4. The derivation for circular straight waveguide

The wave Eqs. (1) and (2) are given in the case of the circular straight waveguide, where

$$\boldsymbol{\varepsilon}(r) = \boldsymbol{\varepsilon}\_0[\mathbf{1} + \boldsymbol{\chi}\_0 \mathbf{g}(r)] \text{ and } \mathbf{g}\_r(r) = [\mathbf{1}/\boldsymbol{\varepsilon}(r)][\partial \boldsymbol{\varepsilon}(r)/\partial r].$$

The proposed technique to calculate the refractive index for discontinuous problems (Figure 1(f) and (g)) is given in this section for the one dielectric coating (Figure 1(f)) and for three dielectric coatings (Figure 1(g)).

#### 4.1 The refractive index for the circular hollow waveguide with one dielectric coating in the cross section

The cross section of the hollow waveguide (Figure 1(f)) is made of a tube of various types of one dielectric layer and a metallic layer. The refractive indices of The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave… DOI: http://dx.doi.org/10.5772/intechopen.80943

the air, dielectric, and metallic layers are nð Þ <sup>0</sup> ¼ 1, nð Þ AgI ¼ 2, and nð Þ Ag ¼ 10 � j60, respectively. The value of the refractive index of the material at a wavelength of λ = 10.6 μm is taken from the table performed by Miyagi et al. [19]. The refractive indices of the air, dielectric layer (AgI), and metallic layer (Ag) are shown in Figure 1(f).

The refractive index (n(r)) is dependent on the transition's regions in the cross section between the two different materials (air-AgI, AgI-Ag).

The refractive index is calculated as follows:

3.5 The hollow rectangular waveguide with the dielectric material between the

The dielectric profile of the hollow rectangular waveguide with the dielectric material between the hollow rectangle and the metal (Figure 1(e)) is calculated by subtracting the dielectric profile of Figure 1(b) from the dielectric profile of

> <sup>g</sup><sup>00</sup> <sup>g</sup>�<sup>10</sup> <sup>g</sup>�<sup>20</sup> … <sup>g</sup>�nm … <sup>g</sup>�NM <sup>g</sup><sup>10</sup> <sup>g</sup><sup>00</sup> <sup>g</sup>�<sup>10</sup> … <sup>g</sup>�ð Þ <sup>n</sup>�<sup>1</sup> <sup>m</sup> … <sup>g</sup>�ð Þ <sup>N</sup>�<sup>1</sup> <sup>M</sup>

: (12)

gNM … …… … … g<sup>00</sup>

Similarly, the Gx and Gy matrices are obtained by the derivatives of the dielectric profile. These matrices relate to the method that is based on the Laplace and Fourier transforms and the inverse Laplace and Fourier transforms [16]. Laplace transform is necessary to obtain the comfortable and simple input-output connections of the fields. The output transverse fields are computed by the inverse Laplace and Fourier

This method becomes an improved method by using the proposed technique and the particular application also in the cases of discontinuous problems of the hollow rectangular waveguide with dielectric material between the hollow rectangle and the metal (Figure 1(e)), in the cross section of the straight rectangular waveguide. In addition, we can find the thickness of the dielectric layer that is recommended to

Several examples will demonstrate in the next section in order to understand the influence of the hollow rectangular waveguide with dielectric material in the cross section (Figure 1) on the output field. All the graphical results will be demonstrated as a response to a half-sine (TE10) input-wave profile and the hollow rectangular waveguide with dielectric material in the cross section of the straight rectangular waveguide.

The wave Eqs. (1) and (2) are given in the case of the circular straight waveguide,

<sup>ε</sup>ð Þ¼ <sup>r</sup> <sup>ε</sup><sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>0</sup> ½ � g rð Þ and grð Þ¼ <sup>r</sup> ½ � <sup>1</sup>=εð Þ<sup>r</sup> ½ � <sup>∂</sup>εð Þ<sup>r</sup> <sup>=</sup>∂<sup>r</sup> :

The proposed technique to calculate the refractive index for discontinuous problems (Figure 1(f) and (g)) is given in this section for the one dielectric coating

4.1 The refractive index for the circular hollow waveguide with one dielectric

The cross section of the hollow waveguide (Figure 1(f)) is made of a tube of various types of one dielectric layer and a metallic layer. The refractive indices of

g<sup>20</sup> g<sup>10</sup> ⋱⋱ ⋱ ⋮ g<sup>20</sup> ⋱⋱ ⋱ gnm ⋱ ⋱⋱ g<sup>00</sup> ⋮

hollow rectangle and the metallic

Electromagnetic Materials and Devices

The matrix G is given by the form

⋮

obtain the desired behavior of the output fields.

4. The derivation for circular straight waveguide

(Figure 1(f)) and for three dielectric coatings (Figure 1(g)).

coating in the cross section

G ¼

Figure 1(d).

transforms.

where

58

$$m(r) = \begin{cases} n\_0 & 0 \le r < b - \varepsilon\_1/2 \\ n\_0 + (n\_d - n\_0) \exp\left[1 - \frac{\varepsilon\_1^2}{\varepsilon\_1^2 - \left[r - \left(b + \varepsilon\_1/2\right)\right]^2}\right] & b - \varepsilon\_1/2 \le r < b + \varepsilon\_1/2 \\ n\_d & b + \varepsilon\_1/2 \le r < a - \varepsilon\_2/2, \\ n\_d + (n\_m - n\_d) \exp\left[1 - \frac{\varepsilon\_2^2}{\varepsilon\_2^2 - \left[r - \left(a + \varepsilon\_2/2\right)\right]^2}\right] & a - \varepsilon\_2/2 \le r < a + \varepsilon\_2/2 \\ n\_m & \text{else} \end{cases}$$

where the internal and external diameters are denoted as 2b, 2a, and 2(a+δm), respectively, where δ<sup>m</sup> is the metallic layer. The thickness of the dielectric coating (d) is defined as [a � b], and the thickness of the metallic layer (δm) is defined as [(a+δm) � a]. The parameter ε is very small [ε = [a � b]/50]. The refractive indices of the air, dielectric, and metallic layers are denoted as n0, nd, and nm, respectively.

### 4.2 The refractive index for the circular hollow waveguide with three dielectric coatings in the cross section

The cross section of the hollow waveguide (Figure 1(g)) is made of a tube of various types of three dielectric layers and a metallic layer. The internal and external diameters are denoted as 2b, 2 b1, 2 b2, 2a, and 2(a + δm), respectively, where δ<sup>m</sup> is the thickness of the metallic layer. In addition, we denote the thickness of the dielectric layers as d1, d2, and d3, respectively, where d<sup>1</sup> = b<sup>1</sup> � b, d<sup>2</sup> = b<sup>2</sup> � b1, and d<sup>3</sup> = a � b2. The refractive index in the particular case with the three dielectric layers and the metallic layer in the cross section of the straight hollow waveguide (Figure 1(g)) is calculated as follows:

$$n(r) = \begin{cases} n\_0 & 0 \le r < b - \varepsilon/2 \\ n\_0 + (n\_1 - n\_0) \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[r - \left(b + \varepsilon/2\right)\right]^2}\right] & b - \varepsilon/2 \le r < b + \varepsilon/2 \\ n\_1 & b + \varepsilon/2 \le r < b\_1 - \varepsilon/2 \\ n\_1 + (n\_2 - n\_1) \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[r - \left(b\_1 + \varepsilon/2\right)\right]^2}\right] & b\_1 - \varepsilon/2 \le r < b\_1 + \varepsilon/2 \\ n\_2 & b\_1 + \varepsilon/2 \le r < b\_2 - \varepsilon/2 \\ n\_2 + (n\_3 - n\_2) \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[r - \left(b\_2 + \varepsilon/2\right)\right]^2}\right] & b\_2 - \varepsilon/2 \le r < b\_2 + \varepsilon/2 \\ n\_3 & b\_2 + \varepsilon/2 \le r < a - \varepsilon/2 \\ n\_3 + (n\_m - n\_3) \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[r - \left(a + \varepsilon/2\right)\right]^2}\right] & a - \varepsilon/2 \le r < a + \varepsilon/2 \\ n\_m & \varepsilon \le \end{cases}$$

where the parameter ε is very small [ε = ½ � a � b =50]. The refractive indices of the air, dielectric, and metallic layers are denoted as n0, n1, n2, n3, and nm, respectively. In this study we suppose that n<sup>3</sup> > n<sup>2</sup> > n1.

The proposed technique to calculate the refractive indices of the dielectric profile of one dielectric coating (Figure 1(f)) or three dielectric coatings (Figure 1(g)), and the metallic layer in the cross section relate to the method that is based on Maxwell's equations, the Fourier-Bessel series, Laplace transform, and the inverse Laplace transform by the residue method [17]. This method becomes an improved method by using the proposed technique also in the cases of discontinuous problems of the hollow circular waveguide with one dielectric coating (Figure 1(f)), three dielectric coatings (Figure 1(g)), or more dielectric coatings.

#### 5. Numerical results

Several examples for the rectangular and circular waveguides with the discontinuous dielectric profile in the cross section of the straight waveguide are demonstrated in this section according to Figure 1(a)–(g).

Figure 4(a)–(c) demonstrates the output field as a response to a half-sine (TE10) input-wave profile in the case of the slab profile (Figure 1(a)), where a = b = 20 mm, c = 20 mm, and d = 2 mm, for ε<sup>r</sup> = 3, 4, and 5, respectively. Figure 4 (c) shows the output field for ε<sup>r</sup> = 3, 4, and 5, respectively, where y = b/2 = 10 mm.

By increasing only the value of the dielectric profile from ε<sup>r</sup> = 3 to ε<sup>r</sup> = 5, the width of the output field decreased, and also the output amplitude decreased.

Figure 5(a)–(e) demonstrates the output field as a response to a half-sine (TE10) input-wave profile in the case of the rectangular dielectric profile in the rectangular waveguide (Figure 1(b)), where a = b = 20 mm and c = d = 2 mm, for ε<sup>r</sup> = 3, 5, 7, and 10, respectively. Figure 5(e) shows the output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

By increasing only the dielectric profile from ε<sup>r</sup> = 3 to ε<sup>r</sup> = 5, the width of the output field increased, and also the output amplitude increased.

The output fields are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the location, and the dielectric profile, as shown in Figure 4(a)–(c) and Figure 5(a)–(e).

Figure 6(a)–(e) shows the output field as a response to a half-sine (TE10) inputwave profile in the case of the circular dielectric profile (Figure 1(c)), for ε<sup>r</sup> = 3, 5, 7 and 10, respectively, where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 1 mm. Figure 6(e) shows the output field for ε<sup>r</sup> =3, 5, 7, and 10, respectively, where y = b/2 = 10 mm. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

The proposed technique in Section 3.3 is also effective to solve discontinuous problems of periodic circular profiles in the cross section of the straight rectangular

The output field as a response to a half-sine (TE10) input-wave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 1 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm,

The output field as a response to a half-sine (TE10) input-wave profile in the case of the rectangular dielectric material (Figure 1(b)), where a = b = 20 mm and c = d = 2 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are a = b = 20 mm, z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave…

DOI: http://dx.doi.org/10.5772/intechopen.80943

(e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

and β = 58 1=m. (e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

The behavior of the output fields (Figures 5(a)–(e) and 6(a)–(e)) is similar when the dimensions of the rectangular dielectric profile (Figure 1(b)) and the circular profile (Figure 1(c)) are very close. The output field (Figure 5(a)–(e)) is

waveguides, and some examples were demonstrated in Ref. [20].

Figure 6.

61

Figure 5.

Figure 4.

The output field as a response to a half-sine (TE10) input-wave profile in the case of the slab dielectric profile (Figure 1(a)), where a = b = 20 mm, c = 20 mm, and d = 2 mm where (a) ε<sup>r</sup> = 3 and (b) ε<sup>r</sup> = 5. (c). The output field for ε<sup>r</sup> =3, 4, and 5, respectively, where y = b/2 = 10 mm.

The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave… DOI: http://dx.doi.org/10.5772/intechopen.80943

#### Figure 5.

where the parameter ε is very small [ε = ½ � a � b =50]. The refractive indices of the air, dielectric, and metallic layers are denoted as n0, n1, n2, n3, and nm, respectively.

The proposed technique to calculate the refractive indices of the dielectric profile of one dielectric coating (Figure 1(f)) or three dielectric coatings (Figure 1(g)), and the metallic layer in the cross section relate to the method that is based on Maxwell's equations, the Fourier-Bessel series, Laplace transform, and the inverse Laplace transform by the residue method [17]. This method becomes an improved method by using the proposed technique also in the cases of discontinuous problems of the hollow circular waveguide with one dielectric coating (Figure 1(f)),

Several examples for the rectangular and circular waveguides with the discontinuous dielectric profile in the cross section of the straight waveguide are demon-

Figure 5(a)–(e) demonstrates the output field as a response to a half-sine (TE10) input-wave profile in the case of the rectangular dielectric profile in the rectangular waveguide (Figure 1(b)), where a = b = 20 mm and c = d = 2 mm, for ε<sup>r</sup> = 3, 5, 7, and 10, respectively. Figure 5(e) shows the output field for ε<sup>r</sup> = 3, 5, 7, and 10,

By increasing only the dielectric profile from ε<sup>r</sup> = 3 to ε<sup>r</sup> = 5, the width of the

The output field as a response to a half-sine (TE10) input-wave profile in the case of the slab dielectric profile (Figure 1(a)), where a = b = 20 mm, c = 20 mm, and d = 2 mm where (a) ε<sup>r</sup> = 3 and (b) ε<sup>r</sup> = 5. (c). The

output field for ε<sup>r</sup> =3, 4, and 5, respectively, where y = b/2 = 10 mm.

The output fields are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the location, and the dielectric profile, as shown in Figure 4(a)–(c) and Figure 5(a)–(e). Figure 6(a)–(e) shows the output field as a response to a half-sine (TE10) inputwave profile in the case of the circular dielectric profile (Figure 1(c)), for ε<sup>r</sup> = 3, 5, 7 and 10, respectively, where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 1 mm. Figure 6(e) shows the output field for ε<sup>r</sup> =3, 5, 7, and 10, respectively, where y = b/2 = 10 mm. The other parameters are z = 0.15 m, k<sup>0</sup> = 167

output field increased, and also the output amplitude increased.

Figure 4(a)–(c) demonstrates the output field as a response to a half-sine (TE10) input-wave profile in the case of the slab profile (Figure 1(a)), where a = b = 20 mm, c = 20 mm, and d = 2 mm, for ε<sup>r</sup> = 3, 4, and 5, respectively. Figure 4 (c) shows the output field for ε<sup>r</sup> = 3, 4, and 5, respectively, where y = b/2 = 10 mm. By increasing only the value of the dielectric profile from ε<sup>r</sup> = 3 to ε<sup>r</sup> = 5, the width of the output field decreased, and also the output amplitude decreased.

three dielectric coatings (Figure 1(g)), or more dielectric coatings.

strated in this section according to Figure 1(a)–(g).

respectively, where y = b/2 = 10 mm.

1=m, λ = 3.75 cm, and β = 58 1=m.

Figure 4.

60

In this study we suppose that n<sup>3</sup> > n<sup>2</sup> > n1.

Electromagnetic Materials and Devices

5. Numerical results

The output field as a response to a half-sine (TE10) input-wave profile in the case of the rectangular dielectric material (Figure 1(b)), where a = b = 20 mm and c = d = 2 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are a = b = 20 mm, z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. (e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

#### Figure 6.

The output field as a response to a half-sine (TE10) input-wave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 1 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. (e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

The proposed technique in Section 3.3 is also effective to solve discontinuous problems of periodic circular profiles in the cross section of the straight rectangular waveguides, and some examples were demonstrated in Ref. [20].

The behavior of the output fields (Figures 5(a)–(e) and 6(a)–(e)) is similar when the dimensions of the rectangular dielectric profile (Figure 1(b)) and the circular profile (Figure 1(c)) are very close. The output field (Figure 5(a)–(e)) is shown for c = d = 2 mm as regards to the dimensions a = b = 20 mm. The output field (Figure 6(a)–(e)) is shown where the radius of circular profile is equal to 1 mm (viz., the diameter 2 mm), as regards to the dimensions a = b = 20 mm.

Figure 7(a)–(e) shows the output field as a response to a half-sine (TE10) inputwave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 2 mm for ε<sup>r</sup> = 3, 5, 7, and 10, respectively. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. Figure 7(e) shows the output field for ε<sup>r</sup> =3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

By changing only the value of the radius of the circular dielectric profile (Figure 1(c)) from 1 mm to 2 mm, as regards to the dimensions of the cross section of the waveguide (a = b = 20 mm), the output field of the Gaussian shape increased, and the half-sine (TE10) input-wave profile decreased.

The dielectric profile of the hollow rectangular waveguide with the dielectric material between the hollow rectangle and the metal (Figure 1(e)) is calculated by subtracting the dielectric profile of the waveguide with the dielectric material in the core (Figure 1(b)) from the dielectric profile according to the waveguide entirely with the dielectric profile (Figure 1(d)).

Figure 8(a)–(c) shows the output field as a response to a half-sine (TE10) inputwave profile in the case of the hollow rectangular waveguide with one dielectric material between the hollow rectangle and the metal (Figure 1(e)), where a = b = 20 mm, c = d = 14 mm, and d = 14 mm, namely, e = 3 mm and f = 3 mm. Figure 8(a)–(b) shows the output field for ε<sup>r</sup> = 2.5 and ε<sup>r</sup> = 4, respectively. Figure 8(c) shows the output field for ε<sup>r</sup> = 2.5, 3, 3.5, and 4, respectively, where y = b/2 = 10 mm. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

Figure 9(a)–(c) shows the output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(f)), where a = 0.5 mm. Figure 9(a)–(b) shows the output power density for w<sup>0</sup> = 0.15 mm and w<sup>0</sup> = 0.25 mm, respectively. The output power density of the central peak is shown

for w<sup>0</sup> = 0.15 mm, w<sup>0</sup> = 0.2 mm, and w<sup>0</sup> = 0.25 mm, respectively, where y = b/2. The

The output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(e)), where a = 0.5 mm: (a) w<sup>0</sup> = 0.15 mm, and (b) w<sup>0</sup> = 0.25 mm. (c). The output power density of the central peak for w<sup>0</sup> = 0.15 mm, w<sup>0</sup> = 0.2 mm, and w<sup>0</sup> = 0.25 mm, respectively, where y = b/2. The other parameters

The output field as a response to a half-sine (TE10) input-wave profile in the case of the hollow rectangular waveguide with one dielectric material between the hollow rectangle and the metal (Figure 1(e)), where a = b = 20 mm, c = 14 mm, and d = 14 mm, namely, e = 3 mm and f = 3 mm: (a) ε<sup>r</sup> = 2.5, and (b) ε<sup>r</sup> = 4. (c). The output field for ε<sup>r</sup> = 2.5, 3, 3.5, and 4, respectively, where y = b/2 = 10 mm. The other parameters are

The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave…

Figure 1(f) and (g) shows two examples of discontinuous problems for circular

Figure 10(a)–(c) shows also the output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(f)), where a = 0.5 mm, but for other values of the spot size. Figure 10(a)–(b) shows the output power density for w<sup>0</sup> = 0.26 mm and w<sup>0</sup> = 0.3 mm, respectively. The output power density of the

other parameters are z = 1 m, nd = 2.2, and nð Þ Ag = 13.5 - j 75.3.

are z = 1 m, nd =2.2, and nð Þ Ag = 13.5 - j 75.3.

z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

DOI: http://dx.doi.org/10.5772/intechopen.80943

13.5 � j 75.3.

Figure 10.

63

z = 1 m, nd = 2.2, and nð Þ Ag = 13.5 � j 75.3.

Figure 9.

Figure 8.

waveguides. The practical results are demonstrated for Figure 1(f).

central peak is shown for w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm,

respectively, where y = b/2. The other parameters are z = 1 m, nd = 2.2, and nð Þ Ag =

The output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(e)), where a = 0.5 mm: (a) w<sup>0</sup> = 0.26 mm, and (b) w<sup>0</sup> = 0.3 mm. (c). The output power density of the central peak for w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively, where y = b/2. The other parameters are

#### Figure 7.

The output field as a response to a half-sine (TE10) input-wave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 2 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. (e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave… DOI: http://dx.doi.org/10.5772/intechopen.80943

Figure 8.

shown for c = d = 2 mm as regards to the dimensions a = b = 20 mm. The output field (Figure 6(a)–(e)) is shown where the radius of circular profile is equal to 1 mm

Figure 7(a)–(e) shows the output field as a response to a half-sine (TE10) input-

(viz., the diameter 2 mm), as regards to the dimensions a = b = 20 mm.

10, respectively, where y = b/2 = 10 mm.

Electromagnetic Materials and Devices

with the dielectric profile (Figure 1(d)).

and β = 58 1=m.

Figure 7.

62

and the half-sine (TE10) input-wave profile decreased.

wave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 2 mm for ε<sup>r</sup> = 3, 5, 7, and 10, respectively. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. Figure 7(e) shows the output field for ε<sup>r</sup> =3, 5, 7, and

By changing only the value of the radius of the circular dielectric profile (Figure 1(c)) from 1 mm to 2 mm, as regards to the dimensions of the cross section of the waveguide (a = b = 20 mm), the output field of the Gaussian shape increased,

The dielectric profile of the hollow rectangular waveguide with the dielectric material between the hollow rectangle and the metal (Figure 1(e)) is calculated by subtracting the dielectric profile of the waveguide with the dielectric material in the core (Figure 1(b)) from the dielectric profile according to the waveguide entirely

Figure 8(a)–(c) shows the output field as a response to a half-sine (TE10) inputwave profile in the case of the hollow rectangular waveguide with one dielectric material between the hollow rectangle and the metal (Figure 1(e)), where a = b = 20 mm, c = d = 14 mm, and d = 14 mm, namely, e = 3 mm and f = 3 mm. Figure 8(a)–(b) shows the output field for ε<sup>r</sup> = 2.5 and ε<sup>r</sup> = 4, respectively. Figure 8(c) shows the output field for ε<sup>r</sup> = 2.5, 3, 3.5, and 4, respectively, where y = b/2 = 10 mm. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm,

Figure 9(a)–(c) shows the output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(f)), where a = 0.5 mm.

w<sup>0</sup> = 0.25 mm, respectively. The output power density of the central peak is shown

The output field as a response to a half-sine (TE10) input-wave profile in the case of the circular dielectric profile (Figure 1(c)), where a = b = 20 mm, and the radius of the circular dielectric profile is equal to 2 mm: (a) ε<sup>r</sup> = 3, (b) ε<sup>r</sup> = 5, (c) ε<sup>r</sup> = 7, and (d) ε<sup>r</sup> = 10. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and

β = 58 1=m. (e) The output field for ε<sup>r</sup> = 3, 5, 7, and 10, respectively, where y = b/2 = 10 mm.

Figure 9(a)–(b) shows the output power density for w<sup>0</sup> = 0.15 mm and

The output field as a response to a half-sine (TE10) input-wave profile in the case of the hollow rectangular waveguide with one dielectric material between the hollow rectangle and the metal (Figure 1(e)), where a = b = 20 mm, c = 14 mm, and d = 14 mm, namely, e = 3 mm and f = 3 mm: (a) ε<sup>r</sup> = 2.5, and (b) ε<sup>r</sup> = 4. (c). The output field for ε<sup>r</sup> = 2.5, 3, 3.5, and 4, respectively, where y = b/2 = 10 mm. The other parameters are z = 0.15 m, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

Figure 9.

The output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(e)), where a = 0.5 mm: (a) w<sup>0</sup> = 0.15 mm, and (b) w<sup>0</sup> = 0.25 mm. (c). The output power density of the central peak for w<sup>0</sup> = 0.15 mm, w<sup>0</sup> = 0.2 mm, and w<sup>0</sup> = 0.25 mm, respectively, where y = b/2. The other parameters are z = 1 m, nd =2.2, and nð Þ Ag = 13.5 - j 75.3.

for w<sup>0</sup> = 0.15 mm, w<sup>0</sup> = 0.2 mm, and w<sup>0</sup> = 0.25 mm, respectively, where y = b/2. The other parameters are z = 1 m, nd = 2.2, and nð Þ Ag = 13.5 - j 75.3.

Figure 1(f) and (g) shows two examples of discontinuous problems for circular waveguides. The practical results are demonstrated for Figure 1(f).

Figure 10(a)–(c) shows also the output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(f)), where a = 0.5 mm, but for other values of the spot size. Figure 10(a)–(b) shows the output power density for w<sup>0</sup> = 0.26 mm and w<sup>0</sup> = 0.3 mm, respectively. The output power density of the central peak is shown for w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively, where y = b/2. The other parameters are z = 1 m, nd = 2.2, and nð Þ Ag =

13.5 � j 75.3.

Figure 10.

The output power density in the case of the hollow circular waveguide with one dielectric coating (Figure 1(e)), where a = 0.5 mm: (a) w<sup>0</sup> = 0.26 mm, and (b) w<sup>0</sup> = 0.3 mm. (c). The output power density of the central peak for w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively, where y = b/2. The other parameters are z = 1 m, nd = 2.2, and nð Þ Ag = 13.5 � j 75.3.

By changing only the values of the spot size from w<sup>0</sup> = 0.15 mm, w<sup>0</sup> = 0.2 mm, and w<sup>0</sup> = 0.25 mm to w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively, the results of the output power density for a = 0.5 mm are changed as shown in Figure 10(a)–10(c).

respectively, the results of the output power density for a = 0.5 mm are changed as

The Influence of the Dielectric Materials on the Fields in the Millimeter and Infrared Wave…

values of w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively.

The two important parameters that we studied were the spot size and the dimensions of the cross section of the straight hollow waveguide. The output results are affected by the parameters of the spot size and the dimensions of the cross

Department of Electrical Engineering, Sami Shamoon College of Engineering,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: zionm@post.tau.ac.il

provided the original work is properly cited.

The output modal profile is greatly affected by the parameters of the spot size and the dimensions of the cross section of the waveguide. Figure 10(a)–(c) demonstrates that in addition to the main propagation mode, several other secondary modes and symmetric output shape appear in the results of the output power density for the

shown in Figure 10(a)–(c).

DOI: http://dx.doi.org/10.5772/intechopen.80943

section of the waveguide.

Author details

Zion Menachem

Beer Sheva, Israel

65

The output modal profile is greatly affected by the parameters of the spot size and the dimensions of the cross section of the waveguide. Figure 10(a)–(c) demonstrates that in addition to the main propagation mode, several other secondary modes and symmetric output shape appear in the results of the output power density for the values of w<sup>0</sup> = 0.26 mm, w<sup>0</sup> = 0.28 mm, and w<sup>0</sup> = 0.3 mm, respectively.

The proposed technique in Section 4.2 is also effective to solve discontinuous problems of the straight hollow circular waveguide with three dielectric layers (Figure 1(g)), and some examples were demonstrated in Ref. [21].
