## 3 Analysis and Results
To see if the optical performance of the freeform optics manufactured by the \(\upmu\)IM/comp-process meets the criterion to preserve the MTF of the nominal Alvarez-optics, the measured surfaces were used to simulate the performance of the compensated Alvarez-optics. The approach used is the same as described in case of the \(\upmu\)IM process. Table 3 depicts the function used in the enhanced model. Table 4 shows the resulting, simulated MTF based on the surface representation shown in Table 3 compared again with the nominal surface. Figure 5: Surface residuals of the freeform optics manufactured using the \(\upmu\)IM/comp-process (data set on a reduced 48 \(\times\) 48 grid). **Left**: plotted on the concentric measuring path; **right**: Contour plot. Figure 6: PV and RMS values of the individual optics manufactured. Left: \(\upmu\)IM; right: \(\upmu\)IM/comp. The results presented in Table 4 show that the optics manufactured using the \(\upmu\)IM/comp-process performs very closely to the nominal value. The deviations in performance are as small as 2% in maximum. The effect of improvement of the MTF by means of the \(\upmu\)IM/comp-process can also be observed in Figure 7 showing the graphs of the simulated MTFs for the different molding processes (\(\upmu\)IM and \(\upmu\)IM/comp) in comparison with the nominal optics. The graphs are depicted for the central field and an allocation of infinite focal length. It is of no doubt that the values of RMS and PV are very useful to describe surface form deviations and are of importance in the characterization of manufacturing quality. This research was designed to show that PV and RMS errors alone are not sufficient to enable statements of the optical performance of freeform optics. In the example shown even surface accuracies of 8.1 \(\upmu\)m PV and 0.85 \(\upmu\)m RMS do not meet the performance requirements for a clear aperture of 5.0 mm. This was shown in case of the uncompensated microinjection-molded freeform optics (see Figure 3). Using metrology data to enhance the model offers the possibility to predict reasonably the expected optical performance of a manufactured freeform optics. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Focal Length [m]** & **MTF Nominal** & **MTF \(\upmu\)IM/Comp** & **Relative MTF \(\upmu\)IM/Comp [\%]** \\ \hline \(\infty\) & 0.34 & 0.34 & 100 \\ 4 & 0.49 & 0.47 & 98 \\ 0.667 & 0.66 & 0.65 & 98 \\ 0.364 & 0.63 & 0.63 & 101 \\ 0.263 & 0.55 & 0.55 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: MTF of the manufactured optics compared with the nominal design (central field, evaluated at 40 cyc/mm). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**(v, y) = a\({}_{1}\) x\({}^{4}\) + a\({}_{2}\) x\({}^{3}\)y + a\({}_{3}\) x\({}^{2}\)y\({}^{2}\) + a\({}_{4}\) xy\({}^{3}\) + a\({}_{5}\) y\({}^{4}\) + a\({}_{6}\) x\({}^{3}\) + a\({}_{7}\) x\({}^{2}\)y + a\({}_{8}\) xy\({}^{2}\) + a\({}_{9}\) y\({}^{3}\) + a\({}_{10}\) x\({}^{2}\) + a\({}_{11}\) xy + a\({}_{12}\) y\({}^{2}\) + a\({}_{13}\) x + a\({}_{14}\) y \\ \hline
**Surface** & **Parameter** & **RMS [\(\upmu\)]** & **Mean Error [1 \(\times\) 10\({}^{-16}\)\(\upmu\)m]** \\ \hline \multirow{6}{*}{\(\upmu\)IM/comp} & a\({}_{1}=-3.182\times 10^{-6}\) & a\({}_{8}=6.796\times 10^{-5}\) & \multirow{6}{*}{\(\upmu\)IM/comp} & \multirow{6}{*}{\(-0.85\)} \\  & a\({}_{2}=-3.180\times 10^{-5}\) & a\({}_{9}=6.396\times 10^{-3}\) & & \\ \cline{1-1}  & a\({}_{3}=-1.482\times 10^{-4}\) & a\({}_{10}=1.335\times 10^{-4}\) & & \\ \cline{1-1}  & a\({}_{5}=5.137\times 10^{-5}\) & a\({}_{11}=1.235\times 10^{-4}\) & & \\ \cline{1-1}  & a\({}_{6}=-1.911\times 10^{-5}\) & a\({}_{12}=8.923\times 10^{-5}\) & & \\ \cline{1-1}  & a\({}_{7}=0.0195\) & a\({}_{13}=-1.675\times 10^{-5}\) & & \\ \cline{1-1}  & a\({}_{14}=1.235\times 10^{-4}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Function of the fitted point clouds as well as the RMS and mean errors showing the accuracy of the fit.