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Imaging and Aberration Theory

Lecture 1: Paraxial optics

2014-11-06

Herbert Gross

Winter term 2014

2

Preliminary time schedule

1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 06.11. Pupils, Fourier optics, Hamiltonian coordinates

pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates

3 13.11. Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 20.11. Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas

5 27.11. Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 04.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders

7 11.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 08.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum

10 15.01. Sine condition, aplanatism and isoplanatism

Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics

11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 29.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement

13 05.02. PSF and transfer function ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and contrast

14 12.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

1. Cardinal elements

2. Lens properties

3. Imaging, magnification

4. Afocal systems and telecentricity

5. Paraxial approximation

6. Matrix calculus

3

Contents

Modelling of Optical Systems

Principal purpose of calculations:

System, data of the structure(radii, distances, indices,...)

Function, data of properties,

quality performance(spot diameter, MTF, Strehl ratio,...)

Analysisimaging

aberration

theorie

Synthesislens design

Ref: W. Richter

Imaging model with levels of refinement

Analytical approximation and classification

(aberrations,..)

Paraxial model

(focal length, magnification, aperture,..)

Geometrical optics

(transverse aberrations, wave aberration,

distortion,...)

Wave optics

(point spread function, OTF,...)

with

diffractionapproximation

--> 0

Taylor

expansion

linear

approximation

4

Paraxiality is given for small angles relative to the optical axis for all rays

Large numerical aperture angle u violates the paraxiality,

spherical aberration occurs

Large field angles w violates the paraxiality,

coma, astigmatism, distortion, field

curvature occurs

Paraxial Approximation

5

Taylor expansion of the sin-function

Definition of allowed error 10-4

Deviation of the various approximations:

- linear: 5 - cubic: 24 - 5th order: 542

6

Paraxial approximation

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

sin(x)

x []

exact sin(x)

linear

cubic

5th order

x = 5 x = 24 x = 52 deviation 10-4

Law of refraction for finite angles I, I

Taylor expansion

Linear formulation of the law of refraction for small angles i, i

Relative direction error of the paraxial approximation

Paraxial approximation

i0 5 10 15 20 25 30 35 40

0

0.01

0.02

0.03

0.04

0.05

i'- I') / I'

n' = 1.9

n' = 1.7

n' = 1.5

...!5!3

sin53

xx

xx

'sin'sin InIn

1

'

sinarcsin

'

'

''

n

inn

in

I

Ii

'' inin

7

Single surface between two media Radius r, refractive indices n, n

Imaging condition, paraxial

Abbe invariant alternative representation of the

imaging equation

'

1'

'

'

fr

nn

s

n

s

n

y

n'ny'

r

C

ray through center of curvature C principal

plane

vertex S

s

s'

object

surface

image

arbitrary ray

'

11'

11

srn

srnQs

Single Surface

8

P principal point

S vertex of the surface

F focal point

s intersection point

of a ray with axis

f focal length PF

r radius of surface

curvature

d thickness SS

n refrative index

O

O'

y'

y

F F'

S

S'

P P'

N N'

n n n1 2

f'

a'

f'BFL

fBFL

a

f

s's

d

sP

s'P'

u'u

Notations of a lens

9

Focal points: 1. incoming parallel ray

intersects the axis in F 2. ray through F is leaves the lens

parallel to the axis

Principal plane P: location of apparent ray bending

y

f '

u'P' F'

sBFL

sP'

principal

plane

focal plane

nodal planes

N N'

u

u'

Nodal points: Ray through N goes through N and preserves the direction

Cardinal elements of a lens

10

Main notations and properties of a lens:

- radii of curvature r1 , r2 curvatures c

sign: r > 0 : center of curvature

is located on the right side

- thickness d along the axis

- diameter D

- index of refraction of lens material n

Focal length (paraxial)

Optical power

Back focal length intersection length,

measured from the vertex point

2

2

1

1

11

rc

rc

'tan',

tan

'

u

yf

u

yf F

'

'

f

n

f

nF

'' ' PF sfs

Main properties of a lens

11

Different shapes of singlet lenses: 1. bi-, symmetric

2. plane convex / concave, one surface plane

3. Meniscus, both surface radii with the same sign

Convex: bending outside Concave: hollow surface

Principal planes P, P: outside for mesicus shaped lenses

P'P

bi-convex lens

P'P

plane-convex lens

P'P

positive

meniscus lens

P P'

bi-concave lens

P'P

plane-concave

lens

P P'

negative

meniscus lens

Lens shape

12

Ray path at a lens of constant focal length and different bending

The ray angle inside the lens changes

The ray incidence angles at the surfaces changes strongly

The principal planes move For invariant location of P, P the position of the lens moves

P P'

F'

X = -4 X = -2 X = +2X = 0 X = +4

Lens bending und shift of principal plane

13

Bending of a Lens

Bending: change of shape for invariant focal length

Parameter of bending

Principal planes are moving

Incidence angles and most aberrations are changing

12

21

RR

RRX

X = +1

X > +1

X = 0

X = -1

meniscus lensX < -1

biconvex lens

biconcave lens

planconvex lens

planconcave lens

planconvex lens

planconcave lens

meniscus lens

14

Changes of the incidence angles at the front and the rear surface of a bended lens

Figure without sign of incidence angle

Angle at the second surface depends on the refractive index

15

Incidence of Bended Lens

0-0.1

3

-0.08

6

-0.06

9

-0.04

12

-0.02

15

0

18

0.02

21

0.04

24

0.06

27

0.08

30

0.1

i []

i1

c1 [mm-1

]

i2 (n=1.5)

i2 (n=4)

Magnification parameter M: defines ray path through the lens

Special cases: 1. M = 0 : symmetrical 4f-imaging setup

2. M = -1: object in front focal plane

3. M = +1: object in infinity

The parameter M strongly influences the aberrations

1'

21

2

1

1

'

'

s

f

s

f

m

m

UU

UUM

Magnification Parameter

M=0

M=-1

M+1

16

Optical Image formation: All ray emerging from one object point meet in the perfect image point

Region near axis: gaussian imaging

ideal, paraxial

Image field size: Chief ray

Aperture/size of light cone:

marginal ray

defined by pupil

stop

image

object

optical

system

O2field

point

axis

pupil

stop

marginal

ray

O1 O'1

O'2

chief

ray

Optical imaging

17

Imaging with a lens

Location of the image: lens equation

Size of the image: Magnification

Imaging by a Lens

object image

-sf

+s'

system

lens

y

y'

fss

11

'

1

s

s

y

ym

''

18

Single surface imaging equation

Thin lens in air focal length

Thin lens in air with one plane surface, focal length

Thin symmetrical bi-lens

Thick lens in air focal length

'

1'

'

'

fr

nn

s

n

s

n

21

111

'

1

rrn

f

1'

n

rf

12'

n

rf

21

2

21

1111

'

1

rrn

dn

rrn

f

Formulas for surface and lens imaging

19

Imaging by a lens in air: lens makers formula

Magnification

Real imaging: s < 0 , s' > 0

Intersection lengths s, s' measured with respective to the

principal planes P, P'

fss

11

'

1

s'

2f'

4f'

2f' 4f'

s-2f'- 4f'

-2f'

- 4f'

real object

real image

real object

virtual object

virtual image

virtual object

real image

virtual image

Imaging equation

s

sm

'

20

Ranges of imaging Location of the image for a single

lens system

Change of object loaction

Image could be: 1. real / virtual

2. enlarged/reduced

3. in finite/infinite distance

Imaging by a Lens

|s| < f'

image virtual

magnified FObjekt

s

F object

s

Fobject

s

F

object

s

F'

F

object

image

s

|s| = f'

2f' > |s| > f'

|s| = 2f'

|s| > 2f'

F'

F'

F'

F'

image

image

image

image

image at

infinity

image real

magnified

image real

1 : 1

image real

reduced

21

Two lenses with distance d

Focal length distance of inner focal points e

Sequence of thin lenses close together

Sequence of surfaces with relative ray heights hj, paraxial

Magnification

n

FFdFFF 2121

e

ff

dff

fff 21

21

21

k

kFF

k k

kkk

rnn

h

hF

1'

1

kk

k

n

n

s

s

s

s

s

sm

'

''' 1

2

2

1

1

Multi-Surface Systems

22

Focal length e: tube length

Image location

Two-Lens System

lens 1

lens 2

d

e

f'1

f2s'

s

e

ff

dff

fff 21

21

21 ''

''

'''

1

1

21

212

'

')'(

''

')'('

f

fdf

dff

fdfs

23

Lateral magnification for finite imaging

Scaling of image size

'tan'

tan'

uf

uf

y

ym

z f f' z'

y

P P'

principal planes

object

imagefocal pointfocal point

s

s'

y'

F F'

Magnification

24

Afocal systems with object/image in infinity Definition with field angle w angular magnification

Relation with finite-distance magnification

''tan

'tan

hn

nh

w

w

'f

fm

Angle Magnification

h

h'w'

w

25

Axial magnification

Approximation for small z and n = n

f

zmf

fm

z

z

1

1'' 2

'tan

tan2

22

u

um

Axial Magnification

z z'

26

Imaging on axis: circular / rotational symmetry Only spherical aberration and chromatical aberrations

Finite field size, object point off-axis:

- chief ray as reference

- skew ray bundels:

coma and distortion

- Vignetting, cone of ray bundle

not circular symmetric

- to distinguish:

tangential and sagittal

plane

O

entrance

pupil

y yp

chief ray

exit

pupil

y' y'p

O'

w'

w

R'AP

u

chief ray

object

planeimage

plane

marginal/rim

ray

u'

Definition of Field of View and Aperture

27

The Special Infinity Cases

Simple case: - object, image and pupils are lying in a finite

distance

- non-telecentric relay systems

Special case 1: - object at infinity

- object sided afocal

- example: camera lens for distant objects

Special case 2: - image at infinity

- image sided afocal

- example: eyepiece

Special case 3: - entrance pupil at infinity

- object sides telecentric

- example: camera lens for metrology

Special case 4: - exit pupil at infinity

- image sided telecentric

- example: old fashion lithographic lens

28

The Special Infinity Cases

Very special: combination of above cases Examples:

- both sided telecentric: 4f-system, lithographic lens

- both sided afocal: afocal zoom

- object sided telecentric, image sided afocal:

microscopic lens

Notice: telecentricity and afocality can not be combined on the same side of a system

29

Special stop positions: 1. stop in back focal plane: object sided telecentricity

2. stop in front focal plane: image sided telecentricity

3. stop in intermediate focal plane: both-sided telecentricity

Telecentricity: 1. pupil in infinity

2. chief ray parallel to the optical axis

Telecentricity

telecentric

stopobject imageobject sides chief rays

parallel to the optical axis

30

Double telecentric system: stop in intermediate focus Realization in lithographic projection systems

Telecentricity

telecentric

stopobject imagelens f1 lens f2

f1f1

f2f2

31

Field-Aperture-Diagram

0.20 0.4 0.6 0.80

4

8

12

16

20

24

28

32

36

NA

w

40

micro

100x0.9

double

Gauss

achromat

Triplet

micro

40x0.6micro

10x0.4

Sonnar

Biogon

split

triplet

Distagon

disc

projection

Gauss

diode

collimator

projection

Petzval

micros-

copy

collimator

focussing

photographic

projection constant

etendue

lithography

Braat 1987

lithography

2003

Classification of systems with

field and aperture size

Scheme is related to size,

correction goals and etendue

of the systems

Aperture dominated:

Disk lenses, microscopy,

Collimator

Field dominated:

Projection lenses,

camera lenses,

Photographic lenses

Spectral widthz as a correction

requirement is missed in this chart

32

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

33

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 color

incidence 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

34

Linear relation of ray transport

Simple case: free space propagation

Advantages of matrix calculus: 1. simple calculation of component

combinations

2. Automatic correct signs of

properties

3. Easy to implement

General case: paraxial segment with matrix

ABCD-matrix :

u

xM

u

x

DC

BA

u

x

'

'

z

x x'

ray

x'

u'

u

x

B

Matrix Formulation of Paraxial Optics

A B

C D

z

x x'

ray x'

u'u

x

35

Matrix Calculus

Paraxial raytrace transfer

Matrix formulation

Matrix formalism for finite angles

Paraxial raytrace refraction

Inserted

Matrix formulation

111 jjjj Udyy

1 jjjj Uyi in

nij

j

j

j''

1' jj UU

1 jj yy

1

'

''

j

j

j

j

j

jjj

j Un

ny

n

nnU

'' 1 jjjj iiUU

j

jj

j

j

U

yd

U

y

10

1

'

'1

j

j

j

j

j

jjj

j

j

U

y

n

n

n

nnU

y

'

'01

'

'

j

j

j

j

u

y

DC

BA

u

y

tan'tan

'

36

Linear transfer of spation coordinate x and angle u

Matrix representation

Lateral magnification for u=0

Angle magnification of conjugated planes

Refractive power for u=0

Composition of systems

Determinant, only 3 variables

uDxCu

uBxAx

'

'

u

xM

u

x

DC

BA

u

x

'

'

xxA /'

uuD /'

xuC /'

121 ... MMMMM kk

'det

n

nCBDAM

Matrix Formulation of Paraxial Optics

37

System inversion

Transition over distance L

Thin lens with focal length f

Dielectric plane interface

Afocal telescope

AC

BDM

1

10

1 LM

11

01

f

M

'0

01

n

nM

0

1L

M

Matrix Formulation of Paraxial Optics

38

Ideal lens - one principal plane

Aplanatic lens - principal surfaces are spheres

- the marginal ray heights in the vortex plane are different for larger angles

- inconsistencies in the layout drawings

Ideal lens

P P'

P = P'

39

The notation ideal imaging is not unique

Ideal is in any case the location of the image point

The geometrical ray paths can be different for 1. paraxial

2. ideal / linear collineation

3. aplanatic

The photometric properties are different due to non-equidistant sampling

If a perfect lens is idealized in a software as one surface, there are principal

discrepances in the location of the

intersection points

40

What is Ideal ?

optical

axis

ideal

vertex

plane

ellipsoidal

mirror

object

pointimage

pointideal

system

paraxial

system

aperture

angles