Fisher information versus signal-to-noise ratio for a split detector

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 idealized 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 normalized difference in counts to $0.884753...$ gives the highest posterior Fisher information, which 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.

Figure 1

(a) Prior quantities: the signal-to-noise-ratio $R_x$ (dashed, blue) and the scaled Fisher information $g^2{F}_x$ (solid, green) as a function of prior signal-to-noise $R_x$, assuming an ideal detector (note their monotonic relation to one another); (b) Information penalty or “transmittance” ${\eta }_F$ of Fisher information (solid, green) or ${\eta }_R$ of signal-to-noise (dashed, blue) at the split detector when $x_0=0$; (c) Posterior quantities: the signal-to-noise ratio $R_n$ (dashed, blue) and scaled Fisher information $g^2{F}_n$ (solid, green) relating to a well aligned $\left(x_0=0\right)$ split detector. The measures are no longer monotonically related. All quantities are dimensionless.

Figure 2

Dependence of two estimators on the observed normalized difference in counts at a well-aligned $\left(x_0=0\right)$ split detector. The blue, dashed line is proportional to the simple linear estimator, which can become significantly biased when the difference in counts is outside of the linear regime (shaded area). It is a first-order approximant to the solid, orange curve, which is proportional to the maximum likelihood estimator. The latter is always unbiased and has unbeatable performance (mean-squared error) when the number of trials is large. The optimum normalized difference in counts 0.884753... balances the prior Fisher information $F_x$ with the information penalty ${\eta }_F$ to achieve the optimum posterior Fisher information $F_n$. This can be achieved by modulating $\lambda$ or $\sigma$.