3. The proposed system

This section shows a proposed model for cascaded n stages of FBGs. Analysis of this model is done by coupling theory [16]. T matrix 2 � 2 where FBG is divided into sections.

Each section is shown in Figure 2 where T is the length of each section and ᴧ is the space between reflected planes of each section where:

a0: is the incident optical signal. b0: is the reflected optical signal. am: is the output optical signal. bm: is the reflected optical signal at output of grating. m: is number of sections.

The transfer matrix can be expressed by small multiplied matrixes as in:

$$
\begin{bmatrix} a\_0 \\ b\_0 \end{bmatrix} = \begin{bmatrix} T^1 \end{bmatrix} \begin{bmatrix} T^2 \end{bmatrix} \begin{bmatrix} T^3 \end{bmatrix} \begin{bmatrix} T^4 \end{bmatrix} \dots \begin{bmatrix} T^m \end{bmatrix} \begin{bmatrix} a\_m \\ b\_m \end{bmatrix} \tag{14}
$$

Replacing m matrix by whole matrix ½ � T where:

$$\left[\mathbf{T}\right] = \prod\_{j=1}^{m} \left[T^{\bullet}\right] \tag{15}$$

Figure 2. Fiber Bragg grating sections.

The transfer matrix can be written as:

$$
\begin{bmatrix} a\_0 \\ b\_0 \end{bmatrix} = \begin{bmatrix} \mathbf{T} \end{bmatrix} \begin{bmatrix} a\_m \\ b\_m \end{bmatrix} \tag{16}
$$

Where ½ � T can be written as:

$$
\begin{bmatrix} a\_0 \\ b\_0 \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} a\_m \\ b\_m \end{bmatrix} \tag{17}
$$

In case of FBG reflection, There is no reflection at o/p at distance L so bm= 0

#### 3.1 In case of one grating

Transfer matrix can be written as:

$$
\begin{bmatrix} a\_0 \\ b\_0 \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} a\_m \\ \mathbf{0} \end{bmatrix} \tag{18}
$$

Where Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> are:

$$
\mathfrak{a}\_0 = T\_{11}\mathfrak{a}\_m \tag{19}
$$

Then from Eq. (12) we get:

Theoretic Study of Cascaded Fiber Bragg Grating DOI: http://dx.doi.org/10.5772/intechopen.83020

Connection between two cascaded FBG.

3.3 In case of third cascaded gratings

Transfer matrix can be written as:

a<sup>01</sup> T<sup>21</sup> 2

2 4

T<sup>11</sup> 2

b<sup>03</sup>

Where,

13

Figure 3.

<sup>a</sup><sup>01</sup> <sup>¼</sup> <sup>T</sup><sup>11</sup>

We can calculate Reflectivity for the second grating by:

<sup>ρ</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>02</sup> a<sup>01</sup>

The output of second grating b<sup>02</sup> is equal to input of third grating:

3

a<sup>03</sup> ¼ b<sup>02</sup> ¼ a<sup>01</sup>

<sup>5</sup> <sup>¼</sup> <sup>T</sup><sup>11</sup> <sup>T</sup><sup>12</sup>

2 T<sup>21</sup>

<sup>¼</sup> <sup>T</sup><sup>21</sup> <sup>∗</sup> am <sup>∗</sup> <sup>T</sup><sup>21</sup> T<sup>11</sup> <sup>2</sup> ∗ am

2

T<sup>21</sup> 2

T<sup>11</sup>

<sup>T</sup><sup>21</sup> <sup>T</sup><sup>22</sup> � � am

0

T<sup>11</sup>

<sup>ρ</sup><sup>2</sup> <sup>¼</sup> <sup>T</sup><sup>21</sup>

am (26)

<sup>2</sup> (30)

<sup>2</sup> (32)

� � (33)

(29)

b<sup>02</sup> ¼ T21am (27)

<sup>R</sup> <sup>¼</sup> <sup>ρ</sup><sup>2</sup> j j<sup>2</sup> (28)

<sup>∴</sup>ρ<sup>2</sup> <sup>¼</sup> <sup>ρ</sup><sup>1</sup> ð Þ<sup>2</sup> (31)

$$b\_0 = T\_{21} a\_m = \frac{a\_0 T\_{21}}{T\_{11}}\tag{20}$$

Then reflectivity R can be calculated by:

$$\mathbf{R} = |\rho\_1|^2 \tag{21}$$

Where,

$$
\rho\_1 = \frac{b\_0}{a\_0} = \frac{T\_{21}}{T\_{11}} \tag{22}
$$

#### 3.2 In case of two cascaded gratings

Figure 3 shows the connection between two FBGs where the output of the first on is connected to input of the second. In this case, the input optical signal for the second stage of the grating from (7) is:

$$b\_0 = a\_{02} = a\_{01} \frac{T\_{21}}{T\_{11}} \tag{23}$$

Transfer matrix can be written as:

$$
\begin{bmatrix} a\_{01} \frac{T\_{21}}{T\_{11}} \\ b\_{02} \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} a\_m \\ \mathbf{0} \end{bmatrix} \tag{24}
$$

Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> in two cascaded are:

$$a\_{01}\frac{T\_{21}}{T\_{11}} = T\_{11}a\_m \tag{25}$$

Theoretic Study of Cascaded Fiber Bragg Grating DOI: http://dx.doi.org/10.5772/intechopen.83020

The transfer matrix can be written as:

Fiber Optic Sensing - Principle, Measurement and Applications

Where ½ � T can be written as:

Transfer matrix can be written as:

3.1 In case of one grating

Where,

12

a0 b0 � �

a0 b0 � �

a0 b0 � �

Where Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> are:

Then reflectivity R can be calculated by:

3.2 In case of two cascaded gratings

second stage of the grating from (7) is:

Transfer matrix can be written as:

a<sup>01</sup> T<sup>21</sup> T<sup>11</sup> b<sup>02</sup>

2 4 ¼ ½ � T

<sup>¼</sup> <sup>T</sup><sup>11</sup> <sup>T</sup><sup>12</sup> T<sup>21</sup> T<sup>22</sup> � � am

In case of FBG reflection, There is no reflection at o/p at distance L so bm= 0

<sup>¼</sup> <sup>T</sup><sup>11</sup> <sup>T</sup><sup>12</sup> T<sup>21</sup> T<sup>22</sup> � � am

<sup>b</sup><sup>0</sup> <sup>¼</sup> <sup>T</sup>21am <sup>¼</sup> <sup>a</sup>0T<sup>21</sup>

<sup>ρ</sup><sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>0</sup> a0 <sup>¼</sup> <sup>T</sup><sup>21</sup> T<sup>11</sup>

Figure 3 shows the connection between two FBGs where the output of the first on is connected to input of the second. In this case, the input optical signal for the

> T<sup>21</sup> T<sup>11</sup>

> > 0 � �

¼ T11am (25)

b<sup>0</sup> ¼ a<sup>02</sup> ¼ a<sup>01</sup>

<sup>5</sup> <sup>¼</sup> <sup>T</sup><sup>11</sup> <sup>T</sup><sup>12</sup> T<sup>21</sup> T<sup>22</sup> � � am

3

Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> in two cascaded are:

a<sup>01</sup> T<sup>21</sup> T<sup>11</sup>

am bm � �

> bm � �

0 � �

T<sup>11</sup>

a<sup>0</sup> ¼ T11am (19)

<sup>R</sup> <sup>¼</sup> <sup>ρ</sup><sup>1</sup> j j<sup>2</sup> (21)

(16)

(17)

(18)

(20)

(22)

(23)

(24)

Figure 3. Connection between two cascaded FBG.

Then from Eq. (12) we get:

$$a\_{01} = \frac{{T\_{11}}^2}{{T\_{21}}} a\_m \tag{26}$$

$$b\_{02} = T\_{21} a\_m \tag{27}$$

We can calculate Reflectivity for the second grating by:

$$\mathbf{R} = |\rho\_2|^2 \tag{28}$$

Where,

$$\rho\_2 = \frac{b\_{02}}{a\_{01}} = \frac{T\_{21} \ast a\_m \ast T\_{21}}{T\_{11}^{\ast^2} \ast a\_m} \tag{29}$$

$$
\rho\_2 = \frac{T\_{21}^2}{T\_{11}^{-2}} \tag{30}
$$

$$\therefore \rho\_2 = \left(\rho\_1\right)^2\tag{31}$$

#### 3.3 In case of third cascaded gratings

The output of second grating b<sup>02</sup> is equal to input of third grating:

$$a\_{03} = b\_{02} = a\_{01} \frac{T\_{21}^2}{T\_{11}^2} \tag{32}$$

Transfer matrix can be written as:

$$
\begin{bmatrix} a\_{01} \frac{T\_{21}^{-2}}{T\_{11}^{-2}}\\ b\_{03} \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} a\_m \\ \mathbf{0} \end{bmatrix} \tag{33}
$$

Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> in two cascaded are:

$$a\_{01}\frac{T\_{21}}{T\_{11}}^2 = T\_{11}a\_m\tag{34}$$

$$a\_{01} = \frac{T\_{11}^{\ \ \ \ \ \beta}}{T\_{21}^{\ \ \ \ \ \ \beta}} a\_m \tag{35}$$

$$b\_{03} = T\_{21} a\_m \tag{36}$$

We can calculate reflectivity for the third grating by:

$$\mathbf{R} = \left| \rho\_3 \right|^2 \tag{37}$$

Where,

$$\rho\_3 = \frac{b\_{03}}{a\_{01}} = \frac{T\_{21} \ast a\_m \ast T\_{21}}{T\_{11}^3 \ast a\_m} \tag{38}$$

$$
\rho\_3 = \frac{T\_{21}^{\cdot 3}}{T\_{11}^{\cdot 3}} \tag{39}
$$

$$\vdots \rho\_3 = \left(\rho\_1\right)^3 \tag{40}$$

4.2 Hamming apodized cascaded fiber Bragg grating

Reflectivity spectrum for four stage uniform fiber Bragg grating.

Theoretic Study of Cascaded Fiber Bragg Grating DOI: http://dx.doi.org/10.5772/intechopen.83020

fiber Bragg grating as in Figure 5.

but reflectivity is decreased.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Reflectivity spectrum for four stage hamming apodized fiber Bragg grating.

Figure 5.

15

Refelectivity, R

0

Figure 4.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Refelectivity, R

We will simulate the spectral characteristics of the cascaded Hamming apodized

1.5495 1.5496 1.5497 1.5498 1.5499 1.55 1.5501 1.5502 1.5503 1.5504 1.5505

Wavelength, um

1.5495 1.5496 1.5497 1.5498 1.5499 1.55 1.5501 1.5502 1.5503 1.5504 1.5505

Wavelength, um

x 10-6

One FBG Two FBG Three FBG Four FBG

x 10-6

One FBG Two FBG Three FBG Four FBG

In this simulation the modulation index, d<sup>n</sup> ¼ 0:0003 and grating length L = 5 mm. From Figure 5 we noted that as the number of cascade of fiber Bragg grating is increased the bandwidth is decreased and the side lobes are also decreased

From (23) we can prove that the reflectivity of three cascaded FBG at the same parameters is equal to cubic reflectivity of the first one. At n stages of cascaded FBGs have the same parameters and each of them have a reflectivity R; the reflectivity of all n groups is equal to Rn .

## 4. Simulation results and discussion

In this section we will display the simulation results of the cascaded uniform and cascaded different apodized fiber Bragg grating to obtain narrow bandwidth without side lobes and maximum reflectivity.

#### 4.1 Cascaded uniform fiber Bragg grating

We will simulate the spectral characteristics of the cascaded uniform fiber Bragg grating as in Figure 4. In this simulation the modulation index,d<sup>n</sup> ¼ 0:0003 and grating length L = 5 mm. From Figure 4 we noted that as the number of cascade of fiber Bragg grating is increased the bandwidth is decreased and the side lobes are also decreased but reflectivity is decreased.

We have obtained from simulation result for one unit of fiber Bragg Grating the reflectivity, R = 99% and bandwidth =0.23 nm but the side lobes is high, for two cascaded units from fiber Bragg grating the reflectivity, R = 98% and bandwidth =0.18 nm but side lobes in this case is decreased on one unit of fiber Bragg grating, for three cascaded units from fiber Bragg grating the reflectivity, R = 97% and bandwidth =0.17 nm but side lobes is decreased and for four cascaded units from fiber Bragg grating the reflectivity, R = 96%, bandwidth =0.16 nm and approximately no side lobes. Then we concluded that Reflectivity, R = 96%, bandwidth = 0.16 nm and the minimum side lobes is achieved at the fourth unit of cascaded fiber Bragg grating.

Transfer matrix parameters T<sup>11</sup> and T<sup>21</sup> in two cascaded are:

Fiber Optic Sensing - Principle, Measurement and Applications

We can calculate reflectivity for the third grating by:

<sup>ρ</sup><sup>3</sup> <sup>¼</sup> <sup>b</sup><sup>03</sup> a<sup>01</sup>

.

Where,

tivity of all n groups is equal to Rn

4. Simulation results and discussion

out side lobes and maximum reflectivity.

4.1 Cascaded uniform fiber Bragg grating

also decreased but reflectivity is decreased.

fiber Bragg grating.

14

a<sup>01</sup> T<sup>21</sup> 2

T<sup>11</sup>

<sup>a</sup><sup>01</sup> <sup>¼</sup> <sup>T</sup><sup>11</sup>

R ¼ ρ<sup>3</sup> 

3

<sup>¼</sup> <sup>T</sup><sup>21</sup> <sup>∗</sup> am <sup>∗</sup> <sup>T</sup><sup>21</sup>

3

T<sup>11</sup> <sup>3</sup> ∗ am

T<sup>11</sup>

From (23) we can prove that the reflectivity of three cascaded FBG at the same parameters is equal to cubic reflectivity of the first one. At n stages of cascaded FBGs have the same parameters and each of them have a reflectivity R; the reflec-

In this section we will display the simulation results of the cascaded uniform and cascaded different apodized fiber Bragg grating to obtain narrow bandwidth with-

We will simulate the spectral characteristics of the cascaded uniform fiber Bragg grating as in Figure 4. In this simulation the modulation index,d<sup>n</sup> ¼ 0:0003 and grating length L = 5 mm. From Figure 4 we noted that as the number of cascade of fiber Bragg grating is increased the bandwidth is decreased and the side lobes are

We have obtained from simulation result for one unit of fiber Bragg Grating the reflectivity, R = 99% and bandwidth =0.23 nm but the side lobes is high, for two cascaded units from fiber Bragg grating the reflectivity, R = 98% and bandwidth =0.18 nm but side lobes in this case is decreased on one unit of fiber Bragg grating, for three cascaded units from fiber Bragg grating the reflectivity, R = 97% and bandwidth =0.17 nm but side lobes is decreased and for four cascaded units from fiber Bragg grating the reflectivity, R = 96%, bandwidth =0.16 nm and approximately no side lobes. Then we concluded that Reflectivity, R = 96%, bandwidth = 0.16 nm and the minimum side lobes is achieved at the fourth unit of cascaded

<sup>ρ</sup><sup>3</sup> <sup>¼</sup> <sup>T</sup><sup>21</sup>

2

T<sup>21</sup>

<sup>2</sup> ¼ T11am (34)

b<sup>03</sup> ¼ T21am (36)

<sup>2</sup> am (35)

<sup>2</sup> (37)

<sup>3</sup> (39)

<sup>∴</sup>ρ<sup>3</sup> <sup>¼</sup> <sup>ρ</sup><sup>1</sup> ð Þ<sup>3</sup> (40)

(38)

Figure 4. Reflectivity spectrum for four stage uniform fiber Bragg grating.
