## 2 Thermal Stability of the Chromatic Confocal Measurement Setup
### Principle of the Femtosecond Laser Chromatic Confocal Probe
The \(Z\) displacement of the target sample surface can be calculated by the following equation:  
Figure 2.0  
where \(dZ/d\lambda\) corresponds to the \(Z\) displacement detection sensitivity with respect to the change in the focused wavelength \(\lambda_{\text{focused}}\). The sensitivity \(dZ/d\lambda\) can be theoretically calculated, since the relationship between the focal length\(f_{Ai}\) of the chromatic objective lens and the corresponding light wavelength \(\lambda_{i}\) can be calculated based on Equations (2) and (3). According to the specification of the chromatic objective lens employed in the first prototype confocal probe (the details of which can be found in Reference [24]), \(f_{i}\)-\(\lambda_{i}\) sensitivity (\(df\!f\!d\lambda\)) is calculated to be 255 nm/nm. It should be noted that the positive direction of the \(Z\) displacement of the target sample surface is set as the direction the sample approaches the chromatic objective lens; namely, \(df\!f\!d\lambda=(-dZ/d\lambda)\). The sensitivity \(dZ/d\lambda\) is therefore calculated to be \(-\)255 nm/nm. Experimental results have demonstrated that the first prototype confocal probe has an axial resolution of 30 nm [24]. Since the fiber-based dual-detector normalization method makes it possible to measure the axial displacement without the influence of the non-uniform spectrum of the light source, an axial displacement measurement range of 40 \(\upmu\)m has been realized by fully utilizing a spectral range of 160 nm of the femtosecond laser [24]. The displacement measurement is also not influenced by the surface reflectance of the target sample, as well as the transmittance of optical components in the probe. Figure 2.1 Schematics of detected spectra by a fiber-based dual-detector. (**a**) An example of the spectra obtained by the pair of single-mode fibers; (**b**) The normalized axial response curve \(I_{\mathrm{n}}\) obtained by \(I_{\mathrm{mea}}\) and \(I_{\mathrm{ref}}\) shown in (**a**). Figure 2.2 Principle of a method to obtain the \(Z\) displacement from the normalized axial response \(I_{\mathrm{n}}\) and \(Z\)-\(\uplambda\) sensitivity: (**a**) Focused wavelength extracted from the normalized axial response \(I_{\mathrm{n}}\); (**b**) \(Z\) displacement calculated by using the Z-\(\uplambda\) sensitivity. On the other hand, due to the slow measurement throughput of the optical spectrum analyzer, it takes quite a long time (approximately 8 s) to take the spectra \(I_{\text{mea}}\) and \(I_{\text{ref}}\) at a position in a sample plane. Good thermal stability of the probe is thus required to apply the probe for long-term displacement measurement or surface profile measurement with multi-point scanning. For this purpose, it is important to identify the thermal stability of the developed probe in axial displacement measurement. It should be noted that the thermal stability of the overall chromatic confocal measurement setup is contributed by not only the thermal stability of the chromatic confocal probe itself but also the thermal stability of the sample, the sample jig, and the mounting plate (Figure 1). From the point of view of the developed femtosecond laser chromatic confocal probe, it is necessary to separate the thermal stability of the probe from those of the sample, the sample jig, and the mounting plate. Since it is difficult to directly measure the thermal stability of the probe, in the rest of this section, the thermal stability of the overall measurement setup shown in Figure 1 is measured first. Then an analytical work is carried out in the next section to characterize the thermal stability of the first prototype probe. Experimental Verification of the Thermal Stability of the Overall Chromatic Confocal Measurement Setup  
It is difficult to directly obtain the thermal stability of the overall measurement setup for axial displacement measurement (\(dZ/dT\)). Meanwhile, the Z-\(\lambda\) sensitivity (\(dZ/d\lambda\)) can be theoretically calculated as described above.