Photonic band gap materials are dielectrics with a synthetic, three dimensional, multiply periodic microstructure (lattice constant of order the optical wavelength) whose distinguishing feature is a very large modulation depth of refractive index. When appropriately designed, these “photonic crystals” exhibit ranges of optical frequency where light cannot exist — the photonic band gaps 7. The current interest in these materials 1–22 has led us to re-appraise propagation in structures that, while not exhibiting a complete photonic band gap (PBG), nevertheless display anomalous and intriguing propagation effects in the vicinity of their Bragg conditions54–58. In most cases, around each Bragg condition appear incomplete momentum and energy gaps (i.e., ranges of, respectively, wavevector and frequency where propagation is forbidden) with widths that are given approximately by the product of the index difference with, respectively, the vacuum wavevector and h times the optical frequency. With the exception of the multi-layer dielectric stack, most conventional electromagnetic gratings, such as those encountered in holography 27, waveguides 45, distributed feedback lasers 35,37,38, acousto-optic 47 and x-ray 61 diffraction, consist of weak periodic perturbations about a mean refractive index. In these gratings, while strong spatial and temporal dispersion are present around each Bragg condition, the ranges of angles and frequencies over which this occurs are very narrow; and although PBG’s do appear, they are incomplete and mostly very weak.