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Unravelling Sensitivity, Signal to Noise and Dynamic Range – EMCCD vs CC

2. Signal-To-Noise

In this section, it makes sense to first understand the treatment of Signal to Noise in the more simple CCD detector. Then we will progress to a treatment of Signal to Noise in the more complicated EMCCD type detector.

2.1 Signal to Noise – general

2.2 Signal to Noise in CCDs

2.2.1 CCD Noise Sources

2.2.2 QE influences the Shot Noise!

2.2.3 Calculating Signal-to-Noise for CCDs

2.3 Signal to Noise in EMCCDs

2.3.1 Additional Noise considerations for EMCCDs

2.3.2 Calculating Signal-to-Noise for CCDs

2.3.3 Calculating Signal-to-Noise for CCDs

2.4 Effect of Pixel Size on S/N

2.1 Signal to Noise – general

Noise of a detector is constructed from a few principal sources:

• Overall noise combines the main noise sources using the ‘root of sum of squares’ law (also known as ‘summing in quadrature’):

Overall Noise =

• Strictly speaking, it is the read noise and darkcurrent noise that contribute to the detection limit, or the system noise floor, that the signal must overcome. Shot noise is essentially a measure of the intensity variation in the signal itself. When comparing overall Signal to Noise figures, the nature of the relative individual noise contributions must always be borne in mind, to get a realistic idea of how detectable your signal really is! The most important aspect is that the signal should be sufficiently clear of the noise floor. The extent of variation of the signal intensity after that (shot noise) is in many ways a secondary issue. These noise sources are represented pictorially below and are considered in more detail in the following section.

2.2 Signal to Noise in CCDs

2.2.1 - CCD Noise Sources

For simplicity sake, it is very useful to first consider the CCD noise sources separately:

CCD Read Noise –

Read Noise in many instances can be considered the true CCD detection limit, particularly the case in fast frame rate experiments because, (a) short exposures combined with low dark current make the dark current noise contribution negligible and (b) faster pixel readout rates, such as 1 MHz pixel readout and higher, result in significantly higher readout noise. For very long exposure times (even with deep thermoelectric cooling) combined with slower readout (sub-MHz pixel readout), the dark noise can eventually become more prominent than the read noise.

CCD Dark Noise –

Dark Noise is essentially the ‘shot noise’ variation of the dark current and is given by the square root of the darkcurrent generated during a given exposure time, which in turn is readily calculated from the CCD spec sheet value of darkcurrent per unit time. Due to the effective cooling inherent to Andor’s high-end cameras, dark current is minimized and even during use of moderately long exposures it may often be considered practically negligible relative to the read noise. The extent of contribution is dependent on exposure time, darkcurrent generally being quoted in electrons/pixel/sec. Deep cooled CCDs can allow for exposure times up to several hours in practice!

Shot Noise –

Shot Noise is due to the ‘particle’ nature of photons. A popular analogy for shot noise is to consider rain drops falling onto adjacent, identically sized patches of pavement (analogous say to adjacent pixels of a CCD). A light rain relates to a low flux of photons, and a heavy rain relates to greater photon flux (more light!). It is obvious that each patch of pavement is unlikely to receive the exact same number of rain drops during a given period (‘exposure time’). Similarly, adjacent pixels of a CCD under uniform illumination will be unlikely to receive the same number of photons during a given exposure time. The same assumption can be made for the same pixel from frame to frame – shot noise quantifies this uncertainty!

So, it is obvious that shot noise will increase with signal intensity, BUT the Signal/Shot Noise ratio will improve also:

e.g. say a light level is such that an average of 10 photons fall within a pixel during a given exposure time. From pixel to pixel and from frame to frame, this carries a shot noise uncertainty of. Therefore the Signal to Shot Noise ratio is = 3.162

If however, the light intensity or the exposure time was increased to yield an average of 100 photons per pixel, then while the shot noise is now = 10, the Signal/Shot Noise ratio is now i.e. a significant improvement!

Basically, Signal/Shot Noise =

2.2.2 - QE influences the Shot Noise!

Contrary to popular belief, shot noise experienced by the detector IS related to the QE of the detector! Back-illuminated sensors with higher QE yields a better Signal/Shot Noise ratio. There is a simple intuitive explanation for this – shot noise must be calculated from the signal represented by the number of photoelectrons in the sensor (electrons generated from photons falling on the sensor), NOT JUST from the number of incoming photons. Therefore, if an average of 100 photons hit a pixel, but the sensor has a QE of 50% at the wavelength of these photons, then an average of 50 photoelectrons will be created – the shot noise and Signal/Shot Noise must be calculated from this value.

ie:

Similarly,

Signal/Shot Noise =

N.B. QE in these equations is the QE as a percentage (the usual understanding of QE). is simply converting this to fraction form, for use in the equations.

2.2.3 - Calculating Signal-to-Noise for CCDs

The overall noise can be calculated from the ‘root of sum of squares’ equation:

Overall Noise =

Where,
RN = Read Noise (rms)
DN = Darkcurrent Noise
Shot Noise = Shot Noise ‘corrected’ for influence of QE, i.e. shot noise derived from signal in photoelectrons.

If the signal is known in photons:

Therefore,

Overall Signal/Noise =

Note that in this equation, both shot noise and darkcurrent are dependent on exposure time (so must be recalculated for different exposure times), whereas read noise is a fixed value (for a given readout rate, e.g. 1 MHz).

Conveniently, if slower readout speeds are used and the electronics of the system are carefully optimized, readout noise can be as low as 2 to 3 electrons rms (compared to typical 40 electrons rms at 5 MHz readout and upwards). This means that a 2 to 3 electron detection limit exists with ‘slower scan’ imaging cameras, sometimes known as ‘staring cameras’ - even a relatively small amount of photons per pixel (10’s photons/pixel) can overcome this noise floor sufficiently to become ‘shot noise limited’. Furthermore, this shot noise would not be increased by multiplicative noise, since this noise source does not exist for standard CCDs. Again, don’t forget that it is the read noise and darkcurrent noise contributions that contribute to the detection limit. The shot noise relates to the degree of variation of the signal around the average value from frame to frame or pixel to pixel, therefore is more relevant to those interested in the degree of strict quantitative accuracy of a given measurement. Different researchers may have their own Signal/Shot noise lower threshold, depending on the nature of the application.

2.3 Signal to Noise in EMCCDs

2.3.1 – Additional Noise considerations for EMCCDs

Read Noise-

The fundamental advantage of EMCCD technology is that EM gains can be sufficient to effectively eliminate readout noise, therefore eliminating this primary contribution to the instrumental detection limit! Importantly, this is accomplished without significant sacrifices to QE such as are otherwise imposed through use of an intensifier tube in an ICCD!

Multiplicative Noise and its effect on Shot Noise–

This noise source is only present in signal amplifying technologies such as found in EMCCDs or ICCDs and is a measure of the uncertainty inherent to the signal multiplying process. That is to say, during each transfer of electrons from element to element along the gain register of the EMCCD, there exists only a small probability that the process of impact ionization will produce an extra electron during that step. This happens to be a small probability but when executed over > 590 steps, very large potential overall EM gains result. However, the downside to this process results from the probabilities! Due to this, there is a statistical variation in the overall number of electrons generated from an initial charge packet by the gain register. This uncertainty is quantified by a parameter called ‘Noise Factor’ and detailed theoretical and measured analysis has placed this Noise Factor at a value of , or 1.41 for EMCCD technology.

Note that this noise source is significantly greater from the amplifying Multi-Channel Plate (MCP) of ICCDs than from the Gain Register of the EMCCD. ICCDs have noise factors typically ranging from 1.5 to >2.

So, this is an additional form of noise that must be taken into account when calculating Signal/Noise for these detectors

However, one way to better understand the effects of this noise source is in terms of an addition to the shot noise of the system. Extra multiplicative noise has the same form as shot noise in that each noise results in an increase in the variation of number of electrons that are read out of a CCD (under constant uniform illumination). Indeed multiplicative noise can be thought to contribute directly to the overall shot noise, in that one should multiply the Shot Noise by the Noise Factor when calculating overall noise.

Simply put, multiplicative noise does not in any way reduce the average signal intensity or reduce the number of photons that are detected, it simply increases the degree of variation of the signal around the mean value, in addition to the variation that already exists from the shot noise (variation from pixel to pixel or from frame to frame). This additional variation to the signal intensity is represented pictorially below as a signal intensity profile.

Dark Current–

It is particularly important to eliminate darkcurrent with EMCCD technology since even single thermally generated electrons in the silicon will be amplified in the gain register just as a single photoelectron, and will appear in the final signal as a single noise spike. For fast frame rate experiments combined with advanced TE cooling (at least -70 C), darkcurrent contribution can be sufficiently low as to render this noise source negligible. Thus it is important to make sure a high-end EMCCD camera is equipped with effective cooling. Some EMCCDs are being sold into the high-end market with inferior cooling, yielding EM-amplified dark current levels that can be several orders of magnitude higher! Inferior cooling can expose excess darkcurrent even under short exposure times (low ms) as shown below. For sake of simplicity, we will assume effective elimination of darkcurrent in treatment of subsequent EMCCD signal to noise calculations.

(A) shows DARK IMAGES taken at x1000 gain at different cooling temperatures, 29ms exposure time. Vertical shift speed was 0.5s/row to ensure minimal CIC. (B) shows typical line intensity profiles across a row of 512 pixels, taken from such dark images at three different cooling temperatures. The cleanest noise floor is clearly seen under conditions of deep cooling, even for such short exposure times

Clock Induced Charge (CIC) –

CIC, also known as a ‘Spurious Noise’ type, is independent of exposure time and are generally single electron events generated during charge shift (by contrast, ‘EBI’ is the form of spurious noise in ICCDs and is exposure dependent). CIC events are generated in every CCD but are normally completely buried in the readout noise, thus ignored. In the EMCCD however, these single electrons are amplified by the gain register just as a single photoelectron would be. In the EMCCD, CIC can in some ways be considered the true limit of detection, in that at the single photon detection level, a single photon spike will be indistinguishable from a CIC spike. Andor have specialised high-resolution clocking parameters however, that enable this source of noise to be minimized. Furthermore, of vital importance in CIC minimization, is the ability to conduct fast vertical (parallel) shifts when reading out the sensor. Andor Technology exclusively push this parameter to sub-microsecond values with significant benefit towards CIC minimization, as shown below. Again, we will ignore this noise source in sub-sequent treatment of signal to noise in EMCCDs.

(A) shows DARK IMAGES taken at x1000 gain at different vertical shift speeds, 29ms exposure time. Cooling temperature was -85 0C to ensure minimal darkcurrent contribution.
(B) shows typical line intensity profiles across a row of 512 pixels, taken from such dark images at three different vertical shift speeds. The cleanest noise floor is clearly seen under conditions of faster vertical shifts, an exclusive Andor capability.

N.B. In practical terms, whilst it is still desirable to eliminate CIC as far as possible, ultra-weak signals of the single photon nature can be distinguishable from CIC spikes in that one could generally expect to see ‘groupings’ of photon spikes from adjacent pixels, even from diffraction limited single molecule emissions.

2.3.2 - Calculating Signal-to-Noise for CCDs

From the above noise descriptions, it is apparent that in most uses of deep-cooled EMCCD, since read noise, darkcurrent and CIC can be practically eliminated (i.e. the noise sources that would define the detection limit have been eliminated), the only sources of noise that need be considered in this basic treatment are shot noise, noise factor (multiplicative noise).

It is simple to reduce the equation for overall noise to include just these parameters, using:

Shot Noise can be determined if the average signal is measured in electrons - by measuring in electrons, the calculation is independent of the sensor’s QE – i.e. the photons have already been converted to photoelectrons so the QE corrected signal is being measured. If the average signal in photons is already known (e.g. estimated from other measurements with PMTs), the shot noise can be corrected for sensor QE at that wavelength:

i.e. Overall Noise =

Therefore,

What about factoring in darkcurrent and CIC?

Since it is very different in nature to shot noise, it is best to consider darcurrent events and CIC separately. Each EMCCD will have a measured figure for the levels of CIC spikes to be expected during a readout. This will give a figure for the average number of random spurious single electron spikes that will appear within the image. If the measured signal is at the very low photon level (one or two electrons per pixel), this noise source will be more significant. If the signal is slightly more intense than this, it may become less of an issue, and may even be thresholded or filtered out during post-acquisition processing.

2.3.3 – Front vs. Back-illuminated EMCCD sensors

In EMCCD technology where read noise is effectively overcome, then QE is the major specification difference between the front and back-illuminated sensors (again, for most low-light applications, we will treat both readout noise and darkcurrent contributions as negligible, the latter due to deep TE cooling). As such, this QE difference will be manifested purely in signal intensity and the effect on shot noise:

The change in Signal/Shot Noise ratio is a factor of

So, if photons are collected at a wavelength where back-illuminated detectors have a 4x greater QE than a front-illuminated camera of the same pixel size, 4x more photons would be converted. If this is worked though the equations, it results in a = 2x improvement in Signal/Shot Noise ratio. Similarly a 2x change in QE means a = 1.41x difference in Signal/Shot Noise ratio.

2.4 Effect of Pixel Size on S/N

It is easy to forget that pixel sizes vary from camera to camera, but in calculating comparative S/N performance, it is critical to factor in this parameter. For example, a pixel that has four times the area, will accept four times as many photons. Similarly, a 2x2 binned ‘superpixel’ will include four times as many photons (this is why binning can improve S/N – four times the photon signal per one readout). Taking pixel size into account is illustrated in the examples below.

3 Dynamic Range

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