Figure 1: Absolute Flux vs. Wavelength for Ge detector
In order to derive the linearity results discussed below, I divided each of the spectra in Fig. 1 by the spectrum of the ND 0.0 (no ND flux reduction) spectrum. This gives a family of relative flux curves (Figure 2) for these filters and allows one to accurately determine what the flux reduction level at each wavelength is for each of the ND filters. This also provides a neat graphical representation of what the number after "ND" means for each of these filters, and why you really shouldn't believe that number at face value without testing it (vis. the large variations in the ND 2.5 filter). So, at each wavelength, the relative flux ratio values of Fig. 2 can be used as a surrogate for exposure time variations on the x-axis of a detector linearity plot.
Figure 3 shows the behavior of detector 10 (chosen
arbitrarily) measured in vacuum at 800 nm, gain state of 1, and
temperature = -26. The y-axis plots all of the measured detector DN
values, and the x-axis is the relative flux, taken directly from Fig.
2. From 4 to 20 measurements were made at each wavelength through
each ND filter, which is why there are multiple points shown at each
flux level. Each of these points was used to formally derive the
regression curve shown. This detector is, as you can see, behaving
quite nicely over the 40 to 100 DN range, with a linear
goodness-of-fit value of 0.9987.
Figure 4 shows data obtained under identical
conditions as Fig. 3, except that the gain state is now 10 instead of
1. The range of DN values is correspondingly large (up to about 600
DN), but this detector is still behaving quite nicely, with R**2 =
0.9999. Again, all of the measurements were used to derive the fit.
Figure 5 shows data obtained at 1500 nm for the
same detector and at gain state 1, temperature = -26, in vacuum. At
this wavelength, the detector is also behaving linearly.
Figure 6 shows data obtained under identical
conditions as Fig. 5, except that the gain state is now 10. Here, we
have run into detector saturation somewhere above 0.4 relative flux
units, and so the overall linear regression does not fit well. Up to
0.4 relative flux units, though, the detector appears to be in the
linear regime.
512 such plots would have to be made to show the behavior of each
detector over the range of conditions tested. Instead (thankfully), I
have developed a smaller family of just 16 plots similar to that shown
in Figure 7. What I have plotted in this figure
is a "spectrum" of the linear fit coefficients and the R**2 values at
a particular wavelength (800 nm) and gain state (1). Each of the
detectors' regression analyses was performed separately, and the
results have been combined here. For example, the values of slope,
offset, and R**2 for detector 10 come from an analysis identical to
that shown in Fig. 3.
This is how I would use the results in Figure 7: I pay most attention to
the bottom plot, which shows how well each detector's calibration data
is best fit by a linear curve. In this case, I would argue that detectors
1 and 4-32 are behaving well, but that detectors 2 and 3 are slightly
anomalous and so perhaps I should go back and produce plots like Fig. 3
for these detectors to see if there were bad data points or some other
reasons for the poorer fit.
I do not understand why there is a sudden jump in slope and offset between
detectors 1-22 and 23-32. Perhaps this is a peculiarity having to do with
the way the line array is manufactured. Or, perhaps this is a manifestation
of spatial variations in the lamp's input flux across the array. Regardless,
note that there is no substantial change in the R**2 value, so that although
these elements may be behaving differently, they are still behaving well.
This is exactly why accurate flatfielding will be essential once we start
instrument level tests.
Finally, Figure 8 shows an example of rather poor
linearity performance, at 900 nm and gain state 1. Figure 9 shows that the reasons for this poor
performance was a set of highly variable DN measurements obtained near
relative flux 0.35 (through the ND 0.5 filter). If outlier
measurements are discarded from the data, then the linearity is seen
to still be quite good at 900 nm.
Figure 2: Absolute Flux vs. Wavelength for Ge
detectorLinearity calculations, T = -26 degrees
I have looked at the linearity behavior of the Ge detectors starting with
the lowest temperature measurements. As Scott has pointed out, there
are almost 100 different measurement conditions (wavelength, ND filters,
gain state) at each temperature. This means that one must come up with a
clever way to display the linearity results for all 32 detectors or else
face the prospect of sifting through hundreds of plots. I think I've
come up with a way, which I'll show below. First, I will walk through
the procedure for a few specific detectors at a few wavelengths as
representative examples.
Figure 3: Ge detector #10, 800 nm, Gain State = 1
Figure 4: Ge detector #10, 800 nm, Gain State = 10
Figure 5: Ge detector #10, 1500 nm, Gain State = 1
Figure 6: Ge detector #10, 1500 nm, Gain State = 10
Figure 7: Spectrum of Linear Fit Coefficients at 800 nm and
Gain State = 1
Figure 8: Spectrum of Linear Fit Coefficients at 900 nm and
Gain State = 1
