The synthesis of three-dimensional (3-D) light distributions is important f
or many applications. For example, in scanning applications it is preferabl
e that the scanning beam preserve its characteristics over a large distance
to yield elongated scanning range. It is evident that any 3-D light distri
bution must satisfy the wave equation or, in second-order approximation, th
e Fresnel diffraction formula. Thus many desirable 3-D light distributions
may not be realizable. We propose a single optical element (OE) that synthe
sizes a physical beam within a certain 3-D region. The OE provides the opti
mal physical beam in comparison with a desired one in the sense of minimal
mean-square error. (C) 2000 Optical Society of America OCIS code: 140.3300.