November 1, 2007 / Vol. 32, No. 21 / OPTICS LETTERS 3149

Spatial solitons in �„2… planar photonic crystals

Katia Gallo1,3 and Gaetano Assanto2,*1Optoelectronics Research Centre, University of Southampton, SO17 1BJ, United Kingdom

2Nonlinear Optics and Optoelectronics Lab (NooEL), INFN and CNISM, University “Roma Tre,”Via della Vasca Navale 84, 00146 Rome, Italy

3Present address: Nonlinear Optics and Optoelectronics Lab, University “Roma Tre”*Corresponding author: assanto@uniroma3.it

Received June 5, 2007; revised August 9, 2007; accepted September 24, 2007;posted September 28, 2007 (Doc. ID 83853); published October 23, 2007

We analyze light self-confinement induced by multiple nonlinear resonances in a two-dimensional ��2� pho-tonic crystal. With reference to second-harmonic generation in a hexagonal lattice, we show that the systemcan not only support two-color �1+1�D solitary waves with enhanced confinement and steering capabilitiesbut also enable novel features such as wavelength-dependent soliton routing. © 2007 Optical Society ofAmerica

OCIS codes: 190.4410, 190.5530, 230.4320.

Significant energy exchange and coherent interac-tions via three-wave mixing can ultimately result inthe mutual trapping and spatial locking of multifre-quency fields into a solitary beam or spatial soliton[1,2]. Bright two-color spatial solitons based onsecond-harmonic generation (SHG) were first demon-strated in bulk KTP [3] and LiNbO3 waveguides[4,5]; they were later reported in a variety of mediaand configurations, including one-dimensional (1D)quasi-phase-matching (QPM) grating structures inplanar waveguides [6–8].

Higher-dimensionality QPM structures, most nota-bly nonlinearly defined photonic crystals (NPCs) en-tailing a two-dimensional (2D) periodic modulation ofthe quadratic susceptibility ��2�, have attracted in-creasing interest since 1998 when they were first pro-posed [9]. Their unique capabilities in mapping mul-tiple parametric resonances in both space andwavelength have been investigated in the context ofmultifrequency generation [10], wavelength inter-change [11], and ��2� cascading [12]. NPCs are ex-pected to provide exciting scenarios in the soliton re-gime, thus far completely unexplored.

In this Letter we analyze multicolor spatial soli-tons in a planar 2D NPC, detailing the study in thecase of twin-beam SHG (TB-SHG), a configuration re-cently implemented in hexagonally poled LiNbO3(HexLN) waveguides [13]. We demonstrate numeri-cally that this system supports solitary waves withunique spectral and spatial features stemming fromthe interplay of two SHG resonances.

The results derived here apply to any 2D NPCwhere two equal-amplitude reciprocal lattice vectors(G1 and G2, see Fig. 1) are available for the simulta-neous QPM of two noncollinear SHG processes (TB-SHG) [13]. Figure 1 illustrates the interaction in thecase of HexLN, i.e., a 2D NPC in which the ��2� sus-ceptibility is periodically modulated in sign (+��2� or−��2�) according to a hexagonal pattern. The funda-mental frequency (FF) pump, propagating at a smallangle ���� with respect to the symmetry axis of thesystem ���, generates two distinct second-harmonic

��� �2�� �2��

(SH) beams. Denoting � , �1 , and �2 the FF

0146-9592/07/213149-3/$15.00 ©

and SH wave vectors in the planar geometry, the cor-responding SHG mismatches are ��1=�1

�2��−2����

−G1 and ��2=�2�2��−2����−G2, respectively. Such

mismatches have the same frequency dependence(determined by the dispersion of the ��2� medium),but their weight depends on the incidence angle �� ofthe FF pump. For symmetric TB-SHG with ��=0, itis ���1�����2�, and the two QPM processes occur atthe same input frequency �=�1=�2. Conversely,when ���0, then ���1�� ���2�, yielding two distinctQPM-SHG resonances �1��2.

To study the nonlinear wave dynamics in a 2D pla-nar NPC configuration, we model the evolution of theslowly varying envelopes for the FF pump �a� and thetwo SH beams (b1 and b2) with a set of three coupledequations:

�a

��+ i�0

�2a

��2 = − i�a*b1 − i�a*b2,

�b1

��+ 1

�b1

��+ i�1

�2b1

��2 + i�1b1 = − i�a2,

�b2

��+ 2

�b2

��+ i�2

�2b2

��2 + i�2b2 = − i�a2. �1�

The frame of reference is defined by the FF directionof propagation ��̂=���� / ������� and an orthogonal axis

Fig. 1. QPM diagram for twin-beam SHG via the recipro-cal lattice vectors G1 and G2 of a quadratic 2D NPC. QPMcan be obtained in distinct directions and for distinct FF

wavelengths.

2007 Optical Society of America

3150 OPTICS LETTERS / Vol. 32, No. 21 / November 1, 2007

��̂� �̂� in the plane of the NPC. The normalized lon-gitudinal and transverse coordinates are �=z /Ld and�=x /w0, respectively (with the diffraction length Ldand the FF input spot size w0). Model (1) accounts fornoncollinearity of the harmonic beams �1,2� as wellas diffraction of FF and SH (�0 and �1,2, respec-tively). In the realistic assumption of small �� anduniform NPC, equal parametric strengths �=�0 ·Ld · �P0 (with P0 the FF input power) are attrib-uted to the two SHG processes in Eqs. (1). Finally,the detunings �1,2= ���1,2��� ·Ld are the normalizedprojections of ��1,2 on �̂. We integrated numericallyEqs. (1) using a split-step beam propagationalgorithm and with reference to TB-SHG in a HexLNNPC [13], taking 1,2= 1,2 ·Ld /w0� ±1.5 ( 1,2 are theangles between �1,2

�2�� and ����),

�0 = Ld/�2 · ������ · w02� = 1/4,

�1,2 = Ld/�2 · ��1,2�2��� · w0

2� � 1/8.

The relative size (i.e., role) of �1 and �2 offers anadditional degree of freedom with respect to standardsingle-beam SHG. This is illustrated in Fig. 2 with2D maps displaying the normalized spot size [�=wout /w0, Fig. 2(a)] and the lateral deviation [��FF,Fig. 2(b)] of the output FF beam versus �1 and �2.In Fig. 2(a) the parameter regions where two-colorlight localization occurs are easily identified bydarker areas in the plane ��1 ,�2�: nondiffracting��=1� quadratic solitons (or simultons, as they en-compass two or more frequencies) form when eitherSHG process is close to phase matching near �1�0+ and �2�0+ [1], with enhanced self-confinementwhen �1��2�0. The �FF map is symmetric withrespect to �1=�2, the latter identifying a perfectlybalanced TB-SHG ���=0�. The ��FF map in Fig. 2(b),conversely, is antisymmetric with respect to the axis�1=�2 along which ��FF=0. When �1��2, de-pending on the dominant SHG process either positiveor negative FF-beam displacements are obtained(darker vertical and horizontal stripes for �1�0+

and �2�0+, respectively).The TB-SHG detunings �1 and �2 vary with FF

wavelength ���� and angle of propagation ����, with alinearized dependence �1,2��� ,���=�����±����� /2,where ����� is a linear wavelength-proportional con-tribution at ��=0 and ����� is an offset applyingwhen ���0 [13]. In actual experiments (e.g., inHexLN) one would normally operate at a fixed ��

while tuning ��, i.e., moving along diagonal lines inthe ��1 ,�2� diagrams. The dashed lines in Fig. 2 in-dicate the two trajectories, �1=�2 and �2=�1+10�, later considered in Figs. 3 and 4 for symmetricand asymmetric TB-SHG, respectively.

Figure 3 illustrates spatial solitons via symmetric(i.e., balanced) TB-SHG; in this case [Fig. 3(a)] thereis no lateral soliton displacement (i.e., ��FF=0), andthe �FF curve (solid) resembles conventional SB-SHG(dashed) [4]. After significant self-defocusing for �1

−

�0 , the output FF spot abruptly reduces for �1

�0+ until a simulton is generated ��FF=1�; then thebeam size progressively increases with �1 up to thelinear diffraction value [horizontal dotted line, Fig.3(a)]. A direct comparison with SB-SHG for the sameexcitation levels [dashed line in Fig. 3(a)], however,clearly shows that TB-SHG can localize light over abroader range of �. Figures 3(b) and 3(c) display theintensity evolution of the TB-SHG simulton resultingfrom the parametric locking of the FF pump [Fig.3(b)] with the two SH components [Fig. 3(c)] corre-sponding to point A ��FF=1� in Fig. 3(a). As expectedby a cascaded interaction [1–4], some harmonic lightradiates away in the early stages of propagationwhere dramatic upconversion and downconversionprocesses take place before the soliton is born.

The dependence of �FF and ��FF with detuning(s)changes dramatically in the case of asymmetric TB-SHG, as visible, e.g., in Fig. 4(a) for �2=�1+10�:parametric confinement occurs near points B ��2��� or C ��1�� /2�, where one or the other SHGprocess dominates. Correspondingly, the output beam

Fig. 2. (Color online) Contour maps of the FF output ver-sus mismatches �1 and �2, calculated for �=5 and �=6.(a) Normalized beam waist �FF=wout /w0 and (b) lateraldisplacement ��FF with respect to the input ��=0�. Thedashed lines indicate the trajectories corresponding to Figs.3 and 4.

is “pulled” and shifts (i.e., “walks” [5,14,15] nonlin-

1 2

(a) and (c) �1=� /2 ��1=10.5��, i.e., point C in (a).

November 1, 2007 / Vol. 32, No. 21 / OPTICS LETTERS 3151

early) towards either negative or positive �. Figures4(b) and 4(c) show the evolution of the simulton (FFcomponent) in the plane �� as it results in points Band C of Fig. 4(a). As �� is kept fixed and �1 is afunction of ��, two-color self-confinement into spatialsimultons is obtained in the 2D NPC with directionsof propagation that can be adjusted by simply vary-ing ��. Additional excitation-dependent effects areunder investigation and will be presented elsewhere.

In conclusion, we demonstrated that optical spatialsolitons can be obtained in a quadratic 2D NPC ex-hibiting two (or more) in-plane SHG resonances. In asymmetric configuration the results highlight an en-hanced spectral range for quadratic confinement oftwo-color solitons, whereas in asymmetric cases soli-tons can be steered by varying the input (FF) wave-length, hence enabling wavelength-angle demulti-plexing schemes. These findings unveil novel andrich scenarios for soliton processing by tailoring thenonlinear resonances in two-dimensional ��2� lattices.

K. Gallo gratefully acknowledges support from theLeverhulme Trust (Special Fellowship ESF/2004/0401) and INTAS (RG 03-51-6562). G. Aassantothanks the Italian MiUR for partial funding (PRIN200509833).

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Fig. 3. Symmetric TB-SHG for �1=�2. (a) Calculated FFbeam waist ��FF� and transverse displacement ���FF� ver-sus �1 for �=5, �=6 (solid lines). The dashed line refers toconventional single-beam SHG. Inset: FF beam propaga-tion at low power ��=0.01�. Beam propagation of (b) FF and(c) SH components for � =� =2�, �=5 [point A in (a)].

Fig. 4. Asymmetric TB-SHG for �2=�2+10�. (a) Nor-malized FF beam waist ��FF� and transverse displacement���FF� versus �1 for �=5, �=6. (b), (c) Beam (FF) intensitypropagation plots for (b) �1=−9� ��2=��, i.e., point B in

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