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

作者：Stuart, CA^*; Zhou, HS

来源：Calculus of Variations and Partial Differential Equations, 2003, 16(4): 335-373.

DOI：10.1007/s005260100153

• ELECTROMAGNETIC-FIELD
• WAVES
• MEDIA

The first part of this paper establishes the existence of a minimizer of problem:
min{F(eta) : eta is an element of H-o(1)(0, infinity) and integral(o)(infinity)eta(2)dr = d(2)}
where
F(eta) = integral(o)(infinity)rPsi(eta/rootr, 2/rootr [eta'+n/2r]) dr and d > 0.
The essential features of the integrand are that
Y : R-2 --> R is convex and psi : [0, infinity)(2) --> R is concave
where psi(s(1), s(2)) = Psi(root2s(1), root2s(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
B = w(r)cos(kz-wt)i(theta)
when expressed in cylindrical polar co-ordinates (r, theta, z). The amplitude w is given by w(r) = w/ck(kr)(-1)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.

• 出版日期2003-4
• 单位中国科学院武汉物理与数学研究所