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Optimal tuning and calibration of bendable mirrors
with slope measuring profilers
Wayne R. McKinney
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Lawrence Berkeley National Laboratory,
1 Cyclotron Road, M/S 2R0400,
Berkeley, CA 94720-8199, USA
+1-510-495-4395 TEL
+1-510-486-7696 FAX
Email address: WRMcKinney@lbl.gov
Jonathan l. Kirschman
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Lawrence Berkeley National Laboratory,
1 Cyclotron Road, M/S 2R0400,
Berkeley, CA 94720-8199, USA
+1-510-495-4117 TEL
+1-510-486-7696 FAX
Email address: jlkirsc@gmail.com
Alastair A. MacDowell
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
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Lawrence Berkeley National Laboratory,
1 Cyclotron Road, M/S 2R0400,
Berkeley, CA 94720-8199, USA
+1-510-495-4276 TEL
+1-510-486-7696 FAX
Email address: AAMacDowell@lbl.gov
Tony Warwick
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Lawrence Berkeley National Laboratory,
1 Cyclotron Road, M/S 2R0400,
Berkeley, CA 94720-8199, USA
+1-510-495-5919 TEL
+1-510-486-7696 FAX
Email address: T_Warwick@lbl.gov
Valeriy V. Yashchuk
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Lawrence Berkeley National Laboratory,
1 Cyclotron Road, M/S 2R0400,
Berkeley, CA 94720-8199, USA
+1-510-495-2592 TEL
+1-510-486-7696 FAX
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ABSTRACT
We describe a technique to optimally tune and calibrate bendable x-ray optics for sub-
micron focusing. The focusing is divided between two elliptically cylindrical reflecting elements,
a Kirkpatrick-Baez (KB) pair. Each optic is shaped by applying unequal bending couples to each
end of a flat mirror. The developed technique allows optimal tuning of these systems using
surface slope data obtained with a slope measuring instrument, the long trace profiler (LTP). Due
to the near linearity of the problem, the minimal set of data necessary for the tuning of each
bender, consists of only three slope traces measured before and after a single adjustment of each
bending couple. The data are analyzed with software realizing a method of regression analysis
with experimentally found characteristic functions of the benders. The resulting approximation to
the functional dependence of the desired shape provides nearly final settings. Moreover, the
characteristic functions of the benders found in the course of tuning, can be used for retuning to a
new desired shape without removal from the beamline and re-measuring. We perform a ray trace,
using profiler data for the finally tuned optics, predicting the performance to be expected during
use of the optics on the beamline.
1. Introduction
A primary goal of 3rd generation synchrotron light sources has been to achieve small spot
sizes, preserving the brightness of the source all the way along the beam line to the sample. Zone
plates,
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special x-ray lenses,
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and mirrors
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have been used successfully. At the Advanced Light
Source (ALS) the focusing is divided in the tangential and sagittal directions into two elliptically
cylindrical reflecting elements, the so-called Kirkpatrick-Baez (KB) pair.
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Because fabrication of
elliptical surfaces is complicated, the cost of directly fabricated tangential elliptical cylinders is
often prohibitive. This is in contrast to flat optics, that are simpler to manufacture and easier to
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measure by conventional interferometry. The figure of a flat substrate can be changed by placing
torques (couples) at each end. Equal couples form a tangential cylinder, and unequal couples can
approximate a tangential ellipse or parabola.
In Sec. 2, we review the nature of the bending, and propose a new technique for optimal
tuning of bendable mirrors before installation in the beamline. The technique adapts a method
previously used to adjust mirrors on synchrotron radiation beamlines.
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However, in our case,
optimal tuning of a bendable mirror is based on surface slope trace data obtained with a slope
measuring instrument, in our case, the long trace profiler (LTP). We show (Sec. 2) that due to the
near linearity of the bending problem, the minimal set of data necessary for tuning of two
benders, consists of only three slope traces measured before and after a single adjustment of each
bending couple. We provide an algorithm that was used in dedicated software for finding optimal
settings for the mirror benders. The algorithm is based on a method of regression analysis with
experimentally found characteristic functions of the benders. The resulting approximation to the
functional dependence of the desired slope shape provides nearly final settings for the benders.
Moreover, the characteristic functions of the benders found in the course of tuning, can be used
for retuning of the optics to a new desired shape without removing from the beamline and re-
measuring with the LTP. In Sec. 3 we provide a reduced form, but more intuitive implementation
of our method. In this case, we subdivide the mirror into three regions, fit a circle to each sub-
region, and also fit a circle to the entire surface. The near linear dependences of the found
curvatures on settings of the mirror benders allow rapid finding of the optimal settings via a
simple linear extrapolation that can be done just graphically. Even the reduced method allows
rapid iterative adjustment of both bending couples, and is typically much faster and more
accurate than a random walk accomplished by fitting the surface heights to an evolving elliptical
