Analysis of the variance of laser calibrations:

Ref 1: Coles Smith's Web page

Ref 2: CERN report, CERN-LAA-HC/91-CERN-PPE/91, A Simple Light Detector Gain Measurment Technique.

Ref 3: Burle Photomultiplier Handbook.

Definitions:

R_{i} = ith measured value for a photomultiplier tube.

In terms of the pmt anode signal R can be written as

R = G h
_{o} Ng
a
_{o}

G is the pmt gain (# anode electrons/Photo-e)

h
_{o} is the quantum efficiency

Ng
_{ }is the number of incident photons on the cathode

a
_{o } ADC conversion, charge (no of anode electrons) to channels.

Measurements of the response of detectors attached to the pmt's provide an additional conversion constant b
_{o}

b
_{o } = 10 channels/ MeV.

which is the PMT gain adjusted conversion of particle energy deposition per ADC channel. This is based on tests with cosmic MIPs. Gains are adjusted so that cosmic MIPs fall in channel 100 (160) front (back). Assuming 2 MeV/cm and 5cm (8 cm) path lengths (This is coverd on Cole Smith's web page). Energy deposited is 10 MeV (16 MeV).

The analysis involves measuring the variances and the means of light pulses that originate at a nitrogen laser. In order to calculate the total variance from a combination of steps, we follow appendix G of Burle. The principle result relates the final variance in a cascade processes to the variances of the processes at each step in a chain. These are process where the initial score (number of photons) is multiplied by a random factor with a known mean and variance to obtain the final score (number of photo-electrons). For a cascade process A through B to obtain AB;

s
_{AB}^{2 } = <N_{B}^{2}> s
_{A}^{2 } + <N_{A}> s
_{B}^{2}

where N is the number of events or score for each process. Notice that the second term does not involve the square of the number of events.

The processes involved in the generation of a pmt event are:

- Light generation
- Generation of photo-electrons with
- Multiplication through multiple dynodes ( fixed # pe's = 1, stochastic multiplication)

If the laser source produced a pure Poisson distribution for the photons arriving at the cathode one would have:

<Ng
_{ }> = average number of photons

s
g
^{2} = <Ng
_{ }> .

However the laser has an intrinsic resolution. We will follow the same procedure as outlined in Burle for calculating the yield from the photo cathode for a varying number of photons.

If we imagine that the intial laser pulse has 10^{9 }photons (Ng
_{L}) and that

VAR = F^{2} Ng
_{L}^{2}

With F being the resolution of the laser intensity (5-10%).

This flux of photons is converted to a photon on the cathode with an extremely small probability P_{t}. We can think of this as the transmission probability. The resulting variance is therefore the same as for photo-electrons produced from a varying photon flux with the probability h
_{o} replaced by P_{t. } Process A is the generation of Ng
_{L } photons with variance given above. Process B is the transmission of one photon to the photo cathode. Therefore

s
_{B}^{2 }= P_{t} - P_{t}^{2}

<N_{B}> = P_{t}

s
_{A}^{2 }= F^{2} <Ng
_{L}^{2}>

<N_{A}> = < Ng
_{L} > .

Combining these as a cascade (see Burle).

VAR = s
^{2} = <N_{B}^{2}>^{ } s
_{A}^{2 }+ < N_{A}>_{ s
B}^{2}

VAR = s
^{2} = P_{t}^{2} F^{2} <Ng
_{L}^{2} >_{ }+ <Ng
_{L} > (P_{t} - P_{t}^{2})

with

< Ng
>= P_{t}^{ } <Ng
_{L}>

VAR = F^{2} <Ng
^{2}>+ < Ng
>_{ }+ P_{t}^{ }<Ng
>

The first terms shows that the fluctuations in the signal is always limited by the product of the intensity times the laser resolution. At high statistics the laser intensity fluctuation will dominate. The second term is the Poisson variance and the third term is negligible.

h
_{o} = average number of pe's (Quantum eff.)

s
_{pe}^{2} = h
_{o} - h
_{o}^{2}

B multiplies the number of events from the A process. This is another cascade process and the total variance for the A-B process is

s
_{AB}^{2 } = <N_{B}^{2}> s
_{A}^{2 } + <N_{A}> s
_{B}^{2}

s
_{AB}^{2} = h
_{o}^{2} (F^{2} <Ng
^{2}>+ < Ng
>_{ }) + <Ng
_{ }> (h
_{o} - h
_{o}^{2} )

s
_{AB}^{2} = h
_{o}^{2} F^{2} <Ng
^{2}> + <Ng
_{ }> h
_{o}

with

<Ng
_{ }> h
_{o AB} = <N_{pe }>

s
_{AB}^{2} = F^{2} <N_{pe}^{2}> + <N_{pe }>

This expression is taken from Burle with all dynode responses assumed to be identical.

s
_{d}^{2 } is the variance from a dynode.

g is the gain of a dynode.

G = g^{k} for a k stage tube.

As pointed out in Burle the fluctuations from dynodes range from pure Poisson to exponential. A general expression in terms of a parameter b (0 ® 1) is

s
_{d}^{2 } = b g^{2 } + g

where b=0 is the pure Poisson case.

Combining this process by treating it as a cascade. The result is

since g is of order 10.

An additional variance can be added in quadrature to account for electronic noise. Pedestal, for example, have typical widths of 10 channels due to electronic noise.

These quantities are measure in terms of the number of anode electrons. This is the same expression given in ref. 2 if one drops the (1/g) factor in the second term and does not include the additional source fluctuation (term 1).

The relative variance is

using H to represent the constant for the second term. (Units of H are Npe.)

A typical set of parms:

Intrinsic laser resolution is F.

F=10%

Pedestal noise is used to determine s
_{noise}.

s
_{noise} = 3 pe

This is the equivalent variation in pes that would produce a 10 channel s
_{noise.}

The dynode response is assume to follow a Poisson distribution and a have a gain of 10.

b=0

g=10

The number of phot-electrons per channel is assumed to be 0.3 pe/channel based on a 10 channels/MeV and 3 pe/MeV.

The top curve shows the total. The intrinsic laser resolution dominates at large channel number. The bottom curve is the part due to the Poisson fluctuation in the number of events. The same curve for a smaller range of channels is shown below. The contribution due to elctronic noise (pedestal width) can been seen as a horizontal line.

Cole Smith suggested that the optimal way to analyze the data was to plot fractional variance vs 1/Npe. The data from the model over the entire range shows the behaviour below.

Since this is a plot vs 1/Npe, the low channels are on the right hand side. The graph below shows the same data for channels greater than 100.

The data above assumed a pedestal noise of 3pe or 10 channels. If you use the average pedestal sigma for e5 data (3.7 channels or 1.1 pe) the you get the plot on the right. The 1/Npe value of .03 corresponds to channel 100. If you restrict your fit to higher channels you should be able to fit the fractional variance to a straight line with the intercept returning the laser resolution and the slope the parameter H ( units Npe).

In an actual measurement the x-axis can be measures in MeV^{-1}. This is accomplished using the calibration from cosmic rays. The number of pe's is unknown. The measured value R is therefore in MeV. (This is the equivalent amount of energy deposited in the scintillator by a mip to give a peak in a given adc channel).

R(MeV) = G a
_{o} b
_{o} R(Npe) = D_{o} R(Npe)

D_{ o} = G a
_{o} b
_{o}

where D_{ o} is conversion from Npe to MeV (MeV/pe). The data fractional variance plotted against R measured in MeV has a slope.

slope= H/ D_{ o}

The value of H is known from the arguments above if we assume a value of g and b. The value of D_{ o} can then be determined (see Cole's page).

Conclusion

In the derivation all quantities were put in terms of pe's. H as given therefore has units of (pe). Actual measurements are recorded in channels or MeV. Knowing what the slope of the graph should be in terms of (pe) we can use a measured slope in (MeV or channels) and the expected slope in (pe) to determine (MeV/pe) or (channels/pe).