www.iap.uni-jena.de

Medical Photonics Lecture

Optical Engineering

Lecture 6: Diffraction

2016-12-01

Herbert Gross

Winter term 2016

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Gross 20.10. Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Gross 03.11. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Components Kempe 10.11. Lenses, micro-optics, mirrors, prisms, gratings

4 Optical systems Gross 17.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

5 Aberrations Gross 24.11. Introduction, primary aberrations, miscellaneous

6 Diffraction Gross 01.12. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

7 Image quality Gross 08.12. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 15.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 22.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 05.01. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Optic design Gross 12.01. Aberration correction, system layouts, optimization, realization aspects

12 Photometry Gross 19.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

13 Illumination systems Gross 26.01. Light sources, basic systems, quality criteria, nonsequential raytrace

14 Metrology Gross 02.02. 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)

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

Fraunhofer Diffraction at a Rectangle

Diffraction at a square / rectangle shaped aperture

Smaller width: larger diffraction angle

square aperture rectangular aperture

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

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

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

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

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

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%

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

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%

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

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

15

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 )

16

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 for larger radii

Encircled energy:

same behavior

Complicated: Definition of compactness of the central peak: 1. FWHM: Airy more compact as Gauss Bessel more compact as Airy

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

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

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 %

Growing spherical aberration shows an asymmetric behavior around the nominal image

plane for defocussing

20

Caustic with Spherical aberration

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

Low Fresnel Number Focussing

21

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

22

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)

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

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

25

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 dxdyx y

i xk ykx y, , ( , , )

vk 2

vg

1

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

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.

27

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

Location of diffraction orders in the

back focal plane

Increasing grating angle with spatial

frequency of the grating

29

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

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

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

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

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

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

35

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

36

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

Degradation due to

1. loss of contrast

2. loss of resolution

37

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

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

39

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

40

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 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

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

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

Blurred imaging:

- limiting case

- information extractable

Blurred imaging:

- information is lost

- what‘s the time ?

Resolution: Loss of Information

45