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Single-beam three-axis SERF atomic magnetometer based on coordinate system rotation

Jialong Zhang, Tianpeng Chen, Chen Wei, Zhonghua Ou, Huimin Yue, and Yong Liu

Open Access Open Access

Abstract

We propose what we believe to be a new single-beam three-axis spin exchange relaxation free (SERF) vector atomic magnetometer scheme based on coordinate system deflection. A theoretical model for the system response under arbitrary angle deflection was established for the first time, and the system response at different angles was simulated and analyzed. The simulation results show that the system response increases in the direction of the non-sensitive axis and decreases in the direction of the sensitive axis as the deflection angle increases, and the two responses tend to be the same when the angle is deflected to 45-degrees. Experimental measurements were carried out at a deflection angle of 45-degrees and the results showed that the sensitivity of the magnetometer was 55fT/Hz1/2 in the x1-axis, 38fT/Hz1/2 in the y1-axis and 60fT/Hz1/2 in the z1-axis. This single-beam magnetometer can be used to construct a miniaturized and low-cost weak magnetic sensor, which is expected to be used for vector measurement of biomagnetism.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Weak magnetic detection is a hot topic in the field of magnetic field measurement, and there are currently many measurement methods available [1,2]. Atomic magnetometers (AMs) have attracted much attention since they were first proposed. A notable advance in this field is the spin exchange relaxation free (SERF) atomic magnetometer reported in 2002 [3,4]. Since its sensitivity is not limited by spin-exchange relaxation, it can theoretically reach the order of aT [5,6]. This theoretical sensitivity exceeds that of the most successful commercially available superconducting quantum interferometer (SQUID) and promises to be the most sensitive magnetic field measurement device [7,8]. Due to its ultra-high sensitivity, SERF atomic magnetometer is widely used in various areas, including the study of magnetocardiography [9,10], fundamental physics [11,12], and zero-field nuclear magnetic resonance [13].

Although SERF atomic magnetometer have ultra-high sensitivity, due to their vector measuring characteristics, they often only can measure uniaxial or biaxial magnetic field signal, and there exists a measuring blind spot in the direction of the pumped light. For this reason, the three-axis SERF atomic magnetometers have attracted a great deal of attention. Compared with uniaxial or biaxial SERF atomic magnetometers, three-axis atomic magnetometers can obtain more magnetic field information and have broad application prospects in many fields [14–17].

In order to achieve three-axis magnetic field measurement, researchers have proposed many effective solutions [18]. Many of these proposed three-axis magnetometer schemes are based on orthogonal dual laser beam schemes, where one laser is used for pumping and the other for detecting. Seltzer and Romails proposed a three-axis vector SERF atomic magnetometer operating in a feedback system and two orthogonal low-frequency modulation fields, with a sensitivity of 1 pT/Hz1/2 [19]. Huang et al. implemented an all-optical three-axis atomic magnetometer using an intensity modulated pump light and bias field. Under a 2 $\mu \textrm{T}$ bias field, the longitudinal sensitivity is 0.8 pT/Hz1/2 and the transverse sensitivity is 1.5pTHz1/2 [20]. Ding et al. demonstrated a three-axis atomic magnetometer with a sensitivity of 0.3pT/Hz1/2 on the z-axis and 2 pT/Hz1/2 on the x- axis and y-axis by applying a bias magnetic field and a longitudinal radio-frequency modulation field in the pumping direction [21]. Yang et al. proposed an all-electric three-axis atomic magnetometer based on parameter oscillation in an unshielded experimental environment. The three-axis sensitivity is in the range of pT [22]. Although the AMs described above have high sensitivity, the orthogonal pump-probe beam scheme increases the complexity of the optical path and limits the miniaturization of the structures. In recent years, single-beam three-axis atomic magnetometers have also been widely reported. Huang et al. described a single beam three-axis AM with a sensitivity of 0.3 pT/Hz1/2 in the x and y axis and a sensitivity of 3 pt/Hz1/2 in the z axis by applying a rotating field on the x-o-y plane and another modulation field on the z-axis [23]. Dong et al. adopted the single-beam scheme and performed three-axis magnetic field measurements by applying modulation fields of different frequencies in the vertical direction to the pump light or by using three modulation fields [23–26]. However, the sensitivity of the pump light corresponding to this axis is much lower than the other two axis. Tang et al. applied a small DC offset field and a high-frequency modulation field to measure a three-axis magnetic field with a sensitivity of 20 fT/Hz1/2in x-axis, 25 fT/Hz1/2 in y-axis, 30 fT/Hz1/2 in z-axis, but used multiple differential structures [27]. Xiao et al. used a reflector to pass a beam of light perpendicularly through the cell to achieve three-axis magnetic field measurements, a scheme that requires the pump light to pass through the cell twice consecutively [28]. As a result, the reduced intensity of light passing through the cell for the second time affects the magnetic field measurement sensitivity. In addition, Wu et al. pumped two cells orthogonally through the beam splitter and thus achieved three-axis measurements. This method gives an equivalent response in the three-axis direction, but requires two cells and is therefore not easy miniaturized [29].

This paper proposes a single-beam three-axis SERF atomic magnetometer based on coordinate system rotation. By changing the angle between the calibration magnetic field and the pump light to rotate the direction of atomic spin polarization, the projection of atomic spin polarization in all directions can be detected, and thus achieving three-axis magnetic field measurement. Based on the Bloch equation, the theoretical model for the system response under arbitrary angle deflection was established, and the system response at different angles was simulated and analyzed, and finally, corresponding experiments were designed to validate the model.

2. Experimental setup and principle

Figure. 1 shows the configuration of the three-axis magnetometer. This experiment adopts a single-beam scheme, employing a sole 795 nm laser beam for simultaneously pumping and detecting, where the laser used here is a tunable semiconductor laser (TOPTICA DL PRO) with a tuning range of up to 40 nm, we tune the laser to the optimal pump frequency by scanning the absorption spectral lines of alkali metal atoms. The laser emits linearly polarized light which, after passing through a 1/2 wave plate, enters the fiber coupler to ensure that the polarization direction of the laser is aligned with the axis of the polarization maintaining fiber. A fiber collimator with a spot diameter of approximately 2 mm then collimates the light. Next, a polarizer and a quarter-wave plate are used to convert the light into circularly polarized light. The purpose of the polarizer is to reduce the influence of laser polarization and ensure that stable circularly polarized light is produced, which is then used to pump alkali metal atoms into the atomic vapor cell. Finally, the laser transmitted through the atomic vapor cell is received by a photodetector. A transconductance amplifier converts the transmitted optical signal into an electrical signal, which is sent to a phase-locked amplifier (Zurich Instruments HF2LI) for demodulation. The rubidium atomic vapor cell is a glass-blown cubic container with external dimensions of 4mm × 4mm × 3mm and internal dimensions of 3mm × 3mm × 2mm, filled with Rb atoms and 600Torr of N2 as buffering and quenching gas. There are two isotopes of rubidium atoms: 87Rb and 85Rb, due to the larger difference in the hyperfine energy between the two ground states of 87Rb atoms compared to 85Rb, the stability of the 87Rb atoms is higher, thus we adopt pure 87Rb as the working medium. Color filters of 0.26mm and 0.63mm are pasted on the front and back surfaces of the atomic vapor cell respectively, which can absorb the 1550nm laser to realize the heating function, thus achieving the SERF state. And the thickness of the front and back surfaces is designed to promote uniformity of temperature within the atomic vapor cell. This optical heating method described here offers advantages over other electrical heating solutions due to its simplicity and the absence of additional magnetic fields [30,31].

 figure: Fig. 1.

Fig. 1. Experimental setup of the triaxial atomic magnetometer. OFC: Optical Fiber Collimator; P: Polarizer; QWP: Quarter Wave Pate; AVC: Atomic Vapor Cell; PD: Photo Detector; Source: Current source

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The dynamic evolution of atomic spin polarization can be represented by a dynamic density matrix [32], which can be simplified to the Bloch equation in the SERF conditions [33]:

$$\frac{d}{{dt}}{\textbf P} = \gamma {\textbf P} \times {\textbf B} + {R_{\textrm{op}}}({s\widehat z - {\textbf P}} )- \frac{{\textbf P}}{{T1,T2}}$$
where ${\textbf P}$ is the macro electron spin polarization, $\gamma = 2\pi \times 28Hz/nT$ is the gyromagnetic ratio of the electron spin, ${R_{op}}$ is the pumping rate, ${\textbf B}$ is the magnetic field, $\hat{z}$ is the direction of pump light, is the degree of polarization of the pump light, and T1 and T2 are the longitudinal and transverse relaxation time, respectively. When $d{\textbf P}/dt = 0$, the atomic polarization along the pump light direction can be expressed as:
$${P_z} = \frac{{{R_p}}}{{{R_2}}}\frac{{{B_z}^2 + \Delta {B^2}}}{{{B_x}^2 + {B_y}^2 + k({B_z}^2 + \Delta {B^2})}}$$
where ${R_{1,2}} = 1/{T_{1,2}}$ is longitudinal and transverse relaxation rates, $\Delta B = {R_2}\textrm{/}\gamma $ is the magnetic resonance line width, we set the scaling factor $k = {R_1}/{R_2}$, and the transverse relaxation time and longitudinal relaxation time can be approximately equal in the state of spin exchange relaxation free, thus the value of k can be considered as 1. The steady state value of ${P_z}$ can be obtained by detecting the transmission of pump light using a photodetector. We apply three-axis modulation magnetic fields ${B_x}^{\bmod }\sin ({\omega _x}t)\widehat x + {B_y}^{\bmod }\sin ({\omega _y}t)\widehat y + {B_z}^{\bmod }\sin ({\omega _z}t)\widehat z$ in three directions. Taylor expansion is applied to the above equation and the first-order term is taken to obtain an approximate solution for atomic polarization as follows [23]:
$${P_z}{|_x} ={-} \frac{{2{R_p}}}{{{R_2}}} \times \frac{{({B_z}^2 + \Delta {B^2}){B_x}{B_x}^{\bmod }\sin ({\omega _x}t)}}{{{{[{B_x}^2 + {B_y}^2 + k({B_z}^2 + \Delta {B^2})]}^2}}}$$
$${P_z}{|_y} ={-} \frac{{2{R_p}}}{{{R_2}}} \times \frac{{({B_z}^2 + \Delta {B^2}){B_y}{B_y}^{\bmod }\sin ({\omega _y}t)}}{{{{[{B_x}^2 + {B_y}^2 + k({B_z}^2 + \Delta {B^2})]}^2}}}$$
$${P_z}{|_\textrm{z}} = \frac{{2{R_p}}}{{{R_2}}} \times \frac{{({B_x}^2 + {B_y}^2){B_z}{B_z}^{\bmod }\sin ({\omega _z}t)}}{{{{[{B_x}^2 + {B_y}^2 + k({B_z}^2 + \Delta {B^2})]}^2}}}$$

From the above equation, it can be seen that three-axis magnetic field measurement can be achieved by demodulating the three modulated magnetic fields. However, there is a problem with this scheme, the signal in the z-axis direction becomes very weak and difficult to measure when the magnetic field approaches zero. Therefore, this article utilizes the method of a three-axis coordinate system deflection to achieve three-axis measurements, at the expense of some sensitivity. As depicted in the Fig. 2(a), the initial z direction is aligned with the pump direction. By deflecting the coordinate system along the y-axis, the x-axis and z-axis become the x1-axis and z1-axis, as shown in Fig. 2(b). At this point, z direction is no longer along the direction of the pump light.

 figure: Fig. 2.

Fig. 2. Schematic diagram of three-axis coordinate rotation. (a) Original coordinate axis. (b) Rotated coordinate axis.

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In this situation, applying a modulation field ${B_{{x_1}}}^{\bmod }\sin ({\omega _{{x_1}}}t)$ to the x1-axis is equivalent to applying a modulation field ${B_{{x_1}}}^{\bmod }\sin ({\omega _{{x_1}}}t)\cos \theta$ and ${B_{{\textrm{x}_1}}}^{\bmod }\sin ({\omega _{{x_1}}}t)\sin \theta$ in x-direction and z-direction. When the system operates in the SERF regime (${B_{{y_1}}} \approx {B_{{z_1}}} \approx 0$), the response in the x1 axis can be obtained from the Eq. (3) and Eq. (5) as:

$${\textrm{P}_z}{|_{{x_1}}} ={-} \frac{{2{R_p}}}{{{R_2}}}\frac{{{B_{{x_1}}}B_{{x_1}}^m\Delta {B^2}{{\cos }^2}\theta \sin ({\omega _{{x_1}}}t)}}{{{{[B_{{x_1}}^2({{\cos }^2}\theta + k{{\sin }^2}\theta ) + k\Delta {B^2}]}^2}}}$$

Similarly, when a modulation magnetic field ${B_{{z_1}}}^{\bmod }\sin ({\omega _{{z_1}}}t)$ is applied in the z1-axis, we can obtain the system response of z1 in the SERF regime (${B_{{x_1}}} \approx {B_{{y_1}}} \approx 0$) as:

$${\textrm{P}_z}{|_{{z_1}}} ={-} \frac{{2{R_p}}}{{{R_2}}}\frac{{{B_{{\textrm{z}_1}}}B_{{\textrm{z}_1}}^m\Delta {B^2}{{\sin }^2}\theta \sin ({\omega _{{\textrm{z}_1}}}t)}}{{{{[B_{{\textrm{z}_1}}^2({{\sin }^2}\theta + k{{\cos }^2}\theta ) + k\Delta {B^2}]}^2}}}$$

Since the y1-axis is always aligned with the y-axis, its system response is unchanged and remains as Eq. (4).

We simulated the responses of the x1 and z1 axis to balance the influence of the angle deviation on the x and z axis as follows. The system response changes with the change in the deflection angle when we apply the same modulated magnetic field in two directions. As the deflection angle increases, the response of the x1 axis decreases, while the response of the z1 axis increases, as shown in the Fig. 3. To balance the relationship between the two, we choose 45 degrees to balance the sensitivity in both directions, which can minimize the difference in measurement sensitivity in the three-axis direction and maximize the overall system response.

 figure: Fig. 3.

Fig. 3. System response changes with the angle of deflection. (a) Changes in x1-axis system response curve with increasing angle. (b) Change in normalized system response with increasing angle x1-axis and z1-axis

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When the angle is 45 degrees, the response of the three-axis system can be expressed as:

$${P_\textrm{z}}{|_{{x_1}}} ={-} \frac{{4{R_p}}}{{{R_2}}}\frac{{{B_{{x_1}}}B_{{x_1}}^m\Delta {B^2}\sin ({\omega _{{x_1}}}t)}}{{{{[B_{{x_1}}^2(1 + k) + 2k\Delta {B^2}]}^2}}}$$
$${P_z}{|_{{\textrm{y}_\textrm{1}}}} ={-} \frac{{2{R_p}}}{{{R_2}}} \times \frac{{{B_{{\textrm{y}_1}}}{B_{{\textrm{y}_1}}}^{\bmod }\Delta {B^2}\sin ({\omega _{{\textrm{y}_1}}}t)}}{{[{B_{{y_1}}}^2 + k\Delta {B^2}){]^2}}}$$
$${P_\textrm{z}}{|_{{\textrm{z}_1}}} ={-} \frac{{4{R_p}}}{{{R_2}}}\frac{{{B_{{\textrm{z}_1}}}B_{{\textrm{z}_1}}^m\Delta {B^2}\sin ({\omega _{{\textrm{z}_1}}}t)}}{{{{[B_{{\textrm{z}_1}}^2(1 + k) + 2k\Delta {B^2}]}^2}}}$$

At this point, the system can sense the magnetic fields in both the x1-axis and z1-axis, thus enabling the measurement of the three-axis magnetic field.

3. Experimental results

In order to visually demonstrate the three-axis response changes of the system after rotation, we first test the system before rotation. In order to reduce the effect of modulated magnetic field crosstalk, three-axis modulation were applied in sequence, respectively. Figure 4(a) shows the system performance with a 120nT modulation magnetic field at 1kHz applied in the x(y,z) directions. By applying a wave scanning magnetic field from -35 nT to 35 nT, the response curve of the system in the direction of the magnetic field can be obtained. It can be seen that there is a linear relationship between the system response output and the measured magnetic field near zero magnetic field. The conversion coefficient between the system response output voltage and the measured magnetic field can be obtained by linearly fitting this area, with x-axis and y-axis of 29.08 mV/nt, 28.17 mV/nT respectively. Because the z-axis is in the direction of the pump light, it does not respond to the magnetic field, the weak signal on the z-axis in the figure is generated by other crosstalk signals. Figure 4(b) shows the sensitivity of x-axis and y-axis, which was obtained by the measurement of the noise spectrum under a calibrated magnetic field with a frequency of 20 Hz and an amplitude of 100 pTrms. In the end, we achieved the sensitivity of 37 fT /Hz1/2 in the x-axis, 41 fT/Hz1/2 in the y-axis. At this point, there is a similar magnetic field response and sensitivity between the x-axis and y-axis.

 figure: Fig. 4.

Fig. 4. Original system response and measurement sensitivity before rotation. (a)Measured dispersion curves along the x1-axis, y1-axis, and z1-axis. (b) Magnetic field sensitivity along the x-axis and y-axis.

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In an ideal condition, the response of the z-axis and x-axis is expected to be consistent at a 45-degree angle. However, there is a deviation between the two due to angle errors during structural installation in actual results. Figure 5(a) shows the dispersion curve of the system response of the rotated three-axis system along x, y, and z-axis, with conversion coefficients of 13.52 mV/nT, 27.81 mV/nT, and 12.59 mV/nT respectively. Figure 5(b) shows the sensitivity of the three-axis atomic magnetometer. In the end, we achieved the sensitivity of 55 fT /Hz1/2 in the x1-axis, 38 fT/Hz1/2 in the y1-axis, and 60 fT/Hz1/2 in the z1-axis.

 figure: Fig. 5.

Fig. 5. System response and measurement sensitivity at 45-degrees. (a)Measured dispersion curves along the x1-axis, y1-axis, and z1-axis. (b) Magnetic field sensitivity of three-axis atomic magnetometer along the x1-axis, y1-axis, and z1-axis.

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From the system response before and after rotating the coordinate axis, it can be seen that the original system can only achieve similar measurement performance on two axes and cannot achieve three-axis measurement. After rotating the coordinate axis, three-axis measurement was successfully achieved, and the x1-axis and z1-axis achieved similar measurement sensitivity. Compared to the original x-axis, the performance of the x1-axis decreased, but the difference was not significant, while the z1-axis greatly improved compared to the original z-axis. Compared with other single beam triaxial vector atomic magnetometers, this method has a smaller difference in sensitivity measurement in the triaxial direction.

The above measurement results are all time-sharing measurements. By changing the frequency of the modulated magnetic field applied on the three axes, we can simultaneously obtain the three-axis response. However, due to the coupling effect between the axes, the system response decreases, and further research is needed on the coupling effect between the axes.

4. Conclusions

In conclusion, we have proposed a single-beam three-axis vector atomic magnetometer scheme based on coordinate system rotation. A theoretical model of the system response to arbitrary angular deflection has been developed, and experimental measurements have been carried out at a 45-degree deflection angle. The experimental results show that under a calibrated magnetic field of 20 Hz, the sensitivity of the magnetometer is 55 fT/Hz1/2 in the x1-axis, 38 fT/Hz1/2 in the y1- axis and 60 fT/Hz1/2 in the z1- axis. It can be seen that the deviation of the coordinate system sacrifices the sensitivity of a sensitive axis, but effectively solves the problem of the insensitive axis.

Funding

National Natural Science Foundation of China (62075032, 62375041).

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 (5)
Fig. 1.
Fig. 1. Experimental setup of the triaxial atomic magnetometer. OFC: Optical Fiber Collimator; P: Polarizer; QWP: Quarter Wave Pate; AVC: Atomic Vapor Cell; PD: Photo Detector; Source: Current source

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Fig. 2.
Fig. 2. Schematic diagram of three-axis coordinate rotation. (a) Original coordinate axis. (b) Rotated coordinate axis.

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Fig. 3.
Fig. 3. System response changes with the angle of deflection. (a) Changes in x1-axis system response curve with increasing angle. (b) Change in normalized system response with increasing angle x1-axis and z1-axis

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Fig. 4.
Fig. 4. Original system response and measurement sensitivity before rotation. (a)Measured dispersion curves along the x1-axis, y1-axis, and z1-axis. (b) Magnetic field sensitivity along the x-axis and y-axis.

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Fig. 5.
Fig. 5. System response and measurement sensitivity at 45-degrees. (a)Measured dispersion curves along the x1-axis, y1-axis, and z1-axis. (b) Magnetic field sensitivity of three-axis atomic magnetometer along the x1-axis, y1-axis, and z1-axis.

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Equations (10)

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d d t P = γ P × B + R op ( s z ^ − P ) − P T 1 , T 2
P z = R p R 2 B z 2 + Δ B 2 B x 2 + B y 2 + k ( B z 2 + Δ B 2 )
P z | x = − 2 R p R 2 × ( B z 2 + Δ B 2 ) B x B x mod sin ⁡ ( ω x t ) [ B x 2 + B y 2 + k ( B z 2 + Δ B 2 ) ] 2
P z | y = − 2 R p R 2 × ( B z 2 + Δ B 2 ) B y B y mod sin ⁡ ( ω y t ) [ B x 2 + B y 2 + k ( B z 2 + Δ B 2 ) ] 2
P z | z = 2 R p R 2 × ( B x 2 + B y 2 ) B z B z mod sin ⁡ ( ω z t ) [ B x 2 + B y 2 + k ( B z 2 + Δ B 2 ) ] 2
P z | x 1 = − 2 R p R 2 B x 1 B x 1 m Δ B 2 cos 2 θ sin ⁡ ( ω x 1 t ) [ B x 1 2 ( cos 2 θ + k sin 2 θ ) + k Δ B 2 ] 2
P z | z 1 = − 2 R p R 2 B z 1 B z 1 m Δ B 2 sin 2 θ sin ⁡ ( ω z 1 t ) [ B z 1 2 ( sin 2 θ + k cos 2 θ ) + k Δ B 2 ] 2
P z | x 1 = − 4 R p R 2 B x 1 B x 1 m Δ B 2 sin ⁡ ( ω x 1 t ) [ B x 1 2 ( 1 + k ) + 2 k Δ B 2 ] 2
P z | y 1 = − 2 R p R 2 × B y 1 B y 1 mod Δ B 2 sin ⁡ ( ω y 1 t ) [ B y 1 2 + k Δ B 2 ) ] 2
P z | z 1 = − 4 R p R 2 B z 1 B z 1 m Δ B 2 sin ⁡ ( ω z 1 t ) [ B z 1 2 ( 1 + k ) + 2 k Δ B 2 ] 2
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