1. Introduction

The Verdet constant is a fundamental parameter associated with the Faraday Effect describing the polarization rotation of light in certain media caused by a magnetic field [1]. For a constant magnetic field $B$ parallel to the path of the light traveling in a Faraday material free of birefringence with a length of $L$, the Verdet constant ${V_{B0}}\left( \lambda \right)$ at a fixed wavelength $\lambda $ relates the polarization rotation angle $ {\theta _0}$ by [2,3]:

It is important to point out that Eq. (1) assumes that the Faraday material is isotropic free of birefringence. When the light propagates in a Faraday material with birefringence, Eq. (1) no longer holds and much more complicated relationships are found between the magnetic field and the polarization rotation angle in general [3,5–7]. In [5], Tabor and Chen studied the Faraday Effect in an Ytterbium Orthoferrite crystal possessing both Faraday rotation and linear birefringence. They showed that the presence of birefringence can drastically affect the behavior of polarization evolution, which is considerably different from pure Faraday rotation of Eq. (1).

In [3,6–7], Faraday rotations in optical fibers with residual birefringence were investigated. Again, the presence of linear birefringence made the behavior of polarization evolution induced by magnetic field very complicated such that a simple Verdet constant was no longer sufficient to describe the Faraday rotation in general. In other words, the linear relation between the Faraday rotation angle and the magnetic field is no longer preserved with the presence of linear birefringence in the fiber.

The linear birefringence of the fiber generally comes from two sources, one is the imperfection in the circularity of fiber’s core and cladding, and the other is from the external and internal mechanical stresses via photoelastic effect [8], including the bending induced stress when winding the fiber in a loop around a conductor for current sensing applications. The axis orientation and magnitude of the combined birefringence are sensitive to the temperature and fiber arrangement, which leads to measurement fluctuations of the polarization rotation angle induced by the magnetic field. Consequently, large uncertainties will result when the fiber is used as part of an electrical current sensor for measuring the electrical current flowing in a conductor.

To overcome the issues caused by the linear birefringence, the annealed fiber [9], the low-stress fiber [10], and the spun fiber [11] were introduced. Eventually, the spun fiber seems to have gained acceptance [12] in the field for the applications of using Faraday Effect for electrical current and magnetic field sensing, because it is effective in minimizing the problems caused by the linear birefringence by introducing a large amount of circular birefringence and can be consistently produced by multiple specialty fiber manufacturers [13,14]. A spun fiber can be made by rotating a preform while drawing the fiber. This spin and draw process introduces a large amount of circular birefringence uniformly distributed along the fiber [15]. The preform used can be either that for making a single mode (SM) fiber or that for making a polarization maintaining (PM) fiber, however, the latter is more widely adopted in the industry because the resulting spun fiber is much less sensitive to bending induced birefringence than that made with the former [1]. The spun fibers made with the SM fiber preform and PM fiber preform are often referred to as the LoBi and HiBi spun fibers, respectively. When it is not spun during the drawing process, the PM fiber preform can produce a PM fiber with a large linear birefringence (defined as the unspun linear birefringence $\Delta n$). However, with the preform spinning during the drawing process, the average linear birefringence of the resulting spun fiber over a length much larger than the spin pitch is expected to approach zero, although the local linear birefringence still remain at the level of the unspun linear birefringence [16]. In fact, the large local linear birefringence renders the HiBi spun fiber less susceptible to bend and stress perturbations, while the near-zero average linear birefringence helps to preserve the linear relationship between the Faraday rotation angle and the longitudinal magnetic field (as will be shown in this paper), which is critical for electrical current and magnetic field sensing applications.

Different manufacturers may choose preforms of different birefringence of the unspun fiber and different spin twist rates in the manufacturing process. These differences are expected to produce spun fibers with different circular birefringences and residual linear birefringences [1,17,18]. Unfortunately, how these differences affect the Verdet constants of the spun fibers or the sensitivity of the Faraday rotation angles when the fibers are subject to a longitudinal magnetic field is not well understood and requires further investigation both theoretically and experimentally, although the circular and residual linear birefringences of the resulting spun fibers have been studied extensively [1,17,19]. For example, in [19], G. Müller et, al. obtained expressions of the magneto-optic phase shift of their current sensor systems as a function of the spin ratio (twice the unspun fiber beat length over spin pitch) and the imperfection of the quarter-wave retarder in the systems, however, no direct linkage between the spin ratio and Verdet constant of the spun fiber was established. In [1], Laming and Payne introduced a parameter called normalized current sensitivity relating to the spin ratio, however, no measurements were performed to obtain the values for real spun fibers. The understanding of these differences on the Faraday Effect is important for current and magnetic sensing applications.

On the other hand, in order to better understand the Faraday Effect in spun fibers, accurate methods for measuring the polarization variation induced by magnetic field are essential. Early measurements [3,7,20] of the Faraday rotation angle in an optical fiber generally involved using a rotatable polarizer at the input of the fiber for launching different input polarizations, a Babinet-Solei1 compensator (BSC) at the fiber’s output to compensate for the linear birefringence in the fiber, followed by a Wollaston polarization beam-splitter (PBS) oriented to have the equal power in the two output beams for maximum measurement sensitivity. To minimize stress induced linear birefringence in the fiber, the fiber had to be kept straight, limiting the length of the fiber to few meters. Such a configuration required careful and frequent adjustments of the polarizer, the BSC, and the Wollaston PBS, which was complicated, time consuming and susceptible to operational errors.

In this paper, we show that in a spun fiber with a large circular birefringence and a large local linear birefringence, the desired linear relationship between the Faraday rotation angle and the magnetic field can approximately hold, which enables us to introduce a term called effective Verdet constant to describe the sensitivity of Faraday rotation in the spun fiber in response to a longitudinal magnetic field. We show that this effective Verdet constant is the product of the Verdet constant ${V_{H0}}$ of an ideal fiber free of birefringence and a factor $\gamma $ which contains the contributions of spun fiber manufacturing parameters, particularly the unspun fiber retardation per unit length and the spin twist rate. We call $\gamma $ the Faraday Effect reduction factor because it is always less than unity and show that the larger the ratio of the spin twist rate to the unspun fiber retardation per unit length, the closer the $\gamma $ approaches unity, or the closer the effective Verdet constant approaches that of the ideal fiber.

To validate our findings, we measure and obtain the effective Verdet constants of three different spun fibers from three different manufacturers using a highly accurate and fully automated polarization measurement system made with binary polarization rotators [17,21]. In a previous publication [17], we used the wavelength differential method to obtain the linear and circular birefringences by measuring the state of polarization (SOP) variations due to wavelength changes, with a birefringence measurement resolution on the order of 10⁻⁷, corresponding to a polarization rotation of 0.10° per nm of wavelength change (the central wavelength of 1310 nm) with a fiber length of 10 m. However, considering that the circular birefringence induced by the Faraday Effect is on the order of 10⁻¹¹, the corresponding polarization rotation is about $2.2 \times {10^{ - 4}}$ degrees per nm of wavelength change in a fiber of 10 m length, with a fiber coil radius of 70 mm and under a current of 100 A, which is too small to be measured with the wavelength differential method. To overcome the difficulty, here we use a time differential method to measure the tiny polarization rotations caused by electrical current (the Faraday rotation).

The retardation $\psi $ associated with a circular birefringence ${n_c}$ is $\psi = \pi {n_c}L/\lambda $, where $\lambda $ is the wavelength of the laser and L is the fiber length. For a given change $\mathrm{\Delta }{n_c}$ of ${n_c}$, the retardation variation Δ$\; \psi $ (corresponding to the SOP change) caused by changing the wavelength and by switching $\mathrm{\Delta }{n_c}$ on and off are $\mathrm{\Delta }\psi = (\frac{{\pi \mathrm{\Delta }{n_c}L}}{\lambda })\frac{{\mathrm{\Delta }\lambda }}{\lambda }$ and $\mathrm{\Delta }\psi = (\frac{{\pi \mathrm{\Delta }{n_c}L}}{\lambda })$, respectively, where $\mathrm{\Delta }\lambda $ is the wavelength changing range. Therefore, for a given circular birefringence Δn_c induced by the Faraday Effect, the time differential method is $\mathrm{\lambda }/\mathrm{\Delta }\mathrm{\lambda }$ times more sensitive than the wavelength differential method.

In addition, because no wavelength scan is required, such a time differential method is sufficiently fast to effectively minimize the SOP variations caused by the changes of environmental conditions, such as temperature, and thus enable accurate measurement of Faraday rotation angles down to 0.012°, leading to the accurate determination of the effective Verdet constants of the three different spun fibers at 1310 nm to be $1.07 \times {10^{ - 6}}$rad/A (iXblue), $ 1.05 \times {10^{ - 6}}$rad/A (Fibercore), and $1.04 \times {10^{ - 6}}$rad/A (YOFC), respectively. The corresponding Faraday Effect reduction factors $\gamma $ are 0.98, 0.96, 0.95, respectively, which are consistent with the theoretical results and relate well to the spin twist rates and the unspun fiber retardation per unit length specified by the fiber manufacturers. Our work provides a simple parameter to quantify the Faraday rotation sensitivity of different spun fibers and will prove useful for the understanding of Faraday Effect in spun fibers, as well as for the electrical current and magnetic sensing applications.

2. Introduction of the effective Verdet constant

2.1. Theory of the effective Verdet constant

For a polarized light signal launched into a spun fiber, the Jones vector of the signal at the fiber output can be expressed as [1,22]:

In the equations above, m, n are integers, $f$ is the Faraday rotation angle per unit length of an ideal fiber, $\rho $ is the spin twist rate of the spun fiber, and $\mathrm{\Delta }\beta $ is unspun fiber retardation per unit length, which relates to the birefringence $\mathrm{\Delta }n$ of the unspun fiber by $\mathrm{\Delta }\beta = 2\pi \mathrm{\Delta }n/\lambda $ (again the unspun fiber is defined as the PM fiber made with the same preform as the spun fiber, but without spinning the preform during fiber drawing process, as described in the Introduction). In general, the local Faraday rotation per unit length $f$ for a fiber coil around an electrical conductor depends on coil’s radius r, the current I in the conductor and the Verdet constant ${V_{H0}}\left( \lambda \right)$ of the fiber core, as shown in Eq. (3e).

In the absence of electrical current $I = 0$, $\delta \left( {I = 0} \right)$ and $\varphi \left( {I = 0} \right)$ represent the intrinsic linear retardation and circular rotation of a spun fiber of length L, respectively. $\delta \left( {I \ne 0} \right)$ and $\varphi \left( {I \ne 0} \right)$ represent the result of the interaction between the intrinsic linear retardation and circular rotation and Faraday rotation of a spun fiber of length L, respectively. For a spun fiber with a sufficiently large spin twist rate $\left( {2\rho /\mathrm{\Delta }\beta \ge 4,\rho \gg f} \right)$, one obtains:

Note that most commercial spun fibers satisfy the conditions $2\rho /\mathrm{\Delta }\beta \ge 4$ and $\rho \gg f$. Consequently, one obtains the circular retardation caused by the Faraday Effect from Eq. (3b):

As will be shown numerically next that Faraday rotation angle of a spun fiber is equivalent to the circular retardation, which can be related to the electrical current by:

2.2. Numerical analysis and Poincaré sphere presentation

On a Poincaré sphere represented by Stokes parameter (S₁, S₂, S₃), Faraday Effect causes the input linear polarization to rotate on the equator of the Poincaré sphere in the absence of linear birefringence in an optical fiber. Similar to the Jones matrix presentation of Eq. (3a), in the presence of linear birefringence, the Muller matrix representing the SOP of light passing through the fiber can also be decomposed into a pure rotator and a pure retarder [17], which implies that the SOP trace caused by a variation of circular birefringence circles around S₃ axis in a plane parallel to the equator when the linear birefringence is fixed or changes much slower than the circular birefringence during the measurement [23,24]. In other words, in the presence fixed birefringence or extremely slow varying linear birefringence, the polarization rotation angle around S₃ axis corresponds to the Faraday rotation angle. Therefore, one may obtain the Verdet constant by measuring the polarization rotation angle induced by a magnetic field on the (S₁, S₂) plane, where the slow axis of the fiber is assumed to be aligned with the x-axis of the (x, y) coordinates defining (S₁, S₂, S₃). Figure 1(a) shows the trace of polarization variation on Poincaré sphere obtained using Eqs. (2) and (3) for the case that a linearly polarized light passing through a spun fiber of length 10 m coiled around an electrical conductor with different alternating currents (ACs). In the calculation, the input polarization is assumed to be linear and aligned with the fast axis of spun fiber’s residual birefringence ($\theta \left( L \right) = 0$) and the following parameters corresponding to a commercial fiber [13] are used: $\mathrm{\Delta }\beta = 598.5\; rad/m$, $\rho = 1309\; rad/m$, ${V_{H0}}\left( \lambda \right) = 5.97 \times {10^{ - 19}}\pi /{\lambda ^2}$ (Verdet constant of fused silica [3]), $\lambda = 1310\; nm$ and $r = 70\; mm$. Figure 1(b) shows the trace in Fig. 1(a) on the (S₁, S₂) plane and the corresponding arc angles. Similarly, the polarization variation trace induced by different direct currents (DCs) in the (S₁, S₂) plane are shown in Fig. 1(c). Note that corresponding to a particular current value, the polarization variation angle in Fig. 1(b) is $2\sqrt 2 $ times that in Fig. 1(c) because the AC swings between negative and positive values (the ACs in Fig. 1(b) are the rms values), resulting a current swing $2\sqrt 2 $ times larger than that in Fig. 1(c) with the same current label.

 

Fig. 1. (a) Polarization variations induced by ACs with amplitudes of 200 A, 400 A, 600 A, 800 A, 1000 A (rms), respectively. The projection of the polarization traces on the equatorial plane of Poincaréa sphere induced by different ACs (b) and DCs (c) flowing in the conductor. (d) Comparison between the circular retardations caused by the Faraday Effect of the exact solution of Eq. (3b) and the approximate solution of Eq. (4b) at different DCs. The relative errors between the approximate and exact solutions are shown at the corresponding data points.

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It can be seen from Eqs. (2) and (3a) that when the linear birefringence $\delta \left( L \right)$ is zero, the SOP rotation angle equals to the circular retardation $\varphi \left( L \right)$. For spun fibers with a weak linear birefringence, it can be show numerically using Eqs. (2) and (3) that the current induced SOP rotation angle is still approximately the same as the current induced circular retardation change $\mathrm{\Delta }{\varphi _F}$ calculated using Eq. (4b), assuming the input SOP is linear and $\theta = 0$ in Eq. (3a) with the same spun fiber parameters above. Therefore, we can use $\mathrm{\Delta }{\varphi _F}$ to represent the physical SOP rotation angle induced by electrical currents, as shown in Eq. (4c).

Now let’s compare the circular retardation caused by the Faraday Effect $\mathrm{\Delta }{\varphi _F}$ obtained using the exact solution Eq. (3b) and the approximate solution of Eq. (4b) at different current levels, with the results shown in Fig. 1(d). As can be seen, the maximum relative error of the approximate solution is less than 3%, which is defined as the difference between the exact and approximate solutions divided by the exact solution. And it validates that the approximation of Eq. (4b) is sufficient accurate and the effective Verdet constant is valid for representing the Faraday rotation in a spun fiber. Note that the SOP rotation angle $\mathrm{\Delta \Phi }$ on the Poincaré sphere is twice that of the Faraday rotation angle in real space ($\mathrm{\Delta }\mathrm{\Phi } = 2\mathrm{\Delta }{\varphi _F}$). From the data one can observe that the Faraday rotation angles are quite small, on the order of 0.14° for $\mathrm{\Delta }{\varphi _F}$ or 0.28° for $\mathrm{\Delta }\mathrm{\Phi }$ per 100 A current. Therefore, care must be taken to eliminate any polarization fluctuations and drifts caused by temperature or fiber motion during the experiment.

3. Experiment

3.1. Measurement system

In the earlier studies [3,18,20], the measurements of the linear and circular birefringent properties of optical fibers generally involved the careful adjustments of the polarizer, the analyzer, the quarter wave plate, and the Babinet-Soleil compensator (BSC) in the measurement system, as shown in Fig. 2(a). The birefringence and the orientation of the compensator could be adjusted to cancel out and thus obtain the birefringence of the fiber [18]. The length of the fiber under test was limited to 2-3 meters and must be kept straight to minimize stress induced linear birefringence in the fiber. It is not hard to imagine that these methods were tedious and time consuming to adjust, and therefore suffered from fluctuations and drifts caused by temperature variations and fiber motions during the measurement.

 

Fig. 2. (a) Early measurement setup using a Babinet-Soleil compensator for studying the birefringence properties in an optical fiber. (b) Binary MO polarization rotators based measurement system with a PSG to generate 6 distinctive SOPs and a PSA to analyze each SOP passing through the DUT. A computer is used to control the PSG and PSA, and process the data using Jones or Muller matrix analysis.

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In a previous publication [17], we used a highly accurate polarization analysis system [21] consisting of a tunable laser, a binary magneto-optic (MO) polarization state generator (PSG), a binary MO polarization analyzer (PSA) and a control computer shown in Fig. 2(b) to accurately obtained the circular birefringence and the residual linear birefringence as a function of temperature. The PSG and PSA each contains 6 binary MO polarization rotators and each of the rotators is capable of rotating the polarization by ±22.5°. In particular, when a positive and negative magnetic field above a saturation level is applied, the rotator rotates the SOP by a precise angle +22.5° and –22.5°, respectively. Collectively, the 6 binary MO rotators, together with the quarter wave plate, in the PSG can generate 6 distinctive states of polarization (SOPs) across the Poincaré sphere, and for each SOP the PSA can obtain 6 distinctive measurement data. Therefore, the PSG and PSA together are capable of obtaining the full Muller matrices of the device under test (DUT) and analyzing its birefringence properties [21]. Because of the high repeatability of the binary polarization rotators, any imperfections of the components used in the system can be calibrated out for achieving high measurement accuracy [25,26]. In the experiment, we used the same polarization analysis system for measuring the Faraday rotation angle described below.

Our experiment setup is shown in Fig. 3(a), which includes a 1310 nm distributed feedback (DFB) laser at the PSG input, a binary MO rotator based polarization analysis system (PSGA), a polarization controller (PC1) at the PSG output, a polarization controller (PC2) at the PSA input, a coiled spun fiber under test (FUT), a current generator capable of generating AC or DC up to 3 kA. A temperature sensor (PT100) is also included to monitor the temperature around the coiled fiber. The total length of the FUT is 10 m, with about 9.4 m coiled around the conductor and about 0.3 m fiber at each end for fusion splicing to two single mode fibers (SMF) connected to PC1 and PC2. The outputs of PC1 and PC2 are then connected to PSG and PSA, respectively, via two low stress connectors. The two fiber pigtails (0.3 m spun fiber each) and two SMFs (0.3 m each) are protected with gooseneck tubes to prevent fiber vibration or fiber motion due to the airflow.

 

Fig. 3. (a) Polarization measurement setup for obtaining the Faraday rotation angles induced by electrical currents in a coiled spun fiber. The inset shows the fixture to ensure the conductor rod to be perpendicular to the plane of the spun fiber coil and pass through the center of the coil placed on a sponge with a marked circular track. (b) The sketch of the spun fiber coil with a diameter of 138.8 mm and $N$ fiber turns. (c) The photo of the aluminum fixture for making the spun fiber coils with a precise diameter. (d) The coil on an adhesive tape after removed from the aluminum fixture. PC1 and PC2: polarization controllers; PSGA: polarization analysis system constructed with binary polarization rotators in a PSG and a PSA.

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In the setup, PC1 and PC2 are used for 1) re-align the coordinate system of PSG with that of PSA, which were mis-aligned from the SMF pigtails of the PSG output and the PSA input, as well as the FUT, and 2) to compensate for the linear birefringence in the SMF’s and FUT so that the Faraday rotation is along the equator of the PSA coordinate system to simplify data processing. Note that PC1 and PC2 are based on squeeze-and-turn mechanism [27], which is the optical fiber equivalent of Babinet-Soleil compensator [28].

As shown in the inset of Fig. 3(a), care was taken to ensure that the copper rod protected with insulating plastic is placed at the center of the fiber coil and is perpendicular to the coil plane for each measurement using a specially made aluminum frame table for minimizing any inconsistency between measurements which may result from the opening of the coil between the two fiber pigtails, as shown in Fig. 3(b), before connecting the coil to the PSGA instrument (no error will result from the Ampere's law if the coil is perfectly closed). The spun fiber coils are securely placed on a circular track marked on a sponge plate for keeping the coils at the same position in different measurements and for dampening acoustic vibrations.

Three different spun fibers from three different commercial vendors (IxF-SPUN-1310- 125-EC from iXblue Co. Ltd., SHB1250 (7.3/125) from Fibercore Co. Ltd. and SH 1310/12-5/250 from YOFC Co. Ltd.) for operation at 1310 nm were measured with different electrical currents from 0 to 1 kA. To ensure all spun fiber coils to be tested are identical, a fiber winding fixture was made with a diameter of 138.5 mm, as shown in Fig. 3(c). Before winding the fiber onto the fixture, a layer of adhesive tape with a thickness of 0.03 mm was tightly placed onto the winding surface of the fixture, which helped to hold the fiber together after the coil winding is complete. The resulting single-layer coils therefore all had the same diameter of 138.8 mm, which includes the diameter of the winding fixture (138.5 mm), the thickness of the adhesive tape (0.03×2 mm), and the fiber diameter of the spun fiber (0.25 mm), as shown in Fig. 3(d). In addition, all fiber coils were ensured to have the same number of turns (21 full turns, single layer) and the same length (10 m) during winding.

Note that coiling the fiber may induce additional linear birefringence, however, the diameter of the fiber coil is sufficiently large (138.8 mm) such that the linear birefringence introduced by the bending is on the order of ${\mathrm{10^{ - 7}}}$ [12,29], which is a order of magnitude smaller than the residual linear birefringence around 7.75×10⁻⁶ in the spun fiber [17]. Therefore, the bending induced birefringence from coiling the fiber can be safely ignored in our measurement.

3.2. Time differential method

In [17], the wavelength differential method was used to measure the circular birefringence and residual linear birefringence in some spun optical fibers. In particular, during the measurement, the wavelength of the laser is scanned while the SOP of light after passing through the DUT is analyzed. The birefringence of the DUT can be obtained by analyzing the rate of SOPs change with wavelength or frequency, with the SOPs change in the equator plane corresponding to the circular birefringence and perpendicular to the equator plane corresponding to the linear birefringence. Because the birefringence measured in [17] are relatively large, on the order of 10⁻⁷ or more, the wavelength scan induced SOP variation is on the order of 0.10° per nm at the central wavelength of 1310 nm for a fiber with a length of 10 m, sufficiently large to be accurately measured with our PSGA system with an angular measurement resolution of 0.02° (to be shown next).

When the fiber coiled around a conductor is subjected to an axial magnetic field (with an unit of Ampere per meter) induced by a current, the Faraday rotation occurs due to the circular birefringence induced by the Faraday Effect, which is extremely weak (∼10⁻¹³) compared with the intrinsic circular birefringence (∼10⁻⁵) and residual linear birefringence (10⁻⁶) of the spun fiber [17], and thus the corresponding SOPs change resulting from the wavelength change, on the order of $2.2 \times {10^{ - 4}}$ degrees per nm (assuming a fiber length of 10 m, with a fiber coil radius of 70 mm and under a current of 100A) on the Poincaré sphere, is too small to be measured with the wavelength differential method. In addition, the wavelength scanning method of our PSGA system takes about 20 seconds to complete a measurement, which is too slow to avoid polarization drift caused by temperature changes or slight fiber motion due to the airflow. Therefore, in this paper we adopt the time differential method: for Faraday rotation induced by an AC, we directly measure the SOP angle swing corresponding to the positive current maximum and the negative current maximum, as shown in Fig. 1(b); for Faraday rotation induced by a DC, we first turn the current on until it stabilizes and then off, and measure the SOP angle difference between the current “on” and “off” states, as shown in Fig. 1(c). Note that it only takes about 0.1 second to complete a differential SOP angle measurement and the effect of SOP drifts caused by the temperature or fiber motion can be effectively eliminated. As estimated theoretically in Section 2.2, the Faraday rotation is on the order of 0.28° per 100 A on the Poincaré sphere (assuming a fiber length of 10 m coiled around a conductor with a radius of 70 mm at 1310 nm), which is sufficiently large to be detected by our PSGA system.

3.3. System measurement resolution

Before conducting detailed measurements, it is important to understand the minimum detectable SOP rotation angle of our PSGA system. Figure 4(a) shows the measured SOP fluctuation and drift of the system when a spun fiber (Fibercore) of 10 m coiled around a conductor without current flow in an experimental setup shown in Fig. 3(a). In principle, temperature variation, fiber motion caused by airflow, and system electronic noise can all contribute to the SOP fluctuation and drift, however in our experiment, because the temperature was relatively stable during 20 minutes measurement period (brown line in Fig. 4(a)) and the fiber pigtails of the FUT were protected with gooseneck tubes, the SOP drift and fluctuation due to these two factors were minimal, and therefore the SOP measurement accuracy and resolution was limited by the system electronic noise (grey line in Fig. 4(a)). Taking 3-point average of the data, the electronic noise can be significantly reduced and the corresponding SOP fluctuation is less than $2 \times {10^{ - 4}}$ rad or 0.012°. Other digital filtering techniques, including taking more averages, can be used to further reduce the system noise floor. In addition, reducing the electronic noise of the amplifier and digital circuit may also be effective in reducing system noise floor.

 

Fig. 4. (a) System noise floor for differential SOP angle $\mathrm{\Delta }\mathrm{\Phi }$ measurement. (b) Five repeated measurements of $\mathrm{\Delta }\mathrm{\Phi }$ induced by currents of 10 A, 20 A, 50 A, and 100 A, each having a measurement time of $\mathrm{\Delta }t = 10$ seconds. 10 m SHB1250 (7.3/125) fiber from Fibercore was used.

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Figure 4(b) shows the measurement results of SOP change induced by electrical currents. In the measurement, the electrical currents of 10 A, 20 A, 50 A, and 100 A were turned on to reach the predetermined value for 10 seconds and then off periodically. The durations for the current off states in between the current on states in Fig. 4(b) varied from 10 to 13 seconds, depending on the current levels required by the power equipment operation, although they appeared to be the same in the figure. The differential SOP angle $\mathrm{\Delta }{\Phi }$ on the Poincaré sphere induced by the 10 A current was ${{4.6}} \times {{10^{ - 4}}}$ rad or 0.026°, obtained by averaging the data in the 10s period. Therefore a differential angle less than 0.026°, corresponding to a current less than 10 A, can be accurately measured with our PSGA system of Fig. 2(b).

3.4. Experimental results and discussions

Figure 5(a) shows the measured SOP variation of light propagating through a 10 m spun fiber (Fibercore) coiled around a conductor on the (S₁, S₂) plane caused by different ACs flowing in the conductor, with the measurement setup shown in Fig. 3(a). As described in Section 3.1, PC1 and PC2 were so adjusted that the SOP traces induced by electrical current were along the equator of the Poincaré sphere. Note that the SOP measurement rate of our PSGA system is about 30 Hz, while the AC varies at 50 Hz. The SOP traces would move back and forth on the equator in response to the cycles of the AC. It would take multiple cycles of SOP traces to determine the maximum swing of the SOP angle corresponding to the positive and negative maximum currents. In experiment, 250 cycles were taken for each AC current setting, corresponding to a measurement time of 5s, during which the polarization fluctuations induced by the environmental temperature or fiber motion could be noticed in the experiment. Consequently, the measurement results were less repeatable than those of the DC to be discussed next in Fig. 5(b).

 

Fig. 5. Measured differential SOP angles on the equator plane of the Poincaréa sphere in response to different electrical currents of 200 A, 400 A, 600 A, 800 A, 1000 A, respectively. O represents the center of the equatorial circle. (a) Alternating current (AC, rms value), and (b) direct current (DC). 10 m SHB1250 (7.3/125) fiber from Fibercore was used.

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Note that because of the slow SOP drift caused by the environmental temperature or fiber motion, the starting SOP for different current settings were different, however, the differential SOP angle corresponding to each current setting was much less affected by the SOP drift, an advantage of the time differential method. One may notice that the SOP variation traces are not exactly on the equator ($s_1^2 + s_2^2 = 1)$, probably because 1) PC1 and PC2 were not perfectly adjusted, and 2) the influence of the residual linear birefringence contribution in the spun fiber. This should have minimal impact on the measurement accuracy of Faraday rotation angle, because the deviation is extremely small.

Figure 5(b) shows the differential SOP angles induced by different DCs of light propagating in the same spun fiber as in Fig. 5(a). At each current setting, the current was turned on and off periodically to induce SOPs change in the (S₁, S₂) plane. In particular, we first turn on the current and wait 10 s for it to stabilize. We then turn off the current and measure the SOP change between the “on” and “off” states. Because almost no waiting period is required for the current to drop to zero, this differential SOP measurement can be completed in about 0.1s. Consequently, the effect of SOP drifts caused by the temperature or fiber motion can be effectively eliminated. 3-point averaging was used in processing the data (∼40 samples/s). Again, due to the SOP drift caused by the environmental temperature or fiber motion, the starting SOP for different current settings were different, however, the differential SOP angle corresponding to each current setting was not affected by the SOP drift. Note that the currents in Fig. 5(a) are rms values so that the differential SOP angle $\mathrm{\Delta \Phi }$ for each current setting with AC is $2\sqrt 2 $ times larger than that of DC with the same current label, because the AC swings from minus to plus, while DC changes from 0 to plus.

The differential SOP rotation angle $\mathrm{\Delta \Phi }$ as a function of current of three consecutive measurements is shown in Fig. 6(a). At high current setting above 700 A, heat was generated by the current flowing in the conductor, which slightly raised temperature around the spun fiber under test, as shown by the brown line in the figure monitored by a temperature sensor shown in Fig. 3(a). At the maximum current setting of 1000 A, the temperature was raised by only 0.3 °C, which should have negligible effect on the measurement accuracy. Curve-fitting yields a slope $\alpha = 4.62 \times {10^{ - 5}}$ rad/A, with a goodness-of-fit of 0.9998. The effective Verdet constant can be obtained from this slope and Eq. (4c) to be $1.05 \times {10^{ - 6}}$ rad/A using the following relation:

 

Fig. 6. (a) Three repeated measurements of the differential SOP $\mathrm{\Delta \Phi }$ on the equator plane of the Poincaréa sphere induced by different DCs with the 0° linear polarization (1, 0, 0). (b) Measurement results of $\mathrm{\Delta \Phi }$ as a function of DCs with four disctinctive input linear SOPs: 0° (1, 0, 0), 45° (0, 1, 0), 90° (-1, 0, 0), and 135° (0, -1, 0). 10 m SHB1250 (7.3/125) fiber from Fibercore was used.

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One may circumvent directly using $r$ and $L$ for ${V_{eff}}$ calculation by accurately obtaining the number of turns $N$ in the fiber coil from Ampere's law reflected in the last term of Eq. (6). However, complication arises due to the existence of two fiber pigtails (as shown in Fig. 3(b) and Fig. 3(d)) of the coil, which fortunately has been successfully accounted for by A. H. Rose and S. M. Etzel in [30]. Accordingly, the fiber pigtails with a length of ${L_0}$ and a separation of ${W_0}$ shown in Fig. 3(b) can be counted as effective fractional turns so that the total number of fiber turns can be expressed as:

The differential SOP rotation angle $\mathrm{\Delta \Phi }$ induced by electrical current is expected to be the same with different input linear SOPs to the spun fiber, or with different input SOPs on the equator of the Poincaré sphere. Figure 6(b) shows the measurement results of $\mathrm{\Delta \Phi }$ for the cases that the input SOPs generated by PSG are 0°, 45°, 90° and 135°, respectively. Note that although there is a single mode fiber (SMF) connecting the PSG to the spun fiber, the proper adjustment PC1 can undo the effect of the birefringence in the SMFs to ensure the SOP entering the spun fiber is linear. In between the measurements with different input SOPs, PC1 and PC2 were slightly readjusted to balance out the birefringence offsets in the SMFs connecting the spun fiber to the PSG at the input end, as well as to the PSA at the output end of the spun fiber. As one can see from Fig. 6(b) that the four curves corresponding to the four input linear SOPs are almost identical, yielding almost the same slope. The average slope obtained from Fig. 6(b) is ${\mathrm{\bar{{\alpha }}}\textrm{}} = 4.61 \times {10^{ - 5}}$ rad/A.

3.5. Effective Verdet constants of three different spun fibers

Different manufacturers choose different preforms with different birefringence or unspun fiber retardation per unit length, and different spin twist rates to produce spun fibers. These differences make different spun fibers having different magnetic sensitivity characterized by their corresponding Verdet constants. Using the published data from different fiber manufacturers, the effective Verdet constants ${V_{eff}}$ and the corresponding Faraday Effect reduction factor $\mathrm{\gamma }$ defined in Eq. (5b) can be obtained, as shown in Table 1. Note that for the Fibercore fiber, only the parameters of circular beat length ${L_c}$ of 63 - 125 mm and spin pitch ${L_t}$ of 4.8 mm are provided, which can be used to obtain the unspun fiber retardation per unit length $\mathrm{\Delta }\beta $ and the spin twist rate $\rho $ using the following relations [17]: ${L_p} = {\left( {\frac{1}{{L_c^2}} + \frac{1}{{{L_t}{L_c}}}} \right)^{ - 1/2}}$, $\mathrm{\Delta }\beta = 2\pi /{L_p}$, and $\rho = 2\pi /{L_t}$, where ${L_p}$ is the linear birefringence beat length of unspun fiber.

Table 1. The data comparison of effective Verdet constants.

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Figure 7 shows the measured differential SOP rotation angles on the equator plane as a function of applied DCs. The effective Verdet constant ${V_{eff}}$ of each spun fiber can be obtained from its curve-fitting slope using Eq. (6). The corresponding Faraday Effect reduction factor $\gamma $ of each fiber can be obtained by taking the ratio of the effective Verdet constant and the Verdet constant of an ideal silica fiber free of birefringence, which is $1.09 \times {10^{ - 6}}\;\textrm{ rad}/\textrm{A}$. The results are listed in the last two columns in Table 1. It can be seen that the spun fiber from iXblue has the highest effective Verdet constant, followed by that from Fibercore and finally YOFC. The corresponding Faraday Effect reduction factors of the three fibers from an ideal fiber are 0.98, 0.96, and 0.95, respectively.

 

Fig. 7. Measured differential SOP rotation angles induced by different levels of DC. ${\alpha _i} = 4.71 \times {10^{ - 5}}\; \textrm{rad}/\textrm{A}$, ${\alpha _F} = 4.61 \times {10^{ - 5}}\; \textrm{rad}/\textrm{A}$ and ${\alpha _Y} = 4.56 \times {10^{ - 5}}\; \textrm{rad}/\textrm{A}$ are the slopes of the fitting curves of the spun fibers from iXblue (IxF-Spun-1310-125-EC), Fibercore (SHB1250 (7.3/125)) and YOFC (SH1310/125/250), respectively.

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The measurement results are consistent with our theoretical analysis and validate the theoretical conclusion that the higher the ratio of $\rho /\mathrm{\Delta }\beta $, the larger the effective Verdet constant or the more sensitive the Faraday rotation to the applied current or magnetic field. It is expected that the iXblue fiber shall also be less sensitive to local perturbations caused by stress and temperature because of the large local birefringence reflected by the large $\mathrm{\Delta }\beta $.

4. Summary

In this paper, the term “effective Verdet constant” is introduced for the first time to quantitatively characterize the sensitivity of Faraday Effect in practical spun optical fibers and is measured with a highly accurate binary polarization rotators based polarization analysis system. In particular, we obtain the expression of the effective Verdet constant of a spun fiber showing that it is always less than that of an ideal fused silica fiber by a factor relating to the ratio of the spin twist rate to the retardation per unit length of the unspun fiber. The larger the ratio, the closer the effective Verdet constant to that of the ideal fiber is. We measure the Faraday rotation angles induced by different amounts of currents with an ultra-sensitive time differential method having an SOP angle resolution of better than 0.02° and experimentally obtain the effective Verdet constants of three different high birefringence spun fibers from three different manufacturers at 1310 nm, with values of ${{1.07}} \times {{10^{ - 6}}}$ rad/A, ${{1.05}} \times {{10^{ - 6}}}$ rad/A, and ${{1.04}} \times {10^{ - 6}}$ rad/A, respectively, which are 98%, 96%, and 95% of that of the ideal fused silica fiber free of birefringence. These values are consistent with our theoretical estimates using the parameters provided by fiber manufacturers of these fibers.