fdtd maxwell™s equations models for nonlinear electrodynamics and...

364 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997

FDTD Maxwell’s Equations Models forNonlinear Electrodynamics and Optics

Rose M. Joseph,Member, IEEEand Allen Taflove,Fellow, IEEE

Invited Paper

Abstract—This paper summarizes algorithms which extend thefinite-difference time-domain (FDTD) solution of Maxwell’s equa-tions to nonlinear optics. The use of FDTD in this field is novel.Previous modeling approaches were aimed at modeling optical-wave propagation in electrically long structures such as fibers anddirectional couplers, wherein the primary flow of energy is along asingle principal direction. However, FDTD is aimed at modelingcompact structures having energy flow in arbitrary directions.Relative to previous methods, FDTD achieves robustness bydirectly solving, for fundamental quantities, the opticalE andHfields in space and time rather than performing asymptotic analy-ses or assuming paraxial propagation and nonphysical envelopefunctions. As a result, it is almost completely general. It permitsaccurate modeling of a broad variety of dispersive and nonlinearmedia used in emerging technologies such as micron-sized lasersand optical switches.

Index Terms—FDTD methods, nonlinear wave propagation.

I. INTRODUCTION

TODAY’S need for faster and better means of com-munication and data processing has encouraged rapid

growth of the field of nonlinear optics. This has led todevelopment of a variety of new devices for transmitting,switching, and storing data using electromagnetic energy inand near the visible spectrum. These devices include opticalfibers, couplers, switches, and amplifiers. Materials technologyand fabrication methods, particularly for semiconductors, haveadvanced accordingly, allowing devices to be built on asubmicron scale with fine structural details. Pulses as fast asa few tens of femtoseconds are being generated and guidedthrough these new systems at repetition rates approaching theterahertz regime.

One result of this progress is the increased need for accuratemodels for predicting the electromagnetic field behavior ofthese new optical devices to permit their efficient designwithout repeatedly building expensive prototypes. Fortunately,the tremendous increase in computing power has made detailed

Manuscript received May 8, 1996; revised October 10, 1996. This workwas supported in part by NSF Grant ECS-9218494, ONR Contract N00014-93-0133, NASA-Ames University Consortium Joint Research InterchangesNCA2-561, 562 and 727, and Cray Research, Inc.

R. M. Joseph is with the MIT Lincoln Laboratory, Lexington, MA 02173USA.

A. Taflove is with the Department of Electrical and Computer Engineering,McCormick School of Engineering, Northwestern University, Evanston, IL60208 USA.

Publisher Item Identifier S 0018-926X(97)02286-2.

numerical modeling available to the optical-design engineer.An emerging modeling tool in this area is the finite-differencetime-domain (FDTD) solution of Maxwell’s equations [1]–[3].While FDTD is now well accepted in the analysis of linearelectromagnetic problems, especially at microwave frequen-cies, it is still a novel approach in the nonlinear opticscommunity that has long used asymptotic and paraxial waveequation models derived from Maxwell’s equations as theprimary detailed modeling tool. As this paper will show, thefull-wave time-domain features of the FDTD field solver,augmented by recent extensions, are particularly well-suitedto bring new insight to nonlinear optics problems.

To observe nonlinear effects in commonly used materials,a high-intensity light source, such as a laser, is required. Thenonlinear behavior is due to the dependence of the polarization

on the applied electric field, . Assuming an isotropicand frequency-independent medium, the polarization can beexpanded as a power series in[4]

(1)

where and are, respectively, the linear and nonlinearcomponents of the polarization. Here, denotes the linearsusceptibility of the medium, and the quantities , ,

are the nonlinear susceptibilities. The particular nonlineareffect observed depends on which term is dominant in (1).Second- and third-order nonlinear behavior, requiring lessinput power to observe than higher order effects, have beenthe topic of much study. Some second-order nonlinear effectsinclude sum and difference frequency generation, parametricamplification, and the Pockels effect [5]. disappears formaterials with inversion symmetry such as optical-fiber glass.In such materials, the lowest order nonlinear coefficient is.Third-order nonlinear effects include the quadratic electro-optic effect, third-harmonic generation, four-wave mixing,intensity-dependent refractive index, stimulated Raman andBrillouin scattering, and two-photon absorption [5]. In mostoptical materials, the refractive index also varies withfrequency resulting in both linear and nonlinear chromaticdispersion.

Both analytical and numerical methods have been applied tomodel the behavior of light propagating in a nonlinear medium.

0018–926X/97$10.00 1997 IEEE

JOSEPH AND TAFLOVE: FDTD MAXWELL’S EQUATIONS MODELS 365

Typical approaches begin with a wave equation

(2)

derived from Maxwell’s equations, where the relative per-mittivity can be spatially varying, frequency dependent,and nonlinear in . Here, the divergence of the electricfield is assumed to be negligible. The linear and nonlinearcontributions of to the polarization can be treated separatelyto obtain

(3)

where is the linear refractive index andcomprises the nonlinear term of interest.

Here, most models for nonlinear optical processes makeany of several assumptions. First, a preferred direction ofenergy transport may be assumed. In guided-wave problems,the transverse modal profile may be decoupled from thepropagation equation. Second, the variation of the electric-field envelope may be assumed to be small over distancesof the order of one optical wavelength. Thisslowly varyingenvelope approximation(SVEA) permits second derivativesof in the propagation direction to be neglected, reducingthe wave equation of (3) to a simpler propagation equation[6]. Such simplified models may have analytical or numericalsolutions that permit effective simulation of optical-pulsepropagation over very long distances [6], [7]. An examplewhere this is required is long-distance pulse propagation in asingle-mode fiber, where signals can travel many hundreds ofkilometers. Here, the transverse (modal) profile of the lightin the fiber remains constant, reducing the problem to onespatial dimension. The reduced-order equation can then betransformed to a reference frame that travels with the pulse

, where is the propagation constant [6]. Themost common case of this method results in thenonlinearSchrodinger equation(NLSE), which is used to study pulsepropagation in a medium

(4)

Here, is the group velocity dispersion andis the nonlinearcoefficient. For an optical pulse propagating in a fiber, forexample, the first term of (4) gives the propagation behavior,the second term accounts for the pulse spreading due tolinear dispersion, and the third term models the third-ordernonlinearity. For a very short pulse having a spectral widthcomparable to its center frequency, cubic and higher orderdispersion effects become appreciable, and the NLSE modelof (4) is inadequate. Additional terms must be included tomodel such higher order behavior [6]. Reference [8] showedthat a suitable NLSE model can accurately predict the behaviorof pulses as short as 10–15 optical cycles for one-dimensional(1-D) propagation problems.

Clearly, optical fibers are designed such that the transversemode is well confined and radiation losses are low. How-ever, for wave propagation that is not totally confined inthe transverse direction, the possibility exists for transverse

energy flow. Consider, for example, the case of optical-beam propagation in a bulk, nonlinear material. Here,by applying theparaxial approximation(which assumes thatthe beam energy does not diverge much from its primarydirection of propagation [9]), an NLSE model analogous to (4)can be constructed wherein the second term in the equationrepresents transverse beam diffraction rather than longitudi-nal pulse dispersion. Note, however, that the assumption ofparaxial propagation fundamentally limits such NLSE modelsto predictions of small-angle scattering or diffraction loss.

In contrast, the numerical FDTD method for nonlinear opticsis completely general. FDTD is an explicit full-wave finite-difference solution of the time-dependent Maxwell’s equationsthat yields both electric and magnetic fields with a spatialresolution much finer than one wavelength. As a result, FDTDtypically requires tens of Mwords and tens of thousands oftime steps of solution evolution to solve practical problemsin modern nonlinear optics. However, the results often justifythe computational burden since the number of approximationsmade is minimized and detailed field solutions are providedwith an accuracy determined primarily by the grid resolution.In fact, for many of today’s compact optical devices, whichspan just a few optical wavelengths, the computational require-ments of FDTD are not prohibitive. In the following sections,we review selected algorithms for modeling nonlinear opticalphysics via FDTD. The algorithms allow prediction of theoptical-field dynamics in complicated optical structures in thepresence of material dispersions of arbitrary order and bothinstantaneous and time-dependent nonlinearities.

II. A N ALGORITHM FOR LINEAR DISPERSIONADAPTABLE

TO NONLINEAR DISPERSIVEOPTICAL MATERIALS

Consider for simplicity a 1-D problem with electric andmagnetic field components and propagating in thedirection. Assuming first that the medium is linear, nonperme-able, isotropic, and nondispersive, Maxwell’s curl equationsin one dimension are

(5a)

(5b)

where vacuum permeability, , and thepermittivity is independent of frequency. Using Yee’s centraldifferencing in time and space [1], these relations can beexpressed as

(6a)

(6b)

(6c)

and the solution iterated to the desired final observation time.For a general dispersive and nonlinear medium, however, theFDTD computational model can retain its fully explicit natureif (6c) is replaced by the constitutive relation

(7)


where is a function to be determined. Several approacheshave been proposed for this purpose. These may be catego-rized into two primary groups: recursive-convolution methods[3], [10]–[12] and the direct time integration methods [2],[13]–[15]. An advantage of the direct time integration methodsis that they can be generalized to nonlinear dispersive materi-als. This section summarizes a direct time integration methodshown to be adaptable to nonlinear optical modeling [2].

We now consider a linear dispersive material characterizedby Lorentzian resonances. For each vector component of

and , we write

(8)

Here, the polarization of an electric-field vector component isexpressed as a sum of terms (dropping the vector componentsubscript for simplicity)

(9)

where each term is a convolution integral

(10)

and each is a Lorentzian in frequency

with

(11)

In (10), we assume zero values of the electric field and thekernel functions for .

Now consider the key property that drives this formulation,namely, each kernel function satisfies the followinglinear, second-order differential equation:

(12)

where it is assumed that and and. This property of the kernel function makes it

possible to treat the convolution integral as a new dependentvariable. It follows immediately that a second-order ordinarydifferential equation can be derived for the linear convolutionintegral by time-differentiating it. This equation determines thepolarization which can then be used to determine. Knowing

(13)

we can write for each convolution integral the differentialequation

where

(14)

As an example of the application of (14), consider the caseof a material having three Lorentzian relaxations. This resultsin the following system of ordinary differential equations:

(15a)

(15b)

(15c)

Applying a second-order accurate, semi-implicit central-difference scheme centered at time step, this system canbe solved to update , , and by inverting the followingset of equations:

(16a)

(16b)

(16c)

where

(17a)

(17b)

(17c)

With the updated values , , and now avail-able, we can update from (13) as

(18)

Equations (16)–(18) are applied in place of (7), performingthe function . For each componentcalculated in this manner, computer storage must be providedfor two previous values of and two previous values of eachof the convolution functions .

The dispersive FDTD algorithm summarized above wasvalidated in [2] by modeling the reflection of a Gaussian pulseincident on a half-space of a dispersive dielectric medium.Fig. 1 shows results for a hypothetical material having threearbitrarily chosen, moderately undamped Lorentzian reso-nances in the optical range. With the baseline low-frequencyand high-frequency permittivities and ,the controlling parameters of the Lorentzians were assumedto be: ( Hz, , ),( Hz, , ), and (

Hz, , ). The FDTD reflection


(a)

(b)

Fig. 1. (a) Real and imaginary parts of the permittivity of a Lorentz mediumhaving three resonances in the optical range. (b) Comparison of FDTD andexact results from dc to 2.0� 1015 Hz for the magnitude of the reflectioncoefficient of a half-space composed of the medium of (a).�x = 2:65 nm,�t = 8:85 as. Problem size: 20 000 1-D cells, 20 000 time steps. Computerresources: 0.92 MW; 1000 CPU s on a single processor of the Cray J90 [2].

coefficient versus frequency was computed by taking the ratioof the discrete Fourier transforms of the reflected and incidentpulses. These data were then compared to the exact valuesobtained by monochromatic impedance theory. Agreement waswithin 0.1% at all frequency comparison points (literally, fromdc to light).

In addition to the wave reflected from a dispersive halfspace, FDTD permits computing the pulse propagating withinthe dispersive medium at any space–time point. Historically,such pulse dynamics have been obtained only by asymptoticor Laplace transform analyses, classically by Sommerfeld[16] and Brillouin [17], and more recently in [18] and [19].Of particular interest in these papers has been the delicateprecursor fields [20] that can precede the main body of apulse propagating in a Lorentz medium. The computation ofthe precursor for a single-Lorentz medium using a subset ofthe above algorithm is reported in [15]. Here, a sinusoidalsource of frequency Hz was assumed locatedat and switched on at for material parameters

, , Hz, , and. Upon comparing the FDTD-computed Sommerfeld

precursor observed at m to the published asymptotic

Fig. 2. Comparison of FDTD, asymptotic, and Laplace transform results forthe Sommerfeld precursor observed atx = 1�m in a Lorentz-dispersivemedium for a unit-step modulated sinusoidal excitation of!c = 1:0� 10

16

at x = 0. �x = 0:0157 nm, �t = 0:0523 as. Problem size: 80 000 1-Dcells, 80 000 time steps. Computer resources: 1.4 MW; 965 CPU s on a singleprocessor of the Cray J90 [2].

[18] and Laplace transform [19] predictions, it was found thatthe FDTD precursor closely agrees with the Laplace transformcalculation. This is shown in Fig. 2.

III. T HIRD-ORDER NONLINEAR MEDIA

This section reviews approaches based upon the FDTDmethod that permit the direct time integration of Maxwell’sequations for materials having third-order nonlinear behavior.In contrast to NLSE models, the optical carrier is retained.Section III-A reviews a relatively simple model of instanta-neous Kerr-type nonlinearity. Section III-B reviews a muchmore complete model that includes simultaneous linear dis-persion, instantaneous nonlinearity, and nonlinearity with afinite-time response.

A. Instantaneous Nonlinearity

The FDTD model for Kerr-type materials assumes an in-stantaneous nonlinear response. The nonlinearity is modeledin the relation , where

(19)

Here, the linear refractive index is dimensionless, and thenonlinear refractive index has units of m/V . From (19),(7) for the latest value of can be expressed by the followingiteration upon the latest value of and the old value of [21]:

(20)

Knowledge of then permits another updating of and, and the process repeats cyclically until time stepping is

completed.


(a)

(b)

(c)

Fig. 3. Visualization of the FDTD computed electric field of optically narrowmutually interacting spatial solitons in Type-RN Corning glass. (a) Repul-sion for relative carrier phase= �. (b) Single coalescence and subsequentdivergence for relative carrier phase= 0. (c) Periodic beam recoalescenceafter doubling the intensity beamwidth and separation parameters of thesimulated beams, but keeping the wavelength constant.�x = �y = 52:8

nm, �t = 88:1 as. Problem size: 1800� 592 2-D cells, 3600 time steps.Computer resources: 10.9 MW; 1340 CPU s on a single processor of theCray J90 [21].

The two-dimensional (2-D) FDTD modeling ofspatialoptical soliton propagation and mutual deflection in a ho-mogeneous glass medium having a Kerr-type instantaneousthird-order nonlinearity is reported in [21]. Here, the trans-verse spreading of a simulated laser beam caused by lineardiffraction was exactly balanced by the transverse beamwidthsharpening effect due to self-focusing caused by the assumednonlinearity. FDTD calculations were made for two parallelbeams spaced at 1.05m, center to center. The beams wereassumed to be switched on at in Type-RN Corningglass ( and m /W). Eachbeam was assumed to have a carrier frequency of 2.3110 Hz ( m), an initial hyperbolic-secant transverseprofile with an intensity beamwidth (FWHM) of 0.65m,and an initial peak of 6.87 10 V/m. The choiceof these parameters yielded copropagating spatial solitons.Fig. 3(a)—taken from [21]—illustrates a mutual repulsion fora carrier phase difference of between the beams. This isexpected from the NLSE. Fig. 3(b) depicts the results for in-phase carriers. For this case, the NLSE predicts that the twobeams interact by alternately attracting, coalescing, repelling,and then recoalescing. If the two beams have the appropriateamplitudes and spacing, the attraction and repulsion is peri-odic. In fact, [22] states that two in-phase fundamental spatialsolitons having an initial field-amplitude distribution in thetransverse direction of

sech sech

(21)

oscillate in the propagation direction with a period of

(22)

based on the NLSE theory of [23]. Here,is the characteristicwidth of the hyperbolic secant, , is the center-to-center separation of the two beams, andis the soliton period. For the choice of parameters used inthe FDTD simulations of [21], the predicted repetition periodwas m. However, as shown in Fig. 3(b), the FDTDcalculations showed only a single beam coalescence and thensubsequent beam divergence to arbitrarily large separationsfor an effective

It was a goal to understand why the FDTD Maxwell’sequations model did not agree with the NLSE predictionin this case. The first possibility considered was that theFDTD simulation was flawed because of inadequate gridresolution or inadequate decoupling of the beam-interactionregion from the outer-grid boundaries (second-order MurABC’s were used at the time). In a series of exploratorymodeling runs, the space–time resolution of the FDTD gridwas progressively refined and the grid was progressivelyenlarged. These changes gave results identical to those of theoriginal FDTD model. Therefore, the original FDTD modelwas concluded to be numerically converged and sufficientlyfree of the outer boundary artifact to yield plausible results.

The second possibility considered was that the ratio ofbeamwidth to wavelength was below the limit of applicabilityof NLSE. Because it is known that additional terms in theNLSE are required to model higher order effects for temporalsolitons, it was reasoned that NLSE modeling of copropagatingspatial solitons would be more physically meaningful if thetwo beams were widened relative to the optical wavelength,while maintaining the same ratio of beamwidth-to-beam sep-aration. This would reduce linear beam-diffraction effects,hopefully bringing the test case into the region of validity forthe simple NLSE model. To test this possibility, [21] reportedtwo new FDTD simulations where the intensity beamwidth

and separation parameters of the simulated beams wereeach doubled and then doubled again, keeping the dielectricwavelength constant. This FDTD simulation is shown inFig. 3(c).

After the first doubling of beamwidth and beam separation,the FDTD-predicted spatial solitons began to qualitativelyshow the recoalescence behavior predicted by NLSE, but witha 38% longer period of recoalescence than the NLSE value.After the second doubling, the FDTD and NLSE predictionsfor repetition period showed much better agreement, differ-ing by only 13%. A third data point (calculated for this paper)involved, yet again, a 50% increase in beamwidth and beamseparation. This resulted in the difference between the FDTDand NLSE calculations of dropping further to only 5%.Results for these numerical experiments are shown in Table I.

It was concluded that there is a strong likelihood that co-propagating optically narrow spatial solitons have only a singlecoalescence and then indefinite separation. The FDTD model


TABLE IPROGRESSIVEAGREEMENT OF FDTD AND NLSE RESULTS

FOR PERIODICITY OF COPROPAGATING SPATIAL SOLITONS AS

THE RATIO OF THE BEAMWIDTH TO WAVELENGTH INCREASES

appears to properly predict the behavior of these solitonsin nonlinear media both in the regime where the standardNLSE model breaks down ( ) and the regimewhere the standard NLSE model is valid ( ). Theparaxial approximation inherent to NLSE, according to [9],accounts only for zeroth-order linear diffraction effects. Sincethe FDTD model implements the fundamental Maxwell’s curlequations, it makes no assumption about a preferred scatteringdirection. It naturally accounts for energy transport in arbitrarytransverse directions and should be exact for the computedoptical electromagnetic fields up to the limit set by the gridresolution. Reference [24] reported a similar need to use afull-wave Maxwell’s model to properly obtain nonparaxialscattering of energy for a corrugated waveguide beam-steeringdevice.

B. Dispersive Nonlinearity

This section reviews an algorithm reported in [25] and [26]for modeling the simultaneous presence of linear dispersion,instantaneous nonlinearity, and dispersive nonlinearity (seealso [24], [27], and [28]). Consider, again, the 1-D problem of(5a) and (b). Now allow for nonlinearity of the dielectric byassuming that the electric polarization consists of two parts:a linear part and a nonlinear part [6]. Maxwell’sequations still govern, but here, must be related to byaccounting for both the linear and nonlinear components ofthe polarization

(23)

Here, is given by the convolution of and thefirst-order susceptibility function

(24)

where provides the physics of linear dispersion normallyassociated with a frequency-dependent permittivity. Further,

is given by the convolution of and the third-order susceptibility function

(25)

where provides the physics of a nonlinearity with retar-dation or memory (i.e., a dispersive nonlinearity). This kernel

and its convolution provides a macroscopic equivalence ofthe optical material for the quantum effects leading to thenonlinearity. For silica, these effects occur at time scales of1–100 fs. Note that may have different resonances anddampings than .

Consider a material having a single-Lorentzian dispersivesusceptibility characterized by the Fouriertransform pair

(26a)

whereresonant frequencydamping constant.

(26b)

Further, the material nonlinearity is assumed to be charac-terized by the following single time convolution for[29]:

(27a)where is the nonlinear coefficient. The causal responsefunction is normalized so that

(27b)

Equation (27) accounts for only nonresonant third-order pro-cesses, including phonon interactions and nonresonant elec-tronic effects. To model these responses, we let

(28a)

where is a Dirac delta function that models Kerr nonres-onant virtual electronic transitions on the order of about 1 fsor less, and is given by the exponential

(28b)

that models transient Raman scattering. Effectively,models a single Lorentzian line centered on the optical-phonon frequency and having a bandwidth of ,the reciprocal phonon lifetime. Note thatparameterizes therelative strengths of the Kerr and Raman interactions.

Following [25], we now describe the system of couplednonlinear ordinary differential equations that governs the timeevolution of the polarization. Assuming zero values of theelectromagnetic field and the kernel functions for , definethe functions and as, respectively, the convolutions

(29a)

(29b)


Then, by time-differentiating and , it can be shown thatthese functions satisfy the system

(30a)

(30b)

where and Equations(30a) and (30b) are first solved simultaneously forandat the latest time step by using a second-order accurate finite-difference scheme (discussed below) that operates on data forthe current value of and previous values of , , ,and . Then the latest value of can be obtained via aNewton’s iteration of the following equation, using the newvalues of , , and :

(31)

The finite-difference realization of the above is summarizedin the two-step procedure that follows.

Step 1: Apply a second-order accurate central-differencescheme centered at time stepfor the coupled system of (30a)and (b). Here the values of and the convolution functions,

and , at time step are taken in a semi-implicit manneras the average of the respective values at time steps and

. This yields the latest values of the convolutionsand and requires only two time levels of storage

(32a)

(32b)

Upon collecting like terms and simplifying the notation asshown, we obtain the following pair of equations for theupdated convolution integrals and :

(33a)

(33b)

Note that the form of the coupling in (33a) and (33b) resultsin a strong diagonal dominance in the associated matrix. Thisfeature is essential for a stable algorithm.

Step 2: Substitute the values of , , andinto (27) to determine via a Newton iteration proce-dure. Suppressing thesubscript, we obtain

(34)

where denotes the approximation of at the thiteration of the Newton procedure and . Resultsto date indicate that only a single iteration of this procedureis sufficient for converged values of the electric field.

The FDTD modeling of a 50-fs pulsed optical-signal sourceswitched on at at the surface of a materialhalf-space having nonlinear dispersive properties was reportedin [25]. The pulse was assumed to have unity amplitude of


(a)

(b)

Fig. 4. (a) FDTD results for a 50-fs optical carrier pulse after propagating55 �m and 126�m in a Lorentz-dispersive medium (linear case), showingthe attenuation, broadening, and frequency-modulation effects of anomalousdispersion. (b) Corresponding FDTD results for the nonlinear dispersive case,showing the formation of a temporal soliton and a small precursor pulse.�x = 7:299 nm, �t = 12:16 as. Problem size: 20 000 1-D cells, 40 000time steps. Computer resources: 0.5 MW; 630 CPU s on a single processorof the Cray J90 [25].

its sinusoidal-carrier electric field, a carrier frequencyHz ( m), and a hyperbolic secant

envelope function with a characteristic time constant of 14.6fs. Approximately seven cycles of the optical carrier werecontained within the pulse envelope, and the center of the pulsecoincided with a zero crossing of the sinusoid. To demonstratesoliton formation over short propagation spans of less than 200

m, the material’s group velocity dispersion,, and nonlinearcoefficient, , were appropriately scaled by selecting thefollowing parameters:

1) linear dispersion: , ,s , s ;

2) nonlinear dispersion: (V/m) , ,fs, fs.

(a)

(b)

Fig. 5. Visualization of the FDTD computed electric field of a 50-fs opticalcarrier pulse propagating in a 1-�m thick slab waveguide. (a) Pulse spreadingand frequency modulation due to anomalous linear dispersion in the slabwaveguide. (b) Corresponding FDTD results for the nonlinear dispersivecase showing the formation of a temporal soliton.�x = �y = 43:8 nm,�t = 72:99 as. 13 334 time steps. Computer resources: 6.0 MW; 3100 CPUs on a single processor of the Cray J90 [26].

This choice resulted in varying in a wide range from 7to 75 ps /m over the spectral width of the modeled pulse,(1.37 0.2) 10 Hz. Finally, by choosing a uniform FDTDspace resolution of 7.3 nm , the numerical phasevelocity error was limited to about 1 part in 10—very smallcompared to the physical dispersions modeled.

Fig. 4 (taken from [25]) depicts the results of the dispersiveand nonlinear dispersive FDTD computations. In Fig. 4(a), thecomputed rightward propagating pulse for the linear Lorentzdispersive case [ set to zero] is graphed at and

time steps. This corresponds to pulse propagation todepths of m and 126 m at times of 487 and 973 fs,respectively, after initiation. It is clear that the assumed lineardispersion caused substantial broadening of the computedpulse along with diminishing amplitude and carrier frequencymodulation: on the leading side of the pulse, and onthe trailing side of the pulse.

Fig. 4(b) graphs the corresponding pulse propagation whenthe dispersive nonlinearity was actuated. Upon the precisechoice of and the initial pulse amplitude, this yieldeda rightward propagatingtemporal soliton that retained itsamplitude and width. Here, the temporal pulse width spreadingeffect caused by the assumed linear dispersion was exactlybalanced by the temporal pulse width sharpening effect causedby the assumed nonlinearity. The smaller magnitude “daugh-ter” pulses have been identified as transient third harmonicenergy [8].

Fig. 5(a) and (b) (taken from [26]) depicts the FDTD-computed results of the 2-D versions of Fig. 4(a) and (b).Here, the pulse was assumed to have the field components

, , and and be guided in the -direction by a 1- mthick planar dielectric slab with vacuum to either side. Again,it was found possible to obtain a propagating, nondispersing,temporal soliton by the precise choice of and the pulseamplitude.


Fig. 6. FDTD-computed normalized intensities for both the fundamental (� = 1:5�m) and the second-harmonic (� = 0:75�m) fields along the lengthof a 2-D slab waveguide with a QPM grating of period 4.54�m. �x = 20:83 nm, �t = 34:7 as. 10 000 time steps. Computer resources: 6.7 MW;1300 CPU s on a single processor of the Cray J90.

IV. SECOND-ORDER NONLINEAR MEDIA

This section presents a simple FDTD algorithm to modelelectromagnetic wave propagation in materials exhibiting aninstantaneous second-order nonlinearity. (Work is in progressto extend this approach to the case of a dispersive second-ordernonlinearity.) Similar to the Kerr nonlinearity, this is modeledin the relation , but now

(35)

where the nonlinear coefficient has units of m/V. Thelatest value of in (7) can be obtained by iteration, using thenew value of and the old value of

(36)

Knowledge of then permits another updating of and, and the process repeats cyclically until time-stepping is

completed.This nonlinear FDTD model is now applied to study the

well-characterized process of second-harmonic generation ina second-order nonlinear semiconductor waveguide in twodimensions. Second-harmonic generation can occur due to theinteraction of an applied field at frequency with thenonlinear medium, resulting in a copropagating signal oscil-lating at frequency 2 . Due to waveguide modal dispersion,the guided mode of the second-harmonic field travels with aspeed different from that of the guided mode of the appliedfield. The efficiency of the frequency-conversion depends onthe degree of phase matching between the two signals [5].

In the present example, an asymmetric 0.44-m thick slabwaveguide ( substrate, guiding layer,

air) is excited by a CW signal at the fundamentalm. The waveguide is designed to support only a

single mode at this frequency with a mode effective indexof . However, the waveguide can support atleast two distinct modes at the second harmonic. The effectiveindex for the lowest order guided mode at the second-harmonicfrequency is . The phase mismatch is definedas . The nonlinear coefficient is

m/V, and the incident field amplitude is1.0 10 V/m.

As energy at the fundamental frequency propagates downthe guide, energy at the harmonic frequency is generatedand begins to propagate in the same direction. Note that theprocess of harmonic generation occurs continuously along theentire length of the guide. For there to be an appreciablebuildup of harmonic energy, the harmonic energy generatedearly in this process and propagating down the guide mustconstructively interfere with that generated later on. In ahomogeneous nonlinear waveguide, low frequency-conversionefficiency occurs because the generated harmonic field gets outof phase with its driving polarization, and power flows fromthe harmonic field back into the fundamental.

When the condition of perfect phase matching is fulfilled,the generated wave maintains a fixed-phase relation withrespect to the nonlinear polarization and is able to extractenergy most efficiently from the incident wave. Quasi-phasematching (QPM) [30] is a method of effectively reducingthe phase mismatch between the two fields. One approach toQPM, selective-area disordering [31], is to periodically zero


out the nonlinear coefficient along the direction of propagationby using a grating structure that combines layers of linearand nonlinear dielectric materials. Here, the second-harmonicfield is allowed to grow during the nonlinear half-period ofthe grating but remains steady during the linear half-period,whereas in a bulk nonlinear medium it would have continuedto develop out of phase. This results in a stepwise increaseof the second-harmonic field and a corresponding stepwisedecrease of the fundamental field.

In the present example, the period of the QPM grating ism. Although the guide can support more than one

mode at the harmonic frequency, this choice of grating periodensures that only the lowest order mode propagates. Fig. 6shows the intensity of both the fundamental and harmonicfields along the length of the waveguide. The bilateral ex-change of energy from the fundamental to the harmonic andback to the fundamental is evident. Simply by truncating thewaveguide at 40 m, it is possible to maximize the transferof energy to the harmonic.

Since this particular problem involves paraxial wave propa-gation, other less computationally intensive approaches such asthe beam propagation method (BPM) [30], [32] are appropriateand, in fact, offer advantages relative to waveguide length anddimensionality. While the full power of the FDTD Maxwell’sequations solver may not be required for this example, it iseasy to see how a variety of engineered structural features inthis and similar nonlinear optical devices could violate theparaxial assumptions. What has been shown is that FDTDcan readily model the physics of instantaneous second-orderoptical nonlinearities. Work is ongoing to construct a compre-hensive validation versus BPM and to investigate numericalstability issues.

V. CONCLUSION

This paper summarized algorithms which extend the FDTDMaxwell’s equations approach to nonlinear optics. The useof FDTD in this field is novel. Previous analytical and nu-merical methods such as the NLSE and the BPM wereaimed at modeling optical-wave propagation in electricallylong structures such as glass fibers and directional couplers,wherein the primary flow of energy is along a single principaldirection. FDTD approaches, however, are aimed primarily atmodeling compact optical structures spanning in the order ofthe wavelength of light and having energy flow in arbitrarydirections.

Relative to NLSE and BPM, FDTD is certainly very com-putationally intensive. However, it has the advantage of beingsubstantially more robust than either NLSE or BPM becauseit directly solves for fundamental quantities, the optical Eand H fields in space and time. FDTD avoids the simplifyingassumptions that lead to conventional asymptotic and paraxial-propagation analyses. It rigorously enforces the vector fieldboundary conditions at all material interfaces at the time scaleof a small fraction of the optical period, whether or not themedia are dispersive or nonlinear.

For all of the third-order nonlinear cases presented here,the time step was chosen to be at or just below the Courant

stability limit. No stability problems were observed, evenafter several hundred thousand time steps. Numerical stabilityissues for the second-order nonlinear algorithm are beinginvestigated.

Extending the FDTD algorithm to the field of active non-linear optics is an area of current research. It is now possibleto model the electrodynamics of micron-scale laser cavitiesin two and three dimensions, using algorithms for the macro-scopic effects of optical-gain media [33], [34]. An emergingfrontier is the combination of FDTD classical electrodynamicsand the quantum mechanics of the materials being modeled[35].

ACKNOWLEDGMENT

The authors would like to thank P. M. Goorjian of NASA-Ames Research Center, Moffett Field, CA, for his collabora-tion in the development of the nonlinear dispersive algorithm.

REFERENCES

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

[2] A. Taflove, Computational Electrodynamics: The Finite-DifferenceTime-Domain Method. Norwood, MA: Artech House, 1995.

[3] K. S. Kunz and R. J. Luebbers,The Finite Difference Time DomainMethod for Electromagnetics.Boca Raton, FL: CRC, 1993.

[4] P. N. Butcher and D. Cotter,The Elements of Nonlinear Optics.Cam-bridge, UK: Cambridge Univ. Press, 1990.

[5] R. W. Boyd, Nonlinear Optics. Boston, MA: Academic, 1992.[6] G. P. Agrawal,Nonlinear Fiber Optics. New York: Academic, 1989.[7] A. Hasagawa and F. Tappert, “Transmission of stationary nonlinear

optical pulses in dispersive dielectric fibers I: Anomalous dispersion,”Appl. Phys. Lett.,vol. 23, pp. 142–144, Aug. 1973.

[8] C. V. Hile and W. L. Kath, “Numerical solutions of Maxwell’s equationsfor nonlinear optical pulse propagation,”J. Opt. Soc. Amer. B,vol. 13,pp. 1135–1145, June 1996.

[9] M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell toparaxial wave optics,”Phys. Rev. A,vol. 11, p. 1365, Aug. 1975.

[10] R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M.Schneider, “A frequency-dependent finite-difference time-domain for-mulation for dispersive materials,”IEEE Trans. Electromagn. Compat.,vol. 32, pp. 222–227, Aug. 1990.

[11] R. J. Luebbers, F. P. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propa-gation in plasma,”IEEE Trans. Antennas Propagat.,vol. 39, pp. 29–34,Jan. 1991.

[12] R. J. Luebbers and F. P. Hunsberger, “FDTD forN th-order dispersivemedia,” IEEE Trans. Antennas Propagat.,vol. 40, pp. 1297–1301, Nov.1992.

[13] T. Kashiwa, N. Yoshida, and I. Fukai, “A treatment by the finite-difference time-domain method of the dispersive characteristics associ-ated with orientation polarization,”Inst. Electron. Inform. Communicat.Eng. (Japan) Trans.,vol. E-73, pp. 1326–1328, Aug. 1990.

[14] T. Kashiwa and I. Fukai, “A treatment by the FD-TD method ofthe dispersive characteristics associated with electronic polarization,”Microwave Opt. Tech. Lett.,vol. 3, pp. 203–205, June 1990.

[15] R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integrationof Maxwell’s equations in linear dispersive media with absorption forscattering and propagation of femtosecond electromagnetic pulses,”Opt.Lett., vol. 16, pp. 1412–1414, Sept. 1991.

[16] A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierendenmedien,” Ann. Phys.,vol. 44, pp. 177–202, 1914.

[17] L. Brillouin, “ Uber die fortpflanzung des lichtes in disperdierendenmedien,” Ann Phys.,vol. 44, pp. 203–240, 1914.

[18] K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description ofelectromagnetic pulse propagation in a linear dispersive medium withabsorption (the Lorentz medium),”J. Opt. Soc. Amer. A,vol. 6, pp.1394–1420, Sept. 1989.

[19] P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of theprecursor fields in linear dispersive pulse propagation,”J. Opt. Soc.Amer. A,vol. 6, pp. 1421–1429, Sept. 1989.


[20] J. D. Jackson,Classical Electrodynamics, 2nd ed. New York: Wiley,1975.

[21] R. M. Joseph and A. Taflove, “Spatial soliton deflection mechanismindicated by FDTD Maxwell’s equations modeling,”IEEE Photon. Tech.Lett., vol. 6, pp. 1251–1254, Oct. 1994.

[22] J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver,J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatialsoliton interactions,”Opt. Lett.,vol. 16, pp. 15–17, Jan. 1991.

[23] C. Desem and P. L. Chu, “Reducing soliton interaction in single-modeoptical fibers,”Inst. Elect. Eng. Proc.,vol. 134, pp. 145–151, June 1987.

[24] R. W. Ziolkowski and J. B. Judkins, “Nonlinear finite-difference time-domain modeling of linear and nonlinear corrugated waveguides,”J.Opt. Soc. Amer. B,vol. 11, pp. 1565–1575, Sept. 1994.

[25] P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Com-putational modeling of femtosecond optical solitons from Maxwell’sequations,”IEEE J. Quantum Electron., vol. 28, pp. 2416–2422, Oct.1992.

[26] R. M. Joseph, P. M. Goorjian, and A. Taflove, “Direct time integrationof Maxwell’s equations in 2-D dielectric waveguides for propagationand scattering of femtosecond electromagnetic solitons,”Opt. Lett.,vol.18, pp. 491–493, Apr. 1993.

[27] R. W. Ziolkowski and J. B. Judkins, “Applications of the nonlinearfinite difference time domain (NL-FDTD) method to pulse propagationin nonlinear media: Self-focusing and linear/nonlinear interfaces,”RadioSci., vol. 28, pp. 901–911, May 1993.

[28] , “Full-wave vector Maxwell equations modeling of self-focusingof ultra-short optical pulses in a nonlinear Kerr medium exhibiting afinite response time,”J. Opt. Soc. Amer. B,vol. 10, pp. 186–198, Feb.1993.

[29] K. Blow and D. Wood, “Theoretical description of transient stimulatedRaman scattering in optical fibers,”IEEE J. Quantum Electron., vol. 25,pp. 2665–2673, Dec. 1989.

[30] H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlineareffects in optical waveguides using a parallel processing beam propa-gation method,”IEEE J. Quantum Electron., vol. 31, pp. 2107–2113,Dec. 1995.

[31] M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasiphase-matched second-harmonic generation: Tuning and tolerances,”IEEE J.Quantum Electron., vol. 28, pp. 2631–2654, Nov. 1992.

[32] D. Yevick, “A guide to electric field propagation techniques for guidedwave optics,” Opt. Quantum Electron., vol. 26, pp. 185–197, Mar.1994.

[33] R. J. Hawkins and J. S. Kallman, “Lasing in tilted-waveguidesemiconductor laser amplifiers,”Opt. Quantum Electron.,vol. 26, pp.S207–S217, Mar. 1994.

[34] S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electro-dynamics of distributed Bragg reflector microlasers: Results from finitedifference time domain simulations,”Radio Sci.,vol. 31, pp. 931–941,July/Aug. 1996.

[35] R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulseinteractions with two-level atoms,”Phys. Rev. A, vol. 52, pp. 3082–3094,Oct. 1995.

Rose M. Joseph(M’93) was born in Kerala, India,on September 25, 1964. She received the B.S.degree in electrical engineering from the Massa-chusetts Institute of Technology (MIT), Cambridge,MA, in 1986, and the M.S. and Ph.D. degrees inelectrical engineering from Northwestern Univer-sity, Evanston, IL, in 1990 and 1993, respectively.

From 1986 to 1988, she was employed by IBM,Kingston, NY, as a Power Systems Engineer. From1995 to 1997 she was a member of the ResearchStaff at the Xerox Corporation Wilson Center for

Research and Technology, Webster, NY. In April 1997 she joined the technicalstaff at MIT Lincoln Laboratory, Lexington, MA. Her research interestsinclude computational electromagnetics and nonlinear optics.

Allen Taflove (F’90) was born in Chicago, IL, onJune 14, 1949. He received the B.S., M.S., and Ph.D.degrees in electrical engineering from NorthwesternUniversity, Evanston, IL, in 1971, 1972, and 1975,respectively.

Since 1988, he has been a Professor in the De-partment of Electrical and Computer Engineering,McCormick School of Engineering, NorthwesternUniversity, Evanston, IL. Since 1990 he has given60 invited talks and lectures on FDTD and horizonsin computational electromagnetics, in the United

States and abroad, including the 90-minute keynote speech at the 1994Salishan Meeting on High Performance Computing. He authored the text-book/research book,Computational Electrodynamics: The Finite-DifferenceTime-Domain Method,(Norwood, MA: Artech House, 1995). He originatedseveral innovative programs in the McCormick School, including the HonorsProgram in Undergraduate Research (a combined B.S./Ph.D. engineeringdegree) and the Undergraduate Design Competition. His current researchinterests include FDTD simulation of picosecond-regime optical phenomenaand devices, especially microlasers (in collaboration with Prof. S. T. Ho), non-invasive microwave detection and imaging of breast cancers (in collaborationwith Mr. J. E. Bridges), and radio wave propagation in urban microcells (incollaboration with Mr. G. F. Stratis).

Dr. Taflove was named a Distinguished National Lecturer for the IEEEAntennas and Propagation Society in 1990–1991, and in 1992 was Chairmanof the Technical Program of the IEEE Antennas and Propagation SocietyInternational Symposium in Chicago, IL. In 1991, he was named McCormickFaculty Adviser of the Year. He is a member of Eta Kappa Nu, Tau Beta Pi,Sigma Xi, International Union of Radio Science (URSI) Commissions B, D,and K, the Electromagnetics Academy, AAAS, and New York Academy ofSciences. His biographical listings includeWho’s Who in Engineering, Who’sWho in America, Who’s Who in Science and Engineering,andWho’s Who inAmerican Education.

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