Abstract
This chapter is devoted to the well-posedness of the grating problems which are presented in Chap. 2. The scattering problems in periodic structures have been studied extensively and a great number of mathematical results are available [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The general result may be stated as follows: the grating problem has a unique solution for all but possibly a countable sequence of frequencies. Unique solvability for all frequencies can be obtained for gratings, which either contain lossy media with nonzero conductivity or have perfectly electrically conducting surfaces with Lipschitz profiles.
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References
T. Abboud, Etude mathématique et numérique de quelques problémes de diffraction d’ondes électromagnétiques, Ph.D. thesis, Ecole Polytechnique, Palaiseau (1991)
T. Abboud, Electromagnetic waves in periodic media, in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, ed. by R. Kleinman et. al. (SIAM, Philadelphia, 1993), pp. 1–9
H. Ammari, G. Bao, Analysis of the diffraction from periodic chiral structures. C. R. Acad. Sci. Paris Sér. I Math. 326, 1371–1376 (1998)
H. Ammari, G. Bao, Maxwell’s equations in periodic chiral structures. Math. Nachr. 251, 3–18 (2003)
H. Ammari, N. Béreux, J.-C. Nédélec, Resonant frequencies for a narrow strip grating. Math. Methods Appl. Sci. 22, 1121–1152 (1999)
H. Ammari, N. Béreux, J.-C. Nédélec, Resonances for Maxwell’s equations in a periodic structure. Japan J. Appl. Math. 17, 149–198 (2000)
H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, H. Zhang, Mathematical and Computational Methods in Photonics and Phononics, Mathematical Surveys and Monographs, vol. 235 (American Mathematical Society, Providence, 2018)
G. Bao, D. Dobson, On the scattering by a biperiodic structure. Proc. Am. Math. Soc. 128, 2715–2723 (2000)
G. Bao, D. Dobson, J.A. Cox, Mathematical studies in rigorous grating theory. J. Opt. Soc. Am. A 12, 1029–1042 (1995)
A. Bonnet-Bendhia, F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Meth. Appl. Sci. 17, 305–338 (1994)
M. Cadilhac, Some mathematical aspects of the grating theory, in Electromagnetic Theory of Gratings, ed. by Petit (1980), pp. 53–62
X. Chen, A. Friedman, Maxwell’s equations in a periodic structure. Trans. Am. Math. Soc. 323, 465–507 (1991)
D. Dobson, A variational method for electromagnetic diffraction in biperiodic structures. Modél. Math. Anal. Numér. 28, 419–439 (1994)
D. Dobson, A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166, 507–528 (1992)
T.K. Gaylord, M.G. Moharam, Analysis and applications of optical diffraction by gratings. Proc. IEEE 73, 894–937 (1985)
A. Kirsch, Diffraction by periodic structures, in Proceedings of Lapland Conferences Inverse Problems, ed. by L. Päivärinta, E. Somersalo (Springer, Berlin, 1993), pp. 87–102
L. Li, A model analysis of lamellar diffraction gratings in conical mountings. J. Mod. Opt. 40, 553–573 (1993)
L. Li, Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction grating. J. Opt. Soc. Am. A 13, 1024–1035 (1996)
E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997)
G. Schmidt, On the diffraction by biperiodic anisotropic structures. Appl. Anal. 82, 75–92 (2010)
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, 407–423 (2011)
D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983)
J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, New York, 2000)
A. Kirsch, F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell’s Equations: Expansion-, Integral-, and Variational Methods (Springer International Publishing, Switzerland, 2015)
R. Adams, Sobolev Spaces (Academic Press, New York, 1975)
K. Yosida, Functional Analysis, 6th edn. (Springer, New York, 1995)
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, New York, 1983)
S.G. Mikhlin, Variational Methods in Mathematical Physics (Pergamon Press, Oxford, 1964)
R. Li, Galerkin Methods for Boundary Value Problems, Shanghai Scientific Publications Co., China (1988)
G. Bao, Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32, 1155–1169 (1995)
D. Dobson, Optimal design of periodic antireflective structures for the Helmholtz equation. Euro. J. Appl. Math. 4, 321–339 (1993)
J. Elschner, M. Yamamoto, An inverse problem in periodic diffractive optics: reconstruction of Lipschitz grating profiles. Appl. Anal. 81, 1307–1328 (2002)
G. Bao, Numerical analysis of diffraction by periodic structures: TM polarization. Numer. Math. 75, 1–16 (1996)
Ch. Weber, A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2, 12–25 (1980)
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Bao, G., Li, P. (2022). Variational Formulations. In: Maxwell’s Equations in Periodic Structures. Applied Mathematical Sciences, vol 208. Springer, Singapore. https://doi.org/10.1007/978-981-16-0061-6_3
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DOI: https://doi.org/10.1007/978-981-16-0061-6_3
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