Design and Correction of optical Systems

Part 10: Performance criteria 1

Summer term 2012Herbert Gross

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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10.1 Introduction

10.2 Geometrical criteria

10.3 Wave aberrations, Rayleigh and Marechal criteria

10.4 PSF criteria: Strehl ratio, moments

10.5 2-point-resolution

10.6 Focussing

10.7 Miscellaneous

Contents

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Geometrical optical criteria:1. Aberrations2. Spot diagrams3. Uniformity illumination irradiance

Wave front:1. Zernike or other coefficients2. PV- and rms-value

Point spread function:1. Strehl ratio2. Diameters, 2nd order moments, curtosis, threshold-width

Resolution and contrast1. 2-point-resolution2. Contrast, line resolution, modulation depth3. Edge image gradient

Other:1. Encircled energy2. Fidelity, correlation, sharpness, structural content3. M2

Performance Criteria - Overview

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Gaussian Moment Spot

Spot pattern with transverse aberrations xj and yj1. 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 y

NyS j

j 1

M rN

x x y yG j S j Sj

2

2 21

GMD 2

M rN

g x x y yGG

j j S j Sj

2 2 21

y

ymax

xmax

x

rrms

xs,ys

5

Spot Diagram

Variation of field and color Scaling of size:

1. Airy diameter (small circle)2. 2nd moment circle (larger circle, scales with wavelength)3. surrounding rectangle 486 nm 546 nm 656 nm

axis

fieldzone

fullfield

6

Wave Aberration

Definition of the peak valley value

exitaperture

phase front

reference sphere

wave aberration

pv-value of wave

aberration

imageplane

7

Wave Aberrations

Mean root square of wave front error

Normalization: size of pupil area

Worst case / peak-valley wave front error

Generalized for apodized pupils (non-uniform illumination)

dydxAExP

ppppmeanppExP

rms dydxyxWyxWA

WWW 222 ,,1

pppppv yxWyxWW ,,max minmax

ppppw

meanppppExPwExP

rms dydxyxWyxWyxIA

W 2)()( ,,,1

8

Typical Variation of Wave Aberrations

Microscopic objective lens

Changes of rms value of wave aberration with1. wavelength2. field position

Common practice:1. diffraction limited on axis for main

part of the spectrum2. Requirements relaxed in the outer

field region3. Requirement relaxed at the blue

edge of the spectrum

Achroplan 40x0.65

on axis

field zone

field edge

diffraction limit

[ m]0.480 0.6440.562

0

Wrms [ ]

0.12

0.06

0.18

0.24

0.30

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Spatial Frequency of Surface Perturbations

Power spectral density of the perturbation Three typical frequency ranges, scaled by diameter D

limiting lineslope m = -1.5...-2.5

log A2Four

long rangelow frequency

figureZernike

midfrequency

microroughness

1/

oscillation ofthe polishing

machine

12/D1/D 40/D

10

Rayleigh criterion:1. maximum of wave aberration: Wpv < /42. beginning of destructive interference of partial waves3. limit for being diffraction limited (definition)4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil5. 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 fronts3. 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

RayleighrmsW

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PV and Wrms-Values

PV and Wrms values fordifferent definitions andshapes of the aberratedwavefront

Due to mixing of lowerorders in the definitionof the Zernikes, the Wrmsusually is smaller incomparison to thecorresponding Seideldefinition

12

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

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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 = 0.0c = 0.1

c = 0.2c = 0.3

c = 0.4c = 0.5

c = 0.7

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Criteria for measuring the degradation of the point spread function:1. Strehl ratio2. width/threshold diameter3. second moment of intensity distribution4. area equivalent width5. correlation with perfect PSF6.power in the bucket

Quality Criteria for Point Spread Function

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Threshold diameter of intensitye.g. 50% (FWhM)

2nd moment (rms)

Energy contentat threshold (PiB)

Petermann definition

Diameter definition with entropie of intensitydistribution

Beam Spot Diameter Definitions

peakthresh IrI

dydxyxI

dydxyxIyyxxrD ssrmsspot

),(

),()()(22

22

threshrE

0

20

2

2

)(

)(2

drrrrE

drrrEwPeter

SS eD S

E xP

E xP

dx I x I x dx

( )

ln( )

( ) ln ( )2 2

16

0,0

0,0)(

)(

idealPSF

realPSF

S IID

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...1DS = 1 is perfect

Critical in use: the completeinformation is reduced to only one number

The criterion is useful for 'good'systems with values Ds > 0.5

Strehl Ratio

r

1

peak reducedStrehl ratio

distributionbroadened

ideal , withoutaberrations

real withaberrations

I(x )

17

Approximation of Marechal:( useful for Ds > 0.5 ) but negative values possible

Bi-quadratic approximation

Exponential approach

Computation of the Marechalapproximation with thecoefficients of Zernike

2241

rms

sWD

N

n

n

m

nmN

n

ns n

cncD

1 0

2

1

20

2

121

121

Approximations for the Strehl Ratio

22221

rms

sWD

224

rmsW

s eD

defocusDS

c20

exac t

Marechal

exponential

biquadratic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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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

8.08106.082

SD

Strehl Ratio Criterion

aberration type coefficient Marechal

approximated Strehl

exact Strehl

defocus Seidel 25.020 a 7944.0 8106.082

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

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Transverse resolution of an image:- Detection of object details / fine structures- basic formula of Abbe

Fundamental dependence of the resolution from:1. wavelength2. numerical aperture angle3. refractive index4. 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)

sin

n

kx

Point Resolution According to Abbe

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Rayleigh criterion for 2-point resolutionMaximum of psf coincides with zeros of neighbouring psf

Contrast: V = 0.15

Decrease of intensity between peaksI = 0.735 I0

unDx Airy sin

61.021

Incoherent 2-Point Resolution : Rayleigh Criterion

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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-Airydistributions

Used in astronomy

0)(

02

2

xxdxId

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

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Visual resolution limit:Good contrast visibility V = 26 % :

Total resolution:Coincidence of neighbouring zero pointsof the Airy distributions: V = 1

Extremly conservative criterion

Contrast limit: V = 0 :Intensity I = 1 between peaks

AiryDun

x

680.0

sin83.0

unDx Airy sin

22.1

AiryDun

x

418.0

sin51.0

Incoherent 2-point Resolution Criterions

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2-Point Resolution

Distance of two neighboring object points Distance x scales with / sinu Different resolution criteria for visibility / contrast V

x = 1.22 / sinutotalV = 1 x = 0.68 / sinu

visualV = 0.26

x = 0.61 / sinuRayleighV = 0.15

x = 0.474 / sinuSparrow

V = 0

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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

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Incoherent Resolution: Dependence on NA

Microscopical resolution as a function of the numerical aperture

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

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Focussing: Definitions

Possible definitions for ‚focussing‘:1. Maximum intensity value on axis or on reference ray2. Image location with largest contrast3. Image location with highest spatial frequency resolution (at threshold)4. Smallest value of the Rms of the wavefront aberration5. Smallest spot diameter with 2nd order moment definition6. Smallest PSF diameter at given threshold value (e.g. 50%, 90%)7. Largest energy content inside a circle of given diameter

No definition of global validityUsefulness depends on application

Some hidden assumptions for ‚focussability:Polarization, coherence, spectral content, geometry of pupil, apodization,...

Diffraction limits focussability (uncertainty relation)Scaling quantity is

Focussability is a measure of beam quality

sin

n

D

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Depth of Field

Comparison of systems with1. Small depth of field 2. Large depth of field

Ref: M. Seesselberg

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Depth of Focus: Geometrical

z

objectplane

zgeo

pentrance

pupil

imageplane

z'geo

p'

exit pupilsystem

Spot spreading in focus: diameter 2 Detector spatial resolution D Depth of focus: 2 < D

Axial interval of sharpness. calculated by geometrical optics

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Highest resolution:- tangent line on transverse

aberration curve- compact central core

of PSF

Best contrast:- fitted mean straight line

on transverse aberrationcurve

- rms-optimum of PSF

Different criteria delivers different best image planes

Best Image Plane: Geometrical Consideration

0

zWrms

0

zDs

u

y

best matchingin thecentre

minimal difference

ymaxymin

ymax

30

Normalized axial intensityfor uniform pupil amplitude

Decrease of intensity onto 80%:

Scaling measure: Rayleigh length- geometrical optical definition

depth of focus: 1RE

- Gaussian beams:'sin' 2 un

RE

Depth of Focus: Diffraction Consideration

unzdiff 2sin

493.021

2

0 2/2/sin)(

z

zIzI

focalplane

beamcaustic

z

depth of focus

0.8

1

I(z)

+R u

z-R u 0

r

intensityat r = 0

2' oE n

R

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Geometrical spot diagram Depends on wavelength and field position Best compromise:

not trivial

Best Focal Plane

z1 = -100 m z2 = -50 m z4 = +50 m z5 = +100 mz3 = 0

axis

fieldzone

field

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Comparison of different performancecriteria as a function of defocusResidual system aberrations:1. astigmatism 1 2. spherical aberration 1

Different results for optimal imageplane determination

Special problems:Strehl criterion fails due to specialconstellation for astigmatism

Comparison of Performnace Criteria

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Conventional definition of distortion

Special definition of TV distortion

Measure of bending of lines

Acceptance level strongly depends onkind of objects:1. geometrical bars/lines: 1% is still

critical2. biological samples: 10% is not a

problem

Digital detection with image post processing: un-distorted image can be reconstructed

Distortion

y

x

H

H

yreal yideal

yyV

HHVTV

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Distortion

15% barrel5% barrel

5% pincushion

barrel and pincushion

2% pincushion

20% keystoneoriginal

10% pincushion

10% barrel

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Chromatical Difference in Magnification

Typical colorringing

Critical for black-white edges

Human eye is verysensitive for theseeffects

Ref: J. Kaltenbach

36

Chromatical Difference in Magnification

Color rings are hardly seen due to colored imageLateral shift of colored psf positions

Ref: J. Kaltenbach

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Real Image with Different Chromatical Aberrations

Summary of Important Topics

Spot diagram, moments Wave aberration PV and Wrms value important criteria, but only one number Rayleigh criterion PV < /4 Marechal criterion Wrsm < /14 (equaivalent with Rayleigh for defocus) Psf criteria: Strehl ratio, diameter, correlation,... 2-point resolution, Abbe resolution Focussing and depth of focus Lateral color aberration: critical

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Outlook

Next lecture: Part 11 – Performance Criteria 2Date: Wednesday, 2012-06-27

Content: 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

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