Abstract

Afterpulsing of a photon-counting detector is a common problem in fluorescence correlation spectroscopy. We have developed a numerical procedure which eliminates the afterpulsing effect by analyzing the time reversal asymmetry of photon data that are recorded with a time-correlated single photon counting device. This method was applied to experimental data and was compared with a previous method [Rev. Sci. Instrum. 76, 033102 (2005).]. It is demonstrated that the present method can completely eliminate the afterpulsing effect even in the case of a sample solution that contains multiple fluorophores having different fluorescence lifetimes, for which the previous method underestimates the correlation amplitude. We also show a modification of the previous method incorporating the time symmetry analysis.

© 2015 Optical Society of America

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References

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  6. H. C. Burstyn and J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27(2), 1071–1085 (1983).
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  9. S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
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2015 (1)

T. Otosu, K. Ishii, and T. Tahara, “Microsecond protein dynamics observed at the single-molecule level,” Nat. Commun. 6, 7685 (2015).
[Crossref] [PubMed]

2013 (3)

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 1. Principle,” J. Phys. Chem. B 117(39), 11414–11422 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 2. Application,” J. Phys. Chem. B 117(39), 11423–11432 (2013).
[Crossref] [PubMed]

T. Otosu, K. Ishii, and T. Tahara, “Note: Simple calibration of the counting-rate dependence of the timing shift of single photon avalanche diodes by photon interval analysis,” Rev. Sci. Instrum. 84(3), 036105 (2013).
[Crossref] [PubMed]

2012 (3)

K. Ishii and T. Tahara, “Extracting decay curves of the correlated fluorescence photons measured in fluorescence correlation spectroscopy,” Chem. Phys. Lett. 519–520, 130–133 (2012).
[Crossref]

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

2011 (1)

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref] [PubMed]

2010 (3)

R. A. Colyer, G. Scalia, I. Rech, A. Gulinatti, M. Ghioni, S. Cova, S. Weiss, and X. Michalet, “High-throughput FCS using an LCOS spatial light modulator and an 8 × 1 SPAD array,” Biomed. Opt. Express 1(5), 1408–1431 (2010).
[Crossref] [PubMed]

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Resolving inhomogeneity using lifetime-weighted fluorescence correlation spectroscopy,” J. Phys. Chem. B 114(38), 12383–12391 (2010).
[Crossref] [PubMed]

2009 (1)

Y. Ohsugi and M. Kinjo, “Multipoint fluorescence correlation spectroscopy with total internal reflection fluorescence microscope,” J. Biomed. Opt. 14(1), 014030 (2009).
[Crossref] [PubMed]

2007 (2)

I. Gregor and J. Enderlein, “Time-resolved methods in biophysics. 3. Fluorescence lifetime correlation spectroscopy,” Photochem. Photobiol. Sci. 6(1), 13–18 (2007).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

2005 (2)

J. Enderlein and I. Gregor, “Using fluorescence lifetime for discriminating detector afterpulsing in fluorescence-correlation spectroscopy,” Rev. Sci. Instrum. 76(3), 033102 (2005).
[Crossref]

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

2003 (1)

2002 (1)

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

1994 (1)

M. Höbel and J. Ricka, “Deadtime and afterpulsing correction in multiphoton timing with nonideal detectors,” Rev. Sci. Instrum. 65(7), 2326–2336 (1994).
[Crossref]

1983 (1)

H. C. Burstyn and J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27(2), 1071–1085 (1983).
[Crossref]

1980 (1)

H. C. Burstyn, “Afterpulsing effects in photon correlation experiments,” Rev. Sci. Instrum. 51(10), 1431–1433 (1980).
[Crossref]

Becker, W.

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Benda, A.

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

Böhmer, M.

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

Burstyn, H. C.

H. C. Burstyn and J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27(2), 1071–1085 (1983).
[Crossref]

H. C. Burstyn, “Afterpulsing effects in photon correlation experiments,” Rev. Sci. Instrum. 51(10), 1431–1433 (1980).
[Crossref]

Chen, B.

Chen, D.

Colyer, R. A.

Cova, S.

Ding, Y.

Elson, E. L.

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref] [PubMed]

Enderlein, J.

I. Gregor and J. Enderlein, “Time-resolved methods in biophysics. 3. Fluorescence lifetime correlation spectroscopy,” Photochem. Photobiol. Sci. 6(1), 13–18 (2007).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

J. Enderlein and I. Gregor, “Using fluorescence lifetime for discriminating detector afterpulsing in fluorescence-correlation spectroscopy,” Rev. Sci. Instrum. 76(3), 033102 (2005).
[Crossref]

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

Erdmann, R.

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

Felekyan, S.

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Ghioni, M.

Gregor, I.

I. Gregor and J. Enderlein, “Time-resolved methods in biophysics. 3. Fluorescence lifetime correlation spectroscopy,” Photochem. Photobiol. Sci. 6(1), 13–18 (2007).
[Crossref] [PubMed]

J. Enderlein and I. Gregor, “Using fluorescence lifetime for discriminating detector afterpulsing in fluorescence-correlation spectroscopy,” Rev. Sci. Instrum. 76(3), 033102 (2005).
[Crossref]

Gulinatti, A.

Höbel, M.

M. Höbel and J. Ricka, “Deadtime and afterpulsing correction in multiphoton timing with nonideal detectors,” Rev. Sci. Instrum. 65(7), 2326–2336 (1994).
[Crossref]

Hof, M.

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

Ishii, K.

T. Otosu, K. Ishii, and T. Tahara, “Microsecond protein dynamics observed at the single-molecule level,” Nat. Commun. 6, 7685 (2015).
[Crossref] [PubMed]

T. Otosu, K. Ishii, and T. Tahara, “Note: Simple calibration of the counting-rate dependence of the timing shift of single photon avalanche diodes by photon interval analysis,” Rev. Sci. Instrum. 84(3), 036105 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 1. Principle,” J. Phys. Chem. B 117(39), 11414–11422 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 2. Application,” J. Phys. Chem. B 117(39), 11423–11432 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Extracting decay curves of the correlated fluorescence photons measured in fluorescence correlation spectroscopy,” Chem. Phys. Lett. 519–520, 130–133 (2012).
[Crossref]

K. Ishii and T. Tahara, “Resolving inhomogeneity using lifetime-weighted fluorescence correlation spectroscopy,” J. Phys. Chem. B 114(38), 12383–12391 (2010).
[Crossref] [PubMed]

Jin, L.

Kalinin, S.

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

Kapusta, P.

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

Kinjo, M.

Y. Ohsugi and M. Kinjo, “Multipoint fluorescence correlation spectroscopy with total internal reflection fluorescence microscope,” J. Biomed. Opt. 14(1), 014030 (2009).
[Crossref] [PubMed]

Kudryavtsev, V.

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Kühnemuth, R.

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Ma, H.

Macdonald, R.

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

Machán, R.

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

Michalet, X.

Ohsugi, Y.

Y. Ohsugi and M. Kinjo, “Multipoint fluorescence correlation spectroscopy with total internal reflection fluorescence microscope,” J. Biomed. Opt. 14(1), 014030 (2009).
[Crossref] [PubMed]

Otosu, T.

T. Otosu, K. Ishii, and T. Tahara, “Microsecond protein dynamics observed at the single-molecule level,” Nat. Commun. 6, 7685 (2015).
[Crossref] [PubMed]

T. Otosu, K. Ishii, and T. Tahara, “Note: Simple calibration of the counting-rate dependence of the timing shift of single photon avalanche diodes by photon interval analysis,” Rev. Sci. Instrum. 84(3), 036105 (2013).
[Crossref] [PubMed]

Patting, M.

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

Rahn, H.-J.

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

Rech, I.

Ricka, J.

M. Höbel and J. Ricka, “Deadtime and afterpulsing correction in multiphoton timing with nonideal detectors,” Rev. Sci. Instrum. 65(7), 2326–2336 (1994).
[Crossref]

Rüttinger, S.

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

Sanabria, H.

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

Sandhagen, C.

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Scalia, G.

Seidel, C. A. M.

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

S. Felekyan, R. Kühnemuth, V. Kudryavtsev, C. Sandhagen, W. Becker, and C. A. M. Seidel, “Full correlation from picoseconds to seconds by time-resolved and time-correlated single photon detection,” Rev. Sci. Instrum. 76(8), 083104 (2005).
[Crossref]

Sengers, J. V.

H. C. Burstyn and J. V. Sengers, “Time dependence of critical concentration fluctuations in a binary liquid,” Phys. Rev. A 27(2), 1071–1085 (1983).
[Crossref]

Tahara, T.

T. Otosu, K. Ishii, and T. Tahara, “Microsecond protein dynamics observed at the single-molecule level,” Nat. Commun. 6, 7685 (2015).
[Crossref] [PubMed]

T. Otosu, K. Ishii, and T. Tahara, “Note: Simple calibration of the counting-rate dependence of the timing shift of single photon avalanche diodes by photon interval analysis,” Rev. Sci. Instrum. 84(3), 036105 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 2. Application,” J. Phys. Chem. B 117(39), 11423–11432 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 1. Principle,” J. Phys. Chem. B 117(39), 11414–11422 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Extracting decay curves of the correlated fluorescence photons measured in fluorescence correlation spectroscopy,” Chem. Phys. Lett. 519–520, 130–133 (2012).
[Crossref]

K. Ishii and T. Tahara, “Resolving inhomogeneity using lifetime-weighted fluorescence correlation spectroscopy,” J. Phys. Chem. B 114(38), 12383–12391 (2010).
[Crossref] [PubMed]

Valeri, A.

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

Wahl, M.

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

Weiss, S.

Zhao, M.

Appl. Opt. (1)

Biomed. Opt. Express (1)

Biophys. J. (1)

E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101(12), 2855–2870 (2011).
[Crossref] [PubMed]

Chem. Phys. Lett. (2)

M. Böhmer, M. Wahl, H.-J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. 353(5–6), 439–445 (2002).
[Crossref]

K. Ishii and T. Tahara, “Extracting decay curves of the correlated fluorescence photons measured in fluorescence correlation spectroscopy,” Chem. Phys. Lett. 519–520, 130–133 (2012).
[Crossref]

ChemPhysChem (1)

S. Felekyan, S. Kalinin, H. Sanabria, A. Valeri, and C. A. M. Seidel, “Filtered FCS: species auto- and cross-correlation functions highlight binding and dynamics in biomolecules,” ChemPhysChem 13(4), 1036–1053 (2012).
[Crossref] [PubMed]

Int. J. Mol. Sci. (1)

P. Kapusta, R. Macháň, A. Benda, and M. Hof, “Fluorescence Lifetime Correlation Spectroscopy (FLCS): concepts, applications and outlook,” Int. J. Mol. Sci. 13(10), 12890–12910 (2012).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

Y. Ohsugi and M. Kinjo, “Multipoint fluorescence correlation spectroscopy with total internal reflection fluorescence microscope,” J. Biomed. Opt. 14(1), 014030 (2009).
[Crossref] [PubMed]

J. Fluoresc. (2)

P. Kapusta, M. Wahl, A. Benda, M. Hof, and J. Enderlein, “Fluorescence lifetime correlation spectroscopy,” J. Fluoresc. 17(1), 43–48 (2007).
[Crossref] [PubMed]

S. Rüttinger, P. Kapusta, M. Patting, M. Wahl, and R. Macdonald, “On the resolution capabilities and limits of fluorescence lifetime correlation spectroscopy (FLCS) measurements,” J. Fluoresc. 20(1), 105–114 (2010).
[Crossref] [PubMed]

J. Phys. Chem. B (3)

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 1. Principle,” J. Phys. Chem. B 117(39), 11414–11422 (2013).
[Crossref] [PubMed]

K. Ishii and T. Tahara, “Two-dimensional fluorescence lifetime correlation spectroscopy. 2. Application,” J. Phys. Chem. B 117(39), 11423–11432 (2013).
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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of the principle of the time symmetry analysis. (a) A timing chart of photon signals obtained with a TCSPC-FCS setup. T: macrotime, t: microtime. (b,c) 2D emission-delay correlation maps of the true signal (b) and the afterpulsing effect (c). 1D decay curves on top and left are made by integrating the 2D maps along the vertical and horizontal axes, respectively. The asymmetry of the 2D maps is evaluated as the difference of the integrated 1D decay curves (d).
Fig. 2
Fig. 2 Decay functions obtained from the pure TMR solution. (a) Δ I ¯ ( t (i) ) n 1 I ¯ I ¯ ( t (i) ) calculated from ensemble decay data. (b) The asymmetry functions C(∆T; t) evaluated at selected ∆T values. (c) Afterpulsing probability function p(∆T) determined by Eq. (14). The vertical axis is scaled as afterpulsing probability per unit time (52.7 ns).
Fig. 3
Fig. 3 Correlation curves of the TMR solution. Blue solid line: raw autocorrelation without afterpulse correction, red solid line: autocorrelation after afterpulse correction using the time symmetry analysis, black dashed line: cross correlation between two detectors, green dashed line: autocorrelation after afterpulse correction using FLCS.
Fig. 4
Fig. 4 Correlation curves of the mixture solution of Cy3 and TMR. Blue solid line: raw autocorrelation without afterpulse correction, red solid line: autocorrelation after afterpulse correction using the time symmetry analysis, black dashed line: cross correlation between two detectors, green dashed line: autocorrelation after afterpulse correction using FLCS. Inset: differences between the raw autocorrelation and the autocorrelations after afterpulse correction using the time symmetry analysis (red) or FLCS (green).

Equations (34)

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G(ΔT) I(T)I(T+ΔT) I(T) 2 = p,q { 1 ΔTΔΔT/2< T q T p <ΔT+ΔΔT/2 0 otherwise N 2 T max 2 ( T max ΔT)ΔΔT .
G (ΔT)= p(ΔT) I(T) .
M(ΔT; t (i) , t (j) )= I(T; t (i) )I(T+ΔT; t (j) ) = 1 ( T max ΔT)ΔΔT p,q { δ t p t (i) δ t q t (j) ΔTΔΔT/2< T q T p <ΔT+ΔΔT/2 0 otherwise .
C(ΔT; t (i) ) C (ΔT; t (i) ) C (ΔT; t (i) )
C (ΔT; t (i) ) j=1 n M(ΔT; t (i) , t (j) )
C (ΔT; t (i) ) j=1 n M(ΔT; t (j) , t (i) ) .
M(ΔT; t (i) , t (j) )= M 0 (ΔT; t (i) , t (j) )+ M (ΔT; t (i) , t (j) ).
C 0 (ΔT; t (i) ) j=1 n M 0 (ΔT; t (j) , t (i) ) j=1 n M 0 (ΔT; t (i) , t (j) ) =0.
M (ΔT; t (i) , t (j) )= I(T; t (i) ) p(ΔT; t (j) ) = I ¯ ( t (i) ) n 1 p(ΔT)
C (ΔT; t (i) ) j=1 n M (ΔT; t (i) , t (j) ) =p(ΔT) I ¯ ( t (i) ),
C (ΔT; t (i) ) j=1 n M (ΔT; t (j) , t (i) ) =p(ΔT) n 1 I ¯ ,
C(ΔT; t (i) )= C (ΔT; t (i) ) C (ΔT; t (i) ) =p(ΔT){ n 1 I ¯ I ¯ ( t (i) ) }
p(ΔT)= C(ΔT; t (i) ) / { n 1 I ¯ I ¯ ( t (i) ) } C(ΔT; t (i) ) / Δ I ¯ ( t (i) ) .
p(ΔT)= i=1 n C(ΔT; t (i) )Δ I ¯ ( t (i) ) / σ 2 ( t (i) ) i=1 n { Δ I ¯ ( t (i) ) } 2 / σ 2 ( t (i) ) .
G 0 (ΔT)G(ΔT)p(ΔT) I ¯ 1 .
I( t (i) )= α=1 m I α d α ( t (i) ) .
I α = i=1 n f α ( t (i) )I( t (i) ) .
F= { D T I ¯ 1 D } 1 D T I ¯ 1 .
g AB (ΔT) I A (T) I B (T+ΔT) = i,j=1 n I A (T; t (i) ) I B (T+ΔT; t (j) ) = i,j=1 n f A ( t (i) ) I(T; t (i) )I(T+ΔT; t (j) ) f B ( t (j) ) = i,j=1 n f A ( t (i) )M(ΔT; t (i) , t (j) ) f B ( t (j) ) .
g AB (ΔT) = 1 ( T max ΔT)ΔΔT p,q { f A ( t p ) f B ( t q ) ΔTΔΔT/2< T q T p <ΔT+ΔΔT/2 0 otherwise .
d ap ( t (i) )= n 1 .
d fl ( t (i) )= { I ¯ ( t (i) ) I ¯ min } / { I ¯ n I ¯ min } .
d en ( t (i) )= I ¯ ( t (i) ) / I ¯ ,
d ap ( t (i) )= n 1 .
g en, ap (ΔT)= i,j=1 n f en ( t (i) )M(ΔT; t (i) , t (j) ) f ap ( t (j) ) = i,j=1 n f en ( t (i) ) M 0 (ΔT; t (i) , t (j) ) f ap ( t (j) ) + i,j=1 n f en ( t (i) ) M (ΔT; t (i) , t (j) ) f ap ( t (j) ) ,
g ap, en (ΔT)= i,j=1 n f ap ( t (j) )M(ΔT; t (j) , t (i) ) f en ( t (i) ) = i,j=1 n f ap ( t (j) ) M 0 (ΔT; t (j) , t (i) ) f en ( t (i) ) + i,j=1 n f ap ( t (j) ) M (ΔT; t (j) , t (i) ) f en ( t (i) ) .
g en, ap (ΔT) g ap, en (ΔT)= i,j=1 n f en ( t (i) ){ M (ΔT; t (i) , t (j) ) M (ΔT; t (j) , t (i) ) } f ap ( t (j) ) = i,j=1 n f en ( t (i) ){ I ¯ ( t (i) ) n 1 p(ΔT) I ¯ ( t (j) ) n 1 p(ΔT) } f ap ( t (j) ) =p(ΔT) I ¯ .
g AB 0 (ΔT)= i,j=1 n f A ( t (i) ) M 0 (ΔT; t (i) , t (j) ) f B ( t (j) ) = i,j=1 n f A ( t (i) ){ M(ΔT; t (i) , t (j) ) M (ΔT; t (i) , t (j) ) } f B ( t (j) ) = g AB (ΔT)p(ΔT){ i=1 n f A ( t (i) ) I ¯ ( t (i) ) }{ j=1 n n 1 f B ( t (j) ) }.
σ 2 [ C(ΔT; t (i) ) ]( t (i) ;ΔT)= σ 2 [ C (ΔT; t (i) ) ]( t (i) ;ΔT)+ σ 2 [ C (ΔT; t (i) ) ]( t (i) ;ΔT).
C (ΔT; t (i) )= j=1 n M 0 (ΔT; t (j) , t (i) ) + j=1 n M (ΔT; t (j) , t (i) ) j=1 n G 0 (ΔT) I ¯ ( t (j) ) I ¯ ( t (i) ) + j=1 n p(ΔT) n 1 I ¯ ( t (j) ) = j=1 n { G(ΔT)p(ΔT) I 1 } I ¯ ( t (j) ) I ¯ ( t (i) ) + j=1 n p(ΔT) n 1 I ¯ ( t (j) ) = k,l=1 n M(ΔT; t (k) , t (l) ) I ¯ ( t (i) ) I ¯ 1 p(ΔT) I ¯ ( t (i) )+p(ΔT) n 1 I ¯
C (ΔT; t (i) )= j=1 n M 0 (ΔT; t (i) , t (j) ) + j=1 n M (ΔT; t (i) , t (j) ) j=1 n G 0 (ΔT) I ¯ ( t (i) ) I ¯ ( t (j) ) + j=1 n p(ΔT) n 1 I ¯ ( t (i) ) = j=1 n { G(ΔT)p(ΔT) I ¯ 1 } I ¯ ( t (i) ) I ¯ ( t (j) ) + j=1 n p(ΔT) n 1 I ¯ ( t (i) ) = k,l=1 n M(ΔT; t (k) , t (l) ) I ¯ ( t (i) ) I ¯ 1 .
G(ΔT)= I(T)I(T+ΔT) I(T) 2 = k,l=1 n M(ΔT; t (k) , t (l) ) I ¯ 2 .
σ 2 [ C(ΔT; t (i) ) ]( t (i) ;ΔT)= C (ΔT; t (i) )+ C (ΔT; t (i) ) =2 k,l=1 n M(ΔT; t (k) , t (l) ) I ¯ ( t (i) ) I ¯ 1 p(ΔT) I ¯ ( t (i) )+p(ΔT) n 1 I ¯ .
G(ΔT)=1+ δI(T)δI(T+ΔT) I(T) 2 .

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