lifetime op• Data kinds: counts → table
• Call: import photoncount; photoncount.lifetime_phasor(decay, bin_ps=100.0, harmonic=1, background=0.0) (or opsphoton.get("lifetime_phasor"))
Phasor (frequency-domain) representation of a decay — the fit-free view.
FLIM's standard fit-free tool. With `omega = 2*pi*harmonic/window` and bin
centres `t_k`::
g = sum(h_k * cos(omega*t_k)) / sum(h_k)
s = sum(h_k * sin(omega*t_k)) / sum(h_k)
For a single exponential of lifetime `tau` under periodic excitation the
exact analytic phasor is `g = 1/(1+(omega*tau)^2)`,
`s = omega*tau/(1+(omega*tau)^2)`, which traces the **universal
semicircle** `(g - 1/2)^2 + s^2 = 1/4 as tau` runs from 0 to infinity.
A multi-exponential decay falls strictly *inside* that circle — which is why
`semicircle_residual` is returned: it is the honest detector of the
single-exponential assumption that :func:lifetime_fit cannot give you.
Returns a dict: `g · s · modulation m = sqrt(g^2+s^2)` ·
`phase_rad · omega_per_ps · tau_phi_ps = tan(phase)/omega` ·
`tau_m_ps = sqrt(1/m^2 - 1)/omega · semicircle_residual`
`= (g-1/2)^2 + s^2 - 1/4` (0 on the circle, negative inside) ·
`total_counts. tau_phi_ps is None` — not a negative number — when
the phase is not in `(0, pi/2), and tau_m_ps is None` when the
modulation is 0 or >= 1; both mean "this is not a decaying single
exponential", which is information, not a failure.
Honest accuracy: the analytic formula is the *continuous* integral over one
excitation period, while this op sums over bins, so the two differ by the
midpoint-rule error. Measured on an exactly bin-integrated single exponential
(`tau = 2000 ps, 256 bins x 100 ps, i.e. a 25.6 ns period): g` is
0.805809 against the analytic 0.805830 (-2.0e-05), `s` is 0.395653 against
0.395561 (+9.2e-05), `tau_phi_ps` comes back as 2000.52 ps (+0.026%),
`tau_m_ps as 1999.74 ps (-0.013%) and semicircle_residual` is
+6.07e-05. Quadrupling to 1024 bins over the same window divides the residual
by exactly 16.00 (to 3.79e-06) and the `tau_phi` error by 16 — the
`O(bin^2)` midpoint behaviour, not a bias in the op.
And the reason `semicircle_residual` earns its place: the same window with
a two-component decay (equal photon budgets at 500 ps and 4000 ps) gives
a residual of -0.0924, i.e. 1500x further inside the circle than the
single-exponential round-off. :func:lifetime_fit would have returned one
confident number for that same histogram.
Raises `ValueError`: negative, non-finite, non-1-D or all-zero *decay*,
a non-positive *bin_ps*, a *harmonic* outside `[1, bins//2]` (above
Nyquist the phasor is aliased and meaningless), a negative *background*, and
a background subtraction that removes every count.
• photon_timeresolved family guide
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• photon_timeresolved — py -3.11 examples/photon_timeresolved.py
table as input)—
lifetime)*Provenance: photoncount.py — PHOTON operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.