## Abstract

Nonlinear dynamics of ultrashort optical pulses in the vicinity of the second zero-dispersion point of a small-core photonic crystal fiber is visualized and studied using cross-correlation frequency-resolved optical gating. New spectral features observed in the experiments match well with recent theoretical predictions of the generation of new frequencies via mixing of solitons and dispersive waves. Power- as well as length-dependent dynamics is obtained showing strong interaction between solitons and dispersive waves, soliton-soliton interaction, soliton stabilization against Raman self-frequency shift and Cherenkov continuum generation.

©2004 Optical Society of America

Photonic crystal fibers (PCF) continue to display interesting properties in the linear and nonlinear optical domains [1]. When the fiber core size is made sufficiently small, the waveguide contribution to the total dispersion of the fiber dominates the material dispersion. In this regime of ~1*µ*m core diameters a second zero-dispersion (2ZD) point develops in the infrared region of the optical spectrum. Precise control of the position of the 2ZD point on the wavelength axis can be achieved through accurate control of the core size. The dispersion slope in the vicinity of the 2ZD point is negative, whereas it is positive near the first zero-dispersion point, typical for the regular fibers and previous generations of PCFs [2, 3]. Theoretical predictions [4, 5, 6] and previous experiments [7, 4, 8] show unusual soliton dynamics in the vicinity of the 2ZD point. These effects may have far-reaching applications since the wavelength of the 2ZD point can be tuned to the telecommunications region around 1550 nm through careful control of the fiber structure during manufacturing.

Figure 1 shows the dispersion and the optical loss of one of the fibers used in our experiments. The high quality of the structure is manifested through the relatively low loss in the wavelength range of interest. The 2ZD point in this case is located at 1510 nm and the measured dispersion slope is large and negative. The dispersion properties of a regular fiber are shown for comparison in the figure by the dashed curve. The small core of the fiber also leads to high peak intensities in the propagating optical pulses. The nonlinear parameter for our PCF is estimated to be *γ*~100*W*
^{-1}
*km*
^{-1}. Note that in the vicinity of the 2ZD point, anomalous dispersion regime is realized at *shorter* wavelengths whereas normal dispersion lies at *longer* wavelengths. Thus, for the ultrashort optical pulses used in our experiments, the strong self-frequency shift resulting from stimulated Raman scattering (SRS) [9] tends to push solitons closer to the 2ZD point as in [4, 5, 8], rather than driving them away as is the case near the first ZD point [1, 2, 3, 10].

Our experimental results reveal the intricate details of the interaction between solitons and dispersive or continuous waves (cw) in time and frequency. In particular, we show that this interaction can lead to the generation of new spectral lines. All these processes are resonant in their nature and require certain wavevector matching conditions to be satisfied. Very recently it has been shown theoretically [6] that the interaction of a soliton and a weak cw within an optical fiber can lead to the excitation of new spectral components, termed “signal” below, with frequencies determined from special wavevector matching conditions [6]:

Here *β*_{signal}
and *β*_{CW}
are the propagation constants of the fundamental fiber mode at the signal and cw-pump frequencies, respectively. *k*_{sol/signal}
and *k*_{sol/cw}
are the wavevectors of the Fourier components of the soliton at the frequencies of the cw-pump and signal fields, respectively, see Ref. [6] for details. For *J*=0, equation (1) coincides with the condition describing emission of the Cherenkov radiation by a soliton [4, 5, 11, 12, 13] without any additional cw-pump. The *J*=±1 case in Eq. (1) predicts generation of new spectral components inside the fiber. Mixing of a soliton and a cw wave is qualitatively different from standard four-wave mixing, in which all the fields are cw or quasi-cw [9]. In the case of the soliton-cw interaction, energy and momentum are shared between the cw-pump, the signal wave and the soliton as a whole. Therefore conservation of the total field momentum in this case cannot be associated with the wavevector matching condition (1) [5, 6]. As we will see below, condition (1) appears to be noncritically satisfied in our system, leading to the observation of well-pronounced effects.

For our experiments 100 fs pulses tunable in the range of 1400–1600 nm are generated by an optical parametric oscillator at a repetition rate of 82 MHz and average power up to 350 mW. From this pulse train the reference and the signal pulses are derived with controllable power ratios. The signal pulse propagates through the fiber and the reference pulse is appropriately delayed in a self-retracing delay apparatus. To obtain cross-correlation frequency-resolved optical gating (X-FROG) traces [14] the signal and reference pulses are mixed in a 200 *µ*m thick BBO crystal and the sum-frequency (SF) generated pulse is spectrally resolved in a 0.3 nm resolution spectrometer. The phase-matching bandwidth of the crystal exceeds 400 nm at the wavelengths used which is sufficient to upconvert the complete spectrum of the signal pulse at the fiber output.

We assume that the dynamics of the dimensionless amplitude *A*(*t*, *z*) of the fundamental fiber mode is governed by the generalized NLS equation [1]

The variable *t* is the time in a reference frame moving with the group velocity *v*
_{0}=*v*(*ω*
_{0}) and measured in units of *τ*: *t*=[*T*-*z*/*v*
_{0}]/*τ*, where *T* is the physical time, *τ*=100 *f s* is the normalization constant chosen to be equal to the pulse duration and *ω*
_{0} is the reference frequency. The frequency dependence of *A* is assumed in the form *e*
^{-iδ t}, where *δ* is the normalized detuning of the physical frequency *ω* from *ω*
_{0}, i.e. *ω*-*ω*
_{0}=*δ*/*τ*. If *β* (*ω*) is the frequency dependence of the propagation constant of the fundamental fiber mode, then

where *β*_{n}
(*ω*)=${\mathit{\partial}}_{\omega}^{n}$*β* (ω) and *L*_{gvd}
=*τ*
^{2}/|*β*
_{2}(*ω*
_{0})| is the group velocity dispersion (GVD) length. Thus the Fourier image of the operator *D*(*i∂*_{t}
) is, in fact, the properly shifted and normalized propagation constant of the fiber mode. *z*=*Z/L*_{gvd}
, where *Z* is the distance along the fiber. *R*(*t*) is the nonlinear part of the response function of the material, which is silica glass in our case:

Here Δ(*t*) and Θ(*t*) are, respectively, delta and Heaviside functions. Eq. (4) includes the instantaneous electronic and delayed Raman contributions with *θ*=0.18, *τ*
_{1}=12.2 *fs*/*τ*, *τ*
_{2}=32 *fs*/*τ* [9]. The field amplitude *A* is measured in units of [*γL*_{gvd}
]^{-1/2}, where *γ* is the nonlinear parameter of the fiber already defined above. Self-steepening [9] was found to be not significant for the effects considered in this paper and therefore was disregarded. Below the results of modelling Eq. (2) are given next to the corresponding experimental measurements, see Figs. 2, 3, Figs. 5, 6 and 7, 8 and very good agreement can be found in all cases.

Several regimes of pulse propagation were studied, with observed dynamics that differs significantly depending on the pump pulse carrier wavelength at the fiber input. Most interesting are regimes when the fiber is pumped on the anomalous dispersion side and at the 2ZD wavelength. These cases are considered below.

One of the resonances described by Eq. (1) with *J*=1 can be observed when the fiber is pumped exactly at the 2ZD point, λ_{2ZD}=1510 nm, Figs. 2, 3. Under these conditions, soliton and quasi-cw fields with different frequencies emerge from the same pump pulse. As the input power is increased the pump pulse splits in the spectral domain into two parts: The longer-wavelength part evolves in the region of normal dispersion and the shorter-wavelength one, in the region of anomalous dispersion [15, 16].

On the time-frequency X-FROG plots of Fig. 2, 3 the radiation follows a quasi-parabolic trajectory. This is consistent with the wavelength dependence of the group delay in the vicinity of the ZD point. The quasi-solitonic pulse, which gradually appears in the region of anomalous GVD as pump power is increased, is quite long at first and is not influenced by the SRS [9]. This pulse emerges increasingly towards the shorter wavelength side of the spectrum because the splitting between the two spectral halves of the pump gets stronger as the input power is increased for the same fiber length. An important feature to notice here is that the radiation propagating in the normal GVD regime exactly on the opposite side from the soliton, and playing the role of the CW-pump that interacts with the soliton, has a missing section or “hole.” It is clear that the soliton will interact most efficiently with radiation propagating at the same group velocity and therefore overlapping with the soliton inside the fiber. When the soliton and radiation share the same group velocity, using Eq. (1) one can show that no new frequencies are efficiently excited from their resonant interaction, see Fig. 4(a). The soliton, however, creates an effective repulsive potential for the radiation, which is responsible for the “hole”. A similar “hole” has been previously numerically generated, but not discussed, in Ref. [2].

With further increase of power the soliton becomes shorter and intrapulse SRS[9] starts to shift its frequency away from the left branch of the parabola, Figs. 2, 3 (39 mW and up). This destroys the group velocity matching between the soliton and CW radiation. The mixing of the soliton and the radiation then leads to energy transfer into a new resonance, with a wavelength lying between the soliton and the CW-pump. This resonance can be found from Eq. (1) with *J*=1, see Fig. 4(b). The resonance radiation starts to fill the spectral gap between the soliton and the right branch of the parabola, Figs. 2, 3 (41–45 mW). The radiation appearing for higher powers, Figs. 2, 3 (45 mW and up), at an angle and to the right from the right branch of the parabola, is the Cherenkov radiation emitted by the soliton [4]. This is the *J*=0 resonance of Eq. (1). For even higher powers the Cherenkov radiation fills the gap in the right branch of the parabola and starts to interfere with the previously emitted dispersive wave, which creates modulation ripples. Thus, the soliton follows an effective return trajectory starting from the 2ZD point, through anomalous dispersion region and back to the proximity of the 2ZD point as the input power is increased. The evidence of this soliton trajectory was present in the spectral data of Harbold, et al. [7] but was not emphasized by the authors.

Similar behavior is observed along the fiber length, Figs. 5, 6, using the cutback technique. In these experiments the input power was kept fixed at 55mW. Strong spectral splitting is observed already after 10 cm of propagation. The soliton is formed in the anomalous dispersion region shortly after that and a “hole” in cw branch is visible on the normal dispersion side. At the end of the fiber the soliton, driven by SRS, comes back to the 2ZD point thus completing the return trajectory.

Different dynamics occurs when we pump the PCF in the region of anomalous GVD, Figs. 7, 8. Under these conditions most of the pump energy is transformed into a soliton, which subsequently Raman self-frequency shifts to the red and approaches the 2ZD point. When a sufficient portion of the soliton’s spectrum extends across the 2ZD point into the normal dispersion region, efficient generation of Cherenkov continuum occurs [4, 5, 8]. In our X-FROG traces the Cherenkov continuum manifests itself through a long tail at SF wavelengths longer than 750 nm (1560 nm fundamental) in the normal dispersion region of the fiber. Spectral recoil from the continuum stabilizes the soliton at a wavelength near to the 2ZD point, Figs. 7, 8 (14 mW and up) [4, 5]. With further propagation the central wavelength of the soliton remains unchanged due to the appearance of a second “force” acting on the pulse’s spectrum that balances the SRS. This second force is the recoil from continuum radiation. Its origin can be understood from energy conservation: since continuum radiation involves no phonons, radiation of long-wavelength continuum photons must be offset by energy-equivalent emission of short wavelength photons. Hence a recoil force acting to shift the soliton spectrum towards the blue appears. Near the 2ZD point a balance of these two forces occurs [4, 5] and our X-FROG measurements show the exact dynamics of the optical fields under these conditions. Note that there is a continuous energy flow from the soliton to the continuum through uninterrupted spectral overlap [13]. As the input pulse energy is increased further, the length of the continuum tail increases because soliton formation, SRS frequency shift, stabilization and the offset for continuum generation occur earlier in the fiber. Subsequent power increase allows formation of a second soliton which carries less energy and therefore is broader, Figs. 7, 8 (28 mW). This new soliton propagates ahead of the first strong soliton and thus effectively interacts with the Cherenkov continuum emitted by the first soliton. The Cherenkov radiation from the primary soliton then plays the role of the CW-pump for the second weak soliton. The interaction of this radiation with the second soliton leads to the appearance of a clearly observable spectral peak which fills the gap between the solitons and the primary Cherenkov band, Fig. 7, 8 (30 mW). The wavelength of the new spectral band corresponds to the resonance wavelength shown in Fig. 4(b) and derived from Eq. (1). At yet higher powers the dynamics becomes overly complicated with multiple solitons present and a continuum radiation tail extending over 20 ps in time.

Note, that when we finished work on this paper the reference [17] has appeared, where mixing of the solitons and dispersive waves generated in the proximity of the zero GVD point with the positive slope, i.e. opposite to ours, have been discussed. Generation of the second blue shifted with respect to the soliton frequency peak, which is one of the central results in the Ref. [17], has not been explained so far by any kind of the wave-number matching condition. Though effects found in Ref. [17] are different to the ones reported above, we expect that the theory developed in [6] and condition (1) can also be used to explain them.

In conclusion, in our work we discovered new regimes of ultrashort pulse propagation in the vicinity of the second (long wavelength) zero dispersion point of a strongly-guiding small-core PCF. With X-FROG measurements we have experimentally obtained a complete temporal-spectral picture of the soliton and radiation dynamics, of their interaction, and also observed multiple-soliton regimes. We have built up a complete time-spectrally resolved experimental picture of soliton stabilization against the Raman self-frequency shift due to recoil from Cherenkov radiation [4, 5] and partly confirmed recent predictions of the generation of new spectral lines from the four-wave mixing of the solitons and dispersive waves in fibers with higher-order dispersion [6]. Further discoveries can be expected by applying the X-FROG technique to the investigation of ultrafast nonlinear optical processes in new types of photonic crystal fiber structures.

## Acknowledgments

This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences nanoscale science research center operated jointly by Los Alamos and Sandia National Laboratories. Los Alamos National Laboratory is a multi-program laboratory operated by the University of California, for the U.S. Department of Energy under contractW-7405-ENG-36. Work of the Bath group has been supported in part by the Leverhulme Trust.

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