Abstract

The scattering of time-harmonic linear waves by periodic media arises in a wide array of applications from materials science and nondestructive testing to remote sensing and oceanography. In this work we have in mind applications in optics, more specifically plasmonics, and the surface plasmon polaritons that are at the heart of remarkable phenomena such as extraordinary optical transmission, surface-enhanced Raman scattering, and surface plasmon resonance biosensing. In this paper we develop robust, highly accurate, and extremely rapid numerical solvers for approximating solutions to grating scattering problems in the frequency regime where these are commonly used. For piecewise-constant dielectric constants, which are commonplace in these applications, surface formulations are clearly advantaged as they posit unknowns supported solely at the material interfaces. The algorithms we develop here are high-order perturbation of surfaces methods and generalize previous approaches to take advantage of the fact that these algorithms can be significantly accelerated when some or all of the interfaces are trivial (flat). More specifically, for configurations with one nontrivial interface (and one trivial interface) we describe an algorithm that has the same computational complexity as a two-layer solver. With numerical simulations and comparisons with experimental data, we demonstrate the speed, accuracy, and applicability of our new algorithms.

© 2014 Optical Society of America

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    [CrossRef]
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  46. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
    [CrossRef]
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    [CrossRef]
  48. N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
    [CrossRef]
  49. D. P. Nicholls, “Three-dimensional acoustic scattering by layered media: a novel surface formulation with operator expansions implementation,” Proc. R. Soc. A 468, 731–758 (2012).
    [CrossRef]

2013 (1)

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

2012 (3)

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure,” J. Comput. Phys. 231, 3007–3022 (2012).
[CrossRef]

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, 687–696 (2012).
[CrossRef]

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

2011 (2)

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]

D. P. Nicholls, “Efficient enforcement of far-field boundary conditions in the transformed field expansions method,” J. Comput. Phys. 230, 8290–8303 (2011).
[CrossRef]

2010 (4)

B. Hu and D. P. Nicholls, “The domain of analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 140, 367–389 (2010).

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

2009 (2)

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]

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]

2008 (4)

D. P. Nicholls and F. Reitich, “Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings,” J. Acoust. Soc. Am. 123, 2531–2541 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462–493 (2008).
[CrossRef]

2007 (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–62 (2007).
[CrossRef]

2005 (1)

B. Hu and D. P. Nicholls, “Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains,” SIAM J. Math. Anal. 37, 302–320 (2005).
[CrossRef]

2004 (3)

2003 (1)

D. P. Nicholls and F. Reitich, “Analytic continuation of Dirichlet-Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

2001 (2)

D. P. Nicholls and F. Reitich, “A new approach to analyticity of Dirichlet-Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 131, 1411–1433 (2001).

D. P. Nicholls and F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

1998 (3)

O. P. Bruno and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

A. Rakic, A. Djurisic, J. Elazar, and M. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[CrossRef]

1993 (3)

1992 (1)

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).

1985 (1)

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

1957 (1)

N. A. Phillips, “A coordinate system having some special advantages for numerical forecasting,” J. Atmos. Sci. 14, 184–185 (1957).

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. A 79, 399–416 (1907).
[CrossRef]

Abashin, M.

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Agarwal, A.

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–62 (2007).
[CrossRef]

Baker, G. A.

G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University, 1996).

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, International Series in Pure and Applied Mathematics (McGraw-Hill, 1978).

Bonod, N.

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

Brekhovskikh, L. M.

L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics (Springer-Verlag, 1982).

Bruno, O. P.

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Chau, K. J.

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Djurisic, A.

Ebbesen, T.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Ebbesen, T. W.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Elazar, J.

Enoch, S.

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

Fan, S.

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, Vol. 131 of Springer Series in Optical Sciences (Springer, 2007), pp. 169–182.

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Ghaemi, H.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Gottlieb, D.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Vol. 26 of CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1977).

Graves-Morris, P.

G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University, 1996).

He, Y.

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure,” J. Comput. Phys. 231, 3007–3022 (2012).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Holmes, R. J.

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Homola, J.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462–493 (2008).
[CrossRef]

Hu, B.

B. Hu and D. P. Nicholls, “The domain of analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 140, 367–389 (2010).

B. Hu and D. P. Nicholls, “Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains,” SIAM J. Math. Anal. 37, 302–320 (2005).
[CrossRef]

Im, H.

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

Johnson, T. W.

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, 687–696 (2012).
[CrossRef]

Jose, J.

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, 687–696 (2012).
[CrossRef]

Kern, A. M.

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Kuipers, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Kurkcu, H.

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]

Lesuffleur, A.

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

Lezec, H.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Lezec, H. J.

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Lindquist, N. C.

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, 687–696 (2012).
[CrossRef]

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Luhman, W. A.

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Lysanov, Y. P.

L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics (Springer-Verlag, 1982).

Maier, S.

S. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Majewski, M.

Malcolm, A.

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]

Martin, O. J. F.

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Moskovits, M.

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Nicholls, D. P.

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

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure,” J. Comput. Phys. 231, 3007–3022 (2012).
[CrossRef]

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]

D. P. Nicholls, “Efficient enforcement of far-field boundary conditions in the transformed field expansions method,” J. Comput. Phys. 230, 8290–8303 (2011).
[CrossRef]

B. Hu and D. P. Nicholls, “The domain of analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 140, 367–389 (2010).

D. P. Nicholls and F. Reitich, “Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings,” J. Acoust. Soc. Am. 123, 2531–2541 (2008).
[CrossRef]

B. Hu and D. P. Nicholls, “Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains,” SIAM J. Math. Anal. 37, 302–320 (2005).
[CrossRef]

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

D. P. Nicholls and F. Reitich, “Shape deformations in rough surface scattering: improved algorithms,” J. Opt. Soc. Am. A 21, 606–621 (2004).
[CrossRef]

D. P. Nicholls and F. Reitich, “Analytic continuation of Dirichlet-Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

D. P. Nicholls and F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

D. P. Nicholls and F. Reitich, “A new approach to analyticity of Dirichlet-Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 131, 1411–1433 (2001).

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Oh, S.-H.

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, 687–696 (2012).
[CrossRef]

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, International Series in Pure and Applied Mathematics (McGraw-Hill, 1978).

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Vol. 26 of CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1977).

Otto, L. M.

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, 687–696 (2012).
[CrossRef]

Phillips, N. A.

N. A. Phillips, “A coordinate system having some special advantages for numerical forecasting,” J. Atmos. Sci. 14, 184–185 (1957).

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

Raether, H.

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

Rakic, A.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. A 79, 399–416 (1907).
[CrossRef]

Reitich, F.

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]

D. P. Nicholls and F. Reitich, “Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings,” J. Acoust. Soc. Am. 123, 2531–2541 (2008).
[CrossRef]

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

D. P. Nicholls and F. Reitich, “Shape deformations in rough surface scattering: improved algorithms,” J. Opt. Soc. Am. A 21, 606–621 (2004).
[CrossRef]

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

D. P. Nicholls and F. Reitich, “Analytic continuation of Dirichlet-Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

D. P. Nicholls and F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

D. P. Nicholls and F. Reitich, “A new approach to analyticity of Dirichlet-Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 131, 1411–1433 (2001).

O. P. Bruno and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

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]

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]

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]

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).

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Shen, J.

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure,” J. Comput. Phys. 231, 3007–3022 (2012).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Shull, P. J.

P. J. Shull, Nondestructive Evaluation: Theory, Techniques, and Applications (Marcel Dekker, 2002).

Tamma, K.

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

Thio, T.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Veronis, G.

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, Vol. 131 of Springer Series in Optical Sciences (Springer, 2007), pp. 169–182.

Wolff, P.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Xu, T.

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

ACS Nano (1)

H. Im, N. C. Lindquist, A. Lesuffleur, and S.-H. Oh, “Atomic layer deposition of dielectric overlayers for enhancing the optical properties and chemical stability of plasmonic nanoholes,” ACS Nano 4, 947–954 (2010).
[CrossRef]

Ann. Phys. (1)

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, 687–696 (2012).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. C. Lindquist, W. A. Luhman, S.-H. Oh, and R. J. Holmes, “Plasmonic nanocavity arrays for enhanced efficiency in organic photovoltaic cells,” Appl. Phys. Lett. 93, 123308 (2008).
[CrossRef]

Chem. Rev. (1)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462–493 (2008).
[CrossRef]

CMES Comput. Model. Eng. Sci. (1)

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

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

J. Acoust. Soc. Am. (3)

D. P. Nicholls and F. Reitich, “Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings,” J. Acoust. Soc. Am. 123, 2531–2541 (2008).
[CrossRef]

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]

O. P. Bruno and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

J. Atmos. Sci. (1)

N. A. Phillips, “A coordinate system having some special advantages for numerical forecasting,” J. Atmos. Sci. 14, 184–185 (1957).

J. Comput. Phys. (4)

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure,” J. Comput. Phys. 231, 3007–3022 (2012).
[CrossRef]

D. P. Nicholls and F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

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]

D. P. Nicholls, “Efficient enforcement of far-field boundary conditions in the transformed field expansions method,” J. Comput. Phys. 230, 8290–8303 (2011).
[CrossRef]

J. Opt. (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (6)

Nat. Mater. (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

Nature (3)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

T. Xu, A. Agarwal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497, 470–474 (2013).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Numer. Math. (1)

D. P. Nicholls and F. Reitich, “Analytic continuation of Dirichlet-Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

Proc. R. Soc. A (2)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. A 79, 399–416 (1907).
[CrossRef]

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

Proc. R. Soc. Edinburgh Sect. A (3)

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).

D. P. Nicholls and F. Reitich, “A new approach to analyticity of Dirichlet-Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 131, 1411–1433 (2001).

B. Hu and D. P. Nicholls, “The domain of analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh Sect. A 140, 367–389 (2010).

Rev. Mod. Phys. (2)

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Sci. Am. (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–62 (2007).
[CrossRef]

SIAM J. Math. Anal. (1)

B. Hu and D. P. Nicholls, “Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains,” SIAM J. Math. Anal. 37, 302–320 (2005).
[CrossRef]

Other (13)

G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University, 1996).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, International Series in Pure and Applied Mathematics (McGraw-Hill, 1978).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Vol. 26 of CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1977).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

C. Godrèche, ed., Solids Far from Equilibrium (Cambridge University, 1992).

P. J. Shull, Nondestructive Evaluation: Theory, Techniques, and Applications (Marcel Dekker, 2002).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics (Springer-Verlag, 1982).

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

S. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

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

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, Vol. 131 of Springer Series in Optical Sciences (Springer, 2007), pp. 169–182.

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Figures (9)

Fig. 1.
Fig. 1.

Plot of three-layer configuration with illumination from below.

Fig. 2.
Fig. 2.

Plot of Cauchy error, |RNuRN1u|L and |RNwRN1w|L, versus perturbation order N for three-layer, dielectric configuration simulated via (a) FE recursions and (b) TFE recursions.

Fig. 3.
Fig. 3.

Plot of energy defect, δ, versus perturbation order N for three-layer, dielectric configuration (λ=670.5nm): (a) FE recursions and (b) TFE recursions.

Fig. 4.
Fig. 4.

(a) Geometry and model (to scale). Upper region, water (sensing area); middle, gold; lower, polymer substrate. Gold/polymer interface specified by Eq. (15): blue curve displays b=5×105, while green curve is “idealized” shape with b=. (b) Close up of (a) near the gold/polymer interface.

Fig. 5.
Fig. 5.

Plot of the reflectivity map for water/gold/polymer configuration (which can exceed 1 as it is normalized by the flat-interface value). Experimental data are depicted with green diamonds, while numerical simulations (via FE recursions) are shown with red stars.

Fig. 6.
Fig. 6.

Intensities |Hz|2 at the wavelengths (a) λ=631nm and (b) λ=717nm.

Fig. 7.
Fig. 7.

Plot of shift in “peak” and “dip” of reflectivity map as the index of refraction, nwater, in water corresponding to the bulk sensitivity to varying concentrations of glycerol in the water.

Fig. 8.
Fig. 8.

Plot of shift in “peak” and “dip” of reflectivity map as layers of Al2O3 are added to the sensor to test local sensitivity.

Fig. 9.
Fig. 9.

Depiction of a sample multilayer device that should be amenable to our HOPS method.

Equations (111)

Equations on this page are rendered with MathJax. Learn more.

S:=SuSvSw={y>h¯}{g(x)<y<h¯}{y<g(x)}.
Ei(x,y,t)=Aeiαx+iβyiωt,i(x,y,t)=Beiαx+iβyiωt.
E(x,y)=eiωtE(x,y,t),H(x,y)=eiωt(x,y,t),
u=u(x,y),v=v(x,y),w=w(x,y),
Δu+ku2u=0y>h¯,
Δv+kv2v=0g(x)<y<h¯,
uv=0,yuτ2yv=0y=h¯,
Δw+kw2w=0y<g(x),
vw=ζy=g(x),
Nvσ2Nw=ψy=g(x),
ζ(x):=wi(x,g(x))=eiβwg(x)eiαx,
ψ(x):=σ2(Nwi)(x,g(x))=σ2{iβwiα(xg)}eiβwg(x)eiαx,
τ2=ku2kv2=nu2nv2,σ2=kv2kw2=nv2nw2,
b<|g|L,|g|L<a<h¯,
Δv+kv2v=0g(x)<y<a,
yvT[v]=0y=a,
Δw+kw2w=0b<y<g(x),
vw=ζy=g(x),
Nvσ2Nw=ψy=g(x),
ywS[w]=0y=b.
v=v(x,y;ε)=n=0vn(x,y)εn,
w=w(x,y;ε)=n=0wn(x,y)εn,
Δvn+kv2vn=00<y<a,
yvnT[vn]=0y=a,
Δwn+kw2wn=0b<y<0,
vnwn=ζnQny=0,
yvnσ2ywn=ψnRny=0,
ywnS[wn]=0y=b,
ζn=Fn(iβw)neiαx,
ψn=σ2{Fn(iβw)n+1+(xf)Fn1(iβw)n1}eiαx,
Qn=m=0n1Fnm{ynmumynmvm},
Rn=m=0n1Fnm{ynm+1umσ2ynm+1vm},m=0n1(xf)Fn1m{yn1mumσ2ynm+1vm}.
vn(x,y)=p=ξ^n,p{eiβv,py+Dpeiβv,py}eiαpx,
wn(x,y)=p=μ^n,p{eiβw,py+Epeiβw,py}eiαpx,
αp:=α+(2π/d)p,βj,p:={kj2αp2,pU(j)iαp2kj2,pU(j),
U(j)={pZ|αp2<kj2}
Dp=e2iβv,pa(iβv,pT^p)/(iβv,p+T^p).
(1+Dp)ξ^n,pw^n,p=ζ^n,pQ^n,p,
(iβv,p)(1Dp)ξ^n,pσ2(iβw,p)w^n,p=ψ^n,pR^n,p,
σ2(iβw,p)(1+Dp)(iβv,p)(1Dp)0.
x=x,y=a(yg(x)ag(x)),g(x)<y<a,
x=x,y=b(yg(x)b+g(x)),b<y<g(x),
x=x,y=y+g(x)(aya),0<y<a,
x=x,y=y+g(x)(b+yb),b<y<0,
v(x,y):=v(x,y+g(ay)/a),
w(x,y):=w(x,y+g(b+y)/b).
Δv+kv2v=Fv(x,y;g,v)0<y<a,
yvT[v]=Jv(x;g,v)y=a,
Δw+kw2w=Fw(x,y;g,w)b<y<0,
vw=ζQ(x;g,v,w)y=0,
yvσ2yw=ψR(x;g,v,w)y=0,
ywS[w]=Jw(x;g,w)y=b,
Jv=1agT[v].
v(x,y;ε)=n=0vn(x,y)εn,w(x,y;ε)=n=0wn(x,y)εn,
Δvn+kv2vn=Fnv(x,y)0<y<a,
yvnT[vn]=Jnv(x)y=a,
Δwn+kw2wn=Fnw(x,y)b<y<0,
vnwn=ζnQny=0,
yvnσ2ywn=ψnRn(x)y=0,
ywnS[wn]=Jnw(x)y=b,
u(x,y)=p=u^peiβu,pyeiαpx,y>h¯,
w(x,y)=p=w^peiβw,pyeiαpx,yb<|g|L,
eu,p:=βu,pβw|u^p|2,ew,p:=βw,pβw|w^p|2.
pU(u)eu,p+(ku2kw2)pU(w)ew,p=1,
δ:=1pU(u)eu,p(ku2kw2)pU(w)ew,p.
R˜u:=pU(u)eu,p,R˜w:=pU(w)ew,p,
Rw=Rw(λ,h):=R˜w(λ,h)R˜w(λ,0).
C0=C0(λ,h):=|w^0(λ,h)|2|w^0(λ,0)|2.
vvN:=n=0Nvn(x,y)εn,wwN:=n=0Nwn(x,y)εn.
vnvnNx:=p=Nx/2Nx/21ξ^n,p{eiβv,py+Dpeiβv,py}eiαpx,
wnwnNx:=p=Nx/2Nx/21w^n,peiβw,pyeiαpx.
vvN:=n=0Nvn(x,y)εn,wwN:=n=0Nwn(x,y)εn,
vnvnNx:=p=Nx/2Nx/21v^n,p(y)eiαpx,
wnwnNx:=p=Nx/2Nx/21w^n,p(y)eiαpx.
v^n,pv^n,pNy:=l=0Nyv^n,p,lTl(2yaa),
w^n,pw^n,pNy:=l=0Nyw^n,p,lTl(2y+bb),
nu=1.1,nv=2.1,nw=3.5.
f(x)=a{cos(2πxd)+19cos(6πxd)+116cos(8πxd)+18sin(6πxd)},
f(x)=a{tanh(b[(xd/2)+c])tanh(b[(xd/2)c])},
nepoxy=1.56,nwater=1.333.
ϵAu=ϵAu+j=16ΔjAuajAuω2ibjAuω+cjAu,
Δw+kw2w=0b<y<g(x),
Δw̲+kw2w̲=0y<b,
ww̲=0,y(ww̲)=0y=b,
w̲(x,y)=p=^w̲peiβw,pyeiαpx,
p=ψ^peiαpx=ψ(x)=^w̲(x,b)=p=^w̲peiβw,p(b)eiαpx,
w̲(x,y)=p=ψ^peiβw,p(y+b)eiαpx.
yw̲(x,b)=p=(iβw,p)ψ^peiαpx=:S[ψ(x)],
ywS[w]=0.
Δw+kw2w=0b<y<g(x),
ywS[w]=0y=b.
|g|L<a<h¯.
Δu+ku2u=0y>h¯,
Δv̲+kv2v̲=0a<y<h¯,
uv̲=0,y(uτ2v̲)=0y=h¯,
Δv+kv2v=0g(x)<y<a,
v̲v=0,y(v̲v)=0y=a,
U(x):=u(x,h¯),V(x):=v̲(x,h¯),Va(x):=v̲(x,a),
U˜(x):=(yu)(x,h¯),V˜(x):=(yv̲)(x,h¯),
V˜a(x):=(yv̲)(x,a);
UV=0,U˜τ2V˜=0,Va=ψ,
G:UU˜,B:(V,Va)(V˜,V˜a),
B=(BuuBulBluBll),
UV=0,G[U]τ2(Buu[V]+Bul[Va])=0,Va=ψ,
(G+τ2Buu)[V]=τ2Bul[ψ].
T[ψ]=(Blu[V]+Bll[ψ])={τ2(G+τ2Buu)1BulBll}[ψ],
G=(iβu,D),Buu=Bll=(iβv,D)coth(iβv,D(h¯a)),
Bul=Blu=(iβv,D)csch(iβv,D(h¯a)),
T^p=τ2(iβv,p)csch(iβv,p(h¯a))(iβu,p)+τ2(iβv,p)coth(iβv,p(h¯a))(iβv,p)coth(iβv,p(h¯a)).
Δv+kv2v=0g(x)<y<a,
yvT[v]=0y=a.

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