Vortex nucleation and evolution in parametric wave mixing

Abstract

SPAINSingular light beams, that contain topological wave front dislocations are ubiquitous entities that display fascinating properties with widespread important applications. I Screw dislocations, or vortices, are a common dislocation type. They are spiral phase-ramps around a singularity where the phase of the wave is undefined and its amplitude vanishes. The order of the screw dislocation multiplied by its sign is referred to as the winding number, or topological charge of the dislocation. Vortices appear spontaneously in several settings, including in speckle-fields, in optical cavities and in the doughnut laser modes, and otherwise they can be generated with phase masks, or with astigmatic optical components. Vortices also form by self-wave front modulation in nonlinear optical media. In this context, parametric wave mixing of multiple waves propagating in quadratic nonlinear media constitutes a fascinating scenario. In this paper we predict a variety of new phenomena, that includes the spontaneous nucleation of multiple vortex twins, vortex rotation and drift, vortex-antivortex interaction and annihilation, and formation of quasi-aligned patterns of single-charge vortices.We consider cw light propagation in a bulk quadratic nonlinear crystal under conditions for type 1 second-harmonic generation. We restrict ourselves to up-conversion geometries with material and light conditions that yield negligible depletion of the pump fundamental frequency (FF) beam. Then, the second-harmonic (SH) beam is dictated by an inhomogeneous linear partial differential equation whose general solution can be obtained by means of the Green function approach. In the case of un-seeded geometries (i.e., no SH input light), and in absence of Poynting vector walk-off between the FF and SH beams, sum-and difference-charge arithmetic operations have been predicted and observed experimentally. 2-4 However, a new range of phenomena is discovered in seeded geometries and with Poynting vector walk-off. In particular, in the case of seeded schemes without walk-off, our numerical and experimental investigations show the spontaneous nucleation of multiple-vortex twins. In such case, the number of vortices present in the SH beam and its total topological charge varies with the propagation distance inside the crystal. Figure 1 shows a typical example of the vortex pattern that is typically obtained. Walk-off introduces a new range of phenomena. In the case of un-seeded schemes with a Gaussian pump beam with a single-charge vortex nested, one finds that the generated SH beam is given by the expression Here the transverse coordinates are normalized to the FF beam width q, and the propagation coordinate 6 is normalized to twice the diffraction length of the FF beam Ldl=k1q2/2. The parameter p is given by P=klq2Ak, where Ak=2kl-k2 is the