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Soviet Journal of Quantum Electronics

Potential characteristics of a phase-conjugate adaptive system

A B Aleksandrov, Yu G Dolotin

Published Apr 30, 1985• 0 citations
Publisher

Abstract

An analysis is made of the efficiency of compensation of distortions of the angular distribution by a phase-conjugate adaptive system of the heterodyne type. Numerical estimates are given of the time and energy characteristics of adaptive systems obtained for different values of the signal/noise ratio in the measuring scheme. It is shown that when the signal/noise ratio in a homogeneous system is of the order of unity, the use of adaptive correction is not very effective. Analytic and numerical results obtained in the present study make it possible to estimate approximately the potential characteristics of phase-conjugate adaptive systems.

DOI
10.1070/qe1985v015n04abeh006979

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An analysis is made of the efficiency of compensation of distortions of the angular distribution by a phase-conjugate adaptive system of the heterodyne type. Numerical estimates are given of the time and energy characteristics of adaptive systems obtained for different values of the signal/noise ratio in the measuring scheme. It is shown that when the signal/noise ratio in a homogeneous system is of the order of unity, the use of adaptive correction is not very effective. Analytic and numerical results obtained in the present study make it possible to estimate approximately the potential characteristics of phase-conjugate adaptive systems.
An analysis is made of the task of measuring and phase conjugation of the optical field against the background of noise in the case of a priori indeterminacy of the statistical properties of the received signal. Mode expansion of a useful signal is employed to show that the optimal algorithm may be realized in at least two variants: either on the basis of multichannel heterodyne reception and phase conjugation of the filtered signal at a radio frequency or with the aid of "physical" phase conjugation methods directly at an optical frequency. Expressions are obtained for estimating the quality of the measurements and phase conjugation. It is shown that practical realization of the latter variant of the optimal algorithm may result in a significant deterioration of the potential characteristics.
It is not possible to make measurements of the phase of an optical mode using linear optics without introducing an extra phase uncertainty. This extra phase variance is quite large for heterodyne measurements, however it is possible to reduce it to the theoretical limit of log [nbar]/(4[nbar]2) using adaptive measurements. These measurements are quite sensitive to experimental inaccuracies, especially time delays and inefficient detectors. Here it is shown that the minimum introduced phase variance when there is a time delay of log τ is τ/(8[nbar]). This result is verified numerically, showing that the phase variance introduced approaches this limit for most of the adaptive schemes using the best final phase estimate. The main exception is the adaptive mark II scheme with simplified feedback, which is extremely sensitive to time delays. The extra phase variance due to time delays is considered for the mark I case with simplified feedback, verifying the τ/2 result obtained by Wiseman and Killip both by a more rigorous analytic technique and numerically.
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.
The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.
An adaptive correction algorithm is demonstrated based on weak measurement, which introduces a feedback and an additional interaction to the system and can dynamically adjust the operating point in accordance with the condition of the estimated phase change. Two schemes, fast adaptive correction and slow adaptive correction, are proposed for different conditions of the modulation device. Fast adaptive correction scheme can realize a real-time correction and maintain the high sensitivity. Slow adaptive correction scheme, as a supplement, can correct the distortion of the measured parameter by changing the measuring period. These two schemes are useful for high precision phase estimation with time in modern physics and practical applications, including, but not limited to, timing synchronization, accurate distance measurement, and gravity wave detection. Moreover, we discuss the deviation of the adaptive correction for considering system noise in practical measurement.