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Optics Express

Quantum limits of superresolution for imaging discrete subwavelength structures

Mikhail I. Kolobov

Published Jan 1, 2008• 7 citations
Publisher

Abstract

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.

DOI
10.1364/oe.16.000058

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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 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.
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.
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.
We present a detailed quantum-statistical analysis of superresolution for optical systems with circular symmetry based on our previous paper [Opt. Commun. 264, 9 (2006)].We consider as example an imaging system with two Fourier lenses; however, our approach can be easily generalized for more complex paraxial systems with circular symmetry. We present an original and very effective algorithm for numerical evaluation of the circular prolate spheroidal functions, which are the eigenmodes of our system. We investigate in detail the two-dimensional reconstruction point-spread function and provide examples of superresolution for rotationally symmetric and angularly dependent objects. Finally, we investigate the role of the local quantum fluctuations in the superresolution process and evaluate the local noise figure for the reconstruction process.
Optically localizing a single quasimonochromatic source to subdiffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width www of the optical system's point-spread function (PSF) and the sought localization precision, ddd, i.e., as $\ensuremath{\alpha}{(w/d)}^{2}$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\ensuremath{\alpha}$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\ensuremath{\alpha}$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating the quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.