Experimental measurements of polystyrene latex spheres in water corroborate the predicted result ... The heterodyne detection of light scattered from particles.
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Eversoleet al.
Frequency-modulation detection of particle diffusion from heterodyne quasi-elastic light scattering J. D. Eversole, A. D. Kersey, A. Dandridge, and R. G. Priest Optical Sciences Division, Code 6500,Naval Research Laboratory, Washington, D.C.20375 Received August 12, 1986; accepted March 5, 1987
The BrowniarVdiffusivemotion of micrometer-sized, suspended particles has been observed by using-heterodyne light scattering with frequency-modulation (FM) detection and signal processing. The FM output is linear in the coherent velocity of the particles, with the random diffusive velocity component giving rise to background noise in
the FM output. Experimental measurements of polystyrene latex spheres in water corroborate the predicted result for the signal-to-noise ratio. The noise component of the FM output was found to be independent of the particle number and, within the instrumental bandwidth, to be flat. Good quantitative agreement between the predicted and measured noise backgrounds was obtained for particle diameters from 0.5 to 50 gm.
INTRODUCTION The heterodyne detection of light scattered from particles undergoing Brownian/diffusive motion through a fluid has received much theoretical attention. 1 -3 Basic theoretical results have been experimentally verified for particles in both liquids 4 and gases5 and for flowing6 ' 7 as well as station-
ary systems, and over the past two decades the technique has been extended to numerous applications. Beyond singleparticle point measurements of velocity, the purpose of many applications is to provide particle size information.'14 The work reported here is concerned primarily with the motion of a population of uniformly sized particles. Most of the previous studies are restricted to the consideration of processing techniques that are essentially linear in the electronic signal from the photodetector.
However, there is one
nonlinear signal-processing technique that is a natural candidate for application to heterodyne signals, namely, frequency-modulation (FM) discrimination. In this paper we describe a theoretical and experimental investigation of particle light scattering with FM discrimination. If the frequency difference between the sample and reference beams is larger than the bandwidth of interest, the resulting interference beat signal from a square-law detector may be re-
garded as a FM carrier. Consequently, conventional FM discrimination electronics may be thought to offer some advantages in processing signals from such heterodyne light
scattering experiments. In the case of light scattering from a single particle, a straightforward analysis shows that the output of a FM discriminator is proportional to the particle velocity. The FM discriminator offers the advantage of rejecting amplitude
the limit of a large number of particles. The subsequent section details experimental measurements on polystyrene latex spheres suspended in water using a He-Ne laser light source and a commercially available FM discriminator.
A
comparison of the predicted and measured diffusive noise backgrounds is presented in the final section, in which it is shown that the model calculations are basically substantiated.
THEORY The simplest case of interest is that of N identical particles illuminated by a laser beam of uniform cross section. If the reference laser beam is shifted in frequency from the probe beam by w#,the heterodyne signal from the photodiode is
S(t) = A R
[ exp(iqxj(t) - ist)]
-
(1)
The constant A is proportional to the geometric mean of the optical power in the signal and reference arms and includes effects of particle scattering cross section, detector efficiency,and collection solid angle. The magnitude of the scattering wave vector is q, and x is the projection of the position of
the jth particle onto the direction of q. Time is denoted by t. The particle position can be thought of as a superposition of a coherent part and an incoherent part. The coherent part is the same for all the particles and arises from a velocity
superimposed upon the motion of all the particles. This
modulation, which may arise from ex-
velocity may be due, for example, to the passage of a long-
traneous effects. The case of light scattered from many
wavelength sound wave through the fluid. The incoherent part is due to the random diffusive motion of the particles. This decomposition of x (the particle label subscript will be
particles is much more complicated.
If there is a collective
or coherent velocity of the particles superimposed upon their diffusive motion, then the FM discriminator output is linear in the coherent velocity. However, the contribution from the diffusive or incoherent component of the particle
suppressed when there is no possibility of confusion) can be
written as
velocities can be shown to add a noise term to the FM signal,
x=
+,
which puts a lower limit on the ability to measure their coherent velocity. In the following section we describe a theoretical model and a calculation of the FM noise term in
X=
Jdtv(t).
0740-3232/87/071220-08$02.00
© 1987 Optical Society of America
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Eversole et al.
1221
Here, x, is the coherent part and x is the random part of x.
fects of these cancellations will not be observed in the ex-
The coherent velocity is v. By using this decomposition and
periment. To get around this problem we present here an alternative approach by which we can estimate the PSD in a manner that is not unacceptably sensitive to the deviations from ideality of the FM discriminator. The point of departure is the observation that the statistics of X and Ycan be generat-
following the approach of Lawson and Uhlenbeck,15 the expression of Eq. (1) can be algebraically rearranged to give the expression
S(t) = (X2 + Y2)"cos(-wt
+ qx, + 4),
(3)
ed by a quasi-random walk in two dimensions (X, Y space).
with
The walk is not completely random because the normal distribution of X and Y keeps the walk confined to the
4b = tan-'[Y(t)/X(t)],
X(t) =
N E
vicinity of the origin (X = Y = 0). The walk can be generated by a discrete-time Markov process.' 7 For such a process
cos(qlj),
j=' N
(4)
Y(t) = E sin(qx!). j=i
If a bandpass filter is used after the photodetector and before the FM discriminator, the input to the FM discriminator has the same form as Eq. (3). In this case the func-
tions X and Y are the bandpass-filtered variants of Eqs. (4). The output of an ideal FM discriminator, given input of the form of Eq. (3), is the time derivative of the argument of the
cosine. The expression for the output F is
F(t) = qv(t) + dt*
(5)
The constant offset w, is unessential and is usually ignored. The first term in Eq. (5) is the signal, and the second term is noise. The analysis of the noise involves a statistical analy-
sis of d4'/dt. This in turn involves the statistics of X and Y. Since X and Y are sums of random variables, they are
random variables themselves. By virtue of the central-limit theorem, X and Y are normally distributed for large N. It is easy to verify that X and Y are independent and identically distributed
with mean zero and variance (1/2)N.
The re-
maining quantity needed for a complete statistical description of X and Y is the two-time correlation function C(t) = (X(t)X(0)) = (Y(t) Y(O)). This correlation function is'well 6 The form is known for the case of diffusive motion.' C(t) = (X(t)X(0))
= (X 2 )exp(-q 2 DItd),
(6)
to be generated, three things are needed: (1) a rule for selecting the starting point, (2) a choice of the time interval per step, and (3) the conditional probability density for choosing the coordinates of the (i + 1)st point given the location of the ith point. The first requirerment is satisfied by choosing a number at random from a normal distribution to serve as the X coordinate and then choosing another number in the same manner to serve as the Y coordinate. The choice of the size of the time step will be discussed in
more detail below. For the present it is viewed as a parameter, t>. The conditional probability P is a product of two terms, one corresponding to the X coordinate of the point and one corresponding to the Y coordinate. The expression for the term corresponding to the X is'8 P(Xi+l/Xi) = K expl-'/2 (1 _ p 2 Y)l[Xi+, -
pX]21,
(7)
where p = C(tj). Here, K is a normalization constant that ensures that the integral of Eq. (7) over Xj+i is equal to unity. The term corresponding to the Y coordinate is identical in form to Eq. (7). When a very long walk is construct-
ed following this description, the statistics of X and Y will have the following properties:
(1) they will be independent
and normal with zero mean and unit variance, and (2) they will have a two-time correlation function given by Eq. (6);
that is, this process generates the correct statistics for X and Y. This is equivalent to the statement that this type of system is ergodic, that time averages are equivalent to ensemble averages. The phase 4' at each step is given by tan-'(Y/X).
It is a
straightforward matter to implement the Markov process on function of X and Y and their derivatives, it is a straightfor-
a computer and obtain a time sequence of values of 4b. A time sequence of time derivatives of 4' can be constructed by
ward, although lengthy, exercise to calculate any statistic
taking first differences of the 4' sequence:
where D is the diffusion constant.
Since d4'/dt is a rational
associated with d4/dt. One simple result is evident at once. Since 4' involves only the ratio of X and Y, none of the
statistics depends on N. Consequently, the noise floor is independent of the number of particles. In what follows, the variables X and Y have been rescaled so as to have unit
variance. The statistics of principal interest for noise-floor characterization are the two-time correlation function and its Fourier transform, the power spectral density (PSD). The result for the correlation function is contained in Ref. 15. Unfortunately, this result is not of practical utility for dealing with the functional form of Eq. (6) or with bandpass-
filtered forms thereof. In this case the result is strongly influenced by some delicate cancellations, which involve ocThese large values occur casional large values of d4/dt.
when X and Y are simultaneously close to zero. The output of a real FM discriminator deviates from the form of Eq. (5) in such cases because of saturation effects. Hence the ef-
(8)
(dt ) = [Di- ijltu
The statistics of d'/dt can be calculated from this sequence. If tq 2D 85 dB), in the 60-75-dB range the demodulator noise floor rose rapidly, and the demodulated signal was attenuated. A second demodulator (Hewlett-Packard
(D
a
0 LD
CD
.0
0
.
0
1.0 0 0 20
3
o CDo'~ -t
0.1
X.I
model 8901A-002 modulation analyzer), whose performance is shown in Fig. 4 (using a signal generator and a broadband noise source), faithfully demodulated signals down to 40-
dB carrier signal-to-noise ratios and had low-noise performance in the 60-75-dB range. The Hewlett-Packard device was used for the subsequent measurements.
Fig. 4. FM demodulator noise floor as a function of the FM carriersignal-to-noise ratio (SNR), 100-Hz bandwidth; also shown is the
equivalent FM deviation noise.
Vol. 4, No. 7/July 1987/J. Opt. Soc. Am. A
Eversole et al.
1225
and the horizontal scale is from 0 to 1000 Hz, giving an
instrument resolution of 6 Hz. The largest resonance peak located in the center is due to the oscillating mirror's being driven at 500 Hz with a rms amplitude corresponding to 0.07 rad. Between 0 and 350Hz a number of smaller and broader peaks appear, which are produced by various mechanical resonances in the interferometer and which constitute an instrumental signature. For comparison purposes both the total light intensity in the sample arm and the magnitude of the beat signal from the aluminum surface were matched to those observed from the particles, so that the visibility was the same in both cases. The resulting noise floor from the
Fig. 5. Oscilloscopetrace of the time dependence of the spectrum analyzer output fixed at the frequency of the difference between the
aluminum surface therefore corresponds to the instrumental resolution under those conditions. The increased noise floor from the particle scattering is a phase noise limitation imposed by the random diffusive motion of the particles. Since the phase shift from the oscillating mirror is calibrat-
reference and sample beams. The vertical and horizontal scales are 10 dB V per division and 100 msec per division. The large dip on
ed, it is straightforward
the right indicates the magnitude of the random amplitude variation of the carrier signal.
in Fig. 6, this value is 0.35 Hz/Hz 12 in terms of frequency deviation (or 1.44 ± 0.02 mrad/Hz 1/2 rms at 500 Hz). Figure
the mean, so that on a random basis the carrier signal can effectively disappear.
Figure 5 is a photograph showing the
time dependence of one frequency component of the spectrum analyzer output, corresponding to the peak of the opti-
to assign a numerical value to the
particle noise floor. For the 1-gm-diameter particles shown 6(b) displays two output spectra from a similar data set of 1gm-diameter particles under the same conditions. Here the spectrum is taken from 0 to 500 Hz, and the oscillating mirror peak has been shifted to -350 Hz. The two spectra
cal beat signal (455 kHz in this case), as displayed on an oscilloscope. The horizontal scale is 0.1 sec per division, and the vertical scale is again logarithmic with 10 dB V per
division. Toward the right-hand side the carrier signal momentarily drops by more than 2 orders of magnitude. Such dropouts occasionally unlock the phase-locked loop in the FM demodulator, and the rate at which this occurs dictates whether data collection is possible. Since the dropout can be arbitrarily close to zero, electronic limiters may improve,
but cannot eliminate, this situation. Some FM discriminators have been designed with a coasting circuit, which allows the phase-locked loop to continue operating with short interruptions of the input carrier. An attempt was made to evaluate one such device obtained on loan22 and to deter(a)
mine whether it would make data collection easier. Howev-
er, the inherent sensitivity of the instrument was not high enough to resolve the diffusive motion background from the particles. Therefore, for this experiment, processing of the FM discriminator output was accomplished with an audio spectrum analyzer (Hewlett Packard 3582A), which digitizes the in-
coming signal and performs a fast Fourier transform over some selected time interval. Successive intervals can then be summed and averaged.
Since the digitized signal was
displayed, it was possible to determine visually when a particular interval was free from dropout and aliasing effects
before counting it as reliable data. (b)
RESULTS AND DISCUSSION Figure 6(a) is a representative photograph of the displayed spectrum analyzer output obtained in the manner just described. The upper trace was generated from scattered light off 1-gm-diameter particles, and the lower trace was obtained from scattered light off a dull aluminum surface. The vertical scale is logarithmic with 10 dB V per division,
Fig. 6. Display the vertical and Hz per division. between sample
traces of the audio spectrum analyzer output with horizontal scales set at 10 dB V per division and 50 (a) The upper and lower traces show the difference beams from particles and a rough surface, respec-
tively. The visibility was the same, and the phase shift in the calibration peak in the center was 0.07 rad rms. (b) Despite a 6X change in particle density, traces from 1-gtm-diameter particles ap-
pear to be essentially the same. The calibration peak is shifted to -350 Hz.
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J. Opt. Soc. Am. A/Vol. 4, No. 7/July 1987
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Eversole et al.
1.0
Scatteringangle=10°
1 pm
0.5
>
K
0. -70 -
0
_____f_ ---- fA@__
0
C]
E
Z5
IL
0.2 I
7.6 ym
a
-
-80
0.1 1
0.05
-90 104
I 10
10,6
1o5
108
ParticleDensity, cm-3 Fig. 7. Measurements of the audio spectrum analyzer noise floor from particles show no dependence different sizes (1.0 and 7.6 gm) over a range of a factor of 10.
in Fig. 6(b) are essentially identical even though the particle density was changed by a factor of 6. Particles densities
were determined on the basis of laser-beam attenuation measurements and appropriate calculated Mie extinction coefficients. These observations of the noise floor in the demodulated
spectra from particles scattering were in good agreement with theoretical predictions. The demodulated spectrum showed a flat frequency dependence, at least over the band-
on particle concentration
for two
width of the el,ectronic filters used. As implied in Fig. 6(b), the particle nc Iise floor was not dependent
on particle con-
centration. B oth of these characteristics were anticipated from theoretical considerations. Figure 7 is a plot of the
measured nois ; floor for two different particle sizes, 1.0 and 7.6 m in dian ieter, each over a range in particle density of
roughly an ord er of magnitude. For both sizes the instrumental noise f loor was measured to be significantly lower before and aft r (at the lowest and highest concentrations)
Approximate Particle Density, cm -'If
-ou
10
109 I
-,
8
10
I
4
1
106
101. 1.0
10
I
I
-
I
'N 'N
N
ED
_ 0.5
I
-n
-70' _ N
Theoretical
CD
_0.2
.
0
I
M'
0. t
N0
-80 _ 0
E
a a)
0.05
-90
l
l
0.5
1
2
5
10
20
50
Particle diameter, pm Fig. 8. Audio spectrum analyzer measurements of the particle noise floor as a function of particle size. Seven discrete sizes of polystyrene spheres in water were used. The solid line is a least-squares fit to the data points, and the dashed line is the predicted noise floor implicit in Fig. 1.
Vol. 4, No. 7/July 1987/J. Opt. Soc. Am. A
Eversole et al.
by comparison with measurements of a rough surface adjusted to give the same visibility. While the level of the noise
signal from the particles was not dependent on particle density, the absolute value of the interferometer signal (carrier) was. At lower particle concentrations the total scattered light intensity and the beat-frequency signal strength decreased. A lower limit on the particle density is therefore reached when the signal becomes small enough that the instrumental resolution or noise floor becomes equivalent to
the particle noise signal. This lower limit is augmented by the unrelated effect that the rate at which dropouts occur appears to increase as the particle density is lowered,making data collection impractical. There is reasonable quantitative agreement between the predicted and measured values of the particle noise floor.
Using Eq. (11) for the PSD of the particle signal and numerical values of q2D and to yields values of the PSD (0 Hz) as a function of particle diameter. The theoretical and observed values for the square root of the PSD have been plotted together in Fig. 8. The data points represent the noise-floor measurements for specific sizes of latex spheres suspended in water. Although the noise floor is independent of the
particle density, for these measurements the particle concentration
was adjusted in each case to attenuate
a beam
through the cuvette by a factor of approximately Ile, so that the difference between the particle signal and the instrumental noise floor is close to maximum. The theory (shown by the dashed line) predicts a slope-versus-particle-size de-
pendence that is within experimental error of the data. As previously discussed, measurements had to be restricted to time intervals in which, by chance, signal fading, or drop-
outs, did not occur. From Eq. (3) it can be seen that carrier fading is associated with locations near the origin in the X, Y parameter plane. Since the region near the origin is responsible for the largest contributions to the PSD, this restriction on the experiment leads, in effect, to a biased averaging of
the PSD over the X, Y phase space. The biased averaging would result in a lower-than-predicted PSD. This mechanism is the likely reason for the systematic difference between the observed and theoretical PSD values. Although it is not possible to quantify this difference, the overall agree-
ment between the predicted signal behavior and the experimental results is quite satisfactory. ACKNOWLEDGMENTS
1227
REFERENCES AND NOTES 1. Four general reference books are B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976); B. Chu, Laser Light Scattering (Academic, New York, 1974); F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981); H. Z.
Cummins and E. R. Pike (eds.), Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974). 2. H. Z. Cummins and J. L. Swinney, "Light beating spectroscopy," Prog. Opt. 8, 135-200 (1970). 3. G. B. Benedek, "Optical mixing spectroscopy with applications to problems in physics, chemistry, biology, and engineering," in
Polarization, Matter and Radiation (Presses Universitaire de France, Paris, 1969), pp. 49-84. 4. N. A. Clark, J. H. Lunacek, and G. B. Benedek, "A study of Brownian motion using light scattering," Am. J. Phys. 38, 575585 (1970). 5. S. S. Penner, J. M. Bernard, and T. Jerskey, "Power spectra observed in laser scattering from moving polydisperse particle system in flames. I. Theory," Acta Astron. 3, 69-91 (1976); "II. Preliminary experiments," Acta Astron. 3, 93-105 (1976). 6. W. Hinds and P. C. Reist, "Aerosol measurements by laser Doppler spectroscopy. I. Theory and experimental results for aerosols homogeneous," Aerosol Sci. 3, 501-514 (1972); "II. Operational limits, effects of polydispersity and applications," Aerosol. Sci. 3, 515-527 (1972). 7. D. P. Choudhury, C. M. Sorensen, T. W. Taylor, J. F. Merklin, and T. W. Lester, "Application of photon correlation spectroscopy to flowing Brownian motion systems," Appl. Opt. 23,41494154 (1984). 8. G. B. King, C. M. Sorenson, T. W. Lester, and J. F. Merklin, "Photon correlation spectroscopy used as a particle size diagnostic in sooting flames," Appl. Opt. 21, 976-978 (1982). 9. M. E. Weill, P. Flament, and G. Gouesbet, "Diameters and
number densities of soot particles in premixed laminar flat flame propane/oxygen," Appl. Opt. 22, 2407-2409 (1983). 10. P. H. P. Chang and S. S. Penner, "Determinations of turbulent velocity fluctuations and mean particle radii in flames using scattered laser-power spectra," J. Quant. Spectrosc. Radiat. Transfer 25, 97-104 (1981). 11. S. S. Penner and P. Chang, "On the determination of log-normal
particle-size distributions using half widths and detectabilities of scattered laser power spectra," J. Quant. Spectrosc. Radiat. Transfer 20, 447-460 (1978). 12. C. B. Bargeron, "Measurement of a continuous distribution of
spherical particles by intensity correlation spectroscopy: analysis by cumulants," J. Chem. Phys. 61, 2134-2138 (1974). 13. J. Briggs and D. F. Nicoli, "Photon correlation spectroscopy of polydisperse systems," J. Chem. Phys. 72, 6024-6030 (1980). 14. R. G. Renninger, M. K. Mazumder, and M. K. Testerman, "Particle sizing by electrical single particle aerodynamic relaxation time analyzer," Rev. Sci. Instrum. 52, 242-246 (1981). 15. J. L. Lawson and G. E. Uhlenbeck, Threshold Signals (McGraw-Hill, New York, 1950), pp. 369-371. 16. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), p. 59.
We wish to express our thanks to all those who contributed
17. S. Karlin and H. M. Taylor, A First Course in Stochastic Pro-
to the success of the experimental measurements, especially, but not limited to, H.-B. Lin, A. B. Tveten, E. McGarry, and A. Parvulescu. The encouragement and support of A. J.
18. K. Fukunaga,Introduction to Statistical Pattern Recognition
Campillo, B. J. Feldman, and T. G. Giallorenzi was also
appreciated. J. D. Eversole is also affiliated with Potomac Photonics, Incorporated, Engineering Research Center, Building 335, University of Maryland, CollegePark, Maryland 20747. A. D. Kersey is also affiliated with Sachs/Freeman Associates, Incorporated, 1401 McCormick Drive, Landover, Maryland 10785.
cesses (Academic, New York, 1975), pp. 29-33. (Academic, New York, 1972), pp. 22-24.
19. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 585-587. 20. C. P. Wang and D. Snyder, "Laser doppler velocimetry: experimental study," Appl. Opt. 13, 98-103 (1974). 21. Three comprehensive references are H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957); M. Kerker, The Scattering of Light (Academic,New York, 1969); C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983). 22. On loan from the Naval Underwater Systems Center, courtesy of J. Longacre and W. Stachnik.
