Mie theory calculations: new progress, with emphasis onparticle sizing

Gerard Grehan and Gerard Gouesbet

New progress in Mie theory calculations is reported. A computer program is available using the Lentz algo-rithm allowing more extensive calculations, namely, studying of bigger diameters and more important ab-sorption coefficients. Emphasis on particle sizing is considered.

1. Introduction

A well-known tool for measuring mean velocitiesand fluctuating velocities in fluids is laser Dopplervelocimetry (LDV).' In this method, the fluid velocityis deduced from the Doppler shift of laser light scatteredby small particles moving with the flow, as far as fluidand particle velocities can be assumed to be equal.Various optical arrangements have been used for years.In the fringe or dual beam system, two light beams fromthe same cw laser are focused so that they intersect inthe fluid at their focal points to form a 3-D set of fringes.It can be shown, using Doppler as well as fringe con-cepts, that a moving center passing through the beamcrossover region scatters light that is modulated atDoppler frequency f. The measurement of the Dopplerfrequency permits the measurement of a component ofthe velocity (see Ref. 2 for a detailed analysis of Dopplersignal characteristics).

Then, interest began to focus on the simultaneousvelocity and diameter measurements of a particlecrossing through the fringe pattern. Let us consider atypical Doppler signal corresponding to the passage ofa single particle in the set of fringes (Fig. 1). Some havemeasured the diameters from the visibility V = (Vmax-Vmin)/(Vmax + Vmin) of the signal,3-7 others from themean amplitudes V by using calibration methods.8-'0As a consequence, interest increased in calculating theproperties of scattered light using the old Mie theory,with emphasis on more improved programs.

Let us, for example, focus on Chigier's work.8 Cal-culations of mean amplitudes of the signals were carried

The authors are with Universite de Rouen, Laboratoire de Ther-modynamique, L. A. CNRS 230, Faculte des Sciences et des Tech-niques, 76130 Mont-Saint-Aignan, France.

Received 17 January 1979.0003-6935/79/203489-05$00.50/0.C) 1979 Optical Society of America.

out with geometrical optics for the real refractive index.The range of validity of geometrical optics is not welldefined at the present time for such applications, butit is hardly thought that it could be acceptable for di-ameters typically less than 20 ,m. Yet there are resultscorrect only for small observation angles (less thanabout 150) (see also Ref. 11). Furthermore, the rangeis not taken into account for the imaginary part of thecomplex refractive index, preventing measurements forabsorbing particles. For example, even an imaginarypart equal to about a tenth of the real part can changethe scattered intensity by a factor of, say, 10 (see Ref.12).

So, to remove such problems, a new computer pro-gram, the so-called SUPERMIDI for Mie theory calcu-lations, has been evolved. It allows computations ofscattered intensities, efficiency factors, phase anglebetween perpendicular components of the scatteredlight, and specific turbidity up to very large particles,and for all angles, as well as for particles whose refrac-tive index has imaginary parts as high as 105, thus cor-responding to the case of perfectly conducting scattercenters. The main feature of the program is to use therecent Lentz algorithm for Bessel calculations,'3 thusallowing more extensive calculations than previousprograms.

After discussing the SUPERMIDI, we shall focus aboutparticle sizing applications.

11. Basic Theory

The Kerker formalism' 4 is used. Coordinate systemsare given in Fig. 2.

An electromagnetic plane wave, characterized by itselectric field E parallel to the OX, axis, propagatestoward the positive X3 in a nonabsorbing medium. Itis scattered by an isotropic homogeneous sphere of ar-bitrary size located in 0. Its diameter is d, and itscomplex relative refractive index is m. The scattered

15 October 1979 / Vol. 18, No. 20 / APPLIED OPTICS 3489

where P(1)(cosX ) is the associated Legendre functionof the first kind. The scattering coefficients a, and bnread

a=n(a) n(fl) - mn in(I)An (a)

" =n(a'(O-)mn(f3(a) ,(3)b kn (a) ~4W) - n (f) t4n(a) (3

m ;n (a/n () - An (d) An (a))

where a is the size parameter

a = 7rd/X,

and : is the second size parameter

= m,

(4)

(5)Fig. 1. A typical Doppler signal.

E

Xk rectangula system

X k spherical system

Fig. 2. Coordinate systems.

light is observed at point P in the far field. (X1 >> X,where X is the wavelength of the incident light in thesurrounding medium.)

The scattered light in the far field is a transversalwave. The electric field of the first component vibratesin the plane (OX3, OP), the second component in theperpendicular plane. The intensities at the point P perunit of incident intensity are IX'2 and IX'3, respec-tively,

2(Xl)2 cos2 X3 | E +1 [anrn(cosX 2 )47r2(') k2n(n +1) j

+ bn7rn(coSX2 )]| + 62 sin2X n +1 2 +X 3 sin2X' Ii n(n + 1)

X an~rn,(COSX) ± bn7_n(cosX'2 )I})

(1)

=- \2 [t5i (cos2X~) 12 2 + e5i 2snX')JS1 12])=47r2(Xt)2 2 oS2 +3 (sinx)ll2 1

where 6'i is the Kronecker symbol, and Si is the ampli-tude function.

The Legendre functions 7rn (cosX2) and mn (cosX2) aregiven by

7rn(COSX2 ) = P)(cosX 2 )/sinX'2

Tn(COSX2 ) = dX P(')(cosX' 2 )(2)

where n (z) and n (Z) are Ricatti-Bessel functions givenby

An( = - Jnl1/2(Z)

t( = (12) [Jn+1/2(Z) + i(1)nJ-n-1 /2 (Z)] J (6)

where Jn+1/2 (z) is the half-integral order Bessel func-tion.

The phase angle between the IX/2- and Ix'3- Vi-brations is given by

- Re(Si) I. m(S2 ) - Re(S2 ) Im(S)tan = 7Re(S) Re(S2) + Im(Si) I. (S2)

The efficiency factors for scattering, extinction, andabsorption read

Qsca = 2 E (2n + )lan12 + Ibn12 1,

a2 12"

Qext = -E (2n + 1) Rejan + bn}>a2 I

(8)

(9)

Qabs = Qext Qsca- (10)

Finally, the specific turbidity r/c (where c is the massconcentration) is defined (see Ref. 14, p. 329)

Tr 37r-- =- (2n + 1) Rejan + c Xa3 n=

(11)

Ill. Lentz Algorithm

The main feature of the SUPERMIDI program is to usethe Lentz algorithm allowing more accurate intensities,efficiency factors, and wider particle size range calcu-lations. The relations [Eq. (3)] are written in the fol-lowing way:

)Pn(a)Wn(0)/4n(3)] - mn,(Ce)1

n -n()[tn(0)/n(0)1 - n(a) *f. ( 12)m An(a) [n(18)/n (0)- n(a)

According to Lentz,' 3 the ratios (1. 3)/On (13) can beexpressed by

44'n(d) n _______0An (p) = n: = + -/()(13)

1;n () 0 Jn+1/2(0)

3490 APPLIED OPTICS / Vol. 18, No. 20 / 15 October 1979

A continued fraction representation for the ratio ofBessel functions of complex argument is then given:

J,1-1/2() = 2(n + 1/2)/31

Jn+ 1/2(f3)

1 1+

-2 (n+ 3) -1 +2 (n + )

where a continued fraction

al +- a2 a3 ++ .a4 + .

is noted as follows:1 1 1

a2 + a3 + a4+ * ..

Here, we have -

am = (-1)m+l2(n + m -/2)13'

(14)

(15)

(16)

(17)

Let us now define a new notation for the continuedfraction:

Iai,a2, .. .apl = ai +- 1.a2 + .. + ap

Then, it is shown by Lentz that

Jn-1. 2(13) al ... ail aq,_., aiJ.+1/20) 1a2l ... aq-i a2l laqX . a2l

(18)

(19)

where q is determined by the condition that the par-ticular qth numerator and denominator are identicalto the number of digits desired.

The main feature is that the ratio rather than theindividual Bessel function is given in such a way that thevalues of An (,) for a given n are independent of all thepreceding n', allowing reduction of computer storagerequirements and cost of computing time and pre-venting overflows in a more extensive range of Mie pa-rameters.

A similar procedure using the Lentz algorithm hasalso been used to compute 4r, (a), 4"n(al), Pn(a), andPn (a).

IV. Calculation Results and DiscussionThe SUPERMIDI program is available.15 Connected

programs, INTIMI, allow integration of the intensitiesin a solid angle limited by four angular coordinatelines:

X2 = X2(P), X2 X2(P) + AX2,

X3 = X4(P), X= X;(P) + AXI.

Computations have been made on an IBM computer,the CIRCE in Orsay, France.

Comparisons have been made with several authors,namely, Van de Hulst,17 Wickramasinghe,1 8 Plass,19 theSIMMIE program from Cherdron et al.

2 0 in its IBMversion, and Dave2 ' (see Tables I and II). Comparisonswith Lentz (private communication) are also availablefrom Ref. 15. All comparisons are satisfactory, exceptfor Plass in the backward direction.

We have calculated a as high as about 800 andimaginary parts of the complex refractive index as highas 105. The actual limitations of the program have not

Table : m = 1.33

a Van de Hulst SIMMIE-IBM SUPERMIDI

1 F 5.3 (-2) 5.26 (-2) 5.26 (- 2)B 2.1 (-2) 2.12 (-2) 2.12 (-2)

2 F 3.9 3.94 3.94B 4.4 (-2) 4.36 (-2) 4.34 (- 2)

3 F 4.2 (+1) 4.17 (+1) 4.17 (+ 1)B 2.0 (-1) 2.04 (-1) 2.04 (.- 1)

4 F 1.98 (+2) 1.98 (+2) 1.98 (+ 2)B 8.7 (-1) 8.71 (-1) 8.70 (- 1)

5 F 5.86 (+2) 5.86 (+2) 5.86 (+ 2)B 2.16 2.16 2.16

20 F 4.8 (+4) 4.8 (+ 4)B 2.4 (+2) 2.4 (+ 2)

30 F 2.08 (+5) 2.08 (+ 5)B 8.91 (+1) 8.91 (+ 1)

50 F 1.54 (+6) 1.54 (+ 6)B 2.55 (+2) 2.55 (+ 2)

70 F 6.14 (+6) 6.14 (+ 6)B 3.79 (+2) 3.79 (+ 2)

88 F 1.564 (+ 7)B 4.986 (+ 3)

100 F 2.7607 (+ 7)B 5.602 (+ 3)

150 F 1.4167 (+ 8)B 4.634 (+ 3)

190 F3.4828 (+ 8)B 6.9713 (+ 4)

200 F4.2358 (+ 8)B 1.0356 (+ 4)

300 F2.1181(+ 9)B 2.3471 (+ 4)

400 F6.6057 (+ 9)B 5.4072 (+ 4)

500 F 1.6105 (+10)B 1.0445 (+ 5)

610.61 F 3.5309 (+10)(d = 100 gm) B 1.675 (+ 5)

F = forward (X2 = 0).B = backward (X2 = 180°).Amplitude functions S = S2.

Table II. Complex Refractive Index

Wickra-masinghe SIMMIE-IBM SUPERMIDI

a = 12 F 8330 8330 8328m = 1.4-2i B 14.1 14.1 14.1a = 8 Plass sIMMIE-IBM SUPERMIDI

m = 1.33-10- 4 i F 2950 2940 2940B 4.19 3.79 3.79

m = 1.33-10- 2 i F 2770 2760 2765B 2.69 2.39 2.385

m = 1.33-10- 1 i F 1844 1860 1857B 0.41 0.291 0.291

m = 1.33-i F 1524 1590 1585B 4.34 2.97 2.97

m = 1.33-lOi F 1334 1344.1B 23.5 16.163

m = 1.33-lOi F 1120.4B 14.96

m = 1.33-lOOOi F 1106.0B 14.941

m = 1.33-10 4 i F 1104.6B 14.939

m = 1.33-10 5 i F 1104.5B 14.939

Amplitude functions S = S 2 .

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Table ll. Scattering Efficiency Factors

= O'n(a) SUPERMIDIa Van de Hulst n (a) nk = 1000

0.2 0.00538 0.005351 2.5 2.0359 2.04015 2.11958 2.1161 2.1178

10 2.06458 2.0624 2.063620 2.03487 2.0330 2.033930 2.02432 2.0228 2.023640 2.01883 2.0177 2.018470 2.01145 2.0109 2.0115

190 2.00471 2.0048 2.0053610.61 2.00167 2.0020 2.0021

Furthermore, some results from SUPERMIDI calcu-lations concerning intensities are shown in Figs. 3 and4.

Let us discuss some features from these curves. Ascan be seen, the number of lobes N for real indices isapproximately equal to the size parameter a. The maineffect of the imaginary part of the complex index ofrefraction is to smooth drastically the angular changesin the amount of scattered light.

730 235

Fig.3. a =5,m=n(1-ik).

k: 0 (for X2 0,S:S:6.14 10 6)

nk:103 (tfor X;O, SS,:2 6,0716 )

1224 379

Fig. 4. a = 70, m = n(1-ik).

been sought because of the cost of the computationssince the present range of applications was sufficient forour purpose.

Some results are given for efficiency factors in TableIII. For highly absorbing spheres, comparisons aremade with two kinds of approximations. The first isthe empirical formula given by Van de Hulst (Ref. 14,p. 117): Qsca = 2 + 0.50a-8/9. The second is theformula given by Thomson (Ref. 16, p. 90), using:

ar =-, ~(20)An (a)

=n (a) (21)

The scattered light in the backward direction is moreimportant for absorbing than for nonabsorbing parti-cles, probably corresponding to a more importantquantity of reflected light in the geometrical optics in-terpretation of the phenomenon. On the other hand,the scattering diagrams do not depend on the imaginarypart of the index of refraction in the forward directionwhen the diameter is high enough, since this phenom-enon is driven by the diffraction of light from theequivalent circular aperture of the particle.

Let us now focus on particle sizing. The extensionof Chigier's method to particles smaller than 20 Atmneeds to get a one-to-one relationship between scatteredintensities integrated on the surface of a collecting di-

3492 APPLIED OPTICS / Vol. 18, No. 20 / 15 October 1979

100

-10

-100

1000

Total Scattered Power

X'--, AX;: 2 n:1,5 - iX2: 1 AX 2'

1 AX 2 n: 1,5-0,1iX 1 AX 2

:5- AX' 10'5 X 10 n:1,5- Oi

_ X 3 - -5 A X 1 n :1,5 - 0,115 AX 10'

1 2 3 5 10 20 30nm d

Fig. 5. Total scattering power vs diameters. Note that 1000 in the ordinate should read -1000.

aphragm and diameters. Figure 5 shows calculatedrelationships between total collected power and diam-eters. It should be noted that almost one-to-one rela-tionships can be obtained for the real refractive index.But it is worthwhile to point out that the imaginary partas low as 0.1 considerably improves the one-to-onecharacter of the relationships. This feature was ex-pected since imaginary parts smooth the lobes of scat-tering diagrams. These results allow extensions ofChigier's calibration methods to particles smaller than20 jAm and to absorbing particles.

Conclusion

This paper discussed the new SUPERMIDI computerprogram for Mie scattering calculations. Its mainfeature is to use a Lentz algorithm for Bessel functionratio computations, allowing calculations in more ex-tensive ranges of particle diameters and absorptioncoefficients. From a theoretical point of view, the Lentzalgorithms and the computer programs using them willallow new exciting developments in the field of Mietheory calculations of the linear scattering of light.

From a practical point of view, attention has beenfocused on simultaneous velocity and particle sizemeasurements by means of laser Doppler systems. Ithas been shown that one-to-one relationships can beobtained between scattered intensities integrated oversurfaces and diameters of the scatter centers, for par-ticles smaller than 20 ,um and, more easily, for absorbingparticles, allowing extensions of Chigier's presentmethod.

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Radiation (Academic, New York, 1969)15. G. Grehan and G. Gouesbet, Report TTI/GG/79/20/03 SUPER-

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