Optical modeling and design of freeform surfaces using anisotropic Radial Basis Functions EOSAM 2014
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Transcript of Optical modeling and design of freeform surfaces using anisotropic Radial Basis Functions EOSAM 2014
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Optical modeling and design of freeform surfaces using
anisotropic Radial Basis Functions
Milan Maksimovic
Focal - Vision and Optics,
Enschede, The Netherlands
European Optical Society Annual Meeting 2014, TOM 3 – Optical System Design and Tolerancing,Berlin, 15-19. September 2014, Adlershof, Berlin, Germany
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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
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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 ?
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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 yz x y w x y
Kc x y
+= + Φ+ − +
∑
Base Conic Linear combination of
Basis functions
2 2
2 2 2,
( )( , )
1 1 ( )
m nmn
m n
c x yz x y c x y
Kc x y
+= ++ − +
∑
Extended polynomial representation
2 22
2 2 21...8 1..37
( )( , ) ( , )
1 1 ( )
ii j j
i j
c x yz x y r A Z
Kc x yα ρ ϕ
= =
+= + ++ − +
∑ ∑
Even Aspheric
expansionZernike Modes
Modelling
Manufacturing Measurements
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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 !
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• 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
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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.)
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RBFs placement grid
Fibonacci grid:
• Deterministic algorithm based on Fibonacci spiral
• Uniform and isotropic resolution
• Equal area (contribution) per each grid point
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RBF representation:
Example biconic surface
Test function: biconic (aspheric) surface (Cy=0.1, Cx=0.05, Kx=Ky=-2.3) normalized on unit circle
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RBF representation:
RMS error vs. number of grid points
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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
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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
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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
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Adaptive RBF representation (1)
2. Localized perturbation
4. Add localized grid pointsin the area of largest error
3. New optimal basis function !
1. Initial representation on the small grid !
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1e-7
Adaptive RBF representation (2)
Localized grid refinement +
Global grid refinement
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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
�, ! � ;+�% < !%�� < � +� < =�;%+�% < !%� <���"#$� �#�� %#$! !#!� %
�
���
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Example: lens perturbations for
wavefront control
Selective perturbations of expansion coefficients
~51 RBFs on Fibonacci grid
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Example: complex beam shaping
• Re-shaping of input Gaussian beam
• Lens description using 21 RBF on Fibonacci grid
• Merit Function based on real ray position @ image plane!
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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
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Thank you for your attention!