Abstract

Stochastic field propagation in nonlinear dispersive media is studied in the undepleted-pump approximation in both the normal- and the anomalous-dispersion regimes. A statistical description of modulational instability is given in the anomalous-dispersion regime. Nonlinear dispersive effects are present even in the normal-dispersion regime. Analytical results are obtained for the evolution of the power spectrum and the relative intensity noise and are confirmed by numerical simulations. The results are applied to the four-wave mixing of broadband signals in nonlinear dispersive media.

© 1995 Optical Society of America

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  1. J. T. Manassah, Opt. Lett. 16, 1638 (1991).
    [CrossRef] [PubMed]
  2. B. Gross and J. T. Manassah, Opt. Lett. 16, 1835 (1991).
    [CrossRef] [PubMed]
  3. M. T. de Araujo, H. R. da Cruz, and A. S. Gouviea-Neto, J. Opt. Soc. Am. B 8, 2094 (1991).
    [CrossRef]
  4. S. Ryu, Electron. Lett. 28, 2212 (1992).
    [CrossRef]
  5. K. Kikuchi, IEEE Photon. Technol. Lett. 5, 221 (1993).
    [CrossRef]
  6. J. N. Elgin, Opt. Lett. 18, 10 (1992).
    [CrossRef]
  7. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).
  8. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  9. N. Wax, ed., Noise and Stochastic Processes (Dover, New York, 1954).
  10. A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  11. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, Boston, Mass., 1989).
  12. M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
    [CrossRef] [PubMed]
  13. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
    [CrossRef] [PubMed]
  14. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  15. A. H. Gnauck, R. M. Jopson, and R. M. Derosier, IEEE Photon. Technol. Lett. 5, 104 (1993).
    [CrossRef]
  16. K. Kikuchi, IEEE Photon. Technol. Lett. 6, 104 (1994).
    [CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, Boston, Mass., 1989).

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).

Araujo, M. T. de

M. T. de Araujo, H. R. da Cruz, and A. S. Gouviea-Neto, J. Opt. Soc. Am. B 8, 2094 (1991).
[CrossRef]

Carter, S. J.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[CrossRef] [PubMed]

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).

Cruz, H. R. da

M. T. de Araujo, H. R. da Cruz, and A. S. Gouviea-Neto, J. Opt. Soc. Am. B 8, 2094 (1991).
[CrossRef]

Derosier, R. M.

A. H. Gnauck, R. M. Jopson, and R. M. Derosier, IEEE Photon. Technol. Lett. 5, 104 (1993).
[CrossRef]

Drummond, P. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[CrossRef] [PubMed]

Elgin, J. N.

J. N. Elgin, Opt. Lett. 18, 10 (1992).
[CrossRef]

Gnauck, A. H.

A. H. Gnauck, R. M. Jopson, and R. M. Derosier, IEEE Photon. Technol. Lett. 5, 104 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gouviea-Neto, A. S.

M. T. de Araujo, H. R. da Cruz, and A. S. Gouviea-Neto, J. Opt. Soc. Am. B 8, 2094 (1991).
[CrossRef]

Gross, B.

B. Gross and J. T. Manassah, Opt. Lett. 16, 1835 (1991).
[CrossRef] [PubMed]

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1990).
[CrossRef]

Haus, H. A.

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[CrossRef]

Jopson, R. M.

A. H. Gnauck, R. M. Jopson, and R. M. Derosier, IEEE Photon. Technol. Lett. 5, 104 (1993).
[CrossRef]

Kikuchi, K.

K. Kikuchi, IEEE Photon. Technol. Lett. 6, 104 (1994).
[CrossRef]

K. Kikuchi, IEEE Photon. Technol. Lett. 5, 221 (1993).
[CrossRef]

Lai, Y.

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[CrossRef]

Manassah, J. T.

J. T. Manassah, Opt. Lett. 16, 1638 (1991).
[CrossRef] [PubMed]

B. Gross and J. T. Manassah, Opt. Lett. 16, 1835 (1991).
[CrossRef] [PubMed]

Potasek, M. J.

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

Reid, M. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[CrossRef] [PubMed]

Ryu, S.

S. Ryu, Electron. Lett. 28, 2212 (1992).
[CrossRef]

Shelby, R. M.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[CrossRef] [PubMed]

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).

Yurke, B.

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

Other (16)

J. T. Manassah, Opt. Lett. 16, 1638 (1991).
[CrossRef] [PubMed]

B. Gross and J. T. Manassah, Opt. Lett. 16, 1835 (1991).
[CrossRef] [PubMed]

M. T. de Araujo, H. R. da Cruz, and A. S. Gouviea-Neto, J. Opt. Soc. Am. B 8, 2094 (1991).
[CrossRef]

S. Ryu, Electron. Lett. 28, 2212 (1992).
[CrossRef]

K. Kikuchi, IEEE Photon. Technol. Lett. 5, 221 (1993).
[CrossRef]

J. N. Elgin, Opt. Lett. 18, 10 (1992).
[CrossRef]

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

N. Wax, ed., Noise and Stochastic Processes (Dover, New York, 1954).

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1990).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, Boston, Mass., 1989).

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[CrossRef]

A. H. Gnauck, R. M. Jopson, and R. M. Derosier, IEEE Photon. Technol. Lett. 5, 104 (1993).
[CrossRef]

K. Kikuchi, IEEE Photon. Technol. Lett. 6, 104 (1994).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Spectral evolution at different distances in the normal-dispersion region for a symmetric input spectrum. FWM causes the quadratic growth and fringe formations. The distance is normalized to ξ = |A0|2, and the frequency is normalized as ω[|β|/(γ |A0|2)]1/2/(4π). The FWHM of the input noise spectrum is 0.4, and its average intensity is 3.2 × 10−5 times the pump intensity. The vertical axis has a relative unit.

Fig. 2
Fig. 2

Same as Fig. 1, except for the sign of the GVD parameter. MI effects dominate at a large distance.

Fig. 3
Fig. 3

Numerical simulation result corresponding to Fig. 1. The center portion is the cw spectrum subjected to finite resolution that is due to the temporal window for calculating the spectrum.

Fig. 4
Fig. 4

Same as Fig. 3, except for the sign of the GVD parameter.

Fig. 5
Fig. 5

Analytic RIN spectra at different distances in the normal-dispersion region under conditions identical to those of Fig. 1. FWM causes fringe formations.

Fig. 6
Fig. 6

Same as Fig. 5, except for the sign of the GVD parameter. MI effects dominate at a large distance.

Fig. 7
Fig. 7

Numerical simulation result corresponding to Fig. 5. The center portion is the cw residue.

Fig. 8
Fig. 8

Same as Fig. 7, except for the sign of the GVD parameter.

Fig. 9
Fig. 9

Spectral evolution at different distances in the normal-dispersion region for an asymmetric input spectrum corresponding to FWM with a noisy probe. The noise spectrum is centered at 0.05 with a FWHM of 0.02; other parameters are identical to those of Fig. 1.

Fig. 10
Fig. 10

Same as Fig. 9, except that the noise spectrum is centered at 0.15 with a FWHM of 0.04 and the GVD is anomalous.

Equations (23)

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z A = i 2 β 2 t t 2 A + i γ | A | 2 A ,
z δ A = i β t t 2 δ A + i γ ( | A 0 | 2 δ A + A 0 2 δ A * ) .
d z δ A ¯ ( ω , z ) i β ω 2 δ A ¯ ( ω , z ) i γ | A 0 | 2 δ A ¯ ( ω , z ) = i γ | A 0 | 2 A 0 2 δ A ¯ * ( ω , z ) , d z δ A ¯ * ( ω , z ) + i β ω 2 δ A ¯ * ( ω , z ) + i γ | A 0 | 2 δ A ¯ * ( ω , z ) = i γ | A 0 | 2 A 0 2 δ A ¯ ( ω , z ) ,
[ δ A ¯ ( ω , z ) δ A ¯ * ( ω , z ) ] = c 1 [ 1 r + ] exp ( i k + z ) + c 2 [ r 1 ] exp ( i k z ) ,
k ± ( ω ) = ± [ ( γ | A 0 | 2 + β ω 2 ) 2 ( γ | A 0 | 2 ) 2 ] 1 / 2 ,
r + ( ω ) = k + ( ω ) β ω 2 γ | A 0 | 2 γ A 0 2 = γ A 0 * 2 k + ( ω ) + β ω 2 + γ | A 0 | 2 ,
r ( ω ) = γ A 0 2 k ( ω ) β ω 2 γ | A 0 | 2 = k ( ω ) + β ω 2 + γ | A 0 | 2 γ A 0 * 2 ,
[ δ A ¯ ( ω , z ) δ A ¯ * ( ω , z ) ] = [ M 11 ( ω , z ) M 12 ( ω , z ) M 21 ( ω , z ) M 22 ( ω , z ) ] × [ δ A ¯ ( ω , 0 ) δ A ¯ * ( ω , 0 ) ] ,
M 11 = [ exp ( i k + z ) r + r exp ( i k z ) ] / ( 1 r + r ) , M 12 = r [ exp ( i k z ) exp ( i k + z ) ] / ( 1 r + r ) , M 21 = r + [ exp ( i k + z ) exp ( i k z ) ] / ( 1 r + r ) , M 22 = [ exp ( i k z ) r + r exp ( i k + z ) ] / ( 1 r + r ) .
S ( ω , z ) = | A s ¯ + δ A s ¯ | 2 / T = | A s ¯ | 2 / T + A s δ A s ¯ * + A s ¯ * δ A s ¯ / T + | δ A s ¯ | 2 / T = | A 0 | 2 δ ( ω ) + | δ A ¯ ( ω , z ) | 2 / T ,
Δ S ( ω , z ) = | M 11 δ A ¯ ( ω , 0 ) + M 12 δ A ¯ * ( ω , 0 ) | 2 / T = | M 11 | 2 Δ S ( ω , 0 ) + | M 12 | 2 Δ S ( ω , 0 ) ,
Δ S ( ω , z ) = Δ S ( ω , 0 ) + ( γ | A 0 | 2 ) 2 [ Δ S ( ω , 0 ) + Δ S ( ω , 0 ) ] × sin 2 [ k ( ω ) z ] / [ k ( ω ) ] 2 ,
Δ S ( ω , z ) = Δ S ( ω , 0 ) + ( z γ | A 0 | 2 ) 2 × [ Δ S ( ω , 0 ) + Δ S ( ω , 0 ) ] ,
Δ S ( ω , z ) = ( γ | A 0 | 2 ) 2 [ Δ S ( ω , 0 ) + Δ S ( ω , 0 ) ] × exp ( 2 | k | z ) / ( 2 | k | ) 2
δ I ( t , z ) / I = [ A 0 * δ A ( t , z ) + A 0 δ A * ( t , z ) ] / | A 0 | 2 .
RIN ( ω , z ) = | A 0 * δ A ¯ ( ω , z ) + A 0 δ A ¯ * ( ω , z ) | 2 / T ( T | A 0 | 4 ) .
RIN ( ω , z ) = | A 0 * M 11 + A 0 M 21 | 2 | δ A ¯ ( ω , 0 ) | 2 / ( T | A 0 | 4 ) + | A 0 * M 12 + A 0 M 22 | 2 | δ A ¯ ( ω , 0 ) | 2 / ( T | A 0 | 2 ) = | A 0 * M 11 + A 0 M 21 | 2 Δ S ( ω , 0 ) / | A 0 | 4 + | A 0 * M 12 + A 0 M 22 | 2 Δ S ( ω , 0 ) / | A 0 | 4 .
RIN ( ω , z ) = | A 0 | 2 [ Δ S ( ω , 0 ) + Δ S ( ω , 0 ) ] × { 1 2 γ | A 0 | 2 sin 2 [ k ( ω ) z ] 2 γ | A 0 | 2 + β ω 2 } ,
RIN ( ω , z ) = RIN ( ω , 0 ) { 1 2 γ | A 0 | 2 sin 2 [ k ( ω ) z ] 2 γ | A 0 | 2 + β ω 2 } .
RIN ( ω , z ) = RIN ( ω , 0 ) γ | A 0 | 2 exp ( 2 | k | z ) 4 γ | A 0 | 2 + β 2 ω 2 ,
Δ S ( ω , z ) = I + ( z ) δ ( ω ω ) + I ( z ) δ ( ω + ω ) ,
I + ( z ) = I + ( 0 ) { 1 + ( γ | A 0 | 2 ) 2 sin 2 [ k ( ω ) z ] / [ k ( ω ) ] 2 } , I ( z ) = I + ( 0 ) ( γ | A 0 | 2 ) 2 sin 2 [ k ( ω ) z ] / [ k ( ω ) ] 2 .
I + ( z ) = I + ( 0 ) [ 1 + ( z γ | A 0 | 2 ) 2 ] , I ( z ) = I + ( 0 ) ( z γ | A 0 | 2 ) 2 .

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