How Kerr nonlinearity influences polarized electromagnetic wave propagation

Abstract

The propagation of electromagnetic TE waves along boundaries of a plane dielectric layer fi lled with a Kerr medium is studied for all possible cases of the problem's parameters. The layer is located between two half-spaces with constant permittivities. The problem is reduced to a nonlinear transmission eigenvalue problem for Maxwell's equations, in which each eigenvalue is a propagation constant of a guided wave. The exact dispersion equation with respect to the eigenvalues (propagation constants) is derived and studied. It is proven that in the presence of the Kerr eff ect, several novel wave-guiding regimes arise, including regimes that have no counterparts in the linear theory. An infi nite number of eigenvalues arise in the focusing case, even if the corresponding linear problem has no solutions (the linear problem always has no more than a fi nite number of solutions). In the defocusing case, only those solutions arise that tend to linear solutions when the nonlinearity coeffi cient vanishes. It is also proven that in the nonlinear case, an infi nite number of eigenvalues do not reduce to the solutions of the corresponding linear problem, even if the nonlinearity coeffi cient tends to zero. Numerical illustrations for the results obtained are provided.

Nonlinear multi-frequency electromagnetic wave propagation phenomena

“…Theoretically, the existence of purely non-linear PCs is proven only for a plane waveguide filled with Kerr medium and 1-frequency GWs [24,[26][27][28]. However, in the other cases presented below, we find numerical solutions that are similar to the purely non-linear ones.…”

“…The case n=1 corresponds to the propagation of monochromatic TE waves. This case attracts a lot of attention, even for Kerr nonlinearity, and has been solved only recently in [24]; complete results have been found in [27,28]. The case n=2 corresponds to the propagation of a 2-frequency (coupled) TE-TE wave; this problem has been studied in [59] in a slightly different formulation using the perturbation method, and in [65] using the integral dispersion equation method.…”

“…It is worth noting that the perturbation method in such problems is only a first step that allows one to derive the existence results. However, in the simplest case for n=1, the results found in [24,27,28] show that there are many opportunities, in particular:…”

“…a 0 and the corresponding linear problem has solutions (it always has a finite number of PCs), then the non-linear problem has an infinite number of solutions. Only a finite number of 'non-linear' solutions have linear counterparts; the other (infinitely many) solutions are non-perturbative [24,27,28]; (ii) If > a 0 and the corresponding linear problem has no solutions, then the non-linear problem has an infinite number of solutions-all of them, obviously, are nonperturbative [27,28]. In this case there exist infinitely many periodic sign-preserving non-linear guided modes; (iii) If < a 0 then there exists a constant < a 0 0 such that for any negative > a a 0 the linear and non-linear problem has a finite number of solutions.…”

“…As is known in the case of n=1 for TE and TM GWs in a layer with Kerr nonlinearity, infinitely many non-perturbative (or purely nonlinear) solutions arise [24,[26][27][28]. The existence of infinitely many (and, more generally, non-perturbative) solutions in these simpler problems allows one to assume that the perturbation method does not help to find all physically interesting solutions in NMF problems.…”