Linearity testing

Figure 1.  CCD bucket analogy (Howell 2006)

Each pixel of a CCD sensor is like a bucket, but instead of collecting rain water, these buckets collect electrons that are generated by the incoming photons.  Just like a bucket can only hold a finite volume of water, a pixel can only collect a finite number of electrons before it "overflows."  When the bucket overflows, the measurement of  the electrons is no longer linear.  What does that mean?  Consider in the water example, a certain number of rain drops will produce a certain change in volume of the amount of water in the bucket.   This relationship holds up to the point where the bucket overflows and the incoming rain drops cannot be measured because all of the space has been taken up.  The same is true for pixels in the CCD sensor:  There is a linear response between the amount of incoming photons and the measured number of electrons per pixel up to a certain point where the relationship breaks down and the pixel "overflows" or "blooms" (see Figure 2 below).

Figure 2.  Example of blooming (AOweb)

Some types of CCD sensors have what is called an Anti-Blooming Gate (ABG) to try to minimize the effects of blooming.  However, since photometry is concerned with precise counting of the number of photons, it is better, but not absolutely necessary, to have a Non Anti Blooming Gate (NABG) to get an unadulterated measurement.  The manufacturer's CCD specifications will include the Full Well Depth of the sensor, or the expected number of electrons that can be collected before the pixels bloom.  In either case (ABG or NABG), it is best practice to actually test the camera to find out where the pixels saturate and if there is any non-linear behavior before the point of saturation.  This will enable better photometry since the "overflow" point is well established and can be avoided.

In practice, most photometry software packages do not actually output electrons, but instead use Analog to Digital Units (ADU).  ADU is simply a numerical representation of the voltage created by the electrons that accumulated in the pixel.  The ADU can be converted to electrons by multiplying by the CCD's gain, another specification given by the manufacturer.  In the case of TECMO, the SBIG ST-8XE (NABG) has a specified Full Well Depth of 100,000 electrons and a gain of 2.5 electrons per ADU.  Therefore, the pixels should bloom at approximately 100,000/2.5 = 40,000 ADU.  To test this, the telescope and CCD were pointed at a diffuse, uniform light source.  For TECMO, white sheets of paper were used to diffuse the light coming from a light bulb.  A good starting place is when a 10 second exposure yields an average ADU measurement of about 10,000.  In this example, the "brightness" was reduced by adding more sheets of paper over the telescope's aperture.  After this, the exposure time was doubled and the ADU measurement was taken again.  What was expected?  Since the exposure time was doubled, the amount of light reaching the pixels was doubled, so the initial ADU value should double as well to around 20,000.  Now another 10 seconds is added so that the exposure is 30 seconds and the ADU measurement should roughly triple.  Keep adding 10 seconds until this relationship breaks down and the ADU measurement no longer goes up because the pixels have "overflowed".  The figure below shows the results of the TECMO linearity test of its SBIG ST-8XE.

The sensor clearly saturates at just over 60,000 ADU.  However, after fitting a straight line to the data, there also appears to be some non-linear behavior before the point of saturation.  It is not entirely clear from this data exactly where the linearity ends, but a reasonable estimate is about 50,000 ADU.  This is a "deeper" full well depth than reported by the CCD manufacturer (SBIG).  However, it is probably best that their estimate is conservative.  In fact, it is smart to use the more conservative number to calculate exposure times for the targets of interest.  For example, if the goal was to expose a target until it reaches half of its full well depth (a typical practice in photometry), it would be wise to use 40,000/2 = 20,000 ADU.  When in the business of counting photons, it is always safer to err on the side of under-exposure in order to maintain the all important linear relationship between the incoming light and measured electrons.

References

1. AOweb:  http://astronomyonline.org/astrophotography/ccd.asp, accessed 11 November 2014
2. Howell, S. B. 2006, Handbook of CCD Astronomy, Cambridge University Press, UK