## 4 Results and Analysis  
### Experiments  
The RM was measured three times along the diagonal of each sample, top left corner, center, and bottom right corner, for both the focal-plane and imaging optical systems. The RM was then used to focus the incident beam to 1024 different segments of the CCD and record the enhancement of the target segment over the background segments. The maximum enhancement for each sample, \(\eta_{max}\), is the average peak enhancement from each of the three measured RMs. The average enhancement for each sample, \(\eta_{avg}\), is the mean enhancement value over all the segments from all three RM. The \(\eta_{max}\) and \(\eta_{avg}\) for each of the five samples are recorded in Table 1.  
Data trends in Figure 3 show that enhancement decreases as surface roughness increases for the imaging system. The enhancement for the focal-plane system is almost flat with the average enhancement just slightly larger than 1; the target segment intensity is not significantly higher than the background. The difference between constructive and destructive interference is a surface height change of \(\
icefrac{{\lambda}}{{2}}\), thus the rate at which the surface height changes is a better predictor for enhancement. The RMS surface slope indicates the rate of the surface-height fluctuations and was determined from the profile data from each sample. For discrete profile data, the surface slope is given by [22],  
\[s=\left[\frac{1}{K-1}\sum_{k=1}^{K-1}\left(\frac{z_{k}-z_{k-1}}{x_{k}-x_{k-1}} -\hat{z}^{\prime}\right)^{2}\right]^{\frac{1}{2}}, \tag{9}\]  
where \(K\) is the total number of measurements, \(z_{k}\) is the surface height from the profilometer, \(x_{k}\) is the lateral sample distance, and \(\hat{z}^{\prime}=\frac{z_{k}-z_{1}}{K\Delta x}\), the average slope. The RMS slopes of the samples are included in Table 1. In general, the samples with larger RMS slope showed lower enhancement.  
Figure 3: Enhancement (\(\eta\)) vs. Surface Roughness (\(\sigma\)). The surface roughness measured by the Tencor profilometer is plotted with enhancement. Linear trend lines are included. For the focal-plane system, maximum enhancement is shown in red, while average enhancement is shown in magenta. For the imaging system, maximum enhancement is shown in blue, while average enhancement is shown in cyan.  
The correlation length, \(\ell_{c}\) in Table 1, is defined as the shift required to lower the autocorrelation of the surface height profile by a factor of \(e^{-1}\) of the maximum value [22]. The apparent inverse relationship between correlation length and enhancement is counter-intuitive but is not conclusive since the samples with the longest correlation length also have the highest roughness. None of the samples showed Gaussian surface-height distributions, and with regards to reflective inverse diffusion, the standard \(e^{-1}\) definition used for correlation length is overly optimistic due to the small surface-height difference between constructive and destructive interference. For reflective inverse diffusion, the proposed lateral dimension, \(\ell_{\lambda/2}\) in Table 1, is described as the distance required to achieve a \(\
icefrac{{\lambda}}{{2}}\) change in surface height given the RMS slope. The slope is calculated for each profile measurement recorded by the stylus profiler and averaged to produce a single \(\ell_{\lambda/2}\) for each sample. The surface profiles collected from the bead blasted sample showed widely varying slopes up to \(\pm 35\%\) from the average slope used to determine \(\ell_{\lambda/2}\). The enhancement performance from the bead blasted sample is likely inflated by an RM measured in area of the sample with a significantly larger \(\ell_{\lambda/2}\) than calculated from the average slope. Enhancement is plotted with \(\ell_{\lambda/2}\) in Figure 4. The imaging system shows that enhancement increases with \(\ell_{\lambda/2}\). The data is inconclusive in the focal-plane system due to low enhancement values and virtually flat trends.