  We investigate controllable spatial modulation of circular autofocusing Airy
beams, under action of different dynamic linear potentials, both theoretically
and numerically. We introduce a novel treatment method in which the circular
Airy beam is represented as a superposition of narrow azimuthally-modulated
one-dimensional Airy beams that can be analytically treated. The dynamic linear
potentials are appropriately designed, so that the autofocusing effect can
either be weakened or even eliminated when the linear potential exerts a
"pulling" effect on the beam, or if the linear potential exerts a "pushing"
effect, the autofocusing effect can be greatly strengthened. Numerical
simulations agree with the theoretical results very well.
