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Lens Design II
Lecture 12: Mirror systems
2017-01-11
Herbert Gross
Winter term 2016
2
Preliminary Schedule
1 19.10. Aberrations and optimization Repetition
2 26.10. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection
3 02.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres
4 09.11. Freeforms Freeform surfaces
5 16.11. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses
6 23.11. Chromatical correction I Achromatization, axial versus transversal, glass selection rules, burried surfaces
7 30.11. Chromatical correction II secondary spectrum, apochromatic correction, spherochromatism
8 07.12. Special correction topics I Symmetry, wide field systems,stop position
9 14.12. Special correction topics II Anamorphotic lenses, telecentricity
10 21.12. Higher order aberrations high NA systems, broken achromates, induced aberrations
11 04.01. Further topics Sensitivity, scan systems, eyepieces
12 11.01. Mirror systems special aspects, double passes, catadioptric systems
13 18.01. Zoom systems mechanical compensation, optical compensation
14 25.01. Diffractive elements color correction, ray equivalent model, straylight, third order aberrations, manufacturing
15 01.02. Realization aspects Tolerancing, adjustment
1. General properties
2. Image orientation
3. Telescope systems
4. Further Examples
3
Contents
Geometry:
1. bending needs the separation of ray bundles
2. helps in folding systems to more compact size
3. switches image orientation in the plane of incidence
4. for centered usage of mirros: central obscuration,
spider legs for mounting
Correction:
1. astigmatism for oblique incidence
2. no color aberrations
3. positive contribution to Pethval curvature
4. usually more sensitive for off-axis field: coma
Miscellaneous:
1. coating is HR, mostly metallic, no ghost images
2. surface accuracy approximately 4 times more sensitive
3. only option for very large diameter (astronomy)
4. aspherical or freeform shape easier to fabricate
5. preferred as scanning or adaptive component
6. plane bending mirrors often realized as prisms
7. only option for extreme UV due to transmission problems
4
General Properties of Mirror Systems
Mirror inverts the system: left handed into right handed coordinate system
Vectorial calculation with tensor calculus possible
Possible solutions for correct ray tracing:
1. distances negative behind the mirror
only obsvious for normal incidence
2. refractive index negative behind the mirror
seems to be unphysical, only formal solution
For complicated prisms with multiple reflections:
tunnel diagram with unfolded reflections
5
Modelling Problems with Mirrors
Tunnel Diagram
Tunnel diagram:
Unfoldung the ray path with invariant sign of the z-component of the optical axis
Optical effect of prisms corresponds to plane parallel plates
More rigorous model:
Exact geometry of various prisms can cause vignetting
3
1 2
2
3
Modelling a Mirror Surface
Problem in coordinate system based raytracing of mirror systems:
right-handed systems becomes left-handed
Possible solutions:
1. Folding the mirror
- light propagation direction changed
z-component inverted
- tunnel diagram for prism
2. negative refractive index
3. inversion of the x-axis
r
spherical
mirror
F
f'
zC
P=P'
folded mirror
surface
Transformation of Image Orientation
Modification of the image orientation with four options:
1. Invariant image orientation
2. Reverted image ( side reversal )
3. Inverted image ( upside down )
4. Complete image inversion
(inverted-reverted image)
Image side reversal in the
principal plane of one mirror
Inversion for an odd number
of reflections
Special case roof prims:
Corresponds to one reflection
in the edge plane,
Corresponds to two reflections
perpendicular to the edge plane
y
x
y
x
y
x
mirror 1
mirror 2
y - z- folding
plane
z
z
Transformation of Image Orientation
image reversion in the
folding plane
(upside down)
image
unchanged
image
inversion
original
folding planeimage reversion
perpendicular to the
folding plane
primary mirror
focus
corrector plate
y
r
a
marginal rays
field
Telescopic System Types
Cassegrain
Schiefspiegler,
obscuration-free
Ref: F. Blechinger
d1
s'2
M1
M2
p
f1
D1
D2
s2
M1
M2
M3
M1
M2
M3
Kutter Tri-Schiefspiegler Buchroeder Tri-Schiefspiegler
Schmidt
catadioptric
Maksutow
M1
M2
L1
L2L3 L4, L5
Catadioptric Telescopes
Maksutov compact
Klevtsov
M1
M2
L1
L2L3 L4, L5
M1
L1, L2
M2
12
Avoiding Mirror Obscuration
Avoiding the central obscuration in mirror systems
Field bias or aperture offset as opportunities
Ref.: K. Fuerschbach
Astigmatism of Oblique Mirrors
Mirror with finite incidence angle:
effective focal lengths
Mirror introduces astigmatism
Parametric behavior of scales astigmatism
2
costan
iRf
i
Rfsag
cos2
i
Rs
iRsi
iRss ast
cos22
coscos2
sin'
22
i0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
s / R = 0.2
s / R = 0.4
s / R = 0.6
s / R = 1
s / R = 2
s' / R
s'ast
R
focal
line L
i
C
s
s'sag
mirror
For an oblique ray, the effective curvatures of the spherical surface depend on azimuth
Astigmatism of oblique used curved surfaces
In particular large effects in case of mirrors
Propagation of curvature components according to Coddington equations
Non-Axisymmetric Systems: Pilot Axis Ray
z'
x
y
Cy
Cx
surface
Rx
Ry
C‘x
C‘y
R'x
R'y
x'
y'
R
inin
l
in
l
in cos'cos'cos
'
'cos'
tan
2
tan
2
R
inin
l
n
l
n
sagsag
cos'cos'
'
'
14
Astigmatism at Curved Mirrors
22
22
11'
'
131'
'
R
s
R
ys
R
s
R
ys
S
T
Image surfaces for a concave mirror
y‘ : image height
sbar: stop position
Special cases of flat image shells as a
function of the stop position
a) stop a center:
zero astigmatism
b) stop at distance
0.42 R:
T=0
c) stop at distance
0.29 R:
B = 0 (best plane)
d) stop at mirror:
S = 0
15
stop
C
RR/2
P B STstop
C
R
0.42 R
P BS T
stop
C
R
P BS T
stop
C
R
P BS T
a) astigmatism A = 0 b) tangential flat
c) best image flat d) sagittal flat
0.29 R
Telescopes with tilted elements
Anastigmatic solution
for two mirrors
Schiefspiegler-Telescopes
y
y
obj
1
2
3
4
ima
d1
d2 d
3
d4
d5
object
plane
image
plane
mirror
M1, r
1
mirror
M2, r
2
d
image
22
21
dr
rr
21
21
2
2
17
Correction of 3D Mirror Systems
Problem: oblique incidence on curved mirrors
creates macroscopic astigmatism
Different solution approaches as tradeoffs between
performanbce vs cost/complexity
M1
circular
symmetric
asphere
M2
pupil
freeform M3
freeform
image
Setup Correction Drawbacks
1 All spherical Select incidence angles to fulfill
Coddington equations, compensation over
all mirrors
1. only limited solution space
(Korsch)
2. coma remains
3. induced aberrations, only small
aperture possible
2 Confocal conic
section
Perfect on axis, if focal points coincide,
rotations optimized to avoid collisions
1. Correction offaxis hard
2. all aspheres is costly
3 Centered
aspheres
Biased sub-aperture or field, centered
surfaces
1. astigmatism and coma coupled
4 Spherical Toric
surfaces
Astigmatism at any surface corrected 1. all non-spherical is costly
2. coma remains
5 Freeform All but one surface spherical, one freeform
surface compensates all coma and
astigmatism
1. large induced aberrations
2. Simultaneous resolution and field
correction needs 2 freeforms
Finding of Initial Systems
Conic section inital system approach for 4-mirror system
- F1 is common to parabola and hyperbola
- F2 is common to hyperbola and
ellipsoid 1
- F3 is commob to ellipsoid 1 and
ellipsoid 2
- image point F4 is also focal point of
ellipsoid 2
Perfect imaging on axis
H. Zhu, Proc. SPIE 9272 (2014) W1
parabola
ellipsoid 1
ellipsoid 2
hyperbola
image
F1
F2
F3
F4
18
TMA Schiefspiegler vs Freeform Solutions
First approach of a corrected obscuration-free three mirror system:
- coaxial circular symmetric system
- one common optical axis
- used for off-axis field part only (field biased approach)
- typically: astigmatism corrected with incidence angle in
complete system
- coupling of astigmatism and coma at every surface
(one degree of freedom lost)
- overall performance reduced
Second approach of a corrected onscuration-free three mirror system:
- vertex-centered unobscured three freeform mirrors
- low-order correction of freeform surfaces, coma and
astigmatism independent corrected
- overall performance improved in comparison
to first approach
19
K. Thompson, Proc. SPIE 9633 (2015)
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
Mangin Mirror
F
Principle:
Backside mirror, catadioptric lens
Advantages:
Mirror can be made spherical
Refractive surface corrects spherical
System can be made nearly aplanatic
-0.005 -0.0105 -0.0161/r
1
Ssph,
Scoma
40
20
0
-20
-40
corrected
coma
spherical
Mangin Mirror
spherical
coma
astigmatism
curvature
distortion
axial
chromatic
lateral
chromatic
-0.02
0
0.02
-0.01
0
0.01
-5
0
5
-5
0
5
-5
0
5
-0.02
0
0.02
1 2 3 sum-4
-2
0
2
4
Seidel surface contributions of a real
lens:
Spherical correction perfect
Residual axial chromatic unavoidable
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
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
Lithographic Optics
H-Design
I-Design
X-Design
EUV - Mirror System
projection
illumination
wafer
mask
source
System:
Only mirrors
Microscope Objective Lens: Catadioptric Lenses
Catadioptric lenses:
1. Schwarzschild design: first large mirror
2. Newton design: first small mirror
Advantageous:
1. Large working distance
2. Field flattening
3. Colour correction
Drawback:
central obscuration reduces
contrast / resolution
a) Schwarzschild b) Newton
Retro Reflecting Systems
r2
r1
M
a) BK7
c) SF59 / TIF6
10
-1
b) SF59
-1
n3
n2
n1
r3
r2
r1
Solution 2 :
Double hemisphere
Correction with two materials
Combined shells:
Retro Reflecting Systems
r
d = r / 2
3. Solution:
Offner-setup
Only small field angles possible
4. Solution:
Gradient-index ball lens
Only academic
f
Fz
y
R
Retro Reflecting Systems
rm
r2
5°
0°
10°
15°
20°
5. Solution:
Lens-mirror-combination
Relation for plano-convex lens :
Limitation :
Field aberrations
rn
nrm2
1
Retro Reflecting Systems
rsph
rm
max
1.
3.2.
incoming
collimated
beam
Special version: ball lens with mirror
6. Solution: axicon
Useful only on axis
Retro Reflecting Systems
triangular area
plane front surface
hexagonal area
corrugated front
surface
2
1
3
ray path
0 10 20 30 40 50 60 70 80 90
0.2
0.4
0.6
0.8
1
air
n = 1.5
n = 2
0 10 20 30 40 50 60 70 80 90
10-4
10-3
10-2
10-1
100
Log P()/P0P()/P0
0
7. Solution :
Corner-cube mirror
Two possible
realizations :
1. Only mirror
2. Corner filled with
glass
Material enhances
backreflection and
maximum field
3sin max
n