Quantum Theory of Superresolution for Two Incoherent Optical Point Sources

Abstract

Using a semiclassical model of photodetection with Poissonian noise and insights from quantum metrology, we prove that linear optics and photon counting can optimally estimate the separation between two incoherent point sources without regard to Rayleigh's criterion. The model is applicable to weak thermal or fluorescent sources as well as lasers.Lord Rayleigh suggested in 1879 that two incoherent optical point sources should be separated by a diffraction-limited spot size for them to be resolved [1]. This criterion has since become the most influential measure of imaging resolution. Under the modern advent of rigorous statistics and image processing, Rayleigh's criterion remains a curse. When the image is noisy, necessarily so owing to the quantum nature of light [2], and Rayleigh's criterion is violated, it becomes much more difficult to estimate the separation accurately by conventional imaging methods [3][4][5]. Modern superresolution techniques in microscopy [6][7][8] can circumvent Rayleigh's criterion by making sources radiate in isolation, but such techniques require careful control of the fluorescent emissions, making them difficult to use for microscopy and irrelevant to astronomy.Here we show that, contrary to conventional wisdom, the separation between two incoherent optical sources can be estimated accurately via linear optics and photon counting (LOPC) even if Rayleigh's criterion is severely violated. Our theoretical model here is based on the semiclassical theory of photodetection with Poissonian noise, which is a widely accepted statistical model for lasers [2] as well as weak thermal [9,10] or fluorescent [5,11] light in astronomy and microscopy. The semiclassical model is consistent with the quantum model proposed in Ref. [12] for weak incoherent sources and the mathematical formalisms are similar, but the semiclassical model has the advantage of being applicable also to lasers, which are important sources for remote-sensing, testing, and proof-of-concept experiments. The semiclassical theory also avoids a quantum description of light and offers a more pedagogical perspective. Compared with the full semiclassical theory in Ref. [13], the Poissonian model is invalid for strong thermal sources but more analytically tractable.Consider J optical modes and a column vector of complex field amplitudes α = (α 1 , . . . , α J ) ⊤ within one coherence time interval. The amplitudes are normalized such that |α j | 2 is equal to the energy in each mode in units of quanta. The central quantity in statistical optics is the mutual coherence matrix [2,9]

Ultimate Precision Bound of Quantum and Subwavelength Imaging

“…5 shows the quantum Fisher information per photon impinging on the image screen versus s/x R , for the Gaussian point-spread function. For small values of η the light is highly attenuated and we observed the same behavior as highly attenuated incoherent sources, while for small η and relatively large squeezing ξ the state becomes effectively semiclassical and manifests the classical "Rayleigh's curse" [10]. On the other hand, for larger value of η we observe a phenomenon of super-resolution for sub-Rayleigh distances, similarly to the optimal entangled sources obtained in the main body of this paper.…”

“…Our findings generalize the seminal Ref. [10] which has led to several experimental advances in quantum imaging [20][21][22][23].…”

“…[10], which considered highly attenuated incoherent sources, to the case of thermal sources of any intensity. A comparison with Eq.…”

Single plane minimal tomography of double slit qubits

“…We assume quasimonochromatic waves, of frequency ω and wavenumber k, in the paraxial approximation with a given polarization and a single spatial dimension x on the slit and the detection planes. A simple mathematical description in terms of wave functions, recently used in the investigation of optical superresolution [19,18], gives the same result as a second quantized treatment in which only the single photon subspace of the full Fock space is taken into account.…”

“…Recently, ideas originated in quantum estimation have been applied to classical contexts. For example, the quantum Fisher information has been used to show that suitable measurements allow the resolution of incoherent sources separated by distances which violate the Rayleigh criterion [18,19]. The physical system that we consider in this work, a double-slit qubit, corresponds to the classical Young's interference experiment.…”