Azimuthons: Spatially Modulated Vortex Solitons

Recent progress in the experimental study of nonlinear optical effects in bulk dielectric media opens up many novel possibilities in the study of transverse self-trapping of light and the formation of spatial optical solitons (1). Spatial optical solitons are stationary self-trapped localized modes in homogeneous self-focusing nonlinear media (2), and they may possess transverse energy flow associated with the complicated phase structure. Two familiar ex- amples include radially symmetric vortex solitons (3,4) and rotating soliton clusters (5) created by several interact- ing fundamental solitons which rotate opposite to the phase gradient (6). In this Letter, we introduce a novel class of spatially localized self-trapped optical beams in nonlinear media, the so-called azimuthons, which provide an important missing link between the radially symmetric vortices and rotating soliton clusters. We reveal novel physics of non- trivial rotation of self-trapped modulated optical beams and show that the associated angular momentum has two different contributions. The first contribution is due to the internal energy flow; it comes from a nontrivial phase as in the case of the radially symmetric vortex solitons; this reflects the wave nature of the self-trapped beams. The second contribution appears only when the rotating beam is modulated or fragmented, and it has a ''particle'' origin. While the latter contribution is important for the soliton clusters, the former one dominates for strongly overlapping beams when the ''particle identity'' in the beam structure is lost. Surprisingly, as we show below, these two contribu- tions can be of opposite signs, giving birth to the non- rotating modulated singular beams described here as stationary azimuthons. In particular, we find the spatial solitons whose visible rotation can be directed alongside or opposite to the direc- tion of the energy flow. Furthermore, the internal energy flow can be balanced exactly by the ''mechanical'' rota- tion, and, in this case, the truly stationary nonrotating states emerge. The term ''azimuthons'' for these novel self- trapped states reflects their distinctive modulated profile. The intensity of such states is a spatially localized ring modulated azimuthally, and the phase carries a screw-type dislocation; in contrast to the linear vortex phase m', the phase of the azimuthon is a staircaselike nonlinear function of the polar angle '. In other words, higher-order spatial solitons can be described in terms of azimuthal deformations of the vortex solitons. We analyze different families of such solutions in both Kerr and saturable non- linear media and demonstrate that the azimuthons are characterized by two independent integer numbers or azi- muthal indices: the topological charge m and the number of the intensity peaks N. For the soliton clusters (5,7), the number of peaks satisfies the condition N 4m, while the rotating azimuthons with N 2m can exist in saturable media; in particular, we demonstrate truly stationary non- rotating azimuthons with the indices m 1 and N 3. We consider the paraxial propagation of light in an isotropic nonlinear medium with an instantaneous re- sponse governed by the nonlinear Schrodinger (NLS) equation, i @E