## 1 Introduction
These two discoveries have motivated much of the recent PCS research because high directionality and speckle/scintillation resistance are ideal source characteristics for directed energy, free-space optical communications (FSOC), medical, and numerous other applications. Because of their many potential uses, the literature is replete with techniques to synthesize sources with controllable spatial coherence. A survey of the literature reveals two general approaches for generating PCBs. The first starts with a spatially coherent source, commonly a laser, and \"spoils\" the coherence by randomizing the wavefront. This is accomplished using spatial light modulators (SLMs) or rotating ground glass diffusers (GGDs) [5; 14; 15]. The second approach starts with a spatially incoherent source, an LED for instance, and exploits the Van Cittert-Zernike theorem (VCZT) or uses spatial-frequency filters (typically referred to as just spatial filters) to synthesize the desired source [16; 17; 18; 19; 20]. Hybrid approaches have also been developed [11; 12; 21]. The \"coherent\" and \"incoherent\" PCS synthesis methods discussed above have pros and cons that make one more suitable for a specific application than the other. FSOC is a good example and will be the focus of the remainder of this paper because of the author's research interests. In FSOC, the source's wavefront cycling rate, i.e., the rate at which the source produces statistically independent wavefronts, must be at least 10-100 times larger than the communication modulation frequency. This is so the detector integrates many independent intensity realizations per digital bit to reduce scintillation caused by atmospheric turbulence between the transmitter and receiver. Considering that FSOC data rates are gigabits to potentially terabits per second and SLM frame rates and GGD rotation rates are optimistically 10 kHz, spoiling the coherence of a laser source is not a realistic option. On the other hand, the wavefront cycling rate of a spatially incoherent source (e.g., an LED, thermal source, et cetera) is determined by the physics of how the source generates light (for the aforementioned sources, the physical mechanism is spontaneous emission) and is ultimately related to the source's bandwidth [6]. The bandwidths of such sources applicable to FSOC vary from a couple THz to 10 s of THz. Even for sources at the low end of this range, the wavefront cycles so quickly that 100 gigabit per second data rates are possible. Because of this, the incoherent option is the only viable approach for FSOC applications. Of the two incoherent techniques, i.e., VCZT and spatial filtering, the latter has many advantages including simplicity and compactness. Surprisingly, despite these advantages and its common use in optical Fourier processing [22], the spatial filtering of a partially coherent source (PCS) has been discussed in relatively few papers [19; 20; 23; 24]. These references, after presenting the requisite Fourier and statistical optics theory, generally focus on specific PCSs, e.g., Gaussian Schell model [1; 5] or incoherent sources. Simple, physical, \"engineering\" expressions for the PCS at the filter output, or directions for designing/choosing a filter to produce a desired beam are not presented. The goal of this paper is to derive these expressions and demonstrate their use. In the next section, starting with a stochastic source field, the field at the output of a general spatial filter is derived. Taking the autocorrelation of the output field and assuming that the source field can be modeled as a quasi-homogeneous source [1; 6] yields closed-form, approximate expressions for the output SD and SDoC. Inverting the SDoC expression produces a function for the filter in terms of the desired output SDoC. Lastly, Section 3 validates these theoretical expressions and demonstrates how to use them to control coherence and beam shape via Monte Carlo wave-optics simulations.