Time-Resolved Spectroscopy
Time-correlated single photon counting: one- or two-exponential decays convolved with a Gaussian IRF in closed form, the decay left over from earlier pulses at the repetition rate, the centroid shift and the laser's transform-limited bandwidth.
Pulse Period 1/f
12.50 ns
decay left at the next pulse e^(−T/τ) = 0.00193; 0.193 % of the photons arrive after it
Centroid Shift ⟨t⟩ − t₀
2.000 ns
= intensity-weighted ⟨τ⟩ (one pulse, full decay); amplitude-weighted 2.000 ns
Measured Peak After t₀
183 ps
IRF σ = 84.9 ps; τ₁/FWHM = 10.0
Transform-Limited Bandwidth
4.413 THz
9.421 nm at 800.0 nm (Gaussian, Δν·Δt = 0.4413)
IRF: Gaussian R(t) of FWHM w centred at t₀ (σ = w/2.3548). Measured: I(t) = R ⊗ Σ αᵢ e^(−t/τᵢ), computed in closed form as the exponentially modified Gaussian ½ e^(σ²/2τ² − (t−t₀)/τ) erfc((σ²/τ − (t − t₀))/(σ√2)).
At repetition rate f the decay of earlier pulses adds: I_f(t) = Σₙ I(t + n/f). The centroid of a complete single-pulse decay is shifted from the IRF centre by the intensity-weighted lifetime Σαᵢτᵢ²/Σαᵢτᵢ (method of moments).
The IRF combines laser pulse, detector jitter and electronics; the curves are noiseless, without pile-up of photon counts (count rate ≪ f assumed) and without background.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Gaussian IRF and exponential decays, convolved exactly; noiseless histogram without background or count-rate pile-up.
- Grushka E. (1972). Characterization of exponentially modified Gaussian peaks in chromatography. Anal. Chem. 44, 1733–1738. doi:10.1021/ac60319a011
- Becker W. (2005). Advanced Time-Correlated Single Photon Counting Techniques. Springer. doi:10.1007/3-540-28882-1
- Lakowicz J. R. (2006). Principles of Fluorescence Spectroscopy, 3rd ed., ch. 4. Springer.