## 4 Conclusions  
The control of an optical source's SDoC via spatial filtering was demonstrated. Starting with a quasi-homogeneous random field, the optical field at the output of a general spatial filter was derived. Taking moments of the stochastic output field yielded closed-form expressions for the output SD and SDoC. The key relation which resulted from this analysis was a closed-form expression for the filter function \(T\) in terms of the desired SDoC.  
The analytical SD and SDoC equations derived in Section 2.1 were verified by Monte Carlo analysis. First, the \(T\) required to produce a specified SDoC was found using the steps developed and discussed in Section 2.1. Then, simulating a source used in FSOC research (namely, the stochastic field emitted from a MMF excited by a broadband laser source) and a simple spatial filter containing \(T\), the output SD and SDoC were computed from many independent realizations of the source field. Lastly, the simulated output SD and SDoC were compared to the theoretical predictions. The agreement between the two was generally good with some discrepancies between the simulated and theoretical SDs. The reason for these differences was discussed.  
In addition, a beam shaping simulation was also performed to demonstrate another use of coherence control via spatial filtering. Using the analytical relations derived in Section 2.1, an approximate expression for the desired far-zone SD, in terms of the filter function \(T\), was derived. This equation was used to find the \(T\) required to synthesize a beam with a complex shape. The simulated results were in excellent agreement with the derived, analytical SDs.  
The spatial filtering technique presented in this paper has many advantages over other PCS synthesis methods. First and foremost among them is speed. As discussed in the Introduction, the wavefront cycling rate for this method can be 10s of THz, which is \(10^{9}\) times faster than SLM- or GGD-based approaches. When compared against other incoherent synthesis techniques (e.g., VCZT),  
Figure 4: Simulation results demonstrating beam shaping using a spatial filter: (**a**) \(S^{\text{out,thy}}\) obtained from Equation (16); (**b**) \(S^{\text{out,sim}}\); (**c**) absolute value of the difference of (**a,b**); (**d**) \(S^{\text{far,thy}}\) obtained from Equation (22); (**e**) \(S^{\text{far,sim}}\), and (**f**) absolute value of the difference of (**d,e**).  
spatial filtering is simpler to implement. As demonstrated in Section 3, changing the output field SDoC requires only a filter change. These pros make PCS spatial filtering useful in any application where coherence control is advantageous. These include, but are not limited to, FSOC, directed energy, medicine, manufacturing, and remote sensing.  
Supplementary materials can be found at [[http://www.mdpi.com/2076-3417/8/9/1465/s1](http://www.mdpi.com/2076-3417/8/9/1465/s1)]([http://www.mdpi.com/2076-3417/8/9/1465/s1](http://www.mdpi.com/2076-3417/8/9/1465/s1)).  
[PERSON] performed the works of conceptualization, formal analysis, methodology, software, validation, visualization, writing--original draft, and writing--reviewing & editing.  
This research received no external funding.  
The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US government.  
The author declares no conflicts of interest.