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Imaging and Aberration Theory

Lecture 13: Point spread function

2017-01-26

Herbert Gross

Winter term 2016

2

Preliminary time schedule

1 20.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 27.11.Pupils, Fourier optics, Hamiltonian coordinates

pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates

3 03.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 10.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas

5 17.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 24.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 01.12. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 08.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 15.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum

10 05.01.Sine condition, aplanatism and isoplanatism

Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics

11 12.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 19.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement

13 26.01. PSF ideal psf, psf with aberrations, Strehl ratio

14 02.02. Transfer function Transfer function, resolution and contrast

15 09.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

1. Introduction

2. Ideal PSF

3. PSF with aberrations

4. High-NA PSF

5. Two-point resolution

Contents

3

Typical change of the intensity profile

Normalized coordinates

Diffraction integral

4

Fresnel Diffraction

z

geometrical

focus

f

a

z

stop

far zone

geometrical

phase

intensity

a

rr

f

avz

f

au

;

2;

22

1

0

20

/

2

0

2 22

)(2

),(

devJef

EiavuE

ui

uafi

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

5

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

6

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

7

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

8

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

9

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

10

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

11

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%

12

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

13

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%

14

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

Psf with Aberrations

Psf for some low oder Zernike coefficients

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

The peak intensities are renormalized

15

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

Growing spherical aberration shows an asymmetric behavior around the nominal image

plane for defocussing

16

Caustic with Spherical aberration

c9 = 0 c9 = 0.7c9 = 0.3 c9 = 1

PSF with coma

The 1st diffraction ring is influenced very sensitive

W31 = 0.03 W31 = 0.06 W31 = 0.09 W31 = 0.15

Psf for Coma Aberration

Separation of the peak and the centroid position in a point spread function with coma

From the energetic point of view coma induces distortion in the image

Psf with Coma

c7 = 0.3 c7 = 0.5 c7 = 1

centroid

x

y

PSF as a function

of the field height

Interpolation is

critical

Orientation of coma

in the field

Small field area with

approximately shift-

invariant PSF:

isoplanatic patch

Variation of Performance with Field Position

x = 0 x = 20% x = 40% x = 60 % x = 80 % x = 100 %

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

20

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

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

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

FWHM

w

E(w)

1

0.8

0.6

0.4

0.2

03 41 2

Airy

Bessel

Gauss

E95%

Apodisation of the pupil:

1. Homogeneous Airy

2. Gaussian Gaussian

3. Ring illumination Bessel

Psf in focus:

different convergence to zero forlarger radii

Encircled energy:

same behavior

Complicated:Definition of compactness of thecentral peak:

1. FWHM: Airy more compact as GaussBessel more compact as Airy

2. Energy 95%: Gauss more compact as AiryBessel extremly worse

Ring shaped pupil illumination

outer radius aa

innen radius ai

parameter

Depth of focus enlarged

Central peak shrinks lateral

Can not be understood geometrically

Psf of a Ring Pupil

1a

i

a

a

Farfield of a ring pupil:

outer radius aa

innen radius ai

parameter

Ring structure increases with

Depth of focus increases

Application:

Telescope with central obscuration

Intensity at focus

1a

i

a

a

2

121

22

)(2)(2

1

1)(

x

xJ

x

xJxI

r-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(r)

= 0.01

= 0.25

= 0.35

= 0.50

= 0.70

22 1sin

2

unz

Psf of an Annular Pupil

Encircled energy curva:

- steps due to side lobes

- strong spreading, large energy content in the rings

r

E(r)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 0.01

= 0.25

= 0.35

= 0.50

= 0.70

PSF for ring-Shaped Pupil

r

z

I(r,z)

How to quantify the aberrated PSF ?

26

Focal Diameter Determination

27

1 2 3 4 5 6

Pupil shape

definiton of focus

Gauss radius w

no trunc..

Super gauss w,m=6

no trunc.

Gauss 2a=3w

with trunc.

Gauss a=w with

trunc.

Circle homoge-

neous D=2a

Linear / homog. slit width

2a

1. Zero point of intensity

#

#

#

1.426

1.220 ( Airy )

1.00

I=0.5 ( FWHM )

0.375

0.513

0.644

0.564

0.519

0.443

I=0.13534 (1/e2 )

0.637

=2/

0.831

1.059

0.914

0.822

0.697

I=0.01

0.966

1.122

1.491

1.238

1.092

( Peak )

0.908

I=0.001

1.183

2.109

1.695

2.925

1.174

( Peak )

0.969

E=0.86466

0.637

=2/

0.818

1.008

0.890

1.378

0.658

E=0.95

0.779

1.040

1.201

1.104

3.915

1.989

Focal diameter:

Basic scaling factor

Additional factor b

Factor differs in size by

one magnitude

NAD foc

b

wbeamradius

truncationno

w

f

aradiusat

truncationwith

a

f

NA

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 )

28

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

29

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

30

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%

31

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

Low Fresnel Number Focussing

37

Small Fresnel number NF:

geometrical focal point (center point of spherical wave) not identical with best

focus (peak of intensity)

Optimal intensity is located towards optical system (pupil)

Focal caustic asymmetrical around the peak location

a

focal length f

z

physical

focal

point

geometrical

focal point

focal shift

f

wavefront

38

Example: focusing by a microlens

radius: 0.1 mm

wavelength: 1 μm

focal length: 10 mm

NA: 0.01

Fresnel #: 1

Result:

highest intensity in front of

geometrical focus consequence of constructive

interference of edge diffraction

Low Fresnel Number Focussing

Systems with small Fresnel number

Same size of:

1. divergence angle and numerical aperture angle

2. focal length and depth of focus

3. influence of geometrical beam shaping and diffraction

Effect of photometric distance law inside depth of focus

Focal shift depends on Fresnel number, apodization and coherence

For definition of best focus by peak intensity on axis:

Relative focal shift for circular homogeneous illuminated pupil:

2

1)(

zzI

Low NA and Focal Shift

0),0(

z

zI foc

2

2

0

4

4sin

21)(

u

u

N

uIzI

F

f

f NF

1

112

2

1 −𝑢

2𝜋𝑁𝐹=𝑓

𝑧

Low Fresnel Number Focussing

40

System with small Fresnel number:

Axial intensity distribution I(z) is

asymmetric

Explanation of this fact:

The photometric distance law

shows effects inside the depth of focus

Example microscopic 100x0.9 system:

a = 1mm , z = 100 mm:

NF = 18

2

2

0

4

4sin

21)(

u

u

N

uIzI

F

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NF = 2

NF = 3.5

NF = 5

NF = 7.5

NF = 10

NF = 20

NF = 100

z / f - 1

I(z)

Focal shift as a function of

1. Fresnel number

2. Apodization

in linear and logarithmic representation

Low NA and Focal Shift

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

42

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

43

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

44

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

45

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

46

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

47

Incoherent Resolution: Dependence on NA

Microscopical resolution as a function of the numerical aperture

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

48

Depth of Focus

Depth of focus depends on numerical aperture

1. Large aperture: 2. Small aperture:

small depth of focus large depth of focus

Ref: O. Bimber

Depth of Focus

Schematic drawing of the principal ray path in case of extended depth of focus

Where is the energy going ?

What are the constraints and limitations ?

conventional ray path

beam with extended

depth of focus

z

z0

Normalized axial intensity

for uniform pupil amplitude

First zero:

depth of focus: 1 Rayleigh length

peak intensity reduced to 80%

transverse width doubled

- Gaussian beams: similar formula

22

'

'sin' NA

n

unRE

Depth of Focus: Diffraction Consideration

2' o

Rn

z

0z

focal

plane

beam

caustic

z

depth of focus

0.8

1

I(z)

z-Ru/2 0

r

intensity

at r = 0

+Ru/2

-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

2

0

sin)(

z

zIzI z

NAz

2

2

ERz 2

𝑟𝑝

z

r

Axial Point-Spread Function52

Intensity I(z)

𝑟𝑝

z

r

Annular Ring Pupil (Bessel Beams)53

Intensity I(z)

𝑟𝑝

z

r

Engineering the PSF: Simple Toraldo Mask54

Intensity I(z)

𝑟𝑝

z

r

Spherical Aberration (1λ)55

Intensity I(z)

𝑟𝑝

z

r

Gaussian Apodization56

Intensity I(z)