  We predict that Bessel-like beams of arbitrary integer order can exhibit a
tunable self-similar behavior (that take an invariant form under suitable
stretching transformations). Specifically, by engineering the amplitude and the
phase on the input plane in real space, we show that it is possible to generate
higher-order vortex Bessel-like beams with fully controllable radius of the
hollow core and maximum intensity during propagation. In addition, using a
similar approach, we show that it is also possible to generate zeroth order
Bessel-like beams with controllable beam width and maximum intensity. Our
numerical results are in excellent agreement with our theoretical predictions.
