Fundamental Precision Bounds for Three-Dimensional Optical Localization Microscopy with Poisson Statistics

Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here, we prove the (sub)optimality of these approaches by deriving and comparing to the measurement-independent quantum Cramér-Rao bound (QCRB). We show that existing methods for depth localization with single-objective collection exceed the QCRB, and we gain insight into the bound by proposing an interferometer arrangement that approaches it. We also show that for light collection with two opposed objectives, an established interferometric technique globally reaches the QCRB in all three dimensions simultaneously, and so this represents an interesting case study from the point of view of quantum multiparameter estimation.

Figure 1

Single-objective collection schematics. (a) Standard microscope with microscope objective (MO), tube lens (TL) and camera (C). Insets: (i) intensity, (ii) example phase of $\psi$. (b) Engineered microscope with phase element. Two lenses form a $4f$ optical correlator [40], within which a phase retarder (e.g., an SLM) is placed. (c) Proposed interferometer for obtaining ${\sigma }_z^{(\mathrm{QCRB})}$. SLM compensates for defocus accrued downstream. Collected light (i) is split by an annular mirror (AM) with an inner radius $r_{\circ }=0.6326$. The “outer” portion (ii) is relayed to the beam splitter (BS) with two unit-magnification telescopes. The “inner” arm (iii) is demagnified with a telescope of magnification $M=0.22$ (iv), expanded into an annulus with axicon $A_{+}$ of phase $\phi (r_F)=680\times r_F$, passed through two relay lenses, recollimated with axicon $A_{-}$ of phase $\phi (r_F)=-680\times r_F$ (v), then relayed to the BS. Intensities illustrated in (i)–(v) have common color scale, except (iv), which has a $5\times$ scale to avoid saturation. Interferometric signals are detected on two cameras C1 and C2 placed at conjugate Fourier planes. Example images recorded on C1 and C2 for various $z$ are shown in (vi). Exact distances between optical elements and diffraction integrals that describe propagation through the apparatus are detailed in the Supplemental Material [41].

Figure 2

Photon-normalized precision bounds for single-objective collection. For $N$ detected photons, divide the vertical axis by $\sqrt{N}$. (a) Lateral localization bounds. The gray shaded region is bounded above by $\sigma ={\sigma }_x^{(\mathrm{QCRB})}={\sigma }_y^{(\mathrm{QCRB})}$. Blue curve: ${\sigma }_{x,y}^{\text{(CRB)}}$ for standard microscope. Red curves (solid is ${\sigma }_x^{(\mathrm{CRB})}$, dotted is ${\sigma }_y^{(\mathrm{CRB})}$): astigmatic microscope with strength specified in main text. Green curve: ${\sigma }_{x,y}^{\text{(CRB)}}$ of proposed interferometer. (b) Depth localization bounds. Color code corresponds to that in (a).

Figure 3

Dual-objective collection schematics. (a) Signals collected by microscope objectives $a$ and $b$ (${\mathrm{MO}}_a$, ${\mathrm{MO}}_b$) detected on two cameras without recombination. (b) Interferometric detection. Optimal lateral localization requires an additional reflection in one arm, enforced here by an $x$-oriented Dove prism (${\mathrm{DP}}_x$).

Figure 4

Photon-normalized precision bounds for dual-objective collection. For $N$ detected photons, divide the vertical axis by $\sqrt{N}$. (a) Lateral localization bounds. The gray shaded region is bounded above by $\sigma ={\sigma }_x^{(\mathrm{QCRB})}={\sigma }_y^{(\mathrm{QCRB})}$. Blue curve: ${\sigma }_{x,y}^{\text{(CRB)}}$ for non-interferometric detection. Gold line: interferometric detection. (b) Depth localization bounds. The color code corresponds to that in (a).