3480 OPTICS LETTERS / Vol. 31, No. 23 / December 1, 2006

Generation of multicolor vector Kerr solitons bycross-phase modulation, four-wave mixing,

and stimulated Raman scattering

Gil Fanjoux, Jérémy Michaud, Michaël Delqué, Hervé Maillotte, and Thibaut SylvestreDépartement d’Optique PM Duffieux, Institut FEMTO-ST, UMR CNRS—Université de Franche-Comté no. 6174, Route

de Gray, 25030 Besançon Cedex, France

Received June 15, 2006; revised August 28, 2006; accepted September 7, 2006;posted September 12, 2006 (Doc. ID 71902); published November 9, 2006

We numerically and experimentally show the existence of multicolor vector spatial solitons in a Kerr planarwaveguide through the combined effects of cross-phase modulation, four-wave mixing, and stimulated Ra-man scattering. Mutual spatial guiding of the Raman–Stokes, anti-Stokes, and pump waves is achieved inthe high-conversion regime mainly by cross-phase modulation and phase-matched four-wave mixing inducedby a power imbalance between Stokes and anti-Stokes components, leading to the generation of a clear-cutsech-shape three-frequency spatial soliton. © 2006 Optical Society of America

OCIS codes: 190.0190, 190.5530, 190.3270, 190.5650, 190.5890, 190.4380.

Among the large family of optical solitons, vector soli-tary waves are currently the subject of many studiesowing to the additional appeal of their multicompo-nent nature. The existence of such entities relies on abalanced interaction of several field components (dif-ferent frequencies, polarizations, or modes) in such away that shape-preserving propagation is allowed.Each field component is mutually guided by the oth-ers but could not have a solitonic propagation alone.An example of multicomponent vector solitons is themulticolor soliton (MS). Until now, this soliton hasbeen observed to a great extent in quadratic media.1

The coupling processes are fundamentally differentthan those occurring in Kerr media and are based onthe focusinglike phase-front curvature due to the cas-cading process and the beam narrowing due to theparametric gain.2 A multifrequency spatial solitonhas also been reported in a Raman gas mediumstrongly driven near maximum molecular coherenceby two lasers slightly detuned from the Ramanresonance.3 However, in cubic transparent media, thegeneration of the MS has not yet been fullyaddressed.4–7 From a theoretical viewpoint, it wasfirst predicted that the MS could exist in focusingKerr media, assuming only cross-phase modulation(XPM) as a coupling process.4 Now, if the phase mis-match between the interacting fields is low, or, moreinterestingly, when powers are sufficient for produc-ing a nonlinear phase change that can counterbal-ance the linear phase mismatch, coherent couplingby four-wave mixing (FWM) must be considered. Byconsidering self-phase modulation (SPM), XPM, andFWM, Chen and Atai5 later showed that the MS in-volving three-frequency trapped beams can exist be-low a critical frequency shift for which the threewaves are either phase matched or highly phase mis-matched. Another recent work has shown that, abovea threshold power, FWM leads to the instability oftwo-color solitons, and they appear as a three-colorsoliton for which the parametric coupling is exactlybalanced by SPM and XPM.6

In this Letter, we present numerical and experi-

mental results that show the spontaneous generation

0146-9592/06/233480-3/$15.00 ©

of a three-frequency spatial soliton in a Kerr CS2 pla-nar isotropic waveguide when stimulated Ramanscattering (SRS) enters into play. Mutual spatialguiding of the Raman–Stokes, anti-Stokes, and pumpwaves is achieved in the strong-conversion SRS re-gime mainly by XPM and phase-matched FWM in-duced by power imbalance, leading to a clear-cutsech-shape three-frequency spatial Raman soliton.

Let us first consider the following three couplednonlinear Schrödinger scalar equations in the �1+1�D configuration for the pump �p�, the first-orderRaman–Stokes �+�, and the anti-Stokes �−� waves:

�Ep

�z=

i

2�p

�2Ep

�x2 + i�p�Pp + 2P+ + 2P−�Ep

+ 2i�pEp*E+E−e−i��z − g+P+Ep + g−P−Ep, �1�

�E±

�z=

i

2�±

�2E±

�x2 + i�±�P± + 2Pp + 2P��E±

+ i�±E�* Ep

2ei��z ± g±PpE±, �2�

Fig. 1. Numerical simulation of the generation of a three-frequency spatial soliton. Bottom: contour plots of (a) thepump, (b) Stokes, (c) anti-Stokes, and (d) whole beam. Top,transverse beam profiles at the waveguide input (thin solidcurve), output (bold solid curve), and after a 2 cm propaga-tion (dashed curve). The parameters are input width

=40 �m, Psat=0.02 W/m, gr=23.2 cm/MW. Pp=1.6Ps.

2006 Optical Society of America

December 1, 2006 / Vol. 31, No. 23 / OPTICS LETTERS 3481

where Ej�x ,z� is the amplitude of the field j �j=p , + ,−�, Pj= �Ej�2 is defined in watts per square meter, �j=2�n2 /�j is the nonlinear parameter where n2 (n2=+3.5�10−18 m2/W for CS2) is the nonlinear coeffi-cient, and �j is the wavelength of the jth field. ��=2�p−�+−�−=�22 is the linear phase mismatch dueto the chromatic dispersion (assuming single-modeand isotropic waveguide) with �j as the propagationconstant, �2 as the group-velocity dispersion of theCS2, and = �p−+� as the Raman shift. x is the freetransverse spatial dimension. The Raman couplingprocess is included in the form of a gain (loss) for theStokes (anti-Stokes) components. To take the Stokespower saturation due to the Raman cascading intoaccount, the gain for the Stokes component is definedas g+=gr / �1+P /Psat�, while g−=gr with gr as the Ra-man gain factor,8 P as the power integrated along x(watts per meter) of the Stokes component, and Psatas an adjustable parameter. Note that our model doesnot intrinsically take into account the wave mixingprocess with the higher Raman orders that shouldalso lead to the generation of a multifrequency soli-ton.

Numerical simulations of Eqs. (1) and (2) allow oneto clearly identify the underlying nonlinear mecha-nisms for MS generation. Figures 1(a)–1(d) show con-tour plots of the longitudinal evolution of the spatialprofiles of the pump, Stokes, anti-Stokes, and wholebeam, respectively. For this simulation, we assumedan input noise level for the Stokes and anti-Stokescomponents corresponding to the quantum noise. Ascan be seen in Fig. 1(a), the input Gaussian pumpbeam first undergoes self-focusing as the input poweris greater than the scalar fundamental soliton powerPs. Figures 1(b) and 1(c) show the efficient generationof the Stokes and anti-Stokes components beyond2 cm propagation and spatially trapped in the pumpbeam. Thereafter, it is clear that, despite the factthat the three components have varying energies, thetotal beam exhibits a stationary propagation asshown in Fig. 1(d). This MS generation is confirmedby the transverse beam profiles plotted at the top ofFigs. 1(a)–1(d) that show, at the waveguide output,clear reshaping of the input beam into a sech-shapesoliton as for both Raman components. We have veri-fied that for a longer propagation distance the MSstill propagates in a stable condition. It can be under-stood as being a composite soliton with variable com-ponents mutually trapped through cross-phase

Fig. 2. Evolution along the waveguide of (a) powers and(b) FWHM of the pump (bold dashed curve), Raman–Stokes(bold solid curve), Raman–anti-Stokes (thin dashed curve)components, and total output beam (gray curve).

modulation. Figures 2(a) and 2(b) show the inte-

grated power P and the full width at half-maximum(FWHM) of the three components in the function of z,respectively. We can distinguish two steps of the low-and high-conversion regime separated by a dashedline at 2 cm. The first one is characterized by an ex-ponential growth of the Stokes and anti-Stokes com-ponents with a large power imbalance ��20 dB�. In-deed, in spite of a linear phase mismatch between theStokes and anti-Stokes waves due to chromatic dis-persion, FWM occurs because the phase matching isself-induced due to the SRS-induced power imbal-ance between the two sidebands.9 Even if the anti-Stokes is first attenuated owing to the Raman pro-cess, this component subsequently follows theevolution of the Stokes wave through parametricamplification.10 This low-conversion regime corre-sponds to the guiding by XPM of the weak Ramancomponents in the waveguide generated by thepump.11 On the other hand, in the strong-conversionregime, the Raman and pump powers become of thesame order such that they are mutually guided andform a MS. This is confirmed by Fig. 2(b) because thepump and Stokes components exhibit the sameFWHM at the waveguide output, whereas the anti-Stokes have a lower FWHM due to a lower power.Note that the energy transfer through SRS from thepump to the Raman–Stokes beam annihilates the for-mation of a higher-order soliton for the pump andleads to the stabilization of the width of the pumpKerr soliton [Fig. 2(b)].

The experimental setup, depicted in Fig. 3, is madeup of a 3 cm long Kerr planar waveguide made up ofa thin 15 �m liquid CS2 layer sandwiched betweentwo SK5 glass blocks. As a pump laser, we use afrequency-doubled Nd:YAG microchip laser with a7 kHz repetition rate and 600 ps pulse duration. InCS2, the Raman frequency shift and linewidth are20 THz and 15 GHz, respectively.8 To measure thetransverse profiles of the pump at 532 nm and theStokes at 551 nm separately, we use a double-gratingsystem (2400 lines/mm) and a CCD camera, while thewhole beam is analyzed by using the zero order of thefirst grating. Note that the anti-Stokes component at513 nm has very low power and has not been ana-lyzed.

Figures 4(a) and 4(b) show the experimental valuesof the power and the FWHM of the pump and thefirst-order Raman–Stokes beams at the waveguideoutput for increasing input power. The numericalpredictions (dashed curves) are in good agreementwith the experimental measurements. In a similarmanner as for Figs. 2(a) and 2(b), two regimes of low-

Fig. 3. Experimental setup.

3482 OPTICS LETTERS / Vol. 31, No. 23 / December 1, 2006

and strong-conversion are clearly distinguishable.MS is generated as soon as its components have simi-lar profiles therefore corresponding to a balanced mu-tual XPM regime. XPM is then the predominant ef-fect in the shape preservation and also in the MSgeneration. Let us note that the MS does not have anintrinsically stable equilibrium because of the SRS-and FWM-induced power transfer (coherent cou-pling) between the three components of solitons, asobserved in Figs. 2(a) and 4(a). However, the wholeentity, for which the total energy is conserved, pre-serves its spatial shape and therefore propagates as amulticolor soliton.

Figures 5(a)–5(d) show the experimental intensityprofiles for the pump in the linear and nonlinear re-gimes, the whole MS and the Stokes beam, respec-tively. As can be deduced, the total Gaussian input

Fig. 4. Experimental (a) output power and (b) FWHM ver-sus input power for the pump ��� and the Stokes �•� waves.Dashed curves, numerical results. The vertical dotted linesindicate the fundamental soliton mean power �0.8 mW� forwhich scalar soliton behavior has been experimentallyobserved.

Fig. 5. Black solid curves, experimental intensity profilesof (a) the output beam in linear regime, (b) the pump, (c)the total, and (d) Stokes output beams in the nonlinear re-gime. The dashed curves correspond to (a) a Gaussian and(b)–(d) to a sech-shape profile, while the dotted–dashedcurve in (a) represents the experimental input profile.

beam exhibits diffraction in the linear regime, while

a clean sech-shape MS is generated in the strong-conversion regime. The pump [Fig. 5(b)] and Stokes[Fig. 5(d)] components then have mainly the samepower and FWHM. Note that the numerical resultsfit the experimental output profiles for the pump andwhole beams very well [Figs. 5(b) and 5(c)] only in thecentral parts, where beam cleaning due to the spatialsoliton effect occurs. More importantly, Fig. 5(d) re-veals that the Stokes component of the MS has aclean-cut sech-shape profile and is totally smooth.This is because the Stokes beam is only generatedwhen the pump beam attains a solitonlike regimeand does not suffer from noisy nonsolitonic parts.This property can be very useful for a beam cleanupapplication.12

To conclude, we presented numerical and experi-mental results showing the generation in a Kerr pla-nar waveguide of a multicolor vector Raman solitoncomposed of three frequencies (pump, Raman–Stokes, and anti-Stokes beams). Moreover, this workopens new potential applications for simultaneouswavelength conversion and beam cleanup.

This work has been supported by the MinistèreDélégué à la Recherche. The authors thank V. Coud-erc for helpful discussions. G. Fanjoux’s e-mail ad-dress is gil.fanjoux@univ-fcomte.fr.

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