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IEEE Journal of Selected Topics in Quantum Electronics

Field Fluctuations Produced by Parametric Processes in Fibers

Colin J. McKinstrie, J. P. Gordon

Published Jun 29, 2011• 18 citations
Publisher

Abstract

In this tutorial, the optical field fluctuations produced by four-wave mixing processes in fibers are studied in detail. Modulation interaction and phase conjugation amplify input signals, whereas Bragg scattering frequency converts an input signal. If the input states are coherent, the quadrature fluctuations associated with the output states have Gaussian statistics. Formulas are derived for the output quadrature means, variances, and correlations, which are valid for arbitrary pump-induced coupling and dispersion- and pump-induced mismatch coefficients. These formulas allow one to determine the output Wigner distributions, which specify the output fields completely, and the effects of the aforementioned processes on the information carried by signals. The results of this paper are relevant to communication systems and quantum-information protocols based on continuous variables.

DOI
10.1109/jstqe.2011.2159704

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In this tutorial, the optical field fluctuations produced by four-wave mixing processes in fibers are studied in detail. Modulation interaction and phase conjugation amplify input signals, whereas Bragg scattering frequency converts an input signal. If the input states are coherent, the quadrature fluctuations associated with the output states have Gaussian statistics. Formulas are derived for the output quadrature means, variances, and correlations, which are valid for arbitrary pump-induced coupling and dispersion- and pump-induced mismatch coefficients. These formulas allow one to determine the output Wigner distributions, which specify the output fields completely, and the effects of the aforementioned processes on the information carried by signals. The results of this paper are relevant to communication systems and quantum-information protocols based on continuous variables.
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.
In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homo-dyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.
The noise properties of propagation through optical fiber are of interest in optical communications. In recent years much attention has been focused on reducing the quantum fluctuations in one of two complementary field variables below the limit associated with coherent states. Nonlinear optical processes have played a key role in the realization of such behavior. We numerically investigate the noise properties associated with competing four-wave mixing (FWM) processes in optical fiber from a classical perspective. The nonlinear interaction of two pump waves at frequencies ω1 and ω2, which produce sidebands at 2ω2–ω1 and 2ω1–ω2 by multiple FWM, is considered. By introducing known classical fluctuations at the fiber input, one can trace the evolution of the noise in the different frequency components of the FWM spectrum by generating histograms at different fiber lengths and for different input pump powers and frequency separations. Classical behavior resembling squeezing is observed under appropriate conditions.
Two sets of formulas are derived for the field-quadrature and photon-number fluctuations (variances and correlations) produced by parametric amplifiers and frequency convertors that are driven by pulsed pumps and act on pulsed signals. The first set is based on the Green functions for the underlying parametric processes, whereas the second is based on the associated Schmidt coefficients and modes. These formulas facilitate the modeling and performance optimization of parametric devices used in a wide variety of applications.
Near-degenerate four-wave parametric mixing in an optical fiber couples fluctuations in signal and idler frequency bands in such a way that the noise at a particular optical phase is reduced while that in an orthogonal phase quadrature is amplified. We have performed experiments on this effect and explained our results in terms of a theory which incorporates a nonlinear dispersion effect. The nonlinear dispersion alters the mathematical character of the solutions and limits the effective interaction length. For relatively low frequency shifts (<100 GHz) and 100 m of fiber, the heterodyne detected optical noise can be reduced by 5 dB in the squeezed quadrature. Theory predicts that this parametric interaction should reduce the noise level below the standard quantum limit. In a realistic fiber, phase noise due to guided acoustic wave Brillouin scattering (GAWBS) dominates the quanturn noise and cannot be reduced by four-wave mixing. Narrow gaps in the GAWBS noise can be engineered into special fibers. At the frequencies of these gaps, the noise level indeed can be squeezed below the quantum limit.