KAHIBARO
Table of Contents
Measuring very short times
In a time of flight detector, the key idea is simple. A particle travels from one place to another, and the detector measures how long that travel takes. The chapter on time of flight explains the overall method. Here we focus only on the timing measurement itself, meaning how the detector determines the arrival time of a signal as accurately as possible.
The difficulty is that particles move very fast, often close to the speed of light. This means the time differences we want to measure are extremely small, commonly in nanoseconds, picoseconds, or even smaller ranges. A timing system must therefore decide very precisely when a signal starts and when it stops.
Start and stop times
A timing measurement usually needs two time markers. One is a start time, and the other is a stop time. The measured flight time is
$$
t_{\text{flight}} = t_{\text{stop}} - t_{\text{start}}.
$$
The start signal may come from the accelerator, from a trigger detector, or from another detector layer. The stop signal is often produced when the particle reaches the time of flight detector itself.
If both start and stop have uncertainties, then the total uncertainty in the measured time depends on both. If the uncertainties are independent, a common relation is
$$
\sigma_t = \sqrt{\sigma_{\text{start}}^2 + \sigma_{\text{stop}}^2}.
$$
This shows that improving only one side is not enough if the other side remains poor.
Important rule:
The measured flight time is the difference of two time stamps,
$$
t_{\text{flight}} = t_{\text{stop}} - t_{\text{start}}.
$$
If the start and stop uncertainties are independent, the timing resolution combines as
$$
\sigma_t = \sqrt{\sigma_{\text{start}}^2 + \sigma_{\text{stop}}^2}.
$$
What defines the arrival time
A detector does not directly measure the particle's arrival time. It measures an electrical pulse created by the detector material and electronics. The timing system must choose a specific point on that pulse and call it the time of arrival.
Different choices are possible. A common simple method is a threshold crossing. In this method, the time stamp is taken when the pulse first rises above a chosen voltage level. If the threshold is fixed, the method is called leading edge timing.
This is easy to implement, but it has an important problem. A larger pulse crosses the threshold earlier than a smaller pulse, even if both were produced at the same true time. This effect is called time walk.
Time walk
Suppose two pulses begin at the same actual instant, but one pulse is taller. The taller one usually reaches the threshold sooner. Then the detector reports different times for signals that should have matched. This creates a systematic timing error.
To reduce time walk, one may use pulse height corrections or more advanced timing methods such as constant fraction discrimination. In constant fraction timing, the time is defined when the signal reaches a fixed fraction of its own maximum amplitude, not a fixed absolute threshold. This makes the measured time less sensitive to pulse size.
Important statement:
With a fixed threshold, different pulse heights can produce different measured times. This effect is called time walk.
A good timing system must correct or reduce time walk.
Rise time and pulse shape
A fast timing measurement depends strongly on pulse shape. If the signal rises steeply, then a small voltage uncertainty produces only a small time uncertainty. If the signal rises slowly, the same voltage uncertainty causes a larger time error.
A useful idea is that timing uncertainty becomes smaller when the slope of the rising edge is larger. In a simplified form,
$$
\sigma_t \approx \frac{\sigma_V}{dV/dt},
$$
where $\sigma_V$ is the voltage noise and $dV/dt$ is the pulse slope at the timing point.
This relation explains why fast detectors and fast electronics are valuable. They produce sharp rising edges and therefore better timing.
Electronic noise and jitter
Real signals are never perfectly clean. Random fluctuations from the electronics and the detector alter the exact shape of the pulse. This causes jitter, meaning random variation in the measured time from event to event.
Noise may come from the sensor, the amplifier, the cable, or the digitizer. If the pulse is small or noisy, the crossing time becomes uncertain. Better signal to noise ratio usually improves timing.
A rough summary is shown below.
| Factor | Effect on timing |
|---|---|
| Larger signal amplitude | Usually improves timing |
| Faster rise time | Improves timing |
| More electronic noise | Worsens timing |
| Threshold too high or too low | Can worsen timing |
| Pulse shape variation | Worsens timing |
Time resolution
The quality of a timing measurement is often described by its time resolution. This tells us how much the measured times spread around the true value. If repeated measurements produce a narrow distribution, the timing is precise. If the distribution is wide, the timing is poor.
Time resolution is commonly expressed by the standard deviation $\sigma_t$ of the time distribution. For very good time of flight systems, this may be tens of picoseconds.
If several detector channels are combined, the effective resolution can improve. For example, if $N$ independent measurements of equal resolution are averaged, the result can improve approximately as
$$
\sigma_{\text{avg}} = \frac{\sigma}{\sqrt{N}}.
$$
This is one reason why some detector systems use multiple timing layers.
Important formula:
For independent repeated measurements of equal precision, averaging improves the resolution approximately as
$$
\sigma_{\text{avg}} = \frac{\sigma}{\sqrt{N}}.
$$
Calibration of timing
A raw time measurement is not yet a final physical time. Different channels may have different cable lengths, different electronic delays, and different response speeds. Calibration is needed so that all channels share a common time reference.
Typical timing calibration corrects for fixed offsets such as
$$
t_{\text{corrected}} = t_{\text{measured}} - t_{\text{offset}}.
$$
There may also be corrections for pulse amplitude, position along a detector, or signal propagation delay inside the detector element.
Without calibration, even a detector with good intrinsic speed can give poor practical results.
Propagation delay inside the detector
Sometimes the particle creates a signal at one point in a detector bar or strip, and the electrical pulse then travels to the readout electronics. This internal travel time adds to the measured time.
If the signal propagation speed in the detector is $v_{\text{sig}}$ and the distance to the sensor is $x$, then the propagation contribution is
$$
t_{\text{prop}} = \frac{x}{v_{\text{sig}}}.
$$
So the observed time may be written as
$$
t_{\text{measured}} = t_{\text{arrival}} + t_{\text{prop}} + t_{\text{electronics}}.
$$
This is important because two particles arriving at the same true time but at different positions may give different readout times unless the position effect is corrected.
If both ends of a detector bar are read out, the average of the two times can reduce position dependence, while the difference of the two times can help determine the hit position.
Digitization and time extraction
Modern systems often record the pulse digitally. Instead of only checking whether the signal crosses a threshold, the electronics sample the pulse shape in time. A computer can then reconstruct the arrival time more accurately.
There are two common electronic approaches. A time to digital converter measures time intervals directly. A waveform digitizer records many voltage samples and extracts time by fitting or interpolation. The second method can be more powerful because it uses more information from the pulse shape.
Coincidence timing
In many experiments, timing measurements are made in coincidence. This means two detectors are required to register signals within a short allowed time window. Coincidence helps identify events that belong together and reject random background.
If the coincidence window is too wide, many unrelated events are accepted. If it is too narrow, true events may be lost. Good timing resolution allows a narrow coincidence window, which improves event selection.
Practical limits
Even the best timing system faces several physical and technical limits. The detector material may produce signals with natural statistical fluctuations. The electronics may add noise. Cable lengths may drift slightly with temperature. Synchronization between channels may change over time. For this reason, high precision timing is not achieved by one component alone, but by careful optimization of the full chain from detector to electronics to calibration software.
A simple timing chain
A timing measurement often follows this sequence. A particle passes through the detector. The detector produces light or charge. That signal is converted to an electrical pulse. The pulse is amplified and shaped. Electronics assign a time stamp. Calibration corrections are applied. The corrected start and stop times are subtracted to obtain the flight time.
| Step | Role in timing measurement |
|---|---|
| Particle interaction | Creates the initial detector signal |
| Signal formation | Produces charge or light |
| Readout conversion | Changes detector response into an electrical pulse |
| Amplification and shaping | Makes the pulse measurable |
| Time extraction | Assigns an arrival time |
| Calibration | Removes offsets and systematic effects |
| Final subtraction | Gives the measured time of flight |
Core ideas to remember
Timing measurements in time of flight detectors are about determining the arrival time of detector pulses with very high precision. The measured time is usually the difference between a start and a stop signal. Precision depends on fast pulse rise, low noise, small time walk, accurate calibration, and correction of internal delays. Good timing is not just fast electronics, it is the combination of detector physics, signal processing, and careful calibration.
Key summary:
A precise timing measurement requires
$$
\text{fast signal} + \text{low noise} + \text{small time walk} + \text{good calibration}.
$$
The final measured time is only meaningful after correcting fixed offsets and propagation delays.