## 3 Methodology
### Simulations  
The propagation model assumes that the SLM is illuminated by a unit amplitude ideal plane wave. The individual SLM segments are represented by unimodular complex coefficients. The phase of these coefficients represents the phase delay imparted by the SLM segments onto the incident plane wave. At \(\lambda=632.8\) nm, aluminum has a reflectivity greater than 0.9, and a complex index of refraction of \(n\approx 1.4+i7.5\) with a penetration depth of just a few nanometers [18; 19]. Thus, absorption and transmission are neglected when modeling the aluminum rough-surface reflectors and the samples are modeled as unimodular complex numbers. The random phase imparted by the rough surface is related to the surface-height fluctuations or surface roughness. Assuming a Gaussian surface-height distribution with a standard deviation equal to or greater than \(\
icefrac{{\lambda}}{{2}}\), the phase distribution becomes uniform when wrapped to the interval \([-\pi,\pi]\)[15].  
Diffraction-based models were simulated using the MATLAB(r) two-dimensional FFT (fast Fourier transform) and the Rayleigh-Sommerfeld transfer function [20]. For the focal-plane optical setup in Figure 1, the complex field at the observation plane is calculated using the Rayleigh-Sommerfeld transfer function and spatial Fourier transform property of the lens. The field in the observation plane is then given by [7],  
\[U_{CCD}(u,v)=\mathcal{F}^{-1}\{\mathcal{F}\{e^{i\frac{1}{2f}(u^{2}+v^{2})}e^{i \theta(u,v)}\mathcal{F}\{U_{SLM}(x,y)\}H(f_{x},f_{y})|_{z=z_{1}}\}\,H(f_{u},f_ {v})|_{z=z_{2}}\}, \tag{5}\]  
where \(k\) is the wavenumber, and \(H(f_{x},f_{y})\) is the Rayleigh-Sommerfeld transfer function [21]. The coordinate axes of source plane at the SLM are given in \((x,y)\), where the coordinate axes of the sample plane at the lens focus are given in \((u,v)\). The horizontal and vertical spatial frequencies of the source and sample planes are \((f_{x},f_{y})\) and \((f_{u},f_{v})\), respectively. The focal length of lens \(L_{1}\) is \(f\), and \(z_{1}\) and \(z_{2}\) are the distances from the SLM to the lens and the reflective sample to the CCD, respectively. The rough surface is represented by \(e^{i\theta(u,v)}\), where the phase function \(\theta(u,v)\) is a random uniform distribution from \([-\pi,\pi]\).  
Figure 2: Image plane optical setup for reflective inverse diffusion. A vertically polarized HeNe laser is expanded, collimated, and normally incident on the SLM. The lens (\(L_{1}\)) produces a demagnified image of the phase modulated beam onto the rough-surface reflector and the reflected speckle pattern is recorded by the CCD. The mirror (\(M_{1}\)) and the NPBS are used to allow normal incidence with the SLM. The demagnification of the system is \(\frac{s_{i}}{s_{o}}\approx\frac{1}{8}\).  
Ideal imaging is assumed for the configuration in Figure 2. Propagation from the SLM to the rough sample is reduced to a global phase shift and can be ignored. The rough surface then randomizes the phase of the ideal image, which is then propagated to the CCD in the observation plane using the Rayleigh-Sommerfeld transfer function. Thus, the field in the observation plane is given by,  
\[U_{CCD}(u,v)=\mathcal{F}^{-1}\{\mathcal{F}\{e^{i\theta(u,v)}U_{ SLM}(\frac{s_{o}}{s_{i}}u,\frac{s_{o}}{s_{i}}v)\}H(f_{u},f_{v})|_{z=z_{2}}\}, \tag{6}\]  
where \(s_{o}\) is the object distance from the SLM to the lens \(L_{1}\), and \(s_{i}\) is the image distance from the lens \(L_{1}\) to the rough-surface sample.