## 3 Effect of Incident Angle of the Light on the Measurement Sensitivity
### Simulation Assessment of Sensitivity  
The relationship between the incident angle \(\theta\) of the laser beam and the sensitivity was theoretically examined using the 3D ray-tracing method. The laser beam was assumed to have an ideal Gaussian intensity distribution. First, random numbers with a Gaussian intensity distribution were generated using the Box-Muller method [11], and one hundred million rays having this distribution were irradiated from the lens. After that the rays were traced through the stylus shaft, and the number of rays received at the respective photodiode was calculated. Here, the intensities received by respective photodiode were expressed in terms of the relative intensities for all the rays as \(I_{\rm X\_SIM}\) (%). Figure 5 shows schematic diagram of the ray-tracing method. The cross-section of the stylus shaft  
Figure 4: Experimentally measured values of \(I_{\rm X}\) for angles \(\theta=0^{\circ}\), \(10^{\circ}\) and \(20^{\circ}\) (displacement of the stylus shaft along Y-axis = \(\pm 2.5\)\(\upmu\)m). Figure 3: Experimental apparatus for measuring the sensitivity of the optical fiber probe (the stylus shaft is inclined in the X-direction). has an elliptical shape that changes with the incident angle \(\theta\) of the laser beam. Tables 1 and 2 list the conditions of ray-tracing and the parameters used for ray-tracing. The coordinate, P\({}_{3}\)(\(x_{3}\), \(y_{3}\), \(z_{3}\)), of the laser irradiation position on the photodiode can be calculated by Equations (2)-(4). \[x_{3}\ =\ L_{1}+\frac{D\sec\theta}{2} \tag{2}\]  
\[y_{3}\ =\ y_{2}+\left(L_{1}-x_{2}+\frac{D\sec\theta}{2}\right)\tan\theta_{2X} \tag{3}\]  
\[z_{3}\ =\ z_{2}+\left(x_{3}-x_{2}\right)\tan\theta_{2Z} \tag{4}\]  
In the measurement system, when the stylus tip makes contact with the surface being measured and the output signal \(I_{\mathrm{X}}\) is equal to the threshold value, which means that the laser irradiation position of the stylus shaft is displaced by 30 nm, the contact between the stylus tip and the surface being measured is detected and the coordinates of the XYZ precision stage are recorded. The tilt angle of the stylus shaft of the laser irradiation part after this contact detection is \(3.1\times 10^{-3}\) degrees. This angle is considerably smaller than the laser irradiation angle of 10, 20 and 30 degrees, so the effect of stylus tilt can be ignored in the simulation. \begin{table}
\begin{tabular}{c c} \hline
**Light Source** & **Wavelength: 630 nm** \\ \hline Diameter of the stylus shaft, \(D\) & 70 \(\upmu\)m \\ \hline Refractive index of optical fiber (stylus shaft) & 1.457 \\ \hline Length from planes 0 to 1, \(L_{0}\) & 20 \(\upmu\)m \\ \hline Length from planes 2 to 3, \(L_{1}\) & 40 mm \\ \hline Numerical aperture of the lens & 0.13 \\ \hline Displacement of the stylus shaft, \(Y\) & –8–8 \(\upmu\)m \\ \hline Diameter of laser spot, \(D_{S}\) & 10 \(\upmu\)m \\ \hline Region of the dual-element photodiode & \(5\times 10\) mm \\ \hline \end{tabular}
\end{table} 
Table 1: Conditions of ray-tracing. Figure 5: Schematic diagram of the ray-tracing method. Figure 6 shows the results of the ray-tracing when the stylus shaft was displaced from \(-2.5\) to \(2.5\)\(\upmu\)m in the Y-direction with the incident angle \(\theta\) of the laser beam as a parameter. The sensitivity in the range of \(-1\) to \(1\)\(\upmu\)m in the Y-direction is \(17.1\), \(17.5\), \(18.5\) and \(20.1\%\)/\(\upmu\)m for \(\theta\) = \(0^{\circ}\), \(10^{\circ}\), \(20\) and \(30^{\circ}\), respectively. It is enhanced by \(2.3\), \(8.2\), and \(17.5\%\) for \(\theta\) = \(10^{\circ}\), \(20^{\circ}\) and \(30^{\circ}\), respectively, compared to \(0^{\circ}\). It is confirmed here that the sensitivity increases with the increase in the incident angle \(\theta\), which is consistent with the experimental results. However, the position of displacement Y, where the experimental output signal \(I_{\text{X}}\) is minimum or maximum, is slightly different from that of \(I_{\text{X\_SIM}}\) in the simulation.