www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 14: Transfer function
2020-01-31
Herbert Gross
Winter term 2019
2
Schedule - Imaging and aberration theory 2019
1 18.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 25.10.Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 01.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 08.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas
5 15.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 22.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders
7 29.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 13.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spectrum
10 20.12.Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 17.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement
13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio
14 31.01. Transfer function transfer function, resolution and contrast
15 07.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsicand induced aberrations, reversability
Fourier method
Optical Transfer function
Contrast and resolution
Quantitative performance assessment
Incoherent image formation
Cascaded systems
Coherent image formation
3D transfer theory and depth resolution
Contents
3
diffracted ray
direction
object
structure
g = 1 /
/ g
k
kT
Definitions of Fourier Optics
Phase space with spatial coordinate x and
1. angle
2. spatial frequency in mm-1
3. transverse wavenumber kx
Fourier spectrum
corresponds to a plane wave expansion
Diffraction at a grating with period g:
deviation angle of first diffraction order varies linear with = 1/g
0k
kv x
xx
),(ˆ),( yxEFvvA yx
A k k z E x y z e dx dyx y
i xk ykx y, , ( , , )
vk 2
vg
1
sin
Arbitrary object expaneded into a
spatial
frequency spectrum by Fourier
transform
Every frequency component is
considered separately
To resolve a spatial detail, at least
two orders must be supported by the
system
NAg
mg
sin
sin
off-axis
illumination
NAg
2
Ref: M. Kempe
Grating Diffraction and Resolution
optical
system
object
diffracted orders
a)
resolved
b) not
resolved
+1.
+1.
+2.
+2.
0.
-2.
-1.
0.
-2.
-1.
incident
light
+1.
0.
+2.
+1.
0.
+2.
-2.
-2. -1.
-1.
+3.+3.
5
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
6
pp
vyvxi
pppsfyxOTF dydxeyxINvvH ypxp
2),(),(
),(ˆ),( yxIFvvH PSFyxOTF
p
xp
xpxOTF dx
vfxP
vfxPvH
22)( *
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
(general: 2D)
Transfer properties:
PSF: response answer of a point object
OTF: response answer of an extended cosine grating
Absolute value of OTF: modulation transfer function (MTF)
MTF is numerically identical to contrast of the image of a cosine grating at the
corresponding spatial frequency
7
Number of Supported Orders
A structure of the object is resolved, if the first diffraction order is propagated
through the optical imaging system
The fidelity of the image increases with the number of propagated diffracted orders
0. / +1. / -1. order
0. / +1. / -1.
+2. / -2.
order
0. / +1. -1. / +2. /
-2. / +3. / -3.
order
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
9
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
10
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
11
Resolution and Spatial Frequencies
Grating object pupil
Imaging with NA = 0.8
Imaging with NA = 1.3
Ref: L. Wenke
12
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
13
14
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
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
15
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 sagittal
plane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
00.5 1
/ max
17
18
Tangential vs Sagittal Resolution
Asymmetry of the PSF
Coma 3rd creates
a 2:3 diameter pattern
Usually coma oriented
towards the axis,
then MTFS > MTFT
(lower row)
Ref: B. Dube, Appl. Opt. 56 (2017) 5661
spatial frequency
spatial frequency
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
19
Polychromatic MTF
20
Polychromatical MTF:
Cut off frequency depends on
Spectral incoherent weighted
superposition of monochromatic MTF’s
0
)( ),()()( dvgSvg OTF
poly
OTF
0.8
1.0
0.6
0.4
0.2
0
0 max
/2
27.5 Lp/mm
max
55 Lp/mm
gMTF
= 350 nm
= 400 nm
= 450 nm
= 500 nm
= 550 nm
= 600 nm
= 650 nm
= 700 nm
polychromatic
ideal 350 nm
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
21
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
22
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
23
Contrast and Resolution
High frequent
structures :
contrast reduced
Low frequent structures:
resolution reduced
contrast
resolution
brillant
sharpblurred
milky
24
Contrast / Resolution of Real Images
resolution,
sharpness
contrast,
saturation
Degradation due to
1. loss of contrast
2. loss of resolution
25
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 frequencies
CSF: 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
26
Photographic lenses with different performance
27
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
Blurred imaging:
- limiting case
- information extractable
Blurred imaging:
- information is lost
- what‘s the time ?
Resolution: Loss of Information
28
Approximation of the transfer function for large aberrations:
Expansion of the Duffieux-Integral
Approximation fails for
large frequencies
Example photographic lens
Geometrical Approximated Transfer Function
gMTF
0.00 10.00 20.00 30.00 40.00 50.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
ideal
axis
full field sag
full field tan
66% field sag
66% field tan
in Lp/mm
ppyxyxyxGTF dydxyvxviyvxvvvg
2
2
11),(
defocus astigmatism
x
coma
xspherical
coma
5. order
x
astigmatism
5. order x
trefoil
x
spherical
5. order
trefoil
5. order
x
astigmatism
y
coma
y
astigmatism
5. order y
trefoil
y
coma
5. order
y
spherical
7. order
coma
7. order
y
coma
7. order
x
astigmatism
7. order xastigmatism
7. order y
trefoil
5. order
y
tilt
x
tilt
y
MTF for Aberrations
OTF is complex function: decomposition into
absolute value (MTF) and phase (PTF)
Phase transfer function:
- corresponds to a lateral offset of the image point
- PTF describes distortion
- can be used to assess coma aberration
Phase Transfer Function
),(),(),( yxPTF vvgi
yxMTFyxOTF evvgvvg
2
)(')(
vgxIxI PTF
PSFPSF
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-6
-4
-2
0
2
4
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
gMTF
() gPTF
()
max
max
spherical aberration : c40
= -0.25
coma
c31
= 0.0
c31
= 0.1
c31
= 0.2
c31
= 0.3
c31
= 0.4
c31
= 0.5
ideal
Phase Mask with cubic
polynomial shape
Effect of mask:
- depth of focus enlarged
- Psf broadened, but nearly constant
- Deconvolution possible
Problems :
- variable psf over field size
- noise increased
- finite chief ray angle
- broadband spectrum in VIS
- Imageartefacts
sonst
xfürexP
xi
0
1)(
3
Cubic Phase Plate (CCP)
z
y-section
x-section
cubic phase PSF MTF
Cubic Phase Mask : PSF and OTF
defocussed focus
Conventional imaging System with cubic phase mask
defocussed focus
defocus negative focussed defocus positive
EDF Example Pictures
Cubic phase plate
- small changes of MTF with defocus
- oscillation of real- and imaginary part:
strong change of PTF
The phase of the Fourier components is
not properly reconstructed:
artefacts in image deconvolution
MTF / OTF for CPP
ohne CCP
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
CCP mit = 1
CCP mit = 2
CCP mit = 4
MTF
Re[OTF]
Im[OTF]
Im[OTF]
Frequenz Re[OTF]
c4 klein
c4 mittel
c4 groß
Real behavior of MTF:
1. invariant MTF for lower spatial frequencies
but re-magnification also increases noise
2. decrease for higher spatial frequencies:
reduction in resolution
MTF / OTF for CCP
0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
a = const
c4 wachsend
MTF Re[OTF]
Resolution Estimation with Test Charts
0 1
10
2
3
4
5
6
6
5
4
3
2
1
6
5
4
3
2
2 31
2
3
2
4
5
6
Measurement of resolution with test
charts:
bar pattern of different sizes
two different orientations
calibrated size/spatial frequency
37
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
38
Resolution Test Chart: Siemens Star
original good system
astigmatism comaspherical
defocusa. b. c.
d. e. f.
39
Fourier Optics – Point Spread Function
Point spread function amplitude in an optical system with magnification m
Pupil function P,
Pupil coordinates xp,yp
PSF is Fourier transform
of the pupil function
(scaled coordinates)
pp
myyymxxxz
ik
pppsf dydxeyxPNyxyxgpp ''
,)',',,(
pppsf yxPFNyxg ,ˆ),(
object
planeimage
plane
source
point
point
image
distribution
Fourier Theory of Incoherent Image Formation
objectintensity image
intensity
single
psf
object
planeimage
plane
Transfer of an extended
object distribution Iobj(x,y)
In the case of shift invariance
(isoplanatism):
incoherent convolution
Intensities are additive
In frequency space:
- product of spectra
- linear transfer theory
- spectrum of the psf works as
low pass filter onto the object
spectrum
- Optical transfer function
),(*),()','( yxIyxIyxI objpsfima
dydxyxIyyxxIyxI objpsfima
),()',,',()','(
dydxyxIyyxxIyxI objpsfima
),()','()','(
),(),( yxIFTvvH PSFyxotf
),(),(),( yxobjyxotfyximage vvIvvHvvI
Incoherent Image Formation
object
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
resolved not resolved Example:
incoherent imaging of bar pattern near the
resolution limit
Incoherent Image Formation
astigmatism comaspherical
aberrationobject ideal
PSF
Example:
incoherent imaging of pattern near the resolution limit with aberrations
Cascaded Optical Systems
Cascaded systems with individual transfer functions:
1. Diffusing screen in intermediate image plane:
Incoherent convolution of individual Psfs
2. Usual case:
Second system works coherent
Wave aberrations can be balanced
Convolution of amplitude Psfs
)()()()( 12 xIxIxIxI objectpsfpsfincoher
2
21 )()()( xAxAxI psfpsfexact
2
2
2
1 )()()( xAxAxI psfpsfapprox
Cascaded Optical Systems
Cascaded systems with individual transfer functions:
Behaviour of the transfer functions in the limiting cases
MTF1
real
MTF2
real
cdef1
= -cdef2
cdef2
= -cdef1
MTFcoh
real
coherent
MTFincoh
real
incoherent
correction
cdef1
= -cdef2
superposed
broadening
cdef1
, cdef2
a) Single
systems
b) Complete
cascaded
system
perfect
Cascaded Optical Systems
Incoherent case:
1. Psf convolution
2. MTF product
system 1 system 2intermediate
image and
diffusing
screen
object
plane
image
planeIpsf,2
Ipsf,1
)()()()( 12 xIxIxIxI objectpsfpsfincoher
)()()()( 12 vIvIvIvI objectpsfpsfincoher
)()()()( 21 xIxIxIxI psfpsfobjectincoher
)()()()( 21 vIvIvIvI psfpsfobjectincoher
Cascaded Optical Systems
Coherent case:
1. Amplitude Psf convolution
2. MTF product
system 1
cdef,2
cdef,1
system 2intermediate
imageobject
plane
image
planeApsf,2
Apsf,1
)()()()(
)()()(
*
2
*
121
2
21
xAxAxAxA
xAxAxI
psfpsfpsfpsf
psfpsfexact
iWiW
iWiWiWiW
exact
evTevT
evTevTevTevTvI
22
2
2
2
1
2
2
2
1
)()(
)()()()()( 2121
Fourier Theory of Coherent Image Formation
Transfer of an extended
object distribution Eobj(x,y)
In the case of shift invariance
(isoplanasie):
coherent convolution of fields
Complex fields additive
In frequency space:
- product of spectra
- linear transfer theory with fields
- spectrum of the psf works as
low pass filter onto the object
spectrum
- Coherent optical transfer
function
object
plane image
plane
object
amplitude
distribution
single point
image
image
amplitude
distribution
dydxyxEyxyxAyxE psfima ),(',',,)','(
dydxyxEyyxxAyxE objpsfima ),(',')','(
),(,)','( yxEyxAyxE objpsfima
),(),( yxEFTvvH PSFyxctf
),(),(),( yxobjyxctfyxima vvEvvHvvE
2
),(),(),( yxobjyxctfyxima vvEvvHvvI
Ideal coherent transfer function:
- corresponds to scaled pupil function
- cut of at pupil edge
- full contrast until edge
Ideal incoherent transfer function:
- convolution of shifted scaled pupils
- smooth decrease to cut-off frequence
Incoherent case:
- higher spatial frequencies resolved
- lower contrast at low frequencies
f
xPvH
p
ctf
)(
2
21
22arccos
2)(
vvvvHotf
Comparison of OTF and CTF
c
H
0 0.5 1
1
coherent
incoherent
50
Ewald Sphere
grating
koutkobj
kin
kin
kobj
kout
Ewald sphere
2objk
L
Assuming an object as grating with period L
Scattering of a wave at the object with
- conservation of energy
- conservation of momentum
The outgoing k-vector must be on a sphere:
Ewald's sphere for possible scattered wave vectors
in outk k
in obj outk k k
51
McCutchen Formula and Axial Resolution
n/
P(z)
zz
z
x
x
P(x)
x
light
cone
Ewald
sphere
transverse
pupil
axial
pupil
cap
nR
2 2 2 2
sin / /x
xv R n NA n NA
Imaging of a plane wave at a volume object
x: minimum value resolution
: maximum interval
Uncertainty relation: x = 1
Radius of the Ewald sphere
generalized 3D pupil: red area
Transverse resolution due to Abbe
Axial resolution:
- height of the cap of the cone
- McCutchen formula
222
2
2
sin11/
1
cos
11
NA
n
NAnn
nRRvz
z
52
3D Transfer Function
z
x
Ewald
sphereforward
2n/
i
i
s
backward
o-max
obj
obj = s - i
Imaging as 3D scattering phenomen
Only special spatial frequencies are allowed due to energy conservation and
momentum preservation
Green circle: supported spatial
frequencies of the transmitted
wave vector
53
3D Transfer Function - Missing Cone
x
z
missing
cone
transfer function
object
x
z
illumination
scattered
wave
transfer function
object
i
s
obs
Realistic case:
finite numerical aperture
Blue cone:
possible incoming wave direction due
to illumination cone
3D coherent transfer function:
limited green area, that fulfills all
conditions
Missing cone:
certain range of spatial axial spatial
frequencies can not be seen in the image,
Example: interfaces of thin coatings are
not seen
3D Fourier transform of pupil with defocussing
M : magnification
pp
yyxxd
ikyxzMz
d
ik
pp
D
psf dydxeeyxPyxApppp
222
2'
23 ),(),(
3D Point Spread Function
x
y
z z'
x'
y'yp
zp
xp
d
d'
object
t(x,y,z)
pupil
P(xp,yp,zp) field
E(x',y',z' )
General coherent 3D transfer function
Fourier transfer of psf field distribuition
Imaging of an transparent object with transmission function T
Offset 1 / : optical path length in thick object
Special case of a single lens with pupil
function P
with
')'()( '2 rderAvH vri
psfcft
zyx
zMvyvxvi
zyxctfzyxobj
ikz
ima dvdvdvevvvHvvvTezyxEzyx
2)1
(2' 1
,,,,)',','(
2,
2)( r
zrzr
rotsym
ctf
vvvPvvH
0
0sin
1
zv
General 3D Transfer Theory
r
z
0.5
1.0
P(r)
0
z0
HCTF