Metasurface Beam Steering — Generalized Snell's Law

Incident Beam

Incidence angle θᵢ 20°

Wavelength λ 0.60

Metasurface Phase Gradient

Phase step per cell Δφ 30°

Show ordinary-refraction reference (Δφ = 0)

Live Readout

Transmitted angle θₜ

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Phase gradient dΦ/dx

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sin θₜ & status

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How it works

An ordinary flat interface obeys Snell's law: n₁ sinθᵢ = n₂ sinθₜ. A metasurface is a sub-wavelength array of resonant nano-antennas whose orientation or shape imprints an extra, spatially-varying phase Φ(x) onto the wavefront as it crosses the interface. Fermat's principle then gives the generalized Snell's law:

n_t sinθₜ − n_i sinθᵢ = (λ₀ / 2π) · (dΦ/dx)

with n_i = n_t = 1 here, this reduces to:
        sinθₜ = sinθᵢ + (λ₀ / 2π) · (dΦ/dx)

Because the added term does not depend on θᵢ, the beam can be steered to an "anomalous" angle a uniform slab of the same two media could never produce — even bent to the same side as the incident ray, or trapped as an evanescent surface wave when |sinθₜ| > 1.

θᵢ — angle of the incoming beam from the surface normal.
λ — free-space wavelength, in the same length units as the antenna spacing.
Δφ per cell — the phase jump between neighbouring antenna columns; each row of the array cycles through 0→2π and repeats, exactly like a real 8-element "supercell" metasurface. This sets dΦ/dx = Δφ / a.
Each antenna's colour and in-plane rotation encode its local phase Φ(x). For antennas that work by geometric (Pancharatnam–Berry) phase, rotating a resonant element by angle α imprints phase 2α on circularly-polarized light — the rotation you see is one real way these devices are built.
The dashed grey ray is where the beam would go with no phase gradient (ordinary refraction, n_i = n_t ⇒ straight through) — compare it to the solid steered beam to see the anomalous bending directly.

This is the mechanism behind flat metalenses, beam-steering LIDAR metasurfaces, and holographic metasurfaces — bending and focusing light with a wafer-thin patterned surface instead of curved glass.

Incident beam

Anomalously refracted beam

Ordinary refraction (reference)

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