## 2 Principle of the Extension of the Measurement Range
### Expansion of the Measurement Range  
The measurement range of the proposed fiber-based dual-detector chromatic confocal probe is determined by the spectral bandwidth of the mode-locked laser source and the chromatic objective employed in the optical setup. A thin plano-convex lens made of N-SF11 is employed in the developed optical setup as the chromatic objective. According to the thin-lens equation [3; 35], the relationship between a focal length of the chromatic objective \(f_{\lambda}\) and a refractive index \(N_{\lambda}\) is given as follows:  
\[f_{\lambda}=\frac{1}{(N_{\lambda}-1)(1/r_{1}-1/r_{2})} \tag{15}\]  
where \(r_{1}\) and \(r_{2}\) are curvature radii of the chromatic objective lens. The refractive index \(N_{\lambda}\) for the light wavelength \(\lambda\) can be obtained from the following Sellmeier equation [3]:  
\[N_{\lambda}=\sqrt{1+\frac{B_{1}\lambda^{2}}{\lambda^{2}-C_{1}}+\frac{B_{2} \lambda^{2}}{\lambda^{2}-C_{2}}+\frac{B_{3}\lambda^{2}}{\lambda^{2}-C_{3}}} \tag{16}\]  
where \(B_{i}\) and \(C_{i}\) are the parameters associated with N-SF11. The measurement range \(\Delta Z\) of the chromatic confocal probe is therefore given as follows:  
\[\Delta Z=|f_{\lambda 1}-f_{\lambda 2}| \tag{17}\]  
where \(f_{\lambda 1}\) and \(f_{\lambda 2}\) are the longest and shortest working wavelengths of the mode-locked femtosecond laser source, respectively. In the previous work by the authors [3], the measurement range of the developed chromatic confocal probe is derived from the linear relationship between the target position and the peak wavelength detected in the normalized intensity ratio. To further expand the measurement range of the fiber-based dual-detector chromatic confocal probe without any modifications on the optical setup, in this paper, a new method employing both the main-lobes and the side-lobes of the axial responses obtained from the normalized intensity ratio \(I_{\text{a}}\) (\(u\), \(u_{d}\)) and the invert normalized intensity ratio \(I_{\text{n}}\) (\(u\), \(u_{d}\)) is proposed. Figure 3 shows a variation of the normalized intensity \(I_{\text{a}}\) as a function of the Z-directional position of a measurement target at a specific wavelength \(\lambda\), which is referred to as the axial response [3]. Figure 3 also shows the axial response obtained from the invert normalized intensity ratio at the same wavelength \(\lambda\). As can be seen in the figures, not only the main-lobes but also the side-lobes of the axial response proposed in our previous work [3], which cannot be observed in the single axial responses of \(I_{\text{m}}\) and \(I_{\text{r}}\) without the normalization process, can clearly be distinguished in the axial responses. This is another benefit of our newly proposed axial response; the side-lobes are clearer compared with the single axial response expected from a Lorentzian shape, in which the side-lobe is fuzzy and cannot be identified easily. Figure 4a-c show the normalized intensity ratio \(I_{\text{a}}\) (\(u\), \(u_{d}\)) and the invert normalized intensity ratio \(I_{\text{n}}\)(\(u\), \(u_{d}\)) with different defocus \(d\), and Figure 4d shows the side-lobe spacing and the main-lobe to side-lobe spacing with respect to different defocus \(d\). In order to make good use of the side-lobes to extend the measurement range, we set defocus \(d\) to be 110 \(\upmu\)m, in which the side-lobes are more visible and identifiable than the cases where \(d\) is set to be within a range from 50 \(\upmu\)m to 80 \(\upmu\)m. Consequently, the employment of the side-lobes has a possibility of expanding the measurement range of the fiber-based dual-detector chromatic confocal probe, and theoretical calculations are therefore carried out in the following.