Fig. 1. (Colour online) Schematic illustration of a super-Gaussian beam passing through a nonlinear medium.

Fig. 2. (Colour online) The normalized peak intensity of a super-Gaussian beam against the propagation distance for different bandwidths. $n_{0}(\lambda _{0})=1.812$, $n_{2}(\lambda _{0})=2.28\times 10^{ - 6}\ {\rm cm}^{2}/{\rm GW}$, $L=6\ {\rm cm}$, the super-Gaussian beam $I_{0}=12\ {\rm GW}/{\rm cm}^{2}$, $N=4$, $w_{0}=3\ {\rm mm}$.

Fig. 3. (Colour online) Evolution of the normalized peak intensity of a broadband super-Gaussian beam passing through different thicknesses of nonlinear medium along the propagation direction. (a) The incident intensity is fixed, $I_{0} =12\ {\rm GW}/{\rm cm}^{2}$, (b) The $B$ integral is fixed. $I_{0} =36\ {\rm GW}/{\rm cm}^{2}$ when $L=2\ {\rm cm}$; $I_{0} =24\ {\rm GW}/{\rm cm}^{2}$ when $L=3\ {\rm cm}$; $ I_{0} =12\ {\rm GW}/{\rm cm}^{2}$ when $L=6\ {\rm cm}$. $N=4$, $w_{0}=3\ {\rm mm}$, and $\Delta \lambda =40\ {\rm nm}$.

Fig. 4. Intensity distribution of a narrowband beam, $\Delta \lambda =0\ {\rm nm}$ (a) and a broadband beam, $\Delta \lambda =40\ {\rm nm}$ (b) at different distances. $n_{0}(\lambda _{0})=1.812$, $n_{2}(\lambda _{0})=2.28\times 10^{ - 6}\ {\rm cm}^{2}/{\rm GW}$, $ L=6\ {\rm cm}$, $I_{0} =12\ {\rm GW}/{\rm cm}^{2}$, $N=20$, and $w_{0}=3\ {\rm mm}$.

Fig. 5. (Colour online) Evolution of the normalized peak intensity of a super-Gaussian beam for different bandwidths, considering the medium’s front surface to have a defect. The defect size $a=100\ \mathrm {\mu} {\rm m}$, $A=1$, $x_{0}=0$; $n_{0}(\lambda _{0})=1.812$, $n_{2}(\lambda _{0})=2.28\times 10^{- 6}\ {\rm cm}^{2}/{\rm GW}$, $ L=6\ {\rm cm}$, $I_{0}=12\ {\rm GW}/{\rm cm}^{2}$, $N=4$, and $w_{0}=3\ {\rm mm}$.

Fig. 6. The propagation of a narrowband beam, $\Delta \lambda =0\ {\rm nm}$ (a) and a broadband beam, $\Delta \lambda =40\ {\rm nm}$ (b) through a nonlinear medium whose front surface contains a defect. Here, the parameters are the same as in Figure 5.

Fig. 7. The propagation of a narrowband beam (a) and a broadband beam (b) when the defect is not in the centre, $x_{0}=0.001\ {\rm m}$, the other parameters are the same as in Figure 6.