Abstract

We numerically investigate supercontinuum generation using continuous-wave pumping. It is found that energy transfer during collision of solitons plays an important role. The relative influence of Raman gain on spectral broadening is shown to depend on the width of the calculation time window. Our results indicate that increasing the spectral linewidth of the pump can decrease the supercontinuum spectral width. Using a fiber with smaller dispersion at the pump wavelength reduces the required fiber length by decreasing the temporal width of the solitons formed from modulation instability. This also reduces the sensitivity to the pump spectral linewidth.

© 2006 Optical Society of America

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2006 (2)

2005 (4)

2004 (5)

2003 (3)

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

A. V. Avdokhin, S. V. Popov, and J. R. Taylor, "Continuous-wave, high-power, Raman continuum generation in holey fibers," Opt. Lett. 28, 1353-1355 (2003).
[CrossRef] [PubMed]

J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, "Mode areas and field-energy distribution in honeycomb photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2037-2045 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (4)

1996 (1)

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996). http://link.aps.org/abstract/RMP/v68/p423.
[CrossRef]

1995 (1)

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086-4092 (1995). http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

1994 (1)

1992 (1)

J. K. Lucek and K. J. Blow, "Soliton self-frequency shift in telecommunications fiber," Phys. Rev. A 45, 6666-6674 (1992). http://dx.doi.org/10.1103/PhysRevA.45.6666.
[CrossRef] [PubMed]

1991 (3)

1989 (3)

1987 (1)

1986 (1)

1972 (2)

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimenstional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62 (1972).

R. G. Smith, "Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering," Appl. Opt. 11, 2489-2494 (1972).
[CrossRef] [PubMed]

Abeeluck, A. K.

A. K. Abeeluck and C. Headley, "Continuous-wave pumping in the anomalous- and normal-dispersion regimes of nonlinear fibers for supercontinuum generation," Opt. Lett. 30, 61-63 (2005).
[CrossRef] [PubMed]

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Agrawal, G. P.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086-4092 (1995). http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Andersen, T. V.

Avdokhin, A. V.

Bang, O.

Bar-Joseph, I.

Barviau, B.

Birks, T. A.

Bjarklev, A.

Blow, K. J.

J. K. Lucek and K. J. Blow, "Soliton self-frequency shift in telecommunications fiber," Phys. Rev. A 45, 6666-6674 (1992). http://dx.doi.org/10.1103/PhysRevA.45.6666.
[CrossRef] [PubMed]

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086-4092 (1995). http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Chau, A. H. L.

Chemla, D. S.

Chen, Y.

Chi, S.

Coen, S.

Crosignani, B.

de Matos, C. J. S.

Dianov, E. M.

Dudley, J. M.

Eggleton, B. J.

Falk, P.

Finot, C.

François, P. L.

Frosz, M. H.

Fujimoto, J. G.

Gapontsev, V. P.

Genty, G.

Golovchenko, E. A.

González-Herráez, M.

Gordon, J. P.

Grossard, N.

Han, Y.-G.

Hansen, K. P.

Harvey, J. D.

Haus, H. A.

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996). http://link.aps.org/abstract/RMP/v68/p423.
[CrossRef]

Headley, C.

A. K. Abeeluck and C. Headley, "Continuous-wave pumping in the anomalous- and normal-dispersion regimes of nonlinear fibers for supercontinuum generation," Opt. Lett. 30, 61-63 (2005).
[CrossRef] [PubMed]

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Herrmann, J.

Hilligsøe, K. M.

Hsiung, P.-L.

Islam, M. N.

Joannopoulos, J. D.

Johnson, S. G.

Jørgensen, C. G.

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Kaivola, M.

Keiding, S.

Kim, N. S.

M. Prabhu, N. S. Kim, and K. Ueda, "Ultra-Broadband CW Supercontinuum Generation Centered at 1483.4 nm from Brillouin/Raman Fiber Laser," Jpn. J. Appl. Phys. 39, L291-L293 (2000). http://dx.doi.org/10.1143/JJAP.39.L291.
[CrossRef]

Knight, J. C.

Ko, T.

Kobtsev, S. M.

Kodama, Y.

Kristiansen, R.

Lægsgaard, J.

Lantz, E.

Larsen, J. J.

Lee, J. H.

Lee, S. B.

Lehtonen, M.

Leonhardt, R.

Lucek, J. K.

J. K. Lucek and K. J. Blow, "Soliton self-frequency shift in telecommunications fiber," Phys. Rev. A 45, 6666-6674 (1992). http://dx.doi.org/10.1103/PhysRevA.45.6666.
[CrossRef] [PubMed]

Ludvigsen, H.

Maillotte, H.

Malomed, B. A.

B. A. Malomed, "Soliton-collision problem in the nonlinear Schr¨odinger equation with a nonlinear damping term," Phys. Rev. A 44, 1412-1414 (1991). http://link.aps.org/abstract/PRA/v44/p1412.
[CrossRef] [PubMed]

Mamyshev, P. V.

Marshall, W. K.

Martin-Lopez, S.

Mølmer, K.

Mortensen, N. A.

Mussot, A.

Nazarkin, A.

Nicholson, J. W.

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Nielsen, C. K.

Nozaki, K.

Paulsen, H. N.

Pilipetskii, A. N.

Pitois, S.

Popov, S. V.

Prabhu, M.

M. Prabhu, N. S. Kim, and K. Ueda, "Ultra-Broadband CW Supercontinuum Generation Centered at 1483.4 nm from Brillouin/Raman Fiber Laser," Jpn. J. Appl. Phys. 39, L291-L293 (2000). http://dx.doi.org/10.1143/JJAP.39.L291.
[CrossRef]

Provino, L.

Randoux, S.

Ranka, J. K.

Russell, P. St. J.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimenstional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62 (1972).

Smirnov, S. V.

Smith, R. G.

Stentz, A. J.

Sucha, G.

Suret, P.

Sylvestre, T.

Taylor, J. R.

Ueda, K.

M. Prabhu, N. S. Kim, and K. Ueda, "Ultra-Broadband CW Supercontinuum Generation Centered at 1483.4 nm from Brillouin/Raman Fiber Laser," Jpn. J. Appl. Phys. 39, L291-L293 (2000). http://dx.doi.org/10.1143/JJAP.39.L291.
[CrossRef]

Vanholsbeeck, F.

Wadsworth, W. J.

Wegener, M.

Wen, S.

Windeler, R. S.

Wong, W. S.

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996). http://link.aps.org/abstract/RMP/v68/p423.
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Yan, M. F.

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Yariv, A.

Yu, M.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086-4092 (1995). http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimenstional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62 (1972).

Appl. Opt. (1)

Appl. Phys. B (1)

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). http://dx.doi.org/10.1007/s00340-003-1201-z.
[CrossRef]

Appl. Phys. Lett. (1)

C. J. S. de Matos, S. V. Popov, and J. R. Taylor, "Temporal and noise characteristics of continuous-wave-pumped continuum generation in holey fibers around 1300 nm," Appl. Phys. Lett. 85, 2706-2708 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

J. Opt. Soc. Am. B (6)

J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, "Mode areas and field-energy distribution in honeycomb photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2037-2045 (2003).
[CrossRef]

P. L. François, "Nonlinear propagation of ultrashort pulses in optical fibers: total field formulation in the frequency domain," J. Opt. Soc. Am. B 8, 276-293 (1991). http://www.opticsinfobase.org/abstract.cfm?URI=josab-8-2-276.
[CrossRef]

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Numerical analysis of the Raman spectrum evolution and soliton pulse generation in single-mode fibers," J. Opt. Soc. Am. B 8, 1626-1632 (1991). http://www.opticsinfobase.org/abstract.cfm?URI=josab-8-8-1626.
[CrossRef]

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, "Femtosecond distributed soliton spectrum in fibers," J. Opt. Soc. Am. B 6, 1149-1158 (1989). http://www.opticsinfobase.org/abstract.cfm?URI=josab-6-6-1149.
[CrossRef]

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, "Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping," J. Opt. Soc. Am. B 19, 765-771 (2002).
[CrossRef]

S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers," J. Opt. Soc. Am. B 19, 753-764 (2002).
[CrossRef]

Jpn. J. Appl. Phys. (1)

M. Prabhu, N. S. Kim, and K. Ueda, "Ultra-Broadband CW Supercontinuum Generation Centered at 1483.4 nm from Brillouin/Raman Fiber Laser," Jpn. J. Appl. Phys. 39, L291-L293 (2000). http://dx.doi.org/10.1143/JJAP.39.L291.
[CrossRef]

Opt. Express (9)

K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keiding,R. Kristiansen, K. P. Hansen, and J. J. Larsen, "Supercontinuum generation in a photoniccrystal fiber with two zero dispersion wavelengths," Opt. Express 12, 1045-1054 (2004).
[CrossRef] [PubMed]

G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, "Enhanced bandwidth of supercontinuum generated in microstructured fibers," Opt. Express 12, 3471-3480 (2004).
[CrossRef] [PubMed]

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Supplementary Material (1)

» Media 1: AVI (1984 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Left: Calculated dispersion profiles for fibers considered here. Right: Comparison of power spectral density S(λ) for two noise models: CW with one photon per mode (blue, solid) and CW with phase noise corresponding to FWHM linewidth of 30 GHz (~0.1 nm), Lorentzian shaped (green, dashed). Average power of input field is 10W. Red dotted curve: resulting spectrum when the CW is replaced by a super-Gaussian pulse of 30 ps width and 10 W peak power (the spectrum is scaled so integrating over the spectrum gives 10 W to allow comparison with the CW spectrum).

Fig. 2.
Fig. 2.

An illustration of the modeled periodic super-Gaussian input pulse. Top: power variation in time, bottom: phase fluctuation in time. In this example the width of the time window is 59.0 ps, the width of the super-Gaussian pulse is 2T sG=30 ps, and ΔνFWHM=265 GHz.

Fig. 3.
Fig. 3.

Spectrogram movie for one simulation of propagation in the d/Λ=0.65 fiber. Input super-Gaussian pulse width 2T sG=30 ps, ΔνFWHM=30 GHz, time window T max=59 ps. Note that the color scale changes during propagation, as some of the solitons acquire higher peak power. The horizontal white line on the bottom Figs. indicates the Raman Stokes wavelength at 1116 nm. pitch1p72 dL0p65 30GHz.avi (1.93 MB).

Fig. 4.
Fig. 4.

Left: Frequency spectrum of pulse power P(t) at z=9 m calculated for one simulation for the d/Λ=0.65 fiber (blue, solid) and one simulation for the d/Λ=0.378 fiber (green, dashed). The vertical lines indicate the frequency with maximum MI gain in the corresponding fibers. Right: Maximum peak power along the fiber length for the same simulation as shown in Fig. 3. The horizontal black line at 152 Windicates the minimum peak power required for a soliton to red-shift to 1116 nm after 74 m of propagation.

Fig. 5.
Fig. 5.

Left: Red-shift rate dν0/dz scaled by -1/|β 2| in the two regimes where analytical expressions are available: T 0~76 fs [Eq. (16)] (blue, solid) and T 0 76 fs [Eq. (15)] (green, dashed). Right: Illustration of a super-Gaussian pulse with width 2T sG=15 ps in a time domain of T max=29.5 ps (top, case A), 2T sG=15 ps and T max=59 ps (middle, case B), and 2T sG=30 ps and T max=59 ps (bottom, case C).

Fig. 6.
Fig. 6.

Spectra calculated for 10 ensembles differing by the random number seed for the phase noise. Λ=1.72 µm, d/Λ=0.65, and the fibre length L=80 m. Linewidth is 30 GHz, input super-Gaussian pulse width 2T sG=15 ps. Left: 214 points used, T max=29.5 ps, smoothed over 8 points. Right: 215 points used, T max=59 ps, smoothed over 16 points.

Fig. 7.
Fig. 7.

Spectra calculated for 10 ensembles differing by the random number seed for the phase noise generation. All for Λ=1.72 µm, d/Λ=0.65, and a fibre length of 80 m. 215 points used, 2T sG=30 ps, T max=59 ps. Left: Spectral linewidth ΔνFWHM=30 GHz. Right: ΔνFWHM=265 GHz. Smoothed over 16 points. The thin black line indicates the input spectrum for one of the simulations.

Fig. 8.
Fig. 8.

Left: Lorentzian shaped power spectrum for 30 GHz (solid, blue) and 265 GHz linewidth (dashed, green). Right: Power evolution along the fibre at the pump wavelength and the Raman Stokes wavelength of 1116 nm. One simulation for a linewidth of 30 GHz (solid, blue) and one simulation for a 265 GHz linewidth (dashed, green).

Fig. 9.
Fig. 9.

Left: Ensemble average for the same parameters as in Fig. 6(right). Right: Ensemble average for the same parameters as in Fig. 7(left). The averaged input spectrum is indicated as a thin black line.

Fig. 10.
Fig. 10.

Calculated spectra at z=64 m for the Λ=1.72 µm, d/Λ=0.378 fiber, super-Gaussian input pulse width 2T sG=30 ps, time window T max=59 ps, 215 points. Left: 30 GHz linewidth. Right: 265 GHz linewidth. 10 ensembles shown in each case. The thin black line indicates the input spectrum for one of the ensembles.

Equations (16)

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E z = i m 2 i m β m m ! m E T m α 2 E + i γ ( ω ) [ 1 + i ω 0 T ] [ E ( z , T ) R ( T ' ) E ( z , T T ' ) 2 d T ' ] ,
R ( t ) = ( 1 f R ) δ ( t ) + f R h ( t ) = ( 1 f R ) δ ( t ) + f R τ 1 2 + τ 2 2 τ 1 τ 2 2 exp ( t τ 2 ) sin ( t τ 1 ) Θ ( t )
E ( 0 , T ) = P 0 exp [ i δ ϕ ( T ) ]
ν i = ν 0 + 1 2 π d ( δ ϕ ) d t = ν 0 + ν R ( t ) .
δ ϕ ( t ) = 2 π t ν R ( ξ ) d ξ ,
D δ ϕ ( τ ) = 8 π 2 τ 0 Γ ν R ( η ) d η ,
Γ ν R ( τ ) = S ν R ( ν ) exp ( i 2 π ν τ ) d ν ,
Γ ν R ( τ ) = S ν R B 2 B 2 exp ( i 2 π ν τ ) d ν .
σ ν R 2 = Γ ν R ( 0 ) = S ν R B 2 B 2 d ν = S ν R B S ν R = σ ν R 2 B
D δ ϕ ( τ ) = 8 π 2 τ σ ν R 2 B 0 B 2 B 2 exp ( i 2 π ν η ) d ν d η = 4 π 2 τ σ ν R 2 B .
σ ν R 2 = Δ ν FWHM B 2 π
E ( 0 , T ) = P 0 exp [ i δ ϕ ( T ) ] exp [ 1 2 ( T T sG ) 2 m ] ,
E z = i β 2 2 2 E T 2 + i γ E E 2 .
ν max = γ P 0 2 π 2 β 2 ,
d ν 0 d z = 8 β 2 T R 2 π 15 T 0 4 , T 0 76 fs ,
d ν 0 d z = 0.09 β 2 Ω R 2 2 π T 0 , T 0 < 76 fs .

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