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Oxford University Press eBooks

Linear optical amplifiers*

J. C. Garrison, R. Y. Chiao

Published Jun 5, 2008• 1 citation
Publisher

Abstract

This chapter begins by specializing the general definition of optical amplifiers to a special class of linear amplifiers that includes all those in common use. Amplifiers are divided into the categories of phase insensitive or phase sensitive, and each of these categories is subdivided into the categories of traveling-wave or regenerative. Illustrative examples are given for each of the four types. This is followed by a general description of single-input, single-output linear amplifiers that includes the input-output equation, the conditions for phase insensitivity, and the constraints imposed by unitarity. This general theory is used to determine the noise limits for linear amplifiers. The general amplifier uncertainty principle obtained in this way shows that there are noise free amplifiers.

DOI
10.1093/acprof:oso/9780198508861.003.0017

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This chapter begins by specializing the general definition of optical amplifiers to a special class of linear amplifiers that includes all those in common use. Amplifiers are divided into the categories of phase insensitive or phase sensitive, and each of these categories is subdivided into the categories of traveling-wave or regenerative. Illustrative examples are given for each of the four types. This is followed by a general description of single-input, single-output linear amplifiers that includes the input-output equation, the conditions for phase insensitivity, and the constraints imposed by unitarity. This general theory is used to determine the noise limits for linear amplifiers. The general amplifier uncertainty principle obtained in this way shows that there are noise free amplifiers.
Optical parametric amplifiers rely on second-order susceptibility (three-wave mixing) or third-order susceptibility (four-wave mixing) in a nonlinear process where the energy of incoming photons is not changed (elastic scattering). In the latter case, two pump photons are converted to a signal and to an idler photon. Under certain conditions, related to the phase evolution of the waves involved, this conversion can be very efficient, resulting in large amplification of an input signal. As the nonlinear process can be very fast, all-optical applications aside from pure amplification are also possible. If the amplifier is implemented in an optical input-phase-sensitive manner, it is possible to amplify a signal wave without excess noise, i.e., with a noise figure of 0 dB. In this paper, we will provide the fundamental concepts and theory of such amplifiers, with a focus on their implementation in highly nonlinear optical fibers relying on four-wave mixing. We will discuss the distinctions between phase-insensitive and phase-sensitive operation and include several experimental results to illustrate their capability. Different applications of parametric amplifiers are also discussed, including their use in optical communication links.
How much noise does quantum mechanics require a linear amplifier to add to a signal it processes? An analysis of narrow-band amplifiers (single-mode input and output) yields a fundamental theorem for phase-insensitive linear amplifiers; it requires such an amplifier, in the limit of high gain, to add noise which, referred to the input, is at least as large as the half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers, which can respond differently to the two quadrature phases ("$cos\ensuremath{\omega}t$" and "$sin\ensuremath{\omega}t$"), the single-mode analysis yields an amplifier uncertainty principle---a lower limit on the product of the noises added to the two phases. A multimode treatment of linear amplifiers generalizes the single-mode analysis to amplifiers with nonzero bandwidth. The results for phase-insensitive amplifiers remain the same, but for phase-sensitive amplifiers there emerge bandwidth-dependent corrections to the single-mode results. Specifically, there is a bandwidth-dependent lower limit on the noise carried by one quadrature phase of a signal and a corresponding lower limit on the noise a high-gain linear amplifier must add to one quadrature phase. Particular attention is focused on developing a multimode description of signals with unequal noise in the two quadrature phases.
The noise properties of the two main kinds of linear optical amplifier, phase-insensitive and phase-sensitive, are reviewed, with particular emphasis on their minimum-noise limits. An optical communications link is considered, in which identical sections of attenuating fiber alternate with identical linear amplifiers. The limits on the overall signal-to-noise ratios of such chains are derived. More attention is paid to the signal-to-noise ratio appropriate to phase-sensitive detection, but some results for direct detection are also given.
The classical definition of noise figure, based on signal-to-noise ratio, is adapted to the case when quantum noise is predominant. The noise figure is normalized to "uncertainty noise." General quantum mechanical equations for linear amplifiers are set up using the condition of linearity and the requirement that the commutator brackets of the pertinent operators are conserved in the amplification. These equations include as special cases the maser, the parametric amplifier, and the parametric up-converter. Using these equations the noise figure of a general amplifier is derived. The minimum value of this noise figure is equal to 2. The significance of the result with regard to a simultaneous phase and amplitude measurement is explored.
The quantum-limited noise figure (NF) for a general combination of linear optical elements is obtained in a compact form. The entries in this expression are obtained from the classical input–output transmission coefficient of the signal, and of the various transmission coefficients from the noise inputs to the output. The same result is also obtained by means of a semiclassical derivation. The linear elements can be: population-inversion amplifiers, parametric amplifiers and/or wavelength converters, beam splitters, lossy fibers, etc. They form a network with a simple directed graph (no loops). We identify a class of networks for which Friis’ formula for concatenated elements holds. We verify the general formula for a variety of well-known cases. We also verify that the NF for an optical communication link periodically amplified by nondegenerate phase-sensitive parametric amplifiers (PSAs) is only about half of that for a similar system using degenerate PSAs.