# Correlation of Intensity Fluctuations for Scattering of a Partially Coherent Plane-Wave Pulse
## 2 Correlation of Intensity Fluctuations
Figure 2 shows that the scattering becomes more directional when \(\sigma\) increases. And Figure 3 shows that the NCIF increases with the increase of \(\sigma\). Next, let us suppose that the pulse is incident on a quasi-homogeneous random medium. The correlation function of the scattering potential of such a medium has the form:  
\[C_{F}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\omega_{0})=C_{R}(\mathbf{R}^{\prime}, \omega_{0})C_{\mathbf{r}}(\mathbf{r}^{\prime},\omega_{0})=C_{0}\exp\Biggl{[}-\frac{(\bm {R}^{\prime})^{2}}{2\sigma_{R}^{2}}-\frac{(\mathbf{r}^{\prime})^{2}}{2\sigma_{\mathbf{ r}}^{2}}\Biggr{]}, \tag{18}\]  
where \(C_{0}\), \(\sigma_{R}\), and \(\sigma_{r}\) are positive constants with \(\sigma_{R}\gg\sigma_{r}\). The function \(C_{R}(\mathbf{R}^{\prime},\omega_{0})\) varies much more slowly with \(\mathbf{R}^{\prime}=(\mathbf{r}_{1}^{\prime}+\mathbf{r}_{2}^{\prime})/2\) than the function \(C_{r}(\mathbf{r}^{\prime},\omega_{0})\) varies with \(\mathbf{r}^{\prime}=\mathbf{r}_{2}^{\prime}-\mathbf{r}_{1}^{\prime}\). It follows from Equations (7), (10), (13), (14), and (18) that the average intensity and the NCIF of the scattered field are given by the expressions:  
\[ \begin{array}{l}\langle I^{(s)}(\mathbf{r},\mathbf{t})\rangle\\ =\frac{(2\pi\sigma_{R}\sigma_{r})^{3}C_{0}\Gamma_{0}T}{\sqrt{(T^{2}+8\sigma_{R }^{2}\sin^{2}\frac{\theta}{2}/c^{2})(1+2\sigma_{r}^{2}\sin^{2}\frac{\theta}{2} (1/T^{2}+2/T_{c}^{2})/c^{2})}}\\ \times\exp\Biggl{[}-\frac{(t-\tau/c)^{2}}{(T^{2}+8\sigma_{R}^{2}\sin^{2}\frac{ \theta}{2}/c^{2})^{2}}\\ +\frac{\omega_{0}^{2}}{(1/T^{2}+2/T_{c}^{2})(1+2(1/T^{2}+2/T_{c}^{2})\sigma_{ r}^{2}\sin^{2}\frac{\theta}{2}/c^{2})}\Biggr{]},\end{array}  \tag{19}\]  
Figure 3: The normalized correlation of intensity fluctuations (NCIF) of the scattered partially coherent Gaussian Schell-model plane-wave pulse versus \(t_{d}=t_{2}-t_{1}\) for different values of \(\sigma\) of the deterministic spherical scatterer with \(\theta=0.5\). Figure 2: The normalized intensity of the scattered partially coherent Gaussian Schell-model plane-wave pulse versus \(\theta\) for different values of \(\sigma\) of the deterministic spherical scatterer. \[ \begin{array}{l}\frac{\langle\Delta I^{(5)}(rs,t_{1})\Delta I^{(5)}(rs,t_{2}) \rangle}{\langle I^{(5)}(rs,t_{1})\rangle\langle I^{(5)}(rs,t_{2})\rangle}\\ \\ =\exp\left[-\frac{(t_{1}-t_{2})^{2}}{\frac{(t^{2}+8t_{2}^{2}\sin^{2}\frac{\pi}{2 }/c^{2})^{1/(1/(17^{2}+2/T_{F}^{2})+2\sigma_{F}^{2}\sin^{2}\frac{\pi}{2}/c^{2} )}}{(1^{4}/(T_{F}^{2}+2T^{2})+(4\sigma_{F}^{2}-\sigma_{F}^{2})\sin^{2}\frac{ \pi}{2}/c^{2})}}\right].\end{array}  \tag{20}\]  
It follows from the analytical expressions Equations (19) and (20) that for a quasi-homogeneous random medium, one can also derive \(\sigma_{R}\) and \(\sigma_{r}\) from both the average intensity and the NCIF of the scattered field. Figures 4 and 5 illustrate the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave for different values of \(\sigma_{R}\) and \(\sigma_{r}\) of the quasi-homogeneous random medium separately. Other calculation parameters are the same as those in Figures 2 and 3. It is shown that \(\sigma_{R}\) and \(\sigma_{r}\) of the quasi-homogeneous random medium determine both the average intensity and the NCIF together; however, the effect of \(\sigma_{r}\) is more apparent than that of \(\sigma_{R}\). Figure 4 shows that the scattering becomes more directional when \(\sigma_{R}\) or \(\sigma_{r}\) increases. Figure 5 shows that the NCIF increases as \(\sigma_{r}\) increases or \(\sigma_{R}\) decreases. In applications, the troposphere and confined plasmas sometimes can be modeled as quasi-homogeneous random media. Figure 4: The normalized intensity of the scattered partially coherent Gaussian Schell-model plane-wave pulse versus \(\theta\) for different values of \(\sigma_{R}\) and \(\sigma_{r}\) of the quasi-homogeneous random medium.