## 4 Experimental Results and Discussion
### Output Power Fluctuation and Longitudinal Mode Shift of the SLM Laser
#### 4.2.3 Experimental Verification About Optical Path-Length Change and Heat Deposition  
To verify our notion of optical path-length change and heat deposition, we first gauged the centroid position changes of the output beam spot along the x- and y-axes at the laser's free-running operation by a beam profiler (BP209-VIS/M, THORLABS). The centroid was the geometrical center of a beam spot.  
Specifically, we recorded the centroid positions of the output laser spot in the x- and y-axes directions with the pump current 1.8, 2.0, 2.2, 2.4, and 2.6 A. The spot centroid positions along two axes were measured at the same time. We also calculated the standard deviation (SD) and range (R) at every pump current. R was the difference between maximum and minimum values, written in gray in Figure 4a,b.  
Figure 4a,b show that the output-laser spot centroid positions along the x- and y-axes have the same trends. Firstly, with the pump current increasing, the centroid of the laser spot gradually moves away. The movement distance on the x-axis (200 \(\upmu\)m) is about twice that on the y-axis. The centroid position movements corresponded to 0.82 \(\upmu\)m optical path-length variation. The pump power's growth led to more deposition of waste heat in the Nd:YVO\({}_{4}\) crystal and consequent variation in optical path length.  
Secondly, the periodicity of the centroid position, regardless of it being parallel to the x- or y-axis, becomes apparent after the pump current reaches 2.2 A. Every period is around two minutes. The period pertained to the TEC's cooling efficiency and the Nd:YVO\({}_{4}\)'s thermal conductivity.  
Furthermore, the R and SD of the 1.8A pump current exceed those of other pump currents, owing to mode competition near the threshold pump power. At every pump current, the y-axis's range is more extensive than that of the x-axis. This is because the imperfect heat-sink structure impaired heat dissipation for the crystal in the y-axis.  
On the other hand, as the current increased, the variation in the centroid position reduced generally. This trend existed throughout the entire process of the current increase, and we selected several currents to demonstrate this trend.  
Figure 5 provides spot centroid-position changes after the insertion of K9 FP etalons into the oscillator of the Nd:YVO\({}_{4}\) laser. The centroid-position results with 4% (Figure 5a) and 80%-reflectivity (Figure 5b) were recorded at 1.83 A and 2.40 A pump currents. Blue and purple lines signify the output spot's centroid positions in the x-axis and y-axis directions, respectively. We worked out the SD and R of those parameters in Figure 5a,b.  
As demonstrated in Figure 5a, after the K9 FP etalon with 4% reflectivity was inserted into the laser, the x-axis and y-axis centroids of the output laser spot still moved periodically. The periods of those two centroids approximate that of the output power with the insertion of the same FP etalon. In Figure 5b, the spot centroid with the 80%-reflectivity etalon had a trend similar to that of the output power with the same etalon.  
Compared with the cases without any FP etalon, the range and the standard deviation of spot centroid position with FP etalons are higher. This rise shows that after the etalon was  
Figure 4: The centroid position changes of the output beam spot along the (**a**) x- and (**b**) y-axes at the laser’s free-running operation, including standard deviation (right y-axis) and range (gray words) at every pump current.  
Figure 5: The laser’s spot centroid positions after the insertion of 4% (**a**) and 80% (**b**) K9 FP etalons: SD, standard deviation; R, range.  
placed into the laser, the etalon's refraction to the laser and modulation on the longitudinal mode gain aggravated the optical path change.