Post on 05-Jun-2018
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Imaging and Aberration Theory
Lecture 13: PSF and transfer function
2015-02-05
Herbert Gross
Winter term 2014
2
Preliminary time schedule
1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 06.11. Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 13.11. Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 20.11. Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 27.11. Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 04.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 11.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 08.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 15.01. Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 29.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement
13 05.02. PSF and transfer function ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and contrast
14 12.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
1. Ideal point spread function
2. PSF with aberrations
3. Strehl ratio
4. High NA vectorial PSF
5. Two-point-resolution
6. Optical transfer function
7. Resolution and contrast
Contents
3
Diffraction at the System Aperture
Self luminous points: emission of spherical waves
Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave
results in a finite angle light cone
In the image space: uncomplete constructive interference of partial waves, the image point
is spreaded
The optical systems works as a low pass filter
object
point
spherical
wave
truncated
spherical
wave
image
plane
x = 1.22 / NA
point
spread
function
object plane
4
PSF by Huygens Principle
Huygens wavelets correspond to vectorial field components
The phase is represented by the direction
The amplitude is represented by the length
Zeros in the diffraction pattern: destructive interference
Aberrations from spherical wave: reduced conctructive superposition
pupil
stop
wave
front
ideal
reference
sphere
point
spread
function
zero
intensity
side lobe
peak
central peak maximum
constructive interference
reduced constructive
interference due to phase
aberration
5
Fraunhofer Point Spread Function
Rayleigh-Sommerfeld diffraction integral,
Mathematical formulation of the Huygens-principle
Fraunhofer approximation in the far field
for large Fresnel number
Optical systems: numerical aperture NA in image space
Pupil amplitude/transmission/illumination T(xp,yp)
Wave aberration W(xp,yp)
complex pupil function A(xp,yp)
Transition from exit pupil to
image plane
Point spread function (PSF): Fourier transform of the complex pupil
function
1
2
z
rN
p
F
),(2),(),( pp yxWi
pppp eyxTyxA
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
''cos'
)'()('
dydxrr
erE
irE d
rrki
I
6
0
2
12,0 I
v
vJvI
0
2
4/
4/sin0, I
u
uuI
-25 -20 -15 -10 -5 0 5 10 15 20 250,0
0,2
0,4
0,6
0,8
1,0
vertical
lateral
inte
nsity
u / v
Circular homogeneous illuminated
Aperture: intensity distribution
transversal: Airy
scale:
axial: sinc
scale
Resolution transversal better
than axial: x < z
Ref: M. Kempe
Scaled coordinates according to Wolf :
axial : u = 2 z n / NA2
transversal : v = 2 x / NA
Perfect Point Spread Function
NADAiry
22.1
2NA
nRE
7
Ideal Psf
r
z
I(r,z)
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
8
Abbe Resolution and Assumptions
Assumption Resolution enhancement
1 Circular pupil ring pupil, dipol, quadrupole
2 Perfect correction complex pupil masks
3 homogeneous illumination dipol, quadrupole
4 Illumination incoherent partial coherent illumination
5 no polarization special radiale polarization
6 Scalar approximation
7 stationary in time scanning, moving gratings
8 quasi monochromatic
9 circular symmetry oblique illumination
10 far field conditions near field conditions
11 linear emission/excitation non linear methods
Abbe resolution with scaling to /NA:
Assumptions for this estimation and possible changes
A resolution beyond the Abbe limit is only possible with violating of certain
assumptions
9
I(r)
DAiry / 2
r0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16 18 20
Airy function :
Perfect point spread function for
several assumptions
Distribution of intensity:
Normalized transverse coordinate
Airy diameter: distance between the
two zero points,
diameter of first dark ring 'sin'
21976.1
unDAiry
2
1
2
22
)(
NAr
NAr
J
rI
'sin'sin2
ak
R
akrukr
R
arx
Perfect Lateral Point Spread Function: Airy
10
log I(r)
r0 5 10 15 20 25 30
10
10
10
10
10
10
10
-6
-5
-4
-3
-2
-1
0
Airy distribution:
Gray scale picture
Zeros non-equidistant
Logarithmic scale
Encircled energy
Perfect Lateral Point Spread Function: Airy
DAiry
r / rAiry
Ecirc
(r)
0
1
2 3 4 5
1.831 2.655 3.477
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2. ring 2.79%
3. ring 1.48%
1. ring 7.26%
peak 83.8%
11
Axial distribution of intensity
Corresponds to defocus
Normalized axial coordinate
Scale for depth of focus :
Rayleigh length
Zero crossing points:
equidistant and symmetric,
Distance zeros around image plane 4RE
22
04/
4/sinsin)(
u
uI
z
zIzI o
42
2 uz
NAz
22
'
'sin' NA
n
unRE
Perfect Axial Point Spread Function
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(z)
z/
RE
4RE
z = 2RE
12
Defocussed Perfect Psf
Perfect point spread function with defocus
Representation with constant energy: extreme large dynamic changes
z = -2RE z = +2REz = -1RE z = +1RE
normalized
intensity
constant
energy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
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 = 0.0
c = 0.1c = 0.2
c = 0.3c = 0.4
c = 0.5c = 0.7
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 )
15
Approximation of
Marechal:
( useful for Ds > 0.5 )
but negative values possible
Bi-quadratic approximation
Exponential approach
Computation of the Marechal
approximation with the
coefficients of Zernike
2
241
rms
s
WD
N
n
n
m
nmN
n
ns
n
c
n
cD
1 0
2
1
2
0
2
12
1
1
21
Approximations for the Strehl Ratio
22
221
rms
s
WD
2
24
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
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
Criteria for measuring the degradation of the point spread function:
1. Strehl ratio
2. width/threshold diameter
3. second moment of intensity distribution
4. area equivalent width
5. correlation with perfect PSF 6.power in the bucket
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%
18
High-NA Focusing
Transfer from entrance to exit pupil in high-NA:
1. Geometrical effect due to projection
(photometry): apodization
with
Tilt of field vector components
y'
R
u
dy/cosudy
y
rrE
E
yE
xE
y
xs
Es
eEy e
y
y'
x'
u
R
entrance
pupil
image
plane
exit
pupil
n
NAus sin
4 220
1
1
rsAA
2cos11
11
2 22
22
0
rs
rsAA
High-NA Focusing
Total apodization
corresponds to astigmatism
Example calculations
4 22
2222
0
12
2cos1111),(),(
rs
rsrsrArA linx
-1 -0.5 0 0.5 10
0.5
1
1.5
2
-1 -0.5 0 0.5 10
0.5
1
1.5
2
-1 -0.5 0 0.5 10
0.5
1
1.5
2
-1 -0.5 0 0.5 10
0.5
1
1.5
2
NA = 0.5 NA = 0.8 NA = 0.97NA = 0.9
r
A(r)
x
y
Vectorial Diffraction for high-NA
Vectorial representation of the diffraction integral according to Richards/Wolf
Auxiliary integrals
General: axial and cross components of polarization
cos2
2sin
2cos
)','(
1
2
20
2/sin4
00
2
I
Ii
IIi
eE
E
E
E
zrE
iu
z
y
x
o
dekrJzrI ikz
0
cos'
00 sin')cos1(sincos),'(
o
dekrJzrI ikz
0
cos'
1
2
1 sin'sincos)','(
o
dekrJzrI ikz
0
cos'
22 sin')cos1(sincos)','(
Pupil
Vectorial Diffraction at high NA
Linear Polarization
High NA and Vectorial Diffraction
Relative size of vectorial effects as a function of the numerical aperture
Characteristic size of errors:
I / Io
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-6
10-5
10-4
10-3
10-2
10-1
100
NA
axial
lateral
error axial lateral
0.01 0.52 0.98
0.001 0.18 0.68
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
24
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
25
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
26
Visual resolution limit:
Good contrast visibility V = 26 % :
Total resolution:
Coincidence of neighbouring zero points
of the Airy distributions: V = 1
Extremly conservative criterion
Contrast limit: V = 0 :
Intensity I = 1 between peaks
AiryDun
x
680.0
sin
83.0
unDx Airy
sin
22.1
AiryDun
x
418.0
sin
51.0
Incoherent 2-point Resolution Criterions
27
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
28
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
29
Incoherent Resolution: Dependence on NA
Microscopical resolution as a function of the numerical aperture
NA = 0.9NA = 0.45NA = 0.3NA = 0.2
30
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
31
Resolution of Fourier Components
Ref: D.Aronstein / J. Bentley
object
pointlow spatial
frequencies
high spatial
frequencies
high spatial
frequencies
numerical aperture
resolved
frequencies
object
object detail
decomposition
of Fourier
components
(sin waves)
image for
low NA
image for
high NA
object
sum
32
pppp
pp
vyvxi
pp
yxOTF
dydxyxg
dydxeyxg
vvH
ypxp
2
22
),(
),(
),(
),(ˆ),( yxIFvvH PSFyxOTF
pppp
pp
y
px
p
y
px
p
yxOTF
dydxyxP
dydxvf
yvf
xPvf
yvf
xP
vvH
2
*
),(
)2
,2
()2
,2
(
),(
Optical Transfer Function: Definition
Normalized optical transfer function
(OTF) in frequency space
Fourier transform of the Psf-
intensity
OTF: Autocorrelation of shifted pupil function, Duffieux-integral
Absolute value of OTF: modulation transfer function (MTF)
MTF is numerically identical to contrast of the image of a sine grating at the
corresponding spatial frequency
33
MTF and Contrast
Object
Contrast
Object spectrum
Image spectrum
Image
)2cos()( 0xvacxIobj
c
a
acac
acac
II
IIV
)()(
)()(
minmax
minmax
0022
)0()(ˆ)( vva
vva
vcxIFvI objobj
)()()( vHvIvI MTFobjima
)2cos()(
)(2
)(2
)0(
)(2
)(2
)0()(ˆ
)()(ˆ)(ˆ)'(
00
2
0
2
0
00
1
11
00
xvvHac
evHa
evHa
Hc
vvvHa
vvvHa
vvHcF
vHvIFvIFxI
MTF
xiv
MTF
xiv
MTFMTF
MTFMTFMTF
MTFobjimaima
34
x p
y p
area of
integration
shifted pupil
areas
f x
y f
p
q
x
y
x
y
L
L
x
y
o
o
x'
y'
p
p
light
source
condenser
conjugate to object pupil
object
objective
pupil
direct
light
at object diffracted
light in 1st order
Interpretation of the Duffieux Iintegral
Interpretation of the Duffieux integral:
overlap area of 0th and 1st diffraction order,
interference between the two orders
The area of the overlap corresponds to the
information transfer of the structural details
Frequency limit of resolution:
areas completely separated
35
Duffieux Integral and Contrast
Separation of pupils for
0. and +-1. Order
MTF function
Image contrast for
sin-object
I
x
V = 100%
V = 33%V = 50%
V = 20%
minI = 0.33 minI = 0.5 minI = 0.66
1
image contrast
pattern
period
spatialfrequency
100 %
0 NA/ 2NA/
/NA /2NA8
pupil
diffraction order
Ref: W. Singer
36
Optical Transfer Function 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
f
avo
'sin222 0
un
f
navvG
2
000 21
22arccos
2)(
v
v
v
v
v
vvHMTF
),(),(),( yxPTF vvHi
yxMTFyxOTF evvHvvH
/ max
00
1
0.5 1
0.5
gMTF
37
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 sin 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
38
Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the
azimuthal orientation of the object structure
Generally, two MTF curves are considered for sagittal/tangential oriented object structures
Sagittal and Tangential MTF
y
tangential
plane
tangential sagittal
arbitrary
rotated
x sagittal
plane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
00.5 1
/ max
39
Real MTF of system with residual aberrations:
1. contrast decreases with defocus
2. higher spatial frequencies have stronger decrease
Real MTF
z = 0
z = 0.1 Ru
gMTF
1
0.75
0.25
0.5
0
-0.250 0.2 0.4 0.6 0.8 1
z = 0.2 Ru
z = 0.3 Ru
z = 1.0 Ru
z = 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 in
RU
max
= 0.05
max
= 0.1
max
= 0.2
max
= 0.3
max
= 0.4
max
= 0.5
max
= 0.6
max
= 0.7
max
= 0.8
Zernike
coefficients:
c5 = 0.02
c7 = 0.025
c8 = 0.03
c9 = 0.05
40
Resolution Test Chart: Siemens Star
original good system
astigmatism comaspherical
defocusa. b. c.
d. e. f.
41
Contrast vs contrast as a function of spatial frequency
Typical: contrast reduced for
increasing frequency
Compromise between
resolution and visibilty
is not trivial and depends
on application
Contrast and Resolution
V
c
1
010
HMTF
Contrast
sensitivity
HCSF
42
Optical Transfer Function of a Perfect System
Loss of contrast for higher spatial frequencies
contrast
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
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 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ideal
MTF
/max
/max
43
Contrast / Resolution of Real Images
resolution,
sharpness
contrast,
saturation
Degradation due to
1. loss of contrast
2. loss of resolution
44