# Symmetry Criterion and Far-Field Control of Photonic-Crystal Surface-Emitting Lasers
## 2 Calculation Model and Method  
The PCSEL model in this paper is shown in Figure 1a. The structure has four layers of cladding layer I, active layer, PhC layer, and cladding layer II, from top to bottom. The refractive indices of materials in these layers are 3.2, 3.52, 3.55, and 3.28, respectively, which are similar to the refractive index of GaAs/AlGaAs. The PhC adopts a square-lattice structure, and the refractive index of air holes was 1. The red box represents the simulation region, whose boundaries are set as perfect matched layer (PML) in all directions. Two frequency-domain field monitors are placed within the center of the PhC layer, and at the position about 0.6 times the wavelength above the top surface of cladding layer I, respectively. Magnetic dipoles act as the injection light sources that favor TE polarization. They are placed at the antinode position in the PhC layer periodically for the selection of target resonance mode.  
Figure 1: (**a**) Schematic of the photonic-crystal surface-emitting laser (PCSEL) model adopted in the simulation. The orange layer represents active layer, below which is the photonic crystal (PhC) layer. The top and bottom layers are cladding layers. The translucent blue planes represent field monitors. The red box represents simulation region. The yellow arrow represents light emission direction. (**b**) The typical photonic band of the square-lattice PCSEL near \(\Gamma_{2}\) point. Point A is the band edge where PCSELs commonly work at.  
For square-lattice PCSELs, the unidirectional vertical light emission can be achieved at the \(\Gamma_{2}\) point, corresponding to the lasing mode. There are four band edges near the \(\Gamma_{2}\) point, including two dielectric bands and two air bands, among which band edge A, as shown in Figure 1b, generally has the lowest threshold and will preferentially lose. The controlling of mode threshold has been investigated in detail by [PERSON] et al. [21]. Here, we select the fundamental mode operating at band edge A as the studied object in this paper.  
Figure 2 displays the simulation results of the square-lattice PCSEL with circular air holes. The magnetic field distribution in PhC layer, as shown in Figure 2a, is directly obtained from monitor 2. The field intensity is in the range from \(2.5\times 10^{-4}\) to \(3.4\times 10^{-2}\) A/m. It clearly illustrates the large-area resonance characteristics in PhC cavities. Moreover, its distribution in the single unit cell is very similar to that in slab model [26], indicating their similarity in the modulation of the electromagnetic field.  
The near-field pattern (NFP), as shown in Figure 2b, is obtained by integrating the Poynting vector from monitor 1. Its intensity is in the range from \(1.8\times 10^{-8}\) to \(1.3\times 10^{-4}\) W/m\({}^{2}\). It has a low intensity in the middle, which is caused by the negative interference originating from its perfect structure symmetry. The corresponding annular FFP is shown in Figure 2c, which is calculated through the Fourier transform of the near field, and its intensity profile at the center along the x direction is shown in Figure 2d. Apparently, the center has zero intensity. This is because vertical radiation decays due to the negative interference when the in-plane wave vector \(\mathbf{k}\) is equal to 0. However, the finite period number of the photonic crystal extends the wavenumber to the periphery of \(\Gamma_{2}\)[26], so the annular spot appears. Evidently, PhC with circular holes is not a good choice for the design of high-power PCSELs with a single-lobed far-field pattern [27; 28].