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Photon random walk in the frequency domain
OPTICS
Optics Communications 92 (1992) 6-11 North-Holland
COMMUNICATIONS
Photon random walk in the frequency domain K o e n Clays l a n d Andrd Persoons Laboratoryof Chemicaland BiologicalDynamics, UniversityofLeuven, Celestijnenlaan200D, 3001Leuven,Belgium Received 9 March 1992; revised manuscript received 16 April 1992
We report on the absolutedetermination of the transport mean free path length of multiple scatteredphotons by multifrequency phasefluorometry. The time delay between the injection of an ultrashort light pulse and the subsequent detection at a certain separation is measured as a phase-shift induced on the Fourier-comi~onentsof the light pulse. The diffusion approximation enables the calculation of the number of steps and the transport mean free path from this time delay. Advantagesof this technique over the time domain technique [Watson et al., Phys. Rev. Lett. 58 (1987) 945 ] are discussed.
1. Introduction Static light scattering (SLS) is an invaluable tool for the determination of the size and the shape of (macro)molecules [ 1,2]. Quasi-elastic light scattering (QELS) can be used to study dynamic processes [3 ]. The autocorrclation of the intensity fluctuations of the scattered light contains information on the dynamics of the scattering particles. Normally, the SLS and QELS experiments must be performed on dilute samples, since only single scattering events are taken into account in the analysis. The functional form of the intensity autocorrelation function in the case of multiple scattering has been derived [4] based on the following assumptions: first, the frequency change upon the scattering event is negligible (propagation of a monochromatic wave), second, the propagation of the light in the medium is diffusive (a random walk, determined only by the number of steps N with length l, the transport mean free path) and third, the structure factor C(q, to), where q and to are the momentum and frequency transfer [3 ], does not vary rapidly with q (the dimensions of the scattering particle are small compared to the wavelength). Experimental measurements of the intensity autocorrelation function for multiple scattering and an approximate Koen Clays is a Research Associate of the Belgian National Fund for ScientificResearch.
6
treatment of the data have been made [ 5-7 ]. The effect of the short-range interparticle correlation was demonstrated not to affect the validity of the diffusion approximation of light in correlated suspensions [5 ]. The expression for the transport mean free path length l needs only to bc slightly modified when interparticle correlations are present. The form factor F(q) must be replaced by F(q) S(q) with S(q) the local structure factor for a domain of correlated scatterers. The rescaled l*, the transport mean free path for the correlated particles, is no longer proportional to the volume fraction of the scattering particles, because of the concentration dependence of S(q). Until now, this transport mean free path has been determined indirectly by measuring the ratio of the transmission through a sample with unknown l* to the transmission through a stock suspension with known 1" [5,8]. Recently the transport mean free path length was determined directly in the time domain [9]. In this communication, we present an alternative technique for the determination of the transport mean free path length l and the number of steps N for a random walk of the light intensity.
2. Experiment A light pulse takes a time S/v to travel over a distance S with speed v. In our experiments, when no
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scattering particles are present in the medium, the distance S is simply the separation between the two fiber tips used to inject and detect the light pulse in the medium. When the pulse is injected into a highly scattering medium, the total distance over which the photons travel in the medium is larger, because of the multiple scattering by the particles. This scattering can be well described in terms of a random walk of the light intensity, characterized by the transport mean free path length I and the number of steps N [ 5 ]. The relationship between total distance S and step size I for such a walk is given by S 2= Nl2g where N is the number of steps and g the Kirkwood correlation parameter g = ( Y cos Ou) with O~j the angle between the wave vectors k~ and kj for 2 subsequent steps i andj. When the scattering from different particles is uncorrelated, g = 1 and S 2=NI 2. When interparticle correlation is present, only I is rescaled to 1" and g remains 1. A domain of correlated scatterers is considered as a single effective scatterer with its form factor F(q) modified by the local structure factor S(q). When the velocity of the light in the medium of interest is known, measurement of the time delay between launching and detection of the light pulse enables the retrieval of the total length of the diffusion path NI*. The number of steps N and the transport mean free path length l* are then readily derived. The time delay between the launching of the light pulse in the scattering medium and the detection at a separation S has been measured by multifrequency phasefluorometry [ 10,11 ]. In this technique, the finite lifetime of the excited state induces a phase-shift on the frequency components of the exciting light pulse. The analysis of the fluorescence decay involves the sine and cosine Fourier-transform S(to) and G (to) of the decay function F(t). The phase-shift is then 0f=tan -1 [S(to)/G(to) ]. For a single-exponential fluorescence decay F(t)=F(0) e x p ( - (t/ Tf) ), with F(t) the fluorescence intensity at time t and zf the fluorescence lifetime, the phase-shift as a function of frequency is Of= tan- 1(2~fic), where f i s the frequency of the Fourier-component. By frequency-stabilizing the output pulse-train for the excitation [ 12 ], precise phase-shifts have offered the possibility to resolve even complex fluorescence decay into individually emitting species [ 10 ]. In our experiments, the launching and detecting
15 August 1992
fiber bundles (statistically interwoven multimode fibers) were directed with their axes parallel and with a separation of (28_+ 1) mm. The fiber tips were placed in the center of a container of 30 × 30 × 30 cm 3, large enough to eliminate detection of light reflected from the boufidaries. The highly scattering medium consisted of triglycerides, suspended in water (Intralipid 20% soybean oil) at 293 K. Intralipid is a fat emulsion widely used as the scattering component in tissue phantom to investigate the propagation of light in tissue. Optical properties of tissues are important in photodynamic therapy as a treatment for cancer [ i 3,14 ]. The optical properties of this aqueous suspeasion have ,been studied. The results indicate that In~ralipid combines a high scattering coefficient witl~ a low absorption coefficient [ 13 ]. The general shape and the particle size distribution of Intralipid have been determined. Electronmicroscope photographs of an Intralipid preparation show the generally spherical shape of the scattering particles [ 14 ]. The mean and the standard deviation of the particle size distribution of 1436 measured Intralipid particles, determined by transmission electron microscopy, is 97 and 3 nm, respectively [ 14]. Measurements of the time delay Td were performed on suspensions with different volume fractions. When there is only ia constant time delay zd, the light intensity is described as I(t)=I(0)8(t--zd). The phase-shift is then given by ~a,i=2gf(Ta+zi). The instrumental time delay Ti is eliminated by measuring the phase-shift ~i=2~fzi with the injection and detection fiber end-to-end. By subtracting this phase-shift 0~ for Zero time-delay from Ca,i, the phase-shift ~a=21tfzd is obtained. The approximation is made that, to first order, the shape of the pulse is not affected by the multiple scattering. There is no need to know the exact shape of the injected light pulse, since a reference phase-measurement at all frequencies of interest iis performed. Hereby we take advantage of the F0urier-transform method in avoiding the calculation of the convolution-integral. The fluorescence contribution to the detected signal can be expected to be negligible in the case of a suspension of triglyceridos. Additionally, the emission monochromator of the iphasefluorometer ensures that only scattering at the laser wavelength (600.0_+ 0.5 nm) is detected. The linear relationship between the phase-shift ~a and the frequency of the Fouriercom-
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MODULATION FREQUENCY (MHz)
Fig. 1. Phase-shift~das a function of the frequencyfofthe Fourier-componentof the light pulse at different volume fractions of a tdglyceride suspension in water (Intralipid 20% soybean oil) (A) 0.2174, (B) 0.1359, (C) 0.0543, (D) 0.0136. Solid lines are theoretical curves accordingto ed= 2~f~d,with ftruncated at 30 MHz.
ponent is illustrated in fig. 1. This linearity up to at least 90 degrees is a good indication that fluorescence contributions are negligible. Indeed, the phaseshift for fluorescence is a non-linear function of frequency and limited to the range of 0 to 90 degrees. From the slope of the curves in fig. 1, the time delay Zd was calculated. Between two scattering events on a triglyceride particle, the photon can be assumed to travel in water, since the scattering process is caused by density differences. With a value of 1.333 for the refractive index of water at 293 K and the derived time delay %, the total length of the diffusion path Nl* has been deduced. Finally, from this length and the separation between the fiber tips S, the number of steps N and the step size l* have been derived. The obtained values for different volume fractions of triglyceride suspensions in water are given in table 1. In fig. 2 the inverse of the transport mean free path length obtained from the data presented in fig. l, is plotted as a function of the volume fraction.
3. Discussion
For the determination of fluorescence lifetimes with phasefluorometry, there is no need to know the
15 August 1992
exact excitation pulse profile. Presently, in deriving the number of steps N and the transport mean free path length l* for multiple scattering, the assumption has been made that the pulse shape was not significantly altered by the scattering process. Only then is the linear dependence ¢d = 2XfXd valid. In practice however, the pulse shape is distorted, because different photons make different random walks, with a different global length of the diffusion path Nl*, to arrive at the detecting fiber tip. This results in a distribution of time delays and hence, in a different pulse shape at the detecting tip. A detailed analysis of the pulse shape [9 ] shows that the rising edges of the delayed pulses are nearly identical when all the scattering particles have nearly the same diffusion constants. This is true for a monodisperse suspension. The trailing edges are dominated by light absorption. In our experiments, the containing vessel is large enough to eliminate back-reflections into the medium. Hence, the diffusion of photons far away from the detecting fiber tip can be considered as absorption, since the probability to arrive at the detecting tip is considerably reduced. The two effects mentioned are expected to reduce the pulse shape distortion. The influence of the pulse shape has further been reduced by taking the slope over the Fouriercomponents below 30 MHz only, as shown in fig. 1. It has been shown that the lower modulation frequencies are less attenuated than the higher frequencies [ 16 ]. Limiting the data analysis to 30 MHz modulation frequency then ensures a negligible effect of absorption. Measurements over the entire bandwidth of the phasefluorometer, as shown in fig. 3, reveal a deviation of the linearity for the components above 50 MHz. Note that the phase-shift ~a amounts to more than 250 degrees, well above the limit of 90 degrees for the phase-shift in fluorescence analysis #~. Therefore, the large phase-shifts are attributed to the time delay % between the injection and the detection of the light pulses. ~ The limit of 90 degrees phase-shiftin'phasefluorometry can be overcomein the exceptional case of excited state reactions of fluorescent reagentstowards fluorescentreaction products, produced in the excited state also. Additionally, the rate constant of the excited state interconversionmust be of the same order of magnitude of the inverseof the rcagent'sexcitedstate lifetime, to proceed to a detectable extent. These conditions are clearlynot fulfilled.
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15 August 1992
Table 1 Experimentally determined slope 2g% of the phase-shift 0d as a function of frequencyfof the Fourier-component, calculated delay time %, total length NI* of the random walk, number of steps N and transport mean free path length 1" for different volume fractions of triglycerides in water at 293 IC The values between parentheses are the estimated uncertainties on the retrieved values. Volume fraction
2~rrd (deg M H z - 1)
% (ns)
0.0136 0.0543 0.1359 0.2174
0.85 1.60 2.45 3.17
2.37 4.44 6.79 8.80
(0.05) (0.05) (0.05) (0.05)
(0.15) (0.15) (0.15) (0.15)
NI* (mm )
N
535 1000 1530 1990
360 1280 3000 5030
(30) (30) (30) (30)
3.0
l* (mm) (30) (55) (85) (110)
1.47 (0.08) 0.78 (0.02) 0.51 (0.01) 0.395 (0.006)
300 A
2.5
250
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a
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0.25
0
I
i
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150
200
250
300
MODULATION FREQUENCY ( M H z )
Fig. 2. The inverse of the transport mean flee path length of photons 1/1" as a function of volume fraction for Intralipid 20% soybean oil suspension in water. The line shows the trend in the data.
A qualitative analysis of the phase-shifts at the hig,h-frequency Fourier-components involves the sine and cosine Fourier-transform of the solution o f the diffusion equation
O0/Ot =DV2C~-T~+N6(x)6(y)6(z-S)6(t),
50
Fig. 3. Measurements of the phase-shift 0d over the entire bandwidth of the phasefluorime~er, for different volume fractions of the triglyceride in aqueous sqspension. Solid lines are fitted curves according to eq. (2). The inset is shown in fig. 1 at a higher magniftcation, where the solid lines are the theoretical curves according to 0d = 2grid. tector are immersed in an infinite scattering and nonabsorbing m e d i u m [ 16 ],
¢=S[6xf/vll ~/2 (1)
where ¢~(t) is the density of the photons at x, y, z at a time t, ~, is the absorption parameter, D= l'v~3 is the photon diffusion coefficient and N photons are injected at t=O. A solution for eq. ( 1 ) has been given in terms of cylindrical Bessel functions for a different geometry [ 9 ]. An approximate expression has been derived in terms of D, ~, S and t only. An approximate analytical expression for the phase shift has been derived in the case where source and de-
(2)
Figure 3 shows the experimentally obtained phase shifts, together with the best fit to eq. (2). It is clear that, for low volume fractions of Intralipid, the approximation is satisfactory, but, for high volume fractions, there is a large discrepancy between the experimental data and eq. (2). This discrepancy cannot be attributed to absorption, since the deviation is largest for the lower modulation frequencies, where the effect of absorption is lowest [ 16 ]. Furthermore, Intralipid is known to have a negligible absorption
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
coefficient [ 13 ]. Phase measurements as a function of distance S between the fiber tips should then give the same value for the transport mean free path l*. Linearization of the exact analytical expression for the phase shift 0 has been suggested in a theoretical frequency domain study and it has been inferred that this linear phase behaviour is characteristic of a time delay between the excitation pulse and the detected pulse [ 17 ]. The plot of the inverse of the transport mean free path as a function of volume fraction (fig. 2 ) shows a large deviation from linearity. A linear dependence is expected without particle correlation. To explain the deviation, an analytical expression for the hardcore interaction structure factor has been presented earlier [ 15 ]. To simulate such a system, polystyrene spheres are usually suspended in water with salt added to screen the electrostatic repulsion. Good agreement between theory and experiment has been observed [5]. For a suspension of triglycerides, the same hard-core structure factor is probably inappropriate. Although the multifrequency phasefluorimeter used for the present study is a rather complex and expensive pulsed-laser based instrument [ 10 ] with additional phase stabilization [ 12 ], it is clear that the 300 MHz bandwidth of the instrument, or the large number of phase-frequency data points, is not essential for the determination of the slope of the phase, plotted as a linear function of the modulation frequency for frequencies up to 30 MHz. In fact, inexpensive lamp-based, commercially available phasefluorimeters with three modulation frequencies over a limited bandwidth are satisfactory for this type of experiment. Thus, the application of the more compact and less expensive frequency domain phasefluorometry technique in the study of multiple light scattering in tissue can eliminate the complex and expensive single photon counting technique. For diagnostic purposes, the possibility for a continuous reading of the phase shift 0d offers the perspective of continuous monitoring of optical properties of tissue.
4. Conclusion In conclusion, we have shown that the absolute determination of the relevant parameter in the diffu10
15 August 1992
sion approximation of multiple light scattering can be performed with the help of a simple description and readily available instrumentation. In aqueous suspensions of triglycerides, the inverse of the transport mean free path length l* does not show a linear dependence on the volume fraction of the scatterers. The same qualitative behaviour is observed here as for suspensions of hard spheres. This method could be of use in the determination of aggregation numbers of micellar solutions as well as concentrated dispersions, provided they are close to being monodisperse. Finally this work is expected to stimulate the development of analytical expressions for the local structure factor S(q) for various types of interparticle correlation.
Acknowledgements This research was supported by research grants from the Belgian government (GOA 87 / 91-109 ) and from the Belgian National Fund for Scientific Research (FKFO 2.9003.90). The authors wish to thank Dr. L. De Maeyer for stimulating discussions.
References [ 1 ] M. Kerker, The scattering of light and other electromagnetic radiation (Academic, New York, 1969). [2] W. Van De Sande and A. Persoons, J. Chem. Phys. 89 ( 1985 ) 404. [3] B.J. Berne and R. Pecora, Dynamic light scattering (Wiley, New York, 1975 ). [4] M.J. Stephen, Phys. Rev. B 37 (1988) 1. [ 5 ] S. Fraden and G. Maret, Phys. Rev. Lett. 65 (1990) 512. [6 ] P.M. Saulnier, M.P. Zinkin and G.H. Watson, Phys. Rev. B 42 (1990) 2621. [ 7 ] D.J. Pine, D.A. Weitz, P.M. Chaikin and E. Herbolzheimer, Phys. Rev. Lett. 60 ( 1988 ) 1134. [8] M.B. van der Mark, M.P. van Albada and A. Lagendijk, Phys. Rev. B 37 (1988) 3575. [9] G.H. Watson, Jr., P.A. Fleury and S.L. McCall, Phys. Rev. Lett. 58 (1987) 945. [ 10] IC Clays, J. Jannes, Y. Engelborghs and A. Persoons, J. Phys. E22 (1989) 297. [ 11]J.R. Lakowicz, G. Laczko and I. Gryczynski, Rev. Sci. Instrum. 57 (1986) 2499. [ 12] IC Clays and A. Persoons, Appl. Optics 27 (1988) 3601. [ 13] I. Driver, J.W: Feather, P.R. King and'J.B. Dawson, Phys. Med. Biol. 34 (1989) 1027.
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[ 14] H.J. van Staveren, C.J.M. Moes, J. van Made, S.A. Prahl and M.J.C. van Gemert, Appl. Optics 30 ( 1991 ) 4507. [15] M.S. Wertheim, Phys. Rev. Lett. 10 (1965) 321. [16] J. Fishkin, E. Gratton, M.J. vandeVen and William W.
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Mantulin, Proc. Soc. Photo-Opt. Instrum. Eng. 1431 (1991) 122. [ 17] M.S. Patterson, J.D. Moulton, B.C. Wilson and B. Chance, Proc. Soc. Photo-Opt. instrum. Eng. 1203 (1990) 62.
11
Optics Communications 92 (1992) 6-11 North-Holland
COMMUNICATIONS
Photon random walk in the frequency domain K o e n Clays l a n d Andrd Persoons Laboratoryof Chemicaland BiologicalDynamics, UniversityofLeuven, Celestijnenlaan200D, 3001Leuven,Belgium Received 9 March 1992; revised manuscript received 16 April 1992
We report on the absolutedetermination of the transport mean free path length of multiple scatteredphotons by multifrequency phasefluorometry. The time delay between the injection of an ultrashort light pulse and the subsequent detection at a certain separation is measured as a phase-shift induced on the Fourier-comi~onentsof the light pulse. The diffusion approximation enables the calculation of the number of steps and the transport mean free path from this time delay. Advantagesof this technique over the time domain technique [Watson et al., Phys. Rev. Lett. 58 (1987) 945 ] are discussed.
1. Introduction Static light scattering (SLS) is an invaluable tool for the determination of the size and the shape of (macro)molecules [ 1,2]. Quasi-elastic light scattering (QELS) can be used to study dynamic processes [3 ]. The autocorrclation of the intensity fluctuations of the scattered light contains information on the dynamics of the scattering particles. Normally, the SLS and QELS experiments must be performed on dilute samples, since only single scattering events are taken into account in the analysis. The functional form of the intensity autocorrelation function in the case of multiple scattering has been derived [4] based on the following assumptions: first, the frequency change upon the scattering event is negligible (propagation of a monochromatic wave), second, the propagation of the light in the medium is diffusive (a random walk, determined only by the number of steps N with length l, the transport mean free path) and third, the structure factor C(q, to), where q and to are the momentum and frequency transfer [3 ], does not vary rapidly with q (the dimensions of the scattering particle are small compared to the wavelength). Experimental measurements of the intensity autocorrelation function for multiple scattering and an approximate Koen Clays is a Research Associate of the Belgian National Fund for ScientificResearch.
6
treatment of the data have been made [ 5-7 ]. The effect of the short-range interparticle correlation was demonstrated not to affect the validity of the diffusion approximation of light in correlated suspensions [5 ]. The expression for the transport mean free path length l needs only to bc slightly modified when interparticle correlations are present. The form factor F(q) must be replaced by F(q) S(q) with S(q) the local structure factor for a domain of correlated scatterers. The rescaled l*, the transport mean free path for the correlated particles, is no longer proportional to the volume fraction of the scattering particles, because of the concentration dependence of S(q). Until now, this transport mean free path has been determined indirectly by measuring the ratio of the transmission through a sample with unknown l* to the transmission through a stock suspension with known 1" [5,8]. Recently the transport mean free path length was determined directly in the time domain [9]. In this communication, we present an alternative technique for the determination of the transport mean free path length l and the number of steps N for a random walk of the light intensity.
2. Experiment A light pulse takes a time S/v to travel over a distance S with speed v. In our experiments, when no
0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume92, number 1,2,3
OPTICS COMMUNICATIONS
scattering particles are present in the medium, the distance S is simply the separation between the two fiber tips used to inject and detect the light pulse in the medium. When the pulse is injected into a highly scattering medium, the total distance over which the photons travel in the medium is larger, because of the multiple scattering by the particles. This scattering can be well described in terms of a random walk of the light intensity, characterized by the transport mean free path length I and the number of steps N [ 5 ]. The relationship between total distance S and step size I for such a walk is given by S 2= Nl2g where N is the number of steps and g the Kirkwood correlation parameter g = ( Y cos Ou) with O~j the angle between the wave vectors k~ and kj for 2 subsequent steps i andj. When the scattering from different particles is uncorrelated, g = 1 and S 2=NI 2. When interparticle correlation is present, only I is rescaled to 1" and g remains 1. A domain of correlated scatterers is considered as a single effective scatterer with its form factor F(q) modified by the local structure factor S(q). When the velocity of the light in the medium of interest is known, measurement of the time delay between launching and detection of the light pulse enables the retrieval of the total length of the diffusion path NI*. The number of steps N and the transport mean free path length l* are then readily derived. The time delay between the launching of the light pulse in the scattering medium and the detection at a separation S has been measured by multifrequency phasefluorometry [ 10,11 ]. In this technique, the finite lifetime of the excited state induces a phase-shift on the frequency components of the exciting light pulse. The analysis of the fluorescence decay involves the sine and cosine Fourier-transform S(to) and G (to) of the decay function F(t). The phase-shift is then 0f=tan -1 [S(to)/G(to) ]. For a single-exponential fluorescence decay F(t)=F(0) e x p ( - (t/ Tf) ), with F(t) the fluorescence intensity at time t and zf the fluorescence lifetime, the phase-shift as a function of frequency is Of= tan- 1(2~fic), where f i s the frequency of the Fourier-component. By frequency-stabilizing the output pulse-train for the excitation [ 12 ], precise phase-shifts have offered the possibility to resolve even complex fluorescence decay into individually emitting species [ 10 ]. In our experiments, the launching and detecting
15 August 1992
fiber bundles (statistically interwoven multimode fibers) were directed with their axes parallel and with a separation of (28_+ 1) mm. The fiber tips were placed in the center of a container of 30 × 30 × 30 cm 3, large enough to eliminate detection of light reflected from the boufidaries. The highly scattering medium consisted of triglycerides, suspended in water (Intralipid 20% soybean oil) at 293 K. Intralipid is a fat emulsion widely used as the scattering component in tissue phantom to investigate the propagation of light in tissue. Optical properties of tissues are important in photodynamic therapy as a treatment for cancer [ i 3,14 ]. The optical properties of this aqueous suspeasion have ,been studied. The results indicate that In~ralipid combines a high scattering coefficient witl~ a low absorption coefficient [ 13 ]. The general shape and the particle size distribution of Intralipid have been determined. Electronmicroscope photographs of an Intralipid preparation show the generally spherical shape of the scattering particles [ 14 ]. The mean and the standard deviation of the particle size distribution of 1436 measured Intralipid particles, determined by transmission electron microscopy, is 97 and 3 nm, respectively [ 14]. Measurements of the time delay Td were performed on suspensions with different volume fractions. When there is only ia constant time delay zd, the light intensity is described as I(t)=I(0)8(t--zd). The phase-shift is then given by ~a,i=2gf(Ta+zi). The instrumental time delay Ti is eliminated by measuring the phase-shift ~i=2~fzi with the injection and detection fiber end-to-end. By subtracting this phase-shift 0~ for Zero time-delay from Ca,i, the phase-shift ~a=21tfzd is obtained. The approximation is made that, to first order, the shape of the pulse is not affected by the multiple scattering. There is no need to know the exact shape of the injected light pulse, since a reference phase-measurement at all frequencies of interest iis performed. Hereby we take advantage of the F0urier-transform method in avoiding the calculation of the convolution-integral. The fluorescence contribution to the detected signal can be expected to be negligible in the case of a suspension of triglyceridos. Additionally, the emission monochromator of the iphasefluorometer ensures that only scattering at the laser wavelength (600.0_+ 0.5 nm) is detected. The linear relationship between the phase-shift ~a and the frequency of the Fouriercom-
Volume 92, number 1,2,3 I00
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MODULATION FREQUENCY (MHz)
Fig. 1. Phase-shift~das a function of the frequencyfofthe Fourier-componentof the light pulse at different volume fractions of a tdglyceride suspension in water (Intralipid 20% soybean oil) (A) 0.2174, (B) 0.1359, (C) 0.0543, (D) 0.0136. Solid lines are theoretical curves accordingto ed= 2~f~d,with ftruncated at 30 MHz.
ponent is illustrated in fig. 1. This linearity up to at least 90 degrees is a good indication that fluorescence contributions are negligible. Indeed, the phaseshift for fluorescence is a non-linear function of frequency and limited to the range of 0 to 90 degrees. From the slope of the curves in fig. 1, the time delay Zd was calculated. Between two scattering events on a triglyceride particle, the photon can be assumed to travel in water, since the scattering process is caused by density differences. With a value of 1.333 for the refractive index of water at 293 K and the derived time delay %, the total length of the diffusion path Nl* has been deduced. Finally, from this length and the separation between the fiber tips S, the number of steps N and the step size l* have been derived. The obtained values for different volume fractions of triglyceride suspensions in water are given in table 1. In fig. 2 the inverse of the transport mean free path length obtained from the data presented in fig. l, is plotted as a function of the volume fraction.
3. Discussion
For the determination of fluorescence lifetimes with phasefluorometry, there is no need to know the
15 August 1992
exact excitation pulse profile. Presently, in deriving the number of steps N and the transport mean free path length l* for multiple scattering, the assumption has been made that the pulse shape was not significantly altered by the scattering process. Only then is the linear dependence ¢d = 2XfXd valid. In practice however, the pulse shape is distorted, because different photons make different random walks, with a different global length of the diffusion path Nl*, to arrive at the detecting fiber tip. This results in a distribution of time delays and hence, in a different pulse shape at the detecting tip. A detailed analysis of the pulse shape [9 ] shows that the rising edges of the delayed pulses are nearly identical when all the scattering particles have nearly the same diffusion constants. This is true for a monodisperse suspension. The trailing edges are dominated by light absorption. In our experiments, the containing vessel is large enough to eliminate back-reflections into the medium. Hence, the diffusion of photons far away from the detecting fiber tip can be considered as absorption, since the probability to arrive at the detecting tip is considerably reduced. The two effects mentioned are expected to reduce the pulse shape distortion. The influence of the pulse shape has further been reduced by taking the slope over the Fouriercomponents below 30 MHz only, as shown in fig. 1. It has been shown that the lower modulation frequencies are less attenuated than the higher frequencies [ 16 ]. Limiting the data analysis to 30 MHz modulation frequency then ensures a negligible effect of absorption. Measurements over the entire bandwidth of the phasefluorometer, as shown in fig. 3, reveal a deviation of the linearity for the components above 50 MHz. Note that the phase-shift ~a amounts to more than 250 degrees, well above the limit of 90 degrees for the phase-shift in fluorescence analysis #~. Therefore, the large phase-shifts are attributed to the time delay % between the injection and the detection of the light pulses. ~ The limit of 90 degrees phase-shiftin'phasefluorometry can be overcomein the exceptional case of excited state reactions of fluorescent reagentstowards fluorescentreaction products, produced in the excited state also. Additionally, the rate constant of the excited state interconversionmust be of the same order of magnitude of the inverseof the rcagent'sexcitedstate lifetime, to proceed to a detectable extent. These conditions are clearlynot fulfilled.
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
Table 1 Experimentally determined slope 2g% of the phase-shift 0d as a function of frequencyfof the Fourier-component, calculated delay time %, total length NI* of the random walk, number of steps N and transport mean free path length 1" for different volume fractions of triglycerides in water at 293 IC The values between parentheses are the estimated uncertainties on the retrieved values. Volume fraction
2~rrd (deg M H z - 1)
% (ns)
0.0136 0.0543 0.1359 0.2174
0.85 1.60 2.45 3.17
2.37 4.44 6.79 8.80
(0.05) (0.05) (0.05) (0.05)
(0.15) (0.15) (0.15) (0.15)
NI* (mm )
N
535 1000 1530 1990
360 1280 3000 5030
(30) (30) (30) (30)
3.0
l* (mm) (30) (55) (85) (110)
1.47 (0.08) 0.78 (0.02) 0.51 (0.01) 0.395 (0.006)
300 A
2.5
250
2.0
2oo
TE E
,t:-
1.5
-,/O3
1.0
< "r r,
100
v•
o
a.
0.5 50 V 0.0
0.00
i
0.05
i i i 0.10 0.15 0.20 VOLUME FRACTION
a
0
0.25
0
I
i
i
i
1O0
150
200
250
300
MODULATION FREQUENCY ( M H z )
Fig. 2. The inverse of the transport mean flee path length of photons 1/1" as a function of volume fraction for Intralipid 20% soybean oil suspension in water. The line shows the trend in the data.
A qualitative analysis of the phase-shifts at the hig,h-frequency Fourier-components involves the sine and cosine Fourier-transform of the solution o f the diffusion equation
O0/Ot =DV2C~-T~+N6(x)6(y)6(z-S)6(t),
50
Fig. 3. Measurements of the phase-shift 0d over the entire bandwidth of the phasefluorime~er, for different volume fractions of the triglyceride in aqueous sqspension. Solid lines are fitted curves according to eq. (2). The inset is shown in fig. 1 at a higher magniftcation, where the solid lines are the theoretical curves according to 0d = 2grid. tector are immersed in an infinite scattering and nonabsorbing m e d i u m [ 16 ],
¢=S[6xf/vll ~/2 (1)
where ¢~(t) is the density of the photons at x, y, z at a time t, ~, is the absorption parameter, D= l'v~3 is the photon diffusion coefficient and N photons are injected at t=O. A solution for eq. ( 1 ) has been given in terms of cylindrical Bessel functions for a different geometry [ 9 ]. An approximate expression has been derived in terms of D, ~, S and t only. An approximate analytical expression for the phase shift has been derived in the case where source and de-
(2)
Figure 3 shows the experimentally obtained phase shifts, together with the best fit to eq. (2). It is clear that, for low volume fractions of Intralipid, the approximation is satisfactory, but, for high volume fractions, there is a large discrepancy between the experimental data and eq. (2). This discrepancy cannot be attributed to absorption, since the deviation is largest for the lower modulation frequencies, where the effect of absorption is lowest [ 16 ]. Furthermore, Intralipid is known to have a negligible absorption
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
coefficient [ 13 ]. Phase measurements as a function of distance S between the fiber tips should then give the same value for the transport mean free path l*. Linearization of the exact analytical expression for the phase shift 0 has been suggested in a theoretical frequency domain study and it has been inferred that this linear phase behaviour is characteristic of a time delay between the excitation pulse and the detected pulse [ 17 ]. The plot of the inverse of the transport mean free path as a function of volume fraction (fig. 2 ) shows a large deviation from linearity. A linear dependence is expected without particle correlation. To explain the deviation, an analytical expression for the hardcore interaction structure factor has been presented earlier [ 15 ]. To simulate such a system, polystyrene spheres are usually suspended in water with salt added to screen the electrostatic repulsion. Good agreement between theory and experiment has been observed [5]. For a suspension of triglycerides, the same hard-core structure factor is probably inappropriate. Although the multifrequency phasefluorimeter used for the present study is a rather complex and expensive pulsed-laser based instrument [ 10 ] with additional phase stabilization [ 12 ], it is clear that the 300 MHz bandwidth of the instrument, or the large number of phase-frequency data points, is not essential for the determination of the slope of the phase, plotted as a linear function of the modulation frequency for frequencies up to 30 MHz. In fact, inexpensive lamp-based, commercially available phasefluorimeters with three modulation frequencies over a limited bandwidth are satisfactory for this type of experiment. Thus, the application of the more compact and less expensive frequency domain phasefluorometry technique in the study of multiple light scattering in tissue can eliminate the complex and expensive single photon counting technique. For diagnostic purposes, the possibility for a continuous reading of the phase shift 0d offers the perspective of continuous monitoring of optical properties of tissue.
4. Conclusion In conclusion, we have shown that the absolute determination of the relevant parameter in the diffu10
15 August 1992
sion approximation of multiple light scattering can be performed with the help of a simple description and readily available instrumentation. In aqueous suspensions of triglycerides, the inverse of the transport mean free path length l* does not show a linear dependence on the volume fraction of the scatterers. The same qualitative behaviour is observed here as for suspensions of hard spheres. This method could be of use in the determination of aggregation numbers of micellar solutions as well as concentrated dispersions, provided they are close to being monodisperse. Finally this work is expected to stimulate the development of analytical expressions for the local structure factor S(q) for various types of interparticle correlation.
Acknowledgements This research was supported by research grants from the Belgian government (GOA 87 / 91-109 ) and from the Belgian National Fund for Scientific Research (FKFO 2.9003.90). The authors wish to thank Dr. L. De Maeyer for stimulating discussions.
References [ 1 ] M. Kerker, The scattering of light and other electromagnetic radiation (Academic, New York, 1969). [2] W. Van De Sande and A. Persoons, J. Chem. Phys. 89 ( 1985 ) 404. [3] B.J. Berne and R. Pecora, Dynamic light scattering (Wiley, New York, 1975 ). [4] M.J. Stephen, Phys. Rev. B 37 (1988) 1. [ 5 ] S. Fraden and G. Maret, Phys. Rev. Lett. 65 (1990) 512. [6 ] P.M. Saulnier, M.P. Zinkin and G.H. Watson, Phys. Rev. B 42 (1990) 2621. [ 7 ] D.J. Pine, D.A. Weitz, P.M. Chaikin and E. Herbolzheimer, Phys. Rev. Lett. 60 ( 1988 ) 1134. [8] M.B. van der Mark, M.P. van Albada and A. Lagendijk, Phys. Rev. B 37 (1988) 3575. [9] G.H. Watson, Jr., P.A. Fleury and S.L. McCall, Phys. Rev. Lett. 58 (1987) 945. [ 10] IC Clays, J. Jannes, Y. Engelborghs and A. Persoons, J. Phys. E22 (1989) 297. [ 11]J.R. Lakowicz, G. Laczko and I. Gryczynski, Rev. Sci. Instrum. 57 (1986) 2499. [ 12] IC Clays and A. Persoons, Appl. Optics 27 (1988) 3601. [ 13] I. Driver, J.W: Feather, P.R. King and'J.B. Dawson, Phys. Med. Biol. 34 (1989) 1027.
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[ 14] H.J. van Staveren, C.J.M. Moes, J. van Made, S.A. Prahl and M.J.C. van Gemert, Appl. Optics 30 ( 1991 ) 4507. [15] M.S. Wertheim, Phys. Rev. Lett. 10 (1965) 321. [16] J. Fishkin, E. Gratton, M.J. vandeVen and William W.
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Mantulin, Proc. Soc. Photo-Opt. Instrum. Eng. 1431 (1991) 122. [ 17] M.S. Patterson, J.D. Moulton, B.C. Wilson and B. Chance, Proc. Soc. Photo-Opt. instrum. Eng. 1203 (1990) 62.
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