sensors Article

A Linear Birefringence Measurement Method for an Optical Fiber Current Sensor Shaoyi Xu 1 , Haiming Shao 2 , Chuansheng Li 2 , Fangfang Xing 3 , Yuqiao Wang 1 and Wei Li 1, * 1 2 3

*

School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China; [email protected] (S.X.); [email protected] (Y.W.) National Institute of Metrology (NIM), Beijing 100029, China; [email protected] (H.S.); [email protected] (C.L.) School of Mechatronic Engineering, Xuzhou College of Industrial Technology, Xuzhou 221116, China; [email protected] Correspondence: [email protected]; Tel.: +86-516-8388-5829

Received: 27 May 2017; Accepted: 1 July 2017; Published: 3 July 2017

Abstract: In this work, a linear birefringence measurement method is proposed for an optical fiber current sensor (OFCS). First, the optical configuration of the measurement system is presented. Then, the elimination method of the effect of the azimuth angles between the sensing fiber and the two polarizers is demonstrated. Moreover, the relationship of the linear birefringence, the Faraday rotation angle and the final output is determined. On these bases, the multi-valued problem on the linear birefringence is simulated and its solution is illustrated when the linear birefringence is unknown. Finally, the experiments are conducted to prove the feasibility of the proposed method. When the numbers of turns of the sensing fiber in the OFCS are about 15, 19, 23, 27, 31, 35, and 39, the measured linear birefringence obtained by the proposed method are about 1.3577, 1.8425, 2.0983, 2.5914, 2.7891, 3.2003 and 3.5198 rad. Two typical methods provide the references for the proposed method. The proposed method is proven to be suitable for the linear birefringence measurement in the full range without the limitation that the linear birefringence must be smaller than π/2. Keywords: optical fiber current sensor; linear birefringence; multi-valued problem

1. Introduction An optical fiber current sensor (OFCS) has a number of advantages over the conventional current sensor, such as high precision, high sensitivity, wide dynamic range and immunity to electro-magnetic interference. Thus, it has attracted wide attention in the past three decades. For example, an OFCS has been proposed based on the Fabry-Perot interferometer using a fiber Bragg grating demodulation [1]; an OFCS has been reported based on a long-period fiber grating with a permanent magnet [2]; and an OFCS has been designed based on microfiber and chrome-nickel wire [3]. Moreover, the most widely used OFCS mechanism by far, the Faraday effect, which has been explored in many configurations [4–7]. In this case, the light propagating in a sensing fiber experiences a rotation in the angle of polarization in the presence of an external magnetic field. The applications of the OFCS using the Faraday effect include the direct current measurement for process control and protection in electrowinning industry [6], the current measurement for revenue metering as well as for the substation control and protection in electric power industry [7], and the current measurement for buried pipelines protection in an urban rail transit system [8,9]. It is well known that the effect of linear birefringence on OFCS is difficult to eliminate, which may degrade the performance of OFCS seriously. Effective elimination methods have been developed based on two types of optical fibers, including the spun high-birefringent optical fiber and the low-birefringent optical fiber. It is proven that the OFCS with the spun high-birefringent optical fiber has a good thermal stability [10,11], however, the high cost Sensors 2017, 17, 1556; doi:10.3390/s17071556

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of fabrication limits the applications of this optical fiber at present. Moreover, the low-birefringent optical fiber is usually annealed [5], twisted [12] or wound along the designed geometric path [8,13] to minimize the effect of linear birefringence. Among them, it is found that the annealed fiber is easily damaged during the installation process. The twisted optical fiber produces the reciprocal circular birefringence through the shearing stress, which may decrease as time goes on. The sensing fibers wound along the cylindrical [8] and toroidal [13] spiral paths have been proposed based on the geometrical rotation effect. In these sensing fibers, the produced circular birefringence only depends on the geometrical parameters of the path rather than the stress, which provides the effective method to eliminate the linear birefringence in the low-birefringent sensing fiber. In this method, the accurate measurement of the linear birefringence can help to optimize the design of the geometrical path. The first one of the typical measurement methods of the linear birefringence has been proposed by Ren, who applied the isotropy of circularly polarized light to measure the linear birefringence of the low-birefringent sensing fiber [14]. In this method, the circularly polarized input light can eliminate the effect of the azimuth angle between λ/4 plate and the low-birefringent sensing fiber. An accuracy of 4% in the linear birefringence can be achieved by this method. The second typical measurement method of the linear birefringence is the one proposed by Tentori [15]. In this method, the input polarizer is rotated from 0◦ to 180◦ . The polarization state of the output light from the low-birefringent sensing fiber describes a major circle in Poincare sphere. Each point in the major circle can be represented by the Stokes vector [S0 ; S1 ; S2 ; S3 ]. And the linear birefringence can be obtained after determining the maximum or minimum value of the S3 . The last of the typical measurement methods of the linear birefringence has been proposed by Segura [16]. This method, based on the use of the Faraday effect, and used to measure the linear birefringence with an accuracy of 5% seems to be simple, fast and requires short lengths of fiber. It is noted however that this method needs to fix the transmission axis of the Wollaston prism at 45◦ with respect to the input polarization. We can find that the three methods described above are all suitable for the measurement of the linear birefringence of the low-birefringent sensing fiber when the linear birefringence is not greater than π/2. If the linear birefringence is greater than π/2, the above methods cannot determine the accurate value of the linear birefringence due to the multi-valued problem that occurs during the measurement. It is known that the linear birefringence of the low-birefringent sensing fiber in an OFCS is usually not smaller than π/2 due to the bending-induced factor. If the OFCS works under varying temperature conditions, the linear birefringence may be greater. Thus, the multi-valued problem may be one of the difficulties to measure the linear birefringence in the OFCS. Moreover, it is known that the principle axes of the low-birefringent sensing fiber are difficult to determine. This may be another difficulty. In this paper, we propose a linear birefringence measurement method for the OFCS. The configuration of the linear birefringence measurement system is illustrated firstly. The steps of the measurement method are then demonstrated. Among these steps, the elimination method is proposed to eliminate the effect of the azimuth angles between the sensing fiber and the two polarizers. The multi-valued problem of the measurement result is simulated and its solution is presented. Finally, a series of experiments based on the different sensor heads of the OFCS have been conducted to prove the feasibility of our proposed method. When the linear birefringence is not greater than π/2, the typical methods proposed by Ren [14] and Tentori [15] provide the references for our proposed method. 2. Linear Birefringence Measurement Method for the OFCS The configuration of the linear birefringence measurement system is shown in Figure 1, which mainly includes a distributed feedback (DFB) laser diode, a fixed polarizer, three aspheric lenses, a sensor head of the OFCS, a rotated polarizer, and a power meter. The DFB laser diode (model DFB-I-50, LabBang Co. Ltd., Beijing, China) was a central wavelength at 1550 nm and its −20 dB spectral width is about 0.2 nm. Moreover, for the fixed polarizer and rotated polarizer (model LPIREA050-C, Thorlabs Co. Ltd., Newton, NJ, USA), the extinction ratios are both greater than 2000:1 at 1550 nm. Finally, for the sensing fiber (model LB 1550-125, Oxford Electronics Co. Ltd., Hants, UK) in the sensor head, its

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spectral width is about 0.2 nm. Moreover, for the fixed polarizer and rotated polarizer (model LPIREA050-C, Thorlabs Co. Ltd., Newton, NJ, USA), the extinction ratios are both greater than Sensors 17, 1556 of 11 2000:12017, at 1550 nm. Finally, for the sensing fiber (model LB 1550-125, Oxford Electronics Co.3Ltd., Hants, UK) in the sensor head, its inherent linear birefringence is 4°/4 m and its bending radius is defined linear as r inbirefringence this work. During application in radius an urban rail transit system, the rDuring is usually inherent is 4◦ /4 our m and its bending is defined as r in this work. our designed as about 40 cm. application in an urban rail transit system, the r is usually designed as about 40 cm. SF

DFB

CD

I:

POWER

1550nm

291 mA

POWER:

50 mW

FP L1

FR

LASER OUT

PM r

L2

L3 A

20.6

RP

CW-POWER

Probe

B

调零

W mW uW

LP-3A 激光功率计 2

20

200

2

Figure 1. Configuration of the linear birefringence measurement system: L1 to L3 are the aspheric Figure 1. Configuration of the linear birefringence measurement system: L1 to L3 are the aspheric lenses, whose effective focal length is about 11 mm and NA is about 0.25; FP is the fixed polarizer; FR lenses, whose effective focal length is about 11 mm and NA is about 0.25; FP is the fixed polarizer; FR is the Faraday rotator, which is only applied to obtain the P3 and P4 shown in Equations (4) and (5); is the Faraday rotator, which is only applied to obtain the P3 and P4 shown in Equations (4) and (5); SF is the sensing fiber in the sensor head; CD is the conductor; RP is the rotated polarizer; PM is the SF is the sensing fiber in the sensor head; CD is the conductor; RP is the rotated polarizer; PM is the power meter. power meter.

In In Figure Figure 1, 1, the the light light is is first first generated generated by by the the DFB DFB laser laser diode. diode. The The pigtail pigtail of of the the DFB DFB is is the the single mode fiber, whose mode field diameter is about 10 um. Thus, when the light is collimated single mode fiber, whose mode field diameter is about 10 um. Thus, when the light is collimated by by L1, L1, the the spot spot diameter diameter and and the the divergence divergence angle angle of of the the collimated collimated light light are are estimated estimated at at 2.2 2.2 mm mm and and ◦ 0.052 light passes through the the FP to a linearly polarized light. 0.052,, respectively. respectively.Then, Then,the thecollimated collimated light passes through FPform to form a linearly polarized The linearly polarized light continues to enter into the SF through L2. It is assumed that the principle light. The linearly polarized light continues to enter into the SF through L2. It is assumed that the polarization axes of the SF areof x axis axis, Thererespectively. are the linear There birefringence the principle polarization axes the and SF yare x respectively. axis and y axis, are theand linear Faraday rotation in the SF. Among them, the former one is induced by the refractive index difference birefringence and the Faraday rotation in the SF. Among them, the former one is induced by the between and y-component while the latter one is inducedwhile by thethe current theisCD. Thus, refractivex-component index difference between x-component and y-component latterinone induced the output lightin from is elliptically polarized. Thethe elliptically polarizedpolarized. light is collimated by L3 by the current thethe CD.SFThus, the output light from SF is elliptically The elliptically and is then analyzed by the RP. Finally, the light intensity from the RP is detected by the PM. polarized light is collimated by L3 and is then analyzed by the RP. Finally, the light intensity from In Figure 1, there are PM. two azimuth angles of the transmission axes of the FP and RP with respect the RP is detected by the to theInprinciple polarization axes of SF, which formed at the points the Figure 1, there are two azimuth anglesare of the transmission axesAofand theB, FPrespectively. and RP withAnd respect azimuth angles are defined as α and β, respectively. Since the SF in the OFCS is the low-birefringent to the principle polarization axes of SF, which are formed at the points A and B, respectively. And sensing fiber, the orientations of its principle polarization axes are Since usuallythe difficult determine, the azimuth angles are defined as α and β, respectively. SF into the OFCS which is the causes that the azimuth angles α and β to be unknown. We believe that the linear birefringence can be low-birefringent sensing fiber, the orientations of its principle polarization axes are usually difficult measured onlywhich after the effectthat of the α and β has eliminated.We believe that the to determine, causes theazimuth azimuthangles angles α and β tobeen be unknown. In this work, an effective method is proposed to measure the linear birefringence is linear birefringence can be measured only after the effect of the azimuth angles in α the andSF.β Ithas divided into three steps: first, we keep the transmission axes of the FP and RP parallel, which is the been eliminated. feasible operation using the high-precision rotation the effective elimination In this work, an effective method is proposed tomount; measurethen, the linear birefringence in themethod SF. It is on the effect of the azimuth angles α and β is proposed based on the orthogonal modulation a divided into three steps: first, we keep the transmission axes of the FP and RP parallel, whichand is the Faraday rotator shown in Figure 2; finally, the multi-valued problem on the linear birefringence is feasible operation using the high-precision rotation mount; then, the effective elimination method solved. The material for the Faraday rotator iron garnet. is orthogonal noted that the azimuth angles on the effect of the azimuth angles α and βisisbismuth proposed based on It the modulation and a αFaraday and β are the same, which can be both represented by θ in this work. rotator shown in Figure 2; finally, the multi-valued problem on the linear birefringence is According to the configuration shown in Figure 1, it isiron assumed that thenoted complex of solved. The material for the Faraday rotator is bismuth garnet. It is thatamplitude the azimuth the light vector from the FP is E . Thus, the complex amplitude of the input light vector of the SF is in angles α and β are the same, which can be both represented by θ in this work. EA = [Ein cosθ; Ein sinθ]. Moreover, the linear birefringence and the Faraday rotation work together on the SF. Thus, the Jones matrix Jsf of the SF can be obtained as Equation (1) [17]:   Js f = 

cos

q

√ sin F2 +(δ/2)2 F2 + (δ/2)2 + i (δ/2) √ 2 F +(δ/2)2 √ 2 2 sin F +(δ/2) F √ 2 F2 +(δ/2)

√ 2 2 − F sin√ 2F +(δ/22)



F +(δ/2)

cos

q

F2

 √ , (1) F2 +(δ/2)2 + (δ/2) − i (δ/2) √ 2 2 2

sin

F +(δ/2)

where, F is the Faraday rotation angle, F = VNI; V represents the Verdet constant of the SF, V = 0.73 urad/A at 1550 nm; N is the number of turns in the fiber loop; I is the input current in the CD; δ is the phase delay induced by the linear birefringence in the SF. The complex amplitude of

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the output light vector of the SF is EB = Jsf ·EA . On this basis, the complex amplitude of the light vector from the RP is Eout = EBx cosθ + EBy sinθ. The light intensity from the RP can be derived as Equation (2):  P1 ≈

1+(cos 2θ )2 +(sin 2θ )2 cos 2 2



F2 +(δ/2)2



F2 F2 +(δ/2)2

2

sin

q

F2

2

+ (δ/2) (cos 2θ )

2



2 . × Ein

(2)

Then, the orientation of the FP remains the same. The RP is subjected to a 90◦ counter-clockwise rotation with respect to its initial orientation. Thus, the azimuth angle of the transmission axis of the RP with respect to the x axis of the SF is changed to (θ + π/2). The light intensity from the RP can be derived as:   √ q 1−(cos 2θ )2 −(sin 2θ )2 cos 2 F2 +(δ/2)2 2 2 F2 2 2 . 2 P2 ≈ + 2 F + (δ/2) (cos 2θ ) × Ein (3) 2 sin 2 F +(δ/2)

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Figure 2. The working mechanism of the Faraday rotator.

According to configuration in Figure 1, it isthe assumed of Moreover, thethe Faraday rotatorshown is installed between FP andthat thethe L2complex to rotateamplitude the output the light vector from thethe FP FP is Eby in. Thus, the complex amplitude of the input light vector rotator, of the SFthe is polarization state from 45◦ counterclockwise. With the help of the Faraday E A=[Eincosθ; Einsinθ]. Moreover, the linear birefringence and the Faraday rotation work together on azimuth angle of the transmission axis of the FP with respect to the x axis of the SF is changed to the SF. Thus, theazimuth Jones matrix sf of the SF can be obtained as Equation (1) [17]: θ + π/4 and the angle Jof the RP is changed as θ + π/4. In this case, the light intensity from the 2 2 RP can be derived  as:  sin F 2 +  2  sin F 2 +  2  2 2   F 2  cos F +  2   i  √  2 2 q  2 F 2 + 2 2  1+(sin 2θ )2 +(cos 2θ )2 cos 2 F2 +(Fδ/2+)2 2  F2 2   2 . , P3J ≈ − 2 sin2 F2 + (δ/2) (sin 2θ ) × Ein (4) (1) sf   2 2 2  F +(δ/2)2 sin F 2 +  2  sin F 2 +  2    2 2  F cos F +  2 i  2       2 2 F 2 +  2  F 2 +  2   Finally, the azimuth angle of the FP keeps θ + π/4 using the Faraday rotator. The azimuth angle of the RP is changed to (θ + 3π/4). In this case, the light intensity from the RP can be derived as: where, F is the Faraday rotation angle, F = VNI; V represents the Verdet constant of the SF,  √ q loop; I is the input current V = 0.73 urad/Aat 1550 nm; N is the number of turns in the fiber in the CD; 1−(sin 2θ )2 −(cos 2θ )2 cos 2 F2 +(δ/2)2 F2 2 2 + ( δ/2)2 (sin 2θ )2 × E2 . P ≈ + sin F (5) 4 δ is the phase delay induced by of the in 2 the linear birefringence F2 +(δ/2)2 in the SF. The complex amplitude output light vector of the SF is EB = Jsf·EA. On this basis, the complex amplitude of the light vector to=Equations andThe (3), light the heterodyne method applied to process the light fromAccording the RP is Eout EBxcosθ + E(2) Bysinθ. intensity from the RPiscan be derived as Equation (2): intensities P1 and P2 . The2first heterodyne (W ) output is given as Equation (6). Then the heterodyne 1  1   cos 2    sin 2 2 cos 2 F 2   2 2  F2 2 2 method goes (W22) output   E in2 . is given sin 2 heterodyne P1 on F 2   2   cos   to process the light intensities P3 and P42. The second (2) 2   2 F   2  as Equation (7). Finally, the sum of the two heterodyne outputs can be obtained as Equation (8):  q the same. The RP is subjected q to a 90 counter-clockwise Then,P1the 2 − P2 orientation2 of the FP remains F2 2 2 + ( δ/2)2 − 2 = cos 2θ + sin 2θ cos 2 F sin F2 + (δ/2)2 (cos 2θ )2 . (6) ( ) ( ) P1 + Prespect 2 F2 +(δ/2 )2 rotation with to its initial orientation. Thus, the azimuth angle of the transmission axis of the RP with respect to the x axis of the SF is changed to (θ + π/2). The light intensity from the RP can be derived as: W1 =

 1   cos 2 P2    



2

  sin 2



2

2

cos 2 F 2   2 

2



F2 F   2  2

2

sin 2

F 2   2 

2

 cos 2 

2

   E in2  

.

(3)

Moreover, the Faraday rotator is installed between the FP and the L2 to rotate the output

method goes on to process the light intensities P3 and P4. The second heterodyne (W2) output is given as Equation (7). Finally, the sum of the two heterodyne outputs can be obtained as Equation (8): P1  P2 F2 2 2 2 sin 2   cos 2    sin 2  cos 2 F 2   2   2 2 2 P1  P2 F   2  Sensors 2017, 17, 1556 W1 

F 2   2 

2

2  cos 2  .

5 of 11

2

W2 

W2 =

P3 − P4 P3 + P4

P3  P4 F 2 2 2 2 2 sin 2 F 2   2   sin 2    sin 2    cos 2  cos 2 F 2   2   2 2 2 q q P3  P4 F 2  2  2

2

= (sin 2θ ) + (cos 2θ ) cos 2

F2

+ (δ/2)

2

−2 2 F 2 F +(δ/2)

sin

2

(6)

F2

2

.

(7) (7)

2

+ (δ/2) (sin 2θ ) .

2

F2 2 2 q sin 2 F 2   2  . F 2 (8) 2 F2 + (δ/2)2 − 2 F   2  2 sin2 F2 + (δ/2)2 . (8) 2 F + (δ/2) According to Equation (8), the relationship of the F, the δ and the U can be obtained as: According to Equation (8), the relationship of the F, the δ and the U can be obtained as: 2 2 F 2   2  U 2  . sin 2 F 2   2    1  q (9)  0 2 2 2 2 2   F )  22 2F + (δ/2 U 2 + ( δ/2)2 − 1 − sin = 0. (9) F 2 2 2 (δ/2 In Equation (9), theFF + and the) U are known while the δ is unknown. We define the function f(δ)In as:Equation (9), the F and the U are known while the δ is unknown. We define the function f (δ) as: U  W1  W 2  1  cos 2 F 2   2   2

U = W1 + W2 = 1 + cos 2

q

2

f (δ) =

2 F 2   2  U  . 2  q f 2   sin 2 F 2   2   1  2 2 2F + (δ/2 U 2  F 2 )  2 2 2 + ( δ/2) − 1 −2  . sin F 2 2 2

(10) (10)

F + (δ/2) There may be a multi-valued problem to determine δ during the process of solving the equation 0. We conduct twoproblem simulations to demonstrate this problem. the first There f(δ) may= be a multi-valued to determine δ during themulti-valued process of solving the In equation simulation, the N is set as 15 and the input current is set as 40,000 A, which leads to F being about f (δ) = 0. We conduct two simulations to demonstrate this multi-valued problem. In the first simulation, 0.438 theoretical value current of δ is about rad (π/2 rad) [18]. When the input current is about 0 and 40,000 A, F is about 0 and 0.5548 rad. Thus, the U is about 0.8777 and 0.1296, respectively. Similarly, the corresponding functions are defined as f3(δ) and f4(δ), respectively. The {fi(δ)} versus δ curves are shown in Figure 4. Sensors 2017, 17, 1556 6 of 11 Obviously, f3(δ) and f4(δ) are all equal to zero only when the δ is equal to 1.693 rad (point 1). The other nine solutions (point 2 to 10) of f3(δ) = 0 are not the solutions of f4(δ) = 0, which are shown in Figure when the δ 4. is equal to 1.693 rad (point 1). The other nine solutions (point 2 to 10) of f 3 (δ) = 0 are not the solutions of f 4 (δ) = 0, which are shown in Figure 4. 1 F=0.5548 rad & U=0.1296 F=0 rad & U=0.8777

f ( ) 3

f( )

0.5

0

34

1 2

56

78

9 10

-0.5 f 4( ) -1

0

5

10

15

20

25

30

[rad]

Figure TheThe {fi (δ)} versus the δthe in the range 0–30 rad: F3 = rad: 0 radFand = 0.8777 F4 = 0.5548 Figure4. 4. {fi(δ)} versus δ in the ofrange of 0–30 3 = U 0 3rad and while U3 = 0.8777 while rad U4 =rad 0.1296. 0.5548 and U4 = 0.1296. F4 =and

Thus,according accordingtotothe theabove abovesimulation simulationresults, results,for forsolving solvingthe themulti-valued multi-valuedδ δproblem, problem,the the Thus, datasets {F i } and {U i } are firstly obtained based on the different values of the input current. On this datasets {Fi } and {Ui } are firstly obtained based on the different values of the input current. On this basis,the thesets setsofofequations equations{fi{f(δ) i(δ) canbe beobtained obtainedasasEquation Equation(11). (11). basis, ==0}0}can  2 F1 2   2q  U  2  2 F1 2  2 2    1  U11   0  δ/2)2 2 2F1 2 +( 2 2 sin2 sin F + δ/2 − 1 − ( )  F   2 1 22  = 0 F1 2 +(δ/21)2  q2    22 2 2F2 +( 2 F2)   2 2  2 2  UU2 2  22  δ/2 2 F sin + δ/2 − 1 − ( ) sin 2 1 F      00   2  2 2 2  = 2 2 F2 +(δ/2 2   F2)2   2  . ..   .     2 F 22   2q2  2  22 2Fk 2 +( 2 2  UUk k  k) 2  δ/2 sin 2 1 F  sin F + δ/2 − 1 −     00 ( )   k k 2  = 2 2 2 Fk +(δ/2 )2 2    Fk    2  2

                  

.

(11) (11)

InInthis the effective solution of {f = 0} been proposed based on the newer version of thiswork, work, the effective solution ofi (δ) {fi(δ) = has 0} has been proposed based on the newer version Matlab software (e.g., (e.g., Matlab R2016a). It includes that: first, thefirst, particle is appliedis of Matlab software Matlab R2016a). It includes that: the swarm particleoptimization swarm optimization toapplied obtain the local optimal results of the δ; then, in the global to obtain the local optimal results of the the ‘Multistart’ δ; then, thesolver ‘Multistart’ solver optimization in the global toolbox is run from theisrandom initial points that come from thecome local from optimal In thisresults. step, optimization toolbox run from the random initial points that the results. local optimal the solver starts each random point and findspoint a minimum of athe sum of squares of In ‘lsqnonlin’ this step, the ‘lsqnonlin’ solver starts initial each random initial and finds minimum of the sum the functions described in {f (δ)}. Finally, the minimum of the sum of squares in all initial points of squares of the functionsi described in {fi(δ)}. Finally, the minimum of the sum of squares in isall searched and the corresponding δ is the desirable δresult. initial points is searched and the corresponding is the desirable result. 3. Experimental In order to verify the feasibility of our proposed method, a series of experiments are conducted based on the different sensor heads of our OFCS. What these sensors all have in common is that their diameters are all 40 cm. Referring to the configuration shown in Figure 1, the magnetic field is produced directly by the input current in the conductor, which requires a strong current generator. In this experiment, the current generator is substituted by a magnetic field generator that can produce the equivalent magnetic field of a maximum of 50,000 A. Moreover, an optical breadboard and a five-axis kinematic optic mount (KOM) are very useful for the output light from the FP to enter into the sensing fiber. The physical map and its illustration of the experimental system are shown in Figure 5. Finally, it is noted that there are two input current during the experiments, one is 0 A and the other is 40,000 A.

In this experiment, the current generator is substituted by a magnetic field generator that can produce the equivalent magnetic field of a maximum of 50,000 A. Moreover, an optical breadboard and a five-axis kinematic optic mount (KOM) are very useful for the output light from the FP to enter into the sensing fiber. The physical map and its illustration of the experimental system are Sensors 2017, 17, 1556 5. Finally, it is noted that there are two input current during the experiments,7 of 11 shown in Figure one is 0 A and the other is 40,000 A.

Figure Figure 5. 5. The The physical physical map map and and its its illustration illustration of of the the experimental experimental system. system.

In the first experiment, the number of turns of the sensing fiber in the sensor head is about 15. theinput first experiment, number turns of the sensing in the sensor head is about 15. WhenInthe current is 0 the A, the four of light intensities P1 to Pfiber 4 shown in Equations (2) to (5) are When the input current is 0 A, the four light intensities P to P shown in (5) are 11.1, 4 11.1, 5.6, 14.69 and 1.76 mW, respectively, which are1 all detected byEquations the PM. (2) Theto heterodyne 5.6, 14.69 and 1.76 mW, respectively, which are all detected by the PM. The heterodyne outputs W 1 and outputs W1 and W2 are about 0.3293 and 0.7860. The sum U is about 1.1154. Moreover, the Faraday W and 0.7860. The U iscurrent about 1.1154. Moreover, Faraday rotation angle F is 2 are about rotation angle0.3293 F is about 0. When thesum input is 40,000 A, the P1the to P 4 are 10.5, 7.18, 12.97 and about 0. When the input current is 40,000 A, the P to P are 10.5, 7.18, 12.97 and 3.42 mW, respectively. 3.42 mW, respectively. The W1 and W2 are about 10.18784 and 0.5827. The U is about 0.7705. And the F The W and W are about 0.1878 and 0.5827. The U is about 0.7705. And the F is about 0.438 rad. Thus, 1 2 is about 0.438 rad. Thus, the equations in the first experiment can be obtained as: the equations in the first experiment can be obtained as:  sin 2  2   0 .4 42 3  0   2  sin20.38 37) −0.4423 2 =0 (δ/2 2 (12) sin 2 0 .19 1 8   2   0 .6 14 8  0 . q  2 2 . (12)  0 .1 91 8  2 0.3837 +( δ/2   ) 2  2  0.1918 + (δ/2) − 0.6148 = 0 2 sin 0.1918+(δ/2)

Thus, after solving the Equation (12), we can get the linear birefringence δ about 1.3577 rad in Thus, after solving the is Equation wethan can the get theoretical the linear birefringence δ about 1.3577 rad in the first experiment, which slightly (12), larger value (δ = 1.337 rad) shown in the the first experiment, is slightly larger than the theoretical value (δ = 1.337 rad) shown in the first simulation. In thewhich first simulation, we only consider the bending-induced birefringence and first simulation. In the first simulation, consider the bending-induced inherent linear birefringence. However,we in only the first experiment, there may be abirefringence small amountand of the inherent linear birefringence. However, induring the firstthe experiment, may befiber. a small amount of the extra linear birefringence that is produced winding ofthere the sensing Thus, we believe extra linear birefringence that is produced during the winding of the sensing fiber. Thus, we believe this measurement result is right. As mentioned in the Introduction section, the measurement this measurement is right. As mentioned theboth Introduction the measurement methods methods proposedresult by Ren [14] and Tentori [15]inare effective section, for the case that the δ is not larger proposed by (about Ren [14] and Tentori [15]can are provide both effective for the case thatproposed the δ is not largerinthan than π/2 rad 1.5708 rad). They the references for our method the π/2 rad (about 1.5708 rad). They can provide the references for our proposed method in the first first experiment. According to [14], the first reference method proposed by Ren requires a λ/4 plate experiment. According toThorlabs [14], theCo. firstLtd., reference method proposed Ren requires a λ/4 plate (model (model WPQ05M-1550, Newton, NJ, USA) and by a Wollaston prism (model WP10, WPQ05M-1550, Thorlabs Co. Ltd., Newton, NJ, USA) and a Wollaston prism (model WP10, Thorlabs Co. Ltd., Newton, NJ, USA). For the sensor head in the first experiment, the measurement process is shown in Figure 6a. The measurement result of the first reference method is about 0.9763, which is equal to sin(δ). Thus, the value of δ is about 1.3526 rad, which is approximately consistent with the measurement result of our proposed method (δ = 1.3577 rad). Moreover, the second reference method proposed by Tentori [15] is also used to measure the δ of the sensor head in the first experiment. Compared with our configuration shown in Figure 1, in the second reference method, the FP is replaced by a RP and the output light from SF is detected directly by a polarimeter (model DOP-101D, General Photonics Co. Ltd., Chino, CA, USA). It is assumed that the output light from the SF can be expressed by the Stokes vector [S0 ; S1 ; S2 ; S3 ]. Among them, the S3 can be expressed as –sin 2αsin δ, where the α

with the measurement result of our proposed method (δ = 1.3577 rad). Moreover, the second reference method proposed by Tentori [15] is also used to measure the δ of the sensor head in the first experiment. Compared with our configuration shown in Figure 1, in the second reference method, the FP is replaced by a RP and the output light from SF is detected directly by a polarimeter (model Sensors 2017, 17, 1556 DOP-101D, General Photonics Co. Ltd., Chino, CA, USA). It is assumed that8 the of 11 output light from the SF can be expressed by the Stokes vector [S0; S1; S2; S3]. Among them, the S3 can be expressed as –sin 2αsin δ, where the α denotes the multi-parameter model, including the denotesangle the multi-parameter model, azimuth the transmission axes of the RP azimuth of the transmission axesincluding of the RP the with respect angle to theof principle polarization axes of SF. with respect to the principle polarization axes of SF. Since the sin 2α is in the range from − 1 to 1, the Since the sin 2α is in the range from −1 to 1, the –sin 2αsin δ can reach its maximum sin δ or –sin 2αsin –sin δ canδ reach its maximum sinFor δ orthe minimum –sin in δ after rotating the RP. For sensor head minimum after rotating the RP. sensor head the first experiment, thethe measurement in the first experiment, the measurement result of the second reference method is shown in Figure 6b. result of the second reference method is shown in Figure 6b. It can be found that the minimum S3 is It can be found that the minimum S is about − 0.979. Thus, the δ is about 1.3655 rad, which is also about −0.979. Thus, the δ is about 1.3655 rad, which is also approximately consistent with the 3 approximately consistent the measurement ourand proposed method (δ = 1.3577 rad)(δ and measurement results of ourwith proposed method (δ =results 1.3577ofrad) the method proposed by Ren = the method proposed by Ren (δ = 1.3526 rad). 1.3526 rad).

(a)

(b)

Figure Figure6.6.The Themeasurement measurementresult resultofofthe thereference referencemethods methodsininthe thefirst firstexperiment: experiment:(a) (a)the themethod method proposed proposedby byRen; Ren;(b) (b)the themethod methodproposed proposedby byTentori. Tentori.

In the second experiment, the number of turns of the sensing fiber in the sensor head is about In the second experiment, the number of turns of the sensing fiber in the sensor head is about 19. 19. When the input current is 0 A, the P1 to P4 are 9.23, 7.9, 13.46 and 2.93 mW. The W1 and W2 are When the input current is 0 A, the P1 to P4 are 9.23, 7.9, 13.46 and 2.93 mW. The W 1 and W 2 are about about 0.0776 and 0.6425. The U is about 0.7201. The F is about 0. When the input current is 40,000A, 0.0776 and 0.6425. The U is about 0.7201. The F is about 0. When the input current is 40,000A, the P the P1 to P4 are 6.64, 10.49, 10.18 and 5.68 mW, respectively. The W1 and W2 are about −0.2248 and 1 to P4 are 6.64, 10.49, 10.18 and 5.68 mW, respectively. The W 1 and W 2 are about −0.2248 and 0.2837. 0.2837. The U is about 0.059. And the F is about 0.5548 rad. Thus, the equations in the second The U is about 0.059. And the F is about 0.5548 rad. Thus, the equations in the second experiment can experiment can be obtained as: be obtained as:  2 2  sin  sin 0 .6 39 9 =0 0 (δ/2  )2 −   0.6399 q  2 2 . (13) +(56 δ/2 22  ) sin 2 2  0.61 2  0.6156 (13) 0.3078 +(δ/2 0.9705 2  0.97 sin 0.30 78 05  = 0 .0  2) −  2 0.3078 +( δ/2)  0 .3 07 8   2  Thus, after solving Equation (13), we can get that the linear birefringence δ is about 1.8425 rad Thus, after solving Equation (13), we can get that the linear birefringence δ is about 1.8425 rad for the sensor head in the second experiment. It is noted that there is the multi-valued problem for the sensor head in the second experiment. It is noted that there is the multi-valued problem for for the reference methods proposed by Ren [14] and Tentori [15] when the δ is larger than π/2 rad. the reference methods proposed by Ren [14] and Tentori [15] when the δ is larger than π/2 rad. With regard to the sensor head in the second experiment, the measured sin δ using the two reference With regard to the sensor head in the second experiment, the measured sin δ using the two methods are about 0.9693 and 0.946, which are shown in Figure 7. Obviously, it cannot determine the reference methods are about 0.9693 and 0.946, which are shown in Figure 7. Obviously, it cannot accurate value of the δ only based on the sin δ. Thus, the accuracy of our proposed method has to be determine the accurate value of the δ only based on the sin δ. Thus, the accuracy of our proposed evaluated indirectly based on the sin δ in the second experiment. The sin δ is about 0.9633 using our method has to be evaluated indirectly based on the sin δ in the second experiment. The sin δ is proposed method, which is approximately consistent with the results obtained by the two reference about 0.9633 using our proposed method, which is approximately consistent with the results methods (sin δ = 0.9693 and sin δ = 0.946). Moreover, the δ in the second experiment (δ = 1.8425 rad) is really larger than the first experiment (δ = 1.3577 rad) with the increasing the number of turns of the sensing fiber. We continue to test the linear birefringence of the different sensor heads. Among them, the numbers of turns of the sensing fiber are 23, 27, 31, 35 and 39, respectively. It is rare that the number of turns (N) is greater than 40 in our practical application of the stray current measurement [9]. Thus, the sensor head with N ≥ 40 is not tested in this work. We can find that the δ obtained by our proposed method is about 2.0983, 2.5914, 2.7891, 3.2003 and 3.5198 rad when the N is about 23, 27, 31, 35 and

Sensors 2017, 17, 1556 Sensors 2017, 17, 1556

9 of 11 9 of 11

S3

obtained two reference δ =N.0.9693 sin δ = fiber 0.946).is Moreover, the δ ininthe 39. The δbyis the increased with themethods increase (sin of the Since and the sensing manually wound the second (δ = 1.8425 rad) is really larger the firstoperation experiment (δ = 1.3577 the sensorexperiment head, the linear birefringence produced inthan the manual cannot be the rad) samewith for each increasing the number of turns of the sensing fiber. sensor head.

(a)

(b)

Figure measurement result resultofofthethe reference methods in second the second experiment: the Figure7.7. The The measurement reference methods in the experiment: (a) the (a) method method proposed by Ren; (b) the method proposed by Tentori. proposed by Ren; (b) the method proposed by Tentori.

We continue to test the linear birefringence of the different sensor heads. Among them, the Thus, this increase is close to linear, which is shown in Figure 8. Moreover, these sensor heads are numbers of turns of the sensing fiber are 23, 27, 31, 35 and 39, respectively. It is rare that the number all measured by the two reference methods proposed by Ren [14] and Tentori [15]. In the reference of turns (N) is greater than 40 in our practical application of the stray current measurement [9]. method proposed by Ren [14], the results (sin δ) are about 0.8754, 0.5341, 0.3049, −0.0511 and −0.357 Thus, the sensor head with N ≥ 40 is not tested in this work. We can find that the δ obtained by our when the N is about 23, 27, 31, 35 and 39. In the reference method proposed by Tentori [15], the proposed method is about 2.0983, 2.5914, 2.7891, 3.2003 and 3.5198 rad when the N is about 23, 27, results (sin δ) are about 0.8616, 0.5091, 0.4161, −0.0657 and −0.3744 when the N is about 23, 27, 31, 31, 35 and 39. The δ is increased with the increase of the N. Since the sensing fiber is manually 35 and 39. The results (sin δ) obtained by our proposed method are about 0.8641, 0.5229, 0.3452, wound in the sensor head, the linear birefringence produced in the manual operation cannot be the −0.0587 and −0.3693 when the N is about 23, 27, 31, 35 and 39. Similarly with the first and second same for each sensor head. experiments, Sensors 2017, 17, the 1556results obtained by these two reference methods can prove the feasibility of 10our of 11 Thus, this increase is close to linear, which is shown in Figure 8. Moreover, these sensor heads proposed method indirectly. are all measured by the two reference methods proposed by Ren [14] and Tentori [15]. In the reference method proposed by 4 Ren [14], the results (sin δ) are about 0.8754, 0.5341, 0.3049, −0.0511 and −0.357 when the N is about 23,Measurement 27, 31, 35 Results and 39. In the reference method proposed by Tentori 3.5198 rad Fitting Curve [15], the results (sin δ) are about 0.8616, 0.5091, 0.4161, −0.0657 and −0.3744 when the N is about 23, 3.5 3.2003 rad 27, 31, 35 and 39. The results (sin δ) obtained by our proposed method are about 0.8641, 0.5229, 0.3452, −0.0587 and −0.3693 when the N is about 23, 27, 31, 35 and 39. Similarly with the first and 3 2.7891 rad second experiments, the results obtained by 2.5914 theserad two reference methods can prove the feasibility of our proposed method indirectly. 2.5 2

1.5

1 15

2.0983 rad

1.8425 rad

1.3577 rad

19

23

27

31

35

39

Number of Turns

Figure8.8.The Themeasurement measurementresults resultsofofδδwhen whenNNisisabout about15, 15,19, 19,23, 23,27, 27,31, 31,35 35and and39, 39,respectively. respectively. Figure

Conclusions 4.4.Conclusions Inthis thispaper, paper,we wepropose proposeaalinear linearbirefringence birefringencemeasurement measurementmethod methodfor foran anOFCS. OFCS.First, First,the the In optical configuration of the measurement system is presented. Then, the elimination method of the optical configuration of the measurement system is presented. Then, the elimination method of the effect of the azimuth angles between the sensing fiber and the two polarizers is demonstrated based on the orthogonal modulation and a Faraday rotator. The relationship of the linear birefringence, the Faraday rotation angle and the final output is determined based on the Jones matrix calculus and the heterodyne method. Moreover, the multi-valued problem on the linear birefringence is simulated and its solution is illustrated when the linear birefringence is unknown. Finally, the

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effect of the azimuth angles between the sensing fiber and the two polarizers is demonstrated based on the orthogonal modulation and a Faraday rotator. The relationship of the linear birefringence, the Faraday rotation angle and the final output is determined based on the Jones matrix calculus and the heterodyne method. Moreover, the multi-valued problem on the linear birefringence is simulated and its solution is illustrated when the linear birefringence is unknown. Finally, the experiments are conducted to prove the feasibility of the proposed method. When the numbers of turns of the sensing fiber in the OFCS are about 15, 19, 23, 27, 31, 35, and 39, the measured linear birefringence obtained by the proposed method are about 1.3577, 1.8425, 2.0983, 2.5914, 2.7891, 3.2003 and 3.5198 rad. Two typical methods provide the references for the proposed method. According to the comparison results, we find that the proposed method is suitable for the linear birefringence measurement in the full range without the limitation that the linear birefringence must be smaller than π/2. In this work, we only test the bending induced and inherent linear birefringence, which have been used to verify the feasibility of our proposed method. It is known that there is no accurate theory to express the temperature induced linear birefringence. This linear birefringence is absolutely unknown. Thus, in the next work, we will continue to test the temperature induced linear birefringence in the OFCS based on our proposed method, which will very useful for the suppression of the temperature error in OFCS. Acknowledgments: The authors acknowledge the Project Funded by National Key Research and Development Program of China (2016YFF0102400); National Natural Science Foundation of China (NSFC) (51607178); China Postdoctoral Science Foundation (2015M581882); Jiangsu Planned Projects for Postdoctoral Research Funds (1601260C); Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (TAPP); and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) for the financial support to this research. Author Contributions: Shaoyi Xu and Chuansheng Li conceived and designed the measurement method and the experiments; Wei Li, Haiming Shao and Yuqiao Wang performed the simulations and experiments; Fangfang Xing analyzed the data; Shaoyi Xu wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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A Linear Birefringence Measurement Method for an Optical Fiber Current Sensor.

In this work, a linear birefringence measurement method is proposed for an optical fiber current sensor (OFCS). First, the optical configuration of th...
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