IN22 · Time individual photons

TCSPC Fluorescence Lifetime Instrument

How can a decay curve separate fluorescence lifetime from detector timing response? Measure the timing spread of a prompt reference, then convolve that measured response with possible fluorescence decays and compare them with photon arrival times. Keep counts low: recording the first photon in a crowded cycle favors early arrivals, and detector dead time removes further events.

Time individual photons, measure a prompt response and background, then compare low-rate lifetime fits with first-photon pile-up and dead time.

Separate waiting from instrument response

Inferred lifetime $2.94\,\mathrm{ns}$

Recorded photons $3758$

Detector pulses per cycle $0.0150$

Binned prompt FWHM $0.781\,\mathrm{ns}$

Fluorescence lifetime

\tau/\mathrm{ns}

3

Excitation period

T/\mathrm{ns}

50

Potential signal photons per pulse

\mu

0.015

Delay cable length

\ell/\mathrm m

0.4

Drag the gold cable handle, then measure fresh prompt and background references.

A low-rate default record is ready.

Electronics and model

Width of each timing offset

a/\mathrm{ns}

Detector dead time

t_d/\mathrm{ns}

One tag per observed cycleBackground events per cycleExcitation cyclesSeed

The detector is nonparalyzable and acts before the per-cycle gate. The low-rate fit requires fewer than 0.025 detector pulses per cycle and less than 2.5 percent busy time; these are teaching thresholds.

A flat single-exponential specimen, ideal periodic trigger, two uniform timing offsets and stationary background are assumed. Prompt reference at the detector wavelength shares the achromatic optical path. Afterpulsing, dispersion, bleaching and multiple lifetimes are omitted.

The measured IRF is uniform within each time bin. Its finite count noise and background uncertainty are not propagated into a certified lifetime interval. Selected simulated journeys skip empty gaps and stretch each path leg separately.

Model assumptions

A periodic ideal reference clock excites a single-exponential ensemble. Source pulse timing and detector jitter are two independent uniform offsets; this teaching response is calibrated from separate prompt-scatter timestamps. The prompt standard uses the detection wavelength and the same ideal achromatic path. Nonparalyzable detector dead time acts before an optional first-tag-per-observed-cycle gate. Recorded pulse IDs identify the preceding clock pulse, not the hidden excitation event. A periodic convolution of the measured, binned response and a single exponential predicts counts; separate blocked-source counts estimate stationary background. The fit is valid only below the declared detector-rate and busy-fraction limits. Prompt finite counts, within-bin approximation and background uncertainty limit precision; no certified confidence interval is supplied. Selected simulated photon journeys skip empty gaps and slow each optical or electrical leg separately. Reabsorption, afterpulsing, photobleaching, dispersion, reference-clock jitter and multiple decay components are omitted.

Each photon contributes one observed delay

Observed delays and measured prompt response

$t_{\mathrm{phase}}/\mathrm{ns}\quad;\quad \log_{10}(1+N_{\mathrm{tags}})$

Green marks sample counts, violet the background-subtracted prompt normalized to signal area, and gold dashes the low-rate candidate.

Empty cycles and detector busy time

Fit profile, timing convention and saved records

Counts and busy intervals are retained with the full number of reference cycles. Losing late photons does not turn those missing cycles into measured fast decays.

$\tau/\mathrm{ns}\quad;\quad \log_{10}(1+D-D_{\min})$

The periodic reference clock starts at zero. Each saved pulse ID names the immediately preceding reference pulse; old excitations may contribute in later cycles. Time tags are integer ten-picosecond ticks.

Frozen records
Record	Photons	Calibration	Inference

CSV retains every sample, prompt and background pulse ID and absolute timestamp, clock descriptors, hardware counters and processing version. The separate generated reference is labelled in JSON and excluded from the inverse.

Start with the essentials

Focus question: Which part of a measured delay distribution belongs to the specimen lifetime?
One-sentence intuition: Measure a prompt response independently, then compare its periodic convolution with possible fluorescence decays against saved photon timestamps.

Core mathematical model

A single exponential lifetime

p(t)=\frac1\tau e^{-t/\tau},\qquad t\ge0

Lifetime is a mean waiting time, not a fixed delay for each photon. The selected model has one lifetime and ideal independent emitters.

A measured timing response broadens a decay

h(t)=\int_{-\infty}^{\infty}r(u)\,p(t-u)\,\mathrm du

The measured prompt response r is combined with each possible lifetime. Repeated excitation adds contributions from earlier cycles; the fit uses the periodic version of this convolution.

The first detected photon is selected unevenly

p_{\mathrm{first}}(t)=\lambda(t)\exp\!\left[-\int_0^t\lambda(u)\,\mathrm du\right]

This unconditional density includes the probability that no earlier photon arrived. Its missing area represents empty cycles. A crowded cycle favors earlier photons before any detector dead-time losses are added.

Common difficulties

A wide response is not a long lifetime

Typical misconceptionThe width of the arrival-time peak directly equals the lifetime.

Better mental modelThe peak includes source width and detector/electronic response. A prompt reference measures these contributions.

More counts can make a fitted decay wrong

Typical misconceptionA brighter sample always gives a more accurate lifetime.

Better mental modelFirst-photon selection favors early arrivals and detector busy time hides other photons. The low-rate fit is withheld outside its stated limits.

Run the experiment

01
Watch a photon journey
Pause the selected journey and identify excitation, waiting, fluorescence collection and electrical timing.
What to observe: The source, specimen and collector form one causal path. Empty gaps are skipped and each leg is slowed separately.
02
Broaden the instrument response
Choose the fast-decay preset and inspect the independently measured prompt curve.
What to observe: A broad prompt response can conceal a much shorter specimen decay. The declared resolution criterion withholds that short lifetime.
03
Cross the next clock pulse
Choose the folded preset and compare the observed pulse IDs with the sample delay histogram.
What to observe: A photon can be registered after a later reference pulse. The inferred model adds earlier-cycle fluorescence.
04
Crowd the detector
Compare the first-photon and detector-busy presets.
What to observe: A per-cycle gate rejects later detector pulses. Nonparalyzable detector dead time removes arrivals across cycles and acts first.
05
Change the actual cable
Drag the gold cable handle and then measure fresh references.
What to observe: The cable centreline length changes the electrical delay. Old saved records retain their own matching timing calibration.

TCSPC Fluorescence Lifetime Instrument

Each photon contributes one observed delay

Observed delays and measured prompt response

Empty cycles and detector busy time

Calibrate the clock spread before measuring the wait

Core mathematical model

A single exponential lifetime

A measured timing response broadens a decay

The first detected photon is selected unevenly

Common difficulties

A wide response is not a long lifetime

More counts can make a fitted decay wrong

Run the experiment

Watch a photon journey

Broaden the instrument response

Cross the next clock pulse

Crowd the detector

Change the actual cable

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