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Advanced Lens Design

Lecture 8: Field flattening

2013-12-03

Herbert Gross

Winter term 2013

1. Refraction

2. Fresnel formulas

3. Optical systems

4. Raytrace

5. Calculation approaches

2

Contents

ideal

image

plane

tangential

shell

sagittal

shell

image surfacesy'

Field Curvature and Image Shells

Imaging with astigmatism:

Tangential and sagittal image shell sharp depending on the azimuth

Difference between the image shells: astigmatism

Astigmatism corrected:

It remains one curved image shell,

Bended field: also called Petzval curvature

System with astigmatism:

Petzval sphere is not an optimal

surface with good imaging resolution

No effect of lens bending on curvature,

important: distribution of lens

powers and indices

3

Petzval Theorem for Field Curvature

Petzval theorem for field curvature:

1. formulation for surfaces

2. formulation for thin lenses (in air)

Important: no dependence on

bending

Natural behavior: image curved

towards system

Problem: collecting systems

with f > 0:

If only positive lenses:

Rptz always negative

k kkk

kkm

ptz rnn

nnn

R '

''

1

j jjptz fnR

11

optical system

object

plane

ideal

image

plane

real

image

shell

R

4

Correction of Astigmatism and Field Curvature

Different possibilities for the correction of astigmatism and field curvature

Two independend aberrations allow 4 scenarious

ST

-2.5 0 2.5

ST S T S T

-2.5 0 2.5 -2.5 0 2.5 -2.5 0 2.5

a) bended

image plane

residual

astigmatism

b) bended

image plane

corrected

astigmatism

c) flattened

image plane

residual

astigmatism

d) flattened

image plane

corrected

astigmatism

DzDz DzDz

y yy

Petzval Shell

The Petzval shell is not a desirable image surface

It lies outside the S- and T-shell:

The Petzval curvature is a result of the Seidel aberration theory

6

sagtan

petsag

ast

ss

ss

s

DD

DD

D 0(a)

pettan

petsag

ast

ss

ss

s

DD

DD

D 0(b)

T PSTP S

sagtan

petsag

petast

ss

ss

ss

DD

DD

DD

)21(

TP S

(d)

TP S

0

)32(

)32(

D

DD

DD

tan

petsag

petast

s

ss

ss(c)

pettan

sag

petast

ss

s

ss

DD

D

DD

2

0

2

TP S

(e)

2

''3'

tansss

sag

pet

DDD

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

7

Field Curvature of a Mirror

Mirror: opposite sign of curvature than lens

Correction principle: field flattening by mirror

Gaussian

image

planeGaussian

image

plane

lens

mirror

Petzval

surface Petzval

surface

f' > 0 / R > 0f' > 0 / R < 0

New Achromate

An achromate is typically corrected for axial chromatical aberration

The achromatization condition for two thin

lenses close together reads

The Petzval sum usually is negative

and the field is curved

A flat field is obtained, if the following condition is fulfilled

This gives the special condition of simultaneous correction of achromatization

and flatness of field

perfect

image

plane

Petzval

shell

y'

f

RP

mean

image shell02

2

1

1

FF

j jjP fnR

11

02

2

1

1 n

F

n

F

2

1

2

1

n

n

New Achromate

This condition correponds to the

requirement to find two glasses on one

straight line in the glass map

The solution is well known as simple

photographic lens

(landscape lens) K5 F2stop

Flattening Meniscus Lenses

Possible lenses / lens groups for correcting field curvature

Interesting candidates: thick mensiscus shaped lenses

1. Hoeghs mensicus: identical radii

- Petzval sum zero

- remaining positive refractive power

2. Concentric meniscus,

- Petzval sum negative

- weak negative focal length

- refractive power for thickness d:

3. Thick meniscus without refractive power

Relation between radii

Fn d

nr r d'

( )

( )

1

1 1

r r dn

n2 1

1

21

211

'

'1

rr

d

n

n

fnrnn

nn

R k kkk

kk

ptz

r2

d

r1

2

2)1('

rn

dnF

drr 12

drrn

dn

Rptz

11

)1(1

0

)1(

)1(1

11

2

ndnrrn

dn

Rptz

11

Group of meniscus lenses

Effect of distance and

refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n n

d

3020 5010

-3

10-1

10-2

1/Rpet [1/mm]

40r

1

[mm]10

K5 / d=15 mm

SF66 / d=15 mm

K5 / d=25 mm

From : H. Zügge

12

Triplet group with + - +

Effect of distance and

refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n1

n2

r 3

d/2

7050 10010

-3

10-1

10-2

1/Rpet [1/mm]

r1

[mm]

BK7

SF66 / FK3 / SF66

From : H. Zügge

13

imageshell

flatimage

fieldlens

Effect of a field lens for flattening the image surface

1. Without field lens 2. With field lens

curved image surface image plane

Flattening Field Lens

14

Microscope Objective Lens

Possible setups for flattening the field

Goal:

- reduction of Petzval sum

- keeping astigmatism corrected

Three different classes:

1. No effort

2. Semi-flat

3. Completely flat

d)

achromatized

meniscus lens

a)

single

meniscus

lense

e)

two

meniscus

lenses

achromatized

b)

two

meniscus

lenses

c)

symmetrical

triplet

f)

modIfied

achromatized

triplet solution

DS

rel.

field0 0.5 0.707 1

1

0.8

0.6

0.4

0.2

0

diffraction

limit

plane

semi

plane

not

plane

Field Curvature

Correction of Petzval field curvature in lithographic lens

for flat wafer

Positive lenses: Green hj large

Negative lenses : Blue hj small

Correction principle: certain number of bulges

j j

j

n

F

R

1

j

j

jF

h

hF

1

16