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IRE Transactions on Communications Systems

Phase Estimators with Predetection Feedback for Optical Communication

Stanley R. Robinson

Published Nov 1, 1976• 2 citations
Publisher

Abstract

Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."

DOI
10.1109/tcom.1976.1093241

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Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."
We present a homodyne receiver structure for the detection of weak coherent states that uses sequential in-phase and quadrature measurements on the received optical signal. This receiver performs the optical carrier phase tracking requiring only a single balanced homodyne detector, by including a postdetection Costas-loop-type feedback, which additionally allows the use of suppressed carrier modulations in the received field, for efficient transmission. We report an experimental interferometric self-homodyne setup for the sequential detection of low photon number, binary phase-modulated optical signals that consist of strongly attenuated laser pulses by using a reference field as the local oscillator with an alternatively switched phase. A Costas loop postdetection subsystem is implemented in discrete time to perform fast real-time optical phase tracking. We also present the experimental results of the homodyne postdetection statistics for received BPSK signals with very low photon numbers, and compare them with the theoretical uncertainty limit. Finally, we conduct bit error rate measurements over a wide range of signal level, as well as a comparison with the standard quantum limit.
Recent inquiries into optical communication have raised questions as to the validity of classical detection and estimation theory for weak light fields. Helstrom [1] proposed that the axioms of quantum mechanics be incorporated into a quantum approach to optical estimation and detection. In this paper, we discuss two important results, the quantum equivalent of the minimum-mean-square-error (MMSE) estimator and the quantum Cramér-Rao bound for estimation of the parameters of an electromagnetic field. The first result, a new one, is applied to linear modulation. We show that homodyning is the optimal demodulation scheme in that case. Parallels to the classical MMSE estimator are drawn. The Cramér-Rao bound, first derived in the quantum ease by Helstrom [1], is applied to specific estimation problems. Details are left to the references, but interesting results are presented.
The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where NNN is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time ttt is a functional of the photocurrent from time 000 up to time ttt in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.
Single photon count distributions at the outputs of a Mach-Zehnder interferometer are highly dependent on the phase difference between the two arms of the apparatus, so photocount records can be used to estimate an unknown optical phase. Experimental results have shown that phase information can be inferred from the data even at low mean input intensities of ⪅10 photons. Here we consider the optimal Bayesian parameter estimate for the model used by a previous frequentist analysis. In the limit of high input intensity the observation is well-modelled as an intensity measurement of a classical wave in Gaussian measurement noise, however in the low intensity regime this simple model does not apply. It seems that no generally valid closed-form estimator for this problem exists without making the Gaussian approximation. Moreover the 2π radian ambiguity in phase estimates presents a problem in defining cost functions to compare estimator performance, and we consider two reasonable alternatives. We present a numerical study of the experiment along with a novel numerical approach to this problem which out-performs the existing estimators and with a squared error that approaches the Cramer-Rao lower bound previously derived in the literature, in the limit where that bound is valid.
For intermediate values of the photon number the adaptive variances should ideally be even lower [10], but the performance of our experiment is limited by finite feedback bandwidth. For signals with large mean photon number, the adaptive scheme is inferior to heterodyne because of excess technical noise in the feedback loop. As the intrinsic phase uncertainty of coherent states becomes large for very low photon numbers, the relative differences among the expected variances for adaptive, heterodyne, and ideal estimation become small. Accordingly, we have been unable to beat the heterodyne limit for phase estimation variances with adaptive homodyne measurement for mean photon numbers N & 8. (Note that all theoretical curves in Fig. 1(a) correspond FIG. 1 (color). Experimental results from the adaptive and heterodyne measurements. (a) Adaptive (blue circles) and heterodyne (red crosses) phase estimate variance vs pulse photon number. The blue dash-dotted line is a second order curve through the adaptive data, to guide the eye. The thin lines are the theoretical curves for heterodyne detection with (solid) and without (dotted) corrections for detector electronic noise. The thick solid line denotes the fundamental quantum uncertainty limit, given our overall photodetection efficiency. (b) Phase-estimator distributions for adaptive (blue circles) and heterodyne (red crosses) measurements, for pulses with mean photon number 2:5. (c) Polar plot showing the variance of adaptive phase estimates (blue dots) for different signal phases (mean photon number 50). The solid blue line is a linear fit to the data. The double red lines indicate the 1 scatter for our heterodyne data, averaged over initial phase.
We show that the optimal Cramér-Rao lower bound on the mean-square error for the estimation of a coherent signal from photon-limited intensity measurements is equal to the number of signal elements, or the number of signal elements minus one when we account for the unobservable reference phase. Whereas this bound is attained by phase-quadrature holography, we also show that it can be attained through a phase-retrieval system that does not require a coherent reference. We also present the bounds for classic phase-retrieval and ptychography, and show that practical coding strategies can approach optimal performance.