Abstract

Degenerate four-wave mixing in a two-level saturable absorbing medium is calculated on and off resonance for arbitrary input laser intensities and absorption coefficients. The maximum efficiency occurs at laser intensities higher than saturation in an optically thick medium. On resonance, the maximum efficiency can approach ∼4%, and it can be as large as 25% when the laser is tuned off resonance.

© 1993 Optical Society of America

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References

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  1. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  2. R. L. Abrams, R. C. Lind, Opt. Lett. 2, 94 (1978).
    [CrossRef] [PubMed]
  3. W. P. Brown, J. Opt. Soc. Am. 73, 629 (1983).
    [CrossRef]
  4. M. T. Gruneisen, A. L. Gaeta, R. W. Boyd, J. Opt. Soc. Am. B 2, 1117 (1985).
    [CrossRef]
  5. A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
    [CrossRef]
  6. Orthogonal pump polarizations allow a relatively simple analytic expression for the medium polarization.
  7. V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
    [CrossRef]
  8. It is assumed that all the absorbers have the same resonant frequency. The present analysis would not correctly describe atoms with Doppler broadening.
  9. It is assumed that all the absorbers have the same resonant frequency. The present analysis would not correctly describe atoms with Doppler broadening.
  10. It is assumed throughout this Letter that the subscripts i in all variables can assume values of 1–4.
  11. L1 and L2 can be set equal to the grating periods λ/[2 sin(θ/2)] and λ/[2 cos(θ/2)], respectively.
  12. I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), pp. 149, 260, 907.
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

1989

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

1986

A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
[CrossRef]

1985

1983

1978

Abrams, R. L.

Boyd, R. W.

A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
[CrossRef]

M. T. Gruneisen, A. L. Gaeta, R. W. Boyd, J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Brown, W. P.

Chaley, A. V.

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

Gaeta, A. L.

A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
[CrossRef]

M. T. Gruneisen, A. L. Gaeta, R. W. Boyd, J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), pp. 149, 260, 907.

Gruneisen, M. T.

A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
[CrossRef]

M. T. Gruneisen, A. L. Gaeta, R. W. Boyd, J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Kabanov, V. V.

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

Lind, R. C.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

Rubanov, A. S.

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

Ryshik, I. M.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), pp. 149, 260, 907.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

Tolstik, A. L.

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

IEEE J. Quantum Electron.

A. L. Gaeta, M. T. Gruneisen, R. W. Boyd, IEEE J. Quantum Electron. QE-22, 1095 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

V. V. Kabanov, A. S. Rubanov, A. L. Tolstik, A. V. Chaley, Opt. Commun. 71, 219 (1989); V. V. Kabanov, S. Rubanov, IEEE J. Quantum Electron. 26, 1990 (1990).
[CrossRef]

Opt. Lett.

Other

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

It is assumed that all the absorbers have the same resonant frequency. The present analysis would not correctly describe atoms with Doppler broadening.

It is assumed that all the absorbers have the same resonant frequency. The present analysis would not correctly describe atoms with Doppler broadening.

It is assumed throughout this Letter that the subscripts i in all variables can assume values of 1–4.

L1 and L2 can be set equal to the grating periods λ/[2 sin(θ/2)] and λ/[2 cos(θ/2)], respectively.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), pp. 149, 260, 907.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran (Cambridge U. Press, Cambridge, 1992), pp. 254–263.

Orthogonal pump polarizations allow a relatively simple analytic expression for the medium polarization.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of DFWM in a nonlinear medium of two-level saturable absorbers.

Fig. 2
Fig. 2

DFWM efficiency obtained on resonance as a function of αL = 0.25, 0.5, 1, 2, 5, and 10 for curves a–f, respectively, and the total input intensity normalized to the saturation intensity. The intensities of the input beams were equal to each other.

Fig. 3
Fig. 3

DFWM efficiency calculated for a detuning of δ = 5 off resonance as a function of αL = 0.25, 0.5, 1, 2, 5, and 10 for curves a–f, respectively, and the total input intensity normalized to the saturation intensity. The intensities of the input beams were equal to each other.

Equations (10)

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E ( r , t ) = 1 2 E ( r ) exp ( i ω t ) + c . c . ,
P ( r , t ) = 1 2 P ( r ) exp ( i ω t ) + c . c .
P ( r ) = α 2 π k ( i + δ ) E ( r ) [ 1 + | E ( r ) / E s | 2 ] ,
2 E 1 c 2 2 E 2 t = 4 π c 2 P t 2 .
E i ( r ) = A i ( r ) E s exp ( i k i r ) ,
P i ( r ) = 1 L 1 L 2 L 1 / 2 L 1 / 2 d x L 2 / 2 L 2 / 2 d z × exp [ i k i ( r + r ) ] P ( r + r ) ,
P i ( r ) = ( i + δ ) α 2 π k E s R i ( r ) ,
k i A i = k i ( 1 i δ ) α R i .
A 1 z = α 1 A 1 κ 1 A 3 A 4 A 2 * , A 2 z = α 2 A 2 + κ 2 A 3 A 4 A 1 * , A 3 z = α 3 A 3 κ 3 A 1 A 2 A 4 * , A 4 z = α 4 A 4 + κ 4 A 1 A 2 A 3 * ,
α 1 ( r ) = ( 1 i δ ) α [ R 1 ( e 1 , e 2 , e 3 , e 4 ) R 1 ( e 1 , e 2 , e 3 , e 4 ) ] / ( 2 e 1 ) , κ 1 ( r ) = ( 1 i δ ) α [ R 1 ( e 1 , e 2 , e 3 , e 4 ) + R 1 ( e 1 , e 2 , e 3 , e 4 ) / ( 2 e 2 e 3 e 4 ) .

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