A Helmholtz resonator approximation for photonic and phononic crystals

We consider acoustic (or electromagnetic) waves in periodic systems of tightly packed, sound hard (metallic) inclusions in two dimensions. In this limit, and for low frequencies, the voids between the narrow gaps act as Helmholtz-like resonators, so the system is equivalent to a set of masses connected by springs. The calculations are performed for a number of different lattice types and inclusion shapes. We also attempt the problem of similar systems in three dimensions, as wall as of systems with a source located in one of the voids. We argue that such crystals constitute a versatile class of metamaterials as their entire acoustic branch (or branches when the discrete analogue is polyatomic) is squeezed into a subwavelength regime, allowing for very high refractive indices.
The fully analytical predictions agree with finite-element simulations. In particular, we verify that the frequencies scale like the ratio of gap width to lattice constant raised to the power 1/4.