Design and Correction of optical Systems

Part 11: Performance criteria 2

Summer term 2012Herbert Gross

1

Overview

1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Correction of aberrations 1 2012-07-0413. Correction of aberrations 2 2012-07-1114. Optical system classification 2012-07-18

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Contents

11.1 Modulation transfer function11.2 Contrast vs resolution11.3 Special criteria11.4 Field dependence of aberrations11.5 Best focussing11.6 Measurement of image performance11.7 Miscellaneous

3

Contrast

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

The MTF-value corresponds to the intensity contrast of an imaged sin grating Visibility

The maximum value of the intensityis not identical to the contrast valuesince the minimal value is finite too

Concrete values:

minmax

minmax

IIIIV

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

peakdecreased

slopedecreased

minimaincreased

4

MTF 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

favo

'sin222 0

unf

navvG

2

000 21

22arccos2)(

vv

vv

vvvHMTF

),(),(),( yxPTF vvHiyxMTFyxOTF evvHvvH

5

MTF and Contrast

Object

Contrast

Object spectrum

Image spectrum

Image

)2cos()( 0xvacxIobj

ca

acacacac

IIIIV

)()()()(

minmax

minmax

00 22)0()(ˆ)( vvavvavcxIFvI objobj

)()()( vHvIvI MTFobjima

)2cos()(

)(2

)(2

)0(

)(2

)(2

)0()(ˆ

)()(ˆ)(ˆ)'(

00

20

20

001

11

00

xvvHac

evHaevHaHc

vvvHavvvHavvHcF

vHvIFvIFxI

MTF

xivMTF

xivMTFMTF

MTFMTFMTF

MTFobjimaima

6

Real MTF of system with residual aberrations:1. contrast decreases with defocus2. higher spatial frequencies have stronger decrease

Real MTF

z = 0

z = 0.1 Ru

gMTF1

0.75

0.25

0.5

0

-0.250 0.2 0.4 0.6 0.8 1

z = 0.2 Ruz = 0.3 Ru

z = 1.0 Ruz = 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 inRU

max = 0.05max = 0.1max = 0.2max = 0.3max = 0.4max = 0.5max = 0.6max = 0.7max = 0.8

Zernikecoefficients:c5 = 0.02c7 = 0.025c8 = 0.03c9 = 0.05

7

MTF-Curve sagittal / tangential

Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on theazimuthal orientation of the object structure

Generally, two MTF curves are considered

y

tangentialplane

tangential sagittal

arbitraryrotated

x sagittalplane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

0 0.5 1 / max

8

Resolution Estimation with Test Charts

0 1

10

2

3

4

5

6

65

4

3

2

1

6

5

4

3

22 3

1

2

3

2

456

Measurement of resolution with test charts: bar pattern of different sizes two different orientations calibrated size/spatial frequency

9

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

10

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

11

Resolution/contrast criterion:Ratio of contrasts with/without aberrations for one selected spatial frequency

Real systems:Choice of several application relevant frequenciese.g. photographic lens:10 Lp/mm, 20 Lp/mm, 40 Lp/mm

Hopkins Factor

)()()( )(

)(

vgvgvg ideal

MTF

realMTF

MTF

12

Consideration of the complete area under the MTF curve in teh relevant intervalof spatial frequencies

In anisotropic systems:volume under MTF-surface

Quite good correlation with visual perception for visual systems

21

)(vvv

MTFMTFa dvvHKA

MTF-Area-Criteriom

13

Balance between contrast and resolution: not trivial Optimum depends on application Receiver: minimum contrast curve serves as real reference

Most detector needs higher contrast to resolve high frequenciesCSF: contrast sensitivity function

Contrast vs Resolution

gMTF

1 : high contrast

2 :high resolution

threshold contrast a :2 is better

threshold contrast b :1 is better

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Contrast and Resolution

High frequent structures :contrast reduced

Low frequent structures:resolution reduced

contrast

resolution

brillant

sharpblurred

milky

15

Contrast vs contrast as a function of spatial frequency

Typical: contrast reduced forincreasing frequency

Compromise between resolution and visibilty is not trivial and depends on application

Contrast and Resolution

16

Blurred imaging:- limiting case- information extractable

Blurred imaging:- information is lost- what‘s the time ?

Resolution: Loss of Information

17

Contrast / Resolution of Real Images

Degradation due to1. loss of contrast2. loss of resolution

18

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

19

Structural content:Contrast capacity of an image

Equal Strehl ratio DS = 0.8but different structural content:

0.79 - 0.73 - 0.70 - 0.69

Structural Content

2

2

2

2

),(

),(

),(

),(

yxyxotf

yxyxotf

image

image

dvdvvvg

dvdvvvg

dydxyxI

dydxyxIS

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

gMTF()

idealc20 = 0.129 c40 = 0.168 c60 = 0.202 c80 = 0.234

20

Aberrations of a single lens

Single plane-convex lens, BK7, f = 100 mm, = 500 nm

Spot as a function of field position

Coma orientation towards theaxis

x

y

21

x

y

PSF as a functionof the field height

Orientation of comain the field

Variation of Performance with Field Position

x = 0 x = 20% x = 40% x = 60 % x = 80 % x = 100 %

22

Strehl ratio:1. Photo objective lens: strong dependence2. Microscope objective lens: weak dependence

Variation of Performance with Field Position

DS

xfield0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.0

micro objective

468 nm656 nm

587 nm

DS

xfield

0 0.2 0.4 0.6 0.8 1.0

photo objective

468 nm656 nm

587 nm

0

0.2

0.4

0.6

0.8

1.0

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Specifications

1. Functional specificationRelated to application, valid for the complete system. Plays the major role of a basic agreement between customer and development.Here two levels must be distinguished:1.1 Basic data1.2 Image quality

2. Manufacturing specificationDeclaration of several points of the engineering approach to clearify the technological and business goals2.1 Business specification to ensure an economic development2.2 Engineering and technology specifications, especially interface data

3. Component specificationsWarranaty of reacing the theoretical goals in practice. Main concept of, all complicated relationships and dependencies of subsystems and components.

4. Assembly specificationAll parts of assembly and interfaces between mechanical and optical components Signal and image processing software, electronical and digital recording systems

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Hartmann Shack Wavefront Sensor

detector

xDmeas

u

h f

array

spotoffset

Darray

Dsub

refractiveindex n

wave-front

Lenslet array divides the wavefront into subapertures Every lenslet generates a simgle spot in the focal plane The averaged local tilt produces a transverse offset of the spot center Integration of the derivative matrix delivers the wave front W(x,y)

25

Spot Pattern of a HS - WFS

a) spherical aberration b) coma c) trefoil aberration

Aberrations produce a distorted spot pattern Calibration of the setup for intrinsic residual errors Problem: correspondence of the spots to the subapertures

Typical setup for component testing

Lenslet array

Hartmann Shack Wavefront Sensor

subaperture

point spread function

2-dimensional lenslet array

a) b)

test surface

telescope for adjust-ment of the diameter

detector

collimator

lensletarray

fiberillumination

beam-splitter

27

Array Signal

array of phase spot pattern cross section of the spot pattern

original discretized and quantized with noise

Lenslet arrayideal signal

Real signal:1. discretization2. quantization3. noise

28

test surface

beamsplitter

reference surfacehere: flat

illumination

to detector

path difference

mRrm

Test by Newton Fringes

Reference surface and test surface with nearly the same radii Interference in the air gap Reference flat or curved possible Corresponds to Fizeau setup

with contact Broad application in simple

optical shop test Radii of fringes

29

Testing with Fizeau Interferometer

Long common path, quite insensitive setup Autocollimating Fizeau surface quite near to test surface, short cavity length Imaging of test surface on detector Straylight stop to bloc unwanted light Curved test surface: auxiliary objective lens (aplanatic, double path) Highest accuracy

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Testing with Twyman-Green Interferometer

detector

objective lens

beam splitter 1. mode:

lens tested in transmissionauxiliary mirror for auto-collimation

2. mode:surface tested in reflectionauxiliary lens to generate convergent beam

collimated laser beam

stop

Short common path,sensible setup

Two different operation modes for reflection ortransmission

Always factor of 2 betweendetected wave and component under test

31

Interferograms of Primary Aberrations

Spherical aberration 1

-1 -0.5 0 +0.5 +1Defocussing in

Astigmatism 1

Coma 1

32

Moving a knife edge perpendicular through the beam cross section

Relationship between powertransmission and intensity:Abel transform for circularsymmetry

Example: geometrical spot with spherical aberration

dr

drrrIxPx

22

)(2)(

z

beforecaustic zone rays below

near paraxial focus

Knife Edge Method

y

x

beam

x

knife edge

movement

33

Method very similar to moving knife edge Integration of slit length must be inverted:

- inverse Radon transform- corresponds to tomographic methods

Slit-Scan-Method

y

x

beam x

moving slit

d

34

Principle of phase retrieval for metrology: intensity variation during z-propagation due to phase errors due to the wave equation, the phase information can be recovered from a z-stack ill posed inverse problem easy measurement, algorithms complicated

Phase Retrieval

z

x

y

System

Austritts-pupille

Bildebene

WellenflächeW(x,y) : lateral

z

x

3D-Intensitäts-StackI(x,y,z)

35

Aufbau für Systemprüfung

Possible illumination setups:

1. Pinhole- coherence critical- low power- deconvolution for

finite size

2. Epi illumination- only symmetrical

aberrations- problems with finite field position

3. Via calibrated objectivelens

36

Phase retrieval method Image z-stack

Correlation of images Phase in pupil

PSF Phase Retrieval

37

Measurement of an edge image

Evaluating the derivative:Line spread function

Fourier transform:optical transfer function

MTF-Measurement by Edge Spread Function

')'()'(

xdxIdxI ESF

LSF

)'(ˆ)( xIFsH LSFOTF

x

edge object

incoherent illumination

Iin(x)

detector plane

x'diffracted

light

geometricalimage of the edge

I(x')

d I(x') dx'

38

Setup:Imaging of a grating

Possible realizations:1. Density type grating, the sine wave is modelled by gray levels2. Area type gratings, the sine wave is modelled by geometrical sine-shaped structures

Area coded sine grating:

MTF-Measurement by Imaging Gratings

lamp

sensor

slitlens under test

diffusor

gratingobject

39

Ghost Images

Ghost image in photographic lenses:Reflex film / surface

Ref: K. Uhlendorf, D. Gängler

40

Different reasons Various distributions

Straylight and Ghost Images

a b

41

Summary of Important Topics

MTF criteria, Hopkins factor, area criterion Test charts for qualitative fast evaluations Optimization contrast vs resolution: not trivial, depends on application Special criteria for edges, information content Problem in practice: field dependence of aberrations Many possible focussing criteria Different types of specifications usual Measurement of system performance via wavefront sensor (Shack-Hartmann) Quite accurate wave front measurement: interferometry PSF phase retrieval: robust and easy to measure method, algorithms complicated MTF measurement complex: sin-pattern, spatial frequency analysis of edge images Real world problems: ghosts and straylight

42

Outlook

Next lecture: Part 12 – Correction of aberrations 1Date: Wednesday, 2012-07-04Contents: 12.1 Symmetry principle

12.2 Lens bending12.3 Correcting spherical aberration12.4 Coma, stop position12.5 Astigmatism12.6 Field flattening12.7 Chromatical correction12.8 Higher order aberrations

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