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where

is the intensity at the centre w = 0 of the diffraction pattern.

The positions of the minima (zeros) of this intensity function are given by the roots of the equation

for w = 0. The positions of the maxima are given by the roots of

If £ = 0, Eq. (3.455) gives the result for the unobstructed aperture of Table 3.25: the first minimum is at w = kpmw = 3.832, or at w = 0.61A/pm. As £ increases, the first root of (3.455) decreases. With £ = 1/2, it is at w = 3.15 or w = 0.50A/pm. It is shown in Born-Wolf [3.120(c)] that

corresponding to w = 2.40 or w = 0.38A/pm for the first dark ring. The central obstruction therefore increases the resolving power in the Rayleigh sense. However, this gain in sharpness of the central maximum is paid for

Fig. 3.105. The diffraction PSF at an annular aperture showing the effect of increasing the central obscuration factor £ (after Born-Wolf [3.120(c)] and G.C. Steward [3.132])

not only by the price of reduced integral energy in the image, but also by the fact that energy is transferred from the core into the secondary maxima, as shown in Fig. 3.105. As we shall see below in connection with the optical transfer function, this results in a loss of contrast in the image and gives the justification for Schiefspiegler in amateur telescope sizes.

If the transmission function over the pupil is no longer uniform but modified in a systematic way, the intensity distribution in the secondary maxima can be modified. This process is called apodisation. The general possibilities are treated by Marechal and Francon [3.26(a)]. Jacquinot and Roizen-Dossier [3.133] exhaustively analysed the possibilities of 1-dimensional screening and this was extended by Lansraux to 2 dimensions [3.134]. Ref. [3.133] gives a general review.

3.10.5 The diffraction PSF in the presence of small aberrations

The effect of small aberrations on the diffraction PSF is of great importance in establishing specifications and tolerances for telescope optics (see RTO II, Chap. 4). We shall again essentially follow the treatment of BornWolf [3.120(d)]; other excellent accounts with different emphases are given by Welford [3.6], Marechal and Francon [3.26(b)] and Schroeder [3.22(f)].

Born and Wolf show that the intensity at the point Q in the diffraction image associated with the Gaussian image point Q0 13 is given by

where I0 is the intensity in the absence of aberrations, i.e. W = 0. fi and ■ are the azimuth angles in the pupil and image respectively, as defined in § 3.10.3, while u and v are the "optical coordinates" of Q:

u = k^)2 r, v = k (R)(p2 + q2)1/2 , where p, q, r are the coordinates in the image plane, R is the distance from the exit pupil to the image Q0 and k = 2n/A.

In the absence of aberrations, the intensity maximum is at the Gaussian focus. If aberrations are present, there will be an intensity maximum at the diffraction focus, which may not be uniquely defined if the aberrations are large. However, if the aberrations are small, there will be a unique diffraction focus with a clearly defined intensity maximum. In this case, the ratio I(Q)/I0 given by Eq. (3.458) can be used as a measure of the effect of the small aberrations on the image quality. This important concept was introduced by Strehl in 1902 [3.135]. Strehl called the ratio I(Q)/I0 the "Definitionshelligkeit", literally "definition brightness". The commonly accepted term in English is the Strehl Intensity Ratio.

13 For simplicity in the following formulation, we will drop the primes of Q', Q' and W' indicating an image as distinct from an object in Fig. 3.99.

An important simplification of Eq. (3.458) is achieved by applying the so-called displacement theorem: the addition of terms in p2, p sin p cos ^ and p0 (constant) to the aberration function have no effect on the intensity distribution of the diffraction image. Such terms refer to changes of the reference sphere, in radius or in tilt. They are Gaussian (first order), as shown in Table 3.1, and are concerned only with the definition of the 'position of the image, not its quality. This theorem is formally proven in Born-Wolf.

It follows that Eq. (3.458) can be simplified as

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