Far-Field Superresolution of Thermal Electromagnetic Sources at the Quantum Limit

We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.

Figure 1

A spatially invariant imaging system. Point sources at $(\pm d/2,0)$ of the object plane $\mathcal{}O$ have images centered at $(\pm d/2,0)$ of the image plane $\mathcal{}I$ but spread out by the PSF of the system.

Figure 2

The QFI of Eq. (7) (solid lines) and the lower bound of Eq. (12) (dash-dotted lines) on spatially and number-resolved direct imaging (DI) for the Gaussian PSF (10). The plots are normalized to the respective maximum values $N_s/2{\sigma }^2$ of the QFI and are independent of the PSF half-width $\sigma$.

Figure 3

Fin-SPADE. The image-plane field is coupled into a multimode fiber and separated into its components in the Hermite-Gaussian ${\mathrm{TEM}}_{q0}$ modes of order $0\le q\le Q$ by evanescent coupling to single-mode fibers supporting those modes. Detectors record the photon number in each mode.

Figure 4

Fin-SPADE performance. The QFI (solid line), the lower bound (12) on the FI of fin-SPADE (dashed lines) for various $Q$, and of DI (dash-dotted line). The Gaussian PSF (10) is assumed, and $N_s=1.5$ photons. The plots are normalized to the maximum value $N_s/2{\sigma }^2$ of the QFI and are independent of $\sigma$. The DI bound assumes a detector of width $17\sigma$ with $P_d=50$ pixels at 100% fill factor and is stable to increase in $P_d$. Number-resolving unity-quantum-efficiency detectors are assumed for all the measurement schemes.

Figure 5

Pix-SLIVER. The image-plane field is separated into its symmetric and antisymmetric components (13) using a balanced Mach-Zehnder interferometer with an extra reflection in one arm before detecting the two outputs using identical detector arrays of width $W$ pixelated in the $x$ direction.

Figure 6

Pix-SLIVER performance. The QFI (solid line), the lower bound (12) on the FI of pix-SLIVER using on-off detection with various $P$ values (dashed lines), the lower bound (12) for DI (dash-dotted line) with number-resolved detection, and the contributions of the symmetric (sym) and antisymmetric (asym) field components to (12) for $P=40$ (dotted lines). The Gaussian PSF (10) is assumed, and $N_s=1.5$ photons. The plots are normalized to the maximum value $N_s/2{\sigma }^2$ of the QFI and are independent of $\sigma$. The lower bounds assume detector array(s) of width $17\sigma$ and 100% fill factor. The DI bound assumes an array with $P_d=50$ pixels and is stable to increase in $P_d$.