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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 11 — Nov. 1, 2013
  • pp: 2175–2187

A fast and high-order accurate surface perturbation method for nanoplasmonic simulations: basic concepts, analytic continuation and applications

Fernando Reitich, Timothy W. Johnson, Sang-Hyun Oh, and Gary Meyer  »View Author Affiliations

JOSA A, Vol. 30, Issue 11, pp. 2175-2187 (2013)

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In this paper we demonstrate that rigorous high-order perturbation of surfaces (HOPS) methods coupled with analytic continuation mechanisms are particularly well-suited for the assessment and design of nanoscale devices (e.g., biosensors) that operate based on surface plasmon resonances generated through the interaction of light with a periodic (metallic) grating. In this connection we explain that the characteristics of the latter are perfectly aligned with the optimal domain of applicability of HOPS schemes, as these procedures can be shown to be the methods of choice for low to moderate wavelengths of radiation and grating roughness that is representable by a few (e.g., tens of) Fourier coefficients. We argue that, in this context, the method can, for instance, produce full and precise reflectivity maps in computational times that are orders of magnitude faster than those of alternative numerical schemes (e.g., the popular “C-method,” finite differences, integral equations or finite elements). In this initial study we concentrate on the description of the basic principles that underlie the solution scheme, including those that relate to analytic continuation procedures. Within this framework, we explain how, in spite of conventional wisdom to the contrary, the resulting perturbative techniques can provide a most valuable tool for practical investigations in plasmonics. We demonstrate this with some examples that have been previously discussed in the literature (including treatments of the reflectivity and band gap structure of some simple geometries) and extend this to demonstrate the wider applicability of the proposed approach.

© 2013 Optical Society of America

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:
Optics at Surfaces

Original Manuscript: June 17, 2013
Manuscript Accepted: August 22, 2013
Published: October 3, 2013

Virtual Issues
Vol. 9, Iss. 1 Virtual Journal for Biomedical Optics
October 29, 2013 Spotlight on Optics

Fernando Reitich, Timothy W. Johnson, Sang-Hyun Oh, and Gary Meyer, "A fast and high-order accurate surface perturbation method for nanoplasmonic simulations: basic concepts, analytic continuation and applications," J. Opt. Soc. Am. A 30, 2175-2187 (2013)

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  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  2. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
  3. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010). [CrossRef]
  4. M. Mansuripur, A. R. Zakharian, A. Lesuffleur, S.-H. Oh, R. J. Jones, N. C. Lindquist, H. Im, A. Kobyakov, and J. V. Moloney, “Plasmonic nano-structures for optical data storage,” Opt. Express 17, 14001–14014 (2009). [CrossRef]
  5. T. W. Johnson, Z. J. Lapin, R. Beams, N. C. Lindquist, S. G. Rodrigo, L. Novotny, and S.-H. Oh, “Highly reproducible near-field optical imaging with sub-20-nm resolution based on template-stripped gold pyramids,” ACS Nano 6, 9168–9174 (2012). [CrossRef]
  6. F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
  7. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462–493 (2008). [CrossRef]
  8. N. C. Lindquist, P. Nagpal, K. M. McPeak, D. J. Norris, and S.-H. Oh, “Engineering metallic nanostructures for plasmonics and nanophotonics,” Rep. Prog. Phys. 75, 036501 (2012). [CrossRef]
  9. D. Barchiesi, B. Guizal, and T. Grosges, “Accuracy of local field enhancement models: toward predictive models?” Appl. Phys. B 84, 55–60 (2006). [CrossRef]
  10. G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, M. Brongersma and P. Kik, eds. Vol. 131 of Springer Series in Optical Sciences (Springer, 2007), pp. 169–182.
  11. J. Hoffmann, C. Hafner, P. Leidenberger, J. Hesselbarth, and S. Burger, “Comparison of electromagnetic field solvers for the 3D analysis of plasmonic nano antennas,” Proc. SPIE  7390, 73900J (2009).
  12. J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. 6, 763–774 (2009). [CrossRef]
  13. A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys. 226, 477–493 (2007). [CrossRef]
  14. G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett. 34, 2216–2218 (2009). [CrossRef]
  15. M. Huber, J. Schöberl, A. Sinwel, and S. Zaglmayr, “Simulation of diffraction in periodic media with a coupled finite element and plane wave approach,” SIAM J. Sci. Comput. 31, 1500–1517 (2009). [CrossRef]
  16. K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,” Opt. Express 17, 14934–14947 (2009). [CrossRef]
  17. O. Tsilipakos, A. Pitilakis, A. Tasolamprou, T. Yioultsis, and E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011). [CrossRef]
  18. D. Christensen and D. Fowers, “Modeling SPR sensors with the finite-difference time-domain method,” Biosens. Bioelectron. 11, 677–684 (1996). [CrossRef]
  19. H. Sai, Y. Kanamori, K. Hane, and H. Yugami, “Numerical study on spectral properties of tungsten one-dimensional surface-relief gratings for spectrally selective devices,” J. Opt. Soc. Am. A 22, 1805–1813 (2005). [CrossRef]
  20. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007). [CrossRef]
  21. J. M. Montgomery, T.-W. Lee, and S. K. Gray, “Theory and modeling of light interactions with metallic nanostructures,” J. Phys. Condens. Matter 20, 323201 (2008). [CrossRef]
  22. N. C. Lindquist, T. W. Johnson, D. J. Norris, and S.-H. Oh, “Monolithic integration of continuously tunable plasmonic nanostructures,” Nano Lett. 11, 3526–3530 (2011). [CrossRef]
  23. A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011). [CrossRef]
  24. O. P. Bruno and M. C. Haslam, “Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences,” J. Opt. Soc. Am. A 26, 658–668 (2009). [CrossRef]
  25. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009). [CrossRef]
  26. B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]
  27. J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011). [CrossRef]
  28. H. Kurkcu, A. Ortan, and F. Reitich, “An efficient integral equation solver for two-dimensional simulations in nanoplasmonics,” in Proceedings of Waves 2013, Tunis, Tunisia (2013).
  29. B. Alavikia and O. Ramahi, “Limitation of using absorbing boundary condition to solve the problem of scattering from a cavity in metallic screens,” in Antennas and Propagation Society International Symposium (APSURSI) (IEEE, 2010), pp. 1–4.
  30. C.-C. Chao, S.-H. Tu, C.-M. Wang, H.-I. Huang, C.-C. Chen, and J.-Y. Chang, “Impedance-matching surface plasmon absorber for FDTD simulations,” Plasmonics 5, 51–55 (2010). [CrossRef]
  31. M. Wang, C. Engstrom, K. Schmidt, and C. Hafner, “On high-order FEM applied to canonical scattering problems in plasmonics,” J. Comput. Theor. Nanosci. 8, 1564–1572 (2011). [CrossRef]
  32. Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652 (2008). [CrossRef]
  33. A. D. Baczewski, N. C. Miller, and B. Shanker, “Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions,” J. Opt. Soc. Am. A 29, 531–540 (2012). [CrossRef]
  34. H. Kurkcu and F. Reitich, “Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies,” J. Comput. Phys. 228, 75–95 (2009). [CrossRef]
  35. K. Warnick and W. Chew, “Numerical simulation methods for rough surface scattering,” Wave Random Media 11, R1–R30 (2001). [CrossRef]
  36. T. Elfouhaily and C. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Wave Random Media 14, R1–R40 (2004). [CrossRef]
  37. J. A. deSanto, “Overview of rough surface scattering,” in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin and D. J. Lockwood, eds., Nanostructure Science and Technology (Springer, 2007), Chap. 8, pp. 211–235.
  38. J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980). [CrossRef]
  39. J. Chandezon, M. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982). [CrossRef]
  40. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999). [CrossRef]
  41. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996). [CrossRef]
  42. G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1131 (1998). [CrossRef]
  43. R. A. Watts, A. P. Hibbins, and J. R. Sambles, “The influence of grating profile on surface plasmon polariton resonances recorded in different diffracted orders,” J. Mod. Opt. 46, 2157–2186 (1999).
  44. M. Kreiter, S. Mittler, W. Knoll, and J. R. Sambles, “Surface plasmon-related resonances on deep and asymmetric gold gratings,” Phys. Rev. B 65, 125415 (2002). [CrossRef]
  45. W.-C. Liu, “High sensitivity of surface plasmon of weakly-distorted metallic surfaces,” Opt. Express 13, 9766–9773 (2005). [CrossRef]
  46. S. Kahaly, S. K. Yadav, W. M. Wang, S. Sengupta, Z. M. Sheng, A. Das, P. K. Kaw, and G. R. Kumar, “Near-complete absorption of intense, ultrashort laser light by sub-λ gratings,” Phys. Rev. Lett. 101, 145001 (2008). [CrossRef]
  47. N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express 15, 11427–11432 (2007). [CrossRef]
  48. I. S. Spevak, M. A. Timchenko, and A. V. Kats, “Design of specific gratings operating under surface plasmon–polariton resonance,” Opt. Lett. 36, 1419–1421 (2011). [CrossRef]
  49. L. Li and G. Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A 28, 738–746 (2011). [CrossRef]
  50. J. Bischoff and K. Hehl, “Perturbation approach applied to modal diffraction methods,” J. Opt. Soc. Am. A 28, 859–867 (2011). [CrossRef]
  51. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981). [CrossRef]
  52. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]
  53. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
  54. J. Bischoff, “Prospects and limits of the Rayleigh Fourier approach for diffraction modelling in scatterometry and lithography,” Proc. SPIE  7390, 73901E (2009).
  55. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951). [CrossRef]
  56. A. G. Voronovich, “Small-slope approximation in wave scattering by rough surfaces,” Sov. Phys. J. Exp. Theor. Phys. 62, 65–70 (1985).
  57. E. Rodriguez and Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992). [CrossRef]
  58. C.-A. Guérin and A. Sentenac, “Second-order perturbation theory for scattering from heterogeneous rough surfaces,” J. Opt. Soc. Am. A 21, 1251–1260 (2004). [CrossRef]
  59. D. Grieser, H. Uecker, S.-A. Biehs, O. Huth, F. Rüting, and M. Holthaus, “Perturbation theory for plasmonic eigenvalues,” Phys. Rev. B 80, 245405 (2009). [CrossRef]
  60. A. Trügler, J.-C. Tinguely, J. R. Krenn, A. Hohenau, and U. Hohenester, “Influence of surface roughness on the optical properties of plasmonic nanoparticles,” Phys. Rev. B 83, 081412(R) (2011). [CrossRef]
  61. A. A. Maradudin and E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993). [CrossRef]
  62. K. A. O’Donnell, “High-order perturbation theory for light scattering from a rough metal surface,” J. Opt. Soc. Am. A 18, 1507–1518 (2001). [CrossRef]
  63. M. A. Demir and J. T. Johnson, “Fourth- and higher-order small-perturbation solution for scattering from dielectric rough surfaces,” J. Opt. Soc. Am. A 20, 2330–2337 (2003). [CrossRef]
  64. K. A. O’Donnell, “Small-amplitude perturbation theory for one-dimensionally rough surfaces,” in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin and D. J. Lockwood, eds., Nanostructure Science and Technology (Springer, 2007), Chap. 5, pp. 107–126.
  65. H.-Y. Xie, M.-Y. Ng, and Y.-C. Chang, “Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect,” J. Opt. Soc. Am. A 27, 2411–2422 (2010). [CrossRef]
  66. J.-J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988). [CrossRef]
  67. D. R. Jackson, D. P. Winebrenner, and A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988). [CrossRef]
  68. A. Soubret, G. Berginc, and C. Bourrely, “Backscattering enhancement of an electromagnetic wave scattered by two-dimensional rough layers,” J. Opt. Soc. Am. A 18, 2778–2788 (2001). [CrossRef]
  69. O. P. Bruno and F. Reitich, “Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A 122, 317–340 (1992). [CrossRef]
  70. O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993). [CrossRef]
  71. O. P. 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, 2307–2316 (1993). [CrossRef]
  72. O. P. 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, 2551–2562 (1993). [CrossRef]
  73. O. Bruno and F. Reitich, “High-order boundary perturbation methods,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001), Chap. 3, pp. 71–109.
  74. L. Kazandjian, “A discussion of the properties of the Rayleigh perturbative solution in diffraction theory,” Wave Motion 42, 169–176 (2005). [CrossRef]
  75. J.-J. Greffet, “Introduction to surface plasmon theory,” in Plasmonics: From Basics to Advanced Topics, S. Enoch and N. Bonod, eds., Vol. 167 of Springer Series in Optical Sciences (Springer, 2012), Chap. 4, pp. 105–148.
  76. D. Maystre, “Theory of Wood’s anomalies,” in Plasmonics: From Basics to Advanced Topics, S. Enoch and N. Bonod, eds., Vol. 167 of Springer Series in Optical Sciences (Springer, 2012), Chap. 2, pp. 39–83.
  77. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).
  78. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968). [CrossRef]
  79. B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996). [CrossRef]
  80. H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, “Fluorescence imaging of surface plasmon fields,” Appl. Phys. Lett. 80, 404–406 (2002). [CrossRef]
  81. R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968). [CrossRef]
  82. J. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965). [CrossRef]
  83. G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and its Applications (Cambridge University, 1996).
  84. A. Rakic, A. Djurišic, J. Elazar, and M. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998). [CrossRef]
  85. Z. Chen, I. R. Hooper, and J. R. Sambles, “Low dispersion surface plasmon–polaritons on deep silver gratings,” J. Mod. Opt. 53, 1569–1576 (2006). [CrossRef]
  86. D. J. Nash and J. R. Sambles, “Surface plasmon–polariton study of the optical dielectric function of silver,” J. Mod. Opt. 43, 81–91 (1996).
  87. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
  88. D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985), Vol. 1, pp. 275–367.
  89. J. Homola, Surface Plasmon Resonance Based Sensors, Vol. 4 of Springer Series on Chemical Sensors and Biosensors (Springer, 2006).
  90. Z. Chen, “Grating coupled surface plasmons in metallic structures,” Ph.D. dissertation (University of Exeter, 2007).
  91. T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998). [CrossRef]
  92. F. García-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999). [CrossRef]
  93. I. R. Hooper and J. R. Sambles, “Dispersion of surface plasmon polaritons on short-pitch metal gratings,” Phys. Rev. B 65, 165432 (2002). [CrossRef]
  94. E. P. Da Silva, G. A. Farias, and A. A. Maradudin, “Analysis of three theories of scattering of electromagnetic radiation by gratings,” J. Opt. Soc. Am. A 4, 2022–2024 (1987). [CrossRef]
  95. A. J. Jerri, ed., Advances in the Gibbs Phenomenon (Σ Sampling, 2011).
  96. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, and D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995). [CrossRef]
  97. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain method (Artech House, 2005).
  98. A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011). [CrossRef]

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