Abstract:
A system for determining an orientation of a nitrogen vacancy (NV) diamond material is disclosed. The system includes the NV diamond material having a plurality of NV centers, a magnetic field generator that generates a magnetic field, a radio frequency (RF) excitation source that provides RF excitation, an optical excitation source that provides optical excitation, an optical detector that receives an optical signal emitted by the NV diamond material, and a controller. The controller controls the magnetic field generator to generate a control magnetic field and controls the magnetic field generator to successively generate calibration magnetic fields. The controller successively receives light detection signals from the optical detector, stores measurement values based on the successively received light detection signals, and calculates an orientation of the NV diamond material based on the stored measurement values.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of priority to U.S. Provisional Patent Application No. 62/112,079, filed Feb. 4, 2015, the entire contents of which are incorporated by reference herein in its entirety. This application is related to co-pending U.S. patent application filed Jan. 21, 2016, titled “APPARATUS AND METHOD FOR RECOVERY OF THREE DIMENSIONAL MAGNETIC FIELD FROM A MAGNETIC DETECTION SYSTEM” (Attorney Docket. No. 111423-1046), which is incorporated by reference herein in its entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    The present disclosure generally relates to magnetometers, and more particularly, to apparatuses and methods for estimating orientations of absolute axes for a magnetic detection system. 
       BACKGROUND 
       [0003]    A number of industrial applications including, but not limited to, medical devices, communication devices, and navigation systems, as well as scientific areas such as physics and chemistry can benefit from magnetic detection and imaging. Many advanced magnetic imaging systems can operate in limited conditions, for example, high vacuum and/or cryogenic temperatures, which can make them inapplicable for imaging applications that require ambient conditions. Furthermore, low cost, small size, weight and power (CSWAP) magnetic sensors of moderate sensitivity, vector accuracy, and bandwidth are valuable in many applications. 
         [0004]    Atomic-sized nitrogen-vacancy (NV) centers in diamond lattices have been shown to have excellent sensitivity for magnetic field measurement and enable fabrication of small magnetic sensors that can readily replace existing-technology (e.g., Hall-effect, SERF, or SQUID) systems and devices. The sensing capabilities of diamond NV (DNV) sensors are maintained in room temperature and atmospheric pressure and these sensors can be even used in liquid environments (e.g., for biological imaging). Measurement of 3-D vector magnetic fields via (DNV) sensing may be beneficial across a very broad range of applications including communications, geological sensing, two and three dimensional imagery over extended distance, navigation, and attitude determination. 
         [0005]    In order to recover the external magnetic field acting on the diamond NV sensor, the orientation of the axes of the diamond lattice of the sensor system should be known. Currently, methods in establishing the orientation of the axes of the diamond lattice are limited to either pre-manufacturing techniques or visual aid inspection. However, these methods may be time consuming, costly, and/or impractical in some instances. For example, during manufacture of the sensor system, the diamond lattice may be mounted to the sensor in such a way that the orientations of the lattice axes are known and established before use of the sensor system. Such a method requires high accuracy and precision in mounting the diamond lattice to the sensor system and may introduce error during the mounting process. In addition, visual aid inspection, such as X-ray diffraction techniques and the like, may not be feasible in cases where the diamond and/or sensor system is hidden from view. 
       SUMMARY 
       [0006]    According to certain embodiments, a system for determining an orientation of a nitrogen vacancy diamond material may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a magnetic field generator configured to generate a magnetic field that is applied to the NV diamond material, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, and a controller. The controller may be configured to control the magnetic field generator to generate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, and control the magnetic field generator to successively generate a plurality of calibration magnetic fields, each having a predetermined direction. The controller may be further configured to successively receive a plurality of light detection signals from the optical detector based on the optical signals emitted by the NV diamond material, store a plurality of measurement values based on the successively received plurality of light detection signals, and calculate an orientation of the NV diamond material based on the stored measurement values. 
         [0007]    According to one aspect, a plurality of calibration magnetic fields may consist of three weak magnetic fields. 
         [0008]    According to one aspect, a predetermined direction of one of the three magnetic fields may be orthogonal to the predetermined directions of the other two of the three magnetic fields. 
         [0009]    According to one aspect, predetermined directions of the plurality of calibration magnetic fields may be different from one another. 
         [0010]    According to one aspect, a plurality of calibration magnetic fields may be at least three. 
         [0011]    According to one aspect, a magnetic field generator may comprise a coil, and the controller may be configured to control the coil to generate a magnetic field having a predetermined direction. 
         [0012]    According to one aspect, a magnetic field generator may comprise a plurality of coils. Each of the coils may be configured to generate a magnetic field having a predetermined direction, and each of the predetermined directions may be different from one another. 
         [0013]    According to one aspect, a plurality of coils is three and the plurality of coils may be configured to generate a magnetic field having three directions orthogonal to one another. 
         [0014]    According to one aspect, a controller may calculate the orientation of the NV diamond material relative to a predetermined standard orientation of the NV diamond material. 
         [0015]    According to one aspect, a controller may calculate a rotation and/or reflection of the orientation of the NV diamond material from the predetermined standard orientation of the NV diamond material. 
         [0016]    According to one aspect, a controller may calculate the rotation and/or reflection based on a least squares fit between the stored measurement values and the generated plurality of calibration magnetic fields. 
         [0017]    According to one aspect, a controller may calculate the rotation and/or reflection based on the solution to the Orthogonal Procrustes Problem. 
         [0018]    According to other embodiments, a system for determining an orientation of a nitrogen vacancy diamond material may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, and an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation. The system may further include a first magnetic field generator configured to generate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, a second magnetic field generator configured to generate a plurality of calibration magnetic fields, and a controller. The controller may be configured to control the second magnetic field generator to successively generate the plurality of calibration magnetic fields, each having a predetermined direction, successively receive a plurality of light detection signals from the optical detector based on the optical signals emitted by the NV diamond material, store a plurality of measurement values based on the successively received plurality of light detection signals, and calculate an orientation of the NV diamond material based on the stored plurality of measurement values. 
         [0019]    According to one aspect, a first magnetic field generator may be a permanent magnet. 
         [0020]    According to one aspect, a second magnetic field generator may comprise a coil, and the controller may be configured to control the coil to generate a magnetic field having a predetermined direction. 
         [0021]    According to one aspect, a second magnetic field generator may comprise a plurality of coils. Each of the coils may be configured to generate a magnetic field having a predetermined direction. 
         [0022]    According to one aspect, a first magnetic field generator may be affixed to a pivot assembly configured to position the first magnetic field generator to a predetermined orientation such that the first magnetic field generator generates the control magnetic field having a predetermined direction. The controller may be further configured to control the pivot assembly. 
         [0023]    According to one aspect, a second magnetic field generator may be affixed to a pivot assembly configured to position the second magnetic field generator to a predetermined orientation such that the second magnetic field generator generates the control magnetic field having a predetermined direction. The controller may be further configured to control the pivot assembly. 
         [0024]    According to one aspect, a controller may calculate the orientation of the NV diamond material relative to a predetermined standard orientation of the NV diamond material. 
         [0025]    According to one aspect, a controller may calculate a rotation and/or reflection of the orientation of the NV diamond material from the predetermined standard orientation of the NV diamond material. 
         [0026]    According to one aspect, a controller may calculate the rotation and/or reflection based on a least squares fit between the stored measurement values and the generated plurality of calibration magnetic fields. 
         [0027]    According to other embodiments, a system for determining an orientation of a nitrogen vacancy diamond material may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, and an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation. The system may further include a first magnetic field generator affixed to a pivot assembly, the pivot assembly being configured to position the first magnetic field generator to a predetermined orientation such that the first magnetic field generator generates a control magnetic field having a predetermined direction that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, a second magnetic field generator configured to generate a plurality of calibration magnetic fields, and a controller. The controller may be configured to control the pivot assembly to position the first magnetic field generator to the predetermined orientation to generate the control magnetic field, control the second magnetic field generator to successively generate the plurality of calibration magnetic fields, each having a predetermined direction, successively receive a plurality of light detection signals from the optical detector based on the optical signals emitted by the NV diamond material, store a plurality of measurement values based on the successively received plurality of light detection signals, and calculate an orientation of the NV diamond material based on the stored plurality of measurement values. 
         [0028]    According to other embodiments, a method for determining a lattice orientation of a nitrogen vacancy (NV) diamond material may include generating a control magnetic field applied to the NV diamond material, applying the control magnetic field to the NV diamond material comprising a plurality of NV centers, generating a plurality of calibration magnetic fields, each having a predetermined direction, applying the plurality of calibration magnetic fields to the NV diamond material, receiving a plurality of light detection signals from an optical detector configured to receive a plurality of optical signals emitted by the NV diamond material due to the applied control magnetic field and the plurality of calibration magnetic fields, storing a plurality of measurement values based on the received plurality of light detection signals, and calculating an orientation of the NV diamond material based on the plurality of stored measurement values. 
         [0029]    According to other embodiments, a method for recovering a sign value of measurement values based on a fluorescence intensity having a plurality of reduced responses emitted by a magneto-optical defect center material may include assigning the largest measurement value a first sign value, the first sign value being positive or negative, assigning the second largest measurement value a second sign value, the second sign value being an opposite sign value to the first sign value, assigning the third largest measurement value a third sign value, the third sign value being an opposite sign value to the first sign value, and assigning the fourth largest measurement value a fourth sign value, the fourth sign value being a sign value such that the sum of the first, second, third, and fourth sign values approach zero. 
         [0030]    According to one aspect, a first sign value may be assigned a positive sign value. 
         [0031]    According to one aspect, a fourth sign value may be assigned a positive sign value. 
         [0032]    According to one aspect, a fourth sign value may be assigned a negative sign value. 
         [0033]    According to other embodiments, a system for determining an orientation of a magneto-optical defect center material may include a magneto-optical defect center material comprising a plurality of defect centers, a magnetic field generator configured to generate a magnetic field, a radio frequency (RF) excitation source configured to provide RF excitation to the magneto-optical defect center material, an optical excitation source configured to provide optical excitation to the magneto-optical defect center material, an optical detector configured to receive an optical signal emitted by the magneto-optical defect center material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, and a controller. The controller may be configured to control the magnetic field generator to generate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the magneto-optical defect center material, control the magnetic field generator to successively generate a plurality of calibration magnetic fields, each having a predetermined direction, successively receive a plurality of light detection signals from the optical detector based on the optical signals emitted by the magneto-optical defect center material, store a plurality of measurement values based on the successively received plurality of light detection signals, and calculate an orientation of the magneto-optical defect center material based on the stored measurement values. 
         [0034]    According to one aspect, a magneto-optical defect center material may be a nitrogen vacancy (NV) diamond material. 
         [0035]    According to other embodiments, a system for determining an orientation of a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers may include means for providing radio frequency (RF) excitation to the NV diamond material, means for receiving an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, means for generating a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, means for successively generating a plurality of calibration magnetic fields, each having a predetermined direction, means for successively receiving a plurality of light detection signals based on the optical signals emitted by the NV diamond material, means for storing a plurality of measurement values based on the successively received plurality of light detection signals, and means for calculating an orientation of the NV diamond material based on the stored measurement values. 
         [0036]    According to other embodiments, a system for determining an orientation of a nitrogen vacancy (NV) diamond material may include the NV diamond material comprising a plurality of NV centers, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, a first magnetic field generator configured to generate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, a second magnetic field generator configured to generate a plurality of calibration magnetic fields, and a controller. The controller may be configured to control the second magnetic field generator to successively generate the plurality of calibration magnetic fields, each having a predetermined direction, successively receive a plurality of light detection signals from the optical detector based on the optical signals emitted by the NV diamond material, store a plurality of measurement values based on the successively received plurality of light detection signals, and calculate a rotation and/or reflection of a predetermined standard orientation of the NV diamond material to an actual orientation of the NV diamond material based on the stored plurality of measurement values. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0037]      FIG. 1  illustrates one orientation of an NV center in a diamond lattice. 
           [0038]      FIG. 2  is an energy level diagram showing energy levels of spin states for the NV center. 
           [0039]      FIG. 3  is a schematic view illustrating an NV center magnetic sensor system. 
           [0040]      FIG. 4  is a graph illustrating the fluorescence as a function of an applied RF frequency of an NV center along a given direction for a zero magnetic field 
           [0041]      FIG. 5  is a graph illustrating the fluorescence as a function of an applied RF frequency for four different NV center orientations for a non-zero magnetic field. 
           [0042]      FIG. 6  is a schematic diagram illustrating a magnetic field detection system according to an embodiment. 
           [0043]      FIG. 7A  is a unit cell diagram of the crystal structure of a diamond lattice having a standard orientation. 
           [0044]      FIG. 7B  is a unit cell diagram of the crystal structure of a diamond lattice having an unknown orientation. 
           [0045]      FIG. 8  is a schematic diagram illustrating a step in a method for determining the unknown orientation of the diamond lattice of  FIG. 7B . 
           [0046]      FIG. 9  is a flowchart illustrating a sign recovery method for the method for determining the unknown orientation of the diamond lattice of  FIG. 7B . 
           [0047]      FIG. 10  is a schematic diagram illustrating a step in the method for determining the unknown orientation of the diamond lattice of  FIG. 7B . 
       
    
    
     DETAILED DESCRIPTION 
       [0048]    The present disclosure relates to apparatuses and methods for accurately estimating the axes&#39; orientation of a diamond lattice used in a magnetic detection system. The process is reduced to a calibration process that may be performed within the system prior to use without the need to perform visual inspection or perform accurate placement of the lattice structure relative to the system. The process may include the application of a bias field that adequately separates out the frequency responses of the NV diamond produced by optical and RF excitation. Small calibration tests, in the form of weak magnetic fields of varying direction, are then applied to the system to allow the system to calculate a rotation (i.e., a rotation and/or reflection) matrix that rotates and/or reflects a defined orientation of the diamond lattice relative to the coordinate reference frame of the system to an unknown orientation that matches the measurements gathered during the calibration tests. 
       The NV Center, its Electronic Structure, and Optical and RF Interaction 
       [0049]    The NV center in a diamond comprises a substitutional nitrogen atom in a lattice site adjacent a carbon vacancy as shown in  FIG. 1 . The NV center may have four orientations, each corresponding to a different crystallographic orientation of the diamond lattice. 
         [0050]    The NV center may exist in a neutral charge state or a negative charge state. Conventionally, the neutral charge state uses the nomenclature NV 0 , while the negative charge state uses the nomenclature NV, which is adopted in this description. 
         [0051]    The NV center has a number of electrons, including three unpaired electrons, each one from the vacancy to a respective of the three carbon atoms adjacent to the vacancy, and a pair of electrons between the nitrogen and the vacancy. The NV center, which is in the negatively charged state, also includes an extra electron. 
         [0052]    The NV center has rotational symmetry, and as shown in  FIG. 2 , has a ground state, which is a spin triplet with  3 A 2  symmetry with one spin state m s =0, and two further spin states m s =+1, and m s =−1. In the absence of an external magnetic field, the m s =±1 energy levels are offset from the m s =0 due to spin-spin interactions, and the m s =±1 energy levels are degenerate, i.e., they have the same energy. The m s =0 spin state energy level is split from the m s =±1 energy levels by an energy of 2.87 GHz for a zero external magnetic field. 
         [0053]    Introducing an external magnetic field with a component along the NV axis lifts the degeneracy of the m s =±1 energy levels, splitting the energy levels m s =±1 by an amount 2 gμ B Bz, where g is the g-factor, μ B  is the Bohr magneton, and Bz is the component of the external magnetic field along the NV axis. This relationship is correct to a first order and inclusion of higher order corrections is a straightforward matter and will not affect the computational and logic steps in the systems and methods described below. 
         [0054]    The NV center electronic structure further includes an excited triplet state  3 E with corresponding m s =0 and m s =±1 spin states. The optical transitions between the ground state  3 A 2  and the excited triplet  3 E are predominantly spin conserving, meaning that the optical transitions are between initial and final states that have the same spin. For a direct transition between the excited triplet  3 E and the ground state  3 A 2 , a photon of red light is emitted with a photon energy corresponding to the energy difference between the energy levels of the transitions. 
         [0055]    There is, however, an alternative non-radiative decay route from the triplet  3 E to the ground state  3 A z  via intermediate electron states, which are thought to be intermediate singlet states A, E with intermediate energy levels. Significantly, the transition rate from the m s =±1 spin states of the excited triplet  3 E to the intermediate energy levels is significantly greater than the transition rate from the m s =0 spin state of the excited triplet  3 E to the intermediate energy levels. The transition from the singlet states A, E to the ground state triplet  3 A 2  predominantly decays to the m s =0 spin state over the m s =±1 spins states. These features of the decay from the excited triplet  3 E state via the intermediate singlet states A, E to the ground state triplet  3 A 2  allows that if optical excitation is provided to the system, the optical excitation will eventually pump the NV center into the m s =0 spin state of the ground state  3 A 2 . In this way, the population of the m s =0 spin state of the ground state  3 A 2  may be “reset” to a maximum polarization determined by the decay rates from the triplet  3 E to the intermediate singlet states. 
         [0056]    Another feature of the decay is that the fluorescence intensity due to optically stimulating the excited triplet  3 E state is less for the m s =±1 states than for the m s =0 spin state. This is so because the decay via the intermediate states does not result in a photon emitted in the fluorescence band, and because of the greater probability that the m s =±1 states of the excited triplet  3 E state will decay via the non-radiative decay path. The lower fluorescence intensity for the m s =±1 states than for the m s =0 spin state allows the fluorescence intensity to be used to determine the spin state. As the population of the m s =±1 states increases relative to the m s =0 spin, the overall fluorescence intensity will be reduced. 
       The NV Center, or Magneto-Optical Defect Center, Magnetic Sensor System 
       [0057]      FIG. 3  is a schematic diagram illustrating a conventional NV center magnetic sensor system  300  that uses fluorescence intensity to distinguish the m s =±1 states, and to measure the magnetic field based on the energy difference between the m s =+1 state and the m s =−1 state. The system  300  includes an optical excitation source  310 , which directs optical excitation to an NV diamond material  320  with NV centers. The system further includes an RF excitation source  330 , which provides RF radiation to the NV diamond material  320 . Light from the NV diamond may be directed through an optical filter  350  to an optical detector  340 . 
         [0058]    The RF excitation source  330  may be a microwave coil, for example. The RF excitation source  330 , when emitting RF radiation with a photon energy resonant with the transition energy between ground m s =0 spin state and the m s =+1 spin state, excites a transition between those spin states. For such a resonance, the spin state cycles between ground m s =0 spin state and the m s =+1 spin state, reducing the population in the m s =0 spin state and reducing the overall fluorescence at resonances. Similarly, resonance occurs between the m s =0 spin state and the m s =−1 spin state of the ground state when the photon energy of the RF radiation emitted by the RF excitation source is the difference in energies of the m s =0 spin state and the m s =−1 spin state, or between the m s =0 spin state and the m s =+1 spin state, there is a decrease in the fluorescence intensity. 
         [0059]    The optical excitation source  310  may be a laser or a light emitting diode, for example, which emits light in the green, for example. The optical excitation source  310  induces fluorescence in the red, which corresponds to an electronic transition from the excited state to the ground state. Light from the NV diamond material  320  is directed through the optical filter  350  to filter out light in the excitation band (in the green, for example), and to pass light in the red fluorescence band, which in turn is detected by the detector  340 . The optical excitation light source  310 , in addition to exciting fluorescence in the diamond material  320 , also serves to reset the population of the m s =0 spin state of the ground state  3 A 2  to a maximum polarization, or other desired polarization. 
         [0060]    For continuous wave excitation, the optical excitation source  310  continuously pumps the NV centers, and the RF excitation source  330  sweeps across a frequency range that includes the zero splitting (when the m s =±1 spin states have the same energy) energy of 2.87 GHz. The fluorescence for an RF sweep corresponding to a diamond material  320  with NV centers aligned along a single direction is shown in  FIG. 4  for different magnetic field components Bz along the NV axis, where the energy splitting between the m s =−1 spin state and the m s =+1 spin state increases with Bz. Thus, the component Bz may be determined. Optical excitation schemes other than continuous wave excitation are contemplated, such as excitation schemes involving pulsed optical excitation, and pulsed RF excitation. Examples of pulsed excitation schemes include Ramsey pulse sequence, and spin echo pulse sequence. 
         [0061]    In general, the diamond material  320  will have NV centers aligned along directions of four different orientation classes.  FIG. 5  illustrates fluorescence as a function of RF frequency for the case where the diamond material  320  has NV centers aligned along directions of four different orientation classes. In this case, the component Bz along each of the different orientations may be determined. These results, along with the known orientation of crystallographic planes of a diamond lattice, allow not only the magnitude of the external magnetic field to be determined, but also the direction of the magnetic field. 
         [0062]    While  FIG. 3  illustrates an NV center magnetic sensor system  300  with NV diamond material  320  with a plurality of NV centers, in general, the magnetic sensor system may instead employ a different magneto-optical defect center material, with a plurality of magneto-optical defect centers. The electronic spin state energies of the magneto-optical defect centers shift with magnetic field, and the optical response, such as fluorescence, for the different spin states is not the same for all of the different spin states. In this way, the magnetic field may be determined based on optical excitation, and possibly RF excitation, in a corresponding way to that described above with NV diamond material. 
         [0063]      FIG. 6  is a schematic diagram of a system  600  for a magnetic field detection system according to an embodiment. The system  600  includes an optical excitation source  610 , which directs optical excitation to an NV diamond material  620  with NV centers, or another magneto-optical defect center material with magneto-optical defect centers. An RF excitation source  630  provides RF radiation to the NV diamond material  620 . 
         [0064]    As shown in  FIG. 6 , a first magnetic field generator  670  generates a magnetic field, which is detected at the NV diamond material  620 . The first magnetic field generator  670  may be a permanent magnet positioned relative to the NV diamond material  620 , which generates a known, uniform magnetic field (e.g., a bias or control magnetic field) to produce a desired fluorescence intensity response from the NV diamond material  620 . In some embodiments, a second magnetic field generator  675  may be provided and positioned relative to the NV diamond material  620  to provide an additional bias or control magnetic field. The second magnetic field generator  675  may be configured to generate magnetic fields with orthogonal polarizations. In this regard, the second magnetic field generator  675  may include one or more coils, such as Helmholtz coils. The coils may be configured to provide relatively uniform magnetic fields at the NV diamond material  620  and each may generate a magnetic field having a direction that is orthogonal to the direction of the magnetic field generated by the other coils. In some embodiments, only the first magnetic field generator  670  may be provided to generate the bias magnetic field. Alternatively, only the second magnetic field generator  675  may be provided to generate the bias magnetic field. 
         [0065]    The system  600  further includes a controller  680  arranged to receive a light detection signal from the optical detector  640  and to control the optical excitation source  610 , the RF excitation source  630 , and the second magnetic field generator  675 . The controller may be a single controller, or multiple controllers. For a controller including multiple controllers, each of the controllers may perform different functions, such as controlling different components of the system  600 . The second magnetic field generator  675  may be controlled by the controller  680  via an amplifier  660 , for example. 
         [0066]    The RF excitation source  630  may be a microwave coil, for example. The RF excitation source  630  is controlled to emit RF radiation with a photon energy resonant with the transition energy between the ground m s =0 spin state and the m s =±1 spin states as discussed above with respect to  FIG. 3 . 
         [0067]    The optical excitation source  610  may be a laser or a light emitting diode, for example, which emits light in the green, for example. The optical excitation source  610  induces fluorescence in the red from the NV diamond material  620 , where the fluorescence corresponds to an electronic transition from the excited state to the ground state. Light from the NV diamond material  620  is directed through the optical filter  650  to filter out light in the excitation band (in the green, for example), and to pass light in the red fluorescence band, which in turn is detected by the optical detector  640 . The optical excitation light source  610 , in addition to exciting fluorescence in the NV diamond material  620 , also serves to reset the population of the m s =0 spin state of the ground state  3 A 2  to a maximum polarization, or other desired polarization. 
         [0068]    The controller  680  is arranged to receive a light detection signal from the optical detector  640  and to control the optical excitation source  610 , the RF excitation source  630 , and the second magnetic field generator  675 . The controller may include a processor  682  and a memory  684 , in order to control the operation of the optical excitation source  610 , the RF excitation source  630 , and the second magnetic field generator  675 . The memory  684 , which may include a nontransitory computer readable medium, may store instructions to allow the operation of the optical excitation source  610 , the RF excitation source  630 , and the second magnetic field generator  675  to be controlled. That is, the controller  680  may be programmed to provide control. 
       Axes of the Diamond Crystal Lattice 
       [0069]    In deriving the total magnetic field vector impinging on the system  600  from the measurements obtained by the intensity response produced by the NV diamond material  620 , it is desirable to establish the orientation of the axes of the diamond lattice of the NV diamond material  620  to allow for the accurate recovery of the magnetic field vector and maximize signal-to-noise information. However, as discussed above, the NV diamond material  620  may be arbitrarily oriented and, thus, have axes in an unknown orientation. Thus, in such a case, the controller  680  may be configured to compute an accurate estimation of the true orientation of the NV diamond lattice, which can be performed on-site as a calibration method prior to use. This information can be subsequently used to accurately recover the full vector information of an unknown external magnetic field acting on the system  600 . 
         [0070]    To begin, a desired geospatial coordinate reference frame relative to the system  600  by which measurement of the total magnetic field vector will take place is established. As shown in  FIGS. 7A and 7B , a Cartesian reference frame having {x, y, z}orthogonal axes may be used, but any arbitrary reference frame and orientation may be used.  FIG. 7A  shows a unit cell  100  of a diamond lattice having a “standard” orientation. The axes of the diamond lattice will fall along four possible directions. Thus, the four axes in a standard orientation relative to the desired coordinate reference frame may be defined as unit vectors corresponding to: 
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         [0071]    For simplicity, the four vectors of equation (1) may be represented by a single matrix A S , which represents the standard orientation of the unit cell  100 : 
         [0000]      A S =[a S,1  a S,2  a S,3  a S,4 ]  (2)
 
         [0072]     The angle between axis i and axis j may also be given by the (i, j) th  row of the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           A 
                           S 
                           T 
                         
                          
                         
                           A 
                           S 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   3 
                                 
                               
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                     ≈ 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 0 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                           
                           
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 0 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                           
                           
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 0 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                           
                           
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                             
                               
                                 109.47 
                                  
                                 ° 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0073]      FIG. 7B  is a unit cell  100 ′ that represents an arbitrarily placed NV diamond material having unknown axes orientation with respect to the coordinate reference frame. By defining the standard orientation matrix A S  with reference to the established coordinate reference frame, the arbitrary orientation shown in  FIG. 7B  may be obtained through a rotation and/or reflection of the standard orientation matrix. This can be achieved by applying a transformation matrix R, which is defined as a general 3×3 matrix representing the three-dimensional, orthogonal Cartesian space and is, at this stage, unknown. The transformation matrix may be used to obtain our desired matrix A as follows: 
         [0000]      A=RA S    (4)
 
       Deriving the Total Magnetic Field Vector 
       [0074]    As described above with reference to  FIGS. 3-5 , the total magnetic field acting on the system  600  may be measured fluorescently. These measurements may be modeled as a linear system from which the total magnetic field impinging on the sensor may be determined: 
         [0000]        m=|A   T   b+n|   (5)
 
         [0075]    Here, b ε            3×1  represents the magnetic field vector acting inside the sensor system, expressed in Cartesian coordinates relative to the coordinate reference frame; A T b represents the projection of the magnetic field vector onto each of the four, arbitrarily-placed NV center diamond lattice axes; n ε            4×1  represents the sensor noise vector; and m ε            4×1  represents the measurement vector, where the i th  element represents the estimated projection of the magnetic field onto the sensor axis i. In terms of units, it is assumed that the measurement vector has been converted from the DNV sensor&#39;s native units (in terms of microwave resonance frequency) into the units of magnetic field strength. Furthermore, the term |A T b+n| represents the element-wise absolute value of A T b+n, rather than its determinant. 
         [0076]    Given the linear model for the magnetic field measurement of equation (5) a least squares estimate of the total magnetic field acting on the system  600  may be given by: 
         [0000]      {circumflex over ( b )}=( A   T ) +   m    (6)
 
         [0077]    In the above equation, the + superscript denotes the Moore-Penrose pseudoinverse. Because the three four-element columns of A T  are linearly independent, equation (6) may be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           b 
                           ^ 
                         
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 AA 
                                 T 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           Am 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   RA 
                                   S 
                                 
                                  
                                 
                                   A 
                                   S 
                                   T 
                                 
                                  
                                 
                                   R 
                                   T 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           Am 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   4 
                                   3 
                                 
                                  
                                 
                                   RIR 
                                   T 
                                 
                               
                               ) 
                             
                             1 
                           
                            
                           Am 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.75 
                            
                           
                             
                               ( 
                               
                                 RR 
                                 T 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           Am 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0078]    In equation (7), A S A S   T 4/3I (established in more detail below) has been substituted. Because R is an orthogonal matrix, equation (7) can be reduced to: 
         [0000]        {circumflex over (b)}= 0.75( I ) −1   Am= 0.75 Am    (8)
 
         [0079]    In equations (7)-(8), it was assumed that all the measurements were weighted equally. If, however, some of the axes have less variance in their measurements or are preferred for other reasons, then different weightings may be used for each of the axes for a more optimal least squares estimate. If w ε R 4×1  represents the positive weights for each of the measurements and W=diag(w), then the weighted least-squares formulation for the total magnetic field may be written as: 
         [0000]        {circumflex over (b)}=             ∥W   1/2 ( A   T   {circumflex over (b)}−m )∥ 2    (9)
 
         [0080]    Based on equation (9), the generalized least squares solution of equation (6) may now be written as: 
         [0000]        {circumflex over (b)} =( W   1/2   A   T ) +   W   1/2   m =( AWA   T ) −1   AWm    (10)
 
         [0081]    For a perfect NV diamond material  620  having no defects (e.g., lattice misalignments, impurities, etc.) and no sensor noise, {circumflex over (b)} should be equal to b. However, in an imperfect system, it is possible to utilize a performance metric to determine the error associated with the measurement. One possible metric that may be used is a 2-norm of the residual vector minimized by the least squares solution. This metric γ may be given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         γ 
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 
                                   A 
                                   T 
                                 
                                  
                                 
                                   b 
                                   ^ 
                                 
                               
                               - 
                               m 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 
                                   
                                     
                                       A 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         AA 
                                         T 
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 Am 
                               
                               - 
                               m 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           A 
                                           T 
                                         
                                          
                                         
                                           ( 
                                           
                                             AA 
                                             T 
                                           
                                           ) 
                                         
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                      
                                     A 
                                   
                                   - 
                                   I 
                                 
                                 ) 
                               
                                
                               m 
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0082]    Because the residual vector is proportional to the measurement amplitude, the magnitude of the true magnetic field may be used to normalize the metric to give a consistent metric even in the presence of a changing true magnetic field: 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     ′ 
                   
                   = 
                   
                     
                       
                          
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       A 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         AA 
                                         T 
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 A 
                               
                               - 
                               I 
                             
                             ) 
                           
                            
                           m 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         b 
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0083]    If the true magnetic field is not known, the measurement vector magnitude may be used to normalize the metric: 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     ″ 
                   
                   = 
                   
                     
                       
                          
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       A 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         AA 
                                         T 
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 A 
                               
                               - 
                               I 
                             
                             ) 
                           
                            
                           m 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         m 
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
       Estimation of Absolute Axes&#39; Orientation in the NV Diamond Material 
       [0084]    By simple substitution of equation (4) into equation (5), the measurement obtained by the system  600  may be represented in terms of the standard orientation matrix: 
         [0000]        m=|A   T   b+n |=|( RA   S ) T   b+n|   (14)
 
         [0085]    As described above, a permanent magnet (e.g., the first magnetic field generator  670 ) and/or coils (e.g., the second magnetic field generator  675 ) may be used to adequately separate out the Lorentzian dips that correspond to the magnetic field measurements along each diamond axis. However, at this point, the orientations of the sensor&#39;s axes are unknown. Thus, the required bias or control magnetic field, defined as b bias , that will produce the desired dip separation is unknown. 
         [0086]    As will be described in more detail below, there exists a plurality of b bias  vectors that can equally separate out the four Lorentzian dips for adequate measurement purposes. Moreover, for the purposes of determining the unknown orientation of the diamond lattice, it is not necessary to precisely place or apply the bias magnetic field that will result in perfectly equal dip separation, which may be more appropriate during field measurement of an external magnetic field. In this case, any b bias  vector that sufficiently separates the four dips may suffice for the determination of the unknown orientation of the diamond lattice, thus increasing the viable b bias  vectors appropriate for this step. Sufficient spectral dip separation, however, may depend on the width of the dips and the planned magnitude of the calibration magnetic fields (described below). The width of the dips varies, depending on diamond composition and sensor laser and/or RF excitation mechanisms. Based on the resulting widths due to inherent sensor characteristics, the magnitude and orientation should be sufficient to ensure that the anticipated maximum spectral shifts that will occur due to the calibration tests will maintain sufficient separation between neighboring Lorentzian dips. 
         [0087]      FIG. 8  shows a step for determining a viable b bias  vector field. As shown in  FIG. 8 , the first magnetic field generator  670  may be arbitrarily placed in one or more positions and/or orientations such that multiple magnetic fields are applied to the diamond having an unknown orientation  100 ′. Measurements of the fluorescence intensity response are taken for each position and/or orientation of the first magnetic field generator  670 . Once a fluorescence intensity response  800  is produced that adequately separates out the four Lorentzian pairs, the position of the first magnetic field generator  670  is maintained and the process may proceed to calibration tests. In other embodiments, the separation process may be performed by the second magnetic field generator  675 . In this case, the controller  680  may be configured to control the second magnetic field generator  675  to generate multiple magnetic fields until the desired separation is produced. In yet other embodiments, the first and/or second magnetic field generators may be affixed to a pivot assembly (e.g., a gimbal assembly) that may be controlled to hold and position the first and/or second magnetic field generators to a predetermined and well-controlled set of orientations, thereby establishing the desired Lorentzian separation and/or calibration magnetic fields (described below). In this case, the controller  680  may be configured to control the pivot assembly having the first and/or second magnetic field generators to position and hold the first and/or second magnetic field generators at the predetermined orientation. 
         [0088]    After an appropriate calibration b bias  field has been found that adequately separates out the four Lorentzian dips, a measurement vector m bias  of the corresponding bias magnet&#39;s magnetic field projections is collected. The measurement vector may be expressed in a similar manner as the linear model described in equation (5): 
         [0000]        m   bias   =|A   T   b   bias   +n   bias |  (15)
 
         [0089]    As noted above with regard to equation (5), the variables represented in equation (15) are the same, but represented in relation to the applied bias field. 
         [0090]    At this point, it is unknown which of the four Lorentzian dips correspond to which of the sensor axes, which still remain unknown. However, because any possible permutation of the axes&#39; ordering can be captured by applying an appropriate orthogonal matrix to A S , and, because the process described herein is estimating the orthogonal matrix that best represents the data, any permutation of the axes&#39; ordering will be compensated by the transformation. Due to this, the axes may be generally assigned such as, for example, the Lorentzian dip that is closest to the zero-field splitting frequency is assigned as a 1 , the second-closest is assigned as a 2 , and so on. 
       Sign Recovery of Magnetic Field Projections 
       [0091]    Due to the symmetry of the DNV sensor measurements, the obtained m bias  vector has no inherent sign information for each of its four components. However, sign information may be recovered using the following process. 
         [0092]    The projections of the magnetic field vector onto the four axes is given by the vector A T b. The sum of the projections may then be initially presumed to equal zero per the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   A 
                                   T 
                                 
                                  
                                 b 
                               
                               ) 
                             
                             i 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       RA 
                                       S 
                                     
                                     ) 
                                   
                                   T 
                                 
                                  
                                 b 
                               
                               ) 
                             
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                            
                           
                             
                               a 
                               
                                 S 
                                 , 
                                 i 
                               
                               T 
                             
                              
                             
                               R 
                               T 
                             
                              
                             b 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             b 
                             T 
                           
                            
                           R 
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               4 
                             
                              
                             
                               a 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             b 
                             T 
                           
                            
                           R 
                            
                           
                               
                           
                            
                           0 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0093]    In the above equation (16), 0 ε           4×1  represents a vector consisting of all zeros. If the sum of the elements of a vector x ε            4×1  equals zero, then a magnetic field vector b may be found whose projections onto the four axes of a diamond is identical to x. In this regard, the magnetic field vector b may be defined as follow: 
         [0000]      b=0.75 Ax   (17)
 
         [0094]    The projection of the magnetic field vector b onto the four axes of a diamond may be given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             T 
                           
                            
                           b 
                         
                         = 
                           
                          
                         
                           0.75 
                            
                           
                               
                           
                            
                           
                             A 
                             T 
                           
                            
                           Ax 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.75 
                            
                           
                             
                               ( 
                               
                                 RA 
                                 S 
                               
                               ) 
                             
                             T 
                           
                            
                           
                             RA 
                             S 
                           
                            
                           x 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.75 
                            
                           
                               
                           
                            
                           
                             A 
                             S 
                             T 
                           
                            
                           
                             R 
                             T 
                           
                            
                           
                             RA 
                             S 
                           
                            
                           x 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.75 
                            
                           
                               
                           
                            
                           
                             A 
                             S 
                             T 
                           
                            
                           
                             A 
                             S 
                           
                            
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0095]    The values for the A S  matrix from equations (1)-(2) may be plugged into equation (18) to give: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             T 
                           
                            
                           b 
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   0.75 
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   0.75 
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   0.75 
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   
                                     - 
                                     0.25 
                                   
                                 
                                 
                                   0.75 
                                 
                               
                             
                             ] 
                           
                            
                           x 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               I 
                               - 
                               
                                 0.25 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                     
                                     
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                     
                                     
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                     
                                     
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                       
                                         1 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                            
                           x 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           x 
                           - 
                           
                             0.25 
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         4 
                                       
                                        
                                       
                                         x 
                                         i 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         4 
                                       
                                        
                                       
                                         x 
                                         i 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         4 
                                       
                                        
                                       
                                         x 
                                         i 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         4 
                                       
                                        
                                       
                                         x 
                                         i 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0096]    Because it was initially assumed that the sum of all the elements of x equals 0, equation (19) can be reduced to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       T 
                     
                      
                     b 
                   
                   = 
                   
                     
                       x 
                       - 
                       
                         0.25 
                          
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     x 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0097]    Thus, a b vector exists whose projections onto the axes of a diamond is identical to x and the initial presumption of equation (16) is proved. Accordingly, the sum of the axes&#39; projections of any magnetic field impinging on a diamond will be equal to zero, and measurements obtained, in the absence of noise, will sum to zero as well. Thus, sign information for the bias measurements may be recovered following this basic principle. This particular step is especially applicable if the bias magnetic field&#39;s projections are much larger than the expected noise levels. 
         [0098]    With reference to  FIG. 9 , a method to recover sign information from the bias field measurements according to one embodiment will now be described. First, in a step S 10 , the largest of the four measurements is arbitrarily set to a sign value, either positive or negative. Once this is chosen, the next steps are dictated based on this sign choice such that the principles of equation (16) are maintained. For example, as shown in the embodiment of  FIG. 8 , the largest of the four measurements, measurement  810   a,  is assigned as positive. Next, in a step S 11 , the second-largest measurement (e.g., measurement  810   b  shown in  FIG. 8 ) is set to negative. By setting the second-largest measurement to negative, the positive value assigned to the largest measurement may be offset toward zero. In a step S 12 , the third-largest measurement (e.g., measurement  810   c  of  FIG. 8 ) is assigned a negative sign value. Because, by definition, the second-largest measurement is smaller than the largest measurement, a negative sign value for the third-largest measurement will offset the largest measurement further towards zero. Finally, in a step S 13 , the smallest measurement is assigned either a positive or negative value that allows for the sum total of the four measurements to approximately equal zero. In  FIG. 8 , the smallest measurement  810   d  is assigned a positive value. After this process, therefore, an appropriately signed m bias  vector may be obtained. 
         [0099]    After application of the bias field that cleanly separates out the four Lorentzian dips and a measurement of the resulting bias field has been collected, a series of calibration tests may be performed. As shown in  FIG. 10 , a series of p known external magnetic fields, in conjunction with the fixed b bias  field, is applied and the resulting sensor measurements are collected. In some embodiments, a series of at least three p (p≧3) weak magnetic fields are applied. In particular embodiments, at least three non-coplanar p (p≧3) weak magnetic fields are applied. In yet other embodiments, three orthogonally spaced p (p≧3) weak magnetic fields are applied. In particular embodiments, four or more p (i.e., p≧4, 5, . . . ) weak magnetic fields are applied. In certain embodiments, the magnitudes of the applied fields are small relative to the bias magnetic field that is applied to separate the Lorentzian responses in frequency and within the dynamic range of the system as defined by the bias magnetic field. In other embodiments, the magnitudes of the applied magnetic fields are large enough to account for errors in the system. In some embodiments, the ranges of strength of the magnetic fields may be from about 0.5 to about 20 micro-Tesla. Such fields may be applied by the second magnetic generator  675  and, thus, controlled by the controller  680 . The known applied external magnetic fields may be represented by the following matrix: 
         [0000]      B=[b 1  b 2  . . . b p ]  (21)
 
         [0100]    In equation (21), b k  represents the k th  field for k=1 . . . p. The obtained measurements m k  corresponding to each b k  may be represented by the linear model described above as: 
         [0000]        m   k   =|A   T ( b   k   +b   bias )+ n   k |  (22)
 
         [0101]    The portion of m k  that corresponds solely to the external magnetic field b k  can be isolated, along with proper sign values, by: 
         [0000]        {tilde over (m)}   k =( m   k   −|A   T   b   bias |)°sgn( A   T   b   bias )   (23)
 
         [0102]    In the above equation, ° represents the Hadamard (i.e., element-wise) matrix product, while sgn( ) represents the element-wise signum function. At this stage, A T  remains unknown. However, A T b bias  may be estimated. This is possible by substituting {hacek over (m)} bias  for A T b bias  in equation (23): 
         [0000]      {tilde over (m)} k ≈(m k −|{hacek over (m)} bias |)°sgn({hacek over (m)} bias )   (24)
 
         [0103]    Combining equations (22) and (23), the derived calibration measurement can be written as follows: 
         [0000]        {tilde over (m)}   k   =A   T   b   k   +ñ   k    (25)
 
         [0104]    In the above equation (25), ñ k =n k °sgn({hacek over (m)} bias )+n bias . 
         [0105]    By defining the matrices {tilde over (M)}=[{tilde over (m)} 1  {tilde over (m)} 2  . . . {tilde over (m)} o ] and Ñ=[ñ 1  ñ 2  . . . ñ p ], the external magnetic fields and their corresponding measurements may be compactly represented by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       T 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               b 
                               1 
                             
                           
                           
                             
                               b 
                               2 
                             
                           
                           
                             … 
                           
                           
                             
                               b 
                               p 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   n 
                                   ~ 
                                 
                                 1 
                               
                             
                             
                               
                                 
                                   n 
                                   ~ 
                                 
                                 2 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   n 
                                   ~ 
                                 
                                 p 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       m 
                                       ~ 
                                     
                                     1 
                                   
                                 
                                 
                                   
                                     
                                       m 
                                       ~ 
                                     
                                     2 
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       m 
                                       ~ 
                                     
                                     p 
                                   
                                 
                               
                             
                             ] 
                           
                           ⇒ 
                           
                             
                               
                                 A 
                                 T 
                               
                                
                               B 
                             
                             + 
                             
                               N 
                               ~ 
                             
                           
                         
                         = 
                         
                           
                             
                               M 
                               ~ 
                             
                             ⇒ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       RA 
                                       S 
                                     
                                     ) 
                                   
                                   T 
                                 
                                  
                                 B 
                               
                               + 
                               
                                 N 
                                 ~ 
                               
                             
                           
                           = 
                           
                             M 
                             ~ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0106]    Once the known B and the measured {tilde over (M)} have been obtained, equation (26) may be expanded as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             RA 
                             S 
                           
                           ) 
                         
                         T 
                       
                        
                       B 
                     
                     + 
                     
                       N 
                       ~ 
                     
                   
                   = 
                   
                     
                       
                         M 
                         ~ 
                       
                       ⇒ 
                       
                         
                           
                             A 
                             S 
                             T 
                           
                            
                           
                             R 
                             T 
                           
                            
                           B 
                         
                         + 
                         
                           N 
                           ~ 
                         
                       
                     
                     = 
                     
                       
                         
                           M 
                           ~ 
                         
                         ⇒ 
                         
                           
                             
                               A 
                               S 
                             
                              
                             
                               A 
                               S 
                               T 
                             
                              
                             
                               R 
                               T 
                             
                              
                             B 
                           
                           + 
                           
                             
                               A 
                               S 
                             
                              
                             
                               N 
                               ~ 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               A 
                               S 
                             
                              
                             
                               M 
                               ~ 
                             
                           
                           ⇒ 
                           
                             
                               
                                 4 
                                 3 
                               
                                
                               
                                 IR 
                                 T 
                               
                                
                               B 
                             
                             + 
                             
                               
                                 A 
                                 S 
                               
                                
                               
                                 N 
                                 ~ 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 A 
                                 S 
                               
                                
                               
                                 M 
                                 ~ 
                               
                             
                             ⇒ 
                             
                               
                                 
                                   R 
                                   T 
                                 
                                  
                                 B 
                               
                               + 
                               
                                 
                                   3 
                                   4 
                                 
                                  
                                 
                                   A 
                                   S 
                                 
                                  
                                 
                                   N 
                                   ~ 
                                 
                               
                             
                           
                           = 
                           
                             
                               3 
                               4 
                             
                              
                             
                               A 
                               S 
                             
                              
                             
                               M 
                               ~ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0107]    From equation (19), A S A S   T =4/3I was demonstrated and thus substituted into equation (27) above. Because the singular values of A S  are known and equal (i.e., about 1.15), the noise term Ñ will not be colored or largely amplified in the expression 3/4A S Ñ. Thus, we can treat the expression 3/4A S Ñ as a new noise term: 
         [0000]        {tilde over (Ñ)}= 3/4 A   S   Ñ   (28)
 
         [0108]    Combining equations (27) and (28) results in: 
         [0000]        R   T   B+{tilde over (Ñ)}= 3/4 A   S   {tilde over (M)}   (29)
 
         [0109]    Taking the transpose of both sides of equation (29) gives: 
         [0000]        B   T   R+{tilde over (Ñ)}   T =3/4 {tilde over (M)}   T   A   S   T    (30)
 
         [0110]    In the next step, an orthogonal matrix {circumflex over (R)} is desired that provides the least-squares fit between B T  and 3/4{tilde over (M)} T A S   T  in equation (30). Some least-squares formulations may introduce translation and/or angular error into the orthogonal matrix {circumflex over (R)}. For example, error may be introduced when applying the matrix {circumflex over (R)} to the standard orientation matrix A S  in the form of a translation of the center of the axes from the standard orientation to the estimated orientation or in a change in the angles shown in equation (3) between given axes. Thus, a least-squares fit that can substantially maintain the relative orientation of the axes to each other when rotating from the standard orientation to the estimated orientation is preferable. In this regard, the orthogonal matrix may be expressed as: 
         [0000]        {circumflex over (R)} =arg min R ε 0(3)   ∥B   T   R− 3/4 {tilde over (M)}   T   A   S   T ∥ F    (31)
 
         [0111]    Where, in equation (31), 0(3) represents the group of orthogonal 3×3 matrices and ∥ ∥ F  represents the Frobenius norm. 
         [0112]    By defining the orthogonal matrix {circumflex over (R)} as above, the particular problem may be reduced to the Orthogonal Procrustes Problem to solve for {circumflex over (R)}. First, the following is defined: 
         [0000]      Z=3/4B{tilde over (M)} T A S   T    (32)
 
         [0113]    A singular devalue decomposition of Z is performed to obtain: 
         [0000]      Z=UΣV T    (33)
 
         [0114]    Where in equation (33), U is an orthogonal 3×3 matrix that contains the left singular vectors of Z; Σ is an orthogonal 3×3 matrix that contains the singular values of Z; and V T  is an orthogonal 3×3 matrix that contains the right singular vectors of Z. Given the above, the solution to the Orthogonal Procrustes Problem of (33) is given by: 
         [0000]      {circumflex over (R)}=UV T    (34)
 
         [0115]    Accordingly, with equation (34), an estimate {circumflex over (R)} is obtained that may be applied to the standard orientation matrix A S  to give the true axes orientation matrix A. Thus, an estimate Â of A can be obtained by applying equation (4) to yield: 
         [0000]      Â={circumflex over (R)}A S    (35)
 
         [0116]    In the embodiment described above, the Orthogonal Procrustes Problem provides an advantage in reducing translation and/or angular error that may be introduced by the least-squares fit and, thus, provides an accurate estimation of the needed rotation matrix. By accurately estimating the rotation matrix, an accurate estimation of the orientation of an arbitrarily placed lattice structure in a magnetic field detection system having a magneto-optical defect center material may be produced. This, in turn, reduces the process to determining the orientation of a diamond in the magnetic detection system  600  to a simple calibration method that may be calculated and controlled by the controller  680  and performed before sensing begins, without the need for pre-manufacturing processes to orient the lattice structure relative to the sensor or additional equipment for visual aid inspection. Moreover, with the above, an accurate estimate of the true orientation of the axes of the NV diamond material  620  may be obtained and recovery of the external magnetic field for magnetic sensing may be improved. 
         [0117]    The embodiments of the inventive concepts disclosed herein have been described in detail with particular reference to preferred embodiments thereof, but it will be understood by those skilled in the art that variations and modifications can be effected within the spirit and scope of the inventive concepts.