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

Verifying model for statistical hypothesis in detection of noncoherent radiation

Vasile D. Babin, Simion Ersen, Adrian Moldovan, Nicusor V. Iftimia

Published Jul 2, 1998• 0 citations
Publisher

Abstract

A mode of quantic detection for noncoherent optical radiation as noise limit is presented. Using the representation on coherent state of density matrix operators ((rho) ) and of detection operators ((pi) ), the detection probability (Qd), false alarm probability (Qo) and signal-noise ratio (S/Z), are estimated, as function of the number of freedom degrees of temporal oscillation modes (Mt) and of the spatial oscillation modes (Ms). A validation algorithm of statistical hypotheses, resulted from noncoherent optical field analysis, is made using Newman-Pearson criterion.

DOI
10.1117/12.312677

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A mode of quantic detection for noncoherent optical radiation as noise limit is presented. Using the representation on coherent state of density matrix operators ((rho) ) and of detection operators ((pi) ), the detection probability (Qd), false alarm probability (Qo) and signal-noise ratio (S/Z), are estimated, as function of the number of freedom degrees of temporal oscillation modes (Mt) and of the spatial oscillation modes (Ms). A validation algorithm of statistical hypotheses, resulted from noncoherent optical field analysis, is made using Newman-Pearson criterion.
When the phase of the transmittance between a unimodal transmitter and a unimodal receiver is uniformly distributed over(0,2\pi), the density operator of the received signal field commutes with the photon number operator, and the optimum detector of the signal counts photons. A simple formula is given for the probability generating function of the number of photons in terms of the quasi-probability distribution of the transmitted signal. It is used to determine the average error probability of an on-off binary communication system transmitting two-photon coherent-state signals, received in the presence of thermal noise.
Our knowledge about surroundings can be achieved by observations and measurements but both are influenced by errors (noise). Therefore one of the first tasks is to try to eliminate the noise by constructing instruments with high accuracy. But any real observed and measured system is characterized by natural limits due to the deterministic nature of the measured information. The present work is dedicated to the identification of these limits. We have analyzed some algorithms for selection and estimation based on statistical hypothesis and we have developed a theoretical method for their validation. A classic (non-quantic) algorithm for observations and measurements based on statistical strategies of optical field is presented in detail. A generalized statistical strategy for observations and measurements on the nuclear particles, is based on these results, taking into account the particular type of statistics resulting from the measuring process also.
We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.
In this analysis the quantum statistics of the received signal of a frequency modulated laser communication system are derived. In particular the detection statistics for heterodyne and electric field detection of the frequency modulated laser beam are determined. These detection statistics are used to define a measure of the communication system performance in terms of a signal to noise ratio. The development of the quantum electromagnetic field in terms of the photon annihilation eigenstates is the basis for determining these statistics. The derivation used accounts for fluctuations in the carrier, fluctuations due to background radiation introduced after modulation, fluctuations due to the local laser oscillator, and zero point field fluctuations. The results of this analysis indicate the probability distribution for the optical heterodyne detection is not a familiar one of classical communication theory, but it is of the form of probability distribution that satisfies experimentally measured photocount statistics of unmodulated laser radiation. The probability distribution for the electric field intensity detection of the frequency modulated laser beam is the familiar Gaussian distribution. (Author)
It is noted that the fiber propagation loss is a random process along the length of propagation. The stochastic nature of the loss process induces a random fluctuation to the energy of the optical signals, which, as an extra source of noise, could become comparable to the amplified-spontaneous-emission noise of optical amplifiers. The optical noise in random loss/gain has a quantum origin, as a manifestation of the corpuscular nature of electromagnetic radiation. This paper adopts the Schrodinger representation, and uses a density matrix in the basis of photon number states to describe the optical signals and their interaction with the environment of loss/gain media. When the environmental degrees of freedom are traced out, a reduced density matrix is obtained in the diagonal form, which describes the total energy of the optical signal evolving along the propagation distance. Such formulism provides an intuitive interpretation of the quantum-optical noise as the result of a classical Markov process in the space of the photon number states. The formulism would be more convenient for practical engineers, and should be sufficient for fiber-optic systems with direct intensity detection, because the quantity of concern is indeed the number of photons contained in a signal pulse. Even better, the model admits analytical solutions to the photon-number distribution of the optical signals.
We consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein--Uhlenbeck process). We obtain an analytical expression for the generating function of photocounts. In the case of a harmonic coherent signal, we analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise. I. The aim of the present work is to derive and study a formula for the probability of counts under the assumption that the photodetector registers an optical field formed by the superposition of a coherent signal and Brownian noise. This problem is of interest in the study of the count distribution in a detector which absorbs a mixture of coherent and incoherent radiation [i]. In the derivation of the formula for the probability density of the number of counts, one encounters specific mathematical difficulties associated with the fact that the photodetection process is extended in time. The probability P(m, T) that, during a time interval T, m photoelectrons are detected when the superposition of a coherent signal B(t) and noise ~(t) falls on the detector, is given by the Mandel formula [2]