Metasurface Phased Array — Grating Lobes vs Generalized Snell's Law

Incident Beam

Incidence angle θᵢ 20°

Wavelength λ 0.60

Metasurface Phased Array

Phase step per element Δφ 30°

Element pitch a / λ 0.50

Show ordinary-refraction reference

Live Readout

θₜ — generalized Snell (m=0)

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θₜ — measured array-factor peak

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peak-vs-theory agreement

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grating lobes (m ≠ 0) in view

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How it works

This is a genuine 2D wave simulation, not a flattened version of the 3D ray picture. Each of the N meta-atoms is treated as an independent Huygens point source: it re-radiates a cylindrical wavelet carrying the local phase Φᵢ = k·xᵢ·sinθᵢ + i·Δφ (incident phase at that element, plus the metasurface's own imprinted phase step). The near-field canvas sums every wavelet, amplitude ~1/√r, at each pixel — the steered beam you see is genuine constructive interference, not a drawn arrow.

The far-field strip computes the exact array-factor closed form for a uniform linear phased array of pitch a:

Ψ(θ) = k·a·(sinθᵢ − sinθ) + Δφ
|F(θ)| = |sin(NΨ/2) / sin(Ψ/2)|

Main lobes at Ψ = 2πm:
  sinθ_m = sinθᵢ + (λ/2πa)·Δφ − m·(λ/a),  m = 0, ±1, ±2, …

The m = 0 lobe is exactly the generalized Snell's law beam from the ray-optics 3D version — same formula, derived here from wave interference instead of Fermat's principle. But a real phased array only samples the ideal continuous phase gradient at discrete points spaced by the pitch a; whenever a isn't much smaller than λ, that undersampling throws power into m ≠ 0 grating lobes spaced by λ/a in sinθ — spurious extra beams no idealized ray tracing shows, and a real design constraint for metasurface engineers (finer pitch → cleaner single-beam steering).

θᵢ, λ — same incident-beam parameters as the 3D model.
Δφ per element — phase jump between neighbouring meta-atoms.
Pitch a/λ — element spacing relative to wavelength; push it toward 1 to watch grating lobes appear in the far-field strip and as a second faint beam in the near field.
The measured peak angle is a live numerical argmax over the far-field array factor — compared against the m = 0 formula as a running self-check.

Near field — Huygens wavelet interferenceFar field — array factor |F(θ)|

Incident wave Metasurface elements (hue = phase) Ordinary refraction (reference) Grating lobe (m ≠ 0)