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M. I. Molina and Y. S. Kivshar Vol. 26, No. 8 /August 2009 /J. Opt. Soc. Am. B 1545

Two-color surface solitons in two-dimensionalquadratic photonic lattices

Mario I. Molina1,* and Yuri S. Kivshar2

1Departmento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile2Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University,

Canberra ACT 0200, Australia*Corresponding author: mmolina@uchile.cl

Received May 15, 2009; revised May 29, 2009; accepted June 1, 2009;posted June 9, 2009 (Doc. ID 111411); published July 8, 2009

We study two-color surface solitons in two-dimensional photonic lattices with quadratic nonlinear response. Wedemonstrate that such parametrically coupled optical localized modes can exist in the corners or at the edgesof a square photonic lattice, and we analyze the impact of the phase mismatch on their properties, stability,and the threshold power for their generation. © 2009 Optical Society of America

OCIS codes: 190.4350, 190.4410, 190.4420, 190.5530, 190.5940.

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. INTRODUCTIONwo-dimensional surface solitons have been predicted toxist recently as novel types of discrete solitons localizedn the corners or at the edges of two-dimensional photonicattices [1–3]. These theoretical predictions were followedy the experimental observation of two-dimensional sur-ace solitons in optically induced photonic lattices [4] andaveguide arrays laser written in fused silica [5]. Impor-

antly, these two-dimensional surface solitons demon-trate novel features in comparison with their counter-arts in truncated one-dimensional waveguide arrays. Inarticular, in a sharp contrast to one-dimensional surfaceolitons, the threshold power of two-dimensional solitonss lower at the surface than in the bulk, making the modexcitation easier [2].

Surface solitons are usually considered for cubic oraturable nonlinear media. However, multicolor discreteolitons in quadratically nonlinear lattices have beentudied theoretically in both one- and two-dimensionalattices [6–9] regardless of the surface localization effects.nly Siviloglou et al. [10] studied discrete quadratic sur-

ace solitons experimentally in periodically poled lithiumiobate waveguide arrays, and they employed a discreteodel with decoupled waveguides at the second harmon-

cs to model some of the effects observed experimentally.A more elaborated theory of one-dimensional surface

olitons in truncated quadratically nonlinear photonic lat-ices, the so-called two-color surface lattice solitons, haseen developed recently by Xu and Kivshar [11], who ana-yzed the impact of the phase mismatch on the existencend stability of nonlinear parametrically coupled surfaceodes and also found novel classes of one-dimensional,

wo-color twisted surface solitons that are stable in aarge domain of their existence.

In this paper, we extend the analysis of two-color sur-ace solitons to the case of two-dimensional photonic lat-ices. We study, for the first time to our knowledge, two-olor surface solitons in two-dimensional square photonic

0740-3224/09/081545-4/$15.00 © 2

attices with quadratic nonlinear response. We analyzehe effect of mismatch on the existence, stability, and gen-ration of surface solitons located in the corners or at thedges of the nonlinear lattice.

. DISCRETE MODELe consider the propagation of light in a two-dimensional

hotonic lattice of a finite extent imprinted in a quadraticonlinear medium, which involves the interaction be-ween the fundamental frequency (FF) and second-armonic (SH) waves. Light propagation is described byhe following coupled nonlinear discrete equations [6,11]:

idun,m

dz+ Cu�2un,m + 2�un,m

* vn,m exp�+ i�z� = 0,

idvn,m

dz+ Cv�2vn,m + �un,m

2 exp�− i�z� = 0, �1�

here un,m and vn,m are the normalized amplitudes of theF and SH waves, respectively; Cu and Cv are the cou-ling coefficients; � characterizes the second-order nonlin-arity; and � is the effective mismatch between two har-onics. The second-order difference operator �2 is defined

s

�2un,m = �i=±1

�un+i,m + un,m+i�.

quations (1) conserve the total power,

P = Pu + Pv = �n,m

��un,m�2 + 2�vn,m�2�. �2�

. STATIONARY STATESe look for stationary two-mode solutions of Eqs. (1)

n the forms u �z�=U exp�i�z� and v �z�

n,m n,m n,m

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1546 J. Opt. Soc. Am. B/Vol. 26, No. 8 /August 2009 M. I. Molina and Y. S. Kivshar

Vn,m exp�2i�z− i�z�, and we obtain the nonlinear alge-raic equations for the (real) mode amplitudes, Un,m andn,m,

− �Un,m + Cu�2Un,m + 2�Un,mVn,m = 0,

− 2�Vn,m + Cv�2Vn,m + �Vn,m + �Un,m2 = 0, �3�

here for the �N�M� lattice we have n=0,1, . . . ,N, m0,1, . . . ,M, and Un,m=0 and Vn,m=0 if either n�0, or�0, or n�N, or m�M.Equations (3) possess symmetry properties. For in-

tance, the transformation �→−�, Vn,m→−Vn,m leavesqs. (3) invariant.In the anti-continuum limit, i.e., when the couplings in

he lattice vanish, Eqs. (3) imply Vn,m=� /2� and Un,m2

�� /2���2�−�� /�. Thus, if ��0, then 2���. If ��0, then���. Also, localized modes should exist outside the lin-ar spectrum band, ����4Cu. Thus, in the propagationonstant-mismatch space, the region where localizedodes exist is bounded by ��4Cu, ��� /2, or ��−4Cu,�� /2. Also in this limit, the propagation constant is pro-ortional to P2 rather than P, as in the case of the cubiconlinearity.We consider a square lattice of a finite extent with N

M=21 and look for two-dimensional localized modes inhe corner (1,1) and at the edge (11, 1) of the lattice. Wese two sets of the coupling parameters: (i) strong cou-ling, Cu=1 and Cv=0.5, which is similar to the paramet-ic processes in bulk media; and (b) no coupling for theecond harmonic, Cu=1 and Cv=0, similar to the fabri-ated structures employed in the recent experiments [10].or given values of the coupling parameters �Cu ,Cv�, non-

inear parameter �, and mismatch �, we use a standardumerical procedure of continuation from the anticon-inuum limit. Examples of the nonlinear corner and edgeonlinear localized states are shown in Fig. 1, where theelds propagating along the waveguides are presented as

superposition of the waveguide modes, U�x ,y��n,mUn,m��x−n ,y−m�, where ��x ,y� is the (single-ode) guide centered on site �n ,m� and similarly for�x ,y�. In Figs. 1 and 4, we use ��x ,y�=exp�−x2−y2� /2,ith 2=0.05.We focus on the analysis of the power dependencies

haracterizing the families of two-dimensional localizedodes. The typical dependencies of the total power (2) on

he propagation constant � are shown in Fig. 2 for severalalues of the mismatch parameter � marked on the plots.he power curve has a typical minimum corresponding tohe minimum (threshold) power for the localized mode toxist. In the region where the slope of the power P be-

ig. 1. (Color online) Examples of the corner (left) and edgeright) localized modes in two-dimensional quadratic lattice.hown are the amplitudes of the FF components for Cu=1, Cv0.5, �=0, and �=5.

omes negative, the localized modes are unstable, as ob-erved for other types of surface modes and also verifiedere by numerical analysis. These results agree qualita-ively with those for the 1D case [11] (the reader shouldote that our current definition of � is the opposite of thatsed in [11]).Figure 3 shows the dependence of the minimum power

min required to create a stable localized mode versusismatch �, either in the corner or at the edge of the lat-

ice. In general, the results for the corner and edge modesre rather similar. Clearly, the value of the mismatch de-ermines the threshold power needed to generate a local-zed state, with a minimum value observed for positive �arger than �=5. The minimum power depends also uponhe value of the coupling constants. However, the main re-ult is that for a large range of mismatch parameter �8, the corner mode requires less power than the edgeode; this is somewhat similar to the case of the cubiconlinear lattice, both for spatial surface solitons [1–3]nd spatiotemporal surface solitons [12].Using the same numerical approach and starting from

he anti-continuum limit, we find several other families ofwo-dimensional localized modes located in a close vicin-ty of the lattice corners and edges. These modes provide awo-dimensional generalization of the surface modesnown for one-dimensional lattices placed at different dis-ances from the edge and corresponding to a crossover be-ween the surface and bulk discrete solitons as discussedarlier [13]. In addition, we find novel classes of the so-alled two-dimensional twisted modes; an example of onef such modes is shown in Fig. 4 for Cu=1, Cv=0.5, and=5.

ig. 2. (Color online) Total power versus propagation constantf the two-dimensional modes localized in the corner (left) and athe edge (right) of the lattice for Cu=1, Cv=0.5, and for three dif-erent values of the mismatch parameter �.

ig. 3. (Color online) Minimum power versus mismatch for theorner (circle) and edge (square) modes for (a) Cu=1, Cv=0.5 andb) Cu=1, Cv=0. For most values of the mismatch parameter, theorner mode requires less power than the edge mode.

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M. I. Molina and Y. S. Kivshar Vol. 26, No. 8 /August 2009 /J. Opt. Soc. Am. B 1547

. DYNAMICAL GENERATIONo study the generation of these two-color surface modes,e launch a tight beam at one site, un,m�0�U0n,m0

m,m0and vn,m�0�=V0n,n0

m,m0, either in the cor-

er or at the edge of the lattice. We let the system evolverom z=0 up to z=zmax and trace the shape of the beameasuring the partial powers Pu and Pv for both FF andH components, respectively. In general, the results areualitatively similar for the two initial beam positions (inhe corner and at the edge). The only difference is in thectual value of the field amplitude at which the beam self-rapping occurs. On the other hand, significant differ-nces are observed between two sets of the coupling pa-ameters and also whether both fields or only the FF fields initially present.

In general, when both fields are initially excited andhe power is strong enough and in the absence of a mis-atch, a localized surface mode is readily formed. When

he initial FF and SH powers are similar, the FF field cap-ures some of the SH power, and as the evolution unfolds,here is a fast-rate power exchange between them [seeig. 5(a)]. Most of the power exchange occurs at the cor-er site, and as one component increases (decreases) itsower content, its spatial profile width decreases (in-reases). This “breathing” behavior of the power contentf each component is accompanied by very little radiation.resence of a mismatch tends to destroy the localization,ut it is restored if the initial power is high enough. Whenhe SH field is initially absent, we observe that it isarder to excite a localized mode. When it is achieved,

ig. 4. (Color online) Example of a two-dimensional twistedode at the lattice corner. Shown are the amplitudes of the FF

left) and SH (right) fields for Cu=1, Cv=0.5, �=5, and �=0.

ig. 5. (Color online) Excitation of two-color surface modes athe lattice corner. Shown are the dynamics of the intensity at theorner site for FF (solid) and SH (dashed) components for (a, b)u=1, Cv=0.5; �=0 with (a) U0=V0=5 and (b) U0=20, V0=0; and

c, d) Cu=1, Cv=0, with (c) U0=5, V0=0, �=0, and U0=20, V00, �=−5.

oth mode components always seem to oscillate betweenero and a maximum amplitude at a fast rate [see Fig.(b)]. When the SH coupling is absent �Cv=0� and thearmonics are matched ��=0�, it can be proven that theesulting equation for the FF field is transformed into aiscrete equation with a nonlinear loss term:

i�d/dz�un,m + Cu�2un,m + 2i�2un,m* �

0

z

un,m2 �s�ds = 0. �4�

his induces a quick power loss in the FF field and itsapid transfer to the SH field. The numerical evolutionhows that at the outset a large percentage of the FFower is quickly transferred to the SH component [seeig. 5(c)], while the remaining FF fraction starts movingway from the corner site in the diagonal direction. Themplitude of this small mobile FF soliton decreases as itoves and its power is transferred to the SH component,hich grows a small “tail” in the diagonal direction. Afterwhile, when the power content of the FF is really small,

he exchange mechanism breaks down, and the remain-ng power is radiated away. Finally, the addition of mis-

atch and a high enough initial FF field restores self-rapping for both fields, where both fields oscillate at aast rate [see Fig. 5(d)].

In conclusion, we have studied localization of light inwo-dimensional quadratically nonlinear photonic latticesnd determined the conditions for the existence of two-olor surface states localized in the corners or at the edgesf the lattice. We have analyzed the impact of the phaseismatch on the properties and stability of two-

imensional localized modes, as well as the thresholdower for their generation.

This work was supported by Fondecyt Grant 1080374nd by the Australian Research Council.

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