Lens specification
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Summary
Fresnel zone table
What this means
Wrapped phase at the lens edge — outermost zones
Focal-spot cross-section (fixed plane z = f)
Chromatic behavior across the band
Method: target phase is the stigmatic hyperbolic profile
φ(r) = −(2π/λ₀)(√(r²+f²)−f) (Aieta 2012 / Khorasaninejad, Science 2016), with exact zone radii
rm = √(2mλ₀f+(mλ₀)²) and local period Λ(r) = λ₀√(r²+f²)/r (O'Shea et al., SPIE TT62 — the
paraxial √(2mλf) is deliberately not used). N-level efficiency ηm = sinc²(m/N) follows Swanson
(MIT LL TR-854); wavelength detuning η(λ) = sinc²(α−1), α = (λ₀/λ)(n(λ)−1)/(n(λ₀)−1) (Buralli &
Morris 1992); relief depth h2π = λ₀/(n−1) with built-in Sellmeier/Cauchy dispersion. PSF and
Strehl come from the radially symmetric Fresnel/Debye integral U(ρ) ∝ ∫P(r)ei[φ(r)+(2π/λ)(√(r²+f²)−f)]J₀(2πrρ/λ√(r²+f²))·r·dr
(Goodman Ch. 4–6), integrated zone-by-zone analytically at the fixed design focal plane — it reproduces
ηN = sinc²(1/N) at λ₀ and the chromatic focal shift f(λ) = f₀λ₀/λ. Pillar diameters use the
truncated-waveguide LP₀₁ effective index of an isolated cylinder (Lalanne 1999) — a lookup that ignores
near-neighbour coupling. Validity: everything here is scalar theory. Above NA ≈ 0.5 (edge features
< ~2λ) the sinc² efficiencies and pillar map become upper bounds, not predictions — validate in
FDTD/RCWA before tape-out (measured anchors: 73% at NA 0.8/532 nm Harvard; 94% at 940 nm NILT; ~40% avg for
the GaN achromat). The achromatic verdict is the Presutti–Monticone time-bandwidth limit
Δω ≤ κ/ΔT, ΔT = (√(f²+R²)−f)/c (Optica 7, 624, 2020). On-axis, unpolarized, normal incidence only.
Why can't a big, fast metalens be broadband-achromatic?
Focusing a pulse means the ray from the lens edge must arrive at focus at the same
time as the axial ray, so the edge meta-atoms must store light for ΔT = (√(f²+R²)−f)/c — picoseconds for a
millimetre-class NA 0.8 lens. A passive, time-invariant structure only a wavelength thick can delay light
by roughly its Q-limited storage time, capping the bandwidth at Δω ≤ κ/ΔT no matter how clever the design.
That is why the celebrated broadband achromats are tiny and slow (Wang 2018: D = 50 µm, NA = 0.106 spanning
400–660 nm) while every high-NA millimetre metalens is a single-wavelength device — and why "make it
achromatic too" is a physics question, not an engineering request.