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Magneto-refractive properties and measurement of an erbium-doped fiber

Sichen Liu, Yi Huang, Chuanlu Deng, Chengyong Hu, Caihong Huang, Yanhua Dong, Yana Shang, Zhenyi Chen, Xiaobei Zhang, and Tingyun Wang

Open Access Open Access

Abstract

The magneto-refractive properties of an erbium-doped fiber (EDF) are investigated by theoretically analyzing the change in mode characteristics with a magnetic field and experimentally measuring it based on a fiber-optic Mach-Zehnder interferometer (MZI). The numerical results indicate that the mode effective refractive index (RI) increases as the magnetic field increases, and the mode field intensity distribution tends to be more concentrated in the core region with an increasing magnetic field. The variation in the mode effective RI of the fundamental mode with the magnetic field is greater than that of the higher-order modes. A magneto-refractive measurement system based on a fiber-optic MZI is set up to analyze the magneto-refractive effect of the EDF. The changes in the mode effective RI measured with a direct-current (DC) magnetic field and with a 100 Hz alternating-current (AC) magnetic field are 4.838×10−6 and 4.245×10−6 RIU/mT, respectively. The experimental results are in reasonable agreement with the theoretical analysis. Furthermore, the error between the experimental and numerical results is discussed. The magneto-refractive properties of the EDF exhibit potential in all-fiber magnetic field or current sensing area.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The magneto-refractive effect, which implies that the refractive index (RI) changes under the application of an external magnetic field, is a new magneto-optic effect found in recent decades [1–3]. Most studies on the magneto-refractive effect have mainly focused on liquids or magnetic metallic films. In the field of liquids, Baranonova et al. first predicted the effect in liquids comprising mainly chiral molecules, and theoretically explained the mechanism of the magneto-refractive effect [1]. An external magnetic field influences the microstructure of liquids, resulting in a change in the permittivity constant, and finally phenomenologically exhibits a variation in the RI [4]. Magnetic fluid, as a novel functional liquid material owing to its strong magneto-refractive effect, can be widely applied to fabricate magnetic field sensors when integrated with various fiber sensing structures, such as fiber gratings [5], multimode fibers [6], photonic crystal fibers [7], tapered fibers [8] or bent fibers [9]. In addition, the mechanism of magneto-refractive effect has been widely investigated. C. Huang et al. proposed the Langevin function to quantitatively describe the dependence of RI of magnetic fluid on magnetic field and temperature [10]. S. Pu et al. investigated the RI of magnetic fluid based on the retroreflection on the fiber-optic end face [11]. Y. Zhao et al. studied the magneto-refractive properties of magnetic fluid by the Fresnel reflection theory, and the RI of the magnetic fluid increased from 1.3412 to 1.36, with an increase in the magnetic field from 0 to 650 Gs [12]. In the field of magnetic metallic films, Jacquet et al. found that the change in the magnetization structure of magnetic metallic multilayer firms is associated with a significant change in the RI, and they named this phenomenon the magneto-refractive effect [3]. They also reported that the transmission spectrum of multilayer films comprising mainly metal materials exhibits a change with a magnetic field because of the magneto-refractive effect. Since then, most studies on the magneto-refractive effect of magnetic metallic films have been mainly based on optical transmission measurements [13–16]. The change in the reflectance and transmittance of films induced by the magnetic field depends on the real and imaginary parts of the permittivity tensor, respectively. The magneto-refractive effect of liquids and magnetic metallic films has attracted considerable research attention worldwide. However, systematic analysis of the magneto-refractive effect of fibers is lacking. Therefore, in this work, we focus on the magneto-refractive properties and measurement of fibers. According to the studies above, the magneto-refractive effect is described by the change in the permittivity tensor of materials with a magnetic field. In this way, we consider that an external magnetic field will influence the permittivity tensor of fibers, which has an impact on the mode characteristics. Therefore, the magneto-refractive properties of fibers can be characterized by the change in the mode effective RI with a magnetic field. A magneto-optic fiber doped with rare-earth ions exhibits a great magneto-optic effect [17–19] and is a suitable candidate for studying the magneto-refractive properties of fibers.

Fiber-optic interferometric measurement systems have been widely studied in the field of fiber sensing owing to their high sensitivity [20–22]. The sensing signals, such as temperature, pressure, or magnetic field, influence the geometric or optical properties of fibers, changing the phase difference between the light propagated in the sensing fiber and the reference fiber. The variation in the phase difference eventually affects the interference effect between these two light beams, which can be used to measure the sensing signals. F. Lv et al. filled the magnetic fluid into a Sagnac fiber ring and observed a shift in the interference spectrum caused by the magnetic field owing to the magneto-refractive effect of the magnetic fluid, and a minimal detectable magnetic field intensity was demonstrated to be ∼3 Oe in response to a 50 Hz AC magnetic field [23]. F. Chen et al. fabricated a magnetic field sensor based on a fiber-optic MZI using TbDyFe as a sensing element to measure a 200 Hz AC magnetic field [24]. Because of the magneto-strictive effect of TbDyFe, the RI and geometric size of the sensing fiber was affected by magnetic field, which varied the interference intensity, and high sensitivity of 69.83 mrad/µT with a resolution of 2.14 nT/Hz (rms) was achieved. According to these studies, the magnetic field affects the RI or geometric size of the sensing fiber owing to the magneto-optic effect, and causes a phase shift, which influences the interference effect. Thus, if the magneto-refractive effect occurs in the sensing fiber, the variation in the mode effective RI will be modulated to the interference light intensity or interference spectrum. Therefore, the magneto-refractive properties of fibers can be investigated using a fiber-optic interferometric system.

In this work, the magneto-refractive properties of an EDF are theoretically and experimentally studied. The permittivity tensor, which contains the magnetic field influence factor at the off-diagonal position, is introduced into Maxwell’s equations. The partial differential equations of the electromagnetic field components are derived and solved based on the finite-element method (FEM). The mode effective RI and mode field intensity distribution of the EDF under different magnetic fields are numerically calculated. Based on a fiber-optic MZI, the variation of the mode effective RI of the EDF is measured with DC and AC magnetic fields, respectively. The experimental results are basically consistent with the numerical results, proving the feasibility of the theoretical analysis and experimental measurements. The error between the numerical and experimental results is also discussed. The magneto-refractive properties of the EDF can be applied in all-fiber magnetic field or current sensors.

2. Theoretical analysis of magneto-refractive properties

The magneto-optic properties are described by a magneto-optic perturbation of the permittivity tensor [25–27]:

$$\varepsilon = \left( {\begin{array}{{ccc}} {{n^2}}&0&0\\ 0&{{n^2}}&0\\ 0&0&{{n^2}} \end{array}} \right) + K\left( {\begin{array}{{ccc}} 0&{{M_z}}&{{M_y}}\\ { - {M_z}}&0&{{M_x}}\\ {{M_y}}&{ - {M_x}}&0 \end{array}} \right), $$
where n is the RI, which is irrelevant to the magneto-optic effect. Mx, My, and Mz are the magnetization components in the axial direction. K = K'+jK'‘ is a complex parameter that depends on the material, where K'‘ is associated with the saturated specific Faraday rotation by
$${\theta _{F,sat}} ={-} k\frac{{K^{\prime\prime}{M_s}}}{{2n}}, $$
where k is the wave number, and Ms is the saturation magnetization. The real part K’, which determines the Faraday ellipticity, is neglected [26]. The Cotton-Mouton effect, which depends quadratically on the magnetization components, is also omitted [27]. The case where the direction of the magnetic field is parallel to the fiber axis is discussed in this work. We assume that the magnetic field is along the z direction. Therefore, only the magnetization component Mz is not zero, and the permittivity tensor of the fibers can be simplified as follows:
$${\varepsilon _\alpha } = \left( {\begin{array}{{ccc}} {n_\alpha^2}&{j{\delta_\alpha }}&0\\ { - j{\delta_\alpha }}&{n_\alpha^2}&0\\ 0&0&{n_\alpha^2} \end{array}} \right), $$
where nα, α=1, 2, denotes the RI without an external magnetic field, the subscript α=1 represents the core region, and α=2 represents the cladding region. δα is the magnetic field influence factor, which is expressed as
$${\delta _\alpha } = K^{\prime\prime}{M_z} = \frac{{2{n_\alpha }v{B_{sat}}}}{{k{M_s}}} \cdot \frac{{{M_m} \cdot m}}{V}, $$
where v is the Verdet constant, Bsat is the magnetic flux density when the magnetization reaches saturation, Mm is the specific magnetization component in the z direction, and m and V are the mass and volume of the fiber sample, respectively.

The mode calculation of the core region is considered as an example. When the permittivity tensor described in Eq. (3) is introduced into Maxwell’s equations, the scalar electromagnetic equations containing the magnetic field influence factor are obtained as follows:

$$\begin{array}{l} \frac{{\partial {E_y}}}{{\partial x}} - \frac{{\partial {E_x}}}{{\partial y}} = j\omega {\mu _0}{H_z}\\ \frac{{\partial {E_z}}}{{\partial y}} - j\beta {E_y} = j\omega {\mu _0}{H_x}\\ j\beta {E_x} - \frac{{\partial {E_z}}}{{\partial x}} = j\omega {\mu _0}{H_y}\\ \frac{{\partial {H_y}}}{{\partial x}} - \frac{{\partial {H_x}}}{{\partial y}} ={-} j\omega {\varepsilon _0}n_1^2{E_z}\\ \frac{{\partial {H_z}}}{{\partial y}} - j\beta {H_y} ={-} j\omega {\varepsilon _0}({n_1^2{E_x} + j{\delta_1}{E_y}} )\\ j\beta {H_x} - \frac{{\partial {H_z}}}{{\partial x}} ={-} j\omega {\varepsilon _0}({n_1^2{E_y} - j{\delta_1}{E_x}} )\end{array}, $$
where ω is the light frequency, µ0 is the vacuum permeability constant, ɛ0 is the vacuum permittivity constant, β=k0neff is the longitudinal propagation constant, k0 is the vacuum wave number, neff is the mode effective RI. When the longitudinal electric field component Ez, the longitudinal magnetic field component Hz, and the transverse magnetic field components Hx and Hy are represented by the transverse electric field components Ex and Ey, partial differential equations of Ex and Ey are derived as follows:
$$\begin{array}{l} \frac{{{\partial ^2}{E_x}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{E_x}}}{{\partial {y^2}}} + \frac{{j{\delta _1}}}{{n_1^2}}\left( {\frac{{{\partial^2}{E_y}}}{{\partial {x^2}}} - \frac{{{\partial^2}{E_x}}}{{\partial x\partial y}}} \right) + {\omega ^2}{\mu _0}{\varepsilon _0}({n_1^2{E_x} + j{\delta_1}{E_y}} )= {({{k_0}{n_{eff}}} )^2}{E_x}\\ \frac{{{\partial ^2}{E_y}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{E_y}}}{{\partial {y^2}}} + \frac{{j{\delta _1}}}{{n_1^2}}\left( {\frac{{{\partial^2}{E_y}}}{{\partial x\partial y}} - \frac{{{\partial^2}{E_x}}}{{\partial {y^2}}}} \right) + {\omega ^2}{\mu _0}{\varepsilon _0}({n_1^2{E_y} - j{\delta_1}{E_x}} )= {({{k_0}{n_{eff}}} )^2}{E_y} \end{array}$$
neff can be calculated by Eq. (6) based on the FEM technique.

The EDF analyzed in this work was fabricated using modified chemical vapor deposition (MCVD) process in combination with atomic layer deposition (ALD) technique [28,29]. The fabrication procedure is briefly described as follows. Firstly, the porous soot layer was deposited on the inner surface of substrate tube using MCVD process. Secondly, Er2O3 (99.9%-Er, Strem Chemicals Inc.) and Al2O3 doped layers were deposited on the inner surface of porous soot layer, which were carried out with an ALD system (TFS-200, Beneq Inc., Finland). Thirdly, Ge-doped SiO2 materials were deposited as core layers using MCVD process, and the tube was collapsed to form a preform; Finally, the preform was drawn into fiber and coated with buffer.

The RI profile of the EDF was obtained using a digital holographic interferometric system [30], as shown in Fig. 1. The elemental content in the core region of the EDF were analyzed using a scanning electron microscope (JSM-7500F, Japan) combined with an Energy Dispersive Spectrometer (MX80-EDS, Oxford, England). The cross-section and element contents are shown in the top left corner and top right corner of Fig. 1, respectively. The concentration of Er3+ ions is 1.3wt%. The RI of the core and cladding region are 1.4794 and 1.4610, respectively. The diameters of the core and cladding region are 9.87 and 130.22 µm, respectively.

 figure: Fig. 1.

Fig. 1. Refractive index profile of the EDF and element contents in the core region, (inner) cross-section of the EDF.

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The specific magnetization of the EDF was investigated using a physical property measurement system (PPMS) (PPMS-9, Quantum Design, USA) at a temperature of 293 K, as shown in Fig. 2(a). The magnetic field in the PPMS was set parallel to that of the EDF sample. Therefore, the direction of the investigated specific magnetization was along the fiber axis, and the magnetization component Mz can be calculated, as shown in Fig. 2(a). The magnetization reaches saturation when the magnetic field is approximately 857.12 mT and the saturation magnetization is approximately 114.06 A/m. The Faraday rotation of the EDF at 1550 nm wavelength was analyzed by a magneto-optic effect measurement system based on the Stokes polarization parameters method [19,31], which is shown in Fig. 2(b). The slope of the Faraday rotation versus the magnetic field determines the magnitude of the Verdet constant, which is approximately 0.403 rad/Tm. Ultimately, the magnetic field influence factor can be calculated using Eq. (4) based on the parameters investigated above.

 figure: Fig. 2.

Fig. 2. (a) Magnetization curve of the EDF at temperature 293 K. (b) Relationship between Faraday rotation and magnetic field at 1550 nm wavelength.

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The change in the mode characteristics of the EDF with a magnetic field was calculated based on the FEM technique. The permittivity tensor, which contains the magnetic field influence factor at the off-diagonal position, was set in the computational domain of the EDF. A perfect matched layer was used on the cladding boundary to reduce the computational cost [32]. The wavelength was 1550 nm, and the maximum mesh size of the core region was set to one-sixth of the wavelength to improve the accuracy. Parametric scanning of the magnetic flux density was performed.

LP01, LP11, LP21, and LP02 modes were found to exist in the EDF. The mode effective RI of the fundamental mode (LP01 mode) increases as the magnetic field increases, as shown in Fig. 3(a). The change in the mode effective RI is 5.423×10−6 RIU/mT by performing a linear fit, and the goodness of fit is 0.9963. With the increasing magnetic field, the mode field intensity distribution of the LP01 mode is more concentrated in the core region, and the intensity decays faster in the cladding region, as shown in Fig. 4(a). The one-dimensional mode field intensity distribution is captured at the region marked with the horizontal line shown in two-dimensional plot. Similarly, by performing a linear fit, the change in the mode effective RI of the higher-order modes (LP11, LP21, and LP02 mode) with the magnetic field are 5.055×10−6, 4.461×10−6, and 3.969×10−6 RIU/mT, respectively, and the goodness of fit are 0.9959, 0.9962 and 0.9965, respectively, as shown in Fig. 3(b). The lower the mode order, the greater is the change in the mode effective RI with the magnetic field. The field intensity proportion of higher-order modes in the core region also increases as the magnetic field increases, as shown in Figs. 4(b)-4(d). The magnetic field influences the magnetization of the EDF, which affects the off-diagonal components in the permittivity tensor. Thus, the mode solutions of the partial differential equations in Eq. (6) change with the magnetic field, which can be explained for the variation in the mode effective RI. Owing to the dependence on the mode effective RI, the mode field intensity distribution also varies with the magnetic field.

 figure: Fig. 3.

Fig. 3. Relationship between mode effective RI and magnetic field at 1550 nm: (a) LP01 mode (fundamental mode) (b) LP11 mode, LP21 mode and LP02 mode (higher-order modes).

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 figure: Fig. 4.

Fig. 4. Relationship between mode field intensity distribution and magnetic field at 1550 nm wavelength: (a) LP01 mode (b) LP11 mode (c) LP21 mode (d) LP02 mode

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3. Magneto-refractive measurement system

A magneto-refractive measurement system based on a fiber-optic MZI [24,33,34] was designed and established to analyze the magneto-refractive effect of the EDF, whose schematic diagram is shown in Fig. 5. The light emitted from the laser (Koheras ADJUSTIK, NKT Photonics) is divided into two beams by coupler 1, whose wavelength is 1550 nm. An EDF is fused on the sensing arm and placed in the axis of a solenoid (LXG-10, LINKJOIN) comprising copper-wire coil with a length of 30.4 cm. The solenoid is driven by an adjustable current power (IT7626, ITECH), which is used to generate a DC magnetic field or AC magnetic field with different waveforms. The direction of the magnetic field is parallel to the fiber axis. Another EDF is fused on the reference arm, whose length is equal to the sensing arm EDF to achieve a better effect of interference. A piezoelectric ceramic (PZT) is wrapped with the reference arm EDF and connected to a signal generator (DG1062Z, RIGOL), which is used to generate the phase carrier signal. The interference of the two beams occurs at coupler 2, and the interference light intensity is detected by a photodetector (PD) (PDB470C, THORLABS) and collected by an oscilloscope (DS2302A, RIGOL). The collected data can be imported into a computer for post-data processing.

 figure: Fig. 5.

Fig. 5. Magneto-refractive measurement system based on fiber-optic MZI.

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Owing to the fusion splicing of the SMF (SMF-28e, Corning) and the EDF, the mode energy proportion in the EDF is calculated using a Gaussian excitation source, as shown in Fig. 6, and the propagation length is 1 cm. The beam waist width of the Gaussian beam is 10.7 µm, which is equal to the mode field diameter of the SMF. The modes in the coupling region propagate and interfere with each other [35]; only the LP0x mode can satisfy the constructive interference condition, which is excited in the EDF. The LP01 mode occupies 92.04% of the mode energy in the EDF, which is significantly higher than that of the higher-order modes. Therefore, the mode effective RI of the LP01 mode is regarded as the mode effective RI of the sensing arm EDF, which is measured in the experiment.

 figure: Fig. 6.

Fig. 6. Mode energy proportion in the EDF when the propagation length is 1 cm.

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When the DC magnetic field measurement is performed, the signal generator is turned off, implying that only the magnetic field signal is introduced into the interference system. The interference light voltage is expressed as

$$U = {R_p}({{I_s} + {I_r}} )[{1 + V\cos k({{n_s} - {n_r}} )L} ], $$
where Is and Ir are the light intensities of the sensing and reference arms, respectively. $V = {{2\sqrt {{I_s}{I_r}} } / {({{I_s} + {I_r}} )}}$ is the interference visibility. Rp is the gain factor of the PD and L is the length of the interference arm. ns is the mode effective RI of the sensing arm EDF, which is affected by the magnetic field. nr is the mode effective RI of the reference arm EDF, which is regarded as a constant when it is not affected by the environment. The mode effective RI of the sensing arm EDF ns can be derived from the interference voltage U according to Eq. (7).

When the AC magnetic field measurement is performed, the magnetic field is applied on the sensing arm, and the phase carrier signal is applied on the reference arm simultaneously. The interference signal is expressed as [36]

$$U(t )= {R_p}({{I_s} + {I_r}} )[{1 + V\cos (C\cos {\omega_c}t + {\varphi_s}(t ))} ], $$
where φs(t)=k[ns(t)-nr]L is the phase shift caused by the AC magnetic field signal, C is the modulation depth, cosωct is the phase shift caused by the phase carrier signal, and ωc is the carrier frequency, which is outside the bandwidth of the magnetic field signal. The phase-generating carrier differential cross-multiplying (PGC-DCM) technique [37] was used to recover the phase shift φs(t) from the interference signal. A base frequency carrier signal and a double frequency carrier signal are introduced to mix with the interference signal. Subsequently, the mixed signal is filtered by two low-pass filters with a cut-off frequency slightly lower than the phase carrier signal. When the filtered signal is processed by differential cross-multiplication, subtraction, and integration, the demodulated signal is obtained as follows:
$$S(t )= {V^2}{[{{R_p}({{I_s} + {I_r}} )} ]^2}{J_1}(C ){J_2}(C ){\varphi _s}(t ), $$
where J1 and J2 are Bessel functions. The phase shift φs(t) caused by the magnetic field can be derived from the demodulated signal S(t), and the mode effective RI ns(t) can be calculated from φs(t)=k[ns(t)-nr]L.

4. Experimental results and discussion

The magneto-refractive properties of the EDF were first investigated using a DC magnetic field. The interference voltage under a DC magnetic field from 0–120 mT was measured, as shown in Fig. 7(a). By performing a polynomial fit, the relationship between the interference voltage and magnetic field was found to exhibit a sinusoidal form, which can be explained by Eq. (7). Simultaneously, the mode effective RI of the LP01 mode shown in Fig. 3(a) was used to calculate the numerical interference voltage, as shown in Fig. 7(a). The experimental results are basically consistent with the numerical results. The mode effective RI of the EDF was calculated from the experimental interference voltage, and the experimental and numerical results are shown in Fig. 7(b). By performing a linear fit, the change in the mode effective RI of the EDF with a magnetic field is 4.838×10−6 RIU/mT, and the goodness of fit is 0.9936. The error between the numerical and experimental results is 0.585×10−6 RIU/mT.

 figure: Fig. 7.

Fig. 7. Numerical and experimental results of the interference voltage and mode effective RI of the EDF under magnetic field from 0–120 mT: (a) interference voltage (b) mode effective RI.

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The magneto-refractive properties of the EDF were also investigated using a 100 Hz sinusoidal AC magnetic field. The interference signal was collected by an oscilloscope and imported into the computer for signal demodulation. The modulation depth was set to 2.37, to maximize the gain of the demodulation result [38]. The frequency of the phase carrier signal was set to 2000 Hz, which was outside the bandwidth of the magnetic field signal. To generate an AC magnetic field with a magnitude from 0–120 mT, the root mean square of the AC magnetic field and DC bias were set to 42.43 and 60 mT, respectively. When a 100 Hz AC magnetic field signal was applied on the sensing arm EDF and the phase carrier signal was applied on the reference arm EDF simultaneously, the interference signal was observed on the oscilloscope, as shown in Fig. 8. The interference signal waveform can be explained using Eq. (8). The phase shift cosωct caused by the phase carrier signal makes the interference signal approximately periodic with a frequency of 2000 Hz. However, the interference signal is not completely periodic because the magnetic field signal is carried on it.

 figure: Fig. 8.

Fig. 8. Interference signal waveform with the application of 100 Hz AC magnetic field signal and phase carrier signal.

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The phase shift φs(t) caused by the AC magnetic field signal was recovered from the interference signal using the PGC-DCM technique, and the mode effective RI ns(t) of the EDF was calculated from φs(t). The time-domain signal of the 100 Hz AC magnetic field and the mode effective RI ns(t) are shown in Fig. 9(a). Subsequently, the data of the mode effective RI in half a magnetic field period were selected for analysis, as shown in the inset of Fig. 9(a). In this interval, a value of the magnetic field corresponds to a value of the mode effective RI, and the relationship between the mode effective RI and magnetic field was obtained, as shown in Fig. 9(b). By performing a linear fit, the change in the mode effective RI with the magnetic field is 4.245×10−6 RIU/mT, and the goodness of fit is 0.9928. The error between the numerical and experimental results is 1.178×10−6 RIU/mT.

 figure: Fig. 9.

Fig. 9. (a) Time-domain waveform of 100 Hz AC magnetic field and the mode effective RI ns(t), (inner) mode effective RI in a period (b) Numerical and experimental results of the mode effective RI under magnetic field from 0–120 mT.

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The numerical and experimental results are summarized in Fig. 10, where Δneff is the variation in the mode effective RI of the EDF. There is an error between the numerical and experimental results for both DC and AC magnetic field measurements. This is mainly because of two reasons. First, the parameters determined for numerical calculation, such as RI, magnetization, and Verdet constant, are investigated in the experiment, which may cause a measurement deviation and eventually cause an error in the numerical results. Second, the influence of the magnetic field distribution in the solenoid is ignored in the theoretical analysis. In fact, the axial magnetic field distribution is calculated by [31]

$$B(z )= \frac{{{\mu _0}({{N / {{L / t}}}} )I}}{2}\left\{ {\left( {\frac{L}{2} + z} \right)\ln \frac{{{r_o} + \sqrt {r_o^2 + {{\left( {\frac{L}{2} + z} \right)}^2}} }}{{{r_i} + \sqrt {r_i^2 + {{\left( {\frac{L}{2} + z} \right)}^2}} }} + \left( {\frac{L}{2} - z} \right)\ln \frac{{{r_o} + \sqrt {r_o^2 + {{\left( {\frac{L}{2} - z} \right)}^2}} }}{{{r_i} + \sqrt {r_i^2 + {{\left( {\frac{L}{2} - z} \right)}^2}} }}} \right\}, $$
where ri and ro are the inner and outer radii of the solenoid, respectively, L is the solenoid length, t = ro-ri is the coil thickness, N is the total number of coils, and I is the current intensity. The axial magnetic field distribution in the solenoid versus the current intensity is shown in Fig. 11(a). The magnetic field gradually decreases toward the solenoid edge, and only the magnetic field close to the center is uniform. However, we assume that the magnetic field remains uniform and constant along the solenoid axis, which is an ideal distribution. The ideal and actual magnetic field distributions are shown in Fig. 11(b). If the EDF comprise m segments of length Lm, the variation of the mode effective RI Δnm in each segment is different, affecting the overall experimental results. However, the actual magnetic field distribution is not considered in the theoretical analysis, causing an error between the numerical and experimental results.

 figure: Fig. 10.

Fig. 10. Numerical and experimental results of mode effective RI change with magnetic field.

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 figure: Fig. 11.

Fig. 11. (a) Axial magnetic field distribution in the solenoid with different current intensity I. (b) Actual and ideal magnetic field distribution when I=1A.

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There is a difference between the experimental results of the DC and 100 Hz AC magnetic field measurements, which is 0.593×10−6 RIU/mT. The reasons behind this are as follows. In the DC magnetic field measurements, the interference voltage was collected when the magnetic field was stable. However, in the AC magnetic field measurement, the amplitude of the magnetic field changes rapidly. There is a response time for the variation of the mode effective RI with the magnetic field, which leads to the case where the change in the mode effective RI cannot match with the change in the magnetic field. To verify this conclusion, the change in the mode effective RI of the EDF were measured with a 20 Hz and a 200 Hz AC magnetic field, and the difference were 0.259×10−6 RIU/mT and 1.174×10−6 RIU/mT compared with the result of the DC magnetic field measurement, respectively, as shown in Fig. 10. The results indicate that the higher the frequency of the magnetic field, the lower is the change in the mode effective RI with the magnetic field. The response time of the mode effective RI could be attribute to the relaxation process of magnetization, which leads to the dependence of AC susceptibility on the frequency of magnetic field [39,40]. As the frequency of magnetic field increases, the real part of AC magnetic susceptibility decreases according to Debye model [41], resulting in the decrease of magnetization. Owing to the relationship between the magneto-refractive effect and magnetization according to Eq. (1), the change of mode effective RI of the EDF also decreases with the increasing magnetic field frequency.

Both the DC and AC magnetic field measurement have advantages and disadvantages. As shown in Fig. 10, the experimental results of DC magnetic field measurement are closer to the theoretical analysis than that of AC magnetic field measurement, which implies that the DC magnetic field measurement has higher accuracy. The reason could be that the AC magnetic field measurement is affected by the response time of the mode effective RI change under AC magnetic field. However, owing to high sensitivity of MZI, the DC magnetic field measurement is susceptible to environmental noise [37], such as temperature or strain. In AC magnetic field measurement, the phase demodulation technique is used, which can eliminate the phase shift caused by environmental disturbances during signal processing [42]. Therefore, the DC magnetic field measurement could be applied to the case where the environment is stable, and the AC magnetic field measurement could be suitable in noisy environments.

5. Conclusion

The magneto-refractive effect of an EDF is theoretically analyzed by numerically calculating the change in the mode characteristics with a magnetic field, and experimentally measured based on a fiber-optic MZI. The permittivity tensor, which includes the magnetic field influence factor at the off-diagonal position, is introduced into Maxwell’s equations. The partial differential equations of the electromagnetic field components are derived and solved based on the FEM technique. The numerical results indicate that the mode effective RI and mode field intensity distribution change with the magnetic field. With increasing magnetic field, the mode effective RI is found to increase, and the variation of the mode effective RI of the fundamental mode is higher than that of the higher-order mode. Furthermore, the mode field intensity distribution is more concentrated in the core region as the magnetic field increases. Based on a fiber-optic MZI, the change in the mode effective RI of the EDF is measured with a DC magnetic field and 100 Hz AC magnetic field, which is 4.838×10−6 and 4.245×10−6 RIU/mT, respectively. The experimental results are basically consistent with the numerical results, proving that the theoretical analysis and experimental measurements are feasible for studying the magneto-refractive effect of the EDF. The error between the experimental and numerical results is mainly because of the measurement deviation of the simulation parameters and the axial magnetic field distribution in the solenoid. The difference between the experimental results of the DC and AC magnetic field measurements is caused by the response time of the mode effective RI changing with the magnetic field. The magneto-refractive properties of the EDF can be used to design and fabricate all-fiber magnetic field or current sensors.

Funding

Advanced Optical Waveguide Intelligent Manufacturing and Testing Professional Technical Service Platform of Shanghai (19DZ2294000); 111 Project (D20031); National Natural Science Foundation of China (61735009, 61875116, 61875118, 62022053).

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 authors upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. Refractive index profile of the EDF and element contents in the core region, (inner) cross-section of the EDF.

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Fig. 2.
Fig. 2. (a) Magnetization curve of the EDF at temperature 293 K. (b) Relationship between Faraday rotation and magnetic field at 1550 nm wavelength.

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Fig. 3.
Fig. 3. Relationship between mode effective RI and magnetic field at 1550 nm: (a) LP01 mode (fundamental mode) (b) LP11 mode, LP21 mode and LP02 mode (higher-order modes).

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Fig. 4.
Fig. 4. Relationship between mode field intensity distribution and magnetic field at 1550 nm wavelength: (a) LP01 mode (b) LP11 mode (c) LP21 mode (d) LP02 mode

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Fig. 5.
Fig. 5. Magneto-refractive measurement system based on fiber-optic MZI.

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Fig. 6.
Fig. 6. Mode energy proportion in the EDF when the propagation length is 1 cm.

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Fig. 7.
Fig. 7. Numerical and experimental results of the interference voltage and mode effective RI of the EDF under magnetic field from 0–120 mT: (a) interference voltage (b) mode effective RI.

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Fig. 8.
Fig. 8. Interference signal waveform with the application of 100 Hz AC magnetic field signal and phase carrier signal.

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Fig. 9.
Fig. 9. (a) Time-domain waveform of 100 Hz AC magnetic field and the mode effective RI ns(t), (inner) mode effective RI in a period (b) Numerical and experimental results of the mode effective RI under magnetic field from 0–120 mT.

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Fig. 10.
Fig. 10. Numerical and experimental results of mode effective RI change with magnetic field.

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Fig. 11.
Fig. 11. (a) Axial magnetic field distribution in the solenoid with different current intensity I. (b) Actual and ideal magnetic field distribution when I=1A.

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Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ε = ( n 2 0 0 0 n 2 0 0 0 n 2 ) + K ( 0 M z M y − M z 0 M x M y − M x 0 ) ,
θ F , s a t = − k K ′ ′ M s 2 n ,
ε α = ( n α 2 j δ α 0 − j δ α n α 2 0 0 0 n α 2 ) ,
δ α = K ′ ′ M z = 2 n α v B s a t k M s ⋅ M m ⋅ m V ,
∂ E y ∂ x − ∂ E x ∂ y = j ω μ 0 H z ∂ E z ∂ y − j β E y = j ω μ 0 H x j β E x − ∂ E z ∂ x = j ω μ 0 H y ∂ H y ∂ x − ∂ H x ∂ y = − j ω ε 0 n 1 2 E z ∂ H z ∂ y − j β H y = − j ω ε 0 ( n 1 2 E x + j δ 1 E y ) j β H x − ∂ H z ∂ x = − j ω ε 0 ( n 1 2 E y − j δ 1 E x ) ,
∂ 2 E x ∂ x 2 + ∂ 2 E x ∂ y 2 + j δ 1 n 1 2 ( ∂ 2 E y ∂ x 2 − ∂ 2 E x ∂ x ∂ y ) + ω 2 μ 0 ε 0 ( n 1 2 E x + j δ 1 E y ) = ( k 0 n e f f ) 2 E x ∂ 2 E y ∂ x 2 + ∂ 2 E y ∂ y 2 + j δ 1 n 1 2 ( ∂ 2 E y ∂ x ∂ y − ∂ 2 E x ∂ y 2 ) + ω 2 μ 0 ε 0 ( n 1 2 E y − j δ 1 E x ) = ( k 0 n e f f ) 2 E y
U = R p ( I s + I r ) [ 1 + V cos ⁡ k ( n s − n r ) L ] ,
U ( t ) = R p ( I s + I r ) [ 1 + V cos ⁡ ( C cos ⁡ ω c t + φ s ( t ) ) ] ,
S ( t ) = V 2 [ R p ( I s + I r ) ] 2 J 1 ( C ) J 2 ( C ) φ s ( t ) ,
B ( z ) = μ 0 ( N / L / t ) I 2 { ( L 2 + z ) ln ⁡ r o + r o 2 + ( L 2 + z ) 2 r i + r i 2 + ( L 2 + z ) 2 + ( L 2 − z ) ln ⁡ r o + r o 2 + ( L 2 − z ) 2 r i + r i 2 + ( L 2 − z ) 2 } ,
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