Fisher Information as an Operational Metric for Structured Optical Beams

TL;DR

The paper presents Fisher information as an operational metric to assess the metrological content of structured optical beams, arguing that entropic measures alone do not predict estimation performance. By treating the transverse intensity as p(x|θ) and using the displacement-estimation framework, it expresses Fisher information in forms like F_theta = ∫ dx (1/p) (∂p/∂θ)^2 and, for translational symmetry, F = ∫ dx (∂p/∂x)^2 / p, with ψ = sqrt(p) giving F = 4 ∫ dx (∂ψ/∂x)^2. An explicit proposition shows that nodal structure near zeros enhances sensitivity, and analyses of Hermite-Gaussian, Laguerre-Gaussian, and finite-energy Bessel-Gauss beams reveal systematic increases in Fisher information with increasing nodal complexity and local gradients. Across these families, Shannon entropy fails to predict estimation sensitivity, highlighting the value of Fisher information as a task-oriented metric with direct relevance to sensing design and potentially to quantum metrology via the quantum Fisher information concept.

Abstract

Structured optical beams possess rich spatial features that are commonly characterized using entropic measures of field complexity. However, such measures do not directly quantify the operational usefulness of optical structure for parameter estimation and sensing. Here we introduce Fisher information as an operational metric to assess the metrological content of structured optical fields. By treating the measured intensity distribution as a statistical object, we define Fisher information with respect to physically relevant parameters, such as transverse displacement. We demonstrate that optical modes with comparable Shannon entropy can exhibit markedly different Fisher information, revealing sensitivity features associated with nodal structure and local curvature. Using Hermite--Gaussian modes as minimal test cases, we show that increasing modal order systematically enhances Fisher information. We then extend the analysis to two widely used families in structured light: Laguerre--Gaussian vortex beams and finite-energy Bessel--Gauss beams. Across these representative families, Fisher information provides a unified and experimentally accessible criterion for comparing structured optical fields in sensing applications.

Figures (6)

• Figure 1: Normalized intensity profiles of HG$_{00}$ and HG$_{10}$ modes.
• Figure 2: Fisher information for displacement estimation.
• Figure 3: Scaling of normalized Fisher information with Hermite--Gaussian order (normalized to HG$_{00}$).
• Figure 4: Normalized Fisher information for displacement estimation for Laguerre--Gaussian modes versus topological charge $|\ell|$ (normalized to $\ell=0$).
• Figure 5: Normalized Fisher information for displacement estimation for finite-energy Bessel--Gauss beams versus the Bessel parameter $k$ (normalized to the smallest $k$ shown).