# Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser
## 3 Methods Based on the Absolute Distance Measurement  
The angular displacement of an object can be measured by multilateration or triangulation [11; 74; 75] in which the displacements of several points on its surface are measured by interferometric methods [13; 76] or non-interferometric methods such as the time-of-flight (TOF) method [77]. By employing a method with a mode-locked femtosecond laser source capable of measuring the absolute distance of a measuring point [40; 41; 42; 43; 44; 62; 63], absolute angular position measurement can be realized. Figure 13 shows an example of the absolute angular position measurement based on the absolute distance measurement by the TOF method combined with the second harmonic generation (SHG) [63]. In the proposed method, a pair of the mode-locked femtosecond laser sources synchronized with the same reference clock but with slightly different pulse repetition rates is employed; one of the mode-locked femtosecond lasers is employed as a signal laser, while the other is employed as a local laser for down-conversion. In the setup, the femtosecond laser beam from the light source is at first collimated by a collimating lens (CL) and is then divided into four sub-beams by using a diffractive optical element (DOE). The sub-beams are made incident to the target surface where four mirrors are placed in an angular distance of 90\({}^{\circ}\) along the circumference direction to reflect the sub-beams to the DOE. It should be noted that the four mirrors are arranged in such a way that the optical path lengths of the sub-beams become different from each other. The reflected sub-beams are then combined at the DOE and are made to pass through the circulator and to superimpose with the local femtosecond laser in free space. The reflected sub-beams are then converted into electric signals by the balanced cross-correlator based on the principle of the nonlinear optical cross-correlation for measurement of the absolute distance of the four mirrors on the object with respect to the DOE. The novel optical configuration with  
Figure 12: Optical modes captured in the setup based on the chromatic aberrations of a simple lens [58]: (**a**) Variation of the light intensities of optical modes captured by the fiber detector; (**b**) Peak optical frequency detected at each angular position of the mirror reflector. the DOE and the four mirror reflectors makes it possible to carry out simultaneous measurement of the absolute distances of the four points on the object by a single balanced cross-correlation (BCC) electrical signal in which the four corresponding pulse trains can be observed independently. By using the measured absolute distances \(d_{i}\) (\(i\) = 1, 2, 3, 4) of the mirror reflectors with respect to the DOE, absolute angular positions can be obtained through a simple calculation based on the geometric relationship of the optical components; the \(X\)- and \(Y\)-directional absolute normal angles of the surface under inspection \(\theta_{x}\) and \(\theta_{y}\), respectively, with respect to the femtosecond laser beam incident to the DOE can be calculated as \(\theta_{x}\) = sin\({}^{-1}\)[(\(d_{1}-d_{3}\))/\(A\)] and \(\theta_{y}\) = sin\({}^{-1}\)[(\(d_{2}-d_{4}\))/\(A\)], where \(A\) is the distance between the two corresponding mirror reflectors. Figure 14 shows the detected angular motion of an object placed 3.7 m away from the DOE. In the experiment, angular motion with a frequency of 1 Hz was given to the object by using a piezoelectric tilt stage. Figure 14a shows the absolute displacements measured at the four positions, and Figure 14b shows the angular motions about the \(X\)- and \(Y\)-axes calculated from the obtained absolute displacements. As can be seen in the figures, the absolute distances at the four positions were successfully detected simultaneously, and the angular motions about the \(X\)- and \(Y\)-axes were successfully reconstructed from the obtained absolute distances.