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Design and Correction of Optical
Systems
Lecture 10: Correction principles I
2019-06-24
Herbert Gross
Summer term 2019
2
Preliminary Schedule - DCS 2019
1 08.04. BasicsLaw of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 15.04. Materials and ComponentsDispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 29.04. Paraxial OpticsParaxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 06.05. Optical SystemsPupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 13.05. Geometrical AberrationsLongitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 20.05. Wave AberrationsFermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 27.05. PSF and Transfer functionDiffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 03.06. Further Performance CriteriaRayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based
criteria, further options
9 17.06. Optimization and CorrectionPrinciples of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 24.06. Correction Principles ISymmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 01.07. Correction Principles IIField flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 08.07. Optical System ClassificationOverview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
1. Introduction
2. Sensitivity and improvement process
3. Special options for spherical aberration
4. Astigmatism
5. Symmetry
6. Coma
7. Distortion
8. Aspheres and higher orders
3
Contents
System Design Phases
1. Paraxial layout:
- specification data, magnification, aperture, pupil position, image location
- distribution of refractive powers
- locations of components
- system size diameter / length
- mechanical constraints
- choice of materials for correcting color and field curvature
2. Correction/consideration of Seidel primary aberrations of 3rd order for ideal thin lenses,
fixation of number of lenses
3. Insertion of finite thickness of components with remaining ray directions
4. Check of higher order aberrations
5. Final correction, fine tuning of compromise
6. Tolerancing, manufactability, cost, sensitivity, adjustment concepts
4
Effectiveness of correction
features on aberration types
Correction Effectiveness
Aberration
Primary Aberration 5th Chromatic
Sp
he
rica
l A
berr
atio
n
Co
ma
Astig
ma
tism
Pe
tzva
l C
urv
atu
re
Dis
tort
ion
5th
Ord
er
Sp
he
rica
l
Axia
l C
olo
r
Late
ral C
olo
r
Se
co
nd
ary
Sp
ectr
um
Sp
he
roch
rom
atism
Lens Bending (a) (c) e (f)
Power Splitting
Power Combination a c f i j (k)
Distances (e) k
Len
s P
ara
mete
rs
Stop Position
Refractive Index (b) (d) (g) (h)
Dispersion (i) (j) (l)
Relative Partial Disp. Mate
ria
l
GRIN
Cemented Surface b d g h i j l
Aplanatic Surface
Aspherical Surface
Mirror
Sp
ecia
l S
urf
ace
s
Diffractive Surface
Symmetry Principle
Acti
on
Str
uc
Field Lens
Makes a good impact.
Makes a smaller impact.
Makes a negligible impact.
Zero influence.
Ref : H. Zügge
5
Strategy of Correction and Optimization
Usefull options for accelerating a stagnated optimization:
split a lens
increase refractive index of positive lenses
lower refractive index of negative lenses
make surface with large spherical surface contribution aspherical
break cemented components
use glasses with anomalous partial dispersion
6
Operationen with zero changes in first approximation:
1. Bending a lens.
2. Flipping a lens into reverse orientation.
3. Flipping a lens group into reverse order.
4. Adding a field lens near the image plane.
5. Inserting a powerless thin or thick meniscus lens.
6. Introducing a thin aspheric plate.
7. Making a surface aspheric with negligible expansion constants.
8. Moving the stop position.
9. Inserting a buried surface for color correction, which does not affect the main
wavelength.
10. Removing a lens without refractive power.
11. Splitting an element into two lenses which are very close together but with the
same total refractive power.
12. Replacing a thick lens by two thin lenses, which have the same power as the two refracting
surfaces.
13. Cementing two lenses a very small distance apart and with nearly equal radii.
Zero-Operations
7
Lens bending
Lens splitting
Power combinations
Distances
Structural Changes for Correction
(a) (b) (c) (d) (e)
(a) (b)
Ref : H. Zügge
8
Removal of a lens by vanishinh of the optical effect
For single lens and
cemented component
Problem of vanishinh index:
Generation of higher orders
of aberrations
9
Lens Removal
1) adapt second radius of curvature 2) shrink thickness to zero
a) Geometrical changes: radius and thickness
b) Physical changes: index
10
Sensitivity by large Incidence
Small incidence angle of a ray:
small impact of centering error
Large incidence angle of a ray:
- strong non-linearity range of sin(i)
- large impact of decenter on
ray angle
Ref: H. Sun
ideal
surface
location
real surface
location:
decentered
ideal ray
real raysurface
normal
ideal ray
real ray
surface
normal
a) small incidence:
small impact of decenter
b) large incidence:
large impact of decenter
i
i
Sensitivity/relaxation:
Average of weighted surface contributions
of all aberrations
Correctability:
Average of all total aberration values
Total refractive power
Important weighting factor:
ratio of marginal ray heights
1 2 3 4 5 6 7 8 91
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
-50
-40
-30
-20
-10
0
1
0
2
0
3
0
4
0
Sph
1 2 3 4 5 6 7 8 91
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
-50
-40
-30
-20
-10
0
1
0
2
0
3
0
Ko
m
a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-2
-1
0
1
2
3
4
Ast
1 2 3 4 5 6 7 8 91
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
-20
-10
0
1
0
2
0
3
0
4
0
CHL
1 2 3 4 5 6 7 8 91
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
0
1
0
2
0
3
0
4
0
5
0
Inz-
Wi
Sph
Coma
Ast
CHL
incidenceangle
k
j
jj FFF2
1
1h
h jj
Sensitivity of a System
11
Quantitative measure for relaxation
with normalization
Non-relaxed surfaces:
1. Large incidence angles
2. Large ray bending
3. Large surface contributions of aberrations
4. Significant occurence of higher aberration orders
5. Large sensitivity for centering
Internal relaxation can not be easily recognized in the total performance
Large sensitivities can be avoided by incorporating surface contribution of aberrations
into merit function during optimization
Fh
Fh
F
FA
jjj
jj
1
11
k
j
jA
Sensitivity of a System
12
Sensitivity of a System
Ref: H.Züggesurfaces
Representation of wave
Seidel coefficients [l]
Double Gauss 1.4/50
13
Relaxed System
Example: achromate with cemented/splitted setup
Equivalent performance
Inner surfaces of splitted version more sensitive
Ref: H. Zügge
a ) Cemented achromate f = 100 mm , NA = 0.1
b ) Splitted achromate f = 100 mm , NA = 0.1
-15
-10
-5
0
5
10
1 2 3
-15
-10
-5
0
5
10
1 2 3 4
Seidel coefficient
spherical aberration
spot enlargement for
0.2 ° surface tilt
surface
index
surface
index
14
15
Relaxed Design
Photographic lens comaprison
Data:
F# = 2.0
f = 50 mm
Field 20°
Same size and quality
Considerably tigher tolerances in the
first solution
Ref: D. Shafer
Data: Wj, Sj, iMR2 , iCR
2 , Sph, G, A
Relay systems
1. ZeBase M 29 , Takahashi, Olympus
2. Hopkins modified
3. Kidger
Example: Sensitivity of Endoscopic Systems
16
Reality:
- as-designed performance: not reached in reality
- as-built-performance: more relevant
Possible criteria:
1. Incidence angles of refraction
2. Squared incidence angles
3. Surface powers
4. Seidel surface contributions
5. Permissible tolerances
Special aspects:
- relaxed systems does not contain
higher order aberrations
- special issue: thick meniscus lenses
Sensitivity and Relaxation
17
performance
parameter
best
as
built
tolerance
interval
local optimal
design
optimal
design
Possible further criteria for modefied merit function to obtain relaxed systems
1. cos-G-factor of ray bending
2. Squared sum of incidence angles
Target: minimum value for i, i'
3. Optimization of performance and performance change simultaneously
Further Parameter of Sensitivity
18
' 'j j j j j j jn s e n s eG
2 21
1'
2
N
j j
j
i iN
2
1
m
m m
MD p
p
performance
limit
as built
performance
nominal
performance
= 12° = 20°
Relaxed system:
as built performance improved
Typically:
- no or weak correlation to designed
performance
- weak decrease in nominal performance
possible
As Built Performance
19
performance
field size
10 0.5
locally optimized
solution
best as-built
performance
nominal
with
tolerances
average
performance
sensitivity
10 0.5
20
Design Solutions and Sensitivity
Focussing
3 lens with
NA = 0.335
Spherical
correction
with/without
compensation
Red surface:
main correcting
surface
Counterbeding
every lens in
one direction
Microscopic Objective Lens
Incidence angles for chief and
marginal ray
Aperture dominant system
Primary problem is to correct
spherical aberration
marginal ray
chief ray
incidence angle
0 5 10 15 20 2560
40
20
0
20
40
60
microscope objective lens
21
Photographic lens
Incidence angles for chief and
marginal ray
Field dominant system
Primary goal is to control and correct
field related aberrations:
coma, astigmatism, field curvature,
lateral colorincidence angle
chief
ray
Photographic lens
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1560
40
20
0
20
40
60
marginal
ray
22
Variable focal length
f = 15 ...200 mm
Invariant:
object size y = 10 mm
numerical aperture NA = 0.1
Type of system changes:
- dominant spherical for large f
- dominant field for small f
Data:
23
Symmetrical Dublet
f = 200 mm
f = 100 mm
f = 50 mm
f = 20 mm
f = 15 mm
Nofocal
length [mm]
Length [mm]
spherical c9
field curvature c4
astigma-tism c5
1 200 808 3.37 -2.01 -2.27
2 100 408 1.65 1.19 -4.50
3 50 206 1.74 3.45 -7.34
4 20 75 0.98 3.93 2.31
5 15 59 0.20 16.7 -5.33
24
Field vs Aperture Correction
Remote pupil system with decreasing field for one wavelength and 4 lenses
Aperture maximum value for overall diffraction limited correction
Large field angles: lenses bended towards pupil
Old achromate move towards usual achromate
2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
50
w [°]
D [mm]
Effect of bending a lens on spherical aberration
Optimal bending:
Minimize spherical aberration
Dashed: thin lens theory
Solid : think real lenses
Vanishing SPH for n=1.5
only for virtual imaging
Correction of spherical aberration
possible for:
1. Larger values of the
magnification parameter |M|
2. Higher refractive indices
Spherical Aberration: Lens Bending
20
40
60
X
M=-6 M=6M=-3 M=3
M=0
Spherical
Aberration
(a)
(b) (c)
(d)-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Ref : H. Zügge
25
Aplanatic Surfaces with Vanishing Spherical Aberration
Aplanatic surfaces: zero spherical aberration:
1. Ray through vertex
2. concentric
3. Aplanatic
Condition for aplanatic
surface:
Virtual image location
Applications:
1. Microscopic objective lens
2. Interferometer objective lens
s s und u u' '
s s' 0
ns n s ' '
rns
n n
n s
n n
ss
s s
'
' '
'
'
'
s'
0 50 100 150 200 250 300-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
object
location
S
aplanaticconcentricvertex
linear growing aberrations with
deviation of object location
no shift invariance in z
V C A
26
Aplanatic lenses
Combination of two spherical
corrected surfaces:
one concentric and one aplanatic
surface:
zero contribution of the whole
lens to spherical aberration
Not useful:
1. aplanatic-aplanatic
2. concentric-concentric
bended plane parallel plate,
nearly vanishing effect on rays
Aplanatic Lenses
A-A convergence
enhanced
no effectconvergence
reduced
parallel offset
A-C
C-A C-C
A-VC-V field lens I field lens II
27
Impact of aplanatic lenses in microscopy on magnification
Three cases of typical combinations
28
Aplanatic Lenses
A-C mAC = n
ss‘
n
A-V mAV = n2
ss‘
n
A-C A-V m = n3
nn
' '
'
y nsm
y n s
Interferometer Collimator Lens
Example lens
Aperture NA = 0.5
Spherical correction with
one surface
possible surfaces
under test
Wrms
[l]
0.1
0.08
0.06
0.04
0.02
00 0.03 0.06 0.09 0.150.12
w [°]
diffraction limit
-0.02
0
0.02
sph
-4
-2
0
2
4
coma
surfaces
sumL1 L2 L3 L4lenses
1 2 3 4 5 6 7 8
29
Microscope Objective Lens
Examples of large-working-
distance objective lenses
Aplanatic-concentric shell-lenses
in the front group
Large diameter of the lens L1 M2 M3 M
4
30
Correction of spherical aberration:
Splitting of lenses
Distribution of ray bending on several
surfaces:
- smaller incidence angles reduces the
effect of nonlinearity
- decreasing of contributions at every
surface, but same sign
Last example (e): one surface with
compensating effect
Correcting Spherical Aberration: Lens Splitting
Transverse aberration
5 mm
5 mm
5 mm
(a)
(b)
(c)
(d)
Improvement
(a)à(b) : 1/4
(c)à(d) : 1/4
(b)à(c) : 1/2
Improvement
Improvement
(e)
0.005 mm
(d)à(e) : 1/75
Improvement
5 mm
Ref : H. Zügge
31
Splitting of lenses and appropriate bending:
1. compensating surface contributions
2. Residual zone errors
3. More relaxed
setups preferred,
although the
nominal error is
larger
Correcting Spherical Aberration : Power Splitting
2.0 mm
0.2 mm
(a)
(b)
(c)
(d)
(e)
0.2 mm
0.2 mm 0.2 mm
Ref : H. Zügge
32
Better correction
for higher index
Shape of lens / best
bending changes
from
1. nearly plane convex
for n= 1.5
2. meniscus shape
for n > 2
Correcting Spherical Aberration: Refractive Index
-8
-6
-4
-2
0
1.4 1.5 1.6 1.7 1.8 1.9
Δs’
4.0
n
n = 1.5 n = 1.7 n = 1.9 n = 4.0
best shape
best shape
plano-convex
plano-convex
1.686
Ref : H. Zügge
33
Correcting Spherical Aberration: Cementing
0.25 mm0.25 mm
(d)
(a)
(c)
(b)
1.0 mm 0.25 mm
Crown
in front
Filnt
in front
Ref : H. Zügge
Correcting spherical aberration by cemented doublet:
Strong bended inner surface compensates
Solid state setups reduces problems of centering sensitivity
In total 4 possible configurations:
1. Flint in front / crown in front
2. bi-convex outer surfaces / meniscus shape
Residual zone error, spherical aberration corrected for outer marginal ray
34
Better correction
for high index also for meltiple
lens systems
Example: 3-lens setup with one
surface for compensation
Residual aberrations is quite better
for higher index
Correcting Spherical Aberration: Refractive Index
-20
-10
0
10
20
30
40
1 2 3 4 5 6 sum
n = 1.5 n = 1.8
SI ()
Surface
n = 1.5
n = 1.8
0.0005 mm
0.5 mm
Ref : H. Zügge
35
IR Objective Lens
Aperture f/1.5
Spectral 8 - 12 mm
2 lenses aspherical
ideal
[cyc/mm]
MTF
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
axis
field
solid: tan
dashed: sag
36
Compound Systems
System groups:
1. Afocal zoom telescope
2. Scanning group
3. Reimager
Suitable for cooled matrix detector
37
Bending effects astigmatism
For a single lens 2 bending with
zero astigmatism, but remaining
field curvature
Astigmatism: Lens Bending
Ref : H. Zügge
20
-0. 01 0-0.02-0. 03 0.01-0. 04
Surface 2
Surface 1
Sum
Curvature of surface 1
15
10
5
0
-5
Astigmatism
Seidel coefficients
in [l]
ST
-2.5 0 2.5
ST S T ST ST
-2.5 0 2.5 -2.5 0 2.5 -2.5 0 2.5 -2.5 0 2.5
38
Principle of Symmetry
Perfect symmetrical system: magnification m = -1
Stop in centre of symmetry
Symmetrical contributions of wave aberrations are doubled (spherical)
Asymmetrical contributions of wave aberration vanishes W(-x) = -W(x)
Easy correction of:
coma, distortion, chromatical change of magnification
front part rear part
l l
l
2
1
3
39
40
Even Aberrations in Symmetrical Systems
Aberrations with even symmetry are doubled
Spherical aberration, Astigmatism, field curvature, axial chromatical aberration
Ref: M. Seesselberg
spherical aberration in an symmetrical system
4 4 9 9W c Z c Z
4 4 9 92 2W c Z c Z
4 4 9 9W c Z c Z
doubled values
41
Odd Aberrations in Symmetrical Systems
Ref: M. Seesselberg
Aberrations with odd symmetry are vanishing
Coma, distortion, transverse chromatical aberration
coma in an symmetrical system
8 8 15 15W c Z c Z 8 8 15 15W c Z c Z
0W
vanishing values
Symmetrical Systems
skew sphericalaberration
Ideal symmetrical systems:
Vanishing coma, distortion, lateral color aberration
Remaining residual aberrations:
1. spherical aberration
2. astigmatism
3. field curvature
4. axial chromatical aberration
5. skew spherical aberration
42
Radii of curvature of wide angle camera lenses - symmetrical setups
Mostly radii 'concentric' towards the stop losition
Locations zj of surfaces normalized for comparison
Nearly linear trend, some exceptions near to the pupil
Stop position centered
43
Wide Angle Lenses - Symmetrical
zjDouble Gauss
Pleogon
Rj
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-250
-200
-150
-100
-50
0
50
100
150
200
250
Biogonzj
stop
Symmetry Principle
Triplet Double Gauss (6 elements)
Double Gauss (7 elements)Biogon
Ref : H. Zügge
Application of symmetry principle: photographic lenses
Especially field dominant aberrations can be corrected
Also approximate fulfillment
of symmetry condition helps
significantly:
quasi symmetry
Realization of quasi-
symmetric setups in nearly
all photographic systems
44
Offner-System
object
image
M
r2
r1
d1
d2
-0.1
0
0.1
-0.1
0
0.1
-0.2
0
0.2
curvature
astigmatism
distortion
M11
M2 M12
sum
Concentric system of Offner:
relation
Due to symmetry:
Perfect correction of field aberrations in third order
21
212
rr
dd
45
Dyson-System
T S
y
-0.10 0-0.20zmirror
object
image
rL
nr
M
Catadioptric system with m = -1 according Dyson
Advantage : flat field
Application: lithography and projection
Relation:
Residual aberration : astigmatism
ML rn
nr
1
46
47
Mono-Centric Systems
Offner
mirror
object
image
rL
nrM
n3
n2
n1
r3
r2
r1
rsph
rm
max
Offner-
Wynne
Dyson
Sutton ball-
lens
Retrofocus I
Retrofocus II
do-nothing lens
(only Petzval)
r2r 1
n n
48
Monocentric System
Nearly perfect system with 4 monocentric surfaces
Data:
f = 25 mm
NA = 0.98
Unfortunately image plane not easily accessible
Only small field possible
No coma due to monocentric behavior
Ref: D. Shafer Proc. SPIE 5865 (2005)
center
Stop
M1
focal plane(curved)
N-BK7N-F2
Mono-Centric Lenses
NewtonSchwarzschild
Stamenov Schmidt
49
Bending of an achromate
- optimal choice: small residual spherical
aberration
- remaining coma for finite field size
Splitting achromate:
additional degree of freedom:
- better total correction possible
- high sensitivity of thin air space
Aplanatic glass choice:
vanishing coma
Coma Correction: Achromat
0.05 mm
'y0.05 mm
'y0.05 mm
'y
Pupil section: meridional meridional sagittal
Image height: y’ = 0 mm y’ = 2 mm
Transverse
Aberration:
(a)
(b)
(c)
Wave length:
Achromat
bending
Achromat, splitting
(d)
(e)
Achromat, aplanatic glass choice
Ref : H. Zügge
50
Effect of lens bending on coma
Sign of coma : inner/outer coma
Inner and Outer Coma
outer coma
large
incidence angle
for upper coma ray
inner coma
large
incidence angle
for lower coma ray
0.2 mm'y
0.2 mm'y
0.2 mm'y
0.2 mm'y
From : H. Zügge
51
Perfect coma correction in the case of symmetry
But magnification m = -1 not useful in most practical cases
Coma Correction: Symmetry Principle
Pupil section: meridional sagittal
Image height: y’ = 19 mm
Transverse
Aberration:
Symmetry principle
0.5 mm
'y0.5 mm
'y
(a)
(b)
From : H. Zügge
52
Combined effect, aspherical case prevent correction
Coma Correction: Stop Position and Aspheres
aspheric
aspheric
aspheric
0.5 mm
'y
Sagittal
coma
0.5 mm
'y
Sagittal
comaPlano-convex element
exhibits spherical aberration
Spherical aberration corrected
with aspheric surface
Ref : H. Zügge
53
Distortion and Stop Position
positive negative
Sign of distortion for single lens:
depends on stop position and
sign of focal power
Ray bending of chief ray defines
distortion
Stop position changes chief ray
heigth at the lens
Ref: H.Zügge
Lens Stop location Distortion Examples
positive rear V > 0 tele photo lens
negative in front V > 0 loupe
positive in front V < 0 retrofocus lens
negative rear V < 0 reversed binocular
54
Example: Achromate
Balance :
1. zonal spherical
2. Spot
3. Secondary spectrum
Coexistence of Aberrations : Balance
standard achromate
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
compromise :
zonal and axial error
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
spot optimizedsecondary
spectrum optimized
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-3
-2
-1
0
1
2
3
sur 1 sur 2 sur 3 sum sur 1 sur 2 sur 3 sur 4 sum
sur 1 sur 2 sur 3 sur 4 sum sur 1 sur 2 sur 3 sur 4 sum
a)
b) c)
spherical
aberration
axial
color
Ref : H. Zügge
55
Example: Apochromate
Balance :
1. zonal spherical
2. Spot
3. Secondary spectrum
Coexistence of Aberrations : Balance
SSK2 CaF2 F13
0.120
rP
400 nm
450 nm
500 nm
550 nm
600 nm
650 nm
700 nm
-0.04 0.04 0.08
z
axis
field
0.71°
field
1.0°
400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm
56