  Closed formulas are derived for the field in the focal region of a
diffraction limited lens, such that the electric field component in a given
direction at the focal point is larger than that of all other focused fields
with the same power in the entrance pupil of the lens. Furthermore, closed
formulas are derived for the corresponding optimum field distribution in the
lens pupil. Focused fields with maximum longitudinal or maximum transverse are
considered in detail. The latter field is similar, but not identical, to the
focused linearly polarized plane wave.
