## 4 Results and Analysis
### Simulations  
The RM produced by the simulation of the focal-plane system in Figure 1 was used to calculate the enhancement for 1024 different observation plane segments. The rough-surface reflector was  
Figure 4: Enhancement (\(\eta\)) vs. Lateral surface dimension (\(\ell_{\lambda/2}\)). The lateral surface dimension, \(\ell_{\lambda/2}\), is plotted with enhancement. The focal-plane system shows no change in enhancement with \(\ell_{\lambda/2}\). However, the average enhancement of 1 indicates that the target segment has the same intensity as the background. made up of individual segments with a uniform phase distribution. The phase of each segment was independent, which leads to a delta-correlated sample. The segment size is a measure of lateral correlation, but due to the phase independence between each segment, it was not directly comparable to \(\ell_{c}\) of the tested aluminum samples. The phase of the simulated rough-surface segments is a uniform distribution over \([-\pi,\pi]\). If bounded over a single wavelength, the phase distribution gives a surface height range from \([-\frac{\lambda}{2},\frac{\lambda}{2}]\). Since the simulated rough-surface segments are independent, the surface-height difference (\(z_{k}-z_{k-1}\)) in Equation (9) for surface slope is a random variable with the same uniform distribution and an expected value of \(\
icefrac{{\lambda}}{{2}}\). Assuming a mean slope of \(\hat{z}^{\prime}=0\), the RMS surface slope for the simulated rough surface is given by,  
\[s_{sim}=\left[\frac{1}{K-1}\sum_{k=1}^{K-1}\left(\frac{E[z_{k}-z_{k-1}]}{ \Delta x}\right)^{2}\right]^{\frac{1}{2}}=\left[\frac{1}{K-1}\sum_{k=1}^{K-1} \left(\frac{\
icefrac{{\lambda}}{{2}}}{\Delta x}\right)^{2}\right]^{\frac{1}{ 2}}=\frac{\lambda}{2\Delta x}, \tag{10}\]  
where \(\Delta x\) is the dimension of the simulated rough-surface sample segment. For the simulations \(\ell_{\lambda/2}=\Delta x\), the simulated rough-surface segment size. Lateral surface dimensions (\(\ell_{\lambda/2}\)) from 10.3 \(\upmu\)m to 660 \(\upmu\)m were simulated. Maximum enhancement was achieved with \(\ell_{\lambda/2}=41\)\(\upmu\)m, which produced an enhancement of \(\eta_{max}=907\) or 88% of the predicted ideal maximum from Equation (3). The best average enhancement of the 1024 simulated measurements was achieved with \(\ell_{\lambda/2}=82\)\(\upmu\)m, which produced an enhancement of \(\eta_{avg}=708\). Increasing \(\ell_{\lambda/2}\) beyond 82 \(\upmu\)m caused both the maximum and average enhancement to decrease. In the focal-plane system, the individual SLM segments act as individual apertures. The spacing and phase of the SLM segments create fringe patterns in the diffraction-limited spot incident on the rough-surface reflector. The sum of all interferences from the SLM segment pairs produces a wavefront that conjugates the phase imparted by the rough-surface reflector to produce a wave that converges in the observation plane. Segment pairs that are farthest apart produce the narrowest fringe spacing of 41 \(\upmu\)m. This represents the smallest lateral surface dimension (\(\ell_{\lambda/2}\)) that the focal-plane system can conjugate and explains the peak maximum enhancement at 41 \(\upmu\)m for the focal-plane system, see Figure 5. Using the RM to refocus light ensures that the light at the target segment is all in-phase, but it does not guarantee it is the only segment where the light is in-phase. In the focal-plane system, as the correlation length of the simulated sample increased, higher-order fringes are incident on the same simulated sample segment and remain in phase. This produces additional foci in the observation plane that increase background intensity and decrease enhancement as the correlation length is increased (see Figure 5). Simulations of the imaging system shown in Figure 2 assumed a 1-mm\({}^{2}\) ideal image of the SLM projected onto the rough-surface reflector.