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In this work, we study wave propagation in generic Hermitian local periodicbaths, and investigate the effects of anisotropy and quasi-breaking ofperiodicity on resonant emission into the band of the bath. We asymptoticallydecompose the Green's function into long-range travelling waves composed of allwavevectors (near-)resonant at the emitter frequency, and rapidly decayingevanescent waves. Our approximation then converges exponentially withincreasing source-receiver separation ${\rho}$ when resonant wavepackets withgroup velocity parallel to ${\rho}$ exist. In hyperbolic media this conditionmay not be satisfied, and we find that the exponential decay length ofoscillating evanescent waves locally around caustics generally depends as apower law with exponent 3/2 on the angle made between ${\rho}$ and the caustic.For ${\rho}$ beyond the caustic we observe that the Green's function can becomealmost imaginary, which results in exclusively incoherent emitter-emitterinteractions and allows the simulation of purely dissipative systems withshort-range interactions. Here the interaction length is tunable via theseparation vector of the emitters. We finally probe the hyperbolic dispersionbeyond the previous regimes by applying an artificial gauge field on thelattice. We find that emission resonant with the corresponding open orbits inthe Brillouin zone is quasi-one dimensional, in contrast to an isotropicenvironment. The quasi-1D emission is further topologically protected againstlocal and global lattice perturbations and periodically refocussing, offering arobust bi-directional transport of excitations in higher-dimensional media.

Green's functions of and emission into discrete anisotropic and hyperbolic baths