Deterministic multi-tool polishing of large-scale aspheric optics converges figure error efficiently inside the corrective band of the available tool influence functions but leaves a persistent residual plateau at higher spatial frequencies. Du et al. (Optik 216, 2020) combined a figuring-ability (FA) tool-selection metric, low-pass-filtered dwell-time computation, and smoothing polishing, and demonstrated it on a Φ540 mm SiC off-axis asphere (0.099λ to 0.018λ RMS in about 33 h versus about 200 h for experience-driven polishing), yet spectral components above ~15.6 mm⁻¹ never converged, the FA thresholds were calibrated only to one TIF library and one mirror, dwell realization remained sensitive to machine kinematics, and no adaptive in-process re-selection existed. Later literature advanced the tractable strands: dwell-time scheduling under dynamic machine constraints, learned and self-adaptive multi-tool decision-making for mid/low bands on sub-meter optics, staged multi-size-tool correction scaled to a 4 m SiC mirror, and band-targeted mid-spatial-frequency suppression on small hard-material optics. However, no published workflow converges the full mid-to-high frequency band, including content above the tools' corrective cutoff, on a large aspheric within a throughput-constrained schedule, and no selection criterion is validated to generalize across optics, TIF libraries and machine platforms while driving automated full-band adaptive re-selection. That core remains open.

Background

Large telescopes and space optical systems increasingly depend on large-scale aspheric mirrors (for example off-axis paraboloids of half-meter class and above), often made of silicon carbide (SiC), a stiff, lightweight but very hard material that is difficult to polish. Such mirrors are finished by computer-controlled optical surfacing (CCOS): a sub-aperture polishing tool is swept over the surface, dwelling longer where more material must be removed. Under Preston's equation, the removed material is the convolution of the dwell-time distribution with the tool influence function (TIF), the spatial material-removal profile of the tool. Every TIF has a size and hence a corrective cutoff spatial frequency: surface-error components with periods shorter than roughly the tool size are invisible to it. Smaller tools reach higher spatial frequencies but need far longer dwell times, so achievable precision and fabrication throughput trade off directly. Mid-spatial-frequency (MSF) errors - quasi-periodic surface waviness with periods from a few millimeters down to tenths of a millimeter - scatter light out of the image core and degrade encircled energy and contrast of the optical system; they are not only left uncorrected but generated and aggravated by the process itself through tool-path periodicity, machine vibration, tool wear, dwell-time discretization and realization errors, and edge effects. Error content is quantified by the power spectral density (PSD) of measured surface maps, and overall figure by the root-mean-square (RMS) surface error, conventionally quoted in wavelengths of a He-Ne laser (λ=632.8\lambda=632.8 nm). Magnetorheological finishing (MRF) uses a magnetically stiffened fluid spot as a small, deterministic removal function and serves as a common fine-finishing tool. Du, Song, Li, Peng, Shi and Zhao (Optik 216, 2020) addressed the long-standing absence of a quantitative rule for orchestrating tools of different sizes during fabrication. They defined a scalar figuring-ability metric FA=σTkFA=\frac{\sigma}{T}k, where σ\sigma is the simulated RMS convergence ratio a candidate TIF achieves on the current surface, TT the dwell time it requires, and kk the ratio of error RMS inside the TIF's corrective band to the RMS above that band; iterating FA over a TIF library yields a staged multi-tool (multi-TIF) schedule. They combined this with low-pass filtering of the dwell-time target at the TIF's cutoff frequency - so the solver never prescribes correction the tool cannot deliver, avoiding extreme local dwell and velocity fluctuations - and delegated residual mid-frequency content to a smoothing polishing process (SPP), in which an elastic full-contact pad preferentially reduces mid-frequency peak-to-valley error by pressure redistribution. On a Φ540 mm SiC off-axis aspheric mirror this workflow reduced the figure from 0.099λ to 0.018λ RMS in about 33 hours across three stages (CCOS tools of 20 mm and 15 mm diameter, then MRF), against roughly 200 hours for the traditional experience-driven approach, with FA-calibrated RMS thresholds for tool switching: above 0.063λ use the D20 tool, at or below 0.063λ the D15 tool, at or below 0.051λ MRF. The same paper documents what the workflow could not do. PSD analysis showed that spectral components above about 15.6 mm⁻¹ did not converge during processing: they lie above the corrective band of every TIF in the library and smoothing polishing could not remove them, leaving a persistent high-frequency residual plateau. The dwell-time solution still suffered realization error from scanning-speed discretization and machine speed and acceleration limits. The FA metric and its thresholds were calibrated only on this TIF library, mirror size, SiC substrate and coating, with no demonstration of generalization to other optics or tooling and no automation of threshold selection. And the practical sources of mid-to-high frequency content - tool wear, machine vibration, edge effects, material-removal fluctuations - were identified but not quantified as to how they perturb FA rankings or the optimal filter cutoff, leaving automated, real-time monitoring and adaptive re-selection of tools and filter parameters unrealized. These limitations jointly define the open problem: full-band spatial-frequency error control in rapid multi-tool fabrication of large aspheres.

Problem Statement

For large-scale aspheric optics fabricated by deterministic multi-tool polishing, establish a fabrication workflow and tool-selection criterion that together converge the surface-error power spectral density across the full mid-to-high spatial-frequency band — including the components that lie above the corrective cutoff of the available tool influence functions and that smoothing polishing cannot remove (reported as the persistent residual above ~15.6 mm⁻¹ on the Φ540 mm SiC aspheric mirror of Du et al. 2020) — while preserving the rapid-fabrication (throughput) objective of the multi-tool approach. The selection criterion, whether the FA metric of the source or a successor, must generalize beyond the single TIF library, mirror size, substrate and coating on which it was calibrated to other mirror sizes, materials, TIF libraries, tooling and machine platforms, remain valid under the practical error sources of production - tool wear, machine vibration, edge effects, material-removal fluctuations, and dwell-time realization error from speed and acceleration limits and tool-path discretization - and support automated, adaptive in-process re-selection of tools and filter parameters. The convergence requirement is anchored to that reported residual: it applies to the spatial-frequency content that the baseline multi-tool workflow leaves non-convergent on the target optic. The rapid-fabrication objective is likewise a binding condition rather than a quantity to report: the workflow must preserve the multi-tool approach's aim of better surface accuracy in a shorter time than traditional experience-driven polishing, not erode it.

The target object class is the large aspheric of half-meter class or above, represented by the Φ540 mm SiC off-axis aspheric mirror of Du et al. 2020, treated experimentally, analytically or numerically; the baseline surface-error spectrum and the influence functions used must be fixed independently of the desired outcome - taken from reported measurements or a published library and quoted verbatim - and must not be chosen so that a removal function with a cutoff placed above the required content satisfies the criterion by construction. The band that must converge is the full mid-to-high spatial-frequency interval, from the mid-spatial-frequency lower edge (surface periods of a few millimeters) up to the highest spatial frequency resolved by the metrology used, and its load-bearing part is the content above ~15.6 mm⁻¹, the frequency above which the baseline workflow leaves a persistent, non-convergent plateau; the requirement is met when the band-averaged surface-error PSD across that interval, and in particular the plateau above ~15.6 mm⁻¹, is shown to fall measurably relative to the baseline rather than remain flat, with the achieved levels and the reduction reported. Convergence must be realized in the surface-error PSD of the target optic itself; adopting tools or a TIF library whose corrective cutoffs lie above the plateau does not by itself achieve it. The criterion must be stated as a quantitative rule and must hold across optic size, material, TIF library and machine kinematics. The workflow, the generalizable selection criterion, validity under the named production error sources, automated adaptive in-process re-selection, and the preserved throughput objective are jointly required.

References

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