Abstract

The coupled-mode equations for two-wave mixing are extended to include light-scattering noise and are solved using the stochastic noise model. Solutions are given for amplitude, phase, and output-power fluctuations during weak-signal amplification. Minimum power requirements for low-noise signal amplification in Kerr media are determined.

© 1992 Optical Society of America

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References

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  1. R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
    [CrossRef]
  2. D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
    [CrossRef]
  3. R. McGraw, Phys. Rev. A42, 2235 (1990).
  4. P. Yeh, J. Opt. Soc. Am. B 3, 747 (1986).
    [CrossRef]
  5. Y. Silberberg, I. Bar-Joseph, J. Opt. Soc. Am. B 1, 662 (1984).
    [CrossRef]
  6. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chap. 11.
  7. R. McGraw, D. Rogovin, Phys. Rev. A35, 1181 (1987).
  8. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.
  9. N. Davidson, Statistical Mechanics (McGraw-Hill, New York, 1962), Chap. 14.

1990 (1)

R. McGraw, Phys. Rev. A42, 2235 (1990).

1989 (2)

R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
[CrossRef]

D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
[CrossRef]

1987 (1)

R. McGraw, D. Rogovin, Phys. Rev. A35, 1181 (1987).

1986 (1)

1984 (1)

Bar-Joseph, I.

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chap. 11.

Davidson, N.

N. Davidson, Statistical Mechanics (McGraw-Hill, New York, 1962), Chap. 14.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.

Gavrielides, A.

R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
[CrossRef]

D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
[CrossRef]

McGraw, R.

R. McGraw, Phys. Rev. A42, 2235 (1990).

D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
[CrossRef]

R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
[CrossRef]

R. McGraw, D. Rogovin, Phys. Rev. A35, 1181 (1987).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chap. 11.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.

Rogovin, D.

R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
[CrossRef]

D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
[CrossRef]

R. McGraw, D. Rogovin, Phys. Rev. A35, 1181 (1987).

Silberberg, Y.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.

Yeh, P.

Appl. Phys. Lett. (2)

R. McGraw, D. Rogovin, A. Gavrielides, Appl. Phys. Lett. 54, 199 (1989).
[CrossRef]

D. Rogovin, R. McGraw, A. Gavrielides, Appl. Phys. Lett. 55, 1937 (1989).
[CrossRef]

J. Opt. Soc. Am. B (2)

Phys. Rev. (2)

R. McGraw, D. Rogovin, Phys. Rev. A35, 1181 (1987).

R. McGraw, Phys. Rev. A42, 2235 (1990).

Other (3)

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), Chap. 11.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University, New York, 1986), p. 203.

N. Davidson, Statistical Mechanics (McGraw-Hill, New York, 1962), Chap. 14.

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Figures (3)

Fig. 1
Fig. 1

Two-wave mixing geometry for weak-signal amplification showing contributions to signal and to noise.

Fig. 2
Fig. 2

Illustrative calculations obtained using the stochastic noise model. The figure shows two-wave-mixing gain and noise for conditions (a) P2(0) = 1 mW ɛ2 = 1.0 × 10−6 cm3/erg, L = 0.1 cm, λ = 0.5 μm, Ipump = 10 kW/cm2, and T = 300 K with α0 computed from Eq. (19); (b) same conditions as (a), but with α0 = 0; (c) same conditions as (b), but with P2(0) = 10 mW; (d) same conditions as (b), but with a higher gain achieved by setting Ipump = 32 kW/cm2. Pump depletion and nonsaturable background loss are included in Fig. 2(a) and neglected in Figs. 2(b)–2(d). For different sets of conditions resulting in the same gain and figure of merit, scatter plots identical to those of Figs. 2(b)–2(d) (apart from a possible ordinate scale change) will be obtained.

Fig. 3
Fig. 3

Nondimensional coefficient f(hLτ) defined by Eq. (B6). The value of exp(hL)—the maximum signal power gain in the absence of noise—is indicated for each curve.

Equations (43)

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ɛ ( r ) = ɛ 0 + ɛ 2 E 2 ¯ ( r ) ,
δ ɛ ( q ) 2 = 8 π k T ɛ 2 / V s ,
2 E ( r , t ) = ( c - 2 2 / t 2 ) D ( r , t ) ,
D ( r , t ) = [ ɛ 0 + δ ɛ ( r , t ) + ɛ 2 E 2 ¯ ( r , t ) ] E ( r , t )
Δ ɛ ( q , t ) / t = - Γ ( q ) Δ ɛ ( q , t ) + F ( t ) ,
F ( t ) 2 = k T Γ ( q ) .
δ ɛ ( q , 0 ) δ ɛ ( q , t ) = δ ɛ ( q ) 2 exp [ - Γ ( q ) t ]
δ ɛ ( r , t ) = d Ω [ a Ω cos ( q · r - Ω t ) + b Ω sin ( q · r - Ω t ) ] ,
a Ω = b Ω = 0
a Ω 2 = b Ω 2 = ( τ / π ) δ ɛ ( q ) 2 / [ 1 + ( Ω τ ) 2 ] .
G ( Ω ) = ½ [ a Ω 2 + b Ω 2 ] = ( τ / π ) δ ɛ ( q ) 2 / [ 1 + ( Ω τ ) 2 ] ,
G ( Ω ) d Ω = δ ɛ ( q ) 2
E ( r , t ) = e 1 E 1 ( r ) cos [ K 1 · r - ω 1 t + θ 1 ( r ) ] + e 2 E 2 ( r ) cos [ K 2 · r - ω 2 t + θ 2 ( r ) ] ,
Δ ɛ ( r , t ) = a 1 cos ( q · r - Ω t ) + b 1 sin ( q · r - Ω t ) ,
a 1 = ( e 1 · e 2 ) ɛ 2 E 1 E 2 / [ 1 + ( Ω τ ) 2 ]
b 1 = - ( e 1 · e 2 ) ɛ 2 E 1 E 2 Ω τ / [ 1 + ( Ω τ ) 2 ]
D ( r , t ) = ɛ 0 E ( r , t ) + ½ ( a 1 + a Ω ) E 1 ( r ) cos ( K 2 · r - ω 2 t + θ 2 ) e 1 + ½ ( a 1 + a Ω ) E 2 ( r ) cos ( K 1 · r - ω 1 t + θ 1 ) e 2 - ½ ( b 1 + b Ω ) E 1 ( r ) sin ( K 2 · r - ω 2 t + θ 2 ) e 1 + ½ ( b 1 + b Ω ) E 2 ( r ) sin ( K 1 · r - ω 1 t + θ 1 ) e 2 .
( K 1 · ) E 1 = K 2 / 4 ɛ 0 ( b 1 + b Ω ) ( e 1 · e 2 ) E 2 - ( α 0 / 2 ) E 1 ,
( K 1 · ) θ 1 = K 2 / 4 ɛ 0 ( a 1 + a Ω ) ( e 1 · e 2 ) E 2 / E 1 ,
( K 2 · ) E 2 = - K 2 / 4 ɛ 0 ( b 1 + b Ω ) ( e 1 · e 2 ) E 1 - ( α 0 / 2 ) E 2 ,
( K 2 · ) θ 2 = K 2 / 4 ɛ 0 ( a 1 + a Ω ) ( e 1 · e 2 ) E 1 / E 2 ,
α 0 = ( ω 4 / 6 π c 4 ) V s δ ɛ ( q ) 2 = 4 3 ( 2 π υ / c ) 4 k T ɛ 2 ,
P = E · d P / d t = - ( 1 16 π ) b 1 ( e 1 · e 2 ) E 1 E 2 Ω ,
a ¯ Ω = b ¯ Ω = 0
a ¯ Ω 2 = b ¯ Ω 2 = δ ɛ ( q ) 2 / [ 1 + ( Ω τ ) 2 ] = ( 8 π k T ɛ 2 / V s ) / [ 1 + ( Ω τ ) 2 ] ,
δ ɛ = 4 π α δ n
ɛ 2 = 2 π α 2 n / k T .
δ n ( q ) 2 = n 2 k T β T / V s = n / V s ,
δ ɛ ( r , t ) = A ( t ) cos ( q · r ) ,
A ( 0 ) A ( t ) = A 2 exp [ - Γ ( q ) t ] ,
A 2 = 2 δ ɛ ( q ) 2 ,
A ( t ) = d Ω [ α Ω cos ( Ω t ) + β Ω sin ( Ω t ) ] .
α Ω = β Ω = 0
α Ω 2 = β Ω 2 = ( 2 τ / π ) A 2 / [ 1 + ( Ω τ ) 2 ] ,
δ ɛ ( r , t ) = d Ω [ a Ω cos ( q · r - Ω t ) + b Ω sin ( q · r - Ω t ) ] ,
d E 2 / d z = α S E 2 + α N E 1 ,
α S = ( K / 4 ɛ 0 ) ( e 1 · e 2 ) 2 ɛ 2 E 1 2 Ω τ / [ 1 + ( Ω τ ) 2 ] = h Ω τ / [ 1 + ( Ω τ ) 2 ] ,
α N = ( K / 4 ɛ 0 ) ( e 1 · e 2 ) b ¯ Ω .
E 2 ( L ) = exp ( α S L ) E 2 ( 0 ) + ( α N / α S ) [ exp ( α S L ) - 1 ] E 1 .
σ ( b ¯ Ω ) = ( 8 π k T ɛ 2 / V s ) 1 / 2 / [ 1 + ( Ω τ ) 2 ] 1 / 2 .
σ ( P 2 ) / P 2 ( 0 ) = [ ( 2 π ) 1 / 2 g E g E - 1 / ( Ω τ ln g E ) 1 / 2 ] [ k T υ / P 2 ( 0 ) ] 1 / 2 = f ( h L , Ω τ ) [ k T υ / P 2 ( 0 ) ] 1 / 2 ,
F = P 2 ( 0 ) / k T υ ,
σ ( P 2 ) / P 2 ( 0 ) = f ( h L , Ω τ ) F - 1 / 2 .

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