CCD Gain

When discussing the physics of Charge-Coupled Devices (CCDs), I described how photons knocked electrons free, and how one could move electrons via electrodes placed on the silicon surface. On the other hand, when discussing the properties of CCD images, I used the word "counts" to describe the value of each pixel. What's the relationship between electrons and pixel values?

The answer is ... it depends on the chip and the software used to read it out. The steps involved in reading the value of a pixel are something like this:

Electrons transferred to "amplifier"; really a capacitor. Units are coulombs.
The voltage induced by this charge is measured. Units are volts.
An Analog-To-Digital (A/D) unit converts the voltage into some other voltage, which may have only one of several discrete levels. Units are still volts.
The voltage is converted into a number which is passed from the hardware to the computer software as the pixel's value. Units are counts, also called "Data Numbers" (DN) or "Analog-to-Digital Units" (ADUs).

In both steps 3 and 4, one can scale the result by any arbitrary factor and the relative pixel values will remain the same. Some software allows the user to modify the scaling factor dynamically; others have a fixed setting.

The end result is that there is some "fudge factor" which relates the initial number of electrons in a pixel to the final number of counts reported by camera software. The ratio of these two numbers is the gain of the camera:

                      # of electrons per pixel
          gain   =  --------------------------
                      # of counts per pixel

How should one choose the gain factor? There are several criteria.

Full-well depth vs. largest pixel value: Each CCD is designed to hold only so many electrons within a pixel before they start to leak outwards to other pixels. This maximum size of a charge packet on the chip is called the full well depth. There is also a "maximum possible number" in the Analog-to-Digital converter. Most CCDs use 14-bit, 15-bit, or 16-bit A/D units: the corresponding maximum pixel values are 2^14 = 16384, 2^15 = 32768, and 2^16 = 65536.
It is logical to arrange the gain so that very roughly, the number of electrons in the full-well depth corresponds to the maximum pixel value.
Readout noise vs. smallest pixel value: One can look at things the other way around: what are the SMALLEST values that make sense? Amplifiers on CCDs have a certain level of random noise: each time one reads the charge in a packet, one gets a slightly different value. A typical readout noise is 5 or 10 electrons RMS (meaning that the standard deviation of a bunch of measurements of the same packet would be about 5 or 10 electrons). Therefore, if two pixels have values which differ by only 3 electrons, it's not easy to tell the difference between them.
The smallest difference one can represent in an integer image is 1 count. To some extent, it makes sense to arrange the gain so that 1 count corresponds to roughly the readout noise. Any finer measurement of the pixel values would yield differences which would be essentially random.

The SBIG CCD camera at the RIT Observatory has a gain factor which is pre-set and fixed. You can look in the FITS header for the GAIN keyword.

Exercise:

What is the gain factor written into the header of our CCD images?

Measuring the gain factor -- in theory

It is possible for you to check the value provided by a manufacturer. The basic idea is that if you look at a constant source of a LOT of photons (either a star, the twilight sky or a bright piece of cardboard), then the number of photons which strike the CCD in a series of exposures won't be exactly the same. Instead, the number of photons will have some scatter around an average value. The standard deviation of the number of photons should be

        stdev (photons)  =  sqrt(number of photons)

So, if you were to take a series of pictures of a bright light source which averaged 16,000 electrons in each pixel, and calculate the mean and standard deviation of the number of ELECTRONS, you'd find

        mean  (electrons)  =   16,000   per pixel
        stdev (electrons)  =      126   per pixel

However, when you measure pixel values from an image, you are actually measuring counts, not electrons. If the gain were exactly 1 electron per count then

                               mean (electrons)        16,000
        mean (counts)      =   ---------------    =  ------------  =  16,000
                               electrons/count            1
                             
                               stdev (electrons)        126
        stdev (counts)     =   -----------------  =  ------------  =     126
                               electrons/count           1

Note that

        stdev (counts)     =   sqrt( mean counts )

But suppose that the gain was set to 4 electrons per count. Then the conversion from electrons to counts changes things:

                               mean (electrons)        16,000
        mean (counts)      =   ---------------    =  ------------  =   4,000
                               electrons/count            4
                             
                               stdev (electrons)        126
        stdev (counts)     =   -----------------  =  ------------  =      32
                               electrons/count           4

Now, because of the gain factor, the standard deviation of the number of counts per pixel is NOT the same as the sqrt of the number of counts:

        stdev (counts)     =    32    

        sqrt (mean counts) =    sqrt(4,000)  =  64

In fact, you can see that we can figure out the gain by looking at the statistics of the counts:

                                       2
                [  sqrt(mean counts)  ]
        gain  = [ -----------------   ]
                [   stdev (counts)    ]

However, this simple method only works if there are no sources of noise in the CCD, so that the number of electrons is given exactly by photon statistics. Any real CCD has several sources of noise, so this technique won't yield precise results (though it ought to get in the ballpark).

Measuring the gain -- a better technique

A better way to measure the gain is to use pairs of flatfield exposures at different signal levels. The idea is to measure the signal and variance of counts in several pairs of images, at different light levels, and then make a plot of variance vs. signal. In brief, here's how to do it. There's a detailed, step-by-step guide available showing how to use the XVista commands to carry out the operations.

start with a sequence of flatfield exposures
subtract a master dark from all flats
now, pick three or four pairs of successive exposures. For each pair,
- calculate the mean value in each image of the pair inside a box of about 30x30 or 40x40 pixels. Make sure the box doesn't include any obvious image defects.
- make a copy of the first image
- use the div command to multiply the first image by a factor which will give it the same mean level as the second.
- subtract the second image from the copy
- calculate the standard deviation of counts in the subtracted copy
- now, we can derive the variance in the subtracted image by squaring the standard deviation. Because we combined two different flatfield images, the subtracted version has a variance twice as big as the variance in either original. In short, calculate variance = (stdev*stdev)/2.0 .
- add a point to the graph in which the X-value is the mean level in the second image, and the Y-value is the variance just calculated.
The slope of this graph is the reciprocal of the gain of the camera. So the gain of the camera, in electrons per count, is given by gain = 1/slope,

Exercise:

Using several pairs of flatfield frames from Apr 11, 2002, and a small box of about 50 x 50 pixels, determine the gain of the RIT CCD camera.
How close is it to the value stated in the FITS header?

For more details on measuring the gain of a CCD camera, see

Measuring the Gain of a CCD Camera, a tech note from Axiom Research.