Phasor approach refers to a method which is used for vectorial representation of sinusoidal waves like alternative currents and voltages or electromagnetic waves. The amplitude and the phase of the waveform is transformed into a vector where the phase is translated to the angle between the phasor vector and X axis and the amplitude is translated to vector length or magnitude. In this concept the representation and the analysis becomes very simple and the addition of two wave forms is realized by their vectorial summation.

In Fluorescence lifetime and spectral imaging, phasor can be used to visualize the spectra and decay curves. In this method the Fourier transformation of the spectrum or decay curve is calculated and the resulted complex number is plotted on a 2D plot where the X axis represents the Real component and the Y axis represents the Imaginary component. This facilitate the analysis since each spectrum and decay is transformed into a unique position on the phasor plot which depends on its spectral width or emission maximum or to its average lifetime. The most important feature of this analysis is that it is fast and it provides a graphical representation of the measured curve.

## . . . Phasor approach to fluorescence lifetime and spectral imaging . . .

If we have decay curve which is represented by an exponential function with lifetime of τ:

${displaystyle d(t)={d_{0}{e}^{-t/tau }}}$ Then the Fourier transformation at frequency ω of

${displaystyle d(t)}$ (normalized to have area under the curve 1) is represented by the Lorentz function:

${displaystyle D(omega )={frac {1}{1+jomega tau }}={frac {1}{1+jomega tau }}{frac {1-jomega tau }{1-jomega tau }}={frac {1-jomega tau }{1+(omega tau )^{2}}}={frac {1}{1+(omega tau )^{2}}}-j{frac {omega tau }{1+(omega tau )^{2}}}}$ This is a complex function and drawing the Imaginary versus real part of this function for all possible lifetimes will be a semicircle where the zero lifetime is located at (1,0) and the infinite lifetime located at (0,0). By changing the lifetime from zero to infinity the phasor point moves along a semicircle from (1,0) to (0,0). This suggest that by taking the Fourier transformation of a measured decay curve and mapping the result on the phasor plot the lifetime can be estimated from the position of the phasor on the semicircle.

Explicitly, the lifetime can be measured from the magnitude of the phasor as follow:

${displaystyle tau ={frac {1}{omega }}{frac {operatorname {Im} D(omega )}{operatorname {Re} D(omega )}}}$ This is a very fast approach compared to methods where they use fitting to estimate the lifetime.