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Effects of the laser radiation pressure in photon correlation spectroscopy
1 April 1994 OPTICS COMMUNICATIONS Optics Communications 107 (1994) 161-169
ELSEVIER
Full length article
Effects of the laser radiation pressure in photon correlation spectroscopy Yasuhiro Harada, Toshimitsu Asakura Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan
Received 10 September 1993
Abstract
Effects of the radiation pressure of illuminating laser light, even with a low power less than a few tens of roW, on a photon correlation function and a particle size obtained by using photon correlation spectroscopy with the homodyne detection are investigated experimentally for suspensions of polystyrene microspheres. The diffusion coefficient of particle suspension, which is determined from the measured correlation time, decreases with an increase in the power of illuminating laser light. This result stems from suppression of the brownian motion of particles within the probe volume due to a radial component of the radiation pressure of illuminating laser light. It turns out that the particle size evaluated from the correlation time is misled to be larger than the true value under illumination of the laser light. Complemental measurements by photon correlation velocimetry are performed in order to demonstrate the effects of the laser radiation pressure on the particle sizing by means of photon correlation spectroscopy and their dependence on the size of microspheres.
1. Introduction
The dynamic light scattering (DLS) method is one of the most excellent and matured tools for investigating the dynamics and size of particles in suspension such as colloid dispersion of chemical and biological specimens [ 1-3]. In the DLS method, we monitor the statistical time-dependence of intensity fluctuations of light scattered by such dynamic particles undergoing the brownian motion. In general, a temporal correlation function of intensity fluctuations is used to evaluate the statistical nature of scattered light and its decay rate provides information about the hydrodynamic property of particle suspension. To be exact, the diffusion constant of particle suspension can be obtained from a reciprocal of the correlation time. The hydrodynamic diameter of particles can also be evaluated from the measured diffusion coefficient, namely, the correlation time, pro-
vided that the temperature and viscosity of the suspension are known. Most of the optical setups used in the DLS method to investigate the particles system require the use of small-angle coherent detection. In this case, a level of the scattered light at the detector becomes extremely low. Due to the quantum-limited noise performance of single-photon counting and the availability of a digital correlator for processing the number of photoelectron counts, the photon correlation technique is suitable for obtaining the correlation function of weak scattered light fluctuations with the high signal-to-noise ratio. The DLS method with this digital correlation technique is usually referred to as a photon correlation spectroscopy (PCS) [4,5 ]. A primary advantage of the DLS method including PCS is that the diffusion coefficient and size of partides can be rapidly and accurately obtained from measurements of the time-dependence of intensity fluctuations of light scattered by the brownian patti-
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cles. It should be noted that the nondestructive nature of laser light in the scattering process is accepted in the method. In most of the light scattering methods including laser Doppler velocimetry, phase/ Doppler particle analyzer and static light scattering as well as the DLS method, it is considered that the illuminating laser beams do not affect the particles' motion but are merely modulated in their amplitude and phase due to the existence and motion of scattering particles. Since our interest exists in the optical particle sizing by means of the DLS method, a random change in position of the scattering particles, which is governed by the thermodynamic property of the particles system, plays an essential role in the principle of the sizing method. If there is a certain source which distorts or affects the particles' brownian motion, a resultant correlation function of light fluctuations scattered by such particles would also be changed. As a consequence, the evaluated particle size misleads an inaccurate value. Therefore, the nondestructive performance of laser light in the DLS method for investigating the dynamics and size of the particles system is indeed an indispensable requirement. On the other hand, since a proposal and pioneering experimental works of Ashkin et al. [ 6-8 ] for trapping and accelerating small dielectric particles by the single laser beam, the availability of quite new conceptual micromanipulation of small particles including biological cells has been favorably noticed and applied to wide fields from biomechanical researches [ 9 ] to a novel scanning probe microscopy [ 10 ]. This technique is based on the fact that the light itself transports the electromagnetic energy in its propagation direction and that the light also carries the momentum in the same direction. Based on a law of the momentum conservation, a fraction of the momentum of light should be transferred to the particle through scattering phenomena and, therefore, a force called the radiation pressure is exerted on the particle [ 11,12 ] as a source for accelerating it toward the beam propagation direction. From the viewpoint of optical particle sizing based on dynamic light scattering phenomena, in which a random motion of scattering particles plays a principal role, it may be quite natural that a question arises concerning the influence of the laser radiation pressure on the particles' motion and, moreover, on the measured correlation function and the resultant particle size.
In this paper, we investigate experimentally the effects of the radiation pressure of illuminating laser light on the correlation function and the particle size measured for suspensions of polystyrene microspheres by means of PCS with the homodyne detection scheme. The power of the laser light illuminating the particle suspension is employed as a parameter controlling a value of the radiation pressure on the scattering particles. The effects appeared in the homodyne experiment and their dependence on the sample particle size are complemented by another experiment based on photon correlation velocimetry.
2. Effect on particle sizing in photon correlation spectroscopy
2.1. Experimental method Fig. 1 shows the schematic diagram of the experimental setup for investigating the effects of the laser radiation pressure in the system of PCS. While main optical components in the figure are primarily concerned with photon correlation spectroscopy for the MI
. M3
"1:1-
TC
BO
L1 L2 P1~ ' ~ M2~ @
4ND M
/Ar+Laser/.
_ / Ele-Ne
~IM - / Laser /
~
~ ' ~ ....... B-t-IIA~2OF1
SampleCell ,
,,
, IF 2
~
I Photon
OF2~ C o r r e l a t o r
Fig. I. Schematicdiagram of the experimentalsetup. Bo:Ar+ laser beam; BI, B2: HeNe laser beams; M~, M2, M3, M4: mirrors; HM: half mirror; LI, I.~, L3, L4, Ls: lenses; P1, P2, P3, P4: polarizers and analyzers; A], A2, A3, At: apertures and pinholes; HWP: half wavelength plate; IFt, IF2: interference filters; ND: neutral density filter;, OF~, OF2: optical fibers; PMt, PM2: photomultipliers; and TC: thermocouple.
Y. Harada, T. Asakura / Optics Communications 107(1994) 161-169
particle sizing, the whole setup for the experiment of photon correlation velocimetry to be given in the next section is also shown in Fig. 1. It is a common feature of the optical systems in both experiments that a power-controllable Ar ÷ laser is employed as an extra light source for applying the variable radiation pressure on particle suspension in addition to a low-power HeNe laser as an ordinary source for probing the particles' dynamics. This configuration is merely introduced for the purpose of clearing up the appearance of the influence of the laser radiation pressure on experimental results. A similar experimental study may be achieved by using an ordinary optical configuration of the PCS system employing single-beam illumination where the beam power is controllable. An Ar ÷ laser beam Bo with the wavelength of 0.5145 ~tm illuminates a sample cell at its beam waist position with the spot size of 7.32 Ixm. The available power incident on the sample cell ranges from 0 to 60 roW. In addition to this beam, a polarized HeNe laser beam B~ with the wavelength of 0.6328 btm also illuminates the sample cell at its beam waist position, where the spot size is 58 ~tm. The power of this probe beam is decreased and kept at less than 0.1 mW by the neutral density filter ND. In this situation, the light intensity of the beam Bo is, at a maximum, about 4000 times larger than that of the probing beam B~. The beams of Bo and B~ intersect each other in the cell at the fixed angle 0= 85 °. Under dual-beam illumination of Bo and B~, only the probing HeNe laser light scattered by particles in the sample cell is detected by a photomultiplier PM~ (Hamamatsu, R649) through the optical system arranged in the horizontal direction. The scattered and transmitted Ar ÷ laser lights are excluded from entering the detecting aperture A2, by using an interference filter IF~ with the maximum transmission at the wavelength of 0.6328 ~tm. The distance between the apertures A~ and A2 with the diameters of 1 mm and 100 Ixm is 250 mm. Photoelectron pulses from the photomultiplier are pre-amplified, discriminated, and fed into a 64-channel photon correlator (Kowa, ECD-2000) for measuring the photon correlation function of the scattered probe light. Polystyrene microspheres with the diameters of 0.264 ~tm and 2.062 lxm (Duke Scientific) are used as standard particle suspensions. The microspheres of each size are suspended in the deionized purified-water at the volume fraction of O.184
163
ppm within the spectrophotometric sample cell having the thickness of 2 mm. This condition for the sample preparation supports the single scattering event in light scattering by suspended microspheres. The temperature of suspensions is also monitored by the thermo-couple TC to evaluate the particle size from measured photon correlation functions. All measurements and evaluations of photon correlation functions are performed for various values of the power of the Ar + laser beam Bo up to 60 mW, while other parameters including the optical geometry and the temperature of the sample solvent are kept constant for each sample suspension. 2.2. Evaluation method of experimental data
Measured photon correlation functions are evaluated in terms of the normalized photon correlation functions [ 13], g2(T), with the form o f g2(Q = 1 +fl e x p ( - z / z c ) ,
(1)
where r denotes the delay time, fl the relative contrast or the degree of coherence, and Tcthe correlation time defined as the 1/e decay time of an exponential signal component. It is evident from the general theory of dynamic light scattering for particle suspensions undergoing the brownian motion that the correlation time T~ in Eq. ( 1 ) is connected with the diffusion coefficient D of particle suspensions by [ 1-5,13 ] "r~-1 =2q2D,
(2)
where q is the modulus of the scattering vector defined by the optical geometry. According to the experimental setup shown in Fig. 1, it is given by q= (4rt/2)m2 s i n ( 0 / 2 ) ,
(3)
where 2 is the wavelength of laser light in vacuum, m 2 the refractive index of the medium surrounding the particles, and 0 the scattering angle. Therefore, the diffusion coefficient of particles is determined from the correlation time of the photon correlation function with the knowledge of the optical configuration. Furthermore, by using the Stokes-Einstein relation [15] D=kBT/3zt~ld,
(4)
where kB denotes the Boltzmann constant, T the ab-
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Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
solute temperature and t/the viscosity of the solvent, the hydrodynamic particle diameter d of particles can be evaluated from the correlation time or the diffusion coefficient measured. Both the linear least-squares fitting and the cumulant methods [ 14 ] are adopted for reducing the measured correlation function to its normalized version of Eq. ( 1 ), yielding fl, Zc, and the square of expected photon counts ( n ) even though the sample suspensions arc monodisperse. All the digital signal processing for measured correlation functions are performed on a microcomputer (Epson, PC-386GS). A reciprocal value of the correlation time, 1/%, and the consequent particle diameter d are evaluated as a function of the Ar + laser power for investigating the effect of the radiation pressure of the Ar + laser. 2. 3. Results and discussion
Typical dependence of normalized correlation functions on the Ar + laser power is shown three-dimensionally in Figs. 2 and 3 for the particle suspensions of 2.062 ~tm and 0.624 ~tm in diameter, respectively. The sequential results in each figure are obtained under the same thermodynamic conditions for each sample suspension: temperatures T of the solvent are 304.03 K and 300.59 K for Figs. 2 and 3, respectively. Related hydrodynamic and optical parameters and theoretical values of the reciprocal correlation-time for each particle suspension are sum-
09
0
8 o "O
Delay Time [ms] 31.5 0 Laser Power [mW] Fig. 2. Evolutionof normalized photon correlationfunctionsof the light scattered by the particle suspension having the diameters of 2.062 $tmfor six differentpowersof the Ar+ laser.
marized in Table 1. It should be noted that a logarithmic of [g2(z) - 1 ] / p is plotted as a normalized correlation value in Figs. 2 and 3. That is, the slope of each normalized correlation function in these figures just represents the reciprocal of the correlation time, 1/zc, and hence the diffusion coefficient D. Normalized correlation functions in both figures show the tendency of extending to the longer delay time under illumination of the Ar + laser, implying an increase of the correlation time, while this trend is hardly recognized in the correlation function for the sample suspension which contains smaller particles in size (see Fig. 3). To evaluate quantitatively the effect of the Ar + laser illumination on the correlation functions, the reciprocal values of the correlation time % normalized by those without illumination of the Ar + laser are analyzed and shown in Fig. 4 as a function of the Ar + laser power for both particle sizes. As predicted from Figs. 2 and 3, the reciprocal values of the correlation time in both particle suspensions decrease monotonically with an increase in the illuminating Ar + laser power. For the higher power of the Ar + laser, 1/% becomes extremely small in comparison with those in absence of the laser illumination. For example, in the case of larger particles, it becomes less than 60% under the illumination power of 50 mW at most. It is worth pointing out that this effect evidently depends on the size of sample particles. That is, the larger the particle size is, the more remarkable a decrease in the reciprocal of the correlation time becomes. On the other hand, a hydrodynamic diameter of polystyrene microspheres in the suspension can be evaluated from the measured correlation time by using Eqs. (2)- (4). The dependence of the evaluated particle size on the power of the Ar + laser is shown in Fig. 5, where the ordinate value is normalized by the particle diameter obtained without illumination of the Ar + laser. The normalizing factor is 2.424 ttm and 0.636 ttm for particle suspensions with the nominal diameters of 2.062 ~m and 0.624 ~m, respectively. The solid lines are obtained as approximated linear lines for each set of experimental data, whereby the increasing rates of the obtained particle size with regard to the laser power are 1.237%/mW for d=2.062 ttm and 0.217%/mW for d=0.624 ttm, respectively. According to the result in the correlation time, the evaluated particle size is misled to be larger
Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
165
Table 1 Thermodynamic condition for two kinds of suspension of polystyrene mierospheres in the deionized purified water and the resultant reciprocal value of the correlation time of homodyne correlation functions without illumination of the Ar + laser. d [l~m]
T[K]
0.624 2.062
~/[Pas]
300.59 304.03
0.8428× 10 -3 0.7830× 10 -3
m2
1/% [ I / s ]
1.332 1.332
theoretical
experimental
534.52 176.19
530.22 150.53
1 , 6
i
o 1.6
,
i
,
t
d = 2.062
0
"17::~ 1.4
/e
n
._N
o
E O
:~ - 0
-
•
.01
Delay Time [ms] 6.3 0 Laser Power [mW] Fig. 3. Evolution of normalized photon correlation functions of the light scattered by the particle suspension having the diameters of 0.624 ~tm for five different powers of the Ar + laser. '
0
i
'
i
'
[
'
•
9E .hi-: ~ ._~
0.6
' ~ d = 2.062t~m
-8 z
0.6 [
0
,
I
,
I ~
.
20 40 60 80 Laser Power [mW]
Fig. 4. Dependence of the normalized reciprocal of the correlation time on the Ar + laser power: symbols ( • ) for 2.062 ~tm and ( O ) for 0.624 ~tm in particle diameter, respectively.
than the nominal value under the influence of the intense laser beam, and the size-dependence of the increasing rate in the obtained size also exists. This is estimated from the experimental results as the relation of ocd 3/2.
Z
1 ~ I
0
d-- 0.624gm ,
,
,
,
20 40 60 80 Laser Power [mW]
Fig. 5. Dependence of the normalized particle diameter on the Ar + laser power: symbols ( • ) for 2.062 ~tm and ( O ) for 0.624 ~tm in particle diameter, respectively.
The above results can be considered as a result stemming from a change in motion of the brownian particles under illumination of the laser. As seen from Eq. (2), a reciprocal of the correlation time for the scattered probe light has a linear relationship with the diffusion coefficient of particles. The latter quantity denotes physically the rate of a variance in position of the brownian particles. Hence, a decreasing character in the reciprocal value of the correlation time implies that the brownian motion of the particles to be investigated is affected and, in other words, suppressed by the radiation of the Ar + laser within the probing volume. It should be noted that the thermal effect induced by the laser illumination, such as an increase in temperature of the solvent, does not give rise to the present experimental results. In such a case, it turns out as an increase in the reciprocal value of the correlation time [ 16 ], implying the contrary to the present experimental results. Consequently, it is suitable to consider that the suppression of the brownian motion due to the radiation pressure of illuminating laser light occurs in the present PCS study
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as same as the results demonstrated by Ashkin [ 6,7 ] for trapping and accelerating single dielectric particles. As referred in the articles of Ashkin [6,7], the force due to the radiation pressure exerted on particles has two radial and axial components. The radial component of force due to the radiation pressure is closely related with a gradient of the intensity profile of a laser beam. Therefore, it is referred to as a gradient force [ 17 ]. If the refractive index of particles is higher than that of a surrounding medium, the gradient force has the direction toward the optical axis of the illuminating laser beam and drives the particles to approach the beam axis. The particles located at the beam axis experience this force isotropically from all radial directions, and the stable trapping of particles is achieved in the transversal section of the beam. There is also a gradient force in the beam-propagation direction. However, it can be ignored provided that the spot size of the illuminating laser beam is sufficiently greater than the radius of particles. Therefore, the axial component of the force can be explained by using only the momentum transfer associated with scattering phenomena occurring between the propagating laser light and particles undergoing the brownian motion. Finally, being suppressed in the radial random motion, the particles move along the beam axis with the size-dependent constant velocity, called as a terminal velocity, due to a drag force of the surrounding viscous medium. This is in the same condition with the present experimental study on the optical particle sizing of suspensions by means of PCS where the gradient force is indeed a main source for suppressing the brownian motion of particles except for the sample concentration. As is well known, photon correlation spectroscopy with a homodyne detection scheme is insensitive to the uniform motion of ensemble particles, which is induced by the axial component of the radiation pressure, only the suppression of the brownian motion of particles is observed in the photon correlation function. If we can demonstrate the above effect of the axial component of the laser radiation pressure and its dependence on particle size, it will turn out as an evidence of the effect appeared in the present PCS study. Another type of the experimental investigation demonstrating the size-dependent effect due to
the axial component of the radiation pressure has been performed by employing the optical configuration for photon correlation velocimetry, and is described in the following section.
3. Detection of uniform flow due to radiation pressure 3. I. Experimental method
Detection of the uniform flow of many particles due to the radiation pressure of the Ar + laser in the propagation direction of an illuminating beam is accomplished under the same experimental condition for sample suspensions in the previous PCS study. However, there is a different point in the experimental setup that the dual-beam illumination of the HeNe laser and the heterodyne detection of the scattered probe light have been employed. As depicted in Fig. 1, the two HeNe laser beams of B~ and B2 illuminate the sample cell at their beam waist positions intersecting each other at an angle q = 10 °. The spot size of each beam is 58 ttm which is the same in the proceeding section. Scattered probe lights for both beams of Bt and B2by sample particles are collected by using the imaging optical setup consisting of aperture A3, lens L5, polarizer P4, interference filter IF2, and pinhole Ps. Then, they are optically mixed on a photosensitive surface of the photomultiplier PM2 (Hamamatsu, R649), producing the photoelectron pulses. By using a photon corrclator, the time-dependence of photoelectron pulses is evaluated in terms of a photon correlation function which contains the information about the magnitude of velocity v in the same direction with the propagation of the Ar + laser beam as a decay of the envelope and an interval of the oscillating features. An expression of the photon correlation function is given in a normalized version [ 13 ] by g2(z)=exp(-v2z2/2w2)[l
+flcos(ktrr) ] ,
(5)
where Wl is the radius of the probe volume, and k the modulus of the difference between the scattering vectors defined by individual illumination-detection directions. According to the optical configuration of the dual-beam illumination shown in Fig. 1, it is given, independently of the direction of a detecting aperture, as follows:
Y. Harada, T. Asakura / Optics Communications 107(1994) 161-169
k = (4rt/2)m2 s i n ( ~ / 2 ) .
(6)
Therefore, from the knowledge about the optical configuration and the property of the solvent fluid, the velocity v of particles moving in parallel to the beam propagation direction can be evaluated by determining the oscillation interval T¢ of the photon correlation function. The relation which connects the particle velocity v and the oscillation interval T~ is given by 2
v= 2m2T~ sin(~/2) "
(7)
Determinations of the oscillation interval and the relevant velocity are performed by using a fast Fourier transform (FFT) algorithm for photon correlation functions measured with the various values of the Ar + laser beam. ' The described experimental setup which employs the photon counting mode for detecting the scattered light intensity and the digital signal processing by means of a photon eorrelator is usually referred to as a photon correlation velocimeter [ 5 ] for the purpose of discriminating it from a laser Doppler velocimeter performing under the analogue processing mode.
3.2. Results and discussion Fig. 6 shows the three-dimensional representation of normalized photon correlation functions obtained
0.5
s2
Delay Time [ms] 630 0 Laser Power [mW] Fig. 6. Evolutionof normalized photon correlationfunctionsof the light scatteredby particle suspensions having the diameters of 2.062 Ixm for four different powers ofAr+ laser in the use of the optical configurationof photon correlationveloeimetry.
167
for particle suspensions with the diameter of 2.062 I~m for four different powers of the Ar + laser. Apparent oscillations occur with the constant time-interval in the correlation function and the interval Tc may become shorter with an increase in the laser power. This implies that the particles in the probing volume are indeed in an uniform flow in the propagation direction of the laser beam and under the influence of the radiation power. It can be concluded from these results that the axial component of the radiation pressure due to the Ar + laser is a source for inducing the uniform flow of particles. On the other hand, these results in photon correlation functions for particle suspension having the smaller diameter of 0.624 ~tm could not be observed. Note the fact that smaller particles in suspension change their positions with the rate higher than larger particles according to the theory of brownian motion (see Eq. (4)). The radiant power of the Ar + laser is not high enough to suppress the brownian motion of smaller particles in this experimental condition. In other words, the trapping process with regard to the cross sectional direction of the laser beam is not fully achieved for the smaller particles and, therefore, the next process of an acceleration also never happens. Nevertheless, these resuits denote that there exists also the size-dependence in the effect of the axial component of the laser radiation pressure as is the same with the results in the previous section. In order to demonstrate quantitatively the relationship between the radiant power and the velocity of particles and its dependence on the particle size, the evaluated velocity is plotted in Fig. 7 as a function of the Ar + laser power. The two solid lines in this figure indicate the theoretical results for the terminal velocity vter of polystyrene microspheres located on the beam axis at the beam waist position. By taking into account the Stokes' drag force of the surrounding fluid, a theoretical value of the terminal velocity is given by [ 18 ] m2 2P vter= 12t/c nw~ Qprd,
(8)
where c is the speed of light in vacuum, P and Woare the total power and the spot size of the illuminating Ar + laser beam, and Qvr is the efficiency factor of the radiation pressure in light scattering by a single spherical particle with the diameter of d. Eq. (8) de-
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Y. Harada, T. Asakura /Optics Communications 107 (199~) 161-169
150 Theo;etical'
/'
'
100 0
o
> 50 ._o
m
//
0 0
20 40 60 80 Laser Power [mW]
Fig. 7. Dependence of the axial component of the particle velocity on the Ar + laser power for the particle suspension having the nominal diameter of 2.062 ttm, together with the theoretical lines evaluated on the basis of the generalized Lorenz-Mie theory for a single particle.
notes that there exists the linear relation between the laser power and the terminal velocity having the linear coefficient of Qp~d. The calculations of Qpr are successfully achieved on a microcomputer by using the program code provided by Bohren and Huffman [ 19 ] where some modifications for the gaussian nature of the laser profile are adopted [20-22 ]. These values are obtained as 0.17985 and 0.29112 for particle suspensions with the diameters of 0.624 ttm and 2.062 tLm, respectively. This implies that, under a rough estimation, the resultant velocity of particles due to the radiation pressure becomes larger for particles having a size-dependence of d 3/2. As is observed experimentally for larger particles, the linear relationship is obvious in the experimental result although there are certain depressions in velocity from the theoretical values for each value of the powers. This discrepancy may come from an overestimation of the total power of the laser beam in the sample cell and an ignorance of the presence of many particles within the beam for theoretical calculations. By taking into consideration this depression of the values in the measured velocity, it is acceptable that the oscillating feature in photon correlation functions has never appeared for smaller particles in the experiment. Thus, the present experimental results support the fact that the motion of suspended particles under investigation is really affected by the illuminating laser light under the size-dependence as pointed out in the previous section. It should be emphasized again that the laser radiation pressure in PCS
misleads the particle diameter especially in larger sample particles. Since only the two different microspheres ranging in the Mie scattering region were used as sample suspensions at the extremely dilute concentration, the exact value of thc size-dependence in both experimental results has not been provided in this study. For further analysis of the effectof the laserradiation pressure on particle suspensions, it is required that wide ranges in the particle size and concentration as well as a spot size and a wavelength of the laser beam must be taken into account. However, it is worthwhile pointing out that thc size-dependences in a decrease of the reciprocal value of the correlation time and the induced uniform flow in the beam propagation direction may provide an additional novel estimation of the particle size. This description implies that a new diagnostic tool for particle suspensions, like the elcctrophoretic light scattering [23 ], may be available by using the radiation pressure-induced phenomcna positivcly with the PCS method. For this purpose, a furthcr theoreticalstudy is required for the radiation pressure of a laserbeam on suspended partides and the induced dynamics of particles.
4. Conclusion As a summary of the experimental investigations for the effects of the laser radiation pressure on the optical particle sizing by means of photon correlation spectroscopy, it can be concluded that the resultant particle size is misled to be larger than the true value under influence of the laser light even with a power of a few tens ofmW. The difference between the true and obtained sizes becomes larger as either the size of particles to be investigated or the laser power increases. From the supplementary experimental study using photon correlation velocimetry, it is found that these results are due to the two dominant changes in a particle behavior induced by the radiation pressure of the illuminating laser. One, which is primarily concerned with the obtained size to be larger, is the suppression of the brownian motion of particles due to the radial component of the radiation pressure, i.e. the gradient force. Another is the acceleration of particles in the beam propagation direction due to the axial component of the radiation pressure. These
Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
changes i n particle d y n a m i c s f r o m the s t a t i o n a r y b r o w n i a n m o t i o n d u e to the laser r a d i a t i o n pressure are fatal d i s a d v a n t a g e s i n p h o t o n c o r r e l a t i o n spectroscopy u s e d for the optical particle sizing. H o w ever, the n o v e l optical m e t h o d for sizing the particle s u s p e n s i o n m a y be a v a i l a b l e o n the basis o f the sized e p e n d e n c e o f the r a d i a t i o n p r e s s u r e - i n d u c e d phen o m e n a after further e x p e r i m e n t a l investigations with the theoretical w o r k s u p p o r t i n g t h e m .
References [ 1 ] B. Chu, Laser light scattering (Academic, New York, 1974); Laser light scatterin~ basic principles and practice, 2nd Ed. (Academic, New York, 1991 ). [2] B.J. Berne and R. Pecora, Dynamic light scattering with applications to chemistry, biology and physics (Wiley, New York, 1976). [3] ICS. Schmitz, An introduction to dynamic light scattering by macromolecules (Academic, San Diego, 1990). [4] H.Z. Cummins and E.R. Pike, eds., Photon correlation and light beating spectroscopy (Plenum, New York, 1973). [5]H.Z. Cummins and E.R. Pike, eds., Photon correlation spectroscopy and velocimetry (Plenum, New York, 1976). [6] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156.
169
[7] A. Ashkin, Sci. Amer. 226 (1972) 62. [ 8 ] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm and S. Chu, Optics Lett. 11 (1986) 288. [9] A. Ashkin, K. Schii, J.M. Dziedzie, U. Euteneuer and M. Schliwa, Nature (London) 348 (1990) 346. [ 10] L. Malmqvist and H.M. Hertz, Optics Comm. 94 (1992) 19. [ 11 ] H.C. van de Hulst, Light scattering by small particles (Wiley, New York, 1957). [ 12] M. Kerker, Scattering of light and other electromagnetic radiation ( Academic, San Diego, 1969). [ 13 ] IC Schnitzel,Appl. Phys. B 42 (1987) 193. [ 14] D.E. Koppel, J. Chem. Phys. 57 (1972) 4814. [ 15 ] A. Einstein, Investigation of theory of brownian movement (Dover, New York, 1956). [ 16] R.S. Hall, Y.S. Oh and C.S. Johnson Jr., J. Chem. Phys. 84 (1980) 756. [ 17 ] J.P. Gordon, Phys. Rev. A 8 ( 1973 ) 14. [ 18 ] K. Schnitzel,W-G. Neumann, J. Mfiller and B. Materzok, Appl. Optics 31 (1992) 770. [ 19 ] C.F. Bohren and D.R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983). [20] G. Gouesbet, G. Grehan and B. Maheu, J. Optics (Pads) 16 (1985) 83. [21 ] G. Gouesbet, B. Maheu and G. Grehan, J. Optics (Pads) 16 (1985) 239. [22] F. Corbin, G. Grehan, G. Gouesbet and B. Maheu, Part. Part. Syst. Charact. 5 (1988) 103. [23] B.R. Ware and W.H. Flygare, Chem. Phys. Lett. 12 ( 1971 ) 81.
ELSEVIER
Full length article
Effects of the laser radiation pressure in photon correlation spectroscopy Yasuhiro Harada, Toshimitsu Asakura Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan
Received 10 September 1993
Abstract
Effects of the radiation pressure of illuminating laser light, even with a low power less than a few tens of roW, on a photon correlation function and a particle size obtained by using photon correlation spectroscopy with the homodyne detection are investigated experimentally for suspensions of polystyrene microspheres. The diffusion coefficient of particle suspension, which is determined from the measured correlation time, decreases with an increase in the power of illuminating laser light. This result stems from suppression of the brownian motion of particles within the probe volume due to a radial component of the radiation pressure of illuminating laser light. It turns out that the particle size evaluated from the correlation time is misled to be larger than the true value under illumination of the laser light. Complemental measurements by photon correlation velocimetry are performed in order to demonstrate the effects of the laser radiation pressure on the particle sizing by means of photon correlation spectroscopy and their dependence on the size of microspheres.
1. Introduction
The dynamic light scattering (DLS) method is one of the most excellent and matured tools for investigating the dynamics and size of particles in suspension such as colloid dispersion of chemical and biological specimens [ 1-3]. In the DLS method, we monitor the statistical time-dependence of intensity fluctuations of light scattered by such dynamic particles undergoing the brownian motion. In general, a temporal correlation function of intensity fluctuations is used to evaluate the statistical nature of scattered light and its decay rate provides information about the hydrodynamic property of particle suspension. To be exact, the diffusion constant of particle suspension can be obtained from a reciprocal of the correlation time. The hydrodynamic diameter of particles can also be evaluated from the measured diffusion coefficient, namely, the correlation time, pro-
vided that the temperature and viscosity of the suspension are known. Most of the optical setups used in the DLS method to investigate the particles system require the use of small-angle coherent detection. In this case, a level of the scattered light at the detector becomes extremely low. Due to the quantum-limited noise performance of single-photon counting and the availability of a digital correlator for processing the number of photoelectron counts, the photon correlation technique is suitable for obtaining the correlation function of weak scattered light fluctuations with the high signal-to-noise ratio. The DLS method with this digital correlation technique is usually referred to as a photon correlation spectroscopy (PCS) [4,5 ]. A primary advantage of the DLS method including PCS is that the diffusion coefficient and size of partides can be rapidly and accurately obtained from measurements of the time-dependence of intensity fluctuations of light scattered by the brownian patti-
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Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
cles. It should be noted that the nondestructive nature of laser light in the scattering process is accepted in the method. In most of the light scattering methods including laser Doppler velocimetry, phase/ Doppler particle analyzer and static light scattering as well as the DLS method, it is considered that the illuminating laser beams do not affect the particles' motion but are merely modulated in their amplitude and phase due to the existence and motion of scattering particles. Since our interest exists in the optical particle sizing by means of the DLS method, a random change in position of the scattering particles, which is governed by the thermodynamic property of the particles system, plays an essential role in the principle of the sizing method. If there is a certain source which distorts or affects the particles' brownian motion, a resultant correlation function of light fluctuations scattered by such particles would also be changed. As a consequence, the evaluated particle size misleads an inaccurate value. Therefore, the nondestructive performance of laser light in the DLS method for investigating the dynamics and size of the particles system is indeed an indispensable requirement. On the other hand, since a proposal and pioneering experimental works of Ashkin et al. [ 6-8 ] for trapping and accelerating small dielectric particles by the single laser beam, the availability of quite new conceptual micromanipulation of small particles including biological cells has been favorably noticed and applied to wide fields from biomechanical researches [ 9 ] to a novel scanning probe microscopy [ 10 ]. This technique is based on the fact that the light itself transports the electromagnetic energy in its propagation direction and that the light also carries the momentum in the same direction. Based on a law of the momentum conservation, a fraction of the momentum of light should be transferred to the particle through scattering phenomena and, therefore, a force called the radiation pressure is exerted on the particle [ 11,12 ] as a source for accelerating it toward the beam propagation direction. From the viewpoint of optical particle sizing based on dynamic light scattering phenomena, in which a random motion of scattering particles plays a principal role, it may be quite natural that a question arises concerning the influence of the laser radiation pressure on the particles' motion and, moreover, on the measured correlation function and the resultant particle size.
In this paper, we investigate experimentally the effects of the radiation pressure of illuminating laser light on the correlation function and the particle size measured for suspensions of polystyrene microspheres by means of PCS with the homodyne detection scheme. The power of the laser light illuminating the particle suspension is employed as a parameter controlling a value of the radiation pressure on the scattering particles. The effects appeared in the homodyne experiment and their dependence on the sample particle size are complemented by another experiment based on photon correlation velocimetry.
2. Effect on particle sizing in photon correlation spectroscopy
2.1. Experimental method Fig. 1 shows the schematic diagram of the experimental setup for investigating the effects of the laser radiation pressure in the system of PCS. While main optical components in the figure are primarily concerned with photon correlation spectroscopy for the MI
. M3
"1:1-
TC
BO
L1 L2 P1~ ' ~ M2~ @
4ND M
/Ar+Laser/.
_ / Ele-Ne
~IM - / Laser /
~
~ ' ~ ....... B-t-IIA~2OF1
SampleCell ,
,,
, IF 2
~
I Photon
OF2~ C o r r e l a t o r
Fig. I. Schematicdiagram of the experimentalsetup. Bo:Ar+ laser beam; BI, B2: HeNe laser beams; M~, M2, M3, M4: mirrors; HM: half mirror; LI, I.~, L3, L4, Ls: lenses; P1, P2, P3, P4: polarizers and analyzers; A], A2, A3, At: apertures and pinholes; HWP: half wavelength plate; IFt, IF2: interference filters; ND: neutral density filter;, OF~, OF2: optical fibers; PMt, PM2: photomultipliers; and TC: thermocouple.
Y. Harada, T. Asakura / Optics Communications 107(1994) 161-169
particle sizing, the whole setup for the experiment of photon correlation velocimetry to be given in the next section is also shown in Fig. 1. It is a common feature of the optical systems in both experiments that a power-controllable Ar ÷ laser is employed as an extra light source for applying the variable radiation pressure on particle suspension in addition to a low-power HeNe laser as an ordinary source for probing the particles' dynamics. This configuration is merely introduced for the purpose of clearing up the appearance of the influence of the laser radiation pressure on experimental results. A similar experimental study may be achieved by using an ordinary optical configuration of the PCS system employing single-beam illumination where the beam power is controllable. An Ar ÷ laser beam Bo with the wavelength of 0.5145 ~tm illuminates a sample cell at its beam waist position with the spot size of 7.32 Ixm. The available power incident on the sample cell ranges from 0 to 60 roW. In addition to this beam, a polarized HeNe laser beam B~ with the wavelength of 0.6328 btm also illuminates the sample cell at its beam waist position, where the spot size is 58 ~tm. The power of this probe beam is decreased and kept at less than 0.1 mW by the neutral density filter ND. In this situation, the light intensity of the beam Bo is, at a maximum, about 4000 times larger than that of the probing beam B~. The beams of Bo and B~ intersect each other in the cell at the fixed angle 0= 85 °. Under dual-beam illumination of Bo and B~, only the probing HeNe laser light scattered by particles in the sample cell is detected by a photomultiplier PM~ (Hamamatsu, R649) through the optical system arranged in the horizontal direction. The scattered and transmitted Ar ÷ laser lights are excluded from entering the detecting aperture A2, by using an interference filter IF~ with the maximum transmission at the wavelength of 0.6328 ~tm. The distance between the apertures A~ and A2 with the diameters of 1 mm and 100 Ixm is 250 mm. Photoelectron pulses from the photomultiplier are pre-amplified, discriminated, and fed into a 64-channel photon correlator (Kowa, ECD-2000) for measuring the photon correlation function of the scattered probe light. Polystyrene microspheres with the diameters of 0.264 ~tm and 2.062 lxm (Duke Scientific) are used as standard particle suspensions. The microspheres of each size are suspended in the deionized purified-water at the volume fraction of O.184
163
ppm within the spectrophotometric sample cell having the thickness of 2 mm. This condition for the sample preparation supports the single scattering event in light scattering by suspended microspheres. The temperature of suspensions is also monitored by the thermo-couple TC to evaluate the particle size from measured photon correlation functions. All measurements and evaluations of photon correlation functions are performed for various values of the power of the Ar + laser beam Bo up to 60 mW, while other parameters including the optical geometry and the temperature of the sample solvent are kept constant for each sample suspension. 2.2. Evaluation method of experimental data
Measured photon correlation functions are evaluated in terms of the normalized photon correlation functions [ 13], g2(T), with the form o f g2(Q = 1 +fl e x p ( - z / z c ) ,
(1)
where r denotes the delay time, fl the relative contrast or the degree of coherence, and Tcthe correlation time defined as the 1/e decay time of an exponential signal component. It is evident from the general theory of dynamic light scattering for particle suspensions undergoing the brownian motion that the correlation time T~ in Eq. ( 1 ) is connected with the diffusion coefficient D of particle suspensions by [ 1-5,13 ] "r~-1 =2q2D,
(2)
where q is the modulus of the scattering vector defined by the optical geometry. According to the experimental setup shown in Fig. 1, it is given by q= (4rt/2)m2 s i n ( 0 / 2 ) ,
(3)
where 2 is the wavelength of laser light in vacuum, m 2 the refractive index of the medium surrounding the particles, and 0 the scattering angle. Therefore, the diffusion coefficient of particles is determined from the correlation time of the photon correlation function with the knowledge of the optical configuration. Furthermore, by using the Stokes-Einstein relation [15] D=kBT/3zt~ld,
(4)
where kB denotes the Boltzmann constant, T the ab-
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Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
solute temperature and t/the viscosity of the solvent, the hydrodynamic particle diameter d of particles can be evaluated from the correlation time or the diffusion coefficient measured. Both the linear least-squares fitting and the cumulant methods [ 14 ] are adopted for reducing the measured correlation function to its normalized version of Eq. ( 1 ), yielding fl, Zc, and the square of expected photon counts ( n ) even though the sample suspensions arc monodisperse. All the digital signal processing for measured correlation functions are performed on a microcomputer (Epson, PC-386GS). A reciprocal value of the correlation time, 1/%, and the consequent particle diameter d are evaluated as a function of the Ar + laser power for investigating the effect of the radiation pressure of the Ar + laser. 2. 3. Results and discussion
Typical dependence of normalized correlation functions on the Ar + laser power is shown three-dimensionally in Figs. 2 and 3 for the particle suspensions of 2.062 ~tm and 0.624 ~tm in diameter, respectively. The sequential results in each figure are obtained under the same thermodynamic conditions for each sample suspension: temperatures T of the solvent are 304.03 K and 300.59 K for Figs. 2 and 3, respectively. Related hydrodynamic and optical parameters and theoretical values of the reciprocal correlation-time for each particle suspension are sum-
09
0
8 o "O
Delay Time [ms] 31.5 0 Laser Power [mW] Fig. 2. Evolutionof normalized photon correlationfunctionsof the light scattered by the particle suspension having the diameters of 2.062 $tmfor six differentpowersof the Ar+ laser.
marized in Table 1. It should be noted that a logarithmic of [g2(z) - 1 ] / p is plotted as a normalized correlation value in Figs. 2 and 3. That is, the slope of each normalized correlation function in these figures just represents the reciprocal of the correlation time, 1/zc, and hence the diffusion coefficient D. Normalized correlation functions in both figures show the tendency of extending to the longer delay time under illumination of the Ar + laser, implying an increase of the correlation time, while this trend is hardly recognized in the correlation function for the sample suspension which contains smaller particles in size (see Fig. 3). To evaluate quantitatively the effect of the Ar + laser illumination on the correlation functions, the reciprocal values of the correlation time % normalized by those without illumination of the Ar + laser are analyzed and shown in Fig. 4 as a function of the Ar + laser power for both particle sizes. As predicted from Figs. 2 and 3, the reciprocal values of the correlation time in both particle suspensions decrease monotonically with an increase in the illuminating Ar + laser power. For the higher power of the Ar + laser, 1/% becomes extremely small in comparison with those in absence of the laser illumination. For example, in the case of larger particles, it becomes less than 60% under the illumination power of 50 mW at most. It is worth pointing out that this effect evidently depends on the size of sample particles. That is, the larger the particle size is, the more remarkable a decrease in the reciprocal of the correlation time becomes. On the other hand, a hydrodynamic diameter of polystyrene microspheres in the suspension can be evaluated from the measured correlation time by using Eqs. (2)- (4). The dependence of the evaluated particle size on the power of the Ar + laser is shown in Fig. 5, where the ordinate value is normalized by the particle diameter obtained without illumination of the Ar + laser. The normalizing factor is 2.424 ttm and 0.636 ttm for particle suspensions with the nominal diameters of 2.062 ~m and 0.624 ~m, respectively. The solid lines are obtained as approximated linear lines for each set of experimental data, whereby the increasing rates of the obtained particle size with regard to the laser power are 1.237%/mW for d=2.062 ttm and 0.217%/mW for d=0.624 ttm, respectively. According to the result in the correlation time, the evaluated particle size is misled to be larger
Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
165
Table 1 Thermodynamic condition for two kinds of suspension of polystyrene mierospheres in the deionized purified water and the resultant reciprocal value of the correlation time of homodyne correlation functions without illumination of the Ar + laser. d [l~m]
T[K]
0.624 2.062
~/[Pas]
300.59 304.03
0.8428× 10 -3 0.7830× 10 -3
m2
1/% [ I / s ]
1.332 1.332
theoretical
experimental
534.52 176.19
530.22 150.53
1 , 6
i
o 1.6
,
i
,
t
d = 2.062
0
"17::~ 1.4
/e
n
._N
o
E O
:~ - 0
-
•
.01
Delay Time [ms] 6.3 0 Laser Power [mW] Fig. 3. Evolution of normalized photon correlation functions of the light scattered by the particle suspension having the diameters of 0.624 ~tm for five different powers of the Ar + laser. '
0
i
'
i
'
[
'
•
9E .hi-: ~ ._~
0.6
' ~ d = 2.062t~m
-8 z
0.6 [
0
,
I
,
I ~
.
20 40 60 80 Laser Power [mW]
Fig. 4. Dependence of the normalized reciprocal of the correlation time on the Ar + laser power: symbols ( • ) for 2.062 ~tm and ( O ) for 0.624 ~tm in particle diameter, respectively.
than the nominal value under the influence of the intense laser beam, and the size-dependence of the increasing rate in the obtained size also exists. This is estimated from the experimental results as the relation of ocd 3/2.
Z
1 ~ I
0
d-- 0.624gm ,
,
,
,
20 40 60 80 Laser Power [mW]
Fig. 5. Dependence of the normalized particle diameter on the Ar + laser power: symbols ( • ) for 2.062 ~tm and ( O ) for 0.624 ~tm in particle diameter, respectively.
The above results can be considered as a result stemming from a change in motion of the brownian particles under illumination of the laser. As seen from Eq. (2), a reciprocal of the correlation time for the scattered probe light has a linear relationship with the diffusion coefficient of particles. The latter quantity denotes physically the rate of a variance in position of the brownian particles. Hence, a decreasing character in the reciprocal value of the correlation time implies that the brownian motion of the particles to be investigated is affected and, in other words, suppressed by the radiation of the Ar + laser within the probing volume. It should be noted that the thermal effect induced by the laser illumination, such as an increase in temperature of the solvent, does not give rise to the present experimental results. In such a case, it turns out as an increase in the reciprocal value of the correlation time [ 16 ], implying the contrary to the present experimental results. Consequently, it is suitable to consider that the suppression of the brownian motion due to the radiation pressure of illuminating laser light occurs in the present PCS study
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Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
as same as the results demonstrated by Ashkin [ 6,7 ] for trapping and accelerating single dielectric particles. As referred in the articles of Ashkin [6,7], the force due to the radiation pressure exerted on particles has two radial and axial components. The radial component of force due to the radiation pressure is closely related with a gradient of the intensity profile of a laser beam. Therefore, it is referred to as a gradient force [ 17 ]. If the refractive index of particles is higher than that of a surrounding medium, the gradient force has the direction toward the optical axis of the illuminating laser beam and drives the particles to approach the beam axis. The particles located at the beam axis experience this force isotropically from all radial directions, and the stable trapping of particles is achieved in the transversal section of the beam. There is also a gradient force in the beam-propagation direction. However, it can be ignored provided that the spot size of the illuminating laser beam is sufficiently greater than the radius of particles. Therefore, the axial component of the force can be explained by using only the momentum transfer associated with scattering phenomena occurring between the propagating laser light and particles undergoing the brownian motion. Finally, being suppressed in the radial random motion, the particles move along the beam axis with the size-dependent constant velocity, called as a terminal velocity, due to a drag force of the surrounding viscous medium. This is in the same condition with the present experimental study on the optical particle sizing of suspensions by means of PCS where the gradient force is indeed a main source for suppressing the brownian motion of particles except for the sample concentration. As is well known, photon correlation spectroscopy with a homodyne detection scheme is insensitive to the uniform motion of ensemble particles, which is induced by the axial component of the radiation pressure, only the suppression of the brownian motion of particles is observed in the photon correlation function. If we can demonstrate the above effect of the axial component of the laser radiation pressure and its dependence on particle size, it will turn out as an evidence of the effect appeared in the present PCS study. Another type of the experimental investigation demonstrating the size-dependent effect due to
the axial component of the radiation pressure has been performed by employing the optical configuration for photon correlation velocimetry, and is described in the following section.
3. Detection of uniform flow due to radiation pressure 3. I. Experimental method
Detection of the uniform flow of many particles due to the radiation pressure of the Ar + laser in the propagation direction of an illuminating beam is accomplished under the same experimental condition for sample suspensions in the previous PCS study. However, there is a different point in the experimental setup that the dual-beam illumination of the HeNe laser and the heterodyne detection of the scattered probe light have been employed. As depicted in Fig. 1, the two HeNe laser beams of B~ and B2 illuminate the sample cell at their beam waist positions intersecting each other at an angle q = 10 °. The spot size of each beam is 58 ttm which is the same in the proceeding section. Scattered probe lights for both beams of Bt and B2by sample particles are collected by using the imaging optical setup consisting of aperture A3, lens L5, polarizer P4, interference filter IF2, and pinhole Ps. Then, they are optically mixed on a photosensitive surface of the photomultiplier PM2 (Hamamatsu, R649), producing the photoelectron pulses. By using a photon corrclator, the time-dependence of photoelectron pulses is evaluated in terms of a photon correlation function which contains the information about the magnitude of velocity v in the same direction with the propagation of the Ar + laser beam as a decay of the envelope and an interval of the oscillating features. An expression of the photon correlation function is given in a normalized version [ 13 ] by g2(z)=exp(-v2z2/2w2)[l
+flcos(ktrr) ] ,
(5)
where Wl is the radius of the probe volume, and k the modulus of the difference between the scattering vectors defined by individual illumination-detection directions. According to the optical configuration of the dual-beam illumination shown in Fig. 1, it is given, independently of the direction of a detecting aperture, as follows:
Y. Harada, T. Asakura / Optics Communications 107(1994) 161-169
k = (4rt/2)m2 s i n ( ~ / 2 ) .
(6)
Therefore, from the knowledge about the optical configuration and the property of the solvent fluid, the velocity v of particles moving in parallel to the beam propagation direction can be evaluated by determining the oscillation interval T¢ of the photon correlation function. The relation which connects the particle velocity v and the oscillation interval T~ is given by 2
v= 2m2T~ sin(~/2) "
(7)
Determinations of the oscillation interval and the relevant velocity are performed by using a fast Fourier transform (FFT) algorithm for photon correlation functions measured with the various values of the Ar + laser beam. ' The described experimental setup which employs the photon counting mode for detecting the scattered light intensity and the digital signal processing by means of a photon eorrelator is usually referred to as a photon correlation velocimeter [ 5 ] for the purpose of discriminating it from a laser Doppler velocimeter performing under the analogue processing mode.
3.2. Results and discussion Fig. 6 shows the three-dimensional representation of normalized photon correlation functions obtained
0.5
s2
Delay Time [ms] 630 0 Laser Power [mW] Fig. 6. Evolutionof normalized photon correlationfunctionsof the light scatteredby particle suspensions having the diameters of 2.062 Ixm for four different powers ofAr+ laser in the use of the optical configurationof photon correlationveloeimetry.
167
for particle suspensions with the diameter of 2.062 I~m for four different powers of the Ar + laser. Apparent oscillations occur with the constant time-interval in the correlation function and the interval Tc may become shorter with an increase in the laser power. This implies that the particles in the probing volume are indeed in an uniform flow in the propagation direction of the laser beam and under the influence of the radiation power. It can be concluded from these results that the axial component of the radiation pressure due to the Ar + laser is a source for inducing the uniform flow of particles. On the other hand, these results in photon correlation functions for particle suspension having the smaller diameter of 0.624 ~tm could not be observed. Note the fact that smaller particles in suspension change their positions with the rate higher than larger particles according to the theory of brownian motion (see Eq. (4)). The radiant power of the Ar + laser is not high enough to suppress the brownian motion of smaller particles in this experimental condition. In other words, the trapping process with regard to the cross sectional direction of the laser beam is not fully achieved for the smaller particles and, therefore, the next process of an acceleration also never happens. Nevertheless, these resuits denote that there exists also the size-dependence in the effect of the axial component of the laser radiation pressure as is the same with the results in the previous section. In order to demonstrate quantitatively the relationship between the radiant power and the velocity of particles and its dependence on the particle size, the evaluated velocity is plotted in Fig. 7 as a function of the Ar + laser power. The two solid lines in this figure indicate the theoretical results for the terminal velocity vter of polystyrene microspheres located on the beam axis at the beam waist position. By taking into account the Stokes' drag force of the surrounding fluid, a theoretical value of the terminal velocity is given by [ 18 ] m2 2P vter= 12t/c nw~ Qprd,
(8)
where c is the speed of light in vacuum, P and Woare the total power and the spot size of the illuminating Ar + laser beam, and Qvr is the efficiency factor of the radiation pressure in light scattering by a single spherical particle with the diameter of d. Eq. (8) de-
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Y. Harada, T. Asakura /Optics Communications 107 (199~) 161-169
150 Theo;etical'
/'
'
100 0
o
> 50 ._o
m
//
0 0
20 40 60 80 Laser Power [mW]
Fig. 7. Dependence of the axial component of the particle velocity on the Ar + laser power for the particle suspension having the nominal diameter of 2.062 ttm, together with the theoretical lines evaluated on the basis of the generalized Lorenz-Mie theory for a single particle.
notes that there exists the linear relation between the laser power and the terminal velocity having the linear coefficient of Qp~d. The calculations of Qpr are successfully achieved on a microcomputer by using the program code provided by Bohren and Huffman [ 19 ] where some modifications for the gaussian nature of the laser profile are adopted [20-22 ]. These values are obtained as 0.17985 and 0.29112 for particle suspensions with the diameters of 0.624 ttm and 2.062 tLm, respectively. This implies that, under a rough estimation, the resultant velocity of particles due to the radiation pressure becomes larger for particles having a size-dependence of d 3/2. As is observed experimentally for larger particles, the linear relationship is obvious in the experimental result although there are certain depressions in velocity from the theoretical values for each value of the powers. This discrepancy may come from an overestimation of the total power of the laser beam in the sample cell and an ignorance of the presence of many particles within the beam for theoretical calculations. By taking into consideration this depression of the values in the measured velocity, it is acceptable that the oscillating feature in photon correlation functions has never appeared for smaller particles in the experiment. Thus, the present experimental results support the fact that the motion of suspended particles under investigation is really affected by the illuminating laser light under the size-dependence as pointed out in the previous section. It should be emphasized again that the laser radiation pressure in PCS
misleads the particle diameter especially in larger sample particles. Since only the two different microspheres ranging in the Mie scattering region were used as sample suspensions at the extremely dilute concentration, the exact value of thc size-dependence in both experimental results has not been provided in this study. For further analysis of the effectof the laserradiation pressure on particle suspensions, it is required that wide ranges in the particle size and concentration as well as a spot size and a wavelength of the laser beam must be taken into account. However, it is worthwhile pointing out that thc size-dependences in a decrease of the reciprocal value of the correlation time and the induced uniform flow in the beam propagation direction may provide an additional novel estimation of the particle size. This description implies that a new diagnostic tool for particle suspensions, like the elcctrophoretic light scattering [23 ], may be available by using the radiation pressure-induced phenomcna positivcly with the PCS method. For this purpose, a furthcr theoreticalstudy is required for the radiation pressure of a laserbeam on suspended partides and the induced dynamics of particles.
4. Conclusion As a summary of the experimental investigations for the effects of the laser radiation pressure on the optical particle sizing by means of photon correlation spectroscopy, it can be concluded that the resultant particle size is misled to be larger than the true value under influence of the laser light even with a power of a few tens ofmW. The difference between the true and obtained sizes becomes larger as either the size of particles to be investigated or the laser power increases. From the supplementary experimental study using photon correlation velocimetry, it is found that these results are due to the two dominant changes in a particle behavior induced by the radiation pressure of the illuminating laser. One, which is primarily concerned with the obtained size to be larger, is the suppression of the brownian motion of particles due to the radial component of the radiation pressure, i.e. the gradient force. Another is the acceleration of particles in the beam propagation direction due to the axial component of the radiation pressure. These
Y. Harada, T. Asakura / Optics Communications 107 (1994) 161-169
changes i n particle d y n a m i c s f r o m the s t a t i o n a r y b r o w n i a n m o t i o n d u e to the laser r a d i a t i o n pressure are fatal d i s a d v a n t a g e s i n p h o t o n c o r r e l a t i o n spectroscopy u s e d for the optical particle sizing. H o w ever, the n o v e l optical m e t h o d for sizing the particle s u s p e n s i o n m a y be a v a i l a b l e o n the basis o f the sized e p e n d e n c e o f the r a d i a t i o n p r e s s u r e - i n d u c e d phen o m e n a after further e x p e r i m e n t a l investigations with the theoretical w o r k s u p p o r t i n g t h e m .
References [ 1 ] B. Chu, Laser light scattering (Academic, New York, 1974); Laser light scatterin~ basic principles and practice, 2nd Ed. (Academic, New York, 1991 ). [2] B.J. Berne and R. Pecora, Dynamic light scattering with applications to chemistry, biology and physics (Wiley, New York, 1976). [3] ICS. Schmitz, An introduction to dynamic light scattering by macromolecules (Academic, San Diego, 1990). [4] H.Z. Cummins and E.R. Pike, eds., Photon correlation and light beating spectroscopy (Plenum, New York, 1973). [5]H.Z. Cummins and E.R. Pike, eds., Photon correlation spectroscopy and velocimetry (Plenum, New York, 1976). [6] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156.
169
[7] A. Ashkin, Sci. Amer. 226 (1972) 62. [ 8 ] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm and S. Chu, Optics Lett. 11 (1986) 288. [9] A. Ashkin, K. Schii, J.M. Dziedzie, U. Euteneuer and M. Schliwa, Nature (London) 348 (1990) 346. [ 10] L. Malmqvist and H.M. Hertz, Optics Comm. 94 (1992) 19. [ 11 ] H.C. van de Hulst, Light scattering by small particles (Wiley, New York, 1957). [ 12] M. Kerker, Scattering of light and other electromagnetic radiation ( Academic, San Diego, 1969). [ 13 ] IC Schnitzel,Appl. Phys. B 42 (1987) 193. [ 14] D.E. Koppel, J. Chem. Phys. 57 (1972) 4814. [ 15 ] A. Einstein, Investigation of theory of brownian movement (Dover, New York, 1956). [ 16] R.S. Hall, Y.S. Oh and C.S. Johnson Jr., J. Chem. Phys. 84 (1980) 756. [ 17 ] J.P. Gordon, Phys. Rev. A 8 ( 1973 ) 14. [ 18 ] K. Schnitzel,W-G. Neumann, J. Mfiller and B. Materzok, Appl. Optics 31 (1992) 770. [ 19 ] C.F. Bohren and D.R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983). [20] G. Gouesbet, G. Grehan and B. Maheu, J. Optics (Pads) 16 (1985) 83. [21 ] G. Gouesbet, B. Maheu and G. Grehan, J. Optics (Pads) 16 (1985) 239. [22] F. Corbin, G. Grehan, G. Gouesbet and B. Maheu, Part. Part. Syst. Charact. 5 (1988) 103. [23] B.R. Ware and W.H. Flygare, Chem. Phys. Lett. 12 ( 1971 ) 81.