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Medical Photonics Lecture

Optical Engineering

Lecture 5: Optical systems

2019-05-15

Herbert Gross

Speaker: Yi Zhong

Summer term 2019

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Zhong 10.04.Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Zhong 17.04.Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Zhong 24.04.Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 08.05. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Zhong 15.05.Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous

7 Image quality Zhong 29.05. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 05.06.Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 12.06.Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 19.06.Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Photometry Zhong 26.06.Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

12 Illumination systems Gross 03.07.Light sources, basic systems, quality criteria, nonsequential raytrace

13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements

1. Photo objective lens

2. Microscope objective lens

3. Binocular

4. Infrared afocal system

Typical Example Systems 1

5. Relay optics

6. Scan-objective lens

7. Collimator objective lens

Typical Example Systems 2

possible surfaces

under test

8. Projector lens

9. Telescope

10. Lithography projection

lens

Typical Example Systems 3

M1

M2

M3

11. Illumination collector system

12. Illumination condenser system

13. Head mounted display

Typical Example Systems 4

eye

pupil

image

total

internal

reflection

free formed

surface

free formed

surface

field angle 14°

14. Stereo microscope

15. Zoom system

Typical Example Systems 5

eyepiece

tube

system zoom

system object

plane

eye

common

axis

stereo

angle

common

objective

lens

f = 61

f = 113

f = 166

Lithographic Optics

H-Design

I-Design

X-Design

9

Model depth of Light Propagation

Different levels of modelling in optical propagation

Schematical illustration (not to scale)

Ref: A. Herkommer

accuracy

calculation

effort

paraxial optic

geometrical

optic

(raytrace)

scalar waveoptic

(high NA)

paraxial

waveoptic

vectorial waveoptic

rigorous waveoptic

Modelling of Optical Systems

Principal purpose of calculations:

1. Solving the direct problem of

understanding the properties:

analysis

2. Solving the inverse problem:

Finding the concret system data

for a required functionality:

synthesis

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

Function, data of properties,

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

Analysisimaging

aberration

theory

Synthesislens design

inverse

problem

Ref: W. Richter

10

Approximation of Optical Models

Imaging model with levels

of refinement

Paraxial model

(focal length, magnification, aperture,..)

approximation

l à 0

no description of

short pulses

Geometrical

optics

Analytical approximation

(3rd order aberrations,..)

exact geometry

Wave optics

no time dependence

Maxwell equations

Scalar approximation

Helmholtz equation

(PSF, OTF,...)

linear

approximation

no description of

small structures

and polarization

effects

no diffraction

no higher order

aberrations

no aberrations

exact

11

Five levels of modelling:

1. Geometrical raytrace with analysis

2. Equivalent geometrical quantities,

classification

3. Physical model:

complex pupil function

4. Primary physical quantities

5. Secondary physical quantities

Blue arrows: conversion of quantities

Modelling of Optical Systems

ray

tracing

optical path

length

wave

aberration W

transverse

aberrationlongitudinal aberrations

Zernike

coefficients

pupil

function

point spread

function (PSF)

Strehlnumber

optical

transfer function

geometricalspot diagramm

rms

value

intersectionpoints

final analysis reference ray in

the image space

referencesphere

orthogonalexpansion

analysis

sum of

coefficientsMarechal

approxima-

tion

exponentialfunction

of the

phase

Fourier

transformLuneburg integral

( far field )

Kirchhoffintegral

maximum

of the squared

amplitude

Fouriertransformsquared amplitude

sum of

squaresMarechalapproxima-

tion

integration ofspatial

frequencies

Rayleigh unit

equivalencetypes of

aberrationsdifferen

tiationinte-

gration

full

aperture

single types of aberrations

definition

geometricaloptical

transfer function

Fouriertransform

approximation

auto-correlationDuffieux

integral

resolution

threshold value spatial frequency

threshold value spatial

frequency approximationspot diameter

approximation diameter of the

spot

Marechalapproximation

final analysis reference ray in the image planeGeometrical

raytrace

with Snells law

Geometrical

equivalents

classification

Physical

model

Primary

physical

quantities

Secondary

physical

quantities

13

Aberrations

Non-perfect imaging through a real optical system

Only in paraxial approximation of small angles the systems are perfect

A perturbation theory for larger angles of

1. the chief ray for the field size description

2. the marginal ray for the aperture size description

explains the aberrations as non-linear effects

The monochromatic primary aberrations are of 3rd order in transverse representation,

they are:

1. spherical aberration: only on axis, circular symmetric

2. coma, grows linear with the field of view, asymmetric

3. astigmatism

4. field curvature, the image is sharp on a curved surface

5. distortion, geometrical deformation of the image, full resolution

Furthermore there are two types of primary chromatical aberrations as a result of the

material dispersion with wavelength:

1. axial color aberration, dispersion of the marginal ray

2. lateral color, dispersion of the chief ray

14

Spherical aberration

Ray intersection length with the optical axis depends on the ray height

plane of

best focus

zone

paraxial

rim

15

Astigmatism and Field Curvature

Astigmatism:

different focal lengths in sag and tan cross

section

Field curvature:

image is sharp on a curved surface

ideal

image

plane

tangential

shell

sagittal

shell

image surfacesy'

focused in center

(paraxial image plane)focused in field zone

(mean image plane)

focused at field boundary

z

y'

receiving

planes

image

sphere

16

Coma

Asymmetrical shape of the

spot

Centroid no longer peak

intensity

saggital

marginal rays

central

ray

angle 60°

coma

blur

lens / pupil

axis

tangential

marginal

rays

circle

radius ~ r² Tangential

coma CT

Saggital

coma CS

CS ~ CT / 3

c7 = 0.3 c7 = 0.5 c7 = 1

centroid

17

Distortion

Deformation of the geometry

No blurrly

Described by chief ray error on

pincussion

distortion

barrel

distortion

object

D < 0

D > 0

lens

rear

stop

imagex

x

y

y

y'

x'

y'

x'

front

stop

18

Chromatical aberrations

Reason of chromatical aberrations is the dispersion of the materials

Two primary aberrations:

1. marginal ray aberration:

axial color, images of different colors at different z-positions

2. chief ray aberration:

transverse color, images of different colors have different size

object

chief ray

marginal

ray

chief rays

marginal rays

l2

l1

ExP

l2

ExP

l1

transverse

chromatical

aberration

axial

chromatical

aberration

ideal

image

l1

ideal

image

l2

19

Chromatical aberrations

Axial color aberration

Transverse color aberration z

l = 648 nm

defocus

-2 -1 0 1 2

l = 546 nm

l = 480 nm

best image planel

z

y

stop

dispersion

prism effect

image

plane

DyCHV

chief

ray

Achromate

Residual aberrations of an achromate

Clearly seen:

1. Distortion

2. Chromatical magnification

3. Astigmatism

20

System performance:

• Aberrations

• Spot diameter

• Wavefront

• Zernike coefficients

• Contrast (MTF)

• Point spread function

Depends on:

• aperture

• field position

• wavelength

• object distance

10:15:47

ORA 24-Apr-07

40X POWER, 0.70 NA

PLAN-ACHROMAT

RAY ABERRATIONS ( MILLIMETERS )

656.3000 NM

587.6000 NM

546.1000 NM

486.1000 NM

435.8000 NM

-0.005

0.005

-0.005

0.005

0.00 RELATIVE

FIELD HEIGHT

( 0.000 )O

-0.005

0.005

-0.005

0.005

0.75 RELATIVE

FIELD HEIGHT

( -3.64 )O

-0.005

0.005

-0.005

0.005

TANGENTIAL 1.00 RELATIVE SAGITTAL

FIELD HEIGHT

( -4.85 )O

11:16:28

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

MODULATION

300 600 900 1200 1500 1800 2100 2400

SPATIAL FREQUENCY (CYCLES/MM)

40X POWER, 0.70 NA

PLAN-ACHROMAT

DIFFRACTION MTF

ORA 29-Apr-07

DIFFRACTION LIMIT

AXIS

T

R0.5 FIELD ( )-2.43 O

T

R0.8 FIELD ( )-3.64 O

T

R1.0 FIELD ( )-4.85 O

WAVELENGTH WEIGHT

656.3 NM 14

587.6 NM 71

546.1 NM 90

486.1 NM 26

435.8 NM 2

DEFOCUSING 0.00000

Ref : C. Menke

Performance Criteria

Raytrace Through a Lens

R1R2

plane

t

n

D

Tracing of rays throug a lens

Application of the law of refraction at every interface

surface between different indices of refraction

Simple geometrical parameter:

- radii of curvature: R1, R2

- thickness t

- diameter D

- refractive index n

Generalization of paraxial picture:

Principal surface works as effective location of ray bending

for object points near the optical axis (isoplanatic patch)

Paraxial approximation: plane

Can be used for all rays to find

the imaged ray

Real systems with corrected

sine-condition (aplanatic):

principal sphere

The principal sphere can not be

used to construct arbitrary ray

paths

If the sine correction is not

fulfilled: more complicated

shape of the arteficial surface,

that represents the ray bending

Principal Sphere

effective surface of

ray bending P'

y

f'

U'

P

23

System of two separated thin lenses

Variation of the back principal plane

as a function of the distribution of

refractive power

24

Principal Planes

P' L1 L2P F

plate

plate

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 bundles:

coma and distortion

- Vignetting, cone of ray bundle

not circular symmetric

- to distinguish:

tangential and sagittal

planeO

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 Field of View and Aperture

25

Classical measure for the opening:

numerical aperture

In particular for camera lenses with

object at infinity:

F-number

Numerical aperture and F-number are to system properties, they are related to a conjugate

object/image location

Paraxial relation

Special case for small angles or sine-condition corrected systems

26

Numerical Aperture and F-number

'sin' unNA DEnP/2

f

image

plane

object in

infinity

u'

EnPD

fF #

'tan'2

1#

unF

'2

1#

NAF

Meridional rays:

in main cross section plane

Sagittal rays:

perpendicular to main cross

section plane

Coma rays:

Going through field point

and edge of pupil

Oblique rays:

without symmetry

Special rays in 3D

axis

y

x

p

p

pupil plane

object plane

x

y

axissagittal ray

meridional marginal ray

skew raychief ray

sagittal coma ray

upper meridional coma ray

lower meridional coma ray

field point

axis point

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

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

tangential

sagittal

28

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

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

29

The physical stop defines

the 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

uobject

image

stop

EnP

ExP

object

image

black box

details complicated

real

system

? ?

Ref: Julie Bentley

30

Entrance and Exit Pupil

exit

pupil

upper

marginal ray

chief

ray

lower coma

raystop

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

entrance

pupil

31

Relevance of the system pupil :

Brightness of the image

Transfer of energy

Resolution of details

Information transfer

Image quality

Aberrations due to aperture

Image perspective

Perception of depth

Compound systems:

matching of pupils is necessary, location and size

Properties of the Pupil

32

Pupil Aberration

Interlinked imaging of field and pupil

Distortion of object imaging corresponds to spherical aberration of the pupil

imaging

Corrected spherical pupil aberration:

tangent condition must be fulfilled

O O’

stop and

entrance pupil

optical system

exit pupil

objectimage

Object imaging Pupil imaging

Blue rays

Red rays

Marginal rays

Marginal raysChief rays

Chief rays

.tan

'tanconst

w

w

33

Sine condition not fulfilled:

- nonlinear scaling from entrance to exit pupil

- spatial filtering on warped grid, nonlinear sampling of spatial frequencies

- pupil size changes

- apodization due to distortion

- wave aberration could be calculated wrong

- quantitative mesaure of offence against the sine condition (OSC):

distortion of exit pupil grid

Sine Condition

xo xp

sphere distorted exit

pupil surface

object

plane

exit

pupil

optical

system

sx

u

xp

x'o

u'

x'p

x'p

image

plane

entrance

pupil

grid

distortion

1sin

unf

xD

ap

p

34

Spherical aberration of the chief ray / pupil imaging

Exit pupil location depends on the field height

Pupil Aberrations

yobject

sP

chief rays

pupil position

pupil

location

35

Pupil Mismatch

Telescopic observation with different f-numbers

Bad match of pupil location: key hole effect

F# = 2.8 F# = 8 F# = 22

a) pupil

adapted

b) pupil

location

mismatch

Ref: H. Schlemmer

36

Eyepiece with pupil aberration

Illumination for decentered pupil :

dark zones due to vignetting

Kidney beam effect

Pupil Aberration

eyepiecelens and

pupil of

the eye

retina

caustic of the pupil

image enlarged

instrument

pupil

37

Artificial vignetting:

Truncation of the free area

of the aperture light cone

Natural Vignetting:

Decrease of brightness

according to cos w 4 due

to oblique projection of areas

and changed photometric

distances

Vignetting

w

AExp

imaging without vignetting

complete field of view

imaging with

vignetting

imaging with

vignetting

field

angle

D

0.8 Daxis

field

truncation

truncation

stop

38

Truncation of the light cone

with asymmetric ray path

for off-axis field points

Intensity decrease towards

the edge of the image

Definition of the chief ray:

ray through energetic centroid

Vignetting can be used to avoid

uncorrectable coma aberrations

in the outer field

Effective free area with extrem

aspect ratio:

anamorphic resolution

Vignetting

projection of the

rim of the 2nd lens

projection of the

rim of the 1st lens

Projektion der

Aperturblende

free area of the

aperture

sagittal

coma rays

meridional

coma rayschief

ray

39

Vignetting

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

40

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

41

Double telecentric system: stop in intermediate focus

Realization in lithographic projection systems

Telecentricity

telecentric

stopobject imagelens f1 lens f2

f1

f1

f2

f2

42

43

Infinity cases

sample layoutexit pupilentrance

pupilimageobjectcase

finitefinitefinitefinite1

infinity

image

telecentric

finitefinitefinite2

infinity

image

telecentric

infinity

object

telecentric

finitefinite3

finitefiniteinfinityinfinity4

finiteinfinityfinitefinite5

finitefinitefiniteinfinity6

finitefiniteinfinityfinite7

finite

infinity

object

telecentric

infinityfinite8

infinity

image

telecentric

finitefiniteinfinity9

example

relay

metrology lens

lithographic

projection lens

4f-system

afocal zoom

telescopes

beam expander

metrology lens

camera lens

focussing lens

eyepiece

collimator

microscopic lens

infinity metrology

lens

finiteinfinityfiniteinfinity10

infinityfiniteinfinityfinite11

impossible

impossible

finiteinfinityinfinityinfinity12

infinityinfinityfiniteinfinity13

impossible

impossible

infinityfiniteinfinityinfinity14

infinityinfinityinfinityfinite15

impossible

impossible

infinityinfinityinfinityinfinity16 impossible

Systematic of all

infinity cases

Physically impossible:

1. object and entrance

pupil in infinity

2. image and exit

pupil in infinity