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An FBG is sensitive to both strain (ε) and temperature (T) changes. In a simplified approximation, one can write:

Where λ is the FBG-wavelength; λ₀ the wavelength at an arbitrary reference strain ε₀ and a reference temperature T₀ ; S_ε  the strain sensitivity or gage factor; S_T  the temperature sensitivity; Δε = ε – ε₀  the strain change with respect to the reference strain and ΔT = T – T₀  the temperature shift with respect to the reference temperature.

The above formula holds for free FBGs and not for FBGs which are fixed to a structure, like for strain gages. In this case, formula (1) should be modified slightly due to the influence of the thermal expansion of the structure. If the structure has a linear expansion coefficient α_t , then the thermal expansion of the structure is given by α_t . ΔT.  If this expansion exceeds the thermal expansion of the fiber glass, which is α_f  ΔT , the structure exercises an extra strain on the fiber which is given by (α_t – α_f) ΔT. Notice that the difference of the thermal expansion coefficients has to be taken because only the excess expansion will be transferred into strain. This strain gives an additional wavelength shift corresponding to S_ε (α_t – α_f) ΔT. The modified formula thus is given by:

It needs to be noticed that a more accurate approximation of the DTG^® response is given by the following function:

Where λ  is the DTG^® wavelength; λ₀ the nominal wavelength i.e. the wavelength at zero strain and at a fixed reference temperatureT₀   ; S₁ and S₂ the temperature sensitivities and k  the strain sensitivity or gage factor. The strain ε is the mechanical strain that is experienced by the fiber. Due to the quadratic nature, the calibration parameters S₁ and S₂  are depending on the used reference temperature. For strain gauges supplied from FBGS,  the parameters S₁ and S₂  are determined for a reference temperature of 22.5°C.

This more complex formula is important to use when large temperature variations are expected and is standard implemented for strain gauges in the Illumisense software of FBGS.

Typical values for S1, S2 and k are:

S1 = 6.37E-06

S2 = 7.46E-09

k = 0.772

assuming T expressed in °C and ε expressed in strain.

The thermal expansion coefficient of te fiber α_f  is typical around 0.5 μstrain/°C

Notice: Parameters S1, S2 and  α_f do not take into account the influence of the coating.