Magnetic field is one of the fundamental physical quantities characterizing both intensity and direction, which is continuously distributed throughout space. The measurement of magnetic fields, particularly vector magnetic field measurement, is vitally important across diverse disciplines. Acquiring both intensity and orientation information of the magnetic field can improve imaging precision in medical applications [1] and plays a critical role in geophysical exploration, assisting in earthquake and volcanic activity prediction as well as resource mapping [2–5]. Additionally, spacecraft and autonomous vehicles depend on vector magnetic field data for accurate navigation, especially in GPS-denied environments [6–9]. Optical fiber magnetic field sensors, known for their natural immunity to electromagnetic interference and ease of integration, are ideal and have found extensive applications for magnetic field detection. These optical fiber magnetometers are generally based on the Faraday effect [10] or magnetostrictive effect [11]. Unfortunately, the design of magnetic field sensors based on the Faraday effect is often complex. Although magnetostrictive-material-based sensors are better suited for static magnetic field measurements, they pose challenges for integration with optical fibers.

Magnetic fluid (MF) is a stable, gel-like liquid in which magnetic nanoparticles are uniformly dispersed within a carrier fluid [12]. MF exhibits exceptional magneto-optical properties under magnetic field, including variable optical transmission [13], tunable refractive index (RI) [14], tunable birefringence [15], the Faraday effect [16], and magnetochromatics [17]. The MF functionalized optical fiber provides a very neat solution for integrated fiber magnetometers with superior sensitivity for magnetic field detection. Uniquely, the optical fiber magnetic field sensors based on the RI tunability of MF have also demonstrated good linearity [18–20]. However, a significant limitation of such types of sensors is their inability to simultaneously measure magnetic intensity and orientation. To measure the magnetic field’s direction, the fiber structure or the magnetic field must be rotated relative to the other in the experimental setup. Additionally, the fiber structure must generate a non-circular symmetric light field distribution.

To enable the simultaneous measurement of field intensity and orientation, two types of MF-based vector magnetic fiber sensors have been developed. A straightforward approach is to use an asymmetric fiber structure such as misaligned fibers [21], U-bent fibers [22], or side-polished fibers [23]. It is true that dislocation fusion disrupts the circular symmetry of the fiber structure, resulting in axial asymmetry. Such non-circular configuration enables the sensing path to interact directly with the external liquid environment being measured, thereby significantly enhancing sensitivity. However, the MF exhibits strong light absorption properties, leading to considerable light loss when light passes directly through it. In U-bent fiber structures, curvature causes the core mode to leak into the cladding, generating higher-order cladding modes that are sensitive to changes in the external refractive index. Nonetheless, smaller bending radii can result in significant insertion loss. Side polishing exposes the side of the fiber to the external environment, enhancing the interaction between the light field and the MF. Changes in the magnetic field direction can alter the chain state of magnetic nanoparticles in the MF, subsequently causing a variation in the refractive index of the MF near the polished surface. This enables vector magnetic field sensing. However, the side-polished optical fiber vector magnetometers are adversely fragile owing to their mechanical fabrication procedure. The second type of vector magnetic field optical fiber sensor introduces asymmetric RI modulation along the fiber core including a fiber Bragg grating (FBG) and a long period fiber grating (LPFG), hence creating an asymmetric evanescent field that interacts with the MF. In particular, LPFG, as a high-performance all-fiber device, features strong capability in manipulating evanescent field in addition to its ease of fabrication, stability, and compact structure. However, the asymmetric refractive index modulation can lead to high polarization-dependent loss (PDL) in the spectral domain, resulting in environmental instability and complexity of the vectorial sensing system [24,25]. While substantial progress has been made in the development of optical fiber vector magnetic field sensors, further exploration is still necessary for sensing technologies that can simultaneously achieve vector measurements, enhance sensitivity, and minimize losses associated with MF.

In this paper, we demonstrated a high-sensitivity vector magnetic field sensor based on a hexagonal fiber grating (HFG) functionalized with magnetic fluid (MF). The HFG was fabricated by hydrofluoric acid (HF) etching of a seven-core fiber (SCF) helical LPFG. Due to the different corrosion rates of HF on the core and cladding of the SCF, the cladding gradually forms a hexagonal structure as the outer core is corroded. With its unique asymmetric waveguide structure and evanescent field distribution, the grating can distinguish the direction of the magnetic field within a range of 0°–30°. The maximum directional sensitivity reaches 1.17 nm/deg, five times higher than that of conventional grating magnetic field sensors. Furthermore, the reduction in fiber diameter enhances the sensitivity of the cladding modes to the surrounding refractive index (SRI). The order of the cladding modes corresponding to the grating’s resonance dip also decreases as the fiber diameter reduces. When the fundamental mode couples with the lowest-order cladding mode (${\mathrm{LP}}_{0,2}$) near the dispersion turning point (DTP), the grating’s sensitivity to the SRI is significantly improved. Thanks to the excellent magneto-optical properties of the magnetic fluid, the sensor achieves a magnetic field intensity sensitivity of 10.48 nm/mT within the range of 0–20 mT, which is one order of magnitude higher than that of conventional grating magnetic field sensors. The proposed HFG magnetic field sensor offers vector magnetic field measurement capability while maintaining high sensitivity, low insertion loss, and low PDL, showcasing broad application potential.

The geometry of the optical fiber structure can be modified through material dissolution by immersing the fiber in a HF solution. The dissolution rate and changes in the waveguide structure of an SCF were experimentally investigated as a function of chemical etching time in a 10% HF solution at room temperature. The cladding diameter of the SCF is 150 μm, with each core having a diameter of 9 μm and a core spacing of 41 μm. The core and cladding materials in the SCF are different. As the SCF is immersed in HF, the cladding diameter gradually decreases with increased etching time. When the SCF is etched to the outer cores, due to the different dissolution rates of the core and cladding materials in HF, the geometric structure of the SCF changes, forming an asymmetric structure. This results in the generation of a strong asymmetric evanescent field as the fiber diameter decreases. Figure 1 shows the variation in SCF diameter in 10% HF as a function of etching time, with an etching rate of approximately 11.87 μm/h. The inset in Fig. 1 shows the typical cross-sectional structure of the SCF at different stages of the corrosion process. Based on the corrosion time, the SCF can be classified into four states (Fig. 1): outer core intact [panels (A1) and (A2)], partial corrosion [panel (A3)], complete dissolution [panel (A4)], and hexagonal structure formation [panel (A5)]. Compared with other asymmetrically structured devices, such as side-polished fibers, misaligned spliced fibers, and U-shaped fibers, it offers advantages in terms of simpler fabrication processes, lower insertion loss, and higher repeatability.

To help design the HFG, we used the Radio Frequency (RF) module in commercial software (COMSOL Physics) to numerically simulate the modal and periodic parameters of the SCF using the finite element method (FEM). The phase-matching condition determines the resonance wavelength of the HFG: $${\lambda }_{\mathrm{res}}=(n_{\mathrm{co}}^{\mathrm{eff}}-n_{\mathrm{cl},m}^{\mathrm{eff}})\mathrm{\Lambda },$$(1)where ${\lambda }_{\mathrm{res}}$ is the resonance wavelength, $n_{\mathrm{co}}^{\mathrm{eff}}$ and $n_{\mathrm{cl},m}^{\mathrm{eff}}$ are the effective refractive index of the core mode and $m$th cladding, respectively, and $\mathrm{\Lambda }$ is the grating period. The surrounding refractive index (SRI) sensitivity of HFG can be deduced from Eq. (1) as follows [26]: $$\frac{\mathrm{d}{\lambda }_{\mathrm{res}}}{\mathrm{d}{n}_{\mathrm{sur}}}={\lambda }_{\mathrm{res}}·\gamma ·{\mathrm{\Gamma }}_{\mathrm{sur}},$$(2)$$\gamma =\frac{\frac{\mathrm{d}{\lambda }_{\mathrm{res}}}{\mathrm{d}\mathrm{\Lambda }}}{n_{\mathrm{co}}^{\mathrm{eff}}-n_{\mathrm{cl},m}^{\mathrm{eff}}},$$(3)$${\mathrm{\Gamma }}_{\mathrm{sur}}=\frac{u_m^2{\lambda }_{\mathrm{res}}^3{n}_{\mathrm{sur}}}{8\pi r_{\mathrm{cl}}^3{n}_{\mathrm{cl}}(n_{\mathrm{co}}^{\mathrm{eff}}-n_{\mathrm{cl},m}^{\mathrm{eff}})(n_{\mathrm{cl}}^2-n_{\mathrm{sur}}^2{)}^{3/2}},$$(4)where $\gamma$ and ${\mathrm{\Gamma }}_{\mathrm{sur}}$ are the waveguide dispersion factor and sensitivity factor to the surrounding refractive index, respectively, $u_m$ is the $m$th root of the first kind of the zeroth-order Bessel function, $r_{\mathrm{cl}}$ is the radius of the cladding, and $n_{\mathrm{cl}}$ is the RI of the cladding. At the DTP, $|\mathrm{d}{\lambda }_{\mathrm{res}}/\mathrm{d}\mathrm{\Lambda }|\to \infty$; thus, $|\gamma |\to \infty$ and the cladding mode obtains the maximum RI sensitivity [26]. Due to the high sensitivity of the HFG operating at the DTP, it is crucial to accurately determine the period of the HFG. Nevertheless, the resonance wavelength of the HFG can be adjusted to the DTP by reducing the fiber diameter. Using the ${\mathrm{LP}}_{0,10}$ cladding mode as an example, Fig. 2(A) [panels (a1)–(d1)] shows the phase matching curve of the HFG with cladding diameters of 150, 140, 70, and 17 μm. When the fiber diameter is 150 μm and the period of the HFG is 440 μm, the HFG operates near the DTP, with double resonance dips appearing in the spectra. On one hand, with the decrease of the fiber diameter, the DTP of each cladding mode blue shifted. The DTP is blue shifted because the reduction of the cladding diameter reduces the effective RI of the cladding mode [27]. When the fiber diameter is reduced to 140 μm, the DTP appears for ${\mathrm{LP}}_{0,10}$ mode. On the other hand, the decrease in fiber diameter reduces the number of cladding modes supported by the fiber. The cladding mode of the DTP-HFG changes from the ${\mathrm{LP}}_{0,10}$ mode to the ${\mathrm{LP}}_{02}$ cladding mode when the cladding diameter is reduced to 17 μm. Furthermore, Eq. (4) indicates that reducing the cladding diameter significantly improves the SRI sensitivity of the HFG. We simulated the two-dimensional electric field intensity distribution at the resonance wavelength of 1550 nm for the ${\mathrm{LP}}_{04}$ cladding mode with a fiber diameter of 70 μm and the ${\mathrm{LP}}_{02}$ cladding mode with a fiber diameter of 17 μm, as shown in Figs. 2(B) and (C). Regions 1, 2, 3, and 4 represent the interface between the cladding and the MF. As the fiber diameter decreases, significant electric field enhancement occurs in these regions. The longitudinal electric field distribution of the sensing structure was further analyzed. From the one-dimensional electric field distribution diagram in Fig. 2(D), it is evident that as the fiber diameter decreases, the evanescent field at the interface between the cladding and the MF is enhanced. Therefore, maximum sensitivity can be achieved by coupling the fundamental mode with the lowest-order cladding mode (${\mathrm{LP}}_{02}$) operating at the DTP in the HFG with a smaller cladding diameter.

Based on simulation results, a series of SCF-HLPGs with grating periods ranging from 430 to 480 μm, with an interval of 10 μm, was inscribed. The detailed fabrication processes and parameters are provided in Appendix A. Initially, the SCF-HLPG does not operate near the DTP. To address this, we applied HF etching to reduce the cladding diameter and adjust the resonance wavelength to align with the DTP, as indicated by the simulation analysis. The SCF-HLPG with a period of 440 μm was immersed in a 10% HF solution for corrosion experiments. The transmission spectra of HFGs with the reduction of the cladding diameter are shown in Fig. 3(A). As the cladding diameter gradually decreased, resonance dip A exhibited a red shift. Subsequently, resonance dip B of the same-order cladding mode appeared and exhibited a blue shift as the cladding diameter decreased. Eventually, the two resonance dips merged into a single dip. The reduction in fiber cladding diameter caused the DTP of the ${\mathrm{LP}}_{0,10}$ mode to blue shift, consistent with the simulation results. As etching time increased, the resonance dip of the ${\mathrm{LP}}_{0,10}$ mode gradually disappeared, and a lower-order cladding mode appeared. This process continued until the lowest-order cladding mode became prominent. The transmission spectra were recorded when the two resonance dips of each cladding mode merged into a single dip, as shown in Fig. 3(B). Figure 3(C) shows the relationship between cladding diameter and mode order, with the mode pattern of ${\mathrm{LP}}_{0,2}$ shown in the insets. The order of the grating mode gradually decreased with increased corrosion time. The SRI characteristics of HFGs with the same grating period of 440 μm but different cladding diameters were experimentally investigated, as shown in Appendix A.

MF is a stable, gel-like liquid in which magnetic nanoparticles are uniformly dispersed within a carrier fluid and coated with surfactants. The tunable RI characteristics of MF can be described by [28] $$n_{\mathrm{MF}}(H,T)=(n_s-n_0)\left[\coth \right(\alpha \frac{H-H_{c,n}}{T})-\frac{T}{\alpha (H-H_{c,n)}}]n_0,$$(5)where $n_{\mathrm{MF}}(H,T)$ is the RI of the MF under the condition that the magnetic field intensity is $H$ and the temperature is $T$. $H_{c,n}$ is the critical field strength, $n_0$ is the RI of the MF under magnetic field intensity lower than $H_{c,n}$, $n_s$ is the saturated value of the RI of the MF, and $\alpha$ is the fitting parameter. The magnetic nanoparticles in the MF transition from a disordered state to an ordered chain structure under the influence of an external magnetic field. This structure changes in response to variations in both the intensity and direction of the external magnetic field. At a stable temperature, when the strength of the external magnetic field exceeds a critical value, the RI of the MF increases with the intensity of the magnetic field.

Figure 4 shows the schematic diagram of the transverse profile of the HFGs immersed in MF. In the absence of an external magnetic field, the magnetic nanoparticles are disordered and freely distributed around the optical fiber, as illustrated in Fig. 4(A) [29]. When an external magnetic field is applied, the magnetic nanoparticles align into an ordered chain structure along the direction of the magnetic field. They accumulate near the surface of the fiber parallel to the magnetic field direction, while remaining sparse near the surface perpendicular to the magnetic field, as shown in Figs. 4(B) and 4(C). The non-uniform distribution of nanoparticles around the fiber results in a non-uniform RI distribution outside the fiber and introduces geometric asymmetry in the fiber structure, altering the cladding mode field distribution of the HFG. Consequently, the resonance wavelength shift of the HFG cladding mode strongly depends on the intensity and direction of the external magnetic field.

The HFG with the ${\mathrm{LP}}_{02}$ cladding mode was selected for vector magnetic field sensing. (The experimental setup for magnetic field sensing is shown in Appendix B.) In the absence of a magnetic field, the nanoparticles surrounding the fiber are uniformly distributed, resulting in a consistent RI around the fiber and maintaining a stable cladding mode field distribution. When a magnetic field is applied, the nanoparticles align along the field direction, causing the RI around the fiber to become non-uniform. This change alters the cladding mode field distribution and leads to a shift in the grating’s resonance wavelength.

The initial orientation of the magnetic field is defined as $\theta =0°$, parallel to the cross-section of the HFG and directed to the right. The magnetic response of the HFG with the ${\mathrm{LP}}_{02}$ mode was measured as the magnetic field intensity increased, with the magnetic field oriented at $\theta =0°$, 15°, and 30°. Figures 5(A)–5(C) show the transmission response to magnetic field intensity for the three different orientations. The saturation of the MF occurs at approximately 20.7 mT. As the magnetic field intensity increased, the two resonance dips of the HFG shifted in opposite directions. Once the MF reached saturation, the resonance dips no longer shifted. The experimental results in Figs. 5(A)–5(C) show that the maximum wavelength shift of the resonance dip was greatest at $\theta =30°$, while it was smallest at $\theta =0°$. The maximum wavelength shifts of the resonance dips increased gradually as $\theta$ increased from 0° to 30°. The variation of wavelength separation (VWS) of two resonance dips corresponding to the cladding mode was adopted to evaluate the magnetic field sensing characteristics of HFGs. Figure 5(D) presents the relationship between VWS and magnetic intensity for the different orientations ($\theta =0°$, 15°, and 30°). The magnetic field sensitivity increased gradually as $\theta$ increased from 0° to 30°. When the magnetic field orientation was 0°, 15°, and 30°, the magnetic field sensitivities were 2.4 nm/mT, 4.4 nm/mT, and 10.48 nm/mT in the range of 0–20.7 mT, respectively. In this experiment, when the magnetic intensity is 20.7 mT, the magnetic fluid reaches the saturated state. At this moment, as the magnetic field intensity continues to increase, the RI of the magnetic fluid no longer increases significantly. Therefore, this magnetic field sensor has high sensitivity within the range of 0 to 20.7 mT, but its sensitivity is lower within the range of 20.7 to 30.2 mT.

In addition, the HFG with the ${\mathrm{LP}}_{02}$ mode at different magnetic field orientations at a constant magnetic field intensity was experimentally investigated. The orientation of the magnetic field was varied from 0° to 60° in 10° increments under a magnetic field intensity of 20.7 mT. As shown in Fig. 5(E), the two resonance dips of the ${\mathrm{LP}}_{02}$ cladding mode shifted in opposite directions, and the VWS of the two resonance dips increased as the orientation $\theta$ changed from 0° to 30°. Conversely, as shown in Fig. 5(F), the two resonance dips of the ${\mathrm{LP}}_{02}$ cladding mode shifted in opposite directions, and the VWS of the two resonance dips decreased as the orientation $\theta$ changed from 30° to 60°.

Figure 5(G) shows the relationship between the VWS of the HFG and different magnetic field orientations in the polar coordinate system, with the magnetic field intensity fixed at 11.4 and 20.7 mT, respectively. The orientation of the magnetic field was varied from 0° to 360° in 10° increments. The VWS increased as the orientation $\theta$ changed from 0° to 30°, and decreased as the orientation $\theta$ changed from 30° to 60°. This phenomenon repeated with a period of 60° over the range of 0° to 360°. When the magnetic field intensity was fixed at 11.4 mT, the maximum VWS of the sensor was 16 nm. When the magnetic field intensity was fixed at 20.7 mT, the maximum wavelength separation of the sensor was 35.1 nm. The maximum orientation sensitivity was 1.17 nm/deg. Considering the 0.02 nm resolution of the OSA, the minimum detectable magnetic field rotation angle is approximately 0.017°.

When no magnetic field is applied, ferromagnetic nanoparticles are uniformly dispersed around the optical fiber, and the grating is surrounded by a liquid with a uniform RI. After applying a magnetic field, the nanoparticles near the surface of the optical fiber tend to align more readily in the direction parallel to the external magnetic field rather than perpendicular to it. This alignment leads to a non-uniform distribution of the external refractive index, which changes the cladding mode field distribution of the optical fiber and thus causes different wavelength shifts. This explains why the sensitivity of the grating varies with different magnetic field directions [30]. When the magnetic field direction is 30°, the magnetic nano-chain clusters are mainly distributed on both sides of the sensor. The contact area between the grating and the magnetic nano-chain clusters is the largest, the concentration of the magnetic fluid is the highest, the refractive index of the external environment of the grating reaches its maximum value, and thus the largest wavelength shift is induced. When the magnetic field direction is 0°, the contact area between the grating and the magnetic nano-chain clusters is the smallest, the concentration of the magnetic fluid is the lowest, the refractive index of the external environment of the grating reaches its minimum value, and thus the smallest wavelength shift is induced. Therefore, as the orientation angle $\theta$ changes from 0° to 30°, the vector magnetic sensitivity increases, while as the orientation angle $\theta$ changes from 30° to 60°, the vector magnetic sensitivity decreases. Since the hexagonal grating structure still has a 30° symmetry, this phenomenon repeats with a period of 60° in the range of 0° to 360°. These results suggest that, compared to circular waveguide gratings, magnetic field sensors based on HFGs hold significant advantages for vector magnetic field sensing applications. Considering the temperature cross-sensitivity of the sensor, we studied the temperature characteristics of the sensor. Since the capillary is filled with MF, the temperature characteristics were measured by immersing the sensor into the heated water bath. In the temperature range of 25°C–85°C, the temperature sensitivity was measured to be $-0.94\mathrm{nm}/°\mathrm{C}$.

We prepared three samples (S1, S2, and S3) and tested the relationship between the VWS of the samples and the magnetic intensity when the magnetic orientation was 30°. As shown in Fig. 6(A), after linear fitting, the magnetic intensity sensitivities of S1, S2, and S3 were 10.80 nm/mT, 10.84 nm/mT, and 10.55 nm/mT, respectively. Meanwhile, we also tested the relationship between the VWS of the samples and the magnetic orientation when the magnetic intensity was 20.7 mT. As shown in Fig. 6(B), after linear fitting, the magnetic orientation sensitivities of S1, S2, and S3 were 1.17 nm/deg, 1.13 nm/deg, and 1.16 nm/deg, respectively. Therefore, this sensor has good repeatability in terms of magnetic intensity and orientation sensitivities.

In this study, we successfully developed a vector magnetic field sensor based on HFGs packaged with MF, demonstrating its ability to measure both the intensity and direction of magnetic fields with high sensitivity. This represents a significant improvement over traditional magnetic field sensors that are limited to measuring magnetic field intensity alone. The HFG-based sensor benefits from the combination of the asymmetric evanescent field of the hexagonal structure and the tunable RI of the MF, which allows for precise vector measurements.

The sensitivity of the HFG sensor reached 10.48 nm/mT for magnetic field intensities within the range of 0–20.7 mT, a value more than 10 times higher than that of conventional fiber sensors. This impressive sensitivity can be attributed to the DTP achieved through HF etching of the cladding, which enhances the interaction between the evanescent field and the external environment. The effective interaction between the HFG and the magnetic fluid is maximized, resulting in higher sensitivity.

The sensor’s orientation sensitivity was also noteworthy, with a maximum value of 1.17 nm/deg in the range of 0°–30°. The asymmetric evanescent field introduced by the hexagonal fiber structure plays a critical role in enabling the detection of magnetic field direction, providing a compact and highly efficient alternative to conventional magnetic field detection systems that rely on more complex structures.

Although various types of sensor devices have been proven effective for vector magnetic field detection, many still offer considerable potential for performance improvement. Table 1 summarizes the characteristics of different fiber vector magnetic field sensors. When compared to other vector magnetic field sensors, such as those based on U-bent fibers and side-polished fibers, the HFG sensor exhibits superior performance in terms of both insertion loss and ease of fabrication. While SPR-based sensors are known for their high sensitivity, their complex manufacturing process, which involves nanometer-scale metal coatings, poses significant challenges and increases costs. Similarly, sensors based on tilted fiber gratings or polarization-maintaining fiber (PMF) LPFGs often require polarized light sources and additional components to achieve vector magnetic field measurement due to asymmetric refractive index modulation or birefringent fibers, which increases PDL and complicates the sensing system.Table 1.

Comparison on the Characteristics of Different Types of Vector Fiber Magnetic Field Sensors

The proposed HFG sensor holds significant potential for applications in various fields, including aerospace, military, and medical diagnostics, where precise vector magnetic field measurements are crucial. Its high sensitivity and orientation detection capabilities make it well-suited for environments where both the intensity and direction of the magnetic field need to be monitored. Additionally, the sensor’s robustness and resistance to electromagnetic interference make it particularly valuable in harsh environments where traditional sensors might fail.

Despite the promising results, there are some limitations that must be addressed in future research. One area of improvement lies in the complexity of the fabrication process, particularly in ensuring consistent etching to achieve the desired cladding diameter. Additionally, while the sensor performed well within the tested range, further experiments are needed to explore its performance at higher magnetic field intensities and in more varied environmental conditions.