# Reflective Inverse Diffusion
## 3 Materials and Methods  
### Laboratory Experiments  
The experimental setup is shown in Figure 1. The primary components of the laboratory setup include a 30-mW HeNe laser (\(\lambda=632.8\) nm) with a semi-permanent neutral density filter attached. The filter had an optical density of 1.0 that reduced the effective power of the laser to approximately 3 mW, rendering it eye safe. Phase modulation is accomplished with a Boulder nonlinear systems (BNS) liquid crystal on silicon (LCoS) SLM. This reflective SLM is 7.68 mm by 7.68 mm, consisting of 512 by 512 pixels, each capable of 256 (8-bit) phase levels over a 2\(\pi\) phase stroke. Feedback is provided by a Santa Barbara Instruments Group (SBIG) ST-402 ME CCD with 765 by 510 pixels and a pixel pitch of 9 \(\upmu\)m by 9 \(\upmu\)m.  
The HeNe laser is first expanded and collimated to the width of the SLM using a 20\(\times\) microscope objective, spatial filter (SF) and lens \(L1\). The non-polarizing beam splitter (NPBS) is used to allow for normal incidence on the SLM. After phase modulation, the beam is focused onto the reflective sample by lens \(L2\) with focal length, \(f=1\) m. The phase-modulated beam is incident onto the diffusely-reflecting sample at 45\({}^{\circ}\). The reflector then scatters the incident beam onto the CCD placed 40 \(\pm\) 0.5 cm from the reflector and at 45\({}^{\circ}\) from its surface normal. The algorithm used to optimize the SLM is based on the continuous sequential algorithm developed for transmissive inverse diffusion [10]. The SLM is first segmented into 2 \(\times\) 2 segments of equal size. The phase of each segment is then cycled, and the target intensity is recorded by the CCD. The phase value of the segment is then set to produce the maximum target intensity at the desired location before moving on to the next SLM segment [10; 11]. Pre-optimization with a smaller number of segments \((N=2)\) was performed before moving to \(N=\{4,8,16,32\}\) sequentially, as it was shown to increase the signal-to-noise ratio in the transmissive experiments [9; 11]. The reflective inverse diffusion algorithm was used for six different scattering materials: Spectralon(r), brushed aluminum, sandblasted aluminum, Infragold(r), white paint on glass and graphite. The reflector materials were selected based on the differences in the scattering properties. The surface roughness of the samples were measured and compared with the reflected spot enhancement produced by the algorithm. These measurements are shown in Table 1.  
In addition to refocusing light after reflection, phase modulation demonstrated the ability to control the location of the focused spot in the observation plane. This non-mechanical beam steering technique required no initial changes to the experimental setup. The algorithm consecutively optimized and refocused light into each of the four corners of the CCD. The CCD was also translated in two dimensions while maintaining a fixed distance of 40 cm from the reflector to test the ability to refocus light over a larger target surface. For these experiments, the CCD was moved from the specular region of reflection into the diffuse region, defined as a horizontal shift of 66 mm and a vertical shift of 88 mm, or approximately 9.0\({}^{\circ}\) and 12.6\({}^{\circ}\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline
**Reflective Samples** & **Enhancement (n)** & **Roughness (\(R_{\text{rms}}\))** & **Final full width at half max (FWHM)** \\ \hline Brushed aluminum & 122.3 & 1.5 \(\upmu\)m & 36 \(\pm\) 3 \(\upmu\)m \\ Infragold® & 89.9 & 9.4 \(\upmu\)m & 38 \(\pm\) 3 \(\upmu\)m \\ Sandblasted aluminum & 67.7 & 2.3 \(\upmu\)m & 38 \(\pm\) 3 \(\upmu\)m \\ Graphite & 37.3 & 3.5 \(\upmu\)m & 41 \(\pm\) 3 \(\upmu\)m \\ White paint & 36.8 & 1.7 \(\upmu\)m & 41 \(\pm\) 3 \(\upmu\)m \\ Spectralon® & 13.8 & Unprofiled & 45 \(\pm\) 3 \(\upmu\)m \\ \hline
**Transmissive Sample** & & & \\ \hline White paint & 56.4 & 1.7 \(\upmu\)m & 63 \(\pm\) 3 \(\upmu\)m \\ \hline \end{tabular}
\end{table}