Abstract

We report a theoretical investigation of the frequency response of optical interference filters written in photorefractive materials. Counterpropagating coherent beams interact in the volume of a photorefractive crystal through two-beam coupling. The resulting hologram is fixed. The reflectivity of the hologram is calculated as a function of frequency. An analytic solution is obtained for arbitrary grating phase ϕ in the lossless case, α = 0. Numerical solutions are performed for α > 0. Experimental results are compared favorably with the theory.

© 1994 Optical Society of America

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  1. G. Rakuljic, A. Yariv, V. Leyva, “High resolution volume holography using orthogonal data storage,” in Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MD-3.
  2. G. Rakuljic, V. Leyva, A. Yariv, “Optical data storage using orthogonal wavelength multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
    [CrossRef] [PubMed]
  3. V. Leyva, G. Rakuljic, A. Yariv, “Volume holography using orthogonal data storage approach,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper FU-7.
  4. G. Rakuljic, V. Leyva, A. Yariv, “Comparison of angle and wavelength multiplexing in holographic data storage (invited paper),” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WE-2.
  5. J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
    [CrossRef]
  6. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, 1991), p. 496.
  7. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  8. P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
    [CrossRef]
  9. Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
    [CrossRef]
  10. Ref. 6, p. 494.
  11. C. Gu, P. Yeh, “Diffraction properties of fixed gratings in photorefractive media,” J. Opt. Soc. Am. B 7, 2339–2346 (1990).
    [CrossRef]
  12. M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
    [CrossRef]
  13. J. H. Hong, R. Saxena, “Diffraction efficiency of volume holograms written by coupled beams,” Opt. Lett. 16, 180–182 (1991).
    [PubMed]
  14. A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
    [CrossRef]
  15. L. Infeld, T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
    [CrossRef]
  16. D. Zwillinger, Handbook of Differential Equations (Academic, New York, 1989).
  17. G. A. Rakuljic, V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18, 459–61 (1993).
    [CrossRef] [PubMed]
  18. G. Rakuljic, 1653 Nineteenth Street, Santa Monica, Calif. 90404 (personal communication, May1993).
  19. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 15.

1993 (2)

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

G. A. Rakuljic, V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18, 459–61 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

1990 (1)

1987 (1)

A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
[CrossRef]

1985 (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

1983 (1)

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[CrossRef]

1982 (1)

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

1951 (1)

L. Infeld, T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[CrossRef]

Abramowitz, A.

A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 15.

Barut, A. O.

A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
[CrossRef]

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Cronin-Golomb, M.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

Fischer, B.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Gu, C.

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Hong, J. H.

Hull, T. E.

L. Infeld, T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[CrossRef]

Infeld, L.

L. Infeld, T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[CrossRef]

Inomata, A.

A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
[CrossRef]

Ja, Y. H.

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Kewitsch, A.

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

Kwong, S. Z.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

Leyva, V.

G. A. Rakuljic, V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18, 459–61 (1993).
[CrossRef] [PubMed]

G. Rakuljic, V. Leyva, A. Yariv, “Optical data storage using orthogonal wavelength multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
[CrossRef] [PubMed]

G. Rakuljic, A. Yariv, V. Leyva, “High resolution volume holography using orthogonal data storage,” in Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MD-3.

V. Leyva, G. Rakuljic, A. Yariv, “Volume holography using orthogonal data storage approach,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper FU-7.

G. Rakuljic, V. Leyva, A. Yariv, “Comparison of angle and wavelength multiplexing in holographic data storage (invited paper),” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WE-2.

Rakuljic, G.

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

G. Rakuljic, V. Leyva, A. Yariv, “Optical data storage using orthogonal wavelength multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
[CrossRef] [PubMed]

G. Rakuljic, 1653 Nineteenth Street, Santa Monica, Calif. 90404 (personal communication, May1993).

V. Leyva, G. Rakuljic, A. Yariv, “Volume holography using orthogonal data storage approach,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper FU-7.

G. Rakuljic, V. Leyva, A. Yariv, “Comparison of angle and wavelength multiplexing in holographic data storage (invited paper),” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WE-2.

G. Rakuljic, A. Yariv, V. Leyva, “High resolution volume holography using orthogonal data storage,” in Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MD-3.

Rakuljic, G. A.

Saxena, R.

Segev, M.

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

Stegun, I.

A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 15.

White, J. O.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

Wilson, R.

A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
[CrossRef]

Yariv, A.

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

G. Rakuljic, V. Leyva, A. Yariv, “Optical data storage using orthogonal wavelength multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
[CrossRef] [PubMed]

G. Rakuljic, A. Yariv, V. Leyva, “High resolution volume holography using orthogonal data storage,” in Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MD-3.

V. Leyva, G. Rakuljic, A. Yariv, “Volume holography using orthogonal data storage approach,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper FU-7.

G. Rakuljic, V. Leyva, A. Yariv, “Comparison of angle and wavelength multiplexing in holographic data storage (invited paper),” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WE-2.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, 1991), p. 496.

Yeh, P.

C. Gu, P. Yeh, “Diffraction properties of fixed gratings in photorefractive media,” J. Opt. Soc. Am. B 7, 2339–2346 (1990).
[CrossRef]

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[CrossRef]

Zwillinger, D.

D. Zwillinger, Handbook of Differential Equations (Academic, New York, 1989).

Appl. Phys. Lett. (1)

M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, “Self-enhanced diffraction from fixed photorefractive gratings during coherent reconstruction,” Appl. Phys. Lett. 62, 907–909 (1993).
[CrossRef]

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

J. Phys. A (1)

A. O. Barut, A. Inomata, R. Wilson, “Algebraic treatment of second Pöschl–Teller, Morse–Rosen, and Eckart equations,” J. Phys. A 20, 4083–4090 (1987).
[CrossRef]

Opt. Commun. (1)

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Rev. Mod. Phys. (1)

L. Infeld, T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[CrossRef]

Other (9)

D. Zwillinger, Handbook of Differential Equations (Academic, New York, 1989).

G. Rakuljic, 1653 Nineteenth Street, Santa Monica, Calif. 90404 (personal communication, May1993).

A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 15.

Ref. 6, p. 494.

G. Rakuljic, A. Yariv, V. Leyva, “High resolution volume holography using orthogonal data storage,” in Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MD-3.

V. Leyva, G. Rakuljic, A. Yariv, “Volume holography using orthogonal data storage approach,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper FU-7.

G. Rakuljic, V. Leyva, A. Yariv, “Comparison of angle and wavelength multiplexing in holographic data storage (invited paper),” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WE-2.

J. O. White, S. Z. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and Their Applications II, P. Günther, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 4.
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, 1991), p. 496.

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Figures (10)

Fig. 1
Fig. 1

Effect of the photorefractive grating phase ϕ on the intensity coupling of two equal-intensity input beams. For ϕ = 0 the intensity coupling is zero and the beams are purely phase coupled. As ϕ deviates from zero, the beams are coupled more and more strongly near the entrance face of the crystal. The coupling constant is g = 20/cm, and the crystal length is L = 0.2 cm.

Fig. 2
Fig. 2

Magnitude of the index grating formed by the intensity coupled beams of Fig. 1. For ϕ = 0 the index grating is constant, and as ϕ increases, the index grating is more strongly apodized, with its maximum at the entrance face of the crystal.

Fig. 3
Fig. 3

Reflectivity from the fixed index gratings of Fig. 2. The reflectivity maximum occurs at a frequency mismatch given by Δβ = g/2 cos ϕ. The overall reflectivity, as well as the sidelobes, is reduced by the grating apodization for ϕ > 0.

Fig. 4
Fig. 4

Reflectivity from the fixed index gratings for ϕ = 0 and ϕ = π/2 at large values of frequency mismatch. The ϕ = π/2 grating reflectivity has reduced sidelobes whose peaks are approximately 6 dB lower than in the ϕ = 0 case (however, the maximum reflectivity is also 3 dB lower).

Fig. 5
Fig. 5

(a) Intensity of beam 1 incident at z = 0 for various values of ϕ, for a loss constant of α = 6/cm, and for equal-intensity inputs. The coupling constant is g = 20/cm, and the crystal length is L = 0.2 cm. (b) Intensity of beam 2 incident at z = L under conditions identical to those for (a).

Fig. 6
Fig. 6

Index grating formed in the crystal by the coupled beams of Fig. 5. The effect of the loss is to eliminate the no-crossing rule for the two beam intensities. Thus when g/2 sin ϕ ≤ ∼α, the intensities can be equal within the volume of the crystal. This equality leads to a maximum in the magnitude of the index grating located away from the edge of the crystal.

Fig. 7
Fig. 7

Reflectivity for the case α = 6/cm versus frequency mismatch for a beam incident at z = 0 upon the index gratings of Fig. 6. The behavior is complex because the magnitude of the index grating near z = 0 (the most efficient region of reflection) first increases and then decreases with increasing photorefractive grating phase ϕ.

Fig. 8
Fig. 8

Reflectivity for the case α = 6/cm versus frequency mismatch for a beam incident at z = L upon the index gratings of Fig. 6. Note the strong nonreciprocity in comparison with Fig. 7. The behavior for this case is simpler than that of Fig. 7, because the magnitude of the index grating near z = L (the most efficient region of reflection) decreases monotonically with increasing photorefractive grating phase ϕ.

Fig. 9
Fig. 9

Reflectivity of the Accuwave H-α filter (heavy curve) compared with theoretical plots (light curves). For the upper curve, g = 3/cm, ϕ = π/3, and α = 0; for the lower curve, g = 5/cm, ϕ = π/6, αwrite = 4/cm, and αread = 0.5/cm.

Fig. 10
Fig. 10

Magnitude of the index gratings used to calculate the reflectivities of Fig. 9. The index grating in the case of nonzero optical absorption is more strongly apodized.

Equations (81)

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A ( z ) = i π Δ n ( z ) λ e i ϕ B ( z ) ,
B ( z ) = i π Δ n * ( z ) λ e i ϕ A ( z ) ,
n ( z ) = n 0 + ½ [ Δ n ( z ) e i ϕ exp ( i 2 k z ) + c . c . ] .
Δ n ( z ) = n 1 A ( z ) B * ( z ) / I ( z ) ,
A ( z ) = i g | B ( z ) | 2 / I ( z ) e i ϕ A ( z ) ,
B ( z ) = i g | A ( z ) | 2 / I ( z ) e i ϕ B ( z ) ,
A ( z ) = a ( z ) e i ψ 1 , B ( z ) = b ( z ) e i ψ 2 ,
a ( z ) = g sin ( ϕ ) b ( z ) 2 / I ( z ) a ( z ) ,
b ( z ) = g sin ( ϕ ) a ( z ) 2 / I ( z ) b ( z ) ,
ψ 1 ( z ) = g cos ( ϕ ) b ( z ) 2 / I ( z ) ,
ψ 2 ( z ) = g cos ( ϕ ) a ( z ) 2 / I ( z ) .
I 1 ( z ) = ½ [ ( c 2 + υ 2 e Γ z ) 1 / 2 + c ] ,
I 2 ( z ) = ½ [ ( c 2 + υ 2 e Γ z ) 1 / 2 c ] .
c = e Γ L I 1 2 ( 0 ) I 2 2 ( L ) e Γ L I 1 ( 0 ) + I 2 ( L ) ,
υ 2 = 4 I 1 ( 0 ) I 2 ( L ) I 1 ( 0 ) + I 2 ( L ) e Γ L I 1 ( 0 ) + I 2 ( L ) .
ψ 1 ( z ) = ½ ( g cos ( ϕ ) z cot ( ϕ ) coth 1 { [ 1 + ( υ / c ) 2 e Γ z ] 1 / 2 } ) ,
ψ 2 ( z ) = ½ ( g cos ( ϕ ) z + cot ( ϕ ) coth 1 { [ 1 + ( υ / c ) 2 e Γ z ] 1 / 2 } ) ,
Δ n e i ϕ = n 1 2 e + i ϕ exp [ + i g cos ( ϕ ) z ] ( 1 + c 2 e Γ z / υ 2 ) 1 / 2 .
A ( z ) = i g 2 exp [ i g cos ( ϕ ) z + i ϕ ] ( 1 + c 2 e Γ z / υ 2 ) 1 / 2 exp ( 2 i Δ β z ) B ( z ) ,
B ( z ) = i g 2 exp [ i g cos ( ϕ ) z i ϕ ] ( 1 + c 2 e Γ z / υ 2 ) 1 / 2 exp ( + 2 i Δ β z ) A ( z ) ,
A ( z ) = i g ( z ) f ( z ) B ( z ) ,
B ( z ) = i [ g ( z ) / f ( z ) ] A ( z ) ,
ξ = Γ υ 2 c exp ( Γ z / 2 ) [ 1 + ( υ / c ) 2 exp ( Γ z ) ] 1 / 2 d z = sinh 1 [ υ c exp ( Γ z / 2 ) ] .
a ( ξ ) = i κ f ( ξ ) b ( ξ ) ,
b ( ξ ) = i κ * f * ( ξ ) a ( ξ ) ,
f ( ξ ) = [ c υ sinh ( ξ ) ] 4 i Δ β / Γ ,
κ = e i ϕ / ( 2 sin ϕ ) ,
a ( ξ ) = T ( ξ ) exp [ i F ( ξ ) ] , b ( ξ ) = V ( ξ ) exp [ i F ( ξ ) ] ,
F ( ξ ) = 1 / ( 2 i ) ln [ f ( ξ ) ] = Δ β / ( g sin ϕ ) ln [ sinh ( ξ ) ] .
T ( ξ ) + [ 1 4 sin 2 ϕ + η 2 i η ( i η + 1 ) sinh 2 ξ ] T ( ξ ) = 0 ,
V ( ξ ) + [ 1 4 sin 2 ϕ + η 2 i η ( i η 1 ) sinh 2 ξ ] V ( ξ ) = 0 ,
l = 1 2 ( 1 4 sin 2 ϕ η 2 ) 1 / 2 + 1 2 ,
κ T = 1 + i η ,
κ υ = i η .
T ( ξ ) = t 1 sinh i η ξ cosh ( ξ ) F 2 1 ( 1 2 i η 2 β , 1 2 i η 2 + β ; 1 2 i η ; sinh 2 ξ ) + t 2 sinh 1 + i η ξ cosh ( ξ ) F 2 1 ( i η 2 β + 1 , i η 2 + β + 1 ; 3 2 + i η ; sinh 2 ξ ) ,
V ( ξ ) = υ 1 sinh 1 i η ξ cosh ( ξ ) F 2 1 ( 1 i η 2 β , 1 i η 2 + β ; 3 2 i η ; sinh 2 ξ ) + υ 2 sinh i η ξ cosh ( ξ ) F 2 1 ( i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 + i η ; sinh 2 ξ ) ,
β = 1 2 ( 1 4 sin 2 ϕ η 2 ) 1 / 2 = l 1 2 .
A ( z ) = [ 1 + ( υ c ) 2 e Γ z ] 1 / 2 { C 1 F 2 1 [ 1 2 i η 2 β , 1 2 i η 2 + β ; 1 2 i η ; ( υ c ) 2 e Γ z ] + C 2 exp [ ( 1 + 2 i η ) Γ z / 2 ] F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 + i η ; ( υ c ) 2 e Γ z ] } ,
B ( z ) = [ 1 + ( υ c ) 2 e Γ z ] 1 / 2 { C 3 exp [ ( 1 2 i η ) Γ z / 2 ] × F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 i η ; ( υ c ) 2 e Γ z ] + C 4 F 2 1 [ i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 + i η ; ( υ c ) 2 e Γ z ] } ,
C 1 = A ( 0 ) 4 sin 2 ( ϕ ) ( 1 + 4 η 2 ) [ 1 + ( υ / c ) 2 ] 1 / 2 exp [ ( 1 2 i η ) Γ L / 2 ] × F 2 1 [ i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 + i η ; ( υ c ) 2 e Γ L ] / D ,
C 2 = A ( 0 ) [ 1 + ( υ / c ) 2 ] 1 / 2 ( υ c ) 2 F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 i η ; ( υ c ) 2 e Γ L ] / D ,
C 3 = A ( 0 ) { υ ( 2 + 4 i η ) sin ϕ i c e i ϕ [ 1 + ( υ / c ) 2 ] 1 / 2 } exp [ ( 1 2 i η ) Γ L / 2 ] × F 2 1 [ i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 + i η ; ( υ c ) 2 e Γ L ] / D ,
C 4 = A ( 0 ) { υ ( 2 + 4 i η ) sin ϕ i c e i ϕ [ 1 + ( υ / c ) 2 ] 1 / 2 } F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 i η ; ( υ c ) 2 e Γ L ] / D ,
D = 4 sin 2 ( ϕ ) ( 1 + 4 η 2 ) exp [ ( 1 2 i η ) Γ L / 2 ] × F 2 1 [ i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 + i η ; ( υ c ) 2 e Γ L ] F 2 1 [ i η 2 β + 1 2 , i η 2 + β + 1 2 ; 1 2 i η ; ( υ c ) 2 ] ( υ c ) 2 F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 i η ; ( υ c ) 2 e Γ L ] F 2 1 [ i η 2 β + 1 , i η 2 + β + 1 ; 3 2 + i η ; ( υ c ) 2 ] .
Δ n = n 1 [ I 1 ( 0 ) I 2 ( 0 ) ] 1 / 2 / I e igz .
R = κ 2 sinh 2 s L κ 2 sinh 2 s L + s 2 ,
κ = g [ I 1 ( 0 ) I 2 ( 0 ) ] 1 / 2 / I ,
s = g ( κ 2 δ 2 / 4 ) 1 / 2 ,
δ = g 2 Δ β .
I 1 ( z ) = I 2 ( z ) = υ / 2 exp ( Γ z / 2 ) = I 1 ( 0 ) e ( Γ z / 2 ) ,
ψ 1 ( z ) = ψ 2 ( z ) = cos ( ϕ ) g z / 2 ,
Δ n ( z ) = n 1 / 2 exp [ i g cos ( ϕ ) z ] .
A ( z ) = i g 2 e i ϕ exp [ i ( g cos ϕ 2 Δ β ) z ] B ( z ) ,
B ( z ) = i g 2 e i ϕ exp [ i ( g cos ϕ 2 Δ β ) z ] A ( z ) .
A ( z ) = i g | B ( z ) | 2 / I ( z ) e i ϕ A ( z ) ( α / 2 ) A ( z ) ,
B ( z ) = i g | A ( z ) | 2 / I ( z ) e i ϕ B ( z ) + ( α / 2 ) B ( z ) .
a ( z ) = sin ( ϕ ) g | b ( z ) | 2 / I ( z ) [ a ( z ) ] ( α / 2 ) a ( z ) ,
b ( z ) = sin ( ϕ ) g | a ( z ) | 2 / I ( z ) [ b ( z ) ] + ( α / 2 ) b ( z ) ,
ψ 1 = cos ( ϕ ) g | b ( z ) | 2 / I ( z ) ,
ψ 2 = cos ( ϕ ) g | a ( z ) | 2 / I ( z ) .
A ( z ) = A ( 0 ) exp ( α z / 2 ) [ B ( 0 ) 2 exp ( 2 α z ) + A ( 0 ) 2 B ( 0 ) 2 + A ( 0 ) 2 ] ( i g ) / ( 2 α ) ,
B ( z ) = B ( 0 ) exp ( + α z / 2 ) [ B ( 0 ) 2 + A ( 0 ) 2 exp ( 2 α z ) B ( 0 ) 2 + A ( 0 ) 2 ] ( i g ) / ( 2 α ) ,
Δ n ( z ) = n 1 A ( 0 ) B ( 0 ) exp ( igz ) A ( 0 ) 2 exp ( α z ) + B ( 0 ) 2 exp ( α z ) .
A ( z ) = i κ f 1 ( z ) exp ( i δ z ) B ( z ) ,
B ( z ) = ± i κ * f 2 ( z ) exp ( i δ z ) A ( z ) .
A ( z ) = T ( z ) exp [ i F ( z ) ] ,
B ( z ) = V ( z ) exp [ i F ( z ) ] ,
T ( z ) = i κ V ( z ) exp { i [ δ z 2 F ( z ) ] } f 1 ( z ) i F ( z ) T ( z ) ,
V ( z ) = i κ * T ( z ) exp { i [ 2 F ( z ) δ z ] } f 2 ( z ) + i F ( z ) V ( z ) .
T ( z ) = [ i δ 2 i F ( z ) + f 1 ( z ) / f 1 ( z ) ] T ( z ) + { | κ | 2 f 1 ( z ) f 2 ( z ) δ F ( z ) + [ F ( z ) ] 2 + i F ( z ) f 1 ( z ) / f 1 ( z ) i F ( z ) } T ( z ) .
F ( z ) = δ z 2 + ln [ f 1 ( z ) ] 2 i .
F ( z ) = δ z 2 ln [ f 2 ( z ) ] 2 i
f 2 ( z ) = 1 / f 1 ( z ) .
T ( z ) = { | κ | 2 [ F ( z ) ] 2 i F ( z ) } T ( z ) ,
V ( z ) = { | κ | 2 [ F ( z ) ] 2 + i F ( z ) } V ( z ) ,
ψ ( r ) = α 2 [ κ ( κ 1 ) sinh 2 ( α r ) λ ( λ + 1 ) cosh 2 ( α r ) + ( 21 1 ) 2 ] ψ ( r ) .
ψ 1 ( r ) = sinh 1 κ ( α r ) cosh λ + 1 ( α r ) F 2 1 [ ( λ κ 2 l + 1 ) + 1 2 , ( λ κ 2 + l ) + 1 2 ; 3 2 κ ; sinh 2 ( α r ) ] ,
ψ 2 ( r ) = sinh κ ( α r ) cosh λ + 1 ( α r ) × F 2 1 [ ( λ + κ 2 l + 1 ) , ( λ + κ 2 + l ) + 1 2 + κ ; sinh 2 ( α r ) ] ,
F 2 1 [ a , b ; c ; z ] = F 2 1 [ b , a ; c ; z ] = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) z n n ! .
d d z F 2 1 [ a , b ; c ; z ] = a b c F 2 1 [ a + 1 , b + 1 ; c + 1 ; z ] ,
F 2 1 [ a , b ; c ; z ] = ( 1 z ) c a b F 2 1 [ c a , c b ; c ; z ] .

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