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Lens Design II
Lecture 3: Aspheres
2018-11-07
Herbert Gross
Winter term 2018
2
Preliminary Schedule Lens Design II 2018
1 17.10. Aberrations and optimization Repetition
2 24.10. Structural modificationsZero operands, lens splitting, lens addition, lens removal, material selection
3 07.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres
4 14.11. FreeformsFreeform surfaces, general aspects, surface description, quality assessment, initial systems
5 21.11. Field flatteningAstigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses
6 28.11. Chromatical correction IAchromatization, axial versus transversal, glass selection rules, burried surfaces
7 05.12. Chromatical correction IISecondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism
8 12.12. Special correction topics I Symmetry, wide field systems, stop position, vignetting
9 19.12. Special correction topics IITelecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems
10 09.01. Higher order aberrations High NA systems, broken achromates, induced aberrations
11 16.01. Further topics Sensitivity, scan systems, eyepieces
12 23.01. Mirror systems special aspects, double passes, catadioptric systems
13 30.01. Zoom systems Mechanical compensation, optical compensation
14 06.01. Diffractive elementsColor correction, ray equivalent model, straylight, third order aberrations, manufacturing
1. Aspheres
2. Conic sections
3. Forbes aspheres
4. System improvement by aspheres
5. Aspheres in Zemax
3
Contents
4
Aspherical Correction
Correction of spherical aberration by
an asphere
Ref: A. Herkommer
a) spherical
lens
b) aspherical
lens
refraction too
strong
asphere reduces
power
2
2 21 1 1
c rz
c r
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 )
5
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
r
2
2 21 1 1
crz
c r
6
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
7
Aspherical Surface Types
Polynomial Aspherical Surface
Standard rotational-symmetric description
22 4
2 20
(r)1 1 1
Mm
m
m
crz a r
c r
Ref: K. Uhlendorf
Basic form of a conic section superimposed by a Taylor expansion of z
r ... radial distance to optical axis
c curvature
conic constant
am aApherical coefficients
2 2r x y
8
r
z(r)
r 4
r 6
r 8
r 10
r 12
r 14
r 16
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)
9Ref: K. Uhlendorf
9
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
10
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
11
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
12
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
13
Perfect stigmatic imaging on axis:
elliptical front surface
Asphere: Perfect Imaging on Axis
concentric
elliptical
14
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]
15
Xray telescopeWolter type I
Nested shells with gracing incidence
Increase of numerical aperture by several shells
Gracing Incidence-Xray Telescope
detector
hyperboloids Wolter type I
rays
paraboloids
nested cylindrical
shells
towards paraboloid
focus point
Woltertyp
1. Paraboloid
2. Hyperboloid
Gracing Incidence-Xray Telescope
Aspheres - Geometry
z
r
aspherical
contour
spherical
surface
z(r)
height
r
deviation
z
Reference: deviation from sphere
Deviation z along axis
Better conditions: normal deviation rs
18
sphere
z
r
perpendicular
deviation rs
deviation z
along axis
height
r
tangente
z(r)
aspherical
shape
Improvement by higher orders
Generation of high gradients
Aspherical Expansion Order
r
z(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]
19
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
r r
residual spherical
transverse aberrations
Corrected
points
with
r' = 0
paraxial
range
r' = c dzA/dr
zA
perfect
correcting
surface
corrected points
residual angle
deviation
real asphere with
oscillations
points with
maximal angle
error
20
Polynomial Aspherical Surface
Forbes Aspheres - Qcon
New orthogonalization and normalization using Jacobi-polynomials Qm
requires normalization
radius rmax
(1:1 conversion to standard
aspheres possible)
• Mean square slope
24 2
max max2 2
0
(r) / r / r1 1 1
M
m m
m
crz r a Q r
c r
M
m
m ma0
5/
Ref: K. Uhlendorf
21
r
z(r)
r 4Q0(r)
r 4Q1(r)
r 4Q2(r)
r 4Q3(r)
r 4Q4(r)
r 4Q5(r)
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 2
22
max max0
2 2 2 20 max0 0
1
(r) 1 1 1
M
m m
m
r r
r rc r rz a B
rc r c r
Ref: K. Uhlendorf
22
r
z(r)
u = (r/rmax)2
u(u-1)B0(u)
u(u-1)B1(u)
u(u-1)B2(u)
u(u-1)B3(u)
u(u-1)B4(u)
u(u-1)B5(u)
23
Forbes Aspheres
Strong asphere Qcon Mild asphere Qbfs
sag along z-axis difference to best fit sphere
sag along local surface normal
slope orthogonal not slope orthogonal
true polynom not a polynomial due to projection
type Q 1 in Zemax type Q 0 in Zemax
direct tolerancing of coefficients no direct relation of coefficients to slope
2 2
22
max max0
2 2 2 20 max0 0
1
(r) 1 1 1
M
m m
m
r r
r rc r rz a B
rc r c r
24 2
max max2 2
0
(r) / r / r1 1 1
M
m m
m
crz r a Q r
c r
r
z(r)
u = (r/rmax)2
u(u-1)B0(u)
u(u-1)B1(u)
u(u-1)B2(u)
u(u-1)B3(u)
u(u-1)B4(u)
u(u-1)B5(u)r
z(r)
r 4Q0(r)
r 4Q1(r)
r 4Q2(r)
r 4Q3(r)
r 4Q4(r)
r 4Q5(r)
24
Selection of Asphere Types
Correction of Retro focus type camera lens
F# = 2.8 , d=21 , 2w = 94°
Considerably better resukt and faster optimization
by the use of Q aspheres
Ref: C. Menke
a) standard asphere b) Qbfs asphere
Asphere far from pupil:
- ray bundels of field points
separated
- field dependend correction
- also impact on distortion
Asphere near pupil:
- all ray bundels equally affected
- problem field angles: coma
25
Impact of Asphere
surface 2
surface 15
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
26
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
27
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
28
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
29
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
30
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
31
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
32
Selection of one aspherical
surface in a photographic lens
33
Aspherization of a Camera Lens
1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.5
1
1.5
spherical
aberration
coma
astigmatism
distortion
surface
index
surface
index
surface
index
surface
index
S14S 9 S 11S 5S 2
spherical system:
197 nm
surface 2:
196 nmsurface 5:
185 nm
surface 9:
187 nm
surface 11:
278 nm
surface 14:
178 nm
Handy Phone Objective lenses
Examples
Ref: T. Steinich
US 7643225L = 4.2 mm , F'=2.8 , f = 3.67 mm , 2w=2x34°
US 6844989L = 6.0 mm , F'=2.8 , f = 4.0 mm , 2w=2x31°
EP 1357414L = 5.37 mm , F'=2.88 , f = 3.32 mm , 2w=2x33.9°
Olympus 2L = 7.5 mm , F'=2.8 , f = 4.57 mm , 2w=2x33°
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
35
Setting of surface properties
Surface properties and settings
36
Standard spherical and conic sections
Even asphere classical asphere
Paraxial ideal lens
Paraxial XY ideal toric lens
Coordinate break change of coordinate system
Diffraction grating line grating
Gradient 1 gradient medium
Toroidal cylindrical lens
Zernike Fringe sag surface as superposition of Zernike functions
Extended polynomial generalized asphere
Black Box Lens hidden system, from vendors
ABCD paraxial segment
37
Important Surface Types
37