# Symmetry Criterion and Far-Field Control of Photonic-Crystal Surface-Emitting Lasers
## 1 Introduction  
Narrow laser beams are required in many applications, such as light detection and ranging (LiDAR), laser processing, etc. [1, 2]. Sometimes, the requirement for divergence angle is even below 0.1\({}^{\circ}\)[3, 4]. Especially, with the continuous development of autonomous driving technology, LiDAR is expected to play a major role in the future. A light source with ultralow divergence angle is vital for improving the spatial resolution of LiDAR. For example, a divergence angle of 0.1\({}^{\circ}\) implies a spot size of 17 cm at a detection distance of 100 m, which is common in autonomous driving applications. However, traditional semiconductor lasers can hardly meet such requirements. For instance, the divergence angle of edge-emitting lasers is generally in the range of 10-30\({}^{\circ}\)[5, 6]. Collimation and focus are necessary in this case. The alignment process is difficult because its accuracy should be of the order of microns. Moreover, the external lens will inevitably weaken the stability of the system. Thus, researchers hope to find a solution at the on-chip level. Vertical-cavity surface-emitting lasers (VCSELs) are good candidates. Although their divergence angle is also large [7, 8], of the order of 10\({}^{\circ}\), they have better circular spots compared with edge-emitting lasers. This greatly reduces the difficulty of collimation. In addition, due to their planar manufacturing process, they can be easily integrated with a microlens [9] or metalens [10, 11], and it is easier to manufacture large-scale chip arrays, although alignment error is still inevitable. Photonic-crystal surface-emitting lasers (PCSELs), a novel kind of surface-emitting semiconductor lasers, retain the above advantages of VCSELs, and fill the blank of a semiconductor light source with a divergence angle below 10\({}^{\circ}\)[12, 13, 14, 15, 16, 17, 18]. It introduces large freedom into the design of resonator cavity, which can realize large-area single-mode resonance. Recently, Noda's group, the main developer of PCSELs, has experimentallyachieved an impressive performance with continuous output power of 29 W and divergence angle of less than 0.4\({}^{\circ}\) without any lens [19; 20]. They also analytically derive the general conditions for ultra-large-area (3-10 mm) single-mode operation in PCSELs to achieve 100 W to 1 kW lasing [21]. This indeed provides a bright prospect for the development and application of PCSELs in the future. In the face of such an impressive result, we want to know why PCSELs can achieve such a low divergence angle, and what factors determine the far-field patterns. Some important works have been performed on this. [PERSON] et al. first demonstrated that the asymmetry of PCSELs is helpful to realize single-lobed FFP and showed us various tailored beams produced by PCSELs [22]. [PERSON] et al. developed a three-dimensional coupled wave model to calculate the far field of finite-size PCSELs [23]. Besides, the perspective of group theory is first introduced to analyze resonant modes of PCSELs by [PERSON] and [PERSON] [24]. [PERSON] et al. recently utilized group theory to analyze far-field polarization of PCSELs with circular air holes [25]. However, the mechanism of the influence of photonic crystal (PhC) geometry on the shape of far-field pattern is still not clear, especially for PCSELs with more complicated PhC geometry. With the further development of PCSELs, more detailed theoretical studies on these influence mechanisms and their underlying physical origin need to be explored. Our work will focus on this goal and attempt to achieve control of the far field with adjusted PhC geometry and symmetry. In this work, we systematically studied the influence of PhC cavity on the far-field performance of PCSELs by three-dimensional finite-difference time-domain (3D-FDTD) simulation. In Section 2, we present the model setup and explain the analysis method we use.