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 epsilon(l) and a complex Kerr coefficient sigma admits blow-up solutions, provided that the real part of a is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs if epsilon(l) and sigma are continuous functions of the transverse coordinate, and the real part of sigma is bounded above by a negative number. It shows that a recently reported nonlinear optical amplification effect, which relies on the existence of blow-up solutions, persists the presence of losses and transverse inhomogeneities.