# Portable Optical Fiber Bragg Grating Sensor for Monitoring Traffic Density
## 2 Methods  
### Fiber Bragg Grating  
Fiber Bragg grating is formed in the optical fiber core by a periodic change of the refractive index. The structure of a uniform Bragg grating in which there are periodically changing layers of the refractive index of core \(n_{1}\) with a higher refractive index index \(n_{3}\) is shown in Figure 1. The uniform Bragg grating is characterized by a constant size of the induced refractive index \(\delta_{n}=n_{3}-n_{1}\) and a constant period of of changes of refractive index.  
When a broadband spectrum of light passes through the structure of the Bragg grating, selected wavelengths in a narrow part of the spectrum are reflected, while other wavelengths are transmitted. The reflected part of the spectrum is called Bragg wavelength and is expressed by Equation (1):  
\[\lambda_{B}=2n_{eff}\Lambda, \tag{1}\]  
where \(n_{eff}\) is the effective refractive index defined by the structure of the Bragg grating and \(\Lambda\) is the distance between the periodical changes to the refractive index in the core of the optical fiber [30].  
The Bragg wavelength is the geometric and optical properties that change under the influence of mechanical and thermal stresses. The relation of the Bragg wavelength change, the relative deformation and temperature is expressed by Equation (2):  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=k\epsilon+\left(\alpha_{\Delta}+\alpha_ {n}\right)\Delta T, \tag{2}\]  
where \(\Delta\lambda_{B}\) is Bragg wavelength shift, \(k\) is deformation coefficient, \(\epsilon\) is deformation, \(\alpha_{\Delta}\) is the coefficient of thermal expansion, \(\alpha_{n}\) is the thermo-optic coefficient and \(\Delta T\) is change of temperature.  
The relation in Equation (2) suggests that thermal and deformation sensitivity changes with the absolute value of the Bragg wavelength. With regard to the values of coefficients stated in the  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Fiber Optic Technology** & **Sensor Location** & **Sensor Fixing Method** & **Measuring Range** & **Measured Quantity** & **Reference** \\ \hline
**Bragg grating sensors** & installed into or onto the road’s surface & adhesion, mechanical grips and other protection & almost physical contact with the sensed body & axial, radial deformation & [24; 25; 26; 27; 28; 29] \\ \hline
**Distributed systems** & installed along the road or into the road’s surface & adhesion, mechanical grips and other protection & almost physical contact with the sensed body & axial deformation, acoustic and vibration response & [22; 23] \\ \hline
**Interferometric systems** & installed into or onto the road’s surface, along the road & adhesion, mechanical grips and other protection & units of meters from the object & acoustic and vibration response, detection of reflection & [15; 16; 17; 18; 19; 20; 21] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the use of basic types of fiber optic sensors in road transport.  
Figure 1: Structure of uniform Bragg grating.  
above relations, the standardized deformation coefficient at a constant temperature is expressed by Equation (3):  
\[\frac{1}{\lambda_{B}}\frac{\Delta\lambda_{B}}{\Delta\epsilon}=0.78\cdot 10^{-6} \quad\mu\text{strain}^{-1}, \tag{3}\]  
and a standardized deformation coefficient at a constant deformation is expressed by Equation (4):  
\[\frac{1}{\lambda_{B}}\frac{\Delta\lambda_{B}}{\Delta T}=6.678\cdot 10^{-6} \quad^{\circ}C^{-1}. \tag{4}\]  
The uniform Bragg grating has a deformation sensitivity of 1.2 pm/\(\mu\)strain and a temperature sensitivity of 10.3 pm/\({}^{\circ}\)C at a wavelength of 1550 nm [31].