# Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium
## 2 Theory
On substituting Equations (4) and (7) into (10), by interchanging the orders of the integrals, we can derive the formula:  
\[\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2},z)=\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}p_{R}(v_{1})p_{T}(v_{2})H^{*}(\mathbf{r}_{1},t_{1},v_{1},v_{2},z) H(\mathbf{r}_{2},t_{2},v_{1},v_{2},z)dv_{1}dv_{2} \tag{11}\]  
where  
\[H^{*}(\mathbf{r}_{1},t_{1},v_{1},v_{2},z)H(\mathbf{r}_{2},t_{2},v_{1},v_{2},z)=H_{R}^ {*}(\mathbf{r}_{1},v_{1},z)H_{R}(\mathbf{r}_{2},v_{1},z)H_{T}^{*}(t_{1},v_{2},z)H_{T} (t_{2},v_{2},z) \tag{12}\]  
and  
\[ \begin{array}{ll}H_{R}^{*}(\mathbf{r}_{1},v_{1},z)H_{R}(\mathbf{r}_{2},v_{1},z)&= \left(\frac{k}{2\pi i2}\right)^{2}\!\!\int\int_{-\infty}^{\infty}H_{R}^{*}(\bm {\rho}_{1},v_{1})H_{R}(\mathbf{\rho}_{2},v_{1})\\ &\times\exp\biggl{[}-ik\frac{(\mathbf{r}_{1}-\mathbf{\rho}_{1})^{2}-(\mathbf{r}_{2}-\mathbf{ \rho}_{2})^{2}}{2z}\biggr{]}d^{2}\rho_{1}d^{2}\rho_{2},\end{array}  \tag{13}\]  
and  
\[ \begin{array}{ll}H_{T}^{*}(t_{1},v_{2},z)H_{T}(t_{2},v_{2},z)&=\frac{\omega_ {0}}{2\pi i2}\int\int_{-\infty}^{\infty}H_{T}^{*}(t_{10},v_{2})H_{T}(t_{20},v_ {2})\\ &\times\exp\biggl{[}-i\omega_{0}\frac{(t_{1}-t_{10})^{2}-(t_{2}-t_{20})^{2}}{ 2z}\biggr{]}dt_{10}dt_{20}.\end{array}  \tag{14}\]  
By inserting Equations (8) and (9) into (13) and (14), respectively, after a tedious integral calculation, one can obtain:  
\[\begin{array}{ll}H_{R}^{*}(\mathbf{r}_{1},v_{1},z)H_{R}(\mathbf{r}_{2},v_{1},z)&= \frac{w_{0}^{2}}{w^{2}(z)}\exp\biggl{[}\frac{ik}{2z}\!\left(\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\[T^{2}(z)=T_{0}^{2}(1-4\pi v_{2}\beta_{2}z)^{2}+\left(\frac{\beta_{2}z}{T_{0}} \right)^{2} \tag{18}\]  
If we let \(\mathbf{r}_{1}=\mathbf{r}_{2}=\mathbf{r}\) and \(t_{1}=t_{2}=t\) in Equations (15) and (16), respectively, we can obtain:  
\[\left|H_{R}(\mathbf{r},v_{1},z)\right|^{2}=\frac{w_{0}^{2}}{w^{2}(z)}\exp\left[- \frac{\mathbf{r}^{2}}{w^{2}(z)}\right] \tag{19}\]  
\[\left|H_{T}(t,v_{2},z)\right|^{2}=\frac{T_{0}}{T(z)}\exp\left[-\frac{t^{2}}{T^ {2}(z)}\right]. \tag{20}\]  
According to Equation (11), we can get spatial-temporal intensity as:  
\[I(\mathbf{r},t;\mathbf{r},t,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p_{R}(v_ {1})p_{T}(v_{2})\left|H_{R}(\mathbf{r},v_{1},z)\right|^{2}\left|H_{T}(t,v_{2},z) \right|^{2}dv_{1}dv_{2} \tag{21}\]  
Equation (21) is the main formula derived in this paper, which is used to investigate the spatial-temporal intensity evolution of the PCP beams in dispersive medium. If we choose the Gaussian weight function to replace Equations (5) and (6), respectively, i.e.,  
\[{p^{\prime}}_{R}(v_{1})=\frac{\sigma^{2}}{\sqrt{\pi}}\exp\left(-\sigma^{4}v_{ 1}^{2}\right) \tag{22}\]  
\[{p^{\prime}}_{T}(v_{2})=\frac{T_{c}^{2}}{\sqrt{\pi}}\exp\left(-T_{c}^{4}v_{2} ^{2}\right) \tag{23}\]  
and considering Equations (4), (7)-(9), we can obtain:  
\[\Gamma_{0}^{\prime}(\mathbf{\rho}_{1},t_{10};\mathbf{\rho}_{2},t_{20})=R^{\prime}(\bm {\rho}_{1},\mathbf{\rho}_{2})T^{\prime}(t_{10},t_{20}) \tag{24}\]  
where  
\[R^{\prime}\left(\mathbf{\rho}_{1},\,\mathbf{\rho}_{2}\right)=\exp\left(-\frac{\rho_{ 1}^{2}+\rho_{2}^{2}}{2w_{0}^{2}}\right)\exp\left[-\frac{\left(\rho_{2}^{2}- \rho_{1}^{2}\right)^{2}}{\sigma_{c}^{4}}\right] \tag{25}\]  
\[T^{\prime}(t_{10},t_{20})\!=\exp\left(-\frac{t_{10}^{2}+t_{20}^{2}}{2T_{0}^{2} }\right)\exp\left[-\frac{\left(t_{20}^{2}-t_{10}^{2}\right)^{2}}{T_{c}^{4}} \right]\exp\left[-i\omega_{0}\left(t_{20}-t_{10}\right)\right] \tag{26}\]  
where \(\sigma_{c}=\sigma/\sqrt{\pi}\) and \(T_{c}^{\prime}=T_{c}/\sqrt{\pi}\). As can be seen from Equations (25) and (26), the spatial part and temporal parts of HCFs have non-uniform coherence distribution. And spatial and temporal parts of MCFs are the same as those in Ref. [64] and Ref. [56], respectively.