Abstract

The effect of partial spatiotemporal coherence in photorefractive two-beam coupling is studied theoretically and experimentally. Coupled-wave equations and a beam-propagation method are used to study the influence of coupling on the statistical properties of the beams. The spatial coherence is found to improve for amplified and to deteriorate for deamplified beams. A two-beam coupling experiment with light of reduced spatial coherence was performed with photorefractive barium titanate. Qualitative agreement between theory and experiment was obtained.

© 1992 Optical Society of America

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References

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  1. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Devices I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
  2. M. Cronin-Golomb, “Achromatic volume holography using dispersion compensation for grating tilt,” Opt. Lett. 14, 1297 (1989).
    [CrossRef] [PubMed]
  3. B. Culshaw, “Optical fibre sensing and signal process,” (Peter Peregrinus, London, 1984).
  4. V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
    [CrossRef]
  5. H. Kong, C. Wu, and M. Cronin-Golomb, “Photorefractive two-beam coupling with reduced spatial coherence,” Opt. Lett. 16, 1183 (1991).
    [CrossRef] [PubMed]
  6. M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1 (1990).
    [CrossRef]
  7. B. Crosignani and A. Yariv, “Degenerate four-wave mixing in the presence of nonuniform pump wave fronts,” J. Opt. Soc. Am. A 1, 1034 (1984);G. Lera and M. Nieto-Vesperinas, “Phase conjugation by four-wave mixing of statistical beams,” Phys. Rev. A 41, 6400 (1990).
    [CrossRef] [PubMed]
  8. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820 (1977).
    [CrossRef]
  9. P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sect. 10.4.
  11. G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).
  12. M. Cronin-Golomb, “Whole-beam method for photorefractive nonlinear optics,” Opt. Commun. (to be published).
  13. B. J. Thompson, “Image formation with partially coherent light,” in E. Wolf, ed., Progress in Optics (North-Holland, Amsterdam, 1969), Vol. 7, p. 169;A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. 25, 353 (1989).
    [CrossRef]

1991 (1)

1990 (1)

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1 (1990).
[CrossRef]

1989 (1)

1984 (1)

1979 (1)

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

1978 (1)

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

1977 (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820 (1977).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sect. 10.4.

Cronin-Golomb, M.

Crosignani, B.

Culshaw, B.

B. Culshaw, “Optical fibre sensing and signal process,” (Peter Peregrinus, London, 1984).

De Santis, P.

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

Gauttari, G.

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

Gori, F.

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

Kong, H.

Palma, C.

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

Solymar, L.

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820 (1977).
[CrossRef]

Thompson, B. J.

B. J. Thompson, “Image formation with partially coherent light,” in E. Wolf, ed., Progress in Optics (North-Holland, Amsterdam, 1969), Vol. 7, p. 169;A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. 25, 353 (1989).
[CrossRef]

Wang, V.

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sect. 10.4.

Wu, C.

Yariv, A.

Zahid, M.

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1 (1990).
[CrossRef]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820 (1977).
[CrossRef]

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

Opt. Commun. (2)

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1 (1990).
[CrossRef]

P. De Santis, F. Gori, G. Gauttari, and C. Palma, “An example of a Collet-Wolf source,” Opt. Commun. 29, 256 (1979);N. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian-Schell model source,” Opt. Commun. 59, 385 (1986).
[CrossRef]

Opt. Eng. (1)

V. Wang, “Nonlinear optical phase conjugation for laser systems,” Opt. Eng. 17, 267 (1978).
[CrossRef]

Opt. Lett. (2)

Other (6)

B. Culshaw, “Optical fibre sensing and signal process,” (Peter Peregrinus, London, 1984).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sect. 10.4.

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).

M. Cronin-Golomb, “Whole-beam method for photorefractive nonlinear optics,” Opt. Commun. (to be published).

B. J. Thompson, “Image formation with partially coherent light,” in E. Wolf, ed., Progress in Optics (North-Holland, Amsterdam, 1969), Vol. 7, p. 169;A. Cunha and E. N. Leith, “Generalized phase-conjugation system using partially coherent light,” IEEE J. Quantum Electron. 25, 353 (1989).
[CrossRef]

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Devices I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).

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Figures (5)

Fig. 1
Fig. 1

Modulus of the degree of coherence of the output signal beam as a function of |(y1y2)/W|, where y1 = − y2, for positive and negative values of the coupling strength; calculated with the quasi-plane-wave coupled- wave approach in which the pump beam is undepleted (coherence width d/W = 0.095); dashed curve, no coupling.

Fig. 2
Fig. 2

Experimental setup. The signal-to-pump intensity ratio is 1:5.14, the beam diameters W are 0.7 mm, the angle of incidence θ = 18°, the normalized coherence width d/W = 0.27, and the crystal amplitude coupling constant was ∼3.3 cm−1. Geometrical factors indicate that, with the interaction region entirely within the crystal, the interaction length was 2.8 mm; therefore the coupling-constant–length product was γl = 0.92. The laser wavelength was 514.5 nm. The coherence of the amplified beam was examined with the interferometer set up around the second beam splitter. M’s, mirrors.

Fig. 3
Fig. 3

Modulus of the degree of coherence of output signal beam as a function of |(y1y2)/W|, where y1 = y2, for various values of coupling strength in the beam-propagation method. The incident beams have the same intensities and profile; d/W as in Fig. 1; dashed curve, no coupling.

Fig. 4
Fig. 4

Modulus of the degree of coherence of the output signal beam as a function of |(y1y2)/W|, where y1y2, for various values of coupling strength in the beam-propagation method. The ratio of pump-to-signal intensities is 100:1 to permit comparison with the quasi-plane- wave results of Fig. 1; d/W as in Fig. 1; dashed curve, no coupling.

Fig. 5
Fig. 5

Experimentally measured modulus of the degree of coherence of output signal beam; stars, without coupling; circles, with photorefractive coupling (amplification).

Equations (16)

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μ ( r 1 , r 2 ) = A ( r 1 ) A ( r 2 ) * [ I ( r 1 ) I ( r 2 ) ] 1 / 2 ,
x = x sin θ y cos θ , y = x sin θ + y cos θ ,
S ( x , y ) x = γ Q ( x , y ) P ( x , y ) , P * ( x , y ) y = γ Q ( x , y ) S * ( x , y ) ,
Q ( x , y ) = S ( x , y ) P * ( x , y ) P ( x , y ) P * ( x , y ) + S ( x , y ) S * ( x , y ) = S ( x , y ) P * ( x , y ) I P ( x , y ) + I s ( x , y ) ,
S ( x , y ) x = γ Q ( x , y ) P ( x ) ,
S ( x , y ) = S 0 ( x , y ) + γ 0 x Q ( x , y ) P ( x ) d x .
Q ( x , y ) = Q 0 ( x , y ) + γ 0 x Q ( x , y ) P * ( x ) P ( x ) I P ( x ) d x ,
Q 0 = S 0 ( y ) P * ( x ) / I P ( x )
P ( x ) P * ( x ) = Γ P ( x , x )
μ S ( y 1 , y 2 ) = Γ S ( y 1 , y 2 ) [ Γ S ( y 1 , y 1 ) ] [ Γ S ( y 2 , y 2 ) ] 1 / 2 ,
μ S ( y 1 y 2 ) = S 0 ( y 1 ) S 0 * ( y 2 ) + 2 γ r 0 l Q ( x ̂ , y 2 ) Q * ( x ̂ , y 1 ) I P ( x ̂ ) d x ̂ [ I S 0 ( y 1 ) + 2 γ r 0 l | Q ( x ̂ , y 1 ) | 2 I P ( x ̂ ) d x ̂ ] 1 / 2 [ I S 0 ( y 2 ) + 2 γ r 0 l | Q ( x ̂ , y 2 ) | 2 I P ( x ̂ ) d x ̂ ] 1 / 2 ,
μ ( y 1 , y 2 ) = A ( y 1 ) A * ( y 2 ) [ I ( y 1 ) I ( y 2 ) ] 1 / 2 = i = 1 n A i ( y 1 ) A i * ( y 2 ) [ i = 1 n I i ( y 1 ) ] 1 / 2 [ i = 1 n I i ( y 2 ) ] 1 / 2 ,
ρ i = exp [ 1 2 ( θ i θ c σ ) 2 ] ,
A i ( y ) = ρ i exp [ 1 2 ( y W ) 2 i k ( θ i θ c ) y ]
μ ( y 1 , y 2 ) = exp [ ( y 1 y 2 d ) 2 ] .
μ = ν / ν 0 ,

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