SELF-MIXING INTERFEROMETRY

Abstract

A self-mixing interferometer configured to monitor particulate material within a monitored region of space comprising a laser cavity assembly (1A) and an optical assembly (1B) configured to bathe the monitored region with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region. A laser monitoring unit (1C) is configured to acquire an interferometric signal generated by the interferometer in response to light returned to the laser cavity assembly from said wavefronts by said particulate material. A processing module (1D) is configured to determine a property of the particulate material within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal.

Claims

1. A self-mixing interferometer configured to monitor particulate material within a monitored region of space comprising: a laser cavity assembly; an optical assembly configured to bathe the monitored region with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region; a laser monitoring unit configured to acquire an interferometric signal generated by the interferometer in response to light returned to the laser cavity assembly from said wavefronts by said particulate material; a processing module configured to determine a property of the particulate material within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal.

2. The self-mixing interferometer according to claim 1, wherein the processing module is configured to determine a property of the particulate material within the monitored region according to a continuous change in the frequency of said waveform.

3. The self-mixing interferometer according to claim 1, wherein the property of the particulate material comprises a property of the path thereof within the monitored region.

4. The self-mixing interferometer according to claim 3, wherein the property of the path comprises a distance to said particulate material relative from the interferometer.

5. The self-mixing interferometer according to claim 3, wherein the property of the path comprises a speed of said particulate material relative to the interferometer.

6. The self-mixing interferometer according to claim 3, wherein the property of the path comprises a direction of said particulate material relative to the interferometer.

7. The self-mixing interferometer according to claim 1, wherein the processing module is configured to determine a size and/or a size distribution of said particulate material within the region of space.

8. The self-mixing interferometer according to claim 1, wherein the processing module is configured to determine a concentration of said particulate material within the region of space.

9. The self-mixing interferometer according to claim 1, wherein the interferometric signal generated by the interferometer and acquired by the laser monitoring unit comprises a voltage waveform signal at least a part of which continuously changes in frequency and corresponds to a voltage across the electrical drive terminals of a laser cavity of the laser cavity assembly.

10. The self-mixing interferometer according to claim 1, wherein the interferometric signal generated by the interferometer and acquired by the laser monitoring unit comprises an optical output power signal at least a part of which continuously changes in frequency and corresponds to an optical output power of a laser cavity of the laser cavity assembly.

11. The self-mixing interferometer according to claim 1, wherein the optical assembly is configured to bathe the monitored region with a static divergent and/or convergent beam of said laser light possessing a curved wavefront in which the monitored region comprises regions other than a focal region of said laser light.

12. The self-mixing interferometer according to claim 1, wherein the optical assembly is configured to bathe the monitored region with a beam of said laser light possessing a substantially flat wavefront moved across the monitored region to a plurality of different directions.

13. The self-mixing interferometer according to claim 1, in which said laser cavity assembly is configured to output a laser beam in each of two or more different directions, wherein the processing module is configured to determine two or three mutually orthogonal components of a velocity of particulate material through the monitored region according to said changes in the frequency of a waveform within at least a part of the interferometric signals generated respectively by the laser cavity assembly when in each of the two or more different directions and/or according to the number of wave cycles within the respective waveforms.

14. The self-mixing interferometer according to claim 1, wherein said waveform within at least a part of the interferometric signal comprises a chirped waveform.

15. The self-mixing interferometer according to claim 1, wherein said laser cavity assembly is configured to output two or more laser beams comprising different respective cross-sectional beam shapes and/or different beam directions.

16. The self-mixing interferometer according to claim 15, wherein the two or more laser beams are configured to overlap within the monitored region to define an overlap region and the processing module is configured to determine a property of the particulate material within the overlap region in response to light returned to the laser cavity assembly concurrently from said wavefronts of said two or more laser beams.

17. The self-mixing interferometer according to claim 15, wherein the processing module is configured to determine a property of the particulate material within the monitored region according to differences in said respective cross-sectional beam shapes.

18. A portable electronic device comprising the self-mixing interferometer according to claim 1.

19. (canceled)

20. An air purification device comprising the self-mixing interferometer according to claim 1.

21. A method for monitoring particulate material within a monitored region of space using self-mixing interferometry comprising: providing an interferometer comprising a laser cavity assembly and an optical assembly; bathing the monitored region with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region; acquiring an interferometric signal generated by the interferometer in response to light returned to the laser cavity assembly from said wavefronts by said particulate material; by a processing module, determining a property of the particulate material within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal.

22-37. (canceled)

Description

SUMMARY OF THE FIGURES

[0149] Embodiments and experiments illustrating the principles of the invention will now be discussed with reference to the accompanying figures in which:

[0150] FIGS. 1A and 1B show schematic representations of a self-mixing interferometer.

[0151] FIG. 1B shows a schematic representations of a self-mixing interferometer.

[0152] FIGS. 2A-2D each show a schematic representations of a self-mixing interferometer.

[0153] FIG. 3 shows a schematic representations of a self-mixing interferometer.

[0154] FIG. 4 shows a schematic representations of curved wavefronts of a laser beam at either side of a focal region of the beam.

[0155] FIGS. 5A-5C show steps in a method implemented by a self-mixing interferometer.

[0156] FIGS. 6A-6B show schematic representations of a self-mixing interferometer.

[0157] FIGS. 7A-7B show schematic representations of a self-mixing interferometer.

[0158] FIGS. 8A-68 show examples of a self-mixing interferometric signal.

[0159] FIG. 9A shows an example of a self-mixing interferometric signal.

[0160] FIG. 9B shows an example of a Fourier scalogram (e.g., spectrogram) for a self-mixing interferometric signal.

[0161] FIG. 9C shows an example of a wavelet scalogram for a self-mixing interferometric signal.

[0162] FIG. 10 shows an example of a wavelet scalogram for a self-mixing interferometric signal.

[0163] FIGS. 11A-11H show examples of a spectrogram or scalogram for self-mixing interferometric signals.

[0164] FIG. 12A shows a schematic representations of a self-mixing interferometer.

[0165] FIG. 12B and FIG. 12C show, schematically two different but parallel paths of a particle through a laser beam (FIG. 12B), and the resulting self-mixing interferometric signals and the respective wavelet scalograms for the self-mixing interferometric signals.

[0166] FIGS. 13A-13B show graphical representations of stages in the implementation of a Hough transformation in an algorithm implemented by a self-mixing interferometer.

[0167] FIG. 14 shows examples of a spectrogram or scalogram for self-mixing interferometric signals.

[0168] FIG. 15 shows an example of a spectrogram or scalogram for a self-mixing interferometric signal.

[0169] FIG. 16A shows an array of self-mixing interferometers employed in determining a particle position.

[0170] FIG. 16B schematically shows interferometer signals generated by the array of self-mixing interferometers of FIG. 16A.

[0171] FIG. 17A shows an array of self-mixing interferometers employed in determining a particle property.

[0172] FIG. 17B schematically shows a plane view of the array of self-mixing interferometers of FIG. 17A comprising the planar cross-section of the laser beams of each interferometer containing the individual interferometers and the path of the particle through the respective laser beams.

[0173] FIG. 17C schematically shows an array of self-mixing interferometers with crossed beams employed in determining a particle property.

[0174] FIG. 17D schematically shows an array of self-mixing interferometers with differing beam geometries employed in determining a particle property.

[0175] FIG. 17E schematically shows an array of self-mixing interferometers employed in determining a particle property.

[0176] FIG. 17F schematically shows an array of self-mixing interferometers with differing beam directions employed in determining a particle property.

[0177] FIG. 18 shows a flow-chart if steps in an algorithm implemented by a self-mixing interferometer.

[0178] FIG. 19 shows a flow-chart if steps in an algorithm implemented by a self-mixing interferometer.

DETAILED DESCRIPTION OF THE INVENTION

[0179] Aspects and embodiments of the present invention will now be discussed with reference to the accompanying figures. Further aspects and embodiments will be apparent to those skilled in the art. All documents mentioned in this text are incorporated herein by reference.

[0180] FIG. 2A schematically illustrates a self-mixing interferometer device 1 for detecting particulate material, according to an embodiment of the invention.

[0181] The device is configured to monitor particulate material 4 within a monitored region of space. It comprises a laser cavity assembly 1A and an optical assembly 1B configured to bathe the monitored region with laser light 2 of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region. The self-mixing interferometer includes a laser monitoring unit 1C configured to acquire an interferometric signal generated by the interferometer in response to light returned to the laser cavity assembly 1A from the wavefronts by the particulate material. The self-mixing interferometer also includes a processing module 1D configured to determine a property of the particulate material within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal.

[0182] In the example shown, the particulate matter 4 is represented as a spherical particle and is shown in each one of three successive positions along the path 7 of the particle within the laser beam 2. Each one of the three successive positions corresponding to a respective one of three successive times on the motion of the particle 4 along its path. The laser beam, in this example, comprises a conically diverging laser beam possessing conical beam edges 8a and a central beam axis 8b. The angular divergence of the conical laser beam is 2n degrees in full, this corresponding to a divergence such that each conical beam edge 8a subtends an angle n relative to the central axis 8b of the laser beam.

[0183] At the point in time when particle 4 had entered the conical laser beam (8a right hand side, FIG. 2A), and a surface part, or facet, of the particle has become parallel to an advancing wavefront of the laser beam as shown, at that point in time a portion 6 of the laser beam incident upon that surface part or facet is back-reflected and/or back-scattered to the optical resonator cavity 1A of the device 1. As a result of this, and in response to the light 6 returned to the laser cavity assembly 1A from these wavefronts by the particulate material, a self-mixing interferometric signal is generated by the optical resonator cavity 1A, as described above. The laser monitoring unit 1C acquires this interferometric signal. The laser monitoring unit may be a photodiode (e.g., PD of FIG. 1C) configured to generate an electrical signal in response to an optical output from the back mirror (e.g., M.sub.1 of FIG. 1C) of the laser resonator cavity 1A, as discussed above. The laser monitoring unit may be a voltage monitor configured to generate an electrical signal in response to the voltage across electrical drive terminals of the laser resonator cavity 1A, as discussed above.

[0184] The self-mixing interferometer also includes a processing module 1D configured to determine a property of the particulate material within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal. The laser monitoring unit 1C is configured to output the acquired interferometric signal to the processing module 1D for processing thereby, to determine a property of the particulate material. The nature of tis processing will be explained in detail below.

[0185] However, while the embodiment shown in FIG. 2A employs an optical assembly 1B configured to bathe the monitored region with laser light 2 of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region, this is not the only way in which to configure the laser light for bathing the monitored region.

[0186] For example, the self-mixing interferometer described in any embodiment herein may be comprised within or upon a portable electronic device (not shown). The self-mixing interferometer may be comprised within a non-portable unit (not shown) for mounting to a wall, to a ceiling or a floor-standing device such as an air purification device for commercial or domestic areas and uses, or within a non-fixed moveable air purification unit, or within a portable or wearable electronic device. Examples of portable or wearable devices include a wrist-mounted electronic device, a smartphone device, a tablet computer device, a laptop computer device, or a bespoke air quality monitoring device.

[0187] In particular, FIG. 2B, FIG. 2C, FIG. 2D and FIG. 4 show alternative arrangements consistent with aspects of the invention. The schematic diagram of FIG. 2B corresponds to the arrangement shown in FIG. 2A. Here it can be seen that the optical assembly is configured to structure the laser beam such that it possesses curved wavefronts 10. The optical path length between the optical resonator cavity 1A and any point on a wavefront 10 is constant. This means that any spacing , in a direction perpendicular to a given wavefront 10, between a point on that wavefront and a position of the particle at a particular instant of time along its part 7 through the laser beam, is equivalent to an optical path difference between light at the optical wavefront and light having reached the particle and about to be reflected from it back to the laser resonator cavity. Because the speed of light is so very much greater than the speed of the particle, the position of a given wavefront 10 when the particle enters the laser beam may be used as a reference wavefront with which to determine how the value of the spacing, , changes as the particle progresses along its path. In other words, a wavefront corresponding to a particular phase of the electromagnetic wave of the laser light can be considered to be always present at the position of the reference wavefront.

[0188] As shown in FIG. 2B, the value of the spacing, , changes continuously (.fwdarw.A.sub.1.fwdarw..sub.2.fwdarw..sub.3 . . . etc.) as the linear path of the particle 7 first coincides with a reference wavefront upon entering the laser beam, then moves further away from the reference wavefront as the particle approaches the central axis of the laser beam, due to the outward curvature of the reference wavefront. Subsequently, the linear path of the particle 7 moves towards the reference wavefront as the particle moves away from the central axis of the laser beam until it finally coincides with the reference wavefront upon exiting the laser beam.

[0189] As shown in FIG. 2C, it can be seen that the optical assembly is configured to structure the laser beam such that it possesses substantially flat wavefronts 10, corresponding to a collimated laser beam. However, the path 7 of the particle through the laser beam is non-linear. In the example shown the path of the particle is curved with a substantially constant curvature. It can be seen that the value of the spacing, , between the path of the particle and a reference wavefront changes not because of any curvature of the reference wavefrontthere is nonebut because of a curvature of the path of the particle. The spacing, , changes continuously (.fwdarw..sub.1.fwdarw..sub.2.fwdarw..sub.3 . . . etc.) as the curved path 7 of the particle first coincides with a flat reference wavefront upon entering the laser beam, then moves further away from the flat reference wavefront as the particle approaches the centre of the laser beam, due to the outward curvature of the particle path 7. Subsequently, the curved path 7 of the particle moves towards the flat reference wavefront as the particle moves away from the centre of the laser beam until it finally coincides with the flat reference wavefront upon exiting the laser beam. The curvature of the path 7 of the particle may be cause, for example a non-linear flow of gas (e.g., by turbulence within the gas), such as air, through which the particle moves.

[0190] FIG. 2D shows a schematic example of the use of a laser possessing flat wavefronts (e.g., a collimated laser beam) as in FIG. 2C, but in which the path 7a of the particle is linear. The spacing, , between the linear path 7a of the particle and a flat reference wavefront once more changes continuously as the linear path 7a of the particle progresses through the laser beam. The apparently linear path 7a of the particle may be not simply linear, but instead may be a path 7b that fluctuates around an average linear path 7c. This may occur, for example, if the particle is within a highly turbulent gas (e.g., air) and/or if the particle is sufficiently small (e.g., a pollen particle) that its path 7b is significantly buffeted or deviated by gas molecules in the manner of a Brownian motion or the like. In this case, the average speed, (v), of the particle, along its average linear path 7c, may be estimated as being the ratio of the linear distance between the points of entry into, and exit from, the laser beam at sides 8a of the beam, and the time interval between these events. The spacing, , between the non-linear path 7b of the particle and a reference flat wavefront once more changes continuously as the non-linear path 7b of the particle progresses through the laser beam.

[0191] In the examples shown in FIG. 2C and FIG. 2D, the interferometer device 1 possesses an optical assembly 1B configured to bathe the monitored region with laser light of the interferometer possessing wavefronts having the same directions (i.e., substantially flat wavefronts) at different respective locations within the monitored region. In addition, in the examples shown in FIG. 2B, FIG. 2C and FIG. 2D (path 7b), the spacing, , between the path (7, 7b) of the particle and a reference wavefront changes continuously and non-linearly over time as the particle progresses through the laser beam. As will be discussed below, the inventors have realised that a consequence of this is that the frequency of a waveform in the interferometric signal of the self-mixing interferometer 1 is caused to change over time in at least a part of the interferometric signal, often appearing in the form of a chirp or the like, in some part of the signal, sometimes in much of the signal, and sometimes in substantially all of the signal depending upon circumstances.

[0192] The inventors have realised that, in some aspects of the invention, this changing frequency of waveform in the interferometric signal can be used to provide information about properties of the particle causing the signal.

[0193] In addition, the inventors have also realised that, in some aspects of the invention, by providing an optical assembly configured to bathe the monitored region with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region, one may use wavefront geometry to provide information about properties of the particle causing the signal. This configuration has the effect of significantly increasing the likelihood that a changing frequency of waveform in the interferometric signal will be observed. This configuration also has the effect of significantly increasing the volume of the region of space the that can be monitored by the self-mixing interferometer 1. As shown in FIG. 4, the laser beam may be configured to converge, with beam converging edges 8a towards a focal region 11 at some desired distance within the monitored region beyond the self-mixing interferometer 1, and to subsequently, naturally, diverge at distances beyond the focal region 11. In both the converging region and the diverging region, the laser beam possesses curved wavefronts having different directions at different respective locations within the monitored region 12, either side of the focal region 11. Notably, the focal region 11 of the laser beam, by virtue of being a laser beam focus, has a flat wavefront.

[0194] FIG. 3 schematically illustrates a further alternative arrangement for the self-mixing interferometer device 1 for detecting particulate material, according to an embodiment of the invention. The device is configured to monitor particulate material 4 within a monitored diverging region 15 of space of angular range 2. In this embodiment, the optical assembly 1B shown in FIG. 2A is configured to bathe the monitored region with laser light 2 of one collimated laser beam possessing flat wavefronts having different directions at different respective locations within the monitored region. The collimated laser beam is moved through a range 14 of different angular positions (e.g., 14a to 14b to 14c etc.) through the monitored region thereby to move the substantially flat wavefronts through a range of different directions across the monitored region. The range 14 of different angular positions spans, for example, an angular range 2. In other words, the angular divergence of the range of different angular positions of the collimated laser beam is 2 degrees in full. This corresponds to a divergence such that the two extreme angular positions of the beam (14a, 14c) subtend an angle n relative to the central angular position 14b of the laser beam. This angular movement of the laser beam may be achieved by a corresponding angular movement of the optical axis of collimating optics of the optical assembly 1B configured to form a collimated laser beam from laser light received thereby from the laser resonator cavity. In the example shown schematically in FIG. 3, the collimating optics comprises an angularly moveable collimating lens 13, however other collimating optics may be used, such as an angularly moveable mirror (not shown) arranged to receive a collimated laser beam from a static collimating lens, and to reflect the received collimated beam through the range of different angular positions.

[0195] FIG. 4A schematically illustrates a sequence of steps performed by the self-mixing interferometer 1 to determine a property of the particulate material 4 within the monitored region. This method corresponds to those aspects of the invention in which the monitored region is bathed with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region, and in which the property of the particulate material is determined according to changes in the frequency of a waveform within at least a part of the interferometric signal. The method comprises the following steps: [0196] Step S1A: By the laser resonator cavity 1A and optical assembly 1B of the interferometer 1, bathing the monitored region with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region. [0197] Step S2A: By the monitoring unit 1C, acquiring an interferometric signal generated by the laser resonator cavity 1A of the interferometer 1 in response to light returned to the laser cavity assembly from wavefronts of the laser beam by the particulate material 4. [0198] Step S3A: By a processing module 1C, receiving the acquired interferometric signal from the monitoring unit 1C and therewith determining a property of the particulate material 4 within the monitored region according to changes in the frequency of a waveform within at least a part of the interferometric signal.

[0199] The self-mixing interferometer 1 may be configured to implement this method in, for example but without limitation to, any one of the arrangements schematically illustrated in FIG. 2A, FIG. 2B, FIG. 4 or FIG. 3. As will be discussed in more detail below, this methodology promotes the appearance of a waveform within at least a part of the interferometric signal that changes in frequency. The inventors have found that these changes may be used to determine properties of the particle within the laser beam.

[0200] FIG. 4B schematically illustrates a sequence of steps performed by the self-mixing interferometer 1 to determine a property of the particulate material 4 within the monitored region. This method corresponds to those aspects of the invention in which at least a part of the interferometric signal comprises a waveform of changing frequency. This may occur either because the monitored region is bathed with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region, or because the particulate material follows a path having different directions at different respective locations within the monitored region. In either case, a property of the particulate material is determined according to a structure in data describing the interferometric signal in a frequency-space transformation thereof wherein at least a part of the interferometric signal comprises a waveform of changing frequency. The method comprises the following steps: [0201] Step S1B: By the laser resonator cavity 1A and optical assembly 1B of the interferometer 1, bathing the monitored region with laser light of the interferometer. [0202] Step S2B: By the monitoring unit 1C, acquiring an interferometric signal generated by the laser resonator cavity 1A of the interferometer 1 in response to light returned to the laser cavity assembly from wavefronts of the laser beam by the particulate material 4. [0203] Step S3B: By a processing module 1C, receiving the acquired interferometric signal from the monitoring unit 1C and therewith determining a property of the particulate material within the monitored region according to a structure in data describing the interferometric signal in a frequency-space transformation thereof wherein at least a part of the interferometric signal comprises a waveform of changing frequency.

[0204] The self-mixing interferometer 1 may be configured to implement this method in, for example but without limitation to, any one of the arrangements schematically illustrated in FIG. 2A, FIG. 2B, FIG. 2C, FIG. 4 or FIG. 3. As will be discussed in more detail below, this methodology is particularly sympathetic to, or complements, the appearance of a waveform within at least a part of the interferometric signal that changes in frequency. The inventors have found that these changes may create structure in data describing the interferometric signal in a frequency-space transformation, and this structure may be used to determine properties of the particle within the laser beam.

[0205] FIG. 4C schematically illustrates a sequence of steps performed by the self-mixing interferometer 1 to determine a property of the particulate material 4 within the monitored region. This method corresponds to those aspects of the invention in which at least a part of the interferometric signal comprises a waveform of changing frequency. This may occur either because the monitored region is bathed with laser light of the interferometer possessing wavefronts having different directions at different respective locations within the monitored region, or because the particulate material follows a path having different directions at different respective locations within the monitored region. In either case, a property of the particulate material is determined according to a wavelet transformation of the interferometric signal at least a part of which comprises a waveform of changing frequency. The method comprises the following steps: [0206] Step S1C: By the laser resonator cavity 1A and optical assembly 1B of the interferometer 1, bathing the monitored region with laser light of the interferometer. [0207] Step S2C: By the monitoring unit 1C, acquiring an interferometric signal generated by the laser resonator cavity 1A of the interferometer 1 in response to light returned to the laser cavity assembly from wavefronts of the laser beam by the particulate material 4. [0208] Step S3C: By a processing module 1C, receiving the acquired interferometric signal from the monitoring unit 1C and therewith determining a property of the particulate material within the monitored region according to a wavelet transformation of the interferometric signal at least a part of which comprises a waveform of changing frequency.

[0209] The self-mixing interferometer 1 may be configured to implement this method in, for example but without limitation to, any one of the arrangements schematically illustrated in FIG. 2A, FIG. 2B, FIG. 2C, FIG. 2D, FIG. 4 or FIG. 3. As will be discussed in more detail below, this methodology is particularly sympathetic to, or complements, the appearance of a waveform within at least a part of the interferometric signal that changes in frequency. The inventors have found that a wavelet transformation of the interferometric signal is surprisingly effective at accurately and efficiently representing genuine changes in the frequency of the interferometric signal as opposed to misrepresenting noise within the signal as a genuine part of the signal. The inventors have found that a wavelet transformation of the interferometric signal can achieve these benefits and improvement to an extent that other frequency transformations cannot. By using a wavelet transformation of the interferometric signal, the inventors have realised that information may be obtained with which to determine properties of the particle within the laser beam.

[0210] Examples of the application of these general principles, as exemplified by the methodologies defined in FIG. 5A, FIG. 5B and FIG. 5C, will be explained in more detail below. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material, e.g., as disclosed herein.

[0211] As a first example, consider the methodology of FIG. 5C as applied to the circumstances illustrated schematically in FIG. 2D. Here, the optical assembly 1B of the self-mixing interferometer 1 is configured to provide a collimated laser beam 8a possessing substantially flat wavefronts 10. Consider a particle 4 passing through the laser beam along a non-linear path 7b within the laser beam which fluctuates around a time-averaged path 7c which is linear and corresponds to average linear velocity v of the particle 4. In this sense, the time-averaged path is equivalent to a notional linear particle path 7a of the particle 4 with a notional linear velocity v=v. This notional linear velocity has a direction subtending an angle relative to the axis of the laser beam. Both the subtended angle and the notional linear velocity v may be determined as follows.

[0212] The angular frequency, .sub.s, of the interferometric signal corresponds to the rate of change of the accumulated phase .sub.s within the light wave of the laser light (6, FIG. 2A) re-injected into the optical cavity 1A of the laser by reflection or back-scattering from the particle 4. Noting that a change dos in the accumulated phase .sub.s arises because of a change dl in the difference between the position of the particle 4 and a reference wavefront of the laser beam (as discussed in detail above), we may write:

[00008]${}_s=\left(\frac{d{}_s}{\mathrm{dt}}\right)=\frac{4}{}\frac{dl}{dt}=\frac{4.Math.v.Math.}{}\cos$

[0213] Given that the duration T of the transit of the particle 4 across the laser beam of width W is:

[00009]$T=\frac{W}{v\sin }$

[0214] One may combine these two equations to yield:

[00010]$={\tan }^{-1}\left[\frac{4W}{{}_{s}T}\right]$$.Math.v.Math.=\frac{1}{T}\sqrt{\frac{W}{4{\left(1+\left[\frac{4W}{{}_{s}T}\right]\right)}^2}}$

[0215] The quantities W and are properties of the laser system and are known. The quantities T and .sub.s may be accurately estimated by applying a wavelet transformation to the interferometric signal. FIG. 10 shows a schematic example of a wavelet scalogram representation of a wavelet transform applied to the interferometric signal, having axes for time, t, signal frequency (i.e., wavelet scale), .sub.s, and wavelet coefficient value. The wavelet coefficient value corresponds to a dimension perpendicular to the plane of the figure. The scalogram shows the results of applying a wavelet transformation to three separate interferometric signals. A first interferometric signal was dominated by waveform having an average angular frequency, .sub.s=.sub.1, which is the lowest amongst the three signals and a duration which is the greatest amongst the three signals. This corresponds to a slowest average particle speed (NB. .sub.s(v) amongst the three particles. A third interferometric signal was dominated by waveform having an average angular frequency, .sub.s=.sub.3, which is the highest amongst the three signals and a duration which is the lowest amongst the three signals. This corresponds to a fastest average particle speed amongst the three particles. A second interferometric signal was dominated by waveform having an average angular frequency, .sub.s=.sub.2, which is intermediate amongst the three signals and a duration, T.sub.2, which is intermediate amongst the three signals. This corresponds to an intermediate average particle speed amongst the three particles. Because of the fluctuating direction of the path (7b, FIG. 2D) of the particle 4, the frequency of the interferometric signal will also fluctuate. This means that the frequency value, w.sub.s, of data points within the scalogram for any one particle path will fluctuate around the average value, .sub.s, of the interferometric signal frequency in the form of a scatter of data. The inventors have found that the wavelet transformation is able to model these fluctuations in signal frequency and this results in a surprisingly robust representation of data within the scalogram which is much less susceptible to data scatter due to signal noise being misrepresented as a signal frequency fluctuation. This means that a more accurate determination of an average angular frequency, .sub.s and a signal duration, T, can be determined from the data. Consequently, more accurate values of the properties of the particle, such as average speed v, and average path orientation across the beam, may be determined. The advantages of a wavelet transformation may also be applied, if desired, similarly to the methodologies describe above with reference to FIG. 5A and/or FIG. 5B.

[0216] As a second example, consider the methodology of FIG. 5A as applied to the circumstances illustrated schematically in FIG. 2A, FIG. 2B, FIG. 4 or FIG. 3. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material, e.g., as disclosed herein. Here, the optical assembly 1B of the self-mixing interferometer 1 is configured to provide a laser beam 8a possessing wavefronts 10 having different directions at different respective locations within the monitored region. As an example, consider a diverging cone of laser light as illustrated schematically in FIG. 6A and in FIG. 6B. This is presented in order to illustrate principles of certain aspects of the invention, but the principles discussed below apply equally to a laser beam with flat wavefronts moved to different directions at different parts of a monitored region, such as in FIG. 3.

[0217] FIG. 6A shows a diverging cone 25 of laser light emanating from the self-mixing interferometer 1. The cone of light is centred upon a cone axis 8b. The angle of divergence of the laser beam relative to the cone axis is . Consider a particle 4 following a linear path (or a time-averaged linear path) 7 passing through the cone of laser light at speed v (or average speed (v)) the direction of which subtends an angle relative to the cone axis 8b. FIG. 6A shows a generalised situation in which the path 7 of the particle 4 does not necessarily reside in a plane also containing the central cone axis 8b of the laser beam. For example, the particle path 7 may be offset from, but parallel to, a plane containing the central cone axis 8b. In this way, the path 7 of the particle 4 may reside within the plane of a triangular slice 26 of the light cone onto which the cone axis 8b projects to define the axis of symmetry of the triangular slice 26. The divergence of the triangular slice relative to the projected axis is . The direction of the path of the particle subtends an angle = relative to the projected cone axis 8b. Though the following discussion applies equally to the more general situation shown in FIG. 6A, the following analysis will consider the simpler situation schematically illustrated in FIG. 6B in which the path 7 of the particle 4 resides in the plane also containing the central cone axis 8b of the laser beam.

[0218] Referring to FIG. 7B one can see that the following relations exist in relation to a change dl in the distance between the particle 4 and a reference wavefront 10 of the laser beam (8a, 8b), measured in a direction perpendicular to the reference wavefront, as determined from two different points on the path 7 of the particle 4 separated by a linear distance ds.

[00011]$\frac{dl}{ds}=\sin \left(\frac{}{2}+-\right);\frac{1}{l}\frac{dl}{ds}=\cos \left(\frac{}{2}+-\right);\frac{1}{l}\frac{dl}{d}=\tan \left(\frac{}{2}+-\right);v=\frac{ds}{dt}$

[0219] Given that the speed of the particle 4 is v, one may write:

[00012]$ds=\mathrm{vdt}$$\frac{dl}{ds}=\frac{1}{v}\frac{dl}{\mathrm{dt}}=\sin \left(\frac{}{2}+-\right)$$\frac{dl}{\mathrm{dt}}=v\sin \left(\frac{}{2}+-\right)$

[0220] Referring to FIG. 7B, the following relations exist between the geometry of the conical laser beam of divergence angle , the angle subtended by the path 7 of the particle relative to the central axis 8b of the conical laser beam, the times t.sub.1 and t.sub.3 of entering and exiting, respectively, the conical laser beam, the intermediate time t.sub.2 of reaching a tangential position (X) at which the part of 7 of the particle is instantaneously parallel to the reference wavefront such that the rate of change dl/dt of the distance between the particle 4 and a reference wavefront 10 of the laser beam momentarily vanishes:

[00013]$\frac{s^{}}{s}=\left[\frac{t_2-t_1}{t_3-t_1}\right]=\frac{\sin \left(+-\frac{}{2}\right)\sin \left(-\right)}{\sin \left(2\right)}$

[0221] Here, s is the part of the path 7 of the particle 4 extending between the point of entry of the particle into the conical laser beam, when at a distance l.sub.1 from the laser resonator cavity 1A, and the tangential position. The quantity s, illustrated in FIG. 7B, is the part of the path 7 of the particle 4 extending between the point of exit of the particle from the conical laser beam, when at a distance 13 from the laser resonator cavity 1A, and the tangential position. Consequently, the quantity s is the full linear path length of the particle within the conical laser beam, such that:

[00014]$s=s^{}+s^{}$

[0222] One can see that:

[00015]$\frac{l_3}{l_1}=\frac{\cos \left(+-\frac{}{2}\right)}{\sin \left(-\right)}$

[0223] More generally, for a given instant in time, t, we may identify the angular position of the particle, relative to the central beam axis 8b as:

[00016]$={\tan }^{-1}\left\{\frac{\left[\frac{t_2-t}{t_2-t_1}\right]\tan }{1+\left[1-\left(\frac{t_2-t}{t_2-t_1}\right)\right]\tan \tan \left(-/2\right)}\right\}$

[0224] Consequently, we may conclude that the rate of change of the distance between the particle 4 and a reference wavefront 10 of the laser beam (8a, 8b), measured in a direction perpendicular to the reference wavefront, is:

[00017]$\frac{\mathrm{dl}}{\mathrm{dt}}=v\sin \left[{\tan }^{-1}\left\{\frac{\left[\frac{t_2-t}{t_2-t_1}\right]\tan }{1+\left[1-\left(\frac{t_2-t}{t_2-t_1}\right)\right]\tan \tan \left(-/2\right)}\right\}-+\frac{}{2}\right]$

[0225] Using the following known trigonometric relation:

[00018]$\sin \left(x+{\tan }^{-1}\left(A/B\right)\right)=\frac{A}{\sqrt{A^2+B^2}}\cos \left(x\right)+\frac{B}{\sqrt{A^2+B^2}}\sin \left(x\right)$

[0226] And defining

[00019]${\tan }^{-1}\left(A/B\right)=\frac{}{2}-$

[0227] Gives

[00020]$\frac{A}{B}=\tan \left(\frac{}{2}-\right)=$

[0228] Therefore

[00021]$\frac{\mathrm{dl}}{\mathrm{dt}}=v\sin \left[{\tan }^{-1}\left\{\right\}+\frac{}{2}-\right]=\frac{v}{\sqrt{1+{}^2}}\cos \left({\tan }^{-1}\left\{\right\}\right)+\frac{v}{\sqrt{1+{}^2}}\sin \left({\tan }^{-1}\left\{\right\}\right)$

[0229] Here

[00022]$=\frac{\left[\frac{t_2-t}{t_2-t_1}\right]\tan }{1+\left[1-\left(\frac{t_2-t}{t_2-t_1}\right)\right]\tan \tan \left(-/2\right)}$

[0230] Given the known trigonometric relation:

[00023]$\cos \left({\tan }^{-1}\left\{\right\}\right)=\frac{1}{\sqrt{1+{}^2}};\sin \left({\tan }^{-1}\left\{\right\}\right)=\frac{}{\sqrt{1+{}^2}}$

[0231] We may write that:

[00024]$\frac{\mathrm{dl}}{\mathrm{dt}}=\frac{v}{\sqrt{1+{}^2}}.Math.\frac{1}{\sqrt{1+{}^2}}+\frac{v}{\sqrt{1+{}^2}}.Math.\frac{}{\sqrt{1+{}^2}}=v\left[\frac{+}{\sqrt{1+{}^2}\sqrt{1+{}^2}}\right]$

[0232] As discussed above, the response of the laser cavity of the interferometer can be expressed in the following terms:

[00025]$P_{\mathrm{Laser}}=A+B\cos {}_{\mathrm{FB}}$${}_{\mathrm{FB}}={}_s+C\sin \left({}_{\mathrm{FB}}+{\tan }^{-1}\right)$$P_{\mathrm{Laser}}=A+B\cos \left[{}_s+C\sin \left({}_{\mathrm{FB}}+{\tan }^{-1}\right)\right]$

[0233] However, when there is only a weak intensity in the light returned to the laser cavity from the particle the coefficient C becomes negligible, and one may write:

[00026]$P_{\mathrm{Laser}}A+B\cos {}_s$

[0234] We may also define the phase .sub.s as:

[00027]${}_s=\left(\frac{d{}_s}{\mathrm{dt}}\right)t+{}_0$

[0235] Therefore, the laser interferometric signal becomes:

[00028]$P_{\mathrm{Laser}}=A+B\cos \left[\left(\frac{d{}_s}{\mathrm{dt}}\right)t+{}_0\right]=A+B\cos \left({}_{s}t+{}_0\right)$

[0236] In other words, the rate of change of the phase .sub.s is equivalent to an instantaneous value of the frequency of the laser interferometric signal:

[00029]${}_s=\left(\frac{d{}_s}{\mathrm{dt}}\right)$

[0237] It can be seen that this frequency, .sub.s, is proportional to the speed, v, of the particle, as well as aspects of the geometry, , of the laser beam and the orientation, , of the path of the particle relative to the axis of the laser beam. Noting again that:

[00030]${}_s=\frac{4}{}\frac{\mathrm{dl}}{\mathrm{dt}}=\frac{4v}{}\left[\frac{+}{\sqrt{1+{}^2}\sqrt{1+{}^2}}\right]$

we may use the above known trigonometric relation to express the term in square brackets as follows:

[00031]$\frac{+}{\sqrt{1+{}^2}\sqrt{1+{}^2}}=\sin \left(+{\tan }^{-1}\left[1/\right]\right)$

[0238] Consequently, the signal frequency may be alternatively express as succinctly as follows:

[00032]${}_s=\frac{4v}{}\sin \left(+{\tan }^{-1}\left[1/\right]\right)$

[0239] Here:

[00033]$=\frac{\left[\frac{t_2-t}{t_2-t_1}\right]\tan }{1+[1-\left(\frac{t_2-t}{t_2-t_1}\right)\tan };$$=\tan \left(\frac{}{2}-\right)$

[0240] For conciseness, we may write the term as a simple algebraic function of time, t, as follows:

[00034]$=\frac{c_1+c_{2}t}{c_3+c_{4}t}$

[0241] In this way, the term is a simple polynomial function of time. Here, the terms c.sub.1, c.sub.2, c.sub.3 and c.sub.4 take the following form:

[00035]$c_1=t_2;$$c_2=-1;$$c_3=\frac{t_2-t_1}{\tan }+t_1;$$c_4=-$

[0242] In the situation where the particle crosses the laser beam axis in a direction perpendicular to the axis, then =/2 and =0, such that .fwdarw..sub.=(c.sub.1+c.sub.2t)/c.sub.3, and:

[00036]${}_s\left(t\right)=\frac{4v}{}\left[\frac{{}_{}}{\sqrt{1+{}_{}^2}}\right];$${}_{}=\left[\frac{t_2-t}{t_2-t_1}\right]\tan$

[0243] In the situation where the particle crosses the laser beam in a direction which forms a tangent to a wavefront of the laser beam at a point in time t=t.sub.2 whilst within the laser beam, then at that point .sub.=0, and .sub.s(t=t.sub.2)=0.

[0244] In the situation where the particle progresses directly along the laser beam axis in a direction parallel to the axis, then =0 such that =0, and:

[00037]${}_s\left(t\right)=\left(\frac{d{}_s}{\mathrm{dt}}\right).fwdarw.\frac{4v}{}\left[\frac{}{\sqrt{1+{}^2}}\right]=\frac{4v}{}\sin \left(\frac{}{2}-\right)=\frac{4v}{}\cos \left(\right)=\frac{4v}{}$

[0245] Particles crossing the laser beam axis in directions between these two extremes will produce instantaneous values of the frequency of the laser interferometric signal between these two extremes:

[00038]$0{}_s\left(t\right)\frac{4v}{}$

[0246] In the situation where the particle crosses the laser beam axis in a direction which forms a tangent to a wavefront of the laser beam at a point in time t=t.sub.2 whilst within the laser beam, then at that point =0, and 0<<; /2:

[00039]${}_s\left(t=t_1\right)=\frac{4v}{}\left[\frac{}{\sqrt{1+{}^2}}\right]=\frac{4v}{}\sin \left(\frac{}{2}-\right)=\frac{4v}{}\cos$

[0247] FIG. 8A and FIG. 8B show examples of this waveform in the interferometric signal, P.sub.Laser, of the laser:

[00040]$P_{\mathrm{Laser}}=A+B\cos \left({}_{s}t+{}_0\right)$

[0248] Here,

[00041]${}_s=\frac{4v}{}\left[\frac{+}{\sqrt{1+{}^2}\sqrt{1+{}^2}}\right];$$=\frac{c_1+c_2.Math.t}{c_3+c_4.Math.t}a+b.Math.t+c.Math.t^2$

[0249] In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material using an interferometric signal of this form, e.g., as disclosed herein. In this example, we have expanded the term as a polynomial function of time variable, t, including terms of order no higher than the second order. FIG. 8A shows this waveform in which a=1, b=0 and c=2. FIG. 8B shows this waveform in which a=1.5, b=6 and c=2. In both FIG. 8A and FIG. 8B, A=3 and B=1.5. In this way, the value of B represents an amplitude of a modulation of the interferometric signal, P.sub.Laser, of the laser. The different values of a, b and c represent differences in the terms c.sub.1, c.sub.2, c.sub.3 and c.sub.4 defined above which, in turn are determined by the known divergence angle of the laser beam, the measurable times t.sub.1, t.sub.0 of the entry of the particle into the laser beam and the time at which the tangential position is momentarily achieved, and the angle of the particle path 7 relative to the central axis of the laser beam. As described above, the time at which the tangential position is momentarily achieved corresponds to a moment at which the frequency .sub.s of the interferometric signal, P.sub.Laser, becomes zero. This is indicated by the turning point 31 of the waveform 30 of FIG. 8A, and by the turning point 32 of the waveform 32 of FIG. 8B. Because the values of a, b and c for FIG. 8B differ from the values of a, b and c for FIG. 8A, then so do the respective positions of the turning points in the associates waveforms (30, 32). Similarly, the frequency each waveform changes continuously, and the frequency of the waveform in FIG. 8B changes more rapidly than does the frequency of the waveform in FIG. 8A.

[0250] These differences, and other differences, in the shape and structure of the waveform, are the result of differ from the values of a, b and c which, in turn are the result of differences in the speed v of the particle 4 within the laser beam and the angle its path 7 subtends to the central axis 8b of the laser beam. The processing module 1D may be configured to generate an estimate for the speed v of the particle and its path angle , by a process of fitting an equation given above for:

[00042]$P_{\mathrm{Laser}}=A+B\cos \left({}_{s}t+{}_0\right)$

[0251] This may be achieved by applying an optimisation routine which varies the values of the speed v of the particle and its path angle within this analytical expression, iteratively in such a way as to minimise a difference, P, between the measured interferometric signal waveform, P.sub.Laser.sup.M, and the analytical expression for the interferometric signal waveform, P.sub.Laser, given above:

[00043]$P=P_{\mathrm{Laser}}^M-P_{\mathrm{Laser}}$

[0252] Once a sufficiently low value of P is achieved, the corresponding values of v and required to achieve that condition may be identified as acceptable estimates of the true speed of the particle and its path angle, respectively. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material, e.g., as disclosed herein.

[0253] Referring to FIG. 12A, let the velocity vector, {right arrow over (v)}, of a particle within the laser beam may be estimated to be:

[00044]$\stackrel{.fwdarw.}{v}=v_x\widehat{x}+v_y\widehat{y}+v_z\widehat{z}$

[0254] Here, the velocity vector is resolved in to three orthogonal velocity components which comprise components directed along a respective one of three orthogonal unit vectors {circumflex over (x)}, , {circumflex over (z)}. Here, assume that the axis of the laser beam is directed in a first direction .sub.1 which is parallel to a plane containing two of three orthogonal unit vectors {circumflex over (x)}, {circumflex over (z)} such that the unit vector .sub.1 is defined as:

[00045]${\hat{u}}_1=\sin \left({}_1\right)\widehat{x}+\cos \left({}_1\right)\widehat{z}$

[0255] The angle .sub.1 is measured relative to the {circumflex over (z)} direction. Next, assume that the axis of the laser beam is moved so as to be directed in a second direction .sub.2 which is also parallel to the plane containing these two orthogonal unit vectors {circumflex over (x)}, {circumflex over (z)} such that the unit vector .sub.2 is defined as:

[00046]${\hat{u}}_2=\sin \left({}_2\right)\widehat{x}+\cos \left({}_2\right)\widehat{z}$

[0256] Here, the angle .sub.2 is also measured relative to the {circumflex over (z)} direction. In this way, the first and second directions of the axis of the laser beam subtend an angle (.sub.2.sub.1).

[0257] Next, assume that the axis of the laser beam is moved again so as to be directed in a third direction .sub.3 which is not parallel to the plane containing the two orthogonal unit vectors {circumflex over (x)}, {circumflex over (z)} such that the unit vector .sub.2 is defined as:

[00047]${\hat{u}}_3=\sin \left({}_3\right)\widehat{x}+\sin \left({}_3\right)\widehat{y}+\cos \left({}_3\right)\widehat{z}$

[0258] Here, the angle .sub.3 and the angle .sub.3 are both measured relative to the {circumflex over (z)} direction in a plane perpendicular to the plane containing the two orthogonal unit vectors {circumflex over (x)}, {circumflex over (z)}. Taking the dot product of the particle velocity vector and the three different laser beam axis directions gives the cosine of the angle .sub.1,2,3 subtended by the particle path 7 and the laser beam axis at these three different beam directions:

[00048]$\stackrel{.fwdarw.}{v}.Math.{\hat{u}}_1=v_x\sin \left({}_1\right)+v_z\cos \left({}_1\right)=\cos \left({}_1\right)$$\stackrel{.fwdarw.}{v}.Math.{\hat{u}}_2=v_x\sin \left({}_2\right)+v_z\cos \left({}_2\right)=\cos \left({}_2\right)$$\stackrel{.fwdarw.}{v}.Math.{\hat{u}}_3=v_x\sin \left({}_3\right)\cos \left({}_3\right)+v_y\sin \left({}_3\right)\sin \left({}_3\right)+v_z\cos \left({}_3\right)=\cos \left({}_3\right)$

[0259] This may be written in matrix form:

[00049]$\left(\begin{array}{c}\cos \left({}_1\right)\\ \cos \left({}_2\right)\\ \cos \left({}_3\right)\end{array}\right)=\left(\begin{array}{ccc}\sin \left({}_1\right)& 0& \cos \left({}_1\right)\\ \sin \left({}_2\right)& 0& \cos \left({}_2\right)\\ \sin \left({}_3\right)\cos \left({}_3\right)& \sin \left({}_3\right)\sin \left({}_3\right)& \cos \left({}_3\right)\end{array}\right)\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)$

[0260] Thus, by inverting the above equation, a vector estimating the velocity vector for the particle may be determined based on the known values of the laser beam angles .sub.1, .sub.2, .sub.3, .sub.3 and the respective three estimated values .sub.1, .sub.2, .sub.3 of the angle subtended by the particle path and the laser beam axis at these three different beam directions:

[00050]$\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)={\left(\begin{array}{ccc}\sin \left({}_1\right)& 0& \cos \left({}_1\right)\\ \sin \left({}_2\right)& 0& \cos \left({}_2\right)\\ \sin \left({}_3\right)\cos \left({}_3\right)& \sin \left({}_3\right)\sin \left({}_3\right)& \cos \left({}_3\right)\end{array}\right)}^{-1}\left(\begin{array}{c}\cos \left({}_1\right)\\ \cos \left({}_2\right)\\ \cos \left({}_3\right)\end{array}\right)$

[0261] In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material, e.g., as disclosed herein. In this way, by moving the laser beam to re-position its central beam axis on three different directions, or by using three separate laser beams each pointed in a respective direction satisfying these angular requirements, then one may calculate an estimate of the velocity of the particle. The processing module 1D may be configured to generate an estimate for the velocity v of the particle accordingly, given the estimated angles .sub.1,2,3 subtended by the particle path 7 and the laser beam axis, and the known values of the laser beam direction angles .sub.1, .sub.2, .sub.3, .sub.3.

[0262] Notably, if path of particle intersects central beam axis (i.e., =.sub.Max), then one may let .sub.3=0; .sub.3=.sub.2:

[00051]$\left(\begin{array}{c}{v}_x\\ v_z\end{array}\right)={\left(\begin{array}{cc}\sin \left({}_1\right)& \cos \left({}_1\right)\\ \sin \left({}_2\right)& \cos \left({}_2\right)\end{array}\right)}^{-1}\left(\begin{array}{c}\cos \left({}_1\right)\\ \cos \left({}_2\right)\end{array}\right)$$v_y=\sqrt{v^2-\left(v_x^2+v_z^2\right)}$

[0263] Here, v is the estimate of the speed of the particle obtained as described above. In this circumstance it is only necessary to move the laser beam to re-position its central beam axis on two (not three) different directions, or by using two (not three) separate laser beams each pointed in a respective direction satisfying these angular requirements. The processing module 1D may be configured to generate an estimate for the velocity {right arrow over (v)} of the particle accordingly, given the estimated angles .sub.1,2 subtended by the particle path 7 and the laser beam axis, the estimated particle speed v, and the known values of the laser beam direction angles .sub.1, .sub.2.

[0264] FIG. 9A shows a measured waveform, P.sub.Laser.sup.M, for an interferometric signal 34 according to an example of the invention. Here it can be seen that the measured waveform has the same functional form as is shown in the examples of the analytical expression shown in FIG. 8A and FIG. 8B.

[0265] Notably, however, the modulation amplitude of the measured waveform 34, corresponding to the amplitude term B in the analytical expression:

[00052]$P_{\mathrm{Laser}}=A+B\cos \left({}_{s}t+{}_0\right)$

shows a systematic rise from a value B=0 at time t=t.sub.1 when the particle entered the laser beam, to a constant value of B=B.sub.Max before subsequently decaying again to a value B=0 at time t=t.sub.3, when the particle exited the laser beam. In between these two time points, the particle reached a tangential position at time t=t.sub.2 when the frequency of the interferometric signal momentarily falls to zero: .sub.s(t=t.sub.2)=0. The rise and fall of the modulation amplitude B is the result of the distribution of laser light intensity across the laser beam, in which the greater intensity is within the central regions of the laser beam, and the light intensity of the laser beam falls to zero at the edge of the laser beam in a smooth fashion rather than in an instantaneous or abrupt edge as has been assumed in the analytical expression shown in FIG. 8A and FIG. 8B. When in regions of greater light intensity, the optical coupling between the light within the laser resonator cavity and the light re-injected into the laser resonator cavity as reflected/scattered from the particle, is greater and this results in a greater modulation amplitude B. This enables one to identify the times t=t.sub.1, and t=t.sub.3 as those times when the interferometric signal 34 begins and ends. The time t=t.sub.2 can also be identified as the time when the frequency of the interferometric signal momentarily falls to zero: .sub.s(t=t.sub.2)=0. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine these times, e.g., as disclosed herein.

[0266] The inventors have found that by applying a frequency-space transformation to the interferometric signal, such as a Fourier transformation or a wavelet transformation has a number of particular benefits in allowing useful information about properties of the particle 4 to be extracted from the structure of the interferometric signal. This is particularly relevant to the methodology discussed above with reference to FIG. 5B and FIG. 5C in terms of determining a property of the particulate material within the monitored region according to a structure in data describing the interferometric signal in a frequency-space transformation (e.g., a Fourier transformation or a wavelet transformation) thereof wherein at least a part of the interferometric signal comprises a waveform of changing frequency.

[0267] For example, referring to FIG. 9B, here is shown a Fourier scalogram (e.g., spectrogram) of the data describing the result of applying a Fourier transformation to the interferometric signal of FIG. 9A. The scalogram (e.g., spectrogram) is a visual representation of the Fourier transform data, having axes for time (horizontal axis), frequency (vertical axis), and power/frequency value (axis perpendicular to the page of the figure). In aspects of the invention, the processing module of the self-mixing interferometer may be configured to generate data describing a Fourier scalogram (e.g., spectrogram), e.g., as disclosed herein.

[0268] Here, one can see that the times t=t.sub.1, and t=13 are easy to determine accurately in terms of a time point at which the magnitude of the power/frequency value of the Fourier scalogram (e.g., spectrogram) exceeds a pre-set threshold value suitably selected to exclude (or exceed) the effects of signal noise. The time t=t.sub.2 can also be identified as the time when the frequency of the Fourier scalogram (e.g., spectrogram) momentarily falls to zero: @s (t=t.sub.2)=0. Notably, one can see that the Fourier scalogram (e.g., spectrogram) displays a V-shaped structure having two inclined lobes (38, 39) of the distribution of data each extending, in opposite respective directions, away from a common vertex centred upon the time point t=t.sub.2.

[0269] By locating the time position of this vertex, one may accurately estimate a value of t.sub.2. This may be done, for example, by applying a regression algorithm to the data of either lobe (or each lobe separately), and determining a linear regression line for that data. The time point at which the given linear regression line crosses the time axis (i.e., .sub.s=0) provides an estimate of the value of t.sub.2. This may be applied to each of the two data lobes (38, 39) to provide two separate estimates of the value of t.sub.2, and the final estimate be calculated as the average of the two separate estimates.

[0270] It is also noted that the two inclined lobes (38, 39) of the distribution of data, and the respective linear regression lines thereof, subtend respective inclination angles 1 and 2 relative to the time axis.

[0271] FIG. 9C, shows a wavelet scalogram of the data describing the result of applying a wavelet transformation to the interferometric signal of FIG. 9A. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to generate data describing a wavelet scalogram, e.g., as disclosed herein. The scalogram is a visual representation of the wavelet transform data, having axes for time (horizontal axis), frequency (vertical axis), and wavelet coefficient magnitude value (axis perpendicular to the page of the figure). It is also noted that the distribution of data in the wavelet scalogram also has two inclined lobes (40, 41), and a respective linear regression line or data trend line thereof subtends respective inclination angles 3 and 4 relative to the time axis. The wavelet transformation in this example was a continuous wavelet transformation (CWT) of a time-series interferometric signal, s(t), of the form:

[00053]$W_s\left(,\right)=\frac{1}{\sqrt{}}s\left(t\right){}^{*}\left(\frac{t-}{}\right)\mathrm{dt}$

[0272] Here, the quantity W.sub.s(,) represents the wavelet coefficient value, and |W.sub.s(,)| the magnitude of this value. The quantity *([t]/) is the complex conjugate of the so-called mother wavelet and the quantities and are the scale of the scale and translation values. In other examples, a discrete wavelet transformation may be used, or some other form of wavelet transformation readily available to the skilled person. The inventors have found that a wavelet transformation provides a particularly effective means of reliably representing the variable-frequency structure of the interferometric signal in ways that are compact, and robust against misrepresenting noise as a signal component.

[0273] Without wishing to be bound by theory, it is believed that this may be because of the nature of the path 7 of a particle through the laser beam. As mentioned above, if the particle path is fluctuates around a time-averaged path that is linear, then the fluctuating path directions will result in corresponding fluctuations in the instantaneous value of the frequency of the interferometric signal. As discussed above with reference to the scalogram of FIG. 10, this fluctuation in signal frequency will result in a broadening of the estimated signal frequency around an average value (e.g. .sub.2) of the fluctuating frequency. Given the variable scale factor built into the wavelet transformation, these fluctuations can be robustly represented by associated fluctuations in the scale factor. In this way, it is suggested, the wavelet transformation may be better able to represent such signals. The inventors have found that this advantage is surprisingly effective when the modulation amplitude B of the interferometric signal becomes very small. This occurs when the re-injected light reflected/scattered from the particle 4 is very weak or sparse. The inventors have found that other frequency-space transformations, such as the Fourier transformations, do not share this quality and are far more susceptible to mistaking signal noise for signal frequency fluctuation when signal modulation amplitudes B become small.

[0274] It is noted that the scalogram shown in FIG. 10 corresponds to a case in which the particle 4 has a linear path, or linear time averaged path, through a laser beam possessing flat wavefronts 10, as schematically shown in FIG. 2D. The scalogram data for a given particle trajectory, in that case, is distributed along a substantially horizontal line, albeit with scatter in the data around the line, centred on a constant average signal frequency value. This is because the orientation (/2) of the path (or the time-averaged path) of the particle 4 relative to the flat wavefronts is substantially constant and therefore the component (v cos ) of the velocity v of the particle perpendicular to the wavefronts, is constant. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to generate data describing such a scalogram, e.g., as disclosed herein.

[0275] However, the scalogram of FIG. 9C corresponds to a case in which the particle 4 has a linear path, or linear time averaged path, through a laser beam possessing wavefronts 10 that have different directions in different parts of the monitored region, such as schematically shown in FIG. 2A, FIG. 2B, FIG. 4 and FIG. 3. In addition, the scalogram of FIG. 9C also corresponds to a case in which the particle 4 has a curved path through a laser beam possessing flat wavefronts 10 such as schematically shown in FIG. 2C. The scalogram data for a given particle trajectory, in these cases, is distributed along a curve that asymptotically (i.e., as t.fwdarw.) approaches a substantially horizontal line centred on a constant signal frequency value. This is because the orientation (/2) of the linear (or curved) path of the particle 4 relative to the curved/changing (or flat) wavefronts only approaches a condition of being substantially constant as t.fwdarw.. Of course, the particle cannot remain in the beam forever and so the data trend line of the lobes (40, 41) of the distribution of data within the wavelet scalogram of FIG. 9C subtend smaller inclination angles (3 and 4) relative to the time axis when the particle path is closer to satisfying tis asymptotic condition. In other words, the smaller the value if the inclination angle of a data lobe of the wavelet scalogram, the smaller is the value of the particle path angle, , and the closer is the value of v cos to v (i.e., the closer cos is to 1.0). This analysis applies equally to the inclination angles of the data distribution in the Fourier scalogram (e.g., spectrogram) of FIG. 9B, (i.e., 1 and 2) relative to the time axis.

[0276] The inventors have realised that these observations may be used to provide an estimate of the speed v of the particle 4 within the laser beam, as follows.

[0277] Referring to FIG. 11A to FIG. 11H, here is schematically shown a set of eight different wavelet scalograms produced by the self-mixing interferometer 1 of eight different interferometer signals obtained in response to a respective one of eight different paths 7 of a particle 4 through the monitoring laser beam 8a. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to generate data describing any such scalogram, e.g., as disclosed herein.

[0278] A first wavelet scalogram is schematically illustrated in FIG. 11A. The lobes of the data distribution of the scalogram are substantially symmetrical about the intermediate time t=t.sub.2. Either side of this intermediate time, each lobe terminates at a time t=t.sub.1 or a time t=t.sub.3 corresponding to the same interferometric signal frequency. This corresponds to the symmetric geometry of the particle path 7 relative to the cone of the laser beam 8a in which =/2, and is a schematic representation of the wavelet scalogram of FIG. 9C.

[0279] A second wavelet scalogram is schematically illustrated in FIG. 11B. The lobes of the data distribution of the scalogram are asymmetrical about the intermediate time t=t.sub.2. Either side of this intermediate time, each lobe terminates at a time t=t.sub.1 or a time t=t.sub.3 corresponding to different interferometric signal frequencies. This corresponds to an asymmetric geometry of the particle path 7 relative to the cone of the laser beam 8a, in which the particle path 7 subtends an angle /2 but differing from /2. In this example, the particle path is directed slightly away from the self-mixing interferometer 1 such that the time interval (t.sub.2t.sub.1) between entering the laser beam and reaching the tangential position, is less than the time interval (t.sub.3t.sub.2) between entering the tangential position and exiting the laser beam. Consequently, the orientation of the particle path 7 has had a greater time to approach the condition in which the path orientation is perpendicular to the wavefronts of the laser beam 8a, and the interferometric signal frequency is higher as the particle 4 exits the laser beam.

[0280] A third wavelet scalogram is schematically illustrated in FIG. 11C. The data distribution of this scalogram has only one lobe and no intermediate time t=t.sub.2 at which a tangential position is reached (i.e., that position is never reached). The single lobe starts at a time t=t.sub.1 and ends at a time t=t.sub.3 corresponding to different interferometric signal frequencies. This corresponds to a more asymmetric geometry of the particle path 7 relative to the cone of the laser beam 8a, in which the particle path 7 subtends an angle <</2. In this example, the particle path is directed further away from the self-mixing interferometer 1 such that the time interval (t.sub.2t.sub.3) is simply the interval between entering the laser beam and exiting the laser beam. Consequently, the orientation of the particle path 7 has had a greater time to approach the condition in which the path orientation is perpendicular to the wavefronts of the laser beam 8a, and the interferometric signal frequency is higher as the particle 4 exits the laser beam. Notably, the angle of inclination of the single lobe relative to the time axis is lower than the angle of inclination of any of the lobes shown in FIG. 11B and FIG. 11A. The reason is as described above, relating to the condition that the smaller is the value of the particle path angle, , and the closer is the value of v cos to v (i.e., the closer cos is to 1.0).

[0281] A fourth wavelet scalogram is schematically illustrated in FIG. 11D. This corresponds to a similar situation as described with reference to FIG. 11C, but in which the particle path angle is small and 0. Notably, the angle of inclination of the single lobe relative to the time axis is lower than the angle of inclination of any of the lobes shown in FIG. 11A, FIG. 11B or FIG. 11C.

[0282] A fifth wavelet scalogram is schematically illustrated in FIG. 11E. The lobes of the data distribution of the scalogram are asymmetrical about the intermediate time t=t.sub.2. Either side of this intermediate time, each lobe terminates at a time t=t.sub.1 or a time t=t.sub.3 corresponding to different interferometric signal frequencies. This corresponds to an asymmetric geometry of the particle path 7 relative to the cone of the laser beam 8a, in which the particle path 7 subtends an angle /2 but differing from /2. In this example, the particle path is directed slightly away from the self-mixing interferometer 1 such that the time interval (t.sub.2t.sub.1) between entering the laser beam and reaching the tangential position, is less than the time interval (t.sub.3t.sub.2) between entering the tangential position and exiting the laser beam. Consequently, the orientation of the particle path 7 has had a greater time to approach the condition in which the path orientation is perpendicular to the wavefronts of the laser beam 8a, and the interferometric signal frequency is higher as the particle 4 exits the laser beam. The time interval (t.sub.3t.sub.1) is the interval between entering the laser beam and exiting the laser beam, and this is relatively large. This is in conjunction with relatively small values of interferometric signal frequency at each respective end of the two data lobes. This indicated that the speed v of the particle 4 is low. In other words, a low particle speed results in a low maximal interferometric signal frequency and a long time duration for the particle to traverse the laser beam.

[0283] A sixth wavelet scalogram is schematically illustrated in FIG. 11F. This corresponds to the same circumstances described above with reference to FIG. 11E, but with the modification that the particle speed is relatively fast. This fact reveals itself in the smaller duration of the time interval t.sub.3t.sub.1 together with the higher values of maximal interferometric signal frequency at the terminal ends of the data lobes.

[0284] seventh wavelet scalogram is schematically illustrated in FIG. 11G. An eighth wavelet scalogram is schematically illustrated in FIG. 11H. The maximum value of frequency .sub.Max2 present in the interferometric signal waveform, and therefore presented within the data describing the scalogram of that waveform, is indicated in FIG. 11G and FIG. 11H schematically.

[0285] These schematics illustrate how the structure of the data describing the wavelet scalogram of the interferometric signal may be used to estimate properties of the particle (e.g., the orientation of its path 7, particle speed v, particle size). It is to be understood that the same principles apply to structure of the data describing the Fourier scalogram (e.g., a spectrogram) of the interferometric signal, or to scalograms or spectrograms of other frequency transformations of the interferometric signal possessing these structural properties.

[0286] FIG. 12B and FIG. 12C show, schematically two different but parallel paths of a particle through a laser beam (FIG. 12B), and the resulting self-mixing interferometric signals and the respective wavelet scalograms for the self-mixing interferometric signals. The angle (=) at which that the linear path of a particle (7 or 7) approaches, and enters, the conical edge of the diverging beam 8a of laser light determines the frequency of the waveform of the interferometric signal generated by the laser interferometer 1 at that moment (t=t.sub.1) in time. Similarly, the angle (=.sub.n) that the same path of a particle approaches, and exits, the opposite conical edge of the diverging beam 8a of laser light determines the frequency of the waveform of the interferometric signal generated by the laser interferometer at that subsequent moment (t=t.sub.3) in time. The entry and exit angles are independent of the distance of the distance of the particle from the laser interferometer at those times. Consequently, the frequency of the waveform of the interferometric signal at its beginning and at its end, corresponding to the moments of entry and exit, respectively, of the particle into the conical laser beam, are also independent of the distance of the distance of the particle from the laser interferometer. The frequencies present within the interferometric signal waveform (90, 91) are determined by the rate at which a particle traverses successive wavefronts of the laser light that intersect the path (7, 7) of the particle through the laser beam. One full wave cycle within the interferometric signal waveform is achieved upon the particle passing through two successive wavefronts of the laser light (i.e., two wavefronts separated by one wavelength of the laser light).

[0287] Wavefronts, by definition, extend in a direction perpendicular to the direction of forward travel (i.e., the light ray direction) of the light wave. Consequently, a particle path subtending a given angle, , of incidence to the light rays defining the bounding edges of the laser beam also subtend a related angle, , relative to the wavefronts of the laser light at the bounding light ray (i.e., FIG. 12B: =). This related angle determines that initial rate at which the particle, of a given speed, can traverses successive wavefronts of the laser light that intersect the path and, therefore, determined the initial frequency of the interferometric waveform. The same holds true for the exit angle subtended by the particle path relative to the bounding ray of the laser light beam.

[0288] However, a closer particle 4 traverses a shorter path 7 through the conical beam, whereas a more distant particle 4 traverses a longer path within the conical beam. For a given speed v of particle, the time taken to traverse the conical beam increases in proportion to an increase in the distance of the distance of the particle from the laser interferometer. For a given speed v of particle, a more distant particle 4 has a path 7 that enters the cone of the laser beam at time: t=t.sub.1, and exits the cone of the laser beam at time: t=t.sub.3. At the intermediate time: t=t.sub.2, the path 7 become tangential to the wavefronts of the laser light signifying the closest approach of the particle to the laser resonator cavity of the interferometer. For the same given speed v of particle, a nearer particle 4 has a path 7 that enters the cone of the laser beam at time: t=t1, and exits the cone of the laser beam at time: t=t3. At the intermediate time: t=t2, the path 7 become tangential to the wavefronts of the laser light signifying the closest approach of the particle to the laser resonator cavity of the interferometer.

[0289] Each particle path crosses a number of curved wavefronts 10 of the laser light as that path (7, 7) crosses the laser beam. Each wavefront crossing corresponds to a full wave cycle in the waveform (90, 91) of the interferometric signal generated by the laser interferometer. A particle path 7 twice as far from the laser cavity of the interferometer may generate twice as many wave cycles within the interferometric waveform as are produced in the interferometric waveform generated by the same laser interferometer by a particle following a parallel path at half the distance from the laser. For example, in the schematic diagram of FIG. 12B, the more distant path 7 of the particle 4 is approximately (but not exactly) twice as far from the laser resonator cavity 1 as is the less distant path 7 of the particle 4. For this reason, the number of wave cycles in the waveform of the interferometric signal 90 corresponding to the more distant path 7 is approximately twice the number of cycles in the interferometric signal 91 corresponding to the less distant path 7. Each waveform of these two interferometric signals starts (at t=t.sub.1 or t=t.sub.1) with the same initial frequency .sub.Max1 when the particle enters the conical laser beam, reduces in frequency to an intermediate minimum value .sub.Min0 (c and subsequently ends (at t=t.sub.3 or t=t.sub.3) with the same final frequency .sub.Max when the particle exits the beam. Because [t.sub.3t.sub.2]>[t.sub.3t.sub.2], this means that the angle:

[00054]$={\tan }^{-1}\left(\left[{}_{\mathrm{Max}2}-{}_{\mathrm{Min}}\right]/\left[t_3-t_2\right]\right)$

subtended by the distribution of the data describing the parts of the scalogram 92 associated with the more distant particle part 7, is less than the angle:

[00055]${}^{}={\tan }^{-1}\left(\left[{}_{\mathrm{Max}2}-{}_{\mathrm{Min}}\right]/\left[t_3^{}-t_2^{}\right]\right)$

subtended by the distribution of the data describing the parts 94 of the scalogram associated with the less distant particle part 7. The same is true of the angles (not shown) subtended by the distribution of the data describing the parts 93 and 95 of the scalograms associated with the parts of the interferometer signals (90, 91) preceding the times t=t.sub.1 (or t=t.sub.1) and the times t=t.sub.2 (or t=t.sub.2). In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine the value of an angle subtended by the distribution of the data describing a selected part of the scalogram, e.g., as disclosed herein.

[0290] The scalogram data may be normalised, or re-calibrated, in order to mitigate against this effect of different distances of particle paths, for particles of a given speed v, upon the angle subtended by the distribution of the data describing the scalogram. FIG. 14 shows a scalogram (e.g., a wavelet scalogram) in which three data sets are represented concurrently to aid understanding and comparison. A first data set describes the scalogram data distribution 92, also shown in FIG. 12C, corresponding to the interferometric signal 90 arising from the more distant path 7 of the particle 4 traversing the laser beam 8a at speed v. This first data set produces a scalogram data distribution extending in a direction that subtends a first angle relative to the time axis of the scalogram. A second data set describes the scalogram data distribution 94, also shown in FIG. 12C, corresponding to the interferometric signal 91 arising from the less distant path 7 of the particle 4 traversing the laser beam 8a at speed v. This second data set produces a scalogram data distribution extending in a direction that subtends a second angle relative to the time axis of the scalogram.

[0291] As is noted in more detail below, the following relation exists:

[00056]$\frac{T}{d_0}=\frac{1}{v}\left[\frac{\sin \left(2\right)}{\sin \left(-\right)\sin \left(+\right)}\right]=\frac{T^{}}{d_0^{}}$

[0292] Here, T=[t.sub.3t.sub.2] and T=[t.sub.3t.sub.2]. The quantities d.sub.0 and d.sub.0 are the distances of closest approach of the more distant particle path 7 and the less distant particle path 7, occurring at times t=t.sub.2 and t=t.sub.2, respectively. Thus, if we assign the time interval T and particle distance d.sub.0 to be reference values, T.sup.(R) and d.sub.0.sup.(R), then a given interval of time T may be re-scaled to a normalising reference time scale defined by:

[00057]$T^{\left(R\right)}=T^{}\left(\frac{d_0^{\left(R\right)}}{d_0^{}}\right)$

[0293] More generally, a re-scaling may be applied according to:

T.fwdarw.NT

[0294] Here, the quantity N=(d.sub.0.sup.(R)/d.sub.0) is a re-scaling coefficient. In this way, the time interval T over which the second data set 94 extends may be normalised to mitigate against the effect upon the scalogram data distribution caused by the lesser distance d.sub.0. The effect is to stretch or compress (i.e., re-scale) the time interval described by the data such that the re-scaled scalogram data distribution 94B extends in a direction that subtends a re-scaled angle relative to the time axis of the scalogram. This re-scaling may achieve the result: @ relating to the fact that, in this example, the particles in question (4, 4) share the same speed v. In this way, the normalised angles (, ) subtended relative to the time axis of the scalogram that separate data sets produce in a respective scalogram data distribution may be used consistently in an estimation of the speed v of the particles in question. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine the value of a re-scaling coefficient and therewith to normalise of the data describing a selected part of the scalogram, e.g., as disclosed herein.

[0295] An angle subtended by the distribution of the data describing parts of the scalograms may be determined by applying a Hough transformation to the data representing the scalogram data. FIGS. 13A and 13B show graphical representations of stages in the implementation of a Hough transformation in an algorithm implemented by a self-mixing interferometer according to aspects of the invention. The Hough transformation seeks to identify a best-fitted regression line 61 through data points 61 in terms of the perpendicular distance r of the line from the origin of coordinates, and the angle .sub.H subtended by that perpendicular distance and the x-axis of the coordinate system. The output of the Hough transformation is a set of one of more points 63 within the Hough transform space each having a coordinate (r, .sub.H). A threshold value .sub.Thr) may be set as shown in FIG. 13B.

[0296] The processing module 1D may be configured to use structures in the scalograms by implementing the following algorithm in order to generate an estimate of the speed v of the particle passing through the laser beam. Referring to FIG. 18, here is shown a sequence of the following steps in this algorithm. [0297] Step S1: Acquire an interferometric signal corresponding to a new candidate event. Her the term candidate event refers to the event of a particle traversing the laser beam and causing generation of an interferometric signal. [0298] Step S2: Apply a wavelet transformation to the interferometric signal to obtain data describing a wavelet scalogram. [0299] Step S3: Apply a filter to the data describing the wavelet scalogram in order to enhance edge features of structures (e.g., data lobes) within the data describing the wavelet scalogram. For example, a kernel based filter may be used, such as a Canny Edge filter or the like. Extract line features from the filtered data. A Hough transform can be used for this Step S4: purpose, for example, as described below in more detail with reference to FIGS. 13A and 13B. [0300] Step S5: Identify whether the inclination of the extracted linear feature is sufficiently small. For example, one may determine if an angle

[00058]$\left(\frac{}{2}-{}_H\right)$

detected via the Hough transform space has a magnitude value below a pre-set threshold value (i.e., see, FIG. 13B,

[00059]$.Math.\frac{}{2}-{}_H.Math.>{}_{\mathrm{Thr}}).$

It is noted that the angle of inclination (.sub.Inc) of the extracted linear feature is related to the angle (.sub.H) detected in the Hough transform space as follows: [0301] .sub.Inc=/2.sub.H. Thus, a sufficiently large value of .sub.H may correspond to a desirably small value of angle of inclination (.sub.Inc). [0302] Step S6: If only one angle value is detected in the Hough transform space then determine that the scalogram data distribution possesses only one lobe (Monotone Curve). FIG. 11C and FIG. 11D are examples of this. [0303] Step S7: If Monotone Curve is determined at Step S6, then determine if the value of angle of inclination (mnc) is sufficiently small. This may be done by comparing the magnitude of the value of angle of inclination to a threshold inclination angle value (e.g., is |.sub.Inc|<.sub.Thr.sup.Inc ?). [0304] Step S8: If the value of angle of inclination (.sub.Inc) is sufficiently small then . . . , [0305] Step S9: Use the frequency of the interferometric signal as an acceptable frequency estimate with which to calculate the particle speed v. Here it is assumed that the frequency of the interferometric signal corresponds to a condition approaching the asymptotic condition described above. In other words, the smaller the value if the inclination angle of a data lobe of the wavelet scalogram, the smaller is the value of the particle path angle, , and the closer is the value of v cos to v (i.e., the closer cos is to 1.0). [0306] Step S10 Calculate an estimate of the particle speed using the accepted frequency, .sub.s, of the interferometric signal according to:

[00060]$v=\frac{{}_s}{4}$ [0307] Step S11: If the value of angle of inclination (.sub.Inc) is not sufficiently small then . . . , [0308] Step S12: Detect extremes of the event: If the extracted linear feature has a falling inclination, then detect the frequency maximum value at the start of the linear feature (i.e. .sub.s.sup.Max at t.sub.1), and detect the frequency minimum value at the end of the linear feature (i.e. .sub.s.sup.Min at t.sub.3). These correspond to the maximum and minimum frequency values present in the data single lobe of the scalogram. If the data lobe has a rising inclination in time, then detect the frequency maximum value at the end of the linear feature (i.e. .sub.s.sup.Max at t.sub.3), and detect the frequency maximum value at the start of the linear feature (i.e. .sub.s.sup.Min at t.sub.1). [0309] Step S13: If Step 5 reveals that two angle values are detected in the Hough transform space then determine that the scalogram data distribution possesses two lobes (V-Shape). FIG. 11A and FIG. 11B, as well as FIGS. 11E to 11H are examples of this. [0310] Step S14: Detect the vertices of the V-shape. If the extracted linear feature has a falling inclination, then detect the frequency maximum value at the start of the first linear feature (i.e. .sub.s.sup.Max1 at t.sub.1), and detect the frequency minimum value within the V-shape where the two linear features meet linear feature (i.e. .sub.s.sup.Min at t.sub.2), and detect the frequency maximum value at the end of the second linear feature (i.e. .sub.s.sup.Max2 at t.sub.3). These correspond to the maximum and minimum frequency values present in the two data lobes of the scalogram. [0311] Step S15: Select the greater of the two values .sub.s.sup.Max1 and .sub.s.sup.Max2. Select the angle of inclination as the angle associated with the linear feature associated with the selected angle. [0312] Step S16: Normalise the event duration (t.sub.3t.sub.1).fwdarw.N.Math.(t.sub.3t.sub.1), where N is a normalisation factor and recalculate the angle of inclination using the renormalised event duration as:

[00061]${}_{\mathrm{Inc}}={\tan }^{-1}\left[\frac{{}_s^{\mathrm{Max}}-{}_s^{\mathrm{Min}}}{N.Math.\left(t_3-t_1\right)}\right]$ [0313] In this step, the time scale of the extracted linear feature of the scalogram is re-scaled such that all events after this step are expressed in terms of this normalised time scale. This step serves to mitigate against a distance dependence otherwise present in the inclination angle of the extracted linear feature of the data lobe, as discussed above with reference to FIGS. 11A to 11H. The normalisation factor may be implemented using coefficients stored in pre-prepared look-up tables based on .sub.s.sup.Max, for Monotone and V-Shape data distributions. [0314] Step S17: Calculate a correction coefficient, E (where E>1), using a pre-stored correction coefficient look-up table. Each correction coefficient within this table is associated with a corresponding value of an inclination angle. Thus, the value of the new angle calculated at Step S16 may be associated with a corresponding correction coefficient within the table. This is done for both Monotone and V-Shape data distributions. Then, calculate the frequency estimate of the event as:

[00062]${}_s^i=E_i.Math.{}_s^{\mathrm{Max}}$ [0315] Step S18: Calculate a weight coefficient (w.sub.i<1) using pre-stored a weight coefficient look-up table. Each weight correction coefficient within this table is associated with a corresponding value of an inclination angle. Thus, the value of the new angle calculated at Step S16 may be associated with a corresponding correction coefficient within the table. This is done for both Monotone and V-Shape data distributions. [0316] Step S19: Calculate a new frequency estimate as the weighted average of previous frequency estimates of using the weight coefficients w; according to:

[00063]${}_s=\frac{{.Math.}_i{w}_i{}_s^i}{{.Math.}_i{w}_i}$ [0317] This frequency estimate is then used to calculate the particle speed according to:

[00064]$v=\frac{{}_s}{4}$

[0318] Finally, at the end of Steps S10 and S19, the list of terms in the weighted average may be updated. The updating relates to the summation operators used in the weighted average. The list of terms (w.sub.i and w.sub.s.sup.i) employed in these summations may be stored by the processing module of the self-mixing interferometer implemented in hardware using a memory buffer allotted a finite storage capacity. At a certain point in time the allotted storage capacity may be reached such that not enough memory resources exist to store all of the terms previously acquired. In such a case, the processing module of the self-mixing interferometer may be configured to limit the list of terms to a selection of terms, by discarding non-selected terms. For example, the processing module of the self-mixing interferometer may be configured to discard the earliest-acquired terms within the list as in a First In First Out (FIFO) buffer until the remaining number of terms are within the allotted storage capacity. In a different example, the processing module of the self-mixing interferometer may be configured to discard terms associated with a lesser weight (the terms in which we have a least confidence) by discarding terms associated with the lowest weight value from amongst all of the weight values within the list of terms, thereby prioritising terms associated with a greater weight (the terms in which we have a greater confidence). The discarding of lowest-weight terms may continue until the number of remaining terms are within the allotted storage capacity. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to implement such updating, e.g., as disclosed herein.

Particle Cross-Section

[0319] In the frequency domain, an incident plane wave of laser light of angular frequency .sub.L propagating along the z-axis through an isotropic medium of refractive index n.sub.0, can be expanded by Legendre functions and Hankel functions:

[00065]$e^{\mathrm{ikz}}=\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\frac{i^l}{2}\left[h_l^{\left(1\right)}\left(k_{0}r\right)+h_l^{\left(2\right)}\left(k_{0}r\right)\right]P_l\left(\cos \right)$

[0320] Here k.sub.0=w.sub.Ln.sub.0/c is the wavevector of the laser light, and the angle is measured relative to the spatial z-axis. When the plane wave is scattered by a spherical obstacle (i.e., a particle) of radius a located at the origin, the total wave outside of the sphere is axially symmetric about the z-axis:

[00066]$u\left(r,,{}_L\right)=\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\frac{i^l}{2}\left[S_l\left({}_L\right)h_l^{\left(1\right)}\left(k_{0}r\right)+h_l^{\left(2\right)}\left(k_{0}r\right)\right]P_l\left(\cos \right)$

[0321] Here,

[00067]$u\left(r,,{}_L\right)={}_{-}^{}\hat{u}\left(r,,t\right)e^{i{}_{L}t}\mathrm{dt}$

[0322] The coefficient S.sub.l(.sub.L) is determined from the boundary conditions requires of the scattering wave at the surface of the scattering particle, ensuring continuity of the wave across the boundary from within the particle to outside it. At large distances relative to the dimensions of the laser wavelength and the scattering particle, such that k.sub.0r>>1, the asymptotic behaviour of the laser light wave traveling at an angle relative to the spatial axis z, and including the portion of light wave scattered by the particle, is given by:

[00068]$u\left(r,,{}_L\right)\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\frac{1}{2{\mathrm{ik}}_{0}r}\left[S_l\left({}_L\right)e^{{\mathrm{ik}}_{0}r}+{\left(-1\right)}^{l+1}{e}^{-{\mathrm{ik}}_{0}r}\right]P_l\left(\cos \right)$$u\left(r,,{}_L\right)e^{{\mathrm{ik}}_{0}z}+\frac{e^{{\mathrm{ik}}_{0}r}}{r}\frac{1}{2{\mathrm{ik}}_0}\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\left[S_l\left({}_L\right)-1\right]P_l\left(\cos \right)$

[00069]$u\left(r,,{}_L\right)e^{{\mathrm{ik}}_{0}z}+f\left(,{}_L\right)\frac{e^{{\mathrm{ik}}_{0}r}}{r}$

[0323] Here,

[00070]$f\left(,{}_L\right)=\frac{1}{2{\mathrm{ik}}_0}\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\left[S_l\left({}_L\right)-1\right]P_l\left(\cos \right)$

[0324] Also, r is the distance of the wave from the scattering particle, and f(, .sub.L) is the scattering amplitude. The boundary conditions determining the value of S.sub.l(.sub.L) provide continuity of u(r,,.sub.L) and u(r,, .sub.L) on the spherical surface (r=a) which requires that:

[00071]$S_l\left({}_L\right)=\frac{k_0{j}_l\left(k_{1}a\right)h_l^{\left(2\right)}\left(k_{0}a\right)+k_1{j}_l^{}\left(k_{1}a\right)h_l^{\left(2\right)}\left(k_{0}a\right)}{k_1{j}_l^{}\left(k_{1}a\right)h_l^{\left(1\right)}\left(k_{0}a\right)+k_0{j}_l\left(k_{1}a\right)h_l^{\left(1\right)}\left(k_{0}a\right)}$

[0325] The result is that:

[00072]$f\left(,{}_L\right)=\frac{i}{k_0}\frac{k_0{j}_l\left(k_{1}a\right)j_l^{}\left(k_{0}a\right)+k_1{j}_l^{}\left(k_{1}a\right)j_l\left(k_{0}a\right)}{k_0{j}_l\left(k_{1}a\right)h_l^{\left(1\right)}\left(k_{0}a\right)+k_0{j}_l^{}\left(k_{1}a\right)h_l^{\left(1\right)}\left(k_{0}a\right)}$

[0326] Here k.sub.1=w.sub.Ln.sub.1/c is the wavevector of the laser light within the particle comprised of a material of refractive index n.sub.1. The functions j.sub.l, j.sub.l, h.sub.l.sup.(1), h.sub.l.sup.(1), h.sub.l.sup.(2), h.sub.l.sup.(2) are, respectively, a spherical Bessel function of the first kind (order l), the derivative of a spherical Bessel function of the first kind (order l), a spherical Hankel function of the first kind (order l), the derivative of a spherical Hankel function of the first kind (order l), a spherical Hankel function of the second kind (order l), the derivative of a spherical Hankel function of the second kind (order l).

[0327] The key fact to note from this result is that the scattering amplitude f(, .sub.L) is a function of the properties of the scattering particle, as well as the properties of the scattered laser light. In particular, the magnitude of the scattering amplitude depends upon the magnitude of the radius, a, of the scattering particle. The differential scattering cross-section of the scattering particle is given by:

[00073]$\frac{d\left(,{}_L\right)}{d}={.Math.f\left(,{}_L\right).Math.}^2$

[0328] The integral of this quantity over the whole solid angle gives the total scattering cross-section as:

[00074]${}_0\left(,{}_L\right)=\frac{d\left(,{}_L\right)}{d}d={.Math.f\left(,{}_L\right).Math.}^{2}d$

[0329] The inventors have realised that an estimate of the radius, a, of the scattering particle may be obtained from the coefficients of a frequency-domain transformation (e.g., a wavelet transformation) of the self-mixing interferometric signal modulation of the laser:

[00075]$P_{\mathrm{Laser}}=A+B\cos \left({}_s.Math.t+{}_0\right)$

[0330] In other words, the magnitude of the modulating coefficient B, is found to depend upon the differential scattering cross-section |f(, .sub.L)|.sup.2 which, in turn, depends upon the scattering amplitude f(, .sub.L) which is a function of the properties of the scattering particle. The inventors have realised that the modulating coefficient B is a measure of the strength of coupling between the light returned by the scattering particle, and the internal laser cavity of the interferometer, and that the strength of coupling is determined by the amount of returned light and, therefore, by the magnitude of the differential scattering cross-section |f(, .sub.L)|.sup.2 in the back-scattering direction from the particle back to the laser cavity (i.e., the scattering angle = such that |f(, .sub.L)|.sup.2=|f(, .sub.L)|.sup.2).

[0331] The inventors have appreciated that because the scattering amplitude comprises a sum of Legendre functions each having a particular coefficient, then this will be reflected in the coupling of the scattered laser light within the interferometer laser cavity via the modulating coefficient B. Since a frequency-domain transformation of the self-mixing interferometric signal comprises a transformation of the modulating coefficient B, then the coefficients of the frequency-domain transformation captures information regarding the coefficients of the terms forming the scattering amplitude:

[00076]$f\left(,{}_L\right)=\frac{1}{2{\mathrm{ik}}_0}\underset{l=0}{\overset{}{.Math.}}\left(2l+1\right)\left[S_l\left({}_L\right)-1\right]P_l\left(\cos \right)$

[0332] The inventors have realised that an estimate of the radius of a scattering particle may be made from a summation of the squared modulus of the magnitudes of the coefficients (W.sub.k,Laser.sup.j) of the frequency-domain transformation applied to the self-mixing interferometric signal of the laser:

[00077]$\underset{j,k}{.Math.}{.Math.W_{k,\mathrm{Laser}}^j.Math.}^2{.Math.f\left(,{}_L\right).Math.}^2$

[0333] Here, for example, one may use a wavelet transformation, with desired mother wavelet

[00078]${}^{*}\left(\frac{t-b}{a}\right),$

as follows:

[00079]$W_{k,\mathrm{Laser}}^j\left(,\right)=\frac{1}{\sqrt{}}{P}_{\mathrm{Laser}}\left(t\right){}^{*}\left(\frac{t-}{}\right)\mathrm{dt}$

[0334] Here, =2.sup.j and =k2.sup.j. Of course, P.sub.Laser(t)=A+B cos(.sub.st+.sub.0) as noted above. The integer j is the scale parameter and the integer k is the shift parameter. Other wavelet transformations may be used. Indeed, the coefficients (W.sub.Laser) of a frequency transformation of the self-mixing interferometric signal may be used.

[0335] For example, the radius, a, of the scattering particle may be estimated according to:

[00080]$a=\frac{}{T}\underset{j,k}{.Math.}{.Math.W_{k,\mathrm{Laser}}^j.Math.}^2$

[0336] Here, the coefficient is a pre-set calibration coefficient, and the time interval T is the time interval of the self-mixing interferometric signal over which the frequency-domain transformation is applied. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a value of the radius of particulate material, e.g., as disclosed herein. As discussed below, this division by the time interval T is a time normalisation process that ensures a correspondence between the energy in within the self-mixing interferometric signal and the energy within the frequency-domain transformation that is applied to that signal. The time normalisation process takes account of the fact that different energies exist at different time scales of a frequency transformation of the self-mixing interferometric signal. For example, with reference to FIG. 12C, this time interval may be: T=t.sub.3t.sub.2. The value of the pre-set calibration coefficient may be determined by pre-calibration performed in controlled conditions wherein a range of particles of varying known radius and speed are passed through the laser beam at a range of different known path directions/orientations and distances from the laser which produces a laser beam of specified beam geometry and beam intensity. The sum of the squared modulus of the magnitudes of the coefficients (W.sub.k,Laser.sup.j) of the frequency-space transformation of the self-mixing interferometric signal may be measured for each individual combination of: radius; speed; path direction/orientation; distance from the laser, included in the pre-calibration process. A look-up table or other addressable data array may be generated in which an input value of the sum of the squared modulus of the magnitudes of the coefficients (W.sub.k,Laser.sup.j) of the frequency-space transformation of the self-mixing interferometric signal results in an output value, from the look-up table, of the pre-set calibration coefficient associated with that input value. This pre-set calibration coefficient may also be associated with pre-calibrated values of: speed; path direction/orientation; distance from the laser for a particle. Accordingly, a combination of input values comprising two or more of: the sum of the squared modulus of the magnitudes of the coefficients (W.sub.k,Laser.sup.j) of the frequency-space transformation; the associated measured particle speed; the associated measured particle path direction/orientation; the associated measured particle distance from the laser, may be used to return a pre-set calibration coefficient from the look-up table or addressable data array using these measurements.

Windowed Fourier Transform

[0337] The Windowed Fourier transform (WFT), or Short-time Fourier transform (STFT), is an example of a Fourier transformation tool for extracting local-frequency information from a signal x(t). For a continuous-time WFT/STFT, the data to be transformed is multiplied by a window function, w, defining a sliding segment of length T which is nonzero for only a short period of time:

[00081]$\mathrm{WFT}\left[x\left(t\right)\right]=W\left(T,\right)={}_{-}^{}x\left(t\right)w\left(t-T\right)e^{-it}\mathrm{dt}$

[0338] For a discrete-time WFT/STFT, the discrete data x.sub.n to be transformed may be broken up into frames:

[00082]$\mathrm{WFT}\left[x_n\right]=W\left(m,\right)=\underset{n=-}{\overset{}{.Math.}}{x}_{n}w\left[n-m\right]e^{-im}$

[0339] The segments can be windowed with an arbitrary function, (w(tT) or w[nm]), such as a Gaussian window or a boxcar (no smoothing) window. In the present invention the signal x(t)=P.sub.Laser(t), and the discrete data x.sub.n corresponds to discrete data samples [P.sub.Laser].sub.n taken at discrete time points (n) within the self-mixing laser interferometer signal. The process of applying a WFT/STFT to a time-series of data is well known to the skilled person and will not be discussed in detail here. However, it is noted that the Fourier transform is performed on the sliding segment from a time series of time step t and total length Not, thus returning frequencies from (T1) to (2t1) at each time step. The WFT/STFT imposes a scale T into the analysis, and the inventors have realised that this can result in some aliasing of low-frequency and high-frequency components that do not fall within the frequency range of the window. The inventors have found that in aspects of the present invention, analyses using a variable scale rather than a fixed or predetermined scaling is more effective because of the presence of a range of frequencies within the self-mixing interferometer signal. It has been found that a method of time-frequency analysis that is scale independent, such as wavelet analysis, is very effective in the context of the aspects of the present invention. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material using a Windowed Fourier transform (WFT), or Short-time Fourier transform (STFT), e.g., as disclosed herein.

Wavelet Transform

[0340] The wavelet transform may be used to analyse time-series data that contain power at many different frequencies. In a time series, x.sub.n, with equal time spacing St and n=0 . . . . N1, one may employ a mother wavelet function which is a function of a non-dimensional time parameter, , such that:

[00083]$\left(\right);=\frac{t-}{}$

[0341] An example is the Morlet wavelet, consisting of a plane wave, of dimensionless frequency .sub.0, modulated by a Gaussian modulation envelope:

[00084]$\left(\right)={}^{-1/4}{e}^{i{}_0}{e}^{-{}^2/2}$

[0342] The term wavelet basis refers only to an orthogonal set of functions. The use of an orthogonal basis implies the use of the discrete wavelet transform, while a nonorthogonal wavelet function can be used with either the discrete or the continuous wavelet transform. In the present disclosure, only the continuous transform is used in the examples given, however it is to be understood that all of the results and methods involving the continuous transform are applicable to the discrete wavelet transform.

[0343] The wavelet transform of a discrete time sequence of data, x.sub.n, is defined as the convolution of x.sub.n with a scaled and translated version of (), such that:

[00085]$W_n\left(\right)=\underset{n^{}=0}{\overset{N-1}{.Math.}}{x}_n{}^{*}\left(\frac{\left[n-n^{}\right]t}{}\right)$

[0344] Here, N is the number of points in the time series. By varying the wavelet scale, a, and translating it along the localised time index, n, one may determine the wavelet amplitudes of any features vary with scale and time. To approximate the continuous wavelet transform, the above convolution may be done N times for each scale, a. The convolution theorem allows all N convolutions to be calculated in Fourier space using a discrete Fourier transform (DFT). The DFT of x.sub.n is:

[00086]${\hat{x}}_k=\frac{1}{N}\underset{n=0}{\overset{N-1}{.Math.}}{x}_n{e}^{-i2\mathrm{kn}/N}$

[0345] Here, k=0 . . . . N1 is the frequency index. Given that the Fourier transform of a function (t/) is equal to {circumflex over ()}(), and applying the convolution theorem, the wavelet transform is the product of the Fourier transforms as follows:

[00087]$W_n\left(\right)=\underset{k=0}{\overset{N-1}{.Math.}}{\hat{x}}_k{\hat{}}^{*}\left({}_k\right)e^{-i{}_{k}nt}$

[0346] Here,

[00088]${}_k=\{\begin{array}{cc}\frac{2k}{Nt}:& kN/2\\ -\frac{2k}{Nt}:& k>N/2\end{array}$

[0347] Examples of wavelet basis functions suitable for use in aspects of the present invention include, but are not limited to, the following: Morlet; Meyer; Paul; Derivative of Gaussian (DOG); Mexican Hat; Daubechies. The Morlet wavelet basis functions may be complex. The Daubechies wavelet basis functions may be of any suitable order (e.g., order=1 (Haar wavelet); order=2, order=4, order=8, etc.). In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine a property of particulate material using a continuous wavelet transform (CWT), or discrete wavelet transform (DWT), e.g., as disclosed herein.

Transform Energy and Particle Scattering Cross-Section

[0348] Suppose a time series f(t) of data represented by a wavelet basis:

[00089]${}_n^j\left(t\right)=\frac{1}{\sqrt{2^j}}\left(\frac{t-n{2}^j}{2^j}\right)$

[0349] Here j and n are integers. The scale is =2.sup.j and the integer n gives the time location as =n2.sup.j. The range of the time location factor n is 2.sup.j. Here, is the mother wavelet that forms an orthonormal set with respect to j and n. The wavelet transform of f(t), and the reconstruction of f(t) from its wavelet transform coefficients are, respectively, given by:

[00090]${\stackrel{}{f}}_n^j=f\left(t\right){}_n^j\left(t\right)\mathrm{dt}$$f\left(t\right)=f_0+\underset{j=0}{\overset{}{.Math.}}\underset{n=0}{\overset{2^j-1}{.Math.}}{\stackrel{}{f}}_n^j{}_n^j\left(t\right)$

[0350] Here, j0, 1 . . . 2.sup.j1, and f.sub.0 is related to scales greater than the time span of the data. Note that, on a specific scale level j, the reconstruction of the time series is the projection of f(t) onto the subspace containing only features of scale 2.sup.j, and is given by:

[00091]$f^j\left(t\right)=\underset{n=0}{\overset{2^j-1}{.Math.}}{\stackrel{}{f}}_n^j{}_n^j\left(t\right)$

such that:

[00092]$f\left(t\right)=f_0+\underset{j=0}{\overset{}{.Math.}}{f}^j\left(t\right)$

[0351] According to Parseval's relation:

[00093]${}_0^t{\left[f^j\left(t\right)\right]}^2\mathrm{dt}=\underset{n=0}{\overset{2^j-1}{.Math.}}{\left[{\stackrel{}{f}}_n^j\right]}^2$

[0352] The left-hand-side of the above equation represents the energy within j.sup.th component of the time-series signal f(t) associated with features of scale level 2.sup.j. The right-hand-side of the equation is the sum of the 2.sup.j=N parts each of which represents the energy on a small interval centred around the time t.sub.n=n/2.sup.j, having a time duration of t.sub.j=1/N=1/2.sup.j. Noting that the right-hand-side can be expressed as a Riemann sum over N parts, according to the middle value theorem, we have:

[00094]${}_0^t{\left[f^j\left(t\right)\right]}^2\mathrm{dt}=\underset{n=0}{\overset{2^j-1}{.Math.}}{\left[f^j\left(t_n\right)\right]}^2{t}_j$

[0353] Thus, [f.sup.j(t.sub.n)].sup.2 is the energy E.sub.n.sup.j at location n for scale 2.sup.j. Comparing this result with Parseval's relation gives:

[00095]$\underset{n=0}{\overset{2^{j}1}{.Math.}}{\left[f^j\left(t_n\right)\right]}^2{t}_j=\underset{n=0}{\overset{2^{j}1}{.Math.}}{\left[{\stackrel{}{f}}_n^j\right]}^2$

[0354] This means that:

[00096]$E_n^j=\frac{{\left[{\stackrel{}{f}}_n^j\right]}^2}{t_j}$

[0355] Accordingly,

[00097]$\underset{j=0}{\overset{}{.Math.}}\underset{n=0}{\overset{2^j-1}{.Math.}}{\left[{\stackrel{}{f}}_n^j\right]}^2=\underset{j=0}{\overset{}{.Math.}}{t}_j\underset{n=0}{\overset{2^j-1}{.Math.}}{E}_n^j$

[0356] If the value of the energy is approximately evenly distributed across the scales, j, of the features of the signal, then the energy values E.sub.n.sup.j are approximately independent of the scale integer j and one may reverse the order of summation as follows:

[00098]$\underset{j=0}{\overset{}{.Math.}}\underset{n=0}{\overset{2^j-1}{.Math.}}{\left[{\stackrel{}{f}}_n^j\right]}^2=\underset{n=0}{\overset{2^j-1}{.Math.}}{E}_n^j\underset{j=0}{\overset{}{.Math.}}{t}_j=t\underset{n=0}{\overset{2^j-1}{.Math.}}{E}_n^j$

[0357] Therefore,

[00099]$\underset{n=0}{\overset{2^j-1}{.Math.}}{E}_n^j=\mathrm{Total}\mathrm{Energy}=\frac{1}{t}\underset{j=0}{\overset{}{.Math.}}\underset{n=0}{\overset{2^j-1}{.Math.}}{\left[{\stackrel{}{f}}_n^j\right]}^2$

[0358] Here,

[00100]$t=\underset{j=0}{\overset{}{.Math.}}{t}_{j}T$

[0359] The inventors have realised that by summing, over scales and time positions (i.e., all n and j), the square of the modulus of the wavelet coefficients (i.e., let {circumflex over (f)}.sub.n.sup.j=W.sub.n,Laser.sup.j) generated by applying a frequency transformation (e.g., a wavelet transform) to the self-mixing interferometer signal (i.e., let f(t)=P.sub.Laser(t)), and dividing the result by the duration (T) of that interferometer signal, one may generate a consistent estimate of the energy of the time-series signal. The time normalisation factor (T) takes account of the fact that different energies exist at different time scales of a frequency transformation of the self-mixing interferometric signal. From that energy estimate, and noting the correlation between the signal energy and the scattering cross-section of the particle responsible for providing the returned light causing the self-mixing interferometer signal to arise, one may generate an estimate of the particle radius as noted above, as follows:

[00101]$a=\frac{}{T}\underset{j=0}{\overset{}{.Math.}}\underset{n=0}{\overset{2^j-1}{.Math.}}{.Math.W_{k,\mathrm{Laser}}^f.Math.}^2$

[0360] It is noted that self-mixing interferometric signals of relatively longer duration will generally include wavelet coefficients of relatively longer/larger time scales, meaning inclusion of significant coefficient terms with larger values of the scale integer j associated with the scale factor =2.sup.j. The time normalisation factor (T) takes account of this. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this property of particulate material, e.g., as disclosed herein.

Wavelet Coefficients and Particle Size

[0361] As discussed above, the feedback level (C) within the laser resonator cavity of a self-mixing interferometer is given by:

[00102]$C=\frac{{}_{ext}}{{}_{int}}\sqrt{1+{}^2}$

[0362] The optical coupling of the light returned from the particle and the light within the resonator cavity of the laser results in the excess phase equation:

[00103]${}_{\mathrm{FB}}={}_s+C\sin \left({}_{\mathrm{FB}}+{\tan }^{-1}\right)$

[0363] The term .sub.FB represents the phase accumulated by the electromagnetic field of the laser on feed-back (FB) transmission through the external cavity. The term os corresponds to the phase accumulated by transmission through the external cavity if the laser were not experiencing optical feedback. The feedback level C determines the degree of nonlinear coupling within the laser resonator cavity. The disclosures herein are mostly, but not exclusively, concerned with the weak feedback regime in which C.fwdarw.0 such that no nonlinear coupling exist. However, the inventors have realised that use can be made of small but non-negligible values of the feedback level C. The dependence of the phase term .sub.FB and the interferometric signal generated by the laser cavity becomes as follows:

[00104]$\begin{array}{c}{}_{FB}={}_s+C\sin \left({}_{FB}+{\tan }^{-1}\right)\\ P_{\mathrm{Laser}}=A+B\cos \left[{}_s+C\sin \left({}_{FB}+{\tan }^{-1}\right)\right]\end{array}$

[0364] The inventors have found that a small but not insignificant appearance of spiky signal components arise increasingly within a self-mixing interferometer signal when feedback-induced instabilities begin to appear at small but not insignificant feedback levels C. The inventors have realised that the more spiky a mother wavelet is the better it is at representing the these spiky signal features. The inventors have devised a technique by which these spiky signal features can be observed by comparing wavelet transformations of the same signal. The inventors have found that smoother mother wavelets (e.g., Morlet) having a relatively lower maximum value of first derivative and/or second derivative (with respect to time) will give stronger wavelet coefficients at a lower, or negligible, feedback level C while spikier mother wavelets (e.g., Daubechies) having a relatively higher maximum value of first derivative and/or second derivative (with respect to time) will give higher wavelet coefficients at higher, or non-negligible, feedback levels C. A suitable comparison between smoother and spikier mother wavelets has been found to be possible according to the ratio of the moduli of wavelet coefficients generated from two different types of mother wavelet. Preferably, the wavelet coefficients being compared correspond to the same scale and the same time location .

[0365] The inventors have discovered that an estimate of the radius, a, of a particle responsible for providing the returned light causing the self-mixing interferometer signal P.sub.Laser to arise, may be determined from a ratio of the moduli of wavelet coefficients generated from two different types of mother wavelet wherein, for example, each wavelet basis has the general form:

[00105]$\begin{array}{c}{\left[{}_{}^{}\left(t\right)\right]}_{\mathrm{TypeM}}=\frac{1}{\sqrt{}}{}_M\left(\frac{t-}{}\right)\\ {\left[{\stackrel{}{f}}_{}^{}\right]}_{\mathrm{TypeM}}={P_{Laser}\left(t\right)\left[{}_{}^{}\left(t\right)\right]}_{\mathrm{TypeM}}\mathrm{dt}\end{array}$

[0366] The estimate of the radius, a, of a particle may be defined as:

[00106]$a={}_d\frac{{.Math.\left[{\stackrel{}{f}}_{}^{}\right].Math.}_{\mathrm{Type}1}}{{.Math.\left[{\stackrel{}{f}}_{}^{}\right].Math.}_{\mathrm{Type}2}}$

[0367] Here, the coefficient .sub.d is a pre-set calibration coefficient. That is to say, the inventors have discovered that, at a given distance, d, from the laser resonator cavity of the self-mixing interferometer, the radius of a particle, a, may be sufficiently large to cause a non-negligible feedback levels C to occur within the laser resonator cavity. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this property of particulate, e.g., as disclosed herein. The onset of these conditions are determined by the radius of the particle, which influences its scattering cross-section, and its distance from the laser resonator cavity which influences the intensity of the returned light injected into the laser resonator cavity. These two parameters, particle radius and distance, are found to be correlated to the onset and magnitude of the feedback level C which are detectable via a comparison of different types (i.e., spike vs. not spiky) of mother wavelets used to represent the same self-mixing interferometer signal.

[0368] For example, wavelet coefficients, |{circumflex over (f)}.sub.n.sup.j|, may be generated from two different types of mother wavelet and corresponding to the same scale =2.sup.j and the same time location =n2.sup.j, as follows:

[00107]$\begin{array}{c}{\left[{}_n^j\left(t\right)\right]}_{\mathrm{TypeM}}=\frac{1}{\sqrt{2^j}}{}_M\left(\frac{t-n{2}^j}{2^j}\right)\\ {\left[{\stackrel{}{f}}_n^j\right]}_{\mathrm{TypeM}}={P_{Laser}\left(t\right)\left[{}_n^j\left(t\right)\right]}_{\mathrm{TypeM}}dt\end{array}$

[0369] Such that:

[00108]$a={}_d\frac{{.Math.\left[{\stackrel{}{f}}_n^j\right].Math.}_{\mathrm{Type}1}}{{.Math.\left[{\stackrel{}{f}}_n^j\right].Math.}_{\mathrm{Type}2}}$

[0370] The value of the pre-set calibration coefficient .sub.d may be determined by pre-calibration performed in controlled conditions wherein a number of particles each having a different known radius a are separately passed through the laser beam of specified beam geometry and beam intensity, at a number of pre-set distances d (or average distance d) from the laser resonator cavity of the self-mixing interferometer. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this property of particulate, e.g., as disclosed herein. The ratio of the moduli of wavelet coefficients generated from two different types of mother wavelet applied to the same self-mixing interferometric signal may be measured for each individual known particle radius for each individual pre-set distance included in the pre-calibration process. A look-up table or other addressable data array may be generated in which the input values of the measured ratio of the moduli of wavelet coefficients and the measured estimate of the distance d (or average distance d) results in an output value, from the look-up table, of the pre-set calibration coefficient .sub.d associated with those input values. The estimate of the distance d (or average distance d) may be obtained using any of the methods disclosed herein. This pre-set calibration coefficient .sub.d may also be associated with pre-calibrated values of: speed; path direction/orientation. Accordingly, a combination of input values comprising two or more of: the ratio of the moduli of wavelet coefficients; an associated measured particle speed; an associated measured particle path direction/orientation; an associated measured particle distance from the laser, may be used to return a pre-set calibration coefficient .sub.d from the look-up table or addressable data array using these measurements.

[0371] The two different types of mother wavelet differ in the sense that the maximum value of the time derivative of the mother wavelet of one type exceeds the maximum value of the time derivative of the mother wavelet of the other type. In other words, one mother wavelet is spikier than the other. Examples of suitable types of mother wavelets from which to select the include, but are not limited to, the following: [0372] TYPE1 (less spiky): Morlet; Meyer; Paul; DOG; Mexican Hat; . . . [0373] TYPE2 (more spiky): Daubechies1; Daubechies2; Daubechies4; . . . . Daubechies (order: N>4); . . .

[0374] For example:

[00109]$a={}_d\frac{{.Math.\left[{\stackrel{}{f}}_{}^{}\right].Math.}_{\mathrm{Morlet}}}{{.Math.\left[{\stackrel{}{f}}_{}^{}\right].Math.}_{\mathrm{Deubechies}2}}$

Frequency-Space Scalogram Structures and Particle Size

[0375] The inventors have discovered that structures present in the data representing a wavelet scalogram or a Fourier scalogram (e.g., spectrogram) may be used to provide an estimate of the radius a of the particle to which the scalogram relates. FIG. 15 shows a schematic representation of a scalogram, such as a wavelet scalogram corresponding to a self-mixing interferometer signal. The scalogram is shown as comprising a succession of short segments (97, 96, 95, 92) each of which span a frequency range of interval size defining a short frequency bin. Each short segment of the scalogram spans a frequency bin of the same interval size but spanning a unique interval of frequencies in which the end of one interval is contiguous with the start of the next interval.

[0376] The frequency range spanned by the interval size is selected such that the distribution of the data describing the scalogram within that frequency range is approximately linear, or can be sufficiently accurately represented as having a substantially constant (or sufficiently slowly changing) value of slope, S.sub.n, defined as:

[00110]$S_n=\frac{}{T_n}$

[0377] Here, the time interval T.sub.n is the interval of time over which the data describing the scalogram within the n.sup.th interval extends. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this slope value, e.g., as disclosed herein. Here, increasing the integer n corresponds to increasing time. It is found that if the scalogram is a wavelet scalogram then the value of slope, S.sub.n, tends to progressively diminish at locations along the distribution of the scalogram data located at progressively higher frequency, , closer to outer terminal ends of the data distribution in the scalogram. In particular, the slope S.sub.n of the n.sup.th interval defines an angle .sub.n=tan.sup.1(S.sub.n) which is smaller as the time interval T.sub.n in question associated with ever higher frequency values becomes ever longer, while the interval remains constant in size. This arises due to a path of a particle being increasingly closer to an orientation that is perpendicular to the wavefronts of the laser light within the laser beam it traverses. For a given particle velocity, the highest possible frequency component of the self-mixing interferometric signal occurs if path of a particle has an orientation that is perpendicular to the wavefronts of the laser light. If a particle path, for a given speed of particle, is not perpendicular to the wavefronts of the laser light, it may still become closer to achieving (without actually achieving) a perpendicular orientation when it is the furthest from the laser resonator, whilst still being within the laser beam. Of course, for a laser beam of finite beam divergence angle, , there comes a point when the particle is no longer (or not yet) within the beam and this approach towards a perpendicular orientation cannot continue. However, the greater the distance from the laser resonator cavity that the particle can achieve whilst still being able to return to the laser resonator cavity a sufficient amount of reflected/scattered light to induce a self-mixing interferometer signal, then the greater will be the highest frequency component of the induced self-mixing interferometer signal. Accordingly, the smaller will be the slope S.sub.n of the scalogram data distribution associated with the frequency bin containing that highest frequency component. In FIG. 15, one can see a succession of time intervals T.sub.1<T.sub.2<T.sub.3<T.sub.end each associated with data corresponding to a frequency bin centred upon a respective frequency located at successively higher frequency value and defining a respective segment (97, 96, 95, 92) of the data representing the scalogram. The terminal data segment 92 has the smallest value of slope S.sub.n=S.sub.end and the smallest value of segment angle .sub.end=tan.sup.1(S.sub.end). Accordingly, this segment of the data representing the scalogram corresponds to a situation in which the orientation of the particle path had made its closest approach towards a perpendicular orientation relative to the wavefronts of the laser beam bathing it. The inventors have found that larger particles are able to return to the laser resonator cavity a sufficient amount of reflected/scattered light to induce a self-mixing interferometer signal whilst at a greater distance from the laser resonator cavity simply because they have a greater scattering cross-section. This means that the larger a particle is, the smaller will be the value of the slope S.sub.end of the data describing the terminal segment of the scalogram of the self-mixing interferometer signal associated with that particle. The inventors have found that for sufficiently small values of the slope S.sub.end of the data describing the terminal segment of the scalogram, the inverse value of the slope correlates to maximum distance d.sub.Max that a particle can be from the laser resonator cavity while returning a sufficient amount of reflected/scattered light to induce a self-mixing interferometer signal the radius. In turn, that sufficient amount of reflected/scattered light is correlated with the radius, a, of the particle in question such that:

[00111]$a=\frac{T_{end}}{}=\frac{}{S_{end}}$

[0378] Here, is a pre-set calibration coefficient. In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this property of particulate, e.g., as disclosed herein. For example, one may apply:

[00112]$S_{end}={\left[\tan \left(\frac{}{2}-{}_H\right)\right]}_{end}$

[0379] Here, .sub.H is an angle of inclination detected via a Hough transform space as discussed above with reference to FIG. 13A and FIG. 13B. Preferably, one may apply the following thresholding constraint that:

[00113]$S_{end}<S_{\mathrm{Threshold}}$

[0380] Here, the value of S.sub.Threshold may be determined by a calibration process and according to user preferences. Smaller values of S.sub.Threshold will ensure that the permitted values of S.sub.end used to calculate the particle radius, correspond to particle paths oriented closer to being perpendicular to the wavefronts of the light of the laser beam. The value of the pre-set calibration coefficient may be determined by pre-calibration performed in controlled conditions wherein a number of particles each having a different known radius a.sub.Calib are separately passed through the laser beam of specified beam geometry and beam intensity, at a number of pre-set orientations (angle relative to the beam axis) sufficient in value to achieve the condition S.sub.end<S.sub.Threshold in the resulting scalogram, and from that data a value =a.sub.calibS.sub.end may be assigned to that laser.

Signal Duration and Distance

[0381] The inventors have discovered that an estimate, (d), of the distance of a particle from a self-mixing laser interferometer cavity, may be made according to the number, N.sub.Cycles, of wave cycles present in the waveform of the self-mixing interferometric signal, P.sub.Laser(t), generated by that laser cavity as follows:

[00114]$.Math.d.Math.=N_{\mathrm{Cycles}}$

[0382] Here, is a pre-set calibration coefficient. The value of this pre-set calibration coefficient may be determined for the laser interferometer in question, which produces a laser beam of specified wavelength and beam geometry, by pre-calibration performed in controlled conditions wherein particles are passed through the laser beam at a range of different known distances, d, from the laser resonator cavity. The number of wave cycles present in the waveform of the self-mixing interferometric signal may be measured for each individual value of calibration distance d from the laser, included in the pre-calibration process. A look-up table or other addressable data array may be generated in which an input value of the number, N.sub.cycles, of wave cycles in the self-mixing interferometric signal results in an output value, from the look-up table, of the pre-set calibration coefficient associated with that laser. This pre-set calibration coefficient may be a constant value that is independent of the number, N.sub.Cycles, of cycles in the self-mixing interferometric signal, in which case the look-up table may be unnecessary, and the value of the pre-set calibration constant may be explicitly expressed as a constant numerical value in the above equation defining the distance estimate d. Alternatively, if the calibration process determines that, for a given laser bean geometry and laser wavelength, the pre-set calibration coefficient shows a non-negligible variation with the value of N.sub.cycles (i.e., that =(N.sub.Cycles) is a function of N.sub.Cycles), then the look-up table may capture this variation and provide a more accurate calibration coefficient providing a better estimate of the particle distance d in terms of the value of the number, N.sub.Cycles, of cycles in the self-mixing interferometric signal.

[0383] FIGS. 12B and 12C show schematic examples of how the distance (d) of a particle (4, 4) from a laser interferometer cavity 1 influences the number of curved wavefronts 10 of the laser light that the path (7, 7) of the particle must cross as it traverses the laser beam. Each wavefront crossing corresponds to a full wave cycle in the waveform (90, 91) of the interferometric signal generated by the laser interferometer. A particle path 7 twice as far from the laser cavity of the interferometer may generate twice as many (N.sub.Cycles) wave cycles within the interferometric waveform as are produced in the interferometric waveform generated by the same laser interferometer by a particle following a parallel path at half the distance from the laser. For example, in the schematic diagram of FIG. 12B, the more distant path 7 of the particle 4 is approximately (but not exactly) twice as far from the laser resonator cavity 1 as is the less distant path 7 of the particle 4. For this reason, the number (N.sub.Cycles16) of wave cycles in the waveform of the interferometric signal 90 corresponding to the more distant path 7 is approximately twice the number (N.sub.Cycles8) of cycles in the interferometric signal 91 corresponding to the less distant path 7.

[0384] In aspects of the invention, the processing module of the self-mixing interferometer may be configured to determine this property of particulate, e.g., as disclosed herein. In aspects of the invention, this feature may be employed to determine an estimate of a position of a particle in a two-dimensional coordinate system containing two lasers, or an estimate of a position of a particle in a three-dimensional coordinate system containing three lasers.

[0385] Referring to FIG. 16A, there is shown a schematic representation of a three-dimensional coordinate system (x, y, z) of orthogonal axis in which a first laser 1A is located upon the x-axis at coordinate (x.sub.0,0,0), a second laser 1B is located upon the y-axis at coordinate (0,y.sub.0,0), and a third laser 1C is located at the origin of coordinates (0,0,0). Consider a particle 4 located within this coordinate system at position at position (x,y,z). It has a radial distance d.sub.A from the first laser 1A, a radial distance de from the second laser 1B, and a radial distance d.sub.C from the third laser 1C. The coordinate of the particle is therefore the location of intersection of the three circles of radius d.sub.A, d.sub.B, and d.sub.C, centred at the three lasers 1A, 1B and 1C respectively. This condition satisfies the following system of simultaneous equations:

[00115]$\begin{array}{c}{\left(x-x_0\right)}^2+y^2+z^2=d_A^2\\ x^2+{\left(y-y_0\right)}^2+z^2=d_B^2\\ x^2+y^2+z^2=d_C^2\end{array}$

[0386] Solving these equations for x, y and z gives:

[00116]$\begin{array}{c}x=\frac{x_0^2-\left(d_A^2-d_C^2\right)}{2x_0}\\ y=\frac{y_0^2-\left(d_B^2-d_C^2\right)}{2y_0}\\ z=\sqrt{d_C^2-\left(x^2+y^2\right)}\end{array}$

[0387] Here, the distances are known from the measurements, made using each of the three lasers, 1A, 1B and 1C, of the number of wave cycles, [N.sub.Cycles].sub.A, [N.sub.Cycles].sub.B and [N.sub.Cycles].sub.C, in the respective waveforms of the interferometric signals generated by each laser in response to the passing of the particle 4 though each of the three laser beams. In particular:

[00117]$\begin{array}{c}{d}_A={.Math.d.Math.}_A={{}_A\left[N_{\mathrm{Cycles}}\right]}_A\\ d_B={.Math.d.Math.}_B={{}_B\left[N_{\mathrm{Cycles}}\right]}_B\\ d_C={.Math.d.Math.}_C={{}_C\left[N_{\mathrm{Cycles}}\right]}_C\end{array}$

[0388] This distance estimate is substantially independent of the speed, v, of the particle in question because the number of wave cycles produces does not depend on the speed with which the particle passes though the successive wavefronts of the laser beam in question. FIG. 16B schematically shows an example of the three interferometric signals (98, 99, 100) produced by the three lasers, 1A, 1B and 1C due to the passage of one particle 4 through each beam. Here:

[00118]$\begin{array}{c}{\left[N_{\mathrm{Cycles}}\right]}_A=16\\ {\left[N_{\mathrm{Cycles}}\right]}_B=11\\ {\left[N_{\mathrm{Cycles}}\right]}_C=9\end{array}$

[0389] This means that d.sub.A=16.sub.A, and d.sub.B=11.sub.B and d.sub.C=9.sub.C. Each of the three lasers has assigned to it a respective value of calibration constant, .sub.A, .sub.B and .sub.C. The values of x, y and z may be determined as described above, accordingly. It will be appreciated that the above techniques may also be applied to other configurations in which only two lasers (e.g., Laser 1B and laser 1C) are present (i.e., laser 1A is absent) such that a coordinate position of the particle within a two-dimensional coordinate system in the z-y plane, may be determined by setting x.sub.0=0 and d.sub.A=d.sub.C in the above equations, to yield:

[00119]$y=\frac{y_0^2-\left(d_B^2-d_C^2\right)}{2y_0}$$z=\sqrt{d_C^2-\left(y^2\right)}$

[0390] The interferometric signals (98, 99, 100) may be overlapping in time while the particle is within all three laser beams simultaneously, though the interferometric signals need not be entirely overlapping with each other from beginning to end because the particle may enter and exit different laser beams, amongst the three laser beams, at slightly different respective times depending on the geometry of the laser beams and the manner of their overlap.

Laser Beam Shape

[0391] FIG. 17A shows an array of two separate self-mixing interferometers, 1a and 1b, employed in determining a particle property including particle speed v and/or particle distance d.sub.0. Each one of the two self-mixing interferometers is as described herein with reference, for example, to FIG. 2A (or FIG. 3 or FIG. 4 for example) and each is configured to produce a diverging conical laser beam (or a moving laser beam which is moved in such a way as to bathe a region defined by a diverging cone). The cross-sectional shape of each of the two respective laser beam cones, in a direction perpendicular to the longitudinal axis of the laser beam cone, is non-circular. The cross-sectional shape of each of the two laser beam cones defines a respective shape which is longest along a first axis perpendicular to the longitudinal axis of the laser beam cone and is shortest along a second axis that is perpendicular to the first axis and perpendicular to the longitudinal axis of the laser beam cone. The first axis of one of the laser beam cones is non-parallel to the first axis of one of the other of the laser beam cones. In this way, the cross-sectional shape of each one of the two laser beam cones is elongated along the respective first axis and these elongation axes are oblique relative to each other.

[0392] For example, in aspects of the invention in which the self-mixing interferometers, 1a and 1b, each produces a respective static conical laser beam, the cross-sectional shape of each of the two respective laser beam cones may be elliptical. For example, the elliptical cross-sectional shape of a first laser beam cone 8a generated by a first self-mixing interferometer 1a may comprise an ellipse major axis 101a and an ellipse minor axis 102a. Similarly, the elliptical cross-sectional shape of a second laser beam cone 8b generated by a second self-mixing interferometer 1b may comprise an ellipse major axis 101b and an ellipse minor axis 102b. These elliptical cross-sectional shapes may be generated by the use of astigmatic optics within the optical assembly (item 1B, FIG. 2A) of the respective self-mixing interferometer (1a, 1b). The first axis 101a of the laser beam cone produced by the first self-mixing interferometer, 1a, is non-parallel to (e.g., approximately perpendicular to) the first axis 101b of the laser beam cone produced by the second self-mixing interferometers, 1b. In this way, the cross-sectional shape of each one of the two laser beam cones is elongated along the respective first axis and these elongation axes are oblique relative to each other.

[0393] The two laser beam cones (8a, 8b) are configured to overlap such that a region of space exists which is simultaneously bathed by laser light from each one of the two self-mixing interferometers (1a, 1b). This region of overlap defines a mutual sensing region in which the two self-mixing interferometers (1a, 1b) may operate in synergy for determining a particle speed v and/or particle distance do of a particle 4 moving along a path 7 passing through the mutual sensing region, as follows.

[0394] FIG. 17B schematically shows a plane view of the array of two self-mixing interferometers of FIG. 17A. Here, the plane view comprises the planar cross-section of the laser beam cone 8a of the first of the two laser interferometers 1a, that contains both the laser interferometer 1a and the path 7 of the particle 4 through the first laser beam cone 8a. In addition, the plane view also comprises the planar cross-section of the laser beam cone 8b of the second of the two laser interferometers 1b, that contains both the laser interferometer 1b and the path 7 of the particle 4 through the second laser beam cone 8b. The planar cross-section of the laser beam cone 8a of the first of the two laser interferometers 1a, defines a triangle with divergence angle .sub.a. Similarly, the planar cross-section of the laser beam cone 8b of the second of the two laser interferometers 1b, defines a triangle with divergence angle .sub.b. Notably, because the orientation of the path 7 of the particle is relatively less oblique to the major axis 101a of the first laser beam cone 8a and relatively more oblique to the major axis 101b of the second laser beam cone 8b, the consequence is that a greater proportion of the particle path 7 intersects the first laser beam cone 8a than the second laser beam cone 8b. This is the reason for the difference =(.sub.a.sub.b) between the divergence angles of the planar cross-sections of the first and second laser beam cones (8a, 8b). The inventors have realised that this difference may be exploited to allow estimates to be made of the particle speed v and/or particle distance d.sub.0.

[0395] Referring to FIG. 17B, the path 7 of the particle 4 subtends an angle relative to the longitudinal axis of each of the first and second laser beam cones (8a, 8b). Here, the example assumes that the longitudinal axes of each of the first and second laser beam cones (8a, 8b) are substantially parallel such that the angle is common to both, but this need not be the case. Consider that the particle 4 enters and leaves the first laser beam cone 8a at times [t.sub.1].sub.a and [t.sub.2].sub.a, respectively, and enters and leaves the second laser beam cone 8b at times [t.sub.1].sub.b and [t.sub.2].sub.b, respectively. Let the length of the path 7 of the particle within the first laser beam cone 8a be s.sub.a, and let the section of that length of the path 7 within the second laser beam cone 8b be s.sub.b (which is a part of s.sub.a). Consider also, that the particle path 7 forms a tangent to the wavefronts of the laser light in each of the first and second laser beams, 8a and 8b, simultaneously when the particle 4 is at a distance d.sub.0 from the respective lasers 1a and 1b. The distance d.sub.0 corresponds to the closest point of approach of the particle whilst within both laser beam cones. One can see that:

[00120]$v\left({\left[t_2\right]}_a-{\left[t_1\right]}_a\right)=s_a$$v\left({\left[t_2\right]}_b-{\left[t_1\right]}_b\right)=s_b$