The lowest set of points in Figure 22 show the resolution of the
time difference between the pulse height weighted average of the 2 PMTs on the
BC408/R2490 3m counter and the trigger counter time. The resolution is about 110
ps and is nearly independent of where the particle passes through the counter.
Similarly Figure 24 shows a plot of the same quantity
for the same BC408 bar but instrumented with R5946 PMTs. In this case the
resolution is about 135 ps near the middle of the bar and improves somewhat near
the ends. Using these resolutions, the measured trigger counter resolution, the
estimated contribution due to the finite width of the trigger counter, and the
expected 
accuracy in the actual experiment we can estimate the resolution
the final system should attain.
The pulse height weighted time difference plotted in Figure 22 is
t = A_Lt_L + A_Rt_R A_L + A_R -t_C2
where 
and 
are the pulse heights of the Left and Right PMT as
measured with the ADC's, 
and 
are the times measured in the
TDC's for the Left and Right Counter and 
is the time measured in the
trigger counter C2. From this formula we can write down an expression
that includes the various contributions to the observed resolution.
= A_L^2 (A_L + A_R)^2(L^2 +TDC^2) + A_R^2 (A_L + A_R)^2(R^2+TDC^2) +C2^2 +width^2
where 
, 
and 
are the expected time
resolutions of the Left, Right, and trigger (C2) due to photostatistics and
the transit time spread of the tube.
In what follows we assume that the tubes have
identical performance (
) and that the intrinsic
resolution due to photostatistics scale as 
where A is the pulse
height. If we define 
to be the expected time resolution when the
particle passes through the center of the 3 m counter with observed pulse height
then 
and
. Similarly, 
is the
RMS measurement error due to nonlinearity from the LRS 2228 TDC. We also note
that this term does not contribute to the trigger counter time resolution since
the trigger counter C2 was used to provide the common start for the TDC. The
term 
term is included to take into account the time smearing
due to the finite width of the trigger counter. For this case the trigger
counter width was 2 cm, using the measured propagation velocity in the 3 m
counter of 15 cm/ns corresponds to 
38.5 ps. Rewriting the
above equation in terms of 
we get:
= A_C A_L + A_RPS^2 + A_L^2 + A_R^2 (A_L + A_R)^2TDC^2 + C2^2 + width^2
To get a feeling for the contribution due to photostatistics we can examine the
case where the particle passes near the center of the counter so that 
. Again, as mentioned above, we assume that all tubes have equivalent
performance for the same number of observed photons then this equation
simplifies to:
= (1/2)PS^2 + (1/2)TDC^2 + C2^2 +width^2
or
= (1/2)PS^2 +X^2
where now we use the symbol 
represent all contributions to the
resolution except photostatistics. We have measured 
for two cases,
the BC408 bar instrumented with R2490 tubes and with R5946 tubes. With the
assumptions mentioned above, the two measurements should only differ because
the R5946 collects less light than the R2490, then we can use that information
to estimate the contribution to the resolution from photostatistics and from
"everything else". In particular we expect the resolution component due to
photostatistics and tts to scale like the square root of the number of photons
collected and therefore like the square root of the photocathode area.
Using 
110 ps for the R2490 case and 
135 ps for the
R5946 gives us the following two equations:
110^2 = (1/2)PS_2490^2 +X^2
and
135^2 = (1/2)PS_5946^2 +X^2 = (1/2)(area ratio)PS_2490^2 +X^2
The R5946 has a 27 mm diameter photocathode compared to the 38 mm photocathode
of the R2490 so the area ratio is 
.
Solving these equations yields 
ps and
ps.
Solving for "everything else" gives 
ps.
This number is in good agreement with our estimate of 
based upon
adding each measured or estimate contribution to 
in quadrature.
Namely we expect
X^2 = (1/2)TDC^2 + C2^2 +width^2
To measure the trigger counter resolution we have performed the following test.
We fabricated two identical trigger counters and arranged them as show in
Figure 34. Each trigger counter had a 40 
40
50 mm piece of BC408 glued to an R2490-05 tube. The relative timing
of the counters was arranged so the bottom counter determined the coincidence
time. The coincidence provided the START signal for TDC.
Figure 35 is a plot of the arrival time measured with
the 2228A TDC for the top counter when both counters were required to have pulse
heights in the minimum ionizing peak. A gaussian fit gives 
1.7 counts = 85 ps. The contributions to this measured resolution are expected
to be given by 
, where
and 
are the contributions from photostatistics from
trigger counter 1 and 2 respectively. Assuming 
, and using
the measured value of 
17.7 ps gives 
59 ps.
Using 
ps, 
ps, and 
ps
gives 
ps in good agreement with the value estimated above from
Figure 22 and Figure 24.
Figure 34: Setup for trigger counter resolution
Figure 35: Trigger counter resolution. The width of 1.7 counts corresponds to
85 ps.
Finally, we can estimate the resolution we expect for the final system in B0. In this case, assuming we use the R5946 and make no improvements in the light collection efficiency beyond what was obtained in the prototype counter then we expect:
= (1/2)PS_5946^2 + (1/2)TDC^2 + to^2 + Z^2
or
= (1/2) 158^2 + (1/2) 17.7^2 + 34^2 +33^2 = 122^2
Thus assuming we make no improvements beyond the performance obtained in the
prototype, then we expect 
ps using R5946 PMT
from the final system in B0 assuming that charged tracks passed through
in the middle of a TOF counter.
Since the amount of light observed by a phototube depends exponentially
on the Z distance of the PMT from the track, we can substitute in equation
9 
and 
, where 
is the effective attenuation length
(225 cm) (see Figure 26) and Z is a distance from the center of
the TOF counter in unit of cm. Then equation 18 becomes:
= 1 e^Z/eff + e^-Z/effPS^2 + e^2Z/eff + e^-2Z/eff (e^Z/eff + e^-Z/eff)^2TDC^2 + C2^2 + width^2
or
= 1 2(Z/eff)PS^2 + (1/2)(1+(Z/eff))TDC^2 + C2^2 + width^2
Figure 36 shows the expected resolution vs Z position calculated
from this formula for the R5946 (solid line) and for the R2490 (dashed line).
As can be seen, the expected time resolution for a TOF system based upon the
R5946 is always below 122 ps. A TOF counter system with the same light
collection efficiency as the R2490 could achieve a resolution in B0 of 
ps. We are working on schemes to improve the light collection
efficiency of the 1.5 inch tube. With Hamamatsu, we are developing rectangular
1.5 inch PMT's with larger photocathodes This should fit in the available space
and capture 43% of the light vs 37% for the R5946. Similarly, by using the
GUIDE7 optical ray tracing program described above we have designed refractive
Circular Parabolic Concentrators[6] as light focusing systems to
focus more of the useful light on the photocathode of the 1.5 inch tubes. The
idea is to collect more of the small angle "early" photons at the expense of
photons that arrive later in time. We have fabricated concentrators optimized to
collect light that arrives in the first 5 ns. The concentrators are 5 cm long
and made from machined and polished Lucite. With these concentrators we can
collect nearly as many early photons on the R5946 photocathode as are collected
by the R2490. Timing measurements with a 3 m bar using cosmic rays are in
progress. Preliminary results indicate a time resolution with the R5946 and CPC
concentrator that is nearly as good as obtained with the R2490. Finally, we
note that in the final system the time resolution may degrade slightly when the
tube is operated in a magnetic field[7]. However a time resolution
below 125 ps and perhaps as good as 100 ps resolution may be achievable.
Figure 36: Expected resolution vs Z position for TOF system in B0.
Solid line is for TOF counter equipped with R5946 and dashed line is for
TOF counter with same light collection as R2490 tube.