Robust and precise algorithm for aspheric surfaces characterization by the conic section
© The Author(s) 2017
Received: 19 January 2017
Accepted: 3 April 2017
Published: 11 April 2017
A new algorithm for precise characterisation of rotationally symmetric aspheric surfaces by the conic section and polynomial according to the ISO 10110 standard is described.
The algorithm uses only the iterative linear least squares. It uses fitting the surface form in a combination with terms containing its spatial derivatives that represent infinitesimal transformations of form.
The algorithm reaches sub-nanometre residuals even though the aspheric surface is translated and rotated in the space.
he algorithm is computationally robust and an influence of local surface imperfections can be easily reduced by use of a criterion for residuals.
Radius of curvature
Parameters of aspherical surfaces from 
Rotations and translations
The algorithm from previous section and also from e.g.  solves the problem where the vertex of aspheric surface is in the origin of coordinates and the surface is not rotated. However, it is not the case for measurement results (3D data of [x, y, z] coordinates) that are generally in an arbitrary coordinate system. This problem can be solved by the following way for relatively small rotations (up to few tens of degrees) and translations (up to few tenths of optical element size).
The rate of convergence of these iterations is superlinear (better for smaller transformations) and faster for higher degree of even-power polynomial. The convergence is slower for larger data sets. Nevertheless, the polynomial degree of 18 (i = 9) is sufficient for iteration convergence of all cases that have been tested (It also includes parameters of various real aspheric lens from different producers.). In the case 1, which is an extreme hyperbolic case with k equal about -171 (not a real lens case), the position error is larger. However, it can be reduced by reduction of the radial range of data or by increasing of polynomial degree.
It should be also noted that in case of large dataset (millions of coordinates), the iterative process could be carried out with some randomly selected part of data (e.g. every thousandth) due to the low roughness of optical surfaces. It speeds up the calculations because the algorithm complexity is given by the linear least squares and thus its calculation time is linearly proportional to the number of data points. The results of such pre-calculation of transformation and form parameters are used for modification of the initial state of algorithm for subsequent calculation with the full dataset that will describe the form more precisely.
The local imperfections in data arise from measurement outliers or effect of dust, marks and/or scratches. These imperfections influence the fitting algorithm results although the area of such imperfection is often relatively small. The lens parameters should be evaluated more precisely without an influence of these data (e.g. the focal point of lens). The optical aperture is not so affected if such data from optically not usable areas are removed. The following algorithm with the residual criterion can be used to solve this problem effectively and to keep the robustness of the presented algorithm.
All iteration steps for evaluation of transformation parameters that was described in the previous section will contain the criterion for the correct data. If the residuals of given points will be greater than e.g. 10σ (where σ is the standard deviation in z-direction for each iteration step) of least squares residuals then these points will be removed from the data set.
We can clearly see that the error of estimated radius is proportional to the height of such imperfection if it is not filtered out (the filter factor is too large). However, the form is fitted correctly for lower filter factors (such as 10σ in this case). Nevertheless, the filter factor cannot be too small because too many points will be excluded from calculations. Thus some compromise must be made and the corresponding filter factor value is different for different aspheric surfaces and for different imperfections. In the case of min-max algorithm (e.g. in ) the form error is large if 0% of points is excluded. However, it can be significantly smaller if some points are excluded (e.g. about 5%). The parameters obtained from such fitting with a reasonable filter factor can be used to display surface deviations for all points and the standardized form error can be evaluated from these residuals.
Another example of imperfection is measurement noise or roughness of surface. The noise in [x,y,z] data coordinates stops the algorithm convergence. Nevertheless, the resulting error of parameters corresponds to the uncertainty arising from this noise and the algorithm robustness is not affected.
Results and discussion
The results clearly showed that algorithm allows finding positions of aspheric lenses of unknown form with sub-nanometre accuracy and it was achieved only by least squares fitting. The fitted form for various aspheric surfaces reached residuals also at sub-nanometre level. The non-linear least squares methods (such as in ) allows less number of iterations. Nevertheless, linear least squares method, presented here, calculates each step faster and its convergence is robust as it was shown on various examples. All imperfections can be effectively filtered out and precise parameters of lens could be easily obtained. The wide range of algorithm properties such as speed, robustness and effective filtering enables its use in automatic processes.
The algorithm for the evaluation of parameters of rotationally symmetric aspheric surfaces was described and tested. The sub-nanometre precision of fitting was reached with this new robust and fast algorithm. The algorithm also does not need precise estimation of parameters for the initial iteration. It allows ISO 10110 characterization with the conic section parameters of surfaces that are rotated and translated as it is common in the data output from measurement devices and alignment markers on lens are not needed. The additional criterion for residuals can effectively remove unuseful data to keep results for the main part of aperture unaffected by local imperfections. Moreover, the part of algorithm that finds the correct coordinate system and removes outliers can be used independently on the conic section fitting part. Thus the algorithm will be useful for the optical community for precise characterisation and testing of aspheric lens.
I would like to thank Pavel Mašika for comments and help with testing.
This work was done in the EMPIR project 15SIB01 FreeFORM. The EMPIR initiative is co-founded by the European Union’s Horizon 2020 research and innovation programme and the EMPIR Participating States. The CMI participation in the project is co-funded by the Ministry of Education, Youth and Sports of the Czech Republic (8B16008).
The author declares that he has no competing interests.
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