Analytical one-dimensional phase gratings in scalar diffraction theory

Diffraction gratings are a commonplace optical tool which split an incident beam into distinct diffraction orders via the far-field interference of light. However, using scalar diffraction theory to model a phase grating with phase profile ϕ(x) requires Fourier transforming exp⁡[iϕ(x)], which predominantly requires numerical calculations outside of a few standard geometries. We present three designs, the full- and half-wave rectified sine wave and the parabola, which have exact analytical closed-form (non-numerical) intensities in every diffraction order. We also analytically describe the diffraction from a known profile interspersed with finite regions of no phase modulation. We fabricate full- and half-wave rectified gratings and demonstrate their electron diffraction. These designs are of the most use for small-wavelength optics with X-rays, electrons, neutrons, and other massive quanta.