Open Access
Add to BibSonomy
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.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As EUV lithography and the 3rd and 4th generation X-ray synchrotron sources evolve, the mirror smoothness and precision requirement have dramatically increased. Single nanometer-level accuracy is required to avoid destruction of the incoming wave front or keeping imaging capabilities at the diffraction limit [1,2]. In the past decade, ion-beam figuring (IBF) [3–17], as a highly deterministic method for the final ultra-precision finishing of optical surfaces, has been successfully applied to provide both flat and curved X-ray mirrors with surface profiles reaching nanometer-level accuracy [8,9,11–13].
IBF removes material from an optical surface at atomic level by sputtering. IBF’s main advantages are its non-contact nature, no load force, low surface and subsurface damage, and low edge effects over conventional figuring methods [12,15,16]. One-dimensional IBF (1D-IBF) can be applied to figure 1D X-ray optics such as Kirkpatrick-Baez mirrors, and it is much simpler than its two-dimensional (2D) counterpart [7,14,17] since it requires only one degree of freedom. Therefore, as shown in Fig. 1(a), a 1D-IBF system has been easily integrated into the multilayer deposition system [18] designed and developed by the National Synchrotron Light Source II (NSLS-II) optical metrology and fabrication group at Brookhaven National Laboratory (BNL). In order to provide an approximate 1D removal function, as shown in Fig. 1(b), a 10 mm × 40 mm rectangular grid is installed to modify the shape of the ion beam.
Fig. 1 The 1D-IBF hardware used in this study. (a) The multi-layer deposition system at NSLS-II optical metrology and fabrication group. (b) The rectangular grid installed to modify the shape of the ion beam.
Figure 2 schematically illustrates the process used for the 1D-IBF. Prior to any 1D-IBF machining, a beam removal function (BRF), which describes a depth removal rate as a function of radial distance from the ion beam center, must be determined. Several footprints are bombarded on a BRF mirror (a BRF mirror refers to the mirror which is used to estimate BRF based on the bombarded footprints) with different etching times [11–13] and the BRF is derived by curve-fitting the footprints using a Gaussian-like function. The BRF only needs to be calculated once for one experiment thanks to the high stability and high linearity of ion beam source [11]. A desired removal curve is the difference between the measured surface profile and a desired surface profile. From the deconvolution of the removal curve and the BRF, a dwell time vector is calculated, which describes the length of the time the ion beam should reside at each machining point. A Computer Numerical Control (CNC) code is then generated to control the relative motions between the sample mirror and the ion beam. As shown in Fig. 2, the 1D-IBF is always implemented as a measurement-and-fabrication loop until the surface residual error is below a certain threshold.
Fig. 2 Schematic illustration of the flow of the 1D-IBF. The BRF is only calculated once and kept invariant for the entire 1D-IBF process. Steps 1 to 5 are performed in a loop.
Our preliminary research works have attempted to figure both flat and spherical grazing-incidence reflective mirrors using this 1D-IBF system [11–13]. The experiment results demonstrated that the figure errors of the flat mirrors were reduced to < 1 nm root mean square (RMS) while the figure error of the spherical mirror was reduced from 21.5 nm to 1.4 nm RMS [11]. In parallel, a new slope-based figuring method [13] was proposed to perform 1D-IBF based on a mirror’s slope profile. In this study, we further investigate typical features that have significant impacts on the correction ability of the 1D-IBF system, and propose a new effective 1D-IBF method considering the following three key aspects.
First, as IBF must be performed in high vacuum, in-situ measurement of the sample mirror can hardly be achieved with the desirable flexibility. For example, synchrotron mirrors must be used in their beamline orientation horizontally or vertically for gravity compensation. This is why ex-situ metrology is a preferred choice. The ex-situ measurement, however, brings the problem of the misalignment (or uncertainty) of coordinates between the metrology and the 1D-IBF hardware. As shown in Fig. 2, the 1D-IBF process requires iterative transformation between these two coordinate systems. The error accumulation will become disastrous to the IBF machining accuracy if they cannot be well calibrated [19]. In this study, we propose to mount the sample mirror and the BRF mirror into a single plate (see Fig. 7). Therefore, the measured BRF center can be used as a reference to calibrate the coordinate correspondence between the two systems.
Second, dwell time calculation, as shown in step 3 in Fig. 2, is the most significant step in a 1D-IBF process. Dwell time calculation is a deconvolution process, which is an ill-posed inverse problem and may not have a unique solution. Difficulties in obtaining a reasonable dwell time solution include two aspects: i) it must be nonnegative, ii) it should closely and smoothly duplicate the desired removal curve. We found that both requirements can be fulfilled by modeling the deconvolution process as a constrained linear least-squares (CLLS) problem. The nonnegativity is ensured by the lower-bound constraint while the smoothness is guaranteed by setting inequality constraints on the maximum difference between each two consecutive elements in the dwell time vector. Additionally, a coarse-to-fine calculation scheme is added to improve the stability of the CLLS algorithm.
Third, smoother movements of translation stages are preferred in the real 1D-IBF machining process. Therefore, in this study, instead of implementing the dwell time vector in a point-by-point manner, we propose a dwell time slicing strategy to divide each element of the dwell time vector into smaller time slices, e.g. 1 s. The ion beam then dwells only one slice a time at each machining point and moves back and forth to complete the entire process in multiple scans.
The rest of the paper is organized as follows. Section 2 briefly explains the principle of the 1D-IBF process. In Section 3, details of the three key aspects in implementing the proposed 1D-IBF method are described, followed by experimental verification of its effectiveness in Section 4. Section 5 concludes the paper.
2. Principle of the 1D-IBF process
A 1D-IBF process is schematically illustrated in Fig. 3. The dwell time is implemented by scanning the sample mirror along the x axis. At a certain machining point, the desired removal amount can be expressed as
where R(x) is the desired removal curve (see Fig. 2), B(x) is the 1D BRF, and T (x) is the dwell time; “*” refers to the convolution operation. As mentioned in Section 1, both B (x) and R (x) are known. While B(x) is learned using ion beam footprints on a BRF mirror, R(x) is calculated as the difference between the measured surface profile Zm (x) and the desired surface profile Zd (x) asThus, the calculation of the dwell time function T (x) is a deconvolution process. To solve T(x), Eq. (1) is always discretized as
for k = 0, 1, ..., Nr − 1, where Nr is the number of total points on the desired removal curve, xk is the kth machining point, Nt is the total number of points on the dwell time vector, and B (xk − ξi) is the material removed per unit time at point xk when the center of the BRF dwells at point ξi. It can be observed from Eq. (3) that the 1D-IBF machining and the metrology hardware can have different sampling intervals with this kind of discretization, which increase the flexibility of the figuring process [6]. Equation (3) can be re-written in matrix form as Equation (4) indicates that the dwell time vector t can be obtained by solving a system of linear equations. However, B is always ill-conditioned and the nonnegativity of t cannot be guaranteed. The solution for these problems and other concerns in solving Eq. (4) are given in Section 3.2.Fig. 3 Schematic of a 1D-IBF process along the x axis, where B(x) represents the Gaussian-like BRF and R(x) is the desired removal curve.
3. Implementation of the 1D-IBF method
In this section, the method used to calibrate the coordinate correspondence between the metrology and the 1D-IBF hardware is illustrated, followed by a detailed explanation of the CLLS algorithm and the coarse-to-fine scheme used to solve Eq. (4). Afterwards, the dwell time slicing strategy is introduced.
3.1. Calibration of coordinate correspondence between the metrology and the 1D-IBF hardware
The 1D-IBF process shown in Fig. 2 indicates that a successful figuring experiment relies on not only the high accuracy of both the 1D-IBF and the metrology hardware, but also the precise calibration of the coordinate correspondence between them. In the previous 1D-IBF research works [9,11–13], a Fizeau interferometer [12], a long trace profiler (LTP) [9], and a 2D Stitching Shack Hartmann (SSH) slope measuring system [11–13] have been used as the metrology hardware in 1D-IBF for their proven high accuracy in height and slope measurement. Although satisfactory results have been obtained, details of how to handle the coordinate discrepancies between the metrology hardware and the 1D-IBF hardware, which is a very significant error source [19], were not clear.
Basically, markers should be used to ensure that the sample mirror is installed at the same positions of the two hardware during every measurement-and-fabrication cycle, however, it is not always possible to put markers on usually a very expensive mirror. Generally, in IBF, another BRF mirror is used to extract the Gaussian-like BRF from the bombarded footprints. These footprints indeed can potentially be used as good calibration markers.
As schematically shown in Fig. 4, we propose to use a single plate to integrate both a BRF mirror and a sample mirror and use the BRF center as a marker to calibrate the two coordinate systems. In this work, only one footprint is used to extract the BRF due to the proven excellent stability and linearity of the ion beam system.
According to Fig. 4, points A, B, O, and C represent the mirror edges of the BRF mirror and the sample mirror, respectively, and point O is set as the origin of the mirror coordinate. Initially, the center of the footprint bombarded on the BRF mirror is guessed at a distance of gfp with respect to O. The coordinate correspondence can be established by calibrating the difference Δ between gfp and the measured footprint center as
where mfp is the distance between A and the learned BRF center, and mAO is the measured distance between A and O.3.2. CLLS dwell time calculation algorithm with a coarse-to-fine scheme
Numerous dwell time calculation algorithms have been proposed in the literature and they can be divided into four classes, Fourier transform-based algorithm [3, 5], iterative algorithm [4, 20], matrix-based algorithms [6,14,21], and Bayesian-based algorithm [22].
In the Fourier transform-based algorithm [3,5], the deconvolution is transferred to frequency domain by Fourier transform as a pointwise division. However, this algorithm is unstable because the divisor (i.e. the Fourier transform of the BRF) is zero at certain frequencies and the discrete Fourier transform expects the input to be periodic at the boundary [3]. Therefore, a thresholded inverse filter and a Band Limited Surface Extrapolation (BLSE) must be employed to overcome these problems and obtain proper results.
The iterative algorithm keeps updating the dwell time vector until the difference between the calculated removal curve and the desired surface profile is below a certain threshold [4,20]. This algorithm is simple to implement. However, it is also unstable and fails to converge to a unique solution [14].
The matrix-based algorithms are straightforward since the discrete convolution can be transferred to matrix operations (see Eq. (4)) and the devolution can be solved as a linear least-squares problem. The main advantage of the matrix-based algorithms is that it does not require that the data generated by metrology hardware and IBF machining must have the same resolution [6, 14]. Nevertheless, the matrix built in the algorithm is rank-deficient so that the solution is not unique. To solve this problem, Singular Value Decomposition (SVD) was employed to obtain a unique solution in the minimum 1-norm sense [6, 14]. A Truncated-SVD (TSVD) method has also been attempted to achieve a finer control of the removal amount by tuning the truncation parameter k [21], however, SVD is a very computational expensive solver when the matrix scale is large. To solve this problem, a Least-Squares QR (LSQR) algorithm, which has been developed to solve large, sparse, and rank-deficient linear systems, was used [6,14,21].
A significant problem of all the three classes of algorithms is that the nonnegativity of the dwell time vector cannot be automatically guaranteed. In real applications, an additional removal amount is manually added to the calculated dwell time to offset its negative elements. Jiao et al. proposed a Bayesian-based iterative optimization algorithm to satisfy the nonnegativity requirement of the dwell time [22]. The Bayesian-based updating ensures that the dwell time is nonnegative if the initial guess is nonnegative. Moreover, they found that variation of dwell time along the machining direction would introduce an implementation error which was proportional to the square of the derivative of the dwell time function along that direction. Smoothing of the dwell time can reduce this error so that a total variation norm based on the gradient of the dwell time was added to the Bayesian-based optimization.
3.2.1. The CLLS dwell time calculation algorithm
We propose an improved matrix-based dwell time calculation algorithm taking the advantages of both the conventional matrix-based algorithms and the Bayesian-based algorithm. Our goal is to ensure that the nonnegativity and the smoothness of the dwell time function can be fulfilled at the same time while the metrology hardware and the 1D-IBF hardware do not need to have the same resolution.
The solution of Eq. (4) can be modeled as a CLLS problem by introducing the nonnegativity and smoothness requirements as the lower-bound and the inequality constraints of t as
where where bi is the maximum absolute dwell time difference between each two consecutive machining positions i and i + 1 for i = 0, 1, 2, ..., NIBF − 2, where NIBF is the total number of the 1D-IBF machining points. Alternatively, Eq. (6) can be rewritten as a general quadratic programming model by letting H = B⊤B and q = −B⊤r as and a multi-threaded quadratic programming solver called qpOases [23] can be used if the problem size is NIBF is large and speed is a big performance concern.It is worth noting that, however, direct usages of the CLLS algorithm may result in inappropriate solutions when the dwell time difference constraints in b are too strict while the shape changes in the desired removal curve r are large. As an example, Fig. 5(c) shows the dwell time vector calculated with inequality constraints where b = 1⊤. Obviously, the dwell time vector can hardly duplicate the shape of the desired removal curve shown in Fig. 5(a) since the constraints are too strict. We propose a coarse-to-fine scheme to solve this problem.
Fig. 5 Desired removal curve and dwell time vectors calculated by three different methods. (a) Desired removal curve. (b) Dwell time vector calculated without the inequality constraints. (c) Dwell time vector calculated with inequality constraints using b = 1⊤. (d) Dwell time vector calculated with the coarse-to-fine scheme.
3.2.2. Coarse-to-fine scheme
Instead of applying the CLLS algorithm to calculate dwell time directly, we seperate the calculation into two levels. On the coarse level, the inequality constraints in Eq. (4) are disabled and an initial dwell time vector tini is calculated, which is always not smooth (see Fig. 5(b)). Therefore, a polynomial fitting is then performed on tini to smooth it and obtain a new fitted dwell time vector tcoarse. Afterwards, tcoarse is substituted back to Eq. (4) and the corresponding rcoarse is calculated. On the fine level, the residual rfine = r − rcoarse is substituted into Eq. (4) and the corresponding residual dwell time vector tfine can be calculated. At last, the final dwell time vector t is calculated as t = tcoarse + tfine. Figure 5(d) shows the dwell time vector calculated using this coarse-to-fine scheme, which is much better than the other two calculation methods (Figs. 5(b) and 5(c)) in terms of smoothly duplicating the desired removal curve.
3.3. Dwell time slicing strategy
Now we have the nonnegative and smoothed dwell time vector t. However, the smooth implementation of t in the real 1D-IBF machining is also preferred to obtain desired figuring results.
Conventionally, t was implemented in a point-by-point way such that the ion beam was moved to each machining point i and kept there for the entire dwell time ti [11–13]. The entire figuring process is thus completed in just a single scan from one end to the other of the sample mirror. We found that, however, smoother movements of the translation stage are preferred to obtain better figuring results.
Fig. 7 Mechanical Plates used to hold both the BRF mirror and the sample mirror, where A, B, O, and C are mirror edges. The scanning direction is from A to B. (a) Left: the circular flat BRF mirror; right: the circular flat sample mirror. (b) Left: the circular flat BRF mirror; right: the rectangular spherical sample mirror.
Therefore, we propose to divide the dwell time at each machining point into smaller time slices and complete the figuring process in multiple scans to smooth the motion of the translation stage. Assuming that the unit time slice used in this study is τ. Before the machining, the maximum dwell time tmax in t is found and utilized to determine the total number of scans S = ⌈tmax/τ⌉. During each scan, the maximum dwell time at each machining point is thus τ seconds, which makes the motion of the translation stage much smoother. In practice, τ should be chosen based on S, which is often a big number when the shape error to be corrected is large. Therefore, it is recommended to use larger τ in the first several runs of the 1D-IBF process and reduce its value when the measured surface profile is getting closer to the desired surface profile.
4. Experiments
The 1D-IBF experiments are performed on a 100 mm-diameter circular flat sample mirror and a 100 mm × 30 mm rectangular spherical sample mirror (R ≈ 30 m), which are mounted on mirror plates with their corresponding BRF mirrors as shown in Figs. 7(a) and 7(b). Both BRF mirrors are 100 mm-diameter circular flat Si mirrors. These experiments are performed on the same 1D-IBF hardware introduced in [11–13] with Ar ion energy 800 eV and ion current 27 mA.
4.1. BRF learning
One footprint is etched on each of the BRF mirrors in the flat-flat plate and the flat-sphere plate with a dwell time of 20 s. An innovative 1D Nano-accuracy Surface Profiler (NSP) with sub-50 nrad RMS slope measurement accuracy is used to measure the footprint [24]. It is worth noting that the slope measurement results obtained from the NSP are integrated using a cubic spline integration since the height-based figuring method is used in this study.
The BRF model with three unknowns used to fit the footprints is defined as
where a is the maximum removal rate with the unit of nm/s and σ denotes the standard deviation. As an example, the measured and learned BRF parameters for the BRF mirror in flat-sphere plate is shown in Fig. 6, in which a = 0.39 nm/s, mfp = 48.0 mm, and σ = 3 mm.4.2. Calibration of the coordinate correspondence between the NSP and the 1D-IBF hardware
Additionally, one specific advantage of the NSP is that it can provide very accurate measurement of the exact positions of the mirror edges. The edge positions shown in Fig. 7 can be directly read from a high accuracy encoder of the translation stage and mAO that used to calculate Δ in Eq. (5) can be obtained.
The calibration for the two plates are shown in Figs. 7(a) and 7(b), where Δ = −0.1 mm and Δ = 2.1 mm for the flat-flat plate and the flat-sphere plate, respectively.
4.3. 1D-IBF results on flat sample mirror
The first experiment is performed on a 100 mm-diameter circular flat Si mirror as shown in Fig. 7(a) with a clear aperture of 80 mm. Figures 8(b) and 8(d) show that the initial slope error is 0.7 μrad RMS and 2.3 μrad peak-to-valley (PV), and the corresponding initial height error is 8.2 nm RMS and 24.8 nm (PV). Figures 8(a) and 8(c) show the two iterations done on the sample mirror using the 1D-IBF process. The final slope error is reduced to 0.3 μrad RMS and 1.5 μrad PV, and the corresponding final height error is reduced to 0.8 nm RMS (factor of 10 improvement) and 3.3 nm PV.
Fig. 8 Results of two iterations of the 1D-IBF process on a flat sample mirror. (a) The slope errors. (b) The RMS and PV values of the slope errors. (c) The height errors. (d) The RMS and PV values of the height errors.
4.4. 1D-IBF results on spherical sample mirror
The second experiment is performed on a 100 mm-long rectangular spherical Si mirror as shown in Fig. 7(b) with a clear aperture of 80 mm. The initial measured radius of curvature (ROC) is 30.29 m. This measured radius is set as a target for all the following experiments. As shown in Figs. 9(b) and 9(d), the initial slope error is 2.91 μrad RMS and 13.74 μrad (PV), and the corresponding initial height error is 30.11 nm RMS and 98.08 nm (PV). Figures 9(a) and 9(c) show that the sample mirror is processed four times. The final achieved slope error is 0.66 μrad RMS and 3.62 μrad PV, and the corresponding final height error is 0.95 nm RMS (factor of 30 improvement) and 4.72 nm PV. Figure 10 shows the measured ROC of the sample mirror after each 1D-IBF run, indicating that the desired ROC is kept constant after each 1D-IBF process.
Fig. 9 Results of two iterations of the 1D-IBF process on a spherical sample mirror. (a) The slope errors. (b) The RMS and PV values of the slope errors. (c) The height errors. (d) The RMS and PV values of the height errors.
Furthermore, the surface roughness of the rectangular spherical sample mirror before and after the 1D-IBF process is investigated using a Zygo New View white-light microscopic interferometer. Three points of interest (POIs) located at the beginning, the middle, and the end region of the mirror along the machining direction are measured. The results in RMS before and after the 1D-IBF process are 0.31 nm, 0.31 nm, and 0.34 nm versus 0.32 nm, 0.34 nm, and 0.34 nm, respectivly, which demonstrate that the surface roughness of the mirror is almost not affected by the 1D-IBF process.
5. Discussion
We further study the improvement of the surface quality and the removal capability of the proposed 1D-IBF method using Power Spectrum Density (PSD) analysis. Figure 11 shows the PSD curves of the rectangular spherical sample mirror before and after each 1D-IBF process. It can be observed that the PSDs keep decreasing at frequencies f ≤ 10 mm−1 during the 1D-IBF processes, indicating that the 1D-IBF method is capable of figuring surface with errors in low to middle frequencies. If removal of higher-frequency surface errors is required, a narrower BRF should be used.
6. Conclusion
In this study, we propose an effective 1D-IBF method with sub-nanometer RMS. The method contains three key aspects which are significant to obtain the desired figuring results. First, the uncertainty caused by the coordinate system misalignment of the metrology and the 1D-IBF hardware is resolved by integrating a BRF mirror and a sample mirror into one single plate. The extracted BRF center position can then be used as a reference to calibrate the two coordinate systems. A dwell time calculation algorithm based on CLLS combined with a coarse-to-fine scheme is proposed to solve the nonnegativity and smoothness issues of a dwell time function simultaneously. Third, a dwell time slicing strategy is employed to also smooth the implementation of the dwell time function in the 1D-IBF machining process. The proposed 1D-IBF method is examined on one flat mirror and one spherical mirror. The final figure errors of both mirrors reach < 1 nm RMS, which proves the effectiveness of the proposed 1D-IBF method in finishing 1D high-precision optics.
Funding
Department of Energy (DOE) Office of Science (DE-SC0012704); Brookhaven National Laboratory (BNL LDRD 17-016).
Acknowledgments
This research was performed at the Optical Metrology Laboratory at the National Synchrotron Light Source II, a U.S. Department f Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. This work is performed under the BNL LDRD 17-016 “Diffraction limited and wavefront preserving reflective optics development.”
References
1. K. Yamauchi, H. Mimura, S. Matsuyama, H. Yumoto, T. Kimura, Y. Takahashi, K. Tamasaku, and T. Ishikawa, “Focusing mirror for coherent hard x-rays,” in Synchrotron Light Sources and Free-Electron Lasers: Accelerator Physics, Instrumentation and Science Applications, (Springer International Publishing, 2016).
2. K. Yamauchi, H. Mimura, T. Kimura, H. Yumoto, S. Handa, S. Matsuyama, K. Arima, Y. Sano, K. Yamamura, K. Inagaki, H. Nakamori, J. Kim, K. Tamasaku, Y. Nishino, M. Yabashi, and T. Ishikawa, “Single-nanometer focusing of hard x-rays by kirkpatrick–baez mirrors,” J. Physics: Condens. Matter 23, 394206 (2011).
3. S. Wilson and J. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–325 (1987). [CrossRef]
4. L. N. Allen and H. W. Romig, “Demonstration of an ion-figuring process,” Proc. SPIE 1333, 22–34 (1990). [CrossRef]
5. S. R. Wilson, D. W. Reicher, C. Kranenberg, J. R. McNeil, P. L. White, P. M. Martin, and D. E. McCready, “Ion beam milling of fused silica for window fabrication,” Proc. SPIE 1441, 82–86 (1991). [CrossRef]
6. C. L. Carnal, C. M. Egert, and K. W. Hylton, “Advanced matrix-based algorithm for ion-beam milling of optical components,” Proc. SPIE 1752, 54–63 (1992). [CrossRef]
7. T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995). [CrossRef]
8. A. Schindler, T. Hänsel, F. Frost, R. Fechner, A. Nickel, H.-J. Thomas, H. Neumann, and D. Hirsch, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, (Optical Society of America, 2002), p. OTuB5. [CrossRef]
9. I. Preda, A. Vivo, F. Demarcq, S. Berujon, J. Susini, and E. Ziegler, “Ion beam etching of a flat silicon mirror surface: A study of the shape error evolution,” Nucl. Instruments Methods Phys. Res. Sect. A 710, 98–100 (2013). [CrossRef]
10. T. Arnold and F. Pietag, “Ion beam figuring machine for ultra-precision silicon spheres correction,” Precis. Eng. 41, 119–125 (2015). [CrossRef]
11. M. Idir, L. Huang, N. Bouet, K. Kaznatcheev, M. Vescovi, K. Lauer, R. Conley, K. Rennie, J. Kahn, R. Nethery, and L. Zhou, “A one-dimensional ion beam figuring system for x-ray mirror fabrication,” Rev. Sci. Instrum. 86, 105120 (2015). [CrossRef] [PubMed]
12. L. Zhou, M. Idir, N. Bouet, K. Kaznatcheev, L. Huang, M. Vescovi, Y. Dai, and S. Li, “One-dimensional ion-beam figuring for grazing-incidence reflective optics,” J. Synchrotron Radiat. 23, 182–186 (2016). [CrossRef]
13. L. Zhou, L. Huang, N. Bouet, K. Kaznatcheev, M. Vescovi, Y. Dai, S. Li, and M. Idir, “New figuring model based on surface slope profile for grazing-incidence reflective optics,” J. Synchrotron Radiat. 23, 1087–1090 (2016). [CrossRef] [PubMed]
14. J. F. Wu, Z. W. Lu, H. X. Zhang, and T. S. Wang, “Dwell time algorithm in ion beam figuring,” Appl. Opt. 48, 3930–3937 (2009). [CrossRef] [PubMed]
15. M. Demmler, M. Zeuner, F. Allenstein, T. Dunger, M. Nestler, and S. Kiontke, “Ion beam figuring (ibf) for high precision optics,” Proc. SPIE 7591, 75910Y (2010). [CrossRef]
16. M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instruments Methods Phys. Res. Sect. B 267, 1390–1393 (2009). [CrossRef]
17. X. Xie and S. Li, “Ion beam figuring technology,” in Handbook of Manufacturing Engineering and Technology, (SpringerLondon, 2015), pp. 1343–1390.
18. R. Conley, N. Bouet, J. Biancarosa, Q. Shen, L. Boas, J. Feraca, and L. Rosenbaum, “The nsls-ii multilayer laue lens deposition system,” Proc. SPIE 7448, 74480U (2009). [CrossRef]
19. S. Li and Y. Dai, Large and Middle-Scale Aperture Aspheric Surfaces: Lapping, Polishing and Measurement (John Wiley & Sons, 2017).
20. H. Cheng, Independent Variables for Optical Surfacing Systems (Springer, 2016).
21. L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Model and method to determine dwell time in ion beam figuring,” Nami Jishu yu Jingmi Gongcheng 5, 107–112 (2007).
22. C. Jiao, S. Li, and X. Xie, “Algorithm for ion beam figuring of low-gradient mirrors,” Appl. Opt. 48, 4090–4096 (2009). [CrossRef] [PubMed]
23. H. J. Ferreau, C. Kirches, A. Potschka, H. G. Bock, and M. Diehl, “qpoases: a parametric active-set algorithm for quadratic programming,” Math. Program. Comput. 6, 327–363 (2014). [CrossRef]
24. S. Qian and M. Idir, “Innovative nano-accuracy surface profiler for sub-50 nrad rms mirror test,” Proc. SPIE 9687, 96870D (2016). [CrossRef]
