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Optics Express

Schmidt decompositions of parametric processes I: Basic theory and simple examples

Colin J. McKinstrie, Magnus Karlsson

Published Jan 14, 2013• 23 citations
Publisher

Abstract

Parametric devices based on four-wave mixing in fibers perform many signal-processing functions required by optical communication systems. In these devices, strong pumps drive weak signal and idler sidebands, which can have one or two polarization components, and one or many frequency components. The evolution of these components (modes) is governed by a system of coupled-mode equations. Schmidt decompositions of the associated transfer matrices determine the natural input and output mode vectors of such systems, and facilitate the optimization of device performance. In this paper, the basic properties of Schmidt decompositions are derived from first principles and are illustrated by two simple examples (one- and two-mode parametric amplification). In a forthcoming paper, several nontrivial examples relevant to current research (including four-mode parametric amplification) will be discussed.

DOI
10.1364/oe.21.001374

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Parametric devices based on four-wave mixing in fibers perform many signal-processing functions required by optical communication systems. In these devices, strong pumps drive weak signal and idler sidebands, which can have one or two polarization components, and one or many frequency components. The evolution of these components (modes) is governed by a system of coupled-mode equations. Schmidt decompositions of the associated transfer matrices determine the natural input and output mode vectors of such systems, and facilitate the optimization of device performance. In this paper, the basic properties of Schmidt decompositions are derived from first principles and are illustrated by two simple examples (one- and two-mode parametric amplification). In a forthcoming paper, several nontrivial examples relevant to current research (including four-mode parametric amplification) will be discussed.
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.
Simultaneous parametric amplification and frequency conversion is a three-mode parametric process. In this paper, the three-mode equations are solved, and the adjoint (spectral) and Schmidt (singular value) decompositions of the associated transfer matrix are determined. The properties and uses of both decompositions are described briefly. These three-mode results show that Schmidt decompositions are fundamentally different from adjoint decompositions, and illustrate other important features of multiple-mode parametric processes.
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.
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.
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.