The Circular Birefringence-Insensitive FBG Sensor for Weak Pressure

The influence of the circular birefringence on the measurement performance was analyzed based on the Polarization Properties of FBG in this paper. Due to the circular birefringence, the linear relationship between the max of PDL and pressure has been broken down. To estimate the cross sensitivity, a new parameter named relative peak of PDL (RPPDL) is proposed. Under different circular birefringence, the same pressure sensitivity of sensor has been achieved. The theoretical analysis and experiment results prove that transverse strain sensor of FBG is insensitive to the circular birefringence by applying the RPPDL. This research can be provided to useful and practical application.


Introduction
For both telecommunications and sensing purposes, FBG thus becomes important to characterize the polarization properties of FBG and their dependence on the wavelength [1][2][3].Furthermore, this study can lead to the development of a new demodulation technique for FBGbased sensors.Caucheteur et al. have used polarization dependent loss (PDL) for transverse strain measurements [4,5].The polarization dependent properties are also used for the magnetic field sensor [6].
Either way, there exist cross-sensitivity problems.In wavelength detection, the centre wavelength will change not only with the strain, but also with the temperature [7].Similarly, in polarization detection, the polarization properties are influenced by the linear and circular birefringence [8,9].Hence, we must take various kinds of measures to compensate or distinguish the cross-sensitivity problems.A number of techniques addressing this issue, such as dual-wavelength superimposed grating, two FBGs in different diameter fiber, hybrid FBG/long period grating, superstructure FBG and Fabry-Perot cavity method, have been reported [10][11][12].Those methods provided some approaches to distinguish cross-sensitivity effect, but most of them needed specific gratings and special technique.
In this paper, the influence of the circular birefringence on the measurement performance was analyzed based on the polarization properties of FBG.To estimate the cross sensitivity, a new parameter named relative peak of PDL (RP PDL ) is proposed.The simulation and experiment results proved that the relative peak of PDL (RP PDL ) could effectively reduce the influence of circular birefringence.

Principle of Measurement
The polarization property of FBG has been widely used in the field of measurement.Here, we shortly introduce the principle of measurement.
The applied force causes a birefringence Δn, which is defined as the difference in refractive index between two orthogonal polarization modes called x and y modes (or eigenmodes).Due to the Δn, the x and y modes undergo different couplings through the grating.The total transmitted signal is the combination of the x and y mode signals.If there is only the linear birefringence, the Jones vector associated to the FBG transmitted signal is given by [5]: , , , , , , 0 0 Where (E i,x , E i,y ) T is the Jones vector of the input signal and t x(y) denotes the transmission coefficient of x(y) mode FBG [13].
Polarization Dependent Loss (PDL) is defined as the maximum change in the transmitted power by the grating as the input state of polarization is varied over all polarization states.
In this paper, the input signal is the linear state at π/4 between the x(y) mode, E i,x = E i,y .In the case of Bragg gratings, it is easy to show that the PDL for the transmitted signal is given by [17]: ( ) ( ) ( ) 10 log 10 log ( ) ( )

The Effect of Linear Birefringence
The study of Caucheteur et al. demonstrated that there is linear relationship between the peak of PDL and pressure [5].But in practice, the fiber has intrinsic circular birefringence in the manufacturing process.Moreover, the induced circular birefringence can be caused by the shape, twisting and axial magnetic field of the fiber materials.
The effect of circular birefringence must be taken into account when researching the performance of FBG weak pressure sensor based on the polarization properties.Due to the circular birefringence, the Equation (1) will be modified as [18,19]: , , , , , , 0 0 (2 ) Where Φ, 2Ω and Δ are the phase shift of elliptically, circularly and linear polarized light, respectively.And we use the L and R subscripts to indentify the eigenmodes corresponding to the left and circularly polarized light.
According to the Equations ( 2) and ( 3), we will understand more clearly the effects of circular birefringence on the evolutions of PDL.

Simulation Results
We design a FBG with 1.455 of n eff , 535 nm of Λ, 4mm of L and 1 × 10 −4 of Δn.According to the given data, the simulation results can be got to analyze the influence of circular birefringence on the performance of proposed FBG sensor.
For purposes of analysis, some abbreviations were de-fined, such as B PDL (the value of PDL that outside the FBG band), P PDL (the peak value of PDL).
According to the Equations ( 1) and ( 2), Figure 1 presents the PDL as a function of pressure.As expected, the increase of pressure leads to a general increase of PDL amplitudes.The P PDL increase linearity with the pressure without circularly birefringence, which is shown in Figure 2.
Considering the influence of circular birefringence, the evolutions of PDL spectrums with pressure and circular birefringence are shown in the Figure 3. Due to the circular birefringence, the B PDL is no longer zero and changed non-linearly with the circularly birefringence.Hence, the P PDL does not increase linearly with the increasing of pressure, as shown in Figure 3. From Figure 3, it also can be seen that the P PDL is influenced by the combination of circular and linear birefringence.
Up until now, it can be seen clearly that the linear relationship [5] between P PDL and pressure was broken because of circular birefringence.In order to eliminate the effect, a new parameter named the relative peak of PDL (RP PDL ) is defined.The RP PDL refers to the difference between the B PDL and P PDL .Without circular birefringence, the RP PDL and P PDL are the same.Under circular birefringence, the evolutions of RP PDL were shown in Figure 4. From which it can be seen that the four RP PDL lines were almost coincided under difference circular birefringence (solid, dash, dot and dash dot).Figure 4 means that the RP PDL is insensitive to the circular birefringence.Hence, the influence of circular birefringence on the performances of the FBG pressure sensors can be eliminated by using the RP PDL .

Experiment Results
The experimental data were then compared to theoretical evolutions.For that purpose, the experiment system was set up.The optical vector analyzer (OVA) is regarded as a light source, detector and processor.The FBG that used in experiment was designed and fabricated by our project group.The parameters of FBG are: n eff = 1.455,Λ = 535 nm, Δn = 5e −5 and L = 10 mm.The width of glass plate is the same size as the length of FBG.The fixture was used to generate the random circular birefringence by twisted the FBG, in this way, only the qualitative method can be used to analyze the circular birefringence.Due to the experimental condition limitations, the applied pressure was in the range 1 ~ 10 N by steps of 1 N.The results under two random conditions with different circular birefringence were calculated and analyzed in this section.
Using the optical vector analyzer whose precision is 10 −5 (dB) in our experiment [20], the PDL evolutions for different pressure under two cases (different circular birefringence) are got and shown in Figure 5.The P PDL becomes more and more obvious along with the increase of pressure.To observe Figures 3 and 4, the simulation results and experiment results were similar.
Based on different pressure and circular birefringence, the experimental data of the B PDL , P PDL and RP PDL values respectively are given in Table 1.
From the Figure 6, we can find that the two set of data about RP PDL and their fitting curves have a good agreement, which demonstrates that the RP PDL is not influ-   enced by the circular birefringence.In addition, due to the values of the fitting curves are increase monotonically with pressure, the RP PDL can be used to retrieve the pressure.Based on calculations, the pressure sensitivity are same 0.229 dB/N.Figure 6 also presents the theoretical results, the pressure sensitivity of which is 0.291 dB/N.The error between the experiment and theoretical results is caused by the manufacturing error of the FBG, such as the photo-induced birefringence [21,22].The   initial value of RP PDL both are 0.66 under two conditions because of the intrinsic birefringence.The theoretical analysis and experiment results prove that the errors caused by the circular birefringence can be effectively eliminated by using the RP PDL .

Conclusion
By analyzing the influences of the circular birefringence on the performance of the FBG pressure sensor, it is known that the nonlinear relationship between the P PDL and the pressure can be caused by the circular birefringence, which affects the accuracy of the FBG pressure sensor based on the P PDL .For overcoming this weakness, the parameter denoted by RP PDL is proposed in this paper which can ensure the linear relationship between the RP PDL and the pressure under the different circular birefringence, thus the pressure can be retrieved by the values of the RP PDL .The theoretical analysis and experiment results prove that the errors caused by the circular birefringence can be effectively eliminated by using the RP PDL .This research is helpful to the practical application of the FBG weak pressure sensor, such as underwater acoustic and liquid level measurement, etc.

Figure 3 .
Figure 3. the evolutions of PDL at different circularly birefringence and pressure (CB = circular birefringence).

Figure 4 .
Figure 4. the evolutions of RP PDL at different circularly birefringence with same pressure (CB = circular birefringence).

Figure 5 .
Figure 5. PDL versus wavelength at pressure for two cases.

Figure 6 .
Figure 6.Evolution of RP PDL in response to a change of pressure.