MTF studies at IJAF

by Michael I. Andersen and Anton Norup Sørensen
Index
Interferometric MTF characterization of a CCD
Introduction
MTF measurements
Results
Phase voltage effect on MTF
Exposure level influence on MTF
Cosmic hit appearance

Interferometric MTF characterization of a CCD

Introduction

We use an interferometric method to measure the MTF of CCD's. Basically, a double aperture interferometer is used to project fringes onto a CCD. By varying the spacing of the apertures, the fringe frequency is changed. Thus, for each setting of aperture spacing, we get the detector MTF corresponding to the fringe frequency. The method and preliminary results are described in An interferometric method for measurement of the detector MTF presented at the ESO workshop on "Optical Detectors for Astronomy", Garching, October 8-10, 1996. We have since improved our interferometer and reduction methods, allowing us to measure the MTF at higher frequencies. A typical fringe image is shown below:

MTF measurements

We have made extensive measurements on an overthinned 2k x 2k 15mu pixel Loral-Lesser CCD (Copenhagen lot, W7-(0,0)), now operated on DFOSC at the Danish 1.54m, ESO, La Silla). Measurements were obtained at 488nm, 543nm, 670nm and 830nm. As an example of what they look like, we show the MTF at 670nm below, while operating the CCD in MPP mode after UV-flooding. As can be seen, the method breaks down at ~0.9 cy/pix where the modulation is down to 1%, but useful information may in fact be retrieved well beyond 1cy/pix, where the component corresponding to the fringes can again be distinguished from the DC component in the Fourier transform of the fringe image. A direct inversion of the MTF measurements into a line spread function is not possible. The data are therefore fitted with a two component Moffat profile

m = p1 / ( 1 + (nu/p2)^2 )^p3 + (1-p1) / ( 1 + (nu/p4)^2 )^p5


Click figure for postscript version.

Legend: MTF measurements of Loral-Lesser CCD @670nm. Plusses are the measurements, diamonds the 'silicon MTF' after dividing with the pixel MTF (SINC(nu) = SIN( pi * nu ) / ( pi * nu ), shown as a dotted curve). The dashed curve gives the 'Moffat' fit to the silicon MTF.

With the fit in hand, we can invert the MTF into a line spread function (LSF) through a Fourier transform. The result is shown in the figure below. The sharp peak in the silicon LSF is emerging as a result of extrapolating the silicon MTF to high frequencies. This does not change the real detector LSF, but effectively only affects the slope of the small 'jump' at the pixel edge. Having the one dimensional silicon MTF, the 2D silicon and detector point spread functions can be derived, if it is assumed that the silicon MTF is rotationally symmetric.


Click figure for postscript version.
Legend: The LSF corresponding to the MTF in the above plot. Dotted curve corresponds to the pixel LSF, dashed curve to the silicon LSF and full drawn curve to the real detector LSF.

Results

An important figure of merit is the normalized peak intensity of the detector-PSF, which we may call the pixel Strehl ratio (analogous to the Strehl ratio, as used in optics). We can define the effective pixel size as the side length of the square box of unit volume, having a height equal to the pixel Strehl ratio, i.e. effective pixel size = 1 / SQRT( pixel Strehl ratio ). From the detector-PSF the 'ensquared energy' (EE_d) (the energy contained within a square as function of side length) may also be calculated. Likewise, the encircled energy (EE_s) can be derived from the silicon- PSF. Analogous to optics, the 80% EE square/circle gives a good description of how far the wings of the detector- and silicon-PSF extends. For the Loral-Lesser CCD we find the following results (reduction of 488nm data is pending). The LSF for these 3 wavelengths are shown in the figure below. As is expected, the sharp peak in the silicon LSF disappear at 543nm, where the absorption depth is only a couple of microns.

543nm 670nm 830nm
Pixel Strehl ratio 0.199 0.291 0.434
Effective pixel size 2.24 pix 1.85 pix 1.52 pix
80% Ensquared Energy 3.90 pix 3.51 pix 3.33 pix
80% Encircled Energy 4.21 pix 3.74 pix 3.48 pix


Click figure for postscript version.
Legend: Left: The silicon LSF at 543nm, 670nm and 830nm. Right: The real detector LSF at the same wavelengths.


Click figure for postscript version.
Legend: Encircled and ensquared energy at 543nm, 670nm and 830nm.

The silicon-PSF at the 3 wavelengths are remarkably different, as can be seen from the left-hand figure above, with the 543nm silicon-PSF having no sharp core at all. The encircled/ensquared energy is plotted in right-hand figure above and listed in the following table. It is obvious that the extended wings of the PSF's are important at all wavelengths. Note that the EEs cross eachother at a diameter of 6-7 pixels. A corresponding trend in the raw MTF data is present, where there is an apparent cross over at ~0.15cy/pixel, the 543nm MTF being the better at the lowest frequencies.

Encircled Energy Ensquared Energy
Box size 543nm 670nm 830nm 543nm 670nm 830nm
[pixels] [%] [%] [%] [%] [%] [%]
1 16.08 25.75 41.48 16.51 24.27 38.80
2 43.27 50.40 59.93 45.93 53.01 61.06
3 64.41 70.31 74.82 67.85 73.26 76.36
4 77.88 82.45 84.31 81.03 84.71 85.41
5 86.10 88.80 89.68 88.69 90.48 90.23
6 91.20 92.20 92.63 93.25 93.57 92.81
7 94.49 94.32 94.33 96.05 95.47 94.30
8 96.63 95.79 95.44 97.77 96.78 95.31
9 98.01 96.88 96.24 98.82 97.72 96.07
10 98.87 97.70 96.91 99.42 98.41 96.73
11 99.39 98.32 97.49 99.75 98.92 97.29
12 99.69 98.79 98.02 99.92 99.29 97.85

Approximately 8% of the flux seems to be absorbed within the well at 670nm while it is ~25% at 870nm. Using an absorption coefficient of 0.067/mu at 830nm and 0.18/mu at 670nm, this indicate a field free region of ~15mu.

Phase voltage effect on MTF

The size of the depleted region near the electrodes depends on the voltage applied to these. As the Loral 2k is doped for MPP operation, integration is normally made in MPP mode, with all three phases at -8V. This minimizes dark current, but also makes the depleted region small, with adverse effects on the MTF. In non-MPP mode, P1 is at +2V, while P2 and P3 are at -8V. As this was observed to provide a better MTF, experiments were made where P1 and P2 were "high" at various voltages during integration.

The graph above shows the effect on MTF from changing the P1, P2 voltages. The MTF is measured at one spatial frequency only (~0.4 cycle/pixel), and is normalized to the MTF in MPP mode, represented with the single data point outside the linear curve.
The gain in MTF appears to be proportional to the voltage applied to P1, P2.

Using the highest P1, P2 voltage of +12V, which gave the best MTF in the previous graph, the voltage of P3 is now changed. The normalization is the same as before. The effect on MTF is slight, but with P3 above -3V, a linear increase with voltage is seen.

As a relative improvement of 43% in MTF was reached by changing the phase voltages from the standard values was reached, the method seems very attractive.
There are some drawbacks, though:
The higher phase voltages means more dark current. At the highest voltages used, the dark current was about 30e-/h, approximately a 20-fold increase compared to the MPP mode. Still, for several applications, this is an acceptable level.
In the standard setup, the full well is 115.000e- for the CCD tested, regardless of MPP mode. By increasing the P1, P2 voltages, no harm to the well capacity was observed, but increasing P3 also caused problems: At voltages P1=+12V P2=+12V P3=+6V, the full well was only 16.000e-. At higher P3 voltages, the imaging capability was lost. At a P3 voltage of +2V, the full well was 69.000e- and the relative gain in MTF was 38%, which seems like a useful compromise.
The high voltages would cause problems with spurious charge and CTE during pre-clear and read-out. To avoid this, standard voltages were used at these times.

Exposure level influence on MTF

As the potential well of a pixel is filled with photo electrons, the potential is reduced and the field-free region should increase in depth. In principle, MTF should then be better at very low exposure levels than near the full well limit.
Below, this dependence is examined by measuring the MTF for different exposure levels, while keeping all other parameters constant.

Click figure for postscript version.

It is seen that the MTF is constant over the entire measured range, from a few hundred electrons per pixel to about 120.000 electrons, just below full well. The actual level of 0.40 is determined from the selection of a arbitrary, but constant, spatial frequency.
The constancy is explained by the very large field free region always present in the Loral CCDs. Any exposure dependence is completely dominated by the large diffusion.

Cosmic hit appearance

The typical appearance of "cosmic hits" in a dark exposure, using a thinned Loral 2k CCD from the Copenhagen lot.
To the left a hit in a shallow angle appears comet-like, as charge diffusion is large at the back side and no diffusion occurs near the electrodes.
To the right, the stellar-like image from a hit approximately perpendicular to the CCD plane.
The spreading of charge from a particle impact is not only due to diffusion. The large number of charges in a small area also causes self-repulsion.

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Last updated September 28, 1999