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arXiv:0804.2875 · Physical Review A

Sub-Rayleigh-diffraction-bound quantum imaging

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro

Published Jan 28, 2009• 124 citations
arXivHTMLPDF

Abstract

The spatial resolution of an imaging apparatus is limited by the Rayleigh diffraction bound, a consequence of the imager's finite spatial extent. We show some $N$-photon strategies that permit resolution of details that are smaller than this bound, attaining either a $1∕\sqrt{N}$ enhancement (standard quantum limit) or a $1∕N$ enhancement (Heisenberg-like scaling) over standard techniques. In the incoherent imaging regime, the methods presented are loss resistant, since classical light sources suffice. Our results may be of importance in many applications: microscopy, telescopy, lithography, metrology, etc.

DOI
10.1103/physreva.79.013827

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It has been theoretically predicted that N-photon quantum imaging can realize either an N-fold resolution improvement (Heisenberg-like scaling) or a N-fold resolution improvement (standard quantum limit) beyond the Rayleigh diffraction bound, over classical imaging. Here, we report the experimental study on spatial sub-Rayleigh quantum imaging using a two-photon entangled source. Two experimental schemes are proposed and performed. In a Fraunhofer diffraction scheme with a lens, two-photon Airy disk pattern is observed with subwavelength diffraction property. In a lens imaging apparatus, however, two-photon sub-Rayleigh imaging for an object is realized with super-resolution property. The experimental results agree with the theoretical prediction in the two-photon quantum imaging regime.
The spatial resolution of a traditional imaging system is restricted by the Rayleigh diffraction limit. In this paper, two types of classical light sources are generated by modulating the amplitude distribution and wavefront of a laser beam randomly, and the generated light sources can exhibit the features of the superposition of two-photon Fock states and the incoherent mixture of two-photon Fock states, respectively. With the generated light sources, the two-fold coherent and incoherent imaging schemes can be achieved, which lead to spatial resolution enhancement, and exceed the Rayleigh diffraction limit in the imaging system.
The resolution limit of standard imaging techniques is expressed by the Rayleigh criterion, which states that two point-like sources are difficult to resolve if their transverse separation is smaller that the Rayleigh length. While the criterion is useful in the case of direct detection imaging, other measurement techniques may not be subject to this limitation. Here we consider the use of imaging to estimate the distance between two incoherent point sources. In this framework, the Rayleigh criterion manifests itself through the so-called Rayleigh curse. The latter expresses the fact that the statistical error in the estimation increases for sub-wavelength separation. Here we show that, in the regime of weak signals, a structured measurement obtained by concatenating a linear interferometer with on-off photo-detection is immune to the Rayleigh curse. In this way we clarify the relation between imaging and interferometry, and establish the optimality of linear interferometry for an arbitrary number of incoherent sources.
We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffraction-limited, linear imaging device within the paraxial approximation. First, we show that the problem of estimating the separation between two pointlike sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two pointlike sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then, we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be superresolved at the sub-Rayleigh scale. Our results apply to a variety of imaging setups, from astronomical observation to microscopy, exploiting quantum detection as well as source engineering.