# A Second Harmonic Wave Angle Sensor with a Collimated Beam of Femtosecond Laser
## 2 Angle Sensor Principle
However, the optical configuration with the focused fundamental wave made the alignment of the optical setup complicated. Furthermore, due to the limitation of the crystal size and the depth-of-focus of the focusing lens employed in the setup, the working distance of the second harmonic wave angle sensor was also limited. In this paper, a collimated fundamental wave whose light intensity is intensified by reducing the beam diameter is employed to address the issues. Figure 3 shows an example of the optical setup based on the Keplerian configuration that reduces the beam width. The following relationship can be found between the initial beam diameter \(d_{1}\) and the expected beam \(d_{2}\):  
\[d_{2}=\frac{d_{1}f_{2}}{f_{1}} \tag{4}\]  
where the focal lengths of the first lens and the second lens are denoted by \(f_{1}\) and \(f_{2}\), respectively. Figure 3: A schematic of the setup reducing the collimated beam width with a pair of plano-convex lenses. Figure 2: Principle of SHG angle sensor method with Fs laser. By using the intensified collimated fundamental wave, the angular displacement of the NLO crystal is expected to be carried out even when the working distance is set to be lengthy. Through the plane wave approximation, power intensity \({P_{\mathrm{ij}}}^{\prime}\) of the second harmonic wave with the \(i\)th wavelength and the \(j\)th angular position can be estimated by the following equation:  
\[{P_{\mathrm{ij}}}^{\prime}=\frac{8\pi^{2}d_{\mathrm{eff}}{}^{2}L^{2}}{{n_{ \mathrm{e}}}(\lambda_{\mathrm{i}})^{2}{n_{\mathrm{e}}}(\theta,\lambda_{ \mathrm{i}})^{2}{\epsilon_{0}}{\epsilon\lambda_{\mathrm{i}}}^{2}S}{P_{\mathrm{ i}}}^{2}\sin{c}^{2}(|\Delta k(\theta)|L/2);\quad\mathrm{i}=1,2,\ldots,\mathrm{m} \tag{5}\]  
where \(d_{\mathrm{eff}}\) is the effective nonlinear coefficient, \({\epsilon_{0}}\) is vacuum permittivity, \(c\) is the speed of light in vacuum, \(L\) is the length of the crystal, and \(S\) is the cross-sectional area. These quantities are considered to be constant and be ignored to simplify the calculation since they remain the same despite the changes of other physical quantities. It should be noted that restricting factors such as heat effect, walk-off effect, and dispersive spreading are not taken into consideration in the above equation [25]. The FW power intensity \(P_{\mathrm{i}}\) can be determined in the experiments where every mode in the fundamental wave has its deterministic power intensity. Furthermore, it should be noted that \({P_{\mathrm{ij}}}^{\prime}\) is proportional to the quadratic of FW power (\({P_{\mathrm{i}}}^{2}\)) and the quadratic of sinc function. The sinc function describes the behavior of the laser propagation inside the crystal. Taking the birefringent effect and phase mismatching \(\Delta k\) into consideration, it can be expressed as follows:  
\[\sin{c}^{2}(|\Delta k(\theta)|L/2)=\frac{\sin^{2}(|\Delta k(\theta)|L/2)}{\left( |\Delta k(\theta)|L/2\right)^{2}}, \tag{6}\]  
where  
\[n_{\text{o}}{{}^{2}}(\lambda)=4.8762+\frac{0.11554}{\lambda^{2}-0.04674}-\left( 0.033119\times\lambda^{2}\right);n_{\text{o}}{{}^{2}}(\lambda^{\prime})=4.8762 +\frac{0.11554}{\lambda^{\prime 2}-0.04674}-\left(0.033119\times\lambda^{ \prime 2}\right) \tag{2}\]0  
The refractive indices of extraordinary ray \({n_{\mathrm{e}}}\) (\(\theta\), \(\lambda^{\prime}\)) can be calculated by the following equation:  
\[n_{\text{o}}{{}^{2}}(\lambda)=4.8762+\frac{0.11554}{\lambda^{2}-0.04674}-\left( 0.033119\times\lambda^{2}\right);n_{\text{o}}{{}^{2}}(\lambda^{\prime})=4.8762 +\frac{0.11554}{\lambda^{\prime 2}-0.04674}-\left(0.033119\times\lambda^{ \prime 2}\right) \tag{2}\]1  
The matching angle in Equation (1) is satisfied when \(\Delta k(\theta_{\mathrm{pm}})=0\) and \({n_{\mathrm{o}}}(\lambda)={n_{\mathrm{e}}}\) (\(\theta_{\mathrm{pm}}\), \(\lambda^{\prime}\)). After all the calculations are fulfilled, the matrix \(P_{2}\) can be built as follows:  
\[P_{2}=\left(