1. Introduction

The evolution of a continuous wave (CW) into a mixture of solitary and dispersive waves through the process of modulation instability (MI) has been studied in a range of systems [1, 2]. The solitons formed in this way posses a distribution of parameters such as duration, amplitude and frequency, and their dynamics, including collisions and interactions with dispersive waves can lead to a continuum of frequencies. In this paper I consider CW supercontinuum generation in optical fibers [3, 4, 5, 6], where a spectrally narrow continuous input wave can be broadened into an extensive red-shifted Raman-soliton continuum through MI and subsequent soliton dynamics which are strongly determined by the dispersion landscape. I show that if the conditions are correct then the solitons formed in this way can generate and interact with dispersive waves in the normal dispersion region to achieve a blue-shifted continuum. The interaction between individual soliton pulses and dispersive waves has been extensively studied in optical fibers and many interesting effects such as dispersive wave generation, soliton trapping and four wave mixing have been analyzed theoretically and experimentally [7, 8, 9, 10, 11]. The CW regime dynamics have not been studied in such detail, and due to the large number of interacting soli-tons, can exhibit qualitatively different behavior, which can be considered to represent a soliton gas, a concept utilized in a number of systems including plasma, condensed matter and optical fibers [12, 19]; or, can provide insight into rogue waves in hydrodynamics [13].

One interesting problem is why the majority of experimental CW pumped supercontinua only exhibit red-shifting solitons [3, 4], and whether this can be modified in some way to obtain a blue shifted continuum. Of particular interest is the possibility of generating a visible CW super-continuum. Recent experimental results have demonstrated visible or significantly blue shifted CW supercontinua [14, 15, 16]. In this paper, through numerical results, I explain in more detail the mechanism for spectral expansion of CW supercontinua towards shorter wavelengths with the aim of further optimisation of the process and the identification of the temporal dynamics. The mechanism involves the simultaneous creation of blue shifted dispersive waves and red-shifting solitons from the initial CW pump field. From this initial state, the red-shifting solitons can trap the blue shifted dispersive waves and shift them further through the soliton trapping effect described and demonstrated with pulse pump conditions [10, 18 , 17]. The resulting dispersive waves can be shifted to the blue spectral region, from the initial infrared pump, and propagate localized in time in correspondence with their trapping solitons. The experimental parameters required are easily accessible, and due to the flexibility arising from the wide range of available dispersive and nonlinear properties in optical fibers, large control over particular spectral enhancements across the visible spectral region is achievable.

I use numerical simulations to illustrate the following discussion. The complex field envelope E(ω, z) at angular frequency ω and axial fiber position z is calculated using the generalized nonlinear Schrödinger equation, modified to include the dispersion of the mode field profile [20],

with

In Eqn. 1 β is the mode propagation constant, Ω = ω - ω ₀ is the frequency shift with respect to a chosen reference frequency ω ₀, n ₂ = 2.7 × 10^-20 m² W^-1 is the nonlinear refractive index, c the speed of light and A_eff the effective mode area. The first term on the right hand side of the response function R(ω ₁ - ω) = (1 - f_r)+f_rh_r(ω ₁ - ω) represents the Kerr effect and the second term represents the Raman effect, where h_r is the Raman response function in the frequency domain and the factor f_r = 0.19 determines the Raman contribution to n ₂.

High power CW fiber lasers usually have many longitudinal modes and due to four wave mixing and dispersion, such lasers have quite broad spectra and temporal intensity fluctuations. Some recent theoretical work has quantified the average statistical behavior of such systems with concepts of wave turbulence [21]. But for these simulations, we require a representative single shot model of the noise, towards which a number of attempts have been made [22, 23, 6, 24]. However, all of these models have some arbitrary parameters or other problem which complicates a clear understanding of the CW continuum dynamics. Instead I chose the simpler model of a pure CW field with additional one photon per mode quantum noise [25]. Such a field approximately represents a single frequency CW laser and leads to an underestimate of the spectral broadening because intensity fluctuations are neglected. Therefore I use quite high powers of 400 W. Although such lasers are available, the actual spectra resulting from these simulations are closer to those observed with more commonly available 50 W lasers.

I consider the continuum development in three photonic crystal fibers [26], all with a hole diameter to pitch ratio of 0.5, but with pitches of 2.2, 2.8 and 3.4 μm which I label as F22, F28 and F34 respectively. The dispersion and group velocity curves for these fibers are shown in Fig. 1. The zero dispersion wavelength shifts from around 0.94 μm in F22 to 1.04 μm in F34, moving closer to our chosen pump wavelength of 1.07 μm. The nonlinearity at our pump wavelength also decreases in this series of fibers, from 25 to 13 W^-1km^-1.

The process of CW continuum formation begins with the modulation instability of the continuous pump wave in the anomalous dispersion region [1]. Fluctuations of the intensity lead to small local modifications of the refractive index through the Kerr effect which form a localized waveguide which further trap the intensity fluctuations. This process leads to an enhancement of the intensity fluctuations which evolve into a train of pulses. An example of such pulse train formation in F34 is shown in Fig. 2. The intimate relationship between MI and the self-trapping mechanism of solitons naturally leads the intensity fluctuations to evolve into temporal solitons. Depending on the dispersion and nonlinear conditions the solitons formed in this way can have a spectral bandwidth large enough for Raman self-scattering to occur [27], leading to a continuous red-shift. MI is a noise seeded process and the solitons formed will have variable bandwidths and durations (see Fig. 2) and thus a range of self-frequency shifts will occur, creating a smooth red-shifted continuum as can be seen in the plots of spectral evolution in Fig. 3. The long wavelength extent of the continuum is limited by either the dispersion magnitude increasing or nonlinearity decreasing, at lower frequencies, such that the soliton bandwidth is compressed and self-scattering reduced [4]; or a second zero dispersion wavelength leading to dispersive wave generation and spectral recoil of the solitons [28].

 

Fig. 1. (a) The group velocity dispersion and (b) the group velocity curves for the PCFs considered.

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Fig. 2. Evolution from a CW field to intensity pulsations through increasing lengths of F34.

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Fig. 3. Spectral evolution of the continuum in (a) F22, (b) F28 and (c) F34.

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The generation of blue shifted waves depends on both the dispersion landscape at the pump wavelength, and the parameters of the solitons formed from the initial MI process. If the solitons are spectrally broad enough such that their blue tail overlaps with the normal dispersion region, then they can excite a dispersive wave at certain phase-matched wavelength, given in the reference frame of the soliton as the difference between the phase accumulation of a dispersive wave, ϕ_r(ω)/L = β(ω) - ωβ ₁((ω_s), and a soliton ϕ_s/L = β(ω_s) - ω_s β ₁(ω_s)+ n ₂ ω_s P ₀/(2cA_eff(ω_s)), [7]; where β ₁ is the first derivative of β with respect to ω, ω_s is the frequency of the soliton with power P ₀ and L is the fiber length. This process can be understood as a tunneling effect from the self-trapping soliton to free propagating radiation [8]. The phase matched wavelengths for this process are shown in Fig. 4(a) for the fibers we are considering. For fibers which have a zero dispersion point further from the pump wavelength, the phase matched wavelengths are also further. The excitation of dispersive waves with this process has been clearly observed for many pulse pump conditions [7], and evidence of it can be seen in the spectra from some CW pumped systems [5, 6], though direct identification has been hampered by a miss-understanding that it results from the fission of high-order solitons. The requirement for spectral overlap between the soliton and phase-matched wavelength implies that the zero dispersion wavelength should not be too far from the pump. This is in contradiction with the generation of a broad red-shifted continuum, which requires a flat dispersion landscape towards lower frequencies [4]. Nethertheless, fiber profiles supporting both systems can be obtained.

 

Fig. 4. (a) The phase matching curves between a soliton at 1.07 μm and dispersive radiation for the PCFs considered. The solid lines are for a 100 W peak power soliton, the dashed lines for 10 kW peak power, intermediate powers phase match between these curves. (b) The group velocity matching (GVM) curves between a soliton and dispersive wave.

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Figure 3 shows the spectral evolution of CW pumped supercontinua in the three fibers described above. It is clear that in F22, where the phase-matched wavelength is furthest from the pump, no blue-shifted spectral power is generated. In contrast, the results for F34 show a considerable fraction (11%) of the spectral power frequency up-shifted. For F28 the result is somewhere in between (0.5%). This occurs because the spectral powers of the excited dispersive waves decrease exponentially in amplitude as they are phase-matched further and further from the soliton [28].

The excitation of a dispersive wave by this process only transfers power to a fixed wavelength defined by the peak power of each soliton, whereas the continua in F34 and F28 show a continuous blue shift of spectral power. This is achieved through the process of soliton trapping of the dispersive waves [10]. The high intensity red-shifting solitons can cause a modulated refractive index on the leading edge of the dispersive waves. This prevents them from dispersing and, through cross-phase modulation, causes them to blue shift, into regions of lower group velocity and thus they decelerate. Simultaneously, the solitons are red-shifting due to Raman self-scattering, also into regions of lower group velocity (see Fig. 1(b)) and therefore they are also retarded and fall back onto the dispersive radiation [10, 18, 17]. This process can be maintained and the dispersive wave is essentially trapped temporally and continuously blue shifted. This has been observed experimentally in pulse pumped cases [11]. It is important to note that with this process an extensive red-shifted continuum is required to achieve significant blue-shifted spectral power.

 

Fig. 5. Spectrograms of the continuum evolution in F34; (a) 2.25 m, (b) 4.5 m, (c) 9 m and (d)18m.

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Figure 4(b) shows the group velocity matching curves between red-shifting solitons and dispersive waves. From this, we can see that the fibers with shorter zero dispersion wavelengths group velocity match solitons to shorter wavelengths and can therefore be expected to lead to further blue shifted radiation. This is evident in Fig. 3, where the blue shifted spectral power in F28 shifts to significantly shorter wavelengths than in F34. There is therefore a compromise between blue shifted spectral power and maximum blue shift. Further insight can be obtained by considering the spectrograms of the optical field as it evolves through the fiber calculated using a windowed Fourier transform of the optical field. In Fig. 5 I use a 1 ps Gaussian window to illustrate the evolution of the continuum in F34. In Fig. 5(a) the field after a short propagation length is shown, where initial intensity modulations and MI sidelobes are beginning to form. In Fig. 5(b) some dispersive waves at shorter wavelengths have been generated from the MI induced solitons, and in Fig. 5(c) the solitons are starting to red-shift due to Raman self-scattering. In Fig. 5(c and d) it is clear that localized dispersive waves are correlated with particular red-shifting solitons [10, 18, 17]. These waves remain temporally and spectrally localized as they propagate and can therefore be termed solitary. The spread in time and frequency of the solitons observed in these spectrograms, and their collisions, can be related to the idea of the soliton gas [19], and is the main qualitative difference between the CW supercontinuum and short-pulse supercontinua where only a few solitons are generated. Despite the experimental demonstration of visible CW supercontinua [14, 15, 16], no experimental diagnostics have been used, to the best of my knowledge, to study single shot dynamics of a continuous wave supercontinuum, such experiments could confirm these results. Additionaly, the dispersive waves could be independently seeded into the experiment from an external source, providing greater control of the temporal dynamics of the blue shifting continuum.

I thank J. R. Taylor, B. A. Cumberland, A. B. Rulkov, S. V. Popov for many useful discussions and suggestions, and the UK Engineering and Physical Sciences Research Council (EPSRC) for financial support.