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Temperature compensation of optical alternating magnetic field sensor via a novel method for on-line measuring

Hongyuan Wang, Junfeng Jiang, Haofeng Hu, Qun Han, Zezhou Sun, Zheng Zheng, Zhenzhou Cheng, and Tiegen Liu

Open Access Open Access

Abstract

The variation of environment temperature is a crucial problem for optical magnetic field sensors based on the magneto-optical crystal. In this paper, we propose a novel temperature compensation method for optical alternating magnetic field measuring by analyzing the demodulation principle and establishing the temperature compensation model, which can implement the functions of temperature compensation and on-line measuring simultaneously. Both the temperature and the alternating magnetic field flux density can be obtained only by adding two magnet rings on the magnetic field sensor. The experimental phenomenon agrees well with the temperature characteristics of the magneto-optical crystal and the theoretical compensation model. The experimental results demonstrate that this sensor has excellent stability whose max relative fluctuation is only 0.7402% in the range of 0-4 mT under a constant temperature. In the temperature compensation experiment of 0 °C, 20 °C and 40 °C, the sensor shows strong temperature robustness that the max absolute and relative errors are 0.07 mT and 3.50%, respectively. Meanwhile, compensation efficiency reaches 83.968%, which can effectively avoid temperature crosstalk to a large extent. Additionally, it has a better compensation performance whose max absolute and relative errors are 0.15 mT and 1.66% in the broader range of 0-16 mT when the actual temperature is accurately known.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

There has been a significant research interest in magnetic field sensors due to the prospective applications such as current measurement, magnetic memory readout, magnetic compass, traffic control, brain function mapping, astronomy, geological and medical applications [1]. During the past four decades, magnetic field sensors in optical techniques have aroused more and more attractions due to the advantages of immunity to electromagnetic interference, strong resistance to chemical erosion, high accuracy, large bandwidth, zero hysteresis and, by utilizing optical fibers, high flexibility, light weight and compact size [2,3].

To date, a large number of methods have been developed for optical magnetic field measuring based on Faraday effect [4–6], magnetic fluid [7–9], magnetostrictive material [10,11], surface plasmon resonance (SPR) [12], photonic crystal fiber [13–15], fiber ring laser [16] and varieties of microstructures [17,18]. Several novel structures of magnetic field sensors have been proposed as follows: Violakis et al. present a magnetic field sensor by using a ferrofluid encapsulated single-mode side-polished D-shaped optical fiber based on differential loss with a sensitivity of 0.015 dB/Gs [19]. Zhao et al. propose a sensor with temperature compensation by combining the Fabry-Perot (F-P) cavity, the characteristics of magnetic-controlled refractive index and fiber Bragg grating (FBG) with a sensitivity of 0.04 nm/Gs and measurement resolution 0.5 Gs [20]. Zhou et al. also present a magnetic field sensor with temperature calibration by using a silica fiber F-P resonator with a silicone cavity [21]. Sun et al. demonstrate a tapered two-mode fiber (TTMF) sandwiched between two single-mode fibers to realize a low-temperature and high-sensitivity magnetic field sensor [22]. However, on the one hand, most of them are designed for static magnetic field measurement because of the slow response time of magnetic fluid and magnetostrictive materials and also limited by the scanning speed of the spectrometer. On the other hand, temperature crosstalk is another significant issue for the magnetic field sensors, which severely affects the accuracy of the sensors [23].

In this paper, a novel optical alternating magnetic field sensor and a new algorithm to compensate the influence of temperature crosstalk are proposed. Both temperature and alternating magnetic field flux density can be obtained simultaneously by combining the output characteristics of magneto-optical crystal in different temperatures and magnetic fields with the demodulation method of Faraday rotation angle using a dual-channel photodetector. The environment temperature can be calculated according to the direct current (DC) offset output of the sensor by adding a constant magnetic field to the magneto-optical crystal, through which the compensation factor can be obtained to decrease the magnetic field fluctuation caused by variation of the ambient temperature of the sensor. The position, shape, and magnetic field flux density of the magnet are all well designed, and this method can achieve excellent temperature compensation for the alternating magnetic field. Therefore, such a method can be applied in the system which is similar to this kind of sensor and not limited to the alternating magnetic field and alternating current (AC) sensing applications.

2. Principle

The working principle of magneto-optical crystal magnetic field sensors is the Faraday effect. The polarization plane will rotate when a linearly polarized light propagates through the magneto-optical medium along the direction of the magnetic field [24]. For a uniform magnetic field and a homogeneous crystal along the light path, the Faraday rotation angle $\theta $ can be expressed as

$$\theta \textrm{ = }V \cdot \int\limits_L {\vec{B} \cdot \textrm{d}\vec{L}} ,$$
where V is the Verdet constant of the magneto-optical crystal, B is the magnetic flux density along the direction of light propagation, and L is the length of the crystal that light travels.

The influence of temperature on the Faraday rotation of a magneto-optical crystal is mainly from two aspects. One is the variation of Verdet constant through affecting the thermal motion of electrons inside of the magneto-optical crystal, and the other is the length change of the crystal due to the thermal expansion effect. The overall temperature-dependence of the Faraday rotation can be expressed as [25]

$$\frac{1}{\theta }\frac{{\textrm{d}\theta }}{{\textrm{d}T}} = \frac{1}{V}\left( {\frac{{\partial V}}{{\partial \lambda }}\frac{{\textrm{d}\lambda }}{{\textrm{d}T}} + \frac{{\partial V}}{{\partial T}}} \right) + \frac{1}{L}\frac{{\partial L}}{{\partial T}},$$
where T represents temperature, and $\lambda $ represents the wavelength of the light.

It can be seen that both the Verdet constant wavelength-dependence and the wavelength temperature drift have an effect on the Faraday rotation angle. Therefore, the dependence of wavelength on temperature, $\frac{{\partial \lambda }}{{\partial T}}$, almost equals zero when a light source with temperature regulation is used directly or the temperature is kept at a constant. Equation (3) can be simplified to

$$\frac{1}{\theta }\frac{{\textrm{d}\theta }}{{\textrm{d}T}} = \frac{1}{V}\frac{{\partial V}}{{\partial T}} + \alpha \frac{1}{L},$$
where $\alpha = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial T}}} \right.} {\partial T}}$ is the thermal expansion coefficient of the magneto-optical crystal. For the crystal used in this paper, its thickness L is 390 μm and $\alpha $ equals $11 \times {10^{ - 6}}\textrm{ }{\textrm{K}^{ - 1}}$. The thickness of the crystal will change about 0.34 μm when the temperature changes in the range of 293.15 ± 40 K(20 ± 40 °C), which is about $8.8{\raise0.5ex\hbox{$\scriptstyle \circ $}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle { \circ{\circ} \circ }$}}$ of the thickness of the magneto-optical crystal and ultimately changes the Faraday rotation angle. In fact, the temperature-dependence of the Faraday rotation is about 0.065 deg/°C, and it will almost decrease 5.2 deg in the ±40 °C range. These theoretical analysis results agree well with the change of the magneto-optical crystal Faraday rotation angle in the experiment where the measuring range is 2-9 mT, and the temperature range is -20-70 °C in [26]. Therefore, compared with the change of thickness resulted from thermal expansion, the variation of Verdet constant caused by temperature plays a dominant role. It is also confirmed in the experiment of this paper that the Verdet constant of magneto-optical crystal decreases with the increase of temperature. The reason why Verdet constant decreases may be that the electron thermal motion inside the magneto-optical crystal is intensified, which enhances the hindrance to the variation of the magnetic moment caused by the external magnetic field.

Magneto-optical materials are no longer isotropic when they are under a magnetic field. Therefore, a phase delay $\varphi = {{\pi ({n_r} + {n_l})L} \mathord{\left/ {\vphantom {{\pi ({n_r} + {n_l})L} \lambda }} \right.} \lambda }$ and a rotation angle $\theta = {{\pi ({n_r} - {n_l})L} \mathord{\left/ {\vphantom {{\pi ({n_r} - {n_l})L} \lambda }} \right.} \lambda }$ will be generated after a beam of linearly polarized light traveling through magneto-optical materials, where ${n_r}$ and ${n_l}$ represent the refractive indices of right-hand and left-hand circular polarization light decomposed from the linearly polarized light passing through the magneto-optical materials.

The internal structure of the sensor is shown in Fig. 1. The magneto-optical crystal rotates the linearly polarized light under the magnetic field and then split up into two beams ${I_ \bot }$ and ${I_\parallel }$ whose polarizations are perpendicular to each other by the polarizing beam splitter (PBS). Angles between the orientation of the polarizer and the directions of PBS are ${\pm} {45^ \circ }$. Output intensities for the sensor are described as ${I_ \bot }$ and ${I_\parallel }$.

$$\left\{ {\begin{array}{{c}} {{I_ \bot } = {I_0}{{\cos }^2}({{{45}^ \circ } + \theta } )= \frac{{{I_0}}}{2}({1 - \sin 2\theta } )\textrm{ = }\frac{{{I_0}}}{2} - \frac{{{I_0}}}{2}\sin [\frac{{2\pi }}{\lambda }({{n_r} - {n_l}} )L]}\\ {{I_\parallel } = {I_0}{{\cos }^2}({{{45}^ \circ } - \theta } )= \frac{{{I_0}}}{2}({1 + \sin 2\theta } )\textrm{ = }\frac{{{I_0}}}{2} + \frac{{{I_0}}}{2}\sin [\frac{{2\pi }}{\lambda }({{n_r} - {n_l}} )L]} \end{array}} \right.$$

The two output channels of the sensor are complementary and appear the change rule of sinusoidal signal approximately with the variation of wavelength. To filter out the influence of the light source disturbance on the input light intensity, the differential signal demodulation method is used for the Faraday rotation angle. Here, S is defined as the output parameter which is introduced as

$$S = \frac{{{I_\parallel } - {I_ \bot }}}{{{I_\parallel } + {I_ \bot }}} = \sin [\frac{{2\pi }}{\lambda }({{n_r} - {n_l}} )L] = \sin ({2\theta } )= \sin ({2VBL} ).$$

 figure: Fig. 1.

Fig. 1. The internal structure of the sensor.

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According to Eq. (5), it is easily known that the arcsine of the output parameter S represents $2\theta $ which is theoretically proportional to the input magnetic field. It also means that the sensor becomes a linear transformation system between the input magnetic field and the output parameter when calculating the $\arcsin (S)$ as the new output parameter instead of the original output parameter S. It is quite apparent that the sensor will show different output characteristics, here means different ks, under different temperatures T if we define the input magnetic field B as the X-axis, output parameter $2\theta $ as the Y-axis and the slope of $2\theta \textrm{ - }B$ curve as k. Therefore, it can be seen from Fig. 2 that there is a distortion-free linear amplification effect between the output and the input of the sinusoidal signal, and the reason why the sensor has a temperature-sensitive impact can be attributed to this linear amplification effect at different amplitudes.

 figure: Fig. 2.

Fig. 2. Temperature compensation theory based on a distortion-free linear amplification effect.

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Hence, the function as follows can be used to eliminate this amplification effect and reduce the influence of temperature on the measurement results.

$${B_{{T_0}}} \propto {k_{{T_0}}} \cdot {{{{[2{\theta _{AC}}]}_{rms}}} \mathord{\left/ {\vphantom {{{{[2{\theta_{AC}}]}_{rms}}} {{k_T}}}} \right.} {{k_T}}},$$
where rms means root-mean-square which is also the effective value of twice the Faraday rotation θ, ${k_{{T_0}}}$ and ${k_T}$ denote the slopes under the temperatures of T0 and T, ${B_{{T_0}}}$ represents the magnetic field flux density under T0. Here, we multiply ${k_{{T_0}}}$ for normalizing the output of the sensor to a certain standard temperature in order that it can be compared with the original output data before compensation.

Here, a novel temperature compensation method is proposed which can measure the effective value of the alternating magnetic field and the current temperature simultaneously by adding two magnet rings to the original sensor, as is shown in Fig. 3. These two magnet rings can bring a constant DC bias for the magneto-optical crystal and allow its static working point to be affected by ambient temperature for alternating measuring.

 figure: Fig. 3.

Fig. 3. Structure of the novel sensor which can implement the compensation.

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Ambient temperature and magnetic field can be calculated by the DC offset and the AC signal output respectively through the equation set as follows:

$$\left\{ {\begin{array}{{c}} {S(t )= \sin ({2{\theta_{AC}}} )+ \sin ({2{\theta_{DC}}} )}\\ {\int_t^{t + n\tau } {S(t )\textrm{d}t} = n\tau \cdot \sin ({2{\theta_{DC}}} )}\\ {\sin ({2{\theta_{DC}}} )= {{ {{f_1}(T )} |}_{{B_{DC}}}}}\\ {{k_T} = {f_2}(T )}\\ {{B_{A{C_{{T_0}}}}} \propto {k_{{T_0}}} \cdot {{{{\{{\arcsin [{S(t )- \sin ({2{\theta_{DC}}} )} ]} \}}_{rms}}} \mathord{\left/ {\vphantom {{{{\{{\arcsin [{S(t )- \sin ({2{\theta_{DC}}} )} ]} \}}_{rms}}} {{k_T}}}} \right.} {{k_T}}}} \end{array}} \right.,$$
where $S(t )$ denotes the original output parameter, $\tau $ represents the period of the signal, ${ {{f_1}(T )} |_{{B_{DC}}}}$ is the function between the output parameter $\sin ({2{\theta_{DC}}} )$ and the temperature in the presence of the DC magnetic field, and ${f_2}(T )$ is the k-T functional expression. ${ {{f_1}(T )} |_{{B_{DC}}}}$ and ${f_2}(T )$ are both available through measuring in experiments.

The slope of $2\theta \textrm{ - }B$ function in Fig. 2 is a constant under a specific temperature, which can be described as:

$$\frac{{2\textrm{d}\theta }}{{\textrm{d}B}} = {k_T}.$$

When the magnetic field is uniform, and the light travels along the direction of the magnetic field, the formula can be obtained as follows:

$$\frac{{2\textrm{d}\theta }}{{\textrm{d}B}} = 2{V_T}L = {k_T}.$$
${k_T}\textrm{ - }T$ change rules can be studied by plotting ${V_T}\textrm{ - }T$ curves as a result that ${k_T} \propto {V_T}$ when the change of L is ignored, which is analyzed above.

According to the study of electrodynamics theory, there are two types of ratios under different temperature ranges of Verdet constant V and magnetic susceptibility $\chi $ [27]. One is ${{V(T )} \mathord{\left/ {\vphantom {{V(T )} \chi }} \right.} \chi } = A({1 + BT} )$. T represents temperature, A and B are the coefficients related to the materials where A is wavelength dependent. We suppose that the magnetic susceptibility obeys the Curie-Weiss law:

$$\chi = \frac{c}{{T - {T_p}}},$$
where c is Curie constant of the material, and ${T_p}$ is Curie temperature. Therefore, the ${k_T}\textrm{ - }T$ curve can be fitted by the formula:
$$k(T )= A^{\prime} \cdot \left( {1 + \frac{{B^{\prime}}}{{T - {T_p}}}} \right).$$

3. Experiment of AC characteristics under constant temperature

To verify the AC performance of the magnetic field sensor, the following experiments are carried out. The schematic of the experimental setup is shown in Fig. 4. Here, the two magnet rings are not necessary because we do not need to measure the temperature for temperature compensation. The magnetic field is generated by a 4 mT/A electrified solenoid in the center of which the sensor is placed by the clamp. The light from the light source travels through the sensor, and the polarization plane of linearly polarized light changes as a result of the existence of the alternating magnetic field. Changes in the light intensity signal of both outputs of the sensor are detected and converted to the voltage signal by the dual-channel photodetector (PD) and sampled to the digital signal by the data acquisition card (DAQ) and then uploaded to the PC for data processing. The temperature of the sensor is controlled by a large size thermostat.

 figure: Fig. 4.

Fig. 4. Schematic of the experimental setup.

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Two voltage signal outputs (out-1, out-2) of PD, its difference without DC offset (diff), and the output parameter S are shown in Fig. 5(a) when 50 Hz AC is applied to the solenoid under room temperature. Outputs of the sensor are recorded under different amplitudes of the alternating magnetic field. The relationship between the output parameter S and various magnetic fields at different time can be seen in Fig. 5(b). It can be seen that the output signal of the sensor is stable in the period under different amplitudes of the alternating magnetic field.

 figure: Fig. 5.

Fig. 5. Two outputs of PD and the parameter S under 50 Hz alternating magnetic field under constant temperature. (a) Two outputs, its difference and output parameter S. (b) Parameter S under different magnetic fields.

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The magnetic field varies from 0.4 mT to 4 mT (current: 0.1 A-1 A) with a step of 0.4 mT. The sensor’s measuring range is estimated in the next part of the experiment in this paper. To verify the relevance of the output parameter S and the alternating input magnetic field, the effective value of the sinusoidal signal S is used to represent the output. Figure 6(a) depicts the fit curve of average S and the effective value of the input magnetic field where T1, T2 to T7 represent different measuring time. It can be seen that there is a good linear relationship between output and input when the effective value of the alternating input magnetic field is less than 4 mT, and the sensor has good time stability. To analyze it more specifically, the time stability of the sensor is investigated, as is shown in Fig. 6(b) and Fig. 6(c). The maximum absolute and relative fluctuations for the output parameter S of the sensor are 0.00019 and 0.7402%, respectively. With the increase of the input magnetic field, the relative fluctuation of the sensor decreases obviously. Theoretically, the actual absolute and relative errors should be less than the fluctuations which are defined as (max-min) and (max-min)/average in this paper.

 figure: Fig. 6.

Fig. 6. Experiment result analysis of AC characteristics under constant temperature. (a) A linear relationship between the output parameter S and the effective value of the magnetic field at different time. (b) Variation of S at different time. (c) The absolute and relative fluctuations of S.

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4. Experiment of temperature compensation

The temperature characteristics of the sensor are investigated by using the DC power supply in Fig. 7(a). The relationship between output parameter S measured in the experiment and input magnetic field can be well fitted by sinusoidal function in the range of 0-36 mT. The sinusoidal function is a periodic function and the measuring range must be less than half of the period of the sinusoidal function (between the minimum and the maximum value of the sinusoidal function). Because in this case, there is one-to-one mapping between S and the magnetic field. It means that if P is used to represent the period of the sinusoidal function, the maximum DC measuring range is between -P/4 and + P/4. Therefore, considering the errors of device machining and installation, the DC measuring range is set between -40 mT and +40 mT. If the peak value of the magnetic field to be measured exceeds this range, the demodulation method described above will have the ambiguity problem. Another thing to note is that the magnet rings must not be added when measuring DC magnetic field. Here, the reason why the output parameter isn’t equal to zero is the angles between the orientation of the polarizer and the directions of PBS cannot be guaranteed to be ${\pm} {45^ \circ }$ as a result of installation error when there is no magnetic field input. To get the dependencies between Faraday rotation and the magnetic field under different temperatures, we calculate the inverse trigonometric function of S and perform a linear fit to it. It can be seen that the Faraday rotation of the sensor shows different slopes under -20-60 °C in Fig. 7(b). The results of Fig. 7 are in great agreement with the theoretical analysis above.

 figure: Fig. 7.

Fig. 7. Characteristics of the sensor under various temperatures obtained by DC measurement experiment. (a) Characteristics of the output parameter S under different temperatures and the measuring range of the sensor. (b) Dependencies between twice the Faraday rotation angle and magnetic field under various temperatures.

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To verify the compensation model, the compensation experiment under the circumstance that temperature is known accurately for AC signal effective value output of the sensor in the range of -20-60 °C through the characteristic temperature curve based on DC signal in Fig. 7(b) is performed. Figure 8 shows the contrast of the AC magnetic field compensation results in which the input magnetic are 0, 4 mT, 8 mT, 12 mT and 16 mT. There is a noticeable temperature compensation effect, and it can be calculated that the maximum fluctuation before compensation is 0.0657. However, it is reduced to 0.0091 after compensation. Overall, the fluctuation caused by temperature variation can be reduced by about 6-8 times through this method, and the max absolute error is 0.15 mT, and the max relative error is 1.66%.

 figure: Fig. 8.

Fig. 8. The contrast of the AC magnetic field compensation results under different temperatures which is precisely known in the range of -20 °C-60 °C when the input magnetic are 0, 4 mT, 8 mT, 12 mT and 16 mT, respectively.

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To realize the temperature self-compensation, the relationship between k and T has been plotted and fitted with the Eq. (11). It can be seen that the fitting curve of experimental data in Fig. 9 is in good agreement with theoretical derivation. The coefficient of determination ${R^2}$ is as high as 0.9985.

 figure: Fig. 9.

Fig. 9. Relationship between slope k and temperature.

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The DC output parameter S of the sensor with two magnet rings under various temperatures is calibrated before measurement. The blue dots in Fig. 10 represent the output parameter of the sensor under different temperatures, and there is an excellent linearity between temperature and S. This fit curve can be used for measuring current temperature. $2\theta $ is derived from the trigonometric relation, and the result meets the above theoretical model in Eq. (11) again.

 figure: Fig. 10.

Fig. 10. Temperature dependence of DC output parameter S and 2θ in a real-world experiment.

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The experiment of temperature self-compensation alternating magnetic field measuring with two magnet rings is performed under the temperatures of 0 °C, 20 °C and 40 °C. Based on the analysis of Fig. 7(a), the magnetic field flux density of two magnet rings should be set appropriately. The change of the output is difficult to be detected if the DC offset point is set too small, and it will cause the consequence that better compensation effects will not occur. However, the measuring range will be limited if it is set too big. Therefore, the DC offset point should be selected by the balance of compensation effects and measuring range. Temperature and slope measuring results are listed in Table 1. It can be seen from the contrast of the relative errors of T and k that this compensation method has a large redundancy of temperature measurement error. Figure 11 shows the AC magnetic field compensation effect under the circumstance that the input alternating magnetic field is in the range of 0-4 mT. To measure temperature and AC magnetic field simultaneously, the offset of DC magnetic field ${B_{DC}}$ is added to the sensor. Therefore, the maximum AC magnetic field measuring range in the actual experiment should be set between $\textrm{ - }({{P \mathord{\left/ {\vphantom {P 4}} \right.} 4} - {B_{DC}}} )$ and $\textrm{ + }({{P \mathord{\left/ {\vphantom {P 4}} \right.} 4} - {B_{DC}}} )$. The AC magnetic field measuring range depends not only on the period of the sinusoidal function, but also on the offset of the DC magnetic field. In order to evaluate the compensation results more intuitively, the concept of compensation efficiency which is defined as (the difference of fluctuations before compensation and after)/(fluctuations before compensation) is proposed. It can be seen that this concept can be appropriately used to describe how much we have compensated. Here, we use the difference of the maximum and the minimum value of Fig. 11 under a certain magnetic field to represent the fluctuation between these three temperatures. Compensation efficiency, absolute error, and relative error are shown in Fig. 12(a) and Fig. 12(b). It can be seen that the max compensation efficiency is 83.968%, and the max absolute and relative errors are 0.07 mT and 3.50% respectively in the real-world experiment.

 figure: Fig. 11.

Fig. 11. AC magnetic field compensation results of the real-world temperature compensation experiment under the circumstance that temperature is unknown.

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 figure: Fig. 12.

Fig. 12. Evaluation of temperature compensation results. (a) Compensation efficiency at different magnetic fields under the temperatures of 0 °C, 20 °C and 40 °C. (b) Absolute error and relative error of the real-world temperature compensation experiment.

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Tables Icon

Table 1. Temperature and slope measuring results.

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5. Conclusion

In this paper, we have demonstrated a novel optical alternating magnetic field sensor and a new approach to compensate the influence of temperature fluctuation. Compensation mechanism and sensor performance were studied and tested. The fabricated sensor has a great stability of which the relative fluctuation is less than 0.7402% under constant temperature. The max absolute error and the relative error are 0.07 mT and 3.50%, and the max compensation efficiency is as high as 83.968 in the range of 0-4 mT under different temperatures of 0 °C, 20 °C and 40 °C. The experiment results of temperature compensation show that our proposed sensor and method can effectively compensate the influence caused by the temperature fluctuation resulting from the changes in the surrounding environment. With the advantages of self-compensation, on-line measuring, anti-electromagnetic interference, small size, and insensitivity to light source intensity fluctuations, we foresee the practical value of this alternating magnetic field sensor and the approach for the improvement of temperature robustness.

Funding

National Natural Science Foundation of China (61775163); National Key Scientific Instrument and Equipment Development Projects of China (2013YQ030915); Young Elite Scientists Sponsorship Program by CAST (2017QNRC001); Qingdao National Laboratory for Marine Science and Technology (QNLM201717).

Disclosures

The authors declare no conflicts of interest.

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Figures (12)
Fig. 1.
Fig. 1. The internal structure of the sensor.

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Fig. 2.
Fig. 2. Temperature compensation theory based on a distortion-free linear amplification effect.

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Fig. 3.
Fig. 3. Structure of the novel sensor which can implement the compensation.

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Fig. 4.
Fig. 4. Schematic of the experimental setup.

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Fig. 5.
Fig. 5. Two outputs of PD and the parameter S under 50 Hz alternating magnetic field under constant temperature. (a) Two outputs, its difference and output parameter S. (b) Parameter S under different magnetic fields.

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Fig. 6.
Fig. 6. Experiment result analysis of AC characteristics under constant temperature. (a) A linear relationship between the output parameter S and the effective value of the magnetic field at different time. (b) Variation of S at different time. (c) The absolute and relative fluctuations of S.

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Fig. 7.
Fig. 7. Characteristics of the sensor under various temperatures obtained by DC measurement experiment. (a) Characteristics of the output parameter S under different temperatures and the measuring range of the sensor. (b) Dependencies between twice the Faraday rotation angle and magnetic field under various temperatures.

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Fig. 8.
Fig. 8. The contrast of the AC magnetic field compensation results under different temperatures which is precisely known in the range of -20 °C-60 °C when the input magnetic are 0, 4 mT, 8 mT, 12 mT and 16 mT, respectively.

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Fig. 9.
Fig. 9. Relationship between slope k and temperature.

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Fig. 10.
Fig. 10. Temperature dependence of DC output parameter S and 2θ in a real-world experiment.

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Fig. 11.
Fig. 11. AC magnetic field compensation results of the real-world temperature compensation experiment under the circumstance that temperature is unknown.

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Fig. 12.
Fig. 12. Evaluation of temperature compensation results. (a) Compensation efficiency at different magnetic fields under the temperatures of 0 °C, 20 °C and 40 °C. (b) Absolute error and relative error of the real-world temperature compensation experiment.

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Tables (1)

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Table 1. Temperature and slope measuring results.

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Equations (11)

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θ  =  V ⋅ ∫ L B → ⋅ d L → ,
1 θ d θ d T = 1 V ( ∂ V ∂ λ d λ d T + ∂ V ∂ T ) + 1 L ∂ L ∂ T ,
1 θ d θ d T = 1 V ∂ V ∂ T + α 1 L ,
{ I ⊥ = I 0 cos 2 ( 45 ∘ + θ ) = I 0 2 ( 1 − sin ⁡ 2 θ )  =  I 0 2 − I 0 2 sin ⁡ [ 2 π λ ( n r − n l ) L ] I ∥ = I 0 cos 2 ( 45 ∘ − θ ) = I 0 2 ( 1 + sin ⁡ 2 θ )  =  I 0 2 + I 0 2 sin ⁡ [ 2 π λ ( n r − n l ) L ]
S = I ∥ − I ⊥ I ∥ + I ⊥ = sin ⁡ [ 2 π λ ( n r − n l ) L ] = sin ⁡ ( 2 θ ) = sin ⁡ ( 2 V B L ) .
B T 0 ∝ k T 0 ⋅ [ 2 θ A C ] r m s / [ 2 θ A C ] r m s k T k T ,
{ S ( t ) = sin ⁡ ( 2 θ A C ) + sin ⁡ ( 2 θ D C ) ∫ t t + n τ S ( t ) d t = n τ ⋅ sin ⁡ ( 2 θ D C ) sin ⁡ ( 2 θ D C ) = f 1 ( T ) | B D C k T = f 2 ( T ) B A C T 0 ∝ k T 0 ⋅ { arcsin ⁡ [ S ( t ) − sin ⁡ ( 2 θ D C ) ] } r m s / { arcsin ⁡ [ S ( t ) − sin ⁡ ( 2 θ D C ) ] } r m s k T k T ,
2 d θ d B = k T .
2 d θ d B = 2 V T L = k T .
χ = c T − T p ,
k ( T ) = A ′ ⋅ ( 1 + B ′ T − T p ) .
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