Signal-to-noise performance

In the absence of the multiplication register, the signal to noise for imaging with the CCD is determined by the photon-shot noise and the readout noise of the analogue CCD amplifier. For an expectation value of $\left <m_{in} \right >$ detected photons, the RMS photon shot noise is equal to $\sqrt{\left <m_{in} \right>}$. At fast readout rates, the readout noise $r$ typically has an RMS of a few tens of electrons and is not correlated with the photon shot noise. The RMS noise $\sigma$ in determining the detected flux in a CCD pixel thus comes to:

$$\sigma=\sqrt{\left <m_{in} \right>+r^{2}}$$ (4.13)

The signal-to-noise ratio $R$ for conventional imaging is equal to:

$$R=\frac{\left <m_{in} \right>}{\sqrt{\left <m_{in} \right>+r^{2}}}$$ (4.14)

At low light levels the detector readout noise dominates Equation 4.13.

If the multiplication register is enabled, then the expectation value for the number of output electrons will be increased by a factor equal to the total multiplication register gain. The variance in the number of output electrons will also be increased, as described by Equations 4.11 and 4.12 for models 1 and 2 respectively. If the output of the multiplication register is treated in an analogue fashion (in the same way as for a conventional CCD), then this variance acts as a source of additional noise.

The signal-to-noise ratio for a register containing $s$ stages each giving a gain of $\mu$ is:

$$R=\frac{\mu^{s} \left <m_{in} \right>}{\sqrt{\left <m_{in} \right>\mu^{s-1}\left (\mu^{s+1} + \mu^{s} - 1\right )+r^{2}}}$$ (4.15)

if the gain stages are described by model 1 and

$$\mbox{$R$}=\frac{\mu^{s} \left <m_{in} \right>}{\sqrt{\left... ...in} \right >\mu^{s-1}\left (2\mu^{s} + \mu - 2\right)+r^{2}}}$$ (4.16)

if the stages are described by model 2. For both models, if the number of stages $s$ is large and the gain per stage $\mu$ is close to unity then the signal-to-noise is well approximated by:

$$R = \frac{g \left <m_{in} \right >}{\sqrt{2g^{2} \left <m_{in} \right >+r^{2}}}$$ (4.17)

$$= \frac{\left <m_{in} \right >}{\sqrt{2\left <m_{in} \right >+\frac{r^{2}}{g^{2}}}}$$ (4.18)

where $g=\mu^{s}$ is the total gain of the multiplication register.

For large multiplication register gains ( $g\gg r/\sqrt{\left <m_{in} \right >}$) the readout noise of the CCD becomes negligible giving:

$$R = \frac{\left <m_{in} \right >}{\sqrt{2\left <m_{in} \right >}}$$ (4.19)

$$= \sqrt{\frac{\left <m_{in} \right >}{2}}$$ (4.20)

It is interesting to compare the signal-to-noise in Equation 4.20 with the signal-to-noise $R_{ideal}$ for an ideal readout-noise free detector (limited only by photon shot noise):

$$R_{ideal} = \sqrt{\left <m_{in} \right >}$$ (4.21)

The ratio of the signal-to-noise of a device with a multiplication register to that of an ideal readout-noise free detector is called the noise factor $F$. For the L3Vision CCDs operated at high gain this noise factor will be $F\simeq\sqrt{2}$.

In an ideal readout-noise free detector a reduction by a factor of $\sqrt{2}$ in signal-to-noise would be brought about if the quantum efficiency of the detector was halved. The noise performance of the L3Vision devices is thus similar to the performance of a readout-noise free device with half the quantum efficiency (see also Mackay et al. (2001)).

If the photon flux per pixel read out is very low, electron multiplying CCDs operated at high gain can be used as photon-counting devices (rather like an array of avalanche photo-diodes). In this mode of operation, it should be possible to detect individual photons with high quantum efficiency. The performance of the devices for photon counting under these conditions will depend on the probability distribution for the output electrons.