6.4. Diffraction pattern and aberrations

The immediate effect of wavefront deviation from perfect spherical is less efficient energy concentration within diffraction pattern. Constructive wave interference at the central peak, and over most of the central disc diminishes, while increasing at the outskirts of the disc and further out, in the area of rings. In effect, the energy is transferred from the central part of the pattern out. This unfavorable change in intensity distribution makes the entire pattern - and so the point-image it represents - appear larger and less contrasty. Direct consequence is lower quality of both point- and extended object-images, which can be measured as a contrast/resolution loss. As already mentioned, two standard tools for expressing the effect of change in intensity distribution on image quality are the Strehl ratio and modulation transfer function (MTF).

The point-object image formed by a perfect spherical wavefront is diffraction pattern with the highest possible energy concentration (FIG. 94). Diameter of the first minima (approximately the middle of the first dark ring) of diffraction pattern determines the "Airy disc", in honor of the British astronomer Sir George B. Airy, who was the first to mathematically describe diffraction phenomenon back in 1834. Airy disc of the perfect diffraction pattern has linear radius of 1.22λF, angular radius of 1.22λ/D (in radians), and contains 83.8% of the total energy. The second minima linear radius is 2.23λF, the third 3.24λF, and so on. Peak intensity of the first bright ring is 0.00175 of the central intensity, while that of the second bright ring is 0.00042.

This intensity configuration determines limit to resolution of a pair of point objects of similar intensities and near-optimum brightness level as ~λ/D in radians (113.4/D in arc seconds, for λ=0.55μ and D in mm). Although image duality still can be detected at somewhat smaller separations, it is due to elongated shape of the two close point images, rather than due to them being visually separated. Pattern's radial symmetry is not limited to the best focus location: it is preserved in defocused patterns as well. Distinctive quality of a perfect diffraction pattern is its identical in- and out-of-focus pattern for any given amount of defocus.

While the size of the central diffraction disc determines limit to resolution of relatively bright point-objects, it is the amount of energy spread around the disc what critically influences the limit to resolution of details on low-contrast extended objects.

FIGURE 94: Perfect diffraction pattern in its longitudinal (left) and transverse cross-section (right), centered at best focus. Wave interference forms radially symmetric intensity distribution, consisting from the central maxima and a series of successive peaks of rapidly decreasing intensity, separated by intensity minimas (right). Similar successive intensity fall-off also forms axially. First axial minima occurs at 1 wave of defocus from the central maxima, making the central axial peak 16λF2 long (left).

Any wavefront deviation from spherical results in energy being drained from the central disc, only to re-emerge in the area of rings. The main and most damaging effect is brightening of the rings, expanding and softening the point-object images. The enlargement of the central disc itself is comparatively inconspicuous, becoming significant only at large error levels. Intensity distribution within diffraction pattern - either perfect or aberrated - is described by the "point spread function", or PSF (FIG. 95).

FIGURE 95: LEFT: Intensity distribution within diffraction pattern of an aberration-free and aberrated wavefront (1/2 wave P-V of balanced primary spherical aberration). Point spread function (PSF) of a perfect (blue) and aberrated (red) wavefront has generally higher central diffraction maxima, and more energy concentrated within the first minima. Distribution of energy within the aberrated pattern is either radially symmetric for symmetric aberrations, like spherical (shown) and zonal aberrations, or varies with the pupil angle for asymmetric aberrations, such as coma and astigmatism. The effect of low to moderate aberration levels on the FWHM is relatively insignificant. It is the increase in relative brightness of the rings area vs. central maxima that does most of the damage to image contrast. RIGHT: Due to the logarithmic response of the eye to light intensity, visual appearance of diffraction pattern is better approximated with PSF graph with the intensity given on logarithmic (log10) scale. This monochromatic PSF indicates that the perceived drop in intensity of the central maxima due to aberrations is small in comparison to the perceived intensity increase in the (affected) ring area. However, since its base is larger than that of eye's logarithmic response (log2.512), it compresses nominal differences more by a factor of 2.5. Hence plot nearly corresponding to the apparent PSF shows larger perceived differences in brightness for both, aberration-free and aberrated pattern (Inset F).

Just as the shape of wavefront deviation varies with different types of aberrations, so does the pattern intensity distribution. Every aberration leaves its unique fingerprint in the pattern's intra- and extra-focal appearance, as shown in FIG. 96 (patterns generated by Aberrator, Cor Berrevoets). Note that brightness of the pattern - and particularly perceived brightness of the rings - depend on aperture size and stellar magnitude.

FIGURE 96: Simulation of the effect of common aberrations on diffraction pattern of unobstructed (left) and 30% obstructed apertures, for ~0.95 (top) and ~0.80 Strehl, and 4λ defocus; the in focus pattern is magnified 5x relative to defocused pattern.

1 - Perfect diffraction pattern: defocused patterns on either side are identical for any given amount of defocus.

2 - Balanced primary (4th order) spherical aberration, λ/8 (top) and λ/4 P-V, noticeably brightens the first bright ring; the inside pattern is larger and dimmer, with the intensity falling from the center out, opposite to the out-of-focus pattern's intensity distribution. The extra-focal patterns are reversed for the negative (under-corrected) aberration.

3 - Balanced 6th/4th-order spherical aberration, 0.2λ and 0.4λ P-V, often times seen in apo refractors and Maksutov-Cassegrain telescopes, forms more distinctly different out-of-focus patterns than lower-order spherical. The focused pattern has the first two bright rings of nearly equal brightness, distinctly different than a single bright ring of the primary sphrical.

4 - Coma, 0.21λ and 0.42λ P-V. causes asymmetric pattern deformation, with the in-focus intensity extending in the direction opposite to the direction of increasing intensity in either of defocused patterns. With the increase in error level, the central disc of in-focus pattern becomes extended and decentered in the direction of intensity flow.

5 - Astigmatism, 0.18λ and 0.37λ, produces a cross-like in-focus pattern, with elliptical extra-focal patterns oriented perpendicularly one to another.

6 - Turned down edge, 0.50λ, 95% radius, 3.4 produces nearly identical patterns on both sides of defocus, except for the intra-focal pattern being somewhat fainter and less contrasty, with less well defined outside edge.

7 - Pinched optics can produce various pattern deformations. The pattern shown - trefoil - would be caused by a support or retaining elements (clips) having near-perfect 3-sided symmetry (0.18λ and 0.36λ P-V).

8 - A form of tube current where the warmer air accumulates close to the top portion of the tube, has the in-focus intensity increased in the orientation of the air flow, appearing either as a partial brightening of the rings or, with the increase in aberration, nearly continuous intensity extension (0.26λ and 0.64λ).

9 - Atmospheric turbulence causes ever changing random wavefront roughness. Increase in the aberration results first in partial, and then complete disintegration of the diffraction pattern. Diffraction limited level (0.80 Strehl) corresponds to high 8 on the Pickering's 1-10 scale. The 0.95 Strehl level is Pickering's high 9.

Note that the star is fairly bright telescopically, with the aberration-free pattern showing one bright ring clearly, and hints of the second one, even the third. The appearance of the ring structure and size of central maxima of the focused star, as well as intensity pattern of defocused images vary with star's telescopic brightness.

Specific, recognizable effect of different aberrations on the appearance of the diffraction pattern, and its very high sensitivity to even small aberration levels, makes possible testing telescope systems based on the characteristic of diffraction pattern they form.

Looking at the effect of aberrations in a somewhat obscured, 3-D context, is helpful to better understand diffraction images in any vertical image plane. Simulations below show the axial cross section in a range from -8λ (inside focus) to +8λ (out of focus) for clear aberration-free aperture and the five most commonly encountered aberrations.

All aberrations are at their "diffraction-limited" level, for 0.0745λ RMS wavefront error and 0.80 Strehl. For light traveling from left to right, spherical aberration is "overcorrected", i.e. with marginal rays focusing longer than paraxial rays (for "undercorrection" pattern is mirror-reversed, i.e. this pattern shows "undercorrection" for light traveling from right to left). The two asymmetric aberrations, astigmatism and coma, are given in three angles, since their 3-D diffraction image is not rotationally symmetrical.

Presence of central obstruction changes the entire diffraction image, as illustrated below. The overall pattern change is similar to those for clear aperture, so this set of simulations concentrates on the primary spherical aberration.

Again, for light traveling from left to right, shown is "overcorrection". The aberration-free pattern has all rays coming to the same, best focus. Aberrated patterns have their best focus in the middle, with marginal rays focusing farther to the right, and paraxial rays to the left of it (since the paraxial focus location doesn't change, this means that the aberrated patterns are shifted to the left, the larger the aberration, the more so). For 1/4 wave P-V, paraxial and marginal focus are separated by 2 waves of defocus, with the best focus midway between. The separation is proportional to the aberration, so it's half as large with 1/8 wave.

Outer rays focusing beyond the inner rays cause the inside focus (left) side having wider spread of energy, while close diffraction orders make the inside of the converging cone on this side more saturated than on the other one. This feature can be helpful in approximating the magnitude of spherical aberration of obstructed apertures in star testing. Due to the saturation of one side of the focus, the central shadow is softer, breaking out at a larger defocus distance than on the other side. Estimation is inherently difficult because of the lowered contrast on the side with brighter inner area, and because of the gradual formation of the shadow on that side. Somewhat easier, hence more accurate, should be to compare shadow diameter on the two sides for equal amount of defocus, at which are both shadows sufficiently well defined. For 1/4 wave P-V, for instance, the shadow should at 7 waves of defocus be about 60% larger (i.e. nearly 40% smaller) on one side vs. the other one. With 1/8 wave about 1/3 larger, and with 1/3 wave roughly twice larger.

◄ 6.3.2. Aberrations and extended objects ▐ 6.4.1. Star testing telescopes ►

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