Abstract

We present a PC-based multi-tau software correlator suitable for processing dynamic light-scattering data. The correlator is based on a simple algorithm that was developed with the graphical programming language LabVIEW, according to which the incoming data are processed on line without any storage on the hard disk. By use of a standard photon-counting unit, a National Instruments Model 6602-PCI timer–counter, and a 550-MHz Pentium III personal computer, correlation functions can be worked out in full real-time over time scales of ∼5 µs and in batch processing down to time scales of ∼300 ns. The latter limit is imposed by the speed of data transfer between the counter and the PC’s memory and thus is prone to be progressively reduced with future technological development. Testing of the correlator and evaluation of its performances were carried out by use of dilute solutions of calibrated polystyrene spheres. Our results indicate that the correlation functions are determined with such precision that the corresponding particle diameters can be recovered to within an accuracy of a few percent rms.

© 2001 Optical Society of America

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References

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  1. H. Z. Cummins, E. R. Pike, eds., Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).
  2. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  3. H. Z. Cummins, E. R. Pike, eds., Photon Correlation Spectroscopy 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, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital 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, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
    [CrossRef] [PubMed]
  8. K. Schatzel, “New concepts in correlator design,” in Institute of Physics Conference Series No. 77: Session 4 (Hilger, London, 1985), pp. 175–184.
  9. K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
    [CrossRef]
  10. K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
    [CrossRef]
  11. K. Schatzel, E. O. Schulz-DuBois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
    [CrossRef]
  12. K. Schatzel, “Single-photon correlation techniques,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 2.
  13. Measurements and Automation, catalog (National Instruments, Austin, Tex., 2000).
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  15. Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
    [CrossRef]
  16. A. E. Smart, R. V. Edwards, W. V. Meyer, “Quantitative simulation of errors in correlation analysis,” Appl. Opt. 40, 4064–4078 (2001).
    [CrossRef]
  17. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 8.
    [CrossRef]
  18. J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
    [CrossRef]
  19. K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function,” Quantum Opt. 2, 287–305 (2000).
    [CrossRef]
  20. R. Peters, “Noise on photon correlation and its effects on data reduction algorithms,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 3.

2001 (1)

2000 (2)

J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
[CrossRef]

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function,” Quantum Opt. 2, 287–305 (2000).
[CrossRef]

1999 (1)

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

1991 (1)

K. Schatzel, E. O. Schulz-DuBois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

1988 (1)

K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

1987 (1)

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

1970 (1)

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

1969 (1)

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Blagrove, R. J.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Drewel, M.

K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

Edwards, R. V.

Eid, J. S.

J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
[CrossRef]

Foord, R.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gratton, E.

J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
[CrossRef]

Grill, W.

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

Jakeman, E.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Jones, R.

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Kojro, Z.

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

Meyer, W. V.

Muller, J. D.

J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
[CrossRef]

Oliver, C. J.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Peacocke, A. R.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Peters, R.

R. Peters, “Noise on photon correlation and its effects on data reduction algorithms,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 3.

Pike, E. R.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

Riede, A.

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 8.
[CrossRef]

Schatzel, K.

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function,” Quantum Opt. 2, 287–305 (2000).
[CrossRef]

K. Schatzel, E. O. Schulz-DuBois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

K. Schatzel, “New concepts in correlator design,” in Institute of Physics Conference Series No. 77: Session 4 (Hilger, London, 1985), pp. 175–184.

K. Schatzel, “Single-photon correlation techniques,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 2.

Schubert, M.

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

Schulz-DuBois, E. O.

K. Schatzel, E. O. Schulz-DuBois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

Smart, A. E.

Stimac, S.

K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 8.
[CrossRef]

Wood, E.

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

Infrared Phys. (1)

K. Schatzel, E. O. Schulz-DuBois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

IRE Conf. Proc. (1)

R. Foord, E. Jakeman, R. Jones, C. J. Oliver, E. R. Pike, “Measurement of diffusion constants of macromolecules by digital autocorrelation of scattered laser light,” IRE Conf. Proc. 14 (1969).

J. Mod. Opt. (1)

K. Schatzel, M. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

Nature (1)

R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, A. R. Peacocke, “Determination of the diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Quantum Opt. (1)

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function,” Quantum Opt. 2, 287–305 (2000).
[CrossRef]

Rev. Sci. Instrum. (2)

Z. Kojro, A. Riede, M. Schubert, W. Grill, “Systematic and statistical errors in correlation estimators obtained from various digital correlators,” Rev. Sci. Instrum. 70, 4487–4496 (1999).
[CrossRef]

J. S. Eid, J. D. Muller, E. Gratton, “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. 71, 361–368 (2000).
[CrossRef]

Other (11)

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 8.
[CrossRef]

K. Schatzel, “Single-photon correlation techniques,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 2.

Measurements and Automation, catalog (National Instruments, Austin, Tex., 2000).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

K. Schatzel, “New concepts in correlator design,” in Institute of Physics Conference Series No. 77: Session 4 (Hilger, London, 1985), pp. 175–184.

H. Z. Cummins, E. R. Pike, eds., Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

H. Z. Cummins, E. R. Pike, eds., Photon Correlation Spectroscopy and Velocimetry (Plenum, New York, 1974).

E. O. Schulz-DuBois, ed., Photon Correlation Techniques in Fluid Mechanics, Vol. 38 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1983).

W. Brown, ed., Dynamic Light Scattering (Clarendon, Oxford, 1993).

R. Peters, “Noise on photon correlation and its effects on data reduction algorithms,” in Dynamic Light Scattering, W. Brown, ed. (Clarendon, Oxford, 1993), Chap. 3.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the behavior of the count rate n(t) plotted against time t in a DLS experiment. To achieve high accuracy in the measurement of the correlation function, sampling time Δt must be much shorter than lag time τ.

Fig. 2
Fig. 2

Systematic absolute error δg μ introduced in the measure of the correlation function by use of triangular averaging, plotted as a function of the ratio between the sampling time Δt and the (field) decay time t 0 of a single-exponential correlation function [see Eq. (8) in the text]. The various curves represent different values of the ratio α between the lag time τ and the sampling time Δt and were generated with a value of β = 1. It can be seen that, regardless of t 0, if an accuracy of better than 10-3 is desired α must be larger than 7 (the solid curve).

Fig. 3
Fig. 3

Schematic diagram of the multi-tau scheme adopted in the correlator for the particular case of a shift register with P = 9 elements and a binning ratio of m = 3. For each linear correlator of order s, the first P/ m lag times overlap the last one of the (s - 1) correlator and are therefore discarded. We obtained the overall lag-time sequence of the correlation function by merging the lag times of each linear correlator (the shaded boxes). For clarity only the first four linear correlators are represented in the figure.

Fig. 4
Fig. 4

Computational time per point t comp/N 0 required for processing the data in the software correlator plotted as a function of the parameter m for different values of the ratio P/ m. The filled symbols refer to the overall computational time per point, whereas the open symbols, which are independent of P/ m, account for only the averaging of the input data to be fed to the different linear correlators. For the curve that represents P/ m = 7 the minimum computational time per point is approximately 5 µs and falls at m ∼ 4.

Fig. 5
Fig. 5

Schematic diagram of the correlator architecture. The pulse stream exiting the photon-counting unit is sampled at two different times and simultaneously fed to the slow and the fast correlators. The slow correlator operates in full real time and acts as the master correlator that triggers the operation of the slave correlator, the fast correlator, that thus operates as a batch processor.

Fig. 6
Fig. 6

Intensity-correlation function as measured with the correlator working in the slow configuration (see Table 1) for a sample of polystyrene spheres with a certified diameter of d cert = 107 ± 7 nm: (a) Data and their fit to Eq. (15) and (b) relative residuals. The scattering angle was 90°, and the measuring time t was approximately 300 s. The diameter recovered from the fitting was d = 106.73 ± 0.1 nm.

Fig. 7
Fig. 7

Intensity-correlation function as measured with the correlator working in the fast configuration (see Table 1) for a sample of polystyrene spheres with a certified diameter of d sert = 30 ± 1.3 nm: (a) Data and their fit to Eq. (15) and (b) relative residuals. The squares represent the results for the slave correlator, whereas the circles represent those for the master correlator. The scattering angle was 90°, and the measuring time t was approximately 300 s. Under this configuration the duty cycle of the slave correlator is approximately 1%, which corresponds to an effective measuring time of approximately 3 s. The diameter recovered from the fitting was d = 30 ± 0.1 nm.

Fig. 8
Fig. 8

Ratio between the recovered and the certified diameters plotted as a function of qd cert for four different particle diameters that were obtained by the measurement of the correlation function at different wave vectors q. The data are spread about the expected value of 1 with deviations of the order of a few percent rms.

Tables (1)

Tables Icon

Table 1 Parameters of the Correlator for Two Working Configurationsa

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Gnτntnt+τ=limT1T0T ntnt+τdt,
gnτntnt+τnt2,
μt=1Δtt-Δt/2t+Δt/2 ntdt=nt * rectt/ΔtΔt,
Gμτμtμt+τ
gμτ=gnτ * Λτ/ΔtΔt,
gnτ=1+β exp-2τ/t0,
gμτ=1+β exp-2τ/t0sinhΔt/t0Δt/t02, τΔt.
δgμτgμτ-gnτ=β exp-2τ/t0sinhΔt/t0Δt/t02-1, τΔt,
gnτ=1+i ai exp-2τ/ti,
Δts=msΔt0,  s=0, 1, 2,, S-1,
Ns=m-sN0,  s=0, 1, 2,, S-1.
τsmin=P/mΔts.
tcompw1i=0S Ns+w2i=0S NsP=N0w1+w2P1-m-s1-m-1N0w1+w2Pmm-1,  m>1,
tcompN0w1+w2αmmm-1,
mmin=1+1+w1/w2α1/2.
gτ=B+β exp-Γτ,
Γ=2Dq2,
D=kT6πηR,

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