## 3 Theoretical Analysis  
In the microchip cavity, the stable oscillating laser satisfies the standing wave condition and has an optical frequency of:  
\[\
u_{q}=q\frac{c}{2L} \tag{1}\]  
where \(c\) is the velocity of light, \(L\) is the optical path of the cavity, and \(q\) is a natural number. The longitudinal mode interval is \(\frac{c}{2L}\). The optical path of the cavity consists of \(L_{YAG}=n_{YAG}\times l_{YAG}\) and \(L_{LTO}=n_{LTO}\times l_{LTO}\). The refractive index of Nd:YAG is 1.82, and for LTO, the mean value is 2.14. Then we can get a longitudinal mode interval of 52.3 GHz. As the total gain bandwidth of Nd:YAG is estimated to be 120 GHz, this means that the cavity can generate two or three longitudinal modes at most. While, by applying relatively low pump power (about within 1.6 times of the threshold power) in the experiment, single longitudinal mode output could be obtained.  
Since the LTO in our experiment is a-cut, the single frequency can be split into two frequencies with perpendicularly linear polarizations under its natural birefringence (the ordinary light polarized along the x axis, and the extraordinary light polarized along the y axis). The frequency difference of the two polarized light can be derived by the equation below:  
\[\Delta\
u=\frac{2l_{LTO}}{\lambda}(n_{e}-n_{o})\frac{c}{2\mathrm{L}}. \tag{2}\]  
where \(n_{e}=2.1403\), \(n_{o}=2.1363\) are the refractive indices of the ordinary and extraordinary light, respectively.  
As the birefractive indexes of the LTO crystal is affected by temperature, the frequency difference has a correlationship with the cooling temperature of the microchip. The variation of the frequency difference with the cooling temperature is determined by the equation below [17]:  
\[\frac{d\Delta\
u}{dT}=\frac{2l_{LTO}}{\lambda}\bigg{[}\frac{\beta n_{e}}{ \beta T}-\frac{\beta n_{o}}{\beta T}+\alpha(n_{e}-n_{o})\bigg{]}\frac{c}{2L} \tag{3}\]  
where \(\alpha\) is the expansion coefficient of LTO crystal in z direction. The value of parameters used in this paper are \(\frac{\beta n_{o}}{\beta T}=25\times 10^{-6}/\mathrm{K}\), \(\frac{\beta n_{e}}{\beta T}=2.4\times 10^{-6}/\mathrm{K}\), \(\alpha=16\times 10^{-6}/\mathrm{K}\)[17].  
And as an electro-optic crystal, the birefractive indexes can also be tuned by electric filed. In the experiment, a transverse electric field is applied to the LTO crystal along the x axis. Due to the  
Figure 1: Schematic diagram of the experimental setup. LD, laser diode; HR, high-reflective mirror (@1064 nm); PD, photoelectric detector; OSA, optical spectrum analyzer; DPSSL, diode-pumped solid-state laser.  
electro-optical effect, the refractive indexes of the LTO crystal along x and y axes are changed, and the changed value is proportional to the electric field strength and the electro-optic coefficient of LTO crystal.  
\[\Delta n_{o}=-\frac{\gamma_{13}n_{0}^{3}V}{2d}\,\Delta n_{e}=-\frac{\gamma_{33}n_{e}^{3 }V}{2d}. \tag{4}\]  
Here \(\gamma_{13}=8\ \mathrm{pm}/V\), \(\gamma_{33}=33\ \mathrm{pm}/V\), \(d=5\ \mathrm{mm}\), and \(V\) is the applied voltage. The relationship between the voltage, the frequency difference of the ordinary light, and the extraordinary light is:  
\[\frac{d\Delta v}{dV}=\frac{l_{LTO}}{\lambda}\left[\frac{\gamma_{13}n_{0}^{3}- \gamma_{33}n_{e}^{3}}{d}\right]\frac{c}{2L}. \tag{5}\]  
Equations (3) and (5) give the theoretical relationships of \(\Delta v\) with cooling temperature and applied voltage. For prediction of experiments results the slope tuning of the frequency difference versus temperature and voltage are calculated:  
\[\frac{d\Delta v}{dT}=2.36\ \mathrm{GHz/K},\ \ \frac{d\Delta v}{dV}=2.56\ \mathrm{MHz/V}.\]