Accurate generation of small angles is of vital importance for calibrating angle-based metrology instruments used in a broad spectrum of industries including mechatronics, nano-positioning, and optic fabrication. We present a novel, piezo-driven, flexure device capable of reliably generating micro- and nanoradian angles. Unlike many such instruments, Diamond Light Source’s nano-angle generator (Diamond-NANGO) does not rely on two separate actuators or rotation stages to provide coarse and fine motion. Instead, a single Physik Instrumente NEXLINE “PiezoWalk” actuator provides millimetres of travel with nanometre resolution. A cartwheel flexure efficiently converts displacement from the linear actuator into rotary motion with minimal parasitic errors. Rotation of the flexure is directly measured via a Magnescale “Laserscale” angle encoder. Closed-loop operation of the PiezoWalk actuator, using high-speed feedback from the angle encoder, ensures that the Diamond-NANGO’s output drifts by only ∼0.3 nrad rms over ∼30 min. We show that the Diamond-NANGO can reliably move with unprecedented 1 nrad (∼57 ndeg) angular increments over a range of >7000 μrad. An autocollimator, interferometer, and capacitive displacement sensor are used to independently confirm the Diamond-NANGO’s performance by simultaneously measuring the rotation of a reflective cube.

Ultra-precise angle measuring devices, including encoders, multi-beam interferometers, and electronic autocollimators are extensively used in a wide range of scientific and industrial sectors for applications requiring state-of-the-art performance, including mechatronics, nano-positioning, and optic fabrication. For the most demanding optical or nano-positioning applications, metrology sensors provide inputs for active correction of angular errors. However, any internal errors of the metrology sensors will also be fed back into the control loop, thereby reducing overall system performance. Therefore, using a calibrated device to reliably and accurately generate known angles is vitally important for investigating and correcting the response of angle metrology instruments. In the micro- and nanoradian regime, calibration is essential as systematic errors can be a significant proportion of the angles to be measured.

Due to increasing demands for improvements in angle calibration,1–5 a European Metrology Research Programme “SIB58 ANGLES” was recently initiated by a consortium of National Metrology Institutes. The project’s goals include autocollimators and their usage in optical profilometry, optimisation of angle encoders, and the creation and calibration of small angle generating devices.6–9 

Perhaps the most demanding scientific application requiring calibration of micro- and nanoradian angles is surface profilometry of state-of-the-art X-ray mirrors used at 3rd and 4th generation synchrotron light sources,10,11 including Diamond Light Source in the UK. Such X-ray optics can be up to 1 m long and require only nanometre-level deviations from the required profile over their entire surface. However, even with such extreme quality, such optics are still a limiting factor for synchrotron X-ray experiments. This motivates the fabrication of flat or highly curved optics with surface polishing errors of <0.5 nm rms, corresponding to slope error (the spatial derivative of height errors) deviations of <50 nrad rms. To put this into perspective, a typical nano-focussing X-ray mirror with a curvature of 50 m has a change in slope of 4000 μrad over a length of 200 mm. Measuring deviations of only a few nanoradians within a total angle change of several million nanoradians requires an accurately calibrated metrology instrument with exceptionally low noise levels and ultra-high resolution. Traditional, mechanical polishing is unable to produce such extreme quality optics, especially for aspheric or free-form surfaces. To resolve this issue, deterministic polishing techniques have been developed, including ion beam figuring12,13 and elastic emission machining.14 However, the final quality of the optic is strongly dependent on metrology instruments capable of providing accurate feedback for iterative cycles of corrective polishing. Accurately creating small angles to investigate and correct the response of optical surface profilers such as Nanometre Optical Metrology instruments (NOMs)15 and Long Trace Profilers (LTPs)16 is essential to aid production of next-generation X-ray mirrors. These arguments motivate the creation of a novel, piezo-driven, flexure device, capable of reliably generating micro- and nanoradian angles. It should also be noted that many other scientific disciplines could also benefit from such extreme angular calibration using a device which can generate nanoradian steps over a range of thousands of microradians.

As shown in Figure 1, Diamond Light Source’s nano-angle generator (Diamond-NANGO) consists of three major components: a cartwheel flexure, an angle encoder, and a linear piezo actuator. Unlike many such instruments, the Diamond-NANGO does not rely on two separate actuators or rotation stages to provide coarse and fine motion. Instead, a single NEXLINE PiezoWalk actuator provides millimetres of travel with sub-nanometre resolution. The piezo exerts a force on the lever arm of the inner section of a cartwheel-type flexure. This rotates the inner section of the flexure, relative to the fixed, outer section. The PiezoWalk actuator is capable of generating a large holding force of 100 N which, in combination with the stiff cartwheel flexure, provides a very stable angle output. Rotation of the flexure is directly measured via a “Laserscale” angle encoder. A cube with five reflective faces is mounted at the centre of rotation of the flexure.

FIG. 1.

Schematic of the Diamond-NANGO, showing the cartwheel flexure (yellow), angle encoder (green disc), linear piezo actuator (cyan), and reflector cube (magenta) mounted at the centre of rotation.

FIG. 1.

Schematic of the Diamond-NANGO, showing the cartwheel flexure (yellow), angle encoder (green disc), linear piezo actuator (cyan), and reflector cube (magenta) mounted at the centre of rotation.

Close modal

For many decades, sine-bars and gauge blocks have traditionally been used in optical testing laboratories to generate discrete, calibrated angles. Continuous angles can be created by replacing the gauge blocks with a micrometer screw or piezo actuator. Typically, a capacitive sensor or linear interferometer measures either the extension of the actuator or displacement of the sine-bar. The pitch rotation is inferred from the change in linear displacement. However, at the micro- and nano-scales, error sources can cause serious problems for this indirect method, including the following: cosine error due to non-orthogonality of the displacement sensor and the plane of reference; parasitic roll or yaw if the actuator is misaligned relative to the rotation axis; bending or gravitational sagging of long sine-bars; and instability of the effective centre of rotation of the sine-bar. In contrast, the Diamond-NANGO’s angular flexure and direct measurement of rotation at the point of interest significantly reduce the impact of the above issues. For example, the measured pitch angle is not influenced by bending of the lever arm by the actuator. Similarly, the thickness of the flexure ensures that the force applied by the actuator is efficiently converted into pitch rotation, and hence parasitic roll and yaw angle changes are minimised. To aid portability and reduce vibrations, the Diamond-NANGO is firmly clamped to a sturdy base-plate via three kinematic mounts. For versatility, the Diamond-NANGO is designed to operate with its rotation axis vector pointing either horizontally (as shown in Figure 1) or vertically (as shown in Figure 7). Operation of a traditional sine-bar with its rotation axis pointing vertically can be problematic.

FIG. 7.

Diamond-NANGO monitored by independent metrology instruments (two autocollimators and an angular interferometer) to investigate its angular performance.

FIG. 7.

Diamond-NANGO monitored by independent metrology instruments (two autocollimators and an angular interferometer) to investigate its angular performance.

Close modal

To achieve an ultra-stable angular output, it is necessary to select construction materials with suitable mechanical strength, flexibility, and thermal stability. The engineering brief for the cartwheel flexure was to deliver an angular range of 10 000 μrad using a force of 100 N supplied by a piezo actuator. An initial concept for the flexure was to use a material with low thermal expansion, such as Invar. However, it was quickly discovered that the raw material would be prohibitively expensive, the maximum permissible stress would be limited, and there were fears over the effect of the aggressive heat treatment on the thin flexure elements during manufacturing. As a compromise between cost and thermal and mechanical performance, titanium alloy 4 was chosen as it has higher yield strength than stainless steel or Invar, thus enabling a greater range of rotation for a given flexure design. Titanium alloy 4 also has lower stiffness than stainless steel, resulting in lower stress for a given rotation. Finite element analysis using ANSYS software was performed to determine the optimal parameters for the cartwheel flexure, including the number, length, and width of flexure ribs. It was discovered that lengthening the ribs was a key factor to keep the maximum stress below the material safety limit, whilst also ensuring that the required load stayed within the actuator’s force capabilities. For a piezo actuator load of 98 N, the final optimized flexure design had a maximum stress of 320 MPa, which is safely below the material yield stress limit of 484 MPa for titanium alloy 4 (see Figure 2). The cartwheel flexure was fabricated from a monolithic block of titanium alloy 4 using electrical discharge machining/wire erosion.

FIG. 2.

Finite element analysis of the angular flexure using ANSYS software, showing the following: (left) maximum deflection of 1.7 mm at the point of action of the piezo generates an angular rotation of 10 000 μrad and (right) maximum stress of 320 MPa at the hinge point remains below the safety limit of 484 MPa for titanium alloy 4.

FIG. 2.

Finite element analysis of the angular flexure using ANSYS software, showing the following: (left) maximum deflection of 1.7 mm at the point of action of the piezo generates an angular rotation of 10 000 μrad and (right) maximum stress of 320 MPa at the hinge point remains below the safety limit of 484 MPa for titanium alloy 4.

Close modal

Traditional piezo actuators cannot simultaneously provide the travel range, applied force, and positional stability required for our application. Instead, a relatively new type of piezo actuator was chosen, based on the stick/slip translational principle. The NEXLINE N-216.20 PiezoWalk drive from Physik Instrumente (PI), employing an E-755.101 controller, is capable of moving 20 mm with a resolution of 0.03 nm and is ideally suited for small-amplitude dynamic adjustment. Based on a flexure lever arm length of 160 mm, this piezo actuator adequately provides the maximum and minimum translations of 1.6 mm and 0.16 nm necessary to generate 10 000 μrad and 1 nrad angles, respectively. The NEXLINE PiezoWalk drive has pairs of piezo stacks which impart motion to the moving “runner.” Each piezo stack consists of two segments: a contraction actuator for unclamping and a shearing actuator for translating the runner. As illustrated in Figure 3, the piezo actuator has two modes of operation: stepping mode for coarse translation (individual steps of 10 nm to 6 μm), realized by coordinated control of shearing and clamping motions of the piezo stacks and fine/analogue mode (sub-nm to 3 μm) using only shearing of the piezo stacks. Aside from ultra-high resolution, the other major advantage of such a device is positional stability. Preloading the piezo stacks against the runner ensures that nanometre stability is maintained, even when voltages are removed. This self-locking mechanism means the system does not consume power at rest, which minimises heat dissipation for improved thermal stability. The strong pushing force of the piezo actuator was chosen to minimise gradual sagging of the actuator under the effect of gravity and associated rotation drift of the lever arm. A spring opposing the piezo actuator ensures that the actuator stays firmly in contact with lever arm of the flexure. This is especially important when the rotation axis of the Diamond-NANGO is pointing upwards. Since the NEXLINE N-216 has a voltage range of ±250 V, as with any high voltage piezo device, it should not be operated in low atmosphere due to the risk of coronal arcing.

FIG. 3.

Schematic of the NEXLINE PiezoWalk drive demonstrating the coarse stepping mode (upper) and fine/analogue shearing mode (lower). In combination, this actuator has sub-nanometre resolution and a multi-millimetre travel range. Reproduced with permission from Physik Instrumente.

FIG. 3.

Schematic of the NEXLINE PiezoWalk drive demonstrating the coarse stepping mode (upper) and fine/analogue shearing mode (lower). In combination, this actuator has sub-nanometre resolution and a multi-millimetre travel range. Reproduced with permission from Physik Instrumente.

Close modal

For true characterization of rotation, it is preferable to measure angle directly at the point of interest, rather than inference based on displacement measured at a remote location. Traditional encoders are typically limited to angular resolutions of several or tens of nanoradians. To ensure exceptional performance for the Diamond-NANGO, a single, state-of-the-art angle encoder based on grating interferometry was employed. The ultra-high resolution angle encoder from Magnescale comprises a crown-glass Laserscale encoder ring, read-head (BH20-001T88), and interpolator unit (BD96-B1400HC). An infra-red laser beam (790 nm wavelength) from a semiconductor source is split into S and P polarized components. The two separate beams are first diffracted by a holographic sinusoidal grating on the encoder ring, before passing through quarter-wave plates and being reflected by mirrors. The beams are diffracted again by the grating before being re-combined by the polarized beam splitter to create interference patterns which are recorded by photo-detectors (see Figure 4). Since double diffraction occurs, interference is subject to four light/dark inversion cycles for each grating period of movement. Thus, a grating pitch of 1 μm produces a signal wavelength of 250 nm. An interpolation factor of 4000X provides linear resolution of 0.0625 nm, which for a scale of radius 41.723 mm corresponds to an angular resolution of <1.5 nrad. Since both laser beams travel a similar distance through the same region of air, this apparatus is largely insensitive to changes in air temperature, humidity, and pressure. The angle encoder ring was positioned to be concentric with the flexure axis. A marker on the encoder scale enables the encoder to be homed to a known position after the device has been re-energized. Analogue signals from the read-head were displayed in real-time on an oscilloscope, thus enabling the read-head to be aligned to the encoder scale by optimizing the circularity of the Lissajous figure.

FIG. 4.

Operation of the diffraction grating interferometer, fed by an infra-red laser, used in the Laserscale angle encoder of the Diamond-NANGO. Reproduced with permission from Magnescale Co. Ltd., Japan.

FIG. 4.

Operation of the diffraction grating interferometer, fed by an infra-red laser, used in the Laserscale angle encoder of the Diamond-NANGO. Reproduced with permission from Magnescale Co. Ltd., Japan.

Close modal

To ensure stable and reliable performance for the Diamond-NANGO, the PiezoWalk drive was operated in closed-loop using feedback from the angle encoder. Custom-built, Labview scripts running on a real-time controller (Compact-Rio NI9012 from National Instruments) acquired data from the Laserscale encoder at 500 Hz via a digital parallel connection to a National Instruments NI9403 acquisition card. Proportional and integral feedback, based on the arithmetic average of 10 consecutive readings, was used to compute updated positional commands to be sent to the piezo actuator at a rate of 50 Hz. An EPICS server running on the real-time controller transmitted angle data to the network and received inputs, including commands to move to a given angle or dwell at a specified position. In this manner, the Diamond-NANGO can be instructed to move, wait, and collect data in synchronization with other metrology instruments or motion stages. This control feature makes it an ideal tool for independent characterization of metrology instruments.

A glass cube, with 25 mm long sides and 5 metallic-coated reflective faces (λ/20 surface quality), was mounted with one of its faces at the centre of the rotary flexure. This ensures that an external metrology instrument views a fixed position on the cube’s “master face” as the Diamond-NANGO is rotated. The cube’s orientation was aligned parallel to the rotation axis of the Diamond-NANGO using an autocollimator. In this arrangement, several metrology instruments can simultaneously view different faces of the reflective cube. This enables multiple, independent measurement of the pitch angle of rotation, whilst also monitoring parasitic yaw or roll rotations. The final implementation of the Diamond-NANGO is shown in Figure 5. The laser source for the angle encoder was initially attached directly to the top of the flexure block but was subsequently removed to improve temperature stability.

FIG. 5.

Photograph of the Diamond-NANGO, with its rotation axis pointing in the horizontal direction.

FIG. 5.

Photograph of the Diamond-NANGO, with its rotation axis pointing in the horizontal direction.

Close modal

To investigate the performance of the Diamond-NANGO, a series of tests were carried out in the DLS Optics and Metrology cleanroom lab.17 Environmental enclosures and robust opto-mechanics are used in the lab to achieve the exceptional temperature (<10 mK over several days) and vibrational stability necessary to perform nano-metrology. The Diamond-NANGO, oriented with its rotation axis pointing vertically, was positioned such that a SIOS SP200TR 3 beam-interferometer and two Moeller-Wedel Elcomat3000 autocollimators could simultaneously measure the rotation angle of a reflector cube mounted on the Diamond-NANGO. Apertures with a diameter of 4 mm were placed ∼200 mm from the reflector cube to define each autocollimator beam and reduce stray light entering the autocollimator. The autocollimators have an angular range of ∼10 000 μrad, thus covering the full operating range of the Diamond-NANGO but are limited to a resolution of >20 nrad. The 3-beam interferometer is theoretically capable of measuring with a resolution of <10 nrad but is sensitive to environmental drifts. The entire setup was placed inside an environmental enclosure to minimize temperature fluctuations, stray light, excessive air flows, and acoustic noise. A simplified schematic and a photograph of the experimental arrangement are shown in Figures 6 and 7, respectively.

FIG. 6.

Schematic diagram showing a plan (top) view of the experimental setup to simultaneously collect angle data from three independent metrology instruments and the Diamond-NANGO.

FIG. 6.

Schematic diagram showing a plan (top) view of the experimental setup to simultaneously collect angle data from three independent metrology instruments and the Diamond-NANGO.

Close modal

The Elcomat3000 autocollimators at DLS have previously been calibrated at the German National Standards Institute, Physikalisch-Technische Bundesanstalt (PTB) and were shown to exhibit a linear response over <±4000 μrad. The Diamond-NANGO was commanded to rotate in 2 bi-directional runs over an angle range of >6500 μrad with 500 μrad steps. Figure 8 shows the expected linear relationship between the autocollimator’s angle reading (plotted on the Y-axis) and the output from the Diamond-NANGO’s angle encoder (X-axis). Data from both metrology instruments were collected over a 10 s period at each step plateau. The gradient of the best fit line was 1.000 275 which, assuming a perfect response from the autocollimator, corresponds to a linearity error of 0.028% for the Diamond-NANGO over an operational range of >6500 μrad. Since only a single encoder was used, the linearity error could be caused by the eccentricity error of the encoder.

FIG. 8.

Diamond-NANGO’s encoder angle (X-axis) versus a calibrated autocollimator (Y-axis) as the NANGO was rotated in 500 μrad increments over a range of 6500 μrad. The gradient of the best fit line is 1.000 275, showing that the Diamond-NANGO’s linearity error is ∼0.028%.

FIG. 8.

Diamond-NANGO’s encoder angle (X-axis) versus a calibrated autocollimator (Y-axis) as the NANGO was rotated in 500 μrad increments over a range of 6500 μrad. The gradient of the best fit line is 1.000 275, showing that the Diamond-NANGO’s linearity error is ∼0.028%.

Close modal

For nano-positioning applications, one of the most important parameters is the minimal incremental step that a device can reliably deliver. Figures 9 and 10 show the output of the Diamond-NANGO’s internal angle encoder when commanded to move in 10 nrad or 1 nrad steps, respectively, in positive and negative directions. Dwell time at each step plateau was 2 and 3 min for the 10 and 1 nrad steps, respectively. Interestingly, elevated noise levels are seen on the step at 30 nrad in Figure 8 in both directions of movement, perhaps indicating dust or minor defects on the corresponding section of the encoder scale. As seen in Figure 10, the peak-to-valley noise level on each step is ∼0.5 nrad, which is sufficiently small to adequately resolve 1 nrad steps. These data clearly show that the Diamond-NANGO is capable of being controlled with 1 nrad steps.

FIG. 9.

Staircase plot showing that the Diamond-NANGO is capable of reliably making 10 nrad steps in the positive and negative directions with minimal turning errors.

FIG. 9.

Staircase plot showing that the Diamond-NANGO is capable of reliably making 10 nrad steps in the positive and negative directions with minimal turning errors.

Close modal
FIG. 10.

Staircase plot showing that the Diamond-NANGO is capable of reliably making 1 nrad steps.

FIG. 10.

Staircase plot showing that the Diamond-NANGO is capable of reliably making 1 nrad steps.

Close modal

Ideally, the 1 nrad stepping performance needs to be confirmed by independent metrology instruments. However, as with many other angle measuring devices, our autocollimators and angular interferometer cannot measure such a small angle due to limitations of resolution, noise, or environmentally induced drifts. Under the most stable conditions, and with optimal data processing, rapid 20 nrad steps (5 steps up and 5 steps down during a 1 s period) could be resolved by the 3-beam interferometer. Figure 11 shows good agreement between the synchronized output of the Diamond-NANGO (blue, solid curve) and the 3-beam angle interferometer (red, dotted curve) for 20 nrad steps. Further, independent verification of the ultra-small angle stepping performance of the Diamond-NANGO was investigated using a capacitive displacement sensor to measure the translation of the lever arm during rotation. Figure 12 shows good agreement between the Diamond-NANGO (blue, solid curve) and the capacitive sensor (red, dotted curve) for 5 nrad steps. Conversion between the measured capacitive sensor displacement and the corresponding angle was pre-calibrated by making a 100 μrad step and measuring the rotation using an autocollimator. The Diamond-NANGO’s performance for larger (>100 nrad) steps was also confirmed using the autocollimators. Extrapolating these facts provides confidence that the 1 nrad steps measured by the Diamond-NANGO are reliable. The minimum incremental step size of the Diamond-NANGO is smaller than the sensitivity/resolution of many angle measuring instruments. As such, this extreme precision makes it an ideal candidate for use as a calibration tool. However, future calibration of the Diamond-NANGO against a certified, national standard is required to ensure the accuracy of its output.

FIG. 11.

Independent verification of rapid (100 ms) steps of 20 nrad generated by the Diamond-NANGO, as observed by the Diamond-NANGO’s internal angle encoder (blue, solid curve) and an external angle interferometer (red, dotted curve).

FIG. 11.

Independent verification of rapid (100 ms) steps of 20 nrad generated by the Diamond-NANGO, as observed by the Diamond-NANGO’s internal angle encoder (blue, solid curve) and an external angle interferometer (red, dotted curve).

Close modal
FIG. 12.

Independent verification of 5 nrad steps generated by the Diamond-NANGO, as observed by the Diamond-NANGO’s internal angle encoder (blue, solid curve) and an external capacitive displacement sensor (red, dotted curve).

FIG. 12.

Independent verification of 5 nrad steps generated by the Diamond-NANGO, as observed by the Diamond-NANGO’s internal angle encoder (blue, solid curve) and an external capacitive displacement sensor (red, dotted curve).

Close modal

In addition to providing a suitably small minimum incremental step, the Diamond-NANGO’s angle output also needs to be stable for extended periods to reliably calibrate other metrology instruments. Ideally, over several minutes, the angle output should drift less than the minimum incremental step. When commanded to maintain position at a given angle, Figure 13 shows that the Diamond-NANGO’s closed-loop output was 0.312 nrad rms over ∼30 min. Stability data were collected at a rate of 50 Hz, and 10 values were averaged to provide 5 data points per second.

FIG. 13.

Stability of the Diamond-NANGO is 0.312 nrad rms over ∼30 min, as measured by its internal angle encoder.

FIG. 13.

Stability of the Diamond-NANGO is 0.312 nrad rms over ∼30 min, as measured by its internal angle encoder.

Close modal

We have designed and built a state-of-the-art, angle generating device capable of reliably creating 1 nrad steps over a range of 7000 μrad. Closed-loop operation of the PiezoWalk actuator, using high-speed feedback from the angle encoder, ensures that the Diamond-NANGO output drifts by ∼0.3 nrad rms over ∼30 min. Such stability, coupled with ultra-high resolution and an extended range, makes it ideal tool for calibrating a variety of angle measuring instruments, including the Elcomat3000-8 autocollimator used by the Diamond-NOM for characterizing state-of-the-art X-ray mirrors. Although the motivation for developing the Diamond-NANGO was investigating and aiding fabrication of ultra-precision, synchrotron X-ray optics, we hope that such a nano-angle generating device can find useful application in a wide range of industries which demand the most exacting performance, including nano-positioning.

This work was carried out with the financial support of Diamond Light Source Ltd., UK. Figures 3 and 4 are reproduced with permission from Physik Instrumente and Magnescale Co. Ltd., Japan.

1.
M.
Astrua
and
M.
Pisani
,
Metrologia
46
,
674
(
2009
).
2.
T.
Yandayan
,
B.
Ozgur
,
N.
Karaboce
, and
O.
Yaman
,
Meas. Sci. Technol.
23
,
094006
(
2012
).
3.
A.
Just
,
M.
Krause
,
R.
Probst
, and
R.
Wittekopf
,
Metrologia
40
,
288
(
2003
).
4.
T.
Yandayan
,
S. A.
Akgoz
, and
M.
Asar
,
Meas. Sci. Technol.
25
,
015010
(
2014
).
5.
R. D.
Geckeler
,
O.
Kranz
,
A.
Just
, and
M.
Krause
,
Adv. Opt. Technol.
1
(
6
),
427
439
(
2012
).
6.
O.
Kranz
,
R. D.
Geckeler
,
A.
Just
, and
M.
Krause
,
Proc. SPIE
8789
,
87890D-1
(
2013
).
7.
D.
Shu
,
J.
Qian
,
W.
Liu
,
S.
Kearney
,
J.
Anton
,
J.
Sullivan
, and
L.
Assoufid
,
Proc. SPIE
9206
,
92060H-1
(
2014
).
8.
M.
Pisani
and
M.
Astrua
,
Appl. Opt.
45
,
1725
(
2006
).
9.
R. D.
Geckeler
and
A.
Just
,
Meas. Sci. Technol.
25
,
105009
(
2014
).
10.
F.
Siewert
,
J.
Buchheim
,
T.
Zeschke
,
M.
Stormer
,
G.
Falkenberg
, and
R.
Sankarid
,
J. Synchrotron Radiat.
21
,
968
(
2014
).
11.
T.
Yandayan
,
R. D.
Geckeler
, and
F.
Siewert
,
Proc. SPIE
9206
,
92060F-1
(
2015
).
12.
A.
Schindler
,
T.
Haensel
,
A.
Nickel
,
H.
Thomas
,
H.
Lammert
, and
F.
Siewert
,
Proc. SPIE
5180
,
64
(
2004
).
13.
L.
Peverini
,
I. V.
Kozhevnikov
,
A.
Rommeveaux
,
P. V.
Vaerenbergh
,
L.
Claustre
,
S.
Guillet
,
J.-Y.
Massonnat
,
E.
Ziegler
, and
J.
Susini
,
Nucl. Instrum. Methods Phys. Res., Sect. A
616
,
115
(
2010
).
14.
K.
Yamauchi
,
H.
Mimura
,
K.
Inagaki
, and
Y.
Mori
,
Rev. Sci. Instrum.
73
,
4028
(
2002
).
15.
S. G.
Alcock
,
K. J. S.
Sawhney
,
S.
Scott
,
U.
Pedersen
,
R.
Walton
,
F.
Siewert
,
T.
Zeschke
,
F.
Senf
,
T.
Noll
, and
H.
Lammert
,
Nucl. Instrum. Methods Phys. Res., Sect. A
616
,
224
(
2010
).
16.
A.
Rommeveaux
,
M.
Thomasset
, and
D.
Cocco
, in
Modern Developments in X-Ray and Neutron Optics
(
Springer
,
Berlin
,
2008
), Vol.
5
 Chap. X.
17.
S. G.
Alcock
and
K. J. S.
Sawhney
,
Proc. SPIE
6704
,
67040E-1
(
2007
).