# Dependence of Dispersion on Metamaterial Structural Parameters and Dispersion Management
## 2 Theory of Light Pulse Propagations in MMs  
Light is a type of electromagnetic wave and its propagation in MMs obeys [PERSON]'s equations. The wave equation for the light pulse propagating in nonlinear dispersive MMs can then be deduced as follows:  
\[\frac{\partial^{2}E}{\partial z^{2}}=\mu_{0}\varepsilon_{0}\frac{\partial^{2} E}{\partial t^{2}}+\mu_{0}\frac{\partial^{2}P_{NL}}{\partial t^{2}}. \tag{1}\]  
It is assumed that \(E=1/2A(z,t)\exp[i(\beta_{0}z-\omega t)]+c.c\) and \(P_{NL}=\varepsilon_{0}\chi^{(3)}|E(z,t)|^{2}E(z,t)\) with \(\chi^{(3)}\) the third-order electric susceptibility. It is known that the relative dielectric constant \(\varepsilon_{r}\) and relative permittivity \(\mu_{r}\) in a MM have to be dispersive, otherwise the energy density could be negative [15]. Their frequency dispersion can be described by a lossy Drude model [16; 17; 18], as follows:  
\[\varepsilon_{r}=1-\frac{{\omega_{pe}}^{2}}{\omega(\omega+i\gamma_{e})}\ \ \ \ \ \mu_{r}=1-\frac{{\omega_{pm}}^{2}}{\omega(\omega+i\gamma_{m})}, \tag{2}\]  
where \(\omega\) is frequency, \(\omega_{pe}\) and \(\omega_{pm}\) are the respective electric and magnetic plasma frequencies. \(\gamma_{e}\) and \(\gamma_{m}\) are the respective electric and magnetic loss terms, which are very small. The loss appearing in the imaginary part of the Drude model has much of its effect on the light intensity only. In our model of dispersion compensation, the reduced intensity because of loss makes the waveform difficult for one to judge if the dispersion is fully compensated. Thereby, in the following analysis, we ignore loss for simplification. The negative refraction behavior is restricted within a certain range of frequency values. We can transform Equation (1) into the frequency space in order to expand \(\varepsilon_{r}(\omega)\) and \(\mu_{r}(\omega)\) in powers of \(\omega\), thus enabling us to treat the material parameters as a power series, which we can truncate to an appropriate order. Taking the inverse Fourier transform, we also ignore the second derivative of \(z\) and introduce the group velocity reference frame \(T=t-z/V_{g}\equiv t-\beta_{1}z\). Then, we get the propagation equation for light pulse in MMs as follows:  
\[\frac{\partial A}{\partial z}=-\frac{i}{2}\beta_{2}\frac{\partial^{2}A}{ \partial T^{2}}+\frac{1}{6}\beta_{3}\frac{\partial^{3}A}{\partial T^{3}}-\frac {i}{24}\beta_{4}\frac{\partial^{4}A}{\partial T^{4}}+i\Gamma\bigg{|}A\bigg{|} 2A. \tag{3}\]  
The third-order nonlinear self-steepening effect that modulates the front edge and trailing edge of a Gaussian pulse in Equation (3) has been ignored, since what we study in the paper is the variation of the pulse shape due to higher order dispersion. However, the self-phase modulation effect remains. The third-order nonlinear self-phase modulation coefficient \(\Gamma\) and \(i\)th order dispersion \(\beta_{i}\) in Equation (3) can be expressed as follows:  
\[\beta_{i}=\frac{d^{i}k}{d\omega^{i}}|_{\omega=\omega_{0}}\ \ \ \ \ \ (i=1,2,3 \ldots\ldots), \tag{4}\]\[\Gamma=\frac{\varepsilon_{0}\mu_{0}\chi^{(3)}\omega^{2}\mu_{r}(\omega)}{2\beta_{0}}, \tag{5}\]  
with \(k\) asthe wave number.