  Flat optics consisting of nanostructures of high-refractive-index materials
produce lenses with thin form factors that tend to operate only at specific
wavelengths. Recent attempts to achieve achromatic lenses uncover a trade-off
between the numerical aperture (NA) and bandwidth, which limits performance.
Here we propose a new approach to design high NA, broadband and
polarization-insensitive multilayer achromatic metalenses (MAM). We combine
topology optimization and full wave simulations to inversely design MAMs and
fabricate the structures in low-refractive-index materials by two-photon
polymerization lithography. MAMs measuring 20 micrometer in diameter operating
in the visible range of 400-800 nm with 0.5 NA and 0.7 NA were achieved with
efficiencies of up to 42%. We demonstrate broadband imaging performance of the
fabricated MAM under white light, and RGB narrowband illuminations. These
results highlight the potential of the 3D printed multilayer structures for
realizing broadband and multi-functional meta-devices with inverse design.
