## 3 Experiments and Discussions
### Evaluation of the Measurement Range and Resolution to Be Achieved by the Developed Setup  
Experiments were extended to evaluate the measurement range of the femtosecond laser differential chromatic confocal probe based on the off-axis differential method. In experiments, a target mirror was moved along the \(Z\)-direction in a step of 2 \(\upmu\)m by a piezoelectric (PZT) stage equipped with a capacitance displacement sensor (P-621.2 CL, Physik Instrumente, Karlsruhe, Germany). Two confocal outputs were obtained at each \(Z\)-position of the target mirror. Figure 12a shows the normalized  
Figure 11: Variation of the FWHM of the peak in the normalized chromatic confocal output due to the change in the off-axis displacement \(d_{\mathrm{m}}\) of the fiber detector in the reference detector. Figure 10: Normalized chromatic confocal output at each off-axis displacement \(d_{\mathrm{m}}\) of the fiber detector in the reference detector. (**a**) \(d_{\mathrm{m}}=0\)\(\upmu\)m; (**b**) \(d_{\mathrm{m}}=1.0\)\(\upmu\)m; (**c**) \(d_{\mathrm{m}}=2.0\)\(\upmu\)m; (**d**) \(d_{\mathrm{m}}=3.0\)\(\upmu\)m.  
chromatic confocal output \(I_{\text{ITLM}}\) obtained through arithmetic operations based on Equation (6) in a wavelength range of 1550-1610 nm. Figure 12b shows the variation of the peak wavelength detected from the obtained \(I_{\text{ITLM}}\) at each \(Z\)-position of the target mirror. The measurement sensitivity, defined as the ratio of the change in the peak wavelength to the \(Z\)-displacement of the target mirror, was evaluated to be \(-3.233\) nm/\(\upmu\)m with good linearity of \(R^{2}\) = 0.9996 over a \(Z\)-directional range of 46 \(\upmu\)m. It should be noted that the above results contained the uncertainty that came from the PZT stage employed in the experiments. Meanwhile, the PZT stage had a positioning resolution of 0.2 nm, which was much smaller than that obtained by the femtosecond laser chromatic confocal probe and was enough for the experiments. The verification experiments in this manuscript have been done as the first step of the investigation of the proposed off-axis differential method. Analyses on the measurement uncertainty of the femtosecond laser chromatic confocal probe with the proposed off-axis method will be carried out in future work while considering the positioning uncertainty of the PZT stage. Figure 13 shows the centroid wavelength \(\lambda_{\text{ITLM}\_C}\)[30] and peak wavelength \(\lambda_{\text{ITLM}}\) of \(I_{\text{ITLM}}\) encoded from the normalized chromatic confocal output obtained at each \(Z\)-position of the target mirror. As can be seen in the figure, the deviations between the peak wavelengths \(\lambda_{\text{ITLM}}\) and centroid wavelengths \(\lambda_{\text{ITLM}\_C}\) of \(I_{\text{ITLM}}\) were found to be within \(\pm\) 1.0 nm, which is negligibly small compared with the actual measurement resolution, and is much smaller than that observed in the conventional defocus method (maximum \(>\) 40 nm) [30]. These results mean that the peak wavelength \(\lambda_{\text{ITLM}}\) detected in the normalized chromatic confocal output in the newly proposed off-axis differential method can directly be adopted in the wavelength-to-displacement encoding, without the complex and time-consuming arithmetic operations for obtaining the centroid wavelength \(\lambda_{\text{ITLM}\_C}\). Experiments were then carried out to investigate the measurement resolution. In the following experiments, the target mirror was made to move along the \(Z\)-direction in a small step by the PZT stage. Figure 14a-c show the variation of the detected peak wavelength in the normalized chromatic output  
Figure 12: Experimental results of the wavelength-to-displacement encoding in the proposed off-axis differential chromatic confocal probe system.