Attaining the quantum limit of superresolution in imaging an object's length via predetection spatial-mode sorting

We consider estimating the length of an incoherently radiating quasimonochromatic extended object of length much smaller than the traditional diffraction limit, the Rayleigh length. This is the simplest abstraction of the problems of estimating the diameter of a star in astronomical imaging or the dimensions of a cellular feature in biological imaging. We find, as expected by the Rayleigh criterion, that the Fisher information (FI) of the object's length, per integrated photon, vanishes in the limit of small sub-Rayleigh length for an ideal image-plane direct-detection receiver. With an image-plane Hermite-Gaussian (HG) mode sorter followed by direct detection, we show that this normalized FI does not diminish with decreasing object length. The FI per photon of both detection strategies gradually decreases as the object length greatly exceeds the Rayleigh limit, due to the relative inefficiency of information provided by photons emanating from near the center of the object about its length. We evaluate the quantum Fisher information per unit integrated photon and find that the HG mode sorter exactly achieves this limit at all values of the object length. Further, a simple binary mode sorter maintains the advantage of the full mode sorter at highly sub-Rayleigh lengths. In addition to this FI analysis, we quantify improvement in terms of the actual mean-square error of the length estimate using predetection mode sorting. We consider the effect of imperfect mode sorting and show that the performance improvement over direct detection is robust over a range of sub-Rayleigh lengths.

Figure 1

Schematic of an incoherently emitting linear object, modeled as a collection of $M$ equal-intensity equally separated point sources, with $M\to \infty$, onto a detector array (top) versus a SPADE which manipulates the image-plane optical field before photon counting, i.e., detecting intensity in a different spatial mode basis (bottom).

Figure 2

Fisher information per photon for estimating the length $\theta$ of an extended 1D object. (a) Comparison between ideal continuum image-plane direct detection (blue solid line), infinite HG mode sorter (red dashed line), and the quantum limit (black dash-dotted line). (b) Fisher information per photon when the line object is approximated as $M$ equally spaced equal-intensity point sources along its length. As $M$ increases, the FI per photon decreases, since the photons, spread across the length of the object, are less efficient in carrying information about the object's length. Here $M=2$ corresponds to the two-point-source problem, where half of the photons are concentrated at one end of the object and the other half at the other end. In (a), $M=56$ was used to compute the red-dashed line, which explains why it is slightly above the quantum limit FI plot. With HG-sorter performance calculated with $M\to \infty$, the red dashed and black dash-dotted plots in (a) would coincide.

Figure 3

Fisher information per photon for estimating the angular length $\theta$ of the object when all the photon energy is assumed to be emitted from a single point at a fixed angular location $\varphi \in [-\theta /2,\theta /2]$ that is assumed to be a priori known.

Figure 4

(a) Individual FI contributions from the $q=0$, 1, 2, and 3 modes and the total FI from all HG modes (which equals the QFI) and (b) fraction of image-plane photons in the $q\mathrm{th}$ HG mode. Most of the information is contained in the $q=1$ (first antisymmetric) mode, but most of the photons are contained in the $q=0$ (PSF) mode, for small $\theta /\sigma$.

Figure 5

Fisher information attained by a binary SPADE receiver, based on separating either the $q=0$ (green) or the $q=1$ (red) HG mode. The FI attained by the infinite HG mode sorter (which equals the QFI) and that of continuum direct detection are reproduced. Finally, the green and red dashed curves show the performance of the $q=0$ and $q=1$ binary SPADE detectors but with the leakage parameter $\epsilon =0.01$. Photon leakage is seen to degrade the performance as $\theta$ approaches zero, but still outperforms ideal direct detection for $\epsilon =0.01$.

Figure 6

Plots of the $\theta$ at which the FI reaches the threshold of 6 dB below the ideal QFI at $\theta =0$. The burgundy curve show the direct-detection case ($\theta =1.4\sigma$), while the $q=0$ blue curve shows the degradation of this 6-dB point for the binary SPADE as the leakage $\epsilon$ is increased.

Figure 7

Comparison of RMSE (solid lines) with CR bounds (dashed lines): (a) RMSE and (b) normalized RMSE, where the RMSE in estimating $\theta$ is expressed as a fraction of the true value of $\theta$ itself. The lighter shade plots correspond to the image-plane direct measurement and the darker shade plots to a $q=0$ binary SPADE. The blue plots are for $N=50$ and the red plots $N=10000$.