Random matrix theory yields a closed-form noise-error bound for reconstructive spectrometers and predicts a 17.9 µm optimal cavity in on-chip FDTD

In the chaotic or diffusive (Ericson) regime with non-dominant noise, the noise-induced mean squared error of a reconstructive spectrometer is bounded below by σ²_ε Tr[G⁺] (G=AᵀA), and this quantity can be written in closed form in terms of Γ_corr, T₀, M, and N. Prior work lacked a systematic account of the physical determinants of reconstructive spectrometer performance, relying on the heuristic that a short Γ_corr is desirable; this work combines Fisher information with Toeplitz matrix theory to give the explicit analytic form E[Tr(G⁺)]=(M²N/|M−N|)·J(a)/T₀², with a=Γ_corr/Δω. The derivation rests on the Cramér–Rao bound and Szegő's theorem, and matches results computed directly from A matrices in the Mahaux–Weidenmüller random matrix model (500 internal modes) for both under-determined and over-determined regimes (Fig. 2c).

Γ_corr is not the sole determinant of performance; the mean transmittance T₀ also matters, and because both vary with coupling strength, the reconstruction error varies non-monotonically with the normalized correlation length a. This corrects the intuition that a shorter correlation length is always better, identifying an optimal design point set by competition between the spectral-correlation penalty and the throughput gain. Fig. 2(c) shows Tr[G⁺] varies non-monotonically with a; the Supplemental Materials give J(a)→1 as a→0 and the asymptote J(a)≈e^{πa}/(4πa²) for a≳1, so error grows exponentially in the highly correlated regime while increasing T₀ can partially offset that penalty.

When the effective signal-to-noise ratio R is sufficiently large, 'super-resolution' below the Γ_corr scale is achievable, with effective resolution Δω_min≈πΓ_corr/ln R² in the correlated-channel regime, multiplied by √2·M/N in the under-determined case. It reframes resolution from a hard physical limit into a parameter-estimation problem and gives the SNR condition R≈δ_th·√(|M−N|/N)·R₀ required for super-resolution. In random matrix simulations, the same scatterer fails to resolve comb lines spaced at Γ_corr=3Δω when R≈10 but resolves them at R≈3×10³ (Fig. S4a–c); Fig. S4(d) shows Δω_min decreasing monotonically with R² and crossing below Γ_corr.

Combining the theory with diffusive transport scaling laws (Γ_corr∝L⁻², T₀∝L⁻¹) predicts the optimal cavity size of an on-chip device without brute-force search. It turns an abstract variance bound into a quantitative device-sizing guideline and distinguishes two design targets: minimizing noise-induced MSE versus minimizing effective resolution at a given threshold. In 2D FDTD simulations with M=9, N=100, fitted diffusion constant D≈120 µm²/THz, and Δω=0.0439 THz, the analytic result gives L_opt≈17.9 µm, matching the observed minimum of Tr[G⁺] (Fig. 3d); Fig. 3(e)(f) show L_opt differs from the L=25 µm that minimizes Δω_min.

The results target chaotic or diffusive scattering in the Ericson regime (strongly overlapping resonances), with non-dominant noise and no strong priors, for devices that infer spectra from intensity measurements on a fixed set of output channels, including photonic chips, multimode fibers, colloids, and nanostructures. For designers, it places device size, channel counts, and noise level in one analytic framework, giving optimal cavity size and resolution expectations without exhaustive search; under shot-noise-limited detection the transmittance penalty weakens from T₀⁻² to T₀⁻¹, making thicker, more strongly scattering media relatively more favorable.

The theory assumes conventional speckle statistics with Lorentzian spectral correlations, which the authors note is not universal; the Supplemental inverse-design study shows that structures optimized via adjoint methods to minimize Tr[G⁺] develop strongly non-Lorentzian correlations and surpass the predicted bounds, with intensity statistics departing from Rayleigh (second-moment ratio ≈5.2 versus ≈1.7 for the initial structure). How to build theoretical tools for non-Lorentzian, non-fully-developed-speckle systems (e.g., extremal statistics of random matrix models) remains open; when priors such as spectral sparsity or training-data priors matter, Bayesian generalizations such as the van Trees inequality may be needed. The loaded text is an incomplete version, with some figures and Supplemental details referenced only, so specific numbers and derivation steps should be confirmed against the original and its Supplemental Materials.