Abstract

We model the double phase-conjugate mirror (DPCM) as a function of time, the average direction of propagation of the two beams forming the DPCM, and one transverse coordinate. Calculations show that the conjugation fidelity and reflectivity have different dependencies on the photorefractive coupling coefficient times length; the fidelity turns on abruptly with a threshold, whereas the reflectivity increases smoothly. The DPCM behaves as an oscillator at and above threshold: the time required for the reflectivity to reach the steady state dramatically slows down near threshold (like critical slowing down in lasers); above threshold the DPCM is self-sustaining even if the random noise terms used to start the process are set to zero. A decrease in the noise level improves the fidelity but increases the response time. The use of unbalanced input beam ratios results in asymmetric conjugation such that the fidelity obtained on the side of the weaker input beam is significantly reduced. The slowing down diminishes with increasing noise level or unbalanced input intensities.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987); Appl. Phys. Lett. 50, 483 (1987); Opt. Eng. 26, 423 (1987).
    [CrossRef] [PubMed]
  2. Q. B. He, J. Quantum Electron. 24, 2507 (1988).
    [CrossRef]
  3. Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
    [CrossRef]
  4. A. A. Zozulya, Opt. Lett. 16, 545 (1991); V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, J. Opt. Soc. Am. B 8, 2497 (1991); N. V. Bobodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. M. Polozkov, and A. A. Zozulya, J. Opt. Soc. Am. B 9, 1493 (1992); K. D. Shaw, Opt. Commun. 90, 133 (1992); 94, 458 (1992).
    [CrossRef] [PubMed]
  5. A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
    [CrossRef]
  6. M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
    [CrossRef] [PubMed]
  7. O. V. Lyubomudrov and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1027 (1992).
    [CrossRef]
  8. A. V. Mamaev and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1036 (1992).
    [CrossRef]
  9. M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
    [CrossRef] [PubMed]
  10. S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
    [CrossRef] [PubMed]
  11. H. Haken, Synergetics (Springer-Verlag, Berlin, 1983), Chap. 5.
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.
  13. P. Giinter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günther and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 2.
  14. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
    [CrossRef]
  15. Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 (1986).
    [CrossRef]
  16. S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
    [CrossRef] [PubMed]
  17. D. A. Fish, T. J. Hall, and A. K. Powell, Opt. Commun. 85, 85 (1991).
    [CrossRef]
  18. G. C. Valley, J. Opt. Soc. Am. B 9, 1440 (1992).
    [CrossRef]
  19. N. V. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).
  20. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  21. M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265 (1990).
    [CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
[CrossRef]

Engin, D.

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

Fainman, Y.

Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 (1986).
[CrossRef]

Fischer, B.

S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987); Appl. Phys. Lett. 50, 483 (1987); Opt. Eng. 26, 423 (1987).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
[CrossRef]

M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265 (1990).
[CrossRef]

Fish, D. A.

D. A. Fish, T. J. Hall, and A. K. Powell, Opt. Commun. 85, 85 (1991).
[CrossRef]

Fisher, B.

S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
[CrossRef] [PubMed]

Giinter, P.

P. Giinter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günther and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 2.

Gu, C.

Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

Haken, H.

H. Haken, Synergetics (Springer-Verlag, Berlin, 1983), Chap. 5.
[CrossRef]

Hall, T. J.

D. A. Fish, T. J. Hall, and A. K. Powell, Opt. Commun. 85, 85 (1991).
[CrossRef]

He, Q. B.

Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

Q. B. He, J. Quantum Electron. 24, 2507 (1988).
[CrossRef]

Huignard, J.-P.

P. Giinter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günther and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 2.

Klancnik, E.

Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 (1986).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).

Lee, S. H.

Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 (1986).
[CrossRef]

Lyubomudrov, O. V.

O. V. Lyubomudrov and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1027 (1992).
[CrossRef]

Mamaev, A. V.

A. V. Mamaev and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1036 (1992).
[CrossRef]

Mazur, A. P.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

Neurgaonkar, R. R.

Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

Novikov, A. D.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

Odulov, S. G.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

Ophir, Y.

M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265 (1990).
[CrossRef]

Orlov, S.

S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
[CrossRef] [PubMed]

Powell, A. K.

D. A. Fish, T. J. Hall, and A. K. Powell, Opt. Commun. 85, 85 (1991).
[CrossRef]

Segev, M.

M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265 (1990).
[CrossRef]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
[CrossRef] [PubMed]

S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
[CrossRef] [PubMed]

Shkunov, V. V.

A. V. Mamaev and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1036 (1992).
[CrossRef]

O. V. Lyubomudrov and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1027 (1992).
[CrossRef]

Soskin, M. S.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

Sternklar, S.

S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987); Appl. Phys. Lett. 50, 483 (1987); Opt. Eng. 26, 423 (1987).
[CrossRef] [PubMed]

S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
[CrossRef] [PubMed]

Valley, G. C.

S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
[CrossRef] [PubMed]

G. C. Valley, J. Opt. Soc. Am. B 9, 1440 (1992).
[CrossRef]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

Vasnetov, M. V.

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

Weiss, S.

S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987); Appl. Phys. Lett. 50, 483 (1987); Opt. Eng. 26, 423 (1987).
[CrossRef] [PubMed]

S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
[CrossRef] [PubMed]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.

Yariv, A.

S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
[CrossRef]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya, Opt. Lett. 16, 545 (1991); V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, J. Opt. Soc. Am. B 8, 2497 (1991); N. V. Bobodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. M. Polozkov, and A. A. Zozulya, J. Opt. Soc. Am. B 9, 1493 (1992); K. D. Shaw, Opt. Commun. 90, 133 (1992); 94, 458 (1992).
[CrossRef] [PubMed]

Other (21)

S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 114 (1987); Appl. Phys. Lett. 50, 483 (1987); Opt. Eng. 26, 423 (1987).
[CrossRef] [PubMed]

Q. B. He, J. Quantum Electron. 24, 2507 (1988).
[CrossRef]

Q. B. He, P. Yeh, C. Gu, and R. R. Neurgaonkar, J. Opt. Soc. Am. B 9, 114 (1992).
[CrossRef]

A. A. Zozulya, Opt. Lett. 16, 545 (1991); V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, J. Opt. Soc. Am. B 8, 2497 (1991); N. V. Bobodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. M. Polozkov, and A. A. Zozulya, J. Opt. Soc. Am. B 9, 1493 (1992); K. D. Shaw, Opt. Commun. 90, 133 (1992); 94, 458 (1992).
[CrossRef] [PubMed]

A. P. Mazur, A. D. Novikov, S. G. Odulov, M. S. Soskin, and M. V. Vasnetov, J. Opt. Soc. Am. B 10, 1408 (1993) (see app.).
[CrossRef]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

O. V. Lyubomudrov and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1027 (1992).
[CrossRef]

A. V. Mamaev and V. V. Shkunov, Sov. J. Quantum Electron. 22, 1036 (1992).
[CrossRef]

M. Segev, D. Engin, A. Yariv, and G. C. Valley, Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

S. Orlov, M. Segev, A. Yariv, and G. C. Valley, Opt. Lett. 19, 578 (1994).
[CrossRef] [PubMed]

H. Haken, Synergetics (Springer-Verlag, Berlin, 1983), Chap. 5.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.

P. Giinter and J.-P. Huignard, in Photorefractive Materials and Their Applications I, P. Günther and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1989), Chap. 2.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. 20, 12 (1984).
[CrossRef]

Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 (1986).
[CrossRef]

S. Sternklar, S. Weiss, M. Segev, and B. Fisher, Appl. Opt. 25, 4518 (1986).
[CrossRef] [PubMed]

D. A. Fish, T. J. Hall, and A. K. Powell, Opt. Commun. 85, 85 (1991).
[CrossRef]

G. C. Valley, J. Opt. Soc. Am. B 9, 1440 (1992).
[CrossRef]

N. V. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

M. Segev, Y. Ophir, and B. Fischer, Opt. Commun. 77, 265 (1990).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Sketch of the DPCM.

Fig. 2
Fig. 2

Steady-state far-field patterns of the two beams as a function of angle. The angles are measured from the forward c axis.

Fig. 3
Fig. 3

Phase profiles of the two beams as a function of angle (only plane waves that belong to the image portions of the beams, i.e., ΩA, ΩB, ΔA, ΔB, are shown). The left-hand and the right-hand figures illustrate the input and the output beams at z = 0.5 cm and z = 0 cm, respectively. The angles are measured from the forward c axis.

Fig. 4
Fig. 4

Conjugation reflectivity and conjugation fidelity as functions of time in units of dark decay time. The lines indicate the two response times: long dashed, fidelity; short dashed, reflectivity.

Fig. 5
Fig. 5

Conjugation fidelities for both beams and the conjugation reflectivity as functions of gain.

Fig. 6
Fig. 6

Response times of the conjugation fidelity (curve with squares) and of the conjugation reflectivity (curve with points) as functions of gain.

Fig. 7
Fig. 7

Conjugation reflectivity as a function of time for three gain values for parameters that give a threshold value of 1.25

Fig. 8
Fig. 8

Conjugation fidelity as a function of gain (for beam A only) for three different seed levels.

Fig. 9
Fig. 9

Conjugation reflectivity as a function of gain for three different seed levels.

Fig. 10
Fig. 10

Conjugation reflectivity response time as a function of gain for three different seed levels.

Fig. 11
Fig. 11

Conjugation fidelity as a function of gain (strong beam) for four different input beam intensity ratios: 1, 3, 6, 9.

Fig. 12
Fig. 12

Conjugation fidelity as a function of gain (weak beam) for three different input beam intensity ratios: 1, 3, 6.

Fig. 13
Fig. 13

Intensity transmission as a function of gain for four different input beam intensity ratios: 1, 3, 6, 9.

Fig. 14
Fig. 14

Response time of conjugation reflectivity as a function of gain for three different input beam intensity ratios: 1, 3, 6.

Fig. 15
Fig. 15

Conjugation fidelity as a function of gain (for beam A) for three different pseudorandom seed phase profiles.

Equations (33)

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

2 E - μ ɛ 0 ( n b + δ n ) 2 2 E t 2 = 0 ,
E ( x , z , t ) = 1 / 2 [ A ( x , z , t ) exp ( i k z - i ω t ) + B ( x , z , t ) exp ( i k z + i ω t ) + c . c . ] x ^ ,
A z - i 2 k 2 A x 2 = i k n b δ n A ,
B z + i 2 k 2 B x 2 = - i k n b δ n B .
A ( x , z , t ) = m a m ( z , t ) exp ( - i k ɛ m x - i k ɛ 2 m 2 z / 2 ) ,
B ( x , z , t ) = m b m ( z , t ) exp ( i k ɛ m x + i k ɛ 2 m 2 z / 2 ) ,
δ n ( x , z , t ) = m , n δ n m n ( z , t ) exp [ - i k ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] ,
τ m n δ n m n t + ( 1 + I 0 / I dark ) δ n m n ( z , t ) = i γ m n / I dark [ a m ( z , t ) a n * ( z , t ) + b n ( z , t ) b m * ( z , t ) ] ,
m a m a m * + b m b m * m , n { a m a n * exp [ - i k ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] + b m b n * × exp [ i k ( m - n ) x ɛ + i k ɛ 2 ( m 2 - n 2 ) z / 2 ] } .
a p z = i k n b m , n δ n m n a p + n - m exp ( - i k ɛ 2 Δ 1 z / 2 ) ,
b p z = - i k n b m , n δ n m n b p - n + m exp ( i k ɛ 2 Δ 2 z / 2 ) ,
2 i k m a m z exp [ - i k ( ɛ m ) 2 z / 2 ] exp ( - i k m ɛ x ) = - 2 k 2 / n b m , n , l a m δ n n l exp [ - i k ɛ 2 ( m 2 + n 2 - l 2 ) z / 2 ] × exp [ - i k ɛ ( m + n - l ) x ] .
a p ( z , t ) z = i k n b n δ n p n ( z , t ) a n ( z , t ) ,
b p ( z , t ) z = - i k n b n δ n n p ( z , t ) b n ( z , t ) .
T B ( t ) = i Δ A b i ( 0 , t ) 2 i Ω B b i ( L , 0 ) 2 ,
CR B ( t ) = i Δ B a i ( L , t ) 2 i Ω B b i ( L , 0 ) 2 ,
CF B ( t ) = ( i Ω B Δ B ) a i ( L , t ) b i ( L ) [ i Ω B a i ( L , t ) 2 i Ω B b i ( L ) 2 ] 1 / 2 ,
CF = - d 2 r A B ( - d 2 r A 2 - d 2 r B 2 ) 1 / 2 ,
N + t = ( s I + β ) ( N - N + ) - γ n N + ,
j = e μ n E + μ k B T n ,
· j e = n t - N + t ,
· E = - ( e ɛ ɛ 0 ) ( n + N A - N + ) .
I ( x , z , t ) = I 0 + m , n I m n ( z , t ) exp [ - i k ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] ,
n ( x , z , t ) = n 0 ( z , t ) + m , n n m n ( z , t ) exp [ - i k ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] ,
N + ( x , z , t ) = N 0 + ( z , t ) + m , n N m n + ( z , t ) exp [ - i h ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] ,
E ( x , z , t ) = m , n E m n ( z , t ) exp [ - i k ( m - n ) x ɛ - i k ɛ 2 ( m 2 - n 2 ) z / 2 ] ,
N m n + t = s ( N D - N A ) I dark I m n - s I dark ( 1 + I 0 ) N m n + - γ ( n 0 N m n + + N 0 + n m n ) ,
n m n t = - i k m n μ e ( e n 0 E m n + i k m n k B T n m n ) + N m n + t ,
i k m n E m n = ( e ɛ ɛ 0 ) ( n m n - N m n + ) ,
2 E m n t 2 + ( 1 τ R e + 1 τ D m n + 1 τ I e + 1 τ di e ) E m n t + [ ( 1 τ D m n + 1 τ I e ) 1 τ die + 1 τ I e τ D m n ] E m n = - i e s ( N D - N A ) I m n τ D m n k m n ɛ ,
τ m n E m n t + ( 1 + I 0 / I dark ) E m n = - i I m n / I dark E D m n 1 + ( E D m n / E q m n ) ,
δ n m n = - ½ n b 3 r m n E m n ,
γ m n = ( π / λ ) n b 3 r m n E D m n 1 + ( E D m n / E q m n ) .

Metrics