## 3 Characterization of the Thermal Stability of the Femtosecond Laser Chromatic Confocal Probe
From Equation (12), the thermal expansion sensitivity of the jig \(dF\)/\(dT\) was evaluated to be 0.383 \(\upmu\)m/\({}^{\circ}\)C. In addition, \(dZ\)/\(d\lambda\) is already known as \(dZ\)/\(d\lambda=-255\) nm/nm from the theoretical investigation described in the previous section of this paper. Meanwhile, the term \(d\lambda\)/\(dF\) in Equation (15) still remains to be addressed for the evaluation of \(dZ\)/\(dT\). To obtain \(d\lambda\)/\(dF\), experiments were carried out by intentionally changing \(F\) in the measurement setup. A flat mirror was employed as the target sample, and was held stationary in the setup. The experiments were carried out in such a way that the parameter \(d\) in Figure 3, which corresponded to the defocus of Detector2 from the focal plane of the fiber coupling lens, was changed from 0 \(\upmu\)m to 300 \(\upmu\)m in a step of 20 \(\upmu\)m, while the focused wavelength at each step was being observed. In the experiments, the deviation of the parameter \(d\) (\(\Delta d\)) corresponded to \(-dF\) in Equation (15). It should be noted that the positive direction of the parameter \(d\) was set to be the one from the fiber detector to the fiber coupling lens. Figure 9a shows the relationship between \(d\) and the corresponding normalized axial responses \(I_{n}\) obtained in the experiments, and Figure 9b shows the relationship between \(d\) and the focused wavelength \(\lambda_{\text{focused}}\) extracted from the normalized axial responses \(I_{n}\) shown in Figure 9a. From the experimental results, the sensitivity \(d\lambda\)/\(dF\) was evaluated to be 0.5686 nm/\(\upmu\)m. From Equation (15), the contribution of \(F\) in \(P_{4}\) was therefore evaluated as \(dZ\)/\(dT\) = (0.383 \(\upmu\)m/\({}^{\circ}\)C)-(0.5686 nm/\(\upmu\)m)-(-255 nm/nm) = \(-\)55.5 nm/\({}^{\circ}\)C.  
Figure 8: Schematic of the optical setup in the chromatic confocal probe. The contribution of the distance \(L\) between the chromatic objective lens and the fiber coupling lens in \(P_{4}\) is also investigated. \(L\) (=500 mm) corresponds to the sum of the lengths of the mounting plate and Jig-A. To estimate the thermal expansion of \(L\) (\(\Delta L\)), a finite element model (FEM) shown in Figure 10 is employed in this paper. From the result of FEM analysis (by the Autodesk Nastran In-CAD), \(\Delta L\) is estimated to be approximately 10 \(\upmu\)m; this value is small compared with \(L\) (=500 mm), and the influence of \(\Delta L\) is negligibly small. From these results, it can be concluded that the contribution of \(F\) is much larger than that of \(L\) in the contribution of the thermal expansion of optical path length \(P_{4}\). As a result, \(P_{4}\) was evaluated to be \(-\)55.5 nm/\({}^{\circ}\)C. It should be noted that the uncertainty of \(L\) could increase the measurement uncertainty of the femtosecond laser chromatic confocal probe. Therefore, the optical path length \(L\) is expected to be reduced for the stabilization of the femtosecond chromatic confocal probe. Table 3 summarizes the contribution of each factor on the overall thermal stability of the setup. In summary, the thermal stability of the overall measurement setup (\(dZ/dT\))\({}_{\text{Overall}}\) is evaluated to be 4.13 \(\upmu\)m/\({}^{\circ}\)C from the theoretical investigation. A difference of approximately 1 \(\upmu\)m/\({}^{\circ}\)C can be found between the thermal stability estimated in theory (4.13 \(\upmu\)m/\({}^{\circ}\)C) and that obtained in the experiments (5.20 \(\upmu\)m/\({}^{\circ}\)C) described in the previous section of this paper; the difference is considered to be the sum of the small contributions from the other factors not considered in the theoretical calculations. From this result, it can be concluded that the reductions of the line lengths of Jigs-A and -B, and the  
Figure 10: Thermal expansion of each component estimated in the simulation based on a finite element model.