## 3 Results  
### Accuracy and Repeatability  
The resulting power profiles for the three precision calibration lenses are shown in Figure 1. Note that the calibration lenses were measured in air as the power of these lenses is specified by the manufacturer. The continuous blue line of the plots in Figure 1 representsthe average of the radial power profiles measured by the NIMO. The dashed blue lines are the limits of the confidence intervals and are computed by adding (for the upper limit) or subtracting (for the lower one) the standard deviation of the radial power profiles from the average power profile. Note that these dashed lines are not always distinguishable from the average power, as happens in panels 1B and 1C. The red continuous line represents the power computed by Zemax Optical Studio from the constitutive data (curvature radius, center and edge thickness, and refractive index of the materials) of the calibration lenses that are supplied by the lens manufacturer. For their part, the dashed red lines represent the limits of the tolerance interval computing using the tolerance tool of Zemax Optics Studio. The computation of the tolerance intervals is as follows: First, a set of tolerances for the constitutive parameters of the lens is fixed. These tolerances include not only the tolerances of geometrical parameters such as curvature radii or center thickness but also tilt and decentration between the lens surfaces, quality of the lens surface finish, etc. The basic idea is to consider the foreseeable sources of manufacturing errors. Next, Zemax generates a series of lenses presenting random errors in the lens parameters. In this way, we have a set of lenses with different manufacturing errors. Afterward, Zemax computes a parameter related to the image quality (such as the RMS radius of the spot generated by the lens in the focal plane) for each of the lenses of the set. Next, we compute the power profile of the lens of the set that presents the worst quality parameter. Then, we calculate the difference between this profile and the theoretical power profile (computed with no tolerance errors). Finally, we determine the tolerance limits by adding and subtracting the absolute value of this difference and the theoretical power of the lens. Note that, as computing lens tolerances with Zemax takes into account the tilt and decentering of the lens surfaces, the resulting tolerance intervals can also be decentered with respect to the theoretical profile. Therefore, the tolerance intervals represent the possible manufacturing errors. From the theoretical and measured power profiles, the systematic and random errors of the NIMO for the calibration lenses were evaluated, as tabulated in Table 1.  
For both the +10.00 and +20.00 D lenses, the systematic errors were lower than 1% (in absolute value) of the base lens power. The +33.33 D lens had a higher amount of error (around 1.7%). However, the manufacturer guarantees that the focal length of the three lenses is within 1% error. This translates to \(\pm\)0.33 D of error in power for the +33.33 D lens, which is close to the 0.50 D measured with the NIMO. Therefore, we can conclude that, for the lenses of +10.00 and +20.00 D power, the systematic error is within the limits given by the lens manufacturer, while the error of the +33.33 D lens is close to this tolerance error (\(\pm\)0.33 D according to the tolerances specified by the lens manufacturer). Moreover, the shape of the average profile was very similar to the theoretical profile and the errors are below the tolerances set by the relevant ISO standard (ISO 11979-2) [20]. Note that, in Figure 1, although the theoretical profile is symmetrical as corresponds to a lens with well-aligned spherical surfaces, the tolerance limits are not, because these limits are computed introducing small errors in the constitutive parameters of the lens, including tilts and decentrations between the lens surfaces that can occur during the manufacturing process.