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

1989

1985

1984

1983

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

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

1975

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.

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

J. Opt. Soc. Am. B

Opt. Commun.

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.

Prog. Quantum Electron.

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

Other

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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