Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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/10.1016/0022-247X(90)90395-V
  • [2] A. Benaldi, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. Part 1: The continuous problem; Part 2: The discrete problem, Math. Comp. 167 (1984), 29-46, 47-68.
  • [3] N. G. De Bruijn, Asymptotic methods in analysis, North-Holland, 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), 894-937.
  • [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, Springer-Verlag, New York-Heidelberg, 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, Springer-Verlag, Berlin-New York, 1980. MR 609533 (82a:78001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35Q60, 35P25, 45B05, 78A45

Retrieve articles in all journals with MSC: 35Q60, 35P25, 45B05, 78A45


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1010883-1
PII: S 0002-9947(1991)1010883-1
Keywords: Maxwell's equations, transmission, reflection, Fredholm equations
Article copyright: © Copyright 1991 American Mathematical Society