阅读说明：本技术 一种nv轴三维空间指向快速测量方法 (NV axis three-dimensional space pointing rapid measurement method ) 是由 陈国彬 杜关祥 杨会 何文豪 顾邦兴 朱少晨 于 2020-06-16 设计创作，主要内容包括：本发明公开一种NV轴三维空间指向快速测量方法,该方法通过调整永磁体与金刚石颗粒样品之间的相对位置得到具有八个分离谱峰的ODMR谱,并通过分析每对谱峰间的频率宽度得到每个NV轴向上投影的磁场强度；利用亥姆霍兹线圈分别对金刚石样品施加沿着空间直角坐标系Z轴正反方向与Y轴正反方向上的静磁场,同时测量其ODMR谱并分析每对谱峰所对应的频率变化量；利用空间坐标转换原理建立起NV坐标系与空间直角坐标系之间的关系,即各NV轴指向角度。本发明只需施加四种不同的磁场矢量,通过两步测量即可实现NV轴三维空间指向的快速测量,相对于现有技术,其需要采集的数据量更少,标定速度更快。(The invention discloses a quick measurement method for three-dimensional space pointing of an NV axis, which obtains an ODMR spectrum with eight separation spectral peaks by adjusting the relative position between a permanent magnet and a diamond particle sample, and obtains the magnetic field intensity projected in each NV axis by analyzing the frequency width between each pair of spectral peaks; respectively applying static magnetic fields in the positive and negative directions of a Z axis and the positive and negative directions of a Y axis along a space rectangular coordinate system to the diamond sample by utilizing Helmholtz coils, simultaneously measuring an ODMR spectrum of the diamond sample and analyzing the frequency variation corresponding to each pair of spectrum peaks; and establishing a relation between the NV coordinate system and a space rectangular coordinate system by using a space coordinate conversion principle, namely, pointing angles of all NV axes. According to the method, only four different magnetic field vectors are applied, and the rapid measurement of the three-dimensional space pointing of the NV axis can be realized through two-step measurement, so that compared with the prior art, the method has the advantages that the data quantity required to be acquired is less, and the calibration speed is higher.)

1. A quick measurement method for three-dimensional space pointing of an NV axis is realized by adopting an NV axial calibration system, and is characterized by comprising the following steps:

obtaining an ODMR spectrum with eight separated spectral peaks, and analyzing the frequency width between each pair of spectral peaks to obtain the magnetic field intensity projected in each NV axial direction;

respectively applying static magnetic fields along the positive and negative directions of a Z axis and the positive and negative directions of a Y axis of a space rectangular coordinate system to the diamond sample, simultaneously measuring an ODMR spectrum of the diamond sample, and analyzing the frequency variation corresponding to each pair of spectrum peaks;

and establishing a relation between the NV coordinate system and the space rectangular coordinate system to obtain the pointing angle of each NV axis.

2. The NV-axis three-dimensional spatial orientation fast measurement method according to claim 1, wherein establishing the relationship between the NV coordinate system and the spatial rectangular coordinate system further comprises:

establishing a conversion model B from an NV coordinate system to an ideal orthogonal coordinate system_o＝K_NoB_NV，B_oIs a component under an ideal orthogonal coordinate system, B_NVIs a component in the NV coordinate system, K_NoA transformation matrix from the NV coordinate system to an ideal orthogonal coordinate system;

obtaining the conversion relation B between the ideal orthogonal coordinate system and the space rectangular coordinate system_o＝K_eoB_e，B_eIs a component under a space rectangular coordinate system, K_eoA transformation matrix from a space rectangular coordinate system to an ideal orthogonal coordinate system;

four different magnetic field vectors respectively applied to the diamond sample are converted into an ideal orthogonal coordinate system to be operated to obtain a conversion matrix K_eoEstablishing a conversion relation B between the NV coordinate system and the space rectangular coordinate system_NV＝K_oNK_eoB_eAnd obtaining the pointing angle of each NV axis.

3. The NV-axis three-dimensional spatial orientation fast measurement method according to claim 2, wherein the coordinate axes in the ideal orthogonal coordinate system are x, y, and z, the coordinate axes in the NV coordinate system are α, β, γ, and β is located in an xOz plane when z coincides with α, an included angle between β, γ, the three axes and the xOy plane is equal to 19.28 ° according to a crystal structure of diamond, and included angles between a projection of the γ axis and the axes in the xOy plane and the x axis are B-120 ° and c-120 °, respectively, so that a conversion model B from the NV coordinate system to the ideal orthogonal coordinate system is obtained_o＝K_NoB_NVWherein, in the step (A),K_oNfor a transformation matrix from an ideal orthogonal coordinate system to an NV coordinate system, transformation matrix K_oNComprises the following steps:

4. the NV axis three-dimensional space pointing fast measurement method according to claim 3, wherein the space rectangular coordinate system XYZ is rotated three times to obtain an ideal orthogonal coordinate system XYZ, which specifically comprises: firstly, a space rectangular coordinate system XYZ rotates anticlockwise around a shaft Z axis

5. The NV-axis three-dimensional spatial orientation fast measurement method according to claim 4, wherein the conversion formula between the spatial rectangular coordinate system XYZ and the ideal orthogonal coordinate system XYZ is:

wherein, K_ω、K_θ、

6. The NV axis three-dimensional spatial orientation rapid measurement method according to claim 5, wherein the static magnetic fields applied to the diamond sample along the positive and negative directions of the Z axis and the positive and negative directions of the Y axis respectively in the spatial rectangular coordinate system are converted into an ideal orthogonal coordinate system for operation to obtain a conversion matrix K_eoThe parameters theta, omega,

converting two magnetic field vectors which are applied to the diamond sample and have the same strength and opposite directions along the Z axis of a space rectangular coordinate system into an ideal coordinate system, and calculating to obtain a value theta omega;

will be aligned with diamondTwo magnetic field vectors which are applied by the sample and have the same strength and opposite directions along the Y axis of the space rectangular coordinate system are converted into an ideal coordinate system, and the magnetic field vectors are obtained through calculationThe value is obtained.

7. The NV axis three-dimensional spatial orientation rapid measurement method according to claim 1, wherein the NV axial calibration system comprises an NV color center magnetic field sensitive portion and a confocal optical path for NV color center fluorescence collection;

in the NV color center field sensitive part: diamond particles are fixed at the tip of the tapered optical fiber; the spiral antenna is connected with the microwave source and used for generating microwave signals; the Helmholtz magnetic field coil is connected with a high-precision current source and used for generating a uniform adjustable direct-current magnetic field; the diamond particles at the tail end of the optical fiber are fixed in the center of the Helmholtz coil and are close to the strongest point of the magnetic field distribution of the antenna; the neodymium iron boron permanent magnet is placed near the diamond particles to provide a fixed magnetic field for splitting the NV color center ODMR spectrum into eight peaks;

in the confocal light path for NV color center fluorescence collection: the pulse signal generator is used for generating control; laser generated by a laser source enters an acousto-optic modulator after being focused by a lens to perform laser pulse time sequence control, divergent light emitted by the acousto-optic modulator is converted into parallel light through the lens and reflected by a dichroic mirror to enter a long working distance objective lens for focusing, and focused light irradiates a diamond through a cylindrical section end of a conical optical fiber; red fluorescent signals generated by the diamond are converted into parallel light through the long working distance objective lens and pass through the dichroic mirror to enter the objective lens for focusing; the focused red fluorescence is received by an avalanche photodiode and converted into an electric signal; the phase-locked amplifier collects and demodulates the electric signal by using the laser microwave secondary modulation signal generated by the pulse signal generator.

Technical Field

The invention relates to the technical field of vector sensor sensitive axis space direction calibration, in particular to a quick NV axis three-dimensional space pointing measurement method.

Background

Diamond is a mineral composed of carbon atoms connected to each other in a tetrahedron bonding manner, and the NV (Nitrogen-Vacancy) color center is a spin defect having a light emitting property in a diamond structure, which is mainly composed of a Nitrogen atom replacing a carbon atom and its neighboring hole. In recent years, it has been studied more and more widely because it has many excellent properties such as light stability, biocompatibility, chemical inertness, long spin coherence, and relaxation time at room temperature.

Diamond NV color center having C_3VSymmetry, which is a crystal axis formed by nitrogen atoms and holes, is called NV axis. According to the atomic structure of diamond, the symmetry axes of all NV color centers in a diamond sample have only four directions, as shown in fig. 1 (a). The ground state of each axial NV color center contains | ms ═ 0>And | ms ═ 1>The spin state of (a) can be measured by an Optical Detection Magnetic Resonance (ODMR) technique. When subjected to an external static magnetic field, the ODMR spectrum at each NV centre in the axial direction produces a zeeman split whose split width is proportional to the projection of the static magnetic field in the NV axis direction. When affected by an external microwave field, the projection intensity of the microwave vector in each NV axis also affects the ODMR peak height of the NV color center in that axis. Thus, the diamond NV color center can be used to measure both vector static magnetic fields and vector microwave fields based on its properties. Compared with the traditional multi-axis vector sensor (a magnetic resistance sensor, a fluxgate sensor, a SQUID and the like), because the angles among four equivalent crystal axes in the NV diamond structure are fixed, the direction error and the position error among the sensitive axes are almost zero. In addition, the micron-scale or even submicron-scale diamond particle sample can also realize high-resolution vector field imaging measurement.

The result of vector measurement using the above characteristics is that, in an NV coordinate system composed of four NV axes, it is necessary to convert the NV coordinate system into measurement data in a spatial rectangular coordinate system for analysis. Because the NV axis direction is consistent with the diamond crystal phase [111] direction, the accurate 111 crystal direction of the block diamond sample with the millimeter level or more can be obtained by utilizing the existing processing technology, and therefore the relation between the NV axis coordinate system and the space rectangular coordinate system in the diamond can be easily established. However, for micron-scale or even nano-scale diamond particles, when the diamond particles are fixed at a certain position in space, the relative direction between the NV axis direction and the rectangular spatial coordinate system is random, and the specific orientation of each NV axis in space cannot be directly determined.

China, with publication number CN109709128A, specially adapted to 2019, 5, 3, discloses a diamond NV axis direction calibration device and method, which are based on the characteristic that the frequency difference of two ODMR spectral peaks corresponding to each NV axis direction of an NV color center diamond linearly changes along with the projection intensity of a static magnetic field in the NV axis direction, and the direction of the NV axis in the diamond is obtained by establishing the conversion relation between the projection of the static magnetic field in an NV coordinate system and the projection in a space rectangular coordinate system, thereby solving the problem that the NV axis direction of micron or nano-scale diamond particles in the space cannot be determined. However, the method needs a large amount of collected data, and the calibration speed is not ideal enough.

Therefore, the invention provides a diamond particle NV axis orientation method in a three-dimensional space to establish a relation between a diamond particle sample NV axis coordinate system and a space rectangular coordinate system based on Zeeman splitting characteristics of different NV axial ODMR spectral peaks under the action of a static magnetic field, so as to solve the problems of the existing calibration method.

Disclosure of Invention

The invention aims to solve the problems in the prior art and provides a rapid measurement method for the three-dimensional space pointing of an NV axis.

In order to achieve the purpose, the invention adopts the technical scheme that:

a quick measurement method for three-dimensional space pointing of an NV axis is realized by adopting an NV axial calibration system, and comprises the following steps:

obtaining an ODMR spectrum with eight separated spectral peaks, and analyzing the frequency width between each pair of spectral peaks to obtain the magnetic field intensity projected in each NV axial direction;

respectively applying static magnetic fields along the positive and negative directions of a Z axis and the positive and negative directions of a Y axis of a space rectangular coordinate system to the diamond sample, simultaneously measuring an ODMR spectrum of the diamond sample, and analyzing the frequency variation corresponding to each pair of spectrum peaks;

and establishing a relation between the NV coordinate system and the space rectangular coordinate system to obtain the pointing angle of each NV axis.

In the technical scheme, the invention provides a method for orienting the NV axis of diamond particles in a three-dimensional space to establish the relationship between the NV axis coordinate system and the space rectangular coordinate system of the diamond particle sample based on the Zeeman splitting characteristics of ODMR spectral peaks in different NV axes under the action of a static magnetic field. The method comprises the steps of firstly, obtaining an ODMR spectrum with eight separation spectral peaks by adjusting the relative position between a permanent magnet and a diamond particle sample, and obtaining the magnetic field intensity projected on each NV axial direction by analyzing the frequency width between each pair of spectral peaks; secondly, respectively applying static magnetic fields in the positive and negative directions of a Z axis and the positive and negative directions of a Y axis along a space rectangular coordinate system to the diamond sample by utilizing Helmholtz coils, simultaneously measuring an ODMR spectrum of the diamond sample and analyzing the frequency variation corresponding to each pair of spectrum peaks; and finally, establishing a relation between the NV coordinate system and a space rectangular coordinate system by using a space coordinate conversion principle, namely, the pointing angle of each NV axis.

As a further technical solution, establishing a relationship between the NV coordinate system and the spatial rectangular coordinate system further includes:

establishing a conversion model B from an NV coordinate system to an ideal orthogonal coordinate system_o＝K_NoB_NV，B_oIs a component under an ideal orthogonal coordinate system, B_NVIs a component in the NV coordinate system, K_NoA transformation matrix from the NV coordinate system to an ideal orthogonal coordinate system;

obtaining the conversion relation B between the ideal orthogonal coordinate system and the space rectangular coordinate system_o＝K_eoB_e，B_eIs divided into a space rectangular coordinate systemAmount, K_eoA transformation matrix from a space rectangular coordinate system to an ideal orthogonal coordinate system;

four different magnetic field vectors respectively applied to the diamond sample are converted into an ideal orthogonal coordinate system to be operated to obtain a conversion matrix K_eoEstablishing a conversion relation B between the NV coordinate system and the space rectangular coordinate system_NV＝K_oNK_eoB_eAnd obtaining the pointing angle of each NV axis.

As a further technical means, assuming that coordinate axes in an ideal orthogonal coordinate system are x, y and z, coordinate axes in an NV coordinate system are α, β, γ, respectively, and when z and α are overlapped, β is located in an xOz plane, and it is known from the crystal structure of diamond that angles between β, γ, the three axes and the xOy plane are equal, a is 19.28 °, angles between projection of the γ axis and the axes in the xOy plane and the x axis are B is 120 ° and c is-120 °, respectively, and thus a conversion model B from the NV coordinate system to the ideal orthogonal coordinate system is obtained_o＝K_NoB_NVWherein, in the step (A),

K_oNfor a transformation matrix from an ideal orthogonal coordinate system to an NV coordinate system, transformation matrix K_oNComprises the following steps:

as a further technical solution, the spatial rectangular coordinate system XYZ is rotated three times to obtain an ideal orthogonal coordinate system XYZ, which specifically includes: firstly, a space rectangular coordinate system XYZ rotates anticlockwise around a shaft Z axisAngle, to obtain an intermediate coordinate system x₁y₁Z; secondly, let the intermediate coordinate system x₁y₁Z around x₁The axis is rotated counterclockwise by an angle theta to obtain a second intermediate coordinate system x₁yz₁(ii) a Finally, let the intermediate coordinate system x₁yz₁And rotating the y axis by an angle omega counterclockwise to obtain an xyz coordinate system.

As a further technical solution, a conversion formula between the spatial rectangular coordinate system XYZ and the ideal orthogonal coordinate system XYZ is:

wherein, K_ω、K_θ、Respectively representing each angle component of a conversion matrix from a space rectangular coordinate system to an ideal orthogonal coordinate system; b is_ex、B_ey、B_ezRespectively, representing the components of the spatial rectangular coordinate system on the xyz three axes.

As a further technical scheme, static magnetic fields applied to the diamond sample along the positive and negative directions of the Z axis and the positive and negative directions of the Y axis of a space rectangular coordinate system are converted into an ideal orthogonal coordinate system for operation to obtain a conversion matrix K_eoThe parameters theta, omega,The method comprises the following steps:

converting two magnetic field vectors which are applied to the diamond sample and have the same strength and opposite directions along the Z axis of a space rectangular coordinate system into an ideal coordinate system, and calculating to obtain a value theta omega;

two magnetic field vectors which are applied to the diamond sample and have the same strength and opposite directions along the Y axis of a space rectangular coordinate system are converted into an ideal coordinate system, and the magnetic field vectors are obtained through calculation

The value is obtained.

As a further technical scheme, the NV axial calibration system comprises an NV color center magnetic field sensitive part and a confocal optical path for NV color center fluorescence collection;

in the NV color center field sensitive part: diamond particles are fixed at the tip of the tapered optical fiber; the spiral antenna is connected with the microwave source and used for generating microwave signals; the Helmholtz magnetic field coil is connected with a high-precision current source and used for generating a uniform adjustable direct-current magnetic field; the diamond particles at the tail end of the optical fiber are fixed in the center of the Helmholtz coil and are close to the strongest point of the magnetic field distribution of the antenna; the neodymium iron boron permanent magnet is placed near the diamond particles to provide a fixed magnetic field for splitting the NV color center ODMR spectrum into eight peaks;

in the confocal light path for NV color center fluorescence collection: the pulse signal generator is used for generating control; laser generated by a laser source enters an acousto-optic modulator after being focused by a lens to perform laser pulse time sequence control, divergent light emitted by the acousto-optic modulator is converted into parallel light through the lens and reflected by a dichroic mirror to enter a long working distance objective lens for focusing, and focused light irradiates a diamond through a cylindrical section end of a conical optical fiber; red fluorescent signals generated by the diamond are converted into parallel light through the long working distance objective lens and pass through the dichroic mirror to enter the objective lens for focusing; the focused red fluorescence is received by an avalanche photodiode and converted into an electric signal; the phase-locked amplifier collects and demodulates the electric signal by using the laser microwave secondary modulation signal generated by the pulse signal generator.

Compared with the prior art, the invention has the beneficial effects that: the method is based on the Zeeman splitting characteristic of different NV axial ODMR spectral peaks under the action of a static magnetic field, the relationship between the NV axis coordinate system and the space rectangular coordinate system of the diamond particle sample is established, only four different magnetic field vectors are applied, and the rapid measurement of the three-dimensional space pointing of the NV axis can be realized through two-step measurement.

Drawings

FIG. 1(a) is a schematic diagram of four orientations of the NV axis in a diamond according to an embodiment of the present invention;

FIG. 1(b) is a schematic diagram of relative directions between an NV coordinate system and a rectangular spatial coordinate system according to an embodiment of the present invention;

FIG. 2 is a schematic diagram of an ideal orthogonal coordinate system and an NV coordinate system according to an embodiment of the present invention;

FIG. 3 is a diagram illustrating a transformation relationship between an ideal orthogonal coordinate system XYZ and a spatial orthogonal coordinate system XYZ according to an embodiment of the present invention;

FIG. 4 is a schematic diagram of an NV axial calibration experiment system according to an embodiment of the present invention;

FIG. 5(a) is a schematic diagram of the measured ODMR signal when the coil magnetic field is zero according to an embodiment of the present invention;

FIG. 5(B) shows a coil magnetic field B according to an embodiment of the present invention_z+And B_z-Comparing the measured ODMR data with a schematic diagram;

FIGS. 6(a) - (d) are schematic diagrams of the Helmholtz coil magnetic field generation steps according to an embodiment of the present invention.

In the figure: 1. a computer; 2. a pulse signal generator; 3. an acousto-optic modulator driver; 4. a microwave source with modulation function; 5. a neodymium iron boron permanent magnet; 6. a microwave switch; 7. a laser light source; 8. an acousto-optic modulator; 9. a confocal optical path; 10. an avalanche photodiode; 11. a phase-locked amplifier; 12. a tapered optical fiber; 13. a Helmholtz coil; 14. micron-sized diamond particles having NV colour centers; 15. a microwave antenna; 16. an adjustable current source; 100. a Z-axis forward magnetic field generated by the coil; 200. a Z-axis reverse magnetic field generated by the coil; 300. a Y-axis forward magnetic field generated by the coil; 400. the Y-axis opposing magnetic field is generated by the coil.

Detailed Description

The technical solutions of the present invention will be described clearly and completely with reference to the accompanying drawings, and it is obvious that the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

The invention provides a method for orienting an NV axis of diamond particles in a three-dimensional space to establish a relation between an NV axis coordinate system and a space rectangular coordinate system of a diamond particle sample based on Zeeman splitting characteristics of ODMR spectral peaks in different NV axes under the action of a static magnetic field. The method comprises the steps of firstly, obtaining an ODMR spectrum with eight separated spectral peaks by adjusting the relative position between a permanent magnet and a diamond particle sample, and obtaining the magnetic field intensity projected on each NV axial direction by analyzing the frequency width between each pair of spectral peaks; secondly, respectively applying static magnetic fields in the positive and negative directions of a Z axis and the positive and negative directions of a Y axis along a space rectangular coordinate system to the diamond sample by utilizing Helmholtz coils, simultaneously measuring an ODMR spectrum of the diamond sample and analyzing the frequency variation corresponding to each pair of spectrum peaks; and finally, establishing a relation between the NV coordinate system and a space rectangular coordinate system by using a space coordinate conversion principle, namely, the pointing angle of each NV axis.

As an implementation manner, the NV axial calibration method mainly includes the following steps:

step 1, establishing a NV coordinate system and space rectangular coordinate system conversion model

According to fig. 1(a), a diamond NV coordinate system is created with four differently oriented NV axes α, β, γ, with an angle of 109.28 ° between each axis, and the relative orientation to a fixed spatial rectangular coordinate system as shown in fig. 1 (b). If three-component data in which a fixed vector is projected on the NV axis component is to be converted into three-component data in a spatial rectangular coordinate system, a conversion relationship between two coordinates needs to be established, and the conversion relationship can be established in two steps:

first, the NV coordinate system is converted into an ideal orthogonal coordinate system

Assuming that coordinate axes in an ideal triaxial rectangular coordinate system are x, y and z, coordinate axes in an NV coordinate system are α, β and gamma, respectively, assuming that z is coincident with α axis, β is located in an xOz plane, and according to the crystal structure of diamond, angles between β, gamma, the three axes and the xOy plane are equal, and a is 19.28 degrees, angles between projection of the gamma axis and the axes in the xOy plane and the x axis are B is 120 degrees and c is-120 degrees, respectively, as shown in fig. 2_NV＝[B_αB_βB_γB]^TComponent B in ideal triaxial rectangular coordinate system_o＝[B_oxB_oyB_oz]^TThe model of the transformation is shown in equation (1).

B_NV＝K_oNB_o(1)

Wherein, the matrix

Is a transformation matrix from an ideal orthogonal coordinate system to an NV coordinate system.

In addition, the transformation model from NV coordinate system to ideal rectangular coordinate can be obtained from equation (1) as shown in (2)

B_o＝K_NoB_NV(2)

Wherein, the matrix

Secondly, converting the ideal orthogonal coordinate system XYZ into a space rectangular coordinate system XYZ

In the previous step, the z-axis of the defined ideal orthogonal coordinate system coincides with the NV coordinate α -axis. In the particle placement process, the direction of the α axis cannot be known, and it cannot be ensured that the α axis coincides with the Z axis of the rectangular spatial coordinate system, so that a conversion relationship from the orthogonal coordinate system XYZ obtained in the previous step to the rectangular spatial coordinate system XYZ needs to be established, as shown in fig. 3.

The XYZ coordinate system can be obtained by three rotations. Firstly, a space rectangular coordinate system XYZ rotates anticlockwise around a shaft Z axisAngle, to obtain an intermediate coordinate system x₁y₁Z; secondly, let the intermediate coordinate system x₁y₁Z around x₁The axis is rotated counterclockwise by an angle theta to obtain a second intermediate coordinate system x₁yz₁(ii) a Finally, let the intermediate coordinate system x₁yz₁And rotating the y axis by an angle omega counterclockwise to obtain an xyz coordinate system. The formula of the three-component transformation between the two coordinate systems is shown in formula (3).

According to the above two steps, the conversion relationship between the NV coordinate system and the spatial rectangular coordinate can be established, as shown in formula (4)

B_NV＝K_oNK_eoB_e(4)

Since the NV axis orientation in the diamond sample is random in space, the matrix K_eoIn

The angle is unknown. Thus, only need to obtainCan establish a conversion relation between the NV coordinate system and the space rectangular coordinate system.

Step 2, determining model parameters

To analyze in the formula (4)

Obtaining a magnetic field component under a known external magnetic field vector NV coordinate system, and converting the magnetic field component into a three-component B under an ideal three-axis rectangular coordinate system by using the formula (1)_oAnd performing operation analysis.

Two same intensity B along Z axis in space rectangular coordinate system_bMagnetic field vector B with opposite directions_e+＝[0 0 B_b]^TAnd B_e-＝[0 0 -B_b]^TWhich have different component values B in an ideal three-axis rectangular coordinate system_o+And B_o-From equation (3), the following relationship can be obtained:

from equation (5), the value θ ω can be obtained as shown in equation (6).

Similarly, two magnetic field vectors with same strength and opposite directions along the Y axis in the space rectangular coordinate system are three-component B 'in the ideal three-axis rectangular coordinate system'_o+And B'_o-Have similar relationship between them, thereby can be derived

The values are shown in (7).