  Recent advances in photonic optimization have enabled calculation of
performance bounds for a wide range of electromagnetic objectives, albeit
restricted to single-material systems. Motivated by growing theoretical
interest and fabrication advances, we present a framework to bound the
performance of photonic heterostructures and apply it to investigate maximum
absorption characteristics of multilayer films and compact, free-form
multi-material scatterers. Limits predict trends seen in topology-optimized
geometries -- often coming within factors of two of specific designs -- and may
be exploited in conjunction with inverse designs to predict when
heterostructures are expected to outperform their optimal single-material
counterparts.
