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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!