2. Two proposed techniques for discontinuous problems in the cross section of the rectangular and circular waveguides

The wave equations for the components of the electric and magnetic field are given by

$$
\nabla^2 E + \alpha^2 \mu \varepsilon E + \nabla \left( E \cdot \frac{\nabla \varepsilon}{\varepsilon} \right) = \mathbf{0} \tag{1}
$$

ωεð Þ¼ <sup>r</sup> <sup>C</sup>ε<sup>e</sup>

where C<sup>ε</sup> is a constant and Ð

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

<sup>g</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>g</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>g</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>g</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>2</sup>

ab <sup>ð</sup>ð Þ <sup>a</sup>�dþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ a�d�ε =2

> ðð Þ <sup>a</sup>þdþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ aþd�ε =2

ðð Þ <sup>b</sup>þcþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ bþc�ε =2

exp 1 � <sup>ε</sup><sup>2</sup>

given by

8

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

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

8

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

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

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

cos

cos

nπx a � �dx <sup>þ</sup>

mπy b � �dy <sup>þ</sup>

ðð Þ <sup>b</sup>�cþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ b�c�ε =2

g xð Þ¼

and

g yð Þ¼

b 6¼ c by

ðð Þ <sup>a</sup>þd�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ a�dþε =2

ðð Þ <sup>b</sup>þc�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ b�cþε =2

where

þ

þ

55

(

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

ωεð Þr dr ¼ 1:

0 0≤x < ð Þ a � d � ε =2

0 ð Þ a þ d þ ε =2 < x≤a

0 0≤y < ð Þ b � c � ε =2

0 ð Þ b þ c þ ε =2 < y≤b

The elements of the matrices are given according to Figure 1(b), in the case of

exp 1 � <sup>ε</sup><sup>2</sup>

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

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

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

" #

cos

" #

" #

� �dx (

exp 1 � <sup>ε</sup><sup>2</sup>

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

exp 1 � <sup>ε</sup><sup>2</sup>

" #

� �dy (

g<sup>0</sup> ð Þ a � d þ ε =2 < x < ð Þ a þ d � ε =2

g<sup>0</sup> ð Þ b � c þ ε =2 < y < ð Þ b þ c � ε =2

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

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

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

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

" #

" #

" #

" #

� <sup>ε</sup><sup>2</sup>

In order to solve inhomogeneous dielectric profiles (e.g., in Figure 1(a)–(b)) in the cross section of the straight waveguide, the parameter ε is used according to the ωε function (Figure 2(b)), where ε ! 0. The dielectric profile in this case of a rectangular dielectric material in the rectangular cross section (Figure 1(b)) is

<sup>ε</sup>2�j j<sup>r</sup> <sup>2</sup> j jr ≤ ε 0 j jr > ε

;

ð Þ a � d � ε =2 ≤x < ð Þ a � d þ ε =2

ð Þ a þ d � ε =2 ≤x < ð Þ a þ d þ ε =2

ð Þ b � c � ε =2≤y < ð Þ b � c þ ε =2

ð Þ b þ c � ε =2≤y < ð Þ b þ c þ ε =2

cos

mπy b

nπx a

cos

cos

nπx a � �dx)

mπy b � �dy)

,

(6)

(3)

,

(4)

:

(5)

and

$$
\nabla^2 H + \alpha^2 \mu \epsilon H + \frac{\nabla \epsilon}{\varepsilon} \times (\nabla \times H) = \mathbf{0} \tag{2}
$$

where ε<sup>0</sup> represents the vacuum dielectric constant, χ<sup>0</sup> is the susceptibility, and g is its dielectric profile function in the waveguide.

Let us introduce the dielectric profile function for the examples as shown in Figure 1(a)–(g) for the inhomogeneous dielectric materials.

## 3. The derivation for rectangular straight waveguide

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

$$\begin{aligned} \boldsymbol{\varepsilon}(\boldsymbol{\varepsilon}, \boldsymbol{y}) &= \boldsymbol{\varepsilon}\_0(\mathbf{1} + \boldsymbol{\chi}\_0 \mathbf{g}(\boldsymbol{\varepsilon}, \boldsymbol{y})), \mathbf{g}\_\mathbf{x} = [\mathbf{1}/\boldsymbol{\varepsilon}(\mathbf{x}, \boldsymbol{y})][\partial \boldsymbol{\varepsilon}(\mathbf{x}, \boldsymbol{y})/\partial \boldsymbol{\varepsilon}], \text{ and} \\ \mathbf{g}\_\mathbf{y} &= [\mathbf{1}/\boldsymbol{\varepsilon}(\mathbf{x}, \boldsymbol{y})][\partial \boldsymbol{\varepsilon}(\mathbf{x}, \boldsymbol{y})/\partial \boldsymbol{\eta}]. \end{aligned}$$

### 3.1 The first technique to calculate the discontinuous structure of the cross section

Figure 2(a) shows an example of the cross section of the straight waveguide (Figure 1(a)) for g xð Þ function. In order to solve inhomogeneous dielectric profiles, we use the ωε function, with the parameters ε<sup>1</sup> and ε<sup>2</sup> (Figures 1(a) and (b)).

The ωε function [18] is used in order to solve discontinuous problems in the cross section of the straight waveguide. The ωε function is defined according to Figure 2(b) as

Figure 2.

(a) An example of the discontinuous problem of the slab dielectric profile in the straight rectangular waveguide. (b) The ωε function in the limit ε ! 0.

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

$$\rho\_{\varepsilon}(r) = \begin{cases} C\_{\varepsilon} e^{-\frac{r^2}{r^2 - |r|^2}} & |r| \le \varepsilon \\ 0 & |r| > \varepsilon \end{cases},\tag{3}$$

where C<sup>ε</sup> is a constant and Ð ωεð Þr dr ¼ 1:

In order to solve inhomogeneous dielectric profiles (e.g., in Figure 1(a)–(b)) in the cross section of the straight waveguide, the parameter ε is used according to the ωε function (Figure 2(b)), where ε ! 0. The dielectric profile in this case of a rectangular dielectric material in the rectangular cross section (Figure 1(b)) is given by

$$g(\mathbf{x}) = \begin{cases} 0 & 0 \le \mathbf{x} < (a - d - \epsilon)/2 \\ g\_0 \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - (a - d + \epsilon)/2\right]^2}\right] & (a - d - \epsilon)/2 \le \mathbf{x} < (a - d + \epsilon)/2 \\ g\_0 & (a - d + \epsilon)/2 < \mathbf{x} < (a + d - \epsilon)/2 \\ g\_0 \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - (a + d - \epsilon)/2\right]^2}\right] & (a + d - \epsilon)/2 \le \mathbf{x} < (a + d + \epsilon)/2 \\ 0 & (a + d + \epsilon)/2 < \mathbf{x} \le a \end{cases} \tag{4}$$

and

2. Two proposed techniques for discontinuous problems in the cross

The wave equations for the components of the electric and magnetic field are

μεE þ ∇ E �

∇ε

where ε<sup>0</sup> represents the vacuum dielectric constant, χ<sup>0</sup> is the susceptibility, and g

Let us introduce the dielectric profile function for the examples as shown in

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

<sup>ε</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ε</sup><sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>0</sup> ð Þ g xð Þ ; <sup>y</sup> , gx <sup>¼</sup> ½ � <sup>1</sup>=εð Þ <sup>x</sup>; <sup>y</sup> ½ � <sup>∂</sup>εð Þ <sup>x</sup>; <sup>y</sup> <sup>=</sup>∂<sup>x</sup> , and

3.1 The first technique to calculate the discontinuous structure of the cross

Figure 2(a) shows an example of the cross section of the straight waveguide (Figure 1(a)) for g xð Þ function. In order to solve inhomogeneous dielectric profiles, we use the ωε function, with the parameters ε<sup>1</sup> and ε<sup>2</sup> (Figures 1(a) and (b)). The ωε function [18] is used in order to solve discontinuous problems in the cross section of the straight waveguide. The ωε function is defined according to

(a) An example of the discontinuous problem of the slab dielectric profile in the straight rectangular waveguide.

μεH þ

∇ε ε 

¼ 0 (1)

<sup>ε</sup> � ð Þ¼ <sup>∇</sup> � <sup>H</sup> <sup>0</sup> (2)

section of the rectangular and circular waveguides

<sup>∇</sup><sup>2</sup> <sup>E</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

<sup>∇</sup><sup>2</sup><sup>H</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

Figure 1(a)–(g) for the inhomogeneous dielectric materials.

3. The derivation for rectangular straight waveguide

gy <sup>¼</sup> ½ � <sup>1</sup>=εð Þ <sup>x</sup>; <sup>y</sup> ½ � <sup>∂</sup>εð Þ <sup>x</sup>; <sup>y</sup> <sup>=</sup>∂<sup>y</sup> :

is its dielectric profile function in the waveguide.

given by

Electromagnetic Materials and Devices

and

waveguide, where

section

Figure 2(b) as

Figure 2.

54

(b) The ωε function in the limit ε ! 0.

$$g(y) = \begin{cases} 0 & 0 \le y < (b - c - e)/2 \\ g\_0 \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[y - (b - c + e)/2\right]^2}\right] & (b - c - e)/2 \le y < (b - c + e)/2 \\ g\_0 & (b - c + e)/2 < y < (b + c - e)/2 \\ g\_0 \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[y - (b + c - e)/2\right]^2}\right] & (b + c - e)/2 \le y < (b + c + e)/2 \\ 0 & (b + c + e)/2 < y \le b \end{cases} \tag{5}$$

The elements of the matrices are given according to Figure 1(b), in the case of b 6¼ c by

$$\begin{split} g(n,m) &= \frac{g\_0}{ab} \left\{ \int\_{(a-d-\varepsilon)/2}^{(a-d+\varepsilon)/2} \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - (a-d+\varepsilon)/2\right]^2} \right] \cos\left(\frac{n\pi\mathbf{x}}{a}\right) d\mathbf{x} \\ &+ \int\_{(a-d+\varepsilon)/2}^{(a+d-\varepsilon)/2} \cos\left(\frac{n\pi\mathbf{x}}{a}\right) d\mathbf{x} + \int\_{(a+d-\varepsilon)/2}^{(a+d+\varepsilon)/2} \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - (a+d-\varepsilon)/2\right]^2} \right] \cos\left(\frac{n\pi\mathbf{x}}{a}\right) d\mathbf{x} \right\} \\ &\qquad \left\{ \int\_{(b-\varepsilon-\varepsilon)/2}^{(b-\varepsilon+\varepsilon)/2} \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{y} - (b-\varepsilon+\varepsilon)/2\right]^2} \right] \cos\left(\frac{m\pi\mathbf{y}}{b}\right) d\mathbf{y} \\ &+ \int\_{(b-\varepsilon+\varepsilon)/2}^{(b+\varepsilon-\varepsilon)/2} \cos\left(\frac{m\pi\mathbf{y}}{b}\right) d\mathbf{y} + \int\_{(b+\varepsilon-\varepsilon)/2}^{(b+\varepsilon+\varepsilon)/2} \exp\left[1 - \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{y} - (b+\varepsilon-\varepsilon)/2\right]^2} \right] \cos\left(\frac{m\pi\mathbf{y}}{b}\right) d\mathbf{y} \right\}, \end{split} \tag{6}$$

where

$$\int\_{(a-d+\varepsilon)/2}^{(a+d-\varepsilon)/2} \cos\left(\frac{n\pi x}{a}\right) d\pi = \begin{cases} (2a/n\pi)\sin\left((n\pi/2a)(d-\varepsilon)\right)\cos\left((n\pi)/2\right) & n \neq 0\\ d-\varepsilon & n=0 \end{cases}$$

and

$$\int\_{(b-\varepsilon+\varepsilon)/2}^{(b+\varepsilon-\varepsilon)/2} \cos\left(\frac{m\pi y}{b}\right) dy = \begin{cases} (2b/m\pi)\sin\left((m\pi/2b)(c-\varepsilon)\right)\cos\left((m\pi)/2\right) & n \neq 0\\ c-\varepsilon & n=0 \end{cases}$$

The elements of the matrices are given according to Figure 1(a), in the case of b = c by

$$\begin{split} &g\_{\varepsilon}(n,m) = \frac{g\_{0}}{ab} \left\{ \int\_{(a-d-\varepsilon)/2}^{(a-d+\varepsilon)/2} \exp\left[1 - \frac{e^{2}}{\varepsilon^{2} - \left[x - (a-d+\varepsilon)/2\right]^{2}}\right] \cos\left(\frac{n\pi x}{a}\right) dx \\ &+ \int\_{(a-d+\varepsilon)/2}^{(a+d-\varepsilon)/2} \cos\left(\frac{n\pi x}{a}\right) dx + \int\_{(a+d-\varepsilon)/2}^{(a+d+\varepsilon)/2} \exp\left[1 - \frac{e^{2}}{\varepsilon^{2} - \left[x - (a+d-\varepsilon)/2\right]^{2}}\right] \cos\left(\frac{n\pi x}{a}\right) dx \right\} \\ &\qquad \left\{ \int\_{0}^{b} \cos\left(\frac{m\pi y}{b}\right) dy \right\}, \end{split}$$

g nð Þ¼ ; m

Figure 3.

where

g nð Þ¼ ; m

57

g0

The arbitrary profile in the cross section.

8

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

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

g0 4ab

g0 4ab

g0 4ab

8d <sup>k</sup>0ym sin

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

8c

16

g xð Þ¼ ; y

entire cross section

g0 4ab

8 >>>>>>>>>><

>>>>>>>>>>:

g0 4ab

g0 4ab

8a <sup>k</sup>0ym sin

8b

16

<sup>k</sup>0xn sin <sup>k</sup>0xna 2

<sup>k</sup>0xk0ynm sin <sup>k</sup>0xna

<sup>k</sup>0xn sin <sup>k</sup>0xnd 2

<sup>k</sup>0xk0ynm sin <sup>k</sup>0xnd

rectangular waveguide is given by (Figure 1(c))

�

k0ymc 2

� � cos <sup>k</sup>0ymb

� � cos <sup>k</sup>0xna

2

2

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

2

2 � � sin

The dielectric profile for the circular profile in the cross section of the straight

g<sup>0</sup> 0≤r < r<sup>1</sup> � ε1=2 <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � <sup>r</sup><sup>1</sup> � <sup>ε</sup>=<sup>2</sup> <sup>≤</sup><sup>r</sup> <sup>&</sup>lt; <sup>r</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>=<sup>2</sup>

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

g<sup>0</sup> n ¼ 0, m ¼ 0

� � � � <sup>n</sup> <sup>¼</sup> <sup>0</sup>, m 6¼ <sup>0</sup>

� � � � <sup>n</sup> 6¼ <sup>0</sup>, m <sup>¼</sup> <sup>0</sup>

� � � � <sup>n</sup> 6¼ <sup>0</sup>, m 6¼ <sup>0</sup>

k0ymb 2

� � cos <sup>k</sup>0ymb

2

3.4 The dielectric profile for the waveguide filled with dielectric material in the

2

2

2 � � sin

� � cos <sup>k</sup>0xna

� � cos <sup>k</sup>0xna

3.3 The dielectric profile for the circular profile in the cross section

<sup>q</sup>εð Þ¼ <sup>r</sup> <sup>ε</sup><sup>2</sup>

else g xð Þ ; y = 0. The radius of the circle is given by r ¼

The dielectric profile (Figure 1(d)) is given by

k0ymb 2

� � cos <sup>k</sup>0ymb

� � cos <sup>k</sup>0xna

2

<sup>4</sup>ab ð Þ <sup>4</sup>cd <sup>n</sup> <sup>¼</sup> <sup>0</sup>, m <sup>¼</sup> <sup>0</sup>

� � � � <sup>n</sup> <sup>¼</sup> <sup>0</sup>, m 6¼ <sup>0</sup>

� � � � <sup>n</sup> 6¼ <sup>0</sup>, m <sup>¼</sup> <sup>0</sup>

� � � � <sup>n</sup> 6¼ <sup>0</sup>, m 6¼ <sup>0</sup>

k0ymc 2

� � cos <sup>k</sup>0ymb

q

2

,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup>

(9)

(10)

.

(11)

where

$$\int\_0^b \cos\left(\frac{m\pi y}{b}\right) dy = (b/m\pi)\sin\left(m\pi\right) = \begin{cases} b & m = 0\\ 0 & m \neq 0 \end{cases}.$$

#### 3.2 The second technique to calculate the discontinuous structure of the cross section

The second technique to calculate the discontinuous structure of the cross section as shown in Figure 1(a) and (b).

The dielectric profile g xð Þ ; y is given according to εð Þ¼ x; y ε0ð Þ 1 þ g xð Þ ; y . According to Figure 3 and for g xð Þ¼ ; y g0, we obtain

$$\begin{split} \mathbf{g}(n,m) &= \frac{\mathbf{g}\_{0}}{4ab} \Big|\_{-a}^{a} dx \int\_{-b}^{b} \exp\left[-j\left(k\_{\rm x}\mathbf{x} + k\_{\rm y}\mathbf{y}\right)\right] d\mathbf{y} \\ &= \frac{\mathbf{g}\_{0}}{4ab} \Big| \int\_{\mathbf{x}\_{\rm{11}}}^{\mathbf{x}\_{\rm{12}}} dx \int\_{\mathbf{y}\_{\rm{11}}}^{\mathbf{y}\_{\rm{12}}} \exp\left[-j\left(k\_{\rm x}\mathbf{x} + k\_{\rm y}\mathbf{y}\right)\right] d\mathbf{y} + \int\_{-\mathbf{x}\_{\rm{12}}}^{-\mathbf{x}\_{\rm{11}}} dx \int\_{\mathbf{y}\_{\rm{11}}}^{\mathbf{y}\_{\rm{12}}} \exp\left[-j\left(k\_{\rm x}\mathbf{x} + k\_{\rm y}\mathbf{y}\right)\right] d\mathbf{y} \\ &+ \int\_{-\mathbf{x}\_{\rm{22}}}^{-\mathbf{x}\_{\rm{12}}} dx \int\_{-\mathbf{y}\_{\rm{22}}}^{-\mathbf{y}\_{\rm{21}}} \exp\left[-j\left(k\_{\rm x}\mathbf{x} + k\_{\rm y}\mathbf{y}\right)\right] d\mathbf{y} + \int\_{-\mathbf{x}\_{\rm{21}}}^{\mathbf{x}\_{\rm{22}}} dx \int\_{-\mathbf{y}\_{\rm{22}}}^{-\mathbf{y}\_{\rm{21}}} \exp\left[-j\left(k\_{\rm x}\mathbf{x} + k\_{\rm y}\mathbf{y}\right)\right] d\mathbf{y} \end{split} \tag{7}$$

If y<sup>11</sup> and y<sup>12</sup> are functions of x, then we obtain

$$\begin{split} g(\boldsymbol{n},m) &= \frac{\mathcal{G}\_{0}}{abk\_{\mathcal{V}}} \Bigg\int\_{\boldsymbol{x}\_{11}}^{\boldsymbol{x}\_{12}} \Bigg[ \sin\left(k\_{\mathcal{V}}\boldsymbol{y}\_{12}\right) - \sin\left(k\_{\mathcal{V}}\boldsymbol{y}\_{11}\right) \Bigg] \cos\left(k\_{\boldsymbol{x}}\boldsymbol{x}\right) d\boldsymbol{x} \\ &= \frac{\mathcal{B}\_{0}}{am\pi} \Bigg\int\_{\boldsymbol{x}\_{11}}^{\boldsymbol{x}\_{12}} \sin\left[\frac{k\_{\mathcal{V}}}{2}\left(\boldsymbol{y}\_{12} - \boldsymbol{y}\_{11}\right)\right] \cos\left[\frac{k\_{\mathcal{V}}}{2}\left(\boldsymbol{y}\_{12} + \boldsymbol{y}\_{11}\right)\right] \cos\left(k\_{\boldsymbol{x}}\boldsymbol{x}\right) d\boldsymbol{x}. \end{split} \tag{8}$$

The dielectric profile for Figure 1(b) is given by

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 3. The arbitrary profile in the cross section.

ðð Þ <sup>a</sup>þd�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ a�dþε =2

and

b = c by

þ

ðb 0 cos

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

ðð Þ <sup>a</sup>þd�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ a�dþε =2

where

section

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

(

<sup>¼</sup> <sup>g</sup><sup>0</sup> 4ab

þ ð�x<sup>11</sup> �x<sup>12</sup> dx ð�y<sup>11</sup> �y<sup>12</sup>

56

4ab

ð<sup>x</sup><sup>12</sup> x<sup>11</sup> dx ð<sup>y</sup><sup>12</sup> y11

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

abky

<sup>¼</sup> <sup>2</sup>g<sup>0</sup> amπ

ab

cos

mπy b � �

� �

(

ðð Þ <sup>b</sup>þc�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ b�cþε =2 cos

Electromagnetic Materials and Devices

cos

nπx a � �

mπy b � �

ðð Þ <sup>a</sup>�dþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ a�d�ε =2

nπx a � �

dy

ðb 0 cos

tion as shown in Figure 1(a) and (b).

ða �a dx ðb �b

,

dx þ

mπy b � �

According to Figure 3 and for g xð Þ¼ ; y g0, we obtain

exp �j kxx <sup>þ</sup> kyy � � � � dy <sup>þ</sup>

If y<sup>11</sup> and y<sup>12</sup> are functions of x, then we obtain

The dielectric profile for Figure 1(b) is given by

sin kyy<sup>12</sup>

ð<sup>x</sup><sup>12</sup> x<sup>11</sup>

ð<sup>x</sup><sup>12</sup> x<sup>11</sup> sin ky

�

�

dx <sup>¼</sup> ð Þ <sup>2</sup>a=n<sup>π</sup> sin ð Þ ð Þ <sup>n</sup>π=2<sup>a</sup> ð Þ <sup>d</sup> � <sup>ε</sup> cosð Þ ð Þ <sup>n</sup><sup>π</sup> <sup>=</sup><sup>2</sup> <sup>n</sup> 6¼ <sup>0</sup> d � ε n ¼ 0

dy <sup>¼</sup> ð Þ <sup>2</sup>b=m<sup>π</sup> sin ð Þ ð Þ <sup>m</sup>π=2<sup>b</sup> ð Þ <sup>c</sup> � <sup>ε</sup> cosð Þ ð Þ <sup>m</sup><sup>π</sup> <sup>=</sup><sup>2</sup> <sup>n</sup> 6¼ <sup>0</sup> <sup>c</sup> � <sup>ε</sup> <sup>n</sup> <sup>¼</sup> <sup>0</sup> :

cos

<sup>0</sup> <sup>m</sup> 6¼ <sup>0</sup> :

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

" #

�

nπx a � � dx

cos

exp �j kxx <sup>þ</sup> kyy � � � � dy

cosð Þ kxx dx:

) :

(7)

(8)

exp �j kxx <sup>þ</sup> kyy � � � � dy

nπx a � � dx )

The elements of the matrices are given according to Figure 1(a), in the case of

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

exp 1 � <sup>ε</sup><sup>2</sup>

dy <sup>¼</sup> ð Þ <sup>b</sup>=m<sup>π</sup> sin ð Þ¼ <sup>m</sup><sup>π</sup> b m <sup>¼</sup> <sup>0</sup>

ð�x<sup>11</sup> �x<sup>12</sup> dx ð<sup>y</sup><sup>12</sup> y11

3.2 The second technique to calculate the discontinuous structure of the cross

The second technique to calculate the discontinuous structure of the cross sec-

The dielectric profile g xð Þ ; y is given according to εð Þ¼ x; y ε0ð Þ 1 þ g xð Þ ; y .

exp �j kxx <sup>þ</sup> kyy � � � � dy

ð<sup>x</sup><sup>12</sup> x<sup>11</sup> dx ð�y<sup>11</sup> �y<sup>12</sup>

� � � sin kyy<sup>11</sup> � � � � cosð Þ kxx dx

> cos ky

<sup>2</sup> <sup>y</sup><sup>12</sup> <sup>þ</sup> <sup>y</sup><sup>11</sup> � � � �

<sup>2</sup> <sup>y</sup><sup>12</sup> � <sup>y</sup><sup>11</sup> � � � �

exp �j kxx <sup>þ</sup> kyy � � � � dy <sup>þ</sup>

" #

exp 1 � <sup>ε</sup><sup>2</sup>

ðð Þ <sup>a</sup>þdþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ aþd�ε =2

$$\begin{cases} \frac{\mathcal{G}\_0}{4ab}(4cd) & n=0, m=0\\ \frac{\mathcal{G}\_0}{4ab} \left(\frac{8d}{k\_{0\jmath}m} \sin\left(\frac{k\_{0\jmath}mc}{2}\right) \cos\left(\frac{k\_{0\jmath}mb}{2}\right)\right) & n=0, m\neq 0\\ \frac{\mathcal{G}\_0}{\mathcal{G}\_0} \left(\begin{array}{c} 8c & \dots & \binom{k\_{0\jmath}m}{2} \end{array}\bigg/ \begin{array}{c} \begin{array}{c} (k\_{0\jmath}m) \\ (k\_{0\jmath}m) \end{array}\right) \end{array} & n=0, m\neq 0 \end{cases}$$

$$\begin{aligned} \mathcal{G}(n,m) &= \begin{cases} \frac{g\_0}{4ab} \left( \frac{8c}{k\_{0\times n}n} \sin\left(\frac{k\_{0\times n}nd}{2}\right) \cos\left(\frac{k\_{0\times n}nd}{2}\right) \right) & n \neq 0, m = 0\\ \frac{g\_0}{4ab} \left( \frac{16}{k\_{0\times k}nmn} \sin\left(\frac{k\_{0\times n}nd}{2}\right) \cos\left(\frac{k\_{0\times n}nd}{2}\right) \sin\left(\frac{k\_{0\times m}n}{2}\right) \cos\left(\frac{k\_{0\times m}nb}{2}\right) \right) & n \neq 0, m \neq 0 \end{cases} \end{aligned}$$

#### 3.3 The dielectric profile for the circular profile in the cross section

The dielectric profile for the circular profile in the cross section of the straight rectangular waveguide is given by (Figure 1(c))

$$\mathbf{g}(\mathbf{x}, \mathbf{y}) = \begin{cases} \mathbf{g}\_0 & \mathbf{0} \le r < r\_1 - \varepsilon\_1/2 \\ \mathbf{g}\_0 \exp\left[\mathbf{1} - q\_\varepsilon(r)\right] & r\_1 - \varepsilon/2 \le r < r\_1 + \varepsilon/2 \end{cases} \tag{10}$$

(9)

where

$$q\_{\varepsilon}(r) = \frac{\varepsilon^2}{\varepsilon^2 - \left[r - (r\_1 - \varepsilon/2)\right]^2},$$

else g xð Þ ; y = 0. The radius of the circle is given by r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup> q .

#### 3.4 The dielectric profile for the waveguide filled with dielectric material in the entire cross section

The dielectric profile (Figure 1(d)) is given by

$$\begin{cases} \text{g}\_0\\ \frac{\text{g}\_0}{4ab} \left( \frac{8a}{k\_{0j}m} \sin \left( \frac{k\_{0j}mb}{2} \right) \cos \left( \frac{k\_{0j}mb}{2} \right) \right) & n = 0, m \neq 0\\ n = 1, \text{and} \quad (h\_1, \dots) \qquad (h\_2, \dots) \text{\dots} \end{cases} \quad n = 0, m \neq 0$$

$$\mathbf{g}(n,m) = \left\{ \frac{\mathbf{g}\_0}{4ab} \left( \frac{8\mathbf{\dot{b}}}{k\_{0x}n} \sin\left(\frac{k\_{0x}na}{2}\right) \cos\left(\frac{k\_{0x}na}{2}\right) \right)\_{\mathbf{i}} \right. \\ \left. \qquad \qquad \qquad \qquad \qquad n \neq 0, m = 0$$

$$\begin{cases} 4ab \left( k\_{0x} n^{\sin} \left( \frac{\pi}{2} \right)^{\cos} \left( \frac{\pi}{2} \right) \right) & n \neq \upsilon, m = \upsilon \\ \frac{g\_0}{4ab} \left( \frac{16}{k\_{0x} k\_{0y} nm} \sin \left( \frac{k\_{0x} n a}{2} \right) \cos \left( \frac{k\_{0x} m a}{2} \right) \sin \left( \frac{k\_{0y} m b}{2} \right) \cos \left( \frac{k\_{0y} m b}{2} \right) \right) & n \neq 0, m \neq 0 \end{cases} \tag{11}$$
