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Imaging and Aberration Theory

Lecture 13: Point spread function

2020-01-24

Herbert Gross

Winter term 2019

2

Schedule - Imaging and aberration theory 2019

1 18.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 25.10.Pupils, Fourier optics, Hamiltonian coordinates

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

3 01.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 08.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas

5 15.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 22.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 29.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 13.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spectrum

10 20.12.Sine condition, aplanatism and isoplanatism

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

11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 17.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement

13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio

14 31.01. Transfer function transfer function, resolution and contrast

15 07.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsicand induced aberrations, reversability

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

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

PSF by Huygens Principle

Huygens wavelets correspond to vectorial field components:

- represented by a small arrow

- the phase is represented by the direction

- the amplitude is represented by the length

Zeros in the diffraction pattern: destructive interference

Ideal point spread function:

pupil

stop

wave

front

point

spread

function

zero intensity

closed loop

side lobe peak

1 ½ round trips

central peak maximum

constructive interference

single wavelets

sum

PSF by Huygens Principle

Apodization:

variable lengths

of arrows

Aberrations:

variable orientation

of arrows

pupil

stop

wave

front

point

spread

function

apodization:

decreasing length of arrows

homogeneous pupil:

same length of all arrows

rp

I(xp)

pupil

stop

ideal

wave

front

point

spread

function

ideal spherical wavefront

central peak maximum

real

wave

front

real wavefront

with aberrations

central peak reduced

0

2

12,0 I

v

vJvI

0

2

4/

4/sin0, I

u

uuI

Circular homogeneous illuminated aperture:

Transverse intensity:

Airy distribution

Dimension: DAiry

normalized lateral

coordinate:

v = 2 x / NA

Axial intensity:

sinc-function

Dimension: Rayleigh unit RE

normalized axial coordinate

u = 2 z n / NA2

Perfect Point Spread Function

NADAiry

22.1

2NA

nRE

r

z

Airy

lateral

aperture

cone

Rayleigh

axial

image plane

optical

axis

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

0,2

0,4

0,6

0,8

1,0

vertical

lateral

u / v

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

10

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

11

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

12

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%

13

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

14

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%

15

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

16

Growing spherical aberration shows an asymmetric behavior around the nominal image

plane for defocussing

17

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

21

Focal Diameter Determination

22

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=0.1353 (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

23

Focus Spot Size of a Lens

Changing the NA/aperture D of a focussing lens:

- small values of D:

diffraction dominates, Airy formula

- large D:

geometrical aberrations dominate

Total aberrations:

superposition of both effects

Dspot

NA / D

Log Dspot

NA / D

diffraction

Airy

diffraction

Airy

geometrical

aberration

geometrical

aberration

total

total

Small aperture:

Diffraction limited

Spot size corresponds

to Airy diameter

Spot size depends on

wavelength

Large aperture:

Diffraction neglectible

Aberration limited

Geometrical effects not

wavelength dependent

But: small influence of

dispersion

Log Dfoc

Log sinu

f=1000 , 500 , 200 , 100 , 50 , 20 , 10 mm

= 10 m

= 1 m

550 nm

1

0

-1

-2-2-3 -1 0

Focussing by a Lens: Diffraction and Aberration

25

Hybrid model

Calculation of diffraction in optical systems:

- it is assumed, that all edge diffraction effects can be accumulated as occuring in the

boundary of the exit pupil

- diffractino effects at the bundle-limiting boundaries inside the system are neglected

- internal boundaries are geometrical projected in the exit pupil

- the light propagation was performed by raytrace into the exit pupil

- the OPPD and the density of rays was used to compose a complex field

- the transition from the exit pupil into the image plane uses traditional formulations of the

diffraction integral

Typical failure of this approach: mode beam clean up pinholes, very large propagation

paths

exit pupil

plane

image

plane

entrance

pupil planeobject

plane

complex field E

OPD: phase

weighting, ray density:

amplitude

y yp y'p y'

z

optical system

upper coma

ray

image

point

off-

axis

object

point

ray tracingdiffraction integral

chief ray

lower

coma ray

edge

wave

edge

wave

Setup with several stops:

different contributions of diffracted waves can be observed in th efinal point:

1. direct geometrical light

2. diffraction at every direct illuminated stop edge (1, 2 or 3)

3. diffraction at one stop, the second interaction at a following stop lies inside the bright

but modified direct light

4. diffraction at one stop, the second interaction at a following stop lies in the geometrical

dark but finite illumination cone

A rigorous calculation should take all

contributions into account

The requirements on sampling and accuracy

are extrem demanding

Cascaded Diffraction in Systems

D1 D3D2

source

point observation

pointdirect geometrical light

diffracted

at stop 1

diffracted

at stop 3

double

diffracted

at stop 1

and 3 (bright)double

diffracted at

stops 1 and

2 (dark)

stops

Asymmetry effects:

- both stops opened half

- various distances

Asymmetry of PSF in near and far

field grows with decreasing NF

Example Systems II

PSF I(x),I(y)

NF = 20

NF = 5

NF = 1.5

NF = 400

x [mm]

y

x,y [mm]

PSF I(x,y)

a) case I:

ExP diffraction

b) case I:

distributed

diffraction

one stop

fixed

upper

stop

moving

lower

stopchange of distance z

Fresnel number NF

edge

wave

Double Gauss photographic lens

Front surface truncates field bundle

at the bottom

Rear surface truncates field bundle

at the top

Calculation steps:

1. front surface until pupil

2. Pupil until rear surface

3. rear surface into image plane

Fresnel numbers large:

NF = 3036 / 6148

Example Systems III

surface 1 surface 6 surface11

truncating

surfaces for

field bundle

NF = 3036

NF = 6148 11

6 6

1

segment 1

stop 1

segment 2

stop 2 sensor

D1=54

D*2=16.66z*1=58.925

z*2=29.047

D*3=69.204CRyCR1=-14.69

yCR3=+20.12

Comparison exit pupil vs z-distributed diffraction

Quantitative difference: 3 10-5

X cross section one order of magnitude smaller

Example Systems III

a) footprint in exit pupil c) difference I

b) image psf I(x,y)

x section

y section (offset)

x [mm] x/y [mm]

y [mm]

0

10-5

-1

-2

-3

1

-4

d) cross sections I(x) , I(y)

x/y [mm]xp [mm]

yp

[mm]

-0.2 -0.1 0 0.1 0.2-6

-5

-4

-3

-2

-1

0

x section

y section

(offset)

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 )

30

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

31

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

32

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%

33

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

39

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

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

I z I

u

N

u

uF

( )sin

0

2

2

12

4

4

f

f NF

1

112

2

Low Fresnel Number Focussing

41

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

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

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

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

Ring shaped pupil illumination

Depth of focus enlarged

Central peak shrinks lateral

Psf of a Ring Pupil

48

Obscuration / PSF

• Central obscuration

• Spider obscuration

Circular apertureno obscuration

Circular aperture28% central obscuration

Circular aperture28% central obscuration

three spider vanes

Circular aperture28% central obscuration

four spider vanes

Ref: K. Uhlendorf

49

PSF Calculation

System with large grid distortion

Perfect phase on axis due to

4 confocal paraboloids

P1

P2

P3

P4

F3 = F4

F1 = F2

spot at exit

intensity at exit

50

PSF Calculation

System with large grid distortion

Calculation of the PSF with direct integration and with FFT-based algorithm

Large deviation from Airy pattern

Differences in algorithms due to grid distortion,

but Strehl nearly identical

Zemax-error due to y-orientation ?

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

51

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

52

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

53

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

54

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

55

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

56

Incoherent Resolution: Dependence on NA

Microscopical resolution as a function of the numerical aperture

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

57

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

2' o

un

R

udiff Run

z

2

1

sin493.0

2

12

focal

plane

beam

caustic

z

depth of focus

0.8

1

I(z)

z-Ru/2 0

r

intensity

at r = 0

+Ru/2

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

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

EDF with Complex Toraldo Mask

I(r,z)

I(x) I(z)

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

I(z) depth for 80%: 13 RE

-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1