Local dynamic behavior of dark beams in self-defocusing Kerr media

Abstract

We study the local dynamic behavior of dark beams in self-defocusing Kerr media. In this, we are encouraged by the interesting results obtained in Refs. [1,2], although these papers are concerned with the bright beams in self-focusing Kerr media. Similar to these papers, we are interested in the two-dimensional (2D) necklace ring beams, but under certain circumstances. Different from these papers, we are interested in the dark necklace beams in self-defocusing media. When the dimensionless ring radius L is far greater than the ring thickness w, the 2D defocusing nonlinear Schrödinger equation in cylindrical coordinates can be simplified to the standard 1D defocused nonlinear Schrödinger (NLS) equation. Then, an approximate solution of the 1D NLS equation can be constructed by different means, for example by a simple combination of exponential and trigonometric functions. In general, such a necklace solution can be described by two parameters: the ring radius L and the topological charge m. Hence, we study the local dynamic behavior of dark beams with different topological charges m and different ring radii L when they are much larger that the ring thickness w. Different spatial distributions of dark beams are obtained for m zero, positive integer, and half-integer. We use direct numerical simulation to confirm the validity of our analytical approximate solution. Our results show that the structure of dark beams can be well controlled by adjusting the ring radius and the topological charge. Our procedure provides a new method for dealing with high-dimensional nonlinear equations of different origin.