Abstract

An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schrödinger equation that is valid from the early stages of self-focusing until the blowup point. The adiabatic law leads to an analytical formula for the location of the blowup point and can be used to estimate the effects of various small perturbations on self-focusing. The results of the analysis are confirmed by numerical simulations.

© 1996 Optical Society of America

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Errata

Gadi Fibich, "Adiabatic law for self-focusing of optical beams: erratum," Opt. Lett. 22, 194-194 (1997)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-22-3-194

References

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  1. P. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  2. S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
    [CrossRef]
  3. G. Fraiman, Sov. Phys. JETP 61, 228 (1985); M. Landman, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 38, 3837 (1988); B. LeMesurier, G. Papanicolaou, C. Sulem, P. Sulem, Physica D 32, 210 (1988).
    [CrossRef] [PubMed]
  4. M. Feit, J. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]
  5. G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
    [CrossRef] [PubMed]
  6. G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
    [CrossRef] [PubMed]
  7. J. E. Rothenberg, Opt. Lett. 17, 1340 (1992); G. Luther, A. Newell, J. Moloney, Physica D 74, 59 (1994).
    [CrossRef] [PubMed]
  8. V. Malkin, Physica D 64, 251 (1993).
    [CrossRef]
  9. V. Talanov, JETP Lett. 11, 199 (1970).
  10. D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
    [CrossRef] [PubMed]
  11. E. Dawes, J. Marburger, Phys. Rev. 179, 862 (1969); J. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]

1996 (1)

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

1995 (1)

G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

1993 (1)

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

1992 (1)

1988 (1)

1986 (1)

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

1985 (1)

G. Fraiman, Sov. Phys. JETP 61, 228 (1985); M. Landman, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 38, 3837 (1988); B. LeMesurier, G. Papanicolaou, C. Sulem, P. Sulem, Physica D 32, 210 (1988).
[CrossRef] [PubMed]

1970 (1)

V. Talanov, JETP Lett. 11, 199 (1970).

1969 (1)

E. Dawes, J. Marburger, Phys. Rev. 179, 862 (1969); J. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

1966 (1)

S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
[CrossRef]

1965 (1)

P. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Akhmanov, S.

S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
[CrossRef]

Dawes, E.

E. Dawes, J. Marburger, Phys. Rev. 179, 862 (1969); J. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Feit, M.

Fibich, G.

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

Fleck, J.

Fraiman, G.

G. Fraiman, Sov. Phys. JETP 61, 228 (1985); M. Landman, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 38, 3837 (1988); B. LeMesurier, G. Papanicolaou, C. Sulem, P. Sulem, Physica D 32, 210 (1988).
[CrossRef] [PubMed]

Kelley, P.

P. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Khokhlov, R.

S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
[CrossRef]

Malkin, V.

G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

Marburger, J.

E. Dawes, J. Marburger, Phys. Rev. 179, 862 (1969); J. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

McLaughlin, D.

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

Papanicolaou, G.

G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

Rothenberg, J. E.

Sukhorukov, A.

S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
[CrossRef]

Sulem, C.

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

Sulem, P.

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

Talanov, V.

V. Talanov, JETP Lett. 11, 199 (1970).

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

JETP (1)

S. Akhmanov, A. Sukhorukov, R. Khokhlov, JETP 23, 1025 (1966); F. Cooper, C. Lucheroni, H. Shepard, Phys. Lett. A 70, 184 (1992).
[CrossRef]

JETP Lett. (1)

V. Talanov, JETP Lett. 11, 199 (1970).

Opt. Lett. (1)

Phys. Rev. (1)

E. Dawes, J. Marburger, Phys. Rev. 179, 862 (1969); J. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Phys. Rev. A (2)

D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 34, 1200 (1986).
[CrossRef] [PubMed]

G. Fibich, V. Malkin, G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

P. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

Physica D (1)

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

Sov. Phys. JETP (1)

G. Fraiman, Sov. Phys. JETP 61, 228 (1985); M. Landman, G. Papanicolaou, C. Sulem, P. Sulem, Phys. Rev. A 38, 3837 (1988); B. LeMesurier, G. Papanicolaou, C. Sulem, P. Sulem, Physica D 32, 210 (1988).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Relative error in L based on the new adiabatic law [Eq. (12); solid curve], Malkin’s law [Eq. (13); dotted curve], the log–log law [relation (2); dashed–dotted curve], and β [relation (6); dashed curve]. The initial condition is ψ0 = 1.02R(r).

Fig. 2
Fig. 2

Comparison of the pure adiabatic law [Eq. (11); dotted curve] and the adiabatic law [Eq. (12); dashed curve] with the numerical solution of the NLS (solid curve). The initial condition is ψ0 = 1.02R(r).

Fig. 3
Fig. 3

Location of blowup point Zc as a function of beam power p according to adiabatic theory [relation (14); solid curve] and numerical simulations (circles) for A, cR(r); B, c exp(−r2) [the dotted curve is Eq. (15)]; and C, c exp(−r4).

Equations (18)

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i ψ z + Δ ψ + | ψ | 2 ψ = 0 , ψ ( 0 , r ) = ψ 0 ( r )
L ( z ) [ 2 π ( Z c z ) ln ln 1 / ( Z c z ) ] 1 / 2 ,
ψ s ( r , z ) V ( ξ , ζ ) L ( z ) exp ( i ζ + i L z L r 2 4 ) ,
ξ = r L , d ζ d z = 1 L 2 .
Δ R R + R 3 = 0 , R ( 0 ) = 0 , R ( ) = 0
β = L 3 L z z .
β N s N c M , N s = | ψ s | 2 r d r ,
H s M ( L z 2 β L 2 ) , H s = | ψ s | 2 r d r ½ | ψ s | 4 r d r , = / r .
L 3 L z z = β , β β 0 : = β ( 0 ) .
L z 2 = β L 2 + H s M , H s H s ( 0 ) .
L = [ L 0 2 ± 2 ( β + H s L 0 2 M z ) z 1 / 2 + H s M z 2 ] 1 / 2 , L 0 = L ( 0 ) .
Z c = { L 0 2 β + ( β + H s L 0 2 / M ) 1 / 2 L z ( 0 ) 0 L 0 2 β ( β + H s L 0 2 / M ) 1 / 2 L z ( 0 ) > 0 , H s < 0 no blowup L z ( 0 ) > 0 , H s > 0 .
L L 0 ( 1 z 2 Z c 2 ) 1 / 2 , Z c = L 0 2 β 0 .
Z ˜ c = L 0 2 β 0 + L 0 2 / F .
L [ 2 β ( Z c z ) + H s M ( Z c z ) 2 ] 1 / 2 .
L [ 2 β ( Z c z ) ] 1 / 2 .
Z c ( M N c p 1 ) 1 / 2 / | ψ 0 | 2 r d r , p = | ψ 0 | 2 r d r N c .
Z c = 0 . 367 [ ( p 1 / 2 0 . 852 ) 2 0 . 0219 ] 1 / 2 ,

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