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


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.


Luminescence decay Relaxation function Characteristic function Fourier transform Spiral 

Mathematics Subject Classification

42A38 14H81 60E10 78A10 



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


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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

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