  The evaluation of the interaction between objects arranged on a lattice
requires the computation of lattice sums. A scenario frequently encountered are
systems governed by the Helmholtz equation in the context of electromagnetic
scattering in an array of particles forming a metamaterial, a metasurface, or a
photonic crystal. While the convergence of direct lattice sums for such
translation coefficients is notoriously slow, the application of Ewald's method
converts the direct sums into exponentially convergent series. We present a
derivation of such series for the 2D and 3D solutions of the Helmholtz
equation, namely spherical and cylindrical solutions. When compared to prior
research, our novel expressions are especially aimed at computing the lattice
sums for several interacting sublattices in 1D lattices (chains), 2D lattices
(gratings), and 3D lattices. We verify our results by comparison with the
direct computation of the lattice sums.
