2. Periodic rectangular and circular profiles in the cross section of the straight waveguide

In this chapter we introduce two different techniques, and the particular applications allow us to improve the mode model so that we can solve inhomogeneous problems also for periodic profiles in the cross section of the straight waveguide. Thus, in this chapter we introduce two techniques to calculate the dielectric profile, the elements of the matrix, and its derivatives of dielectric profile in the cases of periodic rectangular and circular profiles in the cross section of the straight rectangular waveguide.

The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model method becomes an improved method to solve discontinuous problems in the cross section (and not only for continuous problems).

Three examples of periodic rectangular profiles are shown in Figure 1(a–c), and three examples of periodic circular profiles are shown in Figure 1(d–f) in the cross section of the straight rectangular waveguide.

An example of periodic structure with two rectangular profiles along x-axis is shown in Figure 1(a), where the centers of the left rectangle and right rectangle are located at the points (0.25 a, 0.5 b) and (0.75 a, 0.5 b), respectively. An example of periodic structure with two rectangular profiles along y-axis is shown in Figure 1(b), where the centers of the upper rectangle and lower rectangle are located at the points (0.5 a, 0.75 b) and (0.5 a, 0.25 b).

An example of periodic structure with four rectangular profiles along x-axis and y-axis is shown in Figure 1(c). The center of the first rectangle is located at the point (0.25 a, 0.25 b),

cross section of the straight rectangular waveguide. Thus, we need to introduce a technique and a particular application for the two geometric shapes composed of rectangles or circles in the cross section. We need to calculate the dielectric profile, the elements of the matrix, and its derivatives of the dielectric profile in the cases of periodic rectangular profiles (Figure 1(a–c)) and periodic circular profiles (Figure 1(d–f)) in the cross section of the straight rectangular

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace…

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321

It is very interesting to compare between two different techniques of the two different kinds of the profiles (rectangular and circular) in the cross section of the rectangular straight wave-

The two kinds of the different techniques enable us to solve practical problems for the periodic rectangular profiles (Figure 1(a–c)) and periodic circular profiles (Figure 1(d–f)) in the cross

Figure 2(a) shows one rectangular profile where the center of the rectangle is located at the point (0.5 a, 0.5 b). Figure 2(b) shows one circular profile where the center is located at the point (0.5 a, 0.5 b). The proposed techniques to solve discontinuous problems with two different profiles (rectangular and circular) in the cross section will introduce according to

Figure 2(a) shows one rectangular profile in the cross section, and Figure 2(b) shows one circular profile in the cross section. The dielectric profile g xð Þ ; y is given according to

Figure 2. Examples of the rectangular and circular profiles in the cross section of the straight rectangular waveguide. (a)

eð Þ¼ x; y e0ð Þ 1 þ g xð Þ ; y . According to Figure 3(a and b) and for g xð Þ¼ ; y g0, we obtain

3. The techniques to solve two different profiles in the cross section

section of the straight rectangular waveguide.

One rectangular profile. (b) One circular profile.

waveguide.

Figure 2(a and b).

guide.

Figure 1. Examples of the periodic rectangular profiles (a–c) and periodic circular profiles (d–f) in the cross section of the straight rectangular waveguide.

the center of the second rectangle is located at the point (0.75 a, 0.25 b), the center of the third rectangle is located at the point (0.25 a, 0.75 b), and the center of the fourth rectangle is located at the point (0.75 a, 0.75 b).

An example of periodic structure with two circular profiles along x-axis is shown in Figure 1(d), where the centers of the left circle and right circle are located at the points (0.25 a, 0.5 b) and (0.75 a, 0.5 b). An example of periodic structure with two circular profiles along y-axis is shown in Figure 1(e), where the centers of the upper circle and lower circle are located at the points (0.5 a, 0.75 b) and (0.5 a, 0.25 b).

An example of periodic structure with four circular profiles along x-axis and y-axis is shown in Figure 1(f). The center of the first circle is located at the point (0.25 a, 0.25 b), the center of the second circle is located at the point (0.75 a, 0.25 b), the center of the third circle is located at the point (0.25 a, 0.75 b), and the center of the fourth circle is located at the point (0.75 a, 0.75 b).

The objective of this chapter is to introduce two different techniques that allow us to improve the model so that we can solve nonhomogeneous problems also for periodic profiles in the cross section of the straight rectangular waveguide. Thus, we need to introduce a technique and a particular application for the two geometric shapes composed of rectangles or circles in the cross section. We need to calculate the dielectric profile, the elements of the matrix, and its derivatives of the dielectric profile in the cases of periodic rectangular profiles (Figure 1(a–c)) and periodic circular profiles (Figure 1(d–f)) in the cross section of the straight rectangular waveguide.

It is very interesting to compare between two different techniques of the two different kinds of the profiles (rectangular and circular) in the cross section of the rectangular straight waveguide.

### 3. The techniques to solve two different profiles in the cross section

The two kinds of the different techniques enable us to solve practical problems for the periodic rectangular profiles (Figure 1(a–c)) and periodic circular profiles (Figure 1(d–f)) in the cross section of the straight rectangular waveguide.

Figure 2(a) shows one rectangular profile where the center of the rectangle is located at the point (0.5 a, 0.5 b). Figure 2(b) shows one circular profile where the center is located at the point (0.5 a, 0.5 b). The proposed techniques to solve discontinuous problems with two different profiles (rectangular and circular) in the cross section will introduce according to Figure 2(a and b).

Figure 2(a) shows one rectangular profile in the cross section, and Figure 2(b) shows one circular profile in the cross section. The dielectric profile g xð Þ ; y is given according to eð Þ¼ x; y e0ð Þ 1 þ g xð Þ ; y . According to Figure 3(a and b) and for g xð Þ¼ ; y g0, we obtain

the center of the second rectangle is located at the point (0.75 a, 0.25 b), the center of the third rectangle is located at the point (0.25 a, 0.75 b), and the center of the fourth rectangle is located

Figure 1. Examples of the periodic rectangular profiles (a–c) and periodic circular profiles (d–f) in the cross section of the

An example of periodic structure with two circular profiles along x-axis is shown in Figure 1(d), where the centers of the left circle and right circle are located at the points (0.25 a, 0.5 b) and (0.75 a, 0.5 b). An example of periodic structure with two circular profiles along y-axis is shown in Figure 1(e), where the centers of the upper circle and lower circle are located at the points (0.5 a,

An example of periodic structure with four circular profiles along x-axis and y-axis is shown in Figure 1(f). The center of the first circle is located at the point (0.25 a, 0.25 b), the center of the second circle is located at the point (0.75 a, 0.25 b), the center of the third circle is located at the point (0.25 a, 0.75 b), and the center of the fourth circle is located at the point (0.75 a, 0.75 b). The objective of this chapter is to introduce two different techniques that allow us to improve the model so that we can solve nonhomogeneous problems also for periodic profiles in the

at the point (0.75 a, 0.75 b).

straight rectangular waveguide.

320 Emerging Waveguide Technology

0.75 b) and (0.5 a, 0.25 b).

Figure 2. Examples of the rectangular and circular profiles in the cross section of the straight rectangular waveguide. (a) One rectangular profile. (b) One circular profile.

Figure 3. (a) The arbitrary profile in the cross section. (b) The rectangular profile in the cross section, as shown in Figure 2(a).

$$g(n,m) = \frac{g\_0}{4ab} \left\{ \int\_{x\_{11}}^{x\_{12}} dx \int\_{y\_{11}}^{y\_{12}} \exp\left(-j(k\_x x + k\_y y)\right) dy + \int\_{-x\_{12}}^{-x\_{11}} dx \int\_{y\_{11}}^{y\_{12}} \exp\left(-j(k\_x x + k\_y y)\right) dy \right.$$

$$+ \int\_{-x\_{12}}^{-x\_{11}} dx \int\_{-y\_{12}}^{-y\_{11}} \exp\left(-j(k\_x x + k\_y y)\right) dy + \int\_{x\_{11}}^{x\_{12}} dx \int\_{-y\_{12}}^{-y\_{11}} \exp\left(-j(k\_x x + k\_y y)\right) dy \right\}. \tag{1}$$

If y<sup>11</sup> and y<sup>12</sup> are not functions of x, then the dielectric profile is given by

$$\log(n,m) = \frac{g\_0}{ab} \int\_{x\_{11}}^{x\_{12}} \cos(k\_x x) dx \int\_{y\_{11}}^{y\_{12}} \cos(k\_y y) dy. \tag{2}$$

The particular application is based on the ωε function [25]. The ωε function is used in order to solve rectangular profile, periodic rectangular profiles, circular profile, and periodic circular profile in the cross section of the straight waveguide. The ωε function is defined as

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace…

The technique based on ωε function is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large number of dielectric profiles, as shown in Figure 1(c and f). We will demonstrate how to use with the proposed technique for

Figure 1(a) shows the periodic rectangular profile where the center of the left rectangle is located at (0.25 a, 0.5 b) and the right rectangle is located at (0.75 a, 0.5 b). This dielectric profile

> <sup>g</sup>0exp 1 � <sup>q</sup>1ð Þ<sup>x</sup> � � ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> g<sup>0</sup> ð Þ ð Þ� a=2 d<sup>1</sup> þ ε =2 < x < ð Þ ð Þþ a=2 d<sup>1</sup> � ε =2 <sup>g</sup>0exp 1 � <sup>q</sup>2ð Þ<sup>x</sup> � � ð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>g</sup>0exp 1 � <sup>q</sup>3ð Þ<sup>x</sup> � � ð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> g<sup>0</sup> ð Þ ð Þ� 3a=2 d<sup>2</sup> þ ε =2 < x < ð Þ ð Þþ 3a=2 d<sup>2</sup> � ε =2 <sup>g</sup>0exp 1 � <sup>q</sup>4ð Þ<sup>x</sup> � � ð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

ωεð Þr dr ¼ 1. In the

323

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,

(4a)

ωεð Þ¼ <sup>r</sup> <sup>C</sup>εexp �ε<sup>2</sup><sup>=</sup> <sup>ε</sup><sup>2</sup> � j j<sup>r</sup> <sup>2</sup> h i � � for <sup>∣</sup>r<sup>∣</sup> <sup>&</sup>gt; <sup>ε</sup>, where <sup>C</sup><sup>ε</sup> is a constant and <sup>Ð</sup>

3.1. The technique based on ωε function for the periodic rectangular profile

limit ε ! 0, the ωε function is shown in Figure 4.

in the cross section

g xð Þ¼

8

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

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

is given by

and

all the cases that are shown in the examples of Figure 1(a–f).

0 else

Figure 4. The technique based on ωε function in the limit ε ! 0 to solve discontinuous problems.

The derivative of the dielectric profile in the case of y<sup>11</sup> and y<sup>12</sup> which are functions of x is given by

$$\mathcal{g}\_{\mathbf{x}}(n,m) = \frac{2}{am\pi} \int\_{x\_{11}}^{x\_{12}} \mathcal{g}\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) \sin\left[\frac{k\_y}{2}\left(y\_{12} - y\_{11}\right)\right] \cos\left[\frac{k\_y}{2}\left(y\_{12} + y\_{11}\right)\right] \cos(k\_{\mathbf{x}}\mathbf{x})d\mathbf{x},\tag{3}$$

where gxð Þ¼ <sup>x</sup>; <sup>y</sup> ð Þ <sup>1</sup>=eð Þ <sup>x</sup>; <sup>y</sup> ð Þ <sup>d</sup>eð Þ <sup>x</sup>; <sup>y</sup> <sup>=</sup>dx , eð Þ¼ x; y e0ð Þ 1 þ g xð Þ ; y , kx ¼ ð Þ nπx =a, and ky ¼ ð Þ mπy =b. Similarly, we can calculate the value of gyð Þ n; m , where gyð Þ¼ x; y ð Þ 1=eð Þ x; y ð Þ deð Þ x; y =dy .

For the cross section as shown in Figure 2(a) and according to Figure 3(b), the center of the rectangle is located at (0.5 a, 0.5 b), y<sup>12</sup> = b/2 + c/2, and y<sup>11</sup> = b/2 � c/2. Thus, for this case, y<sup>12</sup> � y<sup>11</sup> = c and y<sup>12</sup> þ y<sup>11</sup> = b. In the same principle, the location of the rectangle should be taken into account.

The particular application is based on the ωε function [25]. The ωε function is used in order to solve rectangular profile, periodic rectangular profiles, circular profile, and periodic circular profile in the cross section of the straight waveguide. The ωε function is defined as ωεð Þ¼ <sup>r</sup> <sup>C</sup>εexp �ε<sup>2</sup><sup>=</sup> <sup>ε</sup><sup>2</sup> � j j<sup>r</sup> <sup>2</sup> h i � � for <sup>∣</sup>r<sup>∣</sup> <sup>&</sup>gt; <sup>ε</sup>, where <sup>C</sup><sup>ε</sup> is a constant and <sup>Ð</sup> ωεð Þr dr ¼ 1. In the limit ε ! 0, the ωε function is shown in Figure 4.

The technique based on ωε function is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large number of dielectric profiles, as shown in Figure 1(c and f). We will demonstrate how to use with the proposed technique for all the cases that are shown in the examples of Figure 1(a–f).

#### 3.1. The technique based on ωε function for the periodic rectangular profile in the cross section

Figure 1(a) shows the periodic rectangular profile where the center of the left rectangle is located at (0.25 a, 0.5 b) and the right rectangle is located at (0.75 a, 0.5 b). This dielectric profile is given by

$$g(\mathbf{x}) = \begin{cases} g\_0 \exp\{1 - q\_1(\mathbf{x})\} & ((a/2) - d\_1 - \varepsilon)/2 \le \mathbf{x} < ((a/2) - d\_1 + \varepsilon)/2 \\ g\_0 & ((a/2) - d\_1 + \varepsilon)/2 < \mathbf{x} < ((a/2) + d\_1 - \varepsilon)/2 \\ g\_0 \exp\{1 - q\_2(\mathbf{x})\} & ((a/2) + d\_1 - \varepsilon)/2 \le \mathbf{x} < ((a/2) + d\_1 + \varepsilon)/2 \\ g\_0 \exp\{1 - q\_3(\mathbf{x})\} & ((3a/2) - d\_2 - \varepsilon)/2 \le \mathbf{x} < ((3a/2) - d\_2 + \varepsilon)/2 \\ g\_0 & ((3a/2) - d\_2 + \varepsilon)/2 < \mathbf{x} < ((3a/2) + d\_2 - \varepsilon)/2 \\ g\_0 \exp\{1 - q\_4(\mathbf{x})\} & ((3a/2) + d\_2 - \varepsilon)/2 \le \mathbf{x} < ((3a/2) + d\_2 + \varepsilon)/2 \\ 0 & \text{else} \end{cases}$$

and

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

322 Emerging Waveguide Technology

by

4ab

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

gxð Þ¼ n; m

taken into account.

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

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

(

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

Figure 3. (a) The arbitrary profile in the cross section. (b) The rectangular profile in the cross section, as shown in Figure 2(a).

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

ky

cosð Þ kxx dx

The derivative of the dielectric profile in the case of y<sup>11</sup> and y<sup>12</sup> which are functions of x is given

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

where gxð Þ¼ <sup>x</sup>; <sup>y</sup> ð Þ <sup>1</sup>=eð Þ <sup>x</sup>; <sup>y</sup> ð Þ <sup>d</sup>eð Þ <sup>x</sup>; <sup>y</sup> <sup>=</sup>dx , eð Þ¼ x; y e0ð Þ 1 þ g xð Þ ; y , kx ¼ ð Þ nπx =a, and ky ¼ ð Þ mπy =b. Similarly, we can calculate the value of gyð Þ n; m , where gyð Þ¼ x; y ð Þ 1=eð Þ x; y ð Þ deð Þ x; y =dy .

For the cross section as shown in Figure 2(a) and according to Figure 3(b), the center of the rectangle is located at (0.5 a, 0.5 b), y<sup>12</sup> = b/2 + c/2, and y<sup>11</sup> = b/2 � c/2. Thus, for this case, y<sup>12</sup> � y<sup>11</sup> = c and y<sup>12</sup> þ y<sup>11</sup> = b. In the same principle, the location of the rectangle should be

If y<sup>11</sup> and y<sup>12</sup> are not functions of x, then the dielectric profile is given by

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

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

gxð Þ x; y sin

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

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

ð<sup>y</sup><sup>12</sup> y11

> cos ky

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

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

) : (1)

cosð Þ kxx dx, (3)

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

cos kyy � �dy: (2)

Figure 4. The technique based on ωε function in the limit ε ! 0 to solve discontinuous problems.

$$g(y) = \begin{cases} g\_0 \exp\{1 - q\_5(y)\} & (b - c - \varepsilon)/2 \le y < (b - c + \varepsilon)/2\\ g\_0 & (b - c + \varepsilon)/2 < y < (b + c - \varepsilon)/2\\ g\_0 \exp\{1 - q\_6(y)\} & (b + c - \varepsilon)/2 \le y < (b + c + \varepsilon)/2\\ 0 & \text{else} \end{cases} \tag{4b}$$

and

by

where

else g xð Þ¼ ; y 0.

The radius of the circle is given by

gy ¼

gxð Þ¼ <sup>n</sup>; <sup>m</sup> <sup>¼</sup> <sup>0</sup> <sup>1</sup>

d

8 >>>><

>>>>:

d

a

þ

þ

þ

g xð Þ¼ ; y

�

qeð Þ¼ r

r ¼

q

Similarly, we can calculate the periodic circular profiles according to their location.

ε1

0 else

in the cross section of the waveguide is given in the limit ε ! 0 by

ðð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� a=2 d1�ε =2

ðð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ a=2 d1�ε =2

ðð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� 3a=2 d2�ε =2

ðð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ 3a=2 d2�ε =2

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>5ð Þ<sup>y</sup> � � � � ð Þ <sup>b</sup> � <sup>c</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>y</sup> <sup>&</sup>lt; <sup>ð</sup><sup>b</sup> � <sup>c</sup> <sup>þ</sup> <sup>ε</sup>Þ=<sup>2</sup>

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace…

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>6ð Þ<sup>y</sup> � � � � ð Þ <sup>b</sup> <sup>þ</sup> <sup>c</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ <sup>b</sup> <sup>þ</sup> <sup>c</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

The derivative of the dielectric profile for m = 0 in the case of the periodic profile (Figure 1(a))

d dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>1ð Þ<sup>x</sup> � � � � dx (

d

d

d

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>2ð Þ<sup>x</sup> � � � � dx

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>3ð Þ<sup>x</sup> � � � � dx

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>4ð Þ<sup>x</sup> � � � � dx)

:

,

cosð Þ kxx

cosð Þ kxx

cosð Þ kxx

cosð Þ kxx

3.2. The technique based on ωε function for one circular profile in the cross section

Similarly, we can calculate the derivative of the dielectric profile for any value of n and m.

The dielectric profile for one circle is given where the center is located at (0.5 a, 0.5 b) (Figure 2(b))

g<sup>0</sup> 0 ≤ r < r<sup>1</sup> � ε1=2

ε1 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>

:

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

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

:

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(6b)

325

(7)

where

$$\begin{aligned} q\_1(\mathbf{x}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - \left(\left(a/2\right) - d\_1 + \varepsilon\right)/2\right]^2} & \qquad \qquad q\_2(\mathbf{x}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - \left(\left(a/2\right) + d\_1 - \varepsilon\right)/2\right]^2} \\ q\_3(\mathbf{x}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - \left(\left(3a/2\right) - d\_2 + \varepsilon\right)/2\right]^2} & \qquad \qquad q\_4(\mathbf{x}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{x} - \left(\left(3a/2\right) + d\_2 - \varepsilon\right)/2\right]^2} \\ q\_5(\mathbf{y}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{y} - \left(b - c + \varepsilon\right)/2\right]^2} & \qquad q\_6(\mathbf{y}) &= \frac{\varepsilon^2}{\varepsilon^2 - \left[\mathbf{y} - \left(b + c - \varepsilon\right)/2\right]^2} & \qquad \end{aligned}$$

The elements of the matrix are given in this case by

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup> ab <sup>ð</sup>ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� a=2 d1�ε =2 exp 1 � <sup>q</sup>1ð Þ<sup>x</sup> � �cos nπx a � �dx<sup>þ</sup> ( ðð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup>1�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� a=2 d1þε =2 cos nπx a � �dx <sup>þ</sup> ðð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ a=2 d1�ε =2 exp 1 � <sup>q</sup>2ð Þ<sup>x</sup> � �cos nπx a � �dx<sup>þ</sup> ðð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� 3a=2 d2�ε =2 exp 1 � <sup>q</sup>3ð Þ<sup>x</sup> � �cos nπx a � �dx<sup>þ</sup> ðð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� 3a=2 d2þε =2 cos nπx a � �dx <sup>þ</sup> ðð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ 3a=2 d2�ε =2 exp 1 � <sup>q</sup>4ð Þ<sup>x</sup> � �cos nπx a � �dx) ðð Þ <sup>b</sup>�cþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ b�c�ε =2 exp 1 � <sup>q</sup>5ð Þ<sup>y</sup> � �cos mπy b � �dy <sup>þ</sup> ðð Þ <sup>b</sup>þc�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ b�cþε =2 cos mπy b � �dy<sup>þ</sup> ( ðð Þ <sup>b</sup>þcþ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ bþc�ε =2 exp 1 � <sup>q</sup>6ð Þ<sup>y</sup> � �cos mπy b � �dyg: (5)

The derivatives of the dielectric profile are given in this case by

$$g\_x = \begin{cases} \frac{d}{dx} \left( \ln \left( 1 + \operatorname{g\_0} \operatorname{q\_1} (\mathbf{x}) \right) \right) & ((a/2) - d\_1 - \varepsilon)/2 \le \mathbf{x} < ((a/2) - d\_1 + \varepsilon)/2 \\\frac{d}{dx} \left( \ln \left( 1 + \operatorname{g\_0} \operatorname{q\_2} (\mathbf{x}) \right) \right) & ((a/2) + d\_1 - \varepsilon)/2 \le \mathbf{x} < ((a/2) + d\_1 + \varepsilon)/2 \\\frac{d}{dx} \left( \ln \left( 1 + \operatorname{g\_0} \operatorname{q\_3} (\mathbf{x}) \right) \right) & ((3a/2) - d\_2 - \varepsilon)/2 \le \mathbf{x} < ((3a/2) - d\_2 + \varepsilon)/2' \\\frac{d}{dx} \left( \ln \left( 1 + \operatorname{g\_0} \operatorname{q\_4} (\mathbf{x}) \right) \right) & ((3a/2) + d\_2 - \varepsilon)/2 \le \mathbf{x} < ((3a/2) + d\_2 + \varepsilon)/2 \\\ 0 & \text{else} \end{cases} \tag{6a}$$

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace… http://dx.doi.org/10.5772/intechopen.76794 325

and

g yð Þ¼

324 Emerging Waveguide Technology

where

q1ð Þ¼ x

q3ð Þ¼ x

q5ð Þ¼ y

8 >>><

>>>:

ε2

ε2

The elements of the matrix are given in this case by

(

ðð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup>1�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� a=2 d1þε =2

ðð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� 3a=2 d2�ε =2

ðð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2�<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� 3a=2 d2þε =2

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

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

The derivatives of the dielectric profile are given in this case by

0 else

(

gx ¼

d

8

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

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

d

d

d

ab

ε2

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

<sup>g</sup>0exp 1 � <sup>q</sup>5ð Þ<sup>y</sup> � � ð Þ <sup>b</sup> � <sup>c</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>y</sup> <sup>&</sup>lt; ð Þ <sup>b</sup> � <sup>c</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> g<sup>0</sup> ð Þ b � c þ ε =2 < y < ð Þ b þ c � ε =2 <sup>g</sup>0exp 1 � <sup>q</sup>6ð Þ<sup>y</sup> � � ð Þ <sup>b</sup> <sup>þ</sup> <sup>c</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>y</sup> <sup>&</sup>lt; ð Þ <sup>b</sup> <sup>þ</sup> <sup>c</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

exp 1 � <sup>q</sup>1ð Þ<sup>x</sup> � �cos

dx þ

dx þ

,

ε2 <sup>ε</sup><sup>2</sup> � ½ � <sup>x</sup> � ð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>2</sup> ,

ε2 <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> :

dxþ

exp 1 � <sup>q</sup>2ð Þ<sup>x</sup> � �cos

exp 1 � <sup>q</sup>4ð Þ<sup>x</sup> � �cos

cos

dyg: (5)

mπy b � �

nπx a � �

nπx a � �

dyþ

,

(6a)

dxþ

dx )

nπx a � �

dxþ

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

ðð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ a=2 d1�ε =2

> nπx a � �

ðð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup>2þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þþ 3a=2 d2�ε =2

dy þ

mπy b � �

mπy b � �

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>1ð Þ<sup>x</sup> � � � � ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>2ð Þ<sup>x</sup> � � � � ð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þþ <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>q</sup>3ð Þ<sup>x</sup> � � � � ð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

dx ln 1 <sup>þ</sup> <sup>g</sup><sup>0</sup> <sup>g</sup>4ð Þ<sup>x</sup> � � � � ð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>&</sup>lt; ð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup>

ε2 <sup>ε</sup><sup>2</sup> � ½ � <sup>x</sup> � ð Þ ð Þþ <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> � <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>2</sup>

(4b)

0 else

<sup>ε</sup><sup>2</sup> � ½ � <sup>x</sup> � ð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>2</sup> , q2ð Þ¼ <sup>x</sup>

<sup>ε</sup><sup>2</sup> � ½ � <sup>x</sup> � ð Þ ð Þ� <sup>3</sup>a=<sup>2</sup> <sup>d</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup> <sup>=</sup><sup>2</sup> <sup>2</sup> , q4ð Þ¼ <sup>x</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> , q6ð Þ¼ <sup>y</sup>

ðð Þ ð Þ� <sup>a</sup>=<sup>2</sup> <sup>d</sup>1þ<sup>ε</sup> <sup>=</sup><sup>2</sup> ð Þ ð Þ� a=2 d1�ε =2

cos

cos

nπx a � �

nπx a � �

exp 1 � <sup>q</sup>5ð Þ<sup>y</sup> � �cos

exp 1 � <sup>q</sup>6ð Þ<sup>y</sup> � �cos

exp 1 � <sup>q</sup>3ð Þ<sup>x</sup> � �cos

$$\mathcal{g}\_y = \begin{cases} \frac{d}{dx} \left( \ln \left( 1 + \mathcal{g}\_0 \mid q\_5(y) \right) \right) & (b - c - \varepsilon)/2 \le y < (b - c + \varepsilon)/2 \\\frac{d}{dx} \left( \ln \left( 1 + \mathcal{g}\_0 \mid q\_6(y) \right) \right) & (b + c - \varepsilon)/2 \le x < (b + c + \varepsilon)/2 \\\ 0 & \text{else} \end{cases} \tag{6b}$$

The derivative of the dielectric profile for m = 0 in the case of the periodic profile (Figure 1(a)) in the cross section of the waveguide is given in the limit ε ! 0 by

$$\begin{split} g\_{\mathbf{x}}(n,m=0) &= \frac{1}{a} \Bigg\{ \int\_{((a/2)-d\_1-\varepsilon)/2}^{((a/2)-d\_1+\varepsilon)/2} \cos(k\_{\mathbf{x}}\mathbf{x}) \frac{d}{d\mathbf{x}} \left[\ln\left(1+g\_0 \begin{array}{c} q\_1 \end{array}\right)\right] d\mathbf{x} \\ &+ \int\_{((a/2)+d\_1-\varepsilon)/2}^{((a/2)+d\_1+\varepsilon)/2} \cos(k\_{\mathbf{x}}\mathbf{x}) \frac{d}{d\mathbf{x}} \left[\ln\left(1+g\_0 \begin{array}{c} q\_2 \end{array}\right)\right] d\mathbf{x} \\ &+ \int\_{((3a/2)-d\_2+\varepsilon)/2}^{((3a/2)-d\_2+\varepsilon)/2} \cos(k\_{\mathbf{x}}\mathbf{x}) \frac{d}{d\mathbf{x}} \left[\ln\left(1+g\_0 \begin{array}{c} q\_3 \end{array}\right)\right] d\mathbf{x} \\ &+ \int\_{((3a/2)+d\_2-\varepsilon)/2}^{((3a/2)-d\_2+\varepsilon)/2} \cos(k\_{\mathbf{x}}\mathbf{x}) \frac{d}{d\mathbf{x}} \left[\ln\left(1+g\_0 \begin{array}{c} q\_4 \end{array}\right)\right] d\mathbf{x} \end{split} \right\}.$$

Similarly, we can calculate the derivative of the dielectric profile for any value of n and m.

#### 3.2. The technique based on ωε function for one circular profile in the cross section

The dielectric profile for one circle is given where the center is located at (0.5 a, 0.5 b) (Figure 2(b)) by

$$g(x,y) = \begin{cases} g\_0 & 0 \le r < r\_1 - \varepsilon\_1/2\\ g\_0 \exp\left[1 - q\_\varepsilon(r)\right] & r\_1 - \varepsilon\_1/2 \le r < r\_1 + \varepsilon\_1/2\end{cases} \tag{7}$$

where

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

else g xð Þ¼ ; y 0.

The radius of the circle is given by

$$r = \sqrt{\left(\mathbf{x} - a/2\right)^2 + \left(y - b/2\right)^2}.$$

Similarly, we can calculate the periodic circular profiles according to their location.

Thus, the derivatives of the dielectric profile for one circle are given where the center is located at (0.5 a, 0.5 b) (Figure 2(b)) in the region r<sup>1</sup> � ε1=2 ≤ r < r<sup>1</sup> þ ε1=2 by

$$\mathbf{g}\_x = \frac{-2 \left. \mathbf{g}\_0 \cdot \cos \theta \exp\left[1 - q\_\mathbf{e}(r)\right] \left[r - (r\_1 - \varepsilon\_1/2)\right] \varepsilon\_1^2}{\left\{1 + \mathbf{g}\_0 \exp\left[1 - q\_\mathbf{e}(r)\right]\right\} \left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^2} \tag{8a}$$

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 their

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace…

http://dx.doi.org/10.5772/intechopen.76794

327

The technique is important to solve discontinuous periodic rectangular and circular profiles in the cross section of the straight waveguide. The graphical results in the next section will be

All the graphical results are demonstrated as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (Figure 1(a–c)) and the periodic circular profiles (Figure 1(d–f)) in

The output fields for the periodic structure with the two rectangular profiles along x-axis (Figure 1(a)) are demonstrated in Figure 5(a and b) for e<sup>r</sup> ¼ 3 and for e<sup>r</sup> ¼ 10, respectively. In this examples, the left rectangle is located at the point (0.25 a, 0.5 b), and the center of the right rectangle is located at the point (0.75 a, 0.5 b). Figure 5(c) shows the output field as the function

Figure 5. The output field as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (a–c) that relate to cross section (Figure 1(a)) and for the periodic circular profiles (d–f) that relate to cross section (Figure 1(d)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 10; (c) e<sup>r</sup> ¼ 3, 5, 7, and 10; (d) e<sup>r</sup> ¼ 3; (e) e<sup>r</sup> ¼ 10; and (f) e<sup>r</sup> ¼ 3, 5, 7, and 10.

4. Numerical results of periodic rectangular and circular dielectric

of x-axis where y = b/2 = 10 mm for four values of e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

demonstrated as a response to a half-sine (TE10) input-wave profile.

the cross section of the straight rectangular waveguide.

inverse transforms [23].

materials

$$g\_y = \frac{-2 \text{ g}\_0 \cdot \sin \theta \exp\left[1 - q\_\mathbf{e}(r)\right] \left[r - (r\_1 - \varepsilon\_1/2)\right] \varepsilon\_1^2}{\left\{1 + \text{g}\_0 \exp\left[1 - q\_\mathbf{e}(r)\right]\right\} \left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^2} \tag{8b}$$

else gx ¼ 0, and gy ¼ 0.

The elements of the matrices for one circle are given where the center is located at (0.5 a, 0.5 b) (Figure 2(b)) by

$$g(n,m) = \frac{g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_0^{r\_1 - \epsilon\_1/2} \cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{n\pi\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] + \right\}$$

$$\int\_0^{2\pi} \int\_{r\_1 - \epsilon\_1/2}^{r\_1 + \epsilon\_1/2} \cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{n\pi\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \exp\left[1 - q\_\epsilon(r)\right] \right\} r dr d\theta,\tag{9}$$

$$g\_x(n,m) = -\frac{2g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_{r\_1 - \epsilon\_1/2}^{r\_1 + \epsilon\_1/2} \frac{\epsilon\_1 \,^2[r - (r\_1 - \epsilon\_1/2)] \exp\left[1 - q\_\epsilon(r)\right] \cos\theta}{\left[\epsilon\_1^2 - \left[r - (r\_1 - \epsilon\_1/2)\right]^2\right]^2 \left[1 + g\_0 \exp\left[1 - q\_\epsilon(r)\right]\right]}\right\}$$

$$\cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \right\} r dr d\theta,\tag{10a}$$

$$g\_y(n,m) = -\frac{2g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_{r\_1 - \mathbf{e}\_1/2}^{r\_1 + \mathbf{e}\_1/2} \frac{\mathbf{e}\_1^2 \left[r - (r\_1 - \mathbf{e}\_1/2)\right] \exp\left[1 - q\_\epsilon(r)\right] \sin\theta}{\left[\mathbf{e}\_1^2 - \left[r - (r\_1 - \mathbf{e}\_1/2)\right]^2\right]^2 \left[1 + g\_0 \exp\left[1 - q\_\epsilon(r)\right]\right]} \right.$$

$$\cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \right\} r dr d\theta,\tag{10b}$$

where 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 .

Similarly, we can calculate the periodic circular profile according to the number of the circles and the locations of their centers.

The matrix G is given by the form

$$\mathbf{G} = \begin{bmatrix} \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \mathcal{g}\_{-20} & \cdots & \mathcal{g}\_{-mn} & \cdots & \mathcal{g}\_{-NM} \\ \mathcal{g}\_{10} & \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \cdots & \mathcal{g}\_{-(n-1)m} & \cdots & \mathcal{g}\_{-(N-1)M} \\ \mathcal{g}\_{20} & \mathcal{g}\_{10} & \ddots & \ddots & \ddots & & \\ \vdots & \mathcal{g}\_{20} & \ddots & \ddots & \ddots & & \\ \mathcal{g}\_{mn} & \ddots & \ddots & \ddots & \mathcal{g}\_{00} & \vdots \\ \vdots & & & & \\ \mathcal{g}\_{NM} & \cdots & \cdots & \cdots & \cdots & \cdots & \mathcal{g}\_{00} \end{bmatrix}. \tag{11}$$

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 their inverse transforms [23].

The technique is important to solve discontinuous periodic rectangular and circular profiles in the cross section of the straight waveguide. The graphical results in the next section will be demonstrated as a response to a half-sine (TE10) input-wave profile.

## 4. Numerical results of periodic rectangular and circular dielectric materials

Thus, the derivatives of the dielectric profile for one circle are given where the center is located

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> , (8a)

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> , (8b)

b 2

þ

exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � �)

<sup>1</sup> <sup>þ</sup> <sup>g</sup>0exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � � � �

<sup>1</sup> <sup>þ</sup> <sup>g</sup>0exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � � � �

rdrdθ, (10a)

rdrdθ, (10b)

: (11)

rdrdθ, (9)

gx <sup>¼</sup> �<sup>2</sup> <sup>g</sup><sup>0</sup> cos<sup>θ</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � �½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>ε</sup><sup>1</sup>

gy <sup>¼</sup> �<sup>2</sup> <sup>g</sup><sup>0</sup> sin<sup>θ</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � �½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>ε</sup><sup>1</sup>

The elements of the matrices for one circle are given where the center is located at (0.5 a, 0.5 b)

a 2

cos mπ

e1

cos mπ

e1

cos mπ

Similarly, we can calculate the periodic circular profile according to the number of the circles

<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>

cos mπ

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>e</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>e</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

<sup>b</sup> <sup>r</sup>sin<sup>θ</sup> <sup>þ</sup>

<sup>b</sup> <sup>r</sup>sin<sup>θ</sup> <sup>þ</sup>

� � ���

� � ���

<sup>b</sup> <sup>r</sup>sin<sup>θ</sup> <sup>þ</sup>

� � � �

<sup>b</sup> <sup>r</sup>sin<sup>θ</sup> <sup>þ</sup>

� � � �

b 2

<sup>2</sup>½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>e</sup>1=<sup>2</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � �cos<sup>θ</sup>

b 2

<sup>2</sup>½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>e</sup>1=<sup>2</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � �sin<sup>θ</sup>

b 2

rcosθ þ

a 2

e1

a 2

e1

a 2

� � � �

rcosθ þ

� � � �

ð<sup>r</sup>1þe1=<sup>2</sup> r1�e1=2

rcosθ þ

ð<sup>r</sup>1þe1=<sup>2</sup> r1�e1=2

rcosθ þ

.

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

gnm ⋱⋱⋱ g<sup>00</sup> ⋮

gNM … … …… … g<sup>00</sup>

� � � �

� � � �

at (0.5 a, 0.5 b) (Figure 2(b)) in the region r<sup>1</sup> � ε1=2 ≤ r < r<sup>1</sup> þ ε1=2 by

<sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � � � � <sup>ε</sup><sup>1</sup>

<sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>eð Þ<sup>r</sup> � � � � <sup>ε</sup><sup>1</sup>

else gx ¼ 0, and gy ¼ 0.

326 Emerging Waveguide Technology

(Figure 2(b)) by

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

ab

ð<sup>2</sup><sup>π</sup> 0

gxð Þ¼� n; m

gyð Þ¼� n; m

and the locations of their centers. The matrix G is given by the form

G ¼

⋮

q

where r ¼

ð<sup>2</sup><sup>π</sup> 0

ð<sup>r</sup>1þe1=<sup>2</sup> r1�e1=2

(

ð<sup>r</sup>1�e1=<sup>2</sup> 0

> cos nπ a

> > 8 ><

> > >:

8 ><

>:

ð<sup>2</sup><sup>π</sup> 0

ð<sup>2</sup><sup>π</sup> 0

2g<sup>0</sup> ab

cos nπ a

2g<sup>0</sup> ab

cos nπ a

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>

cos nπ a

All the graphical results are demonstrated as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (Figure 1(a–c)) and the periodic circular profiles (Figure 1(d–f)) in the cross section of the straight rectangular waveguide.

The output fields for the periodic structure with the two rectangular profiles along x-axis (Figure 1(a)) are demonstrated in Figure 5(a and b) for e<sup>r</sup> ¼ 3 and for e<sup>r</sup> ¼ 10, respectively. In this examples, the left rectangle is located at the point (0.25 a, 0.5 b), and the center of the right rectangle is located at the point (0.75 a, 0.5 b). Figure 5(c) shows the output field as the function of x-axis where y = b/2 = 10 mm for four values of e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

Figure 5. The output field as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (a–c) that relate to cross section (Figure 1(a)) and for the periodic circular profiles (d–f) that relate to cross section (Figure 1(d)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 10; (c) e<sup>r</sup> ¼ 3, 5, 7, and 10; (d) e<sup>r</sup> ¼ 3; (e) e<sup>r</sup> ¼ 10; and (f) e<sup>r</sup> ¼ 3, 5, 7, and 10.

The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, λ ¼ 3:75 cm, β ¼ 58 1=m, k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm.

(0.5 a, 0.25 b), respectively. The output field is shown in Figure 6(d) for x-axis where y = b/2 = 10 mm and is shown in Figure 6(e) for y-axis where x = a/2 = 10 mm and for e<sup>r</sup> ¼ 3, 5, 7, and 10,

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The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, c ¼ 2:5 mm, d ¼ 2:5 mm, λ ¼ 3:75 cm,

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile, as shown in Figure 6(d). The output field in Figure 6(e) demonstrates the periodic structure with the two rectangular

The output fields for the periodic structure with the two circular profiles along y-axis (Figure 1(e)) are demonstrated in Figure 7(a–c) for e<sup>r</sup> ¼ 3, 5, and 10, respectively. The centers of the upper circle and the lower circle are located at the points (0.5 a, 0.75 b) and (0.5 a, 0.25 b), respectively. The output field is shown in Figure 7(d) for x-axis where y = b/2 = 10 mm and is shown in

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile. The output field in Figure 7 (e) demonstrates the periodic structure with the two circular profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

Figure 7. The output field as a response to a half-sine (TE10) input-wave profile for the periodic circular profiles (a–c) that relate to cross section (Figure 1(e)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 5; (c) e<sup>r</sup> ¼ 10; (d) e<sup>r</sup> ¼ 3, 5, 7, and 10; and

Figure 7(e) for y-axis where x = a/2 = 10 mm for e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

respectively.

β ¼ 58 1=m, k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm.

profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

(e) e<sup>r</sup> ¼ 3, 5, 7, and 10.

The output fields for the periodic structure with the two circular profiles along x-axis (Figure 1(d)) are demonstrated in Figure 5(d and e) for e<sup>r</sup> ¼ 3 and for e<sup>r</sup> ¼ 10, respectively. In this examples, the left circle is located at the point (0.25 a, 0.5 b), and the center of the right circle is located at the point (0.75 a, 0.5 b). Figure 5(f) shows the output field as the function of x-axis where y = b/2 = 10 mm for four values of e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 5(a–c)) that relate to Figure 1(a) and in the cases of periodic circular profiles (Figure 5(d–f)) that relate to Figure 1(d), where e<sup>r</sup> ¼ 3 and 10, respectively. The behavior is similar, but not for every er, and the amplitudes of the output fields are different.

The output fields (Figure 5(a–f)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles or the two circular profiles along x-axis, and the distance between the two centers of the rectangular or the circular profiles.

The output fields for the periodic structure with the two rectangular profiles along y-axis (Figure 1(b)) are demonstrated in Figure 6(a–c) for e<sup>r</sup> ¼ 3, 5, and 10, respectively. The centers of the upper rectangle and the lower rectangle are located at the points (0.5 a, 0.75 b) and

Figure 6. The output field as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (a–c) that relate to cross section (Figure 1(b)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 5; (c) e<sup>r</sup> ¼ 10; (d) e<sup>r</sup> ¼ 3, 5, 7, and 10; and (e) e<sup>r</sup> ¼ 3, 5, 7, and 10.

(0.5 a, 0.25 b), respectively. The output field is shown in Figure 6(d) for x-axis where y = b/2 = 10 mm and is shown in Figure 6(e) for y-axis where x = a/2 = 10 mm and for e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, λ ¼ 3:75 cm, β ¼ 58 1=m,

The output fields for the periodic structure with the two circular profiles along x-axis (Figure 1(d)) are demonstrated in Figure 5(d and e) for e<sup>r</sup> ¼ 3 and for e<sup>r</sup> ¼ 10, respectively. In this examples, the left circle is located at the point (0.25 a, 0.5 b), and the center of the right circle is located at the point (0.75 a, 0.5 b). Figure 5(f) shows the output field as the function of x-axis where

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 5(a–c)) that relate to Figure 1(a) and in the cases of periodic circular profiles (Figure 5(d–f)) that relate to Figure 1(d), where e<sup>r</sup> ¼ 3 and 10, respectively. The behavior is

The output fields (Figure 5(a–f)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles or the two circular profiles along x-axis, and

The output fields for the periodic structure with the two rectangular profiles along y-axis (Figure 1(b)) are demonstrated in Figure 6(a–c) for e<sup>r</sup> ¼ 3, 5, and 10, respectively. The centers of the upper rectangle and the lower rectangle are located at the points (0.5 a, 0.75 b) and

Figure 6. The output field as a response to a half-sine (TE10) input-wave profile for the periodic rectangular profiles (a–c) that relate to cross section (Figure 1(b)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 5; (c) e<sup>r</sup> ¼ 10; (d) e<sup>r</sup> ¼ 3, 5, 7, and 10;

similar, but not for every er, and the amplitudes of the output fields are different.

the distance between the two centers of the rectangular or the circular profiles.

y = b/2 = 10 mm for four values of e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm.

328 Emerging Waveguide Technology

and (e) e<sup>r</sup> ¼ 3, 5, 7, and 10.

The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, c ¼ 2:5 mm, d ¼ 2:5 mm, λ ¼ 3:75 cm, β ¼ 58 1=m, k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm.

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile, as shown in Figure 6(d). The output field in Figure 6(e) demonstrates the periodic structure with the two rectangular profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

The output fields for the periodic structure with the two circular profiles along y-axis (Figure 1(e)) are demonstrated in Figure 7(a–c) for e<sup>r</sup> ¼ 3, 5, and 10, respectively. The centers of the upper circle and the lower circle are located at the points (0.5 a, 0.75 b) and (0.5 a, 0.25 b), respectively. The output field is shown in Figure 7(d) for x-axis where y = b/2 = 10 mm and is shown in Figure 7(e) for y-axis where x = a/2 = 10 mm for e<sup>r</sup> ¼ 3, 5, 7, and 10, respectively.

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile. The output field in Figure 7 (e) demonstrates the periodic structure with the two circular profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

Figure 7. The output field as a response to a half-sine (TE10) input-wave profile for the periodic circular profiles (a–c) that relate to cross section (Figure 1(e)). The results are shown for (a) e<sup>r</sup> ¼ 3; (b) e<sup>r</sup> ¼ 5; (c) e<sup>r</sup> ¼ 10; (d) e<sup>r</sup> ¼ 3, 5, 7, and 10; and (e) e<sup>r</sup> ¼ 3, 5, 7, and 10.

The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, c ¼ 2:5 mm, d ¼ 2:5 mm, λ ¼ 3:75 cm, β ¼ 58 1=m, k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm. The radius of the circle is equal to 2:5 mm.

By increasing the parameter e<sup>r</sup> from 1.2 to 1.4, the output dielectric profile of the structure of the periodic rectangular profile increased, the output profile of the half-sine (TE10) profile decreased, and the output amplitude increased. These results are strongly affected by the half-sine (TE10) input-wave profile and the locations of the rectangular profiles along x-axis and along y-axis. The parameters are a ¼ 2 cm, b ¼ 2 cm, z ¼ 15 cm, k<sup>0</sup> ¼ 167 1=m, λ ¼ 3:75 cm,

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The output fields for the periodic structure with four circular profiles along x-axis and y-axis (Figure 1(f)) are demonstrated in Figure 8(c and d) for e<sup>r</sup> = 4 and 10, respectively. The center of the first circle is located at the point (0.25 a, 0.25 b), the center of the second circle is located at the point (0.75 a, 0.25 b), the center of the third circle is located at the point (0.25 a, 0.75 b), and the center of the firth circle is located at the point (0.75 a, 0.75 b). The parameters are a ¼ 2 cm,

These results are strongly affected by the half-sine (TE10) input-wave profile, the locations of the rectangular profiles (Figure 1(c)) or circular profiles (Figure 1(f)) along x-axis and along y-axis, and the distance between the centers of the profiles. By increasing the parameter εr, the

The main objective of this research was to understand the influence of the periodic rectangular and circular profiles in the cross section along the straight rectangular waveguide on the output field. The second objective was to develop the technique to calculate two kinds of the proposed periodic profiles in the cross section. The calculations are based on using Laplace and Fourier transforms, and the output fields are computed by the inverse Laplace and Fourier

The contribution of the technique and the particular application is important to improve the method that is based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous problems of the periodic rectangular and circular profiles in the cross section (and not only for the continuous profiles). The particular application is based on the ωε function. Thus, the proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only

All the graphical results were demonstrated as a response to a half-sine (TE10) input-wave profile and the periodic profiles in the cross section of the straight rectangular waveguide.

Three examples of periodic rectangular profiles are shown in Figure 1(a–c), and three examples of periodic circular profiles are shown in Figure 1(d–f) in the cross section of the straight rectangular waveguide. It is very interesting to compare between two different techniques of the two different kinds of the profiles (rectangular and circular) in the cross section of the

b ¼ 2 cm, z ¼ 15 cm, k<sup>0</sup> ¼ 167 1=m, λ ¼ 3:75 cm, and β ¼ 58 1=m.

Gaussian shape of the output field increased.

according to the specific discontinuous problem.

rectangular straight waveguide.

and β ¼ 58 1=m.

5. Conclusions

transforms.

The output fields (Figure 6(a–e)) and Figure 7(a–e)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles (Figure 1(b)) or circular profiles (Figure 1(e)) along y-axis, and the distance between the two centers of the profiles.

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 6(a–e)) that relate to Figure 1(b) and in the cases of periodic circular profiles (Figure 7(a–e)) that relate to Figure 1(e), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are different.

The output fields for the periodic structure with four rectangular profiles along x-axis and y-axis (Figure 1(c)) are demonstrated in Figure 8(a and b) for e<sup>r</sup> ¼ 1:2 and 1:4, respectively. The center of the first rectangle is located at the point (0.25 a, 0.25 b), the center of the second rectangle is located at the point (0.75 a, 0.25 b), the center of the third rectangle is located at the point (0.25 a, 0.75 b), and the center of the firth rectangle is located at the point (0.75 a, 0.75 b).

Figure 8. The output field as a response to a half-sine (TE10) input-wave profile in the case of four rectangular profiles along x-axis and along y-axis (Figure 1(c)) for (a) e<sup>r</sup> ¼ 1:2 and (b) e<sup>r</sup> ¼ 1:4. The output field as a response to a half-sine (TE10) input-wave profile in the case of four circular profiles along x-axis and along y-axis (Figure 1(f)) for (c) e<sup>r</sup> ¼ 4 and (b). e<sup>r</sup> ¼ 10.

By increasing the parameter e<sup>r</sup> from 1.2 to 1.4, the output dielectric profile of the structure of the periodic rectangular profile increased, the output profile of the half-sine (TE10) profile decreased, and the output amplitude increased. These results are strongly affected by the half-sine (TE10) input-wave profile and the locations of the rectangular profiles along x-axis and along y-axis. The parameters are a ¼ 2 cm, b ¼ 2 cm, z ¼ 15 cm, k<sup>0</sup> ¼ 167 1=m, λ ¼ 3:75 cm, and β ¼ 58 1=m.

The output fields for the periodic structure with four circular profiles along x-axis and y-axis (Figure 1(f)) are demonstrated in Figure 8(c and d) for e<sup>r</sup> = 4 and 10, respectively. The center of the first circle is located at the point (0.25 a, 0.25 b), the center of the second circle is located at the point (0.75 a, 0.25 b), the center of the third circle is located at the point (0.25 a, 0.75 b), and the center of the firth circle is located at the point (0.75 a, 0.75 b). The parameters are a ¼ 2 cm, b ¼ 2 cm, z ¼ 15 cm, k<sup>0</sup> ¼ 167 1=m, λ ¼ 3:75 cm, and β ¼ 58 1=m.

These results are strongly affected by the half-sine (TE10) input-wave profile, the locations of the rectangular profiles (Figure 1(c)) or circular profiles (Figure 1(f)) along x-axis and along y-axis, and the distance between the centers of the profiles. By increasing the parameter εr, the Gaussian shape of the output field increased.

### 5. Conclusions

The parameters of these examples are a ¼ 2 cm, b ¼ 2 cm, c ¼ 2:5 mm, d ¼ 2:5 mm, λ ¼ 3:75 cm,

The output fields (Figure 6(a–e)) and Figure 7(a–e)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles (Figure 1(b)) or circular profiles (Figure 1(e)) along y-axis, and the distance between the two centers of the profiles.

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 6(a–e)) that relate to Figure 1(b) and in the cases of periodic circular profiles (Figure 7(a–e)) that relate to Figure 1(e), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the

The output fields for the periodic structure with four rectangular profiles along x-axis and y-axis (Figure 1(c)) are demonstrated in Figure 8(a and b) for e<sup>r</sup> ¼ 1:2 and 1:4, respectively. The center of the first rectangle is located at the point (0.25 a, 0.25 b), the center of the second rectangle is located at the point (0.75 a, 0.25 b), the center of the third rectangle is located at the point (0.25 a, 0.75 b), and the center of the firth rectangle is located at the point (0.75 a, 0.75 b).

Figure 8. The output field as a response to a half-sine (TE10) input-wave profile in the case of four rectangular profiles along x-axis and along y-axis (Figure 1(c)) for (a) e<sup>r</sup> ¼ 1:2 and (b) e<sup>r</sup> ¼ 1:4. The output field as a response to a half-sine (TE10) input-wave profile in the case of four circular profiles along x-axis and along y-axis (Figure 1(f)) for (c) e<sup>r</sup> ¼ 4 and

β ¼ 58 1=m, k<sup>0</sup> ¼ 167 1=m, and z ¼ 15 cm. The radius of the circle is equal to 2:5 mm.

output fields are different.

330 Emerging Waveguide Technology

(b). e<sup>r</sup> ¼ 10.

The main objective of this research was to understand the influence of the periodic rectangular and circular profiles in the cross section along the straight rectangular waveguide on the output field. The second objective was to develop the technique to calculate two kinds of the proposed periodic profiles in the cross section. The calculations are based on using Laplace and Fourier transforms, and the output fields are computed by the inverse Laplace and Fourier transforms.

The contribution of the technique and the particular application is important to improve the method that is based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous problems of the periodic rectangular and circular profiles in the cross section (and not only for the continuous profiles). The particular application is based on the ωε function. Thus, the proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem.

All the graphical results were demonstrated as a response to a half-sine (TE10) input-wave profile and the periodic profiles in the cross section of the straight rectangular waveguide.

Three examples of periodic rectangular profiles are shown in Figure 1(a–c), and three examples of periodic circular profiles are shown in Figure 1(d–f) in the cross section of the straight rectangular waveguide. It is very interesting to compare between two different techniques of the two different kinds of the profiles (rectangular and circular) in the cross section of the rectangular straight waveguide.

Figure 5(a–c) relates to Figure 1(a) and Figure 5(d–f) relates to Figure 1(d). The output fields (Figure 6(a–f)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles or the two circular profiles along x-axis, and the distance between the two centers of the rectangular or the circular profiles.

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waveguide. Applied Scientific Research. 1960;8:141-148

finite conductivity walls. Radioengineering. 2011;20:472-478

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 5(a–c)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 5(d–f)) that relate to Figure 1(d), where e<sup>r</sup> ¼ 3 and 10, respectively. The behavior is similar, but not for every er, and the amplitudes of the output fields are different.

Figure 6(a–e) relates to Figure 1(b) and Figure 7(a–e) relates to Figure 1(e). The output fields (Figure 6(a–e) and Figure 7(a–e)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles (Figure 1(b)) or circular profiles (Figure 1(e)) along y-axis, and the distance between the two centers of the profiles.

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile, as shown in Figure 6(d) and Figure 7(d). The output fields in Figure 6(e) and in Figure 7(e) demonstrate the periodic structure with the two rectangular and circular profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 6(a–e)) that relate to Figure 1(b) and in the cases of periodic circular profiles (Figure 7(a–e)) that relate to Figure 1(e), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are different.

The results of the periodic structures of the output field along x-axis and y-axis are demonstrated in Figure 8(a–d). Figure 8(a and b) relates to Figure 1(c) and Figure 8(c and d) relates to Figure 1(f). By increasing the parameter er, the output dielectric profile of the structure of the periodic rectangular profile increased, the output profile of the half-sine (TE10) profile decreased, and the output amplitude increased. The results are strongly affected by the halfsine (TE10) input-wave profile and the locations of the rectangular or circular profiles. The results show in general similar behavior of the output field, but not the same results, and also the amplitudes of the output fields are different.

The application is useful for straight waveguides in the microwave and the millimeter wave regimes, with periodic rectangular and periodic circular profiles in the cross section of the straight waveguide.

### Author details

Zion Menachem

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

Department of Electrical Engineering, Sami Shamoon College of Engineering, Beer Sheva, Israel

### References

Figure 5(a–c) relates to Figure 1(a) and Figure 5(d–f) relates to Figure 1(d). The output fields (Figure 6(a–f)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles or the two circular profiles along x-axis, and the

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 5(a–c)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 5(d–f)) that relate to Figure 1(d), where e<sup>r</sup> ¼ 3 and 10, respectively. The behavior is similar, but not for

Figure 6(a–e) relates to Figure 1(b) and Figure 7(a–e) relates to Figure 1(e). The output fields (Figure 6(a–e) and Figure 7(a–e)) are strongly affected by the input-wave profile (TE<sup>10</sup> mode), the periodic structure with the two rectangular profiles (Figure 1(b)) or circular profiles

By changing only the parameter e<sup>r</sup> from 3 to 10, the relative profile of the output field is changed from a half-sine (TE10) profile to a Gaussian shape profile, as shown in Figure 6(d) and Figure 7(d). The output fields in Figure 6(e) and in Figure 7(e) demonstrate the periodic

It is interesting to see a similar behavior of the output results in the cases of periodic rectangular profiles (Figure 6(a–e)) that relate to Figure 1(b) and in the cases of periodic circular profiles (Figure 7(a–e)) that relate to Figure 1(e), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the

The results of the periodic structures of the output field along x-axis and y-axis are demonstrated in Figure 8(a–d). Figure 8(a and b) relates to Figure 1(c) and Figure 8(c and d) relates to Figure 1(f). By increasing the parameter er, the output dielectric profile of the structure of the periodic rectangular profile increased, the output profile of the half-sine (TE10) profile decreased, and the output amplitude increased. The results are strongly affected by the halfsine (TE10) input-wave profile and the locations of the rectangular or circular profiles. The results show in general similar behavior of the output field, but not the same results, and also

The application is useful for straight waveguides in the microwave and the millimeter wave regimes, with periodic rectangular and periodic circular profiles in the cross section of the

Department of Electrical Engineering, Sami Shamoon College of Engineering, Beer Sheva, Israel

(Figure 1(e)) along y-axis, and the distance between the two centers of the profiles.

structure with the two rectangular and circular profiles for e<sup>r</sup> ¼ 3, 5, 7, and 10.

distance between the two centers of the rectangular or the circular profiles.

every er, and the amplitudes of the output fields are different.

output fields are different.

332 Emerging Waveguide Technology

straight waveguide.

Author details

Zion Menachem

the amplitudes of the output fields are different.

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


[16] Soekmadji H, Liao SL, Vernon RJ. Experiment and simulation on TE<sup>10</sup> cut-off reflection phase in gentle rectangular downtapers. Progress in Electromagnetics Research Letters. 2009;12:79-85

**Chapter 18**

Provisional chapter

**A Theoretical Model of the Holographic Formation of**

DOI: 10.5772/intechopen.74838

Rapid development of the integrated optics and photonics makes it necessary to create cheap and simple technology of optical waveguide systems formation. Photolithography methods, widely used for these tasks recently, require the production of a number of precision amplitude and phase masks. This fact makes this technology expensive and the formation process long. On another side there is a cheap and one-step holographic recording method in photopolymer compositions. Parameters of the waveguide system formed by this method are determined by recording geometry and material's properties. Besides, compositions may contain liquid crystals that make it possible to create elements, controllable by external electric field. In this chapter, the theoretical model of the holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. Special attention is paid to localization of waveguides in the

The ability to form waveguide systems for optical and terahertz radiation in photopolymerizable compositions recently is of great interest among researchers: [1–4]. Formed holographically or by photolithography methods, such waveguides are widely used in the integrated optics and photonics devices. Besides, it seems urgent to create the manageable light guides, in which the light propagation conditions can be controlled by external influences, such as an electric field.

> © 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited.

© 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, provided the original work is properly cited.

media caused by light field attenuation during the formation process.

Keywords: photopolymer, liquid crystal, waveguide, holography

A Theoretical Model of the Holographic Formation of

**Controllable Waveguide Channels System in**

Controllable Waveguide Channels System in

**Photopolymer Liquid Crystalline Composition**

Photopolymer Liquid Crystalline Composition

Artem Semkin and Sergey Sharangovich

Artem Semkin and Sergey Sharangovich

http://dx.doi.org/10.5772/intechopen.74838

Abstract

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer Liquid Crystalline Composition** A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer Liquid Crystalline Composition

DOI: 10.5772/intechopen.74838

Artem Semkin and Sergey Sharangovich Artem Semkin and Sergey Sharangovich

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74838

#### Abstract

[16] Soekmadji H, Liao SL, Vernon RJ. Experiment and simulation on TE<sup>10</sup> cut-off reflection phase in gentle rectangular downtapers. Progress in Electromagnetics Research Letters.

[17] Abbas Z, Pollard RD, Kelsall W. A rectangular dielectric waveguide technique for determination of permittivity of materials at W-band. IEEE Transactions on Microwave Theory

[18] Hewlett SJ, Ladouceur F. Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff. Journal of

[19] Binzhao C, Fuyong X. Analysis for new types of waveguide with Fourier's expansiondifferential method. International Journal of Infrared and Millimeter Waves. 2008;29:240-248

[20] Hernandez-Lopez MA, Quintillan M. Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method. Journal of Electro-

[21] Vaish A, Parthasarathy H. Analysis of rectangular waveguide using finite element

[22] Baganas K. Inhomogeneous dielectric media: wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide. Journal of Electromagnetic

[23] Menachem Z, Jerby E. Transfer matrix function (TMF) for propagation in dielectric waveguides with arbitrary transverse profiles. IEEE Transactions on Microwave Theory and

[24] Menachem Z, Tapuchi S. Influence of the spot-size and cross-section on the output fields and power density along the straight hollow waveguide. Progress in Electromagnetics

[25] Vladimirov V. Equations of Mathematical Physics. New York (NY): Marcel Dekker, Inc.,

2009;12:79-85

334 Emerging Waveguide Technology

and Techniques. 1998;46:2011-2015

Lightwave Technology. 1995;13:375-383

magnetic Waves and Applications. 2000;14:1707-1722

Waves and Applications. 2002;16:1371-1392

Techniques. 1998;46:975-982

Research. 2013;48:151-173

1971

method. Progress in Electromagnetics Research C. 2008;2:117-125

Rapid development of the integrated optics and photonics makes it necessary to create cheap and simple technology of optical waveguide systems formation. Photolithography methods, widely used for these tasks recently, require the production of a number of precision amplitude and phase masks. This fact makes this technology expensive and the formation process long. On another side there is a cheap and one-step holographic recording method in photopolymer compositions. Parameters of the waveguide system formed by this method are determined by recording geometry and material's properties. Besides, compositions may contain liquid crystals that make it possible to create elements, controllable by external electric field. In this chapter, the theoretical model of the holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. Special attention is paid to localization of waveguides in the media caused by light field attenuation during the formation process.

Keywords: photopolymer, liquid crystal, waveguide, holography

### 1. Introduction

The ability to form waveguide systems for optical and terahertz radiation in photopolymerizable compositions recently is of great interest among researchers: [1–4]. Formed holographically or by photolithography methods, such waveguides are widely used in the integrated optics and photonics devices. Besides, it seems urgent to create the manageable light guides, in which the light propagation conditions can be controlled by external influences, such as an electric field.

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 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, provided the original work is properly cited.

One of the possible solutions of this problem is a holographic recording of the waveguide channels in a photopolymer composition containing liquid crystals.

The aim of this chapter is to develop the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition with dye sensitizer, also known as polymer-dispersed (PDLC) or polymer-stabilized (PSLC) liquid crystals.

#### 2. Theoretical model

We consider the incidence of two plane monochromatic waves E<sup>0</sup> and E<sup>1</sup> with incidence angles θ<sup>0</sup> and θ<sup>1</sup> on the PDLC (PSLC) sample for two formation geometries: transmission (Figure 1a) and reflection (Figure 1b) ones.

Thus, recording waves in the general case can be described as:

$$\begin{aligned} \mathbf{E}\_{0}(\mathbf{r},t) &= \mathbf{e}\_{0} \cdot E\_{0}(\mathbf{r}) \cdot e^{i \cdot \left(\omega t - \mathbf{k}\_{0} \cdot \mathbf{r} - \varphi\_{0}(\mathbf{r})\right) - a(\mathbf{r},t) \cdot (\mathbf{N}\_{0} \cdot \mathbf{r})}, \\ \mathbf{E}\_{1}(\mathbf{r},t) &= \mathbf{e}\_{1} \cdot E\_{1}(\mathbf{r}) \cdot e^{i \cdot \left(\omega t - \mathbf{k}\_{l} \cdot \mathbf{r} - \varphi\_{1}(\mathbf{r})\right) - a(\mathbf{r},t) \cdot (\mathbf{N}\_{l} \cdot \mathbf{r})}, \end{aligned} \tag{1}$$

<sup>E</sup>0ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup>

<sup>E</sup>1ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup>

intensities are determined by the following expression:

�αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup>

<sup>2</sup> ½ � � ch <sup>α</sup>ð Þ� <sup>r</sup>; <sup>t</sup> <sup>N</sup><sup>m</sup>

�αð Þ� r;t d

local contrasts of interference patterns; <sup>K</sup><sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>m</sup>

reactions and turns to colorless leuco form.

of a mount of absorbed radiation can be written as [5]:

reflection geometries, respectively; I

respectively.

<sup>T</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup>

<sup>R</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup> <sup>m</sup>¼o,e I 0 ð Þ� r e

<sup>m</sup>¼o,e I 0 ð Þ� r e

<sup>T</sup>ð Þ <sup>r</sup>; <sup>t</sup> , <sup>I</sup>

I T

I R

where I T absð Þ r; t , I

absð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup>

absð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>X</sup>

tion geometries is obtained in [5] as follows:

R

<sup>m</sup>¼o,e E2

<sup>m</sup>¼o,e E2

I

I

where I

<sup>m</sup>¼o,e

<sup>m</sup>¼o,e

em

em

<sup>0</sup> � E0ð Þ� r e

<sup>1</sup> � E1ð Þ� r e

<sup>0</sup> <sup>þ</sup> <sup>N</sup><sup>m</sup> 1 � � � <sup>r</sup> � <sup>d</sup>

� � � �

0 ð Þ¼ <sup>r</sup> <sup>E</sup><sup>2</sup>

� <sup>d</sup>αð Þ <sup>r</sup>; <sup>t</sup>

of dye; and Iabsð Þ r; t is the light intensity from Bouguer-Lambert law:

<sup>0</sup>ð Þ� r 1 � e

<sup>0</sup>ð Þ� r 1 � e

<sup>i</sup>� <sup>ω</sup>t�k<sup>m</sup>

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

<sup>i</sup>� <sup>ω</sup>t�k<sup>m</sup>

where m = o corresponds to ordinary waves and m = e corresponds to extraordinary waves,

For two recording geometries (see Figure 1), the spatial distributions of the forming field

<sup>0</sup> <sup>þ</sup>N<sup>m</sup> ð Þ<sup>1</sup> ½ � �<sup>r</sup> � <sup>1</sup> <sup>þ</sup> <sup>m</sup><sup>m</sup>ð Þ<sup>r</sup> cos <sup>K</sup><sup>m</sup> � <sup>r</sup> <sup>þ</sup> <sup>φ</sup>0ð Þ� <sup>r</sup> <sup>φ</sup>1ð Þ<sup>r</sup> � � � �

<sup>R</sup>ð Þ <sup>r</sup>; <sup>t</sup> are the spatial distributions of light field intensity for transmission and

<sup>1</sup>ð Þ<sup>r</sup> , <sup>m</sup><sup>m</sup>ð Þ¼ <sup>r</sup>

2

<sup>0</sup>ð Þþ <sup>r</sup> <sup>E</sup><sup>2</sup>

<sup>0</sup> � <sup>k</sup><sup>m</sup>

Under the influence of light field in photopolymer liquid crystalline composition with dye sensitizer, the dye molecule absorbs a light radiation quantum with the dye radical and the primary radical initiator formation. The radical of dye is not involved in further chemical

Thus, during the waveguide channel system formation, dye concentration decreases. This fact causes the light-induced decreasing of light absorption. The absorption coefficient dependence

where αð Þ¼ r; t α<sup>0</sup> � Kdð Þ r; t is the absorption coefficient with light-induced change taken into account; Kdð Þ r; t is dye concentration; α<sup>0</sup> is the absorption of one molecule; β<sup>q</sup> is quantum yield

> �αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup> <sup>0</sup> ½ � �<sup>r</sup> � � <sup>þ</sup> <sup>E</sup><sup>2</sup>

> �αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup> <sup>0</sup> ½ � �<sup>r</sup> � � <sup>þ</sup> <sup>E</sup><sup>2</sup>

Then, the solution of Eq. (3) for light-induced absorption change for transmission and reflec-

<sup>0</sup> �r�φ<sup>0</sup> ð Þ ð Þ<sup>r</sup> �αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup>

<sup>1</sup> �r�φ<sup>1</sup> ð Þ ð Þ<sup>r</sup> �αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup>

<sup>0</sup> ð Þ�<sup>r</sup> ,

� <sup>1</sup> <sup>þ</sup> <sup>m</sup><sup>m</sup>ð Þ<sup>r</sup> cos <sup>K</sup><sup>m</sup> � <sup>r</sup> <sup>þ</sup> <sup>φ</sup>0ð Þ� <sup>r</sup> <sup>φ</sup>1ð Þ<sup>r</sup> � � � � ,

2E0ð Þ� r E1ð Þr E2 <sup>0</sup>ð Þþ<sup>r</sup> <sup>E</sup><sup>2</sup>

<sup>1</sup> ; d is the thickness of the material.

dt <sup>¼</sup> <sup>β</sup><sup>q</sup> � <sup>α</sup><sup>0</sup> � Iabsð Þ <sup>r</sup>; <sup>t</sup> , (4)

�αð Þ� <sup>r</sup>;<sup>t</sup> <sup>N</sup><sup>m</sup> <sup>1</sup> ½ � �<sup>r</sup> � �

�αð Þ� <sup>r</sup>;<sup>t</sup> <sup>d</sup>�N<sup>m</sup>

<sup>1</sup> ½ � �<sup>r</sup> � � , (5)

<sup>1</sup>ð Þ� r 1 � e

<sup>1</sup>ð Þ� r 1 � e

absð Þ r; t are defined for transmission and reflection geometries, respectively.

<sup>1</sup>ð Þ<sup>r</sup> � <sup>e</sup><sup>m</sup>

<sup>0</sup> � <sup>e</sup><sup>m</sup> 1 � � are the

<sup>1</sup> ð Þ�<sup>r</sup> , (2)

http://dx.doi.org/10.5772/intechopen.74838

(3)

337

where e0, e<sup>1</sup> are the unit polarizations vectors of the beams; E0ð Þr , E1ð Þr are the spatial amplitude distributions; φ0ð Þr , φ1ð Þr are spatial phase distributions; k0, k<sup>1</sup> are the wave vectors; N0, N<sup>1</sup> are wave normals; αð Þ r; t is the absorption coefficient; ω ¼ 2π=λ, λ is the wavelength of the recording radiation.

Investigated material is characterized by optical anisotropy properties; thus, in the material, each of recording wave (Eq. (1)) will be divided into two mutually orthogonal ones called ordinary and extraordinary. So, in the sample, Eq. (1) should be rewritten as:

Figure 1. Waveguide channels holographic formation: (a) transmission recording geometry and (b) reflection recording geometry.

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer… http://dx.doi.org/10.5772/intechopen.74838 337

$$\begin{split} \mathbf{E}\_{0}(\mathbf{r},t) &= \sum\_{m=o\_{\ell}} \mathbf{e}\_{0}^{m} \cdot \mathbf{E}\_{0}(\mathbf{r}) \cdot e^{i\left\{at - \mathbf{k}\_{0}^{\pi} \cdot \mathbf{r} - \varphi\_{0}(\mathbf{r})\right\} - a(\mathbf{r},t) \cdot \left(\mathbf{N}\_{0}^{w} \cdot \mathbf{r}\right)}, \\ \mathbf{E}\_{1}(\mathbf{r},t) &= \sum\_{m=o\_{\ell}} \mathbf{e}\_{1}^{m} \cdot \mathbf{E}\_{1}(\mathbf{r}) \cdot e^{i\left\{at - \mathbf{k}\_{1}^{\pi} \cdot \mathbf{r} - \varphi\_{1}(\mathbf{r})\right\} - a(\mathbf{r},t) \cdot \left(\mathbf{N}\_{1}^{w} \cdot \mathbf{r}\right)}. \end{split} \tag{2}$$

where m = o corresponds to ordinary waves and m = e corresponds to extraordinary waves, respectively.

One of the possible solutions of this problem is a holographic recording of the waveguide

The aim of this chapter is to develop the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition with dye sensitizer, also known as polymer-dispersed (PDLC) or polymer-stabilized (PSLC) liquid

We consider the incidence of two plane monochromatic waves E<sup>0</sup> and E<sup>1</sup> with incidence angles θ<sup>0</sup> and θ<sup>1</sup> on the PDLC (PSLC) sample for two formation geometries: transmission (Figure 1a)

where e0, e<sup>1</sup> are the unit polarizations vectors of the beams; E0ð Þr , E1ð Þr are the spatial amplitude distributions; φ0ð Þr , φ1ð Þr are spatial phase distributions; k0, k<sup>1</sup> are the wave vectors; N0, N<sup>1</sup> are wave normals; αð Þ r; t is the absorption coefficient; ω ¼ 2π=λ, λ is the wavelength

Investigated material is characterized by optical anisotropy properties; thus, in the material, each of recording wave (Eq. (1)) will be divided into two mutually orthogonal ones called

Figure 1. Waveguide channels holographic formation: (a) transmission recording geometry and (b) reflection recording

<sup>i</sup>� <sup>ω</sup>t�k0�r�φ<sup>0</sup> ð Þ ð Þ<sup>r</sup> �αð Þ� <sup>r</sup>;<sup>t</sup> ð Þ <sup>N</sup>0�<sup>r</sup> ,

<sup>i</sup>� <sup>ω</sup>t�k1�r�φ<sup>1</sup> ð Þ ð Þ<sup>r</sup> �αð Þ� <sup>r</sup>;<sup>t</sup> ð Þ <sup>N</sup>1�<sup>r</sup> , (1)

channels in a photopolymer composition containing liquid crystals.

Thus, recording waves in the general case can be described as:

E0ð Þ¼ r; t e<sup>0</sup> � E0ð Þ� r e

E1ð Þ¼ r; t e<sup>1</sup> � E1ð Þ� r e

ordinary and extraordinary. So, in the sample, Eq. (1) should be rewritten as:

crystals.

2. Theoretical model

336 Emerging Waveguide Technology

and reflection (Figure 1b) ones.

of the recording radiation.

geometry.

For two recording geometries (see Figure 1), the spatial distributions of the forming field intensities are determined by the following expression:

$$\begin{split} I^{\mathrm{T}}(\mathbf{r},t) &= \sum\_{m=o\_{\mathrm{r}}\epsilon} I^{0}(\mathbf{r}) \cdot e^{-a(\mathbf{r},t) \cdot \left[ \left( \mathbf{N}\_{0}^{m} + \mathbf{N}\_{1}^{m} \right) \mathbf{r} \right]} \cdot \left[ 1 + m^{m}(\mathbf{r}) \cos \left( \mathbf{K}^{m} \cdot \mathbf{r} + \varphi\_{0}(\mathbf{r}) - \varphi\_{1}(\mathbf{r}) \right) \right] \\ I^{R}(\mathbf{r},t) &= \sum\_{m=o\_{\mathrm{r}}\epsilon} I^{0}(\mathbf{r}) \cdot e^{\left[ \frac{-a(\mathbf{r},t)}{2} \right]} \cdot \operatorname{ch} \left\{ a(\mathbf{r},t) \cdot \left[ \left( \mathbf{N}\_{0}^{m} + \mathbf{N}\_{1}^{m} \right) \cdot \mathbf{r} - \frac{d}{2} \right] \right\} \cdot \left[ 1 + m^{m}(\mathbf{r}) \cos \left( \mathbf{K}^{m} \cdot \mathbf{r} + \varphi\_{0}(\mathbf{r}) - \varphi\_{1}(\mathbf{r}) \right) \right], \end{split} \tag{3}$$

where I <sup>T</sup>ð Þ <sup>r</sup>; <sup>t</sup> , <sup>I</sup> <sup>R</sup>ð Þ <sup>r</sup>; <sup>t</sup> are the spatial distributions of light field intensity for transmission and reflection geometries, respectively; I 0 ð Þ¼ <sup>r</sup> <sup>E</sup><sup>2</sup> <sup>0</sup>ð Þþ <sup>r</sup> <sup>E</sup><sup>2</sup> <sup>1</sup>ð Þ<sup>r</sup> , <sup>m</sup><sup>m</sup>ð Þ¼ <sup>r</sup> 2E0ð Þ� r E1ð Þr E2 <sup>0</sup>ð Þþ<sup>r</sup> <sup>E</sup><sup>2</sup> <sup>1</sup>ð Þ<sup>r</sup> � <sup>e</sup><sup>m</sup> <sup>0</sup> � <sup>e</sup><sup>m</sup> 1 � � are the local contrasts of interference patterns; <sup>K</sup><sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>m</sup> <sup>0</sup> � <sup>k</sup><sup>m</sup> <sup>1</sup> ; d is the thickness of the material.

Under the influence of light field in photopolymer liquid crystalline composition with dye sensitizer, the dye molecule absorbs a light radiation quantum with the dye radical and the primary radical initiator formation. The radical of dye is not involved in further chemical reactions and turns to colorless leuco form.

Thus, during the waveguide channel system formation, dye concentration decreases. This fact causes the light-induced decreasing of light absorption. The absorption coefficient dependence of a mount of absorbed radiation can be written as [5]:

$$-\frac{d\alpha(\mathbf{r},t)}{dt} = \beta\_q \cdot \alpha\_0 \cdot I\_{abs}(\mathbf{r},t),\tag{4}$$

where αð Þ¼ r; t α<sup>0</sup> � Kdð Þ r; t is the absorption coefficient with light-induced change taken into account; Kdð Þ r; t is dye concentration; α<sup>0</sup> is the absorption of one molecule; β<sup>q</sup> is quantum yield of dye; and Iabsð Þ r; t is the light intensity from Bouguer-Lambert law:

$$I\_{\rm abs}^{\rm T}(\mathbf{r},t) = \sum\_{m=o\_{\prime}\epsilon} E\_{0}^{2}(\mathbf{r}) \cdot \left(1 - e^{-a(\mathbf{r},t) \cdot \left[\mathbf{N}\_{0}^{m} \cdot \mathbf{r}\right]}\right) + E\_{1}^{2}(\mathbf{r}) \cdot \left(1 - e^{-a(\mathbf{r},t) \cdot \left[\mathbf{N}\_{1}^{m} \cdot \mathbf{r}\right]}\right)$$

$$I\_{\rm abs}^{\rm R}(\mathbf{r},t) = \sum\_{m=o\_{\prime}\epsilon} E\_{0}^{2}(\mathbf{r}) \cdot \left(1 - e^{-a(\mathbf{r},t) \cdot \left[\mathbf{N}\_{0}^{m} \cdot \mathbf{r}\right]}\right) + E\_{1}^{2}(\mathbf{r}) \cdot \left(1 - e^{-a(\mathbf{r},t) \cdot \left[\mathbf{d} - \mathbf{N}\_{1}^{m} \cdot \mathbf{r}\right]}\right)'\tag{5}$$

where I T absð Þ r; t , I R absð Þ r; t are defined for transmission and reflection geometries, respectively. Then, the solution of Eq. (3) for light-induced absorption change for transmission and reflection geometries is obtained in [5] as follows:

$$\begin{aligned} \alpha\_T^m(\mathbf{r}, t) &= \alpha\_{sub} + \alpha\_0 K\_{d0} \cdot e^{-\boldsymbol{\beta}\_q \cdot \mathbf{a}\_0 \cdot \left[ \left( \mathbf{N}\_0^m + \mathbf{N}\_1^m \right) \cdot \mathbf{r} \cdot t \right]} \\ \alpha\_R^m(\mathbf{r}, t) &= \alpha\_{sub} + \alpha\_0 K\_{d0} \cdot e^{-\boldsymbol{\beta}\_q \cdot \mathbf{a}\_0 \cdot \left[ \left( \mathbf{N}\_0^m \cdot \mathbf{r} + d - \mathbf{N}\_1^m \cdot \mathbf{r} \right) \cdot t \right]} \end{aligned} \tag{6}$$

nst ¼ Mn �

where nM, nLC are the monomer and liquid crystal refractive indices.

of KES can be found as a sum of H spatial harmonics [8]:

Mm

the PDLC (PSLC) refractive index at <sup>τ</sup> <sup>¼</sup> 0; <sup>τ</sup> <sup>¼</sup> <sup>t</sup>=T<sup>m</sup>

concentration harmonics can be obtained for D<sup>m</sup>

ficients) and Kdð Þ¼ r; t Kd<sup>0</sup> (stable absorption) [6]:

8

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

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

∂n<sup>m</sup> <sup>0</sup> ð Þ r; τ ∂τ

8

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

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

∂n<sup>m</sup> <sup>1</sup> ð Þ r; τ ∂τ

∂n<sup>m</sup> <sup>H</sup>ð Þ r; τ ∂τ

∂Mm <sup>0</sup> ð Þ r; τ <sup>∂</sup><sup>τ</sup> <sup>¼</sup> <sup>X</sup>

∂Mm <sup>1</sup> ð Þ r; τ <sup>∂</sup><sup>τ</sup> ¼ �Mm

∂Mm <sup>H</sup>ð Þ r; τ <sup>∂</sup><sup>τ</sup> ¼ �N<sup>2</sup>

H

j¼0

2π Ð π �π Mm

components' diffusion characteristic time.

<sup>M</sup><sup>m</sup>ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>X</sup>

<sup>j</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>1</sup>

where Mm

n2 <sup>M</sup> � 1 n2

<sup>M</sup> <sup>þ</sup> <sup>2</sup> <sup>þ</sup> Ln �

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

Because of periodical character of the forming fields' intensities' spatial distributions, solution

<sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> , n<sup>m</sup>ð Þ¼ <sup>r</sup>; <sup>τ</sup> nst <sup>þ</sup><sup>X</sup>

are the monomer concentration and refractive index harmonics amplitudes, respectively; nst is

By substituting Eq. (13) to the KES (Eqs. (7 and 8)) and using the orthogonality of spatial harmonics, a system of coupled kinetic differential equations for the amplitudes of monomer

<sup>1</sup> ð Þþ <sup>r</sup>; <sup>τ</sup> <sup>X</sup>

<sup>M</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>D</sup><sup>m</sup>

<sup>l</sup> ð Þ r; τ

H

l¼0 am 1,l ð Þ<sup>r</sup> <sup>M</sup><sup>m</sup>

H

l¼0 am N,l ð Þ<sup>r</sup> Mm

<sup>l</sup> ð Þ r; τ

<sup>l</sup> ð Þþ <sup>r</sup>; <sup>τ</sup> <sup>δ</sup>nlcM<sup>m</sup>

<sup>l</sup> ð Þþ <sup>r</sup>; <sup>τ</sup> <sup>δ</sup>nlcH<sup>2</sup>

<sup>H</sup>ð Þþ <sup>r</sup>; <sup>τ</sup> <sup>X</sup>

<sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> <sup>d</sup> <sup>K</sup><sup>m</sup> ð Þ<sup>r</sup> , nm

H

l¼0 am 0,l ð Þ<sup>r</sup> Mm

Mn ¼ �δnp

Mn ¼ �δnp

Mn ¼ �δnp

In equation systems Eqs. (14) and (15), a coefficient matrix is introduced [6]:

::……………………………………

Mm

X H

l¼0 am 0,l ð Þ<sup>r</sup> Mm

X H

l¼0 am 1,l ð Þ<sup>r</sup> Mm

:…………………………………………………

X H

l¼0 am H,l ð Þ<sup>r</sup> Mm

and also a system of differential equations for the amplitudes of refraction index harmonics:

n2 LC � 1 n2 LC þ 2

H

j¼0 nm

> 2π Ð π �π nm

<sup>M</sup> is the relative time; T<sup>m</sup>

LCð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>D</sup><sup>m</sup>

<sup>l</sup> ð Þ r; τ

<sup>l</sup> ð Þ r; τ

<sup>1</sup> ð Þ r; τ

:

Mm <sup>H</sup>ð Þ r; τ

,

<sup>j</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>1</sup>

, (12)

http://dx.doi.org/10.5772/intechopen.74838

<sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> , (13)

<sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> <sup>d</sup> <sup>K</sup><sup>m</sup> ð Þ<sup>r</sup>

Mn� <sup>K</sup><sup>m</sup> j j<sup>2</sup> is the

339

(14)

(15)

<sup>M</sup> <sup>¼</sup> <sup>1</sup> Dm

Mn (stable diffusion coef-

where αsub is the substrate absorption coefficient and Kd<sup>0</sup> is the initial dye concentration.

The process of waveguide channels' holographic formation is described by the kinetic equations system (KES), written for monomer concentration and refraction index [6–8]:

$$\frac{\partial M^{\mathrm{m}}(\mathbf{r},t)}{\partial t} = \mathrm{div}\left[D\_{M}^{\mathrm{m}}(\mathbf{r},t)\mathrm{grad}\,M^{\mathrm{m}}(\mathbf{r},t)\right] - K\_{\mathrm{g}} \cdot \left[\frac{\alpha\_{0}\beta\mathrm{K}\_{d0}\tau\_{0}I^{\mathrm{m}}(\mathbf{r},t)}{K\_{b}}\right]^{0.5}M^{\mathrm{m}}(\mathbf{r},t),\tag{7}$$

$$\frac{\partial \mathbf{u}^{m}(\mathbf{r},t)}{\partial t} = \delta \boldsymbol{n}\_{\mathcal{P}} \cdot \mathbf{K}\_{\mathcal{S}} \cdot \left[\frac{\alpha\_{0} \beta \mathbf{K}\_{d0} \pi\_{0} \mathbf{I}^{m}(\mathbf{r},t)}{\mathbf{K}\_{\mathcal{b}}}\right]^{0.5} \frac{M^{m}(\mathbf{r},t)}{M\_{\text{il}}} + \delta \boldsymbol{n}\_{\mathcal{U}} \text{div}\left[\boldsymbol{D}\_{\mathcal{L}\mathcal{C}}^{m}(\mathbf{r},t) \text{grad}\frac{M^{m}(\mathbf{r},t)}{M\_{\text{il}}}\right],\tag{8}$$

where Mn is the initial concentration of the monomer; Kg, Kb are parameters of the rate of growth and breakage of the polymer chain, respectively; β is the parameter of initiation reaction; τ<sup>0</sup> is lifetime of the excited state of the dye molecule; D<sup>m</sup> <sup>M</sup>ð Þ <sup>r</sup>; <sup>t</sup> , <sup>D</sup><sup>m</sup> LCð Þ r; t are the diffusion coefficients of the monomer and liquid crystal, respectively; δnp, δnlc are weight coefficients of the contribution of photopolymerization and diffusion processes; and I <sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup> is the intensity distribution (Eq. (3)).

Diffusion coefficients can be defined from the following equations:

$$\begin{split} D\_{\mathcal{M}}^{m}(\mathbf{r},t) &= D\_{\mathcal{M}n} \exp\left[ -\mathbf{s}\_{\mathcal{M}} \left( 1 - \frac{\mathcal{M}^{m}(\mathbf{r},t)}{M\_{n}} \right) \right] \\ D\_{\mathcal{L}\mathcal{C}}^{m}(\mathbf{r},t) &= D\_{\mathcal{L}\mathcal{C}n} \exp\left[ -\mathbf{s}\_{\mathcal{L}\mathcal{C}} \left( 1 - \frac{\mathcal{L}^{m}(\mathbf{r},t)}{L\_{n}} \right) \right]' \end{split} \tag{9}$$

where DMn and DLCn are the initial diffusion coefficients, respectively; sM, sLC are rates of reduction in time; Ln, Lmð Þ <sup>r</sup>; <sup>t</sup> are the initial and current concentrations of liquid crystal.

Weight coefficients δnp and δnlc from Eq. (7) are found from the Lorentz-Lorentz formula [8]:

$$
\delta n\_p = \frac{4\pi}{3} \cdot \frac{\left(n\_{st}^2 + 2\right)}{6n\_{st}^2} \cdot \left(\alpha\_M + \frac{\alpha p}{l}\right) \cdot \frac{\rho\_M}{W\_M} \tag{10}
$$

$$
\delta n\_{lc} = \frac{4\pi}{3} \cdot \frac{\left(n\_{st}^2 + 2\right)}{6n\_{st}^2} \cdot \left(\alpha\_M \frac{\rho\_M}{W\_M} + \alpha\_{L\mathcal{C}} \frac{\rho\_{L\mathcal{C}}}{W\_{L\mathcal{C}}}\right),
\tag{11}
$$

where αM, αP, αLC are the polarizability of monomer, polymer, and liquid crystal molecules, respectively; rM, rLC are the density of the monomer and liquid crystal, respectively; WM, WLC are molecular weights; l is the average length of polymeric chains; and nst is the refractive index of the composition prior to the start of the recording process, determined by the Lorentz-Lorentz formula from the refractive indices of the monomer and the liquid crystal [8]:

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer… http://dx.doi.org/10.5772/intechopen.74838 339

$$m\_{st} = M\_n \cdot \frac{n\_M^2 - 1}{n\_M^2 + 2} + L\_n \cdot \frac{n\_{LC}^2 - 1}{n\_{LC}^2 + 2} \tag{12}$$

where nM, nLC are the monomer and liquid crystal refractive indices.

αm

αm

<sup>∂</sup>Mmð Þ <sup>r</sup>; <sup>t</sup>

the intensity distribution (Eq. (3)).

<sup>∂</sup>nmð Þ <sup>r</sup>; <sup>t</sup>

338 Emerging Waveguide Technology

<sup>∂</sup><sup>t</sup> <sup>¼</sup> div <sup>D</sup><sup>m</sup>

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>δ</sup>np � Kg � <sup>α</sup>0βKd0τ0<sup>I</sup>

<sup>T</sup> ð Þ¼ r; t αsub þ α0Kd<sup>0</sup> � e

<sup>R</sup> ð Þ¼ r; t αsub þ α0Kd<sup>0</sup> � e

where αsub is the substrate absorption coefficient and Kd<sup>0</sup> is the initial dye concentration.

tions system (KES), written for monomer concentration and refraction index [6–8]:

<sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup>

<sup>0</sup>, <sup>5</sup> <sup>M</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup>

Kb

reaction; τ<sup>0</sup> is lifetime of the excited state of the dye molecule; D<sup>m</sup>

Diffusion coefficients can be defined from the following equations:

<sup>δ</sup>np <sup>¼</sup> <sup>4</sup><sup>π</sup>

<sup>δ</sup>nlc <sup>¼</sup> <sup>4</sup><sup>π</sup>

<sup>3</sup> � <sup>n</sup><sup>2</sup> st <sup>þ</sup> <sup>2</sup> 6n<sup>2</sup> st

<sup>3</sup> � <sup>n</sup><sup>2</sup> st <sup>þ</sup> <sup>2</sup> 6n<sup>2</sup> st

D<sup>m</sup>

D<sup>m</sup>

The process of waveguide channels' holographic formation is described by the kinetic equa-

<sup>M</sup>ð Þ <sup>r</sup>; <sup>t</sup> grad <sup>M</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup> � Kg � <sup>α</sup>0βKd0τ0<sup>I</sup>

Mn

where Mn is the initial concentration of the monomer; Kg, Kb are parameters of the rate of growth and breakage of the polymer chain, respectively; β is the parameter of initiation

diffusion coefficients of the monomer and liquid crystal, respectively; δnp, δnlc are weight

<sup>M</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> DMnexp �sM <sup>1</sup> � <sup>M</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup>

LCð Þ¼ <sup>r</sup>; <sup>t</sup> DLCnexp �sLC <sup>1</sup> � Lmð Þ <sup>r</sup>; <sup>t</sup>

where DMn and DLCn are the initial diffusion coefficients, respectively; sM, sLC are rates of reduction in time; Ln, Lmð Þ <sup>r</sup>; <sup>t</sup> are the initial and current concentrations of liquid crystal.

Weight coefficients δnp and δnlc from Eq. (7) are found from the Lorentz-Lorentz formula [8]:

� α<sup>M</sup>

where αM, αP, αLC are the polarizability of monomer, polymer, and liquid crystal molecules, respectively; rM, rLC are the density of the monomer and liquid crystal, respectively; WM, WLC are molecular weights; l is the average length of polymeric chains; and nst is the refractive index of the composition prior to the start of the recording process, determined by the Lorentz-

Lorentz formula from the refractive indices of the monomer and the liquid crystal [8]:

� α<sup>M</sup> þ

rM WM

αP l 

þ αLC

� rM WM

> rLC WLC

coefficients of the contribution of photopolymerization and diffusion processes; and I

�βq�α0� <sup>N</sup><sup>m</sup>

�βq�α0� <sup>N</sup><sup>m</sup>

<sup>0</sup> <sup>þ</sup>N<sup>m</sup> ð Þ<sup>1</sup> ½ � �r�<sup>t</sup>

<sup>0</sup> �rþd�N<sup>m</sup>

<sup>1</sup> ½ � ð Þ�<sup>r</sup> �<sup>t</sup> , (6)

LCð Þ <sup>r</sup>; <sup>t</sup> grad Mmð Þ <sup>r</sup>; <sup>t</sup>

<sup>M</sup>ð Þ <sup>r</sup>; <sup>t</sup> , <sup>D</sup><sup>m</sup>

, (10)

, (11)

Mmð Þ <sup>r</sup>; <sup>t</sup> , (7)

LCð Þ r; t are the

<sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup> is

, (8)

Mn

<sup>m</sup>ð Þ <sup>r</sup>; <sup>t</sup>

Kb <sup>0</sup>,<sup>5</sup>

<sup>þ</sup> <sup>δ</sup>nlcdiv <sup>D</sup><sup>m</sup>

Mn

Ln

, (9)

Because of periodical character of the forming fields' intensities' spatial distributions, solution of KES can be found as a sum of H spatial harmonics [8]:

$$M^{\mathfrak{m}}(\mathbf{r},\tau) = \sum\_{j=0}^{H} M\_j^{\mathfrak{m}}(\mathbf{r},\tau) \cos(j \mathbf{K}^{\mathfrak{m}} \mathbf{r}), \quad n^{\mathfrak{m}}(\mathbf{r},\tau) = n\_{sl} + \sum\_{j=0}^{H} n\_j^{\mathfrak{m}}(\mathbf{r},\tau) \cos(j \mathbf{K}^{\mathfrak{m}} \mathbf{r}), \tag{13}$$

where Mm <sup>j</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>1</sup> 2π Ð π �π Mm <sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> <sup>d</sup> <sup>K</sup><sup>m</sup> ð Þ<sup>r</sup> , nm <sup>j</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>1</sup> 2π Ð π �π nm <sup>j</sup> ð Þ <sup>r</sup>; <sup>τ</sup> cos <sup>j</sup>K<sup>m</sup> ð Þ<sup>r</sup> <sup>d</sup> <sup>K</sup><sup>m</sup> ð Þ<sup>r</sup>

are the monomer concentration and refractive index harmonics amplitudes, respectively; nst is the PDLC (PSLC) refractive index at <sup>τ</sup> <sup>¼</sup> 0; <sup>τ</sup> <sup>¼</sup> <sup>t</sup>=T<sup>m</sup> <sup>M</sup> is the relative time; T<sup>m</sup> <sup>M</sup> <sup>¼</sup> <sup>1</sup> Dm Mn� <sup>K</sup><sup>m</sup> j j<sup>2</sup> is the components' diffusion characteristic time.

By substituting Eq. (13) to the KES (Eqs. (7 and 8)) and using the orthogonality of spatial harmonics, a system of coupled kinetic differential equations for the amplitudes of monomer concentration harmonics can be obtained for D<sup>m</sup> <sup>M</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>D</sup><sup>m</sup> LCð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>D</sup><sup>m</sup> Mn (stable diffusion coefficients) and Kdð Þ¼ r; t Kd<sup>0</sup> (stable absorption) [6]:

$$\begin{cases} \frac{\partial \mathcal{M}\_0^m(\mathbf{r}, \tau)}{\partial \tau} = \sum\_{l=0}^H a\_{0,l}^m(\mathbf{r}) M\_l^m(\mathbf{r}, \tau) \\\\ \frac{\partial \mathcal{M}\_1^m(\mathbf{r}, \tau)}{\partial \tau} = -M\_1^m(\mathbf{r}, \tau) + \sum\_{l=0}^H a\_{1,l}^m(\mathbf{r}) M\_l^m(\mathbf{r}, \tau) \\ \dots = \dots = \dots = \dots = \dots = \dots = \dots \\ \frac{\partial \mathcal{M}\_H^m(\mathbf{r}, \tau)}{\partial \tau} = -N^2 M\_H^m(\mathbf{r}, \tau) + \sum\_{l=0}^H a\_{N,l}^m(\mathbf{r}) M\_l^m(\mathbf{r}, \tau) \end{cases} \tag{14}$$

and also a system of differential equations for the amplitudes of refraction index harmonics:

$$\begin{cases} \frac{\partial n\_{0}^{m}(\mathbf{r},\tau)}{\partial \tau}M\_{n} = -\delta n\_{p} \sum\_{l=0}^{H} a\_{0,l}^{m}(\mathbf{r})M\_{l}^{m}(\mathbf{r},\tau) \\\\ \frac{\partial n\_{1}^{m}(\mathbf{r},\tau)}{\partial \tau}M\_{n} = -\delta n\_{p} \sum\_{l=0}^{H} a\_{1,l}^{m}(\mathbf{r})M\_{l}^{m}(\mathbf{r},\tau) + \delta n\_{l}M\_{1}^{m}(\mathbf{r},\tau) \\ \quad \dots = \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots \\ \frac{\partial n\_{H}^{m}(\mathbf{r},\tau)}{\partial \tau}M\_{n} = -\delta n\_{p} \sum\_{l=0}^{H} a\_{H,l}^{m}(\mathbf{r})M\_{l}^{m}(\mathbf{r},\tau) + \delta n\_{l}H^{2}M\_{H}^{m}(\mathbf{r},\tau) \end{cases} \tag{15}$$

In equation systems Eqs. (14) and (15), a coefficient matrix is introduced [6]:

$$a\_{j,l}^{m}(\mathbf{r}) = -\begin{cases} a\_1^{m} & e\_2^{m} & e\_3^{m} & 0 & 0 & 0 & \dots & 0 & 0 & 0 & 0\\ 2e\_2^{m} & e\_{11}^{m} & e\_2^{m} & e\_3^{m} & 0 & 0 & \dots & 0 & 0 & 0 & 0\\ 2e\_3^{m} & e\_2^{m} & e\_1^{m} & e\_2^{m} & e\_3^{m} & 0 & \dots & 0 & 0 & 0 & 0\\ 0 & e\_3^{m} & e\_2^{m} & e\_1^{m} & e\_2^{m} & e\_3^{m} & \dots & 0 & 0 & 0 & 0\\ 0 & 0 & e\_3^{m} & e\_2^{m} & e\_1^{m} & e\_2^{m} & \dots & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & e\_3^{m} & e\_2^{m} & e\_1^{m} & \dots & 0 & 0 & 0 & 0\\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & e\_1^{m} & e\_2^{m} & e\_3^{m} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & e\_3^{m} & e\_2^{m} & e\_1^{m} & e\_2^{m}\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & e\_3^{m} & e\_2^{m} & e\_1^{m}\\ \end{cases} \tag{16}$$

M<sup>m</sup>

where functional dependencies of coefficients λ<sup>m</sup>

� �

9

>>>>>>>>>>>>=

>>>>>>>>>>>>;

<sup>0</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>¼</sup> <sup>0</sup> <sup>0</sup>, n<sup>m</sup>

nm

monomer concentration—M<sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> and refractive index <sup>n</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> .

<sup>j</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> nm

and diffusion mechanism contributions to the process of structure's formation.

�

ð Þ� <sup>r</sup> <sup>λ</sup><sup>m</sup> <sup>l</sup> ð Þr

<sup>2</sup> : λ<sup>m</sup> H

<sup>2</sup> : <sup>λ</sup><sup>m</sup><sup>2</sup> H

<sup>2</sup> : <sup>λ</sup><sup>m</sup><sup>3</sup> H

<sup>2</sup> : <sup>λ</sup><sup>m</sup><sup>4</sup> H

<sup>2</sup> : <sup>λ</sup>mH H

> j,l <sup>¼</sup> <sup>A</sup><sup>m</sup> j,l ð Þ<sup>r</sup> , cm

nm

P H l¼0 am j,l ð Þ<sup>r</sup> <sup>P</sup> H q¼0 A<sup>m</sup> l, <sup>q</sup>ð Þr

tic equation cm

Coefficients A<sup>m</sup>

λm <sup>0</sup> λ<sup>m</sup>

8

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

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

λ<sup>m</sup><sup>2</sup> <sup>0</sup> <sup>λ</sup><sup>m</sup><sup>2</sup>

λ<sup>m</sup><sup>3</sup> <sup>0</sup> <sup>λ</sup><sup>m</sup><sup>3</sup>

λ<sup>m</sup><sup>4</sup> <sup>0</sup> <sup>λ</sup><sup>m</sup><sup>4</sup>

λmH <sup>0</sup> <sup>λ</sup>mH

where λ<sup>m</sup>

where n<sup>m</sup>

j = 0,…,H.

j,l

j,l

111 : 1

<sup>1</sup> λ<sup>m</sup>

<sup>1</sup> <sup>λ</sup><sup>m</sup><sup>2</sup>

<sup>1</sup> <sup>λ</sup><sup>m</sup><sup>3</sup>

<sup>1</sup> <sup>λ</sup><sup>m</sup><sup>4</sup>

: : : ::

<sup>l</sup> <sup>¼</sup> <sup>λ</sup><sup>m</sup>

initial conditions, we get.

p jð Þ¼ r; τ δnp

<sup>1</sup> <sup>λ</sup>mH

<sup>l</sup> ð Þ<sup>r</sup> , Am

� � � <sup>j</sup> ð Þ¼ r; τ Mn

A<sup>m</sup> j,0 A<sup>m</sup> j,1 A<sup>m</sup> j,2 A<sup>m</sup> j,3 A<sup>m</sup> j,4 … A<sup>m</sup> j,H

8

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

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

j,l <sup>¼</sup> cm j,l ð Þr .

X H

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

ð Þ<sup>r</sup> exp <sup>λ</sup><sup>m</sup>

<sup>l</sup> ð Þ� <sup>r</sup> <sup>τ</sup> � �, (20)

http://dx.doi.org/10.5772/intechopen.74838

<sup>l</sup> ð Þr is real, different, and negative.

<sup>l</sup> ð Þr are defined as the roots of the characteris-

l¼0 Am j,l

� <sup>¼</sup> 0. Analysis shows that <sup>λ</sup><sup>m</sup>

9

>>>>>>>>>>>>=

¼ Mn

>>>>>>>>>>>>;

ð Þr are defined as solutions of a system of linear algebraic equations:

8

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

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

Then, by substituting (20) to (15) and by integrating the resulting differential equations with

The general solution for the amplitude of j-harmonic of the refractive index can be found:

<sup>1</sup>�exp <sup>λ</sup><sup>m</sup> <sup>q</sup> ½ � ð Þ� <sup>r</sup> <sup>τ</sup> λm

<sup>1</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>¼</sup> <sup>0</sup> <sup>0</sup>, …n<sup>m</sup>

p jð Þþ <sup>r</sup>; <sup>τ</sup> <sup>n</sup><sup>m</sup>

<sup>q</sup> ð Þ<sup>r</sup> , nm

Thus, Eqs. (13), (20), and (23) are the general solutions of nonlinear photopolymerization diffusion holographic recording of waveguide channels system in PDLCs (PSLCs) in the case of stable diffusion coefficients and stable absorption. They define kinetics of spatial profiles of

In case of high nonlinearity, it is possible to form the specific spatial profile of refractive index. The nonlinearity of recording is achieved by changing the ratio of the photopolymerization

δj,<sup>0</sup> cm j, 0 X H

i0¼0 c m j,i<sup>0</sup> c m i0,0

X H

i1¼0 c m j,i<sup>1</sup> X H

> X H

ið Þ <sup>N</sup>�<sup>2</sup> ¼0

i0¼0 c m i1,i<sup>0</sup> c m i0, 0 :………………………………………

> …X H

lc jð Þ¼� r; τ δnlc � j

i2¼0 c m i3,i<sup>2</sup> X H

i1¼0 c m i2,i<sup>1</sup> X H

<sup>H</sup>ð Þ¼ r; τ ¼ 0 0 (22)

lc jð Þ r; τ , (23)

<sup>2</sup> P H q¼0 Am j, <sup>q</sup>ð Þr <sup>1</sup>�exp <sup>λ</sup><sup>m</sup> <sup>q</sup> ½ � ð Þ� <sup>r</sup> <sup>τ</sup> λm <sup>q</sup> ð Þ<sup>r</sup> ,

i0¼0 c m i1,i<sup>0</sup> c m i0,0 9

341

>>>>>>>>>>>>>>>>>=

,

>>>>>>>>>>>>>>>>>;

(21)

cm j,iN�<sup>1</sup>

where e<sup>m</sup> <sup>1</sup> <sup>¼</sup> ffiffi 2 p bm s <sup>1</sup> <sup>þ</sup> Lm s � �; e<sup>m</sup> <sup>11</sup> <sup>¼</sup> ffiffi 2 p bm s <sup>1</sup> <sup>þ</sup> <sup>3</sup>Lm <sup>s</sup> <sup>=</sup><sup>2</sup> � �; em <sup>2</sup> ¼ ffiffi 2 <sup>p</sup> mm s 4b<sup>m</sup> s ; e<sup>m</sup> <sup>3</sup> <sup>¼</sup> ffiffi 2 p bm s Lm <sup>s</sup> =2; m<sup>m</sup> <sup>s</sup> <sup>¼</sup> mmð Þ<sup>r</sup> ; Lm <sup>s</sup> <sup>¼</sup> <sup>L</sup><sup>m</sup>ð Þ¼� <sup>r</sup> <sup>m</sup><sup>m</sup> ½ � ð Þ<sup>r</sup> 2 =16; b<sup>m</sup> <sup>s</sup> <sup>¼</sup> bmð Þ¼ <sup>r</sup> T<sup>m</sup> <sup>p</sup> ð Þr Tm <sup>M</sup>ð Þ<sup>r</sup> are the parameters that characterize the ratio of polymerization and diffusion rates; and T<sup>m</sup> <sup>p</sup> ð Þr is the characteristic polymerization time:

$$T\_p^m(\mathbf{r}) = \frac{1}{K\_\mathcal{g}} \cdot \left[\frac{2K\_b}{\alpha\_0 \beta K\_{d0} \tau\_0 I^m(\mathbf{r})}\right]^{0.5},\tag{17}$$

T<sup>m</sup> <sup>M</sup>ð Þr is the characteristic diffusion time:

$$T\_M^{\rm m}(\mathbf{r}) = \frac{1}{D\_{Mn}^{\rm m} \cdot \left[\mathbf{K}^{\rm m} \cdot \mathbf{r} + \varphi\_0(\mathbf{r}) - \varphi\_1(\mathbf{r})\right]^2} \,. \tag{18}$$

Coefficients am j,l ð Þr describe the contributions of photopolymerization and diffusion recording mechanisms. However, for analysis of equation systems Eqs. (14) and (15), it is convenient to introduce the coupling coefficients c<sup>m</sup> j,l ð Þ¼ <sup>r</sup> <sup>a</sup><sup>m</sup> j,l ð Þ� r j 2 δj,l (δj,l is the Kronecker symbol), which characterizes the coupling between j and l harmonics. The difference between coefficients cm j,l ð Þr and am j,l ð Þr characterizes the contribution of monomer diffusion to the recording process and it is proportional to the second degree of the harmonic's number. The increase of this contribution according to the harmonic's number is due to grating period decrease and, respectively, the diffusion characteristic time decrease for this harmonic.

For solution of the coupled differential equations system (14), the initial conditions should be introduced:

$$M\_0^m(\mathbf{r}, \tau = 0) = M\_{n\prime} M\_1^m(\mathbf{r}, \tau = 0) = 0, \dots, M\_H^m(\mathbf{r}, \tau = 0) = 0. \tag{19}$$

The solution can be found using the operator method [6]. The general solution for the spatial amplitude profiles of monomer concentration harmonics will be:

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer… http://dx.doi.org/10.5772/intechopen.74838 341

$$M\_j^{m}(\mathbf{r}, \tau) = M\_n \sum\_{l=0}^{H} A\_{j,l}^{m}(\mathbf{r}) \exp\left[A\_l^{m}(\mathbf{r}) \cdot \tau\right],\tag{20}$$

where functional dependencies of coefficients λ<sup>m</sup> <sup>l</sup> ð Þr are defined as the roots of the characteristic equation cm j,l ð Þ� <sup>r</sup> <sup>λ</sup><sup>m</sup> <sup>l</sup> ð Þr � � � � � � <sup>¼</sup> 0. Analysis shows that <sup>λ</sup><sup>m</sup> <sup>l</sup> ð Þr is real, different, and negative. Coefficients A<sup>m</sup> j,l ð Þr are defined as solutions of a system of linear algebraic equations:

$$\begin{Bmatrix} \begin{matrix} 1 & 1 & 1 & \cdot & 1\\ \lambda\_{0}^{m} & \lambda\_{1}^{m} & \lambda\_{2}^{m} & \cdot & \lambda\_{3}^{m}\\ \lambda\_{0}^{m^{2}} & \lambda\_{1}^{m^{2}} & \lambda\_{2}^{m^{3}} & \cdot & \lambda\_{4}^{m^{2}}\\ \lambda\_{0}^{m^{3}} & \lambda\_{1}^{m^{3}} & \lambda\_{2}^{m^{3}} & \cdot & \lambda\_{4}^{m^{4}}\\ \lambda\_{0}^{m^{4}} & \lambda\_{1}^{m^{4}} & \lambda\_{2}^{m^{4}} & \cdot & \lambda\_{4}^{m^{4}}\\ \lambda\_{0}^{m^{4}} & \lambda\_{1}^{m^{4}} & \lambda\_{2}^{m^{4}} & \cdot & \lambda\_{4}^{m^{4}}\\ \end{matrix} \times \begin{Bmatrix} A\_{j,0}^{m}\\ A\_{j,1}^{m}\\ A\_{j,2}^{m}\\ A\_{j,3}^{m}\\ \end{Bmatrix} = \begin{Bmatrix} \delta\_{j,0}\\ \delta\_{j,0}^{m}\\ \delta\_{j,0}^{m}\\ \sum\_{i=0}^{H}\delta\_{i,i}\mathcal{L}\_{i,0}^{m}\\ \sum\_{j=0}^{H}\mathcal{L}\_{j,i}^{m}\mathcal{L}\_{i,0}^{m}\\ \end{Bmatrix},\tag{12}$$

where λ<sup>m</sup> <sup>l</sup> <sup>¼</sup> <sup>λ</sup><sup>m</sup> <sup>l</sup> ð Þ<sup>r</sup> , Am j,l <sup>¼</sup> <sup>A</sup><sup>m</sup> j,l ð Þ<sup>r</sup> , cm j,l <sup>¼</sup> cm j,l ð Þr .

am j,l ð Þ¼� r

340 Emerging Waveguide Technology

where e<sup>m</sup>

Lm

T<sup>m</sup>

Coefficients am

and am j,l

introduced:

j,l

introduce the coupling coefficients c<sup>m</sup>

Mm

<sup>1</sup> <sup>¼</sup> ffiffi 2 p bm s

<sup>s</sup> <sup>¼</sup> <sup>L</sup><sup>m</sup>ð Þ¼� <sup>r</sup> <sup>m</sup><sup>m</sup> ½ � ð Þ<sup>r</sup>

<sup>1</sup> <sup>þ</sup> Lm s � �; e<sup>m</sup>

polymerization and diffusion rates; and T<sup>m</sup>

<sup>M</sup>ð Þr is the characteristic diffusion time:

2 =16; b<sup>m</sup>

em <sup>1</sup> e<sup>m</sup>

8

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

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

2e<sup>m</sup> <sup>2</sup> e<sup>m</sup> <sup>11</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup>

2em <sup>3</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup> <sup>1</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup>

0 e<sup>m</sup>

0 0 e<sup>m</sup>

<sup>11</sup> <sup>¼</sup> ffiffi 2 p bm s

> T<sup>m</sup> <sup>p</sup> ð Þ¼ r

Tm <sup>M</sup>ð Þ¼ r

the diffusion characteristic time decrease for this harmonic.

<sup>0</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>¼</sup> <sup>0</sup> Mn, Mm

amplitude profiles of monomer concentration harmonics will be:

<sup>s</sup> <sup>¼</sup> bmð Þ¼ <sup>r</sup>

<sup>2</sup> e<sup>m</sup>

<sup>3</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup> <sup>1</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup>

0 00 e<sup>m</sup>

<sup>3</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup> <sup>1</sup> e<sup>m</sup>

> <sup>3</sup> e<sup>m</sup> <sup>2</sup> e<sup>m</sup>

0 0 0000 : e<sup>m</sup>

0 0 0000 : e<sup>m</sup>

0 0 0000 : e<sup>m</sup>

<sup>1</sup> <sup>þ</sup> <sup>3</sup>Lm <sup>s</sup> <sup>=</sup><sup>2</sup> � �; em

> 1 Kg

D<sup>m</sup>

j,l

ð Þ¼ <sup>r</sup> <sup>a</sup><sup>m</sup> j,l ð Þ� r j 2

T<sup>m</sup> <sup>p</sup> ð Þr Tm

0 0 0000 : 0 e<sup>m</sup>

� <sup>2</sup>Kb α0βKd0τ0I

mechanisms. However, for analysis of equation systems Eqs. (14) and (15), it is convenient to

is proportional to the second degree of the harmonic's number. The increase of this contribution according to the harmonic's number is due to grating period decrease and, respectively,

For solution of the coupled differential equations system (14), the initial conditions should be

The solution can be found using the operator method [6]. The general solution for the spatial

characterizes the coupling between j and l harmonics. The difference between coefficients cm

: : : : : : :: : : :

<sup>2</sup> ¼ ffiffi 2 <sup>p</sup> mm s 4b<sup>m</sup> s ; e<sup>m</sup> <sup>3</sup> <sup>¼</sup> ffiffi 2 p bm s Lm

<sup>m</sup>ð Þ<sup>r</sup>

� �<sup>0</sup>,<sup>5</sup>

1

ð Þr characterizes the contribution of monomer diffusion to the recording process and it

<sup>1</sup> ð Þ¼ <sup>r</sup>; <sup>τ</sup> <sup>¼</sup> <sup>0</sup> <sup>0</sup>, …, Mm

ð Þr describe the contributions of photopolymerization and diffusion recording

<sup>3</sup> 000 : 0000

<sup>3</sup> 0 0: : 0000

<sup>3</sup> 0 : 0000

<sup>3</sup> : 0000

9

>>>>>>>>>>>>>>>>>>>>>=

, (16)

<sup>s</sup> <sup>¼</sup> mmð Þ<sup>r</sup> ;

j,l ð Þr

>>>>>>>>>>>>>>>>>>>>>;

<sup>s</sup> =2; m<sup>m</sup>

, (17)

δj,l (δj,l is the Kronecker symbol), which

<sup>H</sup>ð Þ¼ r; τ ¼ 0 0: (19)

<sup>2</sup> : 0000

<sup>1</sup> : 0000

<sup>1</sup> e<sup>m</sup> <sup>2</sup> em <sup>3</sup> 0

<sup>2</sup> e<sup>m</sup> <sup>1</sup> em <sup>2</sup> e<sup>m</sup> 3

<sup>3</sup> e<sup>m</sup> <sup>2</sup> em <sup>1</sup> e<sup>m</sup> 2

> <sup>3</sup> em <sup>2</sup> e<sup>m</sup> 1

<sup>M</sup>ð Þ<sup>r</sup> are the parameters that characterize the ratio of

<sup>p</sup> ð Þr is the characteristic polymerization time:

Mn � <sup>K</sup><sup>m</sup> � <sup>r</sup> <sup>þ</sup> <sup>φ</sup>0ð Þ� <sup>r</sup> <sup>φ</sup>1ð Þ<sup>r</sup> � �<sup>2</sup> : (18)

Then, by substituting (20) to (15) and by integrating the resulting differential equations with initial conditions, we get.

$$n\_0^m(\mathbf{r}, \tau=0) = 0, n\_1^m(\mathbf{r}, \tau=0) = 0, \dots \\ n\_H^m(\mathbf{r}, \tau=0) = 0 \tag{22}$$

The general solution for the amplitude of j-harmonic of the refractive index can be found:

$$n\_j^m(\mathbf{r}, \tau) = n\_{pj}^m(\mathbf{r}, \tau) + n\_{lcj}^m(\mathbf{r}, \tau), \tag{23}$$

$$\begin{cases} \text{where } n\_{pj}^{\mathfrak{m}}(\mathbf{r}, \boldsymbol{\tau}) = \delta n\_p \sum\_{l=0}^H a\_{j,l}^{\mathfrak{m}}(\mathbf{r}) \sum\_{q=0}^H A\_{l,q}^{\mathfrak{m}}(\mathbf{r}) \frac{1 - \exp\left[\lambda\_q^{\mathfrak{m}}(\mathbf{r}) \cdot \boldsymbol{\tau}\right]}{\lambda\_q^{\mathfrak{m}}(\mathbf{r})}, \ n\_{\mathbf{k}j}^{\mathfrak{m}}(\mathbf{r}, \boldsymbol{\tau}) = -\delta n\_{\mathbf{k}} \cdot j^2 \sum\_{q=0}^H A\_{j,q}^{\mathfrak{m}}(\mathbf{r}) \frac{1 - \exp\left[\lambda\_q^{\mathfrak{m}}(\mathbf{r}) \cdot \boldsymbol{\tau}\right]}{\lambda\_q^{\mathfrak{m}}(\mathbf{r})},\\ j = 0, \dots, H. \end{cases}$$

Thus, Eqs. (13), (20), and (23) are the general solutions of nonlinear photopolymerization diffusion holographic recording of waveguide channels system in PDLCs (PSLCs) in the case of stable diffusion coefficients and stable absorption. They define kinetics of spatial profiles of monomer concentration—M<sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> and refractive index <sup>n</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> .

In case of high nonlinearity, it is possible to form the specific spatial profile of refractive index. The nonlinearity of recording is achieved by changing the ratio of the photopolymerization and diffusion mechanism contributions to the process of structure's formation.

### 3. Numerical simulations

To investigate the formation processes, numerical simulations of first four refractive index harmonics kinetics for transmission and reflection geometries were made with the following parameters: <sup>λ</sup> <sup>¼</sup> 633 nm and <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>30</sup>� for transmission geometry (Figure 1a) and <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>60</sup>� for reflection geometry (Figure 1b); E0ð Þ¼ r E1ð Þ¼ r 1; φ0ð Þr , ¼ φ1ð Þ¼ r 0; e0, e<sup>1</sup> are oriented with respect to extraordinary waves in material; d ¼ 10 μm; δnp=δni ¼ 0:5; and for two values of parameter <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>e</sup> <sup>¼</sup> <sup>0</sup>:2 and <sup>b</sup> <sup>¼</sup> be <sup>¼</sup> 5 in the absence of absorption. Values b ¼ 0:2 and 5 are chosen as an example to investigate two common cases: b << 1 and b >> 1. Simulations were made by Eq. (23); results are shown in Figure 2.

As shown in Figure 4, absorption of the formation field causes dependence of the first harmonic on z-coordinate (see Figure 1) during the recording process. Thus, the spatial profile of

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

http://dx.doi.org/10.5772/intechopen.74838

343

In Figure 4, some characteristic cases of localization of maximum of refractive index change in the thickness can be seen. In particular, in the case of transmission geometry (Figure 4a) at τ ¼ 2:5, maximum is localized between 1.5 and 3.5 μm of the material thickness. For reflection geometry (Figure 4b), one maximum is localized near 5 μm at τ ¼ 1:25, and there are two

refractive index changes its character in time and space.

Figure 2. Refractive index harmonics kinetics for b ¼ 0:2 (a) and b ¼ 5 (b).

As can be seen from Figure 2, in the case of predomination of polymerization (be <sup>¼</sup> <sup>0</sup>:2, Figure 2a), the structure forms quickly, but amplitudes of higher harmonics are high, so the spatial profile of refractive index changes has an inharmonic character. In another case of diffusion predominance (b<sup>e</sup> <sup>¼</sup> 5, Figure 2b), spatial profile is quasi-sinusoidal, but it is slower. These effects can be explained by the following. When polymerization is rapid, in the area of the maximum of intensity distribution (Eq. (3)), molecules of liquid crystal do not have time to diffuse, so the concentrations' gradient is not high enough; monomer molecules do not diffuse from minimums of intensity distribution. Thus, profile of refractive index distribution becomes inharmonic. So, it can be concluded that ratio of polymerization and diffusion rates bmð Þ <sup>r</sup>; <sup>τ</sup> defines the distribution nmð Þ <sup>r</sup>; <sup>τ</sup> .

Corresponding distributions n<sup>e</sup> ð Þ r; τ ¼ 50 for two examined cases in the absence of absorption are shown in Figure 3.

In Figure 3, a new spatial coordinate is introduced: ξ ¼ y for transmission geometry (see Figure 1a) and ξ ¼ z for reflection geometry (see Figure 1b).

According to Eqs. (17) and (18), parameter bmð Þ <sup>r</sup>; <sup>τ</sup> depends on formation field's amplitude and phase distributions as well as material properties. So, by controlling these parameters one can create any distribution of refractive index nmð Þ <sup>r</sup>; <sup>τ</sup> . As is shown in Figure 3, distributions can be quasi-rectangular (Figure 3a) and quasi-sinusoidal (Figure 3b), so they can be mentioned as waveguides systems.

It should be also noted that the amplitude of refractive index change when b ¼ 0:2 (Figure 3a), due to high amplitudes of higher harmonics (see Figure 2a), is lower than in the case of b ¼ 5.

To investigate the impact of absorption on the spatial profile of structure along the thickness of the sample, numerical simulations were made with the following parameters: λ ¼ 633 nm; <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>30</sup>� for transmission geometry (Figure 1a) and <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>60</sup>� for reflection geometry (Figure 1b); E0ð Þ¼ r E1ð Þ¼ r 1; φ0ð Þr , ¼ φ1ð Þ¼ r 0; e0, e<sup>1</sup> are oriented with respect to extraordinary waves in material; <sup>d</sup> <sup>¼</sup> <sup>10</sup> <sup>μ</sup>m; <sup>δ</sup>np=δni <sup>¼</sup> <sup>0</sup>:5; <sup>b</sup> <sup>¼</sup> be <sup>¼</sup> 5; <sup>α</sup><sup>d</sup> <sup>¼</sup> <sup>2</sup> Np. Simulations were made by Eq. (23) for the amplitude of the first harmonic of refractive index and for two formation geometries. Results are shown in Figure 4.

As shown in Figure 4, absorption of the formation field causes dependence of the first harmonic on z-coordinate (see Figure 1) during the recording process. Thus, the spatial profile of refractive index changes its character in time and space.

3. Numerical simulations

342 Emerging Waveguide Technology

defines the distribution nmð Þ <sup>r</sup>; <sup>τ</sup> . Corresponding distributions n<sup>e</sup>

are shown in Figure 3.

waveguides systems.

<sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>30</sup>�

<sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>60</sup>�

parameters: <sup>λ</sup> <sup>¼</sup> 633 nm and <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>30</sup>�

To investigate the formation processes, numerical simulations of first four refractive index harmonics kinetics for transmission and reflection geometries were made with the following

e0, e<sup>1</sup> are oriented with respect to extraordinary waves in material; d ¼ 10 μm; δnp=δni ¼ 0:5; and for two values of parameter <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>e</sup> <sup>¼</sup> <sup>0</sup>:2 and <sup>b</sup> <sup>¼</sup> be <sup>¼</sup> 5 in the absence of absorption. Values b ¼ 0:2 and 5 are chosen as an example to investigate two common cases: b << 1 and

As can be seen from Figure 2, in the case of predomination of polymerization (be <sup>¼</sup> <sup>0</sup>:2, Figure 2a), the structure forms quickly, but amplitudes of higher harmonics are high, so the spatial profile of refractive index changes has an inharmonic character. In another case of diffusion predominance (b<sup>e</sup> <sup>¼</sup> 5, Figure 2b), spatial profile is quasi-sinusoidal, but it is slower. These effects can be explained by the following. When polymerization is rapid, in the area of the maximum of intensity distribution (Eq. (3)), molecules of liquid crystal do not have time to diffuse, so the concentrations' gradient is not high enough; monomer molecules do not diffuse from minimums of intensity distribution. Thus, profile of refractive index distribution becomes inharmonic. So, it can be concluded that ratio of polymerization and diffusion rates bmð Þ <sup>r</sup>; <sup>τ</sup>

In Figure 3, a new spatial coordinate is introduced: ξ ¼ y for transmission geometry (see

According to Eqs. (17) and (18), parameter bmð Þ <sup>r</sup>; <sup>τ</sup> depends on formation field's amplitude and phase distributions as well as material properties. So, by controlling these parameters one can create any distribution of refractive index nmð Þ <sup>r</sup>; <sup>τ</sup> . As is shown in Figure 3, distributions can be quasi-rectangular (Figure 3a) and quasi-sinusoidal (Figure 3b), so they can be mentioned as

It should be also noted that the amplitude of refractive index change when b ¼ 0:2 (Figure 3a), due to high amplitudes of higher harmonics (see Figure 2a), is lower than in the case of b ¼ 5. To investigate the impact of absorption on the spatial profile of structure along the thickness of the sample, numerical simulations were made with the following parameters: λ ¼ 633 nm;

for transmission geometry (Figure 1a) and <sup>θ</sup><sup>0</sup> ¼ �θ<sup>1</sup> <sup>¼</sup> <sup>60</sup>�

etry (Figure 1b); E0ð Þ¼ r E1ð Þ¼ r 1; φ0ð Þr , ¼ φ1ð Þ¼ r 0; e0, e<sup>1</sup> are oriented with respect to extraordinary waves in material; <sup>d</sup> <sup>¼</sup> <sup>10</sup> <sup>μ</sup>m; <sup>δ</sup>np=δni <sup>¼</sup> <sup>0</sup>:5; <sup>b</sup> <sup>¼</sup> be <sup>¼</sup> 5; <sup>α</sup><sup>d</sup> <sup>¼</sup> <sup>2</sup> Np. Simulations were made by Eq. (23) for the amplitude of the first harmonic of refractive index and for two

b >> 1. Simulations were made by Eq. (23); results are shown in Figure 2.

Figure 1a) and ξ ¼ z for reflection geometry (see Figure 1b).

formation geometries. Results are shown in Figure 4.

for reflection geometry (Figure 1b); E0ð Þ¼ r E1ð Þ¼ r 1; φ0ð Þr , ¼ φ1ð Þ¼ r 0;

ð Þ r; τ ¼ 50 for two examined cases in the absence of absorption

for reflection geom-

for transmission geometry (Figure 1a) and

In Figure 4, some characteristic cases of localization of maximum of refractive index change in the thickness can be seen. In particular, in the case of transmission geometry (Figure 4a) at τ ¼ 2:5, maximum is localized between 1.5 and 3.5 μm of the material thickness. For reflection geometry (Figure 4b), one maximum is localized near 5 μm at τ ¼ 1:25, and there are two

Figure 2. Refractive index harmonics kinetics for b ¼ 0:2 (a) and b ¼ 5 (b).

other maximums at <sup>τ</sup> <sup>&</sup>gt; <sup>2</sup>:5. These effects take place because of dependence <sup>b</sup><sup>m</sup>ð Þ<sup>r</sup> . Absorption causes the dependence on z-coordinate and spatial distributions of amplitude, and the phase of recording field (Eqs. (1) and (2)) causes the dependence on x and y coordinates. In more general cases, there is also a temporal dependence <sup>b</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> caused by photo-induced effects (Eq. (6)).

To illustrate the localization of waveguide channels into the material, in Figure 5, there are some more two-dimensional spatial distributions of refractive index, obtained by Eq. (23) for

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

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345

It follows from the abovementioned that by controlling the spatial distribution of light field and taking the absorption effects into account, the waveguide systems localized into the

transmission and reflection geometries—<sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>e</sup> <sup>¼</sup> 5 and for different values of <sup>τ</sup>.

Figure 4. Impact of light absorption: transmission geometry (a) and reflection geometry (b).

Figure 3. Refractive index distributions for b ¼ 0:2 (a) and b ¼ 5 (b).

345

other maximums at <sup>τ</sup> <sup>&</sup>gt; <sup>2</sup>:5. These effects take place because of dependence <sup>b</sup><sup>m</sup>ð Þ<sup>r</sup> . Absorption causes the dependence on z-coordinate and spatial distributions of amplitude, and the phase of recording field (Eqs. (1) and (2)) causes the dependence on x and y coordinates. In more general cases, there is also a temporal dependence <sup>b</sup><sup>m</sup>ð Þ <sup>r</sup>; <sup>τ</sup> caused by photo-induced effects

(Eq. (6)).

344 Emerging Waveguide Technology

Figure 3. Refractive index distributions for b ¼ 0:2 (a) and b ¼ 5 (b).

Figure 4. Impact of light absorption: transmission geometry (a) and reflection geometry (b).

To illustrate the localization of waveguide channels into the material, in Figure 5, there are some more two-dimensional spatial distributions of refractive index, obtained by Eq. (23) for transmission and reflection geometries—<sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>e</sup> <sup>¼</sup> 5 and for different values of <sup>τ</sup>.

It follows from the abovementioned that by controlling the spatial distribution of light field and taking the absorption effects into account, the waveguide systems localized into the

should be stopped before τ ¼ 0:5, for localization into the center of the sample— 0:5 < τ < 2:5—for transmission and reflection geometries, and for τ > 2:5, waveguides will be localized near the top substrate for transmission geometry and not localized for reflection

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

http://dx.doi.org/10.5772/intechopen.74838

347

In our recent works [9–18], we have seen that the impact of the external electric field on the holographic structure formed in PDLC (PSLC) leads to the refractive index change due to electro-optical orientation mechanisms, that is typical to liquid crystals, and the recorded structure can be "erased". So, it can be supposed that the impact of external electric field on individual waveguides will lead to its "erasing", and it is possible to "switch off" some

Thus, in this chapter, the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. The most general cases are described by the developed model, and numerical simulations were made for plane recording waves and stable absorption cases. Special attention is paid to localization of waveguides in the media caused by light field attenuation during the formation process.

It is shown that parameters of the waveguide system formed by holography's methods are determined by recording geometry and material's properties. Also, by control of these parameters, waveguides can be localized into the sample that makes them independent from substrates. Also, introduced compositions contain liquid crystals that make it possible to create

Obtained results can be used for new photonics devices based on photopolymer liquid crys-

The work is performed as a project that is part of Government Task of Russian Ministry of

Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

geometry. The given values of τ are valid for the formation parameters given earlier.

waveguides or change the system's period, and so on.

elements, controllable by external electric field.

talline composition development.

Education (project No. 3.1110.2017/4.6).

Artem Semkin and Sergey Sharangovich\*

\*Address all correspondence to: shr@tusur.ru

Acknowledgements

Author details

4. Conclusion

Figure 5. Two-dimensional refractive index distributions: transmission geometry (a–c) and reflection geometry (d–f).

material can be holographically formed. Localization is important because of the necessity to create a predetermined refractive index profile independent from properties of possible substrates (electrodes, glass, polymer layers, etc.). In particular, to create the waveguide system localized near the bottom substrate, formation geometry should be transmissive and formation should be stopped before τ ¼ 0:5, for localization into the center of the sample— 0:5 < τ < 2:5—for transmission and reflection geometries, and for τ > 2:5, waveguides will be localized near the top substrate for transmission geometry and not localized for reflection geometry. The given values of τ are valid for the formation parameters given earlier.

In our recent works [9–18], we have seen that the impact of the external electric field on the holographic structure formed in PDLC (PSLC) leads to the refractive index change due to electro-optical orientation mechanisms, that is typical to liquid crystals, and the recorded structure can be "erased". So, it can be supposed that the impact of external electric field on individual waveguides will lead to its "erasing", and it is possible to "switch off" some waveguides or change the system's period, and so on.

## 4. Conclusion

Thus, in this chapter, the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. The most general cases are described by the developed model, and numerical simulations were made for plane recording waves and stable absorption cases. Special attention is paid to localization of waveguides in the media caused by light field attenuation during the formation process.

It is shown that parameters of the waveguide system formed by holography's methods are determined by recording geometry and material's properties. Also, by control of these parameters, waveguides can be localized into the sample that makes them independent from substrates. Also, introduced compositions contain liquid crystals that make it possible to create elements, controllable by external electric field.

Obtained results can be used for new photonics devices based on photopolymer liquid crystalline composition development.

### Acknowledgements

The work is performed as a project that is part of Government Task of Russian Ministry of Education (project No. 3.1110.2017/4.6).

### Author details

material can be holographically formed. Localization is important because of the necessity to create a predetermined refractive index profile independent from properties of possible substrates (electrodes, glass, polymer layers, etc.). In particular, to create the waveguide system localized near the bottom substrate, formation geometry should be transmissive and formation

Figure 5. Two-dimensional refractive index distributions: transmission geometry (a–c) and reflection geometry (d–f).

346 Emerging Waveguide Technology

Artem Semkin and Sergey Sharangovich\*

\*Address all correspondence to: shr@tusur.ru

Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

### References

[1] Keil N, Zawadski C, Zhang Z, Wang J, Mettbach N, Grote N, Schell M. Polymer PLC as an Optical Integration Bench. Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference. 2011;2011:1-3

[12] Semkin A, Sharangovich S. Diffraction characteristics of the PDLC photonic structures under the influence of alternating electric fields. Bulletin of Russian Academy of Sciences:

A Theoretical Model of the Holographic Formation of Controllable Waveguide Channels System in Photopolymer…

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349

[13] Semkin A, Sharangovich S. The analytical model of light beams diffraction on the holographic photonic PDLC structure under the influence of an alternating electric field.

[14] Semkin A, Sharangovich S. The PDLC photonic structures diffraction characteristics managing by the spatially non-uniform electric field. International Conference on Advanced Optoelectronics and Lasers (CAOL), IEEE. 2013:48-49. DOI: 10.1109/CAOL.2013.6657522

[15] Semkin A, Sharangovich S. Analytical model of light beam diffraction on holographic polarization spatially inhomogeneous photonic PDLC structures. Physics Procedia. 2015;

[16] Semkin A, Sharangovich S. Light beam diffraction on inhomogeneous holographic photonic PDLC structures under the influence of spatially non-uniform electric field. Journal of Physics: Conference Series. 2016;735:012030. DOI: 10.1088/1742-6596/735/1/012030 [17] Semkin A, Sharangovich S. Highly effective light beam diffraction on holographic PDLC photonic structure, controllable by the spatially inhomogeneous electric field. Physics

[18] Semkin A, Sharangovich S. Theoretical model of controllable waveguide channels system holographic formation in photopolymer-liquid crystalline composition. Physics Procedia.

Physics. 2013;77:1416-1419. DOI: 10.3103/S1062873813120125

Procedia. 2017;86:160-165. DOI: 10.1016/j.phpro.2017.01.011

2017;86:181-186. DOI: 10.1016/j.phpro.2017.01.020

Pacific Science Review A. 2013;15:118-124

73:41-48. DOI: 10.1016/j.phpro.2015.09.119


[12] Semkin A, Sharangovich S. Diffraction characteristics of the PDLC photonic structures under the influence of alternating electric fields. Bulletin of Russian Academy of Sciences: Physics. 2013;77:1416-1419. DOI: 10.3103/S1062873813120125

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ics2031005

0199-6

DOI: 10.1364/JOSAB.32.000912

[1] Keil N, Zawadski C, Zhang Z, Wang J, Mettbach N, Grote N, Schell M. Polymer PLC as an Optical Integration Bench. Optical Fiber Communication Conference and Exposition

[2] Li H, Qi Y, Mallah R, Sheridan J. Modeling the nonlinear photoabsorptive behavior during self-written waveguide formation in a photopolymer. JOSA B. 2015;32:912-922.

[3] Zhang Z, Felipe D, Katopodis V, Groumas P, Kouloumentas Ch, et al. Hybrid photonic integration on a polymer platform. Photonics. 2015;2:1005-1026. DOI: 10.3390/photon-

[4] Zhang Z, Mettbach N, Zawadski C, Wang J, Schmidt D, Brinker W, Grote N, Schell M, Keil N. Polymer-based photonic toolbox: Passive components, hybrid integration and polarisation control. IET Optoelectronics. 2010;5:226-232. DOI: 10.1049/iet-opt.2010.0054

[5] Dovolnov E, Ustyuzhanin S, Sharangovich S. Formation of holographic transmission and reflecting gratings in photopolymers under light-induced absorption. Russian Physics

[6] Dovolnov E, Sharangovich S. Nonlinear model of record and readout of holographic transmission diffraction gratings in absorbent photopolymers. Part I. Theoretical analysis.

[7] Dovolnov E, Sharangovich S. Nonlinear model of record and readout of holographic transmission diffraction gratings in absorbent photopolymers. Part II. Numerical modelling and experiment. Russian Physics Journal. 2005;48:766-774. DOI: 10.1007/s11182-005-

[8] Dovolnov E, Sharangovich S. Analysis of dynamics of holographic grating formation withinharmonic spatial distribution in photopolymer + liquid crystalcompounds. Pro-

[9] Ustyuzhanin S, Nozdrevatykh B, Sharangovich S. Anisotropic light beam diffraction on electrical controlled holographic gratings in photopolymer-dispersed liquid crystals. International Conference on Advanced Optoelectronics and Lasers (CAOL), IEEE. 2008:

[10] Ustyuzhanin S, Nozdrevatykh B, Sharangovich S. Transfer functions of nonuniform transmission photonic structures in polymer-dispersed liquid-crystal materials. Physics

[11] Ustyuzhanin S, Sharangovich S. Analytical model of light beam diffraction by onedimensional electrically controlled non-uniform transmission photon PDLC structures.

of Wave Phenomena. 2010;18:289-293. DOI: 10.3103/S1541308X10040102

Russian Physics Journal. 2011;54:172-179. DOI: 10.1007/s11182-011-9595-2

Russian Physics Journal. 2005;48:501-510. DOI: 10.1007/s11182-005-0159-1

Journal. 2006;49:1129-1138. DOI: 10.1007/s11182-006-0233-3

ceedings of SPIE. 2005;6023:602301. DOI: 10.1117/12.646947

407-409. DOI: 10.1109/CAOL.2008.4671969

(OFC/NFOEC) and the National Fiber Optic Engineers Conference. 2011;2011:1-3


**Chapter 19**

Provisional chapter

**Application of Numeric Routine for Simulating**

Application of Numeric Routine for Simulating

Afonso José do Prado, Luis Henrique Jus,

Afonso José do Prado, Luis Henrique Jus,

Aghatta Cioqueta Moreira, Juliana Semiramis Menzinger, Caio Vinícius Colozzo Grilo,

Aghatta Cioqueta Moreira, Juliana Semiramis Menzinger, Caio Vinícius Colozzo Grilo,

Melissa de Oliveira Santos, Elmer Mateus Gennaro, André Alves Ferreira, Thainá Guimarães Pereira,

Melissa de Oliveira Santos, Elmer Mateus Gennaro, André Alves Ferreira, Thainá Guimarães Pereira,

Marinez Cargnin Stieler and José Pissolato Filho

Marinez Cargnin Stieler and José Pissolato Filho

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

obtained by the mentioned simulations.

method, transmission line modeling

http://dx.doi.org/10.5772/intechopen.74753

Abstract

**Transients in Power Line Communication (PLC) Systems**

DOI: 10.5772/intechopen.74753

Applying numerical routines based on trapezoidal rule of integration (Heun's method for numerical integration), simple models of transmission lines are used to analyze and simulate the propagation of communication signals in PLC-type systems (power line communication systems). Such systems are shared by the same systems for the transfer of electrical power and signal transmission. For the mentioned routines, the main objectives are: simulate the propagation of electromagnetic transients in these systems and analyze the interference of such phenomena in the transmitted signal. Such simulations are performed with classical structures that represent infinitesimal units of transmission lines. Modifications in the structure of such units are analyzed to improve the results

Keywords: waveguide, electromagnetic transient, eigenvalues and eigenfunctions, linear systems, numerical analysis, simulation, state space methods, numerical integration

> © 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited.

© 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, provided the original work is properly cited.

Transients in Power Line Communication (PLC) Systems

#### **Application of Numeric Routine for Simulating Transients in Power Line Communication (PLC) Systems** Application of Numeric Routine for Simulating Transients in Power Line Communication (PLC) Systems

DOI: 10.5772/intechopen.74753

Afonso José do Prado, Luis Henrique Jus, Melissa de Oliveira Santos, Elmer Mateus Gennaro, André Alves Ferreira, Thainá Guimarães Pereira, Aghatta Cioqueta Moreira, Juliana Semiramis Menzinger, Caio Vinícius Colozzo Grilo, Marinez Cargnin Stieler and José Pissolato Filho Afonso José do Prado, Luis Henrique Jus, Melissa de Oliveira Santos, Elmer Mateus Gennaro, André Alves Ferreira, Thainá Guimarães Pereira, Aghatta Cioqueta Moreira, Juliana Semiramis Menzinger, Caio Vinícius Colozzo Grilo, Marinez Cargnin Stieler and José Pissolato Filho

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74753

Abstract

Applying numerical routines based on trapezoidal rule of integration (Heun's method for numerical integration), simple models of transmission lines are used to analyze and simulate the propagation of communication signals in PLC-type systems (power line communication systems). Such systems are shared by the same systems for the transfer of electrical power and signal transmission. For the mentioned routines, the main objectives are: simulate the propagation of electromagnetic transients in these systems and analyze the interference of such phenomena in the transmitted signal. Such simulations are performed with classical structures that represent infinitesimal units of transmission lines. Modifications in the structure of such units are analyzed to improve the results obtained by the mentioned simulations.

Keywords: waveguide, electromagnetic transient, eigenvalues and eigenfunctions, linear systems, numerical analysis, simulation, state space methods, numerical integration method, transmission line modeling

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 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, provided the original work is properly cited.

#### 1. Introduction

Systems of conductors for signal transmission or power transmission, that are, in general, classified as waveguide systems, systems of protection, coordination and control of the main system, the waveguide system, are as important as the conductors that are used. For projecting these accessory systems, analysis of short-circuit levels, overvoltages, as well as the duration of transient phenomena are very important [1–9]. In several situations, it is not possible to perform tests related to the occurrence of transient electromagnetic phenomena in actual transmission systems [10]. One situation for this is when the systems are in the design phase and have not yet been built or fabricated. Another situation is where the system cannot be shut down for maintenance or testing, for example, in the case of transmission lines responsible for interconnecting great power plants to great consumer centers. Because the theory and equations related to the propagation of electromagnetic fields in systems of conductors can be related to power and signal transmissions, different transmission systems are modeled as transmission lines or waveguides [1–14]. For example, systems with low voltage, low power and very high frequency for signal transmission and systems with very high voltage, very high power and low frequency can be modeled as transmission lines or waveguides. In analyses of these types of electrical systems affected by electromagnetic transient phenomena, timedomain and frequency-dependent models are considered efficient and accurate for applications in this field [1–16]. Ways to improve these models have been researched yet, searching for increasing the accuracy of the results and the efficiency of the applied methods. For the analysis of the propagation of transient electromagnetic phenomena in electrical networks using transmission line theory, the waveguides can be decomposed into infinitesimal parts modeled by π circuits or T circuits [3–11, 17]. Simple numerical routines for this type of analysis can be good tools for undergraduate students to investigate and simulate these types of phenomena [11, 18–20] and to test improvements in the numerical model applied to the mentioned analyses. On the other hand, for more complex numerical or simpler numerical models, to a greater or lesser degree, respectively, numerical routines are influenced by numerical errors [1–23].

alternately in the composition of the cascade used to represent transmission lines or waveguides for analysis and simulations of transient electromagnetic phenomena propagation [29–32]. If the damping resistances are not applied in each π circuit of the cascade that represents the waveguides or the transmission lines, the numerical simulation can be numerically unstable [28–32]. The results of several simulations that will be used to compare and determine which structure is adequate to reduce Gibbs' oscillations without compromising the computational time will be presented. In this case, it is considered cascades with classical π circuits and cascades with damping resistances applied in each π circuit. The results presented for these comparisons are based on output voltage versus time graphs and three-dimensional graphs that establish the relationship between the first voltage peaks with the number of π circuits and the damping

Application of Numeric Routine for Simulating Transients in Power Line Communication (PLC) Systems

http://dx.doi.org/10.5772/intechopen.74753

353

The trapezoidal rule or Heun's method is a numerical integration method based on the transformation of differential equations into their algebraic equivalents. The integral of a function is approximated by the first-degree function related to the original function (the area of a trapezoid) where the endpoints are approximated by points of intersection between the original and the first-degree functions. By improving approximation accuracy, a large range of independent variable values can be subdivided into equally small portions, called integration

resistance values during the first voltage reflection to the end of the line.

Applying the trapezoidal rule, the equation below is obtained:

Figure 1. Schema of trapezoidal rule for numerical integration.

ðt kð Þ <sup>þ</sup><sup>1</sup> t kð Þ

f tð Þdt <sup>≈</sup> <sup>Δ</sup><sup>t</sup>

<sup>2</sup> f tð Þþ <sup>k</sup>þ<sup>1</sup> f tð Þ<sup>k</sup> ½ � (1)

2. Trapezoidal rule

steps (Figure 1).

The time step is

Considering the simplified representation of a transmission line by π circuit cascades, the solution of this system is obtained with the application of trapezoidal integration, and the results are affected by numerical oscillations or Gibbs' oscillations [1–27]. It is possible to minimize the influence of numerical oscillations or Gibbs' oscillations, in the obtained results, by means of structural modifications of these circuits [11, 17]. The proposed modification initially involves adding damping resistors (RD) in all π circuits [11, 17]. These resistors are introduced in parallel with the elements in a series of the π circuits (elements representing the longitudinal parameters of transmission lines or waveguides) [28]. However, in spite of decreasing the effects of numerical oscillations considerably, the incorporation of damping resistances in each circuit of the cascade increases the computational time to perform the analyses and simulations of electromagnetic phenomena propagation. So, an alternative structure for the π circuit cascade is proposed, which involves the absence of damping resistance in half of all π circuits, all of which circuits are grouped in the center of the cascade. In other cases, the different structures of π circuits, with and without damping resistance, are applied alternately in the composition of the cascade used to represent transmission lines or waveguides for analysis and simulations of transient electromagnetic phenomena propagation [29–32]. If the damping resistances are not applied in each π circuit of the cascade that represents the waveguides or the transmission lines, the numerical simulation can be numerically unstable [28–32]. The results of several simulations that will be used to compare and determine which structure is adequate to reduce Gibbs' oscillations without compromising the computational time will be presented. In this case, it is considered cascades with classical π circuits and cascades with damping resistances applied in each π circuit. The results presented for these comparisons are based on output voltage versus time graphs and three-dimensional graphs that establish the relationship between the first voltage peaks with the number of π circuits and the damping resistance values during the first voltage reflection to the end of the line.

### 2. Trapezoidal rule

1. Introduction

352 Emerging Waveguide Technology

ical errors [1–23].

Systems of conductors for signal transmission or power transmission, that are, in general, classified as waveguide systems, systems of protection, coordination and control of the main system, the waveguide system, are as important as the conductors that are used. For projecting these accessory systems, analysis of short-circuit levels, overvoltages, as well as the duration of transient phenomena are very important [1–9]. In several situations, it is not possible to perform tests related to the occurrence of transient electromagnetic phenomena in actual transmission systems [10]. One situation for this is when the systems are in the design phase and have not yet been built or fabricated. Another situation is where the system cannot be shut down for maintenance or testing, for example, in the case of transmission lines responsible for interconnecting great power plants to great consumer centers. Because the theory and equations related to the propagation of electromagnetic fields in systems of conductors can be related to power and signal transmissions, different transmission systems are modeled as transmission lines or waveguides [1–14]. For example, systems with low voltage, low power and very high frequency for signal transmission and systems with very high voltage, very high power and low frequency can be modeled as transmission lines or waveguides. In analyses of these types of electrical systems affected by electromagnetic transient phenomena, timedomain and frequency-dependent models are considered efficient and accurate for applications in this field [1–16]. Ways to improve these models have been researched yet, searching for increasing the accuracy of the results and the efficiency of the applied methods. For the analysis of the propagation of transient electromagnetic phenomena in electrical networks using transmission line theory, the waveguides can be decomposed into infinitesimal parts modeled by π circuits or T circuits [3–11, 17]. Simple numerical routines for this type of analysis can be good tools for undergraduate students to investigate and simulate these types of phenomena [11, 18–20] and to test improvements in the numerical model applied to the mentioned analyses. On the other hand, for more complex numerical or simpler numerical models, to a greater or lesser degree, respectively, numerical routines are influenced by numer-

Considering the simplified representation of a transmission line by π circuit cascades, the solution of this system is obtained with the application of trapezoidal integration, and the results are affected by numerical oscillations or Gibbs' oscillations [1–27]. It is possible to minimize the influence of numerical oscillations or Gibbs' oscillations, in the obtained results, by means of structural modifications of these circuits [11, 17]. The proposed modification initially involves adding damping resistors (RD) in all π circuits [11, 17]. These resistors are introduced in parallel with the elements in a series of the π circuits (elements representing the longitudinal parameters of transmission lines or waveguides) [28]. However, in spite of decreasing the effects of numerical oscillations considerably, the incorporation of damping resistances in each circuit of the cascade increases the computational time to perform the analyses and simulations of electromagnetic phenomena propagation. So, an alternative structure for the π circuit cascade is proposed, which involves the absence of damping resistance in half of all π circuits, all of which circuits are grouped in the center of the cascade. In other cases, the different structures of π circuits, with and without damping resistance, are applied The trapezoidal rule or Heun's method is a numerical integration method based on the transformation of differential equations into their algebraic equivalents. The integral of a function is approximated by the first-degree function related to the original function (the area of a trapezoid) where the endpoints are approximated by points of intersection between the original and the first-degree functions. By improving approximation accuracy, a large range of independent variable values can be subdivided into equally small portions, called integration steps (Figure 1).

Applying the trapezoidal rule, the equation below is obtained:

$$\int\_{t(k)}^{t(k+1)} f(t) \, dt \approx \frac{\Delta t}{2} \left[ f(t\_{k+1}) + f(t\_k) \right] \tag{1}$$

The time step is

Figure 1. Schema of trapezoidal rule for numerical integration.

$$
\Delta t = t\_{k+1} - t\_k \tag{2}
$$

Using Eq. (1), Eq. (3) is obtained:

$$\int\_{t(k)}^{t(k+1)} f(t) \, dt \approx y\_{t+1} - y\_t = \Delta y \quad \rightarrow \quad y\_{t+1} = y\_t + \frac{\Delta t}{2} [f(t\_k) + f(t\_{k+1})] \tag{3}$$

Considering a model of a physical system (or physical phenomenon), x is a vector composed by state variables of the mentioned system. Also, considering that the physical system is described by the first-order differential linear system, Eq. (4) is obtained:

$$\frac{d\mathbf{x}}{dt} = A\mathbf{x} + Bu \quad \rightarrow \quad \stackrel{\bullet}{\mathbf{x}} = A\mathbf{x} + Bu \tag{4}$$

the whole system, there is a need to use a cascade with a large number of π circuits. The

Based on the structure of π circuit and Kirchhoff's laws, the relations of voltages and currents

Each π circuit has two state variables: the transversal voltage (vk) and the longitudinal current (ik). For describing the whole transmission line or the waveguide, it is necessary to use an n-

x ¼ ½ i <sup>1</sup> v<sup>1</sup> i <sup>2</sup> v<sup>2</sup> ⋯ i <sup>k</sup> vk ⋯ in vn �

<sup>C</sup> � <sup>1</sup>

1 <sup>C</sup> � <sup>G</sup>

<sup>0</sup> ⋯ ⋯ <sup>0</sup> <sup>2</sup>

Considering only one voltage source in the initial of the transmission line or the waveguide, the B vector is in Eq. (12). If other sources are connected to the system in different points, the B

> <sup>L</sup> <sup>0</sup> <sup>⋯</sup> <sup>0</sup> � � <sup>T</sup>

If damping resistances are included in π circuits, this is shown in Figure 3. The relations of

0 ⋱⋱⋱⋱⋮

<sup>L</sup> <sup>0</sup> ⋯ ⋯ <sup>0</sup>

1 <sup>L</sup> � <sup>R</sup>

<sup>C</sup> ⋱⋱⋮

<sup>C</sup> � <sup>1</sup>

<sup>C</sup> <sup>0</sup>

<sup>L</sup> � <sup>1</sup> L

<sup>C</sup> � <sup>G</sup> C

d t <sup>¼</sup> vk • ¼ 1

Application of Numeric Routine for Simulating Transients in Power Line Communication (PLC) Systems

<sup>C</sup> ð Þ ik � <sup>G</sup> � vk � ikþ<sup>1</sup> (9)

http://dx.doi.org/10.5772/intechopen.74753

355

<sup>T</sup> (10)

(11)

(12)

<sup>L</sup> ð Þ vk�<sup>1</sup> � <sup>R</sup> � ik � vk and d vk

generic unit of π circuits is shown in Figure 2.

for this generic unit are determined by Eq. (9):

order linear numerical system. So, the x vector is

In this case, the structure of the A matrix is based on Eq. (9):

� R <sup>L</sup> � <sup>1</sup>

⋮ ⋱

⋮ ⋱⋱

<sup>B</sup> <sup>¼</sup> <sup>1</sup>

voltages and currents for the generic unit of π circuits are in Eq. (13):

1 <sup>C</sup> � <sup>G</sup>

A ¼

vector should be adequately changed:

dik d t <sup>¼</sup> ik • ¼ 1

Figure 2. The generic unit of π circuits.

In this case, the A matrix represents the system, the B matrix is related to independent inputs of the system, the u vector is the input vector, and the x vector is the vector of the state variables of the system. For numerical applications, Eq. (5) is considered:

$$\frac{\Delta \mathbf{x}}{\Delta t} \approx \frac{d\mathbf{x}}{dt}, \quad \text{if} \ \Delta t \to 0 \tag{5}$$

From Eqs. (3)–(5), for very small time step, Eq. (6) is obtained:

$$\mathbf{x}\_{t+1} = \mathbf{x}\_t + \frac{\Delta t}{2} \begin{bmatrix} \mathbf{\hat{x}}\_{t+1} + \mathbf{\hat{x}}\_t \end{bmatrix} \quad \rightarrow \quad \mathbf{x}\_{t+1} = \mathbf{x}\_t + \frac{\Delta t}{2} [\mathbf{A}\mathbf{x}\_{t+1} + \mathbf{B}u\_{t+1} + \mathbf{A}\mathbf{x}\_t + \mathbf{B}u\_t] \tag{6}$$

Simplifying Eq. (6) and considering that I is the identity matrix, Eq. (7) is obtained:

$$\begin{cases} \mathbf{x}\_{t+1} = \left[I - \frac{\Delta t}{2}A\right]^{-1} \cdot \left[I + \frac{\Delta t}{2}A\right] \mathbf{x}\_t + \left[I - \frac{\Delta t}{2}A\right]^{-1} \cdot \frac{\Delta t}{2}B[u\_{t+1} + u\_t] \\ \mathbf{x}\_{t+1} = A\_1 A\_2 \mathbf{x}\_t + A\_1 B\_1 [u\_{t+1} + u\_t] \end{cases} \tag{7}$$

In this case, A1, A2, and B1 elements in Eq. (7) are

$$\begin{aligned} y\_{t+1} &= \left[I - \frac{\Delta t}{2}A\right]^{-1} \cdot \left[I + \frac{\Delta t}{2}A\right] y\_t + \left[I - \frac{\Delta t}{2}A\right]^{-1} \cdot \frac{\Delta t}{2}B[u\_{t+1} + u\_t] \\ y\_{t+1} &= A\_1 A\_2 y\_t + A\_1 B\_1[u\_{t+1} + u\_t] \end{aligned} \tag{8}$$

Using Eq. (7), it is possible to determine the next state (xt+1) of the analyzed system, if the current state (xt) is known. This characteristic is very important for numerical applications where the functions that describe the physical system are not known or do not exist.

#### 3. Transmission line equivalent circuit model

Analyzing the propagation of waves in transmission lines or waveguides, these systems can be decomposed into infinitesimal portions that can be represented by π circuits. For representing

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Figure 2. The generic unit of π circuits.

Δt ¼ tkþ<sup>1</sup> � tk (2)

<sup>2</sup> ½ � f tð Þþ <sup>k</sup> f tð Þ <sup>k</sup>þ<sup>1</sup> (3)

• <sup>¼</sup> Ax <sup>þ</sup> Bu (4)

<sup>2</sup> Axtþ<sup>1</sup> <sup>þ</sup> Butþ<sup>1</sup> <sup>þ</sup> Axt <sup>þ</sup> But ½ � (6)

<sup>2</sup> B utþ<sup>1</sup> <sup>þ</sup> ut ½ �

<sup>2</sup> B utþ<sup>1</sup> <sup>þ</sup> ut ½ �

(7)

(8)

dt , if <sup>Δ</sup><sup>t</sup> ! <sup>0</sup> (5)

Δt

Using Eq. (1), Eq. (3) is obtained:

354 Emerging Waveguide Technology

xtþ<sup>1</sup> ¼ xt þ

Δt <sup>2</sup> <sup>x</sup> • <sup>t</sup>þ<sup>1</sup> þ x • t h i

xtþ<sup>1</sup> <sup>¼</sup> <sup>I</sup> � <sup>Δ</sup><sup>t</sup>

ytþ<sup>1</sup> <sup>¼</sup> <sup>I</sup> � <sup>Δ</sup><sup>t</sup>

In this case, A1, A2, and B1 elements in Eq. (7) are

2 A � ��<sup>1</sup>

3. Transmission line equivalent circuit model

ytþ<sup>1</sup> <sup>¼</sup> <sup>A</sup>1A2yt <sup>þ</sup> <sup>A</sup>1B<sup>1</sup> utþ<sup>1</sup> <sup>þ</sup> ut ½ �

ðt kð Þ <sup>þ</sup><sup>1</sup> t kð Þ

f tð Þdt <sup>≈</sup> ytþ<sup>1</sup> � yt <sup>¼</sup> <sup>Δ</sup><sup>y</sup> ! ytþ<sup>1</sup> <sup>¼</sup> yt <sup>þ</sup>

dt <sup>¼</sup> Ax <sup>þ</sup> Bu ! <sup>x</sup>

described by the first-order differential linear system, Eq. (4) is obtained:

Δx <sup>Δ</sup><sup>t</sup> <sup>≈</sup> dx

dx

of the system. For numerical applications, Eq. (5) is considered:

From Eqs. (3)–(5), for very small time step, Eq. (6) is obtained:

2 A � ��<sup>1</sup>

xtþ<sup>1</sup> ¼ A1A2xt þ A1B<sup>1</sup> utþ<sup>1</sup> þ ut ½ �

Considering a model of a physical system (or physical phenomenon), x is a vector composed by state variables of the mentioned system. Also, considering that the physical system is

In this case, the A matrix represents the system, the B matrix is related to independent inputs of the system, the u vector is the input vector, and the x vector is the vector of the state variables

! xtþ<sup>1</sup> ¼ xt þ

Simplifying Eq. (6) and considering that I is the identity matrix, Eq. (7) is obtained:

� I þ Δt 2 A � �

� I þ Δt 2 A � � Δt

2 A � ��<sup>1</sup>

2 A � ��<sup>1</sup> � Δt

� Δt

xt <sup>þ</sup> <sup>I</sup> � <sup>Δ</sup><sup>t</sup>

yt <sup>þ</sup> <sup>I</sup> � <sup>Δ</sup><sup>t</sup>

Using Eq. (7), it is possible to determine the next state (xt+1) of the analyzed system, if the current state (xt) is known. This characteristic is very important for numerical applications

Analyzing the propagation of waves in transmission lines or waveguides, these systems can be decomposed into infinitesimal portions that can be represented by π circuits. For representing

where the functions that describe the physical system are not known or do not exist.

the whole system, there is a need to use a cascade with a large number of π circuits. The generic unit of π circuits is shown in Figure 2.

Based on the structure of π circuit and Kirchhoff's laws, the relations of voltages and currents for this generic unit are determined by Eq. (9):

$$\frac{d\dot{\mathbf{i}}\_k}{dt} = \dot{\mathbf{i}}\_k = \frac{1}{L}(\mathbf{v}\_{k-1} - \mathbf{R} \cdot \dot{\mathbf{i}}\_k - \mathbf{v}\_k) \quad \text{and} \quad \frac{d\mathbf{v}\_k}{dt} = \dot{\mathbf{v}}\_k = \frac{1}{\mathbf{C}}(\dot{\mathbf{i}}\_k - \mathbf{G} \cdot \mathbf{v}\_k - \dot{\mathbf{i}}\_{k+1}) \tag{9}$$

Each π circuit has two state variables: the transversal voltage (vk) and the longitudinal current (ik). For describing the whole transmission line or the waveguide, it is necessary to use an norder linear numerical system. So, the x vector is

$$\mathbf{x} = \begin{bmatrix} i\_1 & \upsilon\_1 & i\_2 & \upsilon\_2 & \cdots & i\_k & \upsilon\_k & \cdots & i\_n & \upsilon\_n \end{bmatrix}^T \tag{10}$$

In this case, the structure of the A matrix is based on Eq. (9):

$$A = \begin{bmatrix} -\frac{R}{L} & -\frac{1}{L} & 0 & \cdots & \cdots & 0\\ \frac{1}{C} & -\frac{G}{C} & -\frac{1}{C} & \ddots & \ddots & \vdots\\ 0 & \ddots & \ddots & \ddots & \ddots & \vdots\\ \vdots & & \ddots & \frac{1}{C} & -\frac{G}{C} & -\frac{1}{C} & 0\\ \vdots & & \ddots & \ddots & \frac{1}{L} & -\frac{R}{L} & -\frac{1}{L}\\ 0 & \cdots & \cdots & 0 & \frac{2}{C} & -\frac{G}{C} \end{bmatrix} \tag{11}$$

Considering only one voltage source in the initial of the transmission line or the waveguide, the B vector is in Eq. (12). If other sources are connected to the system in different points, the B vector should be adequately changed:

$$B = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}^T \tag{12}$$

If damping resistances are included in π circuits, this is shown in Figure 3. The relations of voltages and currents for the generic unit of π circuits are in Eq. (13):

$$\stackrel{\bullet}{i\_k} = \frac{\upsilon\_{k-1} - R \cdot i\_k - \upsilon\_k}{L} \quad \text{and} \quad \stackrel{\bullet}{\upsilon}\_k = \frac{\mathrm{i}\_k - (2G\_D + G)\upsilon\_k + G\_D(\upsilon\_{k-1} + \upsilon\_{k+1}) - \mathrm{i}\_{k+1}}{\mathbb{C}} \tag{13}$$

Based on Figure 3 and Eq. (12), the structure of the B vector is in Eq. (14). In this case, only one voltage source at the initial of the waveguide is considered:

<sup>B</sup> <sup>¼</sup> <sup>1</sup>=<sup>L</sup> ½ � GD=<sup>C</sup> <sup>0</sup>⋯<sup>0</sup> <sup>T</sup> (14)

4. Numerical computation

In Figure 4, the flowchart applied to obtain the results of electromagnetic transient phenomena

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In case of Figure 4, the flowchart is based on Eq. (6) to Eq. (12). The simulations are carried out considering that the analyzed waveguide or the transmission line is connected to an independent step voltage source of 1 pu. The end line is opened, and, because of this, the value of the propagated voltage wave is doubled compared to the voltage value at the initial line. Using the flowchart of Figure 4, the parameter values applied to the obtained results are R' = 0.03 Ω/km,

In Figure 5, the flowchart related to the inclusion of damping resistances in the π circuits for representing the analyzed waveguide or the transmission line is shown. The values of R', L',

simulations without the introduction of damping resistances is shown.

L' = 1.2 mH/km, G' = 0.5 μS/km, C<sup>0</sup> = 10 nF/km, Δt = 50 ns, and d = 5 km.

Figure 4. Flowchart for numerical simulations without application of damping resistances.

Figure 5. Flowchart for numerical simulations with applications of damping resistances.

Also, based on Figure 3 and Eq. (12), the structure of the A matrix is in Eq. (15). In this case, new non-null elements are included because of the application of damping resistances:

$$A = \begin{bmatrix} -\frac{R}{L} & -\frac{1}{L} & 0 & \cdots & \cdots & \cdots & 0\\ \frac{1}{C} & -\frac{(G + 2G\_D)}{C} & -\frac{1}{C} & \frac{G\_D}{C} & \ddots & & \ddots & \ddots & \vdots\\ 0 & \ddots & & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots\\ \vdots & \ddots & & \frac{1}{L} & -\frac{R}{L} & -\frac{1}{L} & & \ddots & \ddots & \vdots\\ \vdots & \ddots & & \frac{G\_D}{C} & \frac{1}{C} & -\frac{(G + 2G\_D)}{C} & -\frac{1}{C} & \frac{G\_D}{C} & \vdots\\ \vdots & \ddots & & \ddots & \ddots & \ddots & \ddots & 0\\ \vdots & \ddots & & \ddots & \ddots & \ddots & \frac{1}{L} & -\frac{R}{L} & -\frac{1}{L}\\ \vdots & \ddots & & \ddots & \ddots & \ddots & \frac{G\_D}{C} & \frac{2}{C} & -\frac{(G + 2G\_D)}{C} \end{bmatrix} \tag{15}$$

The damping resistance is determined by

$$R\_D = k\_D \frac{2L}{\Delta t'}, \quad G\_D = \frac{1}{R\_D}, \quad k\_D = \frac{R\_D \Delta t}{2L} = \frac{\Delta t}{2L G\_D} \tag{16}$$

The R, L, G, and C values are calculated by Eq. (17) where d is the line length and n is the number of π circuits:

$$R = R' \cdot \frac{d}{n'} \qquad L = L' \cdot \frac{d}{n'} \qquad G = G' \cdot \frac{d}{n'} \qquad \mathbb{C} = \mathbb{C}' \cdot \frac{d}{n} \tag{17}$$

Figure 3. Introduction of damping resistances in π circuits.

### 4. Numerical computation

ik •

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� R <sup>L</sup> � <sup>1</sup>

⋮ ⋱

⋮ ⋱

1

A ¼

number of π circuits:

<sup>¼</sup> vk�<sup>1</sup> � <sup>R</sup> � ik � vk

<sup>C</sup> � ð Þ <sup>G</sup> <sup>þ</sup> <sup>2</sup>GD

The damping resistance is determined by

R ¼ R 0 � d n

Figure 3. Introduction of damping resistances in π circuits.

<sup>L</sup> and vk

voltage source at the initial of the waveguide is considered:

<sup>C</sup> � <sup>1</sup>

⋮ ⋱ ⋱⋱⋱

RD ¼ kD

C

1 <sup>L</sup> � <sup>R</sup>

GD C

GD

1

<sup>0</sup> ⋯ ⋯⋯ <sup>0</sup> GD

2L Δt

, L ¼ L

0 ⋱ ⋱⋱⋱ ⋱⋱⋮

⋮ ⋱ ⋱⋱⋱ ⋱⋱ 0

, GD <sup>¼</sup> <sup>1</sup>

0 � d n RD

The R, L, G, and C values are calculated by Eq. (17) where d is the line length and n is the

, G <sup>¼</sup> <sup>G</sup><sup>0</sup>

<sup>L</sup> � <sup>1</sup>

<sup>C</sup> � ð Þ <sup>G</sup> <sup>þ</sup> <sup>2</sup>GD

•

Based on Figure 3 and Eq. (12), the structure of the B vector is in Eq. (14). In this case, only one

Also, based on Figure 3 and Eq. (12), the structure of the A matrix is in Eq. (15). In this case,

<sup>L</sup> <sup>0</sup> ⋯⋯ ⋯⋯ <sup>0</sup>

<sup>C</sup> ⋱ ⋱⋱⋮

<sup>L</sup> ⋱⋱⋮

1 <sup>L</sup> � <sup>R</sup>

C

C

GD <sup>C</sup> <sup>⋮</sup>

2

<sup>2</sup><sup>L</sup> <sup>¼</sup> <sup>Δ</sup><sup>t</sup> 2LGD

, C ¼ C

0 � d

<sup>L</sup> � <sup>1</sup> L

<sup>C</sup> � ð Þ <sup>G</sup> <sup>þ</sup> <sup>2</sup>GD C

<sup>C</sup> � <sup>1</sup>

, kD <sup>¼</sup> RDΔ<sup>t</sup>

� d n

new non-null elements are included because of the application of damping resistances:

<sup>¼</sup> ik � ð Þ <sup>2</sup>GD <sup>þ</sup> <sup>G</sup> vk <sup>þ</sup> GDð Þ� vk�<sup>1</sup> <sup>þ</sup> vkþ<sup>1</sup> ikþ<sup>1</sup>

<sup>B</sup> <sup>¼</sup> <sup>1</sup>=<sup>L</sup> ½ � GD=<sup>C</sup> <sup>0</sup>⋯<sup>0</sup> <sup>T</sup> (14)

<sup>C</sup> (13)

(15)

(16)

<sup>n</sup> (17)

In Figure 4, the flowchart applied to obtain the results of electromagnetic transient phenomena simulations without the introduction of damping resistances is shown.

In case of Figure 4, the flowchart is based on Eq. (6) to Eq. (12). The simulations are carried out considering that the analyzed waveguide or the transmission line is connected to an independent step voltage source of 1 pu. The end line is opened, and, because of this, the value of the propagated voltage wave is doubled compared to the voltage value at the initial line. Using the flowchart of Figure 4, the parameter values applied to the obtained results are R' = 0.03 Ω/km, L' = 1.2 mH/km, G' = 0.5 μS/km, C<sup>0</sup> = 10 nF/km, Δt = 50 ns, and d = 5 km.

In Figure 5, the flowchart related to the inclusion of damping resistances in the π circuits for representing the analyzed waveguide or the transmission line is shown. The values of R', L',

Figure 4. Flowchart for numerical simulations without application of damping resistances.

Figure 5. Flowchart for numerical simulations with applications of damping resistances.

G', C<sup>0</sup> , Δt, and n are the same that were used to simulations without damping resistances. The analyzed system is also connected to the 1 pu step voltage, and the end line is also opened.

6. Improvement of the simulation accuracy with damping resistance

Applying damping resistances and using the kD factor as 2.5, the obtained results are shown in Figure 7. Comparing Figure 7 to Figure 6, if the number of π circuits is increased, numerical oscillations or Gibbs' oscillations are decreased. So, considering a constant value for the kD factor, if an adequate number of π circuits is applied, numerical oscillations can be minimized. Similar results can be obtained changing the value of the kD factor. In this case, in Figure 8, the results are obtained using 200 π circuits for different values of the kD factor. Based on these results, the numerical oscillations are decreased if the factor is decreased. Because in this chapter the kD is the integer, the lower value of kD is 1, and the best reductions of numerical

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Figure 7. The step voltage wave propagation varying the number of circuits for the first reflection at the analyzed end line

Figure 8. The step voltage wave propagation varying the factor kD for the first reflection at the analyzed end line with the

with the application of damping resistances and kD = 2.5.

application of damping resistances and n = 200.

#### 5. Effects of the simulation accuracy without damping resistance

Applying the flowchart of Figure 4, the main obtained results are shown in Figure 6. For these results, the time step (Δt) is 50 ns, and the number of π circuits is changed from 50 to 500 with step variation of 1 unit. In general, when the voltage waves are reflected the first time at the end line, the voltage values reach 2.5 pu, initially. This initial value is due to the influence of numerical oscillations or Gibbs' oscillations. These oscillations cause numerical errors of 25% because, in this case, the exact values should be 2 pu. While the reflected voltage wave is propagated to the line initial, after new reflection at the initial line, Gibbs' oscillations are being damped. In case of the second voltage wave reflection at the end line, the voltage values should be null ones. In this instant time, there are numerical problems again that are also represented by Gibbs' oscillations. So, the results obtained from numerical routine based on the flowchart without the application of damping resistances are highly influenced by numerical oscillations during abrupt changes at voltage related to the energization of the transmission line or the waveguide. For the systems modeled by the transmission line theory concepts, the step voltage source represents the main problems that introduce abrupt changes in voltages in the line.

The numerical routine described by the flowchart is simple. Despite this characteristic, the numerical simulations lead to results with errors of 25% independently that the number of π circuits is applied. The increase of the number of π circuits is not related to a correspondent decrease of the numerical errors and numerical oscillations in the obtained results. A proposed alternative is the introduction of damping resistances for decreasing numerical errors and Gibbs' oscillations in the obtained results. Next, both items show the results obtained with this alternative.

Figure 6. The step voltage wave propagation varying the number of circuits for the first reflection at the analyzed end line without the application of damping resistances.

### 6. Improvement of the simulation accuracy with damping resistance

G', C<sup>0</sup>

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alternative.

without the application of damping resistances.

, Δt, and n are the same that were used to simulations without damping resistances. The

analyzed system is also connected to the 1 pu step voltage, and the end line is also opened.

Applying the flowchart of Figure 4, the main obtained results are shown in Figure 6. For these results, the time step (Δt) is 50 ns, and the number of π circuits is changed from 50 to 500 with step variation of 1 unit. In general, when the voltage waves are reflected the first time at the end line, the voltage values reach 2.5 pu, initially. This initial value is due to the influence of numerical oscillations or Gibbs' oscillations. These oscillations cause numerical errors of 25% because, in this case, the exact values should be 2 pu. While the reflected voltage wave is propagated to the line initial, after new reflection at the initial line, Gibbs' oscillations are being damped. In case of the second voltage wave reflection at the end line, the voltage values should be null ones. In this instant time, there are numerical problems again that are also represented by Gibbs' oscillations. So, the results obtained from numerical routine based on the flowchart without the application of damping resistances are highly influenced by numerical oscillations during abrupt changes at voltage related to the energization of the transmission line or the waveguide. For the systems modeled by the transmission line theory concepts, the step voltage source represents the main problems that introduce abrupt changes in voltages in the line.

The numerical routine described by the flowchart is simple. Despite this characteristic, the numerical simulations lead to results with errors of 25% independently that the number of π circuits is applied. The increase of the number of π circuits is not related to a correspondent decrease of the numerical errors and numerical oscillations in the obtained results. A proposed alternative is the introduction of damping resistances for decreasing numerical errors and Gibbs' oscillations in the obtained results. Next, both items show the results obtained with this

Figure 6. The step voltage wave propagation varying the number of circuits for the first reflection at the analyzed end line

5. Effects of the simulation accuracy without damping resistance

Applying damping resistances and using the kD factor as 2.5, the obtained results are shown in Figure 7. Comparing Figure 7 to Figure 6, if the number of π circuits is increased, numerical oscillations or Gibbs' oscillations are decreased. So, considering a constant value for the kD factor, if an adequate number of π circuits is applied, numerical oscillations can be minimized. Similar results can be obtained changing the value of the kD factor. In this case, in Figure 8, the results are obtained using 200 π circuits for different values of the kD factor. Based on these results, the numerical oscillations are decreased if the factor is decreased. Because in this chapter the kD is the integer, the lower value of kD is 1, and the best reductions of numerical

Figure 7. The step voltage wave propagation varying the number of circuits for the first reflection at the analyzed end line with the application of damping resistances and kD = 2.5.

Figure 8. The step voltage wave propagation varying the factor kD for the first reflection at the analyzed end line with the application of damping resistances and n = 200.

oscillations are obtained for this value. Based on Figures 7 and 8, there are two parameters that can minimize numerical oscillations: the number of π circuits and the value of the kD factor.

### 7. Effects of kD factor variation

Applying damping resistances, from Figures 9–15, the number of π circuits is changed from 50 to 500 for different values of the kD factor. In Figure 9, the results are related to kD = 1. For this value of the kD factor, the numerical oscillations are highly minimized. Low numerical oscillations are observed for the number of π circuits about 50. In Figure 10, with kD = 2.5, the numerical oscillations reach higher values than the results shown in Figure 9, and they are related to a range from 50 to about 100 that is bigger than the range observed in Figure 9. In

Figure 9. Results for different quantities of π circuits and kD = 1.

Figure 10, the second overvoltage peak that is lower than the first overvoltage peak is also observed. Both overvoltage peaks are caused by numerical oscillations. Compared to Figure 9, in Figure 10, for the same time interval, the number of overvoltage peaks is increased showing that the damping of numerical oscillations is not effective as well as when the kD factor is equal to 1. Based on Eq. (16), the kD factor is related to the time step (Δt), and the value of this factor is also related to the frequency of the oscillations that are significantly reduced by the application of damping resistances. So, increasing the value of the kD factor, not all numerical oscillations are damped. Because of this, the overvoltage peaks are increased, and other lower overvoltage

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Figure 11. Results for different quantities of π circuits and kD = 5.

Figure 12. Results for different quantities of π circuits and kD = 7.5.

Increasing the value of the kD factor, the influence of damping resistances is decreased. So, the voltage peaks caused by Gibbs' oscillations are increased, if the kD factor is increased. This

peaks arise.

Figure 10. Results for different quantities of π circuits and kD = 2.5.

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Figure 11. Results for different quantities of π circuits and kD = 5.

oscillations are obtained for this value. Based on Figures 7 and 8, there are two parameters that can minimize numerical oscillations: the number of π circuits and the value of the kD factor.

Applying damping resistances, from Figures 9–15, the number of π circuits is changed from 50 to 500 for different values of the kD factor. In Figure 9, the results are related to kD = 1. For this value of the kD factor, the numerical oscillations are highly minimized. Low numerical oscillations are observed for the number of π circuits about 50. In Figure 10, with kD = 2.5, the numerical oscillations reach higher values than the results shown in Figure 9, and they are related to a range from 50 to about 100 that is bigger than the range observed in Figure 9. In

7. Effects of kD factor variation

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Figure 9. Results for different quantities of π circuits and kD = 1.

Figure 10. Results for different quantities of π circuits and kD = 2.5.

Figure 12. Results for different quantities of π circuits and kD = 7.5.

Figure 10, the second overvoltage peak that is lower than the first overvoltage peak is also observed. Both overvoltage peaks are caused by numerical oscillations. Compared to Figure 9, in Figure 10, for the same time interval, the number of overvoltage peaks is increased showing that the damping of numerical oscillations is not effective as well as when the kD factor is equal to 1. Based on Eq. (16), the kD factor is related to the time step (Δt), and the value of this factor is also related to the frequency of the oscillations that are significantly reduced by the application of damping resistances. So, increasing the value of the kD factor, not all numerical oscillations are damped. Because of this, the overvoltage peaks are increased, and other lower overvoltage peaks arise.

Increasing the value of the kD factor, the influence of damping resistances is decreased. So, the voltage peaks caused by Gibbs' oscillations are increased, if the kD factor is increased. This

Figure 13. Results for different quantities of π circuits and kD = 10.

8. Number of π circuit variation

for values of the kD factor from 1 to about 3 (Figure 17).

Figure 16. Results for different values of the kD factor and n = 50.

Figure 15. Results for different quantities of π circuits and kD = 15.

Setting the number of π circuits, the results are analyzed considering the kD changing from 1 to 10. Applying 50 units of π circuits, the voltage peaks are not damped significantly if the kD is changed. It is observed in Figure 16. For 100 units of π circuits, the voltage peaks are damped

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Considering 150 units of π circuits, the range of the kD factor that the voltage peaks are damped is bigger than the previous results. It is shown in Figure 18. In this case, this kD factor range is from 1 to about 4. For 200 units of π circuits, the higher limit of the mentioned range is increased to about 5 (Figure 19). If 250 units of π circuits are applied, the range is further increased and the higher limit is about 6 (Figure 20). Similar relations are observed in

Figure 14. Results for different quantities of π circuits and kD = 12.5.

effect is observed in Figures 11–15. The highest voltage peak for each value of the kD factor is about 2.5 pu. Another consequence is that the second voltage peak is increased, while the kD factor is decreased. The influence of damping resistances for minimizing the numerical oscillations in proposed transmission line model is more effective for small values for the kD factor, considering the lower limit as 1.

Analyzing the results from Figures 9–15, the highest voltage peak for each value of the kD factor is related to the lowest number of π circuits. Increasing the number of π circuits for the same kD factor value, the voltage peak values can be decreased. Changing adequately the kD factor and the number of π circuits, Gibbs' oscillations can be minimized. The sets of kD values and the numbers of π circuits that minimize the numerical oscillations can be determined by analyzing the first voltage peaks of a great number of simulations.

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Figure 15. Results for different quantities of π circuits and kD = 15.

#### 8. Number of π circuit variation

effect is observed in Figures 11–15. The highest voltage peak for each value of the kD factor is about 2.5 pu. Another consequence is that the second voltage peak is increased, while the kD factor is decreased. The influence of damping resistances for minimizing the numerical oscillations in proposed transmission line model is more effective for small values for the kD factor,

Analyzing the results from Figures 9–15, the highest voltage peak for each value of the kD factor is related to the lowest number of π circuits. Increasing the number of π circuits for the same kD factor value, the voltage peak values can be decreased. Changing adequately the kD factor and the number of π circuits, Gibbs' oscillations can be minimized. The sets of kD values and the numbers of π circuits that minimize the numerical oscillations can be determined by

analyzing the first voltage peaks of a great number of simulations.

considering the lower limit as 1.

Figure 13. Results for different quantities of π circuits and kD = 10.

362 Emerging Waveguide Technology

Figure 14. Results for different quantities of π circuits and kD = 12.5.

Setting the number of π circuits, the results are analyzed considering the kD changing from 1 to 10. Applying 50 units of π circuits, the voltage peaks are not damped significantly if the kD is changed. It is observed in Figure 16. For 100 units of π circuits, the voltage peaks are damped for values of the kD factor from 1 to about 3 (Figure 17).

Considering 150 units of π circuits, the range of the kD factor that the voltage peaks are damped is bigger than the previous results. It is shown in Figure 18. In this case, this kD factor range is from 1 to about 4. For 200 units of π circuits, the higher limit of the mentioned range is increased to about 5 (Figure 19). If 250 units of π circuits are applied, the range is further increased and the higher limit is about 6 (Figure 20). Similar relations are observed in

Figure 16. Results for different values of the kD factor and n = 50.

Figure 17. Results for different values of the kD factor and n = 100.

Figures 21–23. If the quantity of π circuits is increased, the range of the kD factor that is related to the minimization of Gibbs' oscillations is increased. In this case, the increase of this quantity

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Still analyzing Figures 21–23, it is observed that the relation between the number of the π circuits and the influence of the kD factor in minimizing the voltage peaks is not linear. There is a saturation point where the increase of the number of π circuits can no longer minimize significant Gibbs' oscillations and, consequently, the voltage peaks in obtained simulations. Because of this, another type of analysis is shown in Figure 24. In this case, the voltage peaks are related to the correspondent values of the kD factor and the number of π circuits. In case of Figure 24, the results are obtained to a time step (Δt) of 50 ns. A region where the numerical oscillations are critically damped and there are no voltage peaks can be observed. In this case, the voltage value is 2 pu and corresponds to the exact values that can be obtained using a numerical routine of Laplace's transformation. So, a specific type of analysis is related to the

is directly related to the increase of the simulation time.

Figure 21. Results for different values of the kD factor and n = 300.

Figure 20. Results for different values of the kD factor and n = 250.

Figure 18. Results for different values of the kD factor and n = 150.

Figure 19. Results for different values of the kD factor and n = 200.

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Figure 20. Results for different values of the kD factor and n = 250.

Figure 17. Results for different values of the kD factor and n = 100.

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Figure 18. Results for different values of the kD factor and n = 150.

Figure 19. Results for different values of the kD factor and n = 200.

Figures 21–23. If the quantity of π circuits is increased, the range of the kD factor that is related to the minimization of Gibbs' oscillations is increased. In this case, the increase of this quantity is directly related to the increase of the simulation time.

Still analyzing Figures 21–23, it is observed that the relation between the number of the π circuits and the influence of the kD factor in minimizing the voltage peaks is not linear. There is a saturation point where the increase of the number of π circuits can no longer minimize significant Gibbs' oscillations and, consequently, the voltage peaks in obtained simulations. Because of this, another type of analysis is shown in Figure 24. In this case, the voltage peaks are related to the correspondent values of the kD factor and the number of π circuits. In case of Figure 24, the results are obtained to a time step (Δt) of 50 ns. A region where the numerical oscillations are critically damped and there are no voltage peaks can be observed. In this case, the voltage value is 2 pu and corresponds to the exact values that can be obtained using a numerical routine of Laplace's transformation. So, a specific type of analysis is related to the

Figure 21. Results for different values of the kD factor and n = 300.

application of three-dimensional graphics showing the influence of the kD factor values and the number of π circuits on the voltage peaks obtained during the numerical simulations. It is

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Based on the results shown in the previous sections, it is concluded that the numerical oscillations and, consequently, the voltage peaks obtained by the proposed model are influenced jointly by two factors: the damping resistance value and the number of π circuits applied to the numerical simulations of electromagnetic phenomena in waveguides or transmission lines. Because of this, the analyses of this joint influence must be based on three-dimensional graphics. In Figure 24, the highest voltage peaks during the first wave reflection on the transmission end line or the receiving end terminal of the waveguide are shown. These peaks depend on the kD factor and the number of π circuits considering the time step as 50 ns. In Figures 25 and 26, the time steps are 10 ns and 200 ns, respectively. These graphics are used for

Based on the last three sets of obtained results of this chapter (Figures 24–26), for a specific time step, there are sets of the number of π circuits and the kD factor values adequate for minimizing Gibbs' oscillations and, consequently, the voltage peaks in simulations of electromagnetic transient phenomena in transmission lines using the numerical routine proposed in this chapter. The time step choice or determination is related to the fundamental frequency of the simulated phenomena. This choice or determination can be related to the type of the analyzed circuit. For example, transmission lines for power systems or waveguides for data

completing the analyses carried out in the previous sections.

Figure 25. Voltage peaks related to the kD factor and the number of π circuits for Δt = 10 ns.

shown in the next section.

9. Other analyses

transmission can be mentioned.

Figure 22. Results for different values of the kD factor and n = 400.

Figure 23. Results for different values of the kD factor and n = 500.

Figure 24. Voltage peaks related to the kD factor and the number of π circuits for Δt = 50 ns.

application of three-dimensional graphics showing the influence of the kD factor values and the number of π circuits on the voltage peaks obtained during the numerical simulations. It is shown in the next section.

### 9. Other analyses

Figure 22. Results for different values of the kD factor and n = 400.

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Figure 23. Results for different values of the kD factor and n = 500.

Figure 24. Voltage peaks related to the kD factor and the number of π circuits for Δt = 50 ns.

Based on the results shown in the previous sections, it is concluded that the numerical oscillations and, consequently, the voltage peaks obtained by the proposed model are influenced jointly by two factors: the damping resistance value and the number of π circuits applied to the numerical simulations of electromagnetic phenomena in waveguides or transmission lines. Because of this, the analyses of this joint influence must be based on three-dimensional graphics. In Figure 24, the highest voltage peaks during the first wave reflection on the transmission end line or the receiving end terminal of the waveguide are shown. These peaks depend on the kD factor and the number of π circuits considering the time step as 50 ns. In Figures 25 and 26, the time steps are 10 ns and 200 ns, respectively. These graphics are used for completing the analyses carried out in the previous sections.

Based on the last three sets of obtained results of this chapter (Figures 24–26), for a specific time step, there are sets of the number of π circuits and the kD factor values adequate for minimizing Gibbs' oscillations and, consequently, the voltage peaks in simulations of electromagnetic transient phenomena in transmission lines using the numerical routine proposed in this chapter. The time step choice or determination is related to the fundamental frequency of the simulated phenomena. This choice or determination can be related to the type of the analyzed circuit. For example, transmission lines for power systems or waveguides for data transmission can be mentioned.

Figure 25. Voltage peaks related to the kD factor and the number of π circuits for Δt = 10 ns.

Author details

Afonso José do Prado<sup>1</sup>

São Paulo State, Brazil

References

894-903

223-228

150(2):200-204

2005;20(3):2358

Serra, Mato Grosso State, Brazil

Elmer Mateus Gennaro<sup>1</sup>

Aghatta Cioqueta Moreira<sup>1</sup>

\*, Luis Henrique Jus<sup>1</sup>

Marinez Cargnin Stieler<sup>2</sup> and José Pissolato Filho<sup>3</sup>

\*Address all correspondence to: afonsojp@uol.com.br

, André Alves Ferreira<sup>1</sup>

, Juliana Semiramis Menzinger<sup>1</sup>

1 Campus of São João da Boa Vista, São Paulo State University (UNESP), São João da Boa Vista,

2 Campus of Tangará da Serra, The University of Mato Grosso State (UNEMAT), Tangará da

[1] Macías JAR, Expósito AG, Soler AB. A comparison of techniques for state-space transient analysis of transmission lines. IEEE Transactions on Power Delivery. April, 2005;20(2):

[2] Macías JAR, Expósito AG, Soler AB. Correction to "a comparison of techniques for statespace transient analysis of transmission lines". IEEE Transactions on Power Delivery. July,

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[4] Nelms RM, Sheble GB, Newton SR, Grigsby LL. Using a personal computer to teach power system transients. IEEE Transactions on Power Systems. August, 1989;4(3):1293-1294 [5] Mamis MS, Koksal M. Transient analysis of nonuniform lossy transmission lines with frequency dependent parameters. Electric Power Systems Research. December, 1999;52(3):

[6] Mamis MS. Computation of electromagnetic transients on transmission lines with nonlinear components. IEEE Proceeding of Generation, Transmission and Distribution. March, 2003;

[7] Mamis MS¸ Koksal M, Solution of eigenproblems for state-space transient analysis of

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, Melissa de Oliveira Santos1

Application of Numeric Routine for Simulating Transients in Power Line Communication (PLC) Systems

, Thainá Guimarães Pereira<sup>1</sup>

,

,

http://dx.doi.org/10.5772/intechopen.74753

,

369

, Caio Vinícius Colozzo Grilo<sup>1</sup>

Figure 26. Voltage peaks related to the kD factor and the number of π circuits for Δt = 200 ns.

### 10. Conclusions

Modifications on the classical structure of π circuits for modeling transmission lines are presented. These modified π circuits are applied to obtain a cascade that represents the analyzed transmission lines. Based on the electromagnetic basic concepts, very long circuits for power transmission and circuits for data transmission can be analyzed using the theoretical bases of the transmission lines. So, a numerical routine for simulating electromagnetic transient phenomena in waveguides (transmission lines for power systems or data transmission) is obtained.

In the proposed numerical routine, damping resistances for minimizing Gibbs' oscillations or numerical oscillations are included. These oscillations are caused by the numerical integration method applied to the solution of the linear system that describes the waveguide. Applying this proposed numerical routine, several results of simulations varying the number of π circuits, the kD factor, and the time step are obtained. These results are concentrated on threedimensional graphics where the joint influence is shown.

Based on the obtained results, it is observed that there are ranges of the model parameters adequate for the minimization of numerical oscillations that influence these results. The main model parameters that influence the minimization of numerical oscillations are the number of π circuits and the kD factor. The kD factor is applied to calculate the value of damping resistances included in each π circuit of the mentioned cascade.

### Acknowledgements

The authors would like to thank the financial support by FAPESP (The São Paulo Research Foundation). The following processes are related to the results shown in this chapter: 2015/ 21390-7, 2015/20590-2, 2015/20684-7, 2016/02559-3, 2017/05988-5, 2017/05995-1, and 2017/23430-1.
