Unified description of nondiffractingXandYwaves

Abstract

A unified spectral and temporal representation is introduced for nondiffracting waves. We consider a set of elementary broadband X waves that spans the commonly considered nondiffracting wave solutions. These basis X waves have a simple spectral representation that leads to expressions in closed algebraic form or, alternatively, in terms of hypergeometric functions. The span of the X waves is also closed with respect to all spatial and temporal derivatives and, consequently, they can be used to compose different types of waves with complex spectral and spatial properties. The unified description of Bessel-based nondiffracting waves is further extended to include singular Neumann and Hankel waves, or Y waves. We also discuss connections between the different known nondiffracting wave solutions, and their relations to the present unified approach.

Nonlinear Unbalanced Bessel Beams: Stationary Conical Waves Supported by Nonlinear Losses

“…Despite the ideal nature of BBs (they carry infinite power), they not only have revealed to be a paradigm for understanding wave phenomena, but also have found applications as diverse as in frequency conversion, or in atom trapping and alignment [7]. Of particular interest for us is the finding [8] that the BB is describable in terms of the interference of two conical Hankel beams [5], carrying equal amounts of energy towards and outwards the beam axis, and yielding no net transversal energy flux in the BB.What we demonstrate here is that a superposition of inward and outward Hankel beams with unequal amplitudes, i.e., an "unbalanced" Bessel beam (UBB), describes the only possible asymptotic form of the SL, nonsingular solutions of the 2D+1 NLSE with NLL. For this reason, we call the solution "non-linear UBB" (NL-UBB).…”

“…This is the case of the sech-type Townes profile in Kerr media [9], associated to wave vector shift δ < 0, and of weakly localized Bessel beams in linear [6] or Kerr media [10], with infinite power and wave vector shift δ > 0. With NLL, instead, the conditions of refilling (5) and localization [a(r) → 0 as r → ∞] require an inward radial power [lhs of (5)] that monotonically increases [rhs of (5)] with r up to reach, at infinity, a constant value equal to the total NLL, N ∞ , assumed they are finite. Stationarity with NLL is thus supported by the continuous refilling of the more strongly absorbed inner part of the beam with the energy coming from its outer part.…”

“…These SL waves cannot be ascribed to the class of solitary waves, but are instead nonlinear conical waves (as the non-linear X waves [4]) of dissipative type, whose stationarity is sustained by a continuous refilling of the nonlinearly absorbed central spot with the energy supplied by linear, conical tails. These waves are not only robust against NLL, but find their stabilizing mechanism against perturbations in NLL themselves.Among the linear conical waves [5], the simplest one is the monochromatic Bessel beam (BB) [6], made of a superposition of plane waves whose wave vectors are evenly distributed over the surface of a cone, resulting in a nondiffracting transversal Bessel profile. Despite the ideal nature of BBs (they carry infinite power), they not only have revealed to be a paradigm for understanding wave phenomena, but also have found applications as diverse as in frequency conversion, or in atom trapping and alignment [7].…”

“…Among the linear conical waves [5], the simplest one is the monochromatic Bessel beam (BB) [6], made of a superposition of plane waves whose wave vectors are evenly distributed over the surface of a cone, resulting in a nondiffracting transversal Bessel profile. Despite the ideal nature of BBs (they carry infinite power), they not only have revealed to be a paradigm for understanding wave phenomena, but also have found applications as diverse as in frequency conversion, or in atom trapping and alignment [7].…”

“…formed by two nondiffracting Hankel beams of the first and second kind [5], of same cone angle θ = 2δ/k, but different weights, α out and α in , that must be related by…”