Abstract
Ultra-high precision measuring machines enable to measure aspheric shapes with an uncertainty of few tens of nanometres. The resulting clouds of points are then associated to theoretical model at the same level of accuracy so as to obtain parameters that indicate about form error. Minimum zone (MZ), defined as the least value of peak to valley (PV), is widely used to assess form error. Least squares method (L2) is often used to determine MZ but the resulting value is usually overestimated. For this reason, L2 is replaced by L∞ norm because it gives a more accurate value of MZ since it directly minimizes PV. Using L∞ norm results in a non-smooth optimization problem and consequently its resolution becomes more challenging compared to L2.In this paper, a novel minimax fitting method for accurate metrology of aspheres and freeform based on a hybrid trust region algorithm (HTR) is proposed. To assess performance of the introduced method, it was compared to an available minimax fitting algorithm based on a smoothing technique: exponential penalty function (EPF). The choice of EPF is justified by superior performances in comparison to existing techniques. Comparison was conducted on reference data, data available in literature and data gathered form measurements of a real optical high quality asphere. Results show superiority of HTR over EPF in both returned PV values and execution time.
Highlights
Aspheres and freeform optics have replaced spherical components in several optical systems due to their superiority over classical elements especially for eliminating spherical aberrations [1]
Aspherics for instance, data gathered from ultra-high precision CMMs must be treated in a way to give parameters that indicate about tolerance zone
In order to assess the performance of the algorithm proposed in regards to exponential penalty function algorithm (EPF) [23], each generated dataset were submitted at the same time to both algorithms: EPF and hybrid trust region (HTR)
Summary
Aspheres and freeform optics have replaced spherical components in several optical systems due to their superiority over classical (spherical) elements especially for eliminating spherical aberrations [1]. Aspherics for instance, data gathered from ultra-high precision CMMs must be treated in a way to give parameters that indicate about tolerance zone One of these parameters is usually taken as the peak to valley (PV). L2 usually overestimates MZ which causes the rejection of a number of conforming parts In another way, L∞ criterion results in a direct minimization of PV and returns the closest value of MZ to actual. L∞ criterion results in a direct minimization of PV and returns the closest value of MZ to actual In this context, a European project 15SIB01-FreeFORM was launched in 2016 to develop reference L∞ fitting algorithms and traceable metrology for aspheres and freeform optical lenses with below 30 nm accuracy [5]. An investigation of a real case study of a measured high quality optical asphere is illustrated
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