# Reflective Inverse Diffusion
## 2 Background  
### Reflective vs. Transmissive Inverse Diffusion  
Light scattering, whether by transmission through or reflection from a medium, is a linear process. Despite the complexity and unknown material properties, it can be considered deterministic as long as the medium is static [13]. Therefore, the relationship developed for transmissive inverse diffusion for the observed total field in the target area is also valid for the reflective case. The field at the \(m\)-th observation plane segment is given by [8]:  
\[E_{m}=\sum_{n=1}^{N_{T}}t_{mn}A_{n}e^{i\Phi_{n}}, \tag{1}\]  
where \(t_{mn}\) is the \(mn\)-th complex-valued element of the transmission/reflection matrix relating the light from the \(n\)-th spatial light modulator (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. The source field is segmented by the SLM into \(N_{T}\) segments and reflected off the scattering medium. The \(N_{T}\) segments of the SLM are equally sized and arranged \(N\times N\), thus \(N_{T}=N^{2}\). The field at each of the \(M\) segments in the observation plane consists of a linear combination of the fields from the \(N\times N\) segments of the SLM [8]. Normalizing Equation (1) to intensity, \(A_{n}=\frac{1}{\sqrt{N_{T}}}\), and the observed intensity at the \(m\)-th observation plane segment is then:  
\[I_{m}=|E_{m}|^{2}=\frac{1}{N_{T}}\left|\sum_{n=1}^{N_{T}}t_{mn}e^{i\Phi_{n}} \right|^{2}. \tag{2}\]  
Intensity enhancement was defined by [PERSON] for transmissive inverse diffusion as [10]:  
\[\eta=\frac{\left\langle I_{opt}\right\rangle}{\left\langle I_{rnd}\right\rangle}, \tag{3}\]  
which is a simple ratio of the average intensity of the optimized spot divided by the average intensity of the unoptimized random speckle [9]. For transmission through a random medium, the \(t_{mn}\) terms follow a statistically-independent complex circular Gaussian distribution [16] with properties that can be used to simplify Equation (3) and express the ideal intensity enhancement as a function of the total number of SLM segments [10],  
\[\eta_{transmissive}=\frac{\pi}{4}(N_{T}-1)+1\approx\frac{\pi}{4}N_{T}. \tag{4}\]  
The thin films of turbid media in transmissive experiments were modeled using a circular Gaussian distribution [8; 9; 10]. However, the multiple random paths light travels when transmitted through a scattering medium are not present in the reflection model. This lack of complex circular Gaussian statistics produces a different expression for ideal enhancement. Using a simplified geometric approximation, the surface properties are modeled as a constant average surface reflectivity and phase delay that is related to the reflector surface height fluctuations [17]. Using this model, Equation (2) is rewritten for the reflection case as:\[I_{m}=\frac{1}{N_{T}}\left|\sum_{n=1}^{N_{T}}\mathbf{r}\,e^{i\theta_{mn}}e^{i\phi_{ n}}\right|^{2}, \tag{5}\]  
where \(\mathbf{r}\) is the average surface reflectivity and \(\theta_{mn}\) is the phase delay caused by the surface height fluctuations of the material. All of the tested materials have surface height fluctuations spanning multiple wavelengths (see \"Roughness\", Table 1); a Gaussian phase distribution with a standard deviation greater than a wavelength appears uniformly distributed when phase wrapped to the interval \([-\pi,\pi]\). Thus, the phase term of the reflector material, \(\theta_{mn}\), is assumed to have a uniform distribution over \([-\pi,\pi]\). The intensity achieves a maximum value when the SLM phase delay is set to cancel the phase delay imposed by the surface height of the reflector, \(\phi_{n}=-\theta_{mn}\),  
\[\left\langle I_{max}\right\rangle=\left\langle\frac{1}{N_{T}}\left|\sum_{n=1}^ {N_{T}}\mathbf{r}\right|^{2}\right\rangle=\mathbf{r}^{2}N_{T}, \tag{6}\]  
where the angled brackets again denote the ensemble average.