A constrained minimization problem and its application to guided cylindrical TM-modes in an anisotropic self-focusing dielectric

Abstract.

The first part of this paper establishes the existence of a minimizer of problem:

\begin{equation} {\rm min}\left\{F(\eta ):\eta \in H_{0}^{1}(0,\infty )\;\mbox{ and } \;\;\int_{0}^{\infty }\eta ^{2}dr=d^{2}\right\} \end{equation}

where

\[ F(\eta )=\int_{0}^{\infty }r\Psi \left( \frac{\eta }{\sqrt{r}},\frac{1}{ \sqrt{r}}\left[\eta ^{\prime }+\frac{\eta }{2r}\right]\right) dr\;\mbox{ and }\;\;d>0. \]

The essential features of the integrand are that

\[ \Psi :{\Bbb R}^{2}\longrightarrow{\Bbb R}\;\mbox{ is convex and }\;\psi :[0,\infty )^{2}\longrightarrow{\Bbb R}\mbox{ is concave}\; \]

where \psi (s_{1},s_{2})=\Psi (\sqrt{2s_{1}},\sqrt{2s_{2}}). We show that the minimizer satisfies an Euler- Lagrange equation and estimates are given for the Lagrange multiplier as a function of d. In the second part of the paper, we use this result to establish the existence of guided TM-modes propagating through a self-focusing anisotropic dielectric. These are special solutions of Maxwell's equations with a nonlinear constitutive relation of a type commonly used in nonlinear optics when treating the propagation of waves in a cylindrical wave-guide. In TM-modes, the magnetic field has the form

\[ {\bf B}=w(r)\cos (kz-\omega t)i_{\theta } \]

when expressed in cylindrical polar co-ordinates $(r,\theta ,z).$ The amplitude w is given by $w(r)=\frac{\omega }{ck}(kr)^{-1/2}\eta (kr)$ where $\eta$ is a minimizer of the problem (0.1) for a function $\Psi$ which is determined by the constitutive relation through a Legendre transformation.