Abstract.

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high-order perturbation of surfaces/asymptotic waveform evaluation (HOPS/AWE) methods for numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two-layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. This result provides hypotheses under which a rigorous numerical analysis could be conducted on our recently developed HOPS/AWE algorithm.

Keywords

  1. high-order perturbation of surfaces methods
  2. layered media
  3. linear wave scattering
  4. Helmholtz equation
  5. diffraction gratings

MSC codes

  1. 65N35
  2. 78A45
  3. 78B22

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References

1.
T. Abboud and J. C. Nedelec, Electromagnetic waves in an inhomogeneous medium, J. Math. Anal. Appl., 164 (1992), pp. 40–58.
2.
R. A. Adams, Sobolev Spaces, Pure Appl. Math. Ser. 65, Academic Press, New York, 1975.
3.
T. Arens, Scattering by Biperiodic Layered Media: The Integral Equation Approach, habilitationsschrift, Karlsruhe Institute of Technology, 2009.
4.
G. Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal., 32 (1995), pp. 1155–1169.
5.
G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001.
6.
G. Bao, D. C. Dobson, and J. A. Cox, Mathematical studies in rigorous grating theory, J. Opt. Soc. Amer. A, 12 (1995), pp. 1029–1042.
7.
J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185–200.
8.
F. Bleibinhaus and S. Rondenay, Effects of surface scattering in full-waveform inversion, Geophysics, 74 (2009), pp. WCC69–WCC77.
9.
J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, Mineola, NY, 2001.
10.
L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics, Springer-Verlag, Berlin, 1982.
11.
O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), pp. 1168–1175.
12.
O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities, J. Opt. Soc. Am. A, 10 (1993), pp. 2307–2316.
13.
O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries. III. Doubly periodic gratings, J. Opt. Soc. Am. A, 10 (1993), pp. 2551–2562.
14.
J. Chandezon, M. Dupuis, G. Cornet, and D. Maystre, Multicoated gratings: A differential formalism applicable in the entire optical region, J. Opt. Soc. Amer., 72 (1982), p. 839.
15.
J. Chandezon, D. Maystre, and G. Raoult, A new theoretical method for diffraction gratings and its numerical application, J. Opt., 11 (1980), pp. 235–241.
16.
X. Chen and A. Friedman, Maxwell’s equations in a periodic structure, Trans. Amer. Math. Soc., 323 (1991), pp. 465–507.
17.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Appl. Math. Sci. 93, 3rd ed., Springer, New York, 2013.
18.
B. Després, Domain decomposition method and the Helmholtz problem, in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Strasbourg, 1991), SIAM, Philadelphia, 1991, pp. 44–52.
19.
B. Després, Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique. Le théorème de Borg pour l’équation de Hill vectorielle, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, 1991, Thèse, Université de Paris IX (Dauphine), Paris, 1991.
20.
M. O. Deville, P. F. Fischer, and E. H. Mund, High-order Methods for Incompressible Fluid Flow, Cambridge Monogr. Appl. Comput. Math. 9, Cambridge University Press, Cambridge, 2002.
21.
D. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl., 166 (1992), pp. 507–528.
22.
D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, RAIRO Modél. Math. Anal. Numér., 28 (1994), pp. 419–439.
23.
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature, 391 (1998), pp. 667–669.
24.
S. Enoch and N. Bonod, Plasmonics: From Basics to Advanced Topics, Springer Series in Optical Sciences 167, Springer, New York, 2012.
25.
L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, Providence, RI, 2010.
26.
G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, N.J, 1976.
27.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983.
28.
C. Godrèche, Solids Far from Equilibrium, Cambridge University Press, Cambridge, 1992.
29.
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics 26, Society for Industrial and Applied Mathematics, Philadelphia, 1977.
30.
J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts Appl. Math. 54, Springer, New York, 2008.
31.
J. Homola, Surface plasmon resonance sensors for detection of chemical and biological species, Chem. Rev., 108 (2008), pp. 462–493.
32.
Y. Hong and D. P. Nicholls, A rigorous numerical analysis of the transformed field expansion method for diffraction by periodic, layered structures, SIAM J. Numer. Anal., 59 (2021), pp. 456–476.
33.
H. Im, S. H. Lee, N. J. Wittenberg, T. W. Johnson, N. C. Lindquist, P. Nagpal, D. J. Norris, and S.-H. Oh, Template-stripped smooth Ag nanohole arrays with silica shells for surface plasmon resonance biosensing, ACS Nano, 5 (2011), pp. 6244–6253.
34.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.
35.
J. Jose, L. R. Jordan, T. W. Johnson, S. H. Lee, N. J. Wittenberg, and S.-H. Oh, Topographically flat substrates with embedded nanoplasmonic devices for biosensing, Adv. Funct. Mater., 23 (2013), pp. 2812–2820.
36.
T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics 132, Springer-Verlag, Berlin, 1995.
37.
M. Kehoe and D. P. Nicholls, A stable high-order perturbation of surfaces/asymptotic waveform evaluation method for the numerical solution of grating scattering problems, SIAM J. Sci. Comput., 2021, in process.
38.
A. Kirsch, Diffraction by periodic structures, in Inverse Problems in Mathematical Physics (Saariselkä, 1992), Lecture Notes in Phys. 422, Springer, Berlin, 1993, pp. 87–102.
39.
S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Birkhäuser Advanced Texts: Basler Lehrbücher, 2nd ed., Birkhäuser, Boston, MA, 2002.
40.
R. Kress, Linear Integral Equations, 3rd ed., Springer-Verlag, New York, 2014.
41.
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
42.
N. Lassaline, R. Brechbühler, S. Vonk, K. Ridderbeek, M. Spieser, S. Bisig, B. le Feber, F. Rabouw, and D. Norris, Optical Fourier surfaces, Nature, 582 (2020), pp. 506–510.
43.
R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2007.
44.
E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, 2nd ed., American Mathematical Society, Providence, RI, 2001.
45.
N. C. Lindquist, T. W. Johnson, J. Jose, L. M. Otto, and S.-H. Oh, Ultrasmooth metallic films with buried nanostructures for backside reflection-mode plasmonic biosensing, Ann. Phys., 524 (2012), pp. 687–696.
46.
P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, in Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, 1990, pp. 202–223.
47.
S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York, 2007.
48.
D. M. Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves, in Proceedings of SPIE - The International Society for Optical Engineering (San Diego, 1991), vol. 1558, Int. Soc. for Optical Engineering, Bellingham, WA, 1991, pp. 213–221.
49.
D. M. Milder, An improved formalism for wave scattering from rough surfaces, J. Acoust. Soc. Am., 89 (1991), pp. 529–541.
50.
M. Moskovits, Surface-enhanced spectroscopy, Rev. Mod. Phys., 57 (1985), pp. 783–826.
51.
F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, SIAM Monogr. Math. Model. Comput., Society for Industrial and Applied Mathematics, Philadelphia, 2001.
52.
D. P. Nicholls, Three-dimensional acoustic scattering by layered media: A novel surface formulation with operator expansions implementation, Proc. R. Soc. Lond. A, 468 (2012), pp. 731–758.
53.
D. P. Nicholls, Numerical solution of diffraction problems: A high-order perturbation of surfaces/asymptotic waveform evaluation method, SIAM J. Numer. Anal., 55 (2017), pp. 144–167.
54.
D. P. Nicholls, On analyticity of linear waves scattered by a layered medium, J. Differ. Equations, 263 (2017), pp. 5042–5089.
55.
D. P. Nicholls, Numerical simulation of grating structures incorporating two-dimensional materials: A high-order perturbation of surfaces framework, SIAM J. Appl. Math., 78 (2018), pp. 19–44.
56.
D. P. Nicholls and F. Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411–1433.
57.
D. P. Nicholls and F. Reitich, Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys., 170 (2001), pp. 276–298.
58.
D. P. Nicholls and F. Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math., 94 (2003), pp. 107–146.
59.
D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Improved algorithms, J. Opt. Soc. Am. A, 21 (2004), pp. 606–621.
60.
D. P. Nicholls and J. Shen, A rigorous numerical analysis of the transformed field expansion method, SIAM J. Numer. Anal., 47 (2009), pp. 2708–2734.
61.
D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation for Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech., 10 (2008), pp. 238–271.
62.
R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980.
63.
N. A. Phillips, A coordinate system having some special advantages for numerical forecasting, J. Atmos Sci., 14 (1957), pp. 184–185.
64.
H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer, Berlin, 1988.
65.
S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Ser. Comput. Math. 39, Springer-Verlag, Berlin, 2011.
66.
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis, and Appications, Springer Ser. Comput. Math. 41, Springer, Heidelberg, 2011.
67.
J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74 (2009), pp. WCC1–WCC26.
68.
C. H. Wilcox, Scattering Theory for Diffraction Gratings, Springer, Berlin, 1984.

Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 1737 - 1765
ISSN (online): 1095-7154

History

Submitted: 11 February 2022
Accepted: 21 October 2022
Published online: 2 June 2023

Keywords

  1. high-order perturbation of surfaces methods
  2. layered media
  3. linear wave scattering
  4. Helmholtz equation
  5. diffraction gratings

MSC codes

  1. 65N35
  2. 78A45
  3. 78B22

Authors

Affiliations

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 USA.
David P. Nicholls Contact the author
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 USA.

Funding Information

National Science Foundation (NSF): DMS-1813033, DMS-2111283
Funding: The work of the second author was supported by the National Science Foundation grants DMS-1813033 and DMS-2111283.

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