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In this paper, we present a unified Fourier transform (FT) approach to the study of arrayed waveguide grating (AWG), fiber Bragg grating (FBG), and photonic crystal (PC) devices. This methodology allows the design characteristics of transverse (AWG) geometries to be mapped on to longitudinal (or distributed) FBG and PC structures, due to a comprehensive isomorphism between these important passive optical grating-based devices. The unified approach presented here, which is based upon a modified Debye-Waller approach to the analytical solution of the coupled-mode equations, allows intuitive, yet accurate appraisal of arbitrary strength coupled structures. Exploiting this isomorphism, we relate our theoretical work to a number of practical cases. For example, we describe an FBG carousel configuration (analogous to a chirped AWG cascade) offering virtually ripple-free third-order dispersion compensation of 6.61 ps/nm2 over a 100-GHz bandwidth.