Identification of Lasing Modes of Photonic-Crystal Surface-Emitting Lasers by Polarization Dependence of the Far-Field Pattern: An Application of the $\mathrm{k}\cdot \mathrm{p}$ Perturbation Theory

Photonic-crystal surface-emitting lasers (PCSELs) have been studied extensively mainly because we can expect both excellent beam quality and large output power. This idea was applied not only to laser diodes of the direct gap semiconductors but also to quantum cascade lasers (QCLs). In the practical design of the PC (photonic crystal) resonators, it is often the case that more than one resonance modes in the zone center ( $\Gamma$ point) are included in the spectral range of the gain medium, so the lasing mode cannot be identified a priori. However, when the PC resonator possesses a certain spatial symmetry, we can identify it by examining the polarization dependence of the far-field pattern (FFP).