# Dependence of Dispersion on Metamaterial Structural Parameters and Dispersion Management
## 5 Dispersion Management in MMs  
### Design of the Optimal Dispersion  
From the above analyses, it is realized that the second- and third-order dispersions are all serious for that particular MM, which is often exampled in the literature with \(\omega_{pe}=1.3673\times 10^{16}\)Hz and  
Figure 5: The pulse waveform comparison at different propagation distances. (**a**) The first case in Table 2; and (**b**) the second case in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\overline{\omega}\) & \(\beta_{2}\)(ps\({}^{2}\)/km) & \(\beta_{3}\)(ps\({}^{3}\)/km) & \(\beta_{4}\)(ps\({}^{4}\)/km) & \(L_{D}\)(km) & \(L^{\prime}_{D}\)(\(km\)) \\ \hline
0.706587 & \(-7.2648\) & 2.0569 & 0.0037 & 3.44 & 60.77 \\
0.70710 & 7.2532 & 2.0828 & 0.0037 & 3.45 & 60.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Two sets of dispersion data including \(\beta_{4}\). Figure 6: Magnification of the partial propagation waveform at FWHM for \(z=21\) km. (**a**) The first case in Table 3; and (**b**) the second case in Table 3.  
\(\overline{\omega}_{p}=0.8\)[21; 22]. This makes the propagation with \(z>22\) km impossible. However, MMs are such extraordinary materials that derive their properties not only from the properties of the base materials, but from their newly artificially designed structures. Their precise shape, geometry, size, orientation, and arrangement give them their smart properties capable of manipulating electromagnetic waves to achieve benefits that go beyond what is possible with conventional materials. Therefore, we expand on our views to other MMs with different \(\omega_{pe}\) and \(\omega_{pm}\) (i.e., \(\overline{\omega}_{p}\)), and take insight into the dependence of dispersion to the structural parameters \(\omega_{pe}\) and \(\omega_{pm}\). To obtain small second- and third-order dispersions for a reasonably long transmission, we present here the design of an MM by using the nano-inclusions model [23], made of properly arranged collections of plasmonic metallic nano-rings. In our design, the nano-rings are made of a typical metal like aluminum, with its electrical plasma frequency described by \(\omega_{pe}=\sqrt{Nq_{e}^{2}/(m\epsilon_{0})}\), where \(N\) is the density of electrons, \(q_{e}\) is the electron charge, \(m\) is the electron mass, and \(\epsilon_{0}\) is the dielectric permittivity in a vacuum [24]. For aluminum, the number of electrons per cubic meter can be \(N=1.806\times 10^{29}\)m\({}^{-3}\)[16], then, there is \(\omega_{pe}=2.396\times 10^{16}\)Hz. When such aluminum nano-rings are embedded in a host medium, they may provide metamaterials with negative effective permeability at optical frequencies. According to the nano-inclusions model by the authors of [23], we calculated the effective permeability for the periodic arrangement of the aluminum nano-rings. The resonant magnetic dipole collective response of the effective permeability is presented in Figure 7, with \(R\) = 40 nm, \(a\) = 16 nm, \(N\) = 6, and \(N_{d}\) = (108 nm)\({}^{-3}\). Following [25; 26], the damping frequency and high frequency dielectric constant of aluminum have been assumed to follow the Drude model with \(\omega_{\tau}=194\) THz and \(\epsilon_{\infty}=1.65\epsilon_{0}\). The background material in this design is \(\varepsilon_{b}=1.2\epsilon_{0}\). With the effective magnetic permeability, the magnetic plasma frequency \(\omega_{pm}\) can be obtained following the Drude model. Using Equation (6), we also obtained a group of 3D plots in Figure 8 for \(0.1\leq\overline{\omega}\leq 0.9\) and \(0.1\leq\overline{\omega}_{p}\leq 2\), with the fixed \(\omega_{pe}=2.396\times 10^{16}\)Hz. According to the definition, the range of \(\overline{\omega}_{p}\) here, gives out the range of \(\omega_{pm}\). We search in Figure 8 for all of the values of \(\omega_{pm}\) gained from our above design to check up their dispersion parameters, and hope to find the potential figures with lower dispersions. We pick out all points with approximately zero \(\beta_{2}\) and very low \(\beta_{3}\).