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6.7. Coherent transfer function, Fourier transform        7.1.1. Inconsistencies in the theoretical concept?
 

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 point-object image intensity distribution caused by the changes in pupil transmission properties, and its effect on image quality.
 

7.1. Central obstruction effects

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 the 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.

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 (1-ο2), resulting from pupil obscuration,

(2) reduction in the relative peak diffraction intensity - and the energy content of the Airy disc - by a factor (1-ο2)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 near-perfect 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)=(ΣA-ΣAo)2, with ΣA being the wave amplitude sum at a point of radius r in the image plane for unobstructed aberration-free 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 aberration-free aperture vs. that in a clear aberration-free aperture, I(0), normalized to 1 is:

I'(0) = (1-ο2)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)/(1-ο2)2.

   Changes in the incident flux (point-source 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 (1-ο2)2, or EE=(1-ο2)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/(1-ο2)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
 (Source: Aberration Theory Made Simple, Mahajan)

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, and to the first and second ring with obstructions of about 0.5D. 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 P-V) 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 P-V) 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 P-V) actually reduces the energy level beyond that point, and so on.

Nominal change in the normalized peak intensity, given by ΔI = 1-I'(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 bell-shaped 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 P-V wavefront error reduces central intensity by 20%, while shrinking the central maxima less than 3% linearly. The difference in volume (energy) vs. aberration-free 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 1-o2 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 P-V 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 (primarily; the overall pattern is somewhat smaller), 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 aberration-free aperture.

The similarity extends to the contrast drop-off for the range of spatial frequencies below ~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'(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 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 ο of CO and similar in effect RMS wavefront error ώ (in units of the wavelength) with respect to low-contrast detail effect as:

ώ~0.24[-log(1-ο2)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 P-V 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 (1-ο2) for obstructions of ~D/3 and smaller, and by a factor (1-ο2+ο4) for larger obstructions, up to ~0.7D. Good approximation for the 1st minima reduction ratio for any obstruction size is 1-οn, with n=2+[ο2/(1-ο)], or n=a+(1/a), with a=1-ο. 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, point-object-resolution-wise, central obstruction makes the aperture act as larger by a 1/(1-ο2) factor with added equivalent of (1-ο2)2 Strehl of primary spherical aberration.


FIGURE 104
: LEFT: Standard polychromatic* MTF plot by OSLO showing contrast loss vs. perfect unobstructed wavefront (green), for aberrated wavefront with 1/4 wave P-V of spherical aberration (blue) and perfect wavefront with 0.325D central obstruction (red). Despite nearly identical amounts of energy inside and outside the Airy disc, the two imperfect apertures show noticeably different contrast transfer, with the obstructed aperture being generally more efficient, particularly in the high-frequency range. The likely cause of better contrast transfer in mid-to-lower frequencies is less energy beyond the 1st bright ring in the obstructed aperture. In the high frequency range, higher contrast results directly from the smaller central disc (in terms of pupil correlation function - determining MTF as pupil autocorrelation, i.e. the relative overlap area vs. annulus area of two circles of unit diameter whose center separation equals the normalized spatial frequency ν - the spike in contrast level above that of perfect unobstructed aperture results directly from the reduced pupil area, with the contrast transfer given by the ratio of overlapping vs. annulus area, as illustrated at right). Typical resolution threshold for bright low-contrast details (LCB) indicates slight advantage of the perfect over the obstructed aperture in bright low-contrast detail resolution. The 1/4 wave aberrated aperture is more adversely affected in this respect. Resolution difference is more pronounced for dim low-contrast (LCD) details, where both obstructed and aberrated aperture are at roughly 2/3 the resolution limit of a perfect aperture (note that this limit is additionally subject to rod resolution). However, considering that eye acuity and resolution for faint objects are already compromised by the shift toward rods-dominant mode, this may have little or no practical effect.
RIGHT
: Relative contrast transfer of annular aperture is, similarly to clear aperture, given by the overlap pupil area  as a function of the normalized frequency
ν. The only difference is that the overlap area measures up against the annulus area, not that of the full circle. It illustrates quite clearly why contrast transfer of obstructed aperture drops at low frequencies, and then bounces upward toward high frequencies, to exceed that of a perfect clear aperture. However, despite its smaller central maxima, which correspondingly increases its limiting stellar resolution, the theory states that obstructed aperture has cutoff frequency identical to that of a clear aperture, regardless of the size of obstruction. And it seems to be controversial. With the MTF of imaging system being Fourier transform of  its PSF, it cannot produce two or more different outcomes for a given PSF. And that is exactly what cutoff frequency independent of the size of central obstruction would imply. For instance, 1/4 wave P-V of primary spherical aberration and 10% smaller linearly aberration-free aperture with D/3 central obstruction will produce nearly identical diffraction pattern, both in the angular size of central maxima, and angular size and intensity of the first bright ring.

*There is a small difference between polychromatic (photopic) and monochromatic MTF plot even for non-refractive optics, showing as lower contrast transfer in the ~0.4-0.6 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(1-ο2)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 unit-diameter 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 square-wave distribution (i.e. clear dark and bright lines), as shown on FIG. 105.


FIGURE 105: The effect of D/3 c. obstruction on contrast transfer with sinusoidal pattern (left) and square-wave pattern (right). Bottom plots show a direct comparison of contrast transfers for unobstructed (left) and obstructed aperture (right). Central obstruction does lower contrast for both, sinusoidal and square-wave patterns, but it degrades the low-contrast resolution limit only with the former.

The overall effect on contrast transfer here is similar, with probably the most significant difference being that the limiting low-contrast resolution (the pattern in top right corners are high-contrast patterns, for which cutoff frequency lies at the horizontal scale) is somewhat reduced for the standard sinusoidal MTF pattern, but not for the square-wave 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?

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