6.2.1.1 Principles

An LE discriminator is the simplest type of time pick‐off circuits. The principle of operation and basic structure of an LE discriminator is shown in Figure 6.7. It simply produces a logic pulse when the input signal crosses a fixed threshold level, which should be above the noise level to prevent spurious triggering on noise signals. An LE discriminator can be simply built by using a fast comparator that features high switching speed with low transition time. One input of the voltage comparator receives the detector pulse, while the other input of the voltage comparator receives a reference voltage that determines the threshold level. The width of output logic pulse should be short enough to operate the coincidence circuit and is usually arranged by a mono‐stable multivibrator circuit [2–5]. An LE discriminator can be equipped with extra discriminators to select the events that lie in an energy range of interest. The energy selection capability also prevents the device from possible false triggering on the baseline noise.

6.2.1.2 Leading‐Edge Discriminator Errors

In principle, there are two important sources of errors in time pick‐off measurements: jitter and walk. These affect the accuracy of the relation between T₁ and T_∘, as shown in Figure 6.6. Timing errors may result from drift in the electronics that is introduced by component aging and by temperature variations. Jitter is the time uncertainty that is caused by noise and also by statistical fluctuations of the detector signals. The noise can originate from the detector, pulse processing electronics, or the discriminator itself. The effect of jitter on the time pick‐off uncertainty is shown in Figure 6.8 for an ideal LE discriminator. If the input signal is assumed to be approximately linear in the region of threshold crossing, for a noise with Gaussian probability density function and with zero mean value, the noise‐induced uncertainty in threshold‐crossing time is given with reasonable accuracy by the triangle rule as

(6.3)

where σ_v is the rms value of noise and σ_T is the rms value of time uncertainty. Equation 6.3 indicates that the time jitter is inversely proportional to the slope of the input signal at the threshold‐crossing time, which means that the threshold level must be set at the point with greatest slope and the pulse should not be slowed down during transit to the discriminator. It is clear that the minimum value of discriminator level is limited by the level of electronic noise. When the pulses are delivered by a charge‐sensitive preamplifier, due to the unavoidable pole created in the preamplifier, the pulses exhibit an exponentially rising region near the starting point, which is called toe [6]. In this case, the lower limit of the threshold level is limited at the region of the unavoidable toe because in this region the slope is degraded, leading to smaller slope‐to‐noise ratio. Another source of jitter results from the statistical amplitude fluctuations that cause uncertainty in the time at which the discriminator fires the output logic pulse [7]. In addition to jitters related to input pulse, there is generally a residual jitter contribution from the electronics system due to variations in the transit times of pulses [8].

The second source of timing error is walk. In general, time‐walk is the time movement of the output logic pulse relative to the input pulse due to variations in the shape and amplitude of the input pulse. Time‐walk is a serious limitation of LE discriminators. Figure 6.9 illustrates the time‐walk of an ideal LE comparator due to input signal variations. In the upper part of the figure, two pulses of the same risetime but different amplitudes are shown. Both pulses start at the same time, but pulse 1 crosses the threshold level at time T₁, while pulse 2 crosses the threshold at time T₂. The difference in threshold‐crossing time causes the output logic pulse from the LE discriminator to walk along the time axis. It is clear that time‐walk increases for small amplitude pulses and thus is most pronounced for signals with amplitudes that only slightly exceed the threshold level. The lower part of Figure 6.9 shows two pulses of the same amplitude but with different risetimes. The slower pulse crosses the discriminator after the crossing time of the first pulse that results in a time‐walk along the time axis.

In a real LE discriminator, an additional contribution to the time‐walk results from the charge sensitivity of comparators [7, 9]. All physically realizable comparators are charge sensitive in a sense that even though the comparator threshold has been exceeded, a finite amount of charge has to be injected to trigger the comparator. The time‐walk due to charge sensitivity is illustrated in Figure 6.10. For pulse 1, the small amount of charge required to trigger the discriminating element moves the trigger time from T_∘ to T₁, and the effective threshold level of the discriminator changes to V_th + ΔVq. These changes are related to the slope of the input signal as it passes through the threshold. The timing errors introduced by charge sensitivity are greater for signals with longer risetimes and for signals with smaller pulse amplitudes above the threshold. The characteristics of the input device are also a major factor in determining the charge sensitivity of the discriminator. By assuming that the input signal is approximately linear during the time that is required to accumulate the required charge, corresponding to area S indicated in Figure 6.10, the error in the effective sensing time is related to the slope of the input signal by [9]

(6.4)

where v(t) is the input pulse as a function of time and T is the threshold‐crossing time of the pulse. One can see in Figure 6.10 that the time‐walk of an LE discriminator tends to decrease for pulses with shorter risetimes though the effective threshold level increases for pulses with short risetime.

The discussion on the leading‐edge discriminator’s timing errors implies that when this technique is restricted to those applications that involve a very narrow dynamic range of pulse amplitudes, excellent timing results can be obtained. This is most frequently achieved by experimenting with the threshold level. Under the optimum threshold level, timing errors due to charge sensitivity and to jitter are minimized for input signals with the greatest slope at threshold‐crossing time.

6.2.1.3 Optimum Timing Filter

The timing jitter due to electronic noise may be minimized by choosing a proper noise filter. Such filter minimizes the time jitter by maximizing the slope‐to‐noise ratio according to Eq. 6.3 [10–12]. The problem of maximizing the slope‐to‐noise ratio can be treated in the same way as the problem of maximizing signal‐to‐noise ratio that was treated earlier in Chapter 4. An input signal V(jω) with a superimposed noise of power spectrum N_∘(ω) is processed by a filter with transfer function H(jω). The slope of the filter output pulse g(t) is dg(t)/dt, which in the frequency domain corresponds to a multiplication of G(jω) with jω where G(jω) is the output pulse in the frequency domain. Then the slope of the output pulse and the mean‐square value of output noise voltage are given by

(6.5)

and

(6.6)

The mean‐square error in the time measurement is then

(6.7)

By defining ρ_T as inverse of σ_T, we obtain

(6.8)

Minimizing Eq. 6.7 is equivalent to maximizing its inverse ρ_T. Thus, by defining

(6.9)

(6.10)

and using the Schwarz inequality, the maximum value of the mean square of ρ_T is obtained as

(6.11)

This maximum value is obtained only when

(6.12)

where K is a constant and is the complex conjugate of y₂(ω). By substituting Eqs. 6.9 and 6.10 into Eq. 6.12 and solving for H(jω), we obtain

(6.13)

This equation represents the optimum filter for timing for an arbitrary signal and noise and also for an arbitrary measurement time. From our discussion in Chapter 4, one realizes that H(jω) differs only by the factor jω from the filter optimizing the signal‐to‐noise ratio. Since a multiplication with jω in the frequency domain corresponds to the derivative in the time domain, the impulse response of the filter that maximizes the slope‐to‐noise ratio is equal to the derivative of the impulse response of the filter that maximizes the signal‐to‐noise ratio. However, one should note that such optimum filter is related in a unique way to the input signal shape and the noise spectrum, and thus, it may not be always realizable or practical. In practice, filters such as integrator–differentiator filters are commonly used to prepare the signal for time pick‐off (see Section 6.2.2.6). The use of such filters particularly has proved to be useful in timing with semiconductor detectors.

6.2.1.4 Extrapolated Leading‐Edge Timing (ELET)

We already discussed that the output of an LE discriminator suffers from time‐walk errors due to variations in the amplitude and shape of input pulses. Although the time‐walk can be reduced by using small threshold levels, the choice of a sufficiently low discriminating threshold for reducing the walk is again restricted by the noise level and the presence of toe region. Therefore, LE discriminators produce a poor time resolution for detectors with variable pulse shapes and amplitudes. Extrapolated leading‐edge discriminator (ELET) is a modified version of the LE discriminator that compensates for the time‐walk stemmed from the variable pulse risetimes and amplitudes. This method was first introduced by Fouan and Passerieux [13], and a variety of circuits based on the same principles have been built mainly for timing with germanium detectors [6, 14–17]. The principle of this method is shown in Figure 6.11. Two LE discriminators with different threshold levels V_L and V_U are used where V_U is generally a multiple of V_L. For pulse 2, the time‐walk with respect to pulse 1 is equal to T_L2 − T_L1 associated with the discriminator set at V_L. Ideally we are interested to have a zero time‐walk, which corresponds to the measurement of the start time of the pulse T_∘. This time cannot be directly marked, but if the time interval between a reference time T_ref and T_∘ is intended to be measured, then the subtraction of the time T_U − T_L from the time T_L − T_ref gives the correct time interval T_∘ − T_ref, if (T_U − T_L) = (T_L − T_∘). Here T_L and T_U are the crossing times of the pulse at V_L and V_U, respectively. In other words, in this method, the triggering times of two discriminators with different threshold levels are used to extrapolate back to a time origin. The ELET method is very effective for linearly rising pulses that do not exhibit change of slope below the upper threshold. However, the noise jitter of this method is poorer than the standard LE method because jitter is introduced by both discriminators. The ELET method has been implemented in real time by using dedicated circuits or standard nuclear instrument modules and also offline by using software extrapolation of discriminators outputs.

6.2.2.1 Principles

The problem of time‐walk in LE discriminators that results from variations in the pulse amplitudes can be avoided by using a time pick‐off circuit in which the discrimination level varies as a constant fraction of pulse amplitude. Such time discriminator was first reported by Gedcke and McDonald in 1967 [18] and is called constant‐fraction discriminator (CFD), which is in wide use today with all types of detectors due to its good timing performance and relatively simple structures. Figure 6.12a shows a basic functional diagram of a CFD. The pulse is sent into two branches. In one branch it is attenuated by the fraction f of the pulse amplitude, while in the other branch it is delayed by t_d. The two branches are compared and a time mark is developed when two signals are equal. This is equivalent to determining the zero‐crossing time of the difference of two signals as shown in Figure 6.12b. The formation of the difference signals or CFD pulse is an important step of proper operation of a CFD circuit. The CFD shaping steps are shown in Figure 6.13. The transfer function of such CFD shaper is given by

(6.14)

CFD shaping produces a bipolar pulse whose zero‐crossing point occurs at the fraction of the pulse amplitude chosen. The zero‐crossing is detected and used to produce an output logic pulse. Ideally, one can set the fraction f to minimize the jitter, setting the threshold fraction at the point of the signal shape where the slope is at maximum, as it was discussed for an LE discriminator. The minimization of time‐walk is achieved by a proper selection of shaping delay, t_d, depending on the characteristics of the input pulse. The role of shaping delay is explained in the following. For a theoretical linear rising input signal described with

(6.15)

where A is a constant and t is time, the constant‐fraction (CF) zero‐crossing time t_cfd can be calculated from

(6.16)

or

(6.17)

This relation holds for a pulse of continuous rise. In practice, pulse has a finite amplitude, and thus the zero‐crossing time can happen before or when the attenuated input pulse has reached its maximum amplitude. In true constant‐fraction (TCF) discrimination circuits, the zero‐crossing time occurs when the attenuated pulse is at maximum. To ensure this happens, the shaping delay is selected so that

(6.18)

where t_r is the input signal risetime. In this case, the zero‐crossing time is given by [7, 9]

(6.19)

In practice, input pulses have finite widths, and thus the shaping delay must also be made sufficiently short to force the zero‐crossing signal to occur during the time that attenuated signal is at its peak. A simple and useful test to ensure that the CFD is properly adjusting is to simultaneously view the bipolar signal, known as monitor signal, and CFD output on a fast oscilloscope that is triggered by the output of the CFD. A TCF timing fundamentally or theoretically (with perfect comparators, attenuators, and delay lines) produces no walk for signals of variable amplitude but with equal risetimes since it makes the arrival time independent of the amplitude of the input pulse. However, in many detectors, the risetime of pulses also varies that has led to another mode of pulse timing with CFDs. This mode is called amplitude and risetime compensated (ARC) timing that will be discussed in Section 6.2.2.3.

6.2.2.2 CFD Timing Errors

In a TCF the jitter is minimized by setting the CFD to provide a time pick‐off signal at the fraction of the input pulse amplitude with maximum slope‐to‐noise ratio. Similar to an LE discriminator, by assuming the CFD signal to be approximately linear in the region of zero‐crossing, the rms value of noise‐induced jitter in the CF zero‐crossing time, σ_T, is given by

(6.20)

where σ_v is the rms value of the noise on the CF bipolar signal and T_cf is the general zero‐crossing time. The rms value of noise on the CFD bipolar signal can be related to the noise on the input signal by making the simplifying assumptions that the CF circuit is noiseless and has an infinite bandwidth and also the noise on the input signal is time‐stationary type and has a Gaussian probability density function with zero mean value. Then, the rms value of noise on the CFD bipolar pulse can be written as [9]

(6.21)

where σ is the rms value of the noise on the input signal, f is the CFD attenuation factor, e_n(t) is the input noise, φ(t_d) is the autocorrelation function of the input noise, and t_d is the CFD shaping delay. In the case of uncorrelated noise, the relation between the rms values of the noise on the CFD bipolar pulse and on the input signal is simplified to

(6.22)

By using Eq. 6.22 and having the slope of the pulse, one can estimate the timing jitter from Eq. 6.20. For linear rising pulses of amplitude V_∘, the slope of the pulses is given by

(6.23)

Combining Eqs. 6.20, 6.22, and 6.23 leads to the noise‐induced jitter as

(6.24)

A comparison of Eq. 6.23 with the result for a simple LE discriminator indicates that under identical input and noise conditions, the jitter is usually worse for TCF timing than that for simple LE timing by the factor (1 + f²)^l/2. However, TCF timing virtually eliminates time jitter due to statistical amplitude variations of detector signals so that TCF timing often results in less overall jitter. A TCF theoretically eliminates the time‐walk due to the variation in the amplitude of pulses of fixed risetime. However, the time‐walk effect attributed to the charge sensitivity of the zero‐crossing comparator remains. Similar to a simple LE discriminator, in a CFD at the zero‐crossing point, an additional charge is required to actually trigger the comparator. This dependency leads to the variations in the charging time, and consequently, the response time of the comparator is a cause of time‐walk in a CFD. Time‐walk can be in the range of several tens to several hundred picoseconds for input signals with a 40 dB dynamic range and 1 ns risetime [19]. An analysis of CFD timing errors caused by voltage comparators can be found in Ref. [20]. Some supplementary techniques have been used to reduce the time‐walk by using automatic gain control that reduces the amplitude variation in the input signals to a comparator and, consequently, the variation in its response time [20].

6.2.2.3 Amplitude and Risetime Compensated (ARC) Timing

The true CFD method described so far is most effective when used with input signals having not only a wide range of amplitudes but also a narrow range of risetimes and pulse widths like outputs of scintillator detectors. However, the output pulses of many detectors such as semiconductors are subject to risetime variations that make the TCF method unsuitable. A CFD can be made insensitive to both pulse‐height and risetime variations by operating it in ARC timing mode, which is achieved by setting the CFD shaping delay so that the time of zero‐crossing occurs before the attenuated input signal has reached its maximum amplitude. Figure 6.14 illustrates the signal formation in an ideal CFD for ARC timing with linear input signals of different risetimes and amplitudes. For ideal linear input signals that begin at time zero, the zero‐crossing time T_ARC from Eq. 6.17 is given by

(6.25)

When the risetime of input pulses is variable, the criterion for t_d to ensure ARC timing is

(6.26)

where t_r(min) is the minimum expected risetime for any input signal [7, 9]. Opposite to TCF method, in ARC timing, the fraction of the input pulse amplitude at which the time pick‐off signal is generated is not constant. The effective triggering fraction for each input pulse is related to the attenuation fraction by the input signal risetime. For linear input signals, the effective ARC timing triggering fraction is given by [9]

(6.27)

This value is always less than f, while in TCF is always equal to f. One should note that the immunity to risetime variations described for ideal linear rising input pulses does not hold for a general case because in practice deviations from linear rise may occur, introducing timing errors.

For linear rising input signals, the jitter due to electronic noise in ARC timing can be obtained by having the pulse slope and noise. By assuming a linear input pulse of amplitude V_∘ and risetime t_r, the slope of the CF signal at zero‐crossing point is given by

(6.28)

By combining Eqs. 6.3, 6.22, and 6.28, we have

(6.29)

A comparison of this equation with that of TCF indicates that for identical input signal and noise conditions and for the same attenuation fraction, the jitter effect in ARC timing is worse than that for TCF timing due to the smaller slope of the ARC zero‐crossing signal. However, the ARC timing remains beneficial when the time‐walk due to risetime variations dominates the timing performance.

6.2.2.4 Practical CFD Circuits

The design of fast timing circuits requires a high level of designer skill as such circuits are very susceptible to the effects of stray parameters, ringing, and spurious signals. The interaction of active devices with the circuit layout, stray inductance, and capacitances and the presence of interference signals all place real limits on circuit performance. The principal functions that are generally performed in a CFD include attenuation, delay, inversion, summing, arming a zero‐crossing detector, detection of the zero‐crossing of the CFD bipolar pulse, and development of an output logic pulse. Most commercial CFDs have also provisions for adjusting the threshold level, the shaping delay, and the walk. The fraction is generally set at 0.2 or 0.3 that can be adjustable.

The initial CFD circuits were built around a tunnel diode, but nowadays high‐speed monolithic comparators are commonly used. The simplest CFD circuit based on monolithic comparators is shown in Figure 6.15a [7, 9, 21]. The circuit is basically the same as that of Figure 6.12a, but it is equipped with an arming circuit, comprised of an LE discriminator to inhibit the output timing pulse unless the input signal exceeds the arming threshold. This prevents the sensitive zero‐crossing device from triggering the baseline noise events. The lowest threshold setting is generally a few millivolt and is determined by the characteristics of the LE comparator. The CFD bipolar pulse is formed actively in the input differential stage of the lower comparator (CF). A monitor signal can be taken at the output of the CF comparator. A one‐shot circuit is placed in series with the output circuit to prevent multiple output signals from being generated from one input. A disadvantage of this circuit is that large time‐walk can result for small and slow rising input pulses, which is explained as follows: neglecting the propagation delays of the active circuit components, the input signal is applied to the CF circuitry and to the LE discriminator simultaneously. The LE discriminator produces an output pulse that crosses the logic threshold of the zero‐crossing AND gate at time t_LE, while the CFD bipolar signal crosses the logic threshold of the zero‐crossing gate at time t_cf. If t_LE < t_cf, then the time of occurrence of the logic pulse at the output of the zero‐crossing AND gate is t_cf, but for input pulses with amplitudes near the LE discriminator threshold and for input signals with exceptionally long risetimes, the time at which the input signal actually crosses the LE discriminator threshold can come after t_cf. Consequently, the output logic pulse will be based on t_LE and major timing errors due to LE time‐walk can occur. To improve this situation, in the circuit shown in Figure 6.15b, an additional LE discriminator is used. The lower‐level leading‐edge (LLLE) discriminator arms the zero‐crossing AND gate G1, while the upper‐level leading‐edge (ULLE) discriminator determines if the input signal is of sufficient amplitude to produce an output pulse from the instrument. The lower‐level threshold is internally set at one‐half the upper‐level threshold. Consequently, signals that satisfy the upper‐level threshold requirement exceed the lower‐level threshold by a factor of two. This dual comparator arrangement greatly reduces LE time‐walk for pulses that just slightly exceed the upper threshold level [9, 21]. Figure 6.15c shows the block diagram of a CFD in which the CFD bipolar pulse is formed passively by using a differential transformer of wide bandwidth [9]. The arming and zero‐crossing detector circuits are the same as in Figure 6.15a. The monitor output is also picked‐off at the input to the CF comparator. There are several other arrangements and designs for CFDs. In some of the designs, more LE comparators are added to allow the selection of an energy range of interest [9, 22]. Such circuits are called differential CFDs and produce an output timing pulse when the input signal exceeds the lower‐level threshold and does not exceed the upper‐level threshold. Another facility provided in some designs is the possibility of rejecting slow risetime events [23]. Since slow risetime events are inevitably accompanied with some timing errors, the rejection of such events helps to maintain the good time resolution achievable with faster pulses of semiconductor detectors that exhibit large risetime variations. However, this is achievable at the cost of some efficiency. To achieve a good time resolution, it is also necessary to stabilize the zero‐crossing point that may be disturbed due to parameters such as input signal dc level, ac interferences, input pulse shape, and high input frequencies. Special stabilization mechanisms have been used to minimize the time‐walk that results from zero‐crossing point fluctuations [24–26]. The walk error of a CFD is generally used to specify its performance, since it indicates the change in the timing moment as a function of the amplitude of the input timing pulse. The walk error of the CFDs can be measured by detecting the change in the peak location of a prompt coincidence time spectrum as a function of the amplitude of one of the input pulses.

6.2.2.5 Monolithic Time Discriminator Circuits

There are applications such as particle physics experiments and medical imaging in which the large number of detector channels necessitates the use of integrated front‐end circuits with accurate timing capability. To provide such timing signals, both LE and CFD methods have been implemented in ASIC systems based on complementary metal–oxide–semiconductor (CMOS) technology. Implementation of LE method has been rather straightforward by implementing fast comparators [27], while that of CFD method requires monolithic‐compatible solutions to replace the shaping delay that in discrete circuits is simply implemented by using a coaxial cable of a particular length [28–30]. Some proposed solutions for a monolithic‐compatible CFD pulse shaping are shown in Figure 6.16 [30]. Figure 6.16a shows a simple CFD pulse shaping method based on a CR differentiator [31]. Since a derivative of a signal with peak exhibits a sign change, a CR differentiator will produce a bipolar signal while no fraction circuit is required. However, in this method the slope through the zero‐crossing and overdrive is dependent on the trailing edge of the input pulse, which can result in more timing jitters and possibly time‐walks [31]. Figure 6.16b shows the lumped‐element RC shaping network [32], which generates a delay signal with a single RC low‐pass filter. The disadvantages of this method are signal amplitude and slope degradation due to the low‐pass filtration. Figure 6.16c shows the distributed RC delay‐line shaping network [33]. In this method, the delay signal for shaping is realized with a serpentine layer of polysilicon over a second grounded polysilicon plate. The polysilicon line is a lossy transmission line with delay and dispersion. Figure 6.16d shows the Nowlin method [34], which utilizes the CR differentiation with a fraction circuit. The attenuated signal is subtracted from the differentiated signal to produce the CFD bipolar pulse. This shaping circuit can be also optimized to produce ARC timing measurement. There are also other non‐delay‐line CFD configurations based on multiple pole low‐pass filters that are discussed in Ref. [29]. However, one should note that none of these methods can achieve the same performance as the ideal delay‐line method.

Implementation of integrated CFD circuits faces other limitations such as offset variations that can be high in CMOS circuits due to process parameter variations. The presence of offset voltage at the input of the zero‐crossing discriminator can destroy the amplitude invariability, creating residual time‐walk. Therefore, static or dynamic offset compensation schemes are often required to cancel dc offsets. In an integrated CFD circuit developed for use with scintillators in PET systems, the same principles as that in Figure 6.15a are used, but a five‐order low‐pass filter is used to produce the required delay, and both the CF and arming circuits are equipped with continuous‐time baseline restorer circuits to cancel dc errors due to circuit offset and event rate effects [35]. Examples of ASIC CFD systems based on a high‐pass CR circuit for generating a CFD bipolar pulse can be found in Refs. [36, 37].

6.4.1.1 Scintillators Coupled to PMTs

The shape of output pulses from a scintillator coupled to a PMT is affected by the time response of the PMT, and the effect becomes particularly important with fast scintillators. The contribution of a PMT to time resolution is mainly associated with the following [67, 68]:

1. The spread in the transit time of electrons traveling from photocathode to the anode. The amount of transit time spread generally depends upon photomultiplier geometric characteristics and its operating conditions.
2. The numbers of photoelectrons released from the photocathode, which is a function of the photocathode sensitivity and efficiency of photoelectrons collection.
3. The spread in the gain of the electron multiplier that translates to fluctuations in the amplitude of the output pulses.

In addition to these parameters, the frequency bandwidth of the PMT and its output circuitry may affect the timing performance. Figure 6.30 illustrates the time response characteristics of a typical PMT. The electron transit time is defined as the average time difference between the arrival time of a δ‐function light pulse and that of the corresponding output current pulse. The transit time spread is defined as the FWHM of the probability distribution of the fluctuations in the transit times. The larger the spread, the less time resolution the PMT has. The risetime is conventionally defined as the time the anode pulse takes to rise from 10 to 90% of its final value in response to a δ‐function input pulse and in fact indicates the frequency bandwidth limitation of the PMT. In addition to this intrinsic bandwidth limit, the output circuitry can place a frequency limit on the output pulses. The dependence of the frequency response on the external circuitry was already described with Eq. 3.79 and is a function of total capacitance and resistance at the PMT output. The total capacitance represents the combined capacitance of the tube and the stray capacitance of the cables and is generally very small, but the choice of load resistor needs some care. A large load resistor reduces the bandwidth but increases the output voltage. Increasing the load resistance also makes the potential drop between the last dynode and the anode small. The consequence of this is the buildup of space charge at the last dynode and poor collection of charges, leading to nonlinearity in the response. Therefore, one must choose an optimized load resistor value based on the requirements of the particular application.

In the previous section, we discussed the intrinsic time resolution of a scintillator where it was assumed that there was no transit time spread in the photodetector. When the photodetector is a PMT, the effect of PMT time spread can be included by replacing f′(t) in Eq. 6.36 with the following convolution [69]:

(6.38)

where I(t) is the probability density function of arrival times of photons at the photocathode produced by an event at t = 0 and P_pmt(t) is the probability density function of the transit time spread of electrons in the PMT. This modified version of timing theory has been successfully applied to predict the time resolution of scintillator detectors [70–72] with the result that the PMT time spread can be considerable even for slow scintillators. However, with modern fast PMTs used with slow scintillators such as Nal(Tl), the PMT probably has a negligible effect on the final time resolution. But for fast scintillators, achieving minimum time resolution requires a PMT with minimum time spread.

6.4.1.2 Effect of Pulse Processing System

The time resolution contribution coming from the timing electronics circuits is generally very small and may be neglected in many applications. But in fast timing measurements in the range of tens of picoseconds, the intrinsic time resolution of the electronics may be considerable [73]. The time fluctuations may arise from time pick‐off circuits, TACs and ADCs or TDCs, signal dividers, fan‐in/fan‐out circuits, and so on. Therefore, for fast scintillators, it is crucial to reduce the electronics modules as much as possible. The most critical part of a timing setup is the time pick‐off circuit. The popular timing methods with PMTs are the LE and CF timing. The LE method produces very good results in measurements with the narrow dynamic range of input pulse amplitudes [74]. To achieve the optimum time resolution, one needs to set the threshold level of the discriminator at maximum slope‐to‐noise ratio that generally corresponds to between 5 and 20% of pulse amplitudes. The choice of a threshold levels reduces time‐walk, but the risk of noise‐induced triggers increases. Although the time‐walk error from an LE discriminator can be also corrected by using the information of the pulse amplitudes offline, in most of the situations, the CFD method is preferred because it cancels the time‐walk online and thus more suitable for large dynamic range measurements. However, in the measurements with fast scintillators such as LaBr₃(Ce), the time‐walk due to the charge sensitivity of the comparator may still become quite noticeable. This time‐walk is dependent on the amplitude of the pulses, that is, the energy of input pulses, and thus it can be measured as a function of energy. The effect of charge sensitivity time‐walk on the calibration of time spectra and its correlation to PMT operation voltages has been studied in regard to its application in nuclear structure fast timing measurements [75].

A problem encountered in timing with slow scintillators is multiple triggering [26]. This is associated with the fact that as a result of the slow decay time of scintillators such as NaI(Tl), the last portion of each anode pulse consists of individual single‐photon pulses. Therefore, the discriminator will trigger once on the LE of the anode pulse and then multiple times at the end of the anode pulse, which can seriously affect the measurement. Multi‐triggering is also a problem with fast scintillators such as BaF₂ that have a slow component as well. To remedy the multi‐triggering problem, a non‐extending dead time is deliberately imposed to the CFD that makes it insensitive to the following individual single‐photon pulses. The length of the dead‐time duration should be long enough compared with the pulse duration and can be ~1 µs for NaI(Tl) detectors.

6.4.1.3 Fast–Slow Measurements

In many radiation detections systems, a simultaneous extraction of both timing and energy information is required. This also allows the measurement of the time resolution as a function of the energy range of input signals. In such applications, the pulse processing arrangement includes a fast and a slow pulse processing channels that extract, respectively, the timing and energy information [22]. Figure 6.31 shows two arrangements for fast–slow measurements with scintillators coupled to PMTs. In the top panel of the figure, an integral mode CFD is used as the time pick‐off device in each branch, and the slow channel is composed of a preamplifier, a shaping amplifier, and a SCA to select pulses in the energy range for which timing information is desired. The shaping amplifier is generally a delay‐line amplifier due to insignificant effect of electronic noise. If two detected events fall within the selected energy ranges, and if they are coincident within the time widow adjusted on the coincidence unit, then precise timing information related to these events is strobed from the TAC and the resulting signal is stored in an MCA. This system can provide both excellent energy selection and timing characteristics but at high count rates since the TAC must handle all start–stop pairs regardless of their energy or coincidence, a count rate limitation is imposed by the TAC, and also the heating effects in the active circuitry may become serious. In the arrangement shown in the lower panel of Figure 6.31, different CFDs are used that generate the timing information and determine the energy range of interest simultaneously. In this case, a CFD output is generated if two detected events fall within the selected energy ranges, and if they are coincident within the resolving time of the coincidence unit, then TAC is gated on to accept the precise timing information. This reduces the events rate of the TAC as it must only handle start–stop signals for events that satisfy the energy and time restrictions. One should note that in this approach the energy discrimination is generally performed based on the amplitude of PMT current pulses as CFD does not integrate the pulses, yet still the energy selection accuracy is satisfactory for most applications.

6.4.1.4 Scintillators Coupled to Photodiodes and Avalanche Photodiodes

Scintillators coupled to photodiodes have been used in some timing applications mainly due to advantages such as compactness, insensitivity to magnetic fields, and high quantum efficiency. Figure 6.32 shows an arrangement for timing measurement with photodiodes. The signal from photodiode is generally read out by using a charge‐sensitive preamplifier [76]. As it was discussed in the previous chapters, the preamplifier output signal is contaminated with the electronic noise that is determined by the diode leakage current, capacitance, and input transistor of the preamplifier. The time jitter due to the electronic noise imposes a severe limitation on the performance of a time discriminator, and thus a TFA is used to improve the slope‐to‐noise ratio by performing independent integration and differentiation on the preamplifier signal. The output of the filter is then fed into a time discriminator. The output of the charge‐sensitive preamplifier can be also simultaneously processed in a slow channel to extract the energy information. In spite of the noise reduction act of TFA, the timing performance of photodiodes is still insufficient for many applications. For example, the time resolution of a combination of CsI(Tl) and photodiode at ⁶⁰Co energies was reported to be in the range of above 100 ns [77] though better time resolutions may be achieved for very large amounts of energy depositions in the scintillator [77, 78].

In the case of APDs due to their internal gain, the effect of electronic noise is significantly reduced. Modern APDs, in addition to high gain (>200), exhibit small leakage current, low excess noise factor, and high quantum efficiency that make them an attractive option for use in applications involving pulse timing. The principle of deriving a time pick‐off signal from APDs is the same as that from photodiodes as shown in Figure 6.32 [79–82]. If an LE discriminator is used for time pick‐off, then the contribution of electronic noise to the time resolution can be estimated from Eq. 6.3. The slope of the pulse is determined by the charge collection time in the diode, the bandwidth or risetime of the preamplifier attached to the diode, the decay time constant of the scintillator, and the TFA shaping time constants. The charge collection time in the APDs is generally very fast; in the order of a few nanoseconds and by using a preamplifier with large bandwidth, for example, 100 MHz, one can also preserve the pulse risetime. If the absence of preamplifier and shaping effects, for the fully integrated pulse from a scintillator, dV/dt in Eq. 6.3 is proportional to the number of electron–hole pairs N_e−h and inversely proportional to the decay time of the scintillator τ [83]:

(6.39)

The rms value of noise at the input of the time discriminator is the combination of noise due to the electronic noise from APD‐preamplifier and photon noise due to the light collection process [84]. The noise behavior of an APD‐preamplifier system in most respects is similar to that of semiconductor detectors. The electronic noise rms value can be calculated from the electronic noise power spectral density, referring to the input of the APD and the frequency response of the entire preamplifier–shaper system. In regard to photon noise, however, since the current due to scintillation light, I_photon(t), decays exponentially with time, the noise is not stationary. Therefore, the mean‐square noise due to the photon flux should be calculated in the time domain from Campbell’s theorem:

(6.40)

where w(t) is the weighting function or the time domain response of the system, F is the excess noise factor, and q is the electric charge. The total value of root‐mean‐square noise voltage at the discriminator is then

(6.41)

where v_e is the rms value of electronic noise. By having dv/dt and v_e, the time jitter can be estimated from Eq. 6.3. The effect of electronic noise can be considerably reduced by cooling the device, thereby improving the time resolution [85]. In practice, the timing performance may be also affected with the nonuniformity and fluctuations in APDs gain and statistical error of the primary electron–hole pairs. The results of timing with LSO crystals coupled to APD show that subnanosecond timing is possible [83] for gamma rays though it is still inferior to that obtainable with PMTs. This is probably because PMTs provide an almost noiseless gain of around 10⁶, while an APD provides only a gain of around 10², and thus with APDs, many photoelectrons must be collected before sufficient charge has been collected to trigger a time discriminator. Nevertheless, the previously mentioned advantages of APD over PMTs have led to the use of such systems for PET imaging applications for which several ASIC signal processing systems have been developed to address several thousands of signal channels [36, 86].

6.4.1.5 Scintillators Coupled to SiPM

SiPMs have fast response and an amplification factor similar to PMTs (around 10⁶) that make them very attractive for use in timing applications. Studies have shown that the contribution of SiPM to the time resolution of an SiPM‐based scintillator detector is mainly determined by its photon detection efficiency (PDE), single‐photon timing jitter, noise (mainly dark counts and optical cross talk), time transit delay, and single photoelectron gain [87–89]. A circuit of time pick‐off from a SiPM generally involves the use of a fast amplifier followed by an LE discriminator. The properties of the amplifier that convert the SiPM current pulse to a voltage pulse play a very important role in the timing performance [90, 91]. A dedicated, high‐bandwidth, low input impedance front‐end electronics with low noise is required to deliver the SiPM pulses to the discriminator with maximum slope. In particular, the combination of SiPM output capacitance with a large input impedance of the amplifier can form an RC circuit with long decay time constant that consequently slows down the pulses. Fast amplifiers suitable for SiPMs have been built based on monolithic components [91], ASIC systems [92], and commercially available RF amplifiers [90]. Excellent results have been reported for LE timing of SiPM coupled to fast scintillators. For example, Schaart et al. [91] reported 100 ps time resolution for 3 × 3 × 5 mm³ LaBr₃(Ce) crystals coupled to Hamamatsu SiPMs at 511 keV gamma‐ray energy. However, these results have been achieved by using waveform digitizers and performing LE discrimination in digital domain. This has been attributed to the accuracy of baseline determination in digital domain that enables setting lower triggering thresholds. The recently developed digital SiPM employs a different approach of time pick‐off with that of analog SiPM. In such devices, each cell has its own readout circuit that is connected to an on‐chip TDC via a configurable trigger network to start the TDC at the first photon or higher photon levels. Time resolutions equivalent or better than analog SiPMs have been reported for digital SiPMs [93, 94].

6.4.2.1 Timing with Germanium Detectors

Timing measurement with germanium gamma‐ray detectors has been of interest particularly for nuclear physics experiments. With germanium detectors, the best time resolution can be achieved by deriving the timing signal from the output of a fast charge‐sensitive preamplifier. In principle, the timing performance of a germanium detector is limited by both time jitter and walk. The jitter results from the electronic noise at the output of the preamplifier, while the time‐walk stems from the fact that the shape of the charge pulses strongly varies with the interaction location of gamma rays. The time‐walk error normally overwhelms the noise jitter effect, and thus the timing technique must be focused on overcoming the time‐walk error. There have been several methods to minimize the effect of time‐walk including extrapolated LE method [13–17], ARC timing [6, 95], and more recently digital pulse timing techniques. Among the analog methods, the ARC timing method is the most widely used one. A system for the measurement of the time resolution of germanium detectors using ARC timing is shown in Figure 6.33 [21, 96]. The germanium detector is placed against a fast scintillator detector such as BaF₂ or LaBr₃(Ce). Since the time resolution of fast scintillators is much less than that of the germanium detector, the measured time resolution essentially reflects the time resolution of germanium detector. To derive a time pick‐off signal, the preamplifier output is presented to a TFA that performs integration and differentiation on the pulses to reduce the duration of pulses and optimize the slope‐to‐noise ratio. The TFA also provides the required gain before the timing discriminator. The TFA output is then used for time pick‐off with a CFD that is operated in ARC timing mode by setting the delay to ~30% of the minimum risetime of TFA output pulses. As it was previously discussed, ARC timing generates a timing marker independent of amplitude and risetime, provided that each pulse has a constant slope throughout its LE. However, real pulses from germanium detectors deviate from linear rise with slope changes before the time of zero‐crossing, and thus ARC timing cannot completely remove the time‐walk. Due to this limitation, the time resolution of large volume HPGe detectors is generally limited to about 8–10 ns.

6.4.2.2 Timing with Silicon and Diamond Detectors

Silicon detectors are widely used for timing measurements of charged particles, and depending on the experimental conditions, time resolutions below a few hundreds of picoseconds are quite common. The selection of suitable preamplifier depends on the delivery of a large signal with small risetime (or equivalently large slew rate dV/dt), and thus both charge‐ and current‐sensitive preamplifiers have been used, depending on the situations [97, 98]. If a charge‐sensitive preamplifier is used, the output is processed with a TFA followed by a CFD. The time resolution will basically depend on the charge collection time, noise, and uniformity of signals shapes. The minimum risetime of charge pulses is equal to the collection time of the electrons and holes in the detector sensitive region, which is determined by the distribution of the generated electron–hole pairs and of the electric field. The minimization of detector thickness reduces the charge collection time, but it increases the capacitance, thereby increasing noise. The effect of capacitance is particularly acute when very thin transmission silicon detectors are used in ΔE–E and time‐of‐flight measurements. An approach to reduce capacitance can be to segment the detector’s sensitive area while each segment is connected to its own readout electronics. The risetime of the pulses may be also affected with the plasma effect discussed in Chapter 1. A current‐sensitive preamplifier can be used when the energy deposition is large, and thus the slope‐to‐noise ratio is inherently high [99–101]. With current‐sensitive preamplifiers, the signals can be directly processed with a CFD. Although a current‐sensitive preamplifier can potentially lead to better time resolutions, the energy resolution will be inferior to that obtainable with charge‐sensitive preamplifiers, while in most experiments in nuclear physics, both good time and energy resolution are simultaneously required. In such cases, charge‐ and current‐sensitive preamplifiers can be simultaneously employed. The current pulse is used to pick off the time signal, while the charge pulse is used for energy measurements. There have been several methods for coupling a detector to both charge‐ and current‐sensitive preamplifiers [97, 102–104]. A common method of coupling a detector to two preamplifiers is shown in Figure 6.34 where a contact of the diode is grounded not directly but via the input of a fast low noise current‐sensitive preamplifier.

Diamond detectors have attracted considerable interest for use in timing measurements of charged particles. The high mobility of charge carriers in conjunction with very high breakdown electric fields and a low dielectric constant provides detector signals of fast risetimes suitable for timing measurements. Moreover, the wide bandgap of diamond produces very low leakage current noise. Diamond detectors are generally used with current‐sensitive preamplifiers in timing measurements, and the main challenge is to preserve the slope of fast current pulses. In this regard, the bandwidth of the preamplifier and the total capacitance at the detector–preamplifier connection play a very important role [105, 106]. With the simplifications that the output current pulse of the preamplifier has a linear rising edge, the slope of the pulse can be estimated as dv/dt = 0.8V_∘/t_r where V_∘ is the amplitude of the output pulse and t_r is the preamplifier risetime (10–90% amplitude level). For a single pole preamplifier, the bandwidth (BW) is related to risetime with BW = 0.35/t_r. Then, the slope of the pulse can be written as dv/dt = 2.28Q·BW/C where Q is the total charge deposited in the detector and C is the total input capacitance. By assuming that the noise results only from the bias resistor and the preamplifier is noiseless, the time jitter from Eq. 6.3 can be obtained as

(6.42)

where the noise rms value has been estimated from Eq. 2.75. This simple relation reflects the importance of the reduction of the total capacitance including the diamond capacitance, the preamplifier input capacitance and parasitic capacitances, and the role of the bandwidth of the system and the amount of charge deposition. This discussion can be extended to a more accurate definition of the noise contribution of the preamplifier that can be found in Ref. [105]. Many studies have reported time resolutions in the range of sub 100 ps for diamond detectors [107].

6.4.2.3 Timing with Compound Semiconductor Detectors

Timing with semiconductor detectors such as CdTe, CdZnTe, TlBr, and HgI₂ has been of interest for use in PET imaging systems. Compound semiconductors are available in different structures such as planar, coplanar, strip, and pixelated. For planar detectors, the conventional ARC timing method as discussed for germanium detectors is commonly used, resulting in time resolutions of 10 ns for 1 mm thick CdTe detectors at 511 keV energy [108]. The poor timing performance stems from the significant variations in the shape of pulses that are due to the large difference in the mobility of electrons and slow drifting holes and electronic noise mainly due to the leakage current of the detectors. Improvement in the timing performance has been achieved by cooling the detectors. For compound semiconductor detectors employing single polarity charge‐sensing technique, in spite of improvement in the energy resolution, the timing performance is generally poor due to the limit imposed by the pulse generation mechanisms. For example, in coplanar detectors, the output pulse is obtained by subtracting the induced pulses on collecting electrode and non‐collecting electrodes, eliminating the hole component of the pulse, which is the major cause of the degradation of energy resolution. However, the subtracted pulse appears with a delay with respect to the time of the gamma‐ray interaction in the detector because electrons generated by the gamma‐ray interaction need to travel from their creation point to the vicinity of the anode grids. Therefore, the variation in the interaction locations sets a basic limit on the timing performance of the detectors. Similarly, in pixelated detectors, the electrons need to travel from the interaction location to the vicinity of the anode to give rise to charge pulses, and thus, in spite of pulses of relatively fast risetime, the time resolution remains inherently limited, larger than 10 ns [109, 110].