Diffraction-free Propagation Research Articles

Symmetry-breaking instabilities of spatial parametric solitons

For three decades optical spatial solitons confined in the transverse plane were commonly believed to be a prerogative of media with cubiclike nonlinearities @1#. A remarkable exception is the work carried out in Ref. @2#, where the possibility to achieve diffraction-free propagation via threephoton interactions in quadratic media ~henceforth, parametric solitons! was first pointed out. The field of quadratic solitons has acquired importance only recently @3#, also stimulated by experiments in second-harmonic generation ~SHG! in bulk media ~211 dimensions! and planar waveguides ~111 dimensions! @4#. Since the parametric solitons are strictly speaking solitary waves ~the model equations are not integrable!, a crucial issue is their stability. Two main types of instabilities can be distinguished: ~i! longitudinal instability against perturbations that share the soliton symmetry @5#; ~ii! symmetrybreaking instabilities ~reminiscent of modulational instabilities of plane-waves @6#!, that take place whenever the solitons are embedded in a higher dimensional ‘‘space’’ with respect to the subspace in which they are localized @7,8#. For the former type of problem, stability criteria have been recently developed @5#, through asymptotic techniques @9#: both ~111!and ~211!-dimensional parametric solitons are stable in the largest portion of their existence domain in the parameter space. Moreover, the global stability ~no collapse! of ~211!and ~311!-dimensional parametric solitons and bullets is supported by the Liapunov-type stability analysis @10#. Conversely, the symmetry breaking of parametric solitons is still an open issue, even though the problem has been widely studied for cubic media @7#. Here we investigate the stability of the whole one-parameter families of ground-state planar SHG solitons. We anticipate that the development of the instability leads either to the formation of lattices of higher dimensional solitons, or to the complete disintegration ~radiative decay! of the soliton. The transverse instability of soliton stripes belongs to the former case, whereas the dynamics of temporal instabilities of both ~111!and ~211!dimensional solitons depends on the dispersive regime. Our results are of great importance for recent experiments in transverse pattern formation occurring via SHG @11,12#. In particular, the filamentation of beams with strongly elliptical cross sections ~i.e., pseudostripe! was already observed @11#, using nonsoliton input conditions ~i.e., SHG from the funda-

A Linearized Approach for Diffraction-free Propagation in a Planar Nonlinear Waveguide

Abstract The aim of this paper is to show that the superposition of three waves (one strong and two weak), under suitable conditions, can create diffraction free propagation in a nonlinear planar waveguide. The propagation is described by the nonlinear Schrodinger equation, which is suitably linearized.