## 2 Materials and Methods  
The monolithic GaN-InGaN core-shell nanorod array was fabricated via a series of pattern-and-regrowth processes. The detailed process flow can be found in [25]. The scanning electron microscopy (SEM), cross-sectional transmission electron microscopy (TEM), and monochromatic cathodoluminescence (CL) images are shown in Figure 1. The core diameter before regrowth was 400 nm wide with a period of 900 nm. After regrowth, the hexagon diagonal was estimated to be 850 nm with a minimum rod spacing of 50 nm. The shell region consisted of five pairs of InGaN/GaN multiple-quantum-wells (MQWs). The composition of the quantum wells was 3-4%. The quantum well (QW) thicknesses ranged from 0.8 nm to 1.8 nm and the barrier thickness ranged from 6 nm to 8 nm according to its position in the rods. The QW/barrier thicknesses were found to be thinner at the rod-top facets and thicker on the vertical sidewalls, which was attributed to the inherently different growth dynamics among the crystallography planes in wurtzite crystals. The 376 nm CL emission in Figure 1c came from the InGaN quantum wells in the shell region, the 365 nm emission was from the core, and the 410 nm emission was attributed to the parasitic segregation at the top corners. The sample was excited by 355 nm Nd:YVO\({}_{4}\) pulse solid-state lasers for the photoluminescence (PL) measurement at room temperature. The excitation source was focused by a 100 times magnification objective lens. The spot size was controlled by tuning the position of the sample along the optical axis.  
Two-dimensional mode simulations were conducted by commercially available software (COMSOL Multiphysics), which solved the wave equation of the electric field (E) with finite element methods. The eigen-solution can be expressed in the following form:  
\[\text{E(x,y,t)}=\text{E}_{0}\text{ (x,y)}^{\ast}\text{exp}(\text{i}\omega\text{t}) \tag{1}\]  
where \(\text{E}_{0}\) stands for the amplitude profile of the standing wave and \(\omega\) is the complex eigen-frequency:  
Figure 1: (**a**) The bird-view scanning electron microscopy; (**b**) cross-sectional transmission electron microscopy; (**c**) top view monochromatic cathodoluminescence images of the GaN-InGaN core-shell array.  
\[\omega=\omega_{\rm res}+\rmi\gamma \tag{2}\]  
where \(\omega_{\rm res}\) is the resonant frequency and \(\gamma\) is the damping rate of the electric field. Then, the Q-factor can be estimated by:  
\[\rmQ=\mid\omega_{\rm res}/2\gamma\mid \tag{3}\]  
Finally, the mode intensity profile (U) can be plotted by:  
\[\rmU(x,y)=\mid\rmE_{0}(x,y)\mid^{\,2} \tag{4}\]  
We adopted the geometry nanorods in Figure 1 as the forming unit in the PM simulations. A single 50 nm InGaN layer was used instead of the actual 5-pair MQW structure in the shell region as a reasonable simplification for a semi-quantitative simulation study. The refractive indices for GaN and InGaN were 2.56 and 2.62, respectively. The extinction coefficients were first assumed to be zero without a loss of generality (i.e., the cold cavity case). Three-dimensional mode simulations were also conducted with the same methodology. Due to limited computational capacity, 3D simulations were only conducted for the single rod case. For the mode simulations that consisted of multiple rods, only 2D simulations were conducted. In a monolithic wurtzite crystal, the perimeters of the hexagon were surrounded by crystallographic planes, so they must be parallel to each other. Therefore, the relative orientation in the crystal between the hexagons is an essential parameter that does not exist in photonic molecules with circular disks.