# Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium
## 3 Spatial-Temporal Self-Focusing of PCP Beams in Dispersive Medium
### Water Case
With increasing \(z\), the pulses focus first on the temporal dimension, then on the spatial dimension. However, at some critical distance, the pulses focus on temporal dimension and spatial dimension simultaneously (see Figure 5c \(z=0.2\) km and Figure 6c \(z=0.04\) km). Figure 5: Contour graph of normalized spatial-temporal intensity distribution \(I(x,\,t,\,z)\) of PCP beams with rect weight function at \(z\) plane with different values of \(z\): (**a**) \(z=0\), (**b**) \(z=0.1\) km, (**c**) \(z=0.2\) km and (**d**) \(z=0.4\) km in the \(x\)-\(t\) plane. Figure 6: Contour graph of normalized spatial-temporal intensity distribution \(I(x,\,t,\,z)\) of PCP beams with Gaussian weight function at \(z\) plane with different values of \(z\): (**a**) \(z=0\), (**b**) \(z=0.03\) km, (**c**) \(z=0.04\) km and (**d**) \(z=0.2\) km in the \(x\)-\(t\) plane. The spatial-temporal self-focusing behavior shown in Figures 5 and 6 can be interpreted as follows. There is some kind of relationship between spatial parameters and temporal parameters, which can be easily seen from the beam spatial width in Equation (17) and pulse duration in Equation (18) of elementary modes of PCP beams; this is because Equations (17) and (18) imply that there are minima for the beam width and pulse duration, respectively. Hence, there are intensity maximum during propagation, when beam widths or pulse durations arrive minima at the same time. More specifically, Equations (17) and (18) can be expanded and re-written as:  
\[w^{2}(z)=a_{1}z^{2}+b_{1}z+c_{1} \tag{27}\]  
\[T^{2}(z)=a_{2}z^{2}+b_{2}z+c_{2} \tag{28}\]  
\[a_{1}=\frac{16\pi v_{1}^{2}w_{0}^{4}+1}{k^{2}w_{0}^{2}},\;b_{1}=-\frac{8\pi v_ {1}w_{0}^{2}}{k},\;c_{1}=w_{0}^{2} \tag{29}\]  
\[a_{2}=\frac{(16\pi v_{2}^{2}T_{0}^{4}+1)\beta_{2}^{2}}{T_{0}^{2}},\;b_{1}=-8 \pi v_{2}T_{0}^{2}\beta_{2},\;c_{1}=T_{0}^{2} \tag{30}\]  
From Equations (27) and (28) we see that \(w^{2}\)(z) and \(T^{2}\)(z) are quadratic functions of \(z\), and will reach minima when \(z_{1}=-b_{1}\)/\(2a_{1}\) and \(z_{2}=-b_{2}\)/\(2a_{2}\), respectively, i.e., the intensity maximum appears at distances:  
Figure 2:0  
Figure 2:1  
where \(z_{1,\min}\) depends on the \(v_{1}\) and incident beam width \(w_{0}\). \(z_{1,\min}\) has an internal relationship with spatial coherence length \(\sigma\) by the rect weight function \(p_{\text{R}}(v_{1})\) shown in Equation (5), while \(z_{2,\min}\) depends on the \(v_{2}\) and the pulse duration. And, \(z_{2,\min}\) is in connection with temporal coherence length \(T_{\text{c}}\) by the rect weight function \(p_{\text{T}}(v_{2})\) shown in Equation (6). Hence, when the self-focusing takes place at spatial and temporal dimensions simultaneously, the relationship between spatial parameters and temporal parameters can be obtained as follows:  
Figure 2:2  
After a comparison between rect weight function and Gaussian weight function, it was seen that the intensity profiles of PCP beams with rect weight function are similar to the case of Gaussian weight function. However, specifically, the spatial-temporal self-focusing effect of the latter is more noticeable and the focal spot is much small--the value of peak intensity can reach 17.4 for the Gaussian weight function case. Nevertheless, the value is 9.3 for the rect weight function. In addition, the positions of focal spots where self-focusing in the spatial and temporal dimensions takes place simultaneously are different from each other. The positions of focal spot are \(z=0.2\) km and \(z=0.04\) km for the rect weight function and Gaussian weight function, respectively. That is to say, the spatial-temporal self-focusing effect of the latter takes place more early.