Abstract

Using the Fourier modal method (FMM) we report our analysis of the transmission resonances of a plasmonic grating with subwavelength period and extremely narrow slits for wavelengths of the incoming, transverse magnetic (TM)-polarized, radiation ranging from 240nm to 1500nm and incident angles from 0° to 90°. In particular, we study the case of a silver grating placed in vacuo. Consistent with previous studies on the topic, we highlight that the main mechanism for extraordinary transmission is a TM-Fabry-Perot (FP) branch supported by waveguide modes inside each slit. The TM-FP branch may also interact with surface plasmons (SPs) at the air/Ag interface through the reciprocal lattice vectors of the grating, for periods comparable with the incoming wavelength. When the TM-FP branch crosses an SP branch, a bandgap is formed along the line of the SP dispersion. The gap has a Fano-Feshbach resonance at the low frequency band edge and a ridge resonance with extremely long lifetime at the high frequency band edge. We discuss the nature of these dispersion features, and in particular we describe the ridge resonance in the framework of guided-mode resonances (GMRs). In addition, we elucidate the connection of the coupling between the TM-FP branch and SPs within the Rayleigh condition. We also study the peculiar characteristics of the field localization and the energy transport in two topical examples.

© 2011 Optical Society of America

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  50. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
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2010 (1)

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]

2009 (1)

2008 (3)

2007 (3)

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1286 (2007).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

2006 (2)

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: The s-polarization case,” J. Opt. A Pure Appl. Opt. 8, S94–S97 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

2005 (3)

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

F. Marquier, J.-J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13, 70–76 (2005).
[CrossRef] [PubMed]

2004 (2)

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004).
[CrossRef] [PubMed]

2003 (1)

I. R. Hooper and J. R. Sambles, “Surface plasmon polaritons on thin-slab metal gratings,” Phys. Rev. B 67, 235404 (2003).
[CrossRef]

2002 (4)

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

W. C. Liu and D. P. Tsai, “Optical tunneling of surface plasmon polaritons and localized surface plasmon resonance,” Phys. Rev. B 65, 155423 (2002).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of the guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

2001 (3)

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5604 (2001).
[CrossRef] [PubMed]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

2000 (2)

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

S. P. Astilean, Ph. Lalanne, and M. Palamary, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

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]

1998 (1)

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

1996 (4)

1995 (1)

1992 (1)

R. Magnusson and S. S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[CrossRef]

1985 (1)

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[CrossRef] [PubMed]

1981 (1)

1979 (1)

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop band,” Appl. Opt. 20, 345–351 (1979).

1978 (1)

1968 (1)

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]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

1907 (1)

Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402(1902).

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 266802 (2008).
[CrossRef]

Arakawa, E. T.

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]

Astilean, S.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Astilean, S. P.

S. P. Astilean, Ph. Lalanne, and M. Palamary, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, “Influence of losses on the superresolution performances of an impedance-matched negative-index material,” J. Opt. Soc. Am. B 25, 236–246 (2008).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[CrossRef]

Cai, W.

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Chettiar, U. K.

Collin, S.

F. Marquier, J.-J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13, 70–76 (2005).
[CrossRef] [PubMed]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

Collins, R. E.

R. E. Collins, Field Theory of Guided Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, New York, 1991).

Cowan, J. J.

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]

D’Aguanno, G.

G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, “Influence of losses on the superresolution performances of an impedance-matched negative-index material,” J. Opt. Soc. Am. B 25, 236–246 (2008).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

de Ceglia, D.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Ding, Y.

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Drachev, V. P.

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]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 454, 39–46 (2007) (and references therein).

Engheta, N.

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 266802 (2008).
[CrossRef]

Fan, S.

S. Fan and J. D. Joannopoulos, “Analysis of the guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Felbacq, D.

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonance in nanoscale structures,” available at http://arxiv4.library.cornell.edu/abs/0902.3014v1.

Friedrich, H.

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[CrossRef] [PubMed]

García de Abajo, F. J.

F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1286 (2007).
[CrossRef]

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]

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: The s-polarization case,” J. Opt. A Pure Appl. Opt. 8, S94–S97 (2006).
[CrossRef]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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.

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 454, 39–46 (2007) (and references therein).

Ghaemi, H. F.

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

Granet, G.

Greffet, J.-J.

Guizal, B.

Hamm, R. N.

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]

Hooper, I. R.

I. R. Hooper and J. R. Sambles, “Surface plasmon polaritons on thin-slab metal gratings,” Phys. Rev. B 67, 235404 (2003).
[CrossRef]

Hugonin, J. P.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Joannopoulos, J. D.

S. Fan and J. D. Joannopoulos, “Analysis of the guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Kildishev, A. V.

Kivshar, Y. S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonance in nanoscale structures,” available at http://arxiv4.library.cornell.edu/abs/0902.3014v1.

Knop, K.

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]

Lalanne, P.

Lalanne, Ph.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

S. P. Astilean, Ph. Lalanne, and M. Palamary, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Lee, K. G.

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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

Li, L.

Liu, W. C.

W. C. Liu and D. P. Tsai, “Optical tunneling of surface plasmon polaritons and localized surface plasmon resonance,” Phys. Rev. B 65, 155423 (2002).
[CrossRef]

Magnusson, R.

Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004).
[CrossRef] [PubMed]

R. Magnusson and S. S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Marquier, 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]

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: The s-polarization case,” J. Opt. A Pure Appl. Opt. 8, S94–S97 (2006).
[CrossRef]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

Mattiucci, N.

G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, “Influence of losses on the superresolution performances of an impedance-matched negative-index material,” J. Opt. Soc. Am. B 25, 236–246 (2008).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonance in nanoscale structures,” available at http://arxiv4.library.cornell.edu/abs/0902.3014v1.

Moharam, M. G.

Möller, K. D.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Moreno, E.

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: The s-polarization case,” J. Opt. A Pure Appl. Opt. 8, S94–S97 (2006).
[CrossRef]

Morf, R. H.

Morris, G. M.

Nevière, M.

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop band,” Appl. Opt. 20, 345–351 (1979).

Novotny, L.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

Palamaru, M.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Palamary, M.

S. P. Astilean, Ph. Lalanne, and M. Palamary, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1991).

Pardo, F.

F. Marquier, J.-J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13, 70–76 (2005).
[CrossRef] [PubMed]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

Park, Q.-H.

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

Pelouard, J. L.

Pelouard, J.-L.

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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]

Petit, R.

R. Petit, “The homogeneous problem,” in Electromagnetic Theory of Gratings, Topics in Current Physics, M.Nevière, ed. (Springer-Verlag, 1980), Chap. 5, Vol. 22.
[CrossRef]

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

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]

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H. Raether, “Surface plasmons in gratings,” Surface Plasmons, Springer Tracts in Modern Physics (Springer-Verlag, 1988), Chap. 6, Vol. 111.

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[CrossRef]

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M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Sambles, J. R.

I. R. Hooper and J. R. Sambles, “Surface plasmon polaritons on thin-slab metal gratings,” Phys. Rev. B 67, 235404 (2003).
[CrossRef]

Scalora, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Shalaev, V.

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

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A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arthech House, 1995).

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Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5604 (2001).
[CrossRef] [PubMed]

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J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[CrossRef]

Teissier, R.

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

Thio, T.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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

Tsai, D. P.

W. C. Liu and D. P. Tsai, “Optical tunneling of surface plasmon polaritons and localized surface plasmon resonance,” Phys. Rev. B 65, 155423 (2002).
[CrossRef]

Vincent, P.

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop band,” Appl. Opt. 20, 345–351 (1979).

Vincenti, M. A.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Wang, S. S.

R. Magnusson and S. S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

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H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[CrossRef] [PubMed]

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

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402(1902).

Yala, H.

Yuan, H.-K.

Zheltikov, A. M.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Appl. Opt. (1)

P. Vincent and M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop band,” Appl. Opt. 20, 345–351 (1979).

Appl. Phys. Lett. (1)

R. Magnusson and S. S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

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

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: The s-polarization case,” J. Opt. A Pure Appl. Opt. 8, S94–S97 (2006).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A: Pure Appl. Opt. 4, S154–S160 (2002).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

Nature (London) (1)

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

Opt. Commun. (1)

S. P. Astilean, Ph. Lalanne, and M. Palamary, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Philos. Mag. (2)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402(1902).

Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

Phys. Rev. (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Phys. Rev. A (2)

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[CrossRef] [PubMed]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2007).
[CrossRef]

Phys. Rev. B (7)

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with sub-wavelength scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of the guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[CrossRef]

W. C. Liu and D. P. Tsai, “Optical tunneling of surface plasmon polaritons and localized surface plasmon resonance,” Phys. Rev. B 65, 155423 (2002).
[CrossRef]

I. R. Hooper and J. R. Sambles, “Surface plasmon polaritons on thin-slab metal gratings,” Phys. Rev. B 67, 235404 (2003).
[CrossRef]

Phys. Rev. E (2)

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with a negative-index-material cladding,” Phys. Rev. E 71, 046603 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: A comprehensive analysis,” Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Phys. Rev. Lett. (7)

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 266802 (2008).
[CrossRef]

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5604 (2001).
[CrossRef] [PubMed]

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]

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]

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

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]

F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1286 (2007).
[CrossRef]

Other (9)

R. Petit, “The homogeneous problem,” in Electromagnetic Theory of Gratings, Topics in Current Physics, M.Nevière, ed. (Springer-Verlag, 1980), Chap. 5, Vol. 22.
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

H. Raether, “Surface plasmons in gratings,” Surface Plasmons, Springer Tracts in Modern Physics (Springer-Verlag, 1988), Chap. 6, Vol. 111.

Special Issue on “Optics on the Nanoscale: Principles, Instrumentation and Applications,” Appl. Phys. B 84 (2006).

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1991).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arthech House, 1995).

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonance in nanoscale structures,” available at http://arxiv4.library.cornell.edu/abs/0902.3014v1.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 454, 39–46 (2007) (and references therein).

R. E. Collins, Field Theory of Guided Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, New York, 1991).

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Figures (11)

Fig. 1
Fig. 1

Metallic grating made of silver with grating thickness d, slit aperture a, and grating period Λ. The incident field is a plane, monochromatic, TM-polarized wave where k 0 = 2 π / λ is the wave vector, λ is the wavelength, and ϑ is the incident angle. In this case we consider air as the incident and the output medium. The Cartesian right-handed system ( x , y , z ) has the z coordinate in the propagation direction and the x coordinate along the periodicity of the grating. The y coordinate is the direction along which the magnetic field H is polarized. The input grating surface is located at z = 0 and the output surface at z = d . Inset, real ( ε r ) and imaginary ( ε i ) part of the electric permittivity of silver from data reported in [13].

Fig. 2
Fig. 2

(a) Log 10 ( T ) versus λ (incident wavelength) and d (grating thickness) for normal incidence. The grating parameters are Λ = 320 nm and a = 32 nm . The various TM-FP branches are put into evidence. The dashed white lines represent the various FP branches calculated using the FP-etalon formula with the complex effective index of the guided mode. (b) Effective index n eff versus λ for the fundamental TM mode of a planar waveguide Ag/air/Ag for different air core thicknesses. The various dimensions of the air core for the different curves from the top to the bottom are reported at the right side of the figure. The last curve on the bottom represents instead the effective index of the SP at a single air/Ag interface. Inset, schematic representation of the MIM waveguide.

Fig. 3
Fig. 3

T versus ω / ω ref and k x / k 0 = sin ( ϑ ) . ω ref is a reference frequency which corresponds to λ = 1 μm . The ω / ω ref scale goes from ω / ω ref = 0.667 ( λ = 1.5 μm ) to ω / ω ref = 2.5 ( λ = 400 nm ). The grating parameters are Λ = 256 nm , d = 400 nm , a = 32 nm . The dashed line corresponds to the dispersion of the equivalent FP-etalon for which the nearly vertical dashed line is related to the classic Brewster condition of a dielectric slab.

Fig. 4
Fig. 4

T versus ω / ω ref and k x / k 0 . The grating parameters common to both figures are Λ = 320 nm , a = 32 nm . The thickness is, respectively, (a) d = 150 nm and (b) d = 250 nm . In both figures the thin-dashed line represents the dispersion of the SPs phase matched with the first reciprocal lattice vector of the grating: Ω ( k SP + G ) . The thick-dashed line is the dispersion of the equivalent FP-etalon: ω FP ( k x ) .

Fig. 5
Fig. 5

(a) T versus ω / ω ref and k x / k 0 for the case of Fig. 4a. This is a magnification near the SP dispersion line. (b) R (reflection) versus ω / ω ref and k x / k 0 . (c) Log 10 ( T ) versus ω / ω ref and k x / k 0 . Note that the SP dispersion line (dashed line) follows exactly the transmission minima.

Fig. 6
Fig. 6

T versus ω / ω ref at k x / k 0 = sin ( 80 ° ) for the case of Fig. 4a. The dispersion of the SP that crosses the TM-FP branch forms a bandgap. The low frequency band edge resonance has the form of a Fano-Feshbach resonance, while the high frequency band edge resonance has the form of a “ridge” resonance with extremely long lifetime. Numerical calculation (continuous line), comparison with the Fano-Feshbach (F-F) resonance (dashed-dotted line), and with the ridge resonance (dashed line). The F-F resonance and the ridge resonance are calculated respectively according to Eqs. (7, 8). Ω = 1.5513 is the SP frequency at k SP - G = k 0 sin ( 80 ° ) . From the numerical calculation we can extract the parameters to fit the resonances which are Ω FF = 1.4885 (resonance frequency for the F-F), Γ FF = 0.06 (lifetime of the F-F), Ω ridge = 1.5708 (resonance frequency of the ridge corresponding to λ = 637 nm ), and Γ Ridge = 0.0045 (lifetime of the ridge) corresponding to Δ λ 1 nm .

Fig. 7
Fig. 7

T versus wavelength for TM polarization (dotted line) and TE polarization (continuous line) at normal incidence in the case of a dielectric grating with refractive index n = 4.13 , thickness d = 96 nm , period Λ = 320 nm , and slit aperture a = 32 nm . The dashed line refers to a uniform layer with same thickness and same refractive index. The arrows indicate the wavelengths of the guided modes of the single, uniform layer ( TE 0 , TE 1 , TM 0 , TM 1 ) coupled with the reciprocal lattice vectors of the grating. The ridge resonances (guided-mode resonances) are located nearby these wavelengths.

Fig. 8
Fig. 8

Log 10 ( T ) at normal incidence versus grating period (Λ) and incident wavelength (λ) for a silver grating with thickness d = 150 nm and slits aperture a = 32 nm . Note that the first branch of transmission minima follows exactly the plasmonic law, λ SP = Λ , not the Rayleigh condition λ = Λ . Also visible is the second branch λ SP = Λ / 2 which departs quite evidently from the line λ = Λ / 2 .

Fig. 9
Fig. 9

T versus wavelength (λ) and incident angle (ϑ) for the case described in Fig. 4a. (a) Topographic view. (b) 3-D view from a different perspective. The Roman numbers and the black dots indicate the resonances that we analyze.

Fig. 10
Fig. 10

(a) Topographic view of | H | 2 versus x and z in an elementary cell of the grating for the resonance I at λ = 640 nm and ϑ = 0 ° . Superimposed (black-dashed line) is the position of the metal. The white lines that cut the figure over the medians represent the sections over which the fields are represented, respectively, in (c) (vertical section) and in (d) (horizontal section). (b) Topographic view of S z versus x and z. (c) Left axis, sections of | E x | 2 (continuous-dotted line) and S z (dashed line) along the x axis, both taken at the center of the slit ( z = 75 nm ). Right axis, section of | H | 2 (continuous line) along the x axis taken at the center of the slit ( z = 75 nm ). (d) Left axis, sections of | H | 2 (continuous line) and | E x | 2 (continuous-dotted line) along the z axis, both taken at the center of the slit ( x = 160 nm ). Right axis, section of S z (dashed line) along the z axis taken at the center of the slit ( x = 160 nm ).

Fig. 11
Fig. 11

(a) Same as in the caption of Fig. 10 for the resonance II at λ = 637 nm and ϑ = 80 ° . (b) Topographic view of S z versus x and z. (c) Left axis, sections of | H | 2 (continuous line) and S z (dashed line) along the x axis, both taken at the center of the slit ( z = 75 nm ). Right axis, section of | E x | 2 (continuous-dotted line) along the x axis taken at the center of the slit ( z = 75 nm ). (d) Sections of | H | 2 (continuous line), | E x | 2 (continuous-dotted line), and S z (dashed line) along the z axis, all three taken at the center of the slit ( x = 160 nm ).

Equations (13)

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T ( λ , ϑ ) = 0 Λ S z ( x , z = d , λ , ϑ ) d x 0 Λ S z input ( x , z = 0 , λ , ϑ ) d x .
T ( λ , ϑ ) = n inc k 0 n out 2 cos ϑ m | t m | 2 Re [ n out 2 k 0 2 α m 2 ] ,
α m = k 0 n inc sin ϑ + 2 m π Λ m = 0 , ± 1 , ± 2 , .
t TM ( k x , ω ) = 2 2 cos ( n ^ eff k 0 2 k x 2 n ^ eff 2 d ) i ( n ^ eff k 0 2 k x 2 n ^ eff 2 ε ^ eff k 0 2 k x 2 + ε ^ eff k 0 2 k x 2 n ^ eff k 0 2 k x 2 n ^ eff 2 ) sin ( n ^ eff k 0 2 k x 2 n ^ eff 2 d ) .
n eff , SP = Re ε ^ Ag 1 + ε ^ Ag ,
ω ph ( k x ) = ω FP ( k x ) ,
ω ph ( k x ) = Ω ( | k SP ± G m | ) m = 0 , 1 , 2 , .
T F - F ( ω ; Ω , Ω FF , Γ FF , T MAX ) = T MAX 1 + q 2 ( ε + q ) 2 ε 2 + 1 ,
T Ridge ( ω ; Ω Ridge , Γ Ridge , T MAX ) = T MAX 2 [ T B - W ( ω ; Ω Ridge , Γ Ridge , g = 1 ) + 1 π arctan [ 4 Γ Ridge ( ω Ω Ridge ) ] + 1 2 ] ,
T B - W ( ω ; Ω Ridge , Γ Ridge , g ) = g ( Γ Ridge / 2 ) 2 ( ω Ω Ridge ) 2 + ( Γ Ridge / 2 ) 2 .
k SP ( Ω ) = G m m = 1 , 2 , .
λ SP = Λ m m = 1 , 2 , ,
λ = Λ ( Rayleigh condition ) .

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