Abstract

In current experiments studying cw optical-wavefront conjugation by degenerate four-wave mixing, the effects of atomic motion are not negligible. I summarize a calculation, which includes such effects from the beginning, of the small-signal phase-conjugate-reflection coefficient for both Doppler-broadened and homogeneously broadened resonant transitions.

© 1979 Optical Society of America

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References

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  1. A. Yariv, D. M. Pepper, Opt. Lett. 1, 16 (1977).
    [CrossRef] [PubMed]
  2. R. L. Abrams, R. C. Lind, Opt. Lett. 2, 94 (1978),3,205 (1978).
    [CrossRef] [PubMed]
  3. F. W. Bloch, Phys. Rev. 70, 460 (1946).
    [CrossRef]
  4. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  5. R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
    [CrossRef]
  6. R. P. Feynman, Statistical Mechanics (Benjamin, Reading, Pa.1972).
  7. This procedure seems to be justifiable only when the lower of the two resonant states is the true ground state. I am indebted to M. Sargent for clarification of this point.
  8. A. J. Palmer, Hughes Research Laboratories, Malibu, California 90265, personal communication.
  9. P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
    [CrossRef]
  10. R. P. Feynman, Phys. Rev. 76, 769 (1949).
    [CrossRef]
  11. B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).
  12. P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
    [CrossRef]

1978 (2)

R. L. Abrams, R. C. Lind, Opt. Lett. 2, 94 (1978),3,205 (1978).
[CrossRef] [PubMed]

P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
[CrossRef]

1977 (1)

1975 (1)

P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
[CrossRef]

1957 (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

1949 (1)

R. P. Feynman, Phys. Rev. 76, 769 (1949).
[CrossRef]

1946 (1)

F. W. Bloch, Phys. Rev. 70, 460 (1946).
[CrossRef]

Abrams, R. L.

Berman, P. R.

P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
[CrossRef]

Bloch, F. W.

F. W. Bloch, Phys. Rev. 70, 460 (1946).
[CrossRef]

Bloom, D. M.

P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
[CrossRef]

Brewer, R. G.

P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
[CrossRef]

Conte, S. D.

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

Economou, N. P.

P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
[CrossRef]

Feynman, R. P.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

R. P. Feynman, Phys. Rev. 76, 769 (1949).
[CrossRef]

R. P. Feynman, Statistical Mechanics (Benjamin, Reading, Pa.1972).

Fried, B. D.

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Levy, J. M.

P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
[CrossRef]

Liao, P. F.

P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
[CrossRef]

Lind, R. C.

Palmer, A. J.

A. J. Palmer, Hughes Research Laboratories, Malibu, California 90265, personal communication.

Pepper, D. M.

Vernon, F. L.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Yariv, A.

A. Yariv, D. M. Pepper, Opt. Lett. 1, 16 (1977).
[CrossRef] [PubMed]

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Appl. Phys. Lett. (1)

P. F. Liao, D. M. Bloom, N. P. Economou, Appl. Phys. Lett. 32, 813 (1978).
[CrossRef]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (2)

F. W. Bloch, Phys. Rev. 70, 460 (1946).
[CrossRef]

R. P. Feynman, Phys. Rev. 76, 769 (1949).
[CrossRef]

Phys. Rev. A (1)

P. R. Berman, J. M. Levy, R. G. Brewer, Phys. Rev. A 11, 1668 (1975).
[CrossRef]

Other (5)

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

R. P. Feynman, Statistical Mechanics (Benjamin, Reading, Pa.1972).

This procedure seems to be justifiable only when the lower of the two resonant states is the true ground state. I am indebted to M. Sargent for clarification of this point.

A. J. Palmer, Hughes Research Laboratories, Malibu, California 90265, personal communication.

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Figures (3)

Fig. 1
Fig. 1

Amplitude reduction factor m(γ1,θ) versus angle (for homogeneously broadened system).

Fig. 2
Fig. 2

Interpolated reflectivity versus angle (normalized to θ = δ = 0) for Doppler-broadened system. γ1γ2 = 10−2.

Fig. 3
Fig. 3

Reflectivity lineshapes (Doppler-broadened system) for various angles. γ1γ2 = 10−2

Equations (24)

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R = κ 2 4 α r 2 [ 1 - exp ( - 2 α r L ) ] 2 ,
t ρ ( t ) = - i h [ H ( t ) , ρ ( t ) ] + relaxation ( T 1 , T 2 ) terms ,
H = ω 0 2 ( 1 0 0 - 1 ) - μ E ( t ) ( 0 1 1 0 ) .
ρ ( t ) ρ ( x , v , t ) , E ( t ) E ( x , t ) , t d d t t + v · .
( t + v · + 1 T 1 ) × [ ρ 3 ( x , v , t ) - ρ 3 ( 0 ) ( v ) ] = 2 μ h E ( x , t ) ρ 2 ,
( t + v · + 1 T 2 ) ρ 1 ( x , v , t ) = - ω 0 ρ 2 ,
( t + v · + 1 T 2 ) ρ 2 ( x , v , t ) = ω 0 ρ 1 + 2 μ h E ( x , t ) ρ 3 ,
ρ 3 ( 0 ) ( v ) = ρ 0 ( π u 2 ) 3 / 2 exp ( - v 2 u 2 ) ,
P ( x , t ) = μ d 3 v ρ 1 ( x , v , t ) .
κ * = α 0 E 1 E 2 E s 2 γ 1 γ 2 2 0 1 d x 0 1 d y × ( { - i δ γ 2 1 d 2 ( x , θ ) Im Z [ ( 1 - x ) ( i + δ ) γ 2 + i x γ 1 d ( x , θ ) ] - 1 - x f 3 ( x , y , θ ) Z [ ( 1 - x ) ( i + δ ) γ 2 + i x γ 1 f ( x , y , θ ) ] } + ( θ π - θ ) ) ,
γ 1 , 2 1 T 1 , 2 1 k u ,
d ( x , θ ) ( 1 - 2 x cos θ + x 2 ) 1 / 2 ,
f ( x , y , θ ) [ ( 1 - x ) 2 ( 1 - 2 y ) 2 cos 2 ( θ / 2 ) + ( 1 + x ) 2 sin 2 ( θ / 2 ) ] 1 / 2 ,
d ( x , θ ) d ( 1 , θ ) ,
f ( x , y , θ ) f ( 1 , y , θ )
κ = - 2 α 0 ( 1 + δ 2 ) ( i + δ ) E 1 E 2 E 0 2 m ( γ 1 , θ ) ,
m ( γ 1 , θ ) = π 2 [ s + exp ( s + 2 ) erfc ( s + ) + s - exp ( s - 2 ) erfc ( s - ) ] ,
s + = γ 1 2 sin θ / 2 ,
s - = γ 1 2 cos θ / 2 .
κ 2 i π α 0 E 1 E 2 E s 2 γ 1 γ 2 2 sin 2 θ ;             θ 0.
κ - α 0 2 E 1 E 2 E s 2 π γ 2 i + δ ;             θ = 0.
sin 4 θ 0 16 γ 1 2 γ 2 2 ( 1 + δ 2 ) .
( Δ ω ) 2 min [ Γ d 2 , ( sin 4 θ 16 γ 1 2 + γ 2 2 ) Γ d 2 ] ,
R γ 1 2 γ 2 2 sin 4 θ + 16 γ 1 2 γ 2 2 ( 1 + δ 2 ) × ( E 1 E 2 E s 2 ) 2 [ 1 - exp ( - 2 α r L ) ] 2 .

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