Abstract

We present a detailed description of a computationally efficient, semi-analytical method (SAM) to calculate the electomagnetic field distribution in a 1D-periodic, subwavelength-structured metal film placed between dielectric substrates. The method is roughly three orders of magnitude faster than the finite-element method (FEM). SAM is used to study the resonant transmission of light through nanoplasmonic structures, and to analyze the role of fundamental and higher-order Bloch surface plasmons in transmission enhancement. The method is also suitable for solving the eigenvalue problem and finding modes of the structure. Results obtained with SAM, FEM, and the finite-difference time-domain method show very good agreement for various parameters of the structure.

©2008 Optical Society of America

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2007 (3)

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. 9, 490–495 (2007).
[Crossref]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788 (2007).
[Crossref]

2006 (2)

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[Crossref] [PubMed]

E. Popov and M. Nevière, “Analytical model of the optical response of periodically structured metallic films: Comment,” Opt. Express 14, 6583–6587 (2006).
[Crossref] [PubMed]

2005 (4)

2004 (2)

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004).
[Crossref]

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[Crossref]

2003 (3)

S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B 67, 035424 (2003).
[Crossref]

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[Crossref]

N. Bonod, S. Enoch, L. Li, E. Popov, and M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482–490 (2003).
[Crossref] [PubMed]

2002 (1)

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[Crossref] [PubMed]

2001 (2)

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

2000 (1)

1999 (1)

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[Crossref]

1996 (3)

1995 (1)

1990 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Prop. 14, 302–307 (1966).
[Crossref]

Benabbas, A.

Bigot, J.-Y.

Bonod, N.

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Cai, W.

W. Cai, D. A. Genov, and V.M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[Crossref]

Cao, Q.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[Crossref] [PubMed]

Chatterjee, A.

J. L. Volakis, A. Chatterjee, and J. L. Kempel, Finite Element Method for Electromagnetics (IEEE Press, New York, 1998).
[Crossref]

Darmanyan, S. A.

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[Crossref]

S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B 67, 035424 (2003).
[Crossref]

A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, “Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates,” J. Opt. Soc. Am. B (submitted).

de Fornel, F.

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004).
[Crossref]

Dykhne, A. M.

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[Crossref]

Ebbesen, T. W.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Enoch, S.

Etchegoin, P. G.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[Crossref] [PubMed]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. 9, 490–495 (2007).
[Crossref]

Garcia-Vidal, F. J.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

García-Vidal, F. J.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[Crossref]

Gaylord, T. K.

Genov, D. A.

W. Cai, D. A. Genov, and V.M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[Crossref]

Gérard, D.

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004).
[Crossref]

Granet, G.

Grann, E. B.

Guizal, B.

Halté, V.

Kempel, J. L.

J. L. Volakis, A. Chatterjee, and J. L. Kempel, Finite Element Method for Electromagnetics (IEEE Press, New York, 1998).
[Crossref]

Kim, T. J.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Kobyakov, A.

A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, “Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates,” J. Opt. Soc. Am. B (submitted).

Krishnan, A.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Lalanne, P.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[Crossref] [PubMed]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]

Lezec, H.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Li, L.

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Lyndin, N. M.

Mafi, A.

A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, “Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates,” J. Opt. Soc. Am. B (submitted).

Maier, S. A.

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

Mansuripur, M.

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Markovic, M. I.

Martin-Moreno, L.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Meerbergen, K.

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

Meyer, M.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[Crossref] [PubMed]

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Moharam, M. G.

Moloney, J. V.

Morris, G. M.

Nevière, M.

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. 9, 490–495 (2007).
[Crossref]

Parriaux, O.

Pendry, J.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Pendry, J. B.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[Crossref]

Pommet, D. A.

Popov, E.

Porto, J. A.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[Crossref]

Rakic, A. D.

Ru, E. C. L.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[Crossref] [PubMed]

Salomon, L.

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004).
[Crossref]

Sarychev, A. K.

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[Crossref]

Shalaev, V. M.

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[Crossref]

Shalaev, V.M.

W. Cai, D. A. Genov, and V.M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[Crossref]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Thio, T.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Tishchenko, A. V.

Tisseur, F.

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
[Crossref]

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Volakis, J. L.

J. L. Volakis, A. Chatterjee, and J. L. Kempel, Finite Element Method for Electromagnetics (IEEE Press, New York, 1998).
[Crossref]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Wolff, P. A.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Xie, Y.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Prop. 14, 302–307 (1966).
[Crossref]

Zakharian, A. R.

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13, 4485–4491 (2005).
[Crossref] [PubMed]

A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, “Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates,” J. Opt. Soc. Am. B (submitted).

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004).
[Crossref]

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[Crossref]

S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B 67, 035424 (2003).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Antennas and Prop. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Prop. 14, 302–307 (1966).
[Crossref]

J. Chem. Phys. (1)

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[Crossref] [PubMed]

J. Opt. A: Pure Appl. Opt. (1)

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. 9, 490–495 (2007).
[Crossref]

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

Opt. Commun. (1)

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1–7 (2001).
[Crossref]

Opt. Express (4)

Phys. Rep. (2)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

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Supplementary Material (2)

» Media 1: AVI (2631 KB)     
» Media 2: AVI (1997 KB)     

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Figures (11)

Fig. 1.
Fig. 1. A schematic of the structure. A thin metal film between two dielectric layers is illuminated by a normally incident plane TM wave. The permittivity of the film is a step periodic function.

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Fig. 2.
Fig. 2. Real (a) and imaginary (b) part of permittivity of several metals as a function of wavelength. Data from [30].

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Fig. 3.
Fig. 3. Choosing the correct sign s ± for the square root of (18) (left) and of (22) (right). The values of (η ± n )2 are shown for evanescent (red) and propagating (blue) harmonics in a lossy dielectric with ℑ(ε ±)>0. In the yellow quadrant, (i) evanescent harmonics decay away from the interface and (ii) for propagating harmonics, the phase front moves in the direction of the wave vector (positive refractive index).

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Fig. 4.
Fig. 4. Transmittance in decibels (10log10 ��) vs. wavelength calculated with the SAM (with N=15), FEM, and FDTD for several thicknesses h of a planar bimetallic grating with Λ=600 nm, Δε=20, ρ=0.5, and γ p=105 nm, ε +=ε -=2.31; (a) h=100 nm, (b) h=72 nm, (c) h=68 nm.

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Fig. 5.
Fig. 5. Transmittance in dB vs. wavelength for a low-contrast, lossless rectangular-profile bimetallic grating. Parameters: ε T=-35, ε B=-40; Λ=600 nm, h=98 nm, ρ=0.5, ε +=ε -=2.31. Only one harmonic term N=1 is kept in the field expansion in the SAM.

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Fig. 6.
Fig. 6. Transmittance of a high-contrast phase grating as a function of wavelength. Permittivity of the film is given by ε(x)=ε 0+2ε 1 cos(2πx/Λ) with ε 1=13, ε 0(λ) is given by (2) with γ p=107 nm; Λ=600 nm, h=100 nm, ε +=ε -=2.31. The simulation shows that N>1 terms in the field expansion is needed even for a purely sinusoidal modulation of the film.

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Fig. 7.
Fig. 7. Scaled value of log10 |detM| on the complex wavelength plane (a) and the corresponding transmittance of the planar bimetallic grating (b). The dispersion of both segments is given by (2) with λp=155 nm and λp=135 nm for the first and the second segment, respectively, ε ∞ T=ε ∞ B=1.53, γ p=4×104 nm, h=95 nm, Λ=600 nm, ρ=0.4, ε +=ε -=2.2, N=15.

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Fig. 8.
Fig. 8. A snapshop of the movie FBSP.avi (size 2.6 MB) demonstrating excitation of the fundamental BSP. Top: transmittance vs. wavelength together with log10 magnitude of the Poynting vector and its direction. Bottom: |E x|, |E z| (V/m), |H y| (A/m) in the structure; h=90 nm, Δε=10, Λ=500 nm, ρ=0.5, γp =5×105 nm, ε +=ε -=2.2. The structure is illuminated by a normally incident TM plane wave with A in x=1 V/m at the wavelength λ=786 nm (red dashed line). [Media 1]

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Fig. 9.
Fig. 9. Same as Fig. 8 for silicon substrate and superstrate, ε +=ε -=13.2. A fourth-order BSP is excited at λ=777.4 (red dashed line). The animation file 4order-BSP.avi (size 2.0 MB) shows the field evolution as a function of light wavelength. [Media 2]

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Fig. 10.
Fig. 10. Transmittance in dB calculated with the SAM (N=15), FEM, and FDTD for the wavelength region of the third-order plasmonic resonance. Parameters of the structure: h=100 nm, Λ=600 nm, ρ=0.5, Δε=10, γp=107 nm, ε +=ε -=9. Dotted, dashed, and solid FDTD curves are obtained with the grid size of 1 nm, 0.5 nm, and 0.25 nm, respectively.

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Fig. 11.
Fig. 11. Transmittance in dB vs. wavelength calculated with the SAM (N=40 and 100), FEM, and FDTD for a true slit lamellar grating with ε T=1. The other component of the grating is low-loss gold with ε B(λ) given by (2) and γp=105 nm. Geometric parameters of the grating: h=130 nm, Λ=600 nm, ρ=0.2, ε +=ε -=2.2. The dashed black curve shows SAM calculations with N=40 and γp=106 nm.

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Equations (68)

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ε ( x ) = { ε T , for 0 ≤ x < ρ Λ 2 and Λ ( 1 − ρ 2 ) < x ≤ Λ , ε B , for ρ Λ 2 ≤ x ≤ Λ ( 1 − ρ 2 ) ,
ε T , B ( λ ) = ε T , B ∞ − [ ( λ p λ ) 2 + i λ p 2 ( γ p λ ) ] − 1 ,
ε ( x ) = ε 0 + 2 ∑ m = 1 M ε m cos ⁡ ( mgx ) ,
ε 0 = ε B ( 1 − ρ ) + ε T ρ , ε m = ε T − ε B π m sin ( π m ρ ) .
∇×E=-μ∂H/∂t, ∇×H=ε∂E/∂t,
K 2 ε ( x ) E z + ∂ 2 E z ∂ x 2 − ∂ 2 E x ∂ x ∂ z = 0 ,
K 2 ε ( x ) E x + ∂ 2 E x ∂ z 2 − ∂ 2 E z ∂ x ∂ z = 0 ,
H y ( x , z ) = 1 i K Z ( ∂ E x ∂ z − ∂ E z ∂ x ) ,
E σ = ( A σ + ∑ n = 1 N [ P σ n cos ⁡ ( ngx ) + Q σ n sin ⁡ ( ngx ) ] ) e k z
  + ( a σ + ∑ n = 1 N [ p σ n cos ⁡ ( ngx ) + q σ n sin ⁡ ( ngx ) ] ) e − k ( z + h ) ,
U N ξ = 0 ,
V N ζ = 0 ,
ζ ≡ 0 ,
U N = K 2 Y N + u N ,
y 11 = ε 0 , y 1,2 n = ε n , y 2 n , 1 = 2 ε n , y 2 n , 2 m = f m B ( n , N ) , y 2 n + 1,2 m + 1 = f m A ( n , N ) ,
f m A , B ( n , N ) = ε ∣ m − n ∣ ∓ Θ ( N − n − m ) ε m + n .
u 11 = u 2 n , 2 n = k 2 , u 2 n + 1 , 2 n + 1 = − n 2 g 2 , u 2 n , 2 n + 1 = − kg , u 2 n + 1,2 n = kg .
det U N ( k ) = 0 ,
det ( A k 2 + B k + C ) = 0 ,
E x ( x , z ) = ∑ j = 1 N + 1 [ A x ( j ) + ∑ n = 1 N P x n ( j ) cos ⁡ ( ngx ) ] e κ j z + ∑ j = 1 N + 1 [ a x ( j ) + ∑ n = 1 N p x n ( j ) cos ⁡ ( ngx ) ] e − κ j ( z + h ) ,
E z ( x , z ) = ∑ j = 1 N + 1 ∑ n = 1 N Q z n ( j ) sin ⁡ ( ngx ) e κ j z + ∑ j = 1 N + 1 ∑ n = 1 N q z n ( j ) sin ⁡ ( ngx ) e − κ j ( z + h ) .
( η n + ) 2 = n 2 g 2 − ε + K 2
E x + ( x , z ) = A x in e − i ε + Kz + R x e i ε + Kz + ∑ n = 1 N P x n + cos ⁡ ( ngx ) e s + η n + z ,
E z + ( x , z ) = ∑ n = 1 N Q z n + sin ⁡ ( ngx ) e s + η n + z
s + = − sgn ( n 2 g 2 − ε + K 2 ) .
P x n + = − η n + Q z n + ( ng ) .
( η n − ) 2 = n 2 g 2 − ε − K 2 ,
E x − ( x , z ) = T x e − i ε − K ( z + h ) ∑ n = 1 N P x n − cos ⁡ ( ngx ) e s − η n − ( z + h ) ,
E z − ( x , z ) = ∑ n = 1 N Q z n − sin ⁡ ( ngx ) e s − η n − ( z + h ) ,
s − = sgn ( n 2 g 2 − ε − K 2 ) .
P x n − = η n − Q z n − ( ng ) .
E x + ( x , 0 ) = E x ( x , 0 ) ,
ε + E z + ( x , 0 ) = ε ( x ) E z ( x , 0 ) ,
H y + ( x , 0 ) = H y ( x , 0 ) , i . e . ( ∂ E x + ∂ z − ∂ E z + ∂ x ) z = 0 = ( ∂ E x ∂ z − ∂ E z ∂ x ) z = 0 .
A x in + R x = ∑ j = 1 N + 1 ( A x ( j ) + a x ( j ) e − κ j h ) ,
P x n + = ∑ j = 1 N + 1 ( P x n ( j ) + p x n ( j ) e − κ j h ) .
ε + Q z n + = ∑ m = 1 N f m A ( n , N ) ∑ j = 1 N + 1 ( Q z m ( j ) + q z m ( j ) e − κ j h ) .
− i ε + K ( A x in − R x ) = ∑ j = 1 N + 1 κ j ( A x ( j ) + a x ( j ) e − κ j h ) .
T x = ∑ j = 1 N + 1 ( A x ( j ) e − κ j h + a x ( j ) ) ,
P x n − = ∑ j = 1 N + 1 ( P x n ( j ) e − κ j h + p x n ( j ) ) ,
ε − Q z n − = ∑ m = 1 N f m A ( n , N ) ∑ j = 1 N + 1 ( Q z m ( j ) e − κ j h + q z m ( j ) ) ,
− i ε + K T x = ∑ j = 1 N + 1 κ j ( A x ( j ) e − κ j h + a x ( j ) ) .
P x n ( j ) = α n j A x ( j ) , Q z n ( j ) = β n j A x ( j )
W N ( j ) ( κ j ) ρ ′ = r ,
α ~ nj = p x n ( j ) a x ( j ) , β ~ nj = q z n ( j ) a x ( j ) .
a = ( A x ( 1 ) , A x ( 2 ) , . . . , A x ( N + 1 ) , a x ( 1 ) , a x ( 2 ) , . . . , a x ( N + 1 ) ) T .
∑ j = 1 N + 1 ( α nj A x ( j ) + α ~ nj a x ( j ) e − κ j h ) = − η n + ng ε + ∑ m = 1 N f m A ( n , N ) ∑ j = 1 N + 1 ( β mj A x ( j ) + β ~ mj a x ( j ) e − κ j h ) ,
∑ j = 1 N + 1 [ ( K ε − − i κ j ) A x ( j ) e − κ j h + ( K ε − + i κ j ) a x ( j ) ] = 0
∑ j = 1 N + 1 [ ( K ε + + i κ j ) A x ( j ) + ( K ε + − i κ j ) a x ( j ) e − κ j h ] = 2 K ε + A x in .
M · a = v ,
M = ( m I ⋮ m II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m III ⋮ m IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ε − K − i κ q ) exp ( − κ q h ) ⋮ ε − K + i κ q ε + K + i κ q ⋮ ( ε + K − i κ q ) exp ( − κ q h ) ) .
m pq I = α pq + η p + ε + pg ∑ j = 1 N f j A ( p , N ) β jq ,
m pq II = ( α ~ pq + η p + ε + pg ∑ j = 1 N f j A ( p , N ) β ~ jq ) exp ⁡ ( − κ q h ) ,
m pq III = ( α pq - η p − ε − pg ∑ j = 1 N f j A ( p , N ) β jq ) exp ⁡ ( − κ q h ) ,
m pq IV = α ~ pq − η p − ε − pg ∑ j = 1 N f j A ( p , N ) β ~ jq .
E x ( x , z ) = ∑ j = 1 N + 1 [ A x ( j ) e κ j z + a x ( j ) e − κ j ( z + h ) ] + ∑ j = 1 N + 1 ∑ n = 1 N [ α nj A x ( j ) e κ j z + α ~ nj a x ( j ) e − κ j ( z + h ) ] cos ⁡ ( ngx ) ,
E z ( x , z ) = ∑ j = 1 N + 1 ∑ n = 1 N [ β nj A x ( j ) e κ j z + β ~ nj a x ( j ) e − κ j ( z + h ) ] sin ⁡ ( ngx ) .
E x + ( x , z ) = A x in e − i ε + Kz + R x e i ε + Kz + ∑ n = 1 N ∑ j = 1 N + 1 ( α nj A x ( j ) + α ~ nj a x ( j ) e − κ j h ) cos ⁡ ( ngx ) e s + η n + z ,
E z + ( x , z ) = − g ∑ n = 1 N ∑ j = 1 N + 1 ( α nj A x ( j ) + α ~ nj a x ( j ) e − κ j h ) sin ⁡ ( ngx ) e s + η n + z η n + ,
E x − ( x , z ) = T x e − i ε − K ( z + h ) + ∑ n = 1 N ∑ j = 1 N + 1 ( α nj A x ( j ) e − κ j h + α ~ nj a x ( j ) ) cos ⁡ ( ngx ) e s − η n − ( z + h ) ,
E z − ( x , z ) = g ∑ n = 1 N ∑ j = 1 N + 1 n ( α nj A x ( j ) e − κ j h + α ~ nj a x ( j ) ) sin ⁡ ( ngx ) e s − η n − ( z + h ) η n − ,
λ > Λ ε ± ,
R x = ∑ j = 1 N + 1 ( A x ( j ) + a x ( j ) e − κ j h ) − A x in .
𝒯 = ε − ∣ T x ∣ 2 ( ε + ∣ A x in ∣ 2 ) ,
ℛ = ∣ R x ∣ 2 ∣ A x in ∣ 2 .
𝒯 = 1 ∣ A x in ∣ 2 ε + [ ∣ T x ∣ 2 ε − + 1 2 ∑ n = 1 L ε − − n 2 g 2 K 2 ∣ P x n − ∣ 2 + g 2 K ℜ ( i ∑ n = 1 L n P x n − ( Q z n − ) * ) ] ,
𝒯 = 1 ∣ A x in ∣ 2 ε − ε + [ ∣ T x ∣ 2 + 1 2 ∑ n = 1 L ∣ P x n − ∣ 2 ( 1 − n 2 g 2 ε − K 2 ) − 1 2 ] ,
ℛ = 1 ∣ A x in ∣ 2 [ ∣ R x ∣ 2 + 1 2 ∑ n = 1 L ∣ P x n + ∣ 2 ( 1 − n 2 g 2 ε + K 2 ) − 1 2 ] ,

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