## 2 Background  
Transmissive and reflective scattering are both linear, deterministic processes as long as the scattering medium is static. In both cases, the resultant scattered field can be considered a linear combination of the inputs. Phase modulation using an SLM divides up the incident light into several individual segments, each with a unique phase. Every sampled location in the observed speckle field is a linear combination of the field from the individual SLM segments. The field at the \(m^{th}\) segment of the observed field is given by [4],  
\[E_{m}=\sum_{n=1}^{N}t_{mn}A_{n}e^{i\phi_{n}}, \tag{1}\]  
where \(t_{mn}\) is the \(mn^{th}\) complex-valued element of the transmission/reflection matrix relating the light from the \(n^{th}\) SLM segment to the \(m^{th}\) segment in the observation plane, and \(A_{n}e^{i\phi_{n}}\) represents the amplitude and phase of the light from the \(n^{th}\) SLM segment. Normalizing Equation (1) to intensity, by setting \(A_{n}=\
icefrac{{1}}{{\sqrt{N}}}\), the observed intensity is then,  
\[I_{m}=\left|E_{m}\right|^{2}=\frac{1}{N}\left|\sum_{n=1}^{N}t_{mn}e^{i\phi_{n}} \right|^{2}. \tag{2}\]  
Intensity enhancement (\(\eta\)) is a measure of performance for controlling scattered light, defined in transmissive experiments as the ratio of the average intensity of the optimized observation plane segments that comprise the refocused spot, \(\left\langle l_{opt}\right\rangle\), to the average intensity of the unoptimized background segments, \(\left\langle I_{md}\right\rangle\)[6]. Assuming the surface-height imperfections of the reflector are on the order of a wavelength and follow a Gaussian distribution, the reflector can be modeled as an average reflectivity with a uniform phase [15]. The reflector statistics can be used to show that the expected maximum ideal enhancement is equal to \(N\), and the total number of SLM segments [7],  
\[\eta=\frac{\left\langle l_{opt}\right\rangle}{\left\langle I_{md}\right\rangle }=N. \tag{3}\]  
Ideal enhancement neglects the effects of polarization due to the linearly polarized input to the SLM. The input is linear \(s\) polarized (perpendicular to the plane of incidence), where pure \(s\) (or pure \(p\)) polarized light maintains a higher level of polarization after reflection from a rough surface [16]. The rotation caused by cross polarization is expected to be less than \(25^{\circ}\), resulting in 90% of the reflected field being polarized in the same direction as the input and accounting for more than 98% of the measure intensity [16]. The method used to measure the RM was based on work by [PERSON] et al. for measuring transmission matrices of turbid media [PERSON] et al. (2017). [PERSON]'s method was based on parallel wavefront optimization method by [PERSON] and expanded to measure the entire transmission matrix, rather than optimize to a single point. Parallel wavefront optimization uses interference between reference and signal fields produced by the SLM to extract the phase information of the RM [PERSON] et al. (2017); [PERSON] et al. (2017). The SLM segments are divided into two groups, the Group 1 segments are each modulated at a separate frequency to produce the signal field, while the Group 2 segments are held static to produce the reference field. The intensity of a given segment in the observation plane is then a combination of the reference and signal fields given by [PERSON] et al. (2017),  
\[ \begin{split} I_{m}&=\left|E_{m}\right|^{2}=\left| \sum_{n=G+1}^{N}t_{mn}+\sum_{p=1}^{G}t_{mp}e^{i\omega_{p}t}\right|^{2}=\left|R_ {m2}+\sum_{p=1}^{G}t_{mp}e^{i\omega_{p}t}\right|^{2}\\ &=\left|R_{m2}\right|^{2}+R_{m2}^{*}\sum_{p=1}^{G}t_{mp}e^{i \omega_{p}t}+R_{m2}\sum_{p=1}^{G}t_{mp}^{*}e^{-j\omega_{p}t}+\sum_{p=1}^{G} \sum_{q=1}^{G}t_{mp}^{*}t_{mq}e^{j(\omega_{q}-\omega_{p})t},\end{split}  \tag{4}\]  
where \(R_{m2}\) is the sum of the static Group 2 segments that produce the reference field. The second term of Equation (4) shows that the matrix coefficients, \(t_{mp}\), occur at discrete frequencies.