Analysis and Suppression of the Cross-Axis Coupling Effect for Dual-Beam SERF Atomic Magnetometer

Abstract

Spin-exchange relaxation-free (SERF) atomic magnetometers operated under a near-zero magnetic field are used for vector magnetic field measurements with high sensitivity. Previously, the cross-axis coupling error evoked by a nonzero background magnetic field has been verified to be adverse in modulated single-beam magnetometers. Here, in a dual-beam unmodulated SERF magnetometer, we propose a somewhat different solution model for the cross-axis coupling effect where the field of interest couples with the interference field. Considering two cases where the transverse or longitudinal background field exists, the cross-axis coupling effect dependence on multiple factors is investigated here based on the dynamic response under a background magnetic field within ±5 nT. The theoretical and experimental investigation suggests that it has an adverse impact on the output response amplitude and phase and tilts the sensitive axis by several degrees, causing a measurement error on the dual-beam magnetometer. To suppress this effect, the background magnetic field is compensated through the PI closed-loop control. The coupling effect is effectively suppressed by 1.5 times at the 10–40 Hz low-frequency band and the sensitivity reaches 2.4 fT/Hz^1/2.

1. Introduction

Optically pumped atomic magnetometers (OPMs) have raised extensive development due to their advantages of high sensitivity [1,2]. Thereinto, atomic magnetometers operated in the spin-exchange relaxation-free (SERF) regime have realized sub-fT measurement sensitivity, becoming the most promising magnetometers for application in biomagnetic measurement, materials characterization, and fundamental physics research [3,4,5,6,7,8,9].

Among various application fields of SERF magnetometers, high sensitivity and low measurement error are essential factors for accurate information acquisition of the magnetic field, while their vector property will bring about the cross-axis coupling error due to the nonorthogonal angles and nonzero background magnetic field [10,11]. SERF magnetometers often work under the near-zero magnetic field environment with background magnetic fields shielded and compensated before measurement. However, the background magnetic fields are difficult to maintain at zero due to the magnetic field drift, which will lead to measurement error [12,13,14], as it also occurs in magnetometers of other principles, such as fluxgate and non-SERF atomic magnetometers [15,16]. In modulated single-beam SERF magnetometers, the gain error and source localization error due to the longitudinal (pump-axis) background magnetic field have been theoretically and experimentally studied [17,18]. Borna et al. [19] proposed the definition of the cross-axis coupling error and they explored the consequent localization and calibration inaccuracy of OPM-based magnetoencephalography systems, while for the dual-beam unmodulated configuration that possess greater potential to achieve higher sensitivity, a complete solution model for this adverse effect is still lacking. Previously, Jiang et al. revealed the interference effect when there existed a longitudinal background field and transverse (perpendicular to the pump-axis) interference field simultaneously, but they did not explain the potential factors that could influence the output signal, and the condition under a transverse background field was not referred [20]. Actually, we find that in a dual-beam unmodulated SERF magnetometer, there is a cross-axis coupling effect (the output signal contains the magnetic field information of the two orthogonal fields of interest and interference) aroused both along the longitudinal and transverse axes when there are background fields. It has an adverse effect on magnetometers’ performance, causing measurement error.

To suppress the cross-axis coupling effect, the triaxial magnetic field should be compensated to zero through active magnetic field compensation [21,22,23,24] along with the triaxial magnetic field modulation technology [25,26,27]. However, the modulation magnetic field of hundreds of nanotesla usually introduces extra spin exchange relaxation and deteriorates the sensitivity of the magnetometer. Hence, it is essential to suppress the coupling effect without damaging the sensitivity. Seltzer et al. proposed a triaxial SERF magnetometer working based on the quasi-static response. However, the sensitivity was of the order of 1 pT/Hz^1/2 for operation under the unshielded environment [28]. Hereinafter, the quasi-static response is used for suppression of the cross-axis coupling effect under the SERF regime with magnetic shielding, which is realized through the closed-loop control technique.

In this study, the performance of the orthogonal dual-beam SERF magnetometer under the background and interference magnetic fields is analyzed. The cross-axis coupling effect dependence on multiple factors, such as the magnitude and frequency of the background and interference magnetic fields, is measured and the adverse influence on the output signal of the magnetometer is verified. Based on the analysis, the coupling effect can be effectively suppressed by 1.5 times at 10–40 Hz through the closed-loop control of the background magnetic field.

2. Methods

For atomic magnetometers working in the SERF regime, the electron spin vector S evolution of the ground state alkali-metal atoms can be described by the Bloch equation [29]:

$$\frac{d}{dt}\mathbf{S} = \frac{1}{q}\left[{\gamma }_e\mathbf{B} \times \mathbf{S} + R_{\mathrm{o}\mathrm{p}}\left(\frac{1}{2}\mathbf{s} - \mathbf{S}\right) - \mathrm{\Gamma }\mathbf{S}\right],$$

(1)

where q denotes the nuclear slowing down factor, which is polarization-related, B is the magnetic field vector, γ_e ≈ 2π × 28 Hz/nT is the gyromagnetic ratio of the electron, R_op is the optical pumping rate, s is the photon spin vector along the pump laser (s = ±1 for a σ ± pumping laser), and Γ is the spin relaxation rate.

Assign that the pump and probe lasers are oriented along the z- and x-axes, respectively. The magnetic field of interest is along the y-axis. Denote S = (S_x, S_y, S_z)^T and B = (B_x(t), B_y(t), B_z(t))^T = (B_x0 + δB_x, B_y0 + δB_y, B_z0 + δB_z)^T, where B_x0, B_y0, and B_z0 are static background magnetic fields. δB_x, δB_y, and δB_z denote the dynamic fields. Under the low-frequency near-zero magnetic field (γ_e|B| << (R_op + Γ) (γ_e × 3.6 nT in our system)), the atomic spin vector along the x-axis detected by the probe laser is given by the steady-state solution of Equation (1):

$$S_x = \frac{R_{\mathrm{o}\mathrm{p}}\left[{\mathrm{\Gamma }}^{\prime }{\gamma }_e{B}_{y0} + {\mathrm{\Gamma }}^{\prime }{\gamma }_e\delta B_y + {\gamma }_e^2{B}_{x0}{B}_{z0} + {\gamma }_e^2{B}_{x0}\delta B_z + {\gamma }_e^2{B}_{z0}\delta B_x + {\gamma }_e^2\delta B_x\delta B_z\right]}{2{\mathrm{\Gamma }}^{\prime }\left[{\mathrm{\Gamma }}^{\prime 2} + {\gamma }_e^2{\left(B_{x0} + \delta B_x\right)}^2 + {\gamma }_e^2{\left(B_{y0} + \delta B_y\right)}^2 + {\gamma }_e^2{\left(B_{z0} + \delta B_z\right)}^2\right]},$$

(2)

where Γ^′ = R_op + Γ are introduced. Assuming B_x0, B_y0, and B_z0 are zeroed and there are no other interference fields imposed along the x- and z-axes, we can obtain:

$$S_x \approx G_{y0}\delta B_y,$$

(3)

where G_y0 is defined as the scale factor for the input δB_y. It can be expressed as

$$G_{y0} = {\gamma }_e{R}_{\mathrm{o}\mathrm{p}}/2{\mathrm{\Gamma }}^{\prime 2}.$$

(4)

However, if the magnetometer works under the background magnetic fields B_x0 or B_z0, the Larmor precession frequency due to γ_eB_x0 and γ_eB_z0 is nonnegligible compared to Γ^′. The B_x0- and B_z0-relative terms in Equation (2) cannot be ignored, which will have an unexpected effect on the output signal. Figure 1a shows the atomic spin vector evolution under zero magnetic field. An applied δB_y precesses the initial spin vector S_z0 into the x-z plane, inducing spin vector projection S_x along the x-axis. Hence, the magnetometer is sensitive to δB_y; in Figure 1b, suppose that there is a background magnetic field B_x0 and biaxial input δB_y and δB; then, S_x will contain the information of both input fields. We call this the cross-axis coupling effect between y- and z-axes, accounting for the term (Γ^′γ_eδB_y + γ_e²B_x0δB_z) in the numerator of Equation (2); the same coupling effect is aroused between y- and x-axes (term γ_e²B_z0δB_x) when B_z0 exists, as shown in Figure 1c.

In this study, we consider the coupling terms (Γ^′γ_eδB_y + γ_e²B_x0δB_z + γ_e²B_z0δB_x + γ_e²δB_xδB_z) in Equation (2) when the background and interference fields alternate between the x- and z-axes. Hereinafter, the cross-axis coupling effects between z- and y-axes (under B_x0), and x- and y-axes (under B_z0) are analyzed.

2.1. Cross-Axis Coupling Effect between z-and y-Axes under B_x0

First, we consider the case where the background magnetic field is along the x-axis. The vector magnetic field is expressed as B = (B_x0, δB_y, δB_z)^T. With B_y0 and B_z0 zeroed, Equation (2) can be simplified as

$$S_x = G_{y1}\delta B_y + G_z\delta B_z.$$

(5)

It can be seen that both δB_y and δB_z contribute to the output S_x. δB_y is viewed as the field of interest and δB_z is the interference field. G_y1 and G_z are the scale factors for the two fields, respectively, and they are given as

$$\left\{\begin{array}{l}{G}_{y1} = \frac{R_{\mathrm{o}\mathrm{p}}{\gamma }_e}{2\left[{\mathrm{\Gamma }}^{\prime 2} + {\gamma }_e^2{B}_{x0}^2 + {\gamma }_e^2{\left(\delta B_y\right)}^2 + {\gamma }_e^2{\left(\delta B_z\right)}^2\right]}\\ G_z = \frac{R_{\mathrm{o}\mathrm{p}}{\gamma }_e^2{B}_{x0}}{2{\mathrm{\Gamma }}^{\prime }\left[{\mathrm{\Gamma }}^{\prime 2} + {\gamma }_e^2{B}_{x0}^2 + {\gamma }_e^2{\left(\delta B_y\right)}^2 + {\gamma }_e^2{\left(\delta B_z\right)}^2\right]}\end{array}\right..$$

(6)

Seen from the denominator of G_y1 in Equation (6), B_x0 and δB_z result in the decrease in the scale factor for δB_y. Moreover, even if there is no input magnetic field along the sensitivity axis, i.e., δB_y = 0, there is a nonzero output for S_x, causing measurement error for the magnetometer. Hence, we find that the B_x0 and δB_z affect the output by changing the sensitivity and tilting the sensitive axis of the magnetometer. The tilting degree is expressed as [19]

$${\phi }_{z,y} \approx \arctan \left(G_z/G_{y1}\right).$$

(7)

To obtain the output signal under the periodic input signal, the dynamic response is analyzed, and the dynamic fields δB_x, δB_y, and δB_z are expressed with a cosinusoidal signal. Assign B = (B_x0, B_ycos(ω_yt + φ_y0), B_zcos(ω_zt + φ_z0))^T, where φ_y0 and φ_z0 are initial input phases. The dynamic solution along the x-axis is given as:

$$S_x^{AC} = A_y\sin \left({\omega }_{y}t + {\phi }_{y0} + {\theta }_y\right) + A_z\sin \left({\omega }_{z}t + {\phi }_{z0} + {\theta }_z\right) + o\left[\omega \left({\omega }_y,{\omega }_z\right)\right],$$

(8)

where $A_y = \frac{S_z{}_0{\gamma }_e{B}_y}{\sqrt{\mathrm{\Gamma }{\prime }^2 + q^2{\omega }_y{}^2}}$ and $A_z = \frac{S_z{}_0{\gamma }_e{}^2{B}_z{B}_x{}_0\sqrt{q^2{\omega }_z{}^2/\mathrm{\Gamma }{\prime }^2 + 1}}{\left(\mathrm{\Gamma }{\prime }^2 + q^2{\omega }_z{}^2\right)}$ denote the amplitude responses corresponding to the fields of interest and interference. Θ_y = arctan(Γ′/qω_y) and θ_z = arctan[Γ′/(qω_z)] denote the output phase delay relative to input initial phase. O[ω(ω_y, ω_z)] are the higher-order terms in the other frequency component that is not included in this discussion. S_z0 is the steady-state spin vector obtained when the transverse magnetic field has little effect on the atomic spin polarization (γ_eB_x0 << Γ′). By extraction of the frequency-related components for y- and z-axes from the output signal, the cross-axis coupling coefficient for z-and y-axes responses is defined as

$$C_{z,y} = A_z/A_y = {\gamma }_e{B}_{x0}\frac{B_{z1}}{B_{y1}}\frac{\sqrt{{\mathrm{\Gamma }}^{\prime }{}^2 + q^2{\omega }_y{}^2}}{{\mathrm{\Gamma }}^{\prime }{}^2 + q^2{\omega }_z{}^2}.$$

(9)

Actually, when γ_eB_x0 is increased and comparable with Γ′, its influence on the spin polarization cannot be ignored anymore. On this occasion, the analytic solution for the Bloch equation is no longer accessible, and there will be nonlinear terms related with B_x0 that are added in S_z0 and A_y, which will be verified in the experiments, while they are not given here with analytic expression.

2.2. Cross-Axis Coupling Effect between x- and y-Axes under B_z0

Here, the background magnetic field is assumed to be along the z-axis. The vector magnetic field is expressed as B = (δB_x, δB_y, B_z0). With B_x0 and B_y0 zeroed, there is

$$S_x = G_{y2}\delta B_y + G_x\delta B_x,$$

(10)

The magnetometer is simultaneously sensitive to δB_y and δB_x. The scale factors are given as

$$\left\{\begin{array}{l}{G}_{y2} = \frac{R_{\mathrm{o}\mathrm{p}}{\gamma }_e}{2\left[{\mathrm{\Gamma }}^{\prime 2} + {\left({\gamma }_e^2{B}_x\right)}^2 + {\left({\gamma }_e^2{B}_y\right)}^2 + {\left({\gamma }_e^2{B}_{z0}\right)}^2\right]}\\ G_x = \frac{R_{\mathrm{o}\mathrm{p}}{\gamma }_e^2{B}_{z0}}{2{\mathrm{\Gamma }}^{\prime }\left[{\mathrm{\Gamma }}^{\prime 2} + {\left({\gamma }_e^2{B}_x\right)}^2 + {\left({\gamma }_e^2{B}_y\right)}^2 + {\left({\gamma }_e^2{B}_{z0}\right)}^2\right]}\end{array}\right..$$

(11)

The magnetic fields B_z0 and δB_x result in the decrease in the scale factor for δB_y, as seen from the denominator of G_y2. It also reveals that even if there is no input δB_y, the output of the magnetometer is not zero, causing measurement error. The tilting degree is

$${\phi }_{x,y} \approx \arctan \left(G_x/G_{y2}\right).$$

(12)

For solution of the dynamic response, set the total magnetic field as B = (B_xcos(ω_xt + φ_x0), B_ycos(ω_yt + φ_y0), B_z0), where φ_x0 and φ_y0 are initial input phases. The dynamic solution is [20]

$$S_x^{AC} = A_x\sin \left({\omega }_{x}t + {\phi }_{x0} + {\theta }_x\right) + A_y\sin \left({\omega }_{y}t + {\phi }_{y0} + {\theta }_y\right),$$

(13)

where $A_x = \frac{S_z{}_0{\gamma }_e{}^2{B}_z{}_0{B}_x}{\sqrt{{\left(\mathrm{\Gamma }{\prime }^2 - {\gamma }_e{}^2{B}_z{{}_0}^2 + q^2{\omega }_x{}^2\right)}^2 + 4\mathrm{\Gamma }{\prime }^2{\gamma }_e{}^2{B}_z{{}_0}^2}}$ and $A_y = \frac{S_z{}_0{\gamma }_e{B}_y\sqrt{\mathrm{\Gamma }{\prime }^2 + q^2{\omega }_y{}^2}}{\sqrt{{\left(\mathrm{\Gamma }{\prime }^2 - {\gamma }_e{}^2{B}_z{{}_0}^2 + q^2{\omega }_y{}^2\right)}^2 + 4\mathrm{\Gamma }{\prime }^2{\gamma }_e{}^2{B}_z{{}_0}^2}}$ denote the amplitude responses to the fields of interference and interest. ${\theta }_x = \arctan \left(\frac{\mathrm{\Gamma }{\prime }^2 + {\gamma }_e{}^2{B}_z{{}_0}^2 - q^2{\omega }_x{}^2}{2q{\omega }_x\mathrm{\Gamma }\prime }\right)$ and ${\theta }_y = \arctan \left[\frac{\mathrm{\Gamma }\prime \left(\mathrm{\Gamma }{\prime }^2 + {\gamma }_e{}^2{B}_z{{}_0}^2 + q^2{\omega }_y{}^2\right)}{q{\omega }_y\left(\mathrm{\Gamma }{\prime }^2 - {\gamma }_e{}^2{B}_z{{}_0}^2 + q^2{\omega }_y{}^2\right)}\right]$ denote the output phase delay relative to the input initial phase. The cross-axis coupling coefficient is defined as

$$C_{x,y} = A_x/A_y = \frac{{\gamma }_e{B}_{z0}}{\sqrt{{\mathrm{\Gamma }}^{’2}+q^2{\omega }_y{}^2}}\frac{B_{x1}}{B_{y1}}\frac{\sqrt{{\left({\mathrm{\Gamma }}^{\prime 2} - {\gamma }_e^2{B}_{z0}{}^2 + q^2{\omega }_y{}^2\right)}^2 + 4{\mathrm{\Gamma }}^{\prime 2}{\gamma }_e^2{B}_{z0}{}^2}}{\sqrt{{\left({\mathrm{\Gamma }}^{\prime 2} - {\gamma }_e^2{B}_{z0}{}^2 + q^2{\omega }_x{}^2\right)}^2 + 4{\mathrm{\Gamma }}^{\prime 2}{\gamma }_e^2{B}_{z0}{}^2}}.$$

(14)

According to Equation (2), the two cross-axis coupling terms (γ_e²B_x0δB_z + γ_e²B_z0δB_x) reveal that the background and interference fields render the cross-multiplication relationship. Hence, the coupling effect can be suppressed by compensating the background magnetic fields B_x0 and B_z0 to zero and the coupling terms can be eliminated. Then, Equation (2) is evolved into

$$S_x{}^{\prime } = \frac{R_{\mathrm{o}\mathrm{p}}\left[{\mathrm{\Gamma }}^{\prime }{\gamma }_e\delta B_y + {\gamma }_e^2\delta B_x\delta B_z\right]}{2{\mathrm{\Gamma }}^{\prime }\left[{\mathrm{\Gamma }}^{\prime 2} + {\gamma }_e^2{\left(\delta B_x\right)}^2 + {\gamma }_e^2{\left(\delta B_y\right)}^2 + {\gamma }_e^2{\left(\delta B_z\right)}^2\right]}.$$

(15)

Equation (15) reveals that the output signal is still influenced by the interference magnetic field (seen from the term γ_e²δB_xδB_z) even if the background magnetic field is compensated, presenting a better suppression consequence under a smaller interference magnetic field. Subsequently, the background magnetic field will be compensated through closed-loop control and the verification experiments will be carried out.

3. Experimental Setup and Procedure

The schematic for the experimental setup is shown in Figure 2. A spherical vapor cell with an outer diameter of 10 mm is used as the sensing element, containing potassium (K), 10 Torr N₂ as the quenching gas, and 700 Torr ⁴He as the buffer gas. To maintain a high-temperature environment, the cell is placed in a ceramic oven, which is attached with flexible polyimide-twisted coils driven by a 100 kHz alternating current. Here, the operating temperature is 18 °C, and the corresponding K atoms density in the cell is about 6.1 × 10¹³/cm³. The magnetometer is integrated into a compact configuration residing in a cylindrical magnetic shield. The magnetic shield consists of an outer four-layer μ-metal to attenuate the Earth’s magnetic field and an inner one-layer ferrite shield with higher resistivity to further reduce the magnetic noise, providing a low-field environment with a residual field of several nanotesla and magnetic noise below 1 fT/Hz^1/2 [30,31]. A set of triaxial coils (x-axis: Helmholtz coils; y-and z-axes: saddle coils) are mounted inside the shield. The coils are driven by function generators (33500B, Keysight), through which the residual magnetic field is compensated and the desired field is applied. Due to the magnetic eddy interference effect caused by the close distance between the coils and ferrite magnetic shield, the coils constants are recalibrated using a fluxgate before operation.

Both the pump and probe lasers are generated by distributed Bragg reflector (DBR) lasers with a collimated 1/e² waist diameter of 2.7 cm. The circularly polarized pump laser used to polarize the K atoms is detuned to the center of the pressure-broadened D1 line of 770.10638 nm and the linearly polarized laser used to probe the atomic polarization is 120 GHz blue detuned from the D1 line.

For the magnetic field measurement and control, the polarization change underwent by the probe laser along the x-axis is measured via balanced polarimetry. Thereafter, a trans-impedance amplifier is used to amplify the detected photodetector signal and transfer it into the voltage signal. Then, the voltage signal is collected and processed by a DAQ and demodulated via a LIA and used for subsequent PI closed-loop control.

The cross-axis coupling effect between z- and y-axes (under B_x0), and x-and y-axes (under B_z0) is studied in the background magnetic field of 0–5 nT (the typical operation range for SERF magnetometers). By collection of the output phase and the amplitude of the sensitive field δB_y, and interference fields δB_x and δB_z via a LIA, the performance of the magnetometer is evaluated. Then, the amplitude and tilting degree for δB_y dependence on the frequency is measured. Finally, the cross-axis coupling effect is suppressed through the dynamic compensation of the background magnetic field.

4. Results and Discussion

4.1. Cross-Axis Coupling Coefficient and Tilting Degree Measurement

To illustrate how the background magnetic fields B_x0 and B_z0 influenced the output signal, a static magnetic field ranging from −5 to +5 nT was successively applied along the x- and z-axes, respectively. The experimental parameters were set as f_y = ω_y/2π = 30.5 Hz, f_i = ω_i/2π = 20 Hz, φ_i0 = 0 (i = x, z), and B_j-rms = 10 pT (root-mean-square value for dynamic sinusoidal signal, j = x, y, z). Through a low-noise LIA, the response amplitude A_j and phase θ_j of the output signal at 30.5 Hz and 20 Hz were extracted, respectively.

Figure 3a shows the biaxial input signals (along the y- and x-axes) and the output signal under the background magnetic field B_z0. Seen from the frequency domain, both input signals contributed to the output signal. The same was true under B_x0. For the signal of interest B_y, it can be seen from Figure 3b that the response amplitude A_y showed an approximate absorption curve with increasing |B_x0|, which was mainly caused by the atomic spin polarization S_z0 precession, while it would not come up under the longitudinal field B_z0, because the spin polarization precession only occurred when there was an orthogonal magnetic field component. When B_z0 was applied, A_y increased, indicating a typical magnetic resonance tendency evoked by B_z0 and f_y. The increased coupling coefficients C_z,y and C_x,y indicated a stronger cross-axis coupling effect with increasing background field, which was consistent with Equations (9) and (14).

The phase delay was also influenced by the background field. Except for the phase reversals of θ_z and θ_x occurring at B_x0 = 0 and B_z0 = 0, respectively, the rates of change in θ_y were 20.2°/5 nT (with B_x0 applied) and 18.8°/5 nT(with B_z0 applied). Referring to the influence factors related to the phase of the magnetometer signal, except for the background magnetic field, there are other factors including the coils’ inductance effect and the coupling between the coil and the magnetic shielding system. These factors are negligible under a small-magnitude and low-frequency magnetic field. Only the phase delay caused by the background magnetic field was considered.

The combined effect of the amplitude and phase delay would adversely influence the property of the magnetometer, including the sensitivity and the measurement accuracy of the magnetic field. At the same time, the phase delay is especially at a disadvantage for the capture of the instantaneous dynamic field and the magnetic source localization under the multi-sensor condition. Subsequently, only the positive-field measurement is given for the sake of brevity of the data.

To demonstrate how the interference field influenced the output signal, the dynamic response amplitude, coupling coefficient, and tilting degree variation with the background magnetic field were measured under a varied interference field amplitude. In Figure 4a, A_y decreased with increasing B_x0 under different B_z-rms. C_z,y and φ_z,y showed a stronger cross-axis coupling effect with increasing B_x0 and B_z-rms, while it was violated when B_z-rms was larger than 4 nT. When B_z0 was applied as shown in Figure 4b, A_y tended to increase with B_z0 under small B_x-rms, while the rule was violated under larger B_x-rms. It was found that the tilting degree φ_z,y (B_x0 = 0.22 nT, B_z-rms = 0.01 nT, and φ_z,y = 8.49°; B_x0 = 0.22 nT, B_z-rms = 0.2 nT, and φ_z,y = 71.48°) increased faster than φ_x,y (B_z0 = 0.22 nT, B_x-rms = 0.01 nT, and φ_x,y = 2.61°; B_z0 = 0.22 nT, B_x-rms = 0.2 nT, and φ_x,y = 42.84°) with increasing interference field, indicating that a background magnetic field of the order of a hundred picotesla would cause severe tilting degrees of the axis with the field of interest.

In addition, the frequency response was measured and is shown in Figure 5. The frequency ranged from 4 to 200 Hz, which covered the typical bandwidth of the SERF magnetometer. Figure 5a,b show that when B_x0 was applied, A_y decreased and the cross-axis coupling effect enhanced monotonously with higher f_y. A_y tended to be less affected by f_y when the frequency was above 200 Hz, which was consistent with the derivation in Equation (8). The coupling effect was weaker with higher f_z and it tended to be less affected by f_z when the frequency was above 200 Hz. In addition, a larger interference field B_z-rms gave rise to a smaller A_y and stronger coupling effect at a certain frequency. When B_z0 was applied, A_y and the cross-axis coupling effect showed a nonmonotonic variation with frequency due to the magnetic resonance. With increased f_y, A_y tended to increase first and then decrease, and it was less affected by f_y over 200 Hz, while the coupling effect did the opposite. With increased f_x, the coupling effect increased first and then decreased. It was less affected by f_x over 200 Hz.

According to C_z,y and C_x,y in Figure 5, the tilting degree of the sensitive axis was extracted, as shown in Figure 6. Φ_z,y was more affected by the higher frequency of the sensitive field while being less affected by the higher frequency of the interference field, indicating a more severe measurement error for the high-frequency signal of interest. For φ_x,y, it increased monotonically with higher f_y and decreased with higher f_x under small B_x-rms (<1 nT), while it was violated when B_x-rms was larger.

4.2. Suppression of the Cross-Axis Coupling Effect

Here, the in-phase components of the demodulated amplitude were used for PI closed-loop control and feedback of the compensation value to the axis with background field drift. The control flow diagram is given in Figure 7. To visibly compare the suppression effect before and after compensation, a magnetic field of nT order was applied. The background magnetic field B_x0 = 1 nT (root-mean-square value, the same below) was set at 0.3 Hz and the interference field B_z-rms = 1.2 nT was set at f_z = 110 Hz, φ_z0 = 0; the background magnetic field B_z0 = 3 nT was set at 0.3 Hz and the interference field B_x-rms = 3.5 nT was set at f_x = 110 Hz, φ_x0 = 0.

The output voltage noise spectral density of the magnetometer was recorded in Figure 8a. Comparing the voltage signal before and after compensation, the cross-axis coupling effect was effectively suppressed by 1.5 times at 10–40 Hz. By transferring the collected voltage noise spectral density into magnetic field sensitivity, as shown in Figure 8b, the sensitivities for B_y improved from 3.9 to 2.7 fT/Hz^1/2 with B_x0 applied and improved from 2.5 to 2.4 fT/Hz^1/2 with B_z0 applied at 20–40 Hz. It showed a more obvious suppression effect for the background magnetic field along the x-axis, which was consistent with the results in Figure 4. As for the 0.3 Hz dynamic background field drift, it was compensated with the PI control phase delay of 2.36° (x-axis) and 2.97° (z-axis), while for a higher-frequency field drift at 18 Hz, there was a larger phase delay with 6.06° (x-axis) and 9.82° (z-axis) due to the higher bandwidth and higher order of the low-pass filter, giving rise to a poorer suppression effect.

For static field compensation with no drift, the method illustrated above is also available. It is worth noting that the background field compensation only solves the cross-axis coupling problem aroused by the background field, while it did not restrain the A_y variation, due to the interference field, as shown in the denominator of Equation (15) and Figure 4 (see curves under B_x0 = 0, B_z0 = 0), which is a question worth further investigation.

5. Conclusions

In summary, the background-magnetic-field-evoked cross-axis coupling effect for a dual-beam SERF atomic magnetometer was analyzed and verified under the background field within ±5 nT (typical SERF regime), indicating the adverse impact on the output signal. For the cross-axis coupling effect between z- and y-axes under a transverse background field B_x0, the coupling enhanced with increased B_x0 and f_y, and the response amplitude A_y of the signal of interest tended to decrease with increased B_x0, so did the sensitivity, while for the effect between x- and y-axes under longitudinal background field B_z0, the coupling variation was nonmonotonic due to the magnetic resonance evoked by B_z0 and B_y. Through the cross-axis magnetic field compensation method, the coupling effect was effectively suppressed by 1.5 times at 10–40 Hz. The method is applicable for static and low-frequency background field compensation, while the frequency of the interference and background fields should satisfy the condition of basic modulation and demodulation regulation because the two are multiple items, which limits the compensation bandwidth to dozens of Hz. The triaxial magnetic field closed-loop control technique with large-field modulation is efficient for field compensation with a wider frequency band and higher dynamic range, but it will sacrifice the sensitivity. An appropriate method should be selected according to practical application.