Fourier representation of the energy distribution of anelectromagnetic field scattered by spherical particles

Reiner Eiden

A solution to the scattering problem for spherical particles is presented in terms of a Fourier series expan-sion for the electromagnetic field as well as for the corresponding energy distribution. The basic assump-tions are the same as for the Mie solution. The Fourier coefficients of the energy distribution show a sys-tematic behavior as functions of the size parameter a = 27rR/A (R = radius of particle, X = wavelength ofscattered light), and they depend significantly on the complex index of refraction N = n - in K of the par-ticles (n = real index of refraction, K = absorption index). These coefficients are approximately propor-tional to the volume of the sphere superimposed with an oscillation of Bessel function type with argument(n - 1) a. They can be measured directly without any intricate Fourier analysis; and since they depend ina distinct way on the quantities n and K, they can be used to determine these optical properties of particlesof an unknown material. The scattering influence of a gaseous carrier medium can be eliminated withoutany experimental arrangement just by using an appropriate Fourier coefficient of the scattered energy dis-tribution.

1. IntroductionTo study the influence of natural and man-made

atmospheric aerosol particles on the energy budget ofthe atmosphere, the climate, etc., the optical proper-ties of these particles have to be known. Since theaerosol particles absorb radiation energy while sus-pended in a gaseous medium and while exchangingenergy with this medium, the most adequate methodto determine the optical properties seems to be toleave the particles in this natural environment.Other methods that require the separation and depo-sition of the particles may cause obscure changes intheir optical structure.

The properties of suspended and dispersed parti-cles can only be determined indirectly, i.e., by inves-tigating the energy distribution of light scattered bythese particles. The analytical relations between thescattered light and the particle properties are usuallyrepresented by the Mie equations.' But the electro-magnetic field or the resulting angular energy distri-bution of the scattered field, which alone can be mea-sured, does not depend very significantly on theproperties of the scatterers. A general inversion ofthe problem is not known to me.

Therefore a different approach to the problem istried in this paper. The energy distribution of the

The author is with the National Center for Atmospheric Re-search, Boulder, Colorado, 80302.

Received 19 July 1974.

scattered field has been expanded in a Fourier series.The coefficients of these expansions are used to getthe information about the optical properties of theaerosol particles. The Mie solution and the Fourierrepresentation of the scattered electromagnetic ener-gy are based on the assumption of a spherical scatter-er. This is certainly not true for all atmosphericaerosol particles. But a randomly orientated densepopulation of particles with different shapes but thesame equivalent radius Re = (3V/47r)1/ 3(V = particlevolume) may be assumed to act like a population ofspherical particles with a radius equal to the equiva-lent radius. This assumption is supported to someextent by an investigation of Plass and Kattawar 2

based on measurements of Holland and Gagne. 3

Bearing this in mind, the following considerationsmay be applied to single spherical particles as well asto populations of particles with different shapes butwith the same equivalent radius, as long as indepen-dent scattering is ensured.

Since it is our intention to apply the Fourier ex-pansion of the scattered energy finally to atmospher-ic aerosol particles, the appropriate interval for thereal refractive index n chosen according to our pres-ent knowledge (Eiden4 ) for the theoretical calcula-tions presented here are 1.33 < n 1.7. The absorp-tion coefficient n K is assumed to be smaller than 0.1.A method to split up the broad atmospheric aerosolsize distribution in small size ranges, to which theconsiderations of this paper can be applied, is atpresent undergoing tests (Bullrich5).

2486 APPLIED OPTICS / Vol. 14, No. 10 / October 1975

er

Incident Wave

-- Y

are the amplitudes of the electric field components ofthe incident wave (index i). S and Tj are propor-tional to the Fourier coefficients of the scattered en-ergy. They are functions of the coefficients Pm andQm,

S = S(a ,N) = 2 + /2 EP(Pm-J* + P J*)

2Q + m=1(2)

of the corresponding components of the scatteredelectric field (index sc),

E0(sc) Ei)(h+ in' cosms) exp (-x)

X"P(/Plane of Observation

Fig. 1. Schematic representation of the scattering geometry.

II. Fourier PresentationThe far-field solution of the scattering problem in

terms of a Fourier expansion is based on the same as-sumptions as the Mie solution of the problem. A ho-mogeneous absorbing sphere is embedded in a non-absorbing medium, the refractive index of which maydiffer from vacuum. The plane electromagneticwave incident on the sphere, which is centered in theorigin of a coordinate system (Fig. 1), is divided intotwo linearly polarized components perpendicular toeach other. The wave is assumed to travel in thenegative direction of the y axis of the Cartesian coor-dinate system, with one component oscillating per-pendicular (I) and one parallel () to the plane ofobservation (x-y plane). The deduction of the gener-al solution follows the derivation of the Mie expres-sions as given, for instance, by Stratton! 3 Hence adetailed derivation can be omitted. It must be men-tioned, however, that the incident electromagneticwave differs from the known Mie solution in that ithas to be expanded in a Fourier series to get the de-sired result. This expansion of the incident wave inFourier series with respect to the scattering angle c)

(Fig. 1) is given in the Appendix.The energy distribution of the electromagnetic

field scattered in the plane of observation originatingfrom the electric field component of the incidentwave perpendicular (I,) and parallel (2) to theplane of observation is

I = I [Eo"]2 ( + Isj cosjd),

12 rAIEo( 0 12 ( + ZTJ cosja),x \ 2 i 1 /

(1)

where x = 2r/X, with r the distance from the centerof the sphere and X the wavelength of the incident ra-diation.

-Eo (i), E. (i)

(3)

ItEesc) = ttE 0i o + A Q. cosm) x Pm..1 x

with

[2(m + 2E + 1) + 1](2 + 1)![2(m + ) + 11!X (m + 2E + 1)(m + + 2)[(m + )!]2(E!Y2 4 6 a+ 26+

2 [2(m + 2 + 1](2E)![2(m + )]! 4

(m + 2)(m + 2 + 1)[(m + )!]2(E!) 224 6 m+26>

Qm = E2-

{ [2(m + 2 + 1) + 1](2 + 1)![2(m + E) + 1]!(m + 2E + 1)(m + 2E + 2)[(m + )!]2(E!)2246

+M 2 [2(m + 2) + 1(2E)![2(m + )1! 4(m + 2E)(m + 2 + 1)(m + )!]2(E!) 22" m+2e(

The star denotes the complex conjugate value of Pm,Qm; and am +2, am +2c+l, bm+2e, bm+2,+1 are the well-known Mie coefficients in the notation given by Vande Hulst.7 These Mie coefficients are functions ofthe complex index of refraction N = n (1 - iK) of thesphere and the size parameter a = 2rR /A, where n isthe real index of refraction, K the absorption index,and R the radius of the sphere. Since only the far-field is considered, r >> R.

111. Properties of the Fourier Coefficients

A. Physical Meaning and Experimental DeterminationThe general mathematical definition of the Fourier

coefficients, e.g.,

(5)fo I, cosj(d& = .rT[1Eoi)12s,

also reveals their physical meaning. They representthe total energy flux scattered into the plane of ob-servation, weighted by the function cosj]b. Hence,Sj is the ratio of the weighted total energy flux scat-tered into the plane of observation to the proper inci-dent energy flux.

Specifically, S is the ratio of the total scattered

October 1975 / Vol. 14, No. 10 / APPLIED OPTICS 2487

zA

-e__

0 2 4 6 8 10 12 14 16 18 20a

Fig. 2. Coefficient SO of the energy distribution I (of the scat-tered electric field component oscillating perpendicular to theplane of observation) as a function of the size parameter a and for

different real refractive indices n.

energy flux unweighted, since cosjb = 1; whereas S,is the ratio of the difference of the total energy fluxscattered in the forward ( = 0) and backward direc-tion ( = 7-) to the incident energy flux. S, can beused as a measure of the asymmetry of the scatteringphenomenon. All the above remarks are, of course,valid for Tj.

The formal relation of the Fourier coefficients, Eq.(5), can be used to develop a direct measuring meth-od, avoiding a sometimes intricate Fourier analysis.To solve this experimental problem, the measuredsignal of the scattered radiation has to be weightedby the factor cos] 4), dependent on the scatteringangle 4). This can be achieved by applying this fac-tor to the signal within the electronic measuring andintegration system. Splitting up and processing thesignal in several parallel channels, which differ onlyin the applied weighting function, allows the syn-chronous measurement of a multitude of successiveFourier coefficients. The number of channels used isonly a question of technical potentiality.

B. Dependence on Particle Size a and RefractiveIndex N

The coefficients Sj (and Tj) can be approximatedby the relatively simple function

S (aN) = (a2 a )2(EfJakl2ky (fj(aN) + C1{1

+ nic/[4( + nfi)]}a + C2(n + l)cPJ[(n-1)a]

exp(-2anK)). (6)

The first term in the boldface parentheses is impor-tant in the size range a < 2. It gives the Rayleigh ordipole case where a << 1; for instance, f (a,N) =21 [(N 2 - 1)/(N 2 + 2)121. In the size range a > 2, thevolume of the particle prevails (a3), superimposed byan oscillation represented by a squared Bessel func-tion of the first kind. The minima of this oscillationare the zeros of the Bessel function and are deter-mined by the refractive index and the size parameter[(n - 1)a].

Figure 2 shows the dependence of SO on the sizeparameter a and the real refractive index n. Figure3 compares the theoretical curve SO with the approxi-mated curve so(p) [Cl = 23/(3,7r2), C2 = r/3, aoOa, =1,nK=O].

With increasing absorption coefficient n K, the gen-eral feature of the curve is retained. The disturbingripples (see Fig. 3) not reproduced by the approxima-tion vanish more and more (Fig. 4). The curve be-comes smooth. But simultaneously, the maxima ofthe oscillation tend to decrease with increasing ab-sorption. This decrease is ensured by the factorexp(-2anK) in the oscillation term of Eq. (6). It re-sembles strongly the absorption term for the propa-gation of electromagnetic radiation in an absorbingmedium.8 The quantities

C1, C2 , jak

are hardly affected by changes of n and n K, at leastin the range of the values used here, and can be treat-ed as constants.

The multiplicative function

( Ej aa Z JakC'2k)_1

controls the changes with varying index j. It repre-sents an S shaped function with values tending to 0with a 0 and to 1 with a - . The point of half

1101

10 _ / _ _ /

2545 678 1011 2 1 151 1X 1. I 1) 01 23 24 25 26 22 30

Fig. 3. Comparison of the theoretical (solid line) and the approx-imated (dashed line) curves of SO and So(P), respectively, for the

real refractive indices n = 1.33 and n = 1.7.

2488 APPLIED OPTICS / Vol. 14, No. 10 / October 1975

' 345678905I I I 4 I.........I5 I I2It ~ ~ ~ ~ 1 It 32 43 5 6 IS S 1 9 3t ot ot 20 2t 22 23 24 25 26 2 28 29 30

Fig. 4. Coefficient SO as a function of the size parameter a for areal refractive index n = 1.5 and the absorption coefficients n K =

0.02, n K = 0.05, n K = 0.08.

value moves with increasing index j to higher valuesof the size parameter a. So the coefficients Sj are,compared with SO, diminished predominantly in thesmaller size range and approach SO asymptoticallywith increasing a. The a value at which Sj reachesan assumed distance from SO will increase with in-creasing index j. The described dependence of Sjon a is shown in Fig. 5. It can be best demonstratedfor absorbing particles, since in this case the curvesare not disturbed by ripples.

In Fig. 5, the equivalent set of curves for Tj also isplotted. They show the same behavior as Sj. Ingeneral it can be stated, that between Sj and Tj existonly quantitative differences. The same approxima-tion can be used to describe Sj and Tj as functions ofthe size parameter a and the complex index of refrac-tion N. The difference between Sj and Tj can be ex-pressed by a different set of constants C1, C2, as inEq. (6). Equation (6) has been compared with theo-retical values up to a = 100. A deviation of less than4% was found, for instance, in the size range 3 < a <100 for SO and a complex refractive index N = 1.5 -0.05i using

CI = 0.27, C2 = 1.042, OaO = =1.

Around a 1, the approximation was not as good,and the deviation amounted to 10%. In this sizerange the term fj((a,N) prevails, and the expressionfo(aN) = 21 [(N2 - 1)/(N2 + 2)]21 used in the compar-ison and certainly correct for a <<1 may be too simplefor a c1.

C. Determination of the Real Refractive Index nA method suggesting itself to determine the real

part of the refractive index of airborne particles is tomeasure a relative minimum of one of the Fouriercoefficients. The minimum is defined by J 1

2 [(n -

1)a] = 0, and the zeros of this Bessel function aretabulated. So n can be calculated immediately if theorder of the zero is known. Unfortunately, the defin-

ing equation is not unique. But this difficulty can beavoided if an adjoining zero is measured simulta-neously or if there is a rough tentative estimation.

Since the actual curve may be disturbed by ripplesor measuring errors it is preferable to measure simul-taneously more than one Fourier coefficient and tosmooth the curves by fitting to the squared Besselfunction. This method has the advantage of beingindependent of the number of scattering particlesand of the knowledge of the irradiance of the incidentelectromagnetic wave.

D. Influence of the Carrier GasThe molecules of the gas in which the aerosol par-

ticles are embedded affect the amount and distribu-tion of the scattered energy. The field scattered bythe aerosol particles is superimposed by the Rayleighor dipole field (a << 1) caused by the molecules. Adeduction of this disturbing energy distribution re-quires the knowledge of the pressure, temperature,nature of, and mixing ratio of the molecular medium.

On the other hand Mie already demonstrated thatonly the first of the Mie coefficients, a1, has to betaken into account to represent a Rayleigh field.Considering this in Eqs. (4) and (2), it follows imme-diately that Rayleigh particles (a << 1) contributeconsiderably only to the coefficients SO, To, T 2 Sothe influence of the molecules of the carrier gas canbe eliminated, if necessary, without any experimentalarrangements simply by avoiding these coefficients.

IV. SummaryIn this paper an exact derivation of the distribu-

tion of energy scattered by single homogeneousspheres in the form of a Fourier series has been de-

I'l 2 31 , I6 I891I I 12 3 4 92, 2 4 s6 7 9 0 11 12 13 14 15 16 17 18 9 021 22 23 24 25 26 27 28 29

Fig. 5. The coefficients S, and Tj as functions of the size parame-ter a for the indices j = 0,3,6,9 and complex index of refraction N

= 1.5 - 0.05i.

October 1975 / Vol. 14, No. 10 / APPLIED OPTICS 2489

scribed. The properties of the resulting Fouriercoefficients have been discussed. They show the ad-vantage that they can be measured directly and si-multaneously. The number of coefficients that canbe obtained simultaneously is only limited by thetechnical possibilities. A somewhat troublesomeFourier analysis can be avoided. Beside the fact thatthe Fourier coefficients can be determined by a sim-ple procedure, the Fourier expansion of the energydistribution makes it easier, for instance, to modelthe atmospheric radiation field.

The Fourier coefficients can be approximated by asimple expression that depends in a distinct way onthe size parameter (0 • a • 100), the real index of re-fraction (1.33 n 1.7), and the absorption coeffi-cient (0.0 < K • 0.1). The most noticeable featurerevealed by this approximation is that for a a > 2 thecoefficients are proportional to the volume of thesphere and to a Bessel type oscillation. The wave-length of this oscillation depends crucially on the realindex of refraction n, whereas the amplitude is expo-nentially dampened depending on the absorptioncoefficient n K. The matching with the Rayleigh sizerange provides an additional function that prevailsincreasingly in the size range a < 3 and depends on Nand c.

Moreover the Fourier representation provides theadvantage of eliminating the influence of a carriergas without any experimental arrangement simply byavoiding use of the first coefficients, S,, T, T2 .

This work was supported by the Deutsche For-schungsgemeinschaft SFB 73 Spurenstoffe der At-mosphaere. It was carried out in part at the Nation-al Center for Atmospheric Research in Boulder, Col.The author is particularly indebted to the ClimateProject group at NCAR and personally to Philip D.Thompson, William W. Kellogg, and John J. DeLuisi.He would like to thank Martial L. Thiebaux, whocritically read the manuscript, and Eileen Workman,who carefully typed the manuscript.

R. Eiden is a Scientific Visitor from the Institut furMeteorologie, Johannes Gutenberg-Universitiit, D 65Mainz, Germany. The National Center for Atmo-spheric Research is sponsored by the NationalScience Foundation.

Appendix: Fourier Representation of anElectromagnetic Plane Wave

The amplitude of the electric field vector of theplane wave incident in the negative y direction of aCartesian coordinate system is as follows.

(a) Oscillating perpendicular to the plane of ob-servation (x-y plane, Fig. 1):

E= exp(iky), 3 ;

M 7 Z 0 XT m ' i)O (2=oOm X m+el(s)

X :Jm+ 2 6 ,3 / 2(X) cosm4

er + (2 %,m)

(m + 2E + 1)(m + 2 + 2) 1d-[X=/2Jm+ 2e+3/2(T)]

x dP lm+ +1kcosO) m2 am +2e

dO (m + 2E)(m + 2E + 1) sinO

XP pm

+ 2 (COS0)&m+2+1/2 (i) }COSm

ma2 + 2 dPim,, + 2 e(COS 0) (

2 (m + 2E)(m + 2 + 1) dO m + 2e + 1/2

- 12 a .- + L Zm + 2 + (!cos )-2(m + 2 + i)(m + 2E + 2) I Asin0d

x [x 1 \m+2+3/2] inmeJ ). (7)

(b) Oscillating parallel to the plane of observa-tion:

"E = exp(iky)e,;

= E E (i)M-1 (2m am +,, P.+ 2 '(COS0)X m=O e=O X

X~m + e + 12(° Sim(>e~ + {2m( _ )a, + 2 x ai2Jsinmwe, + 2m 1°2( +M + 2E)(m + 2E +1

X d[m 2. )] 32°}s dPnM

+ 2 COS 0)

+2m +2+12X

+ m(m + E + 1) (m + 2E + 2) sin9 +AS0

Xjmn+ 26+ 3/ 2(V)}sinmb,6e +(2- 6, J,

x n+2e+I(m + + )(m + 2 + 2

dP, + 2 + (COSO)

') dO Jm+2+3/2M

+2 (m + m2 m m+2e6 + + 2E)(M + 2 + 1)-+2tm s(coO)

x-[X 1 2Jn+2.+T 2(X)] >CosmbeeI.)-dx +2 12X (8)

The Fourier expansion is given with respect to thescattering angle 4), which is related to the regularspherical coordinate 0 by 4) = 0 + (7r/2). The ampli-tudes are normalized to 1. er, e , e, are the unitvectors of the spherical coordinate system modifiedby using b instead of 0 (see Fig. 1).

Pem(co0s) are associated Legendre functions;Je+1/2(x) are Bessel functions of the first order; 'bomthe Kronecker symbol; and k = 27r/X.

To deduce Eqs. (1) and (3), the following knownrelations have been used:

2490 APPLIED OPTICS / Vol. 14, No. 10 / October 1975

exp(iky) = exp(-ix sinO cost);

= E (2 - 6,,)(i)iJm(i sinO) cosmcIr; (9)

J( Rsin0) = .72 am+ 2 P(m + 2 e(cOsO)Jn+ 2 1/2(X); (10)

x31 cosO 6

X am+2 e+ lPm+ 2 e+ I(cosO)Jm+ 26 +3 /2 (X; (11)

with

a +2e = /(7r/2)({f2(m + 2) + 1 (2)!}/

X [2 +26(m +

a..2,. t= VGr/2)(-[2(m + 2 + 1) + 1](2 + 1)!}/

X[2n+ 26(m + )!E!])

References

1. G. Mie, Ann. Phys. 25, 377 (1908).2. G. N. Plass and G. W. Kattawar, Appl. Opt. 10, 1172 (1971).3. A. C. Holland and G. Gagne, Appl. Opt. 9,1113 (1970).4. R. Eiden, Z. Geophys. 39,189 (1973).5. K. Bullrich, Ed., Arbeitsbericht des Sonderforschungsbereichs

73, "Atmosphdrische Spurenstoffe" (Deutsche Forschungs-gemeinschaft, Frankfurt/M.-Mainz, 1974), pp. 74-76.

6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, NewYork, 1941), p.'563.

7. H. C. Van de Hulst, Light Scattering by Small Particles(Wiley, London, 1962), p. 123.

8. M. Born and E. Wolf, Principles of Optics (Pergamon, NewYork, 1964), p. 614.

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October 1975 / Vol. 14, No. 10 / APPLIED OPTICS 2491