## Abstract

Experiments and numerical simulation were performed for verification of the role of femtosecond pulse chirp for supercontinuum generation in photonic crystal fiber. We demonstrate that injection of high power negatively chirped pulses near zero dispersion point brings an advantage over positively chirped pulses resulting in additional collision between solitons and in development of a significantly broader spectrum. Coupling between Raman induced solitons and dispersive waves generated by higher order dispersion was proven to be the key mechanism behind the results.

© 2010 OSA

## 1. Introduction

Supercontinuum generation (SC) manifests itself as the formation of broad continuous spectra by the propagation of laser pulses through the nonlinear media. Since the first observation in 1970 in bulk glass [1], it has been the subject of numerous investigations in a wide variety of nonlinear media, including solids, liquids, gases, and various types of waveguiding structures. The supercontinuum does not originate from a specific single phenomenon but rather from a plethora of nonlinear effects, such as self phase modulation (SPM), high order soliton fission, modulation instability, stimulated Raman scattering (SRS), self-steepening (SS), four wave mixing, which combine to produce an extreme pulse broadening, with the widest and most homogeneous SC spectrum obtained when the pump pulses are launched close to the zero-dispersion wavelength and in the anomalous regime. Therefore the introduction of photonic crystal fibers (PCF) [2], with their elevated nonlinearity, together with the zero dispersion wavelengths located within the generation range of Ti: Sapphire femtosecond lasers, led to a boom in supercontinuum experimental and theoretical studies [3-5]. The effects of input pulse parameters such as pulse energy, peak power, pulse duration, and central wavelength on the SC generation have been investigated thoroughly [3]. A related problem is the influence of input pulse chirp on the SC characteristics, as it can affect the mechanism of soliton fission in a significant way. Corwin et. al. [6] examined the effect of chirp by imposing quadratic spectral phase on the input pulses, while maintaining constant pulse spectral bandwidth. In this case, the pulse energy remains fixed, but increased chirp is associated with increased pulse duration and reduced peak power. As a result it was found that the maximum bandwidth of the generated SC was obtained for the shortest pulses with zero chirp, corresponding to the maximum input peak power. The results obtained with the negatively chirped pulses were found to be similar to the positively chirped case. Other numerical simulations [7,8] demonstrated an increase in the SC bandwidth when the initial frequency chirp grows towards an optimal positive value. However these results were obtained at the relatively low input peak power of only few kW. Recently the propagation of pulses with large positive chirp generated by Ti: Sapphire chirped pulse oscillator was also investigated [9, 10]. In the present work we demonstrate by experimental observation, confirmed by the numerical simulations, that for input pulses with a power of about hundred kW, and with a negative chirp, the SC spectrum tends to be significantly broader than for the pulses with a positive chirp.

## 2. Experimental setup

A Ti:sapphire laser provided 25-fs pulses centered at 790 nm with spectral width of 45 nm FWHM was used in the experiment. The whole experimental setup is shown in Fig. 1 .

The spatial properties of the output laser beam were found to be excellent, demonstrating symmetrical Gaussian profile with M^{2} close to 1. Mirrors introducing negative dispersion of about – 40 fs^{2}/bounce (Layertec GmbH) were used to manage the chirp of the pulses. The mirrors were specially designed as a pair with the aim to compensate the oscillation of GVD and to minimize the introduced third order dispersion (TOD). The beam could experience up to 12 bounces within the single NGVD mirror-pair. To increase the amount of introduced negative dispersion another mirror-pair could be inserted into the beam path. 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 zero dispersion wavelength of λ*D* = 790 nm. The throughput of the PCF exceeds 60%, what was used to calculate peak power insight the PCF. The highly divergent output radiation from the PCF was collimated by an aspheric condensing lens with a 4.6 mm focal length (NA = 0.55). The input pulse duration and phase were measured by Spider (APE Berlin). The original laser pulses showed a positive chirp after passing through the focusing optics with the pulse duration extended from initial 25 fs up to 150 fs. This excess of positive dispersion can be to compensated by introducing 24 reflections from NGVD mirror. By inserting or removing one NGVD mirror-pair the pulse duration could be stretched up to 60 fs. The chirps in these two cases manifest the opposite signs with additionally introduced GVD of about ± 480 fs^{2}. 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. Comparison of the spectra was performed for equal pulse energies at the level of 20 dB, while the peak power decreased as the chirp grew. Surprisingly, we observed a drastic reduction of the spectral bandwidth for pulses with positive chirp compared to unchirped or negatively chirped pulses. At the same time the spectra for the unchirped and negatively chirped pulses demonstrated very similar dynamics showing development of fundamental solitons at much lower input powers compared to the pulses with the positive chirp. From here on we will analyse the pulses with the same duration of 60 fs and peak intensity, but with the opposite chirps, to clarify only the impact from different sign of chirp on SC dynamics.

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

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, τ is the time in a reference frame travelling with the pump light. β

*m*is the

*m*th-order dispersion coefficient at the central frequency ω

_{0}, α is the fiber loss. The nonlinear coefficient is given by

*γ = n*, where

_{2}ω_{0}/(cA_{eff})*n*

_{2}= 2.4*10

^{−20}m

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

*A*is the effective mode area of the fiber. The response function

_{eff}*R(τ) = (1-f*contains both instantaneous and delayed Raman contributions, where

_{R})δ(τ) + f_{R}h_{R}(τ)*fR*= 0.18 is the fraction of Raman contribution to the nonlinear polarization, and

*h*(τ) is the Raman response function of silica fiber, which can be approximated by the expression [11]:

_{R}*hR*(τ) = (τ

_{1}

^{2}+ τ

_{2}

^{2})/(τ

_{1}τ

_{2}

^{2})exp(-τ/τ

_{2})sin(τ/τ

_{1}), with τ

_{1}= 12.2 fs and τ

_{2}= 32 fs. Equation (1) was solved numerically by using the split-step Fourier method [12]. We consider a PCF design which is typical for broadband SC generation near λ = 800 nm. The parameters describe a circular silica rod of diameter of 2.5 μm. The ZDW is λ

*D*= 790 nm. At a wavelength of 800 nm, the nonlinear coefficient is estimated to be γ = 80 W

^{−1}km

^{−1}, and the coefficients of the chromatic dispersion up to seventh order are: β2 = −2.1 fs

^{2}/mm, β3 = 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. The fiber loss is neglected (α = 0) since only a short length of the fiber is considered in the simulations. The input pulses are assumed to have the form:

*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.

Temporal and spectral evolutions of 60-fs (FWHM) input pulses with the opposite chirp signs are displayed in Fig. 2
and Fig. 3
as they propagate through the 38 cm PCF. The simulations were performed for moderate input peak power of 10kW (Fig. 2) and high input peak power of 100 kW (Fig. 3). For the 10 kW input power the dispersion length L_{D} = T_{0}
^{2}/|β_{2}
*|* is about 55 cm, while the nonlinear length L_{NL} = (γP_{0})^{−1}≈1.25 mm is much shorter, thus the pulses launched into the fiber correspond to 21th-order soliton. Through the combination of SPM and SRS on the initial stage of propagation the spectra of the input pulses with opposite chirps are modified differently and distributed in a different way into the frequency ranges with normal and anomalous GVD. For negatively chirped pulses more radiation is boosted into the anomalous GVD region so the initial soliton order increases and then broader SC is produced. After the initial compression stage of few cm, this soliton undergoes fission due to the higher-order dispersive and nonlinear effects, and the pulse breaks up into the multiple fundamental solitons [13] accompanied by emission of non-soliton radiation [4, 14] as shown at Fig. 2(a), 2(c). Phase-matched interaction of solitons with this dispersion waves is another source for generation of new frequency components [4]. Spectra of these solitons are shifted toward longer wavelengths by Raman induced frequency shift [15] (RIF) producing spectral broadening at the red edge of the spectrum in a similar way for the pulses with both chirp signs [Fig. 2 (b), 2(d)]. However, the difference in the pulse dynamics between the positively and the negatively prechirped pulses becomes more significant as more power is launched into the PCF. Figure 3(a), 3(c) displays that the separated and delayed red-shifted Raman solitons interact with radiation and between themselves in a different way for different initial chirp signs. The negatively prechirped pulse undergoes stronger RIF evolving in a smaller number of fundamental Raman solitons, compared with it’s positive counterpart, but these solitons are more intense and exhibit a stronger frequency shift. The negative chirp case provides better coupling between the most delayed soliton and the radiation. The emitted dispersive wave interacts with the next Raman-induced soliton and causes an additional relative deceleration between the two solitons [4] after the propagation of about 300 mm [Fig. 3(c)]. At a distance of about 300 mm [Fig. 3(c)] the solitons from the negatively chirped pulse collide experiencing strong energy exchange with the generation of new frequencies [Fig. 3(d)] [16]. Finally there occurs additional interaction between the two solitons with the following fission and The effects are presented at the long wavelength side of the spectrum in Fig. 3(b, d). The spectral widths at −20 dB level are calculated to be 830 nm for the positive chirp and more than 130 nm broader (up to 960 nm) for the negative chirp.

A comparison between experimental data (solid curve) and numerical simulations (dashed curve) for the input peak powers of 2, 10, 50, 100 kW’s are presented in Fig. 4 .

The experiments reveals an about three times lower threshold for formation of Raman soliton for the pulses with negative chirp (input pulse peak power P_{0} = 1.2 kW) as compared with the positively chirped pulses (input pulse peak power P_{0} = 3.4 kW). We can clearly see stronger broadening for the pulses with negative chirp at 50 and 100 kW input power, compared with the positively chirped pulses with the same input power. Although the calculated spectra differ slightly from the experimental ones, a good consistency can be recognized in the spectrum width and shape. Figure 5(a)
demonstrates the spectral bandwidth measured at 10^{−2} in the experiment and calculated by the numerical modeling as a function of increasing input peak power. From the input peak power exceeding 20 kW the difference in spectral broadening between negatively and positively prechirped pulses starts to manifest itself clearly and grows up to 130 nm for P_{0} = 100 kW.

As already mentioned above, the SC generation is a complicated process originated from the combined action of SPM, SRS, SS and high order dispersion (HOD). According to [7, 10] the positively chirped pulses launched within the anomalous dispersion region undergo initial compression and more rapid spectral broadening than the negatively chirped pulses. In our experiments λZD = 790 nm is located in the middle of the input pulse spectrum, thus the dispersive compression operates in a same way for both chirp signs. The simulations support this assumption as can be seen in Fig. 2(a), 2(c) and Fig. 3(a), 3(c) after 2-3 cm pulse propagation into PCF. To evaluate the contribution from the different phenomena we simulated SC generation by omitting in turn the above effects Fig. 5(b), 5(c). Interplay between Raman-induced solitons and radiation created by HOD [17] was found as the dominant mechanism responsible for the difference in the two chirp signs spectra. We can clearly observe almost identical spectra broadening for both signs of prechirp if Raman term *(f _{R} = 0)* [Fig. 5(b)] or higher order dispersion from 4th order [Fig. 5(c)] are neglected.

## 4. Conclusion

In conclusion, we have studied experimentally and numerically an effect of initial chirp sign of input pulses on pulse dynamics and spectral broadening in PCF. Various input peak power ranging from few kW up to 0.1GW were examined. For the inputs pulse power in a range of 2-10 kW the pulse dynamic is similar for both signs of initial chirp, resulting in the similar bandwidth of output spectrum. However, when the pulses with higher input power injected in PCF, the negative chirp gives rise to stronger red-shifted Raman solitons, colliding with each other and broadening the spectrum in a more efficient way. The effect of pulse prechirping influences of dynamics through the entire propagation length, even on the latest stages of propagation where the highly complicated multiple interactions occur between Raman induced red-shifted solitons and radiation waves.

## Acknowlegements

We gratefully acknowledge Prof. Fedor Mitschke for helpful discussions. Special appreciation is addressed to reviewer 2 for helpful suggestions that led to a clarification of the discussion.

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