Abstract

In this work we recognize new strategies involving optical wave-breaking for controlling the output pulse spectrum in nonlinear fibers. To this end, first we obtain a constant of motion for nonlinear pulse propagation in waveguides derived from the generalized nonlinear Schrödinger equation. In a second phase, using the above conservation law we theoretically analyze how to transfer in a simple manner the group-velocity-dispersion curve of the waveguide to the output spectral profile of pulsed light. Finally, the computation of several output spectra corroborates our proposition.

© 2013 Optical Society of America

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References

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  14. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
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2012

2011

2009

2008

2006

2004

2002

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

1996

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

1993

1992

1989

1985

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–70 (1972).

1967

F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett.19, 1097–1100 (1967).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic, 2007).

Anderson, D.

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Andrés, P.

Bartelt, H.

Boppart, S. A.

Bosman, G. W.

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice Hall, 1995).

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

Desaix, M.

Finot, C.

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

Gorbach, A. V.

Goto, T.

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

Hartung, A.

Hedekvist, P.-O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Heidt, A. M.

Hori, T.

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

Johnson, A. M.

Karlsson, M.

Kibler, B.

Knight, J. C.

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

Krok, P.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Lisak, M.

Liu, Y.

Miret, J. J.

Nishizawa, N.

Provost, L.

Quiroga-Teixeiro, M. L.

Rohwer, E. G.

Rothenberg, J. E.

Schwoerer, H.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–70 (1972).

Shimizu, F.

F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett.19, 1097–1100 (1967).
[CrossRef]

Silvestre, E.

Skryabin, D. V.

Stolen, R. H.

Stone, J. M.

Takayanagi, J.

Tomlinson, W. J.

Tu, H.

Wabnitz, S.

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–70 (1972).

Adv. Comput. Math.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett.19, 1097–1100 (1967).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–70 (1972).

Other

G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic, 2007).

L. Cohen, Time-Frequency Analysis (Prentice Hall, 1995).

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Figures (4)

Fig. 1
Fig. 1

Plot of the evolution of the functions N L 1 (dashed curve) and D 1 (solid curve) for a 5 ps Gaussian input pulse centered at 1550 nm (ω0 = 1215 rad ps−1) and 100 W peak power, which propagates throughout two fibers with γ0 = 400 W−1km−1 and dispersion behavior defined by: (a) β2 = 20 ps2km−1 and βk = 0 for k > 2, i.e., flat GVD profile; and (b) β2 = 20 ps2km−1, β3 = 0, β4 = 1 ps4km−1, and βk = 0 for k > 4, i.e., parabolic GVD profile.

Fig. 2
Fig. 2

Sketch for the interpretation of the FWM processes considered here. We assume that the schematic plots of the instantaneous frequency and instantaneous power correspond to the distance zc. Thick lines denote instantaneous frequencies and their corresponding instantaneous power at the pulse tails (blue and red regions). For two cases (in the central region, tc, and in the trailing edge, tt), we represent an arbitrary pump wave, δωp, interacting with a signal wave, δωs, and producing a certain idler wave, δωi.

Fig. 3
Fig. 3

Normalized output spectrum in dB for: (a) constant dispersion profile; (b) parabolic one; and linear dispersion variation with (c) β3 > 0 and (d) β3 < 0. See input pulse details and dispersion fiber values in the text. The small arrow corresponds to the location of the carrier frequency.

Fig. 4
Fig. 4

(a) Plot of the evolution of the functions N L 1 (dashed curve) and D 1 (solid curve) for a FWHM 250 fs Gaussian input pulse and parabolic β2(ω)-fiber profile; and (b) normalized output spectrum in dB. The rest of input pulse details and dispersion fiber values are discussed in the text. The small arrow corresponds to the location of the carrier frequency.

Equations (19)

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z A = i k = 2 i k β k k ! k t k A + i γ 0 | A | 2 A ,
d d z ( γ 0 | A ( z , t ) | 4 d t 2 | A ( z , t ) | 2 d t + β p ( ω ) | A ˜ ( z , ω ω 0 ) | 2 d ω + | A ˜ ( z , ω ω 0 ) | 2 d ω ) = 0 ,
N L 1 ( z ) = γ 0 | A ( z , t ) | 4 d t 2 | A ( z , t ) | 2 d t ,
D 1 ( z ) = β p ( ω ) | A ˜ ( z , ω ω 0 ) | 2 d ω + | A ˜ ( z , ω ω 0 ) | 2 d ω ,
N L 1 ( z ) + D 1 ( z ) = N L 1 ( 0 ) + D 1 ( 0 ) = C ,
D 1 ( z out ) N L 1 ( 0 ) C .
+ ( ω ω 0 ) 2 ( 1 2 ! β 2 + 1 3 ! β 3 ( ω ω 0 ) + 1 4 ! β 4 ( ω ω 0 ) 2 + ) | A ˜ ( z out , ω ω 0 ) | 2 d ω 1 2 γ 0 | A ( 0 , t ) | 4 d t ,
A ( t ) = | A ( t k ) | e i φ ( t k ) e i ( φ ( t ) / t | t = t k ) ( t t k ) ,
P ( δ ω , z c ) = P 0 2 exp [ 1 2 𝒲 l ( 1 e δ ω 2 δ ω max 2 ) ] ,
P i ( z ) = P s ( z c ) γ 0 2 P p 2 ( z c ) sin 2 ( | g | z ) | g | 2 ,
P i ( z ) 1 2 | g | 2 P s ( z c ) γ 0 2 P p 2 ( z c ) .
δ P p ( z ) 2 P i ( z ) d ω s = P s ( z c ) γ 0 P p ( z c ) Δ β [ 1 + Δ β 4 γ 0 P p ( z c ) ] d ω s .
δ P p ( z ) P p ( z c ) / β 2 ( ω p ) .
S ( ω , z out ) ~ ( ω , z out ) β 2 ( ω ) + 𝒩 ( ω , z out ) ,
z A ˜ ( z , ω ω 0 ) = i β p ( ω ) A ˜ ( z , ω ω 0 ) + i γ 0 ( | A ( z , t ) | 2 A ( z , t ) ) ,
z | A ˜ | 2 = 2 [ A ˜ * i γ 0 ( | A | 2 A ) ] ,
z ( β p ( ω ) | A ˜ | 2 ) = 2 [ γ 0 ( | A | 2 A ) z A ˜ * ] .
z ( β p ( ω ) | A ˜ | 2 ) = 2 [ γ 0 ( | A ( τ ) | 2 A ( τ ) z A * ( t + τ ) d τ ) ] .
z ( 1 2 π β p ( ω ) | A ˜ | 2 d ω + γ 0 2 | A | 4 d t ) = 0 .

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