SIGNAL TO NOISE RATIO FOR SINGLE ELECTRON EVENTS

There are two types of noise present during single electron counting: the noise per event and the noise per pixel. As was mentioned above, the gain process is not exact and there is some noise (of unknown amplitude) on it. Due to charge spreading this noise may be split between several pixels. We will call this noise N_g and the fraction of charge collected in the pixel directly under impact, f. Thus the noise per event in the pixel under impact is fN_g.

3.1 Optical noise

Unwanted optical signal can be a problem in some applications. To determine whether it will be or not, one has to have some idea of the optical spectrum of the scene being imaged, the transmission of the faceplate and photocathode (if present) as a function of wavelength, the transmission of any intervening electrodes and the sensitivity of the CCD to each wavelength. Then the total signal generated optically in the CCD can be expressed as:

S_opt = º [I_opt(l)] [T_f(l)] [T_pc(l)] [E_opt] [QE_CCD(l)] dl                                  (1)

where S_opt = total optically generated signal

I_opt(l) = optical intensity of scene being imaged (as a function of wavelength)

T_f(l) = transmission of faceplate (as a function of wavelength)

T_pc(l) = transmission of photocathode (as a function of wavelength)

E_opt = optical transmission of intervening electrodes

QE_CCD(l) = quantum efficiency of the CCD (as a function of wavelength)

In all but the simplest systems (such as proximity focussed image intensifiers) Eopt will be the most difficult factor to determine. Scattering, diffraction and the angle of the incoming light can cause a very spatially non-uniform optical signal.

Meanwhile, the electron signal is given by:

S_e = º [I_opt(l)] [QE_pc(l)] [E_e] [ G ] / [M]² dl                                                    (2)

where QE_pc(l) = photocathode quantum efficiency (as a function of wavelength)

E_e = electron transmission of intervening electrodes

G = electron gain of CCD (after backscattering and charge loss is taken into account)

M = electron-optical magnification of tube

For good optical rejection the ratio S_opt / S_e should be small. This is generally the case except at wavelengths where the CCD has good QE, but photocathodes do not, i.e. 900-1000 nm. (see figure 1).

Figure 1. Relative sensitivity of Tektronix backside CCDs (with no coatings) and S-25 photocathodes. Problems with unwanted optical signal usually occur at wavelengths where CCD sensitivity is high but photocathode sensitivity is low, i.e. 900-1000 nm.

Tektronix has developed a coating that is opaque to light, yet transmissive to keV electrons for use in situations where unwanted optical signal is a problem. This "light shield" is stable at vacuum bake temperatures of up to at least 300¡ C. There is some reduction in electron energy and hence EBS gain when passing through the shield. Typical gain curves for shielded and unshielded parts are shown in figure 2.

Figure 2. Typical EBS gain versus electron energy for Tektronix backside CCDs with and without a light shield.

Thermally generated dark current follows a Poisson distribution, thus the noise per pixel (in electrons) associated with it goes as the square root of the number of dark current electrons in that pixel. For room temperature applications and longer (~ 1 second) integration times, this can be significant. Also pixel to pixel dark current generation is not uniform, causing a noise that is linearly dependent on integration time. While it is possible to remove this fixed pattern noise by frame subtraction, doing so will increase the dark current shot noise by a factor of the square root of 2.

Read noise is often the largest noise contributor for single electron detection. CCDs are capable of exhibiting less than 10 electrons of read noise at cryogenic temperatures and slow readout speeds (<50 kpixels/second). However, amplifier noise itself is a function of temperature. At video rates the amplifier settling time is necessarily short and noise increases accordingly. A read noise of 100-200 electrons is realistic at 5-10 Mpixels/second readout rates.

Finally we can put all the noises sources present during single electron detection together.
We have:

N_t = [ (fN_g)² + (ö(i*t)*s)² + (N_r)² ] ^1/2 + G* S_opt / S_e + k(i*t)                              (3)

where: N_t = total noise

f = fraction of charge captured in pixel under impact

N_g = gain noise discussed in section 1 above

i = dark current generation rate

t = integration and readout time

s = ö2 if background subtraction is used, 1 if not

N_r = read noise

G * S_opt / S_e = optical noise; the gain of a single electron times the ratio of the optical to electrical signal (see discussion above).

k = 0 if background subtraction is used, or the pixel to pixel dark current non-uniformity if not.

Despite the large number of noise sources, it is likely that only one or two will dominate in a given application. For example, suppose we are operating with a gain of 1000 and suppose the gain noise is the square root of the gain, and that we have a read noise of 150 electrons, are doing background subtraction to remove dark fixed pattern noise, and do not have any unwanted optical signal. Then, depending on the dark current generation rate, and the integration and readout time, the noise will be dominated by read noise, or by a combination of read noise and dark current shot noise. Figure 3 shows total noise in this example, as a function of integration and readout time for several dark current generation rates.

Figure 3. Total noise versus integration and readout time for several dark current generation rates. Read noise is assumed to be 150 electrons per pixel and all other noise sources are considered negligible. 10000 electrons per pixel per second corresponds to about 300 pA/cm² dark current generation rate in a 24 µm pixel. 1000 e/pixel/s would be the generation rate if the device were cooled from 20¡ to 0¡ C.

3.5 Signal

Now we have enough information to determine the gain necessary to achieve a given SNR for single  electron detection. The signal in the pixel under impact is just f, the fraction of charge collected in that pixel, times G, the EBS gain. So for example if we wanted to be sure of having a SNR of 1 we would use f = .25. Thus if our total noise was 200 electrons, we would need a gain of 800.