The existing theoretical equations cannot provide an excellent guidance for developing four-wave mixing (FWM)-based optical logic devices, though the experiments have been done in several researches. The optimization of noise figure performances of such devices should be further investigated. In the paper, the universal analytic expressions for the amplitude and phase of the idler in degenerate or non-degenerate FWM process under pump depletion are derived in detail from the nonlinear coupled-mode equations for guiding optical waves propagation in highly nonlinear fiber. The universal analytic expressions are obtained by the first-and the third-kind of elliptic integrals. By using equivalent infinitesimal to calculate the limit of phase sensitive amplification, we find out the initial phase relationship between the idler and the input guided wave for phase-independent amplification, which is crucially important for explaining the operating principles of the FWM-based adder and subtracter. As an example, the configuration of non-degenerate FWM-based hybrid arithmetic device with three logic functions of A+B-C, A+C-B, and B+C-A for QPSK signals is presented, and then the noise transfer characteristics in terms of signal-to-noise ratio (SNR) and error vector magnitude (EVM) are taken into account by adjusting the fiber length, input wavelength, and optical power. The calculation results show as follows. 1) This kind of arithmetic device has a noise figure of about 1.1 dB and an input SNR of more than 24 dB is necessary for the symbol error rate of 10-3 without forward error correction, corresponding to an output EVM of 23.2%. 2) The length of highly nonlinear fiber used in the hybrid arithmetic device may be taken flexibly, provided that the variation of FWM conversion efficiency is controlled in a range of 1 dB relative to the maximum, with an EVM fluctuation of less than for the idlers. 3) The hybrid arithmetic device has an operating optical bandwidth of about 16 nm for the SNR degradation of 1.3 dB. 4) The output EVM increases with the increase of input power, and the allowable input power should be no more than 100 mW for an input SNR of 28 dB, noting that the larger the input SNR, the higher the allowable input power is.

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.