2890 OPTICS LETTERS / Vol. 29, No. 24 / December 15, 2004

Observation of discrete gap solitons in binarywaveguide arrays

Roberto Morandotti

Institute National de la Recherche Scientifique, Université du Québec, Varennes, Québec J3X 1S2, Canada

Daniel Mandelik and Yaron Silberberg

Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

J. Stewart Aitchison

Department of Electrical and Computer Engineering, University of Toronto, Toronto M5S 3G4, Canada

Marc Sorel

Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow G12 8QQ, Scotland

Demetrios N. Christodoulides

School of Optics/Center for Research and Education in Optics and Lasers, University of Central Florida, Florida 32816-2700

Andrey A. Sukhorukov and Yuri S. Kivshar

Nonlinear Physics Centre and Centre for Ultra-high bandwidth Devices for Optical Systems, Research School of Physical Sciencesand Engineering, Australian National University, Canberra, ACT 0200, Australia

Received April 28, 2004

We report an experimental study of discrete gap solitons in binary arrays of optical waveguides. We observeself-focusing indicating soliton generation when the inclination angle of an input beam is slightly above theBragg angle and show that the propagation direction of the emerging gap soliton is inf luenced by the effectof interband momentum exchange. © 2004 Optical Society of America

OCIS codes: 190.4390, 190.4420.

Nonlinear periodic photonic structures such as arraysof optical waveguides have attracted a lot of interest be-cause of the unique ways they offer for controlling light.Periodic modulation of the refractive index breaks thetranslational invariance and produces an effective dis-creteness in a continuous system, opening up manynovel possibilities for manipulating light propagation,including light localization in the form of discrete op-tical solitons.1,2

Recently it was suggested that a novel type of dis-crete optical soliton, the so-called discrete gap soliton,can be generated in a binary waveguide array by asingle inclined beam3 or two input beams.4 Binaryarrays of optical waveguides are specially engineeredphotonic structures consisting of periodically alternat-ing wide and narrow waveguides [see the example inFig. 1(a)]. Discrete gap solitons in such structures canbe considered as a nontrivial generalization of the spa-tial gap solitons that were recently observed in wave-guide arrays5 and optically induced photonic lattices.6

Gap solitons in binary arrays are associated with fun-damental modes that are strongly confined in narrowwaveguides, whereas in the conventional arrays gapsolitons are based on radiation or higher-order boundmodes. In this Letter we report on the experimentalobservation of discrete gap solitons in binary arrays ofoptical waveguides and demonstrate the effect of inter-band momentum exchange on the soliton steering.

0146-9592/04/242890-03$15.00/0

We investigate spatial beam self-action and solitonformation in etched arrays of 5-mm-long AlGaAswaveguides with an effective refractive-index contrast0.0035.2 The binary arrays are made from wide(4-mm) and narrow (2.5-mm) waveguides with 4-mmedge-to-edge spacing, and accordingly the full periodis d � 14.5 mm, as illustrated in Fig. 1(a). The beampropagation in this structure can be described by anormalized nonlinear Schrödinger equation, i≠E�≠z 1

D≠2E�≠x2 1 v�x�E 1 jEj2E � 0, where E�x, z� is thenormalized envelope of the electric field; x and zare the transverse and the propagation coordinatesnormalized to the characteristic values xs � 1 mmand zs � 1 mm, respectively; D � zsl��4pn0xs

2� isthe beam diffraction coeff icient; n0 � 3.3947 is theaverage medium refractive index; l � 1.5 mm is thevacuum wavelength; v�x� � 2Dn�x�pn0�l; and Dn�x� isthe effective modulation of the optical refractive index.

In waveguide arrays the wave spectrum has abandgap structure with gaps separating bands of thecontinuous spectrum associated with the spatially ex-tended Bloch waves, Ek,n�x, z� � ck,n�x�exp�ikx 1 ibz�,where propagation constant b is proportional to thelongitudinal wave-vector component and n is theband number. In Fig. 1(b), we plot the Bloch-wavedispersion curves and mark two types of spectrumgap that are due to total internal ref lection and Braggscattering.

© 2004 Optical Society of America

December 15, 2004 / Vol. 29, No. 24 / OPTICS LETTERS 2891

Fig. 1. (a) Normalized refractive index in a binary wave-guide array (gray) and Bloch-wave prof iles at the top offirst (solid curve) and second (dashed curve) bands. (b),(c) Dispersion of Bloch waves and the Bloch-wave excita-tion coeff icients versus the transverse wave number, forthe first (solid curves) and second (dashed curves) bands.The Fourier spectrum of an input Gaussian beam witha � 1.2aB and xf � 20 mm is shown in (c) by the shad-ing. (d) Characteristic profile of a discrete gap soliton inthe binary array.

For a self-focusing nonlinearity, discrete gap solitonsappear near the upper edge of the second band4 andare localized at the narrow waveguides, as shown inFig. 1(d). Here, we investigate gap-soliton generationby a single Gaussian beam.3 In our numerical simu-lations the input profile is specified as E�z � 0� �E0 exp�22 ln�2�x2��xf 2� 1 i�a�aB �px�d�, where E0 isthe normalized peak amplitude, xf is the FWHM, anda is the beam inclination relative to Bragg angle aB .An input beam profile can be represented as a Fourierintegral of plane waves with transverse wave numbersk. In periodic lattices there exists a set of eigenmodesin the form of Bloch waves belonging to different bandsbut having the same wave number k. Accordingly, in-side a waveguide array each Fourier component of theinput beam excites a superposition of Bloch waves withmatching wave number k. The amplitudes of the ex-cited Bloch waves are proportional to the coefficientsCn�k� �

Rd0 ck,n�x���

Rd0 jck,n�x�j2, where n is the num-

ber of the band (see details in Ref. 7). In Fig. 1(c) weplot the absolute values of the excitation coefficients.

When the input beam is inclined slightly above theBragg angle, and additionally the beam width is largerthan the lattice period (xf . d), its spectrum fallsinside a section of the Brillouin zone where the ex-

citation of the second band becomes dominant sincejC2�k�j . jC1�k�j [see Fig. 1(c)]. This allows the for-mation of a gap soliton associated with the second-bandcomponents, the propagation direction of which followsthe inclination of the input beam. Bloch waves are si-multaneously excited in the first band; however, theypropagate in the opposite direction and naturally tendto separate from the forming gap soliton. We notethat all Bloch-wave components may become trappedtogether into a multiband breather if a normally inci-dent narrow beam is used.8

Fig. 2. Experimentally measured output intensity distri-bution versus the inclination angle of the input Gaussianbeam with xf � 40 mm FWHM. The thin lines indicatethe positions of narrow waveguides.

Fig. 3. Experimentally measured intensity distributionat the output versus the average input power for incidentangles (a) 1 and (b) 2 in Fig. 2 and xf � 20 mm beamFWHM. 1 mW of average power corresponds to 200 W ofpeak power.

2892 OPTICS LETTERS / Vol. 29, No. 24 / December 15, 2004

Fig. 4. Top, numerically calculated output field intensityversus peak input intensity (I0) for a beam inclination20% above the Bragg angle, corresponding to Fig. 3(b).(a)–(d) Beam propagation along the binary waveguidearray for different input intensities: (a) diffraction in thelinear regime (I0 � 0); (b) formation of a moving discretegap soliton at I0 � 0.5; (c), (d) excitation of slow or sta-tionary gap solitons at I0 � 0.6 and I0 � 0.7. The dashedlines mark the boundary of the experimental sample.

In experiments, first we studied the linear beampropagation (see Ref. 2 for a description of the experi-mental setup). The dependence of the output f ielddistribution on the input angle is shown in Fig. 2.We note that, in a binary waveguide array, the Blochwaves corresponding to the f irst and second bands canbe easily distinguished, since they are primarily local-ized on the wide and narrow waveguides, respectively.The band 2 excitation becomes dominant above theBragg angle, in agreement with the theoretical predic-tions (higher bands are excited as the angle is furtherincreased toward 2aB).

Next we analyze the nonlinear self-action of beamswith inclination angles below and above the Braggangle, as marked in Fig. 2. In the former case theBloch waves associated with the bottom of band 1and experiencing anomalous diffraction are primarilyexcited. This results in beam self-defocusing as non-linearity grows.9 Indeed, we observed broadening of

the output beam with an increase in laser power [seeFig. 3(a)]. At even higher powers, self-focusing of theband 2 components becomes visible.

Strong self-focusing of the band 2 componentsthat are localized at narrow waveguides, indicatingformation of a gap soliton, is observed when theinput beam inclination angle is above the Braggangle [see Fig. 3(b)], whereas band 1 Bloch wavesare much weaker compared with the previous case[Fig. 3(a)]. Additionally, the output position of theband 2 beam fraction depends on the input power.Numerical results for beam propagation (Fig. 4,top) represent the key features observed in the ex-periment. Intensity I0 is in gigawatts per squarecentimeter, assuming the value of effective nonlinearsusceptibility n2 � 2 3 1024 cm2�GW. In the linearregime [Fig. 4(a)], band 2 modes localized at narrowwaveguides move to the right, whereas band 1 modespropagate to the left of the input beam position. Agap soliton forms from the band 2 modes at largerpowers [Figs. 4(b)–4(d)]; the soliton velocity and, ac-cordingly, its output position can be modified alreadyat the initial stage, where it experiences dragging bythe band 1 modes. Additionally, at high powers thesoliton becomes strongly localized [Fig. 4(d)], and itsmotion across the array can become inhibited, simi-larly to discrete solitons in homogeneous waveguidearrays as a result of a self-induced Peierls–Nabarropotential barrier.10

In conclusion, we have generated discrete gap soli-tons in binary waveguide arrays and discussed theeffect of the interband momentum exchange on thepower-dependent soliton steering.

A. A. Sukhorukov’s e-mail address is ans124@rsphysse.anu.edu.au.

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