Optimal Observables and Estimators for Practical Superresolution Imaging

Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramér-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, cross talk, and detector noise.

Figure 1

Schematic representation of the estimation procedure. In the data acquisition block, the image of the two sources enters into an (eventually misaligned) demultiplexing device that performs a mode decomposition affected by cross talk. The intensity of each mode is then measured with noisy detectors. Parameter estimation is performed (in postprocessing) linearly combining the measured intensities with optimal coefficients, and comparing the result with a calibration curve.

Figure 2

Measurement sensitivity $M[d,\theta ,\widehat{\mathbf{N}}]$ for intensity measurements into HG modes $u_{nm}(\mathbf{r})$ with $n,m\le Q=1$, 2 [$K=(Q+1{)}^2$] including (a) misalignment ($d_s/2w=0.01$, ${\theta }_s=\pi /4$), (b) cross talk ($⟨\overline{|c_{ij}{|}^2}⟩=0.0017$), (c) dark counts (${\sigma }_k=0.001\forall k$), and (d) all three imperfections combined. In panels (b) and (d) solid lines and bands represent the mean and 1 standard deviation computed over 500 cross-talk matrices. Dashed lines show the results for ideal measurements, while the green solid line describes ideal direct imaging results [39]. For all plots, we assumed $N\kappa =1.5$ and $\theta =\pi /4$.

Figure 3

Smallest source separation $d$ at which ideal direct imaging outperforms demultiplexing [into HG modes $u_{nm}(\mathbf{r})$ with $n$, $m\le 2$] as a function of the (a) misalignment, (b) cross talk, and (c) dark count strengths for a fixed finite value of the other two imperfections (same as in Fig. 2), and different brightnesses $N\kappa =1.5$ (blue), 5 (red), 10 (green).

Figure 4

Optimal coefficients $m_{ij}$ for measurements in the HG basis $u_{ij}(\mathbf{r})$ with $i$, $j\le 1$. The modes’ intensity distributions are plotted below the corresponding coefficients. Different noise sources are considered: (blue) none, (red) misalignment ($d_s/2w=0.01$, ${\theta }_s=\pi /4$), (green) cross talk ($⟨\overline{|c_{ij}{|}^2}⟩=0.0017$), and (orange) dark counts (${\sigma }_k=0.001\forall k$). Green bars for cross talk are averaged over 500 cross-talk matrices. All plots correspond to $N\kappa =1.5$ and $\theta =\pi /4$.