1. Introduction

Detection of weak magnetic fields is becoming increasingly important for biomedicine, earthquake monitoring, and underwater rescue [1]. Compared to existing electric-based magnetic field sensors, magnetic field sensors based on optical fibers have emerged as a research hotspot for weak magnetic field detection due to advantages such as small size, high magnetic sensitivity, wide spectral response range, and strong anti-interference capability [2]. Common optical fiber magnetic field sensors mainly exploit the Faraday rotation effect type [3] and magnetostrictive properties [4]. However, they are plagued by signal demodulation, low sensitivity, and limited magnetic field detection ranges. In contrast, magnetic fluids (MF) have magnetically controllable refractive indexes [5] giving rise to enhanced interactions between the evanescent field and MF when combined with functional optical fibers to improve the magnetic field sensitivity [6,7].

Due to the limited penetration depth, the evanescent wave is mostly confined near the fiber core. Therefore, it is common to use side polishing or tapering techniques to reduce the distance between the core and MF to improve the magnetic field sensitivity. X. Lei et al. have utilized a femtosecond laser to side-polish a polarization-maintaining fiber to produce a D-shape magnetic field sensor with a maximum magnetic field sensitivity of 0.0823 nm/mT [8]. However, the low magnetic field sensitivity limits the detection of magnetic fields at the mT level. The photonic crystal fiber (PCF) [9] offers a highly flexible structure, and the air holes provide natural channels for material filling [10–13]. Filling the specific air holes with a metal and MF can excite surface plasmon resonance (SPR) [14–16], which not only enhances energy coupling, but also improves magnetic field sensitivity. S. Yao et al. have proposed a PCF-SPR magnetic field sensor with air holes arranged hexagonally to detect magnetic fields in the range of 50-130 Oe with a maximum magnetic field sensitivity of 590 pm/Oe [17]. D. Wang et al. have described a PCF-SPR sensor with a double-ring channel for simultaneous detection of the temperature and magnetic field with a magnetic field sensitivity of 308.3 pm/Oe [18]. However, although the sensitivity of PCF-SPR magnetic field sensors has been improved, the resolution is still around tens nT, thus only providing the ability of preliminary detection of weak magnetic fields. A gold film-modified D-shape PCF-SPR magnetic field sensor [19] in which the MF is filled into an air hole with a diameter of 2.8 µm near the gold film shows a maximum magnetic field sensitivity of 21,750 pm/mT, which is two orders of magnitude higher than that of previously reported sensors, besides a resolution of 4.598 nT. However, since the precise filling of MF in such a small-diameter air hole is challenging in practice, it is still challenging to design a structurally simple and analyte-fillable sensor that is capable of detecting pT-level weak magnetic fields.

Herein, an ultra-high sensitivity MF SPR sensor based on single-hole fiber (SHF) for detecting weak magnetic field is proposed and systematically analyzed. This SHF has a very simple structure consisting of only one large air hole in the cladding. The SHF-based sensor is constructed by filling the air hole with hydrated Fe₃O₄ MF and suspending a metal wire close to the core on the inner wall of the air hole. The finite element method (FEM) is employed to optimize the sensing properties by investigating the effects of the fiber core diameter, air hole diameter, metal wire diameter, core-to-hole spacing, the relative refractive index difference between the core and cladding, and metal type. The results reveal that the SHF-SPR magnetic field sensor can detect magnetic fields in a range from 3.5 mT to 17 mT with a maximum magnetic field sensitivity of 451,000 pm/mT, which is more than one order of magnitude higher than that of the sensor reported in Ref. [19] and three orders of magnitude higher than other similar sensors [17,18,35–38]. Most importantly, the SHF-SPR magnetic field sensor achieves an optimal resolution of 221.73 pT and is capable of detecting weak magnetic fields at the pT level. In addition, a detailed analysis of the fabrication tolerance of each structural parameter is conducted. The SHF-SPR magnetic field sensor features a remarkably simple structure, high magnetic field sensitivity, and the capability to detect weak magnetic fields at the pT level, thus having enormous potential in many applications such as geological exploration and weak biological signal detection.

2. Sensor structure and principle

In order to simplify the sensor structure and enhance the interactions between the core mode and SPP mode, only one large air hole with a diameter of d₃ is introduced to the fiber cladding as shown in Fig. 1, in which a metal wire close to the fiber core is embedded onto the air hole and the MF is filled throughout the entire air hole. It should be noted that the centers of the core, large air-hole and metal wire should be located on the same horizontal line. The core of the SHF is composed of Ge-doped silica with a diameter of d₁. It has a larger refractive index than the pure silica cladding n_cladding, and the refractive index is determined by the Sellmeier equation [20]. The refractive index of the fiber core n_core is derived by Eq. (1) [21]:

 

Fig. 1. Cross-section of the SHF-SPR magnetic field sensor

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The structure of the SHF is very simple as it contains only a single large air hole in the cladding, which can be easily fabricated by drilling methods. Moreover, the large air hole makes it easier for analyte filling and metal embedding. The MF is hydrated Fe₃O₄ with a concentration of 0.68 emu/g. The relationship between the refractive index n_MF(H,T) and magnetic field and temperature is shown in Eq. (2) [22]:

When the light propagates along the SHF, an evanescent wave transmits along the interface between the core and cladding. When light reaches the surface of the metal, it causes uneven distribution of electron density in the metal. Concurrently, a part of electrons are attracted to the region with superfluous positive charge because of the Coulomb force. As the number of electrons increases, the repulsive force between these electrons compels the clustered electrons to disperse from that region. The repetition of this process gives rise to collective oscillations of the entire electron system, manifested in the form of waves, known as surface plasmon wave (SPW), and the surface plasmon polaritons (SPPs) are excited. When the propagation constant of the evanescent wave near the core and SPW on the metal surface is equal, the phase matching condition described in Eq. (3) [24] is satisfied:

At this point, the fundamental mode (FM) and SPP mode exhibit the same electric field variation, resulting in the same real part of the refractive index. As a result, most of the energy in the core is coupled to the surface of the metal, leading to a dramatic increase in the FM loss, and a loss peak appears. The FM loss can be calculated by Eq. (4) [25]:

The essence of SPR is the energy exchange between the FM and the SPP mode, which can be characterized by the mode field distributions and refractive index change curves in Fig. 2(a). As the wavelength increases, the effective refractive indexes of both the FM (blue curve) and SPP modes (red curve) decrease overall. Since the refractive index of the SPP mode decreases faster than that of the FM, there exists an intersection point that satisfies the phase-matching condition in the dispersion curves. The wavelength corresponding to the intersection point is defined as the resonance wavelength (RW). When the wavelength is smaller than the RW, the refractive index of the FM is smaller than that of the SPP mode, and the photon energy is mainly concentrated in the core. As the wavelength increases, the photon energy continues to couple from the FM to the SPP mode, causing a significant increase in the loss of FM. When λ > 705 nm, the repulsion between the free electrons on the metal surface becomes stronger, and the energy is driven back to the core, so that the refractive index of the FM becomes higher than that of the SPP mode and the loss decreases accordingly. Therefore, there is a loss peak in the FM loss curve (green curve). It should be noted that when λ is exactly equal to RW, the FM and SPP modes exhibit the same transmission speed and maximum coupling strength. Additionally, different magnetic fields correspond to different phase-matching points and RWs. When the external magnetic field intensity changes, the refractive index of the FM will vary, resulting in a certain displacement in the phase matching point. Figure 2(b) shows that when the magnetic field goes up from 6 mT to 7 mT, and 8 mT, the RW shifts toward longer wavelengths, and each magnetic field intensity corresponds to a unique RW. Therefore, the magnetic field can be detected by measuring the RW of the loss curve.

 

Fig. 2. (a) Effective indexes of the FM (blue curve) and SPP mode (red curve) and FM loss (green curve) as a function of the wavelength. The inset shows the field distributions at 5 mT; (b) Loss peaks at 5 mT, 6 mT, 7 mT, and 8 mT for the initial structural parameters

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The important properties of the SHF-SPR magnetic field sensor include the magnetic field sensing characteristics, magnetic field sensitivity, resolution, and figure of merit (FOM). In order to accomplish weak magnetic field detection, the magnetic field sensitivity of the sensor should be as high as possible. The magnetic field sensitivity can be calculated by Eq. (6) [27]:

The resolution refers to the ability of the sensor to recognize and measure the smallest variation in the magnetic field. It is inversely proportional to the sensitivity of the sensor and obtained by Eq. (7) [28]:

Another important parameter is the FOM which describes the overall performance of the SHF-SPR magnetic field sensor and is expressed by Eq. (8) [29]:

3. Sensor optimization

The characteristics of the SHF-SPR magnetic field sensor are influenced by the fiber structure, metal type, and relative refractive index difference between the core and cladding. Therefore, a systematic and comprehensive optimization of the structural parameters and metals of the SHF-SPR magnetic field sensor is carried out for a magnetic field intensity of 5 mT. Since the resolution, FOM, and other important characteristics of the sensor are related to the magnetic field sensitivity, the magnetic field sensitivity is utilized as the primary evaluation criterion in the optimization. The initial structural parameters of the sensor are: d₁ = 10.0 µm, d₂ = 2.0 µm, d₃ = 36.8 µm, L = 0.5 µm, and △ = 1.0%, and Al is chosen as the metal.

3.1 Polarization mode

Due to the structural asymmetry of the designed SHF-SPR magnetic field sensor, there are two orthogonal FMs called the x-pol and y-pol modes. It is possible for both modes to couple with the SPP mode near the metal surface when the phase match condition is satisfied. Therefore, the coupling behavior of the two polarization modes is investigated. Figure 3 shows the loss curves and optical field distributions of the x-pol and y-pol modes for magnetic fields of 5 mT and 6 mT. The x-pol mode exhibits a distinct and sharp loss peak arising from coupling with the SPP mode, and the loss peak changes as the magnetic field intensity goes up from 5 mT to 6 mT. According to Eq. (6), the magnetic field sensitivity is 4,700 pm/mT when H = 5 mT. Regarding the y-pol mode, the loss curve is approximately parallel to the x-axis, indicating that the SPP mode has a slight effect on the y-pol mode. The field distributions of the x-pol and y-pol modes at 600 nm further illustrate this point. For the x-pol mode, photons not only concentrate in the core, but have a significant energy distribution near the metal wire, while y-pol is mainly concentrated in the core with little energy coupled to the metal surface. Therefore, x-pol should be chosen for magnetic field sensing.

 

Fig. 3. Loss curves of FM at 5 mT and 6 mT, and field distribution of x-pol and y-pol at 5 mT.

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3.2 Optimization of structural parameters

The effects of core diameter d₁ on the magnetic field sensitivity and loss are investigated and shown in Fig. 4. Figure 4(a) shows that the peak loss decreases and the RW blueshifts with increasing d₁. This is because a larger diameter leads to a higher photon energy in the core and an increase in the refractive index of the FM. The phase velocity declines and the loss peak blueshifts. Figure 4(b) supports this point. As d₁ increases, both the refractive indexes of the FM and SPP modes increase. When the refractive index of the SPP mode decreases at a significantly faster rate than the FM with increasing wavelength, the phase-matching point shifts toward a shorter wavelength. When d₁ = 8 µm, the evanescent wave and SPW have the same transmission speeds. As the wavelength increases, the accelerated SPW reacts with the electron beam to produce an “S-shape” [30] dispersion curve. Figure 4(c) summarizes the magnetic field sensitivities and peak losses for different core diameters within the optimization range. When d₁ decreases to 8.0 µm, although the magnetic field sensitivity increases by 1.3 times, the peak loss increases by 2.37 times, indicating that decreasing d₁ is not recommended. Additionally, when d₁ ≥ 10 µm, the magnetic field sensitivity first increases and then decreases showing a maximum value of 5,300 pm/mT at d₁ = 12 µm. The corresponding resolution is 18.87 nT, illustrating the ability to detect weak magnetic fields. All in all, a core diameter of 12 µm shows better magnetic field sensing characteristics. Moreover, it is worth mentioning that with the increase of d₁, high-order modes may exist in SHF. However, it is still possible to detect the magnetic field by utilizing the variation of FM. On one hand, the high-order modes in the SHF will quickly attenuate or disappear because the loss of the high-order modes is about two orders of magnitude higher than FM. On the other hand, single mode fibers (SMFs) will be fused at both ends of the SHF in practical sensing applications. Through adjusting the splice alignment between SMF and SHF, it can be avoidable to excite high-order modes in the SHF. Therefore, the magnetic field can still be detected by the changes of the FM.

 

Fig. 4. (a) FM loss; (b) Refractive index curves of the FM and SPP modes for different d₁ and (c) Summary of the magnetic field sensitivity and peak loss of the SHF-SPR magnetic field sensor.

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For d₁ = 12 µm, optimization of the metal wire diameter d₂ is performed as shown in Fig. 5. Based on the initial parameters, the magnetic field sensitivity for 5 mT is 5,300 pm/mT. If the metal wire diameter is increased to 2.4 µm, the magnetic field sensitivity decreases to 4,700 pm/mT. To improve the magnetic field sensitivity, d₂ should be reduced. When d₂ < 0.3 µm, the SPP mode disappears and therefore, the range of the metal wire is 0.3 µm ≤ d₂ ≤ 2.4 µm. Figure 5(a) presents the loss spectra of FM for different values of d₂ for two magnetic field intensities and the mode field distributions near the metal wire for d₂ = 2.4 µm and 0.3 µm. When d₂ is large, the evanescent wave cannot penetrate the metal wire, and so the FM can only undergo energy exchange on the part of the metal wire surface near the core resulting in a weaker SPR effect. When d₂ is relatively small, the evanescent wave can interact with the SPW on the entire metal wire surface, causing stronger coupling of the FM and SPP modes. Therefore, as d₂ increases, the RW shows a blueshift for the same reason as d₁. Figure 5(b) summarizes the impact of the metal wire diameter on the magnetic field sensitivity and peak loss. When d₂ decreases, the magnetic field sensitivity improves continuously, while the peak loss gradually increases. When d₂ = 0.3 µm, the magnetic field sensitivity is 33,400 pm/mT, which is approximately 7.11 times that of the initial parameter. The peak loss is 66.68 dB/cm, which is approximately 1.56 times that of the initial parameter. Therefore, we choose to tolerate the increase in peak loss and set d₂ as 0.3 µm. In addition, it is noteworthy that the sensor can measure a minimum weak magnetic field variation of 2.99 nT boding well for higher precision in the detection of weak magnetic fields.

 

Fig. 5. (a) Loss curves of the FM for different d₂ and magnetic fields of 5 mT and 6 mT; (b) Magnetic field sensitivities and peak losses as a function of d₂

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After setting d₁ = 12 µm and d₂ = 0.3 µm, the pore diameter is optimized. Figure 6(a) describes the RWs for different d₃ when H = 5 mT and H = 6 mT. It is evident that with a large increase of the diameter of the air hole, the offset of RW remains almost invariant. Figure 6(b) also shows that a significant variation of d₃ has little effects on the magnetic field sensitivity and peak loss. This is mainly because d₃ has little effects on the contact area between the metal and MF. Hence, the coupling strength between the FM and SPP modes does not change significantly, resulting in stable peak loss and magnetic field sensitivity. In addition, a large pore diameter not only increases the filling speed of the MF, but reduces the difficulty of metal wire modification. Therefore, the value of d₃ is chosen to be 41.8 µm, at which the magnetic field sensitivity and peak loss are 33,300 pm/mT and 66.19 dB/cm, respectively.

 

Fig. 6. (a) Variation of RW with d₃ for H = 5 mT and H = 6 mT; (b) Magnetic field sensitivities (blue) and peak losses (purple) as a function of d₃ for H = 5 mT.

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In addition to the diameter of the core, metal wire, and pore, the spacing L between the core and metal wire is an important factor affecting the characteristics of the SHF-SPR magnetic field sensor. Figure 7 shows the adjusted L based on d₁ = 12 µm, d₂ = 0.3 µm, and d₃ = 41.8 µm. When L is increased from 0.5 µm to 0.6 µm, the magnetic field sensitivity decreases to 31,700 pm/mT, indicating that L should be reduced. When the metal wire is tangent to the core, L = 0. Therefore, the optimization range of the spacing between the core and metal wire is 0 µm - 0.6 µm. As L is shortened, the metal wire gets closer to the core, thereby enhancing the coupling between the FM and SPP modes. At the same time, the RWs redshift. When L = 0 µm, the magnetic field sensitivity at 5 mT reaches 80,500 pm/mT. Obviously, L cannot be further reduced and the optimal value of L is 0 µm. Overall, after systematic and comprehensive optimization of the structural parameters, the magnetic field sensitivity of the SHF-SPR magnetic field sensor is 17.13 times that of the original sensor. Moreover, the resolution is improved to 1.24 nT enabling better detection of weak magnetic fields.

 

Fig. 7. Optimization of the distance between the core and metal wire at a magnetic field of 5 mT by analyzing the loss peaks when H = 5 mT and 6 mT

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3.3 Optimization of △ between the core and cladding

The relative refractive index difference between the core and cladding is another important factor affecting weak magnetic field detection. In actual fabrication of the SHF, the value of Δ typically ranges from 0.5% to 1.5% [31]. Figures 8(a)-(b) present the SPR spectra and magnetic field sensitivities of the sensor for different Δ. As Δ increases, Eq. (1) shows that the refractive index of the fiber core also increases to enhance the confinement ability of the photon energy. Meanwhile, the RW redshifts and the magnetic field sensitivity decreases. Although the magnetic field sensitivity is the highest at Δ = 0.5% (reaching 145,300 pm/mT), choosing Δ = 0.5% increases the FWHM of the FM loss curve. Equation (8) indicates that the FOM is reduced due to the increase of the FWHM. Therefore, to balance the magnetic field sensitivity and overall characteristics of the sensor, Δ is set to be 0.75% corresponding to a magnetic field sensitivity of 104,300 pm/mT and resolution of 958.77 pT at 5 mT, consequently enabling the detection of weak magnetic fields at the pT level.

 

Fig. 8. (a) Loss curves of different △ and (b) Magnetic sensitivities.

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3.4 Optimization of metals

As the main source of the SPR effect, the choice of metals has a significant impact on the characteristics of magnetic field sensors. The common metals are Au, Ag, Cu, and Al. The relative dielectric constants are expressed by Eq. (5) and the values of λ_c and λ_p are listed in Table 1.

Table 1. λ_c and λ_p of different metals.

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Figure 9(a) presents the FM loss spectra at 5 mT and 6 mT for the four metals. Although all four metals exhibit the SPR effect, the magnetic field sensitivity is quite different. The magnetic field sensitivities of Au, Ag, Cu, and Al are 13,000 pm/mT, 28,000 pm/mT, 37,000 pm/mT, and 104,300 pm/mT, respectively, because of the difference in the real part of the relative permittivity as shown in Fig. 9(b). The real part of the relative dielectric constant of Al is significantly larger than that of the other metals, and the refractive index of the metal wire surface is also higher, thus attracting more photons to the surface. The coupling between the FM and SPP modes is further enhanced, leading to better magnetic field sensitivity and resolution. Therefore, aluminum is chosen as the medium to excite SPR.

 

Fig. 9. (a) Loss curves of the sensor with different metals and (b) Real parts of the dielectric constants as a function of the wavelength for different metals.

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In summary, the SHF-SPR magnetic field sensor uses Al to produce the SPR effect, and the relative refractive index difference between the core and cladding is set to 0.75%. The optimized structural parameters are d₁ = 12.0 µm, d₂ = 0.3 µm, d₃ = 41.8 µm, and L = 0 µm. The magnetic field sensitivity and resolution are 104,300 pm/mT and 958.77 pT respectively, which is 22.19 times higher than that of the sensor without optimization. Moreover, the magnetic field sensitivity is inversely proportional to d₂, L, and Δ. The value of d₃ has a slight effect on the magnetic field sensitivity. Our results reveal that it is beneficial to choose a metal with a larger real part of the refractive index to improve the magnetic field sensitivity by exploiting the SPR effect. Our findings provide guidance for the design of weak magnetic field sensors.

4. Results and discussion

4.1 Magnetic sensing characteristics

Although it is challenging to hang a metal wire on the inner wall of the air hole, the fabrication feasibility has been demonstrated by relevant experimental researches [33,34]. Firstly, under the microscope, an Al wire longer than the sensor is passed through the air hole and adjusted to the closest position to the core. Then, the Al wire is immobilized onto the end of the SHF by a small amount of cured glue. Thirdly, the air hole is filled with MF by capillary effect, and the ends are sealed with the cured glue, which can further fix the position of the Al wire to prevent deviation. Finally, the excess Al wire outside the air hole is removed, completing the manufacturing of the SHF-SPR magnetic field sensor.

The optimized SHF-SPR magnetic field sensor can detect weak magnetic fields as shown in Fig. 10. The SMFs are fusion-spliced to both sides of the SHF-SPR magnetic field sensor. One end is connected to a broadband light source, and the other end is connected to an optical spectrum analyzer (OSA). The sensor is placed on a temperature-controlled platform at a constant temperature of 24.3°C. The magnetic field generator and broadband light source are then turned on. The light passes through the SMF and enters the SHF-SPR magnetic field sensor to excite SPR. The magnetic field generator generates a magnetic field in the range of 3.5 mT to 17 mT, and the magnetic field variation changes the refractive index of the magnetic fluid to produce a shift in the SPR resonance peak. Finally, the OSA and a computer are used to measure the RW for magnetic field detection.

 

Fig. 10. Schematic diagram of the apparatus for the detection of magnetic fields.

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In order to detect weak magnetic fields, the sensor must have a high sensitivity. Figure 11(a) presents the partial loss peaks of the SHF-SPR magnetic field sensor for different applied magnetic field intensities and Fig. 11(b) shows the peak losses and magnetic field sensitivities at different magnetic fields. According to Eq. (4), the peak losses diminish gradually. The losses of FMs are below 174.74 dB/cm, and the minimum is 153.41 dB/cm (H = 17 mT). The sensor can detect magnetic fields in the range of 3.5 mT - 17 mT, and the average magnetic field sensitivity is 60,777.78 pm/mT. Moreover, the highest magnetic field sensitivity of 451,000 pm/mT is achieved when H = 3.5 mT, which is significantly higher than other similar magnetic sensors. As shown in Fig. 11(c), the refractive index of the MF rises when the magnetic field increases, thereby causing energy transfer from the core to the metal surface and decreasing the effective refractive index of the FM. At the same time, the phase velocity of light transmission increases giving rise to redshifting RW.

 

Fig. 11. (a) FM loss curves at different magnetic field intensities; (b) Magnetic sensitivities and peak losses; (c) Refractive index curves of the FM and SPP modes.

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Figures 12(a) and (b) present the optimal modeling (fitting) of RW and R_n for the SHF-SPR magnetic field sensor, respectively, according to Eqs. (6-7). The R² values of the two fitted curves are 99.985% and 99.204%, respectively. The sensor exhibits very high precision, with a minimum detectable magnetic field intensity variation of 221.7 pT. This characteristic is significantly better than those reported in Refs. [17–19,35–38].

 

Fig. 12. Fitted curves: (a) RW and (b) Resolution.

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In addition, it should be noted that the refractive index of the MF is also influenced by the temperature [32,33]. Therefore, the magnetic field sensitivity variation of the proposed SHF-SPR magnetic field sensor at 3.5 mT is analyzed within the range of 20.3°C to 60.3°C. Figure 13(a) demonstrates that with the rising temperature, the RW blue-shifts. This is because the refractive index of the MF decreases with an increase in the temperature, making the refractive index of the FM increased. Figure 13(b) indicates that the magnetic field sensitivity of the sensor is inversely proportional to the temperature. When the temperature exceeds 24.3°C, although the magnetic field sensitivity decreases, the temperature compensation by the temperature control platform can ensure the ultra-high magnetic field sensitivity detection characteristics of the sensor. Additionally, the magnetic field sensitivity is minimal (65,200 pm/mT) when T = 60.3°C, but it is still nearly three times the highest magnetic field sensitivity currently reported [19]. Consequently, the proposed SHF-SPR magnetic field sensor retains excellent capability for detecting weak magnetic field when the temperature varies relatively large.

 

Fig. 13. (a) FM loss peaks of 3.5 mT and 4 mT at the range of 24.3°C to 60.3°C; (b) the summary of the magnetic field sensitivities

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4.2 FOM

The curves of FOM with magnetic field intensity are shown in Fig. 14. It is evident that the FWHM does not exhibit a regular variation with increasing magnetic field intensities, but rather increases first before stabilizing. The FOM is inversely proportional to the magnetic field intensity, as the change in magnetic field sensitivity is much larger than the change in FWHM. The FOM of the SHF-SPR magnetic field sensor reaches a maximum of 15.033 mT^-1 at H = 3.5 mT.

 

Fig. 14. FOM versus magnetic field intensities.

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4.3 Fabrication feasibility and tolerance

Although it is challenging to hang a metal wire on the inner wall of the air hole, the fabrication feasibility has been demonstrated by relevant experimental researches [34,35]. Firstly, under the microscope, an Al wire longer than the sensor is passed through the air hole and adjusted to the closest position to the core. Then, the Al wire is immobilized onto the end of the SHF by a small amount of cured glue. Thirdly, the air hole is filled with MF by capillary effect, and the ends are sealed with the cured glue, which can further fix the position of the Al wire to prevent deviation. Finally, the excess Al wire outside the air hole is removed, completing the manufacturing of the SHF-SPR magnetic field sensor.

The fabrication tolerance describes the variations of the structural parameters that can be tolerated in the fabrication of fiber sensors. Since the SHF-SPR magnetic field sensor is sensitive to variations in the structural parameters, it is important to evaluate the fabrication tolerance of the sensor. Figures 15(a)-(c) show the variations of the FM loss peaks and magnetic field sensitivity when the core diameter, Al wire diameter, and pore diameter change within certain ranges. The variation range for d₁ and d₃ is ±5%, while that for d₂ is ±1%. This is because the manufacturing of Al wires is quite mature, and the error range is small. With regard to d₁ and d₂, the changes in the magnetic field sensitivity within the error ranges are 13.20 nm/mT and 11.78 nm/mT, which are acceptable for manufacturing. When the pore diameter varies within ±5%, the change of the magnetic field sensitivity is only 0.9 nm/mT, indicating excellent tolerance. Additionally, although it is unlikely that all parameters of the sensor will change simultaneously, Fig. 15(d) shows the worst-case scenario within a range of ±1%. As the structural parameters increase, the RW blueshifts, and the change in the magnetic field sensitivity is 14.8 nm/mT, which is still within the acceptable range. Consequently, the SHF-SPR magnetic field sensor can tolerate reasonable manufacturing errors.

 

Fig. 15. Loss curves and changes of magnetic sensitivities at 5 mT and 6 mT: (a) d₁, (b) d₂, (c) d₃, and (d) Entire SHF changing within the corresponding range.

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5. Performance comparison

In order to highlight the main advantages of the SHF-SPR magnetic field sensor, Table 2 compares the performance with that of similar sensors reported recently. The comparison includes the fiber type, maximum magnetic field sensitivity, resolution, magnetic field detection range, and FOM. SPR magnetic field sensors are mainly based on PCF with a complex structure and small channels for MF filling, thereby making it difficult to fill with the analytes. Our SHF-SPR magnetic field sensor has a very simple structure with only one large pore, and filling of the MF can be achieved by the capillary effect. Compared to Refs. [36–39], the SHF-SPR magnetic field sensor with significantly higher magnetic field sensitivity showing an improvement about three orders of magnitude. Meanwhile, the magnetic field sensitivity is about 21 times higher than that reported in Ref. [19]. The most important achievement is that this SHF-SPR magnetic field sensor can detect pT-level weak magnetic fields.

Table 2. Comparison of different magnetic field sensors.

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6. Conclusion

An SPR magnetic field sensor based on SHF is designed and analyzed. The cladding of the sensor has only one large air hole. By suspending an Al wire in the air hole and filling with MF, high sensitivity and wide range magnetic field detection are achieved. Based on the magnetic field sensitivity and peak loss at 5 mT, the structural parameters of the SHF-SPR magnetic field sensor are systematically optimized, and the influence of the metals and refractive index difference between the core and cladding on the magnetic field detection characteristics are discussed. The optimal structural parameters are d₁ = 12.0 µm, d₂ = 0.3 µm, d₃ = 41.8 µm, and L = 0 µm. When Δ = 0.75% and the SPR effect is excited by Al, the sensor performance is the best. The magnetic field sensing characteristics, resolution, FOM, and fabrication tolerance of the SHF-SPR magnetic field sensor are derived. Numerical analysis shows that the sensor can detect magnetic fields in the range of 3.5 mT - 17 mT, and the magnetic field sensitivity is above 10,000 pm/mT. The average magnetic field sensitivity is 60,777.78 pm/mT, and the maximum magnetic field sensitivity is 451,000 pm/mT, which is more than one order of magnitude higher than that of the best sensor reported so far and three orders of magnitude higher than that of other similar magnetic field sensors. The SHF-SPR magnetic field sensor can detect a minimum magnetic field change of 221.73 pT and weak magnetic fields at the pT level for the first time. When H = 3.5 mT, FOM reaches the maximum of 15.033 mT^-1, which is better than the values reported previously. Analysis of the fabrication tolerance reveals that when the structural parameters of the sensor vary within the allowed error ranges during fabrication, the change in the magnetic field sensitivity is within an acceptable range. This SHF-SPR magnetic field sensor boasting a wide magnetic field detection range, ultra-high magnetic field sensitivity, excellent FOM, and good fabrication tolerance has broad prospects in the detection of weak magnetic fields in biomedical and other applications.