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Abstract
Pseudo-random paths are a useful tool to reduce mid-spatial frequency errors created in the processing of optical surfaces by sub-aperture polishing tools. Several types of patterns have been proposed, including hexagonal, square and circular, but prior literature has largely focused on flat and gently curved surfaces. Here, an extension of the circular-random path to strongly curved aspheric and freeform surfaces is proposed. The main feature of the algorithm is to cover the entire surface to be polished with a uniformly distributed tool path. Aspheric condenser lenses are then polished with a regular raster and circular-random path. Analysis of the optical performance shows that the random path can reduce the amplitude of mid-spatial frequency errors and relative intensity of satellite images. These features are particularly desirable in short wavelength applications, such as mirrors for EUV and X-ray.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In CNC sub-aperture polishing and compliant grinding [1], raster and spiral tool paths are widely used [2]. These simple paths can cover almost any type of flat or curved surface area with a regular pattern, over which a surface feed profile may be scheduled to achieve regional modifications of the surface profile [3]. But since these paths consist of a repeated pattern, an identical periodic pattern also appears on the processed surface at the cusps of tool footprint overlaps [4]. The generation of such regular pattern, often referred to as Mid-Spatial Frequency (MSF) error, can cause serious degradation of the optical performance due to interference and diffraction within the light beam propagating through the system, even when the amplitude of MSF content is only a few nm.
To address this issue, a variety of pseudo-random path generation methods have been proposed in previous literature. Working in a 2D projection plane, such algorithms include a hexagonal pattern proposed by Dunn et al. [5], square pattern proposed by Wang et al. [6], circular pattern proposed by Takizawa et al. [7], six-directional pattern proposed by Zhao et al. [8], hyper-crossing tool path based on randomized epicyclical motion proposed by Li et al. [9], and even a fractal-like path proposed by Dong et al. [10]. In each of these works, quasi-planar surfaces were polished by the pseudo-random path method and showed lowered MSF content when compared with a conventional raster path or Hilbert curve. However, the demand for low MSF content is not restricted to planar and gently curve optical surfaces. For instance, aspheric and freeform mirrors are used in Extreme Ultra-Violet (EUV) and X-ray application such as lithography, high-energy beams and astronomy. The long focal length typical in such systems makes them very sensitive to slope variations associated with excessive MSF content.
When dealing with aspheric and freeform surfaces, the conventional approach consists of projecting a 2D tool path onto the target 3D surface. While simple, the method has a major drawback in that the density of tool path tracks becomes uneven, depending on the slope of target surface. Some research aiming to reduce the unevenness of such projected tool paths has been conducted. Han et al. modified the intervals of spiral paths place-by-place to avoid overlapping of polishing marks [11]. Chaves-Jacob et al. [12] demonstrated the use of a trochoid pattern on freeform surface, while Tam et al. [13] created Peano curve tool paths on spherical surfaces to achieve uniform density of tool path points distribution. Also of note, Li et al. [14] demonstrated the use of a 2-dimensional vibration actuated polishing head to replace periodic polishing marks with a random-like Lissajous pattern. However, these methods still feature a set of parallel or quasi-concentric curved lines in the construction stage. It should also be noted that Tam’s work is only applicable to axisymmetric shapes. Therefore, a general solution for creating pseudo-random paths on freeform surfaces has not been proposed yet.
In this paper, a method for generating uniformly distributed circular pseudo-random paths on aspheric and freeform surface is proposed which addresses the issue highlighted above. A comparison of MSF content distribution is carried out by simulating the polishing of a spherical lens by raster and circular-random path. Experimental validation is then carried out by polishing an aspheric condenser lens by raster and circular-random path, and measuring the distribution of light in the focal plane. Both in simulations and experiments, the random path is shown to reduce the intensity of satellite spots, by spreading their power across multiple broad directions.
2. Algorithm for generation of circular-random paths on freeform surfaces
2.1 Tool path element
The proposed pseudo-random tool path is created by connecting elements. Figure 1 shows an element of circular-random path [7]. The shape of element is defined on a 2D plane, but three reference points are added at the center of each circle. To create a random path in 3D, it is necessary to place these reference points directly on the freeform surface.
2.2 Generation of path on near-spherical surfaces
To achieve constant removal in the polishing process, path density should be uniform. Therefore, intervals between reference points should also be constant. For spherical surfaces, a structure called ‘icosphere’ can be used to procedurally place vertices with almost uniform density [15]. Figure 2 shows an example of icosphere generation. It is created by iterative splitting from a regular icosahedron inscribed in a sphere, as shown in Fig. 2(a). Each side is divided into two segments at its center. After that, center points are projected to the sphere. Each face is divided into four small regular triangles using these extra center points. This operation can be done repeatedly. Each iteration increases the number of points by a factor of 4. The spacing of points is even, although the 20 original points are surrounded by only 5 vertices while all other points are surrounded by 6 vertices.
Figure 3 shows the procedure for placing a random path onto an aspheric surface with gentle departure from a sphere. Viewpoints of the figure are from the top, except for (f). First, vertices from an icosphere are projected radially to the target surface. The distance between reference points is generally not constant. To adjust the distance, they are regarded as point charges and an associated repelling force is calculated. A point charge qi is calculated for each point, following the equation:
After adjusting the distance between points, triangles are created from these vertices such that no triangle shares an edge with another, as shown in Fig. 3(c). Based on these triangles, the path elements are placed by matching triangle vertices and reference points, as shown in Fig. 3(d). Finally, overlapping parts are trimmed as shown in Fig. 3(e).
Figure 4 details the procedure for removing overlapping parts of the tool path. First, one of the points is selected and the surrounding path elements are picked up. A best-fit plane is calculated for the reference points contained within the path elements. The path elements are then projected in the normal direction to the plane. One of the projected elements is selected, and it is judged whether parts of the other elements are inside this element or not. The parts of path element inside the selected one are removed. Next, another path element is selected, and the same algorithm is repeated.
2.3 Generation of path on general freeform surfaces
In Computer Aided Design (CAD) and Manufacture (CAM) software packages, general surfaces are often represented with B-Splines or Non-Uniform Rational B-Splines (NURBS) [16]. These are piecewise polynomials, built from basis functions by means of the Cox-de Boor recursion formula, taking in 3D the general expression [x,y,z] = S(u,v). The UV coordinate space is typically normalized, and the function S transforms coordinates from UV to the 3D Cartesian XYZ space. By assigning UV coordinates to path points, a circular-random path can be created with relatively low calculation cost. As an example of such NURBS surface, Fig. 5 shows the functional parts of a prosthetic knee joint in Cartesian XYZ and normalized UV coordinates. It should be noted here that the density of surface area is not necessarily maintained when transforming from UV to XYZ space. In the case of the knee joint, the arms extending from the main body appear compressed in the U-direction.
Figure 6 shows the creation procedure of a random path on this NURBS surface. First, starting points are uniformally distributed in the UV plane and tranformed into their associated 3D coordinates, as shown in Fig. 6(a). The charges qi and force vectors Fi are then computed in the Cartesian space. Next, the points are moved by projecting the force vectors back into the UV space. Positions of the points in the Cartesian space are updated iteratively according to their motion in the UV space. This operation is repeated several times to equalize the distance between points in the Cartesian space. Next, path elements are attached to the base points in the UV space, as shown in Fig. 6(c). Overlapping element parts can then be trimmed, as shown in Fig. 6(d). Finally, the tool path is converted back from the UV space into the 3D Cartesian space, as shown in Fig. 6(e).
3. Simulation of spherical optic polishing by circular-random path
3.1 Polishing footprint
In sub-aperture CNC polishing, the effective material removal distribution can be computed by convolution of a dwell time map (reciprocal of feed profile) and polishing footprint [4]. For the purpose of simulation, a Fluid Jet Polishing (FJP) [3] footprint was generated on an optical window of BK7 glass. The process parameters are summarized in Table 1. The polished footprint was measured with a Fizeau interferometer (Wyko RTI4100) and normalized to 1 minute of polishing, as shown in Fig. 7(a). A representative 2D profile was computed by taking the rotational average of several section profiles, as shown in Fig. 7(b).
Table 1. Parameters of FJP footprint.
3.2 Workpiece and tool paths
The workpiece shape used in polishing simulations is a spherical concave lens with 38.6 mm radius of curvature and 40 mm aperture diameter. It sharply deviates from a planar surface, reaching a maximum slope angle of 31.2°. Raster and circular-random paths were generated with a track spacing of 0.8 mm (i.e. the total path length was identical). As shown in Fig. 8(a), the raster track spacing is constant in the XY plane. When projected onto the spherical surface, the actual track spacing as measured along the surface steadily increases towards the top and bottom of the workpiece. By comparison, the circular patterns shown in the XY plane progressively shrinks from the center to the edge of workpiece. Consequently, the actual pattern spacing as measured along the surface is constant in all areas of the workpiece.
A constant surface feed rate was specified along the paths, in order to simulate the polishing of a 1 µm deep layer of material. The algorithm is a modified version of the classic problem on convolution of dwell-time and tool removal function [2]. On the one hand, form error is calculated along the surface normal of workpiece and referenced against the X/Y plane (as measured using for example a Fizeau laser interferometer with transmission sphere). Since the workpiece is strongly curved, the appearance of the removal footprint becomes elongated as the slope in both X and Y-direction increases. Therefore, the material removal prediction algorithm distorts the removal function shape place-by-place along the tool path, and scales it against the equivalent dwell-time computed for each iteration of the optimization routine aiming to reduce residual form error [3]. Figure 9 shows the simulated residual waviness on the workpiece surface after subtraction of the nominal 1 µm material removal layer. In the case of raster path, Fig. 9(a) shows a non-uniform distribution of material removal with excess removal occurring across the center line. The discrepancy between center and edge removal depth, caused by the progressive increase in track spacing along the surface, represents over 10% of total removal depth. Horizontal raster marks, with an amplitude around 15 nm are also visible. In the case of circular-random path, Fig. 9(b) shows a more uniform distribution of material with a slight excess removal at the top and bottom. The discrepancy represents around 2% of total removal depth. The 3σ of surface waviness is 88.0 nm for raster and 22.7 nm for circular-random path. It should be noted here that a method to equalize raster track spacing was recently published in the literature [17]. But while this algorithm is useful for reduction of residual form error on surfaces with constant curvatures (spheres), the principle is not applicable to truly freeform surfaces (for example, containing saddle points).
2D Fourier transform analysis was carried out on the residual maps in order to identify the location and amplitude of peak waviness. Figure 10 shows the resulting data in a disk corresponding to wavelengths up-to 1 mm. A strong peak is visible around 0.8 mm wavelength at 90° direction in Fig. 10(a), which is associated with the raster lines observed in Fig. 9(a). By comparison, the spectral amplitude of circular-random path shown in Fig. 10(a) consists of 6 areas around the 0.8mm wavelength, broadly distributed across the (-180°:180°) range. Consequently, the peak amplitude is around 1 order of magnitude less than that of the raster path. This analysis is confirmed with a comparison of 1D power spectral density for section profiles in the vertical direction, as shown in Fig. 10(c), in which the peak at 0.8mm is again significantly suppressed. By making the spectral power less concentrated on a specific direction, it is expected that the random path should suppress the appearance of undesirable interference and diffraction patterns during light propagation through an optical system.
4. Actual polishing of aspherical optic by circular-random path
4.1 Experimental setup
Polishing experiments by raster and circular-random path were carried out on commercial aspherical condenser lenses (Thorlabs AL2550G with aspheric surface accuracy < 55nm Peak-to-Valley) using the 7-axis FJP machine (Zeeko IRP200) shown in Fig. 11(a). The same nozzle and processing conditions were used as in the generation of the polishing footprint in section 3.1. Raster and circular-random paths were generated with a track spacing of 0.35 mm (i.e. the total path length was identical), while a constant surface feed of 10 mm/min resulted in a 170 min total processing time for each lens. Such long polishing time was specifically selected to accentuate the amplitude of MSF content relative to the original surface accuracy of the lens.
In order to assess the optical performance of polished lenses, light from a 520 nm laser source was expanded to an aperture of 20 mm diameter and sent with normal incidence to the aspherical condenser, as shown in Fig. 11(b). A Charged Coupled Device (CCD) was placed in the back focal plane to measure the distribution of light.
4.2 Results and analysis
The intensity of light recorded on the CCD was normalized and plotted in Fig. 12. For both lenses, the majority of light was condensed within a Gaussian shaped spot about 40 µm wide. Overall, the fraction of total light intensity inside this central spot was 53.9% for the raster polished lens, and 51.0% for the circular-random path polished lens.
Fig. 12. Normalized intensity of light collected by the aspheric condensers in the focal plane, with associated diffraction satellites order.
However, the lens polished by raster path features very strong 1st order satellites in the top-left and bottom-right vicinity of the central area. These attain an average fraction of total intensity of 8.9%. Further out in the same direction, 2nd and 4th order satellites are also significant with an average 1.4% and 0.6% fraction of total intensity. The rest of the light is diffusively distributed across an area 3 times the central spot size.
By comparison, the lens polished by random path shows a more nebulous distribution of stray light. 1st order satellites are observed in 6 evenly spaced directions, attaining an average fraction of total intensity of 1.3%. 2nd order satellites are observed in another 6 directions rotated 30° from the 1st order. They attain an average fraction of total intensity of 0.7%. Meanwhile, 3rd and 4th order satellites are barely visible and collect an insignificant fraction of light.
All results were compiled in Table 2. In summary, while the fraction of stray light is similar for both lenses, the raster polished lens concentrates the majority of diffracted light inside 2 satellites of 1st order. By comparison, about half the equivalent fraction of light is distributed across 12 satellites of 1st and 2nd order for the circular-random polished lens, with the rest diffusely distributed around the central spot. It can thus be said that MSF related light is more diffuse, less concentrated, in the case of circular-random polished lenses. In practical application, such lens is less likely to produce noticeable imaging artifacts and satellites.
Table 2. Fraction of total light intensity in regions of interest.
5. Conclusion
A novel algorithm that extends circular pseudo-random tool paths to non-planar surfaces was proposed and demonstrated. Starting from an initial point distribution, generated either with an icosphere (for near-spherical surfaces) or in the UV space of a NURBS (for general freeforms), a virtual force field was used to equalize the point distribution and place tool path elements across the entire surface.
In polishing simulations on a spherical surface, the non-repeating circular paths were found to reduce the relative residual error from 10% down to 2% of total material removal depth, and peak spectral amplitude by 1 order of magnitude, when compared to raster paths. Actual polishing was carried out on an aspheric condenser lens, showing a significant reduction in the intensity fraction of 1st order satellites from 8.9% down to 1.3%.
Circular-random paths can therefore be considered a promising method for final corrective finishing with sub-aperture polishing tools, especially in the case of next generation EUV and X-ray mirrors for application to lithography, high energy beams and astronomy, in which suppression of MSF related artifacts is a crucial technological requirement.
Funding
Japan Society for the Promotion of Science (20K04192).
Acknowledgments
The authors acknowledge support from a donation fund by DMG Mori Seiki Co., and the loaning by Zeeko Ltd. of the IRP200 polishing machine on which experiments were conducted.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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