Abstract

We present a new method for analyzing the propagation of short optical pulses through a dispersive medium. The idea is to decompose the pulse Fourier spectrum into an infinite set of elementary wave groups, each propagating with its own group velocity. Such a method may be generalized to the case of nonlinear self-phase-modulation and points out the asymmetry observed in the white-light continua generated in the femtosecond regime.

© 1993 Optical Society of America

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References

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  1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
    [CrossRef]
  2. C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
    [CrossRef]
  3. R. L. Fork, C. H. Brito-Cruz, P. C. Becker, C. V. Shank, Opt. Lett. 12, 483 (1987).
    [CrossRef] [PubMed]
  4. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  5. J. M. Combes, A. Grossmann, Ph. Tchamitchian, Wavelets, Time–Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1987).
  6. C. Hirlimann, J.-F. Morhange, Appl. Opt. 31, 3263 (1992).
    [CrossRef] [PubMed]
  7. J. Paye, in Digest of Ultrafast Phenomena VIII (Ecole Nationale Supérieure des Techniques Avancées, Paris, 1992), p. 112.
  8. D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).
  9. C. Hirlimann, M. May, J-F. Morhange, Ann. Phys. Colloq. 2 (Suppl. 16), 181 (1991).
  10. W. H. Knox, N. M. Pearson, K. D. Li, C. A. Hirlimann, Opt. Lett. 13, 574 (1988).
    [CrossRef] [PubMed]
  11. S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, Berlin, 1977), p. 156.
  12. E. Washburn, ed., International Critical Tables of Numerical Data Physics, Chemistry and Technology (McGraw-Hill, New York, 1929), Vol. VI, pp. 341–344.
  13. L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  14. H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
    [CrossRef]
  15. R. L. Fork, C. V. Shank, C. Hirlimann, R. T. Yen, W. J. Tomlinson, Opt. Lett. 8, 1 (1983).
    [CrossRef] [PubMed]

1992 (1)

1991 (1)

C. Hirlimann, M. May, J-F. Morhange, Ann. Phys. Colloq. 2 (Suppl. 16), 181 (1991).

1988 (1)

1987 (1)

1983 (1)

1981 (1)

H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

1980 (1)

L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1970 (1)

C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
[CrossRef]

1969 (1)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

1946 (1)

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Ballant, A. C.

H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Becker, P. C.

Brito-Cruz, C. H.

Combes, J. M.

J. M. Combes, A. Grossmann, Ph. Tchamitchian, Wavelets, Time–Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1987).

Fork, R. L.

Gabor, D.

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Garrett, C. G. B.

C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
[CrossRef]

Gordon, J. P.

L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Grischkowsky, D.

H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Grossmann, A.

J. M. Combes, A. Grossmann, Ph. Tchamitchian, Wavelets, Time–Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1987).

Hirlimann, C.

Hirlimann, C. A.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Knox, W. H.

Li, K. D.

May, M.

C. Hirlimann, M. May, J-F. Morhange, Ann. Phys. Colloq. 2 (Suppl. 16), 181 (1991).

McCumber, D. E.

C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
[CrossRef]

Mollenenauer, L. F.

L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Morhange, J.-F.

Morhange, J-F.

C. Hirlimann, M. May, J-F. Morhange, Ann. Phys. Colloq. 2 (Suppl. 16), 181 (1991).

Nakatsuka, H.

H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Paye, J.

J. Paye, in Digest of Ultrafast Phenomena VIII (Ecole Nationale Supérieure des Techniques Avancées, Paris, 1992), p. 112.

Pearson, N. M.

Shank, C. V.

Stolen, R. H.

L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Tchamitchian, Ph.

J. M. Combes, A. Grossmann, Ph. Tchamitchian, Wavelets, Time–Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1987).

Tomlinson, W. J.

Treacy, E. B.

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Yen, R. T.

Ann. Phys. Colloq. (1)

C. Hirlimann, M. May, J-F. Morhange, Ann. Phys. Colloq. 2 (Suppl. 16), 181 (1991).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

J. Inst. Electr. Eng. (1)

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Opt. Lett. (3)

Phys. Rev. A (1)

C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
[CrossRef]

Phys. Rev. Lett. (2)

L. F. Mollenenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Ballant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Other (5)

J. Paye, in Digest of Ultrafast Phenomena VIII (Ecole Nationale Supérieure des Techniques Avancées, Paris, 1992), p. 112.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

J. M. Combes, A. Grossmann, Ph. Tchamitchian, Wavelets, Time–Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1987).

S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, Berlin, 1977), p. 156.

E. Washburn, ed., International Critical Tables of Numerical Data Physics, Chemistry and Technology (McGraw-Hill, New York, 1929), Vol. VI, pp. 341–344.

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Figures (2)

Fig. 1
Fig. 1

Amplitude of the maxima of the wave groups, seen at a particular point z of the quartz as a function of the time and the frequency: (a) z = 0, (b) z = 2 cm.

Fig. 2
Fig. 2

Amplitude of the wave groups’ maxima for a 3-fs pulse calculated for a phase shift at the peak of the pulse. z = 4 cm, Ψ(ω0) = π.

Equations (14)

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δ S w ( Ω , t ) = O ( Ω , t ) δ Ω ,
O ( Ω , t ) = S ( ω ) exp ( i ω t ) exp [ τ a 2 4 ( Ω ω ) 2 ] d ω .
+ O ( Ω , t ) d Ω = s ( t ) ,
O ( Ω , t ) = s ( θ ) exp [ ( t θ ) 2 τ a 2 ] exp [ i Ω ( t θ ) ] d θ .
δ S w ( Ω , t , z ) = δ Ω S ( ω ) × exp { i [ ω t Φ ( ω ) ] } exp [ τ a 2 4 ( Ω ω ) 2 ] d ω ,
δ S w ( Ω , t , z ) = δ Ω exp { i [ Ω t g Ω Φ ( Ω ) ] } O ( Ω , t t g Ω ) ,
O nl ( Ω , t ) = s ( θ ) exp [ ( t θ ) 2 τ a 2 ] × exp [ i Ω ( t θ ) ] exp [ i Ω c z n 2 | s ( θ ) | 2 ] d θ .
O nl ( Ω , t ) = p 0 ( i ) p p ! [ Ψ ( Ω ) ] p F 2 p + 1 ( ω ω 0 ) × exp ( i ω t ) exp [ τ a 2 4 ( Ω ω ) 2 ] d ω ,
Ψ ( Ω ) = Ω c n 2 ( A ) 2 z
δ S w ( Ω , t , 0 ) = exp ( t 2 τ 2 + τ a 2 ) × exp [ τ 2 τ a 2 4 ( τ 2 + τ a 2 ) ( Ω ω 0 ) 2 ] × exp [ i ( ω 0 τ 2 + Ω τ a 2 τ 2 + τ a 2 ) t ] δ Ω .
M ( Ω , 0 ) = exp [ τ 2 τ a 2 4 ( τ 2 + τ a 2 ) ( Ω ω 0 ) 2 ] exp [ τ 2 4 ( Ω ω 0 ) 2 ] .
( δ S w ) out = δ Ω exp [ ( t t g Ω ) 2 τ a 2 ] exp { i [ Ω t Φ ( Ω ) ] } × p 0 ( i ) p p ! [ Ψ ( Ω ) ] p exp [ τ 2 ( Ω ω 0 ) 2 4 ( 2 p + 1 ) ] .
M ( Ω , z ) = p 0 ( i ) p p ! [ Ψ ( Ω ) ] p exp [ τ 2 ( Ω ω 0 ) 2 4 ( 2 p + 1 ) ] ,
( Δ t ) p = t g [ ω 0 + ( Δ Ω ) e 2 ] t g [ ω 0 ( Δ Ω ) e 2 ] ,

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