## Abstract

Collisions of femtosecond solitons in silica core photonic crystal fibers are investigated experimentally and theoretically. Clear spectral signatures of the significant energy exchange between the interacting pulses are reported. Two primary and competing effects causing energy exchange are *inter*pulse Raman scattering, which is insensitive to the phase difference of the colliding solitons, and the phase sensitive interaction via the Kerr nonlinearity.

©2006 Optical Society of America

## 1. Introduction

Nonlinear optical effects in photonic crystal fibers (PCFs) is currently an active research area. Optical soliton propagation in these fibers has attracted particular attention. Solitons in PCFs have many features which distinguish them from the solitons in telecommunications fibers. For example, solitons can propagate in PCF over a much wider spectral range [1] extending to visible frequencies. Typical soliton durations investigated in solid-core PCFs are 100fs or less: in this regime intrapulse Raman scattering causes a rapid continuous red shift of the soliton frequency, which often plays dominant role in the soliton dynamics [2, 3, 4]. Solitons and associated dispersive radiation can also play a substantial role in shaping supercontinuum spectra generated in PCFs [5, 6, 7]. Nonlinear coupling in the two-core PCF has been observed recently [8] prompting hopes for switching applications and discrete solitons. Hollow core PCFs allow for observation of fiber-guided solitons at previously unprecedent powers [9] and in gaseous media [9, 10]. Soliton effects at the interface between the fiber core and photonic crystal cladding have also been reported [11].

The interaction of colliding solitons is a problem of long-standing interest because of the richness of the non-trivial physical effects and applications to such areas as optical switching and signal transmission at multiple frequencies, see, e.g., [12]–[23]. In switching applications the interest lies in maximizing the energy exchange between colliding solitons, while in transmission applications it should be minimized. Most of the research reported on switching with interacting solitons has been done in the context of spatial solitons, see [24] for a review, while energy exchange between solitons in telecom fibers transmitting at multiple frequencies [20]–[23] has been considered as a parasitic effect. In this work we investigate collisions between solitons propagating in solid-core PCFs, and report strong energy exchange between the colliding solitons. The third-order dispersion and the Raman effect are significant in our experiments and destroy the integrability of the underlying model, and so we anticipate significant departure from the elastic collisions predicted by the idealized nonlinear Schrodinger (NLS) equation [12]. In the next two sections we proceed with description of our results, finding it logically simpler to include a section dedicated to the discussion of previous research just before the summary.

## 2. Experimental results

The fiber used in our experiment is the same as that used in Ref. [4] and similar to the one from Ref. [3]. It is a PCF with a small core (around 0.8mm) and a high air fraction in the cladding. The measured group index (speed of light c divided by the group velocity 1/*β*
^{(1)}: *n*_{g}
=*cβ*
^{(1)}) and group velocity dispersion (*D*=*∂β*
^{(1)}/*∂λ*) of the fundamental mode along the two orthogonal fiber polarisation axes are shown in Fig. 1(a). The fiber exhibits two wavelengths (560nm and 1210nm) at which the group velocity dispersion (GVD) is zero, with anomalous (*D*>0) GVD in between. To avoid any polarization instabilities during soliton propagation [4], all our experiments were performed using the slow axis of the fibre.

To study soliton collisions in our fibre, we use the fact that *intra*pulse Raman scattering red-shifts the soliton into a spectral region of increased group index (see Fig. 1(a)), with greater Raman shifts therefore giving slower propagation. As a result, a higher-energy soliton travels at a slower average speed than a soliton with smaller energy. Through proper choice of pulse delay and appropriate adjustment of differential soliton energy we can thus cause soliton collisions in a fiber of given length.

Our experimental setup is shown in Fig. 1(b). Attenuated 200fs pulses from a mode-locked Titanium-Sapphire laser were rotated by 45 degrees and split equally into the two arms of a polarising interferometer. Each arm of the interferometer contained a quarter wave plate to attenuate the output pulse power independently. One arm of the interferometer is equipped with a movable mirror to adjust the initial delay between the two pulses. At the output of the interferometer, the polarisations of the pulses were aligned using a polarizer and then rotated using a half wave plate to align with the slow axis of the PCF. The fiber length used in our experiments was about 2 meters.

In our first set of experiments we used two pulses with slightly different energies, and scanned the delay between them. We studied the fiber output spectrally using an optical spectrum analyzer. The pulse energy coupled into the fiber from the fixed arm was set at 19pJ and the pulse energy coupled from the movable arm was set to 15pJ. At the start of the experiment the pulse with the smaller energy was delayed in the interferometer by 10ps. Then we decreased the delay through zero using the movable mirror. The transmitted spectrum was recorded for each value of delay and the spectral dependence on delay is shown in Fig. 2(a). It is apparent that when the delay between the two pulses is greater than 6ps there is no interaction between the pulses and transmission spectra are independent of the delay. In this limit, the fiber length was not sufficient for the delay to be overcome through the soliton velocity difference. When the delay is smaller than about 6.5ps, the solitons collide inside the fiber, just before the exit end. As the delay is further decreased this collision point will shift back along the fiber toward the input end. After the collision the differential self-frequency shift increases. This is interpreted as due to transfer of energy between the colliding pulses. The longer-wavelength pulse gains some energy from the shorter-wavelength pulse and becomes shorter in time. As a result, this soliton experiences a more rapid self-frequency shift after the collision. As the delay is decreased and the collision point occurs closer to the input end, the solitons propagate further after the collision resulting in greater final spectral separation. The slower-moving (longer wavelength) soliton is also seen to broaden spectrally. This can be attributed to two factors. First, the energy exchange leads shortening of the stronger soliton and second, the decreasing GVD shortens it further. When the delay is reduced further (less than 0.5ps), the spectrum gets messy due to the strong interference between the input pump pulses. When the delay turns negative, i.e. the pulse with smaller energy is at the front, no soliton interaction was observed.

In a second set of experiments, we fixed the delay between the two pulses at 1.5ps and the energy of the second pulse at 18pJ. We increased the energy of the front pulse from 18pJ to 37pJ in 18 steps and record the transmitted spectrum at each step. The result is shown in Fig. 2(b). As expected, no evidence of collisions is observed when the two pulses have similar energies, and they appear on top of one another in the spectrum. As the energy of the first pulse increases the collision point occurs initially at the fiber output end. Again, there is an interpulse transfer of energy in the collision, with the self-shift of the faster soliton being reduced and that of the slower soliton increased. For front pulse energies greater than 34 pJ the frequency of the slower more energetic soliton stabilizes before the zero-GVD wavelength is reached. At this moment the soliton starts to emit red-detuned Cherenkov radiation into the normal-dispersion region close to 1.3nm [3]. Emission of radiation in this case can be considered as induced by the collision, because without it the radiation would appear only for larger propagation distances.

## 3. Theoretical results

To model and understand the observed soliton dynamics we use the generalised nonlinear Schrodinger (NLS) equations, see, e.g., [4, 12],

Here *γ* is the parameter measuring the strength of the Kerr nonlinearity, *θ*=0.18 characterizes the relative strength of the Kerr and Raman nonlinearities and $R\left(t\right)=\left({\tau}_{1}^{2}+{\tau}_{2}^{2}\right)\u2044\left({\tau}_{1}{\tau}_{2}^{2}\right){e}^{-t\u2044{\tau}_{2}}\mathrm{sin}(t\u2044{\tau}_{1})$ is the Raman response function [12]: τ_{1}=12.2fs, τ_{2}=32fs, ${\int}_{0}^{\infty}$
*R*(*t*)*dt*=1. [2*πτ*
_{1}]^{-1}≃13THz corresponds to the maximum of the Raman gain, see Fig. 3(a). *D*(*i∂*_{t}
)=*β*
^{(2)}/2![*i∂*_{t}
]^{2}+*β*
^{(3)}/3![*i∂*_{t}
]^{3}+… is the dispersion operator matching the entire dispersion profile in the relevant range of frequencies. Eq. (1) is renowned for reliably reproducing experiments on short pulse propagation in PCFs.

First we present some analytical results confirming the existence of energy exchange between the interacting solitons induced by the Raman effect. We assume that the envelope function *A* can be presented as a superposition of the two pulses having different carrier frequencies:

*A*
_{1,2} are assumed to be slowly varying relative to the oscillations with frequency *δ*=*δ*
_{1}-*δ*
_{2}. Fixing *δ*
_{1}>*δ*
_{2}, i.e. *δ*>0, implies that the pulse 2 is red shifted with respect to the pulse 1 and the group velocities of the pulses are related as 1/${\beta}_{1}^{\left(1\right)}$>1/${\beta}_{2}^{\left(1\right)}$. In handling the Raman term we use the fact that *R*(*t*) is localized in time over such an interval that the integral can be evaluated using the expansion |*A*
_{1,2}(*t*-*t′*)|^{2}=|*A*(*t*)|^{2}-*t′∂*_{t}
|*A*
_{1,2}(*t*)|^{2}+…. Thus we find for the nonlinear part of Eq. (1)

$${A}_{1}{e}^{-i{\delta}_{1}t}\left(i\gamma {\mid {A}_{1}\mid}^{2}+i2\gamma \left(1-\theta \right){\mid {A}_{2}\mid}^{2}-f\left(\delta \right){\mid {A}_{2}\mid}^{2}+i\gamma \left(1-\theta \right){A}_{1}{A}_{2}^{*}{e}^{-i\delta t}\right)+$$

$${A}_{2}{e}^{-i{\delta}_{2}t}(i\gamma {\mid {A}_{2}\mid}^{2}+i2\gamma \left(1-\theta \right){\mid {A}_{1}\mid}^{2}-{f}^{*}\left(\delta \right){\mid {A}_{1}\mid}^{2}+i\gamma \left(1-\theta \right){A}_{1}^{*}{A}_{2}{e}^{i\delta t})+\dots $$

Here the complex function *f*(*δ*) is

Now we discuss in detail what has been accounted for and what has been neglected in Eq. (3). The first terms in the both brackets are the usual self- and cross-phase modulations. These terms are slowly varying in time and the coupling between the fields induced by the cross-phase modulation does not depend on the relative phase of the two pulses. The third term proportional to *f*(*δ*) is the dominant part of the *inter*pulse Raman scattering, which is also independent of the relative phase of the two fields. *Ref*(*δ*) is the Raman gain, i.e. it controls transfer of photons from the blue detuned field *A*
_{1} to the red detuned *A*
_{2}, and *Imf*(*δ*) contributes to the cross-phase modulation. The last terms in the both brackets express the Kerr-mediated coupling, which depends on the relative phase of the two fields and contains fast oscillations in time. The phase sensitive and phase insensitive parts of the nonlinear coupling are usually referred to as coherent and incoherent interactions, respectively. What has been omitted from Eq. (3)? The first group of the neglected terms, are all the coherent, i.e. fast oscillating, terms in the Raman part of the nonlinearity. These terms appear to be smallest in their order of magnitude after the splitting into the separate equations in *A*
_{1} and *A*
_{2} is performed. The primary aim of our analytical analysis is to estimate the rate of the energy transfer from one pulse to another over the relatively small propagation length. Therefore the terms responsible for the red shift of the soliton frequencies have also been disregarded. Obviously, the generic Raman term in (1) used in our numerical modeling takes full account of all the intra- and inter-pulse Raman processes.

Under the assumption that the remaining strongly oscillating (~*e*
^{±iδt}) coherent terms in Eq. (3) can be neglected and separating the exponents with different frequencies we derive the incoherently coupled NLS equations:

$$\left[{\partial}_{z}+{\beta}_{2}^{\left(1\right)}{\partial}_{t}+\frac{i}{2}{\beta}_{2}^{\left(2\right)}{\partial}_{t}^{2}\right]{A}_{2}=i\gamma \left({\mid {A}_{2}\mid}^{2}+2\left(1-\theta \right){\mid {A}_{1}\mid}^{2}\right){A}_{2}-{f}^{*}(\delta ){\mid {A}_{1}\mid}^{2}{A}_{2}.$$

Here 1/${\beta}_{1,2}^{\left(1\right)}$ are the local group velocities and ${\beta}_{1,2}^{\left(2\right)}$ are the local GVD coefficients of the two pulses. The interpulse Raman term in Eqs. (5) does not violate the conservation of the total energy in the system. However it affects evolution of the energy difference of the two fields:

From the point of view of the full model (1), the above equation (6) is valid for incoherent interactions and over small propagation intervals.

Assuming that the two pulses are well separated so that nonlinear coupling via Kerr and Raman nonlinearities are negligible we can find two soliton solutions ${A}_{n}=\sqrt{2{q}_{n}\u2044\gamma}{e}^{i{q}_{n}z}\mathit{sech}(\sqrt{2{q}_{n}\u2044\mid {\beta}_{n}^{\left(2\right)}\mid}\left[t-z{\beta}_{n}^{\left(1\right)}-{x}_{n}\right]){e}^{i{\psi}_{n}}$, where *n*=1, 2, *q*_{n}
are the shifts of the soliton wavenumbers, ψ
_{n}
are the constant phases and *x*
_{1}–*x*
_{2} is the initial time delay between the pulses. When the distance between the pulses in time starts to decrease via the mechanisms described in section 2, the rate of the power transfer given by (6) grows exponentially in *z*. This is because the exponential tails of the solitons move towards one another, resulting in rapid growth of the overlap integral in (6).

The energy transfer rate in the right-hand side of Eq. (6) is parameterized by the frequency detuning *δ* between the two solitons, which enters the Raman gain coefficient *Re*[*f*(*δ*)], see Fig. 3(a). Reproducing the conditions of our experiments in our numerical model shows that the collisions occur near to the maximum of the Raman gain, ensuring efficient energy exchange between the pulses. For example, the numerical results shown in Figs. 3(b,c) correspond to the initial pulse delay 5ps and the initial energies of the colliding pulses as in Fig. 2(a). At the moment of collision (*z*≃1.63m), see Figs. 3(b,c), the spectral separation of the pulses can be estimate at 12THz, which is practically at the maximum of the Raman gain (compare Figs. 3(a) and (b)).

An extensive series of numerical calculations corresponding to our experimental conditions, together with the above analytical estimate, unambiguously indicate that the efficient energy exchange between the colliding solitons can be primarily attributed to interpulse Raman scattering. The impact of the Raman effect could actually be two-fold. First, the energy transfer via the Raman channel, and second, the enabling of the energy transfer via the coherent Kerr terms, i.e. essentially via the Kerr four-wave mixing of the solitons, which is disabled in the integrable idealization. Numerical modeling of the soliton collisions with the Raman effect switched off (*θ*=0) excludes higher-order dispersions as a possible trigger for the energy transfer between the pulses. Direct quantitative comparison of the experimental results with numerical modeling remains a challenging problem. One of the reasons for this is that despite appreciable frequency detunings between the colliding solitons our numerical results are still sensitive to the relative phases of the interacting pulses. Figs. 4(a,b) are the results of our numerical modeling corresponding to the experimental measurements in Fig. 2. The intensity and energy of the pulses have been calculated using the nonlinear parameter of the fiber *γ*=0.1/W/m. One can see good qualitative and reasonable quantitative agreement. The agreement gets worse when the initial conditions are changing in such a way that the frequency detuning between the pulses at the collision point decreases. Most strikingly this can be seen by comparing the parts of the Figs. 2(a) and 4(a) corresponding to the small initial delays. Figs. 4(a,b) have been obtained for the zero initial phase difference of the interacting pulses. Extensive series of numerical computations have unambiguously confirmed that the irregularities of the spectral trajectories in Figs. 4(a,b) are due to the phase sensitivity of the energy partitioning between the emerging solitons.

Experimentally (Fig. 2), we do not observe the strongly oscillating spectral traces shown in Fig. 4. However, one can see traces of oscillations of the spectral maxima of the solitons in the bottom half of the Fig. 2(a), for approximately the same delays for which the numerical spectra in Fig. 4(a) oscillate strongly. Likewise, small effects can be observed if one compares the observed (Fig. 2(b)) and modeled (Fig. 4(b)) bending of the soliton spectral trace as function of the pulse energy. Interpreting these discrepancies one should take into account that, while the modeling in Figs. 4(a,b) reflects results of a single collision event, the spectral measurements are averaged over the collisions of the pulse trains, which have fluctuating amplitudes. (The acquisition of a single experimental spectrum takes many seconds, while the series used to produce one of Fig. 2A, B would take the best part of an hour). Though the phase difference of the pulses emerging from the interferometer is not changing from pulse to pulse, even small amplitude fluctuations of the pulses entering the fiber change (via the intrapulse Raman process) the phase difference of the colliding solitons significantly. Therefore the resulting energy division fluctuates from one collision event to another, which translates on the fluctuations of the soliton frequencies at the fiber end. We have not reproduced this effect in detail because of substantial computational demands and no reliable information about amplitude fluctuations. Instead, we have chosen an equivalent and computationally less demanding approach of averaging over the phase difference. Figs. 4(c,d) show the numerically obtained spectral traces averaged over the initial phase difference varying from 0 to 2*π*. Loosely speaking, the experimentally measured spectra fit somewhere in between the averaged and non-averaged calculations. Another notable difference is that in the modeling plots the Cherenkov radiation near 1.3*µ*m is stronger and appears for smaller input powers than in the experiment. This can be attributed to the spectrally dependent loss not accounted in the modeling.

The balance equation (6) is insensitive to the phases of the interacting pulses and therefore does not describe the coherent energy exchange. Formally, separation of Eq. (1) into the coupled equations for *A*
_{1,2} accounting for the coherent interaction is possible only under the assumption that the overlap of the two pulses is weak [25], i.e. the time delay is large relative to the pulse width. The strongly incoherent model, see Eqs. (5), remains valid for arbitrary overlap. The equations including the coherent terms oscillating with frequency δ and neglecting the oscillations with higher frequencies are

$$\left[i2\gamma \left(1-\theta \right)-f\left(\delta \right)\right]{\mid {A}_{2}\mid}^{2}{A}_{1}+i\gamma {A}_{1}^{2}{A}_{2}^{*}{e}^{-i\delta t}+i2\gamma \left(1-\theta \right){A}_{2}{\mid {A}_{1}\mid}^{2}{e}^{i\delta t}+\dots $$

$$\left[i2\gamma \left(1-\theta \right)+{f}^{*}\left(\delta \right)\right]{\mid {A}_{1}\mid}^{2}{A}_{2}+i\gamma {A}_{2}^{2}{A}_{1}^{*}{e}^{i\delta t}+i2\gamma \left(1-\theta \right){A}_{1}{{e}^{-i\delta t}\mid {A}_{2}\mid}^{2}+\dots $$

The rate of the energy transfer between the two pulses obtained from Eqs. (7),(8) is

$$2{\int}_{-\infty}^{\infty}\mid {A}_{1}\mid \mid {A}_{2}\mid \left(Re\left[f\left(\delta \right)\right]\mid {A}_{1}\mid \mid {A}_{2}\mid +2\gamma \left(1-\theta \right)\left({\mid {A}_{1}\mid}^{2}+{\mid {A}_{2}\mid}^{2}\right)\mathrm{sin}\left[\delta t+{\varphi}_{2}-{\varphi}_{1}\right]\right)dt,$$

where *ϕ*
_{1,2} are the phases of the two pulses: ${A}_{\mathrm{1,2}}=\mid {A}_{\mathrm{1,2}}\mid {e}^{i{\varphi}_{\mathrm{1,2}}}$. Thus in addition to the inter-pulse Raman scattering there exists the phase sensitive correction controlling the energy exchange. If *T* measures the width of the interacting pulses, then the boundary between the coherent (the sin-term in Eq. (9) is not averaged to zero) and incoherent (the sin-term is averaged to zero) regimes is estimated from *Tδ*≃2*π*. Fig. 6 shows the dependencies of the coherent (strongly oscillating black line) and incoherent (smooth blue line) parts of the integral in the right-hand side of Eq. (9) on the frequency detuning between the solitons, *δ*/(2*π*). The chosen soliton parameters are typical of those used in our experiments (duration - 100fs, energy - 16pJ) and the pulse separation is 400fs. The smooth line in Fig. 6 corresponds to the incoherent interaction and it reproduces the Raman gain line. We have found that the incoherent interpulse Raman scattering dominates the process of the energy transfer for *Tδ* close and above 2*π*, while for *Tδ* below 2*π* the coherent Kerr mechanisms are dominant. In physical units this means that for and above the detunings ≃10THz at the collision moment the interaction is practically incoherent, while for smaller detunings the coherent effects are substantial, see Fig. 6. This finding agrees with the spectral dynamics in Figs. 2 and 4.

## 4. Discussion

Soliton interactions in telecommunications fibers has been the subject of extensive theoretical and substantial experimental efforts, see, e.g. [12]–[23]. An important difference between our study and the original experiments on the soliton interactions in telecom fibers [13, 14] is that we have studied strong collisions, i.e., collisions when one of the solitons hits the other one at speed. Refs. [13, 14] studied the initially identical solitons placed one next to the other, when the interaction was entirely due to interference effects. This demonstrated the crucial role played by the phase difference, with the in-phase solitons attracting one another and the out-of-phase repelling. Thus [13, 14] focused on the strongly coherent regime, while our work spans a broader range of parameters covering coherent, incoherent and mixed cases of the soliton interaction. No spectral measurements have been reported in [13], while [14] has reported shifts of the soliton frequencies, which have been interpreted as due to power exchange. The spectral separation of the pulses measured in [14] varied from 0.5 to 3.5THz, which is far from the spectral maximum for energy exchange and therefore should result from relatively low power transfer. Also these values have been achieved after long propagation distances of 10km. We, however, routinely observed strong energy exchange with the resulting detunings between the emerging solitons up to 50THz only after 2m of the fiber length, see Fig. 2. Such a strong effect implies attention could be directed towards switching applications of PCF solitons. Since the soliton amplitude is inversely proportional to its width our results also prompt the idea that successive or even a single soliton collision in PCFs can be considered as a method for pulse compression. A review of theoretical studies of the coherent collisions of the fiber soliton, can be found in [12]. We explicitly mention here only few of the numerous papers, which emphasize the role of the Raman effect [15, 16, 17, 18].

Incoherent soliton collisions in fibers have been particularly actively studied in the context of wavelength division multiplexing, see, e.g. [19]–[23] and [22] for a review. Many of these studies either ignore the Raman effect [19]–[22], or consider it as parasitic and focus on the problems very different from the ones discussed here [23]. We also note that a widely applied approach has been to replace the Raman term in Eq. (1) with its approximation ~*A∂*_{t}
|*A*|^{2}, see, e.g. [15, 18, 23]. This is often an adequate approximation for the intrapulse Raman scattering, but it is clearly not satisfactory for description of the interpulse interaction, because it gives an unbounded linear increase in the Raman gain, see Fig. 3(a).

## 5. Summary

We have investigated collisions of femtosecond solitons with different group velocities propagating in photonic crystal fibers and demonstrated good qualitative and reasonable quantitative agreement between experimental and numerical results. The collisions have been found to result in strong energy exchange between the solitons and associated strong frequency shifts. The energy exchange is mediated by the phase-insensitive interpulse Raman scattering and by the phase-sensitive contribution from the Kerr nonlinearity, while the frequency shifts are due to intrapulse Raman scattering. Our studies call for more theoretical and experimental investigations of the complex interplay between the coherent (phase sensitive) and incoherent (phase in-sensitive) mechanisms of the energy exchange and point in the direction of potential use of highly nonlinear PCFs for switching applications with solitons.

## Acknowledgement

This work has been supported by the Leverhulme Trust.

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