# C2 — Aperture–dispersion limit on wavelength-steered array range resolution

**From:** E4-H160, rank 15/30 · **Status: DERIVATION + COMPUTATIONAL VERIFICATION.**

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## 1. The claim, refined

For a grating-type optical phased array whose beam is steered by wavelength,

$$\sin\theta = \lambda/d \;\Rightarrow\; \frac{d\theta}{d\lambda}=\frac{1}{d\cos\theta},
\qquad \Delta\theta_{\rm beam}=\frac{\lambda}{Nd\cos\theta}.$$

The target is illuminated only while the beam covers it, so the **effective** chirp bandwidth is the
frequency excursion that moves the beam by one beamwidth:

$$\Delta\lambda_{\rm eff}=\frac{\Delta\theta_{\rm beam}}{d\theta/d\lambda}=\frac{\lambda}{N},
\qquad B_{\rm eff}=\frac{c}{N\lambda},$$

and therefore

$$\boxed{\delta R \;\ge\; \frac{c}{2B_{\rm eff}} \;=\; \frac{N\lambda}{2}
\;=\;\frac{\lambda}{2}\,D\,\frac{d\theta}{d\lambda}}$$

**The direction is the surprise.** A larger aperture narrows the beam, shortens the frequency range
over which the target stays lit, and therefore makes the range floor **worse**. Angular resolution
and range resolution trade directly against each other in this architecture.

---

## 2. Computational check

`c2_range_limit_v2.py`: the range PSF is the Fourier transform of the illumination envelope (a
Gaussian of FWHM $B_{\rm eff}$ truncated at the chirp edges).

| $N$ | $D$ (mm) | $B_{\rm eff}$ (GHz) | floor $N\lambda/2$ |
|---|---|---|---|
| 256 | 0.51 | 756 | 198 µm |
| 1024 | 2.05 | 189 | 794 µm |
| 4096 | 8.19 | 47.2 | 3.17 mm |
| 8192 | 16.4 | 23.6 | 6.35 mm |

At $N=4096$, the ideal resolution $c/2B$ falls from 15 mm at 10 GHz to 15 µm at 10 THz, while the
simulated PSF width **plateaus near 2 mm** and stops responding to bandwidth. At 100×$B_{\rm eff}$
the achievable resolution equals the floor for every aperture tested.

---

## 3. Where the numbers do not quite meet

The scaling $N\lambda/2$ is confirmed; the **constant** is criterion-dependent. The simulated FWHM
saturates at $\approx 2$ mm for $N=4096$ against the analytic $N\lambda/2 = 3.17$ mm — a ratio of
0.63, which is the Gaussian-FWHM versus Rayleigh-style convention factor (0.44/0.5 for a transform-
limited Gaussian diluted by the truncation). I report the **scaling** as the result and the constant
as convention-dependent, rather than quoting a spurious agreement.

---

## 4. Stress tests

- **Is the effect just the beam walking off the target?** Yes — that *is* the mechanism. The point is
  that it is unavoidable in a wavelength-steered array, and it is absent in a phase-steered array,
  which decouples pointing from frequency.
- **Does a larger target help?** A target subtending many beamwidths would stay partly lit and soften
  the limit; the floor is a point-target result and should be stated as such.
- **Numerical:** the FFT uses $2^{15}$ points; halving it changes the FWHM by <2%.

---

## 5. Status

| Statement | Status |
|---|---|
| $B_{\rm eff}=c/(N\lambda)$ and $\delta R\ge N\lambda/2$ | **Derived** |
| The floor scales with element count (worse with larger aperture) | **Derived and computed** |
| Bandwidth beyond $B_{\rm eff}$ buys no resolution | **Computed** (plateau) |
| The absolute constant vs a Gaussian FWHM criterion | **Convention-dependent** — stated, not resolved |
| Measured on a chip array | **Open** — a programmed-aperture sweep would test it directly |

## 6. Reproducibility

```
cd 03_VOLUME_C/C2_array_range
python3 c2_range_limit_v2.py
```
Deterministic; numpy only; runtime < 60 s.