## 2 Thermal Stability of the Chromatic Confocal Measurement Setup
### Principle of the Femtosecond Laser Chromatic Confocal Probe
It should be noted that one of the fiber detectors (Detector1) is placed at the focal plane of L1, while the other (Detector2) is placed at the position with a defocus \(d\) (\(\
eq\)0) from the focal plane of L2. The sub-beams are then transmitted to an optical spectrum analyzer through the single-mode fibers to obtain the optical spectra of the measurement beam \(I_{\text{mea}}\) and the reference beam \(I_{\text{ref}}\), and then the normalized axial response \(I_{\text{n}}\) is analyzed. Figure 3: Principle of the mode-locked femtosecond laser chromatic confocal probe (BS: Beam splitter, PBS: Polarized beam splitter, QWP: Quarter-wave plate, L1 and L2: Fiber coupling lenses). In the proposed chromatic confocal probe, a fiber-based dual-detector is introduced to eliminate the influence of the non-smooth spectrum of the mode-locked laser source [20; 24]. Based on the spectra \(I_{\text{mea}}\) and \(I_{\text{ref}}\) obtained by Detector1 and Detector2, respectively, a normalized axial response \(I_{\text{n}}\) expressed by the following equation can be obtained:  
\[I_{\text{n}}=\frac{I_{\text{mea}}}{I_{\text{ref}}} \tag{4}\]  
Figure 4a shows an example of the spectra obtained by the pair of single-mode fibers. Each of the obtained spectra (\(I_{\text{mea}}\) and \(I_{\text{ref}}\)) is the consequence of the convolution of the axial response of the confocal setup and the spectrum of the light source. It should be noted that the influences of the reflectance of the target surface as well as the transmittances of optical components in the setup also influence the obtained spectra. Since the femtosecond laser source has the non-uniform spectrum, the obtained spectra \(I_{\text{mea}}\) and \(I_{\text{ref}}\) also become non-uniform. This means that it is difficult to directly extract the peaks in the light spectra, which are required to obtain the axial position information of a measurement target surface under inspection. Meanwhile, by introducing the normalized axial response \(I_{\text{n}}\) expressed by Equation (4), the influences of the non-uniformity in the spectrum of the light source as well as the surface reflectance can be canceled through the normalization process. Furthermore, by employing identical fiber coupling lenses and single-mode fibers for obtaining both \(I_{\text{mea}}\) and \(I_{\text{ref}}\), influences of the transmittances of these optical components can also be minimized through the normalization process. Figure 4b shows the normalized axial response \(I_{\text{n}}\) obtained by \(I_{\text{mea}}\) and \(I_{\text{ref}}\) shown in Figure 4a. As can be seen in this figure, the valley in \(I_{\text{ref}}\) is detected as the peak in the normalized axial response. It should be noted that the peak in the normalized axial response is sharper than those in \(I_{\text{mea}}\) and \(I_{\text{ref}}\). In the confocal system, the sharper peak in the axial response contributes to the realization of higher accuracy in identifying the focused wavelength \(\lambda_{\text{focused}}\), and hence the higher resolution in measurement of the axial displacement. Figure 5 shows the procedure of how to obtain the axial position information of the measurement target surface under inspection from the normalized axial response. Figure 5a shows the normalized axial response \(I_{\text{n}}\). The wavelength \(\lambda_{\text{focused}}\) at the peak in the normalized axial response, which is referred to as the focused wavelength, can be obtained by the following equation:  
\[\lambda_{\text{ focused}}=\frac{\sum(I_{\text{n}}(\lambda_{i})\times\lambda_{i })}{\sum I_{\text{n}}(\lambda_{i})} \tag{5}\]  
Figure 5b shows a schematic of the relationship between the focused wavelength \(\lambda_{\text{focused}}\) and the \(Z\) position information of the measurement target surface.