Toward the diffraction limit with transmissive x-ray lenses in astronomy

Christoph Braig; Peter Predehl

doi:10.1364/AO.51.004638

Applied Optics
Vol. 51,
Issue 20,
pp. 4638-4659
(2012)
•https://doi.org/10.1364/AO.51.004638

Toward the diffraction limit with transmissive x-ray lenses in astronomy

Christoph Braig and Peter Predehl

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Christoph Braig and Peter Predehl, "Toward the diffraction limit with transmissive x-ray lenses in astronomy," Appl. Opt. 51, 4638-4659 (2012)

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Abstract

We develop an analytical approach to refractive, blazed diffractive, and achromatic x-ray lenses of scalable dimensions for energies from 1 to 20 keV. Based on the parabolic wave equation, their wideband imaging properties are compared and optimized for a given spectral range. Low-Z lens materials for massive cores and rugged alternatives, such as polycarbonate or Si for flat Fresnel components, are investigated with respect to their suitability for diffraction-limited high-energy astronomy. Properly designed “hybrid” combinations can serve as an approach to x-ray telescopes with an enhanced efficiency throughout the whole considered band, nearly regardless of their inherent absorption.

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$Z$	Element	$ϱ [g {cm}^{- 3}]$	$α \pm σ_{α}$ in Units of [ ${keV}^{2}$ ]
3	Li	$5.34 \times 10^{- 1}$	$9.72 \times 10^{- 5} \pm 1.38 \times 10^{- 14}$
4	Be	1.85	$3.49 \times 10^{- 4} \pm 5.08 \times 10^{- 13}$
5	B	2.34	$4.66 \times 10^{- 4} \pm 1.52 \times 10^{- 12}$
6	C	2.20	$4.77 \times 10^{- 4} \pm 1.53 \times 10^{- 12}$
7	N	$1.25 \times 10^{- 3}$	$2.72 \times 10^{- 7} \pm 2.45 \times 10^{- 19}$
8	O	$1.43 \times 10^{- 3}$	$3.08 \times 10^{- 7} \pm 2.75 \times 10^{- 19}$

	Be			Li
$E$	$N$	$\frac{ρ_{HEW}}{R}$	$P_{N}$	$N$	$\frac{ρ_{HEW}}{R}$	$P_{N}$
2 keV	108	$3.0 \times 10^{- 3}$	0.47%	290	$1.1 \times 10^{- 3}$	0.66%
3 keV	245	$1.3 \times 10^{- 3}$	0.54%	700	$4.5 \times 10^{- 4}$	0.73%
4 keV	450	$7.2 \times 10^{- 4}$	0.56%	1380	$2.3 \times 10^{- 4}$	0.65%
5 keV	725	$4.4 \times 10^{- 4}$	0.56%	2345	$1.4 \times 10^{- 4}$	0.50%
6 keV	1080	$3.0 \times 10^{- 4}$	0.52%	3650	$9.2 \times 10^{- 5}$	0.30%

$2 m / N_{0}$	0.1	0.5	1.0	2.0	3.0	4.0	5.0	6.0	8.0
$T_{\min}$ (%)	90.5	60.7	36.8	13.5	4.98	1.83	0.67	0.25	0.03
$T_{tot}$ (%)	95.2	78.7	63.2	43.2	31.7	24.5	19.9	16.6	12.5

$\frac{Δ E}{E} [\frac{1}{N}]$	0.2	0.6	1.0	1.4	1.8	2.2	2.6	3.0	3.4
$Δ ϵ \|_{g = 1}$	1.01	1.10	1.29	1.67	2.25	—	—	—	—
$Δ ϵ \|_{g = 2}$	1.00	1.05	1.12	1.22	1.40	1.70	2.17	—	—
$Δ ϵ \|_{g = 4}$	1.00	1.03	1.07	1.15	1.25	1.40	1.61	1.95	—
$Δ ϵ \|_{g = 5}$	1.00	1.03	1.07	1.14	1.24	1.39	1.57	1.86	2.19
$Δ ϵ \|_{g = 10}$	1.00	1.03	1.06	1.12	1.21	1.34	1.52	1.76	2.06

Zone # $N$	$1 \times 10^{2}$	$5 \times 10^{2}$	$1 \times 10^{3}$	$5 \times 10^{3}$	$1 \times 10^{4}$
Rel. error $Δ Q$	$- 2.11 %$	$+ 0.77 %$	$+ 1.13 %$	$+ 1.42 %$	$+ 1.46 %$

Ratio $N / N_{0}$	0.00	1.00	2.51	5.00	10.0
Rel. error $Δ Q$	$- 0.33 %$	$- 0.32 %$	$- 0.32 %$	$- 0.40 %$	$- 0.82 %$

$s$ -ratio	0.0	1.0	2.0	3.0	4.0	5.0	5.5	6.0	6.5	7.0
$Q (s)$	1.00	1.01	1.04	1.08	1.13	1.20	1.25	1.31	1.40	1.56

Parameter	Massive Parabolic	Fresnel ( $m = 1$ )	Achromatic
$F$	$7.79 \times 10^{5} m$	$7.79 \times 10^{5} m$	$3.25 \times 10^{6} m$
$\emptyset_{PSF}$	$4.9 \times 10^{- 4} m$	$4.0 \times 10^{- 4} m$	$1.8 \times 10^{- 3} m$
$Δ ϵ$	$0.13 mas$	$0.11 mas$	$0.11 mas$
$Δ E (⋆)$	$0.012 keV$	$0.024 keV$	$1.605 keV$
$A_{eff} \times Δ E$	$2.89 {cm}^{2} keV$	$29.76 {cm}^{2} keV$	$1149 {cm}^{2} keV$

Symbol	Description	Usage or [Unit]
$R$	Outer, total lens radius	$[R] = m$
$E_{c}$	Fresnel blaze energy	$[E_{c}] = keV$
$ψ$	Relative energy	$ψ \equiv E / E_{c}$
$Δ E$	Absolute spectral width	$[Δ E] = eV$ or keV
$F$	Focal length at energy $E_{c}$	$[F] = m$ or km
$ζ$	Fractional focal distance	$ζ \equiv z / F$
$k$	Wavenumber/diffraction order	$k = 2 π / λ$ or $k \geq 1$
$N$	# (Geometrical Fresnel zones)	$R^{2} = N λ F$
$N_{0}$	Critical zone number	$N_{0} = δ / (2 π β)$
$s$	Zone ratio	$s \equiv N / N_{0}$
$\emptyset_{PSF}$	Focal spot size (HEW)	$[\emptyset_{PSF}] = mm$
$Δ ϵ$	Angular resolution	$Δ ϵ = \emptyset_{PSF} / F$
$T = T_{tot}$	Total lens transmission	$T_{m} (ψ)$ or $T_{(tot)} (s)$
$G$	Achromatic gain	$G = G_{N} (E)$
$A_{eff}$	Effective lens area	$A_{eff} = π R^{2} T (s)$
$f$	$f$ -ratio or $f$ -number	$f = F / 2 R$
$Z$	Atomic order number	$Z \geq 1$
$ξ$	Radius of curvature	$ξ = F δ$
$Δ z_{0}$	Focal DOF	$Δ z_{0} = \pm λ / 2 {(NA)}^{2}$
$κ$	Relative lens radius	$κ = σ / ξ$
$F_{Z}$	Diffractive focal length	$F_{Z} = R^{2} / N λ$
$F_{L}$	Refractive focal length	$F_{L} = ξ / δ$
$A$	Refractive aspect ratio	$A = R / ξ_{L}$

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Applied Optics