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Liq~id AtomiZation and S'pray Systems'

MAY 17-19, 1993

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Effects of finite beam width on elastic light scattering from droplets

INTRODUCTION

J. T. Hodges and C. Presser

Chemical Science and Technology Laboratory National Institute of Standards and Technology,

Gaithersburg, Maryland 20899

A number of experimental techniques based on elastic light scattering are used to measure the size of micron-sized spherical particles and droplets. Some examples include, phase Doppler anemometry\ laser diffraction methods2

, light intensitr deconvolution3 as well as ensemble scattering measurements from axi-symmetric Gaussian and Gaussian sheet beams5

• In the interpretation of these measurements, the classical Lorenz-Mie theory (LMT) ·of elastic light scattering or principles of diffraction and ray optics are invoked to predict the phase, amplitude and intensity of the scattered radiation as a function of particle size, refractive index and scattering angle. The reliability and accuracy of these measurements, thus depend on the degree to which the invoked theory predicts the obsetved features of the scattered field. In this context, it is therefore important to consider realistic situations for which departures from the LMT are significant enough to warrant the use of more exact theories of light scattering.

The LMT encompasses diffraction and ray optics regimes and is an exact formulation of light scattering by homogeneous spheres irradiated by monochromatic infmite plane waves. However, there are a number of influential factors (often present in real systems) which are not included in the LMT. Among the most important effects are particle inhomogeneity, asphericity and non plane-wave irradiation. In practice, the incident field often corresponds to that of a focused laser beam with lateral dimensions comparable to the particle diameter, as with phase Doppler anemometry and the ensemble scattering polarization ratio5

• Since the LMT is strictly valid in the limiting case of infmite plane-wave illumination, it is expected that the accuracy of the LMT will be seriously compromised under experimental configurations where relatively narrow laser beams are utilized.

An extension of the LMT, known as the generalized Lorenz-Mie theory (GLMT), has recently been developed to describe the elastic light scattering from homogeneous spheres illuminated by beams of finite width, arbitrary profile and wavefront cUIVature6

• The GLMT has been incorporated into the interpretation of some phase Doppler data although the exploitation of this theory is in its infancy7

• Some measurements and predictions have been reported for droplets irradiated by axi-symmetric Gaussian beams8

• However, an exhaustive comparison of finite beam light scattering and the GLMT does not yet exist Furthermore, despite the prevalence of small probe beams in many configurations, the GLMT is not applied to most experimental droplet/particle sizing diagnostics.

In this investigation, angular light scattering profiles from droplets illuminated by focused axi-symmetric laser beams were obsetved. The measurements were carried out to provide direct experimental evidence of the sensitivity of angular scattering profiles on beam profile and particle position within beams of finite width. Another goal of this research was to experimentally validate the GLMT. Direct comparison of the data and GLMT predictions will be presented elsewhere. Only the obsetved scattering profiles and some qualitative interpretation are presented herein.

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SCATTERING THEORY

The intensity of light scattered by a homogeneous spherical particle is proportional to the irradiance of the beam, 10 and the intensity functions, ia and i,. The terms, i9, i, correspond to the scattered radiation polarized parallel and perpendicular to the scattering plane, respectively. The coordinates 9 and <P represent the polar and azimuthal angles, respectively of the field point with respect to the origin centered at the particle. The LMT predicts that the intensity functions depend on scattering angle, 9, complex particle refractive index, m, and particle size parameter, a= 1td!A. Thus,

i9LMT(9,m,a) i,LMT(9,m,a)

According to the GLMT theory, the respective intensity functions depend additionally on the azimuthal angle, q,, the particle coordinates (xP,yP,zP) and a set of coefficients, designated by g:Z, which are unique to a .given beam. Their functional dependencies are therefore

In the GLMT formulation, the spherical polar coordinates are fixed relative to the origin located at the incident beam waist.

The explicit dependence of the GLMT intensity functions on particle coordinates and beam coefficients g:Z embodies the fundamental difference between generalized and classical scattering theories. Some features predicted by the GLMT which can not be accounted for in the LMT approximation include asymmetric and position-dependent scattering profiles. For a more detailed explanation of the preceding formulae and derivation of the GLMT see Refs. 6-9.

EXPERIMENTAL APPARATUS

A system allowing precise positioning of individual scattering particles was required due to the extreme sensitivity of scattering pattern to particle size, location and beam profile. This was achieved by levitating a single electrically charged liquid droplet within a hi-hyperboloidal electrodynamic balance. The experimental configuration is illustrated in Fig. 1.

Individual droplets of dioctyl phthalate, an extremely low volatility transparent liquid of refractive index 1.4845, were trapped in the levitator. A precise three-dimensional translation stage with 2 J.Lm resolution was used to displace the droplet/levitator system with respect to the stationary laser beam. The levitated droplets were observed and photographed using a strobe light and a long-distance microscope/CCD camera assembly. A linearly polarized Ar+ laser beam at 514.5 nm was used to illuminate the droplets. The CCD video output was interfaced to a 386 computer with an 8-bit frame grabber. For each droplet position, a mosaic of images spanning over a 40° interval in scattering angle was obtained by scanning the camera horizontally exactly one frame width over the field of view. Other details about the experimental configuration are given in Ref. 9.

RESULTS

For axi-symmetric illumination, the scattered intensity pattern (within a given scattering plane) is symmetric about the beam axis when the particle is located along the beam centerline. This property was used to determine the relative position of the droplet with respect to the beam axis. By displacing the suspended droplet in the direction orthogonal to the beam axis, the droplet position at which the state of symmetry occurred was found by observing the scattering pattern. With the centerline coordinate so determined, relative displacements of the particle were measured using the calibrated translation stage to move the levitator/droplet system.

The origin of the coordinate system, OG(x,y,z) was at the beam waist with the incident beam linearly polarized in the direction of thex axis and propagating along the z axis. For all measurements, detection was in the yz plane and hence the azimuthal scattering angle, 4> was 90° for y d > 0 and 270° for Yd < 0. The detector coordinates are denoted by OG(xd,yd,zd).

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A pair of typical angular scattering profiles proportional to i,(S) are presented in Fig. 2. The

dashed curve depicts the scattering profile in the yz half-plane YJ > 0 and the solid line is that for

YJ < 0. The incident beam was an axi-symmetric Gaussian with a 1/e2 radius of 79 J.lm at the longitudinal particle location. The droplet, located off-axis at coordinates 0 c;( -60,-80,-83 00 )Jlm had a diameter of 95 Jlm. A distinctly asymmetric angular distribution in the scattered intensity is apparent in Fig. 2. One can see that the lobes in the half-plane opposite the particle location are of high visibility and well defined. The angular frequency of the "clean" lobes agree with what LMT predicts for 95 Jlm diameter particle. In contrast, the lobes located on the same side of the beam axis as the particle are significantly attenuated and distorted in a fashion which can not be predicted by LMT. Such asymmetric scattering patterns occur because of the non-uniform illumination by the beam. These data indicate that scattering patterns from off-axis scatterers within beams of fmite width have information content regarding particle position.

A strong variation of the scattering profile with radial position of the droplet is indicated by Figs. 3a-d. For these data, the droplet diameter was 90 Jlm and the beam was an axi-symmetric Gaussian of local radius equal to 79 J.lm. <P = 90°: dashed lines; <P = 270°: solid lines. In Fig. 3a, the angular profile of scattered intensity was nearly identical for complementary azimuthal angles. It was concluded therefore that at this position, the droplet was situated along the axis of the beam. The symmetric character of these data furthermore established that the droplet was quite spherical and that the beam irradiance was symmetric about the x -axis.

In order to obtain the data shown in Figs. 3b-d, the droplet of Fig. 3a was displaced perpendicular to the beam axis. Droplet coordinates were; Fig. 3b: OG(0,40,-8300) Jlm; Fig. 3c: 0 c;(0,1 00,-8300) Jlm; Fig. 3d: 0 G(O, -40,-8300) Jlm. As the particle was moved off axis by 40 J.lm, the fringe maxima shifted by approximately 0.15°. (Note that the angular shifts for the complementary azimuthal angles were in opposite directions.) With the droplet located 100 Jlm off axis in the +y direction (dashed curve), Fig. 3c illustrates that the fringes in the half-plane on the side of the droplet were severely distorted. By comparison with Fig. 2 where asymmetry occurred with the droplet coordinates having non-zero x and y values, Fig. 3c illustrates that asymmetric scattering profiles in the yz plane can be induced by a radial offset solely in the y direction. The mirror image symmetry of the scattering pattern for off-axis particles is illustrated by comparing Figs. 3b and 3d. When the droplet was shifted off axis by 40 J.Lm, (Fig. 3b) asymmetric profiles were recorded. Movement of the droplet in the opposite direction (Fig. 3d) by the same amount gave nearly identical lobe patterns when comparing complementary azimuthal angles.

Side scattering measurements of a 110 J.Lm diameter droplet in the angular interval 80 -100° are shown in Figs. 4a-d. The incident beam had a local radius equal to 160 Jlffi. The droplet coordinates were; Fig. 4a: OG(0,0,-16500) Jlm; Fig. 4b: OG(0,20,-16500) Jlm. The large-scale distribution of scattered intensity varied slightly with a 20 J.lm displacement in the particle position (see Figs.4a-b). The overall distribution was similar to that predicted by LMT. However, further displacement of the droplet modified the detailed features of the scattering pattern as indicated in Figs. 4c-d. In Fig. 4c, the solid line represents the data of Fig. 4a, and the dashed line corresponds to droplet coordinates of; OG(0,-60,-16500) Jlm. The data show that fringe shifts of the order of 0.15° occurred for displacements of less than one droplet diameter. The data for a pair of coordinates equidistant and orthogonal with respect to the beam axis, as shown in Fig. 4d, have similar angular extrema but differ non-uniformly in intensity. The coordinates are; solid line: OG(0,-60,-16500) Jlffi; dashed line: OG(0,60,-16500) Jlm.

154

CONCLUSIONS

Measurements of elastic light scattering from transparent droplets illuminated by focused Gaussian laser beams showed a strong sensitivity to particle position. As the particle was moved off axis, the angular maxima of the fringes shifted. Large droplet displacements destroyed the angular symmetry of the scattering and degraded the fringe visibility in the scattering half-plane on the droplet side of the beam axis. These effects, caused by the non plane-wave nature of the particle illumination, occurred for beam radii as much as three times the droplet diameter. Consequently, the classical Lorenz-Mie theory (as well as principles of ray optics) may not adequately predict the distribution of scattered light for such systems. In this case, one may therefore need to employ the generalized Lorenz-Mie theory to account for the effects of finite beam width.

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ACKNOWLEDGEMENTS

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The authors would like to thank the American Society of Engineering Education for funding the post-doctoral appointment of Dr. Joseph T. Hodges.

REFERENCES 1. W.D. Bachalo and M.J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop

size and velocity distributions," Opt Eng., Vol. 23, No.5, pp. 583-590, 1984.

2. J. Swithenbank, J.M. Beer, D.S. Taylor, D. Abbot and G.C. McCreath, "A laser diagnostic technique for the measurement of droplet and particle size distributions," Experimental Diagnostics in Gas Phase Combustion Systems. B.T. Zinn, ed., Prog. Astro. and Aero, Vol. 53, pp. 421-447, 1977.

3. D.J. Holve and K.D. Annen, "Optical particle counting, sizing and velocimetry using intensity deconvolution", Opt. Engr., Vol. 23, No.5, p. 591, 1984.

· 4. F. Beretta, A. Cavaliere and A. D' Alessio, "Drop size and concentration in a spray by sideward laser light scattering measurements," Comb. Sci. Tech., Vol. 36, pp. 19-37, 1984.

5. J.T. Hodges, T.A. Baritaud and T.A. Heinze, "Planar liquid and gas fuel and droplet size visualization in a D.I. Diesel Engine," SAE 910726, 1991.

6. B. Maheu, G. Gouesbet and G. Grehan, "A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile," J. Opt. (Paris), Vol. 19, No.2, pp. 59-67, 1988.

7. G. Grehan, G. Gouesbet, A. Naqwi and F. Durst, "Trajectory ambiguities in phase Doppler systems: use of polarizers and additional detectors to suppress the effect," in (Jh Inti. Symp. on Appl. of Laser Tech. to Fluid Mech., Lisbon Portugal, pp. 12-19, 1992.

8. F. Guilloteau, G. Grehan and G. Gouesbet, Optical levitation experiments to assess the validity of the generalized Lorenz-Mie theory," Appl. Opt., Vol. 21, No.15, pp. 2951, 1992.

9. J.T. Hodges, G. Grehan, C. Presser and H. G. Semerjian, "Elastic scattering from spheres under non plane-wave illumination," SPIE Proc., Vol. 1862, No. 33, 1993.

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