# Beam Formation and Vernier Steering Off of a Rough Surface
## 3 Non-Mechanical Beam Steering
### Results and Discussion  
Figures 4 and 5 show the relative reduction in enhancement of the beam at the detector as a function of the scan angles in two dimensions while Figure 6 demonstrates the individual relationships between horizontal and vertical scans and enhancement. These data are averaged over multiple trials where the GA was optimized to different initial detector channels. The SLM phase mask was shifted in 16-pixel increments, which is 240 \(\upmu\)m on the detector. The resultant scan angle for each shift was calculated trigonometrically from the displacement of the refocused beam on the detector. The beam's displaced position was determined by centroiding a windowed region of the detector around the beam. Coarse displacement estimates derived from Equation (4) set the window boundaries. Table 2  
Figure 3: Computed phase tilts at the scatterer due to circular shifts of the SLM phase mask in the focal plane setup. The phase at the scatterer is simulated for a given phase mask, and then the phase mask is shifted, creating phase tilts at the scatterer. The SLM phase mask is circularly shifted 8 pixels in (**a**) and 32 pixels in (**b**). What is shown is the the phase differences at the scatterer’s surface between the unshifted and shifted cases. shows the modeled and experimentally observed scan angle increments due to incremental shifts of 240 \(\upmu\)m of the SLM phase mask. Figure 4: Experimental enhancement of displaced spot as a two-dimensional function of displacement size after a circular shift of the SLM phase map of a focal plane system shown in Figure 1 for 600-grit polished nickel. Results were averaged over five trials. The maximum value of \(\eta\) prior to normalization was \(89.9\pm 1.5\). Figure 5: Experimental enhancement of displaced spot as a two-dimensional function of displacement size after a circular shift of the SLM phase map, of focal plane system shown in Figure 1 zinc oxide paint, on glass. Results were averaged over five trials. The maximum value of \(\eta\) prior to normalization was \(80\pm 3\). The nickel sample experiences a quasi-linear decrease in enhancement as a function of scan angle over the range of angles tested. The scan angle at which the enhancement decreases by 50% along either the the horizontal or vertical axis is approximately 0.2\({}^{\circ}\)\(\pm\) 0.02\({}^{\circ}\). The paint sample is notably asymmetrical between the axes. Horizontal scans on the paint sample appear to experience a quasi-linear decrease in enhancement over the range of scan angles tested; however, it is clear from Figure 6 that the behavior of vertical scans cannot adequately be described with a linear relationship over the entire range of angles tested. In the paint sample, scans drop the enhancement to 50% at 0.033\({}^{\circ}\)\(\pm\) 0.015\({}^{\circ}\) along the horizontal axis and by 0.034\({}^{\circ}\)\(\pm\) 0.017\({}^{\circ}\) along the vertical axis. The subsequent decrease by 75% occurs at 0.22\({}^{\circ}\)\(\pm\) 0.02\({}^{\circ}\) along the horizontal axis and at 0.15\({}^{\circ}\)\(\pm\) 0.02\({}^{\circ}\) along the vertical axis. The likely explanation for the difference between the behavior of the nickel and paint samples is that most of the reflection from nickle sample is singly scattered from the surface while only a small fraction of the reflection from the paint is singly scattered. The phase tilts of the electric field incident on the paint sample are not likely to be maintained for the portion of light transmitted past the paint's surface due to the multiple scattering that occurs. This would account for the seemingly discontinuous dropoff that occurs when any shift is made from the center, as seen in Figure 5. With this in mind, it is possible that the  
Figure 6: Scatterplot views of enhancement along the horizontal and vertical cross sections of Figures 4 and 5, normalized to each cross section’s maximum central enhancement. The metal sample’s quasi-linear decrease for both horizontal and vertical scans is shown in (**a**,**b**).