Abstract

We study two-beam interaction in photorefractive sillenite crystals in reflection geometry and derive analytic expressions for the signal gain and its polarization direction in the presence of a strong pump beam. The crystal is cut normal to one of the crystallographic axes, and no external electric field is applied. Crystal absorption and its natural optical activity are taken into account. The coupling effects are strongly dependent on the polarization directions of the interacting beams. We determine the optimal polarization directions and optimal crystal thickness that give maximum signal gain. Experimental results are in excellent agreement with theoretical calculations.

© 1997 Optical Society of America

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References

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  1. P. Günter and J.-P. Huignard, Photorefractive Materials and their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
    [CrossRef]
  2. P. Yeh, “Contradirectional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
    [CrossRef]
  3. M. D. Ewbank, R. A. Vazquez, R. R. Neurgaonkar, and F. Vachss, “Contradirectional two-beam coupling in absorptive photorefractive materials: application to Rh-doped strontium barium niobate,” J. Opt. Soc. Am. B 12, 87–98 (1995).
    [CrossRef]
  4. D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
    [CrossRef]
  5. T. Y. Chang and R. W. Hellwarth, “Optical phase conjugation by backscattering in barium titanate,” Opt. Lett. 10, 408–410 (1985).
    [CrossRef] [PubMed]
  6. G. C. Valley, “Competition between forward and backward stimulated scattering in BaTiO3,” J. Opt. Soc. Am. B 4, 14–19 (1987).
    [CrossRef]
  7. A. V. Mamaev and V. V. Shkunov, “Interaction of counterpropagating waves and phase self-conjugation in a BaTiO3 crystal,” Sov. J. Quantum Electron. 19, 1199–1203 (1989).
    [CrossRef]
  8. A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
    [CrossRef]
  9. S. Mallick, D. Rouède, and A. G. Apostolidis, “Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
    [CrossRef]
  10. V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
    [CrossRef]
  11. J. R. Goff, “Polarization properties of transmission and diffraction in BSO—a unified analysis,” J. Opt. Soc. Am. B 12, 99–116 1995.
    [CrossRef]
  12. A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
    [CrossRef]
  13. S. Mallick and D. Rouède, “Influence of the polarization direction on two-beam coupling in photorefractive Bi12SiO20: diffusion regime,” Appl. Phys. B 43, 239–245 (1987).
    [CrossRef]
  14. 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]
  15. Y. H. Ja, “Beam coupling and decoupling in degenerate two-wave mixing in a reflection geometry with photorefractive Bi12GeO20 crystals,” Opt. Quantum Electron. 16, 399–404 (1984).
    [CrossRef]
  16. N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
    [CrossRef]
  17. M. Miteva and L. Nikolova, “Polarization-dependent self-induced changes in the optical rotation and optical transmittance in doped crystals of the sillenite type,” J. Mod. Opt. 43, 1801–1809 (1996).
    [CrossRef]
  18. N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
    [CrossRef]
  19. M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
    [CrossRef]

1996 (1)

M. Miteva and L. Nikolova, “Polarization-dependent self-induced changes in the optical rotation and optical transmittance in doped crystals of the sillenite type,” J. Mod. Opt. 43, 1801–1809 (1996).
[CrossRef]

1995 (2)

1993 (1)

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

1990 (1)

V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
[CrossRef]

1989 (1)

A. V. Mamaev and V. V. Shkunov, “Interaction of counterpropagating waves and phase self-conjugation in a BaTiO3 crystal,” Sov. J. Quantum Electron. 19, 1199–1203 (1989).
[CrossRef]

1988 (1)

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

1987 (4)

G. C. Valley, “Competition between forward and backward stimulated scattering in BaTiO3,” J. Opt. Soc. Am. B 4, 14–19 (1987).
[CrossRef]

S. Mallick, D. Rouède, and A. G. Apostolidis, “Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
[CrossRef]

S. Mallick and D. Rouède, “Influence of the polarization direction on two-beam coupling in photorefractive Bi12SiO20: diffusion regime,” Appl. Phys. B 43, 239–245 (1987).
[CrossRef]

1986 (1)

1985 (1)

1984 (2)

Y. H. Ja, “Beam coupling and decoupling in degenerate two-wave mixing in a reflection geometry with photorefractive Bi12GeO20 crystals,” Opt. Quantum Electron. 16, 399–404 (1984).
[CrossRef]

N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
[CrossRef]

1983 (1)

P. Yeh, “Contradirectional 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]

1981 (1)

M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
[CrossRef]

Apostolidis, A. G.

Chang, T. Y.

Chen, B.

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

Dovgalenko, G.

N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
[CrossRef]

Erbschloe, D.

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

Ewbank, M. D.

Goff, J. R.

Hellwarth, R. W.

Ja, Y. H.

Y. H. Ja, “Beam coupling and decoupling in degenerate two-wave mixing in a reflection geometry with photorefractive Bi12GeO20 crystals,” Opt. Quantum Electron. 16, 399–404 (1984).
[CrossRef]

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]

Johnson, R. V.

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
[CrossRef]

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

Klein, M.

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
[CrossRef]

Mallick, S.

S. Mallick and D. Rouède, “Influence of the polarization direction on two-beam coupling in photorefractive Bi12SiO20: diffusion regime,” Appl. Phys. B 43, 239–245 (1987).
[CrossRef]

S. Mallick, D. Rouède, and A. G. Apostolidis, “Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

Mamaev, A. V.

A. V. Mamaev and V. V. Shkunov, “Interaction of counterpropagating waves and phase self-conjugation in a BaTiO3 crystal,” Sov. J. Quantum Electron. 19, 1199–1203 (1989).
[CrossRef]

Mandel, A. E.

V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
[CrossRef]

Marrakchi, A.

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
[CrossRef]

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

Miteva, M.

M. Miteva and L. Nikolova, “Polarization-dependent self-induced changes in the optical rotation and optical transmittance in doped crystals of the sillenite type,” J. Mod. Opt. 43, 1801–1809 (1996).
[CrossRef]

Neurgaonkar, R. R.

Nikolova, L.

M. Miteva and L. Nikolova, “Polarization-dependent self-induced changes in the optical rotation and optical transmittance in doped crystals of the sillenite type,” J. Mod. Opt. 43, 1801–1809 (1996).
[CrossRef]

Pencheva, T.

M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
[CrossRef]

Petrov, M.

M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
[CrossRef]

Rouède, D.

S. Mallick and D. Rouède, “Influence of the polarization direction on two-beam coupling in photorefractive Bi12SiO20: diffusion regime,” Appl. Phys. B 43, 239–245 (1987).
[CrossRef]

S. Mallick, D. Rouède, and A. G. Apostolidis, “Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

Shandarov, S. M.

V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
[CrossRef]

Shepelevich, V. V.

V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
[CrossRef]

Shkunov, V. V.

A. V. Mamaev and V. V. Shkunov, “Interaction of counterpropagating waves and phase self-conjugation in a BaTiO3 crystal,” Sov. J. Quantum Electron. 19, 1199–1203 (1989).
[CrossRef]

Slamo, G.

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

Solymar, L.

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

Starkov, V.

N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
[CrossRef]

Stepanov, S.

M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
[CrossRef]

Takacs, J.

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

Tanguay , Jr., A. R.

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
[CrossRef]

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

Vachss, F.

Valley, G. C.

Vazquez, R. A.

Venkateswarlu, P.

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

Wilson, T.

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

Yeh, P.

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

Appl. Phys. A (1)

N. Kukhtarev, G. Dovgalenko, and V. Starkov, “Influence of the optical activity on hologram formation in photorefractive crystals,” Appl. Phys. A 33, 227–230 (1984).
[CrossRef]

Appl. Phys. B (1)

S. Mallick and D. Rouède, “Influence of the polarization direction on two-beam coupling in photorefractive Bi12SiO20: diffusion regime,” Appl. Phys. B 43, 239–245 (1987).
[CrossRef]

Ferroelectrics (1)

V. V. Shepelevich, S. M. Shandarov, and A. E. Mandel, “Light diffraction by holographic gratings in optically active photorefractive piezocrystals,” Ferroelectrics 110, 235–249 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., “Polarization properties of enhanced self-diffraction in sillenite crystals,” IEEE J. Quantum Electron. QE-23, 2142–2151 (1987).
[CrossRef]

D. Erbschloe, L. Solymar, J. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 820–826 (1988).
[CrossRef]

J. Mod. Opt. (1)

M. Miteva and L. Nikolova, “Polarization-dependent self-induced changes in the optical rotation and optical transmittance in doped crystals of the sillenite type,” J. Mod. Opt. 43, 1801–1809 (1996).
[CrossRef]

J. Opt. (Paris) (1)

M. Petrov, T. Pencheva, and S. Stepanov, “Light diffraction from volume phase holograms in electrooptic photorefractive crystals,” J. Opt. (Paris) 12, 287–292 (1981).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

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

N. Kukhtarev, B. Chen, P. Venkateswarlu, G. Slamo, and M. Klein, “Reflection holographic gratings in [111] cut Bi12TiO20 crystal for real time interferometry,” Opt. Commun. 104, 23–28 (1993).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

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]

Y. H. Ja, “Beam coupling and decoupling in degenerate two-wave mixing in a reflection geometry with photorefractive Bi12GeO20 crystals,” Opt. Quantum Electron. 16, 399–404 (1984).
[CrossRef]

Sov. J. Quantum Electron. (1)

A. V. Mamaev and V. V. Shkunov, “Interaction of counterpropagating waves and phase self-conjugation in a BaTiO3 crystal,” Sov. J. Quantum Electron. 19, 1199–1203 (1989).
[CrossRef]

Other (1)

P. Günter and J.-P. Huignard, Photorefractive Materials and their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Crystal orientation and geometry for contradirectional two-wave mixing. The pump beam R0 makes a small angle with the z axis such that the light reflected at the external face z =L does not mix with the outcoming signal. Δn=n03rEsc/2.

Fig. 2
Fig. 2

Variation of the signal gain γ as a function of the polarization direction θ of the incident signal beam for BSO crystal (ρ=-0.39 mm-1; thickness, 2.07 mm). The polarization direction of the pump as it enters the crystal at z=L is θ+ρL. The solid curve is the theoretical result obtained by the numerical integration of Eq. (12) and by use of Eqs. (13) and (14). The circles represent experimental values. The best fit between theoretical and experimental results is obtained for β =0.11mm-1.

Fig. 3
Fig. 3

Numerically calculated γθ curves for a BSO crystal (ρ=0.39 mm-1, β=0.11 mm-1) for different values of ρL (45°, 90°, 135°, 180°, 225°, 270°, and 360°).

Fig. 4
Fig. 4

γθ curve for the BTO crystal (ρ = -0.11 mm-1, L =4.2 mm). The solid curve represents the numerically calculated theoretical result; the circles indicate experimentally measured values. The best fit between theoretical and experimental results is obtained for β=0.133 mm-1.

Fig. 5
Fig. 5

Polarization direction of the amplified signal as a function of θ. Here ϕ is (the polarization direction of the signal at z=L in presence of the pump beam) minus (the polarization direction of the signal at z=L without the pump). The experimental results (circles) are in excellent agreement with the numerically calculated results (solid curves). The theoretical and experimental results in this figure are for the situation in which the pump-beam polarization at z=L is set at θ and not at θ +ρL.

Fig. 6
Fig. 6

φ-θ curves for the optimal situation in which the pump-beam polarization is set at θ+ρL.

Equations (36)

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m(z)=2SR/(R2+S2)2S/R,
dm=2dS/R-2SdR/R2.
Rx=R cos(θ+ρz),
Ry=R sin(θ+ρz).
dUx=R cos(θ+ρz)(π/λ)mΔnmaxdz=2Sβ cos(θ+ρz)dz,
dUy=-2Sβ sin(θ+ρz)dz,
dS(diffraction)=2Sβdz cos(2θ+2ρz).
dS(absorption)=S[exp(-αdz/2)-1]=-Sαdz/2,
dS=Sdz[2β cos(2θ+2ρz)-α/2].
dR/dz=d{R0 exp[-α(L-z)/2]}/dz=αR/2
dR=αRdz/2,
dm=mdz[2β cos(2θ+2ρz)-α].
m=m0 exp[-αz+(2β/ρ)cos(2θ+ρz)sin ρz].
dUx(z)=R0 exp[-α(L-z)/2]cos(θ+ρz)βmdz=2S0β cos(θ+ρz)exp[-αz/2+(2β/ρ)×cos(2θ+ρz)sin ρz]dz,
dUy(z)=-2S0β sin(θ+ρz)exp[-αz/2+(2β/ρ)×cos(2θ+ρz)sin ρz]dz.
dUx(L)=exp[-α(L-z)/2]{dUx(z)cos ρ(L-z)+dUy(z)cos[ρ(L-z)+(π/2)]}=2S0β cos(θ-ρL+2ρz)dz exp[-αL/2+(2β/ρ)cos(2θ+ρz)sin ρz],
dUy(L)=-2S0β sin(θ-ρL+2ρz)dz exp[-αL/2+(2β/ρ)cos(2θ+ρz)sin ρz].
Sx(z=L)=S0 exp(-αL/2)cos(θ+ρL)+Ux,
Sy(z=L)=S0 exp(-αL/2)sin(θ+ρL)+Uy.
γ=(Sx2+Sy2)/S02 exp(-αL).
γ=1+4(β/ρ)A sin ρL cos(2θ+ρL)×[1+(β/ρ)cos ρL sin(2θ+ρL)]+4(β/ρ)2A2 sin2(ρL),
A=exp[-(β/ρ)sin(2θ)].
4(β/ρ)A sin ρL cos(2θ+ρL).
γmax={1+(2β/ρ)sin ρL exp[(β/ρ)sin ρL]}2.
γ=1+(β/ρ)4 exp[-(2β/ρ)sin 2θ]k2π2.
dUx(L)=2S0β cos θ exp[-αL/2+2βz cos 2θ],
dUy(L)=-2S0β sin θ exp[-αL/2+2βz cos 2θ].
γ=1+(1/cos2 2θ)[exp(2βL cos 2θ)-1]2+2[exp(2βL cos 2θ)-1].
γmax=exp(4βL).
dUx(L)=2S0β cos(θ-ρL+2ρz)exp[-αL/2-(β/ρ)sin 2θ]exp[(β/ρ)sin(2θ+2ρz)]dz.
exp[(β/ρ)sin(2θ+2ρz)]
1+(β/ρ)sin(2θ+2ρz)+(β2/2ρ2)
×{0.5[1-cos(4θ+4ρz)]}.
Ux=2S0(β/ρ)exp[-αL/2-(β/ρ)sin 2θ]{[1+(β2/4ρ2)]×sin φ cos θ+(β/4ρ)[2φ sin(θ+φ)+sin 2φ sin(3θ+φ)]-(β2/8ρ2)[sin φ cos(3θ+2φ)+(1/3)sin 3φ cos(5θ+2φ)]},
Uy=-2S0(β/ρ)exp[-αL/2-(β/ρ)sin 2θ]×{[1+(β2/4ρ2)]×sin φ sin θ+(β/4ρ)[2φ cos(θ+φ)-sin 2φ cos(3θ+φ)]-(β2/8ρ2)[-sin φ×sin(3θ+2φ)+(1/3)sin 3φ sin(5θ+2φ)]}.
γ=1+4A2C2(1+C2/4)2 sin2 φ+A2C4(φ2+(sin2 2φ)/4)+(A2C6/16)[sin2 φ+(1/9)sin2 3φ]+4AC(1+C2/4)sin φ cos(2θ+φ)+AC2 sin 2φ sin(4θ+2φ)-(AC3/2)×[sin φ cos(2θ+φ)+(1/3)sin 3φ cos(6θ+3φ)]+2A2C3(1+C2/4)sin φ sin(2θ+φ)[2φ+sin 2φ]-A2C4(1+C2/4)sin φ cos(4θ+2φ)×[sin φ+(1/3)sin 3φ]-A2C4φ sin 2φ cos(4θ+2φ)+(A2C5/2)φ[sin φ sin(2θ+φ)-(1/3)sin 3φ sin(6θ+3φ)]+(A2C5/4)sin 2φ[(1/3)sin 3φ sin(2θ+φ)-sin φ sin(6θ+3φ)]+(A2C6/24)sin φ sin 3φ cos(8θ+4φ)],

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