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Design and Correction of optical Systems

Part 7: Geometrical aberrations

Summer term 2012

Herbert Gross

1

Overview

1. Basics 2012-04-18

2. Materials 2012-04-25

3. Components 2012-05-02

4. Paraxial optics 2012-05-09

5. Properties of optical systems 2012-05-16

6. Photometry 2012-05-23

7. Geometrical aberrations 2012-05-30

8. Wave optical aberrations 2012-06-06

9. Fourier optical image formation 2012-06-13

10. Performance criteria 1 2012-06-20

11. Performance criteria 2 2012-06-27

12. Measurement of system quality 2012-07-04

13. Correction of aberrations 1 2012-07-11

14. Optical system classification 2012-07-18

2012-04-18

2

7.1 Representations

7.2 Power expansion of aberrations

7.3 Seidel aberrations

7.4 Surface contributions

7.5 Primary monochromatic aberrations

7.6 Chromatical aberrations

Part 7: Contents

3

2012-05-30

Optical Image Formation

� Perfect optical image:

All rays coming from one object point intersect in one image point

� Real system with aberrations:

1. transverse aberrations in the image plane

2. longitudinal aberrations from the image plane

3. wave aberrations in the exit pupil

4

� Longitudinal aberrations ∆s

� Transverse aberrations ∆y

Representation of Geometrical Aberrations

Gaussian imageplane

ray

longitudinalaberration

∆ s'

optical axis

system

U'reference

point

referenceplane

reference ray

(real or ideal chief ray)

transverse

aberration∆∆∆∆y'

optical axis

system

ray

U'

Gaussianimageplane

reference ray

longitudinal aberrationprojected on the axis

∆∆∆∆l'

optical axis

system

ray

∆∆∆∆l'o

logitudinal aberrationalong the reference ray

5

Representation of Geometrical Aberrations

ideal reference ray angular aberration∆∆∆∆U'

optical axis

system

real ray

x

z

s' < 0∆∆∆∆

W > 0

reference sphere

paraxial ray

real ray

wavefront

R

C

y'∆∆∆∆

Gaussianreference

plane

U'

� Angle aberrations ∆u

� Wave aberrations ∆W

6

Transverse Aberrations

� Typical low order polynomial contributions for:

Defocus, coma, spherical, lateral color

� This allows a quick classification of real curves

linear:

defocus

quadratic:

coma

cubic:

spherical

offset:

lateral color

7

Transverse Aberrations

∆∆∆∆y ∆∆∆∆x

xp

yp

1

-1

1-1

5 µµµµm5 µµµµm

λλλλ= 486 nm

λλλλ= 588 nm

λλλλ= 656 nm

Dy

Dx

tangential sagittal

� Classical aberration curves

� Strong relation to spot diagram

� Usually only linear sampling along the x-, y-axis

no information in the quadrant of the aperture

8

Spot Diagram

� All rays start in one point in the object plane

� The entrance pupil is sampled equidistant

� In the exit pupil, the transferred grid

may be distorted

� In the image plane a spreaded spot

diagram is generated

9

Spot Diagram

� Table with various values of:

1. Field size

2. Color

� Small circle:

Airy diameter for

comparison

� Large circle:

Gaussian moment

486 nm 546 nm 656 nm

axis

fieldzone

fullfield

10

Aberrations of a Single Lens

� Single plane-convex lens,

BK7, f = 100 mm, λ = 500 nm

� Spot as a function of

field position

� Coma shape rotates according

to circular symmetry

� Decrease of performance with

the distance to the axis

x

y

11

Polynomial Expansion of the Aberrations

� Paraxial optics: small field and aperture angles

Aberrations occur for larger angle values

� Two-dimensional Taylor expansion shows field

and aperture dependence

� Expansion for one meridional field point y

� Pupil: cartesian or polar grid in xp / yp

field

point

opticalaxis

axispoint

entrance

pupil

coma rays

outer rays of aperture cone

chief

ray

objectheight y

xp

O

xp

yp

r

θθθθ

yp

ray

objectplane

meridionalplane

sagittalplane

12

∑ ⋅⋅⋅=∆mlk

ml

p

k

klmp ryaryy,,

cos'),,'( θθ

isum number of terms

Type of aberration

2 2 image location

4 5 primary aberrations, 3rd/4th order

6 9 secondary aberrations, 5th/6th order

8 14 higher order

Polynomial Expansion of Aberrations

� Taylor expansion of the deviation:

y' Image height index k

rp Pupil height index l

θ Pupil azimuth angle index m

� Symmetry invariance: selection of special combinations of exponent terms

� Number of terms: sum of

indices in the exponent isum

� The order of the aperture function

depends on the aberration type used:

primary aberrations:

- 3rd order in transverse aberration ∆y

- 4th order in wave aberration W

Since the coupling relation

changes the order by 1

px

W

Ry

∂

∂⋅−=∆

1

13

Power Series Expansion of Aberrations

� General case : two coordinates in object plane and pupil

� Rotational symmetry: 3 invariants

1. Scalar product of field vector and pupil vector

2. Square of field height

3. Square of pupil height

� Therefore:

Only special power

combinations are

physically meaningful

Object

Pupilx

y

P

F

y

x

xp

yp

z

xp

yp

F

P

222 yxFFF +==⋅rr

222

pp yxPPP +==⋅rr

yyxxFPFP pp ⋅+⋅=−⋅⋅=⋅ )cos(PF

ϕϕrr

14

Polynomial Expansion of Aberrations

� Representation of 2-dimensional Taylor series vs field y and aperture r

� Selection rules: checkerboard filling of the matrix

� Constant sum of exponents according to the order

Field y

Spherical

y0 y 1 y 2 y3 y 4 y 5

Distortion r 0y y3

primary

y 5

secondary

r 1r 1

Defocus

Aper-

ture

r r 2r2yComa

primary

r 3r 3

Spherical

primary

r4

r 5r 5

Spherical

secondary

DistortionDistortionTilt

Coma Astigmatism

Image

location

Primary

aberrations /

Seidel

Astig./Curvat.

cos

cos

cos2

cos

Secondary

aberrations

cos

r 3y 2cos2

Coma

secondary

r 4y cos

r 2y3cos

3

r2y3

cos

r1y 4

r1y 4

cos2

r 3y 2

r 12yr 12y

15

Primary Aberrations

Dy

PryAry

CryrSry

⋅+

⋅⋅+⋅⋅⋅+

⋅⋅⋅⋅+⋅=∆

3

222

223

cos

cos

θ

θ

� Expansion of the transverse aberration ∆y on image height y and pupil height r

� Lowest order 3 of real aberrations: primary or Seidel aberrations

� Spherical aberration: S

- no dependence on field, valid on axis

- depends in 3rd order on apertur

� Coma: C

- linear function of field y

- depends in 2rd order on apertur with azimuthal variation

� Astigmatism: A

- linear function of apertur with azimuthal variation

- quadratic function of field size

� Image curvature (Petzval): P

- linear dependence on apertur

- quadratic function of field size

� Distortion: D

- No dependence on apertur

- depends in 3rd order on the field size

16

Transverse Aberrations of Seidel

� Transverse deviations

� Sum of surface

contributions

( ) ( ) ( )[ ]

( ) ( )

( )'

''2

'''''

'''2

''''''

''

'''''''

'''2

''''''''''2'

''2

'''''

3

322

3

2222

3

22

3

322

3

422

DRn

ssyxy

PRn

ssyxyA

Rn

ssyyxxy

CRn

ssyxyyyxxyS

Rn

syxyy

p

p

p

pppp

p

ppp

p

pppppp

p

ppp

+−

++

++

+++−

+=∆

( ) ( ) ( )[ ]

( ) ( )

( )'

''2

'''''

'''2

''''''

''

'''''''

'''2

''''''''''2'

''2

'''''

3

322

3

2222

3

22

3

322

3

422

DRn

ssyxx

PRn

ssyxxA

Rn

ssyyxxx

CRn

ssyxxyyxxxS

Rn

syxxx

p

p

p

pppp

p

ppp

p

pppppp

p

ppp

+−

++

++

+++−

+=∆

∑=

=k

j

jSS1

'

∑=

=k

j

jCC1

'

∑=

=k

j

jAA1

'

∑=

=k

j

jPP1

'

∑=

=k

j

jDD1

'

17

Surface Contributions

� Spherical aberration

� Coma

� Astigmatisms

� Field curvature

� Distortion

−=

jjjj

jjjsnsn

QS1

''

124ω

−⋅⋅

−=

11

124 111

''

1

pjj

pjpj

jjjj

jjjssQ

Qn

snsnQC

ω

ωω

18

2

11

2

124 111

''

1

−⋅

⋅

−=

pjj

pjpj

jjjj

jjjssQ

Qn

snsnQA

ω

ωω

−⋅−

−⋅

⋅

−=

jjrpjj

pjpj

jjjj

jjjnnrssQ

Qn

snsnQP

1

'

11111

''

12

11

2

124

ω

ωω

−⋅

−⋅−

−⋅

⋅

−=

11

1

2

11

2

124 111

'

11111

''

1

pjj

pjpj

jjrpjj

pjpj

jjjj

jjjssQ

Qn

nnrssQ

Qn

snsnQD

ω

ω

ω

ωω

Surface Contributions of Chromatical Aberrations

� Axial chromatical aberration

� Transverse chromatical aberration

� Total chromatical errors

of a system

Hs s

s sQ

n

nj

CHV p

p

j pj pj F

j

j j

=⋅

−⋅

−

1 1

1 1

1ω ω

ν∆

∆ y y HCHV n j

CHV

j

n

' '= ⋅=

∑1

∆ ss

nK

h

fCHL

n

n n

Linse

CHL'

'

' '= − ⋅ = −

⋅

2

2

2

ω ν

j

j

j

CHL

jKν

ωΦ

= 2)(

19

Surface Contributions

� Abbreviations:

1. height ratio of marginal rays

2. height ratio of chief rays

3. Surface invariant of Abbe

4. Abbe invariant of the pupil imaging

ω j

jh

h=

1

ω pj

pj

p

h

h=

1

Q nr s

j j

j j

= ⋅ −

1 1

Q nr s

pj j

j pj

= ⋅ −

1 1

20

Surface Contributions of Seidel

� In 3rd order (Seidel) :

Additive contributions of all surfaces of a system to the total aberration

� Spherical aberration

� Coma

� Astigmatisms

� Field curvature

� Distortion

SRs

psRC ⋅

−

+−⋅=

'

''4

SRs

psRA ⋅

−

+−⋅=

2

'

''4

21

42

2'

'

'

''

'

11

8

)'('

+−⋅

−⋅

−−=

p

s

s

nn

R

n

sRn

nnnS

2

2

'4

)'('

'

''2

nRp

nnnS

Rs

psRP

−−⋅

−

+−⋅=

Rs

psR

nRp

nnnS

Rs

psRE

−

+−⋅

−−⋅

−

+−⋅=

'

''

'2

)'('

'

''4

2

3

Surface Contributions: Example

22

� Seidel aberrations:

representation as sum of

surface contributions possible

� Gives information on correction

of a system

� Example: photographic lens

12

3 45

6 8 9

10

7

Retrofocus F/2.8

Field: w=37°

SI

Spherical Aberration

SII

Coma

-200

0

200

-1000

0

1000

-2000

0

2000

-1000

0

1000

-100

0

150

-400

0

600

-6000

0

6000

SIII

Astigmatism

SIV

Petzval field curvature

SV

Distortion

CI

Axial color

CII

Lateral color

Surface 1 2 3 4 5 6 7 8 9 10 Sum

Lens Contributions of Seidel

� In 3rd order (Seidel) :

Additive contributions of lenses to the total aberration value

(stop at lens position)

� Spherical aberration

� Coma

� Astigmatisms

� Field curvature

� Distortion 0=lensD

+−

−

+⋅= MnX

n

n

fnsClens )12(

1

1

'4

12

2'2

1

sfAlens

⋅−=

2'4

1

snf

nPlens

⋅

+−=

23

⋅

+

−−

⋅

+

−−⋅

−

++

−−= 2

22

23

32

)1(

2

)1(2

1

2

1)1(32

1M

n

nnM

n

nX

n

n

n

n

fnnSlens

Spherical Aberration

� Spherical aberration:

On axis, circular symmetry

� Perfect focussing near axis: paraxial focus

� Real marginal rays: shorter intersection length (for single positive lens)

� Optimal image plane: circle of least rms value

paraxial

focus

marginal

ray focusplane of the

smallest

rms-value

medium

imageplane

As

plane of the

smallest

waist

2 A s

24

Spherical Aberration

� Single positive lens

� Paraxial focal plane near axis,

Largest intersection length

� Shorter intersection length for

rim ray and outer aperture zones

25

� Spherical aberration and focal spot diameter

as a function of the lens bending (for n=1.5)

� Optimal bending for incidence averaged

incidence angles

� Minimum larger than zero:

usually no complete correction possible

Spherical Aberration: Lens Bending

objectplane

imageplane

principalplane

26

Aplanatic Surfaces

� 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=

+=

+=

+'

' '

'

'

'

27

� Aplanatic lenses

� Combination of 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 : parallel offset

A-C :

convergence enhanced

C-C :no effect

C-A :convergence reduced

28

� Reason for astigmatism:

chief ray passes a surface under an oblique angle,

the refractive power in tangential and sagittal section are different

� A tangential and a sagittal focal line is found in different

distances

� Tangential rays meets closer to the surface

� In the midpoint between both focal lines:

circle of least confusion

Astigmatism

29

� Beam cross section in the case of astigmatism:

- Elliptical shape transforms its aspect ratio

- degenerate into focal lines in the focal plane distances

- special case of a circle in the midpoint: smallest spot

y

x

z

tangentialfocus

sagittalfocuscircle of least

confusion

tangentialaperture

sagittalaperture

Astigmatism

30

Imaging of a polar grid in different planes

sagittal line

tangential line

entrance

pupil

exit

pupil

object

circlesagittal

focus

tangential

focus

best focus

image space

Astigmatism

31

Field Curvature and Image Shells

� Imaging with astigmatism:

Tangential and sagittal image shell depending on the azimuth

� Difference between the image shells: astigmatism

� Astigmatism corrected:

It remains a curved image shell,

Bended field: also called Petzval curvature

� System with astigmatism:

Petzval sphere is not an optimal

surface with good imaging resolution

� Law of Petzval: curvature given by:

� No effect of bending on curvature,

important: distribution of lens

powers and indices

1 1

rn

n fp k kk

= − ⋅⋅

∑'

32

Astigmatisms and Curvature of Field

∆∆ ∆

ss s

best

sag'

' 'tan

=+

2

( )∆ ∆ ∆ ∆s s s spet sag pet' ' ' 'tan − = ⋅ −3

∆ ∆ ∆s s sast sag' ' 'tan

= −

� Image surfaces:

1. Gaussian image plane

2. tangential and sagittal image shells (curved)

3. image shell of best sharpness

4. Petzval shell, arteficial, not a good image

� Seidel theory:

� Astigmatism is difference

� Best image shell

2

''3'

tanss

ssag

pet

∆−∆=∆

z

y'

Petzval

surface

sagittalsurface

tangentialsurface

Gaussianimageplane

mediumsurface

∆∆∆∆s'sag

∆∆∆∆s'tan

∆∆∆∆s'pet

∆∆∆∆∆∆∆∆

∆∆∆∆

33

� Focussing into different planes of a system with field curvature

� Sharp imaged zone changes from centre to margin of the image field

focused in center

(paraxial image plane)focused in field zone

(mean image plane)

focused at field boundary

z

y'

receiving

planes

image

sphere

Field Curvature

34

Blurred Coma Spot

� Coma aberration: for oblique bundels and finite aperture due to asymmetry

� Primary effect: 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 rayzone 1

zone 3

zone 2comablur

lens / pupil

axis

35

Distortion Example: 10%

Ref : H. Zügge

� What is the type of degradation of this image ?

� Sharpness good everywhere !

36

Distortion Example: 10%

Ref : H. Zügge

� Image with sharp but bended edges/lines

� No distortion along central directions

37

� 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

barreldistortion

object

D < 0

D > 0

lens

rear

stop

imagex

x

y

y

y'

x'

y'

x'

front stop

ideal

idealreal

y

yyD

'

'' −=

38

� Non-symmetrical systems:

Generalized distortion types

� Correction complicated

General Distortion Types

original

anamorphism, a10

x

keystone, a11

xy

1. orderlinear

2. orderquadratic

3. order

cubic

line bowing, a02

y2

shear, a01

y

a20

x2

a30

x3 a21

x2y a12

xy2 a03

y3

39

Axial Chromatical Aberration

∆ s s sCHL F C' ' '' '

= −

white

H'

s'F'

s'

s'

e

C'green

red

blue

� Axial chromatical aberration:

Higher refractive index in the blue results in a shorter intersection length for a single lens

� The colored images are defocussed along the axis

� Definition of the error: change in image location /

intersection length

� Correction needs several glasses with different dispersion

40

s∆∆∆∆

λλλλ

e C'F'

Axial Chromatical Aberration

� Simple achromatization / first order

correction:

- two glasses with different dispersion

- equal intersection length for outer

wavelengths (blue F', red C')

- residual deviation for middle wavelength

(green e)

� Residual erros in image location:

secondary spectrum

� Apochromat:

- coincidence of the image location

for at least 3 wavelengths

- three glasses necessary, only with

anomal partial dispersion

(exceptions possible)

white

H'

s'F'

s'

s'

e

C'

green

redblue

secondary

spectrum

41

Axial Chromatical Aberration

� Longitudinal chromatical aberration for a single lens

� Best image plane changes with wavelength

Ref : H. Zügge

42

stop

red

blue

reference

image

plane

y'CHV

'' ''' CFCHV yyy −=∆

e

CF

CHVy

yyy

'

''' '' −=∆

Chromatic Variation of Magnification

� Lateral chromatical aberration:

Higher refractive index in the blue results in a stronger

ray bending of the chief ray for a single lens

� The colored images have different size,

the magnification is wavelength dependent

� Definition of the error: change in image height/magnification

� Correction needs several glasses with different dispersion

� The aberration strongly depends on the stop position

43

� Impression of CHV in real images

� Typical colored fringes blue/red at edges visible

� Color sequence depends on sign of CHV

Chromatic Variation of Magnification

original

withoutlateral

chromaticaberration

0.5 % lateralchromatic

aberration

1 % lateralchromatic

aberration

44

Summary of Important Topics

� There are different representations for aberrations

� Transverse aberrations are the most useful type

� Aberrations can be expanded into a Taylor series for field and aperture coordinate

� There are 7 primary aberrations (Seidel) of 3rd order; spherical aberration, coma,

astigmatism, field curvature, distortion, axial chromatical, lateral chromatical

� In the Seidel definition, surface and lens contributions are additive

� Lens bending strongly changes aberration contribution due to variation of the incidence

angles

� There are special setups with vanishing spherical aberration,

important are aplanatic surfaces

� The image surface has two different equivalent representations:

1. tangential and sagittal image shell

2. astigmatism (difference) and field curvature (mean)

� Chromatic aberrations: axial and lateral, same size as monochromatic primary aberrations

45

Outlook

Next lecture: Part 8 – Wave optical aberrations

Date: Wednesday, 2012-04-25

Contents: 8.1 Introduction, phase and optical path length

8.2 Definition of wave aberrations

8.3 Power expansion of the wave aberration

8.4 Zernike polynomials

8.5 Measurement of wave aberrations

8.6 Special aspects

46

Outlook

Next lecture: Part 7 – Geometrical aberrations

Date: Wednesday, 2012-06-06

Contents: 7.1 Representations

7.2 Power expansion of aberrations

7.3 Surface contributions

7.4 Primary monochromatic aberrations

7.5 Chromatical aberrations

47