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Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research, A
journal homepage: www.elsevier.com/locate/nima
Computer-aided assembly of X-ray parabolic cylinder laterally graded
multilayer reflector to collimate divergent beam in XRD
Yiyun Yao a , Zhong Zhang a , Shengzhen Yi a , Zhe Zhang a , Chun Xie b ,∗
a
Key Laboratory of Advanced Micro Structural Materials MOE, Institute of Precision Optical Engineering, School of Physics Science and Engineering, Tongji
University, 200092, Shanghai, China
b
Sino-German College of Applied Sciences, Tongji University, 200092, Shanghai, China
ARTICLE
INFO
Keywords:
Computer-aided assembly
Beam collimation
Laterally graded multilayer
Parabolic cylinder reflector
ABSTRACT
In X-ray diffraction, it is essential to irradiate the sample with a parallel beam to avoid severe diffraction pattern
aberration. A parabolic cylinder laterally graded multilayer reflector is a good choice for this purpose. However,
its assembly is critical. Usually, reflector performance is simulated during design, and the effects of assembly
errors are qualitatively computed before assembly. However, for micro-scale assembly processes, the qualitative
error results in a time-assuming procedure and an unsatisfactory data. A Computer-aided assembly process,
details of a quantitative guide of adjustment, is hereby presented for a precision adjustment. Firstly, a highefficiency setup to collimate the divergent beam and evaluate the collimation beam is proposed. Secondly, the
performance of the reflector used in the experiment is simulated according to the parameters of each unit in
the setup. Thirdly, during the adjustment process, each essential adjustment step is analyzed, and the resetting
value of each error is quantitatively computed and used to guide the adjustment. Lastly, the assembled reflector
is highly effective with a divergent angle of less than 0.004 deg.
1. Introduction
X-ray diffraction is a widely-used technique for high resolution
structural investigation, which is utilized in numerous areas including
physics, chemistry, biology, and other subjects [1–4]. In the traditional
Bragg–Brentano geometry X-ray diffractometer, the incident beam on
the sample is divergent. As a result, a sample with slight misalignment
will cause severe line broadening, line shift, and intensity loss in the
diffraction pattern [5]. The application of a parabolic cylinder laterally
graded multilayer reflector, to collimate the divergent X-ray beam
significantly reduces the effect of sample misalignment on the measured
result [6–8].
Although the use of this component can improve the quality of Xray diffraction results, perfect performance of the reflector is primarily
based on the quality of the assembly. The traditional assembly process is
a qualitative procedure that is highly reliant on experience, and is therefore user-dependent and time-consuming. From the 1970s onwards,
computer-aided assembly processes have been applied to the adjustment
of optics in the optical path of systems. The approach is a combination
of optical measurement and computer simulation technology and has
achieved notable advancements in the field of precision optical system
assembly [9–12]. Generally, the main steps involved in the computeraided assembly process are as follows: (1) simulation of the optical
performance and evaluation of the influence of assembly errors prior to
assembly, (2) installation of the optics system using coarse adjustment,
(3) measurement of the image quality and a direct comparison of the
measurement result with the theoretical result, (4) utilization of the
simulation program to analyze the adjustment direction and to compute
specific values, and (5) repetition of steps 3 and 4 until the performance
of the optical system is close to the theoretical design target.
In the area of equipment assembly in X-ray diffraction and particularly for X-ray parabolic cylinder laterally graded multilayer reflectors, computer-aided design is always used to simulate the reflector’s
performance in the design process and to evaluate assembly errors. Rio
simulated the influence of the reflector–source distance deviation on
the degree of collimation of the beam, but did not calculate the reflectivity of the reflector [13]. Toraya simulated the effect of the incident
angle deviation and reflector–source distance deviation on intensity,
but not describe the details of the approach during the adjustment
process [14]. Gobel considered that the incident angle error affected the
parallelism of the collimated beam and limited this parameter according
to the Full Width at Half Maximum (FWHM) of the multilayer [15].
Michaelsen established a simple formula to analyze the relationship
between the divergent angle of the collimated beam and the incident
angle error [16]. In a previous study, we simulated the influence of
the assembly errors, and limited the maximum value of the incident
angle deviation and the offset of the source–reflector distance [17].
These published reports examined the effects of assembly errors on the
∗ Corresponding author.
E-mail address: xc0522@tongji.edu.cn (C. Xie).
https://doi.org/10.1016/j.nima.2018.07.052
Received 15 March 2018; Received in revised form 4 June 2018; Accepted 18 July 2018
Available online xxxx
0168-9002/© 2018 Elsevier B.V. All rights reserved.
Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 1. Setup of the experiment, with a Cu k X-ray tube as the source, a slit to control the flux, the reflector to collimate the divergent beam, a CCD camera to generate a digital intensity
image of the reflected beam and position stages to adjust the relative position of each unit.
Fig. 2. (a) Sketch of the pinhole imaging (b) Pattern and profile of the pinhole imaging accepted by CCD camera.
collimating beam before assembly, thereby providing qualitative results
or identifying experimental limitations that can guide the assembly
process. However, computer simulations with resetting values during
assembly were not performed. As such, the assembly is time-assuming,
and precision adjustment is difficult to achieve.
In this paper, we present a computer-aided assembly process for
the assembly of an X-ray parabolic cylinder laterally graded multilayer
reflector for divergent beam collimation, to quantitatively guide the
adjustment procedure. A high efficiency setup is proposed to facilitate
both collimation and the evaluation of the collimated beam. Then, the
performance of the reflector used in the experiment is simulated according to the size of the source, the surface topography of the parabolic
cylinder reflector substrate, and the period thickness distribution of the
laterally graded multilayer coating on the reflector. Each unit is then
adjusted by analyzing each essential adjustment step and quantitatively
computing the adjustment value to guide the adjustment. Finally, an
effective reflector is assembled with a divergent angle of less than 0.004
deg, which is significantly smaller than the 0.02 deg that is typically
required in X-ray crystallography.
incident beam into a digital image which is transferred to a computer
for processing. Each of the main components is equipped with a position
stage with a rotation resolution of 0.01 deg and a step resolution of
25 μm.
2.1. Source
The X-ray is produced with an electric current of 12.0 mA and a
voltage of 15.0 kV. Since the width of the source significantly influences
the performance of the reflector, it is critical to know the exact size of
the source, in order to estimate the performance of the reflector. The
width of the source is obtained by pinhole imaging with a pinhole size
of 15 μm in diameter made of Tantalum. The source width  can be
calculated as follows:
 =  ×  ∕ = 90 ± 25 μm,
(1)
where i = 135 μm is the width of the image, s = 70 mm is the pinhole–
source distance, and i = 105 mm is the pinhole–detector distance, as
shown in Fig. 2. Since the pinhole is insufficiently small, there is a
tolerance related to the geometrical relationship, and the effect of the
tolerance on the simulation will be discussed in 2.3.
2. Experimental setup
The schematic setup for the beam collimation and collimation
evaluation is shown in Fig. 1. The Cu k X-ray tube provides a line
focus source positioned at the focus of the reflector. The slit controls the
flux of the incident beam. As previously indicated, the reflector converts
the divergent X-rays into a parallel beam. The CCD camera converts the
2.2. Slit
To ensure the entire reflector is irradiated, the slit after the source
is slightly enlarged, with the output angle of the source larger than the
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Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 3. (a) Beam-path diagram. (b) Image accepted by the CCD camera.
Fig. 4. Parabolic geometry of the reflector.
Fig. 5. Grazing angle and period thickness of the reflector.
theoretical capture angle of the reflector. Therefore, some rays are not
reflected by the mirror but are directly incident on the receiving surface
of the CCD camera as shown in Fig. 3. In Fig. 3(b), the upper beam
represents the pattern of the direct beam, and the lower one represents
the pattern of the collimated beam. Since the direct beam is divergent
with a spatial angle, the bottom side of the pattern of the direct beam is
not flat.
point, in order to obtain a high intensity. The reflector is designed for Cu
k radiation, and the material combination is W/Si with a thickness ratio
W ∕(W + Si ) of 0.4. Therefore, the period thickness of the multilayer
should vary from 3.104 nm to 3.864 nm according to the variation of
the grazing angle from 1.474 deg to 1.204 deg, as shown in Fig. 5. To
obtain a saturation reflectance, the number of bilayers required is 100.
In this instance, the graded multilayer is deposited using direct current
(DC) magnetron sputtering technology [19–22].
The experimental period thickness distribution of the multilayer
is shown in Fig. 5 (dots), and the variation between this result and
the theoretical data is smaller than 1%. This factor will therefore not
significantly influence the collimation system since the main effect of
the thickness variation is a decrease of the flux when the variation is
larger than 7% [17].
The FWHM bandwidth of the multilayer is about 0.05 deg, and the
reflector–source distance is about 100 mm. Therefore, the acceptable
source tolerance is about 87 μm, while the measured tolerance of the
source is 25 μm.
The substrate of the reflector was fabricated by Zeiss with a tangential slope error of 4.59 μrad and a sagittal error of 1.63 μrad.
According to the slope error, the surface profile ( ) of the reflector
can be described by a series of sinusoidal components with different
2.3. Reflector
Since the real part of the refractive index in the X-ray spectrum is
extremely small, the reflector can only operate in a total reflection mode.
To increase the grazing angle of the multilayer reflector, an artificial
lattice with a tailored shape and lattice constant is utilized.
To collimate the outgoing divergent beam from the source, the shape
of the reflector is a parabolic cylinder with the source at the focus. The
parabola is defined by the equation:
2 = 2 ×  × ,
(2)
with  = 0.106 mm as the parabola parameter [18]. The center of the
reflector is 100 mm away from the source with a size of 40 mm along
in the -axis and 20 mm in the -axis, as shown in Fig. 4.
Since the grazing angle varies along the -axis of the reflector, a
laterally graded multilayer is applied to satisfy Bragg’s Law at each
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Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 6. Intensity distributions of collimated beams for different reflector–detector distances.
Fig. 7. Pattern tilt angle with relative  tilt angle.
frequencies  :


( ) ∑
(
)
  =
 × sin 2 ×  ×  ×  +  ,
(3)
Fig. 8. Pattern with unparalleled -axis between the reflector and the source. (a)
experiment result; (b) simulation result.
=1
∕2
where  = ∕0 + 1, the amplitude is represented as  = 0  , the
phase as  = 2,  = log ( ) ∕ log( ),  is the total length of the
reflector, 0 is the sampling frequency of the measurement equipment,
0 is the ratio of the number of points for the measurement frequency
over real space, and  is a random number between 0 and 1 that is
generated by random seed  [17,23,24]. Since the slope error is an order
of magnitude smaller than 17.4 μrad which only affords a divergence of
0.001 deg and no intensity variation, it is not expected to significantly
influence the collimation performance of the reflector. Therefore, the
main contribution of the beam divergence is from the source size.
distances are shown in Fig. 6. The period thickness at the middle of
the mirror has the smallest variation, therefore, the intensity at the
middle of the collimated beam is highest. The disappearance of the peak
intensity at the center of the distribution for large distances is due to
the divergence of the beam. The width of the collimated beam is about
0.887 mm, and the divergent angle is approximately 0.003 deg. This is
much smaller than 0.02 deg.
3. Adjustment
2.4. Prospect of collimation performance
Initially, it is necessary to ensure that the pattern accepted by the
CCD camera is not unusual, therefore, the -axis of the X-ray tube and
the reflector should be parallel. Secondly, the -axis of the CCD camera
and reflector should be parallel to accurately determine the width of
the pattern of the beam on the -axis of the acceptance plane of CCD
camera. Thirdly, the propagation direction of the collimated beam and
Since the size of the source, the slope error of substrate and the period thickness distribution of the multilayer are known, the collimation
performance of the reflector can be predicted by computer simulation
programming in Matlab [17,25]. The simulated results for the intensity
distributions of the collimated beam for different reflector–detector
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Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 9. (a) Normal. (b) The -axis is unparalleled.
the -axis of the CCD camera should be parallel to obtain an accurate
width of the pattern on the -axis of the acceptance plane of the CCD
camera. Finally, the relative distance and the relative angle of the Xray tube and reflector should be adjusted to achieve ideal collimation
performance.
3.1. -axis of reflector and -axis of X-ray tube
A parallel calibration of the -axes of the source and reflector should
be performed to ensure that the width of the collimated beam along the
-axis is uniform. If the two -axes are not parallel, the relative position
of the reflector and the source along the -axis will be different, and an
oblique pattern will be recorded by the CCD camera. The profile of this
pattern is challenging to identify. The tilt angle of the pattern is linearly
related to the relative angle between the two -axes, and has a value
that is 0.65 of this angle, as shown in Fig. 7. Fig. 8(a) represents an
oblique pattern that was recorded by the CCD camera and the tilt angle
is 1.75 deg. Therefore, there is a tilt of 1.15 deg of the -axis of the two
equipment as simulated in Fig. 8(b).
Fig. 10. (a) Normal. (b) The -axis is unparalleled.
3.2. -axis of X-ray reflector and -axis of CCD camera
negative position of the -axis similar to the green beam. Irrespective
of whether or not the grazing angle is small or large, the reflectivity
will be reduced, leading to a decrease of the flux. The curve in Fig. 12
shows the simulation of relative flux with grazing angle deviation. The
maximum flux is obtained without a grazing angle. Therefore, when the
total intensity has a maximum value, the grazing angle is well-adjusted.
The FWHM is about 0.07 deg for the bandwidth and  of the multilayer
is about 0.05 deg–0.08 deg. Since the reflectivity curve of the W/Si
multilayer is not symmetrical, the curve in Fig. 12 is also asymmetrical.
Its distribution is determined by the reflectivity curve of the multilayer,
so the curve can be simulated using two cubic functions: (1) left:
A parallel calibration of the -axis of the reflector and the CCD
camera should also be performed to obtain a clear and accurate image.
If the edge of the beam is outside of the pixel point array as shown in
Fig. 9(a), the reading will have an error of at least 2 pixels as shown in
Fig. 9(b) since the beam pattern is recorded using the pixel point array
of the CCD camera. Each pixel is 0.009 mm in width. Therefore, this will
give rise to a 0.01 deg error for a reflector–detector distance of 100 mm.
3.3. Propagation director of collimated beam and -axis of CCD camera
The acceptance plane of the CCD camera and the propagation direction of the collimated beam should be vertical, as shown in Fig. 10(a),
otherwise, the width of the collimated beam will be large, as shown in
Fig. 10(b). The error should be smaller than 8.13 deg to keep the beam
width error smaller than one pixel.
 = 0 + 1 ×  + 2 × 2 + 3 × 3 ,
with 0 = 1, 1 = −4.27, 2 = −4.38 ×
right:
 = 0 + 1 ×  + 2 × 2 + 3 × 3 ,
3.4. Grazing angle
(4)
102
and 3 = 4.52 ×
103 ,
and (2)
(5)
with 0 = 1, 1 = 4.59, 2 = −6.16 × 102 and 3 = −7.36 × 103 .
Fig. 13 shows the intensity distribution of the patterns accepted by
the CCD camera, it is clear that the red line is a line with a grazing
angle deviation. The left peak represents the intensity distribution of
the collimated beam pattern while the right peak represents the intensity
distribution of the pattern of the direct beam. The total intensity of the
peak of the collimated beam is 0.81 of the normal flux, so the grazing
angle is smaller than the normal one of 0.015 deg.
The deviation of the grazing angle will diminish the flux of the
collimated beam for a departure from Bragg’s Law, and change the
reflection angle in accordance with the law of reflection, as represented
in Fig. 11 by the green and blue beams. Because of the latter, a
collimated beam with a smaller grazing angle will bias to the positive
position of the -axis similar to the blue beam. In the same way, the
collimated beam with a larger grazing angle will be biased to the
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Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 11. Grazing angle deviation diagram.
Fig. 14. Reflector–source distance deviation diagram. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this
article.)
Fig. 12. Simulation result of the intensity with grazing angle deviation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web
version of this article.)
Fig. 15. Simulation result of the divergence with various reflector–source distance
deviations. (For interpretation of the references to colour in this figure legend, the reader
is referred to the web version of this article.)
Fig. 13. Profile of the image accepted by the CCD camera.
3.5. Reflector–source distance
Fig. 16. Profile of image received at the CCD camera.
The reflector–source distance deviation mainly determines the divergence of the collimated beam. The capture angle becomes smaller
when the reflector is closer to the source, and the collimated beam
is then divergent. On the contrary, the capture angle becomes larger
when the detector is farther away from the source, and as such the
collimated beam becomes convergent. As shown in Fig. 14, the yellow
beam is parallel for the normal reflector–source distance, the blue beam
is divergent for a shorter reflector–source distance and the green beam
is convergent for a longer reflector–source distance. The curve in Fig. 15
shows the simulation result of the divergence with reflector–source
distance deviation. The larger the reflector–source distance deviation,
the faster the change velocity of both the capture angle and the
collimated beam’s divergent angle. So, it can be fitted with a cubic
27
Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Fig. 17. Images at different reflector–detector distances: (a) 90 mm, (b) 140 mm, (c) 190 mm, (d) 240 mm.
function:
y = 0 + 1 ×  + 2 × 2 + 3 × 3 ,
the collimated beam is only divergent in one dimension (-axis), since
it is collimated in the other direction (-axis) by the reflector. It is
important that the loss caused by the absorption of the reflector should
also be considered. Therefore, the intensity of the pattern of the direct
beam accepted by the CCD camera is:
(6)
with 0 = 2.98 × 10−3 , 1 = 2.97 × 10−3 , 2 = 5.57 × 10−5 and 3 =
−7.74 × 10−6 .
To calculate the divergence angle of the collimated beam, we
determined the widths of the patterns of the collimated beam at different
reflector–detector distances. The lines in Fig. 16 represent the intensity
distributions of collimated beam patterns at different reflector–detector
distances during adjustment. It is clear that the FWHM value of the
relative intensity distribution of 340 mm reflector–detector distance is
larger than that of 90 mm reflector–detector distance, meaning that the
collimated beam is divergent, so the reflector–source distance should be
large. The divergence angle can be calculated as:
)
(
2 − 1
× 2 = 0.011 deg,
(7)
 = 
 × 2
  =  0 ∗  ∗  ∗  ,
(8)
and the intensity of the pattern of collimated beam accepted by the CCD
camera is:
 = 0 ∗  ∗  ∗ ,
(9)
with  is the absorption coefficient of air,  is the divergent coefficient
in the -axis direction,  is the divergent coefficient in the -axis
direction, and  is the reflectivity of the reflector. The absorption
coefficient  and  increase with the expansion of the reflector–
detector distance, however, the reflector reflectivity  is constant.
Therefore, the intensity of the pattern of the direct beam decreases faster
than that of the collimated beam.
To calculate the divergent angle of the collimated beam, the intensity distribution is normalized, as shown in Fig. 19. The dotted
line represents the theoretical intensity distribution, and the solid line
represents the experimental data. It is seen that the distributions are
almost the same. The difference between the FWHM of the lines is
smaller than 0.003 mm, however, the size of the pixels of the CCD
camera is 0.009 mm. The accuracy of the measurement is given by:
where 1 = 0.909 mm is the width of the collimated beam at 90 mm,
2 = 0.954 mm is the width of the collimated beam at 340 mm and
2 = 250 mm is the difference between the two reflector–detector
distances. According to the fitting function above Eq. (6), the reflector–
source distance is about 2.262 mm smaller, so the relative distance of
the X-ray tube and reflector should be larger with this value.
4. Collimation performance
+
,
(10)

with  is the pixel size and  is the distance interval, so the divergent
angle is smaller than 0.004 deg.
In the application of X-ray reflectometry, a beam with a divergence
of 0.004 deg is acceptable for the analysis of layers less than 500 nm.
In the application of X-ray diffractometry, the broadening due to such a
small divergence is only 0.008 deg.
 = 2 × atan
After adjusting the relative position of the source, reflector, and CCD
camera, the pattern of the collimated beam can be accurately acquired
at the acceptance plane of the CCD camera, which facilitates the
determination of the intensity distribution and width of the collimated
beam at the collimating plane. The divergent angle of the collimated
beam can be determined by combining multiple sets of data at different
reflector–detector distances.
The four photos in Fig. 17 are acquired by the CCD camera at
different reflector–detector distances of 90 mm, 140 mm, 240 mm
and 340 mm (above: direct beam; bottom: collimated beam). Fig. 18
shows the intensity distributions of the patterns accepted by the CCD
camera at the different reflector–detector distances (left: direct beam;
right: collimated beam). As the reflector–detector distance increases,
the change of the FWHM of the pattern of the collimated beam is only
0.005 mm while the change of the FWHM of the pattern of the direct
beam is 1.45 mm, which is 300 times bigger.
The intensity decreases with the increase of the reflector–detector
distance, and the intensity of the pattern of the direct beam reduces
faster than that of the collimated beam. The decrease results from the
divergence of the beam and absorption by air. The absorption coefficient
of the direct beam and the collimated beam is the same. However, the
direct beam is divergent in two dimensions (-axis and -axis), whereas
5. Conclusion
An X-ray parabolic cylinder laterally graded multilayer reflector is
designed and fabricated to collimate a divergent beam emitted from Cu
k source for X-ray diffraction. The experimental device is made of a
Cu k X-ray tube, a slit, a reflector, an X-ray CCD camera, and position
stages.
To achieve excellent collimation performance, the computer-aided
assembly process for the assembly of the X-ray parabolic cylinder
laterally graded multilayer reflector to collimate divergent beam, is used
to quantitatively guide the adjustment.
Firstly, a high efficiency setup for collimating the divergent beam
from the X-ray tube and evaluating the parallelism of the collimation
beam is proposed.
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Y. Yao et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 22–29
Acknowledgments
This work is supported by the National Natural Science Foundation
of China (No. 61621001, No. 11505129), National Key Scientific Instrument and Equipment Development Project (No. 2012YQ13012505,
2012YQ24026402), Shanghai Pujiang Program (No. 15PJ1408000).
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Fig. 18. Profiles of patterns of the collimated beam at different reflector–detector
distances.
Fig. 19. Relative intensity distribution of each pattern of the collimated beam showing
the divergence.
Secondly, the performance of the reflector used in the experiment
is simulated according to the size of the source, the surface topography
of the parabolic cylinder reflector substrate and the period thickness
distribution of the laterally graded multilayer coating on the reflector.
The divergent angle was determined to be approximately 0.003 deg.
Then, the adjustment is performed by analyzing every essential
adjustment step and quantitatively computing the resetting value, including the -axis of the reflector and the -axis of the X-ray tube,
the -axis of the X-ray reflector and the -axis of the CCD camera, the
propagation director of the collimated beam and the -axis of the CCD
camera, the grazing angle and the reflector–source distance.
Finally, the reflector is effectively assembled with a divergent angle
smaller than 0.004 deg. With this small divergence, it is possible to
measure layer thicknesses of less than 500 nm and only broadens the
reflection widths by 0.008 deg. Moreover, the intensity distribution is
very similar to the uniform theoretical intensity distribution.
29
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