Comparison of exact and approximate optical designs for graded-interface distributed Bragg reflectors

The reflectance of a distributed Bragg reflector (DBR) is maximized when multiple reflections within the structure are phase-matched. DBRs with graded interfaces are often designed using approximations based on optical thickness or coupled mode theory: The former addresses phase-matching only approximately, while the latter does not explicitly address phase-matching at all. By introducing the concept of a generalized interface, we demonstrate how to exactly phase-match Fabry-Perot cavities and DBRs containing arbitrary grades. The theoretical reflectance of exact and approximate mirror designs is compared for linear and asymmetric parabolic grades in a 25 period GaAs-AlAs DBR. For short, symmetric grades, the performance differences are quite small, but for longer, asymmetric grades, they can be significant. Our comparison suggests a simple modification of the optical thickness approach that optimizes its accuracy for asymmetric grades. Our generalization of reflectance phase-matching also has applicability beyond DBR design, since it applies to nonperiodic structures.