## 3 Results
On the other hand, Figure 4b illustrates the profile when all rods were strongly coupled to each other and the mode intensities of all rods were similar. For convenience, we named Figure 4a as the \"weakly-coupled\" PM mode and Figure 4b as the \"strongly-coupled\" PM mode. To simulate the response of the PM modes to the different excitation scenarios, three different optical gain settings were applied. For the cold cavity case as depicted in Figure 4c, the optical gain of all rods was set at zero as the reference. However, because the material gain spectrum actually covered the PM modes' eigen frequencies, the material gain is a function of pumping density. We assumed a pumping power density that gave the excited rods an optical gain of g = 800 cm\({}^{-1}\). Without a loss of generality, we also assumed that the rods were weakly absorbing and the absorption coefficient was \(\alpha=-\)g = 800 cm\({}^{-1}\) if it was not sufficiently pumped. Simultaneously, Figure 4d simulated the case when only the center rod of the PM was pumped, i.e., the small spot case. Figure 4e shows the case when all rods were equally pumped under a large spot. The simulated Q-factors of two PM modes under three pumping scenarios are summarized in Table 1.  
Figure 3: The side-view mode intensity profile of the (**a**) 368 nm and (**b**) 376 nm mode in a single rod whose geometry followed the parameters extracted from SEM and TEM images of Figure 1; (**c**) the mode intensity profile acquired from the white dashed line cross-sections in (**a**,**b**). The Q-factors under the cold cavity condition (Q\({}_{\text{cold}}\)) of the weakly-coupled and strongly-coupled PM was 241 and 448, respectively. Since the PM mode simulations were only two dimensional, the absolute quantity of Qs might not be in full agreement with those three-dimensional simulations in Figure 3. However, the Q-factor response under different excitation scenarios still provided a significant insight. Under the small spot condition, the Q-factor (Q\({}_{\text{small}}\)) of the weakly-coupled PM was reduced by 1% when compared to Q\({}_{\text{cold}}\), while that of the strongly-coupled PM was reduced by 35%. On the other hand, the Q factor of the strongly-coupled PM under the large spot (Q\({}_{\text{large}}\)) was enhanced by 495%, while the Q factor of the weakly-coupled one was only enhanced by 75%. In a nutshell, coupling with adjacent excited or unexcited rods will enhance or decrease the overall Q-factor respectively, as depicted in Figure 4f, and the response was stronger for the strongly-coupled PM modes. A similar argument can also be applied to the PM consisting of more rods. As the excitation area increases, the number of paired interactions between the excited rod will also increase, which yields a greater enhanced overall Q-factor. Since a higher Q-factor is an equivalent statement of a lower threshold pumping density, the significant enhancement from Q\({}_{\text{small}}\) to Q\({}_{\text{large}}\) explains the dramatic P\({}_{\text{th}}\) reduction from Figure 2a to Figure 2b. As the 376 nm mode and 368 nm mode had the same transverse mode profile, this simulation was applied to both lasing wavelengths. To explain the dominant-mode switch phenomena, we needed to consider the different coupling strengths among the different modes. As the 376 nm mode was in the fundamental axial mode and its maximum intensity plane lay in the rod cross-section with a maximum-width, its mode interaction to adjacent rods will be stronger than that of the 368 nm mode. Therefore, we can analogize the 368 nm and 376 nm lasing to the weakly-coupled and strongly-coupled PM modes in Figure 4, respectively. Consider when the 376 nm and 368 nm supermodes have very close P\({}_{\text{ths}}\) under a given diameter of the excited area, D\({}^{\prime}\). When D < D\({}^{\prime}\), the 368 nm supermode will dominate due to a smaller decrement of the Q-factor. Vice versa, when D > D\({}^{\prime}\), the 376 nm supermode will dominate due to a more dramatic increment of the Q-factor.