  We introduce the concept of spatial spectral phase gradient, and demonstrate,
both theoretically and experimentally, how this concept could be employed for
generating single- and multi-path self-accelerating beams. In particular, we
show that the trajectories of the accelerating beams are determined a priori by
different key spatial frequencies through direct spectrum-to-distance mapping.
In the non-paraxial regime, our results clearly illustrate the breakup of Airy
beams from a different perspective, and demonstrate how circular, elliptic or
hyperbolic accelerating beams can be created by judiciously engineering the
spectral phase. Furthermore, we found that the accelerating beams still follow
the predicted trajectory also for vectorial wavefronts. Our approach not only
generalizes the idea of Fourier-space beam engineering along arbitrary convex
trajectories, but also offers new possibilities for beam/pulse manipulation not
achievable through standard direct real-space approaches or by way of
time-domain phase modulation.
