## 5 Oblique Incidence on a Metamaterial: A Positive to Negative Index Refraction  
Consider the 2D motion of the H-polarized field given by \(\mathbf{H}=[0,0,H_{z}]^{T}\) and \(\mathbf{E}=[E_{x},E_{y},0]^{T}\). In the vacuum, the third component of the incident H field is given by  
\[H_{z}^{in}=B_{0}e^{i(k\xi_{i}\cdot\mathbf{x}-kct)}, \tag{30}\]  
where \(k\) is the wave number in vacuum (\(x<0\)), \(\mathbf{x}=(x,y)\), the incident wave propagation direction \(\xi_{i}=(\cos\theta_{i},\sin\theta_{i})\) and \(\theta_{i}\) is the angle of incidence between the incident wave and the normal vector \(\hat{n}=(1,0,0)\) of the interface (\(x=0\)). After the incident wave hits the interface (\(x=0\)), the reflected wave has the form of  
\[H_{z}^{rc}=-\mathcal{R}(ck)B_{0}e^{i(k\xi_{i}\cdot\mathbf{x}-kct)}, \tag{31}\]  
Figure 11: H field at the interface between the vacuum (\(x<0\)) and the metamaterial with negative index of refraction (\(x>0\)) from the bottom to the top for \(\omega_{p}t\) ranging from 5 to 9 in the steps of 2. Figure 10: Transmitted wave motion from the bottom to the top for \(\omega_{p}t\) ranging from 5 to 7 in the steps of 1. The phase velocity has the backward direction. where \(\mathcal{R}(ck)\) is the reflection coefficient, the reflected wave propagation direction  
\(\hat{\kappa_{r}}=(-\cos\theta_{r},\sin\theta_{r})\) and \(\theta_{r}=\theta_{i}\) is the reflective angle between the reflective wave and the interface normal. In the negative index metamaterial (\(x>0\)), the transmitted wave has the form of  
\[H_{z}^{tr}=\mathcal{T}(ck)\frac{B_{0}}{\mu_{eff}(ck)}e^{i(k_{m} \kappa_{t}\cdot\mathbf{x}-kct)}, \tag{32}\]  
where \(\mathcal{T}(ck)\) is the transmission coefficient, \(k_{m}\) is the wave number in the media, the propagation direction in the effective media \(\hat{\kappa_{t}}=(\cos\theta_{t},\sin\theta_{t})\), \(\theta_{t}\) is the refractive angle between the refractive wave and the interface normal and by the Snell's Law \(\sin\theta_{i}=n_{eff}\sin\theta_{t}\). From the Maxwell equation, at the interface (\(x=0\)), we have the transmission conditions  
\[[\hat{n}\times\mathbf{H}]=0\text{ at }x=0 \tag{33}\]  
and  
\[[\hat{n}\times\mathbf{E}]=0\text{ at }x=0. \tag{34}\]  
The transmission condition (33) shows  
\[H_{z}^{in}+H_{z}^{re}=H_{3}^{tr}\text{ at }x=0 \tag{35}\]  
which implies that  
\[B_{0}e^{i(k\hat{\kappa_{t}}\cdot\mathbf{x}-kct)}-\mathcal{R}(ck )B_{0}e^{i(k\hat{\kappa_{r}}\cdot\mathbf{x}-kct)}=\mathcal{T}(ck)\frac{B_{0}} {\mu_{eff}(ck)}e^{i(k_{m}\hat{\kappa_{t}}\cdot\mathbf{x}-kct)}\text{ at }x=0. \tag{36}\]  
Since \(k\sin\theta_{i}=k_{m}\sin\theta_{t}\) by Snell's Law, (36) becomes  
\[1-\mathcal{R}(ck)=\frac{\mathcal{T}(ck)}{\mu_{eff}(ck)}. \tag{37}\]  
We also have  
\[H_{z}^{rc}=-\mathcal{R}(ck)B_{0}e^{i(k\xi_{i}\cdot\mathbf{x}-kct)}, \tag{31}\]0  
So the transmission condition (34) shows  
\[H_{z}^{rc}=-\mathcal{R}(ck)B_{0}e^{i(k\xi_{i}\cdot\mathbf{x}-kct)}, \tag{31}\]1  
Here  
\[\epsilon(\mathbf{x})= \begin{cases}1&x<0,\\ \epsilon_{eff}(ck)&x>0.\end{cases} \]  
Then we have  
\[\left.\partial_{x}(H_{z}^{in}+H_{z}^{re})\right|_{x=0}=\epsilon_{ eff}^{-1}\left.\partial_{x}H_{z}^{tr}\right|_{x_{1}=0} \tag{40}\]which implies  
\[\left(1+\mathcal{R}(ck)\right)k\cos\theta_{i}=\frac{\mathcal{T}(ck)e_{ eff}^{-1}(ck)}{\mu_{eff}(ck)}k_{m}\cos\theta_{i}. \tag{41}\]  
The linear system (37) and (41) has the solution  
\[\mathcal{R}(ck)=\frac{\frac{k_{m}}{k}\cos\theta_{i}-\epsilon_{ eff}\cos\theta_{i}}{\frac{k_{m}}{k}\cos\theta_{i}+\epsilon_{eff}\cos\theta_{i}} \tag{42}\]  
and  
\[\mathcal{T}(ck)=\frac{2\mu_{eff}(ck)\epsilon_{eff}\cos\theta_{i}}{ \frac{k_{m}}{k}\cos\theta_{i}+\epsilon_{eff}\cos\theta_{i}} \tag{43}\]  
Collecting results we have  
**Proposition 2**. : **Waveforms in vacuum and effective material in 2D.