# Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser
## 1 Introduction
### Methods Based on the Chromatic Aberrations of a Simple Lens
Denoting the focal length of the simple lens for the \(i\)th mode in the mode-locked femtosecond laser as \(f_{i}\), the lateral displacement of the focused mode \(d_{i}\) with respect to the optical axis of the simple lens due to the angular displacement \(\theta\) of the target reflector can be expressed as follows [58]:  
\[d_{i}=f_{i}\tan 2\theta \tag{3}\]  
On the assumption that the mode-locked femtosecond laser has a uniform spectrum over its spectral range, and the optical frequency of the \(j\)th mode is the peak frequency in the spectrum of the light rays captured by the fiber detector at the condition where \(\theta=\theta_{0}\) (\(\
eq\)0), the following relationship should be satisfied [58]:  
\[d_{\text{fiber}}=d_{j}=f_{j}\tan 2\theta_{0} \tag{4}\]  
In the same manner, the following Equation should be satisfied in the case with the angular displacement \(\theta_{0}+\Delta\theta\) and the corresponding lateral displacement \(d_{k}\) of the \(k\)th mode at the peak in the spectrum of the captured light rays [58]:  
\[d_{\text{fiber}}=d_{k}=f_{k}\tan 2(\theta_{0}+\Delta\theta) \tag{5}\]  
From Equations (4) and (5), the angular displacement of the target reflector \(\Delta\theta\) can be obtained as follows [58]:  
\[\Delta\theta=\frac{1}{2}\tan^{-1}\left(\frac{f_{j}}{f_{k}}\tan 2\theta_{0} \right)-\theta_{0} \tag{6}\]  
Since \(\theta_{0}\) is known as the design parameter in the optical setup, \(\Delta\theta\) can be obtained based on Equation (6) by calculating \(f_{j}\) and \(f_{k}\) based on the lens equation by using the detected peak frequencies  
Figure 11: Optical setup for measurement of the small angular displacement of an object based on the chromatic aberrations of a simple lens [58]. \(v_{j}\) and \(v_{k}\) of the \(j\)th and \(k\)th modes, respectively. It should be noted that the setup shown in Figure 11 is similar to that shown in Figure 7 based on the dispersive characteristics of a grating reflector. One of the advantages of the optical setup shown in Figure 11 is that a reflective-type grating reflector is not required for measurement; this contributes to building up the optical setup to a compact size. In addition, an arbitrary reflective surface on an object can be measured by the setup shown in Figure 11. It should also be noted that the angle sensor with the diffraction grating shown in Figure 7 can be employed for the evaluation of an object with a reflective surface without grating pattern structures by modifying the optical setup as shown in Figure 2b. However, in this case, attention should be paid to misalignments of the optical components, as well as the alignment of an object under inspection. Figure 12a shows the change in the light intensity of each mode captured by the fiber detector analyzed in optical frequency domain. In the figure, only several modes from 185 THz to 200 THz in a frequency difference of 5 THz are plotted for the sake of clarity. As can be seen in the figure, the light intensity of each mode has been changed by the angular displacement of the mirror reflector. Figure 12b shows the peak frequencies observed at each angular position of the mirror reflector. By detecting the peak frequency in the spectrum of the light rays captured by the fiber detector, the angular displacement of the mirror reflector can thus be detected.