Abstract

We present an experimental and theoretical study of the optical properties of metal-dielectric-metal structures with patterned top metallic surfaces, in the THz frequency range. When the thickness of the dielectric slab is very small with respect to the wavelength, these structures are able to support strongly localized electromagnetic modes, concentrated in the subwavelength metal-metal regions. We provide a detailed analysis of the physical mechanisms which give rise to these photonic modes. Furthermore, our model quantitatively predicts the resonance positions and their coupling to free space photons. We demonstrate that these structures provide an efficient and controllable way to convert the energy of far field propagating waves into near field energy.

© 2010 OSA

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2010 (1)

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

2009 (4)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

2008 (2)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

2007 (4)

2006 (4)

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73(15), 153405 (2006).
[CrossRef]

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
[CrossRef] [PubMed]

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt. 8(4), S87–S93 (2006).
[CrossRef]

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

2005 (1)

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
[CrossRef]

2004 (3)

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67(8), 085415 (2003).
[CrossRef]

2002 (1)

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

2001 (1)

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(3), 033107 (2001).
[CrossRef]

2000 (2)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

1998 (1)

1982 (1)

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(6), 2907–2916 (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 (Lond.) 28, 1103–1106 (1981).
[CrossRef]

1935 (1)

R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935).
[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 (Lond.) 28, 1103–1106 (1981).
[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 (Lond.) 28, 1103–1106 (1981).
[CrossRef]

Andrews, A. M.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

Andrews, S. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

Bahriz, M.

Barbieri, S.

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Barnes, W. L.

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt. 8(4), S87–S93 (2006).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Beere, H. E.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Beltram, F.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

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 (Lond.) 28, 1103–1106 (1981).
[CrossRef]

Brown, J. R

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Capasso, F.

Chassagneux, Y.

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Cho, A. Y.

Collin, S.

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(3), 033107 (2001).
[CrossRef]

Colombelli, R.

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

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 (Lond.) 28, 1103–1106 (1981).
[CrossRef]

Crisafulli, O.

Davies, A. G.

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Faist, J.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Long-wavelength (λ ≈ 8–11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. 23, 1366–1368 (1998).
[CrossRef]

Fernández-Domínguez, A. I.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

García-Vidal, F. J.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

Gmachl, C.

Gordon, R.

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73(15), 153405 (2006).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

Hobson, P. A.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

Hu, Q.

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
[CrossRef]

Hutchinson, A. L.

Iotti, R. C.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Khanna, S. P.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

Klang, P.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

Köhler, R.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Kumar, S.

Lalanne, P.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
[CrossRef]

Lawrence, C. R.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

Lee, A. W.

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

Linfield, E. H.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Y. Chassagneux, R. Colombelli, W. Maineults, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Predictable surface emission patterns in terahertz photonic-crystal quantum cascade lasers,” Opt. Express 17(12), 9491–9502 (2009).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Liu, H.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

Mahler, L.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Maier, S. A.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
[CrossRef] [PubMed]

Maineult, W.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

Maineults, W.

Martín-Moreno, L.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

McPhedran, R. C.

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

Minot, C.

Moreau, V.

Murray, W. A.

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

Painter, O.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Pardo, F.

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(3), 033107 (2001).
[CrossRef]

Pelouard, J.-L.

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(3), 033107 (2001).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Qin, Q.

Reno, J. L.

Ritchie, D. A.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Rodier, J. C.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
[CrossRef]

Rossi, F.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Sagnes, I.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

Sanda, P. N.

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(6), 2907–2916 (1982).
[CrossRef]

Sarrazin, M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67(8), 085415 (2003).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Sheng, P.

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(6), 2907–2916 (1982).
[CrossRef]

Sirtori, C.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Long-wavelength (λ ≈ 8–11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. 23, 1366–1368 (1998).
[CrossRef]

Sivco, D. L.

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

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(6), 2907–2916 (1982).
[CrossRef]

Strasser, G.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

Teissier, R.

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(3), 033107 (2001).
[CrossRef]

Todorov, Y.

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

Y. Todorov and C. Minot, “Modal method for conical diffraction on a rectangular slit metallic grating in a multilayer structure,” J. Opt. Soc. Am. A 24(10), 3100–3114 (2007).
[CrossRef]

Tredicucci, A.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Tyler, J.

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

Vahala, K. J.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Vigneron, J.-P.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67(8), 085415 (2003).
[CrossRef]

Vigoureux, J.-M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67(8), 085415 (2003).
[CrossRef]

Walther, C.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Wedge, S.

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[CrossRef]

Wiersma, D. S.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Williams, B. S.

Williams, C. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wood, R. W.

R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935).
[CrossRef]

Xu, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

Yariv, A.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, P. A. Hobson, and C. R. Lawrence, “Thin resonant structures for angle and polarization independent microwave absorption,” Appl. Phys. Lett. 94(4), 041913 (2009).
[CrossRef]

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

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt. 8(4), S87–S93 (2006).
[CrossRef]

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

Nat. Photonics (3)

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010).
[CrossRef]

Nature (4)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Opt. Acta (Lond.) (1)

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

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. (1)

R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935).
[CrossRef]

Phys. Rev. B (5)

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(3), 033107 (2001).
[CrossRef]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67(8), 085415 (2003).
[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(6), 2907–2916 (1982).
[CrossRef]

A. P. Hibbins, W. A. Murray, J. Tyler, S. Wedge, W. L. Barnes, and J. R. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
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R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73(15), 153405 (2006).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
[CrossRef]

Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. 92, 143904 (2004).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
[CrossRef] [PubMed]

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Other (12)

As the thickness L is increased beyond 1µm the higher order guided modes should be taken into account in order to obtain quantitative results. Nevertheless, the qualitative behaviour described in the text remains the same.

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Figures (9)

Fig. 1
Fig. 1

(a) Schematics of the structure. A semiconductor slab with thickness L is sandwiched between a metallic plane and a lamellar metallic grating. The inset shows a top view of a sample with typical dimensions. The direction and the polarization of the incident plane wave is also indicated. (b) Experimental spectra for incident angles θ = 10° and 45° (dotted curves) and simulation at θ = 45° (continuous line). (c) Resonant frequencies of the reflectivity dips as a function of the inverse strip width c/s. (d) Effective index of all the resonances as deduced from Eq. (2) (dots). The red continuous line is the bulk semiconductor index ns. For all plots, the restrhalen band of GaAs is indicated in grey.

Fig. 2
Fig. 2

(a) Plot of the tree components (Ez, Hy and Ex) of the electromagnetic field for the first (K = 1) resonance, under normal incidence θ = 0°. The maximal values are in red and minimal in blue. The induced charges and currents are obtained trough Eq. (9). (b) Electric field Ez for the first odd resonances (K = 1,3). The arrow indicates the dipole moment of the induced charge distribution on the metallic stripe. (c) Expected charge distribution of the K = 2 mode at normal incidence θ = 0°, and net induced dipole at oblique incidence.

Fig. 3
Fig. 3

(a) Illustration of the impedance mismatch between the TM0 guided mode L and the radiation continuum in the single metal region. (b) Plot of the module and the argument of the complex reflectivity ρ (Eq. (13)), as a function of the thickness L for a wavelength λ = 100µm (frequency ν = 3 THz), in the single-mode approximation.

Fig. 4
Fig. 4

(a) Effective index of the K = 1 and K = 3 resonances (dots) compared with theoretical predictions from Eq. (16) (black lines). The effective index ng of TM0 mode of a planar double-metal waveguide of thickness L = 0.8µm is also indicated. (b) Quality factors as deduced from experiment (dots), full model (red continuous line) and the simplified model from Eq. (17) (dashed curve).

Fig. 5
Fig. 5

Dispersion diagrams for structures with increasing slab thickness L and identical gratings (s = 11.5µm, a = 5.5µm) . The blue continuous line is the air light cone. The blue dotted lines are the folded dispersion of the guided TM modes of a planar double metal waveguide with the same thickness as the grating structures. The two insets show the magnetic field distributions Hy for structures with L = 0.8µm and L = 10µm (not at the same scale).

Fig. 6
Fig. 6

(a) Structure with a square patch grating. The inset shows a top view of a sample with typical dimensions. The incident direction and the possible polarizations of the incident beam are also shown. (b) Reflectivity spectra of a sample with dimensions L = 1.5µm, p = 17µm and s = 14.3µm for both o- and p- polarized incident beam. The incident angle is θ = 45° with respect to the grating normal.

Fig. 7
Fig. 7

Electric field Ez distribution (Eq. (23)) and induced charge distribution (through Eq. (9)) for the first excited modes. Here k// and (E)// are the projections of, respectively, the wavevector and the electric field of the incident wave on the grating surface (xy-plane). For each mode, we have indicated with red arrows the net dipole induced under these experimental conditions.

Fig. 8
Fig. 8

(a) Two reflectivity spectra (dotted curves) for two different values of the slab thickness (L = 0.85µm and L = 1.5µm) and the corresponding lorenzian fits (continuous lines). The grating parameters are p = 17µm and s = 13.0µm. The coupling efficiency 1-Rmin is defined as the amplitude of the lorenzians profile Eq. (29). (b) Experimental values of the coupling efficiency as a function of the strip widths s, for various parameters L and p. The dashed lines are guide for the eye. (c) Simulated coupling efficiency and Poynting flux (Eq. (33)) as a function of the slab thickness, for a grating with p = 17µm and s = 13.0µm. The dots indicate the experimental values. (d) Simulated coupling efficiency and Poynting flux as a function of the grating period, for a slab thickness L = 0.85µm and s = 13.0µm.

Fig. 9
Fig. 9

(a) Experimental configuration for probing the structure reflectivity in the mid-infrared spectral range. (b) Typical reflectivity spectra for a structure with parameters p = 20µm and a = 7.3µm. The CO2 atmospheric absorption is also indicated. (c) Dispersion diagram for the observed resonances, obtained from multiple reflection measurements. The dots are experimental measurements and the continuous lines are numerical computation. The electric field Ey for various mode orders is also plotted.

Equations (36)

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ε ( ν ) = 1 1.52 × 10 30 ν ( ν i 1.6 × 10 13 ) ,
ν K = K 2 n M c s
E z ( x ) E z cos ( π K s x ) , H y ( x ) H y 0 sin ( π K s x ) , x ( 0 , s ) .
E x ( x , z = 0 ) E z ( x = 0 , z = 0 ) δ ( x ) E z ( x = s , z = 0 ) δ ( x s ) ,
E x 0 ( x , z > 0 ) = ( R 0 e i γ 0 z e i γ 0 z ) e i α 0 x ,
α 0 = k 0 sin θ = 2 π λ sin θ ,                               γ 0 = k 0 2 α 0 2 .
R 0 1 = 1 p 0 p e i α 0 x E x ( x , z = 0 ) d x .
I K ( θ ) = 1 ( 1 ) K e i 2 π s λ K sin θ = 1 ( 1 ) K e i π K n M sin θ .
σ = E . n , j = H × n .
( L z 0 ;   < x < 0 ) , E z T M = A exp ( i β x x ) + B exp ( i β x x ) ,
( L z , 0 < x < ) , E z ( x , z ) = 0 + C ( k z ) exp ( i k x x + i k z z ) d k z , k x = n s 2 k 0 2 k z 2 .
ρ = B / A .
ρ = n s n ˜ g n s + n ˜ g , n ˜ g = n g 2 π φ 1 + sinc 2 ( t φ / 2 ) 1 t 2 d t .
1 ρ 2 exp ( 2 i β x s ) = 0.
ν ˜ K = c 2 s n g ( K Arg ( ρ ) π ) + i c 2 s n g ln | ρ | π .
n M = n g 1 Arg ( ρ ) π K .
Q K = Re ( ν ˜ K ) 2 Im ( ν ˜ K ) = π K Arg ( ρ ) 2 ln | ρ | .
C ( k z ) = E z ( x = 0 ) 2 π i 1 e i k z L k z n s 2 k 0 2 k z 2 .
E x ( x , z ) = E z ( x = 0 ) 1 2 π i + e i k z z + i n s 2 k 0 2 k z 2 x n s 2 k 0 2 k z 2 d k z .
ν 2 = ( c K 2 n M s ) 2 + c 2 4 π 2 n s 2 k y 2 .
ν 2 = ( K c 2 n M s ) 2 n s 2 n s 2 sin 2 ϕ .
ν N M = c 2 n M s N 2 + M 2 ,
E z ( x , y ) E z 0 cos ( π N s x ) cos ( π M s y ) ,
k // = ( k 0 sin θ cos ϕ , k 0 sin θ sin ϕ ) ,
( p -polarization ) E // 00 = k // γ k 0 2 ( R 00 e i γ z e i γ z ) e i k // . r
( o -polarization ) E // 00 = ( k // u // ) γ k 0 2 ( R 00 e i γ z e i γ z ) e i k // . r
E // ( x , y , z = 0 ) [ E z ( 0 , y ) δ ( x + s ) + E z ( s , y ) δ ( x ) ] u x + [ E z ( x , 0 ) δ ( y + s ) + E z ( x , s ) δ ( y ) ] u y .
I p r e s o n a t o r e d g e k // . E // k // 2 e i k // r d x d y , I o r e s o n a t o r e d g e k // E // k // 2 e i k // r d x d y .
I p = ( 1 ( 1 ) N e i Ω N M ) δ 0 M , I o = ( 1 ( 1 ) M ) 1 ( 1 ) N e i Ω N M π 2 N 2 Ω N M 2 Ω N M .
R ( ν ) = 1 1 R min 1 + ( ν ν K ) 2 / Δ ν 2 ,
H y = n e i α n x ( P n e i γ n S z + Q n e i γ n S z ) , Q n = r n P n e 2 i γ n S L ,
α n = k 0 sin θ + 2 π p n , γ n S = n s 2 k 0 2 α n 2 .
P n , Q n 0 s H y 0 sin ( π K s x ) e i α n x d x sin ( α n s / 2 ) α n 2 s 2 / 4 1 .
S z = n ( e v ) | P n | γ n S 2 e 2 γ n S L Im r n .
ν M K = c 2 n ' M M 2 a 2 + ( K 1 / 2 ) 2 ( L + h ) 2 .
I M = 1 ( 1 ) M e i Φ π 2 M 2 Φ 2 π M , Φ = 2 π a λ M , K sin θ .

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