www.iap.uni-jena.de

Medical Photonics Lecture

Optical Engineering

Lecture 3: Diffraction

2019-04-24

Herbert Gross

Speaker: Yi Zhong

Summer term 2019

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Zhong 10.04.Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Zhong 17.04.Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Zhong 24.04.Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 08.05. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Zhong 15.05.Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous

7 Image quality Zhong 29.05. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 05.06.Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 12.06.Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 19.06.Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Photometry Zhong 26.06.Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

12 Illumination systems Gross 03.07.Light sources, basic systems, quality criteria, nonsequential raytrace

13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements

▪ Diffraction at an edge in Fresnel

approximation

▪ Intensity distribution,

Fresnel integrals C(x) and S(x)

scaled argument

▪ Intensity:

- at the geometrical shadow edge: 0.25

- shadow region: smooth profile

- bright region: oscillations

−+

−=

22

)(2

1)(

2

1

2

1)( tStCtI

FNxz

xz

kt 2

2=

=

=

Fresnel Edge Diffraction

t-4 -2 0 2 4 6

0

0.5

1

1.5

I(t)

3

▪ Diffraction at a slit:

intensity distribution

▪ Angles with side maxima:

constructive interference

▪ Diffraction pattern

d

0. order

1st diffraction

order

2

sin

sinsin

)(

=

d

d

I

+=

2

1sin max m

d

Fraunhofer Farfield Diffraction

order: -3rd -2nd -1st 0th +1st +2nd +3rd

4

Fraunhofer Diffraction at a Rectangle

▪ Diffraction at a square / rectangle shaped aperture

▪ Smaller width: larger diffraction angle

square aperture rectangular aperture

5

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

6

Huygens Principle

wave fronts

Huygens-Wavelets

propagation direction

stop

geometrical shadow

7

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

8

▪ Typical change of the intensity profile

▪ Normalized coordinates

▪ Diffraction integral

9

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

▪ Focussed beam with various wavelengths

▪ Focal region given by Rayleigh length Ru = /NA2

▪ Geometrical cone outside focal region

▪ Diffraction structure in focal range

10

Geometrical vs Diffraction Ranges

= 1 mm = 2.5 mm = 5 mm = 10 mmA

1/2

geometrical

range

diffraction

focal range

▪ The LOmmel approach is valid for large Fresnel numbers

and takes the small NF-effects not into account

▪ Typical intensity distribution in logarithmic representation

11

Fresnel Diffraction

u

I(u,0)

I(u,0)

v

Log I(u,0)

a

a

zau 2;2

32

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 −

−=−

12

( )( )

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

=

13

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

14

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%

15

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

16

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%

17

▪ Normalized axial intensity

for uniform pupil amplitude

▪ Decrease of intensity onto 80%:

▪ Scaling measure: Rayleigh length

- wave 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

18

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

19

( )( )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 )

20

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

2. Gaussian

3. 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

21

▪ 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 m

m

= 1 m

550 nm

1

0

-1

-2-2-3 -1 0

Focussing by a Lens: Diffraction and Aberration

22

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 %

23

▪ Growing spherical aberration shows an asymmetric behavior around the nominal image

plane for defocussing

24

Caustic with Spherical aberration

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

Low Fresnel Number Focussing

25

▪ 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

Low Fresnel Number Focussing

26

▪ 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)

▪ Diffraction at an edge in Fresnel

approximation

▪ Intensity distribution,

Fresnel integrals C(x) and S(x)

scaled argument

▪ Intensity:

- at the geometrical shadow edge: 0.25

- shadow region: smooth profile

- bright region: oscillations

−+

−=

22

)(2

1)(

2

1

2

1)( tStCtI

FNxz

xz

kt 2

2=

=

=

Fresnel Edge Diffraction

t-4 -2 0 2 4 6

0

0.5

1

1.5

I(t)

27

▪ Line image: integral over point sptread function

LSF: line spread function

▪ Realization: narrow slit

convolution of slit width

▪ But with deconvolution, the PSF can be reconstructed

?= dyyxIxI PSFLSF ),()(

Integration

intens

ity

x

Line spread function

PSF

= dyyxIxI PSFLSF ),()(

Line Image

28

ESF, PSF and ESF-Gradient

▪ Typical behavior of intensity of an edge image for residual aberrations

▪ The width of the distribution roughly corresponds to the diameter of the PSF

▪ Derivative of the edge spread function:

edge position at peak

location

y'0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1 -0.05 0 0.05 0.1 0.15

derivation of

edge spread

function

edge spread

function

point spread

function

29

diffracted ray

direction

object

structure

g = 1 /

= / g

k

kT

Definitions of Fourier Optics

▪ Phase space with spatial coordinate x and

1. angle

2. spatial frequency 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 = 1/g

0k

kv x

xx ==

),(ˆ),( yxEFvvA yx =

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

i xk ykx y, , ( , , )=− +

vk 2=

vg

== 1

sin

30

▪ 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

==

=

sin

sin

off-axis

illumination

NAg

=

2

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

+1.

0.

+2.

+1.

0.

+2.

-2.

-2. -1.

-1.

+3.+3.

31

Number of Supported Orders

▪ 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. / -1. order

0. / +1. / -1.

+2. / -2.

order

0. / +1. -1. / +2. /

-2. / +3. / -3.

order

32

▪ Location of diffraction orders in the

back focal plane

▪ Increasing grating angle with spatial

frequency of the grating

33

Abbe Theory of Microscopic Resolution

+1st

-1st

+1st

-1st

+1st

-1st

objectback focal

planeobjective

lens

0th order

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

34

( )

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

35

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

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

'sin u

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

▪ 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

39

▪ 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

40

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

▪ Degradation due to

1. loss of contrast

2. loss of resolution

41

Modulation Transfer

Convolution of the object intensity distribution I(x) changes:

1. Peaks are reduced

2. Minima are raised

3. Steep slopes are declined

4. Contrast is decreased

I(x)

x

original image

high resolving image

low resolving image

Imax-Imin

42

Test: Siemens Star

Determination of resolution and contrast

with Siemens star test chart:

▪ Central segments b/w

▪ Growing spatial frequency towards the

center

▪ Gray ring zones: contrast zero

▪ Calibrating spatial feature size by radial

diameter

▪ Nested gray rings with finite contrast

in between:

contrast reversal, pseudo resolution

43

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

44

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

45

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

(isoplanasy):

incoherent convolution

▪ Intensities are additive

dydxyxIyyxxgyxI psfinc

−

−

−−= ),()','()','(2

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

dydxyxIyyxxgyxI psfinc

−

−

= ),(),',,'()','(2

46

Fourier Theory of Incoherent Image Formation

object

intensity

I(x,y)

squared PSF,

intensity-

response

Ipsf

(xp,y

p)

image

intensity

I'(x',y')

convolution

result

object

intensity

spectrum

I(vx,v

y)

optical

transfer

function

HOTF

(vx,v

y)

image

intensity

spectrum

I'(vx',v

y')

produkt

result

Fourier

transform

Fourier

transform

Fourier

transform

47

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

-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

resolved not resolved▪ Example:

incoherent imaging of bar pattern near the

resolution limit

48

▪ Blurred imaging:

- limiting case

- information extractable

▪ Blurred imaging:

- information is lost

- what‘s the time ?

Resolution: Loss of Information

49

▪ Typical Psf of the human eye

▪ Change with pupil diameter size:

D = 2 mm : diffraction limited, smallest value

D < 2 mm : broadened due to diffraction

D > 2 mm : broadened due to aberrations (spherical)

50

Human Eye Performance

Ref. G. Yoon

Testchart Acuity of the Eye

Testchart with reduced resolution

51

Visual Acuity

▪ Recognition of simple geometrical

shapes :

1. Landolt ring with gap

2. Letter 'E'

▪ Blur of image on retina with distance

a)

5a

a

a

3a

3a

a

3a

b)

distance 6.096 m

block

letter

E 8.9

mm

image

height

25 mmeye

original blur : a/2 blur : a blur : 2a blur : 3a blur : 4a

3a

52