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Simultaneous magnetic field and temperature measurement with high resolution based on cascaded microwave photonic filters

Naihan Zhang, Muguang Wang, Pufeng Gao, Mengyao Han, Bin Yin, Shiyi Cai, Beilei Wu, Yan Liu, and Desheng Chen

Open Access Open Access

Abstract

A simultaneous magnetic field and temperature sensing scheme based on cascaded microwave photonic filters (MPFs) with high resolution is proposed and experimentally demonstrated. A polarization maintaining fiber bonded with a giant magnetostrictive material acts both as a magnetic field sensing probe and an important unit of a dispersion-induced MPF. A 500 m single mode fiber in a two-tap MPF is used to perform temperature compensation. The power fading frequency of the dispersion-induced MPF and the dip frequency of the two-tap MPF are selected to monitor the magnetic field and temperature changes. When temperature changes, both power fading frequency and dip frequency will change. While only power fading frequency shifts as magnetic field changes. Consequently, dual parameter sensing can be achieved by monitoring the characteristic microwave frequencies of the two MPFs. The temperature cross-sensitivity is well resolved in this way. In the experiment, the microwave frequency changes 5.84 MHz as external magnetic field increases by 1 mT. The corresponded theoretical resolution can reach 0.17 nT, which is only limited by the minimum resolution of vector network analyzer.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Magnetic field sensing is a vital technology in the industrial and scientific fields, such as electric power systems, electromagnetic compatibility and medical instrumentation system [1]. Some electronic schemes have been developed to measure magnetic field, such as fluxgate magnetometer, Hall effect sensor, magnetoresistive magnetometer and superconducting quantum interference (SQUID) device magnetometer [2]. Although these technologies are mature, they still have some problems. The fluxgate magnetometer has high power consumption. Hall effect sensor and magnetoresistive magnetometer are susceptible to environmental temperature. And the SQUID magnetometer normally has complicated operation and large footprint. Compared with other types of magnetic field sensors, fiber optic magnetic field sensors have attracted widespread attraction in recent years, since they have obvious advantages of miniature size, corrosion free and remote measurement capability. Besides, the dominating base material of telecom fiber is high pure SiO2, which is insulative. Thus, fiber optic magnetic field sensor is only influenced by magnetic field but almost unaffected by other electromagnetic signals. Many types of fiber structures, such as fiber Bragg grating (FBG) [3], Fabry-Perot (FP) cavity [4] and modal interferometer [5], have been demonstrated to achieve magnetic field sensing based on Faraday effect [6], magnetostrictive effect [3] and tunable refractive index of magnetic fluid [7]. Giant magnetostrictive material (GMM), whose length can change proportionally with the applied magnetic field, has been widely used in fiber optic magnetic field sensor due to its very large magnetostriction coefficient and short response time [8]. In order to overcome the problem of temperature crosstalk, many fiber optic sensing schemes are developed to achieve simultaneous magnetic field and temperature sensing [9–11]. Generally, most fiber optic sensors demodulate the measurand by monitoring the optical spectrum change using an optical spectrum analyzer (OSA). Due to the poor resolution and slow scanning rate of OSA, the OSA-based demodulation methods have relatively low measurement speed and resolution. High resolution magnetic field measurements are in demand in many fields. For example, for the subsea magnetic field monitoring, the analysis of weak magnetic anomalies can help to clarify regional geological features, such as the distribution of fracture zones and the location of volcanic rocks [12]. And underwater weak magnetic field measurement is also used to detect subsea pipelines [13]. Therefore, it is imperative to make a measurement resolution improvement of magnetic field sensors.

Microwave photonics (MWP) [14], a technology to use the advantages of photonic technologies to realize functions that cannot be directly performed in the radiofrequency domain, is considered an effective way to improve the demodulation performance of the fiber optic sensing. MWP techniques translate the optical spectrum changes induced by measurand variations to microwave signal changes [15,16]. Due to the mature monitoring devices in microwave domain, MWP techniques can be used to achieve high-speed and high-resolution sensing demodulation. Recently, various MWP based sensing methods have been developed to achieve high interrogation performance using optoelectronic oscillator (OEO) [17–19], microwave photonic filter (MPF) [20,21] and microwave photonic interferometry techniques [22,23]. And microwave photonic interrogation has been demonstrated in novel magnetic field sensors based on OEO [24–26]. All these OEO-based sensors have solved the problem of temperature crosstalk. In OEO system, only the mode whose frequency satisfies phase matching relationship can oscillate. For the OEO-based sensors which use MPF to select the oscillating frequency, when measurand changes, the oscillating frequency cannot be varied continuously and the minimum frequency change is a mode interval (i.e., free spectral range (FSR)). Therefore, the resolution (minimum detectable measurand change) of this type of OEO-based sensor is limited by the FSR of OEO. A smaller FSR means a longer OEO fiber loop length, which will increase the probability of mode hopping and affect the oscillating frequency stability of OEO. Therefore, this type of OEO-based sensor requires a trade-off between the resolution and OEO frequency stability. The magnetic field sensing scheme performed by changing the time delay of the OEO loop can address this issue [26]. Nevertheless, this scheme requires a trade-off between measurement range and sensitivity, because the oscillating frequency shift is limited within an FSR. Although some approaches can improve one of these two performances separately, for example, the sensitivity can be increased by special packaging for the sensor probe, and the integration technology can further reduce the loop time delay and thus increase the FSR. But it is still difficult to ensure high sensitivity and resolution at the same time.

In this paper, we develop a high sensitivity and high resolution magnetic field measurement scheme based on cascaded MPFs. The cascaded MPFs can be closed to form an OEO, which can generate high quality microwave signal without external microwave source. The OEO system has been developed to perform current sensing [27]. However, the oscillation of OEO require a high loop gain, which will increase the design difficulty and cost. Besides, the OEO-based sensor in [27] only achieved single parameter sensing, the current sensor unit was embedded in dispersion-induced MPF, but the two-tap MPF was not been used as a sensing part. As mentioned above, the resolution of the sensing system in [27] is also limited by FSR of OEO. The proposed scheme can solve the above problems. The magnetic field sensor is embedded in dispersion-induced MPF and the temperature sensor is embedded in two-tap MPF. The power fading frequency of dispersion-induced MPF can be changed as the phase difference between the optical carrier and modulated sidebands changes, which is designed to be sensitive to both external magnetic field and temperature changes. The dip frequency of the two-tap, which is determined by difference of time delay between the two fiber paths, is only sensitive to temperature. In the proposed system, a tiny phase difference change can cause a large microwave frequency change and the resolution is no longer limited by the mode interval. The power fading frequency shifts 5.84 MHz as external magnetic field changes 1 mT, corresponding to a measurement resolution of 0.17 nT. Compared with the OEO-based sensors, the resolution of proposed sensor improved by three orders of magnitude. In addition, the most crucial problem of fiber optic sensors, temperature crosstalk, has been solved in proposed sensor.

2. Principle

Figure 1(a) illustrates the configuration of the proposed simultaneous magnetic field and temperature sensing system. Figure 1 (b) shows the evolution schematic of optical polarization state and optical spectrum at different position. The light from a tunable laser source (TLS) is transmitted into the polarization maintaining fiber (PMF) through a polarization controller (PC1). The polarization plate of the light after the PC1 has a 45° angle with respect to the principal axis of the PMF. Thus, two orthogonal x, y polarized lights are excited on the two principal axes of the PMF and they have same amplitude. The PMF bonded with a GMM acts as the sensing probe. When the magnetic field applies on the GMM, the magnetostrictive effect occurs and this effect will convert into strain on the PMF bonded on the GMM. The magnetostriction induced GMM length change is caused by small magnetic domains rotation, as shown in Fig. 1 (d). The rotation and reorientation lead to an internal strain in the GMM structure. The strain in the structure stretch the GMM in the magnetic field direction with the increase of magnetic field [28]. Since the birefringence in the PMF will be changed with the strain and temperature variations, the phase difference change between x and y polarized lights is [29,30]

$$\Delta \varphi \textrm{ = }\frac{{\textrm{2}\pi }}{\lambda }({B_0}\Delta L + \Delta B{L_0}) = \frac{{\textrm{2}\pi }}{\lambda }[{({b_\mathrm{\varepsilon }} + {B_0})k\Delta H{L_0} + ({b_\textrm{T}} + {\alpha_M})\Delta T{L_0}} ],$$
where λ is the wavelength of the optical carrier, B0 and L0 are initial birefringence and length of the PMF glued to the GMM, ΔB and ΔL are birefringence and length variations of the PMF, bε and bT are the strain- and temperature-coefficient of birefringence, αM is thermal expansion coefficient of the GMM, k is a coefficient related to the strain transfer efficiency and magnetostrictive constant of GMM, and ΔH and ΔT represent magnetic field and temperature variations.

 figure: Fig. 1.

Fig. 1. (a) Configuration of the cascaded MPF-based magnetic field and temperature sensing system (MF: magnetic field, Temp: temperature). (b) Evolution schematic of optical polarization state and optical spectrum at different position (MZMP or MZMO: polarization plate which is parallel or orthogonal to the MZM principal axis, PBSc or PBSd: polarization plate which aims at the principal axis of the PBS of c or d port). (c) The real image of the magnetic field probe. (d) The principle of the GMM. (e) The principle of simultaneous magnetic field and temperature sensing.

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Then, a Mach–Zehnder modulator (MZM) receives the x, y polarized lights. The x polarized light is aligned to the principal axis of the MZM (MZMP). Since the LiNbO3 MZM has polarization-dependent properties, only x polarized light will be modulated. In the proposed system, the microwave signal used for modulation is generated from port 1 of the vector network analyzer (VNA). While the y polarized light directly passes through the MZM without modulation (MZMO). Since the bias voltage MZM is adjusted to null point for carrier suppressed double sideband modulation, the output optical carrier and generated ±1st-order sidebands have orthogonal polarization states. Suppose the input modulation microwave signal is Vin(t)=Vmcos(ωmt), the output optical signal of the MZM can be represented as

$${E_{\textrm{MZM}}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{{c}} {{E_0}{J_1}(\gamma )({e^{j({\omega_c} + {\omega_m})t}} + {e^{j({\omega_c} - {\omega_m})t}})}\\ {{E_0}{e^{j({\omega_c}t - {\varphi_0} - \Delta \varphi )}}} \end{array}} \right],$$
where E0 is the optical carrier amplitude, J1 is the first-order Bessel function of the first kind, γ=πVm/Vπ is the modulation index, Vπ is the half-wave voltages of the MZM. ωm and ωc represent the angular frequency of the microwave signal and optical carrier, φ0 is initial phase difference between optical carrier and sidebands. The modulated light wave is reflected by a linearly chirped fiber Bragg grating (LCFBG) through an optical circulator (OCir), in which the LCFBG is used as a dispersive element. The reflected signal from the LCFBG is given by
$${E_{\textrm{LCFBG}}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{{c}} {{E_0}{J_1}(\gamma )({e^{j({\omega_c} + {\omega_m})t}}{e^{j{\theta_1}}} + {e^{j({\omega_c} - {\omega_m})t}}{e^{j{\theta_{ - 1}}}})}\\ {{E_0}{e^{j({\omega_c}t - {\varphi_0} - \Delta \varphi )}}{e^{j{\theta_0}}}} \end{array}} \right],$$
where θ0 = zβc, θ1 = zβc + zβc’ωm + zβc’’ωm2/2 and θ-1 = zβc-zβc’ωm + zβc’’ωm2/2 are the phase shifts of the optical carrier and ±1st sidebands which are caused by dispersion effect, z is the traveled distance, βc is the propagation constant at ωc, βc’ and βc’’ are the first- and second-order derivatives of βc. Subsequently, a two-tap MPF is cascaded, which is composed of a PC2, a polarization beam splitter (PBS), two different length fiber paths and a polarization beam combiner (PBC). The polarization direction of the light after PC2 has a 45° angle with respect to the principal axis of the PBS, so interference of the x, y polarized lights will occur on two principal axes of the PBS (PBSc and PBSd), respectively. The interference signal at points c and d have orthogonal polarization state. The output optical signal after PBC can be written as
$$\begin{aligned} {E_{\textrm{PBC}}} &= \left[ {\begin{array}{{c}} {E_{\textrm{PBC}}^c}\\ {E_{\textrm{PBC}}^d} \end{array}} \right]\\& = \frac{1}{2}\left[ {\begin{array}{{c}} {{E_0}{e^{j{\omega_c}(t + \Delta \tau )}}[J_1(\gamma)\left(e^{j\left[\omega_m(t+\Delta \tau)+\theta_1\right]}+e^{-j\left[\omega_m(t+\Delta \tau)-\theta_{-1}\right]}\right)-e^{j\left(\theta_0-\varphi_0-\Delta \varphi\right)}]}\\ {{E_0}{e^{j{\omega_c}t}}[J_1(\gamma)\left(e^{j\left(\omega_m t+\theta_1\right)}+e^{-j\left(\omega_m t-\theta_{-1}\right)}\right)+e^{j\left(\theta_0-\varphi_0-\Delta \varphi\right)}]} \end{array}} \right], \end{aligned}$$
where Δτ is the time delay difference between two fiber paths. The recovered microwave signal after a photodetector (PD) can be written as
$$\begin{aligned} {V_{\textrm{PD}}}(t) &\propto {|{E_{_{PBC}}^c(t)} |^2} + {|{E_{_{PBC}}^d(t)} |^2}\\& = \frac{1}{2}{E_0}^2{J_1}(\gamma )\cos ({\lambda ^2}{\omega _m}^2D/4\pi c - {\varphi _0} - \Delta \varphi )\\& \times \{ \cos [{\omega _m}(t + {\tau _g})] - \cos [{\omega _m}(t + {\tau _g} + \Delta \tau )]\} , \end{aligned}$$
where D = -zβc’’2πc/λ2 is the average dispersion of the LCFBG (in ps/nm), c is the light velocity in vacuum, τg = zβc’. The expression of cos(λ2ωm2D/4πc−φ0−Δφ) on the right-hand side of Eq. (5) is contributed from the dispersion-induced MPF, the amplitude of the microwave signal recovered by the PD is related to the microwave frequency and dispersion of LCFBG. The reason is that sidebands modulated by microwave signals with different frequency have different wavelengths, they will get different phase shifts after LCFBG. It is known that when the phase difference between the optical carrier and sidebands is different, the amplitude of the beat signal generated between them is also different. The expression of cos[ωm(t+τg)]- cos[ωm(t+τg+Δτ)] on the right-hand side of Eq. (5) is contributed from the two-tap MPF, whose production principle is that the time delays of the beat signals from the polarization planes of PBSc and PBSd are different. By applying a Fourier transform to Eq. (5), we have
$$V{^{\prime}_{\textrm{PD}}}(\omega ) \propto \frac{1}{2}{E_0}^2{J_1}(\gamma )\cos ({\lambda ^2}{\omega _m}^2D/4\pi c - {\varphi _0} - \Delta \varphi )V{^{\prime}_{in}}(\omega ){e^{j\omega {\tau _g}}}(1 - {e^{j\omega \Delta \tau }}).$$

Therefore, the frequency response of the cascaded MPF is

$$H({\omega _m}) = \frac{{|{V{^{\prime}_{PD}}({\omega_m})} |}}{{|{V{^{\prime}_{in}}({\omega_m})} |}} = G\cos ({\lambda ^2}{\omega _m}^2D/4\pi c - {\varphi _0} - \Delta \varphi ) \cdot \sin ({\omega _m}\Delta \tau /2),$$
where G represents the effective open loop gain. H1(ωm)=cos(λ2ωm2D/4πc−φ0−Δφ) corresponds to the dispersion-induced MPF. The power fading frequency is at the minimum value of the dispersion-induced MPF, corresponding to cos(λ2ωm2D/4πc−φ0−Δφ) = 0. The first power fading frequency fm1 can be calculated according to λ2ωm12D/4πc−φ0−Δφ=π/2, and expressed as
$$\begin{aligned} {f_{\textrm{m1}}} &= {f_\textrm{c}}\sqrt {\frac{{{\varphi _\textrm{0}}\textrm{ + }\Delta \varphi \textrm{ + }\pi \textrm{/2}}}{{\pi Dc}}} \\& \textrm{ = }{f_\textrm{c}}\sqrt {\frac{{\textrm{2}[{({b_\mathrm{\varepsilon }} + {B_0})k\Delta H{L_0} + ({b_\textrm{T}} + {\alpha_M})\Delta T{L_0}} ]}}{{\lambda Dc}} + \frac{{{\varphi _\textrm{0}}\textrm{ + }\pi \textrm{/2}}}{{\pi Dc}}} , \end{aligned}$$
where fc is the frequency of the optical carrier. Therefore, the relationship between the power fading frequency and magnetic field and temperature is established. H2(ωm)=sin(ωmΔτ/2) corresponds to the two-tap MPF. The free spectral range (FSR) of the two-tap MPF can be expressed as FSR = 1/Δτ=c/nΔl, where n is the effective refractive index of the single mode fiber (SMF), Δl is the fiber length difference between two optical paths. A section of SMF in one of the fiber paths is used for temperature compensation. As a result, the FSR change induced by external temperature variation ΔT will lead to a dip frequency shift. The relationship between the dip frequency fm2 and ΔT can be expressed as
$${f_{\textrm{m2}}} = \frac{{{f_{\textrm{02}}} \cdot FS{R_1}}}{{FS{R_0}}}\textrm{ = }\frac{{{f_{\textrm{02}}} \cdot n\Delta {l_0}}}{{n\Delta {l_0} + \Delta T \cdot n{L_\textrm{s}}({\alpha _\textrm{s}} + \xi )}},$$
where f02 is the selected original dip frequency, Δl0 is the initial fiber length difference between two optical paths, Ls is the SMF length used as the temperature sensing probe, αs is thermal expansion coefficient of the SMF, ξ is the thermo-optic coefficient.

As can be seen from Eq. (8) and Eq. (9), the magnetic field and temperature variations can be simultaneously measured by monitoring the power fading frequency of the dispersion-induced MPF and dip frequency of the two-tap MPF. The frequency response changes of cascaded MPFs versus different measurands are shown in Fig. 1(e). When magnetic field changes, the power fading frequency will shift and the dip frequency of two-tap MPF will be constant. While, both the power fading frequency and dip frequency of two-tap MPF will change as temperature varies.

3. Experiment setup and results

A verification experiment is conducted according to the setup shown in Fig. 1. The devices in the experiment have following parameters: 1. LCFBG (central wavelength: 1549.3 nm, 3-dB bandwidth: 0.35 nm, dispersion: 2645 ps/nm). 2. TLS (Agilent 8164A, wavelength: 1549.3 nm). 3. MZM (JDS Uniphase, 3-dB bandwidth: 10 GHz). 4. PD (Multiplex, MTRX192L, 3-dB bandwidth: 10 GHz, responsivity: 0.8 A/W). The detailed fabrication process of magnetic field sensing probe is as follows: 1. 4 cm long coating is stripped from PMF to reduce the media between GMM and fiber, and then a higher strain transfer efficiency can be obtained. 2. The PMF is wiped with alcohol, and UV glue is dropped on two ends of the GMM (TbDyFe, 4 × 8 × 40 mm) to fix the PMF. The glue between the GMM and the PMF should be slightly thinner for a high transfer efficiency. The PMF should be slightly stretched before being bonded with the GMM. Thus, when magnetic field is applied on the GMM, strain can be efficiently transferred to fiber. 3. The glue is left to dry in room temperature for 24 hours, so that it can be totally cured. The real image of sensing probe is shown in Fig. 1 (c).

In the proposed sensing system, the magnetic field sensor is sensitive to both the magnetic field and temperature. Therefore, two MPFs are cascaded to achieve the simultaneous magnetic field and temperature sensing. A temperature sensor, a 500 m SMF without any complicated design, is embedded in one of the fiber paths of the two-tap MPF to calibrate the temperature influence. And the other fiber path of two-tap MPF is 2.8 m shorter than the sensing path. Firstly, the frequency response of the cascaded MPFs is measured by a VNA (Keysight, E5063A), the sweep range is set from 4.4 GHz to 9 GHz. Figure 2 (a), (b) and (c) illustrate simulated frequency responses of the two-tap MPF, dispersion induced MPF and cascaded MPFs, respectively. While the measured frequency response of cascaded MPFs is shown in Fig. 2 (d). The response of the cascaded MPFs is composed of an envelope of the dispersion-induced MPF and a comb spectrum realized by the two-tap MPF. Due to a 2.8 m fiber length difference between two paths, the two-tap MPF has a FSR of 36.8 MHz. It can be seen from Fig. 2 (c) and Fig. 2 (d) that the measurement results are in good agreement with the simulation results. The reason why the measured frequency response is not very smooth is due to the response of electro-optical devices used in our experiment. Ideally, the amplitude-frequency responses of electro-optical devices, such as MZM and PD, are completely flat. Therefore, these devices do not affect the overall response of the MPF. Actually, the response amplitudes of these devices are not exactly the same at each frequency, which will cause a slight attenuation at some frequencies in the MPF response [31,32]. However, this attenuation will not change as the MPF response shifts.

 figure: Fig. 2.

Fig. 2. The simulated frequency response of (a) two-tap MPF, (b) dispersion-induced MPF and (c) cascaded MPFs (D = 2645 ps/nm, fc = 193.502 THz, φ0+Δφ=1.007 π, Δτ=27.94 ns). (d) The measured frequency response of cascaded MPFs.

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In the magnetic field sensing experiment, a solenoid is applied as a magnetic field source and the strength of magnetic field is adjusted using a DC regulated power supply (5A-100 V, accuracy: 0.1%). The magnetic field probe is placed in the center of the solenoid, and the generated magnetic field is uniform around the sensing probe and its direction is parallel to the sensing probe. In order to measure the strength of the magnetic field, the changes of the first power fading frequency of the dispersion-induced MPF are monitored. The measured frequency responses under different magnetic field strengths are shown in Fig. 3 (a). The response of the dispersion-induced MPF is extracted from the cascaded MPFs response to easily observed the power fading frequency. The red curve in Fig. 3(b) shows the extracted response. The measurement range is from 36 to 93.6 mT. An increasing trend can be observed in the power fading frequency as the magnetic field increases (Fig. 3(c)). The relationship between the magnetic field and power fading frequency can be fitted as y = (0.05x + 19.58)1/2, as shown in Fig. 3 (d). When the magnetic field increases by 1 mT, the frequency varies 5.84 MHz. GMM has a high magnetostrictive constant and fast response speed. However, the magnetic hysteresis effect of GMM will cause a response hysteresis when the magnetic field decreases, the hysteresis curve of the same GMM has been measured in [25]. In order to eliminate the influence of hysteresis effect on measurement results, compensation algorithms can be adopted to compensate the nonlinear phenomenon [33].

 figure: Fig. 3.

Fig. 3. (a) Measured frequency responses of cascaded MPFs at different magnetic fields. (b) The extracted envelope from response of cascaded MPFs when magnetic field is 36 mT. (c) The extracted responses at different magnetic fields. (d) The relationship between the power fading frequency and magnetic field.

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In addition to the magnetic field, the PMF is also sensitive to the temperature. When the temperature increases, the birefringence of the PMF will decrease. In other words, the temperature also has influence on the power fading frequency. Therefore, it is essential to evaluate the temperature dependence of magnetic field sensor. The magnetic field sensing probe is placed in a temperature-controlled water tank, and the temperature is adjusted from 32 to 37 °C with a step of 1 °C. The measured frequency responses are shown in Fig. 4 (a), when magnetic field sensor is under different temperatures. The extracted frequency response of the dispersion-induced MPF is a red curve shown in Fig. 4 (b). It can be seen from the Fig. 4 (c), the power fading frequency gradually decreases when the temperature increases. The temperature response can be fitted as y = (-1.38x + 69.75)1/2, as shown in Fig. 4 (d). When the temperature increases by 1 °C, the frequency changes 139.37 MHz.

 figure: Fig. 4.

Fig. 4. (a) Measured frequency responses of cascaded MPFs at different temperatures. (b) The extracted envelope from response of cascaded MPFs when temperature is 34 °C. (c) The extracted responses at different temperatures. (d) The relationship between the power fading frequency and temperature.

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After that, the calibration curve of the temperature sensor in the two-tap MPF is measured independently. The temperature sensor is placed in a temperature-controlled water tank. When the around temperature changes, the length and refractive index of the SMF will change. And then the FSR of the two-tap MPF varies, which causes a shift of the dip frequency. When the temperature increases from 22 to 43 °C, the dip frequency decreases, as shown in Fig. 5 (a). The relationship between dip frequency and temperature can be fitted as y = 21.21/(0.028x + 4). When the temperature varies 1 °C, the dip frequency shifts 3.6 MHz. Since the sensing fiber is evenly wound on the spool with a very thin layer (about 250 µm) and temperature change in the tank is uniform, the heat conduction is very fast. In this case, a long sensing fiber will not affect the response time. The dip frequency of two-tap MPF is determined by optical path difference between the two optical paths. Since the lengths of the SMFs in the two optical paths are very close, the optical path difference changes little when the two SMFs are affected by the same external condition. Therefore, the frequency stability in our experiment is high, even if we use a 500 m long optical fiber as the sensing probe. It can be seen from Fig. 3 (a), The dip frequency of the two-tap MPF does not change during the 15 minutes of magnetic field measurement. In addition, a quadrupole symmetrical winding method can be used to wind hundreds of meters optical fiber on a ring skeleton with a diameter of only 30 mm [34], and the thickness of the fiber loop is only a few millimeters. Some special doped fiber [35,36] can also be used to enhance temperature sensitivity. Thus, the length of the sensing fiber used can be greatly reduced.

 figure: Fig. 5.

Fig. 5. (a) The measured dip frequencies of two-tap MPF at different temperatures. (b) The relationship between dip frequency and temperature.

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To date, the relationships between different monitored frequencies of two MPFs and measurands have been established. The changes of the magnetic field and temperature can be obtained using the following equation set

$$\left\{ {\begin{array}{*{20}{c}} {{f_{m1}} = \sqrt {0.05 \cdot \Delta H - 1.38 \cdot \Delta T + {f_{01}}^2} }\\ {{f_{m2}} = \frac{{4.1 \cdot {f_{02}}}}{{4 + 0.028 \cdot \Delta T}}} \end{array}} \right.,$$
where f01 and f02 are the original power fading frequency and dip frequency at original measurands. In the practical application, the original frequencies need to be calibrated before measurement. When the original frequencies are determined, the magnetic field and temperature can be demodulated using the given equation set.

In the proposed system, the frequency changes continuously when the magnetic field changes, which is different from the OEO-based sensing schemes. The resolution depends on the minimum resolution of the VNA, but is no longer limited by the FSR of the OEO. In the experiment, the frequency resolution of the VNA is set to 10 kHz. Consequently, the theoretical magnetic field resolution can be calculated to 1.7 µT. The minimum frequency resolution of the VNA used in our experiment can reach 1 Hz, thus the proposed sensing system has a potential to achieve a magnetic field resolution of 0.17 nT. Nevertheless, higher resolution means lower scanning speed. So, it is necessary to make a trade-off between measurement resolution and scanning speed. The measurement resolution and range of our proposed scheme and other OEO-based magnetic field sensing methods are compared in Table 1.

Tables Icon

Table 1. Theoretical resolution and measurement range comparison with OEO-based sensors

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It is obvious that the theoretical measurement resolution of 0.17 nT has improved to a large extend compared with OEO-based methods. And the proposed MPF-based sensing system has achieved a large measurement range. These properties endow the proposed sensing scheme with capability in weak magnetic field detection applications, such as the subsea crustal structure exploration and subsea pipeline detection, etc.

In the proposed system, the frequency instability caused by external environment fluctuations will cause measurement errors. And there is a measurement uncertainty which comes from the frequency reading deviations caused by system noise and the limited sampling point number and resolution of the VNA. All the above factors together contribute to the final measurement error. The frequency response of dispersion-induced MPF is a cosine function, whose power fading frequency is related to the frequency of the optical carrier, average dispersion of the LCFBG and phase difference between optical carrier and sidebands. Theoretically, the response curve at the power fading frequency is a sharp dip. However, the observed dip curve of the microwave response has a certain width due to the noise in the system. Although the smooth algorithm in VNA is applied to reduce the noise on the frequency response, there is still a deviation between the read power fading frequency and the real value. In the future, the Monte Carlo simulation in [37] can be referred to establish an approximate relationship between the standard deviation and noise, line width of response curve and related parameters in the smooth algorithm. Thus, the measurement uncertainty of the proposed system can be quantified. Besides, the deviation can be effectively reduced by using low-noise light source, EDFA and PD in practical application.

4. Conclusion

In conclusion, we have developed and experimentally demonstrated a simultaneous magnetic field and temperature measurement system based on cascaded MPFs. One of the MPFs is induced by the dispersion effect of an LCFBG, and the other is formed by two fiber paths with slightly different lengths. The selected microwave frequencies of the two MPFs are observed to monitor the magnetic field and temperature variations. Because the minimum monitored frequency change of the MPF is no longer limited by mode interval, the theoretical sensing resolution has been greatly improved to 0.17 nT in the proposed system. The proposed sensing system also has a large measurement range of 58 mT. Due to the PMF in sensing probe, the effects of random variations in the polarization state are alleviated. In order to further reduce the cost, a low frequency sweeping source and a power detection module can be used to replace the VNA. In practical application, the PMFs can be used to replace all SMF links, and the function of the PCs can be replaced by 45°-splice. Therefore, the stability of the polarization state can be improved and the system complexity can be reduced. In the future, integration technology can be researched to achieve a portable sensing device. Besides, machine learning can also be used to simplify demodulation process and reduce the hardware cost. Since the proposed system improves the magnetic field measurement resolution by three orders of magnitude, the application field of the MWP-based magnetic field sensors is widened.

Funding

National Natural Science Foundation of China (U2006217, 62371035, 61975009); Beijing Municipal Natural Science Foundation (4212009).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. (a) Configuration of the cascaded MPF-based magnetic field and temperature sensing system (MF: magnetic field, Temp: temperature). (b) Evolution schematic of optical polarization state and optical spectrum at different position (MZMP or MZMO: polarization plate which is parallel or orthogonal to the MZM principal axis, PBSc or PBSd: polarization plate which aims at the principal axis of the PBS of c or d port). (c) The real image of the magnetic field probe. (d) The principle of the GMM. (e) The principle of simultaneous magnetic field and temperature sensing.

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Fig. 2.
Fig. 2. The simulated frequency response of (a) two-tap MPF, (b) dispersion-induced MPF and (c) cascaded MPFs (D = 2645 ps/nm, fc = 193.502 THz, φ0+Δφ=1.007 π, Δτ=27.94 ns). (d) The measured frequency response of cascaded MPFs.

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Fig. 3.
Fig. 3. (a) Measured frequency responses of cascaded MPFs at different magnetic fields. (b) The extracted envelope from response of cascaded MPFs when magnetic field is 36 mT. (c) The extracted responses at different magnetic fields. (d) The relationship between the power fading frequency and magnetic field.

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Fig. 4.
Fig. 4. (a) Measured frequency responses of cascaded MPFs at different temperatures. (b) The extracted envelope from response of cascaded MPFs when temperature is 34 °C. (c) The extracted responses at different temperatures. (d) The relationship between the power fading frequency and temperature.

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Fig. 5.
Fig. 5. (a) The measured dip frequencies of two-tap MPF at different temperatures. (b) The relationship between dip frequency and temperature.

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Tables (1)

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Table 1. Theoretical resolution and measurement range comparison with OEO-based sensors

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Equations (10)

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Δ φ  =  2 π λ ( B 0 Δ L + Δ B L 0 ) = 2 π λ [ ( b ε + B 0 ) k Δ H L 0 + ( b T + α M ) Δ T L 0 ] ,
E MZM = 2 2 [ E 0 J 1 ( γ ) ( e j ( ω c + ω m ) t + e j ( ω c − ω m ) t ) E 0 e j ( ω c t − φ 0 − Δ φ ) ] ,
E LCFBG = 2 2 [ E 0 J 1 ( γ ) ( e j ( ω c + ω m ) t e j θ 1 + e j ( ω c − ω m ) t e j θ − 1 ) E 0 e j ( ω c t − φ 0 − Δ φ ) e j θ 0 ] ,
E PBC = [ E PBC c E PBC d ] = 1 2 [ E 0 e j ω c ( t + Δ τ ) [ J 1 ( γ ) ( e j [ ω m ( t + Δ τ ) + θ 1 ] + e − j [ ω m ( t + Δ τ ) − θ − 1 ] ) − e j ( θ 0 − φ 0 − Δ φ ) ] E 0 e j ω c t [ J 1 ( γ ) ( e j ( ω m t + θ 1 ) + e − j ( ω m t − θ − 1 ) ) + e j ( θ 0 − φ 0 − Δ φ ) ] ] ,
V PD ( t ) ∝ | E P B C c ( t ) | 2 + | E P B C d ( t ) | 2 = 1 2 E 0 2 J 1 ( γ ) cos ⁡ ( λ 2 ω m 2 D / 4 π c − φ 0 − Δ φ ) × { cos ⁡ [ ω m ( t + τ g ) ] − cos ⁡ [ ω m ( t + τ g + Δ τ ) ] } ,
V PD ′ ( ω ) ∝ 1 2 E 0 2 J 1 ( γ ) cos ⁡ ( λ 2 ω m 2 D / 4 π c − φ 0 − Δ φ ) V i n ′ ( ω ) e j ω τ g ( 1 − e j ω Δ τ ) .
H ( ω m ) = | V P D ′ ( ω m ) | | V i n ′ ( ω m ) | = G cos ⁡ ( λ 2 ω m 2 D / 4 π c − φ 0 − Δ φ ) ⋅ sin ⁡ ( ω m Δ τ / 2 ) ,
f m1 = f c φ 0  +  Δ φ  +  π /2 π D c  =  f c 2 [ ( b ε + B 0 ) k Δ H L 0 + ( b T + α M ) Δ T L 0 ] λ D c + φ 0  +  π /2 π D c ,
f m2 = f 02 ⋅ F S R 1 F S R 0  =  f 02 ⋅ n Δ l 0 n Δ l 0 + Δ T ⋅ n L s ( α s + ξ ) ,
{ f m 1 = 0.05 ⋅ Δ H − 1.38 ⋅ Δ T + f 01 2 f m 2 = 4.1 ⋅ f 02 4 + 0.028 ⋅ Δ T ,
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