**2. Operating principle**

The resonant wavelength (*λB*) of a Bragg grating depends on the effective refractive index at the fiber core (*neff* ) and the period of the interference pattern (Λ) [15],

*λ<sup>B</sup>* ¼ 2 ∙ *neff* ∙Λ (1)

In turn, these parameters are affected by temperature variations or mechanical deformations. From **Eq. (1)**, the deviation of the Bragg wavelength due to mechanical deformations Δ*l*, or temperature variations Δ*T* [16] are:

$$
\Delta\lambda\_B = 2\left(\Lambda\frac{\delta n\_{\rm eff}}{\delta l} + n\_{\rm eff}\frac{\delta\Lambda}{\delta l}\right)\Delta l + 2\left(\Lambda\frac{\delta n\_{\rm eff}}{\delta T} + n\_{\rm eff}\frac{\delta\Lambda}{\delta T}\right)\Delta T \tag{2}
$$

The first term of **Eq. (2)** represents the dependence of *λ<sup>B</sup>* as a function of Δ*l*, caused by variations in the grating period and the change in the effective refractive index. This term can be described by:

*Bragg Grating Tuning Techniques for Interferometry Applications DOI: http://dx.doi.org/10.5772/intechopen.106735*

$$
\Delta\lambda\_B = (\mathbf{1} - \rho\_\epsilon) \bullet \varepsilon\_{rel} \bullet \lambda\_B,\tag{3}
$$

where *εrel* is the relative elongation and *ρ<sup>e</sup>* is the effective elasto-optic coefficient. For a silica optical fiber, *ρe*≈0*:*22 [17].

The second term of expression **(2)** represents the effect of temperature on changing the Bragg wavelength. For a Bragg grating with maximum reflectivity at 1550 nm, the typical values of sensitivity to temperature and mechanical stress are, respectively, 13.25 pm/°C and 1.2 pm/με [16].

It is therefore predictable that any change in wavelength associated with external disturbances is the result of the sum of the effects of temperature and mechanical deformations. Therefore, the individual discrimination in the spectral response of the FBG for each source of disturbance needs a method of separating the measurements. Briefly, the existing methods are extrinsic or intrinsic and are reported in detail in [16]. In Section 4.1, a set of tuning experiments is described, based on stretching or compressing the fiber, and which take place in a laboratory in a controlled temperature environment. Under these conditions, the influence of temperature on the wavelength variation is very small compared to the variations induced by the mechanical compression or stretching methods.

#### **2.1 Mechanical linear tuning**

**Figure 1** illustrates the elements that make up a tuning system by mechanical tension on the fiber. The ends of the optical fiber are fixed on two supports, where at least one must be able to move along the axial axis of the fiber.

The relative elongation along the axial axis is defined by

$$
\varepsilon\_{rel} = \frac{\Delta \mathbf{z}}{L}, \mathbf{L} \neq \mathbf{0}, \tag{4}
$$

where Δ*z* is the displacement and *L* is the length of the fiber under the effect of a mechanical deformation [18]. Substituting this result into **Eq. (3)**, we get:

Δ*λ<sup>B</sup>* ¼ 1 � *ρ<sup>e</sup>* ð Þ∙ Δ*z <sup>L</sup>* <sup>∙</sup> *<sup>λ</sup>B*,*<sup>L</sup>* 6¼ <sup>0</sup> (5)

**Figure 1.** *Schematic of the structure of a tuning system for fiber optic Bragg networks.*

The result is an equation that relates the variation of the Bragg wavelength as a function of the normalized fiber length. The change in wavelength will be positive or negative if the Bragg grating undergoes stretching or compression, respectively. Considering that the displacement is much smaller than the length of the grating, **Eq. (5)** is approximately linear.

#### **2.2 Mechanical tuning by bending**

**Figure 2** shows the scheme of a tuning system, in which a force is applied to a flexible blade. Horizontal displacement Δ*z* causes the blade to bend in an arc with angle *θ*. The relationship between the angle arc *θ* and the displacement Δ*z* is given by:

$$
\Delta z = L \left[ 1 - \frac{\sin\left(\frac{\theta}{2}\right)}{\left(\frac{\theta}{2}\right)} \right], \tag{6}
$$

where *L* is the length of the blade in the initial state, with no force applied to it [19, 20].

For an optical fiber embedded in an elastic material at a distance *d* from the blade, the relative elongation is given by [19–21]:

$$
\varepsilon\_{rel} = \mp \frac{d \bullet \theta}{L}, L \neq 0 \tag{7}
$$

The negative sign corresponds to a compressive force when the blade is bent upwards, while a positive sign in the second term of **Eq. (7)** indicates a pulling force on the fiber, bending the blade downwards.

**Figure 2.** *Schematic of the structure of a tuning system based on bending a flexible blade.*

Conjugating **(3)**, **(6)**, and **(7)** into a single equation,

$$\frac{\Delta z}{L} = 1 - \operatorname{sinc} \left( \frac{\Delta \lambda\_B \bullet L}{2 \bullet \pi \bullet d \bullet (1 - \rho\_\epsilon) \bullet \lambda\_B} \right), L \neq 0 \tag{8}$$

The relationship between the spectral tuning *λ<sup>B</sup>* and the displacement is not linear, contrary to the result in the procedure in Section 4.1.

#### **2.3 Temperature tuning**

The second term of **Eq. (2)** provides an explicit relationship of the dependence of the Bragg wavelength on temperature variations and is expressed by:

$$
\Delta\lambda\_B = (a\_T + \xi\_T) \bullet \Delta T \bullet \lambda\_B,\tag{9}
$$

where *α<sup>T</sup>* is the thermal expansion coefficient and *ξ<sup>T</sup>* is the thermo-optical coefficient. In the case of silica, the mentioned constants have the following values: *α<sup>T</sup>* ¼ <sup>0</sup>*:*<sup>55</sup> � <sup>10</sup>�<sup>6</sup>*K*�<sup>1</sup> and *<sup>ξ</sup><sup>T</sup>* <sup>¼</sup> <sup>8</sup>*:*<sup>0</sup> � <sup>10</sup>�<sup>6</sup>*K*�<sup>1</sup> .

A possible tuning scheme is to place the Bragg grating over a thermoelectric module, also called a Peltier cell, and envelop the set-in thermal mass. The objective is to standardize the temperature in the network and increase the thermal conductivity between the optical fiber and the Peltier module [22, 23].

It is also possible to increase the thermal sensitivity of the optical fiber by fixing it on a metallic surface. If a zinc (Zn) sheet is used, it is possible to increase the sensitivity to 40.8 pm/K. By fixing two Peltier modules at the ends of the zinc sheet, a temperature gradient is created that causes a linear variation of the grating period, which is used to tune the Bragg wavelength, maintaining a constant group delay. For this, the temperature difference between the two ends is kept constant, while the temperature at both ends is increased or decreased by the same amount [24].
