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A high-sensitivity single-light-source triaxial atomic magnetometer with double-cell and orthogonally pumped structure

Zinan Wu, Jialong Zhang, Mengyang He, Bokang Ren, Zilong Wang, Chen Wei, Zhonghua Ou, Huimin Yue, Xiaojun Zhou, and Yong Liu

Open Access Open Access

Abstract

We first report a single-light-source orthogonally pumped triaxial atomic magnetometer with a double-cell structure. By using a beam splitter to equally allocate the pump beam, the proposed triaxial atomic magnetometer is responsive to magnetic fields in all three directions, and without sacrificing system sensitivity. The experimental results indicate that, the magnetometer achieves a sensitivity of 22 fT/Hz1/2 in x-direction with a 3-dB bandwidth of 22 Hz, a sensitivity of 23 fT/Hz1/2 in y-direction with a 3-dB bandwidth of 23 Hz, and a sensitivity of 21 fT/Hz1/2 in z-direction with a 3-dB bandwidth of 25 Hz. This magnetometer is useful for the applications that require the measurements of the three components of the magnetic field.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Atomic magnetometers are the most sensitive magnetic field measurement devices [1] in existence, and according to the different measurement mechanisms, they can be classified as scalar atomic magnetometers [2,3] and vector atomic magnetometers [4,5]. In the case of scalar atomic magnetometers, it is common to extract magnetic field signals by measuring the Larmor procession frequency of atomic spins [6], and an RF signal is applied to lock the Larmor procession frequency by using the lock-in technique [7]. Since this procedure measures the transient response of the atomic spins, it is only related to the amplitude of the measured magnetic field. Unlike the scalar atomic magnetometer, the vector atomic magnetometer measures the steady-state response of the atomic spin polarization after the Larmor procession [8], and the magnetic field measurement can be realized by measuring the paramagnetic rotation angle [9,10] or resonance absorption [11]. Moreover, the atomic magnetometers can be further operated in spin-exchange relaxation free (SERF) state [12,13], where the rate of spin exchange is considerably faster than that of Larmor procession, thus greatly improving the system sensitivity. With the advancement of technology, the vector information of magnetic field has been paid more attention, such as magnetic field source localization [14], magnetic navigation [15], etc.

For the biosensors [16–20], multi-axis measurement has become a popular research topic. The triaxial atomic magnetometer can measure the components of the measured magnetic field in three orthogonal directions, thus enabling the acquisition of vectorial information. As for the reported triaxial atomic magnetometers, several modulation magnetic fields are usually used to achieve triaxial measurements [21,22]. In 2004, Seltzer [23] has presented a triaxial atomic magnetometer which achieved a magnetic field measurement sensitivity of 1pT/Hz1/2 in gradiometer mode. The pump-probe scheme is used in their experiment, a circularly polarized pump beam is applied in the z-direction and a linearly polarized probe beam is applied in the x-direction. And then, the dc response of the atomic magnetometer is used to extract the ${B_y}$ while the lock-in output referenced to the modulation magnetic field in the x- and z-direction are used to extract the ${B_z}$ and ${B_x}$, respectively. However, the pump-probe scheme is not conducive to the system miniaturization. Consequently, the single-beam configuration based on the Hanle effect is becoming of interest [24]. To do so, researchers simultaneously applied modulation magnetic fields in x-, y- and z-directions, followed by three independent lock-in amplifiers to demodulate magnetic field signals [8]. Unfortunately, the big problem is that the system is not sensitive to magnetic field in the direction of pumping light beam, so a rotational modulation field is applied in the x-o-y plane to replace the independently modulation field [25]. Nevertheless, due to the weak interaction between the atomic spin polarization and the magnetic field in the pump beam direction, the sensitivity is naturally poorer in this direction, usually by about an order of magnitude than that of the other directions. By putting a mirror at the back of the atomic vapor cell, the propagation direction of the transmitted pump beam is deflected by 90°, enabling one single laser beam to pump two independent atomic vapor cells at the same time, thus making the atomic magnetometer responsive to magnetic fields of all three directions [26]. A drawback of this configuration is that the light beam intensity absorbed by the two atomic vapor cells cannot be guaranteed to be identical, making the system difficult to optimize to the optimal performance. In addition, there are some other ways to realize triaxial measurements, such as introducing a bias magnetic field [27].

In this paper, a single-light-source triaxial atomic magnetometer is proposed, in which the atoms of two separate atomic vapor cells are polarized at two orthogonal directions, and similar sensitivity is achieved in all three directions. The rest of this paper is organized as follows. Sec.2 describes the system structure as well as the theoretical principle, and the response of the triaxial atomic magnetometer is derived based on the Bloch equation. In Sec.3 the experimental results are presented and the system performance is analyzed. Finally, we summarize this paper with a conclusion in Sec.4.

2. Experimental setup and principle

The experimental setup is presented in Fig. 1. A 795 nm linearly polarized pumping light beam is generated by a tunable semiconductor laser (TOPTICA DL PRO), and divided into two light beams by a fiber beam splitter. Two optical fiber collimators placed at the end of the polarization-maintaining fibers for collimating the transferred light beams, with an output beam diameter of approximately 2 mm. And the two light beams pass through the polarizer and the quarter-wave plate in turn to form circularly polarized light beam, where one light beam travels along the y-direction and the other along the z-direction. An atomic vapor cell is placed on each optical path and the two cells are spaced 2 cm apart in the x-direction. The atomic vapor cells are the key part of the whole system, which determines the basic sensitivity of the system. The larger the size, the weaker the spin depolarization effect due to collisions between the atoms and the cell walls, thus substantially increasing the sensitivity of the system. However, the large size atomic vapor cell is not beneficial for miniaturization and integration of the magnetometer system. Hence, we design a miniature atomic vapor cell, which is a glass blowing cube that has an outer dimension of 4.0mm × 4.0mm × 3.0 mm and an inner dimension of 3.0mm × 3.0mm × 2.0 mm. Inside the atomic vapor cells, it is filled with a sufficient amount of 87Rb atoms and 650 Torr of nitrogen. The cell temperatures are controlled at 160° and the number density of 87Rb atoms is about 1.64 × 1014 /cm3. Due to the rapid collisions between the alkali metal atoms, the spin-exchange relaxation between the atoms is eliminated, allowing the atoms to operate in the SERF region, where the system transverse relaxation is reduced, thus improving the system sensitivity. On the light beam path of the atomic vapor cell, we have pasted a 0.26 mm color glass (SCHOTT RG-9) on the incident side and a 0.63 mm color glass on the exiting side. This kind of color glass can absorb 1550 nm light beam and heat up the atomic vapor cell. In comparison with electric heating technology, the laser heating technology has a simple structure and has no introduction of any magnetic field noise [28,29]. In the experiment, we use a Basik series laser from NKT Photonics as the heating source with a central wavelength of 1550 nm, and amplify it with an erbium doped fiber application amplifier (EDFA). The power required to heat the atomic vapor cell to 160°C is about 650 mW. After the atomic vapor cells, two photodetectors with a 5.76mm2 photosensitive surface area are used for detecting the two transmitted light beams, respectively. These components are housed in a 3D printed optical bench made by Polyetheretherketone (PEEK), which is not only non-magnetic but also has excellent resistance to high temperatures.

 figure: Fig. 1.

Fig. 1. Experimental setup of the proposed triaxial atomic magnetometer (no heating light path is illustrated in the figure). OFC: Optical Fiber Collimator; QWP: Quarter Wave Pate; P: Polarizer; PD: Photo Detector; AVC: Atomic Vapor Cell.

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A magnetic shielding cylinder (MSC) is used to shield the geomagnetic field, which consists of five layers of Permalloy and two layers of aluminum with a shielding coefficient of 106. And a triaxial coil system is designed to offset remanence as well as to produce modulation magnetic fields, in which a single pair of Lee-Whiting coil and a couple of saddle coils are involved. The Lee-Whiting coil produces the longitudinal magnetic field (x-direction) and the saddle coils produce the transverse magnetic field (y- and z-directions). The compensation magnetic fields of the triaxial coil are driven by three independent current sources, and the modulation magnetic fields are generated by lock-in amplifier. Outside the MSC, two lock-in amplifiers are used for demodulating the output signals of PD1 and PD2, respectively.

The density matrix [30] can be used to characterize the dynamics of atomic spin,, and in the SERF condition, the density matrix is then simplified to the Bloch equation [31]

$$\frac{{d{\mathbf P}}}{{dt}} = \frac{1}{q}[{\gamma _e}{\mathbf B} \times {\mathbf P} + {R_{op}}({s\widehat z - {\mathbf P}} )- {R_{rel}}{\mathbf P}]$$
where ${\gamma _e} = 2\pi \times 28\textrm{Hz/nT}$ is the gyromagnetic ratio of bare electron, q is the nuclear slowing-down factor, ${R_{op}}$ is the optical pumping rate, and ${R_{rel}}$ is the system relaxation rate. Here, $s \simeq 1$ is the degree of polarization of the optical pumping beam, $\widehat z$ is the pumping beam transmission direction. Under the condition of $d{P}/dt = 0$, we have the projection of ${P}$ in the z-direction as
$${P_z} = \frac{{{R_{op}}}}{{{R_{op}} + {R_{rel}}}}\frac{{{\gamma _e}^2{B_z}^2 + {{({R_{op}} + {R_{rel}})}^2}}}{{{{({R_{op}} + {R_{rel}})}^2} + {\gamma _e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)}}$$

Perform a Taylor expansion on ${P_z}$ and consider its 1st order component, the system responses in all three directions can be expressed as

$${P_z}\left|{_{{B_x}} \propto{-} {P_0}\frac{{2{\gamma_e}^2[{\gamma_e}^2{B_z}^2 + {{({R_{op}} + {R_{rel}})}^2}]{B_x}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$
$${P_z}\left|{_{{B_y}} \propto{-} {P_0}\frac{{2{\gamma_e}^2[{\gamma_e}^2{B_z}^2 + {{({R_{op}} + {R_{rel}})}^2}]{B_y}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$
$${P_z}\left|{_{{B_z}} \propto {P_0}\frac{{2{\gamma_e}^4({B_x}^2 + {B_y}^2){B_z}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$
where ${P_0} = {R_{op}}/({R_{op}} + {R_{rel}})$ is the equilibrium atomic spin polarization in the absence of magnetic field.

From Eq. (3),(4), it can be seen that the system response to ${B_x}$ and ${B_y}$ is similar to the absorption curve, even in the SERF regime (${B_y} \simeq {B_z} \simeq 0$ or ${B_x} \simeq {B_z} \simeq 0$), the solution is still valid owing to the ${({R_{op}} + {R_{rel}})^2}$ term. However, by observing the Eq. (5), we find that when the system operates in the SERF regime (${B_x} \simeq {B_y} \simeq 0$), its response to ${B_z}$ is almost zero. Thus, the system is insensitive to the ${B_z}$. To solve this problem, we implement another atomic vapor cell in the system, and it is pumped in y-direction. Fixing the coordinate system unchanged, the system responses of AVC2 gives

$${P_y} = \frac{{{R_{op}}}}{{{R_{op}} + {R_{rel}}}}\frac{{{\gamma _e}^2{B_y}^2 + {{({R_{op}} + {R_{rel}})}^2}}}{{{{({R_{op}} + {R_{rel}})}^2} + {\gamma _e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)}}$$

Similarly, we can obtain the system responses of AVC2 in three directions as

$${P_y}\left|{_{{B_x}} \propto{-} {P_0}\frac{{2{\gamma_e}^2[{\gamma_e}^2{B_y}^2 + {{({R_{op}} + {R_{rel}})}^2}]{B_x}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$
$${P_y}\left|{_{{B_y}} \propto {P_0}\frac{{2{\gamma_e}^4({B_x}^2 + {B_z}^2){B_y}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$
$${P_y}\left|{_{{B_z}} \propto{-} {P_0}\frac{{2{\gamma_e}^2[{\gamma_e}^2{B_y}^2 + {{({R_{op}} + {R_{rel}})}^2}]{B_z}^2}}{{{{[{{({R_{op}} + {R_{rel}})}^2} + {\gamma_e}^2({B_x}^2 + {B_y}^2 + {B_z}^2)]}^2}}}} \right.$$

Thus, the first atomic vapor cell is sensitive to the ${B_x}$ and ${B_y}$, the second atomic vapor cell is sensitive to the ${B_x}$ and ${B_z}$, then we can realize a triaxial magnetic field measurements.

3. Experimental results

In order to verify the above theoretical derivation, we impose a slowly varying magnetic field to be measured from -50nT to 50nT, and record its resonance absorption curves at the same time. Figure 2 (a) shows the measured resonance absorption curves of the two AVCs (AVC1 in red and AVC2 in black) for magnetic field in x-direction. It is clear that both the two AVCs are able to respond towards the magnetic field at x-direction, and the two curves are closely similar to each other by delivering almost identical pump light and controlling the temperature of both AVCs to be identical. Obviously, the slope on resonance is zero, which means that it is insensitive to the variations of the magnetic field. To solve this problem, we usually apply a bias magnetic field in the system so that the system operates on the side of the resonance, which is called the DC mode. Here, due to the full width at half maximum (FWHM) of the two curves are 34nT and 40nT, we set the amplitude of the bias magnetic field to 18.5nT in order to consider the response of the two AVCs at the same time. By adjusting the pumping beam intensity and the cell temperature, the FWHM of the curve can be further optimized. Within the system’s dynamic range, the system output voltage can be considered as linearly dependent on the magnetic field to be measured, and its slope, or the system magnetic field voltage conversion factor, can be obtained by performing a linear fit to it [32]. The conversion factors of the two AVCs are 37.80 mV/nT and 42.05 mV/nT, respectively. We calibrate the magnetometer system by applying a calibration field of 100pTrms at a frequency of 20 Hz into the system, and Fig. 2 (d) shows the results. It can be seen that the sensitivity of the two AVCs is quite comparable, with both being about 60 fT/Hz1/2 around 15 Hz. In contrast to the AVC2, the AVC1 has a 25 Hz noise source, and we assume it comes from the transmission line.

 figure: Fig. 2.

Fig. 2. (top row) Measured resonance absorption curves along the (a) x-direction, (b) y-direction, and (c) z-direction. (bottom row) The corresponding system sensitivity.

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Analogously, Fig. 2 (b) and (c) illustrates the measured resonance absorption curves for magnetic fields at y- and z-directions, respectively. The resonance absorption curve of AVC1 in the y-direction has a FWHM of 38nT, and that of AVC2 in the z-direction has a FWHM of 40nT. And the conversion factors in the y- and z-direction are 42.35 mV/nT and 33.55 mV/nT, respectively. The reason that the conversion factor in the y-direction is larger than that of in the z-direction is that the transmitted light intensity in the z-direction is stronger, which we can visualize it from the Fig. 2. And the corresponding sensitivities are 40 fT/Hz1/2 and 46 fT/Hz1/2 around 15 Hz. It should be noted that, in agreement with the previous theories, the AVC1 has no response in the z-direction and the AVC2 has no response in the y-direction.

For the purpose of reducing the effect of low-frequency noise, people usually introduce a modulation magnetic field ${B_m}\cos (\omega t)$ to the system [33,34], which shifts the detected magnetic field signals to higher frequency, and the modulated atomic spin polarization components can be obtained at harmonics of ω [9]. In the experiment, we apply the modulation magnetic field in the direction of the magnetic field to be measured, which is generated by the lock-in amplifier, as shown in Fig. 1. The system response of the AVC1 to the magnetic field in the x-direction can be described as follows [35]

$${P_{z - \omega }} = \frac{{{\gamma _e}{R_{op}}{J_0}(M){J_1}(M){B_x}}}{{{{({R_{op}} + {R_{rel}})}^2} + {{({\gamma _e}{B_x})}^2}}}\sin (\omega t)$$
where J is the first kind of Bessel function, and $M = {\gamma _e}{B_m}/(q\omega )$ represents the modulation index. The Eq. (10) indicates that the atomic spin polarization shows a similar dispersion relationship with the magnetic field in the x-direction. And when the magnetic field is zero, the slope of the dispersion curve is the largest, which means that the system response is also the most sensitive. By optimizing the amplitude ${B_m}$ and frequency $\omega $ of the modulation magnetic field, we can change the value of the Bessel term in the system response, and thus optimize the system sensitivity. In addition, the system responses of the atomic magnetometer system to the magnetic field in other directions can be obtained in the same way.

Figure 3 shows the system performance with a 180nT modulation magnetic field at 1kHz applied in the corresponding directions. The left part, the middle part, and the right part represents the system response in the x-, y-, and z-direction, respectively. When compared with the absorption curve in Fig. 2, the dispersion-like curve in Fig. 3 is approximately linear in the near-zero region, making it more suitable for magnetic field measurements. As we can see, the magnetic field in the x-direction can be responded by both the AVC1 and the AVC2, the magnetic field in the y-direction can only be responded by the AVC1, and the magnetic field in the z-direction can only be responded by the AVC2, which are similar to the above system responses without the modulation magnetic field. The system sensitivities in the x-direction are 22 fT/Hz1/2 for cell-1 and 30 fT/Hz1/2 for the cell-2 around 20 Hz, and the system sensitivities in the y- and z-directions are 23 fT/Hz1/2 and 21 fT/Hz1/2 around 20 Hz, respectively, which are better than the performance without the modulation magnetic field.

 figure: Fig. 3.

Fig. 3. (top row) Measured dispersion curves along the (a) x-direction, (b) y-direction, and (c) z-direction. (bottom row) The corresponding system sensitivity.

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Figure 4 shows the measured frequency responses in the three directions of the DC-mode and the magnetic-field-modulation mode. By keeping the amplitude of the calibration magnetic field fixed at 100pTrms, varying its frequency and recording its system response, we use the first order low pass filter function to fit the measured data [36]. The solid lines represent the DC-mode while the dotted lines represent the magnetic-field-modulation mode. To facilitate comparison, we keep the temperature of the two AVCs at 160° and the total incident pumping light beam intensity at 400µW during the procedure. In the DC-mode, the 3-dB bandwidths of the three directions are 16.71 Hz (AVC1 in the x-direction), 17.54 Hz (AVC2 in the x-direction), 17.10 Hz (AVC1 in the y-direction), and 17.00 Hz (AVC2 in the z-direction). In the magnetic-field-modulation mode, the 3-dB bandwidths of the three directions are 21.84 Hz (AVC1 in the x-direction), 22.10 Hz (AVC2 in the x-direction), 22.84 Hz (AVC2 in the y-direction), and 24.65 Hz (AVC2 in the z-direction).

In comparison with the performance of the DC-mode and the magnetic-field-modulation mode, we find that the magnetic-field-modulation mode has not only higher sensitivity but also higher 3-dB bandwidth. We attribute this result to the following two reasons. One is the limit of the low-frequency technical noise, which is suppressed by the phase-locked amplification technology. The other one is due to the modulation magnetic field, which increases the magnetic resonance linewidth and the influence is in a quadratic relationship with the amplitude of the modulation magnetic field [37]. It is worth noting that the modulation magnetic field also increases the optical depth (OD) of the system, which reduces the magnetic resonance linewidth to some extent. Hence, we need to adjust the amplitude of the modulation magnetic field carefully. When compared to the work in [26], the 3-dB bandwidth here is significantly smaller, which might be caused by the higher operating temperature or the weaker pumping light intensity, yet this results in a narrower magnetic resonance linewidth and higher sensitivity. In the future, we intend to study the effects of these two parameters on the system performance, to improve the system 3-dB bandwidth at no expense to sensitivity.

 figure: Fig. 4.

Fig. 4. The normalized frequency responses in three directions. The diamonds represent the measured data in DC mode, and the solid lines represent the fitted curves in DC mode. The down triangles represent the measured data in magnetic-field-modulation mode, and the dashed lines represent the fitted curves in magnetic-field-modulation mode.

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4. Conclusion

In this paper, we have proposed a SERF triaxial atomic magnetometer. With the use of optical fiber beam-splitter, we can pump the two AVCs simultaneously and polarize the atomic spins of these two AVCs in mutually orthogonal directions. Based on this structure, AVC1 can respond to magnetic fields in the x- and y-directions, and AVC2 can response to magnetic fields in the x- and z-directions, thus enabling the measurements of three-axis magnetic fields. As a result, the sensitivities of the three directions are 22 fT/Hz1/2, 23 fT/Hz1/2, and 21 fT/Hz1/2, respectively. The configuration proposed in this paper has a simple structure and high sensitivity, offering a way to design a compact triaxial atomic magnetometer. Predictably, this triaxial atomic magnetometer has the potential to be used in biomedical applications such as magnetoencephalography construction.

Funding

National Natural Science Foundation of China (62075032, 61875033).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. Experimental setup of the proposed triaxial atomic magnetometer (no heating light path is illustrated in the figure). OFC: Optical Fiber Collimator; QWP: Quarter Wave Pate; P: Polarizer; PD: Photo Detector; AVC: Atomic Vapor Cell.

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Fig. 2.
Fig. 2. (top row) Measured resonance absorption curves along the (a) x-direction, (b) y-direction, and (c) z-direction. (bottom row) The corresponding system sensitivity.

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Fig. 3.
Fig. 3. (top row) Measured dispersion curves along the (a) x-direction, (b) y-direction, and (c) z-direction. (bottom row) The corresponding system sensitivity.

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Fig. 4.
Fig. 4. The normalized frequency responses in three directions. The diamonds represent the measured data in DC mode, and the solid lines represent the fitted curves in DC mode. The down triangles represent the measured data in magnetic-field-modulation mode, and the dashed lines represent the fitted curves in magnetic-field-modulation mode.

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Equations (10)

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d P d t = 1 q [ γ e B × P + R o p ( s z ^ − P ) − R r e l P ]
P z = R o p R o p + R r e l γ e 2 B z 2 + ( R o p + R r e l ) 2 ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 )
P z | B x ∝ − P 0 2 γ e 2 [ γ e 2 B z 2 + ( R o p + R r e l ) 2 ] B x 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P z | B y ∝ − P 0 2 γ e 2 [ γ e 2 B z 2 + ( R o p + R r e l ) 2 ] B y 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P z | B z ∝ P 0 2 γ e 4 ( B x 2 + B y 2 ) B z 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P y = R o p R o p + R r e l γ e 2 B y 2 + ( R o p + R r e l ) 2 ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 )
P y | B x ∝ − P 0 2 γ e 2 [ γ e 2 B y 2 + ( R o p + R r e l ) 2 ] B x 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P y | B y ∝ P 0 2 γ e 4 ( B x 2 + B z 2 ) B y 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P y | B z ∝ − P 0 2 γ e 2 [ γ e 2 B y 2 + ( R o p + R r e l ) 2 ] B z 2 [ ( R o p + R r e l ) 2 + γ e 2 ( B x 2 + B y 2 + B z 2 ) ] 2
P z − ω = γ e R o p J 0 ( M ) J 1 ( M ) B x ( R o p + R r e l ) 2 + ( γ e B x ) 2 sin ⁡ ( ω t )
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