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L. Dong and H. Li Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1179

Surface solitons in nonlinear lattices

Liangwei Dong1,* and Huijun Li2

1Institute of Information Optics of Zhejiang Normal University, Jinhua 321004, China2Department of Physics of Zhejiang Normal University, Jinhua 321004, China

*Corresponding author: donglw@zjnu.cn

Received February 17, 2010; revised April 8, 2010; accepted April 9, 2010;posted April 9, 2010 (Doc. ID 123918); published May 11, 2010

We investigate the existence of spatial optical solitons supported by an interface between two nonlinear latticeswith different saturation parameters. Dipole, quadrupole, and vortex solitons are found in the nonlinear sur-face lattices. The slight different saturation degree between two sides of the interface leads to a remarkableasymmetry of solitons with higher power. We reveal that multipole and vortex solitons are stable when theirpower exceeds a threshold value, and stable localized surface nonlinear modes with very high peaks arepossible. © 2010 Optical Society of America

OCIS codes: 190.0190, 190.6135.

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urface solitons are self-trapped nonlinear modes guidedy an interface across which media with different opticalroperties are presented. Surface waves have drawn areat deal of attention in nonlinear optics these days1–4]. The presence of an interface may enrich solitonamilies, change their existence domain, and alter theirtability [1–10]. Diverse types of surface solitons wereredicted or observed in a variety of materials and set-ings [1–10].

Recently, theoretical and experimental studies demon-trated that the transverse nonlinearity modulation ofhe optical material can substantially affect the proper-ies of solitons. For example, soliton shape transforma-ions can be controlled by spatially modulated nonlinear-ty [11], bound states with an arbitrary number of solitonsan be found in systems with spatially inhomogeneousonlinearities [12], and stable light bullets in Bessel opti-al lattices with spatially modulated nonlinearity [13] andn optical tandems [14] are possible.

Solitons supported by nonlinear lattices were first pre-icted in [15], where narrow matter-wave solitons canasily form stable complexes in nonlinear optical lattices.he spatially transverse periodic modulation of the non-

inearity can be used to stabilize solitons [16]. In one-imensional nonlinear lattices, families of optical vectorolitons with complex multihumps were reported [17].undamental, dipole, and vortex solitons have wide sta-ility domains in two-dimensional purely nonlinear lat-ices made up of a periodic array of cylinders [18].

Note that, by its nature, a nonlinear lattice is differentrom the discrete waveguide or a linear lattice at least forhe following three reasons. First, at low soliton power,he linear lattices are periodic refractive index structuresut the nonlinear lattices have almost no influence on theefractive index. Second, the ability of nonlinear latticesor trapping beams becomes stronger at higher power, buthe trapping ability of linear lattices remains unchange-ble. Third, a linear lattice has a bandgap structure but aonlinear lattice does not have. Nonlinear lattices mighte realized in optics as large-period photonic crystals

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hose holes are filled with an index-matching liquid [19],r in photorefractive materials where the periodicallyarying nonlinearity may be created by indiffusion of dop-nts [20].However, thus far, researches on a localized wave in

onlinear lattices mainly focused on the matter-wave soli-ons supported by the combination of linear and nonlinearattices with a few exceptions, e.g., [17,18], where opticalolitons in purely nonlinear lattices were considered.roperties of solitons supported by an interface between

wo purely nonlinear lattices have not been explored yet.n the present paper, we report on the properties of soli-ons in two nonlinear lattices with different saturationegrees separated by an interface. Comparing with theases in uniform nonlinear lattices [18], where compo-ents of multipole and vortex solitons are uniformly dis-ributed, we found surface multipole solitons exhibitingtrong asymmetry and surface vortex solitons with non-anonical phase distribution when their power is rela-ively high. The phase structure of a vortex soliton gradu-lly changes from a uniform stair-like distributionupported by a uniform medium to a nonuniform stair-ike one with the growth of the propagation constant.

We start our analysis by considering beam propagationlong an interface between two periodic nonlinear latticesith different saturable focusing nonlinearities, describedy the nonlinear Schrödinger equation for the dimension-ess complex amplitude of the light field q,

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1 + S�q�2q = 0, �1�

here the transverse x ,y and longitudinal z coordinatesre scaled to the input beam width and diffraction length,espectively. The function ��x ,y� describes the distribu-ion of nonlinear lattices. We assume that the nonlinearattices are composed of a square array of cylinders withadius wR where the saturable nonlinearity exists. Inach cylinder ��x ,y�=1 and ��x ,y�=0 in the spacing be-ween them. The distance between two nearest lattice

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1180 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 L. Dong and H. Li

ites was set as ws=3 without loss of generality. We inves-igate the dynamics of solitons when the saturable non-inearity of nonlinear lattices at the left side of the inter-ace is different from the right side, i.e., Sl�Sr. Equation1) admits several conserved quantities, including theower or energy flow: U=�−�

� �−�� qq�dxdy.

We search for stationary solutions of Eq. (1) in the formf q�x ,y ,z�= �wr�x ,y�+ iwi�x ,y��exp�ibz�, where wr and wire real and imaginary parts of the solution profiles and bs a nonlinear propagation constant. The twisted phasetructure of the stationary solutions can be defined by=�−�

� �−�� arctan�wi /wr�dxdy /2�, where m is the so-called

topological charge” or “winding number” of vortex soli-ons. Substituting the expression into Eq. (1), we obtain

1

2� �2

�x2 +�2

�y2�wr,i − bwr,i +��wr

2 + wi2�

1 + S�wr2 + wi

2�wr,i = 0. �2�

he soliton profiles are found numerically by a two-imensional relaxation algorithm. Obviously, there is aonzero wi only for vortex solutions.To comprehensively understand the stability character-

stics of the surface solitons, we write the perturbed solu-ions of Eq. (1) in the form of q�x ,y ,z�= �wr+ iwi+ �uriui�exp��z��exp�ibz� [9], where ur and ui are the real and

maginary parts of the perturbations, respectively. Substi-uting the above expression into Eq. (1) and linearizingr,i around wr,i, we obtain

− �ur =1

2� �2

�x2 +�2

�y2�ui − bui

+��3wi

2 + wr2 + S�wr

2 + wi2�2�

�1 + S�wr2 + wi

2��2ui

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�1 + S�wr2 + wi

2��2ur,

�ui =1

2� �2

�x2 +�2

�y2�ur − bur +��3wr

2 + wi2 + S�wr

2 + wi2�2�

�1 + S�wr2 + wi

2��2ur

+2�wrwi

�1 + S�wr2 + wi

2��2ui, �3�

hich is a system of coupled Schrödinger-type equationsor perturbation components ur,i and can be solved nu-

erically. The perturbation ur,i may grow with a complexate � during propagation. The solitons are stable onlyhen all real parts of � equal zero.First, we address the properties of multipole mode sur-

ace solitons. Typical soliton profiles are displayed in Fig.. For a small propagation constant, the components of di-oles repel each other due to their out-of-phase distribu-ion and deviate their locations from the center of latticeites slightly. The difference between the saturable non-inearities at the left and right sides can be ignored [Fig.(a)] since the intensity of soliton peaks is low. Two peaksf dipoles have almost the same value [Fig. 1(a)]. How-ver, with the increment of the propagation constant, theeak values of solitons grow rapidly and the role of theaturable nonlinearity becomes important which enlarges

he difference between the two peaks [Fig. 1(b)]. Note thathe power of soliton is 311.3 in Fig. 1(b) and 11.6 in Fig.(a). Surface dipoles residing on the two adjacent diago-al lattice sites were also found. The size of soliton com-onents expands with the decrement of the propagationonstant [Fig. 1(c)] and shrinks with the increment of theropagation constant [Fig. 1(d)]. Again, the difference be-ween the two peaks of dipole solitons increases with therowth of the propagation constant.

Figure 1(e) shows an example of an on-site quadrupoleoliton whose “symmetrical center” resides on one of theattice sites. The soliton component at the right side of thenterface is weaker than those at the left side. We havelso found off-site quadrupole-mode solitons with a sym-etrical center residing on the symmetrical center of the

our neighboring lattice sites [Fig. 1(f)]. The above analy-is demonstrates that a stronger saturable nonlinearityan guide soliton components with higher peaks whichonstitute one of our central results, that is, the highsymmetrical solitons can be induced by nonlinear lat-ices with different saturation parameters beside the op-osite sides of the interface.The general properties of dipole and quadrupole soli-

ons are presented in Fig. 2. The power of dipole solitonsesiding on the nearest and diagonal lattice sites is shownn Fig. 2(a). The power versus propagation constant is aonmonotonic function. This means that there are thresh-ld power values for both types of surface solitons. Dipolesan be found only when the power of the laser beam ex-eeds the critical value which is necessary for sustaininghe soliton solutions. Note that the power of both familiesf dipole solitons is very close and one can distinguishhem only for small propagation constants [inset plot inig. 2(a)]. The power of on-site quadrupoles is higher thanhat of the off-site ones when the propagation constant isot very small. The differences of power between the on-ite and off-site quadrupoles gradually increase with theropagation constant. The main reason is that the num-er of stronger components in on-site quadrupoles is threend in off-site quadrupoles the number is two.

ig. 1. (Color online) Field modulus for surface (a),(b) dipolesesiding on the nearest lattice sites; (c),(d) dipoles residing on theiagonal lattice sites; (e) on-site quadrupole; and (f) off-site quad-upole. b=0.2 in (a) and (c), and 7.0 in (b) and (d)–(f). Here and inhe following, Sr=0.08 for dipoles and 0.09 for quadrupoles andortices; Sl=0.1,wR=3 unless stated otherwise.

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L. Dong and H. Li Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1181

Although there are points of inflection on the powerurves, the instability of multipole surface solitons cannote predicted by the Vakhitov–Kolokolov (V–K) criterionince this criterion is valid only for fundamental solitons.hus, a comprehensive linear instability analysis basedn Eqs. (3) should be performed on the multipole mode so-utions. Perturbation growth rates of two families of sur-ace dipole solitons are displayed in Fig. 2(c). Surface di-oles will be stable when their power (or propagationonstant) exceeds a threshold value. Both families of sur-ace dipole solitons have wide stability domains. At small

ig. 3. (Color online) (a),(b),(e) Stable and (c),(d) unstable propac) and (d). z=512 in (a), (b), and (e); 0 in (c); and 90 in (d). Othe

ig. 2. (Color online) Properties of surface multipole solitons. (orrespond to the solitons shown in Figs. 1(a) and 1(f). Inset: poipole solitons, (d) off-site quadrupoles, and (e) on-site quadrupolearest lattice sites and off-site quadrupoles (dashed line).

ropagation constants, dipoles residing on the diagonalattice sites and on-site quadrupoles are more stable thanipoles residing on the nearest lattice sites and off-siteuadrupoles, respectively. Comparing the instabilitynalysis results [Figs. 2(c)–2(e)] with the power diagramhown in Figs. 2(a) and 2(b), one can find that the V–Kriterion fails to predict the instability of multipole soli-ons.

To elucidate the effects of the interface, we plot the dif-erences of peak values �peak of surface dipole and off-siteuadrupole solitons in Fig. 2(f). For both dipoles and qua-

of multipole solitons at b=6.84 in (a), (b), and (e), and b=1.2 inmeters are the same as in Fig. 1.

Power of dipole and quadrupole solitons versus b. Dashed linesrsus b near points of inflection. Perturbation growth rates of (c)he differences of peak value versus b for dipoles residing on the

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1182 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 L. Dong and H. Li

rupoles, �peak grows with the propagation constant. Thepeak of dipoles is larger than that of quadrupoles due tohe different �S�Sl−Sr� between dipoles and quadrupoles.lso it should be noted that the logarithmic scale of �peak

n Fig. 2(f) may lead to the misunderstanding of the dif-erence between two curves. In fact, the difference be-ween two curves gradually increases with the propaga-ion constant. We can see that the presence of an interfacean increase dramatically the difference of the peak val-es of multipole mode solitons residing on its oppositeides, especially for solitons with higher power.

All results discussed above are confirmed by the out-ome of the direct numerical integration of Eq. (1) withhe input condition of q�x ,y ,z=0�=wr�x ,y��1+��x ,y��,here ��x ,y� is a random noise function. Typical stablend unstable propagation examples of dipole and quadru-ole solitons are presented in Fig. 3. The surface quadru-ole soliton shown in Fig. 3(c) is unstable although nooise is added into it.Besides the surface multipole solitons, we also find sur-

ace vortex solitons in nonlinear lattices. Such solitonsave asymmetrical profiles and noncanonical phase struc-ures. Examples of vortex solitons and their phase distri-utions are shown in Fig. 4. Vortex solitons at lowerower exhibit strong asymmetry. The difference of solitoneaks at the left and right sides of the interface increasesith the propagation constant. While the phase structuref the surface vortex at lower power is similar to that ofhe canonical vortex [Fig. 4(d)], it gradually evolves to aoncanonical one with the increment of the propagationonstant [Fig. 4(e)]. One can see that the noncanonicalhase is a nonuniform stair-like structure which differsrom the canonical one in which the phase is a uniformncreasing function around the phase dislocation. A sur-rising result in the surface model is that only vorticesith highly nonuniform noncanonical phase structures

an be stable [Fig. 4(c)] which is different from the case of

ig. 4. (Color online) Field modulus for surface vortices at b= �aart corresponds to the unstable solutions. (d),(e) Noncanonical purface vortex soliton at b=7.4 and z=512.

he nonsurface model [18]. For example, a vortex solitonith a power of U=1104 is stable after a very long propa-ation distance [Fig. 4(f)].

To shed more light on the properties of surface solitons,e consider the case of a nonlinear lattice with a corner

nterface and different spacings between the lattice sites.umerical analysis reveals that the nonlinear lattice withcorner interface can still support solitons. An example of

ff-site quadrupole solitons in the nonlinear lattices with=0.09 in the third quadrant and 0.1 in the other quad-

nd (b) 7.0. (c) Power versus b for vortex solitons, where dashedstructures corresponding to (a) and (b). (f) Stable propagation of

ig. 5. (Color online) Field modulus for corner surface quadru-oles at (a) b=7.0,ws=3 and (b) b=5.7,ws=2.5. (c) Power and in-tability growth rate of corner surface solitons versus propaga-ion constant. (d) Differences of peak values between strong andeak soliton components at w =2.5 and 3.

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L. Dong and H. Li Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1183

ants is displayed in Fig. 5(a). There are three compo-ents with same absolute value of amplitude. The exis-ence and stability domains of quadrupoles expand withhe growth of the spacing between the nearest lattice sitesFig. 5(c)]. The differences between the weak and strongomponent peaks of quadrupole solitons at ws=2.5 and 3re displayed in Fig. 5(d).Finally, we expect that when the difference of satura-

ion parameters between the two sides of the interface isarger (e.g., �S=0.04), the asymmetry of multipole andortex solitons will be improved rapidly. At a propagationonstant near its cutoff value, the soliton components onhe side with a weak saturation parameter will vanish en-irely and the interface may lost its role in the influence ofhe soliton formation.

In summary, we investigate the existence, stability, andropagation dynamics of surface solitons supported by theurely nonlinear lattices with an interface. We reveal thaturface multipole solitons can be completely stable inide parameter windows. Stable surface vortex solitonsith strong asymmetry are also possible. Our results re-orted here enrich the concept of surface solitons, espe-ially for surface vortex solitons with noncanonical phasetructures.

CKNOWLEDGMENTS

his work is supported by the National Natural Scienceoundation of China (NSFC) (grant no. 10704067).

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