# Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser
## 4 Experimental Investigation Results for the Tracking Local Minimum Method
### Alignment Method of the Optical System
According to the results of \(P(0,\,-200)\) and \(P(1050,\,-200)\) in Figure 8b, the peak transition of the confocal response should be symmetric about the axis of \(\theta_{X}=0\) degree to satisfy \(y_{0}=0\,\upmu\)m.  
Figure 8: Numerical calculation results regarding the amount of absolute z-position change with the misalignment factor \(y_{0}\) and \(z_{0}\); (**a**) the ideal focusing \(z\)-position in normalized \(I_{\text{M}}\) without consideration about confocal response shift related to angle change \(\theta_{X}\); (**b**) the ideal focusing \(z\)-position in normalized \(I_{\text{M}}\) with consideration of confocal response shift related to angle change \(\theta_{X}\). Figure 7: A schematic diagram of the misalignment amount \(y_{0}\) and \(z_{0}\) of the measurement object; (**a**) focusing initial point on the surface of the measurement object with the misalignment amount \(y_{0}\) and \(z_{0}\); (**b**) focusing point on the surface of the measurement object after rotating by an angle \(\theta_{X}\). Based on the above considerations, an experiment was conducted to investigate the effect of the misalignment \(y_{0}\) between the rotation axis and the optical axis. The measurement light spectrum \(I_{\text{Mea}}\) was acquired by scanning the \(z\)-stage from \(-200\)\(\upmu\)m to \(+200\)\(\upmu\)m at 2\(\upmu\)m intervals with \(y_{0}=0\)\(\upmu\)m and \(y_{0}=\) about 1050\(\upmu\)m at each \(\theta_{X}\), respectively. Figure 9a,b shows the measurement light response curve \(I_{\text{M}}\) of the measurement light spectrum \(I_{\text{Mea}}\) obtained at the source center wavelength of 1560 nm when \(y_{0}=0\)\(\upmu\)m, and Figure 9c,d shows that under the condition of \(y_{0}=1050\)\(\upmu\)m, respectively. Here, the results obtained for each \(\theta_{X}\) in Figure 9a, normalized to 1, are shown in Figure 9b. Figure 9c,d are similar to Figure 9a,b, respectively. The results in Figure 9a,b are in good agreement with those in Figure 8b. Based on the above results, the Figure 9a,b condition was defined as the state in which the measurement object was ideally aligned with the center of rotation of the mirror without any misalignment \(y_{0}\) and \(z_{0}\) between the rotation axis of the measurement object and the optical axis. Experimental Analysis of the Influence of the Tilt Angle of the Measurement Object on the Tracking Local Minimum Method  
Finally, the influences of the tilt angle \(\theta_{X}\) of the measurement object on the tracking local minimum method for \(z\)-position measuring were evaluated. Figure 10 shows the obtained optical spectra ranging from 1.48\(\upmu\)m to 1.64\(\upmu\)m at \(\theta_{X}=0\) degree under the above alignment conditions. Figure 10a,b show the results of the measurement light spectrum  
Figure 9: Experimental results regarding the existence of the misalignment factors \(y_{0}\) and \(z_{0}\); (**a**) the relationship between \(\theta_{X}\) and \(z\)-position in \(I_{\text{M}}\) at \(y_{0}=0\)\(\upmu\)m, (**b**) the relationship between \(\theta_{X}\) and \(z\)-position in normalized \(I_{\text{M}}\) at \(y_{0}=0\)\(\upmu\)m, (**c**) the relationship between \(\theta_{X}\) and \(z\)-position in \(I_{\text{M}}\) at \(y_{0}=1050\)\(\upmu\)m, (**d**) The relationship between \(\theta_{X}\) and \(z\)-position in normalized \(I_{\text{M}}\) at \(y_{0}=1050\)\(\upmu\)m.  
and the reference light spectrum \(I_{\text{Ref}}\), respectively, and the normalized intensity ratio \(I_{\text{TLM}}\) calculated from the \(I_{\text{Mea}}\) and \(I_{\text{Ref}}\) based on the TL method. Note that, unlike Figure 4a,c, Figure 10a,b show the \(I_{\text{Mea}}\) and \(I_{\text{Ref}}\) including the light source spectrum. Figure 10c shows the cross-sections at \(z=30~{}\upmu\)m in Figure 10a,b.