Vectorial and random effects in self-focusing and in multiple filamentation

Abstract

The standard explanation for multiple filamentation of laser beams is that breakup of cylindrical symmetry is initiated by noise in the input beam. In this study we propose an alternative deterministic explanation based on vectorial effects. We derive a scalar equation from the vector Helmholtz equation that describes self-focusing in the presence of vectorial and nonparaxial effects. Numerical simulations of the scalar equation show that when the input beam is sufficiently powerful, vectorial effects lead to multiple filamentation. We compare multiple filamentation due to vectorial effects with the one due to noise, and suggest how to decide which of the two leads to multiple filamentation in experiments. We also show that vectorial effects and nonparaxiality have the same effect on self-focusing of a single filament, leading to the arrest of catastrophic collapse, followed by focusing-defocusing oscillations. The magnitude of vectorial effects is, however, significantly larger than that of nonparaxiality.

Femtosecond Filamentation in Air

“…Yet, modulational instability also leads to a selection of a specific transverse length scale which should appear as a main feature of the filamentation patterns. Recently, alternative explanations for multiple filamentation were proposed by Fibich and coworkers [26][27][28]. They showed that deterministic vectorial effects could prevail over the amplification of noise in the process of multiple filamentation.…”

Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

“…A yet more comprehensive model would be the vector NLH that also accounts for the vectorial nature of the electric field while still taking care of the phenomena of nonparaxiality and backscattering. Note that the vectorial effects, nonparaxiality, and backscattering are all of the same order of magnitude; see [10] for more detail.…”

On the limits of validity of nonparaxial propagation equations in Kerr media

“…In order to generalize Eq. (3) beyond the paraxial approximation one has to rely on the (2+1)-D nonparaxial propagation equations present in [1,2,3,4]. The distinction between TE and TM field component is no longer meaningful since the propagation equation is a fully vectorial one.…”

“…Many attempts aimed at generalizing, as a first approximation, the paraxial approach by adding on the RHS of the standard (paraxial) nonlinear Schroedinger equation (NLSE) higher-orders terms of the order of the square of the smallness parameter = λ/σ (where λ and σ are the wavelength and the beam typical transverse dimension, respectively) have resulted in different equations which do not agree among themselves. While some of these nonparaxial nonlinear Schroedinger equations (NNLSE) contain intrinsic scalar approximations, two general approaches, [1,2]- [3,4], preserve the fully vectorial nature of the problem. The first approach is based on the analysis of light propagation in Fourier space, where dealing with nonparaxial effects is both simpler and more natural.…”