Abstract

Optical pulses, which propagate under the combined effects of nonlinearity, dispersion, and diffraction, may collapse in space and time. The standard method for analyzing these collapses is the aberrationless paraxial ray approximation. This method is known to give a quantitatively correct, although not particularly accurate, picture of most properties of the pulse dynamics. However, it is found that the predictions for some of the important pulse parameters are qualitatively wrong and could lead to incorrect conclusions. An alternative variational approach is suggested that remedies these deficiencies and gives results in good agreement with numerical results.

© 1991 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  2. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62–69 (1972).
  3. J. H. Marburger, Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  4. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  5. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  6. J. Manassah, P. L. Baldeck, and R. R. Alfano, Opt. Lett. 13, 1090–1092 (1988).
    [CrossRef] [PubMed]
  7. A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
    [CrossRef]
  8. Y. Silberberg, Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef] [PubMed]
  9. M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  10. W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
    [CrossRef]
  11. D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
    [CrossRef]
  12. D. Anderson, Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  13. M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
    [CrossRef] [PubMed]
  14. P. L. Kelley, Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]

1990 (2)

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Y. Silberberg, Opt. Lett. 15, 1282–1284 (1990).
[CrossRef] [PubMed]

1989 (1)

M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
[CrossRef] [PubMed]

1988 (2)

1983 (1)

D. Anderson, Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

1979 (1)

D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

1975 (1)

J. H. Marburger, Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62–69 (1972).

1968 (1)

W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
[CrossRef]

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Alfano, R. R.

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
[CrossRef] [PubMed]

D. Anderson, Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Baldeck, P. L.

Bonnedal, M.

D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Chen, Y.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Desaix, M.

M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
[CrossRef] [PubMed]

Feit, M. D.

Fleck, J. A.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Haus, H. A.

W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Lisak, M.

M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
[CrossRef] [PubMed]

D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Manassah, J.

Marburger, J. H.

J. H. Marburger, Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Mitchell, D. J.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Poladian, L.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62–69 (1972).

Silberberg, Y.

Snyder, A. W.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Wagner, W. G.

W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Electron. Lett. (1)

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Electron. Lett. 26, 643–644 (1990).
[CrossRef]

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

Opt. Lett. (2)

Phys. Fluids (1)

D. Anderson, M. Bonnedal, and M. Lisak, Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Phys. Rev. (1)

W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Phys. Rev. A (2)

D. Anderson, Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, Phys. Rev. A 40, 2441–2445 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

P. L. Kelley, Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Prog. Quantum Electron. (1)

J. H. Marburger, Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62–69 (1972).

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Figures (5)

Fig. 1
Fig. 1

Comparison of the radial profiles of the self-trapped solutions for the variational solutions (dashed curves) and numerical solutions (solid curves) when the amplitude on the axis is normalized to unity. The dimensionality d is indicated by the labels 1–3.

Fig. 2
Fig. 2

Pulse width as a function of distance of propagation for different values of the parameter ν/μ when d = 2.

Fig. 3
Fig. 3

Intensity as a function of collapse distance for d = 2. The solid curve represents the numerical result [relation (19)], the short-dashed curve represents the variational Gaussian [Eq. (20a)], and the long-dashed curve represents the variational sech [Eq. (20b)].

Fig. 4
Fig. 4

The potential function Π, for the three-dimensional case.

Fig. 5
Fig. 5

Pulse width as a function of distance of propagation for different values of the parameter ν/μ when d = 3.

Tables (2)

Tables Icon

Table 1 Comparison between Predictions Made by the Aberrationless Paraxial Ray Approximation and the Variational Method with Numerical Results for Wave-Number Shift δ and Integrated Pulse Intensity P

Tables Icon

Table 2 Coefficients α, β, and γ Defined in Eqs. (11) and (13)a

Equations (33)

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i u ζ + 1 2 ( 2 u ξ 2 + 2 u η 2 + 2 u τ 2 ) + u 2 u = 0 ,
i u ζ + 1 2 1 ρ d - 1 ρ ( ρ d - 1 u ρ ) + u 2 u = 0 ,
u 2 ζ + ( ρ + d - 1 ρ ) ( u 2 arg u ρ ) = 0 ,
arg u ζ + 1 2 ( arg u ρ ) 2 - ( 2 ρ 2 + d - 1 ρ ρ ) u 2 u - u 2 = 0.
u 2 = u o 2 f d ( ζ ) exp [ - ρ 2 a o 2 f 2 ( ζ ) ] ,
arg u = 1 2 ρ 2 f ( ζ ) d f ( ζ ) d ζ + ϕ ( ζ ) .
d 2 f d ζ 2 = 1 a o 4 f 3 - 2 u o 2 a o 2 1 f d + 1 ,
d ϕ d ζ = - d 2 a o 2 1 f 2 + u o 2 f d .
1 2 ( d f d ζ ) 2 + V ( f ) = 0 ,
V = 1 2 a o 4 [ ( 1 f 2 - 1 ) - 4 a o 2 u o 2 d ( 1 f d - 1 ) ] ,
1 a o 2 = 2 u o 2 ,
arg u = ( 1 - d ) u o 2 ζ .
L = i 2 ( u u * ζ - u * u ζ ) ρ d - 1 + 1 2 | u ρ | 2 ρ d - 1 - 1 2 u 4 ρ d - 1 .
u ( ρ , ζ ) = A ( ζ ) sech ( ρ a ( ζ ) ) exp [ i b ( ζ ) ρ 2 ] ,
L = 0 L d ρ = i 2 ( A d A * d ζ - A * d A d ζ ) a d α d - 1 + A 2 a d + 2 ( d b d ζ + 2 b 2 ) α d + 1 + 1 2 A 2 a d - 2 γ d - 1 - 1 2 A 4 a d β d - 1 ,
α m = 0 x m sech 2 ( x ) d x ,             β m = 0 x m sech 4 ( x ) d x , γ m = α m - β m .
u ( ρ , ζ ) = A ( ζ ) exp [ - ρ 2 2 a 2 ( ζ ) + i b ( ζ ) ρ 2 ] ,
α m = 0 x m exp ( - x 2 ) d x ,             β m = 0 x m exp ( - 2 x 2 ) d x , γ m - 1 = α m + 1 .
A 2 a d = I o ,
b = 1 2 d ln a d ζ ,
d arg A d ζ = i 2 ( A d A * d ζ - A * d A d ζ ) / A 2 = - γ d - 1 α d - 1 1 a 2 + ( 1 + d 4 ) β d - 1 α d - 1 A 2 ,
d 2 a d ζ 2 - γ d - 1 α d + 1 1 a 3 + I o d β d - 1 2 α d + 1 1 a d + 1 = 0 ,
1 2 ( d y E ζ ) 2 = Π ( y ) = 0 ,
Π ( y ) = μ y 2 + ν y d - ( μ + ν ) ,
μ = γ d - 1 2 α d + 1 a o 4 ,
ν = - β d - 1 A o 2 2 α d + 1 a o 2 ,
1 a = A ( β d - 1 d 2 γ d - 1 ) 1 / 2 ,
δ = A 2 β d - 1 α d - 1 ( 1 - d 4 ) ,
y ( ζ ) = [ ( ζ 2 μ ) 2 ( 1 + ν μ ) + 1 ] 1 / 2 .
ζ c 2 μ = 1 [ - ( 1 + ν μ ) ] 1 / 2 .
P P 2 = { 0.852 + [ 0.0219 + 0.135 ( a o 2 / ζ c ) 2 ] 1 / 2 } 2 { 1 + 0.912 ( a o 2 / ζ c ) 2 ζ c 0.135 ( a o 2 / ζ c ) 2 ζ c 0 ,
P P 2 { 1.074 + 1.074 ( a o 2 / ζ c ) 2 Gaussian trial function 1 + 0.895 ( a o 2 / ζ c ) 2 sech trial function .
ζ 2 μ = 2 y + 3 4 y ( 1 - y ) - 3 4 arcsin ( y ) + 3 π 8 .

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