Ecological Research Series
A DIFFRACTION TECHNIQUE
MEASURE SIZE DISTRIBUTION OF
LARGE AIRBORNE PARTICLES
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal
species, and materials. Problems are assessed for their long- and short-term
influences. Investigations include formation, transport, and pathway studies to
determine the fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living organisms
in the aquatic, terrestrial, and atmospheric environments. .
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/3-76-073
June 1976
A DIFFRACTION TECHNIQUE TO MEASURE SIZE
DISTRIBUTION OF LARGE AIRBORNE PARTICLES
by
A. MeSweeney
Georgia Institute of Technology
Engineering Experiment Station
Atlanta, Georgia 30332
Grant No. R-802214
Project Officer
T. G. Ellestad
Aerosol Research Branch
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental. Sciences
Research Laboratory, U.S. EnvironmentalProtection Agency, and approved
for publication. Approval does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection
Agency, nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
11
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ABSTRACT
The purpose of the work described in this report was to test and demonstrate
a coherent optical diffraction technique for measuring the size distribution
of large particles. This technique is based on the generation of a trans-
formation matrix which is used to relate the measured diffraction patterns
to the size distribution of the samples that produced the patterns.
Four different types of samples were considered: 1) pinholes in opaque discs,
2) photographic transparencies with opaque circular spots, 3) particles
deposited on microscope slides, and 4) aerosols. Computer simulations were
performed to assess the accuracy of determining particle size distributions
by the optical diffraction pattern technique under both ideal and nearly
ideal conditions. The results of the computer simulations indicated that
it should be possible to resolve the size range from 5 to 100 pm diameter
into eight subintervals.
Although good results were obtained with an array of circular apertures in
an opaque background, experimental difficulties limited the precision of
this technique applied to particles in a transparent medium. Expected im-
provements based on a reduction of system noise and an increase in detector
sensitivity are discussed, and applied to the requirements on number density
and size range of particles in a transparent medium.
The results obtained on this program will facilitate the design of an effective
system for measuring the size distribution of particles by either of two
techniques: 1) diffraction pattern analysis, or 2) Hodkinson's (ratio of
measurement at two angles) technique.
iii
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iv
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CONTENTS
ABSTRACT iii
FIGURES vi
TABLES vii
ABBREVIATIONS AND SYMBOLS ix
I. INTRODUCTION 1
II. SUMMARY 10
III. CONCLUSIONS 11
IV. RECOMMENDATIONS 12
V. EXPERIMENTAL PLAN 13
VI. METHODS, PROCEDURES, RESULTS, AND DISCUSSIONS .... 15
REFERENCES 41
BIBLIOGRAPHY 42
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FIGURES
Number Page
1. Basic Optical System for Producing and Measuring
Diffraction Patterns ' 3
2. Complete Diffraction Pattern Particle Sizing System 16
vi
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TABLES
Number
1. Circular-Aperture Diffraction Pattern Parameters 5
2. Particle Sizes for Which the First Minimum in the
Diffraction Patterns Occurred at Successive Detector Rings .... 22
3. Ordered Eigenvalues of Covariance Matrix Obtained from
the First Criterion 23
4. Particle Sizes for Which Successive Diffraction Pattern
Minima and Maxima Occurred at Detector Ring 32 24
5. Ordered Eigenvalues of Covariance Matrix Obtained from
the Second Criterion 25
6. Metal Pinhole Diameters and the Ordered Eigenvalues of
the Covariance Matrix 25
7. Uniformly Spaced Aperture Diameters and the Associated
Ordered Eigenvalues 26
8. Particle Diameters for Uniformly Spaced Diffraction
Maxima and the Associated Ordered Eigenvalues 27
9. Results of Normalization of the Uniformly Spaced
Diffraction Patterns 28
10. Results of the Sixth Criterion 29
11. Results of Normalization on the Sixth Criterion 30
12. Specified and Measured Metal Pinhole Diameters 31
13. Metal Pinhole Diameters Measured by Location of
Diffraction Minima 33
14. Experimental Results from Metal Pinholes 34
15. Results from Normalized Experimental Data 34
16. Results of Elimination of Diffraction Pattern Data
for One Pinhole 35
vii
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TABLES (Continued)
Number^ Page
17. Comparison of Experimental and,Computer Simulation
Results .... . '. V'.'V."'." '.4 ."' .:.: .' . V-V-. ::-.' .''^'.--. '.". .^<3$
18. Experimental Results 'fr,om Photographic Samples' . . 'i ."i5".' . . --.' 38
19. Application of Experimentally Deriyed Inversion Matrix
to Measured Diffraction Pattern Data . . ~. : . r.' . ': . . .: .: . . -i' 39
viii
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ABBREVIATIONS AND SYMBOLS
a - radius of particle or aperture
AR - anti-reflection
_2
cm - centimeter, 10 meter
d - diameter of particle or aperture
EFL - effective focal length
f - focal length
G() - kernel function in integral equation
T
G - transpose of matrix G
I - irradiance (power/area)
J () - first order Bessel function of the first kind
k - wavenumber, y-
X - wavelength
m - parameter associated with diffraction patterns
See Table 1.
_3
mm - millimeter, 10 meter
urn - micrometer, 10 meter
n - number density (particles per unit volume)
N - number of particles or apertures
-9
nm - nanometer, 10 meter
PIN - positive - intrinsic - negative
r - radial distance from optic axis
RSI - Recognition Systems, Inc.
UDT - United Detector Technology
W - Watt
ix
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SECTION I
INTRODUCTION
The purpose of the work reported here has been to quantitatively evaluate
a technique for measuring the size distribution of large particles in air.
The technique is relatively new and not widely used. Major advantages
are that the particle number densities can be relatively large and that
the size distribution can be obtained in real-time. In general the evalua-
tion has been directed toward obtaining quantitative values for such parameters
as the optimum particle size and particle number-resolution of the technique,
and then to compare these with the requirements of one specific problem: the
measurement of the number vs. size of large particles in air. There may be
other problems for which this technique is better suited but the purpose of
this work has been to evaluate the performance in terms of the problem stated
above.
The most important element in the technique is the use of coherent laser
radiation to form the diffraction pattern of a group of particles exposed
to the laser beam. The complicated diffraction pattern is made up of the
superposition of all the diffraction patterns due to the individual particles.
This conglomerate diffraction pattern is then measured and analyzed to yield
the size distribution of the particles in the sample. This is referred to
as an indirect measurement. The corresponding direct measurement would
consist of measuring the size of each particle individually and then in-
ferring the size distribution from the accumulated results. This could
be accomplished with an optical microscope and adequate time. The indirect
measurement consists of inferring the size distribution from the measured
superposition of all the diffraction patterns. This can be done essentially
in real-time but with some loss in precision.
This indirect method is analogous to Fourier transform spectroscopy, in
which the power spectrum is measured indirectly. The resultant of the controlled
superposition of all the components of the spectra is first measured and
then inverted to yield the power spectrum. The advantage of Fourier
spectroscopy over conventional spectroscopy is that during the measurement
interval information is collected on every component in the spectrum. In
conventional spectroscopy each element of the spectrum is scanned in sequence
so that the effective integration time, and hence signal-to-noise ratio, is lower
for the same total measurement time.
The mathematical operation of Fourier transformation is also inherent to
the particle sizing technique because the measured diffraction patterns
(the spatial distributions of light intensity) are proportional to the
square of the Fourier transformation of the complex amplitude of the light
in the sample region. However, the operation of finding the size distribution
of the particle sample from the measured diffraction pattern can be reduced
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to a straightforward multiplication of matrices. So, once an inverse
matrix has been obtained, the process .consists of measuring a diffraction pattern
and multiplying this pattern by the inverted matrix. The inverted matrix
remains constant and is calculated only once. The result of the multiplica-
tion is a set of numbers representing the size distribution of the particles
in the sample. The matrix multiplication can be performed essentially in
real-time. -. . ; .... _r .
A line drawing of>the section of the optics, directly,.involved in the produc-
tion and measurement of the diffraction patterns is shown..in Figure 1. A
collimated laser beam illuminates the sample space> .The light transmitted
through the sample volume,-is transformed by the lens, and the Fraunhofer
diffraction patterns are formed in the back focal plane of. the lens* A '
detector array located in this plane measures the spatial, distributions of
irradiance in the diffraction .patterns. .'-..-... . ,
The diffraction pattern produced by a small circular aperture in an opaque . .
screen in the front focal plane of the lenses is described by the following
equation [!]: .: .:.'..
Kr)
2
kd" ~"i
8f
2J(krd/2f),
.kfd72
where I(r) is the diffraction pattern irradiance a distance r from the optic
axis .'.....'". - ' '' "'''','
I is the-irradiance of the uniform/incident beam"
o . _ r ' " ~ '
k is 2if/A, where X is the" wavelength of the laser ra'diation
d is the diameter of the circular aperture, , . . .
f is the focal length of the lens
-' . ^ i . - ; i . .," j
J () is the first order Bessel. f uric tio.n. of firs.t kind with argument
; (krd/2f). , - ... - ... ;;'..,...-....
This, pattern consists of a bright .circular, disk .- ., . .
where the value of m is associated with the ring brightness as indicated in
2
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SAMPLE
REGION
LENS
DETECTOR
ARRAY
LASER
BEAM
Figure 1. Basic Optical System for Producing and Measuring Diffraction Patterns.
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Table 1 [2], The values of m associated with the dark rings are derived
from the zeros of the Bessel function. The values of m associated with
the bright rings give the radii to the peak intensity regions.
The important functional relationships are the direct relation between the
radius of a chosen ring and either the focal length of the lens or the wave-
length of the radiant energy, and the inverse relation between the radius
of the ring and the diameter of the aperture. An increase in either focal
length or wavelength produces an increase in radius of the rings in the
diffraction pattern; that is, the diffraction pattern is expanded. An
increase in the aperture diameter compresses the size of the diffraction
pattern.
The reason for'beginning this discussion of the theory with the above descrip-
tion of the Fraunhofer diffraction pattern produced by a pinhole aperture in
an opaque screfen is that this case is discussed in most textbooks on optics
and hence is most familiar. However, the simplest case of the actual problem
of interest would consist of the complementary situation of an opaque circular
disk in a transparent screen. By Babinet's principle we know that the
diffraction patterns of complementary obstacles are the same, except near
the pd.int where the optic axis intercepts the pattern ,[3]. So the properties
of the circular-aperture diffraction pattern apply as well to the patterns
produced by spherical particles. ,
Also, from scattering theory we know that the scattering efficiency is two
for spherical particles of radius much larger than the wavelength. The
scattering efficiency is defined as the ratio of the total amount of light
scattered to the amount of light incident on the geometrical cross-section
of the particle. A scattering efficiency of two for large particles implies
that the total light scattered by a spherical particle is twice the amount
of light incident on the,geometric cross section of the particle.
Another way of thinking of this is that the light directly incident on
a large spherical particle and the light passing around the particle within
a ring of area equal to the particle cross-section are both scattered. In
the case of an opaque particle the light directly incident on the particle
cross-section is either reflected or absorbed and only the light passing
through the ring around the particle contributes to the particle diffraction
pattern. But this amount of light is the same as that incident on a circular
aperture of the same diameter as the particle. In the case of a circular
aperture the light passing through the aperture produces the diffraction
pattern. So, the total amount of light in the diffraction patterns produced
by large, opaque particles of circular cross-section is the same as the
amount of light in the diffraction pattern of a circular aperture in an
opaque screen. By Babinet's principle we know that the two patterns are
the same both in shape and irradiance.
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TABLE 1
CIRCULAR APERTURE DIFFRACTION PATTERN PARAMETERS [2]
Diffraction
Ring m
1st Dark 1.220
2nd Bright 1.635
2nd Dark 2.233
3rd Bright 2.679
3rd Dark 3.238
4th Bright 3.699
4th Dark 4.241
5th Bright 4.710
5th Dark 5.243
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Before going further we must point out a consideration that limits
the minimum number of particles whose diffraction pattern can actually be
measured with no more than the reasonable amount of precision available.
The diffraction pattern of a pinhole1 in an 'opaque screen is relatively
simple. Only the light that passes through the pinhole goes into the
diffraction pattern arid all of it is accounted for by thepattern. For
the case of an opaque, spherical particle, however,*there are actually two
diffraction patterns present and superimposed. One is the' pattern charac-
teristic of the particle. The" other'pattern present "l"s characteristic
of a circular aperture whose diameter is the same as either the laser beam
diameter or the entrance pupil diameter of the lens, whichever is smaller.
The dimensions of this second pattern are greatly compressed in comparison
with the pattern due to the particle because of the usually much larger
effective diameter of the laser beam. Because of this compression the second
diffraction pattern may be-significant only in a negligibly small region at
the optic axis, as indicated in the statement of Babinet's .principle above.
However, the irradiances in the two patterns must be compared as well as
the geometrical dimensions. -Practically all of the light iri the laser beam
goes into the diffraction pattern due to the limiting.aperture. Only a
relatively small portion of the light goes into the pattern characteristic
of a single particle. In order to increase the irradiarice':in the particle dif-
fraction pattern, a larger number of particles of the :same size must be placed in
the laser beam. The minimum number of particles of one size that produces
a pattern distinguishable from the laser beam pattern determines the
minimum number of particles o~f that size that can be re'sdlved. Results of
calculations based on these ideas will be presented later.
Another aspect of the diffraction pattern produced by a particle must now
be discussed. This concerns the location of the particle in the laser beam.
Light must be thought of as a complex quantity involving both amplitude
and phase. However, the physical parameter that is actually measured is
proportional to the square of the amplitude of the resultant of all the
superimposed components. The phase information is not measured. When a
single particle is moved in the sample plane, only the phase of the field
vector is changed at any point in the diffraction pattern. The amplitude,
and hence the irradiance, stays constant. Thus, there is no detectable
change in the diffraction pattern when the particle is moved provided that
the irradiance of the incident beam is uniform. This means that a
particle fixed on a substrate will produce the same diffraction pattern
as a particle falling through the laser beam. This is the justification
for using permanent samples to simulate, time-varying aerosol samples.
When two or more particles are in the beam simultaneously, the resultant
amplitude in the diffraction pattern is a function of the phase of each
component, and constructive and destructive interference effects may be
observed. However, when the number of particles is very large and the
positions of the particles are random, the irradiances rather than the
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amplitudes of the diffraction patterns add linearly. Therefore, a random
monodisperse sample of N particles would produce a diffraction pattern with
N times the irradiance of a single particle.
If the sizes of the particles in a sample vary over a wide range, then
the distinctive bright and dark circles in the diffraction pattern of a
monodisperse sample become smeared and washed out. However, provided the
measurements are made with adequate precision, the inversion process
transforms the pattern into the size distribution of the particles that
produced the pattern. The precision with which the diffraction measurements
are made determines the particle size resolution achievable. This will be
discussed in more detail later.
So far we have been discussing the diffraction patterns of opaque particles.
If the particles are transparent then the refractive index is important
in determining the shape of the diffraction pattern. The process can be
thought of as consisting of a combination of diffraction and refraction.
The light passing around the particle through a ring whose area is equal
to the cross-sectional area of the particle produces the diffraction pattern
we have been considering. The light transmitted through the particle is
refracted at two surfaces and, since the phase relation between the diffracted
and refracted components remains constant in time, interference will occur
between the two components. If the amplitude of the transmitted light is
comparable to that of the diffracted light, the interference will be significant.
This occurs, for instance, in water droplets and is referred to as anomalous
diffraction [4]. This effect can be accounted for in the diffraction
pattern technique of particle sizing provided the refractive index of the
particles is the same for all particles.
Particles of shapes other than spherical can produce dramatically different
diffraction patterns. If the particles are all the same shape this effect
may be at least partially compensated for in the inversion process, particu-
larly if allowance is made for measuring the angular variation of the
diffraction patterns as well as the radial variation.
The functional relation between the size-distribution of the particles in
a sample and the diffraction pattern produced by that pattern can be written
as an integral equation:
ra2
I(r) = I Z n(a) G(r,a) da
al
where I(r) represents the irradiance at a ring of radius r in the diffraction
pattern
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a., is the radius of the smallest particle in the sample
a» is the radius of the largest particle in the sample
n(a) represents the number of particles per unit volume as a function
of particle'radius.
Jj-CkarYf)
kar/f
is the kernel function of the integral equation. In this case, the kernel
function is the expression for Fraunhofer diffraction by an aperture of
radius a. The integral equation above is,referred to ,as a Fredhplm inte-
gral equation of the first kind. .The variables .whose values are known are
the irradiance I(r) and the kernel function ,G(r,a). The unknown,quantity.,
is n(a), the size distribution of the particles, producing the measured
diffraction pattern.
Anderson and Beissner [5] haveoformulated.the problem in .matrix notation.
as follows:
I '=,GN
where I is a column matrix whose elements,are.the irradiance measurements
at a set of radial distances.
. G is a'rectangular (in general.not square) matrix whose-elements are
discrete values 'of the-kernel-function described earlier.
N is a column matrix whose elements are the number of .particles^as
a function of size.
The solution of the above matrix equation f.qr.N, the discrete values of
the particle size distribution, is" [6]
T -1 T
N = (G^) G I
T ' . ' -
where G is the transpose of G. This solution minimizes the sum of the
squares of the errors.
The above "solution will yield unstable results if the columns of'G are
almost linearly dependent. This will occur when the experimental error in
8
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the diffraction pattern measurement is significant compared with the
actual difference between diffraction patterns produced by particles
differing in size by the resolution limit. This error limits the size-
resolution achievable by the indirect method of particle sizing. The particle
size intervals must be large enough to insure that the diffraction pattern
produced by particles in adjacent intervals differ by quantities that are
large compared with the experimental error in the irradiance measurements.
If the error in the irradiance measurements is too large, one or more
columns of G will be linear combinations of other columns of G. Under
this condition G will not have a generalized inverse and there will not
be a solution. When the columns of G are almost linearly dependent,
small errors in the measured data cause large variations in the solution.
When this is true the matrix is said to be ill-conditioned.
Twomey and Howell [7] have suggested a procedure for calculating the
number of independent pieces of information available in the solution. It
is based on an analysis of the eigenvalues associated with the covariance
matrix of the kernel functions. For a given matrix G of kernel functions
the covariance matrix is G^G. For the diffraction pattern technique of
particle size measurement the covariance matrix is positive, definite, and
symmetric, but not well-conditioned.
The eigenvalues of the covariance matrix are calculated and placed in the
order of decreasing values. These ordered eigenvalues are compared with
a measure of the error in the experimental data. The number of eigenvalues
above this measure is the number of independent inferences that can be drawn
from the data with that level of error. So long as the number of measure-
ments is larger than the number of independent inferences, the most direct
way to increase the number of inferences is to increase the precision of
the measurements rather than to increase the number of measurements.
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SECTION II
SUMMARY
The work on this .grant was directed toward the study of aerosol dynamics
by applying a diffraction pattern analysis technique to the problem of ,
measuring the size distribution of large particles in air- The technique
consisted of deriving a transformation, matrix 'Which was used to convert
measured diffraction patterns :to the size distribution .of the samples that
produced the patterns. The greatest success achieved was. in the measure-
ment of the size distribution of an array of circular apertures;in an ,
opaque background. The more difficult case,, that of analyzing the diffrac-
tion pattern produced by an array of particles in a transparent medium,
yielded only moderate success. The use of.a better quality detector .array
and ..lens would improve'the results,, but the technique will probably be
limited to particle samples of large.number, densities. , , ,. ,..
10
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SECTION III
CONCLUSIONS
This report presents the results of an application of diffraction pattern
analysis to the measurement of the number and size of large particles in
air. The conclusions drawn from the results of this effort are presented
below.
(1) A diffraction pattern inversion technique for sizing
particles was demonstrated successfully for a polydisperse
array of circular apertures in an opaque background.
(2) This technique appears to be best suited to the problem
of counting and sizing apertures in opaque materials.
(3) Extension of the technique to the problem of sizing particles
in air appears to be feasible for samples of large number
density.
(A) This technique may be applicable to the problem of monitoring
the particulate emissions as a function of size from industrial
smokestacks. The advantages of this inversion technique are
the large sample volume and the real-time output of the size
distribution.
(5) The literature search done in conjunction with the work reported
here indicated that the inversion process has been applied to
a wide variety of problems in the areas of remote sensing,
spectroscopy, geology, and particle sizing.
11
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SECTION IV
RECOMMENDATIONS
The further development of the technique described in this report should
consist of the following steps:
(1) Design a computer simulation of the application based on the
results indicated in this report. Include a complete eigenvalue
analysis as described by Twomey and Howell [7].
(2) Design and-purchase a detector array and transform lens
optimized for the application. Two important modifications to
the RSI array would consist of antireflection-coating'the
face of the detector and drilling a hole thrbugh Tth'e center
element to allow the intense "dc component" to pass through
without being reflected between the'detector !and the transform
lens. .
(3) A minicomputer should be used to take and process the data
from the detector array.
(4) Modify the computer program for accurately simulating'diffraction
patterns from both apertures and particles by incl'udirig correction
factors to account for the observed detector performance.
(5), Verify the accuracy by comparing computer generated arid
experimentally.measured patterns.
(6) Generate an inversion matrix from accurately computed patterns
for monodisperse samples.
(7) Measure the diffraction patterns from known polydisperse
samples and apply the inversion matrix in order to compare
the measured and the known values.
12
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SECTION V
EXPERIMENTAL PLAN
The main phases of this project consisted of:
A. Modification of computer programs for simulating the technique
and inverting data.
B. Analysis of the diffraction patterns from single pinholes in
metal disks.
C. Analysis of the diffraction patterns from arrays of circular
apertures and opaque spots on photographic transparencies.
D. Analysis of the diffraction patterns from actual particle
samples on microscope slides.
E. Analysis of the diffraction patterns from actual aerosol samples.
Each of these phases will be discussed briefly now, and in more detail
in Section VI.
A. Three main computer programs were utilized. The first program,
labelled POWER, calculated the diffraction pattern data to be measured
by a detector with the geometry of our Recognition Systems, Inc. (RSI)
array. Parameters in this program are the particle diameter, transform
lens focal length, and laser wavelength. The output consisted of sets of
numbers proportional to the expected output of the RSI array, one set of
31 values for each particle diameter. Each of the 31 values in a set
corresponded to the output of one of the 31 ring elements concentric
with the center element of the RSI array. This was the starting point
for a computer simulation of the particle sizing technique.
The second program was one that utilized diffraction pattern data to
calculate the inversion matrix. This matrix was then used to transform an
arbitrary diffraction pattern to the size distribution of the sample that
produced the pattern. The diffraction pattern could be either computer-
generated or actual data. This program also calculated the eigenvalues
and eigenvectors of the covariance matrix. A "smoothing function" [8]
was included in order to provide a controlled amount of smoothing to
oscillating solutions in the case of an ill-conditioned matrix.
The third program performed a matrix multiplication between the previously
calculated inversion matrix and any set of diffraction pattern data. The
13
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result of this multiplication was a measure of the size distribution of
the particle sample that produced the diffraction pattern. Both the
inversion matrix and the diffraction data could be either computer
generated or actual measured data. This flexibility was used to provide
pure computer simulation with noise-free data, a measure of the effect
of error in either the inversion matrix or the diffraction pattern data,
and a comparison between results obtained by a pure 'simulation and a
completely experimental set of data.
B. The second phase of the project consisted of measuring and analyzing
the diffraction patterns from single pinholes in metal disks. The reason
for doing this was that it provided us "with experimental data under nearly
ideal conditions. The experimental data differ from the computer'generated
diffraction patterns both in precision (the number_of significant digits)
and accuracy. The conditions'were"regardedJas ideal in comparison with an
experimental measure of the diffraction patterns Jof actual particles in that
only the diffraction pattern characteristic of the.pinhole diameter was
present. As pointed out earlier the diffraction pattern'due to an aerosol
sample has superimposed on it the diffraction pattern due to the laser beam
diameter.
C. The third phase of this project consisted of measuring and analyzing
the diffraction patterns from arrays of circular apertures and opaque spots
on photographic transparencies. This was'brie step closer1 to tli£ actual
problem of measuring the patterns produced by aerosol samples, but it
also provided a technique for^generating permanent, monodisperse samples.
D. The fourth phase of this project consisted'of measuring'and analyzing
the diffraction patterns of samples of particles deposited on microscope^
slides. 'Again, this was"one step closer to the goal of measuring aerosol
samples, and yet it provided the convenience of "fairly permanent samples
that could be examined under a microscope.
E. The fifth phase of this project consisted of measuring and analyzing
the diffraction patterns of actual' aerosol samples. This'was the goal
of the project to which the previous phases'were'directed.1"
14
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SECTION VI
METHODS, PROCEDURES, RESULTS, AND DISCUSSIONS
Figure 1 in the discussion of the theory illustrated only the Fourier
transform lens segment of the complete optical system. Figure 2 shows
the complete system as normally used on this project. The argon ion laser
was a Coherent Radiation Model 53 capable of producing about 1.5 W at a
wavelength of 488 nm. The laser output was focused on a pinhole filter
to eliminate undesired components and then allowed to expand to a diameter
of 50 mm before being recollimated.
The irradiance of the pinhole filter/collimator was found to fluctuate
even though the laser had a built-in light sampling and controlling system.
The internal sampling was replaced by an external detector and amplifiers
in the form of a United Detector Technology (UDT) Model 40A radiometer.
The output of the radiometer amplifier was fed into the laser power supply
control circuit to close the control loop.
Only the 3 mm diameter center segment of the 50 mm diameter expanded beam was
passed by the aperture of the beam expander. This was done to insure uniformity
of the irradiance across the beam in the sample region and to minimize the
amount of unnecessary light on the detector. Most of the work was done
with a Fourier transform lens sold by Tropel, Inc. This lens was supposedly
designed to be used at a wavelength of 488 nm corresponding to one of the
lines emitted by the argon-ion laser. The effective focal length (EFL)
of the lens is 59.89 cm, and the diameter of the entrance pupil is 58.4 mm.
The overall transmission through the 6-element lens is supposed to be
greater than 99% at 488 nm. However, the measured transmission turned
out to be only about 82%. Unfortunately, the 17% difference appears to be
due largely to reflection at each air-glass interface even though the
elements are anti-reflection (AR) coated. A portion of the internally
reflected light is superimposed on the diffraction pattern to be measured
at the detector array. This reduced the accuracy and also the usefulness
of the Tropel lens. Photographic and television lenses were also used with
good results when shorter focal lengths were required.
When there were no samples in the laser beam, the light in the beam was
focused at the center of the detector array. If the undiffracted light
were allowed to impinge upon the face of the detector, a portion of it
would be reflected back toward the lens. Because of the relatively high
reflection from the lens surfaces some of the light would be reflected
toward the detector again. In order to reduce this component of the
reflected light , an optical filter was used to trap the undiffracted light
and guide it away from the detector region.
15
-------
Pinhole Filter Beam
"and Aperture
Beam Expander
Argon Ion Laser
:o:
i
Amplitude Stabilization
.Loop
Laser
Power
UDT
Radiometer
...
Fiber-optic
Light trap
Transform
Lens
RSI
Detector
Array
RSI
Readout
Figure 2. Complete Diffraction Pattern -Particle Sizing System.
-------
The detector array that was used to measure the majority of the diffraction
patterns was a Recognition Systems, Inc. diffraction pattern sampling unit.
The geometry of the detector array is such that one half of the 3.18 cm
diameter circular area is covered by 31 concentric ring elements and 1
center element. The other half is covered by 32 wedge shaped elements.
The signal output of the ring elements was proportional to the integral
of the irradiance in the diffraction pattern over the area of each element.
The ideal detector geometry for circularly symmetric diffraction patterns
would consist of concentric rings that integrated around a full 360° circle
rather than the 160° almost-half-rings of the RSI array. However, the
difference is probably not significant in comparison with the potential
improvement from other possible modifications. These ideas were discussed
under RECOMMENDATIONS.
Another detector array that became available for a short time during the
fourth and fifth phases of this project was a fiber-optic/photodiode-array
built at Georgia Tech for NASA [9]. This consisted of an array of 90,000
optical fibers of 76 ym diameter arranged at one end in 168 rings concen-
tric with the center fiber. At the other end of the 60 cm long fiber bundle
each ring of fibers was terminated with a PIN photodiode. The fibers
collected the light from concentric circles and guided it to individual
photodetectors for each circle. The photodiodes produced signals propor-
tional to the integral of the irradiance over each circle. The output
of the array of 169 photodiodes was electronically scanned, digitized,
and fed into a minicomputer. This provided much faster accumulation of
data and allowed real-time subtraction of background levels from the
signal-plus-background levels to yield signals proportional to the
irradiances in the particle diffraction patterns. These diffraction
patterns were displayed in real-time which provided immediate feedback
concerning the results of any sample changes or system adjustments.
As was indicated in Section V the first phase of this project consisted
of modifying the computer programs that had been used at Georgia Tech for
earlier work on this particle sizing technique. The first program modified
for this project was one labelled POWER. This program calculated values
proportional to the power (integrated over the area) incident on each
photodetector in a diffraction pattern. The flux density (power/area)
at a radius r in the Fraunhofer diffraction pattern of a uniformly illuminated
circular aperture is:
Kr)
kd2
8f
L
ZJ^krd/Zf)
krd
2f
17
-------
where I is the uniform flux density (power/area) incident on the aperture,
The units of I(r) are determined by the units in which I is given.
k = 2ir/X where A. is the wavelength of the .radiant energy'
d = diameter of the aperture . . .-..'...
f = effective focal length of the transform lens
J () = first order Bessel function of the first kind.
r = radial distance in the plane of the .diffraction pattern from the
optic axis. The center of the diffraction pattern occurs at the
optic axis of. .the transform lens independent of the. location of
the aperture with respect to the optic axis. . ~ ,/
The flux density was integrated over the area of each detector element
in order to obtain a value proportional to the signal out' of each element.
The proportionality constant was the responsiyity '(voltage/power) of the
detector element. The total flux "or power incident on detector 'element
N is indicated as
P (N) = ",;' .
where r.T = radial distance-to outer edge of'ring element N
N.max ;. .. -.!,-> -j ... -. *.*. . : . - ; -, . ,: .. . . ._
r.T '. = radial distance to :inner edge\6f ring element, N
N,min
dA = infinitesimal element: of area. . . . - :. ;
The geometry of the ring, elements of the RSI detector array was such that
the rings subtended an angle of only 160° at' the center of the array. The
infinitesimal area elements in the, above integration were taken as strips
of length equal to the arc forme'd"by the detector and thickness equal to dr:
.'-. i. ''>"/' >-..,
'" :\vi;l60 '" " ': "" ' .-.
180
Substituting for I(r) and dA,the expression for the power in ring N
became: -' ' .
18
-------
POO =
4 r rN> ^
2(TTd) IQ I
9ir(fA)2 y
rN> min
irrd
fX
dr.
The constant Io was set equal to 1 in the computer program. Because the
dimensions of d, f, and r were converted to microns in the computer program,
the values of power calculated by the program must be multiplied by 10~° to
yield the power in milliwatts for an incident irradiance in the laser beam
of 1 mW/cm^.
In the earlier discussion of the mathematics of the diffraction pattern-to-
particle size distribution transformation, we indicated that the transforma-
tion matrix was
W = (G1 G)
T
G .
This is the least-squares solution for N of the matrix equation
GN
I.
That is, N = WI = (GT G)"1 GT I.
A somewhat more sophisticated solution which includes a parameter for
smoothing oscillations in the solution for an ill-conditioned matrix is
given by [8]
W
U(A + YI)
where W is the "inversion matrix" which can be used to multiply a set of
diffraction pattern data to obtain the size distribution of the
particles that produced the diffraction pattern.
T
U is the matrix whose columns are the eigenvectors of the G G matrix
ordered to correspond to the eigenvalues in descending order.
A is the elementary matrix of ordered eigenvalues on the diagonal
and zeros in all other positions.
19
-------
Y is the value of the smoothing parameter.
I is the identity matrix of the same order as A.
T
U is the transpose of U.
The program SMLS calculated the above jSMoothed L,east jJquares solution. First
it allowed the columns of G to be weighted by any distribution function.
This was done in order to test the effect of normalizing the columns of G
such that the diagonal elements of GTG were unity. The program then cal-
culated the eigenvalues and eigenvectors of G^G and then ordered both the
eigenvalues and eigenvectors in the order of decreasing eigenvalues. The
smoothing parameters were usually set to zero in this project. This had
the effect of reducing the solution to the simpler form:
T 1 T
' W = (G1 G) G1.
It was felt that at this stage unsmoothed solutions could be compared
more meaningfully than solutions with various amounts of smoothing. The
results were always tested by multiplying the sum of the columns of G by
the inversion matrix to see if the result was the sum of the size distribu-
tions used to calculate G.
1 ' . ' : " ' '.
The DIST program calculated size distributions by multiplying columns of
diffraction pattern data by an inversion matrix calculated by the SMLS
program. The sets of calculated diffraction pattern data were summed and
then inverted. In this way sets of diffraction patterns from known mono-
disperse samples were read in and summed. The result of the matrix multi-
plication was compared to the sum of the known distribution of the data
entered. This was done primarily in testing computer simulations of
particle sizing, experiments. Sets of measured diffraction pattern data '
were entered one column at a' time and not summed with other data before
being inverted. The results were then compared with the particle size
distribution if it was known. This was done to simplify the comparison
between the inversion results and the relatively poorly known size distributions.
In order to perform a computer simulation of the technique for particle
sizing by diffraction pattern analysis, the first step was the selection of
a set of particle size intervals. This was done by applying one of several
criteria relating the geometry of the photodetector array with the charac-
teristics of the diffraction patterns produced with 0.488 ym wavelength
light and a transform, lens of 59.89 cm focal, length. These particle sizes
were then ,used in conjunction with the computer program labeled, POWER to
generate the data for calculating an inversion matrix.
In the process of calculating an inversion matrix, as was indicated earlier,
the eigenvalues of the covariance matrix were calculated. Based on the
20
-------
paper of Twomey and Howell [7] the range of the eigenvalues was used as a
measure of the success in the selection of the particle size intervals. A
wide range of values (usually covering several decades) indicated a degree
of dependence among the basis set of diffraction patterns associated with
the set of particle sizes. This implied that within the limits of experi-
mental error at least one of the diffraction patterns could be approximated
by a linear combination of the other patterns. When this condition occurred
the matrix inversion yielded an unstable or worthless solution.
The various criteria that have been tried will first be described. Then
the results of the eigenvalue analysis will be presented and compared.
The following criteria were tested and will be described here:
A. First diffraction minimum at successive detector rings.
B. Successive diffraction nulls and antinulls at detector ring
number 32.
C. Metal pinhole sizes available.
D. Equally spaced particle diameters.
E. Equally spaced diffraction peaks.
F. Diffraction pattern intersections at 95% of peak.
G. Diffraction peak at first null of next larger size.
A. First diffraction minimum at successive detector rings.
This criterion was the most overly optimistic in terms of the expected
resolution. The idea was based on the fact that if the position of the
first null were known, then the diameter of the particle could be "calculated
from
mfX
d =
where m = 1.220 for the first null
f = focal length of lens
A = wavelength of light
r = radial distance from optic axis to first null.
Particles could, easily be sized one at a time by this criterion with a
resolution determined primarily by the spacing between adjacent detector
elements. The particle sizes resulting from the application of this criterion
are listed in Table 2 along with the detector ring number and radius at
which the first null occurred.
21
-------
TABLE 2
PARTICLE SIZES FOR WHICH THE FIRST MINIMUM IN 'THE
DIFFRACTION PATTERNS OCCURRED AT SUCCESSIVE DETECTOR RINGS
Particle Diameter
(urn)
i
23.02
24.32
25.75
27.30
29.02
30.90
32.96
35.24
37.78
40.55
43.71
47.27
51.39
55.98
61.25
67.33
74.40
82.68
92.48
104.24
Ring
Number
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
'17
16
15
14
13
Average Radius
(mm)
15.48
14.66
13.85-
13.06
12.29
11.54
10.82
10.12
9.44
8. '78 '
8.15.
7.53
6.94
6.37
1 5.82
" 5 . 30
4.79
4.32
3.86
3.42
22
-------
This set of particle sizes ranged from 23 to 104 ym diameter. Either the
lens focal length or the laser wavelength could be changed to cover the
range from 5 to 100 ym diameter. For instance, a lens of 13.0 cm focal
length would extend the size range down to 5 ym diameter particles. How-
ever, it was decided that all the preliminary work would be performed with
the 59.89 cm focal length Tropel lens at the 0.488 ym wavelength. Under
these conditions the 23 to 104 ym particle range was divided into 20 sub-
intervals with the greatest resolution at the small end of the range.
It should be noted that the particle sizes tabulated corresponded to the
particle size at the center of each subinterval. The upper and lower
limits of each subinterval were not calculated in most cases.
The most interesting data associated with this set of particle sizes are
the set of ordered eigenvalues associated with the covariance matrix G*G of
the diffraction patterns. This set of eigenvalues is listed in Table 3.
Note that the computer format for indicating powers of 10 is used, e.g.
1.74 + 6 = 1.74 x 106.
TABLE 3
ORDERED EIGENVALUES OF COVARIANCE MATRIX
OBTAINED. FROM THE FIRST CRITERION
1 1.74 + 6
2 1.25 + 5
3 2.66+4
4 8.87 + 3
5 3.70 + 3
6 1.79 + 3
7 9.44+2
8 5.28+2
9 2.78+2
10 5.79+1
11 1.15 + 1
12 2.41 - 2
13 9.14-5
14 2.75-5
15 1.16 - 5
16 4.34-6
17 1.97 - 6
18 6.72-7
19 2.09-7
20 6.52 - 8
23
-------
The range of the eigenvalues covered about 14 decades. The implication of
this was that the hoped for resolution of 20 subintervals is much too fine.
The diffraction patterns characteristic of the 20 particle sizes were not
sufficiently independent to yield a meaningful solution. ' ' . ;
B. Successive diffraction pattern nulls arid antiriulls at the-outermost
detector element. . -' ' -'
The outermost ring element of the detector array was labeled ring'#32.
The criterion discussed here is somewhat analogous to the resolution
criterion of Fourier transform spectroscopy and was studied for this reason.
Table 4 lists the particle diameters 'that satisfy this criterion. ' : '
*" * " - . 1 * .. '
TABLE 4 ,
PARTICLE SIZES FOR -WHICH SUCCESSIVE DIFFRACTION. PATTERN.., ; :
MINIMA AND MAXIMA 'OCCURRED AT DETECTOR RING .32 .
Condition at Particle Diameter
Ring #32 '; " (ym)
1st min. '
2nd max.
2nd min.
3rd max.
3rd min. _
4th max. _, . r
4th min.
5th max,.
5th min.
' -"'' ; '- -23,0
- - . _. . 30.8
42.1
50.5
61.1
69.8
80.0
88.9
98.9
Several things should be noted from the data in Table 4. This criterion
yielded a more uniform spacing between subintervals than did the first
criterion. Also, there are now only 9 subintervals spanning the size
range from 22 to 98.8 urn diameter. The particle diameters listed here were
selected from the results of the computer program POWER in which the diffrac-
tion patterns to be measured by the RSI array were simulated.
The results of the eigenvalue analysis are listed in Table 5.
24
-------
TABLE 5
ORDERED EIGENVALUES OF COVARIANCE MATRIX OBTAINED FROM
THE SECOND CRITERION
Order
Number Eigenvalue
1
2
3
4
5
6
7
8
9
1.32 + 6
6.25 + 4
1.15 + 4
3.81 + 3
1.42 + 3
8.02 + 2
4.19 +2
2.17 + 2
7.43 + 1
The eigenvalues in Table 5 range over only about five decades rather than the
14 decades indicated in the results for the first criterion. Even though
this was a significant improvement it still indicated probable difficulty in
obtaining meaningful data from the inversion of experimental data.
C. Metal Pinhole Sizes Available
This third "criterion" was used just as a comparison with the other criteria
that were based on physical reasoning. The set of pinhole diameters is
listed in Table 6. This set of six pinholes spanned the range from 25 to 76
pm diameter. These sizes were comparable to the first seven sizes selected
by the second criterion except for the absence of a pinhole near 42.1 ym
diameter.
TABLE 6
METAL PINHOLE DIAMETERS AND THE ORDERED EIGENVALUES OF
THE COVARIANCE MATRIX
(pm) Ordered Eigenvalues
25
29
52
62
68
76
5.21 + 3
2.41 + 2
1.83 + 1
3.07 + 0
8.47 - 1
1.65 - 1
25
-------
The ordered eigenvalues for this set of particle sizes is also listed in
Table 6. It is seen that the .rate/of decrease in the eigenvalue is comparable
for the two cases. This was not./too surprising because the particle sizes
were similar.
D. Uniformly spaced particle diameters
' .'''i
This criterion, like the one just preceding, was- not based on any'likely
principle. It was selected as a matter of convenience to allow a comparison
of the results of the analysis of the ordered eigenvalues. The selected
particle diameters are listed in-Table 7. The ordered eigenvalues of the
covariance matrix associated with.this set of particle sizes are also listed
in Table 7. The range of the eigenvalues was still too large to be acceptable.
V TABLE -7
1 ' l'
UNIFOEMLY SPACED APERTURE DIAMETERS'''AND THE
ASSOCIATED ORDERED EIGENVALUES
Particle Diameter ... .. Ordered Eigenvalues
(lam) . -..,.. . .
10. 6 " ''"- ' -
20.0
30.0
40.0
50.0
'. 60vO
70.0 -;
80.0
-'-' -.-. 4.27 '+ .5-,.., . ., ' . .'. ....
2.03+4
3v66 +;3 . . ; ;
1.17 + 3
:- ' 4.88. + 2 ...
2.27 + 2 . . . .'
: 8.20 +1
- 3.27 + 0 .
E. Equally Spaced Diffraction Peaks "'''
This criterion was applied by selecting particle diameters that..produced
diffraction pattern maxima at detector rings numbered 32, 29, 26, 23, 20,
17, 14, 11, and 8. The resultant particle sizes are listed in Table-8.
26
-------
TABLE 8
PARTICLE DIAMETERS FOR UNIFORMLY SPACED DIFFRACTION
MAXIMA AND THE ASSOCIATED ORDERED EIGENVALUES
Detector Ring
Number
32
29
26
23
20
17
14
11
8
Avg . Radius
(mm)
15.5
13.1
10.8
8.8
6.9
5.3
3.9
2.6
1.6
Particle Diameter
(ym)
9.5
11.5
14.0
17.5
22.0
29.0
40.0
55.0
94.0
Ordered
Eigenvalues
4.91 + 5
2.34 + 4
2.57 + 3
2.55 + 2
1.36 + 2
1.62 + 1
2.17 - 2
1.21 - 6
3.60 - 12
The results of calculating the eigenvalues associated with these particle
sizes are also shown in Table 8. It is seen that the eigenvalues span about
17 decades, an enormous range. There was about one decade difference between
adjacent eigenvalues for the first six. Then the values dropped off rapidly.
This indicated that a better selection would have consisted of the set of
six particle sizes that produced diffraction maxima at uniformly spaced
detector ring numbers on 32, 27, 22, 17, 12 and 7.
This set would have spanned about the same particle size range (9.5 - 100 ym)
with the advantage of less overlap of the diffraction patterns due to adjacent
particle sizes. This reduction of the attempted particle size resolution
would have yielded a set of eigenvalues that covered a much smaller range
because the diffraction patterns would have been more nearly independent.
One idea that was tried was to normalize the matrix of diffraction patterns
such that the product of G transpose and G had one's along the main diagonal.
In other words the dot product of each diffraction pattern with itself was
set equal to one by multiplying each diffraction pattern by a suitable constant.
This was done in order to make the diffraction patterns of the smaller particles
comparable with those of the larger particles. However, this was accomplished
by sacrificing particle-number resolution for the smaller particles especially.
It was felt that a trade-off of particle-number resolution for particle-size
resolution might be justified in many applications of this technique.
27
-------
The normalization constants, the particle diameters and the ordered eigen-
values are listed in Table 9. The effect" on :the range of the eigenvalues
has been somewhat beneficial indicating that it should be easier to classify
particles according to the sizes'indicated provided the\number density of
smaller particles was greater' than that of larger, particles in order to
compensate for the less intense diffraction pattern produced by small
particles. This sort of distribution is characteristic of many actual samples:
- ... TABLE 9
'RESULTS OF NORMALIZATION OF THE UNIFORMLY SPACED
DIFFRACTION PATTERNS . ,
Normalization ' Particle r.
Constant ' '"" Diameter (ym). .
\ f \ '
3.25
-1.81
1.06
6.21
" 3.74
r2.03
-' 9.89
' , 4.86
1.47
- 1
- 1
- 1
- 2
-2
-.2
- 3
-'3 '
- 3
: 9.5
- - 11,5
14.0
17.5
" ; -'' '22.-0 :- . :. v,
:'- -'*: : -..'. 29:0 :
' ' :' '-' ' 40:.0 .'- . , . .;. :.-
-" " - ' 55;0,- . :.:.:- -.,'. ,
; " - "' '94 .-0' ' --. '*r . -..
'"' ^ ' -:'- -' '- I ...'... ...
Ordered
Eigenvalues
6.03
1.96
6.50
2.49
:7>71
r-2.79,
1.51
s 2 . 00,
: 1.97,
+ 0
+ 0
_ 1 - - --
- 1
- 2 ,- , ,,,. ..;
' ' " - * '.
^ u
-'V'. .'; '";:..;.]
- 10. : :-.:; - -
F. Diffraction pattern intersections at 95% of peak : ,- ^
This criterion was applied by finding the largest particle; diameter; such
that the'maximum value of the- detector output occurred at the detector
ring furthest from the optic axis (ring i#32) . This condition was met by. .
the 9.5 ym diameter particle when the Tropel lens was used at the 488 nm
argon laser line. Note that because of'the detector geometry (area of r.ing
elements increased with ring "number or distance from optic axis) particles.
of any diameter less than 9.15 ym yield measured diffraction patterns of very
similar shape. The predominant change^ occurs in the. intensity of the pattern,
rather than the shape, when the particle diameter is varied ;from 0 to 9..5 ym.
Since the intensity data were used.to calculate the number of particles in a
specific size range and the intensity was1 a very strong ifunction of :particle
size in this range., the number calculated for this range by the-inversion process
was not unique. That is, essentially the'sarae pattern would be produced by
a very large number of subraicron particles as by a much smaller number of
28
-------
larger particles less than 9.5 ym diameter. Therefore the results in the
smallest particle size interval obtained by the inversion process should
be ignored.
The ring number at which the measured diffraction pattern was 95% of the
value at ring #32 was then found. This was ring #29 for this lens, wavelength,
and detector. Next a larger particle diameter was selected such that at
ring #29 its measured diffraction pattern was about 95% of its peak value.
A particle diameter of 14 ym satisfied this criterion. Continuing in this
fashion, the set of seven particle diameters listed in Table 10 was selected.
TABLE 10
RESULTS OF THE SIXTH CRITERION
Particle
Diameter
(ym)
9.5
14
20
28
38
54
84
Ordered
Eigenvalues
3.17 + 5
1.66 + 4
2.18 + 3
3.77 + 2
8.52 + 1
5.42 + 0
1.01 - 3
The ordered eigenvalues associated with the covariance matrix produced by
the diffraction patterns of those particles is also listed. It is noted
that the first six eigenvalues differed by about one order of magnitude
between adjacent eigenvalues. However, the seventh eigenvalue was about
three orders of magnitude smaller than the sixth. This indicated that the
desired particle size resolution was probably too fine and that the size
range from 9.5 to 84 ym diameter can actually be resolved into only
six subintervals rather than the attempted seven.
The normalization procedure described under the preceding criteria was
applied to this case also. The normalization constants and the resulting
ordered eigenvalues are listed in Table 11.
29
-------
.' :': - v, .y- r. TABLE. 11 - . ,,
RESULTS OF NORMALIZATION" ON THE'SIXTH' CRITERION
Particle
: --Diameter
; ' ' ' /" ; (ym)- " '"-
20.0
28.0
38.0
54.0
84.4
''''[' i
" ( . . ' i .' . *
Normalization . ...
. ; > f Constant, ,.
>.- . j ,
' -'-- '.. ' ^3,25
4.61
2.19
1.11
5.07
.".A" .1.8:7,
-*.-!. i : . . '
- 2
- 2
-2
- 3
-r3 i « >.- -.
Ordered-" :: '
r . , Eigenvalues' : ;':' ! -
, ....'.' 4.00
1.24
4.90
1.87
6.73
'.+ho ' : '' :. -'
'+ o" '-1 il'; '; "'
- 1
- 1
- 2
- 2
Also, it should, be noted that the diffraction pattern for the 14.0 vim"
diameter particle was pmitte'd vfrom the basis set. This: was done to remove
the linear dependence indicated by the previous eigenvalue analysis. The
14.0.ym diameter particle was selected for removal oh'. the basis of the
results of forming the covariance matrix of the full set. This set of.
ordered eigenvalues was the most nearly uniform obtained to date. The
implication is that this is probably a valid criterion for selecting
the basis sets of particle sizes.
G. Diffraction peak at first null of next larger size.
1 -(
This criterion is analogous to .Rayleigh's criterion for diffraction-limited
resolution. The basis for its consideration was the fact that the inner
product of diffraction patterns for "adjacent size intervals might be .
minimized by : the fact that at the peak of one pattern the pattern for
the next larger particle -interval was zero. , However, application 'of this
pattern led to* selection of , the following particle sizes: ".- '' :' ' '->
-9.5 ym diam. - peak @ ring #32 . '"'.'"
"23.0 ym diam.1- 1st null '@ ring #32 .. _,
"' ' pk @ ^approximately #19. 4 f
54 ym diam. - 1st null @' 19. 4 -1 --;
,. .- ....-.-. ..pk.@. .#11 ,,3.
129 ym 'diam. --1st null @ #11.3' :;.,: . : ,
Even if this set of basis sizes yielded a favorable set of eigenvalues
we have already achieved better particle size resolution with the set of
six paxticle sizes derived in the preceding criterion. A set of only
30.
-------
four particle sizes that spans a larger range is not as useful. The set
of eigenvalues for this set of particle sizes was therefore not analyzed.
A set of pinhole apertures was purchased from Optimation, Inc. These
apertures are centered in 9.5 mm diameter 300 series stainless steel discs.
Table 12 lists the nominal and measured sizes of the apertures purchased.
A scanning electron microscope was used to measure the diameters of the
apertures 35 ym and smaller. These measurements were made from photographs
of the magnified images produced by the electron microscope at a known magni-
fication. The magnification was measured by comparing the image size of a
known grid pattern with the dimensions of the original. The 1 and 2.5 ym
diameter apertures were not measured because they were not needed and because
the difficulty of finding the aperture increased as the diameter decreased.
TABLE 12
SPECIFIED AND MEASURED METAL PINHOLE DIAMETERS
Nominal Diameter (van) Measured Diameter
1
2.5
5
7.5
10
12.5
15
17.5
25
35
50
60
70
80
90
100
200
300
400
500
600
700
800
900
1,000
_
-
5.2
7.6
9.8
12
14
18
25
29
52
62
68
76
84
104
184
295
377
507
598
684
769
916
1,000
The apertures 50 ym and larger in diameter were measured with a traveling
microscope. These sizes were large enough that the measurement error was
only a small part of the measured diameters.
31
-------
The majority of the computer simulation experiments had been done assuming
ideal conditions and precise data. The diffraction patterns produced by,
the metal pinhole.apertures were measured primarily to obtain experience
with the apparatus under the simplest conditions'and. to obtain data that
contained actual measurement error.
The irradiance of the beam.had to be high in order to obtain a measurable
diffraction pattern particularly for the smallest apertures." This was
achieved.-by .using the beam, from the Isise'r as was,'Without filtering and-
expansion. .The illuminated"aperture was located in the entrance plane
of the Tropel Fourier transform lens. ' ' ' ; ;. .
' -' . , c . -... _ . ~ * * - -'.'.
The diffraction patterns of the nominally 10y 25, 35, 50, '60, 70,; 80 and
90 ym diameter apertures were measured by manually selecting the ring-
elements in the RSI array and recording the signal levels indicated.
The results of calculating the aperture sizes from the measured diffraction
patterns are indicated in Tab'le 13. These calculation's were based on
the relation between the radii of the diffraction minima and the laser
wavelength, lens focal length, and the aperture diameter. This relation
is represented by the expression:
. mfX
Q = , '
where d is the aperture diameter
m = 1.220 for 1st minima -;
2.233 for 2nd minima
3.238 for 3rd minima
f = lens focal length
A = wavelength of light used
r = radial distance from the optic axis to the diffraction minima,
measured in the plane of the diffraction pattern'.
The focal length was that of the Tropel Fourier transform lens, 59.89 cm.
The wavelength was 0.488 um for the light from the argon-ion laser.
The blanks in the data Table indicate that those diffraction minima were
located beyond the RSI detector array and were not measured. The average
values were calculated from the one, two or three diameters indicated and
were rounded for presentation in the Table. These averages should be
compared with the microscope measurements indicated in the next column. The
agreement is good in general and could be improved with greater effort
if that were required. The fact that the nominally 70 and 80 ym diameter
apertures yielded the same results is due in part to the coarseness of the
diffraction pattern sampling. This was determined by the geometry of the
detector array.
32
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TABLE 13
METAL PINHOLE DIAMETERS MEASURED BY
LOCATION OF DIFFRACTION MINIMA
Nominal
Aperture
Diameter
(uai)
25
35
50
60
70
80
90
Diameter Calculated from Position of
1st
Minimum
24.3
29.0
43.8
56.0
74.4
74.4
82.6
2nd
Minimum
_
-
44.5
56.5
74.3
74.3
80.1
3rd
Minimum
_
-
-
-
72.5
72.5
82.0
Average
24.3
29.0
44.2
56.3
73.7
73.7
81.6
Microscope
Measurement
25
29
52
62
68
76
84
Four sets of measured diffraction patterns from the set of metal pinhole
apertures in the nominal size range from 10 through 90 ym diameter were
averaged together before the inversion matrix and eigenvalues were calculated.
The sizes of the apertures and the ordered eigenvalues are indicated in
Table 14. It should be remembered that each eigenvalue is characteristic
of the whole set of diffraction patterns. So, even though the eigenvalues
are listed next to the aperture diameters it cannot be said that any one
eigenvalue is associated with any one aperture size. In other words, if
only one aperture size is changed the values of all the eigenvalues will
be changed, in general. The only direct association between the aperture
diameters and the eigenvalues is that the number of each is the same.
From Table 14 it is seen that there was a difference of almost two decades
between the largest and next largest eigenvalues. The difference between
adjacent eigenvalues then dropped off to less than one decade until the last
two values. Here the change increased to about one decade between adjacent
values. These results should be compared with those described earlier for
the computer simulation testing of the various criteria applied to the
selection of the particle size intervals. The normalization procedure
also described above was not applied when these results were obtained.
Table 15 shows the results of normalizing the input data so that the inner
product of each diffraction pattern with itself was unity. This improved
the uniformity of the eigenvalues, but it made the apparent particle-number
resolution worse, especially for the smaller particles. This is the same
trade-off described earlier when the normalization procedure was introduced.
33
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TABLE 14
EXPERIMENTAL RESULTS FROM METALr PINHOLES.
Pinhole Diameter
Nominal
(ym)
- 10
25
35
50
60
70
80
90
Measured
(ym)
9.8
25
29
52
62
68
76
84
,.. Ordered ,
Eigenvalues
"2.44 +'3
7.63 -+ 1
. 1.70 + 1 ..
1.26 +,-1
. 5.01 +.0 .
2.44 +. 0
,,2.65 -U
3.54 -,,2,
TABLE 15
RESULTS FROM 'NORMALIZED EXPERIMENTAL DATA
'Ordered Eigenvalues:,
Before .- , After '
Normalization Normalization
2.44 + 3
7.63 +1
1.70 +1
1.26 + 1 '
5.01 + 0
2.44 + 0
2.65 - 1
3.54 - 2
5.83
1.37
5.41
1.86
3.75
. 2 . 01
1.69
4.28
H- 0 .
+ 0
- 1 ..
r 1
- 2 ,
?
- 2
- 3
' \
The smallest eigenvalue for the normalized case was then eliminated by
removing the diffraction data for one of the ap.erture sizes. The selection
of the data to be removed was again based on an examination of the covariance
matrix G^G. This indicated that the data for the 80 ym aperture were more
nearly linear combinations of the data for the other aperture sizes than
any of the other data. Therefore, the diffraction data for the 80 ym aperture
were removed and the eigenvalues of the normalized data were calculated again.
34
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The results are indicated in Table 16 along with the preceding results.
These eigenvalues indicate that the selection of basis sizes is good in-
asmuch as the range of the eigenvalues is minimized and the size interval
from 10 to 90 ym diameter is resolved into the largest number of size
intervals compatible with minimizing the range of eigenvalues.
TABLE 16
RESULTS OF ELIMINATION OF DIFFRACTION PATTERN DATA
FOR ONE PINHOLE
Ordered Eigenvalues of Normalized Data
Including 80 pm Data Excluding 80 ym Data
Aperture
Diameter
(ym)
10
25
35
50
60
70
80
90
Ordered
Eigenvalues
5.83 +
1.37 +
5.41 -
1.86 -
3.75 -
2.01 -
1.69 -
4.28 -
0
0
1
1
2
2
2
3
Aperture
Diameter
(ym)
10
25
35
50
60
70
90
Ordered
' Eigenvalues
4.98
1.21
5.25
1.77
3.64
2.00
1.51
+ 0
+ 0
- 1
- 1
- 2
- 2
- 2
In Table 17 the above results are compared with the computer-simulation
results described above for the selection of particle size intervals based
on the criterion that the diffraction patterns of adjacent particle sizes
intersect at 95% of their peak value. This table represents a direct
comparison of the results obtained under ideal conditions by computer simu-
lation with the results obtained by experimental measurements. Both results
are compatible and indicate that the diffraction pattern technique will
work theoretically and experimentally for sizing apertures. These results
also indicate that the optimum condition under which the technique will
work is that the size distribution of the sample to be measured should be
similar to the graph of the normalization constants vs. aperture diameter.
This will assure optimum size- and number-resolution.
The photographic samples consisted of transparencies with black circular
spots located randomly on the transparencies. The spots were made by
photographing monochromatic light passing through small holes drilled
35
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.TABLE 17
COMPARISON; OF EXPERIMENTAL AND
COMPUTER-SIMULATION RESULTS
Computer Simulation
Particle
Diameters Normalization Ordered
(ym)
9.5
20.0
28.0
38.0
54.0
84.4
Constants
3
4
2
1
5
1
.25 -
.61 -
.19 -
.11 -
.07 -
.,87 -
1
2
2
2
3
3
Eigenvalues
4.00'+
l.;24 +
"4.! 90 -
1,87'..-
6.73 -
1.67 -
0
0
l'-!:
1
2
2 ' :
Measured Pinhole Data
Aperture
Diameter Normalization Ordered
(ym)
10
25
35-
50
60
70
90
Constants
3
3
. - 2
.1,
5
3
' ' '3
.7.7 +.
.52 -
.32'-.
.08 -
.35 -
.83''-
.56'-
0
1
± '.
Is
2 "
2
2
Eigenvalues
3.98
1.21
5.25
.1.77
3; 64
2.00
1.51
+ 0
+ 0
- 1
- 1
- 2
- 2
- 2:
through a thin sheet of metal. The monochromatic light was obtained by
using a green filter between an intense light source and the sheet
metal original. .The distance between the source and the sheet metal was
large enough to. insure uniform illumination of all the circular holes.
The distance from the original to the camera was adjusted to provide the
desired spot size on the transparency. Thus, only one original was needed.
to produce a number of monodisperse samples of any.-size in the range from,
17 to 100 ym. A significant reduction in spot size wpuld have required
more effort than could be justified for this initial investigation. Each
time more holes were drilled through the sheet metal plate, a set of
transparencies was made with the camera at various distances to cover the
size range of interest. -'After development the spots were checked under
a microscope for size and quality before more holes were drilled in the
original. .
A few polydisperse samples were produced by drilling lar.ger holes .through
the original after the desired monodisperse samples had been obtained. Also,
both mono- and polydisperse samples of transparent apertures in a black
background were produced by a similar technique. Stick-on opaque circular
paper discs of different sizes were obtained and placed on sheets of clear
Mylar to serve as originals. The negative transparencies obtained from
these originals then consisted of arrays of transparent circular spots
on a black background. .
Many attempts were made to measure the diffraction patterns produced by
the transparency samples consisting of black spots on a clear background.
However, the influence of the undiffracted incident light predominated
36
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to such an extent that no data were obtained for calculating an inversion
matrix and the associated eigenvalues. The basic difficulty was that a
large portion of the incident light was not diffracted by the simulated
particles and so was focused at the center of the otherwise weak diffraction
pattern. Reflections of even small percentages of this undiffracted
light predominated over the useful signal at the detector. The largest
reflection occurred at the detector itself, because not all of the incident
energy is absorbed by the detector material. Also, this detector array
was fabricated by a technique that required a window between the incident
light and the detector elements. Reflections from this window produced
a strong halo of light incident on the detector elements. The correction
procedure suggested by the detector manufacturer was found to be of no help.
Their more recent detector arrays are fabricated without windows, which
eliminates the problem of halos.
Other techniques for eliminating the undesired light were tried. The most
successful was the use of an optical fiber at the focal po.int of the
undiffracted light. However, even this method was only partly successful.
The fiber collected most of the undesired light and directed it away from
the detector, but there was still too much undesired light incident on the
detector array. A part of this was found to be generated by reflections
from the supposedly antireflection-coated Fourier transform lens. Tropel
claimed that over 99% of the light incident on the lens should be transmitted.
Our measurements indicated that only about 84% of the incident light at the
design wavelength of 0.488 ym was transmitted and at least a large part
of the other 16% was reflected at lens-air interfaces inside the multi-
element transform lens. The lens was returned to Tropel and found by them
to be defective. At any rate, our initial attempts to measure the diffrac-
tion patterns of low number-density opaque particles on a transparent
background have not yielded data suitable for calculating an inversion
matrix. The simpler case of transparent circular apertures on an opaque
background did yield data which were used to calculate an inversion matrix
and the related eigenvalues. The resulting eigenvalues are indicated in
Table 18. The first column indicates the diameter of the pinholes in
the monodisperse samples. The second column indicates the number of
pinholes of each size that produced the measured diffraction patterns.
The third column consists of the ordered- eigenvalues calculated from the
data before the final normalization. These data were already partially
normalized by the use of larger numbers of pinholes for smaller pinhole
diameters. The most interesting feature in this Table is the comparison
of the results before and after normalization. It is seen that before
normalization the smallest eigenvalue was three decades smaller than the
next larger eigenvalue. After normalization all six eigenvalues occurred
within a range of about two decades. This indicated again that it should
be possible to trade off particle-number resolution at the small end of
the particle size range in order to improve the particle-size resolution.
37
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TABLE 18
EXPERIMENTAL RESULTS FROM PHOTOGRAPHIC SAMPLES
Results Before Normalization After Normalization
Aperture ' ; / ; . . . ,
Diameter
(ym)
14
20
28
38
54
84
Number
114
50
24
12
6
2
Ordered
Eigenvalues
3.56 + 2 ,
5.98 + 1
6.83 + 0
4.51 + 0
2.93 + 0
2.86-3
Ordered
Eigenvalues
. 4.17 + 0 '"'
'"'1.28 + 0
3.86 - 1
1.00 - 1
5.20 - 2
. 1.37 - 2
Normalization
Constants
3.26"
0.362
0.121
' 0.120
0.148
0.065
Only one test of the inversion matrix technique for counting and sizing
particles was completed with experimental data. 'The results to be described
should be taken more as an example of the procedure used rather than;as
an example of the accuracy of the technique. With more experience in apply-
ing the technique, and improvements in the optical system,, the accuracy
should be improved. ;
The data from the monodisp'erse pinhole arrays described above were used
to calculate an inversion matrix. However,' the diffraction data for the
smallest particles (14 ym diameter) were removed because of the unusually
large value required for normalization (see Table 18). The inversion matrix
was calculated and used as a multiplier to invert the diffraction pattern
produced by a polydisperse pinhole array. The results of the matrix multi-
plication and the conversions required to account for the normalization
are shown in Table 19. The lefthand column indicates the aperture diameter
at the center of each size interval of the inversion matrix. The second
column is the result of multiplying the measured diffraction pattern data
by the previously calculated inversion matrix. The third and fourth columns
indicate the number of pinholes in the monodisperse samples and the con-
stants used to normalize the monodisperse data for calculating the inver-
sion matrix. The fifth column, the number of pinholes calculated, is
the product of the second, third, and fourth columns. As a comparison,
the known data for the polydisperse sample are indicated in the two right-most
columns.
The first thing to be noted is that the matrix inversion technique should
work best for samples in which the particle number density is large and for
which the size distribution is a continuous function that is parallel to
38
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TABLE 19
APPLICATION OF EXPERIMENTALLY DERIVED INVERSION
MATRIX TO MEASURED DIFFRACTION PATTERN DATA
Monodisperse Data
Aperture Inversion Matrix Number Normali- Number of Polydispersion
Diameter x of zation Pinholes No.of Diameter
(ym) Diffraction Pattern Pinholes Constant Calculated Pinholes (pm)
20
28
38
54
84
0.819
1.61
3.02
4.67
2.19
50
24
12
6
2
0.362
0.121
0.120
0.148
0.065
14.8
4.68
4.35
4.15
0.28
10
5
2
1
14.6
36.6
57.1
76.4
39
-------
.a graph of the product of the normalization .constants multiplied by the
number of apertures in the monodisperse samples. These conditions were
not met in this test, which may account for. some of the discrepancy between
the "number of pinholes calculated" and the "number of pinholes" known
to be in the polydispersion. ''"'' " l
Another point that should be noted is the fact that the "number of pinholes
calculated" was always positive. For several analogous computer simulation -
tests this was not true, particularly when the eigenvalues related to the
inversion matrix covered a wide range. As indicated earlier, the'eigenvalues
associated with this experimentally derived inversion matrix lay within a
relatively small range. Remembering that this, was the first experimental test
of the matrix inversion technique, the results were quite satisfactory.
At the end of the experimental work on this Grant a unique detector array
became available for a short time. This was the fiber-optic/photodiode
array constructed by Georgia Tech for NASA [9] and described earlier. The
minicomputer associated with this array was able to store the background
signal levels and subtract these from signal-plus-background data to yield
the corrected diffraction pattern in real-time. This greatly facilitated
the examination of diffraction patterns from a variety of samples.
Even though an inversion matrix was not derived for this detector array
the size of dense monodisperse samples was calculated from the measured
position of the first null in the diffraction patterns. The samples studied
consisted of those described earlier as well as an assortment of materials
deposited on microscope slides. The spray from an aerosol can will also produce
a measurable diffraction pattern. However, this pattern was characteristic
of a polydispersion with a broad size distribution. This would have re-
quired an inversion matrix for analysis of the size distribution.
40
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46
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
2.
3. RECIPIENT'S ACCESSIOWNO.
4. TITLE AND SUBTITLE
A DIFFRACTION TECHNIQUE TO MEASURE SIZE
DISTRIBUTION OF LARGE AIRBORNE PARTICLES
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
A. McSweeney
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Georgia Institute of Technology
Engineering Experiment Station
Atlanta, GA 30332.
10. PROGRAM ELEMENT NO.
110302
11. CONTRACT/GRANT NO.
R-802214
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final Report 12/73 - 12/75
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The purpose of this project is to test and demonstrate a coherent optical .
diffraction technique for measuring the size distribution of large particles. This
technique is based on the generation of a transformation matrix which is used to
relate the measured diffraction patterns to the size distribution of the samples
that produced the patterns.
Four types of samples are considered: 1) pinholes in opaque discs, 2) photo-
graphic transparencies with opaque circular spots, 3) particles deposited on micro-
scope slides, and 4) aerosols. Computer simulations are performed to assess the
accuracy and resolution of the techniques.
Although good results are obtained for pinholes in opaque discs, experimental
difficulties limit the precision of this techniques applied to particles in a trans-
parent medium. Improvements based on a reduction of system noise and an increase
in detector sensitivity are discussed and applied to the requirements on number
density and size range of particles in a transparent medium.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b. IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
* Air pollution
* Aerosols
* Diffraction
* Particle size distribution
.Lasers
13B
07D
20F
20E
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
21. NO. OF PAGES
57
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
47
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