## 3 Selected Results of Optical Metrology Techniques
### Multiple Simultaneous Viewing Directions Using Four Digital Holographic Microscopes (MuVie)
#### 3.1.2 Robust Interferometric Inspection Using Computational Shear Interferometry (COSI)  
Shear interferometry has proven to be very robust against mechanical vibrations. Shear interferometers are imaging systems, which produce two identical images across the camera plane. The images are mutually shifted and the shift is referred to as the shear. In this configuration, no reference wave is required. However, the interference pattern formed by the two images provides information about finite differences (differential approach) across the surface of the specimen with interferometrical precision. Since the two interfering wave fields associated with the images travel almost identical optical paths, the system is intrinsically immune towards vibrations. An additional benefit of shear interferometry arises from its low demands with respect to both the temporal and the spatial coherence of light. This enables the use of eye-safe and cheap light sources, such as light emitting diodes or even arc lamps. In an industrial environment, where safety issues are critical, this is a decisive argument for shear interferometers. Finally, shear interferometers can also investigate objects with steep slopes, which would cause high fringe densities in common interferometric setups, and--as will be shown in the next section--they can operate with multiple light sources at the same time, which solves the major problem of a limited acceptance angle in interferometry. One of the methods we use is Computational Shear Interferometry (CoSI) [28], which seeks to determine the underlying wave field associated with the two images. The benefit of CoSI is that it provides a wave field as a result. This enables almost any interferometric investigation, such as phase shifting interferometry, digital holography and quantitative phase contrast imaging, but with the advantage of using a robust shear interferometer [29]. Figure 4: Digital holographic microscopy inside of cavities: (**a**) Digital holographic microscope, which is capable of testing one part per second. The inset shows a sketch of a typical object the system is designed for. The highlighted surface inside the cylinder is the region of interest. (**b**) A composed image of the same measurement result. The lower half depicts the measured profile, while the upper half shows the deviation from the known shape. We can clearly see a groove with a depth of approximately 5 \(\upmu\)m, which indicates that this part is damaged. Figure 5a shows the setup based on a liquid crystal spatial light modulator (SLM) [30]. The imaging system consists of a 4\(f\)-arrangement with the SLM in the Fourier domain. It images any wave field incident in the input plane to the sensor plane where the camera is located. The SLM is birefringent, so that only light oriented along the slow axis will be modulated, whereas any other light will be reflected from the silicon back panel of the device [31]. We select the incoming light to be polarized exactly between the slow and the fast axis and program the SLM to display a blazed grating, which exhibits a single diffraction order. In this situation, two laterally separated images will appear across the sensor plane, where the separation, i.e., the shear, can be selected by the orientation and the magnitude of the blazed grating. The polarizers ensure that the two wave fields are able to form an interference pattern. The setup enables fast, precise and highly reproducible adjustment of the shear and ensures almost identical optical paths for both images. If we denote the wave field in the input plane as above by \(u\) and the shear by \(\vec{s}\), we find that the intensity of the interference pattern in the camera domain is given by  
\[I(\vec{x})=|u(\vec{x})|^{2}+|u(\vec{x}+\vec{s})|^{2}+\mathcal{R}\{u^{*}(\vec{ x})u(\vec{x}+\vec{s})\}, \tag{5}\]  
where \(\mathcal{R}\) means the real part. Using phase shifting techniques [32], we can isolate the argument of the real part and arrive at the cross amplitude \(M(\vec{x})=u^{*}(\vec{x})u(\vec{x}+\vec{s})\).