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

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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.

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References

  1. B. Valeur, M.N. Berberan-Santos, Molecular Fluorescence. Principles and Applications, 2nd edn. (Wiley-VCH, Weinheim, 2012)

    Book  Google Scholar 

  2. M.N. Berberan-Santos, B. Valeur, J. Lumin. 126, 263 (2007)

    Article  CAS  Google Scholar 

  3. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962)

    Google Scholar 

  4. D.M. Jameson, Introduction to Fluorescence (CRC Press, Boca Raton, 2014)

    Google Scholar 

  5. G. Weber, J. Phys. Chem. 85, 949 (1981)

    Article  CAS  Google Scholar 

  6. W. Feller, An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. (Wiley, New York, 1971)

    Google Scholar 

  7. D.M. Jameson, E. Gratton, R.D. Hall, Appl. Spectrosc. Rev. 20, 55 (1984)

    Article  CAS  Google Scholar 

  8. M. Berberan-Santos, J. Lumin. 50, 83 (1991)

    Article  CAS  Google Scholar 

  9. M. Itagaki, K. Watanabe, Bunseki Kagaku 43, 1143 (1994)

    Article  CAS  Google Scholar 

  10. M. Itagaki, M. Hosono, K. Watanabe, Anal. Sci. 13, 991 (1997)

    Article  CAS  Google Scholar 

  11. P.J. Verveer, P.I.H. Bastiaens, J. Microsc. 209, 1 (2003)

    Article  CAS  Google Scholar 

  12. A.H.A. Clayton, Q.S. Hanley, P.J. Verveer, J. Microsc. 213, 1 (2004)

    Article  CAS  Google Scholar 

  13. G.I. Redford, R.M. Clegg, J. Fluoresc. 15, 805 (2005)

    Article  CAS  Google Scholar 

  14. M.A. Digman, V.R. Caiolfa, M. Zamai, E. Gratton, Biophys. J. 94, L14 (2008)

    Article  CAS  Google Scholar 

  15. A.H.A. Clayton, J. Microsc. 232, 306 (2008)

    Article  CAS  Google Scholar 

  16. Y.-C. Chen, R.M. Clegg, Photosynth. Res. 102, 143 (2009)

    Article  CAS  Google Scholar 

  17. Y.-C. Chen, B.Q. Spring, C. Buranachai, G. Malachowski, R.M. Clegg, Proc. SPIE 7183, 718302 (2009)

    Article  Google Scholar 

  18. C. Stringari, A. Cinquin, O. Cinquin, M.A. Digman, P.J. Donovan, E. Gratton, Proc. Natl. Acad. Sci. USA 108, 13582 (2011)

    Article  CAS  Google Scholar 

  19. M. Stefl, N.G. James, J.A. Ross, D.M. Jameson, Anal. Biochem. 410, 62 (2011)

    Article  CAS  Google Scholar 

  20. N.G. James, J.A. Ross, M. Stefl, D.M. Jameson, Anal. Biochem. 410, 70 (2011)

    Article  CAS  Google Scholar 

  21. E. Hinde, M.A. Digman, C. Welch, K.M. Hahn, E. Gratton, Microsc. Res. Tech. 75, 271 (2012)

    Article  Google Scholar 

  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)

  23. F. Menezes, A. Fedorov, C. Baleizao, B. Valeur, M.N. Berberan-Santos, Methods Appl. Fluoresc. 1, 015002 (2013)

    Article  Google Scholar 

  24. E. Hinde, M.A. Digman, K.M. Hahn, E. Gratton, Proc. Natl. Acad. Sci. USA 110, 135 (2013)

    Article  CAS  Google Scholar 

  25. Y. Engelborghs, A.J.W.G. Visser (eds.), Fluorescence Spectroscopy and Microscopy (Humana Press, New York, 2014)

  26. M.N. Berberan-Santos, Chem. Phys. 449, 23 (2015)

    Article  CAS  Google Scholar 

  27. A. Stuart, K. Ord, Kendall’s Advanced Theory of Statistics, vol. 1, 6th edn. (Hodder Arnold, London, 1994)

    Google Scholar 

  28. F.G. Teixeira, Traité des Courbes Spéciales Remarquables Planes et Gauches, Tome II (Coimbra University Press, Coimbra, 1908)

    Google Scholar 

  29. J.D. Lawrence, A Catalogue of Special Plane Curves (Dover, Mineola, 1972)

    Google Scholar 

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Acknowledgments

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

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Correspondence to Mário N. Berberan-Santos.

Appendix: Shift property of causal functions

Appendix: Shift property of causal functions

If a given causal function \(f(t)\) is shifted to the right by \(\Delta t\), it becomes another causal function \(g(t)=f(t-\Delta \hbox {t})\). The Fourier cosine transform of \(g(t)\) is, successively,

$$\begin{aligned} G[g]=\int \limits _0^\infty {\cos (\omega t)g(t)dt=} \int \limits _0^\infty {\cos (\omega t)f(t-\Delta t)dt=} \int \limits _0^\infty {\cos (\omega t^{\prime }+\omega \Delta t)f(t^{\prime })dt^{\prime }},\nonumber \\ \end{aligned}$$
(25)

hence

$$\begin{aligned} G[g]=\cos (\omega \Delta t)G[f]-\sin (\omega \Delta t)S[f]. \end{aligned}$$
(26)

Similarly, it is obtained that

$$\begin{aligned} S[g]=\sin (\omega \Delta t)G[f]+\cos (\omega \Delta t)S[f], \end{aligned}$$
(27)

and therefore

$$\begin{aligned} \left[ {\begin{array}{l} G[g] \\ S[g] \\ \end{array}} \right] =\left[ {{\begin{array}{ll} {\cos \left( {\omega \Delta t} \right) }&{} {-\sin \left( {\omega \Delta t} \right) } \\ {\sin \left( {\omega \Delta t} \right) }&{} {\cos \left( {\omega \Delta t} \right) } \\ \end{array} }} \right] \left[ {\begin{array}{l} G[f] \\ S[f] \\ \end{array}} \right] \end{aligned}$$
(28)

A time shift \(\Delta t\) is tantamount to a counter-clockwise rotation by \(\omega \Delta t\) in the phasor space.

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Berberan-Santos, M.N. Probabilistic view of the luminescence phasor plot and description of the universal semicircle as the sum of two spiraling curves. J Math Chem 53, 1207–1219 (2015). https://doi.org/10.1007/s10910-015-0481-y

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  • DOI: https://doi.org/10.1007/s10910-015-0481-y

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