www.iap.uni-jena.de

Imaging and Aberration Theory

Lecture 13: PSF and transfer function

2015-02-05

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. Ideal point spread function

2. PSF with aberrations

3. Strehl ratio

4. High NA vectorial PSF

5. Two-point-resolution

6. Optical transfer function

7. Resolution and contrast

Contents

3

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

4

PSF by Huygens Principle

Huygens wavelets correspond to vectorial field components

The phase is represented by the direction

The amplitude is represented by the length

Zeros in the diffraction pattern: destructive interference

Aberrations from spherical wave: reduced conctructive superposition

pupil

stop

wave

front

ideal

reference

sphere

point

spread

function

zero

intensity

side lobe

peak

central peak maximum

constructive interference

reduced constructive

interference due to phase

aberration

5

Fraunhofer Point Spread Function

Rayleigh-Sommerfeld diffraction integral,

Mathematical formulation of the Huygens-principle

Fraunhofer approximation in the far field

for large Fresnel number

Optical systems: numerical aperture NA in image space

Pupil amplitude/transmission/illumination T(xp,yp)

Wave aberration W(xp,yp)

complex pupil function A(xp,yp)

Transition from exit pupil to

image plane

Point spread function (PSF): Fourier transform of the complex pupil

function

1

2

z

rN

p

F

),(2),(),( pp yxWi

pppp eyxTyxA

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

''cos'

)'()('

dydxrr

erE

irE d

rrki

I

6

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 z n / NA2

transversal : v = 2 x / NA

Perfect Point Spread Function

NADAiry

22.1

2NA

nRE

7

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

8

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 to /NA:

Assumptions for this estimation and possible changes

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

assumptions

9

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

'sin'sin2

ak

R

akrukr

R

arx

Perfect Lateral Point Spread Function: Airy

10

log I(r)

r0 5 10 15 20 25 30

10

10

10

10

10

10

10

-6

-5

-4

-3

-2

-1

0

Airy distribution:

Gray scale picture

Zeros non-equidistant

Logarithmic scale

Encircled energy

Perfect Lateral Point Spread Function: Airy

DAiry

r / rAiry

Ecirc

(r)

0

1

2 3 4 5

1.831 2.655 3.477

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2. ring 2.79%

3. ring 1.48%

1. ring 7.26%

peak 83.8%

11

Axial distribution of intensity

Corresponds to defocus

Normalized axial coordinate

Scale for depth of focus :

Rayleigh length

Zero crossing points:

equidistant and symmetric,

Distance zeros around image plane 4RE

22

04/

4/sinsin)(

u

uI

z

zIzI o

42

2 uz

NAz

22

'

'sin' 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/

RE

4RE

z = 2RE

12

Defocussed Perfect Psf

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%

13

Psf with Aberrations

Psf for some low oder Zernike coefficients

The coefficients are changed between cj = 0...0.7

The peak intensities are renormalized

spherical

defocus

coma

astigmatism

trefoil

spherical

5. order

astigmatism

5. order

coma

5. order

c = 0.0

c = 0.1c = 0.2

c = 0.3c = 0.4

c = 0.5c = 0.7

14

0,0

0,0)(

)(

ideal

PSF

real

PSFS

I

ID

2

2),(2

),(

),(

dydxyxA

dydxeyxAD

yxWi

S

Important citerion for diffraction limited systems:

Strehl ratio (Strehl definition)

Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity

DS takes values between 0...1

DS = 1 is perfect

Critical in use: the complete

information is reduced to only one

number

The criterion is useful for 'good'

systems with values Ds > 0.5

Strehl Ratio

r

1

peak reduced

Strehl ratio

distribution

broadened

ideal , without

aberrations

real with

aberrations

I ( x )

15

Approximation of

Marechal:

( useful for Ds > 0.5 )

but negative values possible

Bi-quadratic approximation

Exponential approach

Computation of the Marechal

approximation with the

coefficients of Zernike

2

241

rms

s

WD

N

n

n

m

nmN

n

ns

n

c

n

cD

1 0

2

1

2

0

2

12

1

1

21

Approximations for the Strehl Ratio

22

221

rms

s

WD

2

24

rmsW

s eD

defocusDS

c20

exac t

Marechal

exponential

biquadratic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16

In the case of defocus, the Rayleigh and the Marechal criterion delivers

a Strehl ratio of

The criterion DS > 80 % therefore also corresponds to a diffraction limit

This value is generalized for all aberration types

8.08106.08

2

SD

Strehl Ratio Criterion

aberration type coefficient Marechal

approximated Strehl

exact Strehl

defocus Seidel 25.020 a 7944.0 8106.08

2

defocus Zernike 125.020 c 0.7944 0.8106

spherical aberration

Seidel 25.040 a 0.7807 0.8003

spherical aberration

Zernike 167.040 c 0.7807 0.8003

astigmatism Seidel 25.022 a 0.8458 0.8572

astigmatism Zernike 125.022 c 0.8972 0.9021

coma Seidel 125.031 a 0.9229 0.9260

coma Zernike 125.031 c 0.9229 0.9260

17

Criteria for measuring the degradation of the point spread function:

1. Strehl ratio

2. width/threshold diameter

3. second moment of intensity distribution

4. area equivalent width

5. correlation with perfect PSF 6.power in the bucket

Quality Criteria for Point Spread Function

d) Equivalent widtha) Strehl ratio b) Standard deviation c) Light in the bucket

h) Width enclosed areae) Second moment f) Threshold width g) Correlation width

SR / Ds

STDEV

LIBEW

SM FWHM

CW

Ref WEAP=50%

18

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 to 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

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)','(

Pupil

Vectorial Diffraction at high NA

Linear Polarization

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

Transverse resolution of an image:

- Detection of object details / fine structures

- basic formula of Abbe

Fundamental dependence of the resolution from:

1. wavelength

2. numerical aperture angle

3. refractive index

4. prefactor, depends on geometry, coherence, polarization, illumination,...

Basic possibilities to increase resolution:

1. shorter wavelength (DUV lithography)

2. higher aperture angle (expensive, 75° in microscopy)

3. higher index (immersion)

4. special polarization, optimal partial coherence,...

Assumptions for the validity of the formula:

1. no evanescent waves (no near field effects)

2. no non-linear effects (2-photon)

sinn

kx

Point Resolution According to Abbe

24

Rayleigh criterion for 2-point resolution

Maximum of psf coincides with zeros of

neighbouring psf

Contrast: V = 0.15

Decrease of intensity

between peaks

I = 0.735 I0

unDx Airy

sin

61.0

2

1

Incoherent 2-Point Resolution : Rayleigh Criterion

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

x / rairy

I(x)

PSF2PSF1

sum

of

PSF

25

Criterion of Sparrow:

vanishing derivative in the center between two

point intensity distribution,

corresponds to vanishing contrast

Modified formula

Usually needs a priory information

Applicable also for non-Airy

distributions

Used in astronomy

0)(

0

2

2

xxd

xId

Incoherent 2-Point-Resolution: Sparrow Criterion

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

x / rairy

I(x)

Rayleigh

AirySparrow

x

Dun

x

770.0

385.0sin

474.0

26

Visual resolution limit:

Good contrast visibility V = 26 % :

Total resolution:

Coincidence of neighbouring zero points

of the Airy distributions: V = 1

Extremly conservative criterion

Contrast limit: V = 0 :

Intensity I = 1 between peaks

AiryDun

x

680.0

sin

83.0

unDx Airy

sin

22.1

AiryDun

x

418.0

sin

51.0

Incoherent 2-point Resolution Criterions

27

2-Point Resolution

Distance of two neighboring object points

Distance x scales with / sinu

Different resolution criteria for visibility / contrast V

x = 1.22 / sinu

total

V = 1x = 0.68 / sinu

visual

V = 0.26

x = 0.61 / sinu

Rayleigh

V = 0.15x = 0.474 / sinu

Sparrow

V = 0

28

2-Point Resolution

Intensity distributions below 10 % for 2 points with different x (scaled on Airy)

x = 2.0 x = 1.22 x = 0.83

x = 0.61 x = 0.474

x = 1.0

x = 0.388 x = 0.25

29

Incoherent Resolution: Dependence on NA

Microscopical resolution as a function of the numerical aperture

NA = 0.9NA = 0.45NA = 0.3NA = 0.2

30

Comparison Geometrical Spot – Wave-Optical Psf

aberrations

spot

diameter

DAiry

exact

wave-optic

geometric-optic

approximated

diffraction limited,

failure of the

geometrical model

Fourier transform

ill conditioned

Large aberrations:

Waveoptical calculation shows bad conditioning

Wave aberrations small: diffraction limited,

geometrical spot too small and

wrong

Approximation for the

intermediate range:

22

GeoAirySpot DDD

31

Resolution of Fourier Components

Ref: D.Aronstein / J. Bentley

object

pointlow spatial

frequencies

high spatial

frequencies

high spatial

frequencies

numerical aperture

resolved

frequencies

object

object detail

decomposition

of Fourier

components

(sin waves)

image for

low NA

image for

high NA

object

sum

32

pppp

pp

vyvxi

pp

yxOTF

dydxyxg

dydxeyxg

vvH

ypxp

2

22

),(

),(

),(

),(ˆ),( yxIFvvH PSFyxOTF

pppp

pp

y

px

p

y

px

p

yxOTF

dydxyxP

dydxvf

yvf

xPvf

yvf

xP

vvH

2

*

),(

)2

,2

()2

,2

(

),(

Optical Transfer Function: Definition

Normalized optical transfer function

(OTF) in frequency space

Fourier transform of the Psf-

intensity

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

Absolute value of OTF: modulation transfer function (MTF)

MTF is numerically identical to contrast of the image of a sine grating at the

corresponding spatial frequency

33

MTF and Contrast

Object

Contrast

Object spectrum

Image spectrum

Image

)2cos()( 0xvacxIobj

c

a

acac

acac

II

IIV

)()(

)()(

minmax

minmax

0022

)0()(ˆ)( vva

vva

vcxIFvI objobj

)()()( vHvIvI MTFobjima

)2cos()(

)(2

)(2

)0(

)(2

)(2

)0()(ˆ

)()(ˆ)(ˆ)'(

00

2

0

2

0

00

1

11

00

xvvHac

evHa

evHa

Hc

vvvHa

vvvHa

vvHcF

vHvIFvIFxI

MTF

xiv

MTF

xiv

MTFMTF

MTFMTFMTF

MTFobjimaima

34

x p

y p

area of

integration

shifted pupil

areas

f x

y f

p

q

x

y

x

y

L

L

x

y

o

o

x'

y'

p

p

light

source

condenser

conjugate to object pupil

object

objective

pupil

direct

light

at object diffracted

light in 1st order

Interpretation of the Duffieux Iintegral

Interpretation of the Duffieux integral:

overlap area of 0th and 1st diffraction order,

interference between the two orders

The area of the overlap corresponds to the

information transfer of the structural details

Frequency limit of resolution:

areas completely separated

35

Duffieux Integral and Contrast

Separation of pupils for

0. and +-1. Order

MTF function

Image contrast for

sin-object

I

x

V = 100%

V = 33%V = 50%

V = 20%

minI = 0.33 minI = 0.5 minI = 0.66

1

image contrast

pattern

period

spatialfrequency

100 %

0 NA/ 2NA/

/NA /2NA8

pupil

diffraction order

Ref: W. Singer

36

Optical Transfer Function of a Perfect System

Aberration free circular pupil:

Reference frequency

Cut-off frequency:

Analytical representation

Separation of the complex OTF function into:

- absolute value: modulation transfer MTF

- phase value: phase transfer function PTF

'sinu

f

avo

'sin222 0

un

f

navvG

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

/ max

00

1

0.5 1

0.5

gMTF

37

I Imax V

0.010 0.990 0.980

0.020 0.980 0.961

0.050 0.950 0.905

0.100 0.900 0.818

0.111 0.889 0.800

0.150 0.850 0.739

0.200 0.800 0.667

0.300 0.700 0.538

Contrast / Visibility

The MTF-value corresponds to the intensity contrast of an imaged sin grating

Visibility

The maximum value of the intensity

is not identical to the contrast value

since the minimal value is finite too

Concrete values:

minmax

minmax

II

IIV

I(x)

-2 -1.5 -1 -0.5 0 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Imax

Imin

object

image

peak

decreased

slope

decreased

minima

increased

38

Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the

azimuthal orientation of the object structure

Generally, two MTF curves are considered for sagittal/tangential oriented object structures

Sagittal and Tangential MTF

y

tangential

plane

tangential sagittal

arbitrary

rotated

x sagittal

plane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

00.5 1

/ max

39

Real MTF of system with residual aberrations:

1. contrast decreases with defocus

2. higher spatial frequencies have stronger decrease

Real MTF

z = 0

z = 0.1 Ru

gMTF

1

0.75

0.25

0.5

0

-0.250 0.2 0.4 0.6 0.8 1

z = 0.2 Ru

z = 0.3 Ru

z = 1.0 Ru

z = 0.5 Ru

-100 -50 0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

gMTF

(z,f)

z in

RU

max

= 0.05

max

= 0.1

max

= 0.2

max

= 0.3

max

= 0.4

max

= 0.5

max

= 0.6

max

= 0.7

max

= 0.8

Zernike

coefficients:

c5 = 0.02

c7 = 0.025

c8 = 0.03

c9 = 0.05

40

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

41

Contrast vs contrast as a function of spatial frequency

Typical: contrast reduced for

increasing frequency

Compromise between

resolution and visibilty

is not trivial and depends

on application

Contrast and Resolution

V

c

1

010

HMTF

Contrast

sensitivity

HCSF

42

Optical Transfer Function of a Perfect System

Loss of contrast for higher spatial frequencies

contrast

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ideal

MTF

/max

/max

43

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

Degradation due to

1. loss of contrast

2. loss of resolution

44