Analysis of the Rigorous Coupled Wave Approach for s-polarized light in gratings

Benjamin J. Civiletti, Akhlesh Lakhtakia, Peter B. Monk

Research output: Contribution to journalArticle

Abstract

We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for s-polarized monochromatic incident light. The RCWA is widely used to solve electromagnetic boundary-value problems where the relative permittivity varies periodically in one direction, i.e., scattering by a grating. This semi-analytical approach expands all the electromagnetic field phasors as well as the relative permittivity as Fourier series in the spatial variable along the direction of periodicity, and also replaces the relative permittivity with a stairstep approximation along the direction normal to the direction of periodicity. Thus, there is error due to Fourier truncation and also due to the approximation of grating permittivity. We prove that the RCWA is a Galerkin scheme, which allows us to employ techniques borrowed from the Finite Element Method to analyze the error. An essential tool is a Rellich identity that shows that certain continuous problems have unique solutions that depend continuously on the data with a continuity constant having explicit dependence on the relative permittivity. We prove that the RCWA converges with an increasing number of retained Fourier modes and with a finer approximation of the grating interfaces. Numerical results show that our convergence results for increasing the number of retained Fourier modes are seen in practice, while our estimates of convergence in slice thickness are pessimistic.

Original languageEnglish (US)
Article number112478
JournalJournal of Computational and Applied Mathematics
Volume368
DOIs
StatePublished - Apr 1 2020

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Permittivity
Light polarization
Gratings
Periodicity
Approximation
Fourier series
Slice
Truncation
Galerkin
Unique Solution
Convergence Properties
Convergence Results
Electromagnetic fields
Electromagnetic Fields
Boundary value problems
Expand
Finite Element Method
Boundary Value Problem
Scattering
Vary

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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Analysis of the Rigorous Coupled Wave Approach for s-polarized light in gratings. / Civiletti, Benjamin J.; Lakhtakia, Akhlesh; Monk, Peter B.

In: Journal of Computational and Applied Mathematics, Vol. 368, 112478, 01.04.2020.

Research output: Contribution to journalArticle

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