Upload
others
View
9
Download
1
Embed Size (px)
Citation preview
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
2
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
3
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
4
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
9
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
11
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
13
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
14
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
15
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
Incoherent Resolution: Dependence on NA
Microscopical resolution as a function of the numerical aperture
NA = 0.9NA = 0.45NA = 0.3NA = 0.2
26
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
27
Depth of Field
Comparison of systems with1. Small depth of field 2. Large depth of field
Ref: M. Seesselberg
28
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
29
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
31
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
32
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
33
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
34
Distortion
15% barrel5% barrel
5% pincushion
barrel and pincushion
2% pincushion
20% keystoneoriginal
10% pincushion
10% barrel
35
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
37
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
39
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
40