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Abstract
We demonstrate a single-beam atomic magnetometer (AM) capable of measuring a three-axis magnetic field with high-sensitivity, achieved by applying a small DC offset field and a high frequency modulation field. To satisfy the miniaturization demand of AMs, an elliptically polarized light detuned by 50 GHz from the resonance transition center is employed. The circularly polarized component is used to polarize the alkali-metal atoms, while the linearly polarized light is used to detect the dynamics of the polarized spin under a magnetic field. Based on theoretical analysis, parameters that significantly affect the performance are optimized, and a sensitivity of 20 fT/Hz1/2 in x-axis, 25 fT/Hz1/2 in y-axis, 30 fT/Hz1/2 in z-axis is achieved with a miniature 4 × 4 × 4 mm 87Rb vapor cell. Moreover, we also verify that the operation principle of AMs can be used to null background magnetic fields in-situ with isotropic compensation resolution of 6.7 pT, which provides an effectively precise method for zeroing ambient magnetic field. The high-sensitivity operating of an elliptically-polarized-laser-based magnetometer provides prospective futures for constructing a compact, low-cost AM, which is particularly applicable for non-invasive bio-magnetic imaging such as array-based magnetoencephalography (MEG) and magnetocardiography (MCG).
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Atomic magnetometers (AMs) operate with a net spin polarization by optical pumping. The perturbation of such spin polarization by the external magnetic field modifies the light-atom interaction, and a measurement of the magnetic field can be achieved either through the absorption or dispersive properties of atoms in a probe beam [1–4]. Based on this operation principle, AMs operating in the spin-exchange relaxation-free (SERF) regime is the most sensitive magnetometer with the sensitivity of 0.16 $\textrm{fT/Hz}^{1/2}$ in a gradiometer arrangement [5] and the sensitivity of 0.54 $\textrm{fT/Hz}^{1/2}$ as a scalar magnetometer [6]. And benefit from the intrinsic advantage of noncryogenic cooling and the potential for miniaturization, AMs are more promising as a viable alternative for superconducting quantum interference devices (SQUIDs) in a wide range of applications such as biomedical imaging including magnetoencephalography (MEG) [7–10] and magnetocardiography (MCG) [11,12], searches for the neutron electric dipole moment (nEDM) [13], tests of CPT and Lorentz symmetries violation [14,15], dynamical detections of geomagnetic field [16].
Generally, AMs, relying on detection of Larmor spin precession under the external magnetic field with atomic medium naturally tend to operate in the scalar mode [17,18], measuring the total magnitude of the magnetic field. While SQUIDs are single-axis vector magnetometers that measure the field projection along one direction or can be configured for direct measurement of the magnetic field gradients [19]. In practice, many applications will be benefit from the sensor that can detect all three orthogonal components, because the three-axis magnetic field measurement can provide a more comprehensive information [20,21]. Seltzer and Romalis demonstrated a three-axis SERF magnetometer with cross-feedback system in an unshielded environment [17]. By applying a high frequency modulation field in the longitudinal direction, a two-axis AM with a sensitivity of 60 $\textrm{fT/Hz}^{1/2}$, and a three-axis magnetometer with a longitudinal field sensitivity of 0.3 $\textrm{pT/Hz}^{1/2}$ and a transverse field sensitivity of 2 $\textrm{pT/Hz}^{1/2}$ were achieved [22,23]. Huang et al. realized an all-optical three-axis spin precession AM using an amplitude-modulated Bell-Bloom method to detect the main field, and the transverse magnetic fields were measured by the optical rotation of the probe beam [24]. Qiu et al. proposed a three-axis AM for nuclear magnetic resonance gyroscopes (NMRGs) based on the magnetic resonance and the nonlinear magneto-optical rotation, which achieved the sensitivities of $20~\textrm{fT/Hz}^{1/2}$ and $100~\textrm{fT/Hz}^{1/2}$ in the longitudinal and the transverse directions, respectively [25]. Although high sensitivity can be realized in the above AMs, while there may be some difficulty for the construction of a miniature AM because the necessary orthogonal pump-probe beams is essential in these schemes, which will increase the complexity of the optical path. In this condition, the single-beam configuration presents a more attractive design for the compact AM [26,27], and a commercially available sensor that can be used by the medical research community is developed by QuSpin [28,29]. However, the detailed operation principle is not demonstrated. Furthermore, the QuSpin AM operates with dual-axis rather than three-axis magnetic field measurement. Huang et al. described a single-beam three-axis AM with the sensitivities of 0.3 $\textrm{pT/Hz}^{1/2}$ in x- and y-axis and 3 $\textrm{pT/Hz}^{1/2}$ in z-axis, which was achieved by applying a rotating field on the x-O-y plane and another modulation field in the z-axis [30]. Combining with magnetic field modulation and laser frequency modulation techniques, Pradhan established a three-axis magnetic field measurement scheme utilizing a single elliptically polarized light beam, achieving the sensitivity of better than 20 $\textrm{pT/Hz}^{1/2}$ in 400 mHz in any direction [31]. These two methods show that a single-beam AM is capable of measuring all three magnetic field components. Actually, it is a challenging task to measure all components of the magnetic field with a high sensitivity in the single-beam scheme, especially in the laser beam direction because the polarization rotation or the light intensity absorption is naturally insensitive to the light propagation direction [32]. Meanwhile, for the purpose of improving the sensitivity, the magnetic field modulation technique is an effective detection method that is widely used to suppress the low frequency technical noise [33]. The alkali-metal atoms are driven to spin precession by applying a high-frequency oscillating magnetic field, and a modulation of the light polarization will be appeared owing to the magneto-optical interaction with atoms. However, the multi-axis magnetic field measurement of previous schemes either have low sensitivity or cannot achieve three-axis measurement, which need to be further research.
In this paper, a compact, fiber-coupled single-beam AM for three-axis magnetic field measurement is proposed. Our configuration is similar to those previously demonstrated single-beam SERF magnetometer [34,35]. A single elliptically polarized laser beam is employed in the system, where the circularly polarized component is used to polarize the alkali-metal atoms and the linearly polarized component is used to detect the optical rotation of atomic spin. However, the main difference is the two mentioned AMs operating in zero-field resonance, below pT. While in this paper, with the purpose of three-axis magnetic field measurement, the presented AM operates in near zero-field condition where an orthogonal DC offset magnetic field is superimposed to the modulation field. After optimization, the sensitivity of 20 $\textrm{fT/Hz}^{1/2}$, 25 $\textrm{fT/Hz}^{1/2}$, 30 $\textrm{fT/Hz}^{1/2}$ is achieved in x-, y-, and z-axis, respectively. The high-sensitivity operating as a three-axis magnetic field sensor proposed here provides a possible approach for multi-dimensional magnetic field measurement, which will be applicable for building a new generation of MEG and MCG system.
2. Methods
The overall dynamical state of the atomic ensemble is given by the evolution of the density matrix. However, in the regime of high alkali vapor density and low magnetic field that the spin-exchange rate is sufficiently larger corresponding to the atomic precession. The equilibrium state of atomic spin can be well characterized by a spin-temperature distribution, and in this case, a convenient simplification over the complicated density matrix equation can be modelled by the phenomenological Bloch equation [36]
Consider a spin evolution of atomic ensemble in the magnetic field $\mathbf {B}=\left (B_x+B_x^\textrm{mod}\textrm{cos}\left (\omega t\right )\right ){\hat {x}}+B_y{\hat {y}}+B_z{\hat {z}}$, where $B_x^\textrm{mod}$ is the modulation magnetic field along the x-axis, and $B_x$, $B_y$, $B_z$ are quasi-static magnetic fields, which are far smaller than $B_x^\textrm{mod}$. Based on the perturbation-iteration method, the perturbation series for the case of a general magnetic field has been deduced in Ref. [38]. And the first harmonic component of the response can be simplified to [39]
Dramatically, when a DC bias magnetic field $B_{z0}$ that is comparable to the linewidth is applied in z-axis, the response to $\delta B_y$ will obtain in the following form
Equation (4) – Eq. (6) indicate that a three-axis magnetic field measurement can be achieved by applying an appropriate bias magnetic field in a particular direction. Furthermore, all responses to the sensitive field are dispersion Lorentzian curves and by sweeping three-axis fields, the remanence can be obtained from the demodulated output of the LIA. Therefore, a method for in-situ triaxial magnetic field compensation can be proposed by adjusting the zero-crossing of the dispersion curve. We will illustrate the detailed operation of magnetic field compensation in the following experiment.
3. Experimental setup
The experimental setup is displayed in Fig. 1. A cubic glass cell with $4~\textrm{mm}$ outer length and $0.5~\textrm{mm}$ thickness is the key sensing unit of the magnetometer. The glass cell is evacuated, baked, and filled with a small droplet of isotopically enriched $^{87}{\textrm{Rb}}$. The vapor cell is also filled with $\textrm{700~torr}~^4\textrm{He}$ as the buffer gas and $\textrm{50~torr N}_2$ for quenching. A boron nitride ceramic oven is used to hold the vapor cell and is heated to the optimal operation temperature of $433~\textrm{K}$ (see subsection 4.2) with a twisted-pair winding resistor driven by a $150~\textrm{kHz}$ AC electronic current. The oven temperature is monitored by a non-magnetic $\textrm{Pt}~1000$ for real-time closed-loop controlling and a temperature fluctuation less than $30~\textrm{mK}$ can be achieved. At this temperature, a dense alkali-metal vapor density of roughly $1.6 \times 10^{14}~{\textrm{cm}}^{-3}$ is realized, offering a rapid spin-exchange rate for suppression of spin-exchange broadening.
Fig. 1. Schematic of the experimental setup. A polarization maintaining fiber (PMF) is used to deliver the distributed feedback laser (DFB) to the AM and the laser beam is expanded to $2.7~\textrm{mm}$ by a set of collimating lenses (C). The elliptically polarized light propagating along z-axis with an ellipticity of $22.5^{\circ }$ is used to polarize Rb atoms and detect the optical rotation angles. The incident light intensity is monitored by the photodiode (PD0) before the vapor cell and the output differential signal is analyzed with a balanced polarimeter, consisting of a polarization beam splitter (PBS) and two photodiodes ($\textrm{PD1}$ and $\textrm{PD2}$). The field coils 1 is a set of compensation coils and field coils 2 is the calibration coils. $\lambda /2$: half-wave plate. $\lambda /4$: quarter-wave plate. R: reflecting prism. TIA: transimpedance amplifier. LIA: lock-in amplifier. DAQ: date acquisition.
A commercially available distributed feedback (DFB) laser generates linear polarization light at 795 nm. The pumping beam is coupled to the magnetometer through a single-mode polarization maintaining fiber (PMF) and a set of collimating lenses are used to produce a $2.7~\textrm{mm}$ Gaussian beam. The elliptically polarized light is created by a quarter-wave plate with its optic axis oriented at an angle $\pi /8$ towards to the polarization beam splitter (PBS). The circularly polarized component is used to spin-polarized alkali-metal atoms, while the linearly polarized component is used to detect the optical rotation angles. After the cell, the transmitted beam is analyzed by the balanced polarimetry technique with a transimpedance amplifier. The half-wave plate after the cell is used for nulling the differential signal between the two detection photodiodes ($\textrm{PD1}$ and $\textrm{PD2}$) before the magnetometer operating.
A multi-layer cylinder magnetic shields consisting of three-layer $\mu$-metal and an aluminum shield with a quasi-static shielding factor at least of $10^5$ provides a quiet magnetic environment with a residual magnetic field below 1 nT. The outermost aluminum shield is used to attenuate high-frequency magnetic noise. Two sets of triaxial coils (one Lee-Whiting coil and two Saddle coils are included in each of the triaxial coils) are placed inside the shielding system. The field coils 1 is used to further suppress the residual magnetic field and apply the modulation magnetic field while the field coils 2 generates the oscillating field for calibration.
4. Result
The remanence, especially the light shift fictitious magnetic field due to the off-resonance of pumping beam will broaden the magnetic-resonance linewidth. Therefore, it is necessary to zero the residual magnetic field before detecting the weak signal. Conventional methods such as the field modulation method based on the orthogonal pump-probe scheme or the non-modulated method employed the single-beam arrangement are either unsuitable for single-beam operation or their compensation resolution in the pumping direction are not high enough [17,40]. In the following, we illustrated the residual magnetic field can be cancelled in-situ utilizing the demodulated output of the LIA and then we determined the sensitivity of the AM under the optimal operation condition.
4.1 In-situ magnetic compensation towards off-resonance pumping beam
As mentioned in Section 2, a $170~\textrm{nT}$ modulation field with frequency of $1~\textrm{kHz}$ was applied in x-direction to shift the signal from DC to the modulation frequency. As shown in Fig. 2(a), we first swept the x magnetic field to obtain the zero-crossing of the dispersion curve from the demodulated output of the LIA, and by adjusting the x field component to move the zero-crossing. When the zero-crossing exactly located on the origin, the residual magnetic was zeroed. Afterward, a $15~\textrm{nT}$ (approximately equal to the magnetic-resonance linewidth ) bias field was added in y-axis, and by sweeping the z magnetic field, the remanence including the light shift fictitious magnetic field was observed [see Fig. 2(b)]. The remanence in z-axis was also compensated by adjusting the z offset field to move the zero-crossing approaching the origin, after that, the y bias field was removed. Similarly, the y residual magnetic field can be zeroed with the same method used in z-axis considering the commutativity of y- and z-axis component in Eq. (3). After several iteration, the residual magnetic field was restored to zero. The compensation resolution was derived from Fig. 2(c). When the magnetic field was swept from −0.1 nT to 0.1 nT, the most scanned points were 31, from which the compensation resolution of $6.7~\textrm{pT}$ was obtained, and considering the coil constant and the sweep rate were equal for three-axis, therefore, the same compensation resolution of $6.7~\textrm{pT}$ was achieved in y-axis.
Fig. 2. The experimental compensation method for (a) x-axis and (b) z-axis. To illustrate the compensation process, a residual field of $\pm ~30~\textrm{nT}, \pm ~20~\textrm{nT}, \pm ~10~\textrm{nT}$ was applied in x- and z-axis, respectively. When the remanence was nulled, the zero-crossing of the dispersion curve would locate on the origin. (c) A magnification of the experimental data in (a) and (b). The solid circles (squares) indicated the measured x (z) data, and the red (blue) line, overlapping the data points was the linear fitting for visual guidance. The resolution of $6.7~\textrm{pT}$ compensating result was observed.
4.2 Performance optimization of three-axis sensitivity
In the experiment, the performance of the elliptically-polarized-laser-based magnetometer is influenced by the DC offset magnetic field, optical parameters, operation temperature, and etc. In general, these parameters have an overall effect on AMs and we will investigate them through the scale factor and the magnetic-resonance linewidth.
Firstly, we studied the influence of the DC bias field on the scale factor of the magnetometer. After nulling the residual magnetic, a sequence of DC bias field was applied on y (z) direction, and the scale factor can be obtained by fitting the linear part of the response curve. The experimental measured scale factor was shown in Fig. (3), the red (blue) line represented the theoretical prediction with Eq. (5) and Eq. (6). The corresponding R-square (the goodness of fit) of y- and z-fitting were 0.9779 and 0.9996, indicating that the fitting curves were in good agreement with the measured data. The R-square of y-fitting was slightly poorer compared to the R-square of z-fitting, probably caused by the magnetic field drift in z direction due to insufficient axial shielding performance. When the bias field was $15~\textrm{nT}$ the maximum scale factor would be achieved, which was agreement with the theoretical prediction according to the half-width $12~\textrm{nT}$ and the modulation index 0.9.
Fig. 3. Dependence of the scale factor on DC offset field. The optimal scale factor was attained when the DC offset field is $15~\textrm{nT}$.
In order to enhance the signal response, the influence of light detuning and the pumping intensity need to be considered. The incident light is completely absorbed when the optical frequency is near resonant while the incident light will pass the vapor cell with little absorption when the light is far detuned. Consequently, an optimal frequency is desired, and in the experiment, the scale factor realized the maximum value when the light was blue detuned by $50~\textrm{GHz}$ [see Fig. 4(a)]. As shown in Fig. 4(b), the response was also improved with the increase of the pumping intensity, however, the transimpedance amplifier would overload when the pumping intensity was over $15~\textrm{mW/cm}^2$ and in the experiment, the pumping intensity of $12.56~\textrm{mW/cm}^2$ was adopted. Meanwhile, the half-width of the magnetic resonance was measured. As the frequency is closer to the D1 resonance transition center or the pumping intensity increases, the pumping rate will become larger, which will broaden the magnetic-resonance linewidth (see Fig. 4).
Fig. 4. Scale factor and half-width under different optical parameters. (a) The scale factor and half-width under different frequency detuning when the pumping intensity was $12.56~\textrm{mW/cm}^2$. (b) The scale factor and half-width under different pumping intensity when the pumping beam was blue detuned by $50~\textrm{GHz}$ from the center of the optical absorption.
Having determined the DC bias magnetic field, pumping frequency and intensity, now we focus on the operation temperature. As we mentioned in Section 2, high alkali vapor density and low magnetic field are crucial to reach SERF regime. However, in the presence of the residual magnetic field, a higher alkali-metal density is demanded to increase the spin-exchange rate for further suppressing spin-exchange relaxation. Figure 5(a)-(c) showed the response of each sensitive axis on the temperature of $363~\textrm{K}$ to $443~\textrm{K}$, covering the typical operating conditions including non-SERF and SERF regime. As shown in Fig. 5(d), compared to the operation at $363~\textrm{K}$ (non-SERF), the accomplishment of the SERF resulted in an improvement in the scale factor over $10~\textrm{dB}$ when the temperature was higher than $413~\textrm{K}$, meanwhile, an appreciable narrowing of magnetic-resonance linewidth was observed as a result of eliminating the spin-exchange relaxation. Although higher temperature can achieve greater response, however, the increase in scale factor seemed to reach saturation when the temperature was high than $433~\textrm{K}$, and considering the heating capacity of the heater, the operation temperature was chosen at $433~\textrm{K}$.
Fig. 5. Response and half-width under the temperature between $363~\textrm{K}$ and $433~\textrm{K}$. The scale factor in SERF was enhanced by more than $10~\textrm{dB}$ compared to non-SERF and the half-width was also narrowed.
Ultimately, choosing the optimal operating parameters mentioned above, the performance of the SERF magnetometer is characterized by the sensitivity and bandwidth. A white noise signal with a bandwidth of $5~\textrm{kHz}$ and an amplitude of $1~\textrm{nT}$ was applied to field coils 2. The AM output was recorded $300~\textrm{s}$, and a comparison of power spectral density between the input and the output was shown in Fig. 6. The green dot line is the input white noise signal and the blue line shows the response signal while the red line is a fit based on a single-pole low-pass filter model with a frequency response function of $1/\sqrt {1+\left (f/f_{\textrm{3dB}}\right )^2}$ [3], where $f$ is the frequency and $f_{\textrm{3dB}}$ is the bandwidth of the magnetometer. The measured 3 dB bandwidths were $60~\textrm{Hz}$, $103~\textrm{Hz}$ and $45~\textrm{Hz}$ for x-, y- and z-axis, respectively. The bandwidth can be further improved by lowering the temperature of the vapor or increasing the pumping intensity, with a sacrifice of sensitivity.
Fig. 6. The normalized frequency response of AM. The green dot line was the white noise input and the blue line represented the response output. A single-pole low-pass filter model was used to fit the frequency response and the bandwidths were 60 Hz in x-axis, 103 Hz in y-axis and 45 Hz in z-axis.
Fig. 7. Noise spectral density of the elliptically-polarized-laser-based magnetometer. The oscillating magnetic field with $100~\textrm{pTrms}$ calibration peaks of $35~\textrm{Hz}$ in x-axis, $25~\textrm{Hz}$ in y-axis and $15~\textrm{Hz}$ in z-axis was applied by field coils 2. The power spectrum density was calculated and averaged for 1-Hz bin, and then divided by the corresponding frequency response function to obtained absolute field sensitivity. The sensitivity of x-, y- and z-axis were $20~\textrm{fT/Hz}^{1/2}$, $25~\textrm{fT/Hz}^{1/2}$ and $30~\textrm{fT/Hz}^{1/2}$, respectively. (Dash line) Magnetic noise obtained by the out-of-phase of the LIA after switching off the modulation field.
Figure 7 shows the magnetic field sensitivity working at the optimal response. To calibrate the sensitivity, an AC magnetic field with a constant calibration peak of $100~\textrm{pTrms}$ was applied at different frequencies. Then the noise spectrum density was normalized by dividing the corresponding frequency response, and therefore, a flat-frequency response within the bandwidth was achieved, with a sensitivity of $20~\textrm{fT/Hz}^{1/2}$ in x-axis, $25~\textrm{fT/Hz}^{1/2}$ in y-axis and $30~\textrm{fT/Hz}^{1/2}$ in z-axis. Besides, the noise at $50~\textrm{Hz}$ was caused by power frequency and other spike noise was the harmonics of the calibration magnetic field. The magnetic noise was measured by recording the out-of-phase of the LIA after removing the modulation magnetic field, and the noise floor of $11~\textrm{fT/Hz}^{1/2}$ observed was quite consistent with the intrinsic noise of $\mu$-metal shield [41]. The sensitivity of proposed magnetometer was limited by technical noise, especially the fluctuations of the laser source.
5. Conclusion
In this paper, we have presented a compact, fiber-coupled atomic magnetometer based on elliptically polarized light. A novel operation mode by superimposing an orthogonal DC offset magnetic field to the modulation field was studied, and several crucial parameters including DC offset magnetic field, optical parameters, operation temperature were optimized based on the theoretical analysis. Furthermore, we also proved that the proposed operation mode can be applied for in-situ magnetic compensation in a single-beam atomic magnetometer, with a high resolution of $6.7~\textrm{pT}$. By a combination of the magnetic compensation and the performance optimization, we achieved a magnetic field sensitivity of 20 $\textrm{fT/Hz}^{1/2}$ in x-axis, 25 $\textrm{fT/Hz}^{1/2}$ in y-axis and 30 $\textrm{fT/Hz}^{1/2}$ in z-axis with a miniature 4 $\times$ 4 $\times$ 4 mm $^{87}\textrm{Rb}$ vapor cell. Sensitivity may be substantially enhanced by suppressing the fluctuations of the laser source. For better common noise cancellation from power lines and other nearby sources, the gradient mode is a preferable method [42]. By performing gradient differential detection, the atomic gradiometer is more promising for applications in challenging environment outside of a laboratory even in the Earth’s ambient environment [43].
The observed frequency responses of x-, y- and z-axis were 60 Hz, 103 Hz and 45 Hz, which is sufficient for the typical MEG signal. However, in the case of epileptic spike activity, high frequency oscillation (HFO) beyond the current AM bandwidth often occurs [44]. As bandwidth and sensitivity are inherent trade-offs in AMs, further improvement in the frequency response can be achieved by implementing the closed-loop scheme to decrease the gain at the cost of lowing sensitivity.
Of particularly value here is the simplified ptical configuration that requires only a single low-power laser. Combined with fiber-optic coupling and microfabricated technique, a miniaturized atomic magnetic sensor with millimetre-scale is feasible in future, which will provide a cost-effective magnetometer arrays, opening up new possibilities for the non-invasive bio-magnetic imaging.
Funding
Beijing Municipal Natural Science Foundation (4191002); National Natural Science Foundation of China (61627806); National Key Research and Development Program of China (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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