Verallgemeinerte Optische Abbildungssysteme, VopSys

®
MacroNano
Generalized Optical Imaging Systems
(Verallgemeinerte Optische Abbildungssysteme, VopSys)
In this project we investigate strategies for the design and optimization of optical imaging systems including freeform
optical elements. To this end we perform research on generalized and linearized models for specific imaging
geometries. In analogy to the paraxial or collinear models for rotationally symmetric systems the ultimate goal could
be an analytical theory of aberrations for non-rotationally symmetric imaging systems. Such generalized modells will
be helpful to decide on the potential of freeform surfaces for the specific application as well as to find appropriate
starting systems for the optimization.Furthermore we investigate innovative algorithms which are capable to handle
the optimization of such generalized systems with a largely increased number of optimization parameters. In order to
handle the tremendously increased complexity of the optimization problem for such cases we consider separable
optimization strategies related to the semi-infinite optimization.
meridional
curvature; radius Rm
P‘m
mal
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ce
surfa
P
plan-symmetric optical system
symmetry-free optical
Pphotograph of a freeform mirror fabricated
by ultraprecision micromilling.
system
reference ray
centre of
entrance pupil
astigmatic ray.tracing and the parabasal
approach lead to an analytic description of
the image position and orientation.
Diffractive-refractive Freeform mirror fabricated
by ultraprecision micromilling and laser ablation.
Gefördert von der Deutschen Forschungemeinschaft DFG
FKZ: SI 573/9-1, HO 2667/1-1
Kontakt:
Prof. Dr. Stefan Sinzinger, Prof. Dr. Armin Hoffmann
Institut für Mikro- und Nanotechnologien
®
IMN MacroNano der Technischen Universität Ilmenau
Helmholtzring 1, 98693 Ilmenau, Deutschland
Telefon +49 3677 69-2490
Telefax +49 3677 69-1281
www.tu-ilmenau.de/to
www.tu-ilmenau.de/orsto/
www.macronano.de
An optimization method for radial NURBS surfaces
J. Werner, M. Hillenbrand, M. Zhao, S. Sinzinger
Fachgebiet Technische Optik, Technische Universität Ilmenau
®
MacroNano
NURBS curves
Introduction
Initially, NURBS have been developed for the (computer
aided) design of freeform surfaces such as car bodies.
Base sphere + NURBS
Few applications in optical design:
- Chase explored the influence of degree and number of
internal knots on rms spot size and optimization time
for a Cassegrain-type telescope [1].
- Ott designed a head-up display [2].
Instead of directly representing an axially symmetric
surface by a NURBS curve, we add such a curve to a base
sphere.
We extend the work by Zhao [3] on an optimization
method based on the local structure of NURBS.
We implement radial NURBS surfaces
- as a user defined DLL for Zemax for use with Zemax’s
built-in optimization and
- in our own optical simulation framework for use with
the new optimization method.
Results
Piecewise optimization
Idea: Optimize only a subset of variables using only a part
of the merit function at a time. Feasible because of the
limited local support of the control points.
For convenience the principal algorithm is shown for case
p = 2 and n = 6. Details of the inner optimization step
are given below.
REPEAT
FOR Step FROM 1 TO 5
OPTIMIZE [only with respective variables and
merit function parts]
UNTIL CONVERGENCE
One of the systems used for evaluation:
U.S. 5,636,065 (1st embodiment)
Systems for evaluation are U.S. patents 5,636,065 (1st
embodiment), 5,754,347 (2nd and 4th) and 6,028,713
(1st) with the aspheric surface replaced by our radial
NURBS surface. Only the parameters of the NURBS
surface are set as variables. We vary p from 2 to 4 and n
from 3 to 8.
Optimization fails in most cases when starting with a
planar surface. In the following the representation with a
base sphere is used.
The table below shows parameters of the 5,636,065 (1st)
system with p=2 and n=4 after optimization.
Because of the strong convex
hull property of NURBS curves,
it is immediately observable
that the sag maximum
departure from the base
sphere is less than 0.0178 mm.
Optimizing all four systems with Zemax for different p
and n. Many configurations lead to a bad local minimum.
Method 1 (used by Zhao [3]) does not attain a local
minimum. It reaches its final result after only one
iteration of the REPEAT-loop.
Method 2 and 3 attain a local minimum. A convergence
proof for such block coordinate descent methods,
applicable to method 3, is given by Bertsekas [5].
When using NURBS to represent axially symmetric
surfaces, the proposed representation with base sphere
makes the sag maximum departure from the sphere
directly visible.
An application for such surfaces might be off-axis
systems where an appropriate choice of the control
points might enable higher quality in the area of interest.
Optimizing all four systems with piecewise optimization
method 2 for different p and n. Using approximation
leads to a good local minimum for all configurations.
References
[1] H. Chase: Optical design with rotationally symmetric NURBS. In Proc. SPIE 4832, 2002.
[2] P. Ott: Optic Design of head-up displays with freeform surfaces specified by NURBS. In Proc. SPIE 7100, 2008.
[3] M. Zhao: Optimierungsmethode für optische Systeme mit lokalen Flächenbeschreibungen. Bachelor thesis, 2012.
[4] L. Piegl and W. Tiller: The NURBS book. 2nd ed., 1997.
[5] D. P. Bertsekas: Nonlinear Programming. 2nd ed., 1999.
www.tu-ilmenau.de
Conclusions
Piecewise optimization with approximation is suitable for
optimizing such surfaces. A drawback is that it cannot be
used to simultaneously optimize other surfaces.
Methods 2 and 3 can be “approximated” when for each
OPTIMIZE, instead of performing a complete
optimization, only the first iteration is performed.
For a control point, y, z and w are not independent,
therefore only two of the three should be used as a
variable for optimization.
These are some optimization results:
Technische Universität Ilmenau
IMN MacroNano®
Fachgebiet Technische Optik
Jürgen Werner
Acknowledgements
The authors would like to thank the DFG for the financial support through the project “Verallgemeinerte optische
Abbildungssysteme” (FKZ: HO 2667/1-1).
Telefon: +49 3677 69-1411
Fax:
+49 3677 69-1281
[email protected]
www.tu-ilmenau.de/optik

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