Re: Linearity of SiPM's (fwd from George)

[Eugene Chudakov <gen@jlab.org>]

Hi,
I would like to discuss the numbers again. 
George wrote:
> Even in the completely unrealistic case of 10,000 photons incident on
> a single SiPM array and an equally unrealistic case of 18% PDE, Nocc
> = 1675.

The maximum photon energy the BCAL can see in GlueX is 2 GeV, released
in the most downstream part. Let us take the Elton's normalization of
1e4 photons/side/1.GeV for showers at the BCAL center. Such a shower
at the downstream end will make 2e4 photons/side/1.GeV.  From the
Irina's calculations I see that for 0.2 GeV photons at 20 degrees, the
maximum energy per a calorimeter cell (the fibers only) is 20 MeV.
The full energy for the cell is 20MeV/0.12=160MeV. So, there are cases
when nealy all the energy of the shower is absorbed in one cell. For
10 deg photons, the maximum energy might be a factor of 2 higher. On
the other hand a 2 GeV shower is deeper, so I assume that for a 2 GeV
shower at 10 deg the maximum energy is 160MeV*2000MeV/200MeV=1600MeV.
The number of photons is 1600MeV/1000MeV*2e4=32000 (about 3 times
higher than the George's upper limit) . The PDE*packing_factor is
about 0.18 (Hamamatsu).  This makes 5760 photoelectrons. The
number of pixels fired is 5480 (see the formula in the George's
mail). The non-linearity correction is 5760/5480=1.05. Let us assume
that the detector is calibrated at 500MeV. The appropriate correction
is 1.2%. So, we will have a 4% shift at 2 GeV. At 2 GeV one expects a
resolution of about 4%. So, the shift is not negligible. It can be
corrected well, unless we combine several SiPMs to one ADC. One should
point out that at 8 GeV the correction would be about 15%. In general,
such a limitation is annoying, since the calorimeter itself can cover
a much wider energy range.  Still it seems to be good enough for
GlueX.

Eugene

------------------------------------------------------
Eugene Chudakov
http://www.jlab.org/~gen
phone (757) 269 5352  fax (757) 269 5703
Thomas Jefferson National Accelerator Facility
12000 Jefferson Ave, Suite #4
Newport News, VA 23606 USA

On Thu, 16 Jul 2009, Elton Smith wrote:

>
>
> ---------- Forwarded message ----------
> Date: Tue, 14 Jul 2009 16:20:44 -0600
> From: George Lolos <gjlolos@uregina.ca>
> To: Hall-D Calorimetry <halld-cal@jlab.org>
> Subject: Linearity of SiPM's
>
> Hi all:
>
> During yesterday's video conference on the preparations for the read
> out review, the issue of SiPM non-linearity cane up.  The number of
> occupied pixels, Nocc is given by the relationship:
>
> Nocc = M x [1-exp(-PDE x Nph/M)]
>
> where M = # pixels in the array (typically 58,240 for the 16 cell A35
> type)
> Nph =  number of photons incident
> PDE = the array photo detection efficiency
>
> Even in the completely unrealistic case of 10,000 photons incident on
> a single SiPM array and an equally unrealistic case of 18% PDE, Nocc
> = 1675.   Now, one has to compute the probability that two photons
> are incident on the same pixel resulting in non-linearity.   The
> number is well below 1%.
>
> I hope this clarifies this issue.  One should also keep in mind that
> the numbers are more likely Nph not more than 4,000 maximum and the
> PDE something like 12%.
>
> George
>