**2.1 Fiber Bragg grating sensors**

The first FBG, fabricated using a visible laser propagating along with the fiber core, was proposed by Ken Hill in 1978 [64]. OFS based on FBGs has been widely applied in the measurement of physical, chemical, biomedical, and electrical parameters, especially for structural health monitoring in civil infrastructures, aerospace, energy, and healthy areas [65].

Classically, an FBG sensor consists of a small segment of a single-mode optical fiber (with a length of a few millimeters) with a photoinduced periodically modulated index of refraction in the fiber core. The FBG resonant wavelength is related to the effective refractive index of the core mode (*neff*) and the grating period (*Λ*). When the grating is illuminated with a broadband optical source, the reflected power spectrum presents a peak (with a full width at half maximum of a few nanometers), which is produced by the interference of light with the planes of the grating and can be described through Eq. (1) [66].

#### **Figure 3.**

*Different types of OFS used to track critical parameters in LIBs.*

*Tracking Li-Ion Batteries Using Fiber Optic Sensors DOI: http://dx.doi.org/10.5772/intechopen.105548*

$$
\lambda\_B = \mathbf{2} n\_{\rm eff} \Lambda \tag{1}
$$

where *λ<sup>B</sup>* is the so-called Bragg wavelength. When the fiber is exposed to external variations of a given measurand (such as strain, temperature, stress, or pressure, among others), both *neff* and *Λ* can be changed, causing an alteration in the Bragg wavelength, as shown in **Figure 4** [66]. In addition to the common advantages of fiber sensors, this wavelength interrogation method offers robustness to noise and power oscillations and also enables wavelength division-multiplexing, by recording numerous FBGs with diverse grating periods in the same optical fiber (see **Figure 5**). This permits the monitorization of different spots in one structure/surface with only one sensor line, decreasing in this way, the total interrogation costs. The FBG sensitivity toward a given parameter is obtained simply by subjecting the sensor to predetermined and controlled variations of such parameters and measuring the Bragg wavelength for each step.

In the case of a linear response, the sensitivity (*k*) is given by the slope of the linear fit obtained from the experimental data. The effects of temperature are accounted for in the Bragg wavelength shift by differentiating Eq. (2),

#### **Figure 4.**

*Scheme and operation mechanism of an FBG sensor to external strain and temperature perturbations.*

#### **Figure 5.**

*A) FBGs network inscribed in the same optical fiber. B) Optical spectrum of a network of 10 FBGs inscribed in the same fiber, where different wavelength peaks can be observed.*

$$
\Delta\lambda = 2\lambda\_B \left( \frac{\mathbf{1}}{n\_{\rm eff}} \frac{\partial n\_{\rm eff}}{\partial T} + \frac{\mathbf{1}}{\Lambda} \frac{\partial \Lambda}{\partial T} \right)
\Delta T = \lambda\_B (a + \xi)\Delta T = k\_T \Delta T,\tag{2}
$$

where *α* and *ξ* are the thermal expansion and thermo-optic coefficient of the optical fiber material, respectively. On the other hand, if the fiber is subjected to strain variations, its response can be determined by differentiating Eq. (3),

$$
\Delta\lambda = \lambda\_B \left(\frac{1}{n\_{\text{eff}}} \frac{\partial n\_{\text{eff}}}{\partial \varepsilon} + \frac{1}{\Lambda} \frac{\partial \Lambda}{\partial \varepsilon}\right) \Delta \varepsilon = \lambda\_B (1 - p\_\varepsilon) \Delta \varepsilon = k\_\varepsilon \Delta \varepsilon,\tag{3}
$$

where *pe* is the photoelastic constant of the fiber (�0.22) and *Δε* is the applied strain. The strain variations can be determined using the equation *Δε = ΔL/L* where *ΔL* is the length variation and *L* is the fiber length over which strain is applied. On a single measurement of the Bragg wavelength shift, it is not possible to decouple the effect of variations in strain and temperature (for example). Normally, a reference is used for temperature measurement, by using another fiber strain-free or other FBGs and sensing heads that have different strain and temperature sensitivities. Different strategies are being used in the literature, such as FBGs recorded in different fiber thicknesses, FBGs recorded in special microstructured fibers, and FBGs cascaded with other optical fiber sensing configurations (FPI, MZI) [66–68]. The discrimination of both variables is performed, through the matrixial method by using all the sensitivities of both sensors to each variable. In this way, a sensitivity matrix (4) for simultaneous measurement of strain and temperature can be derived as:

$$
\begin{bmatrix}
\Delta \varepsilon \\
\Delta T
\end{bmatrix} = \frac{1}{D} \begin{bmatrix}
k\_{FBG1\_r} & -k\_{FBG2\_r}
\end{bmatrix} \begin{bmatrix}
\Delta \lambda\_{FBG1} \\
\Delta \lambda\_{FBG2}
\end{bmatrix},
\tag{4}
$$

where D = *K*FBG1<sup>ε</sup> x *K*FBG2T – *K*FBG1T x *K*FBG2<sup>ε</sup> is the determinant of the coefficient matrix, which must be nonzero for possible simultaneous measurement.

The Bragg gratings can be inscribed in an optical fiber core through side exposure; two main types of techniques can be implemented: interferometric and noninterferometric techniques. In the noninterferometric technique, the phase mask method is one of the most commonly used (see **Figure 6A**). Generally, it is associated with longer laser pulses (near the nanoseconds) in the ultraviolet (UV) region. The phase mask consists of a diffraction grating shaped by small depressions in a silica substrate, separated by a predefined period (phase mask pitch, *ΛPM*), which will define the modulation pattern linked to the resonant Bragg wavelength of the fabricated FBG (see **Figure 6B**).

Depending on the incident angle of the laser beam on the phase mask surface, different diffraction orders will be predominantly transmitted: the pairs +1/0 or + 1/�1. To attain different wavelength peaks, phase masks with different *ΛPM* can be used. Typically, when using a UV laser, a better inscription efficiency is expected for doped or hydrogenated optical fibers [67].

#### **2.2 Tilted FBG sensors**

Compared to the normal FBG sensors, TFBG sensors have a special configuration, which leads to enhanced sensitivity to the surrounding refractive index (SRI). Thus,

#### **Figure 6.**

*A) Schematic representation of the phase mask inscription method using a pulsed laser. +1 and* �*1 indicate the laser beam diffraction orders used to inscribe the Bragg grating in the optical fiber core. B) Typical phase masks used on FBG sensors recording (from Ibsen®).*

#### **Figure 7.**

*Schematic diagram of a TFBG where Λ<sup>g</sup> is the grating period and* θ *is the tilt angle (adapted from [68]).*

this type of sensor has been employed in many parameters sensing, such as temperature, liquid level, RI, and relative humidity, in biochemical research. TFBGs are short-period gratings in which the modulation of the RI is purposely tilted concerning the longitudinal axis of the fiber, to improve the light coupling between the forward-propagating core mode and the backward-propagating cladding modes (see **Figure 7**) [68].

The wavelength of the coupled i-th cladding mode *λcl(i)* can be expressed as (Eq. 5):

$$
\lambda\_{cla(i)} = \left( n\_{\sharp \overline{f}}^{cor} + n\_{\sharp \overline{f} \overline{f}(i)}^{cla} \right) \Lambda = \frac{\left( n\_{\sharp \overline{f}}^{cor} + n\_{\sharp \overline{f} \overline{f}(i)}^{cla} \right) \Lambda\_{\overline{g}}}{\cos \Theta} \tag{5}
$$

where *ncore eff* and *ncla eff i*ð Þare the effective RIs of the fiber core and i-th cladding mode, respectively. *Λ* and *Λ<sup>g</sup>* are the grating periods along with the fiber longitudinal axis and perpendicular to the grating plane, respectively. The excited cladding modes are limited in the fiber cladding by total internal reflection on the cladding-surrounding medium interface. Each of the cladding modes propagates with a corresponding effective RI value. When the RI value of the surrounding medium spreads the one of a specific cladding mode, the cladding mode will be coupled out of the fiber cladding, resulting in a variation in the grating transmission spectrum. Therefore, the shifts of the SRI can be quantitatively tracked by detecting the variations in the grating transmission spectrum of the TFBG [69]. These TFBGs can also be fabricated in line with other normal FBGs to simultaneously decouple different parameters, such as RI and temperature, as they have different sensitivities.

### **2.3 Interferometric sensors**

Since the first study, published in 1897 by Charles Fabry and Alfred Perot, about the FPI principle [70], the OFS based on this methodology was used in numerous applications, such as biological, chemical, and various physical parameters, including temperature, strain, pressure, and RI [63, 71]. Literature shows that these sensors are used also like candidates to improve the discrimination of strain and temperature in batteries [44]. An FPI sensor is performed by considering two parallel reflecting surfaces divided by a certain physical length of the cavity (*L*). FPI sensors can be classified as extrinsic or intrinsic, as can be seen in **Figure 8a** and **b**, respectively. The intrinsic FPI sensor has reflecting components inside the fiber itself [70]. In the extrinsic FPI, the air cavity is designed by an auxiliary structure. Due to the optical phase difference between two reflected signals, the reflection spectrum of an FPI can be defined as the wavelength-dependent intensity modulation of the incident signal spectrum. The phase difference of the FPI (*δFP*) can be given as (Eq. 6):

$$
\delta\_{\rm FP} = \frac{4\pi nL}{\lambda} \tag{6}
$$

where *n* is the RI of the cavity material, and *λ* is the wavelength of the output signal. Consequently, an external perturbation to the FPI sensor (such as strain, temperature, or IR), will promote a length variation of the FPI cavity, resulting in wavelength changes. By tracking the wavelength shift of the spectrum, and after an experimental pre-calibration to each specific parameter, to determine their sensitivities, a linear conversion of the data signals of the measured parameter values can be performed by analyzing the spectrum produced.

Another type of interferometer is the MZI sensor. They are usually applied for sensing parameters such as temperature, strain, curvature, and RI, among others [71], due to their advantages of high RI sensitivity and flexible designs, as shown in **Figure 9**.

An MZI is designed due to the formation of an optical path difference between the fundamental core mode and the higher-order cladding modes in optical fiber. Subsequently, in the interference spectrum, dips or peaks can appear [72]. These peaks or

#### **Figure 8.**

*a) Extrinsic FP sensor performed by forming an external air cavity, and b) intrinsic FP sensor formed by two reflecting components, R1 and R2 (adapted from [70]).*

#### **Figure 9.**

*Different types of MZIs configurations; using: (A) a pair of LPGs, (B) core misalliance, (C) air-hole collapsing of PCF, (D) MMF section, (E) small SMF core, and (F) tapering fiber regions (adapted form [71]).*

dips values are used as tracking signals because they change with external perturbations (such as temperature, strain, pressure, and RI). For simplicity and spectral data analysis, only the core mode (*I*1) and one cladding mode (*I*2) are considered. The transmitted interference signal, *I,* can be expressed as (Eq. 7) [73]:

$$I = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos\left(\phi\right) \tag{7}$$

where *<sup>ϕ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>n</sup>core eff* �*ncla eff* � �*<sup>L</sup> <sup>λ</sup>* , is the phase difference, being *ncore eff* and *ncla eff* the effective RIs of the fiber core and cladding mode, respectively. The *λ* is the input optical wavelength in vacuum, *L* is the interferometric MZI length, and *ϕ = 0* is the initial interference phase. When *I1 = I2*, the fringe visibility reaches its maximum value. From Eq. (7), when *<sup>ϕ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>n</sup>core eff* �*ncla eff* � �*<sup>L</sup> <sup>λ</sup><sup>m</sup>* ¼ ð Þ 2*m* þ 1 *π*, the output intensity dips will appear, where *m* is an integer. Specifically, the phase difference between two adjacent minimum intensity dips is <sup>2</sup>*π*Δ*neff <sup>L</sup> <sup>λ</sup>m*þ<sup>1</sup> � <sup>2</sup>*π*Δ*neff <sup>L</sup> <sup>λ</sup><sup>m</sup>* ¼ 2*π*. Therefore, the difference between two adjacent interference wavelengths, as well known as the free spectral range (FSR) can be calculated as *FSR* <sup>¼</sup> *<sup>λ</sup><sup>m</sup>* � *<sup>λ</sup><sup>m</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>mλm*þ<sup>1</sup> <sup>Δ</sup>*neff <sup>L</sup>* , and the theoretical cavity length is *L* ¼ *λmλm*þ<sup>1</sup> <sup>Δ</sup>*neff*ð Þ *<sup>λ</sup>m*�*λm*þ<sup>1</sup> . Using these formulas, the theoretical values of MZI length can be compared with experimental results to reduce errors, and also, it can be seen that when the *L* increases or the *Δneff* increases, the *FSR* decreases. Note that when *Δneff* changes, it indicates that RI of the external environment changes, promoted by the external parameters, while RI of the optical fiber core is constant. When the external parameters, such as temperature or RI, around the MZI is different, which will lead to the changes of *Δneff*, the wavelength of interference dip will also shift. So, the surrounding environment can be analyzed through spectra after a pre-calibration process to each external parameter to which the optical sensor will be submitted.

#### **2.4 Evanescent wave sensors**

Other types of OFS, which are being used to track specific parameters in Li-ion batteries, were the evanescent wave sensors. This type of sensor is created on the interaction of the evanescent field in the cladding with the fiber surroundings, resulting in fluctuations of the transmitted spectrum. It follows that they hold the capability of translating a discrepancy of the target analyte into optical signals so that they are widely applied to chemical and biosensing [74]. As shown in **Figure 10**, the evanescent field *Eew*(d) decays exponentially as (Eq. 8):

$$E\_{ew}(d) = E\_0 \exp\left(\frac{-d}{d\_p}\right) \tag{8}$$

where *E0* is the magnitude of the field at the fiber core-cladding interface, *d* is the distance from the core-cladding interface, and *dp* is the distance where the evanescent field decreases to *E0/e* and is described as the penetration depth which is given by (Eq. 9):

$$d\_p = \frac{\lambda}{2\pi\sqrt{n\_{core}^2\sin^2\theta - n\_{data}^2}}\tag{9}$$

**Figure 10.**

*Design of a fiber evanescent wave spectroscopy sensor with the standing wave pattern and exponentially decaying evanescent wave.*

where *λ* is the wavelength of the incident light, *θ* is the angle of incidence at the fiber core-cladding interface, and *ncore* and *ncla* are the RIs of the fiber core and cladding, respectively.

This optical fiber methodology of sensing can also be modified by depositing specific film materials (metal-dielectrics) in the fiber cladding surface and interacting between them. In this way, the surface plasmon resonance (SPR) technique can be used. The SPR is a collective oscillation of free electrons excited by light at the metaldielectric interface. The electromagnetic field decays exponentially into both metal and dielectric, the propagation constant of SPR can be given as (Eq. 10):

$$k\_{sp} = \frac{\alpha}{c} \sqrt{\frac{\varepsilon\_m \varepsilon\_d}{\varepsilon\_m + \varepsilon\_d}}\tag{10}$$

where *ω* is the angular frequency of the incident light, *c* is the speed of light in space, and *ε<sup>m</sup>* and *ε<sup>d</sup>* are the dielectric constants of the metal and dielectric, respectively. The propagation constant of the evanescent wave parallel to the planar metal film surface can be expressed as *kew* <sup>¼</sup> *<sup>ω</sup> c* ffiffiffiffiffiffiffiffiffi *εfiber* p *sinθ*, where *εfiber* is the dielectric constant of the fiber. The SPR occurs when both propagation constants are equal, it exhibits high sensitivity to even slight oscillations in the dielectric constant of the dielectric material. Therefore, SPR-based sensors can successfully track diverse variables due to the location of the resonance shifts with the varying the RI of the nearby dielectric.
