Abstract

In this paper, we show that the propagation distance of the long range plasmon wave mode guided by a finite thickness gold metal film can be extended several orders of magnitude longer if we place intermediate dielectric layers on both sides of the metal film and choose the layer thickness properly. The propagation distance goes to infinite if the intermediate layer thickness approaches a critical thickness.

© 2006 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin Heidelberg, 1988).
  2. E. N.  Economou, "Surface plasmons in thin films," Phys. Rev.  182, 539-554 (1969).
    [CrossRef]
  3. J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  4. D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
    [CrossRef]
  5. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000).
    [CrossRef]
  6. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 63, 125417 (2001).
    [CrossRef]
  7. P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
    [CrossRef]
  8. S. J.  Al-Bader, "Optical transmission on metallic wires - fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).
    [CrossRef]
  9. B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
    [CrossRef]
  10. A. Degiron and D. Smith, "Numerical simulations of long-range plasmons," Opt. Express 14, 1611-1625 (2006).
    [CrossRef] [PubMed]
  11. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
    [CrossRef]
  12. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005).
    [CrossRef] [PubMed]
  13. K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, "Long-range surface plasmon polariton nanowire waveguides for device applications," Opt. Express 14, 314-319 (2006).
    [CrossRef] [PubMed]
  14. 4. L. Holland, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1966).
  15. G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983).
    [CrossRef]
  16. F. Y. Kou and T. Tamir, "Range extension of surface plasmons by dielectric layers," Opt. Lett. 12, 367 (1987).
    [CrossRef] [PubMed]
  17. L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986).
    [CrossRef]
  18. J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
    [CrossRef]
  19. E. Anemogiannis, E. N Glytsis, and T. K Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
    [CrossRef]
  20. A. Papoulis, Circuits and Systems (Holt, Rinehart and Winston, Inc., New York, 1980).
  21. P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  22. G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton like waves guided by thin, lossy metal films," Opt. Lett. 8, 383-386 (1983).
    [CrossRef] [PubMed]
  23. J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  24. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
    [CrossRef]

2006

2005

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005).
[CrossRef] [PubMed]

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

2004

S. J.  Al-Bader, "Optical transmission on metallic wires - fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).
[CrossRef]

2001

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 63, 125417 (2001).
[CrossRef]

2000

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
[CrossRef]

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

1999

1992

J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
[CrossRef]

1986

L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986).
[CrossRef]

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

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

1983

G. I. Stegeman, J. J. Burke, and D. G. Hall, "Surface-polariton like waves guided by thin, lossy metal films," Opt. Lett. 8, 383-386 (1983).
[CrossRef] [PubMed]

G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983).
[CrossRef]

1981

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

1972

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

1969

E. N.  Economou, "Surface plasmons in thin films," Phys. Rev.  182, 539-554 (1969).
[CrossRef]

Al-Bader, S. J.

S. J.  Al-Bader, "Optical transmission on metallic wires - fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).
[CrossRef]

Anemogiannis, E.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Aussenegg, F.R.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Berini, P.

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005).
[CrossRef] [PubMed]

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 63, 125417 (2001).
[CrossRef]

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

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
[CrossRef]

Berolo, E.

Boltasseva, A.

Bozhevolnyi, S. I.

Burke, J. J.

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

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy 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, lossy metal films," Opt. Lett. 8, 383-386 (1983).
[CrossRef] [PubMed]

G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983).
[CrossRef]

Charbonneau, R.

Christy, R. W.

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

Degiron, A.

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Ditlbacher, H.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Economou, E. N.

E. N.  Economou, "Surface plasmons in thin films," Phys. Rev.  182, 539-554 (1969).
[CrossRef]

Felidj, N.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Gaylord, T. K

Glytsis, E. N

Hall, D. G.

Haupt, R.

L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986).
[CrossRef]

Johnson, P. B.

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

Jordan, A. K.

J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
[CrossRef]

Kong, J. A.

J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
[CrossRef]

Krenn, J.R.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Lahoud, N.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
[CrossRef]

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005).
[CrossRef] [PubMed]

Lamprecht, B.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Leitner, A.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Leosson, K.

Lisicka-Shrzek, E.

Mattiussi, G.

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005).
[CrossRef] [PubMed]

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
[CrossRef]

Nikolajsen, T.

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Salerno, M.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Sarid, D.

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

Schider, G.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

Smith, D.

Stegeman, G. I

G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983).
[CrossRef]

Stegeman, G. I.

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

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy 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, lossy metal films," Opt. Lett. 8, 383-386 (1983).
[CrossRef] [PubMed]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Tamir, T.

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

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

Wendler, L.

L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986).
[CrossRef]

Xia, J.

J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
[CrossRef]

Appl. Phys. Lett.

B.  Lamprecht, J.R.  Krenn, G.  Schider, H.  Ditlbacher, M.  Salerno, N.  Felidj, A.  Leitner, and F.R.  Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.  79, 51-53 (2001).
[CrossRef]

G. I Stegeman and J. J. Burke, "Long-range surface plasmons in electrode structures," Appl. Phys. Lett. 43, 221-223, (1983).
[CrossRef]

IEEE J. Quantum Electron.

S. J.  Al-Bader, "Optical transmission on metallic wires - fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).
[CrossRef]

J. Appl. Phys.

L. Wendler and R. Haupt, "Long-range surface plasmon-polaritons in asymmetric layer structures," J. Appl. Phys. 59, pp. 3289-3291 (1986).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A.

J. Xia, A. K. Jordan, and J. A. Kong, "Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model," J. Opt. Soc. Am. A. 9, 740-748 (1992).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

E. N.  Economou, "Surface plasmons in thin films," Phys. Rev.  182, 539-554 (1969).
[CrossRef]

Phys. Rev. B

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

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

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 63, 125417 (2001).
[CrossRef]

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

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005).
[CrossRef]

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

Phys. Rev. Lett.

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

Other

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin Heidelberg, 1988).

4. L. Holland, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1966).

F. Y. Kou and T. Tamir, "Range extension of surface plasmons by dielectric layers," Opt. Lett. 12, 367 (1987).
[CrossRef] [PubMed]

A. Papoulis, Circuits and Systems (Holt, Rinehart and Winston, Inc., New York, 1980).

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Figures (10)

Fig. 1.
Fig. 1.

The plasmon waveguide structure of a thin metal film in layered inhomogeneous dielectric materials.

Fig. 2.
Fig. 2.

(a) The phase of t 22 versus the real part of the mode index; (b) the derivative of the phase of t 22 versus the real part of the mode index.

Fig. 3.
Fig. 3.

(a) The phase of t 22 versus the real part of the mode index; (b) the derivative of the phase of t 22 versus the real part of the mode index.

Fig. 4.
Fig. 4.

The magnetic field Hy (in the unit of A/m) profiles of the guided plasmon modes. (a) is the longest range mode; (b) is one short range mode; (c) is the second long range mode; and (d) is the second short range mode.

Fig. 5.
Fig. 5.

(a) The real part of the mode index for varying intermediate dielectric layer thickness; (b) the imaginary part of the mode index for varying intermediate dielectric layer thickness.

Fig. 6.
Fig. 6.

Propagation distances of the guided modes versus intermediate layer thickness for the gold film thickness of 20 nm.

Fig. 7.
Fig. 7.

(a) The real part of the guided mode index versus the intermediate layer thickness; (b) the imaginary part of the guided mode index versus the intermediate layer thickness.

Fig. 8.
Fig. 8.

The propagation distances of the two guided plasmon wave modes versus the intermediate dielectric layer thickness.

Fig. 9.
Fig. 9.

Magnetic field mode profile (in the unit of A/m) of the long range mode at the intermediate layer thickness d=10.5 nm.

Fig. 10.
Fig. 10.

Magnetic field mode profile (in the unit of A/m) of the short range mode at the intermediate layer thickness of d=10.5 nm.

Tables (1)

Tables Icon

Table 1. Guided plasmon wave modes for the intermediate layer thickness of 1000 nm

Equations (5)

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

H yi = { A i exp [ k xi ( x x i 1 ) ] + B i exp [ k xi ( x x i 1 ) ] } exp [ j γ z ] , i = 0 , 1 , 2 , 3 , 4 .
( A 4 B 4 ) = T 3 T 2 T 1 T 0 ( A 0 B 0 ) = [ t 11 t 12 t 21 t 22 ] ( A 0 B 0 ) ,
T i = 1 2 ( [ 1 + ε i + 1 ε i k xi k x ( i + 1 ) ] exp ( k xi d i ) [ 1 ε i + 1 ε i k xi k x ( i + 1 ) ] exp ( k xi d i ) [ 1 ε i + 1 ε i k xi k x ( i + 1 ) ] exp ( k xi d i ) [ 1 + ε i + 1 ε i k xi k x ( i + 1 ) ] exp ( k xi d i ) ) , i = 0 , 1 , 2 , 3 .
( A 4 0 ) = [ t 11 t 12 t 21 t 22 ] ( 1 B 0 ) .
R = B o A o = t 21 t 22 .

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