# Dependence of Dispersion on Metamaterial Structural Parameters and Dispersion Management
## 3 Dispersion in a Regularly Exampled MM  
We plot the refractive index, dispersion coefficients, and third-order nonlinear coefficient in Figure 1 based on Equations (6) and (7). Although the MM structural parameters \(\omega_{pe}\) and \(\omega_{pm}\) can be tuned, we employ the regularly exampled parameters with third-order electric susceptibility \(\chi^{(3)}\) as \(1.9\times 10^{-9}\) W\({}^{-1}\)[20], \(\omega_{pe}\) as \(1.3673\times 10^{16}\) Hz [21], and \(\overline{\omega}_{p}\) as 0.8 [22].  
In Figure 1a, the normalized frequency can be divided into three domains, with \(0<\overline{\omega}<0.8\) as the negative refractive index domain and \(\overline{\omega}>1.0\) as the positive refractive index domain, respectively. The intervening region \(0.8\leq\overline{\omega}\leq 1.0\) is the so called anomalous absorptive domain in the MM, in which the electromagnetic wave cannot propagate. It should be mentioned that (1) in Figure 1d \(\beta_{2}\) increases from negative to positive in the negative refractive index region, with its zero point at \(\overline{\omega}_{D}=0.706844\); (2) in Figure 1e \(\beta_{3}\) is always greater than zero in both the positive and negative refractive index regions, and becomes very smooth while it is far away from the anomalous absorptive domain; (3) in Figure 1f \(\beta_{4}\) is always positive in the negative refractive index region.  
Let's compare Figure 1 to the traditional media listed in Figure 2. Figure 2a,c, obtained from Thorlabs website [4,5], show the \(\beta_{2}\) and \(\beta_{3}\) of four conventional glass as a function of wavelength. In order to compare Figure 2a,c with Figure 1d,e, we need to unify the different units of abscissa in these figures. We thereby obtain Figure 2b,d by redrawing Figure 2a,c with modified abscissa and ordinate units. From Figure 2b it can be seen that \(\beta_{2}\) of these conventional glass are all very small, with the order of magnitude at only \(10^{-1}\). However, \(\beta_{2}\) in Figure 1d is \(10^{4}\) nearby \(\overline{\omega}_{D}\). It can be estimated if extending the \(\beta_{2}\) curves in Figure 2b to \(\overline{\omega}_{D}\), following the current trend, the absolute value \(|\beta_{2}|\) should still be smaller than that in Figure 1d. Thus, we may say that \(|\beta_{2}|\) of the MM, with the regularly used \(\omega_{pe}\) and \(\omega_{pm}\) is much larger than that in the conventional glass. Similarly, from Figure 2d, it can be seen that \(\beta_{3}\) of these conventional glass are all very small, with the order of magnitude at only \(10^{-4}\). However, \(\beta_{3}\) in Figure 1e is \(10^{0}\), nearby \(\overline{\omega}_{D}\). It can be estimated that, if extending the \(\beta_{3}\) curves in Figure 2d to \(\overline{\omega}_{D}\), following the current trend, the absolute value \(|\beta_{3}|\) should still be smaller than that in Figure 1e. Therefore, we may say that \(|\beta_{3}|\) of the MM with the regularly used \(\omega_{pe}\) and \(\omega_{pm}\) is much stronger than that in the conventional glass. What is the impact of such high dispersions in the MM? We find out these effects by studying the light pulse propagation in the MM.  
Figure 1: Variations of refractive index, third-order nonlinear coefficient, first-, second-, third-, and forth-order dispersion on \(\overline{\omega}\). Insets are the magnification nearby \(\overline{\omega}=0.7\). (**a**) Refractive index versus \(\overline{\omega}\); (**b**) third-order nonlinear coefficient versus \(\overline{\omega}\); (**c**) first-order dispersion versus \(\overline{\omega}\); (**d**) second-order dispersion versus \(\overline{\omega}\); (**e**) third-order dispersion versus \(\overline{\omega}\); and (**f**) forth-order dispersion versus \(\overline{\omega}\).  
Figure 2: (**a,c**) Second- and third-order dispersions in several conventional glass verse wavelength; (**b,d**) redrawing (**a,c**) to the normalized frequency.