## 4 Improvement of the Thermal Stability of the Femtosecond Laser Chromatic Confocal Probe  
Aiming to improve the thermal stability of the chromatic confocal probe, a second prototype measurement setup was newly designed and developed. 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 stabilities of the sample, the sample jig, and the mounting plate. This paper is focused on the improvement of the thermal stability of the chromatic confocal probe itself, since the thermal stabilities of the sample, the sample jigs, and the mounting plate change when the probe is employed for different applications. To improve the thermal stability of the femtosecond laser chromatic confocal probe, attentions were paid to choose a material having a low thermal expansion coefficient for the major mechanical component (Jig-B) in the chromatic confocal probe, while the optical path length was designed to be as short as possible. Figure 11 compares the previous first prototype measurement setup and the newly developed second prototype measurement setup. As can be seen in the figure, the size of the second prototype measurement setup is reduced to be 1/6 of that of the first prototype measurement setup. For a fair comparison, optical components identical to those employed in the first prototype measurement setup were employed in the second prototype measurement setup, while the material of Jig-B was switched from aluminum to Super Invar having a far lower thermal expansion coefficient compared with aluminum [35; 36]. All the optical components and jigs were mounted on an aluminum optical breadboard. At first, the improvement of the thermal stability of the second prototype measurement setup was verified in theoretical calculations. By the modifications described above, the optical path length in the second prototype measurement setup was reduced to be approximately 1/6 of that of the previous first prototype measurement setup. The thermal stability of the newly designed second prototype measurement setup is estimated in the same manner as the previous first prototype measurement setup described in the previous section of this paper. Table 4 summarizes the contribution of each factor in the thermal stability of the probe in the second prototype measurement setup. The thermal stability of the chromatic confocal probe in the second prototype measurement setup (0.0197 \(\upmu\)m\({}^{\circ}\)C) is far better than that in the first prototype measurement setup (4.86 \(\upmu\)m/\({}^{\circ}\)C). This result implies that the sub-micrometric stability of the femtosecond laser chromatic confocal probe can be achieved by the second prototype measurement setup. \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Factor** & **value** \\ \hline \multirow{4}{*}{Probe} & Refractive index instability of the chromatic objective lens & \(P_{1}=-0.0151\)\(\upmu\)m\({}^{\circ}\)C \\  & Refractive index instability of the surrounding air & \(P_{2}=-0.0247\)\(\upmu\)m\({}^{\circ}\)C \\  & Thermal expansion of mechanical jigs & \(P_{3}=4.96\)\(\upmu\)m\({}^{\circ}\)C \\  & Thermal expansion of optical path length & \(P_{4}=-0.0555\)\(\upmu\)m\({}^{\circ}\)C \\  & Thermal stability of the probe & \(P=P_{1}+P_{2}+P_{3}+P_{4}\) \\  & Thermal stability of the probe & \(=4.86\)\(\upmu\)m\({}^{\circ}\)C \\ \hline \multirow{4}{*}{(Sample unit)} & (Thermal expansion of a sample) & (\(S_{1}\)) \\  & (Thermal expansion of sample jig) & (\(S_{2}\)) \\  & (Thermal stability of the sample unit) & (\(S=S_{1}+S_{2}\)) \\ \hline \multirow{2}{*}{Mounting plate} & Thermal expansion of the mounting plate, corresponding to the thermal stability of the mounting plate & \(M=-0.735\)\(\upmu\)m\({}^{\circ}\)C \\ \hline \multirow{2}{*}{Thermal stability of the overall setup} & \(O=P+S+M\) \\  & Thermal stability of the overall setup & \(=4.13\)\(\upmu\)m\({}^{\circ}\)C \\ \hline \end{tabular}
\end{table}