Nonlinear guided waves and spatial solitons in a periodic layered medium

772 J. Opt. Soc. Am. B/Vol. 19, No. 4 /April 2002 A. A. Sukhorukov and Y. S. Kivshar

Nonlinear guided waves and spatial solitons in aperiodic layered medium

Andrey A. Sukhorukov and Yuri S. Kivshar

Nonlinear Physics Group, Research School of Physical Sciences and Engineering, Australian National University,Canberra, ACT 0200, Australia

Received May 23, 2001; revised manuscript received October 31, 2001

We present an overview of the properties of nonlinear guided waves and (bright and dark) spatial optical soli-tons in a periodic medium created by linear and nonlinear waveguides. First we consider a single layer witha cubic nonlinear response (a nonlinear slab waveguide) embedded in a periodic layered linear medium anddescribe nonlinear localized modes (guided waves and Bragg-like localized gap modes) and their stability.Then we study modulational instability as well as the existence and stability of discrete spatial solitons in aperiodic array of identical nonlinear layers, a one-dimensional model of nonlinear photonic crystals. We em-phasize both similarities to and differences from the models described by the discrete nonlinear Schrodingerequation, which is derived in the tight-binding approximation, and the coupled-mode theory, which is valid forshallow periodic modulations. © 2002 Optical Society of America

OCIS codes: 190.5940, 130.2790.

1. INTRODUCTIONSpatially localized waves (or intrinsic localized modes) innonlinear lattices have been an active research topic dur-ing the past several years. In application to the prob-lems of nonlinear optics, such modes are known as dis-crete spatial solitons, and they have been describedtheoretically1–6 and recently observed experimentally7–10

in periodic arrays of nonlinear single-mode opticalwaveguides. A standard approach to the study of dis-crete spatial solitons in optical superlattices is to employthe properties of an effective discrete nonlinear Schro-dinger (NLS) equation, which can be derived under someassumptions similarly to the tight-binding approximationin solid-state physics.1 However, nonlinear localizedwaves in a system with a weakly modulated optical re-fractive index are known as gap solitons.11–14 Similarproblems and methods of their solution appear in otherfields, such as the nonlinear dynamics of the Bose–Einstein condensates in optical lattices.15

However, real experiments in guided-wave optics areconducted in the structures of more-complicated geom-etries and, therefore, the applicability of the tight-bindingapproximation and the corresponding discrete equationsbecomes limited. Moreover, one of the main features ofwave propagation in periodic structures (which followsfrom the Floquet–Bloch theory) is the existence of forbid-den transmission bandgaps; therefore, nonlinearly in-duced wave localization can also be possible in the form ofso-called gap solitons located in each of these gaps. How-ever, the effective discrete equations derived in the tight-binding approximation describe only one transmissionband surrounded by two semi-infinite bandgaps, andtherefore the real fine structure of the bandgap spectrumassociated with wave transmission in a periodic mediumis lost. The coupled-mode theory of gap solitons,14 how-ever, describes only the modes that are localized in an iso-lated narrow gap, and it does not allow the gap modes and

0740-3224/2002/040772-10$15.00 ©

the conventional guided waves that are localized as a re-sult of total internal reflection to be considered simulta-neously. However, the complete band structure of thetransmission spectrum and the simultaneous existence oflocalized modes of different types is important in theanalysis of the stability of nonlinear localized modes.Such an analysis is especially important for the theory oflocalized modes and waveguides in realistic models ofnonlinear photonic crystals (see, e.g., Ref. 16 and refer-ences therein).

In this paper we consider a simple model of a nonlinearperiodic layered medium in which the optical superlatticeis formed by a periodic sequence of two linear layers ofdifferent dielectric susceptibilities and whose nonlinearwaveguides are described by thin-film layers embedded init17 (see also Ref. 18). In such a case, the effects of thelinear periodicity and the bandgap spectrum structureare taken into account explicitly, and nonlinearity entersinto the corresponding matching conditions only, permit-ting a direct analytical study. The model looks drasti-cally simplified, but at the same time it includes all therichness of the physics of periodic media.

In the framework of this simplified model, we considertwo different problems. In the first case (Section 2) westudy nonlinear guided waves supported by a thin, iso-lated nonlinear layer with Kerr-type nonlinear responsethat is embedded in a linear periodic structure composedof two layers with different linear dielectric constants,i.e., a nonlinear waveguide in a periodic medium. Weconsider the case for which the layer describes an opticalwaveguide that supports guided waves in a homogeneousmedium of the averaged susceptibility and assume thatthe cubic nonlinear response may be either positive (self-focusing) or negative (self-defocusing). We describe twodifferent types of nonlinear localized mode supported bythis waveguide and also analyze the mode stability. Inthe second case (Section 3) we study the nonlinear local-

2002 Optical Society of America

A. A. Sukhorukov and Y. S. Kivshar Vol. 19, No. 4 /April 2002 /J. Opt. Soc. Am. B 773

ized modes in an infinite structure consisting of a periodicarray of nonlinear layers, similar to the geometry of theexperiments with discrete optical solitons.7–10 In thiscase we study modulational instability in both the self-focusing and the self-defocusing regimes and then discussthe properties of different types of nonlinear localizedmodes such as bright and dark spatial solitons. We em-phasize both similarities to and differences from the mod-els described by the discrete NLS equation and the con-tinuum coupled-mode theory.

2. NONLINEAR WAVEGUIDE IN APERIODIC MEDIUMA. ModelWe consider electromagnetic waves propagating along theZ direction of a slab-waveguide structure created by a pe-riodic array of thin-film nonlinear waveguides. Assum-ing that in the Y direction the field structure is defined bythe linear guided mode of the slab waveguide E(Y), weseparate the dimensions and represent the electric field inthe form E(X, Z)E(Y). Then the evolution of the com-plex field envelope E(X, Z) is governed by the NLS equa-tion

i]E

]Z1 D

]2E

]X2 1 e~X !E 1 g~X !uEu2E 5 0, (1)

where D is the diffraction coefficient (D . 0). The weakmodulation of linear refractive index in the transverse di-rection is defined by the function e(X), whereas g(X)characterizes the Kerr-type nonlinear response. We as-sume that the function e(X) or g(X) (or both of them) isperiodic in X; i.e., it describes a periodic layered struc-ture similar to the so-called transverse Bragg waveguidescreated by the thin-film multilayer structures7–10,19–24 orthe impurity band in a deep photonic bandgap.25

To reduce the number of physical parameters we nor-malize Eq. (1) as follows: E(X, Z) 5 c (x, z)3 E0 exp(ieZ), where e is the mean value of the functione(X), x 5 X/d and z 5 ZD/d2 are dimensionless coordi-nates, and d and E0 are the characteristic transversescale and field amplitude, respectively. Then the normal-ized nonlinear equation has the form

i]c

]z1

]2c

]x2 1 F~I; x !c 5 0, (2)

where the real function F(I; x) 5 d2D21@e(X) 2 e1 g(X)IuE0u2# describes both nonlinear and periodicproperties of the layered medium and I [ u cu2 is the nor-malized local wave intensity. We note that Eq. (2) isHamiltonian, and for the spatially localized solutions thepower

P 5 E2`

1`

u c ~x, z !u2dx

is conserved.At this point it is important to mention that Eq. (2) de-

scribes the beam evolution in the framework of the so-called parabolic approximation that is valid for wavespropagating mainly along the z direction. In otherwords, the characteristic length of the beam distortion

that is due to both diffraction and refraction along the zaxis should be much larger than the beam width in thetransverse direction x. This restriction leads to the con-dition of weakly modulated periodicity, ue(X)2 eu ! u eu. Moreover, the nonlinearity-induced changesin the electric field profile E(Y) should be negligible, acondition that is generally satisfied for realistic physicalsystems operating in a low-absorption regime,26 such asAlGaAs planar waveguide structures with the light fre-quency below the half-gap.

We look for stationary localized solutions of normalizedEq. (2) in the form

c ~x, z ! 5 u~x;b!exp~ibz !, (3)

where b is the propagation constant and the real functionu(x;b) satisfies the stationary nonlinear equation:

2bu 1d2u

dx2 1 F~I; x !u 5 0. (4)

We assume that the basic optical superlattice is linearand that nonlinearity appears only through the propertiesof a thin-film waveguide (or an array of such waveguides).Then, if the corresponding width of the wave envelope ismuch larger than that of the waveguide, we assume thatonly the lowest-order guided waves, which do not containnodes inside the waveguide, can be exited. Then, simi-larly to the Kronig–Penney approximation in solid-statephysics, the thin-film waveguide can be modeled by adelta function, and, in the simplest case of a single non-linear layer, we can write

F~I; x ! 5 n~x ! 1 d ~x !G~I !, (5)

where n(x) [ n(x 1 h) describes an effective potential ofthe superlattice with spatial period h and the functionG(I) characterizes the properties of the nonlinear thin-film waveguide. Specifically, G . 0 if the guiding layerhas an effectively higher refractive index than the sur-rounding structure and G , 0 if the index is lower.

Earlier studies of wave transmission through nonlinearmultilayer structures indicate that the delta-functionmodel allows all generic features of a nonlinear system tobe identified.27 Approximate model (5) makes it possibleto perform a complete analytical treatment of the prob-lem. Additionally, this approximation can easily be ex-tended to the case of a periodic array of thin-film nonlin-ear waveguides (see Section 3 below).

B. Bandgap Spectrum and Localized WavesIf the effective periodic potential n(x) is approximated bya piecewise-constant function, the solution can be decom-posed into a pair of counterpropagating waves with am-plitudes a(x;b) and b(x;b):

ub~x;b! 5 a~x;b!exp@2m~x;b!x#

1 b~x;b!exp@1m~x;b!x#, (6)

where m(x;b) 5 @b 2 n(x)#1/2 is the local wave number.As follows from the Floquet–Bloch theory, for a Bloch-wave solution the reflection coefficient r(x;b)5 b(x;b)/a(x;b) is a periodic function, i.e., r(x;b)5 r(x 1 h;b), and it satisfies the eigenvalue problem


T~x;b!F 1r~x;b!G 5 t~b!F 1

r~x;b!G , (7)

where T(x;b) is a transfer matrix that describes a changeof the wave amplitudes $a,b% after one period (x, x 1 h).The transfer-matrix approach for optical waves is pre-sented in Sec. 1.6 of Ref. 28 [see also Ref. 29, where thenotation corresponds to that of Eq. (7)]. It can be dem-onstrated that det T [ 1; therefore, two linearly indepen-dent solutions of Eq. (7) correspond to a pair of the eigen-values, t and t 21. Relation t(b) determines a bandgapstructure of the superlattice spectrum: the waves arepropagating if utu 5 1, and they are localized if utu Þ 1.19

In the latter case a nonlinear waveguide can support non-linear localized waves as bound states of Bloch-wavesolutions21 with the asymptotics uu(x → 6`)u → 0. Thewave amplitude at the thin-film nonlinear waveguide isdetermined from the continuity condition at x 5 0; i.e.,I0 [ I(01) 5 I(02) and @du/dx#x502

011 G(I0)ux50 5 0.

Then we use Eq. (6) to express the latter conditionthrough the superlattice characteristics

G0 [ G~I0! 5 z~b!, (8)

where z 5 (z1 1 z2)x50 and

z6 5 m6~1 2 r6!

~1 1 r6!;

1 and 2 stand for the characteristics on the right- andleft-hand sides of the layer waveguide, respectively; i.e.,n(x) 5 n1(uxu) for x . 0 and n(x) 5 n2(uxu) for x , 0.

It can be demonstrated29 that, inside the transmissiongaps, Im z(b) [ 0, and one can use this condition to calcu-late the band diagram of a layered structure, as illus-trated in Fig. 1. We notice that the first bandgap (semi-infinite white region at b . 19; see Fig. 1) corresponds tothe condition of total internal reflection (IR), and, there-fore, the Bloch-wave diffraction properties should be simi-lar to those of conventional guided waves. Indeed, it canbe clearly seen from Fig. 1 that G0 5 z(b) . 0 in the IRgap, and therefore localization is possible only at a high-index waveguide.

In contrast, additional bandgaps appear at smaller b asa result of resonant Bragg reflection (BR) by the periodicstructure. In these gaps the spatial localization can alsooccur at a low-index antiwaveguide,19,20 provided thatG0 5 z(b) , 0 (shaded region in Fig. 1). The locations

Fig. 1. Characteristic dependence of z on b for localized states.Waveguiding (white areas, z . 0) and antiwaveguiding (dottedarea, z , 0) localization regimes inside the band gaps are shown.The lattice parameters are h 5 1 and n(x) 5 0 for n 2 1/2, x/h , n and n(x) 5 30 for n , x/h , n 1 1/2, where n is aninteger, a 5 0.5, and g 5 1.

of the waveguiding and the antiwaveguiding regions inthe BR gaps are determined by the position of the guidinglayer inside the layered structure. If the layer is locatedsymmetrically, only one localization regime will be ob-served in a BR waveguide, as was found in earlierstudies.30 This restriction does not hold for higher-orderlocalized modes, which are not described by the delta-function model [Eq. (5)].

One of the key properties of nonlinear localized modesis linear stability, which determines the character of themode dynamics under the action of small perturbations.In general, two different situations of the perturbation-induced mode dynamics are possible. In the first case, anonlinear mode can acquire only small distortions to itssteady-state profile, and the parameters of a nonlinearmode oscillate in the vicinity of its stationary state. Wecall such a nonlinear mode linearly stable. However, un-der the influence of small perturbations the initial devia-tions of the nonlinear mode parameters from their sta-tionary values can grow exponentially; in this case wedefine the nonlinear mode as linearly unstable.

To study linear stability of the localized solutions, weconsider the evolution of small-amplitude perturbationsof the localized state, present the solution in the form

c ~x, t ! 5 @u~x ! 1 v~x !exp~iGz !

1 w* ~x !exp~2iG* z !#exp~ibz !, (9)

and obtain the linear eigenvalue problem for the pertur-bations v(x) and w(x):

2~b 1 G!v 1d2v

dx2 1 n~x !v 1 d~x !@G1v

1 ~G1 2 G0!w] 5 0,

2~b 2 G!w 1d2w

dx2 1 n~x !w 1 d ~x !@G1w

1 ~G1 2 G0!v] 5 0, (10)

where G1 [ G0 1 I0G8(I0). Intensity I0 is calculatedfor an unperturbed solution, and the prime stands for thederivative.

We find that the localized eigenmode solutions of Eqs.(10) exist only for the particular eigenvalues that satisfythe solvability condition Y(G) 5 0, where

Y~G! 5 @G1 2 z~b 1 G!#@G1 2 z~b 2 G!#

2 ~G1 2 G0!2.

The function Y(G) is also known as the Evans function.31

In general, each of the eigenmode solutions falls into oneof the following categories: (i) internal modes, with realeigenvalues that describe periodic oscillations (breathing)of the localized state, (ii) instability modes that corre-spond to purely imaginary eigenvalues with Im G , 0,and (iii) the oscillatory instabilities that can occur whenthe eigenvalues are complex (and Im G , 0). Inasmuchas the linear spectrum has symmetry G → 6 G* , eigen-modes with Im G Þ 0 always indicate instability, and, inwhat follows, we consider only solutions with Re G > 0and Im G > 0 with no lack of generality.


To demonstrate the basic stability results, we considera localized waveguide that possesses a cubic (or Kerr-type) nonlinear response, G(I) 5 a 1 gI. At a properscaling, the absolute value of the nonlinear coefficient gcan be normalized to unity. We assume that the nonlin-ear response is self-focusing, so g 5 11. Therefore weconsider below two qualitatively different examples ofwaveguides that correspond to the different signs of lin-ear coefficient a.

C. High-Index WaveguideWe consider the properties of spatially localized wavessupported by a thin-film waveguide that has a high re-fractive index at small intensities, i.e., a . 0. Then lo-calization can occur only in the wave-guiding regime be-cause z 5 G0 . 0. In examining Fig. 1 we see thatlocalized waves exist in both the IR and the BR gaps, andin the linear limit (when I0 → 0) the correspondingpropagation constants bb are defined by the equationz(bb) 5 a. At higher intensities, the propagation con-stants of the BR and the IR modes increase, and we char-acterize the families of solution by the power dependen-cies presented in Fig. 2, top. Localized waves in the IRband resemble guided waves weakly modulated by the pe-riodic structure [Fig. 2(b)]. Because the wave profile doesnot contain zeros, it is a fundamental eigenmode. There-fore the conditions of the Vakhitov–Kolokolov stabilitytheorem (see the pioneering paper32 and also a recent re-view paper33) are satisfied, and the IR states are unstableif and only if dP/db , 0. At the critical point, dP/db5 0, the linear eigenvalue passes through zero and be-

comes imaginary, as illustrated in Fig. 2 (middle, b. 27.5.)

In contrast, the localized waves in the BR gap (atsmaller b) are similar to the gap solitons composed of mu-tually coupled backward- and forward-propagating waves[Fig. 2(a)]. For the BR states, the Vakhitov–Kolokolovcriterion provides only a necessary condition for stability,because the higher-order localized states can also exhibitoscillatory instabilities. Indeed, we notice that in the lin-ear limit there always exists an internal mode that corre-sponds to a resonant coupling between the BR and IRbandgaps, as Y@bb

(IR) 2 bb(BR)# [ 0. We performed exten-

sive numerical calculations and found that this modeleads to an oscillatory instability of BR waves when thevalue (b 2 Re G) moves outside the band gap; it occurswhen the wave intensity exceeds a threshold value (seeFigs. 2 and 3).

D. Low-Index AntiwaveguideNext we study the case of a thin-film linear anti-waveguide with a low refractive index, a , 0. At smallintensities, I0 , ua/bu, the localized waves can exist onlyin the antiwaveguiding regime, which is possible only inthe BR gap, where z(b) , 0; see Fig. 1. As intensity isincreased, the propagation constant is shifted toward thebandgap edge; see Fig. 4 (top, dotted region). When theintensity exceeds the critical value, the effective refrac-tive index of the guiding layer becomes effectively higherthan in the surrounding structure (G0 . 0), and the lo-calized waves bifurcate from the edges of the wave-guiding regions, as shown in Fig. 4 (top). The character-

istic mode profiles are presented in Figs. 4(a)–(c).Stability properties of such waves are similar to those de-scribed in Subsection 3.C. In particular, localized wavesin the BR gap can exhibit oscillatory instabilities.

We find that an oscillatory instability appears at highpowers such as those shown in Fig. 4(b) for the waves inthe antiwaveguiding regime, whereas these modes arestable in the linear case. Such threshold dependence ofthe wave dynamics on the input power was observed innumerical simulations, where the localization at a finite-width nonlinear waveguide was investigated.21 Quite re-markably, this phenomenon was also observedexperimentally.34 This example demonstrates that thewave properties determined in the delta-function model[Eq. (5)] are indeed quite general.

Fig. 2. Top, power versus propagation constant for the nonlin-ear localized states: solid curves, stable; dashed curve, un-stable; dotted curve, oscillatory unstable. Middle, real (dottedcurves), and imaginary (solid curves) parts of the eigenvalues as-sociated with wave instability. Waveguiding (white areas) andantiwaveguiding (dotted area) localization regimes inside theband gaps are shown. Bottom, localized states that correspondto points marked a and b at the top. The lattice parameters arethe same as in Fig. 1.

Fig. 3. Example of the resonance that occurs between an inter-nal mode of the localized state and a bandgap edge, leading tooscillatory instability.


3. PERIODIC ARRAY OF NONLINEARWAVEGUIDESA. Model and Discrete EquationsNow we analyze both localized and extended nonlinearwaves supported by a periodic array of thin-filmwaveguides with the Kerr-type nonlinear response. Thecorresponding model is a simplified version of the nonlin-ear layered medium that is usually studied inexperiment.7–10 In our case we assume again that thenonlinear layers are thin, so the problem can be studiedanalytically, because the nonlinearity enters into thematching conditions only.

To simplify our analysis further, we assume that thelinear periodicity is associated only with the presence ofthe thin-film waveguides and define the response functionin the model [Eq. (2)] as follows:

F~I; x ! 5 (n

~a 1 gI !d ~x 2 hn !, (11)

where h is the distance between the neighboringwaveguides and n is an integer. As above, the responseof the thin-film waveguides is approximated by the deltafunctions, and the real parameters a and g describe linearand nonlinear properties, respectively. Nonlinear coeffi-cient g can be normalized to unity, so g 5 11 correspondsto self-focusing and g 5 21 corresponds to self-defocusing nonlinearity. Coefficient a (a . 0) definesthe linear response (i.e., the response at the low intensi-ties), and it characterizes the corresponding couplingstrength between the waveguides.

As was demonstrated in Ref. 29, the stationary waveenvelopes defined by Eqs. (3) and (4) can be expressed interms of the wave amplitudes at the nonlinearwaveguides, un 5 u(hn), that satisfy a stationary form ofthe discrete NLS equation:

hUn 1 ~Un21 1 Un11! 1 xuUnu2Un 5 0. (12)

Here Un 5 Aujguun are the normalized amplitudes, x5 sign(jg), and

h 5 22 cosh~hm! 1 aj,

j 5 sinh~hm!/m, m 5 Ab. (13)

Fig. 4. Top, power versus propagation constant. (a)–(c) Local-ized mode profiles for a 5 25 and g 5 11. Notation is thesame as in Fig. 2.

Localized solutions of Eq. (12) with exponentially de-caying asymptotics can exist for uhu . 2, and this condi-tion defines the bandgap structure. Characteristic de-pendencies of parameters h and j versus propagationconstant b are presented in Fig. 5, where the bandgapsare shown as white stripes. The first (semi-infinite)bandgap corresponds to the regime of total IR. Atsmaller b the spectrum bandgaps appear as a result ofresonant BR from the periodic structure. Both thesetypes of bandgap were discussed above, in Subsection 2.B,for a general layered medium.

B. Modulational InstabilityFirst we analyze the properties of the extended plane-wave solutions of the model [Eqs. (2) and (11)], whichhave equal intensities at the nonlinear layers, I0 5 uunu2

5 constant, and correspond to the first transmissionband. These solutions have the form of the so-calledBloch waves (BWs), un 5 u0 exp(iKn), where wave num-ber K is selected in the first Brillouin zone, uKu < p. Us-ing Eq. (12), we find the dispersion relation as K5 6cos21@h(b;a)#, where a 5 (a 1 gI0

2) defines the lay-ers’ response modified by nonlinearity. Because thetransmission bands are defined by the condition uhu , 2,the edge of the band and the band itself are shifted whenthe intensity increases. Indeed, by resolving the disper-sion relation we determine a relation between the propa-gation constant and the wave intensity:

I0~b! 5 2@2 cos K 1 h~b!#

gj~b!. (14)

It can be demonstrated from Eq. (13) that in the firsttransmission band b . 2(p/h)2, so j(b) . 0 (see alsoFig. 5), and it follows that propagation constant b in-creases at higher intensities in a self-focusing medium(g . 0) and decreases in a self-defocusing medium(g , 0).

One of the main problems associated with the nonlin-ear BW modes is their instability to periodic modulationsof a certain wavelength, known as modulational instabil-ity. To describe the stability properties of the periodic

Fig. 5. Characteristic dependencies of parameters h and j onpropagation constant b. White areas mark bandgaps. The lat-tice parameters are h 5 0.5 and a 5 10.


BW solutions we analyze the linear evolution of weak per-turbations in the form of Eq. (9). Now the functions v(x)and w(x) describe the small-amplitude perturbation thatcorresponds to eigenvalue G. Substituting Eq. (9) intothe original model [Eqs. (2) and (11)], we obtain an eigen-value problem that has periodic solutions in the form ofBloch functions, v(x 1 h) 5 v(x)exp@i(q 1 K)# and w(x1h) 5 w(x)exp@i(q 2 K)#, provided that the followingsolvability condition is satisfied:

@h~b 1 G! 1 2gj~b 1 G!I0 1 2 cos~q 1 K !#

3 @h~b 2 G! 1 2gj~b 2 G!I0 1 2 cos~q 2 K !#

5 g2j~b 1 G!j~b 2 G!I02. (15)

The spectrum of the possible eigenvalues G is determinedfrom the condition that the spatial modulation frequen-cies q, which are found from Eq. (15), are all real.

In what follows, we consider two characteristic cases ofstationary BW profiles: unstaggered (K 5 0) and stag-gered (K 5 p). Some particular cases of modulationalinstability of the periodic BW solutions were recentlystudied for the Bose–Einstein condensates in opticallattices35–37 in the mean-field approximation based on theGross–Pitaevskii equation, which is mathematicallyequivalent to Eq. (2) with F(I; x) 5 n(x) 1 xI. It wasdemonstrated that the unstaggered solutions are alwaysmodulationally unstable in a self-focusing medium (x5 11) and stable in a self-defocusing medium (x5 21). It can be proved that similar results are validfor our model. We note that at small intensities themodulational instability in the self-focusing case corre-sponds to long-wave excitations, as illustrated in Fig. 6.In numerical simulations we used a discrete version ofEq. (2), which we obtained by performing an integrationover the intervals (xm 2 Dx/2, xm 1 Dx/2), where xm arethe node locations. Step size Dx was chosen smallenough that u(x) . um inside the integration intervals,and spatial derivatives could be approximated in a stan-dard way.38

We found that the staggered BW modes in a self-defocusing medium (x 5 21) are always modulationallyunstable.35 The properties of such modes in a self-focusing medium (x 5 11), however, are less trivial. Wefound that, in this case, the oscillatory instabilities (i.e.,those with complex G) can appear as a result of reso-nances between the modes of different bands. Such in-stabilities appear in a certain region of the wave intensi-ties in the case of shallow modulations when a is below acertain threshold value, as shown in Fig. 7.

It is interesting to compare these results with the re-sults obtained in the framework of the continuumcoupled-mode theory that is valid for the case of a narrowbandgap, i.e., for small a and small I0 . As was foundearlier,39 in such a case the oscillatory instability appearsabove a certain critical intensity, which is proportional tothe bandgap width. In our case, we found the followingasymptotic expression for the low-intensity instabilitythreshold:

gI0~cr! . a 1 2A2ha3/2/p 1 O~a2!. (16)

Because the band-gap width is 2a/h 1 O(a3/2), we ob-served perfect agreement with the results of the coupled-mode theory (Fig. 7, dashed curve).

In the limit of large a (or large I0), the BW solution iscomposed of weakly interacting waves, each localized atthe individual waveguide. Then the collective dynamicsof the nonlinear modes in such a system can be studiedwith the help of the tight-binding approximation that isvalid for the model of weakly coupled oscillators. Thisapproach leads to the effective discrete NLS equation,which predicts the stability of the staggered modes in aself-focusing medium.40 Numerical and analytical re-sults confirm that our solutions are indeed stable in thecorresponding parameter regions.

We have found that in a self-focusing medium the stag-gered waves are always stable with respect to low-frequency modulations; see Fig. 8. At larger intensitiesunstable frequencies are shifted toward the edge of theBrillouin zone, q 5 p, whose instability growth rate hasthe largest value. The corresponding modula-

Fig. 6. Development of modulational instability in a self-focusing medium for a slightly perturbed unstaggered BW solu-tion with I0 . 0.44. The lattice parameters are a 5 3, h5 0.5, and g 5 11.

Fig. 7. Modulationally unstable staggered BW modes (shadedarea) in a self-focusing medium, shown as intensity (I0) versuslattice parameter a (at h 5 0.5 and g 5 11). Dashed curve,analytical approximation (16) for the low-intensity instabilitythreshold.


tional instability manifests itself through the develop-ment of the period-doubling modulations, as shown inFig. 9.

C. Bright Spatial SolitonsStationary localized modes in the form of discrete brightsolitons can exist with the propagation constant insidethe bandgaps when uhu . 2. Additionally, such solutionscan exist only if the signs of nonlinearity and of discretediffraction are different, i.e., when hx , 22. It followsfrom Eqs. (13) that b . 2(p/h)2 and j . 0 in the IR gapand in the first BR gap (see also Fig. 5), so the type of non-linear response is fixed by the medium’s characteristics,because x 5 sign(g). Therefore, self-focusing nonlinear-ity can support bright solitons in the IR region whereh , 22, i.e., in the conventional wave-guiding regime.In the case of the self-defocusing response, bright solitonscan exist in the first BR gap because the sign of the effec-tive diffraction is inverted (h . 2). In the latter case,mode localization occurs in the so-called antiwaveguidingregime.

Let us now consider the properties of two basic types oflocalized mode: odd, centered at a nonlinear thin-filmwaveguide, and even, centered between the neighboringwaveguides. These two symmetries are U unu5 xsU2unu2s , where s 5 0, 1. For discrete lattices, suchsolutions have already been studied in the literature (see,e.g., Ref. 41), and it has been found that the mode profileis unstaggered (i.e., Un . 0) if h , 22. Equation (12)possesses a symmetry transformation Un→ (21)nUn , h → 2h, and x → 2x, which means thatthe solutions become staggered at h . 2.

To study the linear stability of the localized modes inour model we consider the evolution of perturbations inthe form of Eq. (9) and, using Eqs. (2) and (11), obtain thelinear eigenmode problem for small v(x) and w(x):

2~b 1 G!v 1d2v

dx2 1 (n

@~a 1 2gun2!v 1 gun

2w#

3 d ~x 2 hn ! 5 0,

Fig. 8. Unstable modulation frequencies (shaded area) versusintensity I0 of the staggered BW modes (a 5 3 and h 5 0.5).Dark, solid curve, instability frequency with the largest growthrate.

2~b 2 G!w 1d2w

dx2 1 (n

@~a 1 2gun2!w 1 gun

2v#

3 d ~x 2 hn ! 5 0,

where G stands for the instability growth rate (see Sub-section 2.B above).

Our analysis reveals that even modes are always un-stable with respect to a translational shift along the xaxis. Odd modes are always stable in the self-focusingregime (see Fig. 10) but can exhibit oscillatory instabili-ties in the self-defocusing case when the power exceeds acertain critical value (see Fig. 11).

Owing to periodic modulation of the medium refractiveindex, solitons can form bound states.42 In particular,the so-called twisted localized mode43 is a bound state oftwo out-of-phase bright solitons. Such solutions do nothave their continuous counterparts, and they can existonly when the discreteness effects are strong, i.e., foruhu . hcr . The properties of the twisted modes dependon the separation between the individual solitons that

Fig. 9. Development of the instability-induced period-doublingmodulations. The initial profile corresponds to a slightly per-turbed staggered solution with I0 . 29.87. The parameters arethe same as in Fig. 8.

Fig. 10. Top, power versus propagation constant for odd (darkcurve) and even (light curve) localized modes in a self-focusing(g 5 11) regime: solid curve stable; dashed curve, unstable.Bottom, profiles of the localized modes that correspond tomarked points a and b at the top. The lattice parameters arethe same as in Fig. 5.


form a bound state. The stability properties of thetwisted modes in the IR gap under the conditions of self-focusing can be similar to those identified earlier in theframework of a discrete NLS model.43–45 The character-istics of the twisted modes in the BR gap can differ sub-stantially from those in the former case, because thevalue of h is limited from above (2 , h , hmax) andtherefore some families of twisted modes with small sepa-ration between the solitons may not exist.17

D. Dark Spatial SolitonsSimilarly to the continuous NLS equation with self-defocusing nonlinearity46 or the discrete NLS model,47

our model can support dark solitons, i.e., localized modesthat exist on a BW background. Such dark stationary lo-calized modes can exist in a periodic medium for bothsigns of nonlinearity. To be specific, let us consider thecase of a background corresponding to the BW solutionswith K 5 0, p introduced in Subsection 3.B. Dark-modesolutions can appear at the bandgap edge whereh 5 2x 5 2 sign(g), as in such a case nonlinear and dis-crete diffraction terms have the same signs.46

Similarly to the case of bright solitons discussed above,two basic types of dark spatial soliton can be identified aswell, namely, odd localized modes centered at a nonlinearthin-film waveguide and even localized modes centeredbetween the neighboring waveguides. All such modessatisfy the symmetry condition, U unu1s5 2(2x)s11U2unu21 , where s 5 0, 1 for even and odd

modes, respectively. The background wave is unstag-gered for x 5 21, and it is staggered for x 5 11; the cor-responding solutions can be constructed with the help of asymmetry transformation Un → (21)nUn . However,the stability properties of these two types of localizedstate can be quite different. Indeed, as was demon-strated in Subsection 3.B, the staggered BW backgroundcan become unstable in a self-focusing medium (x5 11). The unstaggered background, however, is al-

ways stable if x 5 21.The two types of dark spatial soliton that exist in our

model are shown in Figs. 12 and 13 for staggered and un-staggered background waves, respectively. We charac-

Fig. 11. Top, power versus propagation constant in the self-defocusing (g 5 21) regime. Notation is the same as in Fig. 10;dotted curve, oscillatory unstable modes.

terize the families of dark solitons by the complimentarypower, defined as

Pc 5 limn→1`

E2nh

1nh

@ uu~x 1 2nh !u2 2 uu~x !u2#dx,

where n is an integer. The localized solutions shown inFig. 13 are similar to those found earlier48 in the contextof the superflow structures in a periodic potential.

4. CONCLUSIONSIn the framework of a simple one-dimensional model of aperiodic layered medium, we have analyzed nonlinearguided waves and discrete spatial solitons of two types,i.e., nonlinear guided waves localized owing to total inter-nal reflection and Bragg-like localized gap modes. Wehave considered two specific geometries of the nonlinearperiodic structures that find their application in experi-mental realizations, namely, a single nonlinear layer em-bedded in a periodic linear medium and a periodic arrayof identical nonlinear layers, the so-called one-

Fig. 12. Top, complementary power versus propagation con-stant for odd (dark curve) and even (lighter curve) dark localizedsolitons in a self-focusing (g 5 11) regime. Notation is thesame as in Fig. 10.

Fig. 13. Top, complementary power versus propagation con-stant in the self-defocusing (g 5 21) regime. Notation is thesame as in Fig. 12.


dimensional nonlinear photonic crystal. We have ana-lyzed the existence and stability of the nonlinear localizedmodes in both models and have described also the modu-lational instability of homogeneous states in a periodicstructure of the nonlinear layers. In particular, we havediscussed both similarity to and difference from the mod-els described by the discrete NLS equation derived in thefrequently used tight-binding approximation, and the re-sults of the coupled-mode theory that is applicable forshallow modulations and a narrow gap. We believe thatour results may be useful for other fields, such as the non-linear dynamics of the Bose–Einstein condensates in op-tical lattices.

ACKNOWLEDGMENTSThe authors are indebted to O. Bang, P. L. Christiansen,and C. M. Soukoulis for collaboration at the initial stageof this project and to A. S. Kovalev, A. A. Maradudin, andY. Silberberg for useful and encouraging discussions.This research has been supported by the Performance andPlanning Fund of the Institute of Advanced Studies, bythe Australian National University, and by the AustralianResearch Council.

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Nonlinear guided waves and spatial solitons in a periodic layered medium