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Characteristics of a fiber Bragg grating (FBG) transverse force sensor based on Stokes parameters are presented. Real-time force measurement is achieved through direct measurement of the Stokes parameters at single wavelength. A proportional relationship and linear fit are found between Stokes parameters and applied force. The sensitivity and dynamic range dependence on the state of polarization (SOP) of the incident light is investigated theoretically and experimentally. A maximum sensitivity of 0.037/N is experimentally achieved and it can be improved further by adjusting the incident SOP. This design significantly reduces system complexity and improves data processing speed, which has great practical value in real-time FBG sensing applications.
© 2015 Optical Society of America
1. Introduction
The sensor based on a fiber Bragg grating (FBG) can satisfy the requirement in sensing a number of physical parameters such as strain and temperature, which changes the center wavelength of the reflected spectrum when the sensing parameters cause grating effective index or grating period variation [1–3]. They provide significant advantages such as small size, geometric flexibility and distributed sensing possibilities [4–6]. Up to now, for most applications of FBGs, the research efforts have concentrated on the features of the spectral behavior of the gratings. The reflection spectrum is then monitored for shifts in wavelength.
In practice, the characteristics of a FBG under transverse load are quite different from axial load [7,8]. In the case of transverse load, stress induced birefringence effects will cause the unique Bragg grating condition to break down and even produce two distinct Bragg wavelengths [9, 10].
Actually this stress induced birefringence is hardly perceived in the amplitude spectral response of the grating written into standard single-mode fiber due to its low sensitivity. R. B. Wagreich et al [10] studied the effect of transverse load on FBG fabricated in low birefringent fiber. The results showed that in the reflected spectrum evolution, the peaks cannot be distinguished until ~40N of load was applied since they overlap. These results demonstrate that the amplitude spectrum is not suited to measure transverse forces lower than a few hundreds of Newton [11].
However, it will lead to significant polarization parameters such as polarization dependent loss (PDL) and differential group delay (DGD) within the grating, which can provide more effective information and therefore lead to the potential development of new types of FBG-based optical sensors [11–15]. In these reports, the evolutions of the PDL or DGD peak amplitudes increase monotonically with applied force in dynamic range. Thus, by measuring PDL or DGD peak amplitudes the amount of applied force is determined. Although the structure is simple, this design requires a complex measuring system to determine the peak values. Large amounts of data points over a wide range of spectrum must be continually analyzed in high resolution, resulting in a rather slow processing speed. Therefore, these systems are not suitable for real-time low-cost applications. We have proposed a novel fiber Bragg grating sensor for weak pressure measurement based on the first Stokes parameter (s1) [16]. In this report one particular wavelength at stop band was chosen for measurement, which will cause more fluctuation of the SOP than that at transmitted band, resulting in more uncertainty and errors. And then a dual-wavelength method was presented to resolve this problem, while this method is not suitable for real-time measurement.
In this paper, we demonstrate a compact real-time transverse-force sensing system through Stokes parameters measurement at single wavelength. The measuring system in our approach is simple and does not require processing of massive amount of spectral data, enabling real-time transverse force monitoring. A proportional relationship and linear fit are observed between the applied force and the Stokes parameters from the FBG.
2. Principle
When the FBG is subjected to a transverse load, the difference between the effective refractive indices of the two orthogonal modes of the fiber within the loaded region will be produced. For simplicity the direction of the transverse force is assumed as y (fast axis), another direction perpendicular to y-axis is x direction (slow axis), z is along the fiber axial direction, as shown in Fig. 1.
The refractive index changes within the loaded zone are given by Eqs. (1) and (2) [7,17]:
Where E and v are the Young’s modulus and Poisson’s coefficient of the optical fiber respectively, p11 and p12 are the strain-optic coefficients. For typical optical fiber, E = 74.52 GPa, v = 0.17, p11 = 0.121, and p12 = 0.270. σx,σy and σz are the stress components in the grating in the x, y and z directions, respectively.
When the FBG is transversely compressed, as shown in Fig. 1, stress in the direction of the applied force σy is negative due to compression. Stresses in the two orthogonal directions, σx and σz are positive. For a given compressive force, these stresses are expressed in the Hertz equations [7,10]:
Where D is the fiber diameter, F is the applied force, and L is the length of the region under stress, u is the correction coefficient which is related to the smoothness of the optical fiber and compression platform. The case of u = 0 denotes that the contact plane is perfectly smooth (no friction) and the fiber is in plane stress state (ends-free). In order to simplify the model of fiber stresses, we limit ourselves to this case.
The refractive indexes in the x and y-directions are then given by and , respectively, with birefringence:
Due to, the x and y modes undergo different couplings through the grating. The total transmitted signal is, therefore, the combination of the x and y mode signals. In a Cartesian coordinate system whose reference axes match the FBG eigenmodes, the Jones matrix of the grating is diagonal and the Jones vector of the transmitted signal is then [11,12]:In Eq. (5), is the Jones vector associated with the input signal, denotes the transmission coefficient of the FBG corresponding to the x(y) mode:Where anddepend on and grating parameters (such as index modulation). L is grating length [18].
The Stokes parameters represent the state of polarization. They can be deduced from the Jones vectors by means of the follows equations:
The normalized Stokes parameters (s1, s2 and s3) can then be computed using the relationship:So the relationship between birefringence and the normalized Stokes parameters are found using Eqs. (1)-(8) [19].
Simulated s1, s2 and s3 evolutions for different values of birefringence are shown in Fig. 2. These evolutions are simulated with grating length of 10 mm and incident angle of 45◦. It can be seen that all the Stokes parameters change with birefringence.
As indicated in Fig. 2, the Stokes parameters change as the applied force increases. Thus, the amount of applied force can be determined by measuring the Stokes parameters at single wavelength. This is achieved by launching a single wavelength laser light into the FBG and measuring the corresponding Stokes parameters using a polarimeter. The sensitivity, linearity and dynamic range are affected by the SOP and wavelength of incident light.
Next we compare the sensitivity and dynamic range for different SOPs in Fig. 3 atwavelength of 1549.3 nm. The incident SOP is supposed to be linear polarized and seven incident angles from 10◦ to 80◦ are studied. From Fig. 3 it is demonstrated that the SOP of light has significant influence on the sensitivity, linearity and dynamic range of these Stokes evolution.
Fig. 3 The influence of incident angles on (a) s1 (b) s2,and (c) s3 evolutions with respect to different values of birefringence.
3. Experiment and discussion
We will now describe our experimental set-up and present some experimental results that we have obtained when transversal loading a uniform FBG. Figure 4 shows the measurement set-up for this experiment.
Fig. 4 (a) Measurement set-up. (b) Transmission spectrum of FBG measured by the laser source and an optical power meter.
A tunable laser source (Agilent 8163B lightwave Multimeter) was used as the laser beam for launching into the FBG. A polarization controller (PC) was used to modify the polarization state of the light. The transmitted signal from FBG was directed into a polarimeter (General Photonics’s in-line polarimeter POD-15-SS-02) which measured the Stokes parameters at a certain wavelength. After A/D conversion and signal process the experimental results were displayed and recorded in a computer.
Compared with conventional sensing systems based on OSA, our approach does not require capturing the entire spectrum for determining the maximum value of polarization parameters. Thus, our approach is simple, compact, consumes less computation power.
During the measurements, the FBG was not strained and the ambient temperature was kept constant. Moreover, the fiber connectors were fixed to avoid polarization instability. The FBG used here was written into hydrogen-loaded standard single-mode fiber by means of the phase mask technique. The length of fiber grating is 10 mm and was placed parallel to a balance reference fiber. Figure 4(b) records the transmission spectrum measured by the laser source and an optical power meter. The center wavelength is 1550.06 nm and the maximum insertion loss of the whole system (most from the FBG) is about 9 dB, remains unchanged during measurement.
In the experiment it is found that the wavelength at stop band of FBG will cause more SOP fluctuation than that at transmitted band. The main reasons may be that the instabilities of optical source and other factors will be enlarged at the slope of the spectrum, resulting in more uncertainty and errors. While in transmitted band, the SOP is much more stable. So the wavelengths at transmitted band were chosen in the measurement, for example 1549.7nm.
In the experiment, the force to be measured was applied to the FBG by placing a weight on a plastic piece against the fiber. The weight applied was increased gradually from 0 to 1200 g (0–11.76 N), stepped by 200g. Experimental results were recorded and saved.
3.1 Effect of the incident SOP
Three SOPs of incident light were chosen randomly by adjusting PC. In the first case, the SOP of light which launches into polarimeter was (−0.35,0.27,0.852). The Experimental evolutions of three Stokes parameters during the process of placing weights were recorded in Fig. 5(a). It can be seen that s1 and s2 increase as applied weight increases while s3 is not so sensitivity in this case.
Fig. 5 The first incident SOP case: (a) Experimental evolutions of three Stokes parameters during the process of placing weights. (b) s1 and (c)s2 amplitude under different amount of applied weights.
The evolutions of s1 and s2 amplitude under different amount of applied weights are presented in Figs. 5 (b) and 5(c), respectively, ranging from 0 to 1200 g (11.76N). It can be seen that in the range of 0 g (0N) to 1200 g (11.76N), the amplitudes of s1 and s2 increase monotonically with applied force. A proportional relationship is found with applied force in the range of 0 to 600 gf. By linear fit, the sensitivities of about 0.037/N and 0.018/N for s1 and s2 are obtained, respectively.
The initial SOPs of light launches into polarimeter were (0.816, −0.083, −0.612) and (−0.453, 0.924, −0.013) for the second and third case, respectively. The Experimental evolutions of three Stokes parameters under force were also recorded in Fig. 6 (a) and Fig. 7 (a), respectively.
Fig. 6 The second incident SOP case: (a) Experimental evolutions, (b) s1 and (c) s2 amplitude under different amount of applied weights.
Fig. 7 The third incident SOP case: (a) Experimental evolutions, (b) s1 and (c) s2 amplitude under different amount of applied weights.
For the second case, with an increasing applied force, s1 and s2 amplitude gradually decrease monotonically, as shown in Figs. 6(b) and 6(c). Both of s1 and s2 have the proportional relationship with applied force in the range of 0 to 800gf. By linear fit, we obtain sensitivities of about 0.029/N and 0.0134/N for s1 and s2, respectively.
For the third case, monotonically decreasing curves are also obtained under increasing applied force, as shown in Figs. 7(b) and 7(c). In this case, for the transverse force smaller than 400gf, the relationships between the applied force and the two Stokes parameters are not so linear. And proportional relationship and linear fit are observed ranged from 400 to 1200gf. The sensitivities of about 0.022/N and 0.016/N for s1 and s2 are obtained, respectively.
The proportional relationship and linear fit allow simple conversion from Stokes parameters to the amount of applied force, enabling fast signal processing speed and real-time measurement. The sensitivity and the linear range are affected by the incident SOP significantly.
It is noted that in simulations as plotted in Fig. 1(a), s1 has little response to transverse force at transmitted band of FBG. The difference between the simulated and experimented results is most likely caused by the supposition that the reference axes match the direction of the transverse force in the simulations, whereas the intrinsic and optical induced birefringence of FBG is not taken into account. This inherent polarization axis may not be consistent with force direction. Hence the theoretical mode can be modified in the further research according to the experiment results.
3.2 Repeatability error
Next, we investigated the repeatability of the measurements. For the third SOP case mentioned above, the measurements were performed four times. The monotonically decreasing curves are plotted in Figs. 8(a) and 8(b) for s1 and s2, respectively. It can be seen that the shape of the four curves are very close to each other. In the range of 400 to 1200gf, all the curves can be linear fitted.
Fig. 8 Measured (a) s1 and (b) s2 amplitude with respect to applied weights for the four measurements.
For s1 the slopes are −0.000218, −0.000228, −0.000245 and −0.00022 for the four curves, respectively. For s2 the slopes are −0.000156, −0.0001625, −0.0001755 and −0.0001495 for the four curves, respectively. The corresponding sensitivity of s1 and s2 are listed in Table 1.
Table 1. The sensitivity of s1 and s2 parameters for the four measurements.
The differences of sensitivity are mainly due to the fact that the processes of placing weights are not exactly the same for the four times, resulting in the slight difference of applied force. At the same time it is should be noted that the initial SOPs with no force for the four measurements are different. That is mainly due to the slow excursion of SOP with respect to time, even without any human interference. Figure 9 is the excursion of SOP during 408 seconds in laboratory, with no human interference. It can be seen that s1 increases by ~0.01 (i.e. from −0.38 to −0.37), s2 decreases by ~0.005 (i.e. from 0.945 to 0.94) and s3 decreases by ~0.05 (i.e. from −0.155 to −0.205). The random excursion of SOP is one of the main reasons that cause repeatability errors.
3.3 Real-time measurement
We applied a mobile telephone (Iphone 6 plus) to FBG and set it to vibrate mode to verify the performance of real-time measurement. The experimental results were recorded in Fig. 10(a). It can be seen that in this state, all of the Stokes parameters can respond to the vibration of phone. And s2 presents the highest sensitivity. At 11s, the sudden change of Stokes curves is due to the applying of the phone. Then vibrate mode was set 9 times. The first vibration of s2 is enlarged as shown in Fig. 10(b) as an example. The amplitude is about 0.055. The period is about 4.5ms, meaning a vibration frequency of about 222 Hz. So the proposed method has a great practical value in real-time FBG sensing applications.
Fig. 10 (a) Experimental evolutions of Stokes parameters response to a vibrated phone and (b) the enlarged responded curve of s2.
Another experiment was performed by applying a little sound box on FBG. The Stokes parameters of light responded to the sound pressure of music well. The corresponding Stokes data were recorded as shown in Fig. 11. It is shown that in this state s2 contain more information than s1 and s3. We can easily recover the music by s2 data, with some noise of course.
3.4 Discussion
From the simulations and experiments, it is shown that the sensitivity of the sensor is affected by many factors such as incident SOP and the chosen wavelength. Of course by careful calibration and adjustment, the highest sensitivity can be gained. For the optimal sensitivity the laser needs to be tightly lined up with the most sensitive region of the Stokes evolution with wavelength. However with the three SOPs and wavelength chosen in the experiments, obvious responses to transverse load have been observed in each case. That means it is not very difficult to obtain a good or satisfying response (of course it may not be the optimal value) by simple adjustment and try.
The experimental results show good linearity between the Stokes values and transverse load in a certain range. At the same time, due to the fact that the whole amplitude spectrum will shift in wavelength when subject to transverse strain, as shown in Fig. 2, we have recognized that the wavelength shift of spectrum may affect the linearity and dynamic range of the measurement system. So we are looking forward to find some solutions to improve the system performance in the further researches, such as special design of the grating structure.
4. Conclusion
In summary, a novel real-time FBG transverse force sensor is proposed and experimentally demonstrated. We demonstrate here that Stokes parameters amplitude can be advantageously used to obtain transverse force measurements with uniform FBGs written into standard single mode fiber. We achieve a simple, compact, and real-time force measurement design through direct Stokes parameters measurement. A proportional relationship and linear fit are found between Stokes parameters and applied force. The sensitivity and linear dynamic range are mainly decided by incident SOP and the wavelength of light. A maximum sensitivity of 0.037/N is experimentally achieved and it can be improved further by adjusting the incident SOP. This design significantly reduces the system complexity and improves data processing speed, which has a great practical value in real-time FBG sensing applications.
Acknowledgments
This work was supported by the Natural Science Foundation of Jiangsu Province of China (Grants No BK2012508) and the National Natural Science Foundation of China (NSFC) (Grant No.61371121).
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