# Beam Formation and Vernier Steering Off of a Rough Surface
## 2 Refocusing Reflected Light from an Intermediate Surface  
### Materials and Methods  
The laboratory setup in the experiment was called a focal plane setup, shown in Figure 1. Previous works discuss the merits of different optical setups for achieving reflective inverse diffusion [12; 13; 20]. The optical source is a frequency-stabilized CW laser at 632.8 nm, expanded and collimated to fill the SLM at near-normal incidence. The SLM imparts a phase mask on incident, collimated light. That light, still approximately collimated, is then focused onto a sample scatterer. The scatterer is placed at the beam waist and is angled to separate incident and reflected light. The Charge Coupled Device (CCD) is placed at the center of reflection's main specular lobe. The SLM used was a Meadowlark Optics Reflective XY series model P512 SLM, which has \(512\times 512\) square pixels, with a pixel pitch of \(15\)\(\upmu\)m for a total effective aperture \(7.68\) mm \(\times\)\(7.68\) mm. The device has a reported fill factor of \(83.4\%\). The device controller was a PCIe 16-bit controller providing 16-bit phase discretization between 0 and \(2\pi\); however, a manufacturer-provided lookup table provided correct digital to phase outputs for 16,384 levels. The detector used was a Thorlabs model 4070M-GE-TE, which has \(2048\times 2048\) square pixels, with a pixel pitch of \(7.4\)\(\upmu\)m. Both devices were controlled by a desktop computer in Matlab(r) using manufacturer-provided libraries. The scattering samples used were polished Nickel and white paint on glass. Surface properties are listed in Table 1. As a metal, the polished Nickel sample was a pure surface scatterer, as no transmission of light into the medium occurred, whereas transmission did occur for the white paint sample. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Roughness** & **Correlation, \(\mathbf{l}_{c}\)** & **Slope** & **Correlation \(\mathbf{l}_{\lambda 2}\)** \\ \hline
600-grit Nickel, average & 0.45 \(\upmu\)m & 16 \(\upmu\)m & 0.066 & 4.79 \(\upmu\)m \\ \hline White Paint on Glass, average & 0.71 \(\upmu\)m & 20 \(\upmu\)m & 0.086 & 3.69 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Surface properties of scattering samples, from a profilometer. Roughness is the standard deviation of surface height. \(l_{c}\) is the distance required to shift the autocorrelation of the surface height profile by \(1/e\). Slope is the RMS surface slope. \(l_{\lambda/2}\) is the distance required to create a height change of \(\lambda/2\) assuming the surface has a linear slope with the value of the RMS slope. Iterative feedback-based wavefront shaping algorithms were used to achieve reflective inverse diffusion. In these algorithms, the SLM was partitioned into \(N\times N\) square segments. As in previous work, it was assumed that the scattering sample would act as a lens with focal length \(Z_{2}\) after optimization and that the minimum spot size at the detector would be based on the diffraction limited beam waist of a single SLM segment propagating through the system [12]. In that work, it was shown that the spot size at the detector would be inversely proportional to \(N\). The detector pixels were accordingly partitioned into square channels such that the width of each channel was as close as possible to the diffraction limited beam waist for the chosen \(N\). Prior to any wavefront shaping, the reflection of the light off of the sample scatterer created a diffuse speckle pattern centered around the main specular lobe of reflection from the sample surface. A single detector channel was chosen to be the target of the refocused, optimized beam, and the remaining channels measured the background speckle intensity over the extent of the detector.