3 Gamma-ray diffractive optics

MAXIM will offer major advances in the X-ray band and is technically so challenging that it may seem premature to consider how comparable resolution might be obtained at still higher energies, where performance is currently far inferior to that possible at lower energies. But this paper aims to show that this may be feasible and that, although there are some disadvantages of working at higher energies, in other respects things actually become simpler.

Fresnel Zone Plates (FZPs, Fig. 1a), as invented by Soret (1875), are in some respects related to the Laue lens mentioned in Sect. 2 as each small region of an FZP can be thought of as a small grating that diffracts the incoming radiation towards the focus, the pitch of the diffraction grating becoming smaller as higher deflection angles are needed at larger radii.

Figure 1: a) A Fresnel Zone Plate (FZP), b) A Phase Zone Plate (PZP), c) A Phase Fresnel Lens (PFL).

A simple FZP has a maximum efficiency of $1/\pi^2 =10.1\%$. This results from the fact that apart from the first order (n=1) focus, energy also passes unfocussed (n=0), to higher orders $(n=3, 5\,...)$ and to virtual focii $(n=-1,-3,-5\,...)$.

Figure 2: The thickness, $t_\pi$, of example materials necessary to produce a phase shift of $\pi$, and the corresponding absorption losses. The materials represented are Be, Si, Ni and Au (from top to bottom and from bottom to top respectively in the two plots). Based on data from Chantler (1995).

As well as independently proposing the FZP, Rayleigh pointed out the possibility, demonstrated by Wood (1898), of obviating the zero order loss in what can be termed the Phase Zone Plate (PZP, Fig. 1b). The opaque regions are replaced by ones that allow the radiation to pass, but which have a thickness of refractive material such as to impose a phase shift of $\pi$. The theoretical efficiency limit is raised to $4/\pi^2 =40.4\%$.

Ideally the thickness, and hence the phase change, within each zone should be a continuous function of radius. This leads to the kineform, or Phase Fresnel Lens (PFL) (Miyamoto 1959) (Fig. 1c), in which ideally all of the power is directed into order n=1, giving an efficiency of up to 100%. The same effect can be achieved with a graded refractive index instead of varying thickness (Fujisaki & Nakagiri 1990; Yang 1993). Stepped approximations to a PFL (di Fabrizio & Gentili 1999) can still have efficiencies approaching this value. Effectively the graded diffraction grating is blazed using refraction to concentrate all of the power into order n=1.

FZPs, PZPs and PFLs are increasing being used for X-ray microscopy and other applications at electron synchrotron facilities. Some example parameters for lenses that are based on these principles and that have been used for laboratory applications are given in Table 1.

In view of the need for high efficiency, we will concentrate here on PFLs, and hence the refractive index of the material from which the lens is made is a critical parameter. The refractive index of a material at X-ray/gamma-ray energies is usually expressed as

$$% n*=(1-\delta) - i\beta.$$ (1)

The real part of n* is slightly less than unity, corresponding to values of $\delta$ which are small and positive. $\delta$ can be expressed in terms of the real component f₁ of the form factor,

$$% \delta={r_{\rm e}\over{2\pi}}~n_{\rm a} \left({hc\over{E}} \right) ^2 f_1,$$ (2)

where $r_{\rm e}$ is the classical electron radius, $n_{\rm a}$ the number density of atoms, and E the energy. Similarly the imaginary component of n*, which corresponds to absorption, can be written

$$% \beta={r_{\rm e}\over{2\pi}}~n_{\rm a} \left({hc\over{E}} \right) ^2 f_2,\quad f_2 = E {\mu_{E}\over{2hcr_{\rm e}}},$$ (3)

where $\mu_{E}$ is the absorption cross-section and E the photon energy.

Well above absorption edges, and apart from small relativistic and nuclear corrections, f₁ is essentially constant and equal to the number of electrons per atom, Z. It follows that the thickness $t_\pi$ necessary to provide a phase shift of $\pi$ is approximately proportional to energy. The thickness of a PZP or the mean thickness of a PFL will be equal to $t_\pi$. Values of $t_\pi$ for some example materials are shown in Fig. 2. Much of the present work depends on the fact that as one moves up in energy from the X-ray band, although $t_\pi$ increases, the associated absorption becomes smaller and over a wide range of energy the losses fall to a low level (Fig. 2).

   

Table 1: Properties of some diffractive lenses used in laboratory systems and of some example systems of the type proposed here for gamma-ray astronomy.

	Example reported laboratory systems			Example diffractive lens
Reference	(a)	(b)	(c)	telescopes
Energy (keV)	0.4	8, 20	8	200	500	847
Type	FZP	PZP	Stepped PFL	PFL
Material	Ge	Au	Ni	Al
Maximum thickness, t ($\mu$m)	0.18	1.6, 3	4.5	450	1200	1900
Shortest period, p ($\mu$m)	0.06	0.5	2	2500	1000	590
Aspect ratio (t/p)	3	3.2, 6	2.25	0.5	1.2	2.0
Focal length f (m)	$8\times 10^{-4}$	3, 7.5	1	109
Diameter (d) (mm)	0.08	0.19	0.15	5000
f-number $(F_{\rm no})$	1	1600, 4000	6700	$2\times 10^8$
Theoretical diffraction-limited
resolution:     Spatial ($\mu$ m)	0.035	0.3	1.2
Angular (arc sec)				$0.3\times 10^{-6}$	$0.12\times 10^{-6}$	$0.07\times 10^{-6}$

References: (a) Spector et al. (1999) (b) Chen et al. (1999) (c) di Fabrizio & Gentili (1999).

The focal length, f, the finest pitch of the pattern, p, and the lens diameter, d, are inter-related; we review first some of the considerations which dictate the choice of these parameters.

For microscopy, very fine pitch patterns are sought because the diffraction limited spatial resolution in the object plane is $s_{\rm d}=1.22\lambda f/d=1.22\lambda F_{\rm no}=0.61p$. Here the resolution is that given by the Rayleigh critereon, which corresponds to the 60% power diameter.

For astronomy, the diffraction limited image spot size is equal to $s_{\rm d}$, as given by the above expressions, but it is the angular resolution $\theta_{\rm d}=s_{\rm d}/f$ that is important. From the points of view of the angular resolution of a telescope and of flux concentration, there is no reason to make p much smaller than the detector spatial resolution. Considerations of electron scattering and stopping distance suggest that the detector spatial resolution is likely to be limited to about 1 mm at 1 MeV, scaling approximately as E^0.8.

Pitches finer than the detector resolution may nevertheless sometimes be useful. They may allow larger lenses or shorter focal length to be used because the finest pitch dictates the maximium diffraction angle and hence relates f and d:

$$% f=0.403\times 10^6 \left({p\over{\rm 1~mm}}\right) \left({d... ...\right) \left({E\over{\rm 1~MeV}}\right) \: \: \: \: {\rm km}.$$ (4)