identification of non-homogeneous spherical … · 2004. 1. 12. · identification of...

Pergamon

J. Quant. Spectrosc. Rodiat. Tramfir Vol. 56, No. 4, 591-608, 1996 pp. Copyright 6 1996 Elsevier Science Ltd

PII: !30022-4073(96)00077-5 Printed in Great Britain. All rights reserved

0022-4073/96 $15.00 + 0.00

IDENTIFICATION OF NON-HOMOGENEOUS SPHERICAL PARTICLES FROM THEIR SCATTERING MATRIX

ELEMENTS

D. BHANTI, S. MANICKAVASAGAM, and M. P. MENGuct College of Engineering, University of Kentucky, Lexington, Kentucky, U.S.A.

(Received 21 July I995; received for publication 31 May 1996)

Abstract-In this paper, a procedure is described to identify the structure of inhomogeneous spherical particles using polarized light. The scattering (Mueller) matrix elements are shown to provide valuable information about the spatial variation of the optical properties of a scattering particle. Two non-homogeneous systems are considered, i.e., a water droplet with soot accumulating at its surface and a coal particle surrounded by a soot cloud. For both systems, the results show that the Mueller matrix elements are sensitive to the thickness of soot layer and soot volume fraction on the surface layer. If the angular variation of the scattering matrix elements can be accurately measured, it would provide an important diagnostic tool in detecting the presence of inhomogeneity in the scattering particle. Since the elements vary considerably with changing soot volume fraction, an accurate measurement would also provide important quantitative information about the soot present in the shell layer. Copyright 0 1966 Elsevier Science Ltd

INTRODUCTION

Absorption and scattering of an electromagnetic wave by a spherical particle can be determined

exactly using the Lorenz-Mie theory, which was formulated over a century ago by Mie and Lorenz, and outlined in detail by van de Hulst,’ Kerker,* and Bohren and Huffman. Over the years, this theory was also extended to several other shapes and different approximations for various physical systems were proposed (see Wiscombe and Mugnai,4 and Hage et a15). During the last two decades, the predictions based on these approaches have guided several researchers to understand new physical phenomena or to design new particle diagnostic systems.

Even though the Lorenz-Mie theory and its derivatives proved very useful for particle sizing and particle property identification purposes, they were not expanded for identifying the variations in particle optical properties. The main reason for this is the fact that most of the applications of the Lorenz-Mie theory are based on measurements of scattered intensity distribution, which can be used to recover only a few independent parameters of the scatterers. If, for example, it were possible to detect the contamination level of a droplet in clouds or the soot volume fraction distribution around a burning coal particle, several other diagnostic and control technologies can be developed. (Note that it is possible to use resonance techniques for some of these problems, if a single particle can be isolated.)

One of the most powerful, yet underused features of light, i.e., the tool employed for particle characterization, is its wave nature. The plane of oscillations of the light (or, its polarization) can be altered externally before and after it interacts with a scatterer. By defining the relation between the properties of the incident and scattered intensities (i.e., measurable quantities) in terms of the physical characteristics of the scatterer, one can arrive at these characteristics using a rigorous inverse analysis.

Here, the key word is the “measurable quantities”, as defined by Stokes in his 1852 paper.6 Before

tTo whom all comspondencc should be addressed.

591

592 D. Bhanti et al

him, most of the attempts to express polarized or unpolarized light were based on the electric field amplitudes, and none of them were successful. Stokes preferred to use a definition in terms of the quantities that can be measured in experiments, and this novel idea helped him to describe both polarized and unpolarized light easily. In the field of optics, the idea of observables was not used for 100 years, until Wolf considered it in his 1954 paper (see Collett’ for detailed discussion of the polarized light, with historical notes). On the other hand, Stokes’ observables were first adapted by Chandrasekhar’ in 1950 to account for the effects of polarization in the radiative transfer equation.

Once the definition of light is given in terms of four Stokes parameters, then any scatterer which alters the properties of incident wavefront can be defined by a 4 x 4 property matrix, which is known as the scattering or the Mueller matrix.

THEORETICAL BACKGROUND

In order to outline this idea better, consider a plane wave incident on a spherical particle. The relation between the incident and scattered electric fields is given by3

Here S’, S2, S3 and S4 are the elements of the amplitude scattering matrix, which depend on the scattering angle and the orientation of the particle with respect to the incident field vector. El, and El represents the parallel and perpendicular components of the electric field. The incident and scattered electric fields are difficult to measure. Instead, it is preferable to follow Stokes6 and write a relation in terms of measurable quantities, such as irradiances:

The 4 x 4 S, matrix is known as the Mueller (scattering) matrix and its components are related to the elements of amplitude scattering matrix (Si) of Eq. (1).3*6 Muller matrix elements S’,, S’X, S33 and S34 contain important phase information that can be used as a tool to accurately determine the optical and geometrical properties of a scatterer. Although there are 16 S, elements, only 7 of them are independent. %” Depending on the symmetry of a scatterer, this matrix can be simplified even further.e”

The knowledge of the matrix elements does not only help in the accurate characterization of homogeneous spherical particles but also assists in detecting the presence of a species with different properties on a homogeneous particle. Interestingly, there is no study available in the literature related to the scattering matrix elements of non-homogeneous particles, although several papers reported these elements for homogeneous spheres and cylinders. Relations between the matrix elements were studied by Abhyankar and Fymat,’ Fry and Kattawar, ‘O and Hovenier et al.” Several different experimental works were reported about measuring the elements of the scattering matrix for spherical and irregular shaped particles. “-I6 The nature of Mueller matrix elements of small spherical particles, as the size approaches the Rayleigh and Rayleigh-Gans limits were investigated by Bickel et al.” A modified version of the Lorenz-Mie theory was used by Videen et alI8 to examine scattering by a system composed of a sphere and a surface. Hofer and Glatter19 performed a numerical analysis for randomly oriented rotationally symmetric objects. Recently, Manick- avasagam and Mengii$’ investigated the Mueller matrix elements of fractal-like agglomerates and the possibility of recovering their structural details from angular measurements. All these relatively recent studies as well as the earlier ones have been reviewed by Govindan,*’ who developed an experimental system to study the structure and properties of soot particles in flames.

Our objective in this paper is to show that the Mueller matrix elements of radially non-homogeneous spheres are sensitive to the radial variation of their optical properties. If these

Identification of non-homogeneous spherical particles 593

elements can be measured accurately, it will be possible to identify the particle characteristics more accurately. We will apply this idea to two systems, a droplet covered by pollutants, and a burning char particle surrounded by a soot cloud. A parametric study will be presented to highlight the importance of pollutants (or soot) volume fraction and the size of the overall system on the scattering matrix elements.

ANALYSIS

Consider a stratified sphere, which is comprised of a series of concentric spheres, each characterized by a size parameter xI = 2xa,/l and a refractive index ml = nr + ik,, 1 = 1, 2 . . . L, where I is the wavelength of the incident radiation and L is the total number of concentric layers. The incident, internal, and scattered electric and magnetic fields can be expressed in terms of vector spherical harmonics. ” The internal electric field for the Ith layer of the particle is

E I,

= cos qi sine 00 P2 nT, 4n + l)~“~“(~)[brn~“(p)+dm~~@)l

E IO = y 5 En[n,(e)[a,ll/.(p)+cax,(p)] - k,(e)[br”~:(P)+d/“X,:(P)ll n-1

E/r+ = y g En[-rn(e)[a~nJ/n(p)+c~n~.(p)l + inn(e)[bh~,:(p)+dr,X:(P)]] (5) n-1

where p = 2nmrr/12 and I(/, and xn are the Ricatti-Bessel functions of the first and second kind of order n respectively. E. = i”(2n + l)E,/n(n + l), G, and T, are defined in terms of the associated Legendre function P,!(cos 0) as

Z” = dP: (cos f3)

d9 (6)

The scattered field from the stratified particle becomes:

’ E, = ’ ‘OS t2 sin ’ 2 n(n + i)E,a,n.(B)t,(p) ?!=I

(7)

(8)

E SB = y 5 E,[ia,T.(e)r:(p)-b,A,(e)r”(P)] (9) n-l

Es+ = -y 2 En[ia.~n(8)5:(p)-b,~,(e)~,(p)] n-l

Table 1. Comparison of absorption efficiencies for a 3-layered particle.

Q .bS.CO~ Q.w x lo5

02 Ocm) a3 (rm) X3 Ref. 22 Ref. 24 Present Ref. 22 Ref.24 Present

4.0 4.01 5.05 29.93 1.642 1.658 1.642 2.748 2.676 2.764 4.0 4.1 5.10 30.23 1.493 1.538 1.493 3.688 3.623 3.675 4.0 4.3 5.22 30.94 1.114 1.125 1.113 3.415 3.688 3.418 4.0 5.68 33.67 0.967 0.967 0.967 4.628 4.528 4.737 4.0 6.43 38.11 1.021 1.021 1.021 7.394 7.566 7.347

594 D. Bhanti et al

1 .OE+05

4.6E+03 C

L 2.2E+02

1 .OE+O 1 -I

1

t..‘.‘.‘..‘...“‘..“.“.’ 4.OE+03

1.3E+03 ol

5; - 1 .E+03

-4.E+03

6.OE+04

3.7E+04

1.3E+04

-l.E+04_(. . 9 . , . . . . I. . . a I 9 u . r I a . a . 1

10 20 30 40 50 60

Scattering Angle (8)

Fig. 1. Scattering matrix elements for a water droplet surrounded by soot cloud; x.,,~ = 100 and xIoot = 5.

where &@) = &,@)+&&I), and the incident field is given as

(11)

(12)

(13)


Here, a/,, bm, cl” and d,, are the internal field coefficients of the Ith layer and a, and b, are the scattered field coefficients of the stratified sphere. These coefficients can then be determined by developing a system of boundary conditions by matching the tangential components of the electric and the magnetic fields at each interface as listed in the Appendix.

Once the scattering coefficients a, and b,, for the stratified sphere are calculated using the procedure described in the Appendix, the elements of the amplitude scattering matrix, S, and Sz can thus be determined using

s, = f 2n+ 1 - (unn, + bnr.) n= I 0 + 1)

(14)

1 .OE+04

4.6E+02 c-

G 2.2E+O 1

7 .OE+OO

1 .OE+02

-2.E+02 N

5; -4.E+02

-7.E+02

6.OE+03

3.7E+03 r-J

8

1.3E+O3

-l.E+OZ

R xwotw = 100; e = 45 I -fv=o I------ . . . . . . . . fv IO.001

-----* tv-0.010 ---' fv-0.030 -- fv=O.loo

Total Size Parameter Fig. 2. Scattering matrix elements for a water droplet surrounded by soot cloud at 0 = 45” for .G,.~~ = 100

and varying fv.uat.

596 D. Bhanti et al

1 .olz+o;

1 .OE+O: -

v, 1.0E+O

1.0E+0c

4.OE+O.

2.3E+O; cv

z 6.7E+O

- 1 .E+O.

1 .OE+O

-3.E+O r)

2

-2.E+O

-3_E+O

3

2

1

3

2

2

1

2

2

1

l-

1 ,

2-

2 -I-

I “. ‘I.. . .‘. - ”

xwotw = 100; B = 165

100 110 120 130 740

Total Size Parameter Fig. 3. Scattering matrix elements for a water droplet surrounded by soot cloud at 0 = 165’ for xWaM = 100

and varying fL.,t.

The Mueller matrix elements for the composite sphere can then be calculated as

Sr = t [IS,l’ + lS,12J (16)

s12 = t [IS2J2 - IS,l’] (17)

SJ3 = Re[&S;)] (18)

& = Im[&SP] 09)


S,, is the differential scattering cross section of the scatterer and its magnitude changes with the scattering angle. If normalized by the area under the S,,-8 curve, SI1 becomes the phase function of the scatterer (which is essential in estimating the distribution of radiation intensity in a medium containing the scatterers). The scattering phase function for the sphere is a function of only S,, and is defined as3

@(~)=LsII(B) C, k2 (20)

where C, = 5 S,,(0) de, and k = 277/l is the propagation constant. S12 is a measure of linearly polarized scattered light for an incident unpolarized light. In general, the angular pattern of SL2 shows how much a particle deviates from Rayleigh spheres. S3s indicates how much of an obliquely polarized light (45”) is transformed to circularly polarized light because of the scatterer. S,., has the similar angular pattern as S 33. For symmetric particles, SIP = St4 = &3 = S2, = $1 = $2 = ~$1 = ~$2 = 0.

RESULT AND DISCUSSION

A FORTRAN code based on the algorithm presented above was written and executed on an IBM 3090 computer. The code was checked by computing absorption efficiencies Qabs for homogeneous, 2-layered and 3-layered particles and comparing the results with those published ear1ieT22,24 which yielded excellent agreement (see Table 1).

Water droplet surrounded by a soot cloud

The first physical system considered is a water droplet-soot stratified system. Soot particles are assumed to be present as a concentric layer around a homogeneous water droplet. The knowledge

1 .OE+04

2.2E+02

4.6E+OO

l.OE-01

l.OE+Ol

-2.E+Ol hl

G

-SE+01

0 30 60 90 120 150 180

Scattering Angle (8) Fig. 4. Scattering matrix elements for a water droplet surrounded by soot cloud; xvSlrr = 5 and xraol = 5.

598 D. Bhanti et al

1 .OE+02 -

1 .OE+Ol -,

Xwater = 5; 0 = 45 \

- fv=O ‘~oE+ooEl, , , . , . . ( , . . ;

1__1*-1* fv * 0.00, -----s fv 9 0.010

1 .OE-0 1 ---* fv - 0.050 -- fv-0.100

4.OE+OO -

2.3E+OO

6.7E-01

- 1 .E+OO

1.3E+Ol -

/ /w

/ \

8.7E+OO - ,//----,~

. &A& ___----------mm_ d

K’----

4.3E+OO - \ ‘\

\ ‘Y \

\, O.OE+OO . . . . , . . . . , . . . , . . . . , . 1 . .

5 6 7 8 9 10

Total Size Parameter

Fig. 5. Scattering matrix elements for a water droplet surrounded by soot cloud at tl = 45” for x,.*, = 5 and varying xtOl~.

of scattering properties of a water particle contaminated with soot on its surface will aid in understanding the scattering behavior of such atmospheric aerosols.

The effective refractive index of the outer inhomogeneous soot layer m,~ is determined using the Maxwell-Garnett theory. rnti is expressed as a function of the refractive index of the absorbing soot particles, rnab, the refractive index of the non-absorbing air rnmd and the volume fraction of the absorbing soot particles fv as3

2

mEfl = m& 1 + 3fv [

s-6 mf + 2mL (

1_f mfbr--mL -’ 'm$ + 2mL >I (21)

Identiikation of non-homogeneous spherical particles 599

The Zlayered particle was composed of an inner homogeneous water layer (ml = 1.33 + O.Oi) and an outer layer composed of a mixture of soot (mz = 1.75 + 0.7%) and air (m3 = 1.0 + O.Oi).

The size of cloud droplets in the atmosphere range from 5 to 50 pm. Thus the size parameter of a cloud droplet corresponding to visible wavelength of solar radiation is 40 c x < 400. The Mueller matrix elements of a water droplet with size parameter xW.* = 100 and x*+8 = 5 are depicted in Fig. 1 for several volume fractions of soot (subscripts s and a refer to soot and air, respectively). Here xr + a = 2nAr/l where Ar is the difference in the radius between the outermost layer and the core. The scattering behavior is presented for angles between 10 and 60”, the regime where the effect of soot volume fraction is noticeable.

As sootf, is increased, the fine angular oscillations in SII, &, Ssj are damped due to increasing absorption by the outer soot layer. However, measuring the Mueller matrix elements at various

1 .OE+O 1

1 .OE+OO

1 .OE-0 1

1 .OE-02

3SE-01

2.1 E-01

7.OE-02

-7.E-02

O.OE+OC!

-4.E-01

-9.E-01

- 1 .E+OC

-0-0____-d--- P\

‘;“-__;y _~q----:

\ .__

‘\J \ \ / 5;e= 165 \/

Xwatof =

6 7 8 9 10

Total Size Parameter Fig. 6. Scattering matrix elements for a water droplet surrounded by soot cloud at ff = 165” for xncn = 5

and varying x,d.

600 D. Bhanti et al

1 .OE+03 4

2.2E+02 T

z 4.6E+Ol

l.OE+Ol 1 , , , . , , , , , , , , , , , , . , .

(v

G

f7

r;:

O.OE+OO

- 1 .E+02

-2.E+02

-3.E+02

IL- L

7

xcom = 50

- Xc+r+o = 50 -----1-1 xc+*+0 = 60

-----* Xo+r+o = 70 ---I x0+8+0 = 80 -- Xc+s+a = 90 -- Xc+.+0 = 100 I ‘. . ’ I. ‘. .

,.,,,,,;” 60 100 140 1;

-1 .E+Ol

-2.E+Ol

-3.E+Ol

-4.E+Ol

-5.E+Ol

-6.E+ol II

60 90 120 150 180


Fig. 7. Scattering matrix elements for a coal particle surrounded by soot cloud for x., = 5 and varying Xloot.

angles will not indicate decisively the presence of soot, especially if the volume fraction of soot is less than 0.01. Figure 1 shows that the curve for fv = 0.01 is almost identical to that of the homogeneous water droplet. Therefore, the presence of soot can be identified only if the volume fraction of soot is in the order of 0.1 or greater.

The matrix elements S1,, Slz and & at constant scattering angles 8 = 45 and 8 = 165 are depicted in Figs. 2 and 3 for different thicknesses of the outer soot layer. For the water droplet with x,,,, = 100, the Mueller matrix elements S,,, Su and & change considerably in both the forward and backward scattering directions as the soot layer is increased. There is a considerable difference in the Mueller matrix elements due to the accumulation of soot on the surface of the water droplet at both the angles considered. Thus, placing a detector at 45 and 165” to measure the matrix elements will provide a diagnostic tool in detecting the accumulation of soot on a pure water


droplet, particularly if soot 5 increases to 0.01. Any one Mueller matrix element cannot be identified as the best indicator for the presence of inhomogeneity in the scatterer. However, the combined knowledge of &, Slz and & matrix elements would provide a powerful diagnostic tool in detecting the presence of soot on cloud droplets.

For experiments conducted in a laboratory, laser light of longer wavelength than visible light can be used to measure the Mueller matrix elements. In such cases, the size parameter x of the stratified particle will be greatly reduced. The matrix elements of a homogeneous water droplet of size parameter xWater = 5 with constant thickness of the outer soot layer x, + a = 5 and varying fV is shown in Fig. 4. S1, is sensitive to the presence of the absorbing soot layer particularly in the backward direction. The oscillatory pattern of S1, as a function of angle gets progressively damped as the volume fraction of soot is increased. Also, S2 changes considerably due to the presence of

1 .OE+O4

1

1 .OE+Ol -

1.. ‘, I * “7 1 *. . ’

-9.OE+02 -

20 60

-I Xc+8+a

100 140 180

O.OE+OO

-5.OE+Ol

- 1 .OE+02

- 1.5E+02

-2.OE+02

-2.5iz+02 I 60 90 120 150 180


Fig. 8. Scattering matrix elements for a coal particle surrounded by soot cloud x,. = 100 and varying Xlwt.

602 D. Bhanti et al

1 .OE+03 1

-2.E+02 -

-3-E+02 1 20 60 100 140 180

----________-______________ :;;;;;p:: 1 ---.....-.............................. - 1 .E+02 -

-2.E+02 -

- f” = 0.1 -2_E+02 -

-3.E+02- . . . . I - 9 . - I . 60 90 120 150 180


Fig. 9. Scattering matrix elements for a coal particle surrounded by soot cloud for x,~~ou~ = 70 and different volume fractions of soot.

the absorbing soot layer, especially in the forward direction (6 = O-30“). Slz is a good indicator of the presence of inhomogeneity at 8 = 1:“. Slz decreases sharply as the volume fraction of the absorbing soot increases. For instance, its value at fv = 0.5 is more than twice that for fv =5 1.0. Thus, the Mueller matrix elements cannot only provide information to the presence of soot but also give information about the volume fraction of soot present in the outer layer. The behavior of SW is not presented here as it was not found to be sensitive to the presence of soot and did not give any additional information.

Figures 5 and 6 respectively depict the Mueller matrix elements at 8 = 45 and 8 = 165” for different thicknesses of the soot layer and soot fv for a water droplet with x,.,, = 5. At 8 = 45”, the Mueller matrix element Sn seems to be more sensitive to the presence of soot than the other


matrix elements. In the backward scattering direction (6 = 165”), the Mueller matrix elements S,,, S,Z and & show considerable variation with soot fy especially if fv is greater than 0.05.

Coal particle surrounded by a soot cloud

In this section, we will present the scattering matrix elements determined for a burning coal particle surrounded by a soot cloud (outer layer). The refractive index for coal is taken to be 1.8 + 0.08~ For soot and air mixture forming the outer layer, the Maxwell-Gamett relation is applied to find an effective index of refraction. The refractive indices of soot and air are assumed

8.X-O 1

4.OE-01

8.OE-01

4.OE-01

o~oE+oo~ . . , . , . . . , l_E$ji ,,,E+ooZjJ 60 100 140 1;

8.OE-01 - #,*r._*_ _____B ___-____-_-_____ ________

----_ __-LI----

4.OE-01 -

O.OE+OO , . . . . , . . , . , . . . .

60 90 120 150 180

Scattering Angle (8) Fig. 10. Normalized scattering matrix elements for a coal particie surrounded by soot cloud for xm, = 50

and varying x-.

604

1 .ZE-tOO i I

_____“__..__..‘.‘.“~-..*..-.~-....~....-..

8,OE_0 1 .______ _a_____

)__-_IIIC _cIcLII

4.OE-v , #_-IL--- - _---

_--- ---“-.xc+s+B = 140 , -----Xc+s+a = 160 -- xc+s+a = 180 ---Xc+s+e = 200

Scattering Angle (8) Fig. 1 I. Nomtalized scattering matrix elements for a coal particle surrounded by SUOC cloud xb = 100

and varying xmr.

to be 1.75 + 0.75~’ and I.0 + O.Oi respectively. The volume faction of soot is varied radially in the outside layer, and is given as:

“C(r) = l/4(1 - r/a& (23

where3 = 0.1, r is the radial location, and aL is the maximum radius of the multilayered sphere. To account for the radial inhomogeneity, the outer layer is divided into 100 sub-layers, in each of which the refractive index is assumed to be utiform.

In the numerical cakulations, the size parameter of the core (x-) is taken as tither 50 or 100. Soot and air which surrounds the coal (core) is assumed to be present in such an extent to yield a total size parameter ix C +,+J of between 50 and 100 in the first case, and between 100 and 200 in the second case (subscripts c, s, and a correspond to cd, soot and air, respectively). Qnly the

Identification of non-homogeneous spherica particles 605

Mueller matrix elements, S,,, &, &, and & are obtained, as it is not necessary to evaluate other elements due to symmetry.

In subsequent discussions, S, represent the normalized Mueller matrix elements. The angular variation of normalized S,, (= $2) & ( = SZ~), & (= &), and & (= $3) as predicted by the algorithm are shown in Figs. 7 and 8.

Figure 7 depicts the angular variation of Sir as a function of x, + ‘L+ a, for a given x,,, (= 50). For the case of xcorr = 50, it is seen that beyond 40”, the magnitude of S, I remains constant. As x, + s + a is increased from 60 to 100, the value of S,, decreases by 50%. It is to be noted that an increase in x,, s+B corresponds to an increase in the amount of soot which surrounds the coal. Similar behavior is observed for the case of x,,, = 100 (see Fig. 8). There is an order of magnitude increase in S,,, as x~+~+~ is doubled in both the cases. This shows that if scattering measurements are made

F

5; 4.6E+Ol - ----.-.........................~

l*OE+O1 1 O.OE+OO -

_..-. ._....--

/- _..=

- 1 .E+02 - ,.@*=‘--

(\I A”

5;

5(

-2.E+02 -

-3,E+02 - -=

1 .‘. . 1’. . . 1.. *.

20 60 100 140 180

O.OE+OO

-1 .E+Ol

-2.E+Ol

-3.E+Ol

-4.E+Ol

-SE+01

-6.E+Ol ! * * . * I ’ . ’ ’ I . . 7 1 ’ . * ’ I 60 90 120 150 180


Fig. 12. Scattering matrix elements for a size distribution of coal particles surrounded by soot cloud.

606 D. Bhanti et al

carefully for a single coal particle, the rate and amount of soot growth around the core can be predicted.

The angular variation of SIZ for various X, + 8 + a values are shown in Figs. 7 and 8. For x,, = 50, the magnitude of & decreases as the amount of soot surrounding the coal is increased. A similar trend is observed for x,,, = 100. In both the cases, at angles greater than 30”, there is sufficient variation in the magnitude of &, which can be meausured in experiments. The magnitude of Ssa remains constant beyond 60”. Increase in x e + s + a results in decrease in the magnitude of & for both the cases of x,, = 50 and 100. These observations suggest the possibility of predicting growth of soot layer around a burning coal particle, by measuring angular variations of &, & and &.

Calculations were also performed to determine the effect of variation of soot concentration around a coal particle on the total scattering characterization. Figure 9 depicts the effect of change in volume fraction of soot around the coal particle on the Mueller matrix elements, for a fixed size parameter xc + s + a (= 70). The volume fraction is changed by changing 5 in Eq. (22) to 0.1, 0.2, and 0.5. It is observed from the figure that beyond 50”, S,, decreases about 50%, ifS, is increased from 0.1 to 0.5. Similar trend is also seen in the behavior of &. On the other hand, & is sensitive to changes in the volume fraction of soot only at angles less than 70”.

When a coal particle is burnt or if it is surrounded by soot cloud, its scattering behavior is expected to change due to a change in refractive index and size parameter of the newly evolved composite particle (coal + soot + air). An investigation of scattering behavior of this new particle as compared to that of the parent coal particle will be valuable in experimental observation of soot formation. Calculations are made by assuming the parent coal particle size parameter to be either 50 or 100. This forms the core of the composite particle. Soot cloud is then added to the parent particle such that the total size parameter of the composite particle is in the range of 50-100 (for X - 50) and in the range of 100-200 (for x,,, = core - 100). Mueller matrix elements are then obtained for these various cases and the results are normalized with the corresponding Mueller matrix elements of the parent coal particle (i.e., core). These results are depicted in Figs. 10 and 11. It is observed that as the size of the soot cloud (volume) around the coal particle is increased, the scattered intensity beyond 20” drops to about 40% (for xcO, = 50) or 10% (for x,,,, = 100) of the magnitude of the parent coal particle. Similar behavior is observed in change of & with respect to that of parent particle. In case of &, the decrease is either 50% (xcom = 50) or 20% (x~~ = 100) beyond 60”. At angles lower than 60”, the behavior is not predictable and hence not shown in the figure.

Even though most of the analysis presented so far is for single-particle systems, it is worthwhile to investigate the effect of particle size of variation in the thickness of the outer soot layer on the matrix elements. This is important as in practical systems, one may encounter different size particles. Figure 12 shows a set of results obtained using a size distribution for a cloud of particles. It is assumed that the diameter of main coal particle remains constant (xmrr = 50), however overall size parameter varies uniformly from 50 (no soot layer) to 100. It is obvious that such a size distribution has an impact on the angular variations of matrix elements. For example, at 30”, Su decreases almost two times, whereas at backward angles the absolute value of S,, decreases almost 20% with increasing soot layer. Note that polydispersity effects are important, and need more investigation than can be covered here.

CONCLUSIONS

In this work, we have shown that Mueller matrix elements provide valuable information about the spatial variation of the optical properties of the scattering particle. For both the non-homogeneous systems considered, i.e., a water droplet with soot accumulating at its surface and a coal particle sourrounded by a soot cloud, the Mueller matrix elements are sensitive to the thickness of soot layer and soot volume fraction on the surface layer. If the angular variation of the Mueller matrix elements can be accurately measured, it would provide an important diagnostic tool in detecting the presence of inhomogeneity in the scattering particle. Since, the elements vary considerably with changing soot volume fraction, an accurate measurement would also provide important quantitative information about the soot present in the shell layer. The quantitative measurements allow the qualitative observation of the variations in particle properties, even if there


are several particles with different layer thicknesses. In a parallel work, we have been developing experimental techniques for measuring the Mueller matrix elements of particles in dynamic systems, such as flames.2’*2s The details of these techniques can be found in these references.

Acknowledgemenr-This work is supported by the DOE-PETC Advanced University Coal Research Program Grant No: DE-FG22-PC92533.

1. 2.

3.

4.

5. 6.

I. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

REFERENCES

H. C. van de Hulst, Light Scattering by Small Particles, Wiley, New York (1957). M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York (1969). C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York (1983). W. J. Wiscombe and A. Mugnai, Single Scattering From Nonspherical Chebyshev Particles: A Compendium of Calculations, NASA Reference Publication 1157 (January 1986). J. 1. Hage, M. J. Greenberg, and R. T. Wang, Appl. Opt. 30, 1141 (1991). G. G. Stokes, Trans. Camb. Phil. Sot. 9,399 (1852); Reprinted in Math. Phys. Papers 3,233, Cambridge University Press, London (1901). E. Collett, Polarized Light: Fundamentals and Applications, Marcel Dekker, New York (1992). S. Chandrasekhar, Radiative Transfer, Oxford Press (1950); Dover Publications, New York (1960). K. D. Abhyankar and A. L. Fymat, J. Math. Phys. 10, 1935 (1969). E. S. Fry, and G. W. Kattawar, Appl. Opt. 20, 2811 (1981). J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, Astron. Astrophys. 157, 301 (1986). A. C. Holland, and G. Gagne, Appl. Opt. 9, 1113 (1970). A. J. Hunt, and D. R. Huffman, Reu. Sci. Znstrum. 44, 1753 (1973). R. J. Perry, A. J. Hunt, and D. R. Huffman, Appl. Opt. 17, 2700 (1978). R. C. Thompson, J. C. Bottiger, and E. S. Fry, Appl. Opt. 19, 1323 (1980). F. Kuik, P. Stammes, and J. W. Hovenier, Appl. Opt. 30, 4872 (1991). W. S. Bickel, A. J. Watkins, and G. Videen, Am. J. Phys. 55, 559 (1987). G. Videen, W. L. Wolfe, and W. S. Bickel, Opt. Engng 31, 341 (1992). M. Hofer, and 0. Glatter, Appl. Opt. 28, 2389 (1989). S. Manickavasagam and M. P. Mengiic, Appl. Opt. (in press). R. Govindan, Master’s Thesis, Univ. of Kentucky, Lexington, KY (1996). D. W. Mackowski, R. A. Altenkirch, and M. P. Mengiic, Appl. Opt. 29, 1551 (1990). R. Bhandari, Appl. Opt. 24, 1960 (1985). M. Sitarski, Langmuir 3, 85 (1987). R. Govindan, S. Manickavasagam, and M. P. Mengiic, in Radiative Transfer I: Proceedings of the International Symposium on Radiative Heat Transfer, M. P. Mengiic, ed., Begell House, New York (1996).

APPENDIX

The coefficients for the internal and scattered fields are listed here for the sake of completeness. For the core (I = 1) and the first layer (I = 2)

m2a&(m1x,) = m,[a&(m,x,) + cax”(mzx,)] (Al)

~I&(~IXI) = ah~.‘(m2xl)+cbil:(m2x,) (A2)

mh&(m~x~) = ml[bb~:(m2xl)+d2nX:(m2x,)] (A3)

b&.(mlxl) = bh~n(m2xI)+dZnXn(m2xI) (A4)

and for components 2 to L, with j = I+ 1

m,[ahk(m,x,) + chx.(mfxf)l = mJa,$.(mjx,) + cfix.(mir)] (W

ah& (mm) + CI.J& (m/a) = a,& (m,s) + cj& (m,xl) W)

m,[br&(mtxt) + d&(mtxt)] = mt[b,& (m,xt) + dBy..‘(mixt)l (.47)

b&(mal) + dhXJm/xt) = b&(m,xl) + d&m,xO 648)

and for the Lth layer and the scattered field coefficients as

a,9.(mrxr)+c~,X.(mLxr) = m&n(xL)-b&(x4 (A9)

a&(mrxr) + c& (m,xL) = $: (XL) - b& (XL) (Al’))

b~,l(l:(mLxL)+d~~~(mrxr) = mJ&(xL)-a&(xL)] (Al 1)

bLn~“(rnLxL)+dLnX”(rnLxL) = $.(xL)-a& (A121

These boundary conditions are however not in a form suitable for machine calculation of the field coefficients. Mackowski et al’* redefined the field coefficients as a,& = a&(m,x,), CL = c&.(mm), bi = b&(mfx,), and dk = d&m,x/) and converted Eqs. (Al)-(AIZ) in terms of the numerically stable logarithmic functions D,!(z) = $:(z)/&(z), D:(z) = $(t)/x.(r), and the

608 D. Bhanti et al

ratio of the Bessel functions &#I, zz) = $&)/ljl&~) and g( ZI, ZZ) = r.(z~)/x.(a). To overcome errors due to loss of Elision, Eqs. (3)~(5) were redefined in terms of &, and the field coefficients as ak = ui, + I&#& cb = -i&&/x., b =AA + id&./w& L = -id&,/X.. In their original formulation, Mackowski et aILl committed an error in defining the a~!,, bb, c,+, and dh coefficients. Consequently, our expressions are slightly different than theirs.

Using the above expressions and the recursion method developed by Bhandari,= the field coefficients can be expressed in the form

6413)

6414)

(A15)

6416)

(4417)

6418)

where

ForI=l,...,L-I

Ao.=Bon= 1 6419)

Co, = D, = 0 6420)

and for I = L,

A,,, = -@$ [fi"A~- I,, + 113;“c,- ($1 6425)

c,,, = @$& [l$"Ar - I,, + fi”cr - I,] (A26)

B,.n = -e [GE3’Br - ,c + GE”‘Dc - ,,I

D1, = @$$ [Gti”Br - ,.n + Gt”Dr. - ,n]

The functions Ff” and Gb are defined as

FykJ = m,a(m/x/)-ml+ ,D!(mt+ 1x11

GY1’ = ml+ OH-mrD’(mr+ la) ” I

6427)

(AW

6439)

6430)

identification of non-homogeneous spherical … · 2004. 1. 12. · identification of...

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