Model

The main challenge in the simulation of the Kerr-lens mode-locked lasers is a formulation of the model, which would adequately describe the laser dynamics and at the same time would not be too complicated to remain physically meaningful. The soliton approach, which fulfills the latter requirement, does not take into account the strong nonlinear behavior of the real-world femtosecond solid-state lasers. Therefore we make use of the relatively simple, but sufficiently general numerical model, first presented in Ref. [34]. The split-step scheme for the simulations of the slowly varying field amplitude $a(t)$ evolution can be represented as:

$$a_{III}(t)=$$ (3)

$$a_{out}(t)=$$ (5)

where the different steps within the full cavity round-trip describe: I - the self-phase modulation action (Eq. ()), where $\vert a\vert^{2}$ means the field intensity, $\beta =2\pi n_{2}x/\lambda n$ (here $n_{2}$ and $n$ are the nonlinear and linear refraction indices, respectively; $x$ is the double length of the active medium; $\lambda$ is the central oscillation wavelength); II - the Kerr-lens induced fast saturable absorber action defining the self-amplitude modulation of the field, $\gamma$ is the modulation depth, $\sigma$ is the inverse intensity of the loss saturation (Eq. (); note, that consideration of the strong modulation is necessary in our case owing to the low critical self-focusing power in ZnSe, which is only $\sim$0.2 MW [26] as a result of the large value of $n_2$); III - the spectral filter action with the inverse bandwidth $t_{f}$, which coincides with the gain bandwidth, $\omega_0$ is the field carrier frequency coinciding with the minimum of the spectral loss (Eq. ()); IV - the homogeneously saturated gain and the output loss action with coefficients $\alpha$ and $\rho$, respectively (Eq. ()); and, finally, the net-GDD action with the dispersion coefficient , where $\phi(\omega)$ is the linear phase retardation of the field (Eq. ()). The full round-trip results from setting $a_{in}(t)$ = $a_{out}(t)$. In practice, when the field change over the round-trip is small (this is valid in our case), the scheme represents the distributed dynamical equation of Ginzburg-Landau type in the presence of an arbitrary strong loss saturation:

$$\frac{{\partial a(z,t)}}{{\partial z}} =$$ (6)

$$\times a(z,t),$$

where $z$ is the formal longitudinal coordinate (the cavity round-trip number). The scheme (-) or, equally, Eq. () has to be supplemented with an equation for the gain coefficient evolution, which for the quasi-two level amplification scheme has the following form:

where $I_{p}$ is the pump intensity, $\nu _{a}$ and $\nu$ are the pump and the lasing field frequencies, respectively; $\alpha _{max}$ is the maximal gain for the full population inversion; $\sigma _{a}$ and $\sigma _{g}$ are the absorption and the gain cross-sections, respectively; is the gain relaxation time. When the pulse is much shorter than the cavity round-trip period $T_{cav}$, which is obviously true in our case, and the dynamical gain saturation is negligible (i. e. there is no time dependence of $\alpha$ on the scale of the pulse duration), which is correct for the sub-picosecond solid-state lasers, Eq. () can be easily integrated to eliminate the time dependence:

where $P$ = $\left( {I_{p}\sigma _{a}/h\nu _{a}}\right) \, T_{cav}$ is the dimensionless pump intensity, $E$ is the pulse energy flux, $E_{s}$ = $h\nu /\sigma _{g}$ is the gain saturation energy flux. The integration of Eq. () over $z$ yields [17]:

It is convenient to use dimensionless quantities in the calculations and we shall normalize the time to $t_{f}$, and the field intensities to $\beta ^{-1}$. However, we will keep the dimensional values, where it is necessary for the physical interpretation. Then the pulse energy flux is normalized to $t_f/\beta$, resulting in the expression for the dimensionless inverse saturation energy flux $\epsilon$ = . This parameter plays a crucial role in our model since it describes the contribution of the gain saturation in the lasing dynamics in comparison to the contribution of the self-phase modulation. Other key parameters are $P$, which has the meaning of the pump energy stored during the cavity period (in the units of absorption saturation energy), and the dimensionless parameter $\sigma$, which describes the strength of the self-amplitude modulation relatively to the self-phase modulation. With this normalization, the field intensity $\left\vert {a(t)}\right\vert ^{2}$ has the physical meaning of the dimensionless $\Phi(t)/\Phi_{crit}$ ratio, where $\Phi(t)$ is the instantaneous power in the laser beam and [33] is the critical power of self-focusing. In the present form the model is valid not only for the description of the Kerr-lens mode-locked solid-state lasers but also for other laser systems with the fast saturable absorber, viz., the additive-pulse and self-polarization rotation mode-locked solid-state and fiber lasers. However, we have to note that in such schemes the strong loss saturation can invert into self-darkening, which produces the passive negative feedback and influences the pulse stability [35]. This effect can not be described by the Eq. () and the results of this paper can be applicable to additive-pulse and self-polarization rotation mode-locked systems only as a week nonlinear approximation.

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