Abstract

Soliton initiation and formation in stimulated Raman scattering, using a π phase shift, is treated numerically. The π phase shift is produced in the Stokes seed by modulating its envelope through zero. We find that by modulating the envelope slowly compared with the coherence decay time T2, a soliton pulse is generated that is much shorter than T2. Analytical approximations for the formation rate are also presented.

© 1987 Optical Society of America

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  1. F. Y. F. Chu and A. C. Scott, “Inverse scattering transform for wave–wave scattering,” Phys. Rev. A 12, 2060 (1975).
    [Crossref]
  2. K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
    [Crossref]
  3. R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
    [Crossref]
  4. K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
    [Crossref]
  5. K. Druhl and G. Alsing, “Effect of coherence relaxation on the propagation of optical solitons: an analytical and numerical case study on asymptotic perturbation theory,” Physica 20D, 429 (1986).
  6. H. Steudel, “Stimulated Raman scattering with an initial phase shift: the pre-stage of a soliton,” Opt. Commun. 57, 285 (1986).
    [Crossref]
  7. D. J. Kaup, “Creation of a soliton out of dissipation,” Physica 19D, 125 (1986).
  8. D. J. Kaup, “The method of solution for stimulated Raman scattering and two-phonon propagation,” Physica 6D, 142 (1983).
  9. J. R. Ackerhalt and P. W. Milonni, “Solitons and four-wave mixing,” Phys. Rev. A 33, 3185 (1986).
    [Crossref] [PubMed]
  10. J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661 (1986).
    [Crossref] [PubMed]
  11. R. Meinel, “Backlund transformation and N-solitons solutions for stimulated Raman scattering and resonant two-photon propagation,” Opt. Commun. 49, 224 (1984).
    [Crossref]
  12. H. Steudel, “Solitons in stimulated Raman scattering and resonant two-photon propagation,” Physica 6D, 155 (1983).
  13. M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
    [Crossref]
  14. W. K. Bishel and M. T. Dyer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677 (1986).
    [Crossref]
  15. W. K. Bishel and M. T. Dyer, “Temperature dependence of the Raman linewidth and line shift for Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113 (1986).
    [Crossref]
  16. A. Z. Grasyuk, “Raman lasers (review),” Sov. J. Quantum Electron. 4, 269 (1974).
    [Crossref]
  17. V. A. Gorbunov, “Stimulated Raman scattering in a field of ultrashort light pulses,” Kvant. Elektron. (Moscow) 9, 152 (1982).

1986 (7)

K. Druhl and G. Alsing, “Effect of coherence relaxation on the propagation of optical solitons: an analytical and numerical case study on asymptotic perturbation theory,” Physica 20D, 429 (1986).

H. Steudel, “Stimulated Raman scattering with an initial phase shift: the pre-stage of a soliton,” Opt. Commun. 57, 285 (1986).
[Crossref]

D. J. Kaup, “Creation of a soliton out of dissipation,” Physica 19D, 125 (1986).

J. R. Ackerhalt and P. W. Milonni, “Solitons and four-wave mixing,” Phys. Rev. A 33, 3185 (1986).
[Crossref] [PubMed]

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661 (1986).
[Crossref] [PubMed]

W. K. Bishel and M. T. Dyer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677 (1986).
[Crossref]

W. K. Bishel and M. T. Dyer, “Temperature dependence of the Raman linewidth and line shift for Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113 (1986).
[Crossref]

1985 (2)

R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
[Crossref]

K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
[Crossref]

1984 (1)

R. Meinel, “Backlund transformation and N-solitons solutions for stimulated Raman scattering and resonant two-photon propagation,” Opt. Commun. 49, 224 (1984).
[Crossref]

1983 (3)

H. Steudel, “Solitons in stimulated Raman scattering and resonant two-photon propagation,” Physica 6D, 155 (1983).

K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
[Crossref]

D. J. Kaup, “The method of solution for stimulated Raman scattering and two-phonon propagation,” Physica 6D, 142 (1983).

1982 (1)

V. A. Gorbunov, “Stimulated Raman scattering in a field of ultrashort light pulses,” Kvant. Elektron. (Moscow) 9, 152 (1982).

1979 (1)

M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
[Crossref]

1975 (1)

F. Y. F. Chu and A. C. Scott, “Inverse scattering transform for wave–wave scattering,” Phys. Rev. A 12, 2060 (1975).
[Crossref]

1974 (1)

A. Z. Grasyuk, “Raman lasers (review),” Sov. J. Quantum Electron. 4, 269 (1974).
[Crossref]

Ackerhalt, J. R.

J. R. Ackerhalt and P. W. Milonni, “Solitons and four-wave mixing,” Phys. Rev. A 33, 3185 (1986).
[Crossref] [PubMed]

Alsing, G.

K. Druhl and G. Alsing, “Effect of coherence relaxation on the propagation of optical solitons: an analytical and numerical case study on asymptotic perturbation theory,” Physica 20D, 429 (1986).

Bishel, W. K.

W. K. Bishel and M. T. Dyer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677 (1986).
[Crossref]

W. K. Bishel and M. T. Dyer, “Temperature dependence of the Raman linewidth and line shift for Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113 (1986).
[Crossref]

Bowden, C. M.

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661 (1986).
[Crossref] [PubMed]

Carlsten, J. L.

K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
[Crossref]

R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
[Crossref]

K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
[Crossref]

M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
[Crossref]

Chu, F. Y. F.

F. Y. F. Chu and A. C. Scott, “Inverse scattering transform for wave–wave scattering,” Phys. Rev. A 12, 2060 (1975).
[Crossref]

Druhl, K.

K. Druhl and G. Alsing, “Effect of coherence relaxation on the propagation of optical solitons: an analytical and numerical case study on asymptotic perturbation theory,” Physica 20D, 429 (1986).

K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
[Crossref]

K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
[Crossref]

Druhl, K. J.

R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
[Crossref]

Dyer, M. T.

W. K. Bishel and M. T. Dyer, “Temperature dependence of the Raman linewidth and line shift for Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113 (1986).
[Crossref]

W. K. Bishel and M. T. Dyer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677 (1986).
[Crossref]

Englund, J. C.

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661 (1986).
[Crossref] [PubMed]

Gorbunov, V. A.

V. A. Gorbunov, “Stimulated Raman scattering in a field of ultrashort light pulses,” Kvant. Elektron. (Moscow) 9, 152 (1982).

Grasyuk, A. Z.

A. Z. Grasyuk, “Raman lasers (review),” Sov. J. Quantum Electron. 4, 269 (1974).
[Crossref]

Kaup, D. J.

D. J. Kaup, “Creation of a soliton out of dissipation,” Physica 19D, 125 (1986).

D. J. Kaup, “The method of solution for stimulated Raman scattering and two-phonon propagation,” Physica 6D, 142 (1983).

Meinel, R.

R. Meinel, “Backlund transformation and N-solitons solutions for stimulated Raman scattering and resonant two-photon propagation,” Opt. Commun. 49, 224 (1984).
[Crossref]

Milonni, P. W.

J. R. Ackerhalt and P. W. Milonni, “Solitons and four-wave mixing,” Phys. Rev. A 33, 3185 (1986).
[Crossref] [PubMed]

Mostowski, J.

M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
[Crossref]

Raymer, M. G.

M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
[Crossref]

Scott, A. C.

F. Y. F. Chu and A. C. Scott, “Inverse scattering transform for wave–wave scattering,” Phys. Rev. A 12, 2060 (1975).
[Crossref]

Steudel, H.

H. Steudel, “Stimulated Raman scattering with an initial phase shift: the pre-stage of a soliton,” Opt. Commun. 57, 285 (1986).
[Crossref]

H. Steudel, “Solitons in stimulated Raman scattering and resonant two-photon propagation,” Physica 6D, 155 (1983).

Wenzel, R. G.

K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
[Crossref]

R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
[Crossref]

K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Stat. Phys. (2)

R. G. Wenzel, J. L. Carlsten, and K. J. Druhl, “Soliton experiments in stimulated Raman scattering,” J. Stat. Phys. 39, 621 (1985).
[Crossref]

K. Druhl, J. L. Carlsten, and R. G. Wenzel, “Aspects of soliton propagation in stimulated Raman scattering,” J. Stat. Phys. 39, 615 (1985).
[Crossref]

Kvant. Elektron. (Moscow) (1)

V. A. Gorbunov, “Stimulated Raman scattering in a field of ultrashort light pulses,” Kvant. Elektron. (Moscow) 9, 152 (1982).

Opt. Commun. (2)

R. Meinel, “Backlund transformation and N-solitons solutions for stimulated Raman scattering and resonant two-photon propagation,” Opt. Commun. 49, 224 (1984).
[Crossref]

H. Steudel, “Stimulated Raman scattering with an initial phase shift: the pre-stage of a soliton,” Opt. Commun. 57, 285 (1986).
[Crossref]

Phys. Rev. A (4)

J. R. Ackerhalt and P. W. Milonni, “Solitons and four-wave mixing,” Phys. Rev. A 33, 3185 (1986).
[Crossref] [PubMed]

F. Y. F. Chu and A. C. Scott, “Inverse scattering transform for wave–wave scattering,” Phys. Rev. A 12, 2060 (1975).
[Crossref]

M. G. Raymer, J. Mostowski, and J. L. Carlsten, “Theory of stimulated Raman scattering with broad-band lasers,” Phys. Rev. A 19, 2304 (1979).
[Crossref]

W. K. Bishel and M. T. Dyer, “Temperature dependence of the Raman linewidth and line shift for Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113 (1986).
[Crossref]

Phys. Rev. Lett. (2)

K. Druhl, R. G. Wenzel, and J. L. Carlsten, “Observation of solitons in stimulated Raman scattering,” Phys. Rev. Lett. 51, 1171 (1983).
[Crossref]

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661 (1986).
[Crossref] [PubMed]

Physica (4)

D. J. Kaup, “Creation of a soliton out of dissipation,” Physica 19D, 125 (1986).

D. J. Kaup, “The method of solution for stimulated Raman scattering and two-phonon propagation,” Physica 6D, 142 (1983).

K. Druhl and G. Alsing, “Effect of coherence relaxation on the propagation of optical solitons: an analytical and numerical case study on asymptotic perturbation theory,” Physica 20D, 429 (1986).

H. Steudel, “Solitons in stimulated Raman scattering and resonant two-photon propagation,” Physica 6D, 155 (1983).

Sov. J. Quantum Electron. (1)

A. Z. Grasyuk, “Raman lasers (review),” Sov. J. Quantum Electron. 4, 269 (1974).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Stokes-seed intensity with AMPS used in the calculation. The π phase shift occurs rapidly at the center of the hole. The dashed curve shows the Stokes pulse before the AMPS.

Fig. 2
Fig. 2

Pump pulse that evolves with (solid curves) and without (dashed curves) an AMPS in the Stokes seed at z = 0.1, 0.15, and 0.35 m. Also shown is the input pump. (a) The pump first depletes when the product of the pump and Stokes fields is the largest, leaving a pulse whose width is characteristic of the Pockels-cell switching time. (b) The pulse narrows because of steady-state gain. (c) The pulse becomes much narrower than the switching time. The peak amplitude is unchanged owing to photon conservation.

Fig. 3
Fig. 3

Difference between the pump pulse with and without and AMPS. The pulse first forms with a width that depends on the duration of the AMPS and then narrows rapidly. When the pulse width becomes narrower than the dephasing time (1/Γ), its width changes slowly.

Fig. 4
Fig. 4

Pulse width from numerical calculations (×’s), pulse width predicted by steady-state theory (solid curve), and exponential narrowing predicted by steady-state theory after pump depletion (dashed curve). The steady-state theory works well when the width is much larger than the damping time.

Fig. 5
Fig. 5

Reciprocal of the pulse width squared from numerical calculations (×’s), steady-state prediction (dashed curve), and hyper-transient prediction (solid curve). Surprisingly, the hypertransient prediction works well even when the pulse width is comparable with the damping time. At z = 35 cm the half-width is 22 psec, compared with the damping time of 60 psec. The steady-state prediction narrows too rapidly because it assumes that all the energy stored in Q is lost; thus it does not allow for energy transfer from Q to the fields.

Fig. 6
Fig. 6

Pump and Stokes intensities (solid curves); polarization Q (dashed curve) in the three characteristic regions at z = 0.35 m. In region I normal Raman scattering occurs. In region II inverse Raman scattering occurs, and in region III oscillations between normal and inverse Raman scattering develop.

Fig. 7
Fig. 7

Polarization Q at various locations in the Raman cell. At z = 0.2 m, Q is characteristic of steady-state evolution, as it nearly follows the product of the driving fields. At z = 0.75 m, Q has its characteristic soliton shape. In the later stages Q changes slowly.

Fig. 8
Fig. 8

Numerical calculation of the pump pulse (solid curve) and corresponding exact soliton solution (dashed curve) at z = 0.75 m.

Fig. 9
Fig. 9

Schematic of the apparatus need to generate the AMPS. The Pockels cell and the quarter-wave plate (λ/4) are rotated about the axis of propagation by 45° with respect to the polarization of the incident beam and transmitting axis of the polarizers.

Equations (30)

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d / d z E p = - E s Q ,             d / d z E s = E p Q ,
d / d τ Q = - Γ Q + Γ 0 E p E s .
E p = 4.57 × 10 6 ( V / m ) exp [ - ( τ / 15 nsec ) 2 / 2 ] ,             E s = E p / 28.
Γ Q = Γ 0 E p E s .
Q = ( Γ / Γ 0 ) E p E s ,
d / d z I p = - 2 ( Γ 0 / Γ ) I s I p ,             d / d z I s = 2 ( Γ 0 / Γ ) I p I s .
I p ( τ ) = Y 0 ( τ ) I p 0 ( τ ) / { I p 0 ( τ ) + I s 0 ( τ ) exp [ 2 Y 0 ( τ ) z Γ 0 / Γ ] } ,
Y 0 ( τ ) = I p 0 ( τ ) + I s 0 ( τ )
I p ( τ ) = I p 0 / [ 1 + I s 0 / I p 0 exp ( 2 I p 0 z Γ 0 / Γ ) ]
τ 1 / 2 = I p 0 / I 0 exp ( - 2 I p 0 z Γ 0 / Γ ) .
d U q = K Q 2 ( z , τ ) d z .
U p = K - τ E p 2 ( z , τ ) d τ .
( d / d z ) U p d z = K d z - τ 2 E p ( d / d z ) E p d τ .
d U p = - K - τ 2 E p E s Q d τ .
( d / d z ) U p = - K Γ 0 - τ 2 Q ( d / d τ ) Q d τ .
d U p = ( K Γ 0 ) d z Q 2 .
K / K = Γ 0 .
E p ( τ , z ) = E 0 f ( τ / w ) .
E s ( τ , z ) = E 0 [ 1 - f 2 ( τ / w ) ] 1 / 2 .
U f = K E 0 2 w A f ,
A f = - f 2 ( x ) d x .
Q = Γ 0 E 0 2 - τ exp [ - Γ ( τ - τ ) ] f ( τ / w ) [ 1 - f 2 ( τ / w ) ] 1 / 2 d τ .
( d / d τ D ) d U = 2 K d z Q ( d / d τ D ) Q ,
d U ( ) = - 2 K d z Γ 0 2 E 0 4 w 3 A q ,
A q = - { - x f ( x ) [ 1 - f ( x ) ] 1 / 2 d x } 2 d x .
( 1 / w 2 ) = 4 Γ Γ 0 E 0 2 ( A q / A f ) ( z - z 0 ) + 1 / w 0 2 .
E in = E 0 sin ( ϕ ) x ^ ,
ϕ = k z - w t .
E = [ E 0 sin ( ϕ + π / 4 + s π / 2 ) x ^ + E 0 sin ( ϕ - π / 4 - s π / 2 ) y ^ ) ] / 2 ,
E = E 0 sin ( ϕ ) cos ( π / 4 + s π / 2 ) x ^ .

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