## 1 Introduction  
Following the nomenclature and notation of [PERSON] [1], a partially coherent source (PCS) is defined by its autocorrelation function, better known as the mutual coherence function (MCF):  
\[\Gamma\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2};t_{1},t_{2}\right)=\langle U\left(\bm {\rho}_{1},t_{1}\right)U^{*}\left(\mathbf{\rho}_{2},t_{2}\right)\rangle, \tag{1}\]  
where \(U\left(\mathbf{\rho},t\right)\) is an instance of a random optical field evaluated at position \(\mathbf{\rho}=\hat{\pi}x+\hat{y}y\) and time \(t\), \(*\) is the complex conjugate, and \(\langle\rangle\) is the average over the ensemble of \(U\) realizations. The first and second moments of many optical sources vary slowly with time such that they can be considered static. This characteristic is referred to as wide-sense stationary and implies that the MCF defined in Equation (1) depends temporally only on the time difference \(\tau=t_{1}-t_{2}\). The temporal Fourier transform of the MCF is the cross-spectral density (CSD) function:  
\[W\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2};\omega\right)=\frac{1}{2\pi}\int_{-\infty}^ {\infty}\Gamma\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2};\tau\right)\exp\left(\mathrm{ j}\omega\tau\right)\mathrm{d}\tau, \tag{2}\]  
where \(\omega\) is the radian frequency. Letting \(\mathbf{\rho}_{1}=\mathbf{\rho}_{2}=\mathbf{\rho}\) yields the spectral density (SD), namely,  
\[S\left(\mathbf{\rho};\omega\right)=W\left(\mathbf{\rho},\mathbf{\rho};\omega\right), \tag{3}\]which is the optical intensity at frequency \(\omega\). Normalizing the CSD produces the spectral degree of coherence (SDoC), i.e.,  
\[\mu\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2};\omega\right)=\frac{W\left(\mathbf{\rho}_{1}, \mathbf{\rho}_{2};\omega\right)}{\sqrt{S\left(\mathbf{\rho}_{1};\omega\right)S\left(\bm {\rho}_{2};\omega\right)}}, \tag{4}\]  
which is a measure of the spatial coherence of the source at frequency \(\omega\). Working with the CSD, as opposed to the MCF, has two significant advantages. The first is the separation of spatial and temporal coherence. In the MCF, time and space are coupled in such a way that determining whether the phenomenon is caused by temporal or spatial coherence can be difficult. On the other hand, the CSD provides a measure of the spatial coherence of an optical source at a certain frequency \(\omega\). In this way, the CSD is much more physical than the MCF, and this physical utility has led to the prediction and subsequent experimental validation of many counter-intuitive optical phenomena, most notably, coherence-induced spectral changes [1; 2] and coherence-induced polarization changes [3; 4; 5]. The second advantage is analytical efficiency. The expression for the propagation of mutual coherence is a complicated four-dimensional (4D) integral over space, where space and time are coupled [6]. The typical approach for dealing with this is to assume that the source is narrowband and quasi-monochromatic. These assumptions simplify the mutual coherence propagation expression significantly; however, they restrict the applicability of the results. On the other hand, the expression for the propagation of CSD--in its most general, paraxial form--is a 4D Fresnel transform [1; 5]. In many practical scenarios, the 4D Fresnel transform simplifies to a 4D Fourier transform. The results obtained using these propagation integrals are accurate regardless of the source's bandwidth. [PERSON]'s CSD formalism has led to the understanding and prediction of many optical phenomena. Among these, [PERSON] and others showed that a spatially partially coherent beam (PCB) can be highly directional similar to a laser [1; 7; 8; 9]. Other research soon followed, showing that the reduced spatial coherence of PCBs makes them less susceptible to scintillation or speckle [5; 10; 11; 12; 13].