Analytical method for four wave mixing in space-frequency multiplexing optical fibers

Abstract

In recent years, the transmission capacity of wavelength division multiplexing (WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing (SDM) has become one of the most concerned technologies. In this paper, the four-wave mixing process (FWM) in fibers is considered from the frequency domain to the mode division multiplexing (MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate four-wave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations, thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems. 3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the few-mode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation (DBP) algorithm, our algorithm has the advantage of lower complexity.