Quantifying classical and quantum bounds for resolving closely spaced, non-interacting, simultaneously emitting dipole sources in optical microscopy

TL;DR

This work addresses the problem of resolving two closely spaced dipole emitters in optical microscopy under high-NA collection, showing that the vectorial nature of dipole emission must be included. Using parameter-estimation theory, it derives the quantum Fisher information $\mathcal{K}(l;\Theta,\Phi)$ and the quantum CRB $\sigma^2_{\text{QCRB}}=1/\mathcal{K}$, and compares with classical FI-based CRBs for direct imaging. Focusing on two limiting orientation scenarios—fixed equal orientations $(\Theta,\Phi)$ and isotropic sampling—the study finds that polarization-filtered image inversion interferometry (III) can saturate the QCRB in special cases, and for general orientations requires filtering into the azimuthal component $\hat{\phi}$ (or a dual-polarization setup) to approach the quantum limit. These results guide practical passive super-resolution strategies in high-NA microscopy, clarifying when parity-based approaches suffice and when vectorial polarization filtering is essential.

Abstract

Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.

Figures (11)

• Figure 1: Overview schematic. We consider two non-interacting, mutually incoherent dipolar sources (orange arrows) of orientation $(\Theta,\Phi)$ symmetrically displaced from the optical axis and separated by a distance, $l$, to be estimated. The polar coordinates $(r,\phi)$ are defined at the back focal plane. Some of the measurements we consider require the placement of additional optical elements (e.g. beam splitters, polarizers, etc.) in the space between the back focal plane and the tube lens which forms the image(s) on the detector(s). Direct imaging is obtained without additional optics in this region. The optical axis is taken to be parallel to $\hat{z}$.
• Figure 2: Schematic of the image inversion interferometer (III). Light collected by the microscope objective is injected from the left in this picture. VHWP: vortex half-wave plate, PBS: polarizing beam splitter, BS: 50:50 beam splitter, M: mirror, C1: camera 1, C2: camera 2. For clarity we have suppressed the tube lenses placed at both outputs of the second beam splitter which form the images at the detector planes. The portion enclosed in the dashed black box (VHWP and PBS) are removed for the unpolarized III measurement. The combination of the VWHP and PBS effectively resolves the collected field into its radially and azimuthally polarized components. In this work we consider measurements in which either or both of these polarized components are injected into a separate III. The III pictured here corresponds to the azimuthally polarized fraction, while the III handling the radially polarized part lies beyond the borders of the picture. The axes labeled in the lower left refer to the Cartesian coordinates just before the first beam splitter, as transformed from the object plane.
• Figure 3: Computed CRBs vs. separation for the special cases $(\Theta=\pi/2,\Phi=0)$ (left) and $\Theta=0$ (right). The gray boxes are bounded above by the QCRBs. $N$ is the number of photons collected. Labels in the legends refer to direct imaging (blue solid), unpolarized III (red solid), $\hat{\phi}$-polarized III (yellow dashed), $\hat{r}$-polarized III (purple dashed), and the combination of separate $\hat{\phi}$- and $\hat{r}$-polarized III measurements (green dotted). The gold dashed line does not appear in the right plot because all of the light collected in this case is $\hat{r}$-polarized.
• Figure 4: High-resolution simulated images for $(\Theta=\pi/2,\Phi=0)$ and fixed $l\approx10$ nm. (a) Direct imaging. (b) Unpolarized III Outputs 1 (b.i) and 2 (b.ii). (c) $\hat{r}$-polarized III Outputs 1 (c.i) and 2 (c.ii). (d) $\hat{\phi}$-polarized III Outputs 1 (d.i) and 2 (d.ii). Scale bar: 250 nm.
• Figure 5: High-resolution simulated images for $\Theta=0$ and fixed $l\approx10$ nm. (a) Direct imaging. (b) Unpolarized III Outputs 1 (b.i) and 2 (b.ii). (c) $\hat{r}$-polarized III Outputs 1 (c.i) and 2 (c.ii). (d) $\hat{\phi}$-polarized III Outputs 1 (d.i) and 2 (d.ii). Scale bar: 250 nm.