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Journal of the Optical Society of America B

Statistical analysis of an incoherently coupled, steady-state optical amplifier

Terence J. Shepherd, Eric Jakeman

Published Nov 1, 1987• 29 citations
Publisher

Abstract

The inverted population model of a linear optical amplifier is extended to incorporate explicit input and output photon fluxes. The input field is assumed to be incoherently coupled to the internal field of the amplifier, thus permitting a population statistical description of the complete system, including photon detection. Steady-state behavior and intensity amplification are simultaneously realizable. The integrated statistics of the photoelectron count at the detector are obtained and employed to evaluate a suitably defined signal-to-noise ratio for the system. The conditions under which signal enhancement can occur are derived and discussed for the case in which the input field photocount is Poisson distributed.

DOI
10.1364/josab.4.001860

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The inverted population model of a linear optical amplifier is extended to incorporate explicit input and output photon fluxes. The input field is assumed to be incoherently coupled to the internal field of the amplifier, thus permitting a population statistical description of the complete system, including photon detection. Steady-state behavior and intensity amplification are simultaneously realizable. The integrated statistics of the photoelectron count at the detector are obtained and employed to evaluate a suitably defined signal-to-noise ratio for the system. The conditions under which signal enhancement can occur are derived and discussed for the case in which the input field photocount is Poisson distributed.
The population-statistical description of a linear optical amplifier, extended to include both a resonant optical cavity and explicit input and output photon fluxes, is shown to yield three useful noise relationships. Despite the treatment's plainly particulate character, it provides an expression for the variance of the time-integrated output-photon-detection process in which terms associated with interferometrically generated beat noise are prominent. For systems whose photodetector electrical bandwidth B is much smaller than the amplifier's optical bandwidth Delta v, as is almost invariably the case for semiconductor amplifiers in lightwave-communication systems, the time-integrated photocurrent variance can be written in a simple, physically intuitive form. Two quotients are of central importance: ( Delta nu /B) and the ratio (/) of mean amplified-signal power to mean amplified-spontaneous-emission power. In the usual case of interest for optical amplifiers in communication systems where>>, the signal-to-noise ratio of the integrated photocurrent, on which the error probability in an on-off-keyed system depends, is approximately=1/2( Delta v/B) ((n/sub S/)/(n/sub ASE/)).>
We calculate the input-output characteristics of an optical amplifier that consists of an inverted-population medium placed inside a high-Q Fabry-P\'erot cavity. Expressions are derived for the output second- and fourth-order spectral and temporal correlation functions, and more generally for the output characteristic functional, in terms of the corresponding input quantities. The photocount first and second factorial moments are obtained for both homodyne and direct detection of the amplifier input and output. The general results are applied to several cases of practical interest, including inputs that have coherent or chaotic coherence properties. Particular attention is paid to the effects of amplification on specific nonclassical varieties of input light. It is shown that a maximum of only twofold amplification is permitted if any squeezing present in the input is to survive as squeezing in the output light. Similarly, for the preservation of photon antibunching in amplification, we show, by consideration of a kind of free-space number-state input, that only small gains are allowed. The amplifier model treated here provides a detailed example of the limitations imposed by quantum mechanics on the minimum noise generated by a multimode linear amplifier. In particular, we show that minimum noise occurs in a cavity that is asymmetric with respect to mirror reflectivities, for which we derive the corresponding conditions. In addition, we demonstrate the ability of the amplifier to improve upon signal-to-noise ratio, otherwise limited by low detector quantum efficiency.
The properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.
This chapter reviews and extends the work on the particular class of linear amplifiers known as inverted population amplifiers. It discusses first the fundamental requirements demanded by quantum mechanics from any theory of linear amplifiers, and then describes the basic model for the atomic amplifier in a closed cavity. The chapter considers the extension of this model to allow explicit incoherent coupling of the internal cavity field to external fields, and treats finally an open system with coherent field coupling. It investigates in particular the conditions under which the amplified field retains quantum properties such as squeezing. The chapter examines the ability of the phase-insensitive systems to enhance the signal-to-noise ratio of an injected signal, and finds that in general any signal is degraded by the amplification process. Phase-insensitive atomic amplifiers destroy all quantum statistical properties of the field if the gain exceeds the low value of two.
It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.