Dynamic light scattering ~DLS! is a well-establishedtechnique that has been used for decades in manyfields of basic and applied science, from physics tochemistry, biology, and medicine. DLS measuresthe time-averaged correlation function of the fluctu-ations of the intensity scattered by the investigatedsample and provides information on the decay or therelaxation time ~or times! that characterizes its un-derlying dynamics. Examples of the applications ofDLS are countless, the most prominent ones probablybeing laser Doppler velocimetry and sizing of submi-crometer particles through the measurement of thetranslational diffusion coefficient associated withparticle Brownian motion. In particular, the lattertechnique has experienced continual growth over re-cent years and is now routinely used in many labo-ratories worldwide, with applications ranging fromindustrial production control to the fundamental
study of interacting-particle systems. Reviews ofthe historical applications of DLS as well as of itsrecent developments and experimental research canbe found in Refs. 1–5 and references therein.
In a DLS experiment the signal-correlation func-tion is usually carried out by means of a digital cor-relator, i.e., a device capable of performing on linedigital signal processing ~DSP! of the stream of countpulses coming from a photodetector, usually a photo-multiplier. Given the intrinsic digital nature of theinput signal, a digital correlator appears, therefore,to be the ideal instrument for tackling such a task,free of noise and other distortion effects that are un-avoidable in any analog-type analysis. Digital cor-relators were first developed in the 1970s by Pike andcolleagues,6,7 and ever since they have been continu-ally developed and improved. But it was in the late1980s that their growth started to increase exponen-tially, thanks to the work of Schatzel and co-workers8–12 who invented the so-called multi-taucorrelator. With the help of the multi-tau scheme itbecame possible to access, within the same correla-tion function, a huge dynamic range of time scalesthat extend from tens of nanoseconds to hours.Since then DLS has become the ideal tool for studyingthe dynamics of many interesting new systems thatwere not previously accessible because they werecharacterized by a large polydispersity of diffusioncoefficients, such as colloidal or polymeric gels, com-plex fluids, foams, and granular materials.
20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4011
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Many commercial digital correlators are availablenowadays on the market, the most popular ones prob-ably being those manufactured by Laser Vertriebsge-sellschaft mbH ~ALV! ~Langen, Germany!, byBrookhaven Instruments ~Holtsville, New York!,nd, in more recent years, by Correlator.com ~Bridge-ater, New Jersey!. These correlators are powerfulSP devices that are based on a multi-tau scheme of
he sampling times and are capable of computingorrelation functions in full real time across a hugeange of different lag times from ;10 ns to hours.hey are easily installed on a personal computer
PC!, are run by user-friendly software, and are pro-ided with nontrivial software libraries for data anal-sis. However, they have several limitations orrawbacks because they are designed and manufac-ured for carrying out the whole ~but only! task ofeasuring the correlation function of a digital signal.hey are not very flexible ~for example, for some of
hem the delay-time grid and the number of bits perhannel are fixed!, are rather expensive, and, abovell, cannot be implemented easily with the advent ofew available technologies.In this paper, we propose an alternative approach
hat is based on the use of commercially available,omparatively inexpensive, general-purpose elec-ronic devices only and on the development of a soft-are algorithm that was written with a modern,
nherently multitasking, graphical programminganguage. For this purpose we used LabVIEW13
~National Instruments!, which has become one of themost popular and powerful programming tools forinteractive data acquisition and DSP. At the sametime the stream of count pulses from the photomul-tiplier were counted with a fast standard counterwhose output was a 32-bit integrated count. Prob-ably the main feature of the counter ~and of its Lab-VIEW drivers! is the possibility of storing its 32-bitoutput data in a so-called double buffer from whichthe numbers can be transferred in blocks to the PC’smemory, asynchronously with the sampling rate.This decoupling between sampling and processing,together with the huge size of this buffer ~limited byonly the RAM memory available in the host PC!,offers great flexibility and allows the data analysis tobe optimized easily for high-speed performance.
But clearly the main limitation of a PC-based soft-ware correlator still remains speed because the timerequired for processing each datum is expected to bemuch higher than the corresponding time required bya hardware correlator ~;10 ns!. This fact limits theminimum lag time attainable in the correlation func-tion because full real-time operation is possible onlywhen the ~average! processing time per point is lesshan the sampling time of the data. If one forgoeseal-time performance, higher sampling rates areossible, but batch processing is required.12 In
other words, the data are sampled and processed inbatches with a reduction of the correlator efficiency orduty cycle ~the ratio of the effective measuring to theelapsed time! that is approximately equal to the ratiobetween the sampling and the processing times.
012 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001
In this research, we tackled the speed problem bydesigning a correlator that is based on a master–slave scheme in which two correlators, one slow andone fast, work in parallel to process the same streamof count pulses that are sampled with two very dif-ferent gate times. The slow correlator operates infull real-time and behaves as the master and triggersthe operation of the fast correlator ~the slave!, whichconsequently operates as a batch processor. Bothcorrelators adopt a flexible multi-tau scheme of lagtimes in which the sampling times of the two corre-lators are programmable by means of software andthe sequence of incoming data is grouped ~or binned!n blocks of m points, with m an integer that we call
the binning ratio. Then the binned data are fed to aset S of linear correlators that compute the correla-tion function on P evenly spaced lag times. Clearly,the correlator performance, such as the accuracy ofthe measured correlation function, the processingtime per point, the minimum and the maximum lagtimes, depend on the parameters m, P, and S and onthe two fast and slow sampling times as well. Bytuning these parameters properly, one can easily op-timize the correlator functioning, and this is a usefulflexibility that is missing in most of the current hard-ware correlators.
This paper is organized as follows: In Section 2,we recall the basic concepts of DLS, and, in particu-lar, we discuss triangular averaging, which deter-mines the ultimate accuracy attainable in acorrelation function. Section 3 is devoted to illus-trating the multi-tau scheme adopted in the correla-tor and provides estimates of the processing time perpoint. Section 4 describes the correlator architec-ture and the master–slave scheme used in our design.The experimental data, taken on solutions of polysty-rene spheres, are reported in Section 5 and are usedto ascertain the correlator performance. Finally, inSection 6 a summary and further developments ofthis research are presented and discussed.
2. Theory
In a DLS experiment the intensity-correlation func-tion of the scattered light is obtained by the measure-ment of the correlation function Gn~t! of the pulsesstreaming out from a photomultiplier2
Gn~t! ; ^n~t!n~t 1 t!& 5 limT3`
1T *
0
T
n~t!n~t 1 t!dt,
(1)
here t is the delay or the lag time and n~t! is thehoton-counting rate. Equation ~1! is more com-only represented in its normalized form of
gn~t! ;^n~t!n~t 1 t!&
^n~t!&2 , (2)
for which gn~`! 5 1 and gn~0! 5 1 1 sn2y^n&2, where
n& is the average count rate, and sn is the standarddeviation. For a stochastic Gaussian process14 in
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which the scattered light is detected within a singlecoherence area, sn 5 ^n&, and, consequently, the am-plitude of the correlation function is b 5 gn~0! 2 gn~`!
1. However, in many experimental situationsore than one coherence area is collected, leading todampening of the fluctuations and a consequent
eduction of that amplitude b to values ,1.Equations ~1! and ~2! are based on the assumption
that the time resolution used for counting the pulsestream is infinite. However, in a real experimentthe pulse stream is always integrated over a finitegate or sampling time Dt ~see Fig. 1!. Consequently,we must replace n~t! as it appears in Eqs. ~1! and ~2!with the average count rate m~t!, given by
m~t! 51Dt *
t2Dty2
t1Dty2
n~t9!dt9 5 n~t! prect~tyDt!
Dt, (3)
where rect~x! is the rectangle function that is equal to1 for uxu # 0.5 and zero elsewhere and the asteriskdenotes a convolution product. Thus the measuredcorrelation function
Gm~t! ; ^m~t!m~t 1 t!& (4)
depends on Dt, and, because ^m& 5 ^n&, it is straight-forward to show that its normalized version can bewritten as
gm~t! 5 gn~t! pL~tyDt!
Dt, (5)
where gn~t! is given by Eq. ~2! and L~x! is the trian-ular function, defined as L~x! 5 1 2 uxu on the sup-ort uxu # 1 and zero elsewhere. Equation ~5! showshat gm~t! is a smoothed version of the true correla-ion function gn~t! and represents a good approxima-
tion of it, provided that gn~t! does not varyignificantly over a time scale that is comparableith Dt. Note that, in the limit of Dt 3 0, the con-
volution function tends to the delta function andtherefore gm~t! 3 gn~t!.
To give an estimate of the approximation intro-duced by Eq. ~5!, which is often called triangularaveraging, we examine the case of a single-exponential decay
gn~t! 5 1 1 b exp~22tyt0!, (6)
Fig. 1. Schematic diagram of the behavior of the count rate n~t!plotted against time t in a DLS experiment. To achieve highaccuracy in the measurement of the correlation function, samplingtime Dt must be much shorter than lag time t.
which is characterized by an amplitude b and a decayrate t0y2. By the insertion of Eq. ~6! into Eq. ~5! it isasy to show that, for t $ Dt, the measured correla-
tion function results in
gm~t! 5 1 1 b exp~22tyt0!Fsinh~Dtyt0!
Dtyt0G2
,
t $ Dt. (7)
It should be stressed that Eq. ~7! is valid for only t $Dt. When t , Dt the result is significantly different~see, for example, Ref. 15! but is not considered herebecause in a multi-tau correlator we always have t $Dt. A comparison of Eqs. ~6! and ~7! shows that theerror dgm is
dgm~t! ; gm~t! 2 gn~t!
5 b exp~22tyt0!HFsinh~Dtyt0!
Dtyt0G2
2 1J ,
t $ Dt, (8)
hich depends on Dtyt0 and decays to zero as approx-imately exp~22tyt0!. Thus the error is always pos-itive and exhibits its maximum for t 5 Dt with anmplitude that depends on b and on the relative mag-itude of the three times t, Dt, and t0. The behavior
of dgm as a function of the ratio Dtyt0 for differentvalues of the ratio a 5 tyDt is shown in Fig. 2 for thecase of b 5 1. All the curves exhibit a maximum forvalues of the ratio Dtyt0 in the range of approximately0.1–1 and decay to zero for Dtyt0 ,, 1 and Dt/t0 .. 1.
As was mentioned above, the errors are maximumfor the curve with a 5 1 and decrease with increasing
. In particular, if an accuracy of 1023 is wanted thevalue of a must be $7, as is shown by the solid curve
Fig. 2. Systematic absolute error dgm introduced in the measureof the correlation function by use of triangular averaging, plottedas a function of the ratio between the sampling time Dt and the~field! decay time t0 of a single-exponential correlation function @see
q. ~8! in the text#. The various curves represent different valuesof the ratio a between the lag time t and the sampling time Dt andwere generated with a value of b 5 1. It can be seen that, re-gardless of t0, if an accuracy of better than 1023 is desired a mustbe larger than 7 ~the solid curve!.
20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4013
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that corresponds to a 5 7. This is the level of accu-racy according to which our correlator was designed~see Section 3!. It should be pointed out that, al-though the curves of Fig. 2 were worked out for thespecific case of a single-exponential decay with a de-cay time t0, the conclusion drawn above is quite gen-eral. Indeed, suppose that more decay times aresimultaneously present in the correlation function,i.e., that
gn~t! 5 1 1 (i
ai exp~22tyti!,
in which (i
ai 5 1, because gn~0! 5 2. Thus, by useof Eq. ~5! the error dgm becomes a sum of differentterms, each one weighted by ai and equal to expres-sion ~8! but with ti instead of t0. If we now plot dgm
against Dt, we obtain curves whose maxima are al-ways lower ~for the same a! than the maximum of thecurves shown in Fig. 2 ~this is because each decaytime gives a contribution that is centered at differentvalues of Dt!. Thus, in conclusion, regardless of thefact that gn~t! may be characterized by one or moredecay times, triangular averaging introduces system-atic deviations that can always be upper bounded, inaccord with the curves of Fig. 2. Moreover, thesecurves were drawn for the case of b 5 1, and thus thedeviations they represent have to be rescaled by am-plitude b when this is less than 1. Finally, it isworth mentioning that a systematic error of the orderof 1023 is rather small and can easily be masked byshot noise if the measuring time, the count rate, orboth are not high enough.
3. Multi-Tau Scheme
A multi-tau correlator is based on the simple consid-eration that is given at the end of the Section 2: Aslong as the lag time is much longer than the samplingtime, there is no point in sampling the signal overtime scales that are orders of magnitude less than thelag time. Thus before computing the correlationfunction at a lag time t the signal is averaged over anntegration time Dt that is chosen so that the ratio
5 tyDt is higher than a given value. This choicensures that the correlation function is computedith the desired accuracy ~see Fig. 2! and sensibly
educes the number of operations to be carried out
Fig. 3. Schematic diagram of the multi-tau scheme adopted in theand a binning ratio of m 5 3. For each linear correlator of orderand are therefore discarded. We obtained the overall lag-time seqcorrelator ~the shaded boxes!. For clarity only the first four linea
014 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001
hen long lag times are considered. Ideally, aulti-tau correlator would have the ratio a equal for
ll lag times, but, in practice, many of the hardwareorrelators that are commercially available are real-zed by the division of the channels of the correlationunction into blocks ~typically made up of eight chan-els! and by doubling the integration time that cor-esponds to each block. In this way lag times thatpan many decades ~from tens of nanoseconds toours! can be covered by use of only a few hundred
channels.Our multi-tau correlator is designed in a similar
way but is more flexible than hardware correlators.It is based on a set S of linear correlators whoseintegration times Dts increase in a geometric progres-sion given by
Dts 5 msDt0, s 5 0, 1, 2, . . . , S 2 1, (9)
here Dt0 is the integration time of the fastest linearcorrelator and is equal to the gate time of the counter.The integer m represents the binning ratio as giveny the ratio between the sampling times of two adja-ent linear correlators. Thus, if N0 denotes the
number of data points fed to the zeroth correlator, thenumber of points Ns handled by the sth correlator is
Ns 5 m2sN0, s 5 0, 1, 2, . . . , S 2 1. (10)
Each correlator computes the correlation functionof the pulse stream @averaged over the gate time Dts,see Eq. ~3!# on a set of equally spaced lag times ts~p!5 pDts, with p 5 0, 1, . . . , P 2 1, where P is the sizeof an array called shift register that is equal for allthe linear correlators. The lag-time overlap be-tween two adjacent correlators s and s 1 1 corre-sponds to the first Pym points of the ~s 1 1!thcorrelator, as sketched in Fig. 3 for the particularcases of m 5 3 and P 5 9. Thus the first Pym pointsof each linear correlator can be discarded ~withoutcomputing the correlation function! and the lowestlag time for each correlator remains
~ts!min 5 ~Pym!Dts. (11)
Because we require that the ratio between the min-imum lag time and the sampling time must be largerthan the factor a, Pym is constrained to the valuePym 5 a. The actual values of P and m can be
lator for the particular case of a shift register with P 5 9 elementse first Pym lag times overlap the last one of the ~s 2 1! correlator
of the correlation function by merging the lag times of each linearrelators are represented in the figure.
corres, thuencer cor
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chosen in accord with the speed performance, whichcan vary depending on experimental conditions. In-cidentally, we note that most of the hardware corre-lators work with values of P 5 16 and m 5 2; thuseach linear correlator uses eight different lag times,and half of the points are discarded.
The software algorithm used for computing the cor-relation function of each single linear correlator isquite simple and resembles the scheme used in hard-ware correlators. After an initial transient in whichthe shift register is loaded with the first P points, thestream of incoming data is processed one by one inthe following way: First, the last ~most recent! da-um is used for updating the shift register, whicheans that all the components of the shift register
re moved to the right ~see Fig. 3!, the last components discarded, and the first component ~with a lag time
of t 5 0! is replaced with the last datum. Then allthe components are multiplied by the last datum, andthe results are summed up into an array, which rep-resents the correlation function. Although, in prin-ciple, all the incoming data could be treated asinteger numbers, all the operations that involve theshift register were carried out with single-precisionaccuracy. This approach was used because wechecked that there was no significant gain in reduc-ing the processing time when integers were comparedwith single-precision numbers. We normalized thecorrelation function by following either Eq. ~2! or di-iding for Gm~t 3 `! or using the symmetrical-ormalization procedure proposed by Schatzel et al.10
In any case to save time the normalization as well asthe other data reductions, such as merging all thelinear correlators, discarding the overlapped data,and data plotting, were carried out only once in awhile ~approximately every 5–10 s! and at the end ofthe measurement.
A qualitative estimate of the time required for com-puting the overall correlation function can be given asfollows: Let us start by considering that each linearcorrelator is fed with Ns points @see Eq. ~10!# and hasto handle a shift register with P components. Thetime necessary for carrying out this task can be di-vided into two parts. The first one is due to theoperations that involve only the number of points~averaging, sums, updating the first component of theshift register, etc.! and therefore scale as approxi-
ately Ns. The second one is due to the number ofoperations carried out with the shift register ~rotat-ng the shift register, multiplications and sums of itsomponents, etc.! and therefore scales as approxi-ately NsP. Here we neglect the Pym points to be
discarded, for which the correlation function does notneed to be computed. This approach is valid becausewe checked that it is faster to compute the correlationfunction over the entire shift register ~and to discardthe Pym points at the end of the process! rather thansplit the shift register into two subarrays and processonly one of them. The time saved in carrying out theabove operations on an array of reduced size @the sizeis reduced by a factor of ~m 2 1!ym# does not com-pensate for the effort of handling the array itself, at
least for arrays of small sizes, as with the shift reg-isters of a standard multi-tau correlator ~approxi-mately #50 components!.
If we indicate with w1 and w2 the weights that areassociated with the operations that scale as Ns andNsP, respectively, and sum up over the S stages, withNs given by Eq. ~10!, we find that the overall compu-ation time tcomp scales as
tcomp , w1 (i50
S
Ns 1 w2 (i50
S
Ns P
5 N0~w1 1 w2 P!1 2 m2s
1 2 m21
, N0~w1 1 w2 P!m
m 2 1, m . 1, (12)
in which the last approximation is valid because usu-ally S .. 1. Expression ~12! shows that tcomp isindependent of S and is determined mostly by thefirst ~s 5 0! correlator. The remaining stages con-tribute for only a factor of my~m 2 1!, which is of theorder of unity for m .. 1. If we now insert intoxpression ~12! the constraint that P 5 am, we obtain
tcomp , N0~w1 1 w2am!m
m 2 1, (13)
which exhibits a minimum for
mmin 5 1 1 S1 1w1yw2
a D1y2
. (14)
It is interesting to note that, when the time requiredfor the multiplications is predominant ~w2 .. w1! forvery long shift registers ~a .. 1!, we have mmin ; 2,which corresponds to the value used in many hard-ware correlators.
However, in our software correlator the situation isquite different because the number of operations nec-essary for handling the data, besides the multiplica-tions carried out with the shift register, is fairly high.Because it is almost impossible to estimate the rela-tive weight w1yw2, we directly measured the compu-tational time required for the program to build acorrelation function with given values of P, m, and S.The result is shown in Fig. 4 in which the computa-tional time per point tcompyN0 is reported as a func-tion of m for different values of the ratio a 5 Pym~solid symbols!. The test was carried out on a givenset of simulated data that was generated in accor-dance with Ref. 16 ~and stored on the hard disk! sohat correlation functions with different values forarameters P and m were obtained by processing thexact same data. The value of S was chosen to have,or each pair of P and m, approximately the sameaximum lag time ~;2 s!. To avoid unreliable re-
ults caused by interference from the operative sys-em under which the host PC works ~Windows ’98!equired that no other applications be active at theime of the test. Figure 4 shows unambiguouslyhat the curves ~solid symbols! with a higher Pym
20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4015
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require a higher tcompyN0 and that their minimaclearly fall at values of m that are larger than 2. Inparticular, the curve representing Pym 5 7 ~whichorresponds to the value used in this study! exhibitsts minimum for m ; 4; the corresponding computa-ional time per point is ;5 ms. For comparison, welso reported in Fig. 4 the computational time peroint for obtaining, from the original counts at theastest sampling time Dt0, the average counts for allhe other gate times Dts. This is shown by the dataopen symbols! that are plotted in the bottom of the
graph and indicates that this time is independent ofP and is of the order of 0.5–1 ms, i.e., approximately10% of the overall computational time. Its depen-dence on m is rather mild, and after an initial decayakes place for the lower values of m it levels out to aonstant value of ;0.5 ms, consistent with what isredicted by expression ~12! when w2 5 0.
4. Correlator Architecture
The correlator design was based on a simple algo-rithm written in LabVIEW,13 a graphical program-ming language ~National Instruments! that isparticularly suited to interactive data acquisition andDSP. LabVIEW is based on a data-flow program-ming model in which the execution order is not de-termined by sequential lines of text ~as in traditionaltext-based languages! but rather is decided by theflow of data between block diagrams. Thus manydiagrams can be executed in parallel within a singleprogram, and, as matter of fact, LabVIEW behaves asa multitasking system with a high level of modularityand hierarchy. Moreover, LabVIEW can easily ad-dress almost the entire RAM available on the PC and,consequently, operations on large arrays or matricescan be carried out at high speeds.
Fig. 4. Computational time per point tcompyN0 required for pro-essing the data in the software correlator plotted as a function ofhe parameter m for different values of the ratio Pym. The filledymbols refer to the overall computational time per point, whereashe open symbols, which are independent of Pym, account for only
the averaging of the input data to be fed to the different linearcorrelators. For the curve that represents Pym 5 7 the minimumcomputational time per point is approximately 5 ms and falls atm ; 4.
016 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001
The only hardware used by the correlator are adetector and a counter. The light detector used inour setup was a photon-counting unit ~HamamatsuModel H6180-01! whose output is a stream oftransistor–transistor logic pulses, with a pulse widthof 9 ns and a maximum operation rate of 30 MHz.Its dark count at room temperature is ;10 countsys.The pulse stream was counted by a 32-bit counter–timer board ~National Instruments Model PCI-6022!equipped with eight input channels and has a maxi-mum input rate of 80 MHz. The minimum source-pulse duration is 5 ns, and the minimum gate-timeperiod is 10 ns. The first-in–first-out ~FIFO! bufferfor the direct memory access ~DMA! transfer of the
ata to the host PC is a 16 3 32 bit buffer. The ratet which data are transferred depends on both the PCpecifications and the operation mode of the counter,ut is ultimately limited by the maximum speed ofhe peripheral–component interconnect ~PCI! bus;33 MHz!. In our system this was of the order of200 ns and was due mainly to the rather shallowIFO buffer of the counter board.The counter can be set to work in the so-called
ouble-buffer ~or circular-buffer! acquisition mode inhich the counter counts continuously and stores theata in a data buffer whose maximum size is limitedy the memory that is available on the host PC. Be-ause each datum is a 32-bit number, for a PC with56-Mbyte RAM this maximum size has to be some-hat smaller than 6.4 3 107. From the data buffer
the data can be read in blocks and transferred to theLabVIEW program by means of a reading bufferwhose size can be chosen at will but that obviouslymust be smaller than the data-buffer size.
The time required for transferring these data de-pends on the host PC and for a 550-MHz Pentium IIIis of the order of 0.1 msypoint, a time much shorterhan typical processing time per point ~see Fig. 4!.he reading–writing procedure from and into theouble data buffer takes places asynchronously, ands the old data are retrieved at one location in theuffer, the new data are stored in a different locationo that still-unread data are not overwritten by theewer data. As long as the backlog between theritten and the read data is less than the double-uffer size, data handshaking can continue. Ofourse, the procedure can keep going indefinitely onlyf the average rate at which the data are written isqual to or less than the average reading rate.The output of the counter is an integrated count,
.e., the number of pulses counted from the beginningf the count up to the current time. This require-ent offers the advantage that the average count
orresponding to an integration time Dts that isonger than the sampling time is carried out with aimple difference between two integrated counts. Ithould be noted that the integrated count is limited to2 bits, i.e., ;4.2 3 109. Thus if an average count
rate of 1–10 MHz is used, the counter goes into over-flow after approximately 430–4300 s, a time that islong enough for typical DLS measurements ~if longer
smcmts
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times are needed, the overflow can easily be takeninto account!.
As was reported in Section 3, the minimum com-putational time for our system under the scheme P 528, m 5 4, and S 5 8 is approximately tcomp ; 5msypoint. Thus the correlator can work in real timeto gate times as high as Dt0 ; 5 ms. To achieveshorter times, we implemented a second fast correla-tor, which works in parallel with the first slow corre-lator, as shown schematically in Fig. 5. The slowcorrelator processes the data that arrive from acounter operating with a long gate time, whereas thefast correlator processes the same data acquired byuse of a counter operating at a much higher samplingrate. The slow correlator works continually,whereas the fast one works only when there is somespare time left over from the slow correlator. In thisscheme the slow correlator is the master correlatorthat works in real time and triggers the operation ofthe slave correlator, which works as a batch proces-or, i.e., intermittently, asynchronously with theaster, and, obviously, not in real time. The slave
orrelator is set to work in a way similar to that of theaster, i.e., with a multi-tau scheme of sampling
imes and an adjustable reading buffer, but with aingle buffer rather than a double buffer.On a software trigger ~called OCCURRENCE in Lab-
VIEW language! from the master correlator, thelave correlator acquires data, stores them in its datauffer, and processes the first reading buffer. Thenn the next OCCURRENCE it processes another reading
buffer, and so on for each OCCURRENCE, until all theata present in its data buffer are processed. At theext OCCURRENCE it starts again from the beginning,cquiring new data. The OCCURRENCE from the mas-
ter correlator arrives whenever, at the end of theprocessing of each reading buffer, the backlog be-tween writing and reading is less than a given frac-tion ~typically ;50%! of the double-buffer size.Clearly, the duty cycle of the slave correlator, i.e., theratio between the effective measuring time and the
Fig. 5. Schematic diagram of the correlator architecture. The putimes and simultaneously fed to the slow and the fast correlators.correlator that triggers the operation of the slave correlator, the f
elapsed time, depends on the settings of the two cor-relators and can be tuned with a certain freedom.The program keeps track of all these times so that itcan take them into account for estimating the errorbars that are associated with the channels of thecorrelation function ~not implemented here!.
In Table 1, we report two typical configurationsnder which our correlator can work. In both caseshe parameters P and m were chosen so that a ratiof a 5 tyDt $ Pym 5 7 was obtained for all the lag
times except the shortest ones that belong to the firststage ~for which t $ Dt0!. Thus aside from these firstlag times ~in which shot noise is usually predomi-nant! the systematic error that is due to triangularaveraging is always ,1023. The slow configuration,in which only the master correlator is active, works infull real time ~with a duty cycle of 100%! and has a
inimum lag time of 5 ms. Its maximum lag time is2 s but can be made much longer without affecting
he correlator performance. Note that the data- andhe reading-buffer sizes are quite large. This size ismportant to keep the computing time per point shortecause in LabVIEW many operations are carried outn a vectorial basis; consequently, the longer the ar-ay to be processed, the shorter the elaboration timeer point. In the fast configuration, in which thelave correlator is also active, the master correlatororks with longer gate times ~;12 ms! and with auch smaller reading buffer. These differences ex-
st for two reasons: On the one hand, there shoulde some time left at the end of each reading buffer forriggering the fast correlator; on the other hand, toeep the duty cycle of the slave from being smalleans that this triggering has to occur frequently,
nd thus the reading buffer of the master has to beeasonably small. From the table one can see that,ecause the data acquisition of the slave occurs ap-roximately every 24 ms ~12 ms 3 2000! and lasts for300 ms ~300 ns 3 1000!, the duty cycle is ;1%, as
eported. This value is rather small, but, in princi-le, at these short lag times a not very long measur-
tream exiting the photon-counting unit is sampled at two differente slow correlator operates in full real time and acts as the masterorrelator, that thus operates as a batch processor.
lse sTh
ast c
20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4017
gattu
wb
Ihcftfti3
3lan;
t
dw
ptsa;
i
Table 1. Parameters of the Correlator for Two Working Configurationsa
4
ing time is necessary for accumulating good intensitystatistics. However, this assumption is true for onlyhigh count rates, when the photon noise ~the noiseassociated with the detection process! can be ne-lected with respect to the signal noise ~the noisessociated with the finite measuring time!. Unfor-unately, in many experimental situations, this rela-ion is not true, and higher duty cycles should besed.To increase the duty cycle, one could operate in twoays: ~a! Further reduce the size of the readinguffer of the master and so increase the rate of OC-
CURRENCE sent to the slave. However, this approachwould increase heavily the elaboration time per pointof the master, soon reaching a value that would be tooclose to its gate time and leaving no spare time. ~b!ncrease the size of the reading buffer of the slave;owever, because of the small FIFO buffer of theounter, this approach would cause DMA transferailures with a subsequent loss of data. Obviously,he higher the rate at which the data are to be trans-erred to the PC memory, the smaller the buffer haso be, and, as a matter of fact, a data buffer of ;1000s the maximum size allowable with a gate time of00 ns.Finally, it should be pointed out that a gate time of
00 ns for the slave represents almost an ultimateimit for our system. If we relax this requirementnd set, for example, the gate time of the slave to 500s, the reading buffer of the slave can be increased to104, and its duty cycle grows immediately to ;10%.
5. Experimental Results
We tested the correlator on real samples by studyingaqueous suspensions of calibrated polystyrenespheres. The particle samples were from Duke Sci-entific Corporation, Palo Alto, California, with diam-eters d between 19 and 107 nm. According to themanufacturer, the particle diameters were certifiedby means of DLS ~d # 40 nm! and transmission elec-ron microscopy ~d $ 50 nm!, and their accuracy was
of the order of 5% or less. All the samples werediluted in distilled water and filtered through a0.22-mm Millipore membrane. Their concentrations
Parameter
Slow Configuration
Master
Gate time Dt0 5 msShift-register length P 28Binning ratio m 4Number of stages S 8Data-buffer size 1 3 107, doubleReading-buffer size 1 3 105
Minimum lag time 5 msMaximum lag time ;2 msDuty cycle 100%
aIn the slow configuration, only the master is active and the corrtime of 5 ms. In the fast configuration, both the master and theof 12 ms, whereas the slave behaves as a batch processor with a d
018 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001
were chosen to produce a count rate of ;105 countsysat 90°. The scattering cell was a glass tube with a15-mm inner diameter that was clamped onto theaxis of a homemade goniometer ~the angular resolu-tion was ;0.1°!. No index-matching vat was used,and the measurements were taken at room temper-ature ~20 6 1 °C!. The light source was a frequency-
oubled 100-mW Nd:YAG laser emitting at 532 nmhose TEM00 beam was mildly focused into the cell
with a waist of 2w0 ; 200 mm. The scattered lightwas detected by use of a standard optical fiber re-ceiver ~ALV, Langen, Germany! that was designed tobe monomode for a wavelength of 633 nm. As aconsequence, approximately two modes could propa-gate through the fiber,17 and the amplitude of thecorrelation function was b ; 0.4.
Figure 6 shows a typical result obtained with ourcorrelator working in the slow configuration, i.e., bysetting the correlator parameters equal to the onesreported in the second column of Table 1. Thus thelag-time span was from 5 ms to ;2 s. The particlediameter was dcert 5 107 6 7 nm; the scatteringangle, 90°; and the measuring time, ;300 s. Thedata were normalized by use of the so-calledsymmetrical-normalization procedure proposed bySchatzel et al.10 and were fitted to the function
g~t! 5 B 1 b exp~2Gt!, (15)
where the baseline B, the amplitude b, and the decayrate G were the fitting parameters. The solid curve
lotted in Fig. 6~a! shows the fitted curve, whereashe relative residuals between the data and the fit arehown in Fig. 6~b!. The match is fairly good withlmost nonsystematic deviations of the order of1023 peak to peak.As is known, the decay rate G depends on the scat-
tering angle u and the particle-diffusion coefficient Dthrough the relation2
G 5 2Dq2, (16)
where q is the magnitude of the scattering wave vec-tor and q 5 ~4pyl!n sin~uy2!, with n as the refractivendex of the medium and l as the laser wavelength.
Fast Configuration
Master Slave
12 ms 300 ns28 284 48 3
1 3 107, double 1 3 103, single2 3 103 1 3 103
84 ms 300 ns;5 s 129.6 ms100% ;1%
r works in full real time ~duty cycle of 100%! with a minimum lagare active. The master works in full real time with a gate timeycle of ;1% and has a gate time of 300 ns.
elatoslaveuty c
E
aIms6np
larabw
dweoddt
ni;i
n
Thus by using Eq. ~16! together with the classicalinstein–Stokes relation, we have
D 5kT
6phR, (17)
nd it is possible to determine the particle radius R.n Eq. ~16!, the parameters k, T, and h are the Boltz-ann constant, the absolute temperature, and the
olvent viscosity, respectively. From the data of Fig., we recovered a particle diameter of d 5 106.7 6 0.1m, which is in excellent agreement with the ex-ected certified value.A typical result that was obtained with the corre-
ator working in the fast configuration ~see columns 3nd 4 of Table 1! is illustrated in Fig. 7. As for theesults shown in Fig. 6, the scattering angle was 90°,nd the measuring time t was approximately 300 s,ut in the case depicted in Fig. 7 the particle diameteras dcert 5 30 6 1.3 nm. The data processed by the
master correlator are represented by the circles,whereas those processed by the slave correlator areindicated by the squares. As is shown in the resid-ual plot of Fig. 7~b!, the data are fitted excellently tothe single-exponential decay function of Eq. ~15! with
eviations that are quite different depending onhether the master or the slave correlator is consid-
red. For the master, the situation is similar to thene shown in Fig. 6~b!, and the amplitude of theeviations is of the order of ;1023. Conversely, theeviations of the slave are almost an order of magni-ude higher, and this is clearly due to the shot noise’s
Fig. 6. Intensity-correlation function as measured with the cor-relator working in the slow configuration ~see Table 1! for a sampleof polystyrene spheres with a certified diameter of dcert 5 107 6 7nm: ~a! Data and their fit to Eq. ~15! and ~b! relative residuals.The scattering angle was 90°, and the measuring time t was ap-proximately 300 s. The diameter recovered from the fitting wasd 5 106.73 6 0.1 nm.
ot being sufficiently averaged out. We recall that,n this configuration, the duty cycle of the slave is1%; thus the effective measuring time of the slave
n this measurement was ;3 s. The bump at veryshort lag times ~#1 ms! is probably due to afterpulseeffects of our phototube ~we observed it also on stillsamples!, and the corresponding data were not con-sidered in the fit. The final result was an estimateddiameter of d 5 30.2 6 0.2 nm, in excellent agree-ment with the expected certified value.
To test the reproducibility and the reliability of theresults shown in Figs. 6 and 7, we carried out severalother measurements on different samples and at dif-ferent scattering angles around 90° ~60° # u # 120°!.Except for the 107-nm particles, the correlator wasoperated in the fast configuration, and all other pa-rameters were similar to those of Fig. 7. All themeasured correlation functions were fitted accuratelywith Eq. ~15! with relative residuals similar to thoseof Fig. 7~b!. The overall results are summarized inFig. 8 in which the ratio of the recovered d to thecertified dcert diameter is reported as a function ofqdcert for four different particle diameters. Asshown, the data are spread around the expectedvalue of 1 with deviations of the order of a few percentrms. This trend is an indication of the overall reli-ability of our measurements, which, however, werecarried out with a homemade setup that must still be
Fig. 7. Intensity-correlation function as measured with the cor-relator working in the fast configuration ~see Table 1! for a sampleof polystyrene spheres with a certified diameter of dsert 5 30 6 1.3
m: ~a! Data and their fit to Eq. ~15! and ~b! relative residuals.The squares represent the results for the slave correlator, whereasthe circles represent those for the master correlator. The scatter-ing angle was 90°, and the measuring time t was approximately300 s. Under this configuration the duty cycle of the slave corre-lator is approximately 1%, which corresponds to an effective mea-suring time of approximately 3 s. The diameter recovered fromthe fitting was d 5 30 6 0.1 nm.
20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4019
aelawbt
4
optimized for optical misalignments and stray-lightrejection. Thus some of the deviations exhibited bythe data of Fig. 8 are likely to be attributed to theapparatus, and the actual accuracy attainable withour software correlator is probably even better thanthat shown in the figure.
6. Conclusions
In this paper, we have shown that a PC-based multi-tau software correlator can be implemented success-fully by use of commercially available electronicequipment, such as a standard photon-counting unit,a fast counter working on the PCI bus, and a PC.The correlator operates online without any storage ona hard disk and is capable of measuring correlationfunctions over time scales of ;5 ms in full real timeand of ;300 ns with batch processing. The softwarelgorithm was developed by use of LabVIEW, a mod-rn graphical programming language that is a popu-ar and powerful tool for interactive data acquisitionnd DSP. The proper functioning of the correlatoras ascertained by use of dilute solutions of cali-rated polystyrene spheres whose diameters were de-ermined to within an accuracy of a few percent.
Compared with the hardware correlators that areavailable on the market ~which can reach time scalesof the order of 10 ns!, our software correlator appearsto be rather slow and not competitive. However, itsspeed limitations are set by the current performanceof today’s electronic and PC technology. For exam-ple, the limit of 300 ns is imposed by the speed ofDMA data transfer over the PCI bus between thePC’s memory and the ~small! FIFO buffer on theboard of the counter. Considering the incrediblerate at which both the PC and data-acquisition tech-nology are growing, we believe that this limit will bereduced in the near future and that the gap withhardware correlators will be progressively shortened.Moreover, hardware correlators are in many casesrather expensive and offer little flexibility because
Fig. 8. Ratio between the recovered and the certified diametersplotted as a function of qdcert for four different particle diametersthat were obtained by the measurement of the correlation functionat different wave vectors q. The data are spread about the ex-pected value of 1 with deviations of the order of a few percent rms.
020 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001
they return only the correlation function of the in-coming data. Conversely, a software correlator ismuch more flexible and gives access to the entiredetected photon sequence, allowing a more sophisti-cated data analysis to be carried out. For example,higher-order correlation functions can be computed,or the distribution of the time intervals between suc-cessive photon events can be retrieved. The lattertype of analysis is particularly convenient when thephoton-counting rate is less than the sampling rate,as was pointed out in Ref. 18. This approach couldopen up a way to overcome the speed limitations ofour correlator and push its minimum time scale to-ward values at which the threshold is determined byonly the time resolution of the photodetector or thecounter.
Finally, it should be noted that, in this paper, wedid not address the issue of the statistical noise as-sociated with the different channels of the measuredcorrelation function, also called correlation estima-tors. The estimate of the variances and the covari-ances associated with the correlation estimators is arather difficult problem that becomes even more dif-ficult for a multi-tau correlator in which two estima-tors at different lag times may be derived from datasampled with two different integration times. Thusmodels that include both random photon noise andcorrelated intensity noise are to be utilized, and thevariance–covariance matrix is estimated by use ofonly the correlation estimators and the informationon the average count rate and the measuringtime.14,19,20 However, and also by consideration ofthe fact that sometimes the predictions of such mod-els do not agree ~as in the cases of Refs. 14 and 19!, itwould be nice to provide estimates of the covariantmatrix independently of the adopted model and ob-tained directly from the analysis of the incoming pho-ton sequence. In this respect a software correlatoroffers a great opportunity because it allows the mea-surements of higher-order correlation functions thatcan be related directly to the covariance matrix of thecorrelation estimators. This approach is obviouslydemanding in terms of the computational load if theentire covariance matrix is wanted. However, onlythe high lag-time estimators are expected to be seri-ously correlated, and, at least for them, this taskshould not be too overwhelming. It is along theselines that we are currently focusing our scientificefforts.
We thank G. Arcovito, M. De Spirito, M. Rocco, andU. Perini for the loan of some of the equipment usedin this study and A. Bailey, E. Paganini, and F. Tres-pidi for helpful discussions. We also acknowledgethe help and the suggestions received from A. Smartand D. S. Cannell for the generation of simulatedDLS data. E. Paganini and F. Trespidi are alsothanked for having introduced us to the world ofLabVIEW, without which we would probably neverhave undertaken this project. This research waspartially supported by funds from the ASI ~the ItalianSpace Agency! and from the INFM ~the Istituto Na-
measurements at large lag times: improving statistical accu-
zionale di Fisica della Materia! with a fellowship to D.Magatti.
References1. H. Z. Cummins and E. R. Pike, eds., Photon Correlation and
Light Beating Spectroscopy ~Plenum, New York, 1974!.2. B. J. Berne and R. Pecora, Dynamic Light Scattering ~Wiley,
New York, 1976!.3. H. Z. Cummins and E. R. Pike, eds., Photon Correlation Spec-
troscopy and Velocimetry ~Plenum, New York, 1974!.4. E. O. Schulz-DuBois, ed., Photon Correlation Techniques in
Fluid Mechanics, Vol. 38 of Springer Series in Optical Science~Springer-Verlag, Berlin, 1983!.
5. W. Brown, ed., Dynamic Light Scattering ~Clarendon, Oxford,1993!.
6. R. Foord, E. Jakeman, R. Jones, C. J. Oliver, and E. R. Pike,“Measurement of diffusion constants of macromolecules by dig-ital autocorrelation of scattered laser light,” IRE Conf. Proc. 14~1969!.
7. R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove,E. Wood, and A. R. Peacocke, “Determination of the diffusioncoefficients of haemocyanin at low concentration by intensityfluctuation spectroscopy of scattered laser light,” Nature 227,242–245 ~1970!.
8. K. Schatzel, “New concepts in correlator design,” in Institute ofPhysics Conference Series No. 77: Session 4 ~Hilger, London,1985!, pp. 175–184.
9. K. Schatzel, “Correlation techniques in dynamic light scatter-ing,” Appl. Phys. B 42, 193–213 ~1987!.
10. K. Schatzel, M. Drewel, and S. Stimac, “Photon correlation
racy,” J. Mod. Opt. 35, 711–718 ~1988!.11. K. Schatzel and E. O. Schulz-DuBois, “Improvements of pho-
ton correlation techniques,” Infrared Phys. 32, 409–416~1991!.
12. K. Schatzel, “Single-photon correlation techniques,” in Dy-namic Light Scattering, W. Brown, ed. ~Clarendon, Oxford,1993!, Chap. 2.
13. Measurements and Automation, catalog ~National Instru-ments, Austin, Tex., 2000!.
14. J. W. Goodman, Statistical Optics ~Wiley, New York, 1985!.15. Z. Kojro, A. Riede, M. Schubert, and W. Grill, “Systematic and
statistical errors in correlation estimators obtained from var-ious digital correlators,” Rev. Sci. Instrum. 70, 4487–4496~1999!.
16. A. E. Smart, R. V. Edwards, and W. V. Meyer, “Quantitativesimulation of errors in correlation analysis,” Appl. Opt. 40,4064–4078 ~2001!.
17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics~Wiley, New York, 1991!, Chap. 8.
18. J. S. Eid, J. D. Muller, and E. Gratton, “Data acquisition cardfor fluctuation correlation spectroscopy allowing full access tothe detected photon sequence,” Rev. Sci. Instrum. 71, 361–368~2000!.
19. K. Schatzel, “Noise on photon correlation data: I. Autocor-relation function,” Quantum Opt. 2, 287–305 ~2000!.
20. R. Peters, “Noise on photon correlation and its effects on datareduction algorithms,” in Dynamic Light Scattering, W.Brown, ed. ~Clarendon, Oxford, 1993!, Chap. 3.
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