3. Electromagnetic wave interaction with metamaterials

In this section, the behavior of electromagnetic wave interaction with metamaterials is studied both analytically and numerically. Analytical formulation of electromagnetic field behavior on a one-dimensional artificial lossless and isotropic

metamaterial slab due to an external plane wave excitation is presented first. Numerical demonstration of two-dimensional electromagnetic wave interaction with artificial DNG, MNG, and ENG metamaterial slabs is then illustrated and discussed.

## 3.1 Normal plane wave interaction with metamaterials

We consider the problem of a uniform plane wave that is traveling along the +zdirection in free space and is incidental normally on an infinitely large metamaterial slab of thickness, h. The metamaterial slab is placed between z = 0 and z = h, as shown in Figure 8, and is made infinitely large along x and y directions. For convenience, the incident electric field of the uniform plane wave will be assumed to be in the +x -direction, while its associated magnetic field will be in +y -direction.

Without loss of generality, the phasor notation will be used to express the total electric and magnetic fields in each region of the problem geometry in Figure 8. In region 1, the electric and magnetic fields are given as:

$$\mathbf{E\_0 = \left(E\_0^+ e^{-jk\_{0x}} + E\_0^- e^{jk\_{0x}}\right) \ \mathbf{a\_x}} \tag{17}$$

$$\mathbf{H}\_0 = \left( H\_0^+ e^{-jk\alpha x} + H\_0^- e^{jk\alpha x} \right) \ \mathbf{a}\_\uparrow \tag{18}$$

where E<sup>þ</sup> <sup>0</sup> , E� <sup>0</sup> , H<sup>þ</sup> = E<sup>þ</sup> <sup>0</sup> =η0, and H� = � E<sup>0</sup> �=η<sup>0</sup> are the electric and magnetic <sup>0</sup> <sup>0</sup> field amplitudes in the forward and backward directions in region 1 (free space), <sup>p</sup> pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi respectively. The parameters k0 <sup>=</sup> <sup>ω</sup> <sup>μ</sup>0ε0, and <sup>η</sup><sup>0</sup> <sup>=</sup> <sup>μ</sup>0=ε<sup>0</sup> are the wavenumber and wave impedance in free space, respectively.

In the second region, in which the metamaterial slab is located, the phasor form of total electric and magnetic fields is given as

$$\mathbf{E\_1 = \left(E\_1^+ e^{-jk\_1x} + E\_1^- e^{jk\_1x}\right) \ \mathbf{a\_x}} \tag{19}$$

$$\mathbf{H}\_1 = \left( H\_1^+ e^{-jk\_1x} + H\_1^- e^{jk\_1x} \right) \ \mathbf{a}\_\uparrow \tag{20}$$

where E<sup>þ</sup> <sup>1</sup> , E� <sup>1</sup> , H<sup>þ</sup> = Eþ=η1, and H� = � E� 1 1 1 1 =η<sup>1</sup> are the electric and magnetic field amplitudes in the forward and backward directions in region 2 (metamaterial slab),

Figure 8. Normal incidence of a uniform plane wave on an infinitely large metamaterial slab of thickness, h.

<sup>p</sup> pffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi respectively. The parameters k1 <sup>=</sup> <sup>ω</sup> <sup>μ</sup>1ε<sup>1</sup> and <sup>η</sup><sup>1</sup> <sup>=</sup> <sup>μ</sup>1=ε<sup>1</sup> are the wavenumber and wave impedance in the metamaterial slab, respectively.

The electric and magnetic fields in region 3, which represents free space, are

$$\mathbf{E}\_3 = E\_3^+ e^{-jk\_0 x} \mathbf{a}\_\mathbf{x} \tag{21}$$

$$\mathbf{H}\_3 = \mathbf{H}\_3^+ e^{-jk\_0 x} \,\mathbf{a}\_\circ \tag{22}$$

where E<sup>þ</sup> and H<sup>þ</sup> = Eþ=η<sup>0</sup> are the electric and magnetic field amplitudes in the <sup>3</sup> 3 3 forward and backward directions in region 3 (free space), respectively.

After applying the boundary conditions at the metamaterial slab walls, z = 0 and z = h, the following four relations summarizing the relationship between the reflected/transmitted electric field components, E� <sup>0</sup> , E<sup>þ</sup> , E<sup>1</sup> �, and E<sup>þ</sup> <sup>2</sup> , respectively, <sup>1</sup> and the incident electric field amplitude, E<sup>þ</sup> <sup>0</sup> , are obtained:

$$E\_{\mathbf{0}}^{-} = \frac{j\left(\eta\_1^2 - \eta\_0^2\right)\sin\left(k\_1 h\right)}{2\eta\_0\eta\_1\cos\left(k\_1 h\right) + j\left(\eta\_1^2 + \eta\_0^2\right)\sin\left(k\_1 h\right)}\,\mathbf{E}\_{\mathbf{0}}^{+}\tag{23}$$

$$E\_1^+ = \frac{\eta\_1(\eta\_1 + \eta\_0) \ e^{-jk\_1h}}{2\eta\_0\eta\_1\cos\left(k\_1h\right) + j(\eta\_1^2 + \eta\_0^2)\sin\left(k\_1h\right)} \ \mathbf{E\_0^+}\tag{24}$$

$$E\_1^- = \frac{\eta\_1(\eta\_0 - \eta\_1) \, e^{jk\_1h}}{2\eta\_0\eta\_1\cos\left(k\_1h\right) + j(\eta\_1^2 + \eta\_0^2)\sin\left(k\_1h\right)} \, E\_0^+ \tag{25}$$

$$E\_2^+ = \frac{2\eta\_0\eta\_1\ e^{-jk\_0h}}{2\eta\_0\eta\_1\cos\left(k\_1h\right) + j(\eta\_1^2 + \eta\_0^2)\sin\left(k\_1h\right)}E\_0^+\tag{26}$$

To simplify Eqs. (23)–(26) further, we define a normalized wave impedance in pffiffiffiffiffiffiffiffiffiffi the metamaterial slab, as <sup>η</sup><sup>m</sup> <sup>=</sup> <sup>η</sup>1/η<sup>0</sup> <sup>=</sup> <sup>μ</sup>r=ε<sup>r</sup> and Eqs. (23)–(26) are read as

$$E\_{\mathbf{0}}^{-} = \frac{j\left(\eta\_{m}^{-2} - 1\right)\sin\left(k\_{1}h\right)}{2\eta\_{0}\eta\_{1}\cos\left(k\_{1}h\right) + j\left(\eta\_{m}^{-2} + 1\right)\sin\left(k\_{1}h\right)}E\_{\mathbf{0}}^{+}\tag{27}$$

$$E\_1^+ = \frac{\left(\eta\_m^{-2} + \eta\_m\right)e^{-jk\_1h}}{2\eta\_m\cos\left(k\_1h\right) + j(\eta\_m^{-2} + 1)\sin\left(k\_1h\right)}E\_0^+\tag{28}$$

$$E\_1^- = \frac{\left(\eta\_m - \eta\_m^2\right)e^{jk\_1h}}{2\eta\_m\cos\left(k\_1h\right) + j(\eta\_m^2 + 1)\sin\left(k\_1h\right)}E\_0^+\tag{29}$$

$$E\_2^+ = \frac{2\eta\_m e^{-jk\_0h}}{2\eta\_m \cos\left(k\_1 h\right) + j(\eta\_m^2 + 1)\sin\left(k\_1 h\right)} E\_0^+ \tag{30}$$

where η<sup>m</sup> denotes the normalized wave impedance in the metamaterial slab.

Eqs. (28)–(29) can be used to compute the fractional electric field components in the forward and backward directions, i.e., along the three media. In this analytical study, the electric field strength inside an artificial lossless metamaterial slab is computed using Eq. (19), at a frequency of 10 GHz. We consider the case of a very thin subwavelength homogeneous DNG metamaterial slab, where real parts of permittivity and permeability are both negative at 10 GHz. One possible implementation of such artificial DNG medium can be realized with sufficient number of repeated patterns of composite AMM (SRRs) along with periodic arrangement of metallic rods or planar metallic strips, as demonstrated in [21, 22]. Since evanescent (non-propagating) waves are expected to exist inside single-negative media with exponentially decaying electric field, such cases are not considered here.

Figure 9 shows the analytically computed electric field strength as a function of the DNG metamaterial slab of thickness, h. In this study, the overall DNG slab thickness is considered as h = 10 mm. The effect of increasing the effective magnetic permeability of the DNG slab from ˜1 to ˜10 is also presented, as shown in Figure 9, where higher mismatch along the interface of DPS-DNG is observed as the effective permeability is increased. The case of matched DNG constitutive parameters with those of DPS (air medium) shows zero reflection from such an interface, as expected. For convenience, the 2D structure is illuminated with an x-polarized normal incident plane wave that originates from z = 0 plane. The plane wave has an electric field amplitude peak of 1.5 kV/m and phase of 0°.

For validation purposes, this problem of interest was numerically modeled and simulated using ANSYS HFSS simulator [41]. Figure 10 presents the developed structure to study the problem of normal plane wave incidence on a onedimensional DNG metamaterial slab. One possible excitation of such plane wave can be numerically realized using a set of periodic boundary conditions; in other words, using perfect electric conductor (PEC) and perfect magnetic conductor (PMC) symmetry planes along four sides of the geometry, x-axis and y-axis walls, respectively, ensure plane wave excitation along with proper excitation of the DNG metamaterial slab (see Figure 10). Good agreement can be seen between analytically and numerically computed electric field strength inside the DNG metamaterial slab, as shown in Figures 9 and 11.

## 3.2 Numerical demonstration of electromagnetic wave interaction with artificial DNG, MNG, and ENG metamaterial slabs

In this section, numerical demonstration of electromagnetic cylindrical wave interaction with various artificial isotropic and lossy metamaterial slabs is illustrated. Figure 12 depicts the numerical full-wave simulation model, where an artificial lossy DNG slab was placed between two natural, lossy DPS slabs. In this numerical study, DPS, DNG, and MNG media are all considered as lossy and isotropic, where for the case of DNG slab, effective constitutive parameters are ε<sup>r</sup> = ˜1 and μ<sup>r</sup> = ˜2.2, with dielectric and magnetic losses of 0.002, while the constitutive parameters are ε<sup>r</sup> = 2.2 and μ<sup>r</sup> = ˜1 for the MNG medium case, and ε<sup>r</sup> = ˜2.2 and μ<sup>r</sup> = 1 for the ENG medium slab case, with similar losses as those

#### Figure 9.

Analytical computation of electric field strength inside a lossless DNG metamaterial slab. Note that in this study, ε<sup>r</sup> was set as ˜1.

Electromagnetic Field Interaction with Metamaterials DOI: http://dx.doi.org/10.5772/intechopen.84170

Figure 10.

The numerical full-wave model used to study the normal incidence of plane wave on a one-dimensional DNG lossless metamaterial slab.

#### Figure 11.

Numerical computation of electric field strength inside a lossless DNG metamaterial slab. Note that in this study, ε<sup>r</sup> was set as -1.

#### Figure 12.

Numerical model used to study the electromagnetic wave interaction with DNG medium.

considered in the DNG slab. The constitutive parameters of the DPS medium are ε<sup>r</sup> = 2.2 and μ<sup>r</sup> = 1, with dielectric and magnetic losses of 0.002. Cylindrical waves were excited from a point source that was placed 7.5 mm away from all the aforementioned slabs. For the electromagnetic wave interaction with MNG medium, the DNG medium in Figure 12 is replaced with MNG medium. The same is also applied to ENG medium. This numerical demonstration was carried out using the numerical full-wave simulator of ANSYS HFSS.

Figure 13 presents the electric field intensity distribution for the DNG, MNG, and ENG media that were captured at a phase of 0 degree and compared against the

#### Figure 13.

Numerically computed electric field intensity distribution for the studied lossy isotropic homogenized slabs of (a) DNG medium, (b) MNG medium, (c) ENG medium, and (d) reference DPS medium.

#### Electromagnetic Field Interaction with Metamaterials DOI: http://dx.doi.org/10.5772/intechopen.84170

reference case of lossy DPS slab. In Figure 13(a), which represents the DNG case, several interesting features can be observed, including negatively refracted waves inside the DNG slab and backward wave propagation (see animated figure, Figure 14(a) as compared against the DPS case of Figure 14(b)), and focusing phenomena of the original electromagnetic cylindrical waves can be seen at the middle of the DNG slab and also at the DPS slab next to the DNG exit face, in which focusing depends on the selection of the refractive indices along with the DNG medium slab thickness.

While electromagnetic wave propagation inside DNG medium is permissible, since both permittivity and permeability are negative and hence result in a positive real wavenumber, evanescent (non-propagating) waves exist in the MNG medium, as shown in Figure 13(b), that only propagate along the interface and decay exponential away from the MNG slab. This is because in the MNG medium, only the permeability is negative, which results in an imaginary negative wavenumber (see Eq. (7)). Similar behavior to the MNG case is also expected for the single-negative ENG medium slab, as shown in Figure 13(c), where evanescent decaying waves are only present in such single-negative medium. For comparison, the case of all DPS media showed normal forward electromagnetic propagation in such lossy media, as shown in Figure 13(d).

A one-dimensional plot of electric field profile inside the aforementioned artificial metamaterials DNG, MNG, and ENG slabs is also presented as shown in Figure 15 and compared against the normal dielectric DPS medium case. As can be seen from Figure 15, propagating electromagnetic field inside the DNG medium

Figure 14.

Animated snapshots for the electric field distribution for (a) DNG medium and (b) DPS medium cases.

#### Figure 15.

A one-dimensional plot of electric field strength inside the (a) DNG medium, (b) MNG medium, (c) ENG medium, and (d) DPS medium.

slab is visible through the recorded electric field profile at a frequency of 10 GHz. Unlike electric field behavior inside the DNG medium, exponentially decaying electric field profile is recorded inside the single-negative MNG and ENG media slabs. Despite the fact that non-propagating (evanescent) waves existed in both MNG and ENG media slabs, the electric field profile is much stronger at the DPS-MNG interface than the strength at the DPS-ENG interface.
