Development of new apertures for coherent X-ray experiments.

Abstract

When one performs a coherent small-angle X-ray scattering experiment, the incident beam must be spatially filtered by slits on a length scale smaller than the transverse coherence length of the source which is typically around 10 microm. The Fraunhofer diffraction pattern of the slit is one of the important sources of background in these experiments. New slits which minimize this parasitic background have been designed and tested. The slit configuration apodizes the beam by the use of partially transmitting inclined slit jaws. A model is presented which predicts that the high wavevector tails of the diffraction pattern fall as the inverse fourth power of the wavevector instead of the inverse second power that is observed for standard slits. Using cleaved GaAs single-crystal edges, Fraunhofer diffraction patterns from 3 and 5.5 keV X-rays were measured, in agreement with the theoretical model proposed. A novel phase-peak diffraction pattern associated with phase variations of the transmitted electric field was also observed. The model proposed adequately accounts for this phenomenon.

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Journal of
Synchrotron
Radiation
ISSN 0909-0495
Editors: G. Ice, A˚. Kvick and T. Ohta
Development of new apertures for coherent X-ray
experiments
Eric M. Dufresne, Steven B. Dierker, Z. Yin and Lonny Berman
J. Synchrotron Rad. (2009). 16, 358–367
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J. Synchrotron Rad. (2009). 16, 358–367Eric M. Dufresne et al. · New apertures for coherent X-ray experiments

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research papers
358
doi:10.1107/S0909049509003720
J. Synchrotron Rad. (2009). 16, 358–367
Journal of
Synchrotron
Radiation
ISSN 0909-0495
Received 22 October 2008
Accepted 30 January 2009
# 2009 International Union of Crystallography
Printed in Singapore – all rights reserved
Development of new apertures for coherent X-ray
experiments
Eric M. Dufresne,a*‡2Steven B. Dierker,b‡ Z. Yinband Lonny Bermanb
aThe Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA, andbThe
National Synchrotron Light Source, Brookhaven National Laboratory, Upton, NY 11973, USA.
E-mail: dufresne@anl.gov
When one performs a coherent small-angle X-ray scattering experiment, the
incident beam must be spatially filtered by slits on a length scale smaller than the
transverse coherence length of the source which is typically around 10 mm. The
Fraunhofer diffraction pattern of the slit is one of the important sources of
background in these experiments. New slits which minimize this parasitic
background have been designed and tested. The slit configuration apodizes the
beam by the use of partially transmitting inclined slit jaws. A model is presented
which predicts that the high wavevector tails of the diffraction pattern fall as the
inverse fourth power of the wavevector instead of the inverse second power that
is observed for standard slits. Using cleaved GaAs single-crystal edges,
Fraunhofer diffraction patterns from 3 and 5.5 keV X-rays were measured, in
agreement with the theoretical model proposed. A novel phase-peak diffraction
pattern associated with phase variations of the transmitted electric field was also
observed. The model proposed adequately accounts for this phenomenon.
Keywords: XPCS; SAXS; Fraunhofer diffraction; coherence.
1. Introduction
In a small-angle X-ray scattering (SAXS) experiment, one
may be sensitive to parasitic scattering from the imperfections
of beamline optical components. The apertures used to limit
the illuminated sample volume and collimate the beam can
become one of the important sources of parasitic background
in the far-field diffraction plane. The imperfections of the
edges cause random scattering and the finite opening of the
aperture diffracts the X-rays. This is particularly important
when performing a SAXS experiment with X-ray photon
correlation spectroscopy (XPCS) (Dierker et al., 1995), espe-
cially when the scattering from the sample is weak and at low
wavevectors (Dufresne et al., 2002) (for a recent review of
XPCS, see Livet, 2007). For example, when observing critical
fluctuations in a binary fluid mixture, it was important to
reduce the background from parasitic scattering near q = 0
(Dufresne et al., 2002). The transverse coherence of a light
source is characterized by a transverse coherence length, lt=
?R/d, where ? is the X-ray wavelength, R is the source–
experiment distance and d is the source size (Goodman, 1985).
For a wavelength ? = 4.0 A˚, a distance R = 20 m and a hori-
zontal source size d = 600 mm, the transverse coherence length
is lt= 13 mm. By limiting the illuminated sample area to linear
dimensions smaller than lt, the sample can be illuminated with
a beam of coherent X-rays. This has been done in the past with
small pinholes, made by laser drilling of a thin Pt foil. The far-
field diffracted intensity of a circular pinhole with a radius r is
I(q) / [J1(qr)/(qr)]2(Goodman, 1985). For large wavevectors,
since J12(qr) / 1/q, I(q) decreases as q?3. In comparison, a
one-dimensional slit has q?2tails, since the intensity follows
I(q) / [sin(q?/2)/(q?/2)]2, where ? is the slit opening.
Circular pinholes were first used in XPCS (Dierker et al., 1995;
Brauer et al., 1995). The advantage of using pinholes is that
they are relatively convenient to align, and their tails fall more
sharply than a one-dimensional slit. On the other hand, they
are produced by laser ablation of a thin metallic foil, which
cannot control the shape of the hole accurately, and their
diameter is fixed.Fig. 1 shows a typical diffraction pattern for a
typical 10 mm pinhole. The beam was filtered with a pair of
WB4C multilayers with a 27 A˚period and a 1.5% bandwidth
(Berman et al., 1997). Their diffraction pattern is typically non-
symmetric, showing large flares and long tails, thus the decay
of their tails is not easy to control. Recently, a casting method
has also been used to make these small pinholes but their
diffraction pattern still remained irregular (van der Veen et al.,
1997). It has been shown recently that pinholes made for
electron microscopy work well, so it is possible to overcome
fabrication difficulties (Livet, 2007).
Rectangular apertures made with either a roller-blade
design (Libbert et al., 1997) or with highly polished tungsten
edges (Vlieg et al., 1997) have been shown to produce well
controlled diffraction patterns. They are now the method of
choice for coherent illumination of a sample. One of the
‡ Formerly at the Department of Physics, University of Michigan, Ann Arbor,
MI 48109, USA.
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advantage of using slits is that their opening is variable, so they
can be set easily to match the smallest transverse coherence
length in the experiment, or easily closed to increase the
limiting wavevector ql, where the speckle contrast is reduced
owing to the loss of longitudinal coherence (see x3). An
important gain in parasitic background reduction can be made
by using a square aperture if one chooses a scattering direction
which is along the diagonal of the square, pointing 45?from
any of the edges. The tails then fall off as q?4, i.e. faster than
for a circular aperture. If smooth polished edges are made,
then an important reduction in parasitic background from the
aperture diffraction pattern can be achieved.
For coherent small-angle X-ray scattering, one typically
reduces the parasitic scattering from the slit blades by setting
up a guard aperture downstream of the coherence slit (Livet,
2007). One would typically set the guard aperture within the
near-field of the coherence slit, with an opening largerthan the
coherence slit. Another recent approach reduces the parasitic
background by adding a channel-cut monochromator between
the guard slit and the sample (Xiao et al., 2006). This novel
approach reduces the background more than with only a
guard slit.
In this paper we will describe a new technique which can
reduce the parasitic diffraction more than a 45?geometry. The
idea is based on the technique of apodization in optics (Born
& Wolf, 1970). By making a soft transmitting edge where the
intensity varies gradually over some distance, one can reduce
the amplitudes of the Fourier components at high wavevec-
tors. Such an edge is shown in Fig. 2. This figure shows how, by
using an inclined slit jaw, one can produce an intensity profile
in a plane behind the aperture where the intensity decreases
more slowly along the x-direction than for an opaque slit jaw.
We chose to investigate an exponential profile in the jaw
because it was simpler to make such a jaw experimentally, but
clearly other geometries could be further investigated
(Libbert et al., 1997). Although refraction effects from the
edges of slits have been discussed recently, they were
measured using a laboratory source with incoherent X-rays
(Nikulin & Davis, 1998). This paper gives a detailed theory
of the effect where diffraction and refraction effects from
coherent X-rays are included. We will show in x2 that the tails
of the diffracted intensity falls as q?4along the x-axis, a
substantial improvement over a circular aperture. The
experimental method and the results sections will describe the
first observation of a peculiar phase grating diffraction
pattern. Finally, we will discuss the implications of these
observations for coherent and incoherent small-angle scat-
tering experiments.
2. Theory: X-ray phase grating aperture
Fig. 2 shows a one-dimensional slit. To prevent the jaws from
colliding, we purposefully offset the slit jaw planes of motion.
research papers
J. Synchrotron Rad. (2009). 16, 358–367Eric M. Dufresne et al.
? New apertures for coherent X-ray experiments
359
Figure 2
(a) The slits geometry and its coordinate system. The beam propagates
along y. (b) Theoretical intensity profile I(x,y0) in the near-field
diffraction region for several effective absorption lengths.
Figure 1
Diffraction pattern of a circular 10 mm-diameter pinhole generated with
3.1 keV X-rays and measured with a CCD having 9 mm pixels, placed
approximately 1 m from the pinhole. The logarithmic grey scale uses
black as the high intensity. The beam stop (a cross) is used to extend the
dynamic range in the tails of the diffraction pattern.
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Two perpendicular slit assemblies can be placed one after the
other to define a two-dimensional aperture. We assume that
linearly polarized X-rays are incident on the slits. This
condition is a good approximation for synchrotron radiation,
especially for radiation emitted by an undulator. For experi-
mental considerations, the wedge angle ? = 90?because our
GaAs blades cleaved well for this cleavage plane. As shown in
Fig. 2, the path length in the slit material along the beam
direction is dy = ? dx, where ? = tan? + 1/tan? = 2/sin(2?). The
function ?(?) has a minimum at ? = ?/4, where ? = 2, and
diverges when ? ! ?/2 or ? ! 0, making the slit jaws opaque.
Several values of ? are given in Table 1. To produce a more
slowly decreasing intensity profile, one could choose a wedge
with an angle ? < 90?. In that case, one would replace the
inclination coefficient by ? = tan?1(?) ? tan?1(? + ?). To
simplify the theoretical treatment, we assume that the incli-
nation angles of the two edges are identical. The slit blades are
positioned at x = ?a, thus the opening of the slit is ? = 2a.
Assuming that a plane wave is incident with wavevector k =
k0y, where k0= 2?/?, the field just behind the slit in the near
field diffraction region at y = y0<< ?2/? is
(
E0exp½iðn ? 1Þk0?dx?
Here, E0 is the incident electric field amplitude and we
neglected any constant phase factor in deriving E(x). The
horizontal distance with respect to a given edge is dx = x ? a
for x > a, and dx = ?x ? a for x < ?a. For X-rays, the index of
refraction n = 1 ? ? + i?, where ? is the index of refraction
decrement, and ? is the absorption coefficient. Some values of
? and ? for GaAs are shown in Table 2. As opposed to visible
light, X-ray photons have an index of refraction slightly lower
than unity, and the absorption correction is typically smaller
than the refraction correction. A surprising aspect of these
slits is that they still define an aperture, even when the two
jaws overlap. The electric field when the slit is closed is similar
to the previous equation, apart from a multiplicative factor
exp[ik0(n ? 1)?2|a|], where a here is negative. This factor
comes from the fact that, when the slit is closed, the path
length difference through the overlapping edges is a constant
equal to 2?|a| for |x| < |a|.
By inspecting the previous equation for |x| > a, one notices
that the phase of the electric field varies with a wavevector
kp= k0?? along the x-axis. From this periodic variation, one
EðxÞ ¼
E0
for jxj<a;
for jxj>a:
ð1Þ
can expect a peak in the diffraction pattern at kp. For later
discussions, let us define the effective slit opening ?eff= ? +
1/(g?), where (2g?)?1is the distance from x = ?a where
|E(x)|2falls by a factor 1/e and g = ?k0. Some typical values of
?effwith ? = 0 are shown in Table 2 for GaAs. The effective
slit opening increases with energy. To observe good fringe
contrast in a Fraunhofer diffraction pattern, the effective
opening must be smaller than the smallest of the transverse
coherence lengths. GaAs is best as a jaw material near 3 keV
because the contribution of the jaws 1/(g?) is smaller than the
typical physical slit opening ? ’ 10 mm, thus the slit can be set
to match the transverse coherence condition. In principle,
GaAs could be used at higher energies by increasing g, setting
the inclination angle ? near 90?. In practice, it may be easier to
select a more absorbing material than to precisely align ?.
Recently, we have used GaAs near 11.0 keV with ? ’ 0?just
below the Ga and As absorption edges. The large absorption
just below the edge is sufficient to make the edges opaque
enough for a standard slit (Dufresne et al., 2002).
The electric field in the far-field diffraction plane is the
Fourier transform of E(x). Defining the Fourier transform of
E(x) as E(k) =Rexp(ikx)E(x)dx, one can show that
EðkÞ ¼ E0
k2þ g2ð?2? ?2Þ þ i2g2??
? g? þ i?
k
2gð? þ i?Þ
sinðkaÞ þ cosðkaÞ
??
:
ð2Þ
For infinite absorption when ? ! 1, the previous equation
yields the same results as a simple one-dimensional slit with
E(k) / sin(ka)/(ka). The field is also proportional to sin(ka)/
(ka) when ? ! ?/2 or ? ! 0 since g ! 1. It is interesting to
note that E(k) is quite similar to the form factor of a blazed
transmission grating (Michette & Buckley, 1993). Recalling
that the intensity is just I(k) = |E(k)|2, one finds that
IðkÞ=I0¼
4g2ð?2þ ?2Þ
½k2þ g2ð?2? ?2Þ?2þ 4g4?2?2
? g2ð?2þ ?2Þsin2ðkaÞ
ð3Þ
?
k2
þ cos2ðkaÞ þ g?sinð2kaÞ
k
?
;
where I0= E02. A similar equation can be derived easily when
the slit is closed. It differs from the previous equation by only
a numerical factor exp(?4g?|a|).
Setting a = 0 in equation (3), I(k) is maximized when
k2+ g2(?2? ?2) = 0, thus the phase peak wavevector is kp=
research papers
360
Eric M. Dufresne et al.
? New apertures for coherent X-ray experiments
electronic reprint
J. Synchrotron Rad. (2009). 16, 358–367
Table 1
Phase peak position versus ? for 3 and 8 keV X-rays in GaAs.
The phase peak position increases with increasing ?. At 3 and 8 keV the
magnitude of kpis in the range where one would typically observe small-angle
scattering. For constant ?, kp(3 keV) > kp(8 keV), as expected from the
wavelength dependence of kp.
? (?)
?
kpat 3 keV (A˚?1)kpat 8 keV (A˚?1)
45
60
80
85
89
2.0
2.3
5.8
11.5
57.3
3.3 ? 10?4
3.8 ? 10?4
9.7 ? 10?4
1.9 ? 10?3
9.5 ? 10?3
1.2 ? 10?4
1.4 ? 10?4
3.3 ? 10?4
6.8 ? 10?4
3.4 ? 10?3
Table 2
Typical values of ? and ? at several energies for GaAs (from Henke et al.,
1993).
The effective slit opening ?eff= ? + 1/(g?) is also shown at several energies
for ? = 60?and for a slit opening ? = 0 mm. GaAs slit jaws work well at 3 keV.
Energy (keV)
??
?eff
3
4
6
8
1.1 ? 10?4
6.1 ? 10?5
2.7 ? 10?5
1.5 ? 10?5
9.1 ? 10?6
1.7 ? 10?5
5.9 ? 10?6
1.3 ? 10?6
4.5 ? 10?7
1.9 ? 10?7
1.7
3.6
11
24
45
10

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?g(?2? ?2)1/2. Since typically ? >> ? for hard X-rays, kp’
?g?. Note that a phase peak would be absent if ? < ?. This
situation typically occurs in materials in the ultraviolet and the
soft X-ray energies (Henke et al., 1993). The phase peak is a
simple refraction effect whose position can be estimated easily
from the symmetric prism formula with the deflection angle
between the main beam and refracted beam ’ = 2?tan(?/2),
with ? = ?/2 ? ?/2 (Born & Wolf, 1970). In our case, the prism
angle ? = ?/2, and ? = ?/4, thus ’ = 2?. For ? = ?/4, the
inclination parameter ? = 2, kp= 2?k0, thus the deflection
angle is also 2?.
The half width at half-maximum of this peak occurs when
k2+ g2(?2? ?2) = ?2g2??. The full width at half-maximum
(FWHM) of the phase peak is ?k = g(?2? ?2+ 2??)1/2?
g(?2? ?2? 2??)1/2’ 2g?, when ? >> ?. By inspection, the
previous equation falls as I(k) ! k?4, for |k ? kp| >> 2g?.
Since ? and ? are typically around 10?5(see Table 2), and
g ranges between 1 and 10 A˚?1, I(k) will fall as k?4in the
range of interest for a small-angle scattering experiment
(>10?4A˚?1). For finite a and small k ! 0, the sin2(ka)/k2term
dominates in equation (3).
In Figs. 3 and 4 we show the behavior of equation (3) for
GaAs edges illuminated with 3 keV X-rays. Fig. 3 shows I(q)
for several slit openings ? = 2a. In Figs. 3–5, we normalized
I(k) so thatRI(k)dk = I0. Perhaps the most surprising aspect of
even when the slits are closed. This peak remains fixed in
reciprocal space as the opening is increased. This is the phase
peak, centered around kp= 3.8 ? 10?4A˚?1, in agreement with
the computed value in Table 1. For wavevectors much larger
than kp, I(k) ’ k?4as discussed earlier. As expected, the
periodicity of the fringes in reciprocal space also increases as
the slit opening is increased.
Fig. 4 shows the diffracted intensity for a fixed opening ? =
5 mm illuminated with X-rays, for various inclination angles ?.
As ? is increased, the phase peak moves to higher wavevectors
this figure is the fact that a peak is observed, away from q = 0,
and more diffracted intensity is found at higher wavevectors
near 10?3A˚?1. Since kp= k0??, this peak moves to higher
wavevector when ? increases since d?/d? > 0 for ?/4 < ? < ?/2.
This dependence is shown in Table 1 for several values of ?.
The positions of the phase peak in Fig. 4 agree with the values
calculated in Table 1 for 3 keV X-rays. Note that, for ? = 89?,
the diffracted pattern for small wavevector (q < 10?2A˚?1)
becomes closer to sin2(ka)/k2.
Fig. 5 shows the calculated intensity for several energies at
fixed opening ? and fixed ?. The position of the phase peak
moves to lower wavevectors as the energy is increased. This is
expected for X-rays given that ? / ?2(Michette & Buckley,
1993), thus kp= (2?/?)?? / ?. Some values of kpare shown in
Table 1 for GaAs at 3 and 8 keV. In Fig. 5, the intensity of the
phase peak becomes higher than the intensity of the direct
beam I(k = 0) for 6 and 9 keV X-rays. This is due to the fact
that at higher energies the contribution of the electric field
research papers
J. Synchrotron Rad. (2009). 16, 358–367Eric M. Dufresne et al.
? New apertures for coherent X-ray experiments
361
Figure 3
Calculated diffracted intensity I(q) for several openings ? = 2a at 3 keV
using GaAs edges with ? = 60?.
Figure 4
Diffracted intensity I(q) for a fixed opening ? = 5 mm for several values
of ?. Increasing the inclination angle moves the phase peak at larger
wavevector.
Figure 5
Diffracted intensity I(q) for three different energies with a 5 mm aperture
and ? = 60?. The phase peak wavevector is inversely proportional to
energy.
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