Abstract

We demonstrate that the phase conjugation of a beam during nondegenerate four-wave mixing is accompanied by a spatial shift relative to the degenerate conjugate-beam location. Experiments with a photorefractive phase-conjugate mirror reveal that the phase-conjugate beam shifts have a nonmonotonic dependence on the probe’s detuning frequency and comprise both lateral displacements of up to 218 μm and angular tilts of up to 34 arc sec. An approximate theory based on spatial dispersion coefficients is in partial agreement with the experimental results.

© 1998 Optical Society of America

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References

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  1. G. C. Papen, B. E. A. Saleh, “Lateral and focal shifts of phase-conjugated beams in photorefractive materials,” Opt. Lett. 14, 745–747 (1989).
    [CrossRef] [PubMed]
  2. F. Goos, H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 6(1), 333–346 (1947); “A new measurement of the displacement of the beam at total reflection,” Ann. Phys. (Leipzig) 6(5), 251–252 (1949).
  3. K. Artmann, “Calculation of the displacement of a totally reflected ray,” Ann. Phys. (Leipzig) 6(2), 87–102 (1948).
  4. I. Newton, Opticks (Dover, New York, 1952).
  5. H. M. Lai, F. C. Cheng, W. K. Tang, “Goos–Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  6. N. H. Tran, L. Dutriaux, Ph. Balcou, A. Le Floch, F. Bretenaker, “Angular Goos–Hänchen effect in curved dielectric microstructures,” Opt. Lett. 20, 1233–1235 (1995).
    [CrossRef] [PubMed]
  7. W. J. Tomlinson, J. P. Gordon, P. W. Smith, A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041–2051 (1982).
    [CrossRef] [PubMed]
  8. O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
    [CrossRef] [PubMed]
  9. For a review, see A. Puri, J. L. Birman, “Goos–Hänchen beam shift at total internal reflection with application to spatially dispersive media,” J. Opt. Soc. Am. A 3, 543–549 (1986).
  10. See, for example, M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1980), Chap. 1, p. 49; M. Young, Optics and Lasers, 3rd ed. (Springer-Verlag, New York, 1986), Chap. 8, pp. 178–180.
  11. A.-A. R. Al-Rashed, B. E. A. Saleh, “Modes of resonators with dispersive phase-conjugate mirrors,” Appl. Opt. 36, 3400–3412 (1997).
    [CrossRef] [PubMed]
  12. Preliminary reports of this research were presented at the 1997 Annual Meeting of the Optical Society of America in Long Beach, California; B. M. Jost, A.-A. R. Al-Rashed, B. E. A. Saleh, “Observation of the Goos–Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
  13. J.-P. Huignard, J. P. Herriau, G. Rivet, P. N. Günter, “Phase-conjugation and spatial frequency dependence of wave-front reflectivity in Bi12SiO20 crystals,” Opt. Lett. 5, 102–104 (1980).
    [CrossRef] [PubMed]
  14. J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
    [CrossRef] [PubMed]
  15. K. R. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
    [CrossRef] [PubMed]
  16. B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
    [CrossRef] [PubMed]
  17. R. A. Fisher, ed., Optical Phase Conjugation (Academic, San Diego, Calif., 1983).
  18. B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
    [CrossRef]
  19. M. Gower, D. Proch, eds., Optical Phase Conjugation (Springer-Verlag, New York, 1994).
  20. P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
    [CrossRef]
  21. A. D. Meigs, B. E. A. Saleh, “Spatial fidelity of photorefractive image correlators,” IEEE J. Quantum Electron. 30, 3025–3032 (1994);“Spatial and temporal fidelity of photorefractive image correlators,” J. Opt. Soc. Am. B 11, 1848 (1994); “Measurement of the spatio-temporal response of photorefractive conjugation,” Opt. Commun. 115, 121 (1995).
    [CrossRef]
  22. D. Magerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195–2198 (1990).
    [CrossRef]
  23. B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
    [CrossRef]
  24. G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Analysis of transient phase conjugation in photorefractive media,” J. Opt. Soc. Am. B 5, 1763–1774 (1988).
    [CrossRef]

1997 (2)

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

A.-A. R. Al-Rashed, B. E. A. Saleh, “Modes of resonators with dispersive phase-conjugate mirrors,” Appl. Opt. 36, 3400–3412 (1997).
[CrossRef] [PubMed]

1996 (1)

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

1995 (2)

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

N. H. Tran, L. Dutriaux, Ph. Balcou, A. Le Floch, F. Bretenaker, “Angular Goos–Hänchen effect in curved dielectric microstructures,” Opt. Lett. 20, 1233–1235 (1995).
[CrossRef] [PubMed]

1994 (1)

A. D. Meigs, B. E. A. Saleh, “Spatial fidelity of photorefractive image correlators,” IEEE J. Quantum Electron. 30, 3025–3032 (1994);“Spatial and temporal fidelity of photorefractive image correlators,” J. Opt. Soc. Am. B 11, 1848 (1994); “Measurement of the spatio-temporal response of photorefractive conjugation,” Opt. Commun. 115, 121 (1995).
[CrossRef]

1990 (1)

D. Magerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195–2198 (1990).
[CrossRef]

1989 (1)

1988 (1)

1986 (3)

1985 (1)

K. R. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

1982 (1)

1980 (2)

1948 (1)

K. Artmann, “Calculation of the displacement of a totally reflected ray,” Ann. Phys. (Leipzig) 6(2), 87–102 (1948).

1947 (1)

F. Goos, H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 6(1), 333–346 (1947); “A new measurement of the displacement of the beam at total reflection,” Ann. Phys. (Leipzig) 6(5), 251–252 (1949).

Al-Rashed, A.-A. R.

A.-A. R. Al-Rashed, B. E. A. Saleh, “Modes of resonators with dispersive phase-conjugate mirrors,” Appl. Opt. 36, 3400–3412 (1997).
[CrossRef] [PubMed]

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

Artmann, K.

K. Artmann, “Calculation of the displacement of a totally reflected ray,” Ann. Phys. (Leipzig) 6(2), 87–102 (1948).

Balcou, Ph.

Birman, J. L.

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1980), Chap. 1, p. 49; M. Young, Optics and Lasers, 3rd ed. (Springer-Verlag, New York, 1986), Chap. 8, pp. 178–180.

Bretenaker, F.

N. H. Tran, L. Dutriaux, Ph. Balcou, A. Le Floch, F. Bretenaker, “Angular Goos–Hänchen effect in curved dielectric microstructures,” Opt. Lett. 20, 1233–1235 (1995).
[CrossRef] [PubMed]

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

Cheng, F. C.

Dai, J.-H.

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Dutriaux, L.

Emile, O.

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

Feinberg, J.

D. Magerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195–2198 (1990).
[CrossRef]

K. R. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
[CrossRef] [PubMed]

Fischer, B.

Galstyan, T.

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

Goos, F.

F. Goos, H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 6(1), 333–346 (1947); “A new measurement of the displacement of the beam at total reflection,” Ann. Phys. (Leipzig) 6(5), 251–252 (1949).

Gordon, J. P.

Günter, P. N.

Hänchen, H.

F. Goos, H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 6(1), 333–346 (1947); “A new measurement of the displacement of the beam at total reflection,” Ann. Phys. (Leipzig) 6(5), 251–252 (1949).

Hellwarth, R. W.

Herriau, J. P.

Huignard, J.-P.

Jost, B. M.

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

Kaplan, A. E.

Lai, H. M.

Le Floch, A.

N. H. Tran, L. Dutriaux, Ph. Balcou, A. Le Floch, F. Bretenaker, “Angular Goos–Hänchen effect in curved dielectric microstructures,” Opt. Lett. 20, 1233–1235 (1995).
[CrossRef] [PubMed]

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

MacDonald, K. R.

K. R. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Magerefteh, D.

D. Magerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195–2198 (1990).
[CrossRef]

Meigs, A. D.

A. D. Meigs, B. E. A. Saleh, “Spatial fidelity of photorefractive image correlators,” IEEE J. Quantum Electron. 30, 3025–3032 (1994);“Spatial and temporal fidelity of photorefractive image correlators,” J. Opt. Soc. Am. B 11, 1848 (1994); “Measurement of the spatio-temporal response of photorefractive conjugation,” Opt. Commun. 115, 121 (1995).
[CrossRef]

Newton, I.

I. Newton, Opticks (Dover, New York, 1952).

Papen, G. C.

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[CrossRef]

Puri, A.

Rivet, G.

Saleh, B. E. A.

A.-A. R. Al-Rashed, B. E. A. Saleh, “Modes of resonators with dispersive phase-conjugate mirrors,” Appl. Opt. 36, 3400–3412 (1997).
[CrossRef] [PubMed]

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

A. D. Meigs, B. E. A. Saleh, “Spatial fidelity of photorefractive image correlators,” IEEE J. Quantum Electron. 30, 3025–3032 (1994);“Spatial and temporal fidelity of photorefractive image correlators,” J. Opt. Soc. Am. B 11, 1848 (1994); “Measurement of the spatio-temporal response of photorefractive conjugation,” Opt. Commun. 115, 121 (1995).
[CrossRef]

G. C. Papen, B. E. A. Saleh, “Lateral and focal shifts of phase-conjugated beams in photorefractive materials,” Opt. Lett. 14, 745–747 (1989).
[CrossRef] [PubMed]

G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Analysis of transient phase conjugation in photorefractive media,” J. Opt. Soc. Am. B 5, 1763–1774 (1988).
[CrossRef]

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[CrossRef]

Smith, P. W.

Tang, W. K.

Tataronis, J. A.

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Analysis of transient phase conjugation in photorefractive media,” J. Opt. Soc. Am. B 5, 1763–1774 (1988).
[CrossRef]

Tomlinson, W. J.

Tran, N. H.

Wang, P.-Y.

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1980), Chap. 1, p. 49; M. Young, Optics and Lasers, 3rd ed. (Springer-Verlag, New York, 1986), Chap. 8, pp. 178–180.

Wu, J.-L.

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Xie, P.

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[CrossRef]

Zhang, H.-J.

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Ann. Phys. (Leipzig) (2)

F. Goos, H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 6(1), 333–346 (1947); “A new measurement of the displacement of the beam at total reflection,” Ann. Phys. (Leipzig) 6(5), 251–252 (1949).

K. Artmann, “Calculation of the displacement of a totally reflected ray,” Ann. Phys. (Leipzig) 6(2), 87–102 (1948).

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

A. D. Meigs, B. E. A. Saleh, “Spatial fidelity of photorefractive image correlators,” IEEE J. Quantum Electron. 30, 3025–3032 (1994);“Spatial and temporal fidelity of photorefractive image correlators,” J. Opt. Soc. Am. B 11, 1848 (1994); “Measurement of the spatio-temporal response of photorefractive conjugation,” Opt. Commun. 115, 121 (1995).
[CrossRef]

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

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

Opt. Commun. (2)

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[CrossRef]

P. Xie, J.-L. Wu, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, “Observation of transverse spatial modulation in probe-pump configurations in BaTiO3:Ce,” Opt. Commun. 126, 255–259 (1996).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. Lett. (3)

K. R. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

D. Magerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195–2198 (1990).
[CrossRef]

O. Emile, T. Galstyan, A. Le Floch, F. Bretenaker, “Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams,” Phys. Rev. Lett. 75, 1511–1513 (1995).
[CrossRef] [PubMed]

Other (6)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, San Diego, Calif., 1983).

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[CrossRef]

M. Gower, D. Proch, eds., Optical Phase Conjugation (Springer-Verlag, New York, 1994).

I. Newton, Opticks (Dover, New York, 1952).

See, for example, M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1980), Chap. 1, p. 49; M. Young, Optics and Lasers, 3rd ed. (Springer-Verlag, New York, 1986), Chap. 8, pp. 178–180.

Preliminary reports of this research were presented at the 1997 Annual Meeting of the Optical Society of America in Long Beach, California; B. M. Jost, A.-A. R. Al-Rashed, B. E. A. Saleh, “Observation of the Goos–Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).

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Figures (6)

Fig. 1
Fig. 1

Illustration of phase conjugation by photorefractive FWM. A photorefractive crystal with the crystal (c) axis oriented at an angle θ c from the normal is pumped by counterpropagating forward and backward pumps with frequencies ω. A probe beam with amplitude A p and frequency ω ± Ω, where Ω is the detuning frequency, and angular spread Δθ interacts with the pump beams within the crystal to create a displaced (by a distance a) conjugate beam A p with frequency ω ± Ω.

Fig. 2
Fig. 2

Nondegenerate photorefractive phase-conjugate mirror experiment. See text for details.

Fig. 3
Fig. 3

Diagram showing the experimentally measured quantities. The values of the lateral displacement a and the angular tilt Φ can be geometrically determined from measurements of the conjugate beam centroid locations ζ (total beam shifts) obtained at two different values of the longitudinal camera distance d. The pumps are not shown.

Fig. 4
Fig. 4

Experimental phase-conjugate beam centroid locations ζ as a function of the probe detuning frequency Ω.

Fig. 5
Fig. 5

Decomposition of the experimental phase-conjugate beam shifts shown in Fig. 4 into lateral and angular components. (a) Experimental (×’s) and theoretical (solid curve) lateral displacement a at the crystal. The theoretical parameters are pump ratio r = 1.2, absorption coefficient α = 2 cm-1, effective electro-optic coefficient r 13 = 10 pm V-1, ordinary refractive index n 0 = 2.4921, and extraordinary index n e = 2.4247; however, the theoretical curve is scaled by a factor of 1.3 × 103. (b) Angular tilt Φ relative to the degenerate (Ω = 0) phase-conjugate beam.

Fig. 6
Fig. 6

Experimental (filled circles) and theoretical (solid curve) dependence of the probe beam detuning frequency required for the maximum phase-conjugate beam shift on the total incident irradiance I. Theoretical parameters are the same as those used in Fig. 5.

Equations (8)

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

A c x = R Ω ,   θ A p * x ,
R Ω ,   θ = | R Ω ,   θ | exp i φ R Ω ,   θ ,
φ R Ω ,   θ = φ R θ c + ka Ω θ - θ c + ,
a Ω = 1 k φ R Ω ,   θ θ θ = θ c
A c x     A p * x + a Ω ;
R Ω ,   θ = - 2 r   γ Ω ,   θ l γ Ω ,   θ l r - 1 + r + 1 α l + κ Ω ,   θ coth κ Ω ,   θ ,
κ Ω ,   θ = α l + γ Ω ,   θ l 2 r - 1 r + 1 2 + γ Ω ,   θ l 2 r r + 1 2 1 / 2 ,
γ Ω ,   θ = γ 0 θ 1 + i Ω τ g θ ,

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