Abstract

The modal method is applied to the problem of conical diffraction on a rectangular slit metallic grating lying on an arbitrary multilayer medium. In the approximation of the surface impedance boundary condition on the grating walls, a single matrix equation is obtained, whose coefficients are expressed simply by the reflectivities on the different layers. A simple and comprehensive treatment is thus obtained for virtually any multilayer system. The method is illustrated for the case of a cavity formed by a planar metallic mirror and a grating, as well as the system formed by a doped layer with Drude susceptibility in a substrate below the grating. The method could be useful for the design of near- and far-infrared devices.

© 2007 Optical Society of America

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2007 (2)

S. Kumar, B. S. Williams, Q. Qin, A. W. M. Lee, and Q. Hu, "Surface-emitting distributed feedback terahertz quantum cascade lasers in metal-metal waveguides," Opt. Express 15, 113-128 (2007).
[CrossRef]

Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
[CrossRef]

2006 (4)

2005 (3)

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]

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Y. Todorov, I. Abram, and C. Minot, "Dipole emission into rectangular metallic gratings with subwavelength slits," Phys. Rev. B 71, 075116 (2005).
[CrossRef]

2003 (5)

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
[CrossRef]

A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
[CrossRef]

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
[CrossRef]

2002 (4)

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (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]

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]

D. Delbeke, P. Bienstman, R. Bockstaele, and R. Baets, "Rigourous electromagnetic analysis of dipole emission in periodically corrugated layers: the grating-assisted resonant-cavity light-emitting diode," J. Opt. Soc. Am. A 19, 871-880 (2002).
[CrossRef]

2001 (1)

A.-L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 4280-4282 (2001).
[CrossRef]

2000 (2)

H. Rigneault, F. Lemarchand, and A. Sentenac, "Dipole radiation into grating structures," J. Opt. Soc. Am. A 17, 1048-1058 (2000).
[CrossRef]

P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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]

1999 (3)

J. A. Porto, F. J. Garcia-Vidal, and J. P. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

R. Amos and W. Barnes, "Modification of spontaneous emission lifetimes in the presence of corrugated metallic surfaces," Phys. Rev. B 59, 7708-7714 (1999).
[CrossRef]

R. K. Lee, O. J. Painter, B. D'Urso, A. Scherer, and A. Yariv, "Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths," Appl. Phys. Lett. 74, 1522-1524 (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 (2)

T. Suzuki and P. K. L. Yu, "Experimental and theoretical study of dipole emission in the two-dimensional photonic band structure of the square lattice with dielectric cylinders," J. Appl. Phys. 79, 582-594 (1996).
[CrossRef]

H. Lochbihler, "Diffraction from highly conducting lamellar gratings in conical mountings," J. Mod. Opt. 43, 1867-1890 (1996).
[CrossRef]

1993 (1)

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

1992 (2)

W. J. Li and B. D. McCombe, "Coupling efficiency of metallic gratings for excitation of intersubband transitions in quantum-well structures," J. Appl. Phys. 71, 1038-1040 (1992).
[CrossRef]

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced Raman scattering," J. Phys.: Condens. Matter 4, 1143-1212 (1992).
[CrossRef]

1989 (1)

P. T. Leung, Y. S. Kim, and T. F. George, "Decay of molecules at corrugated thin metal films," Phys. Rev. B 39, 9888-9893 (1989).
[CrossRef]

1987 (1)

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

1986 (1)

D. Heitmann and U. Mackens, "Grating-coupler-induced intersubband resonances in electron inversion layer of silicon," Phys. Rev. B 33, 8269-8283 (1986).
[CrossRef]

1985 (2)

K. W. Gossen and S. A. Lyon, "Grating enhanced quantum well detector," Appl. Phys. Lett. 47, 1257-1259 (1985).
[CrossRef]

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

1982 (2)

G. H. Agarwal and C. V. Kunasz, "Dipole radiation in the presence of a rough surface. Conversion of a surface-polariton field into radiation," Phys. Rev. B 26, 5832-5842 (1982).
[CrossRef]

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2917 (1982).
[CrossRef]

1981 (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "Highly conducting lamellar diffraction grating," Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

1980 (1)

S. S. Jha, J. R. Kirtley, and J. C. Tsang, "Intensity of Raman scattering from molecules adsorbed on a metallic grating," Phys. Rev. B 22, 3973-3982 (1980).
[CrossRef]

1965 (1)

1935 (1)

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
[CrossRef]

Abram, I.

Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
[CrossRef]

Y. Todorov, I. Abram, and C. Minot, "Dipole emission into rectangular metallic gratings with subwavelength slits," Phys. Rev. B 71, 075116 (2005).
[CrossRef]

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "Highly conducting lamellar diffraction grating," Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Aellen, T.

Agarwal, G. H.

G. H. Agarwal and C. V. Kunasz, "Dipole radiation in the presence of a rough surface. Conversion of a surface-polariton field into radiation," Phys. Rev. B 26, 5832-5842 (1982).
[CrossRef]

Akemann, W.

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced Raman scattering," J. Phys.: Condens. Matter 4, 1143-1212 (1992).
[CrossRef]

Amos, R.

R. Amos and W. Barnes, "Modification of spontaneous emission lifetimes in the presence of corrugated metallic surfaces," Phys. Rev. B 59, 7708-7714 (1999).
[CrossRef]

Andrewartha, J. R.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "Highly conducting lamellar diffraction grating," Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Astelean, S.

P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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]

Aubard, J.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

Aussenegg, F. R.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

Baets, R.

Balkanski, M.

M. Balkanski and R. F. Wallis, Semiconductor Physics and Applications (Oxford U. Press, 2000).

Barbara, A.

A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
[CrossRef]

Barnes, W.

R. Amos and W. Barnes, "Modification of spontaneous emission lifetimes in the presence of corrugated metallic surfaces," Phys. Rev. B 59, 7708-7714 (1999).
[CrossRef]

Beck, M.

Beere, H. E.

Behroozi, P.

Belkin, M. A.

Beltram, F.

Bienstman, P.

Blaser, S.

A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
[CrossRef]

Bockstaele, R.

Bonnet, E.

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "Highly conducting lamellar diffraction grating," Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Buchanan, M.

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
[CrossRef]

Bustarret, E.

A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
[CrossRef]

Capasso, F.

L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, "Microfluidic tuning of distributed feedback quantum cascade lasers," Opt. Express 14, 11660-11667 (2006).
[CrossRef]

S. Khanna, M. Lachab, A. G. Davies, E. H. Linfield, J. A. Fan, M. A. Belkin, and F. Capasso, "Surface emitting terahertz quantum cascade laser with a double-metal waveguide," Opt. Express 14, 11672-11680 (2006).
[CrossRef]

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

Cho, A. Y.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

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]

Colombelli, R.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

Coron, N.

Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
[CrossRef]

Coutaz, J. L.

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "Highly conducting lamellar diffraction grating," Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Davies, A. G.

Delbeke, D.

Demichel, O.

Diehl, L.

D'Urso, B.

R. K. Lee, O. J. Painter, B. D'Urso, A. Scherer, and A. Yariv, "Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths," Appl. Phys. Lett. 74, 1522-1524 (1999).
[CrossRef]

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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]

Enoch, S.

A.-L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 4280-4282 (2001).
[CrossRef]

Esaev, D. G.

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
[CrossRef]

Faist, J.

A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
[CrossRef]

L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, "Microfluidic tuning of distributed feedback quantum cascade lasers," Opt. Express 14, 11660-11667 (2006).
[CrossRef]

Fan, J. A.

Fehrembach, A.-L.

A.-L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 4280-4282 (2001).
[CrossRef]

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N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

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A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
[CrossRef]

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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. A. Porto, F. J. Garcia-Vidal, and J. P. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Garet, F.

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Geddes, C. D.

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
[CrossRef]

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Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
[CrossRef]

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P. T. Leung, Y. S. Kim, and T. F. George, "Decay of molecules at corrugated thin metal films," Phys. Rev. B 39, 9888-9893 (1989).
[CrossRef]

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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]

Gini, E.

A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
[CrossRef]

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A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
[CrossRef]

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C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

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R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
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K. W. Gossen and S. A. Lyon, "Grating enhanced quantum well detector," Appl. Phys. Lett. 47, 1257-1259 (1985).
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A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced Raman scattering," J. Phys.: Condens. Matter 4, 1143-1212 (1992).
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Greffet, J.-J.

Gryczynski, I.

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
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J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
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D. Heitmann and U. Mackens, "Grating-coupler-induced intersubband resonances in electron inversion layer of silicon," Phys. Rev. B 33, 8269-8283 (1986).
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Hofstetter, D.

L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, "Microfluidic tuning of distributed feedback quantum cascade lasers," Opt. Express 14, 11660-11667 (2006).
[CrossRef]

A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
[CrossRef]

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N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
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Hugonin, J. P.

P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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).
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A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
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S. S. Jha, J. R. Kirtley, and J. C. Tsang, "Intensity of Raman scattering from molecules adsorbed on a metallic grating," Phys. Rev. B 22, 3973-3982 (1980).
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Kim, Y. S.

P. T. Leung, Y. S. Kim, and T. F. George, "Decay of molecules at corrugated thin metal films," Phys. Rev. B 39, 9888-9893 (1989).
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S. S. Jha, J. R. Kirtley, and J. C. Tsang, "Intensity of Raman scattering from molecules adsorbed on a metallic grating," Phys. Rev. B 22, 3973-3982 (1980).
[CrossRef]

Krenn, J. R.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
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Lakowicz, J. R.

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
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P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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).
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Lee, R. K.

R. K. Lee, O. J. Painter, B. D'Urso, A. Scherer, and A. Yariv, "Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths," Appl. Phys. Lett. 74, 1522-1524 (1999).
[CrossRef]

Leitner, A.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
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Lemarchand, F.

Leung, P. T.

P. T. Leung, Y. S. Kim, and T. F. George, "Decay of molecules at corrugated thin metal films," Phys. Rev. B 39, 9888-9893 (1989).
[CrossRef]

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N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

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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).
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L. Landau and L. Lifchitz, Electrodynamics of Continuous Media (Mir, 1969).

Linfield, E. H.

Liu, H. C.

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
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H. Lochbihler, "Diffraction from highly conducting lamellar gratings in conical mountings," J. Mod. Opt. 43, 1867-1890 (1996).
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A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
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Lyon, S. A.

K. W. Gossen and S. A. Lyon, "Grating enhanced quantum well detector," Appl. Phys. Lett. 47, 1257-1259 (1985).
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D. Heitmann and U. Mackens, "Grating-coupler-induced intersubband resonances in electron inversion layer of silicon," Phys. Rev. B 33, 8269-8283 (1986).
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Mahler, L.

Malicka, J.

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, "Radiative decay engineering: the role of photonic mode density in biotechnology," J. Phys. D 36, R240-R249 (2003).
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Marquier, F.

Martin-Moreno, L.

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).
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D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
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W. J. Li and B. D. McCombe, "Coupling efficiency of metallic gratings for excitation of intersubband transitions in quantum-well structures," J. Appl. Phys. 71, 1038-1040 (1992).
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Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
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Y. Todorov, I. Abram, and C. Minot, "Dipole emission into rectangular metallic gratings with subwavelength slits," Phys. Rev. B 71, 075116 (2005).
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P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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).
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A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced Raman scattering," J. Phys.: Condens. Matter 4, 1143-1212 (1992).
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J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Oliner, A. A.

Otto, A.

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced Raman scattering," J. Phys.: Condens. Matter 4, 1143-1212 (1992).
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Painter, O.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

Painter, O. J.

R. K. Lee, O. J. Painter, B. D'Urso, A. Scherer, and A. Yariv, "Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths," Appl. Phys. Lett. 74, 1522-1524 (1999).
[CrossRef]

Palmaru, M.

P. Lalanne, J. P. Hugonin, S. Astelean, M. Palmaru, 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).
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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).
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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]

Parriaux, O.

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Pelouard, J. L.

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]

Pendry, J. P.

J. A. Porto, F. J. Garcia-Vidal, and J. P. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Perera, A. G. U.

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
[CrossRef]

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J. A. Porto, F. J. Garcia-Vidal, and J. P. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Qin, Q.

Quémerais, P.

A. Barbara, P. Quémerais, E. Bustarret, T. López, and T. Fournier, "Electromagnetic resonances of sub-wavelength rectangular metallic grating," Eur. Phys. J. D 23, 143-154 (2003).
[CrossRef]

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Rigneault, H.

Rinzan, M. B. M.

D. G. Esaev, S. G. Matsik, M. B. M. Rinzan, A. G. U. Perera, H. C. Liu, and M. Buchanan, "Resonant cavity enhancement in heterojunction GaAs/AlGaAs terahertz detectors," J. Appl. Phys. 93, 1879-1873 (2003).
[CrossRef]

Sagnes, I.

Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
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P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2917 (1982).
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R. K. Lee, O. J. Painter, B. D'Urso, A. Scherer, and A. Yariv, "Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths," Appl. Phys. Lett. 74, 1522-1524 (1999).
[CrossRef]

Schinder, G.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schinder, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003).
[CrossRef]

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A.-L. Fehrembach, S. Enoch, and A. Sentenac, "Highly directive light sources using two-dimensional photonic crystal slabs," Appl. Phys. Lett. 79, 4280-4282 (2001).
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H. Rigneault, F. Lemarchand, and A. Sentenac, "Dipole radiation into grating structures," J. Opt. Soc. Am. A 17, 1048-1058 (2000).
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R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

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P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2917 (1982).
[CrossRef]

Sivco, D. L.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2917 (1982).
[CrossRef]

Stinivasan, K.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

Straub, A.

C. Gmachl, A. Straub, R. Colombelli, F. Capasso, D. L. Sivco, A. M. Sergent, and A. Y. Cho, "Single-mode, tunable distributed feedback and multiple wavelength quantum cascade laser," IEEE J. Quantum Electron. 38, 569-581 (2002).
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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]

Thio, T.

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]

Tishchenko, A. V.

J. L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating diffraction effects in the THz domain," Acta Phys. Pol. A 107, 26-37 (2005).

Todorov, Y.

Y. Todorov, I. Sagnes, U. Gennser, N. Coron, C. Minot, and I. Abram, "Spontaneous emission enhancement in quantum cascade structures in the terahertz domain," Phys. Status Solidi C 4, 524-527 (2007).
[CrossRef]

Y. Todorov, I. Abram, and C. Minot, "Dipole emission into rectangular metallic gratings with subwavelength slits," Phys. Rev. B 71, 075116 (2005).
[CrossRef]

Trediccuci, A.

Trennant, D. M.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

Troccoli, M.

R. Colombelli, K. Stinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Trennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003).
[CrossRef] [PubMed]

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S. S. Jha, J. R. Kirtley, and J. C. Tsang, "Intensity of Raman scattering from molecules adsorbed on a metallic grating," Phys. Rev. B 22, 3973-3982 (1980).
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Wittmann, A.

A. Wittmann, M. Giovannini, J. Faist, L. Hvozdara, S. Blaser, D. Hofstetter, and E. Gini, "Room temperature, continuous wave operation of distributed feedback quantum cascade lasers with widely spaced operation frequencies," Appl. Phys. Lett. 89, 141116 (2006).
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Figures (12)

Fig. 1
Fig. 1

Multilayer stack starting from an infinite substrate B. There is no incoming field from the side of substrate B.

Fig. 2
Fig. 2

Multilayer stack bounded by a metallic grating. The semi-infinite substrate B has arbitrary complex constant ϵ B , whereas medium A above the grating is considered nonabsorbing with a real dielectric constant ϵ A . The origin of the coordinates is taken at the upper side of the grating, on the left side of a metallic stripe.

Fig. 3
Fig. 3

Feymanlike diagram describing the interaction between modes m and m (arrows) through the Rayleigh order n (wavy line), propagating in the stack between layer k and substrate B. The upward direction of the mode arrow corresponds to the overlap integral with a “+” superscript (i.e., I n m + ), and the downward direction corresponds to the “−” superscript of the overlap integral.

Fig. 4
Fig. 4

A 1 μ m thick, 5 × 10 18 cm 3 n + doped GaAs layer sandwiched between a Au grating and an undoped GaAs substrate. The substrate will be considered infinite in the z direction for simplicity. This figure is also a definition of the incident angles ϕ and θ used in the text.

Fig. 5
Fig. 5

Plot of 0th-order reflectivity R 0 as a function of the frequency for the incident angles ϕ = 0 ° and θ = 6 ° . The inset zooms on the small reflectivity features around 4 THz .

Fig. 6
Fig. 6

Frequency–wave vector plot of the 0th-order reflectivity for incidence perpendicular to the grating slits, ϕ = 0 ° . The incident wave vector is expressed in Brillouin zone units 2 π d .

Fig. 7
Fig. 7

Zengerle diagrams for the 0th-order TM reflection R 0 at frequencies 13.4 THz and 16.6 THz . The diffracted orders are highlighted by dotted curves. Only a quarter of the full diagram is plotted because of the evident symmetry properties R 0 ( α , β ) = R 0 ( α , β ) and R 0 ( α , β ) = R 0 ( α , β ) .

Fig. 8
Fig. 8

Metal–semiconductor–grating cavity with thickness L. The metallic mirror is infinite in the z direction.

Fig. 9
Fig. 9

(a) Plot of the 0th-order reflectivity spectrum of the cavity for both polarizations for the incident angles ϕ = 0 ° and θ = 30 ° . For p polarization, the frequencies of the first two resonant features ( 3.5 THz and 6.4 THz ) are indicated. (b) Plots of the H y field in the x z plane for the frequencies 3.5 THz and 6.4 THz , in arbitrary units. The dashed lines indicate the frontiers between the metal–SC–metal and metal–SC–air regions, across which an impedance mismatch appears.

Fig. 10
Fig. 10

Plot of the modal index defined in Eq. (81) (symbols) as a function of frequency. The solid curve represents the variation of the material index of the semiconductor.

Fig. 11
Fig. 11

Zengerle diagram of the 0th-order reflectivity R 0 at frequency 3.5 THz of the first resonance in the spectrum of Fig. 9a.

Fig. 12
Fig. 12

Evolution of the first resonant frequency as a function of the semiconductor slab thickness L. Gray curve, first mode of an open Fabry–Perot cavity coupled with the incident wave. Dashed curve, coupling efficiency ( 1 R 0 ) in percent.

Equations (110)

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E z = e i k p r p ( P i e e i γ i z + Q i e e i γ i z ) ,
H z = e i k p r p ( P i h e i γ i z + Q i h e i γ i z ) .
E z = e i k p r p T B e e i γ B z , H z = e i k p r p T B h e i γ B z ,
E = ± i γ i E z i ω μ 0 z ̂ × H z ϵ i k 0 2 γ i 2 ,
H = ± i γ i H z + i ω ϵ 0 ϵ i z ̂ × E z ϵ i k 0 2 γ i 2 .
P 2 e i γ 2 L 2 = 1 t 21 [ P 1 + ρ 21 Q 1 ] ,
Q 2 e i γ 2 L 2 = 1 t 21 [ ρ 21 P 1 + Q 1 ] .
t 21 e = 2 γ 2 ϵ 1 γ 2 ϵ 2 + γ 1 ϵ 1 , ρ 21 e = γ 2 ϵ 2 γ 1 ϵ 1 γ 2 ϵ 2 + γ 1 ϵ 1 ,
t 21 h = 2 γ 2 γ 2 + γ 1 , ρ 21 h = γ 2 γ 1 γ 2 + γ 1 .
P 1 e i γ 1 L 1 = 1 t 1 B ρ 1 B T B , Q 1 e i γ 1 L 1 = 1 t 1 B T B , P 1 = ρ 1 B Q 1 e 2 i γ 1 L 1 .
P 2 e i γ 2 L 2 = 1 T 2 B R 2 B T B , Q 2 e i γ 2 L 2 = 1 T 2 B T B , P 2 = R 2 B Q 2 e 2 i γ 2 L 2 .
R 2 B = ρ 21 + ρ 1 B e 2 i γ 1 L 1 1 + ρ 21 ρ 1 B e 2 i γ 1 L 1 , T 2 B = t 21 e i γ 1 L 1 t 1 B 1 + ρ 21 ρ 1 B e 2 i γ 1 L 1 .
P k = R k B Q k e 2 i γ k L k .
1 R A B ( ω , k p ) = 0 .
u e i ( k p r p γ A z ) , γ A = ϵ A k 0 2 k p 2 .
k p = ( α , β ) .
Δ ψ + k 0 2 ϵ ( x , y , z ) ψ = 0 .
ψ ( x + d , y , z ) = e i α d ψ ( x , y , z ) .
ψ ( x , y , z ) = e i β y ψ ( x , z ) .
ψ ( x , y , z ) = e i β y n = e i α n x ( P i n e i γ i n z + Q i n e i γ i n z ) ,
α n = α + 2 π n d , γ i n = ϵ i k 0 2 α n 2 β 2 .
ψ ( x , y , z ) = e i β y n = e i α n x ( R n e i γ A n z + u δ 0 n e i γ A 0 z ) ,
ψ ( x , y , z ) = e i β y n = T B n e i ( α n x γ B n z ) .
[ 0 x a ] : ( P r e i γ r x + Q r e i γ r x ) e i μ z + i β y ,
γ r = ϵ r k 0 2 μ 2 β 2 ,
[ a x d ] : ( P M e i γ M x + Q M e i γ M x ) e i μ z + i β y , γ M = ϵ M k 0 2 μ 2 β 2 .
P M e i α d = 1 t r M ( P r e i γ r a + ρ r M Q r e i γ r a ) ,
Q M e i α d = 1 t r M ( ρ r M P r e i γ r a + Q r e i γ r a ) .
( ρ r M ) 2 1 ( ρ r M ) 2 + 1 [ cos ( α d ) cos ( γ r a ) cos ( γ M ( d a ) ) ] = sin ( γ M ( d a ) ) sin ( γ r a ) ,
sin ( γ r a ) = 0 ,
γ r = ν m = π m a , μ m = ϵ r k 0 2 ν m 2 β 2 , m = 0 , 1 , 2 , .
E x [ 0 x a , y , z ] = e i β y m = 0 cos ( ν m x ) ( A m e e i μ m z + B m e e i μ m ( h z ) ) ,
H x [ 0 x a , y , z ] = e i β y m = 1 sin ( ν m x ) ( A m h e i μ m z + B m h e i μ m ( h z ) ) .
H y = m = 0 e i β y cos ( ν m x ) ϵ r k 0 2 ν m 2 [ ( i β ν m A m h + ω ϵ 0 ϵ r μ m A m e ) e i μ m z + ( i β ν m B m h ω ϵ 0 ϵ r μ m B m e ) e i μ m ( h z ) ] .
A m e = i β ν m A m h + ω ϵ 0 ϵ r μ m A m e ϵ r k 0 2 ν m 2 ,
B m e = i β ν m B m h ω ϵ 0 ϵ r μ m B m e ϵ r k 0 2 ν m 2 .
H y [ 0 x a , y , z ] = e i β y m = 0 cos ( ν m x ) ( A m e e i μ m z + B m e e i μ m ( h z ) ) ,
H x [ 0 x a , y , z ] = e i β y m = 1 sin ( ν m x ) ( A m h e i μ m z + B m h e i μ m ( h z ) ) .
E = Z n × H ,
Z = μ 0 ϵ 0 ϵ M = η Z 0 .
H ¯ = Z 0 H .
k ¯ p = k p k 0 , γ ¯ = γ k 0 .
ρ M d e η ϵ d γ ¯ d η ϵ d + γ ¯ d , ρ M d h 1 η γ ¯ d 1 + η γ ¯ d ,
n d λ ϵ M .
ϵ M ( ω ) = 1 ω M 2 ω ( ω + i δ M ) .
0 d e i ( α n α n ) x d x = 0 d e i { [ 2 π ( n n ) x d ] } d x = d δ n , n ,
0 a cos ( ν m x ) cos ( ν m x ) d x = a k m e δ m , m { m = 0 , k m e = 1 m 0 , k m e = 1 2 } ,
0 a sin ( ν m x ) sin ( ν m x ) d x = a k m h δ m , m , k m h = 1 2 .
I n m ± = 1 a 0 a e ± i α n x cos ( ν m x ) d x ,
J n m ± = 1 a 0 a e ± i α n x sin ( ν m x ) d x .
( M ̿ 1 + M ̿ S M ̿ 2 ) [ A m e B m e A m h B m h ]
= 1 α ¯ 2 + β ¯ 2 [ 0 I 0 m + ( α ¯ ϵ A u e ( 1 ρ M A e ) + β ¯ γ ¯ A u h ( 1 + ρ M A h ) ) 0 J 0 m + ( β ¯ ϵ A u e ( 1 ρ M A e ) + α ¯ γ ¯ A u h ( 1 + ρ M A h ) ) ] .
M ̿ S = M ̿ S ( m m ) = [ S B ( m m ) e e 0 S B ( m m ) e h 0 0 S A ( m m ) e e 0 S A ( m m ) e h S B ( m m ) h e 0 S B ( m m ) h h 0 0 S A ( m m ) h e 0 S A ( m m ) h h ] .
S B ( m m ) e e = a 2 d η n = I n m + I n m α ¯ n 2 + β ¯ 2 [ α ¯ n 2 ( 1 + R M B e n ) + β ¯ 2 ( 1 R M B h n ) ] .
R M B e n ( α , β ) = R M B e ( α n , β ) , R M B h n ( α , β ) = R M B h ( α n , β ) ,
1 + R M B e n 2 η = 1 2 η ( 1 + ρ M k e n + R k B e n e 2 i γ k n L k 1 + ρ M k e n R k B e n e 2 i γ k n L k ) = 1 2 η ( 1 + ρ M k e n ) ( 1 + R k B e n e 2 i γ k n L k ) 1 + ρ M k e n R k B e n e 2 i γ k n L k ,
1 + ρ M k e n 2 η = ϵ k η ϵ k + γ ¯ k n ,
Σ B n = 1 d 0 a e i α n x ( E η n × H ¯ ) z = 0 + d x ,
Σ A n = 1 d 0 a e i α n x ( E η n + × H ¯ ) z = h d x .
[ Σ B n x Σ A n x Σ B n y Σ A n y ] = a d [ I n m 0 0 0 0 I n m 0 0 0 0 J n m 0 0 0 0 J n m ] M ̿ 2 [ A m e B m e A m h B m h ] .
R n e = ρ M A e u e δ 0 n η t M A e n α ¯ n Σ A n x + β ¯ Σ A n y 2 ,
R n h = ρ M A h u h δ 0 n t M A h n β ¯ Σ A n x α ¯ n Σ A n y 2 ,
T B n e = η T M B e n α ¯ n Σ B n x + β ¯ Σ B n y 2 ,
T B n h = T M B h n β ¯ Σ B n x + α ¯ n Σ B n y 2 .
t M A e n = 2 η ( γ ¯ A n + η ϵ A ) , t M A h n = 2 1 + η γ ¯ A n .
1 d 0 d S z A d x = S z inc + n = S n z A .
S z inc = ω ϵ 0 k 0 Re ( γ ¯ A ) ϵ A u e 2 + u h 2 α ¯ 2 + β ¯ 2 + ω ϵ 0 k 0 Im ( γ ¯ A ) 2 Im ( ϵ A u e R A 0 e * + u h R A 0 h * ) α ¯ 2 + β ¯ 2 ,
S n z A = ω ϵ 0 k 0 Re ( γ ¯ A n ) ϵ A R A n e 2 + R A n h 2 α ¯ n 2 + β ¯ 2 .
R n = S n z A S z inc .
S n z B = ω ϵ 0 k 0 Re ( γ ¯ B n ) ϵ B T B n e 2 + T B n h 2 α ¯ n 2 + β ¯ 2 .
T n = S n z B S z inc .
n = ( R n + T n ) 1 ,
ψ = d β d α u ( α , β ) e i ( α x + β y γ A z ) .
ψ = d β π d π d d α k u ̃ k ( α , β ) e i ( α k x + β y γ A k z ) ,
u ̃ k ( α , β ) = u ( α k , β ) .
R ̃ k ( α ) = n R n ( α k n ) .
ϵ ( ω , N P ) = ω TO 2 ( ϵ s ϵ ) ω TO 2 ω 2 i ω δ TO + ϵ [ 1 ω P 2 ω ( ω + i δ P ) ] .
ω P = N P e 2 ϵ 0 ϵ s m * ,
α = k 0 sin θ cos ϕ , β = k 0 sin θ sin ϕ .
ω 2 c 2 = k SPP 2 ϵ ( ω , N P ) ϵ 1 ϵ ( ω , N P ) + ϵ 1 ,
Re ( ω ) c 1 ϵ 1 k SPP ,
Im ( ω ) c δ P ω P 2 ϵ k SPP 2 .
k SPP ( ω ) = ( α + 2 π n d , β ) .
ω ± 2 π c d 1 ϵ 1 ± sin θ .
α 2 + β 2 k 0 2 = ω 2 c 2 .
ν m = c m 2 n M ( d a ) .
ω n = c n 1 π 2 L 2 ( n 1 2 ) 2 + k P 2 ,
ν 0 = ω n = 0 2 π = c 4 L 1 n 1 2 sin 2 θ .
0 d e i α n x ( E η n + × H ¯ ) z = h + d x = 0 a e i α n x ( E η n + × H ¯ ) z = h d x ,
n + = ( 0 , 0 , 1 ) ,
0 a H ¯ y z = h + cos ( ν m x ) d x = 0 a H ¯ y z = h cos ( ν m x ) d x ,
0 a H ¯ x z = h + sin ( ν m x ) d x = 0 a H ¯ x z = h sin ( ν m x ) d x .
0 d e i α n x ( E η n × H ¯ ) z = 0 d x = 0 a e i α n x ( E η n × H ¯ ) z = 0 + d x ,
n = ( 0 , 0 , 1 ) ,
0 a H ¯ y z = 0 cos ( ν m x ) d x = 0 a H ¯ y z = 0 + cos ( ν m x ) d x ,
0 a H ¯ x z = 0 sin ( ν m x ) d x = 0 a H ¯ x z = 0 + sin ( ν m x ) d x .
1 α ¯ n 2 + β ¯ 2 [ α ¯ n I n m + 0 β ¯ I n m + 0 0 α ¯ n I n m + 0 β ¯ I n m + β ¯ n J n m + 0 α ¯ n J n m + 0 0 β ¯ J n m + 0 α ¯ n J n m + ] [ ϵ k ( Q k n e + P k n e ) ϵ A ( R n e + u e δ 0 n ) γ ¯ k n ( Q k n h P k n h ) γ ¯ A n ( R n h u h δ 0 n ) ] = M ̿ 1 [ A m e B m e A m h B m h ] .
M ̿ 1 = M ̿ 1 ( m m ) = δ m m [ ( k m e g m k m e g m k m e k m e ) ( 0 0 0 0 ) ( 0 0 0 0 ) ( k m h g m k m h g m k m h k m h ) ] ,
g m = e i μ m h .
M ̿ 2 = M ̿ 2 ( m m ) = δ m m [ ( ν m e g m ν m e + g m ν m e + ν m e ) ( ν m 0 g m ν m 0 g m ν m 0 ν m 0 ) ( ν m 0 g m ν m 0 g m ν m 0 ν m 0 ) ( ν m h g m ν m h + g m ν m h + ν m h ) ] ,
ν m 0 = i β ν m μ m ϵ r , ν m e ± = ϵ r ν m 2 μ m ϵ r ± η , ν m h ± = ϵ r β 2 μ m ϵ r ± η .
ν m e ν m e + = μ m η ϵ r μ m + η ϵ r , ν m h ν m h + = 1 η μ m 1 η μ m ,
S B ( m m ) e e = a 2 d η n = I n m + I n m α ¯ n 2 + β ¯ 2 [ α ¯ n 2 ( 1 + R M B e n ) + β ¯ 2 ( 1 R M B h n ) ] ,
S A ( m m ) e e = a 2 d η n = I n m + I n m α ¯ n 2 + β ¯ 2 [ α ¯ n 2 ( 1 + ρ M A e n ) + β ¯ 2 ( 1 ρ M A h n ) ] ,
S B ( m m ) h h = a 2 d η n = J n m + J n m α ¯ n 2 + β ¯ 2 [ α ¯ n 2 ( 1 R M B h n ) + β ¯ 2 ( 1 + R M B e n ) ] ,
S A ( m m ) h h = a 2 d η n = J n m + J n m α ¯ n 2 + β ¯ 2 [ α ¯ n 2 ( 1 ρ M A h n ) + β ¯ 2 ( 1 + ρ M A e n ) ] ,
S B ( m m ) e h = a 2 d η n = I n m + J n m α ¯ n 2 + β ¯ 2 α ¯ n β ¯ ( R M B e n + R M B h n ) ,
S A ( m m ) e h = a 2 d η n = I n m + J n m α ¯ n 2 + β ¯ 2 α ¯ n β ¯ ( ρ M A e n + ρ M A h n ) ,
S B ( m m ) h e = a 2 d η n = J n m + I n m α ¯ n 2 + β ¯ 2 α ¯ n β ¯ ( R M B e n + R M B h n ) ,
S A ( m m ) h e = a 2 d η n = J n m + I n m α ¯ n 2 + β ¯ 2 α ¯ n β ¯ ( ρ M A e n + ρ M A h n ) .

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