research papers
522 doi:10.1107/S0909049511009083 J. Synchrotron Rad. (2011). 18, 522–526
Journal of
Synchrotron
Radiation
ISSN 0909-0495
Received 21 May 2010
Accepted 9 March 2011
# 2011 International Union of Crystallography
Printed in Singapore – all rights reserved
X-ray collimation by crystals with precise parabolic
holes based on diffractive–refractive optics
Peter Oberta,
a,b
* Peter Mikulı
´
k,
c
Martin Kittler
a
and Jaromir Hrdy
´
a
a
Institute of Physics, Academy of Sciences of the Czech Republic, v.v.i., Na Slovance 2,
CZ-18221 Praha 8, Czech Republic,
b
Swiss Light Source, Paul Scherrer Institut, CH-5235
Villigen, Switzerland, and
c
Department of Condensed Matter Physics, Faculty of Science,
Masaryk University, Kotla
´
r
ˇ
ska
´
2, CZ-61137 Brno, Czech Republic.
E-mail: peter.oberta@psi.ch
Two crystals with precise parabolic holes were used to demonstrate sagittal
beam collimation by means of a diffractive–refractive double-crystal mono-
chromator. A new approach is introduced and beam collimation is demon-
strated. Two Si(333) crystals with an asymmetry angle of =15
were prepared
and arranged in a dispersive position (+, , , +). Based on theoretical
calculations, this double-crystal set-up should provide tunable beam collimation
within an energy range of 6.3–18.8 keV (
B
= 71–18.4
). An experiment study
was performed on BM05 at ESRF. Using 8.97 keV energy, the beam profile at
various distances was measured. The experimental results are in good
agreement with theoretical predictions. Owing to insufficient harmonic
suppression, the collimated (333) beam was overlapped by horizontally
diverging (444) and (555) beams.
Keywords: collimation; diffractive–refractive optics; X-ray optics; parabolic hole.
1. Introduction
Beam shaping based on diffractive–refractive optics has been
previously studied and demonstrated by our group at the
Institute of Physics of the Academy of Sciences of the Czech
Republic. Both Bragg (Hrdy
´
, 1998) and Laue (Hrdy
´
et al.,
2006) diffraction approaches have been demonstrated. Based
on theoretical calculations (Hrdy
´
& Oberta, 2008) and ray-
tracing simulations (Artemiev et al., 2004), various applica-
tions for diffractive–refractive optics were proposed and
tested at synchrotron facilities. Recently, the smallest focal
spot size by diffractive–refractive optics (Oberta et al., 2010)
was achieved and a novel method of higher harmonics
separation in space was proposed at the Swiss Light Source
(SLS; Hrdy
´
et al., 2011). In addition to beam focusing, another
application of diffractive–refractive optics is beam collima-
tion. In this paper a new approach is introduced and beam
collimation is demonstrated using a dispersive arrangement of
two crystals with precise parabolic holes.
There are a number of methods used to collimate X-rays.
Mirrors or asymmetrically cut crystals can be used (Mori &
Sasaki, 1995; Renninger, 1961), but these approaches change
the direction of beam propagation and additional optics are
needed to correct this deviation. Our method is based on two
asymmetrically cut crystals into which we machined holes with
a precise parabolic profile: two channel-cut crystals. The
dispersive arrangement has the advantage, compared with
other methods, that it conserves beam direction: the entrance
and exit beam positions are fixed. The same advantage holds
for refractive lenses (Snigirev et al., 1996); however, they can
only be used for hard X-ray radiation. Another advantage of
the diffractive–refractive method is lower flux loss compared
with other collimation methods. The Si(333) reflection loses
just about 10% of flux after diffraction.
2. Theoretical description
Imagine a crystal into which a hole is drilled. This hole is
further machined to reach an ideal parabolic shape. The
crystal is cut in such a way that the crystallographic planes of
the (111) orientation form an angle with the crystal surface
(asymmetric crystal), Fig. 1. If one takes a point source,
located at a distance S from the crystal, then the diffracted
radiation from the parabolic surface profile of the drilled hole
in the crystal will be sagittally focused to a point at a distance f
from the crystal. The formula describing the parabolic profile
can be expressed as
y ½mm¼a ½mm
1
x
2
½mm
2
; ð1Þ
where a is the parabola parameter. Then the relationship
between S, f and a can be expressed as (Hrdy
´
& Oberta, 2008;
Snigirev et al., 1996)
f ¼ S=ð2aNK
0
S 1Þ; ð2Þ
where
K
0
¼ Kð2 þ b þ 1=bÞ=4 cos ; ð3Þ
b ¼ sinð Þ= sinð þ Þ; ð4Þ
K ¼ð2r
e
F
0
=VÞ d
hkl
; ð5Þ
where for silicon
K ¼ 1:256 10
3
d
hkl
½nm ½nm: ð6Þ
Here r
e
is the classical electron radius, F
0
is the structure
factor, V is the unit-cell volume, is the wavelength, d
hkl
is the
atomic plane spacing, is the asymmetry angle ( > 0 for the
grazing-incidence case), is the Bragg angle, N is the number
of diffraction events and b is the asymmetry factor. A plot of
formula (2) for d
hkl
= (111) shows the dependence of the
focusing distance over energy or Bragg angle (Oberta et al.,
2010). By using only one crystal with a parabolic shaped hole,
Fig. 1, the number of diffracting events is two, N = 2. As shown
by Hrdy
´
& Siddons (1999), such a crystal arrangement cannot
achieve a sharp focal spot, because of the combined vertical
and horizontal spread of the beam. To cancel the vertical and
horizontal beam spreads, a Bartels crystal arrangement must
be used. For this case the number of diffracting events is four,
N = 4 (Fig. 2). A single crystal with two diffracting events, N =
2, is a non-dispersive system, but the two-crystal system used
in the experiment represents a dispersive system, with N =4
diffracting events (Fig. 2).
The focusing condition is sensitive to dispersion and thus
depends on the reflection. Fig. 3 shows the focusing distance
over a Bragg angle of a dispersive arrangement of two crystals
with a parabolic-shaped drilled hole for the Si(333) crystal
orientation. The crystal–source distance was S = 33613 mm as
used at BM05 at ESRF where the experiment was performed.
The parabola parameter of our crystals is a = 0.15 mm
1
.As
can be seen from Fig. 3, for certain
Bragg angles (energies) the focusing
distance is several hundred meters from
the crystal or becomes infinite (inclined
dashed area). Under these conditions
the crystal arrangement is a collimator.
The collimation is therefore indepen-
dent of the divergence of the impinging
beam. This is the working principle of
the diffractive–refractive X-ray colli-
mator.
Fig. 3 plots the calculated focal
distance for three different harmonics. The black full line
represents the Si(333) focusing versus Bragg angle depen-
dence. We can see higher harmonics, plotted as red dash-
dotted [Si(444)] and blue dashed [Si(555)] lines. A negative
focusing distance means a divergence of the beam with a focal
point placed before the crystals. A positive focusing distance
represents focusing with the focal point placed after the
crystals. If a focusing distance of an optical set-up, like the
proposed crystal arrangement, is from several hundred meters
up to kilometers; such a set-up can be described as collimating.
For the case of the diffractive–refractive X-ray collimator this
value was set to 500 m. The focusing distance is very
sensitive to the crystal–source distance S (Fig. 4). By changing
S and moving the crystals over a range of 10 m, we can achieve
a collimation effect over an angular range of
B
= 71–18.4
,
that corresponds to an energy range of 6.3–18.8 keV. Because
the proposed crystal collimator is based on the diffractive–
refractive effect, which is much stronger for longer wave-
lengths, the working range of the collimator lies in the soft X-
ray region. In Fig. 4 we find two collimating angular ranges
(inclined dashed area). The range for larger
B
(smaller
energies) is broader than that for smaller
B
(larger energies).
Both ranges are approaching each other with increasing
distance S.
research papers
J. Synchrotron Rad. (2011). 18, 522–526 Peter Oberta et al.
X-ray collimation 523
Figure 2
The two-crystal dispersive system with four diffraction events. The four
diffraction events are in reality separated in space (1–4). The theory
simplifies them into one point in space.
Figure 3
Focusing distance as a function of Bragg angle for diffractions Si(333)
(black full line), Si(444) (red dash-dotted line) and Si(555) (blue dashed
line). The gray inclined dashed areas represent the working range of the
collimator for a crystal–source distance of 33.6 m.
Figure 1
A silicon Si(111) asymmetrically cut crystal with a parabolic-shaped drilled hole and double-bounce
beam path inside the crystal.
Table 1 includes the tabulated angular and energy ranges
for eight different crystal–source distances, along with the
focal distances of some harmonics. Table 2 also includes the
tabulated broadening of the beam compared with an ideal
collimation as a percentage of the impinging beam dimensions.
The additional broadening of the
Si(333) diffraction for a focal distance of
500 m is within 0.18% of ideal, which
justifies our approximation approach. In
the case of the higher harmonics the
share to the beam broadening is higher,
but still in the range between 5.8%
and 14%.
As one can see from Fig. 4 and Table
2, harmonic contamination will project
itself as an additional beam broadening
and will decrease the multiwavelength
collimation. One can overcome this
problem by using a special crystal
geometry to separate the higher
harmonics in space without deforming
the reflectivity curve, as carried out
by traditional harmonics separation
methods (Hrdy
´
et al., 2011), or by
introducing a slit system between the
two crystals.
3. Experimental results
The experiment was performed on a
vertical diffractometer at the BM05
beamline at ESRF. We set two crystals with
precise machined parabolic holes into a
dispersive position and set the beamline
monochromator to 8.97 keV.
First we set only one crystal into the
diffraction position (
B Si(333)
= 41.4
) and
detected the diffraction. Then we removed
the crystal and set the second crystal. After
adjusting the second crystal we set both
crystals in-line. At different distances we
recorded diffraction images on X-ray films
and on a digital camera.
3.1. Single-crystal arrangement
In a set-up with only one crystal diffracting we detected
higher harmonics. As can be seen in Fig. 5, there are two
parabolic-shaped diffraction spots on the X-ray film. The full
parabolic diffraction spot corresponds to the Si(333) diffrac-
tion, which copies the shape of the parabolic groove. Based on
the distance of 7 m of the X-ray film and the angular separa-
tion of 0.7
of the two diffractions one can calculate the
corresponding energy of the other diffraction. Based on this
approach we found out that the partial parabolic diffraction
corresponds to the Si(444) diffraction energy.
After diffraction from one crystal the higher harmonics are
separated in space (Hrdy
´
et al., 2011), and after two crystals
the higher harmonics are assembled again in-line and overlap
each other. Therefore, higher harmonics contamination could
be solved by introducing a slit system between the two crystals.
The schematic in Fig. 5 shows the layout of the experiment.
research papers
524 Peter Oberta et al.
X-ray collimation J. Synchrotron Rad. (2011). 18, 522–526
Figure 4
Focusing distance versus Bragg angle for four different crystal–source distances. With increasing
crystal–source distance the two working ranges are approaching each other.
Table 1
Angular and energy ranges for eight different crystal–source distances, S, along with the focal
distances of some harmonics.
S (m) Collimation range (
) Collimation range (keV) f
333
(m) f
444
(m) f
555
(m)
32.8 18–19/53.5–71 19.2–18.23/7.38–6.27 500 69.29 49.52
34.8 18.4–19.4/47.5–62 18.81–17.86/8.05–6.72 500 77.26 53.73
36.8 19–20/43–56 18.23–17.36/8.7–7.15 500 87.86 58.65
38.8 19–21/38–51 18.23–16.57/9.64–7.63 500 100 63
40.8 20–22/33.5–46.5 17.36–15.85/10.75–8.17 500 97 65
42.8 20–43 17.36–8.7 500 93 70
44.8 21–39.5 16.57–9.33 500 111 72
46.7 22–37 15.85–9.86 500 118 76
Table 2
Broadening of the beam compared with an ideal collimation as a
percentage of the impinging beam dimensions.
Broadening (%)
S (m) Si(333) Si(444) Si(555)
32.8 0.18 9.8 14.0
34.8 0.18 9.0 13.0
36.8 0.18 7.8 11.8
38.8 0.18 7.0 11.0
40.8 0.18 7.2 10.6
42.8 0.18 7.4 10.0
44.8 0.18 6.2 9.6
46.7 0.18 5.8 9.2
Here A is the crystal–source distance (33600 mm), B is the
crystal–X-ray film distance (7000 mm), X is the Si(333)
diffraction spot size [width of the Si(333) parabolic diffraction,
9.05 mm] and Y is the width of the primary beam, 10.9 mm.
These dimensions are related as
X=A ¼ Y=ðA þ BÞ: ð7Þ
From this simple relationship we can calculate the width of the
Si(333) collimated diffraction spot if the collimation is ideal.
For the ideal collimation the Si(333) diffraction spot is
9.02 mm. The error between experiment (9.05 mm) and
calculation (9.02 mm) is just 0.3%.
3.2. Double-crystal arrangement
By assembling the two crystals in-line, higher harmonics,
which were spatially separated by the first crystal, are
assembled again together by the second crystal. This way the
spot after the two crystals is contaminated by higher harmo-
nics, which are not collimated and degrade the apparent
collimation effect. Also the parabolic-
shaped diffraction after one crystal is
canceled by the second crystal. As a
result we restore a line beam (Figs. 6
and 7). By rocking one of the crystals
(Fig. 6), higher harmonics are visible.
The diffraction in Fig. 6 has a FWHM of
10.4 arcsec, which corresponds to the
Si(444) diffraction at 8.97 keV with a
theoretical FWHM of 10.6 arcsec. The
second diffraction spot, visible in frames
3–8, is the diffracted Si(333). Fig. 7
shows an X-ray film image of diffraction
after two crystals at a distance of 7 m
after the crystals. The false colors
represent the different energies. The red
color denotes the diffraction spot of the Si(333) diffraction,
green the Si(444) diffraction, and light blue for the Si(555)
diffraction. The dimensions of the diffraction footprint in Fig. 7
correspond to the theoretical spread of the higher harmonics
calculated from Fig. 3. The theoretical broadening of the beam
after a slit size of 5 mm should be 5.16, 5. 48 and 5.67 mm for
the Si(333), Si(444) and Si(555) diffraction, respectively. The
beam dimensions obtained by experiment were 5.12, 5.44 and
5.70 mm for the Si(333), Si(444) and Si(555) diffraction,
respectively. The theoretical description (Hrdy
´
& Oberta,
2008; Snigirev et al., 1996) supposes that all the diffraction
events occur in one point in space; this way a clear crystal–
source distance is set. In reality there are four different points
of diffraction in the two crystals; the diffracted beam bounces
twice from each crystal. Contrary to theory, we do not have
one spatial point for all four diffracting events and one
crystal–source distance, but four points and four different
crystal–source distances. This situation can be simplified to
one of the three different points in Fig. 2. We can either choose
the first diffraction event at the first crystal (A) as the point
where all diffraction events occur, or we choose the very exit
point of the fourth and the final diffraction on the second
crystal (C). We can also choose a point between the two
research papers
J. Synchrotron Rad. (2011). 18, 522–526 Peter Oberta et al.
X-ray collimation 525
Figure 6
Diffraction after two crystals at a distance of 7 m. We rocked the first
crystal and registered a strong moving (333) diffraction and a weaker
disappearing (444) diffraction.
Figure 7
X-ray film at a distance of 7 m from the crystal with a (333) diffraction
and the (444) and (555) diffractions in background. The original film spot
(top) and false colors description (bottom).
Figure 5
Diffractions after one crystal at a distance of 7 m. The horizontal line spot is the primary beam
propagating through the crystal, the full parabolic spot is the Si(333) diffraction, and the partial
parabolic double-spot is the Si(444) diffraction. Owing to the diffractive–refractive effect the
harmonics are separated in space.
crystals (B). At the end we can calculate using three different
points, which gives us three different crystal–source distances
(Fig. 2). This situation leads to aberrations and thus dis-
agreements between calculation and experiment (Hrdy
´
et al.,
2005).
The jagged shape of the left-hand side of the diffraction
spot in Fig. 6 is due to crystal subsurface imperfections. The
same imperfection can be seen in Fig. 7.
The horizontal divergence of the bending magnet at BM05
is 2.4 mrad. We observed a 5.16 mm Si(333) diffraction spot at
a distance of 7 m after the crystal from a 5 mm impinging
beam. This means a divergence of the collimator of only
4.7 arcsec in the horizontal plane, which is much smaller
compared with other crystal collimation methods (Ferrari &
Korytar, 2001). The horizontal divergence is not coupled to
the vertical divergence. The vertical divergence is set by the
dispersive setting of the crystals, the cross section of the
DuMond diagram, and is independent of the divergence of the
impinging beam.
4. Conclusions
X-ray collimation as a novel application of diffractive–
refractive optics was studied theoretically and experimentally.
A set-up with two crystals in a dispersive arrangement
successfully demonstrated X-ray collimation in the energy
range 6.3–18.8 keV. The experimental results are close to the
theoretical calculations of the spectral dependence of the
focusing distance. To reach a better agreement between the
theory of beam shaping through diffractive–refractive optics
and our experiment, further theoretical and experimental
research is necessary.
A minor drawback of the device is the higher harmonics
contamination, which overlaps on the primary beam. We can
overcome this problem by using a special crystal set-up (Hrdy
´
et al., 2011) or by placing a slit between the two crystals. The
widths of the diffraction spots in Fig. 5 are experimental proof
of the collimating optics. An interesting feature of this colli-
mation method is its energy range. Being able to collimate
lower energies (below 20 keV) can supplement compact
refractive lenses. The degree of collimation is adjustable with a
shifting of the value of the focusing distance at which we
approximate collimation.
This work was supported by the following funds: Institu-
tional Research Plan AS CR (No. AVOZ 10100522), MPO CR
(FR-TI1/412) and MSMT CR (INGO LA 10010 and MSM
0021622410). We would like to thank Mr Luka
´
s
ˇ
and the
company Polovodic
ˇ
e a.s. which provided the crystals.
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