Abstract

A frequent source of errors in dynamic light-scattering experiments is partial heterodyning caused by scattering on large particles or imperfections of the sample cell. With a conventional two-pinhole receiver it is impossible to distinguish its effect from the effects of a finite detector area and detector nonlinearity. However, an accurate data analysis is feasible when a single-mode light receiver is employed. We present formulas for single-mode autocorrelation and cross-correlation functions that include a local oscillator and an incoherent background of arbitrary strength and take into account detector nonlinearity (e.g., dead time) up to second order. A simple but accurate method for the determination of the nonlinearity parameters and the effective number of receiver modes is also provided. The success of the data-evaluation procedure is demonstrated by the measurement of the hydrodynamic radius of latex in the presence of deliberately added local-oscillator or incoherent-background contributions.

© 1997 Optical Society of America

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References

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  1. G. A. Casay, W. Wilson, “Laser scattering in a hanging drop vapor diffusion apparatus for protein crystal growth in a microgravity environment,” J. Crystal Growth 122, 95–101 (1995).
    [CrossRef]
  2. R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
    [CrossRef]
  3. J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. 79, 45–55 (1994).
  4. J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).
  5. F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
    [CrossRef]
  6. L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
    [CrossRef]
  7. S. E. Bursell, P. C. Magnante, L. T. Chylack, “In vivo uses of quasi-elastic light scattering spectroscopy as a molecular probe in the anterior segment of the eye,” in Noninvasive Diagnostic Techniques in Ophthalmology, B. R. Masters, ed. (Springer, Berlin, 1990), Chap. 18.
    [CrossRef]
  8. R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
    [CrossRef]
  9. M. Dieckmann, K. Dierks, “Diagnostic methods and tissue parameter investigations together with measurement results (in vivo),” in Ophthalmologic Technologies IV, J.-M. Parel, Q. Ren, eds., Proc. SPIE2126, 331–345 (1994).
    [CrossRef]
  10. P. Stephanek, “Data analysis in dynamic light scattering,” in Dynamic Light Scattering, W. Brown, ed. (Oxford Science, Oxford, 1993), Chap. 4, pp. 177–241.
  11. R. G. W. Brown, “Dynamic light scattering using monomode optical fibers,” Appl. Opt. 26, 4846–4851 (1987).
    [CrossRef] [PubMed]
  12. K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
    [CrossRef]
  13. K. Schätzel, R. Kalström, B. Stampa, J. Ahrens, “Correction of detection-system dead-time effects on photon-correlation functions,” J. Opt. Soc. Am. B 6, 937–947 (1989).
    [CrossRef]
  14. J. Rička, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
    [CrossRef] [PubMed]
  15. B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), p. 40.
  16. T. Gisler, H. Rüger, S. U. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Rička, “Mode-selective dynamic light scattering: theory versus experimental realization with single-mode and multimode receivers,” Appl. Opt. 34, 3546–3553 (1995).
    [CrossRef] [PubMed]
  17. F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
    [CrossRef]
  18. H. C. Burstyn, J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27, 1071–1085 (1983). Burstyn and Sengers have treated the problem of afterpulsing: their final formula (4.16) contains a printing error and should read g˜2t=1+α1tn2δ12+β1+α1Γγ1-α2Γγ2×exp-2Γt-2βα1Γγ1exp-2γ1t, where g̃(2) is the normalized intensity correlation function. αi(t) = 2αi γie-2γit are the model functions describing the probability of finding a correlated afterpulse at the time t after an initial pulse, and αi are the integrated afterpulsing probabilities. Γ is the decay rate of the electric-field correlation function, which is assumed to be exponential.
  19. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).
  20. D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
    [CrossRef]

1997 (1)

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

1996 (2)

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[CrossRef]

1995 (3)

G. A. Casay, W. Wilson, “Laser scattering in a hanging drop vapor diffusion apparatus for protein crystal growth in a microgravity environment,” J. Crystal Growth 122, 95–101 (1995).
[CrossRef]

F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[CrossRef]

T. Gisler, H. Rüger, S. U. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Rička, “Mode-selective dynamic light scattering: theory versus experimental realization with single-mode and multimode receivers,” Appl. Opt. 34, 3546–3553 (1995).
[CrossRef] [PubMed]

1994 (1)

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. 79, 45–55 (1994).

1993 (1)

1989 (1)

1987 (1)

1986 (1)

K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
[CrossRef]

1983 (1)

H. C. Burstyn, J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27, 1071–1085 (1983). Burstyn and Sengers have treated the problem of afterpulsing: their final formula (4.16) contains a printing error and should read g˜2t=1+α1tn2δ12+β1+α1Γγ1-α2Γγ2×exp-2Γt-2βα1Γγ1exp-2γ1t, where g̃(2) is the normalized intensity correlation function. αi(t) = 2αi γie-2γit are the model functions describing the probability of finding a correlated afterpulse at the time t after an initial pulse, and αi are the integrated afterpulsing probabilities. Γ is the decay rate of the electric-field correlation function, which is assumed to be exponential.

1972 (1)

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

1971 (1)

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Ahrens, J.

Ansari, R. R.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
[CrossRef]

Arabshahi, A.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Bray, T. L.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

Brown, R. G. W.

Bursell, S. E.

S. E. Bursell, P. C. Magnante, L. T. Chylack, “In vivo uses of quasi-elastic light scattering spectroscopy as a molecular probe in the anterior segment of the eye,” in Noninvasive Diagnostic Techniques in Ophthalmology, B. R. Masters, ed. (Springer, Berlin, 1990), Chap. 18.
[CrossRef]

Burstyn, H. C.

H. C. Burstyn, J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27, 1071–1085 (1983). Burstyn and Sengers have treated the problem of afterpulsing: their final formula (4.16) contains a printing error and should read g˜2t=1+α1tn2δ12+β1+α1Γγ1-α2Γγ2×exp-2Γt-2βα1Γγ1exp-2γ1t, where g̃(2) is the normalized intensity correlation function. αi(t) = 2αi γie-2γit are the model functions describing the probability of finding a correlated afterpulse at the time t after an initial pulse, and αi are the integrated afterpulsing probabilities. Γ is the decay rate of the electric-field correlation function, which is assumed to be exponential.

Campbell, M. C. W.

R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
[CrossRef]

Casay, G. A.

G. A. Casay, W. Wilson, “Laser scattering in a hanging drop vapor diffusion apparatus for protein crystal growth in a microgravity environment,” J. Crystal Growth 122, 95–101 (1995).
[CrossRef]

Chylack, L. T.

S. E. Bursell, P. C. Magnante, L. T. Chylack, “In vivo uses of quasi-elastic light scattering spectroscopy as a molecular probe in the anterior segment of the eye,” in Noninvasive Diagnostic Techniques in Ophthalmology, B. R. Masters, ed. (Springer, Berlin, 1990), Chap. 18.
[CrossRef]

Corti, M.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Degiorgio, V.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Dellavecchia, M. A.

R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
[CrossRef]

DeLucas, L. J.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

Dhadwal, H. S.

R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
[CrossRef]

Dieckmann, M.

M. Dieckmann, K. Dierks, “Diagnostic methods and tissue parameter investigations together with measurement results (in vivo),” in Ophthalmologic Technologies IV, J.-M. Parel, Q. Ren, eds., Proc. SPIE2126, 331–345 (1994).
[CrossRef]

Dierks, K.

M. Dieckmann, K. Dierks, “Diagnostic methods and tissue parameter investigations together with measurement results (in vivo),” in Ophthalmologic Technologies IV, J.-M. Parel, Q. Ren, eds., Proc. SPIE2126, 331–345 (1994).
[CrossRef]

Donati, S.

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Egelhaaf, S. U.

Fankhauser, F.

L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[CrossRef]

Flammer, I.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

Frenz, M.

F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[CrossRef]

Gisler, T.

Kalström, R.

Könz, F.

F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[CrossRef]

Koppel, D. E.

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Leutz, W.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

Magnante, P. C.

S. E. Bursell, P. C. Magnante, L. T. Chylack, “In vivo uses of quasi-elastic light scattering spectroscopy as a molecular probe in the anterior segment of the eye,” in Noninvasive Diagnostic Techniques in Ophthalmology, B. R. Masters, ed. (Springer, Berlin, 1990), Chap. 18.
[CrossRef]

Ricka, J.

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[CrossRef]

F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[CrossRef]

T. Gisler, H. Rüger, S. U. Egelhaaf, J. Tschumi, P. Schurtenberger, J. Rička, “Mode-selective dynamic light scattering: theory versus experimental realization with single-mode and multimode receivers,” Appl. Opt. 34, 3546–3553 (1995).
[CrossRef] [PubMed]

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. 79, 45–55 (1994).

J. Rička, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
[CrossRef] [PubMed]

Rovati, L.

L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[CrossRef]

Rüger, H.

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), p. 40.

Schätzel, K.

Schurtenberger, P.

Sengers, J. V.

H. C. Burstyn, J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27, 1071–1085 (1983). Burstyn and Sengers have treated the problem of afterpulsing: their final formula (4.16) contains a printing error and should read g˜2t=1+α1tn2δ12+β1+α1Γγ1-α2Γγ2×exp-2Γt-2βα1Γγ1exp-2γ1t, where g̃(2) is the normalized intensity correlation function. αi(t) = 2αi γie-2γit are the model functions describing the probability of finding a correlated afterpulse at the time t after an initial pulse, and αi are the integrated afterpulsing probabilities. Γ is the decay rate of the electric-field correlation function, which is assumed to be exponential.

Stampa, B.

Stephanek, P.

P. Stephanek, “Data analysis in dynamic light scattering,” in Dynamic Light Scattering, W. Brown, ed. (Oxford Science, Oxford, 1993), Chap. 4, pp. 177–241.

Suh, K. I.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

Tschumi, J.

Wilson, W.

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

G. A. Casay, W. Wilson, “Laser scattering in a hanging drop vapor diffusion apparatus for protein crystal growth in a microgravity environment,” J. Crystal Growth 122, 95–101 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

K. Schätzel, “Dead time correction of photon correlation functions,” Appl. Phys. B 41, 95–102 (1986).
[CrossRef]

J. Chem. Phys. (1)

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

J. Crystal Growth (2)

G. A. Casay, W. Wilson, “Laser scattering in a hanging drop vapor diffusion apparatus for protein crystal growth in a microgravity environment,” J. Crystal Growth 122, 95–101 (1995).
[CrossRef]

R. R. Ansari, K. I. Suh, A. Arabshahi, W. Wilson, T. L. Bray, L. J. DeLucas, “A fiber optic probe for monitoring protein aggregation, nucleation and crystallization,” J. Crystal Growth 168, 216–226 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

Makromol. Chem. (1)

J. Rička, “Brownian dynamics in strongly scattering porous media: dynamic light scattering with single-mode matching,” Makromol. Chem. 79, 45–55 (1994).

Opt. Commun. (1)

F. T. Arecchi, M. Corti, V. Degiorgio, S. Donati, “Measurements of light intensity correlations in the subnanosecond region by photomultipliers,” Opt. Commun. 3, 284–288 (1971).
[CrossRef]

Opt. Eng. (1)

F. Könz, J. Rička, M. Frenz, “Dynamic light scattering in the vitreous: performance of the single-mode fiber technique,” Opt. Eng. 34, 2390–2395 (1995).
[CrossRef]

Phys. Rev. A (1)

H. C. Burstyn, J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27, 1071–1085 (1983). Burstyn and Sengers have treated the problem of afterpulsing: their final formula (4.16) contains a printing error and should read g˜2t=1+α1tn2δ12+β1+α1Γγ1-α2Γγ2×exp-2Γt-2βα1Γγ1exp-2γ1t, where g̃(2) is the normalized intensity correlation function. αi(t) = 2αi γie-2γit are the model functions describing the probability of finding a correlated afterpulse at the time t after an initial pulse, and αi are the integrated afterpulsing probabilities. Γ is the decay rate of the electric-field correlation function, which is assumed to be exponential.

Prog. Colloid Polym. Sci. (1)

J. Rička, I. Flammer, W. Leutz, “Single-mode DLS: colloids in opaque porous media,” Prog. Colloid Polym. Sci. 104, 49–58 (1997).

Rev. Sci. Instrum. (1)

L. Rovati, F. Fankhauser, J. Rička, “Design and performance of a new ophthalmic instrument for dynamic light-scattering measurements in the human eye,” Rev. Sci. Instrum. 67, 2615–2620 (1996).
[CrossRef]

Other (6)

S. E. Bursell, P. C. Magnante, L. T. Chylack, “In vivo uses of quasi-elastic light scattering spectroscopy as a molecular probe in the anterior segment of the eye,” in Noninvasive Diagnostic Techniques in Ophthalmology, B. R. Masters, ed. (Springer, Berlin, 1990), Chap. 18.
[CrossRef]

R. R. Ansari, H. S. Dhadwal, M. C. W. Campbell, M. A. Dellavecchia, “A fiber optic sensor for ophthalmic refractive diagnostics,” in Fiber Optic Medical and Fluorescent Sensors and Applications, D. R. Hansmann, F. P. Milanovich, G. G. Vurek, D. R. Walt, eds., Proc. SPIE1648, 83–105 (1992).
[CrossRef]

M. Dieckmann, K. Dierks, “Diagnostic methods and tissue parameter investigations together with measurement results (in vivo),” in Ophthalmologic Technologies IV, J.-M. Parel, Q. Ren, eds., Proc. SPIE2126, 331–345 (1994).
[CrossRef]

P. Stephanek, “Data analysis in dynamic light scattering,” in Dynamic Light Scattering, W. Brown, ed. (Oxford Science, Oxford, 1993), Chap. 4, pp. 177–241.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), p. 40.

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Figures (7)

Fig. 1
Fig. 1

Deviation of the mean photocount number from the ideal case (left-hand side) and the PCACF amplitudes (right-hand side) independently of the strength of the local oscillator or the incoherent background. The bottom curves correspond to the highest ∊ values. The solid curves are calculated with no incoherent-background contribution (jb = 0), and the abscissa denotes the strength of the local-oscillator contribution jl. The dashed curves are calculated with no local-oscillator contribution (jl = 0), and the abscissa denotes the strength of the incoherent-background contribution jb.

Fig. 2
Fig. 2

Setup for light-scattering experiments with a single-mode fiber: L1 and L2 are lenses, and P is a polarizer that ensures that our receiver is truly single mode; otherwise we would receive two polarization modes.

Fig. 3
Fig. 3

PCACF amplitudes from autocorrelation measurements. Measured amplitudes versus measured mean count rate r are shown for three single-mode fiber receivers and two detector systems. The solid curves are the best fits with Eq. (24). The number of effective modes is determined to be 1.002 ± 0.002, and the dead-time parameters θ are 32.1 ± 0.8 ns for system 1 and 58.9 ± 1 ns for system 2. The updating parameter ϕ is found to be 0.49 for detection system 1 and 0.40 for detection system 2.

Fig. 4
Fig. 4

Amplitudes from cross-correlation measurements. Measured amplitudes are shown versus measured the mean count rates of both channels. The solid curve is the best quadratic fit. The number of effective modes was determined to be 1.002 ± 0.002.

Fig. 5
Fig. 5

Apparent hydrodynamic radii of latex particles with rh = 45.5 nm, measured with a varying local-oscillator contribution and ∊ = 0.04. The strength of the local-oscillator contribution is estimated by the fit parameter jl. Three fit models are compared: second-order partial heterodyning formula, Eq. (19) (filled circles); partial heterodyning formula with ∊ = 0 (open circles); the homodyne model (open squares).

Fig. 6
Fig. 6

Apparent hydrodynamic radii of latex particles with rh = 45.5 nm, measured with a varying local-oscillator contribution. The dead-time parameter has been increased to ∊ = 0.07. Three fit models are compared: partial heterodyning formula, Eq. (19) (filled circles); partial heterodyning formula with ∊ terms up to first order only (open triangles); partial heterodyning formula with ∊ = 0 (open circles).

Fig. 7
Fig. 7

Apparent hydrodynamic radii of latex particles with rh = 45.5 nm, measured with a varying incoherent-background contribution and constant ∊ = 0.05. The strength of the incoherent-background contribution is estimated by the fit parameter jb. Two fit models are compared: our second-order heterodyning formula, Eq. (19) (filled circles); homodyne model n0nτ¯/n¯2=1+fg1τ2 (open squares).

Equations (37)

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

Jt=t*t,
t=st+bt+l.
Jt=Jl+Jbt+Jst+l*st+ls*t+l*bt+lb*t+b*tst+bts*t.
Jt=Jl+Jb+Jst+l*st+ls*t.
gτ=1+2jljsg1τ+js2g1τ2,
Js=JsJl+Jb+Js, Jl=JlJl+Jb+Js, jb=1-jl-js.
g1τ=s0s*τ/Js.
JMt=i=1MJit.
gMτ=1+1Mg1τ2.
n¯t=αtsJt,  Jt=1tstt+tsJtdt.
ntnt+τ¯=n¯tn¯t+τ.
n¯t=αtsJt1-θαJt+ϕθ2α2Jt2+.
ntnt+τ¯=n¯tn¯t+τ,  τ>τlimit,  n¯<n¯limit.
n¯t=αtsJt1-Jt2Jt2+ϕ2Jt3Jt3+O3,
n0nτ¯=αtsJt2J0JτJt2-2 J02JτJt3+2J02Jτ2Jt4+2ϕ2J03JτJt4+O3.
n¯t=αtsJt1-1+1-jb2-jl2+ϕ26-12jb+9jb2-2jb3+6jbjl2-9jl2+4jl3+O3,
n¯t=αtsJt1-2+6ϕ2+O3.
n¯=αtsJ1-+ϕ2+O3.
n0nτ¯n¯2=1+i=14fi, ϕ, jl, jbg1τi.
n¯1t=α1tsJt1-θ1α1Jt+ϕ1θ12α12Jt2+, n¯2t=α2tsJt1-θ2α2Jt+ϕ2θ22α22Jt2+,
n10n2τ¯n¯1n¯2=1+i=14fxi1, 2, ϕ1, ϕ2, jl, jbg1τi.
n¯=αtsJt1-1+1M+ϕ21+3M+2M2+O3,
n0nτ¯n¯2=1+f2g1τ2+f4g1τ4,
f2=1M1-21+1M+24ϕ-1+12ϕ-2M+8ϕ-1M2,f4=221+MM3.
limτ0n0nτ¯n¯2-1=1M1-21+1M+24ϕ-1+12ϕM+8ϕ+1M2.
=αJtθ=rθ+r2θ21+1M+Oθ3.
=rθ+r2θ21+1-jb2-jl2.
n0nτ¯n¯2=1+i=14fi, ϕ, jl, jbg1τi,
f1=21-jb-jljl1-21+1-jl2-jb2+2-1+4ϕ+21-jb-jl-1+jb-jl+6ϕ+1-jb-jl21+jb-jl-1+3jb-3jl+42+jb-2jlϕ, f2=1-jb-jl21-22+jl2-jb2+2-1-2×1-jb2+10jl2+1-jb+jl1-jb-jl×-1+2jb+3jb2-3jl2+4ϕ6-6jb+jb3-3jbjl2-2jl3, f3=1621-jb-jl3jl, f4=421-jb-jl4, =rθ+r2θ21+1-jb2-jl2.
n0nτ¯n¯2=1+21-jljlg1τ1-21+1-jl2-24-8jl2+8jl3-3jl4-24ϕ+36ϕjl-24ϕjl2+8ϕjl3+1-jl2g1τ2×1-22+jl2-24-8jl2-3jl4-24ϕ+8ϕjl3+162jl1-jl3g1τ3+421-jl4g1τ4,
=rθ+r2θ22-jl2.
n0nτ¯n¯2=1+1-jb2g1τ21-22-jb2-24-8jb+4jb2+4jb3-3jb4-24ϕ+24ϕjb-4ϕjb3+421-jb4g1τ4,
=rθ+r2θ21+1-jb2.
n0nτ¯n¯2=1+1-4-42+24ϕ2×g1τ2+42g1τ4,
=rθ+2r2θ2.
n10n2τ¯n¯1n¯2=1+i=14fxi1, 2, ϕ1, ϕ2, jl, jbg1τi.
x=1+22, -2=12-12+22, E0=12-2, E1=12+22-2, E2=12+12+22-2, px=12ϕ1+22ϕ22-2, fx1=21-jb-jljl1-2x1+1-jl2-jb2+-2-1+4px+21-jb-jl×-1+E0jb-jl+6px+1-jb-jl2×-1+2E0jb-jl+E2jb-jl2+42+jb-2jlpx, fx2=1-jb-jl21-2x2+jl2-jb2+-2-4-4E0jb2-2jl2+2E1jb2-jb2+jl2+E2jb2-jl22+4px6-6jb+jb3-3jbjl2-2jl3, fx3=16121-jb-jl3jl, fx4=4121-jb-jl4, 1=r1θ1+r12θ121+1-jb2-jl2, 2=r2θ2+r22θ221+1-jb2-jl2.

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