The application of cross-beam Laser Doppler Velocimeter (LDV) for sizing small particles was investigated from September 1973 to August 1974. Theoretical results were obtained by analyzing the scattering characteristics of small particles in a cross-beam LDV system. Theoretical calculations based on scalar diffraction theory and Mie scattering theory were performed. Experimental results were also obtained to compare with theoretical predictions. It is concluded that the forward scattering characteristics of small particles in a cross-beam LDV system can be used for particle sizing

Chu, W. P.

English

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(NASA-CR-140446) COHERENT OPTICAL N74-34192
INSTRUMENTATION FOR MEASUREMENTS OF
PARTICLE PARAMETERS Final Technical
Report (Old Dominion Univ. Research Unclas
Foundation) 18 p HC $4.00 CSCL 20H _ G3/24 50412 _
COHERENT OPTICAL INSTRUMENTATION FOR MEASUREMENTS
OF PARTICLE PARAMETERS
FINAL TECHNICAL REPORT
By
William P. Chu
Prepared for the
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Langley Research Center
Hampton, Virginia 23665
Under Q
NASA Research Grant C
NGR 47-003-088 -
Submitted by the /,
Old Dominion University Research Foundation ,
* P.O. Box 6173
Norfolk, Virginia 23508
October 1974
https://ntrs.nasa.gov/search.jsp?R=19740026079 2020-03-23T04:52:10+00:00Z
COHERENT OPTICAL INSTRUMENTATION FOR MEASUREMENTS
OF PARTICLE PARAMETERS
By
William P. Chul
SUMMARY
The objective of this research effort was to investigate
the application of cross-beam laser doppler velocimeter (LDV)
for sizing small particles. This report is a brief summary
of research findings during the period from September 1973 to
August 1974.
This report contains theoretical results obtained from
analyzing the scattering characteristics of small particles in
a cross-beam LDV system. Theoretical calculations based on
scalar diffraction theory and Mie scattering theory have been
performed. Experimental results are also obtained to compare
with theoretical predictions. It is concluded that the forward
scattering characteristics of small particles in a cross-beam
LDV system can be used for particle sizing.
INTRODUCTION
Light scattering properties of small particles have been
used extensively as a diagnostic tool for the measurements of
particle size. In this report, we will investigate the scattering
properties of small particles by two coherent cross-beams. The
1 Research Assistant Professor in Physics, School of Sciences,
Old Dominion University, Norfolk, Virginia 23508.
problem is closely related to defining the characteristics of
the doppler signals generated from a cross-beam LDV system.
Previous investigation by Farmer [1] indicated that the modu-
lation of the doppler signals is a unique function for different
particle shapes and depends only on one parameter given by
the ratio of particle geometric dimension to the fringe spacing
formed in the probe volume of the LDV. In this report, we will
show that Farmer's result is corrected only under some res-
trictive conditions. Only by measuring the forward scattering
with a sufficiently large aperture system can particle size
information then be obtained.
THEORETICAL ANALYSIS
Two approaches to analyzing the scattering of a spherical
particle by two coherent cross-beams have been performed. The
first approach is based on the scalar diffraction theory, which
is an approximate theory valid for particle dimensions much
larger than-a wavelength. The second approach is based on an
exact Mie scattering calculation which takes into account effects
of index of refraction of the particles. The theoretical calcu-
lations will result in a doppler signal with the following
general form
I(t) = Const (1 + V cos 2 wfdt) (1)
where I(t) is the doppler signal intensity, V is the visi-
bility (or the modulation), and fd is the doppler frequency.
What we are interested in is the dependence of the visibility
function V on parameters such as particle size, index of
refraction, cross-beam angle, and the size of the receiving
aperture.
2
1. Scalar Diffraction Theory
Consider in the input plane we have a spherical particle
of radius a being illuminated by two unit amplitude plane
waves with cross-beam angle of 2y. The input field Uinput(X,y)
can then be written as
Uinput(x,y) = (eik sin yx + e-ik sin yx) (1 - P(x - vt,y) (2)
where P(x,y) is the transmission function of the particle,
and v is the velocity assuming along the x-axis. For a
spherical particle of radius a, P(x,y) = Circ (r/a) = 1
for r<a, zero otherwise. The diffracted wave at the far-field
plane (z = zf) will then be given by the following Fourier
transform expression [2],
Udiff(xfyf) = (Const phase)/Xzf
Sf Uinput(x,y) e-(2fi/Xzf)(xfx + y fy) dxdy (3)
Let us consider the integrated total scattering due to the
particle in the forward half plane. As is well known in diff-
raction theory, most of the diffracted energy from a scattering
object is contained within the first forward lobe of the
diffraction pattern. This forward lobe extends to an angle
approximately equal to A/D, where D is the dimension of the
object. Thus the total scattering signal should be approximately
equal to the signal obtained by collecting the forward scattering
within a cone of full angle greater than 2(y + A/D).
The total scattering intensity I(t) can be calculated
from equation (3). Using Parseval's theorem, we obtained the
following result:
I(t) = 2ff dxdy (1 + cos 2xk sin y) P2 (x - vt,y) (4)
3
Equation (4) is a very interesting result. It illustrates
the.fact that the integrated scattering signal is equivalent to
the signal obtained by scanning a set of sinusoidal fringes of
spacing 6 = X/2 sin y with a small aperture of a shape the
same as the particle geometrical cross-section. For a spherical
particle of radius a, we have P2 (x,y) = Circ (r/a). After
some calculations, we then arrive at the visibility function of
the doppler signal as
Vspherical particle = 2J, (2a/6)/(2a/6) (5)
where J, is the first order Bessel function. Figure 1 shows
the visibility function versus the parameter 2a/6. These
results are similar to Farmer's results; however, they are
corrected only for the total scattering signals.
Next we will consider the case of collecting the scattering
along the forward direction with a finite size aperture. The
resulting signals in this case can be calculated by integrating
the scattering intensity across the aperture area. These can
be done directly from equation (3), and the results are summarized
in Figure 2. It shows the visibility function for spherical
particles versus the size of the aperture for different values
of 2a/6. Notice that as the aperture sustains a cone of full
angle greater than 2(y + X/2a), the values of V approach
the limiting values as given by equation (5).
2. Mie Scattering Theory
The problem here is to solve .the'Maxwell's equations under
the appropriate boundary conditions. For single beam scattering,
the calculation is the well documented Mie calculation based on
the multipole expansion of the electro-magnetic field. For the
present case, we have two linearly polarized incident plane waves
of the same polarization with cross-beam angle of 2y. Assuming
a coordinate system symmetric to the two incident plane waves,
4
each plane wave can then be expanded into the following form [3]
6 ik-xE = ie
= 47 i j (kr) Ym (' YZm(y , 7/2) (6)
t m
where YZm is the spherical harmonic, and j is the spherical
Bessel function of order Z. Using equation (6) as the expansion
for the incident plane waves, we can then construct the scattering
fields and apply the appropriate boundary conditions for a
spherical scattering center with radius a and index of refrac-
tion n = n - iO, where n is the real index and -8 is
r r
the imaginary index. The complete calculation is sufficiently
lengthy and will not be presented here. We will try to summarize
some of the results.
For total scattering, where we integrated over 4n steradian,
the visibility function has the following interesting expression
V (2t + 1) a[a2 re (cos 2y) + bt 2 TZ (cos 2y)]V = Z(t + 1)
Z (2 g + 1)(la 12 + lb 12) (7)
where aZ and bt are the standard Mie A and B coefficients,
nr and Tt are similarly the Mie angular function. Notice the
similarity of equation (7) to the single beam Mie scattering field
expressions [4].
Numerical solutions to equation (7) have been obtained and
some of the results are shown in Figures 3 to 6. Figures 3 and
4 are results obtained for a cross-beam angle of 0.2 radian,
whereas Figures 5 and 6 are for a cross-beam angle of 0.1 radian.
The visibility functions show strong oscillation for no absorp-
tion. The oscillation decreases rapidly as absorption is intro-
duced or as the cross-beam angle is decreased. The general
5
shape of the visibility functions is very similar to Figure 1
with results obtained from scalar diffraction theory.
Calculations of the visibility function for a detector
receiving a small cone angle of the forward scattering have
also been done. The results for varying particle size with a
full cone angle of 0.5 radian are shown in Figure 7. The results
are similar to those shown in Figure 3.
Figure 8 shows the change of the visibility function for
a fixed diameter particle with different receiver's cone angles
in both the forward and backward directions. For forward
scattering, the similarity with results from scalar diffraction
calculations is obvious. However, in the back scattering case,
no information on sizing can be obtained from measurements of
the visibility function.
EXPERIMENTAL RESULTS
A standard cross-beam LDV system has been constructed to
measure the visibility function for spherical particles of
different diameters. The glass particles are deposited on a
good quality AR-coated glass plate. The glass plate with diff-
erent particles on it was then slowly scanned across the probe
volume of the LDV system. Measurements of the visibility functions
have been done with this LDV system for different fringe spacings,
particles with different sizes, and different detector's receiving
cone angles. All experiments were performed with detector's
optics in the forward scattering direction. Figure 9 shows the
resulting values of visibility versus 2a/6 for a receiver's
full cone angle of 0.64 radian. Equation (5) is also shown for
comparison. Figure 10 shows the effect of varying the receiving
cone angle on the value of the visibility function for a fixed
size particle. Theoretical results identical to those shown in
Figure 8 are also shown for comparison.
6
CONCLUSION
Theoretical calculations based on both scalar diffraction
theory and the exact Mie scattering theory show that the forward
scattering characteristics of a small particle in a cross-beam
LDV system can be used for particle sizing. The exact Mie cal-
culations indicate that effects due to index of refraction are
small provided the cross-beam angle is small. Both theories
predict similar scattering characteristics with varying receiver's
aperture size.
The materials in this report are being prepared for publi-
cation in technical journals. The journal articles will provide
a detailed description of the calculations involved in deriving
most of the results presented in this report.
REFERENCES
1. Farmer, W.M.: Applied Optics, 11, 2603 (1972).
2. Goodman, J.W.: Introduction to Fourier Optics, McGraw-Hill
New York, 1968, p. 61.
3. Jackson, W.D.: Classical Electrodynamics, John Wiley & Sons,
Inc., New York, 1962, p. 567.
4. Kerker, M.: The Scattering of Light and Other Electromagnetic
Radiation, Academic Press, Inc., New York, p. 47.
7
1.0
.8
.8~q ROJJOUCII
.6
.5
s4-
I *I II4I
-J
-" ° i I i t ll i ti l I 1 11 1 li l i t
S03
-4E
-O .R
-.
-. 3
Figure 1. Visibility function versus the parameter
R = 2a/6 for spherical particles of
radius a, and 6 is the fringe
spacing.
x 10-1
10
a
a) 2a/8 = 0.2
- b) 1.o
, C) 1.4
d) 2.0
7 \e) 3.0
6e
5-
SC
-
-2
3
Figure 2. The visibility functions for spherical
particle versus the detector's aperture
size for five different values of 2a/6.
U = 2arf/lz where r is the radius
of the detector's aperture.
1.nr =1.33.3.
. , .- -=0.0,
7 _-0.1
0.5
"6 1.0
.6
-
2
e3-
. -.
Figure 3. Visibility functions calculated from Mie
scattering theory versus parameter 2a/6
for real index n = 1.33 and four diff-
erent imaginary ifidices. The cross-beam
angle 2y = 0.2 radian.
10
,00
S. =1nr=1.5
S=0.0
.8 -0.1
0.5
.7 1.0
= 0.1.
.6
-J
-. 3-
S.1m
• 4 4
Figure 4. Visibility functions calculated from Mie
scattering theory versus parameter 2a/6.
for real index n = 1.5 and four diff-
erent imaginary indices. The cross-beam
angle 2y = 0.2-radian.
t1
1.0 nr = 1.3 3
• &- = 0.0
. .. .......... 0 .1
. - 0.5
1.0
.7 v = 0.05
.6 -
.5-
03-"-- .3 . ...
I
r,, .2 - II II
- .3
Figure 5. Visibility functions calculated from Mie
scattering theory versus parameter 2a/6
Sfor real index n = 1.33 and-four diff-
erent imaginary iAdices. The cross-beam
angle 2y = 0.1 radian.
-- 12
nr=1.5
p= 0.0
8 0.1
0.5
7 1.0
- = 0.05
.6-
-3J"
03
0l 1 2 3
-. 1 R
-. 2-
-. 3-
Figure 6. Visibility functions calculated.from Mie
scatteriog theory versus parameter 2a/6
for real index nr = 1.5 and four diff-
erent imaginary indices. The cross-beam
angle 2y = 0.1 radian.
13
1.0
.0 n 1.33
.= 0.1
.0-- o= 0.5
.
0.5
CI
L .
.6.9-J*
Figure.7. The visibility function calculated from
Mie theory for a detector with full cone
angle of 0.5 radian in the forward direc-
tion versus the parameter R = 2a/6.
The real index is 1.33 and imaginary
index is zero.
.14
* -.1
- .3
Figure 7. Th visibility function calculated from
Mie theory for a detector with full cone
angle of 0.5 radian in the forward direc-
tion versus the parameter R = 2a/<S.
The real index is 1. 33 and imaginary
index is zero.
14
1.0
a) FORWARD
b) BACKWARD
a
.8 -a = 12.0
---- 0.1
.7 . n=1.5
.6 b
.51
'-
*b
6-4
.__
-. 2m
>.
Od .2 .3 . .
-. 1k _- _dian
Figure 8. The visibility function versus collecting
optics half cone angle in radian for both
the forward and back scattering directions
with a spherical particle of ka = 12.0
and index of refraction n = 1.5. The
cross-beam angle in this case is 2y = 0.2.
15
1..
0 0
o0
0
0
100
0
100
o- 0
.O
.. O
0
__" IIII
o I R J 4 5 6 7
R
Figure 9. Experimental results from the measurements
of the absolute visibility for .different
values of R = 2a/6. Equation (5) is shown 16
for comparison.
n =1.5
.8 Y= .1
,g _ 7=.1
2a =5.8 pm
.7- A=.63pm
a = 27ra/A
.6 = 28
.5
3-
cr* . .
S.1
.I .2 .3 .q .
-.1 RADIAN
- .2 FORWARD HALF ANGLE COLLECTION
- .3
Figure 10. Experimental results on the change of
visibility as a function of receiving
optics half cone angle in radian.
17


Coherent optical instrumentation for measurements of particle parameters

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