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Abstract
Triaxial magnetic field compensation is crucial for a zero-field optically pumped magnetometer (OPM) in pursuit of a zero-field environment. In this work, we demonstrate a triaxial magnetic field compensation method for zero-field OPM based on single-beam configuration. It consists of two procedures: (1) pre-compensation to preliminarily cancel out ambient residual magnetic field by low-frequency magnetic field modulation; and (2) precise compensation to further compensate the residual magnetic field by high-frequency magnetic field modulation. This scheme enables rapid and precise compensation of a large-scale magnetic field and supports real null-point acquisition of the triaxial residual magnetic fields with simple processes. The experimental results show that the compensation resolution on the sensitive axis is better than 1 pT and significantly less than the fluctuation of experimental environments. Our work targets on the quick generation of a zero-field environment for high precision OPM, which is especially advantageous for emerging applications including magnetocardiography (MCG) and magnetoencephalography (MEG).
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a viable alternative to superconducting quantum interference device (SQUID) [1,2], the low-cost zero-field optically pumped magnetometer (OPM) operating in the spin-exchange relaxation-free (SERF) regime can provide comparable measurement sensitivity to the magnetic field without the need for cryogenics [3–6]. Meanwhile, the convenience of OPM miniaturization allows for being arranged in an array, improving the flexibility of sensor layout and the spatial resolution of imaging scanning [7,8]. In addition, the improved proximity to the subject contributes to reducing the distance attenuation of the signal. These capabilities make the zero-field OPM a new generation of magnetocardiography (MCG) and magnetoencephalography (MEG) systems [9–12]. And commercially-integrated zero-field OPM has enabled moving MEG applications toward real-world applications with a wearable system [13,14].
The extremely low magnetic field environment can effectively suppress the spin relaxation from spin-exchange collisions, which enables the zero-field OPM to operate in the SERF regime and achieve ultra-high sensitivity [15,16]. Its realization often goes through two processes: passive magnetic shielding and active magnetic field compensation [13]. The former can greatly attenuate the ambient magnetic field to the nT level by using multi-layer high permeability soft magnetic materials. But further reduction is hindered by the remanence of the shielding material. Consequently, the latter is needed to further compensate the residual magnetic field around the vapor cell to near-zero field, which is implemented by triaxial magnetic field compensation coil.
In previous studies, various triaxial magnetic field compensation schemes for zero-field OPM have been proposed. Nevertheless, these schemes, mainly based on cross-field modulation and electron paramagnetic resonance, are generally applied to the dual-beam (pump-probe) configuration [17,18]. However, the widely used miniaturized atomic magnetometers currently employ a single-beam configuration in which only single beam is used to play the dual roles of optical pumping and probing. It has the advantages of simple structure and compactness. For magnetometers with single-beam configuration, Fang et al. proposed a scheme based on the inherent dependence of the steady-state longitudinal polarization on the triaxial static magnetic fields [19,20]. The scheme is simple to operate and can be applied to a large residual magnetic field range. But the compensation resolution is relatively low, only achieving sub-nT level. Tang et al. presented a compensation method by employing an off-resonance elliptically polarized light, which was implemented by superimposing an orthogonal DC offset field to the modulation field [21], and achieved a magnetic compensation resolution of 6.7 pT for all three axes. However, this method does not strictly consider the coupling effect of the triaxial magnetic field, and lacks an evaluation of the real magnetic null point. Therefore, it’s essential to propose a new triaxial precise compensation scheme specifically applicable to the miniaturized single-beam zero-field OPM.
In this work, we propose a high-precision triaxial magnetic field compensation method for zero-field OPM based on single-beam configuration. Firstly, the response behavior of the longitudinal spin polarization under the combined action of a modulated magnetic field and triaxial quasi-static magnetic fields is theoretically investigated. Then, the decoupling of the three components of the residual magnetic field and the approximation of real null point of the triaxial residual magnetic fields are demonstrated based on the analysis results. Finally, in order to evaluate the effectiveness of the compensation scheme, experiments are conducted by using a single-beam zero-field OPM with a 4 mm $\times$ 4 mm $\times$ 4 mm $^{87}$Rb atomic vapor cell, and a compensation resolution of sub-pT on the sensitive axis is achieved. The experimental results show that this scheme enables precise compensation of residual magnetic field to zero field, which is especially advantageous for applications including MCG and MEG systems.
2. Principle
For the zero-field OPM operating in the SERF regime, the overall evolution of the alkali-atoms spin polarization vector can be described by the Bloch equation [22,23]:
In practice, a high-frequency modulated magnetic field $B_{\rm {mod}}$ cos($\omega t$) is often applied along the transverse x-axis, i.e. the x-mode modulation [24,25], which helps suppress low-frequency noise and improve the sensitivity of atomic magnetometers. The amplitude of the three orthogonal components $B^x_0$, $B^y_0$, $B^z_0$ of residual magnetic field are very small relative to $B_{\rm {mod}}$ after the passive magnetic shielding and preliminary coarse active magnetic field compensation, and the evolution of $\boldsymbol {P}$ mainly occurs in the y-z plane. Based on this approximation, the fundamental frequency response signal $R^{\omega }_z$ can be expressed in the following form [26]:
In order to explore the response characteristics of $R^{\omega }_z$ under the combined action of $B^x_0$, $B^y_0$ and $B^z_0$, and to realize the precise magnetic field compensation of the three axes, the coupling effects of non-sensitive axes on $R^{\omega }_z$ should be included. By introducing the perturbation-iteration method, Dupont-Roc made a systematic derivation on this issue, and gave an exact calculation of $R^{\omega }_z$ [27]. The expression of $R^{\omega }_z (B^x_0, B^y_0, B^z_0)$ for the universal case containing the three components of an arbitrary vector magnetic field is given by
Obviously, the zero crossing of $R^{\omega }_z$ is affected by $B^x_0$, $B^y_0$ and $B^z_0$ simultaneously. To more intuitively reflect the dependence of $R^{\omega }_z$ on the three components of residual magnetic field in the linear range, Eq. (4) needs to be further analyzed. Taking the first-order terms of Taylor expansion of Eq. (4) at $B^x_0 = 0$, $B^y_0 = 0$, $B^z_0 = 0$ respectively yields
where $k_x = \frac {R_{\rm {op}} \gamma ^e J_0 (\beta ) J_1 (\beta )}{\Gamma ^2 + \gamma ^{e 2} \left ( {B^y_0}^2 {+ B^z_0}^2 \right ) J^2_0 (\beta )}$, $b_x = \frac {R_{\rm {op}} \gamma ^{e 2} J^3_0 (\beta ) J_1 (\beta ) {B^y_0 B^z_0}}{\Gamma \left [ \Gamma ^2 + \gamma ^{e 2} \left ( {B^y_0}^2 {+ B^z_0}^2 \right ) J^2_0 (\beta ) \right ]}$; where $k_y = \frac {R_{\rm {op}} J^3_0 (\beta ) J_1 (\beta ) \gamma ^{e 2} {B^z_0}}{\Gamma \left [ \Gamma ^2 + \gamma ^{e 2} {B^x_0}^2 + \gamma ^{e 2} {{B^z_0}^2} J^2_0 (\beta ) \right ]}$, $b_y = \frac {R_{\rm {op}} J_0 (\beta ) J_1 (\beta ) \gamma ^e {B^x_0}}{\Gamma ^2 + \gamma ^{e 2} {B^x_0}^2 + \gamma ^{e 2} {B^z_0}^2 J^2_0 (\beta )}$; where $k_z = \frac {R_{\rm {op}} J^3_0 (\beta ) J_1 (\beta ) \gamma ^{e 2} {B^y_0}}{\Gamma \left [ \Gamma ^2 + \gamma ^{e 2} {B^x_0}^2 + \gamma ^{e 2} {{B^y_0}^2} J^2_0 (\beta ) \right ]}$, $b_z = \frac {R_{\rm {op}} J_0 (\beta ) J_1 (\beta ) \gamma ^e {B^x_0}}{\Gamma ^2 + \gamma ^{e 2} {B^x_0}^2 + \gamma ^{e 2} {B^y_0}^2 J^2_0 (\beta )} .$Many pieces of instructive information are implied in Eqs. (5)–(7). First, there is a linear relationship between $R^{\omega }_z$ and $B_0^x$ (or $B^y_0$, $B^z_0$). Second, the slope $k_x$ is obviously higher than $k_y$ and $k_z$, which means $R^{\omega }_z$ is more sensitive to $B_0^x$ than $B_0^y$ and $B_0^z$. And $k_y$ and $k_z$ show a dispersion relationship with $B^z_0$ and $B^y_0$, respectively, and their half width at half maximum (HWHM) are both determined by $\Gamma$. Third, the specific form of intercept $b_x$ indicates that the zero crossing of $R^{\omega }_0 (B^x_0)$ is the real magnetic null point of $x$-axis unless on the condition that $B^y_0 = 0$ or $B^z_0 = 0$, otherwise none of it could happen. Moreover, both $b_y$ and $b_z$ exhibit a dispersion dependence on $B_0^x$. And in the vicinity of the near-zero field, the decrease of $B_0^x$ will make the zero crossing of $R^{\omega }_0 (B^y_0)$ or $R^{\omega }_0 (B^z_0)$ closer to the magnetic null point. In turn, the reduction of $B^y_0$ and $B^z_0$ will further promote $B^x_0$ to approach the magnetic null point. Based on the above analysis results, the specific steps to realize the triaxial precise magnetic field compensation by observing $R^{\omega }_z (B^x_0, B^y_0, B^z_0)$ can be determined.
Specifically, as shown in Fig. 1(a), in the case where both $B^y_0$ and $B^z_0$ are not completely zeroed, taking typical case $B^y_0 = B^z_0 = 2$ nT as an example, 0.4 nT residual magnetic field still exists on the $x$-axis when $R^{\omega }_z (B^x_0) = 0$. After realizing that $R^{\omega }_z (B^x_0 = 0.4\ \rm {nT}) = 0$, the next step is to zero the residual magnetic field of $y$-axis. As depicted in Fig. 1(b), when there are different static residual or DC-excitation magnetic fields $B^z_{\rm {DC}}$ on the z-axis, the zero crossing of $R^{\omega }_z (B^y_0)$ is closer to the real magnetic null point of y-axis as $B^z_{\rm {DC}}$ increases, which is indicated by the expression of $b_y$. The scale factor reflects the sensitivity of $R^{\omega }_z$ to the variation of ambient magnetic field. Excessive excitation magnetic field should be avoided, for the reason that a larger excitation magnetic field will introduce additional relaxation rate to the atomic ensemble [28]. Here, taking the excitation magnetic field $B^z_{\rm {DC}} = 8$ nT as an example, $B^y_0 = 0.5$ nT when $R^{\omega }_z (B^y_0) = 0$. Comparing Eq. (6) with Eq. (7), it is obvious that $R^{\omega }_z (B^y_0)$ and $R^{\omega }_z (B^z_0)$ share the similar form. When compensating the residual magnetic field of the non-sensitive axis, the effects of $B^y_0$ and $B^z_0$ on $R^{\omega }_z$ are equivalent. Consequently, only the analysis process of y-axis magnetic field compensation is presented here. And the same goes for z-axis, i.e., under the condition of $B^y_{\rm {DC}} = 8$ nT, $B^z_0 = 0.5$ nT when $R^{\omega }_z (B^z_0) = 0$. After the next iteration, as shown in Fig. 1(c) and Fig. 1(d), the zero crossing of the triaxial response $R^{\omega }_z$ will further approximate the real magnetic null point.
Fig. 1. Analytical results from Eq. (4) about the dependence of $R^{\omega }_z$ on $B^x_0$ and $B^y_0$ with the initial condition $R_{\rm {op}} = R_{\rm {rel}} = 500\ s^{- 1}$ and $B_{\rm {mod}} = 200$ nT, $\omega = 1\ \rm {kHz}$. (a) The zero-crossing sinal appears at $B^x_0 = 0.4$ nT when $B^y_0 = B^z_0 = 2$ nT. (b) Under the condition of $B^x_0 = 0.4$ nT, different longitudinal excitation fields $B^z_{\rm {DC}}$ induce different zero crossings. (c) and (d) is the second round of magnetic compensation, indicating the zero crossing is closer to the real magnetic null point.
3. Experimental setup
The experimental setup of the zero-field OPM is illustrated in Fig. 2. A cubic vapor cell (external length: 4 mm, wall thickness: 0.5 mm) containing a droplet of $^{87}$Rb and roughly 2.5 amg of N$_{2}$ buffer and quenching gas is placed at the center of the triaxial coil. The vapor cell is heated to 150 $^{\circ }$C by 200 kHz AC-driven electronic heaters in an oven which is made of boron nitride. The temperature is monitored by a non-magnetic Pt1000 for real-time closed-loop control.
Fig. 2. Experimental setup of the single-beam zero-field OPM. PMF: polarization maintaining fiber; OFC: optical fiber collimator; $\lambda$/4: quarter-wave plate; TIA: transimpedance amplifier; Ref: reference signal; DAQ: data acquisition system.
The pump light tuned to $^{87}$Rb D1 line is emitted by a distributed feedback (DFB) laser with fiber coupling output, and propagates along the z-axis. It is delivered by a non-magnetic fiber collimator with an output light intensity of about 12 mW/cm$^{2}$, and then passes through a linear polarizer and a zero-order quarter-wave plate to generate a circularly polarized light. After the cell, the transmitted pump light is detected by a photodiode and converted into voltage signal by the ultra-low noise photodiode amplifier. The lock-in amplifier (Zurich Instruments, MFLI) generates a modulated magnetic field with a frequency f = 1 kHz along the x-axis, and synchronously demodulates the magnetic field information at specific frequency. The response signal of the magnetometer is recorded by a data acquisition system.
The magnetic shield consists of four layers of mu-metal, which keeps the residual magnetic field below 2 nT. A set of triaxial coil system in the magnetic shield is used to zero the residual magnetic field at the location of the vapor cell. The triaxial coil consists of two saddle coils placed orthogonally on the x- and y-axis, and a Lee-Whiting coil placed on the z-axis [29,30], and is driven by function generators (Keysight, 33522B). The coil constants are calibrated by using a fluxgate magnetometer. The modulation magnetic field, DC excitation magnetic field, and calibration magnetic field are added to the coil through a summing amplifier.
4. Results and discussion
In the following, we illustrate that the residual magnetic field can be eliminated through the proposed combined magnetic field compensation, and determine the magnetic field compensation resolution.
4.1 Pre-compensation
In practical applications such as MEG system, the residual magnetic field in the central area of the magnetic shielding room and the magnetic shielding cylinder is often within the range of several nT to 50 nT [31,32], which is sometimes beyond the typical linear region of the SERF magnetometer. Therefore, before performing precise magnetic field compensation, a pre-compensation process must be implemented to cancel out the majority of residual magnetic to below 1 nT. A low-frequency oscillating magnetic field of amplitude $B^x_{\rm {mod}}$ = 3 nT and angular frequency $\omega$ = 3 Hz is applied along the x-axis, i.e. $B_x = B^x_0 + B^x_{\rm {mod}} \sin (\omega t)$. Based on Eq. (2), the quasi-static response of $P^{x - \rm {mod}}_z$ is given by
Fig. 3. The time-domain oscillatory response signal of pre-compensation. (a) The blue dashed line and the red solid line represent the response of the x-axis before and after compensation respectively. (b) The cyan solid line and the yellow dot-dashed line represent the near zero-field response of the z-axis with $B^y_0 = 0$ nT and $B^y_0 = 5$ nT respectively.
Different from x- and y-axis, when the low-frequency oscillating magnetic field is applied along the z-axis, i.e. $B_z = B^z_0 + B^z_{\rm {mod}} \sin (\omega t)$, the quasi-static response of $P^{x - \rm {mod}}_z$ is
Similarly, the $\omega$-component contribution to $P^{z-\rm {mod}}_z (t)$ is affected by $B^z_0$, and only the 2$\omega$-component is dominant when $B^z_0 = 0$ nT. However, in order to observe a distinct time-domain oscillatory signal, it is necessary to add an offset magnetic field on the transverse directions, as well as increase $B^z_{\rm {mod}}$ appropriately. Specifically, as presented in Fig. 3(b), we deliberately increase $B^y_0$ to 5 nT and set $B^z_{\rm {mod}} = 10$ nT, $\omega$ = 3 Hz. Otherwise, no obvious fundamental frequency oscillation signal could be observed even after completing the z-axis compensation.
Taking the observed dominant 2$\omega$ signal as the criterion, the pre-compensation of the triaxial magnetic field is achieved, and $B^x_0$, $B^y_0$, $B^z_0$ are all near the zero-field resonance. The pre-compensation will ensure that the magnetometer operates in the SERF regime.
4.2 Precise compensation
When performing precise magnetic field compensation, a high-frequency modulated magnetic field ($\omega$ = 1 kHz) with an amplitude of 100 nT is applied along the x-axis, keeping the magnetometer response away from the technical noise sources at lower frequency. As presented in Fig. 4(a), the 2$\omega$ signal (2 kHz) is still prominent in the frequency-domain response, which is consistent with the condition of low-frequency modulation. While the insets (I) and (II) show that the 2$\omega$-signal is far less sensitive to magnetic field variation than the $\omega$-signal, so $R^{\omega }_z$ is selected as the object of observation.
Fig. 4. Experimental process of the precise compensation. (a) FFT of the magnetometer response before (black) and after (red) precise magnetic compensation with their detailed zoom-in around the peaks of 1 kHz and 2 kHz in insets (I) and (II). (b), (c) and (d) represent the compensation process of the x, y and z-axis respectively, and the excitation magnetic fields $B^z_{\rm {DC}} = B^y_{\rm {DC}} =$ 3 nT. The zero crossing of the second round $R^{\omega }_z$ (cyan thin diamond) is closer to the real magnetic null point, the zero crossing of the last round (magenta square), than the one of the first round (olive dot). When $B^z_{\rm {DC}} = 0$ nT (or $B^y_{\rm {DC}} = 0$ nT), $R^{\omega }_z (B^y_0)$ (or $R^{\omega }_z (B^z_0)$) is too weak to distinguish its variation (blue triangle).
According to the results in section 2, we first perform a sweep of the magnetic field along the sensitive axis, and demodulate $R^{\omega }_z$ to get the dispersion curve $R^{\omega }_z (B^x_0)$. After that, the sweeping field needs to be set to the value corresponding to $R^{\omega }_z (B^x_0) = 0$. Before acquiring the dispersion response curve $R^{\omega }_z (B^y_0)$ or $R^{\omega }_z (B^z_0)$, it is necessary to apply a sufficiently large DC excitation magnetic field $B_{\rm {DC}}$ along the other non-sensitive axis. Here, the magnitude of $B^z_{\rm {DC}}$ and $B^y_{\rm {DC}}$ are all set to 3 nT, otherwise $R^{\omega }_z$ is too weak to distinguish, just as the blue triangle depicted in Fig. 4(c) and 4(d). Similarly, it is necessary to set the sweeping field at the zero crossing before proceeding to the next step. Compared with the first round, Fig. 4(b–d) show that the zero crossings of the three axes in the second round are closer to the real magnetic null point. After 4-6 rounds, the zero crossing of $R^{\omega }_z (B^x_0, B^y_0, B^z_0)$ hardly changes at all and can be regarded as the real magnetic null point.
4.3 Compensation resolution
After the triaxial precise magnetic field compensation, the magnetic field compensation capability of the zero-field OPM are evaluated. As shown in Fig. 5(a), a saw-tooth wave magnetic field generated by the triaxial coil is applied along the $x$-axis, with a peak-to-peak value of 1 pT and a frequency of 1 Hz. The response of the magnetometer shows that resolvable changes can be observed even in the case of sub-pT variation. And the compensation resolution of the magnetometer on the x-axis is apparently better than 1 pT. Figure 5(b) describes the response fluctuation within 60 seconds under the ambient magnetic field. It is obvious that the fluctuation contains high-frequency components and low-frequency components, and the long-term fluctuation range is greater than the compensation resolution of the magnetometer, which indicates that the precise compensation is capable of coping with the long-term fluctuation. The fluctuation of the response is mainly caused by factors such as the power noise of the pump light, the change of the background magnetic field, and the system noise which are mainly the low-frequency noise. It is worth noting that the determination of the compensation resolution of non-sensitive axes are not investigated here. The necessity of appropriately increasing $B^z_{\rm {DC}}$ ($B^y_{\rm {DC}}$) to increase $k_y$ ($k_z$) has been clarified by Eq. (6) and Eq. (7). However, the optimal $B^y_{\rm {DC}}$ and $B^z_{\rm {DC}}$ are both affected by the values of $R_{\rm {op}}$ and $R_{\rm {rel}}$. Further research on triaxial scale factor optimization and evaluation of compensation resolution for non-sensitive axes will be carried out in the next work.
Fig. 5. Evaluation of the precise compensation. (a) The excitation magnetic field (light green) has a saw-tooth waveform with a frequency of 1 Hz and a peak-to-peak value of 1 pT. The magenta line represents the resultant response of the magnetometer. (b) The magnetometer response signal at zero magnetic field within 60 s. The colored regions in (a) and (b) represent the variation range of $R^{\omega }_z$.
5. Conclusion
In conclusion, we demonstrate a triaxial precise magnetic compensation scheme of zero-field OPM based on single-beam configuration. The response characteristics of the longitudinal response signal under the combined action of the three-axis residual magnetic fields accompanied by a low-frequency or a high-frequency modulated magnetic field are analyzed. When the 2$\omega$-component is dominant in the time-domain response, it means that the pre-compensation process is completed, which ensures that the zero-field OPM works in the linear region. In the process of precise compensation, the response curves $R^{\omega }_z (B^x_0)$, $R^{\omega }_z (B^y_0)$ and $R^{\omega }_z (B^z_0)$ are acquired by sweeping the magnetic field of the specific axis, and the non-equivalent relationship between the zero crossing of $R^{\omega }_z (B^x_0, B^y_0, B^z_0)$ and the real magnetic null point is indicated. In particularly, when performing precise magnetic field compensation on a non-sensitive axis, it is necessary to apply an DC excitation magnetic field with a suitable magnitude along the other non-sensitive axis to increase the scale factor of the response. Only a few rounds are needed in the process of triaxial precise magnetic field compensation, and the zero crossing can quickly approach the magnetic null point. The development of this work can be directly used to realize the zero-field environment of the OPM and cope with the fluctuation of low-frequency technical noise, which is of great significance to the MCG and MEG systems.
Funding
National Natural Science Foundation of China (No. 61903013); National Key Research and Development Program of China (No. 2018YFB2002405).
Disclosures
The authors declare no conflicts of interest.
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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