Picosecond Timing-Planes for Future Collider Detectors

In addition to the general goal of achieving timing precision that is commensurate with the spatial precision of trackers, picosecond timing bears directly on physics applications in a future collider. We summarize some of these here, and address them in more detail in a later section.

In prior work we have reported picosecond-level timing precision of high-energy particle showers in a beam test using dielectric-loaded microwave rectangular waveguides, which generate band-limited microwave impulses when excited by a charge passing transversely through the waveguide ACE16 . The fast-impulse excitation of the lowest order $TE_{10}$ mode of the waveguide occurs because the charged bunch induces guided microwave Cherenkov emission in the dielectric, in our case $Al_{2}O_{3}$ or alumina, a very radiation-hard insulating material which is also extremely transparent to microwaves, with a dielectric constant of $\epsilon\simeq 10$ and a loss tangent $\tan\delta\leq 10^{-4}$. We use standard WR51 waveguide, with an inside cross section of $6.35\times 12.7$ mm.

Without dielectric loading, WR51 is normally used in the 15-22 GHz range, but after accounting for the dielectric loading, the single mode $TE_{10}$ center frequency is 6.5 GHz, with a bandwidth of $\sim\pm 1.5$ GHz. The resulting signals have risetimes below 40 ps, nearly an order of magnitude better than the fastest SIPM or LGAD risetime. Since these microwave impulses also couple directly to the current of the transiting particle bunch, there is no induced timing jitter due to any intermediate avalanche process.

The significant limitation of this microwave timing method is the relatively high turn-on threshold due to thermal noise limitations. Unlike optical systems where photon or shot noise presents the detection noise floor, in radio and microwave systems the effective system temperature $T_{sys}$ is well above the broadband quantum limit $T_{q}=h\nu/k_{B}$ for Planck’s constant $h$, frequency $\nu$, and Boltzmann’s constant $k_{B}$. In our case at 6.5 GHz, $T_{q}=0.25$K, and thus unless cooled well below 1K, microwave systems cannot achieve single-photon sensitivity. Rather, each mode of the electromagnetic field has a relatively high occupancy of thermal microwave photons. As a result, the induced field from a single transiting charge is undetectable above noise, but since the intensity grows with $(Ze)^{2}$ for a bunch charge number of $Z$, once this level is exceeded, a signal-to-noise ratio (SNR) adequate for picosecond timing precision is soon achieved.

For these reasons, our method is currently applied to electromagnetic showers, which rapidly convert single-particle momentum to a compact, many-particle shower bunch, dominated by $e^{+}e^{-}$ pairs, with a total number proportional to the primary particle energy. The Askaryan effect Ask62 , which selectively enhances the electrons over the positrons, then guarantees a charge excess of typically $\sim 25$% of the electron+positron number.

Our previous work ACE16 established the methodology and achieved timing precision approaching 1 ps, but due to limitations in our calibration of the number of electrons per bunch, the charge per bunch was not well-established. In the beam tests we report here, we have used a calibrated silicon pixel detector to precisely measure the transiting charge per bunch, establishing both the number of electrons and the bunch spatial distribution, an important parameter for determining the sensitivity of the method. We have cooled our system to liquid helium (LHe) temperatures to reduce thermal noise and better establish limitations imposed by it. We report also on the waveguide response to off-center charge transits. Combining various measures from these data with GEANT4 GEANT simulations, we then update our simulations of picosecond timing plane configurations that may be appropriate to the FCC-hh or other future colliders. We find that timing planes based on these methods are most appropriate for use in the high-rapidity regions of the detector, where radiation levels are most challenging, and a large fraction of the collision-induced particles retain very high energies.

In this report we first review the background theory and methods employed. We then detail our experimental setup for the beam test, and present the results of that test. These are analyzed to yield scaling relations which then anchor detailed modeling, including both GEANT results and microwave simulations. We then adopt a working design for timing planes for electromagnetic and hadronic showers, focusing our study on the forward region with pseudorapidity $|\eta|\geq 2.5$. We conclude our report by explore via simulation the timing capability and signal efficiency for several physics cases appropriate to the FCC-hh.

The cutoff frequency of rectangular waveguide is given by

where $a$ is the width of the waveguide, $\mu,~{}\epsilon$ are the permeability and permittivity of whatever fills the waveguide, and $c^{\prime}=c/\sqrt{\epsilon}$ is the speed of light in a pure dielectric where the relative permeability $\mu=\mu_{0}$. In our application, the length of the current element through the waveguide is thus $\lambda/4$ at the cutoff, and about $\lambda/2$ an octave higher in frequency where the next order modes begin to develop. This means the Cherenkov radiation always occurs in the limit of subwavelength track length, and the usual considerations of angular dependence for VC radiation are largely suppressed.

Under these conditions we can gain insight into the electric field strengths and time-domain response with analysis based on vector potential methods which have been used with success to describe radio Cherenkov emission from particle cascades FORTE ; PracticalAccurate .

Following reference FORTE , we start by modeling the transiting charge bunch as a point charge moving at the speed of light along the $z$-axis, with the $x$-axis along the longitudinal waveguide axis and $y$ the transverse axis. The current density is thus

with $J_{x}=J_{y}=0$. We express the current density in the frequency domain via a Fourier over frequency $\omega$:

where the tilde indicates the Fourier dual, and factor of two is required for consistency with the definition of the frequency domain electric field $\tilde{E}(\omega)$. The vector potential in the frequency domain can be determined from the Helmholtz equation

where we have assumed the relative permeability is unity. Since the current has only $z$ components, $\tilde{A}_{x}=\tilde{A}_{y}=0$; the wavenumber $k=n\omega/c$.

In the waveguide only plane waves propagate, with propagation constant $\beta=k\sqrt{1-(f_{c}^{2}/f^{2})}$, where we ignore waveguide attenuation here. A solution for observation point $\mathbf{x}$ along the waveguide longitudinal axis is

Assuming the observation point is some distance down the waveguide so that $x\gg b$, where $b=a/2$ is the height of standard rectangular waveguide, is satisfied,

and the resulting vector potential for the current traversing the waveguide from $z=-b/2$ to $z=b/2$ is

We assume that the charge profile is constant during the waveguide transit, thus $q(z)=q_{0}$. The result of the integral is:

From the frequency domain vector potential, we perform an inverse Fourier transform to recover the time domain vector potential $A(t)$:

This integral does not have a closed analytic solution, although it can be expressed in terms of a linear combination of exponential integrals. We thus integrate it numerically via a discrete Fourier transform, and the resulting electric field is then determined from a numerical derivative:

.

Fig. 1 shows the results of a comparison between these analytic results for the time-domain electric field, and a finite-difference time domain (FDTD) simulation using Remcom’s XF7.9 XF7 , for a single relativistic electron transiting alumina-loaded WR51 waveguide. In order to create the simulation, the relativistic electron is introduced as a series of short discrete current elements, typically 0.5 mm or less in length for this case, and arranged end-to-end transverse to the waveguide, with sequential delays corresponding to the average Lorentz beta for electrons at the critical energy in Alumina, $\sim 50$ MeV. These current elements are individually excited by a short current pulse with a peak current equal to the electron’s apparent current $-ec$. Since the results of the theory are only valid for the lowest-order $TE_{10}$ mode, we have applied a causal infinite-impulse-response filter (8th order) to isolate the signal in the 5-8 GHz portion of the band. These results have only a single degree of freedom in the comparison, and this is not arbitrary as we discuss below. The observed agreement in absolute amplitude is therefore also not arbitrary.

The FDTD results show a flat $TE_{10}$ passband signal, but also strong resonant response near the waveguide cutoff, and also at one of the higher-order mode cutoff frequencies. The vector potential theory, which assumes only propagating modes along the waveguide axis (eg., the $TE_{10}$ mode in the passband) does not show these effects. In the simulation, although the length of the current element due to the transiting electron is subwavelength, there is still power injected preferentially at the Cherenkov angle, about $70^{\circ}$ for alumina, and this power enters the waveguide about $20^{\circ}$ off-axis, where it can excite non-propagating modes. In addition, the TR at entry and exit, while exciting the $TE_{10}$ mode in part, also has forward and backward components that are far from the axis of propagation, thus also exciting these non-propagating modes near the cutoff frequencies. The additional low frequency power observed in the FDTD simulation but not in the theory is likely from these effects.

The vector potential theory predicts a rising spectrum within the $TE_{10}$ mode, and an extended high-frequency component (even after the filter) which is not observed in the FDTD simulation results. A spectrum that rises with frequency is characteristic of coherent Cherenkov emission in unbounded dielectrics SLAC01 ; Tak2000 ; thus we tentatively conclude that the flat spectrum observed in the simulation is a limitation of the FDTD application here. Requirements to maintain the validity of the FDTD simulation introduce one external ansatz: the shape of the current pulse. It can be a relatively short Gaussian, approximating a delta function, but computational gridding and meshing requirements require a minimum width to avoid artifacts. Thus the width of the Gaussian is determined by the limitations of the simulation space, and leads directly to an attenuation of high frequencies in the results. It is unclear whether this explains the differences; further work on higher fidelity simulations may elicit an answer.

Despite these differences, the agreement with the theory is remarkably good given the simplicity of the vector potential approach, and it does provide confidence that we understand the physics of the relativistic current coupling to the waveguide modes to first order. Varying the width of the current-element Gaussian within a moderate range, consistent with the gridding, results in variations of the FDTD simulation amplitude by typically 1-2 dB; we have used a mean value for the Gaussian width here. While it may be possible to refine the theory and simulations to get better agreement, the timing properties are largely unaffected by precise knowledge of the amplitude, as we will see later on.

It is interesting that, contrary to the usual analysis of Cherenkov detectors, the Cherenkov emission angle in this case plays only a secondary role, since only modes consistent with axial propagation are allowed. The dominant coupling to the waveguide can be understood completely in terms of the vector potential of a simple transverse current, truncated at the waveguide edges. In this case, the dielectric performs the role of creating a more compact geometry due to the increased refractive index, and the corresponding lowering of the waveguide frequency range. In the 5-8 GHz range where excellent low-noise amplifiers are available and attenuation losses are relatively low, an unloaded waveguide would be three times larger in its transverse dimensions, making a timing plane rather unwieldy, with correspondingly lower spatial resolution.

In our 2018 beam test, we had two primary goals: (1) to measure with better accuracy the number of electrons per bunch, to calibrate the response of ACE at low beam currents, and (2) to use ultra-low noise amplifiers (Low Noise Factory model number LNF_LNC4_8C with of order 2 K noise figure at LHe ambient) to determine the practical limits of ACE sensitivity. This latter goal required the use of liquid helium as a cryogen to achieve a 4-5K ambient temperature in which the entire detector was immersed, but in fact we expect that improvements in cold-head technology in cryogenics would allow for point-cooling of the LNA if the sensitivity afforded by this approach is deemed necessary. In practice we believe that even with liquid nitrogen or liquid argon cooling, ACE’s sensitivity is useful and effective for a range of future collider applications.

Fig. 2 shows a rendering of the cryostat and ACE elements, along with dimensions for the experiment. Included are the locations of the relevant components of the experimental setup (indicated in yellow boxes on the schematic):

1. 1.

   A silicon integrating pixel detector, the ePix 100 ePix ;

2. 2.

   A custom multi-pixel photon counting system (MPPC) coupled to a Cherenkov radiator;

3. 3.

   A 15mm thick tungsten-alloy block (alloy EF-17, 90% W, 7% Ni, 3% Fe, radiation length about 4.7mm) to pre-shower the bunches entering the ACE-3 elements;

4. 4.

   The cryostat containing the elements;

5. 5.

   The 3 sequential WR51 waveguide elements used in this experiment;

6. 6.

   The three original waveguide elements used in our first ACE experiment, which were placed to the rear of the dewar and used at room temperature only at high beam currents;

7. 7.

   A closed air-filled waveguide-short centered on the beam axis, coupled to a C-band (3.4-4.2 GHz) low-noise amplifier block.

Each of these was measured carefully for materials and thicknesses so that we could create an accurate GEANT4 model of showers that would develop.

Fig. 2(bottom) shows an example of a typical simulated shower: in this case a compact bunch containing ten 14.5 GeV electrons propagate through air from the beampipe exit to the tungsten pre-shower block and the resulting shower passes through the ACE elements within and external to the dewar. The tungsten-alloy block was about 3.2 radiation lengths thick resulting in efficient conversion of most of the electron energy into showers with minimal losses.

All of the detectors which appear on the downstream side of the ACE elements within the dewar were used primarily for triggering and diagnostics and their signals and waveforms are similar to our previous studies. They are not part of the scope of this report.

Originally we intended to submerge the tungsten pre-shower block in the cryogen to avoid the 20 cm gap between it and the ACE elements, but this was deemed impractical because of the large loss of liquid helium that would have been necessary to cool such a large mass. The showers that develop from the tungsten acquire large emittance, and are thus are not fully contained in the ACE elements, and we rely on GEANT4 simulations to estimate the effects of the shower development in this case. Fig. 3 shows the results of the average of many such showers in the configuration shown. Here we have plotted a slice through the radially symmetric distribution of excess electrons passing through each element, and they are fitted to a double-Lorentzian function, which was found to provide a reasonable empirical match, providing a type of “core + halo” combination that represented the data well:

where $N(r)$ is the radial density at distance $r$ from the beam center, here centered on the waveguide center axis, and the $p_{i}$ are the double-Lorentz parameters.

This shape will affect the emission properties of the radiation that is coupled to the waveguide, since it is clearly not consistent with our original assumption of a line-current on the center axis of the waveguide. These effects can be accounted for by a convolution of this form factor with the fields of a single charged particle (or line current) on axis; the coherently summed field, including phase factors for the offset positions of the tracks, is given by

Here $\mathbf{E}_{j}$ is the field from the $j$th particle, $\mathbf{x}_{j}$ is its position, and $\hat{r}$ the unit vector in the direction of observation.

The double Lorentzian function estimated here is specific to our setup, which included the large gap and additional cryogen between the pre-shower block and the ACE elements. We are interested in establishing the shower energy threshold for the case where the pre-shower block location is optimized, and to do this we also simulated a setup where the pre-shower block appears just upstream of the ACE elements, allowing for a much larger fraction of the shower to be captured, and for lower divergence of the shower itself. Fig. 4 shows the results of this GEANT4 simulation and the fitted functions, and it is very evident that the shower containment is much better, and that the compactness of the shower is and order of magnitude higher than previously. We will use these results to scale from our measured data to the more ideal case.

The choice of three sequential ACE elements rather than a more populous array was dictated by constraints on our procurement of ultra-low-noise microwave amplifiers; those acquired represent the current state of the art for such LNAs in an immersed LHe bath with noise temperatures of $\sim 2$ K over the 4-8 GHz band. These same LNAs have equivalent noise figures of around 8K at 77K liquid nitrogen (LN₂) temperatures; we utilized LN₂ for a portion of our tests. Since our detector elements have outputs on either end, to capture both propagating modes we shorted one end to produce a reflection of one of the propagating modes to the opposite output, as we had done successfully in prior studies. This has the effect of combining the two opposite propagating signals into a single waveform

Our previous studies ACE16 had showed good agreement between FDTD simulations and data, especially when all details of the as-built detectors were modeled carefully in the simulation. Thus we created a precise XF model of the 30 cm waveguides, with waveguide-to-coaxial adapters on either end, and a short at the output of one of the waveguides to produce a reflection from one end of the element, which is measured at the other end, and contributes to the timing measurements by adding the phase-inverted reflection to the detected waveform with a propagation delay proportional to the length of the element.

Any realization of a waveguide-to-coaxial adapter for a dielectric loaded waveguide must account for the much lower waveguide impedance due to the presence of the dielectric, and we thus developed a custom adapter to provide the necessary impedance matching. The coupler had a design efficiency of about 90% in power over the $TE_{10}$ band, but this does mean that the amplitude of in-band reflections from either end is of order 30%, which leads to some ringing of the system as the modes damp out. The coaxial coupling for higher-order modes is even less efficient, and thus such modes tend to ring more than the primary mode, although they represent out-of-band power that can be filtered to retain only the $TE_{10}$ mode as needed.

These effects are evident in the results of our FDTD simulation, as shown in Fig. 5. In the left column of the figure panes, we show the intrinsic field strength profiles as measured by a sensor probe placed in the simulation on the waveguide axis just prior to the coaxial adapter. This shows that the induced signal has a very strong initial impulsive broadband component, which in fact extends far above the nominal single-mode portion of the $TE_{10}$ spectrum. The trailing signals are due to strongly dispersive frequencies near the turn-on of two or more modes, including $TE_{10}$ at around 4.7 GHz. Very close to the waveguide turn-on, these modes are effectively non-propagating as their delays extend out far past the region of interest. The blue curve shows the primary transverse electric signal $E_{y}$, and red curve a subdominant $E_{z}$ mode that is excited by the field component of the Cherenkov emission that lies in the longitudinal direction.

For comparison, we also show the $E_{y},E_{z}$ waveforms after filtering to isolate the nominal $5-8$ GHz $TE_{10}$ operational passband. The sharp high-frequency impulse is now significantly reduced by the low-pass filter edge, and the high-pass edge has suppressed the non-propagating modes near the waveguide cutoff. In addition, the parasitic longitudinal modes are also suppressed. In Fig. 5 left bottom, the power spectral density of each of the waveforms above them is shown for both the unfiltered and filtered waveforms. The flat, broadband power of the initial spike is evident, continuing above 12 GHz, and the parasitic mode turn-on is also evident at about 8.2 GHz. With the 8th order time-domain causal filter used, both the parasitic and non-propagating modes are seen to be removed.

On the right-hand column of Fig. 5 we show simulated signals which are representative of what can be measured in our experiment, as the model includes a realistic coaxial adapter, necessary to couple the waveguide modes to a practical system using coaxial cables. We also model the shorted end, producing a reflection which must propagate about 3 times the distance of the prompt signal, and which thus acquires additional frequency dependent group-delay, changing the shape of the reflected waveform compared to the prompt pulse, which is itself somewhat dispersed as well. Because of impedance-matching imperfections, the signals now pick up some structure due to reflections at the couplings. The sequence of panes for the right column is the same as the left, with the exception now that there is no measurement of the parasitic $E_{z}$ mode, since the coaxial coupler can only respond to transverse modes. The strong modulation of the frequency domain power spectral density is due to the Fourier beating effects of the two time-domain peaks. As we will see in the next section, these results closely match what we have measured.

Our experiments were performed during the period from 14-24 October 2018 in the End Station A hall of the SLAC National Accelerator laboratory. Beam parameters were determined by the needs of primary users of SLAC at the time, the Linac Coherent Light Source (LCLS), for which bunches were supplied at bunch charges of around 0.25 nC ($N_{e}=1.6\times 10^{9}$), at a rate of about 100 Hz and beam energies of 14.5 GeV. End Station A’s test beam experiments received about 5 Hz of these same bunches, and beam tuning requirements are dictated by the primary users, leading to periods when we were unable to operate with stable conditions; data was not taken during these conditions to avoid uncertainties in the analysis.

As noted above, a key goal of these tests was to establish the absolute scale of the microwave emission, tied to the total energy of the transiting shower, which is the composite energy of the electron bunch. To do this we used the ePix camera ePix to determine $N_{e}$. The ePix 100, with a read noise of order $50$ e^- rms, gives an SNR of $>40$ for $\sim 8~{}$KeV photons, and detects single high-energy electrons with effectively no background apart from ambient cosmic rays or radiation. In practice its accuracy in the electron beam count measurement far exceeded our requirement for about 3% accuracy, given other experimental limitations.

Fig. 6 shows a composite camera frame for one of our runs during the test, approximately 2500 beam shots. Due to limitations in the collimation of the beam upstream of End Station A, the beam arrived with a non-uniform rectangular pattern as seen in the figure, with a Full-Width at Half-Maximum (FWHM) in each dimension of about $1\times 1.75$ mm. This shape was therefore encoded for use in estimating the effect of phase factors in reducing the waveguide response; the orientation of the waveguide side walls is shown in the figure with dashed red lines.

Fig. 7 shows the individual frame counts of electrons per bunch integrated over the detector frame, along with the distribution, for which the mean and standard deviation for this run is $N_{e}=121\pm 62.6$ electrons per bunch. The electron energy was 14.5 GeV, leading to a total shower energy of $1754\pm 907$ GeV.

Limitations of our data acquisition system prevented us from keeping up with the 5 Hz rate of bunch crossings in the detectors, and while the ePix detector was able to record every bunch, we were unable to synchronize our data to provide a precise measurement of the bunch energy for each of our detected events in ACE. The ePix data are thus used to estimate statistical quantities related to shower energy during the runs, thus achieving calibration of the response for the statistical ensemble of events rather than individual events.

The three ACE elements used for shower detection were fabricated with a newly designed waveguide-to-coaxial adapter, which was fully tested at 77K using liquid nitrogen prior to the SLAC experiment, but could not be tested to liquid helium (LHe) temperatures prior to the beam test due to constraints on LHe availability in Hawaii. The adapters were designed to accommodate to the thermal strains expected, and all survived at 4.2K, but two of the three elements experience a loss of transmission efficiency at the adapter interfaces, leading to reduced signal in the main pulse and larger degrees of ringing in the tails of the measured pulses. Fig. 8 shows the coherently averaged profile of the received signals for a range of runs similar to that shown in Figures 6 and 7.

Tests of the impedance matching of the third ACE element before and after LHe immersion showed that its adapters on either end remained within the acceptable range of at least -10 dB return loss ($>90$% power transmission efficiency) within the passband. The first two elements show factors of $\sim 2$ and $\sim 5$ lower amplitude coupling efficiency however, and the larger reflection amplitudes in the tails are also correlated to this effect. While these elements still preserve pulse shape and timing information, we do not use them for absolute scaling, and all amplitude-dependent quantities are referred only to the third element which retained its design efficiency.

The shape of these pulses, which include a 5-8GHz bandpass filter, compare well to predictions from the FDTD simulations in Fig. 5 above, reproducing the primary and reflected pulse as well as their dispersion, and the ringing in the tail which is due both to residual non-propagating modes and multiple reflections. These coherently averaged measurements provide effective template waveforms for estimating the location of the pulses for individual waveforms using matched filter or cross-correlation methods, under conditions where the signal may be otherwise buried in the noise. (In a practical timing plane application each detector would come with a calibration archive covering its entire active area; such archives would be developed using both direct measurements and validated time-domain models.)

We can approximate the observed SNR $S_{obs}$ in a single ACE element as a combination of the intrinsic SNR $S_{0}$ for the ideal case, and loss factors from two sources:

where $F_{\phi}$ is the loss factor due to partially destructive interference from the spatially dispersed form factor of the shower as it enters the ACE element, and $G_{gap}$ is the amplitude loss factor due to the dilution of the shower peak current density, arising mainly from the gap between the tungsten and the detector, but also affected by the additional intermediate material. The ideal case assumes that the shower is produced in a 15mm tungsten absorber just upstream of the ACE elements.

To determine these factors, we utilize the results from GEANT4 simulations of the ACE3 as-built system, shown in Fig. 3 above, compared to the simulations, shown in Fig. 4, where we remove the gaps and intermediate materials. These results are then also be convolved with the shape of the electron distribution as observed by the ePix camera, although it turns out that this convolution leads to only a few percent loss in amplitude.

To estimate the phase factors, we use a Monte Carlo (MC) numerical integral which distributes the electrons across the convolution of the GEANT4 shower distribution and the input electron distribution. The MC computes the electric field phase factor for the Alumina-loaded WR51 $TE_{10}$ mid-band frequency 6.5 GHz using:

where $E_{0}$ is the electric field per single electron centered in the guide, $k=(\omega n/c)\hat{z}$ is the wave vector along the waveguide axis, for frequency $\omega=2\pi f$ and index of refraction $n$, and $x_{j}$ is the position vector of the $j^{th}$ transiting electron at its internal trajectory midpoint as it passes through the waveguide and normal to it. The cosine term applies the half-cosine factor to the amplitude depending on the transverse location (taken as the $\hat{y}$ direction) in the guide, which has half-width $w$.

For the ACE CH3 element, which we use as the reference for all of these calculations because it had the best SNR and efficiency, the fitted values are shown in Figures 3 and 4 above. With the fitted parameters, the MC can then generate the relevant distributions of excess electrons. When this is done, we find that the amplitude factor for no gaps is $E_{net}^{(n)}=0.442$ and with the gaps and intervening dewar $E_{net}^{(g)}=0.232$, each with negligible statistical error. The ratio of the two gives the relative loss factor for the observed SNR:

The amplitude factor $G_{gap}$ is estimated using the ratio of GEANT4 estimate of the total excess charge for the two cases: using the slices above (which have adequate statistics), the ratio gives $G_{gap}=851/3223=0.264\pm 0.01$. (For later use, we note that our GEANT4 results for ACE CH1 and ACE CH2 give $G_{gap}=0.142,~{}0.191$ for those channels respectively).

Combining these we have for the observed SNR for the as-built system as a function of the idealized SNR for a close-packed system:

The combined loss factors reduce the observed amplitude by about a factor of 7 compared to an ideal, close-packed system.

To estimate the effective shower detection threshold, we first must estimate the observed typical signal-to-noise ratio (SNR) per event in our reference channel 3 for a known mean value of $N_{e}$ and electron energy. To get the observed SNR, we need to average over many events since the individual SNR per event is relatively low due to the loss factors. To ensure that our coherent signal averaging did not introduce additional unmodeled losses, we need to compensate for the trigger jitter, which was of order 1 ns since the trigger was based on the MPPC Cherenkov detector, and the risetime was quite slow compared to the ACE signals.

We have done this iteratively by first constructing templates with approximate averages, cross-correlating and averaging events to then improve on the templates, and repeating this process until the SNR of the combined average did not increase. To illustrate the importance of the template cross-correlation in ACE we show an example of a single event from run 312, where the waveform SNR in channel 3 is estimated to be $\sim 5.2$, in Fig. 9. On the left-hand side the raw, thermal-noise-dominated waveforms are shown, with the signal arrival time expected to be around -39 ns on the time scale shown. On the right side the results of the template cross-correlation are shown for each channel, and the signal is detected in each case at a substantially improved SNR, typically 50% or more. This step is also fundamental to the arrival time estimate, which is provided by the peak of the cross-correlation function.

For the sequence of runs used in this estimate, a total 2475 events were averaged for each waveform – this was the complete set of waveforms within the run set. No rejection of individual events was done so as to avoid any bias in the resulting SNR, and to facilitate comparison with the results of the ePix camera.

The result of the coherent sum for all three ACE channels is shown in Fig. 8 above, and the SNR per channel, measured by taking the ratio of the peak amplitude of the pulse to the RMS voltage of the waveform before the onset of the pulse, is 39.82, 22.65, and 106.43 for channels 1,2, and 3 respectively.

For thermal-noise dominated signals, the SNR grows as $\sqrt{N}$ for $N$ events, and we have confirmed this behavior for our data. For the reference channel 3, the average SNR per event is thus ${\bar{S}}_{obs}=(106.43/\sqrt{2475})=2.14$. For this observed SNR the implied idealized SNR $S_{0}$ for the third ACE element is

This value represents our estimate of the SNR per event in channel 3 for a close-packed system. Because our minimal timing layer design is a 3-element combination, we also estimate the equivalent SNR for the two upstream elements, if they had retained their full coupling efficiency.

Using the GEANT4 for the double Lorentzian for the two upstream elements in the ideal case, fits, shown in Figures 3 and 4 above, and recomputing $F_{\phi}$ and $G_{gap}$ for those elements, we find slightly higher expected SNRs for both of these than for the 3rd element, by factors of 1.430 and 2.010 for the 2nd and 1st elements, respectively. For a three-element coherent sum with independent LNAs (and thus additional thermal noise), the expected measurement SNR at a mean shower energy of $E=1754$ GeV is thus $(1+1.430+2.010=4.44)/\sqrt{3}$ higher than the single-element SNR, and thus

Since the signal depends linearly on the excess charge which in turn depends linearly on shower energy, the threshold for a $5\sigma$ amplitude measurement for a timing plane with three ACE elements in the shower is:

This assumes LHe cooling, and a pre-shower similar to what we have modeled here, with the three ACE elements in close proximity to the pre-shower block. The statistical error in this estimate, given the large number of events averaged, is at the several percent level and omitted here. The systematic error is at this point uncertain.

This threshold value is specific to the measurement of the signal amplitude, not the timing of the shower arrival. For shower arrival timing, we utilize template cross-correlation functions, as described in more detail below (section III.8), and this leads to a significant improvement in the energy threshold for timing. For 25 runs where the average amplitude SNR was $>2$ and the CCF to amplitude ratio could be accurately measured, we find the SNR improvement to be SNR(CCF)/SNR(V)$=1.53\pm 0.17$, where the error a convolution of both statistical errors (with 25 samples) and systematics associated with the non-linear CCF process.

Combining this factor with the amplitude threshold results in a $5\sigma$-level timing threshold of

where we quote an average standard error between the two asymmetric upper (+18 GeV) and lower (-14 GeV) errors.

We have one additional set of contiguous runs, enumerated 311 to 330, taken 24 Oct. 2018, for which we can repeat the amplitude analysis to get an independent estimate. For these runs, the shape of the electron spot as determined by the collimators was unchanged, but the average number of electrons was significantly higher per frame: $\langle N_{e}\rangle=263$.

The averaged waveforms are shown in Fig. 10 and the SNR for each averaged waveform was: ACE CH1: 47.0, ACE CH2: 30.4, ACE CH3: 154.8. The number of events averaged was 1,980 over the 20 runs.

For ACE channel 3, since the SNR grows as $\sqrt{N}$ for $N$ events, the average SNR per event is

.

For this observed SNR, since the loss factor terms $F_{\phi}$ and $G_{gap}$ are the same, the implied idealized SNR $S_{0}$ for the third ACE element is

The shower energy per bunch is now $E_{bunch}=14.5~{}{\mathrm{G}\mathrm{e}\mathrm{V}})\langle N_{e}\rangle=14.5\times 263=3814~{}{\mathrm{G}\mathrm{e}\mathrm{V}}$. Using the same GEANT4 results as the previous section, the idealized 3-element detector SNR per average event is thus:

The implied threshold for a $5\sigma$ amplitude measurement for these data, for a timing plane with three ACE elements in the shower is:

and the corresponding CCF timing threshold is

These two results taken together give a first-order estimate of the systematic error in both amplitude , and taking the arithmetic mean, and the variance, we find the ideal $5\sigma$ thresholds, for liquid Helium cooling on the whole detector, are

The equivalent ideal 3-element $1\sigma$ thresholds, or least-count energies $E_{\ell c}$ are

The equivalent ideal single-element $1\sigma$ least-count energies are

While the $1\sigma$ energy threshold values may be of only academic interest from the detection point-of-view, we will later address configurations of the timing planes that can better optimize for lower effective thresholds, and these values will facilitate scaling to these new configurations. In addition, since in practical applications timing planes will be used to estimate transit times for particles or showers with independently established parameters, useful information can be retrieved for SNRs well below the $5\sigma$ level.

Rectangular waveguide $TE_{10}$ mode electric fields have a characteristic half-cosine amplitude relative to the centerline of the waveguide, that is, if $x$ is the transverse coordinate (along the wider dimension $a$ of the rectangular section) relative to the center, the field strength $E(x)=E(0)\cos(\pi x/(2a))$. Since the shower current and corresponding vector potential is aligned with the $TE_{10}$ field, we may expect that the coupling coefficient would also have a similar dependence on the transverse offset of the current relative to the waveguide axis.

We investigated this response function during a portion of our tests when the beam was relatively stable in both the transverse bunch profile, and in beam current. Fig. 11 shows results of these measurements, with the top left pane displaying a two-dimensional image of the amplitude vs. offset and time for the main shower pulse. Waveforms have been aligned via cross-correlation, since they come from separate runs where phase alignment could not be maintained. Thus while the waveform shape is very similar for all of the offsets, we cannot yet confirm that the timing for such offsets is the same as for the center timing.

Related to the question of possible timing offsets associated with off-center beam centroids is the question of the degree of correlation between off-center-generated waveforms and the on-axis waveform. This may for example suggest that complete calibration of a waveguide element may require calibration for off-center waveforms as well as on-axis waveforms. Fig. 12 shows the normalized cross-covariance coefficient of all CH3 waveforms from the offset scans with an on-axis template waveform. The covariance coefficient saturates at about 0.83 in the central region, consistent with a slight decrease in covariance due to the residual thermal noise present. The covariance measure maintains this high degree of correlation for about 60% of the width, and then declines near the edges. Note that this normalized measure is to first order independent of the amplitude of the signals, but some of the decline is likely due to the lower SNR relative to thermal noise of these waveforms. This result suggests that in practice templates for offset positions could be used to advantage in a template library for shower measurements, but it also indicates that centerline templates would perform reasonably well even for showers near the waveguide edges.

Fig. 13 shows a summary of the amplitude variation with offset, along with a half-cosine curve for comparison. We have attempted to deconvolve the asymmetric shape function of the beam spot combined with the GEANT4 shower shape, but the results indicate that the deconvolution was only partially successful, as there remains an asymmetry in the response function. While the expectations are generally confirmed, precise details will likely require tests with a more symmetric and compact beam spot.

The falloff of the amplitude of the response of the waveguide elements near the edges motivates an improved design that includes layers with positions that alternate the centers and edges of the waveguides to ensure more uniform coverage. In a later section we will introduce such an updated design and assess results via simulations which are informed by these beam test results.

We have addressed the context within which our SLAC timing-plane measurements have been made; each of these amplitude-dependent issues sets scales which will be useful later in applying these results to simulations. For timing, the temporal shape of the waveform is the more critical parameter. To establish that we first use high-beam-current data where the signal is relatively strong, and create a waveform template by averaging many events over a portion of the observations where the beam and experimental configuration are stable. The resulting templates for each channel are then used to generate cross-correlation functions between the template and an individual event, as shown in the example from run 312 above (Fig. 9 in section III.6 above).

To extract timing information from these individual beam shots, the events and templates must be resampled to a finer scale than the original sampling, which is typically 20-40 Gs/second, or 25 to 50 ps/sample. This is done using numerical interpolation methods to sampling rates that will support picosecond precision, and is feasible and numerically stable since the signal is fully band-limited to $<10$ GHz. The other requirement is that the analog-to-digital converter system used to sample the signals must have a jitter level which is below the picosecond level, and a bandwidth that exceeds the $TE_{10}$ single-mode bandwidth. For sampling we have continued to use a Tektronix TDS6154C digital realtime oscilloscope with 15 GHz bandwidth and sub-picosecond jitter specifications. We have also cross-calibrated this scope with two other TDS6804B 8 GHz bandwidth oscilloscopes to verify the timing results.

The normalized CCF computed between a measured event waveform $w(t)$ and the template $\mathcal{C}(t)$, evaluated over a window length $T$ is given by

where $\tau$ is called the lag, or time offset of the product of the functions, and $\mathcal{A}_{w}(0)$, $\mathcal{A}_{\mathcal{C}}(0)$ are the autocorrelations of the waveform and template, respectively, at the zero lag $\tau=0$, which provide the normalization:

with a similar equation for $\mathcal{A}_{\mathcal{C}}$. In practice these equations are discretized at the sample period of the data after interpolating to a suitably fine reference grid, typically 1 ps/sample or finer. The peak of the CCF within the window then determines the absolute delay of the waveform. Normalization of the waveforms is not essential to the timing of the peak, but it provides an estimate of the correlation coefficient which can be used to determine a standard error for the delay time estimate.

Fig. 14 shows the results of single-element timing in the 3rd ACE element for all of the data for which we have good estimates of the composite bunch energy. The energy is scaled relative to the threshold energy, which for our data is affected by the loss factors detailed above. In the ideal case the single-element CCF timing threshold is $\sim 60$ GeV; in the non-ideal conditions of our experiment geometry the operating threshold was about a factor of 7 higher.

In the upper panel of Fig. 14 we show a two-dimensional histogram of the counts vs. relative energy vs. estimated delay, where the relative shower energy in each case is determined by the ePix camera scaling, and the timing is estimated via the CCF process described above. Since the distribution shows evidence for a shift in the mean value as a function of shower energy, the bottom one-dimensional histogram of the data is fit to a sum of two Gaussians, which matches the data well and shows a 1.2 ps offset that appears at higher energies, possibly due to slightly different beam steering for those bunches. The single-element standard deviation in the timing is 1.9 ps at the higher energies and grows to 4.1 ps at the low-energy end.

Fig. 15 shows the dependence of the single-element timing precision on the relative shower energy including a fitted function analogous to what is often used in estimating calorimeter performance:

where in our case the constant term $\alpha$ is consistent with zero, and the coefficients $\beta,\gamma$ for the square root and linear terms in energy are comparable. Since the timing precision is dominated by thermal noise, we can also expect the timing precision to improve with the number of elements combined as $N^{-1/2}$. Fig. 15 may thus be used in combination with the timing threshold energies reporting in Table 1 above to estimate timing vs. shower energy. Thus for example, for the 6-element stack (detailed in the next section) shown in Table 1, the least count energy in the ideal case with fill LHe cooling is 9.8 GeV, implying a timing precision of $\sim 4$ ps for $\sim 25$ GeV ($2.5\sigma$) showers, and under 2 ps for 100 GeV ($10\sigma$) showers. This analysis is oversimplified however, since it does not address the detection efficiency for showers, which will drop significantly below the $5\sigma$-level, and will also depend on the type of shower, whether hadronic or electromagnetic, as well as the location and pre-shower column density of the detector. We address these details in the next section with more advanced modeling and simulations.

Ultimately it is the single-element sensitivity which limits the energy threshold for our detectors or timing planes, since in any practical deployment, the thickness available to the timing plane will be subject to significant constraints. The single-element sensitivity is in turn constrained by the path length of charged particles transverse to it – the induced Cherenkov field strength in the waveguide grows linearly with the path length. However, waveguides cannot be of arbitrary dimensions while retaining acceptable modal structure.

In our previous report, we discussed the possibility of using square waveguides that are the same width (12.7 mm) but twice the height of normal WR51 waveguides. This geometry would support both a $TE_{10}$ and orthogonal $TE_{01}$ mode with twice the pathlength for particles within the waveguide.

We have further explored this option, and while it is feasible from the point of view of the waveguide modes, there are reasons why it is less than optimal in practice. Fig. 16 shows a plot of cutoff frequencies as a function of the waveguide height for Alumina-loaded waveguides with width fixed at the 12.7mm inside width of WR51. The cutoff frequency for a $TE_{nm}$ mode is given by

where the speed of light in the medium is given in terms of the absolute permeability $\mu$ and permittivity $\epsilon$, $a,b$ are the width and height of the waveguide, and we assume $b\leq a$. It is evident from the figure that increasing the height of the waveguide causes the mode frequencies to compress together in their range. At $b=a=12.7$ mm in our case, while we have doubled the pathlength, the single-mode bandwidth is now more than a factor of two smaller, with the $TE_{11}$ mode now infringing on that bandwidth from above. It is unclear whether this will in fact cause problems in practice, since the transverse current element source geometry presented by a shower are in general very inefficient in exciting these other modes.

To avoid this potential bandwidth issue, we have investigated two other ways to increase the shower pathlength in an element: the first is to simply use larger standard waveguide sizes, for example WR62 or WR75, which increase the shower pathlength by 22% and 47% respectively. The corresponding center frequencies decrease by a similar amount, as do the single-mode bandwidths available for detection. In addition, the lower center frequency leads to coarser timing precision due to the slower risetime. However, since the gain in sensitivity is linear with increasing shower pathlength (since the signal is coherent), and the loss of sensitivity goes only as the square root of bandwidth (since the noise is incoherent), there is still an improvement in sensitivity to be gained by this approach. Such trades may be acceptable for a given application, if sensitivity is more critical than timing.

A second option which we found to be more effective is to use a 2:1 E-plane waveguide coupler to combine the signal from a pair of stacked WR51 waveguides into a single LNA receiver at either end. This gains a factor of two in pathlength with no loss of bandwidth or change of center frequency. The two signals generated in the upstream and downstream waveguides are slightly out of phase due to the travel-time delay of the shower bunch as it passes between waveguides. At the center frequency of $\sim 6$ GHz the phase difference for the $\sim 23$ ps shower delay between elements leads to about a 10% loss in the combined amplitude compared to perfect phasing, an acceptable value relative to the alternative of two independent waveguides, where the signal gain improves only as $\sqrt{2}$. Although efficient combiners can be designed for more than two elements, the phase difference rapidly overcomes any gain in the combiner, and for more than two elements coherent combination is not possible, at least for standard rectangular waveguide elements.

Fig. 17 shows the geometry of such a combiner at one end of the stacked waveguide pair, in a section view from the side, with a FDTD simulation overlain showing the magnitude of the upper and lower E-fields as they arrive from the right and merge in the combiner. The combiner gives a good match over the $TE_{10}$ bandwidth of the system, with the output coupling to an SMA microwave coaxial 50$\Omega$ connector at the left. The slight delay of the lower compared to the upper waveguide signal is evident in the figure. This type of tapered combiner is also quite efficient at capturing the $TE_{10}$ mode of a square waveguide, and since it transitions to a normal WR51 section prior to the coaxial adapter, it may also be effective in suppressing unwanted square-waveguide modes.

Fig. 18 shows a decibel-scale plot of the relative voltage coupling in this case. In the simulation we were able measure the combined output, or to turn off the stimulus for the upper or lower waveguide respectively, or also to turn off the septum wall that separated the upper and lower waveguides, effectively creating a square waveguide operating in the $TE_{10}$ mode. This plot compares those four cases. The combined signals shows a relative strength that is about 5dB higher than the individual rectangular waveguide sections, consistent with the loss due to imperfect phasing. The amplitude from the square waveguide is quite comparable to the combiner amplitude, showing that the two methods largely arrive at similar outcomes. For now, the wider availability of standard WR51 waveguide leads us to use the dual-waveguide combiner as a reference design.

For purposes of this study we adopt a full timing layer thickness of $\sim 10$ cm as our reference design. This thickness is relatively small compared to either both electromagnetic and hadronic calorimeter designs now under consideration for the FCC-hh. A reference design using six layers of the combined dual-waveguide elements, with a depth of $10.2$ cm is shown in Fig. 19. Overlain on the figure is an electron-initiated shower, aligned with the interstices of some of the elements, showing how this shower still is fully captured by a minimum of three of the dual-waveguide elements.

The stackup of this timing layer consists of the paired waveguides as the basic element, with three such pairs aligned in the shower direction, and three more pairs with a half-width offset making up the complete stack of six pairs (12 single elements). This longitudinal base section is tiled and repeated along the transverse coordinate from the beam axis to fill in the region desired. The offsets are required due to the half-cosine response of each waveguide element, which diminishes their efficiency at the edges.

To minimize the column depth of material in the stack, we assume waveguides with 1mm thick aluminum walls. The balance of the material is made up by the alumina loads, and the total depth for electromagnetic showers is thus $1.35~{}X_{0}$, and for nuclear showers, $0.37~{}\lambda_{N}$. Copper waveguide walls would double the radiation length depth to $2.75~{}X_{0}$, and the nuclear interaction depth to $0.47~{}\lambda_{N}$. For the $\sim 13\%$ of tracks that align with the interstitial gaps, the depths would be significantly larger with copper walls; for aluminum, the depths actually decrease compared to those paths through the detectors because of the higher density of $Al_{2}O_{3}$.

The overlapping sub-layers guarantee measurement by at least three of the paired dual waveguides for showers aligned with the waveguide walls of one set, and better sampling for other showers at intermediate locations. A minimum configuration with four paired-dual elements is also possible with some increase in turn-on threshold, but for less than four, the sampling will be incomplete and the threshold higher as well. However, a configuration with several four element stacks arranged at longitudinal intervals to capture separate portions of a shower is also a viable possibility and could yield useful partial calorimetry which might supplement a nearby calorimeter, especially at the high-energy end where saturation of other detectors may be an issue.

From our cross-correlation analysis above, we find the effective $5\sigma$ CCF timing threshold for this stackup is

for events along the waveguide edges, and for the average event which showers within all six paired elements,

For particles above this energy showering in the timing plane, we expect to measure the shower arrival time to $\sigma_{\tau}\leq 3$ ps. At liquid nitrogen or liquid argon temperatures, the $5\sigma$ threshold will increase by about a factor of two.

Lower energy showers will be detected with some efficiency down to around twice the thermal noise level, thus even down to 20 GeV for this detector configuration, yielding timing resolution still better than 10 ps. These thresholds are still high compared to the sub-GeV least-count values for most collider detector instruments. For the FCC-hh, even at 100 TeV COM energies, there will be many particles of interest that will fall below detectability for these ACE timing planes, especially in the barrel region at low rapidity values.

However, in the forward, higher-rapidity portions of the detector, particles are far more likely to carry much of the primary beam energy, and thus we believe that these timing layers will be most effective as augmentations to forward physics measurements. The forward region is also that portion of the detector system where radiation exposure is most extreme. ACE timing planes, consisting almost completely of passive, radiation-hard materials, are also very well-suited to deployment in these regions.

One of the critical issues for our methodology is the variance of the delay of the effective electromagnetic centroid of the excess charge distribution of the shower relative to the arrival time of the initiating particle. We term this as the “electromagnetic centroid” since the observed microwave pulse is proportional to a sum of the field amplitude from each particle weighted by its phase factor, which itself is determined primarily by its arrival time delay as it passes through a given waveguide element.

In future beam test experiments the delay between the bunch EM centroid and the resulting shower EM centroid is observable if we use an upstream waveguide element to capture the bunch transit prior to the shower onset; this was however not the case for our 2018 experiment. For now we have done detailed GEANT4 simulations of the timing profile of the showers to measure the particle time of passage in simulation. We take this for now as a proxy for the EM centroid delay and its variance.

Fig. 21 shows the results of simulated shower particle exit times for 100 GeV photon showers in the pre-shower configuration described above. Results are shown for each of the six elements, indicated alphanumerically as 1A, 1B,…,3A,3B, with the number indicating the order in the 3-paired-element stack, and the letter indicating whether the element is first or second in the pair. In each case the light-travel-time offset has been removed to set the origin of the plot. We find the resulting arrival times are described reasonably well by a Wiebull distribution, which is fitted and shown as an overlain curve, with the fitted mean of the distribution indicated in the legend.

The probability density function for the Wiebull distribution Wiebull , here given as a function of time $t$ is given by:

where $\lambda>0$ is the scale parameter and $k>0$ is the shape parameter. The mean of the distribution, which gives the offset delay of the shower relative to the primary particle, is given by

where $\Gamma$ is the gamma function.

While the fitted mean value for the offset delay may be removed from the timing as a nuisance parameter, the standard error on the mean of these distributions is irreducible and represents the noise floor on timing measurements using the shower centroid as a proxy for the primary particle. To determine the mean and its standard deviation, we have used a Monte Carlo method which generates Wiebull random deviates based on the fits to $\lambda,~{}k$, and then using equation 13 to estimate the corresponding mean and its variance. For the photon showers, the typical standard deviation is around 0.1 ps, small compared to the picosecond timing goals here.

Fig. 22 shows the results of simulated shower particle exit times for 100 GeV pion showers, now measured in a timing plane between the EMCAL and HCAL detectors, thus with $1.7$ nuclear interaction lengths upstream of it. While the shower initiation is governed by the nuclear cross sections, the shower development rapidly becomes dominated by electromagnetic cross sections, and the resulting showers are much broader in time delays than for the photon-initiated showers with a relatively small material burden. The resulting fits show standard deviations of around 1 ps, which is comparable to our timing goals, and thus will appear in quadrature with it, and defines the timing noise floor for a layer at this location in the system.

One other issue regarding these particle delays must be addressed: the phase delay distribution of the secondaries leads to a partial loss of coherence in the summation of the individual particle electric fields. We can estimate this effect by looking at the Fourier spectrum of the delay distribution. Fig.  23 shows the effect of these delay distributions on the resulting field amplitude as a function of frequency, in the microwave range of interest. For the photon-initiated showers in the pre-shower region, the effect is completely negligible, due to the fact that all particles are quite prompt on the scale of 8 GHz (a cycle time of 125 ps), the upper frequency band limit for our detector.

For the pion showers, with a significantly longer tail, the effects are not completely negligible: a slope of about 0.5-0.7 dB across the 5-8 GHz band is evident, corresponding to an amplitude loss of 5-8% across the band. This effect is still quite moderate given the large material burden ahead of this timing plane, and does not significantly degrade the efficiency of the timing plane in this configuration.

The possible range of particle showers to be studied for timing purposes is very large, both in particle type and particle energy. We limit ourselves in this study to a small number of variations which are indicative of the expected results. For the FCC-hh, the dynamic range of desire measurements of outgoing particle momenta for a typical event is enormous: from below a GeV/c up several tens of TeV/c, four orders of magnitude or more. It is improbable that any single detector will be able to accommodate this entire dynamic range without several different operating modes.

For ACE, the limiting element in dynamic range is likely to be first-stage low-noise amplifier compression, but it is possible to achieve LNA dynamic range of order $10^{4}$ in amplitude ($\gtrsim 80$ dB) with careful design. For example the LNAs used in our 2018 tests (Low Noise Factory LNF-LNC4_8C models) have a noise temperature of 2.2 K at LHe ambient temperatures, and an output compression point of -12 dBm. The thermal noise of the LNA is equivalent to -100.4 dBm, which implies a nominal dynamic range of $\sim 88$ dB. As per our estimates above, the $1\sigma$ thermal noise level corresponds to 15-30 GeV signal amplitude depending on $T_{sys}$, and thus the ratio of kinematic limit of 50 TeV to our least-count energy is as high as 3300, a 70 dB range in microwave amplitude. It is evident that even current LNA technology easily covers the full dynamic range of events expected for the FCC-hh, with significant margin.

For our current study, the range of $\eta=2.5-6$ for the forward calorimeters where we consider our timing planes, results in significant boosts of the particles and the resulting showers observed in those regions. For a given transverse momentum $p_{T}$, the energy deposited in the calorimeters will be of order $E(\eta)\simeq p_{T}/\cosh{\eta}$, corresponding to factors of between 6^-1 and 200^-1. For our $5\sigma$ shower energy threshold of 50 GeV, the equivalent threshold in particle transverse momentum varies between $\sim 8$ GeV ($\eta=2.5$), to a fraction of a GeV for the highest pseudorapidities. In practice this means that our efficiency for yielding timing results will be much higher in the forward direction that it would in the barrel region.

We consider two different incident particles: photons and pions, at two different total energies, 100 GeV, and 1 TeV. Photons are of interest because they are not observed in the trackers, but will be efficiently observed via a preshower layer, and pions are ubiquitous in virtually every collision, creating hadronic showers, which will be observed in our second timing layer.

Fig. 24 displays a complete set of simulated data for a single shower event, for the case of a 100 GeV photon-initiation shower, in a timing plane placed after the preshower block and just prior to the EMCAL. For each event the simulation uses measured GEANT4 particle distributions in the center of a 1 meter long waveguide stack, as shown on the left figure column for the three dual layers. The simulation then convolves in the electromagnetic response based on our FDTD results as validated by our test beam data. The expected thermal noise floor is imposed on the data, and the signal is received at both ends (denoted left and right) of the waveguide. We then perform a cross-correlation using the response template waveform on each of the left and right received signals. The received waveforms and corresponding CCF are shown in rows for each of the three dual layers, followed by a coherently summed version for the left and right signals, and the corresponding CCF, at the bottom row.

For this event, in which the photon shower had deposited about 35 GeV in the preshower block, the individual element timing was above threshold for the second and third layers as the shower grew, but the signal in the first layer fluctuated below threshold. The coherently combined data showed a significant boost in SNR as expected, and the CCF shows another SNR increase as well.

We have demonstrated that picosecond timing of collider secondary particles, via a $\sim 10$ cm longitudinal sample of their electromagnetic or hadronic showers, is possible above a threshold energy – at least several tens of GeV or more – which is relatively high compared to the least-count energy of current collider detectors. While the FCC-hh will present an entirely new domain of extreme energy particles and showers, there will be many collision products that fall below our least-count energy. Optimizing the design for timing layers in the forward region mitigates this issue to a large degree, but it is still important to explore the strengths and limitations of our methods using examples that are tied to the physics goals of the FCC-hh. We will thus look briefly at several applications here.