ABSTRACT
Precision optics have been widely required in many advanced technological applications. X-ray mirrors, as an example, serve as the key optical components at synchrotron radiation and free electron laser facilities. They are rectangular silicon or glass substrates where a rectangular Clear Aperture (CA) needs to be polished to sub-nanometer Root Mean Squared (RMS) to keep the imaging capability of the incoming X-ray wavefront at the diffraction limit. The convolutional polishing model requires a CA to be extended with extra data, from which the dwell time is calculated via deconvolution. However, since deconvolution is very sensitive to boundary errors and noise, the existing surface extension methods can hardly fulfill the sub-nanometer requirement. On one hand, the figure errors in a CA were improperly modeled during the extension, leading to continuity issues along the boundary. On the other hand, uncorrectable high-frequency errors and noise were also extended. In this study, we propose a novel Robust Iterative Surface Extension (RISE) method that resolves these problems with a data fitting strategy. RISE models the figure errors in a CA with orthogonal polynomials and ensures that only correctable errors are fit and extended. Combined with boundary conditions, an iterative refinement of dwell time is then proposed to compensate the errors brought by the extension and deconvolution, which drastically reduces the estimated figure error residuals in a CA while the increase of total dwell time is negligible. To our best knowledge, RISE is the first data fitting-based surface extension method and is the first to optimize dwell time based on iterative extension. An experimental verification of RISE is given by fabricating two elliptic cylinders (10 mm × 80 mm CAs) starting from a sphere with a radius of curvature around 173 m using ion beam figuring. The figure errors in the two CAs greatly improved from 204.96 nm RMS and 190.28 nm RMS to 0.62 nm RMS and 0.71 nm RMS, respectively, which proves that RISE is an effective method for sub-nanometer level X-ray mirror fabrication.
ABSTRACT
With the rapid evolution of synchrotron x-ray sources, the demand for high-quality precision x-ray mirrors has greatly increased. Single nanometer shape accuracy is required to keep imaging capabilities at the diffraction limit. Ion beam figuring (IBF) has been used frequently for ultra-precision finishing of mirrors, but achieving the ultimate accuracy depends on three important points: careful alignment, accurate dwell time calculation and implementation, and accurate optical metrology. The Optical Metrology Group at National Synchrotron Light Source II has designed and built a position-velocity-time-modulated two-dimensional IBF system (PVT-IBF) with three novel characteristics: (1) a beam footprint on the mirror was used as a reference to align the coordinate systems between the metrology and the IBF hardware; (2) the robust iterative Fourier transform-based dwell time algorithm proposed by our group was applied to obtain an accurate dwell time map; and (3) the dwell time was then transformed to velocities and implemented with the PVT motion scheme. In this study, the technical aspects of the PVT-IBF systems are described in detail, followed by an experimental demonstration of the figuring results. In our first experiment, the 2D RMS in a $ 50\;{\rm mm} \times 5\;{\rm mm} $50mm×5mm clear aperture was reduced from 3.4 to 1.1 nm after one IBF run. In our second experiment, due to a 5 mm pinhole installed in front of the source, the 2D RMS in a $ 50\;{\rm mm} \times 5\;{\rm mm} $50mm×5mm clear aperture was reduced from 39.1 to 1.9 nm after three IBF runs, demonstrating that our PVT-IBF solution is an effective and deterministic figuring process.
ABSTRACT
With the rapid evolution of synchrotron X-ray sources, the demand for high-precision X-ray mirrors has greatly increased. Single nanometer profile error is required to keep imaging capability at the diffraction limit. Ion Beam Figuring (IBF), as a highly deterministic surfacing technique, has been used for ultra-precision finishing of mirrors. One crucial step that guides the IBF process is dwell time calculation. A valid dwell time solution should be non-negative and duplicate the shape of the desired removal map. Another important aspect is to minimize the total dwell time. In this study, we propose a Robust Iterative Fourier Transform-based dwell time Algorithm (RIFTA) that automatically fulfills these requirements. First, the thresholded inverse filtering in Fourier transform-based deconvolution is stabilized and automated by optimizing the threshold value using the Nelder-Mead simplex algorithm. Second, a novel two-level iterative scheme is proposed to guarantee the minimized total dwell time with its non-negativity at each dwell point. Third, a bicubic resampling is employed to flexibly adapt the calculated dwell time map to any IBF process intervals. The performance of RIFTA is first studied with simulation, followed by a comparison with the other state-of-the-art dwell time algorithms. We then demonstrate with an experiment that, using the dwell time calculated by the RIFTA, the total dwell time is shortened by a factor of two and the RMS in a 5 × 50 mm clear aperture was reduced from 3.4 nm to 1.1 nm after one IBF run, which proves the effectiveness and the efficiency of the proposed algorithm.
ABSTRACT
Stitching interferometry is performed by collecting interferometric data from overlapped sub-apertures and stitching these data together to provide a full surface map. The propagation of the systematic error in the measured subset data is one of the main error sources in stitching interferometry for accurate reconstruction of the surface topography. In this work, we propose, using the redundancy of the captured subset data, two types of two-dimensional (2D) self-calibration stitching algorithms to overcome this issue by in situ estimating the repeatable high-order additive systematic errors, especially for the application of measuring X-ray mirrors. The first algorithm, called CS short for "Calibrate, and then Stitch", calibrates the high-order terms of the reference by minimizing the de-tilted discrepancies of the overlapped subsets and then stitches the reference-subtracted subsets. The second algorithm, called SC short for "Stitch, and then Calibrate", stitches a temporarily result and then calibrates the reference from the de-tilted discrepancies of the measured subsets and the temporarily stitched result. In the implementation of 2D scans in x- and y-directions, step randomization is introduced to generate nonuniformly spaced subsets which can diminish the periodic stitching errors commonly observed in evenly spaced subsets. The regularization on low-order terms enables a highly flexible option to add the curvature and twist acquired by another system. Both numerical simulations and experiments are carried out to verify the proposed method. All the results indicate that 2D high-order repeatable additive systematic errors can be retrieved from the 2D redundant overlapped data in stitching interferometry.
ABSTRACT
Ion-beam figuring (IBF) is a precise surface finishing technique used for the production of ultra-precision optical surfaces. In this study, we propose an effective one-dimensional IBF (1D-IBF) method approaching sub-nanometer root mean square (RMS) convergence for flat and spherical mirrors. Our process contains three key aspects. First, to minimize the misalignment of the coordinate systems between the metrology and the IBF hardware, a mirror holder is used to integrate both the sample mirror and the beam removal function (BRF) mirror. In this way, the coordinate relationship can be calculated using the measured BRF center. Second, we propose a novel constrained linear least-squares (CLLS) dwell time calculation algorithm combined with a coarse-to-fine scheme to ensure that the resultant nonnegative dwell time closely and smoothly duplicates the required removal amount. Third, considering the possible errors induced by the translation stage, we propose a dwell time slicing strategy to divide the dwell time into smaller time slices. Experiments using our approaches are performed on flat and spherical mirrors as demonstrations. Measurement results from the nano-accuracy surface profiler (NSP) show that the residual profile errors are reduced to sub-nanometer RMS for both types of mirrors while the surface roughness is not affected by the figuring process, demonstrating the effectiveness of the proposed 1D-IBF method for 1D high-precision optics fabrication.
ABSTRACT
In this paper, a two-dimensional stitching interferometry system using two tiltmeters is proposed. During the scanning and the measurement, one tiltmeter stays with the interferometer and the other one is attached to the translation stage where the surface under test is placed. The differences of the x- and y-tilt readings between these two tiltmeters are recorded as the relative tilt between interferometer and surface under test. The relative tilt in both x- and y-directions are used to correct the surface tip/tilt of each subset, and then the piston is adjusted to get the final stitching surface map. As an example, a stitching result of a 125mm-long mirror surface is presented. The repeatability of our current stitching system is about 1.48 nm RMS. The stitching result is compared to the result of a one-dimensional angular-measurement-based stitching method to discuss the merits and limitation of the proposed method.
