Differentiable Fast Far-Field Transform in Cylindrical Coordinates for Large-Area Cascaded Metalens Optics
We present a fully differentiable far-field transform in cylindrical coordinates for full-area point spread function (PSF) evaluation and optimization of large axisymmetric metalenses. The method computes wave-optical responses of apertures spanning thousands to tens of thousands of wavelengths in diameter (millimeter scales in the visible, centimeter scales in the infrared) in seconds, achieving three to four orders of magnitude speedup over Green's function integration while avoiding the prohibitive memory of two-dimensional FFTs. The approach decomposes vectorial near fields into parallel angular-momentum channels, applies FFTLog-accelerated Hankel transforms, and uses Graf's addition theorem to recenter focal fields under oblique illumination. Analytic adjoint gradients enable optimization with only ~65% overhead relative to a forward simulation. For a 4 mm-diameter aperture (~8000 wavelengths, ~12,600 azimuthal modes) at 30-degree incidence, a forward-adjoint iteration requires only ~12 s on a 350-thread CPU, making oblique optimization practical without ray-tracing approximations. Applied to polychromatic RGB (446/530/650 nm) metalens design at normal incidence, full-area PSF evaluation exposes efficiency limits hidden by conventional cropped-focal-spot analysis: a mono-pillar metalens that appears diffraction-limited achieves only ~6% average absolute focusing efficiency, while direct far-field optimization raises this to 37% (locally periodic approximation) and 51% (zoned discrete axisymmetry). A cascaded double-metasurface design reaches 63%, while a four-metasurface architecture attains 96% average relative efficiency. We also demonstrate millimeter-scale, oblique-incidence optimization of single-surface and doublet architectures; cascaded doublets enable partial coma correction inaccessible to a single rotationally symmetric surface.