Propagation dynamics of modulated vortex beams in nonlocal nonlinear media

The nonlocal nonlinearity allows the refractive index of material at a particular point to be related the beam intensity at all other material points [1], and the media can be found in various system, such as nematic liquid crystals [2], thermal media [3] and atomic vapors with diffusion of charge carriers [4]. Beams may break into multiple fundamental beams in local media, and the nonlocality can prevent the catastrophic collapse of high-dimensional beams and suppress modulation instability and transverse instability in self-focusing media [5], as a result, a range of solitons is supported due to the long-ranged interaction, including multipole [6], [7] and dipole [8], [9] solitons, vector solitons [10], [11] and necklace solitons [12]. Moreover, the bounding states between out-of-phase bright and dark solitons can be formed with providing attractive force [13], [14].

The vortex, presenting a phase singularity [15], [16], causes the splitting of beams in nonlinear media because of the azimuthal instability [17], [18], and multiple beams are generated. The nonlocality, suppressing the azimuthal instability of orbital angular momentum beams, supports the stable propagation of multiple beams called as vortex solitons, which has attracted considerable attention, such as the stationary vortex solitons and azimuthons formation [19], [20], [21]. The existence and stability of vortex solitons or soliton pairs in nonlocal media are solely determined by the nonlocal response function [22], [23]. Higher-order vortex solitons are presented to be stable in nonlocal media with competing focusing and defocusing nonlinearities [24]. When the initial field is considered as the vector vortex beam or the vortex pair, the instability is suffered, and the beam breaks up into several parts, and then multiple fundamental solitons are generated [25], [26]. In fact, multiple solitons generation is important due to the potential applications in optical splitting and controlling. From foregoing researches, the main mechanism of multiple solitons attributes to the intrinsic characteristics of the media. Recently, the effect of the azimuthally modulated phase on beams propagation has been discussed [27], and the beam splits into several beam spots. This phenomenon indicates the generation method of multiple solitons by utilizing the azimuthally modulated vortex structure. To the best of our knowledge, the propagation of modulated vortex beams in nonlocal media has not been investigated.

In this paper, the aim is to investigate the propagation of modulated vortex beams in nonlocal media. The diffraction properties are discussed, and the beam splitting is clearly presented with controllable beam spots, as a result, these beam spots can focus into multiple solitons under the nonlocality. With adjusting the vortex structure and the nonlocality, the number of multiple solitons can be controlled and even presents a linear region with the number of vortex modulation phase folds. These features of the vortex structure present potential applications in optical splitting and optical controlling.

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