We describe and demonstrate the extension of permittivity mode expansion, which is also know as generalized normal mode expansion (GENOME), to open and lossy periodic structures. The resulting expansion gives a complete spatial characterization of any open periodic structure, via the quasi-periodic Green’s tensor, by a complete, discrete set of modes rather than a continuum. The method has been validated by comparing our expansion of an open waveguide array with a direct scattering calculation. Good agreement was obtained regardless of the source location or detuning from resonance.