## 3 Results  
Figure 2a,b illustrated the lasing spectrum under two different excitation spot sizes, and their light-light curves are plotted in Figure 2c. Under a relatively small excitation area (diameter, D \(\sim\) 12 \(\rm\upmu m\)), the threshold pumping density (P\({}_{\rm th}\)) was as high as 847.85 kW/cm\({}^{2}\), and the dominant lasing mode resided near 368 nm. The P\({}_{\rm th}\) was lowered to 313.7 kW/cm\({}^{2}\) while the D was 80 \(\rm\upmu m\). Simultaneously, the dominant modes shifted to 376 nm under a large excitation area. In other words, the P\({}_{\rm th}\) was reduced and the dominant lasing mode switched when more rods were included in the collective lasing. According to the CL in Figure 1c, the gain medium of 376 nm lasing was mostly found in the MQW shell region, while the gain medium of 368 nm was mostly in the core. Therefore, both modes could laste simultaneously if the pumping density was sufficiently high. A more detailed argument about the gain profile evolution with the pumping density can be found in our previous publication [26]. We attributed this excitation-area-dependent collective lasing behavior to the different extent of the side-coupling effects among the different lasing modes [25]. In our previous study, we qualitatively proved this mechanism with a truncated rod experiment where we removed the rods mechanically  
Figure 2: The lasing spectrum under 1.5 times of the threshold pumping density (P\({}_{\rm th}\)), where the diameter of the laser spot was (**a**) 12 \(\rm\upmu m\) and (**b**) 80 \(\rm\upmu m\). The P\({}_{\rm th}\) value and optical microscopic image are the insets. (**c**) The light–light curves with D \(\sim\) 12 \(\rm\upmu m\) (blue) and D \(\sim\) 80 \(\rm\upmu m\) (red). The intensity under small spot excitation was multiplied 25 times intentionally for figure clarity. by blades and then attached them manually to a foreign SiO\({}_{2}\) template by gently smearing them on it. The SiO\({}_{2}\) template was thermally oxidized from Si substrates. While the rods were put on SiO\({}_{2}\), the 376 nm lasing mode was strongly suppressed. As the long wavelength mode had a stronger coupling to the SiO\({}_{2}\) template, its optical energy leaked out more easily than the short wavelength modes [25]. In this publication, we performed a semi-quantitative analysis to prove this mechanism with mode simulations. A 3D model was built according to the actual rod geometry acquired in Figure 1. The energy profile of the two representative modes is illustrated in Figure 3. The 368 nm mode (Figure 3a) had a one node plane along the rod axis while the 376 nm mode (Figure 3b) had none. In the cross-section planes with local maximum intensity (white dashed lines), the energy profiles were all similar to that in Figure 3c. As the Q-factors of either mode were similar (Q = 533 or 542), both modes were competitive to each other in the single rod case. Therefore, the P\({}_{\text{th}}\) variation and dominant mode selection phenomena could not be explained by their original Q-factors. In the collective lasing, the coupling effects between adjacent rods need to be taken into account. To study the side coupling effects to the Q-factors, we formed a 2D seven-rod photonic molecule as depicted in Figure 4a-e. Six rods surrounded the central one with a six-fold rotational symmetry. The rods were aligned along the hexagon diagonals with a 50 nm corner-to-corner distance. Assuming the intensity profile of the central rod is invariant to Figure 3c, where two different supermodes resulted due to the interaction between the outer rods to the central one. Figure 4a is the intensity profile when the outer rods were weakly coupled to the central one. The photons leaking out of the central rod were partly recycled by the outer rods, but did not oscillate in the same way. Therefore, the optical energy was mostly restored only in the central rod.