Fiberized diamond-based vector magnetometers
Georgios Chatzidrosos, Joseph Shaji Rebeirro, Huijie Zheng, Muhib Omar, Andreas Brenneis, Felix M. Stürner, Tino Fuchs, Thomas Buck, Robert Rölver, Tim Schneemann, Peter Blümler, Dmitry Budker, Arne Wickenbrock
FFiberized diamond-based vector magnetometers
Georgios Chatzidrosos,
1, 2
Joseph Shaji Rebeirro,
1, 2
Huijie Zheng,
1, 2
Muhib Omar,
1, 2
Andreas Brenneis, Felix M. St¨urner, Tino Fuchs, Thomas Buck, Robert R¨olver, Tim Schneemann,
1, 2
Peter Bl¨umler, Dmitry Budker,
1, 2, 4 and Arne Wickenbrock
1, 2 Helmholtz-Institut, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany Johannes Gutenberg-Universit¨at Mainz, 55128 Mainz, Germany ∗ Corporate Sector Research and Advance Engineering,Robert Bosch GmbH, D-71272 Renningen, Germany Department of Physics, University of California, Berkeley, CA 94720-7300, USA (Dated: March 3, 2021)We present two fiberized vector magnetic-field sensors, based on nitrogen-vacancy (NV) centersin diamond. The sensors feature sub-nT/ √ Hz magnetic sensitivity. We use commercially availablecomponents to construct sensors with a small sensor size, high photon collection, and minimalsensor-sample distance. Both sensors are located at the end of optical fibres with the sensor-headfreely accessible and robust under movement. These features make them ideal for mapping magneticfields with high sensitivity and spatial resolution ( ≤ mm). As a demonstration we use one of thesensors to map the vector magnetic field inside the bore of a ≥
100 mT Halbach array. The vectorfield sensing protocol translates microwave spectroscopy data addressing all diamonds axes andincluding double quantum transitions to a 3D magnetic field vector.
INTRODUCTION
Nitrogen-vacancy (NV) centers in diamond have at-tracted attention as magnetic field sensors with high spa-tial resolution [2–4] and sensitivity [5, 6]. The range oftheir application includes, but is not limited to, singleneuron-action potential detection [5], single protein spec-troscopy [7], as well as in-vivo thermometry [8]. Advan-tages of NV-based magnetometers, compared to othermagnetic field sensors, include their ability to operate inwide temperature and magnetic field ranges [9]. The abil-ity to operate them also without the use of microwaves [9–11], has recently enabled a variety of new applications inenvironments where microwaves (MW) would be detri-mental [12].The NV consists of a substitutional nitrogen and an ad-jacent vacant carbon site. It can appear in different ori-entations along the crystallographic axes of the diamondlattice. This enables vector measurements of magneticfields [13]. Vector magnetometry itself can be useful inmagnetic navigation applications [14], magnetic anomalydetection, current and position sensing, and the measure-ment of biological magnetic fields [5]. Vector measure-ments near a background field of ∼
100 mT, where theNV’s ground state level anti-crossing (GSLAC) occurs,are of particular interest [11]. Some of the challenges ofvector measurements near the GSLAC field include thenecessity to precisely align the NV and the bias magneticfield axes for optimum sensitivity of the microwave-freemethod [9] or the need to account for transversal field re-lated nonlinearities of the NV gyromagnetic ratio whenperforming microwave spectroscopy. ∗ corresponding authors:[email protected], [email protected] One of the challenges NV magnetometers face is theirlow photon-collection efficiency. Approaches to increasethe efficiency include, use of solid immersion lenses [15–17], or employment of infrared absorption [6, 18–21].Photoluminescence (PL) for fiberized sensors is preferen-tially collected with the same fiber delivering the pumplight but detected on the input side of the fiber [22].Despite considerable effort, even modern sensors typ-ically just feature a PL-to-pump-light ratio of about0.1% [5, 22].In this paper, we discuss the construction of two fiber-ized NV-based vector magnetic field sensors one con-structed in Helmholtz Institute Mainz, referred to asMainz sensor in the following and the other one is asensor demonstrator of Robert Bosch GmbH, referredto as Bosch sensor in the following. They both achievesub-nT / √ Hz magnetic sensitivity. The sensor con-structed in Mainz achieved a sensitivity of 453 pT/ √ Hz,limited by intensity noise of the pump laser, with11 pT/ √ Hz photon-shot noise sensitivity and 0.5% PL-to-pump power ratio taking the fiber coupling efficiencyinto account. The sensor constructed by Bosch achieveda sensitivity of 344 pT/ √ Hz, which was approximatelyone order of magnitude larger than the expected photon-shot noise limited sensitivity [1]. The PL-to-pump powerratio was 0.3% taking the fiber-coupling efficiency intoaccount. The components used for the constructionof both sensors are commercially available. They al-low for a small sensor size (11 ×
40 mm for Mainz and15 ×
25 mm for Bosch) with maximized PL-to-pump-light ratio, as well as robustness to movement, whichalso makes the sensors portable. All of this makes thesensors ideal for mapping magnetic fields and measuringin regions that are not easily accessible. The magne-tometers are constructed in such a way that they allowclose proximity of the sensors to magnetic field sources, a r X i v : . [ qu a n t - ph ] M a r n m pu m p E A - n m s ω L 1 A E c)b)a) PDPD Lens
XYRef in ALIAGem 532nm A O M PIDDichroic MirrorFG Amp d) SensorHeadHelmholtzcoilsfiberPD cableMW cable
FIG. 1. a) Relevant energy levels of the NV center. The states as well as their spin projection m s are labeled. Solid greenlines represent pump light, solid gray lines phononic de-excitation, dashed red lines represent PL, curved gray lines representinter-system crossing. b) Experimental setup in Mainz (the experimental setup by Bosch can be found in Ref. [1]). c) Schematicof the fiberized sensor head. d) Photograph of the fiberized sensor by Robert Bosch GmbH. The abbreviations used in the figureare; PD: Photo-diode, AOM: acousto-optic modulator, PID: proportional-integral-derivative controller, LIA: lock-in amplifier,FG: function generator, Amp: amplifier, MW: microwave. thus allowing for high spatial resolution when mappingmagnetic fields. After demonstrating the sensitivity ofthe sensors and explaining the principles of vector mag-netometry with NV centers, we used the Mainz sensorto perform spatially resolved optically detected magneticresonance (ODMR) measurements covering an area of20 ×
30 mm inside a Halbach-magnet array. The Hal-bach array itself provides a highly homogeneous mag-netic field around 100 mT which makes it ideal for near-GSLAC magnetic field measurements and studies withNV centers. We present the analysis for the translationof the extracted frequency measurements into magneticfield. Details about the construction of the highly ho-mogeneous Halbach array is subject of another publica-tion [23]. EXPERIMENTAL SETUP
The diamond sample used for the Mainz sensor isa 2.0 × × type Ib, (100)-cut, high-pressurehigh-temperature (HPHT) grown sample, purchasedfrom Element Six. The initial [N] concentration of thesample was specified as <
10 ppm. The sample was irra-diated with 5 MeV electrons at a dose of 2 × cm − and then annealed at 700 ◦ C for 8 hours. The diamondsample used for the Bosch sensor is a 0 . × . × . (111)-cut, 99.97 % C enriched, HPHT grown diamond.The sample was irradiated with 2 MeV electrons at a doseof 2 × cm − at room temperature and then annealedat 1000 ◦ C for 2 hours in vacuum. The [NV − ] concentra-tion was determined by electron spin resonance to be0.4 ppm [1].Figure 1 (a) shows the relevant energy levels of the neg-atively charged NV center, which we use for magnetome- try. The ground and excited spin-triplet states of the NVare labeled A and E, respectively [Fig. 1 (a)], the tran-sition between them has a zero-phonon line at 637 nm,but can be exited by more energetic photons (of shorterwavelengths) due to phonon excitation in the diamondlattice. The lower and upper electronic singlet states are E and A , respectively. While optical transition ratesare spin-independent, the probability of nonradiative in-tersystem crossing from E to the singlets is several timeshigher [18] for m s = ± m s = 0. As a conse-quence, under continuous illumination with green pumplight (532 nm), NV centers accumulate mostly in the A , m s = 0 ground state sublevel and in the metastable Esinglet state. For metrology applications, the spins in the A ground state can be coherently manipulated with mi-crowave fields.Figure 1 (b) shows the experimental setup used for themeasurements conducted in Mainz. To initialize theNV centers to the ground-triplet state, we use a 532 nm(green) laser (Laser Quantum, gem532). To reduce in-tensity noise from the laser source, the light intensityis stabilized using an acousto-optic modulator (AOM,ISOMET-1260C with an ISOMET 630C-350 driver) con-trolled through a proportional-integral-derivative con-troller (PID, SIM960), in a feedback loop. The greenlaser is then coupled into a 2 m long, high-power multi-mode fiber cable (Thorlabs, MHP365L02).The MW to manipulate the NV spins are generatedwith a MW function generator (FG; SRS, SG394). Theyare amplified with a 16 W amplifier (Amp; Mini-Circuits,ZHL-16W-43+) and passed through a circulator (Mini-Circuits, CS-3.000) before they are applied to the NVcenters using a mm-sized wire loop. The other side ofthe wire is directly connected to ground. The radius ofthe wire used for the wire loop is 50 µ m. The wire is f res f mod f depth Detection limit2880.4 MHz 3 kHz 100 kHz 3.7 ± √ Hz2894.3 MHz 2 kHz 100 kHz 3.1 ± √ Hz2873.5 MHz 5 kHz 100 kHz 2.7 ± √ HzTABLE I. Settings and detection limit for the three differentNV axis used in the Bosch sensor. then attached to the optical fiber allowing the two partsto move together. For the magnetic resonance measure-ments presented in this paper, the MW frequency wasscanned between 3800 MHz and 5000 MHz. Informationon the experimental setup used to conduct the measure-ments with the Bosch sensor can be found in Ref [1].A schematic of the Mainz fiberized magnetic field sen-sor head is shown in Fig. 1 (c). The diamond is glued toa parabolic condenser lens, which itself is glued onto a11 mm focal length lens and attached to a fiber collima-tor (Thorlabs, F220SMA-532) to which the high-powermulti-mode fiber (Thorlabs, MHP365L02) is connected.The MW wire loop is attached to the other side of the di-amond to provide the rapidly oscillating magnetic fieldsrequired for this magnetic field detection scheme. Thelight is delivered to the diamond via the parabolic con-denser, lens, collimator and the high-power fiber. Thesame components collect the spin-state-dependent redPL of the NV ensemble. On the other side of the fibre thePL is filtered using a longpass dichroic filter (Thorlabs,DMLP605) which is also used to couple the incominggreen light into the fiber. After the dichroic filter residualreflected green light is removed by a notch filter (Thor-labs, NF533-17) and the PL is focused onto a photodi-ode (PD; Thorlabs, PDA36a2). The detected signal withthe PD is connected to a a lock-in amplifier (LIA; SRS,SR830). With this setup we were able to achieve 0.5%PL-to-pump-light ratio, which is an order-of-magnitudeimprovement compared to other fiberized sensors [22]. Aphotograph of the Bosch sensor can be seen in Fig. 1 (d).The sensor head, containing a microwave resonator, acustom designed balanced photodetector, and the dia-mond, which was glued to the collimated output of asingle-mode fiber, is located inside a custom designedHelmholtz coil. The Helmholtz coil was used to generatea magnetic bias field of 1.07 mT and the collimation ofthe laser beam was achieved with a gradient-index lens(GRIN). Further details on the setup used for the Boschsensor can be found in Ref. [1]. The final sensor head, ofthe Mainz sensor, has a diameter of ≤
11 mm and a heightof 40 mm, in a configuration that allows for ≤ µ maverage distance between sample and sensor. The 2 mlong fiber with the attached MW wire and the fiberizedsensor head for the Mainz sensor (shown in Fig. 1 (c))can be moved independently of the other components.The footprint of the sensor head of the Bosch sensor was15 ×
25 mm . The smallest distance between the centerof the diamond and the outer surface is roughly 2.4 mm,the sensor can move independently as well. To filter low- ● ● ● ● ● ●▲ ▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■ ■
100 200 300 400 5000100200300400 Frequency(Hz) M a g n e t i c fi e l d ( n T ) ■●▲ FIG. 2. Amplitude of magnetic test signal, measured alongthree different NV axes with the Bosch sensor. An oscillating,external magnetic field of about 0.7 µ T was applied to thesensor. frequency systematic noise components, e.g. laser-powerfluctuations, the MW-frequency ω L is modulated and thedetected PL is demodulated with the LIA. To producethe magnetic field maps presented here, the Mainz sensorassembly was mounted on a computer-controlled motor-ized 3D translation stage (Thorlabs, MTS25/M-Z8) inthe center of the Halbach array (not shown in Fig. 1). MAGNETIC FIELD SENSITIVITY
Figure 2 depicts the response of the Bosch sensor forthree different NV axes upon application of an AC mag-netic field with varying frequencies and a constant ampli-tude of 0.7 µ T. The three axes see different effective am-plitudes due to the different angles of the NV axes withrespect to the magnetic field vector as expected. The sen-sor response slightly decreases with increasing frequencyof the test field independent of the axis. This decreasemight stem from the fact, that with increasing frequency,the excitation field becomes under-sampled meaning thatthe full reconstruction of the sinusoidal excitation field isnot possible anymore leading to a decrease in measuredamplitude.To estimate the magnetic sensitivity of the two sen-sors we follow another method. When the MW field isresonant with the ground-state m s = 0 → m s = ± ≈
92 kHz and a contrast of 0.6 %. We modulate the MWfrequency around a central frequency f c , and record the -12 -11 -10 -9 -8 -7 -6 Frequency [Hz] M agne t i c f i e l d f l u c t ua t i on s [ T / H z ] / Unshielded magnetically sensitive MainzUnshielded magnetically insensitive MainzUnshielded magnetically sensitive BoschShielded magnetically sensitive BoschUnshielded magnetically insensitive Bosch
FIG. 3. Magnetic field fluctuations measured with the fiber-ized NV magnetometers. The blue and orange traces depictmagnetically sensitive and insensitive data of the Mainz sen-sor. The purple and green traces correspond to magneticallysensitive spectra of the Bosch sensor outside and inside a mag-netic shield, finally, the red trace corresponds to unshieldedmagnetically insensitive data. The average sensitivity in the60-90 Hz band is sub-nT/ √ Hz. The Bosch data are the re-sult of 100 averages, while the Mainz data is from a singleacquisition. first harmonic of the transmission signal with a LIA. Thisgenerates a demodulated signal, which together with theNV gyromagnetic ratio of | γ / π | ∼ .
024 GHz T − canbe used to translate PL fluctuations to effective mag-netic field noise. The optimized parameters resulting inthe best magnetic field sensitivity for the Mainz sensorwere: a modulation frequency of f mod = 13.6 kHz andmodulation amplitude of f amp = 260 kHz. The sensitiv-ity of the Bosch sensor was optimized for 3 different NVaxes as summarized in table I.Figure 3 shows the magnetic-field-noise spectrum of thesensors. The blue trace of Fig. 3 corresponds to magnet-ically sensitive data of the Mainz sensor, the orange tomagnetically insensitive. The purple and green trace cor-respond to magnetically sensitive spectrum of the Boschsensor outside and inside a magnetic shield, respectively,finally the red trace corresponds to the insensitive plot.The magnetically insensitive spectrum can be obtainedif f c is selected to be far from the ODMR features. Thepeak at 50 Hz is attributed to magnetic field from thepower line in the lab. The average sensitivity in the60-90 Hz area is 453 pT/ √ Hz and 344 pT/ √ Hz for theMainz and Bosch sensors, respectively. The noise tracesfor the Bosch sensor are based on continuous data series,that are recorded with a sampling rate of 1 kHz for 100 s.To calculated the amplitude spectral density (ASD), thedata series was split in 100 consecutive intervals, eachwith a duration of 1 s. For each interval the ASD wascalculated. The depicted is the average of the 100 ASDs. b) B N V n m p u m p NV axis 2NV axis 3NV axis 4NV axis 1 ω L a) B B = (B M , θ, φ)φzx yθB FIG. 4. a) The orientation of the diamond sensor in the Hal-bach array. The orientation of the magnetic field is givenin spherical coordinates with angle θ being the latitude an-gle with respect to the x-axis and φ the longitude. Themain component of the magnetic field points along the x-axis.b) Illustration of the diamond lattice structure indicating thefour different NV axes 1,2,3,4 and their relation to the appliedmagnetic field B. Microwaves (MW) with frequency ω L / (2 π )are applied to the sample. VECTOR MAGNETIC FIELD SENSINGData Acquisition
As a demonstration of the robustness and portabilityof our sensors as well as the ability to produce highlyresolved magnetic field maps, we select the Mainz sen-sor to characterize the homogeneity of a custom-madeHalbach-array magnet constructed in the Mainz labora-tory. The schematic of the magnet is shown in Fig. 4 (a)with more details found in Ref. [23]. It is a double ring ofpermanent magnets arranged to generate a homogeneousmagnetic field in its inner bore along the radius of therings. The field outside the construction decays rapidlywith distance. We performed ODMR measurements in a30 ×
20 mm plane nearly perpendicular to the main mag-netic field direction in the center of the Halbach array insteps of 1 mm and 1.5 mm in z and y -direction, respec-tively. The experimental procedure to characterize thismagnet involves reconstruction of the 3D magnetic fieldfrom these ODMR measurements, which we describe inthe next part of this paper. The measurements confirmedthe homogeneity of the magnetic field of the magnet tobe consistent with Hall-probe and NMR measurements,but with a threefold improved field strength resolution,vector information of the magnetic field and sub-mm spa-tial information of these quantities. The orientation ofthe fiberized sensor in the Halbach magnet can be seenin Fig. 4 (a). The magnetic field is given in spherical co-ordinates with respect to the (100) axis of the diamond (cid:126)B = ( B M , θ, φ ). The angle between the magnetic fieldvector and the yz-plane is θ , and φ is the angle betweenthe projection of (cid:126)B in the yz-plane and the y-axis. Thediamond sensor was oriented in the magnetic field suchthat different resonances were visible. Lo ck - i n A m p li f i e r ou t pu t ( V ) MW frequency (MHz) MW frequency (MHz) MW frequency (MHz) MW frequency (MHz)NV axis 4 - DQ NV axis 3 - DQ NV axis 2 NV axis 1
FIG. 5. Example data traces of four optically detected resonances at different position near the center of the Halbach magnet.The resonances appear dispersive due to the use of a modulation technique. The resonances are fitted and their centerfrequencies are extracted. The four different plots correspond to different transitions indicated above the respective plot. Thecolor of the traces indicate the positions at which the data was taken, as seen in Fig. 8.
For each position in the 30 ×
20 mm plane a scan of theapplied MW frequency between 3800 MHz and 5000 MHzwas performed in steps of 1 MHz. This range was limitedby the bandwidth of the MW components. At each fre-quency we recorded the demodulated PL from the LIAx-output with a data acquisition system and stored eachdata set with its respective position. To speed up theacquisition and after determining that no features wereleft out, we just acquired in total 164 frequency valuesaround the expected position of the resonances. In totala frequency scan to determine the four center frequencieslasted 46 s. This can be dramatically improved by for ex-ample applying a frequency lock on the four transitions,respectively.Figure 5 shows an example of collected data in fourdifferent points of the scan. The points are noted inFig. 8 (b) with different colors. The four different fre-quency regions correspond to different kind of transitionsas noted above the figures. The FWHM linewidth ofthe observed features is (11 . ± .
14) MHz and thereforemuch larger than the one given above (0 . ± .
02) MHzfor a small background field along one of the NV axis.This is due to the strong transverse field component butalso caused intentional via MW power broadening. Thissimplified the lineshape by suppressing the hyperfine fea-tures in the spectrum and therefore the analysis routine.We additionally like to note, that the two first of the fea-tures are double quantum transitions (DQ), i.e. magnetictransitions from the m s = − m s = +1 state, nor-mally forbidden, but allowed when transverse magneticfield components are present.Overall the sensor was moved in steps of 1 mm and1.5 mm in z and y -direction, respectively, such that thewhole area was covered. The resulting frequency maps of the chosen four resonances can be seen in Fig. 8 (a);they show structures corresponding to the strength of thelongitudinal (along the NV axis) and transverse (perpen-dicular to the NV axis) component of the magnetic fieldfor a given diamond lattice axis. The information con-tained in these plots is more than sufficient to reconstructall three vector components of the magnetic field at theposition of the sensor. Frequency to vector field conversion
After acquiring MW frequency scans for different po-sitions, the data were fitted with the sum of four deriva-tives of Lorentzians. The four center frequencies, fouramplitudes and a combined linewidth were fit parame-ters, an example of the data and fits can be seen in Fig. 5.By matching these frequencies to the positions of the 3Dtranslation stages we can make frequency maps as shownin Fig. 8 (a).To proceed with the analysis and construction of themagnetic field maps we note that the four different fea-tures presented in Fig. 5 originate from the different NVorientations in the diamond crystal. The positions ofthese features are related to the alignment of the mag-netic field to the NV axis, as well as its amplitude. If themagnetic field is along the NV axis (longitudinal) we ob-serve the m s = 0 → m s = ± m s = − m s = +1 are allowed. We call these double quantum(DQ) transitions. For a given magnetic field direction atthe position of the sensor, both, the longitudinal andtransverse components of the magnetic field are present. T r an s i t i on f r equen cy ( M H z ) Projection of B M same among all NV axis, ϕ = 0 o and θ = 0 o Magnetic field magnitude, |B M | (mT)a) Transition frequencies vs |B M | for all NV axis T r an s i t i on f r equen cy ( M H z ) Magnetic field magnitude, |B M | (mT)b) B M field alligned to NV axis 1, ϕ = 35.27 o and θ = 45 o T r an s i t i on f r equen cy ( M H z ) M | (mT)c) B M field at z = 0, y =15 mm of the Halbach arrray, ϕ = -2.43 o and θ = 35.46 o NV axis 4NV axis 3NV axis 2NV axis 1NV axis 4NV axis 3NV axis 2NV axis 1NV axis 4NV axis 3NV axis 2NV axis 1 B M B M B M FIG. 6. Transition frequencies as a function of | B M | . Thetraces of the four different NV axes are depicted with coloursconsistent with those of Fig. 4. The gray area represents ourscanning range, the vertical dashed line shows the mean fieldstrength inside the Halbach magnet. a) For a magnetic fieldperpendicular to the (100) plane: all transitions overlap sincethe magnetic field projection is the same among all axes. b)For a magnetic field perpendicular to the (111) plane of adiamond: the magnetic field is aligned along NV axis 1, andthe magnetic field projection is the same for the other threeaxes. The slanted dashed line for NV axis 1 corresponds to thefrequency of the DQ transition which in this case due to theabsence of transverse magnetic fields is not allowed. c) For amagnetic field configuration inside the Halbach magnet. - 50 0 500100020003000400050006000 (°)- 150 - 100 - 50 0 50 100 15015002000250030003500400045005000 ϕ (°) T r an s i t i on f r equen cy ( M H z ) T r an s i t i on f r equen cy ( M H z ) Transition frequencies vs θ, ϕ for all NV axis at B M = 104.5 mT NV axis 4NV axis 3NV axis 2NV axis 1a)b) θ FIG. 7. Simulated transition frequencies as a function ofangles θ and φ for the average magnetic field magnitude | B M | = 104 . θ for a fixedangle φ = − . ◦ . b) Transition frequencies as a function ofangle φ for a fixed angle θ = 35 . ◦ . Due to mixing caused by the transverse magnetic fieldcomponent, the NV gyromagnetic ratio depends on thebackground field strength.As a result, to match the above frequencies to magneticfields we create four vectors describing the NV axes. Weuse parameters β to describe the angle between NV axisand magnetic field and Ω for the field strength. Withthese parameters we derive a formula which describesthe transition frequencies of the spin states, from m s = 0to m s = -1, as well as m s = -1 to m s = +1 [24]. Here, weneglect strain and electric field effects.The frequency of the resonances depends on the mag-nitube of the magnetic field B M , as well as the angles θ and φ of the diamond orientation with respect to themagnetic field. Figure 6 shows the resonance frequencydependence as a function of magnetic field magnitude fordifferent fixed angles θ and φ . Figures 6 (a,b) representa magnetic field perpendicular to common diamond sur-face cuts, (100) plane for (a) and (111) plane for (b), (c)depicts a configuration inside our Halbach magnet. Fig-ure 7 shows the dependence of the transition frequencieson the angles θ and φ for a magnetic field matching themean magnitude inside the Halbach magnet. In Fig. 7 (a) φ = -2.43 ◦ and θ is varied. In Fig.7 (b) θ = 35.46 ◦ and φ is varied.In both Fig. 6 and Fig. 7 the four different NV axes aredepicted with colours matching those in Fig. 4. We ob-serve that in Fig. 6 (a) all the transitions overlap, as theprojection of the magnetic field is the same among allaxes. In Fig. 6 (b) the magnetic field is aligned along theNV axis 1, and its projection is the same for the otherthree axes, which appear overlapped in the figure. Thedashed line for NV axis 1 corresponds to the frequencyof the DQ transition which in this case is not allowed,as there is no transverse field on NV axis 1. Figure 6 (c)represents a case we encounter while measuring the fieldinside our Halbach magnet. In Fig. 7, we note the sym-metry between NV axes 1,2 and 3,4 which arises becauseof the crystallographic structure of the diamond. We canuse the information of the above plots to translate thefrequency into magnetic field.After identifying the possible transition frequencies, weuse a numerical method to translate the frequencies intomagnetic field direction and strength. For this methodwe restrict the magnetic field strength values to within10% of what the estimated magnetic field produced byour Halbach array is. Once the magnetic field is cal-culated for all the different frequencies we express it inpolar coordinates. When the initial values for directionand magnitude of the field are approximately known, wefind that the reconstruction is unique. We choose B M to describe the strength of the field, angle θ the lat-itude angle with respect to the x-axis and φ the lon-gitude angle as shown in Fig. 4 (a). Finally, we makea list of these parameters and plot them as shown inFig. 8 (b). In Fig. 8 (b) we included four different colouredcircles to represent the positions at which data for Fig. 5are taken. The measured averages of angles θ and φ in the central 10 × (y-x) of the Halbach mag-net are ( − . ± . ◦ and (35 . ± . ◦ , respec-tively, which is consistent with the orientation of the dia-mond lattice (cf. Fig. 4) relative to the coordinate systemof the magnet. CONCLUSION AND OUTLOOK
We constructed two fiberized NV based magnetic fieldsensors. The sensors feature sub-nT/ √ Hz magnetic fieldsensitivity with high PL-to-pump-light ratio. The com-ponents used to make the sensors are commercially avail-able which makes their construction easy and repro-ducible. The design for the Mainz sensor was made insuch a way as to allow free access to the front side of thediamond as well as for robustness, portability, and small size.With one of the main challenges for NV magnetome-try, especially fiberized, being the low photon-collectionefficiency, which leads to poor photon-shot-noise limitedsensitivity, the 0.5% PL-to-pump-light ratio collection ef-ficiency, demonstrated in the Mainz sensor, can be usedto achieve even higher sensitivity, if the effect of othernoise sources is minimized. This ratio could also be fur-ther optimized by implementing side collection on thediamond [17, 25], as well as highly reflective coatings onthe opposite (front) side of the diamond. These featureswere not implemented in this sensor in order to keep allof its components commercial and easily accessible.The access to the front side of the diamond allows forclose proximity to magnetic field sources, which in turnleads to high spatial resolution. The proximity is cur-rently limited to about 300 µ m by MW wire on top of thediamond. It could be optimized further by using a thin-ner diamond sample, a sample with shallow implantedNV centers or a thinner MW wire such as, for exam-ple, a capton-tape printed circuit board, all dependingon the intended application, as these changes influencethe sensitivity of the sensor. Such optimization would beespecially beneficial for measurements of dipole fields ormeasurements that require high spatial resolution.The robustness and portability of the sensors also makethem attractive for mapping larger areas, or on-the-movemeasurements of magnetic fields. These features com-bined with the small size of the sensors suggest a use formeasuring in hard-to-access places or small areas wheremaneuverability of the sensor would be required, e.g. en-doscopic measurements inside the human body.As a demonstration of the Mainz sensor we measuredthe magnetic field of a custom-designed Halbach array.For these measurements, the small fiberized diamond sen-sor showed a mm-scale spatial resolution, angular reso-lution of the magnetic field vector extracted in the mea-surements were 9 × − degrees and 18 × − degreesfor angles θ and φ , respectively. We note here, that thecorresponding displacement due to such a rotation of thediamond sensor (considering a (0 . × .
5) mm squarecross section) could optically not be resolved. ACKNOWLEDGMENT
This work is supported by the EU FET-OPEN Flag-ship Project ASTERIQS (action 820394), the GermanFederal Ministry of Education and Research (BMBF)within the Quantumtechnologien program (grants FKZ13N14439 and FKZ 13N15064), the Cluster of Excellence“Precision Physics, Fundamental Interactions, and Struc-ture of Matter” (PRISMA+ EXC 2118/1) funded by theGerman Research Foundation (DFG) within the GermanExcellence Strategy (Project ID 39083149). We acknowl-edge Fedor Jelezko for fruitful discussions. z - po s ( mm ) y-pos (mm) y-pos (mm) y-pos (mm) z - po s ( mm ) z - po s ( mm ) Field angle θ in º/1000 Field angle ϕ in º/1000b) y-pos (mm) z - po s ( mm ) NV axis 2 frequency (MHz) NV axis 3 - DQ frequency (MHz) y-pos (mm) NV axis 4 - DQ frequency (MHz) y-pos (mm) z - po s ( mm ) z - po s ( mm ) y-pos (mm)NV axis 1 frequency (MHz) a) 1050-5-10-15 10 -5 0 5 10 15 46504649464846474646464546441050-5-10 1050-5-101050-5-10 z - po s ( mm ) |B M | (mT) Reconstruction of magnetic field parameters FIG. 8. Figure adapted from Ref. [23]. (a) Spatially resolved frequency maps of the individual resonances from Fig. 5 (a). (b)The data collected are used to reconstruct the vector magnetic field in spherical coordinates as defined in Fig. 4 (a), the fourcolored circles note the positions at which data for Fig. 5 are taken.[1] F. M. St¨urner, A. Brenneis, T. Buck, J. Kassel, R. R¨olver,T. Fuchs, A. Savitsky, D. Suter, J. Grimmel, S. Henges-bach, M. F¨ortsch, K. Nakamura, H. Sumiya, S. Onoda,J. Isoya, and F. Jelezko, “Integrated and portable mag-netometer based on nitrogen-vacancy ensembles in dia-mond,” arXiv:2012.01053 , 2020.[2] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. A.Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-mer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-chitsch, F. Jelezko, and W. J., “Nanoscale imaging mag-netometry with diamond spins under ambient condi-tions,”
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