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pm D

where D = 2pm is the diameter of the aperture. The diameter of the Airy disk is therefore

For A = 500 nm and an aperture D of 5 m diameter, the diameter 2w0 is 0.0503 arcsec. This value is small compared with classical values of "good

Fig. 3.103. Fraunhofer diffraction at a circular aperture 6 mm in diameter, magnification 50 x, A = 579 nm. The central maximum has been overexposed to reveal the weak subsidiary maxima. (After Born-Wolf [3.120(b)] andLipson, Taylor and Thompson, courtesy Brian Thompson)

seeing" due to atmospheric turbulence of about 1 arcsec. However, the potentialities of excellent modern sites, if extreme care is taken with regard to local air conditions, are such that diffraction must be taken into account in scientific specifications for the optical quality of modern telescopes for ground-based use. This will be discussed further in RTO II, Chap. 4. For space telescopes, the diffraction limit will anyway be the reference for the optical specification.

Another important criterion is the encircled energy as the fraction of the total energy contained within a given diameter. Expressing this fraction as Lw within the radius w, this is given by

a formula originally derived by Rayleigh [3.130]. Figure 3.104 shows the function (3.448), in which 1, 2, 3 mark the positions of the dark minima from Table 3.25.

Rayleigh defined the resolving power or resolution limit of a telescope observing two incoherent point objects close to each other (e.g. a double star) by the condition that the central maximum of the first diffraction pattern coincides with the first minimum of the second pattern (the Rayleigh criterion) [3.131], originally introduced by Rayleigh as a criterion for the resolution of spectral lines whose intensity distribution follows the slit function of Eq. (3.438). In that case, the Rayleigh criterion gives a minimum intensity between the two peaks of 81% of the maximum. In the case of the circular aperture, the minimum intensity between the two peaks is 74%, more favourable. There is no special physical basis for considering this to be the resolution limit. Rayleigh considered experimental results agreed with an intensity minimum of about 80%. The Strehl criterion, considered below, is

lw

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