Abstract

Two superimposed gratings act to couple light to surface plasmon modes at a metal-air interface. A surface plasmon standing wave is created by generating two counter propagating plasmon waves. The wavelength and angle of incidence of the light that generates the surface plasmon standing wave can be selected by selecting the grating spacing of the couplers. The standing wave can then be out-coupled via the same gratings. In addition to affecting the transmission and reflection signals of the film the structure also enhances the light coupled into the forward- and the back-scattered direction.

© 2006 Optical Society of America

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  1. J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
    [Crossref]
  2. C. Lenaerts, F. Michel, B. Tilkens, Y. Lion, and Y. Renotte, “High transmission efficiency for surface plasmon resonance by use of a dielectric grating,” Appl. Opt. 44, 6017–6022 (2005).
    [Crossref] [PubMed]
  3. T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
    [Crossref]
  4. Y. Wang, “Wavelength selection with coupled surface plasmon waves,” Appl. Phys. Lett. 82, 4385–4387 (2003).
    [Crossref]
  5. W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [Crossref] [PubMed]
  6. P. Sheng, R.S. Stepleman, and P.N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
    [Crossref]
  7. C. Duschl and W. Knoll, “Structural characterization of Langmuir-Blodgett multilayer assemblies by surface plasmon polariton field-enhanced Raman spectroscopy,” J. Chem. Phys. 88, 4062–4069 (1988).
    [Crossref]
  8. H. Knobloch and W. Knoll, “Raman-imaging and spectroscopy with surface plasmon light,” J. Chem. Phys. 94, 835–841 (1991).
    [Crossref]
  9. A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
    [Crossref]
  10. 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]
  11. W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
    [Crossref]
  12. L. Lévesque and P. Rochon, “Surface plasmon photonic bandgap in azopolymer gratings sputtered with gold,” J. Opt. Soc. Am. A 22, 2564–2568 (2005).
    [Crossref]
  13. S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
    [Crossref]
  14. W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (2000).
    [Crossref]
  15. Anne Sentenac and Anne-Laure Fehrembach, “Angular Tolerant resonant grating filters under oblique incidence”, J.O.S.A. A22, 475–480 (2005)
  16. F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
    [Crossref]
  17. A. Natansohn and P. Rochon, “Photoinduced motion in Azo-Containing Polymers” Chem. Rev. 102, 4139–4175 (2002).
    [Crossref] [PubMed]
  18. W. Knoll, “Interfaces and thin films as seen by bound electromagnetic waves,” Annu. Rev. Phys. Chem. 49, 569–638 (1998).
    [Crossref]
  19. P. Uznanski and J. Pecherz, “Surface plasmon resonance of Azobenzene-incorporated polyelectrolyte thin films as an H+ indicator,” J. Appl. Poly. Sci. 86, 1459–1464 (2002).
    [Crossref]

2005 (3)

2004 (1)

T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

2003 (3)

Y. Wang, “Wavelength selection with coupled surface plasmon waves,” Appl. Phys. Lett. 82, 4385–4387 (2003).
[Crossref]

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

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

2002 (2)

P. Uznanski and J. Pecherz, “Surface plasmon resonance of Azobenzene-incorporated polyelectrolyte thin films as an H+ indicator,” J. Appl. Poly. Sci. 86, 1459–1464 (2002).
[Crossref]

A. Natansohn and P. Rochon, “Photoinduced motion in Azo-Containing Polymers” Chem. Rev. 102, 4139–4175 (2002).
[Crossref] [PubMed]

2000 (1)

W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (2000).
[Crossref]

1999 (1)

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

1998 (2)

W. Knoll, “Interfaces and thin films as seen by bound electromagnetic waves,” Annu. Rev. Phys. Chem. 49, 569–638 (1998).
[Crossref]

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 (2)

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
[Crossref]

1994 (1)

A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[Crossref]

1991 (1)

H. Knobloch and W. Knoll, “Raman-imaging and spectroscopy with surface plasmon light,” J. Chem. Phys. 94, 835–841 (1991).
[Crossref]

1988 (1)

C. Duschl and W. Knoll, “Structural characterization of Langmuir-Blodgett multilayer assemblies by surface plasmon polariton field-enhanced Raman spectroscopy,” J. Chem. Phys. 88, 4062–4069 (1988).
[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, 2907–2916 (1982).
[Crossref]

Barnes, W.L.

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

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
[Crossref]

Bozhevolnyi, S.

T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

Cambril, E.

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

Dereux, A.

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

Duschl, C.

C. Duschl and W. Knoll, “Structural characterization of Langmuir-Blodgett multilayer assemblies by surface plasmon polariton field-enhanced Raman spectroscopy,” J. Chem. Phys. 88, 4062–4069 (1988).
[Crossref]

Ebbesen, T.W.

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

Fehrembach, Anne-Laure

Anne Sentenac and Anne-Laure Fehrembach, “Angular Tolerant resonant grating filters under oblique incidence”, J.O.S.A. A22, 475–480 (2005)

Fernandez, U.

A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[Crossref]

Ghaemi, H.F.

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

Giovannini, H.

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

Kim, P-S.

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

Kitson, S.C.

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
[Crossref]

Knobloch, H.

H. Knobloch and W. Knoll, “Raman-imaging and spectroscopy with surface plasmon light,” J. Chem. Phys. 94, 835–841 (1991).
[Crossref]

Knoll, W.

W. Knoll, “Interfaces and thin films as seen by bound electromagnetic waves,” Annu. Rev. Phys. Chem. 49, 569–638 (1998).
[Crossref]

A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[Crossref]

H. Knobloch and W. Knoll, “Raman-imaging and spectroscopy with surface plasmon light,” J. Chem. Phys. 94, 835–841 (1991).
[Crossref]

C. Duschl and W. Knoll, “Structural characterization of Langmuir-Blodgett multilayer assemblies by surface plasmon polariton field-enhanced Raman spectroscopy,” J. Chem. Phys. 88, 4062–4069 (1988).
[Crossref]

Lee, G.

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

Lemarchand, F.

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

Lenaerts, C.

Leosson, K.

T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

Lévesque, L.

Lezec, H.J.

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]

Lion, Y.

Michel, F.

Natansohn, A.

A. Natansohn and P. Rochon, “Photoinduced motion in Azo-Containing Polymers” Chem. Rev. 102, 4139–4175 (2002).
[Crossref] [PubMed]

Nemetz, A.

A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[Crossref]

Nikolajsen, T.

T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

Oh, C-H

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

Pecherz, J.

P. Uznanski and J. Pecherz, “Surface plasmon resonance of Azobenzene-incorporated polyelectrolyte thin films as an H+ indicator,” J. Appl. Poly. Sci. 86, 1459–1464 (2002).
[Crossref]

Priest, T.W.

W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (2000).
[Crossref]

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

Renotte, Y.

Rochon, P.

Sambles, J.R.

S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
[Crossref]

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

Sambles, R.J.

W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (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, 2907–2916 (1982).
[Crossref]

Sentenac, A.

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

Sentenac, Anne

Anne Sentenac and Anne-Laure Fehrembach, “Angular Tolerant resonant grating filters under oblique incidence”, J.O.S.A. A22, 475–480 (2005)

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, 2907–2916 (1982).
[Crossref]

Song, S. Ho

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

Stepleman, R.S.

P. Sheng, R.S. Stepleman, and P.N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

Tan, W.C.

W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (2000).
[Crossref]

Thio, T.

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

Tilkens, B.

Uznanski, P.

P. Uznanski and J. Pecherz, “Surface plasmon resonance of Azobenzene-incorporated polyelectrolyte thin films as an H+ indicator,” J. Appl. Poly. Sci. 86, 1459–1464 (2002).
[Crossref]

Wang, Y.

Y. Wang, “Wavelength selection with coupled surface plasmon waves,” Appl. Phys. Lett. 82, 4385–4387 (2003).
[Crossref]

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]

Yoon, J.

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

Annu. Rev. Phys. Chem. (1)

W. Knoll, “Interfaces and thin films as seen by bound electromagnetic waves,” Annu. Rev. Phys. Chem. 49, 569–638 (1998).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

T. Nikolajsen, K. Leosson, and S. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

Y. Wang, “Wavelength selection with coupled surface plasmon waves,” Appl. Phys. Lett. 82, 4385–4387 (2003).
[Crossref]

Chem. Rev. (1)

A. Natansohn and P. Rochon, “Photoinduced motion in Azo-Containing Polymers” Chem. Rev. 102, 4139–4175 (2002).
[Crossref] [PubMed]

J. Appl. Phys. (2)

A. Nemetz, U. Fernandez, and W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[Crossref]

J. Yoon, G. Lee, S. Ho Song, C-H Oh, and P-S. Kim, “Surface plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[Crossref]

J. Appl. Poly. Sci. (1)

P. Uznanski and J. Pecherz, “Surface plasmon resonance of Azobenzene-incorporated polyelectrolyte thin films as an H+ indicator,” J. Appl. Poly. Sci. 86, 1459–1464 (2002).
[Crossref]

J. Chem. Phys. (2)

C. Duschl and W. Knoll, “Structural characterization of Langmuir-Blodgett multilayer assemblies by surface plasmon polariton field-enhanced Raman spectroscopy,” J. Chem. Phys. 88, 4062–4069 (1988).
[Crossref]

H. Knobloch and W. Knoll, “Raman-imaging and spectroscopy with surface plasmon light,” J. Chem. Phys. 94, 835–841 (1991).
[Crossref]

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

F. Lemarchand, A. Sentenac, E. Cambril, and H. Giovannini, “Study of the resonant behavior of waveguide gratings: increasing the angular tolerance ofguided-mode filters,” J. Opt. A:Pure Appl. Opt. 1, 545–551 (1999).
[Crossref]

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

J.O.S.A. (1)

Anne Sentenac and Anne-Laure Fehrembach, “Angular Tolerant resonant grating filters under oblique incidence”, J.O.S.A. A22, 475–480 (2005)

Nature (2)

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]

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

Phys. Rev. B (3)

P. Sheng, R.S. Stepleman, and P.N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

W.L. Barnes, T.W. Priest, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

W.C. Tan, T.W. Priest, and R.J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11,134–11138 (2000).
[Crossref]

Phys. Rev. Lett. (1)

S.C. Kitson, W.L. Barnes, and J.R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 13, 2670–2673 (1996).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1.

Generation of a surface plasma standing wave by counter propagating waves produced by two coupling gratings.

Fig. 2.
Fig. 2.

An incident beam generates an SP standing wave that re-emits in four directions.

Fig. 3.
Fig. 3.

Doubly corrugated surface spacings.

Fig. 4.
Fig. 4.

Reflection spectrum at an angle of incidence of 2°.

Fig. 5.
Fig. 5.

Reflectivity curves as a function of wavelength from 550 nm to 1000 nm for various incident angles. Counter propagating plasmons are generated at 8.5° and for λ~ 633 nm

Fig. 6.
Fig. 6.

Backscattered signal as a function of the incident at 633 nm.

Fig. 7.
Fig. 7.

p-polarized light at 633 nm scattered in the forward direction.

Equations (4)

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

h ( x ) = h 0 + Δ h cos ( K 1 + K 2 2 x ) cos ( K 1 K 2 2 x )
k 1 x + = ω 1 c sin ( θ ) + K 1 k 1 x = ω 2 c sin ( θ ) K 1 k 2 x + = ω 3 c sin ( θ ) + K 2 k 2 x = ω 4 c sin ( θ ) K 2
k sp = ω c ε gold ε air ε gold + ε air
k 0 sin ( θ out ) = k 0 sin ( θ i ) + ( n i + n o ) K 1 + ( m i + m o ) K 2

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