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Journal of Lightwave Technology

Output moments and photon statistics in fiber raman amplification

R. W. Davies, Paul Melman, Walter Nelson, Mark L. Dakss, B. Foley

Published Jan 1, 1987• 5 citations
Publisher

Abstract

The problem of output moments, photon statistics, and signal-to-noise ratio in fiber Raman amplification is considered using a master equation approach, including absorption in the signal and pump modes. In the parametric approximation for the pump, results are obtained for the output moments which can be cast in a general form derived by Loudon and Shepherd [23] for phase-insensitive quantum amplifiers and attenuators. Beyond the parametric approximation, it is shown that the output moments of the signal may be obtained through iteration from lower order moments in an approximation where signal-pump correlation is ignored. Comparison with other work is discussed.

DOI
10.1109/jlt.1987.1075617

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The problem of output moments, photon statistics, and signal-to-noise ratio in fiber Raman amplification is considered using a master equation approach, including absorption in the signal and pump modes. In the parametric approximation for the pump, results are obtained for the output moments which can be cast in a general form derived by Loudon and Shepherd [23] for phase-insensitive quantum amplifiers and attenuators. Beyond the parametric approximation, it is shown that the output moments of the signal may be obtained through iteration from lower order moments in an approximation where signal-pump correlation is ignored. Comparison with other work is discussed.
We present a semi-classical approach for predicting the quantum noise properties of fiber optical parametric amplifiers. The unavoidable contributors of noise, vacuum fluctuations, loss-induced noise, and spontaneous Raman scattering, are included in the analysis of both phase-insensitive and phase-sensitive amplifiers. We show that the model agrees with earlier fully quantum approaches in the linear gain regime, whereas in the saturated gain regime, in which the classical equations are valid, we predict that the amplifier increases the signal-to-noise ratio by generating an amplitude-squeezed state of light. Also, in the same process, we analyze the quantum noise properties of the pump, which is difficult using standard quantum approaches, and we discover that the pump displays complicated dynamics in both the linear and the nonlinear gain regimes.
Using Caves's theory on the quantum-noise limit in an amplifier, we consider the quantum limit on noise in a Raman amplifier. We show explicitly that the full quantum equations for the Raman interaction can be transformed into the general form for a quantum amplifier. In addition, we show that in the steady-state limit the Raman amplifier becomes a phase-insensitive amplifier. At this limit we find that the output signal-to-noise ratio is unity when one photon per mode is input. Thus we conclude that the Raman amplifier, in the phase-insensitive limit, will operate at the quantum-noise limit.
The stagnating increase in data transmission capacity in optical communication systems combined with the ever growing demand of transmission bandwidth is leading to an impending capacity crunch, referring to the point in time after which the available bandwidth of the individual user starts to decrease. To postpone this point in time, existing technologies in terms of data transmission through optical fibers must be optimized and new degrees of freedom must be introduced to continue the exponential increase in available bandwidth; space-division multiplexing is believed to be the strongest candidate for another degree of freedom in transmission fibers. This thesis is two-fold: firstly, starting at Maxwell’s equations and basic principles of quantum mechanics, a semi-classical model of the noise properties of fiber optical parametric amplifiers and frequency converters is presented. The model accounts for multiple effects present in nonlinear fibers such as four-wave mixing, Raman scattering, distributed loss, and dispersion, and it is valid in the depleted pump regime. After validating the model against well-known results of quantum models, the model is used to predict the impacts of Raman noise, loss, and pump depletion on the noise properties of parametric frequency conversion and phase-insensitive and phase-sensitive parametric amplification. An important part of realizing space-division multiplexing is the ability of optical signal processing so the second part of this thesis addresses few-mode Raman fiber amplifiers and parametric amplifiers and frequency converters. A model of weak random linear mode coupling in the pump of a two-mode distributed Raman fiber amplifier is presented and it is shown that an amplification noise figure induced by mode coupling increases with the degree of mode coupling and that this tendency increases as the pump depletes. Also, a very low mode-dependent gain of 0.25 dB per 10 dB gain is experimentally demonstrated in a two-mode distributed Raman fiber amplifier by exciting the pump in a combination of two modes. A comprehensive model of four-wave mixing in two-mode fibers acvi counting for six simultaneous processes is derived, and the conversion efficiency from signal to idler in the four-wave mixing processes of phase conjugation and Bragg scattering in two two-mode fibers with different phase matching properties are experimentally investigated. A conversion efficiency of > −2.70 dB is demonstrated for Bragg scattering in the conversion of a signal in the LP01-mode to the idler in the LP11-mode; the signal-to-idler separation is ∼ 25 nm. Good qualitative agreement between experiments and theory is found for both processes in both fibers.
The concept of noise figure F and noise measure M applicable to radio frequency and microwave amplifiers is reviewed and extended to cover optical amplifiers. Two noise figures are defined in the optical regime: one gives the ASE output, while the other gives the ratio of signal-to-noise ratios at the input and output. In the quantum regime, the lower limit on these optimum values is set by a generalization of Heisenberg's uncertainty relation. A Raman back-pumped fiber link with post-amplification with an erbium-doped fiber amplifier is used to illustrate the use of noise measure.
Wavelength dependence of quantum noise in nondegenerate phase-sensitive amplification in an optical fiber, where signal and its phase-conjugated idler lights are incident into a fiber together with pump light, is theoretically investigated. An analytical formula of the quantum-limited noise figure, that takes Raman scattering into account as well as parametric amplification, is derived based on full quantum mechanical treatment. Using the obtained formula, a calculation example of the noise figure spectrum is also presented, where the pump light phase is adjusted so that the signal gain is maximum at a phase-matched wavelength, supposing multichannel amplification.