Abstract

Spectral superbroadening of ultrashort-pulse propagation in a nonlinear medium is calculated by solving the nonlinear-wave equation more exactly. The effects of four-wave mixing and pulse deformation on phase modulation are included. The results yield an asymmetric Stokes–anti-Stokes broadening in fair agreement with the experimental observation.

© 1984 Optical Society of America

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References

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  1. R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 584 (1970);Phys. Rev. Lett. 24, 592 (1970).
    [Crossref]
  2. G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
    [Crossref]
  3. N. Bloembergen, Opt. Commun. 8, 285 (1973);W. L. Smith, P. Liu, N. Bloembergen, Phys. Rev. A 15, 2396 (1977).
    [Crossref]
  4. A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
    [Crossref]
  5. F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967);T. K. Gustafson, J. P. Taran, H. A. Haus, J. F. Lifsitz, P. L. Kelley, Phys Rev. 177, 306 (1969).
    [Crossref]
  6. R. A. Fisher, W. K. Bischel, J. Appl. Phys. 46, 4921 (1975);R. A. Fisher, B. R. Suydam, D. Yevick, Opt. Lett. 8, 611 (1983).
    [Crossref] [PubMed]
  7. R. L. Fork, C. V. Shank, C. Hirlimann, R. Yen, Opt. Lett. 8, 1 (1983).
    [Crossref] [PubMed]
  8. P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), Chap. 1.
  9. F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
    [Crossref]

1983 (1)

1975 (1)

R. A. Fisher, W. K. Bischel, J. Appl. Phys. 46, 4921 (1975);R. A. Fisher, B. R. Suydam, D. Yevick, Opt. Lett. 8, 611 (1983).
[Crossref] [PubMed]

1973 (3)

G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
[Crossref]

N. Bloembergen, Opt. Commun. 8, 285 (1973);W. L. Smith, P. Liu, N. Bloembergen, Phys. Rev. A 15, 2396 (1977).
[Crossref]

A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
[Crossref]

1970 (1)

R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 584 (1970);Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

1967 (2)

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967);T. K. Gustafson, J. P. Taran, H. A. Haus, J. F. Lifsitz, P. L. Kelley, Phys Rev. 177, 306 (1969).
[Crossref]

Alfano, R. R.

R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 584 (1970);Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

Bischel, W. K.

R. A. Fisher, W. K. Bischel, J. Appl. Phys. 46, 4921 (1975);R. A. Fisher, B. R. Suydam, D. Yevick, Opt. Lett. 8, 611 (1983).
[Crossref] [PubMed]

Bloembergen, N.

N. Bloembergen, Opt. Commun. 8, 285 (1973);W. L. Smith, P. Liu, N. Bloembergen, Phys. Rev. A 15, 2396 (1977).
[Crossref]

Busch, G. E.

G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
[Crossref]

DeMartini, F.

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Fisher, R. A.

R. A. Fisher, W. K. Bischel, J. Appl. Phys. 46, 4921 (1975);R. A. Fisher, B. R. Suydam, D. Yevick, Opt. Lett. 8, 611 (1983).
[Crossref] [PubMed]

Fork, R. L.

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), Chap. 1.

Gustafson, T. F.

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Hirlimann, C.

Jones, R. P.

G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
[Crossref]

Kaiser, W.

A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
[Crossref]

Kelley, P. L.

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Laubereau, A.

A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
[Crossref]

Penzkofer, A.

A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
[Crossref]

Rentzepis, P. M.

G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
[Crossref]

Shank, C. V.

Shapiro, S. L.

R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 584 (1970);Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

Shimizu, F.

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967);T. K. Gustafson, J. P. Taran, H. A. Haus, J. F. Lifsitz, P. L. Kelley, Phys Rev. 177, 306 (1969).
[Crossref]

Townes, C. H.

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Yen, R.

Chem. Phys. Lett. (1)

G. E. Busch, R. P. Jones, P. M. Rentzepis, Chem. Phys. Lett. 18, 178 (1973).
[Crossref]

J. Appl. Phys. (1)

R. A. Fisher, W. K. Bischel, J. Appl. Phys. 46, 4921 (1975);R. A. Fisher, B. R. Suydam, D. Yevick, Opt. Lett. 8, 611 (1983).
[Crossref] [PubMed]

Opt. Commun. (1)

N. Bloembergen, Opt. Commun. 8, 285 (1973);W. L. Smith, P. Liu, N. Bloembergen, Phys. Rev. A 15, 2396 (1977).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

F. DeMartini, C. H. Townes, T. F. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Phys. Rev. Lett. (3)

R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 584 (1970);Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973);A. Penzkofer, Opt. Commun. 11, 265 (1974);A. Penzkofer, A. Seilmeier, W. Kaiser, Opt. Commun. 14, 363 (1975);A. Penzkofer, W. Kaiser, Opt. Quantum Electron. 9, 315 (1977).
[Crossref]

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967);T. K. Gustafson, J. P. Taran, H. A. Haus, J. F. Lifsitz, P. L. Kelley, Phys Rev. 177, 306 (1969).
[Crossref]

Other (1)

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964), Chap. 1.

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Figures (3)

Fig. 1
Fig. 1

Maximum Stokes (Δω < 0) and anti-Stokes (Δω+ > 0) shifts calculated with different models: usual theory of self-phase modulation (— — —), our theory without pulse distortion (— · —), our theory with pulse distortion (—).

Fig. 2
Fig. 2

Pulse distortion during propagation in a nonlinear medium at various values of Q = n2A2z/cτ, with x = (tn0z/c)/τ.

Fig. 3
Fig. 3

Frequency shift due to phase modulation as a function of x = (tn0z/c)/τ at Q = 2.0.

Equations (12)

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( 2 z 2 n 0 2 c 2 2 t 2 ) E = 4 π c 2 2 t 2 P ( 3 ) ,
i 2 k 0 D + = D ( D + ) + = m = 0 [ ( D i 2 k 0 ) m ] .
i 2 k 0 D + m = 0 ( 1 i ω 0 t ) m = 4 π ω 0 2 c 2 χ ( 3 ) ( 1 + i ω 0 t ) | | 2 2 ω 0 2 n 0 c 2 n 2 | | 2 ( 1 + i ω 0 t ) ,
[ z + n 0 c ( 1 + n 2 n 0 | | 2 ) t ] = i n 2 ω 0 c | | 2 .
[ z + n 0 c ( 1 + n 2 n 0 | | 2 ) t ] | | = 0 ,
[ z + n 0 c ( 1 + n 2 n 0 | | 2 ) t ] ϕ = n 2 ω 0 c | | 2 .
| ( z , t ) | 2 = A 2 / cosh [ ( t n 0 c ) / τ ] .
ϕ = ω 0 τ x ω 0 τ sinh 1 [ sinh x ( n 2 / c τ ) A 2 z ] ,
Δ ω / ω 0 = ( 1 / ω 0 ) ( ϕ / t ) = [ 1 + ( Q 2 2 Q sinh x ) / cosh 2 x ] 1 / 2 1
Δ ω / ω 0 Q sinh x / cosh 2 x ,
Δ ω ± / ω 0 = ½ [ ( Q 2 + 4 ) 1 / 2 ± | Q | ] 1 ,
| ( z , t ) | 2 = A 2 / cosh [ x Q | ( z , t ) | 2 / A 2 ] .

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