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-
www.iap.uni-jena.de
Design and Correction of Optical
Systems
Lecture 7: PSF and Optical transfer function
2017-05-02
Herbert Gross
Summer term 2017
-
2
Preliminary Schedule - DCS 2017
1 07.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 21.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 28.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 05.05. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 12.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 19.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 26.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 02.06. Further Performance Criteria Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based
criteria, further options
9 09.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 16.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 23.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 30.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
13 07.07. Special System Examples Zoom systems, confocal systems
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1. Ideal point spread function
2. PSF with aberrations
3. Strehl ratio
4. Two-point-resolution
5. Optical transfer function
6. Resolution and contrast
Contents
3
-
Huygens Principle
wave fronts
Huygens-Wavelets
propagation direction
stop
geometrical shadow
Every point on a wave is the origin
of a new spherical wave (green)
The envelope of all Huygens wavelets
forms the overall wave front (red)
Light also enters the geometrical shadow
region
The vectorial superposition principle
requires full coherence
4
-
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
-
PSF by Huygens Principle
Huygens wavelets correspond to vectorial field components:
- represented by a small arrow
- the phase is represented by the direction
- the amplitude is represented by the length
Zeros in the diffraction pattern: destructive interference
Ideal point spread function:
pupil
stop
wave
front
point
spread
function
zero intensity
closed loop
side lobe peak
1 ½ round trips
central peak maximum
constructive interference
single wavelets
sum
-
PSF by Huygens Principle
Apodization:
variable lengths
of arrows
Aberrations:
variable orientation
of arrows
pupil
stop
wave
front
point
spread
function
apodization:
decreasing length of arrows
homogeneous pupil:
same length of all arrows
rp
I(xp)
pupil
stop
ideal
wave
front
point
spread
function
ideal spherical wavefront
central peak maximum
real
wave
front
real wavefront
with aberrations
central peak reduced
-
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
8
-
0
2
12,0 Iv
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
r
z
lateral
aperture
cone
axial
image plane
optical
axis
-
Ideal Psf
r
z
I(r,z)
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
10
-
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
11
-
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
12
-
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%
13
-
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
220
4/
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
14
-
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%
15
-
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
16
-
17
Axial and Lateral Ideal Point Spread Function
Comparison of both cross sections
Ref: R. Hambach
-
18
Annular Ring Pupil
Generation of Bessel beams
Ref: R. Hambach
𝑟𝑝
z
r
-
19
Spherical Aberration
Axial asymmetrical distribution off axis
Peak moves
Ref: R. Hambach
-
20
Gaussian Illumination
Known profile of gaussian beams
Ref: R. Hambach
-
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 )
21
-
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
rmss
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
rmss
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
22
-
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
23
-
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 / DsSTDEV
LIBEW
SM FWHM
CW
Ref WEAP=50%
24
-
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
25
-
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
26
-
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
27
-
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
28
-
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
29
-
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
30
-
Incoherent Resolution: Dependence on NA
Microscopical resolution as a function of the numerical aperture
NA = 0.9NA = 0.45NA = 0.3NA = 0.2
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
-
),(ˆ),( yxIFvvH PSFyxOTF
*
2
( ) ( )2 2
( )
( )
x xp p p
OTF x
p p
f v f vP x P x dx
H v
P x dx
Optical Transfer Function: Definition
Normalized optical transfer function (OTF) in frequency space: Fourier transform of the Psf- intensity
Absolute value of OTF: modulation transfer function MTF Gives the contrast at a special spatial frequency of a sine grating
OTF: Autocorrelation of shifted pupil function, Duffieux-integral Interpretation: interference of 0th and 1st diffraction of the light in the pupil
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
-
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
34
-
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
35
-
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
/ max0
0
1
0.5 1
0.5
gMTF
36
-
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
37
-
Modulation Transfer
Convolution of the object intensity distribution I(x) changes:
1. Peaks are reduced
2. Minima are raised
3. Steep slopes are declined
4. Contrast is decreased
I(x)
x
original image
high resolving image
low resolving image
Imax-Imin
-
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 sagittalplane
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/contrast criterion:
Ratio of contrasts with/without aberrations for one selected spatial frequency
Real systems:
Choice of several application relevant
frequencies
e.g. photographic lens:
10 Lp/mm, 20 Lp/mm, 40 Lp/mm
Hopkins Factor
)(
)()(
)(
)(
vg
vgvg
ideal
MTF
real
MTFMTF gMTF
ideal
real
gMTF
real
gMTF
ideal
1
0.5
0
41
-
Photographic lenses with different performance
42
Modulation Transfer Function
10 c/mm
20 c/mm
40 c/mm
Objektiv 1 f/ 3.5 Objektiv 2
000 0000
10
20
30
40
50
60
70
80
90
100
-25 -20 -15 -10 -5 0 5 10 15 20 25
max. MTF Bildhöhe [mm] max. MTF
MT
F [
%]
b
ei 1
0 , 2
0 , 4
0 L
p/m
m ....... ta
n _
__
sa
g
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lens 1 f/3.5 Lens 2
Image height
-
43
Calculation of MTF – Some more examples
1-dim case
circular pupil
Ring pupil =
central obscuration
(75%)
Apodization =
reduced transmission
at pupil edge
(Gauss to 50%)
The transfer of frequencies depends on
transmission of pupil
Ring pupil higher contrast near
the diffraction limit
Apodisation increase of contrast at
lower frequencies
1
0,5
MTF
Ref: B. Böhme
-
Resolution Test Chart: Siemens Star
original good system
astigmatism comaspherical
defocusa. b. c.
d. e. f.
44
-
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
45
-
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
46
-
Contrast / Resolution of Real Images
resolution,
sharpness
contrast,
saturation
Degradation due to
1. loss of contrast
2. loss of resolution
47