5. Simulation results

In this section, simulation results of a few examples of metasurface are demonstrated. All examples are simulated by modal analysis (in MATLAB). Simulation results are repeated by finite element method (FEM) (in HFSS). In the following simulations, it is supposed that two media are extended into infinity, and the reference planes are located 10 mm away from discontinuity in both sides. For demonstrating the results on the Smith chart, all parameters are calculated on the metasurface plane.

The first structure is a strip grating, made from metal and illuminated by a wave with parallel polarization (Figure 3).

A unit cell of this structure is shown in Figure 4. Table 1 specifies the geometrical characteristics of metasurface element. The frequency variations in reflection and transmission coefficients are plotted in Figure 4 for this structure.

> Also, this figure shows the variations in reflection and transmission coefficients versus the width of strip for 10 GHz. According to this figure, the more the parameter w increases, the closer to �1 the parameter |s11| moves. In the limit, when each strip covers all areas of the unit cell, s11 goes to �1. In general, with increasing the area of metasurface element, the parameters s11 and s22 move to �1 along the corresponding circles (θ ! π in Eq. (16)). Also, the parameters s21 and s12 move to origin (Γ = 0) along the corresponding circle (in counterclockwise direction). Note that with decreasing frequency, the parameter θ goes to π for this structure, so that

a w Medium I Medium II

2.5 mm 0.5 mm Air Air

Table 1. Geometrical properties of the metal strip grating (Figure 5) [7].

Figure 4. Frequency variations in S parameters for metal strip grating (a) on the Smith chart (b) [7].

Figure 3. Metal strip grating (a), a unit cell of the structure with suitable boundary conditions in peripheral (parallel

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polarization) (b) [7].

According to this figure, the major part of incident power is reflected back in medium I.

The frequency variations in S parameters on the Smith chart are demonstrated in Figure 4, too. The variations in s12 and s22 for this structure are the same as the variations in s21 and s11, respectively. The dash lines on Figure 4 are s11 and s21 circles.

The metallic strips in this structure behave in the same way as parallel inductance in equivalent circuit. Figure 5 displays the frequency variations in equivalent inductance for the metal grating. Investigation into the Behavior of Metasurface by Modal Analysis http://dx.doi.org/10.5772/intechopen.80584 9

((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

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

þ f g Imð Þ s<sup>22</sup>

þ f g Imð Þ s<sup>21</sup>

The point [�1, 0] is a common point between s11 and s22 circles. The s21 circle is located on the

In this section, simulation results of a few examples of metasurface are demonstrated. All examples are simulated by modal analysis (in MATLAB). Simulation results are repeated by finite element method (FEM) (in HFSS). In the following simulations, it is supposed that two media are extended into infinity, and the reference planes are located 10 mm away from discontinuity in both sides. For demonstrating the results on the Smith chart, all parameters

The first structure is a strip grating, made from metal and illuminated by a wave with parallel

A unit cell of this structure is shown in Figure 4. Table 1 specifies the geometrical characteristics of metasurface element. The frequency variations in reflection and transmission coeffi-

The frequency variations in S parameters on the Smith chart are demonstrated in Figure 4, too. The variations in s12 and s22 for this structure are the same as the variations in s21 and s11,

The metallic strips in this structure behave in the same way as parallel inductance in equivalent circuit. Figure 5 displays the frequency variations in equivalent inductance for the metal grating.

According to this figure, the major part of incident power is reflected back in medium I.

, r<sup>1</sup> <sup>¼</sup> <sup>η</sup><sup>00</sup>

η00 � �

� � II

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

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

8 ><

>:

η00 � �

η00 � �

� �

� � I

( )<sup>2</sup>

<sup>I</sup> þ η<sup>00</sup> � � II

> <sup>I</sup> η<sup>00</sup> � � II

<sup>I</sup> þ η<sup>00</sup> � � II

� �<sup>2</sup>

II þ η<sup>00</sup> � � I , r<sup>1</sup> ¼ 1 þ a<sup>1</sup> (16)

9 >=

>;

(17)

(18)

� � I

II þ η<sup>00</sup> � � I

radius of circle,

8 Metamaterials and Metasurfaces

S<sup>11</sup> ¼ a<sup>1</sup> þ r1e

circles corresponding to s21 and s22 are

Reð Þþ s<sup>22</sup>

Reð Þ� s<sup>21</sup>

right-hand side of the Smith chart.

are calculated on the metasurface plane.

cients are plotted in Figure 4 for this structure.

respectively. The dash lines on Figure 4 are s11 and s21 circles.

5. Simulation results

polarization (Figure 3).

8 < : <sup>j</sup><sup>θ</sup> , a<sup>1</sup> <sup>¼</sup> � <sup>η</sup><sup>00</sup>

η00 � � II

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η00 � �

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

<sup>I</sup> η<sup>00</sup> � � II 9 = ;

2

<sup>I</sup> þ η<sup>00</sup> � � II

η00 � �

q

η00 � �

( )<sup>2</sup>

η00 � �

Figure 3. Metal strip grating (a), a unit cell of the structure with suitable boundary conditions in peripheral (parallel polarization) (b) [7].

Figure 4. Frequency variations in S parameters for metal strip grating (a) on the Smith chart (b) [7].


Table 1. Geometrical properties of the metal strip grating (Figure 5) [7].

Also, this figure shows the variations in reflection and transmission coefficients versus the width of strip for 10 GHz. According to this figure, the more the parameter w increases, the closer to �1 the parameter |s11| moves. In the limit, when each strip covers all areas of the unit cell, s11 goes to �1. In general, with increasing the area of metasurface element, the parameters s11 and s22 move to �1 along the corresponding circles (θ ! π in Eq. (16)). Also, the parameters s21 and s12 move to origin (Γ = 0) along the corresponding circle (in counterclockwise direction). Note that with decreasing frequency, the parameter θ goes to π for this structure, so that

The second structure is the same as the first example, metal grating printed on Teflon but perpendicular polarization. Physical dimensions of this structure are available in Table 2. Simulation

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According to Figure 7, the major part of incident power is transmitted through medium II. Figure 7 displays the frequency variations in S parameters on the Smith chart for this structure,

These parameters are located along the straight line [7]. In contrast to the previous structure, the metasurface element in this structure plays the role of parallel capacitance in equivalent circuit. The frequency variations in equivalent capacitance are shown in Figure 8. Figure 9

In contrast to what happened about the previous structure, the imaginary part of reflection coefficient is lower than zero, and it has a minimum. This point happens when the stored energy is maximum around metasurface [7]. The variations in the real part of reflection coefficient are similar to the previous structure. Same as the previous structure, the increment in w causes the parameters s11 and s22 to move to 1 along the corresponding circles. Decreasing and increasing frequency causes s11 to move to ((η00)II (η00)I)/((η00)II + (η00)I) and 1, respectively.

The next example is a metasurface composed of square patches (Figure 10). Let metasurface elements be printed on FR4. Physical dimensions of each element are presented in Table 3. Figure 10 displays simulation results of this structure. The variations in the scattering param-

a w Medium I Medium II 2 mm 1.5 mm Air Teflon

Figure 7. Frequency variations in S parameters for metal strip grating in perpendicular polarization (a) on the Smith

too. The scattering transfer parameters for this structure are shown in Figure 8.

shows the real and imaginary parts of the reflection coefficient in terms of |s11|.

results for this structure are demonstrated in Figure 7.

eters on the Smith chart are presented in the same figure.

Table 2. Geometrical properties of the metal strip grating (Figure 7) [7].

chart (b) [7].

Figure 5. Frequency variations in equivalent inductance (a) and the variation in S parameters versus the width of strip for 10 GHz (b) [7].

when θ equals to π, the value of s11 is 1. It is clearly predictable from the model used for simulating metasurface.

The variations in real and imaginary parts of s11 versus |s11| are shown in Figure 6 for this structure. Clearly seen, with increasing |s11| from zero to one, the real part of s11 uniformly decreases, while the graph of Im(s11) has a maximum where |s11| is equal to 1/√2 (θ = π/2 in Eq. (16)). This point is corresponding to when the excitation of higher-order modes causes the highest stored energy around the structure [7].

Figure 6. The variations in Re(s11) and Im(s11) versus |s11| [7].

The second structure is the same as the first example, metal grating printed on Teflon but perpendicular polarization. Physical dimensions of this structure are available in Table 2. Simulation results for this structure are demonstrated in Figure 7.

According to Figure 7, the major part of incident power is transmitted through medium II. Figure 7 displays the frequency variations in S parameters on the Smith chart for this structure, too. The scattering transfer parameters for this structure are shown in Figure 8.

These parameters are located along the straight line [7]. In contrast to the previous structure, the metasurface element in this structure plays the role of parallel capacitance in equivalent circuit. The frequency variations in equivalent capacitance are shown in Figure 8. Figure 9 shows the real and imaginary parts of the reflection coefficient in terms of |s11|.

In contrast to what happened about the previous structure, the imaginary part of reflection coefficient is lower than zero, and it has a minimum. This point happens when the stored energy is maximum around metasurface [7]. The variations in the real part of reflection coefficient are similar to the previous structure. Same as the previous structure, the increment in w causes the parameters s11 and s22 to move to 1 along the corresponding circles. Decreasing and increasing frequency causes s11 to move to ((η00)II (η00)I)/((η00)II + (η00)I) and 1, respectively.

The next example is a metasurface composed of square patches (Figure 10). Let metasurface elements be printed on FR4. Physical dimensions of each element are presented in Table 3. Figure 10 displays simulation results of this structure. The variations in the scattering parameters on the Smith chart are presented in the same figure.


Table 2. Geometrical properties of the metal strip grating (Figure 7) [7].

when θ equals to π, the value of s11 is 1. It is clearly predictable from the model used for

Figure 5. Frequency variations in equivalent inductance (a) and the variation in S parameters versus the width of strip for

The variations in real and imaginary parts of s11 versus |s11| are shown in Figure 6 for this structure. Clearly seen, with increasing |s11| from zero to one, the real part of s11 uniformly decreases, while the graph of Im(s11) has a maximum where |s11| is equal to 1/√2 (θ = π/2 in Eq. (16)). This point is corresponding to when the excitation of higher-order modes causes the

simulating metasurface.

10 Metamaterials and Metasurfaces

10 GHz (b) [7].

highest stored energy around the structure [7].

Figure 6. The variations in Re(s11) and Im(s11) versus |s11| [7].

Figure 7. Frequency variations in S parameters for metal strip grating in perpendicular polarization (a) on the Smith chart (b) [7].

Figure 8. The scattering transfer parameters for metal strip grating (a) and frequency variations in equivalent capacitance (b) [7].

Figure 9. The variations in Re(s11) and Im(s11) versus |s11| for metal grating in perpendicular polarization [7].

The variations in scattering transfer parameters are demonstrated in Figure 11 for this structure.

An array of square loops comprises the forth metasurface. This array of loops are located in the boundary between air and FR4. Geometrical properties of each element are available in Table 4. Simulation results of this structure are presented in Figure 12. The variations in the scattering parameters on the Smith chart are plotted in Figure 12.

The variations in scattering transfer parameters are demonstrated in Figure 13 for this structure.

element are specified in Table 5. The frequency variations in s11 and s21 for this structure are plotted in Figure 14. Figure 14 shows the variations in S parameters for this structure on the

Figure 11. The variations in scattering transfer parameters for an array of metallic squares [7].

Figure 10. Frequency variations in S parameters for an array of metallic squares (a) on the Smith chart (b) [7].

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a w Medium I Medium II 3 mm 2 mm Air FR4

Table 3. Geometrical properties of the metasurface comprised from metallic squares [7].

Smith chart.

The last structure is square perforated metal plate. A unit cell of this structure is depicted in Figure 14. Suppose that this structure is embedded in air. The geometrical properties of each Investigation into the Behavior of Metasurface by Modal Analysis http://dx.doi.org/10.5772/intechopen.80584 13

Figure 10. Frequency variations in S parameters for an array of metallic squares (a) on the Smith chart (b) [7].


Table 3. Geometrical properties of the metasurface comprised from metallic squares [7].

Figure 11. The variations in scattering transfer parameters for an array of metallic squares [7].

The variations in scattering transfer parameters are demonstrated in Figure 11 for this structure. An array of square loops comprises the forth metasurface. This array of loops are located in the boundary between air and FR4. Geometrical properties of each element are available in Table 4. Simulation results of this structure are presented in Figure 12. The variations in the

Figure 9. The variations in Re(s11) and Im(s11) versus |s11| for metal grating in perpendicular polarization [7].

Figure 8. The scattering transfer parameters for metal strip grating (a) and frequency variations in equivalent capacitance (b) [7].

12 Metamaterials and Metasurfaces

The variations in scattering transfer parameters are demonstrated in Figure 13 for this structure. The last structure is square perforated metal plate. A unit cell of this structure is depicted in Figure 14. Suppose that this structure is embedded in air. The geometrical properties of each

scattering parameters on the Smith chart are plotted in Figure 12.

element are specified in Table 5. The frequency variations in s11 and s21 for this structure are plotted in Figure 14. Figure 14 shows the variations in S parameters for this structure on the Smith chart.

The variations in scattering transfer parameter corresponding to this structure are demonstrated in Figure 15. The variations in S parameters for this metasurface is in the same way of the first structure. The s11 for this structure goes to 1 when w increases or frequency decreases. In contrast, increasing the frequency causes the parameter s11 to move to 0 along circle.


Table 4. Geometrical properties of the array of metal loops [7].

Figure 14. Frequency variations in S parameters for a square perforated metal plate (a) on the Smith chart (b) [7].

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a w Medium I Medium II 3 mm 1 mm Air Air

Table 5. Geometrical properties of the square perforated metal plate [7].

Figure 15. The locus of scattering transfer parameters for the square perforated metal plate [7].

Figure 12. Frequency variations in S parameters for an array of metal loops (a) on the Smith chart (b) [7].

Figure 13. The locus of scattering transfer parameters [7].

Investigation into the Behavior of Metasurface by Modal Analysis http://dx.doi.org/10.5772/intechopen.80584 15

Figure 14. Frequency variations in S parameters for a square perforated metal plate (a) on the Smith chart (b) [7].


Table 5. Geometrical properties of the square perforated metal plate [7].

The variations in scattering transfer parameter corresponding to this structure are demonstrated in Figure 15. The variations in S parameters for this metasurface is in the same way of the first structure. The s11 for this structure goes to 1 when w increases or frequency decreases. In

a l w Medium I Medium II 1 mm 0.565 mm 0.13 mm Air FR4

contrast, increasing the frequency causes the parameter s11 to move to 0 along circle.

Figure 12. Frequency variations in S parameters for an array of metal loops (a) on the Smith chart (b) [7].

Table 4. Geometrical properties of the array of metal loops [7].

14 Metamaterials and Metasurfaces

Figure 13. The locus of scattering transfer parameters [7].

Figure 15. The locus of scattering transfer parameters for the square perforated metal plate [7].
