## 4 Analysis of the Modulational Instability
### The Anomalous-Dispersion Regime
#### 4.1.3 The Impact of Asymmetry Parameters  
The influence of the GVD difference, \(\alpha\), on the instability spectra is presented in Figure 3b. The limit case of the coupler with zero GVD in the cross channel, \(\alpha=0\), is included as well. As seen in the figure, the MI bandwidth of MI is infinite in the limit case. Both the gain and bandwidth of the MI monotonically decrease with the increase of \(\alpha\), with the MI vanishing in the limit of \(\alpha\rightarrow+\infty\). In other words, relatively weak anomalous GVD in the cross channel strongly affects MI bandwidth in the ANLDC. Figure 1: Modulational instability (MI) gain spectra for symmetric nonlinear directional couplers (SNLDC) and asymmetric (ANLDC) couplers in the anomalous group-velocity dispersion (GVD) regime (\(\sigma_{1}=1\)). Parameters of the symmetric system are \(P=\eta=\alpha=\Gamma=\kappa=1\) and \(\rho=\chi=0\). For the asymmetric ones, the parameters are the same, except for \(\alpha=0.1,\Gamma=2\), \(\chi=0.66\) and \(\rho=0.1\). The influence of the difference in effective mode areas of two cores (\(\Gamma\)) is illustrated by Figure 3c. In this case too, we start with the limit case of an extremely asymmetric coupler, in which the second core is purely dispersive, with zero nonlinearity (\(\Gamma=0\)). In this limit, the MI gain vanishes. The MI gain and bandwidth monotonically increase with the growth of \(\Gamma\). This dependence on \(\Gamma\) is opposite to that on \(\alpha\), which is displayed in Figure 3b. Thus, the MI can be effectively controlled by means of the two asymmetry parameters, \(\Gamma\) and \(\alpha\). Next, we study the effect of the group-velocity mismatch (walk-off between the cores), \(\rho\). Figure 4a shows the impact of \(\rho\) when the asymmetry is represented only by the GVD ratio, \(\alpha=0.1\), while the nonlinear coefficients in both cores are equal. The figure demonstrates that the variation of \(\rho\) in the range of \(\rho\lesssim 1\) weakly affects the MI spectrum. The effect is much stronger at larger values of the walk-off. In particular, the MI spectral band splits into two at \(\rho=2\). The latter effect seems interesting even if the value of \(\rho=2\) may be difficult to attain in real couplers. On the other hand, the analysis demonstrates that the variation of \(\rho\) produces almost no effect on the MI gain in the case when the asymmetry is determined by the difference in the nonlinearity coefficients (\(\Gamma\
eq 1\)), while the GVD coefficients are equal (\(\alpha=1\)). The latter result is not shown here in detail, as it does not display noteworthy features. Figure 3: The MI gain spectra in the asymmetric coupler with anomalous GVD (\(\sigma_{1}=1\)). **(a)** The results for different values of the normalized coupling coefficient, \(\kappa\). Parameters of the system are \(P=2,\eta=0.5,\alpha=0.1,\Gamma=2\) and \(\rho=0.01\). **(b)** For different values of the ratio of the GVD coefficients in the cross and bar channels, \(\alpha\), indicated near each curve with \(P=1,\eta=\kappa=0.5,\Gamma=1\) and \(\rho=0.01\). **(c)** For different values of the ratio of the nonlinearity coefficients in the two cores, \(\Gamma\), which are indicated near the curves. Other parameters are \(P=0.5,\eta=0.1,\kappa=0.2,\alpha=1\) and \(\rho=0.1\). Figure 2: Contour plots showing the dependence of the MI gain on the continuous wave (CW) power, \(P\), and perturbation frequency, \(\Omega\), for symmetric and asymmetric couplers in the anomalous-GVD regime (\(\sigma_{1}=1\)). Parameters of symmetric system **(a)** are \(\eta=\alpha=\Gamma=1\), \(\kappa=2\) and \(\rho=\chi=0\). For the asymmetric system **(b)**, \(\eta=\alpha=0.1\), \(\Gamma=2\), \(\kappa=1\), \(\rho=0.1\), and the phase-velocity mismatch is defined in terms of \(P\), in order to produce the largest gain: \(\chi=-4.95+0.985P\). Note the difference in horizontal scales between (a) and (b). It is obviously interesting as well to investigate the effect of the CW asymmetry ratio, \(\eta\) (see Equation (8)), on the MI.