## 3 Characterization of the Thermal Stability of the Femtosecond Laser Chromatic Confocal Probe
 \tag{10}\]  
Modifying the above equation gives the following equation:  
\[\frac{df}{dT}=\frac{n_{\text{lens}}}{n_{\text{air}}(n_{\text{lens}}-n_{\text {air}})}.f.\frac{dn_{\text{air}}}{dT}\left(=-\frac{dZ}{dT}\right) \tag{11}\]  
Here, the air refractive indices of the lens (\(n_{\text{lens}}\)) and the air (\(n_{\text{air}}\)) at the reference temperature are 1.7432 (at \(\lambda=1560\) nm) and 1.00027, respectively. According to [PERSON]'s theoretical formula [33], the refractive index instability of the air (\(dn_{\text{air}}/dT\)) is approximately \(-1.0\times 10^{-6}/^{\circ}\)C. Therefore, from Equation (11), the contribution of the refractive index instability of the surrounding air in the thermal stability of the probe is calculated as \(P_{2}=-24.7\) nm/\({}^{\circ}\)C at \(\lambda=1560\) nm. (c) Contributions of the thermal expansion of mechanical jigs (\(P_{3}\)) and the mounting plate (\(M\))  
Figure 7 shows a detailed schematic of the chromatic confocal lens and the target sample in the measurement setup shown in Figure 2b. Now we consider the variation of the distance between the target sample and the chromatic objective lens in the setup. With the thermal expansion of components in the measurement setup such as the probe, the sample, the sample jig and the mounting plate, the distance between the chromatic objective lens and the sample can be changed. As the first step of the research, the thermal expansion of the sample and that of the sample jig are not considered for clarity in the following of this paper, since their contributions to the probe instability are expected to be relatively smaller than that of Jig-B, which is much longer than the other jigs in the probe. It should be noted that the thermal expansion of the sample and that of the sample jig will be taken into consideration for more accurate characterization of thermal stability in future work. In general, the linear thermal expansion of an object can be described by the following equation:  
\[d\ell=\alpha\cdot\ell\cdot dT \tag{12}\]  
where \(d\ell\) is the change in the length of the object with the original length \(\ell\), and \(\alpha\) is the linear coefficient of thermal expansion of the object [34]. Here, since the object is expected to expand the same amount in both the directions from its center, \(|dZ|\) can be treated to be equal to \(|d\ell/2|\). From this equation, the contribution of the thermal expansion of a component in the measurement setup can be expressed by the following equation: \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline \(D_{0}\) & \(-3.56\times 10^{-6}\) \\ \(D_{1}\) & \(9.20\times 10^{-9}\) \\ \(D_{2}\) & \(-2.10\times 10^{-11}\) \\ \(E_{0}\) & \(9.65\times 10^{-7}\) \\ \(E_{1}\) & \(1.44\times 10^{-9}\) \\ \(\lambda_{\text{TK}}\) & 0.294 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The physical parameters of N-SF11 employed in the theoretical calculations [32]. \[\frac{dZ}{dT}=\pm\frac{d\ell_{A}}{dT}=\pm\alpha_{B}\cdot\frac{\ell_{A}}{2} \tag{13}\]  
where the subscript A is the name of the component, and the subscript B is the kind of the component material. Here, the sign \"\(\pm\)\" in the equation is determined by the location of the object in the measurement setup. When the chromatic objective lens approaches the target sample due to the thermal expansion of the object, the sign is determined as \"positive (+)\". Based on the above equation, the contributions of the thermal expansion of mechanical jigs in the probe (\(P_{3}\)) and the mounting plate (\(M\)) are investigated. At first, the contribution of the thermal expansion of mechanical jigs in the probe (\(P_{3}\)) is investigated. The main contributor in the mechanical jigs is Jig-B, which connects the lens holder and Jig-A, since its length is the longest among the other jigs in the probe. It should be noted that the influences of other contributors such as Jig-A or lens holders are not considered at this stage, since the influence of Jig-B is dominant.