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Robust and precise algorithm for aspheric surfaces characterization by the conic section
K?en Journal of the European Optical Society-Rapid Publications
Robust and precise algorithm for aspheric surfaces characterization by the conic section
Petr K?en 0
0 Czech Metrology Institute , Okruz?ni? 31, CZ63800 Brno , Czech Republic
Background: 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. Methods: 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. Results: The algorithm reaches sub-nanometre residuals even though the aspheric surface is translated and rotated in the space. Conclusion: he algorithm is computationally robust and an influence of local surface imperfections can be easily reduced by use of a criterion for residuals.
Aspheric lens; Robust algorithm; Least squares fitting; Metrology; ISO 10110
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Background
Aspheric surfaces are recently widely used in industry. One
of their applications is aspheric lens that often needs its
precise characterisation of form. The description of the
aspheric surface by the conic section with a polynomial
correction is common in ray tracing software and in producer
specifications of aspheric lens. The conic section surface
fitting with a polynomial correction was addressed by several
authors [1?5]. Also alternative descriptions of aspheric
surfaces were introduced e.g. in [6, 7]. Nevertheless, a simple
and robust algorithm is still needed to evaluate the conic
section from measurement data in the ISO 10110-12 form.
The coordinate system is shown in the Fig. 1.
The design shape of aspheric surfaces is often described
by the z-coordinate as a function of the distance r from
z-axis in the form
X A2ir2i
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffi2ffiffi!
1??1 ? k?R2
? 21R r2 ? 18?R3k r4
where R is the radius of curvature at the vertex and k is the
conic constant (k < -1 hyperbolic, k = -1 parabolic, k > -1
elliptical, k = 0 spherical surfaces). The correction of surface
is given by the even-power polynomial with coefficients A.
Method
Radius of curvature
The expansion of (2) shows that the radius of curvature
R at the vertex (r is small) can be obtained from linear
least squares (i.e. L2-norm) simply using the first
coefficient of an even-power polynomial
The higher-order terms of Taylor series at the vertex are
negligible and the even-power polynomial with the 18th power
? The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
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z ? q~2r2 ? q~4r4 ? q~6r6 ? q~8r8 ? q~10r10 ? q~12r12
is sufficient for reduction of estimation error for R
from data with a given range of r. However, high
degree of polynomial could introduce numerical
errors. They can be reduced by the following way.
The linearity of problem allows making the second fit
for residuals z ? z?~z for initial estimate ~z obtained
from q~2i as
z ? q2r2 ? q4r4 ? q6r6 ? q8r8 ? q10r10 ? q12r12
with relatively small error because the first term of
expansion of (2) also does not depend on k. The error is
below 10?6 in relative for examples from [3] and the 18th
power polynomial, except the case 1 with relative error
0.002 for R. Thus it is a robust way to evaluate radius of
curvature at the vertex. The polynomial with coefficients
q2i also describes the aspheric surface very well for medium
precision applications (i.e. 1? flatness of wavefront).
Nevertheless, the conic section describes the aspheric form better
with less number of coefficients. For example, the Taylor
series of (2) for hyperbolic surface with high k converges
slowly and thus the even-power polynomial must have
more terms for the corresponding precision.
It corresponds to the parabolic solution from the
previous section. The convergence of iterations is worse close
to k = -1 because there is a small contribution from the
conic section (the terms of expansion for function c). In
the next step of algorithm, user selects between the
hyperbolic region (k < -1) and the elliptic region (k > -1). If the
algorithm output has large errors the second option could
be selected automatically. The next values are then R1 = R0
and k1 = -201 or k1 = -0.5 respectively (The algorithm also
works for oblate elliptical surfaces if k1 is set as a larger
positive number.). The value of k1 for hyperbolic region
can be selected closer to the value -1 (e.g. -3 because the
most of commercial aspheric lenses have the conic
constant above -3). Nevertheless, the value -201 was selected
for dem (...truncated)