We apply the recently developed theory of frequency generation by mixing of solitons and dispersive waves [Phys. Rev. E 72, 016619 (2005)] to explain the observed formation, quasi-trapping and frequency shift of the spectral peaks at the blue edge of supercontinua generated in silica-core photonic crystal fibers.
©2006 Optical Society of America
Supercontinuum generation in photonic crystal fibers has recently attracted significant attention due to the relative simplicity of the experimental arrangements and widely spread applications in spectroscopy and metrology , telecommunications  and other areas. It has been found that different physical effects and their interplay can contribute to the supercontinuum formation. Amongst those are self-phase modulation (SPM) , Raman shifted solitons , blue and red detuned resonant radiation of the solitons [4, 5, 6, 7, 8, 9], four wave mixing between dispersive waves and solitons [10, 11, 12, 13], and modulational instability [6, 14]. The above selection of references are far from being exhaustive, and the number of publications on this topic serves to illustrate the richness of the physics involved.
One of the commonly-used experimental setups used to produce supercontinua spanning the infrared to the violet part of the spectrum is based on a photonic crystal fiber (PCF) with a core diameter of a few microns pumped by sub-picosecond or femtosecond pulses, with the pump wavelength close to the zero of the group velocity dispersion (GVD) with positive GVD slope (see, e.g., [4, 5, 6, 9]). The part of the pump spectrum experiencing anomalous GVD forms a sequence of ultrashort soliton pulses which are rapidly red shifted by the intrapulse Raman scattering . The shortwavelength (blue) edge of the supercontinuum is in the normal GVD range, and it develops distinct and unexpectedly intense spectral peaks [4, 5, 9, 11]. Note that here and below by the ‘red/blue’ radiation or ‘red/blue’ edges of the continuum we commonly understand not the actual colors, but rather the detuning of the radiation from some reference frequency, e.g. the zero GVD point. In particular, in the experiment described below, the ‘blue’ edge corresponds to the actual colors from red to orange.
Despite the fact that the generalized nonlinear Schrodinger (NLS) equation has been re-markably successful in reproducing experimental measurements of supercontinua, (see, e.g. [4, 7, 10, 11, 13, 14]) it would not be an overstatement to say that numerical modeling without analytical results is often not more illuminating than experimental measurements. A popular, but not universal, explanation for the origin of the spectral peaks at the blue edge of the continuum is that they are due to the resonant, or Cherenkov, radiation of the solitons emerging in the region of the anomalous GVD, which overlap spectrally with the normal GVD region [7, 8, 15, 16, 17]. This overlap creates suitable conditions for energy transfer from the soliton to a resonant dispersive wave having a specific frequency and wavenumber at the intersection of the dispersion characteristics of the soliton and linear waves. Some time after the ‘Cherenkov’ mechanism [4, 5, 6, 9] became widely accepted, it was found that the most blue-shifted peaks at the edge of the supercontinuum can be formed due to interaction between dispersive waves and solitons [10, 11]. Papers [10, 11] have provided only experimental and numerical observation of this effect in the course of supercontinuum generation and no wavenumber matching conditions or other theory explaining the appearance of this radiation have been presented there. The analytical theory of generation of new frequencies by mixing of solitons and dispersive waves has been developed in [12, 18]. Ref.  indicated that such mixing could explain the blue edge of the supercontinuum spectrum, but no systematic analysis was presented. Experimental investigations of the interaction of an isolated soliton with a frequency detuned continuous wave in PCFs not obscured by supercontinuum spectra have been reported in [19, 20] and fully confirmed the theoretical predictions of Refs. [12, 18]. Mixing of solitons with dispersive waves has been shown to shape not only the edge of the supercontinuum, but also its central part , providing that the dispersion slope is negative.
An intriguing effect accompanying formation of the peaks at the blue continuum edge is that despite residing deep in the region of normal GVD they form quasi-nondispersivewave packets [10, 11, 21]. This surprising effect of radiation trapping originally received substantial attention in the series of papers by Nishizawa and Goto [22, 23, 24] independently of its relevance to supercontinuum. Though it has been proved both experimentally and numerically [21, 22, 23, 24] that the blue radiation is somehow trapped by a co-propagating soliton on the other side of the zero GVD point, no mathematical theory or physical interpretation of this phenomenon has previously been presented now.
The object of this work is to demonstrate how the conditions for FWM resonances between solitons and dispersive waves [12, 18] can be applied to explain evolution of the blue edge of the supercontinuum. First, using these conditions we will argue that mixing of dispersive waves with solitons is at least as, and very likely, more important than Cherenkov radiation during the initial stages of the blue supercontinuum development. Second, we will demonstrate how for longer propagation distances the FWM of solitons and blue radiation transforms from the process involving two blue pulses (pump and signal) to an intrapulse effect, which explains radiation trapping and the continuous shift of the continuum edge towards the blue. Sections 2, 3.1 and 3.2 contain a description of the model, and theoretical and experimental results highlighting those features important for the rest of the paper. Mechanisms of the formation of the continuum blue edge and transition to the intra-pulse regime are described in sections 3.3 and 3.4.
2. Modeling and measurements
Our model is the generalized nonlinear Schrodinger equation
Here γ is the Kerr nonlinearity parameter, θ=0.18 measures the relative strength of the Kerr and Raman nonlinearities and is the Raman response function: τ1=12.2fs, τ2=32fs for the silica being used in the experiments. (i∂t )= β (n)[i∂t ] n /n! is the dispersion operator such that β(δ)= β(n) δn /n! matches the entire dispersion profile of the fiber mode in the relevant range of frequencies. Here δ=ω-ωref and ωref is the reference frequency. z is the distance along the fiber and t is the time in the reference frame moving at the group velocity at ωref . The fiber used in our experiments was a PCF with a 3µm silica core and having a measured dispersion profile close to that of an isolated silica strand, with a zero-GVD wavelength at λ0=820nm. The range of normal/anomalous GVD extends from λ0 towards shorter/longer wavelengths. The value of γ is estimated at 0.024/W/m. The Ti:sapphire laser used in our experiments generated 200fs pulses at 800nm, i.e. slightly into the range of normal GVD. The peak power used for the results presented throughout this paper is 6.2kW. The experiments were performed using a fiber of 2m length. We coupled our pump pulse train into the fiber using an objective lens, and this coupling was kept fixed during the experiment. Light leaving the fiber was coupled into a multimode fiber and analyzed using an optical spectrum analyzer. A series of spectra were recorded at fixed pump power as the fiber was gradually reduced in length by repeated cutback. The resultant spectra were plotted on a single plot to enable one to observe the spectral evolution along the fiber.
Figure 1(a) shows the experimentally measured spectral evolution under the above input conditions as function of the propagation length. Figure 1(b) shows the modeling results for a 3m long fiber and unchirped input pulses, which corresponds to the experimental conditions. One can see that numerical and experimental patterns of spectral broadening agree very well. The continuum develops in three main stages. At the first stage from z=0 to about 18cm one can see roughly symmetric spectral expansion. After this the red edge of the spectrum is determined by the Raman shifted soliton, while the blue edge goes through the two distinct stages in its development. Between 20cm and 1.2m the blue wing has a sharply defined boundary fixed at ≃670nm and for z>1.2m its edge starts extending continuously towards shorter wavelengths. These three stages can be clearly identified in the experimental plot, validating our numerical model and, thereby, the line of arguments presented in the section 3. Increasing energy of the input pulses leads to the qualitatively similar developments at the blue edge of the continuum, with transition to the regime of the blue shifting continuum edge happens for shorter distances.
A powerful and commonly used visualization tool also used here is cross-correlation frequency-resolved optical gating (XFROG) spectrograms, which allow direct association of the spectral peaks with the parts of the signal in the time domain, see, e.g. [4, 13]. Numerically computed spectrograms in this paper are produced by plotting the integral I(t,ω)=| dt′Ep (t′-t)E(t′)e -iωt|. Here Ep is the pump pulse, which is scanned along the generated signal in time domain. XFROG spectrograms for the six selected distances are shown in Fig. 2. Files with the animations of the entire process of the supercontinuum growth and development are linked to Figs. 3(a,b), which show color maps of |I(t,ω)| and log|I(t,ω)|, respectively.
From Figs. 2(c-f) and most clearly from the animations in Fig. 3 one can see that the wave packets at the short-wavelength edge of the continuum first are dispersing in time, but for z>1.2m they start to form quasi-non-dispersive pulses, the temporal and spectral locations and group velocities of which are strongly correlated with those of the solitons at the long-wavelength side [10, 11, 21, 22, 23, 24].
3. Interpretation of the results
3.1. Initial spectral broadening via SPM and soliton formation
For z upto 15cm the initial pulse undergoes SPM-induced spectral broadening, and develops strong spectral side-lobes  (see Fig. 2(a)). With further propagation the side-lobes move further apart in the frequency domain , so that one of them soon crosses into the range of anomalous GVD, see Fig. 2(b). For slightly longer propagation distances, between 15 and 20cm, the SPM dominated stage of the supercontinuum development ends, see Fig. 2(c). Radiation in the normal GVD regime forms a packet of dispersive waves, the frequency of which does not change with further propagation. This wave packet remains a dominant spectral feature on the blue side of the continuum, as seen between 800 and 730nm in Figs. 1(a,b). The part of the spectrum in the anomalous GVD regime forms multisoliton pulses, which subsequently break into quasi-solitons (‘soliton fission’ [5, 6]), see animations in Fig. 3.
3.2. Conditions for the Cherenkov and FWM resonances
For propagation distances between 20 and 30cm, one clearly observes rapid formation of radiation between 730 and 670nm, which constitutes the blue edge of the continuum for distances up to 1.2m. In this subsection we summarize the wavenumber matching conditions for the two main mechanisms of interaction between the solitons and dispersive waves, which are expected to contributed to this radiation. First, is the Cherenkov radiation emitted by the solitons [5, 6, 8, 15, 16, 17]. The frequency of this radiation can be determined using the wavenumber matching condition
Here βsol (δ)= + δ+q is the frequency dependence of the wave numbers of the Fourier harmonics of the solitons. The linear dependence of βsol on δ reflects suppression of the GVD for the soliton. q=γP/2 is the nonlinear correction to the soliton wavenumber, P is its peak power, =β(δ=ωsol -ωref ), =∂δβ(δ=ωsol -ωref ) and ωsol is the carrier frequency of the soliton.
The second mechanism which can contribute to the formation of the spectral band from 730 to 670nm is radiation emitted due to FWM of solitons with dispersive waves [12, 18, 19, 20]. Intense wave packets in the range from 800 and 730nm (formed by SPM) strongly overlap (in the time domain) with emerging solitons and therefore can efficiently generate new frequencies. There are, in general, two wave number matching conditions giving frequencies of the dispersive waves excited as a result of this interaction [12, 18]:
Here δp =ωp -ωref , ωp is the frequency of the continuous dispersive wave hitting the soliton (pump wave) and other notations are clear from Eq. (2). The first of the above conditions corresponds to the scattering of the wave on the square of the absolute value of the soliton field, while the second condition originates from the wave scattering on the square of the soliton field itself. Eq. (4) is found not to be relevant for the blue edge formation and all calculations of the frequencies of the FWM signal used below have been based on Eq. (3).
Resonant values of δ are the roots of the equations (2), (3), which critically depend on the soliton frequency ωsol and the pump frequency ωp . Note, that unlike for the conventionalFWM of all dispersive waves, the conditions (2)–(4) do not express conservation of the total wave momentum and therefore do not imply any symmetry in the location of the roots [12, 18]. The spectral energy due to FWM of dispersive waves, such as modulational instability, grows exponentially in z, while in the cases ofFWM mediated by solitons it is typically a linear growth .
3.3. Primary band of the blue radiation not associated with SPM: Interpulse FWM vs Cherenkov mechanism
From the non-logarithmic plots in Figs. 4(a1,2), one can see that the spectral band from 730 to 670nm is dominated by two large spectral peaks. As seen in the movies, the left most peak emerges for z around 20cm and the second one for z close to 45cm. When the first peak is generated, the spectrum with anomalous GVD is still partly shaped by the SPM and we cannot say anything quantitative about the carrier frequency of the solitons at this stage. For z>40cm, however, one can clearly see that the strongest (red-most) soliton does not substantially overlap with any other parts of the field, while the second strong pulse overlaps with dispersive waves. Assuming that the soliton carrier frequency within this second pulse can vary from 900nm to 960nm, see the green stripe in Fig. 4(a2) and taking ωp at 760nm, which is the maximum of the spectral side-lob created by the SPM, we plot the ranges of the Cherenkov (dashed area) and FWM (gray area) signals. One can see that the second spectral peak belongs to the FWM band, while Cherenkov peaks can also be seen, but only at the logarithmic scale. An example of the wavenumber matching for a particular choice of the soliton frequency is shown in Fig. 4(a 3).
Note, that the amplitude of the Cherenkov peaks for the given detunings between the solitons and their resonances is several orders of magnitude less than the amplitude of the two strongest blue peaks in Figs. 3(a1,2). Although limited in time (see animations in Fig. 3), the strong growth of these peaks suggests that they have been emitted during the optimal overlap of the quasi-solitons with the pump waves. This also suggests the FWM mechanism rather than one due to Cherenkov radiation. Thus the spectral band from 730nm to 670nm can be formed by both Cherenkov and FWM effects, with the FWM mechanism having few extra arguments in its support. The soliton-mediated FWM process at this stage of spectral expansion is clearly an interpulse one, i.e. there are distinct pump and signal waves. As we will see below the differences between the pump and signals pulses will gradually disappear with subsequent scattering events and FWM will gradually transform from an inter-to intrapulse process. Note that the Raman effect does not play a critical role in the formation of this band, because the SPM itself pulls the spectrum apart quite strongly and creates conditions for the efficient interaction of solitons and dispersive waves. Spectral recoil of solitons due to FWM can also be substantial .
3.4. Secondary band of the blue radiation and its trapping: Intrapulse FWM
One can see from Figs. 4(a1,2) and from the movies that the blue radiation corresponding to the two spectral peaks at 0.7 and 0.65nm discussed above initially lags behind the solitons. While the blue radiation has a fixed frequency and therefore moves with a constant group velocity, the solitons are continuously decelerated. This is because their frequency is shifted into a spectral range with smaller group velocity by intrapulse Raman scattering. Thus the blue peaks unavoidably catch up with the solitons (see movies and Fig. 4(b1)). As soon as the radiation and soliton start to overlap the FWM between them generates a new signal. The signal pulse again lags behind the soliton but again will be caught up. However, with each new scattering event the frequency of the generated signal is getting closer to the pump frequency, see Fig. 4(b3) for the corresponding wave number matching diagrams. Practically, neither the signal nor the pump are monochromatic waves, but pulses with close carrier frequencies which closely overlap in the time domain. For sufficiently large z these pulses do not have time to substantially separate before the next scattering event takes place. The result of this is that the FWM process becomes quasi-continuous and intrapulse, i.e. it happens within a quasi-nondispersive pulse propagating under the soliton umbrella. The continuous interaction of the blue pulse with the soliton is ensured by the matching of the group velocities. The overall result of intrapulse FWM is that the trapped blue peaks are dragged further to the blue while the soliton is red shifted by the Raman effect. Since the mechanisms of the blue wing formation described above are intrinsically linked to the solitons, the formation of the signal at the blue edge is governed not only by the local (short-wavelength) dispersion, but also by the dispersion and attenuation into the mid-infra-red as well. We have confirmed (by intentionally altering the dispersion curve) that the FWM cascade stops the moment that group velocity matching between the blue and red edges of the spectrum becomes impossible.
In order to facilitate understanding of this complex effect, we plot in Fig. 1(c) the group index ng (speed of light in vacuum divided by the group velocity) as a function of the wavelength. The group index curve is quasi-parabolic, which ensures group velocity matching across the zero GVD point extending over the relevant spectral range. One can see that the group index as function of the wavelength reproduces with a reasonable degree of accuracy the locations of the blue and red edges of the continuum after the interpulse FWM kicks in at z≃1.2m. To illustrate how the successive FWM mixing events are mapped on the group index plot we take the wavenumber matching diagram in Fig. 4(a 3) and mark the group indexes of the colliding soliton and dispersive wave with dashed and full lines, respectively (see Fig. 1(c)). Then we calculate the wavelength of the FWM signal using Eq. (3) and mark the corresponding group index with another dashed line. New radiation will interact with already slower soliton and the diagram can be extended further (Fig. 1(c)). One can see as the dependence of ng on λ steepens the frequencies of the pump and signal become closer and closer to each other. Simultaneously the group indexes of the soliton, incoming and generated radiation also become closer. Practically it implies that for z>1.2m the process of the FWM becomes effectively intrapulse, in a sense that there is a Raman shifted soliton on the red edge of the spectrum which continuously scatters the group-velocity-matched blue pulse almost into itself. This process manifests itself in the effect of the quasi-trapping of the blue radiation and its continuous frequency shift towards shorter wavelengths.
We have demonstrated how the wave number matching conditions for the FWM between the linear waves and solitons [12, 18] can be used to quantitatively explain and predict the formation of the blue edge of the supercontinuum spectra in photonic crystal fibers pumped with femtosecond pulses in the proximity of the zero GVD point. We have presented the first theoretically-justified arguments explaining the effect of trapping of the blue radiation by the solitons across the zero GVD wavelength. The above described mechanisms of the blue continuum formation also explain the generation of far-detuned violet and ultraviolet spectral peaks in the tapered PCFs  and the expansion of the blue part of the spectrum in some of the dual pump schemes of the continuum generation, see, e.g. .
This work has been supported by EPSRC. JCK is a Research Fellow of the Leverhulme Trust. Discussions with A. Yulin are acknowledged.
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