3. Analysis of a periodic structure with modal analysis

Generally, a metamaterial is a periodic structure that is made from arranging a lot of the same micro-scatterers in a regular lattice/network (Figure 1) [1, 2]. A metamaterial layer is created when micro-scatterers are distributed in a plane. The difference between metamaterials and

Figure 1. A metamaterial structure consisted of micro-scatterers in a regular array (reproduced from [1]).

Impinging a wave into the metamaterial structure induces currents and charges on the metamaterial elements. In other words, the existence of metamaterial elements reinforces the structure in the (electric and magnetic) polarization. There are different methods developed to determine this polarization. In the following, we discuss about these methods. Then the structure is modeled and analyzed by modal analysis to calculate the S parameters of struc-

Some methods are developed to determine the effective parameters of these structures. The electromagnetic properties of an inhomogeneous composite can be determined exactly by solving Maxwell's equations, which relate the local electric and magnetic fields to the local charge and current densities. To solve this set of equations, a relationship must be assumed that relates the four macroscopic field vectors that arise from the averaging or homogenization

On the other hand, many researchers have in practice used an approach based upon the reflection and transmission coefficients of a metamaterial sample of some defined thickness [2]. The Nicolson-Ross-Weir (NRW) approach is then used to obtain the effective material properties of the bulk metamaterials. The solution of equation in this method is dependent on

other periodic structures is mentioned in [2].

4 Metamaterials and Metasurfaces

2. Modeling metamaterial structure

procedure [3].

ture. These parameters show the general behavior of structure.

As mentioned above, a metasurface is a periodic structure that is comprised from distributing a lot of micro-scatterers on a plane (Figure 2a). Therefore, for extracting the behavior of structure, it is sufficient to consider a period of structure with suitable boundary conditions [6]. Figure 2b shows these boundary conditions. These are two perfect electric conductors and two perfect magnetic conductors. On the other hand, these boundary conditions form/constitute a TEM waveguide, and the metasurface element in the middle is a transverse discontinuity in the waveguide. The element divides the inner medium of the waveguide into two media, medium I and medium II (Figure 2c).

Now, this model must be analyzed by a numerical technique such as mode matching, FDTD, or so on [7–12]. Generally, the element (partially) blocks the way of incident power (Pincident). It causes some of the incident power passes through discontinuity toward load (Ptransmitted), and the remaining power reflects back to the source (Preflected). In mathematical form (Eq. 1)

$$P\_{incident} = P\_{reflected} + P\_{transmitted} \tag{1}$$

According to mode-matching method, the reflected and transmitted waves are expanded in terms of different modes supported by TEM waveguide (Eqs. 2–4) [7]. These modes are mentioned in Eqs. 2–5:

$$
\overrightarrow{E}^i = \hat{a}\_y A\_{00} \quad \overrightarrow{H}^i = -\hat{a}\_x \frac{A\_{00}}{\eta\_0} \tag{2}
$$

Figure 2. Front view of a metasurface structure (a) a TEM waveguide with a metasurface element in the middle (b) and incident, reflected, and transmitted waves in the TEM waveguide from side view (c) [7].

E !<sup>r</sup> <sup>¼</sup> ^ax X k¼TE;TM X m;n Er x � �<sup>k</sup> mne <sup>j</sup>βmn<sup>z</sup> <sup>þ</sup> ^ay <sup>a</sup>00<sup>e</sup> <sup>j</sup>β00<sup>z</sup> <sup>þ</sup> <sup>X</sup> k¼TE;TM X m;n Er y � �<sup>k</sup> mne jβmnz ! <sup>þ</sup> ^azE<sup>r</sup> z H !<sup>r</sup> <sup>¼</sup> ^ax a<sup>00</sup> η0 e <sup>j</sup>β00<sup>z</sup> <sup>þ</sup> <sup>X</sup> k¼TE;TM X m;n Er y � �<sup>k</sup> mn Zk mn e jβmnz 0 B@ 1 CA � ^ay X k¼TE;TM X m;n Er x � �<sup>k</sup> mn Zk mn e jβmnz ! <sup>þ</sup> ^azH<sup>r</sup> z (3) E !<sup>t</sup> <sup>¼</sup> ^ax X k¼TE;TM X m;n Et x � �<sup>k</sup> mne �jβmn<sup>z</sup> <sup>þ</sup> ^ay <sup>c</sup>00e�jβ00<sup>z</sup> <sup>þ</sup> <sup>X</sup> k¼TE;TM X m;n Et y � �<sup>k</sup> mne �jβmnz ! <sup>þ</sup> ^azEt z H !<sup>t</sup> <sup>¼</sup> ^ax � <sup>c</sup><sup>00</sup> η0 e �jβ00<sup>z</sup> � <sup>X</sup> k¼TE;TM X m;n Et y � �<sup>k</sup> mn Zk mn e �jβmnz 0 B@ 1 CA � ^ay X k¼TE;TM X m;n Et x � �<sup>k</sup> mn Zk mn e �jβmnz ! <sup>þ</sup> ^azH<sup>t</sup> z (4)

where E !i , E !r , and E !<sup>t</sup> are incident, reflected, and transmitted electric fields, respectively. This is right about H !i , H !r , and H !t , the incident, reflected, and transmitted magnetic fields, respectively.

$$\beta\_{\rm mm} = \sqrt{a^2 \mu\_0 \varepsilon\_0 \varepsilon\_r - \left(\frac{m\pi}{a}\right)^2 - \left(\frac{n\pi}{b}\right)^2}, \quad Z\_{\rm mm}^{\rm TE} = \frac{a\mu\_0}{\beta\_{\rm mm}}, \quad Z\_{\rm mm}^{\rm TM} = \frac{\beta\_{\rm mm}}{a\nu\varepsilon\_0 \varepsilon\_r} \tag{5}$$

$$\begin{aligned} \text{+ } \text{TEM}: \begin{cases} \quad E\_x^i = 0\\ E\_y^i = A\_{00} \end{cases} \text{ } \begin{cases} H\_x^i = -\frac{A\_{00}}{\eta\_0} \\\ H\_y^i = 0 \end{cases} \end{aligned} \tag{6}$$

continuity of electric and magnetic fields in the aperture (Rc

0 Rm

<sup>t</sup> <sup>R</sup><sup>c</sup> m , H!<sup>i</sup> <sup>þ</sup> <sup>H</sup> !<sup>r</sup> � �

Note that the tangential components of an electric field are continuously passing through the

<sup>t</sup> ! cmn <sup>¼</sup> amn <sup>þ</sup> <sup>A</sup>00δm0δn<sup>0</sup> dmn <sup>¼</sup> bmn �

> <sup>a</sup>00. ffiffiffiffiffiffiffiffiffiffi η0 � � I

<sup>A</sup>00. ffiffiffiffiffiffiffiffiffiffi η0 � � I <sup>q</sup> <sup>þ</sup> <sup>1</sup> ! <sup>s</sup><sup>21</sup> <sup>¼</sup>

where equality between s21 and s12 are resulted from reciprocity. Rewriting Eq. 12 for exciting

Pincident ¼ Preflected þ Ptransmitted ! 1 ¼ j j s<sup>11</sup>

ð Þ! <sup>1</sup> <sup>þ</sup> <sup>s</sup><sup>22</sup> <sup>s</sup><sup>22</sup> <sup>¼</sup> <sup>η</sup><sup>0</sup>

þ f g Imð Þ s<sup>11</sup>

Eq. 15 is a relation of a circle on the complex plane of reflection coefficient (Smith chart). The center and radius of this circle are [�(η00)I/((η00)I + (η00)II), 0] and (η00)II/((η00)I + (η00)II), respectively. This circle crosses horizontal axis in two points: [�1, 0] and [((η00)II � (η00)I)/ ((η00)II + (η00)I), 0]. The first point is corresponding to when a metal plate covers the boundary between two media. The second point happens when two media touch each other completely without any element on the boundary. In this case, the reflection coefficient is

� � I η0 � � II

q

E !t � �

(

4. The variation in S parameters for a metasurface

<sup>c</sup>00. ffiffiffiffiffiffiffiffiffiffiffiffi η0 � � II <sup>q</sup>

<sup>A</sup>00. ffiffiffiffiffiffiffiffiffiffi η0 � � I q ¼

> ffiffiffiffiffiffiffiffiffiffiffiffi η0 � � I η0 � � II s

Suppose that the metasurface element is lossless. Rewriting Eq. 1;

<sup>η</sup><sup>00</sup> � � I

<sup>I</sup> <sup>þ</sup> <sup>η</sup><sup>00</sup> � �

waveguide from transmitted media in Figure 2c [7],

s<sup>21</sup> ¼ s<sup>12</sup> ¼

Considering Eqs. 12–14 simultaneously result into [7],

<sup>η</sup><sup>00</sup> � �

II ( )<sup>2</sup>

Reð Þþ s<sup>11</sup>

t ¼

transverse plane except Rm:

E !<sup>i</sup> <sup>þ</sup> <sup>E</sup> !<sup>r</sup> � �

metal transverse discontinuity [7, 13, 21]:

t ¼ E !<sup>t</sup> � �

Considering Eq. 11 for TEM mode (m = n = 0)

E !<sup>i</sup> <sup>þ</sup> <sup>E</sup> !<sup>r</sup> � �

c<sup>00</sup> ¼ a<sup>00</sup> þ A<sup>00</sup> !

<sup>m</sup>). Rc

(

t ¼ <sup>m</sup> is all the remaining parts of a

http://dx.doi.org/10.5772/intechopen.80584

m, n ¼ 0, 1, 2,… (11)

ð Þ¼ 1 þ s<sup>11</sup> s<sup>12</sup> (12)

ð Þ� 1 þ s<sup>11</sup> 1 (13)

<sup>2</sup> (14)

(15)

(10)

7

0 Rm

Investigation into the Behavior of Metasurface by Modal Analysis

ffiffiffiffiffiffiffiffiffiffiffiffi η0 � � II η0 � � I

s

<sup>2</sup> <sup>þ</sup> j j <sup>s</sup><sup>21</sup>

II

<sup>I</sup> <sup>þ</sup> <sup>η</sup><sup>00</sup> � � II ( )<sup>2</sup>

<sup>2</sup> <sup>¼</sup> <sup>η</sup><sup>00</sup> � �

<sup>η</sup><sup>00</sup> � �

<sup>t</sup> Rc m

H !t � �

$$\begin{aligned} \text{TM}: \begin{cases} \left(E\_{\mathbf{x}}\right)\_{mn} = \frac{m\pi}{a} a\_{mn} \varphi\_{mn}(\mathbf{x}, \mathbf{y}) \; , \; \left(H\_{\mathbf{x}}\right)\_{mn} = -\frac{\left(E\_{\mathbf{y}}\right)\_{mn}}{Z\_{mn}^{TM}}\\ \left(E\_{\mathbf{y}}\right)\_{mn} = -\frac{n\pi}{b} a\_{mn} \psi\_{mn}(\mathbf{x}, \mathbf{y}) \; , \; \left(H\_{\mathbf{y}}\right)\_{mn} = \frac{\left(E\_{\mathbf{x}}\right)\_{mn}}{Z\_{mn}^{TM}} \end{cases} \end{aligned} \tag{7}$$

$$\begin{aligned} \text{TE}: \begin{cases} \left(E\_x\right)\_{mn} = \frac{m\pi}{b} b\_{mn} \varphi\_{mn}(\mathbf{x}, \mathbf{y}) \; , \; \left(H\_x\right)\_{mn} = -\frac{\left(E\_y\right)\_{mn}}{Z\_{mn}^{\text{TE}}}\\ \left(E\_y\right)\_{mn} = \frac{m\pi}{a} b\_{mn} \psi\_{mn}(\mathbf{x}, \mathbf{y}) \; , \; \left(H\_y\right)\_{mn} = \frac{\left(E\_x\right)\_{mn}}{Z\_{mn}^{\text{TE}}} \end{cases} \end{aligned} \tag{8}$$

$$\varphi\_{mn}(\mathbf{x},y) \triangleq \sin\left(\frac{m\pi}{a}\mathbf{x}\right) \sin\left(\frac{n\pi}{b}y\right) \; , \; \psi\_{mn}(\mathbf{x},y) \triangleq \cos\left(\frac{m\pi}{a}\mathbf{x}\right) \cos\left(\frac{n\pi}{b}y\right) \; . \tag{9}$$

where <sup>β</sup>mn, <sup>Z</sup>TE mn, and ZTM mn are propagation constant, characteristic impedance of TE mode, and characteristic impedance of TM mode, respectively. For determining the weights of each mode, the boundary conditions must be applied on the transverse plane (Eq. 5) [7, 13–20]. These boundary conditions include vanishing transverse electric field on the metal element (Rm) and continuity of electric and magnetic fields in the aperture (Rc <sup>m</sup>). Rc <sup>m</sup> is all the remaining parts of a transverse plane except Rm:

$$\left(\overrightarrow{E}^{i} + \overrightarrow{E}^{r}\right)\_{t} = \begin{cases} 0 & \mathcal{R}\_{m} \\ \left(\overrightarrow{E}^{t}\right)\_{t} & \mathcal{R}\_{m}^{c} \end{cases} \quad \left(\overrightarrow{H}^{i} + \overrightarrow{H}^{r}\right)\_{t} = \begin{cases} 0 & \mathcal{R}\_{m} \\ \left(\overrightarrow{H}^{t}\right)\_{t} & \mathcal{R}\_{m}^{c} \end{cases} \tag{10}$$

Note that the tangential components of an electric field are continuously passing through the metal transverse discontinuity [7, 13, 21]:

$$\left(\overrightarrow{E}^{i} + \overrightarrow{E}^{r}\right)\_{t} = \left(\overrightarrow{E}^{t}\right)\_{t} \quad \rightarrow \quad \begin{cases} c\_{mn} = a\_{mn} + A\_{00}\delta\_{m0}\delta\_{n0} \\ d\_{mn} = b\_{mn} \end{cases} m, n = 0, 1, 2, \dots \tag{11}$$

#### 4. The variation in S parameters for a metasurface

Considering Eq. 11 for TEM mode (m = n = 0)

E !<sup>r</sup> <sup>¼</sup> ^ax

H !<sup>r</sup> <sup>¼</sup> ^ax

E !<sup>t</sup> <sup>¼</sup> ^ax

H

where E !i , E !r

right about H

where <sup>β</sup>mn, <sup>Z</sup>TE

X k¼TE;TM

> a<sup>00</sup> η0 e

X k¼TE;TM

> η0 e

, and E

βmn ¼

, and H !t

s

TM :

TE :

<sup>φ</sup>mnð Þ <sup>x</sup>; <sup>y</sup> <sup>≜</sup> sin <sup>m</sup><sup>π</sup>

mn, and ZTM

8 >>><

>>>:

8 >>><

>>>:

!i , H !r

!<sup>t</sup> <sup>¼</sup> ^ax � <sup>c</sup><sup>00</sup>

0 B@ X m;n

0 B@

6 Metamaterials and Metasurfaces

X m;n

Er x � �<sup>k</sup> mne

<sup>j</sup>β00<sup>z</sup> <sup>þ</sup> <sup>X</sup>

Et x � �<sup>k</sup> mne

�jβ00<sup>z</sup> � <sup>X</sup>

k¼TE;TM

<sup>ω</sup><sup>2</sup>μ0ε0ε<sup>r</sup> � <sup>m</sup><sup>π</sup>

k¼TE;TM

X m;n

> X m;n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a � �<sup>2</sup>

> 8 ><

> >:

Ei <sup>y</sup> ¼ A<sup>00</sup>

a

mn ¼ � <sup>n</sup><sup>π</sup>

mn <sup>¼</sup> <sup>m</sup><sup>π</sup> a

> sin <sup>n</sup><sup>π</sup> b y � �

TEM : Ei

ð Þ Ex mn <sup>¼</sup> <sup>m</sup><sup>π</sup>

ð Þ Ex mn <sup>¼</sup> <sup>n</sup><sup>π</sup>

Ey � �

Ey � �

a x � � � <sup>n</sup><sup>π</sup> b � �<sup>2</sup>

<sup>x</sup> ¼ 0

,

Et y � �<sup>k</sup> mn Zk mn e �jβmnz

<sup>j</sup>βmn<sup>z</sup> <sup>þ</sup> ^ay <sup>a</sup>00<sup>e</sup>

Er y � �<sup>k</sup> mn Zk mn e jβmnz

�jβmn<sup>z</sup> <sup>þ</sup> ^ay <sup>c</sup>00e�jβ00<sup>z</sup> <sup>þ</sup> <sup>X</sup>

<sup>j</sup>β00<sup>z</sup> <sup>þ</sup> <sup>X</sup>

CA � ^ay

1

1

k¼TE;TM

k¼TE;TM

CA � ^ay

!<sup>t</sup> are incident, reflected, and transmitted electric fields, respectively. This is

, ZTE

Hi

8 ><

>:

<sup>b</sup> amnψmnð Þ <sup>x</sup>; <sup>y</sup> , Hy

bmnψmnð Þ x; y , Hy

characteristic impedance of TM mode, respectively. For determining the weights of each mode, the boundary conditions must be applied on the transverse plane (Eq. 5) [7, 13–20]. These boundary conditions include vanishing transverse electric field on the metal element (Rm) and

<sup>b</sup> bmnφmnð Þ <sup>x</sup>; <sup>y</sup> , Hð Þ<sup>x</sup> mn ¼ � Ey

, ψmnð Þ x; y ≜cos

Hi <sup>y</sup> ¼ 0

amnφmnð Þ <sup>x</sup>; <sup>y</sup> , Hð Þ<sup>x</sup> mn ¼ � Ey

, the incident, reflected, and transmitted magnetic fields, respectively.

mn <sup>¼</sup> ωμ<sup>0</sup> βmn

<sup>x</sup> ¼ � <sup>A</sup><sup>00</sup> η0

� �

� �

mn are propagation constant, characteristic impedance of TE mode, and

X m;n

X k¼TE;TM

> X m;n

X k¼TE;TM

!

!

Er y � �<sup>k</sup> mne jβmnz

X m;n

Et y � �<sup>k</sup> mne �jβmnz

> X m;n

Et x � �<sup>k</sup> mn Zk mn e �jβmnz

!

, ZTM

� � mn ZTM mn

mn <sup>¼</sup> ð Þ Ex mn ZTM mn

> � � mn ZTE mn

mn <sup>¼</sup> ð Þ Ex mn ZTE mn

> mπ a x � �

cos nπ b y � �

mn <sup>¼</sup> <sup>β</sup>mn ωε0ε<sup>r</sup>

Er x � �<sup>k</sup> mn Zk mn e jβmnz

!

<sup>þ</sup> ^azE<sup>r</sup> z

> <sup>þ</sup> ^azH<sup>r</sup> z

> > <sup>þ</sup> ^azH<sup>t</sup> z

> > > (4)

(5)

(6)

(7)

(8)

: (9)

<sup>þ</sup> ^azEt z (3)

$$a\_{00} = a\_{00} + A\_{00} \quad \rightarrow \quad \frac{c\_{00} \Big/ \sqrt{\left(\eta\_{0}\right)\_{\mathrm{II}}}}{A\_{00} \Big/ \sqrt{\left(\eta\_{0}\right)\_{\mathrm{I}}}} = \frac{a\_{00} \Big/ \sqrt{\left(\eta\_{0}\right)\_{\mathrm{I}}}}{A\_{00} \Big/ \sqrt{\left(\eta\_{0}\right)\_{\mathrm{I}}}} + 1 \to s\_{21} = \sqrt{\frac{\left(\eta\_{0}\right)\_{\mathrm{II}}}{\left(\eta\_{0}\right)\_{\mathrm{I}}}} (1 + s\_{11}) = s\_{12} \tag{12}$$

where equality between s21 and s12 are resulted from reciprocity. Rewriting Eq. 12 for exciting waveguide from transmitted media in Figure 2c [7],

$$\mathbf{s}\_{21} = \mathbf{s}\_{12} = \sqrt{\frac{(\eta\_0)\_I}{(\eta\_0)\_{I\bar{I}}}} (\mathbf{1} + \mathbf{s}\_{22}) \to \mathbf{s}\_{22} = \frac{(\eta\_0)\_I}{(\eta\_0)\_{I\bar{I}}} (\mathbf{1} + \mathbf{s}\_{11}) - \mathbf{1} \tag{13}$$

Suppose that the metasurface element is lossless. Rewriting Eq. 1;

$$P\_{incident} = P\_{reflected} + P\_{transmitted} \rightarrow 1 = |\mathbf{s}\_{11}|^2 + |\mathbf{s}\_{21}|^2 \tag{14}$$

Considering Eqs. 12–14 simultaneously result into [7],

$$\left\{ \text{Re}(s\_{11}) + \frac{(\eta\_{00})\_I}{(\eta\_{00})\_I + (\eta\_{00})\_{II}} \right\}^2 + \left\{ \text{Im}(s\_{11}) \right\}^2 = \left\{ \frac{(\eta\_{00})\_{II}}{(\eta\_{00})\_I + (\eta\_{00})\_{II}} \right\}^2 \tag{15}$$

Eq. 15 is a relation of a circle on the complex plane of reflection coefficient (Smith chart). The center and radius of this circle are [�(η00)I/((η00)I + (η00)II), 0] and (η00)II/((η00)I + (η00)II), respectively. This circle crosses horizontal axis in two points: [�1, 0] and [((η00)II � (η00)I)/ ((η00)II + (η00)I), 0]. The first point is corresponding to when a metal plate covers the boundary between two media. The second point happens when two media touch each other completely without any element on the boundary. In this case, the reflection coefficient is ((η00)II � (η00)I)/((η00)II + (η00)I). It is important that the s11 is coincided on this circle (Eq. 15), regardless of the geometrical shape of the element. Rewriting Eq. 15 in terms of center and radius of circle,

$$S\_{11} = a\_1 + r\_1 \varepsilon^{j\theta} \quad , \quad a\_1 = \frac{-\left(\eta\_{00}\right)\_I}{\left(\eta\_{00}\right)\_{II} + \left(\eta\_{00}\right)\_I} \quad , \quad r\_1 = \frac{\left(\eta\_{00}\right)\_{II}}{\left(\eta\_{00}\right)\_{II} + \left(\eta\_{00}\right)\_I} \quad , \quad r\_1 = 1 + a\_1 \tag{16}$$

r1 and a1 are functions of constitutive parameters of two media that surround the metasurface. The geometrical properties of metasurface (size and shape) determine the parameter θ. The circles corresponding to s21 and s22 are

$$\left\{ \text{Re}(s\_{22}) + \frac{(\eta\_{00})\_{\text{II}}}{(\eta\_{00})\_{\text{I}} + (\eta\_{00})\_{\text{II}}} \right\}^2 + \left\{ \text{Im}(s\_{22}) \right\}^2 = \left\{ \frac{(\eta\_{00})\_{\text{I}}}{(\eta\_{00})\_{\text{I}} + (\eta\_{00})\_{\text{II}}} \right\}^2 \tag{17}$$

$$\left\{ \text{Re}(s\_{21}) - \frac{\sqrt{(\eta\_{00})\_I (\eta\_{00})\_{II}}}{(\eta\_{00})\_I + (\eta\_{00})\_{II}} \right\}^2 + \left\{ \text{Im}(s\_{21}) \right\}^2 = \left\{ \frac{(\eta\_{00})\_I (\eta\_{00})\_{II}}{\left( (\eta\_{00})\_I + (\eta\_{00})\_{II} \right)^2} \right\} \tag{18}$$

The point [�1, 0] is a common point between s11 and s22 circles. The s21 circle is located on the right-hand side of the Smith chart.
