that results from (5.2).Interestingly, the difference scheme just outlined maintains the vanishingdivergence of B, provided that for every integration step G12 remains real(this is also a sufficient condition for having G12ZG21 at every step). The matrixG can be augmented with the polarization field and its associated velocity





written as G33 and G44; the linear coupling of G44 with G12 and G21 provides uswith the integrator of two-dimensional problems in dispersive dielectrics.

Finally, we note that our cellular automaton can be improved by using theresults of Suzuki (1990, 1991). Motivated by applications to the Quantum MonteCarlo simulations, Suzuki found approximations to the operator expðKxðACBÞÞin the form

et1A et2B et3A et4B.etmA;

with explicitly constructed tj, such that the method is OðxmC1Þ for small x.Naturally, our problem fits into Suzuki’s scheme well and his technique can beapplied to make the accuracy of our CA still higher.

6. Final remarks

In conclusion, we have proposed an extension of Białynicki-Birula’s algorithmsuitable for the numerical study of propagation in simple metals as well asinhomogeneous and dispersive dielectrics. The algorithm is conceptuallyextremely simple, since it only requires elementary properties of Pauli matricesand it is very easy to implement. Indeed, a FORTRAN 77 code to implement BBAcontains just a few tens of lines, including declarations and input–outputinstructions (see electronic supplementary material). Although from the point ofview of numerical analysis BBA forms just a specific explicit scheme to integratehyperbolic equations, it has distinct advantages in being an example of aquantum cellular automaton and providing within another context a natural linkto the path-integral representations of the Weyl and Dirac particles. BBA seemsto offer an alternative to the Laplace-transformation approach and othermethods of solving the wave propagation problems in dispersive materials, andallows generalizations to multi-dimensional and nonlinear problems where theLaplace transformation becomes inefficient.

We have seen that the application of the algorithm to the case of one dielectricinterface leads to the amplitudes of reflected and refracted waves with smallvalues of the errors. In our future research, we plan to employ the algorithmdescribed above to the analysis of propagation in two-dimensional media and insystems with moving dielectric walls and layers. Work is already underway ontwo-dimensional as well as on nonlinear variants of BBA that are suitable for aninvestigation of forerunners in Maxwell–Bloch systems.

M.W.J. gratefully acknowledges financial support from the Alexander von Humboldt Foundationand J.M.A.A. is supported by a Royal Commission for the Exhibition of 1851 Research Fellowship.This work has also been supported in part (for A.O.) by the Polish State Committee for ScientificResearch, grant no. PBZ-KBN-044/P03/2001.

Appendix A

The cellular automaton described in this paper can be thought of as just a certainexplicit difference scheme to numerically integrate a system of partial differentialequations. This definition raises the very important question of whether thatscheme is actually stable. So far we are only able to provide the (rather simple)





proofs for homogeneous media. We shall use the standard von Neumann &Richtmyer analysis in terms of Fourier series as in Richtmyer (1957). Strictlyspeaking, this method is applicable to the initial-value problem with periodicboundary conditions. However, we can embed the region of x or x for which weobtain our numerical solution in a very large but finite domain L, so that it ispossible to impose periodic boundary conditions at the borders of L. This makesit possible to use the expansion in terms of a Fourier series,

G1ðx; tÞZXk

eikx ~G1ðk; tÞ;

where G1 is a vector equal to (F, B)T in the case of propagation in a vacuumor homogeneous dielectric or conducting medium; (F, B, W)T in the case ofpropagation in Drude-like medium, or (F, B, Y, W)T in the case of propagation inthe single-resonance dielectric medium.

All cellular automata rules which have been defined in the main body of thepaper can be re-written for Fourier coefficients in the form

~G1ðk; tCDtÞZ H ~Gðk; tÞ: ðA 1ÞThe matrix H which appears in the last equation is called the amplificationmatrix.

We start with a formulation of the necessary condition of stability (seeRichtmyer 1957 for details). Letting l(i) be the eigenvalues of H , then theinequality is

jlðiÞj%1COðDtÞ; ðA 2Þwhich must be fulfilled for every Dt, such that 0!Dt!T, for every k, and for all i(where T is the total simulation time and the index i enumerates the eigenvalues)constitutes the von Neumann necessary condition for stability. In the afore-mentioned book by Richtmyer, four sufficient conditions for stability areprovided. We shall mainly use the second one, which can be formulated asfollows. If all the eigenvalues m(i) of the product PZ H

�H of the complex

conjugate of the amplification matrix and amplification matrix itself satisfy theinequality

mðiÞ%1COðDtÞ; ðA 3Þfor all 0!Dt!T and all k, then the difference scheme is stable. In the case ofpropagation in a vacuum or in the homogeneous dielectric, there is actuallynothing to prove: the scheme is exact, and as shown in figure 1, the roundingerrors seem to accumulate very slowly if at all. We therefore address the case ofpropagation in homogeneous conducting media.

The matrix H has the form

H ZeKha cosðkaÞ i sinðkaÞi eKha sinðkaÞ cosðkaÞ

!; ðA 4Þ

where as always aZcDt. The above matrix has the eigenvalues

ð1=2ÞeKhaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1CehaÞ2 cos2ðkaÞK4 eha

qGð1CehaÞcosðkaÞ

� �: ðA 5Þ





The modulus of the eigenvalue with the ‘minus’ sign behind the square root isbounded by either ð1=2Þð1CexpðKhaÞÞ or by expðKha=2Þ (both smaller than 1),depending on the sign of the expression under the square root. The eigenvaluewith the ‘plus’ sign is bounded by either 1 or expðKha=2Þ, again depending onwhether the square root is real or imaginary. Thus, the moduli of botheigenvalues are smaller than 1, so that von Neumann’s necessary condition forstability is fulfilled. In order to prove the sufficient condition, we multiply thematrix H by its complex conjugate and then expand eigenvalues of that productin terms of Dt. Hence, we obtain the following pair:

ð1K2haCOða2Þ; 1COða2ÞÞ;so that both eigenvalues are smaller or equal to 1CO(Dt), uniformly with respectto k, so that the second sufficient condition for stability is fulfilled. This concludesthe proof for the case of homogeneous conducting media.

We turn to the case of Drude-like media. Now, G1ZðF ;B;W ÞT. In terms ofFourier harmonics, equations (3.5)–(3.7) can again be written as

~G1ðk; tCDtÞZ HG1ðk; tÞ; ðA 6Þwhere the matrix H is now the product of three matrices, HZ H 3H 2H 1, where

H 3 Z

1 0 0

0 1 0

0 0 eKGDt=a

0BB@

1CCA; ðA 7Þ

H 2 Z

cosðDt=aÞ 0 KsinðDt=aÞ0 1 0

sinðDt=aÞ 0 cosðDt=aÞ

0B@

1CA; ðA 8Þ

H 1 Z

cosðkDtÞ i sinðkDtÞ 0

i sinðkDtÞ cosðkDtÞ 0

0 0 1

0B@

1CA: ðA 9Þ

The stability condition requires that there exists t0O0, such that for all k, allDt!t0 and 0%mDt%T (T being the simulation time), the matrices H

m, are

uniformly bounded. We observe that the norm of H is smaller or equal to theproduct of the norms of matrices H i, iZ1, 2, 3. However, the latter are normalmatrices; that is, they commute with their Hermitian conjugates. It follows thattheir norms are equal to their spectral radii, i.e. their largest eigenvalues. Themoduli of the largest eigenvalues of H i are exactly equal to 1, for iZ1, 2, 3. Thismeans that the norms of all the matrices H

mare also bounded by the constant 1,

that which was to be proved.We now consider the scheme given by equations (4.4)–(4.6). The kth harmonic

of the vector GZðF ;B;Y ;W ÞT satisfies the equation

~Gðk; tCDtÞZ H 3H 2H 1~Gðk; tÞ; ðA 10Þ





where H i, iZ1, 2, 3 are 4!4 matrices:

H 3 Z

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 eKGDt=a

0BBBB@

1CCCCA; ðA 11Þ

H 2 Z

cosðffiffiffib

pDt=aÞ 0 0 Ksinð

ffiffiffib

pDt=aÞ=

ffiffiffib

p

0 1 0 0

0 0 1 0ffiffiffib

psinð

ffiffiffib

pDt=aÞ 0 0 cosð

ffiffiffib

pDt=aÞ

0BBBB@

1CCCCA; ðA 12Þ

H 1 Z

cosðkDtÞ i sinðkDtÞ 0 0

i sinðkDtÞ cosðkDtÞ 0 0

0 0 cosðDt=aÞ sinðDt=aÞ0 0 KsinðDt=aÞ cosðDt=aÞ

0BBBB@

1CCCCA: ðA 13Þ

In order to estimate the norm of the matrix H , we first observe that H 1 isunitary. Thus, the norm of H

mis bounded by the norm of ðH 3H 2Þm. It is

therefore enough to check the necessary condition and one of the sufficientconditions for stability for the product H 3H 2. We can observe that this producthas the following eigenvalues: 1, 1 and 2=ðAGBÞ, where

AZ ð1CeGDt=aÞcosðffiffiffib

pDt=aÞ;

B ZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK4 eGDt=a CA2

p:

The eigenvalues possess the following expansion with respect to Dt:

ð1; 1; 1Cð1=2aÞðKGKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2K4b

pÞDt; 1Cð1=2aÞðKGC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2K4b

pÞDtCOððDtÞ2ÞÞ;

uniform with respect to k, and for any mDt!T with integer m. Thus, thenecessary condition for stability is satisfied. Multiplication of H 3H 2 by itscomplex conjugate (which is equal to the product itself), the computation ofeigenvalues and their expansion in terms of Dt lead to the result

ð1; 1; 1Kð1=aÞðGCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2K4b

pÞDt; 1Cð1=aÞðKGC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2K4b

pÞDtCOððDtÞ2ÞÞ;

so that our scheme satisfies again the second sufficient condition for stability.We should note here that in the apparently dubious (from the possibility of

exponential growth of the fields in time) case of media having a negative effectivestatic polarizability, we can still prove the stability of our CA—that proof,however, will be presented elsewhere.

Finally, we observe that a theorem due to Lax asserts that for linear systemswith constant coefficients stability implies convergence. This applies to all





difference schemes introduced in this paper. Therefore, making a larger (so thatthe time-step becomes smaller) in our system can only improve the results.

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