Abstract

Frequency-domain fluorescence lifetime imaging microscopy (FD-FLIM) is a fast and accurate way of measuring fluorescence lifetimes in widefield microscopy. However, the resolution of multiple exponential fluorescence decays has remained beyond the reach of most practical FD-FLIM systems. In this paper we describe the implementation of FD-FLIM using a 40MHz pulse train derived from a supercontinuum source for excitation. The technique, which we term multi-harmonic FLIM (mhFLIM), makes it possible to accurately resolve biexponential decays of fluorophores without any a priori information. The system’s performance is demonstrated using a mixture of spectrally similar dyes of known composition and also on a multiply-labeled biological sample. The results are compared to those obtained from time correlated single photon counting (TCSPC) microscopy and a good level of agreement is achieved. We also demonstrate the first practical application of an algorithm derived by G. Weber [1] for analysing mhFLIM data. Because it does not require nonlinear minimisation, it offers potential for realtime analysis during acquisition.

© 2009 Optical Society of America

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  1. G. Weber, "Resolution of the fluorescence lifetimes in a heterogeneous system by phase and modulation measurements," J. Phys. Chem. 85, 949-953 (1981).
    [CrossRef]
  2. S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
    [CrossRef] [PubMed]
  3. H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
    [CrossRef]
  4. X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
    [CrossRef]
  5. X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
    [CrossRef]
  6. A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
    [CrossRef] [PubMed]
  7. M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
    [CrossRef]
  8. P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
    [CrossRef] [PubMed]
  9. W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
    [CrossRef]
  10. E. B. van Munster and T. W. J. Gadella, "Fluorescence Lifetime Imaging Microscopy (FLIM)," Adv. Biochem. Engin. / Biotechnology 95, 143-175 (2005).
  11. T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
    [CrossRef]
  12. A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
    [CrossRef] [PubMed]
  13. Q. S. Hanley and A. H. A. Clayton, "AB-plot assisted determination of fluorophore mixtures in a fluorescence lifetime microscope using spectra or quenchers," J. Microscopy 218, 62-67 (2005).
    [CrossRef]
  14. E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
    [CrossRef]
  15. A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
    [CrossRef]
  16. A. D. Elder, S. C. Schlachter, and C. F. Kaminski, "Theoretical investigation of the photon efficiency in frequency-domain FLIM," J. Opt. Soc. Am. A 25, 452-462 (2008).
    [CrossRef]
  17. A. Esposito, H. C. Gerritsen, and F. S. Wouters, "Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed," J. Opt. Soc. Am. A 24, 3261-3273 (2007).
    [CrossRef]
  18. J. Philip and K. Carlsson, "Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging," J. Opt. Soc. Am. A 20, 368-379 (2003).
    [CrossRef]
  19. A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
    [CrossRef]
  20. J. R. Lakowicz, Principles of fluorescence spectroscopy (Kluwer Academic, New York, 1999).
  21. T. S. Forde and Q. S. Hanley, "Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer," Appl. Spectro. 60, 1442-1452 (2006).
    [CrossRef]
  22. A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
    [CrossRef]
  23. M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
    [CrossRef] [PubMed]
  24. A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
    [CrossRef] [PubMed]
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  26. E. B. van Munster and T. W. J. Gadella, "Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order," Cytometry Part A 58A, 185-194 (2004).
    [CrossRef]
  27. J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
    [CrossRef]
  28. E. B. van Munster and T. W. J. Gadella, "phiFLIM: a new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy," J Microscopy 213, 29-38 (2004).
    [CrossRef]

2008 (3)

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

A. D. Elder, S. C. Schlachter, and C. F. Kaminski, "Theoretical investigation of the photon efficiency in frequency-domain FLIM," J. Opt. Soc. Am. A 25, 452-462 (2008).
[CrossRef]

2007 (4)

A. Esposito, H. C. Gerritsen, and F. S. Wouters, "Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed," J. Opt. Soc. Am. A 24, 3261-3273 (2007).
[CrossRef]

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

2006 (4)

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

T. S. Forde and Q. S. Hanley, "Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer," Appl. Spectro. 60, 1442-1452 (2006).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

2005 (3)

E. B. van Munster and T. W. J. Gadella, "Fluorescence Lifetime Imaging Microscopy (FLIM)," Adv. Biochem. Engin. / Biotechnology 95, 143-175 (2005).

A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
[CrossRef] [PubMed]

Q. S. Hanley and A. H. A. Clayton, "AB-plot assisted determination of fluorophore mixtures in a fluorescence lifetime microscope using spectra or quenchers," J. Microscopy 218, 62-67 (2005).
[CrossRef]

2004 (3)

E. B. van Munster and T. W. J. Gadella, "Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order," Cytometry Part A 58A, 185-194 (2004).
[CrossRef]

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

E. B. van Munster and T. W. J. Gadella, "phiFLIM: a new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy," J Microscopy 213, 29-38 (2004).
[CrossRef]

2003 (3)

J. Philip and K. Carlsson, "Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging," J. Opt. Soc. Am. A 20, 368-379 (2003).
[CrossRef]

H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
[CrossRef]

A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
[CrossRef]

2002 (1)

M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
[CrossRef]

2000 (2)

P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
[CrossRef] [PubMed]

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
[CrossRef]

1993 (1)

T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
[CrossRef]

1984 (1)

E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
[CrossRef]

1981 (1)

G. Weber, "Resolution of the fluorescence lifetimes in a heterogeneous system by phase and modulation measurements," J. Phys. Chem. 85, 949-953 (1981).
[CrossRef]

Bannister, L. H.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

Bastiaens, P. I. H.

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
[CrossRef]

P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
[CrossRef] [PubMed]

Becker, W.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Benndorf, K.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Bergmann, A.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Biskup, C.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Brennan, C. M.

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Caiolfa, V. R.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

Carlsson, K.

Clayton, A. H. A.

Q. S. Hanley and A. H. A. Clayton, "AB-plot assisted determination of fluorophore mixtures in a fluorescence lifetime microscope using spectra or quenchers," J. Microscopy 218, 62-67 (2005).
[CrossRef]

A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
[CrossRef]

Clegg, R. M.

T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
[CrossRef]

Dai, X.

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

Dai, X. W.

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

Day, R. N.

M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
[CrossRef]

Digman, M. A.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

Eccleston, M. E.

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

Elangovan, M.

M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
[CrossRef]

Elder, A. D.

A. D. Elder, S. C. Schlachter, and C. F. Kaminski, "Theoretical investigation of the photon efficiency in frequency-domain FLIM," J. Opt. Soc. Am. A 25, 452-462 (2008).
[CrossRef]

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Esposito, A.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

A. Esposito, H. C. Gerritsen, and F. S. Wouters, "Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed," J. Opt. Soc. Am. A 24, 3261-3273 (2007).
[CrossRef]

A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
[CrossRef] [PubMed]

Fisher, A. C.

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Forde, T. S.

T. S. Forde and Q. S. Hanley, "Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer," Appl. Spectro. 60, 1442-1452 (2006).
[CrossRef]

Frank, J. H.

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Gadella, T. W. J.

E. B. van Munster and T. W. J. Gadella, "Fluorescence Lifetime Imaging Microscopy (FLIM)," Adv. Biochem. Engin. / Biotechnology 95, 143-175 (2005).

E. B. van Munster and T. W. J. Gadella, "phiFLIM: a new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy," J Microscopy 213, 29-38 (2004).
[CrossRef]

E. B. van Munster and T. W. J. Gadella, "Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order," Cytometry Part A 58A, 185-194 (2004).
[CrossRef]

T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
[CrossRef]

Gerritsen, H.

A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
[CrossRef] [PubMed]

Gerritsen, H. C.

Gratton, E.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
[CrossRef]

Hall, R. D.

E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
[CrossRef]

Hanley, Q. S.

T. S. Forde and Q. S. Hanley, "Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer," Appl. Spectro. 60, 1442-1452 (2006).
[CrossRef]

Q. S. Hanley and A. H. A. Clayton, "AB-plot assisted determination of fluorophore mixtures in a fluorescence lifetime microscope using spectra or quenchers," J. Microscopy 218, 62-67 (2005).
[CrossRef]

A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
[CrossRef]

Herman, P.

H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
[CrossRef]

Hink, M. A.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Jameson, D. M.

E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
[CrossRef]

Jeyasekharan, A. D.

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

Jovin, T. M.

T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
[CrossRef]

Kaminski, C. F.

A. D. Elder, S. C. Schlachter, and C. F. Kaminski, "Theoretical investigation of the photon efficiency in frequency-domain FLIM," J. Opt. Soc. Am. A 25, 452-462 (2008).
[CrossRef]

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Konig, K.

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Lakowicz, J. R.

H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
[CrossRef]

Lew, V. L.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

Lin, H. J.

H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
[CrossRef]

Matthews, S. M.

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Mauritz, J. M. A.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

Periasamy, A.

M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
[CrossRef]

Philip, J.

Schlachter, S.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

Schlachter, S. C.

Slater, N. K. H.

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

Squire, A.

P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
[CrossRef] [PubMed]

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
[CrossRef]

Swartling, J.

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Tiffert, T.

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

van Munster, E. B.

E. B. van Munster and T. W. J. Gadella, "Fluorescence Lifetime Imaging Microscopy (FLIM)," Adv. Biochem. Engin. / Biotechnology 95, 143-175 (2005).

E. B. van Munster and T. W. J. Gadella, "phiFLIM: a new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy," J Microscopy 213, 29-38 (2004).
[CrossRef]

E. B. van Munster and T. W. J. Gadella, "Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order," Cytometry Part A 58A, 185-194 (2004).
[CrossRef]

Venkitaraman, A. R.

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

Verveer, P. J.

A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
[CrossRef]

P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
[CrossRef] [PubMed]

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
[CrossRef]

Weber, G.

G. Weber, "Resolution of the fluorescence lifetimes in a heterogeneous system by phase and modulation measurements," J. Phys. Chem. 85, 949-953 (1981).
[CrossRef]

Wouters, F.

A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
[CrossRef] [PubMed]

Wouters, F. S.

Yue, Z. L.

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

Yunus, K.

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

A. D. Elder, S. M. Matthews, J. Swartling, K. Yunus, J. H. Frank, C. M. Brennan, A. C. Fisher, and C. F. Kaminski, "The application of frequency-domain Fluorescence Lifetime Imaging Microscopy as a quantitative analytical tool for microfluidic devices," Opt. Express 14, 5456-5467 (2006).
[CrossRef] [PubMed]

Zamai, M.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

Adv. Biochem. Engin. / Biotechnology (1)

E. B. van Munster and T. W. J. Gadella, "Fluorescence Lifetime Imaging Microscopy (FLIM)," Adv. Biochem. Engin. / Biotechnology 95, 143-175 (2005).

Anal. Chem. (1)

S. M. Matthews, A. D. Elder, K. Yunus, C. F. Kaminski, C. M. Brennan, and A. C. Fisher, "Quantitative kinetic analysis in a microfluidic device using frequency-domain fluorescence lifetime imaging," Anal. Chem. 79, 4101-4109 (2007).
[CrossRef] [PubMed]

Ann. Rev.Biophys. Bioengin. (1)

E. Gratton, D. M. Jameson, and R. D. Hall, "Multifrequency phase and modulation fluorometry," Ann. Rev.Biophys. Bioengin. 13, 105-124 (1984).
[CrossRef]

Appl. Spectro. (1)

T. S. Forde and Q. S. Hanley, "Spectrally resolved frequency domain analysis of multi-fluorophore systems undergoing energy transfer," Appl. Spectro. 60, 1442-1452 (2006).
[CrossRef]

Biophys. Chem. (1)

T. W. J. Gadella, T. M. Jovin, and R. M. Clegg, "Fluorescence lifetime imaging microscopy (FLIM): Spatial resolution of microstructures on the nanosecond time scale," Biophys. Chem. 48, 221-239 (1993).
[CrossRef]

Biophys. J. (3)

A. Esposito, H. Gerritsen, and F. Wouters, "Fluorescence lifetime heterogeneity in the frequency domain by lifetime moments analysis," Biophys. J. 89, 4286-4299 (2005).
[CrossRef] [PubMed]

P. J. Verveer, A. Squire, and P. I. H. Bastiaens, "Global Analysis of fluorescence lifetime imaging microscopy data," Biophys. J. 78, 2127-2137 (2000).
[CrossRef] [PubMed]

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, "The phasor approach to fluorescence lifetime imaging analysis," Biophys. J. 94, L14-L16 (2007).
[CrossRef] [PubMed]

Cytometry Part A (2)

E. B. van Munster and T. W. J. Gadella, "Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order," Cytometry Part A 58A, 185-194 (2004).
[CrossRef]

H. J. Lin, P. Herman, and J. R. Lakowicz, "Fluorescence lifetime-resolved pH imaging of living cells," Cytometry Part A 52A, 77-89 (2003).
[CrossRef]

J Microscopy (2)

J. H. Frank, A. D. Elder, J. Swartling, A. R. Venkitaraman, A. D. Jeyasekharan, and C. F. Kaminski, "A white light confocal microscope for spectrally resolved multidimensional imaging," J Microscopy 227, 203-215 (2007).
[CrossRef]

E. B. van Munster and T. W. J. Gadella, "phiFLIM: a new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy," J Microscopy 213, 29-38 (2004).
[CrossRef]

J. Microscopy (5)

A. H. A. Clayton, Q. S. Hanley, and P. J. Verveer, "Graphical representation and multicomponent analysis of single frequency fluorescence lifetime imaging microscopy data," J. Microscopy 213, 1-5 (2003).
[CrossRef]

M. Elangovan, R. N. Day, and A. Periasamy, "Nanosecond fluorescence resonance energy transfer-fluorescence lifetime imaging microscopy to localize the protein interactions in a living cell," J. Microscopy 205, 3-14 (2002).
[CrossRef]

Q. S. Hanley and A. H. A. Clayton, "AB-plot assisted determination of fluorophore mixtures in a fluorescence lifetime microscope using spectra or quenchers," J. Microscopy 218, 62-67 (2005).
[CrossRef]

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, "Multiple frequency fluorescence lifetime imaging microscopy," J. Microscopy 197, 136-149 (2000).
[CrossRef]

A. D. Elder, J. H. Frank, J. Swartling, X. Dai, and C. F. Kaminski, "Calibration of a wide-field frequency-domain fluorescence lifetime microscopy system using light emitting diodes as light sources," J. Microscopy 224, 166-180 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Phys. Chem. (1)

G. Weber, "Resolution of the fluorescence lifetimes in a heterogeneous system by phase and modulation measurements," J. Phys. Chem. 85, 949-953 (1981).
[CrossRef]

Micro. Res. Tech. (1)

W. Becker, A. Bergmann, M. A. Hink, K. Konig, K. Benndorf, and C. Biskup, "Fluorescence lifetime imaging by time-correlated single photon counting," Micro. Res. Tech. 63, 58-66 (2004).
[CrossRef]

Nanomedicine: Nanotech. Biology Med. (1)

X. W. Dai, Z. L. Yue, M. E. Eccleston, J. Swartling, N. K. H. Slater, and C. F. Kaminski, "Fluorescence intensity and lifetime imaging of free and micellar-encapsulated doxorubicin in living cells," Nanomedicine: Nanotech. Biology Med. 4, 49-56 (2008).
[CrossRef]

Opt. Express (1)

PLoS ONE (1)

A. Esposito, T. Tiffert, J. M. A. Mauritz, S. Schlachter, L. H. Bannister, C. F. Kaminski, and V. L. Lew, "FRET Imaging of Hemoglobin Concentration in Plasmodium falciparum-Infected Red Cells," PLoS ONE 3, e3780 (2008).
[CrossRef] [PubMed]

Polymer (1)

X. W. Dai, M. E. Eccleston, Z. L. Yue, N. K. H. Slater, and C. F. Kaminski, "A spectroscopic study of the self-association and inter-molecular aggregation behaviour of pH-responsive poly(L-lysine iso-phthalamide)," Polymer 47, 2689-2698 (2006).
[CrossRef]

Other (2)

J. R. Lakowicz, Principles of fluorescence spectroscopy (Kluwer Academic, New York, 1999).

Fianium Inc, SC450 datasheet, http://www.fianium.com.

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Figures (9)

Fig. 1.
Fig. 1.

AB plot showing data for three distinct measurements of a (simulated) biexponential decay sampled at one, three and five times the fundamental frequency of 40 MHz (green circles). Single exponential lifetime components lie on the red semicircle. Numbers on the single-exponential semi-circle indicate decay time in nanoseconds for measurements at the fundamental frequency. See text for further details.

Fig. 2.
Fig. 2.

Overview of the mhFLIM instrument. MCP = Multi-Channel Plate, CCD = charge coupled device array, PSU = power supply unit, AOTF = Acousto-Optic Tunable Filter.

Fig. 3.
Fig. 3.

(A) DC response of image intensifier gain vs. photocathode voltage. (B) (simulation) The sinusoid shown in blue applied to the measured photocathode DC response (shown in (A)) produces the image intensifier gain profile show in green.

Fig. 4.
Fig. 4.

Fourier transform of a measured ICCD gain waveform.

Fig. 5.
Fig. 5.

Phase variation across the ICCD at the fundamental frequency of ν0 = 40MHz (A), the third harmonic of 3ν0 = 120MHz (B) and the 5ν0 = 200MHz. (D) The mean phase delay versus frequency; error bars indicate the standard deviation. The data were obtained from a calibration with Rose Bengal dye.

Fig. 6.
Fig. 6.

Phasor plots from mhFLIM measurements at 1ν0, 3ν0, and 5ν0 for (A) Rhodamine 6G in ethanol. (B) Rose Bengal in ethanol. (C) Rhodamine 6G and Rose Bengal mixture in ethanol. (D) The same dye mixture shown with a biexponential estimate from Eq. (6). The numbers along the semicircle indicate the single-exponential lifetimes for measurements at the fundamental frequency.

Fig. 7.
Fig. 7.

Peak-normalized histograms showing lifetime components of Rhodamine 6G and Rose Bengal mixture in ethanol using TCSPC (red), mhFLIM with global analysis (blue) and mhFLIM with Weber algorithm (green)

Fig. 8.
Fig. 8.

AB plot showing mhFLIM data from SH-SY5 cells labeled with Alexa 555 and Alexa 546 dyes.

Fig. 9.
Fig. 9.

(A) Lifetime components derived from TCSPC, mhFLIM processed with GA algorithm and mhFLIM processed with Weber algorithm. The mean lifetime, τm is derived from τm = α 1 τ 1 + (1 - α 1)τ 2. TCSPC images are approximately 140 μm × 140 μm and mhFLIM images are 100 μm × 100μm(B) Peak-normalized histograms of the data shown in (A). TCSPC data is shown in red, mhFLIM processed with GA algorithm in blue and mhFLIM processed with the Weber algorithm in green.

Tables (2)

Tables Icon

Table 1. Processing times for a 256×256 pixel image.

Tables Icon

Table 2. Lifetimes of pure dye samples calculated using different techniques (ns).

Equations (11)

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

F = N σ τ / τ
A n = m n sin θ n B n = m n cos θ n
A 2 + ( B 1 / 2 ) 2 = 1 / 4
A n ( α 1 , τ 1 , τ 2 ) = α 1 n ω 0 τ 1 1 + ( n ω 0 τ 1 ) 2 + ( 1 α 1 ) n ω 0 τ 2 1 + ( n ω 0 τ 2 ) 2
B n ( α 1 , τ 1 , τ 2 ) = α 1 1 + ( n ω 0 τ 1 ) 2 + ( 1 α 1 ) 1 + ( n ω 0 τ 2 ) 2
C ( p ) = n = 1 N { [ A n ( α 1 , p , τ 1 , p , τ 2 , p ) A ̂ n ( p ) ] 2 w n + [ B n ( α 1 , p , τ 1 , p , τ 2 , p ) B ̂ n ( p ) ] 2 w n }
M 0 = h 1 2 B 1 h 2 2 B 2 M 1 = h 1 A 1 h 2 A 2
M 2 = B 1 + B 2 M 3 = A 1 / h 1 + A 2 / h 2
θ 1 = ( M 3 M 0 M 2 M 1 ) / ( M 2 M 0 M 1 ) θ 2 = ( M 3 M 1 M 2 ) / ( M 2 M 0 M 1 )
τ 1 , τ 2 = [ θ 1 / 2 ± ( θ 1 2 / 4 θ 2 ) 1 / 2 ] / ω
α 1 = 1 B 1 ( 1 + h 1 2 τ 1 2 ) 1 ( 1 + h 1 2 τ 2 2 ) 1 ( 1 + h 1 2 τ 1 2 ) 1

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