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Microscopy
Lecture 3: Physical optics of widefield microscopes
2012-10-29
Herbert Gross
Winter term 2012
2 3 Physical optics of widefield microscopes
Preliminary time schedule
No Date Main subject Detailed topics Lecturer
1 15.10. Optical system of a microscope I overview, general setup, binoculars, objective lenses, performance and types of lenses, tube optics Gross
2 22.10. Optical system of a microscope II Etendue, pupil, telecentricity, confocal systems, illumination setups, Köhler principle, fluorescence systems and TIRF, adjustment of objective lenses Gross
3 29.10. Physical optics of widefield microscopes
Point spread function, high-NA-effects, apodization, defocussing, index mismatch, coherence, partial coherent imaging Gross
4 05.11. Performance assessment
Wave aberrations and Zernikes, Strehl ratio, point resolution, sine condition, optical transfer function, conoscopic observation, isoplantism, straylight and ghost images, thermal degradation, measuring of system quality
Gross
5 12.11. Fourier optical description basic concepts, 2-point-resolution (Rayleigh, Sparrow), Frequency-based resolution (Abbe), CTF and Born Approximation Heintzmann
6 19.11. Methods, DIC Rytov approximation, a comment on holography, Ptychography, DIC Heintzmann
7 26.11. Imaging of scatter
Multibeam illumination, Cofocal coherent, Incoherent processes (Fluorescence, Raman), OTF for incoherent light, Missing cone problem, imaging of a fluorescent plane, incoherent confocal OTF/PSF
Heintzmann
8 03.12. Incoherent emission to improve resolution Fluorescence, Structured illumination, Image based identification of experimental parameters, image reconstruction Heintzmann
9 10.12. The quantum world in microscopy Photons, Poisson distribution, squeezed light, antibunching, Ghost imaging Wicker
10 17.12. Deconvolution Building a forward model and inverting it based on statistics Wicker
11 07.01. Nonlinear sample response STED, NLSIM, Rabi the information view Wicker
12 14.01. Nonlinear microscopy two-photon cross sections, pulsed excitation, propagation of ultrashort pulses, (image formation in 3D), nonlinear scattering, SHG/THG - symmetry properties Heisterkamp
13 21.01. Raman-CARS microscopy principle, origin of CARS signale, four wave mixing, phase matching conditions, epi/forward CARS, SRS. Heisterkamp
14 28.01 Tissue optics and imaging Tissue optics, scattering&aberrations, optical clearing,Optical tomography, light-sheet/ultramicroscopy Heisterkamp
15 04.02. Optical coherence tomography principle, interferometry, time-domain, frequency domain. Heisterkamp
1. Point spread function
2. High-NA effects
3. Apodization
4. Defocussing
5. Index mismatch
6. Coherence
7. Partial coherent imaging
3 3 Physical optics of widefield microscopes
Contents of 3rd Lecture
3 Physical optics of widefield microscopes
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
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 p z n / NA2
transversal : v = 2 p x / NA
3 Physical optics of widefield microscopes
Perfect Point Spread Function
NADAiry
22.1
2NA
nRE
3 Physical optics of widefield microscopes
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 of /NA:
assumptions for this estimation and possible changes
A resolution beyond the Abbe limit is only possible with violating of certain
assumptions
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
p
p
'sin'sin2
pak
R
akrukr
R
arx
3 Physical optics of widefield microscopes
Perfect Lateral Point Spread Function: Airy
Axial distribution of intensity
Corresponds to defocus
Normalized axial coordinate
Scale for depth of focus:
Rayleigh length
Zero crossing points:
equidistant and symmetric,
distance of zeros around image
plane 4RE
22
04/
4/sinsin)(
u
uI
z
zIzI o
42
2 uz
NAz
p
22sin NA
n
unRE
3 Physical optics of widefield microscopes
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/p
RE
4RE
z = 2RE
3 Physical optics of widefield microscopes
Defocussed Perfect Point Spread Function
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%
Spherical aberration Astigmatism Coma
c = 0.2
c = 0.3
c = 0.7
c = 0.5
c = 1.0
3 Physical optics of widefield microscopes
Point Spread Function with Aberrations
Zernike coefficients c in
Spherical aberration,
circular symmetry
Astigmatism,
split of two azimuths
Coma,
asymmetric
Intensity distribution I(r,z) for spherical
aberration
Asymmetry of intensity around the image plane
Usually no zero points on axis intra focal
3 Physical optics of widefield microscopes
Point Spread Function with Spherical Aberration
3 Physical optics of widefield microscopes
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
3 Physical optics of widefield microscopes
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
w
E(w)
1
0.8
0.6
0.4
0.2
03 41 2
Airy
Bessel
Gauss
Apodisation of the pupil:
1. Homogeneous
2. Gaussian
3. Bessel
Psf in focus:
different decrease of the focal
intensity for larger radii
Encircled energy:
same behaviour
General features:
Beam profiles and point spread function complicated
No axial symmetry around image plane
No shift-invariance in z-direction
Vectorial effects: axial components Ez and mixing of lateral x-y-field components,
therefore polarization is important
linear polarized input field breaks symmetry
Apodization of the pupil, depending on correction
Spherical shape of the pupil: defocussing is not described by Zernike c4 alone
3 Physical optics of widefield microscopes
High-NA-Systems
3 Physical optics of widefield microscopes
Axial Magnification at High NA
Paraxial case:
Relation between lateral and axial magnification
Systems with sine condition fulfilled:
pupil must have spherical shape
Transfer between entrance and exit pupil
2''m
n
n
z
zmz
'sin
sin
'
n
nm
entrance
pupil
yp
z z'
'
y'p
exit
pupil
sin zy p
p
p
y
ym
'
3 Physical optics of widefield microscopes
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
3 Physical optics of widefield microscopes
High-NA Focusing
Total apodization
corresponds 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
Systems with high numerical aperture:
- distortion of geometry
- nonlinear relationship between aperture angle
and pupil height rp
This distortion corresponds to an apodization
The function a() depends on the system
correction
)()( 222111 gfgfrp
3 Physical optics of widefield microscopes
High-NA-Systems
d
gdga
)(
sin
)()(
rp
f
z
Sine condition
Herschel condition
Lagrange condition
Paraboloidal
Helmholtz-condition
3 Physical optics of widefield microscopes
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)','(
3 Physical optics of widefield microscopes
Vectorial Diffraction for high-NA
Point spread function components Ix, Iy and Iz
Strong differences in the profiles dependent on defocussing
Large differences in size (here: normalized)
Example: initial polarization: x
Ex
Ey
Ez
3 Physical optics of widefield microscopes
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
Vectorial diffraction integral in Fraunhofer representation
All components x,y,z must be considered
In particular due to the large bending angle an axial component EZ occurs
3 Physical optics of widefield microscopes
High-NA-Systeme
E r F P x y e
n u x y
n u y
n u y
y n u x y
n u y
p p
iz n u x y
p p
p
p
p p p
p
p p
' ,
' sin '
' sin '
' sin '
' sin '
' sin '
' ' sin '
1
1
2
2 2
2 2 2
2 2 2
2 2 2 2
2 2 2
2 2 2 2
1
1
1
1
p
3 Physical optics of widefield microscopes
High-NA Point Spread Function
Comparison of Psf intensity profiles
for different models as afunction 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
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
3 Physical optics of widefield microscopes
Depth of Focus: Diffraction Consideration
2
0
sin)(
u
uIuI
focal
plane
beam
caustic
z
depth of focus
0.8
1
I(z)
+Ru
z-R
u 0
r
intensity
at r = 0
2' o
un
Rp
udiff Run
z
2
1
sin493.0
2
12
3 Physical optics of widefield microscopes
Microscope Objective Lens: Defocusing
Defocusing in high-NA lenses
Geometrical: change of NA
d
focal
extra focal
starting
planes intra focal
d
pupil
sinuo
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
d / f = - 0.001
d / f = 0.001
d / f = 0.003
d / f = - 0.003
d / f = - 0.005
d / f = 0.005
d / f = 0.01
d / f = - 0.01
d / f = - 0.02
d / f = 0.02
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
o
oeff
udff
dfuu
222 tan2
2tansin
Relative change factor
1sin
sin
o
eff
u
u
3 Physical optics of widefield microscopes
Microscope Objective Lens: Defocusing
Real systems:
1. Macroscopic change
of ray path
2. Pupil shrinks down
3. Vignetting surface index
changes,
sharp bending of curve
NA
z in
Ru-100 0 100 200 300 4000
0.2
0.4
0.6
0.8
1
rear
stop
front
surface
tube
lens
0.9
focussed 50 m defocussing
macroscopic changes in the ray path
50 m
defocussing
in the object
3 Physical optics of widefield microscopes
Microscope Objective Lens: Defocussing
Change of performance for variation of the
object distance
Strehl increased for shorter object distance
4 6 8 10 12 14 16
Strehl
magnification m
long object
distance
short object
distance
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
nominal
magnification
3 Physical optics of widefield microscopes
Microscope Objective Lens: Index Mismatch
Objective lens with immersion
3 materials : Immersion (I), cover glass (C) and sample (S)
Refraction law:
Problems by index
mismatches with sample
points deep inside
Strong spherical
aberrations for high-NA
first lens
immersion cover
glassprobe
mediumenlarged picture of
the ray caustic
paraxial
focus
marginal
focus
nCG
nM
SSCCII nnnNA coscoscos
3 Physical optics of widefield microscopes
Microscope Objective Lens: Index Mismatch
Calculation of spherical aberration:
comparison of optical path
length with ideal case
( object on back side of glass)
tC
tS
cover
glass
sample
medium
nS
nI
I
C
S
immersion
liquid
nC
tI
s10
dC0
dI0
dI
a
dC
s1
dS
y1
y2
y2
tC
tS
cover
glass
sample
medium
nC
nS
nI
I
C
S
focussed on
underside of cover
glass
defocussing with
mismatch
immersion
liquid
Spherical aberration:
Vanishing effect for nI=nS
Independent from cover glass
1)/(112/12
2
2
S
S
ISS nNA
n
nntW
3 Physical optics of widefield microscopes
Microscope Objective Lens: Defocusing
Aberrations:
1. Zernike coefficients change
approximately linear
2. Psf strongly disturbed
cj in
-0.01 -0.005 0 0.005 0.01-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
c80
c40
c60
z in
[mm]
10RU
z in
Ru
100
00 3-3 r in D
airy
d=0 Ru d=3 Ru d=6 Ru d=9 Ru
d=12 Ru d=15 Ru d=18 Ru d=21 Ru
d=24 Ru d=27 Ru d=30 Ru d=33 Ru
d=36 Ru d=39 Ru d=42 Ru d=45 Ru
3 Physical optics of widefield microscopes
Microscope objective Lens: Defocusing
Peak height of Psf : decreasing
No longer symmetry between
aberrations for object - image
reversal
Imax
t in
Ru0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
cj in
NA0 0,2 0,4 0,6 0,8 1 1,2
0
0,5
1
1,5
2
2,5
3
3,5
4
c20
c40
c60
c80
solid lines : image side
dashed lines : object side
3 Physical optics of widefield microscopes
Microscope Objective Lens: Index Mismatch
Change of aberration with
depth ts, index ratio and
aperture angle
Psf strongly perturbed
z in
Ru
50
00 3-3 r in D
airy
d=0 Ru d=4 Ru d=8 Ru d=12 Ru
d=16 Ru d=20 Ru d=24 Ru d=28 Ru
d=32 Ru d=36 Ru d=40 Ru d=44 Ru
d=48 Ru d=52 Ru d=56 Ru d=60 Ru
W / tS
0 10 20 30 40 50 60 70 80 90-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
S in °
nC = 1.4
nS = 1.4
nC = 1.0
nS = 1.3
nC = 1.4
nS = 1.33
nC = 1.0
nS = 1.7
nC = 1.52
nS = 1.4
nC = 1.52
nS = 1.33
3 Physical optics of widefield microscopes
Microscope Objective Lens: Index Mismatch
I(r)
r in
Dairy
-2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
a = 0...60 Ru
I(z)
z in
Ru
a = 0...60 Ru
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Point spread function:
1. Lateral : only scaling
2. Axial : strong decrease with
ripple
Focussing into an index-mismatched sample
Strong degradation of the
point spread function with
increasing depth
I(r,z)
0.5
5
0
0
1
4
3
2
1
0-5
r in rairy
z in Ru
3 Physical optics of widefield microscopes
Point Spread Function for Index Mismatch
3 Physical optics of widefield microscopes
Coherence in Optics
Statistical effect in wave optic:
start phase of radiating light sources are only partially coupled
Partial coherence: no rigid coupling of the phase by superposition of waves
Constructive interference perturbed, contrast reduced
Mathematical description:
Averagedcorrelation between the field E at different locations and times:
Coherence function G
Reduction of coherence:
1. Separation of wave trains with finite spectral bandwidth
2. Optical path differences for extended source areas
3. Time averaging by moved components
Limiting cases:
1. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length
2. Incoherence: no correlation, light source with independent radiating point like molecules
3 Physical optics of widefield microscopes
Coherence Function
Coherence function: Correlation
of statistical fields (complex)
for identical locations :
intensity
normalized: degree of coherence
In interferometric setup, the amount of describes the visibility V
Distinction:
1. spatial coherence, path length differences and transverse distance of points
2. time-related coherence due to spectral bandwidth and finite length of wave trains
ttrEtrErr ),(),(),,( 2
*
121
z
x
x1
x2
E(x2)
E(x1)x
r r r1 2
)()(
),,(),,()(
21
212112
rIrI
rrrr
)(),( rIrr
3 Physical optics of widefield microscopes
Double Slit Experiment of Young
D
z1
z2
light
source
screen
with slits
detector
x
x
First realization:
change of slit distance D
Second realization:
change of coherence parameter s of the
source
D0
1
V
3 Physical optics of widefield microscopes
Double Slit Experiment of Young
s = 0 s = 0.15 s = 0.25 s = 0.35 s = 0.40s = 0.30
Partial coherent illumination of a double pinhole/double slit
Variation of the size of the source by coherence parameter s
Decreasing contrast with growing s
Example: pinhole diameter Dph = Dairy / distance of pinholes D = 4Dairy
3 Physical optics of widefield microscopes
Spatial Coherence
1
2
starting
plane
receiving
plane
common
area
Area of coherence / transverse coherence length:
Non-vanishing correlation at two points with distance Lc:
Correlation of phase due to common area on source
Radiation out of a coherence cell of
extension Lc guarantees finite contrast
The lateral coherence length
changes during propagation:
spatial coherence grows with
increasing propagation distance
observation
area
O
r2
r1
r r( , )
1 2
Lc
P1
P2
domain of
coherence
3 Physical optics of widefield microscopes
Spatial Coherence
Incoherent source with
diameter D = 2a
Receiver plane indistance z
Cone of observation
Source is coherent in the
distance
Transverse length of coherence
(zeros of -function)
/ a
z a 2 /
Lz
ac
3 Physical optics of widefield microscopes
Partial Coherent Imaging
Complete description of an optical system:
1. Light source
2. Illumination system, amplitude response hill
3. Transmission object
4. Observation / imaging system with amplitude response hobs
sourceillumination
system
observation
system
pupil Psensor
object
plane
image
planeillumination
field
xs , ys
Is hill Iihobs
Io
xp , yp xi , yi
obs
ill
u
u
sin
sins
3 Physical optics of widefield microscopes
Coherence Parameter
Finite size of source : aperture cone with angle uill
Observation system: aperture angle uobs
Definition of coherence parameter s:
Ratio of numerical apertures
Limiting cases:
coherent s = 0
uill << uobs
incoherent
s = 1
uill >> uobs
object imagesource
xi , yixo , yo
lens
uobs
uill
illumination observation
Heuristic explanation
of the coherence
parameter in a system:
1. coherent:
Psf of illumination
large in relation to the
observation
2. incoherent:
Psf of illumination
small in comparison
to the observation
object objective lenscondensersmall stop of
condenser
extended
source
coherent
illumination
large stop of
condenser
incoherent
illumination
Psf of observation
inside psf of
illumination
Psf of observation
contains several
illumination psfs
extended
source
3 Physical optics of widefield microscopes
Coherence Parameter
Simulation of partial coherent illumination:
Finite size of light source
Corresponding finite size of illuminated area in aperture plane
Every point in this area is considered to emit independent (incoherent)
Off-axis point in aperture plane generates an inclined plane wave in the object
Angular spectrum illumination of the object
describes partial coherence
Estimated sampling of illumination points:
3 Physical optics of widefield microscopes
Partial Coherence
condenseraperture
plane
object
plane
source
extension
size of
coherence cell
ys
Ls
angular
spectrum of
object
illumination
2s
fc
c
ss
f
L
2arcsin22
sin
61.0
nys
Superposition of sourve into coherent beams of several source points
Coherent propagation of source point contributions
Incoherent summation of all point contributions in the image plane
Finite size of source:
object illuminated by different directions
3 Physical optics of widefield microscopes
Fourier Model of Image Formation
xa
illumination
aperturecondenser
object
plane
objective
lenspupil
image
plane
tube
lens
Ea(xa)
fc fo ftxo xp xi
aberrations
xo,W(xo)
chief raysource
point S
Eo(xo)
object
mask
A(xo)
E’’o(xo)
pupil
mask
P(xp)
CTFc(xo)
A(xo)
Ep(xp) E'’’p(xp)
CTFo(xp)
P(xp)
CTFt(xt)
Ei(xi)FFT FFT FFT
diffracted
lightdirect
light
3 Physical optics of widefield microscopes
Partial Coherent Imaging
Image intensity: 1. Correlation of two points in the object
2. Integration over all points in the incoherent light source
Simplification:
thin object, transmission does not change for moderate inclination angles
soooosoillsoillsoiobssoiobsssii dxdxdxxOxOxxhxxhxxxhxxxhxIxI 212
*
12
*
12
*
1 ,,,,,,
yp
xp
yo
xo xi
yi
z
ys
xs
light
sourceobject
planepupil image
hill(xL,x1) hobs(x1,x')Is(xs) Ii(xi)O(x1)
P(xp)o(xo1,xo2) 'o(xo1,xo2)
212
*
12
*
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