Direct time integration of Maxwell's equations intwo-dimensional dielectric waveguides for propagationand scattering of femtosecond electromagnetic solitons
Rose M. Joseph and Peter M. Goorjian
National Aeronautics and Space Administration, Ames Research Center, Mail Stop 202A-2, Moffett Field, California 94035-1000
Allen Taflove
Department of Electrical Engineering and Computer Science, McCormick School of Engineering,Northwestern University, Evanston, Illinois 60208-3118
Received November 30, 1992
We present what are to our knowledge first-time calculations from vector nonlinear Maxwell's equations of
femtosecond soliton propagation and scattering, including carrier waves, in two-dimensional dielectric waveguides.The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, andthe nonlinear convolution accounts for two quantum effects, the Kerr and Raman interactions. By retaining theoptical carrier, the new method solves for fundamental quantities-optical electric and magnetic fields in space
and time-rather than a nonphysical envelope function. It has the potential to provide an unprecedented two-
and three-dimensional modeling capability for millimeter-scale integrated-optical circuits with submicrometerengineered inhomogeneities.
Experimentalists have produced all-optical switchescapable of 100-fs responses.' To model such switchesadequately, nonlinear effects in optical materials2
(both instantaneous and dispersive) must be included.In principle, the behavior of electromagnetic fieldsin nonlinear dielectrics can be determined by solv-ing Maxwell's equations subject to the assumptionthat the electric polarization has a nonlinear rela-tion to the electric field. However, until our pre-vious research,3 4 the resulting nonlinear Maxwell'sequations have not been solved directly. Rather,approximations have been made that result in aclass of generalized nonlinear Schr6dinger equations(GNLSE's) that solve only for the envelope of theoptical pulses. 5
Two-dimensional (2-D) and three-dimensional (3-D)engineered inhomogeneities in nonlinear optical cir-cuits will probably be at distance scales on the orderof 0.1-10 optical wavelengths, and all assumptionsregarding slowly varying parameters (which runthroughout GNLSE theory) will be unjustified. Forsuch devices, optical-wave scattering and diffractioneffects relevant to integrated all-optical switches willbe difficult or impossible to obtain with GNLSE,because its formulation discards the optical carrier.The only way to model such devices is to retain theoptical carrier and solve Maxwell's vector-field equa-tions for the material geometry of interest, rigorouslyenforcing the vector-field boundary conditions andthe physics of nonlinear dispersion.
In this Letter we describe what are to ourknowledge first-time solutions of 2-D vector nonlinearMaxwell's equations for material media with linearand nonlinear instantaneous and Lorentz-dispersiveeffects in the electric polarization. We use the finite-difference time-domain (FD-TD) method in an exten-
sion of our previous research in one dimension.3 4
The optical carrier is retained in this approach.The fundamental innovation is the treatment ofthe linear and nonlinear convolution integrals thatdescribe the dispersion as new dependent variables.By differentiating these convolutions in the timedomain, an equivalent system of coupled, nonlinear,second-order ordinary differential equations (ODE's)is derived. These equations together with Maxwell'sequations form the system that is solved to determinethe electromagnetic fields in nonlinear dispersivemedia. Backstorage in time is limited to only thatneeded by the time-integration algorithm for theODE's (two time steps) rather than that needed tostore the time history of the kernel functions of theconvolutions. Thus, a 2-D nonlinear optics modelfrom Maxwell's equations is now feasible.
Now we present the theoretical development. Con-sider a 2-D transverse-magnetic problem. Maxwell'sequations for the electric- and magnetic-field intensi-ties, Ez, H., and Hy, are given by
a t.LoHf, aE,
at ayauoHy _ aE,
at ax
aD. = aHl _ aH.at ax ay
(1)
We allow for dielectric nonlinearity by assuming thatthe electric polarization, P_, consists of the sum of alinear part, pZL, and a nonlinear part, p2 NL 5 Thenwe have
Dz = eoeE. + Pz, (2)p = pI'L + p NL,
where pZL is given by a convolution of E,(x,y, t) and
1 d 2F 3 dF e83-eC. 1CO2 dt2 + wo2 dt + [1 + ex + a(3)E_ 2
1 F
+ (e. - e-F)(1 - a)X(3)E z 1G+ [ eo + aV(3)E.2 j
F e - E lDL eC + aX( 3)E 2
Here (0p2 = w0) 2(e8 - e,,) and v02 = 002 - 62/4. Fur-
ther, we assume a dispersive (memory-type) materialnonlinearity2 characterized by the following time con-volution for PzNL:
PzNL(x,y, t) = eOx(3 Ez(xyt) f g(t - t')E. 2(x,y,t')dt',
(4)
where X(3) is the nonlinear coefficient and fJg(t)dt =1. Equation (4) accounts for phonon interactions andnonresonant electronic effects, as given by
g(t) = a 8(t) + (1 - a)gR(t),
(,2 ± 722gR(t) = 2 )exP(-t/7 2)sin(t/7j). (5)
Here 8(t) is the instantaneous delta function thatmodels Kerr nonresonant virtual electronic transi-tions on the order of -1 fs or less, and gR(t) modelstransient Raman scattering.
We now describe the system of coupled nonlinearODE's that governs the time evolution of the polariza-tion. Assuming zero values of the electromagneticfield and the kernel functions for t c 0, define thefunctions F(t) and G(t) as the convolutions
rtF(t) = eof X(')(t - t')E,(x,y,t')dt',
tG(t) = eo gR(t - t')Ez2(xy, t')dt'. (6)
Then, by time differentiating F and G, these functionssatisfy the following coupled system:
1 d 2 G
-0o02 d t2 8 dG
(&0o2 dt
+ e + E F =, E.x + aY(3)E,2
L1 + (1 - a)X(3)Ez2]Gec. ± ax(3)E,,2 j
[ el+ 3E 2 Dz,e,, + axX(3)Ez,2] (8)
where 8 = 2/T2 and -&o2 = (1/r )2 + (1/W2)2. Equa-
tions (7) and (8) are first solved simultaneously forF and G at the latest time step by using a second-order accurate finite-difference scheme that operateson data for the current value of D_ and previousvalues of Dz, Ez, F, and G. Then, the latest valueof E, can be obtained by a Newton's iteration, usingthe new values for Dz, F, and G:
(9)
The system of Eqs. (7)-(9) determines values ofE, and P, so that Eq. (2) is satisfied. This proce-dure, combined with the usual FD-TD realizationof Maxwell's equations [Eqs. (1)], comprises thecomplete solution method. The FD-TD algorithmused here is a generalization to two dimensions ofthat in Ref. 4.
The modeling capabilities of this new algorithmare demonstrated by 2-D calculations of propagatingand colliding solitons. The calculations are for apropagating pulse with a carrier frequency of 1.37 x1014 Hz (A = 2.19 ,m) and a hyperbolic-secant enve-lope with a characteristic time constant of 14.6 fs.The computational domain for the 2-D dielectricwaveguide is 110 Am X 5 ,m, with the dielectricwaveguide itself 1 Am thick and with 2 Atm ofair on either side. The first calculation simulatesLorentz-medium linear dispersion alone [Eq. (3)].As Fig. 1 shows, the pulse undergoes predicted pulselengthening owing to dispersive effects.5
The second calculation simulates the effects of thefull linear [Eq. (3)] and nonlinear [Eqs. (4) and (5)]
Fig. 1. Electric field of a propagating optical carrier pulse with initial hyperbolic-secant envelope (A = 2.19 ,um, r =14.6 fs) in a 1-,um-thick linear Lorentz-medium dielectric waveguide.
Fig. 2. Electric field of an optical soliton carrier pulse corresponding to that in Fig. 1, including quantum effectssuch as the Kerr and Raman interactions.
Fig. 3. Electric field of colliding counterpropagating solitonsinterfering, constructively interfering, separating).
polarizations. As shown in Fig. 2, the propagatingpulse now has the features of a soliton with theretention of its length. In addition, detailed plotsshow that a second low-amplitude, high-frequency,daughter pulse forms and moves out ahead of thesoliton.
The third calculation (Fig. 3) simulates the col-lision of two equal-amplitude counterpropagatingsolitons. The results show the solitons interactingduring the collisions and then separating with-out general changes. However, by comparing thecarriers of the collided solitons with those of thenoncollided solitons, precise carrier-phase lags ofthe collided solitons are measured.
The novel approach discussed here achieves robust-ness by rigorously enforcing the vector-field boundaryconditions at all interfaces of dissimilar media inthe time scale of the optical carrier, regardless ofwhether the media are dispersive or nonlinear. Asa result, the new approach is almost completely gen-eral. It assumes nothing about (1) the homogeneityor isotropy of the optical medium, (2) the magnitudeof the nonlinearity, (3) the nature of the material'sw0 -,8 variation, and (4) the shape, duration, andvector nature of the optical pulse(s). By retainingthe optical carrier, the new method solves for funda-
corresponding to that in Fig. 2 (approaching, destructively
mental quantities-the optical electric and magneticfields in space and time-rather than a nonphysicalenvelope function. It has the potential to providean unprecedented 2-D and 3-D modeling capabilityfor millimeter-scale integrated optical circuits withsubmicrometer engineered inhomogeneities.
The work at Northwestern University was sup-ported in part by NASA-Ames University ConsortiumJoint Research Interchanges NCA2-561, 562, and727, by Office of Naval Research contract N00014-93-0133, and by Cray Research, Inc. The work atNASA-Ames Research Center was supported in partby Office of Naval Research MIPR contracts N00014-93-MP24017 and 24018.
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