## 3 Experiments and Discussions  
### Experimental Confocal Configuration  
To verify the feasibility of the proposed off-axis differential method, experiments were carried out by using the developed differential chromatic confocal system, a schematic diagram of which is shown in Figure 6. For a fair comparison, the optical components in the configuration are the same as those adopted in the previous work by the authors of [30] based on the defocus setup, except a manual stage for the adjustment of the off-axis detector shift \(d_{\mathrm{m}}\) of the fiber detector in the reference fiber detector unit. In this paper, \(d_{m}\) in the positive \(Y\)-direction was treated to be positive. Figure 5: A schematic of the tracking local minimum (TLM) algorithm. (**a**) The normalized output obtained from the reference output and the measurement output [30]; (**b**) a schematic of the wavelength-to-displacement encoding in the TLM method. At first, the coarse adjustment of the off-axis fiber in the reference detector was carried out, since too much offset \(d_{\mathrm{m}}\) could result in the local minimum in the reference detector output out of the wavelength range of the optical spectrum analyzer employed in the setup. The amount of \(d_{\mathrm{m}}\) given to the fiber detector in the reference detector unit was adjusted so that the first local minimum wavelength of the confocal output \(I_{\mathrm{2 ref\_m}}\) could be captured at each \(Z\)-directional position of a target surface. It should be noted that the single-mode fiber (SMF-28e, Corning Inc.) had a core diameter of 8.2 \(\upmu\)m and an NA of 0.14. The detector was composed of a single-mode fiber and an optical spectrum analyzer. At each \(Z\)-position, it took about 8 s to capture the spectrum by the measurement and reference detectors. A bandwidth of the optical spectrum analyzer was set to be 0.5 nm with a sampling interval of 0.1 nm. Figure 7 shows the noise reduction performances in the comparison between the conventional defocus differential method and the proposed off-axis differential method. As can be seen in the figures, the out-of-focus noises at the ends of the measurement range (positions A1 and B1) could be found in the conventional defocus differential method. Meanwhile, the out-of-focus noises at the ends of the measurement range (positions A2 and B2) were found to be reduced in the newly proposed off-axis differential method; this contributes to achieving a wider measurement range. These results mean that a better signal-to-noise ratio (SNR) of the normalized chromatic output \(I_{\mathrm{ITLM}}\) can be expected by the newly proposed off-axis differential method. It should be noted that this can also contribute to carrying out stable detection of the peak wavelength to be employed for the detection of the displacement of a surface under inspection through the wavelength-to-displacement encoding. Figure 6: A schematic diagram of the proposed off-axis differential chromatic confocal probe configuration with a mode-locked femtosecond laser source [30]. Figure 8a shows the spectrum of the mode-locked femtosecond laser employed in the experiments, and Figure 8b shows the typical spectra of the two confocal outputs, \(I_{\mathrm{1 mea}}\) and \(I_{\mathrm{2 ref\_m}}\), captured by the dual-fiber detector units. As can be seen in Figure 8b, due to the nonuniform spectrum of the utilized mode-locked femtosecond laser source shown in Figure 8a, it is not so easy to figure out peak wavelengths in the obtained spectra. Figure 8c shows the normalized chromatic confocal output \(I_{\mathrm{ITLM}}\) obtained from the spectra shown in Figure 8a,b. As can be seen in Figure 8c, it is much easier to figure out the peak in the spectrum with a much smaller full width at half maximum (FWHM) value. Figure 7: Noise reduction performances in the two methods.