# A Second Harmonic Wave Angle Sensor with a Collimated Beam of Femtosecond Laser
## 2 Angle Sensor Principle  
The proposed angle sensor is based on SHG, which occurs when an NLO crystal is irradiated with a laser beam of a certain frequency \(\omega\) producing radiation at twice the initial frequency \(2\omega\)[17; 25]. Since the light frequency is inversely proportional to the light wavelength, the fundamental wave (FW) having a wavelength \(\lambda\) generates a second harmonic wave having a wavelength \(\lambda^{\prime}=\lambda/2\). SHG can be employed in angle measurement due to the birefringence in the uniaxial NLO crystal that divides the laser beam into two different rays, the ordinary ray (referred to as the o-ray in the following) and the extraordinary ray (referred to as the e-ray in the following). Both the rays experience their respective refractive indices inside the crystal. In a certain condition where the optical path lengths of these beams are identical inside the crystal, phase-matching occurs, and the light intensity of the second harmonic wave is maximized. Note that for any given \(\lambda\), the phase-matching condition could arise in its corresponding matching angle \(\theta_{\text{m}}\), which can be estimated by the following equation [16; 26]. \[\theta_{\text{m}}=\sin^{-1}\!\left(\sqrt{\frac{n_{\text{o}}{{}^{-2}}(\lambda)-n_{ \text{o}}{{}^{-2}}(\lambda^{\prime})}{N_{\text{e}}{{}^{-2}}(\lambda^{\prime})- n_{\text{o}}{{}^{-2}}(\lambda^{\prime})}}\right), \tag{1}\]  
where \(n_{\text{o}}\) denotes the refractive index of an o-ray and \(N_{\text{e}}\) refers to the refractive index of an e-ray. Regarding the usage of the NLO crystal in the proposed angle sensor, the NLO crystal is required to have a large angular dispersion for the wavelength range of an fs laser to be employed in the angle sensor. In this paper, an fs laser whose optical spectrum ranges from 1480 nm to 1640 nm is used. MgO:LiNbO\({}_{3}\) is thus one of the suited crystals that fulfill the requirement [15; 16] regarding the following Sellmeier equations with the parameters associated with MgO:LiNbO\({}_{3}\):  
\[n_{\text{o}}{{}^{2}}(\lambda)=4.8762+\frac{0.11554}{\lambda^{2}-0.04674}-\left( 0.033119\times\lambda^{2}\right);n_{\text{o}}{{}^{2}}(\lambda^{\prime})=4.8762 +\frac{0.11554}{\lambda^{\prime 2}-0.04674}-\left(0.033119\times\lambda^{ \prime 2}\right) \tag{2}\]  
and  
\[N_{\text{e}}{{}^{2}}(\lambda^{\prime})=4.5469+\frac{0.094779}{\lambda^{\prime 2 }-0.04439}-\left(0.026721\times\lambda^{\prime 2}\right) \tag{3}\]  
Figure 1 shows the trend line of the matching angle with respect to \(\lambda\). As can be seen in the figure, an angular dispersion of approximately 2.87\({}^{\circ}\) between the angular positions of 48.50\({}^{\circ}\) and 51.37\({}^{\circ}\) from the optical axis of the crystal, can be observed in the wavelength range from 1480 nm to 1640 nm. Figure 2 shows a schematic of the angle measurement based on the proposed method. Since fs laser has a wide range of wavelength, when a certain wavelength is at a matching angle, in which high SHG intensity is achieved due to high efficiency, other wavelengths in the same angle undergo phase mismatch with lower intensity. The phase matching and phase mismatch features of the SHG will appear simultaneously in optical spectrum analyzer (OSA) in the form of a spectrum [27], as shown in the inset image in Figure 2. The phase-matching of other wavelengths can thus be satisfied when the MgO:LiNbO\({}_{3}\) crystal experiences angular displacement about its \(Y\)-axis. By utilizing this characteristic, angle measurement can be carried out by monitoring the peak wavelength of the second harmonic wave. Figure 1: The phase-matching angle characteristic of MgO:LiNbO\({}_{3}\) crystal. Aside from the NLO crystal preference, high intensity of the fundamental wave is required to generate a second harmonic wave with microwatts to milliwatts power. In previous work by the authors' team, a focused fundamental wave was employed to generate a second harmonic wave [15; 16].