Discussion

Table 1 shows the non-linearity measurements and some details of the CCDs tested for this paper. For CCD10, the non-linearity cannot be accurately quantified using the $\alpha$ parameter due to higher-order effects. The measured value of $\alpha$ varies between about $1.2 \times 10^{-6}$ and $1.5 \times 10^{-6}$, depending on the range of counts. In order for this CCD to perform adequately in terms of high precision observations, it is necessary to apply a non-linearity correction, for instance, by using the 2nd order fit derived by the BRE method (Equation 1).

Table 1: CCD details for the non-linearity measurements. Further information on the CCDs is given by Downing (1999). The correction is normalised so that $N_t$ (true counts) $= N_m$ (measured counts) for $N_m \rightarrow 0$. The measurements were made in 1997 March and 1998 December for CCD10 and CCD17, respectively. The non-linearity for CCD10 was confirmed to be the same, or similar, for setups in 1996 May and 1997 February.

details and serial number	nominal gain (e$^-$/ADU)	binning used	alpha parameter	non-linearity correction: $N_t =$
CCD10 2Kx2K Tek 1509BR24-01	2.0	1x1	$+1.2\times 10^{-6}$	$$\frac{N_m}{1 + 1.68\times 10^{-6} N_m - 8.8\times 10^{-12} N_m^2}$$
CCD17 2Kx4K SITe 6044FCD04-01	2.5	1x2	$-3.5 \times 10^{-7}$	$$\frac{N_m}{1 - 3.5\times 10^{-7} N_m}$$

The $\alpha$ parameter for CCD17 was measured to be $-3.5 \times 10^{-7}$ in the range 0-40000 ADU (Section 2.3). The non-linearity of this CCD is well characterised by one parameter, which means the non-linearity can be easily and accurately corrected when high-precision measurements need to be made. Note that the CCD17 measurements were taken using 1x2 binning to match the spectral data being taken during the observing run. The saturation level was determined to be around 57000 ADU which is approximately double the saturation level when no binning is used. This is because the serial-register pixels of the CCD, where the electrons are combined, have a higher electron capacity. In this sense, the non-linearity is different when the CCD is binned. The $\alpha$ parameter will not depend on the binning if it is only dependent on the conversion of electrons to ADU.

Ideally, CCDs should have a non-linearity parameter of $\vert \alpha \vert < 10^{-7}$ with a gain of about 2e$^-$/ADU. This means that at an ADU level of 50000, the correction is less than 0.5%. John R. Barton measured an $\alpha$ value of $-3 \times 10^{-8}$ for the AAO 1Kx1K Tek #2 chip (commissioned 1992 July) in `normal' mode with a nominal gain of 2.7e$^-$/ADU (see Tinney 1996). The linearity for this Tek CCD is a factor of 40 better than for CCD10 at Mt. Stromlo. Note that, for a given CCD, the $\alpha$ parameter will be proportional to the gain (e$^-$/ADU) if the non-linearity is only a function of the number of electrons and not a function of the number of electrons and the gain.

The measurements in this paper have focused on non-linearities that are independent of pixel position and that are most likely related to the conversion of electrons to ADU. The BRE and ratio measurements were made using several different regions of the CCD with the regions, or pairs of regions, showing similar results. Non-linearity can also occur when charge is trapped in certain pixels during charge transfer (often called deferred charge). This type of non-linearity is column (or pixel) dependent and is more relevant to low light-level measurements. In addition, the $\alpha$ parameter is not a useful parameter and instead a constant addition to ADU is suitable except at the very-lowest light levels. Deferred-charge corrections are described by Baum et al. (1981), Djorgovski (1984) and Gilliland & Brown (1988). Other forms of low light-level non-linearity are not so straightforward, e.g., deviations from linearity increasing with distance from the readout register (Deeg & Ninkov 1995) and a difference between flat-field and sparse-field performance (Smith 1998a,b; Gilliland et al. 1999).

Astronomers tend to avoid near-saturation limits, so that non-linearity due to saturation is generally not an issue. Interestingly, Gilliland et al. (1999) observed that over a group of pixels, the total measured ADU can remain linear even though some pixels are saturated. This is because charge is transferred from the saturated pixels to nearby pixels rather than photon detections being missed. Note this is not true for digital saturation, where the ADU counts have simply reached their highest value (e.g., 65535).

In this paper, we have demonstrated two techniques for measuring or checking the non-linearity of CCDs. The BRE method measures the variation in the intensity of a region on the CCD (using multiple exposures bracketed by single exposures to monitor any changes in the lamp's intensity). The ratio method measures the variation in the ratio between the intensities of two regions on the CCD. This can provide a more accurate non-linearity curve because it is less affected by changes in the lamp's flux and by uncertainties in the exposure time.