Size Scaling Law for Radiation Losses of Modes in Photonic Crystal Surface Emitting Devices

Photonic-crystal surface-emitting lasers (PCSELs) have garnered significant attention due to their ability to generate laser beams with ultra high power and low divergence. This is because they support high power single mode lasing with volumes orders of magnitude larger than those of conventional semiconductor lasers. The finite lateral size in a PCSEL is a primary factor limiting its lasing mode volume and consequently, its output power. We demonstrate that the scaling relation between the total cavity loss $\alpha=\alpha_\perp + \alpha_\parallel$ and the device size $L$ is such that the surface radiation loss scales as $\alpha_{\perp} \sim O(L^{-2})$, while the edge radiation loss $\alpha_{\parallel} \sim O(L^{-3})$. Both scaling relations can be explained by the second order expansions of the complex frequency $\omega(k)$ of the band diagram. Our results constitute an explicit guideline for PCSEL designs to optimize various optical properties.