Abstract

Wave propagation and surface plasmon resonance are examined in four-layer optical systems in slab geometry for an OLED (organic light-emitting diode) with an embedded thin metal film. For this purpose, both leaky and bound modes are examined in all ranges of the propagation constant, which determines how surface and volume waves are allowed. Intensive parametric studies are performed on the thicknesses of the two embedded layers, along with the cathode condition and the metal’s material dispersion. As a way of interpreting the results, the direction of the depthwise wave propagation is examined in connection with possible excitations arising from light sources within the organic electroluminescence layer. Consequently, several new features are observed on the multiple-wave branches, including exchange of the phase speeds and depthwise standing waves for dissipationless systems. By the insertion of a thin metal film, the light extraction is found to be enhanced through leaky waves from the source layer out toward the viewer’s side.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. A. Neyts, "Simulation of light emission from thin-film microcavities," J. Opt. Soc. Am. A 15, 962-971 (1998).
    [CrossRef]
  2. J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
    [CrossRef]
  3. B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
    [CrossRef]
  4. Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
    [CrossRef]
  5. W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
    [CrossRef]
  6. A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
    [CrossRef]
  7. H. Chen, J. Lee, C. Shiau, C. Yang, and Y. Kiang, "Electromagnetic modeling of organic light-emitting devices," J. Lightwave Technol. 24, 2450-2457 (2006).
    [CrossRef]
  8. M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
    [CrossRef]
  9. B. J. Chen, X. W. Sun, and S. C. Tan, "Transparent organic light-emitting devices with LiF/Mg:Ag cathode," Opt. Express 13, 937 (2005).
    [CrossRef] [PubMed]
  10. G. Z. Ran, G. L. Ma, Y. H. Xu, L. Dai, and G. G. Qin, "Light extraction efficiency of a top-emission organic light-emitting diode with an Yb/Au double-layer cathode and an opaque Si anode," Appl. Opt. 45, 5871-5876 (2006).
    [CrossRef] [PubMed]
  11. P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
    [CrossRef]
  12. S. Wedge and W. L. Barnes, "Surface plasmon-polariton mediated light emission through thin metal films," Opt. Express 12, 3673-3685 (2004).
    [CrossRef] [PubMed]
  13. H. Raether, Surface Plasmon on Smooth and Rough Surfaces and on Gratings, Vol. 3 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).
  14. F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
    [CrossRef]
  15. F. Kliewer and J. R. Fuchs, "Collective electronic motion in a metallic slab," Phys. Rev. 153, 498-512 (1967).
    [CrossRef]
  16. G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton-like waves guided by thin, dissipative metal films," Opt. Lett. 8, 383-385 (1983).
    [CrossRef] [PubMed]
  17. G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
    [CrossRef]
  18. J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  19. F. A. Burton and S. A. Cassidy, "A complete description of the dispersion relation for thin metalfilm plason-polaritons," J. Lightwave Technol. 8, 1843-1849 (1990).
    [CrossRef]
  20. F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
    [CrossRef]
  21. T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).
  22. M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
    [CrossRef]
  23. J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
    [CrossRef]
  24. Y. Li and J. W. Y. Lit, "General formulas for the guiding properties of a multilayer slab waveguide," J. Opt. Soc. Am. A 4, 671-677 (1987).
    [CrossRef]
  25. R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
    [CrossRef]
  26. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structure," Phys. Rev. B 61, 10484-10503 (2000).
    [CrossRef]
  27. I. Breukelaar and P. Berini, "Long-range surface plasmon polariton mode cutoff and radiation in slab waveguide," J. Opt. Soc. Am. A 23, 1971-1977 (2006).
    [CrossRef]
  28. R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
    [CrossRef]
  29. R. E. Smith and S. N. Houde-Walter, "Leaky guiding in nontransparent waveguides," J. Opt. Soc. Am. A 12, 715-724 (1995).
    [CrossRef]
  30. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  31. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
    [CrossRef] [PubMed]
  32. H. I. Lee, "Wave propagation and resonance in four-layer systems for waveguides," in preparation.
  33. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

2006

H. Chen, J. Lee, C. Shiau, C. Yang, and Y. Kiang, "Electromagnetic modeling of organic light-emitting devices," J. Lightwave Technol. 24, 2450-2457 (2006).
[CrossRef]

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

G. Z. Ran, G. L. Ma, Y. H. Xu, L. Dai, and G. G. Qin, "Light extraction efficiency of a top-emission organic light-emitting diode with an Yb/Au double-layer cathode and an opaque Si anode," Appl. Opt. 45, 5871-5876 (2006).
[CrossRef] [PubMed]

I. Breukelaar and P. Berini, "Long-range surface plasmon polariton mode cutoff and radiation in slab waveguide," J. Opt. Soc. Am. A 23, 1971-1977 (2006).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

2005

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).

B. J. Chen, X. W. Sun, and S. C. Tan, "Transparent organic light-emitting devices with LiF/Mg:Ag cathode," Opt. Express 13, 937 (2005).
[CrossRef] [PubMed]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

2004

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
[CrossRef]

S. Wedge and W. L. Barnes, "Surface plasmon-polariton mediated light emission through thin metal films," Opt. Express 12, 3673-3685 (2004).
[CrossRef] [PubMed]

2003

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

2002

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

2001

M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
[CrossRef]

2000

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structure," Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

1998

1997

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

1995

1994

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

1992

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
[CrossRef]

1991

F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
[CrossRef]

1990

F. A. Burton and S. A. Cassidy, "A complete description of the dispersion relation for thin metalfilm plason-polaritons," J. Lightwave Technol. 8, 1843-1849 (1990).
[CrossRef]

1987

1986

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

1984

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

1983

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1967

F. Kliewer and J. R. Fuchs, "Collective electronic motion in a metallic slab," Phys. Rev. 153, 498-512 (1967).
[CrossRef]

Asano, T.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Barnes, W. L.

S. Wedge and W. L. Barnes, "Surface plasmon-polariton mediated light emission through thin metal films," Opt. Express 12, 3673-3685 (2004).
[CrossRef] [PubMed]

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

Beierlein, T.

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Beierlein, T. A.

W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
[CrossRef]

Berini, P.

I. Breukelaar and P. Berini, "Long-range surface plasmon polariton mode cutoff and radiation in slab waveguide," J. Opt. Soc. Am. A 23, 1971-1977 (2006).
[CrossRef]

M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structure," Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

Boardman, A. D.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

Bradberry, G. W.

F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
[CrossRef]

Breukelaar, I.

Brongersma, M. L.

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton-like waves guided by thin, dissipative metal films," Opt. Lett. 8, 383-385 (1983).
[CrossRef] [PubMed]

Burton, F. A.

F. A. Burton and S. A. Cassidy, "A complete description of the dispersion relation for thin metalfilm plason-polaritons," J. Lightwave Technol. 8, 1843-1849 (1990).
[CrossRef]

Cassidy, S. A.

F. A. Burton and S. A. Cassidy, "A complete description of the dispersion relation for thin metalfilm plason-polaritons," J. Lightwave Technol. 8, 1843-1849 (1990).
[CrossRef]

Charbonneau, R.

M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
[CrossRef]

Chen, B. J.

Chen, H.

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Chutinan, A.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Dai, L.

Do, Y. R.

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Forbes, G. W.

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
[CrossRef]

Friend, R. H.

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

Fuchs, J. R.

F. Kliewer and J. R. Fuchs, "Collective electronic motion in a metallic slab," Phys. Rev. 153, 498-512 (1967).
[CrossRef]

Fujita, M.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Fukasawa, Y.

T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).

Greenham, N. C.

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

Greffet, J.-J.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

Hall, D. G.

Ho, P. K. H.

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

Hobson, P. A.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

Houde-Walter, S. N.

R. E. Smith and S. N. Houde-Walter, "Leaky guiding in nontransparent waveguides," J. Opt. Soc. Am. A 12, 715-724 (1995).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
[CrossRef]

Ishihara, K.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Karg, S.

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Karppinen, M.

M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
[CrossRef]

Kiang, Y.

Kim, J.

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

Kim, Y.

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

Kliewer, F.

F. Kliewer and J. R. Fuchs, "Collective electronic motion in a metallic slab," Phys. Rev. 153, 498-512 (1967).
[CrossRef]

Konayashi, T.

T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lee, H. I.

H. I. Lee, "Wave propagation and resonance in four-layer systems for waveguides," in preparation.

Lee, J.

Lee, Y.

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

Li, Y.

Lit, J. W. Y.

Ma, G. L.

Maradudin, A. A.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Nakada, H.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Neyts, K. A.

Noda, S.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Ohata, H.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Pincemin, F.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

Qin, G. G.

Raether, H.

H. Raether, Surface Plasmon on Smooth and Rough Surfaces and on Gratings, Vol. 3 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).

Ran, G. Z.

Riel, H.

W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
[CrossRef]

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Riess, W.

W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
[CrossRef]

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Ruhstaller, B.

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Sage, I.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

Sambles, J. R.

F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
[CrossRef]

Scott, J. C.

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

Selker, M. D.

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

Shen, T. P.

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

Shiau, C.

Shimoji, N.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Shkerdin, G.

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

Smith, R. E.

R. E. Smith and S. N. Houde-Walter, "Leaky guiding in nontransparent waveguides," J. Opt. Soc. Am. A 12, 715-724 (1995).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
[CrossRef]

Song, Y.

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton-like waves guided by thin, dissipative metal films," Opt. Lett. 8, 383-385 (1983).
[CrossRef] [PubMed]

Stiens, J.

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

Sun, X. W.

Takahara, T.

T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Tan, S. C.

Tsuji, T.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Ueno, T.

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Veretennicoff, I.

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

Voronko, A.

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

Vounckx, R.

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

Wallis, R. F.

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

Wasey, J. A. E.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

Wedge, S.

Xu, Y. H.

Yang, C.

Yang, F.

F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
[CrossRef]

Zia, R.

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Mode determination for planar waveguides using the four-sheeted dispersion relation," IEEE J. Quantum Electron. 28, 1520-1526 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, "The role of surface plasmons in organic light-emitting diodes," IEEE J. Sel. Top. Quantum Electron. 8, 378-386 (2002).
[CrossRef]

B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, and W. Riess, "Simulating electronic and optical processes in multilayer in organic light-emitting devices," IEEE J. Sel. Top. Quantum Electron. 9, 723-731 (2003).
[CrossRef]

J. Appl. Phys.

Y. R. Do, Y. Kim, Y. Song, and Y. Lee, "Enhanced light extraction efficiency from organic light-emitting diodes by insertion of a two-dimensional photonic crystal structure," J. Appl. Phys. 96, 7629-7636 (2004).
[CrossRef]

J. Kim, P. K. H. Ho, N. C. Greenham, and R. H. Friend, "Electroluminescence emission pattern of organic light emitting diodes: implications for device efficiency calculations," J. Appl. Phys. 88, 1073-1081 (2000).
[CrossRef]

J. Stiens, R. Vounckx, I. Veretennicoff, A. Voronko, and G. Shkerdin, "Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides," J. Appl. Phys. 81, 1-10 (1997).
[CrossRef]

J. Chem. Phys.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

J. Korean Phys. Soc.

T. Takahara, Y. Fukasawa, and T. Konayashi, "Excitation of two-dimensional optical waves by electric currents in thin metal-gap structures," J. Korean Phys. Soc. 47, S43-S47 (2005).

J. Lightwave Technol.

H. Chen, J. Lee, C. Shiau, C. Yang, and Y. Kiang, "Electromagnetic modeling of organic light-emitting devices," J. Lightwave Technol. 24, 2450-2457 (2006).
[CrossRef]

F. A. Burton and S. A. Cassidy, "A complete description of the dispersion relation for thin metalfilm plason-polaritons," J. Lightwave Technol. 8, 1843-1849 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys., Part 1

M. Fujita, K. Ishihara, T. Ueno, T. Asano, S. Noda, H. Ohata, T. Tsuji, H. Nakada, and N. Shimoji, "Optical and clectrical characteristics of organic light-emitting diodes with two-dimensional photonic crystals in organic/electrode layers," Jpn. J. Appl. Phys., Part 1 44, 3669-3677 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Org. Electron.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, "Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods," Org. Electron. 6, 3-9 (2005).
[CrossRef]

Phys. Rev.

F. Kliewer and J. R. Fuchs, "Collective electronic motion in a metallic slab," Phys. Rev. 153, 498-512 (1967).
[CrossRef]

Phys. Rev. B

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structure," Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, "Scattering of a surface plasmon polariton by a surface defect," Phys. Rev. B 50, 15261-15275 (1994).
[CrossRef]

G. I. Stegeman, A. A. Maradudin, T. P. Shen, and R. F. Wallis, "Refraction of a surface polariton by a semi-infinite film on a metal," Phys. Rev. B 29, 6530-6539 (1984).
[CrossRef]

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, dissipative metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range surface modes supported by thin films," Phys. Rev. B 44, 5855-5872 (1991).
[CrossRef]

Phys. Status Solidi

W. Riess, T. A. Beierlein, and H. Riel, "Optimizing OLED structures for a-Si display applications via combinatorial methods and enhanced outcoupling," Phys. Status Solidi 201, 1360-1371 (2004).
[CrossRef]

Proc. SPIE

M. Karppinen, R. Charbonneau, and P. Berini, "Attenuated total reflection modulator based on surface plasmon excitation," Proc. SPIE 4595, 259-267 (2001).
[CrossRef]

Other

H. I. Lee, "Wave propagation and resonance in four-layer systems for waveguides," in preparation.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

H. Raether, Surface Plasmon on Smooth and Rough Surfaces and on Gratings, Vol. 3 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Schematic diagram for a four-layer system D-ND-D-ND stack. Interface 0–1, for instance, is the boundary surface between media 0 and 1. The thicknesses of the metal and OEL layers are given by h 1 and h 2 , respectively. The waves radiating into medium 0 are of principal interest for bottom-emitting OLEDs.

Fig. 2
Fig. 2

Wave regimes on the plane of the phase constant b r versus a parameter in case with ε ̃ 0 < ε ̃ 2 and ε ̃ 1 r , ε ̃ 3 r < 0 . The two wave-regime boundaries b r , 0 and b r , 2 are drawn along with the three resonance lines b r 01 , , b r 12 , , and b r 23 , . There are only 2D waves in the wave regime 2D0123. In the wave regime 2D013-3D2, there are 2D waves in media 0, 1, and 3, whereas 3D waves are established in medium 2. In the wave regime 2D13-3D02, 2D- and 3D-waves are present in the media pairs 1, 3 and 0, 2, respectively.

Fig. 3
Fig. 3

(a) Phase constants on a linear scale and (b) attenuation constants on a logarithmic scale against the varying h 1 in nanometers for the dissipative system. The data for h 1 < 5 nm are not computed. Branch TM1-B continues both leftward and upward as h 1 decreases, as shown in the inset of (a) on the same coordinate units. The three resonance lines are labeled as the respective imaginary parts b i 01 , , b i 12 , , and b i 23 , in (b) and denoted by differently colored (online) horizontal lines.

Fig. 4
Fig. 4

(a) Phase speeds in medium 0 against the varying h 1 in nanometers for the dissipative system, calculated from the data of Fig. 3. The branch TM2-B increases monotonically with smaller h 1 and thus is not shown for V p 0 > 10 . (b) Phase speeds in the extended range of 70 < h 1 < 160 nm for the two branches TM1-B and TM2-B in the narrow range of 7.0 < V p 0 < 8.4 .

Fig. 5
Fig. 5

Phase angle θ p 0 in medium 0 as measured from the surface normal drawn against varying h 1 in nanometers for a few selected branches of leaky-0 mode. The inset is drawn for branch TM1-L0 on the same coordinate units.

Fig. 6
Fig. 6

Real parts of the depth profiles for a few branches. Each profile is normalized against its own maximum, and medium j is labeled [ j ] just over the horizontal axis. The abscissa indicates the depth coordinate for the two embedded layers, where the left- and right-hand portions are occupied by the metal and OEL layers, respectively. Although the OEL layer is five times as thick as the metal layer for the baseline data, the former is drawn on this graph only twice as thick as the latter for visual clarity.

Fig. 7
Fig. 7

(a) Phase constants and (b) attenuation constants against varying h 2 in nanometers for the dissipative system. The data for h 2 < 10 nm are not computed. Branches TM1-B, TM1-L0, TM2-L0, and TM3-L0 continue both leftward and upward as h 2 decreases, as shown in the inset of (a) on the same coordinate units. The various reference lines in Table 1 are drawn as the horizontal lines in (a). The three resonance lines are indicated by the respective imaginary parts b i 01 , , b i 12 , , and b i 23 , in (b).

Fig. 8
Fig. 8

Portions of the wave branches of leaky-0 mode that satisfy the criterion of the light rays being directed away from the system. The h 1 -variation based on Fig. 3 is provided in (a), while the h 2 -variation based on Fig. 7 is shown in (b). The vertical line in each figure indicates the baseline data, and the horizontal dashed lines are the wave-regime boundary b r , 0 = 1.517 .

Fig. 9
Fig. 9

(a) Phase constants and (b) attenuation constants against varying h 2 in nanometers for the dissipative system. Only branches TM2-B and TM1-L0 are compared with the three cathode conditions. The resonance lines b 23 , at interface 2–3 are denoted by the horizontal lines for the respective cathode conditions.

Fig. 10
Fig. 10

(a) Dimensional propagation constant times the OEL-layer thickness, β h 2 , for the dependence on the gold’s material dispersion as a function of wavelength λ in nanometers. Only the two branches of the bound mode are considered. The real and imaginary parts are shown on the same ordinate unit in (a), where the two wave-regime boundaries are defined by the solid curve β r h 2 = 2 π ( 1.517 ) h 2 λ 1 and the dashed curve β r h 2 = 2 π ( 1.703 ) h 2 λ 1 , respectively. For visual simplicity in (a), some computed points are taken away from the slowly varying branch TM2-B. The real and imaginary parts of the inverse group velocity V g 1 ( λ ) are shown in (b1) and (b2), where ( V g 1 ) r and ( V g 1 ) i on branch TM2–B are exaggerated 50 times.

Fig. 11
Fig. 11

(a) Phase constants and (b) attenuation constants against varying h 2 in nanometers for the dissipationless system. The various reference lines in Table 1 are drawn as the horizontal lines in (a). Propagating waves are established in both the longitudinal and the depthwise directions. The three branches TM1-B, TM1-L0, and TM2-L0 continue both leftwards and upwards with smaller h 2 in (a).

Fig. 12
Fig. 12

(a) Phase constants b r with varying h 2 in nanometers for the dissipationless system. Propagating and standing waves are established in the longitudinal and depthwise directions, respectively. The data for b r > b r , 2 = 1.703 are the same as those shown in Fig. 11a. (b) Thicknesses h 2 in nanometers on multiple branches of higher-order modes plotted against the phase constant b r in the non-2D-wave regime, where the mode number q is written on each curve.

Fig. 13
Fig. 13

(a) Phase constants and (b) attenuation constants against varying h 2 in nanometers for the dissipative three-layer system without the embedded metal film. Branches TM1-B and TE3-L0 are absent. The resonance line b i 23 , is indicated by the horizontal line in (b).

Fig. 14
Fig. 14

Comparison of (a) phase constants and (b) attenuation constants against varying h 2 in nanometers for the dissipative four-layer system (labeled “ 4 _ ”) and for the three-layer system (labeled “ 3 _ ”) without the embedded metal film. Only the three branches TM2-B, TM1-L0, and TE1-L0 are compared.

Tables (2)

Tables Icon

Table 1 Two-Layer Resonance Lines b m n , = [ ε ̃ m ε ̃ n ( ε ̃ m + ε ̃ n ) ] 1 2 for Dissipative (DY) and Dissipationless (DL) Systems with Wave-Regime Boundaries b r , j ε ̃ j r 1 2

Tables Icon

Table 2 Phase Constant b r and Attenuation Constant b i on Multiple Branches for Four- and Three-Layer Systems a

Equations (26)

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

S j ± S j 2 D , S j 2 D S j r 2 D i S j i 2 D ,
S j r 2 D 1 2 b 2 ε ̃ j + d j 0 ,
S j i 2 D 1 2 sign ( e j ) b 2 ε ̃ j d j .
U j { S j ε ̃ j : TM S j : TE .
f 3 ( Z ) D 3 exp [ S 3 ( Z δ 1 δ 2 ) ] ,
f 2 ( Z ) C 2 cosh [ S 2 ( Z δ 1 ) ] + D 2 sinh [ S 2 ( Z δ 1 ) ] ,
f 1 ( Z ) C 1 cosh ( S 1 Z ) + D 1 sinh ( S 1 Z ) ,
f 0 ( Z ) exp ( S 0 Z ) .
H 1 tanh ( S 1 δ 1 ) , H 2 tanh ( S 2 δ 2 ) ,
F = ( U 0 U 2 2 + U 1 2 U 3 ) H 1 H 2 + U 2 ( U 1 2 + U 0 U 3 ) H 1 + U 1 ( U 2 2 + U 0 U 3 ) H 2 + U 1 U 2 ( U 0 + U 3 ) = 0 .
exp [ i ( τ b r X S 0 i Z ) ] exp ( b i X ) exp ( S 0 r Z ) ,
exp [ i ( τ b r X + S 3 i Z ) ] exp ( b i X ) exp ( S 3 r Z ) .
V g ( λ ) V g r i V g i 1 c ω β = ( 1 λ ) ( b λ ) ,
V g 1 ( λ ) ( V g 1 ) r i ( V g 1 ) i c β ω = ( b λ ) ( 1 λ ) .
f 3 3 D ( Z ) D 3 3 D exp [ T 3 2 D ( Z δ 1 δ 2 ) ] ,
f 2 3 D ( Z ) C 2 3 D cos [ T 2 3 D ( Z δ 1 ) ] + D 2 3 D sin [ T 2 3 D ( Z δ 1 ) ] ,
f 1 3 D ( Z ) C 1 3 D cosh ( T 1 2 D Z ) + D 1 3 D sinh ( T 1 2 D Z ) ,
f 0 3 D ( Z ) 2 1 2 cos ( T 0 3 D Z ϕ 0 ) .
F 3 D ( b r ) [ ( V 1 2 D ) 2 V 3 2 D V 0 3 D ( V 2 3 D ) 2 ] H 1 2 D H 2 3 D + V 1 2 D [ V 0 3 D V 3 2 D ( V 2 3 D ) 2 ] H 2 3 D + V 2 3 D [ ( V 1 2 D ) 2 + V 0 3 D V 3 2 D ] H 1 2 D + V 1 2 D V 2 3 D ( V 0 3 D + V 3 2 D ) = 0 .
F 2.5 D ( b r ) [ ( V 1 2 D ) 2 V 3 2 D + V 0 3 D ( V 2 2 D ) 2 ] H 1 2 D H 2 2 D + V 1 2 D [ V 0 3 D V 3 2 D + ( V 2 2 D ) 2 ] H 2 2 D + V 2 2 D [ ( V 1 2 D ) 2 + V 0 3 D V 3 2 D ] H 1 2 D + V 1 2 D V 2 2 D ( V 0 3 D + V 3 2 D ) = 0 .
φ 01 tanh [ 2 π h 1 λ b r 2 ε ̃ 1 r + tanh 1 ( V 0 3 D V 1 2 D ) ] , 2 π h 2 λ ε ̃ 2 r b r 2 = q π + tan 1 ( V 3 2 D V 2 3 D ) + tan 1 ( V 1 2 D V 2 3 D φ 01 ) ,
T j 2 D b r 2 ε ̃ j r > 0 for j = 1 , 3 ,
T j 3 D ε ̃ j r b r 2 > 0 for j = 0 , 2 ,
H 1 2 D tanh ( T 1 2 D δ 1 ) , H 2 3 D tanh ( T 2 3 D δ 2 ) ,
( V j 2 D , V j 3 D ) { ( T j 2 D , T j 3 D ) ε ̃ j r : TM ( T j 2 D , T j 3 D ) : TE
F ( b r , b i = 0 ) = ( V 1 2 D ) 2 V 3 2 D i V 0 3 D ( V 2 3 D ) 2 H 1 2 D H 2 3 D + V 1 2 D [ i V 0 3 D V 3 2 D ( V 2 3 D ) 2 ] H 2 3 D + V 2 3 D [ ( V 1 2 D ) 2 + i V 0 3 D V 3 2 D ] H 1 2 D + V 1 2 D V 2 3 D ( i V 0 3 D + V 3 2 D ) = 0 .

Metrics