1,004
edits
Changes
Jump to navigation
Jump to search
mLine 3:
Line 3: − + Line 28:
Line 28: − + − + Line 73:
Line 73: − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
→Results
[[Image:S20photocathode_QE.jpg|frame|HPD Photo-cathode efficiency as a function of wavelength.]]
[[Image:S20photocathode_QE.jpg|frame|HPD Photo-cathode efficiency as a function of wavelength.]]
The quantum efficiency of this device is shown in the adjacent figure. The DEP-supplied device provides a gain factor of 2700 at the HV of 12 kV and additionally requires 40 - 80 V bias to collect the electrons across the diode junction. Built into the package is a transimpedance amplifier with 50 kΩ || 1.5 pF feedback, requiring ±5 V supplies. The amplifier circuit also contains a 51.1 Ω resistor in series with the output. Reading out the HPD signal from the amplifier with 50 Ω termination creates a factor of two (50 Ω/(50 Ω + 51.1 Ω)) on top of the 50 kΩ current to voltage conversion.
The quantum efficiency of this device is shown in the adjacent figure. The DEP-supplied device has an inherent capacitance (''C<sub>HPD</sub>'') of 200 F and provides a gain factor of 2700 at the HV of 12 kV and additionally requires 40 - 80 V bias to collect the electrons across the diode junction. Built into the package is a transimpedance amplifier with 50 kΩ || 1.5 pF feedback, requiring ±5 V supplies. The amplifier circuit also contains a 51.1 Ω resistor in series with the output. Reading out the HPD signal from the amplifier with 50 Ω termination creates a factor of two (50 Ω/(50 Ω + 51.1 Ω)) on top of the 50 kΩ current to voltage conversion.
Thus, in the complex space, this integral is equivalent to the residues of two poles. Depending on the constants provided, the poles are situated either on the positive imaginary axis or along the -ib/2a line. For the physical constant of our circuit, the former set of poles is relevant. The resulting response function is:
Thus, in the complex space, this integral is equivalent to the residues of two poles. Depending on the constants provided, the poles are situated either on the positive imaginary axis or along the ''-ib/2a'' line. For the physical constant of our circuit, the former set of poles is relevant. The resulting response function is:
<math>g(t) = \frac{2\pi\alpha}{\omega_2} e^{i\omega_1 t} \sinh \omega_2 t </math>
<math>g(t) = \frac{2\pi\alpha}{\omega_2} e^{i\omega_1 t} \sinh \omega_2 t </math>
Since the response seen on the HPD circuit output is the convolution of the light pulse shape (f<sub>L</sub>) with the response function of the HPD, the pulse shape can be recovered by deconvolving the measured signal (f<sub>M</sub>) with the g(ω). By the convolution theorem:
Since the response seen on the HPD circuit output is the convolution of the light pulse shape (''f<sub>L</sub>'') with the response function of the HPD, the pulse shape can be recovered by deconvolving the measured signal (''f<sub>M</sub>'') with the ''g''. By the convolution theorem:
<math>f_L(t) = \mathcal{F}^{-1} \left\{ \frac{f_M(\omega)}{g(\omega)} \right\} </math>
<math>f_L(t) = \mathcal{F}^{-1} \left\{ \frac{f_M(\omega)}{g(\omega)} \right\} </math>
\left( \begin{matrix} n\\k \end{matrix} \right)
\left( \begin{matrix} n\\k \end{matrix} \right)
(1-\alpha-\beta)^{m-n} \alpha^{n-k} \beta^{k}
(1-\alpha-\beta)^{m-n} \alpha^{n-k} \beta^{k}
\ G\left( (m-n) + (n-k)p + kr,\ (m-n)\sigma_1^2 + (n-k)\sigma_2^2 + k\sigma_3^2 \right)
\ G\left( (m-n) + (n-k)p + kr,\ \sigma_0^2 + (m-n)\sigma_1^2 + (n-k)\sigma_2^2 + k\sigma_3^2 \right)
</math>
</math>
This model showed good agreement with the data. The fitter, without any constraints on the third Gaussian, pulled toward a solution in which the peak of secondaries showed slight asymmetry, biased to the left in the spectrum.
This model showed good agreement with the data. The fitter, without any constraints on the third Gaussian, pulled toward a solution in which the peak of secondaries showed slight asymmetry, biased to the left in the spectrum.
=== Results ===
[[Image:HPD_OnePEdist.png|thumb|300px|Distributions of primary and secondary photoelectron signals (noise included) for one incident primary photoelectron.]]
The corrected distribution of secondaries, made up of two Gaussians to account for its slight asymmetry, allows the model to fit the measured spectra. First, the following parameters were fixed from HPD/amplifier specifications and no-light distribution ("pedestal") measurement:
* Total gain, including amplifier and oscilloscope waveform mean rescaling: 1.9264 × 10<sup>5</sup> V/p.e.;
* Pedestal position: -3.6606 × 10<sup>-3</sup> V;
The parameters relating to the shape of the primary and secondary electron distributions, as well as an additional Gaussian noise factor (''σ''<sub>0</sub>) were allowed to vary. A fit with these degrees of freedom was performed on three distributions from drastically different light levels (corresponding to roughly 64, 530 and 973 primary detected photons.) The following model parameters produced reasonably good fits (''χ''<sup>2</sup><2.0) for all three distributions.
{| border="0" cellpadding="20"
|
{| style="text-align:center" border="1" cellspacing="0" cellpadding="5"
|- style="background:#F1F2FF"
! colspan="2"| Gaussian 1
! colspan="2"| Gaussian 2
! Units
|-
| ''p'' || 107.0 || ''r'' || 179.5 || p.e.
|-
| ''α'' || .000899 || ''β'' ||.000299 || -
|-
| ''σ''<sub>2</sub> || 35.83 || ''σ''<sub>3</sub> || 119.2 || p.e.
|-
|}
||
Widths of primary peaks and overall noise:
* ''σ''<sub>0</sub> = 13.7 p.e.
* ''σ''<sub>1</sub> = 0.881 p.e.
|}
[[Image:HPD_3dist.png|frame|Distributions on a logarithmic plot from runs with different light levels (roughly 64, 530 and 973 primary detected photons.) Units are arbitrary and histograms have been rescaled for comparison.]]
Fits with far better agreement were found when the above-listed parameters were free to vary individually for each distribution. Similar and more drastic problems occurred when the simpler one-Gaussian model was used for the distribution of secondary photo-electrons. Evidently, the inaccuracy in its shape is compounded with more convolutions necessary in the high-photon count spectra. Thus, we suspect that the two-Gaussian approximation of this asymmetric distribution is yet not full adequate. However, this already computationally-intensive model (note the three summations) will be even more difficult with more Gaussian and intractable for an arbitrary shape since the convolutions will no longer be straightforward. Disagreements between the fit and data in the right-hand tails of the spectra may also be due to the truncation of the summation in ''n'' corresponding to the number of peaks present due to the convolution with primary and secondary distributions.