Abstract

A qualitative as well as quantitative investigation is made of the conditions for avoiding wave breaking during pulse propagation in optical fibers. In particular, it is shown that pulses having a parabolic intensity variation are approximate wave-breaking-free solutions of the nonlinear Schrödinger equation in the high-intensity limit. A simple expression for the compression factor of a fiber-grating compressor based on parabolic pulses is also derived.

© 1993 Optical Society of America

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References

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  1. D. Grischowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
    [CrossRef]
  2. B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
    [CrossRef]
  3. W. J. Tomlinson, R. H. Stolen, C. V. Shank, J. Opt. Soc. Am. B 1, 139 (1984).
    [CrossRef]
  4. H. E. Lassen, F. Mengel, B. Tromborg, N. C. Albertsen, P. L. Christiansen, Opt. Lett. 10, 34 (1985).
    [CrossRef] [PubMed]
  5. W. J. Tomlinson, R. H. Stolen, A. M. Johnson, Opt. Lett. 10, 457 (1985).
    [CrossRef] [PubMed]
  6. J. E. Rothenberg, J. Opt. Soc. Am. B 6, 2392 (1989).
    [CrossRef]
  7. D. Anderson, M. Desaix, M. Lisak, M. L. Quiroga-Teixeiro, J. Opt. Soc. Am. B 9, 1358 (1992).
    [CrossRef]
  8. J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  9. R. Meinel, Opt. Commun. 47, 343 (1983).
    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, London, 1989).

1992 (1)

1989 (1)

1985 (2)

1984 (1)

1983 (2)

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

R. Meinel, Opt. Commun. 47, 343 (1983).
[CrossRef]

1982 (1)

D. Grischowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

1975 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, London, 1989).

Albertsen, N. C.

Anderson, D.

Balant, A. C.

D. Grischowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

Blow, K. J.

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

Christiansen, P. L.

Cotter, D.

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

Desaix, M.

Doran, N. J.

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

Grischowsky, D.

D. Grischowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

Johnson, A. M.

Lassen, H. E.

Lisak, M.

Marburger, J. H.

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

Meinel, R.

R. Meinel, Opt. Commun. 47, 343 (1983).
[CrossRef]

Mengel, F.

Nelson, B. P.

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

Quiroga-Teixeiro, M. L.

Rothenberg, J. E.

Shank, C. V.

Stolen, R. H.

Tomlinson, W. J.

Tromborg, B.

Appl. Phys. Lett. (1)

D. Grischowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

B. P. Nelson, D. Cotter, K. J. Blow, N. J. Doran, Opt. Commun. 48, 292 (1983).
[CrossRef]

R. Meinel, Opt. Commun. 47, 343 (1983).
[CrossRef]

Opt. Lett. (2)

Prog. Quantum Electron. (1)

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

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, London, 1989).

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Figures (6)

Fig. 1
Fig. 1

Peak intensity I0(x) and chirp function b(x) as functions of distance x. The dashed curves are the asymptotic results given by relations (28).

Fig. 2
Fig. 2

Numerical solution of the nonlinear Schrödinger equation for the amplitude A(x,τ), when A ( 0 , τ ) = 10 1 - τ 2 for distances x of 0, 3/16, 3/8, 9/16, and 3/4.

Fig. 3
Fig. 3

Approximate analytical solution for A(x,τ) according to Eqs. (24) and (25), with the same initial condition and propagation distances as in Fig. 2.

Fig. 4
Fig. 4

Evolution of the amplitude spectrum corresponding to the input pulse A ( 0 , τ ) = 10 1 - τ 2 obtained by integration of the nonlinear Schrödinger equation.

Fig. 5
Fig. 5

Parabolic pulse and corresponding Gaussian pulse with the same energy and FWHM.

Fig. 6
Fig. 6

Compression factor Fc as a function of x/x0, where x0 is the soliton period π/2. I0(0) and τ0(0) are chosen so that the parabola has the same energy and FWHM as the soliton N2 sech2τ, i.e., I0(0) ≈ 1.203N2 and τ0(0) ≈ 1.246.

Equations (40)

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Δ τ = x Δ k 0 x k 0 Δ ω x k 0 d ω c d τ Δ τ ,
x wb = - 1 k 0 d ω c / d τ ,
i Ψ x = α 2 Ψ τ 2 + κ Ψ 2 Ψ .
- ϕ x = α [ 1 A 2 A τ 2 - ( ϕ τ ) 2 ] + κ A 2 ,
( A 2 ) x = 2 α τ ( A 2 ϕ τ ) .
ω c x = τ [ α ( 1 A 2 A τ 2 - ω c 2 ) + κ A 2 ] , ( A 2 ) x = - 2 α τ ( A 2 ω c ) .
ω c ( x , τ ) ω c ( 0 , τ ) + τ { α [ 1 A ( 0 , τ ) 2 A ( 0 , τ ) τ 2 - ω c 2 ( 0 , τ ) ] + κ A 2 ( 0 , τ ) } x .
α 1 A ( 0 , τ ) 2 A ( 0 , τ ) τ 2 + κ A 2 ( 0 , τ ) = C 0 + C 1 τ 2 ,
A 2 ( x , τ ) A 2 ( 0 , τ ) - α τ { A 2 ( 0 , τ ) τ [ α 1 A ( 0 , τ ) 2 A ( 0 , τ ) τ 2 + κ A 2 ( 0 , τ ) ] } x 2 .
A 2 ( 0 , τ ) = A 0 2 ( 1 - τ 2 τ 0 2 )
d ( A 0 2 ) d x = 4 α R A 0 2 a 2 - A 0 2 a d a d x = 4 α R A 0 2 a 2 .
A 0 2 a = constant ,
R = - 1 8 α d d x a 2 .
- a 2 d ϕ 0 d x - ρ 2 [ a 2 ( d R d x - 2 R a d a d x ) - 4 α R 2 ] = α f d 2 f d ρ 2 + κ a 2 A 0 2 f 2 .
a 2 d ϕ 0 d x = constant = - C 0 , a 2 ( d R d x - 2 R a d a d x ) - 4 α R 2 = constant = - C 1 ,
α f d 2 f d ρ 2 = C 0 + C 1 ρ 2 ,
| α f d 2 f d ρ 2 | κ a 2 A 0 2 f 2 .
- a d ϕ 0 d x - ρ 2 [ a ( d R d x - 2 R a d a d x ) - 4 α a R 2 ] κ a A 0 2 f 2 .
- a d ϕ a d x = constant , a ( d R d x - 2 R a d a d x ) - 4 α a R 2 = constant .
f 2 = f 2 ( 0 ) ( 1 - ρ 2 / ρ 2 ) .
2 τ 2 2 = 1 r r ( r r )
I x + τ ( I ω c ) = 0 , ω c x + τ ( I + ω c 2 2 ) = 0.
I ( x , τ ) = I 0 ( x ) [ 1 - τ 2 τ 0 2 ( x ) ] ,             ω c ( x , τ ) = b ( x ) τ ,
d I 0 d x + b I 0 = 0 ,
1 I 0 d I 0 d x + 1 τ 0 d τ 0 d x = 0 ,
d b d x + b 2 = 2 I 0 τ 0 2 .
2 x = 1 - I 0 I 0 + ln ( 1 + 1 - I 0 I 0 ) ,
τ 0 ( x ) = 1 I 0 ( x ) , b ( x ) = - d d x ln [ I 0 ( x ) ] = 2 I 0 ( x ) 1 - I 0 ( x ) ,
b max = 4 3 3
x max = 3 4 + 1 2 ln ( 1 + 3 2 ) .
I 0 ( x ) 1 2 x , τ 0 ( x ) 2 x , b ( x ) 1 x .
Ψ ^ 2 = | - Ψ ( x , τ ) exp ( i ω τ ) d τ | 2 = I 0 ( x ) τ 0 2 ( x ) | - 1 1 1 - t 2 × exp { i [ ω τ 0 ( x ) t - 1 2 b ( x ) τ 0 2 ( x ) t 2 ] } d t | 2 ,
Ψ ^ 2 I 0 ( 0 ) π τ 0 ( 0 ) [ 1 - ( ω 2 I 0 ( 0 ) ) 2 ] .
i Ψ z = - 1 2 2 Ψ τ 2 .
A p 1 - τ 2 / τ 0 2 exp ( - i b τ 2 / 2 ) = A G exp ( - τ 2 2 a 2 - i b τ 2 / 2 ) ,
a = τ 0 2 ln ( 2 ) ,             A G 2 = A p 2 4 3 2 ln ( 2 ) π .
a min = a 1 + a 4 b 2 .
F c = { [ I 0 ( x ) I 0 ( 0 ) ] 2 + I 0 ( 0 ) τ 0 2 ( 0 ) [ ln ( 2 ) ] 2 [ 1 - I 0 ( x ) I 0 ( 0 ) ] } 1 / 2 ,
F c max = τ 0 ( 0 ) ln ( 2 ) I 0 ( 0 ) ,
x wb π / 2 = 1 N 6 π .

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