imaging and aberration theory · 2017. 1. 18. · aberrations, herschel condition, relation to...
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www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 13: Point spread function
2017-01-26
Herbert Gross
Winter term 2016
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2
Preliminary time schedule
1 20.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 27.11.Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 03.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 10.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas
5 17.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 24.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders
7 01.12. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 08.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 15.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 05.01.Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 12.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 19.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement
13 26.01. PSF ideal psf, psf with aberrations, Strehl ratio
14 02.02. Transfer function Transfer function, resolution and contrast
15 09.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
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1. Introduction
2. Ideal PSF
3. PSF with aberrations
4. High-NA PSF
5. Two-point resolution
Contents
3
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Typical change of the intensity profile
Normalized coordinates
Diffraction integral
4
Fresnel Diffraction
z
geometrical
focus
f
a
z
stop
far zone
geometrical
phase
intensity
a
rr
f
avz
f
au
;
2;
22
1
0
20
/
2
0
2 22
)(2
),(
devJef
EiavuE
ui
uafi
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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
5
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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
6
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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
7
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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
8
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Ideal Psf
r
z
I(r,z)
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
9
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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
10
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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
11
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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%
12
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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
13
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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%
14
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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
Psf with Aberrations
Psf for some low oder Zernike coefficients
The coefficients are changed between cj = 0...0.7
The peak intensities are renormalized
15
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
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Growing spherical aberration shows an asymmetric behavior around the nominal image
plane for defocussing
16
Caustic with Spherical aberration
c9 = 0 c9 = 0.7c9 = 0.3 c9 = 1
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PSF with coma
The 1st diffraction ring is influenced very sensitive
W31 = 0.03 W31 = 0.06 W31 = 0.09 W31 = 0.15
Psf for Coma Aberration
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Separation of the peak and the centroid position in a point spread function with coma
From the energetic point of view coma induces distortion in the image
Psf with Coma
c7 = 0.3 c7 = 0.5 c7 = 1
centroid
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x
y
PSF as a function
of the field height
Interpolation is
critical
Orientation of coma
in the field
Small field area with
approximately shift-
invariant PSF:
isoplanatic patch
Variation of Performance with Field Position
x = 0 x = 20% x = 40% x = 60 % x = 80 % x = 100 %
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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
20
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Point Spread Function with Apodization
Apodization:
Non-uniform illumination of the pupil amplitude
Modified point spread function:
Weighting of the elementary wavelets by amplitude distribution A(xp,yp),
Redistribution of interference
System with wave aberrations:
Complicated weighting of field superposition
For edge decrease of apodization: residual aberrations at the edge have decreased
weighting and have reduced perturbation
Definition of numerical aperture angle no longer exact possible
For continuous decreased intensity towards the edge:
No zeros of the PSF intensity (example: gaussian profile)
Modified definition of Strehl ratio with weighting function necessary
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
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Point Spread Function with Apodization
w
I(w)
1
0.8
0.6
0.4
0.2
00 1 2 3-2 -1
Airy
Bessel
Gauss
FWHM
w
E(w)
1
0.8
0.6
0.4
0.2
03 41 2
Airy
Bessel
Gauss
E95%
Apodisation of the pupil:
1. Homogeneous Airy
2. Gaussian Gaussian
3. Ring illumination Bessel
Psf in focus:
different convergence to zero forlarger radii
Encircled energy:
same behavior
Complicated:Definition of compactness of thecentral peak:
1. FWHM: Airy more compact as GaussBessel more compact as Airy
2. Energy 95%: Gauss more compact as AiryBessel extremly worse
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Ring shaped pupil illumination
outer radius aa
innen radius ai
parameter
Depth of focus enlarged
Central peak shrinks lateral
Can not be understood geometrically
Psf of a Ring Pupil
1a
i
a
a
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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
Psf of an Annular Pupil
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Encircled energy curva:
- steps due to side lobes
- strong spreading, large energy content in the rings
r
E(r)
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
= 0.01
= 0.25
= 0.35
= 0.50
= 0.70
PSF for ring-Shaped Pupil
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r
z
I(r,z)
How to quantify the aberrated PSF ?
26
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Focal Diameter Determination
27
1 2 3 4 5 6
Pupil shape
definiton of focus
Gauss radius w
no trunc..
Super gauss w,m=6
no trunc.
Gauss 2a=3w
with trunc.
Gauss a=w with
trunc.
Circle homoge-
neous D=2a
Linear / homog. slit width
2a
1. Zero point of intensity
#
#
#
1.426
1.220 ( Airy )
1.00
I=0.5 ( FWHM )
0.375
0.513
0.644
0.564
0.519
0.443
I=0.13534 (1/e2 )
0.637
=2/
0.831
1.059
0.914
0.822
0.697
I=0.01
0.966
1.122
1.491
1.238
1.092
( Peak )
0.908
I=0.001
1.183
2.109
1.695
2.925
1.174
( Peak )
0.969
E=0.86466
0.637
=2/
0.818
1.008
0.890
1.378
0.658
E=0.95
0.779
1.040
1.201
1.104
3.915
1.989
Focal diameter:
Basic scaling factor
Additional factor b
Factor differs in size by
one magnitude
NAD foc
b
wbeamradius
truncationno
w
f
aradiusat
truncationwith
a
f
NA
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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 )
28
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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
29
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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
30
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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%
31
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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
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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
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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)','(
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Pupil
Vectorial Diffraction at high NA
Linear Polarization
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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
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Low Fresnel Number Focussing
37
Small Fresnel number NF:
geometrical focal point (center point of spherical wave) not identical with best
focus (peak of intensity)
Optimal intensity is located towards optical system (pupil)
Focal caustic asymmetrical around the peak location
a
focal length f
z
physical
focal
point
geometrical
focal point
focal shift
f
wavefront
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38
Example: focusing by a microlens
radius: 0.1 mm
wavelength: 1 μm
focal length: 10 mm
NA: 0.01
Fresnel #: 1
Result:
highest intensity in front of
geometrical focus consequence of constructive
interference of edge diffraction
Low Fresnel Number Focussing
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Systems with small Fresnel number
Same size of:
1. divergence angle and numerical aperture angle
2. focal length and depth of focus
3. influence of geometrical beam shaping and diffraction
Effect of photometric distance law inside depth of focus
Focal shift depends on Fresnel number, apodization and coherence
For definition of best focus by peak intensity on axis:
Relative focal shift for circular homogeneous illuminated pupil:
2
1)(
zzI
Low NA and Focal Shift
0),0(
z
zI foc
2
2
0
4
4sin
21)(
u
u
N
uIzI
F
f
f NF
1
112
2
1 −𝑢
2𝜋𝑁𝐹=𝑓
𝑧
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Low Fresnel Number Focussing
40
System with small Fresnel number:
Axial intensity distribution I(z) is
asymmetric
Explanation of this fact:
The photometric distance law
shows effects inside the depth of focus
Example microscopic 100x0.9 system:
a = 1mm , z = 100 mm:
NF = 18
2
2
0
4
4sin
21)(
u
u
N
uIzI
F
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
NF = 2
NF = 3.5
NF = 5
NF = 7.5
NF = 10
NF = 20
NF = 100
z / f - 1
I(z)
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Focal shift as a function of
1. Fresnel number
2. Apodization
in linear and logarithmic representation
Low NA and Focal Shift
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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
42
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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
43
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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
44
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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
45
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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
46
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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
47
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Incoherent Resolution: Dependence on NA
Microscopical resolution as a function of the numerical aperture
NA = 0.9NA = 0.45NA = 0.3NA = 0.2
48
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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
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Depth of Focus
Schematic drawing of the principal ray path in case of extended depth of focus
Where is the energy going ?
What are the constraints and limitations ?
conventional ray path
beam with extended
depth of focus
z
z0
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Normalized axial intensity
for uniform pupil amplitude
First zero:
depth of focus: 1 Rayleigh length
peak intensity reduced to 80%
transverse width doubled
- Gaussian beams: similar formula
22
'
'sin' NA
n
unRE
Depth of Focus: Diffraction Consideration
2' o
Rn
z
0z
focal
plane
beam
caustic
z
depth of focus
0.8
1
I(z)
z-Ru/2 0
r
intensity
at r = 0
+Ru/2
-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
2
0
sin)(
z
zIzI z
NAz
2
2
ERz 2
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𝑟𝑝
z
r
Axial Point-Spread Function52
Intensity I(z)
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𝑟𝑝
z
r
Annular Ring Pupil (Bessel Beams)53
Intensity I(z)
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𝑟𝑝
z
r
Engineering the PSF: Simple Toraldo Mask54
Intensity I(z)
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𝑟𝑝
z
r
Spherical Aberration (1λ)55
Intensity I(z)
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𝑟𝑝
z
r
Gaussian Apodization56
Intensity I(z)