Becker W. Advanced Time-Correlated Single Photon Counting Techniques

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7.9 Counting Loss in TCSPC Systems 335

The probability of

recording a photon in channel j is the product of the probability

of detecting a photon in channel j and the probability of detecting no photons in

the channels 0 to

j1:



tjPePt

jrecorded

eeeePP

''



(7.20)

The first factor is independent of

j. The second factor is the waveform of the re-

corded signal. The waveform can be written as

...

)(

/43/32/2/



''''

WWWW

tjtjtjtj

ePePPeejf (7.21)

That means that the pile-up adds virtual lifetime components of

/2,

/3 ...

/n, to

the recorded waveform. A mean lifetime,

meanc

, of the recorded curve can be de-

fined as an average of the lifetimes,

weighted by their intensity coefficients, a

meanc

(7.22)

The resulting

meanc

.....

...

4!2

3!1



PPP

meanc

WWW

(7.23)

which can be converted into

meanc

1

(7.24)

For a small value of

P the obtained mean lifetime is



2/1 p

meanc

|

(7.25)

The definition used for

meanc

leads to a simple expression of the measured lifetime.

It is, however, not the only possible one. The lifetimes can also be weighted by the

integral intensities of the exponential components:

meani

(7.26)

The intensity-weighted mean lifetime is a better approximation of the lifetime

obtained by a single-exponential fit or by calculations based on the first moments

[308]. The pile-up distorted

meani

.....

...



PPP

meani

(7.27)

336 7 Practice of TCSPC Experiments

Unfortunately the intensity-weighted mean lifetime does not yield a simple re-

sult for the pile-up. For a small value of

P the lifetime is



4/P1|

meani

(7.28)

Figure 7.78 shows the recorded coefficient-weighted and intensity-weighted

mean lifetime as a function of the number of photons per laser period,

0.2 0.4 0.6 0.8 1.0 1.2 1.61.4 1.8 2

Number of Photons per Signal Period, P

Relative

1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 7.78 Coefficient-weighted mean lifetime (a) and intensity-weighted lifetime (b) as a

function of the number of photons detected per signal period, P

The influence of the pile-up on the obtained lifetimes is surprisingly small. A

5% change in the intensity-weighted mean lifetime is often tolerable. The corre-

sponding P is about 0.2, i.e. the detector count rate is 20% of the pulse repetition

rate. With pile-up correction, even higher count rates appear possible. Pile-up is

therefore not a severe problem at pulse repetition rates between 50 to 90 MHz,

typical for titanium-sapphire lasers and diode lasers.

A remark should be made here about multiplexing signals on a pulse-by-pulse

basis. The principle is shown in Fig. 7.79. Several signals, signal 1 and signal 2,

are offset in time and recorded in the same TAC range with a common stop pulse.

At low intensity (curve A), the pile-up is negligible and both signals are recorded

with their correct shape and intensity. If the intensity of signal 1 is increased into

log

counts

channels

Signal 1 Signal 2

pileup

pileup from

signal 1

Fig. 7.79 Effect of classic pile-up on signals recorded with pulse-by-pulse multiplexing

7.9 Counting Loss in TCSPC Systems 337

the pile-up region (curve B), the shape of signal 1 is distorted. Signal 2 is recorded

with its correct shape but with an incorrect amplitude.

In the general case, each signal gets changed in shape only by its own pile-up.

However, it gets changed in amplitude by the pile-up of all earlier signals. Mutual

pile-up effects between multiplexed signals can be entirely avoided by using the

multiplexing features of advanced TCSPC (see Sect. 3.2, page 33, and Sect. 5.5.8,

page 117).

Several solutions have been suggested to detect pile-up and to reject the re-

cording of multiphoton events [389]. However, for the most commonly used de-

tectors, it is difficult if not impossible to distinguish between one and two photons

arriving within the width of the single-electron response of the detector (Fig. 7.80,

left). Multiphoton events at a time scale of 10 to 20 ns could, in principle, be de-

tected by counting the number of CFD threshold transitions within one signal

period. However, in practice the single-electron-response of the detector is not

free of ringing and reflections. Therefore single-photon pulses near the upper end

of the SER amplitude distribution cause multiple threshold transitions as well.

Discarding all events with multiple threshold transitions would result in losing a

large number of good photon pulses.

Nevertheless, there is a way to reduce the classic pile-up. The light is distrib-

uted into several detectors, or into several channels of a multianode PMT. Two

photons arriving within the same signal period are then more likely to hit different

detectors than the same one (Fig. 7.80, right). Pulses that appear at the outputs of

different detectors can easily be identified as multiphoton events. The recording of

both photons in the TCSPC module can be suppressed and pile-up distortion can

be avoided.

Photon 1

Photon 2

Photon 1

Photon 2

Single detector:

Photons not distingushed

Multichannel detector:

Photons distinguished

Fig. 7.80 Multidetector operation reduces pile-up by recognising multiphoton events

The required circuitry is contained in any router for multidetector TCSPC (see

Sect. 3.1, page 29). The router monitors the output signal of all detector channels

by fast discriminators. When a detector delivers a photon pulse the corresponding

discriminator triggers and an encoder generates a digital „detector channel“ num-

ber (Fig. 3.3, page 30). If several detectors deliver pulses within the response time

of the discriminator/encoder circuitry, the encoder delivers a „Disable Count“

signal. This signal suppresses the storing of the event, i.e. of both photons, in the

memory of the TCSPC module. Thus, waveform distortion by classic pile-up is

338 7 Practice of TCSPC Experiments

reduced. The time interval in which several photons are recognised as a mul-

tiphoton event is usually of the order of several tens of ns.

An even more efficient, yet more expensive way to reduce pile-up distortion is

the multimodule technique. In a multimodule system the photons from several

detectors are processed in parallel. Therefore, not only pile-up distortion but also

counting loss are reduced in proportion to the number of TCSPC channels.

7.9.2 Counting Loss

Due to the finite speed of signal processing, a photon counter is unable to detect a

second photon within a certain dead time after the detection of a previous one. For

gated photon counters or multichannel scalers, the dead time can be as short as a

nanosecond. The relatively complicated signal processing sequence in a TCSPC

device leads to a much longer dead time. Older TCSPC devices had dead times of

the order of 10 µs. Newer, more advanced TCSPC modules are much faster but

still have a dead time in the range of 100 to 150 ns.

The probability of losing a photon can be calculated as follows. The photons

are detected at a rate

det

, and recorded at a rate, r

rec

. Each recorded photon causes a

dead time,

. If another photon is detected within this time it is lost, but it does not

cause a noticeable dead time. The fraction of the total dead time,

dead

, within the

total acquisition time,

T, is therefore

drecdead

trTT / (7.29)

The ratio of the number of photons lost in the dead time to the number of

de-

tected

photons is the same as the ratio of the total dead time to the total acquisition

time:

dreclost

trNN

det

/ (7.30)

The recorded rate is

drecrec

trrrr

detdet

 or

rec

det

1

(7.31)

Please note that this equation applies strictly only to systems in which the lost

photons do not cause any appreciable dead time. If the lost photons cause dead

time, the equation applies only for count rates that are small compared to the re-

ciprocal dead time. The relation between the input count rate and the recorded

count rate is shown in Fig. 7.81.

7.9 Counting Loss in TCSPC Systems 339

0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.90.3

0.1

0.2

0.3

0.4

0.5

Detector Count Rate, in units of reciprocal dead time

Rate

Recorded

Count

1/td

Fig. 7.81 Relation between the input count rate and the recorded count rate for a photon

counter with the dead time, t

, for a continuous light signal

For an input count rate 1/t

, i.e. the reciprocal of the dead time, the recorded

count rate is 50% of the input count rate. The recorded count rate for a 50% count-

ing loss is sometimes defined as the „maximum useful count rate“,

dmu

tr /5.0 (7.32)

Data sheets of TCSPC devices sometimes specify a „maximum count rate“ that

is simply the reciprocal signal processing time. This definition is misleading be-

cause for random input pulses it can be reached only for an infinite detector count

rate. The term „saturated count rate“ should better be used instead of „maximum

count rate“.

Equation(7.31) can be used to calculate the counting efficiency, E:

rec

trr

detdet



(7.33)

with

det

= detected count rate, r

rec =

recorded at a rate, t

d =

dead time.

For the discussion above it was assumed that the detected light signal was con-

tinuous. However, signals measured by TCSPC are mostly pulsed signals. More-

over, the detection and therefore the dead time is synchronised with the signal

period. This synchronisation can lead to a different behaviour than predicted by

(7.33). Dead-time-related counting loss in nonreversed start-stop systems is illus-

trated in Fig. 7.82.

340 7 Practice of TCSPC Experiments

start-

Photon

dead time

ends here

time

system is blind system is active

TAC / ADC

dead time

Laser

Fluorescence

signal

Start

pulse

Start

pulse

Start

pulse

stop

start-

Photon

dead time

ends here

time

system is blind

system is

system is blind

TAC / ADC

dead time

active

Laser

Fluorescence

signal

Start

pulse

Start

pulse

Start

pulse

stop

Fig. 7.82 Dead-time-related counting loss in a nonreversed start-stop system. Left: Signal

period longer than dead time. Right: Signal period shorter than dead time

In nonreversed start-stop systems the TAC is started with the laser pulse. When

a photon is detected, the TAC is stopped and the signal processing starts. Signal

processing takes some time, here indicated as „TAC/ADC dead time“. After the

processing is completed the TAC does not convert another photon pulse until it

has received the next start pulse from the laser. For the counting loss, two cases

have to be distinguished:

If the signal period is longer than the sum of the start-stop time and the

TAC/ADC dead time (Fig. 7.82, left) the dead time ends before the next laser

pulse. If a second photon is detected during this signal period it is lost. Conse-

quently, the situation is exactly described by the classic pile-up effect.

If the signal period is shorter than the sum of the start-stop time and the

TAC/ADC dead time the situation is different (Fig. 7.82, right). Of course, if a

second photon is detected in the same signal period, the system loses it. This is the

classic pile-up effect. However, because the dead time extends into the next signal

period, the TAC does not start with the next laser pulse. Consequently, the system

remains blind for the next signal period as well.

Several signal periods can be lost if the TAC/ADC dead time extends over

more than one period. Even worse, the necessary reset of the TAC causes a dead

time even if no photon was detected. If this reset time extends into the next period

the counting efficiency drops to 50% or less.

Figure 7.83 illustrates the situation for reversed start-stop systems and low-

repetition-rate signals. In the reversed start-stop configuration the TAC is started

when a photon is detected. In a correctly designed system the stop pulse of the

TAC comes from the delayed laser pulse. The stop delay is somewhat longer than

the time interval to be recorded. Data processing starts when the stop, i.e. the de-

layed laser pulse, arrives at the TAC. The system is now blind for the TAC/ADC

dead time. However, the pulse period is longer than the sum of the stop delay and

the TAC/ADC dead time. Therefore the TAC/ADC dead time is entirely outside

time interval in which the signal is recorded. Consequently, the only way the sys-

tem may lose photons is through classic pile-up.

7.9 Counting Loss in TCSPC Systems 341

start-stop

TAC / ADC

Photon dead time

ends here

dead timetime

ref. pulse

System is blind System is active

Delayed

ref. pulse

Delayed

Stop delay Stop delay

Fig. 7.83 Dead-time-related counting loss in a reversed start-stop system. The stop is with a

delayed laser pulse. The signal period is longer than the sum of the stop delay and the

TAC/ADC dead time

Figure 7.84 shows what happens if a reversed start-stop system is operated at

low repetition rate without a delay line in the stop line of the TAC.

start-stop

TAC / ADC

Photon

dead time

ends here

dead timetime

System is blind System is active

Fig. 7.84 Dead-time-related counting loss in a reversed start-stop system. The signal period

is longer than the dead time; the stop is with the next laser pulse

The TAC is started when a photon is detected. It is, however, not stopped with

the delayed laser pulse that produced this photon, but with the next one. The signal

processing starts with this pulse. The TAC/ADC dead time turns the system blind

for a large part of the next laser period. The blind interval of the next period is in

fact the interval where a new signal photon is most likely to be expected. Conse-

quently, the system can be considered to be blind for the next signal period after

the detection of a photon. The probability of losing a photon in the blind period is

repreclose

(7.34)

with

rec

= recorded count rate, f

rep

= signal repetition rate.

342 7 Practice of TCSPC Experiments

The relation between the detector count rate

det

and recorded count rate r

rec

(classic pile-up not included) is

)1(/

detdet

Trrr

rec

 (7.35)

with

T = signal period. In other words, the system has an effective dead time of

one signal period.

The situation for reversed start-stop and high repetition rate signals is shown in

Fig. 7.85. The TAC is started when a photon is detected and stopped with the next

laser pulse. Within the time between the start and the stop, the TAC is unable to

record a second photon. The resulting loss is the classic pile-up effect.

start-stop

TAC / ADC

Photon

Stop

dead time

ends here

dead time

time

System is blind System is active

Pulse Pulse Pulse Pulse

Laser

Fluorescence

signal

Stop Stop Stop

Pulse

Stop

Fig. 7.85 Dead-time-related counting loss in a reversed start-stop system. The signal period

is shorter than the dead time. The stop is with the next laser pulse.

Signal processing starts with the stop, i.e. with the next laser pulse. During the

signal processing time (or TAC/ADC dead time) the system is blind. The

TAC/ADC dead time ends somewhere in one of the subsequent signal periods.

How many photons the system loses depends on the number of signal periods

within the TAC/ADC dead time, and to a smaller degree on the distribution of the

detection probability within the signal period where the dead time ends. The effec-

tive dead time of the system is the TAC/ADC dead time plus the average start-stop

time. For a continuous signal the relation between detected and recorded rate is

given in equation (7.33). For a pulsed signal, the counting loss changes in discrete

steps with the number of signal periods within the dead time. However, it follows

the general tendency of (7.33).

Counting loss in reversed start-stop systems operated at high signal repetition

rate can be compensated for by a „dead time compensation“ in the acquisition

time. The idea behind the dead time compensation is to increase the acquisition

time by the sum of all dead time intervals that occurred during the measurement.

The principle is shown in Fig. 7.86.

7.9 Counting Loss in TCSPC Systems 343

Photons

TAC

busy

Clock

Generator

Timer

Clock

deadtime deadtime

Fig. 7.86 Dead time compensation by gating the timer clock

The TAC delivers a „busy“ signal as long as it is unable to accept a new pho-

ton, i.e. from the start of the TAC to the end of the TAC/ADC dead time. A high-

frequency clock is gated with the busy signal, and the gated clock drives the col-

lection timer. Thus the collection timer is stopped in the dead time intervals.

Of course, dead-time compensation works with absolute precision only if the

photons appear randomly. Unfortunately in most TCSPC applications, the signal

is pulsed and most of the photons are detected in only a small fraction of the signal

period. Nevertheless, the compensation works well for repetition rates higher than

the reciprocal dead time. It can, however, correct for only the loss in the recorded

intensity, not for distortion in the recorded pulse shape.

7.9.3 Dead-Time-Related Signal Distortion

For TCSPC with nonreversed start-stop, the end of the dead-time interval is auto-

matically synchronised with the laser pulses. For reversed start-stop this in not the

case. The situation is shown in Fig. 7.87.

start-stop

TAC / ADC

Photon

Stop

dead time

ends here

Average detection probability

dead timetime

System is blind System is active

Pulse Pulse Pulse Pulse

Laser

Fluorescence

signal

Stop Stop Stop

Photon

Fig. 7.87 Dead-time-related signal distortion in high-repetition-rate reversed-start-stop

systems. The dead time ends anywhere in one of the subsequent signal periods, which

causes a step in the detection probability

344 7 Practice of TCSPC Experiments

The TAC/ADC dead time starts with the stop of the TAC. The end of the dead

time can be at any time within one of the subsequent signal periods. Averaged

over a large number of periods, the result is a step in the recording probability and

thus in the recorded waveform. The size of the distortion depends on the ratio of

the count rate to the signal repetition rate and can be estimated as follows.

The probability,

P, to detect a photon in a particular laser period is

P = r

det

/ f

rep

(7.36)

with r

det

= detector count rate at CFD input, f

rep

= signal repetition rate. However, a

photon in the early part of the period can be

recorded only if no photon was de-

tected in the signal period a dead-time interval before. The probability,

early

,of

detecting a photon in the early part of the period is therefore



reprep

early

detdet

1 (7.37)

The relative size of the distortion is

late

– p

early

) / p

late

= r

det

/ f

rep

(7.38)

Interestingly, the size of the dead-time-related distortion does not depend on the

dead time. The absolute size of the distortion is of the same order as the distortion

by classic pile-up. It even counteracts the classic pile-up because it happens in the

early part of the signal period. Although the size of the distortion is predictable,

the actual shape of the distortion is not. It may differ from a clean step because the

dead time may vary due the TAC voltage of the detected photon, and the transition

from the blind into the active state may cause some ripple in the TAC characteris-

tic for a few ns. Moreover, in practice the dead time is determined by CMOS logic

circuits and active delay lines in the control circuitry of the signal processing elec-

tronics. The dead time can therefore be expected to be stable within only a few ns

at best, and it is difficult to correct for dead-time-related distortion.

In practice the distortion can be minimised in the same way as classic pile-up,

i.e. by maximising the signal repetition rate, or, more exactly, maximising the

TAC stop rate. The signal repetition rate should be as high as possible, and the

setup should avoid frequency division in the reference channel and pulse-by-pulse

multiplexing with a common stop pulse (see Sect. 5.5.8, page 117).

A radical cure to avoid pulse distortion by dead-time-related counting loss

would be to synchronise the end of the dead time with the reference pulses. How-

ever, this would result in more mutual influence of start- and stop-related signals

and therefore impair the differential linearity of the time measurement. Extending

the dead time to a full signal period might also be unacceptable for low-repetition

rate experiments.

Some TCSPC modules provide manual control of the dead time so that possible

distortions can be shifted out of the time interval of interest.