Abstract

We present a theoretical framework for field-based dynamic light scattering microscopy based on a spectral-domain optical coherence phase microscopy (SD-OCPM) platform. SD-OCPM is an interferometric microscope capable of quantitative measurement of amplitude and phase of scattered light with high phase stability. Field-based dynamic light scattering (F-DLS) analysis allows for direct evaluation of complex-valued field autocorrelation function and measurement of localized diffusive and directional dynamic properties of biological and material samples with high spatial resolution. In order to gain insight into the information provided by F-DLS microscopy, theoretical and numerical analyses are performed to evaluate the effect of numerical aperture of the imaging optics. We demonstrate that sharp focusing of fields affects the measured diffusive and transport velocity, which leads to smaller values for the dynamic properties in the sample. An approach for accurately determining the dynamic properties of the samples is discussed.

© 2013 Optical Society of America

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References

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  1. B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Wiley, 1976).
  2. W. Brown, Dynamic Light Scattering: The Method and Some Applications (Clarendon, 1993).
  3. K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
    [CrossRef]
  4. T. Tanaka and G. B. Benedek, “Observation of protein diffusivity in intact human and bovine lenses with application to cataract,” Investig. Ophthalmol. Vis. Sci. 14, 449–456 (1975).
  5. Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).
  6. J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).
  7. R. P. Singh, V. K. Jaiswal, and V. K. Jain, “Study of smoke aerosols under a controlled environment by using dynamic light scattering,” Appl. Opt. 45, 2217–2221 (2006).
    [CrossRef]
  8. P. D. Kaplan, V. Trappe, and D. A. Weitz, “Light-scattering microscope,” Appl. Opt. 38, 4151–4157 (1999).
    [CrossRef]
  9. R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. II: experimental applications,” Biophys. J. 87, 1288–1297 (2004).
    [CrossRef]
  10. R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. I: theory,” Biophys. J. 87, 1279–1287 (2004).
    [CrossRef]
  11. C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
    [CrossRef]
  12. W. Choi, C.-C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Field-based angle-resolved light scattering study of single live cells,” Opt. Lett. 33, 1596–1598 (2008).
    [CrossRef]
  13. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005).
    [CrossRef]
  14. C. Joo, C. L. Evans, T. Stepinac, T. Hasan, and J. F. de Boer, “Diffusive and directional intracellular dynamics measured by field-based dynamic light scattering,” Opt. Express 18, 2858–2871 (2010).
    [CrossRef]
  15. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [CrossRef]
  16. H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
    [CrossRef]
  17. O. Krichevsky and G. Bonnet, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
    [CrossRef]
  18. R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, 1985).
  19. P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
    [CrossRef]

2010 (1)

2008 (1)

2006 (1)

2005 (2)

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005).
[CrossRef]

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

2004 (2)

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. II: experimental applications,” Biophys. J. 87, 1288–1297 (2004).
[CrossRef]

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. I: theory,” Biophys. J. 87, 1279–1287 (2004).
[CrossRef]

2002 (1)

O. Krichevsky and G. Bonnet, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[CrossRef]

2000 (1)

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

1999 (1)

1998 (1)

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

1994 (1)

1987 (1)

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

1986 (1)

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

1975 (1)

T. Tanaka and G. B. Benedek, “Observation of protein diffusivity in intact human and bovine lenses with application to cataract,” Investig. Ophthalmol. Vis. Sci. 14, 449–456 (1975).

1969 (1)

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Akkin, T.

Axelrod, D.

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. II: experimental applications,” Biophys. J. 87, 1288–1297 (2004).
[CrossRef]

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. I: theory,” Biophys. J. 87, 1279–1287 (2004).
[CrossRef]

Badizadegan, K.

Benedek, G. B.

T. Tanaka and G. B. Benedek, “Observation of protein diffusivity in intact human and bovine lenses with application to cataract,” Investig. Ophthalmol. Vis. Sci. 14, 449–456 (1975).

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Wiley, 1976).

Bonnet, G.

O. Krichevsky and G. Bonnet, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[CrossRef]

Brown, W.

W. Brown, Dynamic Light Scattering: The Method and Some Applications (Clarendon, 1993).

Bursac, P.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Butler, J. P.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Carlson, F. D.

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Cense, B.

Choi, W.

Cummins, H. Z.

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Dasari, R. R.

W. Choi, C.-C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Field-based angle-resolved light scattering study of single live cells,” Opt. Lett. 33, 1596–1598 (2008).
[CrossRef]

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

de Boer, J. F.

Dzakpasu, R.

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. II: experimental applications,” Biophys. J. 87, 1288–1297 (2004).
[CrossRef]

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. I: theory,” Biophys. J. 87, 1279–1287 (2004).
[CrossRef]

Evans, C. L.

Fabry, B.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Fang-Yen, C.

Feld, M. S.

W. Choi, C.-C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Field-based angle-resolved light scattering study of single live cells,” Opt. Lett. 33, 1596–1598 (2008).
[CrossRef]

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

Fredberg, J. J.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Fujimoto, J. G.

Georgalis, Y.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Hasan, T.

Hee, M. R.

Herbert, T. J.

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Hollenbach, B.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Izatt, J. A.

Jain, V. K.

Jaiswal, V. K.

Joo, C.

Kaplan, P. D.

Krichevsky, O.

O. Krichevsky and G. Bonnet, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[CrossRef]

Lehrach, H.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Lenormand, G.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Lurz, R.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Nishio, I.

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

Ohnishi, S. T.

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

Oliver, M.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Owen, G. M.

Park, B. H.

Pecora, R.

R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, 1985).

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Wiley, 1976).

Peetermans, J.

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

Perelman, L. T.

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

Saenger, W.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Schatzel, K.

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

Scherzinger, E.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Singh, R. P.

Starikov, E. B.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Stepinac, T.

Swanson, E. A.

Tanaka, T.

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

T. Tanaka and G. B. Benedek, “Observation of protein diffusivity in intact human and bovine lenses with application to cataract,” Investig. Ophthalmol. Vis. Sci. 14, 449–456 (1975).

Trappe, V.

Viasnoff, V.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Wanker, E. E.

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

Wax, A.

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

Weitz, D. A.

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

P. D. Kaplan, V. Trappe, and D. A. Weitz, “Light-scattering microscope,” Appl. Opt. 38, 4151–4157 (1999).
[CrossRef]

Woods, G.

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Yang, C.

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

Yu, C.-C.

Appl. Opt. (2)

Appl. Phys. B (1)

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

Biophys. J. (3)

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. II: experimental applications,” Biophys. J. 87, 1288–1297 (2004).
[CrossRef]

R. Dzakpasu and D. Axelrod, “Dynamic light scattering microscopy. a novel optical technique to image submicroscopic motions. I: theory,” Biophys. J. 87, 1279–1287 (2004).
[CrossRef]

H. Z. Cummins, F. D. Carlson, T. J. Herbert, and G. Woods, “Translational and rotational diffusion constants of tobacco mosaic virus from Rayleigh linewidths,” Biophys. J. 9, 518–546 (1969).
[CrossRef]

Investig. Ophthalmol. Vis. Sci. (1)

T. Tanaka and G. B. Benedek, “Observation of protein diffusivity in intact human and bovine lenses with application to cataract,” Investig. Ophthalmol. Vis. Sci. 14, 449–456 (1975).

J. Biomed. Opt. (1)

C. Yang, L. T. Perelman, A. Wax, R. R. Dasari, and M. S. Feld, “Feasibility of field-based light scattering spectroscopy,” J. Biomed. Opt. 5, 138–143 (2000).
[CrossRef]

Nat. Mater. (1)

P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D. A. Weitz, V. Viasnoff, J. P. Butler, and J. J. Fredberg, “Cytoskeletal remodelling and slow dynamics in the living cell,” Nat. Mater. 4, 557–561 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. Nat. Acad. Sci. (2)

Y. Georgalis, E. B. Starikov, B. Hollenbach, R. Lurz, E. Scherzinger, W. Saenger, H. Lehrach, and E. E. Wanker, “Huntingtin aggregation monitored by dynamic light scattering,” Proc. Nat. Acad. Sci. 95, 6118–6121 (1998).

J. Peetermans, I. Nishio, S. T. Ohnishi, and T. Tanaka, “Light-scattering study of depolymerization kinetics of sickle hemoglobin polymers inside single erythrocytes,” Proc. Nat. Acad. Sci. 83, 352–356 (1986).

Rep. Prog. Phys. (1)

O. Krichevsky and G. Bonnet, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[CrossRef]

Other (3)

R. Pecora, Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum, 1985).

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Wiley, 1976).

W. Brown, Dynamic Light Scattering: The Method and Some Applications (Clarendon, 1993).

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Figures (9)

Fig. 1.
Fig. 1.

(a) SD-OCPM setup. A broadband light (or low-coherence) source illuminates a fiber-based common-path interferometer. The light coupled to the sample arm is delivered to a specimen via an integrated inverted microscope. The backscattered light is recoupled to the fiber for the subsequent interference spectrum measurement at the detection arm. (b) SD-OCPM detection scheme. SD-OCPM uses the light back-reflected from the bottom surface of a coverslip as a reference to ensure the phase stability for measuring amplitude and phase of sample light scattered from the focal volume [14].

Fig. 2.
Fig. 2.

Single scatter is positioned at r⃗j relative to the center of beam waist. fi(x,y) and fc(x,y) represent the incident and collection Gaussian mode functions, respectively. The scattering volume is indicated by the red dashed line.

Fig. 3.
Fig. 3.

Comparison of complex-valued temporal autocorrelation functions evaluated with Eqs. (10) and (28). In order to evaluate the correlation function with Eq. (10), the scattered field from the particles was obtained by integrating all the fields with the incident and scattering vectors determined by NA and was used to compute Eq. (10). On the other hand, Eq. (28) was directly calculated with the assumed particle dynamics.

Fig. 4.
Fig. 4.

Calculated F-DLS measurement of scattering samples with Brownian (a), (c), (e) and sub-diffusive (b), (d), (f) random motion. The magnitudes of autocorrelation functions, MSDs and TADs, are shown for both cases. The case with NA=0 (plane wave illumination and collection) produced correct diffusion coefficient and diffusive property descriptor (ν), but the cases with higher NA exhibited deviations from the NA=0 case. The TADs were zero in all cases, as expected for random motion. The insets in (c) and (d) show the TADs on a linear scale.

Fig. 5.
Fig. 5.

ν and D as a function of NA, obtained from the power-law fits to the MSDs for the samples with Brownian and sub-diffusive motion. The measured ν differed from the actual values to within 5% up to an NA of 1.0, while the errors in the estimation of D were larger (up to 20% for NA=1.0).

Fig. 6.
Fig. 6.

(a) The correction factor (γD) for diffusion coefficient as a function of NA. The correction factors were the same to within 1% for the Brownian and sub-diffusive dynamics. (b) and (c) The corrected MSDs for the samples with Brownian and sub-diffusive dynamics, respectively.

Fig. 7.
Fig. 7.

Magnitudes of autocorrelation functions, MSDs and TADs, for samples with Brownian (ν=1.0) and directional motion (a)–(c), and with sub-diffusive (ν=0.5) and directional motion (d)–(f).

Fig. 8.
Fig. 8.

Correction factor γT for TAD as a function of NA for samples with Brownian (ν=1.0) and sub-diffusive (ν=0.5) motion.

Fig. 9.
Fig. 9.

Corrected MSDs and TADs by applying corresponding γD and γT. (a) and (b) show the corrected MSDs and TADs for the sample with both diffusive (ν=1) and directional dynamics. (c) and (d) are the corresponding plots for the sample with sub-diffusive (ν=0.5) and directional dynamics. The correction factors were found based on the NAs.

Equations (32)

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

fi(x,y)=1πw02exp(x2+y2w02),
F˜i(q⃗i)=exp(w02qi2).
E˜s(q⃗s,r⃗j)=F˜i(q⃗i)exp(ir⃗j·(k⃗ik⃗s))d2qi,
Es(r⃗j)=E˜s(q⃗s,r⃗j)F˜c(q⃗s)d2qs.
Es(r⃗j)=E˜s(q⃗s,r⃗j)F˜i*(q⃗s)d2qs.
Es(r⃗j)=F˜i(q⃗i)exp(ir⃗j·(k⃗ik⃗s))F˜i*(q⃗s)d2qid2qs.
Es(r⃗j)=F˜i(q⃗i)exp(ir⃗j·Q⃗si)F˜i*(q⃗s)d2qid2qs.
Es({r⃗j},t)=j=1Nnj(t)Es(r⃗j),
F(t)=ER*Es({r⃗j},t)=ER*j=1Nnj(t)Es(r⃗j).
Γ(τ)=F(t)F*(t+τ),
Γ(τ)=|ER|2j=1Nk=1Nnj(t)nk(t+τ)×exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)×F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd.
Γ(τ)=|ER|2[ΓjknumΓjkph+Γj=knumΓj=kph]
Γjknum=j=1Nk=1Nnj(t)nk(t+τ),
Γjkph=exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)×F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd,
Γj=knum=j=1Nnj(t)nj(t+τ),
Γj=kph=exp(ir⃗j(t)·Q⃗ba)exp(ir⃗j(t+τ)·Q⃗dc)×F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd.
exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)=exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)
exp(ir⃗·Q⃗)=χ⃗(r⃗)exp(ir⃗·Q⃗)d3r.
exp(ir⃗·Q⃗)=1Vpexp(ir⃗·Q⃗)d3r=1Vpδ(Q⃗).
exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)=1Vp2δ(Q⃗ba)δ(Q⃗dc).
Γjkph=exp(ir⃗j(t)·Q⃗ba)exp(ir⃗k(t+τ)·Q⃗dc)×F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd=1Vp2δ(Q⃗ba)δ(Q⃗dc)F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd.
exp(ir⃗j(t)·Q⃗ba)exp(ir⃗j(t+τ)·Q⃗dc)=exp(iΔr⃗j·Q⃗ba)exp(ir⃗j(t+τ)·ΔQ⃗dcba)=exp(iΔr⃗j·Q⃗ba)exp(ir⃗j(t+τ)·ΔQ⃗dcba),
exp(ir⃗j(t)·Q⃗ba)exp(ir⃗j(t+τ)·Q⃗dc)=1Vpexp(iΔr⃗j·Q⃗ba)δ(ΔQ⃗dcba).
Γj=kph=1Vpexp(iΔr⃗j·Q⃗ba)δ(ΔQ⃗dcba)×F˜i(q⃗a)F˜i*(q⃗b)F˜i*(q⃗c)F˜i(q⃗d)d2qad2qbd2qcd2qd.
Γj=kph=exp(iΔr⃗j·Q⃗si)Λ(q⃗s,q⃗i)d2qsd2qi,
Λ(q⃗s,q⃗i)=exp(w02(32qi2+32qs2q⃗i·q⃗s))exp(2w02(q⃗c+(q⃗iq⃗s)2)2)d2qc.
Γj=kph=exp(Qsi2σ2(τ))exp(iμ⃗(τ)·Q⃗si)Λ(q⃗s,q⃗i)d2qsd2qi.
Γ(τ)=|ER|2j=1Nnj(t)nj(t+τ)exp(Qsi2σ2(τ))exp(iμ⃗(τ)·Q⃗si)Λ(q⃗s,q⃗i)d2qsd2qi,
Γ^(τ)=Γ(τ)Γ(0)=Γ^num(τ)exp(Qsi2σ2(τ))exp(iμ⃗(τ)·Q⃗si)Λ(q⃗s,q⃗i)d2qsd2qi
Γ^(τ)R(τ)=exp(γDQm2σ2(τ))exp(iγTμ⃗(τ)·Q⃗m),
μ(τ)=tan1[Im(R(τ))/Re(R(τ))]γTQm,
MSD(τ)=σ2(τ)+μ2(τ)=ln(R(τ)R*(τ))2γDQm2+[tan1[Im(R(τ))/Re(R(τ))]γTQm]2.

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