2. Basic model and analysis

In the present section, the basic model, governing equations and the analysis of the fiber Bragg grating are investigated to obtain a maximum reflectivity and minimum bandwidth, we discuss more than one case of different models of fiber Bragg grating; we will discuss in this section two models of fiber Bragg grating:


#### 2.1 Uniform fiber Bragg grating

#### 2.1.1 Fiber Bragg grating structure

The basic structure of the uniform fiber Bragg grating is illustrated in Figure 1 [7, 9]. As shown in Figure 1, the refractive index of the core is modulated by a period of Λ. When light is transmitted through the fiber which contains a segment of FBG, part of the light will be reflected. The reflected light has a wavelength equals to the Bragg wavelength so that it is reflected back to the input while others are transmitted. The term uniform means that the grating period, Λ, and the refractive index modulation, δn, are constant over the length of the grating. A grating is a device that periodically modifies the phase or the intensity of a wave reflected on, or transmitted through it [10]. The equation relating the grating spatial periodicity and the Bragg resonance wavelength is given by λB= 2neffΛ. Where neff the effective mode is index and Λ is the grating period [11].

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

Figure 1. Basic structure of fiber Bragg grating.

high-efficiency long-range fiber connections, WDM is introduced. Scattering is a key factor limiting the design of long-distance optical links. Several techniques have achieved effective dispersion compensation (DC). The widely used technologies are DCF and broken glass panels (CFBG). Although DCF is a large-scale unit, CFBG is superior to many faces [5]. Due to its excellent multicast capabilities, the fiber Bragg grating (FBG) sensors are particularly attractive for applications where a large number of sensors are desirable such as industrial process control, fire detection systems, and temperature conversion of power, since sensor FBG occupies a narrowband bandwidth that is very narrow and can easily create a distributed sensor matrix by writing several FBG sensors on a single fiber at different locations [6]. The refractive index of the nucleus is permanently changed. Germanium doped silica fiber is used in the manufacture of FBG because it is sensitive which means that the refractive index of the nucleus changes through exposure to light. The amount of change depends on the intensity and duration of exposure. It also depends on optical fiber sensitivity, so the level of fission with germanium should be high for high reflectivity [7]. Fiber Bragg grating nuts are spectral filters based on the Bragg reflection principle. The light usually reflects the narrow wavelength and sends all other wavelengths. When light is spread by periodic rotation of regions of the upper and lower refractive index, it is partially reflected in each interface between those regions [8]. This chapter is organized as follows. After the introduction in Section 1, Section 2 presents a basic model and analysis. In Section 3, we present the proposed system. The simulation results are displayed and discussed in

Section 4. Finally we devoted to the main conclusions.

Fiber Optic Sensing - Principle, Measurement and Applications

In the present section, the basic model, governing equations and the analysis of the fiber Bragg grating are investigated to obtain a maximum

reflectivity and minimum bandwidth, we discuss more than one case of different models of fiber Bragg grating; we will discuss in this section two models of fiber

The basic structure of the uniform fiber Bragg grating is illustrated in Figure 1 [7, 9]. As shown in Figure 1, the refractive index of the core is modulated by a period of Λ. When light is transmitted through the fiber which contains a segment of FBG, part of the light will be reflected. The reflected light has a wavelength equals to the Bragg wavelength so that it is reflected back to the input while others are transmitted. The term uniform means that the grating period, Λ, and the refractive index modulation, δn, are constant over the length of the grating. A grating is a device that periodically modifies the phase or the intensity of a wave reflected on, or transmitted through it [10]. The equation relating the grating spatial periodicity and the Bragg resonance wavelength is given by λB= 2neffΛ. Where neff

the effective mode is index and Λ is the grating period [11].

2. Basic model and analysis

1. Uniform fiber Bragg grating

2. Apodized fiber Bragg grating

2.1 Uniform fiber Bragg grating

2.1.1 Fiber Bragg grating structure

Bragg grating:

8

### 2.1.2 Theory and principle of operation

The study of the spectral characteristics of the uniform fiber Bragg grating is carried out by solving the dual mode equations. Dual mode theory is an important tool for understanding the design of fiber dividers from fiber Bragg grating [7]. FBG can be considered as a weak wave structure so that the pair mode theory can be used to analyze light propagation in a weak waveguide structure such as FBG. Dual-mode equations that describe the propagation of light can be obtained in FBG using the couple mode theory. The theory of marital status was first introduced in the early 1950s to microwave devices and later applied to optical devices in early 1970 [11].

For maximum reflectivity [12]:

$$\text{R}\_{\text{peak}} = \tanh^2(\text{ kL}) \tag{1}$$

Reflective bandwidth, Δλ of uniform FBG is defined as wavelength bandwidth between the first zero reflective wavelength of both sides of peak reflection wavelength. It can be calculated by a general expression of the approximate bandwidth of the grating is:

$$
\Delta\lambda = \lambda\_{B\,s} \sqrt{\left(\frac{\Delta n\_{ac}}{2n\_{\text{eff}}}\right)^2 + \left(\frac{1}{N}\right)^2} \tag{2}
$$

Where λ<sup>B</sup> is the Bragg (center) wavelength, s is a parameter indicting the strength of the gratings (�1 for strong gratings and �0.5 for weak gratings), N is the number of grating planes, Δnac is the change in the refractive index and neff is the effective refractive index.

The forward propagated light is reflected at Bragg wavelength [13]:

$$
\lambda\_{\mathcal{B}} = \mathbf{2n} \Lambda \tag{3}
$$

Where λ<sup>B</sup> is the Bragg wavelength (wavelength of the reflection peak amplitude), n is the effective refractive index of optical mode propagating along the fiber and Ʌ is the period of FBG structure. For a uniform Bragg grating formed within the core of an optical fiber with an average refractive index no. The index of the refractive profile can be expressed as [6, 14]:

Fiber Optic Sensing - Principle, Measurement and Applications

$$n(\mathbf{z}) = n\_0 + \Delta \mathbf{n} \cos \left(\frac{2\pi \mathbf{z}}{\Lambda}\right) \tag{4}$$

4.Nuttall Function:

where, 0 ≤z≤ L

5. Sinc Function:

3. The proposed system

into sections.

Figure 2.

11

Fiber Bragg grating sections.

A zð Þ¼ <sup>0</sup>:<sup>3635819</sup> � <sup>0</sup>:48917755 2<sup>π</sup> <sup>z</sup>

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

6.Proposed ( cos 8) Function [8]:

a0: is the incident optical signal. b0: is the reflected optical signal. am: is the output optical signal.

m: is number of sections.

L

A zð Þ¼ sinc 2π

A zð Þ¼ cos

the space between reflected planes of each section where:

bm: is the reflected optical signal at output of grating.

a0 b0 � �

Replacing m matrix by whole matrix ½ � T where:

� � <sup>þ</sup> <sup>0</sup>:1365996 4<sup>π</sup> <sup>z</sup>

<sup>z</sup> � <sup>L</sup> 2 L

2z <sup>L</sup> � <sup>1</sup> � � � � <sup>8</sup>

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

Each section is shown in Figure 2 where T is the length of each section and ᴧ is

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

½ �¼ <sup>T</sup> <sup>Y</sup><sup>m</sup>

j¼1

<sup>¼</sup> <sup>T</sup><sup>1</sup> � � <sup>T</sup><sup>2</sup> � � <sup>T</sup><sup>3</sup> � � <sup>T</sup><sup>4</sup> � �… <sup>T</sup><sup>m</sup> ½ � am

bm

<sup>T</sup><sup>m</sup> ½ � (15)

� � (14)

L

� �, <sup>0</sup>≤z≤<sup>L</sup> (12)

� � � <sup>0</sup>:0106411 6<sup>π</sup> <sup>z</sup>

, 0≤z≤L (13)

L � �,

(11)

Where Δn is the amplitude of the induced refractive index perturbation, ᴧ is the nominal grating period and z is the distance along the fiber longitudinal axis. Using coupled-mode theory the reflectivity of a grating with constant modulation amplitude and period is given by the following expression [6, 8, 12]:

$$R(l,\lambda) = \frac{k^2 \sinh^2(sl)}{\Delta \beta^2 \sinh^2(sl) + s^2 \cosh^2(sl)}\tag{5}$$

Where R(l,λ) is the reflectivity, which is a function of the grating length L and wavelength λ, Δβ ¼ β–π=Λ is the detuning wave vector, β ¼ 2πn0=λ is the propagation constant and s <sup>2</sup> <sup>¼</sup> <sup>k</sup>2\_Δβ<sup>2</sup> and <sup>ĸ</sup> <sup>¼</sup> <sup>π</sup>Δ<sup>n</sup> <sup>λ</sup> Mpower.

Mpower is ac coupling coefficient, Mpower is the fraction of the fiber mode power contained by the fiber core.

In the case where the grating is uniformly written through the core, Mpower can be approximated by, 1 � <sup>V</sup><sup>2</sup> , where <sup>V</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>λ</sup> <sup>a</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 <sup>c</sup>0�n<sup>2</sup> cl <sup>p</sup> is the normalized frequency of the fiber, a is the core radius, nCO and nCL are the core and cladding indices, respectively. At the center wavelength of the Bragg grating the wave vector detuning is Δβ = 0, therefore the expression for the reflectivity becomes:

$$R(l,\lambda) = \tanh^2(kl) \tag{6}$$

The reflectivity increases as the induced index of refraction change gets larger. Similarly, as the length of the grating increases, so does the resultant reflectivity.

#### 2.2 Apodized fiber Bragg grating

In the present section, we cast the basic model and the governing equation to apodized fiber Bragg grating. Apodized FBG offer significant improvement in side lobe suppression but on the expense of reducing the peak reflectivity. Apodized gratings have variations along the fiber in the refractive index modulation envelope (Δnαc) with constant grating period and constant DC refractive index function. The index of the refractive profile of Apodized can be expressed as [6]:

$$\mathbf{n}(\mathbf{z}) = n\_{c0} + \Delta n\_0 \mathbf{A}(\mathbf{z}) n\_d(\mathbf{z}) \tag{7}$$

Where nc<sup>0</sup> is the core refractive index, Δn<sup>0</sup> is the maximum index variation, n(z) is the index variation function and (z) is the Apodization function. Apodization profiles are [6, 13, 15]:

1. Uniform:

$$\mathbf{A}(\mathbf{z}) = \mathbf{1}, \mathbf{0} \le \mathbf{z} \le \mathbf{L} \tag{8}$$

2. Hamming Function

$$\mathbf{A(z)} = \mathbf{0.54} - \mathbf{0.46} \cos\left(\frac{2\pi z}{L}\right), \text{ where } \mathbf{0} \le z \le L \tag{9}$$

#### 3. Barthan Function:

$$\mathbf{A(z)} = \mathbf{0.62} - \mathbf{0.48} \left| \frac{\mathbf{z}}{L} - \mathbf{0.5} \right| + \mathbf{0.38} \cos \left( \frac{\mathbf{z}}{L} - \mathbf{0.5} \right), \text{where, } \mathbf{0} \le \mathbf{z} \le L \tag{10}$$

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

4.Nuttall Function:

n zð Þ¼ n<sup>0</sup> þ Δncos

tude and period is given by the following expression [6, 8, 12]:

Fiber Optic Sensing - Principle, Measurement and Applications

<sup>2</sup> <sup>¼</sup> <sup>k</sup>2\_Δβ<sup>2</sup> and <sup>ĸ</sup> <sup>¼</sup> <sup>π</sup>Δ<sup>n</sup>

tion constant and s

power contained by the fiber core.

2.2 Apodized fiber Bragg grating

profiles are [6, 13, 15]:

2. Hamming Function

3. Barthan Function:

A zð Þ¼ <sup>0</sup>:<sup>62</sup> � <sup>0</sup>:<sup>48</sup> <sup>z</sup>

1. Uniform:

10

be approximated by, 1 � <sup>V</sup><sup>2</sup>

R lð Þ¼ ; <sup>λ</sup> <sup>k</sup><sup>2</sup>

Δβ<sup>2</sup>

, where <sup>V</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

Where Δn is the amplitude of the induced refractive index perturbation, ᴧ is the nominal grating period and z is the distance along the fiber longitudinal axis. Using coupled-mode theory the reflectivity of a grating with constant modulation ampli-

Where R(l,λ) is the reflectivity, which is a function of the grating length L and wavelength λ, Δβ ¼ β–π=Λ is the detuning wave vector, β ¼ 2πn0=λ is the propaga-

Mpower is ac coupling coefficient, Mpower is the fraction of the fiber mode

of the fiber, a is the core radius, nCO and nCL are the core and cladding indices, respectively. At the center wavelength of the Bragg grating the wave vector detuning is Δβ = 0, therefore the expression for the reflectivity becomes:

R lð Þ¼ ; <sup>λ</sup> tanh<sup>2</sup>

The reflectivity increases as the induced index of refraction change gets larger. Similarly, as the length of the grating increases, so does the resultant reflectivity.

In the present section, we cast the basic model and the governing equation to apodized fiber Bragg grating. Apodized FBG offer significant improvement in side lobe suppression but on the expense of reducing the peak reflectivity. Apodized gratings have variations along the fiber in the refractive index modulation envelope (Δnαc) with constant grating period and constant DC refractive index function.

Where nc<sup>0</sup> is the core refractive index, Δn<sup>0</sup> is the maximum index variation, n(z) is the index variation function and (z) is the Apodization function. Apodization

> L � �

> > <sup>L</sup> � <sup>0</sup>:<sup>5</sup> � �

� <sup>þ</sup> <sup>0</sup>:38 cos <sup>z</sup>

n zð Þ¼ nc<sup>0</sup> þ Δn0A zð Þndð Þz (7)

A zð Þ¼ 1, 0≤z≤L (8)

, where 0≤ z≤L (9)

, where, 0≤ z≤L (10)

The index of the refractive profile of Apodized can be expressed as [6]:

A zð Þ¼ <sup>0</sup>:<sup>54</sup> � <sup>0</sup>:46 cos <sup>2</sup>π<sup>z</sup>

� �

<sup>L</sup> � <sup>0</sup>:<sup>5</sup> � � �

<sup>λ</sup> Mpower.

In the case where the grating is uniformly written through the core, Mpower can

<sup>λ</sup> <sup>a</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 <sup>c</sup>0�n<sup>2</sup> cl

sinh <sup>2</sup>ð Þ sl

sinh <sup>2</sup>ð Þþ sl <sup>s</sup>2cosh <sup>2</sup>ð Þ sl (5)

p is the normalized frequency

ð Þ kl (6)

2πz Λ � �

(4)

$$\mathbf{A(z)} = \mathbf{0.3635819} - \mathbf{0.48917755} \left( 2\pi \frac{z}{L} \right) + \mathbf{0.1365996} \left( 4\pi \frac{z}{L} \right) - \mathbf{0.0106411} \left( 6\pi \frac{z}{L} \right), \tag{11}$$

where, 0 ≤z≤ L

5. Sinc Function:

$$\mathbf{A}(\mathbf{z}) = \text{sinc}\left(2\pi \frac{z - \frac{L}{2}}{L}\right), \mathbf{0} \le z \le L \tag{12}$$

6.Proposed ( cos 8) Function [8]:

$$\mathbf{A}(\mathbf{z}) = \left(\cos\left(\frac{2\mathbf{z}}{L} - \mathbf{1}\right)\right)^8, \mathbf{0} \le \mathbf{z} \le L \tag{13}$$
