# Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser
## 1 Introduction
### Modeling for Theoretical Equations  
The following is the process of deriving the theoretical equation for the numerical calculations. Figure 2 presents the fiber confocal probe [6]. The light intensity obtained by the fiber-detector of the fiber confocal probe is given by the following equation [4; 5; 6]:  
\[I(\mathbf{r}_{\rm s})=|U|^{2}=\left|\int_{-\infty}^{\infty}c(\mathbf{m})O(\mathbf{m})\exp( 2\pi i\mathbf{r}_{\rm s}\cdot\mathbf{m})dm\right|^{2} \tag{1}\]  
where \(\mathbf{r_{\rm s}}\) is the confocal probe scanning position vector (\(\mathbf{x}_{\rm s},y_{\rm s},z_{\rm s}\)) in Figure 2, \(\mathbf{m}\) is the spatial frequency vector (\(m\), \(n\), \(\rm s\)), and \(\mathbf{c(m)}\) is the optical transfer function. Here, \(\mathbf{O(m)}\) is obtained by the Fourier transform of the 3D object function \(\mathbf{o(r)}\)[5; 6]. When the object to be measured is an ideal flat mirror with angles \(\theta_{X}\) and \(\theta_{Y}\), \(\mathbf{o(r)}\) is given by the following equation:  
\[o(\mathbf{r})=\exp\biggl{(}j\frac{2\pi}{\lambda}\cdot 2\tan\theta_{Y}x\biggr{)} \cdot\exp\biggl{(}j\frac{2\pi}{\lambda}\cdot 2\tan\theta_{X}y\biggr{)}\cdot \delta(z) \tag{2}\]  
Figure 1: A schematic flow of the \(z\)-position measurement algorithm based on the tracking local minimum method.  
where \(\delta\) is the delta function. By using Equation (1), Equation (2) is transformed into the following equation.  
\[\begin{split} I(\mathbf{r}_{s})&=\left|\int_{-\infty}^{ \infty}c(\mathbf{m})O(\mathbf{m})\exp(2\pi i\mathbf{r}_{s}\cdot\mathbf{m})d\mathbf{m}\right|^{2}\\ &=\left|\int_{-\infty}^{\infty}c(\mathbf{m})\cdot\int_{-\infty}^{ \infty}o(\mathbf{r})\exp(-2\pi i\mathbf{r}\cdot\mathbf{m})d\mathbf{r}\cdot\exp(2\pi i\mathbf{r}_{s }\cdot\mathbf{m})d\mathbf{m}\right|^{2}\\ &=\left|\int_{-\infty}^{\infty}c(\mathbf{m})\delta\Big{(}m-\frac{2 \tan\theta_{X}}{\lambda}\Big{)}\delta\Big{(}n-\frac{2\tan\theta_{X}}{\lambda} \Big{)}\exp(2\pi i\mathbf{r}_{s}\cdot\mathbf{m})d\mathbf{m}\right|^{2}\\ &=\left|\exp(2\pi i(x_{s}m_{0}+y_{s}  
\[A=\left(\frac{2\pi a\tau_{0}}{\lambda F}\right)^{2} \tag{8}\]  
\(\alpha_{0}\) in Equation (7) is the numerical aperture of the objective lens for the measurement object, and \(r_{0}\) in Equation (8) is the fiber spot size, given by the following equation.  
\[r_{0}=\frac{\rho_{c}}{\sqrt{2\ln V_{f}}} \tag{9}\]  
In Equation (9), \(\rho_{c}\) is the fiber core radius, and \(V_{f}\) is the fiber parameter, given by the following equation using the fiber aperture \(NA_{\text{fiber}}\)[6,22].  
\[V_{f}=\frac{2\pi}{\lambda}\rho_{c}NA_{\text{fiber}} \tag{10}\]  
Using the above equations, it is possible to numerically calculate the effect of the object tilt angle on the TL method. It should be noted that the above equations are theoretical equations for an ideal confocal optical system without the influence of aberrations \(\Phi\) other than axial chromatic aberration [35, 36, 37, 39, 40, 41] of the optical system since it is difficult to accurately predict the overall aberrations \(\Phi\) of the optical system [39]. Therefore, in this paper, numerical calculations are performed without considering the aberrations \(\Phi\), and then, the effect of the object tilt angle on the TL method is estimated. Finally, the prediction of experimental results in a confocal optical system including the aberrations \(\Phi\) is performed to show the validity of the experimental results.