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Microscopy

Lecture 3: Physical optics of widefield microscopes

2012-10-29

Herbert Gross

Winter term 2012

2 3 Physical optics of widefield microscopes

Preliminary time schedule

No Date Main subject Detailed topics Lecturer

1 15.10. Optical system of a microscope I overview, general setup, binoculars, objective lenses, performance and types of lenses, tube optics Gross

2 22.10. Optical system of a microscope II Etendue, pupil, telecentricity, confocal systems, illumination setups, Köhler principle, fluorescence systems and TIRF, adjustment of objective lenses Gross

3 29.10. Physical optics of widefield microscopes

Point spread function, high-NA-effects, apodization, defocussing, index mismatch, coherence, partial coherent imaging Gross

4 05.11. Performance assessment

Wave aberrations and Zernikes, Strehl ratio, point resolution, sine condition, optical transfer function, conoscopic observation, isoplantism, straylight and ghost images, thermal degradation, measuring of system quality

Gross

5 12.11. Fourier optical description basic concepts, 2-point-resolution (Rayleigh, Sparrow), Frequency-based resolution (Abbe), CTF and Born Approximation Heintzmann

6 19.11. Methods, DIC Rytov approximation, a comment on holography, Ptychography, DIC Heintzmann

7 26.11. Imaging of scatter

Multibeam illumination, Cofocal coherent, Incoherent processes (Fluorescence, Raman), OTF for incoherent light, Missing cone problem, imaging of a fluorescent plane, incoherent confocal OTF/PSF

Heintzmann

8 03.12. Incoherent emission to improve resolution Fluorescence, Structured illumination, Image based identification of experimental parameters, image reconstruction Heintzmann

9 10.12. The quantum world in microscopy Photons, Poisson distribution, squeezed light, antibunching, Ghost imaging Wicker

10 17.12. Deconvolution Building a forward model and inverting it based on statistics Wicker

11 07.01. Nonlinear sample response STED, NLSIM, Rabi the information view Wicker

12 14.01. Nonlinear microscopy two-photon cross sections, pulsed excitation, propagation of ultrashort pulses, (image formation in 3D), nonlinear scattering, SHG/THG - symmetry properties Heisterkamp

13 21.01. Raman-CARS microscopy principle, origin of CARS signale, four wave mixing, phase matching conditions, epi/forward CARS, SRS. Heisterkamp

14 28.01 Tissue optics and imaging Tissue optics, scattering&aberrations, optical clearing,Optical tomography, light-sheet/ultramicroscopy Heisterkamp

15 04.02. Optical coherence tomography principle, interferometry, time-domain, frequency domain. Heisterkamp

1. Point spread function

2. High-NA effects

3. Apodization

4. Defocussing

5. Index mismatch

6. Coherence

7. Partial coherent imaging

3 3 Physical optics of widefield microscopes

Contents of 3rd Lecture

3 Physical optics of widefield microscopes

Diffraction at the System Aperture

Self luminous points: emission of spherical waves

Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave

results in a finite angle light cone

In the image space: uncomplete constructive interference of partial waves, the image point

is spreaded

The optical systems works as a low pass filter

object

point

spherical

wave

truncated

spherical

wave

image

plane

x = 1.22 / NA

point

spread

function

object plane

0

2

12,0 I

v

vJvI

0

2

4/

4/sin0, I

u

uuI

-25 -20 -15 -10 -5 0 5 10 15 20 250,0

0,2

0,4

0,6

0,8

1,0

vertical

lateral

inte

nsity

u / v

Circular homogeneous illuminated

Aperture: intensity distribution

transversal: Airy

scale:

axial: sinc

scale

Resolution transversal better

than axial: x < z

Ref: M. Kempe

Scaled coordinates according to Wolf :

axial : u = 2 p z n / NA2

transversal : v = 2 p x / NA


Perfect Point Spread Function

NADAiry

22.1

2NA

nRE


Abbe Resolution and Assumptions

Assumption Resolution enhancement

1 Circular pupil ring pupil, dipol, quadrupole

2 Perfect correction complex pupil masks

3 homogeneous illumination dipol, quadrupole

4 Illumination incoherent partial coherent illumination

5 no polarization special radiale polarization

6 Scalar approximation

7 stationary in time scanning, moving gratings

8 quasi monochromatic

9 circular symmetry oblique illumination

10 far field conditions near field conditions

11 linear emission/excitation non linear methods

Abbe resolution with scaling of /NA:

assumptions for this estimation and possible changes

A resolution beyond the Abbe limit is only possible with violating of certain

assumptions

I(r)

DAiry / 2

r0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10 12 14 16 18 20

Airy function :

Perfect point spread function for

several assumptions

Distribution of intensity:

Normalized transverse coordinate

Airy diameter: distance between the

two zero points,

diameter of first dark ring 'sin'

21976.1

unDAiry

2

1

2

22

)(

NAr

NAr

J

rI

p

p

'sin'sin2

pak

R

akrukr

R

arx


Perfect Lateral Point Spread Function: Airy

Axial distribution of intensity

Corresponds to defocus

Normalized axial coordinate

Scale for depth of focus:

Rayleigh length

Zero crossing points:

equidistant and symmetric,

distance of zeros around image

plane 4RE

22

04/

4/sinsin)(

u

uI

z

zIzI o

42

2 uz

NAz

p

22sin NA

n

unRE


Perfect Axial Point Spread Function

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(z)

z/p

RE

4RE

z = 2RE


Defocussed Perfect Point Spread Function

Perfect point spread function with defocus

Representation with constant energy: extreme large dynamic changes

z = -2RE z = +2REz = -1RE z = +1RE

normalized

intensity

constant

energy

focus

Imax = 5.1% Imax = 42%Imax = 9.8%

Spherical aberration Astigmatism Coma

c = 0.2

c = 0.3

c = 0.7

c = 0.5

c = 1.0


Point Spread Function with Aberrations

Zernike coefficients c in

Spherical aberration,

circular symmetry

Astigmatism,

split of two azimuths

Coma,

asymmetric

Intensity distribution I(r,z) for spherical

aberration

Asymmetry of intensity around the image plane

Usually no zero points on axis intra focal


Point Spread Function with Spherical Aberration


Point Spread Function with Apodization

Apodization:

Non-uniform illumination of the pupil amplitude

Modified point spread function:

Weighting of the elementary wavelets by amplitude distribution A(xp,yp),

Redistribution of interference

System with wave aberrations:

Complicated weighting of field superposition

For edge decrease of apodization: residual aberrations at the edge have decreased

weighting and have reduced perturbation

Definition of numerical aperture angle no longer exact possible

For continuous decreased intensity towards the edge:

No zeros of the PSF intensity (example: gaussian profile)

Modified definition of Strehl ratio with weighting function necessary


Point Spread Function with Apodization

w

I(w)

1

0.8

0.6

0.4

0.2

00 1 2 3-2 -1

Airy

Bessel

Gauss

w

E(w)

1

0.8

0.6

0.4

0.2

03 41 2

Airy

Bessel

Gauss

Apodisation of the pupil:

1. Homogeneous

2. Gaussian

3. Bessel

Psf in focus:

different decrease of the focal

intensity for larger radii

Encircled energy:

same behaviour

General features:

Beam profiles and point spread function complicated

No axial symmetry around image plane

No shift-invariance in z-direction

Vectorial effects: axial components Ez and mixing of lateral x-y-field components,

therefore polarization is important

linear polarized input field breaks symmetry

Apodization of the pupil, depending on correction

Spherical shape of the pupil: defocussing is not described by Zernike c4 alone


High-NA-Systems


Axial Magnification at High NA

Paraxial case:

Relation between lateral and axial magnification

Systems with sine condition fulfilled:

pupil must have spherical shape

Transfer between entrance and exit pupil

2''m

n

n

z

zmz

'sin

sin

'

n

nm

entrance

pupil

yp

z z'

'

y'p

exit

pupil

sin zy p

p

p

y

ym

'


High-NA Focusing

Transfer from entrance to exit pupil in high-NA:

1. Geometrical effect due to projection

(photometry): apodization

with

Tilt of field vector components

y'

R

u

dy/cosudy

y

rrE

E

yE

xE

y

xs

Es

eEy e

y

y'

x'

u

R

entrance

pupil

image

plane

exit

pupil

n

NAus sin

4 220

1

1

rsAA

2cos11

11

2 22

22

0

rs

rsAA


High-NA Focusing

Total apodization

corresponds astigmatism

Example calculations

4 22

2222

0

12

2cos1111),(),(

rs

rsrsrArA linx

-1 -0.5 0 0.5 10

0.5

1

1.5

2

-1 -0.5 0 0.5 10

0.5

1

1.5

2

-1 -0.5 0 0.5 10

0.5

1

1.5

2

-1 -0.5 0 0.5 10

0.5

1

1.5

2

NA = 0.5 NA = 0.8 NA = 0.97NA = 0.9

r

A(r)

x

y

Systems with high numerical aperture:

- distortion of geometry

- nonlinear relationship between aperture angle

and pupil height rp

This distortion corresponds to an apodization

The function a() depends on the system

correction

)()( 222111 gfgfrp


High-NA-Systems

d

gdga

)(

sin

)()(

rp

f

z

Sine condition

Herschel condition

Lagrange condition

Paraboloidal

Helmholtz-condition


Vectorial Diffraction for high-NA

Vectorial representation of the diffraction integral according to Richards/Wolf

Auxiliary integrals

General: axial and cross components of polarization

cos2

2sin

2cos

)','(

1

2

20

2/sin4

00

2

I

Ii

IIi

eE

E

E

E

zrE

iu

z

y

x

o

dekrJzrI ikz

0

cos'

00 sin')cos1(sincos),'(

o

dekrJzrI ikz

0

cos'

1

2

1 sin'sincos)','(

o

dekrJzrI ikz

0

cos'

22 sin')cos1(sincos)','(


Vectorial Diffraction for high-NA

Point spread function components Ix, Iy and Iz

Strong differences in the profiles dependent on defocussing

Large differences in size (here: normalized)

Example: initial polarization: x

Ex

Ey

Ez


High NA and Vectorial Diffraction

Relative size of vectorial effects as a function of the numerical aperture

Characteristic size of errors:

I / Io

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-6

10-5

10-4

10-3

10-2

10-1

100

NA

axial

lateral

error axial lateral

0.01 0.52 0.98

0.001 0.18 0.68

Vectorial diffraction integral in Fraunhofer representation

All components x,y,z must be considered

In particular due to the large bending angle an axial component EZ occurs


High-NA-Systeme

E r F P x y e

n u x y

n u y

n u y

y n u x y

n u y

p p

iz n u x y

p p

p

p

p p p

p

p p

' ,

' sin '

' sin '

' sin '

' sin '

' sin '

' ' sin '

1

1

2

2 2

2 2 2

2 2 2

2 2 2 2

2 2 2

2 2 2 2

1

1

1

1

p


High-NA Point Spread Function

Comparison of Psf intensity profiles

for different models as afunction of

defocussing

paraxial

NA = 0.98

high-NA

scalar

high-NA

vectorial

error ofmodel

NA0 0.2 0.4 0.6 0.8 1.0

paraxial

scalarhigh-NA

low-NA vectorial

Normalized axial intensity

for uniform pupil amplitude

Decrease of intensity onto 80%:

Scaling measure: Rayleigh length

- geometrical optical definition

depth of focus: 1RE

- Gaussian beams: similar formula

22

'

'sin' NA

n

unRu


Depth of Focus: Diffraction Consideration

2

0

sin)(

u

uIuI

focal

plane

beam

caustic

z

depth of focus

0.8

1

I(z)

+Ru

z-R

u 0

r

intensity

at r = 0

2' o

un

Rp

udiff Run

z

2

1

sin493.0

2

12


Microscope Objective Lens: Defocusing

Defocusing in high-NA lenses

Geometrical: change of NA

d

focal

extra focal

starting

planes intra focal

d

pupil

sinuo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

d / f = - 0.001

d / f = 0.001

d / f = 0.003

d / f = - 0.003

d / f = - 0.005

d / f = 0.005

d / f = 0.01

d / f = - 0.01

d / f = - 0.02

d / f = 0.02

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

o

oeff

udff

dfuu

222 tan2

2tansin

Relative change factor

1sin

sin

o

eff

u

u



Real systems:

1. Macroscopic change

of ray path

2. Pupil shrinks down

3. Vignetting surface index

changes,

sharp bending of curve

NA

z in

Ru-100 0 100 200 300 4000

0.2

0.4

0.6

0.8

1

rear

stop

front

surface

tube

lens

0.9

focussed 50 m defocussing

macroscopic changes in the ray path

50 m

defocussing

in the object


Microscope Objective Lens: Defocussing

Change of performance for variation of the

object distance

Strehl increased for shorter object distance

4 6 8 10 12 14 16

Strehl

magnification m

long object

distance

short object

distance

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

nominal

magnification


Microscope Objective Lens: Index Mismatch

Objective lens with immersion

3 materials : Immersion (I), cover glass (C) and sample (S)

Refraction law:

Problems by index

mismatches with sample

points deep inside

Strong spherical

aberrations for high-NA

first lens

immersion cover

glassprobe

mediumenlarged picture of

the ray caustic

paraxial

focus

marginal

focus

nCG

nM

SSCCII nnnNA coscoscos



Calculation of spherical aberration:

comparison of optical path

length with ideal case

( object on back side of glass)

tC

tS

cover

glass

sample

medium

nS

nI

I

C

S

immersion

liquid

nC

tI

s10

dC0

dI0

dI

a

dC

s1

dS

y1

y2

y2

tC

tS

cover

glass

sample

medium

nC

nS

nI

I

C

S

focussed on

underside of cover

glass

defocussing with

mismatch

immersion

liquid

Spherical aberration:

Vanishing effect for nI=nS

Independent from cover glass

1)/(112/12

2

2

S

S

ISS nNA

n

nntW



Aberrations:

1. Zernike coefficients change

approximately linear

2. Psf strongly disturbed

cj in

-0.01 -0.005 0 0.005 0.01-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

c80

c40

c60

z in

[mm]

10RU

z in

Ru

100

00 3-3 r in D

airy

d=0 Ru d=3 Ru d=6 Ru d=9 Ru

d=12 Ru d=15 Ru d=18 Ru d=21 Ru

d=24 Ru d=27 Ru d=30 Ru d=33 Ru

d=36 Ru d=39 Ru d=42 Ru d=45 Ru


Microscope objective Lens: Defocusing

Peak height of Psf : decreasing

No longer symmetry between

aberrations for object - image

reversal

Imax

t in

Ru0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

cj in

NA0 0,2 0,4 0,6 0,8 1 1,2

0

0,5

1

1,5

2

2,5

3

3,5

4

c20

c40

c60

c80

solid lines : image side

dashed lines : object side



Change of aberration with

depth ts, index ratio and

aperture angle

Psf strongly perturbed

z in

Ru

50

00 3-3 r in D

airy

d=0 Ru d=4 Ru d=8 Ru d=12 Ru

d=16 Ru d=20 Ru d=24 Ru d=28 Ru

d=32 Ru d=36 Ru d=40 Ru d=44 Ru

d=48 Ru d=52 Ru d=56 Ru d=60 Ru

W / tS

0 10 20 30 40 50 60 70 80 90-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

S in °

nC = 1.4

nS = 1.4

nC = 1.0

nS = 1.3

nC = 1.4

nS = 1.33

nC = 1.0

nS = 1.7

nC = 1.52

nS = 1.4

nC = 1.52

nS = 1.33



I(r)

r in

Dairy

-2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

a = 0...60 Ru

I(z)

z in

Ru

a = 0...60 Ru

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Point spread function:

1. Lateral : only scaling

2. Axial : strong decrease with

ripple

Focussing into an index-mismatched sample

Strong degradation of the

point spread function with

increasing depth

I(r,z)

0.5

5

0

0

1

4

3

2

1

0-5

r in rairy

z in Ru


Point Spread Function for Index Mismatch


Coherence in Optics

Statistical effect in wave optic:

start phase of radiating light sources are only partially coupled

Partial coherence: no rigid coupling of the phase by superposition of waves

Constructive interference perturbed, contrast reduced

Mathematical description:

Averagedcorrelation between the field E at different locations and times:

Coherence function G

Reduction of coherence:

1. Separation of wave trains with finite spectral bandwidth

2. Optical path differences for extended source areas

3. Time averaging by moved components

Limiting cases:

1. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length

2. Incoherence: no correlation, light source with independent radiating point like molecules


Coherence Function

Coherence function: Correlation

of statistical fields (complex)

for identical locations :

intensity

normalized: degree of coherence

In interferometric setup, the amount of describes the visibility V

Distinction:

1. spatial coherence, path length differences and transverse distance of points

2. time-related coherence due to spectral bandwidth and finite length of wave trains

ttrEtrErr ),(),(),,( 2

*

121

z

x

x1

x2

E(x2)

E(x1)x

r r r1 2

)()(

),,(),,()(

21

212112

rIrI

rrrr

)(),( rIrr


Double Slit Experiment of Young

D

z1

z2

light

source

screen

with slits

detector

x

x

First realization:

change of slit distance D

Second realization:

change of coherence parameter s of the

source

D0

1

V


Double Slit Experiment of Young

s = 0 s = 0.15 s = 0.25 s = 0.35 s = 0.40s = 0.30

Partial coherent illumination of a double pinhole/double slit

Variation of the size of the source by coherence parameter s

Decreasing contrast with growing s

Example: pinhole diameter Dph = Dairy / distance of pinholes D = 4Dairy


Spatial Coherence

1

2

starting

plane

receiving

plane

common

area

Area of coherence / transverse coherence length:

Non-vanishing correlation at two points with distance Lc:

Correlation of phase due to common area on source

Radiation out of a coherence cell of

extension Lc guarantees finite contrast

The lateral coherence length

changes during propagation:

spatial coherence grows with

increasing propagation distance

observation

area

O

r2

r1

r r( , )

1 2

Lc

P1

P2

domain of

coherence


Spatial Coherence

Incoherent source with

diameter D = 2a

Receiver plane indistance z

Cone of observation

Source is coherent in the

distance

Transverse length of coherence

(zeros of -function)

/ a

z a 2 /

Lz

ac


Partial Coherent Imaging

Complete description of an optical system:

1. Light source

2. Illumination system, amplitude response hill

3. Transmission object

4. Observation / imaging system with amplitude response hobs

sourceillumination

system

observation

system

pupil Psensor

object

plane

image

planeillumination

field

xs , ys

Is hill Iihobs

Io

xp , yp xi , yi

obs

ill

u

u

sin

sins


Coherence Parameter

Finite size of source : aperture cone with angle uill

Observation system: aperture angle uobs

Definition of coherence parameter s:

Ratio of numerical apertures

Limiting cases:

coherent s = 0

uill << uobs

incoherent

s = 1

uill >> uobs

object imagesource

xi , yixo , yo

lens

uobs

uill

illumination observation

Heuristic explanation

of the coherence

parameter in a system:

1. coherent:

Psf of illumination

large in relation to the

observation

2. incoherent:

Psf of illumination

small in comparison

to the observation

object objective lenscondensersmall stop of

condenser

extended

source

coherent

illumination

large stop of

condenser

incoherent

illumination

Psf of observation

inside psf of

illumination

Psf of observation

contains several

illumination psfs

extended

source


Coherence Parameter

Simulation of partial coherent illumination:

Finite size of light source

Corresponding finite size of illuminated area in aperture plane

Every point in this area is considered to emit independent (incoherent)

Off-axis point in aperture plane generates an inclined plane wave in the object

Angular spectrum illumination of the object

describes partial coherence

Estimated sampling of illumination points:


Partial Coherence

condenseraperture

plane

object

plane

source

extension

size of

coherence cell

ys

Ls

angular

spectrum of

object

illumination

2s

fc

c

ss

f

L

2arcsin22

sin

61.0

nys

Superposition of sourve into coherent beams of several source points

Coherent propagation of source point contributions

Incoherent summation of all point contributions in the image plane

Finite size of source:

object illuminated by different directions


Fourier Model of Image Formation

xa

illumination

aperturecondenser

object

plane

objective

lenspupil

image

plane

tube

lens

Ea(xa)

fc fo ftxo xp xi

aberrations

xo,W(xo)

chief raysource

point S

Eo(xo)

object

mask

A(xo)

E’’o(xo)

pupil

mask

P(xp)

CTFc(xo)

A(xo)

Ep(xp) E'’’p(xp)

CTFo(xp)

P(xp)

CTFt(xt)

Ei(xi)FFT FFT FFT

diffracted

lightdirect

light


Partial Coherent Imaging

Image intensity: 1. Correlation of two points in the object

2. Integration over all points in the incoherent light source

Simplification:

thin object, transmission does not change for moderate inclination angles

soooosoillsoillsoiobssoiobsssii dxdxdxxOxOxxhxxhxxxhxxxhxIxI 212

*

12

*

12

*

1 ,,,,,,

yp

xp

yo

xo xi

yi

z

ys

xs

light

sourceobject

planepupil image

hill(xL,x1) hobs(x1,x')Is(xs) Ii(xi)O(x1)

P(xp)o(xo1,xo2) 'o(xo1,xo2)

212

*

12

*

121 ,,, oooooiobsoiobsoooii dxdxxOxOxxhxxhxxxI

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