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Medical Photonics Lecture
Optical Engineering
Lecture 6: Diffraction
2016-12-01
Herbert Gross
Winter term 2016
2
Contents
No Subject Ref Date Detailed Content
1 Introduction Gross 20.10. Materials, dispersion, ray picture, geometrical approach, paraxial approximation
2 Geometrical optics Gross 03.11. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant
3 Components Kempe 10.11. Lenses, micro-optics, mirrors, prisms, gratings
4 Optical systems Gross 17.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting
5 Aberrations Gross 24.11. Introduction, primary aberrations, miscellaneous
6 Diffraction Gross 01.12. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function
7 Image quality Gross 08.12. Spot, ray aberration curves, PSF and MTF, criteria
8 Instruments I Kempe 15.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes
9 Instruments II Kempe 22.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts
10 Instruments III Kempe 05.01. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes
11 Optic design Gross 12.01. Aberration correction, system layouts, optimization, realization aspects
12 Photometry Gross 19.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory
13 Illumination systems Gross 26.01. Light sources, basic systems, quality criteria, nonsequential raytrace
14 Metrology Gross 02.02. Measurement of basic parameters, quality measurements
Diffraction at an edge in Fresnel
approximation
Intensity distribution,
Fresnel integrals C(x) and S(x)
scaled argument
Intensity:
- at the geometrical shadow edge: 0.25
- shadow region: smooth profile
- bright region: oscillations
22
)(2
1)(
2
1
2
1)( tStCtI
FNxz
xz
kt 2
2
Fresnel Edge Diffraction
t-4 -2 0 2 4 6
0
0.5
1
1.5
I(t)
Diffraction at a slit:
intensity distribution
Angles with side maxima:
constructive interference
Diffraction pattern
d
0. order
1st diffraction
order
2
sin
sinsin
)(
d
d
I
2
1sin max m
d
Fraunhofer Farfield Diffraction
order: -3rd -2nd -1st 0th +1st +2nd +3rd
Fraunhofer Diffraction at a Rectangle
Diffraction at a square / rectangle shaped aperture
Smaller width: larger diffraction angle
square aperture rectangular aperture
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
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
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
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
Ideal Psf
r
z
I(r,z)
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
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%
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
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%
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
Psf with Aberrations
Psf for some low oder Zernike coefficients
The coefficients are changed between cj = 0...0.7
The peak intensities are renormalized
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
15
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
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
2. Gaussian
3. Bessel
Psf in focus:
different convergence to zero for larger radii
Encircled energy:
same behavior
Complicated: Definition of compactness of the central peak: 1. FWHM: Airy more compact as Gauss Bessel more compact as Airy
2. Energy 95%: Gauss more compact as Airy Bessel extremly worse
Small aperture:
Diffraction limited
Spot size corresponds
to Airy diameter
Spot size depends on
wavelength
Large aperture:
Diffraction neglectible
Aberration limited
Geometrical effects not
wavelength dependent
But: small influence of
dispersion
Log Dfoc
Log sinu
f=1000 , 500 , 200 , 100 , 50 , 20 , 10 mm
= 10 m
= 1 m
550 nm
1
0
-1
-2-2-3 -1 0
Focussing by a Lens: Diffraction and Aberration
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 %
Growing spherical aberration shows an asymmetric behavior around the nominal image
plane for defocussing
20
Caustic with Spherical aberration
c9 = 0 c9 = 0.7c9 = 0.3 c9 = 1
Low Fresnel Number Focussing
21
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
Low Fresnel Number Focussing
22
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)
Diffraction at an edge in Fresnel
approximation
Intensity distribution,
Fresnel integrals C(x) and S(x)
scaled argument
Intensity:
- at the geometrical shadow edge: 0.25
- shadow region: smooth profile
- bright region: oscillations
22
)(2
1)(
2
1
2
1)( tStCtI
FNxz
xz
kt 2
2
Fresnel Edge Diffraction
t-4 -2 0 2 4 6
0
0.5
1
1.5
I(t)
Line image: integral over point sptread function
LSF: line spread function
Realization: narrow slit
convolution of slit width
But with deconvolution, the PSF can be reconstructed
dyyxIxI PSFLSF ),()(
Integration
intens
ity
x
Line spread function
PSF
dyyxIxI PSFLSF ),()(
Line Image
ESF, PSF and ESF-Gradient
Typical behavior of intensity of an edge image for residual aberrations
The width of the distribution roughly corresponds to the diameter of the PSF
Derivative of the edge spread function:
edge position at peak
location
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
25
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 dxdyx 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.
27
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
Location of diffraction orders in the
back focal plane
Increasing grating angle with spatial
frequency of the grating
29
Abbe Theory of Microscopic Resolution
+1st
-1st
+1st
-1st
+1st
-1st
objectback focal
planeobjective
lens
0th order
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
pppp
pp
vyvxi
pp
yxOTF
dydxyxg
dydxeyxg
vvH
ypxp
2
22
),(
),(
),(
),(ˆ),( yxIFvvH PSFyxOTF
pppp
pp
y
px
p
y
px
p
yxOTF
dydxyxP
dydxvf
yvf
xPvf
yvf
xP
vvH
2
*
),(
)2
,2
()2
,2
(
),(
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
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
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
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
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
35
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
36
Contrast / Resolution of Real Images
resolution,
sharpness
contrast,
saturation
Degradation due to
1. loss of contrast
2. loss of resolution
37
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
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
39
Resolution Test Chart: Siemens Star
original good system
astigmatism comaspherical
defocusa. b. c.
d. e. f.
40
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 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
(isoplanasy):
incoherent convolution
Intensities are additive
dydxyxIyyxxgyxI psfinc
),()','()','(2
),(*),()','( yxIyxIyxI objpsfimage
dydxyxIyyxxgyxI psfinc
),(),',,'()','(2
Fourier Theory of Incoherent Image Formation
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 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
Blurred imaging:
- limiting case
- information extractable
Blurred imaging:
- information is lost
- what‘s the time ?
Resolution: Loss of Information
45