# Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser
## 1 Introduction
### A Method Employing the Dispersive Characteristics of a Diffraction Grating
#### 2.1.2 Light Intensity Detecting-Type Mode-Locked Femtosecond Laser Autocollimator  
In the optical setups shown in Section 2.1.1., the resolution of the angle measurement can be improved by increasing the distance between the grating reflector and the detector. However, in most of the applications, the space available for the optical setup of the angle scale comb is limited; this fact means that there is a trade-off relationship between the resolution of angle measurement and the size of the optical setup. This issue can be addressed by introducing the laser autocollimation into the optical frequency comb. Figure 3a shows an example of the optical setup of a conventional laser autocollimator, which is based on the laser autocollimation [25]. A single-mode laser emitted from a laser source such as a laser diode (LD) is collimated by a collimating lens (CL) and is then made incident to a plane mirror reflector through a polarizing beam splitter (PBS) and a quarter-wave plate (QWP). The reflected beam from the plane mirror reflector goes through the QWP again, is reflected by the PBS, and is then captured by an autocollimation unit composed of a collimator objective (CO) and a photodiode (PD). In the autocollimation unit, the active cell of the PD is placed at the back focal plane of the CO so that the laser beam made incident to the CO can be focused on the PD active cell. The displacement \(\Delta d\) of the focused laser beam on the PD active cell due to the angular displacement \(\Delta\theta\) of the plane mirror reflector can be expressed by the following Equation [4]:  
\[\Delta\theta=\arctan\left(\frac{\Delta d}{2f}\right) \tag{2}\]  
In the case where a single-cell photodiode is employed as the photodetector to detect the spot displacement \(\Delta d\) as shown in Figure 3a, the measuring range of the photodetector is defined by the diameter \(D\) of the focused laser beam on the PD active cell, and the corresponding measuring range of the angular displacement becomes \(\pm\arctan(D/4f)\). The detection sensitivity of \(\Delta d\) by the PD is inversely proportional to the focused spot diameter \(D\)[4]. The decrease of \(D\) for the achievement of the highly sensitive measurement of the angular displacement is thus required; namely, there is a trade-off relationship between the sensitivity and the measuring range in the conventional laser autocollimator with a photodiode and a single-mode laser source. A femtosecond laser autocollimator, which can be realized by combining the angle scale comb with the conventional laser autocollimation, can overcome the aforementioned problem. Figure 3b shows a schematic of the femtosecond laser autocollimator in which a mode-locked femtosecond laser source is employed as the light source. In the optical setup of an angle scale comb shown in Figure 2a, the laser autocollimation unit composed of a collimator objective and a single-cell photodiode is newly employed instead of the sole photodetector. As can be seen in Figure 3b, each  
Figure 2: Applications of the angle scale comb [54]: (**a**) Measurement of the angular displacement of a rotary table; (**b**) Measurement of the surface form of an optical component. of the first-order diffracted beams emanating from the grating reflector is focused onto the focal plane of the collimator objective. As a result, a series of focused diffracted beams aligned in a line can be obtained. The relative position of each of the focused beams is determined by Equations (1) and (2). Since all the focused diffracted beams experience the translational displacement on the PD active cell associated with the angular displacement of the grating reflector, a continuous reading output with a cycle corresponding to the relative position of each of the focused diffracted beams can be obtained.