Abstract

Stimulated photorefractive phase conjugators often exhibit well-defined curved beam paths that cannot be explained by simple beam fanning. We propose a model of these apparently curved paths as a series of straight-line segments, with beams propagating in both directions along these segments. These line segments spring from the amplification of scattered light between regions of the crystal already containing counterpropagating pump beams. As these line segments form, they create new interaction regions that generate new segments, thereby making the final beam path appear to be curved. Application of our model to a single-interaction-region mutually pumped phase conjugator shows that the threshold coupling strength required for the appearance of these new segments is only slightly higher than the threshold for the phase conjugator itself.

© 1992 Optical Society of America

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References

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  1. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [CrossRef] [PubMed]
  2. S. Weiss, S. Sternklar, B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
    [CrossRef] [PubMed]
  3. M. D. Ewbank, “Mechanism for photorefractive phase conjugation using incoherent beams,” Opt. Lett. 13, 47–49 (1988).
    [CrossRef] [PubMed]
  4. M. D. Ewbank, R. A. Vasquez, R. R. Neurgaonkar, J. Feinberg, “Mutually pumped phase conjugation in photorefractive strontium barium niobate: theory and experiment,” J. Opt. Soc. Am. B 7, 2306–2316 (1990).
    [CrossRef]
  5. D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
    [CrossRef]
  6. R. W. Eason, A. M. C. Smout, “Bistability and noncommutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3,” Opt. Lett. 12, 51–53 (1987).
    [CrossRef] [PubMed]
  7. M. Cronin-Golomb, “Almost all transmission grating self-pumped phase-conjugate mirrors are equivalent,” Opt. Lett. 15, 897–899 (1990).
    [CrossRef] [PubMed]
  8. J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); erratum 6, 257 (1981).
    [CrossRef] [PubMed]
  9. M. D. Ewbank, P. Yeh, M. Khoshnevisan, J. Feinberg, “Time-reversal by an interferometer with coupled phase-conjugate reflectors,” Opt. Lett. 10, 282–284 (1985).
    [CrossRef] [PubMed]
  10. M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–29 (1984).
    [CrossRef]
  11. G. Hussain, S. W. James, R. W. Eason, “Observation and modeling of dynamic instabilities in the mutually pumped bird-wing phase conjugator in BaTiO3,” J. Opt. Soc. Am. B 7, 2294–2298 (1990).
    [CrossRef]
  12. D. J. Gauthier, P. Narum, R. W. Boyd, “Observation of deterministic chaos in a phase conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643, (1987).
    [CrossRef] [PubMed]
  13. A. V. Nowak, T. R. Moore, R. A. Fisher, “Observation of internal beam production in a barium titanate phase conjugator,” J. Opt. Soc. Am. B 5, 1864–1878 (1988).
    [CrossRef]
  14. J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
    [CrossRef]
  15. S. K. Kwong, A. Yariv, M. Cronin-Golomb, B. Fischer, “Phase of phase conjugation and its effect in the double phase-conjugate resonator,” J. Opt. Soc. Am. A 3, 157–160 (1986).
    [CrossRef]
  16. A. V. Mamaev, A. A. Zozulya, “Dynamics and stationary states of a photorefractive phase-conjugate semilinear mirror,” Opt. Commun. 79, 373–376 (1990).
    [CrossRef]
  17. V. T. Tikhonchuk, M. G. Zhanuzakov, A. A. Zozulya, “Stationary states of two coupled double phase-conjugate mirrors,” Opt. Lett. 16, 288–290 (1991).
    [CrossRef] [PubMed]
  18. These equations are the same as those used in Ref. 10 but with the subsitution γ→ −γ.

1991 (1)

1990 (4)

1989 (1)

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

1988 (2)

1987 (3)

1986 (1)

1985 (1)

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–29 (1984).
[CrossRef]

1982 (1)

1980 (2)

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[CrossRef]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); erratum 6, 257 (1981).
[CrossRef] [PubMed]

Boyd, R. W.

D. J. Gauthier, P. Narum, R. W. Boyd, “Observation of deterministic chaos in a phase conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643, (1987).
[CrossRef] [PubMed]

Cronin-Golomb, M.

Eason, R. W.

Ewbank, M. D.

Feinberg, J.

Fischer, B.

Fisher, R. A.

Gauthier, D. J.

D. J. Gauthier, P. Narum, R. W. Boyd, “Observation of deterministic chaos in a phase conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643, (1987).
[CrossRef] [PubMed]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[CrossRef]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[CrossRef]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); erratum 6, 257 (1981).
[CrossRef] [PubMed]

Hussain, G.

James, S. W.

Khoshnevisan, M.

Kwong, S. K.

Mamaev, A. V.

A. V. Mamaev, A. A. Zozulya, “Dynamics and stationary states of a photorefractive phase-conjugate semilinear mirror,” Opt. Commun. 79, 373–376 (1990).
[CrossRef]

Moore, T. R.

Narum, P.

D. J. Gauthier, P. Narum, R. W. Boyd, “Observation of deterministic chaos in a phase conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643, (1987).
[CrossRef] [PubMed]

Neurgaonkar, R. R.

Nowak, A. V.

Smout, A. M. C.

Sternklar, S.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[CrossRef]

Tikhonchuk, V. T.

Vasquez, R. A.

Wang, D.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Weiss, S.

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–29 (1984).
[CrossRef]

Yariv, A.

S. K. Kwong, A. Yariv, M. Cronin-Golomb, B. Fischer, “Phase of phase conjugation and its effect in the double phase-conjugate resonator,” J. Opt. Soc. Am. A 3, 157–160 (1986).
[CrossRef]

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–29 (1984).
[CrossRef]

Ye, P.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Yeh, P.

Zhang, S.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Zhang, Z.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Zhanuzakov, M. G.

Zhu, Y.

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Zozulya, A. A.

V. T. Tikhonchuk, M. G. Zhanuzakov, A. A. Zozulya, “Stationary states of two coupled double phase-conjugate mirrors,” Opt. Lett. 16, 288–290 (1991).
[CrossRef] [PubMed]

A. V. Mamaev, A. A. Zozulya, “Dynamics and stationary states of a photorefractive phase-conjugate semilinear mirror,” Opt. Commun. 79, 373–376 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–29 (1984).
[CrossRef]

J. Appl. Phys. (1)

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); erratum 52, 537 (1981).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

A. V. Mamaev, A. A. Zozulya, “Dynamics and stationary states of a photorefractive phase-conjugate semilinear mirror,” Opt. Commun. 79, 373–376 (1990).
[CrossRef]

D. Wang, Z. Zhang, Y. Zhu, S. Zhang, P. Ye, “Observations on the coupling channel of two mutually incoherent beams without internal reflection in BaTiO3,” Opt. Commun. 73, 495–500 (1989).
[CrossRef]

Opt. Lett. (8)

Phys. Rev. Lett. (1)

D. J. Gauthier, P. Narum, R. W. Boyd, “Observation of deterministic chaos in a phase conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643, (1987).
[CrossRef] [PubMed]

Other (1)

These equations are the same as those used in Ref. 10 but with the subsitution γ→ −γ.

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Figures (11)

Fig. 1
Fig. 1

Photomicrograph of stimulated beams inside a BaTiO3 cat conjugator, showing the segmented bent trajectories of the light beams.

Fig. 2
Fig. 2

a, Light beams (stippled arrows) springing up between a pumped crystal and the normal to a mirror. b, Light beams springing up between two pumped crystals. c, Light beam springing up between separate pumped regions of a single crystal, with the pumping beams connected by a photorefractive grating inside the crystal.

Fig. 3
Fig. 3

Schematic of beam paths in a mutually pumped phase conjugator, a, before bifurcation, b, after one bifurcation, and, c, after a second bifurcation. The circles indicate interaction regions.

Fig. 4
Fig. 4

Coupling strength at threshold versus incident beam ratio q for a mutually pumped phase conjugator with no bifurcations (solid curve; see Fig. 3a) and one bifurcation (dotted curve; see Fig. 3b).

Fig. 5
Fig. 5

Assignment of the interacting waves in a four-wave mixing region.

Fig. 6
Fig. 6

Calculated transmission (throughput) of a mutually pumped phase conjugator with no bifurcation (i.e., the one-region geometry of Fig. 3a) and with one bifurcation (i.e., the three-region geometry of Fig. 3b) versus coupling coefficient G, for equal-intensity incident beams put into the device (q = 1). The bifurcated data correspond to seeding values of of 0, 10−4, and 10−3. The inset shows the region near threshold.

Fig. 7
Fig. 7

Diffraction efficiency of the grating in the left-hand (or the right-hand) interaction region (Fig. 3b) versus coupling strength G for q = 1. The curves correspond to seeding values of of 0, 10−4, and 10−3.

Fig. 8
Fig. 8

Appearance of the beams of Fig. 3b for a, small coupling strength G and b, large G.

Fig. 9
Fig. 9

Transmission in the one-region geometry of Fig. 3a, and in the three-region geometry of Fig. 3b versus coupling coefficient Γ for unequal incident beam intensities (q = 2). The curves for the geometry of Fig. 3b correspond to seed values of of 0, 10−4, and 10−3. The inset shows the region near threshold.

Fig. 10
Fig. 10

Diffraction efficiency of the gratings in the left- and the right-hand interaction regions versus Γ for q = 2. Here the stronger beam is incident upon the crystal from the right and the weaker beam is incident from the left. The curves correspond to seeding values of = 0 and = 10−3.

Fig. 11
Fig. 11

Diagram of the three-interaction-region geometry, with the beams used in the analysis labeled. All beams in the right-hand region are primed. Note that the left- and the right-hand regions are laid out as mirror images.

Equations (19)

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T 0 = c 2 ( q 1 / 2 + q - 1 / 2 ) 2 - ( q 1 / 2 - q - 1 / 2 ) 2 4 ,
c = tanh ( G 0 c / 2 ) ,
R right R left 1.
R left = I L I L C [ 1 - exp ( - G left ) ] 2 [ 1 + ( I L / I L C ) exp ( - G left ) ] 2 , R right = I R I R C [ 1 - exp ( - G right ) ] 2 [ 1 + ( I R / I R C ) exp ( - G right ) ] 2 .
d A 1 d x = ν A 4 ,             d A 2 * d x = ν A 3 * , d A 3 d x = - ν A 2 ,             d A 4 * d x = - ν A 1 * , ν = ( γ / I 0 ) ( A 1 A 4 * + A 2 * A 3 ) .
T * = 2 G exp ( - G ) + [ ( 1 - T * ) T * ] 1 / 2 exp ( G ) .
d A 1 d x = ν A 4 ,             d A 2 * d x = ν A 3 * , d A 3 d x = - ν A 2 ,             d A 4 * d x = - ν A 1 * , ν = ( γ / I 0 ) ( A 1 A 4 * + A 2 * A 3 ) .
A 1 ( 0 ) , A 4 ( 0 ) , A 2 ( l ) ,
A 3 ( l ) = 1 / 2 A 2 ( l ) .
z = 0 x d x γ I 0 ( A 1 A 4 + A 2 A 3 ) = 0 x ν d x .
A 1 ( l ) = A 1 ( 0 ) cos ( z l ) + A 4 ( 0 ) sin ( z l ) , A 4 ( l ) = - A 1 ( 0 ) sin ( z l ) + A 4 ( 0 ) cos ( z l ) , A 2 ( 0 ) = A 2 ( l ) [ cos ( z l ) - 1 / 2 sin ( z l ) ] , A 3 ( 0 ) = A 2 ( l ) [ sin ( z l ) - 1 / 2 cos ( z l ) ] ,
tanh ( G a ) = b ,
G γ l , a Δ 2 ( I 1 + I 2 + I 4 ) , b Δ I 4 - I 1 + I 2 + 2 cot ( z l ) [ A 1 ( 0 ) A 4 ( 0 ) + 1 / 2 I 2 ] , Δ 2 ( I 1 + I 4 ) 2 + I 2 2 + 2 I 2 ( I 1 - I 4 ) cos ( 2 z l ) + 4 I 2 A 1 ( 0 ) A 4 ( 0 ) sin ( 2 z l ) + 4 I 2 1 / 2 [ ( I 4 - I 1 ) sin ( 2 z l ) + 2 A 1 ( 0 ) A 4 ( 0 ) cos ( 2 z l ) ] .
A 2 ( l ) = I L 1 / 2 , A 1 ( 0 ) = T 0 1 / 2 A 2 ( 0 ) , A 4 ( 0 ) = A 3 ( 0 ) ;
A 2 ( l ) = I R 1 / 2 , A 1 ( 0 ) = T 0 1 / 2 A 2 ( 0 ) , A 4 ( 0 ) = A 3 ( 0 ) .
A 2 ( l ) = I L 1 / 2 , A 1 ( 0 ) = ( I R T 0 ) 1 / 2 [ cos ( z 2 ) - 1 / 2 sin ( z 2 ) ] , A 4 ( l ) = I R 1 / 2 [ sin ( z 2 ) + 1 / 2 cos ( z 2 ) ] ,
A 2 ( l ) = I R 1 / 2 , A 1 ( 0 ) = ( I L T 0 ) 1 / 2 [ sin ( z l ) - 1 / 2 cos ( z 1 ) ] , A 4 ( l ) = I L 1 / 2 [ sin ( z 1 ) + 1 / 2 cos ( z 1 ) ] ,
A 2 ( l ) = I 1 / 2 , A 1 ( 0 ) = ( T 0 I ) 1 / 2 [ cos ( z ) - 1 / 2 sin ( z ) ] , A 4 ( 0 ) = I 1 / 2 [ sin ( z ) + 1 / 2 cos ( z ) ] .
T = [ A 1 ( 0 ) cos ( z ) + A 4 ( 0 ) sin ( z ) ] 2 I = [ sin 2 ( z ) + T 0 1 / 2 cos 2 ( z ) + 1 / 2 ( 1 - T 0 1 / 2 ) sin ( z ) cos ( z ) ] 2 .

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