# Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser
## 3 Numerical Calculation Results for the Tracking Local Minimum Method  
Based on the equations derived in the previous chapter, numerical calculations of the measurement light confocal response curve \(I_{\text{M}}\) and the reference light confocal response curve \(I_{\text{R}}\) based on our proposed TL method were conducted by using the following Equations (11)-(13):  
\[I_{\text{M}}(z,\lambda,\theta_{X})=|c(l_{0},u)|^{2} \tag{11}\]  
\[I_{\text{R}}(z,\lambda,\theta_{X},z_{d})=|c(l_{0},u+u_{d}/2)|^{2} \tag{12}\]  
\[u_{d}(z,\lambda,z_{d})=-\frac{2\pi}{\lambda}z_{d}\frac{a^{2}}{F^{2}} \tag{13}\]  
In Equation (13), \(z_{d}\) is the offset position on the detector optical axis, which was set to 200 \(\upmu\)m based on previous experimental results [24], \(u_{d}\) is the normalized detector offset position \(z_{d}\), and \(F\) is the focal length of the objective lens for the fiber-detector. Here, values for lens focal length and fiber were taken from the previous study [22]. In addition, considering the circular symmetry of the optical system, only \(\theta_{X}\) was made to change in the calculations. Figure 3: A schematic diagram about a definition of the \(z\)-axis origin. Numerical calculations were performed using MATLAB R2021a for each light source wavelength \(\lambda\) and \(z\)-position. The light source spectral intensity was assumed to be constant regardless of wavelength. Furthermore, the objective lens was assumed to be an optical system aligned under ideal conditions, possessing no aberrations other than an axial chromatic aberration. Due to these assumptions, the effects of the light source spectral shape and the aberrations \(\Phi\) of the optical system on the confocal response cannot be obtained from the numerical results. It should be noted that these numerical simulations were performed to confirm the effect of the object tilt angle \(\theta_{r}\) (=\(\theta_{X}\)) on the response curve of an ideal fiber-detector confocal optical system. Figure 4 shows the results of the calculations for the light source whose spectrum is ranging from 1.48 um to 1.64 um at \(\theta_{X}\) = 0 degree. Figure 4a shows the measurement light confocal response curve \(I_{\text{M}}\) and the reference light spectrum \(I_{\text{Ref}}\), and Figure 4b shows the normalized intensity ratio \(I_{\text{TLM}}\) calculated by the TL method from the \(I_{\text{M}}\) and \(I_{\text{R}}\). It should be noted that the intensities of \(I_{\text{M}}\) and \(I_{\text{R}}\), and \(I_{\text{TLM}}\) were normalized by the maximum intensity for each wavelength. In addition, Figure 4c,d show cross-sections at \(z\) = 16 um in Figure 4a,b, and at the center wavelength \(\lambda\) = 1.56 um in Figure 4a,b, respectively. It should also be noted that Figure 4c shows the confocal response curve when the data are observed as a chromatic confocal probe, while Figure 4d shows the confocal response curve when the data are observed as a scanning confocal probe. From Figure 4c,d, it can be confirmed that the minimum point of the reference light confocal response \(I_{\text{R}}\) appears as a peak in the normalized intensity ratio \(I_{\text{TLM}}\) by the division operation, as shown in Figure 1. Here, for the sake of simplicity, the effect of the object tilt angle on the TL method was discussed with the wavelength fixed as shown in Figure 4d, and then the discussion based on the fixed wavelength extended the considerations to a chromatic confocal probe, as shown in Figure 4c. Then, the effect of the change in the object tilt angle \(\theta_{X}\) on the TL method was estimated with the wavelength \(\lambda\) fixed at 1.56 um. Figure 5a shows only the reference light confocal response curve \(I_{\text{R}}\) for changes in the object tilt angle \(\theta_{X}\), and its contour plot is shown in Figure 5b since the \(z\)-position of the target can be obtained by tracking the local minimum point of the \(I_{\text{R}}\).