Particle Sizing by Interference Fringes and Signal Coherence in Doppler Velocimetry A. R. JONES,M. J. R. SCHWAR BY R. M. FRISTROM," 1-AND F. J. WEINBERG Imperial College London S.W.7. Received 15th January 1973 To supplement another paperf presented at this Symposium (which deals with sizing suspended particles by light scattering) a new optical method is proposed for sizes larger than a few wavelengths. To this end conditions under which the alternating light signal generated by a particle traversing a fringe pattern falls to zero are examined with a view to measuring such particle sizes by varying fringe spacing. The a.c. frequency is a measure of the particle velocity and can be expressed with identical results either as a beat note caused by Doppler shift or in terms of the varying illumination of the particle due to its movement across the light grid.The theory giving the a.c. amplitude for two infinitesimal particles moving with a common velocity as a function of their separation is extended to a variable number of particles moving together. This provides the basis not only for an assessment of how the signal legibility in Doppler velocimetry falls off as the number of scatterers increases but also for the integration which treats particles of finite size as the sum of their infinitesimal elements. A variety of optical systems are proposed for measuring the size of particles droplets fibres etc. under different conditions. For some purposes fringe systems of non-uniform spacing may be advantageous and the use of a dark field schlieren image which shows the particle as a thin circumferential line-in place of the illuminated area produced by normal imaging-is often valuable.Such a schlieren system is used in a simple experimental test of the method ; the results accurately conform to the theoretical predictions. This study was initiated as a potential method for sizing droplets and particles appreciably larger than the wavelength of light during Prof. Weinberg's visit to the Applied Physics Laboratory of the Johns Hopkins University and pursued further in a wider context at Imperial College. The light scattered by a particle depends on the local illumination and when this is in the form of a field of interference fringes which the particle is. traversing an ax.signal will result except under certain conditions which are the subject of this paper. When there are two small particles travelling with a common velocity the signal produced by a photodetector receiving light from both will depend on their separation. Fur certain separations the amplitude of the a.c. signal falls to zero. A similar succession of zeros occurs as the size of one particle increases in relation to the fringe separation or more practically when the fringe separation is decreased in relation to the size of the particle. All this applies to any field of stratified illumination whether produced interfero- metrically or e.g. by projecting the image of a grid. The convenience of interference fringes lies in the ease and precision of varying their separation by changing the angle between the interfering beams.This becomes particularly easy when the light source * Applied Physics Lab. The Johns Hopkins Univ. Silver Spring Maryland U.S.A. 7 now at Paint Research Association Teddington Middx. $ M. D. Carabine and A. P. Moore "A light scattering instrument for kinetic measurements in aerosols with changing particle size distributions". 183 PARTICLE SIZING IN DOPPLER VELOCIMETRY is a laser which is especially suitable for producing interferometrically a fine light grid over a small test area at a very high level of illumination. The frequency of the a.c. signal is a direct measure of the particle velocity which can be described either in terms of traversing a fringe pattern or of the heat note between two frequencies which have experienced a differential Doppler shift due to the interaction of one or both of them with the moving particle.The two descrip- tions are mathematically equivalent both where a fringe pattern moves across a point detector in the image plane and for a fringe pattern in the test space (“ fringe anemometry ”). * The concept is thus relevant to two quite different practical applications. One is a new method of particle sizing based on the disappearance of the a.c. signal at given fringe spacings. The other is the limitation to Doppler velocimetry due to multiple scatterers in the test space. GEN€RAL PRINCIPLES A wide range of optical systems suitable for implementing the various appli- cations of this principle will be discussed.However it will be convenient to consider general properties in terms of the simplest underlyingschem asshownin fig. 1 in which two plane waves are incident at symmetric angles 8 with respect to the x-axis and form a set of interference fringes in the y-z plane. Observation is made of the light scattered into the direction 4 by a particle moving with velocity at an angle a to the y-axis. We shall not concern ourselves with the polar distribution of the scattered light but deal in terms of a constant fractionfof the illumination of the particle. The a.c. frequency is independent of angle at least so long as velocities do not approach the speed of light and for small particlesf is almost independent of angle. This means that light can be collected over a range of angles without affecting the result.The distribution of illumination in the y z plane is given by r = 210(1 +cos (2ky sin O)]. With k* = 27r/rZ* where A* = A/2 sin 8 is the fringe spacing r = 24(1 +cos k*y). ‘t FIG.1 .-Co-ordinate system for scattering of two plane waves by a moving panicle. FRISTROM JONES SCHWAR AND WEINBERG Thus the light scattered at the angle 4 by a point particle situated at Y may be represented by I,, = 2ZOf(+){ 1 +cos k* Y>. (1) Now the particle is moving along the y-axis with speed u = dY/dt = u cos a Thus as the particle moves the scattered intensity flucwates with a frequency vb given by vb = (u cos .)/A*. (2) An alternative approach is based on the Doppler shift.The apparent frequencies of the two waves as seen by the particle are vp,l = tp +; cos (;-a-o)] vp,2 = v[I+-cos -a+8 , c >I Likewise the frequency scattered by the particle as seen by an observer looking along 4is uo = up[ 1-5 cos (;-LY-4)] for both waves. The observer sees a beat frequency v; = I~0,2-~0,1l or V; = 2u cu -cosccsin8 [1-cos(;-LY-cj)] (3) If u/c is small then eqn (3) reduces to eqn (2). Alternatively since the particle is considered as a source emitting at frequency v, in the derivation of eqn (2) if the motion of the particle relative to the observer is taken into account the result is vb = vb 1-sin (a+411 [: which is identical to eqn (3). So both approaches give the same answer. To find the scattered intensity the incident amplitudes of the light wave may be set out as follows Eo,l = E,-,exp(ikYsinO); Eo,2 = E,exp(-ik YsinO).The scattered waves are then ELI = Eo ,/"exp (-ik Y sin 4); Es,2= Eo.2Jfexp (-ik Y sin 4) where k and k2 are the Doppler shifted wave numbers. We have assumed that f(O+ 4) N fC0-4)which is reasonable for small particles. Adding the amplitudes the scattered intensity is I,, = 21,f(l+cos[2kYsin O+(kZ-kl)Ysin 4]}. I86 PARTICLE SlZING IN DOPPLER VELOCIMETRY Now k2-kl 21 (2kuy/c) sin 0 so that Is, N 2Iof( 1 +cos [2k Y sin O( 1+uY/csin 4)]}. (4) For (u,/c)< 1 this equation is identical with eqn (1). Thus the two approaches yield identical results for small u/c. This approximation is inherent in the Doppler equation used i.e.the full relativistic treatment has not been considered. Inclusion of a second particle separated from the first by a distance d gives the scattered intensity as I,, = 2IOf(2+cos ~*Y+cos k*(Y+d)} (5) provided that the detector aperture is sufficiently large for incoherent addition to be used.3 A large aperture integrating over a range of angles can be used sincef for small particles and Vb are almost independent of 4. Eqn (5) can be derived either from eqn (1) or (4) with small u,,/c i.e. either by adding illuminations in two parts of the fringe system or by combining two beat frequencies with a phase difference between them. The beat frequency observed is the same as for one particle and since Is, = 4Iof(1 +COS k*( Y+d/2) cos (k*d/2)} the signal falls to zero whenever d = (2n+ 1)L*/2.In the most general case we may consider N particles having the same velocity < so that their respective separations are constant in time but randomly distributed. If thejth particle is at a distance yj from the first which is situated at Y one may expand eqn (5) into the form N I,, = 210f N+COS k*Y+ C cos k*(Y+yj) I j=2 since y1 = 0. The probable intensity (Isca)is the ensemble average over the cloud of N particles. Since the particles are randomly distributed the probability of yj having any particular value is a constant. Further if the width of the particle cloud is w (less than or equal to the beam width) then -w<yJ<w. Consequently. lW -w j’” ***Sw -w -W Iscady,dy,*.*dy 9 <IsJ = j-w j-w .,.j-w dY2 dY -dY -w -w -w or (Isca) = 210f(N+cos k*Y[l+(N-l)(sin k*w)/k”w]).This is the probable intensity observed at any one time or value of Y. As Y varies the signal varies and the visibility of the observed as. signal is (1sca)max -(Isca>min (1sca)max +(IsJmin’ which gives finally 1 N-1 sin k*w v = -+ -A N Ic*w FRISTROM JONES SCHWAR AND WEINBERCi I87 We note some of the properties of this function. Clearly for N = 1 V = 1 as expected. Also where sin k*w = 0 (i.e. the width of the cloud is a whole number of fringes) or for an infinitely wide cloud V = 1/N. However in general for a large number of particles as N+m V+(sin k*w)/k*w. That this is not zero is explained by the fact that the fringe pattern is repetitive.For every whole fringe the scattered intensity integrates to zero. If there is a fraction of a fringe left over only the effect due to those particles in this section is seen as over a fraction of a fringe the scattered intensity does not integrate to zero. The physical implication is that legibility of the a.c. signal can be assured only in the case of a single scatterer. For any greater number it is always possible that zero a.c. signal will result if the particle distribution is unfavourable. On the other hand it is quite likely that a residual a.c. signal will be obtained for the reasons discussed above even for a large cloud of particles-a result which explains observa- tions in many practical studies when naturally occurring scatterers are used.This applies purely to the a.c. component i.e. under conditions where the detector is not saturated by the d.c. signal which increases with number of scatterers. So far we have considered particles much smaller than the fringe spacing. The relevance to size measurement for which the fringe spacing must be made at least equal to the particle dimensions is by way of regarding infinitesimal particles as elements of a larger particle. We may then take the next step on a model of inte- grating across all the elements illuminated by a sinusoidal fringe pattern or of adding those locating the periphery depending on the optical system used. Consider first a strip of length 21 and width dZ situated with its centre at Y,as seen in fig.2. Taking an elementary area of length dY the element of the scattered intensity from the strip is dZ,, Y+I = 21~fJ (1 +cos k*~) dY dZ Y-1 or wherefis now the scattering efficiency factor. It remains to integrate over 2. We -m Y FIG.2.-Scheme for integration of illumination across a strip situated in an interference pattern. I88 PARTICLE SIZING IN DOPPLER VELOCIMETRY first note that for a longstrip e.g. a fibre or wire parallel to the fringes 1isiodependent of 2 and I,, = 41 fLi(1 +cos k*Y(sin k*l)/k*l) (7) where L is the length of the strip and 21its width. The a.c. component is zero when- ever 21 = nP i.e. no beat frequency will be observed if the strip has a width exactly equal to a whole number of fringes.This is obvious physically since the test object is then exposed to a total illumination which does not vary with change in position and it was this concept which first suggested the method. For a particle with a circular cross-section I = (R2-Z2)' where R is the radius of the particle. Thus Isce= 81,/s1 ((R2-Z2)*+-1 cos k*Y sin (k*JR2-Z2) dZ. 0 k -1 Substituting 2 = R cos # yields Using nJ,(Z) = 2 sin (2 sin 4)sin 4 d# Jal gives where Jl(k*R) is a Bessel function of order one which has zeros at k*R = 3.832 7.016 etc. The a.c. component thus falls to zero for 2R = 1.22 A* 2.24 A* etc. The theory as it stands is restricted to particles large enough for geometrical optics to be applicable i.e. the total flux of radiation extinguished by a particle is proportional to its cross-sectional area.For very smalf particles the scattering efficiency is an oscillatory function of particle size.5 However for transparent particles with size to wavelength ratio D/A;Z60 and for absorbing particles with D/A2 15 the oscillations are effectively damped out. For smaller particles the above concept must be examined in terms of wave theory. Scattering by infinite cylinders parallel to a system of interference fringes has been investigated by Jones.6 It is found that for quite small particles the discrepancy between the zeros as predicted by wave theory and geometrical optics is remarkably small. This is illustrated in fig. 3 which indicates for which particle sizes the two theories agree to within 10 % as a function of refractive index.It should be noted that the wave theory is rigorous and gives the positions of the zeros exactly. A second condition imposed on the above theory is that the detector aperture must be made large enough for the assumption of incoherent superposition to be FRISTROM JONES SCHWAR AND WEINBERG applicable. In fact since large transparent objects scatter very strongly forward it would be advisable to collect as much forward scattered light as possible. The resulting variation would then be expected to be similar to that for the total scattering. The wave theory has shown god agreement between the zeros in the total scattering and in the total forward scattering efficiencies for sizes as low as than 10% L I I I I I c 0 0:2 0.4 0;6 0;8 1.0 1IP FIG.3.4urve approximately indicating particle sizes (D/h)as a function of refractive index p for rigorous wave theory and geometrical optics to agree within 10 % in predicting the position of the first zero of the ax.amplitude (D/A*-l) for cylinders. p = co corresponds to a perfectly conducting cylinder. DliL-2. For small metal particles the scattering is mainly backward which suggests obvious modifications to the optical systems discussed in the next section e.g. a mirror to collect and reflect the backward scattered light. With the exception of fibres and droplets particles are not generally cylindrical or spherical. It is therefore of interest to compare the width deduced with that of the area-mean (since this particular optical system is based on illumination of the entire area-but see below).The correspondence is exact for any shape whose width varies linearly with height. For circular discs the area-mean width is nR2/2R which for the first zero differs from the above value by approximately 5 %. The discrepancy increases for higher-order zeros and though it is calculated easily it may be simplest to use only the first zero in circumstances when mixed shapes are likely to occur. In practice it is convenient to collect only a fraction of the light scattered by the particle into a certain range of solid angles. Moreover the circumstance that the direct light from the source must be cut-off if the photodetector is not to be saturated by an overhwelming d.c.contribution makes any light scattered about the optic axis unavailable. The use of a schlieren system to select a suitable range of scattering angles is not only simple (see fig. 4) but also allows a simpler and more convenient form of the theory to be used. PARTICLE SIZING IN DOPPLER VELOCIMETRY The principle is obvious if we consider the image. Here the luminous regions correspond to those parts of the test objects in which the particular deflections originate which are selected by the schlieren aperture used and which in the absence of refrac- tion are occasioned by diffraction alone.' When this system is applied to transparent lmaginq Leiis -Scattering ScNinen StOD Tat Sdrliercn Space LUIS FIG.4.-Schematic showing basic Schlieren system for the observation of scattered light.droplets it is better to use an opaque dye to avoid complications due to internally refracted rays. The range of angles so defined can be varied by limiting the outer as well as the inner boundary of the schlieren aperture using e.g. circular or double slits of various dimensions. Now the regions around the boundary from which the diffracted light comes are narrowly confined so that except for the very smallest of particles the above theory may be modified as follows. The photodetector is effectively only seeing the edges of the scattering particle crossing the field of fringes. For a cylinder these are two narrow line sources parallel to the fringes separated by the diameter of the cylinder.Eqn (5) can be applied directly giving zeros for 21 = (2n+ l)L*]2. The same result can be arranged for particles of circular cross-section if a strip schlieren stop is used of a width just sufficient to give an image consisting of two points coming from the ends of a diameter. This is the simplest and most direct measurement of width providing enough light is available. For a large number of particles in such a system one has effectively N pairs of particles of fixed separation d. Then eqn (6)takes the form 1 N-1 sin k*w d -+----} cos k* z. N N k"w If a circular schlieren stop is used with a sufficiently large particle of circular cross-section the image takes the form of a ring of radius R. If we take an element d1 of such a ring situated at the angle 8 we have d1 = Rd8 and if 6R is the width of the ring I,, = 4106Rf (1+COS k*y)R d6; s since y = Y+R cos 8 the centre of the ring being at Y then I,, = 410R6Rf [1 +cos k*y cos (k*R cos 8)-sin k*y sin (k*R cos O)] do, f or I,, = 4nR6R10f [1+ Jo(k*R)cos k"~].FRISTROM JONES SCHWAR AND WEINBERG Jo(k*R) is a Bessel function of order zero with zeros at k*R = 2.405 5.520 etc. Hence the a.c. component has zeros for diameters. 2R = 0.766A* 1.757 A* etc. The limitations regarding very small particle sizes which were detailed above for illumination of the whole area also become less serious for the schlieren type of optical system. The dark-field schlieren image when using a strip stop consists of two slivers of illumination at its extremities.These differ from being infinitesimal only to the extent to which the diffracted light derives from regions other than the edge and to the effect on the light pattern of diffraction elsewhere in the system^.^ However since this marking is symmetrical about the particle and since the fringe separation corresponds to the width of the whole particle-at feast for the first extinction-the approximation is a good one provided the schlieren stop is small and the numerical aperture of the optical system is large. In the simplest case of only one imaging lens the detailed structure of the schlieren image depends on the open aperture the stop and the boundaries of the lens (see e.g. Speak and Walters,lo). If this aperture were infinite in extent a point object would be imaged as a point.When a very small schlieren stop is used-and the parallelism of laser beams makes this feasible-the uncertainty in the schlieren edge is in practice limited only by the lens aperture. The dark-field schlieren system is also particularly suitable for very large particles for which the total illumination method is dominated by backward reflection from a three-dimensional body. Although here again the light can be reflected forward and extinction conditions precisely calculated for any known shape the schlieren method can be used directly without modification either to the optical system or to the method of data analysis-irrespective of shape size or reflectivity of the test object. SOME OPTICAL SYSTEMS Several configurations suitable for particular purposes are shown in fig.5. The method was originally intended to select particles of abnormal size (diseased cells) and this exemplifies probably the simplest application possible. Such particles can be held eg on a microscope slide driven at a known velocity or conveyed in a stream of liquid along capillary tubing. If the fringe spacing is set at a value giving zero (or minimum) a.c. amplitude for the standard size any particle of abnormal size will signal its presence by its ax. component. Since the particle velocity can be arranged independently in such applications suitable filter circuits could be used for the known ax. frequency. Under these conditions it would not even be necessary to have only one particle passing through the test space at a time although it would be desirable not to have more than one abnormal particle there.Under less controllable conditions however it is necessary to collect light from only one particle at a time into the photomultiplier otherwise the sizes will "add up " in a manner depending on their separation. This requirement is similar to that though more stringent than in velocimetry (see above) and in a cloud of particles can be arranged most readily by making the area of the fringe field smaller than the minimum particle separation. Using a focused laser beam in a system such as that shown in fig. 5a (similar to the " velocimeter " of Rudd 11) would be suitable for quite dense clouds. In the case of velocimetry there would be no merit in having the separation between the two slits or two apertures adjustable.For present purposes however the two apertures can be mounted on a micrometer screw or pair of callipers which may PARTICLE SIZING IN DOPPLER VELOCIMETRY Lens Lens Schlieren Oetect or slits Beamsplitter \ - - r Detector Space 1 Detector en Oiff 1)sing Screen (4) FIG.5.-Various possible optical systems for the observation of light scattered from particles sub-jected to modulated illumination. FRISTROM JONES SCHWAR AND WEINBERG then be adjusted to the disappearance of the a.c. signal when the particle size can be expressed directly in terms of the separation so measured. The particle velocity may be deduced for each slit separation (other than that giving zero a.c.signal) during this adjustment thereby improving the accuracy of velocimetry. The accuracy of both measurements is limited by the aperture widths which also define the depth of the sampling zone along the optic axis. This system would be well-suited to size determination in monodisperse clouds whether or not the particles move with a uniform velocity component perpendicular to the fringes so that the adjustment of fringe spacing can be carried out during the passage of a succession of individual particles-in which case the velocity of each particle as well as the common size can be deduced. For tenuous clouds extending over appreciable areas a system such as that shown in fig. 5b may be preferable in order to reduce the delays which would occur between successive signals for a small test area.To increase the angle between the beams and decrease fringe spacing conventional interferometers may be used e.g. fig. 5c though the limitations on minimum particle size discussed above still apply and there may be little point in reducing fringe separation to its theoretical minimum of A/2. All the optical systems not using diffused light in fig. 5 are based on schlieren imaging for convenience. If that is not desired it would be advisable to move the photodetector off axis (e.g. in the position shown dotted in 5c) to avoid saturation by light which has not interacted with particles. (b) FIG.6.-(a) Grid of variable spacing for particle sizing.(b)Signal due to a particle traversing the grid. Particle size corresponds to central grid spacing. Many other optical systems may be useful for particular purposes. Probably the simplest method of achieving interference with laser light uses the front and rear reflections from a piece of glass (almost any optical quality if a sufficiently small area is used).12 Fringe spacing can then be varied by varying magnification (fig. 54. The same method of varying spacing can be used when projecting a grid within the test region in which case neither a laser nor indeed monochromatic light s7-7 PARTICLE SIZING IN DOPPLER VELOCIMETRY is required (fig. 5c). As regards this system the " Doppler theory " gives cancellation in pairs at each wavelength when the correct separation is reached.The major difficulty which arises when particles are all of different size is the need to carry out the adjustment in fringe spacing during the passage of each particle unless a spectrum analysis can be carried out on a large number of records. Such adjustment could be carried out mechanically only for particles travelling very slowly and being widely dispersed so that the " null point '' for one could be deter- mined before another entered the test space. An attractive alternative is the use of a grid or fringe system the spacing of which is arranged to vary across the test space. Fig. 6a shows such a grid while fig. 6b shows the signal expected from a particle travelling across it at constant speed whose diameter corresponds to the centre of the range.This system can be contrived by interfering wavefronts the angle between which varies across the field e.g. planar and spherical or by projecting the image of an actual grid made in this form. The use for this purpose of an optical system such as that shown in fig. 5c would allow the overall grid magnification to be varied as well so that a large range of particle sizes could be accommodated. EXPERIMENTAL TESTS The variation of the signal with ratio of fringe spacing to particle size should be much the same whichever optical system is used except that those based on varying magnification also vary the overall illumination level. This however is a trivial point and it was considered that the simplest convenient arrangement could be used for comparison of the signals with the above theory.Preliminary work at the Applied Physics Laboratory was based on an arrangement similar to that shown in GRAT I NGS 50cm. BLIND 87 Line mm? Focal Length (4 I L T s P (61 FIG.7.-Apparatus for particle sizing. (a) Production of the two effective point sources ; (b) the schlieren optics. PLATE1.-Image of wire traversing interference pattern. [Toface page 194 A* =d PLATE2.-Typical photomultiplier output produced by wire traversing a variable interference pattern. FRISTROM JONES SCHWAR AND WEINBERG fig. 54largely because a shadow interferometer based on front and rear reflections at a glass slab had just been set up for fire research.13 These experiments were cursory owing to shortage of time.The work at Imperial College was based on a system similar to fig. 5b and is illustrated in fig. 7a. Fig. 5b is a special case of the system in fig. 7a as there the two sources S1 and S2 are effectively at infinity. The narrowly confined test space of fig. 5b was not required for a test in which the object could be precisely located. In practice the problem is one of being able to adjust the separation of S1 and S2 conveniently and precisely. To achieve this a collimated laser beam passed through a pair of identical gratings having 87 lines mm-l (see fig. 7b). The diffracted beams were then brought to a focus using a 50-cm focal length lens. When both gratings were accurately aligned with respect to each other the diffracted orders coincided exactly.Rotation of one grating in its own plane rotated one set of diffracted orders about their common central maximum. A blind placed in the focal plane selected only one order from each grating so as to produce two point sources corresponding to S and S2 in fig. 7a. The separation of the point images depended only on the relative rotation of the two gratings. The separation of the sources was therefore easily and precisely adjustable and therefore the fringe spacing was readily varied. TABLEFRINGE SPACINGS CORRESPONDING TO MAXIMA AND MINIMA IN THE DOPPLER BEAT FREQUENCY SIGNAL FOR A SINGLE PARTICLE SIZE grating setting min. max. order no. n 1In fringe spacing 1.50 0.5 2.000 0.0730 3.30 1.o 1.ooo 0.0394 5.00 1.5 0.667 0.0275 6.80 2.0 0.500 0.0208 8.80 2.5 0.400 0.0168 The dimensions of the optical system beyond S and S were S2T = 77.0 cm ; TL = 26.2 cm ;LS = 15.0cm ; and SP = 20 cm (see fig.7a). A single stop 2.5 mm wide was used to block off the direct beams. The aperture in front of the photo- multiplier was 3.0 mm long and 0.3 mm wide. The fringe spacing was continuously variable from effectively infinity down to 170pm. The test object was a thin moving wire the average diameter of which was measured with a micrometer as 0.0399+0.0001 cm. It was attached to a fly wheel and driven through the test region by a constant speed electric motor with its axis parallel to the fringes as is illustrated in plate 1. Each time the wire crossed the fringe system the periodic signal picked up by the photomultiplier was displayed on a oscilloscope.Starting with an infinite fringe spacing the oscilloscope traces were observed as the fringe spacing was decreased. A series of readings were made of the grating settings corresponding to the minimum and maximum amplitude in Doppler beat frequency traces. Three examples of the traces recorded are shown in plate 2 and the complete set of data is given in table 1. The grating settings were calibrated by measuring the associated fringe spacings with a travelling microscope. As discussed in the previous section we expect that for this system minima in a.c. amplitude occur when A* = D/(n+3) n = 0,1,2 3.. . s7-7 * PARTICLE SIZING IN DOPPLER VELOCIMETRY Similarly maxima occur whenever A* = Din n = 1,2,3... One interesting result of the experimental test is that the minima in a.c. amplitude approach but never quite reach zero. This is due to the Gaussian modulation of the fringe amplitudes across the field and becomes obvious by reference to fig. 8 which I A t (c) FIG.8.-Effect of adding two unbalanced out of phase a.c. signals. (a) signal produced by leading edge of object ; (b)signal produced by trailing edge of object ; (c)net signal from detector the sum of (a)and (6). 1In FIG.9.-Linear least-squaresfit to the points listed in table 1. FRISTROM JONES SCHWAR AND WEINBERG also explains the shape of the observed traces. The schlieren images of the leading and trailing edges derive from somewhat different regions and the two intensities are therefore never fully matched.Should this become a practical limitation to accuracy it would be necessary to use a wider field of fringes in relation to the width to be measured or to avoid a Gaussian distribution altogether (see fig. 5c). To illustrate the deduction of D and assess variation in individual readings as compared with the mean a graph of A* against n-l was plotted. This is shown in fig. 9 where a linear least-squares fit has been made to the points. Taking all points into account the slope gives d = 0.0377 cm a 5.5 % difference from the micrometer reading. However the error in locating the first minimum is much greater than for the other points.If this first point is omitted one obtains d = 0.0404 which is only 1 % different from the directly measured value. Since the form of the records and analysis is independent of the optical system and of the nature of the test object there seemed little point in extending the experi- mental tests to other configurations. It is concluded that the results conform accurately to the theroretical predictions. M. J. R. Schwar and F. J. Weinberg Proc. Roy. Soc. A 1969,311,469. F. Durst and J. H. Whitelaw Proc. Roy. Soc. A 1971 324 157. L. E. Drain J. Phys. D. Appl. Phys. 1972 5,481. A. ErdClyi Higher Transcendental Functions (McGraw-Hill New York 1955). G. N. Plass Appl. Optics 1966 5 279. A. R. Jones J. Phys. D. Appl. Phys. 1973,6 417. ’F. J. Weinberg Optics of Flames (Butterworths London 1963).M. D. Fox and F. J. Weinberg Brit. J. Appl. Phys. 1960 11 269. K. G. Birch Optica Acta 1968 15 113. lo G. S. Speak and D. J. Walters Aero. Res. Council,Rep. Mem. 1954 2859. M. J. Rudd J. Phys. E. Sci. Instr. 1968 1 723. l2 A. K. Oppenheim P. A. Urtiew and F. J. Weinberg Proc. Roy. SOC. A 1966 291 279. l3 J. E. Creeden R. M. Fristrom C. Grunfelder and F. J. Weinberg J. Phys. D.,Appl. Phys. 1972 5 1063.