Abstract

We revisited the classical Schätzel formulas (K. Schätzel, Quantum Optics 2, 2871990) of the variance and covariance matrix associated to the normalized auto-correlation function in a Dynamic Light Scattering experiment when the sample is characterized by a single exponential decay function. Although thoroughly discussed by Schätzel who also outlined a correcting procedure, such formulas do not include explicitly the effects of triangular averaging that arise when the sampling time Δt is comparable or larger than the correlation time τc. If these effects are not taken into account, such formulas might be highly inaccurate. In this work we have solved this problem and worked out two exact analytical expressions that generalize the Schätzel formulas for any value of the ratio Δt/τc. By the use of extensive computer simulations we tested the correctness of the new formulas and showed that the variance formula can be exploited also in the case of fairly broad bell-shaped polydisperse samples (polydispersities up to ∼ 50 – 100%) and in connection with single exponential decay cross-correlation functions, provided that the average count rate is computed as the geometrical mean of the average count rates of the two channels. Finally, when tested on calibrated polystyrene particles, the new variance formula is able to reproduce quite accurately the error bars obtained by averaging the experimental data.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley,NY1976).
  2. R.G.W. Brown ed., Dynamic Light Scattering (Clarendon,Oxford1993).
  3. K. Schätzel, New Concepts in Correlator Design 2985 (Insr. Phys. Conf. Ser. 77) (Institute of PhysicsBristol) (1985).
  4. E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light”, J. Phys. A: Gen. Phys. 1, 128–138 (1968).
    [Crossref]
  5. V. Degiorgio and J. B. Lastovka, “Intensity-correlations spectroscopy”, Phys. Rev. 4, 2033–2050 (1971).
    [Crossref]
  6. B. E. A. Saleh and M. F. Cardoso, “The effect of channel correlation on the accuracy of photon counting digital autocorrelators”, J. Phys. A: Math. Nucl. Gen. 6, 1897–1909 (1973).
    [Crossref]
  7. K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 287–305 (1990).
    [Crossref]
  8. K. Schätzel, “Erratum - Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 467–468 (1990).
    [Crossref]
  9. M. Molteni, U.M. Weigel, F. Remiro, T. Durduran, and F. Ferri, “Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions”, Opt. Express 22, 28002–28018 (2014).
    [Crossref] [PubMed]
  10. F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
    [Crossref]
  11. K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
    [Crossref]
  12. A. J. F. Siegert, “On the fluctuations in signals returned by many independently moving scatterers“, MIT Rad. Lab. Rep. 465 (1943).
  13. M. Molteni and F. Ferri, “Commercial counterboard for 10 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 87, 113108 (2016).
    [Crossref] [PubMed]
  14. D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering””, Appl. Opt. 40, 4011–4021 (2001).
    [Crossref]
  15. R. Peters, Dynamic Light Scattering, R.G.W. Brown, ed. (Clarendon,Oxford1993), Chap.3
  16. D.E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy”, Phys. Rev. A,  10, 1938–1945 (1974).
    [Crossref]
  17. T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
    [Crossref]
  18. C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
    [Crossref] [PubMed]
  19. G. Dietsche, M. Ninck, C. Ortolf, J. Li, F. Jaillon, and T. Gisler, “Fiber-based moltispeckle detection for time-resolved diffusiong-wave spectroscopy: characterization and application to blood flow detection in deep tissue”, Appl. Opt. 46, 8506–8514 (2007).
    [Crossref] [PubMed]
  20. S. Saffarian and E.L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias”, Biophys. J. 84, 2030–2042 (2003).
    [Crossref] [PubMed]

2016 (1)

M. Molteni and F. Ferri, “Commercial counterboard for 10 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 87, 113108 (2016).
[Crossref] [PubMed]

2014 (1)

2007 (1)

2006 (1)

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

2003 (2)

S. Saffarian and E.L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias”, Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

2001 (2)

D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering””, Appl. Opt. 40, 4011–4021 (2001).
[Crossref]

T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
[Crossref]

1990 (2)

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 287–305 (1990).
[Crossref]

K. Schätzel, “Erratum - Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 467–468 (1990).
[Crossref]

1988 (1)

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
[Crossref]

1974 (1)

D.E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy”, Phys. Rev. A,  10, 1938–1945 (1974).
[Crossref]

1973 (1)

B. E. A. Saleh and M. F. Cardoso, “The effect of channel correlation on the accuracy of photon counting digital autocorrelators”, J. Phys. A: Math. Nucl. Gen. 6, 1897–1909 (1973).
[Crossref]

1971 (1)

V. Degiorgio and J. B. Lastovka, “Intensity-correlations spectroscopy”, Phys. Rev. 4, 2033–2050 (1971).
[Crossref]

1968 (1)

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light”, J. Phys. A: Gen. Phys. 1, 128–138 (1968).
[Crossref]

Berne, B.J.

B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley,NY1976).

Cardoso, M. F.

B. E. A. Saleh and M. F. Cardoso, “The effect of channel correlation on the accuracy of photon counting digital autocorrelators”, J. Phys. A: Math. Nucl. Gen. 6, 1897–1909 (1973).
[Crossref]

Degiorgio, V.

V. Degiorgio and J. B. Lastovka, “Intensity-correlations spectroscopy”, Phys. Rev. 4, 2033–2050 (1971).
[Crossref]

Dietsche, G.

Drewel, M.

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
[Crossref]

Durduran, T.

M. Molteni, U.M. Weigel, F. Remiro, T. Durduran, and F. Ferri, “Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions”, Opt. Express 22, 28002–28018 (2014).
[Crossref] [PubMed]

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Elson, E.L.

S. Saffarian and E.L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias”, Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

Ferri, F.

M. Molteni and F. Ferri, “Commercial counterboard for 10 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 87, 113108 (2016).
[Crossref] [PubMed]

M. Molteni, U.M. Weigel, F. Remiro, T. Durduran, and F. Ferri, “Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions”, Opt. Express 22, 28002–28018 (2014).
[Crossref] [PubMed]

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering””, Appl. Opt. 40, 4011–4021 (2001).
[Crossref]

Furuya, D.

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Gisler, T.

Greenberg, J.H.

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Jaillon, F.

Jakeman, E.

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light”, J. Phys. A: Gen. Phys. 1, 128–138 (1968).
[Crossref]

Koppel, D.E.

D.E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy”, Phys. Rev. A,  10, 1938–1945 (1974).
[Crossref]

Lastovka, J. B.

V. Degiorgio and J. B. Lastovka, “Intensity-correlations spectroscopy”, Phys. Rev. 4, 2033–2050 (1971).
[Crossref]

Li, J.

Magatti, D.

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering””, Appl. Opt. 40, 4011–4021 (2001).
[Crossref]

Molteni, M.

M. Molteni and F. Ferri, “Commercial counterboard for 10 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 87, 113108 (2016).
[Crossref] [PubMed]

M. Molteni, U.M. Weigel, F. Remiro, T. Durduran, and F. Ferri, “Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions”, Opt. Express 22, 28002–28018 (2014).
[Crossref] [PubMed]

Ninck, M.

Ortolf, C.

Pecora, R.

B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley,NY1976).

Peters, R.

R. Peters, Dynamic Light Scattering, R.G.W. Brown, ed. (Clarendon,Oxford1993), Chap.3

Pike, E. R.

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light”, J. Phys. A: Gen. Phys. 1, 128–138 (1968).
[Crossref]

Remiro, F.

Rigler, R.

T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
[Crossref]

Saffarian, S.

S. Saffarian and E.L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias”, Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. F. Cardoso, “The effect of channel correlation on the accuracy of photon counting digital autocorrelators”, J. Phys. A: Math. Nucl. Gen. 6, 1897–1909 (1973).
[Crossref]

Schätzel, K.

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 287–305 (1990).
[Crossref]

K. Schätzel, “Erratum - Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 467–468 (1990).
[Crossref]

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
[Crossref]

K. Schätzel, New Concepts in Correlator Design 2985 (Insr. Phys. Conf. Ser. 77) (Institute of PhysicsBristol) (1985).

Siegert, A. J. F.

A. J. F. Siegert, “On the fluctuations in signals returned by many independently moving scatterers“, MIT Rad. Lab. Rep. 465 (1943).

Stimac, S.

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
[Crossref]

Vogel, H.

T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
[Crossref]

Weigel, U.M.

Wohland, T.

T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
[Crossref]

Yodh, A.G.

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Yu, G.

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Zhou, C.

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Appl. Opt. (2)

Biophys. J. (1)

S. Saffarian and E.L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias”, Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

Boiphys. J. (1)

T. Wohland, R. Rigler, and H. Vogel, “The Standard Deviation in Fluorescence Correlation Spectroscopy”, Boiphys. J. 80, 2987–2999 (2001).
[Crossref]

J. Modern Optics (1)

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurement Large Lag Times: Improving Statistical Accuracy”, J. Modern Optics 35, 711–718 (1988).
[Crossref]

J. Phys. A: Gen. Phys. (1)

E. Jakeman and E. R. Pike, “The intensity-fluctuation distribution of Gaussian light”, J. Phys. A: Gen. Phys. 1, 128–138 (1968).
[Crossref]

J. Phys. A: Math. Nucl. Gen. (1)

B. E. A. Saleh and M. F. Cardoso, “The effect of channel correlation on the accuracy of photon counting digital autocorrelators”, J. Phys. A: Math. Nucl. Gen. 6, 1897–1909 (1973).
[Crossref]

Opt. Express (1)

Opt. Express. (1)

C. Zhou, G. Yu, D. Furuya, J.H. Greenberg, A.G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow cortical spreading depression in rat brain”, Opt. Express. 14, 1125–1144 (2006).
[Crossref] [PubMed]

Phys. Rev. (1)

V. Degiorgio and J. B. Lastovka, “Intensity-correlations spectroscopy”, Phys. Rev. 4, 2033–2050 (1971).
[Crossref]

Phys. Rev. A (1)

D.E. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy”, Phys. Rev. A,  10, 1938–1945 (1974).
[Crossref]

Quantum Optics (2)

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 287–305 (1990).
[Crossref]

K. Schätzel, “Erratum - Noise on photon correlation data: I. Autocorrelation functions”, Quantum Optics 2, 467–468 (1990).
[Crossref]

Rev. Sci. Instrum. (2)

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

M. Molteni and F. Ferri, “Commercial counterboard for 10 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 87, 113108 (2016).
[Crossref] [PubMed]

Other (5)

A. J. F. Siegert, “On the fluctuations in signals returned by many independently moving scatterers“, MIT Rad. Lab. Rep. 465 (1943).

R. Peters, Dynamic Light Scattering, R.G.W. Brown, ed. (Clarendon,Oxford1993), Chap.3

B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley,NY1976).

R.G.W. Brown ed., Dynamic Light Scattering (Clarendon,Oxford1993).

K. Schätzel, New Concepts in Correlator Design 2985 (Insr. Phys. Conf. Ser. 77) (Institute of PhysicsBristol) (1985).

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Figures (7)

Fig. 1
Fig. 1 Comparison between Eqs. (6) and (8). (a) and (b): behaviors of Covg2 (k, l) given by our Eq. (8) as a function of k for fixed Γ = 0.1 and variable l (a) and for variable Γ and fixed l = 5 (b). All the curves were computed by setting M = 105, β = 1, and 〈n〉 = 1. The peaks correspond to the variances (k = l). (c): relative residuals between (6) and (8) for the curves of (a); (d): relative residuals between Eqs. (6) and (8) for the curves of (b).
Fig. 2
Fig. 2 Comparison between Eqs. (7) and (9). (a) and (b): behaviors of σ g 2 2 ( k ) given by Eqs. (7) (open symbols) and (9) (solid small circles) as a function of k for two values of Γ = 0.05 (red circles) and Γ = 0.8 (blue squares) with different statistical accuracies, namely M = 105 (a) and M = 2 × 103 (b). All the four curves were computed by setting β = 1 and 〈n〉 = 1. (c): relative residuals between Eqs. (6) and (8) for the curves of (a); (d): relative residuals between Eqs. (6) and (8) for the curves of (b).
Fig. 3
Fig. 3 Results of the simulation test described in Sect.3.1. (a): behaviors of the average 〈g2(τk)〉 (red circles) and single batch g2(τk) (green triangles) auto-correlation functions obtained by computer simulations for the case of a monodisperse sample with a single exponential decay time τc = 10−4 s. The simulation was carried out by setting β = 1, 〈c.r.〉 = 105 Hz, T = 1 s and N = 104. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g2(τk)〉 have been shifted upwards by 0.5. (b): relative residuals between simulated data and fit; the residuals corresponding to 〈g2(τk)〉 have been zoomed by a factor 100. (c): comparison between the standard deviation σg2(τk) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σg2(τk) and our formula (red circles). In this panel the blue squares represent the residuals when the correlation analysis is carried out by using a different multi-tau scheme with p = 16, m = 2, S = 20.
Fig. 4
Fig. 4 Results of the simulation test described in Sect.3.2. (a): behaviors of the average 〈g2(τk)〉 (red circles) and single batch g2(τk) (green triangles) auto-correlation functions obtained by computer simulations for the case of a polydisperse sample characterized by a Log-normal distribution of decay times with average value 〈τc〉 = 10−4s and relative standard deviation στc/〈τc〉 = 50%. The simulation was carried out by setting β = 1, 〈c.r.〉 = 105 Hz, T = 1 s and N = 104. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g2(τk)〉 have been shifted upwards by 0.5. (b): relative residuals between simulated data and fit. (c): comparison between the standard deviation σg2(τk) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σg2(τk) and our formula (red circles). In this panel the blue squares represent the residuals when a στc /〈τc〉 = 100% polydispersity was considered.
Fig. 5
Fig. 5 Results of the simulation test described in Sect.3.3. (a): behaviors of the average auto- and cross-correlation functions 〈g2(τk)〉AA (blue squares), 〈g2(τk)〉BB (green triangles), 〈g2(τk)〉AB (red circles) and an example of a single batch cross-correlation function g2(τk)AB (orange lozenges) obtained by computer simulations for the case of a monodisperse sample with a single exponential decay time τc = 10−4 s. The simulation was carried out by setting β = 1, 〈c.r.A = 2〈c.r.B = 105 Hz, T = 1 s and N = 104. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fits of the various curves have been shifted upwards by increasing steps of 0.5. (b): relative residuals between cross-correlation simulated data and fit; the residuals corresponding to 〈g2(τk)〉AB have been zoomed by a factor 100. (c): comparison between the standard deviation σg2(τk) associated to the cross-correlation function and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σg2(τk) and our formula (red circles). In this panel we have also reported the residuals obtained for the two auto-correlation functions (green triangles and blue blue squares) and the ones obtained for the cross-correlation when 〈c.r.A = 〈c.r.B = 105 Hz (black dots).
Fig. 6
Fig. 6 Results of the experimental test described in Sect.4. (a): behaviors of the average cross-correlation function 〈g2(τk)〉 (red circles) and an example of a single batch cross-correlation function g2(τk)B (blue lozenges) obtained by DLS data on a aqueous dispersion of d = 41 ±3nm latex spheres whose scattered light was collected at 90° with a mono-mode fiber and coupled to two photo-multipliers whose average count rates were 〈c.r.A ∼ 1.5×104 Hz and 〈c.r.B ∼ 1.0 × 104 Hz. The measuring time was T = 0.7 s and N = 104 independent batch were analyzed. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g2(τk)〉 have been shifted upwards by 1. (b): relative residuals between simulated data and fit; the residuals corresponding to 〈g2(τk)〉 have been zoomed by a factor 50. The orange symbols are the non systematic residuals obtained when 〈g2(τk)〉 is fitted with the with the first two cumulants. (c): comparison between the standard deviation σg2(τk) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σg2(τk) and our formula for the data of (c) (red circles) and when the measurements are taken at a higher count rate (green squares).
Fig. 7
Fig. 7 Comparison between Eqs. (6) and (8). (a): comparison between the behaviors of Covg2(τk, τl) as a function of τk of the auto-correlation function obtained by computer simulations (open symbols) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas for the case of a monodisperse sample with a single exponential decay correlation function with τc = 10−4 s. Different symbols refer to channels acquired with different sampling times indicated on the bottom. The covariances are evaluated at fixed lag-times corresponding to l = 16 for all the shown stages. The peaks correspond to variance (k = l). The measuring time is T = 1 s and the average count rate is 〈c.r.〉 = 105 Hz. (b): absolute residuals between simulations and our formula (open symbols) and the Schätzel formula (solid symbols). The values of Γ corresponding to each stage are reported on top of the panel.

Equations (29)

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g 2 ( τ ) = lim T 1 T 0 T I ( t ) I ( t + τ ) d t [ lim T 1 T 0 T I ( t ) d t ] 2
g 2 ( τ k ) = 1 M k i = 1 M k n i n i + k n 0 n k
n 0 = 1 M k i = 1 M k n i and n k = 1 M k i = k M n i
g 2 ( τ k ) = 1 + β | χ ( τ k ) | 2 .
χ k = exp ( Γ | k | )
Cov g 2 ( k , l ) = β 2 M 1 { exp [ 2 Γ ( k l ) ] [ k l + coth ( 2 Γ ) ] + exp [ 2 Γ ( k + l ) ] [ k + l + coth ( 2 Γ ) ] + + 2 β exp ( 2 Γ k ) [ exp ( 2 Γ l ) ( k l + coth ( 2 Γ ) 2 coth ( Γ ) ) + 2 ( k l + coth ( Γ ) ) ] + + 2 n 1 β 1 ( exp [ 2 Γ ( k l ) ] + exp [ 2 Γ ( k + l ) ] + 2 β exp [ 2 Γ k ] ) + + δ k l n 2 β 2 [ 1 + β exp ( 2 Γ k ) ] }
σ g 2 2 ( k ) = β 2 M 1 { coth ( 2 Γ ) + exp ( 4 Γ k ) [ 2 k + coth ( 2 Γ ) ] + + 2 β exp ( 4 Γ k ) [ 2 k + coth ( 2 Γ ) 2 coth ( 2 Γ ) + 4 β exp ( 2 Γ k ) coth ( Γ ) + + 2 n 1 β 1 [ 1 + exp ( 4 Γ k ) + 2 β exp ( 2 Γ k ) ] + n 2 β 2 [ 1 + β exp ( 2 Γ k ) ] }
Cov g 2 ( k , l ) = β 2 M k { 4 sinh 2 ( Γ ) Γ 2 χ ^ 0 2 e 2 Γ k cosh ( 2 Γ l ) + 8 β sinh 3 ( Γ ) Γ 2 χ ^ 0 e 2 Γ k ( 1 + e 2 Γ l ) + + sin 4 ( Γ ) Γ 4 e 2 Γ k [ ( coth ( 2 Γ ) + d ) e 2 Γ l + ( a coth ( 2 Γ ) + b ) e 2 Γ l + 4 β ( coth ( Γ ) + d ) ] + + 4 β sinh 6 Γ 6 e 2 Γ ( k + l ) [ coth ( Γ ) 1 ] 4 β sin 5 ( Γ ) Γ 5 e 2 Γ ( k + 1 ) [ k + l 4 + 2 coth ( Γ ) ] + + 2 β n [ 2 sinh 2 ( Γ ) Γ 2 e 2 Γ k cosh ( 2 Γ l ) + 2 β sinh 3 ( Γ ) Γ 3 e 2 Γ k ( 1 + e 2 Γ l ) 2 β sinh 4 ( Γ ) Γ 4 e 2 Γ ( k + l ) ] + δ k l β 2 n 2 [ 1 + β sinh 2 ( Γ ) Γ 2 e 2 Γ k ] }
σ g 2 2 ( k ) = β 2 M k { 2 sinh 2 ( Γ ) Γ 2 χ ^ 0 2 e 4 Γ k + 4 β sinh 2 ( Γ ) Γ 2 χ ^ 0 2 e 2 Γ k + 8 β sinh 3 ( Γ ) Γ 3 χ ^ 0 e 4 Γ k + + sin 4 ( Γ ) Γ 4 [ coth ( 2 Γ ) 1 + 4 β e 2 Γ k [ coth ( Γ ) 1 ] + e 4 Γ k [ a coth ( 2 Γ ) + c ] ] + + 4 β sinh 6 ( Γ ) Γ 6 e 4 Γ k [ coth ( Γ ) 1 ] β sinh 5 ( Γ ) Γ 5 e 4 Γ k [ k 2 + coth ( Γ ) ] + χ ^ 0 4 + + 2 β n [ χ ^ 0 2 + sinh 2 ( Γ ) Γ 2 e 2 Γ k ( e 2 Γ k + 2 β χ ^ 0 ) + 2 β sinh 3 ( Γ ) Γ 3 e 4 Γ k 2 β sinh 4 ( Γ ) Γ 4 e 4 Γ k ] + 1 β 2 n 2 [ 1 + β sinh 2 ( Γ ) Γ 2 e 2 Γ k ] }
{ a = 1 + 2 β b = k + l + 2 β k + 2 β l 2 β + 4 β χ ^ 0 2 16 β χ ^ 0 c = 2 k + 4 β k 2 β + 4 β χ ^ 0 2 16 β χ ^ 0 d = k l 2 and χ ^ 0 = 2 Γ 1 + e 2 Γ 2 Γ 2
{ τ k , s = k Δ t s k = 0 , 1 , 2 , , p 1 ( a ) Δ t s = m s Δ t 0 s = 0 , 1 , 2 , , S 1 ( b )
g 2 ( τ k ) = 1 N i = 1 N [ g 2 ( τ k ) ] i
σ g 2 2 ( τ k ) = 1 N i = 1 N [ g 2 ( τ k ) ] i 2 g 2 ( τ k ) 2
g 2 ( τ k ) = B + β exp ( 2 τ k / τ c )
χ ( τ k ) = 0 N ( τ c ) exp ( τ k / τ c ) d τ c
n = n A n B = c . r . A c . r . B Δ t .
I ^ ( t ) = 1 Δ t t Δ t / 2 t + Δ t / 2 I ( t ) d t = 1 Δ t [ I R Δ t ] ( t )
g ^ 2 ( τ ) = 1 Δ t [ g 2 Λ Δ t ] ( τ )
| χ ^ k | 2 = 1 Δ τ ( | χ k | 2 Λ Δ t ) ( τ ) = 1 ( Δ t ) 2 ( k 1 ) Δ t ( k + 1 ) Δ t | χ ( t ) | 2 ( Δ t | t k Δ t | ) d t .
| χ ^ 0 | 2 = 1 ( Δ t ) 2 Δ t Δ t | χ ( t ) | 2 ( Δ t | t | ) d t
| χ ^ ± 1 | 2 = 1 ( Δ t ) 2 0 2 Δ t | χ ( t ) 2 | ( Δ t | t Δ t | ) d t
χ ^ 0 = 1 ( Δ t ) 2 Δ t Δ t e 2 Γ | t | Δ t ( Δ t | t | ) d t = 2 Γ 1 + e 2 Γ 2 Γ 2 ,
χ ^ k = 1 ( Δ t ) 2 ( k 1 ) Δ t ( k + 1 ) Δ t e 2 Γ | t | / Δ t ( Δ t | t k Δ t | ) d t = sinh ( Γ ) Γ e Γ k = sinh ( Γ ) Γ χ k .
g ^ 2 ( τ k ) = 1 + β exp ( 2 Γ k ) sinh 2 ( Γ ) Γ 2 .
i = + | χ ^ i | 2 | χ ^ i + k l | 2 = sinh 4 ( Γ ) Γ 4 e 2 Γ ( k l ) [ k l + coth ( 2 Γ ) ] + 2 sinh 2 ( Γ ) Γ 2 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 2 Γ ( k l ) i = + | χ ^ i | 2 | χ ^ i + k + l | 2 = sinh 4 ( Γ ) Γ 4 e 2 Γ ( k + l ) [ k + l + coth ( 2 Γ ) ] + 2 sinh 2 ( Γ ) Γ 2 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 2 Γ ( k + 1 ) Re ( χ ^ k χ ^ l * i = + χ ^ i χ ^ i + k l * ) = sinh 4 ( Γ ) Γ 4 e 2 Γ k [ k l + coth ( Γ ) ] + 2 sinh 3 ( Γ ) Γ 3 ( χ ^ 0 sinh ( Γ ) Γ ) e 2 Γ k Re ( χ ^ k χ ^ l * i = + χ ^ i χ ^ i + k + l * ) = sinh 4 ( Γ ) Γ 4 e 2 Γ ( k + l ) [ k + l + coth ( Γ ) ] + 2 sinh 3 ( Γ ) Γ 3 ( χ ^ 0 sinh ( Γ ) Γ ) e 2 Γ ( k + l ) Re ( i = + χ ^ i χ ^ i + k * χ ^ i + l * χ ^ i + k + l ) = sinh 4 ( Γ ) Γ 4 e 2 Γ k [ k l + coth ( Γ ) + e 2 Γ l ( coth ( 2 Γ ) coth ( Γ ) ) ] + + 2 sinh 3 ( Γ ) Γ 3 ( χ ^ 0 sinh ( Γ ) Γ ) ( 1 + e 2 Γ l ) e 2 Γ k | χ ^ k | 2 | χ ^ l | 2 i = + | χ ^ i | 2 = sinh 6 ( Γ ) Γ 6 e 2 Γ ( k + l ) coth ( Γ ) + sinh 4 ( Γ ) Γ 4 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 2 Γ ( k + l ) | χ ^ l | 2 Re ( χ ^ k i = + χ ^ i χ ^ i + k * ) = sinh 5 ( Γ ) Γ 5 e 2 Γ ( k + l ) [ k + coth ( Γ ) ] + 2 sinh 4 ( Γ ) Γ 4 ( χ ^ 0 sinh ( Γ ) Γ ) e 2 Γ ( k + l ) | χ ^ k | 2 Re ( χ ^ l i = + χ ^ i χ ^ i + l * ) = sinh 5 ( Γ ) Γ 5 e 2 Γ ( k + l ) [ l + coth ( Γ ) ] + 2 sinh 4 ( Γ ) Γ 4 ( χ ^ 0 sinh ( Γ ) Γ ) e 2 Γ ( k + l )
i = + ( | χ ^ i | 2 ) 2 = sinh 4 ( Γ ) Γ 4 coth ( 2 Γ ) + ( χ ^ 0 2 ) 2 sinh 4 ( Γ ) Γ 4 i = + | χ ^ i | 2 | χ ^ i + 2 k | 2 = sinh 4 ( Γ ) Γ 4 e 4 Γ k [ 2 k + coth ( 2 Γ ) ] + 2 sinh 2 ( Γ ) Γ 2 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 4 Γ k | χ ^ k | 2 i = + | χ ^ i | 2 = sinh 4 ( Γ ) Γ 4 e 2 Γ k coth ( 2 Γ ) + sinh 2 ( Γ ) Γ 2 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 2 Γ k Re ( χ ^ k 2 i = + χ ^ i χ ^ i + 2 k * ) = sinh 4 ( Γ ) Γ 4 e 4 Γ k [ 2 k + coth ( Γ ) ] + 2 sinh 3 ( Γ ) Γ 3 ( χ ^ 0 sinh ( Γ ) Γ ) e 4 Γ k Re ( χ ^ k 2 i = + χ ^ i χ ^ i + k * 2 χ ^ i + 2 k ) = sinh 4 ( Γ ) Γ 4 e 4 Γ k [ coth ( 2 Γ ) coth ( Γ ) ] + sinh 4 ( Γ ) Γ 4 e 2 Γ k coth ( Γ ) + + sinh 2 ( Γ ) Γ 2 e 2 Γ k ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) ( 1 + 2 sinh ( Γ ) Γ e 2 Γ k ) ( | χ ^ k | 2 ) 2 i = + | χ ^ i | 2 = sinh 6 ( Γ ) Γ 6 e 4 Γ k coth ( Γ ) + sinh 4 ( Γ ) Γ 4 ( χ ^ 0 2 sinh 2 ( Γ ) Γ 2 ) e 4 Γ k | χ ^ k | 2 Re ( χ ^ k i = + χ ^ i χ ^ i + k * ) = sinh 5 ( Γ ) Γ 5 e 4 Γ k [ k + coth ( Γ ) ] + 2 sinh 4 ( Γ ) Γ 4 ( χ ^ 0 sinh ( Γ ) Γ ) e 4 Γ k
i = + ( | χ ^ i | 2 ) 2 = i 0 ( | χ ^ i | 2 ) 2 + ( | χ ^ 0 | 2 ) 2
i = + | χ ^ i | 2 | χ ^ i + k l | 2 = i 0 i l k | χ ^ i | 2 | χ ^ i + k l | 2 + | χ ^ l k | 2 | χ ^ 0 | 2 + | χ ^ 0 | 2 | χ ^ k l | 2
Cov g 2 ( τ k , τ l ) = 1 N i = 1 N [ g 2 ( τ k ) ] i [ g 2 ( τ l ) ] i [ 1 N i = 1 N [ g 2 ( τ k ) ] i ] [ 1 N i = 1 N [ g 2 ( τ l ) ] i ]

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