State-vector geometry and guided-wave physics behind optical super-resolution.

We examine the state-vector geometry and guided-wave physics underpinning spatial super-resolution, which can be attained in far-field linear microscopy via a combination of statistical analysis, quantum optics, and spatial mode demultiplexing. A suitably tailored guided-wave signal pickup is shown to provide an information channel that can distill the super-resolving spatial modes, thus enabling an estimation of sub-Rayleigh space intervals ξ. We derive closed-form analytical expressions describing the distribution of the ξ-estimation Fisher information over waveguide modes, showing that this information remains nonvanishing as ξ → 0, thus preventing the variance of ξ estimation from diverging at ξ → 0. We demonstrate that the transverse refractive index profile n(r) tailored to support the optimal wave function ψ(r) for super-resolving ξ estimation encodes the same information about ξ as the entire manifold of waveguide modes needed to represent ψ(r). Unlike ψ(r), n(r) does not need a representation in a lengthy manifold of eigenmodes and can be found instead via adaptive feedback-controlled learning.