## 2 Non-FO Smart Materials: Metamaterials
### Metamaterial Designs
#### 2.2.3 Brief Background of EM Surface Waves  
In general, the plane waves are eigen modes of propagating EM waves in bulk, linear and homogeneous materials. The fields \(\overset{\rightarrow}{E}\) and \(\overset{\rightarrow}{H}\) are phasor vectors of a plane wave in an isotropic dielectric material (such as air), which can be written as Ref. [103].  
\[\overset{\rightarrow}{E}=\overline{\mathrm{E}}e^{i\overset{\rightarrow}{k}.\overset{\rightarrow}{\tau}}\text{ and }\overset{\rightarrow}{H}=\overline{\mathrm{H}}e^{i \overset{\rightarrow}{k}.\overset{\rightarrow}{\tau}} \tag{1}\]  
where \(\overset{\rightarrow}{k}\) (\(=2\pi/\lambda\)) and \(\overset{\rightarrow}{r}=\sqrt{(x^{2}+y^{2}+z^{2})}\) is the distance vector from the input source in space. Furthermore, the cross product  
\[\overset{\rightarrow}{k}\times\overline{\mathrm{E}}=\omega\mu_{o}\overline{ \mathrm{E}}\text{ and }\overset{\rightarrow}{k}\times\overline{\mathrm{H}}=-\omega\varepsilon_{o} \overline{\mathrm{E}} \tag{2}\]  
Figure 3: Representative LSP sensors and output (**a**) radial-LSP sensor, (**b**) longitudinal-LSP sensor, (**c**) probing in Near Field, (**d**) baseline signal of Radial-LSP sensor, and (**e**) baseline signal of longitudinal-LSP sensor.  
wave vector can be related to propagating wave and vertical axis (\(Z\)-axis) as  
\[\begin{array}{c}\stackrel{{\rightarrow}}{{k}}=\kappa\stackrel{{ \rightarrow}}{{u}}_{p}-ip_{m}\stackrel{{\rightarrow}}{{u}}_{Z}\text { in metal }(Z\ <\ 0)\\ \text{and }\stackrel{{\rightarrow}}{{k}}=\kappa\stackrel{{ \rightarrow}}{{u}}_{p}+ip_{d}\stackrel{{\rightarrow}}{{u}}_{Z}\text { in the dielectric }(Z\ >\ 0)\end{array} \tag{3}\]  
where \(\stackrel{{\rightarrow}}{{u}}_{p}=\stackrel{{ \rightarrow}}{{u}}_{X}\cos\psi+\stackrel{{\rightarrow}}{{u}}_{Y}\sin\psi\) is a unit vector, \(\psi\) (0 to 360 degrees) is the angle with respect to the \(X\)-axis, and \(\text{Re}(\kappa)\text{Im}(\kappa)\geq 0\).  
In particular, these SPPs are EM surface waves in the visible/optical range that travel along a metal-dielectric (or air interface). This SPP involves both charge motion in the metal (known as surface plasmon) and EM waves in the air or dielectric material (known as polariton). A SPP will propagate along the interface until its energy is lost either due to absorption in the metal or scattering into the air medium [104; 105]. However, to differentiate between the output signals of the far field in optical range and the near field in the microwave range, we adopted the nomenclature of'spoof SPP' wave instead of SPP wave as suggested in the literature [106; 107]. Generally, these SPP waves and spoof SPP waves are all surface waves with shorter wavelengths than the incident EM waves. However, their confinement in a region (known as sensing zone) and changes to the intensity in the confined zone at a specific location during the course of monitoring, forms the basis for all NDT applications either in sciences or engineering [108; 109; 110]. These waves comprise of electric field and magnetic field, perpendicular to each other.  
The electric fields for 'transverse magnetic wave' can be written in the simplest form of propagating spoof SPP wave, as  
\[\begin{array}{c}\stackrel{{\rightarrow}}{{E}}=\mathbb{E}_{m}e^{ p_{m}Z}e^{i\kappa(x\cos\psi+y\sin\psi)},\ Z\ <\ 0\\ \text{and }\stackrel{{\rightarrow}}{{E}}=\mathbb{E}_{d}e^{-p_{d}Z}e^{ i\kappa(x\cos\psi+y\sin\psi)},\ Z\ >\ 0\end{array} \tag{4}\]  
where the subscripts m and d refer to the copper and air mediums, respectively. Similar expressions can be written for the magnetic field phasor H.  
If \(P_{m}\) and \(P_{d}\) have positive real parts, then both electric fields will decay as \(Z\rightarrow\pm\infty\), indicating localization (within the vicinity) to the interference at \(Z=0\) (decays away from \(Z=0\)). The confinement of such energized surface waves in the \(X\)-\(Y\) plane and the transportation of such energies until they fade out are important in the design of any surface plasmon materials [63; 93; 111].