Abstract

We present a comprehensive theory of dead-time effects on Time-Correlated Single Photon Counting (TCSPC) as used for fluorescence lifetime measurements, and develop a correction algorithm to remove these artifacts. We apply this algorithm to fluorescence lifetime measurements as well as to Fluorescence Lifetime Imaging Microscopy (FLIM), where rapid data acquisition is necessarily connected with high count rates. There, dead-time effects cannot be neglected, and lead to distortions in the observed lifetime image. The algorithm is quite general and completely independent of the particular nature of the measured signal. It can also be applied to any other single-event counting measurement with detector and/or electronics dead-time.

© 2016 Optical Society of America

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References

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  1. U. Kubitscheck, Fluorescence Microscopy: From Principles to Biological Applications (Wiley-VCH, 2013).
    [Crossref]
  2. P. P. Mondial and A. Diaspro, Fundamentals of Fluorescence Microscopy: Exploring Life with Light (Springer, 2014).
    [Crossref]
  3. K. R. Spring and M. W. Davidson, “Introduction to Fluorescence Microscopy,” http://www.microscopyu.com .
  4. T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
    [Crossref] [PubMed]
  5. R. M. Clegg, “Fluorescence resonance energy transfer,” Curr. Opin. Biotechnol. 6, 103–110 (1995).
    [Crossref] [PubMed]
  6. R. M. Clegg, “Fluorescence Resonance Energy Transfer,” in Fluorescence Imaging, Spectroscopy and Microscopy, X. F. Wang and B. Herman, eds. (John Wiley and Sons, 1996), pp. 179–252.
  7. J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).
    [Crossref]
  8. D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).
  9. M. Wahl, “Time-Correlated Single Photon Counting,” http://www.picoquant.com/images/uploads/page/files/7253/technote_tcspc.pdf .
  10. W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Springer, 2005).
    [Crossref]
  11. M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
    [Crossref] [PubMed]
  12. C. Holzapfel, “On Statistics of Time-to-Amplitude converter systems in photon counting devices,” Rev. Sci. Instrum. 45, 894–896 (1974).
    [Crossref]
  13. C. C. Davis and T. A. King, “Photon pile-up corrections in the study of time-varying light sources,” J Phys E: Sci Instrum. 5, 1072 (1972).
    [Crossref]
  14. M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
    [Crossref]
  15. J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
    [Crossref] [PubMed]
  16. L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).
  17. A. McCarthy, R. J. Collins, N. J. Krichel, V. Fernandez, A. M. Wallace, and G. S. Buller, “Long-range time-of-flight scanning sensor based on high-speed time-correlated single-photon counting,” Appl. Opt. 48, 6241 (2009).
    [Crossref] [PubMed]
  18. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2013).

2016 (1)

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

2013 (1)

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

2009 (1)

2007 (2)

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

2000 (1)

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

1995 (1)

R. M. Clegg, “Fluorescence resonance energy transfer,” Curr. Opin. Biotechnol. 6, 103–110 (1995).
[Crossref] [PubMed]

1974 (1)

C. Holzapfel, “On Statistics of Time-to-Amplitude converter systems in photon counting devices,” Rev. Sci. Instrum. 45, 894–896 (1974).
[Crossref]

1972 (1)

C. C. Davis and T. A. King, “Photon pile-up corrections in the study of time-varying light sources,” J Phys E: Sci Instrum. 5, 1072 (1972).
[Crossref]

Arlt, J.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Becker, W.

W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Springer, 2005).
[Crossref]

Buller, G. S.

Clegg, R. M.

R. M. Clegg, “Fluorescence resonance energy transfer,” Curr. Opin. Biotechnol. 6, 103–110 (1995).
[Crossref] [PubMed]

R. M. Clegg, “Fluorescence Resonance Energy Transfer,” in Fluorescence Imaging, Spectroscopy and Microscopy, X. F. Wang and B. Herman, eds. (John Wiley and Sons, 1996), pp. 179–252.

Collins, R. J.

Davis, C. C.

C. C. Davis and T. A. King, “Photon pile-up corrections in the study of time-varying light sources,” J Phys E: Sci Instrum. 5, 1072 (1972).
[Crossref]

Diaspro, A.

P. P. Mondial and A. Diaspro, Fundamentals of Fluorescence Microscopy: Exploring Life with Light (Springer, 2014).
[Crossref]

Enderlein, J.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

Erdmann, R.

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

Fernandez, V.

Fleury, L.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Gregor, I.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

Hecht, B.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Henderson, R. K.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Holzapfel, C.

C. Holzapfel, “On Statistics of Time-to-Amplitude converter systems in photon counting devices,” Rev. Sci. Instrum. 45, 894–896 (1974).
[Crossref]

Kapusta, P.

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

King, T. A.

C. C. Davis and T. A. King, “Photon pile-up corrections in the study of time-varying light sources,” J Phys E: Sci Instrum. 5, 1072 (1972).
[Crossref]

Koberling, F.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

Krämer, B.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

Krichel, N. J.

Kubitscheck, U.

U. Kubitscheck, Fluorescence Microscopy: From Principles to Biological Applications (Wiley-VCH, 2013).
[Crossref]

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2013).

Li, D. D.-U.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Löschberger, A.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

McCarthy, A.

Mondial, P. P.

P. P. Mondial and A. Diaspro, Fundamentals of Fluorescence Microscopy: Exploring Life with Light (Springer, 2014).
[Crossref]

Niehörster, T.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

O’Connor, D. V.

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

Patting, M.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

Pawley, J. B.

J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).
[Crossref]

Phillips, D.

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

Rae, B. R.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Rahn, H.-J.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

Richardson, J. A.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Sauer, M.

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

Segura, J.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Tyndall, D.

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

Wahl, M.

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

Wallace, A. M.

Wild, U. P.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Zumofen, G.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Appl. Opt. (1)

Curr. Opin. Biotechnol. (1)

R. M. Clegg, “Fluorescence resonance energy transfer,” Curr. Opin. Biotechnol. 6, 103–110 (1995).
[Crossref] [PubMed]

J Phys E: Sci Instrum. (1)

C. C. Davis and T. A. King, “Photon pile-up corrections in the study of time-varying light sources,” J Phys E: Sci Instrum. 5, 1072 (1972).
[Crossref]

Nature Methods (1)

T. Niehörster, A. Löschberger, I. Gregor, B. Krämer, H.-J. Rahn, M. Patting, F. Koberling, J. Enderlein, and M. Sauer, “Multi-target spectrally resolved fluorescence lifetime imaging microscopy,” Nature Methods 13(3), 257–262 (2016).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 95, 19–22 (2000).

Proc. SPIE (1)

M. Patting, M. Wahl, P. Kapusta, and R. Erdmann, “Dead-time effects in TCSPC data analysis,” Proc. SPIE 6583, 658307 (2007).
[Crossref]

Rev. Sci. Instrum. (3)

J. Arlt, D. Tyndall, B. R. Rae, D. D.-U. Li, J. A. Richardson, and R. K. Henderson, “A study of pile-up in integrated time-correlated single photon counting systems,” Rev. Sci. Instrum. 84103105 (2013).
[Crossref] [PubMed]

M. Wahl, H.-J. Rahn, I. Gregor, R. Erdmann, and J. Enderlein, “Dead-time optimized time-correlated photon counting instrument with synchronized, independent timing channels,” Rev. Sci. Instrum. 78, 033106 (2007).
[Crossref] [PubMed]

C. Holzapfel, “On Statistics of Time-to-Amplitude converter systems in photon counting devices,” Rev. Sci. Instrum. 45, 894–896 (1974).
[Crossref]

Other (9)

U. Kubitscheck, Fluorescence Microscopy: From Principles to Biological Applications (Wiley-VCH, 2013).
[Crossref]

P. P. Mondial and A. Diaspro, Fundamentals of Fluorescence Microscopy: Exploring Life with Light (Springer, 2014).
[Crossref]

K. R. Spring and M. W. Davidson, “Introduction to Fluorescence Microscopy,” http://www.microscopyu.com .

R. M. Clegg, “Fluorescence Resonance Energy Transfer,” in Fluorescence Imaging, Spectroscopy and Microscopy, X. F. Wang and B. Herman, eds. (John Wiley and Sons, 1996), pp. 179–252.

J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).
[Crossref]

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

M. Wahl, “Time-Correlated Single Photon Counting,” http://www.picoquant.com/images/uploads/page/files/7253/technote_tcspc.pdf .

W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Springer, 2005).
[Crossref]

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2013).

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Figures (10)

Fig. 1
Fig. 1 Schematic of continuous photon detection/registration with dead-time effects of both the electronics (large blue rectangles) and the detector (small violet rectangles). Red balls are successfully detected and recorded photons; blue balls are detected but not recorded photons, because their detection takes place during recovery of the electronics; green balls are photons which hit the detector but are not detected because they fall within the recovery period of the detector. Note that the detector dead-time only occurs after successful detection of a photon (red and blue balls), while the electronics dead-time only occurs after successfully timing a photon (red balls).
Fig. 2
Fig. 2 Schematic of two photon registration events, showing the electronics dead-time (large blue rectangle) and the time region after the electronics dead-time which is still affected by the detector dead-time (small violet rectangle). The double arrow is the time span between the end of the electronics dead-time which started with the last photon registration at time t′ and the next photon registration at time t.
Fig. 3
Fig. 3 Monte Carlo Simulation (MCS, blue) of a TCSPC measurement with an excitation period P = 256, electronics dead-time E = 64, and detector dead-time D = 16. The photon hit rate ε was set to a value so that εP = 2, i.e. that, on average, two photons hit the detector per excitation cycle. The black line shows the underlying perfectly mono-exponential decay curve with a decay time of 30 time units. The green line is the computed h(t) using Eq. (1) and (4) after 10 iterations. The dead-time corrected decay curve, as computed from the simulated h(t), is shown in red. The lower panel shows the relative deviation of the reconstruction (red circles) from the ideal decay (black line), normalized by the square root of the ideal decay.
Fig. 4
Fig. 4 Measured histogram of the inter-photon time distribution (black solid line) extracted from a TCSPC measurement on a fluorescence dye solution. The yellow and red shaded regions on the left are the electronics and detector dead-time intervals, respectively. The cyan bars are the calculated values of mN, Eq. (6), for N = 0,...,5. The red dashed line shows the single-exponential decay of mN. From this fit, one determines an average value of photon hits per excitation cycle of εP = 1.0.
Fig. 5
Fig. 5 Autocorrelation functions for five different photon hit rates ε as indicated in the legend. Measured curves are represented by circles, solid lines show a global fit of Eq. (11) to all five measurements. The yellow and red shaded regions on the left mark the fitted electronics and detector dead-times, respectively. All autocorrelation functions were calculated at evenly spaced time points with 1ns spacing. At very high count rates, fit quality starts to deteriorate due to increasing jitter of the detector dead-time.
Fig. 6
Fig. 6 Results of Monte Carlo simulations of the performance of the recovery algorithm for dead-time corrected decay curves from measured TCSPC. Shown are the mean values (solid lines) and variances (shaded regions) of mono-exponential decay time values which are obtained from fitting the simulated decay curves. Simulations were performed for the same dead-time values as used in Fig. 3, for a range of photon hit values per excitation period, εP, from zero to 2, and for two different values of total number of photon hits, i.e. εP times number of excitation cycles, of 200 (light shaded region) and 1000 (dark shaded region). The corresponding decay curves have smaller number of photons, due to the dead-times of both electronics and detector.
Fig. 7
Fig. 7 TCSPC measurements on Atto655 dye solution at varying excitation power and thus fluorescence intensity. The fluorescence intensity is given here as dead-time corrected values of average number of photon hits per excitation cycle, εP. The inset shows also the relation between the actual average number of detected photons per excitation cycle, ε′P, and εP, showing the increasing dead-time related saturation of the measurement system with increasing intensity. Red symbols show determined lifetime values from uncorrected TCSPC curves, and blue symbols show lifetime values determined from dead-time corrected TCSPC curves. The dashed line shows the average over all lifetime values for all dead-time corrected measurements.
Fig. 8
Fig. 8 Human mesenchymal stem cell with actin filaments labeled by Atto647N, and imaged with a confocal scanning TCSPC microscope. The top row shows the intensity image before (left) and after (right) dead-time correction, where the counts were determined using the calculated hit rates (ε) for each pixel. One can clearly see the significant increase in signal strength after dead-time correction for regions with high fluorescence intensity. The bottom row shows the same for the lifetime images (left before, and right after dead-time correction). The tremendous impact of artifacts on the resulting lifetimes is clearly visible: In regions of high intensity, the lifetime values in the left bottom image always underestimate the true value, as seen in the dead-time corrected image at bottom right. The images are 20 ×20μm2 with a pixel size of 140nm and a dwell time of 5ms per pixel, the yellow scale bar is 5 μm. The highest number of photon hits per excitation cycle in this image is 0.5.
Fig. 9
Fig. 9 Image of the same sample as in Fig. 8, but using a ca. six times lower excitation intensity. Now, the dead-time correction does nearly not change neither the intensity nor the lifetime images, and both lifetime images are close to the dead-time corrected lifetime image of Fig. 8. However, due to the much lower fluorescence signal strength, the lifetime image is much noisier than the dead-time corrected lifetime image in Fig. 8.
Fig. 10
Fig. 10 The upper row shows a line plot through the brightest pixel of the cell measured at high laser power (Fig. 8) for intensity (left) and lifetime (right). The TC-SPC curve of the brightest pixel (εP = 0.50) is given in the lower left plot yielding τuncorrected = (2.87±0.02)ns and τcorrected = (3.42±0.02)ns. Next to it on the right, the TCSPC curve of the same pixel at low laser power (Fig. 9) is shown (εP = 0.09), yielding τuncorrected = (3.36±0.04)ns and τcorrected = (3.47±0.04)ns. For all plots, the gray line represents the uncorrected data and the dashed black line the corrected data.

Equations (11)

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h ( t ) = w ( t ) k ( t )
w ( t ) = t E d t h ( t ) exp [ min ( t + E , t D ) t d τ k ( τ ) ]
w ( t ) = t E D t E d t h ( t ) e t D t d τ k ( τ ) + l = 0 t E D ( l + 1 ) P t E D l P d t h ( t ) e t + E t d τ k ( τ ) .
w ( t ) = e t D t d τ k ( τ ) t E D t E d t h ( t ) + 1 1 e ε P t E D P t E D d t h ( t ) e t + E t d τ k ( τ ) .
g ( T ) 0 P d t h ( t ) exp [ min ( E , T D ) T d τ k ( t + τ ) ] k ( t + T ) .
m N = E + D + N P E + D + ( N + 1 ) P g ( T ) d T = C exp ( N ε P ) ,
f ( t ) = { 0 if t D ε exp [ ε ( t D ) ] if t > D .
a ( t ) = j = 1 t / D f ( j ) ( t )
f ( j ) ( t ) = 0 t d t f ( t ) f ( j 1 ) ( t t )
F ( t ) = { 0 if t E Z 1 a ( t ) if E < t E + D Z 1 a ( E + D ) exp [ ε ( t E D ) ] if t > E + D
A ( t ) = j = 1 t / E F ( j ) ( t )

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