Light propagation in media with a highly nonlinear response: An analytical study

Abstract

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics (2008 Phys. Rev. A 78 021806(R)). Numerous particular cases of media exhibiting hight order nonlinear refractive indexes are considered. We obtain analytical expressions for determining the self-focusing position and new exact expression for calculating the filament intensity. The constructed solutions allowed to revise a so-called self-focusing scaling law, i.e. the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.

On Madelung systems in nonlinear optics: A reciprocal invariance

“…Importantly, it is seen that such gaussons are precluded in the case s = 1. This will be seen to be the case for the 1+1-dimensional Madelung system of [2][3][4].…”

“…Invariance of the Madelung system (4.2)−(4.3) under the two-parameter (δ, ǫ)-class of reciprocal transformations R * given by (4.6) has accordingly been established. This novel reciprocal invariance permits analytic results established in the series of papers [2][3][4] concerning the boundary value problem (4.5) for the Madelung system (4.2)−(4.3) to be extended to a more general class of model (δ, ǫ)-dependent n(I)-relations and associated boundary value problems. Thus, in particular, in the case of the standard nonlinear Kerr medium with n(I) = νI, it is seen that I * dn * = ǫI ν dI and one obtains the associated reciprocal Kerr class of (δ, ǫ)-dependent n * (I) * model relations given by n * = ν 2δǫ 2 (1 − (δ/ǫ)I * ) 2 + const .…”

“…Tatarinova and Garcia in a series of papers [2][3][4] have investigated certain boundary value problems for a Madelung system in nonlinear optics obtained via an eikonal approximation in which the de Broglie-Bohm term is neglected. The respective 1+1-dimensional Madelung system was derived as a reduction of the NLS equation [47] i ∂E ∂z…”

“…This seminal optics system can be encapsulated in a generalised nonlinear Schrödinger equation which likewise involves a de Broglie-Bohm potential. Tatarinova and Garcia in a series of papers [2][3][4] have subsequently investigated a Madelung system in nonlinear optics in which an eikonal approximation was adopted so that the de Broglie-Bohm term therein is neglected. Boundary value problems for this reduced Madelung system were investigated via a hodograph transformation for a variety of nonlinear model optical laws.…”

“…In the case of the reduced Madelung system of [2][3][4] with de Broglie-Bohm term neglected, invariance is established under a two-parameter (δ, ǫ)-class of reciprocal transformations. This allows the analytic results of [2][3][4] to be extended to novel (δ, ǫ)-dependent classes of nonlinear optical model laws.…”