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Physical Optics
Lecture 10: PSF Engineering
2019-01-10
Herbert Gross
Physical Optics: Content
2
No Date Subject Ref Detailed Content
1 18.10. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,
interferometry
2 25.10. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread
function, aberrations
3 01.11. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,
phase imaging
4 08.11.Quality criteria and
resolutionG
Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point
resolution, criteria, contrast, axial resolution, CTF
5 22.11. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,
fluorescence, Jablonski diagram, lifetime, quantum yield, FRET
6 29.11. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,
speckle, OCT-principle
7 06.12. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,
components
8 13.12. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,
pulses, power
9 20.12. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,
CARS microscopy, 2 photon imaging
10 10.01. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,
confocal PSF
11 17.01. Scattering GIntroduction, surface scattering in systems, volume scattering models,
calculation schemes, tissue models, Mie Scattering
12 24.01. Gaussian beams G Basic description, propagation through optical systems, aberrations
13 31.01. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy
beams, applications in superresolution microscopy
14 07.02. Miscellaneous G Coatings, diffractive optics, fibers
K = Kempe G = Gross
PSF engineering
Low Fresnel numbers
High numerical aperture
Apodization
Optical tweezers
Extended depth of focus
3
Contents
0
2
12,0 I
v
vJvI
0
2
4/
4/sin0, I
u
uuI
Circular homogeneous illuminated aperture:
Transverse intensity:
Airy distribution
Dimension: DAiry
normalized lateral
coordinate:
v = 2 p x / l NA
Axial intensity:
sinc-function
Dimension: Rayleigh unit RE
normalized axial coordinate
u = 2 p z n / l NA2
Perfect Point Spread Function
NADAiry
l
22.1
2NA
nRE
l
r
z
Airy
lateral
aperture
cone
Rayleigh
axial
image plane
optical
axis
-25 -20 -15 -10 -5 0 5 10 15 20 250,0
0,2
0,4
0,6
0,8
1,0
vertical
lateral
u / v
5
Axial and Lateral Ideal Point Spread Function
Comparison of both cross sections
Ref: R. Hambach
Low Fresnel Number Focussing
6
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
7
Low Fresnel Number Focussing
Example:
focussing by a micro lens
Data:
- radius of aperture: 0.1 mm
- wavelength: 1 mm
- focal length: 10 mm
- Numerical aperture: NA = 0.01
- Fresnel number: 1
Results:
- highest intensity in front of geometrical
focus
Ref: R. Hambach
Low Fresnel Number Focussing
8
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
Fp
-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)
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
Pupil
Vectorial Diffraction at high NA
Linear Polarization
High-NA Point Spread Function
Comparison of Psf intensity profiles
for different models as a function of
defocussing
paraxial
NA = 0.98
high-NA
scalar
high-NA
vectorial
error ofmodel
NA0 0.2 0.4 0.6 0.8 1.0
paraxial
scalarhigh-NA
low-NA vectorial
12
Gaussian Illumination
Known profile of gaussian beams
Ref: R. Hambach
13
Spherical Aberration
Axial asymmetrical distribution off axis
Peak moves
Ref: R. Hambach
14
Annular Ring Pupil
Generation of Bessel beams
Ref: R. Hambach
𝑟𝑝
z
r
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
l
unz
Psf of an Annular Pupil
Ring pupil illumination
Enlarged depth of focus
Lateral resolution constant due to
large angle incidence
Can not be understood geometrically
Psf for Ring Pupil
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
Point Spread Function with Apodization
Apodisation of the pupil:
1. Homogeneous circle Airy
2. Gaussian Gaussian
3. Ring illumination 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
r
I(r)
1
0.8
0.6
0.4
0.2
00 1 2 3-2 -1
circle Airy
ring Bessel
Gauss Gauss
FWHM
r
E(r)
1
0.8
0.6
0.4
0.2
03 41 2
Airy
Bessel
Gauss
E95%
Special geometry:
1. Pumpbeam Gaussian
2. Erase beam Doughnut
Super resolution in central location
Quelle: P. Török
97 nm
490 nm
104 nm
244 nm
z
z
x
x
1) confocal
2) STED
Stimulated Emission Depletion (STED)
Experimental setup
Quelle: P. Török
Stimulated Emission Depletion
Stimulated Emission Depletion
Example 1
Example 2
Quelle: P. Török
22
Optical Tweezers
Optical tweezers:
optical trap by using gradient forces in 3D for non-conducting particles
The forces on the particles moves towards the region of higher intensity
The change of optical momentums
due to the refracted rays creates
a resulting moment and force
Condition:
charge q in a dipole particle
the coefficient a depends on the
dipole, the refractive index and the
size of the particle
Typical size: femto Newton
Momentum of a photon
Ref: D. Simons
IF
nhpl
23
Optical Tweezers
Basic setup for particle trapping
Branches:
1. laser illumination
2. conventional illumination
3. observation
Ref: D. Simons
sample
beam
steering
mirrors
laser
4Q sensor
lamp for observation
microscopic
lens
observation
camera
24
Optical Tweezers
Gradient forces:
act like a potential
In addition: scattering forces
Non-spherical particles or orbital momentum of the beam:
- additional forces on the particle, optical torque
- windmill effects rotates the particle
Trapping of a particle:
- Brownian motion must be overcome
- typical motion inside the trap
Ref: D. Simons
25
Trapping Forces
Forces in a collimated beam
Forces in a focussed beam
Ref: xxx
26
Optical Tweezers
Optical forces in inhomogeneous fields:
3 different profiles:
a) strong focussed, b) weak focussed, c) ring profile
1. row: profile I(x,y)
2. row: axial forces (weak)
3. row: lateral forces
Ref: D. Simons
27
Optical Tweezers
Optical forces changes locally in the particle as a function of the numerical apertures
Variation of forces inside the
particle can stretch a cell
Ref: D. Simons
28
Optical Tweezers
Applications:
- controlled movement of
cells and particles
- manipulation of particles
or nano-wires
Ref: S. Zwick
Transverse plane :
Width x, gain factor GT
Axial direction :
Width z, gain factor GA
Height of the focus intensity, Strehl ratio DS
Relative height of sidelobes,
Relative height of peak : M = Iside / Ipeak
Absolute transmission of power,
For transmission filter with absorption
Modified criteria for special applications,
Example: intensity squared in case of confocal detection
Characterization of a Caustic Shape
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
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
Approaches for Extended Depth of Focus
1. Psf engineering
- Phase pupil mask
- Amplitude pupil mask
- Complex Pupil mask
2. Phase Pupil mask with digital restoration
3. Lippman imaging with arrays (plenoptical principle, integrated imaging)
4. Time multiplexing of a fast variable system
Considered here: cases 1. and 2.
Pupil Mask Types for Extended Depth of Focus
1. Ring pupil Poon 1987
Ojeda-Castaneda 1988,
2. Toraldo rings Ando 1992, Wang 2001, 3 phase rings
Ben-Eliezer 2008, rings with complex transmission
Demenikov 2009, several rings, fractal
3. Axicon ...
4. Cubic Phase (CCP) Dowski 1994, asymmetric
Takahashi 2008, modified
Bagheri 2008, complex
5. Logarithmic George 2001
6. Polynomial, quartic... Sheppard 1988, Prasad 2003
7. Spoke segments Chang 2010
8. Analytic rational Caron 2008
9. Radial cos-mask Ojeda-Castaneda 1990
10. Diffractive high frequency ...
11. Zernike mask ...
12. Pixelized radial ...
13. Cubic sinusoidal modulated Zhao 2010
14. Combined representations ...
Pupil Mask Types for Extended Depth of Focus
1. Transmission - Phase
- Amplitude
2. Geometry: - asymmetric (cubic,...)
- circular symmetric
- azimuthal periodic
3. Spatial frequency: - smooth (logarithmic, cubic,...)
- mid frequency (Zernike)
- high frequency (diffractive)
4. Surface type: - smooth
- steps in slope (segmented)
- steps in height (binary)
• Uniformity over z
• Axial gain factor of focal depth
• Strehl definition
• Peak height of side lobes
• Transverse resolution
• Power in the bucket, central peak power
• Energy transmission
• Contrast of image
• MTF-requirements
• OTF-requirements
Special EDF - Criteria
• Psf caustic engineering:
Conservation of energy
Exact uniformity ('light sausage') is physically not possible
Sidelobes and contrast reduction ('diffraction noise') physically necessary
Critical transition: inside EDF-interval (uniformity) and outside (no straylight)
• Performance optimization:
Threshhold of detection (eye) suppresses sidelobes
Confocal sidelobe suppression
Improvement by axial destruction of coherence
Digital reconstruction of a nearly perfect image
Selective detection by human brain (?)
• Critical balance between EDF-factor, resolution and contrast necessary
Fundamental Properties and Limits of EDF
Ring shaped masks according to Toraldo :
- discrete rings
- absorbing rings or pure phase shifts
- original setup: only 0 / p values of phase
- special case
Fresnel zone plate
Pure phase rings:
amplitude of psf
Toraldo Ring Masks
0p0
p
0
p
0
p0
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)'(
p
r
1
2
j
n
3 10
0
38
Toraldo Mask for Pupil Engineering
Here: pure phase mask
Ref: R. Hambach
EDF with Complex Toraldo Mask
I(r,z)
I(x) I(z)
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
I(z) depth for 80%: 13 RE
-20 -15 -10 -5 0 5 10 15 200
0.2
0.4
0.6
0.8
1
Absorption filter with profil
Corresponds to a linear sequence
of single psf-peaks
Transmission quite small
Extrem good edf performance
EDF with Chirped Ring Pupil
m
n
n rnm
rP1
22cos)1(2112
1)( p
2)12(
1
mT
xp
yp
y'
x'
pupil
transmission P(r)
image
plane
f
z
array of single
distributions
z
Intensität
-20 -10 0 10 20-4
-3
-2
-1
0
1
2
3
4
K a u s t i k
-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
-20 -10 0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Example with m = 8 peaks
EDF with Chirped Ring Pupil
rp
P(rp)
0 1- 0.5- 1 0.5
Complex Zernike Pupil Mask : EDF
Pupil Zernike, phase and apodization , depth of focus 12 RE
Phase Apodisation
-1 -0.5 0 0.5 1
-6
-4
-2
0
2
4
6
8
W e l l e n f l ä c h e
-1 -0.5 0 0.5 1
-6
-4
-2
0
2
4
6
8
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
-60
-40
-20
0
20
40
A p o d i s i e r u n g
-1 -0.5 0 0.5 1
-60
-40
-20
0
20
40
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10
-3
-2
-1
0
1
2
3
I(x,z) , Dz=12.47849 , Gz=0.142 , Dr=1.22931 , Gr=0.686
-3 -2 -1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Schnitt I(x,z=0)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Schnitt I(x=0,z)
System with EDF can be defined as an enlarged constant axial OTF-distribution.
With clear pupil function
l20 W
Axial
MTF-distribution:
Pupil
manipulation
Enlarged constant
OTF-distribution
With manipulated pupil function
Reduced contrast
Preserved resolution
l20 W
Axial
MTF-distribution:
EDF and MTF-Profile
Ref: S. Förster
MTF and bar pattern as a function of defocus
ideal with Mask
Continuous Phase Mask for Microscope
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)(
3
Cubic Phase Plate (CCP)
z
y-section
x-section
cubic phase PSF MTF
Cubic Phase Mask : PSF and OTF
defocussed focus
Conventional imaging System with cubic phase mask
defocussed focus
Psf traditionalprimary Psf with
phase mask
Psf with phase
mask deconvolved
Cubic Phase Mask : Psf
Defocus
primary
with mask
with mask
deconvolved
System with cubic phase plate
MTF good only along axes
Poor resolution in 45° directions
Orientation dependend resolution with image perturbation
Phase mask PSF MTF
MTF for Cibic Phase Plate
Conventional microscopic image
Image with phase mask
with / without deconvolution
Extended Depth of Focus: Microscopic Imaging
Ref: E. Dowski
defocus negative focussed defocus positive
EDF Example Pictures
Cubic phase plate
- small changes of MTF with defocus
- oscillation of real- and imaginary part:
strong change of PTF
The phase of the Fourier components is
not properly reconstructed:
artefacts in image deconvolution
MTF / OTF for CPP
ohne CCP
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
CCP mit = 1
CCP mit = 2
CCP mit = 4
MTF
Re[OTF]
Im[OTF]
Im[OTF]
Frequenz Re[OTF]
c4 klein
c4 mittel
c4 groß
Real behavior of MTF:
1. invariant MTF for lower spatial frequencies
but re-magnification also increases noise
2. decrease for higher spatial frequencies:
reduction in resolution
MTF / OTF for CCP
0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
a = const
c4 wachsend
MTF Re[OTF]
EDF-Masks
Different types of masks:
1. Shape
2. Phase only - complex
3. Axisymmetry - without
symmetry
Ref: S. Förster
EPI-Fluorescence (Molecular Probes FluoCells)
+10µm 0µm -10µm -20µm
Standard
NA=0.66
EDF
NA=0.66
as measured
EDF
NA=0.66
reconstructed
20µm
µmNA
A 50.061.0
l
Ref: L. Höring
Transverse plane:
Width x, gain factor GT
Axial direction:
Width z, gain factor GA
Height of the focus intensity, Strehl ratio DS
Relative height of sidelobes,
Relative height of peak : M = Iside / Ipeak
Absolute transmission of power,
For transmission filter with absorption
Modified criteria for special applications,
Example: intensity squared in case of confocal detection
Characterization of a Caustic Shape
Invariance of transvere intensity
profile for defocussing
Change of lateral intensity
profile with z
Hilbert space angle
Fisher-Defocus
Numerical EDF - Criteria
0
* ),()(21)( drrzrErEz refp
2222
2*
2
),,(),(
),,(),(1)(
dxdyzyxEdxdyyxE
dxdyzyxEyxEz
ref
ref
dxxIdxzxI
dxxIzxIz
psfpsf
psfpsf
H
)0,(),(
)0,(),()(cos
22
vdz
zvHzJ OTF
2
),()(
Microscope Objective Lens with EDF-Mask
Mechanics:
Lens
Mask
Ref: L. Höring
EDF: Bright field Chromium lines
-4µm
Standard
NA=0.66
EDF
NA=0.66
as measured
EDF
NA=0.66
reconstructed20µm
-2µm
0µm
2µm
-15µm
0µm
µmNA
A 50.061.0
l
-8µm
4µm
Ref: L. Höring
59
Depth of Focus
Highly unrealistic sharpness distribution (computer animation)
Ref.: V. Blahnik
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