◄ 6.3. Effects of aberrations: MTF 3   ▐    7.2. Spider obstruction and apodization ►

 

Any obstruction placed in the light path of an imaging system prevents waves from a portion of the wavefront to reach the focal zone. The consequence is change in wave contribution at every point of the diffraction pattern. The effect is similar to that of wavefront aberrations in that it changes pattern's intensity distribution, the specifics of it depending on the form and size of obstruction. Those common to amateur telescopes are: (1) central obstruction, usually caused by the secondary mirror, (2) secondary holder vanes and (3) apodizing mask. The light loss is significant only with apodizing mask. Real concern is the change in point-object image intensity distribution and its effect on image quality.

7.1. Central obstruction effect

Much has been said about the effect of central obstruction in the amateurs circles, most of it being speculation. The common notion is that it reduces effective linear aperture for low-contrast details by as much as obstruction diameter. Informal attempts have been made to find a theoretical basis for this empirical "rule". Not a few amateurs "tested" it and often concluded that it "works". What tends to be neglected is a pretty obvious fact that in any such comparison there is more than just a single factor - central obstruction - affecting low-contrast performance. Most of these factors - seeing error, overall optical quality, sensitivity to miscollimation and thermal errors, light scatter, baffling - favor smaller unobstructed aperture, usually a high-quality apochromatic refractor, over the larger reflector or catadioptric. Consequently, if the rule "works" empirically, it inadvertently proves it incorrect, as long as it insists on the difference in performance coming from the effect of central obstruction alone.

Central obstruction effect on intensity distribution of the diffraction pattern - or PSF of the obstructed aperture - can be expressed as PSF(r)=[A(p) - υ2A(υp)]2, with A(p) being the normalized amplitude sum at a point with center separation r in the image plane, as given for unobstructed aberration-free aperture (it is a sum of the series in the brackets in the PSF(p) relation), and A(υp) the sum obtained from the same series form within the brackets, with p replaced by υp (in effect, the wave amplitude sum at a given point for the obstruction area is deducted from that for the entire aperture).

Obstruction of the relative size υ in units of the aperture diameter (FIG. 48) affects peak diffraction intensity I normalized to 1 as given by this simple relation:

                                                                 I = (1-u2)2                                                (59)

which is, in effect, the relative annulus area (in units of the clear aperture area) squared.

Nominal change in the normalized peak intensity - given by dI0 = 1-I, I0 being the peak diffraction intensity for perfect wavefront - closely approximates relative amount of energy transferred from the disc to the rings area. This attribute is consistent with the effect of wavefront aberrations, where the relative drop in peak intensity - for relatively small wavefront errors - practically equals relative amount of energy transferred to the rings area.

The similarity extends to the consequential contrast drop-off for the range of spatial frequencies bellow ~0.5 (approximately, left side of the MTF graph), which is the range of resolvable low-contrast details. In other words, for this range of spatial frequencies, the peak intensity I resulting from the central obstruction is comparable to the Strehl ratio for wavefront aberrations. They both indicate relative amount of energy transferred to the rings area - the main factor determining contrast transfer at low- to mid-frequencies of the MTF. Thus, with the RMS wavefront error ω in terms of the Strehl ratio S being given by ω=0.24√-logS, direct relation can be established between the relative linear size u of central obstruction and the similar in effect RMS wavefront error ώ (in units of the wavelength) for the low-contrast detail performance as:

                                            ώ~0.24[-log(1-u2)2]1/2                                     (60)

For u=0.325, this gives ώ~0.075, practically equal to 1/4 wave P-V of spherical aberration level. Comparison with the effect of spherical aberration is most appropriate, due to both effects causing radially symmetric intensity distribution, with the predominant change being brightening of the first bright ring.

However, the actual MTF graph (FIG. 49) indicates that the above formula is somewhat pessimistic in regard to the effect of central obstruction. Probable reason is the effect unique to central obstruction, namely, the reduction in size of the Airy disc caused by it. Linear disc reduction is closely approximated by a factor (1-υ2) for obstructions of ~D/3 and smaller, and by a factor (1-υ2+υ4) for larger obstructions. Apparently, the overall smaller diffraction pattern and brighter central disc give to the obstructed aperture an edge in contrast transfer efficiency with respect to spherical aberration error of near identical nominal energy loss from the Airy disc.

*There is a small difference between polychromatic and monochromatic MTF plot even for non-refractive effects, showing as lower contrast transfer in the highest ~40% frequency range, due to longer wavelengths forming larger Airy disc. However, it doesn't change appreciably plot appearance, due to a perfect aperture being affected in the same manner.

This effect is present at all obstruction sizes. As a result, size of central obstruction causing similar contrast drop to that caused by the amount of low spherical aberration indicated by Eq. 60, is nearly 10% larger, linearly. Thus better approximation for the RMS error of spherical aberration similar in effect to that of the central obstruction u, is given by:

                                                   ώ~0.22[-log(1-u2)2]1/2                              (60.1)

Note that the MTF output varies somewhat from one program to another. For instance, Aberrator and Atmos show more of a contrast drop resulting from central obstruction. It still has an edge over spherical aberration, but the numerical constant in the above approximation is between 0.22 and 0.23.

Analogous to contrast transfer of a perfect aperture (FIG. 45, top right), that of an obstructed aperture is given by the overlapping area of two unit-diameter circles, but this time for the two annuli, normalized to the annulus area.

Since the negative effect of central obstruction is so similar to that of wavefront aberrations, the question of what is its maximum acceptable size can be answered in terms of the conventional aberration limit of 0.80 Strehl. Setting I=0.80 puts the maximum acceptable central obstruction size at ~0.32D according to Eq. 60, and at ~0.35D according to Eq. 60.1. However, it assumes perfect optics. For an actual optical set of the Strehl ratio S smaller than 1, the minimum acceptable obstruction size for the combined ~0.80 Strehl level for the mid MTF frequencies would be obtained from SI=0.80, with the peak intensity factor for the obstruction, as mentioned, I=(1-υ2)2. In this concept, linear obstruction has to be  smaller than 0.35D, so the υ4 factor can be neglected, and the maximum acceptable central obstruction size for the combined ~0.80 Strehl level is:

      umax~√0.5-(0.4/S)  or, adjusted for better contrast transfer,   umax~√0.6-(0.48/S)         (61)

with  umax being the relative obstruction diameter in units of the aperture. More exact calculation would take into account that the RMS wavefront error - hence the resulting Strehl ratio - would likely change due to the presence of obstruction. The RMS wavefront error change can be for the better, or for the worse, depending on the contribution of the obscured central part of the wavefront to its average deviation. The RMS wavefront error of an obstructed aperture diminishes for spherical aberration and defocus, and increases for coma and astigmatism, as given by the following factors for best focus location:

 - spherical aberration: (1-u2)2
     - defocus: (1-u2)
     - coma: (1-u2)(1+4u2+u4)1/2/(1+u2)1/2
     - astigmatism: (1+u2+u4)1/2

Evidently, these factors may have some importance only with larger obstruction sizes and wavefront errors.

The question of the central obstruction size at which its effect becomes insignificant can be answered in a similar manner. For perfect optics, it is determined by any chosen Strehl figure S* considered to be the level of negligible wavefront aberration, from

                                                                   umax~√0.6-(S*/1.7)                                  (62)

adjusted for the better contrast transfer efficiency.

For imperfect optics, with the Strehl S<1, but presumably better than S*, it would be determined from

                                                                   umax~√0.6-(S*/1.7S)                                 (63)

And for an aberrated optics set with S<S*, valid criterion would be how much of an additional contrast loss τ, expressed as a ratio number, is found to be either negligible or acceptable. According to it,

 
                                                                              umax~√τ/1.7                                       (64)

Taking 5% additional average contrast loss (τ=0.05) on low-contrast details as a reasonable level of hard to notice contrast change, we arrive at the size of obstruction likely to produce negligible effect for most people as υ~0.17 of the aperture diameter. Of course, this applies as well to unaberrated apertures.

More detailed insight into the change of intensity distribution and contrast loss caused by central obstruction is given by the PSF and MTF, respectively (FIG. 50).

◄ 6.3. Effects of aberrations: MTF 3   ▐    7.2. Spider obstruction and apodization ►

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