Incomplete Airy beams: finite energy from a sharp spectral cutoff.

We present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting "incomplete Airy beam" is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its fringes. Its propagation can be described explicitly using an incomplete Airy function, from which we derive simple expressions for the beam's total power and mean position. Asymptotic analysis reveals a simple connection between the cutoff and the region of the beam with Airy-like behavior.