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High-Order Perturbation of Surfaces Algorithms for the Simulation of Localized Surface Plasmon Resonances in Two Dimensions

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Abstract

The capability of simulating scattering returns of electromagnetic radiation from bounded obstacles is of overwhelming importance to scientists and engineers. Furthermore, such simulations must be of both surpassing accuracy and high fidelity for many applications of interest. High-Order Spectral methods deliver highly accurate simulations with a relatively small number of degrees of freedom, while interfacial formulations which utilize these discretizations have orders of magnitude smaller execution times and memory requirements. Among these, the High-Order Perturbation of Surfaces algorithms have proved to be a method of choice in layered media applications, and we display here how two of these-the Methods of Field Expansions and Transformed Field Expansions-extend to obstacles of bounded cross-section. In this contribution we provide not only a detailed prescription of the algorithms, but also validate the schemes and point out their benefits and shortcomings. With numerical experiments we show the remarkable efficiency, fidelity, and high-order accuracy one can achieve with implementations of these algorithms.

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References

  1. Baker Jr., George A.: Padé Approximants, 2nd edn. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  2. Bender, Carl M, Orszag, Steven A: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill Book Co, New York (1978). International Series in Pure and Applied Mathematics

    MATH  Google Scholar 

  3. Boyd, John P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications Inc., Mineola (2001)

    MATH  Google Scholar 

  4. Bruno, O., Reitich, F.: Numerical solution of diffraction problems: a method of variation of boundaries. J. Opt. Soc. Am. A 10(6), 1168–1175 (1993)

    Article  Google Scholar 

  5. Bruno, O., Reitich, F.: 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(11), 2307–2316 (1993)

    Article  Google Scholar 

  6. Bruno, O., Reitich, F.: Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings. J. Opt. Soc. Am. A 10(12), 2551–2562 (1993)

    Article  Google Scholar 

  7. Bruno, Oscar P., Reitich, Fernando: Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions. J. Acoust. Soc. Am. 104(5), 2579–2583 (1998)

    Article  Google Scholar 

  8. Burggraf, O.R.: Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151 (1966)

    Article  Google Scholar 

  9. Coifman, R., Goldberg, M., Hrycak, T., Israeli, M., Rokhlin, V.: An improved operator expansion algorithm for direct and inverse scattering computations. Waves Random Media 9(3), 441–457 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  11. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  12. Deville, M.O., Fischer, P.F., Mund, E.H.: High-Order Methods for Incompressible Fluid Flow, volume 9 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  13. Enoch, S., Bonod, N.: Plasmonics: From Basics to Advanced Topics. Springer Series in Optical Sciences. Springer, New York (2012)

    Book  Google Scholar 

  14. El-Sayed, I., Huang, X., El-Sayed, M.: Selective laser photo-thermal therapy of epithelial carcinoma using anti-egfr antibody conjugated gold nanoparticles. Cancer Lett. 239(1), 129–135 (2006)

    Article  Google Scholar 

  15. Evans, Lawrence C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  16. Gottlieb, D., Orszag, S.A.: 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

  17. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, B., Nicholls, D.P.: Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains. SIAM J. Math. Anal. 37(1), 302–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hesthaven, J.S., Warburton, T.: Nodal discontinuous Galerkin methods, volume 54 of Texts in Applied Mathematics. In: Algorithms, analysis, and applications. Springer, New York (2008)

  20. Ihlenburg, Frank: Finite Element Analysis of Acoustic Scattering. Springer, New York (1998)

    Book  MATH  Google Scholar 

  21. Johnson, Claes: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  22. Knupp, P., Salari, K.: Verification of Computer Codes in Computational Science and Engineering. Chapman and Hall/CRC, Boca Raton (2003)

    MATH  Google Scholar 

  23. LeVeque, Randall J.: Finite difference methods for ordinary and partial differential equations. In: Steady-state and time-dependent problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007)

  24. Liu, M., Guyot-Sionnest, P., Lee, T.-W., Gray, S.: Optical properties of rodlike and bipyramidal gold nanoparticles from three-dimensional computations. Phys. Rev. B 76, 235428 (2007)

    Article  Google Scholar 

  25. Loo, C., Lowery, A., Halas, N., West, J., Drezek, R.: Immunotargeted nanoshells for integrated cancer imaging and therapy. Nano Lett. 5(4), 709–711 (2005)

    Article  Google Scholar 

  26. Maier, S.A.: Plasmonics: Fundamentals and Applications. Springer, New York (2007)

    Book  Google Scholar 

  27. Martin, P.A.: Multiple scattering, volume 107 of Encyclopedia of Mathematics and its Applications. In: Interaction of time-harmonic waves with \(N\) obstacles. Cambridge University Press, Cambridge (2006)

  28. Myroshnychenko, V., Carbo-Argibay, E., Pastoriza-Santos, I., Perez-Juste, J., Liz-Marzan, L., Garcia de Abajo, F.: Modeling the optical response of highly faceted metal nanoparticles with a fully 3d boundary element method. Adv. Mater. 20, 4288–4293 (2008)

    Article  Google Scholar 

  29. Milder, D. M.: 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), volume 1558, pp. 213–221. International Society for Optical Engineering, Bellingham (1991)

  30. Milder, D. M.: An improved formalism for wave scattering from rough surfaces. J. Acoust. Soc. Am. 89(2), 529–541 (1991)

  31. Milder, D. M.: An improved formalism for electromagnetic scattering from a perfectly conducting rough surface. Radio Sci. 31(6), 1369–1376 (1996)

  32. Milder, D. M.: Role of the admittance operator in rough-surface scattering. J. Acoust. Soc. Am. 100(2), 759–768 (1996)

  33. Myroshnychenko, V., Rodriguez-Fernandez, J., Pastoriza-Santos, I., Funston, A., Novo, C., Mulvaney, P., Liz-Marzan, L., Garcia de Abajo, J.: Modelling the optical response of gold nanoparticles. Chem. Soc. Rev. 37, 1792–1805 (2008)

    Article  Google Scholar 

  34. Milder, D. M., Sharp, H. Thomas: Efficient computation of rough surface scattering. In: Mathematical and Numerical Aspects of Wave Propagation Phenomena (Strasbourg, 1991), pp. 314–322. SIAM, Philadelphia (1991)

  35. Milder, D. M., Sharp, H.Thomas: An improved formalism for rough surface scattering. ii: Numerical trials in three dimensions. J. Acoust. Soc. Am. 91(5), 2620–2626 (1992)

  36. Novotny, L., Hecht, B.: Principles of Nano-Optics, 2nd edn. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  37. Nicholls, D.P.: Three-dimensional acoustic scattering by layered media: a novel surface formulation with operator expansions implementation. Proc. R. Soc. Lond. A 468, 731–758 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nicholls, D.P.: A method of field expansions for vector electromagnetic scattering by layered periodic crossed gratings. J. Opt. Soc. Am. A 32(5), 701–709 (2015)

    Article  Google Scholar 

  39. Nicholls, D.P.: On analyticity of linear waves scattered by a layered medium. J. Differ. Equ. 263(8), 5042–5089 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nicholls, D.P., Nigam, N.: Exact non-reflecting boundary conditions on general domains. J. Comput. Phys. 194(1), 278–303 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nicholls, D.P., Oh, S.-H., Johnson, T.W., Reitich, F.: Launching surface plasmon waves via vanishingly small periodic gratings. J. Opt. Soc. Am. A 33(3), 276–285 (2016)

    Article  Google Scholar 

  42. Nicholls, D.P., Reitich, F.: A new approach to analyticity of Dirichlet–Neumann operators. Proc. R. Soc. Edinb. Sect. A 131(6), 1411–1433 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Nicholls, D.P., Reitich, F.: Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators. J. Comput. Phys. 170(1), 276–298 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nicholls, D.P., Reitich, F.: Analytic continuation of Dirichlet–Neumann operators. Numer. Math. 94(1), 107–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nicholls, D.P., Reitich, F.: Shape deformations in rough surface scattering: cancellations, conditioning, and convergence. J. Opt. Soc. Am. A 21(4), 590–605 (2004)

    Article  Google Scholar 

  46. Nicholls, D.P., Reitich, F.: Shape deformations in rough surface scattering: improved algorithms. J. Opt. Soc. Am. A 21(4), 606–621 (2004)

    Article  Google Scholar 

  47. Nicholls, D.P., Shen, J.: A stable, high-order method for two-dimensional bounded-obstacle scattering. SIAM J. Sci. Comput. 28(4), 1398–1419 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nicholls, D.P., Tammali, V.: A high-order perturbation of surfaces (hops) approach to Fokas integral equations: vector electromagnetic scattering by periodic crossed gratings. Appl. Numer. Methods 101, 1–17 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Oberkampf, W.L., Trucano, T.G., Hirsch, C.: Verification, validation, and predictive capability in computational engineering and physics. Appl. Mech. Rev. 57(5), 345–384 (2004)

    Article  Google Scholar 

  50. Petit, R. (ed.): Electromagnetic Theory of Gratings. Springer, Berlin (1980)

    Google Scholar 

  51. Raether, H.: Surface Plasmons on Smooth and Rough Surfaces and on Gratings. Springer, Berlin (1988)

    Book  Google Scholar 

  52. Rayleigh, Lord: On the dynamical theory of gratings. Proc. R. Soc. Lond. A79, 399–416 (1907)

    Article  MATH  Google Scholar 

  53. Rice, S.O.: Reflection of electromagnetic waves from slightly rough surfaces. Commun. Pure Appl. Math. 4, 351–378 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  54. Roache, P.J.: Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque (1998)

    Google Scholar 

  55. Roache, P.J.: Verification of codes and calculations. AIAA J. 36(5), 696–702 (1998)

    Article  Google Scholar 

  56. Roache, P.J.: Code verification by the method of manufactured solutions. J. Fluids Eng. 124(1), 4–10 (2002)

    Article  Google Scholar 

  57. Roy, C.J.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205(1), 131–156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Reitich, F., Tamma, K.: State-of-the-art, trends, and directions in computational electromagnetics. CMES Comput. Model. Eng. Sci. 5(4), 287–294 (2004)

    MathSciNet  MATH  Google Scholar 

  59. Strikwerda, John C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2004)

    MATH  Google Scholar 

  60. Xu, H., Bjerneld, E., Käll, M., Börjesson, L.: Spectroscopy of single hemoglobin molecules by surface enhanced raman scattering. Phys. Rev. Lett. 83, 4357–4360 (1999)

    Article  Google Scholar 

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Acknowledgements

D.P.N. gratefully acknowledges support from the National Science Foundation through Grant No. DMS-1522548.

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  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    David P. Nicholls & Xin Tong

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  1. David P. Nicholls
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Correspondence to David P. Nicholls.

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Nicholls, D.P., Tong, X. High-Order Perturbation of Surfaces Algorithms for the Simulation of Localized Surface Plasmon Resonances in Two Dimensions. J Sci Comput 76, 1370–1395 (2018). https://doi.org/10.1007/s10915-018-0665-2

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