## Abstract

We numerically study dispersive wave emission during femtosecond-pumped supercontinuum generation in photonic crystal fibres. We show that dispersive waves are primarily generated over a short region of high temporal compression. Despite the apparent complexity of the pump pulse in this region, we show that the dynamics of dispersive wave generation are dominated by a single fundamental soliton. However, any straightforward application of the theory that is thought to describe the blue emission, considerably underestimates the frequency shift. We show that in fact the red-shift of the soliton, caused by spectral recoil from the growing dispersive wave, causes an additional blue-shift of the resonant frequency which is in good agreement with full simulations.

© 2006 Optical Society of America

## 1. Introduction

Optical supercontinuum (SC) generation [1] is a versatile source of intense ultra-broadband laser light. The interaction of a narrowband laser pulse with dispersion and nonlinearity [2] produces fields of enormous bandwidth that nonetheless retain the coherent nature of their source. One particularly useful and widespread method of achieving SC generation is injecting femtosecond pulses from a Ti:Sapphire mode-locked laser into small-core (≈1*µ*m) photonic crystal fibers [3, 4]. The resulting octave-spanning output spectra are due to the combined action of several processes. One of these is dispersive wave emission [5, 6], which is primarily responsible for wavelengths shorter than the pump. Dispersive waves are important for many applications, such as time frequency metrology [7], optical coherence tomography [8], microscopy [9]. Indeed, in many situations SC generation is merely seen as a means of converting light from 800 nm to the visible. To exploit this ability to its fullest, a complete understanding of the dynamics of dispersive wave emission in SC generation is desirable. In this paper we use numerical simulations to explore the limitations of the present understanding involving fundamental solitons [5, 6]. We aim to obtain a better description of the pump pulse during dispersive wave generation so as to more accurately predict the emitted wavelengths.

Since dispersive wave emission is but one of a series of interrelated processes leading to SC generation, we now summarize the present understanding [2, 10]. The photonic crystal fibers used for SC generation typically have zero-dispersion wavelength slightly shorter than the Ti:Sapphire pump wavelength. The pump pulses experience anomalous dispersion, which, combined with self-phase modulation, leads to higher-order soliton formation. Both higher order dispersion and the non-instantaneous nonlinear response act as perturbations, causing the higher-order soliton to fission into its constituent fundamental solitons. These solitons are considerably shorter than the pump [11], partly explaining the massive spectral width of the SC. In addition, stimulated Raman-scattering causes red-shifting, which continues to act on the fundamental solitons via the soliton self-frequency shift [12] well after the break-up is complete. Finally, higher-order dispersion causes solitons to shed energy to dispersive waves (also known as non-solitonic radiation). The spectrum of dispersive waves is usually blue-shifted relative to the pump because of the positive dispersion slope at the pump wavelength.

The original detailed studies of dispersive wave generation [5, 6, 13] were in contexts other than SC generation. They focused on fundamental solitons, which, when weakly perturbed by higher-order dispersion, shed radiation in the form of a low amplitude temporal pedestal. The radiation frequency *ω*
_{R} is governed by phase matching between the soliton and the low amplitude linear waves; that is *β*
_{S}(*ω*
_{R})=*β*(*ω*
_{R}) where *β*
_{S} is the soliton wavenumber and *β* is the fiber’s linear dispersion curve. The implied assumption that the soliton wavenumber is independent of *z* was approximately true in the original studies [5, 6, 13] because the solitons were both fundamental and weakly perturbed. Thus, besides a small nonlinear phase-shift, the solitons propagated undisturbed at the phase and group velocities of the fiber, leading to a simple expression for *β*
_{S}

where *ω*
_{S} and *P*
_{S} are the soliton’s centre frequency, and peak power, subscripts on *β* denote frequency derivatives, and γ is the fiber’s nonlinear coefficient. The power-dependent term represents the nonlinear phase-shift and is often small compared to the other terms in Eq. (1). Equating *β*
_{S} given in Eq. (1) with the linear wavenumber yields the well-known resonance condition

which is commonly used to predict the location of dispersive wave emission [2, 4, 14, 15].

Disagreement between Eq. (2) and experiment is usually attributed to uncertainty in the fiber dispersion curve. However, we shall show that even in simulations, significant disagreement occurs, especially at high powers where dispersive wave generation is most efficient. Explaining this discrepancy is the motivation for this work. In particular, we show that efficient dispersive wave generation occurs over a narrow range of propagation distances before the emergence of distinct fundamental solitons. Examining the state of the pump pulse during this region, we find that the dispersive wave generation is nonetheless well described by a fundamental soliton-like state, provided a red-shift caused by higher-order dispersion in considered.

We note that another mechanism for dispersive wave blue-shift is cross-phase modulation with one of the solitons [16]. It critically depends on the walk-off dynamics between the two components and produces a distinct splitting of the dispersive wave spectrum. Some cross-phase modulation was observed in our work, especially for high input powers. However, additional blue-shift of the dispersive waves was observed even in those cases where significant cross-phase modulation did not occur. We attribute the diminished influence of cross-phase modulation to differences in the walk-off between the solitons and the dispersive wave. The parameters cited by Genty *et al.* [16] are such that the dispersive wave and ejected soliton have similar group velocities, yielding a gradual walk-off that produces a strong cross-phase modulation effect. Here, in contrast, the group velocity of the dispersive waves is considerably slower than that of the ejected soliton, so they quickly fall behind and the cross-phase modulation is weak.

## 2. Preliminary simulation results

In this section we present the results of a numerical simulation of femtosecond-pumped SC generation using typical experimental parameters. We use the scalar Generalized Nonlinear Schrödinger equation [17, 18], the standard formalism for SC generation:

where *A*(*T, z*) is the time domain envelope, *T*=*t* -*β*
_{1z} is retarded time, *z* is axial propagation distance, *β*_{n}
and *γ* are the *n*
^{th} order dispersion and nonlinear coefficients evaluated at the reference frequency *ω*_{0}
, *τ*
_{shock} models the dispersion of the nonlinearity and *R*(*T*) is the usual silica Raman response [18]. Femtosecond-pumped SC generation occurs in millimetres or centimetres of fiber and losses are therefore ignored. The initial conditions were unchirped hyperbolic-secant pulses i.e.
$A(T,0)=\sqrt{{P}_{0}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\mathrm{sech}\left(\frac{T}{{T}_{0}}\right)$
.

Equation Eq. (3) was solved using the symmetrized Fourier split-step method, using a second order Runge-Kutta algorithm for the nonlinear step. The discretization was sufficient to contain all energy (down to the -40 dB level) in the time and frequency windows, and adaptive step-sizing via step-doubling was used to improve efficiency. The optical photon number was tracked to estimate the global error, which was less than 1%. Although for clarity we here write the reference frequency equal to that of the pump, our code used an internal reference frequency approximately equal to the midpoint of the SC output spectrum so as to most efficiently use the simulated optical bandwidth.

We first consider a single simulation of an *N*=5 pump soliton. The other parameters are given in Table 1. The fiber is modeled on Crystal Fibre NL-2.0-740, a typical high air-fill hexagonal photonic crystal fiber designed for nonlinear experiments at 800 nm. Dispersion coefficients were measured using white-light interferometry [19], whilst the nonlinear coefficient was supplied by the manufacturer. We choose *τ*
_{shock}=1/*ω*
_{0}, ignoring the frequency-dependent effective area which has been shown to have negligible effect for this parameter regime [20]. A summary of the results is given in Fig. 1, whilst an animation of the spectral, temporal and spectrographic evolution is linked to Fig. 2. A broad dispersive wave emerges around 629 nm at *z*=36 mm, indicated by the vertical lines in Figs. 1(b) and 1(d).

## 3. Soliton wavenumber description

We now consider the validity of using Eq. (2) to solve the phase matching condition to estimate the blue shift. First, in the context of SC generation it is unclear what values to assign *ω*
_{S} and *P*
_{S}. For the former, previous authors have implicitly used the pump frequency [14, 15], whilst the latter is either taken to be the pump power, or zero since in any case this term has a only a small effect. Using *ω*
_{S}=*ω*
_{0} and *P*
_{S}=0, a resonance at 659 nm is predicted, 30 nm above the simulation result. Alternatively, putting *P*
_{S}=*P*
_{0} predicts 654 nm, still considerably larger. To show that this discrepancy is not an isolated special case, we repeated the simulation with different pump powers, corresponding to launched soliton numbers 2≤*N*≤10, and extracted the dispersive wave location. The results are shown in Fig. 3(a), and the well-known blue-shift of the dispersive wave with increasing pump power is evident. Here we note that, in addition, the soliton wavenumber prediction becomes increasingly inaccurate. To ascertain whether any amount of nonlinear phase-shift could explain the discrepancy, we computed the maximum power *P*
_{max} over all points (*T*, *z*) in the propagation, and set *P*
_{S}=*P*
_{max} in Eq. (2). Since higher-order soliton evolution involves an initial temporal contraction, we expect this to be significantly higher than the pump peak power and hence cause a larger dispersive wave blue-shift. However, as Fig. 3(a) shows, this is still not sufficient to explain the numerically observed values.

To demonstrate that the nonlinear phase-shift can in no way be wholly responsible for the blue-shift, we inserted the numerically observed blue-shift into Eq. (2) and solved for *P*
_{S}, thus finding the peak power required to achieve the observed nonlinear phase-shift. In Fig. 3(b), this is shown to greatly exceed the actual peak power *P*
_{max} obtained from the simulations. Thus we conclude that no conceivable straightforward application of Eq. (2) is consistent with the numerical results. In the remainder of this paper we thus investigate why this is so, and how this can be improved.

Having shown that Eq. (2) becomes increasingly inaccurate with increasing pump power, we now consider the reason for this discrepancy. The soliton wavenumber Eq. (1), used in deriving Eq. (2), describes a fundamental soliton driving a low-amplitude dispersive wave, which can be treated as a small perturbation. Here, however, we have a fifth-order soliton, which even in the absence of perturbations undergoes complex periodic evolution. In addition, higher-order dispersion significantly alters the propagation, as is clear from the large amount of energy in the dispersive wave. Since analytic solution of Eq. (3) is intractable, we numerically examine the state of the pump pulse as dispersive wave generation is occurring.

## 4. Conditions during dispersive wave generation

As Fig. 1 suggests, efficient dispersive wave generation occurs during a short propagation region. To test this thoroughly, we computed the dispersive wave energy, defined as all energy with wavelength less than 40 nm above the dispersive wave peak, as a function of *z*. The results are shown in Fig. 4(a) for several *N*, and the discontinuous, sudden nature of dispersive wave generation is apparent. We defined the “dispersive wave generation region” (DWGR) as the *z* values over which the dispersive wave energy grows from 20 % to 80 % of its final value. We tested a number of alternative definitions for the dispersive wave energy and DWGR and found that our conclusions were unaffected. For *N*=5, the DWGR is illustrated by the white horizontal lines in Fig. 1, whilst the start and finish are shown for 3<*N*<10 in Fig. 4(b). Also shown is the point *z*=*z*
_{max}, defined as the *z*-coordinate where the pump pulse reaches its maximum temporal intensity *P*
_{max}. For the range of *N* considered here, *z*
_{max} lies within the DWGR, which is 2–8 mm in length. Furthermore, both the DWGR and *z*
_{max} show good agreement with an approximate expression for the soliton fission distance *z*
_{fiss}=*L*
_{D}/N, where *L*
_{D}=${T}_{0}^{2}$/|*β*
_{2}| is the quadratic dispersion length [4].

We now examine the pump pulse in the DWGR. Although Fig. 2 is linked to an animation showing the complete evolution of the SC, the still image shows the conditions at *z*=*z*
_{max}=36 mm. Several temporally overlapping structures are evident, including the beginnings of the dispersive wave around 650 nm and a well defined pulse which goes on to become the most intense ejected soliton.

## 5. Nonlinear wavenumber

To study the phase-matching between linear waves and the evolving pump pulse numerically, we define the nonlinear wavenumber as the spatial frequency including the effects of the nonlinearity. This is easily derived by taking the Fourier transform of Eq. (3) and recasting it in the form

where *ω′*=*ω*-*ω*
_{0}, so that *β*
_{NL}(*ω, z*) and *α*
_{NL}(*ω, z*) are the *z*-dependent nonlinear wavenumber and loss respectively. Equating coefficients we find

where FT denotes the Fourier transform at frequency *ω′*. Using the pre-computed solution *A*(*ω, z*), the nonlinear wavenumber can be evaluated as desired. Clearly, at low intensities linear effects dominate and *β*
_{NL}≈*β*, whilst for a fundamental soliton at the carrier frequency experiencing quadratic dispersion and self-phase modulation only, Eq. (5) reduces to Eq. (1). The *N*-soliton pump pulses are *N*
^{2} times more intense, and therefore would “compensate” for a correspondingly larger amount of quadratic dispersion, if it were present. At *z*=0, this leads to

ignoring Raman and self-steepening. The subsequent evolution of the pump is complex and already well understood in terms of soliton-effect compression [21]. Our aim here is to relate the phase-matching between the compressed pump and linear waves during the DWGR to the dispersive wave growth. Figure 5(a) shows the spectrum at the start and end of the DWGR, whilst Fig. 5(b) shows the linear, soliton and nonlinear wavenumbers at the point of maximum intensity *z*
_{max}. The linear and nonlinear wavenumbers intersect at *ν*=478 THz, close to the dispersive wave frequency of *ν*=476 THz. By contrast, the soliton wavenumber intersects at *ν*=458 THz, underestimating the blue-shift. For plotting, the transformation *β*
_{NL}(*ω*)′=*β*
_{NL}(*ω*)-*β*(*ω*
_{0})-*β*
_{1}(*ω*
_{0})*ω′* (defined similarly for *β* and *β*
_{S}) is applied. This removes the linear phase and group velocities (equivalent to the transformation from the lab to the retarded frame) to make the curvature evident. Also, we plot the spectra and wavenumbers against frequency rather than wavelength as that is the natural domain of the latter. Figure 5(c) shows *β*
_{NL} for various pump powers, and the blue-shift of the intersection point with increasing power is evident. The anomalous behavior of the *N*=7 curve around 450 THz is due the rapid change in phase that occurs when self-phase modulation drives the complex spectral amplitude *A*(*ω, z*) through or close to zero. In general, *β*
_{NL} is ill-defined for any frequency at which *A*(*ω, z*) is close to zero. In Fig. 5(d) we present the resonant wavelength obtained from *β*
_{NL} for the full range of *N*, showing good agreement with the simulation data. The average and maximum differences between the two curves are 7 nm and 18 nm respectively.

The evolution of *β*
_{NL} is shown in the animation attached to Fig. 6(a). The initial condition, as expected from Eq. (6), is roughly parabolic, but after only 5 mm the nonlinear wavenumber has become nearly straight. After soliton fission has occurred, the linear behavior is lost as complex inter-soliton interactions occur. Two simple interpretations are evident from the animation. First for *z*>50 mm, *β*
_{NL}≈*β* for *ν*>470 THz, showing that nonlinear interactions with the dispersive waves have effectively ceased. Second, the straight section of *β*
_{NL} at low *n* for *z*>50 mm corresponds to the emergence of a spectrally distinct fundamental soliton, which gradually red-shifts due to Raman scattering.

To verify that the increase of *β*
_{NL} around *ν*=477 THz is not caused by the influence of the local linear wavenumber, we simulated SC in a hypothetical fibre with *β*
_{3}=0 and *β*
_{4}=4.459×10^{-4} ps^{4}/km, chosen to give the same dispersive wave resonance wavelength as before, i.e. 659 nm. The other parameters were identical. The resulting dispersion curve is symmetric about the pump frequency (in the retarded frame), and the fourth order dispersion creates a second resonance at 1052 nm. Over 3<*N*<10, we found no power-dependent blue-shift of the dispersive wave. Therefore, the blue-shift is not caused merely by the upturn of the linear dispersion curve at high *ν*, or some property of the higher order soliton evolution.

## 6. Spectral recoil of dominant fundamental soliton

Despite the complex multisoliton state of the pump during the DWGR, the nonlinear wavenumber curves are surprisingly linear over a broad frequency range, explaining why the soliton wavenumber has enjoyed reasonable accuracy previously. In Fig. 5(b), the two large deviations of *β*
_{NL} from an approximately straight line occur around the pump frequency (*ν*=370 THz), where the spectral intensity is low since SPM is driving the field through or nearby the origin causing a rapid change in phase. The linear dependence of *β*
_{NL} on frequency is a signature of fundamental soliton propagation, and we attribute it to the dominant effect of the most intense fundamental soliton which forms part of the multisoliton initial state [11]. Since the soliton wavenumber is parallel to the tangent of the linear wavenumber at the soliton’s centre frequency, the slight positive slope of *β*
_{NL} in Fig. 5(c) corresponds to a red-shift of the soliton, resulting in a blue-shift of the intersection point with the linear wavenumber curve. This analysis is consistent with previous work, which attributed pump red-shift, in a general sense, to Raman scattering and third-order dispersion [15]. However, here we show that the pump redshift can be interpreted as the spectral recoil of a fundamental soliton.

We found that the Raman effect has very little influence on the soliton red-shift nor on the dispersive wave phase-matching conditions. In Fig. 7(a), we compare the nonlinear wavenumber at *z*
_{max} with and without the non-instantaneous Raman response included in the simulation. The difference is minor, and it therefore not surprising that the wavelengths of dispersive wave emission are nearly identical, as shown in Fig. 7(b). Despite this, we found that energy transfer to the dispersive wave was significantly reduced by Raman scattering, consistent with previous numerical results by Dudley *et al.* [4], who attributed the reduction to Raman self-frequency shifting of the first ejected fundamental soliton.

The only remaining effects which break the frequency symmetry are odd orders of dispersion, of which only *β*
_{3} is considered here. In addition to dispersive wave emission, third-order dispersion causes a spectral recoil of fundamental solitons away from the zero-dispersion wavelength in order to conserve the spectral “centre ofmass” [5, 13]. In Eq. (3), both self-steepening and Raman scattering violate the conservation of this quantity; however our simulations show that over the DWGR, the spectral centre of mass remains nearly constant (significant red-shifting occurs afterwards due to the soliton self-frequency shift). We therefore attribute the power-dependent red-shift of the dominant pump soliton, and corresponding dispersive wave blue-shift, to spectral recoil caused by the dispersive wave emission. This also explains why no power dependent blue-shift was observed in the *β*
_{3}=0 case mentioned earlier — the two resonances spaced equidistantly (in frequency space) on either side of the pump cancel out any spectral recoil.

To demonstrate the validity of this argument, we solved Eq. (2) simultaneously with the spectral balance condition

treating both *ω*
_{R} and *ω*
_{S} as unknowns. The fraction of energy *f*
_{R} transferred to the dispersive wave is extracted fromthe simulations. We therefore calculate a pair of pump soliton and dispersive wave frequencies which are both phase matched and conserve the spectral centre of mass at the pump frequency. As expected, the recoil of the pump soliton increases with *N*. Furthermore, the calculated resonant frequency agrees to within 15 nm with the observed dispersive wave emission for *N*≤9. Whilst a detailed study of the energy transferred to the dispersive wave is beyond the scope of this work, a rough *a priori* estimate of this quantity would enable Eq. (7) to be used in conjunction with Eq. (2) as an improved prediction of the dispersive wave wavelength.

For larger *N*, the blue-shift is underestimated, which we attribute to the complicating influence of cross phase modulation and four-wave mixing between the solitonic and dispersive wave components [16, 22, 23] and possibly interactions between the solitons themselves [24]. This is seen by examining the N=10 case, attached of Fig. 6(b). By the end of the DWGR at *z*=18 mm, a spectrally distinct dispersive wave is evident at 585 nm, as indicated by a black dashed line. However, the complex series of interactions that occur with subsequent propagation have the net effect of blue-shifting the peak an additional 10 nm to 575 nm, indicated by the second line. The interactions also introduce fine structure into the dispersive wave spectrum, suggesting that a “clean”, near-transform limited dispersive wave pulse can be produced by truncating the fibre just after the DWGR.

We also note that in fibres with a negative dispersion slope, self-frequency shift compensation of fundamental solitons [25, 26] can occur because the soliton’s recoil to the blue, away from the dispersive wave, opposes the Raman red-shift. Our results suggest that novel dynamics may be observed by performing similar experiments using higher-order solitons, such that the initial spectral recoil “overcompensates” for the Raman red-shift, forcing the pulse to the blue until an equilibrium is reached.

## 7. Summary

We first showed that the dispersive waves emitted in femtosecond pumped SC generation experience a power-dependent blue-shift which cannot be explained by the standard picture of phase-matched emission by a fundamental soliton at the pump wavelength, even for the highest peak powers attained by the pump during propagation. Closer examination revealed that the majority of dispersive waves are produced in a narrow region where soliton-effect compression produces maximum intensity of the pump pulse. Although soliton fission has not yet progressed to the stage where distinct fundamental solitons are evident, the spatial frequency of the waves is dominated by a single fundamental soliton which is red-shifted by third-order dispersion. The physical interpretation of this red-shift is spectral recoil from the dispersive waves.

TGB acknowledges support through the Denison Distinguished Visitor scheme. This work was supported by an ARC Discovery Grant.

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