Unitary and singular value decompositions of parametric processes in fibers

摘要

Parametric (four-wave mixing) processes in fibers, driven by one or two strong pumps, couple the evolution of two weak sidebands. These processes are governed by the (spatial) evolution equations d_(z)X velence AX + BX~((dagger)t) and their associated input-output relations X(z) velence M(z)X(O) + N(z)X~((dagger)t)(0), where X is the (sideband) amplitude vector, A and B are coefficient matrices, and M and N are transfer matrices. In principle, one can use unitary decompositions (UDs) to facilitate the mathematical analyses of evolution equations, or singular value decompositions (SVDs) to facilitate the physical interpretations of input-output relations (by diagonalizing them). In these notes, the SVD method is reviewed, and applied to studies of modulation interaction (MI), phase conjugation (PC) and Bragg scattering (BS). The UD method is also reviewed. It works for BS driven by continuous-wave (CW) or pulsed pumps. Although it works for MI and PC driven by CW pumps, it does not work for MI and PC driven by pulsed pumps (because the coefficient matrices are not simultaneously diagonalizable). Another decomposition method, based on eigenvectors of an extended coefficient matrix, and their adjoints, works for CW and pulsed pumps. Further work is required, to clarify the relation between the SVD and adjoint methods, and apply the latter method to processes of current interest.