Abstract

As shown in a recent letter [Nature 452, 728 (2008) ] with a microscopic model, the phenomenon of the extraordinary optical transmission (EOT) is intrinsically due to two distinct surface waves: the surface plasmon polariton and the quasi-cylindrical wave (quasi-CW) that efficiently funnel light into the hole aperture at resonance. Here we present a comprehensive microscopic model of the EOT that takes into account the two surface waves. The model preserves the desirable physical insight of the previous approach, but since it additionally takes into account the quasi-CWs, it provides highly accurate predictions over a much broader spectral range, from visible to microwave radiation. The net outcome is a complete understanding of many aspects of the EOT and especially of the role of the metal conductivity that has largely puzzled the initial interpretations. We believe that the main conclusions of the present analysis may be applied to many Wood-type surface resonances on metallic surfaces.

© 2010 Optical Society of America

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    [CrossRef]

2010 (9)

B. Wang and P. Lalanne, “How many surface plasmons are locally excited on the ridges of metallic lamellar gratings?” Appl. Phys. Lett. 96, 051115 (2010).
[CrossRef]

M. Diwekar, S. Blair, and M. Davis, “Increased light gathering capacity of sub-wavelength conical metallic apertures,” J. Nanophotonics 4, 043504 (2010).
[CrossRef]

G. A. Zheng, X. Q. Cui, and C. H. Yang, “Surface-wave-enabled darkfield aperture for background suppression during weak signal detection,” Proc. Natl. Acad. Sci. U.S.A. 107, 9043–9048 (2010).
[CrossRef] [PubMed]

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nature Mater. 9, 407–412 (2010).
[CrossRef]

H. T. Liu and P. Lalanne, “Light scattering by metallic surfaces with subwavelength patterns,” Phys. Rev. B 82, 115418 (2010).
[CrossRef]

Q. Q. Gan, Y. K. Gao, Q. Wang, L. Zhu, and F. Bartoli, “Surface plasmon waves generated by nanogrooves through spectral interference,” Phys. Rev. B 81, 085443 (2010).
[CrossRef]

L. Cai, G. Y. Li, Z. H. Wang, and A. S. Xu, “Interference and horizontal Fabry–Perot resonance on extraordinary transmission through a metallic nanoslit surrounded by grooves,” Opt. Lett. 35, 127–129 (2010).
[CrossRef] [PubMed]

G. Y. Li, L. Cai, F. Xiao, Y. J. Pei, and A. S. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010).
[CrossRef] [PubMed]

2009 (5)

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

X. Y. Yang, H. T. Liu, and P. Lalanne, “Cross-conversion between surface plasmon polaritons and quasi-cylindrical waves,” Phys. Rev. Lett. 102, 153903 (2009).
[CrossRef] [PubMed]

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64, 453–469 (2009).
[CrossRef]

W. Dai and C. M. Soukoulis, “Theoretical analysis of the surface wave along a metal-dielectric interface,” Phys. Rev. B 80, 155407 (2009).
[CrossRef]

A. Y. Nikitin, S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[CrossRef]

2008 (3)

H. T. Liu, P. Lalanne, X. Y. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

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

2007 (5)

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[CrossRef]

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

2006 (3)

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556 (2006).
[CrossRef]

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express 14, 11823–11832 (2006).
[CrossRef] [PubMed]

2005 (7)

C. Genet, M. P. van Exter, and J. P. Woerdman, “Huygens description of resonance phenomena in subwavelength hole arrays,” J. Opt. Soc. Am. A 22, 998–1002 (2005).
[CrossRef]

C. Liu, V. Kamaev, and Z. V. Vardeny, “Efficiency enhancement of an organic light emitting diode with a cathode forming two-dimensional periodic hole array,” Appl. Phys. Lett. 86, 143501 (2005).
[CrossRef]

F. J. García de Abajo and J. J. Saenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7, 422–426 (2005).
[CrossRef]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

2004 (3)

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

X. G. Luo and T. Ishihara, “Sub-100-nm photolithography based on plasmon resonance,” Jpn. J. Appl. Phys., Part 1 43, 4017–4021 (2004).
[CrossRef]

Y. H. Ye and J. Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84, 2977–2979 (2004).
[CrossRef]

2003 (2)

J. Gómez Rivas, C. Schotsch, P. Haring Bolivar and H. Kurz, “Enhanced transmission of THz radiation through subwavelength holes,” Phys. Rev. B 68, 201306(R) (2003).
[CrossRef]

S. Collin, F. Pardo, and J. L. Pelouard, “Resonant-cavity-enhanced subwavelength metal-semiconductor-metal photodetector,” Appl. Phys. Lett. 83, 1521–1523 (2003).
[CrossRef]

2001 (2)

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

2000 (1)

P. 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]

1999 (1)

M. M. Alkaisi, R. J. Blaikie, S. J. McNab, R. Cheung, and D. R. S. Cumming, “Sub-diffraction-limited patterning using evanescent near-field optical lithography,” Appl. Phys. Lett. 75, 3560–3562 (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 391, 667–669 (1998).
[CrossRef]

1996 (1)

I. I. Smolyaninov, D. L. Mazzoni, and C. C. Davis, “Imaging of surface plasmon scattering by lithographically created individual surface defects,” Phys. Rev. Lett. 77, 3877–3880 (1996).
[CrossRef] [PubMed]

1995 (2)

1991 (1)

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

1987 (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

1985 (1)

E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic, 1985).

1980 (1)

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

1978 (1)

D. A. Hill and J. R. Wait, “Excitation of the Zenneck surface wave by a vertical aperture,” Radio Sci. 13, 969–977 (1978).
[CrossRef]

1973 (1)

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

1941 (1)

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).

’t Hooft, G. W.

O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express 14, 11823–11832 (2006).
[CrossRef] [PubMed]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Aigouy, L.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

Alkaisi, M. M.

M. M. Alkaisi, R. J. Blaikie, S. J. McNab, R. Cheung, and D. R. S. Cumming, “Sub-diffraction-limited patterning using evanescent near-field optical lithography,” Appl. Phys. Lett. 75, 3560–3562 (1999).
[CrossRef]

Alkemade, P. F. A.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Alloschery, O.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

Anderton, C. R.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

Astilean, S.

P. 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]

Atwater, H. A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nature Mater. 9, 407–412 (2010).
[CrossRef]

Baida, F. I.

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

Bardou, N.

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

Bartoli, F.

Q. Q. Gan, Y. K. Gao, Q. Wang, L. Zhu, and F. Bartoli, “Surface plasmon waves generated by nanogrooves through spectral interference,” Phys. Rev. B 81, 085443 (2010).
[CrossRef]

Blaikie, R. J.

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G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

Palamaru, M.

P. 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]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic, 1985).

Pardo, F.

S. Collin, F. Pardo, and J. L. Pelouard, “Resonant-cavity-enhanced subwavelength metal-semiconductor-metal photodetector,” Appl. Phys. Lett. 83, 1521–1523 (2003).
[CrossRef]

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Pei, Y. J.

Pellerin, K. M.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Pelouard, J. L.

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

S. Collin, F. Pardo, and J. L. Pelouard, “Resonant-cavity-enhanced subwavelength metal-semiconductor-metal photodetector,” Appl. Phys. Lett. 83, 1521–1523 (2003).
[CrossRef]

Pendry, J. B.

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

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

Polman, A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nature Mater. 9, 407–412 (2010).
[CrossRef]

Pommet, D. A.

Poujet, Y.

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

Radko, I. P.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Roberts, A.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

Rodier, J. C.

P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7, 422–426 (2005).
[CrossRef]

Rodrigo, S. G.

A. Y. Nikitin, S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[CrossRef]

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Rogers, J. A.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

Rommeluere, S.

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

Saenz, J. J.

F. J. García de Abajo and J. J. Saenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

Salvi, J.

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

Schotsch, C.

J. Gómez Rivas, C. Schotsch, P. Haring Bolivar and H. Kurz, “Enhanced transmission of THz radiation through subwavelength holes,” Phys. Rev. B 68, 201306(R) (2003).
[CrossRef]

Schouten, H. F.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Silberstein, E.

Smolyaninov, I. I.

I. I. Smolyaninov, D. L. Mazzoni, and C. C. Davis, “Imaging of surface plasmon scattering by lithographically created individual surface defects,” Phys. Rev. Lett. 77, 3877–3880 (1996).
[CrossRef] [PubMed]

Soukoulis, C. M.

W. Dai and C. M. Soukoulis, “Theoretical analysis of the surface wave along a metal-dielectric interface,” Phys. Rev. B 80, 155407 (2009).
[CrossRef]

Stewart, M. E.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

Taflove, A.

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

Thio, T.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (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 391, 667–669 (1998).
[CrossRef]

Thompson, L. B.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

Urbach, H. P.

van Exter, M. P.

Van Labeke, D.

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

Vardeny, Z. V.

C. Liu, V. Kamaev, and Z. V. Vardeny, “Efficiency enhancement of an organic light emitting diode with a cathode forming two-dimensional periodic hole array,” Appl. Phys. Lett. 86, 143501 (2005).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Viaris de Lesegno, B.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

Vincent, G.

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

Visser, T. D.

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Wait, J. R.

D. A. Hill and J. R. Wait, “Excitation of the Zenneck surface wave by a vertical aperture,” Radio Sci. 13, 969–977 (1978).
[CrossRef]

Wang, B.

B. Wang and P. Lalanne, “How many surface plasmons are locally excited on the ridges of metallic lamellar gratings?” Appl. Phys. Lett. 96, 051115 (2010).
[CrossRef]

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64, 453–469 (2009).
[CrossRef]

Wang, Q.

Q. Q. Gan, Y. K. Gao, Q. Wang, L. Zhu, and F. Bartoli, “Surface plasmon waves generated by nanogrooves through spectral interference,” Phys. Rev. B 81, 085443 (2010).
[CrossRef]

Wang, Z. H.

Weeber, J. C.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Weiner, J.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Woerdman, J. P.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 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).

Xiao, F.

Xu, A. S.

Yang, C. H.

G. A. Zheng, X. Q. Cui, and C. H. Yang, “Surface-wave-enabled darkfield aperture for background suppression during weak signal detection,” Proc. Natl. Acad. Sci. U.S.A. 107, 9043–9048 (2010).
[CrossRef] [PubMed]

Yang, X. Y.

X. Y. Yang, H. T. Liu, and P. Lalanne, “Cross-conversion between surface plasmon polaritons and quasi-cylindrical waves,” Phys. Rev. Lett. 102, 153903 (2009).
[CrossRef] [PubMed]

H. T. Liu, P. Lalanne, X. Y. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
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Y. H. Ye and J. Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84, 2977–2979 (2004).
[CrossRef]

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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Zhang, J. Y.

Y. H. Ye and J. Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84, 2977–2979 (2004).
[CrossRef]

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G. A. Zheng, X. Q. Cui, and C. H. Yang, “Surface-wave-enabled darkfield aperture for background suppression during weak signal detection,” Proc. Natl. Acad. Sci. U.S.A. 107, 9043–9048 (2010).
[CrossRef] [PubMed]

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Q. Q. Gan, Y. K. Gao, Q. Wang, L. Zhu, and F. Bartoli, “Surface plasmon waves generated by nanogrooves through spectral interference,” Phys. Rev. B 81, 085443 (2010).
[CrossRef]

Appl. Phys. Lett. (5)

B. Wang and P. Lalanne, “How many surface plasmons are locally excited on the ridges of metallic lamellar gratings?” Appl. Phys. Lett. 96, 051115 (2010).
[CrossRef]

S. Collin, F. Pardo, and J. L. Pelouard, “Resonant-cavity-enhanced subwavelength metal-semiconductor-metal photodetector,” Appl. Phys. Lett. 83, 1521–1523 (2003).
[CrossRef]

C. Liu, V. Kamaev, and Z. V. Vardeny, “Efficiency enhancement of an organic light emitting diode with a cathode forming two-dimensional periodic hole array,” Appl. Phys. Lett. 86, 143501 (2005).
[CrossRef]

M. M. Alkaisi, R. J. Blaikie, S. J. McNab, R. Cheung, and D. R. S. Cumming, “Sub-diffraction-limited patterning using evanescent near-field optical lithography,” Appl. Phys. Lett. 75, 3560–3562 (1999).
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Y. H. Ye and J. Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84, 2977–2979 (2004).
[CrossRef]

Chem. Rev. (Washington, D.C.) (1)

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. (Washington, D.C.) 108, 494–521 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. T. Liu, P. Lalanne, X. Y. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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[CrossRef]

J. Mod. Opt. (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
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J. Nanophotonics (1)

M. Diwekar, S. Blair, and M. Davis, “Increased light gathering capacity of sub-wavelength conical metallic apertures,” J. Nanophotonics 4, 043504 (2010).
[CrossRef]

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

P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7, 422–426 (2005).
[CrossRef]

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

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys., Part 1 (1)

X. G. Luo and T. Ishihara, “Sub-100-nm photolithography based on plasmon resonance,” Jpn. J. Appl. Phys., Part 1 43, 4017–4021 (2004).
[CrossRef]

Nano Lett. (1)

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Nat. Phys. (3)

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556 (2006).
[CrossRef]

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. 2, 262–267 (2006).
[CrossRef]

Nature (3)

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[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 391, 667–669 (1998).
[CrossRef]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

Nature Mater. (1)

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nature Mater. 9, 407–412 (2010).
[CrossRef]

New J. Phys. (1)

A. Y. Nikitin, S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[CrossRef]

Opt. Commun. (1)

F. I. Baida, Y. Poujet, J. Salvi, D. Van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-λ annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Philos. Mag. (1)

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

Phys. Rev. B (4)

H. T. Liu and P. Lalanne, “Light scattering by metallic surfaces with subwavelength patterns,” Phys. Rev. B 82, 115418 (2010).
[CrossRef]

W. Dai and C. M. Soukoulis, “Theoretical analysis of the surface wave along a metal-dielectric interface,” Phys. Rev. B 80, 155407 (2009).
[CrossRef]

J. Gómez Rivas, C. Schotsch, P. Haring Bolivar and H. Kurz, “Enhanced transmission of THz radiation through subwavelength holes,” Phys. Rev. B 68, 201306(R) (2003).
[CrossRef]

Q. Q. Gan, Y. K. Gao, Q. Wang, L. Zhu, and F. Bartoli, “Surface plasmon waves generated by nanogrooves through spectral interference,” Phys. Rev. B 81, 085443 (2010).
[CrossRef]

Phys. Rev. Lett. (8)

X. Y. Yang, H. T. Liu, and P. Lalanne, “Cross-conversion between surface plasmon polaritons and quasi-cylindrical waves,” Phys. Rev. Lett. 102, 153903 (2009).
[CrossRef] [PubMed]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, D. L. Mazzoni, and C. C. Davis, “Imaging of surface plasmon scattering by lithographically created individual surface defects,” Phys. Rev. Lett. 77, 3877–3880 (1996).
[CrossRef] [PubMed]

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

F. J. García de Abajo and J. J. Saenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

S. Collin, G. Vincent, R. Haidar, N. Bardou, S. Rommeluere, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104, 027401 (2010).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

G. A. Zheng, X. Q. Cui, and C. H. Yang, “Surface-wave-enabled darkfield aperture for background suppression during weak signal detection,” Proc. Natl. Acad. Sci. U.S.A. 107, 9043–9048 (2010).
[CrossRef] [PubMed]

Radio Sci. (1)

D. A. Hill and J. R. Wait, “Excitation of the Zenneck surface wave by a vertical aperture,” Radio Sci. 13, 969–977 (1978).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[CrossRef]

Science (1)

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

Surf. Sci. Rep. (1)

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64, 453–469 (2009).
[CrossRef]

Other (6)

E. D. Palik, Handbook of Optical Constants of Solids, Part II (Academic, 1985).

Normal modes are defined as waveguide modes that obey an exponential propagation rule exp(ik0neffz) along the invariant z-direction of the waveguide; see details in .

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

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

J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

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Figures (5)

Fig. 1
Fig. 1

Elementary HW scattering events (a)–(c) for building up the EOT phenomenon (d). (a)–(c) HW scattering coefficients at a 1D hole chain under illumination (a) by a HW, (b) by the fundamental mode of the chain, and (c) by a TM-polarized plane wave. This defines six scattering coefficients, ρ, τ, α, β ( k x ) , r, and t ( k x ) , with k x being the x-component of the wave vector of the incident or scattered plane waves. (d) Modal scattering coefficients used in the classical Fabry–Perot equation of the EOT. The arrows on the surface in the chain and in free space denote HWs, fundamental chain modes, and plane waves, respectively. The arrows denoting incident and scattered waves are in red and in green, respectively.

Fig. 2
Fig. 2

Coupled-wave coefficients of a nonperiodic array of 1D hole chains illuminated by a TM-polarized plane wave at oblique incidence. The notation of arrows follows that in Fig. 1. P n , Q n , and c n denote the coefficients of the right-going HW, the left-going HW, and the down-going fundamental chain mode that originate from the n th chain at x = x n ( n = 1 , 2 , , N ) .

Fig. 3
Fig. 3

Comparison between the RCWA data and the model predictions for different incident angles θ and for the near-infrared band. All the data are obtained for a gold membrane in air perforated by a periodic 2D array of square holes; the period is a = 0.94 μ m , the hole side length is D = 0.266 μ m , and the membrane thickness is d = 0.2 μ m . (a) Zeroth-order transmittance obtained with the RCWA (left), the SPP model (middle), and the HW model (right). The dotted-white lines represent the air light lines, at which a diffraction order propagates parallel to the metal surface. (b),(c) Zeroth-order transmittance and reflectance spectra for two incident angles θ = 0 ° (red) and 5° (green), obtained with the RCWA (dotted), the SPP model (dashed), and the HW model (solid). The inset in (b) shows the transmittance in a logarithmic scale and evidences the existence of a deep transmission minimum.

Fig. 4
Fig. 4

Comparison between the fully vectorial data and the model predictions for various wavelength ranges. All the data are obtained for a gold membrane in air perforated by a periodic 2D array of square holes; the hole side length is D / a = 0.28 , and the membrane thickness is d / a = 0.21 , with a being the grating period. (a),(b) Zeroth-order transmittance and reflectance spectra under normal incidence, which are obtained with the RCWA (dotted), the SPP model (dashed), and the HW model (solid) and are shown in the visible (red, a = 0.68 μ m ), the near-infrared (green, a = 0.94 μ m ), and the thermal-infrared (blue, a = 2.92 μ m ) bands. (c),(d) Perfect conductor results under normal and oblique incidence ( θ = 5 ° ) , which show the zeroth-order transmittance T (red) and reflectance R (green) spectra for very low frequencies. The fully vectorial data are shown with dotted curves and the HW-model predictions with dot marks. The inset in (c) shows T in a logarithmic scale.

Fig. 5
Fig. 5

Phase-matching condition under normal incidence. (a) The phase (upper) and the modulus (lower) of τ + ρ (dotted-red lines), 1 / Σ H SP + 1 = exp ( i k SP a ) (dashed-green lines), and 1 / Σ H HW + 1 (solid-blue lines). (b) Transmittance | t A | 2 obtained with the RCWA (dotted-red lines), the pure SPP model (dashed-green lines), and the HW model (solid-blue lines). The inset shows | t A | 2 in a logarithmic scale. The phase-matching condition, which corresponds to the two intersections in (a) for predicting the peak wavelength of | t A | 2 , is labeled by the left and right dashed-dotted vertical lines for the SPP model and for the HW model, respectively.

Equations (26)

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H HW + ( x , 0 ) = H SP + ( x , 0 ) + H CW + ( x , 0 ) = exp ( i k SP x ) + ( 2 π k SP 2 k 0 2 ε d ε m ε m ε d ) 1 ( I m + I d ) ,
I m = exp ( i π / 4 ) ε d ε m ε d 0 + exp ( i k 0 x ε m + i t ) t [ 1 ( ε m + i t ) k 0 2 / k SP 2 ] ε m + i t d t ,
t F ( k x ) = t A 2 ( k x ) exp ( i k 0 n d ) 1 r A 2 ( k x ) exp ( i 2 k 0 n d ) ,
H y ( x , z ) = n = 1 N [ P n H HW + ( x x n , z ) + Q n H HW ( x x n , z ) ] ,
P n = W n β ( k x ) + ( τ 1 ) m = 0 n 1 P m H HW + ( x n x m , 0 ) + ρ m = n + 1 N + 1 Q m H HW ( x n x m , 0 ) ,
Q n = W n β ( k x ) + ρ m = 0 n 1 P m H HW + ( x n x m , 0 ) + ( τ 1 ) m = n + 1 N + 1 Q m H HW ( x n x m , 0 ) ,
c n = W n t ( k x ) + α m = 0 n 1 P m H HW + ( x n x m ) + α m = n + 1 N + 1 Q m H HW ( x n x m ) ,
P n = w P n 1 ,     Q n = w Q n 1 ,     c n = w c n 1 .
P 0 = β ( k x ) + ( τ 1 ) Σ H HW + P 0 + ρ Σ H HW Q 0 ,
Q 0 = β ( k x ) + ρ Σ H HW + P 0 + ( τ 1 ) Σ H HW Q 0 ,
c 0 = t ( k x ) + α Σ H HW + P 0 + α Σ H HW Q 0 ,
H Σ , CW ± ( 2 π k SP 2 k 0 2 ε d ε m ε m ε d ) 1 exp ( i π / 4 ) ε m ε d ε m 0 + t / { exp [ i a ( ± k x k 0 ε d + i t ) ] 1 } [ 1 ( ε d + i t ) k 0 2 / k SP 2 ] ε d + i t d t ,
P 0 = β ( k x ) [ ( 1 / Σ H HW + 1 ) τ ] / Σ H HW + + β ( k x ) ρ / Σ H HW + [ ( 1 / Σ H HW + + 1 ) τ ] [ ( 1 / Σ H HW + 1 ) τ ] ρ 2 ,
Q 0 = β ( k x ) [ ( 1 / Σ H HW + + 1 ) τ ] / Σ H HW + β ( k x ) ρ / Σ H HW [ ( 1 / Σ H HW + + 1 ) τ ] [ ( 1 / Σ H HW + 1 ) τ ] ρ 2 .
t A ( k x ) = t ( k x ) + α β ( k x ) [ ( 1 / Σ H HW + 1 ) ( τ ρ ) ] + β ( k x ) [ ( 1 / Σ H HW + + 1 ) ( τ ρ ) ] [ ( 1 / Σ H HW + + 1 ) τ ] [ ( 1 / Σ H HW + 1 ) τ ] ρ 2 .
r A ( k x ) = r + α 2 [ ( 1 / Σ H HW + 1 ) ( τ ρ ) ] + [ ( 1 / Σ H HW + + 1 ) ( τ ρ ) ] [ ( 1 / Σ H HW + + 1 ) τ ] [ ( 1 / Σ H HW + 1 ) τ ] ρ 2 .
t A ( k x = 0 ) = t + 2 α β ( 1 / Σ H HW + 1 ) ( ρ + τ ) ,
r A ( k x = 0 ) = r + 2 α 2 ( 1 / Σ H HW + 1 ) ( ρ + τ ) ,
ρ = ε m 1 / 2 ρ ( PC ) ,     τ 1 = ε m 1 / 2 ( τ ( PC ) 1 ) ,     α = ε m 1 / 4 α ( PC ) ,     β ( k x ) = ε m 1 / 4 β ( PC ) ( k x ) ,
r = r ( PC ) ,     t ( k x ) = t ( PC ) ( k x ) .
Σ H CW ± = ε m 1 / 2 Σ H CW ± ( PC ) ,     Σ H SP ± = Σ H SP ± ( PC ) .
t A ( k x ) = t ( PC ) ( k x ) + α ( PC ) β ( PC ) ( k x ) [ ( 1 / Σ H CW ( PC ) + 1 ) ( τ ( PC ) ρ ( PC ) ) ] + β ( PC ) ( k x ) [ ( 1 / Σ H CW + ( PC ) + 1 ) ( τ ( PC ) ρ ( PC ) ) ] [ ( 1 / Σ H CW + ( PC ) + 1 ) τ ( PC ) ] [ ( 1 / Σ H CW ( PC ) + 1 ) τ ( PC ) ] ( ρ ( PC ) ) 2 ,
r A ( k x ) = r ( PC ) + ( α ( PC ) ) 2 [ ( 1 / Σ H CW ( PC ) + 1 ) ( τ ( PC ) ρ ( PC ) ) ] + [ ( 1 / Σ H CW + ( PC ) + 1 ) ( τ ( PC ) ρ ( PC ) ) ] [ ( 1 / Σ H CW + ( PC ) + 1 ) τ ( PC ) ] [ ( 1 / Σ H CW ( PC ) + 1 ) τ ( PC ) ] ( ρ ( PC ) ) 2 .
t A ( k x = 0 ) = t ( PC ) + 2 α ( PC ) β ( PC ) ( 1 / Σ H CW ( PC ) + 1 ) ( ρ ( PC ) + τ ( PC ) ) ,
r A ( k x = 0 ) = r ( PC ) + 2 ( α ( PC ) ) 2 ( 1 / Σ H CW ( PC ) + 1 ) ( ρ ( PC ) + τ ( PC ) ) ,
arg ( 1 / Σ H HW + 1 ) = arg ( τ + ρ ) arg ( τ ) mod   2 π .

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