ABSTRACTWe present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one‐dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice‐summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.