Constraints on transmission, dispersion, and density of states in dielectric multilayers and stepwise potential barriers with an arbitrary layer arrangement

Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the nontunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry are imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but occurring within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the potential barriers, only nonperiodic energy intervals can be present in the spectrum of any given structure, and only if the parameters of constituent potential steps are properly chosen. The integral conservation relations derived analytically have also been verified numerically. The relations can be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS.