Abstract

We deduce the signal-to-noise ratio for fluorescence lifetime imaging when using frequency-domain methods. We assume mono-exponential decay and quantum-noise-limited performance. The results are compared with Monte Carlo simulations with good agreement. We also compare our results with previous investigations of time-domain methods for fluorescence lifetime imaging. For a given number of detected photons, we find that frequency-domain and time-domain methods are equally good. The correct choice of detection technique and its parameters is important for obtaining good results.

© 2003 Optical Society of America

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  1. C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).
  2. E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
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    [CrossRef]
  5. R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
    [CrossRef]
  6. A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
    [CrossRef]
  7. M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
    [CrossRef]
  8. J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
    [CrossRef]
  9. C. J. de Grauw, H. C. Gerritsen, “Multiple time-gate module for fluorescence lifetime imaging,” Appl. Spectrosc. 55, 670–678 (2001).
    [CrossRef]
  10. T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
    [CrossRef] [PubMed]
  11. C. J. R. van der Oord, H. C. Gerritsen, F. F. G. Rommerts, D. A. Shaw, I. H. Munro, Y. K. Levine, “Micro-volume time-resolved fluorescence spectroscopy using a confocal synchrotron radiation microscope,” Appl. Spectrosc. 49, 1469–1473 (1995).
    [CrossRef]
  12. P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
    [CrossRef]
  13. A. Draaijer, R. Sanders, H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. Pawley, ed. (Plenum, New York, 1995), pp. 491–505.
  14. K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).
  15. H. Brismar, B. Ulfhake, “Fluorescence lifetime measurements in confocal microscopy of neurons labeled with multiple fluorophores,” Nat. Biotechnol. 15, 373–377 (1997).
    [CrossRef] [PubMed]
  16. T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
    [CrossRef]
  17. A. Squire, P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. (Oxford) 193, 36–49 (1999).
    [CrossRef]
  18. K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
    [CrossRef]
  19. J. R. Lakowicz, H. Szmacinski, “Fluorescence lifetime-based sensing of pH, Ca+2,K+ and glucose,” Sens. Actuators B 11, 133–143 (1993).
    [CrossRef]
  20. G. Krishnamoorthy, A. Srivastava, “Intracellular dynamics seen through time-resolved fluorescence microscopy,” Curr. Sci. 72, 835–845 (1997).
  21. R. D. Spencer, G. Weber, “Measurements of subnanosecond fluorescence lifetimes with a cross-correlation phase fluorometer,” Ann. N.Y. Acad. Sci. 158, 361–376 (1969).
    [CrossRef]
  22. D. V. O’Connor, D. Phillips, Time-Correlated Single Photon Counting (Academic, New York, 1984).
  23. M. Köllner, J. Wolfrum, “How many photons are necessary for fluorescence lifetime measurements?” Chem. Phys. Lett. 200, 199–204 (1992).
    [CrossRef]
  24. R. M. Ballew, J. N. Demas, “An error analysis of the rapid lifetime determination method for the evaluation of single exponential decays,” Anal. Chem. 61, 30–33 (1989).
    [CrossRef]
  25. E. Gaviola, “Ein Fluorometer. Apparat zur Messung von Fluoreszenzabklingungszeiten,” Z. Phys. 42, 853–861 (1927).
    [CrossRef]
  26. K. Carlsson, A. Liljeborg, “Simultaneous confocal lifetime imaging of multiple fluorophores using the intensity-modulated multiple-wavelength scanning (IMS) technique,” J. Microsc. (Oxford) 191, 119–127 (1998).
    [CrossRef]
  27. K. Carlsson, J. Philip, “Theoretical investigation of the signal-to-noise ratio for different fluorescence lifetime imaging techniques,” in Optical Diagnostics of Living Cells V, D. L. Farkas, R. C. Leif, eds., Proc. SPIE4622, 70–78 (2002).
    [CrossRef]
  28. K. Carlsson, A. Liljeborg, “Confocal fluorescence microscopy using spectral and lifetime information to simultaneously record four fluorophores with high channel separation,” J. Microsc. (Oxford) 185, 37–46 (1997).
    [CrossRef]

2001

2000

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

1999

A. Squire, P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. (Oxford) 193, 36–49 (1999).
[CrossRef]

1998

K. Carlsson, A. Liljeborg, “Simultaneous confocal lifetime imaging of multiple fluorophores using the intensity-modulated multiple-wavelength scanning (IMS) technique,” J. Microsc. (Oxford) 191, 119–127 (1998).
[CrossRef]

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

1997

H. Brismar, B. Ulfhake, “Fluorescence lifetime measurements in confocal microscopy of neurons labeled with multiple fluorophores,” Nat. Biotechnol. 15, 373–377 (1997).
[CrossRef] [PubMed]

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

K. Carlsson, A. Liljeborg, “Confocal fluorescence microscopy using spectral and lifetime information to simultaneously record four fluorophores with high channel separation,” J. Microsc. (Oxford) 185, 37–46 (1997).
[CrossRef]

G. Krishnamoorthy, A. Srivastava, “Intracellular dynamics seen through time-resolved fluorescence microscopy,” Curr. Sci. 72, 835–845 (1997).

1996

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

1995

1993

J. R. Lakowicz, H. Szmacinski, “Fluorescence lifetime-based sensing of pH, Ca+2,K+ and glucose,” Sens. Actuators B 11, 133–143 (1993).
[CrossRef]

T. W. J. Gadella, T. M. Jovin, R. M. Clegg, “Fluorescence lifetime imaging microscopy (FLIM): spatial resolution of microstructures on the nanosecond timescale,” Biophys. Chem. 48, 221–239 (1993).
[CrossRef]

T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
[CrossRef] [PubMed]

1992

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

M. Köllner, J. Wolfrum, “How many photons are necessary for fluorescence lifetime measurements?” Chem. Phys. Lett. 200, 199–204 (1992).
[CrossRef]

1990

C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).

1989

R. M. Ballew, J. N. Demas, “An error analysis of the rapid lifetime determination method for the evaluation of single exponential decays,” Anal. Chem. 61, 30–33 (1989).
[CrossRef]

1969

R. D. Spencer, G. Weber, “Measurements of subnanosecond fluorescence lifetimes with a cross-correlation phase fluorometer,” Ann. N.Y. Acad. Sci. 158, 361–376 (1969).
[CrossRef]

1927

E. Gaviola, “Ein Fluorometer. Apparat zur Messung von Fluoreszenzabklingungszeiten,” Z. Phys. 42, 853–861 (1927).
[CrossRef]

Andersson, R. M.

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

Arden-Jacob, J.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Ballew, R. M.

R. M. Ballew, J. N. Demas, “An error analysis of the rapid lifetime determination method for the evaluation of single exponential decays,” Anal. Chem. 61, 30–33 (1989).
[CrossRef]

Bastiaens, P. I. H.

A. Squire, P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. (Oxford) 193, 36–49 (1999).
[CrossRef]

Berland, K. M.

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Brakenhoff, G. J.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Brismar, H.

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

H. Brismar, B. Ulfhake, “Fluorescence lifetime measurements in confocal microscopy of neurons labeled with multiple fluorophores,” Nat. Biotechnol. 15, 373–377 (1997).
[CrossRef] [PubMed]

Buist, A. H.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Buurman, E. P.

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

Carlsson, K.

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

K. Carlsson, A. Liljeborg, “Simultaneous confocal lifetime imaging of multiple fluorophores using the intensity-modulated multiple-wavelength scanning (IMS) technique,” J. Microsc. (Oxford) 191, 119–127 (1998).
[CrossRef]

K. Carlsson, A. Liljeborg, “Confocal fluorescence microscopy using spectral and lifetime information to simultaneously record four fluorophores with high channel separation,” J. Microsc. (Oxford) 185, 37–46 (1997).
[CrossRef]

K. Carlsson, J. Philip, “Theoretical investigation of the signal-to-noise ratio for different fluorescence lifetime imaging techniques,” in Optical Diagnostics of Living Cells V, D. L. Farkas, R. C. Leif, eds., Proc. SPIE4622, 70–78 (2002).
[CrossRef]

Clegg, R. M.

T. W. J. Gadella, T. M. Jovin, R. M. Clegg, “Fluorescence lifetime imaging microscopy (FLIM): spatial resolution of microstructures on the nanosecond timescale,” Biophys. Chem. 48, 221–239 (1993).
[CrossRef]

Coelho-Sampaio, T.

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

de Grauw, C. J.

C. J. de Grauw, H. C. Gerritsen, “Multiple time-gate module for fluorescence lifetime imaging,” Appl. Spectrosc. 55, 670–678 (2001).
[CrossRef]

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

Deimel, M.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Deltau, G.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Demas, J. N.

R. M. Ballew, J. N. Demas, “An error analysis of the rapid lifetime determination method for the evaluation of single exponential decays,” Anal. Chem. 61, 30–33 (1989).
[CrossRef]

Dong, C. Y.

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Draaijer, A.

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

A. Draaijer, R. Sanders, H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. Pawley, ed. (Plenum, New York, 1995), pp. 491–505.

Drexhage, K. H.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

French, T.

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Gadella, T. W. J.

T. W. J. Gadella, T. M. Jovin, R. M. Clegg, “Fluorescence lifetime imaging microscopy (FLIM): spatial resolution of microstructures on the nanosecond timescale,” Biophys. Chem. 48, 221–239 (1993).
[CrossRef]

Gaviola, E.

E. Gaviola, “Ein Fluorometer. Apparat zur Messung von Fluoreszenzabklingungszeiten,” Z. Phys. 42, 853–861 (1927).
[CrossRef]

Gerritsen, H. C.

C. J. de Grauw, H. C. Gerritsen, “Multiple time-gate module for fluorescence lifetime imaging,” Appl. Spectrosc. 55, 670–678 (2001).
[CrossRef]

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

C. J. R. van der Oord, H. C. Gerritsen, F. F. G. Rommerts, D. A. Shaw, I. H. Munro, Y. K. Levine, “Micro-volume time-resolved fluorescence spectroscopy using a confocal synchrotron radiation microscope,” Appl. Spectrosc. 49, 1469–1473 (1995).
[CrossRef]

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

A. Draaijer, R. Sanders, H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. Pawley, ed. (Plenum, New York, 1995), pp. 491–505.

Gijsbers, E. J.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Göbel, F.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

Gratton, E.

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Houpt, P. M.

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

Jovin, T. M.

T. W. J. Gadella, T. M. Jovin, R. M. Clegg, “Fluorescence lifetime imaging microscopy (FLIM): spatial resolution of microstructures on the nanosecond timescale,” Biophys. Chem. 48, 221–239 (1993).
[CrossRef]

Ko, D.-S.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Köllner, M.

M. Köllner, J. Wolfrum, “How many photons are necessary for fluorescence lifetime measurements?” Chem. Phys. Lett. 200, 199–204 (1992).
[CrossRef]

König, K.

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

Krishnamoorthy, G.

G. Krishnamoorthy, A. Srivastava, “Intracellular dynamics seen through time-resolved fluorescence microscopy,” Curr. Sci. 72, 835–845 (1997).

Kusumi, A.

T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
[CrossRef] [PubMed]

Lakowicz, J. R.

J. R. Lakowicz, H. Szmacinski, “Fluorescence lifetime-based sensing of pH, Ca+2,K+ and glucose,” Sens. Actuators B 11, 133–143 (1993).
[CrossRef]

Levine, Y. K.

C. J. R. van der Oord, H. C. Gerritsen, F. F. G. Rommerts, D. A. Shaw, I. H. Munro, Y. K. Levine, “Micro-volume time-resolved fluorescence spectroscopy using a confocal synchrotron radiation microscope,” Appl. Spectrosc. 49, 1469–1473 (1995).
[CrossRef]

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

Lieberwirth, U.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

Liljeborg, A.

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

K. Carlsson, A. Liljeborg, “Simultaneous confocal lifetime imaging of multiple fluorophores using the intensity-modulated multiple-wavelength scanning (IMS) technique,” J. Microsc. (Oxford) 191, 119–127 (1998).
[CrossRef]

K. Carlsson, A. Liljeborg, “Confocal fluorescence microscopy using spectral and lifetime information to simultaneously record four fluorophores with high channel separation,” J. Microsc. (Oxford) 185, 37–46 (1997).
[CrossRef]

Mantulin, W. W.

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

Marx, N. J.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Mitchell, A. C.

C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).

Morgan, C. G.

C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).

Mühlegger, K.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

Müller, M.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Müller, R.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Munro, I. H.

Murray, J. G.

C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).

Norris, T. B.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

O’Connor, D. V.

D. V. O’Connor, D. Phillips, Time-Correlated Single Photon Counting (Academic, New York, 1984).

Oida, T.

T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
[CrossRef] [PubMed]

Philip, J.

K. Carlsson, J. Philip, “Theoretical investigation of the signal-to-noise ratio for different fluorescence lifetime imaging techniques,” in Optical Diagnostics of Living Cells V, D. L. Farkas, R. C. Leif, eds., Proc. SPIE4622, 70–78 (2002).
[CrossRef]

Phillips, D.

D. V. O’Connor, D. Phillips, Time-Correlated Single Photon Counting (Academic, New York, 1984).

Piston, D. W.

D. W. Piston, D. R. Sandison, W. W. Webb, “Time-resolved fluorescence imaging and background rejection by two-photon excitation in laser scanning microscopy,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 379–389 (1992).
[CrossRef]

Rommerts, F. F. G.

Sako, Y.

T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
[CrossRef] [PubMed]

Sanders, R.

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

A. Draaijer, R. Sanders, H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. Pawley, ed. (Plenum, New York, 1995), pp. 491–505.

Sandison, D. R.

D. W. Piston, D. R. Sandison, W. W. Webb, “Time-resolved fluorescence imaging and background rejection by two-photon excitation in laser scanning microscopy,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 379–389 (1992).
[CrossRef]

Sauer, M.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Shaw, D. A.

Siebert, S.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

So, P. T. C.

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Sosnowski, T. S.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Spencer, R. D.

R. D. Spencer, G. Weber, “Measurements of subnanosecond fluorescence lifetimes with a cross-correlation phase fluorometer,” Ann. N.Y. Acad. Sci. 158, 361–376 (1969).
[CrossRef]

Squier, J.

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

Squire, A.

A. Squire, P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. (Oxford) 193, 36–49 (1999).
[CrossRef]

Srivastava, A.

G. Krishnamoorthy, A. Srivastava, “Intracellular dynamics seen through time-resolved fluorescence microscopy,” Curr. Sci. 72, 835–845 (1997).

Sytsma, J.

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

Szmacinski, H.

J. R. Lakowicz, H. Szmacinski, “Fluorescence lifetime-based sensing of pH, Ca+2,K+ and glucose,” Sens. Actuators B 11, 133–143 (1993).
[CrossRef]

Tromberg, B. J.

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

Ulfhake, B.

H. Brismar, B. Ulfhake, “Fluorescence lifetime measurements in confocal microscopy of neurons labeled with multiple fluorophores,” Nat. Biotechnol. 15, 373–377 (1997).
[CrossRef] [PubMed]

van der Oord, C. J. R.

van Veen, J. J. F.

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

Vroom, J. M.

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

Weaver, D. J.

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

Webb, W. W.

D. W. Piston, D. R. Sandison, W. W. Webb, “Time-resolved fluorescence imaging and background rejection by two-photon excitation in laser scanning microscopy,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 379–389 (1992).
[CrossRef]

Weber, G.

R. D. Spencer, G. Weber, “Measurements of subnanosecond fluorescence lifetimes with a cross-correlation phase fluorometer,” Ann. N.Y. Acad. Sci. 158, 361–376 (1969).
[CrossRef]

Wolfrum, J.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

M. Köllner, J. Wolfrum, “How many photons are necessary for fluorescence lifetime measurements?” Chem. Phys. Lett. 200, 199–204 (1992).
[CrossRef]

Yu, W. M.

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

Zander, C.

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

Anal. Chem.

R. M. Ballew, J. N. Demas, “An error analysis of the rapid lifetime determination method for the evaluation of single exponential decays,” Anal. Chem. 61, 30–33 (1989).
[CrossRef]

Ann. N.Y. Acad. Sci.

R. D. Spencer, G. Weber, “Measurements of subnanosecond fluorescence lifetimes with a cross-correlation phase fluorometer,” Ann. N.Y. Acad. Sci. 158, 361–376 (1969).
[CrossRef]

Appl. Spectrosc.

Bioimaging

P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, E. Gratton, “Time-resolved fluorescence microscopy using two-photon excitation,” Bioimaging 3, 49–63 (1995).
[CrossRef]

M. Sauer, J. Arden-Jacob, K. H. Drexhage, F. Göbel, U. Lieberwirth, K. Mühlegger, R. Müller, J. Wolfrum, C. Zander, “Time-resolved identification of individual mononucleotide molecules in aqueous solution with pulsed semiconductor lasers,” Bioimaging 6, 14–24 (1998).
[CrossRef]

Biophys. Chem.

T. W. J. Gadella, T. M. Jovin, R. M. Clegg, “Fluorescence lifetime imaging microscopy (FLIM): spatial resolution of microstructures on the nanosecond timescale,” Biophys. Chem. 48, 221–239 (1993).
[CrossRef]

Biophys. J.

T. Oida, Y. Sako, A. Kusumi, “Fluorescence lifetime imaging microscopy (flimscopy),” Biophys. J. 64, 676–685 (1993).
[CrossRef] [PubMed]

Chem. Phys. Lett.

R. Müller, C. Zander, M. Sauer, M. Deimel, D.-S. Ko, S. Siebert, J. Arden-Jacob, G. Deltau, N. J. Marx, K. H. Drexhage, J. Wolfrum, “Time-resolved identification of single molecules in solution with a pulsed semiconductor diode laser,” Chem. Phys. Lett. 262, 716–722 (1996).
[CrossRef]

M. Köllner, J. Wolfrum, “How many photons are necessary for fluorescence lifetime measurements?” Chem. Phys. Lett. 200, 199–204 (1992).
[CrossRef]

Curr. Sci.

G. Krishnamoorthy, A. Srivastava, “Intracellular dynamics seen through time-resolved fluorescence microscopy,” Curr. Sci. 72, 835–845 (1997).

J. Microsc. (Oxford)

K. Carlsson, A. Liljeborg, “Simultaneous confocal lifetime imaging of multiple fluorophores using the intensity-modulated multiple-wavelength scanning (IMS) technique,” J. Microsc. (Oxford) 191, 119–127 (1998).
[CrossRef]

K. Carlsson, A. Liljeborg, “Confocal fluorescence microscopy using spectral and lifetime information to simultaneously record four fluorophores with high channel separation,” J. Microsc. (Oxford) 185, 37–46 (1997).
[CrossRef]

A. H. Buist, M. Müller, E. J. Gijsbers, G. J. Brakenhoff, T. S. Sosnowski, T. B. Norris, J. Squier, “Double-pulse fluorescence lifetime measurements,” J. Microsc. (Oxford) 186, 212–220 (1997).
[CrossRef]

J. Sytsma, J. M. Vroom, C. J. de Grauw, H. C. Gerritsen, “Time-gated fluorescence lifetime imaging and microvolume spectroscopy using two-photon excitation,” J. Microsc. (Oxford) 191, 39–51 (1998).
[CrossRef]

K. König, P. T. C. So, W. W. Mantulin, B. J. Tromberg, E. Gratton, “Two-photon excited lifetime imaging of autofluorescence in cells during UVA and NIR photostress,” J. Microsc. (Oxford) 183, 197–204 (1996).

T. French, P. T. C. So, D. J. Weaver, T. Coelho-Sampaio, E. Gratton, “Two-photon fluorescence lifetime imaging microscopy of macrophage-mediated antigen processing,” J. Microsc. (Oxford) 185, 339–353 (1997).
[CrossRef]

A. Squire, P. I. H. Bastiaens, “Three dimensional image restoration in fluorescence lifetime imaging microscopy,” J. Microsc. (Oxford) 193, 36–49 (1999).
[CrossRef]

K. Carlsson, A. Liljeborg, R. M. Andersson, H. Brismar, “Confocal pH imaging of microscopic specimens using fluorescence lifetimes and phase fluorometry: influence of parameter choice on system performance,” J. Microsc. (Oxford) 199, 106–114 (2000).
[CrossRef]

Nat. Biotechnol.

H. Brismar, B. Ulfhake, “Fluorescence lifetime measurements in confocal microscopy of neurons labeled with multiple fluorophores,” Nat. Biotechnol. 15, 373–377 (1997).
[CrossRef] [PubMed]

Scanning

E. P. Buurman, R. Sanders, A. Draaijer, H. C. Gerritsen, J. J. F. van Veen, P. M. Houpt, Y. K. Levine, “Fluorescence lifetime imaging using a confocal laser scanning microscope,” Scanning 14, 155–159 (1992).
[CrossRef]

Sens. Actuators B

J. R. Lakowicz, H. Szmacinski, “Fluorescence lifetime-based sensing of pH, Ca+2,K+ and glucose,” Sens. Actuators B 11, 133–143 (1993).
[CrossRef]

Trans. R. Microsc. Soc.

C. G. Morgan, A. C. Mitchell, J. G. Murray, “Nanosecond time-resolved fluorescence microscopy: principles and practice,” Trans. R. Microsc. Soc. 1, 463–466 (1990).

Z. Phys.

E. Gaviola, “Ein Fluorometer. Apparat zur Messung von Fluoreszenzabklingungszeiten,” Z. Phys. 42, 853–861 (1927).
[CrossRef]

Other

K. Carlsson, J. Philip, “Theoretical investigation of the signal-to-noise ratio for different fluorescence lifetime imaging techniques,” in Optical Diagnostics of Living Cells V, D. L. Farkas, R. C. Leif, eds., Proc. SPIE4622, 70–78 (2002).
[CrossRef]

D. V. O’Connor, D. Phillips, Time-Correlated Single Photon Counting (Academic, New York, 1984).

A. Draaijer, R. Sanders, H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy,” in Handbook of Biological Confocal Microscopy, J. Pawley, ed. (Plenum, New York, 1995), pp. 491–505.

D. W. Piston, D. R. Sandison, W. W. Webb, “Time-resolved fluorescence imaging and background rejection by two-photon excitation in laser scanning microscopy,” in Time-Resolved Laser Spectroscopy in Biochemistry III, J. R. Lakowicz, ed., Proc. SPIE1640, 379–389 (1992).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Fluorescence light intensity as a function of time with excitation by a train of Dirac pulses.

Fig. 2
Fig. 2

Normalized relative rms noise F as a function of modulation frequency with lock-in detection. Low F-values represent good performance. Theoretically, F1. Curves shown are for excitation by a train of Dirac pulses and sinusoidal excitation.

Fig. 3
Fig. 3

F-value as a function of modulation frequency for square-wave excitation with different duty cycles a (a=0.0 represents Dirac pulses).

Fig. 4
Fig. 4

Same as Fig. 3, but with a background light intensity of the excitation that is 5% of that of the peak value. Note that the order of the curves with a=0.1 and a=0.2 is reversed compared with those in Fig. 3.

Fig. 5
Fig. 5

Smooth “Gauss-like” kernel. The excitation intensity function was obtained by convolving a square wave with this smoothing kernel.

Fig. 6
Fig. 6

F-value as a function of modulation frequency for excitation with Gauss-like pulses and lock-in detection. a represents the duty cycle for the square wave that was convolved with the smoothing kernel. Compare with Fig. 3.

Fig. 7
Fig. 7

Plot of u versus v from 100 Monte Carlo tests with Dirac excitation and image intensifier detection for τ=0.25 (frequency=0.04/τ*). The cluster uv1 to the right is for ϕ1=230° and ϕ2=280°, and the vertical band uv2 is for ϕ1=60° and ϕ2=260°. Since τ¯=u/v, the spread in the angles to the points gives the spread in τ¯. uv1 is obtained with the angles given in Fig. 8 below. uv2 is an unfortunate choice of ϕ1 and ϕ2.

Fig. 8
Fig. 8

Optimal phase angles for image intensifier detection as a function of modulation frequency. Both sinusoidal and Dirac pulse excitations are shown.

Fig. 9
Fig. 9

F-value as a function of frequency when an image intensifier and sinusoidal or Dirac pulse train excitation are used. The optimal angles according to Fig. 8 were used in the calculations.

Fig. 10
Fig. 10

F-value as a function of frequency with the use of demodulation for determining lifetime. Sinusoidal excitation was assumed in both cases.

Fig. 11
Fig. 11

Results from Monte Carlo simulation and the corresponding theoretical curve. The case shown is Dirac pulse excitation and lock-in detection.

Equations (78)

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

I(t)=I0 exp(-t/τ),t0.
τ=UV=μ1+σ1Y1μ2+σ2Y2,
UV=μ1μ21+κ1Y11+κ2Y2=μ1μ2(1+κ1Y1-κ2Y2-κ1κ2Y1Y2+κ22Y22+).
E{τ}=μ1μ2(1-ρκ1κ2+κ22+),
E{τ2}=μ1μ22(1+κ12+3κ22-4ρκ1κ2).
στ2=D2{τ}=μ1μ22(κ12+κ22-2ρκ1κ2-ρ2κ12κ22-κ24+2ρκ1κ23).
t=2πTt*,τ=2πTτ*.
f(t)=1τexp(-t/τ),t0,
p(t)=(e * f)(t)=exp(-t/τ)τ[1-exp(-2π/τ)],0t<2π.
E{g(t)}=02πg(t)p(t)dt.
pk=02π exp(-ikt)p(t)dt=11+ikτ,-<k<.
E{Xi}=02π sin(t+ϕi)p(t)dt=Im02π exp[i(t+ϕi)]p(t)dt=Im[exp(iϕi)p-1]=Imcos ϕi+i sin ϕi1-iτ=τ cos ϕi+sin ϕi1+τ2.
E{X1}E{X2}=τ cos ϕ1+sin ϕ1τ cos ϕ2+sin ϕ2.
τ¯=s1 sin ϕ2-s2 sin ϕ1-s1 cos ϕ2+s2 cos ϕ1.
E{U}=NE{X1 sin ϕ2-X2 sin ϕ1}=NE{sin ϕ2 sin(t+ϕ1)-sin ϕ1 sin(t+ϕ2)}=N sin(ϕ2-ϕ1)E{sin t}.
E{U}=N sin Δϕ E{sin t}=N sin Δϕ Im(p-1)=N sin Δϕτ1+τ2.
E{V}=N sin Δϕ E{cos t}=N sin Δϕ Re(p-1)=N sin Δϕ11+τ2.
ΔU=[sin(tk+ϕ1)sin ϕ2-sin(tk+ϕ2)×sin ϕ1]p(tk)Δt=sin Δϕ sin(tk)p(tk)Δt,
ΔU2=[sin(tk+ϕ1)sin ϕ2-sin(tk+ϕ2)sin ϕ1]2p(tk)Δt=sin2 Δϕ sin2(tk)p(tk)Δt.
σ12=E{U2}=N sin2 Δϕ E{sin2 t}=N2 sin2 Δϕ E{1-cos 2t}=N2 sin2(Δϕ)[1-Re(p-2)]=N2 sin2(Δϕ)1-11+4τ2,
σ22=E{V2}=N sin2 Δϕ E{cos2 t}=N2 sin2(Δϕ)1+11+4τ2,
ρσ1σ2=E{UV}=N sin2 Δϕ E{sin t cos t}=N2 sin2 Δϕ Im(p-2)=N sin2(Δϕ)τ1+4τ2.
E{τ¯}=τ1+2Nτ2(1+τ2)21+4τ2+.
στ2=D2{τ¯}=τ2(1+τ2)2(1+2τ2)N(1+4τ2).
F=Nσττ=(1+τ2)1+2τ21+4τ21/2.
ha(t)=12πa,|t|<aπ0,else.
a0=1,ak=sin kπakπa11+ikτ,k0.
E{U}=N sin Δϕ Im(a-1)=N sin(Δϕ)sin πaπaτ1+τ2,
E{U2}=N2 sin2(Δϕ)1-Resin 2πa2πa11-i2τ=N2 sin2(Δϕ)1-sin 2πa2πa11+4τ2.
hab(t)=12π[1-(1-a)(1-b)] 1,|t|<aπb,else.
bk=a(1-b)1-(1-a)(1-b)sin kπakπa11+ikτ,k0,
sc(t)=2cπ cos2 ct,|t|<π2c0,else.
c0=1,c±2c=12,
ck=4c24c2-k2sin(kπ/2c)kπ/2c,k0,±2c.
E{U2}=N2 sin2(Δϕ)×1-4c24c2-4sin(π/c)π/ca(1-b)1-(1-a)(1-b)×sin 2πa2πa11+4τ2.
e(t)=12π(1+m sin t),
2πq(t)=1+mτ 0 sin(t-x)exp(-x/τ)dx=1+mτ Im0 exp[i(t-x)-x/τ]dx=1+mτ Imexp(it)1i+1/τ=1+m1+τ2(sin t-τ cos t).
q(t)=12π[1+m cos α sin(t-α)].
q0=1,q1=-i2m cos α exp(-iα),
q-1=i2m cos α exp(iα).
E{Xi}=Im{exp(iϕi)q-1}=m cos α2 cos(ϕi+α)
s1s2=E{X1}E{X2}=cos(ϕ1+α)cos(ϕ2+α).
τ¯=tan α¯=s1 cos ϕ2-s2 cos ϕ1s1 sin ϕ2-s2 sin ϕ1.
E{U}=N{cos ϕ2 Im[exp(iϕ1)q-1]-cos ϕ1 Im[exp(iϕ2)q-1]}=N2m cos α sin Δϕ sin α.
E{V}=N2m cos α sin Δϕ cos α.
E{U2}=N(cos2 ϕ2E{X12}+cos2 ϕ1E{X22}-2 cos ϕ1 cos ϕ2E{X1X2}).
E{Xi2}=12Re[1-exp(i2ϕi)q-2]=12,
E{X1X2}=12 Re{exp[i(ϕ1-ϕ2)]-exp[i(ϕ1+ϕ2)]q-2}=12 cos(ϕ1-ϕ2).
E{U2}=E{V2}=N2 sin2 Δϕ.
E{τ¯}=τ1+2Nm21cos4 α,
στ=τ2N1m cos2 α sin α,
F=Nσττ=2m cos2 α sin α=2(1+τ2)3/2mτ.
E{X}=1+Im[exp(iϕ)p-1]=1+τ cos ϕ+sin ϕ1+τ2.
s1-s3s2-s3=τ cos ϕ1+sin ϕ1τ cos ϕ2+sin ϕ2.
τ¯=s1 sin ϕ2-s2 sin ϕ1-s3(sin ϕ2-sin ϕ1)-s1 cos ϕ2+s2 cos ϕ1+s3(cos ϕ2-cos ϕ1).
D2{U}=N[sin2 ϕ2 D2{X1}+sin2 ϕ1 D2{X2}+(sin ϕ2-sin ϕ1)2].
D2{Xi}=E{Xi2}=1+2 Im[exp(iϕi)p-1]+12 Re[1-exp(2iϕi)p-2]=32+2τ cos ϕi+sin ϕi1+τ2-12cos 2ϕi-2τ sin 2ϕi1+4τ2.
E{UV}=-N[sin ϕ2 cos ϕ2 D2{X1}+sin ϕ1 cos ϕ1D2{X2}+(cos ϕ2-cos ϕ1)×(sin ϕ2-sin ϕ1)].
E{X}=1+Im[exp(iϕ)q-1]=1+m cos α2 cos(ϕ+α).
s1-s3s2-s3=cos(ϕ1+α)cos(ϕ2+α).
τ¯=tan α¯=s1 cos ϕ2-s2 cos ϕ1-s3(cos ϕ2-cos ϕ1)s1 sin ϕ2-s2 sin ϕ1-s3(sin ϕ2-sin ϕ1).
D2{U}=N[cos2 ϕ2 D2{X1}+cos2 ϕ1 D2{X2}+(cos ϕ2-cos ϕ1)2].
D2{Xi}=E{Xi2}=1+2 Im[exp(iϕi)q-1]+12 Re[1-exp(2iϕi)q-2]=32+m cos α cos(ϕi+α).
E{UV}=N[sin ϕ2 cos ϕ2 D2{X1}+sin ϕ1 cos ϕ1D2{X2}+(cos ϕ2-cos ϕ1)(sin ϕ2-sin ϕ1)].
q(t)=12π[1+m cos α sin(t-ϕ0-α)].
U=dets1exp(iϕ1)1s2exp(iϕ2)1s3exp(iϕ3)1,
V=dets1exp(-iϕ1)exp(iϕ1)s2exp(-iϕ2)exp(iϕ2)s3exp(-iϕ3)exp(iϕ3).
E{Xi}=1+Im{exp[i(ϕi+ϕ0)]q-1}=1+m cos α2cos(ϕi+ϕ0+α).
E{U}=iNm cos α2exp[-i(α+ϕ0)][sin(ϕ3-ϕ2)+sin(ϕ1-ϕ3)+sin(ϕ2-ϕ1)],
E{V}=2iN[sin(ϕ3-ϕ2)+sin(ϕ1-ϕ3)+sin(ϕ2-ϕ1)].
|E{U}||E{V}|=m cos α4.
δ(ϕ1, ϕ2, ϕ3)=sin(ϕ3-ϕ2)+sin(ϕ1-ϕ3)+sin(ϕ2-ϕ1).
D2{V}=(E{V})2/N,D2{U}=18D2{V}.
σcos αcos α2=1Nm2 cos2 α[1+sin2 α+(1-m)2 cos2 α].
τ=tan α=(cos-2 α-1)1/2.
σττ=dτd cos αcos ατσcos αcos α=1sin2 ασcos αcos α.
F=Nσττ=1m cos α sin2 α[1+sin2 α+(1-m)2 cos2 α]1/2.
D2{U}(E{V})2/(4N),D2(V)(E{V})2/(2N),

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