N identical antenna elements sit in a line, spaced d apart. Feeding element n with an extra phase nΔφ tilts the combined ("array factor") beam without moving anything mechanically — this is the core idea of a phased array (radar, 5G massive-MIMO, Wi-Fi beamforming):
AF(θ) = Σ w[n]·exp( j·n·2π·(d/λ)·(sinθ − sinθ₀) ), n = 0..N−1
Δφ = −360°·(d/λ)·sinθ₀ (phase step that steers the beam to θ₀)
The top panel renders the real interference wavefield — each element radiates its own circular wave with that phase offset, and the field you see is their sum, exactly like ripples from N synchronized point sources. The bottom polar plot is the exact analytic |AF(θ)|, normalized to its peak.
- Spacing > λ/2 — extra full-cycle solutions of the array-factor equation appear at other angles: grating lobes, as strong as the main beam.
- Amplitude taper — tapering the element weights (here a raised-cosine/Hann taper vs. uniform) trades a wider main beam for lower sidelobes, the classic radar-design tradeoff.
- Directivity uses the standard broadside approximation D₀ ≈ 2(d/λ)·(Σw)²/Σw², valid near broadside for d ≳ λ/2.