2.1. Fundamentals

A FBG consists of a periodic modulation of the index of refraction of the core of an optical fiber [1,2]. Such a modulation can be obtained through holographic techniques [18], phase mask [19] or point-by-point techniques [20]. In the case of a modulation perpendicular to the axis of the fiber, each single core modulation reflects light backwards. By means of coupled-mode analysis, the Bragg condition can be obtained (Equation (1)) [21]:

where Λ is the grating period and neff is the effective index of refraction of the fiber core.

The peak wavelength mentioned above is sensitive to both elongation and temperature changes. On the basis of Equation (1), considering changes in temperature and deformation, the Bragg wavelength changes are given by [21];

From Equation (2), it is possible to obtain Equation (3), which provides the mathematical expression for the variation of the FBG peak wavelength with respect to temperature and strain:

where Δλ is the wavelength shift; λ0 is the base wavelength; ε is the strain; ΔT is the temperature change and αδis the thermo-optic coefficient of the optical fiber;

Finally, k=1−pe is the gage factor, and peis the strain-optic constant defined by Equation (5).

with p11 and p12 components of the strain-optic tensor and υ is the Poisson’s ratio [22] of the optical fiber.

It has to be noted that in Equation (3), the strain is defined by:

where εm is the mechanically induced strain and εT=αglassΔT is the temperature-induced strain, where αglass is the coefficient of thermal expansion per K of the optical fiber. In this case, the optical fiber is made of glass. Equation (6) highlights the cross-sensitivity between strain and temperature. The temperature that induces strain depends on the coefficient of expansion, which, in turn, depends on the temperature.

Combining Equations (3), (5) and (6), Equation (7) can be obtained, which describes the wavelength shift for a FBG due to temperature changes and strain:

In the particular case of an FBG sensor that is not subjected to any mechanical load, there are no mechanically induced deformations in the FBG, and the wavelength shift does not depend on mechanical strain, so that Equation (7) can be expressed as:

For practical purposes, Equation (8) is the mathematical representation of an FBG sensor working as a temperature sensor.

The literature refers to many techniques for performing temperature compensation for strain measurements [9,21,23]. In the present work, a method was used that utilizes two identical FBGs, inscribed in the same type of optical fiber and with the same grating period (Λ). One of the FBGs acts as a strain sensor; this sensor must be temperature-compensated. The strain sensor thus plays the role of the strain-measuring sensor. The other FBG sensor, the compensation sensor, must be located close to the first one. Therefore, it can be assumed that both sensors are subjected to the same temperature variation. The temperature compensation sensor must be isolated from any external mechanical load. If we consider Equation (7) to represent the strain measuring sensor and Equation (8) for the temperature compensation sensor, then by combining both equations, the mathematical expression of temperature-compensated strain(εm) can be obtained:

where ∆λm and ∆λc are the wavelength shifts of measuring and compensation FBGs, respectively; λ0m and λ0c are the base or reference wavelengths for measuring and compensation sensors and k is the calibration factor that is characteristic of the optical bench structure. The factor k is also strain and temperature dependent [22,24].