The diffractive techniques proposed here can be adopted over a very wide range - at least from ${\sim}10$ keV to ${\sim}10$ MeV. To investigate their potential for gamma-ray astronomy, a number of different cases will be considered - a low energy gamma-ray system optimised for 200 keV, a medium energy example (500 keV) and a lens optimised for working in the 847 keV ⁵⁶Co line. Although where possible a lens should be optimised for a particular energy, it will be seen that a given device can in fact function over a relatively wide band by changing the detector position. Figure 2 shows that, even without resorting to exotic materials such as Beryllium, losses in a gamma-ray PFL need only be a few percent. Selecting Aluminium, a low technology, low cost material, one finds $t_{\rm max} (=2t_\pi)$ between 0.45 and 1.9 mm (Table 1) for the energies considered here. The transmission loss is only 1.5 to 2%, even allowing for a 0.25 mm backing for constructional purposes (Fig. 3).

As discussed above, the detector spatial resolution at these energies will be of the order of a millimetre, so from an angular resolution point of view there is no reason to consider very small values of p. This leads to the concept of a very simple lens in which a millimetre scale groove structure is machined into an aluminium plate a millimeter or so thick. The profile for the 500 keV example lens with p=1 mm is illustrated in Fig. 3.

The main problem is that for any reasonable diameter the focal length becomes very long indeed. The 5 m diameter 500 keV lens example (Tables 1, 2, Fig. 3) would have a focal length of just over 10⁶ km at 511 keV. But is this impracticable? The Xeus studies (Bavdaz et al. 1999) have demonstrated the feasibility of having a focussing optic on one spacecraft and a focal plane assembly on another one actively controlled to remain at the focal point. The separation in the case of Xeus is only 50 m but this is in low earth orbit where gravity gradient effects are serious.

Many other mission concepts under study require precision control of the relative positions of spacecraft and the concept of "formation flying'' has been validated by several studies. In fact there are engineering groups at JPL, GSFC and Estec that specialize in it. In particular the LISA gravitational wave observatory mission (LISA Study Team 1998) plans to use three spacecraft with distances of $5\times 10^6$ km between each pair. The entire cluster will orbit the sun at 1 AU. The LISA spacecraft will be actively controlled to hold their position with respect to a proof mass within each to a precision of ${\sim}1$ nm (on a 1 s timescale).

This suggests that a stable baseline a few million km long between two spacecraft is not inconceivable and we will proceed to work through the implications of such a design.

**Table 2:** Expected performance and comparison with other missions.
	Integral SPI	Integral Ibis	Example Diffractive Lens
	Spectrom.	Imager	Telescopes
Focal length (m)	1.7	3.2	10⁹
Band (keV) - fixed configuration	20-8000	15-10000	$200\pm 0.8$	$500\pm 0.9$	$847\pm 1.0$
Band (keV) for 50% response			125-500	325-1200	540-2100
with separation adjusted
Effective area (m²) ⁽¹⁾	0.009	0.05	12.1	6.4	4.4
Angular resolution ( $\mu''$ )
(intrinsic)	10¹⁰	10⁹	0.3	0.12	0.07
(with chromatic aberration)			1.7	0.7	0.5
Sensitivity⁽²⁾
Continuum ⁽³⁾		$1.6\times 10^{-6}$	$2.5\times 10^{-9}$	$4\times 10^{-9}$	$4.5\times 10^{-9}$
Narrow Line ⁽⁴⁾	$2.5\times 10^{-5}$		$8\times 10^{-10}$	$1.5\times 10^{-9}$	$1.8\times 10^{-9}$

⁽¹⁾ At 500 keV for SPI/Ibis; Detector efficiency and 20% provision for lens imperfections taken into account.
⁽²⁾ Sensitivities are for a point source. For the example PFL telescopes, the background taken is based on SPI predictions, scaled to 1 cm², corresponding to 2 $\mu''$ .
⁽³⁾ Photons cm^-2 s^-1 keV^-1 for 5 $\sigma$ in 1 day, E=500 keV, dE=250 keV (Ibis).
⁽⁴⁾ Photons cm^-2 s^-1 for 5 $\sigma$ in 10⁶ s, (SPI figure is for 500 keV).

4 Astronomical applications