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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Fundamental limits on estimation of polarization properties of quasi-monochromatic light

Jerzy Nowakowski, Anthony Carillo

Published Aug 13, 1993• 0 citations
Publisher

Abstract

In this paper we discuss the estimation of parameters describing polarized and partially polarized light when radiation is collected with a receiver consisting of an array of incoherent detectors and registering light levels after passage through an array of retardation plates and analyzers. The results obtained in semiclassical limit include Poisson photon statistics. We obtain fundamental limits on estimation of parameters of interest as a function of the number of photons available, and determine the value of a priori knowledge for achieving the desired estimation accuracy for both signal and background shot noise limited condition. The results presented are applicable to both active and passive systems where measurements of polarization signatures can improve detection and/or discrimination of targets, or when polarization coding is used to transmit information.

DOI
10.1117/12.151040

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In this paper we discuss the estimation of parameters describing polarized and partially polarized light when radiation is collected with a receiver consisting of an array of incoherent detectors and registering light levels after passage through an array of retardation plates and analyzers. The results obtained in semiclassical limit include Poisson photon statistics. We obtain fundamental limits on estimation of parameters of interest as a function of the number of photons available, and determine the value of a priori knowledge for achieving the desired estimation accuracy for both signal and background shot noise limited condition. The results presented are applicable to both active and passive systems where measurements of polarization signatures can improve detection and/or discrimination of targets, or when polarization coding is used to transmit information.
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.
In many applications using light for practical purposes, the accuracy of optical experiment is limited most fundamentally by the finite amount of light utilized in the measurements. Special emphasis should be put on interferometry aimed at measuring the parameters of simple fringe patterns due to the following reasons. First, fringe parameter measurement provides a relatively well defined and tractable example of application of the theory. The desired parameters are easily defined, and methods for their measurement are readily devised based on common sense. Second, fringe parameter measurement is central to all problems involving coherence. The fundamental descriptors of light waves utilized in coherence theory are in fact measurable parameters of fringes. By examining the limitations to fringe parameter measurement, we are actually examining the limitations to the measurability of coherence itself. In this paper, fundamental limitations of estimating the amplitudes and phases of polar-interferograms recorded at low light levels are investigated. By modeling the receiver as a spatial array of photon-counting detectors, results are obtained that permit specification of the minimum number of photoevents required for estimation of fringe parameters to a given accuracy. Both a discrete Fourier-transform estimator and an optimum joint maximum-likelihood estimator are considered to specify the limiting performance of all unbiased estimators in terms of the collected light flux.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.
I present a series of experimental and theoretical advances in the field of quantum state estimation. Techniques for measuring the quantum state of light that were originally developed for distinguishable\nphotons fail when the particles are indistinguishable. I develop new methods for handling indistinguishability in quantum state estimation. The technique I present provides the first complete description of states of experimentally indistinguishable photons. It allows me to derive the number of parameters needed to describe an arbitrary state and to quantify distinguishability. I demonstrate its use by applying it to the measurement of the\nquantum polarization state of two and three-photon systems.\n\nState characterization is optimal when no redundant information is collected about the state of the system. I present the results of the first optimal characterization of the polarization state of a two-photon system. I show an improved estimation power over the previous state of the art. I also show how the optimal measurements lead to a new description of the quantum state in terms\nof a discrete Wigner function.\n\nIt is often desirable to describe the quantum state of a system in terms of\nproperties that are not themselves quantum-mechanical observables. This usually requires a full characterization of the state followed by a calculation of the properties from the parameters characterizing the state. \nI apply a technique that allows such properties to be determined directly, without a full characterization of the state.\nThis allows one such property, the purity, to be determined in a single measurement, regardless of the size of the system, while\nthe conventional method of determining purity requires a number of measurements that scales exponentially with the system size.
We investigate fundamental bounds on the ability to determine photon number distribution and other related quantities from tomographically incomplete measurements with an array of M detectors that can only distinguish the absence or presence of photons. We show that the lower and upper bounds on photon number probabilities can be determined by solving a linear program. We present and discuss numerical results for various input states, including thermal states, coherent states, squeezed states, and highly nonclassical single-photon subtracted squeezed vacuum states. Besides photon number probabilities, we also investigate bounds on the parity of photon number distribution that determines the value of Wigner function of the state at the origin of phase space. Moreover, we also discuss estimation of mean photon number as an example of a quantity described by an unbounded operator. Our approach and results can provide quantitative guidance on the number of detection channels required to determine the photon number distribution with a given precision.
Estimation of the Stokes vector is based on projecting the input light on a number N of polarization analysis states. We address the optimization of the distribution of these analysis states on the Poincaré sphere in the presence of signal-dependent Poisson shot noise for an arbitrary value of N. We show that if this distribution forms a spherical 3 design, the Stokes vector is estimated with minimal equally weighted variance and with estimation variances of the last three Stokes parameters equal and independent of the input Stokes vector. We also demonstrate that in the presence of Poisson shot noise, the estimation signal to noise ratio is independent of N, whereas in the presence of signal independent additive noise, it is proportional to 1/N, which means that there is a precision loss in increasing the number of measurements.