# 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  
Next, efforts were made to reduce the misalignment \(y_{0}\) between the center of rotation of the measurement object and the optical axis and the misalignment \(z_{0}\) between the center of rotation of the measurement object and the surface of the measurement object (Figure 7a). Since the purpose of this study is to investigate the influence of the tilt of the measurement object on the \(z\)-position measurement results, it is important to reduce the influence of the misalignment quantities \(y_{0}\) and \(z_{0}\) on the confocal responses. As can be seen in Equation (4), the TL method defines the object \(z\)-position origin as the position where the light source center wavelength of 1560 nm comes into focus on the surface of the measurement object. Therefore, when the misalignments \(y_{0}\) and \(z_{0}\) exist, the light ray with a wavelength of 1560 nm is aligned to focus on the point \(P(y_{0},z_{0})\), as shown in Figure 7a. In this case, when \(\theta_{X}\) is given to the mirror using a rotary stage in the presence of \(y_{0}\) and \(z_{0}\), the point at which the light of wavelength 1560 nm is focused will change from point \(P\) to point \(P^{\prime}\). Figure 7b shows this situation. The coordinates of point \(P^{\prime}(y_{0}^{\prime},z_{0}^{\prime})\) are given geometrically by the following equation. \[\left\{ \begin{array}{c}{y_{0}}^{\prime}=y_{0}\\ {z_{0}}^{\prime}=\tan\theta_{X}\cdot(y_{0}+\sin\theta_{X}\cdot z_{0})+\cos \theta_{X}\cdot z_{0}\end{array} \right. \tag{14}\]  
From the above equation, the amount of a focusing \(z\)-position change (\(Z\)) at which the confocal response peaks due to the presence of \(y_{0}\) and \(z_{0}\) misalignment between the rotation axis and the optical axis can be expressed as follows. \[Z={z_{0}}^{\prime}-z_{0}+k\sin^{2}\theta_{X} \tag{15}\]  
This leads to the focusing \(z\)-position change (\(Z\)) due to the existence of misalignment factors \(y_{0}\) and \(z_{0}\). The third term, \(k\sin^{2}\theta_{X}\), was added to represent the amount of peak shift caused by aberrations \(\Phi\) in the confocal optics. It should be noted that the reason for adding the term proportional to the square of the \(\sin\) function is that the results are in good agreement with the experimental results, which will be described later. Figure 6: A schematic diagram of the confocal optical system used in the experiment; (**a**) information about the measurement object and fiber-detectors, (**b**) information about the optical components. BS: beam splitter; PBS: polarized beam splitter; QWP: quarter-wave plate. Figure 8a,b show the results of calculating the amount of \(z\)-position change (\(Z\)) at which the confocal response peaks using Equations (14) and (15). Calculations were performed with \(P(y_{0},\,z_{0})=P(0,\,-200)\) and \(P(1050,\,-200)\). Here, results in Figure 8a were calculated under the condition of \(k=0\), while Figure 8b was calculated under the condition of \(k=8000\). The results for \(P(0,\,-200)\) shown in Figure 8a indicate that the focusing \(z\)-position change (\(Z\)) where the confocal response peaks show a transition amount of fewer than \(1\,\upmu\)m within the range of \(\,|\,\theta_{X}\,|=5\) degrees. This suggests that the effect on the confocal response becomes small at \(\,|\,z_{0}\,|<\,200\,\upmu\)m, which is expected from the mechanical installation error. Then, from the results of \(P(1050,\,-200)\) shown in Figure 8a, it can be confirmed that the presence of \(y_{0}\) induces a linear position change in the \(z\)-position where the confocal response peaks with the change in the measurement object angle. This indicates that it is important to set \(y_{0}\) to be \(0\,\upmu\)m to investigate the effect of the object tilt angle on the TL method independently, which is the objective of this study.