Beating Rayleigh’s Curse by Imaging Using Phase Information

Every imaging system has a resolution limit, typically defined by Rayleigh’s criterion. Given a fixed number of photons, the amount of information one can gain from an image about the separation between two sources falls to zero as the separation drops below this limit, an effect dubbed “Rayleigh’s curse.” Recently, in a quantum-information–inspired proposal, Tsang and co-workers found that there is, in principle, infinitely more information present in the full electromagnetic field in the image plane than in the intensity alone, and suggested methods for extracting this information and beating the Rayleigh limit. In this Letter, we experimentally demonstrate a simple scheme that captures most of this information, and show that it has a greatly improved ability to estimate the distance between a pair of closely separated sources, achieving near-quantum-limited performance and immunity to Rayleigh’s curse.

Figure 1

Theory plot of the Fisher information for SPLICE, binary SPADE and IPC vs beam separation $\delta$, normalized to units of $N/4{\sigma }^2$ and $\sigma$, respectively.

Figure 2

Cartoon of experiment. Shown is the experimental apparatus. In the lower right-hand box is a representation of SPLICE, the measurement scheme tested in this experiment. In the upper right-hand box is a sketch of the spatial profile of the electromagnetic field before the measurement. The rest of the figure depicts the device used to simulate the two light sources, which can be displaced around their centroid by the displacement of the top mirror.

Figure 3

Inferred separation vs known actual separation for (a) SPLICE (from lookup on calibration curve) and (b) IPC. Note that the spread in the IPC estimates grows drastically as $\delta \to 0$, while the spread for SPLICE remains essentially constant.

Figure 4

(a) Renormalized standard deviation (SD) and (b) unnormalized root mean-square error in the estimated separation plotted as functions of actual separation for both IPC and SPLICE. We plot the computed SD multiplied by $\sqrt{N}$ to compensate for the scaling of the uncertainty with the size of the data set. The solid and dashed curves are the corresponding Monte Carlo simulations. The dotted curve is the CRB for IPC and the dashed horizontal line represents the absolute fundamental limit of $2\sigma /\sqrt{N}$. The RMSE (unlike SD) is not similarly rescaled. It allows us to gauge absolute error relative to the known value of the parameter being estimated so that biases are accounted for. Note that two methods were used in the fitting of IPC data to Eq. (5); for small $\delta (<0.65\mathrm{mm})$, Eq. (5) was expanded to 2nd order and linear regression was performed whereas for large $\delta (>0.4\mathrm{mm})$, a nonlinear fitting routine built into Mathematica was used.

Figure 5

Averaged measured separations. Mean estimated $\delta$ for IPC and SPLICE plotted against known actual $\delta$. Two methods were used in the fitting of IPC data to equation (5); for small $\delta (<0.65\mathrm{mm})$, equation (5) was expanded to 2nd order and linear regression was performed whereas for large $\delta (>0.4\mathrm{mm})$, a nonlinear fitting routine built into Mathematica was used.