Table A1.
Glossary of variables used in the calculations. All times in ns. All distances in cm.
| a | Fraction of light received by a photodetector when the interaction point is at the same end of the scintillator and the opposite end is absorbing (equations (1a) and (1b)) |
| c | Speed of light in a vacuum (29.979 cm ns−1) |
| DAk | Optical photon time dispersion parameter from photons emitted at depth Zk and either absorbed at rear surface B after being reflected at entrance surface A (equation (8b)), or absorbed directly at entrance surface A (equations (6a) and (9a)) |
| DBk | Optical photon time dispersion parameter from photons emitted at depth Zk and either absorbed at entrance surface A after being reflected at rear surface B (equation (6b)) or absorbed directly at rear surface B (equations (8a) and (9b)) |
| d1 | Constant coefficient for optical photon time dispersion (equations (2a) and (2b)) |
| d2 | Quadratic coefficient for optical photon time dispersion (equations (2a) and (2b)) |
| δAjn | Trigger delay from the arrival of the first possible photon at photodetector A (from an interaction at Zj) to the trigger time TAkn. |
| δBjn | Trigger delay from the arrival of the first possible photon at photodetector B (from an interaction at Zj) to the trigger time TBkn. |
| EAkn | Estimate of the entrance time of the annihilation photon using TAkn and correcting for the depth-dependent annihilation photon, optical photon, and trigger delays (equation (11a)) (should be zero with perfect timing) |
| EBkn | Estimate of the entrance time of the annihilation photon using TBkn and correcting for the depth-dependent annihilation photon, optical photon, and trigger delays (equation (11b)) (should be zero with perfect timing) |
| EABkn | Simple average of the corrected entrance times EAkn and EBkn for double-ended read-out (equation (12)) |
| EWABkn | Inverse-variance weighted average of EAkn and EBkn (equation (13)) |
| fA(Z) | Fraction of photons that reach surface A directly from an ionization event at depth Z (equation (1a)) |
| fB(Z) | Fraction of photons that reach surface B directly from an ionization event at depth Z (equation (1b)) |
| Fn | Fractional trigger level (tabulated from 0 to 1 on index n) |
| Fopt | Fractional trigger level for minimum CRT (fraction of peak amplitude P) (figure B1) |
| H | Number of photoelectrons contributing to the photodetector output pulse at the trigger time (table B1) |
| J | Single photoelectron time jitter of the photodetector (Gaussian fwhm) |
| L | Length of scintillator (distance between surfaces A and B) |
| μ | Attenuation length for 511 keV annihilation photons in the scintillator |
| mAk | Observed number of photoelectrons in photodetector A from an interaction at depth Zk (randomly distributed with mean = NAk, variance = NAk) (equation (4a)) |
| mBk | Observed number of photoelectrons in photodetector B from an interaction at depth Zk (randomly distributed with mean = NBk, variance = NBk) (equation (4b)) |
| n | Index of refraction of the scintillator at the wavelength of the scintillation light |
| Ng | Number of ionization events used in each Monte Carlo calculation |
| Npe | Number of photoelectrons produced in the photodetector (single-ended readout) or the sum in both photodetectors (double-ended readout). This is the product of the scintillator luminosity (photons/MeV), the energy deposited (511 keV), the photon transport efficiency, the photodetector fill factor, and the photodetector quantum efficiency. |
| NAk | Expected number of photoelectrons in photodetector A for an interaction at depth Zk (equation (3a)). |
| NBk | Expected number of photoelectrons in photodetector B for an interaction at depth Zk (equation (3b)). |
| PAk(max) | Maximum amplitude of the photodetector A output pulse for interaction k |
| PBk(max) | Maximum amplitude of the photodetector B output pulse for interaction k |
| P(t) | Photodetector output pulse amplitude at time t summed over all earlier photoelectron responses (equation (7)) |
| Rm | Random number from a set uniformly distributed between 0 and 1 |
| S(t) | Photodetector SER (equation (5)) |
| Sr | Rise time of SER bi-exponential (equation (5)) |
| Sd | Decay time of SER bi-exponential (equation (5)) |
| τd | Decay time of the scintillator |
| τr | Rise time of the scintillator |
| TAkn | Trigger time of photodetector A for interaction k at trigger level Fn PAk(max) (sections 4.2.7, 4.3.7, and 4.4.7) |
| TBkn | Trigger time of photodetector B for interaction k at trigger level Fn PBk(max) (sections 4.2.7, 4.3.7, and 4.4.7) |
| TABkn | Simple average of TAkn and TBkn for double-ended readout. This corrects for the depth-dependent variation in optical photon delay but does not correct for the variations in annihilation photon or trigger delay. (section 4.4.8) |
| Tm | Creation time of the photodetector output pulse from photoelectron m after the arrival of the annihilation photon at the entrance surface (sections 4.2.5, 4.3.5, 4.4.5). |
| VAjn | Variance in the corrected trigger times EAkn averaged over bands of Zj |
| VBjn | Variance in the corrected trigger times EBkn averaged over bands of Zj |
| WSA | CRT for single-sided readout with the photodetector at surface A. Calculated as 1.4142 times the fwhm of the distribution of TAkn at the optimal trigger fraction (section 4.2.8). |
| WSB | CRT for single-sided readout with the photodetector at surface B. Calculated as 1.4142 times the fwhm of the distribution of TBkn at the optimal trigger fraction. |
| WDAB | CRT for double-ended readout using a simple average of the TAkn and TBkn for each interaction k and trigger fraction Fn. Calculated as 1.4142 times the fwhm of the distribution of TABkn at the optimal trigger fraction (section 4.4.13). |
| WEA | CRT for double-ended readout using the optimal trigger time of only photodetector A but correcting for the estimated depth Ẑ of the distribution of entrance time estimates. Calculated as 1.4142 times the fwhm of the distribution of EAkn at the optimal trigger fraction (section 4.4.13) (should be zero with perfect timing). |
| WEB | CRT for double-ended readout using the optimal trigger time of only photodetector B but correcting for the estimated depth Ẑ of the distribution of entrance time estimates. Calculated as 1.4142 times the fwhm of the distribution of EBkn at the optimal trigger fraction (section 4.4.13) (should be zero with perfect timing). |
| WEAB | CRT for double-ended readout using a simple average of EAkn and EBkn. Calculated as 1.4142 times the fwhm of the distribution of EABkn at the optimal trigger fraction (section 4.4.13). |
| WWAB | CRT for double-ended readout using the inverse weighted average of EAkn and EBkn. Calculated as 1.4142 times the fwhm of the distribution of EWABkn at the optimal trigger fraction (section 4.4.13). |
| WZ | fwhm of the distribution of differences between Monte Carlo depth Z and estimated depth Ẑ (section 4.4.15) |
| Zk | Depth of interaction for interaction k. Randomly distributed according to exponential attenuation (section 4.1.1) |
| Ẑ | Depth of interaction estimated from observed photodetector photoelectron counts mAk and mBk (equation (10)) |