## 3 Experiments and Discussions
### Optimum of the Misalignment Value  
The measurement resolution of the femtosecond laser differential chromatic confocal probe based on the TLM algorithm could be affected by the sharpness of the peak in the normalized chromatic confocal output [30]. In this paper, the appropriate off-axis detector shift \(d_{\text{m}}\) was investigated through experiments by using the full width at half maximum (FWHM) of the peak in the normalized chromatic confocal output \(I_{\text{ITLM}}\). At first, numerical calculations [28] were carried out based on Equation (5) to estimate the variation of the FWHM due to the change in \(d_{\text{m}}\). Figure 9 shows the result. As can be seen in the figure, the FWHM was estimated to reduce as there was an increase in the off-axis displacement \(d_{\text{m}}\). It should be noted that the simulation based on Equation (5) is valid under the ideal condition of a simplified optical configuration without any aberrations.  
Figure 8: Schematic of the wavelength-to-displacement encoding in the proposed off-axis differential chromatic confocal probe system. (**a**) A spectrum of the mode-locked femtosecond laser employed in this paper; (**b**) chromatic confocal outputs obtained by the detector units; (**c**) the normalized chromatic confocal output obtained through the arithmetic operation with \(I_{\text{1 mea}}\) and \(I_{\text{2 ref,m}}\).  
Figure 9: Variation of the full width at half maximum (FWHM) of the peak in the normalized chromatic confocal output estimated by the numerical calculations.  
Following the numerical calculations, experiments were carried out. In the experiments, the central wavelength \(\lambda_{0}\) was set to be approximately 1560 nm. The off-axis displacement \(d_{\mathrm{m}}\) of the fiber detector in the reference detector was changed in a step of 1 \(\upmu\)m, while obtaining the normalized chromatic confocal output at each \(d_{\mathrm{m}}\). Figure 10 shows the results. As can be seen in the figure, the FHWM of the peak in \(I_{\mathrm{ITLM}}\) was found to deviate with the change in \(d_{\mathrm{m}}\). Figure 11 summarizes the variation of the FWHM observed in Figure 10. As can be seen in the figures, FWHM values can clearly be figured out when the mismatch displacement (off-axis shift) is smaller than 3 \(\upmu\)m. Due to the effects of the numerical aperture (mismatch) on the normalized chromatic confocal output \(I_{\mathrm{ITLM}}\), it becomes difficult to find out the FWHM value for the case having large \(d_{\mathrm{m}}\). The FWHM value was found to be minimized around \(d_{\mathrm{m}}\) =1 \(\upmu\)m. The root cause of the difference observed between the results of numerical calculations and the experiments is not clear, and further investigation will be carried out in future work. Regarding the experimental results, \(d_{\mathrm{m}}\) was set to be 1 \(\upmu\)m in the following experiments. It should be pointed out that the determination of the appropriate off-axis detector shift \(d_{\mathrm{m}}\) is a less time-consuming task compared with the determination of the appropriate defocus in the conventional setup, where the defocus \(d\) needs to be changed in a wide range up to hundreds of micrometers depending on the confocal setup; this is another advantage of the newly proposed off-axis differential method compared with the conventional defocus method.