When Higher Resolution Reduces Precision: Quantum Limits of Off-Axis Interferometric Scattering Microscopy

TL;DR

Off-axis illumination in interferometric scattering microscopy redistributes information per detected photon, enabling higher localization precision under shot-noise limits. The authors develop a framework based on Fisher information ($\mathrm{FI}$) and quantum Fisher information ($\mathrm{QFI}$) to compute classical CRBs and quantum CRBs ($\mathrm{QCRB}$) for 3D localization, using boundary-element method simulations of the scattered and reflected fields. They find that oblique illumination can boost transverse localization by up to $2.8$× per photon (in the $x$-direction) and improve robustness to defocus, while rotating coherent scattering microscopy (rocs) delivers higher spatial resolution but worse localization precision due to incoherent averaging, requiring more photons for comparable accuracy. The results reveal that higher spatial resolution does not guarantee higher localization precision in coherent off-axis imaging and offer design guidance for next-generation coherent microscopes and related coherent-scattering imaging modalities.

Abstract

Coherent interferometric scattering microscopy (iscat) enables nanoparticle tracking on microsecond timescales and with nanometer precision, and has become a key tool in structural and cellular biophysics. The achievable localization precision in such experiments is fundamentally limited by photon shot noise. Here, we analyze three-dimensional localization precision under oblique illumination in iscat using the framework of (Quantum) Fisher Information. We show that tilting the illumination can enhance localization precision and accuracy per detected photon, while increasing robustness to defocusing. Surprisingly, rotating coherent scattering microscopy (rocs), which incoherently averages oblique illuminations, achieves higher spatial resolution but lower localization precision. Our results establish the quantum limits of off-axis interferometric imaging and reveal that resolution and precision can behave in opposite ways -- a key insight for designing next-generation coherent microscopes.

Figures (4)

• Figure 1: Fisher information (FI) flow (a.u.) in the $x$-$z$ plane for a gold nanosphere (golden circle) near a glass--water interface (orange line), together with the quantum Cramér--Rao bounds (QCRBs) for 3D localization precision under off-axis illumination. (a) FI flow under on-axis illumination, i.e. along the optical axis (gray dashed line). (b) FI flow for an illumination angle of $20^{\circ}$. (c) FI flow for an illumination angle of $40^{\circ}$. (d) QCRBs for localization precision along the $x$ (orange line), $y$ (grey line), and $z$ (purple line) directions. Table S1 in the supporting information reports the measurement-independent gain in localization precision predicted by the QCRBs for off-axis illumination in panel (d).
• Figure 2: Cramér--Rao bounds (CRBs) on the standard deviation of 3D localization precision for a gold nanosphere near a glass-water interface at $\theta = 0^{\circ}$ over a large defocus range and varying numerical aperture ($\mathrm{NA}$). Panels (a--c) show the CRB with respect to the $x$-, $y$-, and $z$-coordinates, respectively. The colorbar represents the value of the CRBs in $\mathrm{nm}$.
• Figure 3: Cramér--Rao bounds (CRBs) on the standard deviation of 3D localization precision for a gold nanosphere near a glass-water interface under off-axis illumination. In panels (a--c), results are shown for a fixed focus plane $z_{\mathrm{f}}=1\,\mathrm{\mu m}$ (dashed line), while the particle position $z_{\mathrm{p}}$ is varied from near the interface up to $3\, \mu\mathrm{m}$. In panels (d--f), the particle position $z_{\mathrm{p}}$ is fixed near the interface (dashed line), and $z_{\mathrm{f}}$ is varied over a large defocus range. The vertical dashed lines in panels (a--f) indicate the Brewster angle $\theta_{\mathrm{B}} \approx 42^{\circ}$. Furthermore, in panels (a--f) the global minimum is denoted by a cross, indicating the best possible bound. Each colorbar represents the CRBs, with values given in $\mathrm{nm}$.
• Figure 4: Interferometric scattering microscopy (iscat) and rotating coherent scattering microscopy (rocs) images for a gold nanosphere on top of a glass substrate. (a) We consider an incoming plane wave with TE polarization and the different angles reported in the insets. (b) iscat images for a focus position of $z_{\mathrm{f}}=500\,\mathrm{nm}$. (c) Corresponding rocs images, which we obtain by rotating the incoming wave around the $z$-axis and summing (incoherently) over the resulting iscat images. The Cramér--Rao bounds (CRBs) obtained from the iscat and rocs images are reported in the respective panels, where $\sigma_{x}$ corresponds to $\sigma^{\theta}_{{\mathrm{CRB}},x}$ with corresponding assignments for $y$ and $z$. (d) The CRBs as a function of defocus show that the localization precision in rocs is consistently worse than in oblique iscat. Since the CRBs for $x$ and $y$ in rocs are identical, they are represented by a single gray dashed line.