  A new class of nonparaxial accelerating optical waves is introduced. These
are beams with a Bessel-like profile that are capable of shifting laterally
along fairly arbitrary trajectories as the wave propagates in free space. The
concept expands on our previous proposal of paraxial accelerating Bessel-like
beams to include beams with subwavelength lobes and/or large trajectory angles.
Such waves are produced when the phase at the input plane is engineered so that
the interfering ray cones are made to focus along the prespecified path. When
the angle of these cones is fixed, the beams possess a diffraction-free Bessel
profile on planes that stay normal to their trajectory, which can be considered
as a generalized definition of diffractionless propagation in the nonparaxial
regime. The analytical procedure leading to these results is based on a ray
optics interpretation of Rayleigh-Sommerfeld diffraction and is presented in
detail. The evolution of the proposed waves is demonstrated through a series of
numerical examples and a variety of trajectories.
