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Physical Review A

Super-resolution quantum interference pattern of a distributed object

Itamar Vidal, E. J. S. Fonseca, J. Miguel Hickmann

Published Oct 18, 2010• 6 citations
Publisher

Abstract

Classically, the spatial resolution of the image of an object is usually determined by the Rayleigh criterion. However, quantum properties of light have been exploited to form images with resolutions exceeding this criterion and beating the standard quantum limit, which allow high measurement sensitivity. We found that it is possible to super-resolve a distributed object made of two parts that are spatially separated but when superposed over each other form an object. A coincidence interference pattern of a nonlocal two-photon system with periodicity proportional to $\ensuremath{\lambda}/2$ was measured even when each photon, with wavelength of $\ensuremath{\lambda}$, has been scattered by different parts of an object, none of them being the object itself.

DOI
10.1103/physreva.82.043827

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Classically, the spatial resolution of the image of an object is usually determined by the Rayleigh criterion. However, quantum properties of light have been exploited to form images with resolutions exceeding this criterion and beating the standard quantum limit, which allow high measurement sensitivity. We found that it is possible to super-resolve a distributed object made of two parts that are spatially separated but when superposed over each other form an object. A coincidence interference pattern of a nonlocal two-photon system with periodicity proportional to \ensuremathλ/2\ensuremath{\lambda}/2\ensuremathλ/2 was measured even when each photon, with wavelength of \ensuremathλ\ensuremath{\lambda}\ensuremathλ, has been scattered by different parts of an object, none of them being the object itself.
High-precision measurements implemented with light are desired in all fields of science. However, light acts as a wave, and the Rayleigh criterion in classical optics yields a diffraction limit that prevents obtaining a resolution smaller than the wavelength. Sub-wavelength interference has potential application in lithography because it beats the classical Rayleigh resolution limit. Here, we carefully study second-order correlation theory to establish the physics behind sub-wavelength interference in photon coincidence detection. A Young's double slit experiment with pseudo-thermal light is performed to test the second-order correlation pattern. The results show that when two point detectors are scanned in different ways, super sub-wavelength interference patterns can be obtained. We then provide a theoretical explanation for this surprising result, and demonstrate that this explanation is also suitable for the results found for entangled light. Furthermore, we discuss the limitations of these types of super sub-wavelength interference patterns in quantum lithography.
Resolution is crucially important for optical imaging, which defines the smallest spatial feature of object that can be delivered by light wave. However, due to the wave nature of light, optical imaging is of limited resolution, widely known as Rayleigh limit or Abbe limit. Nevertheless, this limit can be overcome by considering the loopholes in the derivation of the Rayleigh limit, such as light–matter interaction, structured illumination, and near-field interference. In contrast to the conventional single-photon interference, multi-photon amplitudes responsible for optical high-order interference could be designed to possess a reduced effective wavelength, enabling the breakthrough of the Rayleigh limit. In this review, we will present recently developed super-resolution imaging schemes based on optical high-order interference, and discuss future perspectives.
Imaging systems measuring intensity in the far field succumb to Rayleigh's curse, a resolution limitation dictated by the finite aperture of the optical system. Many proof-of-principle and some two-dimensional imaging experiments have shown that, by using spatial mode demultiplexing (SPADE), the field information collected is maximal, and thus, the resolution increases beyond the Rayleigh criterion. Hitherto, the SPADE approaches are based on resolving the lateral splitting of a Gaussian wavefunction. Here, we consider the case in which the light field originates from a bi-photon source, i.e. spontaneous parametric down-conversion, and a horizontal separation is introduced in one of the two photons. We show that a separation induced in the signal photon arm can be super-resolved using coincidence measurements after projecting both photons on Hermite-Gauss modes. Remarkably the Fisher information associated with the measurement is enhanced compared to the ordinary SPADE techniques by K\sqrt{K}K​, where KKK is the Schmidt number of the two-photon state that quantifies the amount of spatial entanglement between the two photons.
The resolution of optical interference is ultimately limited by the wavelength of light, which is known as Rayleigh criterion. Interestingly, this limitation could be surpassed by considering photonic de Broglie wave, and the resulting subwavelength interference is useful for lithography and super-resolving phase measurementsuper-resolving phase measurement. Usually, quantum entangled state is necessary to realize photonic de Broglie wave. Recent advances show that subwavelength interference could also be realized with a classical light through post-selected operation or by considering light-matter interaction. In this work, a new classical source is designed to achieve subwavelength interference without any post-selected operation in a linear scheme.
Imaging systems measuring intensity in the far field succumb to Rayleigh's curse, a resolution limitation dictated by the finite aperture of the optical system. Many proof-of-principle and some two-dimensional imaging experiments have shown that, by using spatial mode demultiplexing (SPADE), the field information collected is maximal, and, thus, the resolution increases beyond the Rayleigh criterion. Hitherto, the SPADE approaches are based on resolving the lateral splitting of a Gaussian wave function. Here, we consider the case in which the light field originates from a biphoton source, i.e., spontaneous parametric down-conversion, and a horizontal separation is introduced in one of the two photons. We show that a separation induced in the signal photon arm can be superresolved using coincidence measurements after projecting both photons on Hermite-Gauss modes. Remarkably, the Fisher information associated with the measurement is enhanced compared to the ordinary SPADE techniques by K\sqrt{K}K​, where KKK is the Schmidt number of the two-photon state that quantifies the amount of spatial entanglement between the two photons.
We investigate analytically and numerically the role of quantum fluctuations in reconstruction of optical objects from diffraction-limited images. Taking as example of an input object two closely spaced Gaussian peaks, we demonstrate that one can improve the resolution in the reconstructed object over the classical Rayleigh limit. We show that the ultimate quantum limit of resolution in such a reconstruction procedure is determined not by diffraction but by the signal-to-noise ratio in the input object. We formulate a quantitative measure of super-resolution in terms of the optical point-spread function of the system.