I will report onexperiments that were performed with superconducting microwave photoniccrystals, that is, flat resonators containing circular scatterers arranged on atriangular grid, so-called Dirac billiards (DBs). The eigenfrequencies of wavepropagation as function of the two quasimomentum components exhibit a bandstructure which comprises two Dirac points (DPs), where two bands touch eachother conically, and in between them a nearly flat band. This is reminiscent ofa combined Honeycomb-Kagome lattice. The high-precision measurements allowedthe determination of complete sequences of several thousands of eigenfrequencies.Around the DPs the density of states (DOS) of DBs is similar to that ofgraphene and well described by a finite tight-binding model which includesfirst-, second-, and third-nearest-neighbor couplings. At the band edges DBsare governed by the non-relativistic Schrödinger equation of the quantumbilliard, around the DPs by the Dirac equation of the graphene billiard ofcorresponding shape, respectively. We analyzed the spectral properties of DBsof various shapes and compare them to those of graphene billiards andrelativistic & non-relativistic quantum billiards.
