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ARTICLE IN PRESSG ModelJLEO-54624; No. of Pages 6

Optik xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Optik

jo ur nal homepage: www.elsev ier .de / i j leo

odeling and experimental verification of polarization errors inagnac fiber optic current sensor

ihui Wanga,∗, Jianfei Ji b, Qiangsheng Bob, Yubo Yuanb, Jiabin Qiana

Southeast University, School of Instrument Science and Engineering, Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology,inistry of Education, Nanjing 210096, ChinaJiangsu Electrical Power Company Research Institute, Nanjing 211103, China

r t i c l e i n f o

rticle history:eceived 11 September 2013ccepted 15 April 2014vailable online xxx

eywords:iber optic current sensorptical polarization stateirefringenceonreciprocal phase

a b s t r a c t

Sagnac fiber optic current sensor (S-FOCS) is a kind of optical interferometer based on Sagnac structure,optical polarization states of sensing light wave in Sagnac fiber optic current sensor are limited. However,several factors induce optical polarization error, and non-ideal polarized light waves cause the interfer-ence signal crosstalk in sensor, including polarizer, quarter-wave retarder, splice angular, birefringenceand so on. With these errors, linearly polarized light wave in PM fiber and circularly polarized lightwave in sensing fiber become elliptically polarized light wave, then, nonreciprocal phase shift inducedby magnetic field of the current is interrupted by wrong polarization state. To clarify characteristics ofoptical polarization error in fiber optic current sensor, we analyze the evolution process of random opti-cal polarization state, linear optical polarization state and circular optical polarization state in Sagnacfiber optic current sensor by using Poincare sphere, then, build optical polarization error models by usingJones matrix. Based on models of polarization state in Sagnac fiber optic current sensor, we investigate theinfluence of several main error factors on optical polarization error characteristics theoretically, includingextinction ratio in polarizer, phase delay in quarter-wave retarder, splice angular between quarter-waveretarder and polarization maintaining fiber. Finally, we simulate and quantify nonreciprocal phase shift

to be detected in fiber optic current sensor related with optical polarization errors. In the end, we demon-strate S-FOCS in test. The results show that transfer matrix errors are induced by inaccurate polarizationproperties during polarization state conversion, then, the stability and accuracy of the S-FOCS are affected,and it is important to control the polarization properties at each step of the polarization state conversionprecisely.

© 2014 Elsevier GmbH. All rights reserved.

. Introduction

Sagnac fiber optic current sensor (S-FOCS) has following merits,ncluding broad dynamic range, high measuring accuracy, excel-ent insulating performance, and so on, and is attractive for highynamic range current sensing [1,2] In S-FOCS, the nonrecipro-al phase shift is accumulated by circularly polarized light wavesraveling in opposite polarization directions with respect to the

agnetic field produced by the current to be measured, and phaseifferences from 0.1 �rad can be accurately detected by using

Please cite this article in press as: L. Wang, et al., Modeling and expecurrent sensor, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.o

ppropriate phase modulation/demodulation schemes developedor use in the fiber gyroscope [3]. Method of sensing current in-FOCS suffers from a number of difficulties. It is necessary to

∗ Corresponding author.E-mail address: wlhseu@163.com (L. Wang).

ttp://dx.doi.org/10.1016/j.ijleo.2014.04.064030-4026/© 2014 Elsevier GmbH. All rights reserved.

produce and maintain high-quality states of circular polarizationwithin the sensing fiber, and high-quality states of linear polariza-tion within the polarization maintaining (PM) fiber. Exceptionallystable optical components are required for measuring the polariza-tion state changes with the accuracy needed for certain applications[3,15]. In practice, several key components in optic loop showerror characteristics, including polarizer, quarter-wave retarderand sensing head, which induces linearly polarized light waveand circularly polarized light wave in sensing fiber to be ellipti-cally polarized light wave, then nonreciprocal phase shift inducedby current is indistinguishable from those induced by imperfectcomponents. This paper demonstrates polarization models, eval-uates the influence on fiber optic current sensor performance by

rimental verification of polarization errors in Sagnac fiber opticrg/10.1016/j.ijleo.2014.04.064

polarization error factors in imperfect components, then, verifiespolarization errors in Sagnac fiber optic current sensor experi-mentally and proposes methods to suppress optical polarizationerror.

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ematical model of polarization state in S-FOCS. Jones matrix and

Fig. 1. Sagnac fiber optic current sensor.

. Conversion process of optical polarization state in S-FOCS

Current flowing in a wire induces a magnetic field which,hrough the Faraday effect, rotates the plane of polarization ofight traveling in an optical fiber wound around the wire [3]. Fig. 1hows structure of Sagnac fiber optic current sensor, which is aurrent sensing Faraday/Sagnac loop interferometer, commonlysed in gyroscopes and it is sensitive to non-reciprocal effects3,10]. The non-reciprocal phase shift �ϕ between the two counter-ropagating beams is twice the polarization rotation angle induced

n the Faraday rotation scheme, �ϕ is given by

ϕ = 2V · N · I

here V is the Verdet constant of the fiber glass, N is number ofber turns in sensing head, and I is the current in the wire.

Propagation process of light wave in S-FOCS can be described asollows. Light from source is directed through a single-mode fiberoupler to a fiber polarizer. The polarized light is equally split intowo beams, and converted to circularly polarized light waves by auarter-wave retarder just prior to the sensing region. Two beams

ight travel in opposite directions with respect to the magneticeld, and return to linear polarization states on passing through theuarter-wave retarder on the return trip, finally, they back together,hen, they are interfered in polarizer and coupled to the detector. Ahase shift caused by the magnetic field is produced between thewo circularly polarized polarization modes in the sensing region,hich is two times the single-pass Faraday rotation [4,11,12].

Mutual conversion between linear polarization state and cir-ular polarization state is an important issue, and the conversionrocess can be analyzed as follows. Non-linear-polarized light wavemitted from SLD with Jones matrix [Px Py]T travels through polar-zer, and light wave in y axis is filtered, therefore, [Px Py]T becomesinear polarized light wave [Px 0]T. As described upper, light wavePx 0]T experiences 45◦ splice and quarter-wave retarder, the coor-inate transformation process of polarized light-wave is describedy Fig. 2. Fig. 2 shows coordinate of polarization axes between PMber and quarter-wave retarder.

With linearly polarized light wave Px from polarizer and angular between axes of polarizer and PM fiber, components of light wave

n PM fiber’s fast axis and slow axis are given by

Please cite this article in press as: L. Wang, et al., Modeling and expecurrent sensor, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.o

Pxx Pxy]T = [Px cos Px sin ˛]T (1)

Pxx

Fig. 2. Coordinate transformation process of polarized light-wave.

Matrix of light wave component Pxx passing through angular � andquarter-wave retarder is given by

[M]Pxx = [M]ˇ,�/4

[0

Px sin ˛

]=

[[sin cos ˇ(1 − j)]Px sin ˛

[sin2 + j cos2 ˇ]Px sin ˛

](2)

where is angular between axes of PM fiber and quarter-waveretarder, [M]ˇ,�/4 is Jones matrix of quarter-wave retarder. Eq. (2)shows that Pxx is changed to be right hand circularly polarized lightwave.

[M]ˇ,�/4 =[

cos2 + j sin2

sin cos ˇ(1 − j)

sin cos ˇ(1 − j) sin2 + j cos2 ˇ

](3)

Similarly, Matrix of light wave component Pxy passing throughangular and quarter-wave retarder is given by

[M]Pxy = [M]ˇ,�/4

[Px cos ˛

0

]=

[(cos2

+ j sin2 ˇ)Px cos ˛[sin cos ˇ (1 − j)

]Px cos ˛

](4)

Eq. (4) shows that Pxy is changed to be left hand circularly polar-ized light wave.

3. Modeling of optical polarization state in S-FOCS withpoincare sphere

To meet the index requirement of FOCS accuracy within ±0.2%,it is critical to restrain polarization errors, which results from spu-rious interference of light traveling around the optical loop inthe wrong polarization state [3–6,16]. There are several kinds ofpolarization states in S-FOCS, which makes polarization analysis ofS-FOCS complex, as polarization state may change anywhere withinPM fiber and single mode sensing coil, and some light couple to thewrong polarization state by means of polarization cross-coupling,depending on optical components’ performance and fiber’s bire-fringence induced by internally or externally stresses, such asinherent imperfections, temperature, vibration, and so on [3,7–9].Polarization state changes fast enough to introduce a phase dif-ference between counter propagating waves, such as time varyingnonreciprocal. These errors appear as a nonreciprocal phase shiftindistinguishable from that induced by current, and influence thebias and scale factor stability of S-FOCS.

Despite the mathematical complexity of some phenomena isobserved in S-FOCS, the Poincare sphere provides elegant andsimple geometrical interpretations. Focused on polarization char-acteristics in S-FOCS, this section describes polarization evolvingprocess with Poincare sphere and Jones matrix, then, builds math-

rimental verification of polarization errors in Sagnac fiber opticrg/10.1016/j.ijleo.2014.04.064

Poincare sphere are combined, and they offer both a mathemati-cal representation and an intuitive interpretation of nonreciprocalpolarization effects in S-FOCS.

ARTICLE ING ModelIJLEO-54624; No. of Pages 6

L. Wang et al. / Optik xx

Fig. 3. Polarization from linear (Polarizer) to circular (Retarder) on Poincare sphere.

F

Pvaeb

−→

−→Prr

iptapfp

4j

icdut

ig. 4. Polarization from circular (Retarder) to linear (Polarizer) on Poincare sphere.

Polarization state of light wave can be expressed clearly onoincare sphere. By using retardation � and rotation ı, we can con-ert any polarization state into a new polarization state [2,13]. Such

vectorial description of polarization represents three out of fourlements of Stokes vector. Polarization state vector −→

P can be giveny

P (−→P0,−→P45,

−→Pc ) = (cos 2�, sin 2� cos ı, sin 2� sin ı) (5)

0 represents 0◦ linear polarization state with � = 0◦ and ı = 0◦, −→P45

epresents 45◦ linear polarization state with � = 45◦ and ı = 0◦, −→Pc

epresents circular polarization state with � = 45◦ and ı = 90◦.Fig. 3 shows optical polarization evolution from linear polar-

zation state to circular polarization state. Horizontal linearlyolarized light Px0 in PM fiber rotates angle of 2� to Px1 alonghe equator, then Px1 rotates phase retardation angle of ı to Px2long longitude. After transmission in sensor head, the two wavesass through quarter-wave retarder, optical polarization evolutesrom circular polarization state to linear polarization state, and therocess can be described in Fig. 4.

. Modeling of optical polarization state in S-FOCS withones matrix

Poincare sphere describes optical polarization state in S-FOCSntuitively upper, in addition, Jones matrix provides a mathemati-ally rigorous method of tracking polarization [14]. This section, we

Please cite this article in press as: L. Wang, et al., Modeling and expecurrent sensor, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.o

escribe the propagation of the light-wave traveling oppositely bysing Jones matrix of optic components in S-FOCS, and introducehe principle of digital closed-loop signal processing system. The

PRESSx (2014) xxx–xxx 3

transmission matrix with Jones matrix describing the total lightpropagation path is summarized as MCLK and MC CLK

MCLK ={

Mpo · [M�/4−45]−1 · Mloop · M�/4+45 · Mmod 1 · Mpo

}(6)

MC CLK ={

Mpo · [M�/4+45]−1 · Mloop · M�/4−45 · Mmod 2 · Mpo

}(7)

where MCLK represents clockwise light wave transmission matrix,MC CLK represents counterclockwise light wave transmissionmatrix, Mpo is Jones matrix of polarizer, M45 is Jones matrix of45◦ splice, Mmod 1 and Mmod 2 are Jones matrix of birefringencephase modulator by using integrated optic chip (IOC) on differ-ent moment, M�/4+45 is Jones matrix of quarter-wave retarderwith clockwise light wave transmission, M�/4−45 is Jones matrixof quarter-wave retarder with counterclockwise light wave trans-mission, Mloop is Jones matrix of Faraday phase in sensing coil, ϕ(I) isnonreciprocal phase shift between the two waves induced by mag-netic field of current to be measured, Min is Jones matrix of lightwave issuing into polarizer.

Min = [Ex Ey]T (8)

Mpo =[

1 0

0 0

](9)

M�/4+45 =√

22

[1 j

j 1

](10)

M�/4−45 =√

22

[1 −j

−j 1

](11)

Mmod 1 =[

1 0

0 ejϕ(t−�)

](12)

Mmod 2 =[

1 0

0 ejϕ(t)

](13)

Mloop =[

cos ϕ(I) − sin ϕ(I)

sin ϕ(I) cos ϕ(I)

](14)

The detected light intensity resulting from the interference of thetwo returning linear waves is given by

Iout = I0 ·{

1 + cos [ϕ(I) + ϕ(t) − ϕ(t − �)]}

(15)

Feedback control signal processing system in closed loop S-FOCSworks on zero point, the nonreciprocal phase shift is equal andopposite to the phase shift induced by Faraday effect, the currentto be measured can be extracted from feedback electrical signalsapplied to the voltage control phase modulator.

5. Analysis on polarization error factors

In practice, polarization performance of light-wave in S-FOCSis susceptible to several factors, such as inherent defects of opti-cal components, temperature stability, vibration, equipment aging,and current value deviates from the theory value. Polarization errorsources in S-FOCS include linearly polarized error and circularlypolarized error. Linearly polarized error source includes polariza-tion degree error of polarized light-wave traveling from polarizer,integrated optical chip splitting ratio error, 45◦ splice angular align-ment error between PM fiber and retarder, and sensing fiber linearbirefringence error. Circularly polarized error source includes 90◦

phase shift error in retarder.

5.1. Polarization error from imperfect polarizer

rimental verification of polarization errors in Sagnac fiber opticrg/10.1016/j.ijleo.2014.04.064

Polarizer lies in common point with light-wave’s entering andexiting, and has the function to filter polarized light-wave unde-sired, which ensures reciprocity of light-wave in S-FOCS. Polarizer’s

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0 5 10 15 20 25 30 35 40 45-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Polarizer angu lar err or(deg ree )

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43 43 .5 44 44 .5 45 45 .5 46 46 .5 471

1.0005

1.001

1.0015

1.00 2

1.002 5

1.00 3

Splice angular (degree)

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acto

r

ig. 5. Interference intensity error scale curve caused by optical polarization angleeviation.

erformance influence measurement accuracy, imperfect polar-zer will induce polarization state randomly coupling, and expressntensity-type polarization error in S-FOCS [1,3,4].

The output intensity resulting from the interference of twoeturning linear optic waves is given by

= I0 · [1 + cos ϕ(I)] = I0 · [(1 + ς2)2 + (1 − ς2)

2

× cos(ϕmod(t) + 2NVI)] (16)

here � is error factor of polarizer, Mpo is Jones matrix of polarizer.q. (14) shows that the output intensity of light-wave interferenceignal is disturbed by polarizer’s error factor �, then, the capacityf minimum current resolution in S-FOCS is decreased.

With polarized light-wave spindle angular misalignment of �etween PM fiber and polarizer, light-wave interference intensityrror scale is given by

�I

I=

(− 1

2sin 2� + cos2 � sin 2� + cos4 � − 1

4+ 1

4cos2 2�

)2

− 1 (17)

Fig. 5 shows intensity of the interference signal wave even morentense with polarizer axis angle error increases. Nonreciprocalhase shift induced by Faraday magnetic field are interested, then,

ight-wave interference signal intensity (1 − ς2)2I0 has no effect

n phase to be measured by using closed loop control technol-gy, therefore, imperfect polarizer does not affect measurementccuracy, phase to be measured of ϕs(t) still meets the relation-hip of ϕs(t) = −2N · V · I, on condition of meeting minimum currentesolution requirement.

.2. Polarization error from imperfect integrated optical chipplitting ratio

Integrated optical chip (IOC) in S-FOCS outputs two beams ofight wave, named Ix and Iy, as affected by process factors, light-

ave intensity distributes unequally. The output intensity resultingrom the interference of two returning linear optic waves is giveny

= 2√

IxIy{1 + cos[ϕ(I) + ϕmod(t)]} =√

2I0√

cos(2˛)

× {1 + cos[ϕ(I) + ϕmod(t)]} (18)

Please cite this article in press as: L. Wang, et al., Modeling and expecurrent sensor, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.o

hase to be measured of ϕ(I) is not disturbed by integrated opti-al chip splitting ratio error, in spite of light-wave intensity isnfluenced.

Fig. 6. Scale factor with splice angular error.

5.3. Polarization error from imperfect quarter-wave retarder

With Jones matrix of quarter-wave retarder with splice angularof � , the output intensity resulting from the interference of tworeturning linear optic waves is given by

I = I0 · {1 − (1 − sin 2�) cos ϕmod(t)

+ sin 2� cos[ϕ(I) + ϕmod(t)]} (19)

To eliminate stationary bias in signal processing unit, the subtrac-tion is designed between the accumulation of the signal intensityin the positive and negative offset square wave, and we can obtain

�I = 2I0 · {(sin 2� − 1) sin ϕs(t) + sin 2� sin[ϕ(I) + ϕs(t)]} = 0

(20)

ϕs(t) = arctan[− sin 2� sin ϕ(I)

sin 2� − 1 + sin 2� cos ϕ(I)

](21)

It can be seen from Eq. (21) that the relationship between the recov-ered phase shift ϕs(t) and the Faraday phase shift ϕ(I) of S-FOCS withimperfect 45◦ splice angular between PM fiber lead and quarter-wave Retarder is nonlinear, as the relationship between ϕ(I) and Iis linear, therefore, the scale factor of S-FOCS with imperfect 45◦

splice angular is nonlinear.The relationship between scale factor and splice angular is

shown in Fig. 6. It can be seen that the recovered phase shift ϕs(t)in S-FOCS with imperfect 45◦ splice angular between PM fiberlead and quarter-wave retarder is nonlinear, as the relationshipbetween ϕ(I) and I is linear, therefore, the scale factor of S-FOCSwith imperfect 45◦ splice angular is nonlinear. Fig. 7 descripts scalefactor curve with phase angle retardation deviating. With the retar-dation angle gradually deviating from 90◦, scale factor deviatesfrom the ideal value, and linearity error increases.

5.4. Polarization error from imperfect sensing fiber

As linear birefringence of � and circular birefringence of T exitin the sensing coil, light-wave circularly polarization state evolutesto elliptically polarization state. Faraday phase to be measured ismixed with birefringence phase noise bias, and then, measurement

rimental verification of polarization errors in Sagnac fiber opticrg/10.1016/j.ijleo.2014.04.064

precision of R-FOCS is decreased by un-predicated birefringence insensing lead. Phase deviation scale factor SF(,T) in digital closed-loop FOCS is obtained by Eq. (22). Fig. 8 shows scale factor’s

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L. Wang et al. / Optik xxx (2014) xxx–xxx 5

88 88 .5 89 89 .5 90 90 .5 91 91 .5 921

1.000 1

1.000 2

1.0003

1.000 4

1.000 5

1.000 6

1.0007

Retarda tion angu lar(deg ree )

Sca

le f

acto

r

Fig. 7. Scale factor with phase retardation error.

Fw

cc

S

(F(tpaAwfs

6

otcp

Signalprocessing

Currenttransformercalibrator

Signalprocessing

A

Standard current sensor

S-FOCS

Currentsource

Fig. 9. Block diagram of current sensor test system.

ig. 8. Scale factor with linear birefringence within ±1 rad and circular birefringenceithin ±20 rad.

haracteristics with linear birefringence of within ±1 rad andircular birefringence of T within ±20 rad.

F(, 2T) =T2 + 2

[(sin

√2 + T2

)/(√

2 + T2)]

T2 + 2 cos[√

2 + T2] (22)

For the case that the sensing coil contains no linear birefringence = 0), phase deviation scale factor SF(,T)is identically equal to 1.or the case that the sensing coil contains no circular birefringenceT 0), phase deviation scale factor SF(,T) varies as tangent func-ion, and is expressed as SF(, 2T = 0) = (tan)/. To achieve higherformance in bias stability and linearity of S-FOCS, phase devi-tion scale factor SF(,T) must be close to 1 and maintains stable.s analyzed and simulated upper, circular birefringence existingithin the sensing coil yields no phase error, however, linear bire-

ringence yields phase error, and high circular birefringence canuppress the impact of linear birefringence.

. Experiments

To verify optical polarization properties in S-FOCS and its effectn the performance of transformers, we built a current transformer

Please cite this article in press as: L. Wang, et al., Modeling and expecurrent sensor, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.o

est system. The experiment environment of S-FOCS in temperatureontrol box with test temperature from −40 ◦C to 80 ◦C. The princi-le of current transformer test system is described as Fig. 9. Fig. 10

Fig. 10. AC current measurements @ 50 Hz.

shows the results obtained for AC currents with frequency of 50 Hzapplied to S-FOCS.

Experiments are carried out to test the influences on S-FOCS per-formance by imperfect polarized light, including linearly polarizedlight and circularly polarized light. As the fiber optical characteris-tics is difficult to observe after fusion, we joint fiber with a certainamount of error, then, test and compare S-FOCS performance.Results show that the fusion angle between the light source andpolarizer do not affect measurement accuracy of S-FOCS, ±45◦ axialsplice angular alignment error between IOC and retarder affectslight intensity contrast of the interference signal, and accuracy isdisturbed when angular alignment error greater than 0.2◦. Withtemperature changing in temperature control box from −40 ◦C to80 ◦C, birefringence in sensing fiber changes, and linearity error ofS-FOCS appears.

7. Conclusions

Sagnac fiber optic current sensors (S-FOCS) have a number ofinherent advantages over conventional current sensing techniques,and becomes very attractive for metering, control, and protectionin high-voltage substations, especially in harsh environments, Inpractice, environmental disturbances such as time-varying tem-perature gradients and vibration, yield a time varying birefringencein the sensing region, which rotates the plane of polarization,changes optical polarization states, yields an indistinguishable sig-nal among Faraday phase signal induced by current, and produceserroneous results. To maintain a high signal to noise ratio in theoutput signal, it is an urgent thing to restrict polarization errorsand make polarization states constant in S-FOCS. We have carriedout a theoretical investigation and experimental verification onthe polarization error characteristics against the imperfect polar-izer, imperfect quarter-wave retarder, imperfect splice angular,and imperfect sensing fiber in S-FOCS. Results show that imperfectpolarizer with more than 25 dB extinction ratio does not affect mea-surement accuracy, imperfect splitting ratio with 45:55 to 50:50 inIOC has no effect on measurement accuracy. However, the scalefactor of S-FOCS deviates when either the quarter-wave retarderis imperfect, or splice angular between quarter-wave retarder and

rimental verification of polarization errors in Sagnac fiber opticrg/10.1016/j.ijleo.2014.04.064

PM fiber not equal 45◦. Imperfect sensing lead contains linear bire-fringence, which induces linearity error in S-FOCS, and it is urgentto make low linear birefringence, high circular birefringence and

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ARTICLEJLEO-54624; No. of Pages 6

L. Wang et al. / Op

onstant intrinsic birefringence in sensing fiber even in time-arying temperature gradients and vibration environment.

onflicts of interest

The authors declare no conflict of interest.

cknowledgments

The project is supported by the following funds: Fundamen-al Research Funds for the Central Universities (2242013R30016),ational Natural Science Foundation of China (61203192),atural Science Foundation of Jiangsu Province (BK2012326,K20130099), Research Fund of China Ship Industry (13J3.8.4),oundation of Key Laboratory of Micro-Inertial Instrumentnd Advanced Navigation Technology, Ministry of Education201103).

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[[

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