Abstract

We show that the inclusion of the longitudinal component of the induced space-charge electrostatic field during two-beam coupling in a diffusion-dominated photorefractive material such as barium titanate with mismatched boundary conditions can predict enhancement of the higher-order diffraction efficiency, thereby reducing the minimum initial two-beam coupling ratio required for higher-order generation. We perform the analysis using the exact Kukhtarev equations and a rigorous coupled-wave diffraction theory. Expressions for the induced optical permittivity tensor including the presence of the longitudinal field are provided.

© 1996 Optical Society of America

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References

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  1. P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach,” Opt. Eng. 34, 2254–2260(1995).
    [CrossRef]
  2. J. Jarem and P. P. Banerjee, “A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory,” Opt. Commun. 123, 825–842(1996).
    [CrossRef]
  3. J. M. Jarem and P. P. Banerjee, “Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831(1996).
    [CrossRef]
  4. L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials,” Opt. Commun. 51, 76–78(1984).
    [CrossRef]
  5. M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204–2217(1991).
    [CrossRef]
  6. P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
    [CrossRef]
  7. J. C. Kralik and M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405(1994).
    [CrossRef]
  8. P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
    [CrossRef]
  9. B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive materials,” Opt. Lett. 11, 236–238(1986).
    [CrossRef]
  10. M. Cronin-Golomb and A. Yariv, “Plane-wave theory of nondegenerate oscillation in the linear photorefractive passive phase conjugator,” Opt. Lett. 11, 242–244(1986).
    [CrossRef]
  11. D. Wang, Z. Zhang, X. Wu, and P. Yeh, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293(1990).
    [CrossRef]
  12. P. M. Jeffrey and R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped phase-conjugate mirror,” J. Opt. Soc. Am. B 11, 476–480(1994).
    [CrossRef]
  13. W. P. Brown and G. C. Valley, “Kinky beam paths through photorefractive materials,” J. Opt. Soc. Am. B 10, 1901–1906(1993).
    [CrossRef]
  14. M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982(1994).
    [CrossRef]
  15. E. Serrano, V. Lopez, M. Carracosa, and F. Agullo-Lopez, “Recording and erasure kinetics in photorefractive materials at large modulation depths,” J. Opt. Soc. Am. B 11, 670–675(1994).
    [CrossRef]
  16. L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168(1987).
    [CrossRef]
  17. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  18. N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
    [CrossRef]
  19. P. P. Banerjee and J. J. Liu, “Perturbational analysis of steady-state and transient beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423(1993), and references therein.
    [CrossRef]
  20. S. Sochava, K. Buse, and E. Kratzig, “Non-steady state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268(1993).
    [CrossRef]
  21. D. Mahgerefteh and P. Tayebati, “Theory of the photorefractive effect for Bi12SiO20/sub> and BaTiO3/sub> with shallow traps,” J. Opt. Soc. Am. B 8, 1056–1063(1991).
  22. R. S. Cudney, R. M. Pierce, G. D. Bacher, D. Mahgerefteh, and J. Feinberg, “Intensity dependence of the photogalvanic effect in barium titanate,” J. Opt. Soc. Am. B 9, 1704–1713(1992).
    [CrossRef]

1996 (2)

J. Jarem and P. P. Banerjee, “A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory,” Opt. Commun. 123, 825–842(1996).
[CrossRef]

J. M. Jarem and P. P. Banerjee, “Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831(1996).
[CrossRef]

1995 (1)

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach,” Opt. Eng. 34, 2254–2260(1995).
[CrossRef]

1994 (4)

P. M. Jeffrey and R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped phase-conjugate mirror,” J. Opt. Soc. Am. B 11, 476–480(1994).
[CrossRef]

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982(1994).
[CrossRef]

E. Serrano, V. Lopez, M. Carracosa, and F. Agullo-Lopez, “Recording and erasure kinetics in photorefractive materials at large modulation depths,” J. Opt. Soc. Am. B 11, 670–675(1994).
[CrossRef]

J. C. Kralik and M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405(1994).
[CrossRef]

1993 (3)

1992 (1)

1991 (3)

D. Mahgerefteh and P. Tayebati, “Theory of the photorefractive effect for Bi12SiO20/sub> and BaTiO3/sub> with shallow traps,” J. Opt. Soc. Am. B 8, 1056–1063(1991).

M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204–2217(1991).
[CrossRef]

P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
[CrossRef]

1990 (1)

1987 (1)

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168(1987).
[CrossRef]

1986 (2)

1985 (1)

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

1984 (1)

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials,” Opt. Commun. 51, 76–78(1984).
[CrossRef]

1979 (1)

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Agullo-Lopez, F.

Albers, J.

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

Au, L. B.

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168(1987).
[CrossRef]

Bacher, G. D.

Banerjee, P. P.

J. Jarem and P. P. Banerjee, “A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory,” Opt. Commun. 123, 825–842(1996).
[CrossRef]

J. M. Jarem and P. P. Banerjee, “Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831(1996).
[CrossRef]

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach,” Opt. Eng. 34, 2254–2260(1995).
[CrossRef]

P. P. Banerjee and J. J. Liu, “Perturbational analysis of steady-state and transient beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423(1993), and references therein.
[CrossRef]

P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
[CrossRef]

Brown, W. P.

Buse, K.

S. Sochava, K. Buse, and E. Kratzig, “Non-steady state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268(1993).
[CrossRef]

Carracosa, M.

Cronin-Golomb, M.

Cudney, R. S.

Eason, R. W.

Feinberg, J.

Fischer, B.

Gunter, P.

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

Heaton, J. M.

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials,” Opt. Commun. 51, 76–78(1984).
[CrossRef]

Horowitz, M.

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982(1994).
[CrossRef]

M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204–2217(1991).
[CrossRef]

Jarem, J.

J. Jarem and P. P. Banerjee, “A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory,” Opt. Commun. 123, 825–842(1996).
[CrossRef]

Jarem, J. M.

J. M. Jarem and P. P. Banerjee, “Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831(1996).
[CrossRef]

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach,” Opt. Eng. 34, 2254–2260(1995).
[CrossRef]

Jeffrey, P. M.

Kligler, D.

Kralik, J. C.

J. C. Kralik and M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405(1994).
[CrossRef]

Kratzig, E.

S. Sochava, K. Buse, and E. Kratzig, “Non-steady state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268(1993).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Liu, J. J.

P. P. Banerjee and J. J. Liu, “Perturbational analysis of steady-state and transient beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423(1993), and references therein.
[CrossRef]

P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
[CrossRef]

Lopez, V.

Mahgerefteh, D.

R. S. Cudney, R. M. Pierce, G. D. Bacher, D. Mahgerefteh, and J. Feinberg, “Intensity dependence of the photogalvanic effect in barium titanate,” J. Opt. Soc. Am. B 9, 1704–1713(1992).
[CrossRef]

D. Mahgerefteh and P. Tayebati, “Theory of the photorefractive effect for Bi12SiO20/sub> and BaTiO3/sub> with shallow traps,” J. Opt. Soc. Am. B 8, 1056–1063(1991).

Malcuit, M. S.

J. C. Kralik and M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405(1994).
[CrossRef]

Markov, M. B.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Pierce, R. M.

Serrano, E.

Snowbell, M.

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982(1994).
[CrossRef]

Sochava, S.

S. Sochava, K. Buse, and E. Kratzig, “Non-steady state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268(1993).
[CrossRef]

Solymar, L.

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168(1987).
[CrossRef]

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials,” Opt. Commun. 51, 76–78(1984).
[CrossRef]

Song, Q. W.

P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Tayebati, P.

D. Mahgerefteh and P. Tayebati, “Theory of the photorefractive effect for Bi12SiO20/sub> and BaTiO3/sub> with shallow traps,” J. Opt. Soc. Am. B 8, 1056–1063(1991).

Valley, G. C.

Vinetsky, V. L.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

Voit, E.

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

Wang, D.

Wu, X.

Yariv, A.

Yeh, P.

Zha, M. Z.

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

Zhang, Z.

Ferroelectrics (1)

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964(1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168(1987).
[CrossRef]

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

J. Opt. Soc. Am. B (9)

M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204–2217(1991).
[CrossRef]

P. P. Banerjee and J. J. Liu, “Perturbational analysis of steady-state and transient beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423(1993), and references therein.
[CrossRef]

D. Wang, Z. Zhang, X. Wu, and P. Yeh, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293(1990).
[CrossRef]

P. M. Jeffrey and R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped phase-conjugate mirror,” J. Opt. Soc. Am. B 11, 476–480(1994).
[CrossRef]

W. P. Brown and G. C. Valley, “Kinky beam paths through photorefractive materials,” J. Opt. Soc. Am. B 10, 1901–1906(1993).
[CrossRef]

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982(1994).
[CrossRef]

E. Serrano, V. Lopez, M. Carracosa, and F. Agullo-Lopez, “Recording and erasure kinetics in photorefractive materials at large modulation depths,” J. Opt. Soc. Am. B 11, 670–675(1994).
[CrossRef]

D. Mahgerefteh and P. Tayebati, “Theory of the photorefractive effect for Bi12SiO20/sub> and BaTiO3/sub> with shallow traps,” J. Opt. Soc. Am. B 8, 1056–1063(1991).

R. S. Cudney, R. M. Pierce, G. D. Bacher, D. Mahgerefteh, and J. Feinberg, “Intensity dependence of the photogalvanic effect in barium titanate,” J. Opt. Soc. Am. B 9, 1704–1713(1992).
[CrossRef]

Opt. Commun. (6)

S. Sochava, K. Buse, and E. Kratzig, “Non-steady state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268(1993).
[CrossRef]

P. P. Banerjee, Q. W. Song, and J. J. Liu, “Perturbation analysis of transient diffraction efficiency of photorefractive materials,” Opt. Commun. 83, 195–202(1991).
[CrossRef]

J. C. Kralik and M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405(1994).
[CrossRef]

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214(1985).
[CrossRef]

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials,” Opt. Commun. 51, 76–78(1984).
[CrossRef]

J. Jarem and P. P. Banerjee, “A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory,” Opt. Commun. 123, 825–842(1996).
[CrossRef]

Opt. Eng. (1)

P. P. Banerjee and J. M. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach,” Opt. Eng. 34, 2254–2260(1995).
[CrossRef]

Opt. Lett. (2)

Other (1)

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

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Figures (3)

Fig. 1
Fig. 1

Diffracted power in the first order (normalized to total incident power) when the longitudinal electrostatic field is both zero and nonzero. The time step is 2 ms. The number of layers used is NL=160. Inset: the basic geometry of the PR crystal, showing our coordinate system, the crystal’s c axis, the two incident plane waves in a two-beam coupling geometry, and transmitted (T0,1,2) and reflected (R0,1,2) orders.

Fig. 2
Fig. 2

Diffracted power in the second order (normalized to total incident power) when the longitudinal electrostatic field is both zero and nonzero. The time step is 2 ms. The number of layers used is NL=160. The inset of Fig. 1 shows the basis geometry of the PR crystal.

Fig. 3
Fig. 3

Dielectric modulation functions Δx and Δy (proportional to Ex and Ey, respectively) at a time equal to 224 ms. The number of layers used is NL=640.

Equations (21)

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

Esx/t+(eμ/s)nEsx=-(eDs/s)n/x,
Esy/t+(eμ/s)nEsy=-(eDs/s)n/y,
Δx=no2ne2r42Esx, Δy=no2ne2r42Esy;
x¯=k0x, y¯=k0y, n¯=n/NA, t¯=βt, C=β/s,
Δx/t¯+Γ1n¯Δx=-Γ2n¯/x¯,
Δy/t¯+Γ1n¯Δy=-Γ2n¯/y¯,
Γ1=μeNA/βs, Γ2=ek0NADsno2ne2r42/βs.
Γ22n¯/x¯2+Γ1Δxn¯/x¯+[-Γ3(1+I/C)+(Γ1-Γ4)(Δx/x¯+Δy/y¯)-Γ3Γ4(1+n¯)]n¯=Γ3{n¯/t¯-(1+I/C)[ND/NA-1-(1/Γ3)×(Δx/x¯+Δy/y¯)]}-Γ22n¯/y¯2-Γ1Δyn¯/y¯,
Γ3=eNAno2ne2r42/sk0, Γ4=γRNA/β.
xx=nCO2 cos2 θc+nCE2 sin2 θc+Δx(x,y,t)Fxx(x)+Δy(x,y,t)Fxx(y),
xy=(nCO2-nCE2)sin θc cos θc+Δx(x,y,t)Fxy(x)+Δy(x,y,t)Fxy(y)=yx,
yy=nCO2 sin2 θc+nCE2 cos2 θc+Δx(x,y,t)Fyy(x)+Δy(x,y,t)Fyy(y),
Fxx(x)=-FOE[(r13nCO2/r42nCE2)sin θc cos2 θc+2 sin θc cos2 θc+(r33nCE2/r42nCO2)sin3 θc],
Fxy(x)=-FOE[(r13nCO2/r42nCE2)sin2 θc cos θc-cos θc cos 2θc-(r33nCE2/r42nCO2)sin2 θc cos θc],
Fyy(x)=-FOE[(r13nCO2/r42nCE2)sin3 θc-2 sin θc cos2 θc+(r33nCE2/r42nCO2)cos2 θc sin θc].
Fxx(y)=-FOE[-(r13nCO2/r42nCE2)cos3 θc+2 sin2 θc cos θc-(r33nCE2/r42nCO2)sin2 θc cos θc],
Fxy(y)=-FOE[(-r13nCO2/r42nCE2)sin θc cos2 θc-sin θc cos 2θc+(r33nCE2/r42nCO2)sin θc cos2 θc],
Fyy(y)=-FOE[-(r13nCO2/r42nCE2)sin2 θc cos θc-2 sin2 θc cos θc-(r33nCE2/r42nCO2)cos3 θc],
FOE=nCO2nCE2/no2ne2, nCO(E)2=no(e)2-jo(e).
1+rR exp[γL/(sτ+1)]=0
T=2π/Ω=2τ ln(1/rR).

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