# Metamaterial Based Design of Compact UWB/MIMO Monopoles Antenna with Characteristic Mode Analysis
## 2 UWB-MIMO with Characteristic Modes Theory and Metamaterial Structure
### Metamaterial Structure  
Figure 1 illustrates geometry of novel rectangular loop resonator designed on FR4 substrate with a relative permittivity of 4.3, thickness 1.6 mm. The unit cell metamaterial geometry posses an overall size of 3 mm \(\times\) 4.8 mm. The structure is designed based on the analysis called floquet mode [34]. In Figure 1b above, the surface current \(\vec{J}\) is mathematically defined as in the following equation:  
\[\vec{J}\to f(x,y). \tag{8}\]  
Therefore, the \(x\) components of the surface current can be represented as [35]:  
\[\vec{J}_{x}(x,y)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}f(x-m_{a},y -n_{b}), \tag{9}\]  
where \(m\) is the number of unit cell on x-axis, and a is the dimension, while \(n\) is the number of unit cell on y-axis, and \(b\) is the dimension. Equation (9) can also be presented in phase radians as:  
\[\vec{J}_{x}(x,y)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}f(x-m_{a}, y-n_{b})e^{(-jk_{a}m_{a}-jk_{b}m_{b})}. \tag{10}\]  
\(K_{x0}\) and \(k_{y0}\) are the phase radians of x and y axis and can be defined as in Equations (11) and (12):  
\[kx_{o}=k_{o}sin\theta_{o}cos\theta_{o}, \tag{11}\]  
\[ky_{o}=k_{o}sin\theta_{o}sin\theta_{o}. \tag{12}\]  
The unit cell will form a beam in a direction of (\(\theta_{o}\),\(\phi_{0}\)), considering the phase radians along the infinite array. Taking the Fourier transform of the surface current of Equation (10), we have:  
\[\vec{J}_{x}(x,y)=\frac{4\pi^{2}}{ab}\vec{f}(k_{x},k_{y})\sum_{m=-\infty}^{ \infty}\sum_{n=-\infty}^{\infty}\delta(k_{x}-k_{xmn})\delta(k_{y}-k_{ymn}), \tag{13}\]  
\[k_{xmn}=k_{x0}+\frac{2m\pi}{a}, \tag{14}\]  
\[k_{ymn}=k_{y0}+\frac{2m\pi}{b}, \tag{15}\]  
Figure 1: (**a**) Unit cell metamaterial. (**b**) Decomposition of surface current. where \(k_{x}\),\(k_{y}\) represent the phase radians on x and y axis, \(K_{ymn}\), \(K_{xmm}\) shows the phase radians for the incident wave is from port 1 and port 2, and \(a\), \(b\) is the dimension of the unit cell. Transforming the Equation (13) in to time domain, we have:  
\[\vec{J}_{x}(x,y)=\frac{4\pi^{2}}{ab}\vec{f}(x_{y}k_{y})\sum_{m=-\infty}^{ \infty}\sum_{n=-\infty}^{\infty}e^{-}(-jk_{xmm}x-jk_{yny}), \tag{16}\]  
where \(\vec{J}_{x}(x,y)\) is the Fourier transform in time domain, and \(a\) and \(b\) remain the dimensions of the unit cell. From port 1 to port 2, as indicated in Figure 1a, we inject floquet mode and determine a general scattering metric (GSM) for the system:  
\[ \begin{bmatrix}c_{mn}^{TE}\\ c_{mn}^{TM}\\ d_{mn}^{TE}\\ d_{mn}^{TM}\end{bmatrix} = \begin{bmatrix}GSM\\ GSM\\ GSM\\ GSM\\ GSM\end{bmatrix}\begin{bmatrix}a_{mn}^{TE}\\ a_{mn}^{TM}\\ b_{mn}^{TM}\\ b_{mn}^{TM}\end{bmatrix} , \tag{17}\]  
where \(a_{mn}^{TE}\) is the incident wave for transverse electric mode from port 1, \(a_{mn}^{TM}\) is the incident wave for transverse magnetic mode from port 1, \(b_{mn}^{TE}\) is the incident wave for transverse electric mode from port 2, \(c_{mn}^{TE}\) is the scattered transverse electric mode from port 1, \(d_{mn}^{TE}\) is the scattered transverse electric mode from port 2, and \(d_{mn}^{TM}\) is the scattered transverse magnetic mode from port 2. \[ \begin{bmatrix}c_{mn}^{TE}\\ c_{mn}^{TM}\\ d_{mn}^{TE}\\ d_{mn}^{TM}\end{bmatrix} = \begin{bmatrix}S_{11}^{TE,TE}&S_{11}^{TE,TM}&S_{12}^{TE,TE}&S_{12}^{TE,TM}\\ S_{11}^{TM,TE}&S_{11}^{TM,TM}&S_{12}^{TM,TM}&S_{12}^{TM,TM}\\ S_{12}^{TE,TE}&S_{12}^{TE,TM}&S_{12}^{TE,TE}&S_{12}^{TE,TE}&S_{12}^{TE,TM}\\ S_{21}^{TM,TE}&S_{21}^{TM,TM}&S_{22}^{TM,TE}&S_{22}^{TM,TM}\\ b_{mn}^{TM}\end{bmatrix} , \tag{18}\]  
where \(a_{mn}\) is the incident wave from port 1, \(b_{mn}\) is the incident wave from port 2, \(c_{mn}\) is the scattered wave from port 1, and \(d_{mn}\) is the scattered wave from port 2. After placing the ground plane on port two, we will have only port one, and Equation (18) will be as in [36]:  
\[ \begin{bmatrix}c_{mn}^{TE}\\ c_{mn}^{TE}\\ c_{mn}^{TE}\end{bmatrix} =