The following considerations refer to Wolter-I systems, which incorporate the vast majority of astronomical X-ray optics.

Small grazing angles result in large focal lengths. The ratio of aperture size 2r to focal length f is determined by the slope angle a of the first mirror element, which is approximately the grazing angle for paraxial rays:

The ratio 2r/f is typically around 1/10 if the energy band ranges up to 10keV because the grazing angles have to be less than 1°; soft X-ray telescopes, e.g., ROSAT, can work with a ratio of about 1/3.

The full diameter of the field-of-view is limited both by geometrical vignetting, because of the aperture stops, and shape of the reflecting surfaces and by the requirement on the grazing angle of reflection [7]. Also nested mirror shells act as aperture stops. Practically, the usable field-of-view is also limited by the off-axis blurring (see later). Detector sizes are chosen such that the central part of the field-of-view with an acceptable vignetting and blurring is covered. In general, Wolter optics with a lower ratio 2r/ f have smaller fields-of-view than those with large ratios; the ROSAT PSPC, for example, corresponds to field-of-view diameter of 2°, the CCD cameras on XMM-Newton cover about 30'.

Beside the optical axis, the imaging quality degrades continuously with increasing off-axis angles. The off-axis blurring is shown in Fig. 6.5. An empirical formula for the rms blur circle a for a flat detector plane is given by Van Speybroeck and Chase [11]:

Fig. 6.5 Ray tracing calculated point spread function for off-axis angles from 0' to 20' for a nested mirror system with 2rmax/f = 1/10 and I = 1.8rmax with an unshifted detector plane (upperpanel) and with a shifted one - the shift is 0.00125f (lowerpanel); the percentage numbers denote the vignetting

10 f tan a where a again is the slope angle of the first mirror element (the paraboloid), 9 the off-axis angle of the incoming rays, l the length of a mirror element, f the focal length, and Z the ratio of the slope angles of the paraboloid and that of the hyperboloid. Usually, the grazing angles of both optical elements are chosen so that both reflections of an incoming paraxial ray occur under the same angle of incidence thus minimizing the losses due to reflection. The slope angle of the second element, defined with respect to the optical axis, is therefore three times larger than that of the first so that Z becomes 3.

A hyperbolically curved detector surface would be better adapted to the actual focal plane and thus reduce the blur circle by up to 50% because the factor 2 in the third term is then omitted. In practice, a much simpler but less efficient possibility to reduce blurring is a slight shift of the detector toward the mirror system (see Figs. 6.5 and 6.6) [1,7]. The amount of this shift is chosen according to the actually achieved

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Fig. 6.6 Total blur circle diameter vs. off-axis angle in arcminutes, from a ray tracing done on a telescope system with 2r/f = 1/10 and l = 2r; the detector shift is 0.00018f (from Giacconi et al. [7])

Fig. 6.6 Total blur circle diameter vs. off-axis angle in arcminutes, from a ray tracing done on a telescope system with 2r/f = 1/10 and l = 2r; the detector shift is 0.00018f (from Giacconi et al. [7])

on-axis performance of an optical system and with respect to the field-of-view to be optimized.

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