# Dependence of Dispersion on Metamaterial Structural Parameters and Dispersion Management
## 5 Dispersion Management in MMs
### Dispersion Management  
Dispersion compensation is for the circumstance in which \(\beta_{2}\) and \(\beta_{3}\) can both be compensated or \(\beta_{2}\) is compensated, but \(\beta_{3}\) is very small. According to the dispersion compensation theory, to reach the compensation for both \(\beta_{2}\) and \(\beta_{3}\), there should be the following [19]:  
\[\beta_{21}L_{1}+\beta_{22}L_{2}=0\qquad\beta_{31}L_{1}+\beta_{32}L_{2}=0 \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\overline{\omega}\) & \(\overline{\omega}_{p}\) & \(\omega_{pm}\times 10^{16}(Hz)\) & \(\beta_{2}(ps^{2}/km)\) & \(\beta_{3}(ps^{3}/km)\) & \(\beta_{4}(ps^{4}/km)\) & \(L_{D}(km)\) & \(L^{\prime}_{D}(km)\) \\ \hline
0.8794313 & 1.277639 & 3.061222 & 0.002528 & 0.431002 & 0.0003325 & 9889.982 & 290.0218 \\
0.8794314 & 1.277637 & 3.0612173 & 0.000674 & 0.431001 & 0.0003325 & 37111.35 & 290.0227 \\
0.8794315 & 1.277635 & 3.0612125 & \(-\)0.00118 & 0.430999 & 0.0003325 & 21177.19 & 290.0237 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Examples for the lowest second- and third-order dispersions with \(\omega_{pe}=2.396\times 10^{16}\)Hz.  
Figure 8: Dependence of refractive index, third-order nonlinear coefficient, first-, second-, third-, and forth-order dispersion on \(\overline{\omega}\) and \(\overline{\omega}_{p}\), with \(\omega_{pe}=2.396\times 10^{16}\)Hz.  
Figure 9: (a) Gaussian pulse’s 90 km propagation; (b) top view of (a).  
where \(L=L_{1}+L_{2}\) is dispersion period, \(\beta_{2j}\) and \(\beta_{3j}\) (\(j=1,2\)) are second- and third-order dispersion for the MM with length \(L_{j}\).  
Figure 8d shows that \(\beta_{2}\) can be positive and negative, so there is a possibility to reach \(\beta_{2}\) compensation. However, in Figure 8e, \(\beta_{3}\) is always positive, meaning that we cannot find a negative valued \(\beta_{3}\) to reach its compensation. Luckily, for the same normalized frequencies we found two sets of data in Table 5 with an opposite notation and very closed absolute values of \(\beta_{2}\) for compensation. Additionally, both sets of data all have very low \(\beta_{3}\), resulting in \(L^{\prime}D\) as long as 380 km. That is to say, while the broadening due to the GVD is compensated, the effect due to the third-order dispersion has not yet emerged. Thus, we propose a configuration with metamaterials MM1 and MM2, cross-linked periodically with their length \(L_{1}=L_{2}=20\) km. The simulation results for the propagation of the Gaussian pulse in the composite metamaterial MM1 and MM2 is demonstrated in Figure 10.  
Figure 10 shows that when the pulse transmits in MM1, its width spreads with the increase of the propagation distance, and the pulse intensity lowers because of the pulse broadening. At the propagation distance \(L_{1}=20\) km, the pulse intensity reduces 20%. At this time, if we let the pulse enter into MM2 and it continually transmits inside MM2, the reduced intensity starts to be recovered and the broadened width gradually shrinks. At \(Z=L_{1}+L_{2}=20+20=40\) km, the intensity and the width have both been recovered to be equivalent to their initial values. The situation afterwards repeats the first MM1 and MM2 period, until it transmits \(Z=120\) km. While the pulse transmits one MM1 and MM2 period, the impact of \(\beta_{3}\) remains uncompensated and accumulates with the increment of the transmission distance. The simulation shows that while the propagation distance approaches 140 km, the pulse splitting due to \(\beta_{3}\) starts to emerge. Therefore, the MM1 and MM2 cross-linked structure allows a 120 km repeaterless transmission.