Measurement Of Freeform Surfaces Research Articles

Results of round robin form measurements of optical aspheres and freeform surfaces

High-quality aspherical and freeform surfaces are in high demand, and the high-accuracy form measurement of such surfaces is a challenging task. To explore the current status of form measurement systems for complex surfaces such as aspheres and freeforms, interlaboratory comparison measurements are performed. This study presents the pseudonymized results obtained using three different surfaces (metal asphere, glass asphere, toroidal surface) in a total of six different round robins. These results were taken from a total of 13 different measurement instruments based on 9 different measurement principles and operated at 12 different laboratories. They were analyzed using a sophisticated procedure that was first developed in 2018 and then refined and tested on simulated data in 2022 to address the challenges of such a comparison at this level of accuracy. In the current study, we applied these refined methods to data acquired from tactile and optical point measurements as well as from optical areal measurements. As there are no absolutely measured and very well characterized reference standard aspherical and freeform surfaces available at the accuracy level of a few tens of nanometers root-mean-square, the approximated true forms of the surfaces were derived from the measurements and indicate the manufacturing accuracy of the surface forms. Then, the measurement’s differences to the approximated true forms were analyzed, which directly indicate the systematic measurement errors of the instruments. By also comparing the approximated true forms from the two different round robins for each surface, additional insights into the reliability and stability of these so-called virtual reference topographies were gained.

Vortex phase-based adaptive moiré technique accommodating micro displacement measurement of freeform surfaces.

The traditional displacement measurement interferometer (DMI) provides elegant performance by straight interference fringe movement counting to convert a phase calculation into an image motion calculation. However, it cannot be applied to a curve surface displacement measurement. The counting of the movement of irregular fringes is not achievable. We provide an adaptive moiré technique with a vortex phase to realize micro displacement measurement of a freeform surface with any continuous shape. The technique produces straight moiré fringes that rotate in a circle regardless of the shape of interference fringes and tested surface shapes. The vortex phase is used to record only one interferogram before the measurement for subsequent data processing, and then it no longer participates in the displacement measurement process. Therefore, this technology can be employed to remold traditional DMIs. Simulations and experiments validating the method are presented.

Bayesian uncertainty evaluation applied to the tilted-wave interferometer

The tilted-wave interferometer is a promising technique for the development of a reference measurement system for the highly accurate form measurement of aspheres and freeform surfaces. The technique combines interferometric measurements, acquired with a special setup, and sophisticated mathematical evaluation procedures. To determine the form of the surface under test, a computational model is required that closely mimics the measurement process of the physical measurement instruments. The parameters of the computational model, comprising the surface under test sought, are then tuned by solving an inverse problem. Due to this embedded structure of the real experiment and computational model and the overall complexity, a thorough uncertainty evaluation is challenging. In this work, a Bayesian approach is proposed to tackle the inverse problem, based on a statistical model derived from the computational model of the tilted-wave interferometer. Such a procedure naturally allows for uncertainty quantification to be made. We present an approximate inference scheme to efficiently sample quantities of the posterior using Monte Carlo sampling involving the statistical model. In particular, the methodology derived is applied to the tilted-wave interferometer to obtain an estimate and corresponding uncertainty of the pixel-by-pixel form of the surface under test for two typical surfaces taking into account a number of key influencing factors. A statistical analysis using experimental design is employed to identify main influencing factors and a subsequent analysis confirms the efficacy of the method derived.

DYnet++: A Deep Learning Based Single-Shot Phase-Measuring Deflectometry for the 3-D Measurement of Complex Free-Form Surfaces

DYnet++: A Deep Learning Based Single-Shot Phase-Measuring Deflectometry for the 3-D Measurement of Complex Free-Form Surfaces

Spatial-frequency-oriented measurement strategy in two-dimensional slope deflectometry systems.

A single-point-probe-based slope profiler is a common measurement scheme for the measurement of freeform optical surfaces, which has been a challenging research direction. Efficiency is a key issue in two-dimensional scanning-based measurement. This study establishes a measurement system simulation model and reveals that the height reconstruction accuracy of different reconstruction algorithms is primarily correlated with the sampling density. The spatial resolution calibrated of the slope measurement device is also identified to be an essential part of the strategy. Based on a kind of slope profiler, this paper applies variable sampling intervals for different spatial frequency characteristics of the surface under test (SUT). The result shows that the reconstruction accuracy can be controlled by selecting sampling parameters and calibrating the slope measurement device. For objects with different spatial characteristics, targeted optimization of the measurement scheme can be achieved. This strategy also has a certain universality for general scanning slope measurement and height reconstruction, providing a reference for device selection and sampling settings for different spatial frequency measurement requirements.

Influences of fluorescence film thickness and optical aberrations on curved surface metrology in fluorophore-aided scattering confocal microscopy

Fluorescence film thickness and optical aberrations are important systematic error sources of curved surface metrology in fluorescence-aided scattering confocal microscopy (FSCM). In this paper, we established a tilted surface measurement model in FSCM to analyse the influence of fluorescent film thickness and aberrations on the measurement accuracy under different local slope angles of a curved surface. The simulation results demonstrate that the systematic errors caused by the fluorescence film thickness on free-form surface measurements increase with the local slope angle of the surface. The systematic errors caused by optical aberrations will lead to significant deviations between the measured and the actual surface when the local slope angle is greater than the complementary angle of the half aperture angle of the objective lens. Our studies allow for analysing the systematic errors caused by fluorescence film thickness and optical aberrations in different situations, thus contributing to the uncertainty evaluation of free-form surface measurement by FSCM.

Efficient sampling method based on co-kriging for free-form surface measurement

Numerous studies on sampling methods have been conducted to balance the inspection efficiency and accuracy of coordinate measuring machines (CMMs). However, almost all existing methods only take into account the information from the processed surface during the reconstruction process. To address this limitation, an adaptive sampling method based on co-kriging method is proposed, according to the characteristics of the interpolation reconstruction method, we treat certain feature points on the surface as anchor points and determine the distribution of other measurement points through iteration. In the reconstruction process, we employ the co-kriging interpolation method and combine extra information of the model surface to enhance the reconstruction accuracy. Through simulations and actual measurements of processed surfaces, we compare the proposed method with other algorithms. The results indicate that our method is particularly suited for complex surfaces with a limited number of measurement points, and can substantially reduce the reconstruction error when the number of measurement points is the same.

Differential Confocal Optical Probes with Optimized Detection Efficiency and Pearson Correlation Coefficient Strategy Based on the Peak-Clustering Algorithm.

Quantifying free-form surfaces using differential confocal microscopy can be challenging, as it requires balancing accuracy and efficiency. When the axial scanning mechanism involves sloshing and the measured surface has a finite slope, traditional linear fitting can introduce significant errors. This study introduces a compensation strategy based on Pearson's correlation coefficient to effectively reduce measurement errors. Additionally, a fast-matching algorithm based on peak clustering was proposed to meet real-time requirements for non-contact probes. To validate the effectiveness of the compensation strategy and matching algorithm, detailed simulations and physical experiments were conducted. The results showed that for a numerical aperture of 0.4 and a depth of slope < 12°, the measurement error was <10 nm, improving the speed of the traditional algorithm system by 83.37%. Furthermore, repeatability and anti-disturbance experiments demonstrated that the proposed compensation strategy is simple, efficient, and robust. Overall, the proposed method has significant potential for application in the realization of high-speed measurements of free-form surfaces.

An adaptive sampling method for STL free-form surfaces based on the quasi-Gauss curvature grid

ABSTRACT The contact on-machine measurement can be directly used to detect and analyze the computerized numerical control machining quality of parts. To solve the problems of unreasonable distribution of sampling points and omission of sampling area in traditional sampling methods, an adaptive sampling method of the stereolithography (STL) free-form surfaces based on the quasi-Gauss curvature grid is proposed. First, on the quasi-conformal rectangular mapping region of the triangular mesh surface, a globally optimal centroidal Voronoi tessellation is generated with discrete Gauss curvature as the region density. Then, based on the quasi-Gauss curvature grid generated by subdividing the centroid of the centroidal Voronoi tessellation through a quadtree method, several types of feature points are reasonably taken as measuring points to realize the adaptive sampling. Finally, the experiments show that, for the on-machine measurement of STL free-form surfaces, the quasi-Gauss curvature grid sampling strategy can make the distribution of measuring points more reasonable and practical.

On-axis deflectometric system for freeform surface measurement.

We propose an on-axis deflectometric system for the accurate measurement of freeform surfaces with large slope ranges. A miniature plane mirror is attached on the illumination screen to fold the optical path and achieve the on-axis deflectometric testing. Due to the existence of the miniature folding mirror, the deep-learning method is applied to recover the missing surface data in a single measurement. Low sensitivity to the calibration error of system geometry and high testing accuracy can be achieved with the proposed system. The feasibility and accuracy of the proposed system have been validated. The system is low in cost and simple in configuration, and it provides a feasible way for the flexible and general testing of freeform surfaces, with a significant potential of the application in on-machine testing.

An Adaptive Hybrid Sampling Method for Free-Form Surfaces Based on Geodesic Distance

High precision geometric measurement of free-form surfaces has become the key to high-performance manufacturing in the manufacturing industry. By designing a reasonable sampling plan, the economic measurement of free-form surfaces can be realized. This paper proposes an adaptive hybrid sampling method for free-form surfaces based on geodesic distance. The free-form surfaces are divided into segments, and the sum of the geodesic distance of each surface segment is taken as the global fluctuation index of free-form surfaces. The number and location of the sampling points for each free-form surface segment are reasonably distributed. Compared with the common methods, this method can significantly reduce the reconstruction error under the same sampling points. This method overcomes the shortcomings of the current commonly used method of taking curvature as the local fluctuation index of free-form surfaces, and provides a new perspective for the adaptive sampling of free-form surfaces.

Multiple wavelength digital holography for freeform shape measurement and lens alignment.

This paper introduces a technique for freeform optical surface measurements using digital holography with multiple discrete wavelengths or wavelength scans. This experimental arrangement, i.e.,a Mach-Zehnder holographic profiler, is optimized to achieve maximal theoretical precision and is capable of measuring freeform diffuse surfaces. Moreover, the approach can also be used for diagnostics of precise placement of elements within optical systems.

Advances in Measurement and Error Evaluation Technique of Optical Freeform Surfaces

Advances in Measurement and Error Evaluation Technique of Optical Freeform Surfaces

Measurement point layout strategy of free-form surface based on gridding using coordinate measuring machine

The layout of measurement points is the key to the efficient inspection of free-form surfaces. Two algorithms are proposed for the layout of free-form surface measurement points: the free-form surface Gaussian curvature variation grid method and the isoparametric line curvature variation grid method. The former first divides the free-form surface into a uniform grid, determines the number of measurement points for each grid based on the change in the Gaussian curvature of each grid, and selects points within each grid based on a uniform distribution of the Gaussian curvature. The latter is achieved by first taking points from the curvature change on the initial U- and V-direction isoparametric lines, generating isoparametric lines from the points to divide the free-form surface into a grid, and selecting the grid intersections as measurement points. The effectiveness of the algorithm was verified by designing free-form surfaces, performing coordinate measuring machine (CMM) measurement experiments, and comparing the results with those of existing algorithms.

Measurement and Characterization of Position and Posture of Conjunctive Multifreeform Surfaces Using Multisensors

Conjunctive multi-freeform surfaces play an important role in many advanced optical systems. The position and posture accuracy of freeform surfaces are the key to system performance. However, there is no effective method of measuring the positions and postures of multiple freeform surfaces. In this study, a conjunctive multi-freeform surface measurement system based on multi-sensors is developed, and a trust region registration based on the least squares method is proposed, with the aim of registering multiple freeform surfaces at different positions and postures. The performance of the proposed system was analyzed and the errors were calibrated. The calculation accuracy of the registration method was within 20 nm at Φ50 mm in terms of root-mean-of-squares (RMS) values. The positioning accuracy of the freeform surface measurement system proved to be 1.2 μm within the area of 120 mm × 120 mm; the related uncertainty was also analyzed. The proposed measurement approach provides a comprehensive solution to optical conjunctive multi-freeform surfaces.

Real-time 3D measurement of freeform surfaces by dynamic deflectometry based on diagonal spatial carrier-frequency pattern projection

Deflectometry is a three-dimensional optical technique based on structured light projection for measuring and inspecting specular freeform surfaces. Conventional deflectometry method needs complicated calibration and time-consuming phase-extraction algorithm, which are not suitable for industrial metrology requiring dynamic and high-speed measurement. In this paper, a novel dynamic deflectometry method with a simple system setup and calibration is proposed for the three-dimensional measurement of specular freeform surfaces with high speed and high accuracy. We developed a very simple calibration method using a reference flat mirror which helps to save calibration time and computer resources. We also proposed a one-shot method by using a spatial carrier-frequency diagonal fringe pattern for screen projection along with an algorithm to separate a diagonal phase into vertical and horizontal phases. We extensively evaluated the feasibility of proposed calibration and one-shot method by different simulations and measuring several specular surfaces experimentally. Our novel deflectometry gives similar measurement accuracy compared to traditional contact measurement technique while offering much higher measurement speed. Dynamic surface shape measurement of silicon wafer due to temperature’s variation proved the real-time measurement ability of our proposed method.

Fabrication of soft x-ray monolithic Wolter mirror based on surface scanning measurement using touch probe.

The monolithic Wolter mirror is an ideal optical device for focusing soft x rays to a submicron-sized spot, with the advantages of high efficiency, large acceptance, achromaticity, and robustness to alignment error. The fabrication process for this type of mirror has not been established because of the difficulty in highly accurate figure measurement of free-form surfaces with small radii of curvature and steep profiles. In this study, we employed tactile scanning measurement for surface characterization to fabricate a high-precision Wolter mirror. First, it was demonstrated that the touch probe measurement did not leave scratches on the raw surface of the mirror substrate. Next, the measurement capability of the surface profiler was assessed, and the data analysis conditions were determined. Finally, the Wolter mirror was fabricated through repeated figure correction based on the tactile measurement, and the figure error of the final surface was evaluated. Wave-optical simulations that used this error as reference suggested that the size of the beam focused by the mirror was equivalent to the theoretical value at 1000 eV. The reflected image with uniform intensity distribution obtained at SPring-8 also revealed the effectiveness of the present fabrication approach based on tactile measurement.

Robust and accurate measurement of optical freeform surfaces with wavefront deformation correction

The non-null test to detect the modulated wavefront is a widely used method in optical freeform surface measurement. In this study, the wavefront deformation in the non-null test of an optical freeform surface measurement was corrected based on the wavefront propagation model to improve measurement accuracy. A freeform surface wavefront correction (FSWC) measurement system was established to validate the proposed method. Simulation and experimental studies indicated that the proposed method can reduce the influence of freeform surface wavefront deformation in space propagation. Moreover, the freeform surface form accuracy measured by FSWC can reach a root-mean-squared value of 10 nm.

Development of a metrological reference system for the form measurement of aspheres and freeform surfaces based on a tilted-wave interferometer

Aspheres and freeform surfaces play an important role in today’s optics industry. However, the measurement of such complex surfaces is still challenging even with state-of-the-art manufacturing technology, and there is an urgent need in industry for a non-contact, highly accurate reference measurement technique. To meet this demand, at Physikalisch-Technische Bundesanstalt (PTB), a metrological reference system for the contact-free form measurement of aspheres and freeform surfaces is under development. The measurement system is based on a tilted-wave interferometer. Advances in computational capabilities have made it possible to solve the complex inverse problems associated with this measurement system and to develop sophisticated analysis procedures for reconstructing the surface under test from the measured interferogram data. In this paper, we will present the status of the tilted-wave interferometer-based measurement system at PTB, describe the analysis procedures we have designed and show initial measurement results. The benefit of the implementation presented here is that it allows insight to be gained into the performance of the measurement system and enables traceable measurements to be established with low uncertainty.

Measurement of Small-Slope Free-Form Optical Surfaces with the Modified Phase Retrieval.

In this paper, we demonstrate the use of the modified phase retrieval as a method for application in the measurement of small-slope free-form optical surfaces. This technique is a solution for the measurement of small-slope free-form optical surfaces, based on the modified phase retrieval algorithm, whose essence is that only two defocused images are needed to estimate the wave front with an accuracy similar to that of the traditional phase retrieval but with less image capturing and computation time. An experimental arrangement used to measure the small-slope free-form optical surfaces using the modified phase retrieval is described. The results of these experiments demonstrate that the modified phase retrieval method can achieve measurements comparable to those of the standard interferometer.