# Correlation of Intensity Fluctuations for Scattering of a Partially Coherent Plane-Wave Pulse
## 2 Correlation of Intensity Fluctuations
\(\theta\) is the scattering angle, and \(D\) is the domain the scatterer occupies. where the angle brackets denote the average value taken over the ensemble of random medium realizations. We then obtain the formula:  
\[ \begin{array}{ll}W^{(s)}(\textit{rs}_{1},\textit{rs}_{2},\omega_{1},\omega_{2} )&=\int\limits_{D}\int\limits_{D}W^{(i)}(\textit{r}_{1}^{\prime},\textit{r}_{2} ^{\prime},\omega_{1},\omega_{2})C_{F}(\textit{r}_{1}^{\prime},\textit{r}_{2}^{ \prime},\omega_{0})\\ &\times\frac{1}{r^{2}}\exp[ir(k_{2}-k_{1})-i(k_{2}\textit{s}_{2}\cdot\textit{ r}_{2}^{\prime}-k_{1}\textit{s}_{1}\cdot\textit{r}_{1}^{\prime})]d^{3}\textit{r}_{1}^{ \prime}d^{3}\textit{r}_{2}^{\prime}.\end{array}  \tag{13}\]  
The mutual coherence function of the scattered field is obtained by taking the inverse Fourier transform of the cross-spectral density function, which gives:  
\[ \begin{array}{ll}\Gamma^{(s)}(\textit{rs}_{1},\textit{rs}_{2},\omega_{1}, \omega_{2})&=\int\int_{-\infty}^{+\infty}W^{(s)}(\textit{rs}_{1},\textit{rs}_ {2},\omega_{1},\omega_{2})\\ &\times\exp[i(\omega_{1}t_{1}-\omega_{2}t_{2})d\omega_{1}d\omega_{2}.\end{array}  \tag{14}\]  
We now illustrate our analysis by two examples. Suppose first that the pulse is incident on a deterministic spherical scatterer centered at the point \(\textit{r}_{c}\) = (0,0,\(d\)), occupying a finite domain \(D\), with a three-dimensional (soft) Gaussian potential:  
\[F(\textit{r},\omega_{0})=C_{0}\exp[-\frac{x^{2}+y^{2}+(z-d)^{2}}{2\sigma^{2}}], \tag{15}\]  
where \(C_{0}\) and \(\sigma\) are positive constants, which are independent of position but may depend upon \(\omega\). Given the properties of the scatterer and those of the incident field, we may use Equations (7), (10), (11), (14), and (15) to give the following expressions for the intensity statistics. The average intensity of the scattered field is:  
\[ \begin{array}{ll}&\langle I^{(s)}(\textit{rs},t)\rangle\\ &=\frac{(2\pi)^{3}C_{0}^{2}\sigma^{6}\Gamma_{0}T}{r^{2}\sqrt{(1+4\sigma^{2} \sin^{2}\frac{\theta}{2}(1/T^{2}+2/T_{c}^{2})/c^{2})(T^{2}+4\sigma^{2}\sin^{2 }\frac{\theta}{2}/c^{2})}}\\ &\times\exp\left[-\frac{\left(t-(2d\sin^{2}\frac{\theta}{2}+r)/c\right)^{2}}{ T^{2}+4\sigma^{2}\sin^{2}\frac{\theta}{2}/c^{2}}-\frac{\omega_{0}^{2}}{\left[c^{2}/ (4\sigma^{2}\sin^{2}\frac{\theta}{2})+(1/T^{2}+2/T_{c}^{2})\right]}\right], \end{array}  \tag{16}\]  
and the normalized correlation of intensity fluctuations (NCIF) has the form:  
\[ \begin{array}{ll}&\frac{\langle\Delta I^{(s)}(\textit{rs},\textit{I}_{1}) \Delta I^{(s)}(\textit{rs},\textit{I}_{2})\rangle}{\langle I^{(s)}(\textit{ rs},\textit{I}_{2})\rangle\langle I^{(s)}(\textit{rs},\textit{I}_{2})\rangle}\\ &=\exp\left[-\frac{(t_{1}-t_{2})^{2}}{\frac{T_{c}^{2}}{T^{2}}\left[1+4\sigma^{ 2}\sin^{2}\frac{\theta}{2}(1/T^{2}+2/T_{c}^{2})/c^{2}\right]\left[T^{2}+4\sigma ^{2}\sin^{2}\frac{\theta}{2}/c^{2}\right]}\right],\end{array}  \tag{17}\]  
where \(\theta\) is the angle between the observation direction and the incident direction, i.e., \(\textit{s}\cdot\textit{s}_{0}=\cos\theta\). Equations (16) and (17) give the analytical expressions for the average intensity and NCIF of the scattered field. The simple relationships make it easy to derive \(\sigma\) of the medium if we get the intensity, or the correlation of intensity fluctuations which can be performed by Hanbury Brown-Twiss measurements. We calculated in Figures 2 and 3 the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave pulses for different values of \(\sigma\) of the deterministic spherical scatterer, respectively. The calculation parameters are \(T=15\) fs, \(T_{c}=10\) fs, \(\lambda=800\) nm, \(\omega_{0}=2.36\) rad/fs and \(t=\frac{r}{c}\). One finds from Figures 2 and 3 that the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave pulse are closely related to \(\sigma\) of the deterministic spherical scatterer, which may be useful in studying the inverse problem.