Super-resolution microscopy via fluctuation-enhanced spatial mode demultiplexing

TL;DR

The paper tackles diffraction-limited imaging by merging SPADE with stochastic emitter fluctuations, introducing SOFSPADE and SOFIII. Temporal cumulants $\kappa^{(r)}$ enable access to higher spatial moments and allow a simplified III-based path to retrieve the full information content of SPADE. An estimator framework maps temporal moments to spatial moments with a linear ML solution, and a Cramér–Rao analysis provides fundamental limits that are shown to be tight in practice. Simulations demonstrate substantial gains in resolving higher-order moments, with practical implications for confocal and 2D imaging where blinking fluorophores can be leveraged to achieve subdiffraction resolution with reduced experimental complexity.

Abstract

We introduce a superresolution technique that combines spatial mode demultiplexing (SPADE) with emitter blinking. We show that temporal fluctuations not only enhance the precision of SPADE imaging, but also drastically simplify the measurement required to recover full object information -- in the presence of fluctuations, SPADE can be replaced by the much simpler image inversion interferometry. Both gains are enabled by exploiting temporal cumulants of the detected signal.

Figures (4)

• Figure 1: In SPADE technique, image plane field is decomposed into $k$ H-G modes, which allows to estimate lowest $k$ object's even spatial moments. To estimate odd spatial moments, one needs to interfere neighbouring H-G modes, which leads to iSPADE technique.
• Figure 2: Photon emission from quantum dots or dyes used to label an object fluctuates in time---typically, an emitter switches between two states characterized by brightnesses $q_\textrm{off}$ and $q_\textrm{on}$ (a). The resulting image also varies over time (b). In SPADE, the image plane field is sorted into H-G modes (c). Fluctuations of photon counts $n_j$ in different modes provide additional information about object's higher ($\mu \ge 4$) spatial moments through suitable intensity cumulants---this leads to a proposed SOFSPADE technique. In III, the image-plane field is interfered with its spatial inversion (d), yielding a much simpler setup than SPADE, but providing access only to two outcomes corresponding to $0$th and $2$nd spatial moments. However, temporal fluctuations allow all even moments to be recovered from temporal cumulants of the odd-mode intensity $I_-$, which forms the basis of the proposed SOFIII technique.
• Figure 3: Simulations are performed for a subdiffraction object ($\Delta = 0.3$) composed of $k=20$ emitters distributed as in (a). Panel (b) shows the relative errors of even spatial-moment estimates as functions of the number of frames (markers: simulated MSEs; solid lines: C-R bounds) The area corresponding to accuracy 10% or better is shaded in each chart to facilitate comparison. Higher-order moments are harder to estimate, and the benefit of fluctuation-based methods (SOFSPADE, SOFIII) increases with moment order.
• Figure 4: Odd and even moments of an object shown in Fig. \ref{['fig:SPADE_plots']}a are estimated using mean iSPADE method or bin SOFiSPADE, where measurement is simplified, and temporal fluctuations are used. The simulated MSEs and corresponding C-R bounds of both methods are plotted as functions of the number of frames. For even moments, the estimator based on mean iSPADE only becomes efficient for large number of frames, but bin SOFiSPADE method is free of this issue.