Distributed measurement of polarization mode coupling in fiber ring based on P-OTDR complete polarization state detection

Zejia Huang; Chongqing Wu; Zhi Wang; Jian Wang; Lanlan Liu

doi:10.1364/OE.26.004798

1. Introduction

As an important inertial measurement technique, fiber optic gyro (FOG) has aroused great attention in many countries since it was first proposed in 1976 [1]. After years of rapid development, FOG has become a mainstream alternative in the field of inertial technologies and plays a significant role in aircraft surveillance guidance and navigation, attitude control of armored vehicles and tilting trains, spacecraft stabilization, etc [2,3]. As a core component of the FOG, PM fiber ring is made of polarization-maintaining fiber (PMF) with good polarization maintaining performance. However, the inherent polarization-mode coupling (PMC) induced by the fiber structural defects and PMC induced by the external stress and polarization axes misalignment during winding [4–7] may cause critical polarization fluctuations, resulting in random gyro drift which would severely restrict the improvement in measurement accuracy [8–11]. In some reports, polarization fluctuations can be suppressed either by a fiber ring resonator with twin 90°polarization-axis rotated splices or by setting the fiber polarizers at the lead portions of the fiber ring resonator [12–14]. However, to fundamentally improve the performance of FOG, it is significant to characterize the PMC of the PM fiber ring and improve the positioning accuracy of the coupling points along its length.

At present, the mode coupling detection of the PM fiber ring relies mainly on optical fiber white light interferometer [15–20], which maps the mode coupling points by compensating the optical path difference of two orthogonal polarization beams induced by the given coupling points. However, this technology has the following disadvantages. First, the PM fiber ring only resonates at specific wavelengths and the measurements of PMC for every given wavelength are required since PMC is wavelength-dependent, while the mode coupling measured by the optical fiber white light interferometer with a broadband light source is an average over the broad wavelength range, which cannot accurately characterize the PMC of different resonant wavelengths. Second, the polarization axis misalignment between two adjacent fibers is one of the causes of PMC and the misalignment angle is either positive or negative [21], which means that the PMC coefficient should be either positive or negative. However, the mode coupling measured by the white light interferometer is a power coupling coefficient which can only be positive, thus it is not qualified to characterize the real PMC. Third, since the broadband light can cause severe polarization-mode dispersion, the spatial resolution will reduce with the increase in fiber length. Fourth, the second-order coupling may lead to the appearance of ghost coupling points [5]. Fifth, the optical fiber white light interferometer changes the optical path difference between the sample and the reference arms through mechanical scanning and the state of polarization (SOP) of the reflected beam will change randomly during the movement of mirror, which may lead to considerable measurement error. After years of development, the practical and commercialized optical fiber white light interferometer has reached a very high level, capable of measuring ordinary PM fiber ring. However, for higher-quality PM fiber ring of which the PMC is much weaker, the measurement requirements are higher. The optical fiber white light interferometer is limited by its measurement principle and has limited space for further improvement, thus innovative technologies for PMC distributed detection of the PM fiber ring are required to be explored.

Polarization optical time-domain reflectometry (P-OTDR) [22] has been used to obtain the internal optical parameters and the external environmental variation, such as stress, temperature, bending, vibration, etc [23], along the tested fiber by measuring the SOP of Rayleigh backscattered light at the input end. Generally, there are two kinds of P-OTDR. One is based on the power measurement of Rayleigh backscattering in a given polarization direction, namely, the incomplete P-OTDR with relatively low measurement accuracy and spatial resolution (tens of meters) in positioning the mode coupling points [24–28]. Another one is based on the complete SOP measurement of Rayleigh backscattering, namely, the complete P-OTDR [29–32] with higher measurement accuracy because the distributed complete SOP’s (i.e. four Stokes parameters) are obtained along the tested fiber. Up to now, most reports about complete P-OTDR are focusing on the single-mode fiber (SMF) but neglecting the PMF, which is capable of evaluating the polarization maintaining performance of the PM fiber ring. In theory, the pulses width of P-OTDR should be less than 1/4 beat length [33], but the beat length of the polarization maintaining fiber (PMF) is very short, the light source can hardly meet the requirement. However, the pulses, as well as the Rayleigh backscattering, will be transmitted in the direction of the polarization axis once the SOP of the input pulses is aligned with the polarization axis at the input end of the tested PMF. In this case, the pulses width is not required to be less than 1/4 beat length, which means that the complete P-OTDR based on PMF is feasible.

In this paper, a complete P-OTDR scheme based on complete SOP detection is proposed to achieve distributed measurement of the PMC coefficient and evaluate the polarization maintaining performance of the tested PM fiber ring. In Section 2, we prove that the PMC coefficient can be derived from three components of the Stokes vectors at three adjacent points along a fiber using a quaternion-based method. In Section 3, two bow-tie PM fiber rings with different lengths are employed respectively to experimentally verify the above theory, which reveals that the proposed complete P-OTDR scheme is capable to realize the distributed measurement of the PMC characteristics along PM fiber ring with high spatial resolution. In addition, this scheme can also be used to measure the extinction ratio along the PMF simultaneously.

2. The measurement principle of the PMC coefficient

The Stokes vector $S$ rotates on the Poincare sphere with a radius of 1, hence its rate $\partial S / \partial z$ is perpendicular to $S$ and the relationship between them can be expressed as

\partial S / \partial z = B \times S

where

B = | B | \hat{b}

, the unit vector

\hat{b}

denotes the polarization direction of the local principal axis, the magnitude

| B |

, namely the magnitude of the local birefringence, denotes the rate at which the SOP rotates around the principal axis. Considering the impact of PMC on the polarization state transmission, vector

B

should include the PMC coefficient. Next we use a quaternion-based method to derive the relationship between the vector

B

, the transmission constant difference

Δ β = β_{x} - β_{y}

of two orthogonal polarization modes and their PMC coefficient

k

.

Neglecting the loss of the PMF, the coupled mode equations of two orthogonal polarization modes can theoretically be written in the form of Jones vector as follow,

\frac{\partial}{\partial z} [\begin{matrix} {\dot{E}}_{x} \\ {\dot{E}}_{y} \end{matrix}] = [\begin{matrix} i β_{x} & i k \\ i k & i β_{y} \end{matrix}] [\begin{matrix} {\dot{E}}_{x} \\ {\dot{E}}_{y} \end{matrix}]

where

{\dot{E}}_{x}

and

{\dot{E}}_{y}

are the electric field component of the fast and slow axes,

β_{x}

and

β_{y}

are the transmission constants of the fast and slow axes,

k

is the PMC coefficient between the fast and slow axes,

z

is the length of the PMF. Equation (2) can be simplified into the form of quaternion as follow,

\partial J / \partial z = U J

where

J

is the quaternion that corresponds to the Jones vector

{[\begin{matrix} {\dot{E}}_{x} & {\dot{E}}_{y} \end{matrix}]}^{T}

,

U

is the quaternion that corresponds to the Jones matrix

[\begin{matrix} i β_{x} & i k \\ i k & i β_{y} \end{matrix}]

. The Jones matrix can be decomposed as follow,

[\begin{matrix} i β_{x} & i k \\ i k & i β_{y} \end{matrix}] = i \bar{β} [\begin{matrix} 1 \\ 1 \end{matrix}] + i \frac{Δ β}{2} [\begin{matrix} 1 \\ - 1 \end{matrix}] + i k [\begin{matrix} 1 \\ 1 \end{matrix}]

where

\bar{β} = (β_{x} + β_{y}) / 2

is the average transmission constant of the fast and slow axes,

Δ β = β_{x} - β_{y}

is the transmission constant difference, thus the corresponding quaternion of the Jones matrix can be written as

U = i \bar{β} - (Δ β / 2) \hat{i} - k \hat{j}

Substituting Eq. (5) into Eq. (3), Eq. (3) can be written as

\begin{array}{l} \partial J / \partial z = (i \bar{β} - (Δ β / 2) \hat{i} - k \hat{j}) J \\ \partial J^{†} / \partial z = J^{†} (- i \bar{β} + (Δ β / 2) \hat{i} + k \hat{j}) \end{array}}

where

J^{†}

is the Hermitian transpose of

J

, since the Stokes quaternion

S = 2 J J^{†}

, the rate of

S

can be written as

\partial S / \partial z = 2 [(\partial J / \partial z) J^{†} + J (\partial J^{†} / \partial z)]

Substituting Eq. (6) into Eq. (7), Eq. (7) can be written as

\partial S / \partial z = - (Δ β \hat{i} + 2 k \hat{j}) \times i S

The Stokes quaternion $S = s_{0} + i S$ , where $s_{0}$ is the scalar part and $S$ is the vector part, respectively. Substituting it into Eq. (8), then Eq. (8) can be decomposed into

\partial s_{0} / \partial z = 0

\partial S / \partial z = - (Δ β \hat{i} + 2 k \hat{j}) \times S

Compare Eq. (10) and Eq. (1), $B$ is finally derived as

B = - (Δ β \hat{i} + 2 k \hat{j})

It means that once the projections of vector $B$ on the $\hat{i}$ and $\hat{j}$ axes are given, the transmission constant difference $Δ β$ and the PMC coefficient $k$ can be obtained. Furthermore, based on the three points quaternion method we have previously proposed in [32], the projections of vector $B$ on the $\hat{i}$ and $\hat{j}$ axes are calculable as long as the Stokes vectors of three adjacent points along a fiber are obtained. In addition, in [32] we have previously proved that for a fiber shown in Fig. 1, the rotation angles between the polarization quaternion $S_{in} (A)$ , $S_{in} (B)$ and $S_{in} (C)$ of three adjacent points that detected at the input end of the tested fiber are the same as the rotation angles between $S_{A} (A)$ , $S_{A} (B)$ and $S_{A} (C)$ that detected at the point A. Therefore, the rotation angles between the Stokes vectors $S_{in} (A)$ , $S_{in} (B)$ and $S_{in} (C)$ are consistent with the rotation angles between $S_{A} (A)$ , $S_{A} (B)$ and $S_{A} (C)$ on the Poincare sphere.

Fig. 1 Three adjacent points A, B and C along a fiber.

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In conclusion, the PMC coefficient $k$ which characterizes the polarization-maintaining performance of a short section of PMF between A and C can be calculated by using the three points quaternion method as long as the Stokes vectors of three adjacent points A, B and C are obtained at the input end of the tested fiber in a complete P-OTDR system.

3. Experiment results and discussion

The setup of complete P-OTDR based on complete SOP detection is shown schematically in Fig. 2. High peak power pulses (30dBm) of known SOP’s are generated by the distributed feedback (DFB) laser, modulator, polarization beam splitter (PBS), erbium-doped fiber amplifier (EDFA), narrowband filter, and polarization controller (PC) by which the SOP’s are modulated to be aligned with either the fast axis or the slow axis of the tested PM fiber ring at the input end, and launched into the tested PM fiber ring via a circulator. After transmitting through the PM fiber ring, the pulses are received by a polarization analyzer in where the SOP’s are observed in real time, and fed to a digital oscilloscope as a trigger signal. The returning Rayleigh backscattering is coupled by the circulator into the EDFA, narrow-band filter, and PC by which the coordinate system of the tested PM fiber ring is rotated to be consistent with the coordinate system of the following Stokes components detectors, and split into three arms that correspond to three Stokes components (i.e. S1, S2 and S3), respectively. The split Rayleigh backscattering in each arm are modulated, split and detected separately by a PC, PBS and 300MHz bandwidth balanced analog receiver (BAR) and fed to a three-channel digital oscilloscope which averages 10 times the three Stokes components prior to transfer to a computer where the SOP’s and PMC coefficient are calculated and displayed as functions of fiber ring length.

Fig. 2 The setup of complete P-OTDR. DFB: distributed feedback. PPG: programmable pattern generator. PBS: polarization beam splitter. EDFA: erbium-doped fiber amplifier. PC: polarization controller. BAR: balanced analog receiver.

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An 83 m long and an 840 m long bow-tie PM fiber ring (Beijing Glass Research Institute) are tested in the following experiments, respectively. For the 83 meters PM fiber ring, the pulses width, repetition period and the digital oscilloscope sampling frequency are 10 ns, 2 us and 10GHz, respectively. The pulse width of 10 ns determines the spatial resolution of 1 m. In order to verify the credibility of the experimental results, the experiments are conducted when the SOP’s of the incident pulses are aligned with the slow-axis and fast-axis in the forward direction, and with the slow-axis only in the backward direction, respectively. The PMC coefficients obtained with the SOP’s of the incident pulses respectively aligned with the fast and slow axes in the forward direction are compared as shown in Fig. 3(a). The PMC coefficients at the same position measured through two orthogonal polarization axes are opposite, which is explainable by Eq. (2). For example, at some position along the PM fiber ring, the measured PMC coefficient is positive with the SOP’s of the incident pulses aligned with the fast axis, which means at this position the pulses are coupled from the slow axis to the fast axis, so that the measured PMC coefficient at this position is negative with the SOP’s of the incident pulses aligned with the slow axis. In addition, the PMC coefficients obtained with the incident pulses inputted at the opposite ends of the PM fiber ring are compared as shown in Fig. 3(b), from which we can see that at the same position, the measured PMC coefficients in opposite directions are almost consistent. Comparing the curves of slow-axis, fast-axis, forward and backward, the positions A, B, C and D where the PMC are much severe and the position E where the PMC is very weak have very good agreement with positioning error of less than 1 m and spatial resolution of 1 m, which is dependent on the pulses width and the bandwidth of the balanced analog receiver.

Fig. 3 The measured PMC coefficients along the 83 m long PM fiber ring. (a) The SOP’s of the incident pulses are aligned with the fast axis (red) and slow axis (blue) in the forward direction, respectively. (b) The incident pulses, which aligned with the slow-axis, are inputted in the forward direction (blue) and the backward direction (green), respectively.

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In order to visualize the impact of PMC on the polarization state, the distribution of SOP’s at different positions along the PM fiber ring obtained from multiple measurements (1000 times) are depicted on the Poincare sphere in Figs. 4(a) and 4(b) correspond to the positions A and E in Figs. 3(a) and 3(b) respectively. Here the Stokes vector [S1, S2, S3] which aligned with the slow-axis of the tested PM fiber ring is defined as vector [1, 0, 0] on the Poincare sphere. It can be seen that the PMC at position A is relatively large and the corresponding SOP’s deviate from [1, 0, 0] on the Poincare sphere as shown in Fig. 4(a). As a comparison, the PMC at position E is almost zero and the corresponding SOP’s are concentrated around [1, 0, 0] on the Poincare sphere as shown in Fig. 4(b).

Fig. 4 The distribution of SOP’s at different positions along the 83 m long PM fiber ring on the Poincare sphere (1000 measurements). (a) Position A. (b) Position E.

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As shown in Fig. 5, to verify the repeatability of the PMC measurement along PM fiber ring based on the proposed complete P-OTDR scheme, repeated experiments of two opposite directions and two orthogonal polarization axes are respectively conducted with measurement times of 4000 and repeat intervals of 5 s. Comparing the distribution of PMC coefficients measured under different conditions, the positioning results of severe PMC points previously indicated in Fig. 3(e.g., points A, B, C and D in Fig. 5) show a high degree of consistency and stability over a long period of time, which demonstrates that the proposed complete P-OTDR scheme is highly repeatable and reliable for measuring the distribution of PMC coefficient along PM fiber ring.

Fig. 5 2-D heat map of the distribution of PMC coefficients along the 83 m long PM fiber ring by measuring 4000 times repeatedly. (a) Pulses are inputted in the forward direction with SOP’s aligned with the fast-axis. (b) Pulses are inputted in the forward direction with SOP’s aligned with the slow-axis. (c) Pulses are inputted in the backward direction with SOP’s aligned with the slow-axis.

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For the 840 m long PM fiber ring, the pulses width, repetition period and the digital oscilloscope sampling frequency are 50 ns, 20 us and 1GHz, respectively. The pulse width of 50 ns determines the spatial resolution of 5 m. Compare the measured PMC coefficients curves along the PM fiber ring shown in Fig. 6(a) as the pulses inputted in the forward and backward directions, respectively. In addition to the severe PMC induced by splicing at both ends of the PM fiber ring, the points A, B and C with relatively severe PMC measured in two opposite directions also have good agreement. However, at some points such as D, the PMC coefficients measured in opposite directions vary greatly since the PMC coefficient is almost zero in the forward direction while it is relatively large in the backward direction. In order to find out the cause, the variation of Rayleigh backscattered power with time at positions A, B, C and D are studied as depicted in Fig. 6(b). It shows that the Rayleigh backscattered power fluctuates at the same position with time and varies greatly at different positions, especially at position D of which the Rayleigh backscattered power is so weak that it almost exceeds the measurement limit of the detector, thus the measurement errors of SOP’s at this point are relatively large and eventually leading to false alarm.

Fig. 6 (a) The measured PMC coefficients along the 840 m long PM fiber ring with pulses inputted in opposite directions. (b) The variation of Rayleigh backscattered power with time at positions A, B, C and D along the 840 m long PM fiber ring.

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It should be noted that the influential factors of sensing distance and spatial resolution mainly include the pulse peak power, pulse width, the bandwidth and response power of the balanced analog receiver, and the loss of PM fibers. In the above experiments, the incident pulse widths are 10 ns and 50 ns for two PM fibers with length of 83 m and 840 m, respectively. The deciding factor of the pulse widths for different length fibers is the loss of the PM fibers. For the longer fiber with greater loss, larger power pulses are required in order to receive significant Rayleigh backscattering. The power of pulses depend on the peak power as well as the pulse width. Since the high peak power may generate Brillouin backscattering, which could introduce errors, so modulating the pulse width is a better choice. However, in the above experiments the pulse peak power is as high as 30dBm. From the backscattered light spectrum we found that this high peak power lead to Brillouin backscattering, but the Brillouin backscattering power is much weaker than the Rayleigh backscattering power, thus the Brillouin backscattered light introduces very little noise and has little influence on the measurement error.

So far the feasibility of PMC coefficient distributed measurement of PM fiber ring based on complete P-OTDR has been verified by comparing the measurement results of the opposite directions, orthogonal polarization axes and different lengths. In the above experiments, the spatial resolution of the PMC positioning only reaches 1 m and 5 m for the shorter fiber and the longer fiber, respectively, which is mainly limited by the injected pulse width and the bandwidth of the balanced analog receiver used in our experiments system. In addition, the proposed method can also be applied to measure the PMC induced by external factors (such as stress, bending, vibration, etc.) and realize distributed measurement of various parameters. Although the weak Rayleigh backscattered power may lead to false alarm, increasing the Rayleigh backscattered power and the responsivity of the balanced analog receiver are the key to improve the positioning accuracy.

4. Conclusion

Using a quaternion-based method, it is theoretically proved that the PMC coefficient can be obtained from three Stokes components [S1, S2, S3] at three adjacent positions along a fiber. A complete P-OTDR scheme for PMC distributed measurement in PM fiber ring is proposed based on the above theory. By comparing the measurement results of the opposite directions, orthogonal polarization axes and different lengths, the feasibility and high repeatability of the scheme are verified experimentally with positioning spatial resolution of 1 meter. In addition, the method can also be applied to measure the PMC induced by external factors (such as stress, bending, vibration, etc.) and realize distributed measurement of various parameters.

Funding

National Natural Science Foundation of China (NSFC) (61775012 and 61571035).

Acknowledgments

We thank Beijing Glass Research Institute (BGRI) for providing the PM fiber rings.

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Distributed measurement of polarization mode coupling in fiber ring based on P-OTDR complete polarization state detection

Abstract

1. Introduction

2. The measurement principle of the PMC coefficient

3. Experiment results and discussion

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

Optics Express