Maxwell's equations in a periodic structure
Authors:
Xinfu Chen and Avner Friedman
Journal:
Trans. Amer. Math. Soc. 323 (1991), 465507
MSC:
Primary 35Q60; Secondary 35P25, 45B05, 78A45
MathSciNet review:
1010883
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Abstract 
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Abstract: Consider a diffraction of a beam of particles in when the dielectric coefficient is a constant above a surface and a constant below a surface , and the magnetic permeability is constant throughout . is assumed to be periodic in the direction and of the form arbitrary. We prove that there exists a unique solution to the timeharmonic Maxwell equations in having the form of refracted waves for and of transmitted waves for if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the 's, except for a discrete number, there exists a unique solution to the Maxwell equations.
 [1]
Hamid
Bellout and Avner
Friedman, Scattering by stripe grating, J. Math. Anal. Appl.
147 (1990), no. 1, 228–248. MR 1044697
(91g:35197), http://dx.doi.org/10.1016/0022247X(90)90395V
 [2]
A. Benaldi, Numerical analysis of the exterior boundary value problem for the timeharmonic Maxwell equations by a boundary finite element method. Part 1: The continuous problem; Part 2: The discrete problem, Math. Comp. 167 (1984), 2946, 4768.
 [3]
N. G. De Bruijn, Asymptotic methods in analysis, NorthHolland, Amsterdam, 1958.
 [4]
C. A. Coulson and A. Jeffrey, Waves, 2nd ed., Longman, London, 1977.
 [5]
T. K. Gaylord and M. G Moharam, Analysis and applications of optimal diffraction by gratings, Proc. IEEE 73 (1985), 894937.
 [6]
I. S. Gradsteyn and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, 1965.
 [7]
Claus
Müller, Foundations of the mathematical theory of
electromagnetic waves, Revised and enlarged translation from the
German. Die Grundlehren der mathematischen Wissenschaften, Band 155,
SpringerVerlag, New YorkHeidelberg, 1969. MR 0253638
(40 #6852)
 [8]
J. C. Nedelec and F. Starling, Integral equation methods in quasi periodic diffraction problem for the time harmonic Maxwell's equations, Ecole Polytechnique, Centre de Mathematiques Appliquees, Tech. Report #756, April 1988.
 [9]
Roger
Petit (ed.), Electromagnetic theory of gratings, Topics in
Current Physics, vol. 22, SpringerVerlag, BerlinNew York, 1980. MR 609533
(82a:78001)
 [1]
 A. Bellout and A. Friedman, Scattering by strip grating, J. Math. Anal. Appl. (to appear). MR 1044697 (91g:35197)
 [2]
 A. Benaldi, Numerical analysis of the exterior boundary value problem for the timeharmonic Maxwell equations by a boundary finite element method. Part 1: The continuous problem; Part 2: The discrete problem, Math. Comp. 167 (1984), 2946, 4768.
 [3]
 N. G. De Bruijn, Asymptotic methods in analysis, NorthHolland, Amsterdam, 1958.
 [4]
 C. A. Coulson and A. Jeffrey, Waves, 2nd ed., Longman, London, 1977.
 [5]
 T. K. Gaylord and M. G Moharam, Analysis and applications of optimal diffraction by gratings, Proc. IEEE 73 (1985), 894937.
 [6]
 I. S. Gradsteyn and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, 1965.
 [7]
 C. Muller, Foundations of mathematical theory of electromagnetic waves, SpringerVerlag, Berlin, 1969. MR 0253638 (40:6852)
 [8]
 J. C. Nedelec and F. Starling, Integral equation methods in quasi periodic diffraction problem for the time harmonic Maxwell's equations, Ecole Polytechnique, Centre de Mathematiques Appliquees, Tech. Report #756, April 1988.
 [9]
 R. Petit (Editor), Electromagnetic theory of gratings, Topics in Current Phys., vol. 22, SpringerVerlag, Heidelberg, 1980. MR 609533 (82a:78001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110108831
PII:
S 00029947(1991)10108831
Keywords:
Maxwell's equations,
transmission,
reflection,
Fredholm equations
Article copyright:
© Copyright 1991
American Mathematical Society
