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Fig. 3.104. Energy encircled in the radius w of the pattern due to Fraunhofer diffraction at a circular aperture. The fraction of energy is given by Lw = 1 — Jo (w) — J2 (w) with w = kpmw. (After BornWolf [3.120(b)])

a more scientific definition. Nevertheless, the Rayleigh criterion does correspond roughly with practice; though the real detection limits with modern detectors will depend on the signal/noise ratio in the observing mode used. The angular resolution according to Rayleigh is therefore, from (3.446)

and the linear resolution

where N is the f/no. If the angular resolution limit of the human eye is taken as 1 arcmin, then the minimum magnification of a visual telescope to make use of its diffraction limited resolution is given by

In practice, a magnification 2 or 3 times higher can be useful to relax the conditions for the eye: beyond this, the magnification is "empty". Taking A=580 nm for visual use, we get rnmi„ = (0.410)D(mm) (3.452)

Of course, this formula takes no account of the geometrical requirements of the eye pupil relative to the Ramsden disk discussed in § 2.2.4.

3.10.4 The Point Spread Function (PSF) due to diffraction at an annular aperture

The case of an annular aperture is of great importance for reflecting telescopes since all optically centered forms have a central obstruction. In the Cassegrain form the secondary is circular and its presence must lead to an annular aperture. More commonly, the baffle system of a Cassegrain telescope, designed to prevent light outside the field of view reaching the detector, forms the determinant central obstruction. The light distribution in the Fraunhofer diffraction pattern is then given by an integral of the form of Eq. (3.439), but the integration over the pupil radius p now extends over the domain limited by epm < P < Pm, where e is the obscuration factor, a positive number < 1. Eq. (3.441) then becomes

2Ji (kpmw)

2Ji (kepmw) kepmw

The intensity is then given by 