Abstract

The mechanism of resonant perfect optical absorbers is quantitatively revealed by the coupled mode method for the air/grating/dielectric film/air four region system. The sufficient and necessary conditions of the perfect optical absorption are derived from the interface scattering coefficients analyses. The coupling of the Fabry-Perot modes in the grating slits and non-zero order quasi waveguide modes in the dielectric film play a key role for the perfect optical absorption when the light is incident from the grating side. The analytical sufficient and necessary conditions of the perfect optical absorption provide an efficient tool towards geometry design for the perfect optical absorption at the specific wavelengths. The advantages of a widely tunable perfect optical absorption wavelength, a high Q factor and the confined energy loss on metal surfaces make the air/grating/film/air structures promising for applications in sensing, modulation and detection.

© 2016 Optical Society of America

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References

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  1. D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17(2), 196–200 (1976).
    [Crossref]
  2. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976).
    [Crossref]
  3. Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
    [Crossref]
  4. A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
    [Crossref] [PubMed]
  5. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
    [Crossref] [PubMed]
  6. Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
    [Crossref] [PubMed]
  7. Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
    [Crossref]
  8. P. Mandal, “Plasmonic perfect absorber for refractive index sensing and SERS,” Plasmonics 11(1), 223–229 (2016).
    [Crossref]
  9. W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
    [Crossref]
  10. F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
    [Crossref]
  11. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
    [Crossref]
  12. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
    [Crossref]
  13. H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
    [Crossref]
  14. R. L. Chern, Y. T. Chen, and H. Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010).
    [Crossref] [PubMed]
  15. S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
    [Crossref] [PubMed]
  16. J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
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    [Crossref]
  19. J. T. Shen and P. M. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B 70(3), 035101 (2004).
    [Crossref]
  20. F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
    [Crossref]
  21. A. Rahman, K. Vasilev, and P. Majewski, “Analytical solution of the fundamental waveguide mode of one-dimensional transmission grating for TM polarization,” J. Opt. Soc. Am. B 28(12), 2919–2924 (2011).
    [Crossref]
  22. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
    [Crossref] [PubMed]
  23. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
    [Crossref] [PubMed]
  24. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
    [Crossref] [PubMed]
  25. J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
    [Crossref]
  26. I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
    [Crossref]

2016 (1)

P. Mandal, “Plasmonic perfect absorber for refractive index sensing and SERS,” Plasmonics 11(1), 223–229 (2016).
[Crossref]

2015 (1)

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

2014 (4)

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
[Crossref]

2012 (1)

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

2011 (2)

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

A. Rahman, K. Vasilev, and P. Majewski, “Analytical solution of the fundamental waveguide mode of one-dimensional transmission grating for TM polarization,” J. Opt. Soc. Am. B 28(12), 2919–2924 (2011).
[Crossref]

2010 (4)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

R. L. Chern, Y. T. Chen, and H. Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010).
[Crossref] [PubMed]

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

2009 (1)

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

2007 (1)

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

2004 (1)

J. T. Shen and P. M. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B 70(3), 035101 (2004).
[Crossref]

2003 (2)

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

2002 (2)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[Crossref]

2000 (1)

W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
[Crossref]

1999 (1)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[Crossref]

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]

1976 (2)

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17(2), 196–200 (1976).
[Crossref]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976).
[Crossref]

Bardou, N.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Bezuglyi, E. V.

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

Capasso, F.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Chen, S. J.

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

Chen, Y.

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Chen, Y. T.

Chern, R. L.

Collin, S.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Cui, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Degiron, A.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Ding, F.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Dregely, D.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

Ebbesen, T. W.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Gao, H.

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

García de Abajo, F. J.

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[Crossref]

García-Vidal, F. J.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

Giessen, H.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Gu, C.

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

Haïdar, R.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Hao, H. Y.

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

He, S.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

He, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Hentschel, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Hong, M.

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Hutley, M. C.

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976).
[Crossref]

Jin, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Jung, M. J.

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

Kats, A. V.

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

Kats, M. A.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Kong, J.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Kuipers, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

Levchenko, A.

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

Lezec, H. J.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Li, H.

J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
[Crossref]

Li, X.

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Lin, H. Y.

Lin, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Liu, N.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Liu, W.

J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
[Crossref]

Loncar, M.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Luo, X.

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Magnusson, R.

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

Mai, P.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

Maier, S. A.

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Majewski, P.

Mandal, P.

P. Mandal, “Plasmonic perfect absorber for refractive index sensing and SERS,” Plasmonics 11(1), 223–229 (2016).
[Crossref]

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Martín-Moreno, L.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

Maystre, D.

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976).
[Crossref]

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17(2), 196–200 (1976).
[Crossref]

Mesch, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Nie, J.

J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
[Crossref]

Nikitin, A. Y.

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

Pelouard, J. L.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Pendry, J. B.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[Crossref]

Petit, R.

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17(2), 196–200 (1976).
[Crossref]

Platzman, P. M.

J. T. Shen and P. M. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B 70(3), 035101 (2004).
[Crossref]

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[Crossref]

Preist, T. W.

W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
[Crossref]

Rahman, A.

Rommeluère, S.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Sambles, J. R.

W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
[Crossref]

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]

Shankar, R.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Shen, J. T.

J. T. Shen and P. M. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B 70(3), 035101 (2004).
[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]

Song, S. H.

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

Song, Y.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Spevak, I. S.

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[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(6), 2907–2916 (1982).
[Crossref]

Tan, W. C.

W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
[Crossref]

Taubert, R.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

Tittl, A.

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

Vasilev, K.

Vincent, G.

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Weiss, T.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Yang, L.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Yao, Y.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Ye, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Yoon, J. W.

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

Zheng, Z. Y.

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

Zhong, S.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Appl. Phys. B (1)

H. Gao, C. Gu, Z. Y. Zheng, S. J. Chen, and H. Y. Hao, “Nearly perfect absorption in a single-layer metallic grating with rectangular grooves on its front surface,” Appl. Phys. B 117(3), 875–883 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

J. W. Yoon, M. J. Jung, S. H. Song, and R. Magnusson, “Analytic theory of the resonance properties of metallic nanoslit arrays,” IEEE J. Quantum Electron. 48(7), 852–861 (2012).
[Crossref]

IEEE Photonics J. (1)

J. Nie, H. Li, and W. Liu, “Perfect anomalous absorption of TM polarized light in metallic grating situated in asymmetric surroundings,” IEEE Photonics J. 6(6), 4802008 (2014).
[Crossref]

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

Laser Photonics Rev. (1)

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Nano Lett. (3)

A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11(10), 4366–4369 (2011).
[Crossref] [PubMed]

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[Crossref] [PubMed]

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref] [PubMed]

Opt. Commun. (2)

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17(2), 196–200 (1976).
[Crossref]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976).
[Crossref]

Opt. Express (1)

Photonics Res. (1)

Y. Chen, X. Li, X. Luo, S. A. Maier, and M. Hong, “Tunable near-infrared plasmonic perfect absorber based on phase-change materials,” Photonics Res. 3(3), 54–57 (2015).
[Crossref]

Phys. Rev. B (5)

W. C. Tan, J. R. Sambles, and T. W. Preist, “Double-period zero-order metal gratings as effective selective absorbers,” Phys. Rev. B 61(19), 13177–13182 (2000).
[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]

J. T. Shen and P. M. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B 70(3), 035101 (2004).
[Crossref]

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[Crossref]

I. S. Spevak, A. Y. Nikitin, E. V. Bezuglyi, A. Levchenko, and A. V. Kats, “Resonantly suppressed transmission and anomalously enhanced light absorption in periodically modulated ultrathin metal films,” Phys. Rev. B 79(16), 161406 (2009).
[Crossref]

Phys. Rev. Lett. (4)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[Crossref]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003).
[Crossref] [PubMed]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901 (2003).
[Crossref] [PubMed]

S. Collin, G. Vincent, R. Haïdar, N. Bardou, S. Rommeluère, and J. L. Pelouard, “Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane,” Phys. Rev. Lett. 104(2), 027401 (2010).
[Crossref] [PubMed]

Plasmonics (1)

P. Mandal, “Plasmonic perfect absorber for refractive index sensing and SERS,” Plasmonics 11(1), 223–229 (2016).
[Crossref]

Rev. Mod. Phys. (2)

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[Crossref] [PubMed]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 2005).

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Figures (9)

Fig. 1
Fig. 1 Schematic for the air/grating/film/air four region system. The yellow part and white part of layer 2 represents the metal part and the air in slit of the grating respectively. Layer 3 is the dielectric film. Layer 1 and layer 4 are both air. Some interface reflection coefficients and the refractive index for each layer are shown. The p-polarized light is incident from layer 1.
Fig. 2
Fig. 2 The phase dispersion of reflection coefficients (a) ρ 21 , (b) ρ 24 and (c) coefficient u 2 in the grating slits normalized by 2π ) with D = 10 µm, s = 1 µm, Tg =16 µm, Td = 12.5µm.
Fig. 3
Fig. 3 (a) Absorption dispersion calculated by FDTD and (b) modes analysis by the coupled mode method for air/grating/film/air system with D = 10 µm, s = 1 µm, Tg = 16 µm, Td = 12.5 µm under p-polarized incidence. The yellow circles in (a) indicate the POA with abs>95%. In (b), the black lines mark phase( ρ 21 ρ 24 u 2 )=2nπ , the magenta lines are input surface (air/grating) lattice modes, the cyan lines are output surface (grating/film) lattice modes, and the quasi waveguide modes in the dielectric film are plotted in green(−1), blue( + 1) and red lines(−2) by the equation phase( ρ 34,p w p 2 )=2nπ .
Fig. 4
Fig. 4 (a) Absorption/transmittance/reflectance spectra calculated by FDTD (dots) and by the coupled mode method (solid line). (b) The phase spectra of ρ 21 ρ 24 u 2 , ρ 24 , and ρ 24,p w p 2 (p =± 1) calculated by the coupled mode method with D = 10 µm, s = 1 µm, Tg = 16 µm, Td = 12.5µm under p-polarized normal incidence.
Fig. 5
Fig. 5 Geometry and wavelength scan for the POA with abs>99% under p-polarized normal incidence for the air/grating/film/air four region system by FDTD. The fixed parameters are D = 10 µm, s = 1 µm and the POA wavelength range from 10µm to 14µm with different grating thickness and film thickness.
Fig. 6
Fig. 6 (a) Absorption dispersion by FDTD, (b) modes analysis and (c) the schematic of the four region system with the incident light from the film side.
Fig. 7
Fig. 7 (a) Instantaneous electric field distribution and (b) ohmic loss distribution for an air/grating/film/air system with D = 10 µm, s = 1 µm, Tg = 1 µm, and Td = 0.5 µm under normal (θ=0) incident wavelength 10.156 µm. Black lines mark the outlines of the structure.
Fig. 8
Fig. 8 Instantaneous electric field distribution and ohmic loss distribution for (a) wavelength 10.47 µm and (b) wavelength 13.796 µm (POA) under normal (θ=0) incidence. The structure is an air/grating/film/air system with D = 10 µm, s = 1 µm, Tg = 16 µm, and Td = 12.5 µm. Black lines mark the outlines of the structures.
Fig. 9
Fig. 9 Absorption dependence on geometry parameters for the four region system under p-polarized normal incident wavelength 9 µm. The basic geometry parameters are D = 8.85 µm, s = 0.761 µm, Tg = 1 µm and Td = 0.5 µm.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

ρ 14,p = ρ 12,p + τ 12 ρ 24 τ 21,p u 2 1 ρ 21 ρ 24 u 2 .
τ 14,p = τ 12 τ 24,p u 1 ρ 21 ρ 24 u 2 .
phase( ϕ 1 1 ϕ 1 +1 )+2Re( n eff ) k 0 T g =2nπ.
2arctan( ε 1 ε 3 (λ/D) 2 ε 3 (λ/D) 2 ε 1 )+2 k 0 T d ε 3 (λ/D) 2 +π=2nπ.
ϕ= p n eff f g p 2 ( Z s +1/ Y 1,p ) .
ρ 34,p = Y 4,p Y 3,p Y 4,P + Y 3,p .
ϕ 3 = p n eff f g p 2 (1+ ρ 34,p w p 2 ) ( Z s +1/ Y 3,p )[1+ ρ 34,p w p 2 ( Z s 1/ Y 3,p ) / ( Z s +1/ Y 3,p ) .
τ 12 = 2 g 0 (1+ Y 1,0 Z s )( ϕ 1 +1) .
ρ 12,p = δ p,0 1 Y 1,p Z s 1+ Y 1,p Z s n eff f g p Y 1,p 1+ Y 1,p Z s τ 12 .
ρ 21 = ϕ 1 1 ϕ 1 +1 .
τ 21,p = 2 n eff f g p ( Z s +1/ Y 1,p )( ϕ 1 +1) .
ρ 24 = ϕ 3 1 ϕ 3 +1 .
τ 24,p = 2 n eff f g p ( Z s +1/ Y 3,p )( ϕ 3 +1) (1+ ρ 34,p ) w p [1+ ρ 34,p w p 2 ( Z s 1/ Y 3,p ) / ( Z s +1/ Y 3,p )] .
ρ 14,p = ρ 12,p + τ 12 ρ 24 τ 21,p u 2 1 ρ 21 ρ 24 u 2 .
τ 14,p = τ 12 τ 24,p u 1 ρ 21 ρ 24 u 2 .
R P = | ρ 14,p | 2 Re( Y 1,0 Y 1,p ).
T P = | τ 14,p | 2 Re( Y 1,0 Y 4,p ).

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