Bessel beams: a novel approach to periodic structures

Abstract

Bessel and Bessel-like beams in Kerr-like nonlinear materials are numerically investigated. This is conducted with a view to exploiting the behaviour of such profiles for the direct laser writing of periodic structures in highly nonlinear glasses. A highly efficient numerical model is developed for the propagation of radially symmetric profiles based on the quasi-discrete Hankel transform (QDHT), making use of a reconstruction relation to allow the field to be sampled at arbitrary positions that do not coincide with the numerical grid. This Hankel-based Adaptive Radial Propagator (HARP) is shown to be up to 1000 times faster than standard FFT-based methods. The critical self-focusing of the Gaussian beam is reproduced to confirm the accuracy of HARP. Following this the critical self-focusing behaviour of a Bessel-Gauss beam is investigated. It is observed that, for certain parameters, increasing the beam power may prevent blowup in the Bessel-Gauss beam. Below the threshold for self-focusing the Bessel-Gauss beam exhibits periodic modulation in the direction of propagation. The existing equation describing this behaviour is shown to be inaccurate and a modification is proposed based on a power dependent beat-length. This modified beat-length equation is demonstrated to be accurate in both the paraxial and quasi-nonparaxial regime. As the beam decays, the intensity modulation appears negatively chirped. It is demonstrated that this chirp may be controlled through careful shaping of the window. It is also shown that a small Gaussian seed beam may be used to control the positions of the maxima. It is demonstrated that a set of nonlinear Bessel functions exist that exhibit a similar quasi-stationary behaviour in a nonlinear medium to the linear Bessel beam in a linear medium. Furthermore it is shown for the first time that higher-order, Bessel-like, stationary solutions exist for beams with azimuthal phase, and boundary conditions for these functions are derived.