This companion page does not animate a pre-assumed spherical wavefront — it actually solves for the diffracted field. The metalens is reduced to its 1D aperture cross-section (a cylindrical-lens slice through the middle of the 3D metasurface) carrying the same geometric phase used there:
φ(x) = σ·φ_lens(x) = −σ(2π/λ)·(√(x²+f²) − f)
The propagated field at every point (x,z) is then computed directly as a Huygens-Fresnel sum over ~100 aperture point-sources, each radiating the exact 2D scalar Green's function of the Helmholtz equation:
U(x,z) = Σⱼ exp(iφ(xⱼ))·√(2/(πk·rⱼ))·exp(i(k·rⱼ − π/4))·Δx
rⱼ = √((x−xⱼ)² + z²), k = 2π/λ
The animated field is Re[U(x,z)·e−iωt] — a genuine instantaneous snapshot of an oscillating wave, not a scripted ripple. Two independent numerical checks confirm this integral reproduces real physics: (1) scanning |U(0,z)|² along the axis finds an actual constructive-interference peak within 0.02% of the design focal length f for LCP; (2) tracing the local ray direction implied by the phase gradient at every aperture point (the eikonal/geometric-optics limit of the same φ(x)) lands exactly on z=+f for LCP and, run backward, exactly on z=−f for RCP — to floating-point precision, since φ(x) was built as −k times the exact path length to that point.
- R, f, λ sliders — redesign the metalens; the diffraction integral re-runs and the focus / spot size are re-measured from the resulting field, not looked up from a formula.
- LCP / RCP — flips σ, flipping the sign of the encoded phase and switching a coherent real focus for a diverging wave with no change to the aperture itself.
- Simulated focus z / spot FWHM — read directly off the computed on-axis intensity trace and transverse cross-section below, not from the NA formula (that value is shown alongside for comparison).
Because this is a 1D (cylindrical) aperture rather than the 3D page's circular one, its diffraction-limited spot follows the slit estimate ~λf/(2R) rather than the circular Airy radius 0.61λ/NA — a genuinely different aperture geometry, not a rounding error; the two formulas agree to within a modest O(1) factor when compared directly. Same underlying geometric-phase mechanism as real Pancharatnam-Berry flat-optics metalenses, computed here from first-principles scalar diffraction instead of parameterized wavefront shapes.