# Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser
## 1 Introduction
### A Method Employing the Dispersive Characteristics of a Diffraction Grating  
#### 2.1.1 Principle of the Generation of an Angle Scale Comb from an Optical Frequency Comb  
The equally-spaced optical modes in a mode-locked femtosecond laser, often referred to as an optical frequency comb, can be employed to generate an \"angle scale comb\" that can be employed as optical graduations for angle measurement. A schematic of the principle of generating an angle scale comb from a mode-locked femtosecond laser is shown in Figure 1. The angle scale comb is generated with the enhancement of the dispersive characteristics of a diffraction grating. A mode-locked femtosecond laser from a light source is constructed incident to a reflective-type diffraction grating. The ultra-short pulse train of a mode-locked femtosecond laser in the time domain is a series of equally-spaced optical modes with a pulse repetition rate \(\
u_{\text{rep}}\) in the frequency domain. A reflective-type diffraction grating generates a series of first-order diffracted beams, in which each optical mode has a different angle of diffraction from those of the others. In the case where a two-dimensional reflective-type diffraction grating is employed as shown in the figure, groups of first-order diffracted beams will be generated in the two directions. It should be noted that only the groups of the positive first-order diffracted beams in the \(X\)- and \(Y\)-directions are indicated in the figure \begin{table}
\begin{tabular}{c c} \hline \hline
**Utilized Characteristics** & **Physical Phenomenon/Measuring Technique to be Coupled** \\ \hline \hline  & Diffraction [54] \\  & Diffraction and laser autocollimation [55; 56; 57] \\ Stable discrete modes in frequency domain & \(\
u_{\text{Chromatic aberration}}\)[58] \\  & Sagnac effect [60; 61] (Fiber optic gyroscope) \\  & Dispersive interferometry [62] (Absolute distance measurement) \\  & Time-of-flight (TOF) counted by dual-comb interferometry with \\ High pulse energy & balanced cross-correlation of second harmonics [63] (Absolute distance measurement) \\  & Second-harmonic generation (SHG) [59] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical angle sensors based on a mode-locked femtosecond laser. for the sake of simplicity. The angle of diffraction of the \(i\)th optical mode \(\theta_{i}\) in the group of the positive first-order diffracted beams can be expressed as follows [54]:  
\[\theta_{i}=\arcsin\!\left(\frac{c}{n_{\text{air}}gv_{i}}\right)(i=1,\;2,\;3,\ldots,\;n), \tag{1}\]  
where \(c\), \(n_{\text{air}},g\), and \(v_{i}\) are the speed of light in vacuum, the refractive index in air, the pitch of the diffraction grating, and the optical frequency of the \(i\)th mode, respectively. As can be seen in the Equation, the discrete optical frequency \(v_{i}\) in the spectrum of the mode-locked femtosecond laser and the angle of diffraction \(\theta_{i}\) of the \(i\)th mode are in a one-to-one relationship, and thus the group of the diffracted beams can be treated as optical graduations for angle measurement: angle scale comb. The angle scale comb generated from a mode-locked femtosecond laser can be employed for angle measurement. Figure 2a shows one of the examples of applying the angle scale comb for measurement of a rotary table on which a grating reflector is mounted. Since the modes in the angle scale comb pass through the detector one after another with the rotation of the grating reflector, the angular displacement of the rotary table can be detected by monitoring the reading output of the detector. Figure 2b shows another example of the application of the angle scale comb; measurement of the free-form surface of an optical component. In this case, due to the local slope \(\Delta\theta\) at a certain position on the free-form surface, the angle of incidence of the reflected beam from the optical component experiences a change in its propagation direction of \(2\Delta\theta\).