# E4-H160 — Range resolution in wavelength-steered optical arrays

**Project:** Incoprea Research, independent investigation
**Date:** 2026-11-01
**Status:** COMPLETE — hypothesis **corrected and generalized**. The claimed floor is real for a
point target but is *not* a universal physical bound.

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## Research question

Does wavelength-based beam steering impose a fundamental limit on range resolution — and if so,
on what does it depend?

Original claim (Expedition IV, E4-H160): the chirp sweeps the beam, so a point target is illuminated
only while the beam covers it; the effective bandwidth saturates at `B_eff = c/(λD|dθ/dλ|)` and range
resolution floors at `δR = λD|dθ/dλ|/2`, independent of chirp bandwidth.

## Principal findings

1. **The scaling is recovered.** Independent derivation gives `B_eff = c/(Nλ)` at normal incidence and
   `δR = c/(2B_eff)`, growing linearly with aperture at fixed pitch. Confirmed numerically (T1):
   `B_eff = 214 GHz` and `δR = 0.70 mm` for N = 2048, λ₀ = 684 nm; the ratio `δR/D = 0.171` is
   constant across a 32× aperture range.

2. **The limit is an information limit, not a physical one — and it is target-dependent.** The
   Cramér–Rao bound on delay (T2) saturates for a point target at `B > B_eff` but keeps falling as
   `1/B` for an extended target. At `B = 9.35 B_eff` the two differ by a factor **24.8**.

3. **Beyond `B_eff` the point-target bound slightly worsens** (0.0053 → 0.0107 ps from B_eff to
   9.35 B_eff): extra bandwidth adds noise samples where the illumination is weak. This refinement is
   absent from the original claim, which says only that extra bandwidth is useless.

4. **A corrected general statement**: the bound is set by the RMS bandwidth of the
   *illumination-weighted* spectrum, `J(τ) = (2π)² Σ f²|A(f)|²/σ²`, not by the beam width alone.
   The original formula is the special case of a point target with an on-axis Gaussian envelope.

## What is NOT claimed

- Not claimed: that the trade-off itself is novel. A conference paper on swept-source FMCW LiDAR with
  dispersive OPA beam steering was **identified by search but not retrieved** (see `PRIOR_ART.md`).
  It plausibly contains the trade-off.
- Not claimed: experimental confirmation. Nothing here is measured.
- Not claimed: that the estimate attained the bound. The Monte-Carlo check returned RMSE/CRB ≈ 0.75,
  **below** a bound that cannot be beaten — a normalization discrepancy I did not resolve (`RESULTS.md` T3).

## Files

`ORIGINAL_CLAIM.md` · `PRIOR_ART.md` · `PREREGISTRATION.md` · `METHODS.md` · `RESULTS.md` ·
`LIMITATIONS.md` · `DATA_PROVENANCE.md` · `REPRODUCE.md` · `INTEGRITY.md` · `code/` · `results/`

## Next decisive test

A bench measurement with a dispersive OPA and a target at known range, where the same data are
processed twice: (a) matched filter with the measured envelope, (b) unweighted. The prediction is that
(a) and (b) agree below `B_eff` and that (a) beats (b) above it — the point the simulation supports but
could not test empirically.