Fluorescence lifetime: decay ↔ phasor

Fluorescence lifetime imaging records, at every pixel, how long molecules stay excited before they emit. That record can be read as a decay curve or as a single point on a phasor plot. This page puts the two side by side so you can see how one becomes the other.

Drag the sliders, or drag the points on the phasor plot. Everything updates as you move.

An idealized model. Two exponential components, an infinitely short laser pulse and perfect calibration. Real measurements also include the instrument response (IRF); phasor analysis removes it by calibrating against a reference dye of known lifetime.

1Two lifetimes, one phasor

Why phasors: each pixel's decay becomes one point, and mixtures follow simple geometry. A single-exponential decay always lands on the semicircle, sliding from (1, 0) toward the origin as the lifetime grows. A mix of two lifetimes lands on the straight line between them, and where it sits along that line is the fraction of photons from each: the lever rule.

Decay over one laser period
  • τ₁ component
  • τ₂ component
  • Total
  • Light from earlier pulses
Phasor plot (tick labels on the semicircle in ns)
  • τ₁ alone
  • τ₂ alone
  • Mixture
  • Reading τφ and τm
Try a sample:
Harmonic
Phasor (g, s)
Phase lifetime τφ
Modulation lifetime τm
Intensity-weighted mean
Amplitude-weighted mean
Amplitude fraction of τ₁
Light from earlier pulses

Try: move "Photons from τ₁" to 100%. The point jumps onto the semicircle and the phase and modulation lifetimes agree. Anywhere in between, the phase lifetime is the shorter of the two. Compare both with the two mean lifetimes in the table: for a mixture, "the lifetime" depends on which definition you report.

2The laser sets the ruler

Why the repetition rate matters: the phasor is computed at the laser's modulation frequency, and that frequency decides which lifetimes the semicircle spreads out. Phase changes fastest with lifetime where ωτ = 1, about 2 ns at 80 MHz. Much shorter or much longer lifetimes crowd together at the ends of the semicircle. Higher harmonics move that sweet spot to shorter lifetimes.

Long lifetimes also leave light behind when the next pulse arrives. The phasor already includes that light because it is computed over the repeating signal; a fit that assumes each decay ends inside the window does not.

Three laser periods of a single lifetime
  • Decay from each pulse
  • What the detector sees
Where lifetimes land on the semicircle (labels in ns)
  • Your lifetime
  • Reference lifetimes
  • Phase changes at least half as fast as at ωτ = 1
Harmonic
Angular frequency ω
Laser period
ωτ
Lifetime at ωτ = 1
Most sensitive range
Light carried into the next period

Try: set τ to 12 ns at 80 MHz, then lower the repetition rate to 20 MHz. The leftover light drops and the point moves back toward the middle of the arc.

3Photons, noise and background

Why photon counts matter: each pixel's phasor is estimated from a limited number of photons, so real pixels scatter around the true point. The scatter shrinks with the square root of the photon count: four times the photons, half the spread. Uncorrelated background, such as detector dark counts or room light, adds photons that carry no timing. It pulls every phasor straight toward the origin, which leaves the phase lifetime unchanged but makes the modulation lifetime longer.

This section uses the sample from section 1:

Photon histogram of one pixel
  • Photons in this pixel
  • Expected counts
Phasors of 250 simulated pixels
  • Pixels
  • Average of pixels
  • Expected
  • Expected without background
τφ expected
τφ across pixels (middle 50%)
τm expected
τm across pixels (middle 50%)
Effect of background

Try: drop to 50 photons per pixel, then add 20% background. Compare how the phase and modulation lifetimes react.

Going further