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Design and Correction of optical Systems
Part 9: Fourier opticcal image formatioin
Summer term 2012
Herbert Gross
1
Overview
1. Basics 2012-04-18
2. Materials 2012-04-25
3. Components 2012-05-02
4. Paraxial optics 2012-05-09
5. Properties of optical systems 2012-05-16
6. Photometry 2012-05-23
7. Geometrical aberrations 2012-05-30
8. Wave optical aberrations 2012-06-06
9. Fourier optical image formation 2012-06-13
10. Performance criteria 1 2012-06-20
11. Performance criteria 2 2012-06-27
12. Measurement of system quality 2012-07-04
13. Correction of aberrations 1 2012-07-11
14. Optical system classification 2012-07-18
2012-04-18
9.1 Huygens principle of diffraction
9.2 Perfect point spread function
9.3 PSF with perturbations
9.4 Basics of Fourier optics
9.5 Image formation
9.6 Optical Transfer function
Contents:
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
objectpoint
sphericalwave
truncatedspherical
wave
imageplane
∆∆∆∆x = 1.22 λλλλ / NA
point
spread
function
object plane
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
Fλ
),(2),(),( pp yxWi
pppp eyxTyxAπ
⋅=
( ) ( ) ( )
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
+
⋅⋅=∫∫λ
ππ
''cos'
)'()(
'
dydxrr
erE
irE d
rrki
I ∫∫ ⋅−
⋅−=−
θλ
rrrr
rrr
( )( )
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 π z n / λ NA2
transversal : v = 2 π x / λ NA
Perfect Point Spread Function
NADAiry
λ⋅=
22.1
2NA
nRE
λ⋅=
Ideal Psf
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
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 λ/NA:
Assumptions for this estimation and possible changes
� A resolution beyond the Abbe limit is only possible with violating of certain
assumptions
� 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
⋅
⋅=
λ
2
1
2
22
)(
⋅
=
NAr
NAr
J
rI
λ
π
λ
π
'sin'sin2
θλ
πak
R
akrukr
R
arx ====
Perfect Lateral Point Spread Function: Airy
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%
� 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
2u
zNA
z =⋅⋅
=λ
π
'sin'22 un
RE
λ=
Perfect Axial Point Spread Function
Psf with Defocus
� Outside focus :
- Psf is broadened
- a system of rings
takes place
- symmetry around
image plane
Focus
image plane
Defocus
- 2 RE
Defocus
+ 2 RE
Defocus
+ 1 RE
Defocus
- 1 RE
Defocussed Perfect Psf
� Perfect point spread function with defocus
� Representation with constant energy: extreme large dynamic changes
∆∆∆∆z = -2RE ∆∆∆∆z = +2RE∆∆∆∆z = -1RE ∆∆∆∆z = +1RE
normalized
intensity
constant
energy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
Spherical aberration Astigmatism Coma
c = 0.2
c = 0.3
c = 0.7
c = 0.5
c = 1.0
Psf with Aberrations
� Zernike coefficients c in λ
� Spherical aberration,
Circular symmetry
� Astigmatism,
Split of two azimuths
� Coma,
Asymmetric
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 +=
Definitions of Fourier Optics
� Phase space with spatial coordinate x and
1. angle θ
2. spatial frequency ν 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 ν = 1/g
0k
kv x
xx =⋅= λθ
[ ]),(ˆ),( yxEFvvA yx =
( ) ( )A k k z E x y z e dx dyx y
i xk ykx y, , ( , , )=
− +
∫∫
vk π2=
vg
⋅=⋅= λλθ1
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
λ
α
λ
λα
==
⋅=⋅
sin
sin
off-axis illumination
NAg
⋅=
2
λ
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
� 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. order
0. / 1. / 2.
order
0. / 1. / 2. / 3. order
Resolution: Number of Supported Orders
Abbe Theorie of the Microscopic Resolution
� Diffraction of the illumination wave at the object structure
� Occurence of the diffraction orders in the pupil
� Image generation by constructive interference of the supported orders
� Object details with high spatial frequency are blocked by the system aperture and can not
be resolved
-1
1
object planesource
pupil plane
image plane
0
imaginglens
Ref: W. Singer
� Imaging of a crossed grating object
� Spatial frequency filtering by a slit:
� Case 1:
- pupil open
- Cross grating imaged
� Case 2:
- truncation of the pupil by a split
- only one direction of the grating is
resolved
Fourier Filtering
� Improved resolution by oblique illumination in microscopy
� Enhancement only in one direction
Oblique Illumination in Microscopy
Fourier Optical Fundamentals
� Helmholtz wave equation:
Propagation with Green‘s function g,
Amplitude transfer function, impulse response
� For shift-invariance:
convolution
� Green‘s function of a spherical wave
Fresnel approximation
� Calculation in frequency space: product
� Optical systems:
Impulse response g(x,y) is coherent transfer function, point spread function (PSF).
G(νx,νy) corresponds to the complex pupil function
� Fourier transform: corresponds to a plane wave expansion
),(*),','(),','( yxEzyyxxgzyxE −−=
g r rr r
ei k r r
( , ' )'
'r rr r
r r
=−
− −1
4π
g x y zi
ze e
ikz
i
zx y
( , , )( )
= −− +
λ
π
λ2 2
),(),,(),,( yxyxyx vvEzvvGzvE ⋅=ν
)(22
),,( yx vvziikz
yx ezvvG+⋅+−
=λπ
dydxyxEyxyxgzyxE ∫ ∫∞
∞−
∞
∞−
⋅= ),(),,','(),','(
( )f
yxyxP
pp
pp⋅
+⋅−=
λ
π 22
),(
f
yv
f
xv
p
y
p
xλλ
== ,
Fourier Optical Field Propagation through a Lens
� Lens works as a pupil filter,
Phase:
Complex pupil function, transmission t
� Incident plane wave
Field in focal plane,
Far field, Fourier transform
� Image point coordinates correspond to the spatial frequencies
( )( )
∫∫+−
+
⋅⋅= pp
yyxxf
i
pp
yxf
i
dydxeyxAfi
efyxE
pp ''2
''
),(),','(
22
λ
πλ
π
λ
),(),(),( pp yxiP
pppp eyxtyxA ⋅=
Fourier Optics – Point Spread Function
� 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 Coherent Image Formation
� Transfer of an extended
object distribution I(x,y)
� In the case of shift invariance
(isoplanasie):
coherent convolution of fields
� Complex fields are
additive
( )∫∫ ⋅= dydxyxEyxyxgyxE psf ),(',',,)','(
( )∫∫ ⋅−−= dydxyxEyyxxgyxE psf ),(',')','(
( ) ),(,)','( yxEyxgyxE psf ∗=
Fourier Theory of Coherent Image Formation
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 invariance
(isoplanasie):
incoherent convolution
� Intensities are additive
dydxyxIyyxxgyxI psfinc ∫ ∫∞
∞−
∞
∞−
⋅−−= ),()','()','(2
),(*),()','( yxIyxIyxI objpsfimage =
dydxyxIyyxxgyxI psfinc ∫ ∫∞
∞−
∞
∞−
⋅= ),(),',,'()','(2
Fourier Theory of Incoherent Image Formation
objectintensity
I(x,y)
squared PSF,intensity-responseIpsf
(xp,y
p)
imageintensityI'(x',y')
convolution
result
object
intensityspectrum
I(vx,v
y)
opticaltransferfunction
HOTF
(vx,v
y)
imageintensityspectrumI'(v
x',v
y')
produkt
result
Fouriertransform
Fouriertransform
Fouriertransform
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
-0.05 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
-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
Incoherent Image Formation in Frequency Space
� Incoherent illumination:
No correlation between neighbouring object points
Superposition of intensity in the image
� In the case of shift invariance
(isoplanasie):
Incoherent imaging with convolution
� In frequency space:
Product of spectra, linear transfer theory
The spectrum of the psf works as low pass filter onto the object spectrum
Optical transfer function
dydxyxIyyxxgyxI psfinc ∫ ∫∞
∞−
∞
∞−
⋅−−= ),()','()','(2
),(*),()','( yxIyxIyxI objpsfimage =
dydxyxIyyxxgyxI psfinc ∫ ∫∞
∞−
∞
∞−
⋅= ),(),',,'()','(2
[ ]),(),( yxIFTvvH PSFyxotf =
),(),(),( yxobjyxotfyximage vvIvvHvvI ⋅=
( )
pppp
pp
vyvxi
pp
yxOTF
dydxyxg
dydxeyxg
vvH
ypxp
∫ ∫
∫ ∫∞
∞−
∞
∞−
∞
∞−
∞
∞−
+−⋅
=2
22
),(
),(
),(
π
[ ]),(ˆ),( yxIFvvH PSFyxOTF =
pppp
pp
y
px
p
y
px
p
yxOTF
dydxyxP
dydxvf
yvf
xPvf
yvf
xP
vvH
∫ ∫
∫ ∫∞
∞−
∞
∞−
∞
∞−
∞
∞−
−−⋅++
=2
*
),(
)2
,2
()2
,2
(
),(
λλλλ
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
xp
y p
area of
integration
shifted pupil
areas
f x
yf
p
q
x
y
x
y
L
L
x
y
o
o
x'
y'
p
p
light
source
condenserconjugate 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
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
λλ
'sin u
f
avo ==
λλ
'sin222 0
un
f
navvG ===
−
−
=
2
000 21
22arccos
2)(
v
v
v
v
v
vvH MTF
π
),(),(),( yxPTF vvHi
yxMTFyxOTF evvHvvH ⋅=
∆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
decreasedslope
decreased
minima
increased
� 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
ψψψψ
y
tangential
plane
tangential sagittal
arbitrary
rotated
x sagittalplane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
0 0.5 1
νννν / ννννmax
� Circular disc with diameter
D = d x Dairy
� Small d << 1 : Airy disc
� Increasing d:
Diffraction ripple disappear
Incoherent Image of a Circular Disc
Outlook
Next lecture: Part 10 – Performance Criteria I
Date: Wednesday, 2012-06-20
Contents: 10.1 Introduction
10.2 Geometrical aberrations
10.3 Wave aberrations
10.4 Point spread function and point resolution
10.5 Depth of focus