1. Introduction

Now, tiny magnetic structures [1] and magnetic materials [2] are widely used in precision instruments [3], biomedical [4], and fundamental research [5], such as localization and tracking of micro-magnetic robots working in subcutaneous tissues [6], non-destructive detection of weak-magnetic structures in precision instruments [7], and detection of unknown magnetic anomalies in complex environments [8]. However, the blurring of boundary information due to the dispersion of the magnetic field makes it difficult to locate and determine the accurate size information of magnetic anomalies [9,10]. In particular, miniature targets with weak magnetic properties place high demands on sensors’ sensitivity and spatial resolution [11]. Therefore, various applications require effective magnetic detection techniques for spatial anomaly, including not only the 3D dimensions [12] and spatial locations of magnetic anomalies [13] but also the unique magnetic moment directions of magnetic materials [14].

For conventional magnetic detection, the detection of magnetic induction intensity [15] is usually performed. The observed value of the dispersion magnetic field in space is obtained from the magnetic induction intensity. The observed values obtained under different observation coordinate systems are different, and the description of the information is incomplete. In contrast, magnetic tensor imaging [16–18], a new magnetic detection technique, can achieve precise boundary identification of magnetic targets, obtaining higher resolution, accuracy, and clearer detailed features. However, the current magnetic tensor detection is still in the sensing stage [19,20], based on the cross-detector model to approximate the gradient tensor by differential methods, which is widely used in positioning and navigation [21]. Most imaging methods are series point scanning types, with long imaging times and poor data quality. As a result, rapid imaging of the magnetic tensor and interpretation based on magnetic tensor data still present significant difficulties.

Nitrogen vacancy (NV) color center is a quantum sensor [22,23]. Control and detection of electron spin direction by optical means and interpretation of physical quantities concerning the population distribution of electron spin direction using the Hamiltonian equation [24]. It is widely used for multi-physics field detection, such as magnetic [25,26], temperature [27], electric [28], and stress [29]. The diamond's atomic arrangement structure makes NV have vector imaging capability. In comparison to other magnetic detection methods, such as superconducting quantum interference device (SQUID), Lorentz transmission electron microscopy (L-TEM), and probe-based scanning imaging. The advantages of NV magnetic imaging include room-temperature operation, simple detection system, and fast wide-field imaging capability [30]. In addition, the atomic-scale detector size of NV provides high spatial resolution, making it ideal for miniature magnetic structure measurement.

In this paper, we carry out a series of works on the methodology of magnetic tensor imaging and the interpretation of data. First, this paper establishes an NV quantum magnetic detection system for realizing fast vector magnetic imaging of samples. Secondly, this paper divides the space into finite cells to achieve spatial discretization and establish a linear model of the observed data and spatial magnetization intensity. We propose a transformed magnetic tensor imaging method using this model. The Euler deconvolution method [31] was used to interpret magnetic tensor data to achieve the application of 3D boundary identification of magnetic target [32]. Due to the magnetic target's unique N-S pole angle, we built a model for the observation angle transition and a model for the fast magnetic moment judgment to assist the interpretation. To verify the capability of the method, we performed tensor imaging and tensor imaging-based 3D boundary recognition to a 400 µm × 400 µm × 400 µm square permanent magnet. The spatial resolution of the three directions of the experimental results surface is 10 µm, the model accuracy reaches 90.1%, the magnetic declination error is 4.2°, and the magnetic inclination error is 0°, which fully verifies the feasibility of the transformed tensor imaging method and the tensor imaging in the application of boundary recognition.

2. Experimental principles

2.1 NV vector magnetic imaging

The NV color center in diamond is a quantum sensor and consists of a nitrogen (N) atom instead of a carbon (C) atom and a nearby vacancy (V). When the vacancy captures a free electron, it constitutes the $N{V^ - }$, forming an unstable electronic layer structure. In the spin system S = 1 [33], the energy level structure is a triplet state, as shown in Fig. 1(a). ³A₂ is the ground state triplet, ³E is the excited state triplet, and ¹A₁, ¹E is the sub-stable state. The ground state triplet includes states |±1〉 and state |0〉, which are flipped using microwave resonance with zero-field frequency D = 2.87 GHz.

 

Fig. 1. (a) NV energy level structure. (b) NV Axis and Establishment of Coordinate System. (c) Single-pixel point optical detection magnetic resonance (ODMR) data.

Download Full Size | PDF

Optical detection of magnetic resonance (ODMR) is one of the most classical methods for NV detection [34]. For the NV, fluorescence reveals the layout number between states |±1〉 and state |0〉, and the relationship between fluorescence and microwave frequency at this point is obtained by microwave sweeping to determine the microwave resonant frequency. When there is an applied magnetic field, the degenerate states |±1〉 split into state |+1〉, and state |-1〉, and the resonance peaks are separated horizontally due to the Zeeman effect. The relationship between the magnetic intensity and the frequency shift of the microwave resonance frequency can be obtained from the Hamiltonian Eq. (1). µ_B is the Bohr magneton, g is the Langde factor, and S_x,y,z is the projection of the electron spin in the x, y, z plane.

The atomic structure of a diamond is a regular frontal body structure. NV has only four axes, as in Fig. 1(b). When E << D, the transformation of Eq. (1) into Eq. (2) [22], where ${B_i}$ is the projection of magnetic intensity in one axis of NV (i = 1,2,3,4). f_1i and f_2i are the symmetrical resonant peak frequency after frequency shift.

Based on Eq. (2), we can obtain the amount of magnetic intensity projected in the axial direction for each NV. Solving the four axially oriented magnetic projections to obtain triaxial vector information. Establish a coordinate system according to the NV axis as in Fig. 1(b). In the established Cartesian coordinate system, the four axes of NV are $\overrightarrow {{N}{V}_{1}} {= }\left( {{0,0,1}} \right)$, $\overrightarrow {{N}{V}_{2}} {= }\left( {{sin\theta ,0,cos\theta }} \right)$, $\overrightarrow {{N}{V}_{3}} { = }\left( {{-}\displaystyle{{1} \over {2}}{sin\theta ,}\displaystyle{{\sqrt {3} } \over {2}}{sin\theta ,cos\theta }} \right)$, $\overrightarrow {{N}{V}_{4}} { = }\left( {\displaystyle{{1} \over {2}}{sin\theta ,-}\displaystyle{{\sqrt {3} } \over {2}}{sin\theta ,cos\theta }} \right)$, where $\mathrm{\theta }$ is the interior angle of the positive tetrahedron, $\mathrm{\theta }$ = 109.47°. The ODMR data is shown in Fig. 1(c), and the vector information is obtained by solving for the magnetic field strength in each axial direction and mapping it to the axial direction of the coordinate system. The diamond contains many NV color centers, so the NV color centers can be used as a surface array detector to quickly obtain imaging data.

2.2 Magnetic tensor data acquisition

The magnetic induction B is a physical quantity that describes the strength and direction of the magnetic field, and the magnetization M is a physical quantity that describes the strength of the magnetism of a magnetic body. As shown in Eqs. (3) and (4), where H is the magnetic field in the medium, µ₀ is the vacuum permeability, χ is the magnetic induction strength, and in anisotropic matter, the magnetization is a second-order tensor.

The magnetic tensor is the physical quantity describing the anisotropy of the magnetization strength in space. As shown in Fig. 2(a), the magnetic tensor at a single point describes the spatial rate of change of the vector data in three directions. The tensor contains nine components, which can be expressed in the matrix form in Eq. (5) [18]. The following equation is satisfied when the magnetic field B generated by the target magnetic source is a passive field:

 

Fig. 2. (a) The analytic form of the tensor matrix G is shown schematically. The spatial variation of the vector in the direction of the three orthogonal axes has nine components. (b) Schematic diagram of discrete spatial divisions and the relationship of observations to spatial units.

Download Full Size | PDF

$\nabla $ is known as the Hamiltonian operator. Equation (7) is derived from Eq. (6). So only five independent variables need to be detected to obtain the full tensor matrix G.

Magnetic tensor data is high-dimensional information that is difficult to detect directly, and magnetic tensor imaging is a two-dimensional magnetic tensor detection technique. The first step is to obtain the vector magnetic imaging based on the NV. Next, tensor imaging method is developed based on detection data.

Using the equation for density versus volume in space in Eq. (8) and the Poisson distribution equations for gravity and magnetic field, one can obtain Eq. (9) for the magnetization intensity versus magnetic potential data in space. The position of the space element is (ξ, η, ς), and the density is ρ (ξ, η, ς). v is the volume, r is the inter-cell distance from the observation point, and G is the gravitational constant.

M_x, M_y, M_z is the magnetization triple component. The S-N pole direction is taken as the positive direction. We calibrate the angle between the direction of the magnetic moment and the z-axis as the magnetic declination i, and the angle between the projection of the direction of the magnetic moment in the x-y plane and the direction of the y-axis as the magnetic tilt angle $\delta $, abbreviated as (i, $\delta $). The magnetic potential data includes the total magnetic anomaly, the magnetic vector, and tensor data. Using the idea of calculus, the space is divided into finite units, and the integral process is converted into a finite cumulative process to realize the discretization of the space, as shown in Fig. 2(b). We divide the continuous space into a finite number of discrete cells. This completes modeling the linear relationship between the magnetic potential data and the magnetization intensity of the discrete space. See the Supplement 1 for the complete derivation process of formulas.

The linear relationship can be abbreviated as Eq. (10), where d is the observed data (it represents a certain type of magnetic potential data U that we observe), and A is the operator (it represents the linear relationship between M and U in Eq. (9)).

For a space containing magnetic anomalies, both the vector and tensor data satisfy Eq. (10). When the observed data are vector, as shown in Eq. (11), and when the observed data are tensor, as shown in Eq. (12), where i, j is the different directions respectively, as shown in Fig. 2(b).

Therefore, as long as the spatial magnetization intensity M does not change, the tensor data can be calculated directly by Eq. (13). It is worth noting that the orthogonal arithmetic A is not a constant and is affected by the direction of the magnetic moment of the magnetic anomaly. The operator contains two parametric variables: the magnetic declination and the magnetic inclination. Therefore, the correct solutions of the magnetic declination and inclination angles must be calculated from the magnetic vectors in the three directions to obtain the correct orthogonal operator. Therefore, the vector analysis determines the tensor data's accuracy.

We can obtain five independent variables based on the vector data through Eq. (13) and the complete nine components through Eq. (7).

2.3 Interpretation of tensor data and target boundary identification methods

To better explain our approach, we performed simulation experiments. The magnetic anomaly size is shown in Fig. 3(a). The magnetic declination is 90°, the magnetic inclination is 0°. Figure. 3(b) shows a magnetic target's total magnetic anomaly observation. Figures 3(c)-(e) show the magnetic vector data. The magnetic tensor data is obtained based on Eq. (13), as shown in Figs. 3(f)-(k), which are ${B_{xx}}$, ${B_{xy}}$, ${B_{xz}}$, ${B_{yy}}$, ${B_{yz}}$ and ${B_{zz}}$. Magnetic tensor imaging reveals the rate of magnetic change in space. The zero value of ${B_{xx}}\; $ can effectively circle the boundary of the magnetic anomaly in the y-direction, the zero value of ${B_{yy}}$ can effectively circle the boundary of the magnetic anomaly in the x-direction boundary, the polar point of ${B_{xy}}$ corresponds to the corner point of the sample, the polar value of ${B_{xz}}$ indicates the boundary in the y-direction, and the polar value of ${B_{yz}}$ indicates the boundary in the x-direction. Magnetic tensor-based imaging methods have several methods for identifying two-dimensional boundaries, including horizontal resolved signal amplitudes, total resolved signal amplitudes, and tilt angle methods. Tensor images can directly determine the boundary information of the magnetic source to a certain extent. But the boundary information obtained based on one image is not unique and difficult to obtain 3D information. This paper uses the Euler deconvolution to integrate and calculate nine components to obtain 3D boundary information.

 

Fig. 3. (a) Simulation of the observed total magnetic anomaly. (b) Magnetic anomaly location in space and sample size. (c) Simulated ${B_x}$ imaging data. (d) Simulation of ${B_y}$ imaging data. (e) Simulation of ${B_z}$ imaging data. (f)-(k) magnetic tensor imaging data ${B_{xx}}$, ${B_{xy}}$, ${B_{xz}}$, ${B_{yy}}$, ${B_{yz}}$, ${B_{zz}}$ generated based on spatial meshing method with simulated vector data.

Download Full Size | PDF

The Euler equation for the magnetic position data U is shown in Eq. (14) [31]. x, y, z is the sample position, ${x_0}$, ${y_0}$, ${z_0}$ is the observation point position.

Derive Eq. (15) in the x, y, and z directions, respectively:

Organized as Eq. (16), r is the observation position, the cube construction index is N = 3, G is the tensor matrix (9 components), and B is the vector data.

The corner points position of the magnetic anomaly tends to a value of 0, in some of the tensor components. Thus, the magnetic tensor matrix G and its inverse matrix at the location of the corner point become unstable, leading to an increase in the absolute value of the solution result and the appearance of extreme points. Figure 4(a)-(c) show the three-direction localization imaging data in the x, y, and z directions (x, y, z are the results of the solution by Eq. (15). In the case of (90°, 0°), we can mark the corner points by the extreme points of the three images. The black line in the figure marks the boundary case under the 2D plane. We obtain the three-dimensional information through the z-direction in Fig. 4(c). Figure 4(g) shows the x-z side view, and for the gridded space, the spatial cells contribute differently for different observation positions. Therefore, for magnetic anomalies in space, the Euler deconvolution result of the observed edge points can be approximated as the position of the upper-left point of the sample. In contrast, the Euler deconvolution result of the observed point at the center of the sample is the result of the localization of the center point of the sample. From this, we can obtain information about the center of the sample and the dimensions in the vertical direction. As shown in Fig. 4(d), we tested the data at the blue line of Fig. 4(c) for different thicknesses. The results of extracting the center point and edge point are shown in Fig. 4(e), the red line is the result of the z-direction positioning of the center point under different thicknesses, and the black line is the result of z-direction positioning of the edge point under different thicknesses. According to Eq. (17), the longitudinal dimension D is calculated, Z₁ is the positioning result obtained for the edge point (Position B), and Z₂ is the center point (Position A). The thickness results were solved under different thicknesses and the same magnetization are shown as the red line in Fig. 4(f). The thickness results solved under different magnetization strengths and the same thickness are shown as the black line. We find that the results solved by our method are approximately linear with the thickness variation and are not affected by the magnetization strength. However, the distance from the observation plane to the sample affects our results, but the sample is very close to the observation plane, which leads to a small bias in the results.

 

Fig. 4. (a)-(c)Euler deconvolution x, y, ${B_{z1}}$z direction result based on tensor information. (d) z-direction localization results at x = 16 for different thicknesses. (e) Center point z-direction localization results and edge point z-direction localization results at different thicknesses. (f) z-thickness solution results at different thicknesses with the same magnetization intensity, and z-thickness solution results at the same thicknesses with different magnetization intensities. (g) Schematic diagram of the thickness solution method based on the Euler deconvolution method. (h) 3D boundary identification results of the solved magnetic anomaly.

Download Full Size | PDF

Based on the methods, we returned the boundary information of the sample in 3D space, as shown in Fig. 4(h), the 2D information has the result of perfect identification, while the z-direction has the result of small bias due to the distance between the observation surface and the sample, and the result accuracy reaches 96%.

3. Experimental system

Our experimental device consists of an optical module and a synchronous circuit acquisition module, as shown in Fig. 5. The optical module is simple. It consists of a 532 nm laser (Changchun New Industries Optoelectronics Technology, MGL-III-532-1W) and some simple lenses and reflectors. The laser emits a green laser through the excitation optical path on the NV. Fluorescence is collected by an avalanche photodiode (APD) and a charge-coupled device (CCD) camera through the con-focal fluorescence collection system. Two magnets are symmetrically placed around the NV center to build a uniform bias magnetic field. The CCD camera collects the needed imaging data, and the APD monitors fluorescence changes and system noise. The ODMR acquisition requires a unique microwave scanning method. Microwave signals are provided by a microwave source (Rohde & Schwarz, SMA 100 A). The system sets the frequency of the microwave scanning step to 300 MHz and the output power to 30 dBm. The CCD camera must accurately capture the imaging data for each microwave frequency. An arbitrary waveform generator (AWG, Tektronix, AWG 5204) controls the synchronization circuit module. The NV (3 mm × 3 mm × 0.5 mm, type Ib, Element VI) used in this experiment was produced by a high-pressure, high-temperature method. It has a high initial nitrogen concentration (∼50 ppm), and the NV concentration of about 3 ppm. Below the NV is the divided sample space, the size of the sample space in this paper is 1.2 mm × 1.2 mm × 1.2 mm.

 

Fig. 5. Experimental system, optical path map, and control module.

Download Full Size | PDF

4. Experimental results and discussion

To verify the above method, we started a practical test experiment. We chose a square permanent magnet sample (N35) of 400 µm × 400 µm × 400 µm size. Our method contains parameters about the direction of the magnetic moment. So we placed the samples at an angle to demonstrate the feasibility of the method. The details of sample placement are provided in Supplement 1. The magnetic inclination is 60°, magnetic declination is 0°, fixed and encapsulated in a mold. The space above the NV is divided into 120 × 120 × 120 regular cubes. The individual cube size is 10 µm × 10 µm × 10 µm, and the magnetic target is imaged in vector based on the NV wide-field magnetic imaging method. The tensor imaging information is obtained by our proposed linear variation method. As the field is passive, the full tensor matrix is obtained by obtaining five independent components. Finally, the 3D boundary information of the sample is returned by the Euler deconvolution solution and 3D spatial reduction techniques.

Based on the experimental system described above, we used the ODMR method to obtain wide-field magnetic imaging. Based on the coordinate system in Fig. 1(b), we solved the three-component magnetic vector imaging of B_x, B_y and B_z (the unit is Gs), as shown in Fig. 6(a)-(c). The magnetic detection sensitivity is 1 nT/Hz^1/2 which is solved according to Eq. (18), h is the Planck constant, R is the experimental photon rate, C is the contrast of fluorescence intensity change, w is the half-height width, and P is the line-width correction factor related to the spectral line shape [22].

 

Fig. 6. (a)-(c) magnetic vector observation data ${B_x}$, ${B_y}$, ${B_z}$. (d)-(f) magnetic vector observation data ${B_{x1}}$, ${B_{y1}}$,   under the transformed coordinate system (90°,0°)

Download Full Size | PDF

In our tensor imaging approach, the generation of the transformation matrix Aw must include the magnetic target's magnetic moment directions, i.e., magnetic declination and inclination. For this reason, we have developed a feasible and fast method for discriminating the magnetic moment direction. Based on the established grid space, the vector information is approximated as two samples with magnetic moment directions perpendicular to the observation plane and in opposite directions, and the optimal solution is solved by regular inversion. According to this method, the magnetic declination inclination direction of the point is approximated as (64.2°, 0°), and the complete process of the solution is detailed in Supplement 1.

At this point, we have been able to obtain operator A, however, the solution of the Euler deconvolution is largely affected by the magnetic declination and inclination. For this reason, we use a linear transformation method similar to the tensor imaging method to convert the angle of the observation plane and obtain the observation vector data at (90°, 0°), as shown in Fig. 6(d)-(f). The conversion process is detailed in Supplement 1.

Tensor data do not change with transformations of the observation coordinate system. The transformation of the observation plane does not affect the spatial state and position of the object in the absolute coordinate system. After optimizing the magnetic moment direction according to the linear model, we transformed the vector data according to Eq. (13). As shown in Fig. 7, the independent components B_xx, B_xy, B_xz, B_yy, B_yz and B_zz are the rates of change in different directions in space. The unit of the magnetic tensor is Gs/unit, and the unit depends on the spatial unit division of the experimental system, in this paper 1 unit is equal to 10 µm. The results agree with the simulation data, and the transform tensor imaging method is fully justified. Based on the tensor images, we can observe the 2D boundary information of the magnetic anomalies and observe the spatial magnetic field variations and hidden weak magnetic anomalies.

 

Fig. 7. (a) Obtained under the transformed coordinate system. Tensor data ${B_{xx}}$. (b) Tensor data ${B_{xy}}$ obtained under the transformed coordinate system. (c) Tensor data ${B_{yz}}$ obtained under the transformed coordinate system. (d) Tensor data ${B_{xz}}$ obtained under the transformed coordinate system. (e) Tensor data ${B_{yy}}$ obtained under the transformed coordinate system. (f) Tensor data ${B_{zz}}$ obtained under the transformed coordinate system.

Download Full Size | PDF

The Euler deconvolution is solved for the magnetic potential data in Fig. 6 and Fig. 7 according to Eq. (15) and the twisted solution is filtered out. The three-direction solving imaging results shown in Fig. 8(a)-(c) are consistent with Fig. 5(a)-(c), with localized protrusions at the corner locations, and the localized extrema are solved. Imaging in x, y, and z are consistent with the two-dimensional corner localization, so the corner locations in the three images are averaged and connected to obtain the 2D boundary information. The 3D information is solved according to Eq. (17). Figure. 8(d)-(f) are divided into spatial top view, front view, and right-side view, with the black boundary being the actual sample boundary and the red boundary being the solved result boundary. The standard size of our model is 40, and the dimensional accuracy is 98.5% in the X-direction, 80.5% in the Y-direction, and 82.5% in the Z-direction. According to Eq. (19), the positioning accuracy is 92%, where x₁, y₁, z₁ is the true position of the sample, and x₂, y₂, z₂ is the positioning result. position of the sample and x₂, y₂, z₂ is the positioning result. Figure 8(g) shows the spatial model of the inversion results compared with the real space (blue is the real model, red is the inversion result), and Fig. 8(h) shows the results of the magnetic anomaly location compared with the magnetic moment direction (blue is the real direction, red is the inversion result). Solve for the final model accuracy according to Eq. (20). V_real is the volume of the real model, and V is the volume of the overlap between the computational and real models.

 

Fig. 8. (a) Real data Euler deconvolution x-direction localization imaging results. (b) The measured data Euler deconvolution y-direction localization imaging results. (c) Real data Euler deconvolution z-direction localization imaging results. (d) Top view, x-y plane, standard model boundary in black, solution result boundary in red. (e) Front view, x-z plane, standard model boundary in black, solution result boundary in red. (f) Right side view, y-z plane, standard model boundaries in black, solution result boundaries in red. (g) Modeling of spatial samples and computational results. (h) Angle and positioning results.

Download Full Size | PDF

The solution accuracy is 90.1%, and the angular deviation is 4.2°. The experimental results fully demonstrate the method's feasibility in the unknown situation of miniature magnetic target 3D boundary identification method.

5. Conclusion

In this paper, we show computational imaging methods for higher order tensors. We also implement an application based on tensor data, i.e., 3D boundary identification of magnetic targets. In addition, we develop models for quickly solving for the direction of the magnetic moment and for transforming the observation surface to help interpret the tensor data. But it is currently difficult to detect complex magnetic structure. Complex magnetic structures can have multiple extreme points and the direction of the magnetic moment is difficult to determine. In the future, local differential or moving target detection methods can be tried to accomplish magnetic tensor imaging without relying on additional parameters. Detecting tiny structure in 3D space can help us better understand the properties of matter and the operating state of precision instruments. The work in this paper provides new solutions in the field of microscopic magnetic probing.