Abstract

A theoretical study is presented that describes the recording and the readout of transmission volume holograms in dynamic photoanisotropic organic materials. The stationary coupled differential equations of two recording beams that are polarized perpendicular to the plane of incidence are derived and solved analytically for the beam intensities and the phase changes. These equations self-consistently satisfy the recording dynamics of the hologram formation including the effects of fringe curvature and energy transfer. Analytic formulas for the diffraction efficiency are obtained from the readout equations under the assumption of a Bragg-matched weak probe beam of arbitrary polarization and in the small fringe curvature regime. Previously measured parameters of Methyl Orange–polyvinyl alcohol holograms indicate that the optimal hologram thickness is approximately 0.2 mm and predict the decrease of diffraction efficiency with an increase of beam ratio.

© 1993 Optical Society of America

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References

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  1. G. H. Brown, ed., Photochroism (Wiley-Interscience, New York, 1971).
  2. L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
    [Crossref]
  3. I. D. Shatalin, “Mechanism of photoanisotropy in photochemical trans-cis isomerization,” Opt. Spectrosc. (USSR) 66, 209–211 (1989).
  4. V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
    [Crossref]
  5. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [Crossref]
  6. J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
    [Crossref]
  7. T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
    [Crossref]
  8. Sh. D. Kakichashvili, “Regularity in photoanisotropic phenomena,” Opt. Spectrosc. (USSR) 52, 191–194 (1982).
  9. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [Crossref]
  10. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  11. T. Huang, K. H. Wagner, “Coupled mode analysis of dynamic polarization volume holograms,” in Photopolymer Device Physics, Chemistry, and Applications II, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1559, 377–384 (1991).
    [Crossref]
  12. T. Huang, K. H. Wagner, “Photoanisotropic incoherent to coherent conversion using five wave mixing,” in Devices for Optical Processing, D. M. Gookin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1562, 44–54 (1991).
    [Crossref]
  13. Sh. D. Kakichashvili, “Polarization-holographic recording in the general case of a reaction of a photoanisotropic medium,” Sov. J. Quantum Electron. 13, 1317–1319 (1983).
    [Crossref]
  14. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  15. T. Todorov, L. Nikolova, N. Tomova, “Polarization holography. 1: A new high efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309–4312 (1984).
    [Crossref] [PubMed]
  16. J. J. A. Couture, R. A. Lessard, “Modulation transfer function measurements for thin layers of azo dyes in PVA matrix used as an optical recording material,” Appl. Opt. 27, 3368–3374 (1988).
    [Crossref] [PubMed]
  17. R. Richert, “Analysis of non-exponential first-order reactions,” Chem. Phys. Lett. 118, 534–538 (1985).
    [Crossref]
  18. R. Richert, H. Bässler, “Merocyanine ↔ spiropyran transformation in a polymer matrix: an example of a dispersive chemical reaction,” Chem. Phys. Lett. 116, 302–306 (1985).
    [Crossref]

1989 (2)

I. D. Shatalin, “Mechanism of photoanisotropy in photochemical trans-cis isomerization,” Opt. Spectrosc. (USSR) 66, 209–211 (1989).

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

1988 (2)

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

J. J. A. Couture, R. A. Lessard, “Modulation transfer function measurements for thin layers of azo dyes in PVA matrix used as an optical recording material,” Appl. Opt. 27, 3368–3374 (1988).
[Crossref] [PubMed]

1986 (1)

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[Crossref]

1985 (3)

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

R. Richert, “Analysis of non-exponential first-order reactions,” Chem. Phys. Lett. 118, 534–538 (1985).
[Crossref]

R. Richert, H. Bässler, “Merocyanine ↔ spiropyran transformation in a polymer matrix: an example of a dispersive chemical reaction,” Chem. Phys. Lett. 116, 302–306 (1985).
[Crossref]

1984 (2)

T. Todorov, L. Nikolova, N. Tomova, “Polarization holography. 1: A new high efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309–4312 (1984).
[Crossref] [PubMed]

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

1983 (1)

Sh. D. Kakichashvili, “Polarization-holographic recording in the general case of a reaction of a photoanisotropic medium,” Sov. J. Quantum Electron. 13, 1317–1319 (1983).
[Crossref]

1982 (1)

Sh. D. Kakichashvili, “Regularity in photoanisotropic phenomena,” Opt. Spectrosc. (USSR) 52, 191–194 (1982).

1979 (1)

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

Bässler, H.

R. Richert, H. Bässler, “Merocyanine ↔ spiropyran transformation in a polymer matrix: an example of a dispersive chemical reaction,” Chem. Phys. Lett. 116, 302–306 (1985).
[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]

Couture, J. J. A.

Dragostinova, V.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[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]

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]

Heaton, J. M.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Huang, T.

T. Huang, K. H. Wagner, “Coupled mode analysis of dynamic polarization volume holograms,” in Photopolymer Device Physics, Chemistry, and Applications II, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1559, 377–384 (1991).
[Crossref]

T. Huang, K. H. Wagner, “Photoanisotropic incoherent to coherent conversion using five wave mixing,” in Devices for Optical Processing, D. M. Gookin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1562, 44–54 (1991).
[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]

Kakichashvili, Sh. D.

Sh. D. Kakichashvili, “Polarization-holographic recording in the general case of a reaction of a photoanisotropic medium,” Sov. J. Quantum Electron. 13, 1317–1319 (1983).
[Crossref]

Sh. D. Kakichashvili, “Regularity in photoanisotropic phenomena,” Opt. Spectrosc. (USSR) 52, 191–194 (1982).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kukhtarev, N. V.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

Lessard, R. A.

Markovsky, P.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

Mateva, N.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

Mills, P. A.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Nikolova, L.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, “Polarization holography. 1: A new high efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309–4312 (1984).
[Crossref] [PubMed]

Odulov, S. G.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

Paige, E. G. S.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Richert, R.

R. Richert, “Analysis of non-exponential first-order reactions,” Chem. Phys. Lett. 118, 534–538 (1985).
[Crossref]

R. Richert, H. Bässler, “Merocyanine ↔ spiropyran transformation in a polymer matrix: an example of a dispersive chemical reaction,” Chem. Phys. Lett. 116, 302–306 (1985).
[Crossref]

Shatalin, I. D.

I. D. Shatalin, “Mechanism of photoanisotropy in photochemical trans-cis isomerization,” Opt. Spectrosc. (USSR) 66, 209–211 (1989).

Solymar, L.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Soskin, M. S.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

Todorov, T.

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, “Polarization holography. 1: A new high efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309–4312 (1984).
[Crossref] [PubMed]

Tomova, N.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[Crossref]

T. Todorov, L. Nikolova, N. Tomova, “Polarization holography. 1: A new high efficiency organic material with reversible photoinduced birefringence,” Appl. Opt. 23, 4309–4312 (1984).
[Crossref] [PubMed]

Vinetskii, V. L.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

Wagner, K. H.

T. Huang, K. H. Wagner, “Photoanisotropic incoherent to coherent conversion using five wave mixing,” in Devices for Optical Processing, D. M. Gookin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1562, 44–54 (1991).
[Crossref]

T. Huang, K. H. Wagner, “Coupled mode analysis of dynamic polarization volume holograms,” in Photopolymer Device Physics, Chemistry, and Applications II, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1559, 377–384 (1991).
[Crossref]

Wilson, T.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Chem. Phys. Lett. (2)

R. Richert, “Analysis of non-exponential first-order reactions,” Chem. Phys. Lett. 118, 534–538 (1985).
[Crossref]

R. Richert, H. Bässler, “Merocyanine ↔ spiropyran transformation in a polymer matrix: an example of a dispersive chemical reaction,” Chem. Phys. Lett. 116, 302–306 (1985).
[Crossref]

IEEE J. Quantum Electron. (2)

T. Todorov, L. Nikolova, N. Tomova, V. Dragostinova, “Photoinduced anisotropy in rigid dye solutions for transient polarization holography,” IEEE J. Quantum Electron. QE-22, 1262–1267 (1986).
[Crossref]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

J. Mod. Opt. (1)

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, N. Mateva, “Optically-controlled photo-induced birefringence in photo-anisotropic materials,” J. Mod. Opt. 35, 1789–1799 (1988).
[Crossref]

Opt. Acta (1)

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[Crossref]

Opt. Spectrosc. (USSR) (2)

I. D. Shatalin, “Mechanism of photoanisotropy in photochemical trans-cis isomerization,” Opt. Spectrosc. (USSR) 66, 209–211 (1989).

Sh. D. Kakichashvili, “Regularity in photoanisotropic phenomena,” Opt. Spectrosc. (USSR) 52, 191–194 (1982).

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]

Sov. J. Quantum Electron. (1)

Sh. D. Kakichashvili, “Polarization-holographic recording in the general case of a reaction of a photoanisotropic medium,” Sov. J. Quantum Electron. 13, 1317–1319 (1983).
[Crossref]

Sov. Phys. Usp. (1)

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[Crossref]

Other (4)

T. Huang, K. H. Wagner, “Coupled mode analysis of dynamic polarization volume holograms,” in Photopolymer Device Physics, Chemistry, and Applications II, R. A. Lessard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1559, 377–384 (1991).
[Crossref]

T. Huang, K. H. Wagner, “Photoanisotropic incoherent to coherent conversion using five wave mixing,” in Devices for Optical Processing, D. M. Gookin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1562, 44–54 (1991).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

G. H. Brown, ed., Photochroism (Wiley-Interscience, New York, 1971).

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Figures (11)

Fig. 1
Fig. 1

Formation of a photoanisotropic volume hologram.

Fig. 2
Fig. 2

Variation of beam intensity with film thickness, where α = 7.3 mm−1, β″ = − 15 mm/mW, Ir0 = 0.16 mW/mm2, and Is0 = 0.04 mW/mm2.

Fig. 3
Fig. 3

Beam ratio m versus input beam ratio m0 and film thickness d, where α = 7.3 mm−1, β″ = −15 mm/mW, and Ir0 + Is0 = 0.20 mW/mm2.

Fig. 4
Fig. 4

Dependence of the direction of fringe curvature on the sign of β′ when m0 = Ir0/Is0 > 1: (a) β′ > 0, (b) β′ < 0.

Fig. 5
Fig. 5

Phase difference between the two recording beams versus film thickness. The solid curve is for a dynamic medium, where Δϕ = (β′/2β″)ln(m0/m) + ϕr0ϕs0, and the dashed line is for a latent medium, where Δϕ = ϕr0. − ϕs0. β′ = −84 mm/mW and ϕr0. − ϕs0 = 0.05 rad are used in this simulation, in addition to the parameters used in Fig. 2.

Fig. 6
Fig. 6

Phase difference Δϕ between the two recording beams versus input beam ratio m0 and film thickness d, where α = 7.3 mm−1, β′ = −84 mm/mW, β″ = −15 mm/mW, Ir0 + Is0 = 0.20 mW/mm2, and ϕr0ϕs0 = 0.05 rad.

Fig. 7
Fig. 7

Readout of a photoanisotropic volume hologram with a weak probe beam of arbitrary polarization incident at an angle θrθ.

Fig. 8
Fig. 8

(a) Intensity grating vector at the beginning of the exposure (solid chord) and at steady state after fringe curvature (approximated as a tilt) (dashed chord). (b) Readout k-vector diagram with the readout wave vector rotated for perfect Bragg matching.

Fig. 9
Fig. 9

Simulation of diffraction efficiency for m0 = 1, where β′ = −84 mm/mW, β″ = −15 mm/mW, Ir0 = 0.10 mW/mm2, and Is0 = 0.10 mW/mm2.

Fig. 10
Fig. 10

Simulation of diffraction efficiency for m0 = 4, where β′ = −84 mm/mW, β″ = −15 mm/mW, Ir0 = 0.16 mW/mm2, and Is0 = 0.04 mW/mm2.

Fig. 11
Fig. 11

Diffraction efficiency versus beam ratio and total beam intensity, where α = 7.3 mm−1, d = 0.22 mm, β′ = −84 mm/mW, and β″ = −15 mm/mW.

Equations (46)

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

∊; ̂ = = 0 ̂ = + Δ ̂ = ( y , z ) = [ ̂ 0 + κ ̂ I ( y , z ) 0 0 0 ̂ 0 + κ ̂ I ( y , z ) 0 0 0 ̂ 0 + κ ̂ I ( y , z ) ] ,
( · E ) 2 E μ ω 2 ̂ = E = 0 .
2 E + μ ω 2 ( ̂ 0 + κ ̂ I ) E = 0 ,
E = E s + E r = [ A s ( z ) exp ( i k s · r ) + A r ( z ) exp ( i k r · r ) ] exp ( i ω t ) ,
I = ( 0 c n 0 / 2 ) | E | 2 = ( 0 c n 0 / 2 ) { A s A s * + A r A r * + A s A r * × exp [ i ( k s k r ) · r ] + c . c . } .
d A s d z = α 2 A s + i β ̂ 2 0 c n 0 2 ( A s A s * + A r A r * ) A s + i β ̂ 2 0 c n 0 2 A s A r * A r ,
d A r d z = α 2 A r + i β ̂ 2 0 c n 0 2 ( A s A s * + A r A r * ) A r + i β ̂ 2 0 c n 0 2 A r A s * A s ,
d A s d z = α 2 A s i π n 1 λ cos ( θ / 2 ) exp ( i ϕ ) A s A r * A r | A s | 2 + | A r | 2 ,
d A r d z = α 2 A r i π n 1 λ cos ( θ / 2 ) exp ( i ϕ ) A r A s * A s | A s | 2 + | A r | 2 .
d I s d z = α I s β ( I s + 2 I r ) I s ,
d I r d z = α I r β ( I r + 2 I s ) I r
d ϕ s d z = β 2 ( I s + 2 I r ) ,
d ϕ r d z = β 2 ( I r + 2 I s ) ,
I s = I r = I 0 exp ( α z ) 1 + ( 3 β I 0 / α ) [ 1 exp ( α z ) ] ,
ϕ s = ( β / 2 β ) ln { 1 + ( 3 β I 0 / α ) [ 1 exp ( α z ) ] } + ϕ s 0 ,
ϕ r = ( β / 2 β ) ln { 1 + ( 3 β I 0 / α ) [ 1 exp ( α z ) ] } + ϕ r 0 .
m 2 ln m 1 m = m 0 2 ln m 0 1 m 0 + β I s 0 α ( m 0 1 ) 3 m 0 [ 1 exp ( α z ) ] ,
I s = I s 0 ( m m 0 ) ( m 0 1 m 1 ) 3 exp ( α z ) ,
I r = I r 0 ( m m 0 ) 2 ( m 0 1 m 1 ) 3 exp ( α z )
ϕ s = β 2 β ln [ ( m 1 m 0 1 ) 3 ( m 0 m ) ] + ϕ s 0 ,
ϕ r = β 2 β ln [ ( m 1 m 0 1 ) 3 ( m 0 m ) 2 ] + ϕ r 0 ,
ψ ( y , z ) = 2 k ( sin θ ) y + ( β / 2 β ) ln [ m 0 / m ( z ) ] + ϕ r 0 ϕ s 0 .
d y d z = ψ q ( y , z ) / z ψ q ( y , z ) / y = β I s 0 4 k sin θ ( m 0 1 ) 3 [ m ( z ) 1 ] 2 m ( z ) m 0 exp ( α z ) .
I s = I s 0 exp ( α z ) ,
I r = I r 0 exp ( α z ) ,
ϕ s = ϕ s 0 ,
ϕ r = ϕ r 0 ,
I ( y , z ) = I s ( z ) + I r ( z ) + 2 [ I s ( z ) I r ( z ) ] 1 / 2 cos [ ψ ( y , z ) ] ,
d A s d z = [ α s 2 i β ̂ s 2 ( I s + I r ) ] A s + i β ̂ s 2 ( I s I r ) 1 / 2 A r exp { i [ ( δ k ) z δ ϕ ] } ,
d A r d z = [ α r 2 i β ̂ r 2 ( I s + I r ) ] A r + i β ̂ r 2 ( I s I r ) 1 / 2 A s exp { i [ ( δ k ) z δ ϕ ] } ,
d A s d z = [ α s 2 i β ̂ s 2 ( I s + I r ) ] A s + i β ̂ s 2 ( I s I r ) 1 / 2 ( s ̂ · r ̂ ) A r exp { i [ ( δ k ) z δ ϕ ] } ,
d A r d z = [ α r 2 i β ̂ r 2 ( I s + I r ) ] A r + i β ̂ r 2 ( I s I r ) 1 / 2 ( s ̂ · r ̂ ) A s exp { i [ ( δ k ) z δ ϕ ] } ,
δ k = k ( cos θ r cos θ s ) ,
2 sin θ = sin θ r + sin θ s ,
A s ( z ) = A r ( 0 ) exp ( α 2 z ) × exp ( i β ̂ 3 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) × sinh ( i β ̂ 6 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) ,
A r ( z ) = A r ( 0 ) exp ( α 2 z ) × exp ( i β ̂ 3 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) × cosh ( i β ̂ 6 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) ,
A s ( z ) = A r ( 0 ) exp ( α 2 z ) × exp ( i β ̂ 3 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) × sinh ( i β ̂ 6 β ( cos 2 θ ) × ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) ,
A r ( z ) = A r ( 0 ) exp ( α 2 z ) × exp ( i β ̂ 3 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) × cosh ( i β ̂ 6 β ( cos 2 θ ) × ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) .
η = | A s ( z ) A r ( 0 ) | 2 = exp ( α z ) { 1 + ( 3 β I 0 / α ) [ 1 exp ( α z ) ] } 2 / 3 × [ sinh 2 ( 1 6 ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) + sin 2 ( β 6 β ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) ] ,
η = | A s ( z ) A r ( 0 ) | 2 = exp ( α z ) { 1 + ( 3 β I 0 / α ) [ 1 exp ( α z ) ] } 2 β / 3 β × [ sinh 2 ( β 6 β ( cos 2 θ ) ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) + sin 2 ( β 6 β ( cos 2 θ ) ln { 1 + 3 β I 0 α [ 1 exp ( α z ) ] } ) ] .
A s ( z ) = A r ( 0 ) exp ( α 2 z ) exp { i β ̂ 2 β ln [ m 0 m ( m 1 m 0 1 ) 2 ] } × sinh [ i β ̂ 2 β ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] ,
A r ( z ) = A r ( 0 ) exp ( α 2 z ) exp { i β ̂ 2 β ln [ m 0 m ( m 1 m 0 1 ) 2 ] } × cosh [ i β ̂ 2 β ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] ,
A s ( z ) = A r ( 0 ) exp ( α 2 z ) × exp { i β ̂ 2 β ln [ m 0 m ( m 1 m 0 1 ) 2 ] } × sinh [ i β ̂ 2 β ( cos 2 θ ) ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] ,
A r ( z ) = A r ( 0 ) exp ( α 2 z ) × exp { i β ̂ 2 β ln [ m 0 m ( m 1 m 0 1 ) 2 ] } × cosh [ i β ̂ 2 β ( cos 2 θ ) ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] ,
η = | A s ( z ) A r ( 0 ) | 2 = exp ( α z ) m m 0 ( m 0 1 m 1 ) 2 × { sinh 2 [ 1 2 ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] + sin 2 [ β 2 β ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] } ,
η = | A s ( z ) A r ( 0 ) | 2 = exp ( α z ) [ m m 0 ( m 0 1 m 1 ) 2 ] β / β × { sinh 2 [ 1 2 β β ( cos 2 θ ) ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] + sin 2 [ β 2 β ( cos 2 θ ) ln ( m 1 m + 1 m 0 + 1 m 0 1 ) ] } .

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