The dead time of a SPAD is a well known non-ideality within SPAD based systems. It encompases the short recharge time after a sucessful photon detection, when the device is fully insensitive to further photon events.
In the literature, the dead time is often assumed to be fixed, although often with a degree of user control. From experimental work it is clear that the exact dead time of a SPAD varies both between diodes and from event to event, yielding a probability distribution about some user set point.
In the below graph, we look at the action of this dead time variation on the number of counts within a short period. This short period is intended to be the symbol duration in an OOK modulated data stream, say 100Mb/s. The receiver is counting the number of photon detections within that time, and for a given light intensity the number of counts within that period varies due to natural Poisson statistics of light.
Unsurprisingly, both the Poisson statistics of light, and any undesirable channel or receiver behaviour act to produce a statistical distribution for the transmitted 'ones' and a similar, although lower mean distributions for the transmitted 'zeros'. The distance between the distributions and how wide each distribution (standard deviation) is, dictates the error rate of the system.
To investigate the effect, numerically, of only the dead time variation, the below numerical model assumes no poisson variation, so that the only variation observed is that induced by a distribution on the dead time value.
So, on the left in blue, are the photon detections within a short period if the dead time is fixed. Clearly without Poisson statistics and with a fixed dead time, successive symbols or successive time periods show no variation in their observed photon detections. In effect the number of detections has a zero standard deviation distribution, meaning that once Poisson statistics of light are added back into the situation, the receiver would be so called, "quantum limitted".
However, if the dead time has a certain variance or standard deviation, the number of actually detected photons varies wildly from symbol to symbol. Conceptually this is easy to understand, lets say a particular detection has an unusually short dead time (mean - 2 sigma), then within the symbol period there is more time available for detections, and as some photons will arive during the time that the SPAD should have been inactive, the number of photon counts will increase. If however the dead time is unusually long, (say mean + 2 sigma), then the SPAD will be inactive for a greater proportion of the symbol and therefore photons that would otherwise have been detected are now undetected.
On the right of the figure, the histogram of the left hand side's red trace is shown. For completeness it is also shown with an ideal gaussian fit. Clearly despite no Poisson statictical variation in photon arivals, the output exhibits a degree of variation.
So that does this mean? Well in practical terms we can never reduce the variance of the dead time to zero, there will always be some variation, therefore we can never be in a position to state that the natural Poisson statistics of light are the sole contributor to the receiver noise. But can we still use the phrasing of a "quantum limitted" sensor? Yes.
Active quenching has been shown to achieve tighter dead time distributions than passive quenching schemes, however this comes at the cost of circuit area and power disipation.
From a design perspective, the active quenching active reset (AQAR) circuits used within the SPAD array, achieved a minimum dead time of approx 5.5ns and a minimum dead time variance of approx 50ps. As the SPAD used for the tests was read out through a digital signal chain with multiplexers, output buffers, pads, inductive bond wires and non-impedance matched PCB traces, there is the possibility the variance at the actual AQAR crcuit is slightly lower, perhaps 40ps.
This 40ps variation is small in comparison to the overall dead time and is very very small in terms of the exponentially distributed photon interarrival time, even at the SPAD's maximum count rate. As such while the dead time variance effect would contribute to the variation in detected photons, the effect of Poisson statistical arival is far more significant. As such we could safely say the system is quantum limitted.
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