3.5 The hollow rectangular waveguide with the dielectric material between the hollow rectangle and the metallic

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

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

The refractive index (n(r)) is dependent on the transition's regions in the cross

n<sup>0</sup> 0≤ r < b � ε1=2

" #

" #

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

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

n<sup>0</sup> 0≤r < b � ε=2

" #

" #

" #

" #

nm else

<sup>ε</sup><sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>=<sup>2</sup> <sup>2</sup>

n<sup>1</sup> b þ ε=2 ≤r < b<sup>1</sup> � ε=2

<sup>ε</sup><sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>=<sup>2</sup> <sup>2</sup>

<sup>ε</sup><sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>=<sup>2</sup> <sup>2</sup>

n<sup>3</sup> b<sup>2</sup> þ ε=2≤ r < a � ε=2

<sup>ε</sup><sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>=<sup>2</sup> <sup>2</sup>

n<sup>2</sup> b<sup>1</sup> þ ε=2 ≤r < b<sup>2</sup> � ε=2

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup>

nd b þ ε1=2≤ r < a � ε2=2

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup>

b � ε1=2≤ r < b þ ε1=2

,

a � ε2=2 ≤r < a þ ε2=2

b � ε=2 ≤r < b þ ε=2

b<sup>1</sup> � ε2=2≤ r < b<sup>1</sup> þ ε2=2

,

b<sup>2</sup> � ε=2≤ r < b<sup>2</sup> þ ε=2

a � ε=2 ≤r < a þ ε=2

section between the two different materials (air-AgI, AgI-Ag).

ε1

ε2

nm else

The refractive index is calculated as follows:

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

<sup>n</sup><sup>0</sup> <sup>þ</sup> ð Þ nd � <sup>n</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>1</sup>

nd <sup>þ</sup> ð Þ nm � nd exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>2</sup> � <sup>n</sup><sup>1</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>3</sup> <sup>þ</sup> ð Þ nm � <sup>n</sup><sup>3</sup> exp 1 � <sup>ε</sup><sup>2</sup>

coatings in the cross section

Figure 1(f).

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

calculated as follows:

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

n rð Þ¼

59

n rð Þ¼

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 Figure 1(d).

The matrix G is given by the form

$$G = \begin{bmatrix} \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \mathcal{g}\_{-20} & \cdots & \mathcal{g}\_{-nm} & \cdots & \mathcal{g}\_{-NM} \\\\ \mathcal{g}\_{10} & \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \cdots & \mathcal{g}\_{-(n-1)m} & \cdots & \mathcal{g}\_{-(N-1)M} \\\\ \mathcal{g}\_{20} & \mathcal{g}\_{10} & \ddots & \ddots & \ddots & & \\ \vdots & \mathcal{g}\_{20} & \ddots & \ddots & \ddots & & \\ \mathcal{g}\_{mn} & \ddots & \ddots & \ddots & \mathcal{g}\_{00} & \vdots \\\\ \vdots & & & & & \\ \mathcal{g}\_{NM} & \cdots & \cdots & \cdots & \cdots & \mathcal{g}\_{00} \end{bmatrix}. \tag{12}$$

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 transforms.

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 obtain the desired behavior of the output fields.

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.
