---
title: "TCSPC Fluorescence Lifetime Instrument Interactive Lab | Sandbox Physics"
description: "How can a decay curve separate fluorescence lifetime from detector timing response?"
canonical_url: "https://sandboxphysics.com/labs/tcspc-fluorescence-lifetime/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/tcspc-fluorescence-lifetime/"
last_modified: "2026-11-03"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Calibrate the clock spread before measuring the wait

**Background** After a short excitation pulse, different emitters wait different times before emitting. The detector and electronics add their own timing spread.

**Why it matters** A broad arrival-time peak can come from the specimen, the instrument, or both.

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 misconception** The width of the arrival-time peak directly equals the lifetime.

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

#### More counts can make a fitted decay wrong

**Typical misconception** A brighter sample always gives a more accurate lifetime.

**Better mental model** First-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

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

Retain every time tag, reference-cycle count, prompt calibration and background; low-rate assumptions belong with the fitted result.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How individual photons reveal a fluorescence lifetime**](https://sandboxphysics.com/stories/instruments/tcspc-fluorescence-lifetime/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)