## 2 Thermal Stability of the Chromatic Confocal Measurement Setup  
### Principle of the Femtosecond Laser Chromatic Confocal Probe  
A schematic of the design of the overall chromatic confocal measurement setup, which is referred to as the first prototype measurement setup, is shown in Figure 1. The setup is composed of a mode-locked femtosecond laser chromatic confocal probe, which is referred to as the first prototype probe, a sample unit composed of the sample and the sample jigs, and the mounting plate on which the probe and the sample unit are placed. A photograph of the setup is shown in Figure 2. Figure 2a shows the photograph of the overall first prototype measurement setup, and Figure 2b shows the sample unit. The probe and the sample unit are placed on the mounting plate made of stainless steel. Figure 3 shows the principle of the proposed femtosecond laser chromatic confocal probe [24; 25]. In the probe, a mode-locked femtosecond laser source is employed as the light source. The wavelength of \(i\)th mode \(\lambda_{i}\) in a mode-locked femtosecond laser is well known to be expressed by the following equation [26; 27; 28]:  
\[\lambda_{i}=\frac{c}{i\
u_{\text{rep}}+\
u_{\text{CEO}}} \tag{1}\]  
where \(c\) is the speed of light, \(\
u_{\text{rep}}\) is the pulse repetition rate, and \(\
u_{\text{CEO}}\) is the carrier envelope offset frequency. The femtosecond laser beam from the laser source (MenloSystems Inc., Germany, C-Fiber-SYNC100) with an average output power of 30 mW was transmitted to the optical setup by a single-mode fiber. A pulse repetition rate of the femtosecond laser was set to be 100 MHz. A spectral  
Figure 1: A schematic of the first prototype chromatic confocal measurement setup. Figure 2: Photograph of the first prototype chromatic confocal measurement setup: (**a**) Photograph of the overall chromatic confocal measurement setup; (**b**) Photograph of the sample unit composed of a sample and sample jigs. (CL: Collimating lens, BS: Beam splitter, PBS: Polarized beam splitter, QWP: Quarter-wave plate, L1 and L2: Fiber coupling lenses). bandwidth of the femtosecond laser ranged from 1480 nm to 1640 nm. The femtosecond laser beam from the fiber is collimated by a collimating lens (CL), and is made incident to a chromatic objective lens after passing through a polarized beam splitter (PBS) and a quarter-wave plate (QWP). A plano-convex lens made of N-SF11 is employed as the chromatic objective lens. Due to the chromatic aberration of the chromatic objective lens employed in the probe, each mode in the collimated femtosecond laser beam is focused on a different focal plane of the lens. The focal length \(f_{\lambda i}\) of the chromatic objective lens corresponding to the \(i\)th mode is expressed as follows:  
\[f_{\lambda_{i}}=\frac{1}{\Big{(}n_{\lambda_{i}}-1\Big{)}(1/R_{1}-1/R_{2})} \tag{2}\]  
where \(R_{1}\) and \(R_{2}\) represent the curvature radii of the chromatic objective lens [29], and \(n_{\lambda i}\) is the refractive index of the lens material that can be expressed by the following equation:  
\[n_{\lambda_{i}}=\sqrt{1+\frac{B_{1}\lambda_{i}{}^{2}}{\lambda_{i}{}^{2}-C_{1}} +\frac{B_{2}\lambda_{i}{}^{2}}{\lambda_{i}{}^{2}-C_{2}}+\frac{B_{3}\lambda_{i} {}^{2}}{\lambda_{i}{}^{2}-C_{3}}} \tag{3}\]  
where \(B_{k}\) and \(C_{k}\) (\(k\) = 1, 2, 3) are the parameters inherent to the lens material [30]. The relationship between \(\lambda_{i}\) and \(f_{\lambda i}\) can therefore be uniquely identified by Equation (2). The laser beam reflected from the flat mirror surface passes the chromatic objective lens and the PBS again, and is reflected by the PBS. After that, the laser beam is split into two sub-beams (the measurement beam and the reference beam) by a beam splitter (BS). In the probe, a pair of identical single-mode fibers is employed as point detectors. The measurement beam is coupled into one of the single-mode fibers by using a fiber coupling lens (L1), while the reference beam is coupled into the other single-mode fiber by using another fiber coupling lens (L2).