## 4 Normal Incidence on a Half Space of Metamaterial
The 2 component of the \(H\) field is the only nonzero component of the \(H\) field wave form in the metamaterial \((x>0)\) and is_  
\[H_{eff}(k_{0},x,t) = \int_{-\infty}^{\infty}A(k)\mathcal{T}(ck)\frac{B_{0}}{\mu_{eff}( ck)}e^{i(k_{m}(k)x-kct)}\,dk. \tag{19}\]  
\(\bullet\) _Here \(H_{\text{vac}}(k_{0},x,t)=H_{eff}(k_{0},x,t)\) at the interface \(x=0\)_  
_The \(\hat{y}\) component of the \(E\) field is the only nonzero component of the \(E\) field in vacuum \((x<0)\) and is_  
\[E_{\text{vac}}(k_{0},x,t) = \int_{-\infty}^{\infty}A(k)B_{0}\frac{d}{dx}\big{[}e^{i(kx-kct)}- \mathcal{R}(ck)e^{i(-kx-kct)}\big{]}\,dk. \tag{20}\]  
_The \(\hat{y}\) component of the \(E\) field is the only nonzero component of the \(E\) field in the metamaterial \((x>0)\) and is_  
\[E_{eff}(k_{0},x,t) = \int_{-\infty}^{\infty}A(k)\epsilon_{eff}(ck)\mathcal{T}(ck)\frac {B_{0}}{\mu_{eff}(ck)}\frac{d}{dx}\big{[}e^{i(k_{m}(k)x-kct)}\big{]}\,dk.\]  
\(\bullet\) _Here \(E_{\text{vac}}(k_{0},x,t)=E_{eff}(k_{0},x,t)\) at the interface \(x=0\)__where_  
\[\mathcal{R}(ck)=\frac{1-\epsilon_{eff}(ck)\frac{k}{k_{m}(ck)}}{1+\epsilon_{eff}(ck )\frac{k}{k_{m}(ck)}} \tag{21}\]  
_and_  
\[\mathcal{T}(ck)=\frac{2\mu_{eff}(ck)\epsilon_{eff}(ck)}{k}\frac{k_{m}(ck)}{k}+ \epsilon_{eff}(ck)\]  
Now we introduce \(n_{eff}^{2}(ck)=\epsilon_{eff}(ck)\mu_{eff}(ck)\) and \(k_{m}/k=n_{eff}(ck)\) and we can write  
\[\mathcal{R}(ck)=\frac{1-\frac{\epsilon_{eff}(ck)}{n_{eff}(ck)}}{1+\frac{ \epsilon_{eff}(ck)}{n_{eff}(ck)}} \tag{22}\]  
and  
\[\mathcal{T}(ck)=\frac{2n_{eff}^{2}(ck)}{1+\frac{\epsilon_{eff}(ck)}{n_{eff}(ck )}}\]  
Note that \(\mathcal{R}(ck)<1\) if \(n_{eff}<0\) and \(\epsilon_{eff}<0\). In our paper, we consider a signal made of a group of waves with frequencies lying inside a prescribed band of frequencies for which  
\[A(k)=\frac{a}{\sqrt{2\pi}}e^{-\frac{(k-k_{0})^{2}a^{2}}{2}}\hat{\chi}(\omega) \tag{23}\]  
where  
\[\hat{\chi}(\omega)= \begin{cases}1&\text{ for }\omega_{L}\leq\omega\leq \omega_{U},\\ 0&\text{ otherwise }.\end{cases} \]  
In order to make the H-field dimensionless, we can use \(\omega_{p}t\) to denote the time change, and \(\frac{\omega_{p}x}{c}\) the position change. Let \(\bar{\omega}=k_{0}c\) be the center frequency. Then the transmitted wave packet (19) can be written as  
\[\begin{array}{rl}H_{eff}&=\frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-\frac{(\frac{k_{0}}{k_{m}}-\frac{k_{0}^{2}}{k_{m}^{2}})^{ 2}(\frac{\omega_{p}a}{c})^{2}}{2}}\hat{\chi}(\omega/\omega_{p})\mathcal{T}( \omega/\omega_{p})\frac{B_{0}}{\mu_{eff}(\omega/\omega_{p})}e^{i\left(\frac{k \omega_{p}x}{c}-\frac{k_{0}}{k_{m}}(\omega_{p}t)\right)}\,d(\frac{k_{0}}{ \omega_{p}})\\ &=\frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{( \frac{\omega_{p}}{k_{m}}-\frac{\bar{\  
Let \(F(\omega/\omega_{p})=e^{-\frac{(\frac{\omega_{p}}{\omega_{p}}-\frac{\omega_{p}}{ \omega_{p}})^{2}(\frac{\omega_{p}p}{\epsilon})^{2}}{2}}\frac{\mathcal{T}(\omega /\omega_{p})}{\mu_{eff}(\omega/\omega_{p})}e^{i[(n_{eff}\frac{\omega}{\omega_{ p}})(\frac{\omega_{p}p}{\epsilon})-(\frac{\omega}{\omega_{p}})(\omega_{p}p)]}\). Then we have  
\[H_{eff} = \frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty }\hat{\chi}(\omega/\omega_{p})B_{0}F(\omega/\omega_{p})\,d(\frac{\omega}{ \omega_{p}})\] \[= \frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\int_{0.735}^{0.74}B_{ 0}F(\omega/\omega_{p})\,d(\frac{\omega}{\omega_{p}})\] \[\approx \frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\Delta(\frac{\omega}{ \omega_{p}})B_{0}[F(0.735)+F(0.74)],\]  
where \(\frac{\omega_{p}a}{c}\frac{1}{\sqrt{2\pi}}\Delta(\frac{\omega}{\omega_{p}})B_ {0}\) is a constant. Figures 9 and 10 are the graphs for \(F(0.735)+F(0.74)\) for different \(\omega_{p}t\), i.e., the transmitted wave motion in metamaterial (\(x>0\)). Figure 9 shows transmitted wave motion from the bottom to the top for \(\omega_{p}t\) ranging from 5 to 9 in the steps of 2. It can be observed that the group velocity of the transmitted wave in metamaterial has the forward direction (the envelope moves forward).