## 2 Parameter Optimization for Spectral Broadening  
SPM modifies the spectral content of a laser pulse. The result of this modification is influenced by several experimental factors such as the pulse intensity, the wavelength and the nonlinear material properties. When an intense laser pulse propagates through a medium it induces a dynamic refractive index change \(n(t)=n_{0}+n_{2}I(t)\) due to the third-order Kerr nonlinearity \(n_{2}\). This leads to self-focusing, self-steepening and SPM. In the case of SPM, \(n_{2}\) causes a change of the phase of the electric field in the pulse itself. For an input pulse with spatial and temporal intensity profile \(I(r,t)\), this nonlinear phase variation accumulates over the propagation direction \(z\) in a medium with thickness \(l\) according to the \(B\)_integral_:  
\[B\big{(}\ r,t\big{)}=\frac{\omega_{0}}{c}\int_{0}^{l}\ n_{2}\cdot I\big{(}\ z,r,t\big{)}dz \tag{1}\]  
Here \(\omega_{0}\) is the initial circular frequency, \(c\) the speed of light and \(r\) parameterizes the spatial intensity distribution perpendicular to the propagation direction. The accumulated phase shift results in a corresponding frequency shift and thus a broadening or narrowing of the pulse spectrum. A quantification of the maximum frequency shift due to SPM is given by  
\[\Delta\omega_{\textrm{max}}\ =\omega(t)-\omega_{0}\ =-\bigg{(}\ \frac{\partial}{ \partial t}\ B(t)\ \bigg{)}_{\textrm{max}} \tag{2}\]  
where the maximum frequency shift \(\Delta\omega_{\textrm{max}}\) is typically located at the maximum slope of the leading and trailing edge of the pulse intensity. From Equations (1) and (2) one can conclude that high intensities and a long propagation distance in a material with a large positive \(n_{2}\) is favorable to maximize spectral broadening through SPM. However, in bulk materials the large dispersion experienced by a UV pulse has to be taken into account. This leads to a reduction of the pulse intensity and thus the accumulated phase. A further limiting factor is the nonlinear process of self-focusing. As often mentioned in the literature, if the self-focus lies within the crystal length, severe beam distortions, beam breakup [14] or ultimately filamentation occurs [10,18]. To illustrate the influence of the material's dispersion on SPM we calculated the \(B\)_integral_ (Equation (1)) for different thicknesses of a CaF\({}_{2}\) crystal ( \(n_{2}\)\(=1.92\times 10^{-16}\) cm\({}^{2}\)/W [19]). The results for a 25 fs long UV pulse centered at 308 nm with a pulse energy of 3\(\mu\)J are shown in Figure 1. One should note, that for a given crystal thickness we optimize the beam focusing to a minimal beam waist radius \(w\) for which the self-focal length is 1.25 times the crystal length (see orange line in Figure 1). This puts the self-focus outside the crystal and ensures that beam distortions do not occur. The self-focal length for a pulse with a power \(P\) focused to a beam waist \(w\) can be calculated according to Equation (3). \[z_{SF}\left(\ w\right)=\frac{0.183\ \rho_{0}\left(\ w\right)}{\sqrt{\left( \sqrt{P/P_{Cr}\left(\ n_{2}\ \right)}-0.852\ \right)^{2}-0.0219}} \tag{3}\]  
with \(\rho_{0}\) being the Rayleigh length and \(P_{cr}\) the critical self-focusing power threshold [20,21]. In the simulations, the change in the peak intensity due to self-focusing and self-steepening is neglected. The spatial intensity distribution is assumed to be Gaussian and the input pulses are Fourier limited. For the calculation of the _B integral_ we integrate over the whole beam profile. The B integral does not vary with the crystal length when no material dispersion is included in the simulation (Figure 1, dotted line) as the self-focal length is kept 1.25 times the crystal length for the different crystal thicknesses. As soon as 2 nd order dispersion is considered (for a formula see, e.g., [20]), the _B integral_ starts dropping steadily when the crystal thickness exceeds 1 mm and converges to zero for long crystals (Figure 1, dashed line).