Abstract

Aspects of thermally induced nonlinearities in the index of refraction are investigated for the case of degenerate four-wave mixing (DFWM). An expression for the reflectivity is derived within the framework of a coupled thermodynamic–electromagnetic theory, and some implications for the conjugation efficiency are briefly discussed. We also report on experimental measurements of phase-conjugate reflectivities in excess of 200% at 532 nm and comment on the role of saturation effects and pump delays in obtaining the observed efficiencies. Finally, the results of near-field conjugation beam-quality studies are presented with a view toward examining some of the parameters crucial to conjugation fidelity in typical DFWM experimental setups.

© 1983 Optical Society of America

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References

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  1. For reviews on the subject, see, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng.156–183 (1982); C. R. Giuliano, “Appplications of optical phase conjugation,” Phys. Today 34, 27–35 (1981).
    [CrossRef]
  2. M. D. Levenson, K. M. Johnson, V. C. Hanchett, and K. Chiang, “Projection photolithography by wave-front conjugation,” J. Opt. Soc. Am. 71, 737–743 (1981).
    [CrossRef]
  3. J. O. White and A. Yariv, “Spatial information processing and distortion correction via four-wave mixing,” Opt. Eng. 21, 224–230 (1982), and references therein.
    [CrossRef]
  4. G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
    [CrossRef]
  5. R. G. Caro and M. C. Gower, “Amplified phase conjugate reflection of KrF laser radiation,” Appl. Phys. Lett. 39, 855–857 (1981).
    [CrossRef]
  6. R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19, 824–828 (1967).
    [CrossRef]
  7. For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
    [CrossRef]
  8. H. J. Hoffman, “Thermally induced degenerate-four-wave-mixing,” IEEE J. Quantum Electron. (to be published).
  9. J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, and R. K. Jain, “Continuous backward-wave generation by degenerate four-wave mixing in optical fibers,” Opt. Lett. 4, 42–44 (1979).
    [CrossRef] [PubMed]
  10. W. Rother, Z. Naturforsch. 25a, 1120–1135 (1970).
  11. R. C. Caro and M. C. Gower, “Phase conjugation by degenerate four-wave mixing in absorbing media,” IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
    [CrossRef]
  12. H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).
  13. A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” Opt. Commun. 25, 23–25 (1978).
    [CrossRef]
  14. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978).
    [CrossRef] [PubMed]
  15. D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
    [CrossRef]
  16. G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
    [CrossRef]
  17. D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, “Spatial convolution and correlation of optical fields via degenerate four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
    [CrossRef] [PubMed]

1982 (4)

For reviews on the subject, see, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng.156–183 (1982); C. R. Giuliano, “Appplications of optical phase conjugation,” Phys. Today 34, 27–35 (1981).
[CrossRef]

J. O. White and A. Yariv, “Spatial information processing and distortion correction via four-wave mixing,” Opt. Eng. 21, 224–230 (1982), and references therein.
[CrossRef]

R. C. Caro and M. C. Gower, “Phase conjugation by degenerate four-wave mixing in absorbing media,” IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

G. S. Agarwal and E. Wolf, “Theory of phase conjugation with weak scatterers,” J. Opt. Soc. Am. 72, 321–326 (1982).
[CrossRef]

1981 (2)

M. D. Levenson, K. M. Johnson, V. C. Hanchett, and K. Chiang, “Projection photolithography by wave-front conjugation,” J. Opt. Soc. Am. 71, 737–743 (1981).
[CrossRef]

R. G. Caro and M. C. Gower, “Amplified phase conjugate reflection of KrF laser radiation,” Appl. Phys. Lett. 39, 855–857 (1981).
[CrossRef]

1979 (2)

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, and R. K. Jain, “Continuous backward-wave generation by degenerate four-wave mixing in optical fibers,” Opt. Lett. 4, 42–44 (1979).
[CrossRef] [PubMed]

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

1978 (3)

1975 (1)

D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
[CrossRef]

1971 (2)

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
[CrossRef]

1970 (1)

W. Rother, Z. Naturforsch. 25a, 1120–1135 (1970).

1967 (1)

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19, 824–828 (1967).
[CrossRef]

Abrams, R. L.

Agarwal, G. S.

AuYeung, J.

Battra, I. P.

For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
[CrossRef]

Caro, R. C.

R. C. Caro and M. C. Gower, “Phase conjugation by degenerate four-wave mixing in absorbing media,” IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

Caro, R. G.

R. G. Caro and M. C. Gower, “Amplified phase conjugate reflection of KrF laser radiation,” Appl. Phys. Lett. 39, 855–857 (1981).
[CrossRef]

Chiang, K.

Eichler, H.

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

Enns, R. H.

For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
[CrossRef]

Enterlein, G.

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

Fekete, D.

Gower, M. C.

R. C. Caro and M. C. Gower, “Phase conjugation by degenerate four-wave mixing in absorbing media,” IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

R. G. Caro and M. C. Gower, “Amplified phase conjugate reflection of KrF laser radiation,” Appl. Phys. Lett. 39, 855–857 (1981).
[CrossRef]

Gray, M. A.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19, 824–828 (1967).
[CrossRef]

Hanchett, V. C.

Hellwarth, R. W.

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Herman, R. M.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19, 824–828 (1967).
[CrossRef]

Hoffman, H. J.

H. J. Hoffman, “Thermally induced degenerate-four-wave-mixing,” IEEE J. Quantum Electron. (to be published).

Jain, R. K.

Johnson, K. M.

Kuizenga, D. J.

D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
[CrossRef]

Levenson, M. D.

Lind, R. C.

Martin, G.

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Munschau, J.

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

Pepper, D. M.

Phillion, D. W.

D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
[CrossRef]

Pohl, D.

For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
[CrossRef]

Rother, W.

W. Rother, Z. Naturforsch. 25a, 1120–1135 (1970).

Siegman, A. E.

D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
[CrossRef]

Stahl, H.

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

White, J. O.

J. O. White and A. Yariv, “Spatial information processing and distortion correction via four-wave mixing,” Opt. Eng. 21, 224–230 (1982), and references therein.
[CrossRef]

Wolf, E.

Yariv, A.

J. O. White and A. Yariv, “Spatial information processing and distortion correction via four-wave mixing,” Opt. Eng. 21, 224–230 (1982), and references therein.
[CrossRef]

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, and R. K. Jain, “Continuous backward-wave generation by degenerate four-wave mixing in optical fibers,” Opt. Lett. 4, 42–44 (1979).
[CrossRef] [PubMed]

A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” Opt. Commun. 25, 23–25 (1978).
[CrossRef]

D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, “Spatial convolution and correlation of optical fields via degenerate four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

R. G. Caro and M. C. Gower, “Amplified phase conjugate reflection of KrF laser radiation,” Appl. Phys. Lett. 39, 855–857 (1981).
[CrossRef]

D. W. Phillion, D. J. Kuizenga, and A. E. Siegman, “Subnano-second relaxation time measurements using a transient induced grating method,” Appl. Phys. Lett. 27, 85–87 (1975).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. C. Caro and M. C. Gower, “Phase conjugation by degenerate four-wave mixing in absorbing media,” IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

A. Yariv, “Four-wave nonlinear optical mixing as real-time holography,” Opt. Commun. 25, 23–25 (1978).
[CrossRef]

Opt. Eng. (2)

For reviews on the subject, see, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng.156–183 (1982); C. R. Giuliano, “Appplications of optical phase conjugation,” Phys. Today 34, 27–35 (1981).
[CrossRef]

J. O. White and A. Yariv, “Spatial information processing and distortion correction via four-wave mixing,” Opt. Eng. 21, 224–230 (1982), and references therein.
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19, 824–828 (1967).
[CrossRef]

Phys. Status Solidi (1)

For a thorough review of this subject see I. P. Battra, R. H. Enns, and D. Pohl, “Stimulated thermal scattering of light,” Phys. Status Solidi 48, 11–63 (1971).
[CrossRef]

Z. Angew. Phys. (1)

H. Eichler, G. Enterlein, J. Munschau, and H. Stahl, Z. Angew. Phys. 31, 1–4 (1971).

Z. Naturforsch. (1)

W. Rother, Z. Naturforsch. 25a, 1120–1135 (1970).

Other (1)

H. J. Hoffman, “Thermally induced degenerate-four-wave-mixing,” IEEE J. Quantum Electron. (to be published).

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Figures (9)

Fig. 1
Fig. 1

Experimental arrangement for measuring DFWM efficiency and conjugation quality. The cell containing the dye was 0.8mm long with antireflection-coated windows. The lenses L1 and L2 had 90-cm focal lengths, and different lenses were used for L3 in order to vary the probe spot size. In this figure, B.S. designates a beam splitter, D is a time delay, S.F.C. denotes a spatial filter collimator, A is an aberrator, and A′ is an aberrator system (aberrator between two lenses). The subsystems indicated by the dashed lines were used to perform interferometric measurements (symbolized by the dashed circles) on the appropriate beams (designated P, C, and RP for probe, conjugate, and read pump, respectively).

Fig. 2
Fig. 2

Reflectivity as a function of the average pump intensity squared. The straight line represents a best fit to the low-intensity data points.

Fig. 3
Fig. 3

Reflectivity as a function of read-pump delay time. The solid line represents an eyeball fit to the data points (taken at approximately 4-nsec intervals).

Fig. 4
Fig. 4

Reflectivity as a function of average pump intensity. The straight lines are best fits to the data. The data fitted by the dashed line were obtained by using 7-psec delay in the write pump.

Fig. 5
Fig. 5

Reflectivity as a function of average pump-intensity/probe-intensity ratio.

Fig. 6
Fig. 6

Phase profiles obtained with a Zygo wave-front analyzer from interferograms. These profiles represent (a) the input beam, (b) the pump beams at the cell entrance, (c) the aberrated probe, and (d) the corrected conjugate.

Fig. 7
Fig. 7

Phase profiles representing the effect of variations in relative spot size and pump tilt on the conjugate beam quality. The spot sizes are (a) pump, 0.5 mm; probe, 0.3 mm; (b) pump, 0.5 mm; probe, 0.1 mm; (c) pump, 0.8 mm; probe, 0.1 mm. The effect of 1° tilt in the read pump [for case (b)] is represented in (d).

Fig. 8
Fig. 8

Effect of pump curvature on conjugation correction. With the aberrator in the system, (a) is the conjugate for complex conjugate pumps and (b) is the conjugate when the pumps’ curvature is not corrected.

Fig. 9
Fig. 9

Effect of pump aberrations on conjugation quality. In this figure (a) represents the aberrated pump and (b) is the conjugate (taken without an aberrator in the probe’s path).

Equations (12)

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U t = n c α 4 π E 2 + D 2 U - U τ ,
ρ o C v T 1 t - K 2 T 1 - T o F ρ o ρ 1 t = U τ ,
2 ρ 1 t 2 - B T ρ o 2 ρ 1 - η ρ o t 2 ρ 1 - F 2 T 1 = - γ e 8 π 2 ( E 2 ) ,
E i ( r , t ) = 1 2 [ A i ( r , t ) exp ( i ω i t - i k i · r - α i · r / 2 ) + c . c . ] ,
β ρ o ( t + γ τ R ) T 1 - ( γ - 1 ) ρ 1 t = γ γ a q 8 π u A 1 A 3 * exp [ - α 2 z ( 1 + 1 cos θ ) ] ,
( ω B 2 γ + Γ B t + 2 t 2 ) ρ 1 + β ρ o ω B 2 γ T 1 = γ e q 2 8 π A 1 A 3 * exp [ - α 2 z ( 1 + 1 cos θ ) ] ,
A 4 z - n c A 4 t = i k 4 n 2 A 2 exp [ - α 2 ( z - L ) ( 1 - 1 cos θ ) ] × ( γ e ρ o ρ 1 + γ T T o T 1 ) ,
γ e ρ o ρ 1 ( z , t ) + γ T T o T 1 ( z , t ) = q γ a exp [ - α 2 z ( 1 + 1 cos θ ) ] 8 π β ρ o u ( T ) p - t A 1 ( t ) A 3 * ( t , z ) × { exp [ - ( t - t ) / τ R ] - exp [ - Γ B 2 ( t - t ) ] × cos ω B ( t - t ) } d t ,
R = | A 4 | 2 d t | A 3 | 2 d t exp ( - α L ) = D 2 f 2 e - α L ( 1 - e - α L ) 2 F ( t D ) g ( ω B ) Ī 2 ,
D = 2 π ( d n / d T ) p λ ρ o C p t p 3 ,
g ( ω B ) 1 + 3 ω B 2 t p 2 ( 1 2 + 2 cos ω B t p ) ,
F ( t D ) 1 - + 3 t D ( t D + t p ) t p 2 ,