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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 30, Iss. 10 — Oct. 1, 2013
  • pp: 2111–2123

Explicit error bounds for the α-quasi-periodic Helmholtz problem

Natacha H. Lord and Anthony J. Mulholland  »View Author Affiliations


JOSA A, Vol. 30, Issue 10, pp. 2111-2123 (2013)
http://dx.doi.org/10.1364/JOSAA.30.002111


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Abstract

This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

© 2013 Optical Society of America

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1960) Diffraction and gratings : Diffraction theory
(050.2770) Diffraction and gratings : Gratings
(080.2720) Geometric optics : Mathematical methods (general)
(290.0290) Scattering : Scattering
(080.1753) Geometric optics : Computation methods

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 16, 2013
Revised Manuscript: August 27, 2013
Manuscript Accepted: August 27, 2013
Published: September 25, 2013

Citation
Natacha H. Lord and Anthony J. Mulholland, "Explicit error bounds for the α-quasi-periodic Helmholtz problem," J. Opt. Soc. Am. A 30, 2111-2123 (2013)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-10-2111

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