## 4 Experimental Results and Discussion
### Thermally Tunable Frequency Difference  
To roughly measure the frequency differences in a large range (about tens of GHz) by tuning cooling temperature, a LSA IR-L optical spectrum analyzer with 0.02 nm resolution was used. The cooling temperature was continuously increased from 18 \({}^{\circ}\)C to 34 \({}^{\circ}\)C with a step interval of 0.5 \({}^{\circ}\)C. The measurement results are shown in Figure 2.  
Figure 2: Variations of the wavelengths of the dual-frequency components with temperature increasing from 18 \({}^{\circ}\)C to 34 \({}^{\circ}\)C. The points of x- and y-polarization are approximately overlapped between 30 \({}^{\circ}\)C and 32 \({}^{\circ}\)C.  
The x and y polarization correspond to the black and the red curves respectively. By fitting the data points, it can be seen that the laser wavelengths of the two polarizations vary linearly with the temperature increasing. It can be seen that, for y polarized wave, the wavelength curve is monotonic increasing from 1064.26 nm to 1064.58 nm. While, for the x polarized wave curve, there are two saltus steps at the temperature around 25 \({}^{\circ}\)C and 32 \({}^{\circ}\)C. This is because of the longitudinal mode is hopping between neighbor longitudinal modes. It's affected by the relative positions of the gain curve and the cavity longitudinal modes. When the longitudinal mode's optical frequency shifts away from the center of the gain curve, its gain coefficient will drop. Frequency of the neighboring longitudinal mode will move closer to the center of the gain curve and obtain a larger gain coefficient. Due to mode competition, when the net gain of the latter is stronger than that of the former, the former will be extinguished. The mode hopping occurred at 24.5 \({}^{\circ}\)C, the wavelength changed from 1064.49 nm to 1064.31 nm, coming with an approximate 54 GHz frequency difference change, and the second one occurred at 31.5 \({}^{\circ}\)C, the wavelength changed from 1064.56 nm to 1064.39 nm, with a 51 GHz frequency difference change, approximately. They were both very close to the longitudinal mode interval of 52.3 GHz. While, it's interesting that there was no mode hoping for the y polarized wave in such a large frequency variation range, further investigation on this problem is needed in later works. When the cooling temperature was between 18 \({}^{\circ}\)C and 24.5 \({}^{\circ}\)C, the wavelengths of both the ordinary and the extraordinary beam continuously changed with temperature, and no mode hopping occurred. In this temperature range, we used an F-P scanning interferometer to measure the frequency differences more precisely. the F-P scanning interferometer continuously changes the transmission frequency by scanning voltage, ensuring that all frequencies could be transmitted and detected by its photodetector at the back end. Using the F-P scanning interferometer, the number of frequencies and frequency interval of the laser output can be measured. The measurement results are shown in Figure 3a. The results are fitted to obtain a good linear relationship between the frequency difference and the cooling temperature, with an average tuning slope of 4.4 GHz/\({}^{\circ}\)C. It could be seen that there was a reverse between 20.5 \({}^{\circ}\)C and 21 \({}^{\circ}\)C during the temperature's variation. At 18 \({}^{\circ}\)C to 20.5 \({}^{\circ}\)C, it was seen \(v_{x}<v_{y}\), and at 21 \({}^{\circ}\)C to 24.5 \({}^{\circ}\)C, it changed to \(v_{x}>v_{y}\). The tuning slopes in the two temperature ranges were \(-\)4.45 GHz/\({}^{\circ}\)C and 4.62 GHz/\({}^{\circ}\)C, respectively. The average inaccuracy of the frequency difference for each point was about \(\pm\) 50 MHz, due to the \(\pm\) 0.01 \({}^{\circ}\)C accuracy of the temperature controller we used. The maximum frequency difference reached 27 GHz at 24.5 \({}^{\circ}\)C.  
In the experiment, we observed that as the frequency of the two orthogonally polarized beams got quite close, one of them was extinguished, resulting in a single polarization output.