## 3 Characterization of the Thermal Stability of the Femtosecond Laser Chromatic Confocal Probe
It should also be noted that the influence of the displacement of the flat mirror in the \(X\)-direction due to the thermal expansion was expected to be relatively small compared with the main factor regarding a diameter of the focused beam (20 \(\upmu\)m) and out-of-flatness of the flat mirror employed in the following experiment (\(\lambda\)/100\(\theta\)633 nm). When the temperature rises, the chromatic objective lens approaches the target sample due to the thermal expansion of Jig-B, which is made of aluminum having a linear thermal expansion coefficient of \(23.1\times 10^{-6}\)\({}^{\circ}\)C\({}^{-1}\)[34] and its line length is designed to be 430 mm. From Equation (13), the contribution of the Jig-B in the thermal stability of the probe is evaluated as \(P_{3}=4.96\)\(\upmu\)m/\({}^{\circ}\)C.  
The contribution of the mounting plate \(M\) is also investigated. As opposed to the case of Jig-B, the chromatic objective lens moves away from the sample due to the thermal expansion of the mounting plate. The mounting plate is made of stainless steel having a linear thermal expansion coefficient of \(14.7\times 10^{-6}\)\({}^{\circ}\)C\({}^{-1}\)[34], and its line length is designed to be 100 mm. From Equation (13), the contribution of the mounting plate is therefore calculated as \(M=(dZ/dT)_{\text{Mounting plate}}=-0.735\)\(\upmu\)m/\({}^{\circ}\)C.  
(d) Contribution of the thermal expansion of optical path length (\(P_{4}\))  
Figure 8 shows a schematic of the optical setup in the chromatic confocal probe. Here, we define \(dZ_{\lambda i}\) as the axial distance between the focal position of the optical mode with the wavelength \(\lambda_{\text{i+1}}\) and that with the wavelength \(\lambda_{\text{i}}\). According to the geometric relationship in the optical setup, the defocus of the \(i\)th optical mode at the fiber detector \(dZ_{\lambda i}\) corresponding to \(dZ_{\lambda i}\) can be expressed by the following equation:  
\[dZ_{\lambda i}{}^{\prime}=\frac{2F^{2}dZ_{\lambda i}}{2 dZ_{\lambda i}(L-f_{ \lambda i}-F)+f_{\lambda i}{}^{2}} \tag{14}\]  
Figure 7: A detailed schematic of the chromatic confocal lens and the target sample in the chromatic confocal measurement setup shown in Figure 2b. where \(L\) is the distance between the chromatic objective lens and the fiber coupling lens, and \(F\) is the distance between the fiber coupling lens and the fiber detector. It should be noted \(F\) corresponds to the focal length of the fiber coupling lens. In this paper, the sum of \(L\) and \(F\) is treated to be the optical path length. In the first prototype chromatic confocal measurement setup, \(L\) and \(F\) were designed to be 500 mm and 16.6 mm, respectively. As can be seen in the equation, \(dZ_{\lambda i^{\prime}}\) is affected by \(L\) and \(F\), resulting in the change of the axial response \(I_{n}\); namely, the focused wavelength \(\lambda_{\text{focused}}\) is affected by the thermal expansion of the optical path length. However, a theoretical investigation of the contribution of the thermal expansion of optical path length (\(P_{4}\)) is not an easy task since the equation of the normalized axial response \(I_{n}\)[24] is quite complex. In this paper, experiments were therefore carried out to evaluate \(P_{4}\) quantitatively. At first, the contribution of the distance \(F\) between the fiber coupling lens and the fiber detector in \(P_{4}\) was investigated. The contribution of \(F\) in \(P_{4}\) can be calculated from the following equation:  
\[\frac{dZ}{dT}=\frac{dF}{dT}\cdot\frac{d\lambda}{dF}\cdot\frac{dZ}{d\lambda} \tag{15}\]  
In the above equation, \(d\)/\(dT\) and \(dZ\)/\(d\lambda\) can be estimated from theoretical calculations. In the first prototype measurement setup, the fiber coupling lens and the detector were placed on a jig made of aluminum having a linear thermal expansion coefficient of \(23.1\times 10^{-6}\)\({}^{\circ}\)C\({}^{-1}\)[34]. The line length of the jig, which corresponded to \(F\), was designed to be 16.6 mm in the first prototype measurement setup.