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▪ CONTENTS
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6.7. Coherent transfer function, Fourier
transform
▐
7.1.1.
Inconsistencies in the theoretical concept?
► 7. THE EFFECTS OF APERTURE OBSTRUCTION
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, a solid circular obstruction usually caused by
the secondary mirror or its cell, (2) secondary holder vanes and (3) apodizing mask. The light
loss is significant only with apodizing mask. Real concern is the change
in pointobject image intensity distribution caused by the changes in
pupil transmission properties, and its effect on image quality. 7.1. Central obstruction effectsMuch 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 lowcontrast 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 the pretty obvious fact that in any such comparison there is more than just a single factor  central obstruction  affecting lowcontrast 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 highquality 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. While the effect on image contrast is the most important one in most cases, it is not the only one. All of the effects of central obstruction are dependent on its size relative to the aperture. Hereafter, the relative central obstruction diameter in units of aperture diameter D is denoted by ο (left). As it will show ahead, the effects of central obstruction are more closely related to its relative area vs. area of clear aperture (or, alternately, to the relative annulus area), than to its relative linear size. While specialized optical instruments can have central obstruction well in excess of 0.5D, in most of amateur telescopes it is in the 0.2D to 0.5D range. Effects of obstructions significantly larger than 0.5D is not addressed, but it is mostly covered by given general relations. The effects of central obstruction are: (1) reduction in light transmission by a factor of (1ο2), resulting from pupil obscuration, (2) reduction in the relative peak diffraction intensity  and the energy content of the Airy disc  by a factor (1ο2)2, and (3) longitudinal extension of the diffraction pattern by a factor 1+ο2. The relative loss of energy from the Airy disc is well approximated with double the relative obstruction area in the pupil, or 2ο2. Consequences of the latter in regard to intensity distribution within diffraction pattern differ somewhat for nearperfect wavefront on one side, and aberrated wavefronts on the other. Depending on the type and size of wavefront deformation, presence of central obstruction may improve, worsen, or have no appreciable effect on wavefront quality within the annulus, compared to the quality of the entire wavefront. This effect is small to negligible for the usual range of central obstruction sizes.
Central obstruction (CO) effect on the intensity distribution I'(r) of the diffraction pattern  or PSF of the obstructed aperture  normalized to 1 for the peak intensity of clear circular aperture, can be generally expressed as I'(r)=(ΣAΣAo)2, with ΣA being the wave amplitude sum at a point of radius r in the image plane for unobstructed aberrationfree aperture, and ΣAo the amplitude sum for the obstruction area, assumed to be a separate aperture. In effect, the sum of wave amplitudes originating from the obstructed area is deducted from the sum of amplitudes of the aperture without the obstruction, and squared. This new complex amplitude is the squared to obtain energy (illuminance) distribution. Since the amplitude contribution from every point in the pupil equalizes as r approaches zero (i.e. at the central point), the amplitude sum from the obstructed pupil area ΣAo for the central point relates to the clear pupil amplitude sum ΣA as ο2 to 1, and the relative peak diffraction intensity I'(0) of an obstructed aberrationfree aperture vs. that in a clear aberrationfree aperture, I(0), normalized to 1 is: I'(0) = (1ο2)2 (60) which is, in effect, the relative annulus area (in units of the clear aperture area) squared. This implies that the size of CO corresponding to a given peak diffraction intensity I'(0) is ο=(1√I)1/2. With the relative intensity distribution  the basis for contrast transfer  being independent of the actual peak intensity, PSF of an obstructed aperture is conventionally normalized to unity for its actual peak intensity, i.e. given as I'(0)/(1ο2)2. Changes in the incident flux (pointsource intensity) do affect actual pattern brightness, but not the relative intensity distribution within diffraction pattern, which is what determines the efficiency of contrast transfer.
Following table presents intensity distribution
within the fourth minima for unobstructed and selected obstructed
apertures. Pattern radius r is in units of λF, and intensity
I'(r)
is normalized to peak diffraction intensity, to
reflect more clearly relative changes in intensity distribution; EE'
is encircled energy. Analogously to the central intensity, the actual
relative encircled energy of the obstructed aperture is smaller by a factor
(1ο2)2,
or EE=(1ο2)2EE'.
For ο=0.30,
energy encircled within first minima is 0.565, with 0.263 in the rings,
but their values relative to the peak intensity  which is what matters
for contrast transfer  are as shown in the table.
If the photon flux in the annulus would equal that in the unobstructed
aperture, both figures would be larger by a factor of 1/(1ο2)2,
or 0.682 and 0.318, respectively, and the central intensity would be
identical to that of the clear aperture.
TABLE 10: EFFECT OF
CENTRAL OBSTRUCTION ON NORMALIZED PSF INTENSITY DISTRIBUTION Parameter
Radius (r)
Normalized intensity [I'(r)]
Encircled energy (EE') o
► 0 0.10 0.20 0.30 0.40 0.50 0 0.10 0.20 0.30 0.40 0.50 0 0.10 0.20 0.30 0.40 0.50 1ST Max. 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 Min. 1.22 1.21 1.17 1.11 1.06 1 0 0 0 0 0 0 0.838 0.818 0.764 0.682 0.584 0.479 2ND Max. 1.63 1.63 1.63 1.61 1.58 1.54 0.0175 0.0206 0.0304 0.0475 0.0707 0.0963 0.867 0.853 0.818 0.766 0.702 0.618 Min. 2.23 2.27 2.36 2.42 2.39 2.29 0 0 0 0 0 0 0.910 0.906 0.900 0.899 0.885 0.829 3RD Max. 2.68 2.68 2.69 2.73 2.77 2.76 0.0042 0.0031 0.0015 0.0011 0.0033 0.0124 0.922 0.914 0.904 0.902 0.893 0.859 Min. 3.24 3.18 3.09 3.10 3.30 3.49 0 0 0 0 0 0 0.938 0.925 0.908 0.904 0.903 0.901 4TH Max. 3.70 3.70 3.68 3.64 3.66 3.78 0.0016 0.0024 0.0037 0.0028 0.0007 0.0004 0.944 0.936 0.926 0.916 0.905 0.902 Min. 4.24 4.32 4.37 4.22 4.04 4.12 0 0 0 0 0 0 0.952 0.949 0.947 0.929 0.907 0.903
As plots above indicate, most of the energy lost to the central maxima is transferred to the first bright ring with obstructions of 0.4D and smaller, while larger obstructions also brighten the 2nd bright ring. Logarithmic PSF shows much better the effect of obstruction on intensity and pattern of fainter rings (for clarity, given only for the ο values of 0, 0.30 and 0.50). The increase of energy transferred farther away is moderate up to about 0.3D CO. With 0.4D, energy beyond six Airy disc radii from the pattern center is more than 50% greater than for unobstructed aperture, and for 0.5 CO it is more than doubled. This is not unusual; for instance, primary spherical aberration (0.4 wave PV) causing near identical relative drop in the peak PSF intensity as 0.5D CO increases the energy beyond that point less than 20% below the increase due to the obstruction. Coma (2/3 wave PV) more than 40% below, but considering its magnitude in the outer field of fast paraboloids, it will brighten background much more than central obstruction (which is normally significantly smaller than 0.5D). Astigmatism (0.6 wave PV) actually reduces the energy level beyond that point, and so on. Nominal change in the normalized peak intensity, given by ΔI = 1I'(0), closely approximates the relative amount of energy transferred from the disc to the rings area. For instance, 30% CO (ο=0.3) will lower normalized central diffraction intensity produced by a perfect wavefront from 1 to 0.828; at the same time, relative transfer of energy from the Airy disc to the rings area is ~0.17, or 17%. A simple rule of thumb is that the relative loss of energy up to ο~0.4 is closely approximated  as a ratio number  by 2ο2, and for ο~0.5 and larger by 1.9ο2. For instance, 33% linear CO (ο=0.33) effectively reduces normalized central intensity to 0.79. Does that make its effect comparable to that of 0.79 Strehl wavefront error equivalent? Yes and no. There is a distinct difference in the energy distribution change caused by CO compared to, say, primary spherical aberration at the best focus. Looking at the energy in the central maxima as the volume under the bellshaped curve, the main effect of spherical aberration is reducing the height of this volume, with the reduction of volume base (width) being nearly negligible in comparison (for given energy density under the volume, as illustrated on FIG. 2). Specifically, 1/4 wave PV wavefront error reduces central intensity by 20%, while shrinking the central maxima less than 3% linearly. The difference in volume (energy) vs. aberrationfree volume is transferred to the rings area; it is approximately 20%, closely approximated by the relative reduction in central intensity. On the other hand, the volume height corresponding to the central maxima is reduced only by 21% due to the effect of 33% CO, but its base is also reduced by as much linearly. When normalized to 1 for the new (reduced) peak, the relative energy loss comes from its base radius smaller by a 1o2 factor; since it is the square of it that determines the volume for given height, the relative energy loss is also around 21%. Thus, the relative volume reduction  i.e. energy loss to the rings  is nearly identical to that caused by 1/4 wave PV of spherical aberration. Another similarity is that most of this energy goes to the 1st bright ring in either case. As a result, contrast transfer over the range of MTF frequencies where the rings energy is dominant factor  from 0 to about 0.4  is also very similar. However, due to the smaller central maxima, the obstructed pattern rebounds in contrast transfer not only above the aberrated clear aperture level. In the range of MTF frequencies where the dominant factor of contrast transfer becomes the size of central maxima, generally from about 0.4 to 1, contrast transfer for aperture with D/3 CO also rises above that for clear aberrationfree aperture. The similarity extends to the contrast dropoff for the range of spatial frequencies below ~0.5 (approximately, left side of the MTF graph), which is the range of resolvable lowcontrast details. In other words, for this range of spatial frequencies, the peak intensity I'(0) resulting from CO is comparable to the Strehl ratio for wavefront aberrations with respect to the effect on contrast and resolution. They both indicate relative amount of energy transferred to the rings area, the main factor determining contrast level at low to midfrequencies 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 ο of CO and similar in effect RMS wavefront error ώ (in units of the wavelength) with respect to lowcontrast detail effect as: ώ~0.24[log(1ο2)2]1/2 (61) (it can be simplified to an empirical approximation, ώ~0.24ο for ο<0.3 and ώ~0.25ο for 0.15<ο<0.3). For ο=0.325, this gives ώ~0.075, practically equal to 1/4 wave PV of primary spherical aberration level. Comparison with the effect of spherical aberration is most appropriate, due to both CO and spherical aberration causing radially symmetric intensity distribution, with the predominant pattern change being brightening and widening of the first bright ring. However, the actual MTF graph (FIG. 104) indicates that the above formula is somewhat pessimistic in regard to the effect of CO. Obstructed aperture has significantly better contrast transfer  even better than that of a perfect aperture  in the right half of MTF frequency range (i.e. for details smaller than about 2λF linear, or 2λ/D in radians. Even in the left half of the graph (range of resolvable low contrast details), obstructed aperture has an edge. The reason is the effect unique to CO (at least in its extent), namely, the reduction in size of central maxima caused by it. The linear reduction is closely approximated by a factor (1ο2) for obstructions of ~D/3 and smaller, and by a factor (1ο2+ο4) for larger obstructions, up to ~0.7D. Good approximation for the 1st minima reduction ratio for any obstruction size is 1οn, with n=2+[ο2/(1ο)], or n=a+(1/a), with a=1ο. The smaller central maxima, combined with more of the outer energy contained in the 1st bright ring, gives to the obstructed aperture an edge in contrast transfer efficiency with respect to the spherical aberration error of near identical nominal energy loss from the Airy disc. In effect, pointobjectresolutionwise, central obstruction makes the aperture act as larger by a 1/(1ο2) factor with added equivalent of (1ο2)2 Strehl of primary spherical aberration.
*There is a small difference between polychromatic (photopic) and monochromatic MTF plot even for nonrefractive optics, showing as lower contrast transfer in the ~0.40.6 frequency range of the polychromatic MTF, due to the longer wavelengths forming slightly larger Airy disc. However, it doesn't change appreciably plot appearance, due to perfect aperture being affected in the same manner. This effect is present at all obstruction sizes. As a result, size of CO causing similar contrast drop to that caused by the amount of low spherical aberration indicated by Eq. 60 is, for the range of resolvable low contrast details (left half of the MTF graph), nearly 10% larger, linearly. Thus better approximation for the RMS error of spherical aberration similar in effect to that of the central obstruction ο for this frequency range is given by: ώ~0.22[log(1ο2)2]1/2 (61.1) (it can also be simplified to an empirical approximation, ώ~0.21ο; it is within a couple of percentage points from the true value for ο~0.4 and smaller). 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 CO. 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 the contrast transfer of a perfect aperture (FIG. 45, top right), that of an obstructed aperture is given by the overlapping area of two unitdiameter circles, but this time for the two annuli, and relative to the annulus area. As mentioned on the MTF page, the standard MTF plot, as the one above, shows contrast transfer for sinusoidal intensity distribution. As the pattern of intensity distribution and/or detail shape, changes, so does its contrast transfer. This can be illustrated comparing MTF for standard sinusoidal pattern with the contrast transfer function (CTF) for squarewave distribution (i.e. clear dark and bright lines), as shown on FIG. 105.
The overall effect on contrast transfer here is similar, with probably the most significant difference being that the limiting lowcontrast resolution (the pattern in top right corners are highcontrast patterns, for which cutoff frequency lies at the horizontal scale) is somewhat reduced for the standard sinusoidal MTF pattern, but not for the squarewave pattern. Next page addresses seeming inconsistencies and the question of applicable concept in the standard theoretical treatment of the effect of central obstruction. ◄ 6.7. Coherent transfer function, Fourier transform ▐ 7.1.1. Inconsistencies in the theoretical concept? ►
