The Elliptical Polarization of Light Scattered by a Volume of Atmospheric Air

The Elliptical Polarization of Light Scattered by aVolume of Atmospheric Air

Reiner Eiden

The scattering of linearly polarized light by aerosol particles produces partly polarized light whoseellipticity is theoretically and experimentally investigated for the specific case of a continental atmo-spheric aerosol in a volume of air. With the Mie theory and under the assumption of various aerosolsize-distribution models, the ellipticity has been computed as a function of the scattering angle for variouswavelengths. The computations have been based upon complex indices of refraction: M = 1.5-0.0i,M = 1.5-0.Oli, M = 1.5-0.li, M = 1.44-0.0i, M = 1.4-0.0i, M = 1.33-0.0i. The comparison betweencomputed and measured values for the wavelengths of X = 0.45 ju to X = 0.65 ,u discloses for dry aerosol areal part of the refractive index m = 1.50 and an imaginary part 0.01 <5 k < 0.1. With moist aerosol,the real part of the refractive index as well as the imaginary part decreases.

IntroductionUp to the present time, the theoretical and experi-

mental investigations of the properties of aerosol par-ticles have been related to the intensity I(X) and thedegree of polarization P(X) of the scattered radiation indifferent spectral regions.'- 4 However, while thesequantities give significant information on the size distri-bution of the aerosol particles, they do not provide muchinsight into their physical properties. An investigationof the type of polarization, i.e., linear, elliptical, or cir-cular polarization, is much more informative because,in this way, additional information on the complex indexof refraction M = m - ki (m is the real component ofthe refractive index; k is the absorption on coeffi-cient) of the atmospheric aerosol can be obtained.

So far, the elliptical polarization has been studied byFraser5 and Rozenberg 8 and in a preliminary stage bythe author.' However, the relationship between thetype of polarization and the refractive index has notyet been studied in detail.

Theory of Elliptical Polarization

General Considerations

The four Stokes' parameters I = (I,Q,U,V) describefully the polarization 8 of a monochromatic plane wave.They are applicable not only to monochromatic radia-tion, but also to polychromatic radiation if AXo/Xo<<1 (Xois the wavelength range, Xo is the mean wavelength).

The author is with the Meteorological-Geophysical nstitute,Johannes Gutenberg University, Mainz, Germany.

Received 17 June 1965.

Then the radiation is called quasi-monochromatic. Thetheoretical assumption of a plane wave front is approxi-mated by the divergent scattered light, if the radiation,which is scattered into a small or infinitesimal solidangle is considered.

The Stokes' parameters are defined by the followingrelations :'

I(X) = EIIE11* + E±E:*, U(X) = EIIE±* + EjE1*,

Q(X) = EIIE11* - EEI*, V(X) = [EIIEi* - ELE11*]i. (1)

Figure 1(a) illustrates the scattering geometry.The type of polarization is characterized by the ellip-ticity tank = sb/a [Fig. 1(b) ], the ratio of the axes a andb of the ellipse. The positive sign denotes right-handedpolarization and the negative sign denotes left-handedpolarization. can be derived from the equation

sin2k = V/(Q' + U + V12)'/2 (2)

The physical explanation for the occurrence of ellip-tical polarization of the scattered light is the mean-phasedifference of the components of the electric (or magnetic)oscillation of light attributed to scattering by individualparticles. The magnitude of the phase difference de-pends on the particle radius and the refractive index.With decreasing radius, the phase difference also de-creases. Finally, the phase difference, i.e., the ellip-ticity, which results from the scattering on numerousparticles, depends upon the particle size distribution.

The scattered light will be elliptically polarized if thelight incident upon the scattering volume of air is polar-ized and the plane of polarization is inclined to the refer-ence plane.10 The theoretical approach to the problemis simplest for the incident radiation linearly polarizedwith a plane of polarization of 450 inclination to the

April 1966 / Vol. 5, No. 4 / APPLIED OPTICS 569

nc-dent light `: scattering ol.

reference plane i\ E

scatt\\\\er l I Iscatterelih

(Ex)5, t

b

reference plane, so that the incident beam is character-ized by:

Elj(X) = E(X) or I(X) = U)

= 2EIjE±* = 2E1E,1*, Q - V = 0.

It can be assumed that the scattering atmosphericparticles are spherical.' Since their diameter is of thesame order of magnitude as the wavelength of the in-cident light, it is appropriate to use the t\i1ie theory forthe analysis of the scattering process by individual par-ticles.9 "l0 According to this theory, the field intensitiesof the scattered light are functions of the scatteringangle so, the size parameter a = 2rr/X (r is the radius ofthe scattering particle), and the complex index of re-fraction M = m - ki. It may be assumed that thescattering processes on individual particles suspendedin an isotropic volume of air are independent of oneanother. 1

104

\Vw'=3 dN(r) c r v

del A d log r

10 M l

Y * 3

I 0o2 Model B Model C

.,C,, 100 III , vjr -

Fig. 2. Approximate size distribution of the continental atmo-spheric aerosol particles in a volume of air.

The Stokes' parameters for the entire scattering vol-ume of air are obtained by summation of the Stokes'parameters of all individual particles:

I(X)) = I(OdNW;Q(X) = jfr Q(r)dN(r);

U(X) = r U(r)dN(r); V(X) = Jr 2V(r)dN(r). (3)

N(r) [cm-'] is the number of particles of radius 0 to rwithin a unit volume of air. r and r2 are the lower andthe upper limiting radii of the aerosol size distribution.

The following is an approximate relation that has beenestablished by Jungell for the size distribution ofthe continental aerosol particles in the turbid atmo-sphere:

dN(r) = c r- (*)d logr. (4)'

The constant c depends on the turbidity of the air nearthe ground and thus is a measure of the aerosol numberdensity. The exponent v* represents the mass distribu-tion of the particles of various radii; in a log-log dia-gram it gives the slope of the power distribution.

The computations have been based upon three modelsof aerosol size distribution (Fig. 2). Models A and Brepresent pure-power laws with a sharp, lower limit at aradius r = 0.04 ti. These models are, however, unreal-istic. Therefore, an additional, more realistic model Chas been used with v* assumed to be zero within therange r = 0.04 - 0.1 A. The upper limit has beenset at r2 10 A. The theoretical computations havebeen mainly based upon model C, because the compari-son between theoretical and measured values indicatesthat this model fits best with the natural size distribu-tion.

Furthermore, the computations have been based uponthe real part of the refractive index m = 1.50, m = 1.44,m = 1.40, n = 1.33, in order to cover the range that isreasonable for the atmospheric dust (carbon, m = 1.95;water, m = 1.33). The imaginary part of the index hasbeen set k = 0, k = 0.01, k = 0.1 for m 1.50 (carbon, k= 0.66).

Results of Computations

The numerical computations have been performedwith the 2002 C Siemens digital computer* for the scat-tering angle s = 0 (10°) 1800 and the size parametera = 0.2 (0.2) 159.

Figures 3, 4, and 5 present the values of the ellipticitytan/ for the selected aerosol size distribution models andrefractive indices as functions of the scattering angle s.These presentations do not account for the air moleculessuspended in the scattering volume of air.

* The computer is in the Institute for Applied Mathematics ofthe Johannes Gutenberg University, Mainz, Germany.

570 APPLIED OPTICS / Vol. 5, No. 4 / April 1966

Fig. 1. (a) Schematic il-lustration of scattering geo-metry ( = 0 forwarddirection). (b) Coordinatesystem (xj, x) normalto the direction of light

propagation.

0.1

Model A

A=0.45>i= 0.55

= 0.65

M =1.5 -Oi Model

=0.45 = 055

= 0.65

M =15 -i

_.. . ....

0.5 .'c-3~/I ,' * -

0.4 0.4

0.3 0.3

0.2- ~~~~~~~~~~~~~~~~~~~~~~~0.2-

-0.1 - -0.1 -0 ~~~~~~~~~~~~~~~~~~~0

0.1 - ~~~~~~~~~~~~~~~~~~~~~~~~~-0.1-

-Q2- a ) -0.2- b)

---rrTT 1~ T--T -T T TrT I I---T11

00 200 40° 600 80° 1000 1200 140° 1600 180° 00 200 400 600 800 1000 1200 1400 160° 1800

1P -i --- 4

Fig. 3. Computed ellipticity tan3 for different X as function of scattering angle so: (a) model A; (b) model B; complex index ofrefraction M = 1.5 - 0.Oi.

The Effect of the Aerosol Size Distribution Models A,B, and C

The basic trend of the ellipticity is the same for thethree models. Figures 3(a) and 5(a) show that themodels A and C yield similar curves for the same valuesof M; however, they differ in the absolute values oftheir maximum for different N. This effect is due to thediminished number of small aerosol particles in the sizerange r <0.1 u in model C. The curves of model B showa considerable increase in the ellipticity for the scatteringangle in range 5p = 140°-180°. This increase resultsfrom the greater influence of large particles comparedto that for models A and C.

The Dependence on the Real and ImaginaryPart of the Refractive Index

For model C and for k = 0, the variation of m hasfollowing effects: the absolute maximum of tan isshifted from = 1300 for m = 1.50 to o = 140° form= 1.33 [Figs. 4(c) and 5(a) ], and the absolute value ofthe maximum decreases with decreasing m [Figs. 4(a)-(c), and 5(a)], i.e., the curves flatten and the maximaare less marked. The scattering angle for tano = 0,i.e., the point of transition from negative to positivevalues of tano, is o = 850 for m = 1.50 and so = 80°for m = 1.33. Though this displacement is not very

pronounced, it might be helpful in the comparison withmeasurements.

The scattering angle for tan: = 0 varies much morewith the imaginary part k. With increasing k, it isshifted toward greater values [Figs. 5(a)-(c) ]. Furthermore, the ellipticity varies with the wavelength for thereal part of the refractive index kept constant. In-creasing values of k effect a well-marked decrease of thepositive values and a slight increase of the negativevalues.

The Dependence on the Wavelength

1. Model C. In the case of the real refractive indexm = 1.33 and m = 1.4, the positive values of tan:become gradually smaller with increasing wavelengthX [Figs. 4(a)-(c)]. At m = 1.44, this trend starts toreverse: for the scattering angles so from 1100 to 1400,the ellipticity for X = 0.45 A becomes greater than thatfor X 0.4,4 [Fig. 4(a) ]. For m = 1.5, tan: reaches itsmaximum at N = 0.55 y, and the ellipticity at X = 0.65,u is still larger than that at X = 0.4 j.z [Fig. 5(a)]. Anincrease of the imaginary part k [Figs. 5(b) and (c)]reverses the trend of decreasing values of tan3 with in-creasing N.

2. Models A and B. Models A and B also are char-acterized by a decreasing ellipticity with increasing Xfor m = 1.5, except for so between 1550 and 1800 in modelB.


WM1i C 4 144-02400 ,-

=0.45 - _- =055:065 :085 - - --

X, \\ \

I II\

I II

/1,

-01

-02 2 0 2 . I

0° * 70° *0° 60' so, 00 120 10°160 ° 1W 0

09-

0.8

0.7

0.6

a404

63-

02.

0.1

Model C Mzl14-Oi

A=OAO p -.0.45=0.55 ..........:0.65 -_0.05

IX .... ....... ..I - ,.

-0.1

-0 2- b )

z 4 20''- 'C 60- 10 120 io 160 110 U',iv'i

to Mde C 14=133-0i '

. 1:.4 >

09. a4 _

=Q650 8- =0.85 __

07-

06

05.

04

03

02

01

0

-01

-02 ' ')1 20' 40' 60'

/ ' , - _ \ .j/ , .'

/1\. \/I

8 . 1 . .0 . -. IW- W 1 ' ' so° '8

i . a

Fig. 4. Same as Fig. 3 bt model C:-.

(a) M = 1.33 - .i; (b) M = 1.40 -IM -

*0.0i; (c) M = 1.44 - .i.

Model C 8415-0i

1:0401 -

=045 _ _

=055

-065=085

I " 1.01-

00.

07

0.2

0.1

2 0 1

a. 20.-'0 .,fl & >,n- ; i

Model C

=045=0 55:065

=085

M = 5 -QOl.

Fig. 5. Same as Fig. 4 but (a) M = 1.50 - 0.0i;

The greatest dispersion occurs in the case of model Cfor M = 1.44 - .Oi and M = 1.50 - 0.Oli [Figs. 4(a)and 5 (b) ].

The Dependence on the Limiting Radius r2

If tank is plotted as a function of the limiting radiusr2 of the aerosol size distribution [e.g., in Figs. 6(a)-(c) ],the surprising feature is revealed that only particles ofradius r < 3 influence the ellipticity, for particles ofradius r > 3 the value of tank is almost constant. Thisapplies to all scattering angles for all the models, re-fractive indices, and wavelengths upon which the com-putations have been based. Thus, any conclusiondrawn from the comparison between theoretical andexperimental values refers mainly to the particles ofradius r < 3 . The computations for models A and

(b) M = 1.50 - 0.Oli; (c) M = 1.50 - O.li.

C prove that especially the small particles of radius r< 0.1 ,u (because of their great number) pronounce suchstrong effects in various optical phenomena that theirinfluence cannot be neglected.

The author once expressed the expectation7 thatthe magnitude and the position of the maximum ellip-ticity depend on the upper limiting radius: obviously,this is not true.

The Dependence on the Visibility

The molecular scattering in the volume of air underconsideration has not been taken into account in thecomputations of the curves in Figs. 3-5. This is ad-missible only for large turbidity that corresponds to avisibility of V, - 1 km or c - 10-1o (with v* = 3). Theellipticity attributed to Rayleigh scattering is zero,


50

07

06

I 05

~- 04

03

02

01

10

07

06

05

i O-(12

0.1

Model C M =1. -0.1i

=040 X,=0645 - -_=055= 065 -----= 085 _ _ _

is

I/AI

-0.1

- 0'b ) 4 6 , ' 1 1 ' '

08 2b' 40@-~ 60 80@ 100 1 20 140 16 I EO°'S

T

(04 01 04 10 30 50 70 100

Fig. 6. Computed ellipticity tang for different X as function ofthe upper limiting radius r2 of the aerosol size distribution.

Parameter is the scattering angle o.

since V = 0. However, the Stokes' parameters Q =K sin1V and U = 2K cosso do not vanish for Rayleighscattering. K is a function of the wavelength X andthe number N of air molecules within the volume of air.Since the Stokes' parameters for molecular and aerosolscattering have to be summed up, the terms Q and U inEq. (2) are altered; thus, tan: is also of different de-pendence on the scattering angle ap. The magnitude ofthe error in tan, because of the disregard of the molec-ular scattering, depends on the absolute magnitude ofthe Stokes' parameters; i.e., it relies on the turbidity andon their signs, as long as their magnitude is comparableto those of Stokes' parameters for molecular scattering.

Fig. 7. Illustrative representation of the relative error F( s) ofthe ellipticity owing to the neglect of the Rayleigh scattering in

the volume of air (V. = visibility).

In the range of visibilities greater than 1 km, thiseffect of sign produces the smallest relative errorbetween 4 km and 6 km, though the turbidity is greaterbelow 4 km. Figures 7 (a) and (b) show some examplesof the relative error, F (so) = (Atanf/tanl3) 100, for dif-ferent visibilities V, and wavelengths N.

Measurements of Elliptical Polarization

Performance

The measurements of scattering by an undisturbedvolume of air were carried out by equipment mountedoutdoors on the platform of a tower. The arrangementof the measuring system is given in the schematic diagramin Fig. 8. The light source was a xenon high-pressurelamp L (Osram XBO 1001), whose spectral emittanceis almost uniform throughout the visible spectrum.After the passage through the achromatic K and thepolarizer P, a plane wave of polarized light is incidentupon the scattering volume Sc. The plane of polariza-tion for the polarizer P has a 450 inclination to the refer-ence plane, corresponding to the conditions of the the-oretical treatment.

As the scattered light passes through the analyzer A,the retardation plate R, and the interference filter F,its intensity is measured by the photomuliplier M. Thescattering angle can be varied between o = 20° andso = 160°. The scattering volume limited by the aper-ture of the photomultiplier is about 103 cm3 .

When 4 denotes the inclination of the plane of polari-zation of A to the reference plane [Fig. 1(b)], the elec-


Fig. 8. Sehematic diagram of L K 5c

the experimental arrangement.Sc = scattering volume, P = 1polarizer, A = analyzer, R = Rretardation plate, F = inter- A

ference filter, M = photo- Fmultiplier, W = diaphragm, KL = light source, K = achro-matic.

AM

trical-field intensity after the passage of R, A, and F isobtained from the following equation, according toBorn and Wolf:8

B(op,1,,e) = E(tso) cos + I(t,so) sineiE.

And the intensity is expressed, with respect to Eq. (1),as follows:

IMsz,0,e) = E(tsoce) E*(t,o,0,e)

= I cos' + I sin2o + (U cose - V sine) coso sink. (5)

f is the phase shift induced by the retardation plate.The Stokes' parameters are obtained, then, from Eq.

(5) with dependence on the inclination 4) of the plane ofpolarization of the analyzer and the retardation of theplate, as follows:

for = 0 and 7r/2, e = 0:

Q = I(so,0,0) -I[p,(7r/2),0];

for = 7r/4 and (37r/4), e = 0:

U = I[<,(7r/4),0] -I[p(37r/4),0];

for = r/4 and (37r/4), e = o:

V = sin-eoI(,,(r/4),eo) - I(so,(3 r/4),eo)

- coso[I(o,(7r/4),0) - I(so,(37r/4),0)I. (6)

The angle can be selected by choice, whereas theretardation e0 depends on the material of the retardationplate and the wavelength N. (In case the retardationplate is not applied, e = 0.)

The interference filters that have been used have aspectral half-width of A = 0.015 ji. Therefore, thelight that passes through them can be approximated asquasi-monochromatic. Measurements have been takenwithin three wavelength ranges at X = 0.443 u, X =0.548 ui, and X = 0.639 /i, and in the scattering angle so =50° to o = 1600 in 100 intervals. It took almost 1 hto complete one series of measurements.

The consistency of the measurements can be checkedeasily: the intensities related to two planes of polariza-tion of the analyzer that are normal to each other mustbe the same:

I(so,0,0) + Iso,(7r/ 2 ),O] = I[so,(r/ 4 ),O] + I[so,(3 r/ 4 ),01

= I(7r/4)eo] I[sp,(3 ir/ 4),eo].

When the Stokes' parameters have been evaluated bymeans of Eqs. (6), tano can be obtained from Eq. (2).

Experimental Results

Figures 9 and 10 present two series of the measure-ments of the ellipticity. The experimental results thathave been obtained so far can be divided roughly intotwo categories: the relative humidity during the mea-surements either below 70% or above 70%. Junge12

found that the growth of the aerosol particles starts withabout 70% relative humidity, i.e., the water vapor sus-pended in the air starts condensing upon the surface ofthe aerosol particles.

The comparison between the measured and the com-puted curves illustrates this effect. The real part of thecomplex index of refraction is m = 1.50, in the case ofdry aerosol, but m = 1.44 in the case of moist aerosol.The latter has been presented in Figs. 9(a) and 10(a).The dependence of the refractive index on the relative

. 045t p model C M-144-0.A0.443P ml.oured ( Am/65;lMan)

correction orc 1oyleigh-potlclesmodel C M=144-.,k-5-6km

0.

of

A '0.55 model C 14a 144-0.

.1 0S40 meosored ( 0l65, Moinzlcorreclion for RoylIgh-prrltlemodel C M144-Oi,- 5-6km

I

1.0

09-

Q8-

0 -

0.6

0.5

04

Q3-

02

01.

-0

-C

0' 20' 40' 60' 80 10 20' 140' 160' 160 0' 20' 40' 60' 80 100' 120' 10° 160' M-O'

IP - ?

: 0.65 p model C M=144-Oi

1=06390 m-eosd ( l60; M.i z ) ....correclia, for Royleih -poriclesmodel C Ml44 -0oV-g-6 km

/ 'I

12- )

0O 20-' 40- tr 66- 100 120 )~ IO 140- 160 r; 1 80-

W -iFig. 9. Comparison between measured and computed ellipticity tang for different X. 8 March 1965, Mainz, relative humidity 50%

to 60%.


. . .. . . -, . .T . . . . . . . .

I

b )

A-0=45p model C M .5-(0Ii -

A =0443 mesred (365fi MMz) .

correction for Raylegh- -___portclesmodel C M5-(0li'=4-5 km

140' 160' 160'

Same as Fig. 9.

A:0S50V mode C MS-Olli

10.650p model C fM-li ......

1=00639p measured (3136;MfMa)e)

correction for Rayigh- ----paridfemodel C Md15-00 i ,V4-5kan

20' 40' 6 80 '100° 1201 h0' io- 180°' ' 10' 40' 60' 80 100' 120' 140' 160' 180'f -' --

13 March 1965, Mainz, relative humidity 80% to 90%.

humidity has not yet been investigated. The accuracyof the magnitude of the refractive index, which has beenevaluated by means of the method described here, isabout 1%.*

Furthermore, this comparison enables one to makesome estimates of the imaginary part of the refractiveindex M of the aerosol. Figures 9(a), (b), and (c) showthat dry aerosol is characterized by 0.01 < k < 0.1.The values of k increase with increasing wavelength:at X = 0.450 so, k = 0.01, but at X = 0.650 A, k is stillconsiderably less than 0.1. In the present stage of thisinvestigation, it is not possible to give more detailedinformation. Figures 10(a), (b), and (c) indicate thatthe deposit of water causes a decrease towards zero.

Besides the measurements of the ellipticity, therehave been simultaneous measurements of the aerosolsize distribution for the range of particle radii 0.1 <r < 3 , as well as of the visibility. The exponents ofthe aerosol size distribution v* have been found to fallbetween 3.0 and 3.2.

These results are so encouraging that it appears to behighly advantageous to continue this investigation,especially with attention to the absorptive properties ofthe atmospheric aerosol. The index k is importantfor the computation of the diffuse skylight and, thus,for the physical explanation of the results of measure-ments. No straightforward information on the indexk is available so far.

* The potential variation of the real part of the refractiveindex M with wavelength is of the same order of magnitude. Itcan be taken for granted that no anomalous dispersion occurs inatmospheric aerosol.

As for future investigations, it would be advantageousto eliminate the restrictions owing to the actual weathersituation and to work in the laboratory. By means ofartificial accumulation of natural haze, one should at-tempt to establish such a high turbidity that the in-fluence of the air molecules is eliminated.

The author wishes to express his thanks to K. H.Danzer for writing the computer programs used in thisstudy-and for many valuable discussions. The authoris grateful to R. Jaenicke for the measurements of theaerosol size distributions, and he is especially in-debted to K. Bullrich for his helpful suggestions andassistance given during the preparation of this paper.

References

1. E. de Bary and K. Bullrich, Arch. Meteorol. Geophys.Bioklimatol. 12, 254 (1963).

2. E. de Bary and K. Bullrich, ibid. 13, 47 (1963).3. K. Bullrich, Advan. Geophys. 10, 99 (1964).4. K. H. Danzer and K. Bullrich, Appl. Opt. 4, 1500 (1965).5. R. S. Fraser, Sci. Rept. 2, Contract AF 19(604)-2429 (1959).6. G. V. Rozenberg and I. M. Michajlin, Opt. i Spektroskopiya

5, 671 (1958).7. R. Eiden, Z. Meteorol. Suppl. 17, 17 (1965).8. M. Born and E. Wolf, Principles of Optics (Pergamon Press

Ltd., Oxford, 1964), p. 545.9. H. C. van de Hulst, Light Scattering by Small Particles

(John Wiley & Sons, Inc., New York, 1957), p. 41.10. G. Mie, Ann. Physik 25, 377 (1908).11. C. E. Junge, Ber. Deut. Wetterdienstes 35, 261(1952).12. C. E. Junge, Advan. Geophys. 4, 1 (1958).

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The Elliptical Polarization of Light Scattered by a Volume of Atmospheric Air