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( Quantum) FISHER INFORMATION
to what extent one can realize this potential in existing and future nearfield microscopy schemes.
As the light propagates onward and no longer interacts with the particle, its information content about location saturates to the QFI predicted by far-field scattering theory. It is conserved upon free propagation, and its flux density can be calculated from the derivatives of the electric and magnetic fields with respect to the parameters of interest [ 4 ], in close analogy to the Poynting vector describing the energy flux density. For coherent probe light, the QFI is given by a simple expression [ 10 ] and results in QCRBs proportional to λ 2 / N s( or in terms of the minimal standard deviation, σ ∙ λ / √— N s), reflecting the base resolution given by the photon wavelength λ and the detection shot noise over N s independently scattered photons. Crucially, the elastic dipole scattering process is coherent, and the QFI lies not only in the scattered field, but also in its phase with respect to the excitation light. Such considerations are essential when designing a measurement that maximizes FI. For a given measurement, the FI can be calculated from the detected intensities and their derivatives with respect to the parameter of interest. In the absence of technical noise, it is limited by detection shot noise [ 2 ].
These concepts have been applied to various imaging scenarios. In iSCAT, the light scattered by a nanometric object is imaged onto a camera, where it interferes with the light reflected from the microscopy coverslip( see Fig. 1C). The resulting image holds information about the 3D position and the polarizability( mass) of e. g., a protein or nanoparticle [ 5 ]. The QCRBs turn out to be σ x / λ≈0.13 / √— N d in lateral, and
σ z / λ≈0.04 / √— N d in axial direction, with N d < N s detected scattered photons. The excellent axial precision is due to the dependence of the resulting interference pattern on the path length difference between the scattered and reflected light. This advantage is lost in techniques without reflected reference, as the axial QCRB from the scattered light alone is σ z / λ = 0.19 / √— N d. The same formalism also gives an intriguingly simple precision bound for estimating the mass of a scattering protein: σ m / m = 1 / 2√— N d.
In fluorescence microscopy, localization is based on inelastically scattered light. While typical microscopes yield an anisotropic localization precision, FI can guide engineering the wavefronts of the detected light to balance this anisotropy [ 6 ]. It also guides the design of complex imaging schemes that saturate the 2π collection QCRB [ 7 ], which is obtained when collecting light with a single high numerical aperture objective( σ x, y / λ≈0.11 / √— N d, σ z / λ≈0.29 / √— N d). Notably, interferometric detection of light collected with two objectives reaches the 4π collection QCRB( σ x, y / λ≈0.11 / √— N d, σ z / λ≈0.07 / √— N d). The increase in axial precision is now due to the relative phase between the fields collected in the forward and backward direction.
All schemes discussed so far assumed plane-wave illumination. Shaping the excitation beam profile can further boost localization precision, and the optimal shape can be calculated if the scattering matrix of the specimen and its
Figure 1. a) Elastic scattering generates electric and magnetic fields that carry information about the position and polarizability of the scatterer( adapted from [ 3 ]). b) While a light pulse interacts with a scatterer, the QFI is enhanced and oscillates before it ultimately approaches a constant value once the pulse has propagated to the far field( adapted from [ 3 ]). c) In an interferometric scattering microscope, the scattered fields interfere with fields reflected at the coverglass, leading to exceptionally high axial localization precision( adapted from [ 5 ]). d) In MINFLUX, one iteratively adapts the excitation beam shape in order to boost localization precision, which can surpass the QCRB for non-iterative, non-shaped excitation( taken from [ 9 ]).
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