  We use inverse design to discover metalens structures that exhibit broadband,
achromatic focusing across low, moderate, and high numerical apertures. We show
that standard unit-cell approaches cannot achieve high-efficiency high-NA
focusing, even at a single frequency, due to the incompleteness of the
unit-cell basis, and we provide computational upper bounds on their maximum
efficiencies. At low NA, our devices exhibit the highest theoretical
efficiencies to date. At high NA -- of 0.9 with translation-invariant films and
of 0.99 with "freeform" structures -- our designs are the first to exhibit
achromatic high-NA focusing.
