Scattering of laser beams by Mie scatter centers: numerical
results using a localized approximation
Gerard Grehan, B. Maheu, and Gerard Gouesbet
Relying on van de Hulst's localization principle, a localized approximation to the generalized Lorenz-Mie
theory is introduced. The validation of this simple approximation is obtained from numerical comparisons
with the Rayleigh-Gans theory. Other comparisons concerning scattering profiles are carried out first with
theoretical data published in the literature and later with experimental measurements. Original results are
given for coal particles as an example of the versatility of the method.
1. Introduction
For several years research at Rouen University hasbeen aimed at developing laser diagnosis for multi-phase media (particle sizes, concentrations, velocities).This work is still in progress and covers a large range ofdiameters (from under 100 A to over 1 mm) and con-centrations (from 0 to over 10 kg/M3 ).1"2 The basictheoretical background behind this work is the classi-cal Lorenz-Mie theory which describes properties ofthe light scattered by a Mie scatter center (spherical,homogeneous, isotropic, nonmagnetic particle) illumi-nated by a plane wave. However, for some applica-tions, when the particles are illuminated by a laserbeam, the Lorenz-Mie theory is not satisfactory. Agood example is the visibility concept introduced byFarmer3 for simultaneously measuring the size and thevelocity of a particle crossing the control volume of alaser Doppler anemometer. This concept has beenproposed on the basis of a limited scattering theorywhere the illuminated beams are plane waves and hasbeen discussed elsewhere.4'5 It appears that a generaltheory of visibility is lacking and that a significantmove toward such a theory would be the design of ageneralized Lorenz-Mie theory (GLMT) in which theproperties of the light scattered by a Mie scatter centerilluminated by a Gaussian beam would be computed.Such a theory would have a wider range of applicationthan optical sizing, including, for example, the inter-
The authors are with INSA de Rouen, Laboratoire d' Energetiquedes Systemes et Procedes, UA CNRS 230, B.P. 08, 76130 Mont-
pretation of optical levitation experiments. Examplesof such experiments are described by Ashkin,6-8 Roo-sen et al., 9 -12 and also by Gr6han and Gouesbet.13
Roosen et al. 9 -' 2 calculated the radiation pressure forlarge spheres using geometrical optics. More rigorouscalculations would be possible in the framework of aGLMT.
In 198214 two of us presented a GLMT in which thescattered intensity expressions were established usinga Bromwich formalism for the case of Mie scattercenters located on the axis of a laser beam. The laserbeam model was a so-called axisymmetric light profile,in accordance with the Kogelnik and Li work.15 Thistheory (referred to here as order L- of approximation)was not rigorous insofar as the field expressions de-scribing the incident light do not comply with theMaxwell equations, as discussed by Lax et al.16 andDavis.17 Later, we presented an improved theory,18
where the incident light was more precisely described.Again, the field components do not satisfy the Maxwellequations, but it was concluded that the errors in-volved could be detected only in exotic situations (thistheory is referred to here as order L of approximation).Furthermore, the framework of this theory was readyto handle better and better descriptions of the incidentlight, which can be as closed as desired to a completeobedience to the Maxwell equations. The precise de-gree of approximation involved is extensively de-scribed elsewhere.19 Not only were the expressions forthe scattered intensities established in Ref. 18, but alsothe expressions for phase angle, cross sections, effi-ciency factors, and radiation pressure. Order L- ofapproximation was established to be a mere approxi-mation to order L. Because of the high accuracy oforder L, it will be considered here as a rigorous theory(rigorous GLMT).
The rigorous GLMT formalism is a mere generaliza-tion of the LMT as given by Kerker,2 0'2 ' the main
Fig. 1. Geometry of the scattering problem: Mie scatter centerilluminated by an axisymmetric beam with Gaussian amplitude
distribution.
feature being the introduction of a sequence of correct-ing scattering coefficientsgn (n from 1 to + a) involvingthe description of the incident light. The expressionsfor the g are, however, complicated to compute, inthat they contain quadratures of highly oscillatingfunctions. This work is now in progress, and althoughit is incomplete, significant steps have been carriedout. However, the values of gn can be estimated usingan argument which relies on the principle of localiza-tion. The introduction of the correcting scatteringcoefficients, calculated by means of this argument,into the rigorous GLMT, leads to a theory that we callthe localized approximation to the rigorous GLMT.The aim of this paper is to discuss this localized ap-proximation, including the production and the discus-sion of numerical results. Let us emphasize that themethod presented is of particular interest because ofits simplicity, its ability to handle easily a large varietyof incident light profiles, and its straightforward im-plementation by any owner of a Mie computer pro-gram.
Other authors dealing with laser beam scattering bysmall particles have developed several approaches tothe problem. Morita et al.22 have examined the scat-tering of a wave having a Gaussian amplitude distribu-tion by a small sphere located on the axis near thebeam waist. The scattering of a paraxial Gaussianbeam wave by a homogeneous sphere on the beam axishas been studied by Tsai and Pogorzelski23 using seriesexpansions of cylindrical vector wave functions interms of spherical wave functions, a work generalizedby Tam and Corriveau24 for the case of scatteringspheres with arbitrary location. In their most recentpaper, Yeh et a.
2 5 represented a focused beam by itsplane wave spectrum and studied the scattering ofsharply focused beams by spheres and other nonab-sorbing axisymmetric bodies. Finally, Kim and Lee26
published more recent results concerning scattered
energies, absorbed energies, and radiation pressure forspheres arbitrarily located in the beam, using a com-plex-source-point method. When relevant, resultsgiven by these authors are discussed and comparedwith ours. Comparisons with some optical levitationexperiments are also examined.
The paper is organized as follows. Section II recallsthe results from the rigorous GLMT and introducesthe van de Hulst principle of localization to computethe correcting scattering coefficients in the frameworkof the localized approximation. Section III provides avalidation of the localized approximation from com-parisons of its results with those from the Rayleigh-Gans theory for refractive indices close to unity. Sec-tion IV presents comparisons of the localizedapproximation with numerical values available fromother theoretical works published in the literature.Section V presents comparisons with experimental re-sults from optical levitation experiments (to ourknowledge, the only relevant experimental results pre-sented in the literature). Finally, some original resultsare given in Sec. VI with comments concerning theimprovements obtained from the GLMT with respectto the pure plane wave scattering.
11. Localized Approximation to the Generalized Lorenz-Mie Theory
A. Rigorous GLMT
As demonstrations are given elsewhere,148 9 theexpressions for the rigorous GLMT are only brieflymentioned. The Mie scatter center (diameter d, com-plex refractive index m relative to the surroundingnonabsorbing medium) is located at a point 0 of aCartesian coordinate system Oxyz (Fig. 1). The inci-dent Gaussian laser beam (wavelength ) propagatesfrom the negative z to the positive z (the framework ofthe rigorous GLMT remains nevertheless valid formore general incident beams). The aim is to predictthe characteristics of the scattered light observed at apoint P defined by its (r,O,o) spherical coordinates. Inthe far field, it is found that the scattered waves aretransversal (Er = 0) and contain two electric compo-nents Eo and E.
The scattered intensities Io and , associated withthe fields Eo and E, respectively, are given by
IO 22 1S21
2coS
2f,
I= 2 JS'12 sin 2f,'
where the amplitude functions read
S1 = n + I g[arirn(cosO) + b(cos0)],n An(n + 1)
S2 = 2n 1gj[ann(COSO) + b7I-n(CoSO)],n=1 n(n + 1)
(1)
(2)
(3)
(4)
where an and bn are the classical scattering coefficients,
Irn and rn are the Legendre functions (classical nota-tions, see Refs. 14, 18, and 20).
The phase angle y between the perpendicular com-ponents E0 and E<, is given by
Re(S1 ) Im(S 2) - Re(S2 ) Im(Sl)
Re(Sl) Re(S2) + Im(Sl) Im(S2)
The scattering and extinction cross sections Csca andCext, respectively, read
Csca = 3 i (2n + 1)(IanI2 + IbnI2) Ign2, (6)n=l
Cext = -Re (2n + 1)(an + b,)Ign 12. (7)n=1
Finally, the force due to radiation pressure acting onthe sphere can be computed from the pressure crosssection Cpr:
Cpr = Re [ (2n+1) 2 gIl
Cp I7 22n+nn=1
Zn(n +2)n=1
-E n + l (anan+l + bnbn+)gngn+l . (8)n +n=1
These expressions are identical to those in the classi-cal Lorenz-Mie theory except for the appearance of thecorrecting coefficients gn. These coefficients dependonly on the properties of the incident light. They aregiven by
2n + 1l 1 Jt ikr sin2O f * exp(-ikr cosO)7 rn(n + 1) jol~ni
X Al(kr)P (cosO)dOd(kr), (9)
where k is the wave number mw/c, (co being the angularfrequency and c the speed of light). The ' are theBessel spherical functions (ratio of Ricatti-Besselfunctions over argument), and the Pn1 are the associat-ed Legendre polynomials Pnn of order m = 1. Thefunction f is the so-called radial basic function whichcontains all the required information concerning theincident light. At order L of approximation (rigorousGLMT), it reads
f(r sinO, r cosO) = ipo - 2Q r cosO) ' (10)
where %Po is the fundamental mode solution, 1 is the so-called diffraction or spreading length, and Q is given by
1(i + 2z/1)
The aim of the localized approximation is to avoidthe computation of the unpleasant-looking integrals(9), although such computations will be achieved in afurther step still to be completed.
B. Localized Approximation to the GLMT
A plane wave front of infinite extent may be thoughtof as being made up of separate rays of light eachindependently pursuing its own path with the provisothat each ray has a minimal width relative to its lengthand to the wavelength of the light. The representationof a wave front by localized rays is of particular interestand importance in geometrical optics.
In the study of light scattering by spheres having alarge diameter with respect to the light wavelength, aprinciple of localization was formulated by van deHulst (Ref. 27, p. 208) according to which "a term oforder n corresponds to a ray passing the origin at adistance (n + 1/2) (X/27r)." Here, the term of order n isthe term of a Lorenz-Mie- series containing Besselfunctions and spherical harmonics, the integer n rang-ing from 1 to a.
Using localized rays to analyze the scattering of aplane wave in the geometrical optics framework, vande Hulst requires that, in any given case, the width ofthe rays has to be larger than the wavelength X, and, forrays of length 1, it must be larger than :X (Ref. 27, p.103). Consequently, scattering by spherical particlesmay be analyzed by the principle of localization whenthe particles are large enough.
From a mathematical point of view, the principle oflocalization is related to the asymptotic behavior of theBessel functions for large values of integer n. When (n+ 1/2) is greater than the size parameter a = rd/X, thecontribution of the corresponding Bessel functions tothe scattered intensities rapidly becomes negligible.Conversely, the main contribution comes from lowerinteger values corresponding, through the principle oflocalization, to rays hitting the scatter center or pass-ing close to it.
We have extended the principle of localization to thecase of Gaussian beams. Considering the Gaussianbeams at order L- as axisymmetric light profiles, theamplitude function at distance p from the axis is as-sumed to be described by the leading term exp(-p2/w2)for a beam of width w0. Consequently, the amplitudefunction for the term of order n in the Mie series,corresponding to a ray passing at a distancep = (n + 1/2)(X/27r), reads
= exp 22Lrw | | (12)
(11)
At order L- of approximation developed in Ref. 14, theradial basic function is simply taken as o. When theincident light is a mere plane wave, all the gn are foundto be equal to 1 and the rigorous GLMT becomes theclassical Lorenz-Mie theory.
In the localized approximation to the GLMT, weidentify each coefficient gn with the amplitude corre-sponding to the nth ray, i.e., with the amplitude of thewave at a distance (n + 1/2)(X/27r) from the beam axis.For an axisymmetric Gaussian amplitude distribution,the gn consequently read
when w0 - o, the Gaussian beam tends to a planewave, and the gn tend to 1 as they do in the rigorousGLMT.
Substituting for the g [Eq. (13)] in set 1-8, weobtain simple formulas which can be readily handledby Mie calculation programs. The numerical resultswe discuss in subsequent sections were first obtainedusing a CIRCE NAS 90-80 computer with a modifiedversion of the SUPERMIDI program. 2 8'2 9 Later, formost of the results, we also used a personal computerZ100 (8088+8087) with a modified version of the SIM-MIE program.3 0 The SIMMIE program does not use theLentz algorithm3 ' and is more limited than SUPER-MIDI. Its implementation on a personal computerZ100 was however more straightforward.
111. Validation of the Localized Approximation
Although the principle of localization can be justi-fied by mathematical arguments as discussed above, itis possible to consider that its extension (13) to theGLMT remains a somewhat intuitive step. It is conse-quently important to validate it. For this purpose, letus remember that the correcting scattering coefficientsdo not depend on the sphere properties but only on theproperties of the incident light. Hence validation mayrely on any kind of Mie scatter centers and the moreappropriate case will be the simplest one. This case isprovided by a Mie scatter center having a relativerefractive index nearly equal to one for which compu-tations of the light scattered properties can be carriedout independently in the framework of the Rayleigh-Gans theory.
A. Rayleigh-Gans Approximation for Gaussian IlluminationIn the Rayleigh-Gans approach (also called Ray-
leigh-Debye, or Rayleigh-Gans-Rocard), the funda-mental assumption is that the phase shift correspond-ing to any point of the scatter center must benegligible, that is to say,
6 = kdlmu-tMet << 1, (14)
where m is the absolute refractive index of the spherematerial and m is the absolute refractive index of theembedding medium.
An additional assumption concerning the refractiveindices, namely,
I - mee 1, (15)
yields the classical results of the Rayleigh-Gans ap-proximation as given by Rayleigh,3 2 -3 4 Debye,3 5 andGans,36 and summarized in any reference book.20 27
In this section, the Rayleigh-Gans approximation isused to describe the scattering by a sphere illuminatedby an axisymmetric beam with a Gaussian amplitudedistribution (order L-). In this Rayleigh-Gans ap-proximation, due to the basic assumptions, the inci-dent field inside the scatterer is the unperturbed field
and any elementary volume acts as an induced dipoleradiating independently of the others.
The geometry is shown in Fig. 1 (with E, = H = 0).The Gaussian amplitude distribution of the incidentbeam is
E(r,&,p) = E0 exp(_ r2 Sin°)' (16)
where w0 is the amplitude (1/e) radius of the beam andr sinG is the distance from the z axis. An elementaryscattering volume dv = r2 sinOdrd~dsp at point (r,G,(o) inthe sphere is considered an elementary radiating di-pole, emitting a wave defined by the Rayleigh law.With me = 1 [which is equivalent to the case whererelative rather than absolute indices would be intro-duced in (14) and (15)], the elementary amplitude daof the wave emitted by the dipole, observed at pointP(ro,Oo,o0 ), is given by
= 3r 2 2da= - E(r,0,(p) sin(OP,E)dv.
ro,\2 M2 + 2 (17)
In this paper for the sake of simplicity, as is usual inthe Rayleigh-Gans theory, P is chosen in the yOz planeand 1OPI is equal to 1:
r = 1,
so = r/2,
(OP,E) = (OP,Ox) = r/2. (18)
The refractive index m of the scatter center beingclose to unity, the elementary amplitude coming froman induced dv dipole consequently reads
da = r(m 2 ) Eo exp( _ si 2 0 r2 sinOdrd0do.x2 ~~~~~2Wo/(19)
At point P, the phase shift of the elementary waveemitted by the volume dv relative to that emitted by anelementary volume located at point 0, taken as thephase origin, can be shown to be
2 - (27) (2 sin2) (r cos ) sinO sinp - tan - cos)}
(20)
The total amplitude A(00) observed at point P re-sulting from the interference of the elementary wave-lets, is given by integration of the elementary ampli-tudes over the whole scattering particle of diameter dand reads
B. Comparisons Between the Localized GLMT and theRayleigh-Gans Theory
Quadrature (21) has been evaluated by standardnumerical multiple-integration programs. Figures (2)and (3) compare scattered intensities obtained fromEq. (21) as A2(00) with localized GLMT results accord-ing to Eq. (2), for so = 900 theg0 being computed by Eq.(13) for parameter values compatible with Rayleigh-Gans assumptions d = 4 Aim, m = 1.001, and X = 0.5145gm (corresponding to the main green line of an argon-ion laser). The size parameter is 24.4. Figure 2 corre-sponds to wo = 5 gim, that is to say, for a beam diameterdo = 2wo = 10 Am about twice as large as the particlediameter. Only the shape profiles are compared inthese figures (the absolute location of the ordinates isarbitrary). The agreement between both theories issatisfactory. However, there is only a slight differencebetween the present ratio do/d - 2, and the plane wavecase (also presented in the figure) although the differ-ence is significant (note that the logarithmic scalestend to visually wipe out the difference). Figure 3corresponds to wo = 0.5 gim, i.e., to a diameter ratioequal to -0.25. Again, the agreement between thelocalized GLMT and the Rayleigh-Gans theory is sat-isfactory. Furthermore, in this case where the beam-width is much smaller than the particle diameter, itdiverges greatly from the plane wave illumination.Similar computations for other w0 values have beenperformed and also lead to a nearly perfect agreementbetween both theories. These results provide a firmbasis for the validity of the localized approximation tothe evaluation of the correcting scattering coefficientsgn This localized approximation, although it mightbe considered as derived a priori from intuitive argu-ments, is entirely justified by its a posteriori conse-quences. It is expected that, starting from the rigor-ous GLMT and treating the expressions involved byexamining their asymptotic behaviors in the limit ird/X- a, a firm a priori assessment of the localized ap-proximation could be derived. Such work, however, isoutside the scope of this paper.
C. Extension of the Localized GLMT to ArbitraryRefractive Indices
The validation of the localized approximation forlarge particles having a refractive index close to 1 hasbeen chosen because this situation enabled us to com-pute scattering diagrams in the framework of the Ray-leigh-Gans theory, independent of the GLMT frame-work. However, this validation concerns the methodused to compute approximated values for the correct-ing scattering coefficients g, which depend only on theproperties of the illuminating light, independently ofthe properties of the scatter center. The results of Sec.III.B give us a sound and reliable basis for a localizedapproximated description of the incoming beam whichis in fact valid whatever the scattering body. Conse-quently, we are now justified in extending the localizedapproximation to arbitrary diameters and arbitrarycomplex refractive indices. In other words, the rigor-ous GLMT provides us with a rigorous scattering the-
log 1
Rayleigh- Gans^ local. approx.* plane wave
Fig. 2. Comparison of the GLMT (localized approximation) withRayleigh-Gans: m = 1.001; X = 0.5145,um; d =
4Mm; a = 24.4; wo = 5MUm.
Iogl I
-4-
-6.
Rayleigh - Gans, local. approx.* plane wave
I`11~***,** >SX~ ***
,6 50 O(-)
Fig. 3. Comparison of the GLMT (localized approximation) withRayleigh-Gans: m = 1.001; X = 0.5145 Mm; d = 4 m; a = 24.4; wo =
0.5 um.
ory and the Rayleigh-Gans results provide us with avalidation for approximation to the correcting coeffi-cients involved in the rigorous theory.
IV. Comparisons with Other Theoretical Results
A. Tsai and Pogorzelski Results
Figures 4-9 show comparisons between the Tsai andPogorzelski results selected from Ref. 23 and resultsfrom the localized approximation to the GLMT. Thecurves represent the decimal logarithm of scatteredintensities Io or I, vs 0; absolute locations of the loga-rithmic scales will be discussed later. In each case, thecomplex refractive index of the particles is equal to m= 1.5-103i. In fact, Tsai and Pogorzelski merely stat-ed that they considered several conducting spheresand a dielectric one without giving numerical valuesfor the indices. We interpreted this statement bychoosing a very high value (academic) of the imaginarypart of m (103) for the conducting spheres. It has been
Fig. 8. Comparison of the GLMT (localized approximation) withTsai and Pogorzelski Eo polarization: m = 1.5-103i; X = 0.5145 um; d
= 3.087 um; a = 18.8496; wo = 1.029 um.
confirmed that the real part of m has no influence onthe results.
The size parameters 7rd/X chosen by Tsai and Pogor-zelski are 9.2056 (Figs. 4 and 5) and 18.8496 (Figs. 6-9).The beam waists are twice (Figs. 4 and 5 and 8 and 9) orfour times (Figs. 6 and 7) the wavelength. From Fig. 4to Fig. 9 we evolve to situations which become progres-sively more and more different from the pure planewave case. Other results are presented by Tsai andPogorzelski for cases which are near the plane wavecase and have not been taken into account being oflittle interest here. In the localized GLMT calcula-tions, the corresponding parameters chosen are X =0.5145 gm, diameters of 1.5076 and 3.087 gm, andbeam radii (wo) of 1.029 and 2.058 im.
In agreement with Tsai and Pogorzelski, the differ-ence between plane wave scattering (classical Lorenz-Mie theory) and finite beam scattering (GLMT) in-creases when the particle size increases relative to thebeam radius. Although the result is not given in fig-ures, we observed that, when the particle is twentytimes smaller than the beam, the radiation acts like aplane wave. Slight differences begin to appear, main-ly in forward scattering, when the particle dimension isone-half of the beam dimension and evolve to largedifferences when the particle is larger than the beam.
Figs. 4, 6, and 8 compare Tsai and Pogorzelski re-sults with ours for the E0 polarization. Agreement isfairly good for particles with radii of the same order ofmagnitude as or smaller than the beam radius. Forlarger particles, Tsai and Pogorzelski do not fit pre-cisely each variation of the intensity we computed, but
local. approx.
-__ Tsai
--- plane wave
50 100 150 0(0)
Fig. 9. Comparison of the GLMT (localized approximation) withTsai and Pogorzelski E, polarization: m = 1.5-10 3i; X = 0.5145 Am;
d = 3.087 Mm; a = 18.8496; wo = 1.029 am.
the same general trend is observed. We must specifythat, the location of the ordinate origin log Io beingarbitrary, we chose coincidence for the Io intensities(and also, later on, for the I, intensities) in the forwarddirection for plane wave scattering. Other choiceswould be possible, but they would lead to similar con-clusions, namely, that the agreement for the E6 polar-ization between the Tsai and Pogorzelski results andours is, on the whole, satisfactory.
However, the situation is different for E, polariza-tion. Figs.5,7, and 9 show large discrepancies. Theseare probably due to the difference between the trans-verse distributions of the incident beams involved.The simple example used by Tsai and Pogorzelskiassumed a beam possessing two electric field nulls inthe transverse plane corresponding to the 0 polariza-tion. While perhaps somewhat exotic, this beam is arigorous solution of the Maxwell equations. Thepresent example, on the other hand, involves a beamexhibiting no such nulls but which is an approximatesolution of the Maxwell equation.
B. Yeh et al. Results
Figure 10 presents the comparison between the lo-calized approximation and results given by Yeh et al.25
which concern the total intensity (Io + Is) scattered bya transparent sphere of refractive index equal to 1.1,having a size parameter equal to 5.0, located on the axisof a beam with spot size wo equal to 1.5 times the sphereradius. More results are given by Yeh et al. concern-ing off-center locations of the sphere, a geometry forwhich our theory is not yet ready (although we expect
Fig. 10. Comparison of the GLMT (localized approximation) withYeh et al.: m = 1.1; X = 0.5145 ,um; d = 0.81885 Mm; a = 5; wo =
0.61414 um.
the required generalization to be fairly straightfor-ward). Let us recall that Yeh et al. use a plane wavespectrum representation of the incident beam.
Again, the agreement between their approach andours is satisfactory. When compared with the curvefor plane wave scattering, the same trends appear. Weobserve a small shift of the intensity minima towardhigh values of the scattering angle and a general de-crease of the intensities with respect to the plane wavecase.
V. Comparisons with Experimental Measurements
There is an almost total lack of experiment measure-ments of scattering diagrams concerning laser lightscattering by spheres. In fact, we know only one ex-ample, published by Gr6han and Gouesbet, 3 obtainedfrom optical levitation experiments. In these experi-ments, a glass particle is trapped by a focused Ar-ionlaser beam propagating vertically upward. The scat-tered light is collected by an optical fiber with goodangular resolution, and an intensity measurement pro-file is obtained for scattering angles ranging from 15°to 400 in steps of 18 min. The known experimentalparameters are d = 29 + 1 m (optical microscopy), m= 1.5, X = 0.5145 gim. The beamwidth is not knownprecisely but can be estimated.
The comparison between the experimental profileand the Lorenz-Mie (plane wave) scattering profilecomputed with the SUPERMIDI program has been madeby Gr6han and Gouesbet13 with a diameter value equalto 29.479 m (size parameter of 171) compatible with
plane wave
w.= 30 pm
w.= 20 pm
wO= 15 pm
15 21 27 33 39 0 (o)Fig. 11. Scattering pattern for different beam radii: E polariza-
tion: m = 1.5; = 0.5145,um; d = 29.479 Mm; a = 171.
the optical microscopy measurement. This diametervalue ensured a nearly perfect agreement concerningthe locations of extrema in the curves and will be usedin all the results given below. This diameter adjust-ment is very sensitive. A small change in the parame-ter values (Aa = 1 or Ad = 0.16 gim) leads to clearmodifications of the maxima and minima locations.However, the differences between experiments andplane wave theory were significant concerning thescattered intensities, exhibiting a greater decrease ofthe intensity level in experiments than in theory.
In this paper the plane wave case is presented in Fig.11, together with computations in the framework ofthe localized approximation to the GLMT, for differ-ent values of the beam radius wo (30, 20, and 15 gim).Results for a beam diameter equal to about twice theparticle diameter (wo - d) are close to the plane wavecase. For descending values of beam radius wo, thevisibility defined on successive extrema decreases andsimultaneously the intensity level vs the scatteringangle decreases more rapidly. In Fig. 12, the best fit ofthe experimental curve is obtained for w = 20 gm, avalue which makes sense according to the focusinggeometry. Such a fit cannot be obtained by any meansusing the ordinary plane wave case. Consequently,Figs. 11 and 12 show that the localized approximationto the GLMT brings a significant improvement withrespect to pure Lorenz-Mie scattering.
VI. General Comments
The above results confirm the validity and reliabil-ity of the localized approximation to the GLMT. Ap-plications must now be developed for specific pur-poses. Such applications will not be discussed here.However, in this section we discuss just one example ofnew results without any reference to comparisons withexperimental or theoretical results from other authors.
Fig. 12. Comparison between the GLMT (localized approxima-tion) and experiment: E polarization: m = 1.5; X = 0.5145 Mm; d =
29.479jum; a = 171.
This example concerns coal particles which are rep-resentative of a class of strongly absorbing particles weinvestigated in another context.137-39 Calculationswere performed using a Zenith 100 personal computerfor a size parameter a = 100, a refractive index m =1.999-0.6i, and a wavelength X = 0.5145 im. In Figs.13 and 14 we compare the localized approximation andthe plane wave scattering for a diameter d = 16.377,gm.The beam radii are w0 = 24.566 /im (one and a halftimes the particle diameter) and w0 = 8.1885,gm (halfof the particle diameter). Figure 13 shows E0 polariza-tion and Fig. 14 the E polarization. For a givenparticle diameter there is a significant minimum in thewhole scattering pattern at 050' (notwithstandingthe successive minima usually observed). When thebeam narrows, the intensity levels at 0 = 180° remainpractically constant and the minimum in the wholepattern becomes more pronounced, also with a signifi-cant weakening of the forward scattering peak. Inother words, the back scattered light is essentially notmodified while there is a dramatic decrease of lateralscattering. These comments are valid for both polar-izations.
Other applications and comparisons are being devel-oped and will be published in the near future.
VII. Conclusion
We have developed an approximation to a general-ized Lorenz-Mie theory for Gaussian beam scattering.The generalized Lorenz-Mie theory was published pre-viously and this paper details the localized approxima-tion to the GLMT relying on van de Hulst's localiza-tion principle. A firm validation of our approximationhas been obtained by successive comparisons betweenthe results of the localized approximation to theGLMT and (i) predictions of the Rayleigh-Gans the-ory, (ii) theoretical data published by other authors,namely, Yeh et al. and Tsai and Pogorzelski, and (iii)results of experimental scattering profile measure-ments. Finally, choosing the case of coal particles asan example of an application of the localized approxi-
plane wave
w =24.566 m
- -_- WO=8.1885 m
1U 9 18u 0 ()
Fig. 13. Scattering profile of coal particles for different beam radii:E0 polarization: m = 1.999-0.6i; X = 0.5145 um; d = 16.377,um; a =
100.
5
-1
plane wave
w_ = 24.566 Am
_ -- w = 81885 Am
10 90 180 0()
Fig. 14. Scattering profile of coal particles for different beam radii:E, polarization: m = 1.999-0.6i; X = 0.5145 Mm; d = 16.377 Mm; a =
100.
mation to the GLMT, we have outlined the potentiali-ties of this approximation. Because of the simplicityand the versatility of our method, further develop-ments are expected to cover as wide a range of situa-tions as the application field of Mie computer pro-grams.
Computations for this paper have been supported inpart by the Centre Interr6gional de Calcul Electroni-que of Orsay, France, with use of routines from theHarwell library (multiple integration routineQBO1AD).
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2. G. Gouesbet et al., "Laser Optical Sizing from 100 A to 1 mmDiameter and from 0 to 1 kg/M3 Concentration," AIAA paper85-1083, AIAA Twentieth Thermophysics Conference, Wil-liamsburg, VA (19-21 June 1985).
3. W. M. Farmer, "Measurement of Particle Size, Number Density,and Velocity Using a Laser Interferometer," Appl. Opt. 11,2603(1972).
4. 0. Schwebel, G. Gouesbet, J. N. Le Toulouzan, and R. Darrigo,"The G-Scheme of Approximations to the Thermal DiffusionFactors: Explicit Formulae," in Proceedings, InternationalSymposium on Plasma Chemistry, Zurich, 27 Aug.-1 Sept.(1979).
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10. G. Roosen, F. de Saint Louvent, et S. Slansky, "Etude de lapression radiation exercee sur une sphere creuse transparentepar un faisceau cylindrique," Opt. Commun. 24, 116 (1978).
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12. G. Roosen and S. Slansky, "Influence of the Beam Divergence onthe Exerted Force on a Sphere by a Laser Beam and RequiredConditions for Stable Optical Levitation," Opt. Commun. 29,341 (1979).
13. G. Grbhan and G. Gouesbet, "Optical Levitation of a SingleParticle to Study the Theory of the Quasi-Elastic Scattering ofLight," Appl. Opt. 19, 2485 (1980).
14. G. Gouesbet et G. Grehan, "Sur la generalisation de la theorie deLorenz-Mie," J. Opt. 13, 97 (1982).
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19. G. Gouesbet, B. Maheu, and G. Grbhan, "The Order of Approxi-mation in a Theory of the Scattering of a Gaussian Beam by aMie Scatter Center," J. Opt. Paris 16, 239 (1985).
20. M. Kerker, The Scattering of Light and Other ElectromagneticRadiation (Academic, New York, 1969).
21. G. Gouesbet and G. Grbhan, "Corrections for Mie Theory Givenin "The Scattering of Light and Other Electromagnetic Radia-tions: Comments," Appl. Opt. 23, 4462 (1984).
22. N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, "Scatter-ing of a Beam Wave by a Spherical Object," IEEE Trans. Anten-nas Propag. AP-16, 724 (1968).
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37. G. Gouesbet, G. Grbhan, and B. Maheu, "Single ScatteringCharacteristics of Volume Elements in Coal Clouds," Appl. Opt.22, 2038 (1983).
38. B. Maheu, J. N. Letoulouzan, and G. Gouesbet, "Four-FluxModels to Solve the Scattering Transfer Equation in Terms ofLorenz-Mie Parameters," Appl. Opt. 23, 3353 (1984).
39. P. Gougeon, J. N. Letoulouzan, G. Gouesbet, and C. Thbnard,"Optical Diagnosis in Multiple Scattering Media Using a Visi-ble/Infrared Double Extinction Technique," preparation.
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