research papers
Hybrid height and slope figuring method for grazingincidence reflective optics
^{a}National Synchrotron Light Source II (NSLSII), Brookhaven National Laboratory, PO Box 5000, Upton, NY 11973, USA, ^{b}School of Mechanical and Automotive Engineering, Xiamen University of Technology, Xiamen 361024, People's Republic of China, ^{c}James C. Wyant College of Optical Sciences, University of Arizona, 1630 E. University Blvd, PO Box 210094, Tucson, AZ 857210094, USA, ^{d}Large Binocular Telescope Observatory, University of Arizona, Tucson, AZ 85721, USA, ^{e}Council of Scientific and Industrial Research – Central Scientific Instruments Organisation (CSIRCSIO), Chandigarh 160030, India, ^{f}Academy of Scientific and Innovative Research, Chennai, Tamil Nadu 201002, India, and ^{g}Department of Astronomy and Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
^{*}Correspondence email: tianyi@bnl.gov, lhuang@bnl.gov
Grazingincidence reflective optics are commonly used in synchrotron radiation and freeelectron laser facilities to transport and focus the emitted Xray beams. To preserve the imaging capability at the diffraction limit, the fabrication of these optics requires precise control of both the residual height and slope errors. However, all the surface figuring methods are height based, lacking the explicit control of surface slopes. Although our preliminary work demonstrated a onedimensional (1D) slopebased figuring model, its 2D extension is not straightforward. In this study, a novel 2D slopebased figuring method is proposed, which employs an alternating objective optimization on the slopes in the x and ydirections directly. An analytical simulation revealed that the slopebased method achieved smaller residual slope errors than the heightbased method, while the heightbased method achieved smaller residual height errors than the slopebased method. Therefore, a hybrid height and slope figuring method was proposed to further enable explicit control of both the height and slopes according to the final mirror specifications. An experiment to finish an ellipticalcylindrical mirror using the hybrid method with ion beam figuring was then performed. Both the residual height and slope errors converged below the specified threshold values, which verified the feasibility and effectiveness of the proposed ideas.
Keywords: ion beam figuring; synchrotron optics; twodimensional; surface slopes.
1. Introduction
With the rapid evolution of freeelectron lasers and third and fourthgeneration Xray synchrotron light sources, the smoothness and precision requirement on Xray optics have dramatically increased. Grazingincidence mirrors, a key type of Xray optics, are widely used for focusing the emitted hard Xray beams. To preserve the incoming wavefronts and image at the diffraction limit, these mirrors are required to reach the <0.1 µrad root mean square (RMS) level for residual slope errors and <1 nm RMS for residual height errors.
Due to the grazingincidence geometry, dedicated optical metrology systems have been developed to measure such highprecision mirrors. Initially, slope metrology instruments were developed. The long trace profiler (Takacs et al., 1987) and the nanometre optical component measuring machine (NOM) (Siewert et al., 2004) are two well known types of onedimensional (1D) slope profilers widely used at light source facilities around the world (Qian et al., 1995; Ali & Yashchuk, 2011; Siewert et al., 2010; Nicolas et al., 2016; Nicolas & Martínez, 2013; Alcock et al., 2010). With slightly different configurations from the NOM, the nanoaccuracy surface profiler (NSP) was developed (Qian & Idir, 2016) and advanced (Huang, Wang, Nicolas et al., 2020; Huang et al., 2022) using the multipitch technique at National Synchrotron Light Source II (NSLSII), able to reach <0.05 µrad measurement repeatability. Twodimensional (2D) slope metrology has also been attempted, including the use of the NOM in both the x and ydirections (Thiess et al., 2010), stitching ShackHartman (SSH) wavefront sensing (Idir et al., 2014; Adapa et al., 2021) and phase measuring deflectometry (Huang, Su et al., 2015).
As the fourthgeneration synchrotron light sources evolve towards almost fully coherent beam, height metrology has become a necessity in the characterization of synchrotron mirrors. Interferometry is the most widely adopted heightmeasuring technology for mirror characterization. However, since a grazingincidence mirror is longer in its tangential (i.e. x) dimension (100 mm to 1 m) than its sagittal (i.e. y) dimension (5 to 20 mm), stitching interferometry (SI) techniques (Mimura et al., 2005; Yumoto et al., 2016; Vivo et al., 2016; Huang et al., 2019; Huang, Wang, Tayabaly et al., 2020) have been developed to extend the field of view. The main idea is to combine the measurements in individual small fields of view to generate a height map of the entire clear aperture (CA) on the mirror surface. The measurement repeatability can reach <0.5 nm RMS for flat and `shallow' curved mirrors (Huang et al., 2019; Huang, Wang, Tayabaly et al., 2020). Microscopebased SI techniques (Yamauchi et al., 2003; Rommeveaux & Barrett, 2010) have also been attempted to inspect middle to highfrequency errors.
On top of the dedicated metrology, deterministic optics fabrication techniques are required to produce grazingincidence Xray mirrors. Despite both slope and height metrology instruments being available, the current figuring method in deterministic optics fabrication is height based, in which the height removal is modelled as the convolution between the tool influence function (TIF) of a machine tool and its dwell time at different locations on the mirror surface (Jones, 1977). Integration from slope to height is thus required if a slopemeasuring system is used. It was found that this integration in 1D might amplify the errors if the Riemann approximation was used (Zhou et al., 2016). Although more advanced integration methods have been proposed to reduce the integration errors (Huang, Idir et al., 2015; Huang et al., 2017) and higherperformance heightbased optimization algorithms have been developed (Wang, Huang, Kang et al., 2020; Wang, Huang, Choi et al., 2021; Wang, Huang, Vescovi et al., 2021), the control of slope errors is still implicit in the heightbased model, which is inappropriate since even two height error maps can give the same r.m.s. residual level; they may correspond to very different slope errors. This is manifested by the sinusoidal surface error to be corrected, z^{d}(x, y), shown in Fig. 1, defined as
with a = 70 nm. The frequencies, (f_{x}, f_{y}), for Figs. 1(a) and 1(c) are (f_{x}, f_{y}) = (150, 0) mm^{−1} and (f_{x}, f_{y}) = (10, 0) mm^{−1}, respectively. As shown in Figs. 1(b) and Fig. 1(d), it is obvious that, although the two surface height errors have the same RMS value of 50 nm, the one with a higher f_{x} corresponds to a larger slope error in the x direction. In conventional heightbased figuring processes, slope errors were not explicitly monitored and controlled. If there were requirements on residual slope errors, the common solution was to iterate the heightbased optimization, usually with smaller machine tools, expecting that the slope would eventually converge with the height. This method is inefficient, and the residual slope error is unpredictable during the optimization process.
Our preliminary research successfully demonstrated a 1D slopebased figuring model (Zhou et al., 2016), which was further improved by Li & Zhou (2017). However, the model cannot be simply extended to 2D since one height map corresponds to two slope maps, which are slopes in the x and ydirections, respectively, and one dwell time solution should minimize the slope errors in both directions, which is difficult to achieve theoretically. Therefore, in this study, we first formulated the 2D slopebased figuring model and analysed the theoretical difficulty in solving the dwell time from it. Afterwards, an alternating objective optimization method was employed to resolve this difficulty. The analytical simulation indicated that the slopebased optimization achieved smaller residual slope errors than the heightbased optimization, while the heightbased optimization achieved smaller residual height errors than the slopebased optimization. Based on this constatation, a hybrid height and slope figuring method was proposed, which enables the explicit control of both height and slope errors according to the mirror specifications. Finally, an experiment to finish an ellipticalcylindrical mirror with ion beam figuring (IBF) using the hybrid method was demonstrated. Both the residual height and slope errors converged to the specified values (i.e. <1 nm RMS, <0.2 µrad RMS in the xdirection, and <0.5 µrad RMS in the ydirection), which proved the effectiveness of the proposed ideas.
The rest of the paper is organized as follows. Section 2 formulates the 2D slopebased figuring model, followed by an introduction of the alternating objective optimization used to solve the dwell time from 2D slope errors in Section 3. In Section 4 we explain the hybrid height and slope optimization method and study the performances of the heightbased, slopebased and hybrid methods via simulation. The experimental verification is given in Section 5. Section 6 concludes the paper.
2. Twodimensional slopebased figuring model
2.1. From heightbased to slopebased optimization
The classical 2D heightbased figuring model (Jones, 1977) is defined as
where * represents the convolution operator, and the removed height z(x, y) is modelled as the convolution between the TIF b(x, y) and the dwell time t(x, y). After measuring the desired height removal z^{d} and b, the objective is to find a such that
subject to t t_{min}, where z^{r} is the residual errors between z^{d} and z, t_{min} is the minimum dwell time for each dwell point, and RMS(…) represents the RMS value of `…'. According to the property of the convolution, equation (2) can be differentiated as either
or
where g_{x}(x, y) = ∂g/∂x and g_{y}(x, y) = ∂g/∂y represent the slopes of g(x, y) in x and y, respectively.
Equations (4) and (5) indicate that the differentiations can be applied to either the dwell time or the TIF. Although both of them are mathematically correct, equation (4) is difficult to implement in practice, since it requires both the slope (i.e. z_{x} and z_{y}) and height (i.e. b) measurements, and the calculated t_{x} and t_{y} cannot be directly used. On the other hand, equation (5) enables a pure slopebased process, in which two slopebased equations correspond to one dwell time. Therefore, as an analog to the 1D model (Zhou et al., 2016; Li & Zhou, 2017; Zhou et al., 2016), equation (5) is selected as the 2D slopebased figuring model in this study. After measuring the desired slope removals and and the TIF slopes b_{x} and b_{y}, the objectives are to find for which
subject to t , where
and
in which and are the residual slope errors in x and y, respectively.
2.2. Difficulty in 2D slopebased optimization
Assuming that the height error to be corrected is defined by the bisinusoidal equation shown in equation (1), the corresponding desired slope removal in the x and ydirections are
Equation (9) indicates that the slope errors are affected by the frequencies. The higher the values of f_{x} and f_{y}, the more sensitive the slope errors will correspond to small changes of the amplitude a.
The dwell time can be analytically solved from equations (5) and (9). Taking z_{x}(x, y) as an example, equation (9) can be expressed as the imaginary part of a complex number as
where i = , and Im[…] represents the imaginary part of `…'. For convenience, the imaginary operator can be neglected,
By substituting equation (10) into equation (5) and performing Fourier transforms, equation (5) can be transformed to the frequency domain as
where T(u, v) and B(u, v) are the Fourier transforms of t(x, y) and b(x, y), respectively, and δ(u, v) is the delta function. Therefore, the dwell time t(x, y) can be solved from equation (12) as
where B_{x}(f_{x}, f_{y}) and ϕ(f_{x}, f_{y}) are the amplitude and phase of B(f_{x}, f_{y}), respectively. Due to the linear properties of equation (13), the actual dwell time is its imaginary part, which is
Moreover, to make t(x, y) nonnegative, the final t(x, y) should be calculated as
Similarly, the dwell time can be calculated from z_{y}(x, y) in equations (5) and (9) as
Theoretically, equations (15) and (16) should be satisfied at the same time to ensure that equation (5) is valid. This results in
which requires the ratio between each two f_{x} and f_{y} to be equal to the ratio between B_{x}(f_{x}, f_{y}) and B_{y}(f_{x}, f_{y}) at those frequencies. This is extremely difficult to guarantee in practice since a real surface is composed of various f_{x} and f_{y}.
3. 2D slopebased optimization with alternating objectives
Based on the analysis above, we can conclude that optimization by using slope errors is more sensitive to higher frequencies than that using height errors. The slopebased optimization, if it can be solved properly, will thus be more effective in the higherfrequency error ranges, while the heightbased optimization is preferred to correct lowerfrequency errors. However, due to equation (17), it is theoretically difficult to obtain a reasonable dwell time solution from the twoobjective slopebased model. The conventional heightbased optimization methods cannot be directly employed. More advanced optimization strategies are thus necessary in the slopebased optimization.
3.1. Alternating twoobjective optimization
Practically, such a multiobjective optimization problem can only achieve the Pareto optimality (Miettinen, 2012), at which t(x, y) in equation (6) is optimized in a way that f_{2}(t) cannot improve without f_{3}(t) worsening, and vice versa. Every Pareto optimal point resides on the Pareto front; however, it is nontrivial to find out which point is appropriate for a certain slopebased problem in practice.
The most straightforward way of locating a Paretooptimal point is the weighting method (Miettinen, 2012), which combines f_{2}(t) and f_{3}(t) as a single objective, αf_{2}(t) + βf_{3}(t), where α and β are the weights for f_{2}(t) and f_{3}(t), respectively. Nonetheless, it is difficult to determine proper values for α and β. In particular, in a grazingincidence mirror, the slope errors in the ydimension are more difficult to correct than those in the xdimension, since the ydimension is much shorter than the xdimension due to the grazingincidence geometry. The obtained dwell time solution thus usually fails to satisfy the specifications on residual slope errors in both x and ydimensions at the same time. To solve this problem, we employed an alternating objective optimization algorithm (see Table S1 of the supporting information), which progressively approaches the specified residual slope errors by iteratively exchanging the search direction between f_{2}(t) and f_{3}(t) using the estimated residual slope errors obtained from the last iteration. Thus, in each step of the alternating objective optimization, only one objective will be optimized so that any of the heightbased methods can be used.
In detail, the inputs to the algorithm are specified RMSs of the residual slope errors, and , and the desired slope removals, and . The algorithm is initialized with a maximum number of iterations, i_{max}, and the minimum dwell time, t_{min}, which will be added to each dwell point to enforce > 0. We use i_{max} = 10 in this study, since we found that the algorithm usually converged in less than ten iterations, and t_{min} = 0.01 is calculated from the dynamics limits of the translation stages in our IBF system (Wang, Huang, Zhu et al., 2020). The residual slope errors are initialized with the desired slope removals, = and = , and the optimized dwell time is initialized as = 0.
Next, the iterations start with the calculation of the current RMSs of the residual slope errors, and . If both and achieve and , the current iteration stops and is obtained. Otherwise, the objectives f_{2}(t) and f_{3}(t) are alternatively optimized. In this study, we employed an efficient dwell time optimizer proposed in our previous work (Ke et al., 2022) to calculate the intermediate dwell time, which is used to update .
Finally, it is crucial to update and to ensure that the next iteration will be continued on the residual slope errors obtained from the last iteration and thus guarantees the convergence towards and . If the algorithm does not converge in i_{max} iterations, it means that the current TIF is not capable of correcting the remaining slope errors. We thus accept as the solution for the current TIF and select smaller TIFs to repeat the process in pursuit of the eventual convergence to the specifications. With the alternating objective optimization, it is obvious that not only and are being minimized simultaneously but also the convergence can be explicitly controlled according to and .
4. Hybrid height and slope optimization method
In this section, the effectiveness of the alternating objective optimization in the slopebased method is studied. The performances of the heightbased and the slopebased methods are compared by applying them to optimize dwell time from analytical surfaces and an analytical Gaussian TIF. First, the sinusoidal surfaces containing the single frequencies shown in Fig. 1 are tested. Afterwards, the methods are applied to a surface generated with Chebyshev polynomials, which covers a wider range of frequencies. Based on the results, a hybrid height and slope optimization method is then proposed, which enables explicit control of both the residual height and slope errors with respect to the specifications.
4.1. Study on the performances of the slopebased and heightbased methods
4.1.1. Simulation on singlefrequency surfaces
The slopebased and heightbased methods are first studied on the singlefrequency sinusoidal surfaces shown in Fig. 1 to verify their different performances on low to high frequencies. The employed TIF, as shown in Fig. 2, is generated by fitting one of our IBF TIFs with a 2D Gaussian function. The diameter of the TIF is 10 mm, with a full width at halfmaximum (FWHM) of 5 mm. The dwell grid is thus set as larger than the perimeter of the surfaces by the radius of the TIF, i.e. 5 mm. We specify the desired RMSs of the residual height and slope errors as = 1 nm, = 0.2 µrad and = 0.5 µrad, respectively, which are the same as the specifications imposed in the experiment that will be demonstrated in Section 5.
It is worth mentioning that, to make the comparison fair, both the heightbased and slopebased methods use the same optimizer proposed by Ke et al. (2022). Moreover, the heightbased method is performed iteratively on the residual height errors until either all the , and are achieved or i_{max} is reached. The final residual height and slope errors are estimated from and as * , * , * and * , * , * , respectively. These configurations are used in all the simulations demonstrated in this section.
The slopebased and heightbased optimizations are first applied to the lowfrequency surface error shown in Figs. 1(a) and 1(b), where the period of the error is equal to the length of the mirror. The dwell time, residual height errors and residual slope errors estimated from the heightbased and slopebased methods are shown in Figs. 3(a)–3(c) and Figs. 3(d)–(f), respectively.
It is found that, in this lowfrequency example, with the similar total dwell time [Fig. 3(a) and Fig. 3(d)], the heightbased method converges to very small residual errors in both the height and slope [Fig. 3(b) and Fig. 3(c)], while the slopebased method could not achieve the specified residual height error [Fig. 3(e)], though the residual slope error converges to the same level [Fig. 3(f)] as that obtained from the heightbased method. Therefore, in the lowfrequency case, the heightbased method outperforms the slopebased method.
The same operations are then applied to the higherfrequency surface error shown in Figs. 1(c) and 1(d), in which one period of the surface error is equal to the diameter of the TIF. The dwell time, residual height errors and residual slope errors estimated from the heightbased and slopebased methods are shown in Figs. 4(a)–4(c) and Figs. 4(d)–4(f), respectively.
It is obvious that, with similar total dwell time [Figs. 4(a) and 4(d)], as the frequency increases, the slopebased method converges to smaller residual errors in both the height and slope [Figs. 4(e) and 4(f)] than those obtained from the heightbased method [Figs. 4(b) and 4(c)], which indicates that the slopebased method is preferred when correction of relatively highfrequency errors is necessary.
4.1.2. Simulation on a multifrequency surface
Based on the singlefrequency simulation results above, it is clear that the heightbased and slopebased methods have different frequency responses. To futher understand the performance of the slopebased and heightbased methods on real surfaces, which are always composed of various lowtohighfrequency components, they are applied to the surface height and slopes shown in Fig. 5.
The surface height [Fig. 5(a)] and slope errors [Figs. 5(b) and 5(c)] are generated by fitting a measured surface error map with a 10 × 26 = 260order Chebyshev polynomial. The clear aperture of the surface is 20 mm × 150 mm and the is 0.1 mm, which is the same as for our SI system (Huang, Wang, Tayabaly et al., 2020). The initial RMSs of z^{d}, and are 77.0 nm, 4.49 µrad and 3.63 µrad, respectively, which are close to the measurements.
The optimized and are demonstrated in Figs. 6(a) and 6(b), respectively, from which the corresponding residual height and slopes errors are estimated in Figs. 7(a)–7(c) and Figs. 7(d)–7(f), respectively. In terms of the residual height errors, the heightbased method, as shown in Figs. 6(a) and 7(a), outperforms the slopebased method shown in Figs. 6(b) and 7(d). Although both the heightbased and slopebased methods reach the specified , the heightbased method achieves a smaller (about a factor of 0.5) residual height error in a shorter (51 min less) total dwell time.
However, as shown in Fig. 7(c), the heightbased method fails to achieve the specified , which means that the heightbased method does not converge in i_{max} iterations. On the other hand, the slopebased method, as shown in Figs. 7(e) and 7(f), reaches both the specified and in only three iterations. Moreover, the achieved residual slope error in the xdimension is two times smaller than that obtained from the heightbased method shown in Fig. 7(b). In a word, the heightbased method is better at reducing the height errors while the slopebased method is preferred when the residual slope errors are not within the specifications.
From the exploitation of the integrated power spectral density (PSD) distributions of the residual height errors shown in Fig. 8, it is found that the heightbased method achieves a lower PSD in the lowfrequency range while the slopebased method is better at the middlefrequency range, which reveals the different sensitivities of height and slope to the different spatial frequencies of errors.
4.2. Alternating threeobjective optimization
The result shown in Section 4.1 demonstrates that both and are reasonable dwell time solutions for their respective heightbased or slopebased objectives. However, they may fail to minimize the other objectives that are not included in the optimization process. In other words, minimization of f_{1}(t) does not guarantee the minimization of f_{2}(t) or f_{3}(t), and vice versa. Therefore, if there are specifications on both the residual height and slope errors, it is more appropriate to define the entire optimization problem as
subject to t , which contains all the three objectives and should offer a more universal solution that may integrate the merits of both the heightbased and slopebased optimizations. Therefore, we propose a hybrid height and slope method, which extends the idea of the alternating twoobjective optimization (see Table S1) by including the heightbased optimization in the iterative process (see Table S2).
To achieve faster convergence, two key differences from the alternating twoobjective optimization are worth emphasizing here. First, at the end of each objective optimization, both the residual height and slope errors will be estimated using the current , no matter whether it is obtained from the heightbased or the slopebased process. This ensures that z^{r}, and are minimized in the current iteration. Second, the heightbased process (lines 7–12 in Table S2) is first performed in each iteration, since slope errors will always be reduced with a decrease in the height errors.
The performance of the hybrid method is compared with the heightbased and slopebased methods. As shown in Fig. 6(c), after three iterations the hybrid method takes a little longer (14 min more) total dwell time than the heightbased method. However, as shown in Figs. 7(a), 7(b) and 7(c), both the residual height and slope errors greatly outperform those obtained from the heightbased and slopebased methods. By further examining the PSD distribution in Fig. 8, the hybrid method is also found to be superior to both the heightbased and slopebased methods over the entire spatial frequency range.
All these simulation results suggest that either the heightbased or the slopebased method underestimates the removal capability of a TIF, while the hybrid method achieves a more universal dwell time solution. It not only guarantees a more rapid convergence towards the specifications but also enables explicit control of the optimization based on the specifications. It indicates that the hybrid method is necessary and effective in dwell time optimization when there are specifications on height and slope simultaneously.
5. Experiment
To further verify the feasibility of the hybrid method in practice, we applied it to finish one of our grazingincidence mirrors. Recently, we received a request from the In situ and Operando Soft et al., 2020). We thus had the opportunity to test our IBF solutions. The specifications for the ellipticalcylindrical KB mirror are shown in Table 1.
(IOS) beamline at NSLSII to produce a silicon horizontal Kirkpatrick–Baez (KB) mirror using our IBF system (Wang, Huang, Zhu

The size of the CA is 20 mm × 150 mm. The object distance, image distance and grazing angle of the offaxis ellipse, as schematically shown in Fig. 9, are p = 14254.7 mm, q = 2448.8 mm and θ = 1.25°, respectively. The height, tangential slope and sagittal slope errors from the target shape should be less than 1 nm RMS, 0.2 µrad RMS and 0.5 µrad RMS, respectively, and the surface microroughness should not exceed 0.3 nm RMS.
5.1. Experimental setup
To minimize the processing time, we started by finding the bestfit sphere to the target elliptical cylinder. We fit the ellipse parameters in Table 1 to a circle and found that the optimal radius of curvature (ROC) of the circle that achieved the minimal average material removal is 199 m. Therefore, as shown in Fig. 10, we purchased a 30 mm × 160 mm spherical mirror, which had been shaped to be within ±1% of the expected ROC, as the base mirror. Also, the mirror was further pitchpolished to achieve the specified roughness level, since IBF can hardly reduce but introduce minimal damage to the roughness (Mikhailenko et al., 2022). The left and right ends of the mirror were marked as A and B, respectively.
The desired height removal from the initial spherical mirror to the target ellipticalcylindrical mirror is shown in Fig. 11(a), where the initial height error is 183.19 nm RMS. In this experiment, as an example shown in Figs. 11(b) and 11(c), the 2D slope data were generated from the height measurement using a 2.5 mm × 2.5 mm sliding window in the experiment. We found that this method excellently matches the 2.5 mm pinhole used in our NSLSII NSP system (Huang, Wang, Tayabaly et al., 2020). This will be further manifested in the final inspection crossvalidated between our SI and NSP systems in Section 5.4.
The IBF system (see Fig. S1 of the supporting information) is equipped with a KDC10 gridded from Kaufman & Robinson Inc. The working parameters for the are beam voltage U_{b} = 600 V, beam current I_{b} = 10 mA, accelerator voltage U_{a} = −90 V and accelerator current I_{a} = 2 A. One of the most frequently used TIFs of the IBF system is shown in Fig. 12, which is obtained by placing a 5 mm diaphragm in front of the to constrain the shape of the ion beam. The radius of the TIF is 5 mm, with the FWHM equal to 4.4 mm. The peak removal rate is 5.5 nm s^{−1} and the volumetric removal rate is 98.8 nm mm^{2} s^{−1}.
5.2. Heightbased figuring of the mirror
Initially, before we proposed the hybrid method, we started the IBF of the mirror based on the desired height removal and TIF shown in Figs. 11(a) and 12(a), respectively. The heightbased estimation is given in Fig. 13(a), which used a raster tool path with 0.5 mm machining intervals. The total processing time is more than ten hours, which is too long to be completed in a single IBF run. Therefore, to maintain the stability of the ion beam and reduce the nonlinearity in material removal caused by the thermal effect, we divided the dwell time into 50 IBF runs before sending to our IBF system. We monitored the intermediate residual height errors by measuring the mirror every ten cycles. The measured residual height error map after the 50 cycles of IBF is shown in Fig. 13(b), from which the residual slope errors in x and y were calculated from the residual height errors using the method mentioned in Section 5.1 and Fig. 11.
It is worth mentioning that the small `bumps' shown in Fig. 13(b) resulted from the stage failure during the tenth cycle. However, since the damage was rather small compared with the total removal amount, we thus continued the experiment to see whether the mirror could still meet the specification with these artefacts. As shown in Fig. 13(b), it was found that both the residual height and slope errors (i.e. 1.15 nm RMS, 0.26 µrad RMS and 0.68 µrad RMS, respectively) were still larger than the specifications given in Table 1. Conventionally, to further pursue the specifications, we would repeat the heightbased IBF process with a smaller TIF, expecting that the slope will finally converge with the height. However, in this study, to testify the feasibility of the proposed hybrid method, we tried to apply it to finish the mirror using the same TIF shown in Fig. 12.
5.3. Finishing of the mirror with the hybrid method
The estimations obtained from the hybrid method based on the measurement in Fig. 13(b) are shown in Fig. 14(a), which shows that both the height and slope specifications can be achieved in 45.95 min. Similar to the figuring process, the dwell time was divided into two IBF runs, after which the final residual height and slope errors shown in Fig. 14(b) reached 0.69 nm RMS, 0.19 µrad RMS and 0.43 µrad RMS, respectively. Although the measured residual errors were slightly larger than the estimations due to the actual hardware limits, they all achieved the specifications, and the final height and slope convergence ratios are 99.6%, 98.1% and 98.5%, respectively, which verifies the feasibility of the proposed hybrid height and slope optimization method and the efficiency of our IBF system.
It is worth reiterating that the estimations in the real experiment shown in Figs. 13 and 14 are worse than those of the simulation given in Fig. 7 within an order of magnitude. This is due to the unavoidable noise from either the metrology instruments or the IBF processes. From the metrology, our SI and NSP are within the highfrequency uncorrectable noise levels of 0.3 nm and 50 nrad, respectively. During an IBF process, positioning errors, thermal effects and dynamic limits also contribute to the overall uncertainties.
5.4. Final inspection of the mirror
To confirm that the finished mirror had achieved all the specifications given in Table 1, the final inspection was performed using the NSP (Huang, Wang, Nicolas et al., 2020) for slope measurement, the SI for height measurement and a Zygo NewView whitelight interferometer for roughness examination. Each NSP or SI measurement was performed ten times from A to B then B to A. The residual height errors measured with the SI system achieved the same RMS level as Fig. 14(b); however, to guarantee that the height measurements are reliable, we decided to crossvalidate them using our NSP system. As the NSP is a 1D slope profiler, we inspected the centre line of the mirror along the xdirection. The centre lines from the SI measurements were also extracted and converted into slope profiles using the method mentioned in Section 5.1. Fig. 15 demonstrates the slope profiles of the centre line of the mirror measured with both the NSP and SI, with the average of every ten scans highlighted in bold. It was found that the measurements obtained from two different instruments validated each other and the residual slope errors were all below the specifications, which proved that the fabricated mirror had achieved both the residual height and slope specifications (refer to Figs. S2 and S3 for more details of the inspection reports).
6. Conclusion
In this study, we proposed a twodimensional slopebased figuring model, enabling the dwell time to be directly optimized using slope data from measurements. Due to the two objective functions in the slopebased model, we introduced an alternating optimization algorithm to iteratively approach both objectives. From the comparison between the heightbased and slopebased methods, we found that the heightbased method is better at reducing residual height errors while the slopebased method is preferred when there are strict requirements on residual slope errors. Based on this constatation, we proposed the hybrid height and slopebased optimization method which alternatively minimizes both the height and slope objectives. From the simulation result, it was found that the hybrid method outperformed both the heightbased and slopebased methods in the entire range of spatial frequencies, which is especially useful for synchrotron mirrors which have strict specifications on residual height and slope errors simultaneously. Finally, we applied the hybrid method to finish an ellipticalcylindrical mirror using ion beam figuring. The mirror has achieved all the specifications, which proves the effectiveness of the proposed ideas. We thus recommend using this method for synchrotron mirror fabrication.
Supporting information
Tables S1 and S2; Figures S1 to S3. DOI: https://doi.org/10.1107/S160057752201058X/ju5048sup1.pdf
Funding information
This work was supported by the Accelerator and Detector Research Program, part of the Scientific User Facility Division of the Basic Energy Science Office of the US Department of Energy (DOE), under the Field Work Proposal No. FWPPS032. This research was performed at the Optical Metrology Laboratory at the National Synchrotron Light Source II, a US DOE Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory (BNL) under Contract No. DESC0012704. This work was performed under the BNL LDRD 17016 `Diffraction limited and wavefront preserving reflective optics development'.
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