A high-order perturbation of envelopes (HOPE) method for scattering by periodic inhomogeneous media
Author:
David P. Nicholls
Journal:
Quart. Appl. Math. 78 (2020), 725-757
MSC (2010):
Primary 65N35, 78M22, 78A45, 35J25, 35Q60, 35Q86.
DOI:
https://doi.org/10.1090/qam/1568
Published electronically:
March 16, 2020
MathSciNet review:
4148825
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Abstract: The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.
References
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- Gang Bao, Lawrence Cowsar, and Wen Masters (eds.), Mathematical modeling in optical science, Frontiers in Applied Mathematics, vol. 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. MR 1831328
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- Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Co., New York, 1978. International Series in Pure and Applied Mathematics. MR 538168
- L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of ocean acoustics, 2nd ed., Springer Series on Wave Phenomena, vol. 8, Springer-Verlag, Berlin, 1991. MR 1100922
- David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 3rd ed., Applied Mathematical Sciences, vol. 93, Springer, New York, 2013. MR 2986407
- M. O. Deville, P. F. Fischer, and E. H. Mund, High-order methods for incompressible fluid flow, Cambridge Monographs on Applied and Computational Mathematics, vol. 9, Cambridge University Press, Cambridge, 2002. MR 1929237
- D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, RAIRO Modél. Math. Anal. Numér. 28 (1994), no. 4, 419–439. MR 1288506, DOI https://doi.org/10.1051/m2an/1994280404191
- David C. Dobson, Optimal design of periodic antireflective structures for the Helmholtz equation, European J. Appl. Math. 4 (1993), no. 4, 321–339. MR 1251818, DOI https://doi.org/10.1017/S0956792500001169
- 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), no. 6668, 667–669.
- I. El-Sayed, X. Huang, and M. El-Sayed, Selective laser photo-thermal therapy of epithelial carcinoma using anti-egfr antibody conjugated gold nanoparticles, Cancer Lett. 239 (2006), no. 1, 129–135.
- O. G. Ernst and M. J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, Numerical analysis of multiscale problems, Lect. Notes Comput. Sci. Eng., vol. 83, Springer, Heidelberg, 2012, pp. 325–363. MR 3050918, DOI https://doi.org/10.1007/978-3-642-22061-6_10
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
- Xiaobing Feng, Junshan Lin, and Cody Lorton, An efficient numerical method for acoustic wave scattering in random media, SIAM/ASA J. Uncertain. Quantif. 3 (2015), no. 1, 790–822. MR 3394366, DOI https://doi.org/10.1137/140958232
- Xiaobing Feng, Junshan Lin, and Cody Lorton, A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients, Int. J. Uncertain. Quantif. 6 (2016), no. 5, 429–443. MR 3636926, DOI https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016016805
- L. Geli, P. Y. Bard, and B. Jullien, The effect of topography on earthquake ground motion: a review and new results, Bulletin of the Seismological Society of America 78 (1988), no. 1, 42–63.
- David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR 0520152
- Jan S. Hesthaven and Tim Warburton, Nodal discontinuous Galerkin methods, Texts in Applied Mathematics, vol. 54, Springer, New York, 2008. Algorithms, analysis, and applications. MR 2372235
- J. Homola, Surface plasmon resonance sensors for detection of chemical and biological species, Chemical Reviews 108 (2008), no. 2, 462–493.
- Frank Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences, vol. 132, Springer-Verlag, New York, 1998. MR 1639879
- Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1987. MR 925005
- 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), 2812–2820.
- Rainer Kress, Linear integral equations, 3rd ed., Applied Mathematical Sciences, vol. 82, Springer, New York, 2014. MR 3184286
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Randall J. LeVeque, Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Steady-state and time-dependent problems. MR 2378550
- 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, Annalen der Physik 524 (2012), 687–696.
- Andrea Moiola and Euan A. Spence, Is the Helmholtz equation really sign-indefinite?, SIAM Rev. 56 (2014), no. 2, 274–312. MR 3201183, DOI https://doi.org/10.1137/120901301
- M. Moskovits, Surface–enhanced spectroscopy, Reviews of Modern Physics 57 (1985), no. 3, 783–826.
- David P. Nicholls, High-order spectral simulation of graphene ribbons, Commun. Comput. Phys. 26 (2019), no. 5, 1575–1596. MR 3997393, DOI https://doi.org/10.4208/cicp.2019.js60.13
- David P. Nicholls and Fernando Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 6, 1411–1433. MR 1869643, DOI https://doi.org/10.1017/S0308210500001463
- David P. Nicholls and Fernando Reitich, Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys. 170 (2001), no. 1, 276–298. MR 1843612, DOI https://doi.org/10.1006/jcph.2001.6737
- David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math. 94 (2003), no. 1, 107–146. MR 1971215, DOI https://doi.org/10.1007/s002110200399
- D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A 21 (2004), no. 4, 590–605.
- D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Improved algorithms, J. Opt. Soc. Am. A 21 (2004), no. 4, 606–621.
- D. P. Nicholls, F. Reitich, T. W. Johnson, and S.-H. Oh, Fast high-order perturbation of surfaces (HOPS) methods for simulation of multi–layer plasmonic devices and metamaterials, Journal of the Optical Society of America, A 31 (2014), no. 8, 1820–1831.
- David P. Nicholls and Mark Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech. 10 (2008), no. 2, 238–271. MR 2411413, DOI https://doi.org/10.1007/s00021-006-0231-9
- David P. Nicholls, High-order perturbation of surfaces short course: analyticity theory, Lectures on the theory of water waves, London Math. Soc. Lecture Note Ser., vol. 426, Cambridge Univ. Press, Cambridge, 2016, pp. 32–50. MR 3409888
- David P. Nicholls, High-order perturbation of surfaces short course: boundary value problems, Lectures on the theory of water waves, London Math. Soc. Lecture Note Ser., vol. 426, Cambridge Univ. Press, Cambridge, 2016, pp. 1–18. MR 3409886
- Roger Petit (ed.), Electromagnetic theory of gratings, Topics in Current Physics, vol. 22, Springer-Verlag, Berlin-New York, 1980. MR 609533
- Stefan A. Sauter and Christoph Schwab, Boundary element methods, Springer Series in Computational Mathematics, vol. 39, Springer-Verlag, Berlin, 2011. Translated and expanded from the 2004 German original. MR 2743235
- Jie Shen and Tao Tang, Spectral and high-order methods with applications, Mathematics Monograph Series, vol. 3, Science Press Beijing, Beijing, 2006. MR 2723481
- Jie Shen, Tao Tang, and Li-Lian Wang, Spectral methods, Springer Series in Computational Mathematics, vol. 41, Springer, Heidelberg, 2011. Algorithms, analysis and applications. MR 2867779
- P. J. Shull, Nondestructive evaluation: Theory, techniques, and applications, Marcel Dekker, 2002.
- John C. Strikwerda, Finite difference schemes and partial differential equations, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. MR 2124284
- L. Tsang, J. A. Kong, and R. T. Shin, Theory of microwave remote sensing, Wiley, New York, 1985.
- J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics 74 (2009), no. 6, WCC1–WCC26.
- P. Yeh, Optical waves in layered media, vol. 61, Wiley-Interscience, 2005.
References
- T. Arens, Scattering by biperiodic layered media: The integral equation approach, Habilitationsschrift, Karlsruhe Institute of Technology, 2009.
- George A. Baker Jr. and Peter Graves-Morris, Padé approximants, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 59, Cambridge University Press, Cambridge, 1996. MR 1383091
- Gang Bao, Lawrence Cowsar, and Wen Masters (eds.), Mathematical modeling in optical science, Frontiers in Applied Mathematics, vol. 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. MR 1831328
- Gang Bao, David C. Dobson, and J. Allen Cox, Mathematical studies in rigorous grating theory, J. Opt. Soc. Amer. A 12 (1995), no. 5, 1029–1042. MR 1335399, DOI https://doi.org/10.1364/JOSAA.12.001029
- Gang Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal. 32 (1995), no. 4, 1155–1169. MR 1342287, DOI https://doi.org/10.1137/0732053
- Gang Bao, Numerical analysis of diffraction by periodic structures: TM polarization, Numer. Math. 75 (1996), no. 1, 1–16. MR 1417860, DOI https://doi.org/10.1007/s002110050227
- Gang Bao and David C. Dobson, Nonlinear optics in periodic diffractive structures, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993) SIAM, Philadelphia, PA, 1993, pp. 30–38. MR 1227826
- Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Co., New York, 1978. International Series in Pure and Applied Mathematics. MR 538168
- L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of ocean acoustics, 2nd ed., Springer Series on Wave Phenomena, vol. 8, Springer-Verlag, Berlin, 1991. MR 1100922
- David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 3rd ed., Applied Mathematical Sciences, vol. 93, Springer, New York, 2013. MR 2986407
- M. O. Deville, P. F. Fischer, and E. H. Mund, High-order methods for incompressible fluid flow, Cambridge Monographs on Applied and Computational Mathematics, vol. 9, Cambridge University Press, Cambridge, 2002. MR 1929237
- D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, RAIRO Modél. Math. Anal. Numér. 28 (1994), no. 4, 419–439. MR 1288506, DOI https://doi.org/10.1051/m2an/1994280404191
- David C. Dobson, Optimal design of periodic antireflective structures for the Helmholtz equation, European J. Appl. Math. 4 (1993), no. 4, 321–339. MR 1251818, DOI https://doi.org/10.1017/S0956792500001169
- 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), no. 6668, 667–669.
- I. El-Sayed, X. Huang, and M. El-Sayed, Selective laser photo-thermal therapy of epithelial carcinoma using anti-egfr antibody conjugated gold nanoparticles, Cancer Lett. 239 (2006), no. 1, 129–135.
- O. G. Ernst and M. J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, Numerical analysis of multiscale problems, Lect. Notes Comput. Sci. Eng., vol. 83, Springer, Heidelberg, 2012, pp. 325–363. MR 3050918, DOI https://doi.org/10.1007/978-3-642-22061-6_10
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
- Xiaobing Feng, Junshan Lin, and Cody Lorton, An efficient numerical method for acoustic wave scattering in random media, SIAM/ASA J. Uncertain. Quantif. 3 (2015), no. 1, 790–822. MR 3394366, DOI https://doi.org/10.1137/140958232
- Xiaobing Feng, Junshan Lin, and Cody Lorton, A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients, Int. J. Uncertain. Quantif. 6 (2016), no. 5, 429–443. MR 3636926, DOI https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016016805
- L. Geli, P. Y. Bard, and B. Jullien, The effect of topography on earthquake ground motion: a review and new results, Bulletin of the Seismological Society of America 78 (1988), no. 1, 42–63.
- David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR 0520152
- Jan S. Hesthaven and Tim Warburton, Nodal discontinuous Galerkin methods, Texts in Applied Mathematics, vol. 54, Springer, New York, 2008. Algorithms, analysis, and applications. MR 2372235
- J. Homola, Surface plasmon resonance sensors for detection of chemical and biological species, Chemical Reviews 108 (2008), no. 2, 462–493.
- Frank Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences, vol. 132, Springer-Verlag, New York, 1998. MR 1639879
- Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1987. MR 925005
- 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), 2812–2820.
- Rainer Kress, Linear integral equations, 3rd ed., Applied Mathematical Sciences, vol. 82, Springer, New York, 2014. MR 3184286
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
- Randall J. LeVeque, Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Steady-state and time-dependent problems. MR 2378550
- 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, Annalen der Physik 524 (2012), 687–696.
- Andrea Moiola and Euan A. Spence, Is the Helmholtz equation really sign-indefinite?, SIAM Rev. 56 (2014), no. 2, 274–312. MR 3201183, DOI https://doi.org/10.1137/120901301
- M. Moskovits, Surface–enhanced spectroscopy, Reviews of Modern Physics 57 (1985), no. 3, 783–826.
- David P. Nicholls, High-order spectral simulation of graphene ribbons, Commun. Comput. Phys. 26 (2019), no. 5, 1575–1596. MR 3997393, DOI https://doi.org/10.4208/cicp.2019.js60.13
- David P. Nicholls and Fernando Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 6, 1411–1433. MR 1869643, DOI https://doi.org/10.1017/S0308210500001463
- David P. Nicholls and Fernando Reitich, Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys. 170 (2001), no. 1, 276–298. MR 1843612, DOI https://doi.org/10.1006/jcph.2001.6737
- David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math. 94 (2003), no. 1, 107–146. MR 1971215, DOI https://doi.org/10.1007/s002110200399
- D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A 21 (2004), no. 4, 590–605.
- D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Improved algorithms, J. Opt. Soc. Am. A 21 (2004), no. 4, 606–621.
- D. P. Nicholls, F. Reitich, T. W. Johnson, and S.-H. Oh, Fast high-order perturbation of surfaces (HOPS) methods for simulation of multi–layer plasmonic devices and metamaterials, Journal of the Optical Society of America, A 31 (2014), no. 8, 1820–1831.
- David P. Nicholls and Mark Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech. 10 (2008), no. 2, 238–271. MR 2411413, DOI https://doi.org/10.1007/s00021-006-0231-9
- David P. Nicholls, High-order perturbation of surfaces short course: analyticity theory, Lectures on the theory of water waves, London Math. Soc. Lecture Note Ser., vol. 426, Cambridge Univ. Press, Cambridge, 2016, pp. 32–50. MR 3409888
- David P. Nicholls, High-order perturbation of surfaces short course: boundary value problems, Lectures on the theory of water waves, London Math. Soc. Lecture Note Ser., vol. 426, Cambridge Univ. Press, Cambridge, 2016, pp. 1–18. MR 3409886
- Roger Petit (ed.), Electromagnetic theory of gratings, Topics in Current Physics, vol. 22, Springer-Verlag, Berlin-New York, 1980. MR 609533
- Stefan A. Sauter and Christoph Schwab, Boundary element methods, Springer Series in Computational Mathematics, vol. 39, Springer-Verlag, Berlin, 2011. Translated and expanded from the 2004 German original. MR 2743235
- Jie Shen and Tao Tang, Spectral and high-order methods with applications, Mathematics Monograph Series, vol. 3, Science Press Beijing, Beijing, 2006. MR 2723481
- Jie Shen, Tao Tang, and Li-Lian Wang, Spectral methods, Algorithms, analysis and applications, Springer Series in Computational Mathematics, vol. 41, Springer, Heidelberg, 2011. MR 2867779
- P. J. Shull, Nondestructive evaluation: Theory, techniques, and applications, Marcel Dekker, 2002.
- John C. Strikwerda, Finite difference schemes and partial differential equations, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. MR 2124284
- L. Tsang, J. A. Kong, and R. T. Shin, Theory of microwave remote sensing, Wiley, New York, 1985.
- J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics 74 (2009), no. 6, WCC1–WCC26.
- P. Yeh, Optical waves in layered media, vol. 61, Wiley-Interscience, 2005.
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Additional Information
David P. Nicholls
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
MR Author ID:
635039
Email:
davidn@uic.edu
Keywords:
Linear wave scattering,
Helmholtz equations,
Maxwell equations,
inhomogeneous media,
layered media,
high-order spectral methods,
high-order perturbation of envelopes methods
Received by editor(s):
October 9, 2019
Received by editor(s) in revised form:
January 20, 2020
Published electronically:
March 16, 2020
Additional Notes:
The author gratefully acknowledges support from the National Science Foundation through grant No. DMS–1813033.
Article copyright:
© Copyright 2020
Brown University
