## 3 Analysis and Results  
Analysis of the metrology data results in an averaged RMS value of 0.85 \(\upmu\)m and an averaged PV value of 8.1 \(\upmu\)m. Using RMS and PV residuals to evaluate the optical surface quality, it is obvious, that the smaller an RMS value is, the better the nominal geometry of the surface is resembled. The same holds for PV: the higher the PV value the larger the maximum deviation between measured and nominal surface. However, neither of the two values describe the optical performance due to geometrical deviations. It is not possible to derive how this surface deviation will affect the system's optical performance. The absolute PV value of 8.1 \(\upmu\)m seems to be very small for a 5 mm clear aperture. The same is true for the RMS error below 1 \(\upmu\)m. In the event of spherical optics, no one would expect the optical performance to decrease significantly due to these small values. However, using freeform optics an estimation of the impact of geometrical deviations becomes less intuitive. Figure 1: Freeform lens manufactured by the \(\upmu\)IM process. Figure 2: Surface residuals with regard to the nominal surface: plotted on the concentric measuring path (**left**), contour plot (**right**). To analyze the effect of geometric deviations due to the manufacturing processes, the simulation model has to be enhanced by measured surface data. This approach enables simulation and performance analysis of the real manufactured geometry. The surface fit on the measuring points was conducted by means of a polynomial equation, since the nominal surface is also described by a polynomial. The use of higher order polynomial terms is due to the complexity of freeform optics. Surface generation using fourth order polynomials achieved better results than an approximation of the surface based on spline interpolation [20]. Optical surfaces were often fitted by means of Zernike polynomials. Zernike polynomials offer the advantage to describe directly low frequency errors and hence, can be used as direct measure of optical aberrations. Since the individual Alvarez lens part has no meaningful optical effect, optical aberrations cannot be assigned to the individual surfaces directly. Hence, the advantage of using Zernike polynomials does not exist. Alvarez optics generate its optical effect only in conjunction with a second element. Determination of form deviations of freeform surfaces by means of a surface fit is conform to the modified ISO-10110 standard [23]. Table 1 depicts the functions generated by means of the surface fit, the RMS error between the manufactured surface and the fitted function, and the mean error of the surface fit. The mean error as well as the RMS error are small enough to allow a representation of the manufactured optics by the fitted surface function given in Table 1. The simulation model now is enhanced by means of the fitted surface function. A performance simulation based on the enhanced model allows for a comparison of the optical performance with the originally designed nominal freeform surfaces. To quantify the optical performance, the modulation transfer function (MTF) was applied as criterion. The MTF is an accepted criterion to quantify the optical imaging quality of lenses or lens systems [29]. The performance of the manufactured Alvarez-optics was tested in an existing setup in context of the development of the Artificial Accommodation System [8]. The AAS features a modular concept, where housing as well as optical surfaces integrated in the housing is used for different varifocal principles. Originally, the housing is designed for an optical subsystem using a lens moveable along the optical axis to adjust the focal length. Hence, the rear-side optics is a negative aspheric lens. Combining this rear-side optics with an Alvarez-optics, an aspherical inset lens has to be designed to add a positive refraction power to the negative lens of the rear side. The optical performance of the assembly containing the Alvarez-optics suffers from the adaptation of the existing setup.