Fisher information vs. signal-to-noise ratio for a split detector

TL;DR

This paper compares Fisher information and signal-to-noise ratio in a split detector for Gaussian beam displacement estimation, revealing that optimal tuning can significantly improve measurement precision and impact weak-value amplification strategies.

Contribution

It demonstrates the discrepancy between Fisher information and SNR in split detectors and identifies optimal detector tuning for enhanced measurement accuracy.

Findings

Optimal normalized difference tuning at 0.884753 maximizes Fisher information.

Fisher information can be improved by at least 2.5 times over linear operation.

Adaptive realignment mitigates the information penalty from high SNR before detection.

Abstract

We study the problem of estimating the magnitude of a Gaussian beam displacement using a two pixel or 'split' detector. We calculate the maximum likelihood estimator, and compute its asymptotic mean-squared-error via the Fisher information. Although the signal-to-noise ratio is known to be simply related to the Fisher information under idealised detection, we find the two measures of precision differ markedly for a split detector. We show that a greater signal-to-noise ratio 'before' the detector leads to a greater information penalty, unless adaptive realignment is used. We find that with an initially balanced split detector, tuning the normalised difference in counts to 0.884753... gives the highest posterior Fisher information, and that this provides an improvement by at least a factor of about 2.5 over operating in the usual linear regime. We discuss the implications for weak-value amplification, a popular probabilistic signal amplification technique.