Abstract

In parallel-polarized photorefractive four-wave mixing the phase-matching condition can be achieved automatically if the pump beams are allowed to propagate in directions opposite to each other. However, in cross-polarized photorefractive four-wave mixing, the incident angles of the beams should be adjusted to maintain the phase-matching condition. We analyze the phase-matching properties of cross-polarized four-wave mixing with extraordinary writing beams and an ordinary reading beam and obtain the phase-conjugate reflectivity in consideration of phase mismatching. We also perform an experiment in which a phase-conjugate beam is generated in cross-polarized four-wave mixing and confirm the analyses.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  2. T. K. Das, G. C. Bhar, “Phase-conjugate bistability and multistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019–1032 (1994).
    [CrossRef]
  3. R. M. Pierce, R. S. Cudney, “Photorefractive coupling between orthogonally polarized light beams in barium titanate,” Opt. Lett. 17, 784–786 (1992).
    [CrossRef] [PubMed]
  4. M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.
  5. J. B. Norman, “Phase-conjugate Michelson interferometers for all-optical image processing and computing,” Am. J. Phys. 60, 212–220 (1992).
    [CrossRef]
  6. H. Kang, C. X. Yang, G. G. Mu, Z. K. Wu, “Real-time holographic associative memory using doped LiNbO3 in a phase-conjugate resonator,” Opt. Lett. 15, 637–639 (1990).
    [CrossRef] [PubMed]
  7. T. S. Yu, S. Yin, “Applications of photorefractive crystals to signal processing,” Int. J. Opt. Comput. 2, 143–163 (1991).
  8. S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
    [CrossRef]
  9. A. Novikov, S. Odoulov, R. Jungen, T. Tschude, “Spatial subharmonic generation of orthogonally polarized light waves in BaTiO3 by phase-matched nonlinear mixing,” Opt. Lett. 16, 1941–1943 (1991).
    [CrossRef] [PubMed]
  10. A. Novikov, S. Odoulov, R. Jungen, T. Tschudi, “Spatial subharmonic generation in BaTiO3,” J. Opt. Soc. Am. B 9, 1654–1660 (1992).
    [CrossRef]
  11. B. I. Sturman, M. Yu. Goul’kov, S. G. Odoulov, “Phenomenological analysis of the parametric scattering process in photorefractive crystals,” J. Opt. Soc. Am. B 13, 577–583 (1995).
    [CrossRef]
  12. M. Cronin-Golomb, M. P. Tarr, “Applications of birefringent phase matching for photorefractive devices,” Opt. Lett. 20, 2252–2254 (1995).
    [CrossRef] [PubMed]
  13. A. Roy, K. Singh, “Cross-polarization coupling via reflection-type degenerate four-wave mixing in compound semiconductor photorefractive crystals,” J. Appl. Phys. 70, 562–567 (1991).
    [CrossRef]
  14. A. Roy, K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate four-wave mixing in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987–1000 (1994).
    [CrossRef]
  15. A. Roy, K. Singh, “Effects of cross-polarization coupling in DFWM in photorefractive compound semiconductor crystals: transmission geometry,” J. Mod. Opt. 38, 1763–1772 (1991).
    [CrossRef]
  16. H. Kong, C. Lin, A. M. Biernacki, M. Cronin-Golomb, “Photorefractive phase conjugation with orthogonally polarized pumping beams,” Opt. Lett. 13, 324–326 (1988).
    [CrossRef] [PubMed]
  17. 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, 12–31 (1984).
    [CrossRef]
  18. J. E. Ford, Y. Fainman, S. H. Lee, “Enhanced photorefractive performance from 45°-cut BaTiO3,” Appl. Opt. 28, 4808–4815 (1989).
    [CrossRef] [PubMed]
  19. M. H. Garrett, J. Y. Chang, H. P. Jenssen, C. Warde, “Self-pumped phase conjugation and four-wave mixing in 0°- and 45°-cut n-type BaTiO3:Co,” Opt. Lett. 18, 405–407 (1993).
    [CrossRef] [PubMed]

1995 (2)

1994 (2)

T. K. Das, G. C. Bhar, “Phase-conjugate bistability and multistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019–1032 (1994).
[CrossRef]

A. Roy, K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate four-wave mixing in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987–1000 (1994).
[CrossRef]

1993 (1)

1992 (3)

1991 (5)

A. Roy, K. Singh, “Cross-polarization coupling via reflection-type degenerate four-wave mixing in compound semiconductor photorefractive crystals,” J. Appl. Phys. 70, 562–567 (1991).
[CrossRef]

T. S. Yu, S. Yin, “Applications of photorefractive crystals to signal processing,” Int. J. Opt. Comput. 2, 143–163 (1991).

S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

A. Novikov, S. Odoulov, R. Jungen, T. Tschude, “Spatial subharmonic generation of orthogonally polarized light waves in BaTiO3 by phase-matched nonlinear mixing,” Opt. Lett. 16, 1941–1943 (1991).
[CrossRef] [PubMed]

A. Roy, K. Singh, “Effects of cross-polarization coupling in DFWM in photorefractive compound semiconductor crystals: transmission geometry,” J. Mod. Opt. 38, 1763–1772 (1991).
[CrossRef]

1990 (1)

1989 (1)

1988 (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, 12–31 (1984).
[CrossRef]

Bhar, G. C.

T. K. Das, G. C. Bhar, “Phase-conjugate bistability and multistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019–1032 (1994).
[CrossRef]

Biernacki, A. M.

Chang, J. Y.

Cronin-Golomb, M.

Cudney, R. S.

Das, T. K.

T. K. Das, G. C. Bhar, “Phase-conjugate bistability and multistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019–1032 (1994).
[CrossRef]

Fainman, Y.

Fischer, B.

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, 12–31 (1984).
[CrossRef]

Ford, J. E.

Garrett, M. H.

Goul’kov, M. Yu.

Holtmann, L.

S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

Jenssen, H. P.

Jungen, R.

Kang, H.

Kong, H.

Krätzig, E.

S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

Lee, S. H.

Lin, C.

Mu, G. G.

Norman, J. B.

J. B. Norman, “Phase-conjugate Michelson interferometers for all-optical image processing and computing,” Am. J. Phys. 60, 212–220 (1992).
[CrossRef]

Novikov, A.

Odoulov, S.

Odoulov, S. G.

Okamoto, A.

M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.

Pierce, R. M.

Roy, A.

A. Roy, K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate four-wave mixing in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987–1000 (1994).
[CrossRef]

A. Roy, K. Singh, “Cross-polarization coupling via reflection-type degenerate four-wave mixing in compound semiconductor photorefractive crystals,” J. Appl. Phys. 70, 562–567 (1991).
[CrossRef]

A. Roy, K. Singh, “Effects of cross-polarization coupling in DFWM in photorefractive compound semiconductor crystals: transmission geometry,” J. Mod. Opt. 38, 1763–1772 (1991).
[CrossRef]

Saito, M.

M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.

Singh, K.

A. Roy, K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate four-wave mixing in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987–1000 (1994).
[CrossRef]

A. Roy, K. Singh, “Cross-polarization coupling via reflection-type degenerate four-wave mixing in compound semiconductor photorefractive crystals,” J. Appl. Phys. 70, 562–567 (1991).
[CrossRef]

A. Roy, K. Singh, “Effects of cross-polarization coupling in DFWM in photorefractive compound semiconductor crystals: transmission geometry,” J. Mod. Opt. 38, 1763–1772 (1991).
[CrossRef]

Sturman, B.

S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

Sturman, B. I.

Takamura, M.

M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.

Takayama, Y.

M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.

Tarr, M. P.

Tschude, T.

Tschudi, T.

Warde, C.

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, 12–31 (1984).
[CrossRef]

Wu, Z. K.

Yang, C. X.

Yariv, A.

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, 12–31 (1984).
[CrossRef]

Yeh, P.

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

Yin, S.

T. S. Yu, S. Yin, “Applications of photorefractive crystals to signal processing,” Int. J. Opt. Comput. 2, 143–163 (1991).

Yu, T. S.

T. S. Yu, S. Yin, “Applications of photorefractive crystals to signal processing,” Int. J. Opt. Comput. 2, 143–163 (1991).

Am. J. Phys. (1)

J. B. Norman, “Phase-conjugate Michelson interferometers for all-optical image processing and computing,” Am. J. Phys. 60, 212–220 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

S. Odoulov, B. Sturman, L. Holtmann, E. Krätzig, “Nonlinear scattering in BaTiO3 induced by two orthogonally polarized waves,” Appl. Phys. B 52, 317–322 (1991).
[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, 12–31 (1984).
[CrossRef]

Int. J. Opt. Comput. (1)

T. S. Yu, S. Yin, “Applications of photorefractive crystals to signal processing,” Int. J. Opt. Comput. 2, 143–163 (1991).

J. Appl. Phys. (1)

A. Roy, K. Singh, “Cross-polarization coupling via reflection-type degenerate four-wave mixing in compound semiconductor photorefractive crystals,” J. Appl. Phys. 70, 562–567 (1991).
[CrossRef]

J. Mod. Opt. (2)

A. Roy, K. Singh, “Cross and parallel polarization coupling in transmission-type degenerate four-wave mixing in compound semiconductor photorefractive crystals: orientational dependence,” J. Mod. Opt. 41, 987–1000 (1994).
[CrossRef]

A. Roy, K. Singh, “Effects of cross-polarization coupling in DFWM in photorefractive compound semiconductor crystals: transmission geometry,” J. Mod. Opt. 38, 1763–1772 (1991).
[CrossRef]

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

Opt. Lett. (6)

Opt. Quantum Electron. (1)

T. K. Das, G. C. Bhar, “Phase-conjugate bistability and multistability in moving-grating operated orthogonally polarized pump four-wave mixing in photorefractives,” Opt. Quantum Electron. 26, 1019–1032 (1994).
[CrossRef]

Other (2)

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

M. Saito, A. Okamoto, Y. Takayama, M. Takamura, “Phase matching property of cross polarized four wave mixing in BaTiO3 crystal and optical bus application,” paper presented at the Sixth Topical Meeting on Photorefractive Materials, Effects and Devices, Chiba, Japan, 11–13 June 1997, paper WP28.

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 (11)

Fig. 1
Fig. 1

Interaction of the beams.

Fig. 2
Fig. 2

Angular dependence of the coupling coefficient in the case of (a) ordinary rays and (b) extraordinary rays.

Fig. 3
Fig. 3

Four-wave mixing.

Fig. 4
Fig. 4

Phase-matching angle: (a) Incident angle of the forward pump beam. (b) Incident angle of the probe beam.

Fig. 5
Fig. 5

Angle differences from counterpropagation for the case of variation of (a) the incident angle of the backward pump beam and (b) the direction of the c axis.

Fig. 6
Fig. 6

Phase mismatch.

Fig. 7
Fig. 7

Phase-conjugate reflectivity with phase mismatch.

Fig. 8
Fig. 8

Experimental setup: HWP, half-wave plate; PBS, polarizing beam splitter; BS, beam splitter; M, mirror; and PD, photodetector.

Fig. 9
Fig. 9

Experimental measurement of the phase-conjugate reflectivity.

Fig. 10
Fig. 10

Angle between the incident beam and the crystal surface.

Fig. 11
Fig. 11

Relation between the beam’s incident angle outside crystal and the one inside the crystal (a) for ϕ = 0.90°, (b) when the small-angle variation is ϕ = 90°, and (c) when the small-angle variation is ϕ = 0°.

Equations (29)

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

γ ao = ω r eff o 2 n o c   cos   θ a k g k B T / e 1 + k g 2 / k D 2 ,
γ bo = ω r eff o 2 n o c   cos   θ b k g k B T / e 1 + k g 2 / k D 2 ,
γ ae = ω r eff e 2 n ae c   cos   θ a k g k B T / e 1 + k g 2 / k D 2 cos θ a + θ b ,
γ be = ω r eff e 2 n be c   cos   θ b k g k B T / e 1 + k g 2 / k D 2 cos θ a + θ b ,
r eff o = r 13 sin α a + α b 2 ,
r eff e = r 13 cos   α a cos   α b n o 4 + r 33 sin   α a sin   α b n e 4 + 2 r 42 cos 2 α a + α b 2 n o 2 n e 2 sin α a + α b 2 .
k 1 = ω c   n e α 1 - sin   θ 1 x ˆ + cos   θ 1 z ˆ ,
k 2 = ω c   n o sin   θ 2 x ˆ - cos   θ 2 z ˆ ,
k 3 = ω c   n 0 - sin   θ 3 x ˆ - cos   θ 3 z ˆ ,
k 4 = ω c   n e α 4 - sin   θ 1 x ˆ - cos   θ 4 z ˆ ,
α 1 = α + θ 1 ,
α 4 = α - θ 4 ,
n e α j = cos   α j / n o 2 + sin   α j / n e 2 - 1 / 2 .
Δ k = k 1 - k 4 - k 2 - k 3 .
Δ k = 0 ,
θ 2 = θ 3 ,
n e α 1 cos   θ 1 = n e α 4 cos   θ 4 ,
sin   θ 2 + sin   θ 3 n o - n e α 1 sin   θ 1 = n e α 4 sin   θ 4 .
Δ k z = n e α 1 cos   θ 1 - n e α 4 cos   θ 4 + cos   θ 3 - cos   θ 2 n o .
A 1 z = - γ e I 0 A 1 A 4 * + A 2 * A 3   exp - i Δ k z z A 4 ,
A 2 z = - γ o I 0 A 1 * A 4 exp - i Δ k z z + A 2 A 3 * A 3 ,
A 3 z = γ o I 0 A 1 A 4 * exp i Δ k z z + A 2 * A 3 A 2 ,
A 4 z = γ e I 0 A 1 * A 4 + A 2 A 3 * exp i Δ k z z A 1 ,
| A 1 A 4 * |     | A 2 * A 3 | .
A 1 z = - γ e I 0   I 4 A 1 ,
A 2 z = - γ o I 0   A 1 * A 4 A 3 exp - i Δ k z z ,
A 3 z = γ o I 0   A 1 A 4 * A 2 exp i Δ k z z ,
A 4 z = γ e I 0   I 1 A 4 .
R = p q sin 2 γ o 0 L cos Δ k z z 2   cosh 2 γ e z + ln   q 2 d z ,

Metrics