Advanced Fresnel X-ray telescopes for spectroscopic imaging
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DOI: 10.1007/s10686-009-9180-7
- Cite this article as:
- Braig, C. & Predehl, P. Exp Astron (2010) 27: 131. doi:10.1007/s10686-009-9180-7
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Abstract
We present an effective achromatic imaging technique for diffraction-limited X-ray telescopes. For a common focal length, independent optical sections of the aperture are dedicated to two or more spectral bands and optimized with respect to their efficiency. For this purpose, we introduce the “achromatic gain” of an X-ray hybrid lens. Large-scale segmented and nested apertures provide a promising implementation of that scheme. An optimized numerical example and astrophysical simulations prove its capabilities for an energy range between 6 and 16 keV.
Keywords
Fresnel lensesX-ray opticsInstrumentationHigh resolution imaging techniques1 Introduction
Fresnel lenses and their derivatives have been suggested for next generation X-ray telescopes [1, 2]. Being inexpensive and lightweight, diffraction limited optics with an angular resolution down to 10^{ − 3} or even 10^{ − 6} arcsec may be manufactured from low-Z lens materials like beryllium (Be) or even ordinary plastics. Regarding Abbe’s law, aperture diameters of a few decimeters would be sufficient beyond photon energies of about 1 keV. Preliminary calculations [1, 3] were followed up by novel schemes [4] for hard-X and Gamma-ray applications. Recent papers analyzed specific aspects [5] and proposed an optimized design for simple transmissive X-ray telescopes [6].
However, little success has been made in the improvement of the dispersion limited spectral bandwidth. Even the bandwith of achromatic doublets is restricted to few 10^{2} eV below 20 keV. For scientific purposes, multi-band telescopes that cover two or more energy intervals simultaneously are of great interest.
In this paper a possible improvement is discussed. We review fundamentals of diffractive-refractive X-ray lenses and outline principles of segmented large-scale apertures. Such optics provide all conditions for the development of nested two-band achromatic telescopes. We introduce the “achromatic gain” as the central concept of such common-focus instruments. An optimized configuration is designed, briefly analyzed with respect to material insufficiencies and applied to an astrophysical target. The simulation, performed on the central region of a galaxy, demonstrates the usability of our scheme.
2 Hybrid Fresnel lenses
2.1 Fresnel diffraction in the X-ray regime
2.2 Achromatic hybrid optics
Global symbols and definitions used within this work
Symbol | Description | Usage |
---|---|---|
R | Outer lens radius | |
\(r_{\textrm{obs}}\) | Radius of central obstruction | \(r_{\textrm{obs}}\leq R\) |
a | Fractional lens obstruction | \(a\equiv r_{\textrm{obs}}/R\) |
T_{N} | #(segment rings) | |
k | Index of segment rings | 1 ≤ k ≤ T_{N} |
η_{N} | Segmented conversion | \(\eta_{N}(a)\lesssim 3\) |
E_{c} | Fresnel blaze energy | |
ψ | Relative energy | ψ ≡ E/E_{c} |
ΔE | Absolute spectral width | \([\Delta E]=\textrm{eV}\) |
F | Focal length at energy E_{c} | |
ζ | Fractional focal distance | ζ ≡ z/F |
m | Diffraction order | m ≥ 0 |
N | #(geometrical Fresnel zones) | R^{2} = NλF |
N_{ ⋆ } | #(Fresnel zones) per segment | N = T_{N}N_{ ⋆ } |
N_{0} | Critical zone number | N_{0} = δ/(2πβ) |
s | Zone ratio | s ≡ N_{( ⋆ )}/N_{0} |
E_{0} | Basic multi-band energy | |
ψ_{0} | Relative multi-band energy | ψ_{0} ≡ E/E_{0} |
E_{p,q} | Multi-band blaze energies | E_{p,q} ∝ E_{0} |
\(r_{\textrm{lens}}^{(i,o)}\) | Inner, outer partial lens radius | \(r_{\textrm{lens}}^{(i)} < r_{\textrm{lens}}^{(o)}\) |
\(\varnothing_{\textrm{FOV}}\) | Detector diameter | \(\varnothing_{\textrm{FOV}}=2r_{\textrm{FOV}}\) |
\(n_{\textrm{FOV}}\) | #(resol. elements) in FOV | \(n_{\textrm{FOV}}\propto\varnothing_{\textrm{FOV}}\) |
\(\varnothing_{\textrm{PSF}}\) | Focal spot size (HEW) | \(\left[\varnothing_{\textrm{PSF}}\right]=\textrm{mm}\) |
Δϵ | Angular resolution | \(\Delta\epsilon=\varnothing_{\textrm{PSF}}/F\) |
w | #(coh. steps) within segment | |
\(\mathcal{T}\) | Mean lens transmission | \(\mathcal{T}=\mathcal{T}_{(w)}(s)\) |
\(\mathcal{G}\) | Achromatic gain | \(\mathcal{G}=\mathcal{G}\left(N,N_{0}\right)\) |
\(A_{\textrm{eff}}\) | Effective lens area | \(A_{\textrm{eff}}=\pi R^{2}\mathcal{T}(s)\) |
3 Segmented apertures
In case of large apertures with diameters beyond ∼2 m, it is useful to replace common compact lenses by segmented optics. In the following, we review and extend the detailed considerations of [5, 6].
3.1 Conventional segmentation
3.2 Segmentation with coherent stepping
3.3 Angular resolution of segmented lenses
The spatial and angular resolution provided by an incoherently assembled aperture—with or without coherent stepping—is governed by the “mean” segment size, in contrast to common lenses, whose diffraction-limited angular resolution scales with the reciprocal lens radius R^{ − 1}: Since all single segments act independently, an optical coherence is exclusively maintained therein.
The actual angular resolution depends not only on the—slightly varying—size of the lens segments but also on the radial transmission gradient therein. The contribution of such an achromatic prism-like device to the focal point spread function coincides with the squared Fourier transform of its amplitude transmission. The latter depends on the radial coordinate r within the lens according to Fig. 5. As it was shown in detail elsewhere [5, 6], the diffraction pattern in the focal plane can be approximated assuming almost squared shaped segments for ring numbers \(k \gtrsim 3\), averaged along 0 ≤ φ < 2π.
In summary, the focal spot of conventional as well as coherently stepped segmented apertures is a function of the effective geometrical segment size and the non-uniform absorption due to the refractive lens profile. Since both features affect the PSF independently, their contributions can be factorized.
Conversion for segmented apertures
a | 0 | 0.1 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
η_{N} | 2.27 | 2.30 | 2.35 | 2.55 | 2.86 | 3.19 | 3.55 |
Relative angular resolution for segmented hybrid optics without reduction (w = 1) for various s-ratios
s | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Q_{1} | 1.00 | 1.01 | 1.03 | 1.06 | 1.09 | 1.12 | 1.17 | 1.21 | 1.27 |
Relative angular resolution for coherently stepped segments, based on absorption
s | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
---|---|---|---|---|---|---|---|---|---|---|
Q_{5} | 1.00 | 1.02 | 1.04 | 1.06 | 1.10 | 1.15 | 1.21 | 1.28 | 1.36 | 1.48 |
Q_{10} | 1.00 | 1.00 | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 | 1.07 | 1.08 | 1.10 |
It should be noted that “segmentation” does not necessarily mean an incoherent superposition of the focal spots provided by all lens segments—if accurately aligned, the phase condition would be preserved over the whole aperture and the Abbe limit \(\Delta\varepsilon_{\textrm{coh}}\) is still valid. However, this fine adjustment on a scale of microns will cause substantial problems in space and is therefore considered to be practically impossible.
4 Dual band apertures
Many astronomical investigations benefit from multi-wavelength observations. For instance, insights into conversion mechanisms of AGN or X-ray binaries may arise from hardness ratios. Simple achromatic two-band telescopes for such applications use the common-focus concept mentioned above. Its theory is based on the one developed for purely diffractive apertures [5]. We will repeat the main results in short.
5 The achromatic gain
In Fig. 4 the normalized focal lengths of Fresnel lenses are shown for the first three diffraction orders.
5.1 Definition for hybrid lenses
5.2 Optimization of common-focus instruments
Zone numbers of Li and Be optics for a given focal length
E(keV) | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
---|---|---|---|---|---|---|---|---|---|
\(N_{\textrm{opt}}^{(Li)}\) | 690 | 1582 | 2430 | 2898 | 3010 | 2888 | 2688 | 2480 | 2278 |
\(N_{\textrm{opt}}^{(Be)}\) | 218 | 516 | 912 | 1326 | 1668 | 1884 | 1982 | 1994 | 1946 |
6 An optimized design
Based on (22), several possible solutions for the lens radii within the nested aperture can be found. In general, compared to purely diffractive systems [5], the obstruction for the outer lens is enlarged by 50–100 % in case of an achromatic scaling factor. Following (24), the small ring lens for E_{2} must be encircled by the large partial lens for the soft band at E_{1}. So it is desirable to find an optimized arrangement that favors the soft X-ray band.
6.1 Lens parameters
Technical details of the optimized nested two band telescope
Band | Energy | w | a | N_{ ⋆ } | \(\mathcal{G}\) | Mat. | T_{N} | \(r_{\textrm{lens}}^{(o)}\) |
---|---|---|---|---|---|---|---|---|
E_{1} | 6.0 keV | 4 | 0.57 | 4480 | 22.2 | Li | 15 | 1.39 m |
E_{2} | 15.7 keV | – | 0.19 | 2750 | 49.5 | Be | 20 | 0.79 m |
Performance parameters of the optimized two-band telescope design, according to Fig. 11
Parameter | Symb. | E_{1} (6.00 keV) | E_{2} (15.7 keV) |
---|---|---|---|
Resolution | Δϵ | 0.67 mas | 0.54 mas |
Spot size | \(\varnothing_{\textrm{PSF}}\) | 0.94 mm | 0.75 mm |
Field of view | \(\varnothing_{\textrm{FOV}}\) | 39 cm | 30 cm |
Focal length | F | 287 km | |
\(A_{\textrm{eff}}\times\Delta E\) | 1212 cm^{2} keV | 5331 cm^{2} keV |
6.2 Stray light halos
Data of the stray light halos from Fig. 13
Band | Energy | m | ζ_{ − } | ζ_{ + } | \(P_{\textrm{scat}}\) | \(\left(A_{\textrm{eff}}\times \Delta E\right)_{\textrm{scat}}\) |
---|---|---|---|---|---|---|
E_{1} | 6 keV | 2 | 0.28 | – | 1.31% | 1.1×10^{1} cm^{2} keV |
E_{2} | 16 keV | 2,1 | 0.72 | 1.62 | 23.9% | 5.8×10^{3} cm^{2} keV |
7 Material issues
Though only very few empirical studies exist on the effect of elementary and other impurities on (X-ray) photons propagating through the lens medium, theoretical predictions clearly describe how photons are scattered by such inhomogeneities [11]. In this work we restrict to predominant absorption losses caused by atomic substitutions in the regular crystal lattice consisting of Li or Be, respectively.
8 Simulations
We demonstrate the astronomical capability of the presented single-focus dual band instrument. As mentioned at the beginning of Section 4, dual band telescopes might be applied to investigations of hardness ratios, probably the simplest form of imaging spectroscopy. It is convenient to assume an ordinary power law whose photon index may vary within \(1.2\lesssim\gamma\lesssim 1.8\).
8.1 X-ray emission from NGC 4594
The flux density at 15.7 keV amounts to about 20% of the corresponding one at 6 keV. Since the telescope provides an almost inverse sensitivity, we expect similar count rates in both energy bands. Integrated to an exposure time \(\Delta t_{\textrm{obs}}\) of 10^{6} s, photon numbers greater than 2×10^{4} in the soft and hard band may be expected. In general, the target photons will be distributed among the number of pixels corresponding to the resolution of the X-ray source. However, in the case of NGC 4594, the central black hole has been estimated to \(10^{9} M_{\odot}\) [15]. The corresponding gravitational radius R_{s} of \(2.9\times 10^{12}\,\textrm{m}\) implies an overall diameter of the accretion disk of at least 10^{15}-10^{16} m, depending on the theoretical model. The angular resolution of \(\lesssim 10^{-3}\) arcsec yields an estimated length scale of the same order, about 10^{15} m. Though the detailed structure of the accretion disk will hardly be resolved by our telescope, an optimal signal-to-noise ratio (SNR) may be expected for signal photons distributed over a few pixels. In particular, an assumed accretion disk which is extended to \(10^{3}\, R_{s}\) would correspond to an angular diameter of 4.2 mas.
8.2 Signal-to-noise ratio
Photon counts and covered number of pixels \(\#(\textrm{pix})\) for the modeled disk-jet system in the core of NGC 4594
Target region | E_{1} (6.0 keV) | E_{2} (15.7 keV) |
---|---|---|
\(S_{\textrm{disk}}\,/\, \#(\textrm{pix})\) | 8.32×10^{3} / 105 | 1.75×10^{4} / 105 |
\(S_{\textrm{jets}}\,/\, \#(\textrm{pix})\) | 2.04×10^{4} / 185 | 7.13×10^{3} / 185 |
The SNR may be thus as good as 8.3 for 6 keV and 10 in the hard band, as it is confirmed by numerical simulations: For each pixel within the FOV, the background level is superimposed by the presumed original intensity distribution of the target. The result is convoluted with the discrete PSF and “randomized” by an artificial Poisson noise (see Fig. 15).
Obviously, almost nothing is known about the actual length and brightness of the synchrotron jets and the accretion disk might be 10 times smaller than assumed above. However, our simulations prove the feasibility of fruitful observations using two-band achromatic telescopes.
9 Conclusion
An extension to single-band hybrid Fresnel X-ray telescopes for spectroscopic applications is developed. The detectable energy range may be doubled at least, using several partial instruments with a common focal length. An efficient design relies on segmented, nested ring lenses for the simultaneous imaging of two energy bands. The “achromatic gain” is introduced as an essential concept for optimized assemblies of that type. Central detector units minimize the natural as well as artificial background noise. The observational sensitivities of the soft and hard band are very suitable for measurements of hardness ratios for power-law emitting sources like AGN or X-ray binaries.
Further theoretical steps should enhance the luminous power and the number of energy bands. From a technological point of view, dedicated detectors for the spectral structure of stepped hybrid profiles are of great interest. Other challenges point to investigations of low-Z lens materials and an accurate control system for the positioning of lens and detector in space—problems that will be attended to forthcoming publications.
Acknowledgements
This work was financially supported in part by the Heidenhain foundation, Traunreut (Germany). The authors gratefully recognize encouraging discussions with Joachim Trümper. Special thanks are dedicated to Karsten Jessen who carefully read the manuscript.
Open Access
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