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).
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 and 30%
obstructed apertures, for ~0.95 and ~0.80 Strehl level and 4λ
defocus; the in focus pattern is magnified ~9x relative to defocused
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, 0.19λ
produces a cross-like in-focus pattern, with elliptical extra-focal patterns
oriented perpendicularly one to another.
Specific, recognizable effect of different
aberrations on the appearance of diffraction pattern, and its very high
sensitivity to even small aberration levels, makes possible testing
telescope systems based on the characteristic of diffraction pattern