### 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 language | English (US) |
---|---|

Article number | 112478 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 368 |

DOIs | |

State | Published - Apr 1 2020 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational and Applied Mathematics*,

*368*, [112478]. https://doi.org/10.1016/j.cam.2019.112478

}

*Journal of Computational and Applied Mathematics*, vol. 368, 112478. https://doi.org/10.1016/j.cam.2019.112478

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Analysis of the Rigorous Coupled Wave Approach for s-polarized light in gratings

AU - Civiletti, Benjamin J.

AU - Lakhtakia, Akhlesh

AU - Monk, Peter B.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85072869397&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072869397&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2019.112478

DO - 10.1016/j.cam.2019.112478

M3 - Article

AN - SCOPUS:85072869397

VL - 368

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 112478

ER -