equations () are nothing more than exponential Fourier series multiplied by the cell-to-cell propagation factor: The Bloch wave expansions are chosen since any field solution within an infinite periodic volume must have the same periodicity as the medium itself, or stated another way, the fields in neighboring cells must be identical up to a (real or complex) propagation factor. In passbands the propagation factor is an exponential function with purely imaginary argument and in the stop bands (or band gaps) it is a decaying exponential function whose argument has a real component. The wave numbers α0, β0 and γ0 satisfy the relations: and outside of these ranges, the bands are periodic. The Bloch waves are periodic functions of space, with periods lx, ly, lz and the bands are periodic functions of wavenumber, with periods: , and Integral equation for PEC media Substituting equations () into () and () yields the spectral domain Greens function relating the radiated electric field to its source currents: where, is the tensor Green's function in the spectral domain. Note that the spatial domain convolution has been transformed into a simple multiplication in the spectral domain, consistent with the convolution theorem for Fourier transforms. With this equation for the electric field, the electric field boundary condition (requiring that the total tangential electric field be zero on the surface of PEC scatterer) becomes: Since we are seeking characteristic modes (eigenmodes) of the structure, there is no impressed E-field on the RHS of this electric field integral equation (EFIE). Equation () is not strictly correct however, since it is only the tangential components of electric field that are actually zero on the surface of the PEC scatterer. This inexactness will be resolved presently, when we test this equation with the electric current basis functions - defined as residing on the surface of the scatterer. Method of moments (MoM) solution As is usual in the method of moments, the source currents are now expanded in terms of a sum over some known set of basis functions with unknown weighting coefficients Jj: Different structures will have different sets of basis functions for representing the currents on the elements and as in the ordinary spatial domain method of moments, the solution (in this case, the band diagram) is a function of the set of basis functions used. Substituting () into () and then testing the resulting equation with the i-th current