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1412 OPTICS LETTERS / Vol. 16, No. 18 / September 15, 1991
Direct time integration of Maxwell's equations in lineardispersive media with absorption for scatteringand propagation of femtosecond electromagnetic pulsesRose M. Joseph, Susan C. Hagness, and Allen Taflove
Department of Electrical Engineering and Computer Science, McCormick School of Engineering, Northwestern University,Evanston, Illinois 60208-3118
Received May 29, 1991We report the initial results for femtosecond pulse propagation and scattering interactions for a Lorentzmedium obtained by a direct time integration of Maxwell's equations. The computational approach providesreflection coefficients accurate to better than 6 parts in 10,000 over the frequency range of dc to 3 X 1016 Hzfor a single 0.2-fs Gaussian pulse incident upon a Lorentz-medium half-space. New results for Sommerfeld andBrillouin precursors are shown and compared with previous analyses. The present approach is robust and per-mits two-dimensional and three-dimensional electromagnetic pulse propagation directly from the full-vectorMaxwell's equations.
A pulse propagating in a dispersive medium such asan optical fiber exhibits complicated behavior. It isof interest to have an accurate numerical model forthis behavior as well as for other electromagnetic in-teractions with frequency-dependent materials.The finite-difference time-domain (FD-TD)method is a numerical technique for direct time in-tegration of Maxwell's equations.` 4 It is a compu-tationally efficient approach to modeling sinusoidalor impulsive electromagnetic interactions with arbi-trary three-dimensional structures consisting oflinear, possibly anisotropic, lossy dielectric andpermeable media with frequency-independentparameters. It has been used for predicting elec-tromagnetic wave scattering, penetration, and radi-ation for a variety of problems.5 Recently, therange of FD-TD modeling applications has been sub-stantially expanded to include ultra-high-speed sig-nal lines,6 subpicosecond electro-optic switches,7 andlinear optical directional couplers.8Attempts have been made to extend FD-TD tofrequency-dependent materials. Chromatic disper-sion can be expressed in the time domain as a convo-lution integral involving the electric field and acausal susceptibility function. This convolutionintegral can be efficiently incorporated into the FD-TD algorithm for a first-order (Debye) dispersion.9In this Letter we present a more general approachthat permits modeling of media having arbitrary-order dispersions. Our approach is based on asuggestion by Jackson (Ref. 10, p. 331) to relate theelectric displacement D(t) to the electric field E(t)through an ordinary differential equation in time.We consider a one-dimensional problem with fieldcomponents E, and H, propagating in the x direc-tion. If we assume first that the medium is nonper-meable, isotropic, and nondispersive, Maxwell's curlequations in one dimension are
aH,_ 1 aE,at 0 axaD, = aH,at ax
(la)(lb)
Here Dz = eE., where the permittivity s is indepen-dent of frequency. Using central differencing intime and space,' we can express the curl equationsin finite-difference form as the following second-order accurate leapfrog algorithm:
2y(i+ 2) = Hy,-2(i + 2)At+ AA [Ezn(i 1) En(i)]
Ezn+l(i) = En4(i)+ A [HAt(i + I)- H,,n+(i 2)
(2a)
(2b)where Ezn(i)denotes the electric field sampled atspace point x = iAx and time point t = nAt.(Please refer to Ref. 2 for the proper numerical sta-bility criterion.)For many dispersive media of interest, however,e = e(w). We propose to include this frequency de-pendence in the FD-TD model by concurrently inte-grating an ordinary differential equation in timethat relates D,(t) to E2(t). As suggested by Jackson,this equation is derived by taking the inverseFourier transform of the complex permittivityexpression,
)=:(c) (3)For an order-M dispersion, the computational modelnow becomes a three-step recursive process that re-
0146-9592/91/181412-03$5.00/0 ( 1991 Optical Society of America
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September 15, 1991 / Vol. 16, No. 18 / OPTICS LETTERS 1413tains the fully explicit nature of the original disper-sionless FD-TD formulation,
y+2(i + .) = Hyf--(i+ .)+ A[Ezn(i + 1) -E 8n(i)], (4a)
Dzn+l(i)=Dn(i) + At[Hn+2(j + .)-Hn+2(-2] (4b)
Ezn+l(i) f(Dn+l,.. ., Dzn-M1;E;... ,EznM+l).(4c)
At any time step n, this method requires storage ofM previous values of Dz and M - 1 previous valuesof Ez beyond the current field values. The approachwill be made clear by the following examples:Example 1: A first-order (Debye) dispersion canbe specified by=8,. + 8_ =(5)1-j(oT Ez(co)' 5where es= 8(0), e. = 8(oo), and Xis the Debye relax-ation time constant. If we take the inverseFouriertransform of Eq. (5) as defined by
f(t) = f f(co)exp(-jiot)dw,
This differential equation can be easily differencedto solve for Efn' in terms of known values of E, andDz for insertion into Eq. (4c),En,+l [((ooAt2 + 28At 2)Dz+l - 4Dn
+ (6002At2 - 28At + 2)DZn-1 4e.Ezn- (c02 t2 8 - 286Ate. 2e.)Ezn-l]/
(oo02 t288 + 28At8. + 28.). (11)We first demonstrate the accuracy of this methodby computing the wideband reflection coefficient fora planar interface between vacuum and a half-spacemade of the Lorentz medium of Fig. 1. A single0.2-fs duration Gaussian pulse (between the l/epoints) is normally incident upon the interface.This pulse has a spectrum that covers the full rangefrom dc to 3 x 1016Hz. Data are taken every timestep (At = 2.0 x 10-19 ) at a fixed observation pointon the vacuum side of the interface. The FD-TDcomputed complex-valued reflection coefficient isobtained by taking the ratio of the discrete Fouriertransforms of the reflected and incident pulses.Figure 2 compares the magnitude and phase of this10.0
(6)the result is a first-order differential equation intime relating D, and E8,
dD= dEtD_+ ZE +I..dt ~ dt (7)This differential equation can be easily differencedto solve for Ez,"' in terms of known values ofE2 andD8 for insertion into Eq. (4c),
E,,n~~i At+2T li At 2T -(i= ~~Dlj)+D,~i2Ts()= 88At Z 2TE. + 8sAt+ 2-8. + 8At En(i). (8)
Example 2: A second-order (Lorentz) dispersioncan be specified by8(to) = E. - cso (s 8 8E.) Dj(w)02 + 2j1o(a _ (o2 E8(c) (9)
where 8, = E(0), 8. = E(-), coo s the resonant fre-quency, and 8 is the damping coefficient. Figure 1shows the relative permittivity curve for a Lorentzmedium that has the following parameters:= 2.258o, 8. = 8o, 0o = 4.0 x 1016 ad/s,
8 = 0.28 x 1016S-i.Inverse Fourier transformation of Eq. (9) results inthe following second-order differential equation re-lating D, and E,:
2D, + 2dD 8 + d2D 2 E + 23EdEz+ dd2 (10dtd E8+,.t 2 - (10)
8.0 -.- 6.0 -
4.0 -Q)0- 2.0 -6)*S 0.0 -Q)CY2.0 -
-4.0 0 16 20. . . . . .12(units of 10 rad/sec)Fig. 1. Complex permittivity of the Lorentz mediumwith parameters E, = 2.25so, E8. = E0, wo = 4.0 x1016ad/s, and 8 = 0.28 x 10'6 -'.6).80 0.6a). _, 0.4Q)0O 0.20
6)0- 0.0
-0.0 6)cn
--45.0 ' ,02. _ 0Q-)
6)6),
180.0(units of 1016rad/sec)Fig. 2. Comparison of FD-TD and exact results from dcto 3 x 10'6Hz for the magnitude and phase of the reflec-tion coefficient of a half-space made of the Lorentzmedium of Fig. 1.
,I11 Real1I --- ImaginaryIIIII'I'
I
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1414 OPTICS LETTERS / Vol. 16, No. 18 / September 15, 19910.005a,-D
_ 0.003D-- -0.001
LZ.o -0.0030) -0.005Lb 1 .00
- Asymptotic_ Laplace transform-- FD-TD
1.06 1.12O=ct/x 1.18 1.24Fig. 3. Comparison of FD-TD, asymptotic,'3 and Laplace-transform'4 results for the Sommerfeld precursor ob-served at x = 1 u.Lmn the Lorentz medium of Fig. 1 for aunit-step modulated sinusoidal excitation (co,= 1.0 x1016rad/s) at x = 0.
0.12a)-0
.5_LLE
.006)
Ld
0.08
0.04
0.00
-0.04 4-i.4s 1.50 1.55 1.60 1.65Q=ct/xFig. 4. FD-TD results for the total signal (including theBrillouin precursor) at x = 10 jum n the Lorentz mediumof Fig. 1 for the unit-step modulated sinusoidal excitation.
served at x = 1 ,m to the asymptotic"3 and Laplace-transform"4 predictions. Much closer agreementwith the Laplace-transform calculation is noted.Extensive numerical convergence studies of the FD-TD results indicate that the zero crossings of theprecursor converge quickly (at relatively coarse gridresolutions), while the envelope converges moreslowly to a limiting distribution. Overall, webelieve that the FD-TD computed envelope distribu-tion shown in Fig. 3 is within 3% of the limiting dis-tribution obtained at infinitely fine grid resolution.For completeness, Fig. 4 shows the total signal atx = 10 gm in the Lorentz medium computed withthe FD-TD method. This includes the -Brillouinprecursor. These results are again somewhat dif-ferent from the asymptotic results reported inRef. 13, yet the FD-TD calculations here exhibit atleast the same degree of convergence as those ofFig. 3.The method of this Letter should be directly ap-plicable to full-vector electromagnetic pulse propa-gation and scattering effects for inhomogeneousdispersive media in two and three dimensions. Weforesee the possibility of incorporating materialnonlinearity to obtain the dynamics of soliton propa-gation and scattering directly from the time-depen-dent Maxwell's equations.
This research was supported in part by U.S.Office of Naval Research contract N00014-88-K-0475, NASA-Ames University Consortium Joint Re-search Interchange no. NCA2-562, and CrayResearch, Inc. We thank K. E. Oughstun forproviding graphical results of his precursor analy-ses'3 ,14 and P. Goorjian of NASA-Ames for suggest-ing the improved time-stepping scheme of Eq. (11).References
reflection coefficient as a function of frequency tothe exact solution (Ref. 10, p. 282). The deviationfrom the exact solution over the complete range of dcto 3 x 1016Hz is less than 6 parts in 10,000. (This6/10,000 error occurs at the peak of the reflectionmagnitude curve.)The time integration of Maxwell's equations per-mits the computation of a pulse propagating in adispersive medium at any space-time point. His-torically, such pulse dynamics have been obtainedonly by asymptotic analyses, notably by Sommer-feld'1 and Brillouin12 in 1914. More recently, ad-vances in uniform asymptotic analysis for suchproblems have been made by Oughstun and Sher-man'3 and in Laplace transform analysis by Wynset al.'1To demonstrate the integration of Maxwell's equa-tions to obtain pulse dynamics, we now use the FD-TD method to compute the precursor fields for aunit-step modulated sinusoidal signal propagatingin the Lorentz medium discussed in Figs. 1 and 2.Now the signal source is located at x = 0. The car-rier frequency co,is 1016 ad/s. Figure 3 comparesthe FD-TD computed Sommerfeld precursor ob-
1. K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302(1966).2. A. Taflove and M. E. Brodwin, IEEE Trans. Mi-crowave Theory Tech. MTT-23, 623 (1975).3. G. Mur, IEEE Trans. Electromagn. Compat. EC-23,377 (1981).4. K. R. Umashankar and A. Taflove, IEEE Trans. Elec-tromagn. Compat. EC-24, 397 (1982).5. A. Taflove, Wave Motion 10, 547 (1988).6. G. C. Liang, Y W Liu, and K. K. Mei, IEEE Trans.Microwave Theory Tech. MTT-37, 1949 (1989).7. E. Sano and T. Shibata, IEEE J. Quantum Electron.26, 372 (1990).8. S. T. Chu and S. K. Chaudhuri, IEEE J. LightwaveTechnol. 7, 2033 (1989).9. R. Luebbers, F P. Hunsberger, K. S. Kunz, R. B. Stan-dler, and M. Schneider, IEEE Trans. Electromagn.Compat. EC-32, 222 (1990).10. J. D. Jackson, Classical Electrodynamics, 2nd ed.(Wiley, New York, 1975).11. A. Sommerfeld, Ann. Phys. 44, 177 (1914).12. L. Brillouin, Ann. Phys. 44, 203 (1914).13. K. E. Oughstun and G. C. Sherman, J. Opt. Soc. Am.
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