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Design and Correction of optical Systems

Part 9: Fourier opticcal image formatioin

Summer term 2012

Herbert Gross

1

Overview

1. Basics 2012-04-18

2. Materials 2012-04-25

3. Components 2012-05-02

4. Paraxial optics 2012-05-09

5. Properties of optical systems 2012-05-16

6. Photometry 2012-05-23

7. Geometrical aberrations 2012-05-30

8. Wave optical aberrations 2012-06-06

9. Fourier optical image formation 2012-06-13

10. Performance criteria 1 2012-06-20

11. Performance criteria 2 2012-06-27

12. Measurement of system quality 2012-07-04

13. Correction of aberrations 1 2012-07-11

14. Optical system classification 2012-07-18

2012-04-18

9.1 Huygens principle of diffraction

9.2 Perfect point spread function

9.3 PSF with perturbations

9.4 Basics of Fourier optics

9.5 Image formation

9.6 Optical Transfer function

Contents:

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

objectpoint

sphericalwave

truncatedspherical

wave

imageplane

∆∆∆∆x = 1.22 λλλλ / 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

Fλ

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

pppp eyxTyxAπ

⋅=

( ) ( ) ( )

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

+

⋅⋅=∫∫λ

ππ

''cos'

)'()(

'

dydxrr

erE

irE d

rrki

I ∫∫ ⋅−

⋅−=−

θλ

rrrr

rrr

( )( )

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 π z n / λ NA2

transversal : v = 2 π x / λ NA

Perfect Point Spread Function

NADAiry

λ⋅=

22.1

2NA

nRE

λ⋅=

Ideal Psf

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

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 λ/NA:

Assumptions for this estimation and possible changes

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

assumptions

� 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

⋅

⋅=

λ

2

1

2

22

)(

⋅

=

NAr

NAr

J

rI

λ

π

λ

π

'sin'sin2

θλ

πak

R

akrukr

R

arx ====

Perfect Lateral Point Spread Function: Airy

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%

� 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

2u

zNA

z =⋅⋅

=λ

π

'sin'22 un

RE

λ=

Perfect Axial Point Spread Function

Psf with Defocus

� Outside focus :

- Psf is broadened

- a system of rings

takes place

- symmetry around

image plane

Focus

image plane

Defocus

- 2 RE

Defocus

+ 2 RE

Defocus

+ 1 RE

Defocus

- 1 RE

Defocussed Perfect Psf

� Perfect point spread function with defocus

� Representation with constant energy: extreme large dynamic changes

∆∆∆∆z = -2RE ∆∆∆∆z = +2RE∆∆∆∆z = -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

Psf with Aberrations

� Zernike coefficients c in λ

� Spherical aberration,

Circular symmetry

� Astigmatism,

Split of two azimuths

� Coma,

Asymmetric

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 +=

Definitions of Fourier Optics

� Phase space with spatial coordinate x and

1. angle θ

2. spatial frequency ν in mm-1

3. transverse wavenumber kx

� Fourier spectrum

corresponds to a plane wave expansion

� Diffraction at a grating with period g:

deviation angle of first diffraction order varies linear with ν = 1/g

0k

kv x

xx =⋅= λθ

[ ]),(ˆ),( yxEFvvA yx =

( ) ( )A k k z E x y z e dx dyx y

i xk ykx y, , ( , , )=

− +

∫∫

vk π2=

vg

⋅=⋅= λλθ1

sin

� Arbitrary object expaneded into a spatial

frequency spectrum by Fourier

transform

� Every frequency component is

considered separately

� To resolve a spatial detail, at least

two orders must be supported by the

system

NAg

mg

λ

α

λ

λα

==

⋅=⋅

sin

sin

off-axis illumination

NAg

⋅=

2

λ

Ref: M. Kempe

Grating Diffraction and Resolution

optical

system

object

diffracted orders

a)

resolved

b) not

resolved

+1.

+1.

+2.

+2.

0.

-2.

-1.

0.

-2.

-1.

incident

light

� A structure of the object is resolved, if the first diffraction order is propagated

through the optical imaging system

� The fidelity of the image increases with the number of propagated diffracted orders

0. / 1. order

0. / 1. / 2.

order

0. / 1. / 2. / 3. order

Resolution: Number of Supported Orders

Abbe Theorie of the Microscopic Resolution

� Diffraction of the illumination wave at the object structure

� Occurence of the diffraction orders in the pupil

� Image generation by constructive interference of the supported orders

� Object details with high spatial frequency are blocked by the system aperture and can not

be resolved

-1

1

object planesource

pupil plane

image plane

0

imaginglens

Ref: W. Singer

� Imaging of a crossed grating object

� Spatial frequency filtering by a slit:

� Case 1:

- pupil open

- Cross grating imaged

� Case 2:

- truncation of the pupil by a split

- only one direction of the grating is

resolved

Fourier Filtering

� Improved resolution by oblique illumination in microscopy

� Enhancement only in one direction

Oblique Illumination in Microscopy

Fourier Optical Fundamentals

� Helmholtz wave equation:

Propagation with Green‘s function g,

Amplitude transfer function, impulse response

� For shift-invariance:

convolution

� Green‘s function of a spherical wave

Fresnel approximation

� Calculation in frequency space: product

� Optical systems:

Impulse response g(x,y) is coherent transfer function, point spread function (PSF).

G(νx,νy) corresponds to the complex pupil function

� Fourier transform: corresponds to a plane wave expansion

),(*),','(),','( yxEzyyxxgzyxE −−=

g r rr r

ei k r r

( , ' )'

'r rr r

r r

=−

− −1

4π

g x y zi

ze e

ikz

i

zx y

( , , )( )

= −− +

λ

π

λ2 2

),(),,(),,( yxyxyx vvEzvvGzvE ⋅=ν

)(22

),,( yx vvziikz

yx ezvvG+⋅+−

=λπ

dydxyxEyxyxgzyxE ∫ ∫∞

∞−

∞

∞−

⋅= ),(),,','(),','(

( )f

yxyxP

pp

pp⋅

+⋅−=

λ

π 22

),(

f

yv

f

xv

p

y

p

xλλ

== ,

Fourier Optical Field Propagation through a Lens

� Lens works as a pupil filter,

Phase:

Complex pupil function, transmission t

� Incident plane wave

Field in focal plane,

Far field, Fourier transform

� Image point coordinates correspond to the spatial frequencies

( )( )

∫∫+−

+

⋅⋅= pp

yyxxf

i

pp

yxf

i

dydxeyxAfi

efyxE

pp ''2

''

),(),','(

22

λ

πλ

π

λ

),(),(),( pp yxiP

pppp eyxtyxA ⋅=

Fourier Optics – Point Spread Function

� Optical system with magnification m

Pupil function P,

Pupil coordinates xp,yp

� PSF is Fourier transform

of the pupil function

(scaled coordinates)

( ) ( ) ( )[ ]∫∫

−⋅+−⋅⋅−

⋅⋅= pp

myyymxxxz

ik

pppsf dydxeyxPNyxyxgpp ''

,)',',,(

( )[ ]pppsf yxPFNyxg ,ˆ),( ⋅=

object

planeimage

plane

source

point

point

image

distribution

Fourier Theory of Coherent Image Formation

� Transfer of an extended

object distribution I(x,y)

� In the case of shift invariance

(isoplanasie):

coherent convolution of fields

� Complex fields are

additive

( )∫∫ ⋅= dydxyxEyxyxgyxE psf ),(',',,)','(

( )∫∫ ⋅−−= dydxyxEyyxxgyxE psf ),(',')','(

( ) ),(,)','( yxEyxgyxE psf ∗=

Fourier Theory of Coherent Image Formation

Fourier Theory of Incoherent Image Formation

objectintensity image

intensity

single

psf

object

planeimage

plane

� Transfer of an extended

object distribution I(x,y)

� In the case of shift invariance

(isoplanasie):

incoherent convolution

� Intensities are additive

dydxyxIyyxxgyxI psfinc ∫ ∫∞

∞−

∞

∞−

⋅−−= ),()','()','(2

),(*),()','( yxIyxIyxI objpsfimage =

dydxyxIyyxxgyxI psfinc ∫ ∫∞

∞−

∞

∞−

⋅= ),(),',,'()','(2

Fourier Theory of Incoherent Image Formation

objectintensity

I(x,y)

squared PSF,intensity-responseIpsf

(xp,y

p)

imageintensityI'(x',y')

convolution

result

object

intensityspectrum

I(vx,v

y)

opticaltransferfunction

HOTF

(vx,v

y)

imageintensityspectrumI'(v

x',v

y')

produkt

result

Fouriertransform

Fouriertransform

Fouriertransform

Comparison Coherent – Incoherent Image Formation

object

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

incoherent coherent

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

bars resolved bars not resolved bars resolved bars not resolved

Incoherent Image Formation in Frequency Space

� Incoherent illumination:

No correlation between neighbouring object points

Superposition of intensity in the image

� In the case of shift invariance

(isoplanasie):

Incoherent imaging with convolution

� In frequency space:

Product of spectra, linear transfer theory

The spectrum of the psf works as low pass filter onto the object spectrum

Optical transfer function

dydxyxIyyxxgyxI psfinc ∫ ∫∞

∞−

∞

∞−

⋅−−= ),()','()','(2

),(*),()','( yxIyxIyxI objpsfimage =

dydxyxIyyxxgyxI psfinc ∫ ∫∞

∞−

∞

∞−

⋅= ),(),',,'()','(2

[ ]),(),( yxIFTvvH PSFyxotf =

),(),(),( yxobjyxotfyximage vvIvvHvvI ⋅=

( )

pppp

pp

vyvxi

pp

yxOTF

dydxyxg

dydxeyxg

vvH

ypxp

∫ ∫

∫ ∫∞

∞−

∞

∞−

∞

∞−

∞

∞−

+−⋅

=2

22

),(

),(

),(

π

[ ]),(ˆ),( yxIFvvH PSFyxOTF =

pppp

pp

y

px

p

y

px

p

yxOTF

dydxyxP

dydxvf

yvf

xPvf

yvf

xP

vvH

∫ ∫

∫ ∫∞

∞−

∞

∞−

∞

∞−

∞

∞−

−−⋅++

=2

*

),(

)2

,2

()2

,2

(

),(

λλλλ

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

xp

y p

area of

integration

shifted pupil

areas

f x

yf

p

q

x

y

x

y

L

L

x

y

o

o

x'

y'

p

p

light

source

condenserconjugate 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

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

λλ

'sin u

f

avo ==

λλ

'sin222 0

un

f

navvG ===

−

−

=

2

000 21

22arccos

2)(

v

v

v

v

v

vvH MTF

π

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH ⋅=

∆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

decreasedslope

decreased

minima

increased

� 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

ψψψψ

y

tangential

plane

tangential sagittal

arbitrary

rotated

x sagittalplane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

0 0.5 1

νννν / ννννmax

� Circular disc with diameter

D = d x Dairy

� Small d << 1 : Airy disc

� Increasing d:

Diffraction ripple disappear

Incoherent Image of a Circular Disc

Outlook

Next lecture: Part 10 – Performance Criteria I

Date: Wednesday, 2012-06-20

Contents: 10.1 Introduction

10.2 Geometrical aberrations

10.3 Wave aberrations

10.4 Point spread function and point resolution

10.5 Depth of focus