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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 17 — Aug. 22, 2005
  • pp: 6615–6625
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The role of pump incoherence in continuous-wave supercontinuum generation

Frédérique Vanholsbeeck, Sonia Martin-Lopez, Miguel González-Herráez, and Stéphane Coen  »View Author Affiliations


Optics Express, Vol. 13, Issue 17, pp. 6615-6625 (2005)
http://dx.doi.org/10.1364/OPEX.13.006615


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Abstract

Supercontinuum generation can be achieved in the continuous-wave regime with a few watts of pump power launched into kilometer-long fibers. High power spectral density broadband light sources can be obtained in this way. Using a generalized nonlinear Schrödinger equation model and an ensemble averaging procedure that takes into account the partially-coherent nature of the pump laser, we fully explain for the first time the spectral broadening mechanisms underlying this process. Our simulations and experiments confirm that continuous-wave supercontinuum generation involve Raman soliton dynamics and dispersive waves in a way akin to pulsed supercontinua. The Raman solitons are however generated with a wide distribution of parameters because they originate from the random phase and intensity fluctuations associated with the pump incoherence. This soliton distribution is averaged out by experimental measurements, which explains the remarkable smoothness of experimental continuous-wave supercontinuum spectra.

© 2005 Optical Society of America

1. Introduction

Supercontinuum (SC) generation in optical fibers and waveguides has attracted considerable research interest in the last few years [1–8

01. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

]. Applications of SC generation in optical fibers can now be found in fields such as optical frequency metrology [9

09. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

, 10

10. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, Th. Udem, and T. W. Hänsch, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef] [PubMed]

], optical coherence tomography [11

11. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber,” Opt. Lett. 26, 608–610 (2001). [CrossRef]

, 12

12. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Study of an ultrahigh-numerical-aperture fiber continuum generation source for optical coherence tomography,” Opt. Lett. 27, 2010–2012 (2002). [CrossRef]

], and multiple optical carrier generation for wavelength-division multiplexing [13

13. T. Morioka, K. Mori, and M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single laser source using supercontinuum in optical fibres,” Electron. Lett. 29, 862–864 (1993). [CrossRef]

, 14

14. H. Takara, T. Ohara, and K. Sato, “Over 1000 km DWDM transmission with supercontinuum multi-carrier source,” Electron. Lett. 39, 1078–1079 (2003). [CrossRef]

]. This renewed research interest in SC light sources has been made possible thanks to the recent development of special fibers with high nonlinearities, namely photonic crystal fibers [1

01. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

,3

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

] and tapered fibers [2

02. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000). [CrossRef]

]. SC is typically obtained in those fibers by pumping them with high peak-power input pulses with durations ranging from several nanoseconds to several tens of femtoseconds [1

01. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

, 3

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

, 15

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

]. It results from the combined effects of self- and cross-phase modulation (SPM and XPM), four-wave mixing (FWM), and stimulated Raman scattering (SRS), that together lead to multisoliton dynamics and dispersive wave generation [16–19

16. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901/1–4 (2001). [CrossRef] [PubMed]

].

2. Experiment

In order to understand better the results of the numerical simulations that are presented later in this paper, we briefly describe here the experiment that our simulations tend to reproduce. This cw SC generation experiment was recently performed by some of us [6

06. M. González-Herráez, S. Martín-López, P. Corredera, M. L. Hernanz, and P. R. Horche, “Supercontinuum generation using a continuous-wave Raman fiber laser,” Opt. Commun. 226, 323–328 (2003). [CrossRef]

] and has led to the generation of a 200 nm wide SC, covering the S, C and L transmission bands defined by the International Telecommunication Union (ITU).

The pump laser used in our experiment is a cw cascaded Raman fiber laser (RFL) with a depolarized single-mode output tuned at a wavelength of 1455.3 nm [6

06. M. González-Herráez, S. Martín-López, P. Corredera, M. L. Hernanz, and P. R. Horche, “Supercontinuum generation using a continuous-wave Raman fiber laser,” Opt. Commun. 226, 323–328 (2003). [CrossRef]

]. The laser output power can be as high as 2.1 W and, thanks to a feedback control loop, is stabilized to within ±10 mW. The spectral linewidth of the RFL grows as the power increases. For the maximum 2.1 W output power, the full-width-at-half-maximum (FWHM) of the output laser spectrum is Δλ = 1.1 nm (or Δv = 150 GHz), with a RIN < -110 dBc/Hz (measured in the range 0-1 GHz). CW supercontinuum generation is observed by launching the light generated by the RFL in a 7-km-long spool of nonzero dispersion-shifted fiber (NZDSF). The nonlinear coefficient of this fiber, measured with a Sagnac interferometer [28

28. D. Monzón-Hernández, A. N. Starodumov, Y. O. Barmenkov, I. Torres-Gómez, and F. Mendoza-Santoyo, “Continuous-wave measurement of the fiber nonlinear refractive index,” Opt. Lett. 23, 1274–1276 (1998). [CrossRef]

], is found to be γ = 2.7 W-1 km-1. Direct probing with a tunable laser has also provided us with an independent measurement of the Raman gain coefficient, gR = 0.7 W-1 km-1. The dispersion coefficients at the pump wavelength have been obtained by the phase shift method, β 2 = -0.11 ps2/km, β 3 = 0.06 ps3/km, β 4 = -1.55 × 10-4 ps4/km, and β 5 = 1.33 × 10-6 ps5/km. The uniformity of the chromatic dispersion along the fiber length has also been evaluated using a method developed by some of us [29

29. M. González-Herráez, L. Thévenaz, and P. Robert, “Distributed measurement of chromatic dispersion by four-wave mixing and brillouin optical-time-domain analysis,” Opt. Lett. 28, 2210–2212 (2003). [CrossRef]

] and is found to be better than ±0.13 ps2/km at the wavelength of 1555 nm (spatial resolution of ~ 500 m). The fiber absorption is nearly uniform from 1450 to 1620 nm and is about α = 0.2 dB/km. It then increases up to 0.4 dB/km at 1700 nm. The pump wavelength was chosen to lie in a spectral region of small anomalous dispersion (β 2 < 0) in order to favor phase-matched FWM processes. Also, the use of a long fiber with low losses ensures strong FWM-mediated power transfer between the pump and all the generated spectral components as well as a low SRS threshold.

Fig. 1. SC spectrum with different pump powers at room temperature (≃ 20 °C). All the curves have been normalized to the same peak value. Inset shows how increasing pump power leads to a stronger pump depletion.

3. Numerical model

Our cw simulations of SC generation are based on a generalized nonlinear Schrodinger Eq. (GNLSE) for the scalar electric field envelope E(z,t) that is suitable for studying broadband light evolution [3

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

,21

21. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904/1–4 (2003). [CrossRef] [PubMed]

,27

27. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

,30

30. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics Series, 3rd ed. (Academic Press, San Diego, 2001).

,31

31. P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations,” J. Opt. Soc. Am. B 18, 139–152 (2001). [CrossRef]

],

Ez=α2Eiβ222Et2+β363Et3++iγE(tR(t')E(z,tt')2dt'+iΓR(z,t)).
(1)

Here z is the longitudinal fiber coordinate and t is the retarded time. R(t) = (1 - fR)δ(t) + fRhR(t) is the nonlinear response of silica that includes instantaneous Kerr and delayed Raman nonlinearities with fR = 0.18. For hR(t), we used the experimentally measured Raman response of silica fibers. The stochastic variable ΓR(z,t) represents the spontaneous Raman noise and has frequency-domain correlations [21

21. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904/1–4 (2003). [CrossRef] [PubMed]

, 31

31. P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations,” J. Opt. Soc. Am. B 18, 139–152 (2001). [CrossRef]

]

Fig. 2. (a) Temporal intensity profile of the cw input beam used in the simulations for one particular realization of the random initial spectral phase (shown here with an average power of 1.7 W). (b) Intensity autocorrelation of the RFL output at 2.1 W.
ΓR(Ω,z)ΓR*(Ω',z')=(2fRħω0γ)ImhR(Ω)[nth(Ω)+U(Ω)]δ(zz')δ(ΩΩ')
(2)

where n th(Ω) = [exp(ħΩ/(kT)) - 1]-1 is the thermal Bose distribution of the phonons and U is the Heaviside step function. All the other symbols have their usual meaning and are set to values identical to those of the experiment.

To this aim, we adopt a phenomenological approach in which the cw input wave is represented as having the measured spectral density of our pump laser with an underlying random spectral phase. This approach leads to an initial temporal intensity profile that is essentially made up of a random succession of pulses of a few ps durations (about the inverse of the Δv = 150 GHz spectral width of the pump laser), as illustrated in Fig. 2(a) for one particular realization of the random spectral phase. Such strong intensity fluctuations could at first appear unrealistic for a stable cw RFL exhibiting a RIN < - 110 dBc/Hz at 1 GHz. We have however confirmed their existence through an intensity autocorrelation measurement of the RFL output [Fig. 2(b)]. As can be seen, a peak about twice as strong as the cw background is detected, a feature compatible with intensity fluctuations of nearly 100 % contrast. We note that such behavior has been observed with two RFLs from different manufacturers. Although this description might not necessarily represent the real nature of a partially coherent cw beam [26

26. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838 [CrossRef] [PubMed]

,33

33. S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995). [CrossRef] [PubMed]

], strong intensity fluctuations do appear to be present in the input beam and we will show in the following that it leads to a very good agreement with experimental results. We must also point out that these fluctuations should not affect the noise properties of the Raman amplifiers used in wavelength-division-multiplexed telecommunication systems [34

34. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002). [CrossRef]

] because these amplifiers are mostly based on backward pumping schemes and because of typically large walk-off effects between the pump and the signals (that are separated by about 13 THz). Accordingly, fast pump fluctuations are averaged out along the amplifier length.

Fig. 3. (a) Numerical output spectra for various pump power levels and (b) temporal intensity output at P p = 1.7 W for the initial condition shown in Fig. 2(a). (c) Measured intensity autocorrelation of the SC at P p = 1.7 W.

We have calculated numerically the output spectra corresponding to the initial condition shown in Fig. 2(a) after propagation in our NZDSF for various pump power levels. The results are presented in Fig. 3(a). This figure shows that the low power simulations (up to 0.72 W) agree fairly well with the experimental results (Fig. 1) as they reproduce the MI sideband growth and the pump broadening. However, higher pump power simulations depart significantly from the experiments. In particular, the supercontinuum smoothness is not reproduced at all and we can clearly see, for the highest pump power, the generation of soliton-like features red-shifted from the pump beam. The solitonic nature of these spectral features can be readily confirmed by observing the corresponding temporal intensity output [Fig. 3(b)]. In this case, two ultra-short pulses of 220 fs and 250 fs (FWHM) can be clearly seen. Numerical Fourier filtering of the spectrum enables us to associate these two pulses with the two broad spectral features observed at 1570 nm and 1660 nm, respectively. At 1570 nm, chromatic dispersion amounts -6.7 ps2/km while it reaches -12.6 ps2/km at 1660 nm. A straightforward calculation then shows that these two pulses are close to single fundamental solitons [30

30. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics Series, 3rd ed. (Academic Press, San Diego, 2001).

]. While the presence of these fs solitons might seem surprising when considering that we are in principle dealing with a cw laser output, these are not mere numerical artifacts as can be shown by observing the experimental intensity autocorrelation trace of the SC output at 1.7 W pump power [Fig. 3(c)]. This trace exhibits a subpicosecond feature much narrower than the autocorrelation trace of the pump laser [Fig. 2(b)] and confirms that ultrashort features have developed during propagation. The observed pulses are actually Raman solitons whose generation is seeded from the modulation instability (MI)-induced break-up of the temporal fluctuations present in the initial condition. As these solitons propagate down the fiber, they undergo a continuous shift to longer wavelengths because of the Raman self-frequency shift. While shifting their wavelengths, these solitons shed away some blue-shifted radiation in the form of dispersive waves, which explains the generation of the blue-side of the continuum [7

07. A. K. Abeeluck and C. Headley, “Supercontinuum growth in a highly nonlinear fiber with a low-coherence semiconductor laser diode,” Appl. Phys. Lett. 85, 4863–4865 (2004). [CrossRef]

, 8

08. 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]

, 17

17. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901/1–4 (2002). [CrossRef] [PubMed]

, 23

23. P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987). [CrossRef]

, 26

26. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838 [CrossRef] [PubMed]

]. Such femtosecond Raman soliton generation has been previously experimentally observed about two decades ago when pumping fibers with ps pump pulses [22–24

22. E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Zh. Eksp. Teor. Fiz. 41, 242–244 (1985) [JETP Lett. 41, 294–297 (1985)].

] and is one of the key process underlying SC generation in the pulsed regime [16

16. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901/1–4 (2001). [CrossRef] [PubMed]

,17

17. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901/1–4 (2002). [CrossRef] [PubMed]

,27

27. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

]. Theoretical studies have shown that these solitons should also be generated in the cw regime [25

25. 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). [CrossRef]

, 26

26. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838 [CrossRef] [PubMed]

, 35

35. J. Nathan Kutz, C. Lyngå, and B. J. Eggleton, “Enhanced supercontinuum generation through dispersion-management,” Opt. Express 13, 3989–3998 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-3989 [CrossRef] [PubMed]

] but our work, together with Refs. [7

07. A. K. Abeeluck and C. Headley, “Supercontinuum growth in a highly nonlinear fiber with a low-coherence semiconductor laser diode,” Appl. Phys. Lett. 85, 4863–4865 (2004). [CrossRef]

, 8

08. 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]

] is the first to provide a clear experimental evidence of their presence. Quite interestingly, this picture makes clear that nonlinear phenomena associated with the propagation of short pulses also play an important role in the cw regime, and that, after solitons are generated through MI-induced break-up, cw SC generation occurs essentially in an identical way than in the pulsed regime. We must also note that the initial temporal intensity fluctuations due to the beam incoherence are an important ingredient to speed up the spectral broadening since this naturally leads to an enhanced MI gain in comparison to a pure cw input beam.

At this point we have established that the dynamics of Raman solitons significantly contribute to cw SC generation in a way akin to what happens in pulsed-pumped SC. The contribution of these solitons to the spectral broadening, although very important, is however not sufficient to fully account for the experimental results, and in particular for the spectral smoothness of the generated spectra. As we will discuss below, the incoherent nature of the pump beam is actually the key ingredient to reconcile experiments with simulations.

4. Ensemble averaging

The simulation results presented in Fig. 3 have been obtained for one particular realization of the random initial spectral phase. Given the discrete nature of our numerical scheme, it means that we have only simulated the propagation of one particular temporally-limited snapshot of a partially coherent cw beam [as shown in Fig. 3(a)]. Recent studies performed under pulsed pumping conditions have however demonstrated that the SC generation process is typically very sensitive to random input noise [21

21. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904/1–4 (2003). [CrossRef] [PubMed]

,27

27. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

]. Noise should a priori play an even more important role on the SC generation process in the cw regime since a partially coherent cw beam is inherently random from start to end, whereas the pulses emitted by a mode-locked laser have a relatively well defined temporal intensity and phase profile with only a limited superimposed random noise component.

To assess the influence of the random nature of our cw pump beam, we have run additional sets of numerical simulations. Within each set, all the simulations have been performed under identical conditions except for the random input spectral phase. The results obtained for a pump power of P p = 1.7 W are illustrated in Fig. 4. Here we have plotted the initial input intensities and the corresponding output spectra for five different simulations. This figure clearly reveals how different initial fluctuations evolve into Raman solitons exhibiting widely different parameters. Basically, when fluctuations of higher amplitude are present in the initial condition, MI generates and fissions solitons earlier, and these solitons can then undergo a larger self-frequency shift while propagating in the remaining part of the fiber. As can be seen, this effect is rather drastic. The wavelengths of the solitons at the fiber end differ by more than 100 nm between different simulations, and even the number of generated solitons can vary. These observations are even made clearer in the movie associated with Fig. 4 and that shows how the temporal and spectral intensities of the field evolve along the fiber for the five simulations. In particular, by observing the appearance of fully developed MI oscillations on top of the initial fluctuations, we can clearly see that MI occurs at a different position along the fiber in the different simulations.

As a matter of fact, Fig. 4(b) reveals that the randomness of the fluctuations present in the cw pump beam leads to such a broad distribution of Raman solitons that they cover the entire spectral band of the experimentally observed SC [24

24. 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). [CrossRef]

]. We must also point out that the soliton parameters vary on a time scale of the order of the coherence time of the pump laser (about a few picoseconds), i.e., much shorter than the integration time of an optical spectrum analyzer.

Fig. 4. Results of five identical simulations differing only by the random initial spectral phase and performed with a pump power P p = 1.7 W. (a) represents the initial field intensities while (b) shows the corresponding generated spectra. A movie of the simulation can be seen by clicking on the figure. [Media 1]

Accordingly, we have to conclude that the spectral characteristics of the randomly-generated Raman solitons are significantly averaged in the measurements and that the experimental spectra are simply the envelope of the Raman soliton distributions. This explains the smoothness of cw SC spectra and why they exhibit no visible signature of individual solitons. We must stress that this averaging is qualitatively different from what typically occurs in the pulsed regime due to the pump laser noise [21

21. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904/1–4 (2003). [CrossRef] [PubMed]

, 27

27. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

, 36

36. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002). [CrossRef]

]. In that latter case, the fluctuations are not strong enough to completely smooth the spectrum and hide the underlying spectral broadening mechanisms. Accordingly, the signature of Raman solitons is still clearly visible. In other words, the overall spectral shape of pulsed SC in mainly determined by the single-pulse response. This contrasts with the cw regime where the spectral shape is mostly determined by the statistical distribution of the initial fluctuations.

This discussion makes clear that the experimental situation does not correspond to one particular simulation. Instead, an ensemble average over many identical simulations performed with different random initial spectral phases is needed to reach a good agreement [24

24. 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). [CrossRef]

, 27

27. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

]. To verify this interpretation, we have performed such an ensemble average numerically over 100 identical simulations. The result of these calculations is plotted in Fig. 5 (solid curves) for 0.72 and 1.7 W pump power and is compared to the corresponding experimental spectra (dashed curves). As can be seen, the agreement is excellent. The SC smoothness is now reproduced correctly. Even some fine spectral features such as the location of the maximum and some other peaks as well as the shape of the blue shoulder of the SC are correctly simulated. As regards the decreasing slope of the spectrum on the long wavelength side of the pump, we can simply interpret it from the low probability to generate solitons that undergo a large frequency shift. Note that we have not performed any adjustments of the simulation parameters to try to fit better the experimental and numerical results. The larger bandwidth of the simulated spectra observed in Fig. 5(b) can probably be explained by the imperfect modelling of the fluctuations of our pump laser. Also, our simulations do not take into account polarization effects, the frequency-dependent effective area, as well as longitudinal variations of the fiber parameters that typically reduce the efficiency of nonlinear processes. Additionally, some of the fiber parameters might have been imperfectly evaluated.

We must stress that, in addition to the ensemble averaging procedure described above, the complexity of cw SC generation simulations is also compounded by the long fiber length and by the associated huge temporal walk-off. In the case of our experiment, the maximum walk-off occurs between the zero-dispersion wavelength (1453 nm) and the red-side of the continuum (i.e., 1625 nm) and is as high as 4 ns over the 7 km-long fiber. It means that a numerical window wider than 4 ns is in principle required to keep track of all the generated spectral components. Given that our numerical scheme is based on Fourier transforms, the sampling theorem then imposes the use of a very large number of discretization points to describe the SC spectral bandwidth, typically 219 or 220, making the simulations very computationally intensive. In practice, we have tested the robustness of our numerical integration scheme by performing ensemble of simulations with a number of points ranging from 212 to 218 (corresponding to temporal windows from 50 ps to 3.2 ns). After averaging over the ensembles, the simulated spectra obtained in each case are essentially identical. The numerical results shown in this paper have actually been obtained with 214 points, which represents a good trade-off between the number of simulations required for a good averaging and the overall computation time. It is important to note, however, that the split-step Fourier scheme is implicitly based on periodic boundary conditions. Accordingly, if one uses a time window shorter than 4 ns to describe the cw field, a cyclic wrapping of the temporal envelope of the electric field occurs. This means that a Raman soliton travelling at a group velocity different than that of the pump may periodically travel across the simulated window and interact over and over with the same portion of the field. Of course, this is not what happens in the experiment where Raman solitons forever drift away from their source point and interact with other random realizations of the fluctuations. The fact that the simulation results obtained with < 4 ns-wide numerical windows are essentially independent of the number of discretization points indicate however that a limited sample of fluctuations is statistically significant and represents correctly the overall physics.

Fig. 5. Comparison of experimental and simulated SC spectra obtained for a pump power of (a) P p = 0.72 W and (b) P p = 1.7 W. The numerical results have been averaged over an ensemble of 100 simulations differing only by the random initial spectral phase.

5. Conclusion

Acknowledgments

F. Vanholsbeeck and S. Coen acknowledge the support of the New Economy Research Fund of the New Zealand Foundation for Research, Science and Technology. S. Coen is also supported by the Royal Society of New Zealand. We also would like to thank Hector Fernandez and Javier Solís for helping us with the autocorrelation measurements.

References and links

01.

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

02.

T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000). [CrossRef]

03.

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

04.

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]

05.

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 B77, 211–218 (2003). [CrossRef]

06.

M. González-Herráez, S. Martín-López, P. Corredera, M. L. Hernanz, and P. R. Horche, “Supercontinuum generation using a continuous-wave Raman fiber laser,” Opt. Commun. 226, 323–328 (2003). [CrossRef]

07.

A. K. Abeeluck and C. Headley, “Supercontinuum growth in a highly nonlinear fiber with a low-coherence semiconductor laser diode,” Appl. Phys. Lett. 85, 4863–4865 (2004). [CrossRef]

08.

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]

09.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

10.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, Th. Udem, and T. W. Hänsch, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef] [PubMed]

11.

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber,” Opt. Lett. 26, 608–610 (2001). [CrossRef]

12.

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Study of an ultrahigh-numerical-aperture fiber continuum generation source for optical coherence tomography,” Opt. Lett. 27, 2010–2012 (2002). [CrossRef]

13.

T. Morioka, K. Mori, and M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single laser source using supercontinuum in optical fibres,” Electron. Lett. 29, 862–864 (1993). [CrossRef]

14.

H. Takara, T. Ohara, and K. Sato, “Over 1000 km DWDM transmission with supercontinuum multi-carrier source,” Electron. Lett. 39, 1078–1079 (2003). [CrossRef]

15.

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

16.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901/1–4 (2001). [CrossRef] [PubMed]

17.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901/1–4 (2002). [CrossRef] [PubMed]

18.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt. Express 10, 1083–1098 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083 [PubMed]

19.

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12, 4614–4624 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614 [CrossRef] [PubMed]

20.

F. Vanholsbeeck, S. Coen, Ph. Emplit, C. Martinelli, and T. Sylvestre, “Cascaded Raman generation in optical fibers : Influence of chromatic dispersion and Rayleigh backscattering,” Opt. Lett. 29, 998–1000 (2004). [CrossRef] [PubMed]

21.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904/1–4 (2003). [CrossRef] [PubMed]

22.

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Zh. Eksp. Teor. Fiz. 41, 242–244 (1985) [JETP Lett. 41, 294–297 (1985)].

23.

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987). [CrossRef]

24.

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). [CrossRef]

25.

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). [CrossRef]

26.

A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838 [CrossRef] [PubMed]

27.

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

28.

D. Monzón-Hernández, A. N. Starodumov, Y. O. Barmenkov, I. Torres-Gómez, and F. Mendoza-Santoyo, “Continuous-wave measurement of the fiber nonlinear refractive index,” Opt. Lett. 23, 1274–1276 (1998). [CrossRef]

29.

M. González-Herráez, L. Thévenaz, and P. Robert, “Distributed measurement of chromatic dispersion by four-wave mixing and brillouin optical-time-domain analysis,” Opt. Lett. 28, 2210–2212 (2003). [CrossRef]

30.

G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics Series, 3rd ed. (Academic Press, San Diego, 2001).

31.

P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations,” J. Opt. Soc. Am. B 18, 139–152 (2001). [CrossRef]

32.

M. T. de Araujo, H. R. da Cruz, and A. S. Gouveia-Neto, “Self-phase modulation of incoherent pulses in single- ,” J. Opt. Soc. Am. B 8, 2094–2096 (1991). [CrossRef]

33.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995). [CrossRef] [PubMed]

34.

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002). [CrossRef]

35.

J. Nathan Kutz, C. Lyngå, and B. J. Eggleton, “Enhanced supercontinuum generation through dispersion-management,” Opt. Express 13, 3989–3998 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-3989 [CrossRef] [PubMed]

36.

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002). [CrossRef]

37.

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” to be published in Optics Letters.

OCIS Codes
(060.2380) Fiber optics and optical communications : Fiber optics sources and detectors
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Research Papers

History
Original Manuscript: July 6, 2005
Revised Manuscript: August 9, 2005
Published: August 22, 2005

Citation
Frédérique Vanholsbeeck, Sonia Martin-Lopez, Miguel González-Herráez, and Stéphane Coen, "The role of pump incoherence in continuous-wave supercontinuum generation," Opt. Express 13, 6615-6625 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-17-6615


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References

  1. J. K. Ranka, R. S. Windeler, and A. J. Stentz, �??Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,�?? Opt. Lett. 25, 25�??27 (2000). [CrossRef]
  2. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, �??Supercontinuum generation in tapered fibers,�?? Opt. Lett. 25, 1415�??1417 (2000). [CrossRef]
  3. 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]
  4. 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]
  5. 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 B77, 211�??218 (2003). [CrossRef]
  6. M. González-Herráez, S. Martín-López, P. Corredera, M. L. Hernanz, and P. R. Horche, �??Supercontinuum generation using a continuous-wave Raman fiber laser,�?? Opt. Commun. 226, 323�??328 (2003). [CrossRef]
  7. A. K. Abeeluck and C. Headley, �??Supercontinuum growth in a highly nonlinear fiber with a low-coherence semiconductor laser diode,�?? Appl. Phys. Lett. 85, 4863�??4865 (2004). [CrossRef]
  8. 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]
  9. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S.Windeler, J. L. Hall, and S. T. Cundiff, �??Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,�?? Science 288, 635�??639 (2000). [CrossRef] [PubMed]
  10. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S.Windeler, R. Holzwarth, Th. Udem, and T. W. Hänsch, �??Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,�?? Phys. Rev. Lett. 84, 5102�??5105 (2000). [CrossRef] [PubMed]
  11. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, �??Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber,�?? Opt. Lett. 26, 608�??610 (2001). [CrossRef]
  12. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, �??Study of an ultrahigh-numerical-aperture fiber continuum generation source for optical coherence tomography,�?? Opt. Lett. 27, 2010�??2012 (2002). [CrossRef]
  13. T. Morioka, K. Mori, and M. Saruwatari, �??More than 100-wavelength-channel picosecond optical pulse generation from single laser source using supercontinuum in optical fibres,�?? Electron. Lett. 29, 862�??864 (1993). [CrossRef]
  14. H. Takara, T. Ohara, and K. Sato, �??Over 1000 km DWDM transmission with supercontinuum multi-carrier source,�?? Electron. Lett. 39, 1078�??1079 (2003). [CrossRef]
  15. 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]
  16. A. V. Husakou and J. Herrmann, �??Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,�?? Phys. Rev. Lett. 87, 203901/1�??4 (2001). [CrossRef] [PubMed]
  17. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, �??Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,�?? Phys. Rev. Lett. 88, 173901/1�??4 (2002). [CrossRef] [PubMed]
  18. G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, �??Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,�?? Opt. Express 10, 1083�??1098 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083</a> [PubMed]
  19. G. Genty, M. Lehtonen, and H. Ludvigsen, �??Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,�?? Opt. Express 12, 4614�??4624 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614</a> [CrossRef] [PubMed]
  20. F. Vanholsbeeck, S. Coen, Ph. Emplit, C. Martinelli, and T. Sylvestre, �??Cascaded Raman generation in optical fibers : Influence of chromatic dispersion and Rayleigh backscattering,�?? Opt. Lett. 29, 998�??1000 (2004). [CrossRef] [PubMed]
  21. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, �??Fundamental noise limitations to supercontinuum generation in microstructure fiber,�?? Phys. Rev. Lett. 90, 113904/1�??4 (2003). [CrossRef] [PubMed]
  22. E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel�??makh, and A. A. Fomichev, �??Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,�?? Zh. Eksp. Teor. Fiz. 41, 242�??244 (1985) JETP Lett. 41, 294�??297 (1985)].
  23. P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, �??Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,�?? IEEE J. Quantum Electron. QE-23, 1938�??1946 (1987). [CrossRef]
  24. 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). [CrossRef]
  25. 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). [CrossRef]
  26. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, �??Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,�?? Opt. Express 12, 2838�??2843 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838</a> [CrossRef] [PubMed]
  27. J. M. Dudley and S. Coen, �??Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,�?? Opt. Lett. 27, 1180�??1182 (2002). [CrossRef]
  28. D. Monzón-Hernández, A. N. Starodumov, Y. O. Barmenkov, I. Torres-Gómez, and F. Mendoza-Santoyo, �??Continuous-wave measurement of the fiber nonlinear refractive index,�?? Opt. Lett. 23, 1274�??1276 (1998). [CrossRef]
  29. M. González-Herráez, L. Thévenaz, and P. Robert, �??Distributed measurement of chromatic dispersion by four-wave mixing and brillouin optical-time-domain analysis,�?? Opt. Lett. 28, 2210�??2212 (2003). [CrossRef]
  30. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics Series, 3rd ed. (Academic Press, San Diego, 2001).
  31. P. D. Drummond and J. F. Corney, �??Quantum noise in optical fibers. I. Stochastic equations,�?? J. Opt. Soc. Am. B 18, 139�??152 (2001). [CrossRef]
  32. M. T. de Araujo, H. R. da Cruz, and A. S. Gouveia-Neto, �??Self-phase modulation of incoherent pulses in single-mode optical fibers,�?? J. Opt. Soc. Am. B 8, 2094�??2096 (1991). [CrossRef]
  33. S. B. Cavalcanti, G. P. Agrawal, and M. Yu, �??Noise amplification in dispersive nonlinear media,�?? Phys. Rev. A 51, 4086�??4092 (1995). [CrossRef] [PubMed]
  34. M. N. Islam, �??Raman amplifiers for telecommunications,�?? IEEE J. Sel. Top. Quantum Electron. 8, 548�??559 (2002). [CrossRef]
  35. J. Nathan Kutz, C. Lyng°a, and B. J. Eggleton, �??Enhanced supercontinuum generation through dispersion-management,�?? Opt. Express 13, 3989�??3998 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-3989">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-3989</a> [CrossRef] [PubMed]
  36. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O�??Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, �??Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,�?? Opt. Lett. 27, 1174�??1176 (2002). [CrossRef]
  37. A. Sauter, S. Pitois, G. Millot, and A. Picozzi, �??Incoherent modulation instability in instantaneous nonlinear Kerr media,�?? to be published in Optics Letters.

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