# Correlation of Intensity Fluctuations for Scattering of a Partially Coherent Plane-Wave Pulse
## 2 Correlation of Intensity Fluctuations  
We begin with a brief review of correlation of intensity fluctuations in the space-time domain at two space-time points, say (\(\mathbf{r}_{1}\), \(t_{1}\)) and (\(\mathbf{r}_{2}\), \(t_{2}\)) [23; 24]. The intensity fluctuations are defined by the formula:  
\[\langle\Delta I(\mathbf{r}_{j},t_{j})\rangle=I(\mathbf{r}_{j},t_{j})-\langle I(\mathbf{r}_ {j},t_{j})\rangle\ (j=1,2), \tag{1}\]  
where:  
\[I(\mathbf{r}_{j},t_{j})=E^{*}(\mathbf{r}_{j},t_{j})E(\mathbf{r}_{j},t_{j}), \tag{2}\]  
is the instantaneous intensity of the field \(E(\mathbf{r}_{j},t_{j})\). The asterisk denotes the complex conjugate and the angular bracket denotes the ensemble average. Then the correlation of intensity fluctuations at two space-time points (\(\mathbf{r}_{1}\), \(t_{1}\)) and (\(\mathbf{r}_{2}\), \(t_{2}\)) has the form:  
\[\langle\Delta I(\mathbf{r}_{1},t_{1})\Delta I(\mathbf{r}_{2},t_{2})\rangle=\langle I( \mathbf{r}_{1},t_{1})I(\mathbf{r}_{2},t_{2})\rangle-\langle I(\mathbf{r}_{1},t_{1}) \rangle\langle I(\mathbf{r}_{2},t_{2})\rangle. \tag{3}\]  
We note that the first term on the right-hand side of Equation (3) is the fourth-order correlation function. Assuming that the field fluctuations obey Gaussian statistics and the first moment of the field \(E(\mathbf{r}_{j},t_{j})\) is zero, the fourth-order correlation function can be expressed in terms of second-order moments by the Gaussian moment theorem:  
\[ \begin{array}{ll}\langle I(\mathbf{r}_{1},t_{1})I(\mathbf{r}_{2},t_{2})\rangle&= \langle E^{*}(\mathbf{r}_{1},t_{1})E(\mathbf{r}_{1},t_{1})\rangle\langle E^{*}(\mathbf{r} _{2},t_{2})E(\mathbf{r}_{2},t_{2})\rangle\\ &+\langle E^{*}(\mathbf{r}_{1},t_{1})E(\mathbf{r}_{2},t_{2})\rangle\langle E^{*}(\mathbf{r }_{2},t_{2})E(\mathbf{r}_{1},t_{1})\rangle\\ &=\langle I(\mathbf{r}_{1},t_{1})\rangle\langle I(\mathbf{r}_{2},t_{2})\rangle+| \Gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2})|^{2}.\end{array}  \tag{4}\]  
On substituting from Equation (4) into Equation (3) we obtain for the correlation of intensity fluctuations the formula:  
\[\langle\Delta I(\mathbf{r}_{1},t_{1})\Delta I(\mathbf{r}_{2},t_{2})\rangle=|\Gamma( \mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2})|^{2}, \tag{5}\]  
where \(\Gamma(\mathbf{r}_{1},t_{1},\mathbf{r}_{2},t_{2})\) is the mutual coherence function of the field. Its normalized version:  
\[\frac{\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2})}{\sqrt{\Gamma(\mathbf{r}_{1},\mathbf{ r}_{1},t_{1},t_{1})}\sqrt{\Gamma(\mathbf{r}_{2},\mathbf{r}_{2},t_{2},t_{2})}}=\frac{ \Gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2})}{\sqrt{\langle I(\mathbf{r}_{1},t_{1}) \rangle\langle I(\mathbf{r}_{2},t_{2})\rangle}}=\gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2}), \tag{6}\]  
is the complex degree of coherence of the field. From Equations (5) and (6) it follows that:  
\[\frac{\langle\Delta I(\mathbf{r}_{1},t_{1})\Delta I(\mathbf{r}_{2},t_{2})\rangle}{ \langle I(\mathbf{r}_{1},t_{1})\rangle\langle I(\mathbf{r}_{2},t_{2})\rangle}=| \gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2})|^{2}. \tag{7}\]  
The term on the left-hand side of Equation (7) may be defined as the normalized correlation of intensity fluctuations of the field. This formula shows that the normalized correlation of intensity fluctuations is equal to the squared modulus of the degree of coherence at two space-time points (\(\mathbf{r}_{1}\),\(t_{1}\)) and (\(\mathbf{r}_{2}\),\(t_{2}\)). Correlation of Intensity Fluctuations of a Partially Coherent Plane-Wave Pulse Scattered by Deterministic and Random Media  
We began by considering a temporally partially coherent Gaussian Schell-model plane-wave pulse incident upon a linear scatterer, which occupies a finite domain \(D\).