Testing for non-linearities

A CCD consists of an array of elements (pixels) arranged on a very thin silicon layer. Incident photons on each pixel are converted to electron-hole pairs. Up to about 500000 electrons can be stored in each pixel, depending on the CCD. After an exposure is finished, the electrons can be moved around and read out by a controller which converts the electron charge in each pixel (or binned group of pixels) to digital counts (analogue-to-digital units or ADU). The conversion factor (electrons/ADU) is called the gain and is typically in the range 2-5. Non-linearity occurs when the gain varies with the number of electrons.

There will certainly be large non-linearities as saturation is approached, due to the unguided transfer of electrons from saturated or nearly-saturated pixels to neighbouring pixels. Here, we are concerned with smaller but still significant non-linearities that occur well below saturation. The non-linearity corrections described in this paper should be applied after bias subtraction, before any further data reduction.

The most obvious method to test for non-linearities is to measure the intensity of a flat-field as a function of exposure time and then plot count rate (measured counts / exposure time) versus exposure time (e.g., Barton 1986). For a linear CCD, the count rate should be constant. However, this method requires an extremely stable light source, such as a beta light or a stabilised LED (e.g., Smith 1998a,b). In this paper we discuss two methods that can be used with ordinary dome flats: a bracketed repeat-exposure method (an extension of the bracketed method described by Gilliland et al. 1993) and a ratio method (Baldry 1999).

• The bracketed repeat-exposure (BRE) method involves making single and multiple exposures of different lengths. A typical sequence would consist of: bias frame, 2s, 2$\times$2s, 2s, 3$\times$2s, 2s, 4$\times$2s, 2s, 5$\times$2s, 2s, 6$\times$2s, 2s, 7$\times$2s, 2s, 8$\times$2s, 2s, 9$\times$2s, 2s, 10$\times$2s, 2s, bias. Note that the longer exposures are built from multiples of a single shutter-open time, to avoid systematic inaccuracies in the shutter timing¹. The single exposures (bracketing the multiple exposures) are used to calibrate drifts in the intensity of the lamp. In practice, we have found that changes in typical dome lamps are slow enough to be well calibrated by this method. If necessary, only a small region of the CCD need be read out in order to keep the readout time to a minimum.

  By using the single exposures to determine the expected counts for the multiple exposures, it is possible to plot relative gain² (measured-counts / expected-counts) versus measured-counts for a region on the CCD (note that all counts are bias-corrected first). For a linear CCD, the gain does not depend on the measured counts.

• The ratio method is an indirect method and, as the name suggests, involves calculating the ratio between the measured light level on two regions of the CCD. By varying the exposure time, we can then plot the ratio versus measured-counts. For a linear CCD, the ratio should be constant. This method is unaffected by uncertainties in the total light level, except in the case of spectra, where changes in the temperature of the lamp will affect the ratio between some regions. To avoid this problem, half of the length of a slit can be covered with a filter to create a light level difference at each wavelength. Uncertainties in the exposure time do not directly affect the measurements but if the two regions are far apart on the CCD, there could be a systematic error due to the time it takes a shutter to move across a focal plane. The exposure times should be long enough or the two regions should be close together on the CCD to avoid this problem.

• Another method related to the ratio method is described by Leach et al. 1980. It is based on the principal that, for a linear CCD, correcting a flat-field taken at one exposure level with a flat-field at a different exposure level should result in a uniform field (excluding cosmic-ray detections).

• There is also a variance method which we have not used but shall describe briefly. The principal rests on the fact that the statistical variance of a flat field due to photon noise is equal to the number of photons detected. Thus, $g \sigma_{\rm ADU}^2 = N_{\rm ADU}$, where $g$ is the gain in electrons per ADU, $\sigma^2$ is the variance due to photon noise only and $N$ is the average bias-corrected counts. The gain can then be measured for various values of measured counts, which for a linear CCD should be constant. In order for the method to be accurate, consideration must be taken of the readout noise and pixel-to-pixel intrinsic variations of the flat field (e.g., Leach 1987).

In the next sections, we describe some tests and measurements of two different CCDs using the BRE and ratio methods.