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Optics Express

Schmidt decompositions of parametric processes III: simultaneous amplification and conversion

Colin J. McKinstrie

Published Jun 19, 2015• 4 citations
Publisher

Abstract

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.

DOI
10.1364/oe.23.016949

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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.
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.
In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA† and A†A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistently carry out the SVD and the Schmidt decomposition including the phase factors.
A unified treatment of the description of the statistical dynamical properties of two-mode quadratic optical parametric processes with intense coherent or stochastic pumping (parametric amplification, frequency conversion, second subharmonic generation, etc. as special cases are involved) including the lossy mechanism is developed. The Heisenberg-Langevin equations are derived and their solutions are used to obtain the quantum characteristic functions and the corresponding quasidistributions. The Fokker-Planck equation approach is adopted as well and the equivalence of both the approaches is explicitly found.
We develop for the first time a quantum theory of simultaneous amplification and frequency up-conversion of optical images in two coupled three-wave processes. The theory developed takes into account the diffraction effect and the difference of the group velocities of the interacting waves. Due to the degenerate parametric process involved in the nonlinear optical interactions, the up-conversion also possesses phase sensitivity. As a result, the statistical characteristics of the amplified image and the converted one depend on the phase relation between the signal and the pump waves. It is shown that at the output of the nonlinear crystal the signal-to-noise ratios for the amplified image and the converted one may be same as those for the initial image, i.e. transformation of the optical image may be 'noiseless'.