## 3 Selected Results of Optical Metrology Techniques
### Multiple Simultaneous Viewing Directions Using Four Digital Holographic Microscopes (MuVie)
#### 3.1.3 Multiple Aperture Shear Interferometry (MArS)  
A very promising technique based on shear interferometry is _Multiple Aperture Shear Interferometry_ (MArS), which is currently under development in collaboration with the _Physikalisch-Technische Bundesanstalt_ (PTB). The aim of MArS is to compensate for the problem of a limited acceptance angle of current interferometers for lens testing. If lenses with steep slopes are investigated, only a small portion of the light reflected by the surface under test (SUT) is incident into the entrance pupil of the interferometer. Usually, computer generated holograms (CGH) are used to pre-shape the illuminating wave front, so that the reflected light can be imaged [33]. However, CGHs are expensive and not customizable for different lenses. MArS exploits the fact that shear interferometers can be operated with multiple independent light sources, i.e., multiple illumination apertures [34]. In this situation, a large number of light sources can be distributed around the SUT, and each light source will partially contribute to the measurement. Figure 6a shows a sketch of the setup. It consists of a shear interferometer, similar to the arrangement introduced in Figure 5a, but instead of a single light source, we use multiple illumination apertures, where two of them are evenly transported through the observation lens. Figure 6b depicts the intensity in the image plane for the case of a spherical mirror such as SUT [35]. Each light source creates a rectangular spot across the image of the SUT. Together, they cover the entire surface. Figure 6: Multiple Aperture Shear Interferometry (MArS): (**a**) The setup consists of a shear interferometer and multiple independent light sources for illumination (illumination apertures). Some of the illumination apertures have to be guided through the observation lens in order to access the entire surface under test (SUT). (**b**) Intensity image recorded by the camera with a spherical mirror as the SUT. The illumination apertures appear as rectangular areas, where one aperture corresponds to one light source. Reprinted with permission from Reference [35]. When multiple independent wave fields are incident in the object plane, we cannot assign a consistent wave front to the light. However, we can still assign a coherence function  
\[\Gamma(\vec{x}_{1},\vec{x}_{2})=\langle u^{*}(\vec{x}_{1},t)u(\vec{x}_{2},t) \rangle=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}u^{*}(\vec{x}_{1},t)u(\vec {x}_{2},t)dt \tag{8}\]  
to it that depends on the two positions \(\vec{x}_{1}\) and \(\vec{x}_{2}\) in space. Shear interferometers measure the coherence function \(\Gamma(\vec{x},\vec{x}+\vec{s})\). In fact, for the fully coherent case, Equation (8) becomes the cross amplitude \(M\) introduced earlier. For the sum \(u(\vec{x},t)=\sum u_{n}(\vec{x},t)\) of multiple independent wave fields \(u_{n}(\vec{x},t)\), we find after insertion into Equation (8)  
\[\Gamma(\vec{x},\vec{x}+\vec{s})=\sum_{n}u^{*}(\vec{x})u(\vec{x}+\vec{s}). \tag{9}\]  
In MArS, we perform a large number of measurements while varying the shear \(\vec{s}\) along a lattice so that we can interpret Equation (9) as a function in \(\vec{s}\) instead of the spatial coordinate \(\vec{x}\). If we select the shears across a small local area, we can regard Equation (9) as a sum of plane waves, with each of the waves being multiplied with a constant \(u_{n}(\vec{x})\). The plane waves can be associated with wave vectors \(\vec{k}_{n}(\vec{x})\), where each of the superposing wave fields contributes with one wave vector. We can separate the \(k\)-vectors in frequency space using the Fourier transform \(F\{\Gamma(\vec{s})\}\)[36]. The idea is now to assign light rays to the directions given by the aggregate of wave vectors. The light rays can then serve as input to a variational problem, where an inverse ray-tracer is employed to determine the SUT [37], subject to the position of the light sources, which have to be known in advance by means of a calibration process.