2.2 The general theory on cross-phase modulation (XPM) intensity fluctuation

For the general case of two channels, the input optical powers are denoted by P tð Þ, P<sup>0</sup> ð Þt , respectively [28]. Only in the first walk-off length, the nonlinear interaction (XPM) is taken into account; in the remaining fibers, signals are propagated linearly along the fibers, and dispersion acts on the phase-modulated signal resulting in intensity fluctuation. According to [4], the whole length L is separated into two parts 0 < z < Lwo and Lwo < z < L; Lwo is the walk-off length, Lwo ¼ Δt=ð Þ DΔλ . Δt is the edge duration of the carrier wave, D is the dispersion coefficient, and Δλ is the wavelength spacing between the channels. By the smallsignal analysis, the phase modulation in channel 1 originating in dz at z can be expressed as

$$d\phi\_{\rm XPM}(z,t) = \gamma 2P'(z,t-z\beta\_1')e^{-ax}dz\tag{12}$$

This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from z to the receiver. So, at the fiber output, the intensity fluctuation originating in dz in the frequency domain is given by [29].

$$dP\_{\rm{XPM}}(\mathbf{z},\omega) = 2\left[e^{i\alpha\mathbf{z}\boldsymbol{\theta}\_{1}}P(\mathbf{z},\omega)\right]\otimes\left\{e^{-a(L-\boldsymbol{z})}\cdot e^{i\alpha\boldsymbol{\theta}\_{1}(L-\boldsymbol{z})}\sin\left[b(L-\boldsymbol{z})\right]d\boldsymbol{\uprho}\_{\rm{XPM}}(\mathbf{z},\omega)\right\}
$$

$$= 4\gamma\left[e^{i\alpha\boldsymbol{\theta}\_{1}}P(\mathbf{z},\omega)\right]\otimes\left\{e^{-a(L-\boldsymbol{z})}\cdot e^{-\alpha\mathbf{z}}\cdot e^{i\alpha\boldsymbol{\theta}\_{1}^{\prime}\boldsymbol{x}}\cdot e^{i\alpha\boldsymbol{\theta}\_{1}(L-\boldsymbol{z})}P'(\mathbf{z},\omega)\sin\left[b(L-\boldsymbol{z})\right]\right\}d\mathbf{z}\tag{13}$$

<sup>⊗</sup> representing the convolution operation <sup>b</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>Dλ<sup>2</sup> =ð Þ 4πc , where c is the speed of light. At the fiber output, the XPM-induced intensity fluctuation is the integral of Eq. (13) with z ranging from 0 to L:

$$\begin{aligned} \label{eq:SDIM-1} P\_{\text{XPM}} &= \int\_0^L dP\_{\text{XPM}}(z, \omega) dz \\ &= \int\_0^L 4\gamma \left[ \epsilon^{i \text{ou} \beta\_1} P(z, \omega) \right] \otimes \left\{ \epsilon^{-a(L-x)} \cdot \epsilon^{-ax} \cdot \epsilon^{i \text{ou} \beta\_1' x} \cdot \epsilon^{i \text{ou} \beta\_1 (L-x)} P'(z, \omega) \sin \left[ b(L-x) \right] \right\} dx \end{aligned} \tag{14}$$

The walk-off between co-propagating waves is regulated by the convolution operation.
