Coupled-mode equations for Kerr media with periodically modulated linear and nonlinear coefficients

2282 J. Opt. Soc. Am. B/Vol. 19, No. 9 /September 2002 Haus et al.

Coupled-mode equations for Kerr media withperiodically modulated

linear and nonlinear coefficients

Joseph W. Haus and Boon Y. Soon

Electro-Optics Program, University of Dayton, Dayton, Ohio 45469-0245

Michael Scalora

Weapons Sciences Directorate, AMSMI-RD-WS-ST, Research, Development and Engineering Center, U.S. ArmyMissile Command, Building 7804, Redstone Arsenal, Alabama 35898-5000

Cocita Sibilia

Istituto Nazionale di Fisica Materia at Dipartimento di Energetica, Universita di Roma ‘‘La Sapienza,’’ Via A.Scarpa, 16, 1-00161 Rome, Italy

Igor V. Mel’nikov*

Centro Investigaciones en Ingenieria y Ciencias Aplicadas, Universidad Autonoma del Estado de Morelos, AvenidaUniversidad 1001, col. Chamilpa, 62210 Cuernavaca, Morelos, Mexico, and General Physics Institute of the

Russian Academy of Sciences, ul. Vavilova 38, Moscow 117942, Russian Federation

Received December 12, 2001

We apply the multiple-scales formalism to derive a complete set of equations for a finite medium with periodiclinear and nonlinear Kerr optical coefficients. The equations for a single-frequency field reveal three new,nonlinear terms that are related to the difference in the Kerr nonlinearity in two-component media. The non-linear evolution of coupled forward and backward fields in a multilayered film is numerically simulated by aspectral method. We examine the linear stability of the steady-state solution for an infinite medium and ex-tend previous discussions of modulational instabilities to the new set of equations. We find that the inhomo-geneous coefficient can selectively suppress modulational instability in the longitudinal or transverse direc-tion. © 2002 Optical Society of America

OCIS codes: 190.4420, 190.4400, 190.3100.

1. INTRODUCTIONPeriodically layered dielectric films have been explored ina number of photonic applications from polarizers, filters,and beam splitters to laser and detector cavity designs.The so-called photonic-bandgap (PBG) devices are com-pact and can be designed to support a large bandwidth.1–3

One-dimensional systems that consist of periodic, dielec-tric stacks or periodically patterned waveguides offer fab-rication simplicity; hence nonlinear effects in layeredstructures have been studied and analyzed to reveal awide variety of phenomena. Potential device applica-tions include nonlinear optical limiters,4–6 optical diodes,7

pulse generation and propagation,8,9 coherent blue-lightgeneration,10–13 and a high-gain optical parametric am-plifier for nonlinear frequency conversion.14 A number ofpapers have been devoted to the appearance of gap soli-tons, which were identified by Chen and Mills15 and sub-sequently investigated by a number of researchers; re-views of both theoretical and experimental results can befound in Refs. 16 and 17.

In a waveguide geometry the dispersion can be ma-

0740-3224/2002/092282-10$15.00 ©

nipulated by the choice of the waveguide design. Nonlin-ear interactions are promoted because diffraction is re-duced or eliminated and the losses can be managedthrough careful design and fabrication.18,19 Using thisgeometry in conjunction with a PBG structure provides afurther engineering degree of freedom to maintain a con-tinuous interaction of nonlinear waves and preservephase-matching conditions.11,13 The use of band-edge ef-fects serves to slow the waves and concentrate the field inthe nonlinear material. Periodicity in more than one di-mension is also of interest because of the unusual fea-tures present in the band structures of multidimensionalsystems. In addition to being nonmonotonic, they mayform overlapping bands with a different transverse sym-metry for each eigenmode.20 Eigenmode symmetry is animportant factor in the optical characteristics of photoniccrystals.

In this paper, we apply multiple-scales analysis to de-rive a set of nonlinear, coupled-mode equations for coun-terpropagating waves in a finite, periodic medium.Equations are derived for forward- and backward-propagating fields tuned to one wavelength. Multiple-

2002 Optical Society of America

Haus et al. Vol. 19, No. 9 /September 2002 /J. Opt. Soc. Am. B 2283

scales analysis has been used to examine nonlinear opti-cal systems. That method provides a systematic, pertur-bative procedure to incorporate higher-ordereffects.16,17,21,22 The present analysis includes temporaland transverse structure as well as periodic losses and pe-riodic nonlinear effects. A numerical algorithm that werecently developed to simulate systems of this nature23,24

is modified in Section 3 and applied to selected cases inSection 5. In Section 4 we extend the stability analysis ofLichinitser et al.25 to handle our coupled-mode equations.Our analysis of the infinite system is used to indicate theappearance of modulational instabilities.

2. GENERALIZED MODE-COUPLEDEQUATIONSWe begin by considering the propagation of a wave per-pendicular to the surface of a multilayer dielectric stack.The field satisfies the scalar wave equation:

¹2E 21

c2

]2E

]t2 54p

c2

]2PL

]t2 14p

c2

]2PNL

]t2 . (1)

The left-hand side of Eq. (1) contains the usual waveequation for propagation in vacuum. The polarization onthe right-hand side is divided into two parts. The firstterm is the linear polarization, which is a linear functionof the field through the dielectric susceptibility. The di-electric susceptibility is dispersive and position depen-dent, so it depends on both the longitudinal spatial vari-able (denoted z) and time, i.e., x(z, t). The linearpolarization can be written in dispersive, inhomogeneousmedia as

PL 5 xS z, i]

]t DE. (2)

The tilde over the x denotes the Fourier transform in timeof the susceptibility function. The linear susceptibility iscombined with the free-space permittivity to form the di-electric function

eS z, i]

]t D 5 1 1 4pxS z, i]

]t D .

The frequency-dependent dielectric function is complex;the real and the imaginary components are denoted bythe subscripts r and i, respectively:

e~z, v! 5 er~z, v! 1 i e i~z, v!.

The nonlinear susceptibility is chosen to be that of a Kerrmedium, namely,

PNL 5 x~3 !~z !E3. (3)

The functions x and x (3) are assumed to be weaklyvarying functions of the z coordinate. Their expansioncan be written in Fourier series decomposition, as follows:

f~z ! 5 (m52`

`

fm exp~i2pnz/L!, (4)

where the function f(z) denotes either the linear dielec-tric function or the nonlinear susceptibility. The period-icity of the PBG is denoted L.

The multiple-scales method is a systematic approachthat can be applied to a wide variety of problems. Weconsider a wave incident upon a nonlinear medium trav-eling along the positive z axis. The approach consists ofexpanding the space and the time coordinates in a powerseries of a small parameter that we denote m: tn5 mnt0 and zn 5 mnz0 . In the past this method was ap-plied to periodic materials; our procedure parallels thatdevelopment, except that here we truncate the expansionat first-order perturbation theory. The spatial and thetemporal derivatives are

]

]t5

]

]t01 m

]

]t11 m2

]

]t21 ..., (5a)

]

]z5

]

]z01 m

]

]z11 m2

]

]z21 .... (5b)

Similarly, the electric field is also expanded in powers ofthe perturbation parameter:

E 5 E0 1 mE1 1 m2E2 1 .... (6)

The nonlinear terms and the transverse coordinates, aswell as the periodic contributions to the dielectric func-tion, are assumed to be of the order m. The real part ofthe space-averaged, linear dielectric function can be largeand is therefore included nonperturbatively.22 An expan-sion in powers of m is performed, and individual ordersare gathered for analysis. The lowest-order equation is

]2E0

]z02 2

1

c2

]2

]t02 F e0S i

]

]t0DE0G 5 0. (7)

The solution that has a single carrier frequency field hasthe form of a plane-wave expansion:

E0 5 $Ef~z1 , t1 ,... !exp@i~kz0 2 vt0!#

1 Eb~z1 , t1 ,... !exp@i~2kz0 2 vt0!# 1 c.c.%.

(8)

The dispersion relation has the form

k2 5 er~z, v!~v2/c2!. (9)

The field amplitudes depend on the higher-order coor-dinates; they are denoted by a subscript f or b for forwardor backward propagation, respectively, through the me-dium. The first-order terms in m lead to a secular sourceterm for the wave equation. There are 24 terms in thenonlinear polarization that contribute to the single-frequency fields. They are combined with Fourier com-ponents of the Kerr nonlinearity in Eq. (4) [i.e., f 5 x (3)]to yield the coupled-mode equations. The elimination ofthe secular terms leads to the following set of coupledequations for the forward- and the backward-propagatingwaves:

1

v

]Ef

]t5 2

]Ef

]z1

i

F¹2

'Ef 1 S id 2a0

2 DEf

1 S ik1 2a1

2 DEb 1 ih0~ uEfu2 1 2uEbu2!Ef

1 ih1~ uEbu2 1 2uEfu2!Eb 1 ih21Ef2Eb*

1 ih2Eb2Ef* , (10a)


1

v

]Eb

]t5 1

]Eb

]z1

i

F¹'

2Eb 1 S id 2a0

2 DEb

1 S ik21 2a21

2 DEf 1 ih0~ uEbu2 1 2uEfu2!Eb

1 ih21~ uEfu2 1 2uEbu2!Ef 1 ih1Eb2Ef*

1 ih22Ef2Eb* , (10b)

where v 5 ]v/]k is the group velocity; F 5 2k is a dif-fraction parameter, d 5 @k 2 (p/L)# is the detuning ofthe laser emission’s wave number from the center of thestop band, k61 5 (v2/2kc2) er61(v) is the grating couplingcoefficient; a0 5 e i0(v)(v2/kc2) is the average absorptionin the medium, and a61 5 e i61(v)(v2/kc2) is related tothe absorption difference between the dielectrics thatcompose each period of the structure. h0 5 2px0

(3)

3 (v2/kc2) is the homogenized nonlinear Kerr coeffi-cient, and h61 5 2px61

(3)(v2/kc2) and h62 5 2px62(3)

3 (v2/kc2) are, respectively, the first and the secondFourier components of the Kerr nonlinearity function. Ina two-component dielectric structure they are related tothe difference between the Kerr coefficients of the two me-dia. The coefficients h l , a l , and k l are, in general, com-plex. The Fourier amplitudes depend on the underlyingsymmetry of the lattice and are based on a specific choiceof the origin of the sample. For instance, the first layermay be either the highest or the lowest dielectric constantmaterial.

The boundary conditions are applied at opposite ends ofthe sample:

Ef ~x, y, 0, t ! 5 S~x, y, t !,

Eb~x, y, L, t ! 5 0. (11)

We do not consider the possibility here that signals maybe incident at the two ends of the sample. The functionS(x, y, t) is the input pulse applied to the sample. Tak-ing the Fourier transform of the temporal and the trans-verse coordinate variables and using the decomposition ofthe amplitudes with the factors exp(2ivt)exp(iax) allowsthe linear equations to be analytically solved. After al-gebraic simplifications have been made the solutions are

Ef ~q, z, v ! 5 S~q, v !$D cos @D~z 2 L !#

2 iV sin@D~z 2 L !#%/@D cos~DL !

2 iV sin~DL !#, (12a)

Eb~q, z, v ! 5 S~q, v !iK21 sin@D~z

2 L !#/@D cos~DL ! 2 iV sin~DL !#,

(12b)

where the following complex coefficients are defined:K61 5 k61 1 ia61/2, V 5 d 1 v 2 q2/F 1 ia0/2, andD2 5 V2 2 K1K21 . We directly deduce the complex re-flection amplitude from Eq. (12b) by applying the bound-ary condition at z 5 0; it is written as

r 5 iK21 sin~DL !/@D cos~DL ! 2 iV sin~DL !#, (13)

and the complex transmission amplitude deduced fromEq. (12a) by application of the boundary condition at z5 L is

t 5 D/@D cos~DL ! 2 iV sin~DL !#. (14)

These solutions are briefly discussed and illustrated inSection 5 below. They accurately describe the transmis-sion and the reflection of pulses in linear PBGs with in-homogeneous loss.

3. NUMERICAL COMPUTATIONS FORCOUPLED-MODE EQUATIONSEquations (10) with boundary conditions (11) are solvedby a variation of the slowly varying envelope in timemethod used for deep gratings.23 Some of the present au-thors and others previously introduced this method tostudy modulational instabilities in a homogeneous KerrPBG.24 The left-hand sides of Eqs. (10) are integrated byuse of a split-step operator method and fast Fourier trans-forms to diagonalize the derivative operations. In whatfollows, we set the group velocity, v, equal to unity.Equations (10) are rewritten in the following vector form[defining U 5 (Ef , Eb)T, where the superscript T de-notes the transpose of the vector]:

]U

]t5 ~L 1 V !U. (15)

The solution for the incremented time t 1 Dt is expressedin the split-step operator form:

U~t 1 Dt ! 5 exp~DtL/2!exp~DtV !exp~DtL/2!U~t !.(16)

Operator L is a 2 3 2 diagonal matrix whose componentsare

Lff 5 2]

]z1

i

F¹'

2 1 id 2 a0/2, (17a)

Lbb 5]

]z1

i

F¹'

2 1 id 2 a0/2. (17b)

We diagonalize operator L by Fourier transforming the(x, z) variables. The second operator is separated intotwo parts (V 5 N 1 K): One matrix, which we denoteN, has components that depend on the field:

Nff 5 ih0~ uEfu2 1 2uEbu2! 1 2ih1EbEf*

1 ih21EfEb* 1 ih2Eb2Ef* /Ef , (18a)

Nbb 5 ih0~ uEbu2 1 2uEfu2! 1 2ih21Ef Eb*

1 ih1EbEf* 1 ih22Ef2Eb* /Eb , (18b)

Nbf 5 ih21uEfu2, (18c)

Nfb 5 ih1uEbu2. (18d)

The other matrix K has only nonzero off-diagonal compo-nents:

Kfb 5 2ik1 2 a1/2, Kbf 5 2ik21 2 a21/2. (19)

The central operator in Eq. (16) is subsequently decom-posed into a linear portion and nonlinear contributions:


exp~DtV ! 5 exp~DtK/2!exp~DtN !exp~DtK/2!. (20)

The solution of Eq. (20) in this form is further complicatedby the off-diagonal components of N. To deal with thiscomplication we use the exponential form for the diagonalpart of the solution and approximate the off-diagonal partby a first-order difference scheme. The matrix con-structed from diagonal components of N is given the sub-script d, and the matrix of off-diagonal components isgiven the subscript o:

exp~DtN ! ' exp~DtNd! 1 NoDt. (21)

Whereas this decomposition gives good convergence inmost cases, a higher-order scheme could be adopted whenfewer numerical errors or a larger step size is desired. Inthe longitudinal direction, we discretized the lattice into2048 points for simplicity in applying the fast Fouriertransform. We use only one transverse direction in thispaper; the number of discrete points in 256 points.

Because this method steps the solution forward in time,we modify the boundary condition in Eqs. (11) to an initialcondition by displacing the function from the boundary atz 5 0. The relation between the two conditions isE(x, z, 0) 5 S@x, 0, (z 2 z0)/v# because the pulse propa-gates without dispersion in the medium outside the non-linear medium. We also assume that the boundary is im-pedance matched so that, in the absence of the forward–backward-coupling coefficients, no reflected wave isgenerated. The initial pulse has been displaced by z0 toa position sufficiently far outside the medium that the tailof the initial pulse is not appreciable inside the PBG. Weverified the accuracy of the code by comparing resultswith the analytic solutions of the linear equations and byverifying the conservation of energy at each time step.

4. STABILITY ANALYSISThe linear stability analysis is calculated for an infinitemedium. We follow the development used by Lichinitseret al.25 in a recent paper devoted to modulation instabili-ties in systems with homogeneous Kerr nonlinearity, i.e.,the Fourier coefficients h61 and h62 vanish in Eqs. (10).We find the steady-state solution by assuming thatsteady-state solutions have the following form: Ex5 Ax exp@i(Kz 2 Vt)#, where the subscript x 5 f, b. Theamplitudes are real and are defined by two new param-eters, total amplitude a and the parameter that mixes thebackward-propagating amplitude f:

Af 5a

~1 1 f 2!1/2 , Ab 5af

~1 1 f 2!1/2 .

The absorption coefficients are set to zero, and the KerrFourier series is assumed to generate a symmetric func-tion. The amplitudes and the propagation constants sat-isfy the equations

V 5 2k

2 S f 11

f D 23

2h0a2 2

1

2h1

a2

1 1 f 2

3 S 6f 11

f1 f 3D 2

1

2h2a2, (22)

K 5 2d 2k

2 S f 21

f D 21

2h0

a2

1 1 f 2 ~1 2 f 2!

21

2h1

a2

1 1 f 2 S 1

f2 f 3D

21

2h2

a2

1 1 f 2 ~1 2 f 2!. (23)

Note that f , 0 corresponds to the high-frequency branchof the dispersion curve and f . 0 corresponds to the low-frequency branch. The values f 5 61 are points on theband edges. We determine the stability of these solu-tions by linearizing Eqs. (10); we adopt the following no-tation:

Ex 5 ~Ax 1 ex!exp@i~Kz 2 Vt !#. (24)

The linearized equations have the forms

1

v

]e f

]t1

]e f

]z5 ik1eb 2 ifk1e f 1

i

F¹'

2e f

1 ih0

a2

1 1 f 2 @e f 1 e f* 1 2 f~eb 1 eb* !#

1 ih1

a2

1 1 f 2 @~2 1 f 2!eb 1 eb* 2 f~3

1 f 2!e f 1 f 2~eb 1 eb* ! 1 2 f~e f 1 e f* !#

1 ih2

a2

1 1 f 2 ~2f 2e f 1 f 2e f* 1 2 feb!,

(25a)

1

v

]eb

]t2

]eb

]z5 ik1e f 2 i

k1

feb 1

i

F¹'

2eb 1 ih0

a2

1 1 f 2

3 @ f 2~eb 1 eb* ! 1 2 f~e f 1 e f* !#

1 ih1

a2

1 1 f 2 F ~1 1 2 f 2!e f 1 f 2e f*

2~1 1 f 2!

feb 1 e f 1 e f* 1 2 f~eb

1 eb* !G 1 ih2

a2

1 1 f 2 ~2eb 1 eb*

1 2 fe f!. (25b)

The equations that include the terms h0 are identical tothose derived previously by Lichinitser et al.25 We findthe dispersion relation by using linear solutions

ex 5 ax1 exp@i~k • x 2 vt !#

1 ax2 exp@2i~k! • ~x! 2 vt#, (26)

where the new subscripts plus and minus denote thepositive- and the negative-frequency solutions of Eqs.(25). The amplitudes are assumed to be real. The fouralgebraic equations derived from Eqs. (25) are


Fkz 2v

v1 kf 1

k'2

F2 G0 1 G1~2f 1 f 3! 1 G2f 2Gaf1

2 ~G0 1 2 fG1 1 G2f 2!af2 2 @k 1 2 fG0 1 G1~2

1 2 f 2! 1 2 fG2#ab1 2 @2 fG0 1 G1~1 1 f 2!#ab2

5 0, (27a)

2~G0 1 2 fG1 1 G2f 2!af1 1 F2S kz 2v

v D 1 kf 1k'

2

F

2 G0 1 G1~2f 1 f 3! 1 G2f 2Gaf2

2 @2 fG0 1 G1~1 1 f 2!#ab1 2 @k 1 2 fG0

1 2G1~1 1 f 2! 1 2 fG2]ab2 5 0, (27b)

2@k 1 2 fG0 1 2G1~1 1 f 2! 1 2 fG2#af1 2 @2 fG0

1 G1~1 1 f 2!]af2 1 F2S kz 1v

v D 1k

f1

k'2

F

2 G0f 2 1 G1

~1 2 f 2!

f1 G2Gab1

2 ~G0f 2 1 2 fG1 1 G2!ab2 5 0, (27c)

2@2 fG0 1 G1~1 1 f 2!#af1 2 @k 1 2 fG0 1 2G1~1 1 f 2!

1 2 fG2]af2 2 ~G0 1 2 fG1 1 G2!ab1

1 F S kz 1v

v D 1k

f1

k'2

F2 G0f 2

1 G1

~1 2 f 2!

f1 G2Gab2 5 0, (27d)

where the nonlinear coefficients are defined by Gm5 hma2/(1 1 f 2) for m 5 0, 1, 2. The final dispersionrelation is the determinant of the matrix above, whichcan be cast into the form of a polynomial. For the gen-eral case it is too complex to put into compact form, butthere are two special cases that deserve further scrutiny:f 5 21 at the band edge on the high-frequency branchand f 5 1 on the low-frequency branch. The result iswritten in the form of a quadratic equation for v2

v2 1 2v2~A2 1 B2 2 C2 2 D2 2 kz2! 1 kz

4 1 A4 1 B4

1 C4 1 D4 2 8ABCD 1 2kz2~A2 1 D2 2 C2

2 B2! 2 2@A2~B2 1 C2 1 D2! 1 B2~C2 1 D2!

1 C2D2] 5 0, (28)

where the coefficients are defined as

A 5 62G0 1 2G1 , (29a)

B 5 G0 6 2G1 1 G2 , (29b)

C 5 k 1k'

2

F2 G0 1 G2 , (29c)

D 5 k 6 2G0 1 4G1 6 2G2 . (29d)

The upper (plus) sign corresponds to f 5 1, and the lower(minus) sign to f 5 21. A modulational instability is in-dicated by the occurrence of an imaginary frequency con-tribution at nonzero wave-vector values. The growth oftransverse wave vectors contributes to conical emission,and the growth of longitudinal wave vectors leads to pulsebreakup. In the quadratic equations the instability oc-curs either as a result of a negative discriminant fromquadratic equation (28) or simply when the coefficient ofv2 in Eq. (28) becomes smaller than the square root of apositive discriminant.

In general we find that the gain is a sensitive functionof the parameter variables. The challenge is to be able toapply the results of the linear stability analysis in thissection to nonlinear evolution in finite systems. The ana-lytical results serve as a guide to the behavior of finitesystems. Stability in systems tuned far from the bandedges is analyzed by use of the determinant of the coeffi-cients in Eqs. (27) for general values of f.

5. RESULTSThe central result of our paper is found in Eqs. (10). Inthis section, we illustrate the effect that the new, nonlin-ear terms have on the modulational instability and brieflydiscuss the results in absorptive linear PBGs. There aremany parameters in the equations, so our analysis hereprobes certain characteristics of pulse propagation in non-linear PBGs. We confine ourselves to exploring selectedconditions that compare the results of homogeneous non-linearity with that of inhomogeneous nonlinearity. Ourattention in this section is devoted to the role played byG1 in determining the modulational instability. For thenumerical simulations and illustrations of the previousresults we scale the space, time, and field amplitude suchthat the following parameter values are used: k1 5 k215 1 and v 5 1.

A. Linear AnalysisThe linear equations have noteworthy properties that webriefly report. Here we neglect transverse effects, al-though paraxial effects are included in our analysis [Eqs.(12)]. In Fig. 1 the transmission function from Eq. (14) isplotted for several selected cases; two of them are plottedin Fig. 1(a). The solid curve represents the lossless me-dium. The transmission spectrum has a gap centered atd 5 0, and its width is 2. The length of the medium is 5.The transmission maxima are unity for the lossless me-dium. These are Fabry–Perot-like resonances, and thelocal field in the medium at the maxima is resonantly en-hanced. The dashed–dotted curve in Fig. 1(a) is thetransmission spectrum for a medium with homogeneousabsorption, i.e., a0 5 0.1. The transmission maxima aremore strongly affected than are the minima betweenthem. The oscillations are greatly reduced, and the gapedges have a smaller slope. Figure 1(b) illustrates twocases with inhomogeneous absorption profiles, where theFourier coefficient of the absorption is nonzero. The in-homogeneous absorption yields a transmission functionthat is asymmetric. For a1 5 a21 5 0.1 (solid curve) theabsorption on the low-frequency side is larger. Simply


changing the phase of the inhomogeneous absorption co-efficient yields a1 5 a1 5 0.1 (dashed curve). The trans-mission function now has greater absorption on the high-frequency side of the gap.

This asymmetric absorption effect is a result of thefield’s spatial localization in the structure. On the low-frequency side of the gap the field is concentrated in thehigh-dielectric layers, whereas on the high-frequency sideof the gap it is concentrated in the low-dielectric layers.For homogeneous absorption, the absorption effect isevenly spread over the high- and the low-index materials,and the transmission is reduced symmetrically about thestop band. However, when the Fourier coefficient, a1 , isnonzero, the absorption is concentrated either in the high-dielectric material (a1 5 0.1) or in the low-dielectric ma-terial (a1 5 0.1).

The reflection also displays a similar asymmetric tun-ing effect when inhomogeneous absorption is present.Figure 2 illustrates the reflection spectra for the same pa-rameter values used in Fig. 1. Whereas the transmissionwas reduced by at least a factor of 2 at the transmissionresonances, the reflection coefficient displays a much

Fig. 1. Transmission spectra for four cases; in all curves thescaled length is L 5 5. (a) No absorption, solid curve; homoge-neous absorption, a0 5 0.1, dashed–dotted curve. (b) Inhomo-geneous absorption: a0 5 0.1 and a1 5 0.1, solid curve; a05 0.1 and a1 5 20.1, dashed curve.

smaller change owing to the losses because the reflectedfield at the points of high reflectivity is localized near thefront face of the sample.

B. Modulational InstabilityWe apply the linear stability analysis of Section 4 to thestudy modulational instabilities for the modified dynam-ics of a PBG medium with inhomogeneous Kerr coeffi-cients. Lichinitser et al.25 already analyzed in detail thehomogeneous nonlinearity modulational instabilities; in aprevious paper the nonlinear dynamics of the pulses in ahomogeneous nonlinear medium were studied for severalcases.24 In this subsection, we relate the homogeneouscase to the inhomogeneous case. The gain is defined bythe imaginary part of the frequency in Eq. (28):

G 5 uIm~v !u. (30)

When f 5 1, the discriminant of the quadratic equation ispositive for a homogeneous medium, and the frequencybecomes imaginary to signal the instability region. Fig-ure 3 is a plot of the gain at the upper band edge;G0 5 1, and all other nonlinear coefficients vanish. This

Fig. 2. Reflection spectra for the four cases illustrated in Fig. 1.The parameter values are found in the caption of Fig. 1.


value of G0 represents a large intensity; simulations ofthe nonlinear dynamics show strong modulation of thepulse at intensities that are more than 10 times smallerthan this value.24 The gain has two distinct sidelobesalong the line where the transverse wave number van-ishes. The sidelobes become less distinct as the trans-verse wave number increases. The two peaks coalesceand then split again for larger transverse wave-numbervalues. Along the line kz 5 0 there is a maximum valueof the gain near k'

2/F5 2.5. Continuing along the linekz 5 0, the gain eventually decreases and becomes zeronear k'

2/F 5 4.2. The gain has an off-axis maximum forlarger transverse wave numbers.

Figure 4 shows plots of the gain with a nonlinear me-dium that also possesses modulated nonlinear coeffi-cients. In the top figure the Kerr coefficient’s modulationamplitude is a modest 10% of the homogeneous medium.This is sufficient to modify the behavior near kz 5 0.The modulational instability boundary is pushed back toa transverse wave-number value of k'

2/F 5 0.6. Thepeak gain for the line kz 5 0 occurs near k'

2/F 5 2, andits on-axis value vanishes near k'

2/F 5 3.4. The gainalong the line of the vanishing transverse wave numbernearly vanishes. When the modulation is increased to40% the shape of the gain surface is reduced in ampli-tude, with a broad double peak near kz 5 0 and two flatridge lines that appear separated from the gain at smallwave vectors.

For further values of the nonlinear coefficient’s modu-lation in the range 40%–60% the surface undergoes rapidchanges. The top illustration in Fig. 5 is the gain surfacefor a 60% modulation. The broad features indicate that,overall, the linear gain does not favor one wave vectorover another. A small on-axis maximum appears in thetransverse direction, suggesting that the transverse in-stability can occur without strong longitudinal modula-tion. Increasing the Kerr coefficient’s modulation to100% entirely suppresses the modulational instability.The gain curves have a broad transverse gain centered onthe line kz 5 0, and the peak in the longitudinal wavevector is also at kz 5 0. The two features in Fig. 5 are

Fig. 3. Gain profile for f 5 21, G0 5 1, G1 5 0, G2 5 0. Thecurves are labeled with the values of the scaled transverse wavenumber, i.e., k'

2/F, which starts at zero and increases in incre-ments of D(k'

2/F) 5 0.1.

also found in our simulations of the nonlinear equations,and they are discussed in Subsection 5.C below.

Before leaving the linear stability analysis, we mentionthat, for tuning at the lower band edge, i.e., f 5 1, withcomparable values of the parameters studied the modula-tional instability gain has a maximum at kz 5 0, and it isquite flat over large values of the transverse wave num-ber. This behavior has been found for homogeneous andmodulated Kerr coefficients, and our analysis has been re-stricted to positive values of these coefficients, but clearlydifferent results will be found for different signs of the co-efficients. This complexity makes it difficult to catego-rize the behavior of nonlinear PBGs.

C. Simulations of the Nonlinear EquationsWe solve the nonlinear equations by applying the numeri-cal technique outlined in Section 3. We step the fieldsforward in time to solve the equations of motion. Onetransverse dimension is used in the simulations. An ini-tial Gaussian-shaped pulse is launched toward the PBGfrom outside the medium:

F~x, y, z, 0! 5 A exp@2~z 2 z0!2/sz2#exp~2x2/sx

2!.(31)

Fig. 4. Gain profile ( f 5 21): (a) G0 5 1, G1 5 0.1, G2 5 0;(b) G0 5 1, G1 5 0.4, G2 5 0.


The initial displacement z0 of the pulse is chosen to beseveral times longer than the pulse width sz 5 1/4p toprevent any significant overlap between the medium andthe leading edge of the wave. Also, the pulse is shortenough that it sloshes back and forth in the PBG mediumduring the interaction,24 but we worked with very longpulses with this technique and obtained nearly steady-state results. The transverse width was chosen as sx5 1; this value sets the transverse length scale. Thecoupling coefficient is F 5 100, so the transverse couplingis weak and does not noticeably affect the pulse tuning ata selected frequency. The z axis spans 20p, and the xaxis spans 8p. The pulse is tuned to the first transmis-sion maximum in the high-frequency pass band. Wemonitor energy conservation, which is constant to withinapproximately 1% or less.

Figure 6 shows views of the forward- and thebackward-propagating pulses after the initial Gaussianpulse has interacted with the medium. A bar marks themedium on opposite sides of the picture for the forwardfield and for the backward field; the nonlinear medium isin the range z P (0, 6.1). The pulse slows as it enters

Fig. 5. Gain profile ( f 5 21): (a) G0 5 1, G1 5 0.6, G2 5 0;(b) G0 5 1, G1 5 1, G2 5 0.

the medium, and the nonlinearity initially focuses thelight toward the central axis. The snapshots in Fig. 6 arefor an elapsed time of T 5 40. The forward pulse showsstructure in both the transverse and the longitudinal di-rections. The on-axis energy is directed off axis at sev-eral angles. The energy continues to spread as it propa-gates in the far field. The intensity along thepropagation direction also has a high-frequency oscilla-tion, which is attributed to the growth in modulational in-stability. This behavior is expected, based on linear sta-bility analysis; an example with the gain at nonzerowave-vector values is illustrated in Fig. 3. The length ofour medium and the pulse length are sufficient for themodulational instability to be manifest in our simula-tions.

When the medium has an inhomogeneous Kerr coeffi-cient the modulational instability can be suppressed.The sign of the nonlinear coefficient G1 strongly changesthe gain and the shape of the modulational instability.For instance, the gain peaks that are close to the kz 5 0axis in Fig. 4 are eliminated at a value of G1 5 20.1,which indicates the absence of modulational instability.Figure 7 shows snapshots of the pulse at the elapsed timethe T 5 40 when G1 5 0.4. The longitudinal instability,although it is still present, is not so distinct as the purehomogeneous Kerr nonlinearity in Fig. 6. The slow oscil-lations of the pulse along the propagation direction aredue to the sloshing of the pulse back and forth inside thePBG. The transverse direction retains a high-frequencycomponent that leads to excess pulse spreading.

Increasing the inhomogeneous Kerr coefficient to G15 1 completely suppresses the modulational instability

Fig. 6. Forward and backward field profiles for A 5 0.6, d5 1.12, h0 5 1, h1 5 0, h2 5 0, and L 5 6.1 Shown is the ab-solute amplitude of the fields after a time lapse of T 5 40. Barsat the top and the bottom of each photograph indicate the PBG’sposition; the bar extends from z 5 0 to z 5 6.1 in each picture.


in both directions, as suggested by the linear stabilityanalysis in Subsection 5.B. As the pulse propagatesthrough the PBG, the strong nonlinearity concentratesthe pulse energy close to the axis, but it emerges from theregion with a smooth phase profile. The field amplitudein Fig. 8 has a smooth shape. The transmission changeswith intensity; most of the light is reflected in this case,whereas at low launch intensities most of the light istransmitted. The optical limiting action of inhomoge-neous Kerr media can be used as a basic element for logicgates in an all-optical transmission system.5,6

Fig. 7. Forward and backward propagation for A 5 0.6, d5 1.12, h0 5 1, h1 5 0.4, h2 5 0, and L 5 6.1. See the captionto Fig. 6 for further details.

Fig. 8. Forward and backward propagation for A 5 0.6, d5 1.12, h0 5 1, h1 5 1, h2 5 0, and L 5 6.1. See the captionto Fig. 6 for further details.

Equations (10) describe the complicated, nonlinear evo-lution of the fields for a PBG structure. The stabilityanalysis helps by indicating the general instability behav-ior; however, because it assumes an infinite medium inwhich the fields are uniform throughout, it lacks the pre-cision to be a quantitative tool for the finite sample forwhich the fields have local maxima and amplitudes thatare fixed by the boundary conditions. Nevertheless, wehave verified features that were found from the linear sta-bility analysis. This verification means that linear sta-bility analysis a useful design tool for probing the variablespace and subsequently using the nonlinear analysis toexamine the parameter region to obtain the desired phe-nomenon.

6. CONCLUSIONSThe multiple-scales development of the coupled-modeequations for a nonlinear Kerr medium introduces severalnew terms that reflect the periodic character of the non-linearity. In addition, terms with a periodic loss (or gain)have been incorporated into the equations. ComplexKerr coefficients may be introduced for the study of effectsof saturable absorption or reverse saturable absorption indissimilar nonlinear media. The transmission spectrafor nonlinear absorption show an asymmetry about theband edge that is similar to the asymmetry discussed forlinear inhomogeneous absorption in Fig. 2. This effectcan be used to improve the dynamical range in opticallimiting devices and is a topic for future investigation.

To put the scaled numbers into a physical context, wetake the example of a medium with an average index of1.5 and a relative dielectric contrast of De 5 0.01. For awavelength of l 5 1 mm, the coupling coefficient is k5 20 mm21. The subscripts have been dropped here.All the other linear lengths are scaled to parameter k.The medium’s length in the simulations is L 5 300 mm,which is approximately 120 dielectric layers. The coeffi-cients of the nonlinear terms have the same order of mag-nitude, as was defined in the text following Eqs. (10). Inscaled units the nonlinear length is LNL 5 (ha2)21,where a is the amplitude of the optical field; our simula-tions use h 5 1, and modulation instabilities appear neara 5 0.3. For the linear coupling length in our examplethe nonlinear length in physical units is 450 mm. The in-dex change that is due to this nonlinearity is of the orderof 1023. In a medium with a nonlinear index parameterof n2 5 10212 cm2/W, modulational instability occurs atan intensity near 1 GW/cm2. The optimal design of a de-vice based on the optical limiting or the modulational in-stability phenomenon could significantly lower thethreshold, and, if a large index contrast were used, the de-vice could support a large bandwidth in a compact struc-ture.

Linear stability analysis is a useful first tool for explor-ing the range of behavior that can be expected from theequations. We have restricted the presentation of resultsin this paper to positive values of the nonlinear coeffi-cients. However, there are new and complicated gaincurves associated with negative or mixed values of theKerr coefficients, which we intend to examine thoroughlyin subsequent publications.


The nonlinear simulations provide an accurate solutionof the model equations for situations that require counter-propagating waves with diffraction and nonlinearities.Previously, the PBG with homogeneous nonlinearity wasshown to lead to modulational instability, depending onthe side of the band gap to which the pulse was tuned.Using inhomogeneous nonlinearity will result in a muchlarger range of possible phenomena, and the field propa-gation can be tailored to specific desired outcomes.

ACKNOWLEDGMENTSThe research of J. W. Haus was supported by NationalScience Foundation grant ECS-9630068. I. V. Mel’nikovwas partially supported by the Consejo Nacional de Cien-cia y Tecnologıa (Mexico) through its Catedra Patrimonialprogram; he is also grateful to the Electro-Optics Pro-gram of the University of Dayton, Dayton, Ohio, for itshospitality.

*I. V. Melnikov’s current address is the Department ofComputer Engineering, University of Toronto, 10 King’sCollege Road, Toronto M5S 3G4, Ontario, Canada.

J. W. Haus’s e-mail address is [email protected].

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Coupled-mode equations for Kerr media with periodically modulated linear and nonlinear coefficients