Multiparameter estimation for the superresolution of two incoherent sources

TL;DR

The paper addresses simultaneous estimation of three scene parameters—separation $(d)$, centroid $(c)$, and brightness imbalance $(p)$—for two incoherent optical sources in the sub-Rayleigh regime. It introduces a dual-SPADE measurement using two MPLCs, one shifted, to lift degeneracies and maximize information across a broad parameter range, and it derives estimators from calibrated mode responses. Experiments with distinguishable sources show 2–4 orders of magnitude improvement below the diffraction limit and bias reductions when using the dual-MPLC, with estimated errors approaching the CRBs. In the indistinguishable-source scenario, simulated data confirm persistent, though degraded, sensitivity, and the dual-MPLC configuration still substantially reduces estimator bias. Overall, the work demonstrates robust, simultaneous multiparameter superresolution imaging via SPADE, scalable to more sources and potentially extendable to lower photon regimes.

Abstract

We experimentally demonstrate the simultaneous estimation of the three parameters characterizing a pair of incoherent optical sources in the sub-Rayleigh regime, enabling super-resolved scene characterization. Using spatial-mode demultiplexing (SPADE) with two demultiplexers--one deliberately shifted--we determine separations well below the diffraction limit and achieve sensitive joint estimation of separation, centroid, and relative brightness over a broad range of scene configurations in a single experimental setting. We benchmark our performance using Fisher-information-based Cramér-Rao bounds, and discuss the corresponding quantum limits. We investigate two complementary scenarios: a realistic case with slightly non-identical sources, and an idealized case of indistinguishable sources.

Figures (4)

• Figure 1: Cramér--Rao bounds associated with $d$, for one and two MPLC configuration associated with (a) distinguishable and (b) indistinguishable sources. In solid lines we show the average precision on $d$. The corresponding shaded area is the region containing $90\%$ of the computed values. We use an ensemble of scenes defined by $d\in [0.01,0.3]w_0$, $c\in [-0.15,0.15]w_0$ and $p= 0.5$, with $N=10^{11}$ signal photons. For indistinguishable sources we also show the quantum CRB, which exhibits nearly constant value of $\approx 2.3\times10^{-6}$.
• Figure 2: (a) The 1D profile of two Gaussian beams with the same waist $w_0$, separated by a distance $d$ and centered at $c$ (with respect to the first MPLC origin). The brightness imbalance is characterized by the parameter $p$, so that the triplet $\boldsymbol{\theta} = (d, c, p)^\top$ fully defines the scene. (b) Conceptual scheme of the setup. The left part corresponds to source generation, where motorized translation stages (MTS) are used to tune the positions of the two sources and change the scene. The brightness imbalance is controlled by adjusting the power in each source path via attenuators. The light is propagated through a telescope for mode-matching and then detected by the two demultiplexers, with one deliberately transversely shifted relative to the other (right part).
• Figure 3: Multiparameter estimation results on experimental data. Photon flux of $N = 10^{11}$ over 0.1ms. (a) Parameter-wise bias and sensitivity for $d$, $c$, and $p$. For each scene, the bias and sensitivity are computed from the ensemble of time-bin estimates, then averaged over all scenes. Results are grouped by brightness imbalance, and the table also specifies whether the dual-MPLC configuration was used. (b) Detailed results for the case $p = 0.3$. (b.1) Statistical errors $\sigma_d$ and (b.2) $\sigma_c$, obtained for each scene from the variability of the time-bin estimates of the separation $d$ and centroid $c$. Errors are shown as functions of the reference parameters $d_{\mathrm{ref}}$ and $c_{\mathrm{ref}}$ (see colors legend). Solid lines represent the Cramér–-Rao bounds computed from the experimental noise. (b.1.*) Mean estimate of separation $d_{\mathrm{est}}$ versus reference distances $d_{\mathrm{ref}}$. The red line denotes unbiased estimation.
• Figure 4: Estimation results on simulated data for the identical-source case. Photon flux of $N = 10^{11}$ over 0.1ms. (a) Same layout as in Fig. \ref{['fig:exp_results']}(a). (b) Specific case of $p_\mathrm{ref}=0.3$. Left column: Mean parameter estimates (averaged over all time bins) as functions of the reference parameters $d_{\mathrm{ref}}$ and $c_{\mathrm{ref}}$ (see legend). Dashed lines indicate unbiased estimations. Right column: Associated statistical errors, also shown as functions of the reference parameters $d_{\mathrm{ref}}$ and $c_{\mathrm{ref}}$.