rlf~wi

Angular spectrum of light transmitted throughturbid media: theory and experiment

Alexander A. Kokhanovsky, Reiner Weichert, Michael Heuer, and Wolfgang Witt

Measurements of the angular spectrum of light transmitted through turbid slabs with monodispersionsof polystyrene spheres have been performed. The results obtained are compared with theoretical cal-culations, based on the small-angle approximation of the radiative transfer theory. The experimentaldata and the theoretical results coincide with a high accuracy, which allows us to develop the laserdiffraction spectroscopy of optically thick light-scattering layers. © 2001 Optical Society of America

OCIS codes: 010.1290, 010.1110, 010.3310, 010.4450, 120.5820, 290.4210.

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1. Introduction

The analysis of the Fraunhofer diffraction patterns ofdifferent light-scattering media is a standard tool inthe field of particulate media characterization.1Both the size2 and the shape3 of particles can beetrieved from angular dependencies of the scatteredight at small observation angles. For instance, itollows for polydispersions of large spherical particleska .. 1, 2um 2 1uka .. 1, k 5 2pyl, where l is theavelength, m is the relative refractive index, and a

s the radius of particles!,2

I~u! 5 Cu22 *0

`

a2f ~a!J12~kau!da, (1)

where C 5 const, f ~a! is the particle size distribution,I~u! is the measured scattered intensity, u is the scat-tering angle, and J1~kua! is the Bessel function.There are many methods to retrieve the particle sizedistribution f ~a! from Eq. ~1!.2 They have already

When this research was performed, A. A. Kokhanovsky~alex@zege.bas.net.by! was with the Department of Chemical En-gineering and Chemical Technology, Imperial College of Science,Technology and Medicine, London SW7 32BY, UK, and with theInstitute of Physics, 70 Skarina Avenue, Minsk, Belarus. He isnow with the Institute of Environmental Physics, University ofBremen, Bremen D-28334, Germany. R. Weichert is with theInstitute of Particle Technology, Clausthal Technical University,Leibnizstrasse 19, D-38658 Clausthal-Zellerfeld, Germany. M.Heuer and W. Witt are with Sympatec GmbH, D-38644 Goslar,Germany.

Received 26 June 2000; revised manuscript received 2 January2001.

0003-6935y01y162595-06$15.00y0© 2001 Optical Society of America

een incorporated into the software of modern parti-le sizers.1,4–6

Equation ~1! can be applied only in the case of thinturbid layers, when the multiple light scattering isnegligible. However, there is a need for informationon particle size distributions in dense media withstrong multiple light scattering. This is becausesome media ~e.g., in solid state! cannot be diluted orre not under control ~e.g., various aerosols andlouds in the sky!. The inverse problem for multiplycattering media was formulated and solved in Refs.–11.Our task in this paper is the experimental study of

he influence of the multiple light scattering onraunhofer diffraction patterns of polysterene spher-

cal particles. Experiments were carried out withhe standard commercially available Sympatec Par-icle Sizer.5 Particles were almost monodispersed

and characterized by mode radii near 5 and 50 mm.he results of experiments were compared with the-retical solutions, obtained in Ref. 12.

2. Experiment

The experimental setup is presented in Fig. 1. AHe–Ne laser beam with the wavelength l 5 0.6328mm passes through a beam expander and a samplewith monodispersions of polystyrene spheres. Thediffraction pattern is measured with the help of thedetector with 31 semicircular elements of a differentwidth.5 Polystyrene spheres with certified mean di-ameters d0 equal to 100 6 2.0 mm and 9.685 6 0.064mm have been used in the experiment. The coeffi-cient of variation of the particle size distribution wasequal to 0.042 for large particles and 0.014 forsmaller ones. Particles were composed of polysty-rene DVB ~divinylbenzene, 4–8%! and had the spe-

1 June 2001 y Vol. 40, No. 16 y APPLIED OPTICS 2595

1

F

Tb

Ottmm

d

l

dnm

estsntptpw

iT4a

d

a;s

rmpo

2

cific gravity 1.05 gyml. The refractive index ofparticles at the wavelength 0.59 mm was equal to.59.The results of the experiment are presented in

igs. 2 and 3 for different optical thicknesses t andsizes of particles. Diameters of particles were muchlarger than the wavelength of the incident light.Note that it follows in a good approximation for the

Fig. 1. Experimental setup.

Fig. 2. Angular dependence of the light transmitted through asample with polystyrene spherical particles according to the ex-periment and the theory at the optical thickness equal to 0.144,0.463, and 3.108. The diameter of spheres is equal to 9.685 mm.

Fig. 3. Same as Fig. 2 but with diameter of spheres of 100 mm andptical thickness equal to 0.06, 3.257, and 5.345.

596 APPLIED OPTICS y Vol. 40, No. 16 y 1 June 2001

extinction efficiency factor Qext 5 2 in this case.hus the optical thickness is approximately giveny13 t 5 3Lcydef, where where def is the Sauter di-

ameter of particles, L is the geometrical thickness ofa sample, and c is the volumetric concentration ofparticles in the suspension. It follows for the valueof c that

c 5 defty3L.

ne can see that the value of c is simply expressed byhe optical thickness of the medium t. The value of

was measured independently during the experi-ent. The geometrical thickness L was equal to 20m and def ' d0, owing to the high monodispersity of

particles in a sample.The experiment ~symbols! shows that the increase

in the concentration of particles leads to higher val-ues of the transmitted intensity in the region of smallvalues of t. However, the transmitted intensity re-

uces with further growth of the optical thickness.Indeed, the intensity of the transmitted diffused

ight Id is equal to zero if there are no particles in acuvette. It grows with the optical thickness ini-tially. However, it is equal to zero for semi-infinitemedia. Thus it should be a maximum in the depen-dencies Id~t! at some value of t. This is in correspon-

ence with our experimental results. It should beoted that oscillations in Fig. 3 confirm the highonodispersity of particles in a sample.Another interesting feature is related to the broad-

ning of the diffraction pike due to the multiple light-cattering phenomenon. The larger the opticalhickness t, the larger the half-width of the angularpectrum of the transmitted light. It should beoted that the half-widths of the small-angle peaks ofhe scattered light also increase with the decrease ofarticle size @see Eq. ~1!#. Thus the presence of mul-iple light scattering could lead to underestimation ofarticle size in the standard inversion procedures,hich do not account for multiple light scattering.Figures 2 and 3 differ owing to the absence of min-

ma in Fig. 2 for low concentrations of particles.his is due to the limitation of observation angles by° in the experiment. The first minimum shouldppear at the scattering angle

ud 53.83 3 180

px

egrees according to the Fraunhofer theory.14 Herex 5 2payl is the size parameter. The value of ud ispproximately equal to 0.4° at a 5 50 mm. But it is4° at a 5 5 mm, which is almost outside the mea-

ured angular range.It should be noted that multiple light scattering

educes oscillations and makes diffraction patternsore smooth. The same effect could be due to the

olydispersity of particles in a turbid medium.

tp

snsow

mpcv

wqt

aisse

i

m

Tl

3. Theory

The diffused transmitted intensity can be found fromthe solution of the radiative transfer equation.12,13,15–17

This equation is written in the following form in thecase of the normal illumination of a scattering plane–parallel layer with randomly oriented particles by aplane wave,

cos qdI~t, q!

dt5 2I~t, q! 1

v0

2 *0

p

I~t, q9! p~q, q9!

3 sin q9dq9, (2)

where q is the observation angle; v0 5 sscaysext is thesingle-scattering albedo; ssca and sext are scatteringand extinction coefficients; t 5 sextL is the opticalhickness; L is the geometrical thickness of a layer;~q, q9! 5 1⁄2 *0

2p p~u!dw is the phase function, aver-aged on the relative azimuth w; I is the light intensity;and u 5 arccos~cos q cos q9 1 sin q sin q9 cos w! is thecattering angle. It should be noted that Eq. ~2! doesot account for the spatial width and the angularpread of the incident beam. Thus it can be usednly if the beam cross section is sufficiently large,hich is the case for our experiment ~see Fig. 1!.Our primary task here is to study the influence ofultiple light scattering on Fraunhofer diffraction

atterns in optically dense disperse layers. Thus weonsider the behavior of the function I~t, q! at smallalues q and assume that

cos qdI~t, q!

dt<

dI~t, q!

dt.

It follows from Eq. ~2! in this case that

dI~t, q!

dt5 2I~t, q!

1v0

2 *0

p

I~t, q9! p~q, q9!sin q9dq9.

(3)

The integral in Eq. ~3! accounts for the multiple scat-tering of photons from all scattering directions to thedirection specified by the observation angle q. Itshould be noted that the function I~t, q9! is charac-terized by a rather sharp peak at q9 ' 0 ~see, e.g., theresults of our experiment in Figs. 2 and 3!. Thesame is true for the phase function p~u!, which ishighly peaked in the forward-scattering direction forparticles large compared with the wavelength of theincident radiation. This allows us to change thelimit of integration in the integral term of Eq. ~3! fromp to ` and obtain the following approximate trans-port equation,

dI~t, q!

dt5 2I~t, q! 1

v0

2 *0

`

I~t, q9!p~q, q9!q9dq9,

(4)

here we accounted for the fact that sin q9 ' q9 as9 3 0 and the phase function p~u! is normalized byhe following condition:

12 *

0

`

p~u!udu 5 1. (5)

One should expect that approximation ~4! provideshigh accuracy at small values of q, which are of

nterest in this paper. Equation ~4! can be exactlyolved with several different techniques. For in-tance, one can perform the Fourier transform of thisquation and obtain12

I~t, q! 5I0

2p *0

`

exp$2t@1 2 v0 P~s!#%J0~sq!sds,

(6)

assuming that the intensity of the incident wave is

I~0, q! 5 I0d~q!, (7)

where

d~q! 51

2p *0

`

J0~sq!sds (8)

s the delta function and I0 is the constant. HereJ0~sq! is the Bessel function. The function

P~s! 512 *

0

`

p~u!J0~su!udu (9)

in Eq. ~6! is the Fourier–Bessel transform of thephase function p~u!. It is called the space-frequencydecay function. One can see that P~s 5 0! 5 1 as aresult of the normalization condition ~5!. It is easyto verify solution ~6! by means of substituting it intoEq. ~4!.

Thus Eq. ~6! allows us to investigate diffractionpatterns of multiply scattering layers by simplemeans. Let us subtract the coherent component of alight field I0d~q!exp~2t! @see Eq. ~8!# from the generalsolution ~6!. Then it follows for the diffused trans-

itted light Id~t, q! that

Id~t, q! 5I0

2pexp~2t! *

0

`

$exp@v0 P~s!t# 2 1%

3 J0~sq!sds. (10)

his equation can be simplified in the case of thinayers. It follows from Eq. ~10! as t 3 0 that

Id~t, q! 5 ~I0y4p!v0tp~q!, (11)

where @see Eq. ~9!#

p~q! 5 2 *0

`

P~s!J0~sq!sds. (12)

1 June 2001 y Vol. 40, No. 16 y APPLIED OPTICS 2597

dcf

s

eF

dd

a

hl

e

2

Thus the small-angle diffused intensity is determinedby the angular distribution of singly scattered lightas it should be for small values of the optical thick-ness.

Equation ~10! allows us to study the dependence ofiffraction patterns on the optical thickness and con-entration of particles. To do so, we should know theunction P~s! in Eq. ~10!. That function differs for

particles of different shapes and internal structures.For instance, it follows in the framework of theFraunhofer approximation for monodispersed spher-ical particles16 that

v0 512

, p~u! 54J1

2~ux!

u2 . (13)

From Eqs. ~9! and ~13! one can obtain

P~s! 52p HarccosS s

2xD 2s

2xF1 2 S s

2xD2G1y2JuS s

2xD ,

(14)

where u~sy2x! 5 1 at s # 2x and u~sy2x! 5 0 at s .2x. Thus the diffused intensity Id~t, q! can be pre-ented in the following form @see Eq. ~10!#:

Id~t, q! 5 DF~t, a!, (15)

where D 5 2x2I0yp is the constant, which does notdepend on the concentration of particles, and thefunction

F~t, z! 5 exp~2t! *0

1 SexpH t

p@arccos~ y!

2 y~1 2 y2!1y2#J 2 1DJ0~ yz! ydy (16)

depends only on the optical thickness t and the pa-rameter z 5 2qx. The function F~z! at t 5 0.1, 1, 3,and 5 is presented in Fig. 4. One can see that theintensity of the diffused transmitted light increases

Fig. 4. Dependence F~z! at different values of optical thicknessqual to 0.1, 1, 3, and 5.

598 APPLIED OPTICS y Vol. 40, No. 16 y 1 June 2001

with the optical thickness, reaches the maximal val-ues, and starts to decrease. The amplitude of oscil-lations decreases with the optical thickness t.However, the half-width of the small-angle peak in-creases with the optical thickness. All these fea-tures are consistent with the experimental datapresented in Figs. 2 and 3.

The dependence of light intensity @Eq. ~15!# on theoptical thickness at z 5 1.5 is presented in Fig. 5.One can see that the maximum of the intensity at thefixed angle occurs at t ' 1 in this case. It should benoted that the function Id~t, q! in Fig. 5 was normal-ized to 1 at t 5 0.1.

4. Comparisons of Computed and ExperimentalResults

Let us compare results of calculations with the ex-perimental data obtained. To this end results of cal-culations with Eqs. ~15! and ~16! were matched to thexperimental data at u 5 1° for smaller particles ~seeig. 2! and at u 5 0.1° for larger particles ~see Fig. 3!.

The agreement between theory and experiment isexcellent for all optical thicknesses except tails of theangular distributions of transmitted light ~see, inparticular, Fig. 3!. Thus simple Eq. ~16! can be in-

eed used for solution of both the inverse and theirect problems of light-scattering media optics.The poor agreement of the theory and experiment

t q . 2° in Fig. 3 is due to the fact that multiplescattering of light refracted and reflected byparticles15–17 becomes important in this case. Thisis not accounted for in Eq. ~16!, where the geometricaloptics component of the scattered light is simply ne-glected. Accounting for the geometrical optics scat-tering becomes important for larger particles atsmaller scattering angles than it is for smaller par-ticles ~see Figs. 2 and 3!. This is due to smaller

alf-widths of Fraunhofer diffraction patterns forarger scatterers.

Accounting for the geometrical optics scattering is

Fig. 5. Dependence of transmitted diffused light intensity @seeEqs. ~15! and ~16!# on optical thickness at z 5 1.5.

p

T

iT

e1st

oi

rpcwT

iK

rather simple. Indeed, neglecting the interferencebetween diffracted, reflected, and refracted beams,one obtains16

p~u! 5ssca

dpd~u! 1 sscagpg~u!

sscad 1 ssca

g ,

where sscad and ssca

g are parts of the total scatteringcoefficient ssca 5 ssca

d 1 sscag, related to diffraction

and geometrical optics scattering processes, respec-tively.

It follows that sscad 5 ssca

g for large transparentarticles18,19 and that

p~u! 5pd~u! 1 pg~u!

2. (17)

he function pd~u! is described by Eq. ~13!, and thefunction pg~u! represents the geometrical optics con-tribution to the phase function p~u!. Note that func-tions pd~u!, pg~u!, and p~u! are normalized accordingto Eq. ~5!.

Calculations performed in the framework of thegeometrical optics approximation15–17 show that thefunction pg~u! can be approximated by the followingsimple formula:

pg~u! 5 4b exp~2bu2!, (18)

where the value of b depends only on the refractivendex of transparent particles and not on their size.he multiplier 4b in Eq. ~18! is due to the normaliza-

tion condition ~5!.For the Fourier–Bessel transform ~9! of the phase

function ~17! it follows that

P~s! 5Pd~s! 1 P g~s!

2, (19)

where the value of Pd~s! can be found from Eq. ~14!and @see Eqs. ~9! and ~18!#,

P g~s! 5 exp~2s2y4b!. (20)

Fig. 6. Angular dependence of light intensity according to theexperiment and the theory for samples with polystyrene spheres of100-mm diameter with accounting for the geometrical optical scat-tering at optical thickness equal to 3.257 and 5.345.

Results of the comparison of the calculated diffusedtransmitted intensity @see Eqs. ~10! and ~19!# withxperimental data for monodipersed spheres with a00-mm diameter are presented in Fig. 6. One canee that accounting for the geometrical optics scat-ering @see Eqs. ~19! and ~20!# indeed provides higher

accuracy at larger values of q. Note that the valuef b 5 5 in Eq. ~20! was obtained by fitting of exper-mental and theoretical results.

It should be noted that Eq. ~10! is also valid forandomly oriented nonspherical particles and forolydispersed media. However, these media areharacterized by different spectra P~s! as comparedith the simple case of monodispersed spheres.his results in different angular spectra Id~t, q! of the

diffused light. For instance, for disperse media withpolydispersions of spherical particles it follows that16

^P~s!& 5

*0

`

a2f ~a! P~s!da

*0

`

a2f ~a!da

, (21)

where f ~a! is the particle size distribution and P~s! isdescribed by Eq. ~19!.

5. Conclusions

The experimental measurements of the small-anglepatterns of particulate media show that the influenceof multiple light scattering causes the broadening ofthe angular spectrum of the transmitted light andreduces the intensity of the Fraunghofer rings formonodispersed particles. One can see that Eqs. ~15!and ~16! describe the experimental data with a highaccuracy. Thus they can be used as a theoreticalbasis for the laser diffraction spectroscopy of turbidmedia.

The function ^P~s!& in Eq. ~21! can be retrievedfrom Eq. ~10! by the inverse Fourier–Bessel trans-form. On the other hand, the Fourier–Bessel trans-form of the function ^P~s!& @see Eq. ~12!# allows us toobtain the function p~u! @see Eq. ~1!# and reduce theinverse problem for a multiply light-scattering me-dium to the well-studied case of single scattering.

It is interesting to note that integral equation ~1!can be solved analytically by use of the generalizedFourier transform.14 Thus three successive integraltransforms allow us to obtain the analytical solutionfor the particle size distribution from measurementsof the diffused transmitted light in the multiple-scattering regime. Other possibilities for solvingthe inverse problem associated with Eq. ~10! are dis-cussed in Ref. 17.

A. A. Kokhanovsky is grateful to the Alexander vonHumboldt Foundation ~Germany! and the Engineer-ng and Physical Sciences Research Council ~Unitedingdom! for support of this research. He thanks

A. G. Borovoi and E. P. Zege for many fruitful dis-cussions on the subject. Special thanks are ex-

1 June 2001 y Vol. 40, No. 16 y APPLIED OPTICS 2599

9. E. D. Hirleman, “Modelling of multiple scattering effects in

2

tended to A. R. Jones for valuable comments andhelp.

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600 APPLIED OPTICS y Vol. 40, No. 16 y 1 June 2001

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