Quantum Theory of Superresolution for Two Incoherent Optical Point Sources

Rayleigh’s criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern far-field superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measurements with linear optics and photon counting can estimate their separation from the far field almost as precisely as conventional methods do for isolated sources, rendering Rayleigh’s criterion irrelevant to the problem. Our results demonstrate that superresolution can be achieved not only for fluorophores but also for stars.

Popular Summary

In the 19th century, scientists discovered that the wave nature of light causes pointlike sources to appear blurry, where the size of the blurred spot sets the fundamental resolution of an instrument such as a telescope. In a seminal 1879 paper, Lord Rayleigh suggested that a pair of spots corresponding to stars had to be separated by at least the spot size in order to resolve the two stars. Researchers have since found “Rayleigh’s criterion” to be an excellent rule of thumb for both telescopes and microscopes. Here, we show that, contrary to conventional wisdom, Rayleigh’s criterion is not fundamental, and cleverer optical techniques can measure the separation between two light sources much more accurately than previously realized. Our calculations, based on quantum-information theory, reveal that the light actually carries much more information about the source positions than previously thought, and Rayleigh’s criterion is a problem only because conventional imaging techniques are unable to extract all of the information.

By considering two weak sources emitting in the optical regime, we derive the fundamental quantum limit on the precision of separating the sources. We find that Rayleigh’s criterion has no influence on this limit. To reach the limit, our new optical method, called spatial-mode demultiplexing (SPADE), splits the incoming light into specially designed channels. This process is extremely sensitive to the separation between the light sources, allowing the separation to be estimated as accurately as quantum mechanics allows and without regard to Rayleigh’s criterion.

We expect that our findings will pave the way for significant advances in both the study of binary stars and the imaging of single molecules by improving the imaging resolution by orders of magnitude.

Figure 1

(a) Two photonic wave functions on the image plane, each coming from a point source. $X_1$ and $X_2$ are the point-source positions, ${\theta }_1$ is the centroid, ${\theta }_2$ is the separation, and $\sigma$ is the width of the point-spread function. (b) If photon counting is performed on the image plane, the statistics are Poisson with a mean intensity proportional to $\mathrm{\Lambda }(x)=[|{\psi }_1(x){|}^2+|{\psi }_2(x){|}^2]/2$.

Figure 2

Plots of Fisher information versus the separation for a Gaussian point-spread function. ${\mathcal{K}}_{11}$ and ${\mathcal{K}}_{22}$ are the quantum values for the estimation of the centroid ${\theta }_1=(X_1+X_2)/2$ and the separation ${\theta }_2=X_2-X_1$, respectively, while ${\mathcal{J}}_{11}^{(\text{direct})}$ and ${\mathcal{J}}_{22}^{(\text{direct})}$ are the corresponding classical values for direct imaging. The horizontal axis is normalized with respect to the point-spread function width $\sigma$, while the vertical axis is normalized with respect to $N/(4{\sigma }^2)$, the value of ${\mathcal{K}}_{22}$.

Figure 3

The quantum Cramér-Rao bound ($1/{\mathcal{K}}_{22}$) and the classical bound for direct imaging ($1/{\mathcal{J}}_{22}^{(\text{direct})}$) on the error of separation estimation. The bounds are normalized with respect to the quantum value $4{\sigma }^2/N$. Rayleigh’s curse refers to the divergence of the classical bound when ${\theta }_2\lesssim \sigma$, as discovered by Refs. [8, 9, 10].

Figure 4

A multimode-waveguide SPADE with a grating output coupler and far-field photon counting. The photon counter at the end of the multimode waveguide captures any remaining photon in the higher-order or leaky modes.

Figure 5

An alternative design, with evanescent coupling to single-mode waveguides with different propagation constants for phase matching. The photon counter at the end of the multimode waveguide captures any remaining photon in the higher-order or leaky modes.

Figure 6

Binary SPADE with evanescent coupling to only one single-mode waveguide.

Figure 7

An alternative design of binary SPADE with a single-mode waveguide and leaky-mode detection.

Figure 8

Fisher information for separation estimation versus normalized separation ${\theta }_2/\sigma$ for a Gaussian point-spread function. ${\mathcal{J}}_{22}^{(\mathrm{HG})}$ is the information for the ideal Hermite-Gaussian-basis measurement, which is equal to the quantum value ${\mathcal{K}}_{22}$, ${\mathcal{J}}_{22}^{(\text{direct})}$ is for direct imaging, and ${\mathcal{J}}_{22}^{(\mathrm{b})}$ is for binary SPADE. The vertical axis is normalized with respect to ${\mathcal{J}}_{22}^{(\mathrm{HG})}={\mathcal{K}}_{22}=N/(4{\sigma }^2)$.

Figure 9

Fisher information for separation estimation versus normalized separation ${\theta }_2/W$ for the sinc point-spread function. ${\mathcal{K}}_{22}$ is the quantum value, ${\mathcal{J}}_{22}^{(\text{direct})}$ is the numerically computed value for direct imaging, and ${\mathcal{J}}_{22}^{(\mathrm{b})}$ is that for binary SPADE tailored for the sinc function. The vertical axis is normalized with respect to ${\mathcal{K}}_{22}={\pi }^{2}N/(3W^2)$.

Figure 10

A hybrid measurement scheme that splits the optical field by a beam splitter, measures one output port by a photon-counting array, and uses the centroid estimate ${\check{\theta }}_1$ to align SPADE at the other port.

Figure 11

Fisher information for separation estimation with SPADE with misalignment levels $\xi =0,0.1,\dots ,0.5$ (solid curves) and direct imaging (dash-dotted curve). The different solid curves can be distinguished by their decreasing values with larger misalignments.

Figure 12

Fisher information for separation estimation with binary SPADE with misalignment levels $\xi =0,0.1,\dots ,0.5$ (solid curves) and direct imaging (dash-dotted curve). The different solid curves can be distinguished by their decreasing values with larger misalignments.

Figure 13

Cramér-Rao bounds on the mean-square error of estimating $X_1$ or $X_2$ for a 50-50 hybrid scheme (solid line) and direct imaging (dash-dotted line). Note that the log-log scale is used here for clarity, unlike all the other plots in this paper. The vertical axis is normalized with respect to the error of locating an isolated source with direct imaging.

Figure 14

Simulated mean-square errors for SPADE with maximum-likelihood estimation, conditioned on $L$ detected photons. Note that the vertical axis is normalized with respect to the Cramér-Rao bounds $4{\sigma }^2/L$, so the plotted values are the actual errors magnified by $L/(4{\sigma }^2)$. Each error is computed by averaging $10^5$ simulations, and the lines connecting the data points are guides for eyes.

Figure 15

Simulated mean-square errors for binary SPADE with maximum-likelihood estimation, conditioned on $L$ detected photons. Note that the vertical axis is normalized with respect to $4{\sigma }^2/L$, so the plotted values are the actual errors magnified by $L/(4{\sigma }^2)$. Each error is computed by averaging $10^5$ simulations, and the lines connecting the data points are guides for eyes.

Figure 16

Simulated mean-square errors of binary SPADE with misalignment level $\xi =0.1$, conditioned on $L$ detected photons. The maximum-likelihood estimator that assumes no misalignment is used. Note that the vertical axis is normalized with respect to $4{\sigma }^2/L$. Each error is computed by averaging $10^5$ simulations, and the lines connecting the data points are guides for the eye. The errors are substantially below the Cramér-Rao bound for direct imaging (dash-dotted curve).