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www.iap.uni-jena.de
Medical Photonics Lecture
Optical Engineering
Lecture 7: Image Quality
2017-12-07
Herbert Gross
Winter term 2017
2
Schedule Optical Engineering 2017
No Subject Ref Date Detailed Content
1 Introduction Gross 19.10. Materials, dispersion, ray picture, geometrical approach, paraxial approximation
2 Geometrical optics Gross 02.11. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant
3 Diffraction Gross 09.11. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function
4 Components Kempe 16.11. Lenses, micro-optics, mirrors, prisms, gratings
5 Optical systems Gross 23.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting
6 Aberrations Gross 30.11. Introduction, primary aberrations, miscellaneous 7 Image quality Gross 07.12. Spot, ray aberration curves, PSF and MTF, criteria
8 Instruments I Kempe 14.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes
9 Instruments II Kempe 21.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts
10 Instruments III Kempe 11.01. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes
11 Optic design Gross 18.01. Aberration correction, system layouts, optimization, realization aspects
12 Photometry Gross 25.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory
13 Illumination systems Gross 01.02. Light sources, basic systems, quality criteria, nonsequential raytrace
14 Metrology Gross 08.02. Measurement of basic parameters, quality measurements
3
Performance Criteria Overview
Applications
Geometrical
model
Diffraction
model
Longitudinal
aberrations
Transverse
aberration curves
Spot diagrams
Wave aberrations
AdvantagesQuantitative
numbers
scaling on Rayleigh unit
scaling on Airy diameter
rms
pv
scaling on Airy diameters
rms
pv
Zernike decomposition
RepresentationsLimitations
Problems
astigmatism
axial chromatical
field curvature
not useful in the field
not defined for afocal
camera lensessimple direct analysis
possible
1 curve per field point
to be re-defined for afocal
any illustrative analysis complicated
any
direct measurable
scaling on wavelength
all orders separated
only one wavelength
only one field point
normalization radius of Zernikes
Point spread
function
Modulation transfer
function
Strehl ratio
scaling on Airy diameter
Hopkins number
microscopy
astronomy
diffraction limited
direct relation to resolution
easy white light formulation
computational problems for large
aberrations
camera lenses
lithography
projection lenses
direct analysis possible
easy white light formulation
computational problems for large
aberrations
analysis complicated
Spot Diagram
Table with various values of:
1. Field size
2. Color
Small circle:
Airy diameter for
comparison
Large circle:
Gaussian moment
486 nm 546 nm 656 nm
axis
fieldzone
fullfield
4
Gaussian Moment Spot
Spot pattern with transverse aberrations xj and yj
1. centroid
2. 2nd order moment
3. diameter
Generalized:
Rays with weighting factor gj:
corresponds to apodization
Worst case estimation:
size of surrounding rectangle Dx=2xmax, Dy = 2ymax
xN
xS jj
1
yN
yS jj
1
M rN
x x y yG j S j Sj
22 21
rmsG rMD 22
M rN
g x x y yG
G
j j S j S
j
22 21
5
Wave Aberration
Definition of the peak valley value WPV
Reference sphere corresponds to perfect imaging
Rms-value is more relevant for performance evaluation
exit
aperture
phase front
reference
sphere
wave
aberration
pv-value
of wave
aberration
image
plane
6
Wave Aberration Criteria
Mean quadratic wave deviation ( WRms , root mean square )
with pupil area
Peak valley value Wpv : largest difference
General case with apodization:
weighting of local phase errors with intensity, relevance for psf formation
dydxAExP
ppppmeanpp
ExP
rms dydxyxWyxWA
WWW222 ,,
1
pppppv yxWyxWW ,,max minmax
pppp
w
meanppppExPw
ExP
rms dydxyxWyxWyxIA
W2)(
)(,,,
1
7
8
PSD Ranges
Typical impact of spatial frequency
ranges on PSF
Low frequencies:
loss of resolution
classical Zernike range
High frequencies:
Loss of contrast
statistical
Large angle scattering
Mif spatial frequencies:
complicated, often structured
fals light distributions
log A2
Four
low spatial
frequency
figure errormid
frequency
range micro roughness
1/
oscillation of the
polishing machine,
turning ripple
10/D1/D 50/D
larger deviations in K-
correlation approach
ideal
PSF
loss of
resolution
loss of
contrast
large
angle
scattering
special
effects
often
regular
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
4
PVW
Rayleigh Criterion
The Rayleigh criterion
gives individual maximum aberrations
coefficients,
depends on the form of the wave
Examples:
aberration type coefficient
defocus Seidel 25.020 a
defocus Zernike 125.020 c
spherical aberration
Seidel 25.040 a
spherical aberration
Zernike 167.040 c
astigmatism Seidel 25.022 a
astigmatism Zernike 125.022 c
coma Seidel 125.031 a
coma Zernike 125.031 c
11
a) optimal constructive interference
b) reduced constructive interference
due to phase aberrations
c) reduced effect of phase error
by apodization and lower
energetic weighting
d) start of destructive interference
for 90° or /4 phase aberration
begin of negative z-component
Rayleigh criterion:
1. maximum of wave aberration: Wpv < /4
2. beginning of destructive interference of partial waves
3. limit for being diffraction limited (definition)
4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil
5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...)
Marechal criterion:
1. Rayleigh crierion corresponds to Wrms < /14 in case of defocus
2. generalization of Wrms < /14 for all shapes of wave fronts
3. corresponds to Strehl ratio Ds > 0.80 (in case of defocus)
4. more useful as PV-criterion of Rayleigh
Criteria of Rayleigh and Marechal
14856.13192
Rayleigh
rmsW
12
Impression of CHV in real images
Typical colored fringes blue/red at edges visible
Color sequence depends on sign of CHV
Chromatic Variation of Magnification
13
Axial Chromatical Aberration
Special effects near black-white edges
boarder
magenta
blue boarder
Ref: J. Kaltenbach
14
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 )
16
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
17
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
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
Normalized axial intensity
for uniform pupil amplitude
Decrease of intensity onto 80%:
Scaling measure: Rayleigh length
- depth of focus: 1RE
- Gaussian beams: similar formula
22
'
'sin' NA
n
unRE
Depth of Focus: Diffraction Consideration
2
0
sin)(
u
uIuI
20' on
z
Ediff 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
Criteria for measuring the degradation of the point spread function:
1. Strehl ratio
2. Standard deviation
3. Full width half maximum (FWHM)
4. Second moment
5. Correlation with perfect PSF
6. Various energy-based widths
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%
22
The PSF is very sensitive to coma
In a well constructed system, 5-7 diffraction rings are observable by visual inspection
In the case of coma, the asymmetry of the pattern is particularly sensitive
The 1st diffraction ring is visibly influenced by a Zernike coefficient as small as /30
c31 = 0.03 c31 = 0.06 c31 = 0.09 c31 = 0.15
Point Spread Function for Coma Aberration
Differences
between Strehl
and Ipeak,
if the profile
is structured
24
Strehl Ratio and PSF-Peak Height for Aberrations
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
coma c8
astigmatism c5
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
c4 [] c5 []
c9 [] c8 []
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
defocussing c4
Strehl peak
0 0.5 1 1.5 2
spherical aberration c9
0
0.2
0.4
0.6
0.8
1
Incoherent Image Formation
astigmatism comaspherical
aberrationobject ideal
PSF
Example:
incoherent imaging of pattern near the resolution limit with aberrations
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
Encircled Energy
Relative amount of energy passing a variable stop:
- encircled energy
- power in the bucket
General formulation:
Special case of circular symmetry:
E r I x x y y dx dyS S
x r y
x r y
y r
y r
( ) ,
2 2
2 2
E r I r r dr
r
( ) ( ) 20
Encircled Energy Function
The encircled energy function shows structured behavior in case of aberrations
Larger sensitivity for small intensity levels (sidelobes)
Problem: Reference in case of apodization or central obscuration
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ecirc
(r)
r / rAiry
ideal
spherical / 4
ring with
= 0.3
coma / 4
)(
)()(
rE
rErEcirc
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
29
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
30
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
31
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
32
Incoherent Resolution: Dependence on NA
Microscopical resolution as a function of the numerical aperture
NA = 0.9NA = 0.45NA = 0.3NA = 0.2
33
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
34
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 sinusoidal 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
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
36
Consideration of the complete area under the MTF curve in the relevant interval
of spatial frequencies
In anisotropic systems:
volume under MTF-surface
Quite good correlation with visual
perception for visual systems
21
)(vvv
MTFMTFa dvvHKA
MTF-Area-Criterion
gMTF
1
AMTFa
2
37
Photographic lenses with different performance
38
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
Microscope Resolution with Immersion
Imaging of a Chromium mask with 125 nm pitch
Imaging without / with water immersion
Enhancement of resolution and contrast
Ref: W. Osten
150x/0.9 air 200x/1.2 water immersion Lens (Leica)
dydxyxE
dydxyxExx
m
m
2
2
),(
),(
ddE
ddEm
m
2
2
),(
),(
o
xxxx
k
kvu
Quality of Laser Beams: Moments
Conventional criteria of imaging systems are nor useful for laser beams:
1. significant apodization
2. no imaging application
3. status of coherence may be complicated
Description of the complex fields by moments of second order:
1. spatial moments of intensity profile
second moments describes beam width
third moment describes asymmetry
2. angular moment of the direction distribution
second moment describes the divergence
Alternative descriptions of impuls:
1. angle ux
2. spatial frequency x
3. transverse wavenumer kx
Mixed moments: description of twist effects
2222
xxxxx wwM
M wx ox x
2
Quality of Laser Beams: M2
Characterizing beam quality
M2
Special case: definition in waist plane
Properties of M2:
1. Gaussian beam TEM00: M2 = 1
Smallest possible value
2. Paraxial optical systems: M2 remains constant for propagation
3. Real beams: M2 > 1 describes the decrease in quality and focussability
relative to a gaussian beam
Reasons for degradation of beam quality:
1. intensity profile
2. phase perturbation
3. finite degree of coherence
Incoherent mixture of modes: additive composition of M2
General beams: components and mixed terms 4442
2
1xyyx MMMM
a) object
Image quality with Real Objects
b) good image c) defocussed d) axial chromatic
aberration
e) lateral chromatic
aberration
g) chromatical
astigmatism
f) sphero-
chromatism
Real Image with Different Chromatical Aberrations
original object good image color astigmatism 2
6% lateral color axial color 4
Energy Transmission of Microscope Lens
Reduced throughput due to
1. absorption
2. coatings
Strong spectral dependence
T [%]
100
80
60
40
20
0
1 3 5 7 9 11 13surfaces
365 nm
546 nm
400 nm440 nm700 nm644 nm600 nm
absorption
Simple model:
Finite residual reflectivity R at N surfaces
Considering energetic transfer
from signal to false light level,
Multiple reflecting light taken
into account
False light intensity:
Effect of False Light on SNR
NF RRN
RI
1
)1(1
1
I
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
R = 0.995
R = 0.99
R = 0.98
R = 0.97
R = 0.95
N
Transmission
Signal
False light
decreasing
signal
(1-R = 2%)
increasing
false light
Achromate
Residual aberrations of an achromate
Clearly seen:
1. Distortion
2. Chromatical magnification
3. Astigmatism
51
Contrast and Resolution
original 256 x 256 blurr 3 pixel blurr 6 pixel blurr 9 pixel
original straylight 15% straylight 30% straylight 50%
SSim Image Quality Metric
Idea : combined criterion with best correlation to subjective performance
Best modelling of human visual, cortical and neuronal perception system
SSim = structural similarity index measure
Three major aspects taken into account with weighted superposition:
1. Luminence / brightness I(x,y)
2. Contrast C(x,y)
3. Structures / information S(x,y)
Mathematical definition: three autocorrelation-terms with adaptable
parameters Cj:
Experience:
Good correlation with subjective judgement of test persons (visual perception)
Extended version: complex SSim
Better consideration of slight image shift / rotation
3
3
2
22
2
1
22
1 22
),(),(),(),(
C
C
C
C
C
C
yxsyxcyxlyxS
yx
yx
yx
yx
yx
yx
SSim Image Quality Metric
Examples of
SSim-values: a) reference
b) contrast
c) luminance
d) white noise
e) impulsive noise
f) JPEG compression
g) Blurr
h) zooming
i) shift right
j) shift left
k) rotation counter cw
l) rotation clockwise
Ref: Wang / Bovik
SSim - Comparison to MSE-Metric
Classical MSE-metric (rms difference to reference image):
Fails for quite simple degradations
Better significance by SSim
starting
point
optimization
with SSim
Ref: Wang / Bovik
SSim - Comparison to MSE
SSim:
- Can be defined as field
(not a single number)
- Indicates local differences to
the reference image
blurredoriginal
MSE SSim
Ref: Wang / Bovik