<html><body>Hi guys:<div><br></div><div>I have been thinking again about the perceived need for linearity measurements capability in the calibration system. Let's look at the numbers a bit more carefully and, please, correct me if I am wrong:</div><div><br></div><div><b>Nocc</b> = <b>M x [1-exp(-PDE x Nph/M)]</b></div><div><br></div><div>Where: <b>Nocc</b> is the number of pixels occupied by processing incident photons</div><div> </div><div> <b>M</b> is the number of pixels in the array</div><div><br></div><div> <b>Nph </b>is the number of incident photons</div><div><br></div><div>The ratio, then, of <b>Nocc/M</b> is the degree of non-linearity. For example, if <b>M= 56,000</b> pixels, <b>PDE is 0.20</b> and <b>Nph ~ 50,000</b> per side per GeV, we get <b>Nocc/M ~ 8.5%</b>, assuming that ~20,000 photons go into only one array. So, the number of ~10% non-linearity that Elton often quotes is in the right ball park. So, the calculations are correct but is the input correct?</div><div><br></div><div>Let's start with the 56,000 ph/side/GeV. This number comes out of Andrei's calculations based on the number of p.e.'s we get from the fibers using the Sr90 source and Irina's simulations of the energy deposition in the matrix. However, these numbers were obtained with naked fibers, where the full effect of the outer cladding comes into play. In the BCAL, the fibers trap the light in the region between the core and the first to second cladding volume, so the trapping efficiency will be reduced. We will know the number of effective photons trapped in the BCAL when we get the cosmic ray measurements done. In any case, the number will be less.</div><div><br></div><div>Now come the really significant assumptions: </div><div><br></div><div>Even at oblique angles (less than 20 degrees) is it reasonable to expect that all the photons on one side will go into a volume confined in two dimensions within 2 cm x 2 cm? I have a difficult time accepting this and perhaps a real simulation of the shower generated by 1 GeV photon randomly distributed across the 2 cm width of a read-out cell will give us better answers. The Moliere radius alone is over 3 cm so spill over will occur even if the photon was incident dead center on a read-out cell. So, how many photons will be generated in a read-out cell as a fraction of the total energy deposited in the BCAL? We need a better result to draw conclusions on this.</div><div><br></div><div>Even if we take the 56,000 photons per side per GeV, this number corresponds to the number of photons arriving at the interface of the BCAL with the light guide. Here one has to start counting losses: losses crossing the glued interface from fibers to light guide. Then, and this is a significant number, losses from end of light guide to the SiPM. Unless I recall wrong, our measurements here between physical contact with Si grease between light guide and PMT, on one hand, and air gap on the other, the latter was around 50% of the former. In other words, just by the air gap we lose about 50% of the light. So, the realistic maximum number of photons reaching photosensors per side per GeV is much closer to 20,000 than to 56,000. If half of them are confined within the volume of one read-out cell (one SiPM), then <b>Nocc/M</b> is ~ 3.5%.</div><div><br></div><div>Is such non-linearity an issue to try and monitor if it means more complicated calibration system?</div><div><br></div><div>George</div><div><br></div></body></html>