Optical Phased Array — Grating Lobes & Phase Quantization

Beam Steering Controls

Target steering angle θ 30°

Element pitch d 450 nm

Wavelength λ 532 nm

Phase control resolution

Far-Field Readout

Achieved main-lobe angle

0.0°

Steering error

0.0°

Strongest grating / quantization lobe

none

Pitch / λ

0.85

Grating-free limit

0.77

How it works

An optical phased array (OPA) — the on-chip cousin of a metasurface, used in solid-state LiDAR — steers light with no moving parts by giving each emitter its own controllable phase. For N emitters on a line with pitch d, emitter n (centered at x_n) is driven with phase

φ(x_n) = −k0 · x_n · sinθ_target,   k0 = 2π/λ

This linear phase gradient makes every emitter's wavefront arrive in phase along direction θ_target, so the array's far-field intensity — the array factor — peaks there:

AF(θ) = (1/N) |Σ_n exp( i[k0·x_n·sinθ + φ(x_n)] )|

Two real hardware limits show up directly in AF(θ):

Grating lobes. If the pitch is too large relative to λ, AF(θ) has more than one perfect peak. A lobe-free scan up to θ_target requires d/λ < 1 / (1 + |sinθ_target|) — widen the pitch slider past this limit and a second, unwanted beam appears.
Phase quantization. Real phase shifters (thermo-optic or electro-optic) only offer a finite number of discrete phase levels. Rounding φ(x_n) to N_bits levels adds a periodic phase-error ripple across the array, which scatters power into a low-level "quantization lobe" and slightly shifts the true peak away from θ_target — the steering error readout above.

Drag θ target to pick a scan angle, then push pitch past the grating-free limit or drop the phase resolution to 1–2 bits to watch the far-field pattern below the array grow a second lobe.

Emitter phase (hue = φ) Main lobe direction Grating / quantization lobe

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