6.5.1. MTF 3: Aberration compounding, CCD contrast transfer, MTF limitations

Expressing combined effect of two or more aberrations through the MTF can be done, similarly as for the wavefront aberrations, either precisely, through a complex calculation, or as a single number extracted from statistically averaged probability (approximation). As the above graphs show, difference in the effect on performance of various aberrations with a similar RMS wavefront error level becomes more significant as the nominal error increases. For two or more aberrations combined, the effect is, on average, greater than either single aberration components alone. However, they do not add up arithmetically. For relatively small, unrelated or random wavefront errors, the combined effect is given by the square root of the individual errors squared, or

ωc= (Sωi2)1/2 (58)

with ωi being the individual RMS wavefront error. Similarly, the combined effect on peak intensity of the PSF (Strehl) and, thus, average MTF contrast loss can be, for mostly unrelated smaller aberrations, closely approximated as the product of individual Strehl ratios for any number i of different aberrations, Strehl being, in fact, an MTF (contrast) degradation factor:

Sc= S1S2 ... Si (58.1)

The product Sc is a combined Strehl (the peak diffraction intensity ratio) for the errors included; it also expresses the average MTF contrast level in units of the average contrast level of a perfect aperture.

The conventional minimum acceptable level of optical quality - so called diffraction-limited standard - is 1/4 wave P-V wavefront error of spherical aberration or, in more universal form, the associated 1/13.4 wave RMS (0.80 Strehl). That corresponds to 0.42 wave P-V of coma, 0.37 wave P-V of astigmatism, and so on. This criterion is not strict - it is a somewhat lose dividing line between good and bad optics. In fact, since it is, in the commercial environment, habitually applied to the quality level inherent to optical surfaces alone, it is safe to assume that the actual quality minimum is lower, perhaps significantly. In addition to induced aberrations considered earlier, as well as chromatism, one of the factors lowering optical quality in addition to that determined by the wavefront aberration level alone is the effect of central obstruction.

Beside wavefront aberrations and pupil obstructions, contrast degradation can be caused by detector properties. Specifically, contrast transfer of a CCD chip depends on its pixel size, relative to the system Airy disc diameter. Contrast transfer in the final image is a product of contrast transfers of a telescope and CCD chip. As given by Schroeder (Astronomical Optics, p309), for square pixel with the side p in units of λF, contrast transfer (MTF) is given by:

with ct being the telescope MTF, the sinc term the MTF of CCD chip (the angle is in radians) and ν the normalized spatial frequency. For p=2, which for an ƒ/5 imaging system and λ=0.5μ determines the pixel size as 5μ, contrast transfer drops to zero at ν=0.5 (from ct=sinπ/π=sin180°/π), effectively halving the cutoff frequency of a telescope. In addition, contrast level within resolvable range of frequencies is degraded to the level of twice smaller aperture (FIG. 63). Half as large pixel significantly improves contrast transfer efficiency, but in order for it to approach that of a telescope - assuming its optical quality is not significantly bellow the usual standards - pixel size needs to be reduced to 0.5λF, or one half of the system's aberration-free FWHM (Nyquist criterion).

FIGURE 63: Additional contrast degradation in 42% obstructed system resulting from a relative pixel size p, in units of system's aberration-free FWHM (~λF). The combined system+detector MTF is a product of system's and pixel's MTF (i.e. product of their respective contrast transfers at each frequency). For p=2, contrast drops to zero at spatial frequency ν=0.5; it still retains low negative, or reverse contrast (the bright lines become dark, and vice versa), with the final loss of contrast at ν~0.9. For an f/8 imaging system and no significant aberrations present, this loss of contrast would correspond to 8.8μ pixel size, for λ=0.55m. Smaller pixels deliver better contrast transfer, and are not prone to the occurrences of the "negative" effect in the high-frequency range. To nearly preserve contrast transfer of the system in this case, pixel size shouldn't exceed 2.2μ. This requirement is significantly more relaxed in average field conditions, mainly due to the actual FWHM radius at the detector being larger up to several times (depending on the aperture size and length of exposure) than the aberration-free FWHM's, as a result of blurring caused by atmospheric turbulence.

At this pixel size, the sinc factor for ν=0.5 (approximate MTF cutoff frequency for extended bright low-contrast details for aberration-free aperture), is sin45/(π/4)=0.9, degrading telescope's contrast transfer at this frequency by only 10%. Relative contrast degradation at this pixel size does increase for higher frequencies, which will also slightly reduce the point-source cutoff frequency.

Obviously, aberration level substantially degrading actual contrast transfer of a telescope would correspondingly ease the limit to a maximum pixel size needed to nearly maintain this contrast transfer level. By significantly enlarging telescope's FWHM at the detector, the ever present seeing error puts the actual pixel size limit significantly above the formal Nyquist criterion level, for all but very small apertures. For short- and long-exposure imaging, in order to determine the approximate limit to the pixel size for nearly preserving contrast transfer efficiency of a telescope, the λF unit in the Nyquist criterion needs to be replaced with the respective seeing FWHM.

Of course, other factors influencing CCD contrast transfer efficiency also need to be considered, particularly the level of its internal (electron) noise, generally lower with larger pixels.

Finally, a word about limitations of the standardized MTF in projecting its output to the actual observing. One comes from the standardized MTF being strictly exact only for the object used: parallel, contrasty, brightly illuminated dark and bright lines. For different types of objects, both contrast transfer and cutoff frequency (limiting resolution) will deviate. Also, the MTF assumes continuous spread of energy; in other words, every line interacts with all the energy reaching it from both, left and right. Actual observing objects are finite, and the absence of continuous spread results in the outer areas being subjected to less of energy overlapping than the MTF assumes. It is particularly pronounced with small, or narrow objects (FIG. 64).

FIGURE 64: LEFT: Typical continuous MTF pattern exposes every single point to all the energy from all surrounding points that extends to it (gray circles). All lines between A and B will contribute to the energy spread out by diffraction and aberrations to the mid point marked by the cross. RIGHT: With relatively small ( with regard to the diffracted/aberrated energy spread radius) object image, significant portion of the energy lost to the Airy disc will spread out of object area (C), with comparatively less energy affecting the image itself. It would result in better contrast transfer than what MTF indicates for given aberration level.

The consequence is better contrast transfer and resolution than what the MTF (thus, also PSF and the Strehl) would indicate, even for an object identical to the standardized MTF pattern. Specifics of this effect vary with the object type, as well as type and level of aberrations present.

◄ 6.5. Effects of aberrations: MTF 2 ▐ 7. OBSTRUCTION EFFECTS ►

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