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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
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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
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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
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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
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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
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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
23
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
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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
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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
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Interferograms of Primary Aberrations
Spherical aberration 1
-1 -0.5 0 +0.5 +1Defocussing in
Astigmatism 1
Coma 1
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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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