Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations

This paper presents experimental evidence that orthogonally crossed dark soliton stripes form quasi-twodimensional spatial solitons with a soliton constant equal to that of singly charged optical vortices. Besides the pairs of oppositely charged optical vortex solitons, the snake instability of the dark formation at moderate saturation is found to lead to generation of steering mixed edge–screw phase dislocations with zero total topological charges. PACS: 42.65; 42.65.Tg; 42.65.Sf Mathematically, dark optical solitons are particular solutions of the (1+1)-dimensional nonlinear Schrödinger equation for negative nonlinearity [1]. Physically, a spatial soliton forms when the beam interacts with the medium of propagation and changes its refractive index in a manner that leads to an exact compensation for the dark beam diffraction [2]. Onedimensional dark spatial solitons (1D DSSs) in bulk nonlinear media are experimentally generated as dark stripes [3], whereas the only truly two-dimensional DSSs are observed in the form of optical vortex solitons (OVSs) [4, 5]. The experimental formation of dark spatial solitons is inevitably accompanied by self-defocusing and intensity reduction of the finite background beam. As pointed out in the literature [6] dark pulses/beams created on finite backgrounds, even longlived, disappear as soon as the propagation distance becomes large enough. In the strict mathematical sense they are not proper solitons. In real physical systems, losses, saturation, and higher transverse dimensionality result in nonintegrable model equations. Their solitary solutions, however, possess a large number of characteristics in common with the soliton solution of the integrable (1+1)-dimensional nonlinear Schrödinger equation [2]. Despite certain adiabatic relaxation characteristics these solutions are widely denoted with the term ‘soliton’ (see, for example, [7]) and we adopt it with this weaker meaning throughout this work. ∗ Permanent address: Sofia University, Department of Quantum Electronics, 5, J. Bourchier Blvd., BG-1164 Sofia, Bulgaria (Fax: +35-92/9625-276, E-mail: [email protected]) Orthogonally crossed 1D DSSs are found to propagate almost independently and noninteracting thus forming quasi2D DSSs (‘fundamental dark-soliton cross’, [3]). Our interest was aroused by the fact that, similarly to 1D and OV solitons [4, 8], quasi-2D DSSs should obey the guiding properties required for all-optical branching and switching applications [9]. However, these nonlinear propagation schemes will prove to be of technological importance only if they are stable under propagation. It has been shown [10] that moderate saturation of the nonlinearity can effectively suppress the dark soliton’s transverse instability at the expense of weakened soliton steering [11] and reshaping [12]. In the regime of strong saturation, however, another type of instability is to be expected ([13] and Fig. 2 therein). This paper presents comparative experimental data on the soliton constants of 1D DSSs, quasi-2D DSSs, and singlycharged OVSs generated in a thermal nonlinear medium with moderate saturation. The input dark beams with their specific phase distributions ((crossed) edge or screw dislocations) are obtained by computer-generated holograms which are of binary type and have the same grating period. The soliton constant of the quasi-2D DSS was found to be equal to that of an OVS. However, the rate at which the product of the square of the quasi-2D dark beam width and the dark irradiance tends to its asymptotic value, namely the soliton constant, corresponds to that of a 1D DSS. In this sense, the term ‘quasi-2D DSS’ is more precise than ‘dark soliton cross’. Intentional perturbation of the quasi-2D DSS was found to result in a modulational instability and generation of 1D dark beams of finite length and mixed edge–screw phase dislocation. Although the concept of wavefront dislocations was introduced by Nye and Berry almost 25 years ago [14], this observation is, to the best of our knowledge, chronologically the third one [15] and the first to clearly identify mixed phase dislocations in the optical range. 1 Experimental setup The experimental arrangement used is shown in Fig. 1. In order to obtain the desired phase dislocation within the dark