The simplest possible design for a KB microfocus mirror system?
S. P. Collins, R. C. Harwin, S. M. Scott, D. M. Hawkins, F. Fabrizi, B. Moser, G. Nisbet, J. P. Sutter, and W. S.
Harwin
Citation: AIP Conference Proceedings 1741, 040018 (2016); doi: 10.1063/1.4952890
View online: http://dx.doi.org/10.1063/1.4952890
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1741?ver=pdfcov
Published by the AIP Publishing
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The Simplest Possible Design for a KB Microfocus Mirror
System?
S P Collins1,a) , R C Harwin1,2 , S M Scott1 , D M Hawkins1 , F Fabrizi1 , B Moser1 ,
G Nisbet1 , J P Sutter1 and W S Harwin3
Diamond Light Source, Harwell Science & Innovation Campus, Didcot, OX11 0DE, UK
Department of Physics, Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, UK
3
School of Systems Engineering, University of Reading, Whiteknights, Reading, Berkshire, RG6 6AH, UK
1
2
a)
Corresponding author: [email protected]
Abstract. We report a design for a Kirkpatrick-Baez (KB) microfocussing mirror system. The main components are described,
with emphasis on a ‘tripod’ manipulator, where we outline the required coordinate transformation calculations. The merit of this
device lies in its simplicity of design, minimal degrees of freedom, and speed and ease of setup on a beamline. Test results and an
example of the mirrors in use on Diamond Beamline I16, showing a high-resolution polar domain map of KTiOPO4 with a spot
size of 1.25 µm x 1.5 µm, are presented.
Requirements
Dedicated microfocus beamlines are essential elements of the experimental portfolio of all modern synchrotron
sources. Indeed, the available focal spot size continues to shrink, with ∼ 10 nm being current state-of-the-art. However,
there is an increasing demand for ‘add-on’ microfocus capabilities on many - perhaps even a majority - of beamlines
covering a wide range of science and techniques. While the requirements of rapid (temporary) set-up and non-ideal
beamline layout present a challenge, it is often the case that the ultimate in microfocus spot size is not required. Such
is the case for I16 (Materials & Magnetism) at Diamond Light Source [1]: a focal spot size of a few microns is often
ideal, with optics that can be set up and aligned very quickly and do not interfere significantly with the motions of
the large diffractometer or sample cryostat. Specifically, we require optics that have the fewest possible degrees of
freedom in order to be quick to set up, stable, and easy for the non-specialist to use, a device that can be mounted
from the side to avoid collisions with the kappa diffractometer, the ability to change photon energy without affecting
the focus, and options for optimizing for small (a micron or so) focus with reduced intensity or a larger spot with the
full undulator flux.
Design: Optics
The requirement for high efficiency over a wide (∼3-12 keV) energy range dictates that the optic should be based
on a doubly-focusing mirror. Moreover, the highly elliptical shape required for a large demagnification makes the
KB system (consecutive vertical and horizontal one-dimensional focusing mirrors) an attractive option, from the
perspective of ease of manufacture (and cost), and for the option of focusing either vertically, horizontally, or both.
In order to keep the system as simple as possible, we have opted for pre-shaped mirror substrates, avoiding bending
mechanisms. For further simplicity, the two mirrors are fixed with respect to each other and manipulated as a single
rigid object (Fig. 1).
Our requirement for a generous (>80 mm) working distance sets a limit on the mirror focal lengths (Table 1),
but these are adequate to give theoretical focal sizes of below a micron in both directions when the main beamline
focusing mirrors are driven out of the beam, assuming negligible shape errors in KB mirrors or the beamline’s channelcut monochromator. More complex optical modeling was carried out by the SHADOW ray-tracing package, which
Proceedings of the 12th International Conference on Synchrotron Radiation Instrumentation – SRI2015
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showed that a relative alignment error between the two mirrors of < 0.1◦ (yaw) would have a negligible impact on the
focus - a precision that can be accommodated by a carefully-designed manual mirror support.
Ray-tracing was also deployed in order to investigate the optical performance of the KB mirrors used in conjunction with the main beamline mirrors, in their normal focusing configuration, allowing the full undulator beam to be
focused onto the KB mirrors, preserving virtually the entire flux. The predicted spot-size of∼ 6 (V) x 10 (H) µm with
full flux represented a very useful option for the beamline. Moreover, it was expected, and later demonstrated, that
under-focusing with the main mirrors would lead to a smaller microfocus spot size (see Fig. 1).
FIGURE 1. Left: The KB mirror housing, supported by an asymmetric tripod manipulator (Stewart platform). The mirror substrates
and x-ray beam directions are shown below the mirror housing. Right: Three operational modes of the KB mirrors. Top: the KB
mirrors used as the only focusing element, accepting around 1% of the available undulator fan. Middle: The KB mirrors used in
conjunction with the main beamline focusing mirrors, set to their nominal focus. Bottom: The KB mirrors used in conjunction with
the main beamline focusing mirrors, set to under-focus the beam in order to give the best focus with the full undulator flux.
Design: Manipulator
The requirement for an over-hanging side-mounted option suggested severe weight constraints, and the unsuitability of bulky nested translation and rotation stages. A hexapod represented a more attractive solution. However,
commercially-available units tend to be symmetric, giving equivalent angular ranges and precisions in all directions.
The KB mirror system, on the other hand, has a much lower tolerance for errors in either of the two mirror pitch
angles, but is insensitive to modest rotations about the beam direction. A bespoke, asymmetric system was therefore
proposed. A hexapod was rejected in favor of an asymmetric tripod, isomorphic to a 6-3 Stewart platform, consisting
of three fixed-length legs, each supported by a pair of orthogonal slides (Fig. 1). The mirror assembly is connected
via ball-joints (precision bearings), and the slides by flexure hinges. This system was straightforward to manufacture
(given high-precision slides), and provides the required six degrees of freedom (three rotations and three translations)
via the six slides.
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Coordinate transformations
While the asymmetric tripod provides an excellent engineering solution, the calculations necessary to operate the
device are not trivial. However, once these have been coded for the most general case, changing the specific geometry
of the tripod becomes trivial. The problem is essentially one of computing coordinate transformations between the
‘top’ plate, containing the optics, and the ‘bottom’ plate which is fixed. Multiple virtual centres of rotation can be
defined, allowing each mirror to be rotated about its centre. A small but significant change in the pitch angle of one
mirror leads to an equivalent but insignificant yaw rotation of the second, thus the mirrors can be considered to be
effectively independent. Our calculations are described in detail in a separate publication [2], and implemented in
Python code which is available from the authors. The details of the calculations will not be given here.
Two coordinate transformations are required. The forward kinematic transformation calculates the positions and
angles of the top plate for a given set of base slide settings. While it is relatively straightforward to define this problem
mathematically, we have shown [2] that the problem can be mapped on to an eighth-order polynomial, for which there
exists no general closed-form analytical solutions. Thus, we have proved that a numerical solver must be deployed.
While inelegant, such solvers are fast and give excellent convergence.
The reverse kinematic transformation - calculation of the base slide settings for a given set of angles and positions
- is more straightforward and has a closed-form solution.
Laboratory test results
Test measurements carried out under metrology laboratory conditions, with the tripod side-mounted, demonstrated
very precise control and good stability. The main concern for beamline use is drift, and this was found to be significant for around an hour after all slide motions had ceased (see Figure 2) and are strongly correlated with the motor
temperature. The drifts in pitch angles in µrad, correspond to approximately a quarter to a half of the value in focal
spot position, i.e. up to around 2.5 µm in the worst case, while the motor cools. This is consistent with x-ray measurements on the beamline. Subsequent tests have shown that this drift can be reduced by adopting a larger motor holding
current which maintains the motors at a more constant temperature. Laboratory measurements show drifts of less than
a micron in either direction, over 12 hours. Slightly worse drifts of up to 3 µm in the vertical beam position, measured
with x-rays on the beamline, are thought to be dominated by (known) vertical sample/imager position drifts. However,
even the current level of drift is commensurate with the x-ray focal spot size and experiment set-up time. Systematic
errors in the mirror angles can arise from poorly-defined geometrical parameters - errors in hinge orientation, slide
centres etc, although these are of little concern in practice.
In use
The mirrors are extremely easy to set up and use, and cause minimal obstruction of the diffraction experiments. Focal
spot size measurements have been made with the KB mirrors as the only focusing element, with incident beams
of various sizes, and in conjunction with the main beamline focusing mirrors. The best results are summarized in
Table 1. It is noteworthy that that latter set-up, with the beam under-focused by the main mirrors, can give the entire
undulator flux in a spot size of around 2.5 (V) x 6.5 (H) µm - a very useful configuration for mapping spatial domains
in weakly-scattering systems.
TABLE 1. Optical parameters (Diamond Beamline I16). Spot sizes
were measured with ‘knife-edge’ scans.
Source distance
Focus distance (VFM)
Focus distance (HFM)
Spot size (KB only)
Spot size (KB and main mirrors)
50 m
260 mm
150 mm
1.25 (V) x 1.5 (H) µm
2.5 (V) x 6.5 (H) µm
We have demonstrated the use of the mirrors by carrying out measurements on Diamond beamline I16 (Materials
& Magnetism) to image artificial inversion domains in a crystal of KTiOPO4 (Fig. 2). This technique provided an
unprecedented polar contrast of 270 by utilizing two photon energies: one just above the Ti K-edge and one just
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below [3]. The map shows very clearly the result of artificially poling the crystal with an electric field to produce a 10
micron periodic pattern (albeit not over the entire sample). This map, with a resolution of 1.25µm (V) x 1.5µm (H),
gives important insight into the manufacturing quality of artificial non-linear optical materials used as frequency
doublers.
Summary
We demonstrate an extremely simple, low-cost microfocus mirror system, based on pre-shaped substrates and an
asymmetric tripod manipulator. The calculations necessary to operate the device have been coded in Python. The
system has proved to be easy to use and adequately stable, allowing options for focal size and photon flux. The mirrors
work very effectively in conjunction with the main (under-focused) beamline mirrors to provide a small focus with
the full undulator flux. We have demonstrated the mirrors by mapping polar domains in artificially-poled KTiOPO4 .
FIGURE 2. Left: High precision angular stability measurements. The vertical pitch angular drift is strongly correlated with the
motor temperatures (green lines), showing excellent stability once the motors have cooled. The worst-case focus position drifts
(µm) correspond to around the half of the angular drift (µrad). Right: A map of the polar domains in periodically-inverted KTP [3],
with a period of 10µm. The red and blue regions correspond to fully-oriented ‘up’ and ‘down’ domains, whereas the artificiallypoled stripes are incomplete and only partially inverted.
ACKNOWLEDGMENTS
The Authors are grateful to Alex Bugner and Hiten Patel for their valuable assistance with the metrology lab measurements.
REFERENCES
[1]
[2]
[3]
S. P. Collins, A. Bombardi, A. R. Marshall, J. H. Williams, G. Barlow, A. G. Day, M. R. Pearson, R. J.
Woolliscroft, R. D. Walton, G. Beutier, and G. Nisbet, AIP Conf. Proc. 1234, 303–306 (2010).
R. C. Harwin, P. M. Sharkey, W. S. Harwin, S. M. Scott, D. M. Hawkins, and S. P. Collins, IMA Conference
on Mathematics of Robotics (2015).
F. Fabrizi, P. A. Thomas, G. Nisbet, and S. P. Collins, Acta. Cryst. A 71, 361–367 (2015).
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