Measure information in an interference fringe

For a probability law $p(x\mid\phi)$, the score is the derivative of its logarithm with respect to phase. Under regularity conditions, including permission to differentiate the normalization, the score has mean zero. Its second moment is Fisher information. For one Bernoulli outcome the two scores are $p'/p$ and $-p'/(1-p)$, weighted respectively by probabilities $p$ and $1-p$. Adding their squared contributions gives a compact expression that includes both response and binary noise.

The formula is local: it compares distributions for nearby parameter values. It does not certify that two distant phases are distinguishable. Its elementary Bernoulli form is regular when $0<p<1$ and the parameterization is smooth. At deterministic endpoints, divisions by zero require separate limiting and support analysis. In particular, an apparent cancellation of zero factors is not permission to apply every regular unbiased-estimation theorem at a boundary. Specify an interior operating region before interpreting an inverse-information expression as a precision bound.

Consider a calibrated sine fringe $p=(1+V\sin\phi)/2$, where visibility $V$ is a known dimensionless number between zero and one. Differentiating produces a slope $V\cos\phi/2$, while the outcome variance is $(1-V^2\sin^2\phi)/4$. Their ratio yields the information. At the midpoint $\phi=0$, the information is $V^2$: reducing contrast to one half requires four times as many independent outcomes to restore the same local information. This statement assumes that contrast itself is known rather than fitted as a nuisance parameter.

For imperfect visibility, the turning point of the fringe has zero first derivative and zero local Fisher information. Nearby phases there change the mean only to second order. Moving the analysis-pulse phase can shift the operating point back to the midpoint. For unit visibility, the interior phase information is constant, but deterministic fringe extrema remain singular points for the ordinary regular argument. These distinctions explain why maximum signal amplitude and maximum useful local sensitivity need not occur at the same setting.

Multiplying by the repetition count follows because independent scores add and their cross terms have zero expectation. Correlated measurements or a drifting phase do not obey that simple calculation without a joint probability model. The information is an expected property of an experiment, not the observed count in one run. A rare count can produce a surprising estimate while leaving the precomputed Fisher information unchanged. Observed likelihood curvature is another quantity and need not equal expected information for a small dataset.

The Cramér–Rao inequality bounds variance for estimators satisfying its unbiasedness and regularity requirements. Reaching it is a separate question involving estimator choice, available data, and local identifiability. A finite experiment can have bias or multimodal likelihoods even when the inverse information is small. Report the bound as a design benchmark, then check the actual inference procedure. Also retain parameter units: replacing angular frequency by ordinary frequency introduces a factor of two pi in the derivative and its square in information.