_applied sciences_  
_Article_  
Investigation of Angle Measurement Based on Direct Third Harmonic Generation in Centrosymmetric Crystals  
[PERSON]  
[PERSON]  
[PERSON]  
[PERSON]  
[PERSON]  
[PERSON]  
and [PERSON]  
1 Precision Nanometrology Laboratory, Department of Finemechanics, Tohoku University,  
Sendai 980-8579, Japan  
2 Department of Engineering, Graduate School of Engineering, Kyushu Institute of Technology,  
1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi 804-8550, Japan  
* Correspondence: [EMAIL_ADDRESS]; Tel. : +81-022-795-6953  
Footnote 1: email: [EMAIL_ADDRESS]  
Footnote 2: email: [EMAIL_ADDRESS]  
Footnote 3: email: [EMAIL_ADDRESS]  
Footnote 4: email: [EMAIL_ADDRESS]  
Footnote 5:  
different modes was then generated at the grating reflector [8]. As the stage was rotated, the detected intensity would change periodically, which extends the measurement range of angle measurement to a maximum of 15,000 arcseconds. Similarly, the mode-locked femtosecond laser autocollimator also extends the angle measurement range by taking the advantage of the widely spreading spectrum of the femtosecond laser source in the frequency domain [9; 10]. Compared with the optical lever sensor, the angle measurement sensitivity of the autocollimator is not affected by the distance between the detector and the grating reflector, where a compact setup design can be realized. Meanwhile, an extremely high electrical field of femtosecond laser induces the nonlinear polarization field which is the source of nonlinear optical (NLO) phenomena, such as the second harmonic generation (SHG) [11]. SHG is a common NLO process in which the second harmonic of an incident laser is generated by passing some kind of material through it. The produced second harmonic wave (SHW) has the double frequency \(\
u_{2}\) (\(\
u_{2}\) = 2\(\
u_{1}\)) and the half wavelength \(\lambda_{2}\) (\(\lambda_{2}\) = \(\lambda_{1}\)/2) of the incident laser beam (fundamental wave), which has the frequency and the wavelength of \(\
u_{1}\) and \(\lambda_{1}\). The angle dependence of SHG in the birefringent crystal was found in the early 1960s [11]. As the laser propagates through the birefringent crystal in different directions, the phase mismatch between the laser and the second harmonic wave changes due to anisotropy, resulting in the intensity changes of the generated SHW. Based on this principle, several prototypes of the SHG-based angle measurement were developed with the commercial mode-locked femtosecond laser [12; 13; 14]. In the early sensors, a lens was employed to focus the laser beam into a crystal for reducing the diameter of the light spot for SHW production. At first, Beta Barium Borate (\(\beta\)-BBO) crystals with low wavelength dispersion of the SHG output were used and a sub-arcsecond resolution was realized [12]. However, the measurement range of the sensor was narrow and the chromatic aberration in the crystal was unavoidable due to the narrow optical pulse width, resulting in a significant reduction in measurement sensitivity. Then, the wavelength-dependent SHG angle sensor was proposed [13], where a shift of the SHG spectrum in the frequency domain was used to denote the angle position. However, the focused laser beam should be very carefully aligned with the NLO crystal, making it difficult to measure a moving target [15]. Then, to deal with this problem, an SHG angle sensor with a collimated femtosecond laser beam was constructed where a second harmonic wave was successfully generated with the collimated laser beam [14]. SHG is related to the symmetry of the crystal, described by the point group of the crystal. The different point groups usually have different forms of second-order optical nonlinear susceptibility \(\chi^{(2)}\) for generating a second harmonic wave. Theoretically, the SHG signal can only be detected in a crystal without inversion symmetry, called a non-centrosymmetric crystal [16; 17; 18]. Several birefringent crystals have an inversion center, also referred to as centrosymmetric structures.