## Abstract

The propagation of a self-trapped laser beam in a planar waveguide that exhibits a Kerr nonlinearity and a normal chromatic dispersion is considered. We demonstrate experimentally for the first time to our knowledge that such a beam undergoes an undulation responsible for ultrafast transverse oscillations of its axis. This phenomenon, called “snake instability”, was predicted theoretically in 1973 by Zakharov and Rubenchik on the basis of a study of the soliton solutions of the hyperbolic nonlinear Schrödinger equation. The signature of this instability is observed in the spatially resolved temporal spectrum.

© 2008 Optical Society of America

Since the early sixties the phenomenon of laser beam self-trapping in dielectric media has been thoroughly studied for both its fundamental and applied interests [1]. From a theoretical point of view, self-trapped laser beams are described as spatial solitons, a terminology that refers to their robust nature and their particle-like behaviours. These solitons, which result from the balance between diffraction and a nonlinear self-focusing effect, have been observed in several materials [1]. Among these materials semiconductors play a particular role because their instantaneous and large optical Kerr nonlinearity put forward promising potential applications to ultrafast all-optical signal processing. However, theoretical studies taking into account the true dimensionality of the propagation problems (e.g., including time in the models) have shown that soliton laser beams are often unstable. Instabilities leading to wave collapse, beam filamentation, or modulation instabilities have been predicted [2–5] and observed [6–12]. In the present work we report on the so-called “snake instability” theoretically predicted more than 30 years ago [3]. This instability is responsible for the growth of ultrafast transverse undulations of the laser beam axis, in a way analogous to the motion of a snake.

In ideal loss-less Kerr media, in which the refractive index exhibits a linear variation with light intensity (*n*=*n*
_{0}+*n*
_{2}
*I*), spatial solitons with two transverse dimensions [(2+1)D] only exist at one value of the beam power and are therefore unstable against wave collapse, leading in practice to beam filamentation [12]. To the opposite, spatial solitons propagating in planar waveguides, i.e., (1+1)D solitons, have an amplitude that is inversely proportional to their width, which makes them naturally immune to wave collapse. However the canonical (1+1)D nonlinear Schrödinger (NLS) equation commonly used to describe these spatial solitons is inappropriate because it does not include the material chromatic dispersion that was shown to play a central role in the beam dynamics even in the continuous-wave (cw) regime [3]. Indeed, in the anomalous dispersion regime, time plays the same role as a transverse spatial dimension and the space-time coupling then leads to an instability that is analogous to the wave collapse of (2+1)D NLS spatial solitons. Specifically, cw soliton beams that propagate in the anomalous dispersion regime undergo the so-called “neck” instability characterized by a temporal periodic modulation of their intensity, leading to trains of collapsing pulses. The situation is far more intriguing in normally dispersive planar waveguides since time no longer plays the same role as an additional spatial transverse dimension. It was shown theoretically that the space-time coupling in this case leads to the snake instability and thus results in transverse undulations of the laser beam[3]. Anomalous dispersion (i.e., the neck-instability) corresponds to the elliptic (2+1)D NLS equation while normal dispersion (i.e., snake instability) corresponds to the hyperbolic (2+1)D NLS equation. The elliptic NLS equation also rules the propagation of bright and dark soliton stripes in bulk Kerr materials (2 transverse dimensions) and transverse instabilities have been experimentally observed in these systems [6, 7] as well as in similar configurations in Bose-Einstein condensates [13, 14]. The first experimental observation of the snake instability of the hyperbolic NLS bright soliton was reported in the field of hydrodynamics [15] while it has been later reported in optics, on spatially extended femtosecond pulses in semiconductor planar waveguides [8, 9]. The neck-type instability has also been recently observed on self-trapped beams in quadratic media (self-trapping due to wave-mixing processes) [10, 11]. Besides these experiments a large number of authors have reported the observations of spatial soliton beams in Kerr-type planar waveguides [1] but the issue of their stability has never been addressed from an experimental point of view.

In dispersive self-focusing planar waveguide, the slowly varying amplitude *A* of an optical wave obeys the (2+1)D NLS equation:

where the electric field is the real part of *A*exp[*i*(*kz*-*w*
_{0}
*t*)], *w*
_{0} being the central frequency of the laser beam and *k* its wavenumber. The notations here and below are as follows: *z* is the coordinate in the propagation direction of the laser beam, *x* is the in-plane transverse coordinate, *t* is the time, *β*
_{2} is the group velocity dispersion coefficient and *γ*>0 is the nonlinearity coefficient. The soliton solution of Eq.1
*A*=*A*
_{0}sech(*x*/*X*
_{0})exp(*i*ϕ_{NL}), where *ϕ*
_{NL}=*γA*
^{2}
_{0}
*z*/2 and *A*
^{2}
_{0}=1/(*γkX*
^{2}
_{0}), represents a shape-invariant monochromatic laser beam. Yet, in dispersive media (*β*
_{2}≠0), this solution is unstable, which is revealed by the exponential growth of perturbations of the forme *ε*(*x, z, t*)=*ε*
_{0}(*u*
_{1}+*iu*
_{2})exp(*iϕ _{NL}*) where

*u*

_{1,2}=[

*U*

_{1,2}(

*x*)exp(

*iΩt*+Γ

*z*)+

*c.c*.] [3, 16]. When the group velocity dispersion is normal (

*β*

_{2}>0), i.e. when the NLS equation is hyperbolic, the eigenmodes

*U*

_{1,2}(

*x*) are antisymmetric. The net effect of these unstable eigenmodes is therefore to shift the lateral position of the soliton, the direction of the lateral shift being reversed every half-period Δ

*t*=π/

*Ω*(see Fig. 1).

The experimental setup is shown in Fig. 2. The laser source is a picosecond mode-locked fiber laser (10MHz repetition rate, 5 ps full-width at half maximum [FWHM]) amplified by an Erbium/Ytterbium doped fiber amplifier. The output of the fiber amplifier is first collected by a ×16 microscope objective, then coupled into the waveguide with a 16.5*µ*m width (FWHM) by means of a cylindrical lens (f=200mm) and a ×60 microscope objective. The semiconducor planar waveguide is 12mm-long and is made up of a 1.6*µ*m-thick guiding layer of Al_{0.18}Ga_{0.82}As on the top of a 4*µ*m-thick cladding of Al_{0.24}Ga_{0.76}As. In order to avoid any back-reflection into the amplifier, a free space isolator is used. This isolator also enables us to set the polarization horizontally to excite the TE_{0} mode of the waveguide and to control the input power by means of a half-wave plate. At the output of the waveguide, the beam is collected by a ×60 microscope objective. Part of the light is used to image the output laser beam on an infrared vidicon camera. The remaining part of the beam is imaged by a lens (f=140mm) on the entrance of a home made two-dimensional spectrometer made up of a collimating lens (f=200mm), a grating (1200 groves/mm) and an imaging lens (f=80mm). The resolution of this spectrometer is 1.1 nm. The two-dimensional spectrum was recorded by the same infrared vidicon camera. The response time of this camera is in the millisecond range and determines the integration time of the experimental time-averaged spectra. The calibration of the spectrometer was performed by using a tunics source and an optical spectrum analyser.

The carrier frequency of the laser source, being slightly below the half band-gap, ensures a focusing optical Kerr nonlinearity [17] together with a normal group velocity dispersion [18]. At low power, the laser beam that has an input full width at half maximum (FWHM) of 16.5*µ*m broadens up to 190 *µ*m FWHM at the output of the waveguide, which corresponds to approximately 10 diffraction lengths. Figure 3 clearly shows that, as the power increases, the width of the output profile decreases, with the narrowest profile of 40 *µ*m FWHM occurring at the peak power of 780W. Note that the waveguide losses decrease the beam power during propagation, which explains why the optimal output width remains larger than the input width. However, losses do not prevent the snake instability from developing because the snake instability gain, Γ=22 dB/cm, is much larger than the measured loss coefficient *α*=3.8 dB/cm.

Obviously, the transverse undulations can not be directly observed as depicted in Fig. 1, firstly because the intensity profile can only be measured at the output of the waveguide and secondly because of the femtosecond time scale involved in the process. However, due to its spatiotemporal nature the instability can be identified through the spectral characteristics of the output beam. Figure 4 shows the input and output spectra. We observe the existence of two symmetric low amplitude sidebands in the input beam spectrum. These sidebands are generated by modulational instability within the laser source. Their maxima are located 24 nm apart from the central wavelength, which makes them close to the maxima of the instability gain spectrum that, according to theory, are located 27 nm apart from the spectrum centre. This perturbation in the input beam thus provides a natural seed for the snake instability. At high power we indeed clearly observe in Fig. 4 the amplification of the sidebands as well as the generation of their high-order harmonics. This spectral signature constitutes a genuine demonstration of the modulational instability of bright solitons of the hyperbolic NLS equation and confirms that space-time coupling leads to complex behaviours in nonlinear systems (let us recall that, in a (1+1)D system such as an optical fiber, continuous waves are stable when propagating in a normally dispersive focusing Kerr medium[19]).

However, the spectral signature shown in Fig.4 is not sufficient to determine whether the spatial soliton undergoes a transverse undulation or a simple modulation in time. To identify unquestionably the snake instability we have to analyse the spatial profile of the corresponding unstable mode. Because this mode is antisymmetric (see Fig.1), its intensity profile vanishes at the center of the soliton beam and it can therefore be easily identified. In practice we thus have to analyse the spatial profile of the amplified spectral sidebands in a spatially resolved spectrum. The spatially resolved spectrum provides the output spectrum for all values of the transverse coordinate x across the beam width. As shown in Fig. 5, the two sidebands located at 1512 nm and 1560 nm exhibit a dip at the centre of the beam. This result is in good agreement with theory [3, 16] and thus demonstrates that the laser beam undergoes a snake-type modulational instability.

For the sake of clarity, the spatial profiles at the sideband wavelength 1512 nm and at the central wavelength 1530 nm are plotted in Fig. 6. We see that the sideband profile is not totally antisymmetric, in particular, it exhibits a nonzero value at its centre. This apparent discrepancy with theory can be explained by the fact that the seed sidebands are generated by the laser source and their spatial shape is therefore not controllable. Due to beam defects and entrance face imperfections, the profile of the sideband seeds is obviously not perfectly symmetric. This is of major importance if one considers the theory proposed in Ref.[16] where it is shown that, besides antisymmetric unstable modes, symmetric unstable modes also exist in the hyperbolic NLS equation. Due to the lack of symmetry of the seed sidebands, one must expect that both the symmetric and antisymmetric modes are seeded and amplified. But the symmetric mode is less developed than the antisymmetric one because its gain is lower [16], it is thus only responsible for a small distortion of the expected ideal antisymmetric profile. This explains the asymmetry and the non-zero central intensity observed in the sideband spatial profiles.

The originality of our result is to provide unprecedented experimental demonstration of the effect of chromatic dispersion on the stability of self-trapped laser beams propagating in nonlinear optical Kerr media with one transverse dimension (planar waveguide structure). Specifically, we demonstrated that normal chromatic dispersion is responsible for ultrafast in-plane transverse undulations of bright spatial soliton beams, as was first theoretically predicted more than 30 years ago [3]. This demonstration has been performed through the analysis of the spatially resolved spectrum of the beam. Our analysis shows that space-time coupling plays a major role in the dynamics of light self-trapping processes, which is liable to have important consequences as regards their exploitation for practical applications. As our system is ruled by the canonical hyperbolic NLS equation, our results are relevant to other fields of nonlinear sciences such as hydrodynamics [16, 20] and plasma physics [21, 22].

The authors are grateful to R. Baets and D. Taillaert (INTEC, Ghent University) for providing the waveguide. This work was supported by the Fonds de la Recherche Fondamentale Collective under grant F 2.4513.06 and by the Belgian Science Policy office under Grant No. IAP6-10. S.-P. Gorza acknowledges the support of the Fonds de la Recherche Scientifique (F.R.S.-FNRS, Belgium).

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