Based on more than 30 years of research on differential theories of gratings, this book describes developments in differential theory for applications in spectroscopy, acoustics, X-ray instrumentation, optical communication, information processing, photolithography, high-power lasers, high-precision engineering, and astronomy. Introducing the Fast Fourier Factorization approach to improve the convergence of a truncated series, the book examines multilayers, stacked gratings, crossed gratings, photonic crystals, and isotropic and anisotropic materials; techniques and examples in grating design; and Maxwell equations in a truncated Fourier space.