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**Conical diffraction by multilayer gratings: a recursive integral equation approach.**
*(English)*
Zbl 1289.78005

The author treats the solution of a multilayer diffraction gratings problem using a boundary value method.

The optical device is composed of an arbitrary number of different material layers. The interfaces between the layers are assumed to be non-interacting, possible non-smooth, periodic in \(x\)-direction, and translation invariant in \(z\)-direction. The material parameters, the permittivity \(\varepsilon \) and the permeability \(\mu \), can be complex values, that means, also optical metamaterials are included. Considered is the so-called conical or off-plane diffraction, where the direction of the incident wave is not orthogonal to the \(z\)-axis.

The author starts with a short history of the latest progress in the numerical methods for this problem. Based on the papers [ D. Maystre, ” A new general integral theory for dielectric coated gratings”, J. Opt. Soc. Am. 68, 490–495 (1978)], [L. I. Goray and G. Schmidt, J. Opt. Soc. Am. A 27, 585–597 (2010)], and [G. Schmidt and B. H. Kleemann, J. Mod. Opt. 58, No. 5–6, 407–423 (2011, Zbl 1221.78050)] the recursive algorithm, the discretization method, and the analysis of the method, respectively, are described.

Starting from the time-harmonic Maxwell equations the scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients including special transmission conditions at the interfaces of the layers. Applying direct and indirect boundary integral methods gives a system of \(2N \times 2N\) singular integral equations, where \(N\) is the number of interfaces. A recursive procedure is presented that treats on each step the problem for one of the interfaces with \(2 \times 2\) operator matrices allowing to solve the problem on a standard personal computer.

Analyzing the algorithm necessary and sufficient conditions for the applicability of the method are formulated in proved theorems. The equations are unique solvable if and only if the matrix operator is invertible. Especially, the solution is unique if the imaginary parts of the material parameters \(\varepsilon \) and \(\mu \) are positive.

The optical device is composed of an arbitrary number of different material layers. The interfaces between the layers are assumed to be non-interacting, possible non-smooth, periodic in \(x\)-direction, and translation invariant in \(z\)-direction. The material parameters, the permittivity \(\varepsilon \) and the permeability \(\mu \), can be complex values, that means, also optical metamaterials are included. Considered is the so-called conical or off-plane diffraction, where the direction of the incident wave is not orthogonal to the \(z\)-axis.

The author starts with a short history of the latest progress in the numerical methods for this problem. Based on the papers [ D. Maystre, ” A new general integral theory for dielectric coated gratings”, J. Opt. Soc. Am. 68, 490–495 (1978)], [L. I. Goray and G. Schmidt, J. Opt. Soc. Am. A 27, 585–597 (2010)], and [G. Schmidt and B. H. Kleemann, J. Mod. Opt. 58, No. 5–6, 407–423 (2011, Zbl 1221.78050)] the recursive algorithm, the discretization method, and the analysis of the method, respectively, are described.

Starting from the time-harmonic Maxwell equations the scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients including special transmission conditions at the interfaces of the layers. Applying direct and indirect boundary integral methods gives a system of \(2N \times 2N\) singular integral equations, where \(N\) is the number of interfaces. A recursive procedure is presented that treats on each step the problem for one of the interfaces with \(2 \times 2\) operator matrices allowing to solve the problem on a standard personal computer.

Analyzing the algorithm necessary and sufficient conditions for the applicability of the method are formulated in proved theorems. The equations are unique solvable if and only if the matrix operator is invertible. Especially, the solution is unique if the imaginary parts of the material parameters \(\varepsilon \) and \(\mu \) are positive.

Reviewer: Georg Hebermehl (Berlin)

### MSC:

78A45 | Diffraction, scattering |

78M15 | Boundary element methods applied to problems in optics and electromagnetic theory |

45E05 | Integral equations with kernels of Cauchy type |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65R20 | Numerical methods for integral equations |

### Keywords:

diffraction; periodic structure; multilayer grating; singular integral formulation; recursive algorithm### Citations:

Zbl 1221.78050
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\textit{G. Schmidt}, Appl. Math., Praha 58, No. 3, 279--307 (2013; Zbl 1289.78005)

### References:

[1] | B.V. Bazalij, V.Yu. Shelepov: On the spectrum of the potential of a double layer on a curve of bounded rotation. Boundary Value Problems for Differential Equations. Collect. Sci. Works, Kiev, 1980, pp. 13-30. (In Russian.) · Zbl 0462.47034 |

[2] | J. Elschner, R. Hinder, F. Penzel, G. Schmidt: Existence, uniqueness and regularity for solutions of the conical diffraction problem. Math. Models Methods Appl. Sci. 10 (2000), 317-341. · Zbl 1010.78008 |

[3] | J. Elschner, M. Yamamoto: An inverse problem in periodic diffractive optics: Reconstruction of Lipschitz grating profiles. Appl. Anal. 81 (2002), 1307-1328. · Zbl 1028.78008 |

[4] | L. I. Goray, G. Schmidt: Solving conical diffraction grating problems with integral equations. J. Opt. Soc. Am. A 27 (2010), 585-597. |

[5] | L. I. Goray, J. F. Seely, S. Yu. Sadov: Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme-ultraviolet gratings at near normal incidence. J. Appl. Physics 100 (2006). |

[6] | V.Yu. Gotlib: On solutions of the Helmholtz equation that are concentrated near a plane periodic boundary. J. Math. Sci., New York 102, 4188-4194; Zap. Nauchn. Semin. POMI 250 (1998), 83-96. (In Russian.) · Zbl 1071.35512 |

[7] | D. Maystre: A new general integral theory for dielectric coated gratings. J. Opt. Soc. Am. 68 (1978), 490-495. |

[8] | D. Maystre: Electromagnetic study of photonic band gaps. Pure Appl. Opt. 3 (1994), 975-993. |

[9] | R. Petit, ed.:, Electromagnetic theory of gratings. Topics in Current Physics, 22, Springer, Berlin, 1980. · Zbl 1221.78050 |

[10] | G. Schmidt: Integral equations for conical diffraction by coated grating. J. Integral Equations Appl. 23 (2011), 71-112. · Zbl 1241.78015 |

[11] | Schmidt, G.; Laptev, A. (ed.), Boundary integral methods for periodic scattering problems, No. 12, 337-363 (2010), Dordrecht · Zbl 1189.78026 |

[12] | G. Schmidt, B.H. Kleemann: Integral equation methods from grating theory to photonics: An overview and new approaches for conical diffraction. J. Mod. Opt 58 (2011), 407-423. · Zbl 1221.78050 |

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