## 3 Selected Results of Optical Metrology Techniques  
### Techniques Based on Interferometry  
#### 3.1.1 Rapid Holographic Shape Measurement of Micro Parts  
A 100% quality inspection of metallic micro components often fails due to the compromise between precision and speed. We therefore developed a fast measurement detection and quality inspection of micro components [21]. The system is based on digital holography (DH), an interferometric technique that allows to extract shape data from one or more recorded holograms [22]. The great advantage of DH is that the required holograms can be recorded within a fraction of a second, while it allows measuring with interferometric precision [11]. Additionally, the focus can be adjusted subsequent to the measurement, which allows for an extended depth of focus without the requirement of an optical scanning process. Finally, because digital holographic shape measurement is based on the evaluation of the phase of light, it does not require a triangulation base and can therefore be used to inspect, e.g., the bottom of a drill hole or functional surfaces that are surrounded by steep walls [23]. Figure 2 shows the basic setup of a digital holographic microscope for shape measurement of technical objects [24]. It consists of an illumination, a microscope objective, and optics to guide the reference wave. In our systems, we use optical fibers to transport illumination and reference waves in order to achieve maximum flexibility. If refocusing is necessary, it is useful to employ a telecentric microscope objective in order to avoid depth-dependent magnification [25]. Figure 2: Digital holographic microscope setup: The reference wave and the illumination are provided by an optical fiber. The object and reference wave are superposed using a beam splitter between the long distance microscope objective (LDM) and the tube lens. From the recorded interference pattern, it is possible to refocus the object subsequent to the measurement by means of computational methods. For microscopic investigations, it is common that the setup is arranged to image the object onto the camera target. Hence, the interference pattern across the camera plane allows determining the complex amplitude [11]  
\[u(\vec{x})=a(\vec{x})\exp\left[\mathrm{i}\phi(\vec{x})\right] \tag{1}\]  
of the wave field across the object plane with its amplitude \(a(\vec{x})\) and phase \(\phi(\vec{x})\). The shape of the object \(h(\vec{x})\) can be determined from the corresponding phase distribution \(\phi(\vec{x})\), which depends on the optical path difference between the object and the reference wave. With a plane reference wave and normal illumination, we find  
\[\phi(\vec{x})=\frac{2\pi}{\lambda}\cdot 2h(\vec{x}), \tag{2}\]  
with the wavelength \(\lambda\). Because of the ambiguity of the phase, heights can only be uniquely determined within the interval \([-\pi,\pi]\). For configurations corresponding to Equation (2), this means a height range of \([-\lambda/4,\lambda/4]\). If this is too limited, multiple measurements \(\phi_{i}\) associated with different wavelengths \(\lambda_{i}\) can be evaluated in combination [24]. The most prominent approach for this task is the so-called synthetic wavelength method [26], which is based on the phase difference of two phase distributions \(\phi_{1}\) and \(\phi_{2}\) corresponding to measurements with wavelengths \(\lambda_{1}\) and \(\lambda_{2}\)  
\[\phi_{2}(\vec{x})-\phi_{1}(\vec{x})=\frac{2\pi}{\Lambda}\cdot 2h(\vec{x}), \tag{3}\]  
with the synthetic wavelength \(\Lambda=(\lambda_{1}\cdot\lambda_{2})/(\lambda_{1}-\lambda_{2})\). This extends the interval of unique height values to \([-\Lambda/4,\Lambda/4]\), which can be significantly larger if the wavelengths are selected to be close together. Finally, in microscopy, because of the large numerical aperture \(NA\) of the imaging system, the depth of focus \(d_{f}=\lambda/(NA)^{2}\) is often smaller than the size of the object, leaving parts of it out of focus.