Multi-ring magnetic-field source, and associated methods

Abstract

A spin-center-based sensing method includes decreasing an angle between a Halbach magnetic field produced by a dual-Halbach ring and a symmetry axis of a spin center of a crystal, located in a bore of the dual-Halbach ring, by superimposing the Halbach magnetic field with an axial magnetic field produced by a third ring. Said decreasing produces a pre-tilted magnetic field orientated at an oblique angle with respect to a plane of the dual-Halbach ring. The third ring includes a circular array of permanent magnets that is coaxial and coplanar with the dual-Halbach ring.

Description

PATENT Attorney Docket No. QCAD.P2002WO / 00677700MULTI-RING MAGNETIC-FIELD SOURCE, AND ASSOCIATED METHODSCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U. S. Provisional Application No.63 / 749,364, filed on 24 January 2025, the disclosure of which is incorporated herein by reference in its entirety.GOVERNMENT RIGHTS

[0002] This invention was made with government support under grant number 80NSSC24CA146 awarded by the National Aeronautics and Space Administration (NASA). The government has certain rights in the invention.BACKGROUND

[0003] Nitrogen vacancy center (NV) based sensing platforms have applications in microscopy and imaging, magnetometry, and navigation / positioning. All-optical NV-nuclear magnetic resonance (NMR) is one example of such a platform enabling chemical and isotopic analysis.. These platforms are limited by the bias field: the static bias source fixes the instrument footprint and directly constrains optical access, microwave / RF delivery, alignment strategy, and long-term stability. Conventional implementations typically rely on bulky permanent-magnet platforms or high-homogeneity superconducting magnets, often requiring large mechanical stages, shimming hardware, and associated thermal-management overhead to achieve the field uniformity needed for high-quality spectra.SUMMARY OF THE EMBODIMENTS

[0004] Embodiments disclosed herein address this bottleneck by providing a miniaturized, integration-ready bias magnetic array built from permanent-magnet blocks, assembled in tolerance-controlled carriers. The permanent-magnet blocks may be commercially available. In embodiments, a multi-ring magnetic-field source includes a Halbach rings that concentrate magnetic flux into a clear bore while suppressing external fields, enabling a compact, passively stable bias source with no resistive power dissipation and reduced stray-field footprint. Embodiments are based on a multi-Halbach-ring architecture for (i) achieving a compact, high-field, high-homogeneity 1LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700tunable bias source within a clear bore, and (ii) relaxing the alignment of the bias direction to a chosen NV axis without imposing large global tilt angles that complicate optical and RF / MW integration.

[0005] In embodiments, multi-ring magnetic-field sources target the field attributes required for compact, room-temperature magnetic resonance based on defect-center readout: (i) low field strength up to high field strength (e.g., ≥ 180 mT-class) to increase thermal nuclear polarization and separate resonances without relying on complex hyperpolarization workflows, and (ii) sub-100-ppm-class homogeneity over the optical footprint to preserve coherence, avoid sensor dephasing, and suppress peak broadening that otherwise limits spectral resolution and coherent averaging

[0006] Embodiments of multi-ring magnetic-field sources replace laboratory-scale bias-field infrastructure with a compact, low-cost, commercially sourced, mechanically tunable, self-shielding bias module that preserves a clear bore for co-packaging optics, RF, and microwave hardware into a single mechanically registered sensor head.

[0007] Embodiments of multi-ring magnetic-field sources disclosed herein exhibit one or more of the following technical benefits.(a) Enables compact NV-NMR and related quantum sensors by replacing bulky biasfield infrastructure with a miniaturized permanent-magnet bias module.(b) Uses commercially available NdFeB magnet blocks and tolerance-controlled carriers, reducing cost, lead time, and manufacturability risk versus custom or superconducting solutions.(c) Implements a Halbach rings to concentrate flux into a clear bore while suppressing external fields, improving packaging density, and reducing strayfield footprint.(d) Provides a multi-ring architecture that simultaneously targets high field strength and high homogeneity within the same compact mechanical envelope.(e) Achieves strong-class bias fields to increase thermal nuclear polarization and support resonance separation without reliance on hyperpolarization workflows. (f) Delivers sub-100-ppm-class homogeneity on the optical footprint to preserve coherence, reduce dephasing, and limit peak broadening for improved spectral quality.2LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700(g) Provides mechanical tunability (phase / geometry control) to trim field magnitude and curvature without powered shims, minimizing thermal and electrical overhead.(h) Facilitates alignment to a chosen NV axis with reduced global tilt requirements (i) Preserves a clear bore for co-packaging optics, RF coils, and microwave delivery into a single mechanically registered sensor head suitable for deployable systems.

[0008] In a first aspect, a spin-center-based sensing method includes decreasing an angle between a Halbach magnetic field produced by a dual-Halbach ring and a symmetry axis of a spin center of a crystal, located in a bore of the dual-Halbach ring, by superimposing the Halbach magnetic field with an axial magnetic field produced by a third ring. Said decreasing produces a pre-tilted magnetic field orientated at an oblique angle with respect to a plane of the dual-Halbach ring. The third ring includes a circular array of permanent magnets that is coaxial and coplanar with the dual-Halbach ring.

[0009] In a second aspect, a magnetic-field source includes three coaxial rings: an inner Halbach ring, an outer Halbach ring, and an axially magnetized ring magnet. The inner Halbach ring is nested inside the outer Halbach ring. The inner Halbach ring includes a first plurality of permanent magnets. The outer Halbach ring includes a second plurality of permanent magnets.BRIEF DESCRIPTION OF THE FIGURES

[0010] FIGs. 1A-1C show measurement hardware and measurements for a permanent magnet of embodiments of a dual-ring bias array disclosed herein.

[0011] FIGs. 2A-2D illustrate and describe a dual Halbach-ring configuration of a dual-ring bias array disclosed herein.

[0012] FIG. 3 includes plots of spatially-averaged bias-field magnitude and relative field non-uniformity of embodiments of multi-ring magnetic-field sources disclosed herein.

[0013] FIG.4A is a schematic of a dual-ring Halbach array, in an embodiment. FIGs.4B, 4C, and 4D are performance maps of the dual-ring Halbach array of FIG.4A aligned with one selected NV center orientation.

[0014] FIG. 5A is a schematic of a multi-ring magnetic field source, in an embodiment.3LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0015] FIG. 5B illustrates in-plane and axial magnetic fields produced by the multiring magnetic field source of FIG. 5A.

[0016] FIG. 6A is a schematic of a dual-ring Halbach array, in an embodiment. FIGs.6B, 6C, and 6D are performance maps of the dual-ring Halbach array of FIG. 6A aligned with one selected NV center orientation.

[0017] FIG. 7A is a schematic of an embodiment of a multi-ring magnetic field source, which is the dual-ring bias array of FIG. 6A with the addition of a third ring. FIGs.7B, 7C, and 7D are performance maps of the multi-ring magnetic field source of FIG. 7A aligned with one selected NV center orientation.

[0018] FIG. 8A is a schematic of an embodiment of a multi-ring magnetic field source, which is the dual-ring bias array of FIG. 4A with the addition of a third ring. FIGs.8B, 8C, and 8D are performance maps of the multi-ring magnetic field source of FIG. 8A aligned with one selected NV center orientation.

[0019] FIGs. 9A and 9B are CAD renderings of an integrated NV-pNMR instrument that include an embodiment of the multi-ring magnetic-field source of FIG. 8A, in an embodiment.

[0020] FIG. 10 is a flowchart illustrating a method for using a magnetic-field source of FIG. 5A to perform spin-center-based sensing with a crystalline sample containing spin centers, in an embodiment.

[0021] FIG. 11 is a flowchart illustrating a spin-center-based sensing method, which may be implemented with embodiments of the magnetic-field source of FIG. 5A.DETAILED DESCRIPTION OF THE EMBODIMENTS

[0022] Halbach arrays are arrangements of permanent magnets (PMs) in which the magnetization direction rotates around the structure so that flux is reinforced within a target region and suppressed externally (self-shielding). In the bore-forming geometry relevant to compact magnetic resonance, this property concentrates Bo into a clear aperture while reducing fringe fields and magnetic interference with nearby instrumentations, thereby reducing (or eliminating) the need for auxiliary magnetic shielding and enabling tighter packaging. A Halbach ring is a discrete Halbach implementation assembled from nominally identical, uniformly magnetized blocks placed at fixed azimuthal angles (e.g., n = 8, 16, 24, 32, 48,...). As a consequence, part-to-4LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700part variability in remanence, dimensions, squareness, and magnetization direction must be quantified.

[0023] FIGs 1A-1C show the metrology used to validate the "nominally identical magnet." A 3D-printed fixture (Fig. 1A) was designed to hold a single NdFeB block (3.2 mm square cross-section, 19 mm length) in a repeatable socket while providing indexed Hall-probe positions along the block axis, enabling controlled one-dimensional scans of Bi(d) with approximately 3-mm position increments. Using this fixture, more than 280 individual magnets were measured at multiple distances, sampling both near- and intermediate-field regimes with large statistics.

[0024] The resulting ensemble-averaged field decay is consistent with the simulated response for the same block geometry (Fig. 1B): raw measurements (gray points) follow the modeled decay (black curve). At the same time, the distance-binned mean with ±3σ dispersion (black markers) remains within a few mT across the full range.

[0025] FIG. 1C shows the histogram of field amplitudes at the closest probe distance, fitted with a Gaussian distribution (gray line), where μ = 101.5 mT with σ = 1.35 mT (3σ = 4.05 mT). The near-Gaussian profile, characterized by a lack of heavy tails or outliers, suggests a statistically uniform population where deviations are primarily driven by small, uncorrelated part-to-part differences rather than systematic manufacturing defects. This result demonstrates that commercially available NdFeB blocks are sufficiently consistent to serve as practical building elements for the biasarray architecture: a multi-ring magnetic field source.

[0026] Embodiments disclosed herein leverage this modularity: commercially available NdFeB blocks are assembled in fabricated, tolerance-controlled carriers to realize ring-based (multi-ring) Halbach ring architectures that generate strong, homogeneous bias fields inside a clear bore while maintaining a compact external envelope. A single-ring Halbach array can deliver a strong bore field, but it offers limited in situ adjustability once the mechanical stack-up is fixed.

[0027] FIGS. 2A-2D illustrate and describe a dual Halbach ring configuration. FIG.2A is a representative CAD rendering of a dual-ring Halbach array with 16 NdFeB blocks per ring. Ring 1 is fixed to the base support, while Ring 2 is mounted in a rotational ring that enables controlled bi-directional azimuthal rotation (phase P) relative to Ring 1. A5LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700lid plate, anchored to the base via a vacuum-suction interface, clamps the ring carriers to maintain magnet seating and ring registration during rotation.

[0028] FIG. 2B is a plot of Bore-center field magnitude as a function of relative ring rotation angle, comparing experiment and magnetostatic simulation. FIG. 2C is a plot showing scaling of the bore-center field with array diameter across five fabricated designs, with linear fits to experiment and simulation. FIG. 2D is a stray-field map measured —15 cm above the bore (right) compared to simulation (left), demonstrating strong attenuation of the external field relative to the —220-240 mT operating field inside the bore.

[0029] A two-ring architecture (as the design shown in Fig. 2A characterized by 16 magnets each ring) adds a purely mechanical tuning degree of freedom that decouples bias-field trimming from powered shimming: the total bias field is the superposition of the inner- and outer-ring contributions, Bo(P) = BRI + RZ(P) BR2, where |3 is the relative azimuthal phase (rotation) of the outer ring. Varying |3 changes |Bo| while preserving a clear bore and passive operation, enabling post-assembly adjustment of the operating point without extra equipment.

[0030] FIG. 2B benchmarks the tuning principle and the design-to-hardware pipeline. The array geometry is parameterized in CAD, and its performance is evaluated with a magnetostatic forward model. Prototypes are realized with 3D-printed, tolerance-controlled carriers that define magnet seating and enable deterministic rotation of the second ring. The total field magnitude at the bore center is then recorded as a function of the second ring rotational angle |3 and compared with respect to simulations (black and gray curves in panel b, respectively). This analysis yields the characteristic constructive / destructive interference curve expected from the superposition of two discrete dipoles, demonstrating that the field can be mechanically set to a target operating point without powered shims. The residual discrepancy between model and measurement is consistent with fabrication and assembly tolerances (e.g., magnet-to-slot clearance and ring registration).

[0031] FIG. 2C generalizes this validation into a scaling relationship for system integration. Five Halbach-ring assemblies spanning different diameters were fabricated using commercially available, nominally identical cuboidal magnets (see Fig.l) integrated into fully 3D-printed mechanical structures, and their bore-center fields were compared against simulation. Over the explored design space, both experiment 6LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700and simulation exhibit an approximately linear dependence of Bo on diameter with similar slopes, making diameter a direct design knob for setting the target field while preserving internal clearance.

[0032] FIG. 2D reports spatial field mapping used to verify the realized field topology and quantify deviations from ideal symmetry. It demonstrates how the assembled hardware actually distributes magnetic flux in space beyond a single-point reading. Specifically, FIG. 2D compares the simulated (left) and measured (right) out-of-bore field map at the same stand-off plane (—15 cm above the device), showing consistent spatial structure: the field is concentrated into two lobes above the ring locations with peak values on the order of 45-50 mT, while it drops toward —10-20 mT toward the periphery. Both simulation and measurement show that the external field is only on the order of 10-50 mT, whereas the operating field inside the bore is approximately 220 mT. This corresponds to an attenuation of roughly one to two orders of magnitude, depending on position within the map, providing direct evidence of the self-shielding behavior of the Halbach-ring architecture and its ability to confine the high-field region to the sensor bore while substantially reducing the stray-field footprint at instrument-relevant distances.

[0033] A quantitatively constrained design space was established by surveying candidate dual-ring Halbach configurations spanning magnet grade, magnet geometry, and array scale. In total, 36 designs were simulated: 17 configurations using N42-grade NdFeB blocks in a 32-magnet dual-ring architecture, 14 configurations using N52-grade blocks in the same 32-magnet architecture, and (informed by those results) an additional 5 configurations in an enlarged 48-magnet dual-ring architecture. For each design, magnet sizes and grades were selected from commercially available catalog parts. Performance was evaluated using a benchmarked magnetostatic forward-model workflow, computing both bias-field strength and ppm-level homogeneity while simultaneously tracking integration-relevant clearance metrics.

[0034] Across the full set, the same global trends emerged. First, all configurations exhibit a simple field-space trade-off: larger bores provide more room for the sensor head but reduce the achievable bias field. Second, magnet grade shifts these trade curves vertically: for a fixed architecture and comparable geometry, moving from N42 to N52 increases the attainable bias field (typically on the order of 20-25%) without changing the qualitative dependence on clearance, confirming that the dominant design 7LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700lever is still geometry rather than grade. Third, magnet geometry controls where a configuration sits on the trade curve: larger magnet cross-sections and longer blocks increase field at a given clearance but consume bore space, whereas smaller blocks preserve clearance at the cost of reduced field; conversely, extending the magnet length provides a route to recover field at larger bores by increasing the effective magnetic volume without shrinking the in-plane clearance as aggressively. Fourth, homogeneity generally degrades as the design is pushed toward the most compact, highest-field end of the trade space (where gradients are steeper) and improves as the bore grows and the field curvature relaxes. Increasing the magnet count from 32 to 48 further smooths the field distribution and tends to improve uniformity over larger usable regions, enabling more clearance for the chassis while maintaining a bias field closer to the project target (~ 200-240mT).

[0035] FIG. 3 summarizes the survey by projecting each candidate design into a common performance-integration plane. Different curves (labelled from Ci to Cs) correspond to different array configurations as specified in the legend. Specifically, 32 and 48 are the total number of Neodymium Iron Boron (NdFeB) magnets forming the investigated dual-ring configuration. L, W, and T refer to the PM length, width, and thickness in inches. N42 and N52 refer to the magnet grade.

[0036] FIG. 3 includes plots 310 and 320. Plot 310 reports the spatially averaged bias-field magnitude (|B|) as a function of the largest inscribed square side linscribed (bottom axis), with the corresponding bore diameter dbore shown on the top axis. Each curve corresponds to one candidate configuration (see plot legends for more information), and the annotated markers indicate representative operating points for that geometry. Plot 320 displays the corresponding relative field nonuniformity o / p (in ppm) for the same configurations and horizontal axis, providing a direct view of how ppm-level homogeneity evolves as internal clearance increases.

[0037] The key information conveyed by FIG. 3 is the existence of a compact set of "integration-feasible" solutions that simultaneously satisfy the bias-field target and provide sufficient internal space for the sensor head. Based on these results, the survey converged on an optimal configuration for the current sensor-head chassis (characterized by NV diamond, microfluidic reservoir, MW antenna, and RF coils hardware required to perform NV-NMR measurements), delivering (|B|) ~ 248 mT8LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700while preserving approximately 2.8 cm of available internal space using permanent magnets with a cross-section of 6.4 mm and 26 mm long.Dual-Ring Magnetic-Field Source

[0038] The results of FIG. 3 correspond to bias array configurations whose bias field lies in the xy-plane, with an orientation set by the array phase. To meet the fieldalignment requirement for high-contrast ODMR in NV-NMR, embodiments adopt a fixed geometric convention: the diamond is rotated by -TI / 4 about the laboratory z-axis, and the bias array is configured to produce a field directed along the laboratory x-axis and rigidly rotated by 35.264° about the axis a = b0 × ẑ to map the array’s in-plane dipole field onto the chosen NV direction. Here, bO expresses the field orientation before the tilting. The results of this configuration are summarized in FIGs.4A-4D.

[0039] In FIGs.4A-4D, the tilted-bias configuration described above is evaluated as a function of two alignment degrees of freedom: the second-ring rotation p (phase trimming) and the axial position z of the NV plane relative to the bias array mid-plane. For each (p, z) pair, the magnetostatic field is computed on an xy grid spanning a 30×30 μm2spot-sized region (representative of the optical excitation / collection footprint) and reduced to spatially averaged metrics.

[0040] FIG.4A shows a dual-ring Halbach array 400 after the rigid 35.264° tilt, together with the laboratory coordinate system and the target NV-axis direction.Halbach array 400 includes coaxial Halbach rings 410 and 420. Halbach rings 410 and 420 include a plurality of permanent magnets (PM) 412 and a plurality of permanent magnets (PM) 422, respectively. For clarity of illustration, FIG.4A is annotated with respective dashed ellipses on top surfaces of magnets 412 and 422 to denote Halbach rings 410 and 420. Herein, examples of magnetic materials that form permanent magnets such as PMs 412 and 422 include neodymium, samarium-cobalt, ferrite, and PrFeB.

[0041] FIG.4B reports the mean projected bias field along the selected NV axis, ⟨B∥⟩, which remains in the -240-245 mT range across p E [-10°, +10°]. This field is maximized near z - 0, indicating that phase trimming preserves the target operating field at the sensor location.

[0042] FIG.4C reports the corresponding relative inhomogeneity o / p, reported in ppm, over the illuminated region: uniformity is best at the symmetry plane (tens of9LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700ppm) and degrades systematically with axial displacement, reaching O(102) ppm by |z| ≈ 5 mm, with only weak dependence on p over the scanned range.

[0043] FIG.4D isolates the directional requirement for high-contrast ODMR by mapping the mean misalignment angle 0NV between the Halbach-ring field direction and the selected NV axis; the minimum occurs near (p, z) (0, 0), where 0NV is sub-degree, and increases to several degrees at large |z| and |P|, showing that phase errors and axial offsets primarily perturb field direction rather than field magnitude. Together, these maps define an operating window centered at z ~ 0 where the tilted Halbach ring simultaneously delivers the required bias field, sub-100-ppm-class uniformity on the optical footprint, and near-parallel alignment to the selected NV class.Multi-Ring Magnetic-Field Sources

[0044] FIG. 5A is a schematic of a multi-ring magnetic field source 500, hereinafter magnetic-field source 500. Magnetic-field source 500 includes an inner Halbach ring 510, an outer Halbach ring 520, and axially magnetized ring magnet 530, each of which is coaxial about a common axis 501. For sake of brevity, rings 510, 520, and 530 are also referred to as Halbach ring 510, Halbach ring 520, and third ring 530. Halbach rings 510 and 520 include a plurality of permanent magnets (PM) 512 and a plurality of permanent magnets (PM) 522, respectively. Halbach ring 510 is nested inside Halbach ring 520. Outer Halbach ring 520 may be nested inside third ring 530, as illustrated in the embodiment of FIG. 5A. Rings 510, 520, and 530 maybe coplanar in a plane perpendicular to common axis 501. Halbach rings 510 and 520 form a dual-ring Halbach bias structure.

[0045] In embodiments, each PM 512 and each PM 522 has a magnetic moment in a plane perpendicular to common axis 501. Ring 530 may include a plurality of PMs 532 having a magnetic moment parallel to common axis 501. The total number of PMs 512 of Halbach ring 510 may equal the total number of PMs 522 of Halbach ring 520. In embodiments, each of PMs 512 have a same first shape and a same first surface field and each of PMs 522 have a same second shape and a same second surface field. The second surface field may be equal to or differ from the first surface field. The first shape and the second shape may be identical, and may be one of a cube, a rectangular cuboid, and a cylinder.10LEGAL\112473815\9Attorney Docket No. QCAD.P2002WO / 00677700

[0046] Magnetic-field source 500 may include a crystalline sample 540 located within a dipolar magnetic field generated by magnetic-field source 500. Crystalline sample 540 may include one or more color centers and / or spin centers. Examples of spin centers include vacancy defects and paramagnetic point defects. Examples of paramagnetic point defects include vacancy defects, interstitial defects, substitutional defects, antisite defects, Frenkel defects, Schottky defects, complex defects, color centers, charge state defects, impurity complexes, and substitutional defects (e.g., dopants). Additional examples of parametric point defects includes complex point defects (a.k.a. extended point defects), such as clusters.

[0047] A material composition of crystalline sample 540 may include one or more semiconductors, such as GaN, silicon, germanium, and II-VI semiconductors (e.g., ZnTe, ZnSe, ZnS). Crystalline sample 540 may have a bandgap that equals or exceeds 0.6 eV. The semiconductor may be a wide-bandgap material, examples of which include diamond, silicon carbide, and hexagonal boron nitride. Herein, a wide-bandgap material is one having a bandgap that equals or exceeds 2 eV.

[0048] Each of rings 510, 520, and 530 may include a respective mount 511, 521, and 531, each of which includes a plurality of holes that support and position PMs 512, 522, and 532, respectively. For clarity of illustration, FIG. 5A does not include a reference number for each of PMs 512, 522, and 532. PMs 512, 522, and 532 maybe affixed to mounts 511, 521, and 531. Mount 521 maybe axially rotatable with respect to the inner mount 511 about axis 501. FIG. 5A illustrates crystalline sample 540 within a bore of mount 511. The bore may be defined by an inner annular surface 513 of mount 511.

[0049] Embodiments of magnetic-field source 500 include a permanent-magnet assembly that generates a compact, strong, and homogeneous bias field inside a clear bore, enabling integration of the components required to perform microscale (or nanoscale) nuclear magnetic resonance (NMR) applications using spin centers in crystalline sample 540.

[0050] Magnetic-field source 500 generates a strong, tunable, dipole-like transverse magnetic field BR1R2,φwith sub-ppm class spatial homogeneity over the optical / sample region of interest (e.g., a disk or cylinder centered on the laboratory reference frame, which remains collinear with the NV sample, with dimensions set by the objective NA and focal geometry). The disposition of PMs sets the in-plane (f>11LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700orientation of magnetic field 1? RRshown in FIG. 5B. Halbach ring 520 may rotate with respect to ring 510 by an azimuthal angle β. This relative rotation provides a passive, purely geometric tuning knob to fine-tune the magnitude of the transverse bias field |BR1R2|. To preserve the in-plane φ orientation and field homogeneity while rotating β, Halbach ring 510 and Halbach ring 520 may be designed using the same number of magnets, with the same grade, shape, and geometrical features. Having the possibility to have a tunable-bias field allows performing NMR experiments at different magnetic fields using the same device.

[0051] The magnet count (e.g., a multiple of 8), shape, geometrical specifications, and grade of the permanent magnets (PMs) forming Halbach ring 510 and Halbach ring 520 set: (i) The available space of the bore of Halbach ring 510 and (ii) the magnetic field strength and spatial homogeneity ofR

[0052] Third ring 530 adds an axial field component BR3 z, shown in FIG. 5B, that vector-sums with the transverse Halbach component BR1R2,φ. As a consequence, the net bias direction (B̂R1R2R3) is pre-tilted toward a selected target orientation μ̂target. Target orientation μ̂targetmay be any of the 4 NV center orientations of a {100}-oriented diamond. In the following, we will consider the NV with an angle αNV= 35.3° out of the transverse x-y plane. The three-ring assembly has a net bias field equal to Bnet(r) = BR1R2,⊥(r) + BR3,z(r), where r E £1. £1 defines the sample region of interest. The pre-tilt direction is defined by:0pre= tan-1f LR3’Z' (1)Wvwl / and the design of third ring 530 maybe constrained by the condition: |BR3,Z| = |BR1R2,φ| tan θpre. More generally, |BR3,z| may be substantially equal to |BRIR2tan 0pre. For example, the ratio of |BR3,Z| to |BRIR2tan 0premaybe between 0.95 and 1.05.

[0053] The magnitude of 0preis set by:(a) The shape, count, geometrical specifications, grade, and dispositions of the PMs forming Halbach ring 510 and Halbach ring 520.(b) The shape, count, geometrical specifications, and grade of one or more PMs forming third ring 530.12LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0054] Third ring 530 may include multiple PMs arranged as a circular array. The PMs forming third ring 530 must be axially magnetized, and their count, shape, grade, and geometrical features may be different from those forming ring 510 and ring 520. Uniform azimuthal spacing may be selected to suppress transverse components and minimize coupling into the in-plane bias direction. Specifically, for third ring 530 implemented as N3uniformly distributed PMs magnetized along ±z, the axial component at the target region £1 is constrained to be a monotonic function of its magnetic moment and placement. That is, |BR3,Z| = F(N3, Br, Vmag, r3, z3), where Bris remanence (magnet grade), Vmagis the magnet volume (magnet geometrical features), r3is the radius of third ring 530 (lower bound imposed by the dual-ring Halbach design), and z3is the axial offset. A practical design scaling is |BR3,Z|N3 BrVmag / [μ0(r32+ z32)3 / 2] showing that increasing N3, Bror Vmagincrease pre-tilt, while increasing r3or \z31 decreases pre-tilt.

[0055] Once the features of Halbach rings 510 and 520 are fixed, 0premay range from a partial correction up to near-complete alignment of the net bias field with respect to μ̂targetdepending on the specifications of third ring 530. Even if full alignment is not reachable magnetically, magnetic-field source 500 reduces the rigid mechanical rotation 0rigidrequired to align the bias field to the selected direction of a spin center’s symmetry axis, such as an NV axis. Herein, NV may refer to a nitrogen vacancy axis or, more generally to the symmetry axis of a crystal’s spin center.

[0056] In conventional approaches, the alignment between the bias field and the chosen symmetry axis orientation is obtained by mechanically rotating the bias source by an angle αNV= 35.3°. In magnetic-field source 500, θrigid= |θpre− αNV|, improving objective placement and alignment, which is the most critical constraint for NV excitation and fluorescence collection in NV-NMR experiments.

[0057] A method of generating a bias magnetic field aligned to a selected nitrogenvacancy (NV) axis for an NV-center sensor includes at least one of the following enumerated steps.(a) Providing permanent-magnet rings 510 and 520 arranged concentrically about a bore axis, each comprising a plurality of permanent magnets with azimuthally varying magnetization directions configured to generate, at a region of interest £1 within the bore, a transverse bias field componenthaving sub-ppm-class 13LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700spatial homogeneity over £1. Ring 520 may be rotatable relative to ring 510 by an azimuthal angle p to passively tune the magnitude ||BR1R2,φ| without changing the bore aperture, the magnet grade, or the in-plane azimuthal orientation. Rings 510 and 520 may have the same magnet count (e.g., a multiple of 8), the same magnet grade, and the same magnet shape and geometric features, such that an in-plane azimuthal orientation φ of BR1R2,φis preserved when ring 520 is rotated relative to Ring 510.(b) Providing third ring 530 arranged concentrically about the bore axis and radially outside rings 510 and 520. Ring 520 may include a plurality of permanent magnets magnetized substantially parallel or anti-parallel to the bore axis and configured to generate an axial bias field component BR3 z.(c) Selecting one or more parameters of third ring 530, which include at least one of:magnet count N3, magnet remanence Br, magnet volume Vmag, ring radius n such that at £1 an axial-to-transverse ratio satisfies eqn. (1) thereby setting a pre-tilt angle 0preof a net bias field direction BR1R2R3toward a selected target orientation μ̂targetcorresponding to one of the four NV axes of a 100-diamond. (d) Assembling rings 510, 520, and 530 to produce a net bias field Bnet(r) =5R1R2,±(r) + BR3,Z(^) for r E £1.(e) Applying a rigid mechanical rotation of the assembled permanent-magnet assembly by a residual angle 6rigidto align BR1R2R3with the selected NV axis, wherein 6rigidis reduced relative to a rigid rotation required to align the net bias field to the selected NV axis in the absence of third ring 530 (see example below).Pre-Tilting Field Source: Embodiment 1

[0058] In the following, we will discuss embodiments of magnetic-field source 500 capable of generating a net bias field with a 0pre~ aNV. FIG. 6A shows a dual-ring Halbach array 600, which includes Halbach rings 610 and 620, which are respective examples of Halbach rings 510 and 520. Halbach ring 610 includes sixteen PMs 612. Halbach ring 620 includes sixteen PMs 622. For clarity of illustration, FIG. 6A is annotated with respective dashed ellipses on top surfaces of magnets 612 and 622 to denote Halbach rings 610 and 620.14LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0059] Each of PMs 612 and 622 is an N42 grade NdFeB magnet with a cuboidal cross-section of 1 / 8 inches and a length of % inches. The inner bore diameter of Halbach ring 610 is 20.2 mm, and magnets 612 are displaced to set <p = 0. To alignI with the (1,1,1) / 3 NV orientation, we have to rotate dual-ring Halbach array 600 by an angle αNV= 35.3°.

[0060] FIGs. 6B, 6C, and 6D summarize the properties of the bias field array generated by this dual-ring as a function of two alignment degrees of freedom: the second-ring rotation p (phase trimming) and the axial position z of the NV plane, which remains flat in the laboratory frame. FIG. 6B is a heatmap plot of a spatially averaged projected bias field along the selected NV axis, ⟨B∥⟩, computed over a 30×30 μm2spotsized region when aligned with NV (i.e., 35.3°), evaluated versus second-ring rotation p and axial offset z. The region of interest is flat with respect to the laboratory frame. FIG.6C is a heatmap plot showing relative field inhomogeneity o / p (ppm) over the same region of interest (ROI). FIG 6D is a heatmap plot of mean angular misalignment A0NV between the local bias-field direction and the selected NV axis.

[0061] By adding a third ring a pre-tilt angle 0preequal to 34.4° is obtained. The resulting magnetic-field source 700 is shown in FIG. 7A. Magnetic-field source 700 is dual-ring Halbach array 600 with the additional of a third ring 730, which is an example of third ring 530. Third ring 530 includes 9 N52 grade NdFeB magnets with a cuboidal cross-section of % inches and a length of 1.5 inches. For clarity of illustration, FIG. 7A is annotated with dashed curves on top surfaces of magnets that form Halbach rings 610 and 620 and third ring 730.

[0062] The magnetic properties of magnetic-field source 700 full assembly are summarized in FIGs. 7B, 7C, and 7D, using the same two alignment degrees of freedom as in FIGs. 6B, 6C, and 6D, respectively. FIG. 7B is a heatmap plot showing spatially averaged projected bias field along the selected NV axis, ⟨B∥⟩, computed over a 30×30 μm2spot-sized region when aligned with NV (i.e., 35.3°), evaluated versus second-ring rotation p and axial offset z. The region of interest is flat with respect to the laboratory frame. FIG. 7C is a heatmap plot showing relative field inhomogeneity o / p (ppm) over the same ROI. FIG. 7D is a heatmap plot showing the mean angular misalignment A0NV between the local bias-field direction and the selected NV axis.

[0063] FIG. 7A shows the resulting three-ring geometry, where magnets 732 of third ring 730 are placed concentrically outside dual-ring Halbach array 600 and 15LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700magnetized along the z axis. Compared to the dual-ring baseline dual-ring Halbach array 600, the addition of third ring 730 ensures almost full alignment with the chosen NV orientation {0rigid= \0pre— aw| = 0.9°) without the need of any extra mechanical rotation. Moreover, third ring 730 increases the available projected bias amplitude and its spatial homogeneity over the considered ROI. These results are shown in Fig. 7B and FIG. 7C.

[0064] Finally, Fig. 7D shows that the average angular deviation relative to the selected NV-axis direction, ( 0NV, is compressed across the fulldomain with respect to Fig. 6D. Overall, FIGs. 7A-7D demonstrate that introducing third ring 730 enables a near-NV-matched pre-tilt while keeping the NV plane mechanically flat, and simultaneously provides a higher usable bias amplitude, improved projected homogeneity, and reduced residual angular mismatch over the optical / sample region of interest.Pre-Tilting Field Source: Embodiment 2

[0065] FIG. 8A is a CAD rendering of a multi-ring magnetic-field source 800, which is Halbach array 400 with the addition of a third ring 830, which is an example of third ring 530. FIG. 8A also denotes a laboratory axis, and a target NV-axis direction. For clarity of illustration, FIG. 8A is annotated with dashed ellipses on top surfaces of magnets that form Halbach rings 410 and 420 and third ring 830.

[0066] FIGs. 8B, 8C, and 8D show performance maps of bias array 500. FIG. 8B shows spatially averaged projected bias field along the selected NV axis, ⟨B∥⟩, computed over a 30×30 μm2spot-sized region of interest when aligned with NV (i.e., 35.3°), evaluated versus second-ring rotation p and axial offset z of the bore plane. FIG. 8C shows the corresponding relative field inhomogeneity o / p (ppm) over the same ROI. FIG. 8D shows mean angular misalignment 0NV between the local bias-field direction and the selected NV axis.

[0067] Third ring 830 reduces the rigid mechanical tilt required to align Bo with the selected NV axis. The design extends the dual-ring geometry of dual-ring Halbach array 400 by adding a concentric third axial "booster" ring (ring 830) outside Halbach ring 420. Third ring 830 includes twenty-nine permanent magnets 832, not all of which are labeled in FIG. 8A for clarity of illustration. Each magnet 832 is an N52 grade NdFeB block having the same cuboidal dimensions used in Halbach rings 410 and 420, but 16LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700magnetized along the laboratory z direction (long-axis magnetization anti-parallel to z). Third ring 830 introduces a controlled Bzcomponent that pre-tilts the net field at the bore center by 9.1° out of the xy-plane. As a result, the subsequent rigid rotation required to map the array field onto the chosen NV axis is reduced from 35.264° to -26.17°, while preserving the same internal bore (~ 39 mm) and the same chassisrelevant in-plane clearance established in FIGs.4A-D.

[0068] The p-z maps in FIGs. 8B-8D confirm that this added degree of freedom improves practical usability without reducing field quality along the optical footprint. Over p E [-10°, +10°], the mean projected field on the selected NV axis remains in the mid-240 mT range (-240-248 mT in FIG. 8B, versus -240-245 mT in FIG. 4B), indicating that third ring 830 preserves the target operating field while providing an additional alignment handle.

[0069] Uniformity improves modestly but measurably: at z - 0 both designs yield tens-of-ppm o / p on the 30×30 μm2footprint, while at \z\ 5 mm the three-Halbach-ring configuration remains below -110 ppm (FIG. 8C) compared to -140 ppm in FIG.4C over the same p span, consistent with reduced axial curvature. The directional metric also improves slightly, with a larger sub-degree region around (p, z) (0, 0) and a reduced peak misalignment within the scanned window (-3.0° in FIG. 8D versus -3.5° in FIG.4D).

[0070] The primary benefit is experimental: reducing the required tilt directly eases objective and sensor-head alignment, and increases geometric margin for the optical train and surrounding equipment. This functionality comes at the cost of a larger outer diameter and higher magnet count, but it remains a fully passive, commercially sourced architecture that preserves the bore clearance set by the dual-ring design while reducing alignment constraints.

[0071] Importantly, the degree of tilt reduction is specification-dependent. The magnitude of the pre-tilt (and therefore the residual rigid rotation required for NV-axis alignment) is set by the relative contributions of Halbach rings 410 and 420 and third ring 830, which depend on the permanent-magnet specifications used in Halbach rings 410 and 420 (grade, cross-section, length, and resulting in-plane dipole strength), as well as the booster-ring specification and placement. Alternative three-ring configurations were found to reduce the required rigid tilt by up to approximately 18°17LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700relative to the dual-ring baseline, while maintaining 200 mT-class projected fields and sub-100-ppm-class uniformity over the optical footprint.

[0072] Embodiments of multi-ring magnetic-field source 500 enable a compact NV-pNMR architecture in which the bias source no longer dictates the overall footprint, as illustrated in FIGs. 9A and 9B, which are CAD renderings of an integrated NV-pNMR instrument 990 that include an embodiment of multi-ring magnetic-field source 500. FIGs. 9A and 9B are best viewed together in the following description.

[0073] NV-pNMR instrument 990 includes a multi-ring magnetic field source 900, an alignment stage 910, opto-mechanics 920, a fiber collimator 930, and a detector 940. Fiber collimator 930 may be coupled to a laser, not shown, which may emit 520-nm light. The laser may be part of NV-pNMR instrument 990. Field source 900 is an example of bias array 500 and has a bore 902 formed by its Halbach rings. Field source 900 may include each of rings 510, 520, and 530 of magnetic-field source 500.

[0074] Magnetic-field source 900 acts as the central structural element, while a sensor-head 992 is positioned at the center of bore 902 via a compact alignment stage 912 that provides the degrees of freedom required to align the designed Bodirection to the selected NV orientations. Compact alignment stage 912 includes a rotation stage 913 that rotates field source 900 about an axis 915. Axis 915 may be perpendicular to an axis 901 of magnetic-field source 900. Axis 901 is an example of axis 501.

[0075] The optical architecture maybe compact: an objective 922 positioned below the bore supports epi-illumination / epi-collection, with laser illumination (e.g., 520-nm) delivered into objective 922 and NV photoluminescence collected back through the same path and routed to detector 940. This geometry minimizes component count and decouples much of the optical alignment from the bias array once the objective-to-sample spacing is set.

[0076] Embodiments of multi-ring magnetic-field sources disclosed herein exhibit one or more of the following features and / or technical benefits:• Multi-ring Halbach bias array assembled from commercially available NdFeB blocks in tolerance-controlled carriers, providing a high-field clear bore with reduced stray fields.• Dual-ring architecture with mechanical phase tuning p enables post-assembly trimming of Bo without powered shims.18LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700• Down-selected integration-feasible configuration delivers -248 mT while preserving -2.8 cm internal clearance using commercially sourced magnets.• Three-ring architecture adds an axial booster ring to introduce a controlled Bzcomponent, reducing the required global tilt for NV-axis alignment while preserving bore clearance and -240-250 mT-class operating fields.• Enables a compact NV-pNMR sensor-head layout with co-packaged hardware.

[0077] FIG. 10 is a flowchart illustrating a method 1000 using a magnetic-field source to perform spin-center-based sensing with a crystalline sample containing spin centers, the crystalline sample being located within a dipolar magnetic field generated by the magnetic-field source. Magnetic-field source 500 and crystalline sample 540 are respective examples of the magnetic-field source and the crystalline sample. Method 1000 includes at least one of steps 1010 and 1020.

[0078] In embodiments, the magnetic-field source generates the dipolar magnetic field within the bore. The dipolar magnetic field points in a direction that is perpendicular to a rotation axis (e.g., axis 501) of the magnetic-field source. Step 1010 includes rotating the magnetic-field source about the rotation axis to align the direction of the dipolar magnetic field with the axes of the spin centers. In embodiments, the crystalline sample is a diamond sample, the direction of the dipolar magnetic field forms a 54.7-degree angle relative to a surface-normal of a face of the diamond sample.

[0079] Step 1020 includes rotating the outer Halbach ring (e.g., ring 520) of the magnetic-field source, relative to the inner Halbach ring (e.g., ring 510) of the magnetic-field source, to change the magnitude of the dipolar magnetic field. The magnetic field source may be configured such that step 1020 has the effect of suppressing rotation of the direction of the dipolar magnetic field. The dipolar magnetic field may remain fixed during step 1020.

[0080] FIG. 11 is a flowchart illustrating a spin-center-based sensing method 1100, which may be implemented with embodiments of magnetic-field source 500. Method 1100 includes at least one of steps 1110 and 1120. The following description of method 1100 includes parenthetical numbers following terms recited by the method. The parenthetical number indicates that the element associated with the number in parentheses is an example of the term. For example, the description of step 1110 below recites "crystal (540)," which means that crystalline sample 540 is an example of the crystal introduced in step 1110.19LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0081] Step 1110 includes decreasing an angle between a Halbach magnetic field produced by a dual-Halbach ring and a symmetry axis of a spin center of a crystal (540), located in a bore of the dual-Halbach ring, by superimposing the Halbach magnetic field with an axial magnetic field produced by a third ring (530). The dual Halbach ring includes Halbach rings 510 and 520 of magnetic-field source 500. Step 1110 produces a pre-tilted magnetic field orientated at an oblique angle with respect to a plane of the dual-Halbach ring.

[0082] The dual-Halbach ring and the third ring constitute a multi-ring assembly. Step 1120 includes rotating the multi-ring assembly until the pre-tilted field is parallel to the symmetry axis. In step 1120, the amount of rotation is less because of the pretilted magnetic field resulting from step 1110. This decreased rotation results in a more homogenous field in z (lab frame) and less obstruction of an instrument apparatus component, such as objective 922, FIG.9B. The rotation of step 1120 may be about an axis (915) perpendicular to a common axis (501, 901) of the Halbach rings and the third ring. Compact alignment stage 912 of NMR instrument 990 may execute step 1120. Combinations of Features

[0083] Features described above, as well as those claimed below, may be combined in various ways without departing from the scope hereof. The following enumerated examples illustrate some possible, non-limiting combinations.

[0084] Embodiment 1. A spin-center-based sensing method, comprising: decreasing an angle between a Halbach magnetic field produced by a dual-Halbach ring and a symmetry axis of a spin center of a crystal, located in a bore of the dual-Halbach ring, by superimposing the Halbach magnetic field with an axial magnetic field produced by a third ring, thereby producing a pre-tilted magnetic field orientated at an oblique angle with respect to a plane of the dual-Halbach ring, the third ring including a circular array of permanent magnets that is coaxial and coplanar with the dual-Halbach ring.

[0085] Embodiment 2. The method of embodiment 1, the dual-Halbach ring and the third ring constituting a multi-ring assembly, and further comprising: rotating the multi-ring assembly until the pre-tilted field is parallel to the symmetry axis.

[0086] Embodiment 3. The method of either one of embodiments 1 or 2, the symmetry axis being a defect axis of a defect that is one or both of a vacancy defect or a paramagnetic defect.20LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0087] Embodiment 4. The method of embodiment 3, the crystal being a crystalline wide-bandgap material.

[0088] Embodiment 5. A magnetic-field source comprising: an inner Halbach ring comprising a first plurality of permanent magnets; an outer Halbach ring comprising a second plurality of permanent magnets, the inner Halbach ring being nested inside the outer Halbach ring, the inner Halbach ring and the outer Halbach ring being coaxial; and an axially magnetized ring magnet that is coaxial with the inner Halbach ring and the outer Halbach ring.

[0089] Embodiment 6. The magnetic-field source of embodiment 5, the outer Halbach ring being nested inside the axially magnetized ring magnet.

[0090] Embodiment 7. The magnetic-field source of embodiment 6, the inner Halbach ring, the outer Halbach ring, and the ring magnet being coaxial about a common axis, each of the first plurality of magnets and the second plurality of magnets having a magnetic moment in a plane perpendicular to the common axis; the ring magnet comprising a third plurality of permanent magnets each having a magnetic moment parallel to the common axis.

[0091] Embodiment 8. The magnetic-field source of either one of embodiments 6 or 7, the inner Halbach ring, the outer Halbach ring, and the ring magnet being coplanar.

[0092] Embodiment 9. The magnetic-field source of any one of embodiments 5-8, wherein the number of the first plurality of permanent magnets equals the number of the second plurality of permanent magnets.

[0093] Embodiment 10. The magnetic-field source of any one of embodiments 5-9, wherein: all of the first plurality of permanent magnets have the same first shape; all of the first plurality of permanent magnets have the same first surface field; all of the second plurality of permanent magnets have the same second shape; and all of the second plurality of permanent magnets have the same second surface field.

[0094] Embodiment 11. The magnetic-field source of embodiment 10, the first surface field being equal to the second surface field or different from the second surface field.

[0095] Embodiment 12. The magnetic-field source of any one of embodiments 5-11, further comprising a crystalline sample located within a dipolar magnetic field generated by the magnetic-field source, the crystalline sample containing a spin center.21LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700

[0096] Embodiment 13. The magnetic-field source of embodiment 12, the inner Halbach ring, the outer Halbach ring, and the ring magnet being coaxial about a common axis, the spin center having a symmetric axis oriented at an angle with respect to the common axis, the inner Halbach ring and the outer Halbach ring producing a magnetic field, perpendicular to the common axis, having a magnitude B±; the axially magnetized ring magnet producing a magnetic field having a component, parallel to the common axis, substantially equal to B⊥tan θ.

[0097] Embodiment 14. The magnetic-field source of either one of embodiments 12 or 13, the crystalline sample having a material composition that includes one of diamond, silicon carbide, and hexagonal boron nitride.

[0098] Embodiment 15. The magnetic-field source of embodiment 14, the spin center being a vacancy defect and / or a paramagnetic point defect.

[0099] Embodiment 16. The magnetic-field source of any one of embodiments 5-15, inner Halbach ring and the outer Halbach ring being coaxial about a common axis, and further comprising: an inner mount, the first plurality of permanent magnets being affixed thereto; and an outer mount axially rotatable with respect to the inner mount about the common axis, the second plurality of permanent magnets being affixed thereto.

[0100] Embodiment 17. A method comprising using the magnetic-field source of embodiment 5 to perform spin-center-based sensing with a crystalline sample containing spin centers, the crystalline sample being located within a dipolar magnetic field generated by the magnetic-field source.

[0101] Embodiment 18. The method of embodiment 17, the crystalline sample located inside a bore of the magnetic-field source, wherein: the magnetic-field source generates the dipolar magnetic field within the bore, the dipolar magnetic field pointing in a direction that is perpendicular to a rotation axis of the magnetic-field source; and the method further comprises rotating the magnetic-field source about the rotation axis to align the direction of the dipolar magnetic field with axes of the spin centers.

[0102] Embodiment 19. The method of embodiment 18, the crystalline sample being a diamond sample, the direction of the dipolar magnetic field forming a 54.7-degree angle relative to a surface normal of a face of the diamond sample.

[0103] Embodiment 20. The method of embodiment 17, further comprising rotating the outer Halbach ring of the magnetic-field source, relative to the inner 22LEGAL\112473815\9Attorney Docket No. QCAD.P2002WO / 00677700Halbach ring of the magnetic-field source, to change the magnitude of the dipolar magnetic field.

[0104] Changes may be made in the above methods and systems without departing from the scope of the present embodiments. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. Herein, and unless otherwise indicated the phrase "in embodiments" is equivalent to the phrase "in certain embodiments," and does not refer to all embodiments.

[0105] As used in this specification, any appendices thereto, and the appended claims, the singular forms "a," "an," and "the" include plural referents unless the content clearly dictates otherwise. It should also be noted that the term "or" is generally employed in its sense including "and / or" unless the content clearly dictates otherwise. Regarding instances of the terms "and / or" and "at least one of," for example, in the cases of " A and / or B," "at least one of A and B," and "at least one of A or B," such phrasing encompasses the selection of (i) A only, or (ii) B only, or (hi) both A and B. In the cases of " A, B, and / or C, " "at least one of A, B, and C," and "at least one of A, B, or C," such phrasing encompasses the selection of (i) A only, or (ii) B only, or (hi) C only, or (iv) A and B only, or (v) A and C only, or (vi) B and C only, or (vii) each of A and B and C. This may be extended for as many items as are listed.

[0106] The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.23LEGAL\112473815\9

Claims

Attorney Docket No. QCAD. P2002WO / 00677700CLAIMSWhat is claimed is:

1. A spin-center-based sensing method, comprising:decreasing an angle between a Halbach magnetic field produced by a dual-Halbach ring and a symmetry axis of a spin center of a crystal, located in a bore of the dual-Halbach ring, by superimposing the Halbach magnetic field with an axial magnetic field produced by a third ring, thereby producing a pre-tilted magnetic field orientated at an oblique angle with respect to a plane of the dual-Halbach ring,the third ring including a circular array of permanent magnets that is coaxial and coplanar with the dual-Halbach ring.

2. The method of claim 1, the dual-Halbach ring and the third ring constituting a multiring assembly, and further comprising:rotating the multi-ring assembly until the pre-tilted field is parallel to the symmetry axis.

3. The method of claim 1, the symmetry axis being a defect axis of a defect that is one or both of a vacancy defect or a paramagnetic defect.

4. The method of claim 3, the crystal having being a crystalline wide-bandgap material.

5. A magnetic-field source comprising:an inner Halbach ring comprising a first plurality of permanent magnets;an outer Halbach ring comprising a second plurality of permanent magnets, the inner Halbach ring being nested inside the outer Halbach ring, the inner Halbach ring and the outer Halbach ring being coaxial; andan axially magnetized ring magnet that is coaxial with the inner Halbach ring and the outer Halbach ring.

6. The magnetic-field source of claim 5, the outer Halbach ring being nested inside the axially magnetized ring magnet.24LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 006777007. The magnetic-field source of claim 6, the inner Halbach ring, the outer Halbach ring, and the ring magnet being coaxial about a common axis,each of the first plurality of magnets and the second plurality of magnets having a magnetic moment in a plane perpendicular to the common axis;the ring magnet comprising a third plurality of permanent magnets each having a magnetic moment parallel to the common axis.

8. The magnetic-field source of claim 6, the inner Halbach ring, the outer Halbach ring, and the ring magnet being coplanar.

9. The magnetic-field source of claim 5, wherein the number of the first plurality of permanent magnets equals the number of the second plurality of permanent magnets.

10. The magnetic-field source of claim 5, wherein:all of the first plurality of permanent magnets have the same first shape;all of the first plurality of permanent magnets have the same first surface field; all of the second plurality of permanent magnets have the same second shape; and all of the second plurality of permanent magnets have the same second surface field.

11. The magnetic-field source of claim 10, the first surface field being equal to the second surface field or different from the second surface field.

12. The magnetic-field source of claim 5, further comprising a crystalline sample located within a dipolar magnetic field generated by the magnetic-field source, the crystalline sample containing a spin center.

13. The magnetic-field source of claim 12,the inner Halbach ring, the outer Halbach ring, and the ring magnet being coaxial about a common axis, the spin center having a symmetric axis oriented at an angle 6 with respect to the common axis,the inner Halbach ring and the outer Halbach ring producing a magnetic field, perpendicular to the common axis, having a magnitude B±;25LEGAL\112473815\9Attorney Docket No. QCAD. P2002WO / 00677700the axially magnetized ring magnet producing a magnetic field having a component, parallel to the common axis, substantially equal to B⊥tan θ.

14. The magnetic-field source of claim 12, the crystalline sample having a material composition that includes one of diamond, silicon carbide, and hexagonal boron nitride.

15. The magnetic-field source of claim 14, the spin center being a vacancy defect and / or a paramagnetic point defect.

16. The magnetic-field source of claim 5, inner Halbach ring and the outer Halbach ring being coaxial about a common axis, and further comprising:an inner mount, the first plurality of permanent magnets being affixed thereto; and an outer mount axially rotatable with respect to the inner mount about the common axis, the second plurality of permanent magnets being affixed thereto.

17. A method comprising using the magnetic-field source of claim 5 to perform spin- center-based sensing with a crystalline sample containing spin centers, the crystalline sample being located within a dipolar magnetic field generated by the magnetic-field source.

18. The method of claim 17, the crystalline sample located inside a bore of the magnetic- field source, wherein:the magnetic-field source generates the dipolar magnetic field within the bore, the dipolar magnetic field pointing in a direction that is perpendicular to a rotation axis of the magnetic-field source; andthe method further comprises rotating the magnetic-field source about the rotation axis to align the direction of the dipolar magnetic field with axes of the spin centers.

19. The method of claim 18, the crystalline sample being a diamond sample, the direction of the dipolar magnetic field forming a 54.7-degree angle relative to a surface normal of a face of the diamond sample.26LEGAL\112473815\9Attorney Docket No. QCAD.P2002WO / 0067770020. The method of claim 17, further comprising rotating the outer Halbach ring of the magnetic-field source, relative to the inner Halbach ring of the magnetic-field source, to change the magnitude of the dipolar magnetic field.LEGAL\112473815\9

Images