Four-channel optically pumped atomic magnetometer for magnetoencephalography

: We have developed a four-channel optically pumped atomic magnetometer for magnetoencephalography (MEG) that incorporates a passive diffractive optical element (DOE). The DOE allows us to achieve a long, 18-mm gradiometer baseline in a compact footprint on the head. Using gradiometry, the sensitivities of the channels are < 5 fT/Hz 1/2 , and the 3-dB bandwidths are approximately 90 Hz, which are both sufficient to perform MEG. Additionally, the channels are highly uniform, which offers the possibility of employing standard MEG post-processing techniques. This module will serve as a building block of an array for magnetic source localization.


Introduction
The dominant technology for human biomagnetic measurements has been liquid-heliumcooled superconducting quantum interference device (SQUID) magnetometers because their high sensitivity makes them capable of detecting the very weak signals from humans [1]. Common biomagnetic applications include measurement of the magnetic fields from the heart, magnetocardiography (MCG) [2], and from the brain, magnetoencephalography (MEG) [3]. For these applications, though, optically pumped (atomic) magnetometers (OPMs) have emerged as a promising replacement for SQUIDs [4,5]. OPMs measure magnetic fields by probing the interaction of near-resonant laser light with an alkali metal vapor that is in a magnetically sensitive state. The sensitivity and bandwidth of OPMs are sufficiently high to observe MCG and MEG signals of interest, but OPMs operate at or above room temperature, so no liquid helium infrastructure is required. Without a liquid helium dewar, the size of an OPM instrument may be much smaller than SQUID systems, and the OPMs may be reconfigurable. Reconfigurability would allow sensors to be placed closer to the magnetic sources and would be especially advantageous for human subjects with small heads, such as children. There are potential cost savings with an OPM-based system because a smaller system could employ a much smaller magnetic shield, and there would of course be no cost of liquid helium. Additionally, existing tools for analysis and interpretation of SQUID systems may be adapted to an OPM system [6]. The technology of OPMs is being actively developed for both MCG and MEG. Initial work on MCG began with vapor cells pumped by discharge lamps [7,8]. More recently, multiple designs have been employed for laser pumping of macroscopic vapor-cell [9][10][11][12][13][14][15][16][17]  In this work, we report a new OPM sensor for the detection of biomagnetic fields, specifically those of MEG. Our multichannel vapor-cell module is based on the design of Refs [27, 28], which used a two-color pump/probe scheme on a single optical axis. The twocolor scheme enabled approximately independent control of the sensitivity (via the linearlypolarized probe laser) and bandwidth (via the circularly-polarized pump laser) of the OPM [32]. Moreover, having one optical axis allowed the module to be compact. Four magnetometer channels were formed in a square arrangement by detecting the single probe beam on quadrant photodiodes, and four synthetic gradiometers were formed by taking the differences between spatially adjacent magnetometer channels in software. This approach was designed to perform well as a gradiometer, since all four magnetometer channels were in a single module and therefore experienced almost identical operating conditions. Thus, common mode noise was canceled effectively in the gradiometric mode of operation, substantially improving the sensitivity over the magnetometers alone. However, a disadvantage of the previous design was that the gradiometer baseline (the separation between adjacent magnetometer channels) was short, approximately 5 mm, because the four channels were formed from a single, large laser beam.
Our new sensor module incorporates an innovative optical design based on a passive diffractive optical element (DOE), which splits the incoming laser light from one beam into four. Each of the four outgoing beams is used for a magnetometer within a sensor module, so the design is compact. The DOE also allows us to achieve a long gradiometer baseline of 18 mm within a small cross-sectional footprint on a human subject's head. Using gradiometry, the sensitivity of each channel (< 5 fT/Hz 1/2 ) is better than sufficient to observe MEG signals over the frequency range of interest (< 100 Hz). The high degree of similarity among the channels, which enables effective gradiometry, also holds the promise of being able to employ the data post-processing techniques, such as signal-space projection [33] and signalspace separation [34], that are essential to the current practice of MEG. For the sensor module itself, most of the parts are constructed using additive manufacturing (3D printing), which carries multiple advantages, including low production cost, rapid manufacturing (and thus design improvement), availability of non-magnetic materials, and intricate components. The sensor module is designed to serve as a building block of an array for an OPM-based MEG system for source localization.

Principles of optically pumped magnetometers
In our OPM, a vapor-phase alkali metal sample held near zero magnetic field is illuminated with a near-resonant, circularly-polarized laser beam. By the optical pumping process, the laser field imparts its angular momentum on the alkali atoms in the form of an average electron spin polarization, P that is parallel to the laser beam. When exposed to a magnetic field, B, the spin polarization precesses about B: where γ is the gyromagnetic ratio of the electron and q is the nuclear slowing-down factor, which depends on the isotopic nuclear spin [24]. For 87 Rb, which has a nuclear spin of 3/2, the value of q varies from 4 (when the polarization is unity) to 6 (when the polarization is zero). The total rate of loss of polarization, or decoherence, Γ, includes multiple contributions: the optical pumping itself (which reestablishes the direction of P along the pumping axis), collisions of the alkali atoms with each other and with other atoms or molecules (e.g. a buffer gas), and interactions with other laser beams that may be present. We operate the OPM near zero magnetic field to be in the so-called SERF regime, where decoherence due to alkali-alkali spin-exchange collisions is strongly suppressed [21,35].
In the case where the optical pumping happens along the z axis and where the only nonzero magnetic field is along the y direction, the steady-state components of the polarization are given by Ref where P 0 = R/Γ, R is the rate of optical pumping, and Equations (2) and (3) encapsulate the ways in which a magnetic field B y may be measured via the spin polarization. In the simplest case, the circularly-polarized pump beam, oriented along the z axis, is the only beam present. The absorption of the pump depends on P z . It is then apparent from Eq. (2) that, as B y is varied, the absorption exhibits a Lorentzian lineshape with a maximum value proportional to P 0 and a half width at half maximum of ΔB. Furthermore, if a magnetic field modulation is applied in the y direction while the absorption is phase-sensitively detected, then the resulting signal is directly proportional to B y . However, if B y = 0 and B x ≠ 0, then B x may instead be measured by applying a field modulation in the x direction. In this way, either transverse component of the magnetic field may be measured in one experimental configuration.
Another way to measure B y is to add a second, linearly-polarized, off-resonant probe beam to the system. In this case, the polarization of a probe beam oriented along the z or x axis undergoes Faraday rotation that is respectively proportional to P z or P x . Polarimetry of the probe beam, which thus measures P z or P x , allows for the determination of B y . If the probe points in the z direction, then at B y = 0, the atomic vapor has a maximum polarization of P 0 , and the probe experiences the strongest polarization rotation. When a magnetic field transverse to the propagation axis of the light is applied, the atomic polarization, and thus the rotation of the probe polarization, is reduced. Overall, the situation is analogous to the pumpbeam-only case: the Lorentzian resonance may be mapped onto a linear signal by applying a field modulation and performing phase-sensitive detection, and either B y or B x may be measured. If the probe propagates along the x direction, then Eq. (3) is already linear in B y near B y = 0, and this component of the magnetic field may be extracted directly from the dispersive lineshape. However, only the B y component may be determined in this configuration. Likewise, only the B x component may be measured if the probe beam is oriented along the y axis. Consequently, a probe beam oriented along the z direction has the greatest flexibility: The measured field component can be selected by the orientation of the field modulation.

Experimental methods
In this work, we have employed a passive diffractive optical element (DOE) in the OPM design. The DOE (Holoeye Photonics AG) is a two-dimensional diffraction grating that sends most of the incident power out into four first-order beams. The four beams are arranged in a square, exiting the DOE at 10° relative to normal incidence. At 780 and 795 nm, approximately 66% of the input laser power is distributed equally among the first-order spots, while the zeroth-order spot is suppressed. With the DOE in the optical assembly, the gradiometer baseline is set by the distance between the DOE and a collimating lens. In this way, a large gradiometer baseline, 18 mm in the current design, may be achieved while still maintaining a small cross-sectional footprint of the sensor module of 40 mm × 40 mm. Alignment of the optical assembly is simplified because the four first-order beams are well separated. Since the DOE delivers approximately the same laser power to each first-order spot, all four magnetometer channels experience similar conditions, which is desirable for forming synthetic gradiometers, as discussed in Section 1 above. optical element, and is then collimated before entering the Rb vapor cell. Light reflects off of the vapor cell, and the 795-nm pump light is rejected before reaching the polarization analyzer. In this top view, only two of the four laser beams are shown. Abbreviations include 4-CH balanced PD: four-channel balanced (subtracted) photodiodes; PBS: polarizing beam splitter cube; λ/2: half wave plate; λ/4: quarter wave plate. The distance from the midpoints of the laser beams long their paths through the vapor cell to the outside of the module casing is 9 mm. The location of the head of a human MEG subject is noted for reference. (b) Calculated laser beam profiles through the vapor cell. The separation of adjacent spots is 18 mm. The numbers label the magnetometer channels as viewed from the fiber input side of the module.
The sensor design is depicted in Fig. 1, and a picture of the sensor module is shown in Fig. 2. Light at 795 nm (pump) and 780 nm (probe) is brought into the sensor module via a polarization-maintaining optical fiber. The co-propagating beams first pass through a polarizer to ensure linear polarization. A dual-wavelength wave plate makes the pump light circularly polarized while keeping the probe light linearly polarized. Light is then focused and, just after the focus, passes through the DOE. The DOE splits the beams into various diffraction orders, but primarily the four first-order spots. A second, large lens collimates the four first-order beams before they are transmitted to the vapor cell. The vapor cell is coated with a dielectric anti-reflective optical coating on the outer front side and a high-reflective coating on the outer back side, which reflects the pump and probe beams back towards the detectors. A 780-nm half wave plate balances the probe light intensity between the two arms of the polarimeter, and an interference filter rejects the pump light. Probe light then enters the polarizing beam splitter (PBS) cube to measure the polarization angle of the probe beam. Difference signals from the pairs of quadrant photodiodes are brought out of the module electrically using connectors (Hirose DF13 Series, tin pins) that we determined to have a minimal residual magnetic field signature. Sensor module with the top casing removed. Blackout material, which obscures the receiving optics, minimizes optical crosstalk and terminates stray reflections within the module. The optical fiber in the input optical assembly is terminated with a glass ferrule for its nonmagnetic properties. The Kevlar thread across the module is used to actuate the half wave plate on the detection arm while the sensor is sealed and in operation. The collimating lens is obscured by its holder and is not directly visible in the picture. The oven has coils, which are printed on flex circuit material, wrapped around its perimeter. These coils may supply both magnetic field modulation and control, although they are not used for the measurements shown below.
conventionally machined non-magnetic materials, and much more intricate designs are possible. The small sizes of the components, in turn, help to minimize the footprint of the sensor module on the head. Assembly of the sensor modules is performed using plastic hardware and adhesives that we verified to have minimal residual magnetic signatures.
The vapor cell, which is made of borosilicate glass, has an internal optical path length of 4 mm and has inner dimensions of approximately 25 mm × 25 mm. The intersections of the ~2.5-mm diameter (FWHM) laser beams and the 4-mm path length through the vapor cell define the four sensing volumes in each magnetometer module. Another advantage of using a DOE to generate separated beams is that the sensing volume of each magnetometer is small and the position well-defined, which are desirable characteristics for performing magnetic source localization. Because of the 1/r 3 decay of magnetic field strength, where r is the distance from the source, the signal observed from external magnetic sources is significantly increased by minimizing the spatial standoff. Consequently, the centers of the magnetometer's sensing volumes were designed to be as close as possible, about 9 mm, to the outside surface of the module, which rests against the head of the MEG subject. The highreflective coating on the back side of the cell helps minimize the distance between the vapor cell and the wall of the module because an additional mirror does not need to be inserted into the optical path.
The vapor cell has about 600 Torr of nitrogen buffer gas, which serves multiple purposes. The pressure of the gas gives a diffusion length smaller than the size of the beam for times shorter than the decoherence time 1/Γ, and the nitrogen molecules collisionally quench the excited state of the Rb and thus minimize radiation trapping [36]. The buffer gas pressure broadens the optical resonance linewidth to ~18 GHz.
The vapor cell operates at approximately 150 °C, which corresponds to a Rb number density of ~1 × 10 14 cm −3 . The temperature is maintained through continuous electrical heating; ac current at 200 kHz is sent through a twisted pair of high-gauge phosphor-bronze wire (Lake Shore Cryotronics, Inc.). The 200-kHz heater frequency is well above the magnetic field modulation frequency of 1 kHz (see below) to minimize the interference with measurements. However, the presence of the heater current does reduce the amplitude of the magnetometer signal by up to 10%, depending on the channel location. We conjecture that this behavior is due to spin decoherence caused by the ac magnetic field. The heater wire is affixed to the vapor cell with Kapton tape, which was determined using our magnetometer not to have a significant magnetic signature. We did not use thermally conductive epoxy because all that we tested were detectably magnetic. The temperature of the vapor cell is monitored using a Type E thermocouple (Omega Engineering). Although both the chromel and constantan alloys that comprise the Type E thermocouple contain significant fractions of nickel, which by itself is ferromagnetic, the Curie temperature for each alloy is well below room temperature. However, care must be taken not to magnetize the thermocouple, which can easily be done with, for example, a handheld bar magnet.
For biomagnetic applications, the exterior surface of the module must be maintained at a temperature that is comfortable for contact with human skin. This requirement is met using a twofold approach: the vapor cell oven is insulated with a high-quality polyimide blanket, and the module casing itself is actively cooled by flowing air through it. Polyimide insulation (Pyropel MD) was chosen because we found other candidate low-conductivity materials (Pyrogel 2250 aerogel blanket insulation and Zircar Microsil) to have magnetic signatures. For the air cooling of the module casing, cooling channels are 3D printed into the body of the module. Room-temperature air then flows through the length of the casing and between the walls of the oven. We have measured the outer surface of the module to be at 37.5 °C (that is, 0.5 °C above body temperature) when in contact with the human head. This temperature could reduced futher by adding insulation, such as layers of Kapton tape, between the module and the scalp.
To make the magnetometer sensitive to a single component of the transverse magnetic field, a modulation must be applied in that direction (see Section 2 above), which in this case is a ~140 nT modulation amplitude at 1 kHz. The optimal modulation strength is greater than the width of the magnetic resonance [see below] because the 1-kHz driving occurs far from resonance. The magnetic field is detected by de-modulating the subtracted photodiode output signals with a lock-in amplifier. Both of the two linearly independent components of the magnetic field orthogonal to the lasers can be collected in each magnetometer channel simply by changing the direction of the modulation. All measurements were taken in a 4-concentriclayer mu-metal magnetic shield. Coils inside the magnetic shield were used to modulate the magnetic field and to cancel residual fields. When the sensor is transferred to a large shield for MEG measurements, coils wrapped around the sensor oven (shown in Fig. 2) will provide the required field modulation.
We optimize the power and detuning of the pump and probe beams for good magnetic field sensitivity while maintaining a reasonable bandwidth for biomagnetic measurements. The power in the 780-nm probe beam is 2.0 mW per magnetometer channel, and the frequency is detuned from the D 2 resonance by 133 GHz. Approximately 5.7 mW per magnetometer channel of 795-nm pump light, detuned from the D 1 resonance by a magnitude of 10 GHz, optically pumps the 87 Rb atoms inside the vapor cell. A higher pumping rate shortens the response time of the atomic spins, and this power was chosen to achieve a frequency response in our sensor fast enough to observe biomagnetic signals of interest. At zero magnetic field, approximately 15% per channel of the pump light travels through the vapor cell and reaches the interference filter, to be rejected.
The circular polarization of the pump laser generates an ac Stark shift, or light shift, that manifests as a fictitious magnetic field along the longitudinal (optical) axis [37, 38]. We call the pump laser frequency at which this fictitious magnetic field is nulled out the "zero lightshift point." In principle, the largest magnetic response should correspond to the conditions of the largest atomic spin polarization (see Section 2 above) and the narrowest magnetic resonance linewidth. These circumstances occur when the optical pumping laser is tuned to the peak of the optical resonance, which usually coincides with the zero-light-shift point. However, as shown in Fig. 3, the largest magnetic response, as measured by the slope of the lock-in amplifier output, is found at + 10 GHz detuning from the zero light-shift point. In addition, the lock-in output slope versus detuning is asymmetric. To better understand these effects, we performed a full density matrix simulation of the optical pumping under the approximation that the atomic vapor is optically thin. The simulation reveals that the effects can be explained by the nitrogen buffer gas in the vapor cell. The nitrogen buffer gas provides fast quenching of the excited electronic state after optical pumping, and the excited state does not have sufficient time for hyperfine spin relaxation before it decays back to the ground electronic state. With strong spin exchange in the ground state and broadband optical pumping, a spin-temperature distribution of the ground-state hyperfine sublevels is quickly generated. This behavior leads to two important effects: (1) higher spin polarization can be achieved with the pump laser tuned close to the lower ground-state hyperfine manifold (i.e. a positive detuning), and (2) the pump absorption cross section is asymmetric with respect to the pump laser frequency and also favors a pump laser frequency close to the lower hyperfine manifold. The asymmetry of the absorption cross section resembles that of the lock-in slope curve in Fig. 3. We do not provide a simulated curve for comparison to the data because the simulation does not account for the high optical thickness of the vapor cell (which would be computationally challenging to include with the full density matrix treatment). However, there is good qualitative agreement between the simulation and experiment. A detailed explanation of the related physics may be found in Ref [39].

Results
The zero-field magnetic resonance of the Rb atoms in magnetometer channel 4, which is obtained by sweeping one of the transverse fields, is shown in Fig. 4(a). The line shape of the magnetic resonance without the 1-kHz modulation applied (cyan trace) is visibly not Lorentzian. [A Lorentzian line shape would be expected from Eq. (2).] To understand the line shape, we implemented a model to account for the optical thickness and diffusion of the atomic vapor as described in Ref [40]. Because our OPM is well described by fast quenching of the excited state and a spin-temperature distribution, a full density-matrix model can be simplified to a two-level system with a spin-polarization-dependent slowing-down factor, q(P). This approximation greatly reduces the computing resources required to employ a finite element method calculation for the spatial diffusion of the atoms and the pump laser propagation through the vapor cell. Using the operating conditions of our OPM in the model, we find that when the transverse magnetic field is zero, the strong optical pumping power allows the pump laser to highly spin polarize the atoms and bleach the vapor through the cell, even though the optical depth is very high for the unpolarized vapor [ Fig. 4(b)] [41]. When applying a transverse magnetic field, the spins precess away from the optical axis, and the spin polarization along the pump direction is reduced, as seen in the 10-nT case in Fig. 4(b). The pump laser intensity inside the vapor cell also decays more quickly along the pump direction. In addition, since the pump laser beam has a roughly Gaussian intensity profile, the diameter of the pump laser beam inside the vapor cell shrinks more quickly than in the zero-field case. These effects of high optical thickness lead to a spatially varying optical pumping rate that depends on the magnetic field. The probe laser interacts with all of the spin-polarized atoms inside the beam volume, thus integrating over atoms experiencing different optical pumping rates. The atomic spin polarization calculated from the model [black dashed curve in Fig. 4 (a)] agrees well with experimental data.
The red trace in Fig. 4 shows the magnetic resonance when the 1-kHz modulation is applied. The magnetic field sweep was 10 s in duration, which is much longer than the response time of the atoms (below), so the width of the trace at a given magnetic field represents the amplitude of atoms' response to the 1-kHz modulation. The blue trace is the demodulated output of the lock-in amplifier where the phase is tuned to maximize the response to the applied magnetic field. The magnetic field is measured in the linear section around zero field. We find that the slope of the lock-in output drops to 95% of the slope measured at zero field at approximately ± 2 nT.
The magnetic field sensitivities (rms magnetic noise densities) of the four magnetometer channels (see Fig. 5), measured at the same time and normalized by the corresponding frequency responses (see Fig. 6). The noise floor is 10-20 fT/Hz 1/2 over 5-200 Hz, aside from several technical noise peaks. The average noise of channels 1-4 are 17.8, 18.2, 17.1, and 16.1 fT/Hz 1/2 over the same frequency range, and the average over the relative standard deviations of the four noise levels at each frequency is 9%. The approximately 10 fT/Hz 1/2 noise floor of the magnetometer measurements is consistent with noise from the magnetic shield [27]. The slopes of the lock-in outputs of channels 1-4 are 1.2, 0.91, 0.80, and 1.2 V/nT, respectively, which gives a peak variation from the mean of the slope of 22%. We believe this variation in the lock-in output slopes is due to imperfect optical alignment of the probe light with the photodiodes, which is discussed in more detail below. The normalized frequency responses of the magnetometer channels are shown in Fig. 6. Frequency responses were obtained by measuring the spectrum that resulted from sending a linearly-chirped sinewave magnetic field into the magnetometer, where the amplitude of each frequency component was ~2 pT. The frequency response of the magnetometer channels is well described by the combination of (1) the atomic response, which itself is well described by a single-pole low-pass filter [22], and (2) the 4-pole low-pass RC filter of the lock-in amplifier. A fit to the expected amplitude response function, where f is the frequency, f 3dB is the 3-dB bandwidth of the atoms, and τ LI = 0.3 ms is the time constant for the lock-in low-pass filter, gives the bandwidth of the atomic response for a given channel. The response of the lock-in amplifier, given in the first moiety of Eq. (5), has a 3-dB bandwidth of 230 Hz for our time constant. The atomic f 3dB bandwidths are 89.9, 88.9, 90.2, and 87.7 Hz respectively for the four channels. The peak deviation from the mean of the bandwidths is < 2%. Note that the frequency responses of all four channels are plotted in Fig.  6, but the responses are nearly indistinguishable. The bandwidth for each channel may be increased further by increasing the pump laser power or by lowering the temperature of the vapor cell, although signals < 100 Hz, which are well captured by current module performance, are of most interest for biomagnetic applications. Also plotted in Fig. 6 is the response of one of the gradiometers, the difference of channel 1 and channel 2. From this comparison, we determine that the gradiometric cancellation of an applied field is approximately a factor of 100. However, due to slight differences in the bandwidths of the channels, phase shifts between the channels near f 3dB reduce the gradiometric cancellation, causing the differential frequency response to peak at f 3dB . The performance of four synthetic gradiometers is shown in Fig. 7(a). Gradiometer data was measured by taking the difference between two adjacent magnetometer channels and dividing the result by 2 1/2 to show the equivalent single-channel magnetometer sensitivity, which represents the intrinsic performance of the sensor module. The noise levels of the gradiometers are significantly reduced, with three of the four gradiometers having a sensitivity better than 5 fT/Hz 1/2 over 10-100 Hz, aside from a few technical noise peaks. The improvement of the gradiometers over the magnetometers is far from the factor of 100 implied by the field cancellation shown in Fig. 6 because not all of the noise is correlatedwhereas the external field applied in Fig. 6 was. Multiple noise sources, the largest of which is the photon shot noise, limit the gradiometric noise cancellation (see Fig. 8 below). Notably, the gradiometer between channels 2 and 4 has significantly more noise above 5 Hz than the others. This higher noise floor is due primarily to probe laser amplitude noise, which we discuss in detail in the next paragraph below. Figure 7(b) shows the output of the subtracted photodiodes prior to lock-in amplification for all four magnetometer channels. One of the primary reasons to measure the polarization angle of the light by subtracting the photodiode signals is that when the difference of the photodiode outputs is zero, only polarization rotation is detected, and amplitude noise is canceled. However, the behavior of the PBS cube depends on the input angle of incidence; light beams entering the PBS at different angles will require different polarization angles to be split equally between both output ports of the PBS. Additionally, the polarization extinction ratio of the PBS can depend strongly on the input angle of incidence. Each beam from the four magnetometer channels enters the PBS at a 10° angle to the normal of the input face, but in different quadrants. Consequently, each beam requires a different setting of the half wave plate immediately before the PBS to zero the difference of the photodiode outputs. As a compromise, we set the half wave plate so that the mean of the subtracted photodiode outputs is close to zero. Channels 2 and 4 have the largest dc separation. Consequently, when they are subtracted, the magnetic noise still cancels, but the optical amplitude noise tends to add. The net result in this case is increased overall noise. We tested several dielectric PBS cubes, and all exhibited similar angular dependence of the required input polarization for balanced detection. However, the angular dependence of the extinction ratio varied widely between different PBS cube types. We chose the Edmund Optics 780-nm laser line PBS, whose worst of the eight extinction ratios was 25:1 (for the transmitted beam of channel 3) and whose best extinction ratio was 160:1 (for the transmitted beam of channel 1). Calcite beamsplitters (e.g. Glan-Taylor polarizers) exhibited very little angular dependence for our configuration, but they would present additional design challenges due to the angles of the reflected beams and were deemed too expensive for fabricating a large number of sensors. gradiometer (blue trace), the inferred single-channel sensitivity is 3-4 fT/Hz 1/2 between 10 and 100 Hz. Unlike in the magnetometer case, the loss of sensitivity above ~90 Hz, due to the limited bandwidth, is clearly observable. (b) The subtracted photodiode output for each magnetometer channel. With the magnetometer operating at zero field, the second harmonic of the 1-kHz modulation is mainly observed. The dc offsets among the four channels are caused by the angular dependence of the polarizing beamsplitter. The large dc offset between channel 2 (red trace) and channel 4 (blue trace) is responsible for the worst gradiometer performance of the green trace in (a).
The performance of magnetometer channel 4 and of the synthetic gradiometer formed between channels 3 and 4 is shown in more detail in Fig. 8. The total magnetic noise of the magnetometer and gradiometer is the same as that shown in Figs. 5 and 7(a), respectively. In addition to the total noise, the noise from the quadrature output (90° phase shift) of the lockin amplifier, the noise when the pump light is blocked, the electronic noise (noise when no light enters the magnetometer), and the estimated photon shot noise are plotted. The quadrature-output and pump-blocked signals indicate the performance of the optical system because the magnetometer is not magnetically sensitive in these configurations. For the magnetometer [ Fig. 8(a)], these measures indicate that the total noise is not intrinsic to the magnetometer but due to magnetic field noise inside the shield. Likely contributions to this field noise are Johnson noise in the shield material (estimated to be ~10 fT/Hz 1/2 at frequencies below 40 Hz [42]), electrical current noise on the coils internal to the shield, and vibrations in shield. In Fig. 8(b), the total noise of the gradiometer is substantially lower and is much closer to the optical noise levels, particularly at higher frequencies. In the gradiometer, the optical noise level is nearly that of the photon shot noise. In the magnetometer, the optical noise is above the photon shot noise, indicating that for the Ch 3 − Ch 4 gradiometer there is some degree of optical noise cancellation as well as magnetic noise cancellation. We have not studied the noise below 10 Hz, but we conjecture that it is due to drifts in the vapor cell temperature (which is not actively stabilized) and in laser power and frequency. The electronic noise is sufficiently low that it does not contribute to the total noise, except minimally at two specific frequencies. The fundamental limit on the sensitivity for this magnetometer design is set by the atom shot noise [21]. For our Rb number density, laser interaction volume, and atomic 3-dB bandwidths, we estimate the atom shot noise to be ~0.5 fT/Hz 1/2 , approximately a factor of 6 away from our noise floor with gradiometry. Fig. 8. The detailed performance of (a) magnetometer channel 4 and (b) the Ch 3 − Ch 4 gradiometer. To illustrate the contributions of different noise sources to the magnetometer, optical and electronic noise sources are plotted with the total magnetic noise. The sensitivity of the Ch 3 − Ch 4 gradiometer above ~10 Hz is limited primarily by the photon shot noise.

Conclusions and future work
We have developed a new, four-channel OPM sensor whose design incorporates a DOE for the detection of biomagnetic fields, specifically those in MEG. The 18-mm gradiometer baseline is comparable to that of SQUID sensors currently in use. We have achieved gradiometric sensitivities of < 5 fT/Hz 1/2 , which are better than sufficient to observe MEG signals, especially with the current 9-mm sensor-to-head distance. At low frequency, the sensitivity may be improved by actively stabilizing the temperature of the vapor cell, which is in progress. The performance of the gradiometers above ~10 Hz is limited primarily by laser amplitude noise and photon shot noise, which can be improved by better active stabilization and by re-optimizing the module to operate at higher probe laser power, respectively.
The observed bandwidths are near 90 Hz, also sufficient to measure MEG signals of interest. Within the sensor module, the relative uniformity of the bandwidths is better than 2%, which holds promise for the future use of essential MEG signal processing techniques. The uniformity (and, to some extent, the performance of the gradiometers) could be improved further by changing the polarization analyzer so that differences in the PBS angle relative to the beams do not result in different extinction ratios or dc offsets in the subtracted photodiode outputs. Such a modification could be accomplished, for example, by including a collimating lens in the detection assembly.
The compact footprint of this four-channel design will enable multiple sensors to be packed densely around the head in an array. Additionally, the two-color-pump scheme for light shift cancellation demonstrated here will allow for the necessary and simultaneous magnetic compensation of multiple sensor modules. Although not used in the results reported here, the module design accommodates individual field modulation coils on each oven, which will be required for operating an array of sensors. Construction of an array of these OPM sensors for magnetic source localization in human subjects is under way.