Phasor approach refers to a method which is used for vectorial representation of sinusoidal waves like alternating 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 facilitates the analysis; 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. Importantly, the analysis is fast and provides a graphical representation of the measured curve.
Temporal phasor If we have decay curve which is represented by an exponential function with lifetime of τ:
d ( t ) = d 0 e − t / τ {\displaystyle d(t)={d_{0}{e}^{-t/\tau }}}
Then the Fourier transformation at frequency ω of d ( t ) {\displaystyle d(t)} (normalized to have area under the curve 1) is represented by the Lorentz function:
D ( ω ) = 1 1 + j ω τ = 1 1 + j ω τ 1 − j ω τ 1 − j ω τ = 1 − j ω τ 1 + ( ω τ ) 2 = 1 1 + ( ω τ ) 2 − j ω τ 1 + ( ω τ ) 2 {\displaystyle D(\omega )={\frac {1}{1+j\omega \tau }}={\frac {1}{1+j\omega \tau }}{\frac {1-j\omega \tau }{1-j\omega \tau }}={\frac {1-j\omega \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:
τ = 1 ω Im D ( ω ) Re D ( ω ) {\displaystyle \tau ={\frac {1}{\omega }}{\frac {\operatorname {Im} D(\omega )}{\operatorname {Re} D(\omega )}}}
This is a much faster approach than methods where fitting is used to estimate the lifetime.
Multi-exponential cases The semicircle represents all possible single exponential fluorescent decays. When the measured decay curve consists of a superposition of different mono-exponential decays, the phasor falls inside the semicircle depending on the fractional contributions of the components. For a bi-exponential case with lifetimes τ1 and τ2, all phasor values fall on a line connecting the phasors of τ1 and τ2 on the semicircle, and the distance from the phasor to τ1 determines the fraction α. Therefore, the phasor values of the pixels of an image with two lifetime components are distributed on a line connecting the phasors of τ1 and τ2. Fitting a line through these phasor points with slope (v) and interception (u), will give two intersections with the semicircle that determine the lifetimes τ1 and τ2:
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