A new polarizationmodulated light scattering instrumentArlon J. Hunt and Donald R. Huffman Citation: Review of Scientific Instruments 44, 1753 (1973); doi: 10.1063/1.1686049 View online: http://dx.doi.org/10.1063/1.1686049 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/44/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Polarizationmodulated Raman scattering measurements of nematic liquid crystal orientation Rev. Sci. Instrum. 67, 3924 (1996); 10.1063/1.1147293 Orientation dynamics of a polymer melt studied by polarizationmodulated laser Raman scattering J. Rheol. 38, 1101 (1994); 10.1122/1.550586 New polarizationmodulation spectrometer for simultaneous circular dichroism and optical rotatory dispersionmeasurements. II. Design, analysis, and evaluation of a prototype model Rev. Sci. Instrum. 60, 3633 (1989); 10.1063/1.1140467 ENDOR with polarizationmodulated radio frequency fields Rev. Sci. Instrum. 57, 209 (1986); 10.1063/1.1138971 New polarizationmodulation spectrometer for simultaneous circular dichroism and optical rotary dispersionmeasurements (I): Instrument design, analysis, and evaluation Rev. Sci. Instrum. 56, 2237 (1985); 10.1063/1.1138355

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A new polarization-modulated light scattering instrument*

Arion J. Hunt and Donald R. Huffman

Department of Physics, University of Arizona, Tucson, Arizona 85721 (Received 25 June 1973; and in final form, 10 September 1973)

A new light scattering instrument is described for measuring, as a function of scattering angle, the elements of the matrix describing light scattered from small particle systems. The instrument uses a piezo-optical birefringence modulator to modulate the polarization state of the incident light beam, and uses lock-in amplifier detection of the scattered light. The first and second harmonics of the modulation frequency are used with various combinations of filters and orientations to give the separate matrix elements. A treatment of a simple scattering experiment by means of Mueller calculus is carried through to illustrate the interactions of the light with the scattering system and with the various components of the measuring instrument. Evaluation of the instrumental performance is shown through measurements on two systems of monodisperse polystyrene spheres of average radius 550 and 3940 A, compared with Mie calculations. Further measurements on sulfur colloids are presented, and suggestions of the value of this type of instrument in future work are given.

I. INTRODUCTION

In order to completely characterize a system of small particles optically, it is necessary to determine the intensity and polarization of the scattered light for several different polarization states of the incoming light. Various techniques have been developed in the past to obtain the angular variation of intensity of a specific polarization of the scat­tered light. This was done in order to determine the size and/or size distribution of an assembly of spherical particles with known optical constants. In 1950, Kerker and La Merl

developed a method using either the polarization ratio or the phase angle 'of the scattered light as a function of the scattering angle to extract size information. Since that time, several techniques2-4 have been used with greater or lesser success. However, few attempts have been made to experi­mentally characterize the scattered light completely enough to determine more than two elements of the scattering matrix. 6 To do this all four Stokes parameters of the scat­tered light must be measured at each angle for various states of polarization of the incoming beam. Holland and Gagne6

recorded intensity information for 18 combinations of filters and subsequently took differences between the recorded values to obtain the matrix elements. The disadvantage of this technique is that the values needed to obtain the elements appear as small differences between relatively large quantities. Hence, small errors introduced by drift and instabilities are magnified.

In this paper a technique is reported which avoids these problems by the use of a polarization modulator.7.s Briefly, the polarization of the light is periodically varied in such a fashion that the synchronously detected components of the scattered light are proportional to the matrix elements. By proper choice of polarizers, orientations, and harmonics, selected normalized elements of the scattering matrix can be plotted automatically (by pairs or singly) on x-y recorders.

The system records the data in a relatively short time (a scan from 50 to 1680 may be made in a few minutes) and produces an output in a convenient format for direct com­parison with computer generated plots resulting from Mie calculations. The sensitivity of the system is such that it will respond to 1% differences in linear polarization, which is considerably better, than in most subtractive techniques. The

1753 Rev_ Sci. Instrum., Vol. 44, No. 12, December 1973

accuracy of the instrument is about ±3% presently but that figure should be improved with further development of calibration techniques. The angular resolution of the instru­ment is 1.5°. The normalized parameters obtained from the system are in a convenient form to obtain the degree of partial polarization which may be used to give a measure of the polydispersity in size.

Before describing the apparatus in detail, it will be useful to review, in Sec. II, the treatment of polarized light by Mueller calculus. This will serve to introduce the notation and matrix formalism as well as to discuss a simple case that is important in understanding the operation of the instru­ment. Section III contains a detailed discussion and analysis of the operation of the apparatus as well as calibration procedures. In Sec. IV, instrumental performance is evalu­ated by comparing the results of measurements with calcu­lations for two model small-particle systems which have independently known characteristics. The final section con­tains a brief discussion of other applications of the technique.

II. LIGHT SCATTERING, BASIC CONSIDERATIONS

A. Formalism

The polarization state of any beam of noncoherent light may be characterized by a four element Stokes vector_ This vector may be determined phenomenologically9 or its ele­ments may be expressed equivalentlyIo in terms of quadratic functions of the electric fields as

1= (EIEI*+ ErEr *) ;

Q=(EIEI*-ErEr*) ;

U = (EIEr *+ ErEI*) ;

V =(i(EIEr*-ErEI*)).

(1)

The subscripts refer to the components of the electric field parallel 1, and perpendicular T, to the scattering plane which is defined by the direction of the incoming and scattered waves. The brackets refer to time averages and the asterisks denote the complex conjugates.

The matrix representing the linear transformation of the Stokes vector is referred to as the Muellerll matrix and is a

Copyright © 1973 by the American Institute of Physics 1753

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1754 A. J. Hunt and D. R. Huffman: Light scattering

useful tool for manipulating polarized light. Shurcliff9 gives a

list of matrices corresponding to various polarizing and

retarding elements. The most general form of the Mueller

matrix, one with no zero elements, allows for all possible

mixing of polarization states; the general scattering matrix

is of this form. This matrix may be derived from the

amplitude transformation equations for the electric fields

given by

Ez = A 2EIO+ A sEro,

Er = A 4EIO+ A lEro,

1754

(2)

by substitution into Eq. (1) and solving for the elements of the 4X4 intensity transformation matrix.

The general form of the scattering matrix as given by van de Hulst6 is

[

!(m2+m3+m4+m1) !(m2-mS+m4-m1) S2S+S41 Hm2+mS- m4- m1) Hm2- mS- m4+m1) S23- S 41

S24+S 31 S24-SS1 S21+S34 D24+D31 D24-D31 D21+Ds4

-D23-D41 -D23+D41 -D21+D34

S21-S34

All the elements are functions of the scattering angle. The relationships with the amplitude transformation coefficients are

M·= IA-Iz , " Dij =i/2 (AiA/-AjA;*),

Sij=t(AiA/+AjAi*)'

(3)

For the case of spherical particles, no mixing between the land, channels occurs so that A3=A4=O and the matrix simplifies to

o o

[

t(mz+m1) t(m2-m1) 0

Fs= t(m2-m1) t(m2+ m1) 0 o 0 S21 -D21 o 0 D21

B. Analysis of a Simple Scattering Experiment

(4)

In order to determine the results of a typical scattering experiment, it is necessary to multiply the Stokes vector of the incoming light by the matrix representing each optical element or scatterer in order of traversal. Consider a simple scattering experiment in which light, polarized at 45° to the scattering plane, is incident on a system of spherical scat­terers. The resultant Stokes vector for this experiment is calculated to be

[~ 1 l: i:;~::~ 1 u =2 S21 .

V D21

(5)

The quantities m1 and mz are proportional to the light scattered in the vertical and horizontal plane, respectively. Thus, for this polarization state of the incoming light, the four elements of the resultant vector are just the four inde-

FIG. 1. Schematic diagram of the scattering apparatus.

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

pendent elements of the scattering matrix for spheres. This result will be helpful in interpreting later results.

A useful property of Stokes vectors is that they are additive for combining beams of incoherent light. Thus, the scattering from a collection of randomly positioned spheres may be represented by the sum of the Stokes vectors for the individual spheres. The degree of partial polarization of the light is defined in terms of the Stokes vector elements as

(6)

In order to gain a clearer understanding of the scattering example above, it is worthwhile to examine the resultant Stokes vector for the degree of polarization. To do this one must consider the scattering medium more carefully. Deirmendjian12 pointed out that the degree of partial polar­ization resulting from scattering by a collection of identical spheres is equal to unity (if the incoming light is completely polarized). However, if the spheres are not all of equal size, he also noted that the degree of partial polarization is less than one and may be used to help determine the amount of polydispersity in size. Thus in order to determine F, the problem must be solved in detail for the matrix elements or it may be determined directly from the experimental curves.

III. EXPERIMENTAL SYSTEM

A. The Instrument

In this section the experimental system is described in more detail, and examples are given of how the elements of the scattering matrix may be obtained. Figure 1 is a sche­matic diagram of the scattering apparatus and a complete block diagram of it is given in Fig. 2. Light from a high

BIREFRINGENCE

MODULATOR

SCATTERING CELL

PHOTOMULTIPLIER

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1755

EMI 97616

A. J. Hunt and D. R. Huffman: Light scattering

~ ~ \j~ ~~

FILTERS LINEAR ANALYZER

LAMP

lfM0

71 I rl\ ·t-r -Bif FILTERS OR MONCHROMATOR

FIG. 2. Block diagram of the scattering apparatus.

pressure Hg lamp (Osram HBO 100) or a laser is collimated and passed through a polarization modulator before entering the scattering cell. The scattered light is detected by a photomultiplier tube (EMI 9781B) in an arm which may be rotated through an angle from 0° to 1680 by a dc motor. Monochromatic filters and polarizers may be placed in front of the detector. The ac component of the photomultiplier signal is detected by a pair of lock-in amplifiers and displayed on x-y recorders. The dc component of the output is switched into one of two channels depending on the desired mode of operation. It may be measured directly by a logarithmic amplifier and recorder, or it may pass into a picoammeter which drives a control circuit. In the second mode the photomultiplier current is servoed to a constant value with an operational amplifier which controls the phototube high voltage.

The O-drive for the recorders is produced by stepping motors which are triggered by a digitizer circuit. This con­sists of a flip-flop activated by a photo transistor which views a light emitting diode through an occulting disk. The disk is geared to the same shaft that drives the rotating arm so that the total number of holes that pass the photo­transistor determines the x position of the recorder.

To understand the operation of the instrument in more detail first consider the birefringence modulator. The heart

lHm2+m1) Hm2-ml) 0

1 H m2- ml) H m2+m l) 0 "2 0 0 5 21

o 0 D21

If there are no further polarizers in the light path the photomultiplier will respond to the total intensity of the light, given by the first element of the Stokes vector,

I =lk[(ml+m2)+ (m2-ml) cosoJ, (7)

where k is a constant depending on the photo tube sensitivity, light collecting efficiency, etc. The photomultiplier current may be broken into two parts; (ml+rm2) is a constant in time but varies with angle, (m2-ml) coso is a complex time

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

1755

of the device (Fig. 3) is a vibrating block of amorphous quartz which is driven at its natural frequency (50 kHz) by a piece of crystalline quartz of similar proportions. The crystal forms the feedback element of an oscillator circuit which can be adjusted in amplitude. The applied stress in the amor­phous quartz induces a periodic birefringence which, if of the correct amplitude, could modify the polarization state of the incoming light from linear (at 45 0 with respect to modulator axis) to alternately right and left circularly polarized.

To make the discussion more quantitative, Mueller calcu­lus may be used to analyze the light as it progresses through the instrument. Before proceeding, the angular orientation of the polarizer and modulator must be specified. The two cases considered here are:

1. When the modulator axis is at 45° with respect to the scattering plane and the polarizer P oriented ;:tt 0°.

II. When the modulator axis is parallel to the scattering plane and the polarizer at 45°.

The modulator may be described in both cases as a sinusoidially varying phase plate. The eigenvector9 of the phase plate is specified by the polarization state of light which remains unchanged when traversing the element and must be known to calculate the effect of a phase retarder. It depends on the azimuthal angle of the fast axis of the re­tarder and may be specified by the Stokes vector. The eigenvector is (1,0,1,0) and (1,1,0,0) for case I and II, respectively. Once the eigenvector is found, the phase retardance matrix may be calculated from the general form given by Shurcliff.

B. Linear Polarization and IntenSity Measurements

For case I, the Stokes vector of the light as it emerges from the modulator is given by the matrix product

1 1 0 "2 0

o

o 0 coso 0 o 1

sino 0

-~noj l~ ~ ~ ~j ~ o 0 0 0 0 0 cosO 0000 0

where a = instantaneous angle of retardation. The scattering from a system of spherical particles, given by the scattering matrix operating on the above Stokes vector is

varying quantity. Consider how the time variation enters through the phase retardance 0,

27rd 0=-5 sin wt=A sin wt,

A

where d=thickness of the modulator element, A=wave­length, 5=stress coefficient. Hence the phase retardance is a sinusoidal function of time, but it occurs as an argument of

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1756 A. J. Hunt and D. R. Huffman: Light scattering

a trigonometric function. Expanding sinll and cosll in terms of Bessel functions, 8

sin (A sinwt) = 2J 1 (A) sinwt+ 2J 3 (A) sin3wt+ ...

cos(A sinwt)=Io(A)+2I 2(A) cos2wt+···. (8)

The time averages of sinll and cosll are zero and Io(A), respectively. To simplify the data interpretation it would be convenient for Io(A) to be zero. This is accomplished by adjusting the amplitude of the vibration of the modulator so that A = 138°, which corresponds to the first zero of the Io (A) function. This restriction fixes A for each wavelength used.

Another function of the electronics is to force the dc component of the phototube output to be a constant, thus effectively dividing by the total intensity. Normalizing the output in this way accomplishes two useful functions. The unnormalized signals vary over several orders of magnitude, (for example see the tables in Ref. 12) and are inconvenient to record. After normalization, the signals vary between the limits of ± 1 (with one exception, see beloW). Secondly, random fluctuations in the intensity of the lamp are smoothed out, thus obviating the necessity of a second photomultiplier and subtractive circuits.

The circuit for accomplishing this normalization is shown in Fig. 4. The output of the photomultiplier is divided into two channels, the ac part is capacitively coupled to the lock-in amplifiers, while the dc component passes into a picoammeter. The output of the picoammeter is compared with an adjustable reference voltage by an operational amplifier. The difference signal drives a programmable phototube high voltage power supply. The effect of this servo circuit is to keep the dc phototube current at a con­stant value, thus effectively dividing the ac signal by the total light intensity. Provision is made for measuring the total light intensity by switching the dc signal directly to a logarithmic amplifier and x-y recorder while disabling the servo circuit.

After accomplishing these electronic functions, the output of the photomultiplier may be written

or since (ml+m2) is constant,

(9)

By examining the transformation equations [Eqs. (2), (3)J, it can be seen that ml and m2 are proportional to the in­tensities of the scattered light in the rand l directions. Hence, Eq. (9) states that for case I the signal proportional to the second harmonic of the oscillator frequency is just the degree of linear polarization of the scattered light, or

m2-ml p=--.

m2+ml (10)

It may be seen by reference to Eq. (4) that this is just the

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

1758

ACOUSTIC TRANSDUCER

(50 KHz)

I STRAIN INDUCED BIREFRINGENCE

FIG. 3. Birefringence modulator element.

second element of the scattering matrix for spheres, nor­malized by the total light intensity.

The total direct current may be obtained by disabling the servo circuit, since the time average of cos2wt is zero. In this case however, because the total intensity of the light is measured, there must be a correction for the observed scat­tering volume. This is done by dividing the total intensity term by the sine of the scattering angle. For convenience the logarithm of this variable is measured. Then the observed quantity corresponding to the matrix element ml+m2 is given by

[m1(0)+m2(0) sinO 0 ]

L:=log X . sinO ml(00)+m2(00)

(11)

The second term in the logarithm normalizes the output so as to make L: equal to zero for the minimum scan angle 00•

C. Circular and Oblique Polarization Measurements

In order to extract the other two elements of the scat­tering matrix for spheres, we must examine the operation of the system when the modulator axis is parallel to the scattering plane. Using the eigenvector for case II, the Mueller matrix is found to be

[

1 0

M- 0 1 0 -00 cosll

o 0 -sinll

o o o

sinll cos/l

Following the same procedure as for case I, the Stokes vector

PHOTOMULTIPLIER

~_--II--_A_._C_. ~) TO LOCK-IN AMPUFIERS FOR w,2w,ECT.

PICO

AMM.

LOG

AMP 1--------';> TOTAL INTENSITY

PHOTO ~-~ SERVO 1-----1 TUBE

IO.C.=CONST. '-----' PS.

TWO CHANNEL OUTPUT

FIG. 4. Block diagram of normalization circuit.

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1757 A. J. Hunt and D. R. Huffman: Light scattering

for the scattered light is

l Hm1+m2)

1 Hm2-m l)

2" 5 21 cOSO+D21 sino D21 cOSO-521 sino

Now, the time varying components appear only in the third and fourth element. To obtain an ac signal, a linear polarizer is placed at 45° orientation in front of the phototube. This results in a new Stokes vector, which may be calculated in the usual fashion, given by

lHm1+m2)+5~ coso+D21 sino

Hm1+m2)+521 cOSO+D21 sino o

Again expanding the time varying components, taking account of io(A) =0, and normalizing, the output current is given by

10:: G+2J1(A)D sinwt+2J2(A)5 cos2wt)], where

D21 S21 D=---, S=---, (12)

m1+m2 m1+m2

where D and 5 represent the normalized quantities corre­sponding to the measurements. It can be seen from the above that the signals proportional to the first and second harmonic of the driving frequency give the two remaining normalized elements of the scattering matrix for spheres.

The physical significance of these elements may be interpreted by examining the Stokes vector of the scattered light given in Eq. (5). This resulted from the case when the incident light is linearly polarized at 45°. Then 5 21 is pro­portional to the light scattered at ±4So to the scattering plane, and D21 gives the component of the light which is circularly polarized. The normalized quantity 5 should vary from + 1 in the forward direction to -1 for backscatter if the scatterers exhibit no linear dichroism. Again D should be zero in the forward direction for systems exhibiting no circular dichroism. It is important to emphasize here that these quantities are inherent to the scattering matrix and their relation to the Stokes vector of the scattered light depends entirely on the particular experimental situation.

Until now, only spherical scatterers have been considered. The same method may be used to investigate the elements of more general scattering matrices. Perrin13 showed that for isotropic scatterers with no optical activity the scattering matrix takes the following form:

f:;~: t H o 0 -S34 S44j

It can be shown by the same type of analysis that all the elements of this matrix may be obtained. In addition, they will be normalized by the total intensity and result from the same two cases with only the addition of a final circular polarizer. The elements of more complex matrices (see for example the 10 parameter matrix given by Van de Hulst on

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

1757

p. SO) may also be extracted, although not in such a direct manner.

D. Calibration

Calibration of the instrument is relatively easy since most of the measurements are normalized, in the case of total intensity to the intensity at some reference angle, and in the other three cases to unity. It is only necessary to insert a linear polarizer or phase plate of the proper retardance to produce a full scale (100%) calibration. For case I a polarizer may be inserted into the beam when the detector arm is set at 0°. (A set of attenuators reduces the intensity of the beam to normal operating levels in order to avoid damage to the photomultiplier tube.) From Eq. (9), it can be seen that aligning the polarizer in the r and I directions causes either m1 or m2 to be zero. This results in a positive or negative signal of equal value which may be adjusted to give full scale (100%) deflection. For case II, S is adjusted for 100% deflection at 0° for no intervening polarizers. The D signal may be calibrated by inserting a variable retarding stress plate into the beam and adjusting the phase plate for maximum deflection. The variable retarder assembly con­sists of a small clamp which can be rotated about an axis parallel to the beam that applies an adjustable force to a floating jaw. This produces a uniform stress in a carefully machined Plexiglas or quartz rectangle, thereby inducing a birefringent phase shift between light having a polarization parallel and perpendicular to the jaws.

The condition io(A) =0 may be obtained14 by examining the resultant beam for case II with the scattering cell removed. By alternatively placing a polarizer in front of the photomultiplier tube at ±4So, the following current will result:

10 [ 2h(A) cos2wt ] 1 1± .

l±io(A) 1±io(A)

If io(A)~O, switching from the +45° to the -45° polarizer will change the applied photomultiplier tube voltage, which will vary the gain of the tube, in turn causing unequal posi­tive and negative second harmonic signals. Thus to obtain the necessary condition io(A) =0, the modulator amplitude is adjusted until the two signals have the same absolute magnitude.

IV. RESULTS FOR LATEX SPHERES

A. Mie Calculations

Mie16 calculations were performed to obtain the elements of the scattering matrix for the spherical particles of two model systems. The computations were performed on a CDC 6400 with a computing routine developed by Dave.16

The results of the calculations were graphed directly by a Cal Comp 665 plotter in the same format as the experi­mental output to facilitate direct comparisons. A Gaussian size distribution was used that matched the specifications given by the manufacturer of the test particles. The results of the Mie calculations were summed over 21 sizes, ranging more than two standard deviations from the center radius to obtain the curves for a given size distribution. Further

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1758 A. J. Hunt and D. R. Huffman: Light scattering 1758

o o~~~~ .... ~~.

o o

o Lf)

o

o

o Lf)

o

0 FIG. 5. Calculated and experimental POL and S signals for 550 A radius latex particles for light of wavelength 4350 A.

~~~~~~------------------~--------------------~~~4 0

0

(j)

0 "? 0 I

~ g I 0+.OO---+-~20-.0-0 -+---401-.-00---6+0 .-00-+--8-+0-.0-0 -+---<101-0-. 00-0---1+-20-.0-0 -+-=1=+!40!!!!.!!loOO .... ~16 ... 0 .-OO ...... -+I80!OO

THETA

increasing the number of integration steps did not affect the results.

The effect of the spectral distribution of the Hg lamp was calculated by assuming that it had a Gaussian shape and performing numerically a double integration over the two distribution functions. That is, if F(a/X) represents the result of one Mie calculation for a given radius a, and wavelength X, then the average result is given by

where 0"1 and 0"2 give the Gaussian parameters for the distri­bution in wavelength and size, respectively. The spectral distribution, which was somewhat narrower than the size distribution, was summed over 11 wavelengths for the larger test particles. There was very little difference between the results that included both distribution functions and those with only the distribution in size.

B. Experimental Results and Comparison with Theory

In order to evaluate the performance of the instrument, a series of measurements was undertaken of two model

FIG. 6. Calculated and experimental D and SUM signals for 550 A radius latex particles for light of wavelength 4350 A.

0

o o

o Lf)

o

0 0

0

0 "? 0

I

systems. This acted as a test of the theory and also helped locate problem areas. The two systems used were prepared from DOW17 polystyrene latex spheres dispersed in water. According to the manufacturer's specifications, one system had an average radius of 550 A, the other 3940 A radius. Measurements were made at three wavelengths, the 5461 and 4350 A lines of mercury, and at 3250 A using a He-Cd laser. The 550 A spheres were small enough to produce Rayleigh-like scattering. This case is especially easy to visualize since the vertical component of the light is scattered isotropically and the horizontal component varies as the cosO, where 0 is the scattering angle. It is easy to see that the scattering amplitudes are given by

Al=l

A 2 =cosO.

Substituting these values into Eqs. (3), (10), (11), and (12) and normalizing them so as to correspond to the signal output gives

1-cos20 cosO p s= , D=O

1+cos28 (13)

{ 1 +cos28 sin80 }

L= loglo . sinO 1 +cos280

0 0

0

0 "? 0 I

0 C? ,.

:E :=J (j)

0 "? ,.

o 0 o 0

lo+.o-o-+--420-.0-04-~.o~.O-O-o---6+-0.-00-+--6+0-.0-0-+-~IO-0.-00-+---121-0-.0-0~-1+-40~.0~O+--+16~0.=OO-+-~18JOO THETR

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

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1759 A. J. Hunt and D. R. Huffman: Light scattering

A=4350 A :5 N I

fi? N I

0 C?

'" I

0 II!

'" I

~

I: ::::> (f)

1759

FIG. 7. Calculated and experimental D and SUM signals for 4060 A radius latex particles for light of wavelength 4350 A.

+-~--~~~-r--~-+--+-~--~~~~--~=+~~~--~~~f±l~·!OO

Figures 5 and 6 illustrate the computer-produced plots of these four variables (I:.=SUM, P=POL). The results are actually from the Mie calculation for spheres of real refrac­tive index 1.592 in water, but correspond closely to the Rayleigh scattering case. The experimental curves for a wavelength of 4350 A are also displayed in Fig. 5 and 6. It can be seen that the agreement between the calculated and experimental results is quite good for the three normalized quantities, with a slight deviation in magnitude at low and high angles for the total intensity (SUM) curve.

The second scattering system investigated was composed of the larger polystyrene spheres (Dow lot number 2F2}, diameter 0.794,",). The Mie computation must be done for this case, and it predicts complex scattering curves charac­teristic of the higher order electromagnetic modes of a sphere. Figures 7 and 8 illustrate the results for all four measurements. It can be seen that the position of most of the theoretically predicted features are correctly given in the

experimental curves. The angular position of these are indicative of the size of the spheres while the heights are more characteristic of the size distribution. The fact that the experimental positions of the peaks are given correctly is confirmation that the technique is working as predicted. The discrepancy in the height of the peaks can be attributed to multiple scattering, light scattered from the walls of the scattering cell, impurities, and clustering. The effect of multiple scattering on the SUM and D curves was experi­mentally investigated. For larger dilution factors, less light is multiply scattered into the acceptance angle of the detector. This trend may be seen in the sum curves for vari­ous dilutions in Fig. 9.

Another major factor which can lead to extraneous scat­tered light is the internal reflections of the scattered light within the cell.18 This becomes more important for larger colloids which have strong scattering lobes. Several tech­niques were tried in order to reduce the amount of reflected

A=4350A

-I o a..

fi? o I

fi? o

I

~ ~ IO~.OO=-+-~~~--~~--~-+--~~--~--~-+--+-~--~~~~~

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

(f)

FIG. 8. Calculated and experimental POL and S signals for 4060 A radius latex particles for light of wavelength 4350 A.

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1760 A. J. Hunt and D. R. Huffman: Light scattering

FIG. 9. Measurement of the effect of mUltiple scattering on the SUM signal for 4060 A latex particles at 3250 A.

o ~ I

o q N I

o ~ I

1760

o

~+O--;-~20--~--4~O--+--6+0--+--1eo--;---loor--r--I20+--+--~14-0~--~~~o--r--,e~o

scattered light. The most successful scheme was to coat the inside walls of the cell except the viewing area with a black, nonscattering paint,19

Recently it has been pointed out20 that additional parti­cles may be present in the latex sols not within the specified Gaussian size distribution. In addition, any impurities present including clusters of the sol particles,21 would in­crease the general scattered light and cause a decrease in the size of the structure of the curves. The large latex sols were prepared with deionized water and centrifuged to decrease the number of two and three particle agglomerates. This pro­duced a substantial increase in the peak height and brought the experimental results into better agreement with the calculations.

Figures 10 and 11 illustrate the experimental and theo­retical results for the ultraviolet laser measurements at a wavelength of 3250 A. The results are appreciably better with the laser source and allow the study of much more tenuous colloids and aerosols.

~ -~ 0

FIG. 10. Calculated and experimental POL ~ and S signals for 4060 A radius latex particles 0

for laser light of wavelength 3250 A. -.J 0 a...

~ 0 I

THETA

The index of refraction of bulk polystyrene, measured at 3250 A, by use of a spectrophotometer and the method of minimum deviation was determined to be 1.672+13i where 13<0.0001. The indices used at 4350 and 5461 A were 1.61+0.0i and 1.60+0.0i, respectively. With these values of the refractive index, it was found that the experimental results were in better agreement with calculations for slightly larger sphere sizes (at all wavelengths) than that specified by the manufacturer of the latex sols. The best fit was for a radius of 4060 A. Since in this case, the method is able to measure the absolute size of the latex spheres to an accuracy of greater than 1%, it allows a new and sensitive determination of that parameter.

Now it is possible to reconsider the question of the degree of polarization raised in the last part of Sec. II. Equations (10) and (12) indicate that the quantities P, S, and Dare just the original matrix elements divided by the total intensity of scattered light at that angle. However, in Eq. (5) they occur as the last three unnormalized elements of the

A.=3250A ~ -~ 0

~ 0

(/)

5! 0 I

~ ~ lo+.OO--+--120-.0-0~--40~.O-O-r~6+0.-00-+--8~O~.OO-1--~IOO~.OO~~1~2o~.O~O+-~140~.=OO+-~16~O.~OO~~1~!OO

THETA

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

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1761

~ -0 II)

0

A. J. Hunt and D. R. Huffman: Light scattering

A = 3250 A

~ '" I

~

1761

~ 0 .... ~~~~-,~~~~--~~-------------+~Hrl~r---~~~

FIG. 11. Calculated and experimental D and SUM signals for 4060 A radius latex particles for laser light of wavelength 3250 A.

0

0 "? 0 I

~ '" J

~ ~ IO~.OO--+--2~O.-OO-+--4~O.-OO-+--6~O.-OO-+--B~O.-OO-+--I~OO-.O-o+--1~20-.0-0+--1~40-.0-0+--I~~~.OO~--±I~·;O

THETA

Stokes vector. Therefore, the degree of partial polarization defined in Eq. (6) for the experiment in Sec. II is given by

(P2+S2+D2)!=F,

hence the degree of partial polarization is easily determined from the measured quantities. For Rayleigh scattering it is equal to one for all angles. In more complex cases involving size distributions, it decreases nonuniformly from unity in the forward direction to a low at some intermediate angle. An analysis of both the theoretical and experimental results for the small spheres indicates that F is very nearly one for all angles. In the case of the larger latex spheres, F varies considerably with angle, approaching one for forward and backscatter directions while reflecting the complex varia­tions in the other scattering parameters at intermediate angles. The experimental data follows most of the variations but is usually somewhat lower in value. This may be interpreted as due to either a size distribution wider than the manufacturer's specifications or the other nonideal factors mentioned before or both.

0 "? -

0 "? c

~ c

..J C Q.,

0 II] 0 I

V. FURTHER INVESTIGATIONS

The instrument has been used for preliminary investiga­tions of a number of other colloidal and aerosol systems. Rather extensive investigations of the growth of the La Mer sulfur sols22 have already been carried out. Figures 12 and 13 illustrate the agreement between theory and experiment for determining the size of a sulphur sol at one stage in its growth. In general the experimental agreement with theory for the sulfur sols has been better than for the latex spheres. This is probably due to the fact that the sulfur has fewer particles outside the Gaussian size range than do the latex spheres. Small impurity particles might be effectively re­moved if they served as nucleation sites for the sulfur particles.

The instrument seems well suited to study multiple scattering effects experimentally by varying the concentra­tion of a sol with known characteristics. The effect on the scattering matrix of alignment of ferromagnetic particles by an external field allows the variation of the symmetry of the

. 0

A=4350A "? -0 "? c

~ FIG. 12. Calculated and experimental POL and S signals for approximately 3000 A spheri-

c cal sulfur particles at 4350 A.

Vl

~ 0 I

~ ~ 7B~.OO=-+-~W~.OO~--4~O~.OO~--6~O-.OO-+--8~O-.O-O+--+lOO-.-OO+--+l20-.-OO+--+l4-0.-00+--+l~~.-OO~~l';.

THETA

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

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1762 A. J. Hunt and D. R. Huffman: Light scattering 1762

a II>

a

A=4350A

FIG. 13. Calculated and experimental D and SUM signals for approximately 3000 A spheri-cal sulfur particles at 4350 A. ~

ofa--~~~~~~~--__ ~~--------------~----~--~

o a 11? a I

a 11? I'

~ '" I

a 11?

'" I

a "? ., I

~ ~

1: ::J (/)

'o7.oo=-~~20~.~OQ-+--,~o~.O~Q~--6~O.~oo-+--~eo-.o-o~--1+00-.0-04-~12~O-.Q-Qr--l+'O-.-QO~--16~O-.Q-O+--+leo%o THETA

scattering system. Early measurements on smoke aerosols indicate they may have definite polarization signatures in spite of the wide particle size distribution.

Improvements are planned to the scattering cell to reduce the amount of scattered light from internal reflections within the cell by trapping it in an extended Rayleigh horn.18 The possibility of suspending and performing measurements on a single particle will be pursued because the elimination of the smoothing effect produced by the particle size distribution would allow for much more precise measurements.

ACKNOWLEDGMENT

The authors would like to thank Professor James C. Kemp for the use of a polarization modulator, which made possible the development of the present light scattering instrument.

·Supported by the Research Corporation and by the Atmospheric Sciences Section, National Science Foundation.

1M. Kerker and V. K. La Mer, J. Am. Chern. Soc. 72, 3516 (1950). 2W. B. Dandliker, J. Am. Chern. Soc. 72, 5110 (1950).

Rev. Sci. Instrum., Vol. 44, No. 12, December 1973

3A. F. Stevenson, W. Heller, and M. L. Wallach, J. Chern. Phys. 34,1789 (1961).

4J. P. Kratohvil and C. Smart, J. Colloid Sci. 20, 875 (1965). 5H. C. van de Hulst, Light Scattering by Small Particles (Wiley,

New York, 1957). 6 A. C. Holland and G. Gagne, Appl. Opt. 9, 1113 (1970). 7J. C. Kemp, J. Opt. Soc. Am. 59, 950 (1969). 8S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instrum. 40, 761

(1969). 9W. A. Shurcliff, Polarized Light (Harvard University Press,

Cambridge, 1966). 10M. J. Walker, Am. J. Phys. 22, 170 (1954). IIH. Mueller, J. Opt. Soc. Am. 38,661 (1948). 12D. Deirmendjian, Electromagnetic Scattering on Spherical

Polydispersions (American Elsevier, New York, 1969). 13F. Perrin, J. Chern. Phys. 10,415 (1942). 14S. N. Jasperson, private communication. 15G. Mie, Ann. Phys. (Leipz.) 25, 377 (1908). 16J. V. Dave, IBM Scientific Center, Palo Alto, California, Report

No. 320-3236, 1968. 17Dow Chemical Company, 1200 Madison Avenue, Indianapolis,

Indiana 46225. 18W. Heller and J. Witeczek, J. Phys. Chern. 74,4241 (1970). 193M brand velvet coating, 100 series No. 101-CIO black,

Minnesota Mining and Manufacturing Co., St. Paul, Minnesota. 2°D. D. Cooke and M. Kerker, J. Colloid Interface Sci. 42, 150

(1973). 21J. A. Davidson, C. W. Macosko, and E. A. Collins, J. Colloid

Interface Sci. 25, 381 (1967). 221. Johnson and V. K. La Mer, J. Am. Chern. Soc. 69, 1184 (1947).

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