# 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
A collimated mode-locked femtosecond laser is made to pass through a Fabry-Perot etalon with a  
Figure 4: Optical setup of the mode-locked femtosecond laser autocollimator [55]: (**a**) A schematic of the optical setup; (**b**) A photograph of the optical setup. Figure 5: Variation of the reading output of the femtosecond laser autocollimator due to the angular displacement of the grating reflector [55]. free spectral range (FSR) of 100 GHz and is then made incident to a grating reflector, and a part of the first-order diffracted beams emanating from the grating surface is captured by the single-mode fiber detector through the collimator objective. The spectrum of the captured laser beam is analyzed by an optical spectrum analyzer. It should be noted that the etalon is employed in this setup to identify each mode in the angle scale comb in optical frequency domain by using a spectrometer with a limited frequency resolution. Figure 8 shows an example of the angle scale comb observed in optical frequency domain. As can be seen in the figure, each of the comb modes is successfully identified in the optical frequency domain. Although a limited result over an angle range of approximately 400 arc-seconds is indicated in the figure, each of the comb modes is verified to be identified over an angular range of approximately 6\({}^{\circ}\), which is limited by the spectral width of the mode-locked femtosecond laser, as well as the resolving power of the diffraction grating employed in the setup. The expansion of the spectral range of a mode-locked femtosecond laser with a supercontinuum [71] is expected to achieve a further wider measurement range. In addition, the results have demonstrated that a high resolution of 0.03 arc-seconds can be achieved by interpolating the intensity variation of each of the comb modes. Figure 6: Mode-locked femtosecond laser autocollimator based on the optical frequency-domain angle measurement [56]. Figure 7: Femtosecond laser autocollimator based on the principle of the optical frequency-domain angle measurement [56]. Measurement of the absolute angular position of an object is a difficult task for the conventional autocollimator [17; 18; 19; 20; 21; 22; 23; 24] and the laser autocollimator with a single-mode laser source [25; 26; 27; 28], since it is difficult to verify the absolute angular position of a target reflector with respect to the optical axis of the autocollimation unit in a space; namely, the determination of the \"zero-angle\" position. Although a method employing a retroreflector is available for the zero-angle position adjustment [72], this method can only be applied to the applications where the propagation directions of the incident beam and the reflected beams become parallel to each other. Furthermore, the accuracy of the zero-angle position adjustment is dominated by the accuracy of the retroreflector. This issue can be addressed by the optical frequency-domain femtosecond laser autocollimator with the enhancement of an assisted angle sensor [57]. Figure 9 shows a schematic of the high-precision zero-angle position adjustment in the optical frequency-domain femtosecond laser autocollimator. As can be seen in the figure, \(\Delta\theta\), which is the difference of the angles of diffraction of the (\(i+1\))th and \(i\)th first-order diffracted beams with mode frequencies of \(\
u_{i+1}\) and \(\
u_{i}\), respectively, is verified by measuring the angular displacement of the grating reflector by the assisted angle sensor such as a conventional autocollimator or a rotary encoder embedded to the rotary table on which the grating reflector is mounted. By using the obtained \(\Delta\theta\), the absolute angle \(\Phi\) between the femtosecond laser beam and the optical axis of the laser autocollimation unit can be obtained. It should be noted that the final measurand of the method is not \(\Phi\) but \(\theta\), the angle of the normal of the grating reflector with respect to the incident femtosecond laser.