Tagger Time Resolution

The Tagger Microscope is expected to have a time resolution of 200 ps or better. This specification imposes a strict demand on the light pulse rise time uncertainty because it introduces trigger time uncertainty when Constant Fraction Trigger (CFT) point is used.

Scintillation Pulse Time Uncertainty

The scintillations in the fiber are distributed exponentially with a characteristic decay time $\tau$ = 2.7  for BCF-10/20. Since the variance is $\tau ^{2}$ (and is additive over independent events) the time uncertainty for N generated photons becomes $\tau /N^{1/2}$. Thus, in order of priority, this relation implies

1. scintillating fiber with smallest decay time must be procured
2. photon capture and detection efficiency must be optimized

Photon Capture and Detection

Let us evaluate the photon detection choices available: scintillating fibers BCF-10, BCF-20 and photo-sensors SSPM-0606BG4-PCB (SSPM~06) from Photonique and Hamamatsu's MPPC S10362-11-050C (MPPC 050).

Taking the accepted photon yield in plastic scintillator to be 8000γ/MeV, the following considerations control the total photon capture and detection:

• taking the energy deposition in plastic (~1 $g/cm^{3}$) to saturate at ~2 MeV/cm with 2 cm length of scintillator yields 4 MeV
• The critical angle in multi-clad BCF scintillating fibers is 27.4^o C resulting in forward capture of 5.6%
• Frequency-dependent efficiency functions must be taken into account. Using nominal efficiency specifications, SiPM PDE distribution weighted by scintillator (BCF-10 or BCF-20) emission and BCF-98 waveguide transmission spectra is integrated over all wavelengths. These distributions are shown in adjacent figures.
• Assuming for simplicity a uniform light beam profile emerging from the waveguide, the output is rescaled according to active area available on the SiPM (e.g. 1 mm² for Hamamatsu MPPC compared to 4 mm² waveguide cross-section)
• Correction introduced for the saturation effect of the firing pixel population. When an appreciable fraction of available pixels are firing, the possibility of a photon hitting a firing pixel can no longer be neglected. This can be expressed as hollows:

$N_{fired}=N_{total}\left(1-e^{\frac {-N_{\gamma }\cdot PDE}{N_{total}}}\right)$

• Finally, time resolution can be calculated based on the relation from the discussion above. Note that we must use the actual number of the pixels that fired for our N.

Actual number of pixels fired BCF-10 BCF-20 MPPC 188 154 SSPM~05 25 58 SSPM~06 162 299

Leading to time resolution:

Time resolution (ps) BCF-10 BCF-20 MPPC 235 254 SSPM~05 525 353 SSPM~06 211 151

This rough calculation shows that only Photonique's SSPM ~06 with BCF-20 and possibly Hamamatsu's MPPC 050 with BCF-10 will conform to timing requirements.