Analytical electromagnetic solution for Bragg mirrors with graded interfaces and guidelines for enhanced reflectivity

Bragg mirrors, the key element of vertical-cavity surface-emitting lasers, always present graded interfaces when current is pumped through them, so as to reduce their series resistance and related heating problems. We present here an analytical solution for the electromagnetic fields in graded Bragg mirrors. It allows to generate reference reflection and transmission coefficients that are compared with those obtained by the staircase approximation method. Graded mirrors are then analyzed in detail and it is shown that, with an optimized design, higher reflectivity than in the abrupt scheme can be achieved