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Abstract
We propose a three-axis closed-loop optically pumped magnetometer with high sensitivity. The closed-loop magnetometer has a three-axis sensitivity of approximately 30 fT/Hz1/2 using two orthogonal laser beams for pumping and probing the alkali metal atoms. In the closed-loop mode, the dynamic range is improved from ±5 nT to ±150 nT. The bandwidth is increased from about 100 Hz to over 2 kHz with 10 kHz modulation fields in x- and y-axes and another 6 kHz modulation field along the z-axis. Compared with single-axis or dual-axis magnetometers, the proposed magnetometer not only provides the direction and magnitude of the magnetic field but also has high robustness in a challenging environment. The magnetometer has applications in biomagnetic measurements, magnetic resonance imaging, and fundamental physics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
10 November 2022: A correction was made to the author listing.
1. Introduction
Optically pumped magnetometers (OPMs) are widely used to detect magnetic fields in magnetoencephalography(MEG) [1–3], magnetocardiography(MCG) [4], magnetic resonance imaging [5], and fundamental physics [6,7]. Currently, OPMs operating in the spin-exchange relaxation-free (SERF) regime have been proven to have ultra-high sensitivity [8,9]. Owing to their high sensitivity and room-temperature operation [10], OPMs have become the main competitors of superconducting quantum interference device (SQUID) magnetometers in medical diagnostics and imaging [11,12].
With the development of SERF atomic magnetometers [13–15], dual-axis and three-axis magnetometers have been studied in depth in addition to single-axis sensors. Li et al. realized a dual-axis SERF magnetometer using a sinusoidal magnetic field modulation along the pump-light direction based on a pump-probe scheme [16]. Osborne et al. used only one pump laser beam to develop a dual-axis atomic magnetometer with two separate modulation fields [17]. However, these SERF magnetometers can not probe the magnetic field along the optical path direction of the pump beam. Three-axis atomic magnetometers in the SERF regime have been extensively explored to obtain three components of the magnetic field. For example, Huang et al. developed a three-axis SERF atomic magnetometer with a pump laser beam and three low-frequency modulation fields [18]. Boto et al. realized a three-axis magnetometer using two orthogonal pump laser beams and three modulation fields [19]. Furthermore, there are other design schemes for realizing three-axis measurements, including the application of an elliptically polarized laser beam [20], varying the direction of the pump laser [21], and using multiple vapor cells [22]. Although these atomic magnetometers have high sensitivity, their small dynamic range (usually several nanoteslas) and low bandwidth (approximately 100 Hz) restrict their practical applications.
To solve these weaknesses, some research groups have introduced a closed-loop control method for SERF atomic magnetometers. For example, Guo et al. developed a single-axis closed-loop SERF magnetometer without a modulation field and achieved a high sensitivity of 15 fT/Hz1/2 in the frequency range of 30-750 Hz, the dynamic range of which was also improved to ±15 nT [23]. Wang et al. improved the bandwidth of a single-beam SERF magnetometer to over 4 kHz with a sensitivity of 35 fT/Hz1/2 in closed-loop mode [24]. By introducing a rotating modulation field, Tang et al. developed a dual-axis closed-loop atomic magnetometer with a bandwidth of 1.8 kHz and a dynamic range of 90 nT at a sensitivity of 20 fT/Hz1/2 [25]. The single-axis and dual-axis atomic magnetometers in the closed-loop mode mentioned above cannot eliminate the magnetic interference along a non-sensitive axis and thus decrease the performance. Seltzer and Romalis demonstrated a closed-loop three-axis atomic magnetometer using low-frequency modulation fields along two axes [26]. However, this quasi-static atomic magnetometer has limited bandwidth. Therefore, a high-sensitivity three-axis closed-loop optically pumped magnetometer with wide bandwidth and dynamic range should be studied and discussed.
In this study, we developed a three-axis closed-loop SERF magnetometer with high sensitivity, wide bandwidth, and a large dynamic range. Two orthogonal pump laser beams were used to realize a three-axis measurement with magnetic-field modulation [19]. With a feedback system and triaxial coil, the magnetic field detected by the alkali metal atoms is locked to near zero. The bandwidth, dynamic range, and sensitivity of the open-loop and closed-loop modes were investigated and compared. The bandwidth was improved from about 100 Hz to over 2 kHz in the closed-loop mode, which was an enhancement of approximately 20-fold compared with that in the open-loop mode. The closed-loop sensor achieved a large dynamic range of ±150 nT compared with that of ±5 nT in open-loop mode. A three-axis sensitivity of 30 fT/Hz1/2 is obtained in the closed-loop mode.
2. Method
The operation principle of the three-axis SERF magnetometer is based on three modulation fields using two orthogonal pump laser beams used for pumping and probing the alkali metal atoms [19], as shown in Fig. 1. The first light beam along the x-axis polarizes the alkali metal atoms and can measure the magnetic field in the y- and z-axes. The second light beam along the z-axis orthogonal to the first beam can measure the magnetic field in the x- and y-axes. When a rotating field Bmcos(ωmt)$\hat{x}$ + Bmsin(ωmt)$\hat{y}$ is applied, the two magnetic fields along the x- and y-axes can be detected simultaneously by the pump laser along the z-axis. A second pump laser beam along the x-axis with a modulation field Bzmcos(ωzmt)$\hat{z}$ is introduced to detect the magnetic field along the z-axis. All the signals show a dispersion line shape as a function of the magnetic field. To decrease the cross-talk effect, the modulation frequency ωzm along the z-axis is adjusted to be different from those along the x- and y-axes.
Fig. 1. Diagram of the three-axis atomic magnetometer using two orthogonal pump beams and three modulation fields. All signals show a dispersion line shape as a function of the magnetic field.
The SERF atomic magnetometer has limited bandwidth (approximately 100 Hz) and dynamic range (several nanoteslas). In general, the magnetometer bandwidth and dynamic range are dependent on the spin-relaxation rate [26]. In the open-loop mode, it is difficult to improve the bandwidth and dynamic range without a loss of sensitivity.
To increase the bandwidth and dynamic range of the atomic magnetometer, we present a closed-loop method by applying three-axis feedback to maintain the magnetic field detected by the Rb atoms near zero. For a single-beam magnetometer with one modulation field in the open-loop mode, the system can be described with a second-order transfer function, according to our previous work [24]. In this study, the atomic magnetometer behaved as an overdamped second-order system and was approximately described by a first-order system. The magnetometer response along the sensitive axis was modeled as
where ωc is the cutoff angular frequency, and G0 denotes the DC response. This variable is defined by s = jω.The transfer function of a PI controller is given by
where Kp and Ki denote the coefficients of proportional and integral gains, respectively.In the closed-loop mode, three PI modules were applied to control the magnetic field using a triaxial coil. The transfer function of the coil system is described using the scaling coefficient Kcoil. Therefore, the transfer function of the atomic magnetometer in the closed-loop mode is given by
The three feedback loops are independent of each other and can be described by Eq. (3). In the closed-loop mode, three feedback loops are operated simultaneously, the feedback signal of which also serves as a response of the three-axis closed-loop atomic magnetometer.
The spin evolution of the optically pumped alkali metal atoms in the SERF regime is described using the Bloch equation [8]
By applying a modulation field, Bmcos(ωmt) along the x-axis, B0 is the magnetic field to be detected. The first harmonic of the magnetometer response has a dispersion line shape and is approximately proportional to B0 [28]
According to Eq. (7), the electron polarization Pz is nearly linearly proportional to the magnetic field B0, which is reflected by the light intensity of the pump laser detected by a photodiode (PD) [29]
where Sout is the output signal, S0 is the original signal, and OD is the optical depth.3. Experimental setup and procedure
The experimental setup is shown in Fig. 2. A cubic glass cell with an inner length of 8 mm contained a droplet of 87Rb metal, 2100 Torr 4He buffer gas, and 70 Torr N2 quenching gas. Helium and nitrogen gases can suppress relaxation due to wall collisions and eliminate radiation trapping, respectively. The glass cell was heated to 160 °C by two flexible heating films using AC currents at 500 kHz, without affecting the magnetometer operation. The number density was improved to approximately 1014 cm-3 at 160 °C for rubidium atom vapor. The temperature of the glass cell was estimated by a nonmagnetic Pt1000 thermometer, which was controlled by a temperature control program encoded through LabVIEW.
Fig. 2. Schematic of the experimental apparatus. PMF: polarization maintaining fiber; CL: collimator; λ/4: quarter-wave plate; BS: 50:50 beam splitter; PD: photodiode; M: reflection mirror; LIA: lock-in amplifier; TIA: trans-impedance amplifier; PI: proportional-integral controller; DAQ: data acquisition equipment.
An external cavity diode laser (Toptica, DL pro) generated a pump beam with the wavelength (794.94 nm) detuned about 0.04 nm from the 87Rb D1 resonance line. The pump laser was sent to the magnetometer setup through a polarization-maintaining fiber with a collimator. The linearly polarized beam is left-circularly polarized through a quarter-wave plate and then split into two beams by a 50:50 beam splitter. The transmitted light polarized a part of the 87Rb atomic ensemble along the x-axis, whereas the reflected light polarized another part along the z-axis. These two beams were detected by two silicon photodiodes, PD 1 and PD 2.
A triaxial coil system was mounted surrounding the vapor cell of the magnetometer to apply the modulation field and feedback signal driven by waveform generators (Keysight, 33522 B) and the feedback system. The modulation fields Bmcos(ωmt)$\hat{x}$ and Bmsin(ωmt)$\hat{y}$ were applied to measure the magnetic field along the x- and y-axes through the propagation of the reflected pump laser along the z-axis. Similarly, the magnetic field along the z-axis was detected by the transmitted pump laser propagating along the x-axis and the modulation field Bzmcos(ωzmt)$\hat{z}$.
The output signals from the two silicon photodiodes, PD 1 and PD 2, are demodulated by two lock-in amplifiers (Zurich Instruments, MFLI); this is the response to the magnetic field. The in-phase and out-of-phase signals of the demodulated signal from LIA 1 at a reference frequency ωm were the response to the magnetic fields along the x- and y-axes, respectively. Similarly, the response signal along the z-axis was demodulated by LIA 2 at a reference frequency ωzm.
In the closed-loop mode, the three magnetometer response signals were locked at point zero using three PI controllers and a triaxial coil. The output signals from the two LIAs were transmitted to the three PI modules, the feedback signal of which suppressed the three components of the detected magnetic field by the coils. The output signal from the PI modules was acquired by a data acquisition board for sensitivity, dynamic range, and bandwidth analyses.
In the following sections, we described the operation of the three-axis atomic magnetometer in the open-loop and closed-loop modes. The bandwidth, dynamic range, and sensitivity of the atomic magnetometer were investigated and compared for these two modes.
4. Results and discussion
4.1 Bandwidth in the open-loop and closed-loop modes
To investigate the limiting factors of the atomic magnetometer bandwidth, we first applied a modulation field along the x-axis. Several modulation magnetic fields with different frequencies (100 Hz, 300 Hz, 700 Hz, 1 kHz, and 10 kHz) were applied to the x-axis coil. According to Eq. (7), the modulation amplitude should be adjusted with the modulation frequency to obtain a sufficiently strong magnetometer response signal.
The bandwidth of the magnetometer is limited to approximately 100 Hz, owing to the spin relaxation rate, as shown in Fig. 3. The modulation amplitude is set to 4.4-, 7.6-, 12.9-, 18.6- and 80.3 nT corresponding to 100 Hz, 300 Hz, 700 Hz, 1 kHz, and 10 kHz, respectively. The bandwidths are 9.5 Hz, 27 Hz, 48 Hz, 55 Hz, and 82 Hz with modulation frequencies of 100 Hz, 300 Hz, 700 Hz, 1 kHz, and 10 kHz, respectively. With an increase in the modulation amplitude, the spin-exchange relaxation rate is improved according to Eq. (6). Therefore, the bandwidth also increases when the spin-relaxation rate increases.
Fig. 3. Magnetometer frequency response with different modulation parameters. The solid lines (solid boxes) stand for the fitting (experimental) results.
Based on the above results and discussion, we operated the three-axis atomic magnetometer using a rotation modulation field of 10 kHz along the x-y plane and another modulation at a frequency of 6 kHz along the z-axis. The cut-off frequency of the LIA low-pass filter was set to 5 kHz.
The white noise with a bandwidth of 800 Hz and an amplitude of 100 pTrms is applied along the x-, y- and z-axes by the magnetic coils to measure the magnetometer bandwidth [30]. Every set of atomic magnetometer outputs was collected over 180 s in the open-loop and closed-loop modes. The frequency response is usually expressed in logarithmic unit decibels (dB), which provide a wide view range,
where the RdB(f) is the frequency response with a unit of dB, f is the corresponding frequency, R(f) is the measured amplitude and Rr is the reference amplitude. Because the amplitude in the low frequency is almost the same, the amplitude at 1 Hz is used as the reference amplitude.As shown in Fig. 4, the Fourier spectrum of the output signal was used to evaluate the sensor bandwidth. The magnetometer bandwidth is defined as the frequency range where the magnitude of response is greater than -3 dB. In the open-loop mode, the three-axis bandwidth was found to be approximately equal to 100-200 Hz due to the increase of the spin relaxation rate with the three modulation fields. In the closed-loop mode, the magnetometer bandwidth expanded from about 100 Hz to over 2 kHz, which is approximately 20 times higher than that in the open-loop mode. Moreover, the closed-loop sensor displayed a flat frequency response between DC and 1 kHz.
Fig. 4. Frequency response of the OPM in closed-loop and open-loop modes along the (a) x-axis, (b) y-axis, and (c) z-axis. The closed-loop magnetometer bandwidth is improved to over 2 kHz along three axes. The red line represents the open-loop mode and the black line represents the closed-loop mode.
4.2 Dynamic range in the open-loop and closed-loop modes
The typical magnetic resonance signal of an open-loop SERF magnetometer has a dispersion line shape. In the open-loop mode, the output signal of the atomic magnetometer is directly collected by the DAQ as shown in Fig. 5(a). In the closed-loop mode, the feedback signal from the PI controller is used to cancel the magnetic field, which is also collected by the DAQ as the output of the closed-loop magnetometer as shown in Fig. 5(b). The atomic magnetometer response was measured by sweeping the magnetic field along the x-, y-, and z-axes, as shown in Fig. 6. In the open-loop mode, the linear region was limited to approximately ±5 nT. When the amplitude of the magnetic field exceeded the dynamic range, the sensor failed to function properly. With the three-axis magnetic feedback, the sensor displayed high robustness against the changing magnetic field, and thus achieved a wide dynamic range of ±150 nT, as shown in Fig. 6.
Fig. 5. Schematic diagram for the data acquisition method in open-loop and closed-loop mode. (a) The response of the magnetometer was directly collected by the DAQ in the open-loop mode. (b) The feedback signal from the PI controller was collected by the DAQ as the output of the closed-loop magnetometer in the closed-loop mode.
Fig. 6. Responses of the three-axis atomic magnetometer. The dynamic range in the open-loop mode is about ±5 nT, and it is improved to ±150 nT in the closed-loop mode.
The magnitude Af of the feedback signal depends on the coil constant Bcoil (the magnitude of the magnetic field generated by unit current in the unit of nT/mA) and the drive current Id in the unit of mA,
The maximum of the feedback signal can be improved when the coil constant K or drive current Id increases. In other words, we can increase the coil turns to increase coil constant K or use a larger drive current, thus the dynamic range is further improved.
4.3 Sensitivity in open-loop and closed-loop modes
The magnetometer sensitivity was evaluated in the open-loop mode along the x-, y-, and z-axes. We simultaneously applied a magnetic modulation field at a frequency of 1.1 kHz in the x-y plane and another modulation field at 1 kHz along the z-axis. The cut-off frequency of the LIA low-pass filter was set to 100 Hz.
A calibration magnetic field with a frequency of 30.5 Hz and an amplitude of 100 pTrms was employed along three axes in turn to measure the sensitivity of the magnetometer. The response to the magnetic field was collected for approximately 100 s. The corresponding amplitude spectral density is divided by the frequency response to obtain a normalized value in each 1 Hz bin. The calibration field was used to convert the voltage noise to sensitivity. In the open-loop mode, the magnetometer sensitivity is 20 fT/Hz1/2, 15 fT/Hz1/2, and 25 fT/Hz1/2 along the x-, y-, and z-axes, respectively as shown in Fig. 7(a). The sensitivity along the y-axis is superior to that along the x- and z-axes. Owing to the light shift effect, the alkali metal atoms experienced a virtual gradient magnetic field, which reduces the sensitivity along the x- and z-axes.
Fig. 7. Magnetometer sensitivity in the closed-loop and open-loop modes. (a) The three-axis sensitivity in the open-loop mode is 20 fT/Hz1/2, 15 fT/Hz1/2, and 25 fT/Hz1/2 along the x- (black line), y- (blue line) and z-axes (red line), respectively. (b) The three-axis closed-loop magnetometer sensitivity is about 30 fT/Hz1/2 along the x-, y- and z-axes.
In the closed-loop mode, the three-axis sensitivity is attenuated owing to the electronic noise from the closed-loop control system. As shown in Fig. 7 (b), the sensitivity of the closed-loop atomic magnetometer was approximately 30 fT/Hz1/2 along the x-, y-, and z-axes.
5. Conclusion
In summary, we demonstrated a three-axis closed-loop optically pumped magnetometer with wide bandwidth and high sensitivity. In the open-loop mode with a modulation field of approximately 1 kHz, the magnetometer had a high sensitivity of 20 fT/Hz1/2, 15 fT/Hz1/2, and 25 fT/Hz1/2 along the x-, y-, and z-axes, respectively. The closed-loop magnetometer sensitivity decreased to 30 fT/Hz1/2 with a wide dynamic range of ±150 nT. When a modulation field with a higher frequency was applied, the magnetometer bandwidth was improved to over 2 kHz. The proposed three-axis closed-loop atomic magnetometer has good performance in terms of sensitivity, dynamic range, and bandwidth, which is beneficial for applications in biomagnetic measurements, magnetic resonance imaging, and fundamental physics.
Funding
Key Technologies Research and Development Program (2018YFB2002405); National Natural Science Foundation of China (61903013).
Acknowledgments
The authors sincerely thank Ziao Liu and Weiyi Wang for their assistance with the experiments.
Disclosures
The authors declare no conflict 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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