Depolarization of scattered light by externally oriented dipole scatterers

Depolarization of Scattered Light by Externally Oriented Dipole Scatterers R. V. M E H T A , H. S. SHAH, AND J. •.

DESAI

Applied Science Department, Sardar Vallabhbhai Regional College of Engineering and Technology, Surat, and Physical Research Laboratory, Ahmedabad

Received August 18, 1970 Expressions for depolarization of scattered light by externally oriented dipole scatterers are derived using Mfiller matrices. The force field is assumed to be a magnetic field. Three different cases of orienting torque are considered: magnetically anlsotropic spheroids oriented (1) parallel to the field, (2) perpendicular to the field, and (3) polar spheroids having a permanent dipole moment along their axis. INTRODUCTION

direction of the wave scattered by a particle is defined by two angles 0 and ~, 0 being the polar angle with the OZ direction and being the angle of azimuth with respect to a selected direction in a plane perpendicular to OZ. The direction OZ of the incident wave and the direction 0 and q~ of the scattered wave define a scattering plane. The unit vector normal to this plane is referred to as i, and the unit vector in the plane is referred to as f. The sense is so chosen that i X f is the direction of propagation. The intensity and state of polarization of a scattered wave are determined by the Stokes parameters. The Stokes vector of the incident wave defined by Parke is

When a plane electromagnetic wave is incident on a cloud of randomly oriented dipole anisotropic scatterers, the scattered light at 90 ° is not completely polarized. The depolarization factors have been calculated by many authors (l(a), 2, 3). If these dipole scatterers are perfectly oriented, then depolarization factors will be zero. A case of particular interest is the depolarization factors when the dipole scatterers are oriented under a suitable external field. In the present paper, the expressions are derived for the depolarization factors for spheroidal particles oriented under the magnetic field. These expressions are also directly applicable for the electrostatic field. The method used here is similar to that used by van de Hulst (l(b)). Since the transformation matrix used by van de Hulst in connection with the derivation of symm e t r y relations is complicated, the simpler matrix given by Parke (4) is used here.

7 Eo=

r. 0, *0n -

]E,oEt

I 1o ,o I

[1]

L+_I [EroE*] The Stokes parameters defined by van de Hulst are related to the above by the following relation

SCATTERING BY A SINGLE PARTICLE The intensity and the state of polarization for a plane electromagnetic wave propagating along the OZ direction are determined by the two complex amplitude components taken in a plane perpendicular to OZ. The

L20/, =

°LLooJ' IL

3our~al of Colloid and Interface Science, Vol. 36, No. 1, May 1971

8O

I-7 ,

E21

DEPOLARIZATION OF SCATTERED LIGHT

81

The M matrix for the mirror image is

where

:, 0 01 2LOI -i0 ol -OJ"

V JnJ~l -J~2J~* -J11Jl~ -J21Jx* J22Jl* J~i j*l~

110

T-

Jl :l

The electric vector of the scattered wave in terms of the electric vector of the incident wave can be obtained by the relation

FJ11

= JE0;

J12~

J = LJ~I J ~ j "

J~J~*

~J11J~l

-J~J~*

J~J~2 .]

J11J~1 J21Jl~l JllJ2~l J21 J2~l~

I

[3]

J Z2*i|

*I " I JllJ*2 J21J*12 JllJ2* J21J22 OJ12Jl~ J22J~2 J12J~2 J22J2~J

The resultant matrix becomes

[4]

4Jll J*l 0

J12 Jl~l Jll J1"2 J12 J1~2"~

M = IJ21Jl*~ J22Jl* J21Jl*~ J22Jl*2/ J 11J~1 * Jl~ J*~1 JllJ*: Jl~J~[ ' ," ~J~lJ*l J~2J*~I J~l J*~ J ~ J ~*J

-J~J~*

I #

The M matrix for the reciprocal position of the mirror image is

(The J matrix is identical to the S matrix used by van de I-Iulst. ) The Stokes vector of the scattered wave is given by L = MLo

J12J1*2i -J~Jl~*

[5] l

0 4J22 J1*1

0 2(J12J~* + J21J*2) 2(J21J2* ~- J~2Jl*) 0

The corresponding matrix as given by van de Hulst can be obtained by the following transformation

0

2(J12J*2 q- J2iJ*1) 7

4JllJ22* 0

4J22J 2"

F = T-1MT, where T -~ is the reciprocal matrix of T. SCATTERING BY A CLOUD OF PARTICLES In actual practice, one deals with a system containing a large number of particles. Stokes parameters for the scattered wave by such a system can be obtained by using the resultant M matrix obtained by the addition of the M matrix elements of the individuM particles. If there exists a plane of symmetry in such a system, then the scattering matrix for its reciprocal position, mirror image, and reciprocal position of the mirror image can be easily obtained (1 (c)). The corresponding M matrices are given below. The M matrix for the reciprocal position is

r J11J~1

-- J21Jl~

I -Jr~Jll

J~Jll Jl~J~l , j* j-* . --J~ ~ --J~J*2

• J~2Jl~

-- J11J2~l

J21 J2~1 1

.J [6]

The resultant M matrix for a cloud of particles can be obtained by multiplying the above matrix by dn, the angular distribution function of the particle, and by integrating all the elements of the matrix over all possible orientations. By Eq. [4], the Stokes parameters for a scattered wave are obtained as I"

L1 = ] [4J11J*lLlo + 2(J12Jt* + J~lJ*~)L4o] dn; L2 = f [4J22Jl*L2o + 2(J12Jl* + J21J*1)L4o] dn; L3 = f [2 (J12J21 * + J21J*12)L2o

+ 4JllJ*~ L~0] dn; -J~J*l[ * + J21J*12)Llo --j~ij*~ • L4 = f [2 (J21J2i * J~J~ _] + 4J22 J22 L4o] dn. Journal of Collc~id and Interface Science,

Vol.36, Xo. 1, $1ay1971

82

MEHTA, SHAH, AND DESA1 DEPOLARIZATION FACTORS

DIPOLE SCATTERERS

The Stokes parameters for natural light are (L10, 0, 0, L~o), and the corresponding parameters for the scattered light are

The scattering matrix for dipole scatterers is given by (l(d)): J12 7

Jll

L,=f

21 J22A = ik3

[4511J1"1 L~0

FP2~ cos eP12-- P~3 sin 0

* L 40] dn; + 2(Jl~J~* + J2~J21)

L: = 0;

• I_

[7]

L~ = 0; L4 --

f

[11]

P~k = Pkl = C~1C~1al+ C~2Ck2a2+ @3C~3a3,

[2(J~lJ~* + JuJ~2) lO

*L

+ 4J~J2* L4o] dn.

Substituting the values of the parameters for 90 ° in the following equation, the depolarization factors for the natural light can be obtained A-

P12 cos 0 - - P13 sin 0-] Pn ]

L1 L4

where C~i (i, j = 1, 2, 3) are the direction cosines and aj are the optical polarizabilities of the particle. The scattering matrix for 90 ° becomes

J

F--P23 ik3 L P12

--P~3~

[12]

Pll ] "

The particles are assumed to be spheroids with their axis of symmetry along al, and

f (2J11J~l -4- J~J~* + J:~J~*) dn f (2J22J2* + J12Ji* + J:~J2*) dn

0/2 ~-~ 0/3.

[8]

If the incident light is plane polarized in the r direction, then the Stokes parameters of the incident light are (0, 0, 0, L40) and the corresponding parameters for the scattered light are

Let the magnetic field be acting along the OY direction with the axis of symmetry making an angle 7 with the direction of the field. As mentioned earlier, ~ denotes the azimuthal angle with respect to OX. The direction cosines are then given by Vsin~,eos~ C = - cos-y

eos-ycos~ -sin'y

~sin ~, sin ~

cos ~, sin ~

L1 = f 2(J12Jl* + J21J21) * L 40 dn;

--0~1 cos ~ _J [13]

L2 = O;

[9]

L3 = 0;

Now the scattering matrix [12] becomes C(a2 - al) sin ~, cos ~/sin 4, J = ik3 [_(al a2) sin 7 cos ~' cos ~,

L4 = fd (4J2~J22)*L40 dn.

-

Hence the depolarization factor for the incident plane polarized light is

f A' =

[10]

f 2J22J2* dn Journal of Colloid and Interface Science,

[14] "

Now the expressions [8] and [10] for depolarization become

(J~2J~* + J2~J2*) dn

J

(al - a2) sin 2 7 cos ~ sin q~ 1 (al - a2) sin 2 ~' cos2 ~ + a2

Vol. 36, No. 1, May 1971

A--

L1 L4

DEPOLARIZATION OF SCATTERED LIGHT ha

x f f [sin2 3" cos=3" cos~ 4

VII = =

2k--T (x~

q- sin s ~ cos= 4 sin 2 4

fflXsin23"coe4 • (1

q-

sin 2 3" cos = 4)

q- 2Y sin=3' cos = 4 q - 2asa=*] dn

ffx ie A' =

3" cos 6

[15]

• (1 -- sin = 3' cos 2 4) dn

_

Y

=

a,a2

__

~=) (~* - - 2a2a2

-~2

[16]

Case I: Anisotropie Parallel to the Field.

fTf0

sin = 3' cos= ¢ dn;

Nh

cos 2 4 de

fo ='~

4_Eh

•

q - o~=al .

Spheroids

VH2 2 (x~ -- Xs) cos2%

f0

s i n 3 3" e~2 ~o~2~

d3';

_ N h fo ~ sin ~ 3" eh,~ co~, d3". 4Eh

Substituting x = h cos 3", we get I1

Nh 2 £ t ~ ( x 2) - 4_Ehh 1 - - ~2 e~Sdx; --

N 2

-

2

N h 2 fo 1~XSe x2 4E~ ~ dx,

N (w

)

2 2 h S \ E h -- 1 ;

N

Oriented

In this case xl > x2 and x2 = xa, where xi is the magnetic susceptibility along three axes. The potential energy of a particle in the magnetic field is given by U -

e~' do: -- i e~f ( - i h ) ;

*);

The depolarization factors for a particular system can be obtained b y determining the distribution function dn. This function depends upon the magnetic properties of the particles. Three cases are considered here: (1) magnetically anisotropic spheroids oriented parallel to the field, (2) magnetically anisotropic spheroids oriented perpendicular to the field, and (3) polar particles having a permanent dipole moment along their axis.

×=);

-

where £ is the Boltzmann constant and T is the absolute t~mperature. Substituting the above value of dn in expressions [15] and [16], we can determine the values of A and A'. In this ease, the following integrals are to be evaluated. I~ =

A- 2Y sin 2 3' cos= 4 q- 2a~ a=*) dn

JC = (~

fo

E~ =

q- 2 sin = 3" cos 2 3" sin 4] dn

83

-

2

[1 -

F(h)];

1 [heh~ F(h) = ~

[2 =

)

\ E~ -- 1

.

foS~fo ~ sin s 3' cos4 ¢ 4rr __Nh e

[17]

h2 cos2 3'

Ej~

sin 3" d3" d4;

27r

where V is the volume of the particle and H is the applied magnetic field. In this equation the contribution due to anisometry is neglected. Using expression [17] and the Boltzmann theorem we get Nh et~ °0~2 dn 47rE~ sin 3, d3" d4;

[18]

f0

cos~ 4 dq~ - 3~ 4

I2 - 3 N h 16Eh

siI2 3" et'~ °°~ ~ sin 3" d3".

Let h cos 3" = x; therefore 12 -

3N f 16Et~ h

1 - 2x= x4 ~ - -4- ~ e*~ dx;

Journal of Colloid and Interface Science, Vol. 36, No. 1, M a y 1971

:84

MEHTA,

SHAg, AND DESAI

and

where o~ is the dipole moment;

I2 = ~3N

Nil

[ 2Eh -- 4-~ ~1 (heh~_ Eh)

+~

dn

--4he ~+~E,~)

1

;

A =

Here A!

! [ h e h' Hence

eh

cos

27r d' - -

v

e -t~ s i n ~ d~, dq~.

Substituting the above value of dn and proceeding as in the previous case, we obtain

3N [2 - 4F(h) + 2S(h)]. I~ = Tg

1 [he h2 1) F(h) = ~ \ E ~ ;

-

X[1 + 10L(h) -- l l T ( h ) ] 7X ÷ 8Y -- (10X + 8 Y ) L ( h ) ' + 3XT(h) + 16as as* X[1 + 2L(h) -- 3T(h)] 6X[1 -- 2L(h) -t- T(h)] q- 8Y[1 - L(h)] + 16a2 a2*

L(h) = 1

2 e~ q- e-h 2 h eh -- e-h + ~ ;

T(h) = i

4 [ha(e h +

)

e-h)

X[I + 10F(h) -- l l S ( h ) ]

A=

7X q- 8Y - (10X ÷ 8 Y ) F ( h ) ' -ff 3XS(h) q- 16a2 as* k' =

eh _

DISCUSSION

Case II: Anisolropic Spheroids Oriented Perpendicular to the Field.

The general expressions for depolarization factors can be expressed as A=

In this case x2 > x1 and x~ = xa, and proceeding exactly as in Case I we can show that A, =

X[1 + 10M(h) -- l l R ( h ) ] 7X q- 8 Y (10X q- 8Y)M(h) + 3XR(h) q- 16as as* -

A' =

e--h

X[1 + 2F(h) -- 3S(h)] 6X[1 - 2F(h) + S(h)] + 16as a2* -[- 8Y[1 -- F(h)]

A=

6]

eh -t- e-t~ + 6h - -

x[1 +

2¢(h)

-

3~(h)]

-

[19]

.

6X[1 -- 24(h) + ~(h)]

[201

+ 8Y[1 - q~(h)] + 16a2 a2*

X[1 + 2M(h) -- 3R(h)] . 6X[1 -- 2M(h) + R(h)] + 8Y[1 -- M(h)] + 16-s a2*

M ( h ) = 9"9"9"9"9-~ 1

X[1 + 10qs(h) -- l l ¢ ( h ) ] 7X + 8Y -- (10X ÷ 8Y)q~(h) + 3X~(h) + 16a2 a2*.

where

¢(h) = F ( h ) in case

I,

~(h) = S(h) 4(h) = M(h)

Eh' ] '

in case If, ¢(h) = R(h)

h

Eh' = £ e-x~ dx = e~,f (h); and

1 [~

R(h) = ~

he-h' ( 3 + h2)l

El,

"

q~(h) = L ( h )

in case I l l .

J/(h) = T ( h ) Case I I I : Polar Particles Having a Permanent Dipole Moment along Their Axis. In this ease potential energy is given by U = --o~H cos 3' Journal of Colloid and Interface Science, Vol. 36, No. 1, M a y 1971

I t can be easily shown that A'

ZX--A,+I

I1+

10~(h) -- 11¢(h) 1

1+2~(h)

3~

3"

[21]

DEPOLARIZATION OF SCATTERED LIGHT

For random orientation it can be shown that limf,~0 0(h) = 1/3 and limt~o ~(h) = 1/5. Substituting these limits in expressions [19], [20], and [21], we get A = 2 X / ( 4 X + 5 Y + 15a2*a2)

and

85

that limt~-.0 ~ (h) = 1; lim~0# (h) = 1. Consequently, the values of depolarization become zero. It will be interesting to carry out numerical calculations and study how the depolarization factors h and A~ vary with orientations of different elongations of spheroidal particles. I~EFERENCES

~X' = X / (3X + 5 Y + 15a2*a~)

and = 2 ~ ' / ( a ' + 1). These expressions are identical with those given by van de Huls~ (1 (e)). For complete orientation, it can be shown

"Light Scattering by Small Particles." Wiley, New York, 1957. (a) ibid., pp 79; (b) ibid., pp 47; (c) ibid., pp. 49; (d) ibid., pp 79; (e) ibid., pp 80. 2. ATLAS, D., KEaKER, M. L., AND H I T S C H F E L D r W., J. Atmospheric Terrestr. Phys. 3, 108. (1953). 3. KmS~NAN, R. S., Proc. Indian Acad. Sci. 1A, 717, 782 (1935). 4. P~RKE, N. G., J. Math. Phys. 28, 2 (1949). 1. VAN DE I'IuLsT, I-I. C.,

Journal of Colloid and Interface Science, Vol. 36, No. 1, N[ay 1971