Calibration-Free Vector Magnetometry Using Nitrogen-Vacancy Center in Diamond Integrated with Optical Vortex Beam
Bing Chen, Xianfei Hou, Feifei Ge, Xiaohan Zhang, Yunlan Ji, Hongju Li, Peng Qian, Ya Wang, Nanyang Xu, Jiangfeng Du
CCalibration-free vector magnetometry usingnitrogen-vacancy center in diamond integratedwith optical vortex beam
Bing Chen, † Xianfei Hou, † Feifei Ge, † Xiaohan Zhang, † Yunlan Ji, † Hongju Li, † Peng Qian, † Ya Wang, ∗ , ‡ , ¶ , § Nanyang Xu, ∗ , † and Jiangfeng Du ∗ , ‡ , ¶ , § † School of Electronic Science and Applied Physics,Hefei University of Technology, Hefei,Anhui 230009, China ‡ Hefei National Laboratory for Physical Sciences at the Microscale and Department ofModern Physics, University of Science and Technology of China, Hefei 230026, China ¶ CAS Key Laboratory of Microscale Magnetic Resonance, USTC § Synergetic Innovation Center of Quantum Information and Quantum Physics, USTC
E-mail: [email protected]; [email protected]; [email protected]
Phone: +0551 62902751. Fax: +0551 62919106
Abstract
We report a new method to determine the orientation of individual nitrogen-vacancy(NV) centers in a bulk diamond and use them to realize a calibration-free vector magne-tometer with nano-scale resolution. Optical vortex beam is used for optical excitationand scanning the NV center in a [111]-oriented diamond. The scanning fluorescencepatterns of NV center with different orientations are completely different. Thus theorientation information of each NV center in the lattice can be known directly withoutany calibration process. Further, we use three different-oriented NV centers to form a a r X i v : . [ qu a n t - ph ] F e b agnetometer and reconstruct the complete vector information of the magnetic fieldbased on the optically detected magnetic resonance(ODMR) technique. Comparingwith previous schemes to realize vector magnetometry using NV center, our method ismuch more efficient and is easily applied in other NV-based quantum sensing applica-tions. Keywords : NV centers, optical vortex beam, optically detected magnetic resonance(ODMR), vector magnetometry
Introduction
Detection of weak magnetic fields with nano-scale resolution is of great importance in funda-mental physics, medicine technology and material sciences.
Atomic magnetometers candetect magnetic field with extremely high sensitivity, but suffer from low spatial resolution.Superconducting quantum interference devices (SQUIDs) can reach nanoscale resolution,but they have a finite size and act as perturbative probes over a narrow temperature range.Recently, NV center in diamond is emerging as a promising candidate for nano-scale magne-tometry and solid-state quantum information processing. The electron spin in NV center,often with the assistance of proximate nuclear spins, works as a quantum sensor with highsensitivity and long coherence time. It is widely used in room-temperature magnetic sensingand capable of measuring spins in a single-molecular level.
Reconstructing full information of the magnetic field vector ( i.e. , including magnitude andorientation) is often crucial in applications such as biological magnetic field sensing and thecondensed matter physics.
For single NV center, the basic idea of measuring magneticfield is to detect the frequency shift due to the Zeeman effect by the optically detectedmagnetic resonance (ODMR) procedure. In weak magnetic field regime, it measures onlythe magnetic field along the axis defined by the nitrogen atom and the vacancy in thediamond lattice. To realize a real-time vector magnetometer in a three-dimensional space,2 bjective DM SPAD
VR GT Fiber xyz E A A E MW m s = ±1 m s = ±1 m s = m s = CNV (a) (b) (c)
Figure 1: (color online). (a) Schematics of the experimental setup with a home-built confocalmicroscope. The inset shows that the linearly polarized beam is passed through the vortexretarder, which can generate the azimuthally polarized beam. VR: Vortex Retarder; GT:Glan-Taylor Polarizer; DM: dichroic mirror; SPAD: single photon avalanche diode. (b)Scheme of energy levels of the NV center electron spin. Its ground state ( A ) and excitedstate ( E ) are both spin triplets. The ground state ( A ) has a zero-splitting (2.87GHz)between the m s = 0 and m s = ±
1. The excited state ( E ) is governed by spin-orbit andspin-spin interactions, split by 1 .
43 GHz between m s = 0 and m s = ± m s = 0 , ±
1) in the excited state exhibit spontaneous decay by photon emission. (c)Diagram of [111]-oriented diamond lattice containing nitrogen(yellow) adjacent to a latticevacancy(gray).one needs to integrate at least three unparalleled scalar magnetometers together. Becauseof the C ν symmetry of the diamond lattice, there’re four kinds of NV centers with differenttetrahedral orientations referring to the laboratory frame. Thus it is naturally to form anvector magnetometer with high resolution utilizing the four different-oriented NV centersin a bulk diamond. However, NV center is randomly generated in the diamond, it isnot possible to know the position and direction of each NV center in advance. In a vectormagnetometry application, we firstly locate the NV centers using a confocal system and thendo a calibration process to know exactly the orientation of each NV center.One conventional way to calibrate the NV magnetometer is to measure the ODMR spectraat given vector magnetic fields generated by a three-dimension electromagnet. By analyz-ing the ODMR spectra, the orientation of NV center could be figured out in experiment.Recently another optical calibration method is also developed by continuously changing the3olarization of the pumping laser. It is because that the photoluminescence of NV centerdepends on the overlap between the polarization of the laser and the orientation of the NVcenter. However, both these two calibration methods are time-consuming, preventing areal-time application of vector magnetometry based on NV centers.In this letter, we report the realization of a calibration-free vector magnetometer withNV centers by using the optical vortex technique. The optical vortex beam with spatiallyvarying electric filed polarization is a fast-developed technique, and is widely used in atomicphysics and quantum optics.
Azimuthally polarized laser beam, as a kind of vortex beam,is employed to excite the NV centers. The fluorescence pattern of the NV center collectedby the confocal microscopic system is dependent on its orientation. Thus we integrate thelocating process and the calibration process by scanning the fluorescence of the NV center.We realize this process in a [111]-oriented bulk diamond and further measure the magneticfield vector with standard ODMR procedures using three different-oriented NV centers inthe laboratory frame. This new method opens the way towards a real-time nano-scale vectormagnetometer and is generally applicable in other NV-based sensing applications.
Results and discussion
The spin-Hamiltonian of the NV center can be written as the sum of zero-field splittingterm, electron spin and nuclear spin Zeeman splitting terms, the hyper-fine interaction term(hyperfine splitting tensor A ), nuclear quadrupole interaction ( Q ): ˆ H = D ˆ S z + g e µ B (cid:126)B · ˆ S + ˆ S · A · ˆ I + Q ˆ I z + g N µ N (cid:126)B · ˆ I (1)where D =2.87 GHz is zero-field splitting parameter. µ B is the Bohr magneton, µ N is thenuclear magneton. g e and g N are the electron spin and N nuclear spin g-factor, respectively.In the Hamiltonian, the first term denotes the zero field splitting (ZFS), the second termis the electron spin Zeeman splitting, the third term is the hyper-fine interaction between4 X axis[μm] xxyz y
X axis[μm] Y ax i s [ μ m ] Y ax i s [ μ m ] Y ax i s [ μ m ] Y ax i s [ μ m ] (b) (c)(a) NV NV NV NV θ: 0.37°φ: 153.68°θ: 109.84°φ: 20.60°θ: 109.25°φ: 260.51°θ: 109.31°φ: 140.74° Figure 2: (color online). (a) Schematic drawing showing the four NV center orientationsfor [111]-oriented diamond. The yellow sphere represents nitrogen atom that replaces theoriginal carbon atoms, and the gray sphere represents vacancy. (b) The fluorescence patternsof the four different oriented NV centers. (c) The fitted patterns of four different oriented NVcenters. We fit the fluorescence patterns based on pattern matching algorithm by Pythonand obtain the orientation information of four NV centers. The polar angle θ and azimuthangle ϕ of four NV centers are (0.37 ◦ , 153.68 ◦ ), (109.84 ◦ , 20.60 ◦ ), (109.25 ◦ , 260.51 ◦ ), and(109.31 ◦ , 140.74 ◦ ).electron spin and nuclear spin, the fourth term is the nuclear quadrupole splitting ( I > m S = ± A ) are degenerate at zero magnetic field. At external magnetic fieldcondition, the m S = ± m S = 0 and m S = ± m S = 0and m S = ± A ) can be detected by ODMR.In contrast to previous work, we employ the azimuthally polarized beam, which is tightly5ocused by a high numerical-aperture ( NA ) aplanatic objective lens, to scan the NV centers.The polarization near the focal spot region is inhomogeneous. The electric field vector ( E ( r ) )near focus can be expressed as E ϕ ( r, z ) = 2 A (cid:90) α √ cosθ (sin θ ) J ( kr sin θ ) e ikz cos θ dθ. (2)Here J n ( x ) is the Bessel function of the first kind with order n . r = (cid:112) x + y is the positionof the focus spot. k = 2 π/λ is the wave number of the light in the medium. In the integral,the upper bound α of the polar angle ( θ ) is determined by the NA of the objective lens. A is the field strength at the pupil aperture. For azimuthally polarized beam, the total fieldis transverse and we only consider the azimuthal component of the electric field vector nearfocus region. Its intensity is an on-axis null and annular intensity distribution at all distance z from the paraxial focus.The fluorescence intensity distribution is determined by the interaction between the elec-tric field vector and the excitation dipole of the NV center. Strain-dependent opticalmeasurement of NV center has indicated that two orthogonal electric dipoles are in theplane perpendicular to the symmetry axis, which are responsible for the emission fluo-rescence. The fluorescence intensity distribution by scanning NV center can be written as I ( x, y ) ∝ | E ( r ) · µ exc1 | + | E ( r ) · µ exc2 | , where E ( r ) is the position dependent electric fieldand the µ exc1 , is the excitation dipole vector.Because of the inhomogeneous transverse polarization components around the focal spot,the scanning confocal image has characteristic intensity pattern that is different from theGaussian distribution with a linearly polarized beam. The fluorescence pattern with theazimuthally polarized beam is dependent on the three-dimensional orientation of the NVcenter’s excitation dipole. We simulate this process and calculate the photon pattern nu-merically. In Fig.2(c), different patterns indicate different oriented NV centers.The experiment is performed on a purpose-built confocal microscopy system as shown6 .80 2.81 2.82 2.83 2.84 2.850.90.951 2.92 ... Hz Figure 3: (color online). ODMR spectra for different orientation NV centers. In thestatic magnetic field, different orientation NV centers have anisotropic Zeeman splittingand ODMR spectra. Due to hyperfine interaction with the electron spin and N nuclearspin, the transitions between m s = 0 and m s = ± , NV , and NV .in Fig.1(a) to address single NV center in a type-IIa, single-crystal synthetic bulk diamond.A collimated 532 nm pulse laser beam is passed through the single mode optical fiber andshaped into approximately ideal Gaussian beam. The Gaussian beam is sent through theGlan-Taylor polarizer and produced high-quality linear polarization beam with an extinctionratio greater than 100,000:1. Such a beam is then sent to the Zero-Order Vortex Half-Wave Retarder(m=1) which can generate azimuthally polarized beam with the doughnut-like feature as shown in Fig.1(a). The generated azimuthally polarized beam is then focusedon the NV center using an Olympus oil-immersion microscope objective lens ( NA =1.40).A piezoelectric transducer (PZT) stage holds the diamond perpendicular to the beam inorder to implement the scanning in the transverse x-y plane of the sample. The fluorescencephotons ranging from 600 to 800 nm are collected by the same objective lens, and detected byusing the single photon avalanche diode (SPAD). The microwave signal (generated by Rohde& Schwarz) is amplified (Mini-Circuits ZHL-42W+) and delivered to an impedance-matchedcopper slotline (0.1 mm gap and an Ω-type ring in the middle) deposited on a coverslip, andfinally coupled to the NV center. A small permanent magnet (south pole) in the vicinity ofthe diamond generates the static magnetic field to be detected.Experimentally, we scan the diamond using the azimuthally polarized beam with theresult shown in Fig.2(b), where four kinds of patterns are all observed and coincident with7umerical simulation. To figure out the orientation of NV centers in the laboratory frame,we select four different patterns, which are noted as NV , NV , NV , and NV . Thenwe develop an optimization algorithm based on the Nelder-Mead method to determine theangles associated with the scanned patterns. The optimization routine starts from a randomstate of the angles and minimize the difference between simulation result and experimentalpattern by varying the angles iteratively. Note that, there’s still a rotational symmetry of180 ◦ in the azimuth angle, which can be easily broken by using a simple static magneticfield. Fig.2 shows the final result including polar angle θ and azimuth angle ϕ of the NVcenters, where all four patterns match well with experiment.With all the obervations, we then proceed to demonstrate the vector magnetometer withthree different-oriented NV centers. And the full vector information of the field could bereconstructed from the individual measurement results and the NV center orientation. Ineach measurement, the magnitude of magnetic field, the polar angle α between the magneticfield and the NV center axis are obtained by the standard ODMR procedure.The ODMR spectra lines are usually power broadened by laser and microwave (MW)radiation. We employ the normal Gaussian beam to excite the NV centers and get theODMR spectra. To obtain the high resolution ODMR spectra, we replace the CW (con-tinuous wave) ODMR process with the pulsed ODMR process using pulsed laser ( ∼
400 ns)and MW field (electron spin π -pulse). By sweeping MW field frequency, we can observethe electron spin transitions from m S = 0 to m S = ± A ), which areTable 1: The location of NV centers, the polar angles α between the static magnetic fieldand the NV axis, and the magnetic filed magnitude extracted from the ODMR spectra ofNV , NV , NV shown in Fig.3. The location of NV centers in diamond is also shown inscanning confocal image (part 1 of the Supporting Information).NV NV NV (x,y) ( µ m) (16.175, 7.335) (16.450, 1.920) (5.346, 7.678) α ( ◦ ) 117.62 ± ± ± ± ± ± a) α =102.55° x θ B = 8. °φ B =182. =117.62° α =106.96°NV NV B NV xx xyy y yzzz z (c) (d)(b) B Figure 4: (color online). (a)-(c) The possible orientation of the magnetic field. The polarangles α between the magnetic field and the NV − center axes are 117 . ◦ , 106 . ◦ , 102 . ◦ respectively. Which tell the possible magnetic field vector is a cone around the NV centeraxis. (d) The determination of the magnetic field direction with three different-orientedNV centers. The intersections of three cones form a triangle. The least square method isemployed to get the optimal solution, which is θ B = 8 . ◦ , ϕ B = 182 . ◦ .both three peaks owing to the hyper-fine interaction between electron spin and N nu-clear spin ( I N = 1) based on Eq.1. As shown in Fig.3, the middle peaks ( ω and ω )of the three peaks are the transition frequencies of | m s = 0 , m I = 0 (cid:105) → | m s = − , m I = 0 (cid:105) and | m s = 0 , m I = 0 (cid:105) → | m s = +1 , m I = 0 (cid:105) , respectively. For magnetic field in arbitrarydirection, the transition frequencies can be selected to calculate the magnitude according to B = (cid:112) (1 / ω + ω − ω ω − D ) / ( g e µ B / ¯ h ) and the polar angle α ∈ [0 , π ] expressed by α = arccos ( ± (cid:113) (2 ω − ω − D )( ω − ω + D )( ω + ω + D )[9 D ( ω − ω ω + ω − D )] ). Based on the ODMR spectra, we only canget the polar angle α between the magnetic field and the NV axis. But the polar angle α only tells that the possible orientation of the magnetic field is a cone around the NV centeraxis as shown in Fig.4(a)-(c). 9o completely obtain the information about the magnetic field orientation, we need threedifferent orientation NV centers at least. Here, we select NV , NV and NV as magneticsensors. The fitted results of the polar angle and magnitude for NV , NV and NV are shownin Table 1. In theory, the cones around three NV center axes have an intersection, which isthe absolute orientation of the magnetic field in laboratory coordinate system as shown inFig.4. Experimentally, three cones can’t strictly intersect at a point and have deviation inthe overlap. The deviation can be from strain of lattice and temperature fluctuation duringmeasurement process. The intersections of three cones form a triangle in unit sphere asshown in the inset of Fig.4(d). To evaluate the direction of the magnetic field, we employthe least square method to get the optimal solution (see Supporting Information). Thedirection of the magnetic field is θ B = 8 . ◦ , ϕ B = 182 . ◦ with error less than 0 . ◦ . Conclusion
In conclusion, we propose and demonstrate an efficient way to reconstruct vector informationof magnetic field with NV centers in diamond. The optical vortex beam is azimuthallypolarized beam which induces an orientation-dependent image pattern when scanning NVcenters in a confocal system. With the vortex beam, the orientations of the NV centers in abulk diamond could be directly determined from the scanning image. Combining with theODMR spectra, the complete information of the magnetic field including the magnitude andorientation could be reconstructed. Our works provides a calibration-free nano-scale vectormagnetometry and can be easily used in NV-based magnetic field sensing and imaging.
Acknowledgement
The authors are grateful to Heng Shen, FengJian Jiang and Yong Zhou for fruitful discus-sions. This work is supported by the National Key R&D Program of China (Grants No.2018YFA0306600, No. 2018YFF01012505 and No. 2018YFF01012500), the National Natu-10al Science Foundation of China (Grants No. 11604069, No. 61805064, No. 11775209, No.81788101, No. 11761131011 and No. 11904070), the Fundamental Research Funds for theCentral Universities, the CAS (Grants No. GJJSTD20170001, No. QYZDY-SSW-SLH004),the Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000).
References (1) Degen, C. L.; Reinhard, F.; Cappellaro, P. Quantum sensing.
Reviews of ModernPhysics , , 035002.(2) Barry, J. F.; Schloss, J. M.; Bauch, E.; Turner, M. J.; Hart, C. A.; Pham, L. M.;Walsworth, R. L. Sensitivity optimization for NV-diamond magnetometry. Reviews ofModern Physics , , 015004.(3) Kimball, D. F. J.; Budker, D. Optical Magnetometry ; Cambridge University Press,2013.(4) Budker, D.; Romalis, M. Optical magnetometry.
Nature Physics , , 227–234.(5) Vasyukov, D.; Anahory, Y.; Embon, L.; Halbertal, D.; Cuppens, J.; Neeman, L.; Fin-kler, A.; Segev, Y.; Myasoedov, Y.; Rappaport, M. L., et al. A scanning superconductingquantum interference device with single electron spin sensitivity. Nature Nanotechnol-ogy , , 639–644.(6) Doherty, M. W.; Manson, N. B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollen-berg, L. C. The nitrogen-vacancy colour centre in diamond. Physics Reports , , 1–45.(7) Shi, F.; Zhang, Q.; Wang, P.; Sun, H.; Wang, J.; Rong, X.; Chen, M.; Ju, C.; Rein-hard, F.; Chen, H., et al. Single-protein spin resonance spectroscopy under ambientconditions. Science , , 1135–1138.118) Van der Sar, T.; Wang, Z.; Blok, M.; Bernien, H.; Taminiau, T.; Toyli, D.; Lidar, D.;Awschalom, D.; Hanson, R.; Dobrovitski, V. Decoherence-protected quantum gates fora hybrid solid-state spin register. Nature , , 82–86.(9) Abobeih, M.; Randall, J.; Bradley, C.; Bartling, H.; Bakker, M.; Degen, M.;Markham, M.; Twitchen, D.; Taminiau, T. Atomic-scale imaging of a 27-nuclear-spincluster using a quantum sensor. Nature , , 411–415.(10) Chen, B.; Hou, X.; Zhou, F.; Qian, P.; Shen, H.; Xu, N. Detecting the out-of-time-order correlations of dynamical quantum phase transitions in a solid-state quantumsimulator. Applied Physics Letters , , 194002.(11) Grinolds, M. S.; Hong, S.; Maletinsky, P.; Luan, L.; Lukin, M. D.; Walsworth, R. L.;Yacoby, A. Nanoscale magnetic imaging of a single electron spin under ambient condi-tions. Nature Physics , , 215–219.(12) Maze, J. R.; Stanwix, P. L.; Hodges, J. S.; Hong, S.; Taylor, J. M.; Cappellaro, P.;Jiang, L.; Dutt, M. G.; Togan, E.; Zibrov, A., et al. Nanoscale magnetic sensing withan individual electronic spin in diamond. Nature , , 644–647.(13) Chaudhry, A. Z. Detecting the presence of weak magnetic fields using nitrogen-vacancycenters. Physical Review A , , 062111.(14) Wang, P.; Yuan, Z.; Huang, P.; Rong, X.; Wang, M.; Xu, X.; Duan, C.; Ju, C.; Shi, F.;Du, J. High-resolution vector microwave magnetometry based on solid-state spins indiamond. Nature Communications , , 1–5.(15) Glenn, D. R.; Bucher, D. B.; Lee, J.; Lukin, M. D.; Park, H.; Walsworth, R. L.High-resolution magnetic resonance spectroscopy using a solid-state spin sensor. Nature , , 351–354. 1216) Wang, P.; Chen, S.; Guo, M.; Peng, S.; Wang, M.; Chen, M.; Ma, W.; Zhang, R.;Su, J.; Rong, X., et al. Nanoscale magnetic imaging of ferritins in a single cell. ScienceAdvances , , eaau8038.(17) Degen, C. Scanning magnetic field microscope with a diamond single-spin sensor. Ap-plied Physics Letters , , 243111.(18) Casola, F.; van der Sar, T.; Yacoby, A. Probing condensed matter physics with mag-netometry based on nitrogen-vacancy centres in diamond. Nature Reviews Materials , , 1–13.(19) Robledo, L.; Bernien, H.; Van Der Sar, T.; Hanson, R. Spin dynamics in the opticalcycle of single nitrogen-vacancy centres in diamond. New Journal of Physics , ,025013.(20) Chakraborty, T.; Zhang, J.; Suter, D. Polarizing the electronic and nuclear spin of theNV-center in diamond in arbitrary magnetic fields: analysis of the optical pumpingprocess. New Journal of Physics , , 073030.(21) Chen, B.; Geng, J.; Zhou, F.; Song, L.; Shen, H.; Xu, N. Quantum state tomography ofa single electron spin in diamond with Wigner function reconstruction. Applied PhysicsLetters , , 041102.(22) Xu, N.; Tian, Y.; Chen, B.; Geng, J.; He, X.; Wang, Y.; Du, J. Dynamically PolarizingSpin Register of N-V Centers in Diamond Using Chopped Laser Pulses. Physical ReviewApplied , , 024055.(23) Le Sage, D.; Arai, K.; Glenn, D. R.; DeVience, S. J.; Pham, L. M.; Rahn-Lee, L.;Lukin, M. D.; Yacoby, A.; Komeili, A.; Walsworth, R. L. Optical magnetic imaging ofliving cells. Nature , , 486–489.1324) Schirhagl, R.; Chang, K.; Loretz, M.; Degen, C. L. Nitrogen-vacancy centers in dia-mond: nanoscale sensors for physics and biology. Annual Review of Physical Chemistry , , 83–105.(25) McGuinness, L. P.; Yan, Y.; Stacey, A.; Simpson, D. A.; Hall, L. T.; Maclaurin, D.;Prawer, S.; Mulvaney, P.; Wrachtrup, J.; Caruso, F., et al. Quantum measurement andorientation tracking of fluorescent nanodiamonds inside living cells. Nature Nanotech-nology , , 358–363.(26) Balasubramanian, G.; Chan, I.; Kolesov, R.; Al-Hmoud, M.; Tisler, J.; Shin, C.;Kim, C.; Wojcik, A.; Hemmer, P. R.; Krueger, A., et al. Nanoscale imaging mag-netometry with diamond spins under ambient conditions. Nature , , 648–651.(27) Maertz, B.; Wijnheijmer, A.; Fuchs, G.; Nowakowski, M.; Awschalom, D. Vector mag-netic field microscopy using nitrogen vacancy centers in diamond. Applied Physics Let-ters , , 092504.(28) Pham, L. M.; Bar-Gill, N.; Le Sage, D.; Belthangady, C.; Stacey, A.; Markham, M.;Twitchen, D.; Lukin, M. D.; Walsworth, R. L. Enhanced metrology using preferen-tial orientation of nitrogen-vacancy centers in diamond. Physical Review B , ,121202.(29) Weggler, T.; Ganslmayer, C.; Frank, F.; Eilert, T.; Jelezko, F.; Michaelis, J. Deter-mination of the Three-Dimensional Magnetic Field Vector Orientation with NitrogenVacany Centers in Diamond. Nano Letters , , 2980–2985.(30) Alegre, T. P. M.; Santori, C.; Medeiros-Ribeiro, G.; Beausoleil, R. G. Polarization-selective excitation of nitrogen vacancy centers in diamond. Physical Review B , , 165205.(31) Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Ad-vances in Optics and Photonics , , 1–57.1432) Maurer, C.; Jesacher, A.; F¨urhapter, S.; Bernet, S.; Ritsch-Marte, M. Tailoring ofarbitrary optical vector beams. New Journal of Physics , , 78.(33) Kimura, W.; Kim, G.; Romea, R.; Steinhauer, L.; Pogorelsky, I.; Kusche, K.; Fer-now, R.; Wang, X.; Liu, Y. Laser acceleration of relativistic electrons using the inverseCherenkov effect. Physical Review Letters , , 546.(34) Zhan, Q. Trapping metallic Rayleigh particles with radial polarization. Optics Express , , 3377–3382.(35) Huang, L.; Guo, H.; Li, J.; Ling, L.; Feng, B.; Li, Z.-Y. Optical trapping of goldnanoparticles by cylindrical vector beam. Optics Letters , , 1694–1696.(36) Dorn, R.; Quabis, S.; Leuchs, G. Sharper focus for a radially polarized light beam. Physical Review Letters , , 233901.(37) Cardano, F.; Karimi, E.; Slussarenko, S.; Marrucci, L.; de Lisio, C.; Santamato, E.Polarization pattern of vector vortex beams generated by q-plates with different topo-logical charges. Applied Optics , , C1–C6.(38) Ye, Y.-H.; Dong, M.-X.; Yu, Y.-C.; Ding, D.-S.; Shi, B.-S. Experimental realization ofoptical storage of vector beams of light in warm atomic vapor. Optics Letters , , 1528–1531.(39) Karedla, N.; Stein, S. C.; H¨ahnel, D.; Gregor, I.; Chizhik, A.; Enderlein, J. Simultaneousmeasurement of the three-dimensional orientation of excitation and emission dipoles. Physical Review Letters , , 173002.(40) Dolan, P. R.; Li, X.; Storteboom, J.; Gu, M. Complete determination of the orientationof NV centers with radially polarized beams. Optics Express , , 4379–4387.(41) Patra, D.; Gregor, I.; Enderlein, J. Image analysis of defocused single-molecule images15or three-dimensional molecule orientation studies. The Journal of Physical ChemistryA , , 6836–6841.(42) Epstein, R.; Mendoza, F.; Kato, Y.; Awschalom, D. Anisotropic interactions of a singlespin and dark-spin spectroscopy in diamond. Nature Physics ,1