Embed Size (px)
Citation preview
www.iap.uni-jena.de
Advanced Lens Design
Lecture 7: Aspheres and freeforms
2013-11-26
Herbert Gross
Winter term 2013
1. Aspheres
2. Conic sections
3. Forbes aspheres
4. System improvement by aspheres
5. Free form surfaces
2
Contents
22 yxz
222
22
111 yxc
yxcz
22
22 yxRRRRz xxyy
Conic section
Special case spherical
Cone
Toroidal surface with
radii Rx and Ry in the two
section planes
Generalized onic section without
circular symmetry
Roof surface
2222
22
1111 ycxc
ycxcz
yyxx
yx
z y tan
3
Aspherical Surface Types
222
22
111 yxc
yxcz
1
2
b
a
2a
bc
1
1
cb
1
1
ca
Explicite surface equation, resolved to z
Parameters: curvature c = 1 / R
conic parameter
Influence of on the surface shape
Relations with axis lengths a,b of conic sections
Parameter Surface shape
= - 1 paraboloid
< - 1 hyperboloid
= 0 sphere
> 0 oblate ellipsoid (disc)
0 > > - 1 prolate ellipsoid (cigar )
4
Conic Sections
Conic aspherical surface
Variation of the conical parameter
Aspherical Shape of Conic Sections
z
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
22
2
111 yc
cyz
Equation
c : curvature 1/Rs
: eccentricity ( = -1 )
radii of curvature :
22
2
)1(11 cy
ycz
2
tan 1
s
sR
yRR
2
32
tan 1
s
sR
yRR
vertex circle
parabolic
mirror
F
f
z
y
R s
C
Rsvertex circle
parabolic
mirror
F
y
z
y
ray
Rtan
x
Rsag
tangential circle
of curvature
sagittal circle of
curvature
Parabolic Mirror
Simple Asphere – Parabolic Mirror
sR
yz
2
2
axis w = 0° field w = 2° field w = 4°
Equation
Radius of curvature in vertex: Rs
Perfect imaging on axis for object at infinity
Strong coma aberration for finite field angles
Applications:
1. Astronomical telescopes
2. Collector in illumination systems
Simple Asphere – Elliptical Mirror
22
2
)1(11 cy
ycz
F
s
s'
F'
Equation
Radius of curvature r in vertex, curvature c
eccentricity
Two different shapes: oblate / prolate
Perfect imaging on axis for finite object and image loaction
Different magnifications depending on
used part of the mirror
Applications:
Illumination systems
Equation
c: curvature 1/R
: Eccentricity
22
2
)1(11 cy
ycz
ellipsoid
F'
F
e
a
b
oblate
vertex
radius Rso
prolate
vertex
radius Rsp
Ellipsoid Mirror
Aspheres - Geometry
z
y
aspherical
contour
spherical
surface
z(y)
height
y
deviation
z
sphere
z
y
perpendicular
deviation rs
deviation z
along axis
height
y
tangente
z(y)
aspherical
shape
Reference: deviation from sphere
Deviation z along axis
Better conditions: normal deviation rs
Perfect stigmatic imaging on axis:
Hyperoloid rear surface
Strong decrease of performance
for finite field size :
dominant coma
Alternative: ellipsoidal surface on front surface and concentric rear surface
Asphere: Perfect Imaging on Axis
1
1
1
1
1
2
2
2
2
n
ns
r
n
s
n
sz
ns
z
r
F
0
100
50
Dspot
w in °0 1 2
m]
Perfect stigmatic imaging on axis:
elliptical front surface
Asphere: Perfect Imaging on Axis
concentric
elliptical
Improvement by higher orders
Generation of high gradients
Aspherical Expansion Order
r
y(r)
0 0.2 0.4 0.6 0.8 1-100
-50
0
50
100
12. order
6. order
10. order8. order
14. order
2 4 6 8 10 12 1410
-1
100
101
102
103
order
kmax
Drms
[m]
Aspheres: Correction of Higher Order
Correction at discrete sampling
Large deviations between
sampling points
Larger oscillations for
higher orders
Better description:
slope,
defines ray bending
y y
residual spherical
transverse aberrations
Corrected
points
with
�y' = 0
paraxial
range
�y' = c dzA/dy
zA
perfect
correcting
surface
corrected points
residual angle
deviation
real asphere with
oscillations
points with
maximal angle
error
Polynomial Aspherical Surface
Standard rotational-symmetric description
0
0,5
1
1,5
2
0 0,2 0,4 0,6 0,8 1 1,2
h
h^4
h^6
h^8
h^10
h^12
h^14
h^16
M
m
m
mhahc
hhz
0
42
22
2
111)(
Ref: K. Uhlendorf
Basic form of a conic section superimposed by a Taylor expansion of z
h ... Radial distance to optical axis
... Curvature
c ... Conic constant
am ... Apherical coefficients
15
Polynomial Aspherical Surface
Forbes Aspheres
From the standard description no info about maximum deviation and how strong the
the asphere is
Departure over 64.1mm CA ??
* Fig. 2.5 of Advanced Optics Using Aspherical Elements (SPIE Press, 2008)
Editors R. Hentschel, B. Braunecker, H. Tiziani.
Some recently patented aspheres. All lengths are in mm, and Ak in units of mm1–k.
Accuracy / sig. digits ??
16 Ref: K. Uhlendorf
Polynomial Aspherical Surface
Forbes Aspheres - Qcon
New orthogonalization and normalization using Jacobi-polynomials Qm
requires normalization radius hmax
(1:1 conversion to standard aspheres possible)
• Mean square slope
M
m
mm hhQahhhc
hhz
0
2
max
4
max22
2
//111
)(
-1
-0,5
0
0,5
1
1,5
2
0 0,2 0,4 0,6 0,8 1 1,2
h
h^4*Q0
h^4*Q1
h^4*Q2
h^4*Q3
h^4*Q4
h^4*Q5
M
m
m ma0
5/
17 Ref: K. Uhlendorf
Polynomial Aspherical Surface
Forbes Aspheres - Qbfs
Limit gradients by special choice of the scalar product
(1:1 conversion to standard aspheres not possible)
• Mean square slope
M
m
mah0
22
max/1
2
max
022
0
22
0
2
0 /:mit 1
1
11)( hhuuBa
h
uu
h
hhz
M
m
mm
-0,5
0
0,5
0 0,2 0,4 0,6 0,8 1 1,2
h
u(1-u)B0
u(1-u)B1
u(1-u)B2
u(1-u)B3
u(1-u)B4
u(1-u)B5
18 Ref: K. Uhlendorf
Polynomial Aspherical Surface
Forbes Aspheres - Qbfs
Same aspheres in terms of Qbfs. Each am is given in units of nm.
Easier to interpret
usually more efficient in optimization
19 Ref: K. Uhlendorf
Polynomial Aspherical Surface
Other descriptions
2222
6
6
4
4
2
4
4
2
2
2
1
1
02
zyxs
scscsC
sbsbB
kA
CBzAz
M
m
N
n
nm
ij
M
m
m
m zhahahk
hz
0 1
2
0
2
22
2
1110
)(
)(
tgz
tfh
Superconic (Greynolds 2002)
• Implicit z-polynomial asphere (Lerner/Sasian 2000)
• Truncated parametric Taylor (Lerner/Sasian 2000)
20 Ref: K. Uhlendorf
Correction on axis and field point
Field correction: two aspheres
Aspherical Single Lens
spherical
one aspherical
double aspherical
axis field, tangential field, sagittal
250 m 250 m 250 m
250 m 250 m 250 m
250 m 250 m 250 m
a
a a
Reducing the Number of Lenses with Aspheres
Example photographic zoom lens
Equivalent performance
9 lenses reduced to 6 lenses
Overall length reduced
Ref: H. Zügge
a) all spherical
9 lenses
Vario Sonnar 3.5 - 6.5 / f = 28 - 56
b) with 3 aspheres
6 lenses
length reduced
aspherical
surfaces
Reducing the Number of Lenses with Aspheres
Example photographic zoom lens
Equivalent performance
9 lenses reduced to 6 lenses
Overall length reduced
Ref: H. Zügge
436 nm
588 nm
656 nm
xpyp
xy
axis field 22°
xpyp
xy
xpyp
xy
axis field 22°
xpyp
xy
A1A3
A2
a) all spherical, 9 lenses
b) 3 aspheres, 6 lenses,
shorter, better performance
Photographic lens f = 53 mm , F# = 6.5
Reducing the Number of Lenses with Aspheres
Binocular Lenses 12.5x
Nearly equivalent performance
Distortion, Field curvature and pupil aberration slightly improved
1 lens removed
Better eye relief distance
a) Binocular 12.5x, all spherical
b) Binocular 12.5x, 1 aspherical surface
field curvature in dptr distortion in %
-2 0 +2 -5 0 +5
yytan sag
-2 0 +2 -5 0 +5
yytan sag
Lithographic Projection: Improvement by Aspheres
Considerable reduction
of length and diameter
by aspherical surfaces
Performance equivalent
2 lenses removable
a) NA = 0.8 spherical
b) NA = 0.8 , 8 aspherical surfaces
-13%-9%
31 lenses
29 lenses
Ref: W. Ulrich
Location depending on correction target:
spherical : pupil plane
coma and astigmatism: field plane
No effect on Petzval curvature
Best Position of Aspheres
-0.5 0 0.5 1-0.2
-0.1
0
0.1
0.2
0.3
0.4
spherical
coma
astigmatism
distortion
d/p'
aspherical
effect
Example:
Lithographic lens
Sensitivities for aspherical correction
Aspherical Sensitivity
S1 S5 S12S16
S23 S28S4stop
5 10 15 20 25 30 350
1
2
3
5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
5 10 15 20 25 30 350
0.1
0.2
0.3
0.4
5 10 15 20 25 30 350
0.05
0.1
0.15
0.2
spherical
aberration
coma
astigmatism
distortion
surface
index
surface
index
surface
index
surface
index
Strong asphere : Turning points z''=0
Deviation from sphere z
Realization Aspects for Aspheres
asphere
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
-2
-1
0
-0.2
0
0.2
-0.05
0
0.05
r
r
r
profile z(r)
1. derivative z'(r)
2. derivative z''(r)
r
r
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
profile deviation z(r)
1. derivative z'(r)
0
1
2
3
0
0.5
1
Aplanatic Telescope with two aspheres
Point-by-point determination of aplanatic imaging conditions
Aplanatic Aspherical Systems
secondary
mirror
primary
mirror
F
fP
DP
d
uP u'
fS
image
asphere 1asphere 2
ray 1
ray 2
Special correcting free shaped aspheres:
Inversion of incoming wave front
Application: final correction of lithographic systems
Aspheres – Correcting Residual Wave Aberrations
conventional lenslens with correcting
surface
Extended polynomials
classical non-orthogonal monomial representation
Zernike surface
Only useful for circular pupils and low orders
Splines
Localized description, hard to optimize, good for manufacturing characterization
Generalized Forbes polynomials
Promising new approach, two types, strong relation to tolerancing
Radial basis functions
Non-orthogonal local description approach, good for local effect description
Wavelets
Not preferred for smooth surfaces, only feasible for tolerancing
Fourier representation
Classical description without assumptions, but not adapted to aberrations
Smooth vs segmented, facetted, steps, non-Fermat surfaces
Real world is still more complicated
31
Freeform Systems: Surface Representations
Extended polynomials in x,y:
Zernike expansion
Extended Forbes asphere
Expansion in other orthogonal
polynomial systems:
Legendre, Chebychev, ...
Fourier expansion
Expansion into non-orthogonal
local shifted Gaussian
functions (RBF)
Cubic spline, locally in patch j,k defined as polynomials of order 3
32
Freeform Systems: Equations of Description
2,
,2222
22
)1()1(11),(
mn
mn
mn
xxxx
yxyxa
xcxc
ycxcyxz
02222
22
),()1()1(11
),(j
jj
xxxx
yxyxZc
xcxc
ycxcyxz
2
2
1 0
02
200
22
2
2
2
2
22
2
)sin()cos(
1
1
11),(
a
rQmbma
a
r
a
rQa
rc
a
r
a
r
rc
cryxz
m
n
m n
m
n
m
n
m
n
nn
n
k
m
j
m n
jkmnkj yyxxayxz
3
0
3
0
, ),(
mn
w
yy
w
xx
nm
yyxx
yx y
n
x
n
eaycxc
ycxcyxz
,2222
22
22
)1()1(11),(
mn
yikxik
nm
yyxx
yx ymxneBycxc
ycxcyxz
,2222
22
Re)1()1(11
),(
Generalized approach for
orthogonal surface decomposition
Slope orthogonality is guaranteed
and is related to tolerancing
33
Freeform Systems: Forbes Surfaces
Ref: C.Menke/G.Forbes, AOT 2(2013)p.97
2
2
1 0
02
200
22
2
2
2
2
22
2
)sin()cos(
1
1
11),(
a
rQmbma
a
r
a
rQa
rc
a
r
a
r
rc
cryxz
m
n
m n
m
n
m
n
m
n
nn
History:
- exact solutions of Fermat-priciple for one wavelength and only a few field points
corresponding to the number of surfaces
- development of algorithms for illumination tailoring
- mostly methods are applicable for illumination and imaging
Dimension:
- 2D is much easier / 3D is complicated and often not unique
SMS-method of Minano
- construction of the surfaces ray by ray with simple procedure
- approved method in illumination and imaging
Ries tailoring
- method used since longer time
- exact algorithm not known
Oliker-Method for illumination
approximation of smooth surface by sequence of parabolic arcs
Reality:
- due to finite size of surcse sna dbroadband applications tailored methods are only useful
for finding a good starting system for optimization
34
Freeform Systems: Exact Tailoring
Arbitrary variation of performance over the transverse cross section
- one aberration value is not feasible to describe the distribution over the bundle cross section
- uniformity of aberration variation is important property
- single numbers should be defined to summarize the performance by moments, rms,...
- additional uniformity parameters are necessary to be defined in the merit function
New aberration types are hard to interpret
Nodal aberration theory
- vectorial approach on aberration theory describes
more general geometries without
symmetry
- nodal points are locations with corrected
aberrations in the field
35
Freeform Systems: Performance Assessment
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
binodal
points
Basic idea of tailoring:
- point-wise mapping of surfaces
- fulfillment of Fermat principle
- fulfillment of correct photometry
36
Freeform Systems: Exact Tailoring
Ref: Winston/Minano/Benitez, Nonimaging optics
Optimization of systems with freeform surfaces:
- huge number of degrees of freedom
- large differences in convergence according to surface representation
- local vs global influence functions
- definition of performance and formulation of merit function is complicated and
cumbersome
Classical system matrix for local defined splines is ill conditioned
Starting systems:
- still more important as in conventional optics
- only a few well known systems published
- larger archive for starting systems not available until now
- own experience usually is poor
Best location of FFF surfaces inside the system:
- still more important as in the case of circular symmetric aspheres
- no criteria known until now
37
Freeform Systems: Optimization
38
Freeform Optical Systems
Optical system
concept
Optical design
model
Automatic
optimization
Merit function
System
constraints
Technological
conditions
Freeform
surfaces
Surface
metrology
Scattered
point cloud
Fitting model
CAD modelPerformance
measurement
System
integration
CMM
manufacturing
Quality
requirements
Performance
calculationVerification
optical design
surface
manufacturing
and metrology
system
integration and
metrology
M1
M2(stop)
M3
General purpose:
- freeform surfaces are useful for compact systems with small size
- due to high performance requirements in imaging systems and limited technological
accuracy most of the applications are in illumination systems
- mirror systems are developed first in astronomical systems with complicated symmetry-free
geometry to avoid central obscuration
HMD
Head mounted device with extreme size constraints
HUD
Head up display, only few surfaces allowed
Schiefspiegler
- astronomical systems without central obscuration
- EUV mirror systems for next generation lithography systems
Illumination systems
Various applications, smooth and segmented
39
Freeform Systems: Applications
