An asymptotic model based on matching far and near field expansions for thin gratings problems

Monk, Peter B.; Rivas, Cinthya; Rodríguez, Rodolfo; Solano, Manuel E.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S507-S533

In this paper, we devise an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surface-relief grating. Far from the grating, we assume that the solution can be written as a power series in terms of the grating thickness δ, the coefficients of this expansion being smooth up to the grating. However, the expansion approximates the solution only sufficiently far from the grating (far field approximation). Near the grating, we assume that there exists another expansion in powers of δ (near field approximation). Moreover, there is an overlapping zone where both expansion are valid. The proposed model is based on matching the two expansions on this overlapping domain. Then, by truncating terms of order δ² or higher, we obtain explicitly the equations satisfied by the lowest order terms in the power series. Under appropriate assumptions, we prove second order convergence of the error with respect to δ. Finally, an alternative form, more convenient for implementation, is derived and discretized with finite elements to perform some numerical tests.

Classification : 65N30, 74M35
Keywords: diffraction grating, thin layers, asymptotic analysis, finite element method

[1] A. Barnett and L. Greengard, A new integral representation for quasi-periodic scattering problems in two dimensions. BIT 51 (2011) 67–90. | MR | Zbl

[2] M. Born and E. Wolf, Principles of Optics, 6th edition. Cambridge University (1980).

[3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition. Springer (2002). | MR | Zbl

[4] N. Chateau and J.-P. Hugonin, Algorithm for the rigorous coupled-wave analysis of grating diffraction. J. Opt. Soc. Am. A 11 (1994) 1321–1331.

[5] Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41 (2004) 799–826. | MR | Zbl

[6] B. J. Civiletti, A. Lakhtakia and P. B. Monk, Analysis of the rigorous coupled wave approach for $s$-polarized light in gratings. J. Comput. Appl. Math. 368 (2019) 112478. | MR | Zbl

[7] B. Delourme, High-order asymptotics for the electromagnetic scattering by thin periodic layers. Math. Methods Appl. Sci. 38 (2015) 811–833. | MR | Zbl

[8] B. Delourme, H. Haddar and P. Joly, Approximate models for wave propagation across thin periodic interfaces. J. Math. Pures Appl. 98 (2012) 28–71. | MR | Zbl

[9] B. Delourme, H. Haddar and P. Joly, On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems. Math. Models Methods Appl. Sci. 23 (2013) 2433–2464. | MR | Zbl

[10] J. Elschner and G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas. Math. Meth. Appl. Sci. 21 (1998) 1297–1342. | MR | Zbl

[11] A. Gillman and A. Barnett, A fast direct solver for quasi-periodic scattering problems. J. Comp. Phys. 248 (2013) 309–322. | MR | Zbl

[12] G. Granet and B. Guizal, Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization. J. Opt. Soc. Am. A 13 (1996) 1019–1023.

[13] H. Haddar, P. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15 (2005) 1273–1300. | MR | Zbl

[14] P. Lalanne and G. M. Morris, Highly improved convergence of the coupled-wave method for TM polarization. J. Opt. Soc. Am. A 13 (1996) 779–784.

[15] L. Li, Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A 13 (1996) 1870–1876.

[16] L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings. J. Opt. Soc. Am. A 14 (1997) 2758–2767.

[17] E. G. Loewen and E. Popov, Diffraction Gratings and Applications. Marcel Dekker (1997).

[18] D. Maystre, editor, Selected Papers on Diffraction Gratings. SPIE Press (1993).

[19] A. Maurel, J.-J. Marigo and A. Ourir, Homogenization of ultrathin metallo-dielectric structures leading to transmission conditions at an equivalent interface. J. Opt. Soc. Am. B. 33 (2016) 947–956.

[20] M. G. Moharam, E. B. Grann and D. A. Pommet, Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J. Opt. Soc. Am. A 12 (1995) 1068–1076.

[21] P. B. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press (2003). | MR | Zbl

[22] Ö. Özdemir, H. Haddar and A. Yaka, Reconstruction of the electromagnetic field in layered media using the concept of approximate transmission conditions. IEEE Trans. Antennas Propag. 59 (2011) 2964–2972. | MR | Zbl

[23] C. Rivas, M. E. Solano, R. Rodriguez, P. B. Monk and A. Lakhtakia, Asymptotic model for finite-element calculations of diffraction by shallow metallic surface-relief gratings. J. Opt. Soc. Am. A 34 (2017) 68–79.

[24] M. E. Solano, M. Faryad, A. S. Hall, T. E. Mallouk, P. B. Monk and A. Lakhtakia, Optimization of the absorption efficiency of an amorphous-silicon thin-film tandem solar cell backed by a metallic surface-relief grating. Appl. Opt. 52 (2013) 966–979.

[25] M. E. Solano, M. Faryad, P. B. Monk, T.E. Mallouk and A. Lakhtakia, Periodically multilayered planar optical concentrator for photovoltaic solar cells. Appl. Phys. Lett. 103 (2013) 191115.

[26] M. E. Solano, M. Faryad, A. Lakhtakia and P. B. Monk, Comparison of rigorous coupled-wave approach and finite element method for photovoltaic devices with periodically corrugated metallic back reflector. J. Opt. Soc. Am. A 31 (2014) 2275–2284.

[27] M. E. Solano, G. D. Barber, A. Lakhtakia, M. Faryad, P. B. Monk and T.E. Mallouk, Buffer layer between a planar optical concentrator and a solar cell. AIP Adv. 5 (2015) 097150.

[28] M. V. Shuba, M. Faryad, M. E. Solano, P. B. Monk and A. Lakhtakia, Adequacy of the rigorous coupled-wave approach for thin-film silicon solar cells with periodically corrugated metallic backreflectors: spectral analysis. J. Opt. Soc. Am. A 32 (2015) 1222–1230.

[29] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition. Artech House (2005). | MR | Zbl