# Influence of Imperfect Faraday Rotation Mirror on All-Fiber Optic Current Sensor

Authors: Jianhua Wu; Xiaofeng Zhang; Liang Chen
Journal: Journal of Sensors
 (2022)

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## Abstract

The effects of the Faraday rotation mirror on the all-fiber optic current sensor (AFOCS) are studied in this paper. The reflectivity degradation of FRM with the long-term operation, the misalignment of the optical axis between FRM and the sensing fiber, and the influence of temperature fluctuation on FRM are modeled and simulated in this paper. The effect of temperature is the biggest obstacle to the application of the Faraday rotation mirror, while other factors can be easy to overcome by the data processing method. The experiments are designed and realized for verification. The temperature fluctuation range is related to the measured electric current to satisfy the required accuracy for IEC 60044-8 class 0.2S.


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## Body

## 1. Introduction

With many unique advantages compared to traditional sensors, optical fiber sensors have been studied and applied to many different areas [1–5]. As a potential substitute for the traditional electromagnetic current transformer in power industry, the all-fiber optic current sensor (AFOCS) has many advantages, e.g., immunity against electromagnetic interference, high sensitivity, wide dynamic range, measuring AC and DC simultaneous, light weight, and small size [6, 7]. Due to these intrinsic advantages, AFOCS has been widely used in many electric current measurement fields, such as the electrolytic aluminium industry [8], high voltage direct current transmission system [9], international thermonuclear experimental reactor [10], and ship leakage current measurement [11].Accuracy is one of the most important factors restricting the rapid development of AFOCS. Therefore, improving the accuracy has attracted the attention of many researchers. The first factor that affects the accuracy is the imperfect device. The defective manufacture of the polarizer [12] and polarization-dependent crosstalk [13] will affect the extinction ratio and lead to the relative error. The second factor is the principal axes angle offsets, which are the essential factors affecting the output errors of AFOCS [14, 15]. The last factor is the external interference, such as the temperature and vibration, which will affect the relative error and scale factor [16, 17]. However, there is little literature on the reflection unit [18].The influence of the Faraday rotation mirror (FRM) on AFOCS focuses on two aspects. One is the reflected light energy, which affects the light intensity detected by the photodetector, while the light intensity affects the signal-to-noise ratio [19]. The other is to adjust the polarization state of reflected light, eliminate the influence of reciprocal linear birefringence, and improve the system accuracy [20]. However, due to the limitation of the manufacturing process level, FRM is usually imperfect [21]. At the same time, FRM is sensitive to temperature [22], and the polarization characteristics of silver thin film mirror typically change with the ambient temperature [23]. Therefore, the features of FRM must be affected by the temperature. In this paper, the influence of FRM on relative error is investigated by modelling and simulation. Experiments were designed and realized to verify the theoretical analysis and measure the temperature coefficient of FRM. Finally, the comprehensive influence of FRM on AFOCS is obtained.The remainder of the paper is organized as follows. Section2 is about the operation principle. The optical design and the data processing method are briefly introduced in this part. Section 3 is the modelling and simulation. We modeled and simulated the influence of FRM on the system accuracy by reflectivity, the misalignment of the optical axis, and the temperature. The experiment and discussion are in Section 4. The last section is the conclusion.


## 2. The Operating Principle

As the most mature structure of AFOCS, the reflective interferometer current sensor is mainly composed of a light source, an optical system, and a data processing unit [6]. As the key of the optical system, the sensing head usually includes a quarter-wave plate, a sensing fiber ring, and a reflection unit. A phase modulator is introduced to improve the anti-interference ability of the system with phase modulation technology [24]. The structure of AFOCS is shown in Figure 1.Figure 1
Reflective interferometer current sensor.The dotted black line denotes a single-mode fiber for energy transmission. The dot-and-dash line represents the polarization-maintaining fiber used to maintain the polarization state of the light. The solid black line expresses the sensing fiber, which is spun highly birefringent fiber. The arrow indicates the direction of electronic signal transmission. The working principle of the system is based on the Faraday effect and Ampere Circuital Theorem, which is described in Reference24. When the mirror is the FRM, the detected intensity resulting from the interference is given by
(1)Iout=kP01+cosθF‐Δφ2,where k denotes the scale factor related to optical path loss and photoelectric conversion rate of photodetector, P0 represents the superluminescent diode optical power, and Δφ denotes the phase difference modulated by the phase modulator. θF=4NVI+π is the state of the polarization (SOP) rotation angle, where Nexpresses the number of turns of the sensing fiber wrapped around the current-carrying wire, I denotes the electric current in the current-carrying wire, and V represents the Verdet constant of the sensing fiber. The Verdet constant is affected by the operating wavelength, material properties, and temperature [25].The first harmonic method [24] is used for data demodulation, and the Faraday rotation angle is
(2)θF=arcsin−U1J1δ⋅kP0≈−U1J1δ⋅kP0,where U1=−P0⋅sinθF⋅J1δ denotes the result of correlation demodulation, J1δ represents the first-order Bessel function of the first kind, δ is the modulation amplitude, which is usually 1.84 V to get the maximum of J1δ, and k and P0 are the same as Equation (1).


## 3. Modeling and Simulation

The relative error is used to describe the effect of FRM on the AFOCS. The relative errorη is given by
(3)η=ΔθFθF×100%,where ΔθF denotes the difference of the Faraday rotation angle due to the change of the reflection unit and θF denotes the Faraday rotation angle when the reflection unit is perfect. This paper is mainly about the effect of FRM on AFOCS, so it is assumed that the other parts of the optical system are excellent.


### 3.1. Effect of Reflectivity

The reflectivity of FRM will decline after a long-term online operation. The rotation angle of SOP to be measured will drift with the reflectivity of FRM. The drift of SOP rotation angle is(4)ΔθF=k2U1J1δ⋅kP0−U1J1δ⋅kP0,where k2 denotes the reflectivity of FRM after a long-time online operation, k2<1.Substituting Equations (2) and (4) into Equation (3), the relative error is given by
(5)η=k2−1×100%,where η and k2 are the same as Equations (3) and (4).The reflectivity of FRM is 100% and 99% in the perfect and actual conditions, respectively. The relationship between the relative error and the reflectivity is shown in Figure2.Figure 2
Relationship between the relative error and reflectivity.The relative error will increase as the reflectivity of FRM declines, as shown in Figure2. When the reflectivity drift is over 0.2%, AFOCS cannot satisfy the required accuracy for IEC 60044-8 class 0.2S [26].Harmonic division demodulation method could be applied to eliminate the effect of the energy fluctuation on the relative error [27]. The SOP rotation angle is
(6)θF=arctan−J2δ⋅U1J1δ⋅U2,where U2=P0⋅cosθF⋅J2δ and J2δ denotes the second-order Bessel functions of the first kind. The SOP rotation angle has nothing to do with the light intensity, according to Equation (6). The influence of reflectivity is eliminated using this demodulation method.


### 3.2. Effect of the Misalignment of Optical Axis

FRM consists of a nonreciprocal 45 deg rotator (45 deg Faraday rotator) followed by a coated mirror [20, 21], as shown in Figure 3.Figure 3
Faraday rotation mirror.There is a fusion point between FRM and the sensing fiber, as shown in Figure3. A polarization-maintaining fiber fusion splicer carries out the optical axis to axis fusion. The cross-section of alignment is shown in Figure 4.Figure 4
The cross-section of alignment error.Thex-axis and y-axis in solid lines represent the sensing fiber’s slow and fast axis directions, and the x′-axis and y′-axis in dotted lines denote the slow and fast axis directions of FRM. The rotation angle of these two coordinate axes is θ. The Jones matrix of FRM is
(7)M=MθoutMFoutMmirMFinMθin,where Mθout and Mθin are the Jones matrix for the fusion point through an angle θ in the backward and forward direction, respectively, MFout and MFin represent the Jones matrix for the Faraday rotator through a nonreciprocal angle θT backward and forward direction, respectively, and Mmir denotes the Jones matrix for the mirror. The details of the matrix are defined in Appendix.


#### 3.2.1. The Faraday Rotator Is Perfect

In this case,θT=45 deg. The Jones matrix for FRM is
(8)M=01−10.When the Faraday rotator is perfect, the optical axis misalignment between the sensing fiber and FRM has nothing to do with the system accuracy.


#### 3.2.2. The Faraday Rotator Is Imperfect

In this case, the Jones matrix for FRM is(9)M=cos2θTsin2θT−sin2θTcos2θT.When the Faraday rotator is imperfect, the misalignment of the optical axis between the sensing fiber and FRM also has nothing to do with the system accuracy. However, the imperfect Faraday rotator will be the main factor for relative error, which will be discussed in Section3.3.


### 3.3. The Effect of the Faraday Rotator

The imperfect Faraday rotator can be divided into two categories. One is the Faraday rotation angle deviated from 45 deg due to the imperfect manufacturing process, and the other is the Faraday rotation angle will change with temperature.


#### 3.3.1. The Effect of the Imperfect Manufacturing Process

When the Faraday rotator is imperfect, the Jones matrix of FRM is represented in Equation (9). The relative error will become
(10)η=2ΔθTθF×100%,where η and θF are defined as Equation (3). ΔθT=θT‐π/4 is the uncertain angle from π/4 rad. The range of ΔθT is usually from −0.5 deg to 0.5 deg for commercial products [10]. The number of turns of sensing fiber is 100, and the Verdet constant is 1.1×10−6 rad/A. If the relative error equals 0.2%, the relationship between the electric current to be measured and the uncertainty angle is shown in Figure 5.Figure 5
The relationship between the electric current to be measured and the uncertain angle.When the electric current to be measured is on the left and upper of the line, the relative error will meet the measurement requirements. Otherwise, the AFOCS will not meet the system accuracy requirements, as shown in Figure5.It should be noted that when the influence of zero bias is eliminated, the error introduced by FRM due to the manufacturing process can be eliminated. The error generated by the Faraday rotator is a fixed additive noise for the SOP rotation angle, so it can be eliminated by subtracting a fixed coefficient.


#### 3.3.2. The Effect of Temperature

Faraday rotator rotates of the SOP because the ferromagnetic materials provide a constant magnetic field. NdFeB is one kind of ferromagnetic material, which is usually applied to provide a constant magnetic field for a commercial Faraday rotator, but it is sensitive to temperature. When the temperature dependence and the nonlinearity of the magneto-optic material are ignored, the remanence temperature coefficient [28] is given by
(11)aT=ΔBdΔT⋅BdT0,where BdT0 is the magnetic induction intensity when the temperature is T0, ΔT represents the temperature fluctuation, T denotes the operation temperature, ΔT=T−T0, and ΔBd is the fluctuation of the magnetic induction intensity. The magnetic induction intensity can be given by Equation (12) when the operation temperature is T.
(12)BT=ΔBd+BdT0=1+aTΔTBdT0,where BT is the magnetic induction intensity when the operation temperature is T.The rotation angle of the Faraday rotator is 45 deg when the temperature isT0. The Faraday rotation angle is proportional to the magnetic induction intensity, so the Faraday rotator rotation angle can be expressed as Equation (13) when the temperature fluctuation is ΔT.
(13)θT=π41+aTΔT,where θT is the Faraday rotator rotation angle when the temperature is T. The relationship between the Faraday rotator rotation angle and temperature fluctuation is shown in Figure 6.Figure 6
The relationship between the Faraday rotator rotation angle and temperature fluctuation.The Faraday rotator rotation angle is proportional to the temperature fluctuation, as shown in Figure6, and the slope is the remanence temperature coefficient. We can get the relationship between the relative error and the temperature fluctuation by Equations (10), (12), and (13). The turns of sensing fiber and the Verdet constant are the same as Section 3.3.2. When the electric current to be measured is 10 kA, the relationship between relative error and temperature fluctuation is shown in Figure 7.Figure 7
The relationship between relative error and temperature fluctuation.The range of the operation is within 0.2%; the temperature fluctuation range increases with the remanence temperature coefficient decline, as shown in Figure7.


### 3.4. The Effect of the Mirror

To improve the reflectivity and the bandwidth of FRM, the dielectric film mirror composed of alternating films with high and low refractive index is usually applied as the mirror. The influence of temperature on the dielectric film is mainly in two aspects: the thickness and the refractive index. According to Reference [29], as the temperature changes from T0 to T, the thickness and refractive index of the film are given by Equations (14) and (15).
(14)dfT=df01+αf−BfΔT,(15)nfT=nf0+dndTfΔT+1−nf0−dndTfΔTAfΔT1+3αf+AfΔT,where f is variable and f=l and f=h denote the low- and high-index material, respectively, Af=21−2vfαs−αf/1−vf, and Bf=2vfαs−αf/1−vf. αf and αs are the thermal expansion coefficients of the film and the substrate, respectively, vf denotes the Poisson ratio of film, and ΔT=T−T0 represents the temperature fluctuation. dn/dTf is the thermal coefficient of the refractive index of the film material. df0 and nf0 denote the physical thickness and refractive index of the film, respectively. TiO2 and SiO2 are chosen as the high- and low-index material, respectively. Their thermal coefficients of the refractive index of the film material are 65×10−6/K and 40×10−6/K, respectively, and the thermal expansion coefficients of them are 7.5×10−6/K and 0.55×10−6/K [29], respectively. The refractive index and the thickness change with temperature fluctuation are shown in Figure 8.Figure 8
The effect of temperature fluctuation on mirror: (a) the relationship between the relative refractive index and temperature fluctuation and (b) the relationship between the film thickness and temperature fluctuation.
(a)(b)The reflection spectrum of the mirror changes with the thickness and refractive index of the thin film. However, the evolution of the reflection spectrum does not affect the accuracy of the system with the harmonic division demodulation, as described in Section3.1. Therefore, the influence of temperature on the mirror can be ignored.


## 3.1. Effect of Reflectivity

The reflectivity of FRM will decline after a long-term online operation. The rotation angle of SOP to be measured will drift with the reflectivity of FRM. The drift of SOP rotation angle is(4)ΔθF=k2U1J1δ⋅kP0−U1J1δ⋅kP0,where k2 denotes the reflectivity of FRM after a long-time online operation, k2<1.Substituting Equations (2) and (4) into Equation (3), the relative error is given by
(5)η=k2−1×100%,where η and k2 are the same as Equations (3) and (4).The reflectivity of FRM is 100% and 99% in the perfect and actual conditions, respectively. The relationship between the relative error and the reflectivity is shown in Figure2.Figure 2
Relationship between the relative error and reflectivity.The relative error will increase as the reflectivity of FRM declines, as shown in Figure2. When the reflectivity drift is over 0.2%, AFOCS cannot satisfy the required accuracy for IEC 60044-8 class 0.2S [26].Harmonic division demodulation method could be applied to eliminate the effect of the energy fluctuation on the relative error [27]. The SOP rotation angle is
(6)θF=arctan−J2δ⋅U1J1δ⋅U2,where U2=P0⋅cosθF⋅J2δ and J2δ denotes the second-order Bessel functions of the first kind. The SOP rotation angle has nothing to do with the light intensity, according to Equation (6). The influence of reflectivity is eliminated using this demodulation method.


## 3.2. Effect of the Misalignment of Optical Axis

FRM consists of a nonreciprocal 45 deg rotator (45 deg Faraday rotator) followed by a coated mirror [20, 21], as shown in Figure 3.Figure 3
Faraday rotation mirror.There is a fusion point between FRM and the sensing fiber, as shown in Figure3. A polarization-maintaining fiber fusion splicer carries out the optical axis to axis fusion. The cross-section of alignment is shown in Figure 4.Figure 4
The cross-section of alignment error.Thex-axis and y-axis in solid lines represent the sensing fiber’s slow and fast axis directions, and the x′-axis and y′-axis in dotted lines denote the slow and fast axis directions of FRM. The rotation angle of these two coordinate axes is θ. The Jones matrix of FRM is
(7)M=MθoutMFoutMmirMFinMθin,where Mθout and Mθin are the Jones matrix for the fusion point through an angle θ in the backward and forward direction, respectively, MFout and MFin represent the Jones matrix for the Faraday rotator through a nonreciprocal angle θT backward and forward direction, respectively, and Mmir denotes the Jones matrix for the mirror. The details of the matrix are defined in Appendix.


### 3.2.1. The Faraday Rotator Is Perfect

In this case,θT=45 deg. The Jones matrix for FRM is
(8)M=01−10.When the Faraday rotator is perfect, the optical axis misalignment between the sensing fiber and FRM has nothing to do with the system accuracy.


### 3.2.2. The Faraday Rotator Is Imperfect

In this case, the Jones matrix for FRM is(9)M=cos2θTsin2θT−sin2θTcos2θT.When the Faraday rotator is imperfect, the misalignment of the optical axis between the sensing fiber and FRM also has nothing to do with the system accuracy. However, the imperfect Faraday rotator will be the main factor for relative error, which will be discussed in Section3.3.


## 3.2.1. The Faraday Rotator Is Perfect

In this case,θT=45 deg. The Jones matrix for FRM is
(8)M=01−10.When the Faraday rotator is perfect, the optical axis misalignment between the sensing fiber and FRM has nothing to do with the system accuracy.


## 3.2.2. The Faraday Rotator Is Imperfect

In this case, the Jones matrix for FRM is(9)M=cos2θTsin2θT−sin2θTcos2θT.When the Faraday rotator is imperfect, the misalignment of the optical axis between the sensing fiber and FRM also has nothing to do with the system accuracy. However, the imperfect Faraday rotator will be the main factor for relative error, which will be discussed in Section3.3.


## 3.3. The Effect of the Faraday Rotator

The imperfect Faraday rotator can be divided into two categories. One is the Faraday rotation angle deviated from 45 deg due to the imperfect manufacturing process, and the other is the Faraday rotation angle will change with temperature.


### 3.3.1. The Effect of the Imperfect Manufacturing Process

When the Faraday rotator is imperfect, the Jones matrix of FRM is represented in Equation (9). The relative error will become
(10)η=2ΔθTθF×100%,where η and θF are defined as Equation (3). ΔθT=θT‐π/4 is the uncertain angle from π/4 rad. The range of ΔθT is usually from −0.5 deg to 0.5 deg for commercial products [10]. The number of turns of sensing fiber is 100, and the Verdet constant is 1.1×10−6 rad/A. If the relative error equals 0.2%, the relationship between the electric current to be measured and the uncertainty angle is shown in Figure 5.Figure 5
The relationship between the electric current to be measured and the uncertain angle.When the electric current to be measured is on the left and upper of the line, the relative error will meet the measurement requirements. Otherwise, the AFOCS will not meet the system accuracy requirements, as shown in Figure5.It should be noted that when the influence of zero bias is eliminated, the error introduced by FRM due to the manufacturing process can be eliminated. The error generated by the Faraday rotator is a fixed additive noise for the SOP rotation angle, so it can be eliminated by subtracting a fixed coefficient.


### 3.3.2. The Effect of Temperature

Faraday rotator rotates of the SOP because the ferromagnetic materials provide a constant magnetic field. NdFeB is one kind of ferromagnetic material, which is usually applied to provide a constant magnetic field for a commercial Faraday rotator, but it is sensitive to temperature. When the temperature dependence and the nonlinearity of the magneto-optic material are ignored, the remanence temperature coefficient [28] is given by
(11)aT=ΔBdΔT⋅BdT0,where BdT0 is the magnetic induction intensity when the temperature is T0, ΔT represents the temperature fluctuation, T denotes the operation temperature, ΔT=T−T0, and ΔBd is the fluctuation of the magnetic induction intensity. The magnetic induction intensity can be given by Equation (12) when the operation temperature is T.
(12)BT=ΔBd+BdT0=1+aTΔTBdT0,where BT is the magnetic induction intensity when the operation temperature is T.The rotation angle of the Faraday rotator is 45 deg when the temperature isT0. The Faraday rotation angle is proportional to the magnetic induction intensity, so the Faraday rotator rotation angle can be expressed as Equation (13) when the temperature fluctuation is ΔT.
(13)θT=π41+aTΔT,where θT is the Faraday rotator rotation angle when the temperature is T. The relationship between the Faraday rotator rotation angle and temperature fluctuation is shown in Figure 6.Figure 6
The relationship between the Faraday rotator rotation angle and temperature fluctuation.The Faraday rotator rotation angle is proportional to the temperature fluctuation, as shown in Figure6, and the slope is the remanence temperature coefficient. We can get the relationship between the relative error and the temperature fluctuation by Equations (10), (12), and (13). The turns of sensing fiber and the Verdet constant are the same as Section 3.3.2. When the electric current to be measured is 10 kA, the relationship between relative error and temperature fluctuation is shown in Figure 7.Figure 7
The relationship between relative error and temperature fluctuation.The range of the operation is within 0.2%; the temperature fluctuation range increases with the remanence temperature coefficient decline, as shown in Figure7.


## 3.3.1. The Effect of the Imperfect Manufacturing Process

When the Faraday rotator is imperfect, the Jones matrix of FRM is represented in Equation (9). The relative error will become
(10)η=2ΔθTθF×100%,where η and θF are defined as Equation (3). ΔθT=θT‐π/4 is the uncertain angle from π/4 rad. The range of ΔθT is usually from −0.5 deg to 0.5 deg for commercial products [10]. The number of turns of sensing fiber is 100, and the Verdet constant is 1.1×10−6 rad/A. If the relative error equals 0.2%, the relationship between the electric current to be measured and the uncertainty angle is shown in Figure 5.Figure 5
The relationship between the electric current to be measured and the uncertain angle.When the electric current to be measured is on the left and upper of the line, the relative error will meet the measurement requirements. Otherwise, the AFOCS will not meet the system accuracy requirements, as shown in Figure5.It should be noted that when the influence of zero bias is eliminated, the error introduced by FRM due to the manufacturing process can be eliminated. The error generated by the Faraday rotator is a fixed additive noise for the SOP rotation angle, so it can be eliminated by subtracting a fixed coefficient.


## 3.3.2. The Effect of Temperature

Faraday rotator rotates of the SOP because the ferromagnetic materials provide a constant magnetic field. NdFeB is one kind of ferromagnetic material, which is usually applied to provide a constant magnetic field for a commercial Faraday rotator, but it is sensitive to temperature. When the temperature dependence and the nonlinearity of the magneto-optic material are ignored, the remanence temperature coefficient [28] is given by
(11)aT=ΔBdΔT⋅BdT0,where BdT0 is the magnetic induction intensity when the temperature is T0, ΔT represents the temperature fluctuation, T denotes the operation temperature, ΔT=T−T0, and ΔBd is the fluctuation of the magnetic induction intensity. The magnetic induction intensity can be given by Equation (12) when the operation temperature is T.
(12)BT=ΔBd+BdT0=1+aTΔTBdT0,where BT is the magnetic induction intensity when the operation temperature is T.The rotation angle of the Faraday rotator is 45 deg when the temperature isT0. The Faraday rotation angle is proportional to the magnetic induction intensity, so the Faraday rotator rotation angle can be expressed as Equation (13) when the temperature fluctuation is ΔT.
(13)θT=π41+aTΔT,where θT is the Faraday rotator rotation angle when the temperature is T. The relationship between the Faraday rotator rotation angle and temperature fluctuation is shown in Figure 6.Figure 6
The relationship between the Faraday rotator rotation angle and temperature fluctuation.The Faraday rotator rotation angle is proportional to the temperature fluctuation, as shown in Figure6, and the slope is the remanence temperature coefficient. We can get the relationship between the relative error and the temperature fluctuation by Equations (10), (12), and (13). The turns of sensing fiber and the Verdet constant are the same as Section 3.3.2. When the electric current to be measured is 10 kA, the relationship between relative error and temperature fluctuation is shown in Figure 7.Figure 7
The relationship between relative error and temperature fluctuation.The range of the operation is within 0.2%; the temperature fluctuation range increases with the remanence temperature coefficient decline, as shown in Figure7.


## 3.4. The Effect of the Mirror

To improve the reflectivity and the bandwidth of FRM, the dielectric film mirror composed of alternating films with high and low refractive index is usually applied as the mirror. The influence of temperature on the dielectric film is mainly in two aspects: the thickness and the refractive index. According to Reference [29], as the temperature changes from T0 to T, the thickness and refractive index of the film are given by Equations (14) and (15).
(14)dfT=df01+αf−BfΔT,(15)nfT=nf0+dndTfΔT+1−nf0−dndTfΔTAfΔT1+3αf+AfΔT,where f is variable and f=l and f=h denote the low- and high-index material, respectively, Af=21−2vfαs−αf/1−vf, and Bf=2vfαs−αf/1−vf. αf and αs are the thermal expansion coefficients of the film and the substrate, respectively, vf denotes the Poisson ratio of film, and ΔT=T−T0 represents the temperature fluctuation. dn/dTf is the thermal coefficient of the refractive index of the film material. df0 and nf0 denote the physical thickness and refractive index of the film, respectively. TiO2 and SiO2 are chosen as the high- and low-index material, respectively. Their thermal coefficients of the refractive index of the film material are 65×10−6/K and 40×10−6/K, respectively, and the thermal expansion coefficients of them are 7.5×10−6/K and 0.55×10−6/K [29], respectively. The refractive index and the thickness change with temperature fluctuation are shown in Figure 8.Figure 8
The effect of temperature fluctuation on mirror: (a) the relationship between the relative refractive index and temperature fluctuation and (b) the relationship between the film thickness and temperature fluctuation.
(a)(b)The reflection spectrum of the mirror changes with the thickness and refractive index of the thin film. However, the evolution of the reflection spectrum does not affect the accuracy of the system with the harmonic division demodulation, as described in Section3.1. Therefore, the influence of temperature on the mirror can be ignored.


## 4. Experiments and Discussion

### 4.1. The Effect of Temperature on FRM

The Faraday rotator rotation angle changes with temperature and adversely impacts the system accuracy. The experimental system is designed in Figure9.Figure 9
The temperature stability test system of FRM.The dotted line denotes the single-mode optical fiber to transmit energy. The polarization-maintaining fiber is represented by the solid blue line to maintain the state of polarization. The solid red line expresses the electronic signal. The temperature control pipeline of the vacuum high- and low-temperature test box is expressed with a double solid line, as shown in Figure9. The model and main parameters of experimental devices are shown in Table 1.Table 1
Device model and main parameters.
NameBrandModel or main parametersLight sourceThorlabs China Co., Ltd.S5FC1018PPolarizerMC Fiber Optics Co., Ltd.The extinction ratio is no less than 28 dBCirculatorMC Fiber Optics Co., Ltd.Commercially availablePhase modulatorSWT OPTICS Co., Ltd.The half-wave voltage is 4 VPolarization-maintaining fiberYOFC Optical Fiber and Cable Co., Ltd.PM1016-AFaraday rotation mirrorMC Fiber Optics Co., Ltd.Reflectivity is greater than 99%@1310 nmPhotodetectorConquer Co., Ltd.KG-HSPLock-in amplifierZurich InstrumentsMFLI 500 kHzSignal generatorTektronixAFG1062Vacuum high- and low-temperature test boxWuhan Tailunte Century Technology Co., Ltd.TLVB-VHLB50, range: -60°C-100°C,accuracy±0.1°CDC-regulated power supplyRohde & Schwarz, Munich, GermanyR&S®HMP4000The temperature of the vacuum high- and low-temperature test box was set from 0°C to 80°C, and the temperature step interval was 1°C. When the temperature reached the set temperature, keep the temperature for 3 minutes before data acquisition to ensure that the FRM had been evenly heated. The data acquisition time is 1 minute, and the mean value is used as the measurement result to eliminate the influence of random noise. The relationship between correlation demodulation results and temperature obtained by the lock-in amplifier is shown in Figure10.Figure 10
The relationship between correlation demodulation results and temperature.The hollow dots and the solid lines are the observed and fitted values, as shown in Figure9. The correlation coefficient is 0.9939, which shows that the fitted values are in good agreement with the experimental values. At this time, the data fitting equation is given by
(16)RT=acosbT+c=6.029cos0.02968T−1.0222.The remanence temperature coefficient isaT=0.009447 based on Equations (6), (9), (13), and (16). The relationship between the relative error and the temperature fluctuation is shown in Figure 11.Figure 11
The relationship between relative error and temperature fluctuation.When the relative error is within ±0.2% for power system measurement, the temperature fluctuation range is ±0.6°C, far less than the required working range of AFOCS. Therefore, optimizing the Faraday rotator, such as by vacuum thermal insulation packaging, is necessary to maintain the Faraday rotator in a constant temperature range.


### 4.2. The Effect of Temperature on AFOCS

We tested the effect of temperature on AFOCS in this section. We introduced a quarter-wave plate and the sensing fiber on the basis of Figure9, as shown in Figure 12.Figure 12
The temperature stability test system of AFOCS.The quarter-wave plate was made by ourselves, which is shown with the pink line in Figure12. The sensing fiber is the spun highly birefringent fiber made from YOFC Optical Fiber and Cable Co., Ltd. The beat length and spin pitch are 10 mm and 5 mm, respectively, as shown in Figure 12.The temperature of the vacuum high- and low-temperature test box was set from 5°C to 65°C, and the temperature step interval was 2°C. When the temperature reached the set temperature, keep the temperature for 3 minutes before data acquisition to ensure that the sensing head had been evenly heated. The data acquisition time is 1 minute, and the mean value is used as the measurement result to eliminate the influence of random noise. The relationship between the zero drift (the electric current was set to zero) of the system and temperature is shown in Figure13.Figure 13
The relationship between the temperature and output of AFOCT.The hollow dots and the solid lines are the observed and fitted values, as shown in Figure9. The correlation coefficient is 0.9867, which shows that the fitted values are in good agreement with the observed values. At this time, the data fitting equation is given by
(17)R′T=a1sinb1T+c1+a2sinb2T+c2=AT+BT,where a1=2.71, b1=0.02347, c1=0.5561, a2=0.7596, b2=0.1414, and c2=2.261. The remanence temperature coefficient is 0.007113. The fitted function could be regarded as the sum of two functions. One is related to the Verdet constant and the quarter-wave plate (Bt), and the other is related to the FRM (At). The remanence temperature coefficient is a little different from aT obtained in Section 4.1. The reason may be some errors generated during the fusion process of the sensing optical fiber.


## 4.1. The Effect of Temperature on FRM

The Faraday rotator rotation angle changes with temperature and adversely impacts the system accuracy. The experimental system is designed in Figure9.Figure 9
The temperature stability test system of FRM.The dotted line denotes the single-mode optical fiber to transmit energy. The polarization-maintaining fiber is represented by the solid blue line to maintain the state of polarization. The solid red line expresses the electronic signal. The temperature control pipeline of the vacuum high- and low-temperature test box is expressed with a double solid line, as shown in Figure9. The model and main parameters of experimental devices are shown in Table 1.Table 1
Device model and main parameters.
NameBrandModel or main parametersLight sourceThorlabs China Co., Ltd.S5FC1018PPolarizerMC Fiber Optics Co., Ltd.The extinction ratio is no less than 28 dBCirculatorMC Fiber Optics Co., Ltd.Commercially availablePhase modulatorSWT OPTICS Co., Ltd.The half-wave voltage is 4 VPolarization-maintaining fiberYOFC Optical Fiber and Cable Co., Ltd.PM1016-AFaraday rotation mirrorMC Fiber Optics Co., Ltd.Reflectivity is greater than 99%@1310 nmPhotodetectorConquer Co., Ltd.KG-HSPLock-in amplifierZurich InstrumentsMFLI 500 kHzSignal generatorTektronixAFG1062Vacuum high- and low-temperature test boxWuhan Tailunte Century Technology Co., Ltd.TLVB-VHLB50, range: -60°C-100°C,accuracy±0.1°CDC-regulated power supplyRohde & Schwarz, Munich, GermanyR&S®HMP4000The temperature of the vacuum high- and low-temperature test box was set from 0°C to 80°C, and the temperature step interval was 1°C. When the temperature reached the set temperature, keep the temperature for 3 minutes before data acquisition to ensure that the FRM had been evenly heated. The data acquisition time is 1 minute, and the mean value is used as the measurement result to eliminate the influence of random noise. The relationship between correlation demodulation results and temperature obtained by the lock-in amplifier is shown in Figure10.Figure 10
The relationship between correlation demodulation results and temperature.The hollow dots and the solid lines are the observed and fitted values, as shown in Figure9. The correlation coefficient is 0.9939, which shows that the fitted values are in good agreement with the experimental values. At this time, the data fitting equation is given by
(16)RT=acosbT+c=6.029cos0.02968T−1.0222.The remanence temperature coefficient isaT=0.009447 based on Equations (6), (9), (13), and (16). The relationship between the relative error and the temperature fluctuation is shown in Figure 11.Figure 11
The relationship between relative error and temperature fluctuation.When the relative error is within ±0.2% for power system measurement, the temperature fluctuation range is ±0.6°C, far less than the required working range of AFOCS. Therefore, optimizing the Faraday rotator, such as by vacuum thermal insulation packaging, is necessary to maintain the Faraday rotator in a constant temperature range.


## 4.2. The Effect of Temperature on AFOCS

We tested the effect of temperature on AFOCS in this section. We introduced a quarter-wave plate and the sensing fiber on the basis of Figure9, as shown in Figure 12.Figure 12
The temperature stability test system of AFOCS.The quarter-wave plate was made by ourselves, which is shown with the pink line in Figure12. The sensing fiber is the spun highly birefringent fiber made from YOFC Optical Fiber and Cable Co., Ltd. The beat length and spin pitch are 10 mm and 5 mm, respectively, as shown in Figure 12.The temperature of the vacuum high- and low-temperature test box was set from 5°C to 65°C, and the temperature step interval was 2°C. When the temperature reached the set temperature, keep the temperature for 3 minutes before data acquisition to ensure that the sensing head had been evenly heated. The data acquisition time is 1 minute, and the mean value is used as the measurement result to eliminate the influence of random noise. The relationship between the zero drift (the electric current was set to zero) of the system and temperature is shown in Figure13.Figure 13
The relationship between the temperature and output of AFOCT.The hollow dots and the solid lines are the observed and fitted values, as shown in Figure9. The correlation coefficient is 0.9867, which shows that the fitted values are in good agreement with the observed values. At this time, the data fitting equation is given by
(17)R′T=a1sinb1T+c1+a2sinb2T+c2=AT+BT,where a1=2.71, b1=0.02347, c1=0.5561, a2=0.7596, b2=0.1414, and c2=2.261. The remanence temperature coefficient is 0.007113. The fitted function could be regarded as the sum of two functions. One is related to the Verdet constant and the quarter-wave plate (Bt), and the other is related to the FRM (At). The remanence temperature coefficient is a little different from aT obtained in Section 4.1. The reason may be some errors generated during the fusion process of the sensing optical fiber.


## 5. Conclusion

As an essential part of AFOCS, the influence of FRM is analyzed in this paper. The conclusion can be summarized as follows.Firstly, harmonic division demodulation is better than the first harmonic method to copy with the fluctuation of light intensity, which is caused by long-term operation or the effect of the temperature fluctuation on the thin film mirror.Secondly, the optical axis misalignment between the sensing fiber and FRM has nothing to do with the relative error. So the single-mode fusion splicer can be applied in this fusion splicing process.Thirdly, the influence of temperature on the Faraday rotator determines the performance of the reflection unit. The temperature fluctuation range is related to the measured electric current to satisfy the required accuracy for IEC 60044-8 class 0.2S if the FRM does not work at a stable temperature.



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