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Imaging and Aberration Theory
Lecture 7: Coma and Distortion
2013-12-12
Herbert Gross
Winter term 2013
2
Preliminary time schedule
1 24.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 07.11. Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 14.11. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 21.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 05.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 12.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 19.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 09.01. Chromatical aberrations
Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 16.01. Further reading on aberrations sensitivity in 3rd order, structure of a system, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics
11 23.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF
12 30.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement
13 06.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, vectorial aberrations, partial symmetric systems
1. Geometry of coma spot
2. Coma-dependence on lens bending, stop position and spherical aberration
3. Point spread function with coma
4. Distortion
5. Examples
3
Contents
Occurence of coma:
skew chief ray and finite aperture
Asymmetry between upper and lower coma ray
Bended plane of sagittal coma rays
stop
n n'
auxiliary axis lower coma ray
upper coma ray
chief ray
sagittal rays
C
tangential coma
sagittalcoma
astigmatic difference between coma rays
T
S
center of curvature
surfaceR
Coma
lowercoma ray
upper coma ray
sagittalrays
auxiliary axis
chiefray
pupil intersection point
5
Ray Caustic of Coma
only sagittal
rays
bended ray
fan
coma
A sagittal ray fan forms a groove-like
surface in the image space
Tangential ray fan for coma:
caustic
Building of Coma Spot
Coma aberration: for oblique bundels and finite aperture due to asymmetry
Special problem: coma grows linear with field size y
Systems with large field of view: coma hard to correct
Relation of spot circles
and pupil zones as shown
chief ray
zone 1
zone 3
zone 2coma
blur
lens / pupil
axis
Coma
Coma deviation, elimination of the azimuthal dependence:
circle equation
Diameter of the circle and position variiation with rp2
Every zone of the circlegenerates a circle in the
image plane
All cricels together form a comet-like shape
The chief ray intersection point is at the tip of
the cone
The transverse extension of the cone shape has
a ratio of 2:3
the meridional extension is enlarged and gives
a poorer resolution
y'
30°
rp =1.0
sag 90°
tan 0°/180°
45° 135°
rp = 0.75
rp = 0.5
x'
sagittal
coma
tangential
coma
Ray trace properties
Double speed azimuthal growth
between pupil and image
Sagittal coma smaller than tangential coma
Coma
chief ray
sagittal ray
sagittal ray
tangential ray
tangential ray y’
x’
xp
yp
C
S
T
CT
tangential coma
CS
sagittal coma
C
S
T
Pupil Image
Ref: H. Zügge
sagyy 3tan
Coma
Wave aberration
tangential sagittal
22
Primary coma
Transverse ray aberration y x y
Pupil: y-section x-section x-section
0.01 mm 0.01 mm 0.01 mm
Geometrical spot through
focus
0.0
2 m
m
Modulation Transfer Function MTF
MTF at paraxial focus MTF through focus for 100 cycles per mm
tangential
sagittal
tangential
sagittal
z/mm
-0.02 0.020.0-0.01 0.01
1
0
0.5
0 100 200cyc/mm z/mm
1
0
0.5
0.020.0-0.02
Ref: H. Zügge
Typical representations of coma
Cubic curve in wavefront cross section
Quadratic function in transverse
aberrations
Orientation of the coma shape:
disctinction between
1. outer coma, tip towards optical axis
2. inner coma, tip outside
Orientation of the coma spot is always
rotating with the azimuthal angle of the
considered field point
j
y
x
r
Coma Orientation
x'
y'
inner
comaouter
coma
Bending a single lens
Variation of the primary aberrations
The stop position is important
for the for the off-axis
aberrations
Typical changes:
1. coma linear
2. chromatical magnification
linear
3. spherical aberration
quadratically
Bending of a Lens
aberration
p
coma
spherical
aberration
0
distortion
stop at
lens
transverse
chromatical
aberration
Lens with remote stop
Not all of the aberrations
spherical, astigmatism and
coma can be corrected by
bending simultaneously
Zero correction for coma
and astigmatism possible
(depends on stop position)
Spherical aberration not
correctable
12
Lens with Remote Stop
-0.03 -0.014 0.002 0.018 0.034 0.05-100
-70
-40
-10
20
50
80
110
140
170
200
1/R1
spherical
coma
astigmatism
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
Ref: H. Zügge
The lens contribution of coma is given by
if the stop is located at the lens
Therefore the coma can be corrected by bending the lens
The optimal bending is given by
and corrects the 3rd order coma completly
The stop shift equation for coma is given by
with the normalized ratio of the chief ray height
to the marginal ray height
If the spherical aberration SI is not corrected, there is a natural stop position with vanishing
coma
If the spherical aberration is corrected (for example by an aspheric surface), the coma
doesn't change with the stop position
14
Lens Bending and Natural Stop Position
Xn n
nM
( )( )2 1 1
1
MnX
n
n
fnsClens )12(
1
1
'4
12
h
hhE oldnew
IIIII SESS
Example
The front stop position of a
single lens is shifted
The 3rd order Seidel coefficient
as well as the Zernike coefficient
vanishes at a certain position
of the stop
15
Coma-free Stop Position
-550
-4
60
-3
70
-2
80
-1
90
0
100
1
110
2
120
3
130
4
140
5
150 50 60 70 80 90 100 110 120 130 140 150
Ac
t1-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t1
c8
114.3 114.5
t1 = 50
t1 = 114.3
t1 = 150
stop shifted
Influence of Stop Position on Coma
Achromat 4/100, w = 10°, y‘ = 17.6
Tranverse aberr. +- 2.0
w = 0° w= 10°
y‘ y‘ y‘ y‘
Spot
w = 10°
Ref: H. Zügge
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
Cases:
a) simple achromate,
sph corrected, with coma
b) simple achromate,
coma corrected by bending, with sph
c) other glass choice: sph better, coma reversed
d) splitted achromate: all corrected
e) aplanatic glass choice: all corrected
Coma Correction: Achromate
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
Stop Position Influence for Corrected Spherical
Achromat 4/100, w = 10°, y‘ = 17.6, (Asphäre für Sph. Aberr. = 0)
Transverse aberr. +- 2.0
w = 0° w= 10°
y‘ y‘ y‘ y‘
Spot
w = 10° Asphäre
Ref: H. Zügge
Combined effect, aspherical case prevents 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
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)
Ref: H. Zügge
Geometrical Coma Spot
Geometrical calculated spot
intensity
The is a step at the lower circle boundary
The peak lies in the apex point
The centroid lies at the lower circle boundary
The minimal rms radius is
0 1 2 30
1
2
3
4
5
6
7
8
9
10
x
y
y
I(0,y) [a.u.]
0
1
2 1-1
3
shape coma inside else3
1
circlelargestinside3
1
2),(
22
3
22
3
yxRA
aI
yxRA
aI
yxI
c
ExP
c
ExP
a
ARr crms
3
2
PSF with coma
The 1st diffraction ring is influenced very sensitive
W31 = 0.03 W31 = 0.06 W31 = 0.09 W31 = 0.15
Psf for Coma Aberration
yx
1
0
0 0.05-0.05
I(x)
x
W13
= 0.3 W13
= 1.0 W13
= 5.0 W13
= 2.4 W13
= 10.0
Psf with Coma
Ref: Francon, Atlas of optical phenomena
Change of Zernike coma coefficient
- peak height reduced
- peak position constant due to tilt
component
- distribution becomes asymmetrical
Change of Seidel coma coefficient
- peak height reduced
- peak position moving
- distribution becomes asymmetrical
Transversal Psf with Coma
-15 -10 -5 0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(x)
y
W13 = 0.0
W13 = 0.2
W13 = 0.3
W13 = 0.4
W13 = 0.5
Peak
I(x)
-15 -10 -5 0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
W13 = 0.0
W13 = 0.1
W13 = 0.2
W13 = 0.3
W13 = 0.4
Separation of the peak and the centroid position in a point spread function with coma
From the energetic point of view coma induces distortion in the image
Psf with Coma
c7 = 0.3 c7 = 0.5 c7 = 1
centroid
Defocus: centroid moves on a straight
line (line of sight)
Peak of intensity moves on a curve
(bananicity)
Psf with Coma
c8 = 0.3
c8 = 0.5
c15 = 0.5
acoma = 1.7
x I(z)
zz
-8 -6 -4 -2 0 2 4 6 8-0.1
-0.05
0
0.05
0.1
0.15
y
z
solid lines : peak
dashed lines : centroid
c8 = 0.3
c8 = 0.5
c15
= 0.5
acoma
= 1.7
Centroid of the psf intensity
Elementary physical argument:
The centroid has to move on a straight line:
line of sight
Wave aberrations with odd order:
- centroid shifted
- peak and centroid are no longer
coincident
dydxzyxIx
PdydxzyxI
dydxzyxIxzxs ),,(
1
),,(
),,()(
,...5,3,1
1)1(22
)(n
n
ExP
s cnD
zzy
Line of Sight
exit
pupil
image
plane
centroid ray
chief ray
real
wavefront
reference
sphere
z
y
Image Degradation by Coma
Imaging of a bar pattern with a coma of 0.4 in x and y
Structure size near the diffraction limit
Asymmetry due to coma seen in comparison of edge slopes
object Psf with 0.4 x-coma
psf enlarged
image
image x/y section
Psf with 0.4 y-coma
psf enlarged
image
image x/y section
Coma Truncation by Vignetting
without vignettierung
with vignettierung tangential / sagittal
Ref: H. Zügge
Distortion Example: 10%
Ref : H. Zügge
What is the type of degradation of this image ?
Sharpness good everywhere !
Ref: H. Zügge
Image with sharp but bended edges/lines
No distortion along central directions
Distortion Example: 10%
Distortion
y'
x'
ideal
image
real
image
image
height
h
h
h
1
2
3
aberration
1h
3h
2h
Distortion. change of magnification over the field
Corresponds to spherical aberration of the chief ray
Measurement: relative change of image height
No image point blurr
only geometrical shape deviation
Sign of distortion:
1. V < 0 : barrel,
lens with stop in front
2. V > 0: pincushion,
lens with rear stop
Vy y
y
real ideal
ideal
Conventional definition of distortion
Special definition of TV distortion
Measure of bending of lines
Acceptance level strongly depends on
kind of objects:
1. geometrical bars/lines: 1% is still
critical
2. biological samples: 10% is not a
problem
Digital detection with image post
processing: un-distorted image can be reconstructed
y
x
H
H
yreal
yideal
y
yV
H
HVTV
TV Distortion
Purely geometrical deviations without any blurr
Distortion corresponds to spherical aberration of the chief ray
Important is the location of the stop:
defines the chief ray path
Two primary types with different sign:
1. barrel, D < 0
front stop
2. pincushion, D > 0
rear stop
Definition of local
magnification
changes
Distortion
pincussion
distortion
barrel
distortion
object
D < 0
D > 0
lens
rear
stop
imagex
x
y
y
y'
x'
y'
x'
front
stop
ideal
idealreal
y
yyD
'
''
Distortion and Stop Position
Positive, pincushion Negative, barrel
Sign of distortion of a single lens
depends on stop position
Ray bending of chief ray
determines the distorion
Lens Stop Distor-
tion Example
positive lens rear stop D > 0 Tele lens
negative lens front stop D > 0 Loupe
positive lens front stop D < 0 Retro focus lens
negativ lens rear stop D < 0 reversed Binocular
Ref: H. Zügge
Combination of distortion of 3rd and 5th order:
Bended lines with turning points
Typical result for corrected/compensated distortion
Distortion of Higher Order
3rd order negative5th order positiveobject image
Non-symmetrical systems:
Generalized distortion types
Correction complicated
General Distortion Types
original
anamorphism, a10
x
keystone, a11
xy
1. order
linear
2. order
quadratic
3. order
cubic
line bowing, a02
y2
shear, a01
y
a20
x2
a30
x3 a21
x2y a12
xy2 a03
y3
Distortion occurs, if the magnification depends on the field height y
In the special case of an invariant location p‘ of the exit pupil:
the tangent of the angle of the chief ray should be scaled linear
Airy tangent condition:
necessary but not sufficient condition for distortion corection:
This corresponds to a corrected angle of the pupil imaging
form entrance to exit pupil
Reasons of Distortion
O1
O2
O'1
O'2EnP ExP
w2w'2
ideal
real
y2 y2
p p'
y'
wo'2
tan '
tan
w
wconst
Second possibility of distortion:
the pupil imaging suffers from longitudial spherical aberration
The location of the exit pupil than depends on the field height
With the simple relations
we have the general expression
for the magnification
For vanishing distortion:
1. the tan-condition is fulfilled (chief ray angle)
2. the spherical aberration of the pupil imaging is corrected (chief ray intersection point)
Reasons of Distortion
O1
O2
O'1
O'2EnP ExP
w2w'2
ideal
real
y2 y2
p po'p'
y'
w
w
p
yp
p
p
wp
wyp
y
yym o
tan
'tan)(''
tan
'tan)('')(
'tan'',tan wpywpy
Distortion of a Retrofocus System
negativesfront goup
positive rear groupstop
barreldistortion
20 %Retro focus-
systems :
barrel distortion
Keystone distortion
Scheimpflug Imaging
ideal
real
41
'
y
s
s'
h
y'
h'
j
tilted
object
tilted
image
system
optical axis
principal
plane
Distortion
15% barrel5% barrel
5% pincushion
barrel and pincushion
2% pincushion
20% keystoneoriginal
10% pincushion
10% barrel
Visual impression of distortion
on real images
Visibility only at straight edges
Edge through the center are
not affected
Example lens with 210°field of view
Fish-Eye-Lens
100%
y
-100% 0
Distortion types
Fish-Eye-Lens
y' [a.u.]
0
0.5
1
1.5
2
0 10 20 30 40 50 60 70 80 90
w
[°]
stereographicgnomonic
f--projection
aperture
related
orthographic
a b
Head Mounted Display
Commercial system:
Zeiss Cinemizer
Critical performance of distortion due to
asymmetry
Refractive 3D-system
Free-formed prism
Field dependence of coma, distortion and
astigmatism
One coma nodal point
Two astigmatism nodal points
Head-Up Display
eye
pupil
image
total
internal
reflection
free formed
surface
free formed
surface
field angle 14°
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
binodal
points