Abstract:
MEMS magnetometers with optically transduced resonator displacement are described herein. Improved sensitivity, crosstalk reduction, and extended dynamic range may be achieved with devices including a deflectable resonator suspended from the support, a first grating extending from the support and disposed over the resonator, a pair of drive electrodes to drive an alternating current through the resonator, and a second grating in the resonator overlapping the first grating to form a multi-layer grating having apertures that vary dimensionally in response to deflection occurring as the resonator mechanically resonates in a plane parallel to the first grating in the presence of a magnetic field as a function of the Lorentz force resulting from the alternating current. A plurality of such multi-layer gratings may be disposed across a length of the resonator to provide greater dynamic range and/or accommodate fabrication tolerances.

Description:
GOVERNMENT INTERESTS 
     Embodiments of the invention were developed under Contract No. DE-AC04-94AL85000 between Sandia Corporation and the U.S. Department of Energy. The United States Government has certain rights in this invention. 
    
    
     RELATED APPLICATIONS 
     This application is related to U.S. patent application Ser. No. 13/325,683, entitled OPTICALLY TRANSDUCED MEMS GYRO DEVICE, to Nielson, et al., filed even date herewith, the entirety of which is incorporated herein by reference for all purposes. 
     TECHNICAL FIELD 
     The present invention generally relates to Micro Electro-Mechanical System (MEMS) sensors, and more particularly to Lorentz force magnetometers. 
     BACKGROUND 
     Many conventional MEMS force and inertial sensors, such as accelerometers, magnetometers, and magnetometers, etc. include a pair of drive electrodes to which an alternating current (AC) is applied to induce a mechanical resonance in a suspended member. 
     In Lorentz force magnetometers, the alternating current is of a frequency tuned to induce mechanical deformation of the suspended member in a desired resonance mode by passing current through the suspended resonator so that, in the presence of a magnetic field, Lorentz forces form in the resonator inducing the resonator to mechanically deform. A number of techniques have been applied to sense the extent of deformation induced by the Lorentz forces and thereby measure the magnetic field strength. For example, deformation of the resonator is often sensed by capacitive coupling between resonator and the support, as measured by either primary drive electrodes or secondary sensing electrodes. Piezoelectric techniques have also been utilized, for example, by sensing mechanical deformation of the resonator with a piezoelectric material embedded in a resonator. 
     However, the conventional sensing techniques suffer from cross-talk and inherent sensitivity and other limitations. Both capacitive and piezoelectric techniques may also lack dynamic range such that many separate devices are required to conduct field measurements within different magnetic environments or where field strength varies widely and/or rapidly over time. Accordingly, there remains a need for MEMS Lorentz magnetometers that are more sensitive and/or have greater dynamic range. 
     SUMMARY OF THE DESCRIPTION 
     Embodiments of the present invention include MEMS magnetometers with optically transduced resonator displacement. Improved sensitivity, crosstalk reduction, and extended dynamic range may be achieved with devices having a deflectable resonator suspended from a support, a first grating extending from the support and disposed over the resonator, a pair of drive electrodes to pass an alternating current through the resonator, and a second grating in the resonator overlapping the first grating to form a multi-layer grating having apertures that vary dimensionally in response to deflection occurring as the resonator mechanically resonates in a plane parallel to the first grating in the presence of a magnetic field as a function of the Lorentz force. 
     In embodiments, a plurality of multi-layer gratings may be disposed across a length of the resonator to provide greater dynamic range and/or accommodate fabrication tolerances. In certain embodiments, phase between first and second gratings of each multi-layer grating in the plurality is varied to account for fabrication variation and ensure at least one multi-layer grating provides a threshold level of light intensity modulation with resonator displacement. In certain other embodiments, multi-layer gratings are disposed at locations of the resonator which experience different degrees of deflection, enabling a large dynamic range of magnetic field strength to be optically transduced via light intensity modulation satisfying predetermined threshold criteria. 
     In embodiments, a resonator has a bulk substrate thickness substantially equal to that of the support for increased dimensional mass, increased out-of-plane rigidity, and increased conductive cross-section at anchor points. In certain such embodiments, separate crystalline layers of a silicon-on-insulator (SOI) substrate are electrically connected with conductive vias through the insulator layer to form conductive anchors through an entire thickness of the support substrate. 
     In embodiments, methods of sensing a magnetic field include placing embodiments of the MEMS magnetometer described herein in a magnetic field and measuring the amplitude, or another displacement characteristic of resonance induced in the resonator, in response to the Lorentz effect of alternating current tuned to a resonant frequency of the resonator passing through the resonator. In certain such embodiments, light sources interrogate the multi-level gratings with a wavelength of light that is greater than grating feature dimensions (e.g., spaces or line elements) and intensity modulation of zeroth orders reflected or transmitted through the grating are sensed with a photosensitive device. Intensity variations are output for determination of magnetic field strength. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the present invention are illustrated by way of example, and not limitation, in the figures of the accompanying drawings in which: 
         FIGS. 1A and 1B  illustrate plan views of an optically transduced MEMS magnetometer, in accordance with an embodiment of the invention; 
         FIGS. 1C and 1D  illustrate cross-sectional views of the optically transduced MEMS magnetometer illustrated in  FIG. 1A , in accordance with an embodiment; 
         FIG. 2  illustrates an expanded plan view of a multi-layer grating in the optically transduced MEMs magnetometer illustrated in  FIG. 1A , in accordance with an embodiment; 
         FIGS. 3A and 3B  illustrate a system for sensing a magnetic field, in accordance with an embodiment; 
         FIG. 4  is a flow diagram illustrating a method of sensing a magnetic field, in accordance with an embodiment of the invention; 
         FIG. 5  is a flow diagram illustrating a method of operating a MEMS magnetometer having a plurality of multi-layer gratings, in accordance with an embodiment of the invention; and 
         FIG. 6  is a flow diagram illustrating a method of operating a MEMS magnetometer having a plurality of multi-layer gratings, in accordance with an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description of optically transduced MEMS magnetometers, numerous details are set forth, however, it will be apparent to one skilled in the art, that the present invention may be practiced without these specific details. In some instances, well-known methods and devices are shown in block diagram form, rather than in detail, to avoid obscuring the present invention. Reference throughout this specification to “an embodiment” means that a particular feature, structure, function, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. Thus, the appearances of the phrase “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the invention. Furthermore, the particular features, structures, functions, or characteristics may be combined in any suitable manner in one or more embodiments. For example, a first embodiment may be combined with a second embodiment anywhere the two embodiments are not mutually exclusive. 
     The terms “coupled” and “connected,” along with their derivatives, may be used herein to describe structural relationships between components. It should be understood that these terms are not intended as synonyms for each other. Rather, in particular embodiments, “connected” may be used to indicate that two or more elements are in direct physical or electrical contact with each other. “Coupled” my be used to indicated that two or more elements are in either direct or indirect (with other intervening elements between them) physical or electrical contact with each other, and/or that the two or more elements co-operate or interact with each other (e.g., as in a cause an effect relationship). 
     The terms “over,” “under,” “between,” and “on” as used herein refer to a relative position of one material layer with respect to other layers. As such, for example, one layer disposed over or under another layer may be directly in contact with the other layer or may have one or more intervening layers. Moreover, one layer disposed between two layers may be directly in contact with the two layers or may have one or more intervening layers. In contrast, a first layer “on” a second layer is in contact with that second layer. 
       FIGS. 1A and 1B  illustrate plan views of an optically transduced MEMS magnetometer, in accordance with an embodiment of the invention.  FIGS. 1C and 1D  illustrate cross-sectional views of the MEMS magnetometer  100  along the C-C′ and D-D′ lines illustrated in  FIG. 1A , respectively. 
     As illustrated in  FIG. 1A , the MEMS magnetometer  100  includes at least one deformable diffractive optical structure (e.g., a multi-layer grating  124 A) disposed on a top side (or alternatively a bottom side) of the MEMS magnetometer  100  for optical domain transduction of the resonator displacement. Transduction in the optical domain enables greater sensitivity than conventional techniques and also enables the resonator to be bulk machined which enables the MEMS magnetometer  100  to be fabricated simultaneously with the bulk micromachined gyroscope described in the above referenced U.S. Patent Application, entitled OPTICALLY TRANSDUCED MEMS GYRO DEVICE, to Nielson, et al. A multi-axis inertial sensor of superior performance may be provided with both inertial and magnetic sensing functions provided by multiple such optically transduced MEMS devices monolithically integrated on a same substrate. As such, in certain advantageous embodiments, one or more of the MEMS magnetometers described herein are provided in combination with one or more bulk micromachined gyroscopes. Of further advantage, the crosstalk present when Lorentz force drive and capacitive or other electrical sensing is used together is not present when optical displacement sensing resulting from the deformable diffractive optical structures is used. 
     In embodiments, a MEMS magnetometer includes a support and at least one resonator suspended from the support. Examples of more than one resonator include a tuning fork magnetometer. It is understood that based on the illustrative single resonator embodiments provided herein, one of skill in the art may readily adapt a conventional thin film tuning fork magnetometer design, or other multi-mass magnetometer known in the art, to incorporate one or more of the techniques described herein, such as optical domain transduction, bulk machining, etc. 
     In the exemplary embodiment depicted in  FIG. 1A , the support  101  forms a frame completely surrounding a resonator  105 . As further shown in  FIGS. 1C and 1D , both the support  101  and the resonator  105  have at least a bulk substrate thickness, T. As such, the resonator  105  and the support  101  have substantially the same thickness with minor thickness differences between the two arising from differences in the thin film processing over the substrate  102  for one or the other of the resonator  105  and the support  101 . For example, one or more layers of thin film materials such as silicon dioxide, polysilicon, silicon nitride, electrode metallization, etc. utilized in the fabrication of the MEMS magnetometer  100  may be present to differing extents between the support  101  and the resonator  105 . 
     Generally, the bulk substrate thickness, T may vary depending on convention for the material selected for the substrate  102  ( FIG. 1C ). While, the substrate  102  may generally be any material suitable for semiconductor device fabrication, in advantageous embodiments a bulk substrate composed of a poly or single crystalline material such as, but is not limited to, silicon, germanium, silicon-germanium, or a III-V compound semiconductor material is utilized. In embodiments, the substrate  102  has conductivity sufficient for a voltage applied for drive of the resonator to pass through the entire bulk substrate thickness T without detrimental voltage drop. For example, impurity doped substrates may be utilized, as known in the art. In a preferred embodiment, the substrate  102  is a semiconductor (silicon) on insulator (SOI) substrate having a buried insulator layer  109  (e.g., silicon dioxide) spanning the entire area of the substrate  102 , as known in the art. For such embodiments, the bulk substrate thickness T includes both a base semiconductor substrate portion  102 A disposed below the buried insulator layer  109 , and also an overlying semiconductor substrate portion  102 B. Generally, the bulk substrate thickness, T is to be at least 500 μm. In a preferred embodiment where the substrate material is silicon (e.g., a single crystal silicon wafer), the bulk substrate thickness, T will be approximately 700 μm (i.e. between about 650 μm and about 750 μm). 
     As further shown in  FIG. 1A , delineation of the resonator  105  from the support  101  is by the trench  106  that is nearly continuous with the only discontinuities occurring at anchor points (e.g., anchors  115 A and  115 B). As further shown in  FIGS. 1B and 1C , the trench  106  has a lateral gap dimension, G, defining the critical dimension and aspect ratio of the trench  106 . As the trench  106  passes completely through the substrate  102 , the trench  106  is at least 500 μm deep (e.g., as measured from a bottom side of the substrate  102 ) and approximately 700 μm deep for the exemplary silicon substrate embodiments. Because the lateral gap dimension, G, does not limit device displacement transduction sensitivity, the dimension may be whatever is sufficient for the amount of resonator displacement induced during resonance and/or otherwise needed for a given deep trench etch process. Generally, the lateral gap dimension, G, is microns wide. In exemplary silicon substrate embodiments, the lateral gap dimension, G, is between 20 μm and 60 μm for an aspect ratio between about 20:1 and 30:1, which is readily attainable using conventional anisotropic silicon etching techniques known in the art. 
     While a resonator may generally have any shape, in the exemplary embodiment illustrated in  FIG. 1A , the resonator  105  is a beam having a longitudinal beam length extending along a first dimension (e.g., x-dimension in  FIG. 1A ). The bulk substrate thickness, T, is therefore orthogonal to the longitudinal beam length, extending out-of-plane (e.g., in the z-dimension) and a transverse beam length is along the y-dimension. As such, the trench  106  defines faces of the resonator which extend in the z-dimension through the bulk substrate thickness. Separated from the resonator face  110 A by the trench  106  are opposing support faces  110 B and  110 C which each form a portion of electrically independent drive electrodes  130 A,  130 B. The drive electrodes  130 A,  130 B are isolated from surrounding regions of the support  101  by isolation  172  wrapping around the MEMS magnetometer  100  and are to be powered by an AC signal applied via pads  131 A,  131 B, respectively. Each of the two anchor points  115 A and  115 B are electrically coupled to the corresponding ones of the electrodes  130 A,  130 B and conduct current to opposite end portions of the resonator  105 . For the preferred SOI embodiments, current conduction through the anchor points  115 A,  115 B is provided across the entire bulk substrate thickness T with conductive vias  108  passing through the buried insulator layer  109  to electrically couple the base semiconductor substrate portion  102 A with the overlying semiconductor substrate portion  102 B, as shown in  FIGS. 1C and 1D . 
     In the exemplary embodiment illustrated in  FIG. 1A , the first and second electrically independent support faces  110 B and  110 C diverge at a center of the resonator  105  (separated by isolation  172 ) and wrap continuously around the anchors  115 A and  115 B converge back at a center of the resonator on an opposite side of the resonator  105  (again separated from each other by isolation  172 ). With the electrodes  130 A,  130 B contacting both legs of each anchor  115 A and  115 B, current is spread across all four anchor legs. The legs of anchor  115 A and  115 B are ideally located in the vicinity of the nodal point of the desired resonant mode to reduce energy losses in the desired resonant mode of the mechanical resonator. 
     As further illustrated in  FIG. 1B , the resonator  105 , as attached to the support  101  at two anchor points, is to deflect along the longitudinal axis relative to the support  101  in response to an AC signal imparted by the drive electrodes  130 A and  130 B in the presence of a magnetic field. In the exemplary embodiment, the resonator  105  is to oscillate in the plane of the support  101  only (i.e., in an x-y plane) with oscillation modes out of the x-y plane (i.e., in the z-dimension) suppressed by the anchors  115 A,  115 B having a bulk thickness T (e.g., 700 μm) which is many times a critical dimension of the anchor in the x-dimension of approximately 50 μm. Further the resonant beam has an aspect ratio such that its thickness in the z-dimension is thicker than its width in the y-dimension that forces the desired in-plane resonant mode to be the lowest frequency mode. In the exemplary embodiment, in the presence of the magnetic field, current passing through a longitudinal length between the anchors  115 A and  115 B induces the longitudinal, in-plane deflection through the Lorentz effect. With the current applied at an AC frequency approximately equal to a resonant frequency of the resonator  105 , the Lorentz force is amplified by the mechanical Q of the resonator  105 . In a first embodiment, the resonator  105  resonates only in response to the Lorentz force being of sufficient magnitude (i.e., when in the presence of a sufficiently strong magnetic field). In a second embodiment, the driven AC signal alone induces the in-plane resonance with amplitude of the resonance then modulated by the Lorentz force. 
     In an embodiment, the deformable diffractive optical structure employed for optical transduction of resonator displacement is a multi-layer grating. In the exemplary embodiment, each grating (i.e., grating) in the multi-layer gratings  120 A,  120 B,  122 A- 122 C,  124 A- 124 C, and  126 A,  120 B is a sub-wavelength grating, however chirped gratings and other forms known in the art may also be employed. As shown in  FIG. 1C , the multi-layer grating  124 B includes a first grating  144 B rigidly affixed to the support  101  and extending over the trench  160  to be disposed over a second grating  154 B formed on the resonator  105 . Each of the first and second gratings  144 B and  154 B may be formed in either the substrate  102 , or as illustrated in  FIG. 1C , in thin film layers  104 ,  103 , respectively, which are formed on the substrate  102 . In the exemplary embodiment, the first grating  144 B is disposed in a material layer  104  that forms a bridge spanning the transverse length of the resonator  105 . Exemplary thin film materials which can be used for layers  104  and  103  include silicon dioxide, silicon nitride and polysilicon, as well as any other material known in the art having adequate thermal stability and structural integrity for the geometries and functions described herein. 
     As further illustrated in  FIG. 1B , aperture dimensions of a multi-layer grating is to vary with the displacement of the second grating  154 B relative to the first grating  144 B as the resonator deflects along its longitudinal length in the plane of resonance. While first-order gratings with feature dimensions significantly larger than a wavelength of light used for the optical transduction (i.e., the operating wavelength) may be used, near field multi-layer gratings are preferred as being capable of modulating the intensity and polarization of an incident light in response to much smaller displacements of the grating elements (i.e., lines and spaces). Near field multi-layer gratings having line elements of the gratings separated by distances which are less than the operating wavelength are known to modulate the intensity of reflected or transmitted zero orders even when displacement in-plane of the grating is only fractions of a wavelength. For a detailed description of such near field multi-layer gratings, the interested reader is referred to commonly assigned U.S. Pat. Nos. 7,173,764 and 7,339,738 to Carr et al., which are incorporated herein by reference in their entirety for all purposes. The sub-wavelength displacement sensitivity possible with the near field multi-layer grating enables a level of magnetic field sensitivity heretofore unknown. 
     In an embodiment, a MEMS magnetometer includes a plurality of near field multi-layer gratings. As shown in  FIG. 1A , the plurality of multi-layer gratings  120 ,  122 ,  124 , and  126  are disposed in a one dimensional array across the longitudinal beam length. In an embodiment, the plurality of multi-layer gratings is disposed symmetrically about a center of the resonator  105 . In the exemplary embodiment illustrated in  FIG. 1A , the plurality of multi-layer gratings is disposed symmetrically to have pairs of the multi-layer gratings disposed over points of equivalent deflection. For example, multi-layer grating pair  122 A and  124 A are disposed equidistant from a center of the longitudinal beam length (extending along x-axis if  FIG. 1A ). Similarly, multi-layer grating pairs  122 B and  124 B,  122 C and  124 C,  120 A and  126 A,  120 B and  126 B are similarly equidistant from the beam center, beam ends, and anchors  115 A,  115 B. An alternative symmetrical arrangement has a single multi-layer grating centered over the center of the longitudinal beam length. 
     In an embodiment, across the plurality of arrayed multi-layer gratings, line elements in the gratings of each multi-layer grating have a same pitch, but an alignment offset between the gratings  144  and  154  is different across the plurality of multi-layer gratings so that phase between the overlapping gratings is modulated across the plurality of multi-layer gratings.  FIG. 2  illustrates an expanded plan view  200  of a portion of the MEMS magnetometer  100 . The expanded plan view  200  illustrates multi-layer gratings  122 A and  124 A. Inset within  FIG. 2 , cross-sectional views  222 A illustrates line elements in the first grating  122 A having a first pitch P 1 , defining a period of the first grating  144 A. Similarly, line elements in the second grating  154 A have a second pitch P 2 . The pitches P 1  and P 2  may be substantially equal to each other, but are not necessarily so. Alignment between the first and second gratings  144 A and  154 A defines a phase φ 1  for the first multi-layer grating  122 A. 
     In the cross-sectional view  224 B inset within  FIG. 2 , line elements in the first grating  144 A have the first pitch P 1 . Similarly, line elements in the second grating  154 A have the second pitch P 2 . As such, between the two multi-layer gratings  122 A and  124 A, the pitches of respective gratings are equal. Alignment (i.e., overlay) between the first and second gratings of the multi-layer gratings  122 A and  124 A is however different such that a phase φ 2  for the second multi-layer grating  124 A is different from the phase φ 1 . In one exemplary embodiment, the difference in phase between adjacent ones of the arrayed gratings is 10-20 nm. In the preferred embodiment, phase is varied in equal steps from one end of the resonator to the other so that over the ten multi-layer gratings arrayed along the longitudinal beam length of the resonator  105 , phase is varied by 80-150 nm to account for misalignment of the gratings forming each multi-layer grating that could reduce the intensity modulation with displacement. In another embodiment, phase is varied in equal steps from the center of the resonator to each end so that over two sets of five multi-layer gratings arrayed along the longitudinal beam length of the resonator  105 , phase is varied by 40-75 nm. As such, multi-layer grating phase is the same at locations of equivalent degrees of deflection. In another embodiment however, the multi-layer grating phase φ is varied only within the grating pairs. For example, just as the multi-layer grating pairs  122 A and  124 A have differing phases φ 1  and φ 2 , multi-layer grating pairs  122 B and  124 B have the same two differing phases φ 1  and φ 2 , multi-layer grating pairs  122 C and  124 C also have the same two differing phases φ 1  and φ 2 , and so do multi-layer grating pairs  120 A,  126 A, and  120 B,  126 B. As such, phase is varied across two locations having equivalent degrees of deflection permitting a phase selection and a degree of deflection to be independently selected. 
     Varying the multi-layer grating phase φ across an arrayed plurality of multi-layer gratings enables magnetic field sensing system incorporating the MEMS magnetometer  100  to undergo a functional test where multi-layer gratings in the array may be binned out to identify the grating having a phase, φ n , that satisfies a predetermined threshold criteria, e.g., a greatest displacement induced modulation of light intensity (e.g., zeroth order) or an intensity modulation level within a predetermined range. Variation in the fabrication process affecting the gratings may therefore be accounted for to advantageously improve measurement sensitivity. In an alternate embodiment, whether or not multi-layer grating phase is modulated across the plurality, more than one multi-level grating may be utilized for displacement transduction. For example, differential measurements may be performed and/or a comb filter or other filtering may be applied to combine outputs from more than one multi-level grating in each array and remove common mode noise, etc. as known in the art of electrical signal conditioning. 
     As further depicted in the expanded plan view  200  in  FIG. 2 , apertures  145  and intervening line elements  146  are in an x-y plane parallel to the plane of the support, and to the plane of resonance. The major aperture lengths extend along the x-dimension and to define the first grating period along the y-dimension to transduce displacement resulting from the in-plane deflection along the longitudinal beam length. As shown in  FIG. 1C , the second grating  154 B comprises a similar plurality of regularly spaced line elements in a second plane parallel to the plane of the support  101 , and plane of resonance, but separated in the z-dimension from the first line elements. This z-dimension separation between gratings may be less than a wavelength of light utilized for interrogation of the grating. 
       FIGS. 3A and 3B  illustrate a magnetic field sensor  300  employing the MEMS magnetometer  100 , in accordance with an embodiment. For the exemplary embodiment, the magnetic field sensor  300  utilizes the multi-layer grating  124 A as a reflective grating, however a sensor employing a transmissive grating may be similarly designed (e.g., where both the first and second gratings  144 A and  154 A are cantilevered off their respective bulk structures). In the magnetic field sensor  300 , a light source  302 - 1  is provided on a substrate  310 - 1  together with a pair of detectors  304 . Exemplary light sources for interrogating the multi-layer grating  124 A include a VCSEL or an LED. 
     For the exemplary linear grating, light having a specific wavelength (i.e., operating wavelength) is diffracted into modes that emerge along multiple discrete angles. For the preferred near field grating embodiments where the multi-layer gratings are “sub-wavelength,” the light source  302  is to emit an incident light beam  316  at a wavelength that is greater than the pitch of the grating (either the first grating  144 A or second grating  154 A, although in the preferred embodiment where the both the first and second grating have the same pitch, the wavelength is grating than the pitch of both first and second gratings). The sub-wavelength grating has only a zeroth-order mode and a surface wave mode. When optically coupled, varying the offset between the two gratings allows for modulation of the light between transmission and reflection in the zeroth order mode. 
     Exemplary detectors  304  include a PIN photodiode, an active pixel sensor (APS), or the like. In one near field embodiment, the first photodetector  304 - 1  is to measure an intensity of zeroth-order light modulated by the multi-layer grating  124 A, though other orders may also be sensed. The first detector  304 - 1  (i.e., a position sensing detector) is to detect the light portion  318 - 1  reflected off the multi-layer grating  124 A. A second detector  304 - 2  (i.e., a reference detector) is to detect a light portion  318 - 2  of the light  16  reflected off a reference surface outside the multi-layer grating  124 A (i.e., the reference light portion  318 - 2  bypasses the multi-layer grating  124 A). The reference detector  304 - 2  provides an electrical output signal which can be combined with the electrical output signal of the detector  304 - 1  to substantially remove any common-mode noise (i.e., noise due to the light source or other factors which is common to both reflected light portions  318 - 1  and  318 - 2 ) and thereby improve a sensitivity for sensing relative displacement of the second grating  154 A disposed on the resonator. In alternate embodiments of the present invention where a transmitted light portion is detected, a reference detector can be similarly used to detect a portion of the incident light beam  316  which bypasses the multi-layer grating  124 A. Output from first detector  304 - 1  provides a first basis for outputting a displacement measure which is then translated downstream into a sensed magnetic field strength (gauss). 
     For the exemplary embodiments employing a plurality of multi-layer gratings, each multi-layer grating having a particular phase is paired with a light source  302  and at least one optical detector  304 , for example an array of LEDs or VCSELs may be disposed over each array of multi-layer gratings.  FIG. 3B  illustrates a cross-section view of the magnetic field sensor  300  taken through a second multi-layer grating  122 A on the substrate  102  disposed relative to a second light source  302 - 2  that is fabricated in a substrate  310 - 2 , which may be the same substrate as  310 - 1 , or a separate (third) substrate. First detector  304 - 3  and reference detector  304 - 4  operate in the same manner described for detectors  304 - 1  and  304 - 2  to detected light portions  318 - 3  and  318 - 4 , respectively, with the position sensing output from first detector  304 - 3  being a second basis for outputting a sensed magnetic field strength. The intensity modulation generated by one from the plurality of multi-layer gratings, or intensity modulation generated by each multi-layer grating of the array may be collected simultaneously (in parallel) or serially sampled in a time divided (e.g., in round robin fashion) or other multiplexed manner. 
       FIG. 4  is a flow diagram illustrating a method  400  of sensing a magnetic field, in accordance with an embodiment of the invention. The method  400  may be performed with the magnetic field sensor  300  when in an operational mode. At operation  405 , the MEMS magnetometer  100  is placed in a magnetic field. At operation  410 , an AC current is applied to the resonator, the frequency of which is approximately a mechanical resonant frequency of the resonator. At operation  415 , a multi-layer grating, such as multi-layer grating  122 A, is interrogated with light, for example of a wavelength greater than a feature size of the multi-layer grating. At operation  420 , the extent of resonator deflection is determined based on modulation of light intensity by the multi-layer grating in proportion to the Lorentz force resulting from the AC current conducted through the resonator in the presence of the magnetic field. At operation  425 , a determination of the magnetic field strength is made based on the measured resonator deflection. 
       FIG. 5  is a flow diagram illustrating a method  500  of operating a MEMS magnetometer having a plurality of multi-layer gratings, in accordance with an embodiment of the invention. The method  500  may be performed with the magnetic field sensor  300  during an operational mode to select a multi-layer grating from a plurality of multi-layer gratings disposed at locations subject to different degrees of deflection so as to implement the method  400 . Again beginning with operation  405 , the MEMS magnetometer  100  is placed in a magnetic field. At operation  510 , a first multi-layer grating associated with a first degree of resonator deflection, such as the smallest degree of deflection (e.g., multi-layer grating  122 C in  FIG. 1A ), is interrogated. At operation  515 , an intensity of an optical response (e.g., reflected zeroth order light), as modulated by the first multi-layer grating, is determined. If a predetermined intensity modulation threshold is met (e.g., sufficiently large), resonator displacement is transduced with the first multi-layer grating and output as the basis for determining the measured field strength at operation  425 . If the predetermined intensity modulation threshold is not met, the method  500  proceeds to operation  512  where a second multi-layer grating proximate to a point of relatively greater degree of deflection is interrogated. At operation  516 , an intensity of an optical response (e.g., reflected zeroth order light), as modulated by the second multi-layer grating, is determined and output as the basis for determining the measured field strength at operation  425 . Method  500  may proceed to incrementally advance through every multi-layer grating in a plurality with each subsequently selected grating associated with a greater degree of resonator deflection until the multi-layer grating disposed at a point of maximum deflection is interrogated, at which point a measured response of maximum sensitivity is output. 
       FIG. 6  is a flow diagram illustrating a method  600  of operating a MEMS magnetometer having a plurality of multi-layer gratings, in accordance with an embodiment of the invention. The method  600  may be performed with the magnetic field sensor  300  during as a functional test prior to entering an operational mode to select a multi-layer grating that will be used during an operational mode from a plurality of multi-layer gratings having different phases. At operation  605 , the MEMS magnetometer  100  is placed in a magnetic field. At operation  610 , a first multi-layer grating associated with a first phase between first and second gratings is interrogated with light. An intensity of an optical response (e.g., reflected zeroth order light), as modulated by the first multi-layer grating, is then determined. At operation  620 , a second multi-layer grating having a second multi-layer grating phase, different than the first, is interrogated and an intensity of an optical response (e.g., reflected zeroth order light) assessed. At operation  630 , the first or second multi-level grating is selected based on predetermined selection criteria. For example, the first or second multi-level grating satisfying a predetermined intensity modulation threshold is selected. Alternatively, the first or second grating having the greater intensity modulation is selected. The selected multi-layer grating along with the associated light source and detector are then identified for further use, for example as the exclusive displacement transducer output used in method  400 . 
     It is to be understood that the above description is illustrative, and not restrictive. For example, while flow diagrams in the figures show a particular order of operations performed by certain embodiments of the invention, it should be understood that such order is not required (e.g., alternative embodiments may perform the operations in a different order, combine certain operations, overlap certain operations, etc.). Furthermore, many other embodiments will be apparent to those of skill in the art upon reading and understanding the above description. Although the present invention has been described with reference to specific exemplary embodiments, it will be recognized that the invention is not limited to the embodiments described, but can be practiced with modification and alteration within the spirit and scope of the appended claims. The scope of the invention should, therefore, be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled.