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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