Simulation of electron transport in a catoptric photomultiplier tube concept for large rectangular scintillator crystals

Spectroscopic scintillation detector form factors have been guided primarily by the design of commercially available photonic sensors. These devices, such as photomultiplier tubes, silicon photomultipliers, and hybrid photodetectors have underperformed in one or more areas such as size, power consumption, and resolution. A novel photomultiplier tube having a 50.8×152.4 mm2 rectangular window, utilizing a reflection-mode photocathode, and a low-gain, miniaturized dynode set is considered here to improve photosensor packaging while enabling high-efficiency, low-resolution scintillation spectroscopy with large, planar scintillators. Using a phenomenological multiphysics simulation process informed by empirical data, photoelectron collection efficiency, single-photoelectron response, electron transit time, and transit time spread have been modeled over a range of operating potentials. At 750 V between the photocathode and anode, 72.5% of photoelectrons are collected at the first dynode, and the average gain is estimated to be 805. The most probable transit time is 14.9 ns, with a transit time spread of 2.7 ns full-width at half-maximum.


Introduction
The desire to reduce the size of photodetectors for gamma-ray spectroscopy using scintillation crystals has led to development of metal-package photomultiplier tubes (PMTs) such as the Hamamatsu R11265 and the widespread use of silicon photomultipliers (SiPMs) [1,2].These smaller photon-sensing technologies meet the requirements for a small scintillator-crystal readout, but still have limitations in supporting large scintillation crystals or operating at elevated temperatures.SiPMs and small PMTs struggle to compete with large area PMTs with regard to photon sensing area and total photon collection.Tiling a large number of SiPMs can achieve competitive photon collection but suffers increased noise from dark counts and capacitance evident in the low-energy region of the pulse-height spectrum [2].Large scintillation crystals are often rectangular in shape to maximize sensor solid angle, but large PMTs traditionally have a circular photosensitive area.Spectrometer performance is compromised when the crystal optical window dimensions are not similar to the PMT photosensitive area.Common large-area, rectangular crystal geometries with optical-coupling dimensions exceeding the diameter of a typical circular PMT window suffer from resolution losses due to inefficient light collection [3].Additionally, large-area PMTs tend to suffer from photosensitivity nonuniformities [4].This may further increase energy resolution when mating to scintillation crystals.
Considering the challenges of producing a large, uniform photosensitive area while reducing photosensor volume, this work explores a novel PMT design exhibiting a large, rectangular photosensitive area which can more easily mate to rectangular crystals and achieve a smaller form-factor than large cylindrical PMTs.Electron transport in a novel reflective-photocathode PMT design is presented with focus on photoelectron transport toward a miniaturized linear-focus dynode set and the resulting signal amplification.With coarse assumptions made about secondary emissive material performance and uniformity of photosensitivity, estimations on signal and timing performance of the unique PMT design are presented.

PMT design
A challenge in mating optical sensors to high aspect ratio scintillator crystals is achieving uniform response along the width of the crystal.In this "catoptric" PMT design, a long, miniaturized linear-focused dynode set is considered which extends nearly the entire length of a rectangular optical window to maximize response uniformity out to the end of the window where collection losses are often observed.Such is the case in mounting small PMTs to larger windows.A goal in this design is to maintain a focusing electric field that is uniform along the entire length of the dynode set to minimize edge-effects produced by the outer envelope of the PMT.
Given the smaller size of the dynode set, photoelectron collection is reduced which has detrimental effects on energy resolution [5].To compensate, a high-performance reflection-mode photocathode is deposited on the PMT envelope.Reflection photocathodes have historically offered nearly double the quantum efficiency of their transmissive counterparts, while lower resistivity enhances linearity for high intensity and high count rates [6,7].Optimized transmission-mode cathodes must be semitransparent to balance the absorption of photons while limiting the absorption of photoelectrons born within the material.However, reflection-mode photocathodes emit photoelectrons from the side through which the light enters; in a way self-optimizing light-absorption and photoemission.Further quantum efficiency enhancement is observed when depositing a thin, semitransparent reflection-mode photocathode on a reflective substrate by extending the optical path through the material while limiting that of photoelectrons [3,6].Though modern techniques which offer precise control over the cathode growth and activation processes have produced transmissive "ultra-bialkali" photocathodes with performance comparable to, or exceeding, those of reflection mode cathodes, the process is not well described and likely quite difficult to implement with sufficient uniformity to achieve spectrometer resolution enhancement [3,7].However, the application of these advanced processes do not negate the inherent advantages of the reflection-mode photocathode.To the authors' knowledge, no large-area, reflection-mode, ultra-bialkali photocathodes are commercially available, but a catoptric design could provide a unique application space.Given historical comparisons, advanced cathode growth techniques utilized on reflective substrates could produce extremely high QE bialkali photocathodes further improving scintillation spectrometer resolution, limited only by material nonuniformity [5,8,9].
One consequence of high-efficiency radiation detection is that power consumption scales proportionally to count rate.Power consumption also scales with power supply voltage and multiplication within the PMT.An effective way to combat the power consumption increases due to high detector efficiency is to shift the signal burden from the photomultiplier onto a low-power, charge-sensitive amplifier.In the Catoptric PMT, the multiplier gain is reduced by limiting the number of dynode stages present within the device and, to a lesser extent, by reducing the operating voltage held between the photocathode and anode.Limiting the number of dynodes also allows for an overall size reduction, desired for mobile detection applications.
The Catoptric PMT is comprised of four primary components.The rectangular window serves as an optical interface to the scintillation crystal as well as an electrical feedthrough.A metal envelope serves as an optical reflector, electrostatic focusing lens, and reflective photocathode substrate.The curved photocathode helps guide photoelectrons to the first dynode.The envelope design is reminiscent of a Winston cone described and similarly used in other works [6,10].Figure 1 shows a cross-sectional view of the Catoptric PMT componentry.
-2 - Additional focusing electrodes -designed and positioned to limit optical absorption -are held at specific voltages to steer photoelectrons towards the miniaturized first dynode.Table 1

Multiphysics simulation methods
To iterate design and estimate performance prior to fabrication, a number of charged particle simulations were first performed in two-dimensions and then again in three-dimensions using an accurate model of the final geometry imported to COMSOL Multiphysics® [11].Each simulation consisted of two parts.First an electrostatic simulation was used to establish the unique electric field produced by the Catoptric PMT.The 3-dimensional solution of the electrostatic simulation is fed into the charged particle simulation which transports electrons through the supplied electrostatic field solution.
Electrons striking a miniature dynode stage with sufficient energy yields one or more secondary electrons.Charge multiplies and cascades through the dynode chain until they reach the anode where they are collected.A method of simulating electron transport within PMTs for COMSOL previously -3 -developed by the authors was updated to include more accurate photoelectron and secondary electron energy distributions, as well as improved secondary electron yield (SEY) sampling based on standard simulation methods and available data [12][13][14][15][16][17][18][19][20].The improved model better represents single electron gain variability while maintaining accurate timing characteristics.
To capture the device's collection efficiency (CE) and gain performance, a number of simulations were performed with photocathode-anode voltages ranging from 200 V to 1500 V, wherein 1,000 photoelectrons were released from pseudorandom locations across the photocathode and the pertinent values were recorded.For the purposes of this work, CE refers to the ratio of the number of photoelectrons collected at the first dynode to the number released.Gain refers specifically to the photomultiplier gain, which is the average ratio of charge collected at the anode to the charge released from the photocathode.
The photocathode was then divided into 100 regions of equal area.500 photoelectrons were distributed randomly within each region and released simultaneously to quantify the novel design's spatially-dependent CE and gain uniformity.For each simulation, the design's inherent plane of symmetry was utilized to reduce the number of degrees of freedom, thereby reducing simulation time.
Lastly, to quantify transit time at different operating voltages, simulations were performed at 550 V, 750 V, 950 V, 1150 V, and 1500 V.For each, 50,000 photoelectrons were released from pseudorandom locations across the photocathode.Secondary emissive yield was reduced to 1, and each collected electron was regarded as the peak of a single electron pulse providing signal transit time.The rate of electron collection at each operating potential is representative of the distribution of transit times and was normalized at each operating potential for comparison.

Results
The results of 750 V electrostatic and charged particle tracing simulations is shown in figure 2. The electrostatics surface plot illustrates the electric potential distribution, while the overlaid black lines indicate a 20 V/div equipotential distribution within the vacuum.The large spacing between the photocathode and 20 V equipotential line is indicative of a relatively low electric field, and the effects can be seen through the charged particle tracing results, below.

Collection efficiency and gain
The average gain and CE measured over the entire photocathode area monotonically increases with higher operating potential.It is evident that higher electric fields near the photocathode more effectively guide photoelectrons towards the first dynode.Below, in figure 3, simulated gains and CEs are displayed for many operating potentials.To meet gain and power consumption targets, the device should be operated between an estimated 550 V and 900 V.
Using the advanced simulation techniques, single-electron response (SER) has been tabulated for 2,500 unique photoelectron releases.These results, plotted in figure 4, were obtained at 750 V operating potential.A minimum response of 2 electrons was included.The primary peak, mean of 1437, represents the most likely full PMT gain, whereas a collection of smaller pulses between 2-400 are products of backscatter from dynode 1 and imperfect dynode collection efficiency.The excess noise factor (ENF), which helps determine gamma-ray spectrometer resolution, may be calculated -4 -  from the SER distribution: where  is the SER and ⟨⟩ is the photomultiplier gain.The simulated Catoptric device ENF is 1.29; typical PMT ENFs range from 1.1-1.5.A lower ENF is preferable for high-resolution spectroscopy, however large devices typically have higher values attributed to electron-transport and material nonuniformities [3].The single electron charge resolution (FWHM) is 81.8%, which is comparable to or better than other large-area PMTs [3,21], though this value does not account for the spatial nonuniformity of SEY.Catoptric SER and spatial uniformity results will be used in conjunction with photon transport to better characterize scintillation spectrometer response in future work.

Collection efficiency and gain uniformity
Gain and CE dependence on photoelectron release position were quantified over 100 regions of equal area across the photocathode.Simulations were performed at 750 V operating potential.Figure 5 -5 - illustrates the Catoptric device response uniformity.Photoelectrons released near the sidewalls are not as effectively guided towards the first dynode.Additionally, photoelectrons released from the areas with the lowest performance approach the first dynode from the "back" side leading to overshoot, as seen in figure 2. Nearing the corners, both collection and gain are improved as the substrate electrostatically guides photoelectrons on a favorable trajectory towards the center.Contrarily, approaching the midline near the endwall, the low electric field cannot as efficiently steer photoelectrons.

Timing
Transit-time distribution for various operating voltages are displayed in figure 6. Transit time and transit-time spread (TTS) are both reduced at higher K-D 1 voltages.Lower extraction fields near the photocathode periphery (figure 2) result in longer transit times, as seen in the distribution.In each case, the time required to reach the first dynode is approximately 83% of the total transit time, and the TTS -6 -at D 1 ranges from 50-60% of the distribution width at the anode -decreasing with higher voltage.At 750 V operating potential, the peak transit time is 14.9 ns, while the TTS is 2.7 ns full-width at half-maximum (FWHM).Compared to typical 50.8 mm diameter, 8-10 stage linear-focused devices, the transit time is faster due to the reduced number of stages, while TTS is larger -primarily because of the large area photocathode and suboptimal front-end electron-optics.For reference, commercial linearfocus PMTs have transit times of 16-50 ns and TTS ranging from 0.37-1.1 ns at operating voltages between 750-2500 V [7].Although relatively long tails are present in each of the distributions below 1500 V, the Catoptric device's transit time is sufficient for fast inorganic scintillators, such as LaBr 3 (Ce).Higher electric fields near the photocathode at 1500 V greatly reduce transit times for photoelectrons released near the edges.

Conclusions
An advanced multiphysics simulation method has been implemented and used to design a novel photomultiplier tube for use in gamma-ray spectroscopy.The new design incorporates a reflectionmode photocathode to increase QE and a miniaturized linear-focused dynode set with four stages to reduce light absorption at the optical interface.Optically reflective side walls serve to focus photons on the photocathode while the electric field guides photoelectrons to the first dynode.Photon collection is sacrificed for higher QE which negates spectroscopic resolution losses.Having fewer multiplication stages, one may shift the signal burden from the photomultiplier to a charge sensitive amplifier which results in extremely low power consumption.At 750 V, average CE is 72.5%, photomultiplier gain is estimated to be 805, and the ENF is 1.29 for a uniform photoelectron release.The most probable transit time is 14.9 ns, and the TTS is 2.7 ns at FWHM.Compared to typical linear-focus PMTs used for gamma-ray spectroscopy, the Catoptric device is faster and exhibits a typical TTS [7].The miniaturized dynodes and large, reflective photocathode inhibit faster front-end electron optics and are the primary contributions to the timing characteristics.Future work will combine simulated performance metrics with realistic optical simulations and QE nonuniformity to provide an estimate of spectroscopic resolution.

Figure 1 .
Figure 1.A section view of the Catoptric PMT assembly with major components labeled.
describes the equivalent resistive-divider network used to define electrode potential within the simulation.The external dimensions of the Catoptric PMT are 162.6 mm × 61 mm × 52.5 mm for an optical interface of 50.8 mm × 152.4 mm, though further development could yield a range of PMT sizing options.The 8 mm × 150.8 mm dynodes are mounted lengthwise near the center of the optical window.This obstruction is anticipated to marginally reduce photon collection and have minimal performance impact based on comparisons to existing techniques which leverage photodetectors that do not fully cover the crystal face of larger crystal packages.Coupled gamma-ray and optical photon Monte Carlo simulations necessary to quantify the dynode structure's effect on spectrometer resolution are currently in development and considered future work.

Figure 2 .
Figure 2. Left) Electrostatics simulation of the Catoptric PMT at 750 V operating potential plotted on the plane of symmetry.Right) Electron transport and multiplication at 750 V operating potential and 14 ns after release.Note that some photoelectrons released near the edges overshoot the first dynode.

Figure 3 .
Figure 3. Average gain and CE of the Catoptric PMT from 200 V to 1500 V operating potential for photoelectrons pseudorandomly distributed over the entire photocathode area.

Figure 4 .
Figure 4. SER for 2500 unique photoelectron releases.Minimum included response is 2 electrons.

Figure 5 .
Figure 5. Left) Spatial uniformity of collection efficiency.Right) Spatial uniformity of gain.Heat map values correspond to equal-area regions from which photoelectrons were released.

Figure 6 .
Figure 6.Transit-time distributions for single-photoelectron pulses at the anode for various operating voltages.Higher electric fields near the photocathode at 1500 V greatly reduce transit times for photoelectrons released near the edges.

Table 1 .
The equivalent resistive divider used for Catoptric PMT simulations.K represents the photocathode, D  represents the n th dynode, and A represents the anode.A common resistance, R, is used to divide the operating voltage amongst the various electrodes.