# Dependence of Dispersion on Metamaterial Structural Parameters and Dispersion Management
## 4 Light Pulse Propagation in the MM
### Impact of \(\beta_{2}\) and \(\beta_{3}\)  
#### 4.1.1 Impact of \(\beta_{2}\) at \(\overline{\omega}_{D}\)  
The zero dispersion frequency \(\overline{\omega}_{D}=~{}0.706844\) (with \(\beta_{2}=~{}0\)) in Figure 1d means the Gaussian pulse can propagate in the MM forever with no pulse spreading due to \(\beta_{2}\). This is a specific phenomenon in the MM and seems an ideal circumstance. From Figure 1e, there is \(\beta_{3}=2.0698\) ps\({}^{3}\)/km at \(\overline{\omega}_{D}\). This \(\beta_{3}\) is large so that the corresponding dispersion length is only \(L^{\prime}{}_{D}=60.39\) km. To have a fair idea of when and how \(\beta_{3}\) takes effect, we simulate the propagation of the Gaussian pulse in Figure 3.  
It is shown in Figure 3 that Gaussian pulse keeps its original waveform unchanged at \(z<21\) km. However, at around \(z=22\) km, the pulse starts forming a splitting peak under the influence of \(\beta_{3}\). Although the pulse splitting just starts around \(z=22\) km, it develops quickly and has become obvious, while the pulse just transmits one more kilometer. This illustrates that \(\beta_{3}\) starts to take its effect at a much earlier distance than \(L^{\prime}{}_{D}=60.39\) km, and when a pulse transmits 60.39 km, the impact of \(\beta_{3}\) should be very serious, so that the original pulse waveform may be completely ruined. We have pointed out in the former section that \(\beta_{3}\) in the MM is four orders of magnitude stronger than that in ordinary glass. It is such a super larger third-order dispersion that pulse splitting starts at a short distance. Thus, although an ideal circumstance of \(\beta_{2}~{}=~{}0\) predicts an infinite transmission, the strong \(\beta_{3}\) limits the practical transmission to a very short distance. Therefore, we need to alter our perspective by seeking an occurrence with the compensated \(\beta_{2}\) and very small \(\beta_{3}\). In the following two sections, we first discuss the situation nearby the zero-dispersion frequency \(\overline{\omega}_{D}\). Figure 3: Pulse waveforms at different distances when \(\beta_{2}=0\). 1.2 Impact of \(\beta_{2}\) and \(\beta_{3}\) when \(\overline{\omega}<\overline{\omega}_{D}\)  
When \(\overline{\omega}<\overline{\omega}_{D}\) in Figure 1d, there is \(\beta_{2}<0\). Two sets of data are listed in Table 1 with different \(|\beta_{2}|\) but little varied \(\beta_{3}\). In addition, \(\beta_{3}\) is chosen to be close to the data discussed in Section 4.1.1. Thus, \(L^{\prime}{}_{D}\) is closed to that in the last section. Figure 4 shows the waveform plot corresponding to the two cases in Table 1. Similarly, the pulse starts to split around \(z=22\) km, although \(L^{\prime}{}_{D}\) exceeds 60 km. In comparison to Figure 4a,b, we can see that for the second case in Table 1, where \(|\beta_{2}|\) is relatively larger, the secondary peak generated from the pulse splitting is a little far from the main peak and its peak intensity is relatively smaller. Thus, it may say that, to some extent, a larger \(|\beta_{2}|\) can inhibit the influence of pulse splitting. In addition, Figure 4 also show that the secondary peak appears at the front edge of the pulse if \(\beta_{2}~{}<0\). We also noted that the pulse splitting in Figure 4b is not so serious as that in Figure 3, although \(\beta_{3}\) is similar. Again, this is due to the inhabitation of \(\beta_{2}\) to the pulse splitting. 1.3 Impact of \(\beta_{2}\) and \(\beta_{3}\) when \(\overline{\omega}>\overline{\omega}_{D}\)  
When \(\overline{\omega}>\overline{\omega}_{D}\) in Figure 1d, there is \(\beta_{2}>0\). Two sets of data are listed in Table 2, with different \(\beta_{2}\) but little varied \(\beta_{3}\). Similarly, \(\beta_{3}\) is chosen to be close to the data discussed in Section 4.1.1. Figure 5 shows the waveform plot corresponding to Table 2. Similarly, the pulse starts to split around \(z=22\) km, although \(L^{\prime}{}_{D}\) exceeds 60 km.