## 3 Characterization of the Thermal Stability of the Femtosecond Laser Chromatic Confocal Probe  
The possible factors contributing to the thermal stability of the overall measurement setup are shown in Table 1. The contribution of each factor is analyzed for characterizing the thermal stability of the probe itself. (a) Contribution of the refractive index instability of the chromatic objective lens (\(P_{1}\))  
The relationship between the refractive index of a lens placed in a vacuum and the temperature \(T_{0}\) (reference temperature @20 \({}^{\circ}\)C) is described by the following equation [31; 32]:  
\[\frac{dn_{\text{abs}}(\lambda_{i},T)}{dT}=\frac{n^{2}(\lambda_{i},T_{0})-1}{2 n(\lambda_{i},T_{0})}\bigg{(}D_{0}+2D_{1}\Delta T+3D_{2}\Delta T^{2}+\frac{E_{0}+2E_ {1}\Delta T}{\lambda_{i}^{2}-\lambda_{i\text{TK}}^{2}}\bigg{)} \tag{8}\]  
where \(\Delta T\) (=\(T\)\(-\)\(T_{0}\)) is temperature deviation from a reference temperature \(T_{0}\), and \(D_{p}\) (\(p\) = 0, 1, 2), \(E_{q}\) (\(q\) = 0, 1) and \(\lambda_{\text{TK}}\) are the parameters inherent to the lens material. The physical parameters associated with the employed chromatic objective lens in this paper are presented in Table 2[32]. Based on Equation (8), refractive index instability is calculated to be \(-1.0629\times 10^{-6}\)\({}^{\circ}\)C\({}^{-1}\) at \(\lambda_{i}\) = 1560 nm; this value corresponds to thermal instability of the focused wavelength \(d\lambda\)/\(dT\) of 0.0594 nm/\({}^{\circ}\)C from Equation (3). Therefore, the contribution of the refractive index instability of the chromatic objective lens in the thermal stability of the probe is calculated as \(P_{1}\) = (0.0594 nm/\({}^{\circ}\)C)(-255 nm/nm) = \(-\)15.1 nm/\({}^{\circ}\)C from Equation (7). \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Factor** & **Symbol** \\ \hline \multirow{8}{*}{Probe} & Refractive index instability of the chromatic objective lens & \(P_{1}\) = (\(dZ/dT\))\({}_{\text{P1}}\) \\  & Refractive index instability of the surrounding air & \(P_{2}\) = (\(dZ/dT\))\({}_{\text{P2}}\) \\  & Thermal expansion of mechanical jigs & \(P_{3}\) = (\(dZ/dT\))\({}_{\text{P3}}\) \\  & Thermal expansion of optical path length & \(P_{4}\) = (\(dZ/dT\))\({}_{\text{P4}}\) \\  & Thermal stability of the probe & \(P\) = (\(dZ/dT\))\({}_{\text{Probe}}\) \\  & Thermal expansion of the sample & \(S_{1}\) \\ Sample unit & Thermal expansion of the sample jig & \(S_{2}\) \\  & Thermal stability of the sample unit & \(S\) = (\(dZ/dT\))\({}_{\text{Sample unit}}\) \\  & Thermal expansion of the mounting plate (corresponding to & \(=S_{1}+S_{2}\) \\ \hline \multirow{2}{*}{Mounting plate} & Thermal expansion of the mounting plate (corresponding to & \(M\) = (\(dZ/dT\))\({}_{\text{Mounting plate}}\) \\  & the thermal stability of the mounting plate) & \(O\) = (\(dZ/dT\))\({}_{\text{Overall}}\) \\  & Thermal stability of the overall setup & \(=P+S+M\) \\ \hline \end{tabular}
\end{table} 
Table 1: Possible factors contributing to the thermal stability of the overall measurement setup. (b) Contribution of the refractive index instability of the surrounding air (\(P_{2}\))  
When the air refractive index \(n_{\text{air}}\) and the lens refractive index \(n_{\text{lens}}\) are defined, Equation (2) can be rewritten as follows:  
\[f_{\lambda_{i}}=\frac{1}{(n_{\text{lens}}/n_{\text{air}}-1)(1/R_{1}-1/R_{2})} \tag{9}\]  
Differentiating\(f_{\lambda 1}\) with respect to \(n_{\text{air}}\) gives the following equation:  
\[ \begin{array}{l}\frac{df}{dt_{\text{air}}}=\frac{1}{\left(n_{\text{lens}}/n _{\text{air}}-1\right)^{2}(1/R_{1}-1/R_{2})}\frac{n_{\text{lens}}}{n_{\text{ air}}^{2}}\\ \frac{n_{\text{lens}}}{n_{\text{air}}(n_{\text{lens}}-n_{\text{air}})}.f \end{array}