Abstract

In addition to fiber nonlinearity, fiber dispersion plays a significant role in spectral broadening of incoherent continuous-wave light. In this paper we have performed a numerical analysis of spectral broadening of incoherent light based on a fully stochastic model. Under a wide range of operating conditions, these numerical simulations exhibit striking features such as damped oscillatory spectral broadening (during the initial stages of propagation), and eventual convergence to a stationary, steady state spectral distribution at sufficiently long propagation distances. In this study we analyze the important role of fiber dispersion in such phenomena. We also demonstrate an analytical rate equation expression for spectral broadening.

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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2008 (1)

2007 (1)

2006 (3)

2005 (1)

2003 (1)

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. 15(12), 1698–1700 (2003).
[CrossRef]

1992 (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

1991 (1)

1976 (1)

1965 (1)

H. Hodara, “Statistics of thermal and laser radiation,” Proc. IEEE 53(7), 696–704 (1965).
[CrossRef]

Babin, S. A.

Barviau, B.

Bouteiller, J.-C.

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. 15(12), 1698–1700 (2003).
[CrossRef]

Churkin, D. V.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Goodman, J. W.

Hodara, H.

H. Hodara, “Statistics of thermal and laser radiation,” Proc. IEEE 53(7), 696–704 (1965).
[CrossRef]

Horak, P.

Ibsen, M.

Ismagulov, A. E.

Kablukov, S. I.

Manassah, J. T.

McCoy, A. D.

Newell, A. C.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

Podivilov, E. V.

Pushkarev, A.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

Randoux, S.

Richardson, D. J.

Suret, P.

Thomsen, B. C.

Zakharov, V. E.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. 15(12), 1698–1700 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

Opt. Lett. (4)

Physica D (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57(1-2), 96–160 (1992).
[CrossRef]

Proc. IEEE (1)

H. Hodara, “Statistics of thermal and laser radiation,” Proc. IEEE 53(7), 696–704 (1965).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Other (4)

G. P. Agrawal, Nonlinear fiber optics, 3rd ed. (Academic Press, New York, 2001).

M.-A. Lapointe and M. Piche, “Linewidth of high-power fiber lasers,” Proc. Photonics North 7386, 73860S (2009).

D. B. S. Soh and J. P. Koplow, “Analysis of incoherent spectral broadening in optical fibers with nonzero dispersion,” to submit for publication.

V. E. Zakharov, V. Lvov, and G. Falkovich, Wave Turbulence, (Springer-Verlag, New York, 1992).

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Figures (8)

Fig. 1
Fig. 1

Normalized output spectra for various fiber lengths from a fully stochastic numerical simulation. The curves correspond to different propagation distances as indicated. The number of trials in each ensemble average was 1000, and the spectral resolution was 0.02 nm

Fig. 2
Fig. 2

Fully stochastic numerical simulation results. Left: spectral broadening factor as a function of fiber length, for various input powers (numerical annotations indicate power in Watts). Right: spectral broadening factor for passage through 80, 200, and 400 m long spans of fiber, as a function of input power.

Fig. 3
Fig. 3

Fully stochastic numerical simulation results. Spectral broadening factor Θ as a function of propagation distance, for various fiber dispersion values. Simulated with input power of 2 W.

Fig. 4
Fig. 4

The spectral broadening factor as a function of propagation distance predicted by the fully stochastic simulation (red) and the analytical formula for ensemble-averaged spectral power density (blue).

Fig. 5
Fig. 5

Comparison of linear spectra taken at 40 m intervals of propagation distance. Left: Fully stochastic simulation. Right: Numerical evaluation of the analytical derivation.

Fig. 6
Fig. 6

Comparison of logarithmic spectra taken at 40 m intervals of propagation distance. Left: Fully stochastic simulation. Right: Numerical evaluation of the analytical derivation.

Fig. 7
Fig. 7

Spectral broadening factor Θ predicted by fully stochastic simulation (solid curves) and numerical evaluation of the analytical derivation (dotted curves).

Fig. 8
Fig. 8

Relative error in Θ for calculation based on analytical rate equation.

Equations (14)

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d A j d z     =     α 2 A j +     i γ l m n A l * A m A n δ j + l , m + n e i Δ k j l m n z
P ( A j = Z | z = 0 ) = 1 2 π 3 p 0 ( ω j ) exp ( | Z | 2 2 p 0 ( ω j ) )
d p j d z = α p j + 2 Re [ i γ l m n A l * A j * A m A n     δ j + l , m + n e i Δ k j l m n z ] ,
p j ( z ) = A j ( z ) A j * ( z ) ,
d p j ( z ) d z = α p j ( z ) + 4 γ 2 z = 0 z e 2 α ( z z ) l m n { p l p j p m p n [ ( 1 p l + 1 p j ) ( 1 p m + 1 p n ) ] × δ j + l , m + n cos ( Δ k j l m n ( z z ) ) } d z .
Δ k j l m n = k ( ω j ) k ( ω l ) + k ( ω m ) + k ( ω j + ω l ω n ) = β 2 ( ω j ω m ) ( ω l ω m ) ,
2 2 γ P 0 L e f f 1.
κ ( z ) 1 z z = 0 z cos ( β 2 Ω ¯ 2 z ) d z = sinc ( β 2 Ω ¯ 2 z ) ,
| d ρ p k ( z ) d z | 16 γ 2 ρ p k ( 0 )     P 0 2 e α z ( 1 e α z α ) κ ( z ) ,
Λ ( z ) 16 γ 2 P 0 2 e α z ( 1 e α z α ) κ ( z ) .
| d ( ρ p k ( z ) / ρ p k ( 0 ) ) d z | | Λ ( z ) | .
2 2 γ P 0 L e f f 1.
Ξ = λ g R 64 2 π n 2 1.
Ξ = ( 1.0 × 10 6 m )     ( 1.0 × 10 13 m W 1 ) 64 2 π ( 2.7 × 10 20 m 2 W 1 ) = 0.013 1.

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