Abstract

The nonstationary (time-dependent) problem of longitudinal modulation instability of a self-focusing beam in a nonlinear medium with and without saturation is solved for the first time, to our knowledge, by using numerical methods. We present the dependence of instability growth rates on the modulation frequency for different saturation parameters. The radial profile of the complex perturbation function is also calculated. The result of modulation instability in the problem considered is the separation of the beam into a periodic sequence of light clumps (light bullets).

© 1992 Optical Society of America

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    [CrossRef] [PubMed]
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1991

1990

1988

N. N. Akhmediev, V I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65, 193 (1988).
[CrossRef]

S. Manuef, F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

1985

A. Barthelemy, S. Manuev, C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

1980

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

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

1973

V. E. Zakharov, A. M. Rubenchik, Zh. Eksp. Teor. Fiz. 65, 907 (1973).

1965

P. L. Kelley, Phys. Rev. Lett. 15, 1085 (1965).
[CrossRef]

V. I. Bespalov, V. I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 (1965); T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

1964

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

1962

G. A. Askar’yan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962).

Aitchison, J. S.

Akhmediev, N. N.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

N. N. Akhmediev, V I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Askar’yan, G. A.

G. A. Askar’yan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962).

Barthelemy, A.

A. Barthelemy, S. Manuev, C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov, V. I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 (1965); T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Brinkman, W. F.

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Desailly, R.

S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65, 193 (1988).
[CrossRef]

Froehly, C.

S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65, 193 (1988).
[CrossRef]

A. Barthelemy, S. Manuev, C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Gordon, J. P.

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

Hasegawa, A.

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Heatley, D. R.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

Jackel, J. L.

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1085 (1965).
[CrossRef]

Korneev, V I.

N. N. Akhmediev, V I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Leaird, D. E.

Manassah, J. T.

Maneuf, S.

S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65, 193 (1988).
[CrossRef]

Manuef, S.

S. Manuef, F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Manuev, S.

A. Barthelemy, S. Manuev, C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Mitskevich, N. V

N. N. Akhmediev, V I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Mollenauer, L. F.

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

Oliver, M. K.

Reynaud, F.

S. Manuef, F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Rubenchik, A. M.

V. E. Zakharov, A. M. Rubenchik, Zh. Eksp. Teor. Fiz. 65, 907 (1973).

Silberberg, Y.

Smith, P. W E.

Soto-Crespo, J. M.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

Stolen, R. H.

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

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 (1965); T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Weiner, A. M.

Wright, E. M.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov, A. M. Rubenchik, Zh. Eksp. Teor. Fiz. 65, 907 (1973).

IEEE J. Quantum Electron.

A. Hasegawa, W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Opt. Commun.

A. Barthelemy, S. Manuev, C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65, 193 (1988).
[CrossRef]

S. Manuef, F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Opt. Lett.

Phys. Rev. A

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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

P. L. Kelley, Phys. Rev. Lett. 15, 1085 (1965).
[CrossRef]

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Pis’ma Zh. Eksp. Teor. Fiz.

V. I. Bespalov, V. I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 (1965); T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Sov. Phys. JETP

N. N. Akhmediev, V I. Korneev, N. V Mitskevich, Sov. Phys. JETP 67, 89 (1988).

Zh. Eksp. Teor. Fiz.

G. A. Askar’yan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962).

V. E. Zakharov, A. M. Rubenchik, Zh. Eksp. Teor. Fiz. 65, 907 (1973).

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Figures (3)

Fig. 1
Fig. 1

Squared instability growth rate δ2 versus squared modulation frequency γ2. The stars correspond to calculated data. The solid curves are the results of approximation by the formula given by Eq. (6) with different coefficients. The numbers on the curves denote the saturation parameter β.

Fig. 2
Fig. 2

Real and imaginary parts of the perturbation function f(r) for three different modulation frequences: (a) γ2 = 0.0025, (b) γ2 = 2.5, and (c) γ2 = 4.0. The nonlinear mode Ψ0(r) is also shown for comparison. The perturbation functîons are normalized in such a way that Re[f(0)] + Im[f(0)] = Ψ0(0). The saturation parameter β is equal to zero (the case of Kerr nonlinearity).

Fig. 3
Fig. 3

Squared limiting frequency of instability γlim2 versus saturation parameter β.

Equations (7)

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i Ψ z + Δ r Ψ + Ψ τ τ Ψ + | Ψ | 2 Ψ / ( 1 + β | Ψ | 2 ) = 0 ,
Ψ = Ψ 0 ( r ) + μ f ( r , z ) cos ( γ τ ) ,
i f z + Δ r f ( 1 + γ 2 ) f + ( 2 | Ψ 0 | 2 f + β | Ψ 0 | 4 f + Ψ 0 2 f * ) / ( 1 + β | Ψ | 2 ) 2 = 0 .
f ( r , z ) = f ( r ) exp ( δ z ) ,
i δ f + Δ r f ( 1 + γ 2 ) f + ( 2 | Ψ 0 | 2 f + β | Ψ 0 | 4 f + Ψ 0 2 f * ) / ( 1 + β | Ψ | 2 ) 2 = 0 .
δ 2 = 1 . 072 γ ( 5 . 45 γ 2 ) ( γ 2 + 0 . 71 ) 1 / 2 .
δ 4 = 0 . 008422 ( γ 2 ) 6 0 . 160955 ( γ 2 ) 5 + 2 . 23124 ( γ 2 ) 4 14 . 785 ( γ 2 ) 3 + 29 . 4048 ( γ 2 ) 2 + 19 . 2436 ( γ 2 ) .

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