www.iap.uni-jena.de

Medical Photonics Lecture

Optical Engineering

Lecture 7: Image Quality

2017-12-07

Herbert Gross

Winter term 2017

2

Schedule Optical Engineering 2017

No Subject Ref Date Detailed Content

1 Introduction Gross 19.10. Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Gross 02.11. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Gross 09.11. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 16.11. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Gross 23.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Gross 30.11. Introduction, primary aberrations, miscellaneous 7 Image quality Gross 07.12. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 14.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 21.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 11.01. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Optic design Gross 18.01. Aberration correction, system layouts, optimization, realization aspects

12 Photometry Gross 25.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

13 Illumination systems Gross 01.02. Light sources, basic systems, quality criteria, nonsequential raytrace

14 Metrology Gross 08.02. Measurement of basic parameters, quality measurements

3

Performance Criteria Overview

Applications

Geometrical

model

Diffraction

model

Longitudinal

aberrations

Transverse

aberration curves

Spot diagrams

Wave aberrations

AdvantagesQuantitative

numbers

scaling on Rayleigh unit

scaling on Airy diameter

rms

pv

scaling on Airy diameters

rms

pv

Zernike decomposition

RepresentationsLimitations

Problems

astigmatism

axial chromatical

field curvature

not useful in the field

not defined for afocal

camera lensessimple direct analysis

possible

1 curve per field point

to be re-defined for afocal

any illustrative analysis complicated

any

direct measurable

scaling on wavelength

all orders separated

only one wavelength

only one field point

normalization radius of Zernikes

Point spread

function

Modulation transfer

function

Strehl ratio

scaling on Airy diameter

Hopkins number

microscopy

astronomy

diffraction limited

direct relation to resolution

easy white light formulation

computational problems for large

aberrations

camera lenses

lithography

projection lenses

direct analysis possible

easy white light formulation

computational problems for large

aberrations

analysis complicated

Spot Diagram

Table with various values of:

1. Field size

2. Color

Small circle:

Airy diameter for

comparison

Large circle:

Gaussian moment

486 nm 546 nm 656 nm

axis

fieldzone

fullfield

4

Gaussian Moment Spot

Spot pattern with transverse aberrations xj and yj

1. centroid

2. 2nd order moment

3. diameter

Generalized:

Rays with weighting factor gj:

corresponds to apodization

Worst case estimation:

size of surrounding rectangle Dx=2xmax, Dy = 2ymax

xN

xS jj

1

yN

yS jj

1

M rN

x x y yG j S j Sj

22 21

rmsG rMD 22

M rN

g x x y yG

G

j j S j S

j

22 21

5

Wave Aberration

Definition of the peak valley value WPV

Reference sphere corresponds to perfect imaging

Rms-value is more relevant for performance evaluation

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

6

Wave Aberration Criteria

Mean quadratic wave deviation ( WRms , root mean square )

with pupil area

Peak valley value Wpv : largest difference

General case with apodization:

weighting of local phase errors with intensity, relevance for psf formation

dydxAExP

ppppmeanpp

ExP

rms dydxyxWyxWA

WWW222 ,,

1

pppppv yxWyxWW ,,max minmax

pppp

w

meanppppExPw

ExP

rms dydxyxWyxWyxIA

W2)(

)(,,,

1

7

8

PSD Ranges

Typical impact of spatial frequency

ranges on PSF

Low frequencies:

loss of resolution

classical Zernike range

High frequencies:

Loss of contrast

statistical

Large angle scattering

Mif spatial frequencies:

complicated, often structured

fals light distributions

log A2

Four

low spatial

frequency

figure errormid

frequency

range micro roughness

1/

oscillation of the

polishing machine,

turning ripple

10/D1/D 50/D

larger deviations in K-

correlation approach

ideal

PSF

loss of

resolution

loss of

contrast

large

angle

scattering

special

effects

often

regular

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

4

PVW

Rayleigh Criterion

The Rayleigh criterion

gives individual maximum aberrations

coefficients,

depends on the form of the wave

Examples:

aberration type coefficient

defocus Seidel 25.020 a

defocus Zernike 125.020 c

spherical aberration

Seidel 25.040 a

spherical aberration

Zernike 167.040 c

astigmatism Seidel 25.022 a

astigmatism Zernike 125.022 c

coma Seidel 125.031 a

coma Zernike 125.031 c

11

a) optimal constructive interference

b) reduced constructive interference

due to phase aberrations

c) reduced effect of phase error

by apodization and lower

energetic weighting

d) start of destructive interference

for 90° or /4 phase aberration

begin of negative z-component

Rayleigh criterion:

1. maximum of wave aberration: Wpv < /4

2. beginning of destructive interference of partial waves

3. limit for being diffraction limited (definition)

4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil

5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...)

Marechal criterion:

1. Rayleigh crierion corresponds to Wrms < /14 in case of defocus

2. generalization of Wrms < /14 for all shapes of wave fronts

3. corresponds to Strehl ratio Ds > 0.80 (in case of defocus)

4. more useful as PV-criterion of Rayleigh

Criteria of Rayleigh and Marechal

14856.13192

Rayleigh

rmsW

12

Impression of CHV in real images

Typical colored fringes blue/red at edges visible

Color sequence depends on sign of CHV

Chromatic Variation of Magnification

13

Axial Chromatical Aberration

Special effects near black-white edges

boarder

magenta

blue boarder

Ref: J. Kaltenbach

14

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

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

17

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

Decrease of intensity onto 80%:

Scaling measure: Rayleigh length

- depth of focus: 1RE

- Gaussian beams: similar formula

22

'

'sin' NA

n

unRE

Depth of Focus: Diffraction Consideration

2

0

sin)(

u

uIuI

20' on

z

Ediff 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

21

Depth of Focus

Highly unrealistic sharpness distribution (computer animation)

Ref.: V. Blahnik

Criteria for measuring the degradation of the point spread function:

1. Strehl ratio

2. Standard deviation

3. Full width half maximum (FWHM)

4. Second moment

5. Correlation with perfect PSF

6. Various energy-based widths

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%

22

The PSF is very sensitive to coma

In a well constructed system, 5-7 diffraction rings are observable by visual inspection

In the case of coma, the asymmetry of the pattern is particularly sensitive

The 1st diffraction ring is visibly influenced by a Zernike coefficient as small as /30

c31 = 0.03 c31 = 0.06 c31 = 0.09 c31 = 0.15

Point Spread Function for Coma Aberration

Differences

between Strehl

and Ipeak,

if the profile

is structured

24

Strehl Ratio and PSF-Peak Height for Aberrations

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

coma c8

astigmatism c5

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

c4 [] c5 []

c9 [] c8 []

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

defocussing c4

Strehl peak

0 0.5 1 1.5 2

spherical aberration c9

0

0.2

0.4

0.6

0.8

1

Incoherent Image Formation

astigmatism comaspherical

aberrationobject ideal

PSF

Example:

incoherent imaging of pattern near the resolution limit with aberrations

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

Encircled Energy

Relative amount of energy passing a variable stop:

- encircled energy

- power in the bucket

General formulation:

Special case of circular symmetry:

E r I x x y y dx dyS S

x r y

x r y

y r

y r

( ) ,

2 2

2 2

E r I r r dr

r

( ) ( ) 20

Encircled Energy Function

The encircled energy function shows structured behavior in case of aberrations

Larger sensitivity for small intensity levels (sidelobes)

Problem: Reference in case of apodization or central obscuration

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ecirc

(r)

r / rAiry

ideal

spherical / 4

ring with

= 0.3

coma / 4

)(

)()(

rE

rErEcirc

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

29

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

30

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

31

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

32

Incoherent Resolution: Dependence on NA

Microscopical resolution as a function of the numerical aperture

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

33

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

34

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 sinusoidal 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

Resolution/contrast criterion:

Ratio of contrasts with/without aberrations for one selected spatial frequency

Real systems:

Choice of several application relevant

frequencies

e.g. photographic lens:

10 Lp/mm, 20 Lp/mm, 40 Lp/mm

Hopkins Factor

)(

)()(

)(

)(

vg

vgvg

ideal

MTF

real

MTFMTF gMTF

ideal

real

gMTF

real

gMTF

ideal

1

0.5

0

36

Consideration of the complete area under the MTF curve in the relevant interval

of spatial frequencies

In anisotropic systems:

volume under MTF-surface

Quite good correlation with visual

perception for visual systems

21

)(vvv

MTFMTFa dvvHKA

MTF-Area-Criterion

gMTF

1

AMTFa

2

37

Photographic lenses with different performance

38

Modulation Transfer Function

10 c/mm

20 c/mm

40 c/mm

Objektiv 1 f/ 3.5 Objektiv 2

000 0000

10

20

30

40

50

60

70

80

90

100

-25 -20 -15 -10 -5 0 5 10 15 20 25

max. MTF Bildhöhe [mm] max. MTF

MT

F [

%]

b

ei 1

0 , 2

0 , 4

0 L

p/m

m ....... ta

n _

__

sa

g

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lens 1 f/3.5 Lens 2

Image height

Microscope Resolution with Immersion

Imaging of a Chromium mask with 125 nm pitch

Imaging without / with water immersion

Enhancement of resolution and contrast

Ref: W. Osten

150x/0.9 air 200x/1.2 water immersion Lens (Leica)

dydxyxE

dydxyxExx

m

m

2

2

),(

),(

ddE

ddEm

m

2

2

),(

),(

o

xxxx

k

kvu

Quality of Laser Beams: Moments

Conventional criteria of imaging systems are nor useful for laser beams:

1. significant apodization

2. no imaging application

3. status of coherence may be complicated

Description of the complex fields by moments of second order:

1. spatial moments of intensity profile

second moments describes beam width

third moment describes asymmetry

2. angular moment of the direction distribution

second moment describes the divergence

Alternative descriptions of impuls:

1. angle ux

2. spatial frequency x

3. transverse wavenumer kx

Mixed moments: description of twist effects

2222

xxxxx wwM

M wx ox x

2

Quality of Laser Beams: M2

Characterizing beam quality

M2

Special case: definition in waist plane

Properties of M2:

1. Gaussian beam TEM00: M2 = 1

Smallest possible value

2. Paraxial optical systems: M2 remains constant for propagation

3. Real beams: M2 > 1 describes the decrease in quality and focussability

relative to a gaussian beam

Reasons for degradation of beam quality:

1. intensity profile

2. phase perturbation

3. finite degree of coherence

Incoherent mixture of modes: additive composition of M2

General beams: components and mixed terms 4442

2

1xyyx MMMM

a) object

Image quality with Real Objects

b) good image c) defocussed d) axial chromatic

aberration

e) lateral chromatic

aberration

g) chromatical

astigmatism

f) sphero-

chromatism

Real Image with Different Chromatical Aberrations

original object good image color astigmatism 2

6% lateral color axial color 4

USAF Test Target

0 1

10

2

3

4

5

6

6

5

4

3

2

1

6

5

4

3

2

2 31

2

3

2

4

5

6

Siemens Star Test Plate

Color Printer Test Image

Testchart Visual Acuity

Snellen test chart

Energy Transmission of Microscope Lens

Reduced throughput due to

1. absorption

2. coatings

Strong spectral dependence

T [%]

100

80

60

40

20

0

1 3 5 7 9 11 13surfaces

365 nm

546 nm

400 nm440 nm700 nm644 nm600 nm

absorption

Simple model:

Finite residual reflectivity R at N surfaces

Considering energetic transfer

from signal to false light level,

Multiple reflecting light taken

into account

False light intensity:

Effect of False Light on SNR

NF RRN

RI

1

)1(1

1

I

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

R = 0.995

R = 0.99

R = 0.98

R = 0.97

R = 0.95

N

Transmission

Signal

False light

decreasing

signal

(1-R = 2%)

increasing

false light

Achromate

Residual aberrations of an achromate

Clearly seen:

1. Distortion

2. Chromatical magnification

3. Astigmatism

51

Different reasons

Various distributions

Straylight and Ghost Images

a b

Contrast and Resolution

original 256 x 256 blurr 3 pixel blurr 6 pixel blurr 9 pixel

original straylight 15% straylight 30% straylight 50%

Image Contrast

Image processing: contrast enhancement

Ref: T. Sievers

SSim Image Quality Metric

Idea : combined criterion with best correlation to subjective performance

Best modelling of human visual, cortical and neuronal perception system

SSim = structural similarity index measure

Three major aspects taken into account with weighted superposition:

1. Luminence / brightness I(x,y)

2. Contrast C(x,y)

3. Structures / information S(x,y)

Mathematical definition: three autocorrelation-terms with adaptable

parameters Cj:

Experience:

Good correlation with subjective judgement of test persons (visual perception)

Extended version: complex SSim

Better consideration of slight image shift / rotation

3

3

2

22

2

1

22

1 22

),(),(),(),(

C

C

C

C

C

C

yxsyxcyxlyxS

yx

yx

yx

yx

yx

yx

SSim Image Quality Metric

Examples of

SSim-values: a) reference

b) contrast

c) luminance

d) white noise

e) impulsive noise

f) JPEG compression

g) Blurr

h) zooming

i) shift right

j) shift left

k) rotation counter cw

l) rotation clockwise

Ref: Wang / Bovik

SSim - Comparison to MSE-Metric

Classical MSE-metric (rms difference to reference image):

Fails for quite simple degradations

Better significance by SSim

starting

point

optimization

with SSim

Ref: Wang / Bovik

SSim - Comparison to MSE

SSim:

- Can be defined as field

(not a single number)

- Indicates local differences to

the reference image

blurredoriginal

MSE SSim

Ref: Wang / Bovik

59

Beautiful and Ugly Images