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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