Volume-averaged rate equations for planar and disk-cavity lasers

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Abstract
Volume-averaged rate equations are formulated for both planar and disk-cavity lasers. These equations involve not only volume averages of the photon and population number densities, but also volume averages of products of these quantities, and the average over the reflecting surfaces of the incident photon flux. Closure relations for this open set are derived using exact solutions obtained from spatially dependent laser rate equations. The results show that over a wide range of conditions, above threshold, with reflectivities greater than 0.2, the usual spatially uniform rate equations are adequate for symmetric planar cavities. Falling below threshold, or towards smaller values of reflectivity, these equations are shown to be increasingly less accurate. Ultimately, simple closure relations do not suffice. For disk cavities, the exact solutions as well as the volume-averaged rate equation formulations are new. Radial gain profiles, and directional-intensity distributions at the mirror, are presented. Closure conditions for specular and diffuse-reflectivity models are given. As in the planar case, simple closure relations eventually fail to give an adequate description as reflectivities tend towards zero or operation falls below threshold.