Blow-up solutions of Helmholtz equation for a Kerr slab with a complex linear and nonlinear permittivity

Abstract

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing.This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number. *

Blowing up light: a nonlinear amplification scheme for electromagnetic waves

“…[30] establishes the existence of blow-up solutions for the more general situations where ε l and σ are continuous complex-valued functions of z with the real part of σ having a negative upper bound, i.e., there is a real number s max such that Re[σ(z)] ≤ s max < 0 for all z ∈ [0, L]. In particular, the initial values E (0) and E ′ (0) determine a blow-up solution of the Helmholtz equation for such a Kerr slab provided that Re[E (0) * E ′ (0)] > 0 and L ≥ 2.023 × k 2 |s max |Re[E (0) * E ′ (0)] −1/3 , [30]. These results provide the theoretical grounds for comprehensive studies of more realistic applications of the nonlinear resonance phenomenon we have introduced in this article.…”