Existence of guided cylindrical TM-modes in a homogeneous self-focusing dielectric

Abstract

We establish the existence of nontrivial solutions for the nonlinear eigenvalue problem which describes self-trapped transverse magnetic field modes in a cylindrical optical fiber made from a self-focusing dielectric material. It amounts to finding solutions in the Sobolev space H01(0,∞) of a singular second order differential equation which is quasilinear and, in an appropriate sense, asymptotically linear. Solutions are critical points of an energy functional which has a mountain pass structure, although for all relevant parameter values the problem is in resonance at infinity. The quasilinearity complicates the proof of a Palais–Smale condition, and the asymptotic linearity means that the standard methods for showing that a P-S sequence is bounded do not apply. Both the linearization and the asymptotic linearization have only continuous spectrum. The eigenvalues determine the wavelengths of self-trapped modes and our results establish the existence of such modes for the largest possible range of wavelengths.