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We show that it is possible to independently control both the trajectory andthe maximum amplitude along the trajectory of a paraxial accelerating beam.This is accomplished by carefully engineering both the amplitude and the phaseof the beam on the input plane. Furthermore, we show that the width of anaccelerating beam is related only on the curvature of the trajectory.Therefore, we are able to produce beams with predefined beam widths andamplitudes. These results are useful in applications where precise beam controlis important. In addition we consider radially symmetric abruptly autofocusingbeams. We identify the important parameters that affect the focalcharacteristics. Consequently, we can design autofocusing beams with optimizedparameters (such as sharper focus and higher intensity contrast). In all ourcalculations the resulting formulas are presented in an elegant and practicalform in direct connection with the geometric properties of the trajectory.Finally we discuss methods that can be utilized to experimentally realize suchoptical waves.

Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams