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 εl and a complex Kerr coefficient σ admits blow-up solutions, provided that the real part of σ is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs if εl and σ are continuous functions of the transverse coordinate, and the real part of σ 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.