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Physical Optics
Lecture 4 : Fourier Optics
2017-04-19
Beate Boehme
Physical Optics: Content
2
No Date Subject Ref Detailed Content
1 05.04. Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation,
interferometry
2 12.04. Diffraction B Slit, grating, diffraction integral, diffraction in optical systems, point spread
function, aberrations
3 19.04. Fourier optics B Plane wave expansion, resolution, image formation, transfer function,
phase imaging
4 26.04. Quality criteria and
resolution B
Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point
resolution, criteria, contrast, axial resolution, CTF
5 03.05. Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence,
components
6 10.05. Photon optics D Energy, momentum, time-energy uncertainty, photon statistics,
fluorescence, Jablonski diagram, lifetime, quantum yield, FRET
7 17.05. Coherence G Temporal and spatial coherence, Young setup, propagation of coherence,
speckle, OCT-principle
8 24.05. Laser B Atomic transitions, principle, resonators, modes, laser types, Q-switch,
pulses, power
9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations
10 07.06. Generalized beams D Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy
beams, applications in superresolution microscopy
11 14.06. PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping,
confocal PSF
12 21.06. Nonlinear optics D Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,
CARS microscopy, 2 photon imaging
13 28.06. Scattering G Introduction, surface scattering in systems, volume scattering models,
calculation schemes, tissue models, Mie Scattering
14 05.07. Miscellaneous G Coatings, diffractive optics, fibers
D = Dienerowitz B = Böhme G = Gross
Diffraction in optical systems
self luminous point: emission of spherical wave
optical system: only a limited solid angle is propagated
truncation of the spherical wave results in a finite angle light cone
in the image space: uncomplete constructive interference of partial waves
spreaded image point
the optical systems works as a low pass filter limited resolution
Field in the image plane ~ Fourier transformation of the complex pupil function A(xp,yp)
Object plane aperture image plane
truncated
spherical
wave A(xp,yp)
pp
yyxxR
i
pp
ExP
dydxeyxAyxEpp
ExP
''2
,)','('
where A(xp,yp) describes
transmission and phase (wavefront)
2* )','(''')','(' yxEEEyxI
DAiry
E(x,y)
3
Resolution – More incoherent points
more independent self luminous points: emission of N spherical waves
summation of intensities
Object plane
aperture image plane
truncated
spherical
wave
A(xp,yp)
DAiry
3. plane
wave
DAiry
DAiry
In the aperture (pupil plane) we observe a plane wave for each object point
For N points N independent plane waves wit different directions
Diffraction for all waves
Superposition of the point images
4
Resolution – Vice Versa discusssion
Plane waves with different directions in the object plane
Focused, convergent waves in the pupil plane
Coordinate of focus depends on direction of plane wave
Limitation of directions by the aperture
Superposition of the transmitted plane waves in the image
Plane waves can be thought of generated by a grating, illuminated with a plane wave
Far field diffraction pattern in the pupil
Object plane
aperture
image plane
xp, yp
plane wave
superposition
),(ˆ),( yxEFvvA yx
E(x,y)
dydxeyxEkkA yx ykxki
yx ),(,
vx, yy
y
x
vfy
vfx
'
'
5
Sine-grating in the object plane
Two diffraction orders:
0. order = transmitted light
+1. order
- 1. order
Increasing diffraction angle with smaller
period g / increasing spatial
frequency v = 1/g
Location of diffraction orders in the
back focal plane depends on grating
period
The sine-grating can only be
reproduced in the image, if
orders 0, +1 and -1 are transmitted
There is a minimum period
which can be transmitted
6
Plane wave expansion
+1st
-1st
+1st
-1st
+1st
-1st
objectback focal
planeobjective
lens
0th order
Modulation Transfer Function - MTF
Aberration free circular pupil:
Reference frequency
Cut-off frequency:
Analytical representation
'sin' un
f
avo
NAvv
22 0max
2
000 21
22arccos
2)(
v
v
v
v
v
vvHMTF
/ max
00
1
0.5 1
0.5
MTF
Perfect system
Incoherent illumination:
coherent
illumination
The optical system acts as
Low-pass filter with cut-off frequency
7
Calculation of MTF
MTF describes transmission of sine gratings by the optical system
Description in frequency space
Calculation and explanation as description of point image in frequency space
= spectrum of PSF
Alternative calculation: Autocorrelation of pupil function
= overlap integral as function of shift
For 1-dim pupil: autocorrelation of two Top-hat functions = triangle function
For 2-dim circular pupil: autocorrelation of two circles:
proportional to the overlapping surface
triangle-similar at center
slow decrease to zero
2sina/ sina/
2sina/
𝐴(𝑥′) = 𝐹 𝑥 + 𝑥′∗𝐹 𝑥 − 𝑥′ 𝑑𝑥
8
9
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
10
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%)
1
0,5
Transmission of pupil (pupil function A)
= real function, describes:
ideal optical system
no complex part Autocorrelation real
no modification of phase
no aberrations
MTF
11
Calculation of MTF – complex pupil function
T(xp,yp)
Transmission:
circular pupil
Linear wavefront = phase at pupil
Autocorrelation in x-direction
~ frequency response of the system:
𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp −2𝜋𝑖 𝑊(𝑥𝑃, 𝑦𝑃 )
𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎𝑥𝑃)
𝑂𝑇𝐹(𝑣𝑥) = 𝐴 𝑥𝑝 +𝜆𝑓2𝑣𝑥 , 𝑦𝑝
∗𝐴 𝑥𝑝 −
𝜆𝑓2𝑣𝑥 , 𝑦𝑝 𝑑𝑥𝑝𝑑𝑦𝑝
MTF = abs (OTF)
0 0.5 1 0
0,5
1
0 0.5 1
2
PTF = angle(OTF) = phase transmission
12
Calculation of MTF – complex pupil function
T(xp,yp)
Transmission:
circular pupil
Linear wavefront = phase at pupil
Spherical phase: Defocus
Autocorrelation in x-direction
~ frequency response of the system:
𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp −2𝜋𝑖 𝑊(𝑥𝑃, 𝑦𝑃 )
𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎𝑥𝑃)
𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎(𝑥𝑃² + 𝑦𝑃²)
𝑂𝑇𝐹(𝑣𝑥) = 𝐴 𝑥𝑝 +𝜆𝑓2𝑣𝑥 , 𝑦𝑝
∗𝐴 𝑥𝑝 −
𝜆𝑓2𝑣𝑥 , 𝑦𝑝 𝑑𝑥𝑝𝑑𝑦𝑝
MTF = abs (OTF)
0 0.5 1
2
PTF = angle(OTF) = phase transmission
13
OTF – Transfer of sine gratings
Transfer of amplitude sine-grating with
Mean intensity and Contrast V
𝐼
𝐼′
x, x’
1/v PTF(v)
𝑉 =𝐼𝑀𝑎𝑥 − 𝐼𝑀𝑖𝑛𝐼𝑀𝑎𝑥 + 𝐼𝑀𝑖𝑛
Object-Contrast
Image-Contrast 𝑉′ = 𝑀𝑇𝐹 𝑣 ∙ 𝑉
𝐼
𝑂𝑇𝐹 = 𝑀𝑇𝐹 𝑣)exp (−𝑖 𝑃𝑇𝐹(𝑣)
For an aberration-free system
PTF = 0
The image has reduced contrast
With aberrations periodic
structures are transferred with
phase shifts.
Incoherent Image Formation
One illumination point generates a plane wave in the object space
Diffraction of the wave at the object structure
Diffraction orders occur in the pupil
Constructive interference of all supported diffraction orders in the image plane
Too high spatial
frequencies are
blocked
object plane
pupilplane
imageplane
f f f f
u() U (x)1
h()
f f
lightsource
s() U (x)0
T(x)
s
s
Ref: W. Singer
14
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 sine grating
Contrast of an corresponding rectangular grating is higher than for the sine grating
because higher diffraction orders help “Square wave MTF”
The maximum value of the intensity
is not identical to the contrast value
since the minimal value is finite too
Visibility of rectangular grating
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
Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the
azimuthal orientation of the object structure „surface MTF“
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
16
Polychromatic MTF
Cut off frequency depends on
Polychromatic MTF:
Spectral incoherent weighted
superposition of
monochromatic MTF’s
Example: uncorrected axial color
F (486), D(587), C(656nm)
with SF6 instead SF5
0
)( ),()()( dvHSvH MTF
poly
MTF
#
122 0max
F
NAvv
17
contrast decreases with defocus
higher spatial frequencies have
stronger decrease
Zero values in MTF indicate
phase shift of OTF contrast reversal
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
18
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
Phase shift in transfer function
19
Resolution Test Chart: Siemens Star
original good system
astigmatism comaspherical
defocusa. b. c.
d. e. f.
20
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
21
Blurred imaging:
- limiting case
- information extractable
Blurred imaging:
- information is lost
- what‘s the time ?
Resolution: Loss of Information
22
Contrast / Resolution of Real Images
resolution,
sharpness
contrast,
saturation
Degradation due to
1. loss of contrast
2. loss of resolution
23
Contrast as a function of spatial frequency
Compromise between
resolution and visibilty
is not trivial and depends
on application
Contrast and Resolution of Real Applications
V
/c
1
010
HMTF
Contrast
sensitivity
HCSF
Real systems:
Limited contrast sensitivity
of detectors
for instance: 8Bit = 256ct
limit 1/256 for contrast
contrast sensitivity may
depend on direction and
spatial frequency
Image processing with
contrast enhancement
Human eye: about 0,25%
contrast sensitivity v/vreal
24
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
25
OTF – Calculation from Point spread function
Ideal imaging airy distribution = answer of the optical system to an ideal point source
the system transfer is described in space
frequency space Fourier transform of the airy intensity pattern
Real imaging OTF = Fourier Transform of the point spread function (2D)
in general no radial-symmetric function
aperture image plane
A(xp,yp)
DAiry
DAiry
DAiry
26
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:
OTF: in general complex function, describes transfer of amplitude and phase
response answer of an extended cosine grating
MTF = modulation transfer function (MTF) = Absolute value of OTF
MTF is numerically identical to contrast of the image of a cosine grating at the
corresponding spatial frequency
PTF = phase transfer function
distinguish: PSF = response answer of a point object
27
),(),(),( yxPTF vvHi
yxMTFyxOTF evvHvvH
Fourier theory of image formation
Coherent and incoherent image formation
28
Fourier Optics – Point Spread Function
Optical system with pupil function P,
Pupil coordinates xp,yp
PSF is Fourier transform
of the pupil function (scaled coordinates)
Intensity of point image
pp
yyyxxxz
ik
pppsf dydxeyxPyxyxgpp ''
,~)',',,(
pppsf yxPFyxg ,ˆ~),(
object
planeimage
plane
source
point
point
image
distribution
29
psfpsfpsfpsf gggyxI *2
),(
Fourier Theory of Incoherent Image Formation
objectintensity image
intensity
single
psf
object
planeimage
plane
Transfer of an extended
object distribution I(x,y)
In the case of shift invariant PSF
(isoplanatism) = convolution of intensities
In frequence domian: Product of
intensity transfer function Hotf(vx,vy)
and object intensity
Absolute value of
the OTF = MTF
Low pass filter of
intensity distribution
),(*),()','( yxIyxIyxI objpsfimage
dydxyxIyyxxgyxI psfinc
),(),',,'()','(2
30
),(),(),( yxobjyxotfyximage vvIvvHvvI
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
31
Fourier Theory of Coherent Image Formation
Transfer of an extended electric field
distribution in object plane E(x,y)
In the case of shift invariant PSF
(isoplanatism) = convolution of fields
Symbol for convolution
object
plane image
plane
object
amplitude
distribution
single point
image
image
amplitude
distribution
dydxyxEyxyxgyxE psf ),(',',,)','(
dydxyxEyyxxgyxE psf ),(',')','(
32
E(x,y)
),(,)','( yxEyxgyxE psf
Fourier Theory of Coherent Image Formation
Convolution in spatial domain
description of field as sum of frequency (grating) components
transition to frequency domain by Fourier Transformation
Convolution in space corresponds to product of spectra
Coherent optical transfer function
spectrum of the PSF
works as low pass filter
onto the object spectrum
),(),( yxgFTvvH PSFyxctf ),(),(),( yxobjyxctfyxima vvEvvHvvE
2
),(),(),( yxobjyxctfyxima vvEvvHvvI
33
object
plane image
plane
object
amplitude
distribution
single point
image
image
amplitude
distribution
),(,)','( yxEyxgyxE psf
Fourier Theory of Image Formation
object
amplitude
U(x,y)
PSF
amplitude-
response
Hpsf (xp,yp)
image
amplitude
U'(x',y')
convolution
result
object
amplitude
spectrum
u(vx,vy)
coherent
transfer
function
hCTF (vx,vy)
image
amplitude
spectrum
u'(v'x,v'y)
product
result
Fourier
transform
Fourier
transform
Fourier
transform
Coherent Imaging
object
intensity
I(x,y)
squared PSF,
intensity-
response
Ipsf
(xp,y
p)
image
intensity
I'(x',y')
convolution
result
object
intensity
spectrum
I(vx,v
y)
optical
transfer
function
HOTF
(vx,v
y)
image
intensity
spectrum
I'(vx',v
y')
produkt
result
Fourier
transform
Fourier
transform
Fourier
transform
Incoherent Imaging
4.2 Image simulation
34
Comparison Coherent – 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
incoherent coherent
-0.0 5 0 0.0 5
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
-0.0 5 0 0.0 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.0 5 0 0.0 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.0 5 0 0.0 5
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
bars resolved bars not resolved bars resolved bars not resolved
35
Incoherent image:
homogeneous areas, good similarity between object
and image, high fidelity
Coherent image:
Granulation of area ranges, diffraction ripple at
edges
incoherent coherent
Coherent – Incoherent Image Formation
incoherent
coherent
36
Image of an edge for
incoherent illumination
No oscillations, smooth distribution
Ideal position of the edge at 50%
Width of the edge transition depends
on PSF of imaging setup
I(x)
x
z
aky
z
aky
z
akySiyI inc
edge 2
12
cos121
2
1)(
Incoherent Image of an Edge
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
37
Image of an edge for coherent illumination
with integral sine function Si
a: half diameter of aperture
Intensity 25% at edge
Error in detection of edge position for 50%
criterion
Amplitude and intensity distribution:
- oscillations around the edge
- in shadow region hard to resolve
x
E(x)
I(x)
x
2
)( 1
2
1)(
z
akySiyI coh
edge
a
zx
212.0
Coherent Image of an Edge
38
Partial coherent imaging
39
Partial Coherent Imaging
Every object point is illuminated by an angle spectrum due to the finite extend of the source
In the pupil the diffraction orders are broadened additionally
no full constructive interference in the image plane
influence of the illumination onto the image should be considered
object pupil imagelight
source
+ 1
- 1
0
source condenser object lens image
angle shift
40
Heuristic explanation
of the coherence
parameter in a system:
1. coherent:
Psf of illumination
large in relation to the
observation
2. incoherent:
Psf of illumination
small in comparison
to the observation
Coherence parameter s:
describes ratio of
illumination NA to
observation NA
object objective lenscondensersmall stop of
condenser
extended
source
coherent
illumination
large stop of
condenser
incoherent
illumination
Psf of observation
inside psf of
illumination
Psf of observation
contains several
illumination psfs
extended
source
Coherence Parameter
𝜎 =𝑁𝐴(𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛)
𝑁𝐴(𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛)
41
Partial Coherent Imaging - Example Bar-Pattern
12 mm
object
s = 0.08
pupil intensity
imageobject
spectrum
s = 0.50 s = 1.0
coherence function
Coherence of fields at
two object points
transmission by the
optical system
intensity & amplitude
transmission non-linear
Incoherence:
spreading of edges
Higher coherence:
periodic intensity
modulations (compare
slit with coherent illum.)
Optimization of
coherence settings
Optimization of
illumination
42
Resolution and Contrast for Partial Coherent Imaging
HCTF(s)
1
2
light source
light source
a) partial
coherent
HCTF(s)
light source
light
source
b) incoherent
1
2
x
y
x
y
incoherentpartial
coherent coherentpartial coherent
oblique illumination
coherent oblique
illumination
Transfer of spatial frequencies
depend on illumination settings
and directions
Analytical representation
only for circular Symmetry
possible (Kintner)
Transfer capability depends
on integration overlap of
illumination and detection
pupils
Transmission Cross Correlation
Function (TCC)
43
Partial Coherent Imaging of Siemens Star
coherent partial coherent incoherent
frequency
o = sinu /
frequency
o = 2sinu /
44
Pupil Illumination Pattern
Coherent
Off Axis Annular Annular
Dipole Rotated Dipole
Disk s = 0.5 Disk s = 0.8
6 -Channel
Variation of the pupil illumination
Enhancement of resolution
Improvement of contrast
Object specific optimization
Often the best compromise with
partial coherent illumination where
slightly increased intensity at the
edges
In microscopy the adjustment of
Koehler illumination corresponds to
choosing this setup
Ref: W. Singer
45
Lichtquelle
Kollektor Kondensor Objektiv
Objektebene
Bildebene
Leuchtfeld-
blende
Kondensor-
blende
Apertur-
blende
Application in Microscopy
Koehler illumination
to
image
Microscopic
lens condensor
Object
plane Illumination
aperture
imaging
(microscope)
aperture
collector
Light
source
light field
diaphragm
Light field diaphragm to limit illuminated object area
aperture at the condenser to adjust degree of coherence 𝜎𝑖𝑙𝑙𝑢𝑚 𝜎𝑖𝑚𝑎𝑔𝑖𝑛𝑔
46
Procedure:
1. Aperture stop open, field stop closed: center visible rim of field stop
2. Focussing of condensor: axial shift of condensor, sharpen rim of field stop
3. Open field stop until the visible field is illuminated
4. Close aperture stop to about 2/3
5. Adjust brightness of lamp
Kondensor
Objekt-
ebene
Apertur-
blendeLeuchtfeld-
blende Filter
Kollektor
Lampe
Application in Microscopy
Koehler illumination adjustment
Aperture
stop
condensor
Object
field
stop
collector
lamp
47
Imaging of phase objects
48
Pure phase mask as object constant intensity at image
Variation of focus during observation
Phase structure becomes visilbe
Reason: defocus modifies MTF shift of PFT at pupil zones
Imaging of transparent phase objects 49
Quantitative Phasenmikroskopie
Setup :
anschaulich
Abbildungs-
system
Quelle
Quelle
Messebenen
Objekt
1. System-
messung
2. Abbildung
Verteilung durch
Objekt verändert
Imaging of phase objects 50
Pure phase transmission at object
Approximation: small phase
Modification at pupil: phase mask
For 0. diffraction order:
reduced transmssion a
90°phase shift
approximated intensity at image
)()()( xiexBxP
)(1)( xixE
)'()'(' xiiaxE
)'()'(' xaxI
KondensorBild
Bertrand-
linseObjektiv
Ring-
blendeObjekt
Phasen-
platte
Zernike phase contrast
Phase
object
lens
Pupil with
phase plate
lens
image
condensor
ring
Illumination
51
Klassisches DurchlichtPhasenkontrastbild
Phasenkontrast nach Zernike : Beispiel
Zernike phase contrast 52
Axial resolution and depth of focus
53
Depth of Focus: Geometrical
z
2
object
plane
zgeo
p
entrance
pupil
image
plane
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
54
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
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
55
Normalized axial intensity
for uniform pupil amplitude
Decrease of intensity onto 80%:
Scaling measure: Rayleigh length
- geometrical optical definition
depth of focus: 1RE
- Gaussian beams: similar formula
22
'
'sin' NA
n
unRu
Depth of Focus: Diffraction Consideration
2
0
sin)(
u
uIuI
2' o
un
R
udiff Run
z
2
1
sin493.0
2
12
focal
plane
beam
caustic
z
depth of focus
0.8
1
I(z)
z-Ru/2 0
r
intensity
at r = 0
+Ru/2
56
Depth of Focus
Depth of focus depends on numerical aperture
1. Large aperture: 2. Small aperture:
small depth of focus large depth of focus
Ref: O. Bimber
57
Ring pupil illumination
Enlarged depth of focus
Lateral resolution constant due to
large angle incidence
Can not be understood geometrically
Depth of focus for Annular Pupil
58
Farfield of a ring pupil:
outer radius aa
innen radius ai
parameter
Ring structure increases with
Depth of focus increases
Application:
Telescope with central obscuration
Intensity at focus
1a
i
a
a
2
121
22
)(2)(2
1
1)(
x
xJ
x
xJxI
r-15 -10 -5 0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(r)
= 0.01
= 0.25
= 0.35
= 0.50
= 0.70
22 1sin
2
unz
Depth of focus for Annular Pupil
59
Ring shaped masks according to Toraldo :
- discrete rings
- absorbing rings or pure phase shifts
- original setup: only 0 / values of phase
- special case
Fresnel zone plate
Pure phase rings:
amplitude of psf
Extended depth of focus: Toraldo Ring Masks
00
0
0
0
Phasen-
platte
Fokus
n
j j
j
j
j
j
j
i
ukr
ukrJ
ukr
ukrJerE j
1 1
112
1
122
'sin'
'sin'2
'sin'
'sin'2)'(
r
1
2
j
n
3 10
0
60
MTF and bar pattern as a function of defocus
ideal with Mask
Extended depth of focus
61
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)(
3a
Cubic Phase Plate (CCP)
z
y-section
x-section
cubic phase PSF MTF
62
Cubic Phase Plate: PSF and OTF
defocussed focus
Conventional imaging System with cubic phase mask
defocussed focus
63
Conventional microscopic image
Image with phase mask
with / without deconvolution
EdoF: CPP for microscopic Imaging
Ref: E. Dowski
64
Dark field illumination in microscopy
Foucault knife edge method for aberration measurement
Schlieren method for measurement of striae and inhomogeneities in materials
Zernike contrast method in microscopy
Use of apodization for suppression of diffraction rings, resolution enhancing masks
Beam clean up of laser radiation with kepler system and mode stop
Edge enhancement techniques in lithography
Oblique illumination for resolution enhancement in microscopy
Schmidt corrector plate in astronomical telescopes
Pupil filter masks to generate extended depth of focus
Resolution enhancement by structured illumination
Applications of Fourier Filtering Techniques
65
Depth of focus at the human eye –
Correction of presbyopia
66
Human eye and Visual acuity
Distance 5 – 6m
1min for
Visual Acuity
VA 20/20 = 1
1‘ VA = 1
2‘ VA = 0.5
Vitreous
Lens
Cornea
Anterior
chamber
Iris = stop Retina
(Macula)
Intraocular lens:
67
Human eye with monofocal intraocular lens (IOL)
Visus 1 corresponds to
Object @ 1min = 290µm @ 1m (spaces)
paraxial image 4.7µm @ water
period 105 L/mm (rect Bar+space)
corresponds to cone-pitch 2.3µm
@ macula
Pseudophakic eye
With artificial intraocular lens
Correction to refraction = 0
diffraction-limited imaging
Effective focal length 21.7mm @ air
pupil diameter 2 – 6mm
Stop-Diameter 2mm 3mm 4mm 6mm
vg = 2 NA‘ / 260 390 521 780 L/mm
NA‘ = 0.1416
Airy radius 4,8µm 2,4µm 1,6µm
VA = 1
Every day situations a
105L/mm
68
Diffraction-limited PSF @ retina:
Simulation of visual performance
Monofocal imaging
Calculation of imaging to retina:
Airy distribution
Size for 2mm and 4mm pupil:
Axial 291µm 73µm ~1/stop²
radial 4.7µm 2.35µm ~ 1/ stop
distance to zero intensity
Contrast 105L/mm für visual acuity = 1 = 20/20
@ 2 points 99.9% 99.1%
@ sinus 50.8% 75%
3 objects at different distances
F E N at far, near and an intermediate distance
Calculation of intensity at common image plane
Variation of power addition with glasses
69
Monofocal Imaging
Image stack,
Stop 2mm
Inreasing Power of Glasses or addition
70
Monofocal Imaging
Image stack,
Stop 4mm
71
Stop 2mm
Monofocal Imaging
images @ far, intermediate, near with add correction
Stop 4mm
influence on contrast and resolution visible
increase of DOF & less accommodation necessary at small pupils
Simulation of visual performance
72
Far and Near image simultaneously sharp
Power addition 3.75dpt
Separation of light onto
two foci,
greater amount to far
Ideal Bi-focal Imaging
2 Diffraction-limited PSFs in two z-positions
Images for stop 2mm:
Symbol intensities fit to PSF
With a small add power all tree
symbols become visible
simultaneously
Simulation of visual performance
73
Stop 4mm
reduction of contrast barely remarkable (in comparison to monofocal lens)
An ideal bifocal lens would show bad intermediate contrast
An ideal model of real Bifocal lenses is not sufficient
maybe Halos for very high far intensities
Stop 2mm
Ideal Bifocal Imaging
images @ far, intermediate, near with add correction
Simulation of visual performance
74
Stop 4mm
Far Focus Far, symbol for VA 0.5 Add 0.3… 0.4 dpt
Real Bifocal lens
Stop 2mm
Simulation fits
to clinical results
Simulation of visual performance
75
76
MTF: through focus
depends on pupil diameter
With smallest pupil best depth of focus,
No intermediate degradation
two foci, but reduced contrast
MTF-reduction corresponds to
The light propagation to the
other image plane
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400
7
7
3-D imaging
Paraxial monofocal imaging:
Object stack scales to image stack
Influence of the eye by Convolution with incoherent transfer function = PSF
Monofocal IOL: Diffraction-limited PSF
Bifocal IOL: construction of PSF from avaiable data
Object(x,y,z) o PSF(x,y,z) = Image(x,y,z)
Object Image
PSF
Base
PSF
Add Power
@Infinity @4dpt @Infinity @4dpt
monofocal
imaging
O =
77
7
8
3-D imaging
Bifocal Imaging:
Object stack scales to image stack
Influence of the eye by Convolution with incoherent transfer function = PSF
Monofocal IOL: Diffraction-limited PSF
Bifocal IOL: construction of PSF from avaiable data
Object(x,y,z) o PSF(x,y,z) = Image(x,y,z)
Object Image PSF
Base
PSF
Add Power
@Infinity @4dpt @Infinity @4dpt
Bifocal
imaging
O =
2. Image
78
Pure diffractive lens Valle, 2005
symmetric sinus-profile ~cos(r²)
Intensity distribution @ far – near - intermediate
@ 4mm 30% - 30% - 30% sinus-contrast ~ 32%
@ 2mm 40% - 35% - 25% contrast ~ 22%
stop 4mm
stop 2mm
Symmetric intensity distribution
Reduced contrast in comparison to ideal bifocal
lens
Asymmetry for very small pupils
Simulation of visual performance 79
80
Simulation of visual performance
Quartic Axicon Ares, 2005
Wavefont ~ A r4 – B r²
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.020
0.01
0.02
0.03
0.04
0.05
0.06
Punkt - Kontrast K = 0.71382
PSF(r,z) PSF(z) and PSF(r, z = 0)
-1.5 -1 -0.5 0 0.50
0.01
0.02
0.03
0.04
0.05
0.06
Contrast:
2 points 60%
sinus 7%
Visible symbols
for VA = 0.5
80