System and method for determining acceleration

Abstract

An accelerometer and method for determining local acceleration is disclosed. The accelerometer comprises a test mass arranged to freely move, the test mass having a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other and first, second and third interferometer arrangements optically coupled to the respective reflectors for measuring respective interferometric optical power variations associated with movement of the test mass. The accelerometer also comprises a processing arrangement for processing the first, second and third interferometric optical power variations to determine the local acceleration.

Description

SYSTEM AND METHOD FOR DETERMINING ACCELERATION PRIORITY DOCUMENTS

[0001] The present application claims priority from Australian Provisional Patent Application No. 2023902573 titled “SYSTEM AND METHOD FOR DETERMINING ACCELERATION” and filed on 14 August 2023, the content of which is incorporated by reference in its entirety. TECHNICAL FIELD

[0002] The present disclosure relates to the determining of local acceleration. In a particular form, the present disclosure relates to the determining of local acceleration employing an interferometric system. BACKGROUND

[0003] The ability to measure local acceleration precisely and accurately is important in a number of fields such as inertial navigation, measurement systems and motion sensors. A notable example of measuring local acceleration is the measurement of local gravity. Determining local gravity can be important in applications such as hydrology, vulcanology, mineral surveying and subsurface mapping where the local value of gravity may be used to infer properties of the underlying composition and structure of the earth.

[0004] There are broadly two types of accelerometers or gravimeters (ie, an accelerometer configured to measure local gravity) comprising relative systems and absolute systems. A relative accelerometer will rely on there being a scale factor between its acceleration measurement and the true local acceleration with this scale factor having to be periodically calibrated as the acceleration measurement will rely on mechanical or electrical properties of the measurement system that drift over time. An absolute accelerometer will directly measure the kinematics of a test mass and determine the local acceleration on this basis and as a consequence does not require periodic calibration.

[0005] An issue with accelerometers, and in particular absolute accelerometers, is the difficulty of obtaining a necessary level of precision while also retaining sufficiently low size, weight, power, and cost (SWaPc) to be portable. As a result, “portable” commercial devices for precisely measuring acceleration and gravity tend to be costly and will often still not meet size and weight requirements. SUMMARY

[0006] In one aspect, the present disclosure provides an accelerometer for determining a local acceleration, comprising:a test mass arranged to freely move, the test mass having a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other; a first interferometer arrangement optically coupled to the first reflector for measuring a first interferometric optical power variation associated with movement of the test mass; a second interferometer arrangement optically coupled to the second reflector for measuring a second interferometric optical power variation associated with movement of the test mass; a third interferometer arrangement optically coupled to the third reflector for measuring a third interferometric optical power variation associated with movement of the test mass; and a processing arrangement for processing the first, second and third interferometric optical power variations to determine the local acceleration.

[0007] In another form, one or more of the first, second or third reflectors is a retroreflector.

[0008] In another form, the retroreflector comprises an optical centre co-located with a centre of mass of the test mass.

[0009] In another form, the retroreflector is a corner cube reflector.

[0010] In another form, the test mass has a cubic configuration and the first, second and third reflectors comprise first, second and third corner cube reflectors inset into orthogonal faces of the test mass, and wherein optical centres of the first, second and third corner cube reflectors are co-located to the centre of mass of the test mass.

[0011] In another form, the accelerometer further comprises a test mass chamber, wherein the test mass is arranged to freely move within the test mass chamber.

[0012] In another form, the test mass chamber is configured to sustain a residual gas pressure of less than 10-2Pa.

[0013] In another form, one or more of the first, second or third interferometer arrangements comprises: a laser beam split into a reference beam component and an interrogation beam component; an interrogation arm configured so that the interrogation beam component travels along the interrogation arm and is reflected from a respective reflector of the test mass; a reference arm configured so that the reference beam component travels along the reference arm and interferes with the reflected interrogation beam component; and an intensity detector to measure the interferometric optical power variation resulting from the interference of the reference beam component and the reflected interrogation beam component.

[0014] In another form, the processing arrangement comprises: a phase measuring apparatus for determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations; one or more data processors for determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations; and one or more data processors for determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase- based trajectory components.

[0015] In another form, the phase measuring apparatus comprises for one or more of the first, second and third interferometric optical power variations: an analogue-to-digital converter for digitising a respective interferometric optical power variation to produce a digitised fringe pattern; and a data processor for processing the digitised fringe pattern to extract the respective phase variation corresponding to the respective interferometric optical power variation.

[0016] In another form, the phase measuring apparatus comprises for one or more of the first, second and third interferometric optical power variations: a modulator for modulating a respective reference beam component of a respective interferometer arrangement by a modulation frequency, a first signal multiplier for forming an in-phase interferometric optical power variation signal by multiplying a respective interferometric optical power variation by a first periodic modulation signal based on the modulation frequency; a second signal multiplier for forming a quadrature-phase interferometric optical power variation signal by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees; a low pass filter arrangement for filtering both the in-phase interferometric optical power variation signal and the quadrature-phase interferometric optical power variation signal; and a phase determining module operable to process the filtered in-phase interferometric optical power variation signal and the filtered quadrature-phase interferometric optical power variation signal to determine the respective phase variation corresponding to the respective interferometric optical power variation.

[0017] In another form, the one or more data processors for determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components are configured to: determine a respective test mass trajectory component based on the first, second and third phase- based trajectory components; anddetermine the respective local acceleration component from the respective test mass trajectory component.

[0018] In another form, determining a respective test mass trajectory component based on the first, second and third phase-based trajectory components comprises determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component; and determining the respective local acceleration component comprises deriving the local acceleration component from the determined 2ndorder coefficient.

[0019] In another form, determining the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component comprises: representing the test mass trajectory components in polynomial form and substituting polynomial parameterised forms of the test mass trajectory components into a system of equations relating the first, second and third phase-based trajectory components and the test mass trajectory components; and solving the system of equations to determine one or more coefficients of the polynomial parameterised forms of the test mass trajectory components including the 2ndorder coefficients.

[0020] In another form, further representing the phase based trajectory components in polynomial form and determining values for the coefficients of the polynomial parameterised forms of the determined phased based trajectory components.

[0021] In another form, determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component comprises determining values for the coefficients of polynomial representations of the first, second and third phase-based trajectory components based on the determined first, second and third phase-based trajectory components.

[0022] In another form, determining the respective local acceleration component from the respective test mass trajectory component further comprises compensating the determined 2ndorder coefficient of the polynomial representation with a compensation factor characterising an effect of a Coriolis force at a measurement latitude to form a compensated 2ndorder coefficient and determining the respective local acceleration component comprises deriving the local acceleration component from the compensated 2ndorder coefficient.

[0023] In another form, the accelerometer is configured so that a selected direction of the first, second and third directions is aligned to a direction of gravitational acceleration.

[0024] In another form, the accelerometer is configured so that a selected direction of the first, second and third directions is at a predetermined orientation with respect to a direction of gravitational acceleration.

[0025] In another form, the test mass chamber comprises a launching arrangement configured to launch the test mass in the selected direction.

[0026] In another form, the launching arrangement is configured to launch the test mass a predetermined height with respect to the launching arrangement.

[0027] In another form, a mass of the test mass is less than 0.01 kg.

[0028] In a second aspect, the present disclosure provides a method for determining a local acceleration, comprising: configuring a test mass to freely move, the test mass having a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other; measuring a first interferometric optical power variation associated with movement of the test mass by a first interferometer arrangement optically coupled to the first reflector; measuring a second interferometric optical power variation associated with movement of the test mass by a second interferometer arrangement optically coupled to the second reflector; measuring a third interferometric optical power variation associated with movement of the test mass by a third interferometer arrangement optically coupled to the third reflector; and processing the first, second and third interferometric optical power variations to determine the local acceleration.

[0029] In another form, one or more of the first, second or third reflectors is a retroreflector.

[0030] In another form, the retroreflector comprises an optical centre co-located with a centre of mass of the test mass.

[0031] In another form, the retroreflector is a corner cube reflector.

[0032] In another form, the test mass has a cubic configuration and the first, second and third reflectors comprise first, second and third corner cube reflectors inset into orthogonal faces of the test mass, and wherein optical centres of the first, second and third corner cube reflectors are co-located to the centre of mass of the test mass.

[0033] In another form, the test mass is configured to freely move within a test mass chamber.

[0034] In another form, the test mass chamber is configured to sustain a residual gas pressure of less than 10-2Pa.

[0035] In another form, measuring one or more of the first, second or optical power variations comprises: splitting a laser beam into a reference beam component and an interrogation beam component; reflecting the interrogation beam component from a respective reflector of the test mass to form a reflected interrogation beam component; configuring the reference beam component to interfere with the reflected interrogation beam component; and measuring the interferometric optical power variation resulting from the interference of the reference beam component and the reflected interrogation beam component.

[0036] In another form, processing the first, second and third interferometric optical power variations to determine the local acceleration comprises: determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations; determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations; and determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components.

[0037] In another form, determining a respective phase variation comprises: digitising a respective interferometric optical power variation to produce a digitised fringe pattern; and processing the digitised fringe pattern to extract the respective phase variation corresponding to the respective interferometric optical power variation.

[0038] In another form, determining a respective phase variation comprises: modulating a respective reference beam component of a respective interferometer arrangement by a modulation frequency, forming an in-phase interferometric optical power variation signal by multiplying a respective interferometric optical power variation by a first periodic modulation signal based on the modulation frequency;forming a quadrature-phase interferometric optical power variation signal by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees; filtering by a low pass filter both the in-phase interferometric optical power variation signal and the quadrature-phase interferometric optical power variation signal; and processing the filtered in-phase interferometric optical power variation signal and the filtered quadrature-phase interferometric optical power variation signal to determine the respective phase variation corresponding to the respective interferometric optical power variation.

[0039] In another form, determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components comprises: determining a respective test mass trajectory component based on the first, second and third phase-based trajectory components; and determining the respective local acceleration component from the respective test mass trajectory component.

[0040] In another form, determining the respective test mass trajectory component based on the first, second and third phase-based trajectory components comprises determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component; and determining the respective local acceleration component comprises deriving the local acceleration component from the determined 2ndorder coefficient.

[0041] In another form, determining the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component comprises: representing the test mass trajectory components in polynomial form and substituting polynomial parameterised forms of the first, second and third test mass trajectory components into a system of equations relating the first, second and third phase-based trajectory components and the test mass trajectory components; and solving the system of equations to determine one or more coefficients of the polynomial parameterised forms of the test mass trajectory components including the 2ndorder coefficients.

[0042] In another form, the method further comprises representing the phase based trajectory components in polynomial form and determining values for the coefficients of the polynomial parameterised forms of the determined phased based trajectory components.

[0043] In another form, determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component comprises determining values for the coefficients ofpolynomial representations of the first, second and third phase-based trajectory components based on the determined first, second and third phase-based trajectory components.

[0044] In another form, determining the respective local acceleration component from the respective test mass trajectory component further comprises compensating the determined 2ndorder coefficient of the polynomial representation with a compensation factor characterising an effect of a Coriolis force at a measurement latitude to form a compensated 2ndorder coefficient and determining the respective local acceleration component comprises deriving the local acceleration component from the compensated 2ndorder coefficient.

[0045] In another form, the method further comprises aligning a selected direction of the first, second and third directions to a direction of gravitational acceleration.

[0046] In another form, the method further comprises aligning a selected direction of the first, second and third directions at a predetermined orientation with respect to a direction of gravitational acceleration.

[0047] In another form, the method further comprises launching the test mass in the selected direction.

[0048] In another form, the method further comprises launching the test mass to a predetermined height.

[0049] In another form, a mass of the test mass is less than 0.01 kg. BRIEF DESCRIPTION OF DRAWINGS

[0050] Embodiments of the present disclosure will be discussed with reference to the accompanying drawings wherein:

[0051] FIG.1 is a figurative view of an accelerometer for determining local acceleration in accordance with some embodiments;

[0052] FIG.2A is a figurative view of an interferometer arrangement for measuring in a selected direction the interferometric intensity variation associated with movement of a test mass in accordance with some embodiments;

[0053] FIG.2B is a figurative view of an interferometer arrangement for measuring in a selected direction the interferometric intensity variation associated with movement of a test mass in accordance with some embodiments;

[0054] FIG.2C is a figurative view of an interferometer arrangement for measuring in a selected direction the interferometric intensity variation associated with movement of a test mass in accordance with some embodiments;

[0055] FIG.2D is a figurative view of an interferometer arrangement for measuring in a selected direction the interferometric intensity variation associated with movement of a test mass in accordance with some embodiments;

[0056] FIG.2E is a figurative view of a master laser arrangement for generating two or more locked laser beams in accordance with some embodiments;

[0057] FIG.3 is a system overview diagram of a processing arrangement for determining the local acceleration in accordance with some embodiments;

[0058] FIG.4 is a system overview diagram of a phase measuring apparatus for determining a phase variation corresponding to an interferometric optical power variation in accordance with some embodiments;

[0059] FIG.5 is a system overview diagram of a phase measuring apparatus for determining a phase variation corresponding to an interferometric optical power variation in accordance with some embodiments shown as incorporated into an interferometer arrangement similar to FIG.2A;

[0060] FIG.6 is a perspective view of a test mass comprising first, second and third reflectors in accordance with some embodiments;

[0061] FIG.7 is a diagram showing the effect of a rotation of a corner cube reflector about its centre of mass;

[0062] FIG.8 is a figurative view of an interferometer arrangement for measuring in a selected direction the interferometric intensity variation associated with movement of a test mass in accordance with some embodiments;

[0063] FIG.9 is a figurative view of a test mass chamber configured for the measurement of local gravitation and comprising a launching arrangement to launch a test mass in a launch direction opposed to the direction of gravitational acceleration in accordance with some embodiments;

[0064] FIG.10A is a flowchart of a method for determining local acceleration in accordance with some embodiments;

[0065] FIG.10B is a flowchart of a method for measuring an optical power variation in accordance with some embodiments;

[0066] FIG.10C is a flowchart of a method for processing first, second and third interferometric optical power variations to determine a local acceleration in accordance with some embodiments;

[0067] FIG.10D, is a flowchart of a method for determining a phase variation in accordance with some embodiments;

[0068] FIG.10E is a flowchart of a method for determining first, second and third local acceleration components from first, second and third phase-based trajectory components in accordance with some embodiments; and

[0069] FIG.11 is an end-on figurative view of accelerometer deployed in an unmanned aerial vehicle (UAV) or drone configuration in accordance with some embodiments.

[0070] In the following description, like reference characters designate like or corresponding parts throughout the figures. DESCRIPTION OF EMBODIMENTS

[0071] Referring now to FIG.1, there is shown a figurative view of an accelerometer 100 for determining local acceleration in accordance with some embodiments.

[0072] In this example, accelerometer 100 comprises a test mass 120 that is arranged to freely move. The test mass 120 incorporates a reflection arrangement comprising first, second and third reflectors that are configured to reflect in first, second and third directions that are orthogonal with respect to each other (indicated by cartesian coordinates ^, ^ and ^ in FIG.1).

[0073] Accelerometer 100 further comprises a first interferometer arrangement 140 optically coupled to the first reflector for measuring a first interferometric optical power variation associated with movement of the test mass, a second interferometer arrangement 150 optically coupled to the second reflector for measuring a second interferometric optical power variation associated with movement of the test mass, and a third interferometer arrangement 160 optically coupled to the third reflector for measuring a third interferometric optical power variation associated with movement of the test mass 120.

[0074] Accelerometer 100 additionally comprises a processing arrangement 170 for processing the first, second and third interferometric optical power variations (indicated by ^^(^)and ^^(^)in FIG.1) to determining the local acceleration (ie, ^ ).

[0075] In one example, the first, second and third reflectors may be configured as a planar reflecting surface or mirror. In one example, the overall configuration of the test mass may be configured as a cube with three of the orthogonal faces of the cube configured as the first, second and third reflectors.

[0076] In another example, the overall configuration of the test mass may be spherical and the first, second and third reflectors correspond to curved reflection regions on the surface of the test mass.

[0077] In another example, one or more of the first, second or third reflectors is a retroreflector. In another example, the retroreflector comprises an optical centre co-located with a centre of mass of the test mass. In yet another example, the retroreflector is a corner cube reflector where in one embodiment the optical centre of the corner cube reflector is co-located to the centre of mass of the test mass. In a further example, the corner cube reflector is as an open corner cube reflector where again in one embodiment the optical centre of the open corner cube reflector is co-located to the centre of mass of the test mass.

[0078] In various examples, where the environment of the test mass may require conditioning, accelerometer 100 may include a test mass chamber 110 (shown in dotted outline). In one example, test mass chamber 110 is configured to provide a reduced pressure environment to reduce the effects of air drag on the movement of test mass 120. In one embodiment, the test mass chamber 110 is configured to sustain a residual gas pressure of less than 10-2Pa.

[0079] Referring now to FIG.2A, there is shown a figurative view of an interferometer arrangement 200 for measuring in a selected direction the interferometric optical power variation associated with movement of a test mass in accordance with some embodiments. In various examples, interferometer arrangement 200 may correspond to one or more of the interferometer arrangements 140, 150 and 160 shown in FIG.1.

[0080] Interferometer arrangement 200 comprises a laser 210 emitting a beam 240 that is split by beam splitter 220 into a reference beam component 240A and interrogation beam component 240B.

[0081] Interferometer arrangement 200 further comprises an interrogation arm 280 configured so that the interrogation beam component 240B travels along the interrogation arm 280 and is reflected from a respective reflector 290 of the test mass 120 and a reference arm 270 configured so that the reference beam component 240A travels along the reference arm 270 and interferes with the reflected interrogation beam component 240B at beam splitter 220 (in this example). Interferometer arrangement 200 furthercomprises an intensity detector 250 to measure the interferometric optical power variation resulting from the interference of the reference beam component 240A and the reflected interrogation beam component 240B.

[0082] In this example, reference arm 270 includes a reference arm retroreflector 230 so that the reflected reference beam component 240A is offset from the incident reference beam component 240A in order to meet with the reflected interrogation beam component 240B at beam splitter 220 (as viewed by intensity detector 250) where in this example reflected interrogation beam component 240B is reflected from a retroreflector 290 of moving test mass 120.

[0083] In this example, test mass chamber 110 will include an optical window 111 to allow passage of interrogation beam component 240B to be reflected from reflector 290 of test mass 120.

[0084] In this manner, movement of test mass 120, as an example in the ^ direction, will result in an inteferometric optical power variation as measured by intensity detector 250 associated with movement of the test mass 120. As will be appreciated, intensity detector 250 will function to measure the optical intensity integrated over a sensor area to provide an optical power measurement. In one example, intensity detector 250 will be in the form of a photodetector that outputs a voltage (eg, ^^(^)) corresponding to the measured intensity integrated over a sensor area.

[0085] Referring now to FIGS.2B-2D, there are shown alternative interferometer arrangements 201, 202, 203 in accordance in accordance with some embodiments that in various examples may correspond to one or more of the interferometer arrangements 140, 150 and 160 shown in FIG.1. In FIGS. 2A-2D, the same reference characters have been used to designate corresponding parts in each of the depicted interferometer arrangements.

[0086] Considering interferometer arrangement 201 in FIG.2B, it can be seen in this example that retroreflector 230 of FIG.2A has been replaced by a pair of reflectors 231, 232 oriented to reflect the reference beam component 240A to beam splitter 220 where it interferes with the interrogation beam component 240B which has been reflected from reflector 290 of test mass 120.

[0087] In FIG.2C, the interrogation arm 280 of interferometer arrangement 202 comprises a reflection arrangement that in this example includes retroreflector 281 and reflectors 282, 283 to orient the interrogation beam component 240B to interfere with reference beam component 240A at a second beam splitter 225.

[0088] Referring now to FIG.2D, there is shown an interferometer arrangement 203 adopting a polarizing optics arrangement. In this example, interferometer arrangement 203 comprises a laser 210emitting a beam 240 that is split by (in this embodiment) a polarizing beam splitter 225 into a reference beam component 240A that travels through polarizing beam splitter 225 to intensity detector 250 and a beam splitter reflected interrogation beam component 240B that is reflected from polarizing beam splitter 225.

[0089] Due to the operation of polarizing beam splitter 225 the reference beam component 240A and beam splitter reflected interrogation beam component 240B will have orthogonal plane polarizations. As an example, the reference beam component 240A may be P-polarized for a polarizing beam splitter 225 configured to transmit P-polarized light and the beam splitter reflected interrogation beam component 240B will be S-polarized.

[0090] As shown in FIG.2D, the beam splitter reflected interrogation beam component 240B is reflected from the retroreflector 290 of moving test mass 120 to be further reflected by a first polarization modulation arrangement comprising in this example the beam splitter reflected interrogation beam component 240B traversing a ¼ wave plate 287, being reflected from mirror 289 and passing once again through ¼ wave plate 287, as a result changing the polarization of the beam splitter reflected interrogation beam component 240B by 90° to polarization modulated reflected interrogation beam component 240B´.

[0091] As an example, where beam splitter reflected interrogation beam component 240B is S-polarized the polarization modulated reflected interrogation beam component 240B´ will be P-polarized and reflected by retroreflector 290 on the return path and then be transmitted by polarizing beam splitter 225 where it is then reflected by interrogation beam retroreflector 281 to then have its polarization changed again by a second polarization modulation arrangement comprising ¼ wave plate 288 and mirror 289 (in this embodiment) which changes the polarization again by 90° (ie, S polarized) to twice polarized modulated reflected interrogation beam component 240B´´ to be reflected again by interrogation beam retroreflector 281 towards polarizing beam splitter 225. As the polarization state of twice polarized modulated reflected interrogation beam component 240B´´ corresponds to the polarization state that was originally reflected by polarizing beam splitter 225, twice polarized modulated reflected interrogation beam component 240B´´ will be reflected towards intensity detector 250 via polariser 254.

[0092] As would be appreciated, interferometer arrangement 203 now removes any phase contribution from the movement of optical components in the reference arm 270 and the operation of the first and second polarization modulation arrangements switch the polarisation so that the interrogation arm beam component 240B transitions between being reflected and transmitted by the polarizing beam splitter 225. In this manner, retroreflector 281 becomes the inertial reference.

[0093] The optical path length is also increased by a factor of two relative to embodiments without mirror 289 or similar (eg, see also FIG.8 and associated description). In addition, movement ofpolarizing beam splitter 225 and mirror 289 forming part of polarization modulation arrangement, including rotation of mirror 289, will cancel out any phase contribution in interferometer arrangement 203. Rotation of mirror 289 will additionally not cause translation of twice polarized modulated reflected interrogation beam component 240B´´ relative to reference beam component 240A.

[0094] As would also be appreciated, other types of interferometer configurations may be adopted in accordance with the present disclosure. As would also be appreciated, different interferometer arrangement configurations may be adopted for different measurement directions as required. In other examples, the same laser 210 may be adopted as a master laser for each interferometer arrangement. In another example, the laser(s) may be locked to an atomic transition to ensure an absolute frequency reference.

[0095] Referring now to FIG.2E, there is shown a figurative view of master laser arrangement 2000 for generating two or more locked laser beams in accordance with some embodiments that in one example may be used to generate laser beams (eg, laser beam 240 shown in FIGS.2A-2D) for an interferometer arrangement in accordance with the present disclosure.

[0096] Master laser arrangement 2000 comprises in this example a tuneable source laser 2010 and laser locking arrangement 2050 comprising a vapour cell and modulator arrangement 2020 and laser locking electronics 2040 to drive the tuneable source laser 2010. In one example, tuneable source laser 2010 is a semiconductor laser diode whose wavelength may be tuned by temperature and injection current. In various embodiments, laser diode 2010 may be configured in a distributed feedback (DFB) or a distributed Bragg reflector (DBR) configuration to improve robustness to mechanical shocks.

[0097] In operation a secondary laser light component 2070A taken from the output laser beam 2070 by fibre beam splitter 2045 and directed to laser locking arrangement 2050. In locking arrangement 2050, a probe beam component 2070B and a pump beam component 2070C are formed by, in this example, a beam splitter 2021 from the laser light component 2070A. Probe beam component 2070B is directed immediately through the vapour cell 2026 (ie, the probe light) and the pump beam component 2070C (ie, the pump light) is directed through modulator 2025 that is configured to modulate either the phase, frequency, or amplitude of the pump beam component 2070C. The modulated pump beam component 2070D is then directed to vapour cell 2026.

[0098] The non-linear interaction between the modulated pump beam component 2070D and probe beam component 2070B in vapour cell 2026 generates a spectroscopic signal 2070E that may be detected by detector 2028 and processed by laser locking electronics 2040 to generate a locking signal 2040A to lock the laser 2010 to a frequency corresponding to a particular atomic transition of the vapour cell 2026. Theoutput laser beam 2070 of source laser 2010 is then divided into (in this example) three output laser locked beam components by a fibre beam splitter 2055.

[0099] In one example, vapour cell 2026 is a rubidium vapour cell and the laser frequency is locked to a transition of87Rb. In one example, the laser wavelength is locked to a wavelength of 780.241 nm and has an output power of at least 10 mW. In one example, vapour cell 2026 resides in a magnetically shielded, temperature-controlled enclosure to control for any drift in the transition frequency.

[0100] While FIG.2E is directed to a master laser arrangement 2000 involving modulation transfer spectroscopy, in other examples the modulator 2025 could be placed in the probe beam path in a frequency modulation spectroscopy approach. In another example, a time-varying magnetic field may be applied to vapour cell 2026 in a Zeeman modulation approach to the master laser arrangement.

[0101] As would be appreciated, the above described fibre based semiconductor laser diode master laser arrangement 2000 is advantageous to reduce the SWaPc value for an accelerometer constructed in accordance with the present disclosure. In other examples, where the size, weight and power requirements are not as critical then other laser source and associated locking arrangements may be adopted as required.

[0102] It is instructive at this point to review the background optical theory relating to the operation of an accelerometer in accordance with the present disclosure. Consider for present purposes the third interferometric optical power variation in the ^ direction ^^(^) in FIG.1. In general, an interferometer arrangement 160 will comprise a laser beam that may be represented as a Gaussian beam. Accordingly, as an example, the electric field of the laser beam propagating in the ^ direction may be represented as:Eqn.1

[0103] where ^ = (^, ^) so that r^= ^^+ ^^and noting that the test mass 120 may also be translated in the ^ or ^ directions. In Eqn.1, ^^is the electric field amplitude (assumed real), ^(^) is the beam waist with minimum ^^= ^(0), ^(^) is the wavefront curvature, ^(^) is the Gouy phase shift, ^ is the wavenumber, ^ = ^^ is the angular frequency, ^ is the speed of light, and ^ is an arbitrary phase.

[0104] The beam waist, radius of curvature, and Gouy phase may be encapsulated in the beam’s ^-parameter, ie: ^(^) = ^ − ^^^1(^) =1^(^) +2^^^^(^)^Eqn.2

[0105] with Rayleigh range ^^= ^^^^ / 2.

[0106] Taking as an example, the interferometer arrangement 200 depicted in FIG.2A at the intensity detector 250, there will be a combined intensity measure of two laser beam components corresponding to the reference beam component 240A and the interrogation beam component 240B.

[0107] The electric fields of these two beam components may be denoted as the interrogation field ^^(^, ^) and the reference field ^^(^, ^). Suppose that due to motion of the test mass 120 in the ^ or ^ directions (ie, orthogonal to the ^ direction) that the interrogatiom beam component at the intensity detector is displaced from the reference beam component by ^^.

[0108] In the most general case, the reference and interrogation beam components may be at different frequencies denoted ^^and ^^, respectively, where ^^=+ ^!with ^!being an optional modulation frequency applied to the reference beam component to assist in phase determination (eg, see discussion with reference to FIG.5 below).

[0109] The intensity at the photodetector is given by " = 2#^^|^(^, ^)|^where:

[0110] similarly for ^^and ^^. Here, ℓ^(^) is the extra path traversed by the interrogation arm 280 and ℓ^is the extra path traversed by the referencearm 270. The term ℓ9comprises all common paths such as the path from the laser 210 to the beam splitter 220 and the distance from the beam splitter 220 to the intensity detector 250.

[0111] Intensity detector 250 will function to measure the optical power, ie the integrated intensity over the ^ − ^ plane. In practice, intensity detector 250 will provide a time varying output voltage that measures the variation in integrated intensity or optical power. For Gaussian beams, the measured optical power at the detector 250 associated with the vertical ^ axis will be:Eqn.4

[0112] for optical powers in the reference and interrogation arms ^^,^and ^^,^and transverse displacement ^^^(^) = 4[^^(^) + ^^(^)] for test mass 120 having a test mass trajectory defined by test mass trajectory vector @(^) =where ^(^), ^(^) and ^(^) are the test mass trajectory components in ^, ^ and ^ directions respectively. The factor of 4 in ^^is due to the doubling of the transverse displacement of the beam from retroreflection from the associated reflector 290 of test mass 120.

[0113] Referring now to FIG.3, there is shown a system overview diagram of a processing arrangement 300 for determining the local acceleration in accordance with some embodiments. In one example, the processing arrangement 300 may correspond to the processing arrangement 170 of the accelerometer 100 illustrated in FIG.1.

[0114] Processing arrangement 300 comprises a phase measuring apparatus 310 for determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations.

[0115] Referring now to FIG.4, there is shown a system overview diagram of a phase measuring apparatus 400 for determine a phase variation corresponding to an interferometric optical power variation according to an illustrative embodiment. In one example, the phase measuring apparatus 310 illustrated in FIG.3 may comprise three individual phase measuring apparatus 400 operable to process each of the first, second and third interferometric optical power variations. As depicted, phase measuring apparatus400 is operable to determine measured phase corresponding to measured interferometric optical power variation ^^(^).

[0116] In this example, phase measuring apparatus 400 comprises an analogue-to-digital converter (ADC) 410 to digitise interferometric optical power variation signal to produce the corresponding digitised fringe pattern 420 and a data processor 430 to process the digitised fringe pattern 420 to extract the phase variation corresponding to the interferometric optical power variation being processed. In one example, the clock of the ADC is derived from an atomic clock.

[0117] In one example, digitised fringe pattern 420 is processed by data processor 430 to form an in-phase component signal by subtracting from the respective interferometric optical power variation ^(^) its mean value. A Hilbert transform is then applied to the in-phase component signal by data processor 430 to obtain the quadrature component signal corresponding to the in-phase component signal and both in-phase and quadrature component signals are then further processed by the quadrant- preserving atan2(^, ^) function to extract the phase ^A(^) corresponding to the respective interferometric optical power variation ^(^).

[0118] Referring now to FIG.5, there is shown a system overview diagram of a phase measuring apparatus 500 for determine a phase variation corresponding to an interferometric optical power variation in accordance in accordance with some embodiments. In this example, phase measuring apparatus 500 is shown as incorporated into an interferometer arrangement 200 similar to FIG.2A, but as would be appreciated phase measuring apparatus 500 may be incorporated into other types of interferometer arrangements. As would also be appreciated, interferometer arrangement comprising phase measuring apparatus 500 may correspond to one or more of the interferometer arrangements 140, 150 and 160 shown in FIG.1.

[0119] In this example, phase determining apparatus 500 comprises a modulator 515 for modulating a respective reference beam component 240A of a respective interferometer arrangement by a modulation frequency ^!. In one example, phase measuring apparatus 500 may comprise a signal generator 510 for generating a modulation signal 510A having the modulating frequency ^!.

[0120] As referenced above, reference beam component 240A of laser 210 is modulated 515 by modulation signal 510A. In this manner, the total electric field ^(^) at photodetector 250 is the sum of the frequency shifted reference beam component 240A (ie, shifted in frequency by ^!) and the interrogation beam component 240B as follows: ^(^) = ^C(^) + ^D(^)%^EFGEqn.5

[0121] where ^C(^)is the electric field of the interrogation beam component 240B and ^D(^) is the electric field of the reference beam component 240B (prior to frequency shifting). As would be appreciated, the electric fields are complex-valued.

[0122] Considering a simplified form of Eqn.3, the intensity "(^) measured at the photodetector 250 is then:Eqn.6

[0123] and the voltage is then proportional to the integral of "(^) over its transverse extent and hence proportional to the interferometric power variation ^(^). The voltage as generated by photodetector 250 is then: ^(^) = ^^(^)(1 + N(^)cos[^!^ + ^(^)]) Eqn.7

[0124] where the contrast N(^) and the phase O(^) may depend on geometric details of the laser beams in addition to movement of the test mass 120.

[0125] Phase measuring apparatus 500 further comprises a first signal multiplier 525 for forming an in-phase interferometric optical power variation signal 526 by multiplying a respective interferometric optical power variation (represented by ^(^)) by a first periodic modulation signal based on the modulation signal 510A (forming in-phase signal processing path 520) and a second signal multiplier 535 for forming a quadrature-phase interferometric optical power variation signal 536 by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees (ie, 510A + 90°) (forming a quadrature phase processing path 530)

[0126] In one example, the in-phase interferometric optical power variation signal 526 is formed by multiplying ^(^) with first modulation signal oscillating at cos^!^ at signal multiplier 525 and similarly the quadrature-phase interferometric optical power variation signal 536 is formed by multiplying ^(^) with a second modulation signal oscillating at sin^!^ (ie, phase-shifted by 90°) at signal multiplier 535 resulting in:Eqn.8

[0127] Phase measuring apparatus 500 further comprises a low pass filter arrangement for filtering both the in-phase interferometric optical power variation signal 526 and the quadrature-phase interferometric optical power variation signal 536.

[0128] In one example, the low pass filter arrangement comprises a low pass filter module 527 having a cut-off frequency much lower than ^!and a separate low pass filter module 537, again using a cut-off frequency much lower than ^!, in the quadrature phase processing path 530.

[0129] Following this low-pass filtering, the filtered in-phase interferometric optical power variation signal ℐ 528 (ie, multiplied by cos^!^ in this example) and quadrature-phase interferometric optical power variation signal Q 538 (ie, multiplied by sin^!^ in this example) then correspond to:Eqn.9

[0130] Finally, phase measuring apparatus 500 comprises a phase determining module 580 operable to process the filtered in-phase interferometric optical power variation signal 528 and the filtered quadrature-phase interferometric optical power variation signal 538 to determine the respective phase variation ^A(^) corresponding to the respective interferometric optical power variation ^(^).

[0131] In this example, the phase determining module 580 may be implemented using the quadrant-preserving atan2(^, ^) function to extract the phase ^(^). As would be appreciated, the phase including its derivative, may be reconstructed because of the additional information about the sign of ^(^) through the Q component of the signal. Note, also, that neither the time-dependent contrast N(^) nor voltage offset ^^(^) in principle affect the calculation of ^(^) since they are common to both ℐ and Q.

[0132] As would be appreciated, phase measuring apparatus 500 implemented in accordance with the present disclosure results in the phase (and its sign) being extracted directly allowing the direction of the test mass to be discerned. Additionally, the process of multiplying the measuredinterferometric optical power variation with sinusoidally varying signals at the modulation frequency and then low-pass filtering the result is equivalent to applying a bandpass filter with equivalent cut-off frequency to the original raw signal, at least from a noise consideration. In this example, applying a filter cut-off frequency that is much less than the modulation frequency will result in any low frequency noise being significantly attenuated. This can be advantageous because noise at low frequencies is typically much larger than noise at high frequencies.

[0133] In various examples, modulation frequency ^!is chosen to substantially exceed the expected Doppler shift of the test mass but also taking into account that higher frequencies will require more expensive ADCs and signal processing units.

[0134] In one example, the modulation frequency ^!is selected to have a frequency between 20 MHz to 250 MHz. In another example, the modulation frequency ^!is selected to have a frequency between 125 MHz to 250 MHz. In yet another example, the modulation frequency ^!is selected to have a frequency between 200 MHz to 250 MHz. In a further example, the modulation frequency ^!is selected to have a frequency greater than 250 MHz. Typically, the cut-off frequency will be selected to be much less than the modulation frequency but again much larger than the maximum expected Doppler shift. In one example, cut-off frequency is selected to be between 1 MHz and 4 MHz for launch heights of the test mass between 100 um and 1 mm.

[0135] Referring back to FIG.3, following determination of the respective phase variations (ie, ^A^(^) and ^A^(^)) corresponding to the first, second and third interferometric optical powervariations, processing arrangement 300 comprises one or more data processors 320 for determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations.

[0136] Once again, taking the example of the ^ direction, the path-length difference Δℓ^may bedefined through ℓ^(^)− ℓ^ = 2^(^)+ 2Δℓ^, noting again that ^(^)is the ^ component of the trajectoryof test mass 120. Similarly, the phase difference ^^= ^^,^− ^^,^may also be defined. Using similardefinitions for the other coordinates, the measured phases, ^A^(^) andT^ ^(^) may be expressed interms of the test mass trajectory components ^(^), ^(^) and ^(^) as follows:

[0137] In one example, the path length differences Δℓ^,^,^may be assumed to be much smaller than their respective Rayleigh ranges (^^, ^^, ^^). As an example, for a 1 mm beam waist, the Rayleigh range is approximately 4 m, and path length differences between the interrogation and the reference arms may be configured to be 1 mm or less. In various embodiments, test mass 120 may be configured to move a distance in the order of 1 mm. In this regime, the inverse tan term in Eqns.10a, 10b and 10c may be linearised and the term (^(^) − Δℓ^)^+ ^^^in Eqn.10c may be approximated by ^^^(and similarly in Eqns.10a and 10b).

[0138] In this manner, simplified versions of Eqns.10a, 10b and 10b may be written as follows:

[0139] where ^(^), ^(^) and ^(^) are the test mass trajectory components defining the movement of the test mass 120.

[0140] Considering the ^ direction once again, the phase-based trajectory component ^̂(^) corresponding to the phase variation ^A^(^) may be determined in one example by multiplying by a scale factor which converts every 2` radians of phase change to an effective distance corresponding to half a wavelength. In another example, a further correction due to the Gouy phase may be incorporated into the scale factor. Accordingly, ^̂(^) may be determined as follows: ^̂(t) =bT^(G)^dV f^Eqn.12 Y

[0141] and similarly, for ^g(^) and ^g(^).

[0142] Referring again to FIG.3, following determination of the phase-based trajectorycomponents corresponding to phase variations (ie, ^A^(^), ^A^(^) and ^A^(^)), processing arrangement 300comprises one or more data processors 330 for determining first, second and third local accelerationcomponents corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components.

[0143] In one example, the one or more data processors are configured to solve for a respective test mass trajectory component based on the first, second, third phase-based trajectory components and then determine the respective local acceleration component from the respective test mass trajectory component.

[0144] Following Eqn.12, in one example the phase-based trajectory components (^g, ^g, ^̂) may then be expressed in terms of ^(^), ^(^) and ^(^) (ie, the test mass trajectory components in ^, ^ and ^ directions respectively) as follows:^̂^ are defined as offsets that encapsulate constant terms.

[0146] In one example, solving for a respective test mass trajectory component based on the first, second, third phase-based trajectory components comprises determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component and in this case determining the respective local acceleration component then comprises deriving the local acceleration component from the determined 2ndorder coefficient. This recognises that the 2ndorder coefficient of the quadratic term in any polynomial representation of the test mass trajectory component will be related by a factor of two to the local acceleration component.

[0147] In one example, to obtain an expression for the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component the test mass trajectory components are represented in polynomial form and substituted into a system of equations relating the phase-based trajectory components and the test mass trajectory components (eg, Eqns.13a, 13b and 13c) and the system of equations is solved to determine one or more parameters of the polynomial parameterised form of the test mass trajectory components including the 2ndorder coefficients.

[0148] As can be seen from the system of equations defined by Eqns.13a, 13b and 13c, the phase-based trajectory components(^g(^), ^g(^), ^̂(^))are expressed in terms of the test mass component trajectories (^(^), ^(^), ^(^)). In one example, to determine an expression for the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component, the test mass trajectory components (^(^), ^(^), ^(^)) may be expressed as follows: ^(^) =∑lm^ ^l^lEqn. 14a^(^) = ∑lm^ql^lEqn.14b ^(^) = ∑lm^^l^lEqn.14c

[0149] where the polynomials can be up to any order. In one example, the polynomials will be to second order. In another example, directed to also determining the local acceleration gradient, a fourth order polynomial may be used.

[0150] As referred to above, the local acceleration, denoted as ^(t) = (^^(^), ^^(^), ^^(t)) will be directly related to the second order terms (^^, q^, ^^) of any polynomial expansion of the test mass component trajectories (^(^), ^(^), ^(^)) by the relationship (^^, ^^, ^^) = 2(^^, q^, ^^).

[0151] As Eqns.13a-13c contain terms such as ^^(^), ^^(^) or ^^(^) it is noted that the square of any of the polynomials in Eqns.14a, 14b or 14c may be written in the form of:

[0152] The product between a square and another polynomial can also be expressed as follows:

[0153] Substituting the polynomial expansions (to 2ndorder) for ^(^), ^(^) and ^(^) of Eqns. 14a-14c into Eqn.13c (as an example) then ^̂(^) may be rewritten as:Eqn.15

[0154] Eqn.15 may be simplified in a number of ways. In one example, an assumption can be made that when test mass 120 is at rest (^ = 0) then the three pairs of interrogation and reference component beams are aligned for each of the orthogonal directions such that there is no transverse displacement of the beams on the respective intensity detectors.

[0155] This implies that at ^ = 0, ^(0) = ^(0) = ^(0) = 0 which further implies that for the zeroth order coefficients ^^= q^= ^^= 0. Longitudinal displacement is accounted for in Δℓ^and ^̂^(and associated quantities for other dimensions). Following this, Eqn.15 may then be simplified as follows:Eqn.16

[0156] In one example, the phase-based trajectory components (^g(^), ^g(^), ^̂(^)) are also represented in polynomial form: ^g(^) = ∑l^gl^l^g(^) =∑l qAl^l^̂(^) = ∑l^̂l^l

[0157] and the polynomial coefficients of the phase-based trajectory components can be expressed in terms of the polynomial coefficients of the test mass trajectory components. In this example the previous system of equations (including equivalent expansions ^g(^) and ^g(^) similar to Eqn.16) may then be written as:Eqn.17

[0158] As can be seen from Eqn.17, the phase-based trajectory first-order coefficients are equal to the first-order coefficients of the test mass trajectories (ie, ^g<= ^<, qA<= q<and ^̂<= ^<). For thegeneral case, this leaves a system of six equations for ^g^, ^g~, qA^, qA~, ^̂^, and ^̂~ with six unknowns(Δℓ^, Δℓ^, Δℓ^) and (^^, q^, ^^) where in this example the second order coefficients (^^, q^, ^^) aredesired to be determined. In this case, a linear least-squares fitting algorithm (as an example) may be used to fit a 3rd order polynomial to each of (^g(^), ^g(^), ^̂(^)) which will determine values for the coefficients of the polynomial parameterised form of the phase based trajectory components ^gl, qAl, and ^̂l. In one example, the system of six equation is solved numerically by searching for the vector ^ = K^^, q^, ^^, Δℓ^, Δℓ^, Δℓ^L that solves the following vector equation:

[0159] A non-linear equation solver will typically require a starting value of ^, denoted ^^. Inone example, the starting point can be chosen to be ^^ = (^g^, qA^, ^̂^, 0, 0, 0).

[0160] In this example, using the third-order term in the polynomial expansion for the phase- based trajectory components may be sensitive to whether there are sufficiently large accelerations in all three axes such that the third order coefficients {^g~, qA~, ^̂~} are measurably different from zero because otherwise there is no way to infer the path length differences {Δℓ^, Δℓ^, Δℓ^} from Eqn.18.

[0161] As discussed previously, once Eqn.14 has been solved for (^^, q^, ^^) then the localacceleration vector will be given by ^ =K^^, ^^, ^^L= 2(^^, q^, ^^)

[0162] In another example, where the path length differences (Δℓ^, Δℓ^, Δℓ^) are known, then a second-order polynomial may be fitted to each phase based trajectory component (^g(^), ^g(^), ^̂(^)) using a linear least-squares fitting algorithm (as an example) and the values (^^, q^, ^^) may be directly calculated from the estimated values of the coefficients as follows:^Wℓ^ = ^̂^−^[^gf'V^Af']^^h'YEqn.19c

[0163] where again the local acceleration vector will be ^

[0164] Alternatively, while the path length differences may not be precisely known precisely but instead can be bounded, then the maximum error for the different acceleration axes may be estimated. As an example, consider a measurement of gravity in the ^ direction, and let the test mass have some transverse velocity; then the correction to the true value of ^ is given by the following expression:

[0165] In one example, an accelerometer in accordance with the present disclosure is constructed so that the respective path length differences (Δℓ^, Δℓ^, Δℓ^) are configured to be less than 1 mm. In another example, the accelerometer may be constructed so that the respective path length differences (Δℓ^, Δℓ^, Δℓ^) are configured to be less than 100 µm. As can be seen, where the Rayleigh ranges are large enough then the effects of path length differences may be ignored entirely depending on the level of precision required.

[0166] In another example, and depending on the circumstances, an assumption could be made that there is no velocity in the ^ and ^ directions implying ^<= q<= 0, in which case there would be no need to determine the path length difference.

[0167] Referring now to FIG.6, there is shown a perspective view of an example test mass 600 comprising first, second and third reflectors in accordance with some embodiments. In various examples, test mass 600 may correspond to test mass 120 shown in FIG.1.

[0168] In this example, test mass 600 has a cubic configuration or profile with each of the reflectors 610, 620, 630 implemented as a respective open corner cube reflector inset into three orthogonal faces of the test mass 600. A corner cube reflector has the property that any light ray reflected from the corner cube is parallel to its incident ray, although the two rays will be offset from each other. As illustrated in FIG. 6, corner cube reflector 610 will reflect in the ^ direction, corner cube reflector 620 will reflect in the ^ direction and corner cube reflector 630 will reflect in the ^ direction.

[0169] Throughout this specification, the optical centre of a reflector is defined as the point about which the reflector can rotate where no additional phase is added to the optical field. In the example of an open corner cube reflector, the optical centre coincides with the apex of the reflector. In this example, the optical centres of the first, second and third reflectors 610, 620, 630 (ie, their vertices) are configured to be co-located to the centre of mass 650 of the test mass 600 (ie, as shown in FIG.6).

[0170] While example test mass 600 has an overall external cubic configuration or profile it will be appreciated that the corner cube reflectors could be implemented in a test mass having other external configurations where the centre of mass of the test mass is co-located with the optical centre of the corner cube reflectors. In one example, the test mass may have an overall external spherical configuration with the retroreflectors inset into the spherical surface of the test mass. In this example, and in principle, any shape for the overall external configuration may be adopted as long as the requirement that the optical centres coincide with the COM is met. In one example, the external configuration may be a cylinder. In another example, the external configuration may be a triangular prism.

[0171] In another example, where the test mass is expected to travel a relatively small distance relative to its size, the three reflecting faces of the corner cube may be replaced by an equivalent three mirror arrangement comprising three small mirrors whose individual orientations and positions are selected such that a laser beam incident on one of the mirrors from a limited range of incident angles is appropriately retroreflected from the combination of the three mirrors. Where the test mass travels a small distance relative to its size, the laser beams incident on the corner cube retroreflectors will only sample small parts of the reflecting surfaces implying that the rest of these surfaces are extraneous and may be effectively removed. Accordingly, in these circumstances an equivalent three mirror arrangement may be substituted for the corner cube retroreflector where three discrete mirrors are arranged in positions and orientations such that the mirror surfaces are coincident with the equivalent reflecting surfaces of the corner cube retroreflector as a result effectively reproducing the reflective operability of the corner cube retroreflector.

[0172] It is instructive to understand the potential effects of rotation of a test mass, comprising in this example a retroreflector, on any phase measurement. Referring now to FIG.7, there is shown a diagram 700 showing the effect of a rotation of a corner cube reflector 710 about its centre of mass (COM) to rotated position 710´. For the purposes of this discussion, the COM is assumed to be a height ℎ above the apex 720 of the corner cube reflector 710.

[0173] In general, a rotation of a corner cube reflector about an arbitrary point imparts both a path length difference and a translation of the beam. To understand the effects of rotation of a corner cube reflector two properties of these types of reflectors are employed. The first is that the optical path length of any ray reflected from a corner cube reflector is equal to the optical path length of a ray that travelsalong the symmetry axis of the corner cube reflector and reflects from its apex. The second property is that the transverse coordinates of the reflected ray, relative to the incident wavevector, are mirrored across the apparent position of the corner cube reflector apex. Combined, this implies that if a corner cube reflector rotates by an angle ^ about a point ℎ above or below the apex of the CC, then the optical path length changes by: ΔℓD^G= ℎ(1 − cos^) Eqn.21

[0174] and the beam, incident at a distance ^ from the symmetry axis of the unrotated corner cube reflector 710, is translated by: Δ^ = 2ℎsin^ Eqn.22

[0175] relative to the unrotated case.

[0176] In free fall, the corner cube reflector may only rotate about its centre of mass at a fixed angular frequency Ω where ^ = Ω^. There may, therefore, be an optical path length change of: ΔℓD^G= ^ℎ(1 − cosΩ^) ≈^ ^ ^^ Ω ^ Eqn.23

[0177] which gives rise to a “fictitious” acceleration signal of ^^ = ℎΩ^.

[0178] There is also a translation of the beam of: Δ^ ≈ 2ℎΩ^ Eqn.24

[0179] which will affect the phase-based trajectory estimate as a fictitious velocity of ℎΩ.

[0180] In one aspect, an accelerometer in accordance with the present disclosure includes a test mass rotation measuring arrangement that is operative to determine the rotation of the test mass while it is freely moving.

[0181] Referring now to FIG.8, there is shown a figurative view of an interferometer arrangement 800 for measuring in a selected direction the interferometric optical power variation associated with movement of a test mass in accordance with some embodiments. In various examples, interferometer arrangement 200 may correspond to one or more of the interferometer arrangements 140, 150 and 160 shown in FIG.1.

[0182] By comparison with interferometer arrangement 200, interferometer arrangement 800 comprises a reflector 860 that is configured to reflect the interrogation beam component 840B originating from beam 840 emitted by laser 810 that is reflected by the retroreflector 890 back along the incident path along interrogation arm 880 to meet the reflected reference beam component 840A of reference arm 870 which in this example is reflected from reflector 830 at beam splitter 820. In this example, retroreflector 890 is aligned to have a surface that is normal to the interrogation beam component 840B that is reflected by the retroreflector 890 so that the interrogation beam component 840B is reflected and travels back along the exact same path following reflection by reflector 860. This results in there being no displacement of the beam in the transverse direction when there is movement of the test mass 120 in the transverse direction. Additionally, the effective path length of the interrogation arm is now increased and there is a factor of two gain in the phase change for a change in the test body position.

[0183] In the ideal case, motion of the test mass in the axes orthogonal to the measurement direction does not cause translation of the measurement laser beam. Under these conditions, Eqn.4 may be rewritten as: ^(^)^:^^ ,-^^,3^ = ^^,^+ ^^,^− cos (ℓ (^)^ℓ-)';<V 3^^'YEqn.25

[0184] (ie, Eqn.4 with ^^= 0). Similar equations apply to the ^ and ^ directions. Measuring the acceleration proceeds in a similar manner where the test mass is launched and the measured phasevariation are determined. In this case, the measured phases ^A(^), ^A^(^)may be expressedterms of the component trajectories ^(^), ^(^) and ^(^) of the test mass as follows:

[0185] The respective phase variations may then be converted into phase-based trajectory components (^g, ^g, ^̂) in accordance with a modified version of Eqn.12, given by:Eqn.26d

[0186] which then yields: ^g(^) = ^(^) + ^g^Eqn.27a ^g(^) = ^(^) + ^g^Eqn.27b ^̂(^) = ^(^) + ^̂^. Eqn.27c

[0187] As can be seen by inspection, without cross-coupling of the translational motion to the measurement, the respective phase-based trajectory component is equivalent to the true test mass trajectory component with a constant offset. In this example, the acceleration components may be determined from the phase-based trajectory components by fitting a second-order polynomial to each of phase-based trajectory components ^g, ^g, and ^̂ as this will be equivalent to determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component and then multiplying the determined second-order coefficients by a factor of two to calculate the acceleration components.

[0188] Referring back to FIG.1, in other examples the processing arrangement 170 for processing the first, second and third interferometric optical power variations to determine the local acceleration does not involve an intermediate determination by a phase measuring apparatus of the phase variation corresponding to the interferometric optical power variation. In one example an analogue-to- digital converter (ADC) may be employed to digitise the interferometric optical power variation signal to produce the corresponding digitised fringe pattern and a data processor is configured to process the digitised fringe pattern to determine acceleration.

[0189] In one example, the digitised interferometric fringe pattern is processed to determine the time at which the signal crosses the midpoint and then to convert the time differences to a measurement of distance through the wavelength. The pairs of times and spatial distances are then fitted to a parabola using in one example a linear least-squares algorithm to determine acceleration. As would be appreciated, this process is especially applicable where the test mass traverses a significant distance resulting in numerous zero crossings in the digitised fringe pattern.

[0190] In another example, suited to where the test mass is launched and then allowed to fall, the optical power variation ^(^) is fitted to a simplified form of Eqn.4 where the path length of theinterrogation arm is replaced with its parabolic trajectory from which the acceleration may then be determined. As would be appreciated, if the digitised fringe pattern follows a simple chirped interference pattern (ie, minimal low frequency noise) then this will assist in the determination of acceleration.

[0191] Referring now to FIG. 9, there is shown a figurative view of a test mass chamber 900 configured for the measurement of local gravitation and comprising a launching arrangement 950 to launch a test mass 920 in a launch direction opposed to the direction of gravitational acceleration (as indicated by ^) in accordance with some embodiments. In this example, test mass 920 comprises a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other and wherein the optical centres of the first, second and third reflectors are co-located to a centre of mass of the test mass as has been previously discussed. An example test mass that may be employed is shown in FIG.6. In various examples, test mass chamber 900 may be combined with the interferometric and data processing arrangements that have been previously described to determine interferometric intensity arrangements in the three orthogonal axes that correspond to three reflectors associated with the movement of the test mass under the action of gravitational acceleration to determine gravitational acceleration.

[0192] In this example, test mass chamber 900 is arranged so that a selected direction of the first, second and third directions is aligned with a direction of gravitational acceleration. As illustrated in FIG. 9, the selected direction corresponds to the ^ direction. In other examples, test chamber 900 may be oriented in a predetermined orientation with the expected direction of gravitational acceleration. In a further example, test chamber 900 may be oriented so that the expected gravitational components in the ^, ^ and ^ directions are approximately equal.

[0193] In one example, test mass 920 comprises a small cube formed of a suitable metal such as aluminium with side lengths on the order of 1 cm, and a mass that is on the order of 10-3kg. Into three of the orthogonal faces of the cube will be inscribed an open corner cube reflector such as shown in FIG.6. In one example, the reflection faces of the corner cube reflectors will be suitably coated or plated (eg, gold plated) to improve reflectivity. In this example, the apexes of these corner cube reflectors will coincide at roughly the centre of the cube and the opposite three faces will have material removed or added in order to ensure that the COM of the test mass coincides with the coincident apexes of the corner cube reflectors to reduce the effects of rotations on measurements of gravity. In another example, the test mass may be formed of a non-conductive material such as glass or ceramic whose reflection faces may be suitably coated or plated to improve reflectivity. In another example, the open corner cube reflector is replaced by the equivalent three mirror arrangement as described above.

[0194] In one example, test mass 920 comprises a three-point contact 922 on its bottom surface 921 which mates with a complementary contact 952 located on the top surface 951 of launchingarrangement 950. In one example, launching arrangement 950 is configured as an electro-mechanical launcher such as a piezoelectric transducer (PZT) which under the action of a high-voltage pulse applied to the PZT will cause a sudden change in the vertically-oriented strain and launches test mass 920 off the launching arrangement 950 and then enter free fall.

[0195] In various examples in accordance with the present disclosure, the test mass may configured to have a mass less than 0.01 kg. In one example, the test mass is configured to have a mass less than 0.005 kg. In yet another example, the test mass is configured to have a mass between 0.001 kg and 0.005 kg. In yet another examples, the mass of the test mass may be configured to be less than 0.05 kg. In other examples, the mass of the test mass may be greater than 0.05 kg.

[0196] In one example, the test mass is launched to a maximum height of approximately 1 mm which corresponds to a free-fall time of approximately 30 milliseconds. As an example, a test mass of approximately 0.01 kg will require an initial launch speed of 140 mm / s requiring a launching force of approximately 1 N applied over a time of 100 µs. In other examples, the test mass may be launched to a maximum height of approximately 1 cm with a correspondingly increased launching force required.

[0197] As would be appreciated, the combination of a relatively small mass for test mass 920 coupled with a small launch height will reduce the impact forces when the test mass 920 returns to the launching arrangement 950 under the action of gravitational acceleration. As will be appreciated, test mass chamber 900 will have suitable optical windows so that the interferometer arrangements can optically couple to the reflectors of the test mass.

[0198] In other examples, accelerometers and methods for measuring local acceleration in accordance with the present disclosure may be based on dropping a test mass.

[0199] Defining the linear size of the test mass as the quantity ^, then in one example, the test mass may be launched vertically upwards a distance ^ = ^^ where 0 < ^ ≤ 1 (ie, an upwards distance less than or equal to the linear size of the test mass), and the trajectory of the test mass is measured for the entire duration that the test mass is freely falling.

[0200] In another example, the test mass may be dropped a distance ^ = ^^ where again 0 < ^ ≤ 1 (ie, a dropped distance less than or equal to the linear size of the test mass), and the trajectory of the test mass is measured for the entire duration the test mass is freely falling.

[0201] In yet another example, the test mass may be launched vertically a distance ^ > ^, but the trajectory of the test mass is measured only in a region about the apex of the vertical motion where the test mass’s displacement about the apex is ^^ where 0 < ^ ≤ 1.

[0202] In a further example, the test mass may be launched or dropped vertically a distance ^ = ¢^ with ¢ > 1, and the vertical motion is measured over the entire distance ^, but any horizontal trajectory or motion of the test mass is measured in a region about the apex of the vertical motion where the test mass’s displacement about the apex is ^^ where 0 < ^ ≤ 1.

[0203] As would be appreciated, in various examples the measured motion of the test mass may be selectively measured over a distance corresponding to less than the linear size of the test mass.

[0204] As discussed previously, in various examples the test mass chamber may be configured to provide a reduced pressure environment As would be appreciated, residual pressure in the test mass chamber may affect the accuracy of the measurement of acceleration or gravity due to the drag force on the test mass.

[0205] The drag force on a test mass falling through the rarefied gas in a vacuum is £¤^^¥= where:¦¥^§is the mass density of the residual gas, ¨ is the cross-sectional area of the test mass, © is the velocity of the test mass, and Δ©¥^§is the rms velocity spread of the molecules in the gas.

[0206] If the drag force is £¤^^¥= #£¥^^­^G^= #¦^^¤^¨^^ with ¦^^¤^the density of the test mass and # the maximum allowable error in the measurement of gravity due to drag, then the residual vacuum pressure ^ in one example may be configured to be less than

[0207] In one example, the residual gas in the test mass chamber (eg, test mass chamber 900) is predominantly molecular nitrogen and the temperature is 300 K so that Δ©¥^§= 300 m / s, ¦^^¤^= 2700 kg / m~, # = 10&µ, ^ = 10&^, and ^ = 1 cm, implying a residual gas pressure ^ ≤ 0.0045 Pa (ie, a residual gas pressure of less than 4.5 ×10-3Pa).

[0208] In another example, ^ = 10&~, so that residual gas pressure ^ ≤ 0.014 Pa (ie, a residual gas pressure of less than 1.4 ×10-2Pa).

[0209] In yet another example, ^ = 1, so that ^ ≤ 4.4 × 10&4Pa (ie, a residual gas pressure of less than 4.4 ×10-4Pa). In yet another example, ¦^^¤^= 20,000 kg / m~, so that ^ ≤ 0.0033 Pa (ie, a residual gas pressure of less than 3.3 ×10-3Pa).

[0210] Accordingly, in various examples test mass chamber 900 may be configured to have a reduced pressure of less than 10-2Pa as previously described to reduce air drag.

[0211] For satellite applications (see description below), where there is reduced concern about any potential impact between the test mass and any test mass chamber, a much larger mass may be used. In one example, the mass of the test mass may be in the order of 1 kg.

[0212] In one example directed to the measurement of local gravity, the effect of gravitational gradients may be included. The gravitational gradient may be represented by the second-rank tensor Γ with components= {^, ^, ^} and the local gravitational field or acceleration is denoted by » = [^^, ^^, ^^]¼.

[0213] The trajectory of a test mass in a gravitational field with a weak gradient is given by: ^(^) = ^ 1^+ ½^^ + ^ 1 ~ 1 42[» + Γ^^]^ +6Γ½^^ +24Γ»^≈ ^^+ ½^^ + 1 ^ 1 ~ 1 42»^ +6Γ½^^ +24Γ»^Eqn.28

[0214] where » + Γ^^≈ » (ie, is approximated as the local acceleration). Typically, the off- diagonal elements of Γ are negligible for density anomalies that are far from the accelerometer, so in the following only the diagonal elements of Γ need be considered which gives trajectories:

[0215] where ½^= [©^, ©^, ©^] is the initial velocity of the test body on launch, and the trajectory along each axis is independent of the trajectories along the other axes.

[0216] Regardless of whether the simultaneous measurement of all three components of the acceleration vector » reduces systematic errors associated with the translation of the test body in directions orthogonal to any particular measurement, measuring the initial velocity of the test mass allowsfor compensation of the Coriolis effect. For a test body in a uniformly rotating frame with angular velocity ¿, the Coriolis acceleration is: À9= 2¿ × ½.

[0217] Consider the case of an accelerometer being positioned on the surface of the Earth at a latitude of ^, and the axes of the accelerometer are aligned such that the ^ axis is pointed east, the ^ axis is pointed north, and the ^ axis is pointed away from the center of the Earth. Defining ¿ such that it has magnitude Ω and is oriented along the axis of the spinning Earth, the Coriolis acceleration is follows: À9= 2Ω[©^(cos^ − sin^)Ág − ©^sin^Âg + ©^cos^Ãg] Eqn.30a À9= 2Ω^©^Ág + 2Ω^©^Âg + 2Ω^©^Ãg Eqn.30b

[0218] with Ω^= Ω(cos^ − sin^) Eqn.31a Ω^= −Ωsin^ Eqn.31b Ω^= Ωcos^. Eqn.31c

[0219] The trajectory of the test body, including gravitational gradients, is then:

[0220] An initial horizontal velocity of ©^= 1 mm / s at ^ = −35∘on the surface of the Earthwill cause a systematic error of 2Ωcos^©^ ≈ 10&Å m / s2 to the true acceleration in the vertical direction^^. This choice of coordinates is chosen for calculational convenience and is not a requirement on theorientation of the accelerometer as a coordinate transformation to the measured trajectories may beapplied to change to a coordinate system aligned as stated above. An initial vertical launch speed of ©^ =44 mm / s (as an example) will cause a systematic error of 1.6 × 10-6 m / s2 to the measurement of ^^.

[0221] The coefficients of the test mass trajectory components may be read off from Eqn.32 as: ^^ = ^^q^ = ^^^^ = ^^^< = ©^q< = ©^Eqn.33

[0222] In this example, in order to include the gravitational gradient and Coriolis effect, polynomial expansions up to fourth order in Eqn.14 are used. This will then modify Eqn.15, but when the approximation that ^(0) = ^(0) = ^(0) = 0 is applied that yielded Eqn.16 in this case the important change will be that Eqn.17 is modified such that the third order coefficients are now:

[0223] By analogy with Eqn.18, now have the following vector equation incorporating the Coriolis effect and gravitational gradients:

[0224] and once again a search is carried out for the vector ^ =solves this vector equation.

[0225] Solving Eqn.35 requires knowledge of the values of Γ^^, Γ^^, Γ^^, Ω^, Ω^, and Ω^. The quantities Ω^, Ω^, and Ω^are related to the Earth’s rate of rotation and the latitude of the accelerometer, where the rate of rotation is accurately measured by metrology labs around the world and is independent of location and the latitude can be measured accurately using a global navigation satellite system (GNSS) as an example. The gravity gradients are approximately constant across much of the Earth, with Γ^^≈ 3 ×

[0226] Once the solution vector ^ has been calculated, compensated 2ndorder coefficients (^^Æ, q^Æ, ^^Æ) for determining the respective local acceleration may be determined from the measured first and second order coefficients that includes a compensation factor characterising the effect of the Coriolis force at the measurement latitude as follows: ^^Æ= ^^− Ω^^<Eqn.36a q^Æ= q^− Ω^^<Eqn.36b^^Æ= ^^− Ω^^<Eqn.36c

[0227] Ttrue accelerations can then be calculated from the compensated 2ndorder coefficients as follows: ^^= 2^^ÆEqn.37a ^^= 2q^ÆEqn.37b ^^= 2^^ÆEqn.37c

[0228] As discussed previously, using the third-order term in the polynomial expansion for the phase-based trajectory components may be sensitive to whether there are sufficiently large accelerations in all three axes such that the third order coefficients {^g~, qA~, ^̂~} are measurably different from zero because otherwise there is no way to infer the path length differences {Δℓ^, Δℓ^, Δℓ^} from Eqn.35 above.

[0229] Once again if the path length differences (Δℓ^, Δℓ^, Δℓ^) are known, then a second-order polynomial may be fitted to each phase based trajectory component (^g(^), ^g(^), ^̂(^)) using a linear least- squares fitting algorithm (as an example) and the values (^^, q^, ^^) and (^<, q<, ^<) may be directly calculated from the estimated values of the coefficients in accordance with Eqn.19 and then substituted into Eqn.37 above to obtain the gravitational acceleration components.

[0230] Referring now to FIG. 10A, there is shown a flowchart of a method 1000 for determining local acceleration in accordance with some embodiments. In one example, method 1000 may be operated in combination with accelerometer 100 as illustrated in FIG.1.

[0231] At step 1010, method 1000 comprises configuring a test mass to freely move, where the test mass includes a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other. At step 1020, method 1000 comprises measuring a first interferometric optical power variation associated with movement of the test mass by a first interferometer arrangement optically coupled to the first reflector. At step 1030, method 1000 comprises measuring a second interferometric optical power variation associated with movement of the test mass by a second interferometer arrangement optically coupled to the second reflector. At step 1040, method 1000 comprises measuring a third interferometric optical power variation associated with movement of the test mass in the third direction by a third interferometer arrangement optically coupled to the third reflector. At step 1050, method 1000 comprises determining by a dataprocessor the local acceleration based on the first, second and third interferometric optical power variations.

[0232] Referring now to FIG.10B, there is shown a flowchart of a method 3000 for measuring an optical power variation in accordance with some embodiments. In one example, one or more of method steps 1020, 1030, 1040 illustrated in FIG.10A may be implemented using method 3000. Additionally, in various examples, method 3000 may be operated with respect to interferometer arrangements 200, 201, 202 illustrated in FIGS.2A-2C.

[0233] At step 3010, method 3000 comprises splitting a laser beam into a reference beam component and an interrogation beam component. Step 3020 then comprises reflecting the interrogation beam component from a respective reflector of the test mass to form a reflected interrogation beam component and step 3030 comprises configuring the reference beam component to interfere with the reflected interrogation beam component. Finally step 3040 comprises measuring the interferometric optical power variation resulting from the interference of the reference beam component and the reflected interrogation beam component.

[0234] Referring now to FIG.10C, there is shown a flowchart of a method 4000 for processing first, second and third interferometric optical power variations to determine a local acceleration in accordance with some embodiments. In one example, method step 1050 illustrated in FIG.10A may be implemented using method 4000. Additionally, in one example, method 4000 may be operated with respect to processing arrangement 300 illustrated in FIG.3.

[0235] At step 4010, method 4000 comprises determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations. Step 4020 then comprises determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations while step 4030 comprises determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components.

[0236] Referring now to FIG.10D, there is shown a flowchart of a method 5000 for determining a phase variation in accordance with some embodiments. In one example, method step 4010 illustrated in FIG.10C may be implemented using method 5000. Additionally, in one example, method 5000 may be operated with respect to phase measuring apparatus 500 illustrated in FIG.5.

[0237] At step 5010, method 5000 comprises modulating a respective reference beam component of a respective interferometer arrangement by a modulation frequency. Step 5020 then comprises forming an in-phase interferometric optical power variation signal by multiplying a respectiveinterferometric optical power variation by a first periodic modulation signal based on the modulation frequency and step 5030 comprises forming a quadrature-phase interferometric optical power variation signal by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees;

[0238] At step 5040, method 5000 comprises filtering by a low pass filter both the in-phase interferometric optical power variation signal and the quadrature-phase interferometric optical power variation signal and step 5050 comprises processing the filtered in-phase interferometric optical power variation signal and the filtered quadrature-phase interferometric optical power variation signal to determine the respective phase variation corresponding to the respective interferometric optical power variation.

[0239] Referring now to FIG.10E, there is shown a flowchart of a method 6000 for determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components in accordance with some embodiments. In one example, method step 4030 illustrated in FIG.10C may be implemented using method 6000. Additionally, in one example, method 6000 may be implemented using processors 320, 330 illustrated in FIG.3.

[0240] At step 6010, method 6000 comprises determining a respective test mass trajectory component based on the first, second and third phase-based trajectory components and step 6020 comprises determining the respective local acceleration component from the respective test mass trajectory component. In one example, determining the respective test mass trajectory component based on the first, second and third phase-based trajectory components comprises determining a 2nd order coefficient of a polynomial representation of the respective test mass trajectory component. Determining the respective local acceleration component then comprises deriving the local acceleration component from the determined 2nd order coefficient.

[0241] In one example, determining a 2nd order coefficient of the polynomial representation of the respective test mass trajectory component comprises representing the test mass trajectory components in polynomial form and substituting the polynomial parameterised forms of the test mass trajectory components into a system of equations relating the phase based trajectory components and the test mass trajectory components; and solving the system of equations to determine one or more coefficients of the polynomial parameterised form of the test mass trajectory components including the 2nd order coefficients (eg, see Eqn.18)

[0242] In one example, the phase based trajectory components are represented in polynomial form and values for the coefficients of the polynomial parameterised forms are determined from thedetermined phased based trajectory components. In one example, determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component comprises determining values for the coefficients of polynomial representations of the first, second and third phase-based trajectory components based on the determined first, second and third phase-based trajectory components (eg, see Eqn.17).

[0243] In another example, determining the respective local acceleration component from the respective test mass trajectory component further comprises compensating the determined 2ndorder coefficient of the polynomial representation with a compensation factor characterising the effect of the Coriolis force at a measurement latitude to form a compensated 2ndorder coefficient and determining the respective local acceleration component comprises deriving the local acceleration component from the compensated 2ndorder coefficient (eg, see Eqns 37a, 37b, and 37c).

[0244] An accelerometer or method for determining local acceleration formed in accordance with the present disclosure may be readily adopted to a construct a small or portable form factor device while still maintaining a high level of precision. As an example, an accelerometer directed to the measurement of local gravitational acceleration (ie, an absolute gravimeter) may incorporate a test mass having a mass of less than 0.01 kg which is cyclically launched or dropped a distance of less than 1 mm. As would be further appreciated, conventional current laser technology and data processing capability may be utilised to implement an accelerometer in accordance with the present disclosure. In implementations requiring a low pressure test mass chamber, conventional chemical getter or small ion pumps may be employed.

[0245] Referring now to Figure 11, there is shown an end-on figurative view of an accelerometer 1100 deployed in an unmanned aerial vehicle (UAV) or drone configuration in accordance with some embodiments. In this example, the UAV or drone 1150 comprises a propulsion arrangement 1160 in the form of a multicopter having a number of spaced apart propellers 1161 supporting a drone body or housing 1170 which typically contains the communication, power supply and guidance systems of the drone 1150 (not shown). Drone based accelerometer 1100 in this example comprises a test mass chamber 1190 incorporating a test mass 1191 and launching arrangement 1192, interferometer arrangements 1195, laser source arrangement 1196 and associated processing arrangement 1197 in accordance with the present disclosure and configured to reduce the overall weight, size and power requirements.

[0246] In one example, test mass chamber 1190 may be in the form of the test mass chamber 900 illustrated in FIG.9. While in this example, the test mass chamber 1190 is shown oriented with one measurement direct selected to be in the expected direction of gravitational acceleration, in other embodiments test mass chamber 1190 is oriented at a predetermined direction offset from the expected direction of gravitational acceleration.

[0247] In this example, accelerometer 1100 further comprises an optional vibration isolation system 1145 that operates to reduce the transfer of low frequency vibrations in one or more of the orthogonal measurement directions. In one example, the vibration isolation system is a passive vibration system such as a mass-spring damping system. In another example, the vibration isolation system is an active vibration isolation system having a vibration sensor, actuator and feedback control system.

[0248] As would be appreciated, drone base accelerometer 1100 may be flown to a selected location where the drone 1150 may be commanded to hold a substantially stationary position by virtue of the drone control system. As would be appreciated, this provides a cost effective and convenient way of obtaining local gravitation measurements in difficult to access areas. Additionally, multiple drone based accelerometers 1100 may be flown to a given location allowing multiple measurements of the local gravitation to be taken. In other examples, an autonomous drone based accelerometer 1100 may be schedule to take local gravitational measurements in accordance with a predetermined schedule and without operator intervention.

[0249] In another example, an accelerometer in accordance with the present disclosure may be deployed in a satellite based device. Compared to the drone based accelerometer, which may require vibration isolation, it is unlikely that a satellite based accelerometer will require vibration isolation on the basis that the accelerometer is not measuring gravity but local acceleration. Additionally, while weight is important for a drone-based accelerometer device, it is not as important as for the satellite based device.

[0250] By comparison with the drone based accelerometer for measuring local gravitation there is no longer a need for a launch mechanism, as the accelerometer will be in free-fall along with the mass. However, a satellite based accelerometer is also likely be able to rotate freely, which means that it is possible for the test mass chamber to rotate with respect to the test mass in such a way that the interferometer beams may no longer impinge on the reflectors of the test mass.

[0251] In one example, the test mass chamber may be geometrically configured to prevent the test mass from rotating to the extent that the interferometer arrangements are not optically coupled to the reflectors. In one example, this could be achieved by reducing the size of the test mass chamber and / or by having structures inside the test mass chamber that prevent excessive rotation. In another example, in order to prevent the test mass from adhering to the side of the test mass chamber, the entire accelerometer may be attached to one or more piezo-electric transducers that can function to “kick” the device in such a way that the test mass is dislodged from the sides of the test mass chamber. In another example, in order to prevent the test mass from adhering to the side of the test mass chamber, external electric fields may be used to reposition the test mass in the test mass chamber between measurements of acceleration.

[0252] In another example, and as described above, as the impact of test mass on the test mass chamber is no longer a limiting factor in the precision of the accelerometer then the test mass may instead have significant weight such as in the order of 1 kg. A consideration may be if the test mass is too light then the radiation pressure from the interferometer arrangements may impart a non-negligible acceleration onto the test mass giving rise to a fictitious acceleration signal.

[0253] In various embodiments, the satellite measurement of acceleration may be significantly more precise than that of an equivalent terrestrial measurement as the interrogation time is no longer limited by terrestrial gravity. Instead, generally speaking, individual measurements may be carried out over several seconds, as a result improving the precision to about 10-11m / s2in one second of measurement time.

[0254] The methodologies described in the present disclosure may be implemented by various means depending upon applications according to particular examples. For example, such methodologies may be implemented in hardware, firmware, software, optical componentry, or combinations of these implementation aspects. In a hardware implementation, for example, a processing unit may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, electronic devices, signal generators, analogue-to-digital convertors (ADCs) or other devices units designed to perform the functions as described.

[0255] Some portions of the detailed description included in the present disclosure are presented in terms of algorithms or symbolic representations of operations on binary digital signals stored within a memory of a specific apparatus or special purpose computing device or platform. In the context of this particular disclosure, the term specific apparatus or the like includes a general purpose computer once it is programmed to perform particular operations pursuant to instructions from program software. Algorithmic descriptions or symbolic representations are examples of techniques used by those of ordinary skill in the signal processing or related arts to convey the substance of their work to others skilled in the art. An algorithm is here, and generally, is considered to be a self-consistent sequence of operations or similar signal processing leading to a desired result.

[0256] In this context, operations or processing involve physical manipulation of physical quantities. Typically, although not necessarily, such quantities may take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared or otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to such signals as bits, data, values, elements, symbols, characters, terms, numbers, numerals, or the like. It should be understood, however, that all of these or similar terms are to be associated with appropriate physicalquantities and are merely convenient labels. Unless specifically stated otherwise, as apparent from the discussion herein, it is appreciated that throughout this specification discussions utilising terms such as "processing," "computing," "calculating," "determining" or the like refer to actions or processes of a specific apparatus, such as a special purpose computer or a similar special purpose electronic computing device. In the context of this specification, therefore, a special purpose computer or a similar special purpose electronic computing device is capable of manipulating or transforming signals, typically represented as physical electronic or magnetic quantities within memories, registers, or other information storage devices, transmission devices, or display devices of the special purpose computer or similar special purpose electronic computing device.

[0257] The reference to any prior art in this specification is not, and should not be taken as, an acknowledgement or any form of suggestion that such prior art forms part of the common general knowledge.

[0258] It will be understood that the terms “comprise” and “include” and any of their derivatives (e.g. comprises, comprising, includes, including) as used in this specification, and the claims that follow, is to be taken to be inclusive of features to which the term refers, and is not meant to exclude the presence of any additional features unless otherwise stated or implied.

[0259] In some cases, a single embodiment may, for succinctness and / or to assist in understanding the scope of the disclosure, combine multiple features. It is to be understood that in such a case, these multiple features may be provided separately (in separate embodiments), or in any other suitable combination. Alternatively, where separate features are described in separate embodiments, these separate features may be combined into a single embodiment unless otherwise stated or implied. This also applies to the claims which can be recombined in any combination. That is a claim may be amended to include a feature defined in any other claim. Further a phrase referring to “at least one of” a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover: a, b, c, a-b, a-c, b-c, and a-b-c.

[0260] It will be appreciated by those skilled in the art that the disclosure is not restricted in its use to the particular application or applications described. Neither is the present disclosure restricted in its preferred embodiment with regard to the particular elements and / or features described or depicted herein. It will be appreciated that the disclosure is not limited to the embodiment or embodiments disclosed, but is capable of numerous rearrangements, modifications and substitutions without departing from the scope as set forth and defined by the following claims.

Claims

CLAIMS 1. An accelerometer for determining a local acceleration, comprising: a test mass arranged to freely move, the test mass having a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other; a first interferometer arrangement optically coupled to the first reflector for measuring a first interferometric optical power variation associated with movement of the test mass; a second interferometer arrangement optically coupled to the second reflector for measuring a second interferometric optical power variation associated with movement of the test mass; a third interferometer arrangement optically coupled to the third reflector for measuring a third interferometric optical power variation associated with movement of the test mass; and a processing arrangement for processing the first, second and third interferometric optical power variations to determine the local acceleration.

2. The accelerometer of claim 1, wherein one or more of the first, second or third reflectors is a retroreflector.

3. The accelerometer of claim 2, wherein the retroreflector comprises an optical centre co-located with a centre of mass of the test mass.

4. The accelerometer of claim 2 or 3, wherein the retroreflector is a corner cube reflector.

5. The accelerometer of claim 1, wherein the test mass has a cubic configuration and the first, second and third reflectors comprise first, second and third corner cube reflectors inset into orthogonal faces of the test mass, and wherein optical centres of the first, second and third corner cube reflectors are co-located to the centre of mass of the test mass.

6. The accelerometer of any one of claims 1 to 5, further comprising a test mass chamber, wherein the test mass is arranged to freely move within the test mass chamber.

7. The accelerometer of claim 6, wherein the test mass chamber is configured to sustain a residual gas pressure of less than 10-2Pa.

8. The accelerometer of any one of claims 1 to 7, wherein one or more of the first, second or third interferometer arrangements comprises: a laser beam split into a reference beam component and an interrogation beam component;an interrogation arm configured so that the interrogation beam component travels along the interrogation arm and is reflected from a respective reflector of the test mass; a reference arm configured so that the reference beam component travels along the reference arm and interferes with the reflected interrogation beam component; and an intensity detector to measure the interferometric optical power variation resulting from the interference of the reference beam component and the reflected interrogation beam component.

9. The accelerometer of any one of claims 1 to 8, wherein the processing arrangement comprises: a phase measuring apparatus for determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations; one or more data processors for determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations; and one or more data processors for determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase- based trajectory components.

10. The accelerometer of claim 9, wherein the phase measuring apparatus comprises for one or more of the first, second and third interferometric optical power variations: an analogue-to-digital converter for digitising a respective interferometric optical power variation to produce a digitised fringe pattern; and a data processor for processing the digitised fringe pattern to extract the respective phase variation corresponding to the respective interferometric optical power variation.

11. The accelerometer of claim 9, wherein the phase measuring apparatus comprises for one or more of the first, second and third interferometric optical power variations: a modulator for modulating a respective reference beam component of a respective interferometer arrangement by a modulation frequency, a first signal multiplier for forming an in-phase interferometric optical power variation signal by multiplying a respective interferometric optical power variation by a first periodic modulation signal based on the modulation frequency; a second signal multiplier for forming a quadrature-phase interferometric optical power variation signal by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees; a low pass filter arrangement for filtering both the in-phase interferometric optical power variation signal and the quadrature-phase interferometric optical power variation signal; and a phase determining module operable to process the filtered in-phase interferometric optical power variation signal and the filtered quadrature-phase interferometric optical power variation signal todetermine the respective phase variation corresponding to the respective interferometric optical power variation.

12. The accelerometer of any one of claims 9 to 11, wherein the one or more data processors for determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components are configured to: determine a respective test mass trajectory component based on the first, second and third phase- based trajectory components; and determine the respective local acceleration component from the respective test mass trajectory component.

13. The accelerometer of claim 12, wherein: determining a respective test mass trajectory component based on the first, second and third phase-based trajectory components comprises determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component; and determining the respective local acceleration component comprises deriving the local acceleration component from the determined 2ndorder coefficient.

14. The accelerometer of claim 13, wherein determining the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component comprises: representing the test mass trajectory components in polynomial form and substituting polynomial parameterised forms of the test mass trajectory components into a system of equations relating the first, second and third phase-based trajectory components and the test mass trajectory components; and solving the system of equations to determine one or more coefficients of the polynomial parameterised forms of the test mass trajectory components including the 2ndorder coefficients.

15. The accelerometer of claim 14, further comprising representing the phase based trajectory components in polynomial form and determining values for the coefficients of the polynomial parameterised forms of the determined phased based trajectory components.

16. The accelerometer of claim 15, wherein: determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component comprises determining values for the coefficients of polynomial representations of the first, second and third phase-based trajectory components based on the determined first, second and third phase-based trajectory components.

17. The accelerometer of any one of claims 13 to 16, wherein determining the respective local acceleration component from the respective test mass trajectory component further comprises compensating the determined 2ndorder coefficient of the polynomial representation with a compensation factor characterising an effect of a Coriolis force at a measurement latitude to form a compensated 2ndorder coefficient and determining the respective local acceleration component comprises deriving the local acceleration component from the compensated 2ndorder coefficient.

18. The accelerometer of any one of claims 1 to 17, wherein the accelerometer is configured so that a selected direction of the first, second and third directions is aligned to a direction of gravitational acceleration.

19. The accelerometer of any one of claims 1 to 17, wherein the accelerometer is configured so that a selected direction of the first, second and third directions is at a predetermined orientation with respect to a direction of gravitational acceleration.

20. The accelerometer of claim 18 or 19, wherein the test mass chamber comprises a launching arrangement configured to launch the test mass in the selected direction.

21. The accelerometer of claim 20, wherein the launching arrangement is configured to launch the test mass a predetermined height with respect to the launching arrangement.

22. The accelerometer of any one of claims 18 to 21, wherein a mass of the test mass is less than 0.01 kg.

23. A method for determining a local acceleration, comprising: configuring a test mass to freely move, the test mass having a reflection arrangement comprising first, second and third reflectors configured to reflect in first, second and third directions that are orthogonal with respect to each other; measuring a first interferometric optical power variation associated with movement of the test mass by a first interferometer arrangement optically coupled to the first reflector; measuring a second interferometric optical power variation associated with movement of the test mass by a second interferometer arrangement optically coupled to the second reflector; measuring a third interferometric optical power variation associated with movement of the test mass by a third interferometer arrangement optically coupled to the third reflector; and processing the first, second and third interferometric optical power variations to determine the local acceleration.

24. The method of claim 23, wherein one or more of the first, second or third reflectors is a retroreflector.

25. The method of claim 24, wherein the retroreflector comprises an optical centre co-located with a centre of mass of the test mass.

26. The method of claim 24 or 25, wherein the retroreflector is a corner cube reflector.

27. The method of claim 23, wherein the test mass has a cubic configuration and the first, second and third reflectors comprise first, second and third corner cube reflectors inset into orthogonal faces of the test mass, and wherein optical centres of the first, second and third corner cube reflectors are co-located to the centre of mass of the test mass.

28. The method of any one of claims 23 to 27, wherein the test mass is configured to freely move within a test mass chamber.

29. The method of claim 28, wherein the test mass chamber is configured to sustain a residual gas pressure of less than 10-2Pa.

30. The method of any one of claims 23 to 29, wherein measuring one or more of the first, second or optical power variations comprises: splitting a laser beam into a reference beam component and an interrogation beam component; reflecting the interrogation beam component from a respective reflector of the test mass to form a reflected interrogation beam component; configuring the reference beam component to interfere with the reflected interrogation beam component; and measuring the interferometric optical power variation resulting from the interference of the reference beam component and the reflected interrogation beam component.

31. The method of any one of claims 23 to 30, wherein processing the first, second and third interferometric optical power variations to determine the local acceleration comprises: determining first, second and third phase variations corresponding to the first, second and third interferometric optical power variations; determining first, second and third phase-based trajectory components corresponding to the first, second and third phase variations; and determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components.

32. The method of claim 31, wherein determining a respective phase variation comprises: digitising a respective interferometric optical power variation to produce a digitised fringe pattern; and processing the digitised fringe pattern to extract the respective phase variation corresponding to the respective interferometric optical power variation.

33. The method of claim 31, wherein determining a respective phase variation comprises: modulating a respective reference beam component of a respective interferometer arrangement by a modulation frequency, forming an in-phase interferometric optical power variation signal by multiplying a respective interferometric optical power variation by a first periodic modulation signal based on the modulation frequency; forming a quadrature-phase interferometric optical power variation signal by multiplying the respective interferometric optical power variation by a second modulation signal corresponding to the first modulation signal phase shifted by 90 degrees; filtering by a low pass filter both the in-phase interferometric optical power variation signal and the quadrature-phase interferometric optical power variation signal; and processing the filtered in-phase interferometric optical power variation signal and the filtered quadrature-phase interferometric optical power variation signal to determine the respective phase variation corresponding to the respective interferometric optical power variation.

34. The method of any one of claims 31 to 33, wherein determining first, second and third local acceleration components corresponding to the first, second and third orthogonal directions from the first, second and third phase-based trajectory components comprises: determining a respective test mass trajectory component based on the first, second and third phase-based trajectory components; and determining the respective local acceleration component from the respective test mass trajectory component.

35. The method of claim 34, wherein: determining the respective test mass trajectory component based on the first, second and third phase-based trajectory components comprises determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component; and determining the respective local acceleration component comprises deriving the local acceleration component from the determined 2ndorder coefficient.

36. The method of claim 35, wherein determining the 2ndorder coefficient of the polynomial representation of the respective test mass trajectory component comprises:representing the test mass trajectory components in polynomial form and substituting polynomial parameterised forms of the first, second and third test mass trajectory components into a system of equations relating the first, second and third phase-based trajectory components and the test mass trajectory components; and solving the system of equations to determine one or more coefficients of the polynomial parameterised forms of the test mass trajectory components including the 2ndorder coefficients.

37. The method of claim 36, further comprising representing the phase based trajectory components in polynomial form and determining values for the coefficients of the polynomial parameterised forms of the determined phased based trajectory components.

38. The method of claim 37, wherein: determining a 2ndorder coefficient of a polynomial representation of the respective test mass trajectory component comprises determining values for the coefficients of polynomial representations of the first, second and third phase-based trajectory components based on the determined first, second and third phase-based trajectory components.

39. The method of any one of claims 35 to 38, wherein determining the respective local acceleration component from the respective test mass trajectory component further comprises compensating the determined 2ndorder coefficient of the polynomial representation with a compensation factor characterising an effect of a Coriolis force at a measurement latitude to form a compensated 2ndorder coefficient and determining the respective local acceleration component comprises deriving the local acceleration component from the compensated 2ndorder coefficient.

40. The method of any one of claims 23 to 39, further comprising aligning a selected direction of the first, second and third directions to a direction of gravitational acceleration.

41. The method of any one of claims 23 to 39, further comprising aligning a selected direction of the first, second and third directions at a predetermined orientation with respect to a direction of gravitational acceleration.

42. The method of claim 40 or 41, further comprising launching the test mass in the selected direction.

43. The method of claim 42, comprising launching the test mass to a predetermined height.

44. The method of any one of claims 40 to 42, wherein a mass of the test mass is less than 0.01 kg.

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