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arXiv:0812.2142 · Optics Letters

Quantum limits in the measurement of very small displacements in optical images

Claude Fabre, J.-B. Fouet, Agnès Maître

Published Jan 1, 2000• 119 citations
arXivHTMLPDF

Abstract

We consider the problem of the measurement of very small displacements in the transverse plane of an optical image with a split photodetector. We show that the standard quantum limit for such a measurement, which is equal to the diffraction limit divided by the square root of the number of photons used in the measurement, cannot be overcome by use of ordinary single-mode squeezed light. We give the form of possible multimode nonclassical states of light, enabling us to enhance by orders of magnitude the resolution of such a measurement beyond the standard quantum limit.

DOI
10.1364/ol.25.000076

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We consider the problem of the measurement of very small displacements in the transverse plane of an optical image with a split photodetector. We show that the standard quantum limit for such a measurement, which is equal to the diffraction limit divided by the square root of the number of photons used in the measurement, cannot be overcome by use of ordinary single-mode squeezed light. We give the form of possible multimode nonclassical states of light, enabling us to enhance by orders of magnitude the resolution of such a measurement beyond the standard quantum limit.
Using continuous wave superposition of spatial modes, we demonstrate experimentally displacement measurement of a light beam below the standard quantum limit. Multimode squeezed light is obtained by mixing a vacuum squeezed beam and a coherent beam that are spatially orthogonal. Although the resultant beam is not squeezed, it is shown to have strong internal spatial correlations. We show that the position of such a light beam can be measured using a split detector with an increased precision compared to a classical beam. This method can be used to improve the sensitivity of small displacement measurements.
Using continuous wave superposition of spatial modes, we demonstrate experimentally displacement measurement of a light beam below the standard quantum limit. Multimode squeezed light is obtained by mixing a vacuum squeezed beam and a coherent beam that are spatially orthogonal. Although the resultant beam is not squeezed, it is shown to have strong internal spatial correlations. We show that the position of such a light beam can be measured using a split detector with an increased precision compared to a classical beam. This method can be used to improve the sensitivity of small displacement measurements.
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.
Quantum imaging can be defined as an area of quantum optics that investigates the ultimate performance limits of optical imaging allowed by the quantum nature of light. Quantum Imaging techniques possess a high potential for improving the performance in recording, storage, and readout of optical images beyond the limits set by the standard quantum level of fluctuations known as the shot noise. This talk aims at giving an overview of the fundamentals of Quantum Imaging as well as its most important directions. We shall discuss generation of the spatially multimode squeezed states of light be means of a travelling-wave optical parametric amplifier. We shall demonstrate that this kind of light allows us to reduce the spatial fluctuations of the photocurrent in a properly chosen homodyne detection scheme with a highly efficient CCD camera. It will be shown that using the amplified quadrature of the light wave in a travelling-wave optical parametric amplifier, one can perform noiseless amplification of optical images. We shall provide recent experimental results demonstrating a single-shot noiseless image amplification by a pulsed optical parametric amplifier. One of important experimental achievements of Quantum Imaging, coined in the literature as quantum laser pointer, is a precise measurement of position and transverse displacement of a laser beam with resolution beyond the limit imposed by the shot noise. We shall describe briefly the idea of the experiment in which the transverse displacement of a laser beam was measured with resolution of the order of Angstrom. The problem of precise measurement of transverse displacement of a light beam brings us to a more general question about the quantum limits of optical resolution. The classical resolution criterion, derived in the nineteen century by Abbe and Rayleigh, states that the resolution of an optical instrument is limited by diffraction and is related to the wavelength of the light used in the imaging scheme. However, it was for a long time recognised in the literature that in some cases when one has some a priori information about the object, one can improve the resolution beyond the Rayleigh limit using the so-called super-resolution techniques. As a final topic in this talk, we shall discuss the quantum limits of optical super-resolution. In particular, we shall formulate the standard quantum limit of super-resolution and demonstrate how one can go beyond this limit using specially designed multimode squeezed light.
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.
Abstract We present a general method to find the ultimate limits on the resolution of optical image on a surface perpendicular to the light beam propagation. We prove that the optimal sensitivity always obeys the standard quantum limit for a single mode light. In the absence of photon losses, we find the possibility to achieve the Heisenberg limit by using two-mode or multi-mode lights. We also use the variational approach to obtain the lower bound on the sensitivity considering the effects of photon losses.