Now, researchers at Results in Optics report a unified computational framework that tackles this problem head-on. Dahi Ghareab Abdelsalam Ibrahim and Nicholas Devaney have combined three established techniques into a single workflow: a Profile Rotation Model that dissects a two-dimensional surface into one-dimensional diametrical profiles, a Root Bracketing Model that detects curvature sign changes directly from discretely sampled data, and a ray-traced fringe simulation coupled with Zernike polynomial fitting that reconstructs the surface from optical interference patterns. The aim, the authors emphasize, is not to invent new mathematics but to integrate proven tools into a pipeline robust enough to survive realistic metrology conditions, where noise, discretization and incomplete aperture access sabotage straightforward analytical approaches.

The heart of the method is the Root Bracketing Model, a deliberately simple alternative to iterative root-finding techniques such as Newton-Raphson. Instead of solving for exact zeros of a second derivative, the model scans along each sampled profile and flags locations where the second derivative changes sign between adjacent sampling points, effectively bracketing the curvature transition within a known interval. This initial-guess-independent approach sidesteps a major weakness of iterative methods, which can converge to wrong answers or fail outright when noisy data produces false zero-crossings. The accuracy of the bracketing technique is governed by spatial sampling resolution, meaning finer discretization yields sharper localization of inflection points.

To validate the approach, the team simulated a 111-millimeter circular freeform surface described by an eighth-degree polynomial containing both even and odd terms, producing asymmetric sag variations with a maximum of roughly 2.84 millimeters. They then contaminated the surface with realistic imperfections: a 76-nanometer RMS low-spatial-frequency figure error mimicking grinding and polishing deviations, and a 0.5-nanometer RMS surface roughness representing micro-scale texture. The Profile Rotation Model extracted 180 diametrical sag profiles from this noisy surface, and the Root Bracketing Model identified curvature transitions across them, with most profiles exhibiting two dominant inflection regions. These were reassembled into a two-dimensional curvature map revealing precisely where the surface changes its bending character.

Benchmarking against Newton-Raphson refinement showed remarkable agreement. For a representative profile, the bracketing method placed inflection points at 45.44 and 74.00 millimeters, while Newton-Raphson yielded 45.48 and 74.07 millimeters, a difference of just 0.04 to 0.07 millimeters, comfortably within the sampling resolution. The bracketing approach was also faster, averaging 0.000191 seconds per computation compared with 0.000275 seconds for Newton-Raphson, a speed-up of about 1.44 times. Monte Carlo trials further probed robustness: at noise levels of 0.5 micrometers or below, curvature transitions were detected successfully in every trial across sampling intervals from 0.1 to 2.0 millimeters, and detection remained fully reliable at moderate noise when combined with Gaussian smoothing and spatial separation constraints.

The optical side of the framework is equally ambitious. The researchers built a custom ray-tracing code that propagates roughly 20 million rays through a multi-surface system containing the freeform surface, computing reflections, transmissions via Snell’s Law, optical path lengths and phase, ultimately generating intensity and phase fringe patterns at two wavelengths. At the familiar helium-neon wavelength of 632.8 nanometers, the large sag produced approximately 4,488 fringes per sag direction, an unwieldy density for reconstruction. Retracing at 0.284 millimeters, a terahertz-range wavelength used purely as a numerical scaling strategy, collapsed the fringe count to about 20, enabling stable phase extraction. Fringe thinning algorithms then skeletonized the patterns, and Zernike polynomial fitting with 48 coefficients reconstructed surface heights.

A subtle problem emerged during reconstruction: fitting diametrical profiles with Zernike polynomials produced wild errors, yielding a peak-to-valley height of 28.7 millimeters on a surface whose true value was about 3.5 millimeters. The culprit was asymmetry, since fringe counts on left and right halves differed, corrupting the fit. The solution was elegant: divide the intensity map into 360 radial profiles extending from center to edge, reconstruct each independently with Zernike fitting, then combine them through the Profile Rotation Model. This radial approach preserved the surface’s asymmetric geometry and reduced deviations to tens of micrometers, roughly 1.16 percent of the maximum simulated height, though accuracy still depended on having sufficient fringe information in each profile.

Experimental validation followed on a genuinely measured surface. Using a Fizeau interferometer with 1-inch reference optics, the team captured a four-fringe interferogram at 632.8 nanometers from a mildly freeform, asymmetric optical surface with a 25.4-millimeter clear aperture. Zernike reconstruction revealed a clear asymmetry, with the right half shorter than the left, and curvature analysis located a single inflection point along the asymmetric cross-section. Strikingly, when the most symmetric measured profile was identified by mirror-comparison analysis and rotationally replicated, the resulting synthetic surface produced a symmetric ring-like inflection distribution, exactly as geometry dictates. This correspondence between surface shape and detected curvature transitions demonstrates that the framework captures real geometric structure rather than numerical artifacts.

The practical implications extend to manufacturing. By integrating the difference between measured profiles and smooth fits, the workflow quantified surface imperfections across four distinct regions, some exhibiting deviations between 1 and 5 micrometers, others exceeding 5 micrometers. The authors map these findings to corrective strategies: moderate imperfections suit computer-controlled optical surfacing or magnetorheological finishing, while significant ones demand precision machining such as single-point diamond turning or CNC grinding. Inflection maps themselves can guide polishing, since high-curvature transitions near surface edges are natural targets for fine abrasive treatment.

The team is candid about limitations. Inflection-point distributions are not directly measurable observables but computationally derived features, so experimental validation remains indirect, and the detected transition count depends on chosen smoothing scales and sampling resolution. The eighth-degree polynomial used for profile fitting is an empirical choice for these surfaces, not a universal requirement, and computational cost, roughly 90 hours of ray tracing on a standard laptop, invites optimization through parallelization. Still, the framework offers something freeform optics has lacked: an initial-guess-independent, symmetry-agnostic way to extract curvature-transition information from the messy, discrete, partially covered data that real metrology actually produces. As freeform optics continues its march from specialty applications toward mainstream design, tools like this one promise to make these sculpted surfaces not only easier to make, but easier to truly understand.

Subject of Research: Detection and reconstruction of inflection points on freeform optical surfaces using root-bracketing, ray-traced fringe simulation, and Zernike polynomials

Article Title: Inflection point detection and reconstruction on freeform surfaces using the root-bracketing model, ray-traced fringe simulation, and Zernike polynomials

Article References: Ibrahim, D. G. A., & Devaney, N. (2026). Inflection point detection and reconstruction on freeform surfaces using the root-bracketing model, ray-traced fringe simulation, and Zernike polynomials. Results in Optics, 25, Article 101154. https://doi.org/10.1016/j.rio.2026.101154

Image Credits: AI Generated

DOI: 10.1016/j.rio.2026.101154

Keywords: freeform optics, inflection points, curvature transition, Root Bracketing Model, Profile Rotation Model, Zernike polynomials, ray tracing, fringe simulation, Fizeau interferometry, surface metrology, terahertz wavelength, optical surface reconstruction