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Physical Review Letters

Quantum Limits on Optical Resolution

Mikhail I. Kolobov, Claude Fabre

Published Oct 30, 2000• 178 citations
Publisher

Abstract

We discuss the ultimate limit imposed by quantum fluctuations of light for resolution of fine details in optical images. For this purpose, we extend in the quantum domain the classical analysis of the object reconstruction, or superresolution, in terms of prolate spheroidal function basis. We derive the expression for ultimate resolution limit in the reconstructed object using an illumination of the full object plane by a multimode squeezed vacuum. We show that the gain in resolution using multimode squeezed light is maximum when the Shannon number of the imaging system is close to unity.

DOI
10.1103/physrevlett.85.3789

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We discuss the ultimate limit imposed by quantum fluctuations of light for resolution of fine details in optical images. For this purpose, we extend in the quantum domain the classical analysis of the object reconstruction, or superresolution, in terms of prolate spheroidal function basis. We derive the expression for ultimate resolution limit in the reconstructed object using an illumination of the full object plane by a multimode squeezed vacuum. We show that the gain in resolution using multimode squeezed light is maximum when the Shannon number of the imaging system is close to unity.
Summary form only given. To achieve super-resolution one must take into account a priori information about localization of the object. Such an a priori information exists in the case of an object of finite size. Its Fourier transform located in the focal plane of the lens is an entire analytic function, and analytic continuation of the object spectrum outside the spatial-frequency band allows for unlimited resolution. However such a super-resolution is extremely sensitive to the noise in the detected image. The ultimate limit of super-resolution is determined by the quantum fluctuations of light in the detection plane. We present the quantum theory of super-resolution for a one-dimensional coherent optical imaging scheme and discuss possibilities of improving the resolution limit by using multimode squeezed light. For quantum-mechanical description of this scheme we take into account the vacuum fluctuations of the electromagnetic field in the input plane outside the object. Interference of these vacuum fluctuations with the field amplitude of the object in the detection plane is the origin of the standard quantum limit or shot noise of photodetection in the image. To go beyond this limit we suggest to illuminate the object with multimode squeezed light having the spot size much larger than the object area. Then the vacuum fluctuations outside the object can be effectively suppressed resulting in sub-shot-noise detection of the image.
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.
We numerically investigate the role of quantum fluctuations in superresolution of optical objects. First, we confirm that when quantum fluctuations are not taken into account, one can easily improve the resolution by one order of magnitude beyond the diffraction limit. Then we investigate the standard quantum limit of superresolution which is achieved for illumination of an object by a light wave in a coherent state. We demonstrate that this limit can be beyond the diffraction limit. Finally, we show that further improvement of superresolution beyond the standard quantum limit is possible using the object illumination by a multimode squeezed light.
We present the quantum theory of superresolution for discrete subwavelength structures. It allows to formulate, in particular, the standard quantum limit of superresolution achieved for illumination of the structure by light in coherent state. Our theory is based on discrete prolate spheroidal sequences and functions which are the proper basis set of the problem. We demonstrate that the superresolution factor is much higher for discrete structures than for continuous objects for the same signal-to-noise ratio. This result is a clear illustration of the crucial role of a priori information in superresolution problems.
We propose a source of multimode squeezed light that can be used for the superresolving microscopy beyond the standard quantum limit. This source is an optical parametric amplifier with a properly chosen diaphragm on its output and a Fourier lens. We demonstrate that such an arrangement produces squeezed prolate spheroidal waves which are the eigen modes of the optical imaging scheme used in microscopy. The degree of squeezing and the number of spatial modes in illuminating light, necessary for the effective object field reconstruction, are evaluated
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.