## Abstract

The complex mechanism of multiple interactions between solitary and dispersive waves at the advanced stage of supercontinuum generation in photonic crystal fiber is studied in experiment and numerical simulations. Injection of high power negatively chirped pulses near zero dispersion frequency results in an effective soliton fission process with multiple interactions between red shifted Raman solitons and dispersive waves. These interactions may result in relative acceleration of solitons with further collisions between them of quasi-elastic or quasi-plastic kinds. In the spectral domain these processes result in enhancement of certain wavelength regions within the spectrum or development of a new significant band at the long wavelength side of the spectrum.

©2010 Optical Society of America

## 1. Introduction

Conversion of short laser pulses into light with broad continuous spectra by nonlinear optical processes, known as the supercontinuum (SC) generation, has attracted a lot of scientific attention over the past few decades. Since the first observation in 1970 in bulk glass [1], numerous investigations were conducted in a wide variety of nonlinear media, including solids, liquids, gases, and various types of waveguiding structures. The origin of supercontinuum is not due to a specific single phenomenon but rather to a plethora of nonlinear effects, such as self phase modulation (SPM), high order soliton fission, modulation instability, stimulated Raman scattering (SRS), self-steepening (SS), and four wave mixing. The invention of photonic crystal fibers (PCF) [2], with their enhanced nonlinearity, together with the zero dispersion wavelengths located within the generation range of Ti: Sapphire femtosecond lasers, further boosted experimental and theoretical studies [3–5] of supercontinuum generation. The widest and most homogeneous SC spectrum is obtained when the pump pulses are launched close to the zero-dispersion wavelength (ZDW) and in the anomalous regime [3]. The effects of input pulse parameters such as energy, peak power, duration, and central wavelength on the SC generation have been investigated thoroughly [3]. There were also numerous studies dedicated to control of supercontinuum spectrum by means of initial pulse prechirping [6–8]. In the spectral region of long wavelengths where the dispersion is anomalous, solitons play a major role in the supercontinuum formation process [4]. They influence pulse spectral broadening at the initial stage of supercontinuum formation via a fission process [3,9], and at later stages via their interaction with dispersive waves by means of Kerr and Raman nonlinearities [4,10,11]. The aim of the present work is to demonstrate multiple interactions both between solitons and dispersive waves and between adjacent solitons at the advanced stage of supercontinuum formation. These interactions result in relative accelerations between the solitons, resulting in various types of soliton collisions. Effective management of the above processes by variation of input parameters results in modifications of supercontinuum spectral profile and further spectral extension into the long wavelength region.

## 2. Experimental setup

A Ti:sapphire laser, generating 30-fs pulses at 790 nm, was used in the experiment. The spatial profile of the output laser beam was Gaussian with M^{2} close to 1. The beam could experience multiple reflection within the NGVD mirror-pair introducing negative dispersion of about –40 fs^{2}/bounce (Layertec GmbH). The laser radiation was focused by a microscope objective (40X, NA = 0.65) into the photonic crystal fiber (PCF) of 38 cm length, with the diameter of the central rod of 2.5 μm and the zero dispersion wavelength of *λ _{D}* = 790 nm. The throughput of the PCF was about 60%; this was used to calculate peak power inside the PCF. The highly divergent output radiation from the PCF was collimated by an aspheric condensing lens with 4.6 mm focal length (NA = 0.55). The input pulse duration and phase were measured by SPIDER (APE Berlin). More details to the experimental setup can be found in [8]. The excess of positive dispersion from the focusing optics can be compensated by introducing 24 reflections from the NGVD mirrors. By adding some reflections on the NGVD mirror pair the pulse duration could be stretched up to 60 fs with negative chirp. The spectra of the radiation after propagation in the PCF were recorded for different values of the input pulse chirp as a function of input pulse power.

## 3. Prechirped pulse propagation through PCF and supercontinuum generation

In order to study various physical phenomena involved in the experimentally observed results we use a generalized scalar nonlinear Schroedinger equation to model the pulse propagation inside the fiber [3]

*A*propagates along the fiber with longitudinal coordinate z. Time in a reference frame traveling with the pump light is represented by

*Τ*.

*β*

_{m}is the

*m*th order dispersion coefficient at the central frequency

*ω*

_{0},

*α*is the fiber loss. The nonlinear coefficient is

*γ = n*

_{2}

*ω*

_{0}*/(cA*

_{eff}

*)*, where

*n*

_{2}= 2.4∙10

^{−20}m

^{2}/W is the nonlinear refractive index of fused-silica glass, and

*A*

_{eff}is the effective modal area of the fiber. The response function is given by

*R(Τ) = (1-f*

_{R}

*)δ(Τ) + f*

_{R}

*h*

_{R}

*(Τ)*with first and second terms standing for instantaneous and delayed Raman contributions, respectively.

*f*

_{R}= 0.18 is the fraction of Raman contribution to nonlinear polarization, and

*h*

_{R}(

*Τ*) = (

*τ*

_{1}

^{2}+

*τ*

_{2}

^{2})/(

*τ*

_{1}

*τ*

_{2}

^{2})exp(-

*Τ*/

*τ*

_{2})sin(

*Τ*/

*τ*

_{1}) approximates the Raman response function of silica fiber [12] with

*τ*

_{1}= 12.2 fs and

*τ*

_{2}= 32 fs. Equation (1) was solved numerically by using the split-step Fourier method [13]. We consider a typical PCF of core diameter 2.5 μm with zero dispersion wavelength (

*λ*

_{D}) = 790 nm. At a wavelength of 800 nm, where the input pulse is centered, the nonlinear coefficient is estimated to be

*γ*= 80 W

^{−1}km

^{−1}, and the dispersion coefficients up to seventh order are:

*β*

_{2}= −2.1 fs

^{2}/mm,

*β*

_{3}= 69.83 fs

^{3}/mm,

*β*

_{4}= −73.25 fs

^{4}/mm,

*β*

_{5}= 191.95 fs

^{5}/mm,

*β*

_{6}= −727.13 fs

^{6}/mm,

*β*

_{7}= 1549.4 fs

^{7}/mm. Since only a short length of fiber is considered in the simulations, the fiber loss is neglected (

*α*= 0). The input pulses are approximated by

*P*

_{0}is the peak power,

*T*

_{0}is related to the FWHM by

*T*

_{FWHM}≈1.763

*T*

_{0}, and

*C*= − 1.25 is the parameter representing the initial linear frequency chirp as it appears in the experiment. A comparison between experimental data (solid curve) and numerical simulations (dashed curve) for the input peak powers of 2, 10, 50, and 100 kW is presented in Fig. 1 . Although the calculated spectra differ slightly from the experimental ones, a good consistency can be recognized in the spectral width and shape. Figure 1 (b) demonstrates the spectral bandwidth measured at 20 dB below maximum in the experiment and calculated by the numerical modeling as a function of increasing input peak power. The phenomena we wish to address here do not occur for input peak power below 40 kW. Therefore we show the numerically obtained temporal and spectral evolutions of 60 fs (FWHM) input pulses at 50 kW and 100 kW in Fig. 2 and Fig. 3 . The PCF length was 38 cm as in the experiment.

After the initial compression stage of a few cm, the input pulse, as it corresponds to a high order soliton, undergoes fission due to the higher-order dispersive and nonlinear effects, and the pulse breaks up into multiple fundamental solitons [9] accompanied by emission of non-soliton radiation [4,10]. Spectra of these solitons are shifted toward longer wavelengths by Raman induced frequency shift [14] (RIF) producing spectral broadening at the red edge of the spectrum. Comparison of Figs. (2, 3) reveals, unsurprisingly, that the higher the pulse power, the stronger the RIF. Also, for more powerful pulses the initial soliton order increases, more radiation is launched into the anomalous GVD region, and broader SC is produced.

We now concentrate on details of the interaction between solitons and radiation. Figure 4 (a)
shows a zoomed region from Fig. 2 (*P*
_{0}
*=* 50 kW), while Fig. 4(b) corresponds to the case of Fig. 3 (*P*
_{0}
*=* 100 kW). Preceding solitons emit radiation which hits trailing solitons; some of the radiation is absorbed, and the trailing soliton is decelerated. One example of this phenomenon can be observed in Fig. 4(a) after propagation of 240 mm. The soliton undergoes multiple interactions with incident dispersive waves and starts to obtain acceleration. As the result of these is the power in the wavelength region around 1070 nm is noticeably enhanced (see the position highlighted by an arrow in Fig. 2(b) and in Fig. 1(a)). An even more interesting scenario happens when more power (100 kW) is launched Fig. 4(b). The emitted dispersive wave interacts with the next Raman-induced soliton and bends the trailing soliton's trajectory so that both solitons approach each other [4]. At the maximum distance shown, which is the entire fiber length, the solitons are about to collide and experience strong energy exchange with the generation of new frequencies [11]. There occurs additional quasi-elastic collision between the tails of the two solitons with ensuing fission. These effects are presented at the long wavelength side of the spectrum extending the spectrum from 480 nm up to 1420 nm measured at 20 dB below maximum in Fig. 3 (b) and Fig. 1 (a).

To prove that indeed the emitted radiation is responsible, we performed an additional simulation very similar to the one of Fig. 3, but with one difference: We artificially eliminated the radiation in the temporal window between the two solitons. Figure 5(a) demonstrates that the trajectory of the second soliton is indeed modified: it tends to continue moving along the initial trajectory. This is the situation recently described in [15]. To make the difference clearer, we compare in Fig. 5(b) the pulse structure at the fiber end for both the realistic case and the hypothetical case of radiation being absent. Obviously radiation here accounts for a temporal shift of 0.6 ps. Thus we prove that it is indeed the radiation emitted from the first soliton that changes the trajectory of the trailing soliton and enables the generation of new frequencies.

Since it is expected that the dynamics of interactions is strongly dependent on phase matching conditions [4,16], additional simulations were performed to determine the influence of the chirp parameter and higher-order dispersion of the PCF. While the chirp was varied, the total pulse power was kept constant. The interaction of Raman solitons with dispersive waves is extremely sensitive to the initial chirp parameter. We observed in numerical calculations the relative acceleration between the solitons in case of the input pulse of 100kW when the value of chirp parameter *C* was varied between *C =* −1.15 and *C =* −1.32. The process of interaction of solitons with dispersive waves and the resultant collision between them changed considerably within even this small range from a quasi-elastic to a quasi-plastic mode of the collision. Figure 6 (a,b)
demonstrates an example of a plastic collision at *C =* −1.16 when the colliding solitons combine into one giant solitary wave which keeps propagating at large group velocity. In the process a strong spectral band is generated, which extends the SC spectrum beyond 1500 nm.

When the chirp parameter was taken outside the range specified above, no relative acceleration was observed as demonstrated in Fig. 7 (a) . Another important parameter of the system is the dispersion profile of the PCF [4]. We performed an additional simulation excluding higher than 3rd order dispersion and observed that the solitons repelled each other rather than attracted; see Fig. 7 (b).

## 4. Conclusion

We have studied experimentally and numerically the dynamics of negatively prechirped pulses with different input peak powers in PCF with resultant spectral broadening and supercontinuum generation. For higher input powers we observed multiple interactions of Raman-induced red-shifted solitons and dispersive waves. We show that the radiation can be absorbed by trailing solitons, altering their path. This can lead to additional soliton quasi-elastic or quasi-plastic collisions and to an enhancement of the power in certain spectral regions of the generated supercontinuum or, even more interestingly, in the generation of new frequency components. Thus the spectrum of the supercontinuum can be managed and further extended towards longer wavelengths by designing fiber and input chirp parameters.

We acknowledge partial funding by Deutsche Forschungsgemeinschaft.

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