Abstract

Development of scattered radiation in the geometry of a double phase-conjugate mirror is investigated numerically in the framework of a two-dimensional model that accounts for both diffraction and noncollinearity of the interacting beams. The large-scale structure of the scattered beams is found to be distorted because of the convective flow of energy out of the interaction region. We show that the output characteristics of the double phase-conjugate mirror depend strongly on the level of seed radiation in the direction of the scattered beams. The seed radiation may be due to incoherent scattering of the pumping beams inside the medium or to self-broadening of the pumping beams’ spectra because of nonlinear self-interaction.

© 1995 Optical Society of America

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References

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  1. S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
    [Crossref] [PubMed]
  2. B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
    [Crossref]
  3. N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–40 (1965).
    [Crossref]
  4. A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).
  5. A. A. Zozulya, “Double phase-conjugate mirror is not an oscillator,” Opt. Lett. 16, 545–547 (1991).
    [Crossref] [PubMed]
  6. V. T. Tikhonchuk and A. A. Zozulya, “Structure of light beams in self-pumped four-wave mixing geometries for phase conjugation and mutual conjugation,” Prog. Quantum Electron. 15, 231–293 (1991).
    [Crossref]
  7. A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
    [Crossref] [PubMed]
  8. K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
    [Crossref]
  9. K. D. Shaw, “Vector versus scalar theory of the double phase conjugate mirror,” Opt. Commun. 94, 458–468 (1992).
    [Crossref]
  10. K. D. Shaw, “Operation of the double phase conjugate mirror for TE polarization: exact solution and failure of the slowly varying envelope approximation,” Opt. Commun. 103, 326–338 (1993).
    [Crossref]
  11. M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of photorefractive double phase conjugate mirror,” Opt. Lett. 18, 1828–1830 (1993).
    [Crossref] [PubMed]
  12. D. Engin, M. Segev, S. Orlov, A. Yariv, and G. Valley, “Double phase conjugation,” J. Opt. Soc. Am. B 11, 1708–1717 (1994).
    [Crossref]
  13. N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [Crossref]
  14. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [Crossref]
  15. M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
    [Crossref]
  16. L. Sun and G. L. Yip, “Modified finite-difference beam-propagation method based on the Douglas scheme,” Opt. Lett. 18, 1229–1231 (1993).
    [Crossref] [PubMed]
  17. D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [Crossref]
  18. A. A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
    [Crossref]

1994 (2)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
[Crossref] [PubMed]

D. Engin, M. Segev, S. Orlov, A. Yariv, and G. Valley, “Double phase conjugation,” J. Opt. Soc. Am. B 11, 1708–1717 (1994).
[Crossref]

1993 (4)

K. D. Shaw, “Operation of the double phase conjugate mirror for TE polarization: exact solution and failure of the slowly varying envelope approximation,” Opt. Commun. 103, 326–338 (1993).
[Crossref]

M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of photorefractive double phase conjugate mirror,” Opt. Lett. 18, 1828–1830 (1993).
[Crossref] [PubMed]

L. Sun and G. L. Yip, “Modified finite-difference beam-propagation method based on the Douglas scheme,” Opt. Lett. 18, 1229–1231 (1993).
[Crossref] [PubMed]

A. A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[Crossref]

1992 (3)

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
[Crossref]

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[Crossref]

K. D. Shaw, “Vector versus scalar theory of the double phase conjugate mirror,” Opt. Commun. 94, 458–468 (1992).
[Crossref]

1991 (2)

A. A. Zozulya, “Double phase-conjugate mirror is not an oscillator,” Opt. Lett. 16, 545–547 (1991).
[Crossref] [PubMed]

V. T. Tikhonchuk and A. A. Zozulya, “Structure of light beams in self-pumped four-wave mixing geometries for phase conjugation and mutual conjugation,” Prog. Quantum Electron. 15, 231–293 (1991).
[Crossref]

1989 (2)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
[Crossref]

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

1987 (2)

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).

1985 (1)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

1965 (1)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–40 (1965).
[Crossref]

Anderson, D. Z.

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
[Crossref] [PubMed]

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

Cronin-Golomb, M.

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
[Crossref]

Engin, D.

Fischer, B.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
[Crossref]

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

Hermansson, B.

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

Kroll, N. M.

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–40 (1965).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Odoulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Orlov, S.

Saffman, M.

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
[Crossref] [PubMed]

Segev, M.

Shaw, K. D.

K. D. Shaw, “Operation of the double phase conjugate mirror for TE polarization: exact solution and failure of the slowly varying envelope approximation,” Opt. Commun. 103, 326–338 (1993).
[Crossref]

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[Crossref]

K. D. Shaw, “Vector versus scalar theory of the double phase conjugate mirror,” Opt. Commun. 94, 458–468 (1992).
[Crossref]

Silin, V. P.

A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Sternklar, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
[Crossref]

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

Sun, L.

Tikhonchuk, V. T.

V. T. Tikhonchuk and A. A. Zozulya, “Structure of light beams in self-pumped four-wave mixing geometries for phase conjugation and mutual conjugation,” Prog. Quantum Electron. 15, 231–293 (1991).
[Crossref]

A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).

Valley, G.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Weiss, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
[Crossref]

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

Yariv, A.

Yevick, D.

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

Yip, G. L.

Zozulya, A. A.

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
[Crossref] [PubMed]

A. A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[Crossref]

V. T. Tikhonchuk and A. A. Zozulya, “Structure of light beams in self-pumped four-wave mixing geometries for phase conjugation and mutual conjugation,” Prog. Quantum Electron. 15, 231–293 (1991).
[Crossref]

A. A. Zozulya, “Double phase-conjugate mirror is not an oscillator,” Opt. Lett. 16, 545–547 (1991).
[Crossref] [PubMed]

A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals: I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

IEEE J. Quantum Electron. (3)

D. Yevick and B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

A. A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[Crossref]

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989).
[Crossref]

J. Appl. Phys. (1)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–40 (1965).
[Crossref]

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

Opt. Commun. (4)

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[Crossref]

K. D. Shaw, “Vector versus scalar theory of the double phase conjugate mirror,” Opt. Commun. 94, 458–468 (1992).
[Crossref]

K. D. Shaw, “Operation of the double phase conjugate mirror for TE polarization: exact solution and failure of the slowly varying envelope approximation,” Opt. Commun. 103, 326–338 (1993).
[Crossref]

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992).
[Crossref]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave mixing geometries,” Phys. Rev. Lett. 73, 818–821 (1994).
[Crossref] [PubMed]

Prog. Quantum Electron. (2)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[Crossref]

V. T. Tikhonchuk and A. A. Zozulya, “Structure of light beams in self-pumped four-wave mixing geometries for phase conjugation and mutual conjugation,” Prog. Quantum Electron. 15, 231–293 (1991).
[Crossref]

Sov. Phys. JETP (1)

A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, “The theory of phase conjugation during stimulated scattering in a self-intersecting light beam,” Sov. Phys. JETP 65, 443–449 (1987).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the DPCM.

Fig. 2
Fig. 2

Transmissivities (Trl, Tlr) and conjugation fidelities (Hr, Hl) as functions of nonlinearity for levels of random seeding noise (a) = 10−4, (b) 10−6, and (c) 0. (d) Time evolution of the transmissivities and the conjugation fidelities for = 10−4 and γ0l = 8.

Fig. 3
Fig. 3

Fourier power spectra of the interacting fields: (a) input and output scattered beam A3 and counterpropagating input pump beam A4, (b) output scattered beam A1 and input counter-propagating pump beam A2. The width of the depicted region corresponds to an angular window covering 2.3 deg.

Fig. 4
Fig. 4

Intensity distributions of the interacting fields: (a) output scattered beam A3 and counterpropagating input pump beam A4, (b) output scattered beam A1 and input counterpropagating pump beam A2, (c) input and output pump beam A4, (d) input and output pump beam A2.

Fig. 5
Fig. 5

Fourier power spectra of (a) input and (b) output single pump beam A4 in the absence of the second pump and no random seeding for γ0l = 8.0. The downward-pointing arrows indicate the direction of propagation of the corresponding scattered beam.

Fig. 6
Fig. 6

Transmissivities (Trl, Tlr) and conjugation fidelities (Hr, Hl) as a function of the angle between the pumping beams. The dashed–dotted curve is the local coupling coefficient γ(θ)l.

Fig. 7
Fig. 7

Transmissivities Tlr and conjugation fidelities Hr as functions of the nonlinearity for the transversely bounded pumping beams described by Eqs. (7) and for levels of random seeding = 10−5, 10−12. The length of the nonlinear medium equals 1.5 mm.

Fig. 8
Fig. 8

Same as in Fig. 7 but for the transversely unbounded pump beams described by Eqs. (8).

Equations (17)

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( γ p θ ) tanh ( Q l / 2 ) = Q .
( x i 2 k 2 y 2 ) A f = 2 i γ 0 ν A f ,
( x i 2 k 2 y 2 ) A b = 2 i γ 0 ν A b ,
I 1 ( 1 + 1 k θ D ν y ) t ν + ν 1 ( k θ D ) 2 2 ν y 2 × ( 1 + 1 k θ D ν y ) 1 = 1 k θ D y ln I .
γ ( θ ) = γ 0 2 ( θ / θ D ) 1 + ( θ / θ D ) 2 .
A l , in ( y ) = B l , in ( y ) exp [ i θ l k y 4 ( y y l ) 2 / d 2 ] .
H l = | d y A 3 , out ( y ) A 4 , in ( y ) | 2 [ d y | A 3 , out ( y ) | 2 ] [ d y | A 4 , in ( y ) | 2 ] ,
H r = | d y A 1 , out ( y ) A 2 , in ( y ) | 2 [ d y | A 1 , out ( y ) | 2 ] [ d y | A 2 , in ( y ) | 2 ] .
A 4 , in ( y ) = [ 1 + cos ( 10 π y / d ) ] exp [ i θ 4 k y 4 ( y y 4 ) 2 / d 2 ] ,
A 2 , in ( y ) = [ 1 cos ( 14 π y / d ) ] exp [ i θ 2 k y 4 ( y y 2 ) 2 / d 2 ] .
A 4 , in ( y ) = [ 1 + cos ( 10 π y / d ) ] exp ( i θ 4 k y ) ,
A 2 , in ( y ) = [ 1 cos ( 14 π y / d ) ] exp ( i θ 2 k y ) .
( d 2 d x 2 + 2 i k x d d x ) A 1 = 2 i k x γ I ( A 1 A 4 * + A 2 * A 3 ) A 4 ,
( d 2 d x 2 2 i k x d d x ) A 3 = 2 i k x γ I ( A 1 A 4 * + A 2 * A 3 ) A 2 .
κ 1 ( κ 3 κ 2 ) B 2 B 3 + κ 2 ( κ 1 κ 3 ) B 1 B 3 + κ 3 ( κ 2 κ 1 ) B 1 B 2 = 0 ,
B l = ( 1 + q ) q 2 i k x γ ( κ l 2 + 2 i k x κ l 2 i k x γ 1 + q ) exp ( κ l l ) + q ,
A 2 ( 4 ) , in exp [ i k θ 2 ( 4 ) y ] [ 1 + a 2 ( 4 ) cos ( 2 π y / l g ) + b 2 ( 4 ) cos ( 4 π y / l g ) ] ,

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