Device Initialisation

Avalanche detector initialisation

For large (N > ?) arrays of detectors, a renewal process distribution is sampled to determine the times since the last avalanche event.

With a perfect recharge time $\tau_{rc} = 0$ , the distribution of times between detections for a device would be an exponential distribution:

$f_t(t) = \lambda \exp(-\lambda t)$

However, in reality microcells must recharge after detection of a photon, which takes a small but finite amount of time (10s of nanoseconds). The recharging of microcells is assumed to be a capacitor RC recharging process:

$V_{\mu}(t) = V_{\text{over}} \left(1 - \exp \left(-\frac{t}{\tau_{rc}}\right) \right) $

And the photon detection efficiency $\eta$ as a function of the device current overvoltage is approximated as:

$\eta(V_{\mu}) = \eta_{\max} \left ( 1 - \exp \left ( \frac{-V_{\mu}}{V_{\text{chr}}} \right) \right) $

In combination with the fact photons are exponentially distributed in time, the probability density function of times between microcell avalanches becomes a renewal process with the PDF:

$f_t(t) = \eta(V_{\mu}(t)) \lambda \exp(-\lambda * t * p_t(t))$

where

$p_t(t) = \frac{1}{t} \int_0^t \eta(V_{\mu}(t)) dt$

This however is only half of the story - this distribution tells us the PDF of the time between avalanches - NOT the PDF of the times of the devices since their last detection for for some random point in time.

To approximate this, the following is done:

$f_x(t) = \frac {\int_t^{\infty} f_t(t) dt} {\int_0^{\infty} \int_t^{\infty} f_t(t) dt dt}$

Which provices the correct shape of the PDF for some random halting point in time.

NOTE - THIS DISTRUBITON IS NOT CORRECT, FOR CASES WHERE $\lambda$ CHANGES OVER TIME - THIS APPROXIMATION FOR f_x(t) DOES NOT HOLD!!

Nonetheless, this distrubution f_x(t) is randomly sampled for each detector in the array to produce a starting point for the simulation.

A way to test if this method is good is to simulate the full device enough for sufficient statistics with a constant illumination source, and see if there are any transients at the beginning of the simulation. When comparing this renewal process method against the exponential distrubution assumption - there is far less of a transient (if any), so I think this a good place to stop for now.