Abstract

Bounding the errors of measurements derived from correlation functions of light scattered from some physical systems is typically complicated by the ill conditioning of the data inversion. Parameter values are estimated from fitting well-chosen models to measurements taken for long enough to look acceptable, or at least to yield convergence to some reasonable result. We show some simple numerical simulations that indicate the possibility of substantial and unanticipated errors even in comparatively simple experiments. We further show quantitative evidence for the effectiveness of a number of ad hoc aspects of the art of performing good light-scattering experiments and recovering useful measurements from them. Separating data-inversion properties from experimental inconsistencies may lead to a better understanding and better bounding of some errors, giving new ways to improve overall experimental accuracy.

© 2001 Optical Society of America

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References

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  1. V. Degiorgio, J. B. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2050 (1971).
    [CrossRef]
  2. E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
    [CrossRef]
  3. J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
    [CrossRef]
  4. K. Schätzel, “Noise in photon correlation and photon structure functions,” Opt. Acta 30, 155–166 (1983).
    [CrossRef]
  5. M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
    [CrossRef]
  6. E. R. Pike, E. Jakeman, “Photon statistics and photon-correlation spectroscopy,” Adv. Quantum Electron. 2, 1 (1974).
    [CrossRef]
  7. D. S. Cannell, Department of Physics, University of California Santa Barbara, Santa Barbara, Calif. 93106 (private communication, 15March2000).
  8. A. V. Lomakin, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (private communications, 10January2000 and 1May2000).
  9. A. V. Lomakin, “Fitting the correlation function,” Appl. Opt. 40, 4079–4086 (2001).
    [CrossRef]
  10. B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication, Springer Series in Optical Sciences (Springer-Verlag, New York, 1978), p. 18.
  11. D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers (Wiley, New York, 1994).
  12. W. V. Meyer, G. H. Wegdam, D. Feinstein, J. A. Mann, “Advances in surface-light-scattering instrumentation and analysis: noninvasive measuring of surface tension, viscosity, and other interfacial parameters,” Appl. Opt. 40, 4113–4133 (2001).
    [CrossRef]
  13. W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.
  14. R. G. W. Brown, “Homodyne optical fiber dynamic light scattering,” Appl. Opt. 40, 4004–4010 (2001).
    [CrossRef]
  15. B. Chu, Laser Light Scattering (Academic, New York, 1974), p. 104.

2001

1983

K. Schätzel, “Noise in photon correlation and photon structure functions,” Opt. Acta 30, 155–166 (1983).
[CrossRef]

1982

M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
[CrossRef]

1974

E. R. Pike, E. Jakeman, “Photon statistics and photon-correlation spectroscopy,” Adv. Quantum Electron. 2, 1 (1974).
[CrossRef]

1973

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

1971

V. Degiorgio, J. B. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2050 (1971).
[CrossRef]

E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
[CrossRef]

Brown, R. G. W.

R. G. W. Brown, “Homodyne optical fiber dynamic light scattering,” Appl. Opt. 40, 4004–4010 (2001).
[CrossRef]

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

Cannell, D. S.

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

D. S. Cannell, Department of Physics, University of California Santa Barbara, Santa Barbara, Calif. 93106 (private communication, 15March2000).

Chu, B.

B. Chu, Laser Light Scattering (Academic, New York, 1974), p. 104.

Degiorgio, V.

V. Degiorgio, J. B. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2050 (1971).
[CrossRef]

Feinstein, D.

Hughes, J.

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

Jakeman, E.

E. R. Pike, E. Jakeman, “Photon statistics and photon-correlation spectroscopy,” Adv. Quantum Electron. 2, 1 (1974).
[CrossRef]

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
[CrossRef]

Lastovka, J. B.

V. Degiorgio, J. B. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2050 (1971).
[CrossRef]

Lock, J. A.

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

Lomakin, A. V.

A. V. Lomakin, “Fitting the correlation function,” Appl. Opt. 40, 4079–4086 (2001).
[CrossRef]

A. V. Lomakin, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (private communications, 10January2000 and 1May2000).

Mann, J. A.

Meyer, W. V.

W. V. Meyer, G. H. Wegdam, D. Feinstein, J. A. Mann, “Advances in surface-light-scattering instrumentation and analysis: noninvasive measuring of surface tension, viscosity, and other interfacial parameters,” Appl. Opt. 40, 4113–4133 (2001).
[CrossRef]

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

Montgomery, D. C.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers (Wiley, New York, 1994).

Oliver, C. J.

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

Pike, E. R.

M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
[CrossRef]

E. R. Pike, E. Jakeman, “Photon statistics and photon-correlation spectroscopy,” Adv. Quantum Electron. 2, 1 (1974).
[CrossRef]

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
[CrossRef]

Runger, G. C.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers (Wiley, New York, 1994).

Saleh, B.

B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication, Springer Series in Optical Sciences (Springer-Verlag, New York, 1978), p. 18.

Schätzel, K.

K. Schätzel, “Noise in photon correlation and photon structure functions,” Opt. Acta 30, 155–166 (1983).
[CrossRef]

Smart, A. E.

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

Swain, S.

E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
[CrossRef]

Taylor, T. W.

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

Wegdam, G. H.

Adv. Quantum Electron.

E. R. Pike, E. Jakeman, “Photon statistics and photon-correlation spectroscopy,” Adv. Quantum Electron. 2, 1 (1974).
[CrossRef]

Appl. Opt.

J. Phys. A

E. Jakeman, E. R. Pike, S. Swain, “Statistical accuracy in the digital autocorrelation of photon counting fluctuations,” J. Phys. A 4, 517–534 (1971).
[CrossRef]

J. Hughes, E. Jakeman, C. J. Oliver, E. R. Pike, “Photon-correlation spectroscopy: dependence of linewidth error on normalization, clip level, detector area, sample time and count rate,” J. Phys. A 6, 1327–1336 (1973).
[CrossRef]

Opt. Acta

K. Schätzel, “Noise in photon correlation and photon structure functions,” Opt. Acta 30, 155–166 (1983).
[CrossRef]

Phys. Rev. A

V. Degiorgio, J. B. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2050 (1971).
[CrossRef]

Proc. R. Soc. London Ser. A

M. Bertero, P. Boccacci, E. R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. R. Soc. London Ser. A 383, 15–29 (1982).
[CrossRef]

Other

D. S. Cannell, Department of Physics, University of California Santa Barbara, Santa Barbara, Calif. 93106 (private communication, 15March2000).

A. V. Lomakin, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (private communications, 10January2000 and 1May2000).

B. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communication, Springer Series in Optical Sciences (Springer-Verlag, New York, 1978), p. 18.

D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers (Wiley, New York, 1994).

W. V. Meyer, A. E. Smart, D. S. Cannell, R. G. W. Brown, J. A. Lock, T. W. Taylor, “Laser light scattering: multiple scattering suppression with cross correlation, and flare rejection with fiber optic homodyning,” in Proceedings of the Thirty-Seventh AIAA Aerospace Science Meeting and Exhibit (American Institute of Aeronautics and Astronautics, New York, 1999), paper AIAA 99-0962.

B. Chu, Laser Light Scattering (Academic, New York, 1974), p. 104.

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Figures (12)

Fig. 1
Fig. 1

Analytic representations of single-exponential decay (□), double-exponential decay (○), oscillatory decaying cosine (◇), and critically damped decaying cosine (△).

Fig. 2
Fig. 2

Analytic single exponential without noise (solid curves), synthesized correlation function with channel-to-channel correlation (○), and function with uncorrelated shot noise (●). The last two function yield similar errors (approximately 3%).

Fig. 3
Fig. 3

S.D.s of the fits for a single exponential: decay rate (□) and intercept (△) for a fixed zero baseline; decay rate (■), intercept (▲), and baseline (●) for a fitted baseline; decay, intercept, and baseline (solid curves) from theory.

Fig. 4
Fig. 4

Errors from shot noise for a single exponential: decay rate (□) and intercept (△) for a fixed zero baseline; decay rate (■), intercept (▲), and baseline (●) for a fitted baseline.

Fig. 5
Fig. 5

Estimate of the linewidth in the time and the frequency domains from analysis of analytic data in the time domain (■); in the frequency domain with unextended (▲) and extended (●) baselines; for synthesized data in the time domain (□); for synthesized data in the frequency domain with unextended (△) and extended (○) baselines.

Fig. 6
Fig. 6

Percentage errors in the linewidth from fitting in the time domain for analytic (■) and synthesized (□) data; in the frequency domain for analytic (●) and synthesized (○) data with an extended baseline. The symbols for the unextended baseline are omitted to permit reasonable scaling.

Fig. 7
Fig. 7

Covariance plot for a single exponential synthesized over 8192 decay times.

Fig. 8
Fig. 8

Covariance plot for an analytic single exponential.

Fig. 9
Fig. 9

Difference between the synthesized and the analytic covariance plots, showing no overall trend.

Fig. 10
Fig. 10

S.D.s of the fits for an oscillatory cosine: decay rate (□), cosine frequency (○), and intercept (△) for a fixed zero baseline; decay rate (■), cosine frequency (●), intercept (▲), and baseline (◆) for a fitted baseline; decay, cosine frequency, and intercept (solid curves) from theory. Note that the filled symbols overlie and obscure the open symbols.

Fig. 11
Fig. 11

S.D.s of the fits for a critically damped cosine: decay rate (□), cosine frequency (○), and intercept (△) for a fixed zero baseline; decay rate (■), cosine frequency (●), intercept (▲), and baseline (◆) for a fitted baseline; decay, intercept, and baseline (solid curves) from theory.

Fig. 12
Fig. 12

S.D.s of the fits for a double exponential: first decay rate (□), first intercept (△), second decay rate (◇), and second intercept (○) for a fixed zero baseline; first decay rate (■), first intercept (▲), second decay rate (◆), second intercept (●), and baseline (×) for a fitted baseline; solid lines with smaller symbols show theoretical results for the first decay rate (■), second decay rate (▲), and both intercepts (◆, ●), which are indistinguishable on the plot.

Tables (5)

Tables Icon

Table 1 Formulas and Parameter Values for Data Synthesis

Tables Icon

Table 2 Relative S.D.s for the Single Exponential

Tables Icon

Table 3 Relative S.D. Deviations for the Oscillatory Cosine

Tables Icon

Table 4 Relative S.D.s for the Critically Damped Cosine

Tables Icon

Table 5 Relative S.D.s for the Double Exponential

Equations (35)

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

xt-1=xtc+rt,
xt=rt+crt+1+c2rt+2+c3rt+3+= limitNk=0N ckrt+k,
xtxt+M=cM1-c2x2=exp-γM1-c2x2.
I=B+A0 exp-γt02+B+A1 exp-γt12+B+A2 exp-γt22+=n=0B+An exp-nγtn2.
I=n=0exp-2nγ=11-exp-2nγ.
I=11-exp-2γ+11-exp-4γ,
I=121-exp-2γ+121-2 exp-2γcos2ω+exp-4γ1/2.
gnα=gnα0+kgnαkαk-α0k,
2=ndn-gnα0-kgnαkΔαk2,
ndn-gnα0-kgnαkΔαkgnαj=0
ndn-gnα0gnαj=nkgnαjgnαkΔαk.
Bjk=ngnαjgnαk
ndn-gnα0gnαj=k BjkΔαk.
ndn-gnα0j Blj-1gnαj=jk Blj-1BjkΔαk=Δα1,
mndn-gnα0dm-gmα0×j Blj-1k Bpk-1gnαjgmαk=ΔαpΔαl.
Λnm=dn-gnα0dm-gmα0,
mn Λnmj Blj-1k Bpk-1gnαjgmαk=ΔαpΔαl=Cpl.
R0mfkfk+m1-|m|N.
R˜m1Nk=0N-m fkfk+m,
R˜mR0m.
Λmn=R˜m-R0mR˜n-R0n,
Λmn=R˜mR˜n-R0mR0n,Λmn=1N2k=0N-m fkfm+kk=0N-n fkfn+k-R0mR0n,
Λmn=1N2k=0N-mkN-n fkfm+kfkfn+k-R0mR0n.
fkfn+kfkfm+k=R0nR0m+R0k-kR0m-n+k-k+R0m+k-kR0k-k-n.
Λmn=1N2k=0N-mk=0N-nR0k-kR0k-k+m-n+R0m+k-kR0k-k-n.
Λmn=1Np=-p=1-|m|NR0pR0p+m-n+R0p+mR0p-n.
Λmn=1N-R0pR0p+m-n+R0p+mR0p-ndp.
Vars2=2σ22N-1,
S.D.s2σ2=2N-11/2.
fn=B+A exp-λ|n|.
Λexp=A2exp-γ|n+m|1+γ|n+m|+exp-γ|n-m|1+γ|n-m|γN.
fn=B+A exp-λ|n|cosωn.
Λcos=A22Nexp-λ|n-m|2λ2+ω2cosω|n-m|-λω sinω|n-m|λλ2+ω2+ω|n-m|cosω|n-m|+sinω|n-m|ω+exp-λ|n+m|×2λ2+ω2cosω|n+m|-λω sinω|n+m|λλ2+ω2+ω|n+m|cosω|n+m|+sinω|n+m|ω.
fn=B+A1 exp-λ1|n|+A2 exp-λ2|n|.
Λdoub=12Nexp-λ1|n-m|A12λ1-2A1A2λ1-λ2+2A1A2λ1+λ2+A12|n-m|+exp-λ2|n-m|×A22λ2+2A1A2λ1-λ2+2A1A2λ1+λ2+A22|n-m|+exp-λ1|n+m|A12λ1-2A1A2λ1-λ2+2A1A2λ1+λ2+A12|n+m|+exp-λ2|n+m|×A22λ2+2A1A2λ1-λ2+2A1A2λ1+λ2+A22|n+m|.

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