to compute bands in various types of doped and undoped photonic crystals. Not surprisingly, doping photonic crystals with defects provides a means for designing photonic passbands, in just the same way that doping semiconductors with chemical impurities provides a means for designing electronic passbands. For many subsectional basis functions, such as those having a half-sine or triangular shape along a round wire, the FT of the basis function for negative wave numbers -α, -β, -γ is the complex conjugate of the basis function FT for positive wave numbers. As a result, the matrix in eqn. () is Hermitian. And as a result of that, only half the matrix needs to be computed. And a second result is that the determinant is a purely real function of the real-valued wavenumber k. Zeroes generally occur at zero-crossings (inflection points, where curvature is zero), so a simple root-finding algorithm such as Newton's method is usually sufficient to find the roots to a very high degree of accuracy. If may still be useful however to plot the determinant as a function of k, to observe its behavior near the zeros. As a matter of computational convenience, whenever the matrix is larger than 2×2 it's vastly more efficient to compute the determinant either by reducing the matrix to upper triangular form using QR decomposition or to compute the determinant by reducing to echelon form using Gaussian elimination, rather than trying to actually compute the determinant of the matrix directly. Analysis – first principles approaches Spectral domain method of moments (overview and mathematical introduction) Background History Historically, the first approach to solving for fields reflected and transmitted by FSS was the spectral domain method (SDM), and it is still a valuable tool even today [Scott(1989)]. The spectral domain method is known at Ohio State University as the periodic method of moments (PMM). The SDM starts out with an assumed Floquet/Fourier series solution for all fields, currents and potentials whereas the PMM starts out with a single scatterer, then adds in all of the scatterers in the infinite plane (in the spatial domain), then uses a transformation to yield the spectral domain representation of the fields. Both approaches are effectively the same approach, in the sense that they both assume an infinite planar structure which gives rise to a discrete Fourier series representation for the fields. Advantages and disadvantages The spectral domain method has one very important