  We investigate the propagation of light beams including Hermite-Gauss,
Bessel-Gauss and finite energy Airy beams in a linear medium with parabolic
potential. Expectedly, the beams undergo oscillation during propagation, but
quite unexpectedly they also perform automatic Fourier transform, that is,
periodic change from the beam to its Fourier transform and back. The
oscillating period of parity-asymmetric beams is twice that of the
parity-symmetric beams. In addition to oscillation, the finite-energy Airy
beams exhibit periodic inversion during propagation. Based on the propagation
in parabolic potential, we introduce a class of optically-interesting beams
that are self-Fourier beams -- that is, the beams whose Fourier transforms are
the beams themselves.
