Abstract

The coupled equations satisfied by the pump, probe, and signal waves in a four-wave interaction in a two-level saturable absorber are derived, taking into account the effects of absorption and depletion. Numerical solutions of these equations are presented in two limits; one in which the probe-wave amplitude is small compared with that of the pump waves and another in which the probe- and pump-wave amplitudes are comparable. The results indicate that absorption effects are important when the wave amplitudes are less than the saturation amplitude of the absorption process. The importance of the depletion effects depends on the type of four-wave interaction. The most significant effects occur when the undepleted theory predicts probe-wave reflectivities exceeding unity.

© 1983 Optical Society of America

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References

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  1. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); Opt. Lett. 3, 203 (1978).
    [Crossref] [PubMed]
  2. D. E. Watkins, J. F. Figueira, and S. J. Thomas, “Observation of resonantly enhanced degenerate four-wave mixing in doped alkali halides,” Opt. Lett. 5, 169–171 (1980).
    [Crossref] [PubMed]
  3. P. F. Liao and D. M. Bloom, “Continuous-wave backward-wave generation by degenerate four-wave mixing in ruby,” Opt. Lett. 3, 4–6 (1978).
    [Crossref] [PubMed]
  4. S. M. Wandzura, “Effects of atomic motion on wavefront conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Lett. 4, 208–210 (1979).
    [Crossref] [PubMed]
  5. J. F. Lam and R. L. Abrams, “Theory of nonlinear optical coherences in resonant degenerate four-wave mixing,” Phys. Rev. A 26, 1539–1548 (1982).
    [Crossref]
  6. A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 465–467.

1982 (1)

J. F. Lam and R. L. Abrams, “Theory of nonlinear optical coherences in resonant degenerate four-wave mixing,” Phys. Rev. A 26, 1539–1548 (1982).
[Crossref]

1980 (1)

1979 (1)

1978 (2)

Abrams, R. L.

J. F. Lam and R. L. Abrams, “Theory of nonlinear optical coherences in resonant degenerate four-wave mixing,” Phys. Rev. A 26, 1539–1548 (1982).
[Crossref]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); Opt. Lett. 3, 203 (1978).
[Crossref] [PubMed]

Bloom, D. M.

Figueira, J. F.

Lam, J. F.

J. F. Lam and R. L. Abrams, “Theory of nonlinear optical coherences in resonant degenerate four-wave mixing,” Phys. Rev. A 26, 1539–1548 (1982).
[Crossref]

Liao, P. F.

Lind, R. C.

Thomas, S. J.

Wandzura, S. M.

Watkins, D. E.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 465–467.

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Figures (5)

Fig. 1
Fig. 1

Four-wave mixing geometry.

Fig. 2
Fig. 2

Weak-probe reflectivity for parallel polarized pumps with δ = 0 (constant-pump theory, dashed lines; variable-pump theory, solid lines).

Fig. 3
Fig. 3

Weak-probe reflectivity for orthogonally polarized pumps with δ = 0 (constant-pump theory, dashed lines; variable-pump theory, solid lines).

Fig. 4
Fig. 4

Weak-probe reflectivity for parallel polarized pumps, with δ = 10 (constant-pump theory, dashed lines; variable-pump theory, solid lines).

Fig. 5
Fig. 5

Strong-probe reflectivity for orthogonally polarized pumps with δ = 15 (constant-pump theory, dashed lines; variable, depleted-pump theory, solid lines). The curves for the results that include the effect of depletion are labeled by the ratio of the probe-wave irradiance Ip to that of the pumps (If = IbI).

Equations (48)

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P = 2 0 α 0 k i ( 1 + i δ ) 1 + δ 2 E 1 + E 2 / Î s ,
2 E - 1 c 2 2 E t 2 = 1 0 c 2 2 P t 2 ,
E = [ E f exp ( i k · x ) + E b exp ( - i k · x ) + E p exp ( i K · x ) + E s exp ( - i K · x ) ] exp ( - i ω t ) ,
[ ( i 2 k · E f + T 1 2 E f ) exp ( i k · x ) + ( - i 2 k · E b + T 1 2 E b ) exp ( - i k · x ) + ( i 2 K · E p + T 2 2 E p ) exp ( i K · x ) + ( - i 2 K · E s ) + T 2 2 E s ) exp ( - i K · x ) ] exp ( - i ω t ) + k 2 0 P = 0.
( i 2 k · + T 1 2 ) E f + P ˆ f = 0 ,
( - i 2 k · + T 1 2 ) E b + P ˆ b = 0 ,
( i 2 K · + T 2 2 ) E p + P ˆ p = 0 ,
( - i 2 K · + T 2 2 ) E s + P ˆ s = 0 ,
P ˆ i = 1 l 1 l 2 - l 1 / 2 l 1 / 2 d ξ 1 - l 2 / 2 l 2 / 2 d ξ 2 exp ( - i k i · x + i ω t ) k 2 0 P ,
l 1 = λ 2 cos θ 2 ,
l 2 = λ 2 sin θ 2 .
ζ = 2 α 0 z cos θ 2 .
d E f d ζ + 1 - i δ 2 p 1 E 1 = 0 ,
- d E b d ζ + 1 - i δ 2 p 2 E b = 0 ,
d E p d ζ + 1 - i δ 2 ( f 1 E p - g E s * ) = 0 ,
- d E s d ζ + 1 - 1 δ 2 ( f 2 E s - g E p * ) = 0.
q = [ 1 + 2 ( I f + I b ) + ( I f - I b ) 2 ] 1 / 2 ,
p 1 , 2 = 1 q - 1 2 I f , b ( 1 + I f + I b q - 1 ) ,
g = - 2 E f E b q 3 ,
f 1 , 2 = 1 + I f + I b q 3 ,
s = 1 + I f + I b ,
p 1 , 2 = 1 / s ,
g = ( I f I b ) 1 / 2 s 2 ,
f 1 , 2 = 1 + I f , b s 2 .
d E f d ζ + ( 1 - i δ ) ( d 2 - h I s ) E f - ( 1 - i δ ) h E p E s E b * = 0 ,
- d E b d ζ + ( 1 - i δ ) ( d 2 - h I p ) E b - ( 1 - i δ ) h E p E s E f * = 0 ,
d E p d ζ + ( 1 - i δ ) ( d 2 - h I b ) E p - ( 1 - i δ ) h E f E b E s * = 0 ,
- d E s d ζ + ( 1 - i δ ) ( d 2 - h I f ) E s - ( 1 - i δ ) h E f E b E p * = 0 ,
I f s b p = E f E s * + E b * E p ,
d = 1 [ ( 1 + I f + I b + I p + I s ) 2 - 4 I f s b p 2 ] 1 / 2 ,
h = 1 2 I f s b p 2 [ ( 1 + I f + I b + I p + I s ) d - 1 ] .
d R d ζ - 1 2 [ f 1 + f 2 + i δ ( f 1 - f 2 ) ] R + 1 + i δ 2 g R 2 + 1 - i δ 2 g * = 0 ,
R ( ζ 0 ) = 0.
R = i κ sin ( ζ 0 w 2 ) ω cos ( ζ 0 w 2 ) + α r sin ( ζ 0 w 2 ) .
i κ * = ( 1 + i δ ) g 1 + δ 2 ,
α τ = - Re [ ( 1 + i δ ) f 1 + δ 2 ] ,
w = [ κ 2 - α τ 2 ] 1 / 2 .
d I f d ζ + p 1 I f = 0 ,
- d I b d ζ + p 2 I b = 0 ,
d ( ϕ f + ϕ b ) d ζ - δ 2 ( p 1 - p 2 ) = 0 ,
d A f d ζ + ( d 2 - h I s ) A f - ( 1 + δ 2 ) 1 / 2 h A b A p A s × cos ( ϕ s d + ϕ δ ) = 0 ,
- d A b d ζ + ( d 2 - h I p ) A b - ( 1 + δ 2 ) 1 / 2 h A f A p A s × cos ( ϕ s d + ϕ δ ) = 0 ,
d A p d ζ + ( d 2 - h I b ) A p - ( 1 + δ 2 ) 1 / 2 h A s A f A b × cos ( ϕ s d - ϕ δ ) = 0 ,
- d A s d ζ + ( d 2 - h I f ) A s - ( 1 + δ 2 ) 1 / 2 h A p A f A b × cos ( ϕ s d - ϕ δ ) = 0 ,
d ϕ s d d ζ + δ h ( I f - I b - I p + I s ) + ( 1 + δ 2 ) 1 / 2 h × [ ( A b A p A s A f - A f A p A s A b ) sin ( ϕ s d + ϕ δ ) + ( A f A b A s A p - A f A b A p A s ) sin ( ϕ s d - ϕ δ ) ] = 0 ,
ϕ s d = ϕ f + ϕ b - ϕ p - ϕ s ,
ϕ δ = tan - 1 ( δ ) .
- A s [ d ϕ s d ζ + δ ( d - h I f ) ] - ( 1 + δ 2 ) 1 / 2 h A p A f A b × sin ( ϕ s d - ϕ δ ) = 0.