Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T19:10:21.604Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain

Published online by Cambridge University Press:  14 November 2011

Oscar P. Bruno  and
Fernando Reitich
Oscar P. Bruno
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160, U.S.A.
Fernando Reitich
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213–3890, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating

with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables (x, y, δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 317 - 340
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alber, H. D.. A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent. Proc. Roy. Soc. Edinburgh Sect. A 82 (1979), 251–272.CrossRefGoogle Scholar
2Bruno, O. and Reitich, F., Numerical solution of diffraction problems: a method of variation of boundaries (submitted).Google Scholar
3Cadilhac, M.. Some mathematical aspects of the grating theory. In Electromagnetic Theory of Gratings, ed. Petit, R. (Berlin: Springer, 1980).Google Scholar
4Chen, X. and Friedman, A.. Maxwell's equations in a periodic structure. Trans. Amer. Math. Soc. 323 (1991), 465–507.Google Scholar
5Colton, D. and Kress, R.. Integral equation methods in scattering theory (New York: John Wiley, 1983).Google Scholar
6Courant, R. and Hilbert, D.. Methods of mathematical physics (New York: John Wiley, 1962).Google Scholar
7Maystre, D.. Rigorous vector theories of diffraction gratings. In Progress in Optics, ed. Wolf, E. (Amsterdam: North Holland, 1984).Google Scholar
8Meecham, W. C.. On the use of the Kirchoff approximation for the solution of reflection problems. J. Rational Mech. Anal. 5 (1956), 323–334.Google Scholar
9Millar, R. F.. On the Rayleigh assumption in scattering by a periodic surface II. Proc. Cambridge Philos. Soc. 69 (1971), 217–225.CrossRefGoogle Scholar
10Petit, R.. A tutorial introduction. In Electromagnetic Theory of Gratings, ed. Petit, R. (Berlin: Springer, 1980).CrossRefGoogle Scholar
11Rayleigh, Lord. On the dynamical theory of gratings. Proc. Roy. Soc. London Ser. A 79 (1907), 399–416.Google Scholar
12Rayleigh, Lord. Note on the remarkable case of diffraction spectra described by Prof. Wood. Phil. Mag. Ser. 6 14 (1907), 60–65.CrossRefGoogle Scholar
13Sommerfeld, A.. Partial Differential Equations in Physics (New York: Academic Press, 1961).Google Scholar
14Uretsky, J. L.. The scattering of plane waves from periodic surfaces. Ann. Phys. 33 (1965), 400–427.CrossRefGoogle Scholar
15Watson, G. N.. A treatise on the theory of Bessel functions (New York: MacMillan, 1944).Google Scholar
16Wilcox, C. H.. Scattering theory for diffraction gratings (New York: Springer, 1984).CrossRefGoogle Scholar
17Wood, R. W.. On a remarkable case of uneven distribution of light in a diffraction grating spectrum. Phil. Mag. 4 (1902), 396–402.CrossRefGoogle Scholar