Scaling method to classify confined wave states

Wave phenomena represent a major domain in physics that is present at every observational scale. By tuning the properties of the medium in which the waves propagate, the propagation can be modified, controlled, and utilized. In this sense, an especially interesting effect is that of wave confinement, achieved by introducing disorder into an otherwise periodic medium. The interference of waves in such an altered structure may result in confinement of a large portion of energy inside a small volume of the medium, allowing for a variety of applications.

However, fabrication of such media is usually complicated by high costs and practical difficulties, especially at the nano-scale. Thus, it is often necessary to analyze the wave confinement numerically, to optimize the material before it is physically fabricated. To date, however, there is no fundamentally reliable computational method capable of identifying confined wave states within a given medium.

Here, we present such a computational method and illustrate it on a nanophotonic problem of light confinement in photonic crystals. Our method is based on the scaling theory known in condensed matter physics. Our method also provides fundamental constraints to distinguish between different degrees (dimensionalities) of confinement. The method has general application for waves of arbitrary type, in samples with an arbitrary integer dimension.