European Optical Society Annual Meeting 2014, TOM 3 –Optical System Design and Tolerancing, Berlin, 15-19. September 2014, Adlershof, Berlin, Germany

1. 1
Optical modeling and design of freeform surfaces using
anisotropic Radial Basis Functions
Milan Maksimovic
Focal -Vision and Optics,
Enschede,The Netherlands
EuropeanOptical SocietyAnnual Meeting 2014,TOM 3 – Optical System Design andTolerancing,
Berlin, 15-19. September 2014, Adlershof, Berlin,Germany

2. 2
Outline
• Introduction : freeform optics and their mathematical representations
• (Anisotropic ) Radial Basis Functions in (optical) modeling
• Selected numerical and design examples
• Aspherics and freeform surfaces using optimally placed and shaped RBFs
• Grid adaptation strategy
• (Localized) surface perturbations for wavefront control
• Complex beam shaping
• Concluding remarks

3. 3
Introduction
Freeform optics: no rotational invariance, surfaces with arbitrary shape and regular or irregular
global or local structure:
Spherical,R=const.
Rot. symmetry
Aspheric, R=f(y)
Rot. symmetry
Freeform,z=f(x,y)
No symmetry
• enhanced flexibility in design,
• boost in optical performances,
• combining multiple functionalities into
single component,
• simplifying complex optical systems by
reducing element count,
• lowering costs in manufacturing,
• reducing stray-light
• easing system integration and assembly
What is the best way
for optical designer
to optimize/tolerance
freeform optics ?

4. 4
Freeform surface representations
• Traditional analytic aspheric polynomial and extended polynomial
representations
• Global approximants (over entire surface) vs local approximates
• Polynomial representation with orthogonal bases : Q-polynomials , Zernike
polynomials,…
• Spline representations (NURBS), wavelets, …
• Important attributes
• Numerical efficiency, e.g. existence of recurrence
relations for computations
• Robustness to numerical round-off error
• Manufacturability constraints
• Adaptation to arbitrary surface apertures and shapes
( )
2 2
2 2 2
( )
( , ) ,
1 1 ( )
i i
i
c x y
z x y w x y
Kc x y
+
= + Φ
+ − +
∑
Base Conic Linear combination of
Basis functions
2 2
2 2 2
,
( )
( , )
1 1 ( )
m n
mn
m n
c x y
z x y c x y
Kc x y
+
= +
+ − +
∑
Extended polynomial representation
2 2
2
2 2 2
1...8 1..37
( )
( , ) ( , )
1 1 ( )
i
i j j
i j
c x y
z x y r A Z
Kc x y
α ρ ϕ
= =
+
= + +
+ − +
∑ ∑
Even Aspheric
expansion
Zernike Modes
Modelling
ManufacturingMeasurements

5. 5
Radial Basis Functions
and Scattered Data Approximation
• General input is N points of scattered data in 2D region ( , ) with k=1,2,…N
• Linear combination of basis functions | ∙ − | should fit the data on sampled points =
= −
• Interpolation approach gives always non -singular (dense) liner system
= →
| − | ⋯ | − |
⋮ ⋱ ⋮
| − | ⋯ | − |
⋮ = ⋮
• If number of samples is larger than ( M>N) number of basis functions approximate solution can be
obtained by (least squares ) optimization
• Important is to deal with ill-conditioned systems (e.g. Riley’s Algorithm,Tikhonov regularization, etc. )
• Interplay between numerical ill-conditioning (stability ) and accuracy of solution important in practice
• Optimal placement and the choice of basis functions is data dependent
RBFs enable general surface representation with (possibility for) local surface control !

6. 6
• Multi-centric (local) shifted Gaussian function / Anisotropic Gaussian Radial Basis Functions:
, ! = "#$ # %#$! !#! %
Standard isotropic (Sx=Sy) Gaussian RBFs are used in optical design (literature)
• Comparable performance with standard aspheric representations on rotationally symmetric surfaces
• Used for optimization off-axis free-form surfaces outperforms classical representations
New approaches are being proposed in literature:
• Hybrid methods combining local and global approximants ( RBF and φ- polynomials)
• Using compactly supported RBFs
Remaining practical challenge is optimal placement and shape for small number (<500) of basis functions
• Reduction of the number of basis functions required to describe a freeform surface within
manufacturing/measurement accuracy
Anisotropic Gaussian RBF

7. 7
RBF optimal shape parameter using
Leave-One-Out- Cross-Validation
• Split data on training and evaluation data
• RBF interpolation on the training data for fixed (k=1,…,N) and fixed shape parameters (ε)
&' = (Φ* | − ( |
( ((, )
with training data out of {f1,f2,…fk-1,fk+1,…fN} and &' =
• Evaluate error at one validation point - not used to determine interpolation:
. / = − &' -, /
• Optimal parameters are determined through optimization:
0123 = 456789* " / " = . , … , .
• Comparison of the error norms for different values of the shape parameter
→ Optimum is the one which produces the minimal error norm!
• LOOCV attributes
Can be computationally very expensive
Does not require knowledge of exact solution
Easily applicable for multidimensional shape parameters
• Alternative algorithms for speed- up exist in mathematical literature (Rippa algorithm, etc.)

8. 8
RBFs placement grid
Fibonacci grid:
• Deterministic algorithm
based on Fibonacci spiral
• Uniform and isotropic
resolution
• Equal area (contribution)
per each grid point

9. 9
RBF representation:
Example biconic surface
Test function: biconic (aspheric) surface (Cy=0.1, Cx=0.05, Kx=Ky=-2.3) normalized on unit circle

10. 10
RBF representation:
RMS error vs. number of grid points

11. 11
RBF representation of freeform surface:
Zernike mode on Fibonacci grid
• RMS Error~ 4.2e-5 @ 151 RBF basis functions on Fibonacci grid
• ~50k error evaluation points on uniform rectangular grid
• Sx=0.8203;Sy=0.9931

12. 12
RBF representation of freeform surface:
Zernike mode on Chebyshev grid
• RMS Error~ 8.5e-6 @ 176 RBF basis functions on Chebyshev grid
• ~50k error evaluation points on uniform rectangular grid
• Sx=0.85253;Sy=1.0656

13. 13
Parabola C=0.1,K=-1
RBF representation of freeform surface:
perturbed parabolic surface
• RMS Error~ 8.5e-6 @ 251 RBF basis functions
on Fibonacci grid
• ~50k error evaluation points on uniform
rectangular grid
• Sx=3.8401; Sy=5.4277

14. 14
Adaptive RBF representation (1)
2. Localized perturbation
4. Add localized grid points
in the area of largest error
3. New optimal basis
function !
1. Initial representation
on the small grid !

15. 15
1e-7
Adaptive RBF representation (2)
Localized grid refinement
+
Global grid refinement

16. 16
Define target
ray positions
and initialize
surface design
parameters
Initialize pupil
sampling grid
Optimize with
local DLS or OD
optimizer
Optical modeling and design using
RBF representations
Ray-tracing,
Optimization
&Tolerancing
User Defined Surface
RBF representation for
Optical design
Use optimal
grid and
shape
Optically
relevant merit
function
, !
;+ %
< !%
< + < = ;%+ % < !%
< "#$ # %#$! !#! %

17. 17
Example: lens perturbations for
wavefront control
Selective perturbations of expansion
coefficients
~51 RBFs on Fibonacci grid

18. 18
Example: complex beam shaping
• Re-shaping of inputGaussian beam
• Lens description using 21 RBF on Fibonacci grid
• Merit Function based on real ray position @ image plane!

19. 19
Concluding remarks
• Anisotropic RBFs can be used for efficient freeform surface representation
• Optimal grid for placement of RBFs depends required accuracy and expected shape:
• Fibonacci grid can be beneficial to capture complex surface shapes with smallest number of basis
functions due to equal contribution of local surface regions
• Adaptive refinement of the grid is possible and can lead minimal number of RBFs at fixed accuracy
• Optimal RBFs shape parameters can be pre-computed on selected representative surface shapes
• Optics commonly deals with mathematically well defined class of surfaces that can be used to learn
optimal parameters
• Number of RBF terms can be minimized using optimal parameters
• RBF based representation in standard ray-tracing code facilitates:
• Linking RBFs based representation with optically relevant merit function
• Local surface perturbation for tolerancing or wavefront control
• Complex shape parameterization
• Number of terms in representation can be minimized

20. 20
Thank you for your attention!