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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maxwell's equations in a periodic structure

Authors: Xinfu Chen and Avner Friedman
Journal: Trans. Amer. Math. Soc. 323 (1991), 465-507
MSC: Primary 35Q60; Secondary 35P25, 45B05, 78A45
MathSciNet review: 1010883
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Abstract: Consider a diffraction of a beam of particles in $ {\mathbb{R}^3}$ when the dielectric coefficient is a constant $ {\varepsilon _1}$ above a surface $ S$ and a constant $ {\varepsilon _2}$ below a surface $ S$, and the magnetic permeability is constant throughout $ {\mathbb{R}^3}$. $ S$ is assumed to be periodic in the $ {x_1}$ direction and of the form $ {x_1} = {f_1}(s),\,{x_3} = {f_3}(s),\,{x_2}$ arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in $ {\mathbb{R}^3}$ having the form of refracted waves for $ {x_3} \ll 1$ and of transmitted waves for $ - {x_3} \gg 1$ if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the $ \varepsilon $'s, except for a discrete number, there exists a unique solution to the Maxwell equations.

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Keywords: Maxwell's equations, transmission, reflection, Fredholm equations
Article copyright: © Copyright 1991 American Mathematical Society