Advertisement

Journal of Mathematical Chemistry

, Volume 53, Issue 5, pp 1207–1219 | Cite as

Probabilistic view of the luminescence phasor plot and description of the universal semicircle as the sum of two spiraling curves

  • Mário N. Berberan-Santos
Original Paper

Abstract

Luminescence decay functions describe the time dependence of the intensity of radiation emitted by electronically excited species. Decay phasor plots (plots of the Fourier sine transform vs. the Fourier cosine transform, for one or several angular frequencies) are being increasingly used in fluorescence, namely in lifetime imaging microscopy. In this work it is shown that the universal semicircle, locus of all exponential decay functions, can be viewed as the weighted sum of two spiraling phasors, one corresponding to a truncated exponential and the other to a shifted exponential. The geometric details of this recomposition are discussed. With area normalization, the decay functions form a subset in the universe of one-sided probability density functions, the same being valid for the phasor plots, which are parametric plots of the respective characteristic functions.

Keywords

Luminescence decay Relaxation function Characteristic function Fourier transform Spiral 

Mathematics Subject Classification

42A38 14H81 60E10 78A10 

Notes

Acknowledgments

This work was carried out within Projects PTDC/QUI-QUI/123162/2010 and RECI/CTM-POL/0342/2012 (FCT, Portugal).

References

  1. 1.
    B. Valeur, M.N. Berberan-Santos, Molecular Fluorescence. Principles and Applications, 2nd edn. (Wiley-VCH, Weinheim, 2012)CrossRefGoogle Scholar
  2. 2.
    M.N. Berberan-Santos, B. Valeur, J. Lumin. 126, 263 (2007)CrossRefGoogle Scholar
  3. 3.
    A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962)Google Scholar
  4. 4.
    D.M. Jameson, Introduction to Fluorescence (CRC Press, Boca Raton, 2014)Google Scholar
  5. 5.
    G. Weber, J. Phys. Chem. 85, 949 (1981)CrossRefGoogle Scholar
  6. 6.
    W. Feller, An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. (Wiley, New York, 1971)Google Scholar
  7. 7.
    D.M. Jameson, E. Gratton, R.D. Hall, Appl. Spectrosc. Rev. 20, 55 (1984)CrossRefGoogle Scholar
  8. 8.
    M. Berberan-Santos, J. Lumin. 50, 83 (1991)CrossRefGoogle Scholar
  9. 9.
    M. Itagaki, K. Watanabe, Bunseki Kagaku 43, 1143 (1994)CrossRefGoogle Scholar
  10. 10.
    M. Itagaki, M. Hosono, K. Watanabe, Anal. Sci. 13, 991 (1997)CrossRefGoogle Scholar
  11. 11.
    P.J. Verveer, P.I.H. Bastiaens, J. Microsc. 209, 1 (2003)CrossRefGoogle Scholar
  12. 12.
    A.H.A. Clayton, Q.S. Hanley, P.J. Verveer, J. Microsc. 213, 1 (2004)CrossRefGoogle Scholar
  13. 13.
    G.I. Redford, R.M. Clegg, J. Fluoresc. 15, 805 (2005)CrossRefGoogle Scholar
  14. 14.
    M.A. Digman, V.R. Caiolfa, M. Zamai, E. Gratton, Biophys. J. 94, L14 (2008)CrossRefGoogle Scholar
  15. 15.
    A.H.A. Clayton, J. Microsc. 232, 306 (2008)CrossRefGoogle Scholar
  16. 16.
    Y.-C. Chen, R.M. Clegg, Photosynth. Res. 102, 143 (2009)CrossRefGoogle Scholar
  17. 17.
    Y.-C. Chen, B.Q. Spring, C. Buranachai, G. Malachowski, R.M. Clegg, Proc. SPIE 7183, 718302 (2009)CrossRefGoogle Scholar
  18. 18.
    C. Stringari, A. Cinquin, O. Cinquin, M.A. Digman, P.J. Donovan, E. Gratton, Proc. Natl. Acad. Sci. USA 108, 13582 (2011)CrossRefGoogle Scholar
  19. 19.
    M. Stefl, N.G. James, J.A. Ross, D.M. Jameson, Anal. Biochem. 410, 62 (2011)CrossRefGoogle Scholar
  20. 20.
    N.G. James, J.A. Ross, M. Stefl, D.M. Jameson, Anal. Biochem. 410, 70 (2011)CrossRefGoogle Scholar
  21. 21.
    E. Hinde, M.A. Digman, C. Welch, K.M. Hahn, E. Gratton, Microsc. Res. Tech. 75, 271 (2012)CrossRefGoogle Scholar
  22. 22.
    M.A. Digman, E. Gratton, in Fluorescence Lifetime Spectroscopy and Imaging: Principles and Applications in Biomedical Diagnostics (L. Marcu, P.M.W. French, and D.S. Elson eds., CRC Press, Boca Raton, 2012)Google Scholar
  23. 23.
    F. Menezes, A. Fedorov, C. Baleizao, B. Valeur, M.N. Berberan-Santos, Methods Appl. Fluoresc. 1, 015002 (2013)CrossRefGoogle Scholar
  24. 24.
    E. Hinde, M.A. Digman, K.M. Hahn, E. Gratton, Proc. Natl. Acad. Sci. USA 110, 135 (2013)CrossRefGoogle Scholar
  25. 25.
    Y. Engelborghs, A.J.W.G. Visser (eds.), Fluorescence Spectroscopy and Microscopy (Humana Press, New York, 2014)Google Scholar
  26. 26.
    M.N. Berberan-Santos, Chem. Phys. 449, 23 (2015)CrossRefGoogle Scholar
  27. 27.
    A. Stuart, K. Ord, Kendall’s Advanced Theory of Statistics, vol. 1, 6th edn. (Hodder Arnold, London, 1994)Google Scholar
  28. 28.
    F.G. Teixeira, Traité des Courbes Spéciales Remarquables Planes et Gauches, Tome II (Coimbra University Press, Coimbra, 1908)Google Scholar
  29. 29.
    J.D. Lawrence, A Catalogue of Special Plane Curves (Dover, Mineola, 1972)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.CQFM - Centro de Química-Física Molecular and IN - Institute of Nanoscience and Nanotechnology, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

Personalised recommendations