Design and Correction of optical Systems

Part 5: Properties of Optical Systems

Summer term 2012Herbert Gross

Overview

1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Measurement of system quality 2012-07-0413. Correction of aberrations 1 2012-07-1114. Optical system classification 2012-07-18

2012-04-18

2

5.1 Pupil - basic notations- special rays- ray sets- pupil- vignetting

5.2 Canonical coordinates - normalized properties- pupil sphere- aplanatic imaging systems

5.3 Special imaging setups - telecentricity- anamorphotic imaging- Scheimpflug condition- measurement basic system parameters

5.4 Delano diagram - basic idea- examples

Contents Part 53

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

Finite field size, object point off-axis:

- chief ray as reference

- skew ray bundles: coma and distortion

- Vignetting, cone of ray bundlenot circular symmetric

- to distinguish:tangential and sagittalplane

Definition Field of View and Aperture4

Marginal ray :connects the center of theobject field and the rimof the pupil

Chief ray :connects the outer pointof the field of view and thecenter of the pupil

Coma rays:connects the rim of theobject field and the rim ofthe pupil

Special Rays5

Quantitative measures of relative opening / size of accepted light cone

Numerical aperture

F-number

Approximation for smallapertures:

'sin unNA

EXDfF '#

NAF

21#

Size of Aperture

imageplane

marginal ray

exitpupil

chief ray

U'W'DEX

f'

6

Meridional rays:in main cross section plane

Sagittal rays:perpendicular to main crosssection plane

Coma rays:Going through field pointand edge of pupil

Oblique rays:without symmetry

Special Rays in 3D

axis

y

x

p

p

pupil plane

object plane

x

y

axissagittal ray

meridionalmarginal ray

skew raychief ray

sagittal comaray

uppermeridionalcoma ray

lowermeridionalcoma ray

fieldpoint

axis point

7

Off-axis object point:1. Meridional plane / tangential plane / main cross section plane

contains object point and optical axis2. Sagittal plane:

perpendicular to meridional plane through object point

Tangential- and Sagittal Plane

x

y

x'

y'

z

lens

meridionalplane

sagittalplane

objectplane image

plane

8

Transverse aberrations:Ray deviation form ideal image point in meridional and sagittal plane respectively

The sampling of the pupil is only filled in two perpendicular directions along the axes

No information on the performance of rays in the quadrants of the pupil

Ray-Fan Selection for Transverse Aberration Plots

x

y

pupil

tangentialray fan

sagittalray fan

objectpoint

9

Ray fan:2-dimensional plane set of rays

Ray cone:3-dimensional filled ray cone

Ray Fan and Ray Cone

objectpoint

pupilgrid

10

Pupil sampling in 3D for spot diagram:all rays from one object point through all pupil points in 2D

Light cone completly filled with rays

Pupil Sampling 11

Pupil sampling for calculation of tranverse aberrations:all rays from one object point to all pupil points on x- and y-axis

Two planes with 1-dimensional ray fans

No complete information: no skew rays

Pupil Sampling 12

Pupil Sampling13

Artefacts due to regular gridding of the pupil of the spot in the image plane

In reality a smooth density of the spot is true

The line structures are discretization effects of the sampling

Pupil Sampling Spot Artefacts14

Pupil stop defines:1. chief ray angle w2. aperture cone angle u

The chief ray gives the center line of the oblique ray cone of an off-axis object point The coma rays limit the off-axis ray cone The marginal rays limit the axial ray cone

Diaphragm in Optical Systems

y

y'

stoppupil

coma ray

chief ray

marginal ray

apertureangle

u

field anglew

object

image

15

The physical stop definesthe aperture cone angle u

The real system may be complex

The entrance pupil fixes the acceptance cone in the object space

The exit pupil fixes the acceptance cone in the image space

Diaphragm in Optical Systems

Ref: Julie Bentley

16

Relevance of the system pupil :

Brightness of the imageTransfer of energy

Resolution of detailsInformation transfer

Image qualityAberrations due to aperture

Image perspectivePerception of depth

Compound systems:matching of pupils is necessary, location and size

Properties of the Pupil17

Entrance and Exit Pupil

exitpupil

upper marginal ray

chiefray

lower coma ray

stop

field point of image

UU'

W

lower marginal ray

upper coma ray

on axis point of image

outer field point of object

object point

on axis

entrancepupil

18

Optical Image formation:

Sequence of pupil and image planes Matching of location and size of image planes necessary (trivial) Matching of location and size of pupils necessary for invariance of energy density In microscopy known as Köhler illumination

Nested Ray Path19

Pupil Mismatch

Telescopic observation with different f-numbers Bad match of pupil location: key hole effect

20

Artificial vignetting:Truncation of the free areaof the aperture light cone

Natural Vignetting:Decrease of brightness according to cos w 4 dueto oblique projection of areas and changed photometricdistances

Vignetting

w

AExp

imaging without vignetting

complete field of view

imaging with vignetting

imaging with vignetting

fieldangle

D

0.8 Daxis

field

truncation

truncation

stop

21

3D-effects due to vignetting

Truncation of the at different surfaces for the upper and the lower partof the cone

Vignetting

object lens 1 lens 2 imageaperturestop

lower truncation

upper truncation

sagittal trauncation

chief ray

coma rays

22

Truncation of the light cone with asymmetric ray path for off-axis field points

Intensity decrease towardsthe edge of the image

Definition of the chief ray:ray through energetic centroid

Vignetting can be used to avoiduncorrectable coma aberrationsin the outer field

Effective free area with extremaspect ratio:anamorphic resolution

Vignetting

projection of the rim of the 2nd lens

projection of therim of the 1st lens

Projektion der Aperturblende

free area of the aperture

sagittal coma rays

meridional coma rayschief

ray

23

Ideal chief ray concept:- breaks down in case of complicated beam truncation- there is no longer a clear and simple unique pupil plane

Possible criteria:1. midpoint of rays in meridional section2. location of largest sagittal section3. centroid of area of the free pupil

Physical relevant is the energycentroidThis is a general definition andstill works in 3D

Centroid Ray

meridionalplane

center of gravity

sagittal aperture

limitingapertures

tangentialaperture

24

Vignetting

Photographic lens 100mm f/2

1. with strong vignettingrays aimed to boundaries

2. Without vignettingno really transmitted rays shown

Ref: V. Paruchuru

25

Vignetting

Illumination fall off in the image due to vignetting at the field boundary

26

Generalization of paraxial picture:Principal surface works as effective location of ray bending

Paraxial approximation: plane

Real systems with corrected sine-condition (aplanatic):principal sphere

Pupil Sphere27

Pupil sphere:equidistant sine-sampling

Pupil Sphere28

Aplanatic system: Sine condition fulfilled Pupil has spherical shape Normalized canonical coordinates

for pupil and field

Canonical Coordinates

xunx

sin ''sin'' xunx

EP

pp h

xx

EP

pp h

xx

''

'

29

Definition of canonical coordinates of an oblique ray bundle

Canonical Coordinates

O

EN

hTCL

Q

y yp

chief ray

lowertangentialcoma ray

uppertangentialcoma ray

Q'

EX

y'p y'

O'

REN

hTCU

UTCU

h'TCU

h'TCL

UCR

UTCL

U'CRU'TCL

uppertangentialcoma ray

lowertangentialcoma ray

chief ray

REX

30

Normalized pupil coordinates

Aplanatic imaging

Normalized field coordinates

Paraxial imaging

Reference on chied ray

Reduced image side coordinates

EP

pp h

xx

EP

pp h

xx

''

'

xnx

sin ''sin'' xnx

pp xx '

xx '

HSTKOnunNA sinsinsin

xn

x sag sin

yny HS

tansinsin

Canonical Coordinates31

Special stop positions:1. stop in back focal plane: object sided telecentricity2. stop in front focal plane: image sided telecentricity3. stop in intermediate focal plane: both-sided telecentricity

Telecentricity:1. pupil in infinity2. chief ray parallel to the optical axis

Telecentricity

telecentricstop

object imageobject sides chief raysparallel to the optical axis

32

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

Telecentricity33

Anamorphotic imaging:different magnifications in x- and y-cross section,tangential and sagittal magnification

Identical image location in both sections

Anamorphotic factort

sanamophF

ktk

tt un

un

,

1,1

ksk

ss un

un

,

1,1

Anamorphotic Imaging Setup

cylindricallens 1

us

ut

cylindricallens 2

34

Realization of an anamorphotic imaging with cylindrical lenses

Anamorphotic Imaging Setup

fx

fy

x'

xy

y'

35

Object surface is spherical bended with radius R:image is bended by R‘

Paraxial approximation:depth transfer magnification gives

Notice: R and R‘ are bended with the same orientation,This behavior is opposite to the curved image in the Petzval picture

Curved Object Surface

Ryz2

2

2' mzz

mRR '

y

objectsurface

imagesurface

R

z

y'

R'

z'

36

Imaging with tilted object plane

If principal plane, object and image plane meet in a common point:Scheimpflug condition, sharp imaging possible

Scheimpflug equation

tan'tantantan

'

ss

Scheimpflug Imaging37

Derivation of Scheimpflug imagingcondition with depth magnification

Scheimpflug Imaging

yo

z

z'

P P'

F'

s's

'

d

y'o y'

y

2

22 ''

yym

zz

o

38

General property:- Magnification depends on location in the object plane- anamorphotic magnification- corresponds to macroscopic keystone

distortion

Imaging Relation

'

s s'

d

h

h'

Objekt Bild

ss''

tan'tan

sin'1''

yss

x

'sinsin

sin'1''

2

yss

y

'cossin'

fV

Scheimpflug Imaging39

Scheimpflug Imaging

Example for oblique imaging with Scheimpflug condition Sharp image Strong distortion

40

Keystone distortion

Scheimpflug Imaging

ideal

real

41

Collimated incident light Calibrated collimator with focal length fc and test chart with size y Selection of sharp image plane Analysis of image size

Measurement of Focal Length with Collimator

f'c f'

test chart image

y y'

collimator test optic

yyff c'''

42

Setup with distance object-image L > 4f Known location of the principal plane P of the system

distance dP between principal planes Selection of two system locations with sharp image Relative axial shift D between the two setups

Measurement of Focal Length According to Gauss

H

H

dLDdLf

44

2

F F'

L

F F'

dP

Ds

s'

P P'

P P'

position 1

position 2

f f'

43

Telecentric movable measurement microscope with offset y: Abbe focometer Focusing of two different test charts with sizes y1 and y2 Determination of the focal length by

Measurement of Focal Length with Focometer

eyy

fyu 12tan

44

Setup with fiber and plane mirror for autocollimation Change of distance between test lens and fiber Analysis of the recoupled power into the fiber (confocal) gives the focal point

Measurement of Focal Length by Confocal Setup45

Special representation of ray bundles in optical systems:marginal ray height vs. chief ray height

Delano digram gives useful insight intosystem layout

Every z-position in the system corresponds to a point on the line of the diagram

Interpretation needs experience

Delano Diagram

CRyy

lens

y

field lens collimatormarginal ray

chief ray

y

y

y

lens atpupil

position

field lensin the focal

plane

collimatorlens

MRyy

46

Pupil locations:intersection points with y-axis

Field planes/object/image:intersectioin points with y-bar axis

• Pupillenlagen

Construction of focal points by parallel lines to initial and final linethrough origin

Delano Diagram

y

y

objectplane

lens

imageplane

stop andentrance pupil

exit pupil

y

y

objectspace

imagespace

front focalpoint F rear focal

point F'

47

Influence of lenses:diagram line bended

Location of principal planes

Delano Diagram

y

y

strong positiverefractive power

weak positiverefractive power

weak negativerefractive power

y

y

object space image space

principalplane

yP

yP

48

Conjugated point are located on a straight line through the origin

Distance of a system point fromorigin gives the systems half

diameter

Delano Diagram

y

y

objectspace image

space

conjugate line

conjugate linewith m = 1

principal point

conjugate points

y

y

maximum height ofthe coma ray at

lens 2

curve ofthe system

lens 1lens 2

lens 3

D/2

49

Afocal Kepler-type telecope

Effect of a field lens

Delano Diagram Examples

y

y

lens 1objective

lens 2eyepiece

intermediatefocal point

y

y

lens 1objective

lens 2eyepiece

field lensintermediatefocal point

50

Microscopic system

Delano Diagram Example

y

y

eyepiece

microscopeobjective tube lens

object

image at infinity

aperture stop

intermediateimage

exit pupil

telecentric

51

Summary of Important Topics

Chief ray and marginal ray define the field size and the aperture size of a system Special ray sets are defined to exhaust the light cone The aperture gives the light cone angle The pupil gives the light flux through a system In a real system the physical stop, the object space and the image space pupil must be

distinguished Vignetting is a partial blocking of outer rays for off-axis field points and obliques cones Apalantic corrected systems have a spherical shaped pupil Canonical coordinate help to describe systems in a normalized way Telecentric system have a stop in a focal plane and a axis-parallel chief ray The Scheimpflug-condition allows a sharp imaging for tilted object planes Scheimpflug setups suffer from large distortion The measurement of basic system parameters is mainly based on the imaging equation The Delano diagram is defined by a chief-ray vs marginal ray height The Delano diagram helps in understanding complicated combined systems

52

Outlook

Next lecture: Part 6 – PhotometryDate: Wednesday, 2012-05-23Contents: 6.1 Basic notations

6.2 Lambertian radiator6.3 Flux calculation6.4 Light transport in optical systems6.5 Further modeling

53