# Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser
## 5 Conclusions  
In this study, the effect of the object tilt angle on the TL method, which is a method for detecting the object \(z\)-position in our proposed dual-fiber-detection confocal optical system, has been investigated. It should be noted that the proposed TL method uses \(z\)-\(\lambda\) sensitivity to convert from the wavelength of the confocal response curve local minimum point to the \(z\)-position. At first, a theoretical equation was developed to investigate the effect of the object tilt angle on the confocal response curve. Numerical calculations were then performed based on the constructed equation, and the results have revealed that the  
Figure 12: The influence of the tilt angle existence on the \(z\)-\(\lambda\) sensitivity based on the TL method; (**a**) the \(z\)-\(\lambda\) sensitivity (\(dz\)/\(d\lambda\)) and \(z\)-intercept (\(z_{0}\)) obtained in the experiments in each \(\theta_{X}\), (**b**) the linear approximation \(R^{2}\) obtained in the experiments in each \(\theta_{X}\). hem of the sidelobe of the confocal response curve has widened with the object tilt angle \(\theta_{X}\). In conjunction with this, the \(z\)-position intercept of the \(z\)-\(\lambda\) sensitivity line, which detects the \(z\)-position of the measurement object from the wavelength of the confocal response curve minimum, showed a parallel transition in the \(z\)-position positive direction. This indicates that the effect of the object tilt angle \(\theta_{X}\) cannot be ignored in the developed chromatic confocal probe employing the TL method that detects the \(z\)-position from the wavelength in focusing on the surface of the measurement object. Here, the aberrations \(\Phi\) in the optical system were not taken into account when examining the theoretical equation. This is the reason why it is difficult to completely estimate the aberrations in the entire optical system. The inclusion of aberrations should be considered in future studies. In addition, an experimental study was conducted to determine the effect of the object tilt angle on the TL method. At first, experimental investigations realized that the dual-fiber-detector confocal optics contains aberrations, and that the peak wavelength of the confocal response more shifts with the object tilt angle \(\theta_{X}\) due to the effect of these aberrations than estimated in numerical calculations. This factor was considered due to the effect of aberrations \(\Phi\) existing in the confocal optics. These experimental results mean that the \(z\)-\(\lambda\) sensitivity (\(dz/d\lambda\)) and \(z\)-intercept \(z_{0}\) change with the object tilt angle \(\theta_{X}\). In the future, new confocal optics that are not affected by the aberrations \(\Phi\) of the confocal optics will be constructed regarding the aberration correction method described in references [35; 39]. For example, aberrations \(\Phi\) in the optical system can be corrected by changing the optical system tube length [35]. The investigations of the effect of the object tilt angle on the TL method in this paper were conducted without changing the parameters of the lens aperture and beam size of the developed confocal optics in previous works [26; 27]. Future work includes optimization of the lens aperture and beam size parameters of the confocal optics. In addition, improving the \(z\)-position detection method to a new method that can simultaneously measure the \(z\)-position and acquire the angle \(\theta_{X}\) information of the measurement point will be carried out as future work. Conceptualization, W.G. and Y.S. ; methodology, R.S., Y.S. and W.G. ; software, R.S. and Y.S. ; validation, Y.S., W.G. and H.M.; formal analysis, R.S., H.M., Y.S. and W.G. ; investigation, R.S., H.M. and Y.S. ; resources, Y.S. and W.G. ; data curation, Y.S. and W.G. ; writing--original draft preparation, R.S. and Y.S. ; writing--review and editing, Y.S. and W.G. ; visualization, W.G. and Y.S. ; supervision, W.G. ; project administration, W.G. ; funding acquisition, W.G., Y.S.