Unshielded 3-Axis Vector Magnetometer

• Unshielded three-axis vector magnetometers are defined as systems that capture or reconstruct the full magnetic-field vector (B_x, B_y, B_z) in ambient environments without magnetic shielding.
• They employ diverse modalities—such as fluxgate, NV diamond, alkali-vapor, and Hall sensors—to overcome hard-iron biases and soft-iron distortions through methods like ellipsoid fitting and projection inversion.
• Advanced calibration and closed-loop techniques, including machine learning approaches, enhance low noise performance and dynamic range for applications ranging from geomagnetic surveys to biomedical imaging.

An unshielded three-axis vector magnetometer is a magnetometric system that reconstructs the full magnetic-field vector in ambient conditions without enclosing the sensor in magnetic shielding. In current arXiv literature, the term covers low-power three-axis fluxgates calibrated directly in the natural Earth magnetic field, nitrogen-vacancy diamond sensors that recover the field from multiple crystallographic projections, alkali-vapor instruments that encode vector information in harmonic or frequency channels, and compact Hall or spin-orbit-torque devices that transduce orthogonal components electrically (Alimi et al., 2021, Clevenson et al., 2018, Pyragius et al., 2018, Schönau et al., 2024, Jin et al., 17 Feb 2025). Across these modalities, the central task is stable recovery of $B_x$, $B_y$, and $B_z$ in the presence of hard-iron offsets, soft-iron distortion, non-orthogonality, optical or microwave drift, environmental clutter, and axis misalignment.

1. Definition and modality scope

The defining property of the class is simultaneous or reconstructable access to the vector magnetic field under ambient conditions. In the simplest instrumental sense, this means direct readout of three orthogonal channels; in a broader sense, it also includes systems that measure multiple projections or repeated scalar magnitudes and then invert for the vector. The open-source Hall scanner based on the MMC5983MA is explicit on this point: because all three components $B_x$, $B_y$, $B_z$ are recorded at each spatial point, the device provides full vector measurements, and magnitudes can be computed as $B=\sqrt{B_x^2+B_y^2+B_z^2}$ (Dinçer et al., 2023).

The literature also makes clear that unshielded operation is not equivalent to “bias-free” operation. Fluxgate calibration can be performed outdoors or in a ferromagnetic-free environment with no external reference sensors, using the natural, undisturbed Earth’s magnetic field as the calibration field (Alimi et al., 2021). By contrast, NV and optically pumped systems often remain unshielded while deliberately imposing a bias field, such as $B_0 \approx 7.8$ mT in closed-loop NV vector magnetometry or about $730\,\mu\text{T}$ in a strong-bias cesium OPM, because the sensor still operates in ambient laboratory or geomagnetic conditions rather than inside magnetic shielding (Clevenson et al., 2018, Schönau et al., 2024). This distinction is important: “unshielded” describes the magnetic environment, not the absence of internal control fields.

The practical motivation is repeatedly framed against scalar magnetometers. The fluxgate survey study states that scalar magnetometers are expensive, power consuming and bulky, and notes additional drawbacks such as dead zones; in that same comparison, a calibrated three-axis vector fluxgate is described as low-cost, compact, low-power, and able to deliver scalar-equivalent magnitude readings suitable for magnetic surveys after in-situ calibration (Alimi et al., 2021). A related misconception is that vector devices are merely inferior scalar substitutes. The cited work instead presents them as dual-use instruments: they can provide scalar magnitude, preserve vector information, and eliminate scalar dead zones (Alimi et al., 2021).

2. Measurement principles and reconstruction formalisms

A common reconstruction formalism for unshielded vector magnetometers begins with a distorted or projected measurement model and then applies an inversion. For three-axis fluxgates, the raw measurement is modeled as

$m = S R B + b + n,$

where $B_y$0 is a full calibration matrix covering scale factors, non-orthogonality, soft-iron distortion, and frame misalignment, $B_y$1 is the hard-iron bias, and $B_y$2 is measurement noise. Under the constant-magnitude assumption $B_y$3, the locus of $B_y$4 over orientation lies on an ellipsoid,

$B_y$5

with $B_y$6, so the calibrated output is

$B_y$7

In this framework, hard-iron bias is the ellipsoid center, while soft-iron distortion and non-orthogonality are absorbed into the full $B_y$8 transform (Alimi et al., 2021).

A second paradigm uses multiple physical projections and solves a linear inverse problem. In closed-loop NV vector magnetometry, each diamond axis $B_y$9 yields a projection

$B_z$0

and the field vector is reconstructed by least squares from the four tetrahedral projections,

$B_z$1

An analogous projection model appears in wide-field NV imaging, where pixel-wise ODMR fits return four axis projections $B_z$2, and the vector field is reconstructed as

$B_z$3

In both cases, vector information is overdetermined by crystallographic symmetry rather than by three discrete orthogonal sensors (Clevenson et al., 2018, Segura et al., 2022).

A third paradigm is harmonic encoding. In the RF-dressed Voigt-effect alkali magnetometer, the first-harmonic quadratures map transverse fields while the second harmonic maps the longitudinal field. Near the sensitive point, the demodulated observables satisfy

$B_z$4

so triaxial readout is obtained from a single optical axis by synchronous demodulation of polarization homodyne signals (Pyragius et al., 2018). Related phase-geometry inversion is used in double-resonance alignment magnetometry, where the on-resonance phases of the $B_z$5 and $B_z$6 components determine the field orientation (Ingleby et al., 2018).

A fourth paradigm measures a scalar observable under controlled perturbation. In the strong-bias cesium OPM, the measured Larmor frequency is linearized as

$B_z$7

so each head measures, to first order, the projection of the ambient field onto the bias direction $B_z$8 (Schönau et al., 2024). In the geomagnetic-range free-spin-precession alignment magnetometer, one obtains scalar magnitudes with and without a known added field and then reconstructs each component from

$B_z$9

Here the atomic sensor is scalar at the measurement stage, while vector information is recovered by controlled dc perturbation (Jin et al., 17 Feb 2025).

3. Unshielded operating regimes

Unshielded operation is constrained first by environmental stationarity. In the fluxgate calibration work, Earth’s field magnitude $B_x$0 is treated as constant over the few minutes of calibration, and the dominant survey noise is attributed to ambient disturbances from local ferromagnetic clutter, with typical clutter peak-to-peak $B_x$1–$B_x$2 nT in land surveys. The same study therefore requires calibration away from nearby ferromagnetic objects and away from time-varying distortions such as nearby moving steel (Alimi et al., 2021). The open-source Hall scanner adopts an explicit subtraction strategy instead: in “coil mode,” each point is measured with current off and on, separated by about $B_x$3 s, and the source field is extracted from polarity-flipped readings so that the result is more robust to time-varying ambient fields (Dinçer et al., 2023). Wide-field NV current imaging uses differential acquisition at $B_x$4, $B_x$5, and $B_x$6 to cancel common-mode drifts and offsets under room-temperature, unshielded operation (Segura et al., 2022).

Closed-loop frequency tracking is a second recurrent strategy. In the NV closed-loop architecture, the lock-in output is integrated to steer the microwave local oscillator to resonance. The reported transfer analysis states that, at sufficient loop gain, the measurement becomes insensitive to resonance contrast, fluorescence amplitude, and linewidth, and exhibits zero steady-state error to step inputs; the output is the resonance frequency itself rather than a slope-proportional voltage (Clevenson et al., 2018). The strong-bias OPM applies the same systems logic in a different form: a homogeneous bias field of about $B_x$7 defines the sensitive axis, while the difference signal of two opposite-circularly-polarized pump channels is servoed to keep the oscillator on the Larmor frequency (Schönau et al., 2024). In the Voigt-effect alkali system, the authors explicitly state that active feedback on the external field should enable an extension of dynamic range as well as operation in unshielded scenarios (Pyragius et al., 2018).

A third strategy is geometric or spectral self-referencing. The fluxgate method estimates $B_x$8 as the median of $B_x$9 during rotation, without any external reference (Alimi et al., 2021). The warm-$B_y$0Rb microwave–optical double-resonance instrument determines $B_y$1 from the spacing of seven Zeeman-resolved double-resonance features, so the magnitude channel is self-calibrated by the ground-state gyromagnetic ratio rather than by external amplitude calibration (Babaei et al., 11 Jul 2025). This suggests that unshielded robustness is often obtained not from intrinsic immunity to ambient fields, but from a combination of environmental discipline, internal referencing, frequency-domain metrology, and closed-loop control.

4. Representative sensor architectures

The contemporary literature spans several distinct hardware realizations, with markedly different trade-offs in dynamic range, bandwidth, spatial resolution, calibration burden, and achievable noise floor (Alimi et al., 2021, Clevenson et al., 2018, Pyragius et al., 2018, Dinçer et al., 2023, Chen et al., 2022, Schönau et al., 2024, Jin et al., 17 Feb 2025).

Architecture	Core mechanism	Representative reported figures
Fluxgate with in-situ AI calibration (Alimi et al., 2021)	Ellipsoid-to-sphere calibration in Earth’s field	Uncalibrated PTP ≈ 24,055 nT; AI calibrated PTP ≈ 8 nT
Closed-loop NV vector magnetometer (Clevenson et al., 2018)	Frequency locking to $B_y$2 on four NV axes	~1 nT/√Hz; ~4 mT tracking on top of 7.8 mT bias
RF-dressed Voigt-effect alkali magnetometer (Pyragius et al., 2018)	$B_y$3 and $B_y$4 quadratures map $B_y$5	well below 1 pT/√Hz in shielded hot vapour
Open-source Hall scanner (Dinçer et al., 2023)	Three-axis Hall sensor with SET–RESET offset cancellation	±8 G per axis; RMS of the field readout below 0.3 mG
Single-device SOT anomalous Hall magnetometer (Chen et al., 2022)	$B_y$6 senses $B_y$7; $B_y$8 senses $B_y$9	±50 Oe for $B_z$0; ±100 Oe for $B_z$1
Strong-bias Cs OPM (Schönau et al., 2024)	Projection measurement along a homogeneous bias direction	below 60 fT/√Hz; sensor bandwidth of $B_z$2 kHz
Geomagnetic-range FSP alignment magnetometer (Jin et al., 17 Feb 2025)	Scalar FSP plus added dc fields along $B_z$3	5.3, 4.7, and 9.3 pT/√Hz

These figures should not be read as directly interchangeable. The fluxgate numbers refer to scalar-magnitude residuals after in-situ calibration in Earth’s field; the NV closed-loop system emphasizes frequency-tracking dynamic range; the Voigt and strong-bias OPM results are from atomic media with very different bandwidth and field regimes; the Hall and SOT devices prioritize compactness, automation, and low-cost electrical integration (Alimi et al., 2021, Clevenson et al., 2018, Pyragius et al., 2018, Dinçer et al., 2023, Chen et al., 2022, Schönau et al., 2024). Wide-field NV ensemble devices add a further axis of comparison—sub-micrometer spatial resolution and near-surface vector imaging—rather than competing primarily on scalar sensitivity or power (Segura et al., 2022).

5. Calibration, estimation, and inference algorithms

Calibration is the dominant algorithmic differentiator in unshielded vector magnetometry. In the fluxgate case, the deterministic approach is geometric ellipsoid fitting, implemented through the magnitude-constraint objective

$B_z$4

or equivalently by fitting a quadratic surface $B_z$5 with $B_z$6, then extracting the ellipsoid center and shape matrix. The same study also introduces a compact neural estimator that learns the linear map from noisy ellipsoid samples to projected sphere samples. The network has two hidden layers with three nodes each, uses linear activation and mean-squared error, and applies dropout on two neurons every 30 epochs; the reported implementation is intentionally small because the ellipsoid-to-sphere map is intrinsically linear under the constant-magnitude assumption (Alimi et al., 2021).

A separate use of learning appears in warm-$B_z$7Rb microwave–optical double resonance, where the amplitude pattern of seven double-resonance features encodes field direction. The instrument records 800 points across a 3 MHz microwave sweep and feeds the normalized one-dimensional trace into a Conv1D network with $B_z$8 filters, ReLU activations, max pooling, dropout, Adam optimization, and Huber loss with $B_z$9. Using 3000 spectra measured at random orientations in the positive octant at fixed $B=\sqrt{B_x^2+B_y^2+B_z^2}$0, bagging over five CNNs yields mean absolute angle errors of $B=\sqrt{B_x^2+B_y^2+B_z^2}$1 and $B=\sqrt{B_x^2+B_y^2+B_z^2}$2, while $B=\sqrt{B_x^2+B_y^2+B_z^2}$3 is obtained from the feature spacings with an accuracy of $B=\sqrt{B_x^2+B_y^2+B_z^2}$4 nT near $B=\sqrt{B_x^2+B_y^2+B_z^2}$5 (Babaei et al., 11 Jul 2025). In this setting, learning is not used to replace a known linear calibration; it is used because the spectral amplitudes depend on multi-level optical pumping and polarization geometry in a way that is difficult to parameterize compactly.

Offset cancellation and cross-axis calibration are equally central in low-cost electronic sensors. The Hall scanner exploits the MMC5983MA’s polarity flipping and degaussing through a SET–MEASURE–RESET–MEASURE sequence. With the source absent, the two polarities yield

$B=\sqrt{B_x^2+B_y^2+B_z^2}$6

and analogous combinations with the source present isolate the source field while suppressing slow thermal or offset drift (Dinçer et al., 2023). In the single-beam polarimetric rubidium magnetometer, three reflected lock-in outputs are related to $B=\sqrt{B_x^2+B_y^2+B_z^2}$7 by a calibrated $B=\sqrt{B_x^2+B_y^2+B_z^2}$8 mixing matrix $B=\sqrt{B_x^2+B_y^2+B_z^2}$9, and the vector estimate is obtained from $B_0 \approx 7.8$0 after offset removal (Pradhan, 2016). These examples underscore a general point: unshielded vector magnetometry is usually calibration-limited before it is sensor-limited.

6. Performance envelope and application domains

The reported performance envelope is modality-specific rather than hierarchical. At the compact, low-power end, the fluxgate survey system reduces root-mean-squared noise from the order of $B_0 \approx 7.8$1 nT to the order of $B_0 \approx 7.8$2 nT in a full $B_0 \approx 7.8$3 rotation in the natural Earth magnetic field, with AI calibration reporting PTP $B_0 \approx 7.8$4 nT in a typical in-situ run (Alimi et al., 2021). At the closed-loop diamond end, NV sensors report ~1 nT/√Hz sensitivity together with continuous tracking over ~4 mT on top of a 7.8 mT bias (Clevenson et al., 2018). At the atomic end, the strong-bias cesium OPM demonstrates a white noise floor of below $B_0 \approx 7.8$5 between 100 Hz and 600 Hz and a sensor bandwidth of $B_0 \approx 7.8$6 kHz (Schönau et al., 2024), while the geomagnetic-range FSP alignment instrument reports approximately 5.3, 4.7, and 9.3 pT/√Hz on the three axes near $B_0 \approx 7.8$7 (Jin et al., 17 Feb 2025). A magneto-optical iron-garnet design instead emphasizes spatial resolution, claiming simultaneously all three spatial components with high spatial resolution and sensitivity up to $B_0 \approx 7.8$8 (Ignatyeva et al., 2020).

Applications follow these operating envelopes. The low-power fluxgate system is presented as a practical alternative for magnetic surveys, especially where the environmental clutter floor is already $B_0 \approx 7.8$9–$730\,\mu\text{T}$0 nT peak-to-peak (Alimi et al., 2021). Diamond ensemble imaging is explicitly used for full vector magnetic maps and current-density reconstruction in quasi-2D conductors, with sub-micrometer spatial resolution and an NV layer approximately 8–16 nm below the surface (Segura et al., 2022). The Hall scanner is designed for characterization of coils, permanent magnets, and parasitic fields in ultracold-atom experiments, including Helmholtz cages and Zeeman slowers (Dinçer et al., 2023). Atomic and magneto-optical vector magnetometers are repeatedly linked to geomagnetism, navigation, compact sensor arrays, magnetocardiography, magnetoencephalography, and other biomedical or field-deployed settings (Pyragius et al., 2018, Pradhan, 2016, Ignatyeva et al., 2020, Schönau et al., 2024).

The recurring comparison with scalar magnetometers should therefore be interpreted carefully. Scalar cesium systems can achieve PTP $730\,\mu\text{T}$1 nT over $730\,\mu\text{T}$2, but the cited fluxgate work emphasizes that calibrated vector instruments offer reduced size, cost, and power, elimination of scalar dead zones, and simultaneous access to both vector mapping and scalar magnitude (Alimi et al., 2021). A plausible implication is that application fit is often determined less by absolute sensitivity than by deployment geometry, calibration overhead, and whether the environment favors scalar invariance or vector observability.

7. Limitations, misconceptions, and research directions

The dominant limitations are strongly modality-dependent but structurally similar. Fluxgate in-situ calibration assumes a constant-magnitude field and is therefore corrupted by time-varying fields, local anomalies, temperature drift, narrow-band sensor dynamics, and non-uniform rotation coverage (Alimi et al., 2021). NV closed-loop systems mitigate contrast and linewidth drift, but high-field operation remains bounded by resonance crossings, state mixing, and fluorescence suppression near level anti-crossings (Clevenson et al., 2018). RF-dressed alkali systems remain sensitive to misalignment, gradients, RF pickup, and operating-point trade-offs between transverse sensitivity and robustness (Pyragius et al., 2018). Spin-orbit-torque Hall devices show cubic crosstalk terms at larger fields, producing $730\,\mu\text{T}$3 and $730\,\mu\text{T}$4 distortions that must be modeled or subtracted outside the small-field linear regime (Chen et al., 2022). Strong-bias OPMs reduce perpendicular sensitivity but impose stringent requirements on bias-field homogeneity and thermal stability of the permanent-magnet structure (Schönau et al., 2024). Microwave–optical double-resonance direction inference exhibits larger angular errors near symmetry axes where some transitions vanish and the spectral fingerprint becomes less informative (Babaei et al., 11 Jul 2025).

A common misconception is that unshielded vector magnetometry is a single problem with a single optimal solution. The literature instead shows at least four substantially different solution classes: direct three-channel electrical sensing, projection inversion over multiple internal axes, harmonic encoding, and scalar-plus-perturbation reconstruction. Another misconception is that machine learning enters only when the physics is poorly understood. In the cited work, the opposite is often true: the fluxgate neural network is deliberately linear because the ellipsoid-to-sphere map is known to be linear under the calibration assumptions, while the CNN used for microwave–optical double resonance is introduced precisely because the spectral amplitude manifold is known to be structured but cumbersome to invert analytically (Alimi et al., 2021, Babaei et al., 11 Jul 2025).

Future directions are correspondingly heterogeneous. The fluxgate work proposes online or continuous calibration with adaptive AI models, multi-sensor fusion with accelerometer and gyroscope information, temperature-augmented calibration models, and mechanized rotation rigs for more uniform orientation coverage (Alimi et al., 2021). The microwave–optical double-resonance study points to multi-tone microwave excitation and real-time inference as a path beyond sweep-limited bandwidth (Babaei et al., 11 Jul 2025). The strong-bias OPM explicitly outlines three-axis realization through three orthogonal heads, a reorientable bias direction, or small-angle directional modulation around a permanent-magnet bias (Schönau et al., 2024). Across platforms, the consistent research trajectory is toward self-calibrating, low-power, miniaturizable vector sensors that remain quantitatively reliable in ambient magnetic environments rather than merely operable within them.