Abstract

Reflectivity and spectral properties of a thick reflection grating are discussed for the second-order Bragg diffraction regime. Analytical expressions for the reflection and gain coefficients are obtained with the help of the developed theory. A procedure is proposed for determining the wavelength of laser oscillations generated in an optically active grating. The influence of the second harmonic of dielectric permittivity spatial modulation on reflection and spectral characteristics of a grating was also analyzed. Comparison of the approximate results obtained by the proposed approach and by the numerical solution is carried out, and their excellent agreement is demonstrated.

© 2012 Optical Society of America

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References

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  1. A. Saliminia, A. Villeneuve, T. V. Galstian, S. LaRoshelle, and K. Richardson, “First- and second-order Bragg gratings in single-mode planar waveguides of chalcogenide glasses,” J. Lightwave Technol. 17, 837–842 (1999).
    [CrossRef]
  2. T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
    [CrossRef]
  3. T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
    [CrossRef]
  4. Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
    [CrossRef]
  5. M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103508 (2007).
  6. T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
    [CrossRef]
  7. O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
    [CrossRef]
  8. V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. 1. Approximate theory,” J. Opt. Soc. Am. B 29, 691–697 (2012).
    [CrossRef]
  9. A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).
  10. F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
    [CrossRef]
  11. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics (Pergamon, 1960).
  12. M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684–690 (2009).
    [CrossRef]
  13. V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
    [CrossRef]

2012 (1)

2011 (2)

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
[CrossRef]

2009 (2)

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684–690 (2009).
[CrossRef]

2008 (1)

V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

2007 (1)

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103508 (2007).

2002 (2)

Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
[CrossRef]

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

2001 (1)

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
[CrossRef]

1999 (1)

Abbate, G.

Bazhenov, V. Y.

Bruder, F.-K.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Deuber, F.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Fäcke, T.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Fitio, V. M.

V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. 1. Approximate theory,” J. Opt. Soc. Am. B 29, 691–697 (2012).
[CrossRef]

V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Galstian, T. V.

Gersborg-Hansen, M.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103508 (2007).

Hagen, R.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Hönel, D.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Iskra, K. F.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Jurbergs, D.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Kavc, T.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kern, W.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kogure, M.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Kranzelbinder, G.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kriensen, A.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103508 (2007).

Kzianzou, V.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics (Pergamon, 1960).

Langer, G.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

LaRoshelle, S.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics (Pergamon, 1960).

Maeda, M.

Miyamoto, S.

Neger, T.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Oki, Y.

Pogantsch, A.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Richardson, K.

Rölle, T.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Sakhno, O. V.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
[CrossRef]

V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Saliminia, A.

Samuel, I. D. W.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Schade, W.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
[CrossRef]

Scheel, D.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
[CrossRef]

Schrader, S.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Slussarenko, S. S.

Smirnova, T. N.

V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. 1. Approximate theory,” J. Opt. Soc. Am. B 29, 691–697 (2012).
[CrossRef]

O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
[CrossRef]

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Stumpe, J.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
[CrossRef]

Toussaere, E.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Turnbull, G. A.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Vasa, N. J.

Vasnetsov, M. V.

Villeneuve, A.

Voss, T.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
[CrossRef]

Weiser, M.-S.

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Yariv, A.

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Yeh, R.

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Appl. Phys. B (2)

O. V. Sakhno, J. Stumpe, and T. N. Smirnova, “Distributed feedback dye laser holographically induced in improved organic—inorganic photocurable nanocomposites,” Appl. Phys. B 103, 907–916 (2011).
[CrossRef]

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103508 (2007).

Chem. Mater. (1)

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. (1)

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Photopolym. Sci. Technol. (1)

F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, D. Jurbergs, M. Kogure, T. Rölle, and M.-S. Weiser “Full-color self-processing holographic photopolymers with high sensitivity in red—the first class of instant holographic photopolymers,” J. Photopolym. Sci. Technol. 22, 257–260 (2009).
[CrossRef]

Opt. Lett. (1)

Optik (1)

V. M. Fitio, O. V. Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Other (2)

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics (Pergamon, 1960).

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

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Figures (8)

Fig. 1.
Fig. 1.

Dependences of reflectivity on the grating thickness (a) and on the angle of wave propagation in the grating (b); curve 1, d = 1200 Λ ,; curve 2, d = 3000 Λ . Continuous curves are calculated using Eq. (7). The points are the result obtained by the FFE method.

Fig. 2.
Fig. 2.

I ε 20 as a function of the wavelength for θ 0 = θ B = 0 , the Bragg’s wavelength λ B = 500 nm , d = 2000 Λ , R ε 20 = 2.25 , ε ˜ 1 = 0.1 .

Fig. 3.
Fig. 3.

Calculated spectral dependence of reflection coefficient for the optically active grating in a medium with gain. The computation parameter is I ε 20 = 0.0002135 (other parameters are the same as in Fig. 2).

Fig. 4.
Fig. 4.

Spectral dependences of R (a) and (d), I ε 20 (b) and (e), and R g (c) and (f). The parameters of computation are ε ˜ 1 = 0.015 (a)–(c), ε ˜ 1 = 0.054 (d)–(f), I ε 20 = 0.0002979 (c), and I ε 20 = 0.0000432 (f).

Fig. 5.
Fig. 5.

Zero-reflection wavelengths of a passive grating ( λ p ) and lasing wavelengths of an optically active grating ( λ g p ) versus p . Plus (minus) corresponds to the sign before the square root in Eq. (14).

Fig. 6.
Fig. 6.

Reflection coefficient versus grating thickness (a) and wavelength (b). The parameters of computation are ε ˜ 1 = 0 , ε ˜ 2 = 0.01 , (curves 1); ε ˜ 1 = 0.1 , ε ˜ 2 = 0 , (curves 2); ε ˜ 1 = 0.1 , ε ˜ 2 = 0.01 (curves 3).

Fig. 7.
Fig. 7.

I ε 20 as a function of the wavelength for θ 0 = θ B = 0 ; the Bragg wavelength is λ B = 500 nm , and grating parameters are d = 2000 Λ , R ε 20 = 2.25 , ε ˜ 1 = 0.1 , ε ˜ 2 = 0.001 .

Fig. 8.
Fig. 8.

Calculated spectral dependences of the gain. The computation parameter is Im ε 20 = 0.0001273 (other parameters are the same as in Fig. 7).

Equations (15)

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ε 2 ( z ) = R ε 20 + i I ε 20 + ε ˜ 1 2 [ exp ( i K z ) + exp ( i K z ) ] + ε ˜ 2 2 [ exp ( i 2 K z ) + exp ( i 2 K z ) ] ,
2 E ( x , z ) z 2 + 2 E ( x , z ) x 2 + ( 2 π λ ) 2 { R ε 20 + i I ε 20 + ε ˜ 1 2 [ exp ( i K z ) + exp ( i K z ) ] + ε ˜ 2 2 [ exp ( i 2 K z ) + exp ( i 2 K z ) ] } E ( x , z ) = 0 ,
E ( x , z ) = E t ( x , z ) + E r ( x , z ) = t ( z ) exp [ i ( k x x k z z ) ] + r ( z ) exp [ i ( k x x + k z z ) ] .
r ¯ ( m Λ ) = E r ( x , m Λ ) E t ( x , m Λ ) = χ 2 sinh ( s m Λ ) s cosh ( s m Λ ) ( Δ k z + i g / 2 χ 1 ) sinh ( s m Λ ) ,
t ¯ ( m Λ ) = E t ( x , 0 ) E t ( x , m Λ ) = s s cosh ( s m Λ ) ( Δ k z + i g / 2 χ 1 ) sinh ( s m Λ ) ,
s = ± [ χ 2 2 ( Δ k z + i g / 2 χ 1 ) 2 ] 1 / 2 , Δ k z = k z K , g = γ / k z , γ = ( 2 π / λ ) 2 I ε 20 , χ 1 = ( κ 1 2 / 2 k z ) { ( k z 2 + i γ ) 1 [ ( k z + K ) 2 k z 2 i γ ] 1 } , χ 2 = κ 1 2 / 2 k z ( k z 2 + i γ ) 1 , κ 1 = ( 2 π / λ ) 2 ( ε ˜ 1 / 2 ) .
T ( m Λ ) = | t ¯ ( m Λ ) | 2 , R ( m Λ ) = | r ¯ ( m Λ ) | 2 .
K κ 1 2 / 6 K 3 k z K + 5 κ 1 2 / 6 K 3 .
D ( λ , I ε 20 ) = s cosh ( s m Λ ) ( Δ k z + i g 2 χ 1 ) sinh ( s m Λ ) = 0.
s 2 d 2 = [ χ 2 2 ( Δ k z χ 1 ) 2 ] d 2 = p 2 π 2 ;
k z = K + χ 1 ± [ ( p π / d ) 2 + χ 2 2 ] 1 / 2 .
ω g p ω p = c n 20 [ K + χ 1 ± p π d ( 1 + χ 2 2 d 2 p 2 π 2 ) 1 / 2 ] = c n 20 [ K + χ 1 ± ( p π d + 1 2 χ 2 2 d p π ) ] ,
Δ ω g p Δ ω p = c n 20 [ π d + 1 2 χ 2 2 d p ( p + 1 ) π ] .
λ g p λ p = 2 π n 20 K + χ 1 ± [ ( p π / d ) 2 + χ 2 2 ] 1 / 2 .
K κ 1 2 / 6 K 3 + κ 2 / 2 K k z K + 5 κ 1 2 / 6 K 3 κ 2 / 2 K .

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