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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 14435–14447
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Visible supercontinuum generation in photonic crystal fibers with a 400W continuous wave fiber laser

J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 14435-14447 (2008)
http://dx.doi.org/10.1364/OE.16.014435


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Abstract

We demonstrate continuous wave supercontinuum generation extending to the visible spectral region by pumping photonic crystal fibers at 1.07 μm with a 400 W single mode, continuous wave, ytterbium fiber laser. The continuum spans over 1300 nm with average powers up to 50 W and spectral power densities over 50 mW/nm. Numerical modeling and understanding of the physical mechanisms has led us to identify the dominant contribution to the short wavelength extension to be trapping and scattering of dispersive waves by high energy solitons.

© 2008 Optical Society of America

1. Introduction

Continuous wave supercontinua were first generated in the late 1990s [12

12. P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW fibre Raman laser source 1.6–1.9 µm,” Electron. Lett. 32, 1486–1487 (1996). [CrossRef]

, 13

13. 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, 291–293 (2000). [CrossRef]

], but significant development did not occur until high nonlinearity fibers and high power single mode CW lasers became more widely available. The first demonstration in photonic crystal fiber (PCF) [1

1. 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]

] showed that multi-watt average powers are achievable in a simple experimental configuration. Subsequently a large body of work has been produced showing results pumped at 1.55 µm in conventional optical fibers [2

2. J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jorgensen, “Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers,” Appl. Phys. B 77, 211–218 (2003). [CrossRef]

, 3

3. 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]

, 6

6. A. B. Rulkov, A. A. Ferin, J. C. Travers, S. V. Popov, and J. R. Taylor, “Broadband, low intensity noise CW source for OCT at 1800nm,” Opt. Commun. 281, 154–156 (2008). [CrossRef]

, 14

14. S. V. Popov, J. R. Taylor, A. B. Rulkov, and V. P. Gapontsev, “Multi-watt, 1.48–2.05 µm range CWRaman-soliton continuum generation in highly-nonlinear fibres,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2004) paper CThEE4.

] and at 1.06 µm in PCF [1

1. 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]

, 4

4. 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 (2004). [CrossRef]

, 5

5. J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended CW supercontinuum generation in a low water-loss Holey fiber,” Opt. Lett. 30, 3132 (2005). [CrossRef] [PubMed]

, 7

7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

]. In the former case, due to favorable dispersion curves, continua extending to wavelengths short of the pump wavelength have been observed [8

8. A. 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]

], along with the usual Raman-soliton continuum extending to longer wavelengths. In PCFs pumped at 1.06 µm this has not been observed until now.

The reason for this fact is due to competing requirements on the dispersion curves for efficient Raman-soliton continuum generation to long wavelengths, and for short wavelength extension. For the former, it is important that |β 2|/γ (where β 2 is the group velocity dispersion and γ the nonlinear coefficient) stays relatively constant with wavelength, to allow the solitons to shift as far as possible to longer wavelengths before broadening so much that Raman self-scattering no longer occurs. For the latter, we need to pump close to the zero dispersion wavelength, to allow MI or soliton coupling to dispersive waves to transfer power to the normal dispersion region. These two requirements conflict as pumping close to the zero generally implies a steeper dispersion slope.

We have recently taken two approaches to achieving short wavelength generation for 1.07 µm pumping, with the aim of generating a continuous wave visible supercontinuum, which would have numerous applications. In one approach we have optimized the dispersion to precisely accommodate the relevant phase and group velocity matching conditions to reach the visible, sacrificing somewhat the Raman-soliton continuum [15

15. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “Towards visible CW pumped supercontinua,” Opt. Lett. doc. ID 97526 (posted 14 August 2008, in press). [CrossRef]

]. As an alternative, in the present work, we show how by power scaling the pump, using an industrial class continuous wave fiber laser, we can also extend to the visible spectral region.

This paper is organized as follows. In Section 2 we explain the experimental setup and the results we have achieved. In Section 3 we discuss our model of nonlinear propagation in optical fibers, including some details on the initial conditions. In Section 4 we discuss the physical mechanisms involved along with some simulation results and note that further extension to the blue should be possible using cascaded or tapered fibers.

2. Experiment

2.1. Setup

Fig. 1. Experimental setup.

Figure 1 shows the experimental setup. The industrial class pump laser was supplied by IPG Photonics. It emitted up to 432Wof average power at 1.07 µm, with random polarization and a spectral linewidth of 3.6 nm. The single mode output of the laser was interfaced to a collimating unit producing a 7 mm (1/e2) collimated beam. Correspondingly, we were unable to splice the laser output directly to the setup and therefore used a bulk lens in order to couple this beam into a series of mode matching single mode fibers to reduce the mode field diameter, before finally splicing to the PCF we were using. The free-space coupling was typically greater than 70% efficient, the total free-space to PCF efficiency was between 30% and 50% depending on the splice loss to the PCF. The coupling at full power was stable, and did not require constant monitoring or adjustment, at least for time periods exceeding 1 hour, however, in initial experiments we reduced the thermal load on the coupling lens and input fiber by modulating the laser with a duty cycle of between 1 and 40. Final results were taken with no modulation in order to scale the average power. References to equivalent power refer to the peak or equivalent CW power with no modulation.

Table 1. Parametersa of the photonic crystal fibers.

table-icon
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We pumped several PCFs to probe the affect of dispersion profile and nonlinearity on the resulting supercontinuum shape. The PCF parameters and the names we use to refer to them are given in Table 1. HF1050 is similar to PCFs commonly used for visible supercontinuum generation with picosecond and nanosecond lasers at 1.06 µm [16

16. A. B. Rulkov, M. Y. Vyatkin, S. V. Popov, J. R. Taylor, and V. P. Gapontsev, “High brightness picosecond all-fiber generation in 525–1800nm range with picosecond Yb pumping,” Opt. Express 13, 377–381 (2005). [CrossRef] [PubMed]

, 17

17. J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended blue supercontinuum generation in cascaded holey fibers,” Opt. Lett. 30, 3132–3134 (2005). [CrossRef] [PubMed]

]. It has a zero dispersion wavelength near to 1.05 µm and a low anomalous dispersion at the pump wavelength. The dispersion curve for this and the other PCFs we used are shown in Fig. 2.

Fig. 2. Calculated dispersion curves for our fibers. HF1050 (blue), HF840 (red), HFDBL (green).

HF840 has been previously used for continuous wave supercontinuum generation as it has a low water loss value [5

5. J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended CW supercontinuum generation in a low water-loss Holey fiber,” Opt. Lett. 30, 3132 (2005). [CrossRef] [PubMed]

]. It has a single zero dispersion wavelength around 0.84 µm. HFDBL has two zero dispersion wavelengths at around 0.81 and 1.73 µm, and has also been used previously for high power supercontinuum generation [7

7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

].

We pumped a number of lengths of each fiber and show below results for the approximately optimal length, judged by the smoothness and extent of the continuum produced. The output power of each fiber changes considerably depending on their different attenuation curves and the particular spectral extent of each continuum.

2.2. Results

Figure 3 shows the results of pumping 17 m of HFDBL with our setup. A maximum of 170W was coupled into the PCF forming a continuum spanning from 1.06 to 1.9 µm with ~10 dB flatness. The role of the second zero dispersion wavelength is clearly evidenced by the dip in spectral power around 1.7 µm. The effect of water loss is also apparent as the spectral power reduces after 1.4 µm. The qualitative form of the spectra is very similar to those we previously achieved in [7

7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

] with 50Wpumping, but with the higher powers accessible in this work we have significantly increased the power transfered to the dispersive wave formed beyond the second zero dispersion wavelength. The spectral power densities between 1.06 and 1.4 µm are over 50 mW/nm as the output powers were around 27 W.

The supercontinuum spectra obtained at the output of 20 m of HF840 are shown in Fig. 4 for a pump power of 170W. The total average output power was over 50W and the continuum extended from 1.06 to 2.2 µm. The 10 dB width of the spectrum was over 900 nm, between 1.14 and 2.05 µm. Across this range the spectral power exceeds 10 mW/nm, with half of the range between 50 and 100 mW/nm. Water loss causes a significant fall off in spectral power, although the power density at the long edge of the continuum around 2.1 µm is still over 5 mW/nm, which is sufficient for most applications.

These results clearly demonstrate the power scaling potential of using an industrial class laser, but pumping HFDBL and HF840 has not lead to a visible supercontinuum spectrum. The reasons for this are analyzed in detail below, but stem from the fact that the zero dispersion wavelength is too far from the pump wavelength. To generate visible continua there must be a mechanism to transfer power to the normal dispersion region. This can be achieved either through the generation of dispersive waves from the solitons formed from the MI process which initiates the continuum, or directly from widely separated MI sidebands, where one MI side-lobe overlaps with the normal dispersion region. Both of these processes require the pump wavelength to be sufficiently close to the zero dispersion.

Fig. 3. Measured spectra out of 17 m HFDBL for 50 W (blue), 85 W (red) and 170 W (green) equivalent coupled pump power on a relative power scale.
Fig. 4. Measured spectra out of 20 m of HF840 for 170 Wequivalent pump power, normalized to the total average output power of 50 W.

HF1050 has a suitable dispersion curve. The result of pumping 50 m of this fiber with 230W from our setup is shown in Fig. 5. The total average output power was 28 W. It is clear that the supercontinuum extends to the visible spectral region, down to 0.6 µm and the continuum appeared bright red to the eye. The long wavelength edge of the continuum was 1.9 µm. The spectral powers were over 2 mW/nm in the short wavelength side, which is competitive with the highest power pulse pumped visible supercontinua, and over 10 mW/nm in the infrared region, with substantial spectral regions between 20 and 30 mW/nm. This result shows that visible supercontinua similar to those obtained with pulse pumped systems can be achieved with extremely high power CW pump conditions, but with additional benefits due to scalable spectral power and flatness. We should be able to extend the continuum to the blue, and scale the power further by carefully designing the fiber dispersion and pump conditions, as discussed in the following sections.

Fig. 5. Measured spectra out of 50 m of HF1050 for 230 W equivalent pump power, normalized to the total average output power of 28 W.

3. Modelling

3.1. Propagation equation

To gain insight into the supercontinuum dynamics we modelled the propagation of a model CW field through the PCFs. The complex field envelope E(ω, z) at angular frequency ω and axial fiber position z was calculated using the generalized nonlinear Schrödinger equation, modified to include the dispersion of the mode field profile [18

18. J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007). [CrossRef] [PubMed]

, 19

19. P. V. Mamyshev and S. V. Chernikov, “Ultrashort-Pulse Propagation in Optical Fibers,” Opt. Lett. 15, 1076–1078 (1990). [CrossRef] [PubMed]

],

zE(ω,z)=i(β(ω)β(ω0)ωβ(ω)ω0Ω)E(ω,z)
α(ω)2E(ω,z)
in2ωcAeff(1/4)(ω)dω1E¯(ω1,z)R(ω1ω)
×dω2E¯(ω1ω2ω,z)E¯(ω2,z),
(1)

with

E¯(ω,z)=E(ω,z)Aeff(1/4)(ω).
(2)

In Eq. (1) β (ω) is the mode propagation constant, Ω=ω-ω 0 is the frequency shift with respect to a chosen reference frequency ω 0, n 2=2.74×10-20 m2 W-1 is the nonlinear refractive index [20

20. D. Milam, “Review and Assessment of Measured Values of the Nonlinear Refractive-Index Coefficient of Fused Silica,” Appl. Opt. 37, 546–550 (1998). [CrossRef]

], c the speed of light and Aeff the effective mode area. The wavelength dependent loss is included in α(ω). The first term on the right hand side of the response function,

R(ω1ω)=(1fr)+frhr(ω1ω),
(3)

represents the Kerr effect and the second term represents the Raman effect, where h r is the Raman response function in the frequency domain [21

21. D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function,” J. Opt. Soc. Am. B 19, 2886–2892 (2002). [CrossRef]

] and the factor fr=0.19 determines the Raman contribution to n 2. There is some variation in the literature about whether a factor of 2/3 should be included as a prefix to the Raman part of R(ω 1-ω). This was reported in [22

22. 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]

], and arises from ignoring a cross term when deriving Eq. (1). It turns out, however, that a thorough analysis of the ways which n 2 and gr (the Raman gain coefficient) are experimentally measured, along with a self-consistent analysis of their relation leads to Eq. (3) being the correct definition for use in our propagation equation [23

23. J. Lægsgaard, DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Ørsteds Plads 345V, DK-2800 Kgs. Lyngby, Denmark, “Raman term in the nonlinear Schrödinger equation,” (personal communication, 2008).

].

Equation (1) can be integrated directly after a change of variables, or more commonly it is solved using the split-step Fourier method. Here we use the Runge-Kutta in the interaction picture method [24

24. J. Hult, “A Fourth-Order Runge-Kutta in the Interaction Picture Method for Simulating Supercontinuum Generation in Optical Fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]

]. The step size was chosen automatically based on the relative local error [25

25. O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003). [CrossRef]

], which was held below 1×10-6 for the simulations discussed below.

Modelling CW phenomena is made difficult by the time scales involved. To make the simulations tractable we can only simulate a snapshot of the field as it propagates. We therefore have to carefully choose a time window which contains sufficient information to accurately reproduce experimental observations. The consensus from the current literature is that a time window of several hundred picoseconds with the periodic boundary conditions inherent to the split-step Fourier method is sufficient [9

9. F. Vanholsbeeck, S. Martin-Lopez, M. González-Herráez, and S. Coen, “The role of pump incoherence in continuous-wave supercontinuum generation,” Opt. Express 13, 6615–6625 (2005). [CrossRef] [PubMed]

, 26

26. S. M. Kobtsev and S. V. Smirnov, “Modelling of high-power supercontinuum generation in highly nonlinear, dispersion shifted fibers at CW pump,” Opt. Express 13, 6912–6918 (2005). [CrossRef] [PubMed]

, 27

27. A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express 15, 11553–11563 (2007). [CrossRef] [PubMed]

]. In the simulations described below we used 218 grid points over a time window of 256 ps. To visualize the results we use spectrograms or XFROG traces, computed with a windowed Fourier transform of the field envelope:

S(t,ω,z)=10log10Eref(tt)E(t,z)exp[iωt]dt
(4)

Here Eref is an envelope of a reference pulse, in our case a 3 ps Gaussian.

3.2. Initial conditions

There is great difficulty in modelling the initial conditions of the pump CW fields for supercontinuum generation as accurate single-shot diagnostics of CW lasers are difficult to obtain. This problem is compounded when considering CW fiber lasers as the laser cavities are both non-linear and highly dispersive, leading to more complicated field evolution than their bulk laser counterparts. This problem is also of great importance as modulation instability, the precursor of CW supercontinua, is highly dependent on the input noise conditions. We have performed a comparison of the various models previously used with careful experimental results, and designed our own model. These results are reported elsewhere [30

30. J. C. Travers, S. V. Popov, and J. R. Taylor, “A New Model for CW Supercontinuum Generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CMT3.

,31

31. J. C. Travers, Femtosecond Optics Group, Physics Department, Prince Consort Road, Imperial College, London SW7 2AZ, UK, is preparing a manuscript to be called “Modelling the Initial Conditions of Continuous Wave Supercontinuum Generation.”

]. Here we briefly summarize the main points.

One of the simplest models is that of a CW field with no temporal or spectral phase or amplitude fluctuations, with quantum noise [26

26. S. M. Kobtsev and S. V. Smirnov, “Modelling of high-power supercontinuum generation in highly nonlinear, dispersion shifted fibers at CW pump,” Opt. Express 13, 6912–6918 (2005). [CrossRef] [PubMed]

]. This leads to a spectrum with a very narrow spike at the central laser frequency and is approximately comparable to a single frequency laser. Such narrow pump spectra are not comparable to those observed from a high power fiber laser. To improve on this, a number of models have been based on the phase diffusion concept, where the temporal phase is modelled as a Gaussian noise process, which leads to Lorentzian shaped laser spectra [32

32. 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). [CrossRef] [PubMed]

, 33

33. M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006). [CrossRef] [PubMed]

]. This model neglects temporal amplitude fluctuations, which in a nonlinear dispersive cavity with high powers are certain to exist. Also, Lorentzian spectra with the bandwidths equivalent to our pump lasers contain significant power outside of the gain region of the fiber, beyond experimental observation. An alternative model is to represent the laser field as a collection of longitudinal modes with no phase relationship [9

9. F. Vanholsbeeck, S. Martin-Lopez, M. González-Herráez, and S. Coen, “The role of pump incoherence in continuous-wave supercontinuum generation,” Opt. Express 13, 6615–6625 (2005). [CrossRef] [PubMed]

, 34

34. B. Barviau, S. Randoux, and P. Suret, “Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber,” Opt. Lett. 31, 1696–1698 (2006). [CrossRef] [PubMed]

]. This model starts from the measured spectral power of the pump laser with a random spectral phase added to each frequency bin. This leads to very strong intensity fluctuations in the time domain of the order of the inverse spectral width of the pump (coherence time). However the resulting fluctuations in this case are too high as dispersion and self-phase modulation are completely neglected, although they should be significant in a fiber laser cavity. A careful comparison between these models and real experimental results with CW fiber lasers has shown them to be somewhat limited even, in some cases, for qualitative comparison [30

30. J. C. Travers, S. V. Popov, and J. R. Taylor, “A New Model for CW Supercontinuum Generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CMT3.

, 31

31. J. C. Travers, Femtosecond Optics Group, Physics Department, Prince Consort Road, Imperial College, London SW7 2AZ, UK, is preparing a manuscript to be called “Modelling the Initial Conditions of Continuous Wave Supercontinuum Generation.”

].

Our approach was to model the effects of dispersion and nonlinearity in the fiber lasers by modelling the whole laser itself. We start with quantum noise represented as two photons per mode. We then amplify through a fiber with spectrally dependent gain, gain saturation, nonlinearity and dispersion. Bragg gratings are modelled at the end of each amplification pass simply by a suitable spectral filter. We iterate the field through such a cavity until the average output power is at the desired level. The resulting field exhibits many of the expected characteristics: temporal and spectral amplitude and phase fluctuations and a triangular shaped spectrum on a log scale. The power fluctuations are much weaker than with the random spectral phase model described above. Comparisons with experimental results have shown this model to be an improvement on the other proposals [30

30. J. C. Travers, S. V. Popov, and J. R. Taylor, “A New Model for CW Supercontinuum Generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CMT3.

, 31

31. J. C. Travers, Femtosecond Optics Group, Physics Department, Prince Consort Road, Imperial College, London SW7 2AZ, UK, is preparing a manuscript to be called “Modelling the Initial Conditions of Continuous Wave Supercontinuum Generation.”

]. We modelled our 400 W laser in this way and used it to produce the simulated results reported below.

4. Discussion

4.1. Long wavelength generation: the Raman-soliton continuum

The initiation of a continuous wave supercontinuum in the low anomalous dispersion region is due to modulation instability [35

35. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980). [CrossRef]

, 36

36. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986). [CrossRef] [PubMed]

]. Noise fluctuations of an otherwise CW or quasi-CW field become self-trapping due to the Kerr effect. This process is fundamentally linked to the existence of solitons, which are maintained by the same trapping effect. Fundamental solitons are a stable condition—adiabatic amplification simply leads to shorter fundamental solitons—and therefore the process of MI does not naturally lead to higher order soliton solutions.

The early stages of the CW continuum formation are clearly identifiable in the simulated spectrograms of our pump laser in HFDBL, shown in Fig. 6. In Fig. 6(a) is shown the pump field with the modelled initial conditions as described in Section 3. In Fig. 6(b) we see the emergence of solitons, which we attribute to MI. Here we can clearly see that the solitons form earlier and with higher peak power at the peaks of the input power fluctuations. Note also that the solitons being formed are much shorter in duration than the input power fluctuations. The dependence of MI on the noisy initial conditions means that we create a train of solitons with a distribution of energies, this leads to a smooth continuum.

Once formed, the solitons may shift due to Raman self-scattering [37

37. E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stelmakh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294 (1985).

,38

38. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

]. To do so they must have a short enough duration so that their bandwidth is broad enough to self amplify through Raman. The frequency shift of a soliton is proportional to the fourth power of the soliton energy and so the shape of the soliton energy distribution strongly affects the resulting spectrum, which has been called a Raman-soliton continuum [39

39. A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulses of Four Optical Cycles from an Optimized Optical Fibre/Grating Pair/Soliton Pulse Compressor at 1·32 µm,” J. Mod. Opt. 35, 7–10 (1988). [CrossRef]

]. In Fig. 6(c) we see how Raman self-scattering has started to shift the highest energy solitons to longer wavelengths, beginning the continuum formation, while new solitons are forming from the remaining pump field. We should note again that these are single shots of 200 ps duration from a continuous process. The summation of all of these solitons leads to a very smooth spectrum on average.

Fig. 6. Simulated spectrograms of the supercontinuum development through HFDBL for 170Wpump power. The spectrograms are calculated for fiber lengths of: (a) 0 m, (b) 1.5 m, (c) 3.0 m, (d) 7.0 m. The full color scales and axis figure scales change for each sub-figure and range over 40 dB.

In addition to the above process, inelastic collisions between solitons can transmit significant energy from higher frequency solitons to those with a lower frequency. Solitons collide when they overlap in time (i.e. align vertically in these spectrograms) as long as they are close enough spectrally for the Raman process to occur (up to about 40 THz). As solitons with lower frequency have clearly Raman shifted further, they tend to have higher energies and so the highest energy solitons get further excited. This enhances the continuum bandwidth. In the context of CW supercontinua this was discussed in [33

33. M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006). [CrossRef] [PubMed]

], and was originally identified in picosecond pumped supercontinua [40

40. 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]

]. In Fig. 6(d) we see evidence of soliton collisions. The non-solitonic traces left behind by the solitons is the radiation shed by a pair of solitons after an inelastic collision mediated through the Raman process occurred. Both pulses involved will have energies mismatched from the required soliton energy, therefore some energy is shed as the pulse adapts back to the soliton condition.

Limitations to the continuum bandwidth are caused by at least four mechanisms. Firstly, there is a finite length of fiber, and so the maximum shift is that achieved by the most energetic soliton through the necessarily limited nonlinear medium. Secondly, losses become significant, which reduces the soliton energy and can slow or stop their shift. Thirdly, the balance of non-linearity and dispersion, which maintains the soliton shape can change very significantly and thus broaden the soliton temporally so that it no longer has the spectral bandwidth for Raman self-scattering [7

7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

]. Finally, the Raman-soliton continuum can be limited by a second zero dispersion wavelength. As we noted in [7

7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

], HFDBL has a second zero dispersion which prevents the continued shift of the solitons as the anomalous dispersion region is limited [41

41. D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, “Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

]. This is seen clearly in Fig. 6(d) where the solitons gather around 1.6 to 1.7 µm, before the zero dispersion point, and phase-matched dispersive waves are generated beyond this point, in the normal dispersion region. The dispersive waves are clearly being chirped as they propagate through the fiber. These features were clearly identifiable on the experimental spectra shown in Fig. 3.

4.2. Short wavelength generation

There are two main mechanisms available for generating wavelengths short of the pump in the CW regime. Either solitons formed from MI have a spectral overlap with the normal dispersion region, and can therefore directly excite phase-matched dispersive waves [42

42. P. Wai, H. Chen, and Y. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990). [CrossRef] [PubMed]

, 43

43. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

], or the anti-Stokes MI sidelobe is itself in the normal dispersion region causing a growth of power there. The excitation of phase-matched dispersive waves is proportional to the spectral amplitude of the soliton spectrum at the phase-matched wavelength, and as the soliton spectral power drops off exponentially, pumping close to the zero is required. Similarly, for MI extension to the normal dispersion region, a low anomalous dispersion is required for wide Stokes shifts and closeness to the zero dispersion is required to achieve overlap into the normal dispersion region. In fact, these two processes are intimately linked. Therefore it is clear how pumping close to the zero dispersion wavelength is essential, as we observed experimentally in Section 2.

After this initial stage, further extension to the short wavelength region can be gained either through four wave mixing between the soliton continuum and the dispersive waves, or through the soliton trapping of dispersive waves [10

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

, 11

11. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007). [CrossRef]

]. In both cases we must generate significant power in the normal dispersion region for further extension to occur. In the four wave mixing case this is essential to achieve phase-matching to yet shorter wavelengths. In the soliton trapping case it is inherent to the process that the trapped waves are group-velocity matched to the solitons, thus requiring that a dispersion zero must be between the soliton and dispersive wave frequencies. It should be noted that some authors have attributed short wavelength generation from 1.5 µm CW pumped continua to soliton fission processes [8

8. A. 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

9. F. Vanholsbeeck, S. Martin-Lopez, M. González-Herráez, and S. Coen, “The role of pump incoherence in continuous-wave supercontinuum generation,” Opt. Express 13, 6615–6625 (2005). [CrossRef] [PubMed]

]. However, soliton fission requires high order soliton solutions which, as noted above, are not naturally generated from the MI process; we therefore believe that they cannot play a role in the continuum mechanism.

To identify which mechanism (four wave mixing, or soliton trapping) dominates in the case of HF1050 we can plot the phase and group velocity matching curves for the two processes and compare with our experimental results, as shown in Fig. 7. For the four wave mixing phase-matching curve it is assumed that the required pump wavelengths (in the normal dispersion regime), are made available in the initial supercontinuum stages, this was verified by experiment. At small Stokes/anti-Stokes shifts from the pump, the two matching curves are quite similar, preventing unambiguous differentiation between the processes from experimental data, but at large Stokes shifts the curves diverge considerably. Marked by a pair of horizontal and vertical lines on Fig. 7 are the longest Stokes and shortest anti-Stokes wavelengths generated in our experiments. It is clear that they cross almost precisely on the calculated group-velocity matching curve. This implies that the soliton trapping of dispersive waves is the dominant mechanism for our short wavelength extension.

To verify this we ran numerical simulations of our pump laser propagating through HF1050. The results are shown in Fig. 8. The spectrogram of the pump is shown in Fig. 8(a) and exhibits temporal power fluctuations as described in Section 3. After 3 m of propagation (Fig. 8(b)) MI has led to the formation of solitons and dispersive waves have been formed at wavelengths short of the pump. We estimate that the MI sidelobe separation for our pump and fiber conditions is ~20 nm which is not large enough for significant generation of power in the normal dispersion region. Instead the dispersive waves are excited by the very short duration (and hence wide bandwidth) solitons formed through the MI process. Fig. 8(c) shows that as the solitons shift to longer wavelengths via the Raman effect, the dispersive waves maintain their relative positions to individual solitons. This is characteristic of the soliton trapping effect [10

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

, 11

11. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007). [CrossRef]

]. In addition, examination of the precise locations of individual soliton-dispersive wave pairs is in agreement with the group velocity matching curve shown in Fig. 7. The short wavelength extension through this process is understood as described in [10

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

,11

11. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007). [CrossRef]

]. The soliton modulates the refractive index such that the dispersive wave cannot escape in one direction. The soliton then chirps the dispersive wave towards the blue, through cross-phase modulation, where the group velocity is lower. It then shifts to longer wavelengths through Raman, in doing so it is decelerated as longer wavelengths have lower group velocity, and therefore falls back onto the dispersive wave, and thus the process can repeat. This trapping effect therefore leads to a cascade of scattering events for the dispersive wave pushing it further and further towards the blue. In Fig. 8(d) we see that the solitons have pushed the dispersive waves as far short as 0.6 µm, in agreement with our experiments.

Fig. 7. Calculated phase (PM) and group velocity (GVM) matching curves for HF1050. The solid yellow line indicates the long wavelength edge in the supercontinuum output of HF1050 and the solid green line indicates the short wavelength edge.

This process will be limited by either the breaking of the group velocity matching or by the halting of the red-shifting solitons, which can occur by any of the reasons described in Section 4.1. Further extension to the blue should be possible if we use cascaded or tapered fibers to extend the group velocity matching of the anti-Stokes components, as previously demonstrated in the picosecond pump regime [17

17. J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended blue supercontinuum generation in cascaded holey fibers,” Opt. Lett. 30, 3132–3134 (2005). [CrossRef] [PubMed]

, 44

44. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zerodispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006). [CrossRef] [PubMed]

].

5. Conclusion

We have demonstrated the extension of a continuous wave supercontinuum to the visible spectral region and analyzed the physical mechanisms enabling this process. We have identified the two requirements to generate a short wavelength continuum: pumping close to the zero dispersion with sufficient power to generate dispersive waves, and met them by using a selected photonic crystal fiber and pumping with an industrial class 400W, continuous wave, ytterbium fiber laser. The high power available with this laser enabled the scaling of the supercontinuum average power to 50 W and spectral power densities of over 50 mW/nm over wide supercontinua spanning over 1300 nm. Further optimization of the fiber design, and the use of cascaded or tapered fibers should allow extension to the blue spectral region as has been shown with picosecond pump conditions.

Fig. 8. Simulated spectrograms of the supercontinuum development through HF1050 for 170 W pump power. The spectrograms are calculated for fiber lengths of: (a) 0 m, (b) 3 m, (c) 9 m, (d) 25 m. The full color scales and axis figure scales change for each sub-figure and range over 40 dB.

References and links

1.

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]

2.

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jorgensen, “Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers,” Appl. Phys. B 77, 211–218 (2003). [CrossRef]

3.

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]

4.

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 (2004). [CrossRef]

5.

J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended CW supercontinuum generation in a low water-loss Holey fiber,” Opt. Lett. 30, 3132 (2005). [CrossRef] [PubMed]

6.

A. B. Rulkov, A. A. Ferin, J. C. Travers, S. V. Popov, and J. R. Taylor, “Broadband, low intensity noise CW source for OCT at 1800nm,” Opt. Commun. 281, 154–156 (2008). [CrossRef]

7.

B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5954–5962 (2008). [CrossRef] [PubMed]

8.

A. 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.

F. Vanholsbeeck, S. Martin-Lopez, M. González-Herráez, and S. Coen, “The role of pump incoherence in continuous-wave supercontinuum generation,” Opt. Express 13, 6615–6625 (2005). [CrossRef] [PubMed]

10.

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

11.

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007). [CrossRef]

12.

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW fibre Raman laser source 1.6–1.9 µm,” Electron. Lett. 32, 1486–1487 (1996). [CrossRef]

13.

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, 291–293 (2000). [CrossRef]

14.

S. V. Popov, J. R. Taylor, A. B. Rulkov, and V. P. Gapontsev, “Multi-watt, 1.48–2.05 µm range CWRaman-soliton continuum generation in highly-nonlinear fibres,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2004) paper CThEE4.

15.

B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, “Towards visible CW pumped supercontinua,” Opt. Lett. doc. ID 97526 (posted 14 August 2008, in press). [CrossRef]

16.

A. B. Rulkov, M. Y. Vyatkin, S. V. Popov, J. R. Taylor, and V. P. Gapontsev, “High brightness picosecond all-fiber generation in 525–1800nm range with picosecond Yb pumping,” Opt. Express 13, 377–381 (2005). [CrossRef] [PubMed]

17.

J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended blue supercontinuum generation in cascaded holey fibers,” Opt. Lett. 30, 3132–3134 (2005). [CrossRef] [PubMed]

18.

J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007). [CrossRef] [PubMed]

19.

P. V. Mamyshev and S. V. Chernikov, “Ultrashort-Pulse Propagation in Optical Fibers,” Opt. Lett. 15, 1076–1078 (1990). [CrossRef] [PubMed]

20.

D. Milam, “Review and Assessment of Measured Values of the Nonlinear Refractive-Index Coefficient of Fused Silica,” Appl. Opt. 37, 546–550 (1998). [CrossRef]

21.

D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function,” J. Opt. Soc. Am. B 19, 2886–2892 (2002). [CrossRef]

22.

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]

23.

J. Lægsgaard, DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Ørsteds Plads 345V, DK-2800 Kgs. Lyngby, Denmark, “Raman term in the nonlinear Schrödinger equation,” (personal communication, 2008).

24.

J. Hult, “A Fourth-Order Runge-Kutta in the Interaction Picture Method for Simulating Supercontinuum Generation in Optical Fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]

25.

O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003). [CrossRef]

26.

S. M. Kobtsev and S. V. Smirnov, “Modelling of high-power supercontinuum generation in highly nonlinear, dispersion shifted fibers at CW pump,” Opt. Express 13, 6912–6918 (2005). [CrossRef] [PubMed]

27.

A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express 15, 11553–11563 (2007). [CrossRef] [PubMed]

28.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]

29.

V. P. Tzolov, M. Fontaine, N. Godbout, and S. Lacroix, “Nonlinear self-phase-modulation effects: a vectorial first-order perturbation approach,” Opt. Lett. 20, 456–458 (1995). [CrossRef] [PubMed]

30.

J. C. Travers, S. V. Popov, and J. R. Taylor, “A New Model for CW Supercontinuum Generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CMT3.

31.

J. C. Travers, Femtosecond Optics Group, Physics Department, Prince Consort Road, Imperial College, London SW7 2AZ, UK, is preparing a manuscript to be called “Modelling the Initial Conditions of Continuous Wave Supercontinuum Generation.”

32.

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). [CrossRef] [PubMed]

33.

M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006). [CrossRef] [PubMed]

34.

B. Barviau, S. Randoux, and P. Suret, “Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber,” Opt. Lett. 31, 1696–1698 (2006). [CrossRef] [PubMed]

35.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980). [CrossRef]

36.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986). [CrossRef] [PubMed]

37.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stelmakh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294 (1985).

38.

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

39.

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulses of Four Optical Cycles from an Optimized Optical Fibre/Grating Pair/Soliton Pulse Compressor at 1·32 µm,” J. Mod. Opt. 35, 7–10 (1988). [CrossRef]

40.

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]

41.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, “Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

42.

P. Wai, H. Chen, and Y. Lee, “Radiations by solitons at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990). [CrossRef] [PubMed]

43.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

44.

A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zerodispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006). [CrossRef] [PubMed]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(140.3510) Lasers and laser optics : Lasers, fiber

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: August 8, 2008
Revised Manuscript: August 26, 2008
Manuscript Accepted: August 28, 2008
Published: August 29, 2008

Citation
J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, "Visible supercontinuum generation in photonic crystal fibers with a 400W continuous wave fiber laser," Opt. Express 16, 14435-14447 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435


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References

  1. 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]
  2. J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, and C. G. Jorgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear, dispersion-shifted fibers," Appl. Phys. B 77, 211-218 (2003). [CrossRef]
  3. M. Gonzalez-Herraez, S. Mart?n-Lopez, 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]
  4. 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 (2004). [CrossRef]
  5. J. C. Travers, S. V. Popov, and J. R. Taylor, "Extended CW supercontinuum generation in a low water-loss Holey fiber," Opt. Lett. 30, 3132 (2005). [CrossRef]
  6. A. B. Rulkov, A. A. Ferin, J. C. Travers, S. V. Popov, and J. R. Taylor, "Broadband, low intensity noise CW source for OCT at 1800nm," Opt. Commun. 281, 154-156 (2008). [CrossRef]
  7. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, "29 W High power CW supercontinuum source," Opt. Express 16, 5954-5962 (2008). [CrossRef]
  8. A. 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]
  9. F. Vanholsbeeck, S. Martin-Lopez, M. Gonzalez-Herraez, and S. Coen, "The role of pump incoherence in continuous-wave supercontinuum generation," Opt. Express 13, 6615-6625 (2005). [CrossRef]
  10. P. Beaud, W. Hodel, B. Zysset, and H. Weber, "Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber," IEEE J. Quantum Electron. 23, 1938-1946 (1987). [CrossRef]
  11. A. V. Gorbach and D. V. Skryabin, "Theory of radiation trapping by the accelerating solitons in optical fibers," Phys. Rev. A 76, 053803 (2007). [CrossRef]
  12. P. Persephonis, S. V. Chernikov, and J. R. Taylor, "Cascaded CW fibre Raman laser source 1.6-1.9 um," Electron. Lett. 32, 1486-1487 (1996). [CrossRef]
  13. 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, 291-293 (2000). [CrossRef]
  14. S. V. Popov, J. R. Taylor, A. B. Rulkov, and V. P. Gapontsev, "Multi-watt, 1.48-2.05 um range CWRaman-soliton continuum generation in highly-nonlinear fibres," in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2004) paper CThEE4.
  15. B. A. Cumberland, J. C. Travers, S. V. Popov, and J. R. Taylor, "Towards visible CW pumped supercontinua," Opt. Lett. doc. ID 97526 (posted 14 August 2008, in press).
  16. A. B. Rulkov, M. Y. Vyatkin, S. V. Popov, J. R. Taylor, and V. P. Gapontsev, "High brightness picosecond all-fiber generation in 525-1800nm range with picosecond Yb pumping," Opt. Express 13, 377-381 (2005). [CrossRef]
  17. J. C. Travers, S. V. Popov, and J. R. Taylor, "Extended blue supercontinuum generation in cascaded holey fibers," Opt. Lett. 30, 3132-3134 (2005). [CrossRef]
  18. J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schr¨odinger equation," Opt. Express 15, 16110-16123 (2007). [CrossRef]
  19. P. V. Mamyshev and S. V. Chernikov, "Ultrashort-Pulse Propagation in Optical Fibers," Opt. Lett. 15, 1076-1078 (1990).
  20. D. Milam, "Review and Assessment of Measured Values of the Nonlinear Refractive-Index Coefficient of Fused Silica," Appl. Opt. 37, 546-550 (1998). [CrossRef]
  21. D. Hollenbeck and C. D. Cantrell, "Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function," J. Opt. Soc. Am. B 19, 2886-2892 (2002). [CrossRef]
  22. 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]
  23. J. Lægsgaard, DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, ?rsteds Plads 345V, DK-2800 Kgs. Lyngby, Denmark, "Raman term in the nonlinear Schr¨odinger equation," (personal communication, 2008).
  24. J. Hult, "A Fourth-Order Runge-Kutta in the Interaction Picture Method for Simulating Supercontinuum Generation in Optical Fibers," J. Lightwave Technol. 25, 3770-3775 (2007). [CrossRef]
  25. O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightwave Technol. 21, 61-68 (2003). [CrossRef]
  26. S. M. Kobtsev and S. V. Smirnov, "Modelling of high-power supercontinuum generation in highly nonlinear, dispersion shifted fibers at CW pump," Opt. Express 13, 6912-6918 (2005). [CrossRef]
  27. A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, "Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths," Opt. Express 15, 11553-11563 (2007). [CrossRef]
  28. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell??s equations in a planewave basis," Opt. Express 8, 173-190 (2001).
  29. V. P. Tzolov, M. Fontaine, N. Godbout, and S. Lacroix, "Nonlinear self-phase-modulation effects: a vectorial first-order perturbation approach," Opt. Lett. 20, 456-458 (1995).
  30. J. C. Travers, S. V. Popov, and J. R. Taylor, "A New Model for CW Supercontinuum Generation," in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CMT3.
  31. J. C. Travers, Femtosecond Optics Group, Physics Department, Prince Consort Road, Imperial College, London SW7 2AZ, UK, is preparing a manuscript to be called "Modelling the Initial Conditions of Continuous Wave Supercontinuum Generation."
  32. 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). [CrossRef]
  33. M. H. Frosz, O. Bang, and A. Bjarklev, "Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation," Opt. Express 14, 9391-9407 (2006). [CrossRef]
  34. B. Barviau, S. Randoux, and P. Suret, "Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber," Opt. Lett. 31, 1696-1698 (2006). [CrossRef]
  35. A. Hasegawa and W. F. Brinkman, "Tunable coherent IR and FIR sources utilizing modulational instability," IEEE J. Quantum Electron. 16, 694-697 (1980). [CrossRef]
  36. K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986). [CrossRef]
  37. E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stelmakh, and A. A. Fomichev, "Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers," JETP Lett. 41, 294 (1985).
  38. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662-664 (1986).
  39. A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, "Pulses of Four Optical Cycles from an Optimized Optical Fibre/Grating Pair/Soliton Pulse Compressor at 1· 32 um," J. Mod. Opt. 35, 7-10 (1988). [CrossRef]
  40. 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).
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