Laue gamma-ray lenses for space astrophysics:status and prospects

We review feasibility studies, technological developments and astrophysical prospects for Laue lenses devoted to hard X-/gamma-ray astronomy observations.


• Deep study of high energy emission physics in compact Galactic objects and AGNs
A clue to the emission region and mechanism, along with the properties of the hidden black hole, can be obtained with the measurement of the high energy cutoff and its relation with the power-law energy spectrum of the compact objects. The current observational status is far from clear ( see, e.g., Ref. 15,16). Much more sensitive observations are needed, for both AGNs and compact Galactic sources. In the case of blazars, the gamma-ray observations are crucial for the determination of their emission properties given that their energy emission peaks at hundreds of keV. 17 • Establishing the precise role of non-thermal mechanisms in extended objects like Galaxy Clusters The existence of hard tails from Galaxy Clusters (GC) is still matter of discussion, 18 and, should they exist, their origin is also an open issue. Are they the result of a diffuse emission or are they due to AGNs in the GC? In the former case, what is the emission mechanism? What is their contribution to CXB? To answer these questions will require much more sensitive observations, like those achievable with broad band Laue lenses.
• Origin of Cosmic hard X/soft gamma-ray diffuse background Currently, a combination of unobscured, Compton thin and Compton thick radio-quiet AGN populations with different photon index distributions and fixed high energy spectral cutoff (E c ) are assumed in synthesis models of the Cosmic X-ray background (CXB). 19 Is it reasonable to assume a fixed E c for these sources? A photon-energy dependent contribution from radio-loud AGN to CXB, like blazars, is generally assumed. But their real contribution is still matter of discussion. Deep spectral measurements of a significant sample of AGNs beyond 100 keV is needed to solve these issues.

• Positron astrophysics
Positron production occurs in a variety of cosmic explosion and acceleration sites, and the observation of the characteristic 511 keV annihilation line provides a powerful tool to probe plasma composition, temperature, density and ionization degree. The positron annihilation signature is readily observed from the galactic bulge region yet the origin of the positrons remains mysterious. Compact objects -both galactic and extragalactic -are believed to release significant numbers of positrons leading to 511 keV gamma-ray line emission in the inevitable process of annihilation. A recent SPI/INTEGRAL all-sky map 5 of galactic e − e + annihilation radiation shows an asymmetric distribution of 511 keV emission that has been interpreted as a signature of low mass X-ray binaries with strong emission at photon energies >20 keV (hard LMXBs). A claim for an annihilation line from a compact source (Nova Muscae) was reported in the 90s 20 but was never confirmed. Much more sensitive observations are needed to study the annihilation line origin, sources and their nature.

• Physics of supernova explosions
Type Ia supernovae (SNe Ia) are major contributors to the production of heavy elements and hence a critical component for the understanding of life cycles of matter in the Universe and the chemical evolution of galaxies. Because Laue lens telescopes allow the direct observation of radioactive isotopes that power the observable light curves and spectra, gamma-ray observations of SNe Ia that can be performed with this type of instrument are in a position to allow a breakthrough on the detailed physical understanding of SNe Ia. This is important for its own sake, but it is also necessary to constrain systematic errors when using high-z SNe Ia to determine cosmological parameters.

LAUE LENS CONCEPT
Diffraction lenses use the interference between the periodic nature of the electromagnetic radiation and a periodic structure such as the matter in a crystal. For a classical textbook on X-ray diffraction see, e.g., Ref. 21. In a Laue lens, the photons pass through the full crystal, using its entire volume for interacting coherently. In order to be diffracted, an incoming gamma-ray must satisfy the Bragg-condition, relating the spacing of lattice planes d hkl with the energy of incident photons E and with the angle of incidence θ B with respect to the chosen set of planes (hkl) * : where d hkl (inÅ) is the spacing of the lattice planes (hkl), n is the diffraction order, hc = 12.4 keV·Å and E is the energy (in keV) of the gamma-ray photon. An elementary illustration of the Bragg condition, in two different configurations (reflection and transmission), is given in Fig. 1, where it can be seen that the incident waves are reflected by the parallel planes of the atoms in the crystal.
A Laue lens is made of a large number of crystals, in transmission configuration (Laue geometry), that are disposed such that they will concentrate the incident radiation onto a common focal spot. A convenient way to visualize the geometry of a crystal lens is to consider it as a spherical cup covered with crystal tiles having their diffracting planes perpendicular to the sphere (see Fig. 2). The focal spot is on the symmetry axis at a distance f = R/2 from the cup, with R being the radius of the sphere of which the spherical cup is a part; f is called the focal length.
From the Bragg equation, for the first diffraction order (n = 1), it can be seen that the photons incident on a given crystal at distance r (r min ≤ r ≤ r max ) from the lens axis can be reflected toward the lens focus if their energy E is given by where the approximated expression is valid for gamma-ray lenses, given the small diffration angles involved. * The indices h, k, l, known as Miller indices, are defined as the reciprocals of the fractional intercepts which the lattice plane makes with the crystallographic axes. For example, if the Miller indices of a plane are (hkl), written in parentheses, then the plane makes fractional intercepts of 1/h, 1/k, 1/l with the axes, and, if the axial lengths of the unit cell are a, b, c, the plane makes actual intercepts of a/h, b/k, c/l. If a plane is parallel to a given axis, its fractional intercept on that axis is taken as infinity and the corresponding Miller index is zero. If the Miller indices [hkl] are shown in square brackets, they give the direction of the plane with the same indices. Conversely, the lens radius r (see Fig. 2) at which the photon energy E is reflected in the focus is given by Rotation around the lens optical axis at constant r results in concentric rings of crystals (see left panel of Fig. 3), while a uniformly changing value of r gives rise to an Archimedes spiral (right panel of Fig. 3). Assuming that the chosen diffracting planes (hkl) of all the lens crystals are the same, in the first case (constant r) the energy of the diffracted photon will be centered on E for all the crystals in the ring, while in the second case (Archimedes spiral), the reflected energy E will continuously vary from one crystal to the other, as shown in

Energy passband
Any Laue lens will diffract photons over a certain energy passband (E min ,E max ). From eq. 2, at first order diffraction (the most efficient), it results that Given that, for astronomical applications, the lens passband is desired to be covered with the highest effective area † and in a smooth manner as a function of energy, the energy bands of the photons reflected by contiguous crystal rings or, in the case of the Archimedes structure of a lens, by contiguous crystals, have to overlap each   other, like in Fig. 4. Since the full width at half maximum (fwhm) of the acceptance angle δ (known as the Darwin width) of perfect crystals is extremely narrow (fractions of an arcsec to a few arcsec, see Ref. 21), such materials are not suitable for astrophysical Laue lenses. In order to increase the energy passband of individual crystals one uses mosaic crystals or curved crystals (see following section). The mosaicity of mosaic crystals (see Sect. 3.1) and the total bending angle of curved crystals (see Sect. 3.2) govern the flux throughput, the angular resolution and the energy passband of the Laue lenses. The diffracted flux from a continuum source increases with increasing the mosaicity of mosaic crystals or the total bending angle of curved crystals. For a crystal lens telescope, crystals with mosaicities or total bending angles ranging from a few tens of arcseconds to a few arcminutes are of interest.
The bandwidth of a lens for an on-axis source is determined by the mosaicity or total bending angle of the individual crystals and by the accuracy of the alignment of the single crystals. By forming the derivative of the Bragg relation in the small angle approximation (2d hkl θ B ≈ nhc/E), we get If ∆θ B is the mosaicity of the mosaic crystal or the total bending angle of the curved crystal, the corresponding energy passband ∆E of the crystal becomes It is worth pointing out that, whereas the energy passband of a crystal lens grows with the square of energy, the Doppler broadening of the astrophysical lines (e.g. in SN ejecta) increases linearly with energy for a given expansion velocity.

CRYSTAL REFLECTIVITY
Both mosaic crystals and curved crystals are suitable to be used for a Laue lens. We discuss the properties of both of them and their reflectivity.

Mosaic crystals
Mosaic crystals are made of many microscopic perfect crystals (crystallites) with their lattice planes slightly misaligned with each other around a mean direction, corresponding to the mean lattice planes (hkl) chosen for diffraction. In the lens configuration assumed, the mean lattice plane is normal to the surface of the crystals. The distribution function of the crystallite misalignments from the mean direction can be approximated by a Gaussian function: where ∆ is the magnitude of the angular deviation from the mean, while β m = 2.35η is the fwhm of the mosaic spread (called mosaicity).
For the Laue geometry and diffracting planes perpendicular to the cross section of the crystal tile (see, e. g., Fig. 2), the crystal reflectivity R(∆, E) is given by Ref. 21: where I 0 is the intensity of the incident beam, µ is the absorption coefficient corresponding to that energy, γ 0 is the cosine of the angle between the direction of the photons and the normal to the crystal surface, T is the thickness of the mosaic crystal and σ is: where in which r e is the classical electron radius, F hkl is the structure factor, inclusive of the temperature effect (Debye-Waller's factor), V is the volume of the crystal unit cell, λ is the radiation wavelength and θ B is the Bragg angle for that particular energy, while f (A) in Eq. 10 is well approximated by: Here B 0 is the Bessel function of zero order integrated between 0 and 2A, with A defined as follows: in which t 0 is the crystallite thickness, and Λ 0 (extinction length) is defined for the symmetrical Laue case (see e.g. ref. 23) as: In general f (A) < 1 and converges to 1 if t 0 ≪ Λ 0 . In this case we get the highest reflectivity.
The quantity γ 0 σ is known as secondary extinction coefficient and T /γ 0 is the distance travelled by the direct beam inside the crystal.

Curved crystals
Similarly to mosaic crystals, curved crystals have an angular dispersion of the lattice planes and thus a much larger energy passband (see Eq. 7) than perfect crystals. The properties of these crystals and the methods to get them are discussed in Ref. 24. Here we summarize their reflection properties.
The most recent theory of the radiation diffraction in transmission geometry for such crystals, in the case of a large and homogeneous curvature, is now well fixed and has been compared with the experimental results (see Refs. 25,26). In this theory, the distortion of diffracting planes is described by the strain gradient β s , that, in the case of a uniform curvature, is given by: where Ω is the total bending angle and corresponds to the mosaicity of the mosaic crystals, T 0 is the thickness of the crystal and δ is the Darwin width.
When the strain gradient |β s | becomes larger than a critical value β c = π/(2Λ 0 ), it has been shown that, for a uniform curvature of planes, the peak reflectivity R max of a curved crystal is given by: where I peak r is the reflected peak intensity, c p = Ω/T 0 is the curvature of the lattice planes assumed to be uniform across the crystal thickness, and the extinction length Λ 0 = Λ 0 (E) is given by Eq. 14. The reflected intensity profile I r (c p , E) is that of a perfect crystal with the Darwin width replaced with Ω. This profile is shown in Fig. 5.
From the last equation, it can be shown that the highest peak reflectivity is obtained for a curvature of the lattice planes given by: where M = π 2 d hkl Λ 2 0 and N = µΩ cos θB .

Mosaic crystals vs. curved crystals
Both mosaic crystals and curved crystals can be used for a Laue lens, if they can be produced with the needed angular spread. However, in principle, curved crystals can reach a higher efficiency than mosaic crystals. Indeed, while the diffraction efficiency of mosaic crystals is limited to 50%, that of curved crystals can reach 100%. Another advantage of curved crystals is that the diffraction profile of a curved crystal is rectangular with width related to Ω, while that of mosaic crystals is Gaussian with fwhm equal to the mosaicity β m . Given the absence of Gaussian tails, curved crystals concentrate the flux better (see next Section). This better performance of the curved crystals with respect to the mosaic crystals for Laue lenses is discussed in depth in Ref. 27.
Curved crystals can be obtained in various ways. 24 The most feasible techniques to be used for Laue lenses include the elastic bending of a perfect crystal (the technique commonly adopted in synchrotron radiation facilities), the deposition of a coating on a wafer, the growing of a two-component crystal whose composition varies along the crystal growth axis (see, e.g., Ref. 25) or the indentation of one face of a wafer. The last technique is being developed at the University of Ferrara (V. Guidi, private communication) with very satisfactory results (see Fig. 6). Also the deposition of a coating on a wafer is being tested at the same University.  . ∆θ gives the difference between the incidence angle of the monochromatic photon beam and the Bragg angle. Open circles: ratio between measured intensity of the diffracted beam and measured intensity of the transmitted beam (also called diffraction efficiency). Filled circles: difference between transmitted and diffracted intensities. Note that the angle ∆θ, through the Bragg law, is related to the reflected photon energy. Thus the figure also shows the energy bandwidth of the curved crystal. Reprinted from Ref. 28, who tested the crystal sample.

OPTIMIZATION OF A LAUE LENS
The free parameters of a Laue lens are the crystal properties (materials, lattice planes for diffraction, microcrystal size and mosaicity in the case of mosaic crystals, total bending angle in the case of curved crystals, crystal thickness), the lens geometry (ring-like or Archimedes' spiral), its focal length and its nominal energy passband. Many optimization studies of these parameters have been performed and tested. 24, 29-31

Crystal material
Independently of the crystal structure (mosaic or curved), in order to optimize the crystal reflectivity it is important to maximize Q(E) as defined in Eq. 11. This is the integrated crystal reflectivity per unit volume, whose normalization is the ratio |F hkl /V | 2 between the structure factor of the chosen lattice planes F hkl and the volume V of the unit cell. The inverse of V is the atomic density N . Thus a small V (or a high N ) is important for maximizing Q(E). The value of N as a function the element atomic number Z is shown in Fig. 7. Also double-element crystal materials can be used for Laue lenses. Several of them, developed for other applications, are already available, like GaAs, InAs, CdTe, CaF 2 . With some improvements, these crystal materials can be used for Laue lenses (see discussion in Ref. 24).
Clearly the best lattice planes are those that optimize the structure factor |F hkl |, under the condition that the corresponding d hkl is consistent with lens constraints, such as energy passband, lens size and focal length (see below).

Crystallite size and angular distribution in mosaic crystals
From the reflectivity equation (Eq. 9), the crystallite thickness t 0 plays an important role in the reflectivity optimization. For fixed values of the mosaicity and crystal thickness, the highest reflectivity is obtained for a crystallite thickness that satisfies the condition t 0 ≪ Λ 0 . In general, this implies a thickness of the order of 1 µm.
Unfortunately this condition is still not always satisfied. From extended tests performed on Cu(111) supplied by ILL, 32 it was found that the condition above is satisfied in single points, 22 but not when the entire crystal cross section is irradiated (values even higher than 100 µm have been found 24 ). In addition, in Ref. 24 it is found that t 0 is energy dependent, which is a surprising result that requires an interpretation (see discussion therein).  The crystal mosaicity is another crucial parameter for the optimization of the lens performance. It can be seen 11, 29-31 that, even if a higher mosaicity gives a larger lens effective area (see Fig. 9), a higher spread also produces a larger defocusing of the reflected photons in the focal plane and thus a lower lens sensitivity. This can be seen by introducing the focusing factor G(E) of a Laue lens: in which A ef f (E) is the effective area of the lens and A d is the area of the focal spot which contains a fraction f ph of photons reflected by the lens.
Assuming that the detector noise is Poissonian, it can be easily shown that G(E) is inversely proportional to the minimum detectable continuum intensity of a lens: where I min is given in units of photons cm −2 s −1 keV −1 in the interval ∆E around E, n σ is the significance level of the signal (typically n σ =3-5), B is the focal plane detector background intensity (counts cm −2 s −1 keV −1 ), ∆T is exposure time to a celestial source, and η d is the focal plane detector efficiency at energy E. For a lens made of mosaic crystals of Cu(111), Fig. 10 shows the dependence of G on focal length in two different energy bands, for different values of the mosaicity.
As can be seen, in spite of the fact that a higher spread gives a higher effective area, a lower spread gives a higher G and thus a higher lens sensitivity. This effect is small at short focal lengths (10-20 m), but it becomes very significant at long focal lengths (>30 m), especially at high energies (∼ 500 keV).

Crystal thickness
The crystal thickness is another crucial parameter for the reflectivity optimization of a mosaic crystal. The best crystal thickness is given by in the case of a mosaic crystal, and it is given by in the case of a curved crystal, where the involved quantities are defined in the sections above. In the case of a mosaic crystal, the optimum crystal thickness for various materials is shown in Fig. 11.
As can be seen, the best crystal thickness depends on the absorption coefficient µ. A high absorption coefficient implies a low crystal thickness for the reflectivity optimization.

Focal length
The focal length has a key importance in the case of Laue lenses, more than its importance in the case of traditional focusing telescopes. Indeed, given that the energy passband of a single crystal is quite narrow (see e.g., Fig. 4), from the expression of nominal energy passband (E min , E max ) of a lens (see eqs. 4 and 5), it results that, for first order diffraction, the lens radii (r min , r max ), needed to get a given passband, depend linearly on f : Figure 11. The best crystal thickness that maximizes the crystal reflectivity, for various materials. The mosaicity assumed is 1 arcmin, the crystallite thickness is 1 µm, and the crystal plane chosen for all of them is (111).
Given that high energy photons are focused by the innermost part of the lens, to get a given lens inner area, the focal length must be increased (the lens area approximately increases with f 2 ).
A gamma-ray lens with a broad continuum coverage from 300 keV to 1.5 MeV was proposed in the 90's by N. Lund. 33 He assumed mosaic crystals of Copper and Gold. In order to achieve a significant effective area at high energies (350 cm 2 at 300 keV and 25 cm 2 at 1.3 MeV), the focal length proposed was 60 m.

Broad vs. narrow passband Laue lenses
For the lens optimization, the selection criteria of the crystal material and lattice planes can change depending on the requested lens passband. Two classes of Laue lenses can be identified, broad passband and narrow passband. The former covers a broad energy band (e.g., 100-600 keV) for the study of continuum source spectra, while the latter achieves an optimal sensitivity in a relatively narrow energy band (e.g., 800-900 keV) for gamma-ray line spectroscopy. These two classes of lenses require different criteria in the crystal choice and disposition in the lens for its optimization.

Narrow passband Laue lenses
Assuming a ring-like geometry, these lenses use a different crystalline plane (hkl) for every ring in order to diffract photons in only one energy band centered on the energy E 0 . If, for a given focal length, the ring at distance r 0 from the lens axis concentrates photons centered at energy E 0 using crystals with crystalline plane spacing d 0 , a ring with a radius r 1 > r 0 , according to the Bragg law, will concentrate the same photon energies only if the crystalline plane spacing d 1 is smaller than d 0 or if a higher order is used. From Eq. 3, for materials with a cubic structure for which d hkl is inversely proportional to √ h 2 + k 2 + l 2 , the ring radii are proportional to this quantity.
As the diffraction efficiency decreases with increasing diffraction order n, a crystal in an exterior ring will add less efficient area to the lens than a crystal on an inner ring. However, since the number of crystals increases with the ring-radius, all rings will usually contribute about the same amount of efficient area to the lens. Using larger and larger Bragg angles with increasing ring radius allows the instrument to be relatively compact, featuring a shorter focal length than required if the above criterion were not adopted. An example of a narrow passband Laue lens, the balloon telescope CLAIRE, will be discussed below.

Broad passband Laue lenses
These lenses use the best combination of crystalline planes to cover in the most efficent way the lens passband. Lowest order planes, e.g. (111), are preferred because we can exploit, in addition to their optimum diffraction efficiency, also the higher order diffraction of the same planes. The principle used for covering a broad energy band is the following. In the simple assumption that a single crystal material and crystal plane (hkl) are used, assuming a ring-like geometry of the lens, different concentric rings focus slightly different energies because of the varying Bragg angle, and thus with with several rings a broad energy band can be covered with the first order diffraction (lens nominal energy passband) . But, in addition to the first order diffraction, higher order diffraction can be exploited that increases the effective area at higher energies, while also extending the passband. An example of the distribution of the various crystals and lattice planes in different rings for a broad passband lens (100-600 keV) made of mosaic crystals, with 20 m focal length, is studied in Ref. 28, and it is shown in Fig. 12.
Diffraction lenses with broad energy passbands were also developed and tested for low energy X-rays since the sixties (e.g. Lindquist and Webber 34 ). Today, photons up to 80 keV can be efficiently focused thanks to the development of grazing incidence based on multilayer mirrors (see other papers in this issue). Above this upper threshold, the development of Laue lens telescopes becomes crucial for photon focusing.

OPTICAL PROPERTIES OF A LENS
The optical properties of a lens, for both on-axis and off-axis incident photons, have been investigated by means of Monte Carlo (MC) simulations. 11,31 For flat crystal tiles, the Point Spread Function (PSF) depends on the crystal size, on their mosaicity and on the accuracy of their positioning in the lens.
In Fig. 13 we show the calculated on-axis Point Spread Functions for two cases: a ring-shaped lens of 40 m focal length and a spiral-shaped lens of 6 m focal length. In the first case the lens has a 150-600 keV energy passband and a crystal tile cross section of 10×10 mm 2 , while in the second case it has a 70-300 keV passband and a crystal tile cross section of 15×15 mm 2 . In both cases it is supposed that the crystals, made of Cu(111), have a mosaic structure with 1 arcmin spread and are perfectly oriented in the lens. The 6 m focal length lens has been proposed for a balloon experiment. 35 In the case of curved crystals, whose development is giving very satisfactory results (see section 3.3), the expected PSF becomes very sharp, with a great advantage in terms of angular resolution and sensitivity. Assuming a lens of 15 m focal length made of crystals with a mosaic spread of 30 arcsec, the PSF obtained in the case of 15×15 mm 2 flat mosaic crystals and that obtained in the case of curved crystals with a curvature radius of 30 m (2 times the focal length) are shown in Fig. 15. The difference between the two PSFs is outstanding. In the case   of curved crystals we expect an angular resolution of 20 arcsec and a sensitivity higher than the corresponding lens with flat crystals by a factor of about 10.
In Fig. 14, for flat crystal tiles of 15×15 mm 2 , we show the expected PSF of the 40 m focal length lens when three sources are in the Field of View (FOV), with one of the sources on-axis and the other two off-axis. As can be seen, in the case of off-axis sources the image has a ring shape centered in the on-axis source image, with inner radius that increases with the offset angle, and a non uniform distribution of the reflected photons with azimuth. It is found that the integrated number of photons focused by the lens does not significantly vary from an on-axis source to an off-axis source, but they are spread over an increasing area. As a consequence, in principle the FOV of the lens is determined by the detector radius, but, taking into account that the focused photons from sources at increasing offset spread over an increasing area, the lens sensitivity decreases with the source offset. The azimuthal non uniformity of the PSF for off-axis sources can be usefully exploited, because it gives information on the azimuthal source direction. The angular resolution, in addition to the size of the flat crystal tiles, depends on the mosaic spread and on the misalignments of the lens crystal tiles. For the lens image shown in Fig. 14, the angular resolution is of the order of, or even better than, 1 arcmin. Figure 16 shows the cumulative distribution of the on-axis photons with the distance from the lens focus, for different values of the mosaic spread, in the case of a 40 m focal length lens made of mosaic crystals of Cu(111) with 15×15 mm 2 flat crystal tile cross section. As can be seen, for a low mosaic spread, the distribution is driven by the crystal size, while for large spreads, it is mainly driven by this spread.

THE DEVELOPMENT OF LAUE LENSES
Two key issues have to be faced in order build a Laue lens: • Development of technologies for the mass production of suitable crystals (mosaic and/or curved crystals) in a reasonable time, consistent with that of preparation of a space mission; • Development of a technology for assembling, in an equally reasonable time, thousands of crystal tiles in a lens with the proper orientation accuracy, that largely depends on the focal length.
Laue lens developments are being carried out in different institutions. We summarize here the major results obtained in our institutes.

CLAIRE -a narrow passband Laue lens experiment
The objective of the R&D project CLAIRE (French word that means 'clear') 37 Table 1) mounted on eight rings of a 45 cm diameter Titanium frame. In each ring i, the combination of the crystal plane spacing d i and the Bragg angle θ Bi results in the concentration of 170 keV photons onto a common focal spot of 1.5 cm diameter at 279 cm behind the lens.
CLAIRE's crystals were produced by N. Abrosimov at the Institut für Kristallzüchtung (IKZ) in Berlin. The Germanium-rich Ge 1−x Si x crystals (x ≈ 0.02) were grown by a modified Czochralski technique using Silicon feeding rods to replenish the loss of Si in the melt during the growth. The mosaicities of the Ge 1−x Si x crystals range between roughly 30 arcsec and 2 arcmin, leading to a field of view of about 1.5 arcmin and a diffracted energy bandwidth of about 3 keV at 170 keV. A correlative study between crystal structure, mosaicity and diffraction efficiency of the CLAIRE crystals is presented in Abrosimov et al (2005). 39 After cutting the crystal ingots at IKZ Berlin, most of the crystal tiles were characterized (mosaicity) at the Hard X-Ray Diffractometer of ILL Grenoble.
At CESR Toulouse, the individual crystal tiles were mounted on flexible aluminium supports, which in turn are mounted on the lens frame. The reinforced 45 cm diameter titanium frame that holds up to 576 crystals on 8 rings was designed and manufactured at the Argonne National Laboratory, Argonne, USA. The tuning of the lens consisted of tilting each crystal tile to the appropriate Bragg angle so that the diffracting energy was 170 keV for a source at infinity. Instead of directly calibrating the lens for a parallel beam of 170 keV photons, crystal tuning on the 20 m optical bench at CESR used a 150 kV X-ray generator situated on the lens optical axis at a distance of 14.16 m. At this distance, a crystal was correctly tuned (170 keV at infinity) if it diffracted 122.28 keV photons. A co-aligned mask (brass-lead sandwich) of the size of the entire lens, placed on the optical axis just in front of the lens, was used to select an individual crystal for tuning, while shielding already tuned crystals.
The resulting geometric area of the CLAIRE lens was 511 cm 2 , the FOV and the passband were 90 ′′ and ∼ 3 keV, respectively. The photons were focused onto a small 3 × 3 array of high-purity Germanium detectors, housed in a single cylindrical aluminum cryostat. Each of the single Ge bars was an n-type coaxial detector with dimensions of 1.5 cm × 1.5 cm × 4 cm. Focusing onto such a small detector volume already results in very low background noise. The CLAIRE stabilization and pointing system was developed by the balloon division of the French space agency CNES.  Table 1. The crystalline plane, the crystal size, the number of crystals per ring as well as the ring radius and the Bragg angle at 170 keV are listed for each ring of the prototype crystal lens.  Figure 18 shows the recorded spectra for these experiments. The energies of the centroids are in very good agreement with theory, slight departures from theoretical values (less than 0.5 keV) being the consequence of the incident spectrum shape and/or the detector calibration drifts. The measured peak efficiencies of the ground experiments are in fairly good agreement with the Crab observation: when their efficiencies are rescaled for a polychromatic source at infinity, a peak efficiency of 9±1% is obtained. At first glance, this figure may seem rather modest, however when considering the constraints on a compact balloon instrument, the result is actually very positive. The short focal length leads to a lens with the outermost rings occupied by crystals with high reflection orders n (see eq.  of crystals are situated in the outer rings where efficiencies are unfortunately lowest. A future space instrument will allow longer focal lengths and hence only low order crystalline planes with the highest efficiencies would be used. Also, the quality of the CLAIRE crystals was quite heterogeneous: the efficiency of the individual crystals within a ring varied by factors of 2 to 10, depending on the ring! However, the CLAIRE lens contained crystals as efficient as the Darwin model predicts: the best crystals of the lens showed peak efficiencies well above 20%.
CLAIRE's balloon flight provided the first observation of an astrophysical source with a gamma-ray lens. In combination with the long-distance test on the ground, these results validate the theoretical models and demonstrate the principle of Laue lens. Moreover, CLAIRE's stratospheric flight represents a first demonstration of the Laue lens technology in a space environment.

Current crystal development
The crystal development status is discussed in detail in Refs. 24, 40. Mosaic crystals of Copper are developed and produced at ILL, 32 gradient composition crystals of Si 1−x Ge x alloy with x increasing along the crystal growth axis (curved crystal) are produced at IKZ. 39 Crystal developments for Laue lenses are being undertaken in Italy: mosaic crystals at the CNR-IMEM Institute, and curved crystals at the Physics Department, University of Ferrara (see Fig. 6).

Current lens assembly technology development
Currently new lens assembly technologies are being developed at the University of Ferrara, for building broad passband (70/100-600 keV) Laue lenses, and at CESR for building high energy narrow passband (800-900 keV) Laue lenses.
The crystal tile positioning accuracy in the lens is the most critical issue. It depends on the mosaic spread and the focal length. Longer focal lengths require better positioning accuracies, thus the development is more challenging for lenses working at the highest energies.

Assembly technology status for broad passband Laue lenses
A technology for assembling crystal tiles for a moderately short (≤10-15 m) focal length lens is at an advanced stage of development at the University of Ferrara. 36,41,42 It does not require any mechanism for a fine adjustment of the crystal orientation once the crystal is positioned in the lens frame. Using this technology, a first lens prototype with 6 m focal length has already been developed and tested. It makes use of mosaic crystals of Cu(111).
The technique adopted is described in detail in Refs. 36, 41. It makes use of a counter-mask provided with holes, two for each crystal tile. Each tile is positioned on the countermask by means of two cylindrical pins, rigidly glued to the crystal tile, that are inserted in the countermask holes. The pin direction and the axis of the average lattice plane of each crystal tile have to be exactly orthogonal. The hole axis direction constrains the energy of the photons diffracted by the tile, while the relative position of the two holes in the countermask establishes the azimuthal orientation of the axis of the crystal lattice plane. This axis has to cross the lens axis.
Depending on the direction of the hole axes in the countermask, the desired geometry of lens can be obtained. In the case of a lens for space astronomy, the hole axes have to be directed toward the center of curvature of the lens. In the case of the developed prototype, the hole axes were set parallel to the lens axis for the quick test of the lens with an X-ray tube, that provides a divergent X-ray beam.
Once all the crystal tiles are placed on the counter-mask, a frame is glued to the entire set of the crystals. Then the lens frame, along with the crystals, is separated from the counter-mask and from the pins. In the case of the first prototype, instead of using a chemical attack in order to separate the countermask from the lens as foreseen in the project, a mechanical separation was attempted.
The first developed prototype is made of a 36 cm diameter ring of 20 mosaic crystal tiles. The mosaic spread of the used crystals ranges from ∼ 2.5 to ∼ 3.5 arcmin. The tile cross-section is 15×15 mm 2 while its thickness is 2 mm. The lens frame is made of carbon fiber composite with a total thickness of 1 mm.
The X-ray beam used, first to assemble the crystals and then to test the complete lens, is that available in the LARIX (LArge Italian hard X-ray) facility of the University of Ferrara. For a LARIX description see Ref. 43. A view of the experimental apparatus in the current configuration is shown in Fig. 19.
The prototype was thoroughly tested using the polychromatic X-ray beam described above, as reported in Ref. 36. Figure 20 shows the difference between the measured PSF and that obtained from a simulation, in which a perfect positioning of the crystals in the lens was assumed. As can be seen, only the center part of the measured image (i.e., the black region) is subtracted by the simulated image. The corona still visible in the difference image is the result of the cumulative error (mainly that due to the mechanical separation of the lens from the countermask) during the lens assembly process.
The disagreement between the measured and the expected PSF is also apparent by comparing the cumulative distribution of the photons with the distance from the lens focus (see Fig. 20, right panel). As can be seen, the PSF radius at which the expected fraction of focused photons reach the saturation (16 mm) corresponds to ∼60% of the measured fraction.
The spectrum of the photons focused by the developed prototype is shown in Fig. 21, where we compare the measured spectrum of the central region (i.e., photons in the black region of the left panel of Fig. 20) with the spectrum of all reflected photons. As can also be seen from this figure, the centroid of the spectrum of the central region achieves an intensity level 0.8 times that of the peak spectrum of all reflected photons.
A new prototype is being developed that takes into account the experience gained constructing the first one. 42

R&D for a tunable narrow passband lens
Observing in only one narrow energy band might be considered too much of a handicap for a space instrument. In the framework of an R&D project for the French Space Agency CNES, a tunable γ-ray lens prototype (Fig. 22  a) was developed and demonstrated. 44 The capability to observe more than one astrophysical line with a narrow passband Laue lens requires the tuning of two parameters: the Bragg angle θ B and the focal length f . While the length f will have to be controlled to within ∼ 1 cm, the precision of the crystal inclination has to be   better than the mosaic structure of the crystals. In the setup of Ref. 44, each crystal was tuned by using piezodriven actuators to change the crystal inclination, and an eddy-current sensor to determine the current position (Fig. 22a). The resolution of the control-loop permitted an angular resolution of 0.1−0.4 arcsec. The stability was found to be better than 0.8 arcsec per day and the reproducibility of a particular tuning better than 5 arcsec over a 10 day period (Fig. 22b).

PROSPECTS AND CONCLUSIONS
A big effort has already been invested in the development of focusing Laue lenses for gamma-ray astronomy (>70/100 keV). Thanks to the most recent developments, Laue lenses with short focal length (10-15 m) are already feasible. The major task now in progress is the development of the crystals needed to optimize the lens effective area. A project "LAUE", supported by the Italian Space Agency, has just started in Italy (main contractor DTM, Modena) for the development of both suitable Laue lens crystals and an advanced assembly technology for long focal length lenses (up to 100 m). Thus far, the major limit to the launch of a Laue lens gamma-ray telescope has been the need of long focal lengths (20-100 m), that implies the use of two satellites in formation flying, one for the lens, and the other for the focal plane detector. The development of extendable booms up to 20 m, the optimization of the lens effective area and the limitation of the lens passband to lower energies, all make realistic the prospect of broad  band satellite missions that could join together multilayer mirrors and Laue lenses to extend the focusing band up to several hundreds of keV.
As an example, the 10 5 s continuum sensitivity of a Laue lens made of mosaic crystals, that was investigated in Ref. 28 and mentioned above (see sect. 4.5.2), is shown in Fig. 23. The use of curved crystals can further increase the lens sensitivity.
In the framework of formation flying missions, the mission concept DUAL (Ref. 45) is currently under study by a consortium of institutes from Europe, Japan and the USA. The DUAL mission is composed of a Widefield Compton telescope (WCT) which carries out all-sky surveys, and a narrow band (800-900 keV) Laue-Lens Telescope (LLT) that simultaneously performs very deep observations of selected narrow-field targets. Combining a small Compton telescope with a Laue lens will permit the resolution of the apparently contradictory needs for a future gamma-ray mission: to map out large-scale distributions, monitor extreme accelerators, and measure the polarization of gamma-ray bursts with a small medium-sensitivity Compton Camera and simultaneously accomplish the stringent performance requirements for the specific science goal of SN1a thanks to the Laue Lens. A compact, wide field (2-3 π steradian) Compton telescope with a modest geometric area (400-1600 cm 2 ) resulting in a effective area of 40-160 cm 2 can fulfill the needs for the "all-sky science" and simultaneously serve as a focal plane detector for the LLT. A Laue lens that, in 10 6 s, can achieve sensitivities of 10 −6 photons cm −2 s −1 for a 3% broadened line at 847 keV can be made from Au, Ag and Cu crystals, similar to those presently available. The model lens used for the sensitivity shown in Fig. 24 would have a focal length of the order of 68 m, a radius of 40-58 cm and a total mass of 40 kg.
We expect that future broad band X-/gamma-ray missions for the deep study of non thermal astrophysical processes above 100 keV, antimatter annihilation signatures, and nuclear lines from SN explosions will include Laue lenses.