Parabolic refractive X-ray lenses - IUCr Journals

Parabolic refractive X-ray lenses - IUCr Journals
research papers
Parabolic refractive X-ray lenses
Bruno Lengeler,* Christian G. Schroer, Boris Benner,
Achim Gerhardus, Til Florian GuÈnzler, Marion
Kuhlmann, Jannik Meyer and Christiane Zimprich
II. Physikalisches Institut, RWTH Aachen, D-52056 Aachen,
Germany. E-mail:
Parabolic refractive X-ray lenses are optical components, especially
suitable for third-generation synchrotron radiation sources. This
article describes the status of the development of our lenses and
illustrates the possibilities for micrometre and submicrometre
focusing and for X-ray imaging in absorption and phase contrast.
The parabolic lens pro®le ensures distortion-free imaging of high
quality. First characteristics of Be lenses are given. A microscope
based on Be lenses is expected to have a lateral resolution below
80 nm.
Keywords: parabolic refractive X-ray lenses; X-ray microscopy;
X-ray microanalysis; tomography.
1. Introduction
X-ray diffraction, X-ray analysis in the micrometre and submicrometre range and imaging with X-rays have found a broad spectrum of
applications in physics, chemistry, materials science and life sciences.
Mirrors (Kirkpatrick & Baez, 1948; Hignette et al., 2001), multilayers
(Underwood et al., 1986, 1988), capillaries (Bilderback et al., 1994;
Hoffman et al., 1994), Fresnel zone plates (Lai et al., 1992), Bragg±
Fresnel zone plates (Aristov et al., 1989; Chevallier et al., 1995) and
waveguides (Lagomarsino et al., 1997; Feng et al., 1998) are the
standard optical components for generating a microfocus for hard
X-rays in combination with synchrotron radiation sources. Refractive
X-ray optics of various designs (Snigirev et al., 1996; Elleaume, 1998;
Lengeler et al., 1999a,b; Kohmura et al., 1999; CederstroÈm et al., 2000,
2001; Dudchik et al., 2000; Piestrup et al., 2000; Aristov et al., 2000)
which have been developed recently extend the list of these optical
components. In this article we describe the characteristic features of
parabolic refractive X-ray lenses (PRXLs) for microanalysis and
imaging designed and fabricated at the Aachen University of Technology (Lengeler et al., 1999a,b).
Glass lenses are among the oldest optical components, with an
extremely broad spectrum of applications. This is due to the
outstanding properties of glass concerning homogeneity, reproducibility, strong refraction and low absorption for visible light. On the
other hand, for X-rays in condensed matter, refraction is weak and
absorption is strong as expressed in the index of refraction,
n = 1 ÿ ‡ i, by the small refractive decrement of order 10ÿ6 and
the large absorption coef®cient = 4= when compared with the
values for visible light in glass. For that reason, refractive X-ray lenses
were long considered as not feasible. However, refraction of X-rays in
matter is not zero and absorption is not in®nite, and refractive lenses
for hard X-rays with a focal length in the range of 1 m and below have
been developed in the last ®ve years.
short focal distances are to be achieved with a small number of lenses.
As a consequence, the geometrical aperture of the lens is typically
1 mm and happens to match perfectly the beam size of undulator
sources in third-generation synchrotron radiation sources. Second,
many lenses are stacked in a row in order to increase the refractive
power. The focal length, f , in the thin-lens approximation is then
given by
f ˆ R=2N;
where N is the number of lenses in the stack. With N between 10 and
a few 100, a focal length of 1 m is feasible below about 40 keV. Third,
the mass absorption coef®cient, =, decreases as Z3 =E3 by photoabsorption and then levels off below 0.2 cm2 gÿ1 when Compton
scattering dominates = (Lengeler et al., 1999a). As a consequence,
even for hard X-rays, above about E = 5 keV, only the lightest
elements with a low atomic number Z, like Li, Be, B, C and Al, are
suitable as lens material. Ni is a suitable candidate for energies
around 100 keV.
A substantial improvement in the performance of refractive X-ray
lenses was achieved at our institute by the development of a novel
manufacturing technique by which the lenses are shaped as a double
paraboloid of rotation (Fig. 1). Details are given by Lengeler et al.
(1999a,b). Note the concave form of the lenses which is a consequence of the real part 1 ÿ in the index of refraction being smaller
than 1. The merit of this procedure, as compared with the initial row
of drilled holes (Snigirev et al., 1996), is in the suppression of spherical
aberration and in the focusing in both directions. This allows for
achieving a focal spot size in the submicrometre range. This is also a
prerequisite for imaging by means of refractive X-ray lenses. The
transmission of a stack with N lenses is given by (Lengeler et al.,
T ˆ 1 ÿ exp…ÿ2ap † …2ap †ÿ1 exp…ÿNd†;
2ap ˆ
NR20 R20
f 2
Na r0 Z ‡ f 0 …E†
2. Concept of parabolic refractive X-ray lenses
A number of aspects had to be taken into account for manufacturing
refractive X-ray lenses. First, the radius of curvature R of the
refracting surfaces must be small, in fact below 1 mm, if reasonably
J. Synchrotron Rad. (2002). 9, 119±124
Figure 1
Model of an aluminium refractive X-ray lens with double parabolic pro®le.
One quadrant has been removed in order to show the lens pro®le. In reality
the geometric aperture 2R0 is about 1 mm in diameter.
# 2002 International Union of Crystallography
Printed in Great Britain ± all rights reserved
research papers
and where d is the distance between the apices of the parabolas in a
lens. Na is Avogadro's constant, r0 is the classical electron radius, A is
the atomic mass and Z ‡ f 0 …E† is the atomic form factor in the
forward direction. Note that for a given X-ray energy E the transmission depends only on the inverse focal length and on the mass
absorption coef®cient =, which is an atomic property only
dependent on the atomic number Z and independent of the density.
Away from absorption edges of the lens material, C is almost constant
for all elements. A focal length substantially below 30 cm will lead to
a poor transmission even for low-Z elements as lens material. The
form ®delity and the surface ®nish of the lenses are characteristics
which afford special care. With modern CNC machines it is possible
to generate a rotational parabolic shape within 1 mm and a surface
roughness below 0.1 mm. Fig. 2 shows the cross sections through Al,
Ni and Be lenses which demonstrates the high quality of our manufacturing technique concerning form ®delity and surface roughness.
The pro®les were measured with a white-light interferometer. At the
steep ¯anks of the embossed parabola the amount of backscattered
light becomes very weak. Therefore, only the bottom area with a
diameter of about 150±200 mm is shown in Fig. 2. For aluminium
lenses, this area corresponds to the effective aperture of the lens.
Therefore, large ®gure errors would be detected and avoided prior to
X-ray optical tests at the synchrotron radiation source. Small ®gure
errors that arise during the state-of-the-art manufacturing process do
not measurably affect the X-ray optical properties of the lens. For
beryllium lenses, however, only the central part of the effective
aperture can be investigated by white-light interferometry. For these
lenses it may become necessary to quantify the overall ®gure error by
X-ray optical imaging. This will require the systematic investigation
of aberrations.
There is an important point about roughness which should be
mentioned. For a high-quality X-ray mirror surface the root-meansquare roughness must be below a few 0.1 nm. Refractive lenses
give excellent results even when the roughness is up to a few 100 nm.
This much lower sensitivity to roughness of lenses compared with
mirrors makes the manufacturing process of lenses much less
demanding. The effect can be understood in the following way.
Roughness enters in the transmitted intensity in both cases as a
damping factor, exp…ÿ2NQ2 2 †, where N = 1 for a mirror and where
Q is the momentum transfer. For a lens, Q = k1 ÿ nk1 = k1 is of the
Ê ÿ1 whereas for a mirror Q = 2k1 1 ' 2 10ÿ2 A
Ê ÿ1 if the
order 10ÿ5 A
Figure 2
Height pro®les of an aluminium, a nickel and a beryllium lens measured by
white-light interferometry. The pro®les are displaced in the vertical by 10 mm.
The parabolic ®t gives the quoted values of R and a surface roughness far
below 1 mm.
Bruno Lengeler et al.
Parabolic refractive X-ray lenses
angle of total re¯ection 1 is of the order of 0.1 . The much larger
momentum transfer in a total re¯ecting mirror must be compensated
by a much lower surface roughness in order to generate the same
damping factor as in a stack of lenses. Note that the roughnesses in
the individual lenses are independent from one another, so that it is N
rather than N 2 which enters in the damping factor exp…ÿ2NQ2 2 †.
We would like to insist that a parabolic lens pro®le has major
advantages compared with a spherical pro®le as generated, for
example, by impressing a sphere from a ball-bearing in the lens
material. First, parabolic lenses are free of spherical aberration,
resulting in a small spot size and in almost distortion-free imaging, as
demonstrated below. Secondly, parabolic pro®les can be generated
with higher accuracy as explained in Fig. 3. A parabolic tool always
generates a force component normal to the surface reducing the
tendency for microcracks and hence for pro®le errors and surface
roughness. In contrast, a sphere generates at the equator line only
shear forces without compression. Thirdly, as shown in Fig. 3, the
radius of curvature R and the geometrical aperture 2R0 are decoupled from one another for a parabola whereas they are not for a
sphere. For parabolic lenses it is possible to combine a geometrical
aperture of 1 mm with a radius of curvature R of 200 mm and below.
The smaller the value of R, the lower is the focal length f and the
smaller is the number of lenses needed in a stack.
3. Applications of parabolic refractive X-ray lenses
We will now discuss two main ®elds of application for parabolic
refractive X-ray lenses: ®rst, the generation of a micrometre and
submicrometre focus for diffraction, ¯uorescence, absorption and
re¯ectometry; second, magni®ed imaging in absorption and phase
contrast and tomography.
The generation of an X-ray microfocus is based on the demagni®cation of an X-ray source on the sample. The smaller the source size
the smaller will be its image. Therefore, a low- undulator will
generate a smaller focus than a high- undulator at the expense of
¯ux in the microbeam, since the beam divergence is larger for the
low- undulator, reducing the beam intensity at the aperture of the
lens. The demagni®cation m = L2 =L1 = f =…L1 ÿ f † can be a factor of
100 or more. There is no need for an order-sorting aperture as in
Figure 3
Advantages of parabolic lens pro®les as compared with spherical ones. (a) A
parabolic embossing tool generates everywhere normal forces which results in
a smoother surface. (b) For a parabolic pro®le the geometric aperture 2R0 and
the radius of curvature, which determines the focal length, are decoupled.
J. Synchrotron Rad. (2002). 9, 119±124
research papers
Fresnel zone plates. Since the optical path is straight, alignment is
easy and can be done in 15 minutes.
We have demagni®ed the source at beamline ID22 of the ESRF
with a stack of Al lenses. The radiation was monochromated by
means of an Si (111) double-crystal monochromator operating at
18.2 keV. The number of lenses in the stack was N = 220 and the
radius of curvature was R = 209 5 mm. With a source-to-lens
distance of L1 = 41.7 m the optimal image distance was determined
experimentally to be L2 = 331 mm, giving a focal distance of 328 mm.
This results in a demagni®cation of the source by a factor of 126. The
horizontal and vertical microbeam sizes were measured by a ¯uorescence technique with a gold knife-edge deposited on silicon. The
Au L radiation (9.71 keV) was measured by an energy-dispersive
SiLi detector. Fig. 4 shows a scan through the microbeam in the
vertical direction. A vertical beam width of Bv = 480 nm was
measured. This compares well with an expected full width at halfmaximum (FWHM) size of 450 nm including diffraction and roughness. Note that the low background in the intensity pro®le is characteristic of high-quality lenses with a low level of small-angle and
diffuse scattering. The horizontal spot size of Bh = 5.17 mm (FWHM)
is large compared with the vertical spot size Bv, since the horizontal
source size is large in the high- section at beamline ID22 of the
ESRF. At a low- section of the ESRF the undulator source is smaller
by a factor of six, resulting in a six times smaller horizontal spot size.
The measured gain (intensity in the spot as compared with that
behind a pinhole of equal size) was 367.
By using the microbeam produced with the refractive lens, many
X-ray analytical techniques, such as diffraction (Castelnau et al.,
2001), ¯uorescence analysis (Bohic et al., 2001; Simionovici et al.,
2000; Schroer et al., 2000), absorption spectroscopy and small-angle
scattering, can be performed with spatial resolution on the micrometre and submicrometre scale. The small aperture of the lens
compared with the image distance L2 leads to a beam divergence that
is suf®ciently small for most diffraction experiments (k=k < 10ÿ3 ).
We now consider imaging by hard X-ray microscopy based on
refractive lenses. Just as with a normal microscope, the object is
placed slightly outside of the focus of the lens (Lengeler et al., 1999a).
Then a strongly magni®ed picture is generated on a two-dimensional
camera. The magni®cation is given by
Figure 4
Vertical pro®le through a microbeam generated by a parabolic refractive
X-ray lens at ID22 at the ESRF. An error function is ®tted to the data
measured by a knife-edge technique. Its derivative gives the vertical width of
the spot.
J. Synchrotron Rad. (2002). 9, 119±124
M ˆ L2 =L1 ˆ
L1 ÿ f
Fig. 5 shows a micrograph (M = 21) of a gold test pattern taken with
120 Al lenses (R = 0.2 mm) at 25 keV. The period of the test structure
is 2 mm, the line width is 1 mm and the thickness is 2 mm.
The lateral resolution for incoherent illumination of the object is
given by (Lengeler et al., 1999a)
dt ˆ 0:75
ˆ 0:75
The numerical aperture N.A. is given by
N:A: ˆ
where L1 is the object±lens distance and (Lengeler et al., 1999a)
Deff ˆ 2R0 1 ÿ exp…ÿap † =ap
Deff is the effective aperture of the lens, reduced by photon absorption and scattering, compared with the geometrical aperture 2R0.
With aluminium lenses we have reached a resolution of 0.34 mm at
23.5 keV which is close to the theoretical value expected from
photoabsorption in Al.
The lower absorption in Be leads to an expected lateral resolution
of 80 nm and below. The in¯uence of absorption is also clearly
demonstrated in the size of the ®eld of view. Fig. 6 shows two
numerically generated images of a Ni mesh (period 50 mm, width of
grid wires 10 mm, thickness 10 mm) as imaged at 20 keV in a one-toone geometry ( f = 2.35 m, L1 = L2 = 2f ) with a Be and an Al lens
(small inset). The parabolic refractive Be lens with 50 lenses in the
stack gives a ®eld of view larger than 800 mm. For an Al lens (N = 31,
all other parameters equal) the ®eld of view is about 200 mm.
Fig. 7 shows the micrograph of a Ni mesh with a 12.5 mm period
taken at 25.5 keV with 120 Al lenses and a focal length of 1.05 m. The
magni®cation was M = 21. Note the excellent quality of the image
very close to the quality expected from the numerical simulation
Figure 5
Micrograph of a gold test pattern (period 2 mm, thickness 2 mm) taken at
25 keV with an Al lens (N = 120, R = 0.2 mm, f = 1.05 m). The image of the
pattern was magni®ed 21 onto a CCD detector (L1 = 1.10 m, L2 = 23 m).
Bruno Lengeler et al.
Parabolic refractive X-ray lenses
research papers
shown in Fig. 8. All numerical simulations are based on Fresnel
propagation, where the object is illuminated coherently. The partial
coherence of the undulator source is not taken into account. The
parabolic shape of the lenses ensures an image free of distortion by
spherical aberration. A numerical simulation of the in¯uence of
spherical aberration is shown in Fig. 9 for the same nickel mesh. The
spherical lenses show a strong cushion-like distortion and other
artifacts whereas the parabolic lenses generate a distortion-free
image in the simulation (Fig. 8) and in the experiment (Fig. 7).
Today, projection imaging is the most common method in hard
X-ray microscopy at storage rings. Its lateral resolution is limited by
the detector resolution, which is slightly below 1 mm for the best
detectors available today (Koch et al., 1998). This high resolution is
obtained at the expense of detector ef®ciency using thin monocrystalline scintillator ®lms. Since the hard X-ray microscope
produces magni®ed X-ray images, it can alleviate the requirements on
the detector resolution while producing images with a higher lateral
Figure 6
Expected ®eld of view for parabolic refractive beryllium and aluminium X-ray
lenses with E = 20 keV, f = 2.35 m, N = 50 for Be and N = 31 for Al. The test
pattern is a Ni grid with 50 mm and 10 mm grid width and thickness.
Numerically generated X-ray micrograph of the Ni mesh as imaged in a oneto-one geometry at 25 keV using an aluminium lens with N = 120.
Figure 9
Figure 7
Flat-®eld-corrected X-ray micrograph of a Ni mesh (2000mesh) recorded with
an aluminium lens at 25 keV (N = 120, f = 1.05 m, L1 = 1.10 m, L2 = 23 m).
Figure 8
Bruno Lengeler et al.
Parabolic refractive X-ray lenses
Numerically generated X-ray micrograph of the Ni mesh from Fig. 7 imaged
by a stack of spherical Al lenses (N = 120, E = 25 keV). Except for the lens
shape, all parameters have been kept the same as for Fig. 8.
J. Synchrotron Rad. (2002). 9, 119±124
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One major advantage of hard X-rays is their large penetration
depth which allows one to image the interior of a sample. Combining
hard X-ray microscopy with tomographic techniques allows the
reconstruction of the three-dimensional structure of a sample
(Schroer et al., 2001; Rau et al., 2001). For this purpose, a set of X-ray
micrographs is recorded as a function of sample rotation at equidistant steps in an angular interval from 0 to 180 . For standard
tomographic techniques to be applicable, the X-ray micrographs need
to be projection images of the sample that are free of distortion.
Owing to the large depth of ®eld, the X-ray micrographs are sharp
projections. The parabolic shape guarantees an image free of
distortion. For partially coherent illumination of the sample, the
contrast in the image plane depends on the position of the sample in
the ®eld of view. This contrast and how to cope with it for tomographic imaging is described in detail elsewhere (Schroer et al., 2002).
Fig. 10 shows a reconstructed slice through a ®bre reinforced concrete
sample. 250 X-ray micrographs with 10:6 magni®cation were
recorded at 20.65 keV using an Al refractive lens (N = 42, f = 1.89 m).
The object distance was L1 = 2.07 m, and the high-resolution CCD
detector (FReLoN2000; Weitkamp et al., 1999) was placed at an
image distance of L2 = 22.03 m. The slice in Fig. 10 was reconstructed
using ®ltered backprojection. In principle, the three-dimensional
resolution in the reconstructed image can be as high as the lateral
resolution in single projections. Here, however, the resolution was
limited by mechanical instabilities of the rotation and translation
stages. The motion of the rotation axis produces, for instance, the
distortions of the circular ®bres in Fig. 10. The requirements on the
rotation stage will become stronger as the lateral resolution in the
X-ray images can be improved, for example, by using a more transparent lens materials.
Instead of generating a magni®ed image of an object we might also
be interested in demagnifying the image, as, for example, for lithographical purposes. Up to now, there were no ways to demagnify a
mask with hard X-rays. Our lenses can easily demagnify a mask by a
factor of ten, thus alleviating substantially the requirements on
lithography masks. First tests have been very promising. Another
advantage is the large depth of ®eld in X-ray imaging by refractive
lenses. Indeed the longitudinal resolution dl ,
dl ˆ 0:64=…N:A:†2 ;
Figure 10
Magni®ed tomographic slice through a ®bre reinforced concrete sample. The
slice was reconstructed using 250 X-ray micrographs at a magni®cation of 10.6.
J. Synchrotron Rad. (2002). 9, 119±124
is of the order of millimetres (Lengeler et al., 1999a). This allows one
to project a sharp image of a mask into a thick resist and alleviates the
requirements on the longitudinal positioning of mask and resist. As
opposed to the proximity method, mechanical contact between the
delicate mask and the resist can be avoided. However, there are also
serious drawbacks. The ®eld of view will at best be 1 mm resulting in
the need of stitching in order to expose larger structures. Since
refractive lenses show chromatic aberration, quasi-monochromatic
radiation must be used for the lithography process. This requires a
high spectral brilliance of the X-ray source to avoid long exposure
times. Chemically ampli®ed resists, such as SU8, may help to reduce
exposure times but introduce other dif®culties.
We would like to thank A. Snigirev and the beamline staff at ID22
of the ESRF, in particular A. Simionovici, C. Rau and T. Weitkamp
for the collaboration and support during the testing of the refractive
lenses and the joint development of X-ray microscopy techniques
based on them. The ®bre reinforced concrete sample was provided by
B. Banholzer of the IBAC at the University of Technology in Aachen,
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