2.1. Principle of the Interrogation Method
Figure 1 shows the schematic diagram of the FBG interrogation method based on reflective-matched FBG scheme. A light emitted from amplified spontaneous emission (ASE) broadband source enters the sensing FBG (seFBG) through a circulator, and the reflection spectrum of seFBG then propagates back through the circulator again to the 50:50 coupler, where the reflected signal is divided into two parts of equal power. A part of the light is received by the photodiode of channel 1 on the acquisition circuit board, and the other part is filtered and reflected again by the reference FBG (reFBG). The reflection spectrum of the reFBG is received by the photodiode of channel 2 on the same acquisition circuit board.
Figure 1.
Schematic diagram of the FBG interrogation method based on a reflective-matched FBG scheme.
Figure 1.
Schematic diagram of the FBG interrogation method based on a reflective-matched FBG scheme.
The proposed FBG interrogation method based on a reflective-matched FBG sensing interrogation scheme demodulates the central wavelength shift of the seFBG on the basis of the ratio of the overlapping reflection spectrum power of the seFBG and the reFBG to the reflection spectrum power of the seFBG. As shown in
Figure 2, the central wavelength of the reFBG is
, and the initial central wavelength of the seFBG is λ
2. Meanwhile, the initial ratio of their overlapping power to the reflection spectrum power of the seFBG is
. Due to the effect of the external physical parameter, the central wavelength of the seFBG shifts to
and the power ratio becomes
. Moreover, calculating the power ratio can suppress the optical power fluctuation compared with detecting the optical power directly.
Figure 2.
Principle of the proposed FBG interrogation method: is the reflection spectrum power of seFBG; is the initial overlapping reflection spectrum power of the seFBG and the reFBG; is the overlapping reflection spectrum power of the seFBG and the reFBG after the influence of the external physical parameter.
Figure 2.
Principle of the proposed FBG interrogation method: is the reflection spectrum power of seFBG; is the initial overlapping reflection spectrum power of the seFBG and the reFBG; is the overlapping reflection spectrum power of the seFBG and the reFBG after the influence of the external physical parameter.
The FBG’s reflection spectrum can be expressed as [
12]:
where
is the central wavelength of the FBG;
is the full width at half maximum (FWHM) of the FBG’s reflection spectrum.
Hence, the power of the FBG’s reflection spectrum can be expressed as:
According to Equations (1) and (2) above, the overlapping power of the seFBG and reFBG’s reflection spectrum can be expressed as:
where
and
are the central wavelength of the seFBG and reFBG, respectively;
and
are the FWHM of spectra of the seFBG and reFBG, respectively. Therefore, the power ratio of the optical signals received by the acquisition circuit board can be expressed as:
In experiments, reFBGs and seFBGs fabricated under the same conditions are used, and the FWHM of the seFBG and reFBG are approximately equal,
. Hence Equation (4) can be simplified as:
According to the result of Equation (5), the relation between the power ratio and the central wavelength difference of the reFBG and seFBG can be established as shown in
Figure 3. It indicates that the FWHM of the reFBG has an effect on the sensitivity of the interrogation method, and the sensitivity can be improved by configuring reFBG with a narrow FWHM. Considering that the reFBG is unaffected by the measured physical parameter, the shift of the central wavelength difference between the reFBG and seFBG can be simplistically regarded as the central wavelength shift of the seFBG. Therefore, the relation between the power ratio and the central wavelength shift of the seFBG is nonlinear, but there is an approximately linear and most sensitive section. Then the existence of this section is proved, and the condition to optimize the interrogation method working in its approximately linear and most sensitive mode is established.
Figure 3.
Relation between the power ratio and the central wavelength difference between the reFBG and seFBG while the FWHM of the seFBG and reFBG are equal.
Figure 3.
Relation between the power ratio and the central wavelength difference between the reFBG and seFBG while the FWHM of the seFBG and reFBG are equal.
The first order differential of the power ratio
is the gradient of the power ratio induced by an external influence which characterizes the sensitivity of the interrogation method; the second order differential of power ratio
is the gradient of the power ratio’s variation, and the interrogation method works in its optimized mode when
reaches its maximum and
is zero. To obtain the relation between the central wavelength of the seFBG and reFBG, the following calculations are implemented. The first order differential of the power ratio is:
As shown in
Figure 4, the first order differential reaches its extrema twice, corresponding to two optimized modes of the interrogation method.
Figure 4.
Relation between the first order differential of the power ratio and central wavelength difference between the reFBG and seFBG.
Figure 4.
Relation between the first order differential of the power ratio and central wavelength difference between the reFBG and seFBG.
The second order differential of the power ratio is:
Let
, the relation between the central wavelengths of the seFBG and reFBG to optimize the method working in its optimized mode can be expressed as:
Therefore, the optimal initial matching condition to optimize the interrogation method working in an approximately linear and most sensitive mode is established when the initial central wavelengths of the seFBG and reFBG satisfy Equation (8). The central wavelength of the reFBG is known and the constant during the measurement process and the central wavelength variation of the seFBG can thus be demodulated.
It is shown in Equation (5) and
Figure 3 that the nonlinearity of interrogation method is determined by the measurable range. Assuming that the FWHM of the seFBG and reFBG are 0.2 nm, a measurable range of the central wavelength variation of the seFBG with a nonlinearity of less than 1% can reach ±43 pm, and a measuring range of the central wavelength variation of the seFBG with a nonlinearity of less than 5% can reach ±85 pm. It is sufficient for the application of the FBG sensors.
The noise level varies with the central wavelength of the reFBG and seFBG and it is necessary to investigate its influence and the performance extremes of the interrogation method. The noise of the ASE broadband source can be assumed to be a band-limited white noise with a noise power spectrum density (NPSD) of
. Meanwhile, the noise power within the received optical signal of the overlapping reflection spectrum, is
and it reaches
under the optimal initial matching condition as shown in
Figure 5a. The relation between
and the central wavelength difference with a nonlinearity of less than 1% is established in
Figure 5b. It can be seen that the noise level increases with the overlapping reflection spectrum of the reFBG and seFBG and it reaches the maximum at the edge of the linear working range with a minimum central wavelength difference of the reFBG and seFBG that means the interrogation method works at its extreme performance, so the interrogation method is tested under conditions close to the edge of the linear working range, including the minimum detectable optical power.
Figure 5.
Analysis of the noise power introduced by ASE broadband source: (a) Principle of the noise level analysis; (b) the relation between and central wavelength difference in the most sensitive section with the nonlinearity of 1%.
Figure 5.
Analysis of the noise power introduced by ASE broadband source: (a) Principle of the noise level analysis; (b) the relation between and central wavelength difference in the most sensitive section with the nonlinearity of 1%.
2.4. Principle of the Anti-temperature Perturbation of the Interrogation Method
The temperature-sensitivity of FBG sensors influences their accuracy and limits their application for precise measurements. The cross-sensitivity of strain and temperature is the key problem of the FBG sensors. Several approaches have been implemented to compensate or extract the temperature variation, such as FBG sensors with a sandglass-shape [
13], FBG sensors with an additive micro-scale bi-material coating [
14], FBG sensors using a Fabry-Perot laser diode and a dual-stage FBG optical demultiplexer [
15]. However, the proposed interrogation method could reduce the temperature sensitivity of the FBGs without any extra approaches by employing dual mutually referenced FBGs with a same thermo-optic coefficient and thermal expansion in a uniform environment.
In conventional interrogation methods without a reference FBG, the central wavelength variation of the seFBG
for a temperature change
can be written as:
where,
is the thermal expansion coefficient of the fiber (0.55 ×
/°C for a germanium doped silica-core fiber);
is the thermos-optic coefficient (8.6×
/°C for a germanium doped silica-core fiber).
The reFBG adjusted by the mechanical adjustment in the proposed interrogation method compensates the drift caused by the direct effect of temperature variation during the measurement. Meanwhile, the error resulting from the axial strain changes as a result of the different thermal expansion of the silica-core fiber and the mechanical adjustment was introduced. Consequently, the central wavelength variation of the reFBG
due to the temperature change
can be written as:
where,
is the thermal expansion coefficient of the mechanical adjustment (23.6 ×
/°C for aluminium alloy);
L is the fiber length exposed between the two fixtures of the mechanical adjustment which is 3 cm long for the experiment. Considering that the central wavelength of the seFBG and reFBG are close, and the proposed interrogation method demodulates the central wavelength of seFBG according to the shift of the central wavelength difference between the reFBG and seFBG, the effect of temperature on the proposed interrogation method compared with the conventional interrogation method can be expressed as:
Therefore, the proposed interrogation method can effectively reduce the influence of the temperature drift.