 Angular resolution set by transverse spectral noise

Angular resolution set by transverse spectral noise quantifies how sharply an observed spectral vector constrains the orientation of an anisotropic two-coordinate noise model. Use the covariance from transverse-noise limit of a rotating rank-one covariance model, with known longitudinal standard deviation $\sigma$ and transverse standard deviation $0<ϵ<\sigma$. For an observation of length $R>0$ aligned with orientation ${\theta }_0$, the Gaussian density givesThe determinant is independent of orientation, so the angular dependence comes entirely from the covariance-weighted residual.^[ref]

Define a local half-width by a chosen log-density drop $c>0$. For $2c\le K$, the lobe around ${\theta }_0$ has half-widthWhen this angle is small and $ϵ\ll \sigma$, it is approximately $\sqrt{2c}ϵ/R$. If $2c>K$, even an orthogonal orientation fails to reach that drop, so this local-width criterion does not exclude any orientation. At $R=0$ or equal noise scales, the observation has no angular preference in this model.

For an original example, choose $R=2$, $\sigma =0.8$ and $ϵ=0.04$ in consistent signal units. Then $K=2493.75$. A drop $c=1/2$ gives half-width about $0.0200$ radians, or $1.15^{\circ }$. Reducing only the transverse standard deviation to $0.02$ approximately halves the width. Increasing a numerical regularization parameter is not an experimental improvement in this noise scale.

The global orientation ambiguity in a rank-one covariance fit remains: orientations separated by $\pi$ give the same covariance. A narrow local lobe does not distinguish those two orientations. Nor is a fixed log-density drop automatically a confidence interval with a stated coverage probability. Multiple observations, unknown noise scales and experimental nuisance parameters require their corresponding inference model.

Repeated reference spectra under stable conditions can estimate the longitudinal and transverse noise components. Their covariance must include shared calibration effects, as required by the measurement-propagation rule.^[ref] The numerical example is a model calculation in spectral coordinates, rather than a specification of a particular optical instrument.