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where P is the phase function. The autocorrelation function (3.494) must be multiplied by the cosine of the argument of (3.502) in terms of the pupil function for the real part A and the sine for the imaginary part B. For details the reader is referred to Wetherell [3.137]. The two components are given by

Figure 3.110 shows the function L(s) from (3.501). It was one of the basic conditions (see § 3.10.1) of the diffraction integral that the relative aperture of the optical system should not be too large. This condition applies equally to the above derivation of the MTF. However, this theory is applicable to large field angles. This introduces a cosine term into (3.491) in which I(£, n) becomes I(£, n cos upr) if the field angle is in the t-section. Welford [3.6] shows that the cosine term disappears in the autocorrelation function.

Lis)

Lis)

Cosine Aperture Distribution

Fig. 3.110. The MTF for a circular pupil free from aberrations and obstruction, corresponding to the diffraction PSF with incoherent illumination

Fig. 3.110. The MTF for a circular pupil free from aberrations and obstruction, corresponding to the diffraction PSF with incoherent illumination

Some important properties of the Fourier transform relations must be emphasized. If I/ (£,n) is the intensity distribution in the image, I0 (£0 , no) is the intensity distribution in the object (incoherently illuminated) and Iq (£ — £0— n0) is the PSF for some image point Q related to its object point by the magnification m, then

Ii(£,n) = J J Io(&,no)Iq (£ - &,n - no) d£o dno (3.504)

It is shown (for example in Marechal-Francon [3.26(i)]) that it follows from the theorem of Parseval that the Fourier transforms of the above quantities are related by the simple equation ii(s,t) = io(s,t)iQ(s,t) , (3.505)

i.e. the Fourier transform of the image is simply the product of the Fourier transform of the object and the Fourier transform of the PSF, the latter being iQ(s,t) = OTF. It is clearly far simpler to operate with (3.505) than with (3.504), a convolution process usually written in the form

The simple multiplicative form of (3.505) also has the immense advantage that a chain of factors affecting the final image quality can be handled by multiplying their individual OTFs. This can be applied not only to a succession of individual optical systems or elements but also to detectors and disturbing factors such as image motion and manufacturing errors.

An important modification is introduced into the MTF function shown in Fig. 3.110 if a central obstruction is introduced. This problem was first solved in analytical form by O'Neill [3.139]. The form of the MTF function for various values of obstruction ratio £ is shown in Fig. 3.111. The essential feature is that the contrast is increased near the transmission limit, but reduced at low frequencies. This result is exactly what would be expected from the intensity distribution of Steward [3.132] in the diffraction PSF for an annular

Fig. 3.111. MTF for a circular pupil free of aberration with central obstruction £ (after Lloyd [3.140])

aperture (Fig. 3.105). The narrowing of the central maximum corresponds to the increased contrast at the limit frequencies of resolution, while the rapid increase of intensity in the diffraction rings corresponds to the contrast reduction at low frequencies. O'Neill [3.139] points out that Rayleigh [3.141] comments on experiments of Herschel in improving resolution of double stars by introducing a central obstruction. O'Neill also points out the interesting limit case of an infinitely narrow annular aperture as e ^ 1, which gives simply a sharp peak near the limit of resolution and is the two-dimensional analogue of a Young's double slit aperture with the peak occurring at the spatial frequency of the fringes.

The complete formula for the functions of Fig. 3.111 is also given by Wetherell [3.137] including correction of an error in the original O'Neill paper.

The reduction in contrast for intermediate frequencies of about 0.3 is the principal justification for the Schiefspiegler of Kutter and other forms, discussed in § 3.7 above. For an amateur telescope with D = 30 cm, the diameter of the Airy disk with A = 500 nm is 0.83 arcsec. The resolution limit in the Rayleigh sense is 0.42 arcsec, corresponding roughly to our normalized limit frequency 1. If the best atmospheric "seeing" is 1 arcsec, the limit frequency observable in practice is about 0.4 on the normalized scale. With a central obstruction e = 0.35, the loss of contrast from the curves in Fig. 3.111 is of the order of 30%. This loss can be serious for critical planetary observation. For constant atmospheric seeing, the larger the telescope, the more the normalized atmospheric limit frequency is pushed leftwards to small values where the relative contrast loss becomes much less. This is the main reason the Schiefspiegler is of much less interest for large professional telescopes.

The results given in Figs. 3.110 and 3.111 are for the MTF of aberrationfree systems affected only by diffraction. If the aberration in the pupil function F in Eq. (3.494) is non-zero, then the contrast will be reduced. Pioneer work in the solution of (3.494) for basic aberrations with incoherent illumination was done by Hopkins [3.142]. If, for example, the function W in (3.488) represents pure defocus, then we insert

Results for defocus, third order spherical aberration combined with defocus and third order astigmatism were given by Hopkins [3.142], Black and Linfoot [3.143] and De [3.144] respectively, and are quoted by Born-Wolf [3.120(f)]. Figure 3.112 shows the results for defocus. Unlike the Strehl Intensity Ratio, which is only a useful measure for small aberrations, the MTF calculations are valid for large aberrations. The parameter shown on the curves measures the defocus effect and is defined by

where z is the distance of the defined image plane from the Gaussian focus. Values of m > 2 lead to negative contrasts (contrast inversion) at quite low frequencies. The appearance is shown by Marechal-Francon [3.26(j)].

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