# Propagation Property of an Astigmatic sin-Gaussian Beam in a Strongly Nonlocal Nonlinear Media
## 1 Introduction  
In the past decades, self-trapped optical beams in nonlocal nonlinear media, especially in strongly nonlocal nonlinear media (SNNM), have been a topic of considerable interest due to their theoretical importance and many experimental observations [1; 2]. SNNM is a novel kind of media in which the characteristic length of the nonlocal response is much larger than the beam width. Distinct from the conventional local nonlinearity, the nonlocality allows the refractive index of the media at a particular point to be related to the beam intensity at all other points. Note that the nonlocal nonlinearity which can support a variety of nonlocal spatial optical solitons exhibits in many physical systems, and some of them have been observed experimentally [3; 4; 5; 6; 7; 8; 9; 10]. Moreover, it has been reported that a great number of optical beams can steadily propagate in SNNM under sufficient conditions, including Gaussian beams and higher-order Gaussian beams, four-petal Gaussian beams, Lorentz-Gaussian beams, the beams carrying wave front dislocations such as Hermite-, Hermite-cosh- or Laguerre-Gaussian beams, and so on [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. As well known, pure wave front dislocations in a monochromatic wave are divided into two types: one is the longitudinal screw dislocation which is also known as the optical vortex with spiral phase, and the other is the transverse edge dislocation with \(\pi\)-phase shift located along a line in the transverse plane. During the past years, most of the previous investigations on propagation of beams in SNNM have been focused on those beams whose pattern structure remains invariant although the pattern size may change during propagation. Due to the complexity of evolution of beams in the nonlocal domain, it is interesting to find that the changes of wave front dislocations are possible, which turns out to be very important for the propagation characteristics and will modify the pattern structure for a laser beam in SNNM [24; 27; 28].  
Like Hermite-Gaussian beams, sin-Gaussian beams carrying edge dislocations are the special case of the Hermite-sinusoidal-Gaussian beams whose propagation dynamics through various optical systems had been studied widely [29; 30; 31; 32; 33; 34; 35; 36; 37]. However, their propagation in SNNM remains unexplored. In this work, we aim to study the propagation of sin-Gaussian beams with astigmatism in SNNM. A closed-form propagation equation of astigmatic sin-Gaussian beams in SNNM is derived and illustrated with numerical examples. Our results show some novel variations of wave front dislocations occur for astigmatic sin-Gaussian beams propagating in SNNM. The remainder parts of this paper are organized as follows: Section 2 represents the theoretical formulation for astatic sin-Gaussian beams propagating in SNNM. In Section 3 main characters associated with intensity and phase distributions of a propagating sin-Gaussian beam are analytically investigated and a large number of numerical calculations are performed based on the obtained formulae in Section 2. Finally, in Section 4 the main results obtained in this paper are summarized.