# Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser
## 4 Angle Detection Based on the High Pulse Energy of a Mode-Locked Femtosecond Laser  
The small angular displacement of an object can also be measured by utilizing the angle-dependency of the second harmonic generation (SHG) in nonlinear optics, where the second harmonic light with the doubled optical frequency \(\
u_{2}=2\
u_{1}\) of the fundamental light \(\
u_{1}\) incident to a nonlinear optical component is generated. It is well known that an efficient SHG can be accomplished by the procedure referred to as index matching [73], where the intensity of the second harmonic light depends on the angle of incidence of the fundamental light; this angle dependency can be employed for small angular displacement measurement. Owing to the characteristic of a mode-locked femtosecond laser with high pulse energy, effective SHG can be achieved by focusing a mode-locked femtosecond laser beam in a nonlinear crystal. On the assumption that the Rayleigh length \(b\) of the focused fundamental light lay with wavelength \(\lambda_{1}\) is much longer than the length of the negative uniaxial crystal with refractive  
Figure 15: Angle measurement based on the absolute distance measurement by a Michaelson interferometer with a mode-locked femtosecond laser source [62]: (**a**) Optical configuration with two measurement arms; (**b**) Spectrum of the combined beams. indices of \(n_{\mathrm{e}}\) and \(n_{\mathrm{o}}\) for extraordinary and ordinary rays, respectively, the intensity of the second harmonic light \(l_{2}\) with wavelength \(\lambda_{2}\) to be generated by SHG can be expressed as follows [59]:  
\[I_{2}=\frac{8\pi^{2}d_{\mathrm{eff}}^{2}L^{2}}{n_{1}^{2}n_{2}\varepsilon_{0}c \lambda_{1}^{2}}l_{1}^{2}\mathrm{sin}\,c^{2}(\Delta L/2) \tag{7}\]  
where \(l_{1}\) is the light intensity of the fundamental light ray, \(d_{\mathrm{eff}}\) is the effective nonlinear coefficient [32], \(\varepsilon_{0}\) is the vacuum permittivity, \(c\) is the speed of light in a vacuum. In the above Equation, \(\Delta k\) is a \(\theta\)-dependent phase mismatching that can be represented as \(\Delta k=4\pi[n_{\mathrm{o}}(\lambda_{1})-n_{\mathrm{e}}(\theta,\lambda_{2} )]/\lambda_{1}\). The angle \(\theta=\theta_{\mathrm{m}}\) satisfying the sinc-term in the above Equation to be the maximum value of 1 (namely, \(n_{\mathrm{e}}=n_{\mathrm{o}}\) and \(\Delta k\) thus becomes zero) is referred to as the matching angle (Figure 16) that can be obtained by the following Equation [59]:  
\[\theta_{m}=\mathrm{arcsin}\Bigg{(}\sqrt{\frac{\left[n_{\mathrm{O}}(\lambda_{ 1})\right]^{-2}-\left[n_{\mathrm{O}}(\lambda_{2})\right]^{-2}}{\left[n_{ \mathrm{e}}(\lambda_{1})\right]^{-2}-\left[n_{\mathrm{O}}(\lambda_{2})\right] ^{-2}}}\Bigg{)} \tag{8}\]  
According to the above Equations, the intensity of the second-harmonic light \(l_{2}\) decreases rapidly with the small angular displacement of the nonlinear crystal from the matching angle; this characteristic of the SHG can be employed for angle measurement. Figure 17a shows an example of how to apply the characteristic of the matching angle in SHG for measurement of the small angular displacement of an object [59]. In the setup, a nonlinear crystal is mounted on a rotary table. For angle measurement, the optic axis of the nonlinear crystal is aligned to have the angle \(\theta\), which is almost equal to the matching angle \(\theta_{\mathrm{m}}\), with respect to the propagating direction of the fundamental light wave as shown in Figure 18b so that the second harmonic light with enough power can be obtained. A small angular displacement of the rotary table can be measured by detecting the angular displacement of the nonlinear crystal by monitoring the change in the intensity of the second harmonic light with a photodetector. The feasibility of the method described above has been verified in experiments. Figure 18a,b show a schematic and a photograph of the developed setup. An Erbium-doped fiber-based mode-locked femtosecond laser with the spectrum ranging from 1480 nm to 1640 nm has been employed as the light source.