Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data

Enhanced-spatial-resolution optical surface profiler based on focusing deflectometry

Deflectometric slope profiler is an essential technique for accessing the surface metrology of mirrors used in synchrotron radiation beamlines. To increase the upper spatial frequency bandwidth limits of deflectometric slope profiler, reducing the beam spot size on the mirrors is necessary. In this paper, we introduce a profiler system: the focusing long trace profiler (FLTP). It contains a newly developed optical head capable of raising upper spatial frequency bandwidth limits by using a focused beam instead of a collimated beam to scan the sample. This feature has been proven in a numerical simulation experiment, where a spatial resolution of up to around 0.05 mm was reached when the sample is set at focus plane. The system is implemented and characterized in several experiments; calibration of the focusing optical head shows that it can achieve a high angular accuracy of sub-50 nrad root-mean-square (rms) and defocusing of sample under test (SUT) has no effect on the measurement results; the measurement tests also demonstrate the system's advantage in highly curved mirror profile metrology.

Surface shape determination with a stitching Michelson interferometer and accuracy evaluation

Stitching methods are increasingly used for determining the surface shape of large and high precision optical elements used in synchrotron beamlines. They consist in reconstructing the surface topography from multiple measurements on overlapping parts of the optics aperture by various algorithms. This paper is an attempt to investigate how true and accurate such a reconstruction can be. Error sources are identified and evaluated throughout the acquisition and processing steps. The analysis is based on the example SOLEIL Michelson interferometer for nano-topography, a dedicated measurement bench for stitching interferometry. We propose a method for determining the error made on the estimate of the interferometric reference surface from the stitching dataset. This determination is made before and independently of the stitching procedure itself.

Absolute, high-accuracy characterization of a variable line spacing grating for the European XFEL soft X-ray monochromator

The European XFEL is a free-electron laser (FEL) with a superconducting linear accelerator and three beamlines (SASE1, SASE2, and SASE3) covering the energy range from 250 eV to 24 keV. The SASE3 beamline is dedicated to the soft x-ray range (0.25-3 keV) and is designed to operate in both monochromatic and non-monochromatic mode. A variable line spacing - plane grating (VLS-PG) monochromator is placed along the beam transport system for the monochromatic mode. The VLS parameters of the grating profile are challenging from a manufacturing and measuring perspective, especially at the desired length of 530 mm. A shorter grating of 150 mm has been procured to allow early operation of the facility. We describe the characterization method that was used to assess the VLS parameters of the grating using Fizeau interferometry. The method is intrinsically absolute and limited only by the quality of the test grating and the noise level. Further measurements using a white-light-interferometry profilometer are also reported. We discuss the possibility of extending the method to the future 530 mm long grating.

Self-calibration of Fizeau interferometer and planar scale gratings in Littrow setup

A new method, in which the wavefronts of the zero-order and the positive and negative first-order diffracted beams from a planar scale grating in Littrow setup are analyzed by a Fizeau interferometer, is proposed to evaluate the Z-directional out-of-flatness as well as the X- and Y-directional pitch deviations of the planar scale grating over a large area. Meanwhile, the surface profile of the reference optical flat in the Fizeau interferometer can also be determined in a much simpler and more efficient approach than the commonly used liquid-flat reference and three-flat test calibration methods. Simulations are presented to verify the feasibility of the proposed method.

Calibration and optimization of an x-ray bendable mirror using displacement-measuring sensors

We propose a method to control and to adjust in a closed-loop a bendable x-ray mirror using displacement-measuring devices. For this purpose, the usage of capacitive and interferometric sensors is investigated and compared. We installed the sensors in a bender setup and used them to continuously measure the position and shape of the mirror in the lab. The sensors are vacuum-compatible such that the same concept can also be applied in final conditions. The measurement is used to keep the calibration of the system and to create a closed-loop control compensating for external influences: in a demonstration measurement, using a 950 mm long bendable mirror, the mirror sagitta is kept stable inside a range of 10 nm Peak-To-Valley (P-V).

Large aperture Fizeau interferometer commissioning and preliminary measurements of a long x-ray mirror at European X-ray Free Electron Laser

The European XFEL (X-ray Free Electron Laser) is a large facility under construction in Hamburg, Germany. It will provide a transversally fully coherent x-ray radiation with outstanding characteristics: high repetition rate (up to 2700 pulses with a 0.6 ms long pulse train at 10 Hz), short wavelength (down to 0.05 nm), short pulse (in the femtoseconds scale), and high average brilliance (1.6 ⋅ 10(25) (photons s(-1) mm(-2) mrad(-2))/0.1% bandwidth). The beam has very high pulse energy; therefore, it has to be spread out on a relatively long mirror (about 1 m). Due to the very short wavelength, the mirrors need to have a high quality surface on their entire length, and this is considered very challenging even with the most advanced polishing methods. In order to measure the mirrors and to characterize their interaction with the mechanical mount, we equipped a metrology laboratory with a large aperture Fizeau interferometer. The system is a classical 100 mm diameter commercial Fizeau, with an additional expander providing a 300 mm diameter beam. Despite the commercial nature of the system, special care has been taken in the polishing of the reference flats and in the expander quality. We report the first commissioning of the instrument, its calibration, and performance characterization, together with some preliminary results with the measurement of a 950 mm silicon substrate. The intended application is to characterize the final XFEL mirrors with nanometer accuracy.

Absolute flatness testing of skip-flat interferometry by matrix analysis in polar coordinates

A new method utilizing matrix analysis in polar coordinates has been presented for absolute testing of skip-flat interferometry. The retrieval of the absolute profile mainly includes three steps: (1) transform the wavefront maps of the two cavity measurements into data in polar coordinates; (2) retrieve the profile of the reflective flat in polar coordinates by matrix analysis; and (3) transform the profile of the reflective flat back into data in Cartesian coordinates and retrieve the profile of the sample. Simulation of synthetic surface data has been provided, showing the capability of the approach to achieve an accuracy of the order of 0.01 nm RMS. The absolute profile can be retrieved by a set of closed mathematical formulas without polynomial fitting of wavefront maps or the iterative evaluation of an error function, making the new method more efficient for absolute testing.

Characterization of a piezo bendable X-ray mirror

A full-scale piezo bendable mirror built as a prototype for an offset mirror at the European XFEL is characterized. The piezo ceramic elements are glued onto the mirror substrate, side-face on with respect to the reflecting surface. Using a nanometre optical component measuring machine and a large-aperture Fizeau interferometer, the mirror profile and influence functions were characterized, and further analysis was made to investigate the junction effect, hysteresis, twisting and reproducibility.

Absolute measurement of optical flats based on basic iterative methods

Generalized basic iterative methods for absolute measurement of optical flats are presented. They are based on iterative scheme and can be classified into block Jacobi Successive Over-relaxation (SOR) method and block SOR method. Both methods are effective for solving the three-flat problem with pixel-level spatial resolution, without the usage of a fitting procedure. Compared to the block Jacobi SOR method, the block SOR method with an inexpensive choosing of relaxation factor (such as ω = 1.5) converges much faster and saves more computational costs and memory space without reducing accuracy. It has been proved by both simulation results and experimental results. The proposed basic iterative methods are generalized; can correctly reconstruct absolute figures with pixel-level spatial resolution; are easy to understand and implement; and computationally efficient.

Iterative algorithm for absolute planarity calibration in three-flat test

An iterative algorithm is presented to calibrate the shapes of the reference surfaces in three-flat test. Three measurements in three-flat test determine the even part of the flat surfaces. The odd part can be obtained by adding one or more rotation measurements and using analytical recovery algorithm. Here we present an iterative algorithm to obtain the odd part instead of special analytical recovery algorithm. The iterative algorithm is purely numerical, and only involved in data rotation operation. A simulation experiment proves the validity and high accuracy of the proposed algorithm method.

Absolute surface metrology by rotational averaging in oblique incidence interferometry

A modified method for measuring the absolute figure of a large optical flat surface in synchrotron radiation by a small aperture interferometer is presented. The method consists of two procedures: the first step is oblique incidence measurement; the second is multiple rotating measurements. This simple method is described in terms of functions that are symmetric or antisymmetric with respect to reflections at the vertical axis. Absolute deviations of a large flat surface could be obtained when mirror antisymmetric errors are removed by N-position rotational averaging. Formulas are derived for measuring the absolute surface errors of a rectangle flat, and experiments on high-accuracy rectangle flats are performed to verify the method. Finally, uncertainty analysis is carried out in detail.