#### 1. Introduction

An extensive study of femtosecond laser filamentation is up to date stimulated by the numerous applications arising from the highly nonlinear nature of this process. Combined action of the Kerr nonlinearity and plasma defocusing in air allows laser pulse propagation up to kilometers distances giving an opportunity for measuring atmospheric trace species [1]. Extended spatial confinement of femtosecond laser beam and high peak intensity inside the filament can be beneficial for the development of nitrogen-based remote lasers [2]. A complex interplay between self-action processes (self-phase modulation, self-steepening, etc.) and plasma generation results in multi-octave spectral broadening [3]. Laser filamentation in anomalous dispersive media is considered as a powerful way to produce few-cycle pulses in near and mid-IR [4], whereas two-color filamentation is used for generation of THz radiation [5].

Seeding optical parametric devices with supercontinuum (SC) is widely used technique since its greatly reduce amplification threshold and improve energy stability. For this purpose white-light generation stage based on pump pulse spectral broadening in transparent dielectrics is usually applied. YAG crystal is the material of choice for SC generation because its high nonlinear refractive index (compared to sapphire, quartz, etc.) and high optical damage threshold. The SC generation with femtosecond near-IR laser pulses in this crystal was widely studied experimentally and theoretically [6]. However, in several early published papers [7] appearance of the distinct peaks in SC spectrum was observed. The origin of these peaks stays unclear and requires detailed numerical study.

In this paper, we analyze numerical model, which describes pulse propagation under filamentation of femtosecond laser pulses centered at 1240 nm (that corresponds to Cr:Forsterite laser system). We report numerical investigation of the Raman nonlinearity influence on spatio-temporal pulse dynamics. Our results show that this nonlinear term leads to asymmetrical pulse profile: energy shifting toward the leading edge.

#### 2. Numerical simulation

Theoretical model, which describes SC generation process in YAG crystal, is relied on paraxial approximation, according to which wave vector of incident wave is directed along with pulse propagation axis, and also on slowly varying envelope approximation (SVEA), according to which magnitude of electrical field

where

The first term in the right part of (1) describes diffraction in medium, the second one – the second order of dispersion, the third one – the Kerr and Raman nonlinearities (2), the forth one – multiphoton absorption (MPA) (3), the fifth one – nonlinearity induced by plasma generation (4).

where

Kerr and Raman contribution into nonlinearity caused by two factors: “instantaneous” electron response (the Kerr nonlinearity) and slow exiting vibration molecules levels with decrement

The rate equation for electron density includes multiphoton ionization, avalanche ionization and electron recombination and can be expressed as:

where

For numerical calculations we use Gaussian in space (FWHM equals

#### 3. Results and discussion

Firstly, we investigated the influence of incident pulse energy on propagation process. For this purpose, we calculated the distribution of radiation intensity for the different laser pulse energy at the propagation axis. When pulse power exceeds critical power P

Comparison of intensity distribution for Raman nonlinearity level

In Fig.2 there are depicted radiation intensity distributions at the axis of propagation in the dependence on z-coordinate and retarded time t for the various Raman nonlinearity levels. Since pulse propagates through media with normal group velocity dispersion, initially Gauss-shape pulse splits into two subpulses, the leading one propagates with higher and the trailing one – with lower group velocity in comparison with that is for central wavelength. In the absence of the Raman nonlinearity there is deformation of the trailing edge (Fig.2a): it becomes steeper, that is caused by plasma generation. Moreover, when pulse energy is high enough, the radiation intensity of the trailing subpulse becomes so high, that the secondary temporal splitting occurs. If the Raman nonlinearity is taken into account, there is asymmetry in time-domain: the leading subpulse has higher intensity and propagates at longer distance in comparison with the trailing subpulse. In Fig.2b and 2c the first focusing point even has not well-marked trailing edge. It is important to notice qualitative difference between effects induced by plasma generation and the Raman nonlinearity. The first one also contributes to asymmetrical temporal pulse profile, but energy at the trailing edge is absorbed, whereas the second one leads to energy shifting toward the leading edge. Such pattern can be seen by comparison of intensity distribution in Fig.2a and 2c. Another feature consists in pulse center lad in time domain that is clearly seen in Fig.2c. Notice that as follow from Fig.1 when θ = 0.2, 0.4, electron density is less by 1-2 orders of magnitude than that for θ = 0, i.e. there is only the Raman nonlinearity influence on intensity distribution in Fig.2b and 2c.

In Fig.3 spatio-temporal pulse profiles (STPP) at the distances 0.5, 1 и 1.5 cm from the front side of crystal are depicted. Fig 3 (a-c) show the STPP in the absence of Raman nonlinearity. In Fig.3a (corresponds to the beginning of the first pulse splitting event) after passing the subpulse radiation the radial distribution of the laser pulse intensity becomes more complicated, due to self-focusing of the pulse the central time slices suffers from the decreasing of the velocity in comparison with the peripheral ones. During further pulse propagation, new splitting events occur and additional subpulses are formed. Fig.3 (d-f) show the STPP correspond to θ = 0.4, distances from the front side of crystal are also 0.5, 1 и 1.5 cm. For the case of z = 0.5 cm (Fig.3d), pulse intensity increased but there is no pulse splitting. At z = 1 cm (Fig. 3e) pulse splitting has occurred and the central time slices are defocused together with the pulse tail. At z = 1.5 cm (Fig. 3 f) pulse profile considerably becomes more complicated due to additional nonlinear focus and splitting events. But steep leading edge and sloping trailing edge are well noticeable that proves the earlier conclusion about the Raman nonlinearity influence.

#### 4. Conclusion

In conclusion, we have studied nonlinear propagation of femtosecond near-infrared laser pulses in YAG crystal and have revealed the Raman nonlinearity influence on spatio-temporal pulse dynamics. In normal dispersive regime splitting of initial laser pulse into two subpulses occurs. We have shown that the Raman nonlinearity leads to asymmetrical temporal pulse profile: energy shifting toward the leading edge and emerging out of the pulse center. As a result, the number of nonlinear foci along the z-axis increases. Analyzing the dynamics of the radial intensity distributions we have revealed the complicated nature over the propagation coordinate. In the origin of such behavior the prevailing factor is the plasma defocusing of the time slices of the laser pulse. The tandem action of the plasma and the Raman nonlinearities can bring some benefits to the radial pulse profile control.

This work was supported by the Russian Scientific Fund (Grant no. 17-72-20130).