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Magnetic field sensing can be directly (i.e. without requiring magnetic fuilds or magnetostrictive materials) obtained from the estimation of the circular birefringence induced in optical fibers through the so-called Faraday effect. In standard telecommunication-grade optical fiber, the amount of induced circular birefringence is however of the same order of the intrinsic fiber linear birefringence or even below. Hence, whenever uniform fiber Bragg gratings (FBGs) are used to probe this evolution, the resulting accuracy is usually very poor, even in the case of polarization-assisted measurements based on polarization dependent loss (PDL) or differential group delay (DGD). In this work, we demonstrate that the rotation of the diattenuation vector computed from the Mueller matrix of an FBG in transmission mode can be efficiently used as a read-out technique to sense a magnetic field evolution with a resolution of 0.1T.
© 2016 Optical Society of America
1. Introduction
Fiber Bragg gratings (FBGs) are nowadays mainly used for temperature and strain sensing. Besides classical measurements based on amplitude spectrum measurements either in reflection or in transmission, the use of FBG polarization properties such as polarization dependent loss (PDL) and differential group delay (DGD) has multiplied the sensing modalities [1]. In the case of transverse force measurements, it has been shown that the induced linear birefringence can be determined much more accurately by using the FBG polarization properties instead of tracking the shift of their reflection band [2,3], even allowing the determination of non-uniform transverse load profiles [4,5]. A magnetic field also induces birefringence (circular this time) in the fiber, resulting from the so-called Faraday effect [6]. Hence, the FBG response could be used to retrieve the value of the circular birefringence in the fiber core and as a result the corresponding magnetic field [7]. However, the birefringence induced by a magnetic field in a standard telecommunication-grade single-mode optical fiber (SMF-28) is very weak. The Verdet constant of silica is 0.54 rad/Tm at ~1550 nm, yielding a circular birefringence value of 2.7 10−7 for a magnetic field of 1 T [6]. This value is comparable to the intrinsic fiber linear birefringence, not to say below. Hence, classical polarization-assisted metrology (i.e. relying on the PDL and DGD curves of an FBG) suffers from a poor sensitivity in the case of uniform FBGs photo-inscribed in SMF-28 [8–10]. This mainly results from the linear birefringence of the optical fiber. Better results have been demonstrated with dedicated techniques such as the use of the third Stokes parameter [11], the circular-polarization dependent loss (used with an Erbium-doped fiber with a Verdet constant of −11 rad/Tm) [12] or compound phase-shifted FBGs [13]. Magneto-strictive materials and ferro-fluids have also been used [14,15]. The latter can be interrogated by coupling light outside the fiber, for instance by using tilted fiber Bragg gratings [15,16]. However, such materials are subject to saturation for increasing magnetic field values [14,15].
To get rid of the presence of linear birefringence issue while avoiding the use of extra sensitive elements around the fiber, we demonstrate in this work an original demodulation technique that can accurately compute the magnetic field value induced on a standard uniform FBG photo-inscribed in the core of a telecommunication-grade optical fiber. The sensing mechanism is based on the determination of the diattenuation vector rotation, which is computed from the Mueller matrix of the FBG measured in transmission. The orientation of this vector depends on the ratio between the linear and circular birefringences. This method uses the linear birefringence present at the Bragg grating location to determine the circular birefringence created by the magnetic field. The proposed method has a limit of detection of ~0.1T and is not affected by the optical fibers that link the FBG to the measurement device.
In the following, we first recall the Faraday effect. Then, we define the diattenuation vector and show, through numerical simulations, how it is influenced by a magnetic field. Experiments are then reported with uniform FBGs manufactured in SMF-28.
2. Faraday effect
The state of polarization of light rotates by a certain angle α when a magnetic field is applied in the direction of light propagation. This angle α is given by:
with B the magnetic field, l the length traveled in the medium and V the Verdet constant. This rotation results from the creation of circular birefringence in the fiber so that the right-handed and left-handed circular states of polarization do not propagate at the same velocity, therefore inducing a phase shift between the two states. Let us recall that when the fiber is subject exclusively to the circular birefringence, the eigenstates of polarization are the right- and left-handed circular states. The birefringence ∆nc created by a magnetic field B is given by:with λ the wavelength of the light. In the case of a classical optical fiber, the Verdet constant is 0.54 rad/Tm at 1550nm [6]. A magnetic field of 1T induces therefore a birefringence of ~2.7 10−7. For rare earth doped fibers, the Verdet constant can increase up to −11 rad/Tm for high concentration of Erbium [12] or even −30 rad/Tm in the case of Terbium [17,18].In practice, optical fibers also present linear birefringence due to, for example, ovality of the fiber core, bending, transverse load so that both circular and linear birefringences are usually present in the fiber. The linear birefringence has a major impact on the FBG response [10] and its effect must be taken into account, as we will show in section 4.
3. Diattenuation vector
When birefringence impacts the FBG, its spectral response is polarization dependent. PDL and DGD curves are usually used to determine the birefringence since they are directly proportional to its value, for values below 1 10−4 [2,19]. The PDL in transmission is defined as the difference in dB between the maximum and minimum of the transmitted amplitude spectrum relative to the state of polarization. For FBGs, the maximum and the minimum of the transmitted amplitude occur for the eigenmodes and the PDL is given by [19]:
The DGD represents the difference in propagation time of the two eignemodes. It is defined as the absolute value of the difference between the two group delays of the two eigenmodes:In principle, these two parameters can be used to determine the magnetic field [8,10] but the obtained results are strongly impaired in practice due to the non-discrimination between linear and circular birefringence effects, as we will demonstrate in the following. It is the reason why we propose to use in this work the diattenuation vector [20], which allows this discrimination [21]. It is defined as follows:where the M0i (i = 0,…,3) are the first raw elements of the Mueller matrix represented by:The diattenuation is the norm of the diattenuation vector and is given by:where Tmax and Tmin are the maximum and minimum of the transmitted amplitude spectrum, respectively. The orientation of this vector depends on the ratio between the linear and circular birefringences. Furthermore, only the third element of the vector D is not null if there is only circular birefringence in the fiber. To obtain the diattenuation vector, one needs to know the Mueller matrix. In the following, we use the aforementioned formalism to study the influence of the magnetic field on the polarization properties of uniform FBGs.4. Magnetic field influence on the FBG polarization properties
To simulate the full Mueller matrix of the FBG when both linear and circular birefringences are present in the fiber, we solve the coupled mode equations given in [10] for the two orthogonal input states of polarization Ex = (1 0)T and Ey = (0 1)T . The Jones matrix is then retrieved from the solutions of the two boundary value problems for the input states Ex and Ey. The Mueller matrix is finally calculated from the Jones matrix [20].
Simulations are conducted to evaluate the impact of a magnetic field on the FBG polarization properties in the presence of linear birefringence corresponding to a standard telecommunication-grade optical fiber. Two cases are considered (sections 4.1 and 4.2).
4.1 Ideal case where only the fiber section containing the FBG is considered
In the ideal case where no linear birefringence is present in the fiber, the PDL and DGD are well adapted to determine the magnetic field since both parameters increase nearly linearly with the magnetic field. Figure 1 displays the PDL and DGD spectral evolutions for increasing magnetic field values in the range 0-1T. The FBG parameters (FBG length, pitch and modulation amplitude) considered here are: L = 5 mm, Λ = 533.2 nm, δn = 2 10−4, respectively. In this case, the evolution of their maximum amplitude can be tracked to determine the magnetic field value, as done for transverse force sensing [2,3]. However in practice, there is always linear birefringence present in the fiber (most usually in the range [5 10−7 – 5 10−6]) and the observed changes are strongly impacted in this case.
Fig. 1 DGD and PDL spectral evolutions when a magnetic field is applied to the FBG section in the absence of linear birefringence.
Figure 2 shows the PDL and DGD maximum amplitude evolutions when a magnetic field is applied to the FBG, for 3 linear birefringence cases. In the presence of linear birefringence (named ΔnL in the following), the PDL and DGD do no longer increase linearly with B and the amplitude of their modification is strongly decreased compared to the utopic case with no linear birefringence. This lack of sensitivity leads us to consider the diattenuation vector in the following.
Fig. 2 Evolution of the DGD and PDL maximum values when a magnetic field is applied to the FBG section for 3 different linear birefringence values (ΔnL = 0, 2.5 10−7 and 5 10−7).
Figure 3 shows the three elements of the diattenuation vector when the linear birefringence is considered to be 5 10−7. The third element of the diattenuation vector D(3), which is the difference in intensity between the right and left circular birefringences, shows a strong linear dependence on the magnetic field value even when linear birefringence is present in the fiber. In the following, we will therefore focus on this relevant parameter.
Fig. 3 Diattenuation vector elements evolution when a magnetic field is applied to the FBG section for a linear birefringence value of 5 10−7 (a) and evolution of the maximum of D(3) as a function of the magnetic field value (b).
4.2 Practical case where connecting fibers are present at both sides of the FBG
In this section, we analyze the effect of optical fiber links that connect the FBG to the measurement device. To this aim, the Jones matrix of the FBG is multiplied by two Jones matrices of birefringent elements, without considering additional fiber loss (usually very small). Figure 4 presents simulation results obtained when fiber links are considered at each side of the FBG. A beat length of 15 m was considered for the fibers. In Fig. 3(b), LB and LA means the fiber lengths before and after the FBG. Figures 3(a)-3(c) show that the 3 components of the diattenuation vector are now impacted while the sensitivity of D(3) is strongly decreased. Figure 3(b) confirms that in the presence of fiber links, the evolution of the maximum D(3) is strongly impacted by the physical lengths of the fiber sections. Moreover, it can be no longer monotonous with the increase of the magnetic field value. Hence, D(3) is not well suited in practice to determine the magnetic field, which is similar to the third normalized Stokes parameter evolution [11].
Fig. 4 Diattenuation vector elements evolution when a magnetic field is applied to the FBG section for a linear birefringence value of 5 10−7 in the presence of fiber links (a) and evolution of the maximum of D(3) as a function of the magnetic field value for different fiber links lengths (b).
Based on these observations and considering that an optical fiber link rotates all the elements of the diattenuation vector by the same amount on the Poincaré sphere, it was decided to consider the rotation of the diattenuation vector as a parameter to quantify the magnetic field value. Figure 5 shows the evolution of the diattenuation vector on the Poincaré sphere in the two considered cases. This picture confirms that considering the rotation of the diattenuation vector allows to get rid of the optical fiber link influence.
Fig. 5 Evolution of the diattenuation vector on the Poincaré sphere as a function of the magnetic field for a linear birefringence of 5 10−7.
To determine the rotation, the direction of the diattenuation vector is computed with respect to the reference one 0, taken in the absence of magnetic field. The rotation angle is then given by:
This angle is related to the ratio between linear and circular birefringences, as sketched in Fig. 6. The amount of linear birefringence can be determined from the PDL and/or DGD spectrum measured in the absence of magnetic field so that the circular birefringence can be easily computed from the knowledge of the rotation angle.The rotation angle remains constant in the grating spectrum but the wavelengths for which the diattenuation vector takes maximum values are less influenced by the noise. Therefore, we concentrate on that part of the spectrum to determine the rotation angle.
5. Experiments and results
5 mm long uniform FBGs were photo-inscribed in the core of hydrogen-loaded standard single-mode fibers, using a frequency-doubled argon laser and the phase mask technique, as described in [22]. They were positioned in the gap of an electromagnet (6 mm in length, cf. Fig. 7) that was specially designed to allow us to tune the applied magnetic field between 0 and 1 T. The gratings were connected in transmission to an optical vector analyzer (OVA) from Luna Technologies. For each value of the magnetic field, 20 measurements were performed with the OVA and the complex Jones matrix corresponding to the mean of these 20 measurements was used to compute the diattenuation vector. The magnetic field value is adjusted by modifying the current in the electromagnet. It is measured by a hand-held gaussmeter placed in the gap.
Fig. 7 Photograph showing the electromagnet and the optical fiber going through. The inset displays the top view, focusing on the gap of the electromagnet with the fiber in the middle.
In the following, we report experiments obtained on a 5 mm long uniform FBG. A good knowledge of the FBG physical parameters and the linear birefringence is required to use the diattenuation vector as a read-out technique to determine the magnetic field. These parameters were then numerically reconstructed from the transmitted amplitude spectrum and DGD curve using a synthesis algorithm derived from [23]. Figure 8 depicts the experimental and simulated evolutions of the grating. The reconstruction process yields the following parameters: L = 5.05 mm, Λ = 533.42 nm, δn = 1.58 10−4 and ΔnL = 1.40 10−6. There is an uncertainty on the reconstruction of the linear birefringence of the FBG and it will directly affect the magnetic field measurement. However, to get rid of this possible issue, it is possible to calibrate the sensor by applying a known magnetic field since the linear birefringence of the FBG does not change.
Fig. 8 Experimental and reconstructed transmitted amplitude spectrum and DGD curve for the uniform FBG used in this work.
Figure 9 depicts the evolution of the orientation of the diattenuation vector on the Poincaré sphere for 14 different magnetic field values between 0 and 1 T. The values displayed are a mean over a wavelength range of 0.175nm centered on the two peaks of the diatenuation vector.
Fig. 9 Evolution of the orientation of the diattenuation vector on the Poincaré sphere for 14 different magnetic field values between 0 and 1 T.
These data were used to determine the magnetic field using the aforementioned technique. Figure 10(a) presents the reconstructed magnetic field values as a function of the applied ones for increasing and decreasing solicitations. A linear regression of the data yields a slope of 0.97 and 0.93, respectively. Figure 10(b) depicts the relative error computed with respect to the linear regression. It shows that the mean relative error is close to 0.02 T and 0.04 T for increasing and decreasing magnetic field values, respectively.
Fig. 10 Experimental reconstruction of the magnetic field value as a function of the applied one (a) and relative error with respect to the linear regression of the raw data (b).
It is worth mentioning that, from this measurement and others conducted on similar gratings (not shown here), the reconstructed magnetic field values are always slightly underestimated with respect to the applied ones. We attribute this to the fact that the whole grating length is not subject to a constant magnetic field in the gap of the electromagnet. This assumption has therefore been verified by numerical simulations. It turns out that the obtained experimental results are consistent with the case where ~90% of the grating length is subject to the magnetic field. This discrepancy could not be alleviated in our experimental set-up, due to the divergence of the magnetic field lines in the gap of the electromagnet and the difficulty to precisely align the FBG, both axially and in the cross-section of the gap.
Finally, although temperature changes were not considered in our work, we believe that they would not affect the performances of the demodulation technique. Indeed, temperature changes on the FBG induce the same wavelength shift for all polarization states. The influence on connecting fibers is expected to be negligible, especially if they are secured and also considering that temperature changes are usually a slowly varying. It is also worth mentioning that the Verdet constant of silica glass weakly varies with a temperature variation [24].
6. Conclusion
In this article, we have presented a new demodulation technique allowing the use of a standard uniform FBG photo-inscribed in a telecommunication-grade single-mode optical fiber for magnetic field sensing. Our technique is based on the tracking of the diattenuation vector (specifically its rotation with respect to a reference state), which can be easily determined from the Mueller matrix of the FBG measured in transmission mode. Our technique avoids the influence of linear birefringence which is known to have a detrimental effect on other polarization-based measurements reported so far. It does not require any external transducer element so that the proposed technique is not subject to saturation. Experimental results conducted on 5 mm long FBGs confirm the good potential of the technique for magnetic field measurement in the range 0–1 T with a resolution of 0.1T.
Acknowledgments
This work is supported by the European Research Council (ERC) through the starting Independent Researcher Grant PROSPER (Grant agreement no. 280161 – https://www.umons.ac.be/erc-prosper), by the F.R.S.-FNRS (Associate Researcher grant of C. Caucheteur) and by the ARC (Actions de recherche concertée) research programme of the UMONS-ULB Academy (Prediction grant).
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